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PROGRESS IN OPTICS VOLUME 51
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EDITORIAL ADVISORY BOARD
G.S. Agarwal
Stillwater, USA
G. Agrawal
Rochester, USA
T. Asakura
Sapporo, Japan
M.V. Berry
Bristol, England
A.T. Friberg
Stockholm, Sweden
D.F.V. James
Toronto, Canada
F. Gori
Rome, Italy
P. Knight
London, England
G. Leuchs
Erlangen, Germany
J.B. Pendry
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J. Peˇrina
London, England
PROGRESS IN OPTICS VOLUME 51
EDITED BY
E. Wolf University of Rochester, N.Y., U.S.A.
Contributors G. Björk, J.J.M. Braat, E. Brambilla, S. Cattaneo, P. Dirksen, I.R. Gabitov, A. Gatti, A.J.E.M. Janssen, M. Kauranen, A.B. Klimov, A. Lakhtakia, N.M. Litchinitser, L. Lugiato, T.G. Mackay, A.I. Maimistov, C.R. Pollock, L.L. Sánchez-Soto, V.M. Shalaev, S. van Haver
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Preface This volume presents seven review articles that cover a broad range of topics of current research interest in optics. The first article by N.M. Litchinitser, I.R. Gabitov, A.I. Maimistov and V.M. Shalaev presents a review of theoretical as well as experimental research on metamaterials. Such materials have negative refractive index, making it possible to produce unusual effects that cannot be produced with ordinary materials. They include inverse geometrical optics effects, new types of modes in pulse propagation and new kinds of nonlinear interactions, for example. The second article by M. Kauranen and S. Cattaneo reviews recent research on the use of polarization techniques in surface nonlinear optics. The techniques which are described in this article make it possible to determine, with great accuracy, various parameters and also to obtain qualitative information about the sample. The techniques can also be used to separate part of the multi-polar bulk background signal from surface contributions, which has been a longstanding problem in nonlinear surface optics. The third article by T.G. Mackay and A. Lakhtakia is concerned with electromagnetic fields in linear bianisotropic media. In such media the fields D and H are coupled anisotropically to both the fields E and B. The article discusses the properties of the electromagnetic fields in media of this kind, including constitutive relations, plane-wave propagation, Green’s function and the realizations of bianisotropic media by the process of homogenization. The next article by C.R. Polock presents a review in the field of ultrafast optical pulses. There has been great progress made on this area; the generation and applications of ultrafast pulses has grown from nanosecond pulses in the 1960s to attosecond pulses today. In this chapter the general principles behind the generation of ultrashort pulses are reviewed and some of their applications are discussed. The fifth article by A. Gatti, E. Brambilla and L. Lugiato gives an overview of the relatively new field of quantum imagery. The quantum aspects of optical spatial pattern is first explained, followed by discussion of spatial intensity correlations. The so-called ghost imaging is also treated, followed by an account of image amplification by parametric down conversion. The article also included
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discussion of quantum laser pointers, which allow detection of displacement of a laser beam with a precision beyond the standard quantum limit. The article which follows, by J.J.M. Braat, S. van Haver, A.J.E.M. Janssen and P. Dirksen, deals with the analysis and efficient computational techniques for determining point-spread functions of optical systems in the presence of aberrations. The article describes an extension of the Nijboer–Zernike diffraction theory of aberrations, including strong defocusing and vector diffraction effects. The extension makes it possible to determine intensity distribution in the focal region in focusing systems of high numerical aperture. It is shown that analytical expressions for the three-dimensional intensity distribution may be used to solve some inverse imaging problems. Both numerical and experimental examples of the inverse procedure are presented. The concluding article by G. Björk, A.B. Klimov and L.L. Sánchez-Soto is concerned with the discrete Wigner function and with some of its uses. The continuous Wigner function has found numerous applications in many branches of physics. Its discrete counterpart is of more recent origin. It has a rich mathematical structure and has useful applications, for example, in quantum computing, in quantum cryptography and in connection with entanglement. From the above remarks it is clear that this volume presents results of both basic and applied current research in broad areas of optics.
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Emil Wolf Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA September 2007
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Chapter 1. Negative refractive index metamaterials in optics, Natalia M. Litchinitser (Ann Arbor, MI, USA), Ildar R. Gabitov (Tucson, AZ, USA), Andrei I. Maimistov (Moscow, Russian Federation) and Vladimir M. Shalaev (West Lafayette, IN, USA) . . . . . . . . . . . . . . . . . . . . . .
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§ 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Ambidextrous light in a left-handed world . . . . . . . . . . . . . . . 1.2. Negative index: Brief history . . . . . . . . . . . . . . . . . . . . . . . § 2. Optical negative index metamaterials: State of the art . . . . . . . . . . . . 2.1. Plasmonic NIMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Loss management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Alternative approaches to negative refraction . . . . . . . . . . . . . . § 3. Negative refraction and superlens . . . . . . . . . . . . . . . . . . . . . . . 3.1. Negative refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Superlens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 4. Enhanced nonlinearity and its origin in metamaterials . . . . . . . . . . . . § 5. Optical bistability and solitons . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Generalized nonlinear Schrödinger equation . . . . . . . . . . . . . . 5.2. Solitons in plasmonic nanostructures . . . . . . . . . . . . . . . . . . 5.3. Gap solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Optical bistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Ultra-narrow spatial solitons . . . . . . . . . . . . . . . . . . . . . . . § 6. “Backward” phase-matching conditions: Implications for nonlinear optics 6.1. Second-harmonic generation . . . . . . . . . . . . . . . . . . . . . . . 6.2. Optical parametric amplification . . . . . . . . . . . . . . . . . . . . . § 7. Surface polaritons, waveguides and resonators . . . . . . . . . . . . . . . . 7.1. Linear surface polaritons . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Nonlinear surface polaritons . . . . . . . . . . . . . . . . . . . . . . . 7.3. NIM slab as a linear waveguide . . . . . . . . . . . . . . . . . . . . . 7.4. Linear waveguide in nonlinear surroundings . . . . . . . . . . . . . . 7.5. Nano-resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 8. New frontiers: Metamaterials for cloaking . . . . . . . . . . . . . . . . . . § 9. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 2. Polarization techniques for surface nonlinear optics, Martti Kauranen (Tampere, Finland) and Stefano Cattaneo (Eindhoven, The Netherlands) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 2. Polarization effects in the nonlinear response of surfaces and thin films . . . . . . . . 2.1. Functional form of the measured signals . . . . . . . . . . . . . . . . . . . . . . 2.2. Approximation of unity refractive indices . . . . . . . . . . . . . . . . . . . . . . 2.3. Polarization arrangements for the characterization of nonlinear samples . . . . . 2.4. Low-symmetry samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Experimental considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 3. Applications of polarization techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Chirality and circular-difference response . . . . . . . . . . . . . . . . . . . . . . 3.2. Higher-multipole contributions to the surface nonlinearity of isotropic materials § 4. Complete theoretical model including linear optics . . . . . . . . . . . . . . . . . . . 4.1. Geometry and notational conventions . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Second-harmonic field exiting from a thick sample . . . . . . . . . . . . . . . . . 4.3. Limit of zero thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Effect on the susceptibility components . . . . . . . . . . . . . . . . . . . . . . . § 5. Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 3. Electromagnetic fields in linear bianisotropic mediums, Tom G. Mackay (Edinburgh, Scotland, UK) and Akhlesh Lakhtakia (University Park, PA, USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 1. Introduction . . . . . . . . . . . . . . . . . . . . . § 2. The Maxwell postulates and constitutive relations 2.1. Maxwell postulates . . . . . . . . . . . . . . 2.2. Constitutive relations . . . . . . . . . . . . . 2.3. The frequency domain . . . . . . . . . . . . 2.4. 6-vector/6 × 6 dyadic notation . . . . . . . . 2.5. Form invariances . . . . . . . . . . . . . . . 2.6. Constitutive dyadics . . . . . . . . . . . . . . § 3. Linear mediums . . . . . . . . . . . . . . . . . . . 3.1. Isotropy . . . . . . . . . . . . . . . . . . . . 3.2. Anisotropy . . . . . . . . . . . . . . . . . . . 3.3. Bianisotropy . . . . . . . . . . . . . . . . . . 3.4. Nonhomogeneous mediums . . . . . . . . . § 4. Plane-wave propagation . . . . . . . . . . . . . . 4.1. Uniform and non-uniform plane waves . . . 4.2. Eigenanalysis . . . . . . . . . . . . . . . . . 4.3. Isotropic scenarios . . . . . . . . . . . . . . 4.4. Anisotropic scenarios . . . . . . . . . . . . . 4.5. Bianisotropic scenarios . . . . . . . . . . . . 4.6. Nonhomogeneous mediums . . . . . . . . . 4.7. Plane waves with negative phase velocity . . § 5. Dyadic Green functions . . . . . . . . . . . . . .
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Contents 5.1. Definition and properties . . . . . . . . . . . . . . 5.2. Closed-form representations . . . . . . . . . . . . 5.3. Eigenfunction representations . . . . . . . . . . . 5.4. Depolarization dyadics . . . . . . . . . . . . . . . § 6. Homogenization . . . . . . . . . . . . . . . . . . . . . 6.1. Constituent mediums . . . . . . . . . . . . . . . . 6.2. Maxwell Garnett formalism . . . . . . . . . . . . 6.3. Bruggeman formalism . . . . . . . . . . . . . . . 6.4. Strong-property-fluctuation theory . . . . . . . . . 6.5. Anisotropy and bianisotropy via homogenization . § 7. Closing remarks . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 4. Ultrafast optical pulses, Clifford R. Pollock (Ithaca, NY, USA) 211 § 1. Overview of ultrashort optical pulses . . . . . . . . . . . . . . 1.1. Historic developments in short optical pulse development 1.2. Outline of chapter . . . . . . . . . . . . . . . . . . . . . . § 2. Fundamental properties of optical pulses . . . . . . . . . . . . 2.1. Amplitudes, envelopes, and intensity . . . . . . . . . . . 2.2. Phase, frequency, and group delay . . . . . . . . . . . . . 2.3. Time–bandwidth product . . . . . . . . . . . . . . . . . . 2.4. The “zero area” pulse . . . . . . . . . . . . . . . . . . . . § 3. Ultrashort-pulse generation . . . . . . . . . . . . . . . . . . . 3.1. Spectral properties of ultrafast laser materials . . . . . . . 3.2. Modelocking issues . . . . . . . . . . . . . . . . . . . . . 3.3. Active and passive modulation . . . . . . . . . . . . . . . 3.4. Modelocking schemes . . . . . . . . . . . . . . . . . . . § 4. Ultrafast-pulse characterization . . . . . . . . . . . . . . . . . 4.1. Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . 4.2. Frequency-resolved optical gating (FROG) . . . . . . . . § 5. Ultrafast Ti:sapphire lasers and amplifiers . . . . . . . . . . . 5.1. Dispersion control . . . . . . . . . . . . . . . . . . . . . . 5.2. Ultrashort Ti:sapphire lasers . . . . . . . . . . . . . . . . 5.3. Ti:sapphire amplifiers . . . . . . . . . . . . . . . . . . . . § 6. Attosecond pulses . . . . . . . . . . . . . . . . . . . . . . . . . § 7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 5. Quantum imaging, Alessandra Gatti, Enrico Brambilla and Luigi Lugiato (Como, Italy) . . . . . . . . . . . . . . . . . . . . . . . . . . § 1. Introduction to quantum imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Optical parametric down-conversion of type I . . . . . . . . . . . . . . . . . . . . . 1.2. Spatially multimode versus single-mode squeezing . . . . . . . . . . . . . . . . . . 1.3. Spatial structure of squeezed vacuum states in the degenerate optical parametric oscillator below threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. Quantum images in the OPO above and below threshold . . . . . . . . . . . . . . . . 1.5. The interference of signal and idler waves in type I PDC . . . . . . . . . . . . . . .
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§ 2. Quantum spatial intensity correlations in optical parametric down-conversion . . . . . . . 2.1. Degenerate OPO below threshold, spatial quantum correlation and entanglement . . 2.2. Multimode-model for single-pass parametric down-conversion . . . . . . . . . . . . 2.3. Single-pass PDC of type I. Near-field/far-field duality . . . . . . . . . . . . . . . . . 2.4. Single-pass PDC of type II. Simultaneous near-field and far-field spatial correlation 2.5. Detection of sub-shot-noise spatial correlation in the high gain regime of type II PDC. Spatial analogue of photon antibunching . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Detection of weak amplitude objects beyond the standard quantum limit . . . . . . . 2.7. Multimode polarization entanglement in high-gain PDC . . . . . . . . . . . . . . . . § 3. Ghost imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. General theory of ghost imaging with entangled beams . . . . . . . . . . . . . . . . 3.2. Two paradigmatic imaging schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Spatial average in ghost diffraction: Increase of spatial bandwidth and of speed in retrieval. Homodyne detection scheme . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Debate: Is quantum entanglement really necessary for ghost imaging? . . . . . . . . 3.5. Ghost imaging by splitted thermal-like beams: Theory . . . . . . . . . . . . . . . . . 3.6. Resolution aspects, correlation aspects, visibility aspects . . . . . . . . . . . . . . . 3.7. Ghost imaging with splitted thermal beams: Experiment . . . . . . . . . . . . . . . . 3.8. Complementarity between “thermal” ghost imaging and the classic Hanbury-Brown– Twiss (HBT) correlation technique, with respect to spatial coherence . . . . . . . . . § 4. Image amplification by parametric down-conversion . . . . . . . . . . . . . . . . . . . . . 4.1. Twin (quantum entangled) images . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Noiseless amplification of images . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Theory of noiseless amplification of optical images . . . . . . . . . . . . . . . . . . 4.4. Noiseless amplification of optical images: Experiments in the pumped regime . . . . 4.5. Noiseless amplification of optical images: Experiment in the cw regime. Experimental observation of twin images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 5. The quantum laser pointer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. 1D experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. 2D quantum laser pointer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 6. Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Object reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Entangled two-photon microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Quantum-optical coherence tomography . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Quantum ellipsometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5. Transverse distribution of quantum fluctuations in free-space spatial solitons . . . . 6.6. Quantum fluctuations in cavity solitons . . . . . . . . . . . . . . . . . . . . . . . . . 6.7. Quantum holographic teleportation and dense coding of optical images . . . . . . . 6.8. Quantum-optical lithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 6. Assessment of optical systems by means of point-spread functions, Joseph J.M. Braat, Sven van Haver (Delft, The Netherlands), Augustus J.E.M. Janssen and Peter Dirksen (Eindhoven, The Netherlands) § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . 1.1. The optical point-spread function . . . . . . . 1.2. Quality assessment by inverse problem solving § 2. Theory of point-spread function formation . . . . .
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2.1. Field representations and the diffraction integral . . . . . . . . . . . . . . . . . . . 2.2. The Debye integral for focused fields . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. The Rayleigh-I integral for focused fields . . . . . . . . . . . . . . . . . . . . . . . 2.4. Comparison of the various diffraction integrals . . . . . . . . . . . . . . . . . . . . 2.5. The amplitude of the point-spread function produced by an optical system . . . . . 2.6. Analytic expressions for the point-spread function in the focal region (scalar case) 2.7. Analytic expressions for the point-spread function in the vector diffraction case . . 2.8. The point-spread function in a stratified medium . . . . . . . . . . . . . . . . . . . § 3. Energy density and power flow in the focal region . . . . . . . . . . . . . . . . . . . . . 3.1. Expression for the electric energy density . . . . . . . . . . . . . . . . . . . . . . . 3.2. Expression for the Poynting vector . . . . . . . . . . . . . . . . . . . . . . . . . . § 4. Quality assessment by inverse problem solution . . . . . . . . . . . . . . . . . . . . . . 4.1. Intensity measurements and phase retrieval . . . . . . . . . . . . . . . . . . . . . . 4.2. The optical inverse problem for finite-aperture imaging systems . . . . . . . . . . 4.3. Solving the optical inverse problem using phase diversity . . . . . . . . . . . . . . § 5. Quality assessment using the Extended Nijboer–Zernike diffraction theory . . . . . . . 5.1. Scalar retrieval process using the Extended Nijboer–Zernike theory . . . . . . . . 5.2. Pupil function retrieval for high-NA imaging systems . . . . . . . . . . . . . . . . 5.3. Retrieval examples for high-NA systems . . . . . . . . . . . . . . . . . . . . . . . § 6. Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Derivation of Weyl’s plane wave expansion of a spherical wave . . . . . . . . . Appendix B: The Debye integral in the presence of aberrations . . . . . . . . . . . . . . . . Appendix C: Series expansion of the diffraction integral at large defocus . . . . . . . . . . m (r, f ) . . . . . . . . . . . . . Appendix D: Series expansion for the diffraction integral Vn,j m D.1. Expansion using the functions Vn (r, f ) . . . . . . . . . . . . . . . . . . . . . . . D.2. Expansion using the functions Tnm (r, f ) . . . . . . . . . . . . . . . . . . . . . . . Appendix E: The predictor–corrector procedure . . . . . . . . . . . . . . . . . . . . . . . . . Appendix F: Zernike coefficients for circularly symmetric polarization states . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 7. The discrete Wigner function, Gunnar Björk (Kista, Sweden), Andrei B. Klimov (Jalisco, Mexico) and Luis L. Sánchez-Soto (Madrid, Spain) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 1. § 2. § 3. § 4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Wigner function . . . . . . . . . . . . . . . . . . . . . . . . . Discrete finite space and finite fields . . . . . . . . . . . . . . . . . . . . The generalized Pauli group . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Prime-dimensional spaces . . . . . . . . . . . . . . . . . . . . . . . 4.2. Power-of-a-prime-dimensional spaces . . . . . . . . . . . . . . . . . § 5. Mutually unbiased bases . . . . . . . . . . . . . . . . . . . . . . . . . . . § 6. The discrete Wigner function . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Wigner function in prime-dimensional spaces . . . . . . . . . . . . 6.2. Wigner function in composite-dimensional spaces . . . . . . . . . . 6.3. Wigner function for pN -dimensional space . . . . . . . . . . . . . . § 7. Reconstruction of the density operator from the discrete Wigner function 7.1. Lines and rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Marginal probability density and the density operator . . . . . . . .
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7.3. Tomographic reconstruction . . . . . . 7.4. Rotation operators . . . . . . . . . . . . 7.5. The phase of the displacement operator § 8. Applications . . . . . . . . . . . . . . . . . . § 9. Discussion and outlook . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .
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Author index for Volume 51 . . . . Subject index for volume 51 . . . . Contents of previous volumes . . . Cumulative index – Volumes 1–51
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E. Wolf, Progress in Optics 51 © 2008 Elsevier B.V. All rights reserved
Chapter 1
Negative refractive index metamaterials in optics by
Natalia M. Litchinitser Department of Electrical Engineering and Computer Science, University of Michigan, 2200 Bonisteel Boulevard, 3113 ERB1, Ann Arbor, Michigan 48109, USA e-mail:
[email protected]
EBL
Ildar R. Gabitov Department of Mathematics, University of Arizona, 617 North Santa Rita Avenue, Tucson, Arizona 85721, USA
Andrei I. Maimistov Department of Solid State Physics, Moscow Engineering Physics Institute, Kashirskoe sh. 31, Moscow 115409, Russian Federation
Vladimir M. Shalaev School of Electrical and Computer Engineering and Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA
ISSN: 0079-6638
DOI: 10.1016/S0079-6638(07)51001-2 1
Contents
Page § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
§ 2. Optical negative index metamaterials: State of the art . . . . . . . .
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§ 3. Negative refraction and superlens . . . . . . . . . . . . . . . . . . .
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§ 4. Enhanced nonlinearity and its origin in metamaterials . . . . . . . .
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§ 5. Optical bistability and solitons . . . . . . . . . . . . . . . . . . . . .
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§ 6. “Backward” phase-matching conditions: Implications for nonlinear
EBL § 7. Surface polaritons, waveguides and resonators . . . . . . . . . . . . optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38 44
§ 8. New frontiers: Metamaterials for cloaking . . . . . . . . . . . . . .
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§ 9. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2
§ 1. Introduction Recent advances in nanofabrication have led to the demonstration of a novel class of optical materials – metamaterials – that are artificially engineered structures with electromagnetic properties unattainable in nature. One of the most fascinating examples of such structures is negative-index metamaterials (NIMs). The emergence of optical metamaterials opens up fundamentally new regimes of light–matter interaction and prompts unique opportunities for manipulating light. Over the last seven years there has been enormous progress in the field of NIMs from both theoretical and experimental viewpoints (Ramakrishna [2005], Veselago, Braginsky, Shklover and Hafner [2006]). The latest developments in the field of negative refraction and microwave/terahertz NIMs have been reviewed by Pendry [2004], Smith, Pendry and Wiltshire [2004], Boardman, King and Velasco [2005] and others. Recent advancements in the studies of highly unusual linear properties of optical NIMs have been discussed by Veselago and Narimanov [2006], Agranovich and Gartstein [2006], and Klar, Kildishev, Drachev and Shalaev [2006]. Some remarkable nonlinear properties of NIMs have been highlighted in the recent Nature Photonics article by Shalaev [2007]. In this review we intend to present a comprehensive overview of the state of the art in the design and fabrication of NIMs at optical frequencies, their linear and nonlinear properties, and potential applications of optical NIMs and a more general class of optical metamaterials.
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1.1. Ambidextrous light in a left-handed world The propagation of electromagnetic waves in optical media is determined by two material parameters: dielectric permittivity ε and magnetic permeability μ, re√ lated to the refractive index n through n = εμ. Figure 1a shows a diagram of all possible combinations of ε and μ. The majority of conventional materials belongs to region I where ε > 0 and μ > 0 corresponding to n > 0, where propagating waves are allowed. It is noteworthy that the magnetic susceptibility of most natural materials is very small in comparison with their dielectric sus3
4
Negative refractive index metamaterials in optics
[1, § 1
Fig. 1. (a) A diagram of all possible combinations of dielectric permittivity ε and magnetic permeability μ. (b) A conventional material built of regular atoms. (c) A metamaterial built of “meta-atoms”.
ceptibility, thus limiting the interaction of atoms to the electric component of the electromagnetic wave, leaving the magnetic component largely unexploited. Indeed, μ is close to unity for many naturally existing materials. The reason for this difference in the strength of electric and magnetic field coupling to atoms is that the magnetization of any (non-ferromagnetic) material is a relativistic effect, of the order ∼ v 2 /c2 ∼ α 2 1, where v is the velocity of the electrons in the atoms, and α ∼ = 1/137 is the fine-structure constant (Landau, Lifshitz and Pitaevskii [1984]). Magnetism is particularly weak at optical frequencies because the relaxation times of paramagnetic and ferromagnetic processes are significantly longer than the optical period, leaving the electron movement in atoms as the only mechanism for the magnetic response. As a result, the magnetic field component is usually not involved in light–matter interactions. Only at very high intensities of ∼1018 W/cm2 do the effects of the magnetic and electric fields on the electron motion become comparable, giving rise to a number of new effects such as the photon-drag effect (Wegener [2005]) and nonlinear Thomson scattering or Larmor radiation (Wegener [2005], Chen, Maksimchuk and Umstadter [1998]). The second (ε < 0 and μ > 0) and fourth (ε > 0 and μ < 0) quarters of the diagram in fig. 1 correspond to opaque materials that cannot support any propagating waves. However, materials with parameters corresponding to the third quarter of the diagram (ε < 0 and μ < 0) are transparent and allow wave propagation. While the dielectric permittivity of some naturally occurring materials is negative in certain frequency ranges (e.g. metals including gold, silver and aluminum possess negative ε at the visible and near-ultraviolet frequencies), no isotropic materials with negative μ are readily available. Some antiferromagnets and insulating ferromagnets have been shown to provide negative effective magnetic permeability (Campley and Mills [1982], Hartstein, Burstein, Maradudin, Brewer
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1, § 1]
Introduction
5
Fig. 2. (a) Vectors E, H , and k form a “right-handed” triplet in PIMs, and (b) a “left-handed” triplet in NIMs.
and Wallis [1973]). Moreover, the possibility of simultaneously realizing negative effective εeff and μeff in antiferromagnetic metals has been discussed by Campley and Mills [1982]. Nevertheless, no materials with ε and μ being simultaneously negative had been found in nature or demonstrated experimentally prior to the year 2000. But what is so exciting about such materials? In 1968, Veselago showed theoretically that materials with simultaneously negative ε and μ possess a number of unusual and often counterintuitive properties. To understand a lightwave interaction with such materials, let us consider Maxwell’s equations written in the form ω μH, c ω k × H = − εE. c
k×E =
EBL (1.1)
Equations (1.1) suggest that if ε > 0 and μ > 0 in the same frequency range, the vectors E, H and k form a “right-handed” triplet as shown in fig. 2a, and the refractive index is positive. As usual, the Poynting vector, defined as S=
c E × H, 4π
(1.2)
is parallel to the k-vector. However, if ε < 0 and μ < 0 in the same frequency range, the vectors E, H and k form a “left-handed” triplet as shown in fig. 2b, and √ the refractive index is negative, i.e. n = − εμ. The former class of materials is often referred to as “right-handed materials” (RHMs) or positive-index materials (PIMs), while the latter class is referred to as “left-handed materials” (LHMs) or negative-index materials (NIMs). The Poynting vector in NIMs is anti-parallel to the k-vector. Therefore, the phase velocity, defined as vp = ω/k, is co-directed with the energy velocity determined by the Poynting vector in PIMs and counterdirected with the Poynting vector in NIMs. The opposite directionality of the phase velocity and the Poynting vector is taken as the most general definition of NIMs.
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Negative refractive index metamaterials in optics
[1, § 1
Although the LHMs introduced by Veselago have been assumed to have ε, μ and n negative (and real) numbers in the same frequency range, the term “lefthanded materials” is often used in a broader context to include other optical systems that possess antiparallel phase velocity and the Poynting vector. Examples of such materials, including photonic crystals, anisotropic waveguides, and others, will be discussed in § 2. We note that there are two general theoretical approaches to the description of NIMs at optical frequencies. In one approach [used in eq. (1.1)] a set of four fields, E, D, H , and B, is introduced, where D = ε(ω)E, B = μ(ω)H for a monochromatic wave. In another approach the so-called E,B&D picture is used, in which B = H , with μ = 1, and D = εE with ε = ε(ω, k), including both temporal and spatial dispersions (Agranovich, Shen, Baughman and Zakhidov [2004a]). There is, however, a one-to-one correspondence in the formal description of propagation in NIMs by the two approaches. In this chapter the (E, D, H, B) approach will be followed. As mentioned above, a negative refractive index has not been found in naturally occurring materials due to the limitations imposed on their properties by their constituent components − atoms and molecules schematically shown in fig. 1b. On the contrary, metamaterials are built of so-called artificial or “meta”atoms (fig. 1c). Meta-atoms are properly engineered resonant nanostructures that bring new degrees of freedom to the design of functional materials. The electrical permittivity and magnetic permeability of metamaterials can, in principle, be designed positive, negative or even zero at any selected frequency by properly adjusting the dimensions, periodicity and other properties of the meta-atoms. Therefore, metamaterials can fundamentally change both linear and nonlinear light–matter interactions by bringing the magnetic component of the field into play in addition to its electric component and thus making the electromagnetic waves “ambidextrous” in the optical domain. The current state of the art in the design and fabrication of optical metamaterials will be discussed in § 2. The nonlinear properties of metamaterials are also expected to be significantly modified by their design, and in particular by their enhanced interactions with the magnetic field component. The nonlinear response of metamaterials has been analyzed for the particular case of a two-dimensional NIM periodic structure created by arrays of wires and split-ring resonators (SRR) embedded in a nonlinear dielectric host material, and has been shown to have two different contributions (Zharov, Shadrivov and Kivshar [2003]): the first contribution originates from an intensity-dependent part of the effective dielectric permittivity of the nonlinear dielectric host material, and the second contribution comes from the periodic structures of resonators. The magnetic nonlinearity has been found to be much
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1, § 1]
Introduction
7
stronger than the nonlinearity in the dielectric properties due to the field enhancement in the split-ring resonators, as will be discussed in more detail in § 4. Nevertheless, quoting a recent paper by Klein, Enkrich, Wegener and Linden [2006] “a systematic microscopic theory of the nonlinear optical properties of metallic metamaterials would be highly desirable but is currently not available.” Antiparallel vp and S in NIMs have been shown to manifest in many extraordinary optical phenomena, including negative refraction and amplification of the evanescent waves, remarkably changed Manley–Rowe power conservation relations and new “backward” phase-matching conditions (Veselago [1968], Pendry [2000], Zharov, Zharova, Shadrivov and Kivshar [2005], Popov and Shalaev [2006a, 2006b], Popov, Slabko and Shalaev [2006], Shadrivov, Zharov and Kivshar [2006], Maimistov, Gabitov and Kazantseva [2007]), surface and guidedwave regimes unattainable in conventional waveguides, and new types of temporal and spatial solitons (Scalora, Syrchin, Akozbek, Poliakov, D’Aguanno, Mattiucci, Bloemer and Zheltikov [2005], Scalora, D’Aguanno, Mattiucci, Akozbek, Bloemer, Centini, Sibilia and Bertolotti [2005], Lazarides and Tsironis [2005], Kourakis and Shukla [2005], Shadrivov, Zharova, Zharov and Kivshar [2005], Shadrivov and Kivshar [2005], Gabitov, Indik, Litchinitser, Maimistov, Shalaev and Soneson [2006], Marklund, Shukla and Stenflo [2006]; these will be discussed in §§ 4–7. Recently, several novel device applications have been proposed that rely on the negative phase shift (phase advance) introduced by NIMs. These include miniaturized optical resonators, phase compensators/conjugators, sub-wavelength waveguides, couplers, and laser cavities (Engheta [2002], Alu and Engheta [2003, 2004], Caiazzo, Maci and Engheta [2004], Engheta and Ziolkowski [2005], Ziolkowski [2006]). Negative phase shifts have been predicted to have a pronounced effect on optical bistability, potentially facilitating new tools for NIM characterization, and to enable enhanced nonreciprocal transmission that can be utilized in optical diode-like device applications (Feise, Shadrivov and Kivshar [2004, 2005], D’Aguanno, Mattiucci, Scalora and Bloemer [2004], Hegde and Winful [2005a], Jiang, Chen and Zhu [2006], Litchinitser, Gabitov, Maimistov and Shalaev [2007]). These effects and applications will be discussed in §§ 5 and 7. Finally, in addition to the exciting phenomena and potential applications associated with NIMs per se, a more general class of metamaterials, including NIMs, has been shown to provide new ways for the manipulation of an object’s degree of visibility (Alu and Engheta [2005a], Pendry, Schurig and Smith [2006], Leonhardt [2006], Leonhardt and Philbin [2006], Milton and Nicorovici [2006], Schurig, Mock, Justice, Cummer, Pendry, Starr and Smith [2006], Cai, Chettiar, Kildishev
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Negative refractive index metamaterials in optics
[1, § 2
and Shalaev [2007]). The theoretical aspects and first experimental realization of metamaterial-based cloaking will be discussed in § 8.
1.2. Negative index: Brief history A most comprehensive theoretical description of NIMs has been presented by Veselago [1968], who formulated and analyzed the most important properties of NIMs with ε and μ being negative in the same frequency range and also addressed the issue of the practical realization of such materials. However, the possibility of opposite directionality of the phase and energy velocities has been discussed at least as early as in 1904–1905 by Lamb in his paper “On group-velocity” and by Von Laue [1905]. Lamb [1904] acknowledged Schuster for noticing that a group velocity can be negative (i.e., can have a sign opposite to that of the phase velocity) owing to anomalous dispersion. Lamb [1904] and Pocklington [1905] considered some examples of mechanical systems where the phase and group velocities can be in the opposite direction. Schuster [1904] has shown that this property could also be realized in optical systems. The first detailed discussion of negative refraction can be found in Mandelshtam’s [1947] lectures, while the relationship between simultaneously negative ε and μ and negative refraction was first noted by Sivukhin [1957] and studied in more detail by Pafomov [1959]. Finally, negative refraction of a wave propagating across the boundary with a gyrotropic medium was discussed in Agranovich and Ginzburg’s [1984] book.
EBL
§ 2. Optical negative index metamaterials: State of the art The first experimental demonstrations of NIMs were reported at microwave frequencies by Smith, Padilla, Vier, Nemat-Nasser and Schultz [2000] and Shelby, Smith and Schultz [2001]. Following the approach proposed by Pendry, Holden, Robbins and Stewart [1999], these first NIMs were based on pairs of subwavelength concentric SRRs providing negative μ and straight wires responsible for negative ε. This structure can be considered as an electronic circuit consisting of inductive and capacitive elements (an LC-circuit). The rings form the inductances, while the two slits and the gap between the two rings can be considered as capacitors. A magnetic field, oriented perpendicular to the plane of the rings, induces an opposing magnetic field in the loop due to Lenz’s law, which leads to a diamagnetic response resulting in a negative permeability. The capacitors are necessary to guarantee that the wavelength of the resonance is larger than the dimensions of the SRR. In this configuration, a strong magnetic response has been
1, § 2]
Optical negative index metamaterials: State of the art
9
achieved by operating in the vicinity of the LC resonance of the split ring (Pendry, Holden, Robbins and Stewart [1999]). The frequencies of the LC resonances in this case are largely determined by the geometry and size of the split ring rather than by the properties of the metal. At microwave frequencies, metals can be considered as perfect conductors because the skin depth is significantly smaller than the characteristic size of the meta-atom. The same technique of obtaining negative magnetic permeability using SRRs has been implemented in the terahertz frequency range by scaling down the dimensions of the split rings (Linden, Enkrich, Wegener, Zhou, Koschny and Soukoulis [2004]). However, scaling down to the near-infrared and visible frequencies turned out not to be straightforward, not only due to technical challenges related to the fabrication of resonant structures on the nanoscale, but also because metals no longer behave as perfect conductors in this frequency range and electromagnetic fields penetrate into the metal. As a result, magnetic resonance is significantly reduced. Therefore, different approaches have been developed (Shalaev, Cai, Chettiar, Yuan, Sarychev, Drachev and Kildishev [2005], Zhang, Fan, Panoiu, Malloy, Osgood and Brueck [2005], Grigorenko, Geim, Gleeson, Zhang, Firsov, Khrushchev and Petrovic [2005], Zhang, Fan, Malloy, Brueck, Panoiu and Osgood [2006], Dolling, Enkrich, Wegener, Soukoulis and Linden [2006a, 2006b], Dolling, Wegener, Soukoulis and Linden [2007]).
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2.1. Plasmonic NIMs The first metamaterials with a negative index of refraction at optical frequencies have been demonstrated using pairs of metallic nanorods by Shalaev, Cai, Chettiar, Yuan, Sarychev, Drachev and Kildishev [2005] (shown schematically in fig. 3a) and using pairs of dielectric voids in metal by Zhang, Fan, Panoiu, Malloy, Osgood and Brueck [2005] as shown in fig. 3b. In both experiments the NIMs were fabricated in the form of thin films of thicknesses 160 and 120 nm, respectively. In such thin films Snell’s law cannot be used to detect or measure the refractive index. The negative refraction property reveals itself in a phase shift (phase advance) that was measured interferometrically in the experiments (Shalaev, Cai, Chettiar, Yuan, Sarychev, Drachev and Kildishev [2005], Drachev, Cai, Chettiar, Yuan, Sarychev, Kildishev, Klimeck and Shalaev [2006], Zhang, Fan, Panoiu, Malloy, Osgood and Brueck [2005]). It is important that an unambiguous determination of the refractive index requires four parameters to be measured: transmittance T , reflectance R, and the absolute values of the phases ϕt and ϕr of the transmitted and reflected fields. Then the complex refractive index for the NIM
10
Negative refractive index metamaterials in optics
[1, § 2
Fig. 3. Schematics of the structures fabricated and characterized in the first NIM experiments, (a) at 1.5 µm by Shalaev, Cai, Chettiar, Yuan, Sarychev, Drachev and Kildishev [2005], and (b) at 2 µm by Zhang, Fan, Panoiu, Malloy, Osgood and Brueck [2005].
thin film sandwiched between the air and glass can be found from (Kildishev, Cai, Chettiar, Yuan, Sarychev, Drachev and Shalaev [2006]) cos(nkΔ) =
1 − r 2 + ns t 2 , (ns + 1)t + rt (ns − 1)
(2.1)
EBL
where k is the wavevector of light in vacuum, Δ is the thickness of the NIM thin film, r and t √ are complex reflection and transmission coefficients, t = √ T exp(iϕt ), r = R exp(iϕr ), and ns is the refractive index of the substrate. The idea of using pairs of metallic rods to produce effective magnetic permeability was discussed for the first time by Lagarkov and Sarychev [1996]. The possibility of realizing a negative index of refraction at optical frequencies in paired nanorod composites was proposed by Podolskiy, Sarychev and Shalaev [2002] and discussed in detail by Panina, Grigorenko and Makhnovskiy [2002], Podolskiy, Sarychev and Shalaev [2003], and Podolskiy, Sarychev, Narimanov and Shalaev [2005]. The origin of a negative refractive index in a composite material built with such paired nanorods can be understood as follows. The electric resonances of an individual nanorod originate from the excitation of the surface waves on the metal–air interface. While such surface waves, known as surface plasmon polaritons, cannot be excited with the plane wave in a semi-infinite medium, they are excited in the finite size nanorods. In a paired nanorod configuration two types of plasmon polariton waves can be supported: symmetric and anti-symmetric. The electric field, oriented parallel to the nanorods, induces parallel currents (symmetric plasmon polariton wave) in both nanorods, leading to the excitation of a dipole
1, § 2]
Optical negative index metamaterials: State of the art
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Fig. 4. (a) The electric dipole moment of a coupled nanorod pair with the length of the nanorod being a = 162 nm, the separation between two nanorods d = 80 nm, and the nanorod diameter b = 32 nm. (b) The magnetic dipole moment of the system in (a). (Adapted from Klar, Kildishev, Drachev and Shalaev [2006].)
moment. The magnetic field, oriented perpendicular to the plane of the nanorods, excites anti-parallel currents (anti-symmetric plasmon polariton wave) in the pair of nanorods. Combined with the displacement currents between the nanorods, they induce a resonant magnetic dipole moment. The excited moments are codirected with the incident field when the wavelength of an incident light is above the resonance, and they are counter-directed to the incident fields at wavelengths below the resonance, as shown in fig. 4. The excitation of such plasmon resonances for both the electric and magnetic field components results in the resonant response of the refractive index. In particular, the refractive index can become negative at wavelengths below the resonance (Podolskiy, Sarychev and Shalaev [2002, 2003]). On the other hand, the two parallel nanorods can be considered as an open current loop, which acts as a transmission line. Such a loop is closed through the displacement current, and it therefore supports the resonant modes of the electric and magnetic components of the electromagnetic field. Then, a resonance in the pair of nanorods can also be thought of as a resonance in an optical LC-circuit, with the inductance L provided by the metal rods and the capacitance C provided by the dielectric gaps between the nanorods. While the first experiments reported by the Purdue group (Shalaev, Cai, Chettiar, Yuan, Sarychev, Drachev and Kildishev [2005]) and by the New Mexico– Columbia team (Zhang, Fan, Panoiu, Malloy, Osgood and Brueck [2005]) did prove the possibility of NIMs at optical frequencies, the first NIM samples possessed significant losses. It is noteworthy that the condition of both real ε and μ being negative, originally introduced by Veselago [1968], is a sufficient but not a necessary condition for n being negative. Indeed, Depine and Lakhtakia [2004]
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have derived a more general condition that assures a negative real part of n in the form Re(ε)|μ| + Re(μ)|ε| < 0, assuming that ε = Re(ε) + i Im(ε) and μ = Re(μ) + i Im(μ). This condition is valid for passive NIMs and suggests that a negative refractive index can be achieved even when Re(ε) < 0 but Re(μ) > 0 provided that Im(μ) = 0. However, this requirement implies that the material is inherently lossy. A figure of merit (FOM) that can be used to characterize NIM performance is often defined as |Re(n)| (2.2) , Im(n) where Re(n) and Im(n) are the real and imaginary parts of n. The larger the FOM, the better the NIM’s transmission properties are. However, it should be mentioned that another important factor affecting the overall transmission is impedance mismatch. An ultimate goal of NIM design and fabrication is the demonstration of materials with a large negative real part of n, a large FOM, a broad bandwidth corresponding to both ε and μ being negative and optimized impedance matching. The current state of the art for optical NIMs was demonstrated by the Karlsruhe–Iowa State team (Dolling, Enkrich, Wegener, Soukoulis and Linden [2006a]). Using a self-supporting fishnet structure consisting of rectangular dielectric voids in parallel metal films, they achieved a FOM of 3 at λ = 1.4 µm. Recently, the Karlsruhe group has also reported the first 3-layered NIM with a FOM of 2.5 at λ = 1.41 µm. In table 1 we summarize recent experimental results reported by different groups, and compare them in terms of Re(n), the operating wavelength, the FOM and the structure of the sample. Recently, several advanced designs for optimizing NIM properties have been proposed. These include NIMs combining magnetic resonators with metal films (Chettiar, Kildishev, Klar and Shalaev [2006]), impedance-matched low-loss NIMs utilizing pairs of metal strips embedded in gain materials (Klar, Kildishev, Drachev and Shalaev [2006]), NIM structures built of periodic arrays of pairs of short metal wires and continuous wires (Zhou, Zhang, Tuttle, Koschny and Soukoulis [2006]), materials with a unit cell comprising a continuous optically thin metal film sandwiched between two identical optically thin metal strips, where the incorporation of the middle thin-metal film relaxes the requirement on large wavelength/unit cell ratios (Shvets and Urzhumov [2006], Lomakin, Fainman, Urzhumov and Shvets [2006]), metallic horseshoe-shaped or U-shaped nanostructures showing a magnetic plasmon resonance even when the characteristic size of the nanostructure is much smaller than the optical wavelength (Sarychev, Shvets and Shalaev [2006]), and finally, three-dimensional NIMs (Zhang, Fan, Panoiu, Malloy, Osgood and Brueck [2006]). F =
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1, § 2]
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Table 1 |Re(n)|/ Im(n)
Group
Re(n)@λ
Purdue1
−0.3 at 1.5 µm
0.1
New Mexico & Columbia2
−2 at 2 µm
0.5
New Mexico & Columbia3
−4 at 1.8 µm
2
Karlsruhe & Iowa State4
−1 at 1.4 µm −2 at 1.45 µm
3 1.5
Karlsruhe & Iowa State5
−0.6 at 780 nm
0.5
Karlsruhe6
−1 at 1.41 µm
2.5
Purdue7
−1 at 813 nm
1.3
1 2 3 4 5 6 7
Sample
3 layers
Shalaev, Cai, Chettiar, Yuan, Sarychev, Drachev and Kildishev [2005]. Zhang, Fan, Panoiu, Malloy, Osgood and Brueck [2005]. Zhang, Fan, Malloy, Brueck, Panoiu and Osgood [2006]. Dolling, Enkrich, Wegener, Soukoulis and Linden [2006a]. Dolling, Wegener, Soukoulis and Linden [2007]. Dolling, Wegener and Linden [2007]. Chettiar, Kildishev, Yuan, Cai, Xiao, Drachev and Shalaev [2006].
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2.2. Loss management Among other properties, losses remain the most critical problem for the plasmonic NIMs that were fabricated and characterized in the pioneering experiments discussed in the preceding subsection. These losses are the main obstacle for the development of practical applications of optical NIMs. In particular, it has been shown that even very small losses significantly deteriorate the imaging properties of a so-called “superlens” (Podolskiy and Narimanov [2005a]). Shamonina, Kalinin, Ringhofer and Solymar [2001] and Ramakrishna, Pendry, Wiltshire and Stewart [2003] have found that the effect of losses in bulk NIMs can be reduced by replacing a slab of bulk NIM with a layered stack of alternating NIM and PIM layers. A significant improvement can be achieved if the PIM layers are also made out of gain materials such as optically pumped semiconductors (fig. 5a) (Ramakrishna and Pendry [2003]). For example, in order to achieve a low-loss transmission in silver layers in red light, blue or ultraviolet pumped AlGaAs layers can be used as gain elements. A significant improvement in transmission has been demonstrated for the silver/gain material
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Negative refractive index metamaterials in optics
[1, § 2
Fig. 5. (a) NIM/PIM stack comprised of metal and gain material layers (Ramakrishna and Pendry [2003]); (b) schematic of a NIM utilizing pairs of metal strips embedded in gain material; (c,d) the transmission and reflection coefficients as well as the real and imaginary parts of the refractive index as functions of gain calculated for the structure shown in (b). (Adapted from Klar, Kildishev, Drachev and Shalaev [2006].)
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stack at optical frequencies. In addition, Tretyakov has suggested that both loss and bandwidth limitations can be efficiently reduced in active NIMs (Tretyakov [2001]). Klar, Kildishev, Drachev and Shalaev [2006] proposed immersing the NIM’s meta-atoms such as pairs of metal nanorods or double silver strips into the gain medium, as shown in fig. 5b. Coatings made of solutions of dye molecules, for example Rhodamine 6G, or semiconductor nanocrystals, such as CdSe, have been suggested as potential candidate materials for practical implementation of this approach. Figures 5c and d show the transmission and reflection coefficients as well as the real and imaginary parts of the refractive index as functions of gain calculated for the structure shown in fig. 5b. In this case the structure was illuminated with a plane wave at a wavelength of 584 nm from above. It shows that the structure becomes transparent at a gain level of 12,000 cm−1 , while the real part of the refractive index is practically unaffected by the presence of the gain material. Finally, Popov and Shalaev [2006a, 2006b] proposed an alternative way of compensating losses in optical NIMs by efficient optical parametric amplification enabled by new “backward” phase-matching conditions that will be discussed in § 6.2.
1, § 2]
Optical negative index metamaterials: State of the art
15
2.3. Alternative approaches to negative refraction Several alternative approaches to the realization of NIMs have been suggested, including photonic crystals (PC) (Notomi [2000]), anisotropic waveguides (Podolskiy and Narimanov [2005b]), organic and uniaxial gyrotropic crystals and a thin film on a metal substrate (Agranovich, Shen, Baughman and Zakhidov [2004b], Agranovich, Gartstein and Zakhidov [2006]), nanotransmission lines (Engheta, Salandrino and Alu [2005]), frequency-selective surfaces (Alu and Engheta [2005b], Khoo, Williams, Diaz, Chen, Bossard, Werner, Graugnard and Summers [2006]) and three- and four-level atomic systems (Oktel and Müstecaplıoglu [2004], Thommen and Mandel [2006]). 2.3.1. Photonic crystals In this subsection we discuss NIMs properties that can be achieved in PCs (Kosaka, Kawashima, Tomita, Notomi, Tamamura, Sato and Kawakami [1998], Notomi [2000], Gralak, Enoch and Tayeb [2000], Luo, Johnson, Joannopoulos and Pendry [2002], Foteinopoulou, Economou and Soukoulis [2003], Foteinopoulou and Soukoulis [2003], Parimi, Lu, Vodo and Sridhar [2003], Cubukcu, Aydin, Ozbay, Foteinopoulou and Soukoulis [2003a, 2003b], Berrier, Mulot, Swillo, Qiu, Thylén, Talneau and Anand [2004] and Schonbrun, Tinker, Park and Lee [2005]). PCs are dielectric or metallic structures periodically modulated on the scale of a wavelength that allow electromagnetic radiation of certain wavelengths to enter their structure but block other spectral components in a fashion analogous to the semiconductor bandgap (Yablonovitch [1987], John [1987]). PCs composed of dielectric materials have the important advantage of significantly lower loss compared to metallic NIMs at optical frequencies. While many applications of PCs rely on the presence of the photonic bandgap, interesting spatial and temporal effects, including negative refraction, have been predicted and demonstrated at frequencies outside the bandgap (Russell [1991], Lin, Hietala, Wang and Jones [1996], Eggleton, Slusher, de Sterke, Krug and Sipe [1996], Litchinitser, Eggleton and Patterson [1997], Kosaka, Kawashima, Tomita, Notomi, Tamamura, Sato and Kawakami [1998], Notomi [2000], Ozbay, Bulua, Aydina, Caglayana and Guvena [2004]). Notomi [2000] has pointed out that in weakly modulated photonic crystals the effective refractive index cannot be defined because the propagation direction depends on angle of incidence and wavelength, and represents a diffraction phenomenon. In strongly modulated photonic crystals, however, light propagation becomes refraction-like in the vicinity of the photonic band gap, where the effective refractive index can be defined. Its sign and the absolute value can
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Fig. 6. Analogy between effective-mass approximation for Bloch electron bands and effective-index approximation for Bloch photon bands. (Adapted from Notomi [2000].)
be varied by varying the crystal structure, the refractive indices of composing materials, and the wavelength. The effective index can be negative or less than unity. Finally, there is an interesting analogy between the photonic bandgap in PCs and the electronic bandgap in semiconductors, as shown in fig. 6. In a semiconductor, a negative-effective-mass state appears below the energy gap, and a positive-effective-mass state is above the gap. A similar behavior is seen for the effective refractive index in PCs. This is not surprising because the sign of the effective mass in semiconductors and the sign of the effective index in photonic crystals are both derived from the band curvature. Also, the effective-mass approximation is only valid near the bandgap. Similarly, the effective-index approximation is only valid near the photonic bandgap. While a negative refraction can indeed be realized in PCs, and has recently been demonstrated in the near infrared frequency region by Berrier, Mulot, Swillo, Qiu, Thylén, Talneau and Anand [2004], and Schonbrun, Tinker, Park and Lee [2005], not all unusual properties predicted for homogeneous NIMs can be observed straightforwardly in PCs. For instance, the amplification of evanescent field components contributing to super-resolution predicted by Pendry [2000] can only be realized to some extent in PCs. This is due to the fact that for the very large k-vector components responsible for the resolution of the smallest features, the PC cannot be considered as an effective medium and thus there is an upper limit on the transverse wavevector of evanescent components that can be amplified (Luo, Johnson, Joannopoulos and Pendry [2002]).
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Fig. 7. Isofrequency curves and relative directions of the wavevector k and the Poynting vector S for (a) isotropic material, (b) material with εx , εz > 0, (c) material with εx < 0, εz > 0. (Adapted from Alekseyev and Narimanov [2006].)
2.3.2. Anisotropic waveguides Another non-magnetic, non-resonant approach to the realization of negative refraction with low losses utilizes uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis (Belov [2003], Podolskiy and Narimanov [2005a], Podolskiy, Alekseyev and Narimanov [2005], Alekseyev and Narimanov [2006]). Figure 7 shows isofrequency curves and relative directions of the vectors S and k for a lossless isotropic medium and for the two cases of uniaxial anisotropy (εx , εz > 0 and εx < 0, εz > 0). In the anisotropic case the optical axis is pointing along the x axis. Since ordinary waves are not affected by the anisotropy, only extraordinary polarization is considered. In the isotropic case, the wavevector surfaces are circles, and therefore S ∝ ∇k ω(k) ∝ k, i.e. the vectors S and k are collinear (fig. 7a). For εx = εz , εx , εz > 0, the wavevector surfaces become ellipsoidal and, as a result, the angle between S and k is non-zero (fig. 7b). Finally, for a material with negative transverse dielectric permittivity εx < 0 and positive in-plane permittivity εz > 0, the dispersion relation becomes hyperbolic. The curvature of the hyperbola is such that Sz and kz have opposite signs (fig. 7c). The z-component of the Poynting vector for the extraordinary wave can be written as kz Sz = (2.3) H 2. 2ωεx
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Therefore, if εx < 0, Sz is also negative, and thus Sz is opposite to the direction of the wavevector component kz . This opposite directionality leads to a (effective) negative refractive index for refraction at an interface and for light propagation in a waveguide. In particular, an effective negative refractive index has been predicted for a waveguiding system when anisotropic material with εx < 0 and εz > 0 is used as the core of a planar (in the y–z plane) waveguide with metal walls. In this case modes with negative group velocity arise (Podolskiy and
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Fig. 8. Metamaterial structures for the potential realization of an anisotropic dielectric response at IR and optical frequencies: (a) layered structure based on semiconductor quantum wells or layered plasmonic or polar materials; (b) aligned plasmonic nanowires with negative εpl embedded in a dielectric host with positive εh ; (c) composite of spheroidal inclusions in an isotropic host.
Narimanov [2005a]); for these modes the wavevector and the energy flux are antiparallel. Although this approach has not yet been realized experimentally, several metamaterial-based structures have been proposed for the realization of the required anisotropic dielectric response at the IR and optical frequencies, as shown in fig. 8. These include: (a) A layered structure based on multiple semiconductor quantum wells (Podolskiy and Narimanov [2005b]) or on layered plasmonic (Au, Ag, Al) or polar (SiC) materials (Shvets [2003]) (fig. 8a). (b) A two-dimensional structure based on an array of aligned plasmonic nanowires with εpl < 0 embedded in a dielectric host with εh > 0 (Wangberg, Elser, Narimanov and Podolskiy [2006]) (fig. 8b). (c) A composite structure of sub-wavelength anisotropic (e.g. spheroidal) inclusions in an isotropic dielectric host (fig. 8c).
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2.3.3. Other designs In addition, several other promising possibilities of realization of NIMs at optical frequencies have been suggested. Agranovich, Shen, Baughman and Zakhidov [2004a] suggested the possibility of realizing negative refraction at optical frequencies in a number of non-magnetic media including bulk waves in organic crystals and in gyrotropic materials. In organic crystals, the effective mass of the Frenkel exciton can be designed to be negative, leading to a negative spatial dispersion for the exciton. This dispersion in turn leads to the appearance of additional exciton–polariton waves with a negative group velocity (shown in fig. 9a). Negative exciton–polariton dispersion also occurs in cubic or uniaxial gyrotropic crystals. The dispersion curve of excitons splits into two branches that couple with the light of two different circular polarizations to form two exciton–
1, § 2]
Optical negative index metamaterials: State of the art
19
Fig. 9. (a) Dispersion curve of exciton–polaritons near an exciton resonance. The polariton waves near point 2 have a negative group velocity. (b) Dispersion of polaritons in a cubic gyrotropic crystal near exciton resonance (or in uniaxial crystals with the wavevector along the optical axis); curves 1 and 2 correspond to right and left circular polarizations, polaritons in region 3 have a negative group velocity. (c) Surface plasmon dispersion of metal covered by a thin metal film for a particular set of parameters. (Adapted from Agranovich, Shen, Baughman and Zakhidov [2004b].)
polariton branches, one of which (lower branch) has a section with a negative slope corresponding to a negative group velocity as shown in fig. 9b. Opposite directionality of the group and phase velocities can also be realized in a structure consisting of a thin film on a metal substrate. When surface polaritons propagate along a metal surface covered with a thin film with a dipole resonance, gaps open up in the surface plasmon–polariton spectrum. As in the previous two cases, the lower branch of the surface polariton dispersion curve has a negative slope as shown in fig. 9c, and the corresponding surface polaritons exhibit a negative group velocity. However, it should be mentioned that losses associated with this structure may be significant. Engheta, Salandrino and Alu [2005] have shown theoretically that plasmonic and non-plasmonic nanoparticles that are significantly smaller than a light wavelength behave as nanoinductors and nanocapacitors, respectively, as shown in fig. 10a. When fused together, nanoparticles with positive and negative permittivities form parallel and serial nanoelements that can be arranged into optical nanotransmission lines. Two examples of such nanotransmission lines are shown in fig. 10b: series nanoinductors and shunt nanocapacitors providing a conventional transmission line operating in the optical regime (left), and shunt nanoinductors and series nanocapacitors forming a negative-index optical transmission line (right). Khoo, Williams, Diaz, Chen, Bossard, Werner, Graugnard and Summers [2006] have designed a metallo-dielectric frequency selective surface (FSS) that exhib-
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Fig. 10. (a) Nanoparticles behaving as nanocapacitors and nanoinductors. (b) Left: a conventional transmission line operating in the optical regime; right: a negative-index transmission line. (See Engheta, Salandrino and Alu [2005].)
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ited a refractive index with a negative real part ranging from −1.6 to −1.0 and a very small imaginary part, i.e. very low loss in the wavelength range around 3 µm. The FSS structure consisted of a 120 nm-thick silver screen sandwiched between two 0.508 µm-thick polyimide films with the unit cell size of the FSS of 2.75 × 2.75 µm2 . Also, it has been suggested that by incorporating a liquid crystal as an overlay onto these FSS-based structures it is feasible to design NIMs tunable in a wide range of frequencies in the visible and infrared ranges. Finally, the possibility of realization of electromagnetically induced negative refractive properties in three-level and four-level atomic media was proposed by Oktel and Müstecaplıoglu [2004] and Thommen and Mandel [2006], respectively. These systems would have the advantage of being homogeneous and not requiring any periodicity.
§ 3. Negative refraction and superlens 3.1. Negative refraction Negative refraction is one of the basic consequences of negative n. For a plane wave propagating from one medium into another the boundary conditions are
1, § 3]
Negative refraction and superlens
21
Fig. 11. (a) Negative refraction at NIM interface. (b) Experimental setup used for the demonstration of negative refraction. (c) Experimental results: transmitted power as a function of the refraction angle for a Teflon sample (dashed curve) and for a NIM sample (solid curve). (Adapted from Shelby, Smith and Shultz [2001].)
given by (1)
Et
(2)
= Et ,
ε1 Ez(1) = ε2 Ez(2) ,
(1)
Ht
(2)
= Ht ,
μ1 Hz(1) = μ2 Hz(2) .
(3.1)
These equations indicate that while the transverse components of the fields (denoted by a subscript “t”) are not affected by the signs of ε and μ, the longitudinal components (denoted by a subscript “z”) are. Negative refraction and the opposite directionality of the vectors k and S are illustrated in fig. 11a. While the Poynting vector is always directed away from the interface between the two media, the propagation vector k points towards the interface in NIMs. Figure 11b shows the experimental setup used in the first experiment demonstrating negative refraction (Shelby, Smith and Schultz [2001]). In this experiment, the deflection of a beam of microwave radiation was measured as the beam passed through a prism-shaped sample of NIM or a conventional PIM material (Teflon) for comparison. Figure 11c shows the transmitted power as a function of the refraction angle for both the Teflon sample (dashed curve) and the NIM sample (solid curve). Later, negative refraction was also reported by Houck, Brock and Chuang [2003]. Another interesting consequence of the refractive index being negative is the possibility of constructing an imaging system based on a flat slab of NIM with n = −1 surrounded by a conventional medium with n = 1, which was first discovered by Veselago [1968]. A truly remarkable finding was made by Pendry [2000], who noticed that a flat slab of NIM under appropriate conditions would not only focus propagating (or far-field) components emanating from the source, but also recover the evanescent
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(near-field) components that cannot be refocused by a conventional lens. Thus, Pendry named it a “superlens”.
3.2. Superlens The resolution of a conventional optical imaging system is limited by the wavelength of incoming light. The reason for the limited resolution is diffraction and the incapability of conventional imaging devices to re-focus evanescent field components. Consider an object and a lens placed along the z-axis so that the light from the object is propagating in the positive z direction. The field radiated from the object can be written in terms of its angular spectrum as E(x, y, z, t) = A(kx , ky ) exp i(kz z + ky y + kx x − ωt) , (3.2) where kz =
ω2 /c2 − kx2 + ky2
and only the positive square root is taken since the energy is propagating in the positive z direction. As long as (kx2 + ky2 ) < ω2 /c2 , all kz components are real and propagating and, therefore, phase correction is required to re-focus them in the image plane. However, if (kx2 + ky2 ) > ω2 /c2 , kz becomes imaginary, and the wave becomes an evanescent wave decaying along the positive z direction before reaching the image plane. The challenge here is that such waves decay in amplitude and, therefore, in order to focus them, amplitude correction (i.e. amplification) is required. Conventional lenses can only perform phase correction. Thus, evanescent components, responsible for the high-frequency and correspondingly small-scale features of the object, are usually lost in conventional imaging systems as illustrated in fig. 12a. The highest resolution can be estimated in terms of optical wavelength as follows:
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ω 2π (3.3) = , and therefore, xmin ≈ λ. c λ More precisely, the diffraction limit or Abbe limit for a conventional lens is given by x = λ/(2n sin α), where λ, n and α are the wavelength of light, the refractive index of the medium in object space (immersion material), and the semi-aperture angle of the lens, respectively. Therefore, x can be estimated as xmin ≈ λ/(2n), which is limited by the wavelength and the refractive index of available transparent immersion media. As mentioned above, Veselago [1968] noted that a planar slab of material with a refractive index of n = −1 and ε = μ = −1 in air (n = ε = μ = 1) can kmax ≈
1, § 3]
Negative refraction and superlens
23
Fig. 12. (a) Propagating (upper plot) and evanescent (lower plot) field components processed by a conventional lens; (b) propagating (upper plot) and evanescent (lower plot) field components processed by “Pendry’s” superlens.
produce an image of a point source as shown in the upper plot in fig. 12b. In a negative-index slab, diverging rays from an object are negatively refracted at the first surface of the slab, reversing their trajectories so that they form an image first within the slab, and then a second one outside the slab. Moreover, no reflection at the interfaces occurs owing to impedance matching. It is noteworthy that Veselago’s “planar lens” differs from a conventional lens in several respects: (1) it does not require any curvature, (2) it cannot focus parallel rays, (3) it does not have an optical axis, and (4) it cannot provide any magnification, i.e. its magnification is always unity. One of the most unusual properties of a planar slab of NIM was first realized by Pendry [2000], who found that a slab with refractive index n = −1 surrounded by air (n = 1) not only focuses the propagating field components but can also restore the evanescent field components, potentially improving the resolution in the image plane as schematically shown in the lower plot in fig. 12b. Thus, such a NIM slab has been called a “superlens”. The unique properties of such superlenses have stimulated an enormous interest as well as debates among researchers (Pendry [2000], ’t Hooft [2001], Williams [2001], Garcia and NietoVesperinas [2002], Shen and Platzman [2002], Ye [2003], Lagarkov and Kissel [2004], Merlin [2004], Larkin and Stockman [2005], Podolskiy and Narimanov [2005a], Jacob, Alekseyev and Narimanov [2006], Taubner, Korobkin, Urzhumov, Shvets and Hillenbrand [2006], Blaikie, Melville and Alkalsi [2006]).
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Owing to the opposite directionality of the phase velocity and the energy velocity in NIMs, for the waves propagating in the positive z direction kz = − ω2 /c2 − kx2 + ky2 . For the evanescent components the transmission coefficient is given by T = exp(αd), where α= kx2 + ky2 − ω2 /c2 . Therefore, the evanescent waves emerge from the far side of the slab enhanced in amplitude. This does not violate energy conservation because evanescent waves carry no energy. Figure 12b (lower plot) shows schematically an evanescent wave exponentially decaying away from an object and growing exponentially within the planar negative index slab. In the right half-space outside of the slab, the wave decays again until it reaches the image plane. Thus, the resolution limit of the planar negative index slab is determined by the number of evanescent field components from the object that can be recovered, rather than by the diffraction limit. However, in order to refocus those evanescent field components, the thickness d of the slab, the distance a between the object and the slab surface, and the distance b between the image and the slab surface must all be small in comparison to the wavelength. Otherwise, the evanescent wave components from the object decay to a level that their retrieval becomes unattainable due in particular to material losses of the slab. It has been shown that even a very small loss would cancel a superlensing effect (Podolskiy and Narimanov [2005a], Podolskiy, Kuhta and Milton [2005]). Recently, several ideas for converting the evanescent field components into propagating components, potentially increasing the resolution in the far-field zone, have been discussed by Durant, Liu, Fang and Shang [2006], Salandrino and Engheta [2006], and Jacob, Alekseyev and Narimanov [2006]. In addition, Smolyaninov, Hung and Davis [2006] have proposed and experimentally demonstrated a new design for a magnifying superlens in the visible range. An important limit is when the dimensions (a, d, b) of a system are much smaller than the wavelength, the electric and magnetic fields decouple, and the requirement for superlensing of the TM wave is reduced to only εslab = −εh , where εh is the permittivity of the host material surrounding the lens (Pendry [2000]). The corresponding condition for TE-wave superlensing is only μslab = −μh , where μh is the permeability of the host material surrounding the lens. The condition εslab = −εh corresponds to the condition for excitation of the surface plasmon resonance in a metal slab. This plasmon resonance is responsible for the building up of electromagnetic energy, which enables the exponential growth of evanescent
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components. A slab of silver illuminated at its surface plasmon resonance has already been used to demonstrate growth of evanescent waves, submicron imaging, and imaging well beyond the diffraction limit (Liu, Fang, Yen and Zhang [2003], Melville, Blaikie and Wolf [2004], Melville and Blaikie [2005], Fang, Lee, Sun and Zhang [2005]). Although limited to the near-field zone, this kind of nearfield superlens is likely to find many applications, including biomedical imaging and nanolithography. One of the limitations associated with a lens based on a bulk metal is that its operation is limited to a single wavelength satisfying the condition εslab (λ) = −εh . To overcome this limitation a superlens based on metal–dielectric composites has been proposed (Cai, Genov and Shalaev [2005]). The operation wavelength of such lens can be controlled by the metal filling factor of the composite. Pokrovsky and Efros [2003] have proposed another lens configuration, based on a flat slab of NIM. In contrast to the “Veselago” lens described above, the only condition that the new lens must satisfy is n2 = −n1 , but ε2 = −ε1 and μ2 = −μ1 . The most important difference between the new type of lens and the “Veselago” lens is the presence of reflected waves. As a consequence, the new lens produces a periodic array of 3D images of different intensities. Since only one condition has to be satisfied instead of the two (ε2 = −ε1 and μ2 = −μ1 ) in Veselago’s design, this new lens should be easier to fabricate.
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§ 4. Enhanced nonlinearity and its origin in metamaterials As discussed in the introduction, the interactions of the magnetic field component with conventional materials may become significant only at very high intensities of ∼1018 W/cm2 . On the other hand, metamaterials that enable the magnetic field component even in a linear regime are expected to give rise to a plethora of new nonlinear phenomena at moderate intensities used in conventional nonlinear optics. The majority of studies of NIMs to date have focused on linear properties, assuming that the dielectric permittivity ε and magnetic permeability μ are independent of the intensity of the electromagnetic field. The nonlinear properties of NIMs, where one or both material parameters are intensity-dependent, are not fully explored. However, even initial studies have already predicted novel fundamental properties of NIMs and entirely new regimes of nonlinear interactions. In this section we discuss the effects of NIM micro- or nanostructure on their nonlinear response. Zharov, Shadrivov and Kivshar [2003] have investigated the nonlinear response of a two-dimensional NIM periodic structure consisting of arrays of metallic wires
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Negative refractive index metamaterials in optics
[1, § 4
Fig. 13. Schematic of composite metamaterial structure. The lower inset shows a unit cell of the periodic structure. The upper inset shows the SRR equivalent oscillator. (See Zharov, Shadrivov and Kivshar [2003].)
and SRRs embedded into a nonlinear dielectric and operating at microwave frequencies (fig. 13). The metallic wires are responsible for the negative sign of the effective dielectric permittivity, and the SRRs provide the negative sign of the magnetic permeability. Two important contributions to the nonlinear properties of this structure have been identified. The first originates from the intensity-dependent part of the effective dielectric permittivity of the nonlinear dielectric host material, εD = εD (|E|2 ). Then, for the electric field orientation along the wires as shown in fig. 13, the effective nonlinear dielectric permittivity is obtained in the form
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εeff = εD |E|2 −
ωp2 ω(ω − iγe )
(4.1)
,
where ωp ≈ (c/d)[2π/ ln(d/r)]1/2 is the effective plasma frequency, c is the speed of light, d is the lattice constant, r is the wire radius, and γe = c2 /2σ S ln(d/r), with σ the conductivity of the wire metal, and S the effective area of the wire cross-section. The second contribution originates from the periodic structures of resonators, since the SRR capacitance depends on the strength of the local electric field in a narrow slot. The effective magnetic permeability has been found in the form μeff = 1 +
F ω2 ω02 NL (H ) − ω2
+ iΓ ω
,
(4.2)
where ω02 NL (H ) = (c/a)2 dg /πhεD (|Eg (H )|2 ) is the eigenfrequency of oscillations in the presence of the external field, h is the width of the ring, F =
1, § 5]
Optical bistability and solitons
27
πa 2 /d 2 1, Γ = c2 /(2πσ ah) if h is smaller than the thickness of the skin layer δ, and Γ = c2 /(2πσ aδ) for h > δ. Equation (4.2) shows that the resonant frequency of the array of the SRRs depends on the amplitude of the external magnetic field and, consecutively, leads to the intensity-dependent μeff . The magnetic nonlinearity in this case has been found to be much stronger than the nonlinearity in the dielectric properties owing to the field enhancement in the SRRs. The intensity-dependent μeff can be switched from positive to negative values, which in turn changes the material properties from PIM to NIM and back. It has been suggested that nonlinear NIMs can be created by inserting nonlinear elements into the slots of SRRs (Zharov, Shadrivov and Kivshar [2003], Lapine, Gorkunov and Ringhofer [2003]). For example, in the microwave range a nonlinear response has been obtained by inserting diodes into the SRRs (Lapine, Gorkunov and Ringhofer [2003]). This approach allows easy tuning of the NIM properties by an external field. A similar analysis for an SRR array alone and with parameters appropriate for optical frequencies has been performed by O’Brien, McPeake, Ramakrishna and Pendry [2004]; they demonstrated large local fields potentially leading to enhanced nonlinear effects and a bistable switching of the resonance frequency as a function of field strength. In another study a simple theory supporting experimentally observed SHG in magnetic metamaterials has been proposed (Klein, Enkrich, Wegener and Linden [2006]). Experimental results reported in this work indicate that much larger SHG signals are detected when magnetic-dipole resonances are excited, as compared with purely electric-dipole resonances. These results have been shown to be in a qualitative agreement with calculations based on the magnetic component of the Lorentz force exerted on metal electrons.
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§ 5. Optical bistability and solitons Solitons are fascinating nonlinear phenomena ubiquitous in sciences, including mathematics, optics, quantum mechanics, particle physics and molecular biology. In optics, the term “soliton” refers to a localized pulse that travels without changing its shape, and the difference between a “soliton” and a “solitary wave” is often disregarded. A variety of solitons, including temporal, spatial, spatiotemporal, incoherent, gap and discrete solitons, have been predicted and demonstrated in conventional PIM materials. The existence of solitons relies on the balance of nonlinear and dispersive (or diffractive) effects. Owing to the unique relationship between the phase and energy velocities, strong frequency dependence of
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[1, § 5
Fig. 14. Frequency-dependent effective dielectric permittivity (dashed line) and effective magnetic permeability (solid line) of the metamaterial. (Adapted from Shadrivov, Zharov and Kivshar [2006].)
the material parameters, and enhanced nonlinearities, new soliton phenomena are anticipated in NIMs. Recently several aspects of temporal and spatial dynamics and soliton propagation in NIMs have been addressed by Scalora, Syrchin, Akozbek, Poliakov, D’Aguanno, Mattiucci, Bloemer and Zheltikov [2005], Scalora, D’Aguanno, Mattiucci, Akozbek, Bloemer, Centini, Sibilia and Bertolotti [2005], Lazarides and Tsironis [2005], Kourakis and Shukla [2005], Shadrivov, Zharova, Zharov and Kivshar [2005], Shadrivov and Kivshar [2005], Gabitov, Indik, Litchinitser, Maimistov, Shalaev and Soneson [2006], Marklund, Shukla and Stenflo [2006], Kockaert, Tassin, Van der Sande, Veretennicoff and Tlidi [2006], and Tassin, Van der Sande, Veretenov, Kockaert, Veretennicoff and Tlidi [2006].
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5.1. Generalized nonlinear Schrödinger equation Figure 14 shows an example of frequency-dependent ε(ω) and μ(ω). In this example, a negative index of refraction occurs in the frequency range ω0 < ω < ωm , where ω0 is the resonant frequency of the metamaterial and both ε and μ are negative. In the frequency range ω > ωp , where ωp is the plasma frequency, the metamaterial possesses PIM properties. In the absence of losses, both NIM and PIM spectral regions are transparent. In the frequency regions ω < ω0 and ωm < ω < ωp , where dielectric permittivity and magnetic permeability have different signs, the refractive index is imaginary and as a result the material is opaque.
1, § 5]
Optical bistability and solitons
29
Owing to this strong frequency dependence, the optical pulse width is expected to have an important effect on the dynamics of nonlinear pulse propagation in NIMs. The pulse spectrum may overlap with one or more spectral regions in the metamaterial depending on its temporal width and shape (Agranovich, Shen, Baughman and Zakhidov [2004a]). There are two potentially important limits for pulse interactions with NIMs: (1) the wide spectrum of an ultrashort pulse covers frequency regions corresponding to the positive and negative refractive index as well as those where one of the material parameters is negative while the other is positive, and (2) the spectral content of the pulse is very narrow, so that the entire pulse spectrum falls in either the positive- or the negative-index region. We note here that the strong frequency dependence of the dielectric permittivity and magnetic permeability is one of the most fundamental properties of NIMs; it can be understood as follows: the energy flux density (Poynting vector) in a variable electromagnetic field is defined by eq. (1.2) regardless of whether dispersion is present. Using Maxwell’s equations, the rate of change of the energy in a unit volume can be written as 1 ∂D ∂B −∇ · S = (5.1) E +H . 4π ∂t ∂t
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In a dielectric medium without dispersion, when ε and μ are real constants, eq. (5.1) can be regarded as the rate of change of the electromagnetic energy W = εE 2 + μH 2 ,
(5.2)
which would be negative given that both ε and μ are less than zero. However, in a transparent dispersive medium the mean value of the electromagnetic energy is defined as
= ∂(ωε) E 2 + ∂(ωμ) H 2 . W (5.3) ∂ω ∂ω Equation (5.3) shows that in order for the total energy to be positive when both ε and μ are negative, it is necessary that ∂(ωε) ∂(ωμ) (5.4) > 0, > 0, ∂ω ∂ω which implies that ε(ω) and μ(ω) must be frequency dependent. Scalora, Syrchin, Akozbek, Poliakov, D’Aguanno, Mattiucci, Bloemer and Zheltikov [2005, see also the Erratum] derived a generalized nonlinear Schrödinger equation taking into account the frequency dependence of both dielectric permittivity and magnetic permeability and describing the propagation of ultrashort pulses in NIMs and, in fact, in a more general class of magnetically active
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[1, § 5
metamaterials:
iβμχ (3) ik ∂ 2 E μχ (3) ∂E 2 + |E| ≈ 1 − |E|2 E ∂z 2 ∂t 2 2n 4n2
γ + μ ∂t (|E|2 E) μ (3) − , +χ 2n ∂t 2Vg n2
(5.5)
where E is the electric field envelope function (not necessarily slowly varying), χ (3) is the cubic nonlinearity coefficient, k is the group velocity dispersion coefficient given by ∂ εγ + μα 1 1 1 k = (5.6) − αγ − β , = ∂ω Vg βn Vg2 4π and the coefficients α, α , γ , and γ are defined as ˜ ω)] ˜ ∂ 2 [ωε( ∂[ωε( ˜ ω)] ˜ , , α = ∂ ω˜ ∂ ω˜ 2 ∂[ωμ( ˜ ω)] ˜ ˜ ω)] ˜ ∂ 2 [ωμ( γ = , , γ = 2 ∂ ω˜ ∂ ω˜ Vg is the group velocity, β = 2π ω, ˜ ω˜ = ω/ωp , ωp is the plasma frequency. Equation (5.5) uncovers several new features that are distinct from a standard NLSE derived for non-magnetic materials. The linear and nonlinear coefficients can be tailored through the linear properties of the medium to attain any combination of signs unachievable in conventional materials, thus having a significant potential for the realization of a wide class of solitary waves. In particular, the sign of the group-velocity dispersion can be positive or negative and is determined by the particular choice of ε and μ parameters [eq. (5.6)]. Also, assuming a positive χ (3) , the sign of the leading nonlinear coefficient is always positive because the ratio μ/n is positive. Therefore, both bright and dark soliton solutions have been predicted to emerge depending on the sign of the group velocity. On the other hand, the pseudo-χ (5) correction term is proportional to (−μ2 /n3 ), which makes the coefficient positive. Therefore, its effect is to enhance the nonlinearity, which is in sharp contrast to the case of conventional materials where the effect of this term is to quench the nonlinearity. α=
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5.2. Solitons in plasmonic nanostructures Gabitov, Indik, Litchinitser, Maimistov, Shalaev and Soneson [2006] have considered short-pulse propagation in the simplest case of a nanostructure consisting
1, § 5]
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31
Fig. 15. (a) Schematic of transitions in a nanostructure consisting of metallic nanoparticles embedded in a glass host (ωr is the frequency of plasmonic oscillations, ω0 is the light carrier frequency, ωh is the resonance frequency of the host material); (b,c) soliton amplitude and phase for three different soliton velocities.
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of metallic nanoparticles embedded in a glass host, such that the resonance frequencies of the host medium are well separated from those of the nanoparticles, as schematically shown in fig. 15a. Examples include silver or gold spherical or spheroidal nanoparticles embedded in SiO2 . In these cases, the plasmonic resonance frequencies are in the visible part of the spectrum, whereas the resonance of the host is in the ultraviolet. Although this structure possesses only an electric resonance (i.e. resonance of ε) and, therefore, does not produce a negative refractive index, it has been found that it provides a good simplified model providing important insight into the nonlinear optical properties of more complex structures such as plasmonic NIMs. The origin and magnitude of third-order nonlinearity in such nanostructures are well characterized theoretically and experimentally (Uchida, Kaneko, Omi, Hata, Tanji, Asahara, Ikushima, Tokizaki and Nakamura [1994], Rautian [1997], Drachev, Buin, Nakotte and Shalaev [2004]). In particular, it has been shown that quantum effects in metal nanoparticles driven by a resonant optical field play an important role in inducing a strong nonlinear response (Drachev, Buin, Nakotte and Shalaev [2004]). Light interaction with metal nanoparticles can be described by a system of equations consisting of Maxwell’s equation for the electric field and an oscil-
32
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[1, § 5
lator equation describing the displacement of conduction electrons in the metal nanoparticles from equilibrium (plasmonic oscillations). Assuming pulse dynamics to vary on a scale that is much slower than the plasmonic, host atom, and carrier wave oscillations, the interaction of the electric field with plasmonic oscillations in nanoparticles in the presence of this cubic nonlinearity can be described by the forced Duffing equation in the slowly varying envelope approximation, i
∂Q 3γ e |Q|2 Q = − E, + (ωr − ω0 )Q + ∂t 2ω0 2mω0
(5.7)
where E and Q are the slowly varying envelopes of the electric field and plasmonic oscillations, respectively, t is time, γ is the coefficient of nonlinearity, and e and m are the electron charge and rest mass. The equation for the electric field envelope is 2πω0 Np e ∂E 2πω0 Na |d|2 1 ∂E i Q − E + =− ∂z vg ∂t cn0 cn0 h¯ a −
2πiω0 Na |d|2 ∂E , ∂t cn0 h ¯ 2a
(5.8)
where z is the propagation coordinate, vg is the group velocity, c is the speed of light, n0 is the refractive index evaluated at the carrier frequency ω0 , and Np is the product of conduction electron density N and the metal-filling factor p. The parameter Na is the concentration of host atoms, d is the projection of the dipole matrix element in the direction of the electric field polarization, and a = ωh −ω0 is the detuning from the resonance frequency of the host atoms. The last two terms in eq. (5.8) are the additional corrections to the refractive index and group index accounting for the off-resonance interaction with the host medium (taken as an ensemble of two-level atoms). Equations (5.7)–(5.8) represent a generalization of the classical Maxwell–Lorentz model and are referred to as the Maxwell–Duffing model. Solitary-wave solutions of eqs. (5.7)–(5.8) have been found analytically by Gabitov, Indik, Litchinitser, Maimistov, Shalaev and Soneson [2006]. These are four-parameter solutions characterized by solitary wave velocity, frequency, phase shift, and the initial pulse position. Electric field amplitudes and phases of the solitary waves are shown in figs. 15b and c for different soliton velocities. Numerical simulations of the Maxwell–Duffing system of equations have indicated that input Gaussian pulses with sufficient amplitudes evolve to stable solitary waves and exhibit behavior analogous to the self-induced transparency in the Maxwell–Bloch system. The simulations have also revealed that the collision dynamics is strongly dependent on initial solitary wave parameters, leading to principally different regimes of interactions: in one regime the collisions are
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Optical bistability and solitons
33
Fig. 16. Electric field and medium excitation corresponding to an inelastic collision resulting in the formation of a “hot spot” – a region where the energy of the solitary wave is transferred to the material excitation.
quasi-elastic, in another regime the collisions are inelastic and at certain parameters result in the development of “hot spots” – regions where the energy of the solitary wave is transferred to the material excitation shown in fig. 16. The formation of such hot spots represents a form of light trapping and, therefore, has high potential for developing phase-controlled optical storage applications. As already mentioned, understanding the properties of light propagation in a nanostructure consisting of metallic nanoparticles embedded in a glass host is useful for studying wave dynamics in NIMs that simultaneously possess electric and magnetic resonances. A particular case of a nanostructure with a nonlinear electric response and a linear magnetic response was analyzed by Gabitov, Indik, Litchinitser, Maimistov, Shalaev and Soneson [2006]. It was found that the governing nonlinear equations for the electric and magnetic fields decouple, resulting in a system of equations very similar to that derived for the simplest nanostructures composed of metallic nanoparticles when μ = 1. Therefore, many of the results obtained for the simple structure are relevant for NIMs with a nonlinear electric response and a linear magnetic response. Although only the effects of nonlinear polarization have been analyzed, the contribution of magnetic nonlinearities may be even more pronounced than that of electric nonlinearities (Zharov, Shadrivov and Kivshar [2003]). Therefore, a potentially important extension of this work would take into account nonlinear polarization and nonlinear magnetization as well as the effects of a particular nanostructure.
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5.3. Gap solitons Conventional one-dimensional layered periodic structures composed of two PIMs possess a photonic bandgap, also called a Bragg gap (Joannopoulos, Meade and
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Negative refractive index metamaterials in optics
[1, § 5
Winn [1995]). The intensity-dependent refractive index of the constituent layers allows tuning the properties of the photonic bandgap structure, resulting in optical switching, hysteresis, and bistability. Also, in the nonlinear regime, nonlinearity combined with a strong dispersion owing to the periodic structure may lead to the formation of gap solitons (Winful, Marburger and Garmire [1979], Chen and Mills [1987a], Eggleton, Slusher, de Sterke, Krug and Sipe [1996]). Gap solitons are intense pulses that can propagate inside the photonic bandgap. An important property of gap solitons is their ability to travel at group velocities lower than the speed of light in the medium, making them promising candidates for optical information storage applications. In addition to a conventional photonic bandgap, a one-dimensional periodic structure consisting of alternating layers of PIM and NIM displays a new type of photonic bandgap, a so-called zero-refractive-index or “zero-n” gap (Li, Zhou, Chan and Sheng [2003]). This new gap is unique to NIM/PIM periodic structures, as it has been shown to exist in a range of frequencies corresponding to a zero (volume) averaged refractive index. The gap due to n¯ = 0 possesses several unique properties that distinguish it from a Bragg gap: While the Bragg frequency corresponding to the center of the Bragg gap is strongly dependent on the period of the structure (or the lattice constant), the zero-n¯ gap is independent of periodicity and remains invariant with the scaling of the lattice constant. In contrast to the Bragg gap, the zero-n¯ gap is robust in the presence of the disorder introduced by varying the layer thicknesses and is insensitive to input angle. In the presence of Kerr nonlinearity, a zero-n gap has been shown to switch from low transmission to a perfectly transmitting state and to form a novel kind of gap soliton (Hegde and Winful [2005a, 2005b]). This zero-n gap soliton has been shown to possess several superior properties in comparison with conventional gap solitons, including relative insensitivity to the angle of incidence and robustness in the presence of structural disorder and material loss. These features significantly simplify the excitation of these new solitons and the realization of their potential practical applications. Gap solitons have also been predicted even in a single optically thick slab of NIM (D’Aguanno, Mattiucci, Scalora and Bloemer [2004]). The gap in a single slab of NIM originates from the frequency-dependent NIM parameters ε and μ that, for example, can take the form (Ziolkowski and Hayman [2001], D’Aguanno, Mattiucci, Scalora and Bloemer [2004])
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(ωpm /ωpe )2 1 (5.9) , μ(ω) ˜ =1− , ω( ˜ ω˜ + iγ˜e ) ω( ˜ ω˜ + iγ˜m ) where ω˜ = ω/ωpe is the normalized frequency, ωpe and ωpm are the respective electric and magnetic plasma frequencies, and γ˜e and γ˜m are the respective ε(ω) ˜ =1−
1, § 5]
Optical bistability and solitons
35
Fig. 17. (a) Refractive index as a function of normalized frequency. (b) Linear transmittance of the NIM slab. (See D’Aguanno, Mattiucci, Scalora and Bloemer [2004].)
normalized electric and magnetic loss terms. Figure 17 shows the frequencydependent refractive index and linear transmittance of a single nonlinear NIM slab. The spectral width of the gap ωg and the center-gap frequency ωgc depend on the electric and magnetic plasma frequency as follows: ωg = |ωpe − ωpm | and ωgc = (ωpe + ωpm )/2. While it is expected that a single slab of dispersive material with a cubic nonlinearity supports soliton waves in general, a remarkable feature of the NIM slab is that it supports both bright and dark gap solitons. In the case of self-focusing nonlinearity and ωpm /ωpe < 1, bright solitons are excited near the high-frequency edge of the gap where n > 0, while dark solitons are excited near the low-frequency edge where n < 0.
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5.4. Optical bistability Optical bistability is a class of optical phenomena in which a nonlinear system can exhibit two steady transmission states for the same input intensity (Gibbs [1985]). The input–output characteristic of such a system forms a hysteresis loop. Optical bistability has been predicted and experimentally realized in various settings, including a Fabry–Perot resonator filled with a nonlinear material (Gibbs [1976]), layered periodic structures (Winful, Marburger and Garmire [1979]), and a nonlinear slab (Chen and Mills [1987b], Basharov [1988], Trutschel and Lederer [1988], Trutschel, Lederer and Langbein [1989]). A nonlinear film surrounded by a linear dielectric with a high refractive index is known to exhibit bistability and more generally multistability when illuminated at an angle θin ,
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Negative refractive index metamaterials in optics
[1, § 5
such that θres < θin < θTIR , where θres is the angle corresponding to the resonant peak nearest to the angle of total internal reflection (TIR) θTIR in the linear transmission curve (Trutschel, Lederer and Langbein [1989]). In this configuration, transmission in the linear regime is low. However, as the incident intensity increases, in the case of self-focusing Kerr nonlinearity, the nonlinear refractive index increases, resulting in a shift of both θTIR and θres to larger values. Simultaneously, the transmission coefficient becomes a multi-valued function of the input flux, leading to bistable behavior. Optical bistability has been predicted and numerically studied in various NIM structures, including symmetric and asymmetric layered periodic structures with multiple NIM layers or a NIM defect (Feise, Shadrivov and Kivshar [2004, 2005], Hegde and Winful [2005b]), a Fabry–Perot resonator containing a nonlinear NIM material (D’Aguanno, Mattiucci, Scalora and Bloemer [2004]). Since currently optical NIMs are available only in the form of a thin film, the nonlinear transmission properties of a layered structure consisting of a nonlinear slab and a sub-wavelength layer of NIM (with a width similar to the first experimentally demonstrated NIM films) have been investigated (Litchinitser, Gabitov, Maimistov and Shalaev [2007]). It has been shown that even such a thin film of NIM could significantly modify the nonlinear response of such a bilayer. The optical bistability phenomenon in these structures may be used as a novel tool for NIM characterization, owing to the very high sensitivity of the hysteresis width and depth to the changes of the NIM parameters, as shown in fig. 18. In addition, nonreciprocal transmission with enhanced operational range has been demonstrated numerically, enabling novel device applications such as optical memory and the optical diode.
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5.5. Ultra-narrow spatial solitons Boardman, Velasco, King and Rapoport [2005] and Boardman, Egan, Velasco and King [2005] have numerically studied interactions of so-called “optical needles” with NIMs. An optical needle is a form of spatial soliton or stable self-channeling with a channel width smaller than the linear wavelength of the light in a medium with a self-focusing nonlinearity. Owing to their extremely high concentration of optical power, optical needles are attractive from both fundamental and applied standpoints. Figure 19a presents the example of a soliton lens comprised of a slab of linear NIM and a linear PIM, which has been shown to enable the distortionless delivery of an ultranarrow soliton from one nonlinear medium to another through an entirely linear region. This soliton lens is based on the concept of ‘complementary
1, § 5]
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37
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Fig. 18. Schematics of a bilayer structure comprising (a) PIM thin film and (b) NIM thin film. (c,d) Transmission coefficient versus input flux for fixed ε and varying μ of (c) the PIM thin film and (d) the NIM thin film.
media’ introduced by Pendry and Ramakrishna [2003], which are known to optically cancel out one another at a specific frequency. The simulation results based on the finite-difference time-domain method illustrating the process of diffraction cancellation in a soliton lens are shown in fig. 19b. The possibility of delivering optical needle-like beams from one point in space to another may be important for the development of future sub-wavelength all-optical chips.
Fig. 19. (a) A soliton lens comprised of a slab of linear NIM and PIM; (b) diffraction cancellation in a soliton lens.
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Negative refractive index metamaterials in optics
[1, § 6
Other novel soliton phenomena predicted in NIMs include single- and multihump, symmetric and antisymmetric, and asymmetric spatial solitons (Shadrivov and Kivshar [2005]) and nonlinearity-induced wave transmission through an opaque NIM slab accompanied by the generation of spatiotemporal solitons (Shadrivov, Zharova, Zharov and Kivshar [2005]).
§ 6. “Backward” phase-matching conditions: Implications for nonlinear optics Wave mixing represents one of the most general and important processes in nonlinear optics. If two electromagnetic waves with carrier frequencies ω1 and ω2 propagate in a quadratic nonlinear medium characterized by a nonlinear (elec(2) tric) susceptibility χp , a number of new frequencies can be generated, including ω1 ± ω2 (sum- and difference-frequency generation), 2ω1 and 2ω2 (secondharmonic generation, SHG). However, the relative efficiency of these processes varies in dispersive media. The so-called phase-matching condition determines which process dominates. The equations describing the three wave interactions in a χ (2) -medium in the slowly varying envelope and phase approximation can be written in the following form (Shen [1984]): 2πω12 μ(ω1 ) NL ˆk1 ∂ + 1 ∂ A1 = i P (ω1 ) exp(−ik1 z), ∂z v1 ∂t c 2 k1 2πω22 μ(ω2 ) NL ∂ 1 ∂ P (ω2 ) exp(−ik2 z), + A2 = i kˆ2 ∂z v2 ∂t c 2 k2 2πω32 μ(ω3 ) NL ˆk3 ∂ + 1 ∂ A3 = i (6.1) P (ω3 ) exp(−ik3 z), ∂z v3 ∂t c 2 k3 where kj2 = (ωj /c)2 ε(ωj )μ(ωj ), kˆj is the sign of ε(ωj )μ(ωj ), n2j = ε(ωj )μ(ωj ) is the refractive index squared, and P NL (ω1 ) = χ (2) (ω1 ; ω3 , −ω2 )A3 A∗2 exp iz(k3 − k2 ) , P NL (ω2 ) = χ (2) (ω2 ; ω3 , −ω1 )A3 A∗1 exp iz(k3 − k1 ) , P NL (ω3 ) = χ (2) (ω3 ; ω1 , ω2 )A1 A2 exp iz(k1 + k2 ) . (6.2)
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The right-hand sides of eqs. (6.1) are not affected by the signs of ε and μ; how ever, the direction of wave propagation is defined by the sign of ε(ωj )μ(ωj ). Equations (6.1) describe the process of SHG if ω1 = ω2 ≡ ω, A1 ≡ A2 ≡ Aω are the frequency and the amplitude of the fundamental wave, and ω3 = 2ω1 ≡ 2ω,
1, § 6]
“Backward” phase-matching conditions: Implications for nonlinear optics
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A3 ≡ A2ω are the frequency and the amplitude of the second harmonic. The process of parametric amplification is described by eqs. (6.1) if ω1 ≡ ωs , A1 ≡ As correspond to the frequency and the amplitude of the signal wave, ω2 = ωi , A2 ≡ Ai are those of the idler wave, and ω3 = ωp , A3 ≡ Ap correspond to the pump wave. Also, it is convenient to re-define ε(ωj ) ≡ εj and μ(ωj ) ≡ μj .
6.1. Second-harmonic generation Second-harmonic generation is a nonlinear process, in which photons at fundamental frequency ω interacting with a nonlinear material are combined to create new photons with a doubled (“second harmonic”) frequency 2ω. In NIMs fundamentally new regimes of SHG can be realized owing to the inherent frequency dependence of material properties and their unique property of antiparallel phase and energy velocity directions (Zharov, Zharova, Shadrivov and Kivshar [2005], Popov and Shalaev [2006a], Popov, Slabko and Shalaev [2006], Shadrivov, Zharov and Kivshar [2006]). Note that the SHG process is considered here for the case of nonlinear polarization instead of that of nonlinear magnetization, as in Shadrivov, Zharov and Kivshar [2006], and Popov and Shalaev [2006a], in order to compare to the conventional case of SHG in PIMs. The equations for the electric and magnetic field components are very similar, with the exception that the factor ε(ω) entering the equations for the magnetic field is replaced by the factor μ(ω) when the equations are re-written for the electric field. Figure 20a shows a schematic of the SHG process in a NIM slab. It is assumed that the material is a NIM at the fundamental frequency ω and it is a PIM at the second-harmonic frequency 2ω. The energy flow of the fundamental frequency is assumed to be from left to right. Since in NIMs the directions of the Poynting vector (energy flow) and the phase velocity are opposite, the phase of the wave at the fundamental frequency travels from right to left. In order to satisfy the phase-matching requirement k2ω = 2kω , the k-vector of the generated second harmonic should also travel from right to left. Since the second harmonic propagates in the PIM, its energy flow is co-directed with the phase velocity and, therefore, the energy propagates from right to left as well, as shown in fig. 20a. This new type of phase-matching condition, “backward phase-matching”, has been shown to enable a fundamentally new SHG regime. For comparison a schematic of the SHG process in a PIM is shown in fig. 20b.
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Fig. 20. (a,b) Schematic of the SHG process in (a) NIMs and (b) PIMs. (c,d) Fundamental and second-harmonic field amplitudes squared versus z in (c) NIMs and (d) PIMs. (Adapted from Popov and Shalaev [2006a].)
For SHG in a NIM eqs. (6.1) take the form 2πω2 μω NL 1 ∂ ∂ P (ω) exp(−ikω z), Aω = i 2 − + ∂z vω ∂t c kω 2π(2ω)2 μ2ω NL ∂ 1 ∂ P (2ω) exp(−ik2ω z). + A2ω = i ∂z v2ω ∂t c2 k2ω
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(6.3)
Assuming the continuous-wave case and using the symmetry properties of the tensor χ (2) (Shen [1984], Boyd [1992]), eqs. (6.3) can be written in the form ∂Aω 2Kω2 μω A2ω A∗ω exp[−i kz], = −i 2 ∂z c kω ∂A2ω 4Kω2 μ2ω 2 Aω exp[i kz], =i 2 ∂z c k2ω where k = 2kω − k2ω , K =
2π (2) χ (2ω) c2
k2ω d|A2ω |2 kω d|Aω |2 − = 0, μω dz 2μ2ω dz
=
(6.4)
π (2) χ (ω), c2
(6.5)
or for the energy flow d|Sω | d|S2ω | − = 0. (6.6) dz dz Assuming that the phase-matching condition k2ω = 2kω is satisfied (implying that εω = −ε2ω and μω = −μ2ω ), the spatially invariant Manley–Rowe relations take
1, § 6]
“Backward” phase-matching conditions: Implications for nonlinear optics
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the form |Aω |2 − |A2ω |2 = C 2 = const.
(6.7)
In the conventional PIM case, the Manley–Rowe relations require that the sum of the squared amplitudes is constant. The unusual form of the Manley–Rowe relations in NIMs is explained by the fact that the Poynting vectors for the fundamental and the second harmonic are antiparallel, while their wavevectors are parallel. The equations for the real amplitudes and phases, introduced as Aω,2ω = eω,2ω exp(iϕω,2ω ), take the form deω = κeω e2ω sin(θ ), dz dϕω = −κe2ω cos(θ ), dz
de2ω = κeω2 sin(θ ), dz dϕ2ω e2 = κ ω cos(θ ), dz e2ω
(6.8)
where θ = ϕ2ω −2ϕω , κ = 4Kω2 μ2ω /(c2 k2ω ). Taking into account the equations for ϕω and ϕ2ω , it is found that θ = 3π/2. Then, de2ω = −κeω2 = −κ C 2 + e2ω (z)2 . (6.9) dz Note that in the PIM case the right-hand sides of eqs. (6.8) for the amplitudes would have opposite signs, while in the case of NIMs the signs are the same. Also, the boundary conditions in the NIM case are specified at opposite interfaces of the slab of a finite length L and are given by eω (0) = e10 and e2ω (L) = 0, while in PIMs both conditions are specified at the front interface and are given by eω (0) = e10 , e2ω (0) = 0. Then, the solutions for eω,2ω in a lossless NIM slab of length L are obtained in the form
EBL
eω (z) =
C , cos[Cκ(L − z)]
e2ω (z) = C tan Cκ(L − z) ,
(6.10)
where CκL = arccos(C/e10 ).
(6.11)
Therefore, the spatially invariant intensity difference given by eq. (6.7) depends on the slab thickness. Equation (6.7) shows that the incoming radiation at the fundamental frequency can be converted to the second harmonic frequency propagating in the opposite direction with an efficiency approaching 100% in a semiinfinite lossless NIM slab. Thus, the NIM slab acts as a nonlinear mirror. Owing to the boundary conditions for the second harmonic at the rear interface of the slab, the conversion at any point within the NIM slab depends on the total thickness of
42
Negative refractive index metamaterials in optics
[1, § 6
the slab. In contrast, in the PIM case the solutions for eω,2ω are C z z , e2ω (z) = C · tanh eω (z) = , z0 cosh z0 1 . where C = e10 , z0 = (6.12) ke10 Figures 20c and d compare the solutions for the NIM and PIM cases. Another unusual case of SHG in NIMs is realized when the second-harmonic wave corresponds to the NIM’s transparent spectral region, while the fundamental wave belongs to the opaque spectral region (Agranovich, Shen, Baughman and Zakhidov [2004a], Zharov, Zharova, Shadrivov and Kivshar [2005]). While the fundamental wave does not propagate in such a metamaterial slab, the material is transparent for the second harmonic frequency. It has been shown that this slab may operate as a nonlinear lens that would provide an image of the source at the second harmonic with sub-wavelength resolution and, therefore, provides an additional means of loss mitigation in NIMs (Zharov, Zharova, Shadrivov and Kivshar [2005]). SHG in NIMs has also been studied in cavity environments such as a Bragg grating made of alternating NIM/PIM layers (D’Aguanno, Mattiucci, Bloemer and Scalora [2006]). Improved conversion efficiencies, owing to the presence of NIM layers, have been predicted. Finally, Maimistov, Gabitov and Kazantseva [2007] have considered the steady-state propagation of coupled wave packets in the quadratic nonlinear NIM and have found that under some conditions selftrapping of the interacting wave packets takes place. As a result, a two-frequency wave packet (simulton) can be generated with distinct properties as compared with those of solitons in quadratic nonlinear PIMs.
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6.2. Optical parametric amplification The backward phase-matching condition has been shown to have a profound effect on the parametric interactions in NIMs. Figures 21a,b show schematics of optical parametric amplification processes in NIMs and PIMs. Assuming that the pump ω3 and idler ω2 frequencies belong to the PIM range and the signal ω1 frequency belongs to the NIM range in undepleted pump approximation, i.e. A3 = Ap = const, eqs. (6.1) can be written as follows: 4πω12 χ (2) μ1 dA1 A3 A∗2 exp(i kz), = −i dz c 2 k1 4πω22 χ (2) μ2 dA2 A3 A∗1 exp(i kz), =i dz c 2 k2 where k = k3 − k1 − k2 , ω2 = ω3 − ω1 .
(6.13)
1, § 6]
“Backward” phase-matching conditions: Implications for nonlinear optics
43
Fig. 21. (a,b) Schematics of optical parametric amplification in (a) NIMs and (b) PIMs. (c,d) The phase-matched amplification factor for the signal wave, and the conversion factor for the idler wave in (c) NIM and (d) PIM with absorption (α1 L = 1, α2 L = 1/2); (d) represents the dependence schematically. (Adapted from Popov and Shalaev [2006b]).
Since loss is one of the major problems in existing NIMs, and especially in those operating at optical frequencies, optical parametric amplification has been proposed as a means to overcome dissipative losses (Popov and Shalaev [2006a]). Then, eqs. (6.13) can be re-written to include the effect of losses α1 and α2 as α1 da1 = −iga2∗ exp(i kz) + a1 , dz 2 da2 α 2 = iga1∗ exp(i kz) − a2 , dz 2
EBL
where aj =
4
εj Aj √ μj ω j
and
g=
(6.14)
√ ω1 ω2 4π (2) . χ A3 √ 4 c ε1 ε2 /μ1 μ2
Note that the boundary conditions for a1 are defined at the opposite side of the slab (of thickness L), as compared with a2 and are given by a1 (L) = a1L and a2 (0) = a20 . With these boundary conditions, the solutions to eqs. (6.14) take the form (Popov and Shalaev [2006a, 2006b]) a1 (z) = C1 exp β1+ z + C2 exp β2+ z , a2∗ (z) = κ1 C1 exp β1− z + κ2 C2 exp β2− z , (6.15) where i k α1 − α2 , β1,2 = + iR, 2 4 ∗ exp(β + L) a1L κ2,1 − a20 2,1 , C1,2 = ± D (±R + is) κ1,2 = , D = κ2 exp β1+ L − κ1 exp β2+ L , g ± β1,2 = β1,2 ±
44
Negative refractive index metamaterials in optics
[1, § 7
α1 + α2 i k − . 4 2 Solutions (6.15) describe several unusual properties of parametric amplification in NIMs. In particular, the amplification factor for the signal wave and the conversion efficiency for the idler wave show oscillatory behavior even at k = 0, which is in sharp contrast to those in PIM as illustrated in figs. 21c and d. The reason for this behavior is that the signal and the idler are determined by the boundary conditions on opposite sides of the slab and they increase in opposite directions. Recently, parametric amplification has been demonstrated experimentally in negative-index nonlinear transmission line media (Kozyrev, Kim and van der Weide [2006]). Finally, it is also noteworthy that the possibility of oscillations without a cavity has been found under certain conditions in the NIM slab geometry considered in this section (Popov and Shalaev [2006b]). R=
g2 − s 2,
s=
§ 7. Surface polaritons, waveguides and resonators Most of the fascinating properties of NIMs reveal themselves when NIMs are combined with PIMs either in the frequency domain, as discussed in the preceding section, or in the spatial domain. A number of new surface and waveguiding phenomena have been predicted for NIMs and PIMs combined spatially. For example, surface waves (or surface plasmons) are responsible for the improved resolution of a superlens as discussed above. Therefore, a detailed analysis of their properties is of paramount importance.
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7.1. Linear surface polaritons A number of unique and fascinating properties of NIM-based guiding structures have been discovered by Ruppin [2000], Darmanyan, Neviere and Zakhidov [2003, 2005], Shadrivov, Sukhorukov, Kivshar, Zharov, Boardman and Egan [2004], and Boardman, Egan, Velasco and King [2005], Boardman, Velasco, King and Rapoport [2005], D’Aguanno, Mattiucci, Scalora and Bloemer [2005]. Surface polaritons, first predicted by Sommerfeld [1909], are electromagnetic waves propagating along the interface between two media, such that their amplitudes decay exponentially away from the interface. In this subsection both media are assumed to be linear and isotropic, with material parameters (ε1 , μ1 ) and (ε2 , μ2 ) that can be either positive or negative. The dispersion relation for the TM (TE) type of surface polaritons can be obtained from the solution of the wave equation
1, § 7]
Surface polaritons, waveguides and resonators
45
for the y component of the magnetic (electric) field ∂ 2 Hy ∂ 2 Hy + + k02 εμHy = 0 for the TM wave, ∂z2 ∂x 2 ∂ 2 Ey ∂ 2 Ey + + k02 εμEy = 0 for the TE wave, (7.1) ∂z2 ∂x 2 where k0 = ω/c, ω is the wave frequency, c is the speed of light in vacuum, ε = ε1 , μ = μ1 for x < 0 and ε = ε2 , μ = μ2 for x > 0. Surface polariton solutions of eqs. (7.1) can be written as Hy = A0 exp iβz − κ1,2 |x| , (7.2) Ey = A0 exp iβz − κ1,2 |x| , where A0 is the wave amplitude at the interface, β is the propagation constant, and κ1,2 = [β 2 − ε1,2 μ1,2 k02 ]1/2 are the transverse wave numbers. The dispersion relations for the two types of surface polaritons are obtained from the requirement that the tangential components of the fields should change continuously across the interface (Ruppin [2000]) κ1 ε2 + κ2 ε1 = 0 for the TM wave, κ1 μ2 + κ2 μ1 = 0 for the TE wave.
EBL
(7.3)
It is assumed here that ε1 , ε2 , μ1 , μ2 are real, and therefore, κ1 , κ2 have to be real and positive so that the field amplitudes decay exponentially away from the interface. Then, the following conditions can be found: β 2 > ε1 μ1 k02 , ε2 μ2 k02 .
(7.4)
Following Shadrivov, Sukhorukov, Kivshar, Zharov, Boardman and Egan [2004] the dispersion relations for the two cases can be written as ε1 ε2 (ε2 μ1 − ε1 μ2 ) X(X − Y ) for the TM case, = ε1 μ1 k02 2 2 (X 2 − 1) (ε2 − ε1 ) μ1 μ2 (ε1 μ2 − ε2 μ1 ) Y (Y − X) β 2 = k02 for the TE case, (7.5) = ε1 μ1 k02 (Y 2 − 1) (μ22 − μ21 )
β 2 = k02
where X = |ε2 |/ε1 and Y = |μ2 |/μ1 . Equations (7.5) define the existence conditions for the surface polaritons. Equations (7.3) and (7.5) for the TM case reduce to the well-known dispersion relation for surface polaritons in conventional materials if both media are non-magnetic. The existence of a TM surface polariton at a PIM/PIM interface requires negative dielectric permeability in one of the semiinfinite media. A TE surface polariton is not supported on such interface of two non-magnetic materials. The existence of a TE surface polariton at a PIM/PIM interface requires negative magnetic permeability on one side of the interface. As
46
Negative refractive index metamaterials in optics
[1, § 7
Fig. 22. (a) Existence regions of surface waves on the parameters X and Y . The inset shows the schematics of the geometry and surface wave profile. (b–e) Surface wave profiles for interfaces of different combinations of linear/nonlinear NIMs/PIMs.
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until recently negative-ε materials were available but negative-μ materials were not, most studies addressed the TM case only. A TE surface polariton at the interface of two conventional dielectrics has been shown to occur only if at least one of these materials is nonlinear (Litvak and Mironov [1968]). The existence conditions for surface polaritons that can be supported by a NIM/PIM interface are more complex. One distinctive property of a NIM/PIM interface in comparison with a regular PIM/PIM interface is that both TM and TE surface polaritons can be supported by the same interface in the linear regime, although not simultaneously (i.e. not for the same set of parameters as shown in fig. 22a). However, since a fundamental property of NIMs is the frequency dependence of their material parameters, TE and TM polaritons can exist at different frequencies. Finally, the energy fluxes in the PIM and NIM half-spaces are given by (Shadrivov, Sukhorukov, Kivshar, Zharov, Boardman and Egan [2004]) A2 c2 β 1/ε1 for TM, P = 0 if x < 0, 16πωκ1 1/μ1 for TE, P =
A20 c2 β 16πωκ2
1/ε2 1/μ2
for TM, for TE,
if x > 0.
(7.6)
1, § 7]
Surface polaritons, waveguides and resonators
47
Equations (7.6) show that if ε1 , μ1 > 0 and ε2 , μ2 < 0 the energy fluxes are in opposite directions at the NIM and PIM sides of the interface. This vortex-like behavior is another unique feature of a linear NIM/PIM interface. Depending on the material parameters ε1,2 and μ1,2 (or equivalently depending on the relative parameters X and Y ) the total energy flux along the interface can be positive (forward waves) or negative (backward waves) as indicated in fig. 22a.
7.2. Nonlinear surface polaritons Nonlinear optics of surface polaritons have been extensively studied at conventional PIM/PIM interfaces for various system configurations. Second-harmonic generation, difference-frequency generation and nonlinear wave-mixing processes are examples of the nonlinear surface effects that have been observed, and they have found numerous applications in the field of surface spectroscopy (Shen [1984]). A number of unique features of nonlinear surface polaritons have been discovered to exist at a NIM/PIM interface, provided that one or both materials are nonlinear (Shadrivov, Sukhorukov, Kivshar, Zharov, Boardman and Egan [2004]). As an example, in this subsection we examine in detail the properties of TE-polarized surface polaritons at the interface of a nonlinear NIM and a linear PIM. The material parameters are given by for x < 0, μ1 for x < 0, ε1 and μ = ε= (7.7) ε2 + εNL |E|2 for x > 0, μ2 for x > 0.
EBL
Equation (7.1) for the TE case can be re-written as ∂ 2E ∂ 2E + + k02 ε1 μ1 E = 0 for x < 0, ∂z2 ∂x 2 ∂ 2E ∂ 2E + + k02 ε2 μ2 + μ2 εNL |E|2 E = 0 for x > 0. (7.8) 2 2 ∂z ∂x For the case of self-focusing NIM μ2 εNL > 0, the stationary solutions can be searched for in the form E1,2 = Φ1,2 exp(iβz). Then, the transverse profile of the stationary surface polariton is given by E0 exp(η1 x) x < 0, Φ(x) = (7.9) 1/2 (2/μ2 εNL ) η2 sech[η2 (x − x0 )] x > 0, where E0 and x0 are parameters that can be determined from the continuity conditions for the tangential components of the fields at the boundary between the two media. Then, the dispersion relation and the relationship between the electric
48
Negative refractive index metamaterials in optics
[1, § 7
field amplitude at the interface and the position of the center of x0 of the sech function are found in the form tanh(η2 x0 ) = μ2 η1 /(μ1 η2 ), E0 = (2/μ2 εNL )1/2 η2 sech(η2 x0 ).
(7.10)
Note that a surface polariton always has a maximum at the interface of a nonlinear NIM and a linear PIM, as shown in fig. 22b, which contrasts with the case of a nonlinear/linear PIM/PIM interface where the electric field has a maximum shifted to the self-focusing nonlinear medium, as shown in fig. 22c (Boardman, Velasco, King and Rapoport [2005]). It can be shown that the same result is obtained in the case of surface polaritons at the interface between a linear NIM and a nonlinear PIM. Figures 22d and e show the structure of the surface polaritons propagating along the nonlinear NIM/nonlinear PIM interface. In this case, depending on the material parameters, the polariton profile consists of either one or two maxima. Finally, analysis of surface polaritons of finite spatial and temporal extent indicates that the group velocity of nonlinear surface waves can be controlled by changing the intensity of the electromagnetic field. This property provides a new way for manipulating energy flow in such structures, in particular allowing the switching of the surface wave from forward-propagating to backwardpropagating. The possibility of controlling surface waves is particularly important for imaging applications based on NIM flat lenses.
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7.3. NIM slab as a linear waveguide The linear properties of guided waves in the NIM slab configuration can be studied using the standard approaches developed for asymmetric or symmetric waveguides (Marcuse [1991]). The dispersion relations for the TE-polarized waves propagating in an asymmetric waveguide described by ⎧ ⎨ ε1 > 0, μ1 > 0 for x < 0, ε2 < 0, μ2 < 0 for 0 < x < L, (7.11) ⎩ ε3 > 0, μ3 > 0 for x > 0, can be derived as follows. The electric fields that satisfy the linear wave equation (7.1) in each layer can be written as E1,2,3 = Φ1,2,3 exp(iβz). Then, the solutions in the cladding layers, i.e. for x < 0 and x > 0, are given by Φ1 (x) = A1 exp(κ1 x) and Φ3 (x) = A3 exp(−κ3 x),
1, § 7]
Surface polaritons, waveguides and resonators
49
respectively, where κ12 = β 2 − k02 ε1 μ1 > 0, κ32 = β 2 − k02 ε3 μ3 > 0, with κ1 , κ3 > 0. In the core layer, i.e. for 0 < x < L, two cases should be considered: κ22 = β 2 − k02 ε2 μ2 > 0
and κ22 = β 2 − k02 ε2 μ2 < 0.
The former case corresponds to the surface polaritons discussed by Ruppin [2001], also referred to as “slow waves” by Shadrivov, Sukhorukov and Kivshar [2003a], while the latter case corresponds to the slab or “fast” waves. Note that both types of waves are guided waves in the case of NIMs. The solution is given by Φ2 (x) = A2 exp(−κ2 x) + B2 exp(κ2 x). The corresponding dispersion relation is found in the form μ2 κ1 μ3 κ2 1− exp(−κ2 L) 1− μ2 κ3 μ1 κ2 μ2 κ1 μ3 κ2 1+ exp(κ2 L) = 0, + 1+ (7.12) μ2 κ3 μ1 κ2 which reduces to the results of Ruppin [2001] for the symmetric case ε1 = ε3 > 0, μ1 = μ3 > 0: κ2 μ3 = −μ2 κ3 tanh(κ2 h/2),
κ2 μ3 tanh(κ2 h/2) = −μ2 κ3 .
(7.13)
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The corresponding dispersion relations for a TM-polarized wave propagating in a generalized three-layer waveguide can be written as ε2 κ1 ε3 κ2 1− exp(−κ2 L) 1− ε2 κ3 ε1 κ2 ε2 κ1 ε3 κ2 1+ exp(κ2 L) = 0, + 1+ (7.14) ε2 κ3 ε1 κ2 which in the case of symmetric waveguide reduces to κ2 ε3 = −ε2 κ3 tanh(κ2 L/2),
κ2 ε3 tanh(κ2 L/2) = −ε2 κ3 .
(7.15)
In conventional non-magnetic dielectric or metallic slab waveguides, with frequency-dependent dielectric permittivity, surface polaritons exist only in the stop band, corresponding to negative permittivity. In magnetic material slabs with constant dielectric permittivity and a frequency-dependent magnetic permeability, surface polaritons also exist in the stop band corresponding to negative permeability. In sharp contrast, a NIM slab supports surface localized modes (slow waves) within the frequency pass band (i.e. they are guided waves) corresponding to simultaneously negative permittivity and permeability. The solution for the guided waves can be written in the form Φ2 (x) = A2 exp(−iκ2 x) + B2 exp(iκ2 x).
50
Negative refractive index metamaterials in optics
[1, § 7
Therefore, in order to re-write the dispersion relations (7.12)–(7.14) for the guided waves, κ2 can be re-defined as κ2 → iκ2 , leading, for example in the case of the TE-polarized wave, to the following dispersion relation: μ2 κ1 iμ3 κ2 1− exp(−iκ2 L) 1− μ2 κ3 iμ1 κ2 iμ3 κ2 μ2 κ1 + 1+ (7.16) 1+ exp(iκ2 L) = 0, μ2 κ3 iμ1 κ2 which can be written as exp[i(2κ2 h + ϕ1 + ϕ3 )] = 1, where ϕ1 and ϕ3 are defined as iμ1 κ2 −1 iμ1 κ2 ≡ exp(iϕ1 ) and 1− 1+ μ2 κ1 iμ2 κ1 iμ3 κ2 iμ3 κ2 −1 1+ (7.17) ≡ exp(iϕ3 ). 1− μ2 κ1 iμ2 κ3 As a result, the well-known condition for the guided modes of a linear waveguide is obtained: 2κ2 h + ϕ1 + ϕ3 = 2πm,
m = 0, 1, 2, . . . .
EBL
(7.18)
Equation (7.18) implies that the total 2πm-phase shift of the zigzagging wave propagating between the two waveguide boundaries includes a propagation phase shift 2κL, as well as phase shifts ϕ1 and ϕ3 owing to the total internal reflection at the interfaces (Tamir [1988], Marcuse [1991]) often referred to as the linear Goos– Hänchen shift (Qing and Chen [2004], Shadrivov, Zharov and Kivshar [2003]). These phase shifts can be re-written as ϕ3 μ1 k02 ε2 μ2 − β 2 μ3 k02 ε2 μ2 − β 2 ϕ1 and tan . = = tan 2 μ2 β 2 − k02 ε1 μ1 2 μ2 β 2 − k02 ε3 μ3 (7.19) Equations (7.19) show that in contrast to the conventional PIM/PIM interface, in the case of the NIM/PIM interface the Goos–Hänchen shift possesses a negative sign. Detailed analysis of the dispersion relations for the symmetric NIM slab has revealed several unusual properties of such waveguides (Shadrivov, Sukhorukov and Kivshar [2003a]), which can be summarized as follows. NIM slab waveguides support slow modes that can be symmetric (node-less) or antisymmetric (containing one zero). In contrast to the conventional PIM waveguides that can only support fast guided modes if the refractive index of the core is higher than that of the cladding, guided modes can be supported by both a low-index
1, § 7]
Surface polaritons, waveguides and resonators
51
Fig. 23. Double-vortex structure of energy flow for localized surface wave. (See Shadrivov, Sukhorukov and Kivshar [2003a].)
and a high-index NIM slab. The fundamental node-less mode that always exists in PIM slab waveguides does not exist at all in the NIM slab. While PIM waveguides also support higher-order modes with well-defined properties, their
number being determined by the so-called V -parameter (V = (2h/λ) n22 − n21 for a symmetric waveguide), in NIM waveguides the properties of higherorder modes are more complex. For instance, the first-order mode disappears as the waveguide width exceeds some critical value. Also, two modes with the same number of nodes can be simultaneously supported. Finally, as discussed above, the energy fluxes are counter-directed at the interface of the NIM and PIM. Results of numerical simulations indeed show a unique double-vortex structure of the energy flow for the localized surface wave in the NIM slab (fig. 23).
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7.4. Linear waveguide in nonlinear surroundings In this subsection we consider a linear NIM slab waveguide surrounded by a nonlinear dielectric. Similar to the linear case discussed above, this waveguide supports bounded surface (slow) waves κ22 = β 2 − k02 ε2 μ2 > 0 and slab (fast) modes κ22 = β 2 − k02 ε2 μ2 < 0. The waveguide material parameters are given by ⎧ (1) ⎪ ⎨ ε1 + εNL > 0, μ1 > 0, (7.20) ε2 < 0, μ2 < 0, ⎪ ⎩ (3) ε3 + εNL > 0, μ3 > 0. For the surface waves the solutions in the slab (0 < x < L) are found in the form Φ2 (x) = A2 exp(−κx) + B2 exp(κx). The solutions in the claddings are given
52
Negative refractive index metamaterials in optics
[1, § 7
by Φj (x) =
2κj2
−1/2
(j ) k02 μj εNL
sech κj (x − xj ) ,
(j )
for |x| > L and μj εNL > 0, 2κj2 −1/2 Φj (x) = cosech pj (x − xj ) , (j ) 2 k0 μj εNL (j )
for |x| > L and μj εNL < 0,
(7.21)
(7.22)
= − = − and xj are the integration conwhere stants corresponding to the maxima of the electric field, j = 1 or 3. Depending (j ) on the signs of μj εNL , four solutions are possible. Here, as an example we con(j ) sider the case of μj εNL > 0. Then, the dispersion relations are found in the form κ1 μ2 κ3 μ2 tanh(κ1 x1 ) 1 + tanh κ3 (L − x3 ) exp(2κ2 L) 1 + κ2 μ1 κ2 μ3 κ1 μ2 κ3 μ2 = 1− (7.23) tanh(κ1 x1 ) 1 − tanh κ3 (L − x3 ) . κ2 μ1 κ2 μ3 κ12
β2
k02 ε1 μ1 ,
κ32
β2
k02 ε3 μ3 ,
EBL
Defining φ1 and φ3 by κ1 μ 2 tanh(κ1 x1 ), tanh(φ1 /2) = κ2 μ1 κ3 μ 2 tanh κ3 (L − x3 ) , tanh(φ3 /2) = κ2 μ3 Equation (7.23) takes the form 2κ2 L + φ1 + φ3 = 0,
(7.24)
(7.25)
and can be satisfied if either (φ1 < 0 or φ3 < 0) or (φ1 and φ3 < 0). Finally, if the electric field amplitudes at the interfaces x = 0 and x = L are defined as −1/2 2κ12 sech κ1 (x − x1 ) and A12 = (1) 2 k0 μ1 εNL −1/2 2κ32 A23 = (7.26) sech κ3 (x − x3 ) (3) k02 μ3 εNL then the relationship between the parameters x1 and x3 can be found from A23 = A12 cosh(κ2 L) − (μ2 κ1 /κ2 ) sinh(κ2 L) , (7.27)
1, § 7]
Surface polaritons, waveguides and resonators
53
that is, a relationship between x1 and x2 . Then the dispersion relation (7.23) contains x1 as a parameter. On the other hand, eqs. (7.26) determine A12 in terms of x1 . Therefore, the electric field amplitude at x = 0 can be chosen as a parameter instead of x1 . Then, (7.23) and (7.26) provide an implicit relationship between the propagation constant β and the frequency ω, β = β(ω; A212 ). Corresponding dispersion relations and their analysis for the case of slab (fast) modes have been performed by Shadrivov [2004] and will not be detailed here. In particular, symmetric, asymmetric, and antisymmetric, forward and backward modes have been found.
7.5. Nano-resonators Optical resonators are the essential components of many optical systems, including lasers, parametric oscillators, interferometers, optical delay lines, and filters. The simplest form of resonator is a plane mirror Fabry–Perot cavity. The resonance condition in a Fabry–Perot cavity of length L is given by 2L = mλ, where m is an integer number and λ is the optical wavelength in the medium filling the resonator. Therefore, the lower limit on the length of the cavity that supports the lowest-order resonant mode is L = λ/2 = λ0 /(2n), where λ0 is the optical wavelength in vacuum and n is the index of refraction of the resonator medium. Engheta [2002] has shown that significantly more compact sub-wavelength cavities can be realized using NIM layers inside the resonator. The dispersion relation for a resonator formed by two perfectly conducting plates filled with two materials of thicknesses d1 and d2 with corresponding material parameters (ε1 , μ1 ) and (ε2 , μ2 ) shown in fig. 24, can be obtained as follows. In layer 1 (0 x d1 ), the electric and magnetic field components can be written as n1 k0 (1) (1) Hy(1) = E cos(n1 k0 z), Ex(1) = E0 sin(n1 k0 z), (7.28) iωμ1 0
EBL
and in layer 2 (d1 x d1 + d2 ) they are given by Ex(2) = E0(2) sin n2 k0 (d1 + d2 − z) , n2 k0 (2) Hy(2) = − E0 cos n2 k0 (d1 + d2 − z) . iωμ2 From the boundary conditions, (1)
(7.29)
(2)
E0 sin(n1 k0 d1 ) = E0 sin(n2 k0 d2 ), n1 (1) n2 (2) E0 cos(n1 k0 d1 ) = − E0 cos(n2 k0 d2 ), μ1 μ2
(7.30)
54
Negative refractive index metamaterials in optics
[1, § 7
Fig. 24. Sub-wavelength compact cavity resonator formed by a pair of NIM and PIM layers sandwiched between two reflectors. (Adapted from Engheta [2002].)
the dispersion relations are obtained in the form n2 n1 tan(n1 k0 d1 ) + tan(n2 k0 d2 ) = 0. μ2 μ1
(7.31)
Note that up to this point the signs of n1 , n2 , μ1 , and μ2 have not been specified. Assuming layer 1 is the PIM layer (ε1 > 0, μ1 > 0) and layer 2 (ε2 < 0, μ2 < 0) is the NIM layer, eq. (7.31) can be re-written as
EBL
|n2 | n1 tan |n1 |k0 d1 − tan |n2 |k0 d2 = 0, |μ2 | |μ1 |
(7.32)
which finally leads to tan(|n1 |k0 d1 ) |μ2 ||n1 | = . tan(|n2 |k0 d2 ) |μ1 ||n2 |
(7.33)
Equation (7.33) shows that in this case (PIM/NIM configuration) there are no limitations on the total length of the resonator, i.e. on L = d1 + d2 , but rather on the ratio of d1 and d2 . This conclusion becomes more obvious if d1 , d2 and ω are chosen such that the small-argument approximation can be used for the tangent function, in which case eq. (7.33) reduces to d1 ∼ |μ2 | . = d2 |μ1 |
(7.34)
In contrast, if both layers 1 and 2 are made of PIMs (PIM/PIM configuration), eq. (7.31) is still valid, but the tangent functions should have opposite signs in order to satisfy eq. (7.31). Then, if d1 < π/(2n1 k0 ) for tan(n1 k0 d1 ) to be positive, d2 must exceed π/(2n2 k0 ) in order to have tan(n2 k0 d2 ). As a result, there is a limitation on the total length L = d1 + d2 .
1, § 8]
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The idea of sub-wavelength cavity resonators based on NIM/PIM pairs has been further extended to the case of ultracompact waveguides (Alu and Engheta [2004]), backward couplers (Engheta and Ziolkowski [2005]) and lasers (Ziolkowski [2006]). The development of such ultrasmall photonic components is essential for future nano- and microphotonic integrated circuits.
§ 8. New frontiers: Metamaterials for cloaking While the first optical metamaterials were demonstrated only two years ago (Shalaev, Cai, Chettiar, Yuan, Sarychev, Drachev, Kildishev [2005], Zhang, Fan, Panoiu, Malloy, Osgood, Brueck [2005]) numerous potential applications are envisioned. These include nanoscale lithography, biological imaging, beam steering, optical memory, sensing, improved light coupling to solar cells and light out-coupling from organic light-emitting devices and optical nanocomponents, and even the possibility of controlling an object’s degree of visibility. Recently, three approaches have been proposed to achieve a nearly perfect shielding effect, or “cloaking”: one based on a plasmonic or metamaterial cover proposed by Alu and Engheta [2005a], a superlens-based approach introduced by Milton and Nicorovici [2006], and a coordinate-transformation or wave-redirection approach proposed by Pendry, Schurig and Smith [2006] and independently by Leonhardt [2006]. Of these three approaches, the coordinate-transformation approach appears to be the most general in the following sense: the design of the cloak is not “object-specific”, i.e. the same cloak can be used for different objects and, importantly, microscopic as well as macroscopic objects can be made invisible using this approach. The theory behind this approach originates from Ward and Pendry’s paper (Ward and Pendry [1996]), where the problem of computational studies of complex systems involving several length scales was addressed. An example of such scales is the wavelength of light in free space, which can be on the order of a few thousand Ångströms, compared to the skin depth in metals, which is usually a few tens of Ångströms. It has been shown that instead of using a nonuniform mesh, the conventional approach to this kind of problem, details of the mesh structure can be included into an effective ε and μ. With respect to cloaking applications, the idea of coordinate transformation can be utilized as follows. To cloak an object in a certain volume of space, the space should be transformed to create a concealment volume. Since actual space transformation is not feasible, one could use the fact that in Maxwell’s equations the material parameters enter in such a way that the same effect can be achieved by transforming the material properties. Therefore, the space outside the conceal-
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Fig. 25. (a,b) Ray trajectories in the cloak assuming that R2 λ: (a) two-dimensional cross section; rays are diverted within the annulus of cloaking material contained within R1 < r < R2 and return to their original path on the far side of the cloak; (b) three-dimensional view of the same system. (c) A point charge located near the cloaked sphere. (Here it is assumed that R2 λ, the near-field limit. The electric displacement field is plotted. The field is excluded from the cloaked region, but emerges on the far side of the cloaking sphere undisturbed.) (Adapted from Pendry, Schurig and Smith [2006].)
ment volume can be replaced by a material in which light rays travel the exact same paths they would have traveled in the transformed space. Maxwell’s equations in a system of Cartesian coordinates are given by μ ∂H ε ∂E (8.1) , ∇ ×H = , c ∂t c ∂t where both ε and μ may depend on position. Ward and Pendry proved that the transformation to a general system defined as ∇ ×E =−
q1 (x, y, z),
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q2 (x, y, z),
(8.2)
q3 (x, y, z)
does not change the form of Maxwell’s equations but changes the definitions of ε and μ (Ward and Pendry [1996], Leonhardt [2003]). Maxwell’s equations in the new system of coordinates can be written as εˆ ∂ Eˆ (8.3) , c ∂t where εˆ and μˆ are in general tensors, and Eˆ and Hˆ are renormalized electric and magnetic fields. An example of a cloaking device is shown in fig. 25. The object, chosen to be a sphere of radius R1 for simplicity, is concealed with a metamaterial that is designed to deflect the rays that would have hit the object, guide them around the object, and return them to their original path. The cloaking region is contained within the annulus R1 < r < R2 . A coordinate transformation that takes all fields in the region r < R2 and compresses them into the region R1 < r < R2 is given by ∇q × Eˆ = −
μˆ ∂ Hˆ , c ∂t
∇q × Hˆ =
r = R1 + r(R2 − R1 )/R2 ,
θ = 0,
φ = φ.
(8.4)
1, § 8]
New frontiers: Metamaterials for cloaking
57
Fig. 26. Cancellation of the overall dipole moment through an induced negative polarization vector. (Adapted from Alu and Engheta [2005a].)
The corresponding values for the electric permittivity ε and magnetic permeability μ can take any value in the region r < R1 , and are given by R2 (r − R1 )2 , R2 − R1 r R2 = . R2 − R1
εr = μ r = εφ = μ φ
εθ = μ θ =
R2 , R2 − R1 (8.5)
Finally, for r > R2 all components of ε and μ are equal to 1. A cloak designed using eqs. (8.4)–(8.5) will exclude all fields from the central region. On the other hand, no fields can leave the region r < R1 . Several issues are associated with a practical realization of such a cloaking device: (1) very large or very small values of ε and μ can be difficult to achieve, and thus, cloaking might be imperfect if some values prescribed by eq. (8.5) cannot be realized; (2) in the example considered by Pendry, Schurig and Smith [2006] the effect is only achieved at one frequency. Obviously, broadband cloaking would be desirable for most practical applications. Two alternative schemes for the cloaking of objects have been discussed by Alu and Engheta [2005b] and by Milton and Nicorovici [2006]. The basic idea of Alu and Engheta is shown schematically in fig. 26. In this scheme the invisibility results from the cancellation of the overall dipole moment through an induced negative polarization vector in the cloaking region. The cloak and the object form a composite such that the total scattering cross-section is reduced. Some of the current limitations of this approach are (1) single-wavelength operation, (2) object size is limited to microscopic objects, and (3) the cloak is objectspecific. Finally, the third approach, proposed by Milton and Nicorovici [2006], relies on the use of a superlens or, more precisely, on the regions of anomalous localized resonance occurring near superlenses. Nicorovici, McPhedran and Milton [1994] have found that a coated cylinder, now called a cylindrical superlens, with a core
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of dielectric constant εc = 1 and radius rc and a shell with dielectric constant εs = −1 + iεs and outer radius rs , in the limit εs → 0 would be invisible to any applied quasistatic transverse magnetic (TM) field. As an extension of that earlier work, Milton and Nicorovici [2006] have recently shown that not only is the lens invisible in this limit, but cylindrical objects are as well, or at least any finite collection of polarizable line dipoles that lie within a certain radius of the cylindrical superlens. The main idea of this approach is as follows. When the resonant field generated by a polarizable line or point dipole acts back on the polarizable line or point dipole it effectively cancels the field acting on it from outside sources, resulting in cloaking. Cloaking has been proven in the quasistatic limit for finite collections of polarizable line dipoles that all lie within a specific distance from a coated cylinder with a shell permittivity εcl = −εc = −εext , where εc is the core permittivity and εext is the permittivity of the surrounding medium. Also, cloaking extends outside the quasistatic regime for a plane-parallel superlens. When a polarizable line dipole is located less than a distance d/2 from the lens, where d is the thickness of the lens, it will be cloaked due to the presence of a resonant field in front of the lens. The first experimental demonstration of cloaking at microwave frequencies has been reported recently (Schurig, Mock, Justice, Cummer, Pendry, Starr and Smith [2006]). In that experiment the object, a copper cylinder, was concealed by a cylindrical cloak built using SRRs positioned with their axes along the radial direction as shown in fig. 27. Two main design parameters of the SRRs, the length of the split and the radius of the corners, shown in the inset in fig. 27, were used to tune the material parameters to those prescribed by the design equations
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εz =
b b−a
2 ,
μ r =
r −a r
2 ,
μ θ = 1.
(8.6)
Experimental results showing the fields impinging on the bare conducting cylinder and the cloaked conducting cylinder are shown in fig. 27. Although the invisibility demonstrated in these experiments is not perfect owing to simplifications made at the design stage, the feasibility of the electromagnetic cloaking mechanism on the basis of metamaterial design has been demonstrated. Very recently, the first theoretical design of a non-magnetic cloak operating at optical frequencies was proposed by Cai, Chettiar, Kildishev and Shalaev [2007]. It may take some time before optical cloaking becomes a reality due to the more difficult fabrication of metamaterials at optical frequencies, but basic ideas of cloaking are applicable to all frequencies.
1, § 9]
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59
Fig. 27. (a) Two-dimensional experimental cloaking structure. Snapshots of time-dependent, steady-state electric field patterns. The cloak lies in the annular region between two circles and surrounds a conducting Cu cylinder at the inner radius. (b) Experimental measurement of the bare conducting cylinder. (c) Experimental measurement of the cloaked conducting cylinder. (Adapted from Schurig, Mock, Justice, Cummer, Pendry, Starr and Smith [2006].)
§ 9. Summary
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The emergence of optical metamaterials, and NIMs in particular, has given rise to numerous unusual linear and nonlinear optical phenomena that cannot be realized in conventional materials. The constituent components of these materials, meta-atoms, can be designed to empower optical magnetic activity, to enable fundamentally new regimes of linear and nonlinear light–matter interaction, and even to manipulate an object’s degree of visibility at optical frequencies. The fabrication of optical NIMs is quite challenging and requires approaches going beyond a straightforward scaling of meta-atom sizes from microwave to near-infrared and visible frequencies. Several proof-of-principle experiments reported over the last two years form the basis for building functional NIMs and the realization of new photonic devices. Nevertheless, there is a lot of room for improvement and, therefore, for both experimental and theoretical fundamental and applied studies. The ultimate goal from the experimental viewpoint is the fabrication of three-dimensional, low-loss, tunable and broadband NIMs. Owing to the complexity of NIMs, theoretical predictions and numerical analysis are the essential components of NIM research. The demonstration of NIMs and magnetic activity at optical frequencies has motivated re-consideration of almost all well-established linear and nonlinear op-
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tical phenomena that reveal themselves in unusual and often counter-intuitive ways in NIMs. Optical metamaterials, without doubt, constitute one of the most exciting areas of research in modern optics, that is likely to result in discoveries of new phenomena and the development of novel device applications for nanophotonics.
Acknowledgements The authors gratefully acknowledge the support of the Army Research Office through Grants W911NF-07-1-0343 and 50342-PH-MUR, of the National Science Foundation through Grant DMS-050989, and of the Russian Foundation for Basic Research through Grant 06-02-16406.
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E. Wolf, Progress in Optics 51 © 2008 Elsevier B.V. All rights reserved
Chapter 2
Polarization techniques for surface nonlinear optics by
Martti Kauranen Institute of Physics, Tampere University of Technology, FI-33101 Tampere, Finland e-mail:
[email protected]
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Stefano Cattaneo Philips Research, High Tech Campus 34, 5656 AE Eindhoven, The Netherlands e-mail:
[email protected]
ISSN: 0079-6638
DOI: 10.1016/S0079-6638(07)51002-4 69
Contents
Page § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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§ 2. Polarization effects in the nonlinear response of surfaces and thin films
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§ 3. Applications of polarization techniques . . . . . . . . . . . . . . . . .
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§ 4. Complete theoretical model including linear optics . . . . . . . . . .
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§ 5. Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . .
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Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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§ 1. Introduction Nonlinear optical processes depend on the symmetry of the medium in which they occur. Such dependence is reflected in the tensorial nature of the nonlinear optical susceptibility, which is the key quantity describing the processes. Symmetry considerations are particularly important for second-order nonlinear effects, that is, for effects that depend quadratically on the intensity of illumination. This is evident, for example, in the well-known selection rule: within the electric-dipole approximation, second-order processes can occur only in materials that lack a center of symmetry. Techniques based on second-order effects are therefore attractive tools for studying effects associated with the structure of materials, particularly surface effects (Shen [2000]) or the orientation of molecules on surfaces (Heinz, Tom and Shen [1983]). The tensorial nature of nonlinear processes gives rise to a sensitive dependence of the processes on the polarization of light (Shen [1984], Boyd [2003]). The polarization dependence therefore yields valuable information on the sample structure and symmetry (Heinz, Tom and Shen [1983], Kauranen, Verbiest and Persoons [1998]). A more detailed analysis of the polarization dependence allows the susceptibility tensor of even complicated samples to be determined (Kauranen, Maki, Verbiest, Van Elshocht and Persoons [1997], Kauranen, Van Elshocht, Verbiest and Persoons [2000]). Over the years, considerable effort has been put into refining such polarization techniques to improve their precision and reliability. The field of surface nonlinear optics is not limited to the study of bare surfaces or interfaces between two materials. An important part of the field is the study of thin molecular films ordered on a substrate, leading to samples that consist of three distinct layers. On the other hand, for samples of proper interfaces between two materials, the surface nonlinearity can often be assumed to occur only in a very thin surface layer, effectively giving rise also to a three-layer system. The same formalism can therefore be used to describe both types of samples. Surfaces are unavoidably associated with strong gradients in the electromagnetic fields and material properties. Such gradients are expected to favor electricquadrupole (or even higher multipole) contributions to the nonlinear response (Bloembergen and Pershan [1962], Guyot-Sionnest, Chen and Shen [1986]).
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On the other hand, electric-quadrupole and magnetic-dipole effects are known to lead to a second-order response in the bulk of centrosymmetric materials (Bloembergen, Chang, Jha and Lee [1968]). This has two important consequences. First, it can be shown that the surface multipoles and part of the bulk response contribute to an effective electric-dipole-type surface susceptibility (Guyot-Sionnest, Chen and Shen [1986]) with the only difference that such effective surface nonlinearities may not fulfill all the symmetry rules of true electric-dipole nonlinearities. Second, the existence of multipolar bulk contributions means that great care must be taken to exclude such contributions from the measured signals when nonlinear techniques are used as surface probes. Important advances in nonlinear polarization techniques have been made in the past 10–15 years. For example, several techniques have been developed to investigate chiral properties of thin films (for reviews, see Kauranen, Verbiest and Persoons [1998], Hicks [2002], Sioncke, Verbiest and Persoons [2003], Fischer and Hache [2005], Belkin and Shen [2005]). In addition, it has been shown very recently that an important component of the bulk nonlinear response of isotropic materials can be unambiguously separated from surface contributions by its polarization properties (Cattaneo and Kauranen [2005], Figliozzi, Sun, Jiang, Matlis, Mattern, Downer, Withrow, White, Mochan and Mendoza [2005]). This is important whenever second-order techniques are used as surface probes. This chapter provides an overview of second-order nonlinear optical polarization techniques as applied to the study of surfaces and thin films. We will emphasize the common underlying principles of the techniques, their practical implementation, and their most important applications. Our explicit discussions will be based on second-harmonic generation (SHG). However, we will treat cases with one and two input beams at the fundamental frequency. It is evident that the basic principles are also applicable to sum-frequency generation (SFG) and difference-frequency generation (DFG). We also note that the number of independent components of the various susceptibility tensors can be very large for samples of low symmetry. Although the basic physical principles remain unchanged in such cases, they give rise to very long mathematical expressions that are difficult to decipher. We will therefore occasionally give explicit results only for the simple case of achiral materials whose surface layer has in-plane isotropy. The review is structured as follows: In § 2 we present the basic ideas behind nonlinear optical polarization techniques. For this purpose, the nonlinearity is assumed to be of electric-dipole origin and limited to a very thin surface layer with in-plane isotropy. In addition, the nonlinear layer and the media on both sides of it are assumed to have unity refractive indices. Several methods for the experimental characterization of nonlinear optical samples are discussed.
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Section 3 presents two important applications of polarization techniques in surface and thin-film nonlinear optics. The first application is to the study of chiral thin films. In fact, such samples have been intimately connected with the development of some of the most comprehensive polarization techniques presently in use. The second application is the use of polarization techniques to distinguish between the surface and bulk contributions of isotropic materials. Although the model of unity refractive indices is useful for understanding the operation principles of the polarization techniques, the linear optical properties of the various materials play an important role in the interpretation of the experimental results (Heinz [1991], Reider and Heinz [1995], Cattaneo, Miettinen, Vuorimaa, Lemmetyinen and Kauranen [2006]). While more detailed models are well understood and conceptually straightforward, they are often tedious to implement, in particular when propagation effects in the nonlinear layer become important. In § 4 we expand the theoretical treatment to include the linear optical properties of the samples and discuss the implications for thick samples and for ultrathin samples. Finally, the conclusions and an outlook for nonlinear optical polarization studies are presented in § 5.
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§ 2. Polarization effects in the nonlinear response of surfaces and thin films The first demonstrations of the intrinsic surface sensitivity of SHG (Brown and Matsuoka [1969], Rudnick and Stern [1971], and Chen, Bower, Wang and Lee [1973]) opened the door to its application in the study of surfaces and interfaces, as well as molecular adsorption and thin molecular films on surfaces (Chen, Heinz, Ricard and Shen [1981], Heinz, Chen, Ricard and Shen [1981]). Such studies suggested that the second-harmonic response depends not only on the nonlinearity of the adsorbate molecules, but also on their alignment. Polarization studies of SHG were subsequently used to determine the components of the second-harmonic susceptibility tensor and to obtain information on the molecular orientation (Heinz, Chen, Ricard and Shen [1982], Heinz, Tom and Shen [1983]). Since these pioneering experiments, polarization techniques have become very important tools in surface nonlinear optics. They have been applied to measure, for example, the orientation of molecules at a liquid–air interface (Rasing, Shen, Kim, Valint and Bock [1985]), molecular reorientation during a twodimensional liquid phase transition (Rasing, Shen, Kim and Grubb [1985]), polar ordering at the surface of liquid crystals (Guyot-Sionnest, Hsiung and Shen [1986], Feller, Chen and Shen [1991]), as well as other molecular orientation effects (Marowsky, Gierulski, Steinhoff, Dorsch, Eidenschnik and Rieger [1987],
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Fig. 2.1. Schematic representation of single-beam SHG measurements. A fundamental beam e(ω) is incident on a nonlinear sample. For thin films and surfaces, second-harmonic light is produced with approximately equal efficiency in the reflected [E1 (2ω)] and transmitted [E2 (2ω)] directions. The fields are most naturally divided into their p- and s-polarized components (parallel and normal to the plane of incidence, respectively) as shown.
Marowsky, Steinhoff, Chi, Hutter, Wagnière [1988], Marowsky and Steinhoff [1988], Marowsky, Lüpke, Steinhoff, Chi and Möbius [1990]). Similar methods were used to determine the symmetry of crystal surfaces (Tom, Heinz and Shen [1983], Heinz, Loy and Thompson [1985], Andersson, Schanne-Klein and Hache [1999]). In the following, we will present the basic features and the underlying principles of such polarization techniques.
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2.1. Functional form of the measured signals When characterizing a sample with nonlinear optical techniques, it is important to make the distinction between parameters that are directly accessible in an experiment and more fundamental properties of the material, such as the nonlinear susceptibility tensor. As we will see later, the tensor cannot be measured directly but must be extracted from experimental data with a theoretical model that accounts for the linear optical properties of the material. Any nonlinear measurement, on the other hand, can be described in terms of a few macroscopic parameters. We consider a situation where a laser beam at the fundamental frequency ω and with field amplitude e(ω) is incident on an interface or thin-film sample with a second-order nonlinear optical response but otherwise arbitrary symmetry (fig. 2.1). At this point, we assume for simplicity that the nonlinear response of the sample can be described in the electric-dipole approximation. The input beam then induces in the sample a polarization at the second-harmonic frequency P(2ω) that depends quadratically on the amplitude of the applied fundamental
2, § 2]
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field: P(2ω) = χ(2ω; ω, ω) : e(ω)e(ω),
(2.1)
where χ(2ω; ω, ω) is the second-harmonic susceptibility tensor (Shen [1984], Boyd [2003]). Note that here and in the rest of the chapter we use a lowercase symbol to indicate a quantity that refers to the fundamental frequency and an uppercase symbol to indicate one that refers to the second-harmonic frequency. The polarization acts as a source for the second-harmonic field, which depends linearly on P(2ω). In a surface geometry, the field vectors are naturally divided into p- and s-polarized components (parallel and normal to the plane of incidence, respectively). For sufficiently thin films, polarization effects due to linear light propagation in the film can be neglected (Kauranen, Verbiest and Persoons [1999]). Regardless of sample symmetry, a given second-harmonic field will then be of the functional form (Kauranen, Verbiest, Maki and Persoons [1994]) E j (2ω) = f j ep (ω)2 + g j es (ω)2 + hj ep (ω)es (ω),
(2.2)
where the superscript j = p, s denotes the polarization of the second-harmonic signal. This result is quite general, as eq. (2.2) includes all quadratic combinations of the components of the fundamental field. Equation (2.2) is valid for samples of arbitrary symmetry and is valid even when multipolar contributions to the nonlinear response are included. The symmetry properties of the nonlinear medium impose additional restrictions on the functional form of a particular second-harmonic signal. In the simplest possible case of achiral thin films with in-plane isotropy (symmetry group C∞v ) and within the electric-dipole approximation, the coefficients f s , g s and hp vanish (Kauranen, Verbiest and Persoons [1998]), giving
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E p (2ω) = f p ep (ω)2 + g p es (ω)2 ,
(2.3)
E (2ω) = h e (ω)e (ω).
(2.4)
s
s p
s
Equation (2.4) expresses the well-known fact that an s-polarized second-harmonic signal can be obtained from a sample of C∞v symmetry only when the input field contains both p and s polarization components. The expansion coefficients in the above equations depend on the susceptibility components of the sample and on its linear optical properties (Kauranen, Verbiest, Maki and Persoons [1994], Maki, Kauranen and Persoons [1995]). In fact, the coefficients can be interpreted as effective susceptibility components in the (p, s) reference system. However, they are not pure material constants, as they also depend on experimental details such as the angles of incidence and, for samples with in-plane anisotropy, the azimuthal orientation of the sample (Kauranen, Van
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Fig. 2.2. Detailed geometry and notation for single-beam SHG measurements in the limit of unity refractive indices. The subscripts 1 and 2 as well as + and − are used to describe the second-harmonic fields to make the notation consistent with the detailed theory of § 4.
Elshocht, Verbiest and Persoons [2000]). Nevertheless, the coefficients are the quantities that can be measured most directly in an experiment. Since they represent the starting point for any successive data analysis, special attention should be given to their precise determination.
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2.2. Approximation of unity refractive indices
Although the inclusion of linear optical effects is crucial for relating the expansion coefficients to the components of the susceptibility tensor, it is instructive to calculate the coefficients by assuming that the linear refractive indices of the nonlinear layer and the media surrounding it are equal and unity. This assumption allows neglecting the complicated details of the problem but accounts for its salient features. A complete treatment including the indices of refraction of the various materials will be developed in § 4. We consider the geometry shown in fig. 2.2. The fundamental beam is incident ˆ In the on the sample with an angle of incidence θ and wavevector k = (ω/c)k. approximation of unity refractive index, we need not distinguish between fields inside and outside the medium. We use a reference system in which x and y are the coordinates in the plane of the sample, and z is along the sample normal. For samples with in-plane isotropy, the directions of x and y can be chosen freely. For convenience, the x and y directions are defined to be parallel and perpendicular to the plane of incidence, respectively. The propagation direction of the fundamental beam is then given by the unit vector kˆ = sin θ xˆ − cos θ zˆ . The unit vectors of the (p, s) reference system are then defined as pˆ = sˆ × kˆ = sin θ zˆ + cos θ xˆ and sˆ = −ˆy.
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The Cartesian components of the nonlinear polarization P(2ω) are, from eq. (2.1), P i (2ω) = χij k (2ω; ω, ω)ej (ω)ek (ω),
(2.5)
where the indices ij k refer to the x, y, z coordinates and summation over repeated indices is implied. In our simple model, we can describe SHG from the layer by considering the radiative properties of elementary electric dipoles, which state that the field emitted by a dipole p in the far (radiation) zone is proportional to (ˆo × p) × oˆ , where oˆ is the direction of observation (Jackson [1975]). The planar extension of the nonlinear material provides a phase-matching condition in x and y directions for the second-harmonic radiation from elementary dipoles, which translates into conservation of the wavevector component along the surface. SHG then occurs only along two well-defined directions, specified by wavevectors ˆ ± with K ˆ ± = sin θ xˆ ± cos θ zˆ (upper sign: reflection, lower sign: K± = (2ω/c)K transmission). When the thickness of the nonlinear layer is much smaller than the coherence length of SHG, the two contributions are of comparable strength. The ˆ ± is then proportional to second-harmonic field amplitude emitted in direction K ˆ ± = P(2ω) − P(2ω) · K ˆ± K ˆ ±. ˆ ± × P(2ω) × K E(2ω) ∼ K (2.6)
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The second part of the equation shows that the radiated field is proportional to the ˆ ± = 0, we obtain transverse part of P(2ω). In terms of vectors nˆ that satisfy nˆ · K (Heinz [1991]) nˆ · E(2ω) ∼ nˆ · P(2ω).
(2.7)
The vectors e(ω) and E(2ω) describing the amplitudes of the fundamental and second-harmonic fields are most naturally expressed in the (p, s) reference sysˆ ± = sin θ zˆ ∓cos θ xˆ . tem. Similarly to the fundamental field, we define Pˆ ± = sˆ × K ˆ ± , we conclude that Since Pˆ ± and sˆ are by definition perpendicular to K E p (2ω) ∼ Pˆ ± · P(2ω),
(2.8)
E (2ω) ∼ sˆ · P(2ω).
(2.9)
s
The p and s components of the second-harmonic field amplitude are therefore proportional to the projection of P(2ω) along the corresponding unit vector, which shows the convenience of the (p, s) representation. Writing the Cartesian components of P(2ω) explicitly, we obtain E p (2ω) ∼ P z (2ω) sin θ ∓ P x (2ω) cos θ,
(2.10)
E (2ω) ∼ −P (2ω).
(2.11)
s
y
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Equation (2.10) shows that only the components P x and P z of the nonlinear polarization contribute to E p , while E s is directly proportional to P y . The detailed form of the polarization components depends on the symmetry of the sample through the second-harmonic susceptibility tensor χij k . For samples of C∞v symmetry, the independent non-vanishing components of the tensor are χzzz , χzxx = χzyy , and χxxz = χxzx = χyyz = χyzy . Equation (2.5) then implies that E p arises only from the components χzzz , χzxx = χzyy and χxxz = χxzx . Similarly, eqs. (2.11) and (2.5) show that E s originates from the susceptibility components χyyz = χyzy . By rewriting the expressions in terms of the (p, s) components of the fundamental field e(ω), one can show that the components of E(2ω) are of the functional forms anticipated in eqs. (2.3) and (2.4), with expansion coefficients f p = (χzxx ∓ 2χxxz ) sin θ cos2 θ + χzzz sin3 θ,
(2.12)
g = χzxx sin θ,
(2.13)
h = 2χxxz sin θ.
(2.14)
p
s
As before, the upper (lower) sign corresponds to the reflected (transmitted) second-harmonic signal. This calculation shows that, when the linear optical properties of the sample are neglected, the expansion coefficients can be directly related to the components of the susceptibility tensor. For samples of C∞v symmetry, for example, the coefficients g p and hs yield directly the components χzxx and χxxz of the susceptibility tensor [eqs. (2.13) and (2.14)]. The third component χzzz can then be extracted from the coefficient f p [eq. (2.12)].
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2.3. Polarization arrangements for the characterization of nonlinear samples 2.3.1. Fixed polarization measurements Over the years, various SHG techniques have been proposed to characterize the nonlinear response of a sample (Heinz [1991], Reider and Heinz [1995]) or, in other words, to determine the expansion coefficients of eq. (2.2). The expansion coefficients can be addressed separately by properly selecting the polarizations of the fundamental and SHG beams (Heinz [1991]). As we saw in the previous section, such measurements also yield directly the components of the susceptibility tensor when the approximation of unity refractive indices is justified. In general, several susceptibility components contribute to the measured second-harmonic signals (Shen [1984]). To see how the components can be separated by controlling the polarization state of the beams, we consider further the
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case of samples of C∞v symmetry, with independent susceptibility components χzzz , χzxx , and χxxz . The analysis outlined in the previous section implies that the last two components can be addressed separately by a proper choice of polarizations. The s component of the second-harmonic signal depends in fact only on χxxz [eqs. (2.4) and (2.14)], and requires the fundamental beam to have both p and s components. When the fundamental beam is s-polarized, on the other hand, only χzxx contributes to the second-harmonic signal [eqs. (2.3) and (2.13)], which will be p-polarized. To isolate χzzz , the fundamental field would need to be polarized along the surface normal. This is difficult to achieve in practice, as it would require the fundamental beam to propagate in the plane of the surface. Fortunately, the component χzzz can be determined from additional polarization measurements. One method to do this, proposed by Heinz, Tom and Shen [1983], involves using an analyzer to minimize the SHG output. Similar information can also be obtained by varying the incident angle of the incoming beam (Boyd [1987], Heinz [1991]). Although frequently used in the past, the latter method is less convenient in practice, particularly when a reflected second-harmonic signal is detected. The susceptibility components are real when the material is transparent at both the fundamental and the second-harmonic frequency (Boyd [2003]). In general, however, the components are complex, and phase differences can occur between them. In a typical experiment based on fixed polarizations, the intensity of a second-harmonic signal is measured. This provides direct information on the magnitude of the susceptibility components, but contains little information on their phase. A well-known method of determining the phase is based on measuring interference between the second-harmonic signal from the sample and that produced by the residual fundamental beam in a well-characterized optical crystal (Chang, Ducuing and Bloembergen [1965], Kemnitz, Bhattacharyya, Hicks, Pinto, Eisenthal and Heinz [1986], Stolle, Marowsky, Schwarzberg and Berkovic [1996]). A variable optical phase delay between the two second-harmonic sources is introduced, e.g., by varying the distance between them or by means of a gas cell with variable pressure. The phase of the susceptibility components of the sample can then be extracted from the interference pattern by extrapolating to zero phase delay (Chang, Ducuing and Bloembergen [1965]), or by replacing the sample with a second reference crystal (Kemnitz, Bhattacharyya, Hicks, Pinto, Eisenthal and Heinz [1986], Chen, Machida and Yamanoto [1998]). In their simplest implementations, however, techniques that use only certain combinations of fixed linear polarizations do not possess phase sensitivity and are therefore not suitable for the determination of complex-valued susceptibilities. Extensions of the techniques have been proposed to determine phase differences
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between the susceptibility components (Heinz [1991]). Although systematic, such procedures are difficult to implement. 2.3.2. Continuous polarization measurements An alternative approach to characterize the SHG response of a sample is based on continuous measurements of the second-harmonic signal as a function of the polarization of the incoming fundamental beam. Compared to interference methods, such techniques provide only the relative phase of the components, but are much more direct. Curiously, in the first demonstration of such a technique (Felderhof, Bratz, Marowsky, Roders and Sieverdes [1993]), the components of the susceptibility tensor were assumed to be real. In a subsequent refinement (Roders, Befort, Marowsky, Möbius and Bratz [1994]), the components were allowed to assume arbitrary complex values. In these early experiments, a half-wave plate was used to rotate the polarization of the fundamental beam, and multiple solutions had to be eliminated by measurements at different angles of incidence (Roders, Befort, Marowsky, Möbius and Bratz [1994]). Moreover, Fourier analysis of the recorded polarization patterns was used to extract the expansion coefficients. Wave-plate techniques were also applied to study low-symmetry chiral samples with a large number of independent tensor components (Kauranen, Verbiest, Maki and Persoons [1994], Verbiest, Kauranen, Persoons, Ikonen, Kurkela and Lemmetyinen [1994]). In these studies, the half-wave plate was replaced by a quarterwave plate. This choice proved to be more effective in determining complexvalued coefficients, because the quarter-wave plate introduces a nontrivial phase difference between the polarization components of the fundamental beam (Maki, Verbiest, Kauranen, Van Elshocht and Persoons [1996], Maki, Kauranen, Verbiest and Persoons [1997]). In addition, a more direct method was used to extract the expansion coefficients, rather than the unnecessarily complicated Fourier method. For a given experimental geometry, full characterization of the nonlinear response of a sample requires determining the expansion coefficients for two orthogonal second-harmonic polarization components (e.g., p and s). The relative complex values of coefficients belonging to a given set can be determined directly by detecting the corresponding second-harmonic polarization and modulating the polarization of the incoming beam. The problem then consists of relating the p- and s-polarized coefficients. Their relative magnitudes can be determined by calibrating the second-harmonic signals. However, besides being impractical and subject to errors, absolute calibration does not yield the relative phases of the coefficients (Kauranen, Van Elshocht, Verbiest and Persoons [2000]). More reliable results are obtained by mixing p- and s-polarized second-harmonic components with an analyzer placed at ±45◦ with respect to the plane of incidence
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(Cattaneo and Kauranen [2003], Mitchell [2006]). The detected signal is then 2 I ±45 (2ω) = E p (2ω) ± E s (2ω) .
(2.15)
This introduces interference between p- and s-polarized coefficients and allows determining their relative complex values very precisely. An alternative approach consists of introducing complex scaling factors unique to each measurement. Despite increasing the number of unknowns in the problem, this approach has some advantages for low-symmetry samples (Kauranen, Maki, Verbiest, Van Elshocht and Persoons [1997], Kauranen, Van Elshocht, Verbiest and Persoons [2000], Siltanen, Cattaneo, Vuorimaa, Katz, Phillips, Lemmetyinen and Kauranen [2004], Siltanen and Kauranen [2006]). Wave-plate techniques have been applied to determine the susceptibility components of nonlinear thin films and surfaces of various symmetries. Important applications of these techniques include the study of optical activity (see § 3.1) and magnetic-dipole contributions in isotropic chiral thin films (Maki, Kauranen and Persoons [1995], Kauranen, Maki, Verbiest, Van Elshocht and Persoons [1997], Van Elshocht, Verbiest, Kauranen, Persoons, Langeveld-Voss and Meijer [1997], Kauranen, Verbiest and Persoons [1998, 1999], Schanne-Klein, Hache, Roy, Flytzanis and Payrastre [1988]). Using a quarter-wave plate method, it was demonstrated that optical activity also occurs in achiral thin films when the reflection symmetry of the experimental arrangement is broken by in-plane anisotropy of the samples (Verbiest, Kauranen, Van Rompaey and Persoons [1996], Verbiest, Kauranen and Persoons [1998]), by the alignment of an analyzing polarizer (Stolle, Marowsky, Schwarzberg and Berkovic [1996]), or by a second fundamental beam (Verbiest, Kauranen and Persoons [1999]). In subsequent experiments, the contributions of chirality and anisotropy to optical activity were unambiguously separated (Verbiest, Van Elshocht, Kauranen, Hellemans, Snauwaert, Nuckolls, Katz and Persoons [1998], Kauranen, Van Elshocht, Verbiest and Persoons [2000], Sioncke, Van Elshocht, Verbiest, Persoons, Kauranen, Phillips and Katz [2000]). A quarter-wave plate technique, but again based on Fourier analysis, was also applied to determine the susceptibility components of isotropic adsorbates (Geiger, Stolle, Marowsky, Palenberg and Felderhof [1995]). Other examples include the determination of symmetry and phase of second-harmonic signals from crystal surfaces (Andersson, Schanne-Klein and Hache [1999]), the study of molecular alignment at a polymer/glass interface (Park, Yoo, Sakai, Hoshi, Ishikawa and Takezoe [1998]) and in organic thin films (Simpson and Rowlen [2000]).
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2.3.3. Two-beam second-harmonic generation Traditionally, SHG polarization measurements have been performed in a collinear geometry, with a single beam at the fundamental frequency. However, such geometry suffers from two significant problems. The measured polarization patterns do not always allow the expansion coefficients f, g, and h to be determined in a precise way. Furthermore, the geometry does not offer direct ways of verifying the quality of the experimental data or the determined susceptibility components. Such consistency tests are desirable, as they can be used to address possible systematic errors in the experimental setup or in the theoretical model used to interpret the results. Noncollinear arrangements are commonly used in SFG experiments, as these naturally involve two separate light sources. Such arrangements have been shown to offer some important advantages. As the signal is spatially separated from the input beam, the need for spectral filtering is eliminated (Shen [1989]). Clearly, this feature is also shared by SHG with two noncollinear fundamental beams. For instance, Muenchausen, Keller and Nogar [1987] demonstrated that the noise due to broadband fluorescence and scattered light is effectively reduced in two-beam SHG geometry by spatially filtering the signal. Provencher, Côté and DenariezRoberge [1993], and Provencher, Denariez-Roberge, Suau, Tian, Munger and Leblanc [1995] realized that separately controlling the polarization states of the two input beams increases the number of independent measurements and facilitates the experimental characterization of the nonlinear response of a sample. Noncollinear arrangements have also been used for continuous polarization measurements of the second-harmonic response (Cattaneo and Kauranen [2003], Cattaneo, Vuorimaa, Lemmetyinen and Kauranen [2004], Cattaneo [2004]). A given second-harmonic field produced jointly by the two fundamental beams a(ω) and b(ω) will be of the functional form (fig. 2.3)
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E j (2ω) = f j a p (ω)bp (ω) + g j a s (ω)bs (ω) + hj a p (ω)bs (ω) + k j a s (ω)bp (ω)
(2.16)
with j = p, s. As is evident from eq. (2.16), in two-beam SHG four expansion coefficients are needed to specify a given second-harmonic signal. This may at first appear as an unnecessary complication as opposed to the single-beam arrangement, where only three coefficients appear [eq. (2.2)]. However, simpler polarization patterns can be recorded by keeping the polarization of one beam (control beam a) fixed while continuously modulating the polarization of the other beam (probe beam b).
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Fig. 2.3. Schematic representation of two-beam SHG measurements. Two fundamental beams a(ω) and b(ω) are incident on a nonlinear sample and produce jointly a second-harmonic signal in transmission or reflection. For clarity, only the transmitted second-harmonic beam is shown.
A detected signal is then of the functional form 2 I m (2ω) = α m bp + β m bs
(2.17)
where the index m denotes a particular measurement, i.e., a particular choice of control and signal polarizations. This simple dependence allows a very precise determination of the experimental parameters α m and β m . By using different polarization combinations, all eight expansion coefficients f j , g j , hj , and k j for j = p, s can conveniently be determined pairwise. This procedure also yields extra experimental information, which can be used to address the internal consistency of the determined expansion coefficients (Cattaneo and Kauranen [2003], Cattaneo, Vuorimaa, Lemmetyinen and Kauranen [2004], Cattaneo [2004]). To elucidate the advantages of two-beam SHG, we again consider samples of C∞v symmetry. For a given experimental geometry, complete characterization of such a sample with single-beam SHG requires the relative complex values of three expansion coefficients f p , g p , and hs [eqs. (2.3) and (2.4)]. Detecting the ppolarized second-harmonic signal allows f p and g p to be determined. Detecting the s-polarized second-harmonic signal does not yield any information, since hs only appears as an absolute scaling constant. With an analyzer placed at ±45◦ with respect to the plane of incidence, the detected second-harmonic signal is 2 2 2 I ±45 (2ω) = f p ep + g p es ± hs ep es . (2.18)
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A single measurement (at +45◦ or −45◦ ) is then sufficient to determine all three expansion coefficients. As the measurements are insensitive to absolute phase, one of the expansion coefficients can be taken as real. Nevertheless, the pattern
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depends simultaneously on no less than five real fitting parameters and is therefore often insensitive to small changes in their values. In principle, the values of f p and g p from the p-polarized measurement could be used when fitting the pattern measured at ±45◦ . This, however, can result in increased uncertainty, as possible errors in the first measurement propagate to the second one. The high C∞v symmetry results in simplified functional forms of the secondharmonic signals also in the case of two fundamental beams a(ω) and b(ω). The p and s polarization components of the second-harmonic field are then given by (Cattaneo and Kauranen [2003]): E p (2ω) = f p a p (ω)bp (ω) + g p a s (ω)bs (ω),
(2.19)
E (2ω) = h a (ω)b (ω) + k a (ω)b (ω).
(2.20)
s
s p
s
s s
p
Similarly to what was done for the single-beam case (§ 2.2), it can be shown that the coefficients in the approximation of unity refractive indices are given by f p = ∓2χxxz sin θ a cos θ b + cos θ a sin θ b cos Θ + 2χzxx cos θ a cos θ b sin Θ + 2χzzz sin θ a sin θ b sin Θ, g = 2χzxx sin Θ, p
h = 2χxxz sin θ , s
a
k = 2χxxz sin θ , s
θa
b
θb
(2.21) (2.22)
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(2.23) (2.24)
where and are the incident angles of the two fundamental beams, Θ is the angle of propagation of the second-harmonic beam, and the upper (lower) signs refer to reflected (transmitted) second-harmonic signals (fig. 2.4). For a complete characterization, the four expansion coefficients f p , g p , hs , and k s must be determined. However, once the polarization of the control beam is fixed, a given second-harmonic signal is specified by only two parameters α m and β m [eq. (2.17)], which can be determined very precisely. Different polarization choices for control and second-harmonic beams cause different pairs of coefficients to interfere with each other (Cattaneo, Vuorimaa, Lemmetyinen and Kauranen [2004]). Although three measurements are sufficient to determine the relative values of all four expansion coefficients, the scheme naturally leads to the four measurements shown in table 2.1. Combining the results of all these measurements allows a given coefficient to be determined in two independent ways. Comparison of the two values is a measure of the quality of experimental data and, in particular, of possible systematic errors in the experimental setup, such as misalignments or bad optical components (Cattaneo and Kauranen [2003]). Inconsistency between the coefficients can also indicate that the symmetry assumed for the sample (in this case C∞v ) is not correct.
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Fig. 2.4. Detailed geometry and notation for two-beam SHG measurements in the limit of unity refractive indices. The subscripts 1 and 2 as well as + and − are used to describe the second-harmonic fields to make the notation consistent with the detailed theory of § 4. Table 2.1 Polarization combinations for determining the expansion coefficients by two-beam SHG for a sample of C∞v symmetry Measurement
Control
1 2 3 4
±45◦ ±45◦ p s
Signal
αm
βm
p s ±45◦ ±45◦
fp
±g p hs ±hs gp
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±k s fp ±k s
2.3.4. Ellipsometric measurements As we have already discussed, the starting point of any precise characterization of the second-order response of a surface or a thin film should be the accurate and reliable determination of the experimental parameters f, g, h and k for a number of detected second-harmonic signals. Above, we have emphasized the use of wave plates to modulate the state of polarization of the fundamental beam and detecting various polarization components of the second-harmonic signal. The experimental coefficients can also be determined, at least for the case of the onebeam geometry, by modulating the polarization properties of the second-harmonic signal. This is because SHG is a coherent process leading to a signal with a welldefined state of polarization. Such approaches are often referred to as nonlinear ellipsometry (Plocinik and Simpson [2003], Plocinik, Everly, Moad and Simpson [2005], McHugh, Atkinson and Kubrakov [2005]) due to their similarities to traditional ellipsometry. For the specific case of null-ellipsometry, the state of polarization of the secondharmonic beam is modulated by a half-wave and a quarter-wave plate until the
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signal can be extinguished by an analyzer (Plocinik and Simpson [2003]). By repeating this for a number of input polarizations, the relative complex values of the expansion coefficients can be determined. Compared to the rotating quarterwave plate technique, null ellipsometry involves several search procedures for the extinction. On the other hand, only signals for fixed settings of the polarization components need to be recorded, thereby potentially speeding up the experiment. The search procedures can be avoided by rotating one of the polarization components in the second-harmonic signal path (Plocinik, Everly, Moad and Simpson [2005], McHugh, Atkinson and Kubrakov [2005]), leading to a similar predetermined procedure as the quarter-wave plate technique except that now the state of polarization of the detected second-harmonic light is modulated. In addition, null ellipsometry may become important in kinetic studies of biological binding processes where the suppression of background signals is important (Polizzi, Plocinik and Simpson [2004]).
2.4. Low-symmetry samples So far, we have described the main ideas behind the polarization techniques by assuming that the samples are achiral and isotropic in the plane of the substrate. However, the techniques themselves are not limited to such high-symmetry samples. Chiral and/or anisotropic samples have an increased number of independent components of the susceptibility tensor. While the functional forms of the secondharmonic signal in the single- and two-beam geometries [eqs. (2.2) and (2.16), respectively] remain valid, the detailed forms of the expansion coefficients [eqs. (2.12)–(2.14) and (2.21)–(2.24)] are different. For the case of chiral and isotropic surfaces (symmetry group C∞ ), reflection symmetry with respect to planes containing the surface normal is broken. This introduces additional components (χxyz = χxzy = −χyzx = −χyxz ) already in the electric-dipole-allowed susceptibility tensor (Byers, Yee, Petralli-Mallow and Hicks [1994]). The appearance of these components gives rise to effects that can be associated with chirality (see § 3.1). However, the in-plane coordinate directions x and y can still be chosen freely because of isotropy. In-plane anisotropy complicates the situation even further as the coordinates x and y must now be chosen in a system tied to the sample. Contrary to the isotropic case, different choices for the in-plane directions result in different values of the susceptibility components. On the other hand, the experimental geometry is still most conveniently described in the (p, s) system. Depending on the azimuthal orientation of the sample in the setup, the in-plane coordinates then contribute
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differently to the various expansion coefficients, i.e., the coefficients are functions depending also on the azimuthal orientation angle (Kauranen, Van Elshocht, Verbiest and Persoons [2000]). To completely understand the nonlinear response of a low-symmetry sample, one should determine all the independent components of the susceptibility tensor. To do this, it is necessary to measure at least the same number of independent experimental coefficients as there are unknown tensor components. The coefficients can then be related to the tensor components using an appropriate theoretical model. Depending on the symmetry of the sample, the number of independent measurements can be increased, e.g., by detecting second-harmonic signals in reflection and transmission (Kauranen, Maki, Verbiest, Van Elshocht and Persoons [1997]), or by varying the azimuthal orientation (Verbiest, Van Elshocht, Kauranen, Hellemans, Snauwaert, Nuckolls, Katz and Persoons [1998], Kauranen, Van Elshocht, Verbiest and Persoons [2000], Siltanen, Cattaneo, Vuorimaa, Katz, Phillips, Lemmetyinen and Kauranen [2004], Siltanen and Kauranen [2006]).
2.5. Experimental considerations
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In single-beam measurements, the fundamental beam is applied to the sample, and transmitted or reflected second-harmonic light is detected (fig. 2.5a). In continuous polarization measurements, the second-harmonic signal is monitored continuously while the polarization of the fundamental beam is varied by a rotating wave plate. While important information can be obtained using a half-wave plate, in the case of complex-valued coefficients the best results are obtained using a quarterwave plate (Maki, Verbiest, Kauranen, Van Elshocht and Persoons [1996], Maki, Kauranen, Verbiest and Persoons [1997]). Before the wave plate, the polarization state of the fundamental beam has to be cleaned with a linear polarizer. A similar analyzer can be used to select a particular second-harmonic polarization component. In certain ellipsometric measurements, both half- and quarter-wave plates are used to control the state of polarization of the second-harmonic light before detection through an analyzer (Plocinik and Simpson [2003], Plocinik, Everly, Moad and Simpson [2005]). For two-beam SHG measurements (fig. 2.5b), the fundamental beam is split into two beams of nearly the same intensity (control and probe), which are applied to the same spot of the film. The second-harmonic signal generated jointly by the control and probe beams in either transmission or reflection is recorded continuously as a function of the probe polarization. During the measurements, the polarization of the control beam is kept fixed. As in the single-beam setup,
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Fig. 2.5. Schematic representations of the arrangement for the one- (a) and two- (b) beam SHG experiments. An important difference is that spectral filters, which may compromise the polarization purity of the setup, are not needed in the two-beam setup. HWP, half-wave plate; QWP, quarter-wave plate; PMT, photomultipler tube.
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polarizers are used to clean the probe polarization before the quarter-wave plate and to select a particular component of the control and second-harmonic beams. Clearly, in order to obtain reliable results with nonlinear polarization techniques, it is essential to use high-quality polarization components and to take due care in their alignment in the experimental setup. High-quality Glan polarizers (extinction ratio ∼ 10−6 ) are usually used in all cases to define the linear states of polarization. Unfortunately, such polarizers often lead to a considerable deviation in the propagation direction of the transmitted beam. This problem is particularly critical for the two-beam geometry when the control and signal polarizations are varied between different measurements. One should therefore select polarizers that minimize beam deviation to maintain the precision of two-beam experiments. For precise polarization control, zero-order wave plates with the highest possible retardation tolerance (at least λ/500) should be used. This is particularly important as the laser sources that are typically used have a short pulse length and consequently a relatively broad bandwidth. Compound zero-order wave plates and achromatic wave plates consist of two crossed high-order pieces. We and others (e.g., Mitchell and McAloney [2004]) have found that the mutual alignment of
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the two pieces of such wave plates can be surprisingly poor. Deviations of several degrees in the alignment are not uncommon. One should therefore take great care in verifying that the wave plates behave as expected. The alignment problems of compound wave plates can of course be resolved by true zero-order wave plates. In the single-beam setups, the second-harmonic light has to be separated from the strong fundamental beam by filtering with a short-pass filter and/or an interference filter. In addition, a long-pass filter before the nonlinear sample has to be used to block unwanted second-harmonic light generated by the preceding optical components. An isotropic filter oriented at normal incidence should not have any influence on light polarization. However, stress-induced birefringence in the filter can noticeably change the polarization of the transmitted beam. Such filters can therefore compromise the polarization purity of the arrangement. In the two-beam setups, the fundamental and second-harmonic beams propagate along different directions and no spectral filters are needed. Two-beam arrangements, however, are more difficult to align than one-beam arrangements. It is particularly important to verify that the two input beams and the output beam remain in the same plane of incidence when their polarization is varied by reorienting the polarization components. Here, the most critical components are the calcite Glan polarizers. The precision of the experimental characterization of a thin film can also be influenced by the substrate (Cattaneo, Vuorimaa, Lemmetyinen and Kauranen [2004]). In general, the measured second-harmonic signal includes contributions from both surfaces of the substrate. While the contributions from the film and the film-side surface of the substrate are in phase or have a constant phase difference, the relative phase of the contribution from the second surface of the substrate depends on its exact thickness, which is usually different for each sample. Such effects can have a dramatic impact on the measured signals for thin films (Cattaneo, Vuorimaa, Lemmetyinen and Kauranen [2004]). While the contribution from the second surface of the substrate can be eliminated by proper experimental geometry and by using thick and/or wedged substrates, eliminating the contribution of the surface supporting the thin film is not straightforward (Berkovic, Shen, Marowsky and Steinhoff [1989]). Besides dipole-allowed surface contributions, the response of the sample also includes, in general, dipole-forbidden contributions from the bulk of the substrate (Bloembergen, Chang, Jha and Lee [1968], Guyot-Sionnest, Chen and Shen [1986], Guyot-Sionnest and Shen [1987]). It is common to neglect all substrate contributions when they are expected to be much weaker than the response of the nonlinear film. For molecules with large nonlinearity, this may be already be valid in the case of a monolayer. For molecules with a relatively weak second-harmonic response, however, this assumption
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may be justified for sufficiently thick films, but should not be made for thin films (Cattaneo, Vuorimaa, Lemmetyinen and Kauranen [2004]).
§ 3. Applications of polarization techniques The most direct application of nonlinear polarization techniques is the accurate determination of the second-order susceptibility tensor of the samples. As explained in § 2, the tensor yields important information on the structure of the sample. This is of relevance both for the characterization of new nonlinear materials and for fundamental studies of surface and interface effects. Valuable information on the investigated samples can also be extracted from the general features of the polarization response, without the need to extract the susceptibility components. In this section, we review two examples of such applications.
3.1. Chirality and circular-difference response Chiral molecules have no reflection or inversion symmetry and occur in two forms (enantiomers) that are mirror images of each other. In linear optics, such molecules are known for their optical activity effects, which arise from the different interaction of the molecules with left- and right-hand circularly polarized light (Barron [2004]). The differences in the real parts of the refractive indices lead to the rotation of the plane of linear polarization as light traverses a chiral medium. The differences in the imaginary parts, on the other hand, lead to circular dichroism, i.e., different absorbances for the two circular polarizations. In isotropic solutions of chiral molecules, optical activity effects arise from magnetic contributions to the linear optical response. Linear optical activity effects are usually relatively weak, with a relative change n/n < 10−3 for the refractive index. Chiral molecules are necessarily non-centrosymmetric and possess an electricdipole-allowed second-order response. This response survives even in isotropic collections of chiral molecules although SHG is forbidden (Giordmaine [1965]). SFG from sugar solutions was observed already in 1966 (Rentzepis, Giordmaine and Wecht [1966]) and the result later confirmed by Shkurinov, Dubrovskii and Koroteev [1993]. However, the relative strength of such effects has recently been reinvestigated (Fischer, Wiersma, Righini, Champagne and Buckingham [2000], Belkin, Han, Wei and Shen [2001]) and possible reasons for the discrepancy between the old and new results have been presented (Cattaneo, Siltanen, Wang and Kauranen [2005]).
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Thin films of chiral molecules are often isotropic in the plane of the substrate but have net molecular orientation along the substrate normal. Thin films can also be chiral due to a twist in the orientational distribution of achiral molecules (Burke, Moad, Polizzi and Simpson [2003]). In the case of in-plane isotropy, chiral thin films belong to the symmetry group C∞ . Compared to isotropic films of achiral molecules (symmetry C∞v ), chirality introduces new components (χxyz = χxzy = −χyzx = −χyxz ) into the second-order susceptibility tensor already in the electric-dipole approximation (Byers, Yee, Petralli-Mallow and Hicks [1994], Byers and Hicks [1994], Reider and Heinz [1995]). On the level of the expansion coefficients of the single-beam geometry, all three coefficients f , g, h are then allowed for both p- and s-polarized second-harmonic light. In addition, the expansion coefficients can be classified as achiral (f p , g p , hs ) or chiral (f s , g s , hp ), depending on whether they are allowed for both C∞v and C∞ symmetries or only for C∞ (Maki, Kauranen and Persoons [1995]). It is then straightforward to show that the SHG efficiency from chiral thin films is different for left- and right-hand circularly polarized light. In the one-beam geometry of fig. 2.2, the two circular polarizations of the fundamental beam are characterized by ep (ω) = ±ies (ω).
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The intensity of second-harmonic light then becomes from eq. (2.2) 4 I (2ω) = |−f + g ± ih|2 es (ω) .
(3.1)
(3.2)
The circular-difference effects therefore arise from interference between the achiral and chiral parts of the second-order response provided that a phase difference between the two types of coefficients exists. Such an effect was first observed from a binaphthyl (Petralli-Mallow, Wong, Byers, Yee and Hicks [1993]) and soon thereafter from a dipeptide (Crawford, Haslam, Probert, Gruzdkov and Frey [1994]) and polyisocyanide (Kauranen, Verbiest, Maki and Persoons [1994]). The circular-difference effects have been explained within the electric-dipole approximation of the light–matter interaction (Byers, Yee, Petralli-Mallow and Hicks [1994], Byers and Hicks [1994]) and by including magnetic contributions to the second-order response (Kauranen, Verbiest, Maki and Persoons [1994], Maki, Kauranen and Persoons [1995], Kauranen, Maki, Verbiest, Van Elshocht and Persoons [1997], Schanne-Klein, Hache, Roy, Flytzanis and Payrastre [1988]). The circular-difference response was conveniently accessed by using a rotating quarter-wave plate to control the state of polarization of the fundamental beam, which subsequently led to the development of the more general techniques described in § 2.3.2 (Maki, Kauranen, Verbiest and Persoons [1997]).
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For isotropic solutions of chiral molecules, all linear polarizations are equivalent. However, in a surface or thin-film geometry, the p and s components are not equivalent. As a consequence, chiral properties of thin films can also be addressed by linear fundamental polarizations that are oriented at ±45◦ from the p direction ep (ω) = ±es (ω).
(3.3)
In this case, the intensity of the second-harmonic signal becomes 4 I (2ω) = |f + g ± h|2 es (ω) .
(3.4)
The intensities are therefore different for the ±45◦ linear polarizations. However, such linear-difference effect arises from the in-phase parts of the achiral and chiral expansion coefficients. Such effects were first predicted by Hecht and Barron [1994] and observed by Verbiest, Kauranen, Maki, Teerenstra, Schouten, Nolte and Persoons [1995]. The circular- and linear-difference effects are not the only ways to probe chiral properties of thin films. For example, it can be shown that in certain cases, the rotation of the direction of polarization of the second-harmonic beam away from the direction of the input polarization can be taken as a probe of chirality (Byers, Yee and Hicks [1994]). In addition, when the full polarization lineshape [e.g., eq. (2.2)] is recorded by rotating a quarter-wave plate through −180◦ to +180◦ , the lineshape must possess reflection symmetry with respect to 0◦ if the thin film is achiral (Kauranen, Verbiest and Persoons [1998]). An asymmetry is therefore also a probe of chirality and does not require any specific phase relation between the achiral and chiral parts of the nonlinearity. A more comprehensive review of the various chiral signatures of thin films in the one-beam geometry has been presented by Kauranen, Verbiest and Persoons [1998]. Additional collinear techniques based on the polarization analysis of the second-harmonic signal have been analyzed by Hecht and Barron [1996]. Recently, Kriech and Conboy [2004] demonstrated a new technique to probe chirality based on two-beam second-harmonic generation with symmetrically applied fundamental beams. As seen in § 2.3.3, achiral films with in-plane isotropy (symmetry C∞v ) are characterized by SHG coefficients f p , g p , hs and k s . Chirality of the films leads to three additional coefficients f s , hp and k p (g s = 0 even for C∞ symmetry), which depend on the tensor components χxyz = χxzy = −χyzx = −χyxz . When the fundamental beams are incident symmetrically with respect to the sample normal (i.e., θ a = −θ b in the geometry of fig. 2.4), hs = −k s (proportional to χxxz ) and hp = −k p (proportional to χxyz ) while all other expansion coefficients vanish. The surviving coefficients lead to a
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second-harmonic signal along the sample normal when mixed s and p input polarizations are used. Most importantly, the p component of the second-harmonic signal in this geometry only arises from the chiral part of the response, while the s-polarized signal is completely achiral. This remarkable feature has been applied by the same group to obtain microscopic images of chiral surfaces (Kriech and Conboy [2005]). It is important to note that all the above chiral-sensitive techniques are based on the in-plane isotropy of the thin-film sample. For samples with in-plane anisotropy, chiral signals can arise if the orientation of the sample makes the setup chiral (Verbiest, Kauranen, Van Rompaey and Persoons [1996]). As a consequence, great care must be exercised to separate the chiral and anisotropic properties of samples with very low symmetry. The safest way to do that is through a complete tensor analysis of the nonlinear response of the sample (Verbiest, Van Elshocht, Kauranen, Hellemans, Snauwaert, Nuckolls, Katz and Persoons [1998], Kauranen, Van Elshocht, Verbiest and Persoons [2000], Siltanen, Cattaneo, Vuorimaa, Katz, Phillips, Lemmetyinen and Kauranen [2004]). To conclude this subsection, we note that nonlinear probes of chirality are not limited to second-order effects and thin-film samples. Other reviews of chirality and chiral nonlinear optics have been presented by Hicks [2002], Sioncke, Verbiest and Persoons [2003], Fischer and Hache [2005], as well as Belkin and Shen [2005].
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3.2. Higher-multipole contributions to the surface nonlinearity of isotropic materials Being second-order effects, SFG and SHG are electric-dipole-forbidden in the bulk of centrosymmetric materials. At the surface of such materials, however, the symmetry is broken and the processes are dipole-allowed. In addition, contributions can arise from the bulk of the material as a consequence of higher-multipole processes. SHG in centrosymmetric media was examined theoretically for the first time by Bloembergen, Chang, Jha and Lee [1968]. Their approach included an electricdipole-forbidden bulk contribution and a thin polarization sheet at the surface, a phenomenological model that is still used nowadays to describe the process. The surface contribution was estimated in terms of bulk parameters, and the secondharmonic response was considered to arise mainly from the field discontinuity at the interface, whereas the effect of the structural discontinuity was considered to be negligible. This interpretation was consistent with the experimental investigations of the time, which indicated a lack of sensitivity to surface conditions
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(Ducuing and Bloembergen [1963], Brown, Parks and Sleeper [1965], Bloembergen, Chang and Lee [1966, 1968]). The first indications of the intrinsic surface sensitivity of SHG were provided by the experimental results of Brown and Matsuoka [1969], Rudnick and Stern [1971], and Chen, Bower, Wang and Lee [1973]. However, only after the experimental work of the early 1980s (Chen, de Castro and Shen [1981], Heinz, Chen, Ricard and Shen [1981], Chen, Heinz, Ricard and Shen [1983], Tom, Mate, Zhu, Crowell, Heinz, Somorjai and Shen [1984]) did it become clear that the prevalent model for the nonlinear response of centrosymmetric media, based on their bulk properties, had to be extended to include surface electric-dipole contributions. Guyot-Sionnest, Chen and Shen [1986] expanded the theory to show that both structural and field discontinuities separately contribute to the nonlinearity of an interface, giving rise to quadrupole (and possibly higher multipole) effects. The two contributions can, however, be combined into an effective electric-dipole-type surface susceptibility. In a subsequent experimental study, Guyot-Sionnest and Shen [1987] separately addressed the contributions to the nonlinearity of a solid–liquid interface due to structural discontinuities and to field discontinuities by varying the refractive index of the liquid. When the refractive indices of the media are matched, there is no field discontinuity and only the structural discontinuity is present. The authors concluded that the nonlinearity of a glass–air interface arises essentially from the field discontinuity, which has an electric-quadrupole character. In addition, they observed that the second-harmonic response of the same molecular monolayer could show an electric-dipole or -quadrupole character, depending on the fundamental frequency used.
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3.2.1. Theoretical treatment At the interface of two isotropic and centrosymmetric media, the inversion symmetry is broken and SHG is electric-dipole-allowed (fig. 3.1). In addition, nonlocal (electric-quadrupole) contributions arise from the rapid variation of the electric field and of the material properties across the layer. All these effects can, however, be described by an effective surface polarization (Guyot-Sionnest, Chen and Shen [1986]) Ps (2ω) = χ s : e(ω, z = 0)e(ω, z = 0),
(3.5)
where e(ω, r) is the total fundamental field, x and y are the coordinates in the plane of the surface, and z is along the surface normal. For surfaces of C∞v symmetry, the effective susceptibility tensor χ s has only three independent compos = χ s = χ s = χ s , χ s = χ s , and χ s . nents: χxxz xzx yyz yzy zxx zyy zzz
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Fig. 3.1. Geometry and notation to discuss the surface and bulk contributions to the nonlinear response. The figure displays two fundamental beams a(ω) and b(ω), which are required to access the separable bulk contribution. The total fundamental field used in § 3.2.1 is then a b e(ω, r) = a(ω)eik ·r + b(ω)eik ·r .
The effective polarization at frequency 2ω in the bulk of an isotropic medium is given by (Pershan [1963], Adler [1964], Guyot-Sionnest and Shen [1987]) Peff = P − ∇ · Q + i
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c ∇ × M. 2ω
(3.6)
To first order in the magnetic-dipole and electric-quadrupole interactions, the nonlinear polarization at 2ω is given by (Heinz [1991], Reider and Heinz [1995], Kauranen, Verbiest and Persoons [1998]) j k eem j k P i (2ω) = χijeee k (2ω, ω, ω)e (ω)e (ω) + χij k (2ω, ω, ω)e (ω)b (ω) eeq
+ χij kl (2ω, ω, ω)ej (ω)∇ k el (ω),
(3.7)
where e(ω) and b(ω) are, respectively, the electric field and magnetic induction field at the fundamental frequency. The other two terms on the right-hand side of eq. (3.6) are, respectively, the nonlinear quadrupolarization qee
Qij (2ω) = χij kl (2ω, ω, ω)ek (ω)el (ω),
(3.8)
and the nonlinear magnetization j k Mi (2ω) = χijmee k (2ω, ω, ω)e (ω)e (ω).
(3.9)
As indicated by the number of subscripts, χ eee , χ eem and χ mee are all thirdrank tensors. However, χ eem and χ mee are associated with two polar vectors (electric quantities) and an axial vector (magnetic quantity), while χ eee is associated
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with three polar vectors. As a consequence, the tensors transform differently under improper operations (reflection or inversion) (Arfken and Weber [1995]). Under proper operations (rotations), however, similar transformation rules apply. In particular, for three-dimensional isotropic materials, each tensor has only one independent component (Shen [1984], Boyd [2003]): χxyz = χyzx = χzxy = −χxzy = −χyxz = −χzyx .
(3.10)
For SHG, the two last indices of χ eee and χ mee can be freely interchanged and the tensors vanish identically. In the case of χ eem , however, the indices are associated with distinguishable fields and therefore the tensor is, in general, non-vanishing. With eq. (3.10) and using Maxwell’s equations, the contribution of χ eem to P i can be rewritten as (Cattaneo [2004]) ic eem (3.11) χ (δj k δil − δj l δik )ej ∇ k el . ω xyz Clearly, the χ eem contribution acts as an effective quadrupole contribution. The contribution of Q to the effective nonlinear polarization of eq. (3.6) can also be reduced to a similar form. With eq. (3.8), we obtain qee qee ∇ j Qij = ∇ j χij kl ek el + χij kl ek ∇ j el + el ∇ j ek . (3.12) eem χxyz [e × b]i =
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For the purpose of characterizing the nonlinear response of homogeneous bulk materials, the first term in eq. (3.12) can be neglected (Heinz [1991], Sipe, Mizrahi and Stegeman [1987]). Rearranging the dummy indices j, k and l, we obtain qee qee ∇ j Qij = χiklj + χikj l ej ∇ k el . (3.13) The effective polarization is therefore given by Q
i = χij kl ej ∇ k el , Peff Q
eeq
qee
(3.14) qee
eem with χij kl = χij kl − χiklj − χikj l + ic ω χxyz (δj k δil − δj l δik ). For isotropic materials, the fourth-rank tensor χ Q has 27 non-zero components, of which only 3 are independent (Boyd [2003]): Q
Q Q Q δij δkl + χxyxy δik δj l + χxyyx δil δj k . χij kl = χxxyy
(3.15)
The effective polarization [eq. (3.14)] then becomes i Q Q Q = χxxyy ei ∇ j ej + χxyxy ej ∇ i ej + χxyyx ej ∇ j ei , Peff
(3.16)
or, equivalently 1 Q Q Q Peff = χxxyy e(∇ · e) + χxyxy ∇(e · e) + χxyyx (e · ∇)e. 2
(3.17)
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This reduces to the form usually given in the literature for the effective bulk polarization (Bloembergen, Chang, Jha and Lee [1968], Guyot-Sionnest, Chen and Shen [1986], Guyot-Sionnest and Shen [1987], Sipe, Mizrahi and Stegeman [1987]): Peff (2ω, r) = βe(ω, r) ∇ · e(ω, r) + γ ∇ e(ω, r) · e(ω, r) + δ e(ω, r) · ∇ e(ω, r), (3.18) with β = χxxyy , γ = χxyxy /2, and δ = χxyyx . The parameter δ = δ + β + 2γ , originally introduced by Bloembergen, Chang, Jha and Lee [1968], then becomes Q δ = χxxxx . In terms of the original tensors χ eeq , χ qee and χ eem the parameters are Q
eeq
Q
qee
Q
qee
β = χxxyy − χxyyx − χxyxy ,
(3.19)
eeq qee eem , γ = χxyxy /2 − χxxyy − (ic/2ω)χxyz eeq qee qee eem δ = χxyyx − χxyxy − χxyyx + (ic/ω)χxyz , eeq qee δ = χxxxx − 2χxxxx .
(3.20) (3.21) (3.22)
By integrating over the portion of space occupied by the bulk, eq. (3.18) can be modified into a form that explicitly shows the dependence of the bulk contributions on the coherence length (Heinz [1991], Shen [1999], Wei, Hong, Lvovsky, Held and Shen [2000], Held, Lvovsky, Wei and Shen [2002]). For homogeneous media, the first term in eq. (3.18) can be neglected, since ∇ · e(ω, r) = ε −1 ∇ · d(ω, r) vanishes according to the Maxwell equations. As realized by several authors (Sipe, Mizrahi and Stegeman [1987], Guyot-Sionnest and Shen [1988], Heinz [1991], Reider and Heinz [1995]), the second term in eq. (3.18) is indistinguishable from surface contributions in experiments that allow no modification of the surface. In particular, the dependence of the term on coherence length is always compensated by geometrical factors. Its contribution s and χ s of the effective surface susceptican be included in the components χzxx zzz bility (Guyot-Sionnest and Shen [1988], Heinz [1991], Reider and Heinz [1995]). The third term in eq. (3.18) vanishes when a single fundamental beam is present in the material (Heinz [1991], Sipe, Mizrahi and Stegeman [1987]), but can be accessed using two input beams. Contrary to surface-like contributions, it leads to radiation that builds up in the medium and therefore depends on the coherence length of the nonlinear process (Heinz [1991], Shen [1999], Wei, Hong, Lvovsky, Held and Shen [2000], Held, Lvovsky, Wei and Shen [2002]). As we will show later, this term also displays a polarization dependence that is completely different from that of surface contributions. The two contributions can therefore be separated by appropriate polarization measurements of the second-harmonic response.
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Considering the intrinsic symmetry properties of the tensors χ eeq and χ qee and their relation to χ eem , the above derivation suggests that the parameters β, γ , and δ are often of the same order of magnitude (Held, Lvovsky, Wei and Shen [2002]). The separable δ contribution has therefore been used as a guide to the strength of the inseparable γ contribution. When the dominant nonlocal contribution occurs at the fundamental frequency, for example, eqs. (3.20) and (3.21) yield γ = δ /2 for the electric-quadrupole part and γ = −δ /2 for the magneticdipole part of the response. When the dominant nonlocal contribution occurs at the second-harmonic frequency, the relation between γ and δ is less trivial and depends on the system under investigation. However, a relation can be found by applying microscopic or quantum-mechanical models of the interaction. For example, numerical calculations for non-interacting spherical molecules and for C60 (fullerene) molecules treated within a LCAO (Linear Combination of Atomic Orbitals) calculation both yielded γ = −δ /3 (Koopmans, Janner, Wierenga, Rasing, Sawatzky and van der Woude [1995]). These arguments are not valid when δ vanishes, as for example under completely off-resonant excitation or in the freeelectron approximation (Bloembergen, Chang, Jha and Lee [1968]). 3.2.2. Identification of bulk contributions to surface nonlinear optics
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Whenever second-order processes are used as surface probes, it is desirable to separate the surface and bulk contributions to the nonlinear response of the sample. In some cases, such as for isotropic materials, this can be done by a proper choice of polarizations (Guyot-Sionnest and Shen [1987], Sipe, Mizrahi and Stegeman [1987]). In general, however, there are limitations to the extent to which this is possible. Wang [1969] showed that the reflected second-harmonic signal produced by a linearly polarized plane wave incident on an isotropic surface depends on the γ bulk coefficient only through a particular linear combination with a surface coefficient. Guyot-Sionnest, Chen and Shen [1986] came to the same conclusion when considering other special classes of SHG experiments. Thereafter, a number of sophisticated experimental arrangements involving noncollinear input beams were proposed to unambiguously separate the surface and bulk contributions (Guyot-Sionnest, Chen and Shen [1986], Sipe, Moss and van Driel [1987]). However, such suggestions proved to be misleading, as Sipe, Mizrahi and Stegeman [1987] showed that the result obtained by Wang remains valid in the case of noncollinear beams of arbitrary polarizations and propagation directions. As a consequence, an important component of the bulk contribution is indistinguishable from surface contributions in experiments that do not allow the surface to be modified. This result was then confirmed by other authors (Guyot-Sionnest and Shen [1988], Heinz [1991], Reider and Heinz [1995]). The
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calculation of Sipe, Mizrahi and Stegeman [1987] shows that in the case of a single fundamental beam incident on a homogeneous, isotropic material, s-polarized SHG only arises from surface contributions. However, additional bulk contributions to s-polarized SHG occur if one takes into account possible variations of the bulk parameters (Guyot-Sionnest and Shen [1988]). When two noncollinear input beams are used (fig. 3.1), the bulk response includes the δ component that leads to radiation which builds up in the medium and therefore depends on the coherence length of the nonlinear process (Heinz [1991], Sipe, Mizrahi and Stegeman [1987]). Such dependence can be used to separate this component from surface contributions. In principle, this could be done by performing measurements with different angles of incidence. In practice, however, this is difficult to achieve as the coherence length does not usually vary strongly over the accessible angles (Heinz [1991]). The situation is somewhat simpler when measurements in both reflection and transmission geometries can be performed. In the latter case the phase-matching condition is better approximated than in the former, leading to an increased coherence length and, consequently, a larger bulk contribution (Heinz [1991], Shen [1999]). Absolute comparison of reflected and transmitted signals relies, however, on accurate calibration of the experimental setup and requires detailed knowledge of the linear optical properties of the sample. For these reasons, the bulk and surface contributions have usually been separated by comparing reflected and transmitted spectra measured using visible–infrared SFG (Shen [1999], Wei, Hong, Lvovsky, Held and Shen [2000], Held, Lvovsky, Wei and Shen [2002]). In addition to being limited to systems where the bulk and surface spectra are clearly different, the alignment of such experiments is quite involved (Wei, Hong, Lvovsky, Held and Shen [2000]). It was recently shown that the surface and bulk contributions to the nonlinear response of isotropic materials can be addressed by detailed polarization measurements of the two-beam second-harmonic response (Cattaneo and Kauranen [2005]). Contrary to methods relying on coherence length dependence, this technique does not require comparison of signals measured in different geometries. Moreover, it can be applied to any isotropic surface, since the polarization signatures follow directly from symmetry and depend only weakly on the linear optical properties of the material. In the following, we therefore derive expressions for the polarization signatures of surface and bulk contributions in the limit of unity refractive indices. We consider two fundamental beams a(ω) and b(ω) incident on the sample (fig. 3.1). The second-harmonic field arising from the effective surface polarization Ps [eq. (3.5)] is obtained by following the procedure described in § 2.2. As is evident from eqs. (2.19), (2.21) and (2.22), the p component of the second-
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harmonic field includes contributions from all three independent components of χ s . The s component of the second-harmonic field, on the other hand, arises s alone [eqs. (2.20), (2.23), and (2.24)]: from χxxz s sin θ a a p (ω)bs (ω) + sin θ b / sin θ a a s (ω)bp (ω) . E s (2ω) = 2χxxz (3.23) As only appears as an overall scaling factor, the polarization dependence of E s (2ω) is completely specified by the term in square brackets in eq. (3.23), which depends only on the incident angles θ a and θ b of the fundamental beams. To obtain the effective polarization due to bulk contributions, we have to consider the propagation of the fundamental beams in the bulk medium. The total fundamental field in the bulk can be expressed as s χxxz
a ·r
e(ω, r) = a(ω)eik
+ b(ω)eik ·r , b
(3.24)
where ka and kb are the wavevectors of the beams (fig. 3.1). The separable bulk contribution is specified by the third term in eq. (3.18). In the present geometry, we obtain a b Pδ (2ω, r) = iδ b(ω) · ka a(ω) + a(ω) · kb b(ω) ei(k +k )·r , (3.25)
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whose s-component is found to be ω P δ s (2ω, r) = iδ cos θ a sin θ b − sin θ a cos θ b c a b × a p (ω)bs (ω) − a s (ω)bp (ω) ei(k +k )·r .
(3.26)
The reflected and transmitted s-polarized second-harmonic fields are obtained by integrating this source over the portion of space occupied by the bulk medium (Heinz [1991], Shen [1999], Wei, Hong, Lvovsky, Held and Shen [2000], Held, Lvovsky, Wei and Shen [2002]). This only affects the signal strength but not its polarization dependence, which is specified by the term in square brackets in eq. (3.26). Clearly, the differences between the polarization signatures of eqs. (3.23) and (3.26) are more pronounced when θ a and θ b have the same sign. The different polarization signatures of bulk and surface contributions allow the contributions to be separated in a direct and unambiguous way. The technique was demonstrated by measuring reflected and transmitted polarization patterns from a poled polymer film and a glass (BK7) surface (Cattaneo and Kauranen [2005]). The response of the polymer film is dominated by electric-dipole contributions (Bosshard, Sutter, Prêtre, Hulliger, Flörsheimer, Kaatz and Günter [1995]) and should follow the surface model of eq. (3.23). As expected, the reflected and transmitted polarization patterns were found to be essentially identical
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and entirely determined by the experimental geometry. The polarization patterns of the glass surface, on the other hand, were shown to be very different for the reflected and transmitted directions. The bulk contribution was found to dominate in transmission, whereas surface contributions were found to dominate in reflection. The results obtained in the two geometries were in excellent agreement with coherence-length arguments. Two-beam polarization studies will allow the nonlinear response of isotropic, centrosymmetric materials to be investigated in great detail. For example, theoretical models could be used to derive a quantitative relation between the measurable bulk component δ and the component γ that is undistinguishable from surface contributions. Alternatively, surface-like bulk contributions could be addressed by determining the components of the effective surface susceptibility tensor and comparing them with theoretical predictions. Similar studies are also expected to be particularly important for the case of nanostructured materials where strong gradients favoring the quadrupole response can occur (Dadap, Shan, Eisenthal and Heinz [1999], Dadap, Shan and Heinz [2004]). In fact, the first results reporting the isolation of bulk contributions in the response of nanostructured materials have recently been reported (Figliozzi, Sun, Jiang, Matlis, Mattern, Downer, Withrow, White, Mochan and Mendoza [2005], Sun, Figliozzi, An and Downer [2005]).
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§ 4. Complete theoretical model including linear optics In surface and thin-film SHG experiments, the expansion coefficients f, g, h, and k [eqs. (2.2) or (2.16)] are the fundamental quantities that can be determined experimentally. To extract the components of the susceptibility tensor of the sample, the relationship between the coefficients and the tensor components must be calculated within a proper theoretical model. In particular, the theoretical model must include linear optical effects such as refraction, multiple reflections and propagation effects. The resulting geometric (or macroscopic) local-field corrections were first considered by Bloembergen and Pershan [1962], who determined the second-harmonic field emerging from a finite slab of a nonlinear material. The solution for a surface or ultrathin film can be obtained from their result in the limit of zero thickness. This was the approach used in the early studies of surface SHG (Chen, Heinz, Ricard and Shen [1981], Heinz, Chen, Ricard and Shen [1981]). Heinz [1982] developed an alternative and more intuitive model based on electromagnetic boundary conditions across a polarization sheet, whose complete derivation was published much later (Heinz [1991]). The incorporation of linear
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optical effects in the theoretical model has also been discussed, e.g., by Felderhof and Marowsky [1987], and Aktsipetrov, Akhmediev, Baranova, Mishina and Novak [1985]. These treatments clearly show the importance of the linear properties of the sample in the theoretical description of SHG. Particularly important is the refractive index of the nonlinear layer, which may be different from that of the bulk material and should be determined experimentally (Zhuang, Miranda, Kim and Shen [1999]). This, however, is difficult to achieve for ultrathin layers of one or a few monolayers. Often, the refractive index of the layer is identified with that of the surrounding media (Heinz, Tom and Shen [1983], Felderhof, Bratz, Marowsky, Roders and Sieverdes [1993], Verbiest, Kauranen, Van Rompaey and Persoons [1996], Felderhof and Marowsky [1987]). When not properly justified, such simplifications greatly limit the reliability of conclusions based on the theoretical model. For ultrathin molecular layers of thickness much smaller than the wavelength, the inclusion of the linear properties of the layer does not represent a fundamental problem. This is because the tangential component of the electric field is continuous across the layer, while its normal component is simply screened by the dielectric constant of the layer (Heinz [1991]). When the model is used to extract the components of the susceptibility tensor from experimental data, different choices for the linear properties are equivalent to rescaling the components by appropriate dielectric constants (Heinz [1991], Guyot-Sionnest, Shen and Heinz [1987]). This also applies to different conventions adopted in the literature, which relate the nonlinear polarization to optical fields in different regions of the sample (Heinz [1991], Mizrahi and Sipe [1988], Zhang, Zhang and Wong [1990]). For thicker samples, however, the inclusion of linear effects is more complicated because of propagation effects in the nonlinear layer. Sipe [1987] developed a Green-function formalism to calculate the second-harmonic fields for various multilayer geometries, which allows the linear properties of layers of arbitrary thickness to be included in a direct and physically appealing way. The model generalizes the work of Bloembergen and Pershan [1962] and, in the limit of zero thickness, agrees with the results of the polarization sheet model (Mizrahi and Sipe [1988]). Most studies aimed at determining average molecular orientation are based on the assumption of simple molecules with only one or a few non-vanishing components of the molecular hyperpolarizability tensor (Heinz, Tom and Shen [1983], Heinz [1991]). This assumption results in stringent requirements for the components of the macroscopic susceptibility tensor (Heinz [1991]). Such requirements
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have been used to determine a posteriori proper values for the linear properties of the layer, when these could not be determined by independent measurements (Dick, Gierulski and Marowsky [1985], Girling, Cade, Kolinsky, Jones, Peterson, Ahmad, Neal, Petty, Roberts and Feast [1987], Zhang, Zhang and Wong [1990], Sieverdes, Pinnow and Marowsky [1992], Zhuang, Miranda, Kim and Shen [1999]). Other studies (Felderhof, Bratz, Marowsky, Roders and Sieverdes [1993], Roders, Befort, Marowsky, Möbius and Bratz [1994]), although based on the assumption of rod-like molecules, reported results that are in disagreement with the assumption. The authors suspected the disagreement to be caused by intermolecular interactions, while in fact it might have been partly due to the failure to include the linear properties of the layer. In addition to the macroscopic local-field factors, microscopic factors due to interactions between molecules in the nonlinear material need to be considered. Such factors are much more difficult to handle than macroscopic ones, as they require detailed knowledge of the system on the molecular level. For threedimensional isotropic or cubic media with isotropic refractive index, Lorentz– Lorenz local-field factors can be used (Aktsipetrov, Akhmediev, Baranova, Mishina and Novak [1985]). Such factors only rescale the susceptibility components but do not affect their relative values. In the case of highly ordered molecular systems, where isotropy cannot be assumed, explicit calculations need to be performed at the molecular level. Unfortunately, the calculations are sensitive to parameters such as the linear polarizability tensor and the intermolecular distances, which are usually known only within certain limits (Eisert, Dannenberger and Buck [1998]). Moreover, different microscopic models yield considerably different results (Ye and Shen [1983], Cnossen, Drabe and Wiersma [1992]). In addition to the detailed form of the local-field factors, even the necessity to include them is ambiguous in the literature (Eisert, Dannenberger and Buck [1998]). In some experimental studies, the factors have been explicitly shown to be negligible (Chen, Bower, Wang and Lee [1973]), while in other cases they have been shown to have considerable effects, unless the system is dilute (Hayden [1988]). A literature survey reveals, however, that in the vast majority of studies aimed at the determination of the relative values of the susceptibility components, the microscopic local-field factors are neglected. In the following subsections we present the basic framework for including the linear optical properties of nonlinear optical samples. We limit our discussion to geometric (macroscopic) local-field corrections. After explaining the notation, we discuss the implications for thick samples and for ultrathin samples.
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Fig. 4.1. Geometry and detailed notation to discuss two-beam SHG from a surface or thin film. The sample is divided into 3 regions with different refractive indices. Two input beams are incident on the sample from region 1 and joint second-harmonic beams are produced in reflection and transmission. The coordinate system (x, y, z) associated with the sample is also shown.
4.1. Geometry and notational conventions To describe SHG from a surface or a thin nonlinear film, space is divided into three distinct regions (fig. 4.1): the nonlinear optical layer (i.e., the region with a nonzero nonlinear response, denoted as region 3) and two linear media surrounding it (regions 1 and 2). A complete description assigns a different linear optical constant to each region (Heinz [1991], Maki, Kauranen and Persoons [1995]). Since most parameters characterizing an optical beam depend on the linear properties of the medium, it is important to use an approach that includes the properties in a natural way (Sipe [1987], Maki, Kauranen and Persoons [1995]). Unfortunately, this inevitably results in notational complications. In the following, we present such a formalism in the limit of isotropic linear properties of all materials. We also assume that the linear constants change step-wise at the interface between two media. We will use similar notational conventions for both the fundamental and second-harmonic fields. As previously, we use lowercase symbols for quantities at the fundamental frequency and uppercase symbols for quantities at the secondharmonic frequency. Material quantities (such as the second-harmonic polarization of the nonlinear layer) are naturally expressed in a Cartesian reference system (x, y, z) associated with the sample. We take the nonlinear layer to be parallel to the x–y plane and to cover the region from z = 0 to z = −D. Beam parameters (such as the field amplitude) are most conveniently expressed using a reference system (p, s, k) specific to each beam, which will be introduced below.
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4.1.1. Fundamental fields In each region m = 1, 2, 3, we consider fundamental monochromatic fields of the form 1 e˜ m (r, t) = Re em (ω, r)e−iωt = em (ω, r)e−iωt + e∗m (ω, r)eiωt . (4.1) 2 We consider the general situation in which two beams a(ω) and b(ω) at the fundamental frequency are incident on the nonlinear layer (fig. 4.1). In this case, the total fundamental field is em (ω, r) = am (ω, r) + bm (ω, r).
(4.2)
In the following, we limit our discussion to the fundamental beam a(ω). Similar expressions apply to beam b(ω), provided that index a is replaced by b. Note that the results derived also apply to the traditional case of one-beam second-harmonic generation when a = b = e/2. We assume that the fundamental beams are plane waves. Since reflections occur at interfaces, a field consists in general of a downward (denoted by a minus sign) and an upward propagating part (+ sign) am (ω, r) = am+ (ω)eikm+ ·r + am− (ω)eikm− ·r , a
a
EBL= (ωn /c)kˆ with r = (x, y, z). The wavevectors k
(4.3)
az ˆ are ≡ va ± wm specific to each region of the samples, but their component in the x–y plane is conserved. For a given in-plane component va , the component of kam± along the sample normal is (Sipe [1987]) 2 a wm = (ωnm /c)2 − (v a )2 , with v a ≡ va · va . (4.4) a m±
m
a m±
Since the components of kam± in the plane of the sample are equal, the z dependence can be separated explicitly, a ·R
am (ω, r) = am (ω, z)eiv
(4.5)
,
with R = (x, y) and am (ω, z) = am+ (ω)eiwm z + am− (ω)e−iwm z . a
a
(4.6)
For simplicity, we take the beam to propagate in the x–z plane. The vector va is then along the x direction (va = v a xˆ ), and the unit vectors kˆ am± can be defined as kˆ am± = sin θma xˆ ± cos θma zˆ .
(4.7)
When the refractive index nm is real, the angle θma corresponds to the propagation angle of the fundamental beam. For complex nm , the angle is complex and
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therefore does not have a tangible geometrical interpretation. Nevertheless, all equations remain valid even in this general case (Sipe [1987]). The field vectors are naturally expanded in terms of their p and s components as p
s (ω)ˆs. am± (ω) = am± (ω)pˆ am± + am±
(4.8)
While the s direction is uniquely defined (ˆs = −ˆy), the p direction depends on the propagation direction of the beam (pˆ am± = sˆ × kˆ am± = sin θma zˆ ∓ cos θma xˆ ). As easily verified, the Cartesian components of am± (ω) are given by p
x (ω) = ∓am± (ω) cos θma , am± y am± (ω) z (ω) am±
= =
s −am± (ω), p am± (ω) sin θma .
(4.9) (4.10) (4.11)
4.1.2. Source polarization Under the influence of the fundamental field, each unit (atom or molecule) of the nonlinear layer develops an oscillating dipole moment. Due to nonlinearities in the microscopic response, the dipole moment contains a component at frequency Ω = 2ω. On the macroscopic level, the material response is represented by the polarization ˜ t) = Re P(Ω, r) exp(−iΩt) P(r, 1 = P(Ω, r) exp(−iΩt) + P∗ (Ω, r) exp(iΩt) , (4.12) 2 which is defined as the average dipole moment per unit volume. Although alternative conventions have been used (Heinz [1991], Heinz, Chen, Ricard and Shen [1981], Sipe, Mizrahi and Stegeman [1987]), the most natural one relates the components of the polarization to the components of the total fundamental field e3 (ω, r) present inside the nonlinear layer (Heinz [1991, 1982], Aktsipetrov, Akhmediev, Baranova, Mishina and Novak [1985], Dick, Gierulski and Marowsky [1985], Girling, Cade, Kolinsky, Earls, Cross and Peterson [1985]):
EBL
j
P i (Ω, r) = χij k (Ω; ω, ω)e3 (ω, r)e3k (ω, r),
(4.13)
where χij k (Ω; ω, ω) is the second-harmonic susceptibility tensor, the indices ij k refer to the x, y, z coordinates, and summation over repeated indices is implied. Using eqs. (4.2) and (4.5) and keeping only the terms in which both fundamental beams contribute, we obtain P i (Ω, r) = P i (Ω, z)eiV·R ,
(4.14)
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with V = va + vb and j
P i (Ω, z) = 2χij k (Ω; ω, ω)a3 (ω, z)b3k (ω, z),
(4.15)
where intrinsic permutation symmetry χikj (Ω; ω, ω) = χij k (Ω; ω, ω) has been used. The susceptibility tensor χij k (Ω; ω, ω) depends on the symmetry of the sample. In general, the tensor has 27 independent components. For samples of C∞v symmetry (Shen [1984]), the susceptibility tensor has only three independent components χzzz , χzxx = χzyy and χxxz = χxzx = χyyz = χyzy . From eq. (4.15), the components of P(Ω, z) are then P x (Ω, z) = 2χxxz a3x (ω, z)b3z (ω, z) + a3z (ω, z)b3x (ω, z) , (4.16) y y z z y P (Ω, z) = 2χxxz a3 (ω, z)b3 (ω, z) + a3 (ω, z)b3 (ω, z) , (4.17) x z z z (2) x P (Ω, z) = 2χzzz a3 (ω, z)b3 (ω, z) + 2χzxx a3 (ω, z)b3 (ω, z) y y + a3 (ω, z)b3 (ω, z) . (4.18) 4.1.3. Generated second-harmonic field The polarization at frequency Ω = 2ω acts as a source of electromagnetic radiation at Ω. Detailed understanding of the radiative process requires a proper theoretical model (Sipe [1987]). Here we derive the functional form of the secondharmonic field based on momentum conservation, while its amplitude will be derived in the next subsection. By itself, each atom or molecule radiates at Ω according to the well-known dipole pattern (Jackson [1975]). The planar extension of the nonlinear layer provides a phase-matching condition in the x and y directions, which requires the component of the wavevector parallel to the plane of the layer to be conserved (Shen [1984]). Any second-harmonic signal arising from a polarization of the form of eq. (4.14), with P i (Ω, z) unspecified, will therefore be characterized by a wavevector with the same in-plane component V = va + vb . For a given choice of V there are two possibilities, corresponding to upward- and downwardpropagating second-harmonic components (Sipe [1987]):
EBL
ˆ m± ≡ V ± Wm zˆ , Km± = (ΩNm /c)K
(4.19) ˆ ≡ V · V. Note that, in general, Km± is not
= (ΩNm − and with parallel to kˆ am± + kˆ bm± . Following the notation used for the fundamental beam, the emitted secondharmonic field present in region m is Wm2
/c)2
V2
V2
Em (Ω, r) = Em+ (Ω)eiKm+ ·r + Em− (Ω)eiKm− ·r .
(4.20)
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Clearly, it is convenient to separate the z-dependence and rewrite eq. (4.20) as Em (Ω, r) = Em (Ω, z)eiV·R ,
(4.21)
where Em (Ω, z) = Em+ (Ω)eiWm z + Em− (Ω)e−iWm z .
(4.22)
For fundamental beams propagating in the x–z plane, V = V xˆ , with V = v a +v b . ˆ m± as We therefore define the unit vectors K ˆ m± = sin Θm xˆ ± cos Θm zˆ . K
(4.23)
Θm is the propagation angle of the second-harmonic beam only for real Nm , but its definition remains valid for complex Nm (Sipe [1987]). The angle Θm satisfies nm sin Θm = (4.24) sin θma + sin θmb . 2Nm The second-harmonic field can be expanded as p s (Ω)ˆs, Em± (Ω) = Em± (Ω)Pˆ m± + Em±
(4.25)
ˆ m± = sin Θm zˆ ∓ cos Θm xˆ . with Pˆ m± = sˆ × K In the above treatment, we have neglected terms corresponding to processes driven by each fundamental beam separately, which lead to coherent secondharmonic radiation propagating approximately along kˆ am± and kˆ bm± .
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4.2. Second-harmonic field exiting from a thick sample Due to reflections, the fundamental field in region 1 has both downward- and upward-propagating parts. The upward-propagating part, however, has no effect on the nonlinear interaction. The fundamental field in region 1 is therefore taken to be e1 (ω, r) = a1 (ω, r) + b1 (ω, r) = a1 (ω)eik1− ·r + b1 (ω)eik1− ·r . a
b
(4.26)
As stated in eq. (4.15), the fields that drive the nonlinear polarization are the fundamental fields a3 (ω, r) and b3 (ω, r) inside the nonlinear layer. Their calculation is a standard problem of thin-film linear optics (Born and Wolf [1980]). We again review the results of the calculation for the fundamental beam a(ω) only. In region 3, we must consider both upward- and downward-propagating parts of the fundamental fields [eqs. (4.3), (4.5), and (4.6), with m = 3], a3 (ω, r) = a3+ (ω)eik3+ ·r + a3− (ω)eik3− ·r = a3 (ω, z)eiv a
a
a ·R
,
(4.27)
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where a3 (ω, z) = a3+ (ω)eiw3 z + a3− (ω)e−iw3 z . a
a
As before, we expand the amplitudes as a3± (ω) = find (Maki, Kauranen and Persoons [1995]) j
s (ω)ˆs, and + a3±
a
aj aj
j
a3+ (ω) = a1 (ω)
(4.28) p a3± (ω)pˆ a3±
t13 r32 e2iw3 D aj aj
a
,
(4.29)
a
,
(4.30)
1 − r31 r32 e2iw3 D aj
j
j
a3− (ω) = a1 (ω)
t13 aj aj
1 − r31 r32 e2iw3 D
as (r as ) is the Fresnel amplitude coefficient (Sipe with j = p, s and where, e.g., t13 13 [1987]) for transmission (reflection) of the s component of beam a(ω) at the interface between regions 1 and 3. The source polarization is then readily determined by inserting eqs. (4.28)– (4.30) [and the corresponding expressions for beam b(ω)] in eqs. (4.9) and (4.11) and further in eqs. (4.14) and (4.15). The source leads to downward- and upwardpropagating second-harmonic waves inside the layer. As there are no sources outside the nonlinear layer, only an upward-propagating second-harmonic wave is present in region 1,
EBL
E1 (Ω, r) = E1 (Ω)eiK1+ ·r ,
(4.31)
and only a downward-propagating wave is present in region 2, E2 (Ω, r) = E2 (Ω)eiK2− ·r .
(4.32) p Em (Ω)Pˆ m±
+ We again expand the field vectors in the p, s basis as Em (Ω) = s (Ω)ˆs, where the upper sign is to be used with m = 1 and the lower sign with Em m = 2. When the linear optical properties of the sample are included, the determination of the amplitudes E1 (Ω) and E2 (Ω) is not a trivial problem. A solution can be found by solving the Maxwell equations with proper boundary conditions (Heinz [1991], Bloembergen and Pershan [1962]). Here, we use an alternative approach based on a Green-function formalism (Sipe [1987]) and summarize the main results of the calculation in the following. Initially, we assume that the source of SHG consists of an infinitely thin polarization sheet at z = z0 , P(Ω, z) = P(Ω)δ(z − z0 ).
(4.33)
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In this case, it can be shown that the components of the reflected second-harmonic field E1 (Ω) are given by (Maki, Kauranen and Persoons [1995]) p
T31 Ω 2 −1 W p p 2iW3 D c2 3 1 − R31 R32 e x p iW (2D+z ) 3 0 × P R32 e − e−iW3 z0 cos Θ3 p + P z R32 eiW3 (2D+z0 ) + e−iW3 z0 sin Θ3 ,
p
E1 (Ω) = 2πi
E1s (Ω)
s T31 y = −2πi 2 W3−1 s s e2iW3 D P c 1 − R31 R32 s iW (2D+z ) 0 + e −iW3 z0 , × R32 e 3
(4.34)
Ω2
(4.35)
and the components of the transmitted second-harmonic field E2 (Ω) are given by p
T32 Ω 2 −1 W3 p p 2iW3 D 2 c 1 − R31 R32 e x iW z p × P e 3 0 − R31 e−iW3 z0 cos Θ3 p + P z eiW3 z0 + R31 e−iW3 z0 sin Θ3 ,
p
E2 (Ω) = 2πi
E2s (Ω) =
EBL
s T32 Ω2 −2πi 2 W3−1 P y eiW3 z0 s s 2iW D c 1 − R31 R32 e 3
(4.36) s −iW3 z0 . + R31 e (4.37)
Equations (4.34)–(4.37) include the Fresnel coefficients evaluated at the secondharmonic frequency. A solution for a general polarization source can be constructed from these particular solutions [Sipe, 1987]. In fact, an arbitrary source P(Ω, z) satisfies P(Ω, z) = P(Ω, z0 )δ(z − z0 ) dz0 , (4.38) where the integrand is of the form of eq. (4.33). Therefore, the general solution can be obtained by replacing P(Ω) with P(Ω, z0 ) in the expressions above and integrating from z0 = −D to z0 = 0. Even when one fully accounts for the linear optical properties of the sample, the components of the second-harmonic field have the functional form given in eq. (2.16). For C∞v symmetry, the expansion coefficients [eqs. (2.19) and (2.20)] become p
fm = F1m χxxz + F2m χzxx + F3m χzzz ,
(4.39)
p gm
(4.40)
= Gm χzxx ,
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hsm = Hm χxxz ,
(4.41)
= Km χxxz ,
(4.42)
s km
where m defines the reflected or transmitted second-harmonic signal. The factors F1m , F2m , F3m , Gm , Hm , and Km depend on the linear optical properties and, for films of finite thickness, are best evaluated numerically. 4.3. Limit of zero thickness Analytical expressions for the expansion coefficients can be obtained for the simpler case of an infinitely thin nonlinear layer (Maki, Kauranen and Persoons [1995]). We start by calculating the fundamental fields present in the layer in this limit and, as before, give the results only for beam a(ω). In the limit of zero thickness, 2w3a D 1 and w3a z 1, and eqs. (4.27) and (4.28) can be simplified to a3 (ω, r) = a3+ (ω)eik3+ ·r + a3− (ω)eik3− ·r = a3 (ω)eiv a
a
a ·R
,
(4.43)
with a3 (ω) = a3+ (ω) + a3− (ω).
(4.44)
EBL
Similarly, eqs. (4.29) and (4.30) can be simplified to aj aj
j
j
a3+ (ω) = a1 (ω)
t13 r32 aj aj
1 − r31 r32
,
(4.45)
,
(4.46)
aj
j
j
a3− (ω) = a1 (ω)
t13 aj aj
1 − r31 r32
with j = p, s. These expressions [and the corresponding expressions for beam b(ω)] can then be inserted in eq. (4.15) to calculate the source polarization. The components of the reflected and transmitted second-harmonic fields are then, respectively, of the functional forms j
j
j p p T31 Ω 2 −1 j f1 a1 b1 + g1 a1s b1s W3 j j 2 c 1 − R31 R32 j p s j p + h1 a1 b1 + k1 a1s b1 ,
E1 (Ω) = 2πi
j
E2 (Ω)
j j p p T32 Ω2 f2 a 1 b1 = 2πi 2 W3−1 j j c 1 − R31 R32 j p j p + h2 a1 b1s + k2 a1s b1 ,
(4.47)
j
+ g2 a1s b1s (4.48)
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where j = p, s and the expansion coefficients have been renormalized in view of the common prefactor. The expansion coefficients for C∞v can be shown to be (Cattaneo [2004]) ap bp
p
fm =
p ∓2χxxz 1 − R3i cos Θ3 ap ap bp bp (1 − r31 r32 )(1 − r31 r32 ) ap bp × 1 + r32 1 − r32 sin θ3a cos θ3b ap bp + 1 − r32 1 + r32 cos θ3a sin θ3b p ap bp + 2χzxx 1 + R3i 1 − r32 1 − r32 cos θ3a cos θ3b sin Θ3 p ap bp + 2χzzz 1 + R3i 1 + r32 1 + r32 sin θ3a sin θ3b sin Θ3 , t13 t13
(4.49)
gm =
as t bs t13 p 13 as bs 1 + r32 sin Θ3 , 2χzxx 1 + R3i 1 + r32 as as bs bs (1 − r31 r32 )(1 − r31 r32 )
hsm =
ap bs t13 t13 ap s bs 2χxxz 1 + R3i 1 + r32 1 + r32 sin θ3a , ap ap bs bs (1 − r31 r32 )(1 − r31 r32 )
p
(4.50)
bp
s km =
as t t13 13
EBL 1+R 2χ
as r as )(1 − r bp r bp ) (1 − r31 32 31 32
xxz
s 3i
(4.51)
as
bp
1 + r32 1 + r32 sin θ3b , (4.52)
where for the reflected beam m = 1, i = 2 and the upper sign must be used, and for the transmitted beam m = 2, i = 1 and the lower sign must be used. In the limit of unity refractive indices for all regions of the sample, eqs. (4.49)–(4.52) reduce to eqs. (2.21)–(2.24). The results can be greatly simplified, and their physical meaning becomes more apparent, by working in the x, y, z reference system (Heinz [1991]). Inserting eqs. (4.45) and (4.46) in eqs. (4.9)–(4.11), we find ap
ap a ap ap 1 − r32 cos θ3 , 1 − r31 r32 as t13 y as a3 (ω) = −a1s (ω) 1 + r32 , as as 1 − r31 r32 ap t13 p ap a a3z (ω) = a1 (ω) ap ap 1 + r32 sin θ3 , 1 − r31 r32
a3x (ω)
=
p a1 (ω)
t13
(4.53) (4.54) (4.55)
and analogous expressions for beam b(ω). By expressing also the field components on the right-hand side of the equations in the x, y, z basis and rewriting the
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factors involving Fresnel coefficients, we obtain (Heinz [1991]) cos θ2 ap x n1 ap t12 a1 (ω) = t21 a1x (ω), cos θ1 n2 y y as y a1 (ω), a3 (ω) = a2 (ω) = t12 a3x (ω) = a2x (ω) =
a3z (ω) =
n22
a z (ω) n23 2
=
n22 sin θ2 ap z n1 n2 ap t a (ω) = 2 t12 a1z (ω). n23 sin θ1 12 1 n3
(4.56) (4.57) (4.58)
Equations (4.56)–(4.58) can be derived directly from the electromagnetic boundary conditions, which require the tangential component of the electric field and the normal component of the electric displacement to be continuous across the interface. Clearly, the linear optical properties of the layer (indicated by n3 ) only play a role in the z component of the field [eq. (4.58)]. Equations (4.56)–(4.58) can be rewritten in matrix form as (Heinz [1991]) a3 (ω) = fa12 (ω)a1 (ω),
(4.59)
a,xx where fa12 (ω) is a diagonal matrix with elements f12 ap a,zz and f12 = (n1 n2 /n23 )t12 .
=
ap a,yy (n1 /n2 )t21 , f12
as = t12
From eq. (4.15), we obtain the polarization in the nonlinear optical layer a,jj
EBL j
b,kk P i (Ω) = 2f12 f12 χij k (Ω; ω, ω)a1 (ω)b1k (ω).
(4.60)
Similarly to eq. (2.7), the reflected second-harmonic field E1 (Ω) can be expressed in terms of a vector nˆ 1 that satisfies nˆ 1 · K+ = 0 (Heinz [1991]): nˆ 1 · E1 (Ω) ∼ nˆ 3 · P(Ω),
(4.61)
where the vector nˆ 3 is obtained by correcting nˆ 1 for propagation in the nonlinear layer: nˆ 3 = F12 (Ω)nˆ 1 .
(4.62)
The diagonal matrix F12 (Ω) is defined analogously to fa12 (ω) with all the Fresnel factors and refractive indices evaluated at the second-harmonic frequency. The transmitted second-harmonic field E2 (Ω) can be calculated similarly by interchanging indices 1 and 2 in eqs. (4.61) and (4.62).
4.4. Effect on the susceptibility components The components of the susceptibility tensor can be extracted from the experimentally determined expansion coefficients f j , g j , hj , and k j by comparing them with their corresponding theoretical expressions (Cattaneo and Kauranen [2003],
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Cattaneo, Vuorimaa, Lemmetyinen and Kauranen [2004]). The choice of the theoretical model is very important, since its details have strong influence on the results for the susceptibility components. In particular, the results depend sensitively on the linear optical properties of the sample. For layers with thickness d comparable to the optical wavelength λ, propagation effects within the layer also have to be considered. Such effects, however, can be neglected for ultrathin layers with d λ. On the basis of the results of the previous section, it can be shown that the dielectric constants ε of the layer appear only as factors that rescale the effective susceptibility components (Guyot-Sionnest, Shen and Heinz [1987]): χijeffk (2ω; ω, ω) = χij k (2ω; ω, ω)/ε δiz (2ω)ε δj z (ω)ε δkz (ω),
(4.63)
where δij is the Kronecker delta. For thicker samples with a thickness comparable to the optical wavelength, the effect of the linear properties is more complicated. In this case, the model of § 4.2 and the susceptibility components are best evaluated numerically. Clearly, it is essential to be able to verify the reliability of the results, in particular when the linear properties of the sample cannot be measured directly and have to be assumed. For simple nonlinear molecules, the results for the tensor components can, to some extent, be compared to the values predicted by the theory. For example, it is common to assume that the molecular hyperpolarizability tensor of rod-like molecules is dominated by a single component along the molecular axis (Heinz, Tom and Shen [1983]). As can be shown by a straightforward calculation, the components of the macroscopic susceptibility tensor are then expected to satisfy (Shen [1984])
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χzxx = χxxz = χzzz /r,
(4.64)
with r a real constant. We note that the first part of the relation (χzxx = χxxz ) is formally similar to Kleinmann’s symmetry, which is only valid under completely nonresonant conditions (Boyd [2003]). For rod-like molecules, however, the relation is a fundamental symmetry property of the susceptibility tensor and therefore remains valid in all spectral ranges. Equation (4.64) can be used to verify the validity of the susceptibility components when rod-like molecules are used. In the general case of molecules with several non-vanishing hyperpolarizability components, however, the susceptibility components can assume arbitrary complex values. In this case, there is no direct way of verifying whether the results obtained are correct.
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Recent measurements on samples of rod-like molecules have shown that the inferred susceptibility components are in agreement with eq. (4.64) only when the linear optical properties of the sample are included in the proper way, for example using the model of § 4.2 (Cattaneo, Miettinen, Vuorimaa, Lemmetyinen and Kauranen [2006]). Consistency was achieved without considering microscopic local-field factors, suggesting that such effects are relatively unimportant for the material investigated.
§ 5. Conclusions and outlook We have presented a review of the use of polarization techniques in second-order surface nonlinear optics. These techniques possess significant advantages for detailed studies of the nonlinear and structural properties of surfaces and thin films. They also allow an important component of the bulk nonlinear response to be unambiguously separated from the effective surface response. The polarization techniques are based on the dependence of nonlinear optical processes on the structure and symmetry of materials. The symmetry arguments are particularly important for second-order processes, which are forbidden in centrosymmetric materials within the electric-dipole approximation of the light– matter interaction. This selection rule justifies the use of second-order processes as a tool to study surfaces and interfaces, where the symmetry is necessarily broken. On the macroscopic level, the structure and symmetry of a sample influence the nonlinear susceptibility tensor, which therefore contains important information about the sample. The tensor is most conveniently defined in a coordinate system tied to the sample. Experiments, however, are usually more convenient to describe in a different coordinate system, and the susceptibility tensor can rarely be measured directly. The determination of the tensor therefore consists of two parts: measurement of the experimental parameters [e.g., the expansion coefficients defined in eqs. (2.2) or (2.16)] and extraction of the components of the susceptibility tensor from the parameters. The tensorial character of the nonlinear response leads to a polarization dependence of the measured signals. For a precise determination of the experimental parameters, detailed polarization techniques are essential. The choice of the technique itself may not be the most important issue, and we expect techniques based on continuous polarization modulation of the input beams and techniques based on ellipsometry both to have their advantages. However, great care is required in the experimental work as the measured signals are sensitive to the smallest
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details of the experiment. The polarization dependence of the signals can then in some cases be used to draw qualitative conclusions about the properties of a sample. The extraction of the components of the susceptibility tensor from the experimental parameters relies on a theoretical model. The details of the model can greatly influence the results for the tensor and thereby the conclusions regarding, e.g., molecular orientation in thin films. For reliable results, the most complete theoretical models accounting for all the linear optical effects should be used. In certain cases, the results can also be compared to theoretical predictions regarding the values of the susceptibility components. Multipolar background signals from centrosymmetric bulk materials have been a long-standing problem in surface nonlinear optics. Two-beam polarized SHG allows an important part of the bulk response to be separated from surface-like contributions, based on their different polarization signatures. This opens the door for detailed studies of bulk and surface nonlinearities of various materials and for the development of nanostructured materials with enhanced multipolar response. An outstanding problem is the quantitative separation of the effective surface response into its electric-dipole-allowed surface part, multipolar surface part, and the part with bulk origin. Several of the detailed polarization techniques have been developed over the past 10–15 years and are well understood. Additional benefits can be expected from the combination of the overall symmetry arguments regarding the experimental parameters with optimized experimental geometries, which can make certain nonlinear signals highly selective to the desired properties of the sample. We therefore expect polarization techniques to undergo further development and become more widespread in a number of applications of nonlinear optics.
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Acknowledgements We acknowledge discussions and/or collaborations on various aspects and stages of this work and over several years with L.D. Barron, Ch. Bosshard, R.W. Boyd, B.K. Canfield, M.C. Downer, C. Flytzanis, P. Günter, F. Hache, L. Hecht, T.F. Heinz, J.M. Hicks, T.J. Katz, H. Lemmetyinen, J.J. Maki, E.W. Meijer, R.J.M. Nolte, A. Persoons, Th. Rasing, F.J. Rodriguez, J.J. Saarinen, M.C. Schanne-Klein, Y.R. Shen, A.P. Shkurinov, M. Siltanen, G.J. Simpson, J.E. Sipe, Yu. Svirko, T. Verbiest, E. Vuorimaa, and F.X. Wang. This work has been supported by the Academy of Finland (53961 and 107009), by the Center for International Mobility in Finland, and the Graduate School of Modern Optics and Photonics in Finland.
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E. Wolf, Progress in Optics 51 © 2008 Elsevier B.V. All rights reserved
Chapter 3
Electromagnetic fields in linear bianisotropic mediums by
Tom G. Mackay School of Mathematics, The University of Edinburgh, Edinburgh EH9 3JZ, Scotland, UK e-mail:
[email protected]
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Akhlesh Lakhtakia Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, Pennsylvania 16802-6812, USA e-mail:
[email protected]
ISSN: 0079-6638
DOI: 10.1016/S0079-6638(07)51003-6 121
Contents
Page § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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§ 2. The Maxwell postulates and constitutive relations . . . . . . . . . . .
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§ 3. Linear mediums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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§ 4. Plane-wave propagation . . . . . . . . . . . . . . . . . . . . . . . . .
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§ 5. Dyadic Green functions . . . . . . . . . . . . . . . . . . . . . . . . .
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§ 6. Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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§ 7. Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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References
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§ 1. Introduction An isotropic medium has electromagnetic properties which are the same in all directions. However, the notion of isotropy, as encountered in elementary treatments of electromagnetics, is an abstraction which requires qualification when applied to real materials. For example, liquids and random composite mediums may be isotropic on a statistical basis, while cubic crystals are isotropic when viewed at macroscopic length-scales. Electromagnetically isotropic mediums are characterized simply by scalar constitutive parameters which relate the induction field phasors D and H to the primitive field phasors E and B. Often, naturally occurring materials and artificially constructed mediums are more accurately described as anisotropic rather than isotropic. Anisotropic mediums exhibit directionally dependent electromagnetic properties, such that D and E are not aligned or H and B are not aligned. Dyadics (i.e., second-rank Cartesian tensors) are needed to relate the primitive and the induction field phasors in anisotropic mediums. Bianisotropy is the natural generalization of anisotropy. In the electromagnetic description of a bianisotropic medium, both D and H are anisotropically coupled to both E and B. Hence, in general, a linear bianisotropic medium is characterized by four 3×3 constitutive dyadics. Though seldom described in standard textbooks, bianisotropy is commonplace. Suppose a certain medium is characterized as an isotropic dielectric medium by an observer in an inertial reference frame Σ. The same medium generally exhibits bianisotropic properties when viewed by an observer in another reference frame that translates at uniform velocity with respect to Σ. Aside from relativistic scenarios, bianisotropic effects are observed at low frequencies and temperatures in a host of naturally occurring minerals (O’Dell [1970], Schmid [2003]). Furthermore, the phenomenon of bianisotropy looks set to play an increasingly important role in the rapidly burgeoning fields relating to complex composite mediums. In particular, bianisotropic mediums may be readily conceptualized through the process of homogenization of a composite of two or more constituent mediums. Thereby, metamaterials may be realized that exhibit novel electromagnetic properties which are not exhibited (or at least not exhibited to the same degree) by the constituent mediums (Walser [2003]).
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A broad overview of electromagnetic bianisotropy is provided in the following sections. Our survey includes anisotropic mediums as an important subcategory of bianisotropic mediums. Where appropriate, the presentation of bianisotropic and anisotropic properties is preceded by a brief description of the corresponding properties for isotropic mediums. We are confined to passive mediums throughout this chapter. In § 2 the constitutive relations for bianisotropic mediums, and their general properties, are discussed. Commonly encountered classifications of bianisotropic mediums are presented in § 3. The propagation of plane waves in bianisotropic mediums is considered in § 4. The Green-function method is widely used in source-field problems. Representations of dyadic Green functions for bianisotropic mediums are described in § 5. The conceptualization of bianisotropic mediums as homogenized composite mediums is the subject of § 6. A few closing remarks are provided in § 7. The following notation is adopted: 6-vectors (3-vectors) are in bold (normal) face and underlined, whereas 6 × 6 (3 × 3) dyadics are in bold (normal) face and twice underlined. The caret symbol ˆ placed over a 3-vector signifies a unit vector, and the unit Cartesian vectors are denoted by x, ˆ yˆ and zˆ . The adjoint, determinant, inverse, transpose and trace of a dyadic Q are represented by adj Q, det Q, Q−1 , QT and tr Q, respectively. Furthermore, ⎫ diag(Qx , Qy , Qz ) = Qx xˆ xˆ + Qy yˆ yˆ + Qz zˆ zˆ ⎪ ⎬ (1.1) . Q 0 ⎪ 1 Diag(Q , Q ) = ⎭ 1 2 0 Q
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The 6×6 (3×3) identity dyadic is represented by I (I ); the 4×4 identity matrix is written as I. The tilde symbol ˜ placed over a field dyadic or vector indicates that the quantity is in the time domain; the same quantity in the frequency domain is denoted without a tilde. Contraction of indexes for dyadics and vectors is denoted by the dot product. The permittivity and permeability of free space (i.e., vacuum in the absence of any gravitational field) are written as 0 and μ0 , respectively, and c0 = (0 μ0 )−1/2 represents the speed of light in free space. The angular frequency is ω. The real and imaginary parts of z ∈ C are specified as Re{z} and Im{z}, respectively, whereas z∗ denotes the complex conjugate of z.
§ 2. The Maxwell postulates and constitutive relations The action of complex assemblies of matter on electromagnetic waves lies at the heart of this chapter. The atomic (and sub-atomic) nature of matter does not di-
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rectly concern us here, since we adopt a macroscopic viewpoint wherein the electromagnetic wavelengths are large compared with interatomic distances. Thus, we refer in this chapter to ‘mediums’ and ‘materials’ rather than ‘assemblies of atoms and molecules’. The essential theoretical basis is provided by the Maxwell postulates for macroscopic fields combined with constitutive relations, but we keep in mind that electromagnetism is fundamentally a microscopic science. In this section, we present the salient features of macroscopic electromagnetic theory which underpin the remainder of the chapter.
2.1. Maxwell postulates The basic framework for our description of electromagnetic anisotropy and bianisotropy is constructed in terms of the four macroscopic electromagnetic fields ˜ ˜ ˜ t), D(r, t), B(r, t) and H˜ (r, t). These are piecewise differentiable vector E(r, functions of position r and time t which arise as spatial averages of microscopic ˜ ˜ t) and B(r, t) are fields and bound sources (Jackson [1999]). The fields E(r, directly measurable quantities which produce the Lorentz force. Accordingly, ˜ ˜ ˜ E(r, t) and B(r, t) are viewed as the primitive fields. The fields D(r, t) and H˜ (r, t) develop within a medium in response to the primitive fields; hence, they ˜ ˜ are considered as induction fields. Conventionally, E(r, t) and D(r, t) are called the electric field and the dielectric displacement, respectively. The conventional ˜ terms for B(r, t) and H˜ (r, t), namely the magnetic induction and magnetic field, respectively, are confusing and are avoided in this chapter. ˜ ˜ ˜ The physical principles governing the behaviour of E(r, t), D(r, t), B(r, t) ˜ and H (r, t) are encapsulated by the Maxwell curl postulates ⎫ ∂ ˜ ⎪ t) = J˜ e (r, t) ⎪ ∇ × H˜ (r, t) − D(r, ⎬ ∂t (2.1) ⎪ ∂ ˜ ⎪ ˜ ∇ × E(r, t) + B(r, t) = −J˜ m (r, t)⎭ ∂t and divergence postulates ˜ t) = ρ˜e (r, t) ∇ • D(r, . (2.2) ˜ t) = ρ˜m (r, t) ∇ • B(r,
EBL
The terms on the right sides of (2.1) and (2.2) represent sources of fields. Whereas J˜ e (r, t) and ρ˜e (r, t) are the externally impressed electric current and electric charge densities, respectively, the magnetic current and magnetic charge densities – denoted by J˜ m (r, t) and ρ˜m (r, t) – do not represent physical quantities
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[3, § 2
but are incorporated for mathematical convenience (Lakhtakia [1995]). In consonance with our macroscopic viewpoint, the source terms are also piecewise differentiable and satisfy the continuity relations ⎫ ∂ ⎪ ∇ • J˜ e (r, t) + ρ˜e (r, t) = 0 ⎪ ⎬ ∂t (2.3) . ⎪ ∂ ⎪ ˜ ∇ • J m (r, t) + ρ˜m (r, t) = 0⎭ ∂t A redundancy is implicit in eqs. (2.1)–(2.3), from the macroscopic viewpoint. The continuity relations (2.3), when combined with the Maxwell curl postulates (2.1), yield the Maxwell divergence postulates (2.2). Therefore, under the presumption of source continuity, there is no need for us to consider explicitly the divergence postulates (2.2).
2.2. Constitutive relations The Maxwell curl postulates (2.1) provide us with a system of two linear vec˜ tor differential equations in terms of the two primitive vector fields E(r, t) and ˜ ˜ ˜ B(r, t) and the two induction vector fields D(r, t) and H (r, t). In order to solve these differential equations, further information – in the form of constitutive relations relating the induction fields to the primitive fields – is needed. It is these constitutive relations which characterize the electromagnetic response of a medium. The constitutive relations may be naturally expressed in the general form ˜ ˜ ˜ D(r, t) = F E(r, t), B(r, t) (2.4) , ˜ ˜ t), B(r, t) H˜ (r, t) = G E(r,
EBL
˜ ˜ wherein F and G are linear/nonlinear functions of E(r, t) and B(r, t) for linear/nonlinear mediums. In general, the electromagnetic response of a medium is nonlocal with respect to both space and time. Thus, the constitutive relations of a linear medium should be stated as (Weiglhofer [2003]) ⎫ ⎪ ˜ ˜ − r , t − t ) D(r, t) = ˜ EB (r , t ) • E(r ⎪ ⎪ ⎪ ⎪ ⎪ t r ⎪ ⎪ • ˜ 3 ⎪ ˜ ⎬ + ξ (r , t ) B(r − r , t − t ) d r dt ⎪ EB (2.5) , ⎪ ⎪ ⎪ ˜ − r , t − t ) H˜ (r, t) = ζ˜ (r , t ) • E(r ⎪ ⎪ EB ⎪ ⎪ ⎪ ⎪ t r ⎪ • ˜ 3 ⎭ + ν˜ (r , t ) B(r − r , t − t ) d r dt EB
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127
where ˜ EB (r, t), ξ˜ (r, t), ζ˜ (r, t) and ν˜ EB (r, t) are constitutive dyadics EB EB (i.e., second-rank Cartesian tensors) that can be interpreted as 3 × 3 matrixes. While spatial nonlocality can play a significant role when the wavelength is comparable to some characteristic length-scale in the medium (Ponti, Oldano and Becchi [2001]), it is commonly neglected and lies outside the scope of this chapter. On the other hand, temporal nonlocality is almost always a matter of central importance, because of the high speeds of electromagnetic signals. We therefore concentrate here upon linear, spatially local, constitutive relations of the form ⎫ ˜ ˜ ˜ ˜ EB (r, t ) • E(r, t) = t − t ) + ξ˜ (r, t ) • B(r, t − t ) dt ⎪ D(r, ⎪ ⎪ EB ⎪ ⎬ t ⎪. ˜ ˜ ⎪ t − t ) + ν˜ EB (r, t ) • B(r, t − t ) dt ⎪ H˜ (r, t) = ζ˜ (r, t ) • E(r, ⎪ ⎭ EB t
(2.6) 2.3. The frequency domain The convolution integrals appearing in the constitutive relations (2.6) result in mathematical complexities when those relations are substituted in the Maxwell postulates. A widely used technique to circumvent those difficulties (often without loss of essential physics) is to introduce the temporal Fourier transforms via
EBL
∞ Z(r, ω) =
˜ t) exp(iωt) dt, Z(r,
(2.7)
−∞
with Z standing in for EB , ξ , ζ , ν EB , E, D, B and H , while ω is called the √ EB EB angular frequency and i = −1. After taking the temporal Fourier transforms of (2.6) and implementing the convolution theorem (Arfken and Weber [1995]), the frequency-domain constitutive relations emerge as D(r, ω) = EB (r, ω) • E(r, ω) + ξ (r, ω) • B(r, ω) EB . (2.8) H (r, ω) = ζ (r, ω) • E(r, ω) + ν EB (r, ω) • B(r, ω) EB
˜ ω) is partnered with H˜ (r, ω) rather than Often in electromagnetic theory, E(r, ˜ B(r, ω); for example, in the formulation of boundary conditions and the definition of the Poynting vector (Chen [1983]). Consequently, it is convenient and widespread practice to express frequency-domain constitutive relations as D(r, ω) = EH (r, ω) • E(r, ω) + ξ (r, ω) • H (r, ω) EH . (2.9) B(r, ω) = ζ (r, ω) • E(r, ω) + μ (r, ω) • H (r, ω) EH
EH
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The names Boys–Post and Tellegen are often associated with the constitutive relations (2.8) and (2.9), respectively (Weiglhofer [1998a]). A one-to-one correspondence between the Boys–Post representation and the Tellegen representation is straightforwardly established via (Weiglhofer [2003]) EB (r, ω) = EH (r, ω) − ξ ξ
EB
ζ
(r, ω) = ξ
EB
EH
EH
(r, ω) • μ−1 (r, ω) • ζ
(r, ω) • μ−1 (r, ω)
EH
EH
(r, ω) = −μ−1 (r, ω) • ζ EH
ν EB (r, ω) = μ−1 (r, ω)
EH
⎫ (r, ω)⎪ ⎪ EH ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(r, ω)
(2.10)
EH
and EH (r, ω) = EB (r, ω) − ξ ξ
EH
ζ μ
EH
(r, ω) = ξ
EB
EB
(r, ω) • ν −1 (r, ω) EB
(r, ω) = −ν −1 (r, ω) • ζ EB
EH
(r, ω) • ν −1 (r, ω) • ζ EB
(r, ω) = ν −1 (r, ω) EB
EB
⎫ (r, ω)⎪ ⎪ EB ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(r, ω)
,
(2.11)
EBL
wherein the invertibility of ν EB (r, ω) and μ (r, ω) has been assumed. We EH largely make use of the Tellegen representation in this chapter, but with occasional recourse to the Boys–Post representation where necessary. The mathematical simplicity of the frequency-domain formulation as compared to the time-domain formulation is gained at a cost in terms of physical interpretation. The frequency-dependent constitutive dyadics EB,EH (r, ω), (r, ω), ζ (r, ω), ν EB (r, ω) and μ (r, ω) are complex-valued quanξ EB,EH EB,EH EH tities; so also are the frequency-dependent field phasors E(r, ω), D(r, ω), B(r, ω) and H (r, ω). The real-valued physical entities they represent surface only indirectly upon subjecting them to the inverse temporal Fourier transform. In this chapter, phasors are also called fields – in keeping with widespread usage. The corresponding frequency-domain Maxwell curl postulates develop as ∇ × H (r, ω) + iωD(r, ω) = J e (r, ω)
∇ × E(r, ω) − iωB(r, ω) = −J m (r, ω)
,
(2.12)
where the source terms J e,m (r, ω) are the Fourier transforms of J˜ e,m (r, t), defined as in (2.7) with Z = J e,m . The constitutive relations (2.9) – or equally (2.8) – together with the Maxwell curl postulates (2.12) form a self-consistent system into which anisotropy and bianisotropy are incorporated.
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2.4. 6-vector/6 × 6 dyadic notation The use of a 6-vector/6 × 6 dyadic notation allows the Tellegen constitutive relations (2.9) to be expressed very compactly as C(r, ω) = K EH (r, ω) • F(r, ω),
(2.13)
with the 6-vectors T C(r, ω) = D(r, ω), B(r, ω) and
(2.14)
T F(r, ω) = E(r, ω), H (r, ω)
(2.15)
containing components of the electric and magnetic fields, while the 6 × 6 constitutive dyadic
EH (r, ω) ξ (r, ω) EH . K EH (r, ω) = (2.16) ζ (r, ω) μ (r, ω) EH
EH
The result of combining the constitutive relations (2.9) with the Maxwell curl postulates (2.12) is thereby succinctly expressed as L(∇) + iωK EH (r, ω) • F(r, ω) = Q(r, ω), (2.17)
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with the linear differential operator 0 ∇ ×I , L(∇) = −∇ × I 0
(2.18)
and the source 6-vector T Q(r, ω) = J e (r, ω), J m (r, ω) .
(2.19)
In a similar fashion, the four 3 × 3 dyadics EB (r, ω), ξ (r, ω), ζ (r, ω) EB EB and ν EB (r, ω), which specify the constitutive properties in the Boys–Post representation (2.8), may be represented by the 6 × 6 constitutive dyadic
EB (r, ω) ξ (r, ω) EB . K EB (r, ω) = (2.20) ζ (r, ω) ν EB (r, ω) EB
The transformations (2.11) and (2.10) may then be expressed in terms of the invertible 6 × 6 dyadic function τ which we define through the following relationships: K EB (r, ω) ≡ τ K EH (r, ω)
EH (r, ω) − ξ (r, ω) • μ−1 (r, ω) • ζ (r, ω) ξ (r, ω) • μ−1 (r, ω) EH EH EH EH EH = −1 −1 −μ
EH
(r, ω) • ζ
EH
(r, ω)
μ
EH
(r, ω)
(2.21)
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and K EH (r, ω) ≡ τ −1 K EB (r, ω)
EB (r, ω) − ξ (r, ω) • ν −1 (r, ω) • ζ (r, ω) EB EB EB = −1 −ν EB (r, ω) • ζ
EB
ξ
EB
(r, ω)
(r, ω) • ν −1 (r, ω) EB ν −1 (r, ω) EB
.
(2.22) 2.5. Form invariances The Maxwell postulates retain their form under certain linear coordinate-and-field transformations, thereby leading to the concepts of spatial and temporal invariances as well as spatiotemporal covariance. While spatiotemporal covariance is of immense theoretical importance, invariances with respect to spatial and temporal transformations are commonly applied in many practical situations. Chiral invariance, which captures the non-uniqueness of the Maxwell postulates under linear field transformations, is also significant. An invariance of the Maxwell postulates in the frequency domain to a certain transformation involving complex conjugates has recently been reported. Finally in this subsection, implications of various transformations on electromagnetic energy and momentum are outlined.
EBL
2.5.1. Time reversal We denote the operation of time reversal by T , i.e., T {t} = −t. Under the presumption that electric and magnetic source densities transform as (Kong [1972]) T ρ˜e (r, t) = ρ˜e (r, −t) (2.23) , T ρ˜m (r, t) = −ρ˜m (r, −t) the continuity relations (2.3) yield T J˜ e (r, t) = −J˜ e (r, −t) , T J˜ m (r, t) = J˜ m (r, −t) and the electromagnetic fields must therefore transform as ˜ ˜ ˜ ˜ t) = E(r, −t), T {D(r, t)} = D(r, −t) T E(r, ˜ ˜ t) = −B(r, −t), T H˜ (r, t) = −H˜ (r, −t) T B(r,
(2.24)
(2.25)
in order to preserve the form of the Maxwell postulates. From the definition of the temporal Fourier transform (2.7), we see that the frequency-domain counterparts
3, § 2]
The Maxwell postulates and constitutive relations
of (2.23), (2.24) and (2.25) are T ρe (r, ω) = ρe∗ (r, ω), ∗ (r, ω), T ρm (r, ω) = −ρm ∗ T E(r, ω) = E (r, ω), T B(r, ω) = −B ∗ (r, ω),
⎫ T J e (r, ω) = −J ∗e (r, ω)⎪ ⎪ ⎪ ⎪ T J m (r, ω) = J ∗m (r, ω) ⎬ , T D(r, ω) = D ∗ (r, ω) ⎪ ⎪ ⎪ ⎪ ⎭ T H (r, ω) = −H ∗ (r, ω)
131
(2.26)
where the superscript ∗ indicates the complex conjugate. Therefore, under time reversal, the constitutive dyadics transform as ⎫ T EB,EH (r, ω) = ∗EB,EH (r, ω) ⎪ ⎪ ⎪ ⎪ ⎪ (r, ω) = −ξ ∗ (r, ω) ⎪ T ξ ⎪ ⎪ EB,EH EB,EH ⎬ ∗ (r, ω) = −ζ (r, ω) , T ζ (2.27) EB,EH EB,EH ⎪ ⎪ ⎪ ⎪ ⎪ T ν EB (r, ω) = ν ∗EB (r, ω) ⎪ ⎪ ⎪ ⎭ ∗ T μ (r, ω) = μ (r, ω) EH
EH
by virtue of (2.8) and (2.9). The time-reversal asymmetry which is exhibited by (r, ω) and ζ (r, ω) origithe magnetoelectric constitutive dyadics ξ EB,EH EB,EH nates from irreversible physical processes, such as can develop through the application of quasistatic biasing fields or by means of relative motion (Post [1997]). We enlarge upon these matters in § 3 in the context of Faraday chiral mediums and Lorentz-transformed constitutive dyadics.
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2.5.2. Spatial inversion Let us turn now to the inversion of space, denoted by the operator P as P{r} = −r. Similarly to the time-reversal scenario, if we assume that the electric and magnetic charge densities transform as (Kong [1972]) P ρ˜e (r, t) = ρ˜e (−r, t) , (2.28) P ρ˜m (r, t) = −ρ˜m (−r, t) then
P J˜ e (r, t) = −J˜ e (−r, t) P J˜ m (r, t) = J˜ m (−r, t)
(2.29)
by virtue of the continuity relations (2.3); furthermore, the form invariance of the Maxwell postulates enjoins the relationships ˜ ˜ ˜ ˜ P E(r, t) = −E(−r, t), P D(r, t) = −D(−r, t) . (2.30) ˜ ˜ t) = B(−r, t), P H˜ (r, t) = H˜ (−r, t) P B(r,
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The action of the spatial-inversion operator P on the field quantities is not altered upon switching from the time domain to the frequency domain. Thus, from the constitutive relations (2.8) and (2.9) we find ⎫ P EB,EH (r, ω) = EB,EH (−r, ω) ⎪ ⎪ ⎪ ⎪ ⎪ (r, ω) = −ξ (−r, ω) ⎪ P ξ ⎪ ⎪ EB,EH EB,EH ⎬ (r, ω) = −ζ (−r, ω) . P ζ EB,EH EB,EH ⎪ ⎪ ⎪ ⎪ ⎪ P ν EB (r, ω) = ν EB (−r, ω) ⎪ ⎪ ⎪ ⎭ P μ (r, ω) = μ (−r, ω) EH
(2.31)
EH
2.5.3. Lorentz covariance Suppose an inertial reference frame Σ moves with constant velocity v = v vˆ with respect to an inertial reference frame Σ. The Lorentz transformation (Chen [1983]) ⎫ r = Y • r − γ vt ⎪ ⎬ , r •v t =γ t− 2 ⎪ ⎭ c0
EBL
(2.32)
relates the space–time coordinates (r , t ) in Σ to the space–time coordinates (r, t) in Σ, wherein ⎫ Y = I + (γ − 1)ˆv vˆ ⎪ ⎪ ⎬ 1 2 −2 , γ = 1−β ⎪ ⎪ ⎭ β = c0−1 v
(2.33)
The Maxwell postulates are Lorentz covariant, which means that they retain their form under the spatiotemporal transformation (2.32). The Lorentz covariance of the Maxwell postulates has far-reaching implications for the constitutive relations that develop in uniformly moving reference frames, as mentioned in § 3.3.1. 2.5.4. Chiral invariance As well as being form-invariant under spatial, temporal and spatiotemporal transformations described in the previous subsections, the Maxwell postulates do not
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133
change under the following transformation of fields (Lakhtakia [1995]) ⎫ ˜ ˜ t) = E(r, t) cos ψ − Z H˜ (r, t) sin ψ Rψ E(r, ⎪ ⎪ ⎪ ⎪ −1 ˜ ˜ ˜ Rψ H (r, t) = Z E(r, t) sin ψ + H (r, t) cos ψ ⎬ ˜ ˜ ˜ ⎪ t) = B(r, t) cos ψ + Z D(r, t) sin ψ Rψ B(r, ⎪ ⎪ ⎪ ⎭ −1 ˜ ˜ ˜ Rψ D(r, t) = −Z B(r, t) sin ψ + D(r, t) cos ψ
(2.34)
and source densities ⎫ Rψ ρ˜e (r, t) = ρ˜e (r, t) cos ψ − Z −1 ρ˜m (r, t) sin ψ ⎪ ⎪ ⎪ ⎪ ⎬ Rψ ρ˜m (r, t) = Z ρ˜e (r, t) sin ψ + ρ˜m (r, t) cos ψ . Rψ J˜ e (r, t) = J˜ e (r, t) cos ψ − Z −1 J˜ m (r, t) sin ψ ⎪ ⎪ ⎪ ⎪ ⎭ Rψ J˜ m (r, t) = Z J˜ e (r, t) sin ψ + J˜ m (r, t) cos ψ
(2.35)
Here ψ is a complex-valued angle and the scalar Z is an impedance required to maintain dimensional integrity. When ψ ∈ R (the set of all real numbers), the transformation operator Rψ represents a rotation of the fields. Hence, the Maxwell postulates are said to possess the property of chiral invariance. An especially interesting transformation arises for ψ = π/2: The electric and magnetic fields, and similarly the electric and magnetic charge densities, interchange under Rπ/2 , which is often called the duality transformation (Jackson [1999]). Owing to the duality of the electric charge and the magnetic charge, it is merely a matter of convention whether a particular charged particle is said to possess an electric charge or a magnetic charge. Furthermore, the question of the existence of magnetic monopoles is more fundamentally the question of whether all charged carriers possess the same proportion of electric charge and magnetic charge. If the answer to this question is in the affirmative, then – by applying a Rψ transformation with the appropriate choice of ψ – either the magnetic monopole or the electric monopole could be said to not exist. Let us also note that duality is best considered globally (i.e., for all mediums, at all times, and everywhere) so that the appropriate choice of ψ is made for physical certainty; however, that choice does not preclude the later application of duality in a local context for mathematical convenience. The constitutive relations (2.6) retain their form under the transformation of fields (2.34) provided that the constitutive dyadics transform as
EBL
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⎫ Rψ ˜ EB (r, t) = cos2 ψ ˜ EB (r, t) + Z −2 sin2 ψ μ ˜ (r, t) ⎪ ⎪ EB ⎪ ⎪ ⎪ − Z −1 sin ψ cos ψ ξ˜ (r, t) + ζ˜ (r, t) ⎪ ⎪ ⎪ EB EB ⎪ ⎪ ⎪ −1 ⎪ ˜ (r, t) ⎪ Rψ ξ˜ (r, t) = sin ψ cos ψ Z ˜ EB (r, t) − Z μ ⎪ ⎪ EB EB ⎪ ⎪ 2 ˜ 2 ˜ ⎪ + cos ψ ξ (r, t) − sin ψ ζ (r, t) ⎬ EB EB . ⎪ ˜ (r, t) ⎪ Rψ ζ˜ (r, t) = sin ψ cos ψ Z ˜ EB (r, t) − Z −1 μ ⎪ ⎪ EB EB ⎪ ⎪ 2 ˜ 2 ˜ ⎪ + cos ψ ζ (r, t) − sin ψ ξ (r, t) ⎪ ⎪ EB EB ⎪ ⎪ ⎪ 2 2 2 ⎪ Rψ μ ˜ (r, t) = Z sin ψ ˜ EB (r, t) + cos ψ μ ˜ (r, t) ⎪ ⎪ EB EB ⎪ ⎪ ⎪ + Z sin ψ cos ψ ξ˜ (r, t) + ζ˜ (r, t) ⎭ EB
(2.36)
EB
2.5.5. Conjugate invariance The frequency-domain Maxwell curl postulates (2.12) are invariant under a further transformation, namely the conjugate transformation denoted by C. The conjugate-transformed fields are defined as (Lakhtakia [2004a]) C H (r, ω) = H ∗ (r, ω) C E(r, ω) = E ∗ (r, ω), , (2.37) C D(r, ω) = −D ∗ (r, ω), C B(r, ω) = −B ∗ (r, ω)
EBL
whereas the source densities transform as ∗ (r, ω) C ρe (r, ω) = −ρe∗ (r, ω), C ρm (r, ω) = −ρm . C J e (r, ω) = J ∗e (r, ω), C J m (r, ω) = J ∗m (r, ω)
(2.38)
When applied to linear materials, the Maxwell postulates remain invariant, provided that the constitutive dyadics undergo the following transformations: C EB (r, ω) = − ∗EB (r, ω), C ξ (r, ω) = ξ ∗ (r, ω) EB EB (2.39) C ζ (r, ω) = ζ ∗ (r, ω), C ν EB (r, ω) = −ν ∗EB (r, ω) EB
and
EB
C EH (r, ω) = − ∗EH (r, ω), C ζ (r, ω) = −ζ ∗ (r, ω), EH
EH
C ξ (r, ω) = −ξ ∗ (r, ω) EH EH . (2.40) C μ (r, ω) = −μ (r, ω) EH
EH
The conjugation symmetry represented in eqs. (2.37)–(2.40) arises as a generalization of the transformation which reverses the sign of the real permittivity and permeability scalars for isotropic dielectric–magnetic mediums (Lakhtakia [2004a]). The significance of this transformation is discussed further in § 4, in the context of negative-phase-velocity mediums. The effect of the conjugate transformation (2.39) would be observable in, for example, plane-wave propagation through a material slab.
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The Maxwell postulates and constitutive relations
135
2.5.6. Energy and momentum It is of practical interest to consider how the spatial, temporal and field transformations influence measurable quantities. To this end, let us draw attention to the energy flow density, as quantified via the instantaneous Poynting vector ˜ t) = E(r, ˜ S(r, t) × H˜ (r, t);
(2.41)
the total energy density 1 ˜ ˜ ˜ W˜ (r, t) = D(r, t) • E(r, t) + B(r, t) • H˜ (r, t) ; 2 and the Maxwell stress dyadic 1 ˜ ˜ ˜ t) • E(r, t) + B(r, t) • H˜ (r, t) I T˜ (r, t) = − D(r, 2 ˜ ˜ ˜ + D(r, t)E(r, t) + B(r, t)H˜ (r, t).
(2.42)
(2.43)
By the straightforward application of the field transformations (2.25) and (2.30), respectively, we see that ⎫ ˜ t) = −S(r, ˜ −t)⎪ T S(r, ⎬ (2.44) T W˜ (r, t) = W˜ (r, −t) ⎪ ⎭ ˜ ˜ T T (r, t) = T (r, −t)
EBL
and
⎫ ˜ ˜ t) = −S(−r, t)⎪ P S(r, ⎬ P W˜ (r, t) = W˜ (−r, t) . ⎪ ⎭ P T˜ (r, t) = T˜ (−r, t)
(2.45)
The chiral invariance of the Maxwell postulates carries over to measurable quantities. From (2.34), it follows immediately that ⎫ ˜ t) = S(r, ˜ t) ⎪ Rψ S(r, ⎬ (2.46) Rψ W˜ (r, t) = W˜ (r, t) . ⎪ ⎭ ˜ ˜ Rψ T (r, t) = T (r, t) Therefore, the electromagnetic fields cannot be uniquely determined from measurements of electromagnetic energy and/or momentum.
2.6. Constitutive dyadics Let us now look more closely at the constitutive dyadics which characterize the electromagnetic response of a medium. In the most general linear scenario, the
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Electromagnetic fields in linear bianisotropic mediums
[3, § 2
6 × 6 constitutive dyadic K EH (r, ω) comprises 36 complex-valued parameters. This vast parameter space may be reduced through the imposition of physical constraints which require the constitutive dyadics to satisfy certain symmetries. Also, we often have occasion to limit our attention to special cases and idealizations which manifest as symmetries of the constitutive dyadics. 2.6.1. Constraints 2.6.1.1. Causality and Kramers–Kronig relations The formulations of constitutive relations must conform to the principle of causality; i.e., an ‘effect’ must appear after its ‘cause’. So, neither can a cause and its effect be simultaneous, nor can an effect precede its cause. The principle of causality is most transparently implemented in the time domain for constitutive relations of the form (2.4). ˜ t) and H˜ (r, t) develop in response to the primitive The induced fields D(r, ˜ ˜ fields E(r, t) and B(r, t), such that ˜ ˜ t) = 0 E(r, t) + P˜ (r, t) D(r, , (2.47) ˜ ˜ t) − M(r, t) H˜ (r, t) = μ−1 B(r, 0
where 0 = 8.854 × 10−12 F m−1 and μ0 = 4π × 10−7 H m−1 are the permittivity and permeability of free space, respectively. The polarization P˜ (r, t) and the ˜ t) indicate the electromagnetic response of a medium, and magnetization M(r, must therefore be causally connected to the primitive fields. Therefore, with regard to the time-domain linear constitutive relations (2.5)– (2.6), causality dictates that ⎫ ˜ EB (r, t) − 0 δ(r)I ≡ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ξ˜ (r, t) ≡ 0 EB (2.48) for t 0, ⎪ ζ˜ (r, t) ≡ 0 ⎪ ⎪ EB ⎪ ⎪ ⎭ μ−1 0 δ(r)I − ν˜ EB (r, t) ≡ 0
EBL
where δ(·) is the Dirac delta function.1 When translated into the frequency domain, the causality requirement (2.48) gives rise to integral relations between the real and imaginary parts of the frequency-dependent constitutive parameters, as follows. Let the scalar function f˜(r, t) represent an arbitrary component of a Boys–Post constitutive dyadic; i.e., f˜(r, t) is a component of ˜ EB (r, t) − 0 δ(r)I , ξ˜ (r, t), EB
1 The Dirac delta function is defined in (5.5).
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˜ ζ˜ (r, t) or μ−1 0 δ(r)I − ν˜ EB (r, t). The temporal Fourier transform of f (r, t) EB may be expressed as ∞ f (r, ω) =
f˜(r, t) exp(iωt) dt,
(2.49)
0
wherein the causality constraint (2.48) has been applied to set the lower limit of integration equal to zero. The analytic continuation of f (r, ω) in the upper complex ω plane is provided by the Cauchy integral formula f (r, ω) =
1 2πi
f (r, s) ds, s−ω
(2.50)
where the integration contour extends around the upper half plane. The integrand in (2.50) vanishes as |s| → ∞ for Im{s} > 0 due to the exp(iωt) factor occurring in the integral representation (2.49). Hence, the contour integral (2.50) reduces to an integral along the real axis. Counting the single pole on the real axis at ω = s as a half-residue, we have
EBL
1 f (r, ω) = P πi
∞
−∞
f (r, s) ds, s−ω
(2.51)
where P indicates the Cauchy principal value. Therefore, we see that the real and imaginary parts of f (r, ω) are related as the Hilbert transforms 1 Re f (r, ω) = P π
∞
−∞
1 Im f (r, ω) = − P π
Im{f (r, s)} ds s−ω
∞
−∞
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ Re{f (r, s)} ⎪ ⎪ ds ⎪ ⎪ ⎪ ⎭ s−ω
.
(2.52)
Finally, since f˜(r, t) is real-valued, the symmetry condition f (r, −ω) = f ∗ (r, ω)
(2.53)
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relates f to its complex conjugate f ∗ . Thus, (2.52) yield the Kramers–Kronig relations (Post [1997])2 ⎫ ∞ ⎪ ⎪ s Im{f (r, s)} 2 ⎪ ⎪ Re f (r, ω) = P ds ⎪ ⎪ 2 2 ⎪ π s −ω ⎬ 0 (2.54) . ∞ ⎪ ω Re{f (r, s)} ⎪ 2 ⎪ ⎪ ds ⎪ Im f (r, ω) = − P ⎪ ⎪ π s 2 − ω2 ⎭ 0
Although the relations (2.54) are presented here for components of the Boys– Post constitutive dyadics, analogous relations hold for components of the Tellegen constitutive dyadics by virtue of eqs. (2.11). The Kramers–Kronig relations are a particular example of dispersion relations that apply generally to frequency-dependent, causal, linear systems (Hilgevoord [1962]). These may be usefully employed in experimental determinations of constitutive parameters (Bohren and Huffman [1983]). 2.6.1.2. Post constraint A structural constraint, known as the Post constraint, is available for those linear mediums which exhibit magnetoelectric coupling (Post [1997]). The Post constraint may be expressed as tr ζ (r, ω) − ξ (r, ω) = 0, (2.55) EB
or equivalently as tr μ−1 (r, ω) • ζ EH
EB
EH
EBL
(r, ω) + ξ
EH
(r, ω) = 0.
(2.56)
As a result, only 35 independent complex-valued parameters are needed to characterize the most general linear medium. The origins of the Post constraint lie in the microscopic nature of the primitive electromagnetic fields and the Lorentz covariance of the Maxwell equations (Lakhtakia [2004b]). Post established his eponymous constraint more than 40 years ago (Post [1997]). Two additional independent proofs, one based on a uniqueness requirement (Lakhtakia and Weiglhofer [1996a, 1996b]) and another based on multipole considerations (de Lange and Raab [2001]), further secured the standing of the Post constraint. However, there is recent experimental evidence that the Post constraint is violated at low frequencies (Hehl, Obukhov, Rivera and Schmid [in press]), but there is no microscopic understanding as yet of this evidence. The incorporation of the hitherto-undiscovered axion will lead to a re2 An alternative approach to the derivation of the Kramers–Kronig relations, exploiting the properties of Herglotz functions, has recently been reported by King [2006].
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evaluation of the Post constraint even for free space (Hehl and Obukhov [2005], Lakhtakia [2006]). 2.6.1.3. Onsager relations The Onsager relations are a set of reciprocity relations which are applicable generally to coupled linear phenomena at macroscopic length-scales (Onsager [1931a, 1931b], Casimir [1945]). They were originally established for instantaneous phenomenons, but their scope was extended by means of the fluctuation–dissipation theorem (Callen and Greene [1952]) to include time-harmonic phenomenons as well (Callen, Barasch and Jackson [1952]). Central to the Onsager relations is the assumption of microscopic reversibility. Consequently, in order to apply the Onsager relations to electromagnetic constitutive relations, the contribution of free space must be excluded because microscopic processes cannot occur in free space. The frequency-dependent quantities P (r, ω) and M(r, ω), which are the temporal Fourier transforms of the polar˜ ization P˜ (r, t) and the magnetization M(r, t), represent the electromagnetic response of a medium in relation to the electromagnetic response of free space. For linear homogeneous mediums, the constitutive relations (2.8) reduce to ⎫ ⎬ P (r, ω) = EB (ω) − 0 I • E(r, ω) + ξ (ω) • B(r, ω) EB . M(r, ω) = − ζ (ω) • E(r, ω) + ν (ω) − μ−1 I • B(r, ω) ⎭
EBL
EB
EB
0
(2.57) In an external magnetostatic field B dc , the application of the Onsager relations to the constitutive relations (2.57) yields the constraints (Lakhtakia and Depine [2005]) ⎫ (η = , ν)⎬ η (ω)B = ηT (ω)−B EB EB (2.58) , dc dc ⎭ ξ (ω)B = ζ T (ω)−B EB
dc
EB
dc
T
where the superscript denotes the transpose operation. The equivalent constraints for the Tellegen constitutive dyadics follow immediately from (2.10) as ⎫ (η = , μ)⎬ η (ω)B = ηT (ω)−B EH EH (2.59) . dc dc ⎭ = −ζ T (ω) ξ (ω) EH
B dc
EH
−B dc
2.6.2. Specializations 2.6.2.1. Lorentz reciprocity The issue of Lorentz reciprocity – which is closely related to the topics of time reversal and the Onsager relations – crops up frequently in theoretical studies involving complex mediums (Altman and Suchy
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[3, § 2
[1991]). It is often described in terms of the interchangeability of transmitters and receivers. Let us consider two frequency-domain electric source current densities, namely J ae (r, ω) and J be (r, ω), and two frequency-domain magnetic source current densities, namely J am (r, ω) and J bm (r, ω). The sources labelled a generate fields described by the electric and magnetic fields E a (r, ω) and H a (r, ω), whereas the sources labelled b generate fields described by the electric and magnetic fields E b (r, ω) and H b (r, ω). The interaction of the a sources with the field generated by the b sources is quantitated as the reaction (Kong [1986]) a a, b = (2.60) J e (r, ω) • E b (r, ω) − J am (r, ω) • H b (r, ω) d3 r, Va
where the integration region Va contains the a sources. Similarly, the interaction of the b sources with the field generated by the a sources is represented by a,b (r, ω) and the reaction b, a . If the medium which supports J a,b e,m (r, ω), E a,b H (r, ω) is such that a, b = b, a ,
(2.61)
then it is called Lorentz-reciprocal. Combining the Tellegen constitutive relations (2.9) with the Maxwell curl postulates (2.12) and integrating thereafter, we obtain the reaction difference
EBL
a, b − b, a b = −iω E (r, ω) • EH (r, ω) − TEH (r, ω) • E a (r, ω) Va ∪Vb
+ H a (r, ω) • μ (r, ω) − μT (r, ω) • H b (r, ω) EH EH + E b (r, ω) • ξ (r, ω) + ζ T (r, ω) • H a (r, ω) EH EH + H a (r, ω) • ζ (r, ω) + ξ T (r, ω) • E b (r, ω) d3 r, EH
EH
(2.62)
where the integration region Va ∪ Vb contains both the sources a and b. Thus, Lorentz reciprocity is signalled by (Krowne [1984]) ⎫ η (r, ω) = ηT (r, ω) (η = , μ)⎬ EH EH (2.63) . ⎭ ξ (r, ω) = −ζ T (r, ω) EH
EH
The corresponding symmetries for the Boys–Post representation follow immediately from (2.11) as ⎫ η (r, ω) = ηT (r, ω) (η = , ν)⎬ EB EB . (2.64) ⎭ ξ (r, ω) = ζ T (r, ω) EB
EB
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The Lorentz-reciprocity conditions (2.63) and (2.64) coincide with the Onsager relations (2.59) and (2.58), respectively, in the absence of a magnetostatic field. Many widely studied anisotropic and bianisotropic mediums are required to satisfy the Lorentz-reciprocity conditions. Lorentz-reciprocal mediums arise commonly as dielectric and magnetic crystals, whereas plasmas are not Lorentzreciprocal, as becomes clear from § 3. 2.6.2.2. Neglect of dissipation By virtue of the principle of causality, no medium – with the unique exception of free space, which is not a material – responds instantaneously to an applied electromagnetic field (Chen [1983], Weiglhofer and Lakhtakia [1996]). Therefore, dissipation is exhibited by all material mediums. However, on occasion it may be expedient to neglect dissipation, particularly if attention is confined to a narrow ω-range wherein dissipation is tiny over length scales of interest. In order to focus upon dissipation in the frequency domain, we introduce the time-averaged Poynting vector ! 1 S(r, ω) t = Re E(r, ω) × H ∗ (r, ω) 2
(2.65)
EBL
of a monochromatic field. Of particular relevance to us here is the divergence of (2.65). Utilizing the Maxwell curl postulates (2.12) in the absence of sources (i.e., J e,m (r, ω) ≡ 0), together with the Tellegen constitutive relations (2.9), we find that eq. (2.65) provides (Kong [1986]) ! iω ∗ ∇ • S(r, ω) t = E (r, ω) • EH (r, ω) − +EH (r, ω) • E(r, ω) 4 + H ∗ (r, ω) • μ (r, ω) − μ+ (r, ω) • H (r, ω) EH EH + E ∗ (r, ω) • ξ (r, ω) − ζ + (r, ω) • H (r, ω) EH EH + H ∗ (r, ω) • ζ (r, ω) − ξ + (r, ω) • E(r, ω) , EH
EH
(2.66)
with the superscript + indicating the conjugate transpose. A medium is nondissipative provided that ∇ • S(r, ω) t = 0. Thus, dissipation is neglected by enforcing the equalities (Chen [1983]) η ξ
EH EH
⎫ (r, ω) = η+ (r, ω) (η = , μ)⎬ EH
(r, ω) = ζ + (r, ω) EH
⎭
,
(2.67)
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Table 1 The conditions imposed upon the Tellegen constitutive parameters by Lorentz reciprocity and by the neglect of dissipation; three forms of K EH are represented K EH
Lorentz-reciprocal
Nondissipative
I ξ I
ξ = −ζ
= ∗ ξ = ζ∗ μ = μ∗
ξ = −ζ
∗ = ∗ ξ = ζ
ζ I μI
⎡%
11 0 0 22
0 0
&
%
⎥ ⎥ &⎥ 0 ⎥ ⎦
⎢ ⎢ 0 0 33 ⎢% ⎢ ζ11 0 0 & ⎣
%
&
%
11 21 31
12 22 32
13 23 33
⎢ ⎢ ⎢ ⎢% ⎢ ζ11 ζ12 ζ13 & ⎣ ζ21 ζ31
ζ22 ζ32
ζ23 ζ33
μ = μ∗
μ11 0 0 μ22 0 0 0 μ33
0 ζ22 0 0 0 ζ33
⎡%
&⎤
ξ11 0 0 0 ξ22 0 0 0 ξ33
ξ11 ξ12 ξ13 ξ21 ξ22 ξ23 ξ31 ξ32 ξ33
&⎤
⎥ ⎥ ⎥ ⎥ & μ13 ⎥ ⎦ μ
%μ
m = m ξ m = −ζm μ m = μm
11 μ12 μ21 μ22 23 μ31 μ32 μ33
∗ m = m ∗ ξ m = ζm
μ m = μ∗m
Notice that for the medium represented in the first example, the Lorentz-reciprocity condition ξ = −ζ must be satisfied in order to comply with the Post constraint.
EBL ⎫
or equivalently η ξ
EB EB
(r, ω) = η+ (r, ω) (η = , ν)⎬ EB
(r, ω) = −ζ + (r, ω) EB
⎭
.
(2.68)
The distinction between the conditions for the neglect of dissipation and for Lorentz reciprocity must be noted. These are summarized in table 1 for three commonly encountered forms of the constitutive dyadic K EH (r, ω).
§ 3. Linear mediums The notions of anisotropy and bianisotropy may be simply characterized in terms of what they are not: the adjectives ‘anisotropic’ and ‘bianisotropic’ describe those mediums which are not isotropic. Therefore, prior to discussing anisotropic and bianisotropic mediums in this section, we first establish our terms of reference by briefly considering isotropic mediums. We then present a survey of the commonly encountered classifications of anisotropy and bianisotropy, as expressed in terms of their constitutive relations and constitutive dyadics.
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3.1. Isotropy The primitive fields and induced fields in an isotropic medium are co-directional so that its constitutive dyadics reduce to scalars. 3.1.1. Free space Let us begin with the most fundamental medium in electromagnetics: free space, otherwise referred to as vacuum. From our viewpoint of classical physics, free space is devoid of all matter. Quantum-electrodynamical processes, through which energetic fluctuations may become interwoven with free space, are not relevant to us here. Thus, free space represents the reference medium, relative to which the electromagnetic responses of all mediums are gauged (Weiglhofer [2003]). Our discussion of the electromagnetic properties of free space in this section pertains to flat space–time exclusively. The subject of free space in generally curved space–time is taken up in § 3.4.2. By definition, free space is clearly both isotropic and homogeneous. Its constitutive properties are specified by the scalar permittivity 0 and scalar permeability μ0 . Free space holds the unique distinction of being the only medium for which there is an exact, spatiotemporally local relationship between the induction fields and the primitive fields. The time-domain constitutive relations of free space are ˜ ˜ t) = 0 E(r, t) D(r, , (3.1) ˜ H˜ (r, t) = μ−1 B(r, t)
EBL
0
and its frequency-domain constitutive relations have the same form: D(r, ω) = 0 E(r, ω) . H (r, ω) = μ−1 0 B(r, ω)
(3.2)
If axions are ever discovered, eqs. (3.1) and (3.2) shall have to be modified by the incorporation of magnetoelectric terms that violate the Post constraint (Hehl and Obukhov [2005]). 3.1.2. Dielectric–magnetic mediums The simplest material mediums are the isotropic, homogeneous, dielectric–magnetic mediums. Their electromagnetic properties are characterized in terms of the scalar permittivity (ω) and scalar permeability μ(ω) by the frequency-domain Tellegen constitutive relations
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D(r, ω) = (ω)E(r, ω) B(r, ω) = μ(ω)H (r, ω)
[3, § 3
(3.3)
.
Causality demands that both and μ be ω-dependent, complex-valued parameters with Im{} > 0 and Im{μ} > 0. The placement of and μ in the upper half of the complex plane follows from the sign convention for the exponent of Fourier transform kernel (2.7). Parenthetically, we remark that though isotropic dielectric–magnetic mediums have been very widely studied since the earliest days of electromagnetics, their properties are still the subject of ongoing research. For example, negative refraction – which follows from the condition Re{(ω)} Re{μ(ω)} + <0 (3.4) Im{(ω)} Im{μ(ω)} being satisfied at a particular frequency ω – is a topic of intense current interest (Lakhtakia, McCall and Weiglhofer [2003], Litchinitser, Gabitov, Maimistov and Shalaev [2008]); see § 4.7 for further details. 3.1.3. Isotropic chirality and biisotropy A homogeneous, isotropic chiral medium is described by the frequency-domain Tellegen constitutive relations D(r, ω) = (ω)E(r, ω) + χ(ω)H (r, ω) . (3.5) B(r, ω) = −χ(ω)E(r, ω) + μ(ω)H (r, ω)
EBL
Three complex-valued, ω-dependent scalars – permittivity , permeability μ and the chirality parameter χ – characterize it (Lakhtakia [1994]). Notice that, in view of the definition presented in § 2.6.2, isotropic chiral mediums are Lorentz-reciprocal. The corresponding non-reciprocal isotropic medium, as would be described by the Tellegen constitutive relations D(r, ω) = (ω)E(r, ω) + χ1 (ω)H (r, ω) , (3.6) B(r, ω) = χ2 (ω)E(r, ω) + μ(ω)H (r, ω) with χ1 = −χ2 , is in violation of the Post constraint (Lakhtakia [2004b]). Such a medium is called a biisotropic medium.
3.2. Anisotropy The primitive fields and induced fields in isotropic mediums are co-directional. In contrast, the defining characteristic of anisotropic mediums is that E and D are not aligned and/or B and H are not aligned. Hence, whereas scalars provide the
3, § 3]
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constitutive characterizations for isotropic mediums, the constitutive properties of anisotropic mediums are described using dyadics. Numerous examples of dielectric anisotropy crop up within the context of crystal optics (Born and Wolf [1980], Nye [1985]). Nematic and smectic liquid crystals fall into the anisotropic dielectric category (Chandrasekhar [1992], de Gennes and Prost [1993]). Magnetic anisotropy is an important characteristic of many diamagnetic and paramagnetic substances (Nye [1985]). 3.2.1. Uniaxial anisotropy At macroscopic length scales, a single distinguished axis identifies a uniaxial medium. The distinguished axis may originate from microscopic attributes, such as the underlying crystalline structure or particulate geometry. Let the unit vector uˆ point in the direction of the distinguished axis. The frequency-domain constitutive relations of a uniaxial, homogeneous, dielectric medium are then given in the Tellegen representation as D(r, ω) = uni (ω) • E(r, ω) (3.7) , B(r, ω) = μ0 H (r, ω)
EBL
where the ω-dependent permittivity dyadic uni may be expressed as uni (ω) = (ω)(I − uˆ u) ˆ + u (ω)uˆ u. ˆ
(3.8)
The permittivity scalars and u are complex-valued quantities; causality requires that Im{} > 0 and Im{u } > 0. Without any loss of generality, we may choose to orient our coordinate system in such a way that the distinguished direction coincides with the z axis. Thereby, uni (ω) acquires the diagonal matrix form uni (ω) = diag (ω), (ω), u (ω) .
(3.9)
Similarly, a uniaxial, homogeneous magnetic medium may be described by the Tellegen constitutive relations D(r, ω) = 0 E(r, ω) (3.10) , B(r, ω) = μ (ω) • H (r, ω) uni
with μ
uni
(ω) = μ(ω)(I − uˆ u) ˆ + μu (ω)uˆ u. ˆ
(3.11)
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The ω-dependent parameters μ and μu of causal mediums lie in the upper half of the complex plane. The choice uˆ = zˆ leads to the permeability matrix representation μ (ω) = diag μ(ω), μ(ω), μu (ω) . (3.12) uni
A uniaxial dielectric–magnetic medium is described by the Tellegen constitutive relations D(r, ω) = uni (ω) • E(r, ω) . (3.13) B(r, ω) = μ (ω) • H (r, ω) uni
3.2.2. Biaxial anisotropy The natural generalization of the uniaxial dielectric medium specified by eqs. (3.7)– (3.9) is the orthorhombic biaxial dielectric medium described by the Tellegen constitutive relations (ω) • E(r, ω) D(r, ω) = ortho bi , (3.14) B(r, ω) = μ0 H (r, ω) where the permittivity dyadic ortho (ω) has the diagonal form bi (ω) = diag x (ω), y (ω), z (ω) . ortho bi
EBL
(3.15)
Causality dictates that the imaginary parts of the ω-dependent permittivity scalars x , y and z are positive-valued. While the diagonal dyadic representation (3.15) can be mathematically convenient, it is not particularly insightful from a physical perspective. The equivalent representation (Chen [1983]) (ω) = a (ω)I + b (ω)(uˆ m uˆ n + uˆ n uˆ m ) ortho bi
(3.16)
highlights the biaxial symmetry via the two unit vectors uˆ m and uˆ n (Weiglhofer and Lakhtakia [1999]). In the absence of dissipation, the ω-dependent permittivity scalars a and b are real-valued and the unit vectors uˆ m and uˆ n are aligned with the optic ray axes. In a general direction through a biaxial medium, electromagnetic radiation may propagate in one of two different modes; the two modes are distinguished from each other by their different rates of energy flow. However, in the two privileged directions provided by the optic ray axes, only one energy velocity is permissible (Born and Wolf [1980]).3 For dissipative mediums, both a 3 The optic ray axes should not be confused with the optic axes that identify the two privileged directions in which electromagnetic waves may propagate through a biaxial medium with only one phase velocity. See § 4.4.2.
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and b are necessarily complex-valued, in which case Re{a , b } and Im{a , b } are generally associated with different pairs of (uˆ m , uˆ n ) axes (Mackay and Weiglhofer [2000]). More general biaxial, homogeneous, dielectric mediums may be characterized by the Tellegen constitutive relations ⎫ ⎪ mono (ω) ⎪ bi • E(r, ω)⎬ D(r, ω) = tri (3.17) , bi (ω) ⎪ ⎪ ⎭ B(r, ω) = μ H (r, ω) 0
where the 3 × 3 permittivity dyadics have the symmetric forms ⎛ ⎞ x (ω) α (ω) 0 (ω) = ⎝ α (ω) y (ω) mono 0 ⎠ bi 0 0 z (ω) and
⎛
x (ω) ⎝ α (ω) tri (ω) = bi β (ω)
α (ω) y (ω) γ (ω)
⎞ β (ω) γ (ω) ⎠ . z (ω)
(3.18)
(3.19)
EBL
The constitutive permittivity dyadics which specify biaxial anisotropy are summarized in table 2. As in the orthorhombic scenario, the ω-dependent permittivity scalars x , y and z are complex-valued, with causality dictating that Im{x , y , z } > 0. The off-diagonal ω-dependent scalars α , β and γ are real-valued. Biaxial mediums belonging to the monoclinic and triclinic crystal systems (Nye [1985]) are described by (3.18) and (3.19), respectively. The distinction between the three Table 2 The constitutive permittivity dyadics which describe the three biaxial crystal systems Crystal system Orthorhombic
Constitutive dyadic form % &
Complex-valued and real-valued scalars x , y , z ∈ C
Monoclinic
%
&
x , y , z ∈ C α ∈ R
Triclinic
% & x α β
x , y , z ∈ C α , β , γ ∈ R
x 0 0 0 y 0 0 0 z
x α 0 α y 0 0 0 z α y γ β γ z
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biaxial crystal systems stems from the symmetries of the primitive unit cell belonging to the underlying Bravais lattice. If all three basis vectors of the primitive unit cell are orthogonal then the crystal structure is orthorhombic; only two basis vectors are orthogonal in the monoclinic system; whereas there are no orthogonal basis vectors for triclinic crystals (Ashcroft and Mermin [1976]). Biaxiality is not restricted to dielectric mediums. In a precisely analogous manner, biaxial magnetic mediums are classified. Thus, the Tellegen constitutive relations ⎫ D(r, ω) = 0 E(r, ω) ⎪ ⎪ ⎪ ⎫ ⎪ ortho ⎪ ⎬ (ω) ⎪ μ ⎪ bi ⎬ (3.20) , mono (ω) • H (r, ω)⎪ B(r, ω) = μ ⎪ ⎪ bi ⎪ ⎪ ⎪ ⎪ ⎭ μtri (ω) ⎭ bi
describe orthorhombic, monoclinic and triclinic biaxial magnetic mediums, respectively, where the corresponding permeability dyadics have the symmetric forms ⎫ μortho (ω) = diag μx (ω), μy (ω), μz (ω) ⎪ ⎪ bi ⎪ ⎛ ⎞⎪ ⎪ ⎪ μx (ω) μα (ω) 0 ⎪ ⎪ ⎪ mono ⎪ ⎝ ⎠ ⎬ (ω) = μα (ω) μy (ω) μ 0 bi (3.21) 0 0 μz (ω) ⎪ , ⎛ ⎞ ⎪ ⎪ ⎪ μx (ω) μα (ω) μβ (ω) ⎪ ⎪ ⎪ μtri (ω) = ⎝ μα (ω) μy (ω) μγ (ω) ⎠ ⎪ ⎪ ⎪ bi ⎭ μβ (ω) μγ (ω) μz (ω)
EBL
with μx , μy , μz ∈ C and μα , μβ , μγ ∈ R. A biaxial dielectric–magnetic medium is described by the Tellegen constitutive relations D(r, ω) = bi (ω) • E(r, ω) (3.22) , B(r, ω) = μ (ω) • H (r, ω) bi
wherein the symmetric permittivity and permeability dyadics bi (ω) and μ (ω), bi respectively, may be of the orthorhombic, monoclinic or triclinic type. 3.2.3. Gyrotropy The uniaxial and biaxial manifestations of anisotropy discussed in §§ 3.2.1 and 3.2.2 are represented mathematically in terms of symmetric permittivity and
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permeability dyadics. Uniaxial and biaxial mediums are therefore Lorentz-reciprocal. Let us turn now to a fundamentally different type of anisotropy, namely gyrotropy, which is characteristic of mediums that are not Lorentz-reciprocal. Manifestations of gyrotropy are found within the realm of magneto-optic mediums (Mansuripur [1995], Gersten and Smith [2001]). As an illustrative example, we consider an incompressible plasma of electrons in thermal motion (Felsen and Marcuvitz [1994]). A quasistatic magnetic field B 0 = B0 uˆ of strength B0 is applied in the direction of the unit vector u. ˆ Adopting a macroscopic viewpoint, we assume that the plasma with charge density −n0 q and mass density n0 m is homogeneously distributed in free space. Additionally, the plasma density is taken to be sufficiently low that collisions between electrons may be neglected. The average electron velocity is v˜ (r, t). The governing equations are provided by the time-domain Maxwell curl postulates ⎫ ∂ ˜ ⎪ t) = −n0 q v˜ (r, t)⎪ ∇ × H˜ (r, t) − 0 E(r, ⎬ ∂t (3.23) , ⎪ ∂ ˜ ⎪ ˜ ⎭ ∇ × E(r, t) + μ0 H (r, t) = 0 ∂t along with the equation of motion
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∂ ˜ −n0 q E(r, (3.24) t) + v˜ (r, t) × B 0 = n0 m v˜ (r, t). ∂t Using eq. (3.24) to eliminate v˜ (r, t) from eqs. (3.23) and thereupon comparing with the source-free versions of the Maxwell curl postulates (2.1), we find that the constitutive relations ˜ ˜ t) = ˜ gyro E(r, t) D(r, (3.25) ˜ t) H˜ (r, t) = μ0 −1 B(r, emerge (Weiglhofer [2003]). The permittivity dyadic operator ˜ gyro {•} is defined formally by −1 , ˜ gyro = 0 I + 0 ωp2 I ∂tt − ωc (uˆ × I )∂t (3.26) √ with ωp = q n0 /m0 being the plasma frequency and ωc = qB0 /m being the gyrofrequency. The shorthand notation ∂t ≡ ∂/∂t and ∂tt ≡ ∂ 2 /∂t 2 has been introduced in eq. (3.26), and the inverse of the temporal differential operator may be interpreted as ∂t−1 f (t)
t ≡ −∞
f (t ) dt .
(3.27)
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By straightforward dyadic manipulations, eq. (3.26) may be expressed in the more convenient form (Chen [1983])
ωp2 ωc ωp2 ˜ gyro = 0 1 + 2 ˆ + 2 uˆ × I (I − uˆ u) ωc + ∂tt (ωc + ∂tt )∂t ωp2 (3.28) + 1+ uˆ uˆ , ∂tt where the interpretation g(t) =
1 f (t) C + ∂tt
⇔
(C + ∂tt )g(t) = f (t)
(3.29)
is to be understood. The differential operators appearing in the representations (3.26) and (3.28) signify temporal dispersion. Upon applying the temporal Fourier transform (2.7), the differential operator ∂t transforms to −iω. Thus, the time-domain constitutive relations (3.25) transform into the frequency-domain constitutive relations D(r, ω) = gyro (ω) • E(r, ω) (3.30) B(r, ω) = μ0 H (r, ω)
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in the Tellegen representation. The ω-dependent permittivity dyadic gyro is specified as
iωp2 ωc ωp2 gyro (ω) = 0 1 + 2 uˆ × I − u ˆ u) ˆ + (I ωc − ω 2 (ωc2 − ω2 )ω ωp2 (3.31) + 1 − 2 uˆ uˆ . ω Without loss of generality, let us orient our coordinate system such that the quasistatic biasing magnetic field is directed along the z axis; i.e., we choose uˆ ≡ zˆ . Thereby, the constitutive dyadic (3.31) may be reformulated as ⎛ ⎞ ωp2 iωp2 ωc − (ω2 −ω 0 1 + ω2 −ω 2 2 )ω c c ⎜ ⎟ 2 ⎜ ⎟ ωp2 p ωc gyro (ω) = 0 ⎜ iω (3.32) ⎟. 1 + 0 ωc2 −ω2 ⎝ (ωc2 −ω2 )ω ⎠ 0
0
1−
ωp2 ω2
The gyrotropic form (3.32) is characterized by a uniaxial anisotropy in the direction of the quasistatic biasing magnetic field, together with antisymmetric terms associated with the plane perpendicular to that direction.
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With reference to the results of § 2.6.2, notice that the gyrotropic medium described jointly by eqs. (3.30) and (3.31) is nondissipative. A more realistic plasma representation may be developed through taking account of electron collisions. This may be achieved formally via the substitution ω → ω − iΩ, where Ω ∈ R is a suitably chosen damping parameter (Weiglhofer [2003]). Finally, we mention that gyrotropy also arises in magnetic mediums, for example in ferrites (Lax and Button [1962]). The Tellegen constitutive relations for a gyrotropic magnetic medium may be expressed as D(r, ω) = 0 E(r, ω) , (3.33) (ω) • H (r, ω) B(r, ω) = μ gyro
with the ω-dependent permeability dyadic μ
gyro
(ω) = μ(ω)(I − uˆ u) ˆ + iμg (ω)uˆ × I + μu (ω)uˆ u; ˆ
(3.34)
μ, μu , μg ∈ R for nondissipative mediums and μ, μu , μg ∈ C for dissipative mediums. When uˆ = zˆ , eq. (3.34) may be recast in matrix notation as ⎛ ⎞ μ(ω) −iμg (ω) 0 (ω) = ⎝ iμg (ω) μ (3.35) μ(ω) 0 ⎠. gyro 0 0 μu (ω)
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3.3. Bianisotropy Bianisotropy arises from the combination of anisotropy with magnetoelectric coupling. Thus, in a bianisotropic medium, D is anisotropically coupled to both E and B, and H is anisotropically coupled to both E and B as well (Weiglhofer [2003]). In general, a linear bianisotropic medium is described by four 3 × 3 constitutive dyadics. 3.3.1. Mediums moving at constant velocity Isotropy and anisotropy are not invariant under the Lorentz space–time transformation (2.32). A medium which is isotropic or anisotropic in one inertial reference frame is generally bianisotropic in all other inertial reference frames. However, a homogeneous medium that is spatially local, but not temporally local, in one reference frame is generally spatiotemporally nonlocal in another reference frame (Lakhtakia and Weiglhofer [1996a], Censor [1998]). This fact confines analysis on simply moving, physically realistic materials to a consideration only of plane waves.
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3.3.2. Uniaxial and biaxial bianisotropy The unixial and biaxial classifications introduced in §§ 3.2.1 and 3.2.2, respectively, for dielectric and magnetic mediums carry over to bianisotropic mediums in a straightforward way. Thus, a general biaxial structure for a homogeneous bianisotropic medium may be specified by the Tellegen constitutive relations (Mackay and Weiglhofer [2001a]) T T D(r, ω), B(r, ω) = K EH (ω) • E(r, ω), H (r, ω) , (3.36) where the 6 × 6 constitutive dyadic K EH is assembled from four symmetric 3 × 3 dyadics as
EH (ω) ξ (ω) EH K EH (ω) = ζ (ω) μ (ω) EH EH ⎡ % (ω) (ω) (ω) & % ξ (ω) ξ (ω) ξ (ω) & ⎤ x
α
β
⎢ α (ω) y (ω) γ (ω) ⎢ β (ω) γ (ω) z (ω) =⎢ ⎢ % ζx (ω) ζα (ω) ζβ (ω) & ⎣ ζα (ω) ζy (ω) ζγ (ω) ζβ (ω) ζγ (ω) ζz (ω)
x
α
β
ξα (ω) ξβ (ω) % μx (ω) μα (ω) μβ (ω)
ξy (ω) ξγ (ω) ⎥ ξγ (ω) ξz (ω) ⎥ &⎥, μα (ω) μβ (ω) ⎥ ⎦ μy (ω) μγ (ω) μγ (ω) μz (ω) (3.37) with x,y,z , ξx,y,z , ζx,y,z , μx,y,z ∈ C and α,β,γ , ξα,β,γ , ζα,β,γ , μα,β,γ ∈ R. The orthorhombic bianisotropic specialization follows from setting ⎫ α,β,γ (ω) = 0 ⎪ ⎪ ⎬ ξα,β,γ (ω) = 0 (3.38) . ζα,β,γ (ω) = 0 ⎪ ⎪ ⎭ μα,β,γ (ω) = 0
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3.3.3. Faraday chiral mediums Faraday chiral mediums (FCMs) are conceptualized as homogenized composite mediums (Engheta, Jaggard and Kowarz [1992]). Their bianisotropic nature arises through combining • natural optical activity, as exhibited by isotropic chiral mediums (Lakhtakia [1994]), with • Faraday rotation, as exhibited by gyrotropic mediums (Lax and Button [1962], Chen [1983], Collin [1966]). Two examples of FCMs are the chiroferrite medium which develops from the homogenization of an isotropic chiral medium with a magnetically biased ferrite (Weiglhofer, Lakhtakia and Michel [1998]), and the chiroplasma medium which
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develops from the homogenization of an isotropic chiral medium with a magnetically biased plasma (Weiglhofer and Mackay [2000]). The frequency-domain constitutive relations of a homogeneous FCM are rigorously established as (Weiglhofer and Lakhtakia [1998]) ⎫ (ω) • H (r, ω) ⎬ D(r, ω) = FCM (ω) • E(r, ω) + iξ FCM (3.39) , (ω) • E(r, ω) + μ (ω) • H (r, ω)⎭ B(r, ω) = −iξ FCM
FCM
in the Tellegen representation, where the 3 × 3 constitutive dyadics (ω) = η(ω)I + ηu (ω) − η(ω) uˆ uˆ + iηg (ω)uˆ × I (η = , ξ, μ), η FCM (3.40) all have the same form. The constitutive parameters , u , g , ξ , ξg , ξu , μ, μg and μu are complex-valued in general, but are real-valued for a nondissipative FCM. The FCM structure described by eq. (3.40) is associated with the homogenization of particulate composite mediums based on spherical particulate geometry. More general FCM forms develop through the homogenization of composite mediums containing nonspherical particles (Weiglhofer and Mackay [2000], Mackay, Lakhtakia and Weiglhofer [2001a]).
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3.4. Nonhomogeneous mediums So far in this section, the descriptions of anisotropy and bianisotropy have been provided in terms of constitutive dyadics which are independent of the position vector r. However, the macroscopic properties of many important classes of complex mediums cannot be adequately described without accounting for the dependency of the constitutive properties upon r. Nonhomogeneity increases the complexity of our mathematical analyses; nevertheless, theoretical descriptions of electromagnetic properties are well-established for certain nonhomogeneous anisotropic mediums. 3.4.1. Periodic nonhomogeneity One of the most extensively studied examples of anisotropy in nonhomogeneous mediums is provided by cholesteric liquid crystals (CLCs). The periodic nonhomogeneity of CLCs stems from their helicoidal arrangement of aciculate molecules. In essence, CLCs are periodically twisted, uniaxial, dielectric mediums. Their frequency-domain constitutive relations are given by (Chandrasekhar
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[1992], de Gennes and Prost [1993]) D(r, ω) = CLC (r, ω) • E(r, ω)
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B(r, ω) = μ0 H (r, ω)
(3.41)
,
where the permittivity dyadic ˆ u(r) ˆ CLC (r, ω) = a (ω)I + b (ω) − a (ω) u(r)
(3.42)
is described in terms of the complex-valued, ω-dependent, scalars a and b and the unit vector u(r) ˆ which represents the local optic axis. After choosing the z axis to coincide with the helicoidal axis, the unit vector u(r) ˆ may be expressed as πz πz ≡ u(z) ˆ = xˆ cos + yˆ sin , u(r) ˆ (3.43) Ω Ω where Ω is the dielectric periodicity along the z direction. The helicoidal conformation of CLCs also forms the basis of the recently developed chiral sculptured thin films (CSTFs) (Venugopal and Lakhtakia [2000]). These comprise parallel helical nanowires deposited on a substrate. The helical shape engenders structural chirality, while the nanowires on the average are characterized by local orthorhombic symmetry. CSTFs are of considerable technological interest because of their optical properties. The constitutive relations of a CSTF may expressed as D(r, ω) = CSTF (r, ω) • E(r, ω) (3.44) , B(r, ω) = μ0 H (r, ω)
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with the permittivity dyadic
CSTF (z, ω) = S z (z) • S y (ψ) • diag b (ω), c (ω), a (ω) •
S −1 (ψ) • S −1 (z). y z
(3.45)
Here, a,b,c are three complex-valued, ω-dependent parameters. The tilt dyadic ⎛ ⎞ cos ψ 0 − sin ψ S y (ψ) = ⎝ 0 (3.46) 1 0 ⎠ sin ψ 0 cos ψ and rotation dyadic ⎛ −h sin πz cos πz Ω πzΩ S z (z) = ⎝ h sin πz cos Ω Ω 0 0
⎞ 0 0⎠, 1
(3.47)
specify the nanowire morphology wherein Ω is the half-pitch of the helical nanowires, h = ±1 denotes the structural handedness, and ψ ∈ (0, π/2]. For further
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details on the optical characteristics of CSTFs, the reader is referred to the specialist literature (Lakhtakia and Messier [2005]). The generalization of the CSTF form (3.44)–(3.45) leads to (Lakhtakia and Weiglhofer [1995, 1997a]) D(r, ω) = HBM (z, ω) • E(r, ω) + ξ (z, ω) • H (r, ω) HBM (3.48) , (z, ω) • E(r, ω) + μ (z, ω) • H (r, ω) B(r, ω) = ζ HBM
HBM
which are the constitutive relations for a helicoidal bianisotropic medium (HBM). Herein, all four of the 3×3 nonhomogeneous constitutive dyadics have the general form ⎛ ⎞ η11 (z, ω) η12 (z, ω) η13 (z, ω) (z, ω) = ⎝ η21 (z, ω) η22 (z, ω) η23 (z, ω) ⎠ (η = , ξ, ζ, μ), η HBM η31 (z, ω) η32 (z, ω) η33 (z, ω) (3.49) and are factorizable as η
HBM
(z, ω) = S z (z) • η
HBM
(0, ω) • S −1 (z) z
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(η = , ξ, ζ, μ).
(3.50)
3.4.2. Gravitationally induced bianisotropy
In the preceding subsections of § 3, the backdrop for our description of electromagnetic phenomena has been flat space–time. We now digress in order to show that the classical vacuum in curved space–time is not isotropic but bianisotropic instead. In the general-relativistic literature, the space–time curvature induced by a gravitational field is conventionally characterized using a symmetric covariant tensor of rank 2, known as the metric (d’Inverno [1992]). In keeping with the notational practices implemented throughout this chapter, we represent the space– time metric by the symmetric, real-valued, 4 × 4 matrix g˜ (r, t). The components of g˜ (r, t) are the space–time metric components. This matrix is spatiotemporally nonhomogeneous and local, in general. Following a standard procedure wherein the covariant Maxwell equations are expressed in noncovariant form (Landau and Lifshitz [1975], Skrotskii [1957], Plébanski [1960]), the electromagnetic response of vacuum in curved space–time may be represented by the time-domain constitutive relations (Schleich and Scully [1984], Lakhtakia and Mackay [2004a]) ⎫ ˜ ˜ D(r, t) = 0 γ˜ (r, t) • E(r, t) − c0−1 Γ˜ (r, t) × H˜ (r, t) ⎬ (3.51) ˜ ˜ t) = μ0 γ˜ (r, t) • H˜ (r, t) + c0−1 Γ˜ (r, t) × E(r, t)⎭ B(r,
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in Tellegen form. Herein, the 3 × 3 dyadic γ˜ (r, t) has components [− det g˜ (r, t)] γ˜ (r, t) m = − [˜g(r, t)]1,1
1/2
g˜ −1 (r, t) +1,m+1
, m ∈ {1, 2, 3} , (3.52)
and the 3-vector Γ˜ (r, t) is given as −1 T Γ˜ (r, t) = g˜ (r, t) 1,1 g˜ (r, t) 1,2 , g˜ (r, t) 1,3 , g˜ (r, t) 1,4 ,
(3.53)
with [˜g(r, t)] ,m denoting the ( , m)th element of g˜ (r, t). Thus, the mathematical description of electromagnetic fields in gravitationally affected vacuum is isomorphic to the description of electromagnetic fields in a fictitious, instantaneously responding medium described by the constitutive relations (3.51). Consequently, analytical techniques commonly used to investigate electromagnetic problems (without considering the effects of gravitational fields) may be applied to the study of gravitationally affected vacuum – courtesy of the medium described by eqs. (3.51). This fictitious medium is generally bianisotropic, the symmetries of its constitutive dyadics being determined by those of the underlying metric. The constitutive dyadics are both spatially nonhomogeneous and time-varying. The medium is neither Lorentz-reciprocal in general (cf. § 2.6.2.1) nor dissipative (cf. § 2.6.2.2), but it does satisfy the Post constraint (2.56). As the matrix g˜ (r, t) is real symmetric, the permittivity and permeability dyadics 0 γ˜ (r, t) and μ0 γ˜ (r, t), respectively, are orthorhombic and have the same eigenvectors. Finally, let us note that in flat space–time the matrix g˜ (r, t) simplifies to diag(1, −1, −1, −1) and the constitutive relations (3.51) reduce to the familiar form (3.1).
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§ 4. Plane-wave propagation From a theoretical perspective, the passage of electromagnetic signals through a complex medium is generally a complicated issue – and perhaps it is best broached by considering the propagation of plane waves. Although plane waves themselves are idealizations, being of limitless spatial and temporal extents and possessing infinite energy, they can provide a reasonable understanding of fields far away from their sources. Furthermore, realistic signals may be represented as superpositions of plane waves. Descriptions of plane-wave propagation in isotropic dielectric mediums, as well as in certain relatively simple, anisotropic dielectric mediums, are widely
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available in the literature (Born and Wolf [1980], Chen [1983], Nye [1985], Kong [1986]). This is especially true when the effects of dissipation are neglected, the topic then often coming under the heading of ‘crystal optics’. In this section, a survey of plane-wave solutions of the Maxwell curl postulates in anisotropic and bianisotropic mediums, including dissipative and nonhomogeneous mediums, is presented. We begin with a general description of plane waves. Next, as a precursor to the later subsections on anisotropic and bianisotropic mediums, a brief outline of plane-wave propagation in isotropic mediums is provided. Thereafter, we discuss the modes of uniform plane-wave propagation which are supported by various anisotropic and bianisotropic mediums, including nonhomogeneous mediums. Plane-wave propagation in isotropic mediums is independent of direction of propagation, which contrasts sharply with plane-wave propagation in anisotropic and bianisotropic mediums.
4.1. Uniform and non-uniform plane waves Let us recall from § 2 that the electric and magnetic field phasors E(r, ω) and H (r, ω), respectively, can be conveniently combined in the 6-vector field phasor F(r, ω) defined in eq. (2.15). Electromagnetic plane waves may be represented mathematically by field phasors of the form F(r, ω) = F 0 (ω) exp i(k • r − ωt) , (4.1)
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where F 0 (ω) ∈ C6 is independent of r. The wavevector k ∈ C3 , in general, and may be expressed as k = kR kˆ R + ikI kˆ I ,
(4.2)
wherein the scalars kR,I ∈ R and the unit vectors kˆ R,I ∈ R3 . Parenthetically, note that although k varies with ω, for convenience we do not express this dependence explicitly. In light of eq. (4.2), the field phasor (4.1) may be written as F(r, ω) = F 0 (ω) exp −kI kˆ I • r exp i kR kˆ R • r − ωt . (4.3) On planes in R2 specified by kˆ R • r = constant,
(4.4)
we see from eq. (4.3) that F(r, ω) has constant phase. In other words, a propagating plane of constant phase is described by the field phasor (4.1). The planes of
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constant phase propagate in the direction of kˆ R with velocity vp =
ω ˆ k . kR R
(4.5)
This velocity is called the phase velocity, and its magnitude is the phase speed. Plane waves may be classified as either uniform or nonuniform as follows: • If kˆ R = ±kˆ I then the wavevector k may be stated as ˆ k = k k,
(4.6)
with the complex-valued wavenumber k = kR ± ikI and the real-valued unit vector kˆ ≡ kˆ R . Thus, on planes of constant kˆ • r, F(r, ω) has both constant phase and constant amplitude. Accordingly, these plane waves are known as uniform plane waves. Uniform plane waves also arise in the nondissipative scenario characterized by kI = 0. • If kˆ R = ±kˆ I and kI = 0, then the amplitude of F(r, ω) is generally not uniform on planes of constant kˆ R • r. Accordingly, these plane waves are known as nonuniform plane waves. The wavevector k has inherently complex-valued components for non-uniform plane waves, whereas for uniform plane waves k may have either complex- or real-valued components. Uniform plane waves with real-valued k propagate without attenuation. For practical applications, uniform plane waves are encountered much more often than nonuniform plane waves. For this reason, we concentrate on uniform plane waves in the remainder of this section.
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4.2. Eigenanalysis The starting point for our analysis is provided by the frequency-domain Maxwell curl postulates in the absence of sources; i.e., ∇ × H (r, ω) + iωD(r, ω) = 0 . (4.7) ∇ × E(r, ω) − iωB(r, ω) = 0 Let us consider the most general linear homogeneous medium – namely, a homogeneous bianisotropic medium, whose Tellegen constitutive relations are expressed as D(r, ω) = EH (ω) • E(r, ω) + ξ (ω) • H (r, ω) EH . (4.8) B(r, ω) = ζ (ω) • E(r, ω) + μ (ω) • H (r, ω) EH
EH
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The combination of (4.7) and (4.8) provides us with L(∇) + iωK EH (ω) • F(r, ω) = 0,
159
(4.9)
wherein the 6-vector/6 × 6 dyadic notation of § 2.4 has been implemented, with the constitutive dyadic
EH (ω) ξ (ω) EH K EH (ω) = (4.10) . ζ (ω) μ (ω) EH
EH
On restricting our attention to plane-wave solutions, as represented by the field phasor (4.1), the differential dyadic equation (4.9) simplifies to L(ik) + iωK EH (ω) • F 0 (ω) = 0. (4.11) The algebraic dyadic equation (4.11) is generally amenable to eigenanalysis. The existence of nonzero solutions to (4.11) immediately leads us to the dispersion relation det L(ik) + iωK EH (ω) = 0. (4.12) For uniform plane waves, i.e., k = k kˆ with k ∈ C and kˆ ∈ R3 , the dispersion relation (4.12) yields a polynomial of the fourth degree in k. Accordingly, it has four solutions k = k , ∈ [1, 4]. The dispersion relation (4.12) is quadratic in k 2 for all Lorentz-reciprocal mediums and all Lorentz-nonreciprocal mediums with null-valued magnetoelectric constitutive dyadics (such as gyrotropic mediums). Hence, for these mediums two solutions, k = k1 and k = k2 , say, hold for propagation co-parallel with k and two solutions, k = k3 and k = k4 , say, hold for propagation anti-parallel to k, wherein k1 = −k3 and k2 = −k4 . If k1 = k2 (or, equivalently, k3 = k4 ), the medium is supposed to possess birefringence. In contrast, the term unirefringence is used to describe those instances in which k1 = k2 (or, equivalently, k3 = k4 ). For bianisotropic Lorentz-nonreciprocal mediums (such as Faraday chiral mediums), four distinct roots – that do not come in positive/negative pairs – generally emerge from the dispersion relation (4.12). Equation (4.11) yields F 0 (ω) = adj L(ik) + iωK EH (ω) • N, (4.13)
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where N is an arbitrary 6-vector. Without the boundary/initial conditions being specified, only the relative orientations of E(r, ω) and H (r, ω) may be deduced in general. It follows from eq. (4.1) that E(r, ω) = E 0 (ω) exp i(k • r − ωt) (4.14) , H (r, ω) = H 0 (ω) exp i(k • r − ωt)
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wherein E 0 (ω), H 0 (ω) ∈ C3 . Furthermore, it is helpful to decompose these phasors as E 0 (ω) = E 0R (ω) + iE 0I (ω) (4.15) , H 0 (ω) = H 0R (ω) + iH 0I (ω) with E 0R,0I (ω), H 0R,0I (ω) ∈ R3 . The relative orientations and magnitudes of E 0R (ω) and E 0I (ω) determine the polarization state of the electric field: • if E 0R (ω) × E 0I (ω) = 0 then the electric field is linearly polarized; • if E 0R (ω) • E 0I (ω) = 0 and |E 0R (ω)| = |E 0I (ω)| then the electric field is circularly polarized; • otherwise the electric field is elliptically polarized. The polarization state of the magnetic field is similarly determined by the relative orientations and magnitudes of H 0R (ω) and H 0I (ω). In optics, the ‘polarization state’ of a plane wave is that of its electric field. Although the derivation of the k roots of eq. (4.12) and the associated eigenvectors F 0 (ω) is mathematically straightforward, the process can involve unwieldy expressions, especially for bianisotropic mediums. Symbolic manipulations packages – such as Mathematica™ and Maple™ – can be usefully employed in the analysis.
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4.3. Isotropic scenarios 4.3.1. Dielectric–magnetic mediums Isotropic dielectric–magnetic mediums, as characterized by the Tellegen constitutive relations (3.3), are unirefringent. The corresponding dispersion relation (4.12) yields two roots: 1/2 k1 = k2 = ω (ω)μ(ω) (4.16) 1/2 . k3 = k4 = −ω (ω)μ(ω) As regards the orientations of E(r, ω) and H (r, ω), a straightforward manipulation of the Maxwell curl postulates (4.7) delivers the orthogonality relations k • E(r, ω) = k • H (r, ω) = 0.
(4.17)
Plane waves in isotropic dielectric–magnetic mediums can be arbitrarily polarized: linearly, circularly, or elliptically. The time-averaged Poynting vector, defined in eq. (2.65), simplifies to 2 ! exp(−2kI kˆ • r) S(r, ω) t = E 0 (ω) kR kˆ 2ωμ(ω)
(4.18)
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for isotropic dielectric–magnetic mediums, and therefore is always aligned with the wavevector k. 4.3.2. Isotropic chiral mediums Isotropic chiral mediums, characterized by the Tellegen constitutive relations (3.5), are birefringent. The wavenumbers which emerge from the dispersion relation (4.12) are 1/2 k1 = −k3 = ω (ω)μ(ω) − iχ(ω) (4.19) 1/2 . + iχ(ω) k2 = −k4 = ω (ω)μ(ω) To determine the associated fields, it is mathematically convenient to introduce the Beltrami fields (Lakhtakia [1994])
⎫ 1 μ(ω) 1/2 ⎪ ⎪ Q 1 (r, ω) = H (r, ω) ⎪ E(r, ω) + i ⎬ 2 (ω) (4.20)
⎪ . 1/2 1 (ω) ⎪ ⎪ Q 2 (r, ω) = E(r, ω) ⎭ H (r, ω) + i 2 μ(ω) Thereby, a simplified representation of the Maxwell curl postulates (4.7) is delivered in the form of the two uncoupled first-order differential equations ∇ × Q 1 (r, ω) − k1 Q 1 (r, ω) = 0 . (4.21) ∇ × Q 2 (r, ω) + k2 Q 2 (r, ω) = 0
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Thus, we find that plane waves in an isotropic chiral medium must be circularly polarized. The left circular polarization state is represented by Q 1 (r, ω), and the right circular polarization state by Q 2 (r, ω). Furthermore, since k1 = k2 , the left and right circularly polarized plane waves propagate with different phase velocities and with different attenuation rates. Accordingly, isotropic chiral mediums are said to be circularly birefringent. A Beltrami description is also possible for plane waves in homogeneous biisotropic mediums characterized by eqs. (3.6), as delineated by Chambers [1956].
4.4. Anisotropic scenarios 4.4.1. Uniaxial mediums We commence our survey of uniform plane-wave propagation in homogeneous anisotropic mediums with the uniaxial dielectric medium, characterized by the Tellegen constitutive relations (3.7) with permittivity dyadic (3.8). The distin-
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[3, § 4
guished axis uˆ in eq. (3.8) is parallel to the sole crystallographic axis of the medium. The wavenumbers are extracted from the dispersion relation (4.12) as ⎫ 1/2 ⎪ k1 = −k3 = ω (ω)μ0 ⎪ ⎬
1/2 . (ω)u (ω)μ0 ⎪ k2 = −k4 = ω ⎪ ⎭ kˆ • (ω) • kˆ
(4.22)
uni
Two distinct modes of plane-wave propagation are supported: ordinary as specified by the wavenumber k1 = −k3 , and extraordinary as specified by the wavenumber k2 = −k4 . The phase speed of an extraordinary plane wave depends on the direction of propagation, whereas all ordinary plane waves have the same phase speed. ˆ we see from For the special propagation direction specified by kˆ = u, eqs. (4.22) that k1 = k2 (and, equivalently, k3 = k4 ). Therefore, parallel to ±uˆ both plane waves propagate with only one phase speed. The unique direction specified by uˆ is called the optic axis. The optic axis for any uniaxial dielectric medium coincides with the crystallographic axis. The relative orientations of k, E(r, ω) and H (r, ω) may be determined from eq. (4.13). For both ordinary and extraordinary plane waves, H (r, ω) • k = 0, D(r, ω) • k = 0 and B(r, ω) • k = 0. Whereas E(r, ω) • k = 0 for ordinary plane waves, E(r, ω) • k = 0 for extraordinary plane waves in general. The time-averaged Poynting vector for the uniaxial dielectric medium with permittivity dyadic (3.8) is given by (Chen [1983])
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2 ! exp(−2kI kˆ • r) Re E 0 (ω) k ∗ − E 0 (ω) • k ∗ E ∗0 (ω) . S(r, ω) t = 2ωμ0 (4.23) Therefore, for ordinary plane waves the time-averaged Poynting vector is directed ˆ whereas it lies in the plane formed by kˆ and Re{[E 0 (ω) • k ∗ ]E ∗ (ω)} for along k, 0 extraordinary plane waves. Mathematically, plane-wave propagation in uniaxial magnetic mediums, as described by the Tellegen constitutive relations (3.10) with permeability dyadic (3.11), is isomorphic to plane-wave propagation in uniaxial dielectric mediums. Significantly different plane-wave properties are exhibited by uniaxial dielectric–magnetic mediums, as specified by the Tellegen constitutive relations (3.13) with permittivity and permeability dyadics (3.8) and (3.11), respectively. The dis-
3, § 4]
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163
persion relation (4.12) yields the four solutions 1/2 ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ uni
1/2 . ⎪ (ω)μ(ω)μu (ω) ⎪ ⎪ k2 = −k4 = ω ⎪ ⎭ ˆk • μ (ω) • kˆ
k1 = −k3 = ω
(ω)u (ω)μ(ω) kˆ • (ω) • kˆ
(4.24)
uni
All wavenumbers are dependent on the direction of propagation – i.e., unlike uniaxial dielectric and uniaxial magnetic mediums, there is no ‘ordinary’ plane-wave ˆ mode which propagates with the same phase speed in all directions. When kˆ = u, we see from eqs. (4.24) that k1 = k2 (and, equivalently, k3 = k4 ). Thus, as is true for uniaxial dielectric and uniaxial magnetic mediums, incidental unirefringence occurs for propagation parallel to the crystallographic axis u. ˆ In addition, we observe that pathological unirefringence, characterized by k1 = k2 (and, equivaˆ arises in the special case, identified by (Lakhtakia, lently, k3 = k4 ) for all k, Varadan and Varadan [1991]) μ(ω) (ω) = . u (ω) μu (ω)
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(4.25)
4.4.2. Biaxial mediums Next, let us consider a biaxial dielectric medium characterized by the Tellegen constitutive relations D(r, ω) = bi (ω) • E(r, ω) , (4.26) B(r, ω) = μ0 H (r, ω) where the symmetric permittivity dyadic bi (ω) may be of the orthorhombic, monoclinic or triclinic type, as listed in table 2. The corresponding dispersion relation (4.12) delivers the biquadratic polynomial (Michel [1997]) abi k 4 + bbi k 2 + cbi = 0,
(4.27)
with ⎫ ⎪ abi = kˆ • bi (ω) • kˆ ⎪ ⎬ 2 ˆ ˆ ˆ ˆ • • • • • bbi = ω μ0 k bi (ω) bi (ω) k − k bi (ω) k tr bi (ω) . (4.28) ⎪ ⎪ ⎭ c = ω4 μ2 det (ω) bi
0
bi
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Electromagnetic fields in linear bianisotropic mediums
Two, generally independent, wavenumbers emerge from eq. (4.27): ⎫
2 − 4a c )1/2 1/2 −bbi + (bbi ⎪ bi bi ⎪ ⎪ k1 = −k3 = ⎬ 2abi .
1/2 2 − 4a c )1/2 ⎪ −bbi − (bbi ⎪ bi bi ⎪ ⎭ k2 = −k4 = 2abi
[3, § 4
(4.29)
For further understanding of plane-wave propagation in biaxial dielectric mediums, it is illuminating to confine ourselves to the realm of crystal optics, wherein the components of the permittivity dyadic bi (ω) are real-valued. By manipulating eq. (4.11) and its complex conjugate, we find that (Chen [1983])
k4 ˆ ˆ • (ω) k k • E 0 (ω) × E ∗0 (ω) = 0 I− (4.30) det bi (ω) bi and
I−
k 4 kˆ • bi (ω) • kˆ det bi (ω)
kˆ kˆ
•
H 0 (ω) × H ∗0 (ω) = 0.
(4.31)
Hence, E 0 (ω) × E ∗0 (ω) = 0 and H 0 (ω) × H ∗0 (ω) = 0 unless the determinants of the coefficient dyadics on the left sides of eqs. (4.30) and (4.31), respectively, are null-valued. The following two central attributes of plane-wave propagation in nondissipative biaxial dielectric mediums may thus be deduced: • Plane waves associated with the distinct wavenumbers k1 and k2 (or, equivalently, k3 and k4 ) are linearly polarized with mutually orthogonal planes of polarization. • For two directions of propagation, specified implicitly by
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2 bbi − 4abi cbi = 0,
(4.32)
the wavenumbers k1 and k2 (and, equivalently, k3 and k4 ) coincide. Along these two directions – known as the optic axes – plane waves can have any polarization state. Unlike uniaxial dielectric mediums and uniaxial magnetic mediums, the optic axes of a biaxial medium do not coincide with its crystallographic axes. The time-averaged Poynting vector (Chen [1983]) ! S(r, ω) t =
ω(k • E 0 )2 k • bi (ω) • k k + ω2 μ0 bi (ω) 2 2 2 det[ω μ0 bi (ω) − k I ] • (ω) − tr (ω) I + k 2 (ω) • k (4.33) bi bi bi
is not generally aligned with the wavevector k.
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The mathematical description of plane-wave propagation in biaxial magnetic mediums, as described by the Tellegen constitutive relations (3.20) with permeability dyadic being of the orthorhombic, monoclinic or triclinic type, as specified in eqs. (3.21), is equivalent to that for biaxial dielectric mediums. The Tellegen constitutive relations (3.22) specify a biaxial dielectric–magnetic medium. The increased parameter space associated with such mediums, as compared with biaxial dielectric mediums or biaxial magnetic mediums, gives rise to plane-wave propagation with a richer palette of attributes. For example, under certain pathological conditions, biaxial dielectric–magnetic mediums may exhibit only one optic axis (Shen, Win, Chen, Fan, Ding, Wang, Tian and Ming [2005]). 4.4.3. Gyrotropic mediums Turning now to an example of a Lorentz-nonreciprocal anisotropic medium, we focus on the gyrotropic dielectric medium characterized by the Tellegen constitutive relations (3.30) wherein the components of the antisymmetric permittivity dyadic ˆ + ig (ω)uˆ × I + u (ω)uˆ uˆ gyro (ω) = (ω)(I − uˆ u)
(4.34)
are complex-valued, in general. Thus, the dispersion relation (4.12) simplifies to the biquadratic polynomial (Chen [1983])
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agyro k + bgyro k + cgyro = 0, 4
with
2
⎫ ⎪ agyro = kˆ • gyro (ω) • kˆ ⎪ ⎪ ⎬ 2 ˆ ˆ bgyro = ω μ0 k • adj gyro (ω) − tr adj gyro (ω) I • k . ⎪ ⎪ ⎪ ⎭ cgyro = ω4 μ20 det gyro (ω)
Two, generally independent, wavenumbers emerge therefrom:
⎫ 2 −bgyro + (bgyro − 4agyro cgyro )1/2 1/2 ⎪ ⎪ ⎪ k1 = −k3 = ⎬ 2agyro .
2 −bgyro − (bgyro − 4agyro cgyro )1/2 1/2 ⎪ ⎪ ⎪ ⎭ k2 = −k4 = 2agyro
(4.35)
(4.36)
(4.37)
The following two special scenarios are worthy of particular mention: • For propagation either co-parallel or antiparallel to the biasing magnetic field (cf. § 3.2.3), kˆ • uˆ = ±1; hence, 1/2 k1 = −k3 = ω2 μ0 (ω) − g (ω) (4.38) 1/2 . k2 = −k4 = ω2 μ0 (ω) + g (ω)
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When dissipation is negligibly small, we may set , g , u ∈ R; and the upper and lower wavenumbers specified by eqs. (4.38) then correspond to left and right circularly polarized plane waves, respectively. The difference in phase speeds associated with the wavenumbers k1 and k2 (or, equivalently, k3 and k4 ) results in a rotation of the plane of polarization known as Faraday rotation. • For propagation perpendicular to the biasing magnetic field, the wavenumbers ⎫
2 (ω) − g2 (ω) 1/2 ⎪ 2 ⎬ k1 = −k3 = ω μ0 (ω) (4.39) ⎪ 1/2 ⎭ 2 k2 = −k4 = ω μ0 u (μ) emerge by using kˆ • uˆ = 0 in eqs. (4.37). Whereas k1 and k3 depend upon the strength of the biasing magnetic field, k2 and k4 do not. On ignoring dissipation (i.e., setting , g , u ∈ R), it transpires that the upper and lower wavenumbers specified by eqs. (4.39) correspond to elliptically and linearly polarized waves, respectively. The mathematical description of plane-wave propagation for gyrotropic magnetic mediums is equivalent to that for gyrotropic dielectric mediums. 4.4.4. Voigt waves
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As described in §§ 4.4.1–4.4.3, two distinct plane waves generally propagate in each direction in anisotropic mediums. We close our consideration of plane-wave propagation in anisotropic mediums with an anomalous example wherein the two plane waves coalesce to form a Voigt wave. Experimental observations of these waves were reported by Woldemar Voigt [1902] more than 100 years ago. While the theoretical basis for Voigt waves was developed in the 1950s (Pancharatnam [1958]), this topic has attracted renewed attention lately as a consequence of advances in metamaterial technologies. Recent work has demonstrated the possibility of constructing complex composite materials which support the propagation of Voigt waves, using component materials which do not themselves support Voigtwave propagation (Mackay and Lakhtakia [2003]). Let us take a general anisotropic dielectric medium, specified by the Tellegen constitutive relations D(r, ω) = aniso (ω) • E(r, ω) , (4.40) B(r, ω) = μ0 H (r, ω) with complex-valued permittivity dyadic ⎛ ⎞ 11 (ω) 12 (ω) 13 (ω) aniso (ω) = ⎝ 21 (ω) 22 (ω) 23 (ω) ⎠ , 31 (ω) 32 (ω) 33 (ω)
(4.41)
3, § 4]
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as the setting for our analysis. For definiteness – and without loss of generality – we examine uniform plane-wave propagation along the direction of the z axis; i.e., kˆ = zˆ . Thus, focussing on the electric field phasor E(r, ω), we consider plane waves of the form ⎡ ⎤ ⎡ ⎤ E0x (ω) Ex (z, ω) E(r, ω) ≡ ⎣ Ey (z, ω) ⎦ = ⎣ E0y (ω) ⎦ exp i(kz − ωt) . (4.42) Ez (z, ω) E0z (ω) Combining eqs. (4.40) and (4.11), we eliminate E0z (ω) to obtain
ω2 μ0 δ11 δ12 E0x (ω) E0x (ω) • = k2 , E0y (ω) E0y (ω) 33 (ω) δ12 δ22 where
⎫ δ11 = 11 (ω)33 (ω) − 13 (ω)31 (ω)⎪ ⎪ ⎪ ⎪ δ = (ω) (ω) − (ω) (ω)⎬ 12
12
33
13
32
δ21 = 21 (ω)33 (ω) − 23 (ω)31 (ω)⎪ ⎪ ⎪ ⎪ ⎭ δ22 = 22 (ω)33 (ω) − 23 (ω)32 (ω)
.
(4.43)
(4.44)
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The matrix on the left side of eq. (4.43) has only one eigenvalue but two linearly independent eigenvectors for isotropic dielectric mediums. The normal situation for anisotropic mediums is that of birefringence, wherein the matrix in eq. (4.43) has two distinct eigenvalues and two independent eigenvectors. The general solution for the x and y components of E(r, ω) may then be expressed as
Ex (z, ω) Ex1 (ω) = C1 exp(ik1 z) Ey (z, ω) Ey1 (ω)
Ex2 (ω) exp(ik2 z) exp(−iωt). + C2 (4.45) Ey2 (ω) Herein, the eigenvectors [Ex1 , (ω), Ey1 (ω)]T and [Ex2 , (ω), Ey2 (ω)]T correspond to the two wavenumbers ⎫ 1/2 1/2 ⎪ μ0 2 ⎪ ⎪ (δ11 + δ22 ) + (δ11 − δ22 ) + 4δ12 δ21 k1 = ω ⎬ 233 (ω) 1/2 ⎪ , 1/2 μ0 ⎪ ⎪ ⎭ k2 = ω (δ11 + δ22 ) − (δ11 − δ22 )2 + 4δ12 δ21 233 (ω) (4.46) respectively, while C1,2 are amplitude coefficients which may be determined from boundary/initial conditions.
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Anomalously, the matrix in eq. (4.43) can have only one independent eigenvector – and therefore only one eigenvalue k. Accordingly, the matrix cannot be a scalar matrix. In this case, the general solution for the x and y components of E(r, ω) may be expressed as
Ex1 (ω) Ex2 (ω) Ex (z, ω) = C1 + ikzC2 exp i(kz − ωt) , Ey (z, ω) Ey1 (ω) Ey2 (ω) (4.47) which represents a Voigt wave. A prominent distinguishing feature of the solution (4.47) is that the plane-wave amplitude has a linear dependence on propagation distance. Sufficient conditions for Voigt-wave propagation are (Gerardin and Lakhtakia [2001]) • (δ11 − δ22 )2 + 4δ12 δ21 = 0, and • |δ12 | + |δ12 | = 0. These conditions cannot be satisfied by uniaxial dielectric mediums, but can be satisfied by certain non-orthorhombic biaxial dielectric mediums and gyrotropic mediums (Agranovich and Ginzburg [1984]). Furthermore, in connection with the remaining subsections in this section, we note that Voigt-wave propagation has also been investigated in certain bianisotropic (Berry [2005]) and periodically nonhomogeneous mediums (Lakhtakia [1998a]).
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4.5. Bianisotropic scenarios As the parameter space associated with bianisotropic mediums is much larger than that associated with anisotropic mediums, an extremely rich and diverse array of plane-wave propagation characteristics can be exhibited by bianisotropic mediums. We consider two examples of plane-wave propagation in bianisotropic scenarios. The first is plane-wave propagation in a bianisotropic medium which is isotropic dielectric–magnetic when viewed by an observer in a co-moving reference frame. As indicated in § 4.5.1, this bianisotropic medium is in fact unirefringent. The second is the Faraday chiral medium, which can support plane waves with four distinct wavenumbers for propagation along an arbitrary axis. Both examples are of Lorentz-nonreciprocal mediums. The dispersion relation (4.12) reduces to a polynomial that is quadratic in k 2 for all Lorentz-reciprocal mediums, whether bianisotropic or not. Consequently, Lorentz-reciprocal bianisotropic mediums support plane waves with at most two independent wavenumbers for propagation along any axis.
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4.5.1. Mediums moving at constant velocity Suppose an isotropic dielectric–magnetic medium is characterized by the Tellegen constitutive relations D (r , ω ) = Σ (ω )E (r , ω ) (4.48) B (r , ω ) = μ Σ (ω )H (r , ω ) in an inertial reference frame Σ , which moves at a constant velocity v = v vˆ relative to the inertial reference frame Σ. A plane wave, described by the phasor F (r , ω ) = F 0 (ω ) exp i(k • r − ω t ) (4.49) with respect to the reference frame Σ , is described by eq. (4.1) with respect to the reference frame Σ. Herein {t, r} and {t , r } are related via the Lorentz transformations (2.32), and (Chen [1983]) ωv • k = γ k vˆ − 2 vˆ + (I − vˆ vˆ ) • k, (4.50) c0 ω = γ (ω − k • v).
(4.51) Σ ,
Since the medium is unirefringent with respect to it is clear from eq. (4.50) that the medium is also unirefringent with respect to Σ. Furthermore, even though the plane wave may be taken to be uniform with respect to Σ (i.e., k = k kˆ with kˆ ∈ R3 ), we see from eq. (4.50) that the plane wave is generally nonuniform with respect to Σ.
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4.5.2. Faraday chiral mediums The concept of a Faraday chiral medium (FCM) was presented in § 3.3.3. In the Tellegen representation, the frequency-domain constitutive relations of a FCM are given by eqs. (3.39) with the 3 × 3 constitutive dyadics (3.40). The corresponding dispersion relation (4.12) yields a quartic polynomial in k with four distinct roots. These may be extracted by standard algebraic or numerical methods (Abramowitz and Stegun [1965]). The mathematical description of plane-wave propagation in FCMs simplifies considerably in the particular case that kˆ = u; ˆ i.e., propagation along the direction of the biasing magnetic field. The four wavenumbers 1/2 1/2 ⎫ k1 = ω (ω) + g (ω) μ(ω) + μg (ω) − ξ(ω) − ξg (μ) ⎪ ⎪ ⎪ 1/2 1/2 ⎪ ⎪ μ(ω) + μg (ω) − ξ(ω) − ξg (μ) ⎬ k2 = ω − (ω) + g (ω) 1/2 1/2 ⎪, μ(ω) − μg (ω) + ξ(ω) − ξg (μ) ⎪ k3 = ω (ω) − g (ω) ⎪ ⎪ 1/2 1/2 ⎪ ⎭ μ(ω) − μg (ω) + ξ(ω) − ξg (μ) k4 = ω − (ω) − g (ω) (4.52)
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[3, § 4
then emerge from the dispersion relation (4.12). Lorentz-nonreciprocity is evident from the fact that k1 + k3 = 0 and k2 + k4 = 0. The orientations of the electromagnetic field phasors are determined by analyzing eq. (4.13): thereby, it is found that there is no component of the electric field in the direction of u, ˆ whereas the time-averaged Poynting vector is aligned parallel to uˆ (Mackay and Lakhtakia [2004]).
4.6. Nonhomogeneous mediums Let us now discuss propagation in two types of nonhomogeneous mediums. The first is the general class of helicoidal bianisotropic mediums (HBMs). These represent a wide range of technologically important materials such as CLCs and CSTFs. The second is gravitationally affected vacuum, which is of particular interest in view of the recent discovery that certain space–time metrics can support propagation with negative phase velocity – a topic that will be discussed further in § 4.7. The mediums discussed in §§ 4.6.1 and 4.6.2 are atypical of nonhomogeneous mediums insofar as analytical (or semi-analytical) methods may be fruitfully deployed to explore the characteristics of plane-wave propagation therein. Generally, wave-propagation studies for nonhomogeneous mediums pose formidable challenges to theoreticians, particularly if closed-form solutions are sought.
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4.6.1. Periodic nonhomogeneity Let us recall from § 3.4.1 that the constitutive relations for a HBM with its helicoidal axis aligned with the Cartesian z axis may stated in the Tellegen representation as eqs. (3.48), with constitutive dyadics (3.49) and (3.50). On combining the constitutive relations (3.48) and the Maxwell curl postulates (4.7) we find ∇ × E(r, ω) = iω ζ (z, ω) • E(r, ω) + μ (z, ω) • H (r, ω) HBM HBM . (z, ω) • H (r, ω) ∇ × H (r, ω) = −iω HBM (z, ω) • E(r, ω) + ξ HBM
(4.53) As a step towards eliminating the z-dependence on the right side of (4.53) for propagation along the helicoidal axis, the Oseen transformation (Oseen [1933]) (z) • E(r, ω) E (r, ω) = S −1 z , (4.54) H (r, ω) = S −1 (z) • H (r, ω) z
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171
is implemented. Thereby, eqs. (4.53) are recast as S −1 (z) • (∇ × I ) • S z (z) • E (r, ω) z (z, ω) • E (r, ω) + μ = iω ζ
⎫ ⎪ ⎪ ⎪ ⎬ (z, ω) • H (r, ω) ⎪
HBM HBM . −1 ⎪ • • • S z (z) (∇ × I ) S z (z) H (r, ω) ⎪ ⎪ ⎪ ⎭ (z, ω) • H (r, ω) = −iω HBM (z, ω) • E (r, ω) + ξ HBM
(4.55)
Upon introducing the Fourier representations E (r, ω) = ex (z, ω)xˆ + ey (z, ω)yˆ + ez (z, ω)ˆz exp(iκx) , H (r, ω) = h x (z, ω)xˆ + h y (z, ω)yˆ + h z (z, ω)ˆz exp(iκx)
(4.56)
wherein κ may be interpreted as a wavenumber in the xy plane, eqs. (4.55) generally reduce to the 4 × 4 matrix differential equation ∂ F (z, ω) = M (z, ω)F (z, ω). ∂z
(4.57)
Herein,
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T
F (z, ω) = ex (z, ω), ey (z, ω), h x (z, ω), h y (z, ω)
(4.58)
is a column 4-vector, and the 4 × 4 matrix function M (z, ω) – which is too cumbersome to reproduce explicitly here – may be conveniently expressed as
M (z) = A + κ
C exp 1,1
iπz Ω
+ C 1,−1 exp
iπz − Ω
i2πz i2πz + C 2,0 + C 2,−2 exp − , (4.59) + κ 2 C 2,2 exp Ω Ω where the 4 × 4 matrixes A , C 1,±1 , C 2,0 , and C 2,±2 are independent of z and κ but not of Ω and ω. For axial propagation, κ = 0 and the closed-form solution to eq. (4.57) arises as (Lakhtakia and Weiglhofer [1995]) F (z, ω) = exp(iA z)F (0, ω).
(4.60)
For nonaxial propagation, κ = 0 and the solution to eq. (4.57) may be expressed in terms of a power series in z through exploiting the representation (4.59) (Lakhtakia and Weiglhofer [1997a]). Alternatively, a piecewise uniform approximation may be implemented (Lakhtakia and Messier [2005]).
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[3, § 4
4.6.2. Gravitationally affected vacuum As described in § 3.4.2, the electromagnetic properties of vacuum in generally curved space–time are equivalent to those of a fictitious, instantaneously responding, bianisotropic medium. The time-domain constitutive relations for the fictitious medium (3.51) are expressed in terms of the 3×3 dyadic γ˜ (r, t) and 3-vector Γ˜ (r, t), which are derived from the space–time metric and specified in eqs. (3.52) and (3.53), respectively. The nonhomogeneous nature of the equivalent medium represented by eq. (3.51) may be effectively dealt with by implementing a piecewise uniform approximation (Lakhtakia, Mackay and Setiawan [2005]). Attention is focussed on a space– time neighbourhood which is sufficiently small so that the non-uniform quantities γ˜ (r, t) and Γ˜ (r, t) may be replaced by their uniform equivalents γ¯ and Γ¯ , respectively. Thereby, the propagation of plane waves within the neighbourhood may be analysed using the methods described in § 4.2. Global solutions may then be developed by stitching together the plane-wave solutions from adjacent space–time neighbourhoods. This piecewise uniform approximation technique is widely used in solving differential equations with nonhomogeneous coefficients (Hoffman [1992]) and also for nonaxial propagation in CLCs and CSTFs (Lakhtakia and Messier [2005]). Let us now proceed to plane-wave solutions ˜ t) = Re E 0 (ω) exp i(k • r − ωt) E(r, (4.61) , H˜ (r, t) = Re H 0 (ω) exp i(k • r − ωt)
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within a neighbourhood which is sufficiently small that the non-uniform constitutive relations (3.51) may be replaced by their uniform counterparts (Lakhtakia, Mackay and Setiawan [2005]) ⎫ ˜ ˜ t) = 0 γ¯ • E(r, t) − c0−1 Γ¯ × H˜ (r, t) ⎬ D(r, (4.62) . ˜ ˜ B(r, t) = μ0 γ¯ • H˜ (r, t) + c0−1 Γ¯ × E(r, t)⎭ The phasor amplitudes E 0 (ω) and H 0 (ω) in eq. (4.61) are determined by the boundary/initial conditions. Combining the source-free Maxwell curl postulates ⎫ ∂ ˜ ⎪ t) = 0⎪ ∇ × H˜ (r, t) − D(r, ⎬ ∂t (4.63) , ⎪ ∂ ˜ ⎪ ˜ ∇ × E(r, t) + B(r, t) = 0 ⎭ ∂t with the constitutive relations (4.62), we obtain the eigenvector equation
2 ω det(γ¯ ) − p • γ¯ • p I + pp • γ¯ · E 0 (ω) = 0, (4.64) c0
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where p=k−
ω Γ¯ . c0
(4.65)
The dispersion relation
2 2 ω • • p γ¯ p − det(γ¯ ) = 0 c0
(4.66)
thereby arises, from which four roots emerge: k1 = −k3 =
k2 = −k4 =
⎫ kˆ • γ¯ • Γ¯ + (kˆ • γ¯ • Γ¯ )2 ⎪ ⎪ ⎪ ⎪ c0 kˆ • γ¯ • kˆ ⎪ ⎪ 1/2 ⎪ ⎪ ⎪ ˆ ˆ ¯ ¯ • • • • ⎬ − k γ¯ k(Γ γ¯ Γ − det γ¯ ) ω
. ⎪ kˆ • γ¯ • Γ¯ − (kˆ • γ¯ • Γ¯ )2 ⎪ ⎪ ⎪ ⎪ c0 kˆ • γ¯ • kˆ ⎪ ⎪ 1/2 ⎪ ⎪ ⎭ ˆ ˆ ¯ ¯ • • • • − k γ¯ k(Γ γ¯ Γ − det γ¯ ) ω
(4.67)
Just as described in § 4.5.1, the wavenumbers k1 and k2 (or, equivalently, k3 and k4 ) given by eqs. (4.67) are not independent, and the medium represented by eqs. (3.51) is unirefringent accordingly. The general solution to eq. (4.64) may be expressed as the sum
EBL
E 0 (ω) = Aa (ω)ˆe a (ω) + Ab (ω)ˆe b (ω),
(4.68)
wherein the complex-valued amplitude scalars Aa,b (ω) are determined from boundary/initial conditions, and the unit vectors eˆ a,b (ω) are taken as ⎫ ˆ γ¯ −1 • w ⎪ ⎪ ⎪ eˆ a (ω) = −1 ⎪ ⎪ ⎬ |γ¯ • w| ˆ (4.69) . γ¯ −1 • [p × eˆ a (ω)] ⎪ ⎪ ⎪ ⎪ eˆ b (ω) = −1 ⎪ |γ¯ • [p × eˆ a (ω)]| ⎭ ˆ • p = 0, but is otherwise The unit vector w ˆ in eqs. (4.69) is orthogonal to p, i.e., w arbitrary. In a similar vein, H 0 (ω) = Aa (ω)hˆ a (ω) + Ab (ω)hˆ b (ω),
(4.70)
with the unit vectors hˆ a,b (ω) =
1 −1 γ¯ • p × eˆ a,b (ω) . ωμ0
(4.71)
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From eqs. (4.68) and (4.70), we find that the corresponding time-averaged Poynting vector ! 1 Aa (ω)2 eˆ (ω) • γ¯ • eˆ (ω) S(r, ω) t = a a 2ωμ0 det γ¯ 2 + Ab (ω) eˆ b (ω) • γ¯ • eˆ b (ω) γ¯ • p (4.72) is aligned with γ¯ • p. Note that a local observer is cognizant only of a flat space–time, and will therefore choose a space–time coordinate system such that γ¯ = I and Γ¯ = 0 (Lakhtakia, Mackay and Setiawan [2005]). 4.7. Plane waves with negative phase velocity We bring our survey of plane-wave propagation to a close by considering the relationship between the phase velocity v p , as defined in eq. (4.5), and the timeaveraged Poynting vector S(r, ω) t , as defined in eq. (2.65). The phase velocity may be classified as being (Mackay and Lakhtakia [in press-a]) • positive if v p • S(r, ω) t > 0, and • negative if v p • S(r, ω) t < 0. Conventional materials support plane-wave propagation with positive phase velocity (PPV). However, due to recent progress in the development of novel materials and metamaterials, attention is turning increasingly towards parameter regimes giving rise to negative phase velocity (NPV). Furthermore, the concept of infinite phase velocity – which arises at the boundary between NPV and PPV propagation in isotropic dielectric–magnetic mediums – has come up (Lakhtakia and Mackay [2004b]). For isotropic dielectric–magnetic mediums, the phenomenon of NPV propagation is closely associated with negative refraction (Lakhtakia, McCall and Weiglhofer [2003], Ramakrishna [2005]). An explosion of interest in this topic took place at the beginning of this century, following an experimental report of microwave negative refraction in an effectively homogeneous metamaterial which comprised a periodic array of metallic wire and ring inclusions embossed on circuit boards (Shelby, Smith and Schultz [2001]). Subsequent efforts by experimentalists and theorists have been directed towards higher-frequency regimes, with optical negative refraction being achieved most recently, albeit with considerable associated dissipation (Dolling, Wegener, Soukoulis and Linden [2007], Shalaev [2007]). The prospect of fabricating lenses with extremely high resolving power from negatively refracting materials has fuelled much of this work (Pendry [2004]).
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3, § 5]
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175
A simple construction method for isotropic dielectric–magnetic metamaterials which support NPV propagation has recently been proposed. The method is based on the homogenization of a random assembly of two different types of spherical particle. The two types of constituent particles, type a and type b, can each be made of an isotropic, homogeneous, dielectric–magnetic material, with permittivities a and b , and permeabilities μa and μb , respectively. Provided that a,b and μa,b lie within certain parameter ranges, with the real parts of a,b being negativevalued and the real parts of μa,b being positive-valued (or vice versa), the bulk constituent materials do not support NPV propagation whereas the corresponding homogenized composite medium does. Whether or not NPV propagation is supported by the homogenized composite medium depends upon the relative proportions of the constituents, as well as the size (Lakhtakia and Mackay [2006]) and spatial distribution (Mackay and Lakhtakia [2006]) of the their particles. As compared with isotropic dielectric–magnetic mediums, anisotropic and bianisotropic mediums offer greater scope for achieving NPV propagation on account of their larger parameter spaces (Hu and Lin [2003], Mackay and Lakhtakia [2004]). In particular, bianisotropic homogenized composite mediums which support NPV propagation may be conceptualized – as Faraday chiral mediums (Mackay and Lakhtakia [2004]), for example – which arise from constituent mediums which do not themselves support NPV propagation. There are also noteworthy manifestations of NPV in astrophysical scenarios, with potentially important consequences for observational astronomy. The property of supporting NPV propagation is not Lorentz covariant (cf. § 2.5.3). Thus, a medium may support PPV propagation from the perspective of one inertial observer, but support NPV propagation from the perspective of another inertial observer (Mackay and Lakhtakia [in press-a]). In addition, vacuum itself – which does not support NPV propagation in any inertial reference frame in flat space– time – can support NPV propagation for certain curved space–times, such as the Kerr space–time (Mackay, Lakhtakia and Setiawan [2005a]) and Schwarzschild– anti-de Sitter space–time (Mackay, Lakhtakia and Setiawan [2005b]). This distinction essentially arises from the differences between covariant and noncovariant formalisms (Mackay and Lakhtakia [in press-b], McCall [2007]).
EBL
§ 5. Dyadic Green functions What is the (frequency-domain) electromagnetic field generated by a given distribution of sources immersed a specific medium? Due to the linearity of the Maxwell postulates, this fundamental question may be tackled by means of dyadic Green functions (DGFs) (Tai [1994], Weiglhofer [1995]). The explicit delineation
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[3, § 5
of DGFs for complex mediums poses formidable challenges to theoreticians. Closed-form representations of DGFs are available for isotropic mediums and for some classes of relatively simple anisotropic and bianisotropic medium, but not for general anisotropic and bianisotropic mediums. However, integral representations of DGFs can always be found. For certain applications – such as in the determination of the scattering response of electrically small particles, utilized in homogenization studies – approximative solutions may be constructed using only the singular part of the DGF, which gives rise to the depolarization dyadic. In this section, we survey DGFs and depolarization dyadics for unbounded anisotropic/bianisotropic mediums. The mediums described are homogeneous, with the exception of HBMs in § 5.3.2.
5.1. Definition and properties Let Q(r, ω) defined in eq. (2.19) be the source 6-vector immersed in the homogeneous bianisotropic medium characterized by the Tellegen 6 × 6 constitutive dyadic K EH (ω). The relationship between Q(r, ω) and F(r, ω) is dictated by the frequency-domain Maxwell curl postulates L(∇) + iωK EH (ω) • F(r, ω) = Q(r, ω). (5.1)
EBL
Since (5.1) is a linear differential equation, its solution may be expressed in terms of a 6 × 6 DGF G(r − r , ω). Thus, the field at position r is given as F(r, ω) = Fcf (r, ω) + G(r − r , ω) • Q(r , ω) d3 r , (5.2) V
wherein the source points r are confined to the region of integration V . The 6-vector Fcf (r, ω) denotes the corresponding complementary function as per L(∇) + iωK EH (ω) • Fcf (r, ω) = 0. (5.3) By construction, the DGF is the solution of the differential equation L(∇) + iωK EH (ω) • G(r − r , ω) = δ(r − r )I,
(5.4)
where 1 δ(r) = 2π 3
∞ exp(iq • r) d3 q
(5.5)
−∞
is the Dirac delta function. Hence, the DGF may be viewed informally as representing the ‘response’ of the medium to a point ‘source’. Furthermore, the DGF
3, § 5]
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177
is required to satisfy the Sommerfeld radiation condition (Felsen and Marcuvitz [1994]). The 6 × 6 DGF G(r − r , ω) may be expressed in terms of its four component 3 × 3 DGFs as
ee G (r − r , ω) Gem (r − r , ω) G(r − r , ω) = (5.6) . Gme (r − r , ω) Gmm (r − r , ω) These 3 × 3 DGFs satisfy the differential equations (Weiglhofer [1995]) ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ em ⎪ L e (∇, ω) • G (r − r , ω) ⎪ ⎪ −1 ⎪ • ⎬ = − ∇ × I + iωξ (ω) μ (ω)δ(r − r )⎪ L e (∇, ω) • Gee (r − r , ω) = iωI δ(r − r )
EH
EH
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
L m (∇, ω) • Gme (r − r , ω) = ∇ × I − iωζ (ω) • −1 (ω)δ(r − r ) EH EH
•
mm
L m (∇, ω) G
(r − r , ω) = iωI δ(r − r )
(5.7)
,
wherein the linear differential operators L e (∇, ω) = ∇ × I + iωξ
(ω) μ (ω) EBL
⎫ (ω) ⎪ ⎪ ⎪ ⎪ ⎬ − ω2 EH (ω) −1 . L m (∇, ω) = ∇ × I − iωζ (ω) • EH (ω) • ∇ × I + iωξ (ω) ⎪ ⎪ ⎪ EH EH ⎪ ⎭ 2 − ω μ (ω) •
EH
−1 EH
•
∇ × I − iωζ
EH
Thus, if as
Gee (r
− r , ω)
is known, then
Gme (r − r , ω) =
EH
(5.8) Gme (r
− r , ω)
may be calculated directly
1 −1 μ (ω) • ∇ × I − iωζ (ω) • Gee (r − r , ω), EH iω EH (5.9)
in lieu of solving the differential equation for Gme (r−r , ω) in eq. (5.7). Similarly, Gem (r − r , ω) may be determined as Gem (r − r , ω) = −
1 −1 EH (ω) • ∇ × I + iωξ (ω) • Gmm (r − r , ω), EH iω (5.10)
provided that Gmm (r − r , ω) is known. Also, once one of the four 3 × 3 DGFs in eq. (5.6) known, the others can often be deduced from symmetry considerations.
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5.2. Closed-form representations A variety of ingenious methods, commonly based on dyadic operator manipulations and/or field transformations, have been devised in the pursuit of closed-form representations for DGFs (Weiglhofer [1993], Olyslager and Lindell [2002]). Often such methods have been developed on an ad hoc basis. In contrast, the following dyadic scalarization and factorization technique is generally applicable, at least in principle: Formally, the 6 × 6 DGF may be expressed as (Weiglhofer [1993]) † G(r − r , ω) = L(∇) + iωK(ω) σ (r − r , ω). (5.11) Herein, the adjoint operation indicated by the superscript † is defined implicitly via † L(∇) + iωK(ω) • L(∇) + iωK(ω) = H(∇, r, ω)I, (5.12) whereas the scalar Green function σ (r − r , ω) is the solution of H(∇, r, ω)σ (r − r , ω) = δ(r − r ),
(5.13)
EBL
with H(∇, r, ω) being a scalar fourth-order differential operator. Straightforward, albeit lengthy, matrix-algebraic operations result in the construction of the adjoint operator [L(∇) + iωK(ω)]† for anisotropic and bianisotropic mediums. However, solutions to the fourth-order partial differential equation represented by eq. (5.13) are generally elusive. The existence of closed-form expressions for DGFs is closely linked to the factorization properties of H(∇, r, ω). In many instances where H(∇, r, ω) is expressible as a product of two second-order differential operators, closed-form representations for DGFs have been established (Weiglhofer [2000], Olyslager and Lindell [2002]). A different approach for the determination of DGFs is provided by spatial Fourier transformations. Upon transforming the spatial coordinates in eq. (5.4), the differential equation is converted into a soluble algebraic equation. However, the process of inverse Fourier transformation, which is required to extract an explicit DGF representation, is generally problematic. 5.2.1. Isotropic mediums 5.2.1.1. Dielectric–magnetic mediums The 3 × 3 DGF Gee (r − r , ω) for the iso isotropic dielectric–magnetic medium described by the Tellegen constitutive relations D(r, ω) = (ω)E(r, ω) and B(r, ω) = μ(ω)H (r, ω) emerges as a result of
3, § 5]
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179
applying ∇ • from the left to both sides of eq. (5.7). Thereby, we find that
1 ee ∇∇ giso (r − r , ω), (5.14) G iso (r − r , ω) = iωμ(ω) I + 2 ω (ω)μ(ω) with the scalar Green function giso (r − r , ω) satisfying the scalar differential equation 2 ∇ + ω2 (ω)μ(ω) giso (r − r , ω) = −δ(r − r ). (5.15) The solution of eq. (5.15), namely 1/2 1 |r − r | , exp iω (ω)μ(ω) (5.16) 4π|r − r | is known very well (Chen [1983]). At locations outside the source region (i.e., r = r ), the explicit representation (Chen [1983], Van Bladel [1991])
ee ˆ iso (R, ω) G iso (R, ω) = iωμ(ω) (I − Rˆ R)g (5.17) giso (r − r , ω) =
i ˆ iso (R, ω) (I − 3Rˆ R)g ω[(ω)μ(ω)]1/2 R 1 ˆ R)g ˆ iso (R, ω) (I − 3 R − 2 ω (ω)μ(ω)R 2
+
EBL
(5.18)
follows from combining eqs. (5.14) and (5.16), with R = R Rˆ = r − r . At locations inside the source region the DGF is singular, as discussed in § 5.4. (r − r , ω), the dual DGF In view of the expression (5.14) for Gee iso (ω) ee G (r − r , ω) μ(ω) emerges immediately, whereas the expressions (r − r , ω) = Gmm iso
(5.19)
Gem (r − r , ω) = −Gme (r − r , ω) = ∇ × g iso (r − r , ω)I iso iso
(5.20)
follow from eqs. (5.9) and (5.10). 5.2.1.2. Isotropic chiral mediums The study of fields and sources in isotropic chiral mediums is efficiently carried out via the introduction of the Beltrami fields Q1 (r, ω) and Q2 (r, ω) defined in eqs. (4.20) and the corresponding Beltrami source current densities (Lakhtakia [1994])
⎫
⎪ 1 μ(ω) 1/2 ⎪ W 1 (r, ω) = J e (r, ω) − J m (r, ω) ⎪ i ⎬ 2 (ω) (5.21)
⎪ . 1 (ω) 1/2 ⎪ ⎪ W 2 (r, ω) = J m (r, ω) ⎭ J e (r, ω) − i 2 μ(ω)
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[3, § 5
Then the Maxwell curl postulates can be set down as ∇ × Q (r, ω) + (−1) k Q (r, ω) = W (r, ω) ( = 1, 2),
(5.22)
with wavenumbers 1/2 − iχ(ω) k1 = ω (ω)μ(ω) 1/2 . k2 = ω (ω)μ(ω) + iχ(ω)
(5.23)
The solution of eq. (5.22) is Q (r, ω) = Qcf (r, ω) +
G (r − r , ω) • W (r , ω) d3 r
( = 1, 2),
V
(5.24) where
Qcf (r, ω)
is the complementary function satisfying the equation
∇ × I + (−1) k I
•
Qcf (r, ω) = 0 ( = 1, 2).
(5.25)
The dyadic Beltrami–Green functions
EBL
G (r − r , ω) = ∇ × I − (−1) k I 1 • I + ∇∇ g (r − r , ω) ( = 1, 2), k 2
(5.26)
employ the scalar Green functions g (r − r , ω) =
exp(ik |r − r |) 4π|r − r |
( = 1, 2),
(5.27)
that are isomorphic to giso (R, ω). A similar procedure is useful for biisotropic mediums characterized by eqs. (3.6), as shown by Monzon [1990]. 5.2.2. Uniaxial dielectric–magnetic mediums Let us now turn to the anisotropic dielectric–magnetic mediums characterized by a single distinguished axis, oriented arbitrarily along the direction of the unit vector u. ˆ These mediums are characterized by the constitutive relations (3.13) with permittivity and permeability dyadics (3.8) and (3.11), respectively. By a process involving the diagonalization of the linear differential operator L e (or, equivalently, L m ) combined with dyadic factorizations, the 3 × 3 DGFs for
3, § 5]
Dyadic Green functions
181
the uniaxial dielectric–magnetic medium are determined as (Weiglhofer [1990]) ⎫ ⎪ ⎪ (r − r , ω) = iωμ(ω) −T (r − r , ω) Gee ⎪ ⎪ uni ⎪
⎪ ⎪ ⎪ ⎪ ∇∇ −1 ⎪ + u (ω) uni (ω) + 2 guni (r − r , ω) ⎪ ⎪ ⎪ ω (ω)μ(ω) ⎪ ⎪ ⎪ ⎪ em −1 ⎪ G uni (r − r , ω) = −(ω) uni (ω) • (∇ × I ) ⎪ ⎪ ⎪ ⎪ μ ⎪ −1 • μu (ω)g ⎬ (r − r , ω)μ (ω) + T (r − r , ω) uni uni
⎪ Gme (r − r , ω) = −μ(ω)μ−1 (ω) • (∇ × I ) ⎪ ⎪ uni uni ⎪ ⎪ ⎪ ⎪ • −u (ω)g (r − r , ω) −1 (ω) + T (r − r , ω) ⎪ uni ⎪ uni ⎪ ⎪ ⎪ ⎪ mm ⎪ ⎪ G uni (r − r , ω) = iω(ω) T (r − r , ω) ⎪ ⎪ ⎪
⎪ ⎪ ⎪ ∇∇ ⎪ μ −1 ⎭ + μu (ω)μ (ω) + 2 guni (r − r , ω) ⎪ uni ω (ω)μ(ω)
.
(5.28) The scalar Green functions η
guni (R, ω) =
,μ guni (r
− r , ω)
in eqs. (5.28) are defined by
EBL
exp{iω[(ω)μ(ω)]1/2 [ηu (ω)R • η−1 (ω) • R]1/2 } uni
4π[ηu (ω)R • η−1 (ω) • R]1/2 uni
(η = , μ),
(5.29)
while the dyadic T (r − r , ω) is specified as4 T (R, ω) =
ˆ × u) ˆ u (ω) (R × u)(R μu (ω) μ (R, ω) − (R, ω) g g (ω) uni μ(ω) uni (R × u) ˆ 2
1 ˆ × u) ˆ 2(R × u)(R + I − uˆ uˆ − iω[(ω)μ(ω)]1/2 (R × u) ˆ 2 (R × u) ˆ 2 1/2 (ω) • R × guni (R, ω) u (ω)R • −1 uni 1/2 μ . − guni (R, ω) μu (ω)R • μ−1 (ω) • R (5.30) uni
The specializations appropriate to uniaxial dielectric mediums and uniaxial magnetic mediums follow straightforwardly from eqs. (5.28), by implementing the substitutions μ (ω) = μ0 I and uni (ω) = 0 I , respectively. uni
4 The T (r, ω) defined by eq. (5.30) is not related to the Maxwell stress dyadic T˜ (r, t) of eq. (2.43).
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5.2.3. More complex mediums For gyrotropic dielectric mediums and gyrotropic magnetic mediums, i.e., those with permittivity and permeability dyadics of the form η
gyro
(ω) = η(ω)(I − uˆ u) ˆ + iηg (ω)uˆ × I + ηu (ω)uˆ uˆ (η = , μ), (5.31)
respectively, DGFs in terms of one-dimensional integrals involving cylindrical functions are available, but closed-form representations are not (Weiglhofer [1994]). Similarly, some analytical progress towards closed-form DGFs for general uniaxial bianisotropic mediums described by the Tellegen constitutive dyadics η
uni
(ω) = η(ω)(I − uˆ u) ˆ + ηu (ω)uˆ uˆ
(η = , ξ, ζ, μ),
(5.32)
has been reported, but has not yet reached a satisfactory degree of conclusion (Weiglhofer and Lindell [1994], Olyslager and Lindell [2002]). A noteworthy example of a bianisotropic scenario for which a closed-form DGF is available arises within the context of gravitationally affected vacuum. As discussed in §§ 3.4.2 and 4.6.2, the electromagnetic properties of vacuum in generally curved space–time are equivalent to those of a fictitious, instantaneously responding, nonhomogeneous, bianisotropic medium. Within a space–time neighbourhood which is small compared to the curvature of space–time, in a noncovariant formalism the nonhomogeneous bianisotropic medium may be approximated by the homogeneous bianisotropic medium with time-domain constitutive relations (4.62). On using the frequency-domain equivalent of eqs. (4.62), the corresponding DGFs may be derived by the following two-step procedure. First, by applying the field-source transformations F (r, ω) = F(r, ω) exp −iω(0 μ0 )1/2 Γ¯ • r (5.33) , Q (r, ω) = Q(r, ω) exp −iω(0 μ0 )1/2 Γ¯ • r
EBL
the problem simplifies to that for an orthorhombic biaxial dielectric–magnetic medium. Second, under the affine transformations F (r, ω) = Diag γ¯ 1/2 , γ¯ 1/2 • F γ¯ 1/2 • r, ω (5.34) , Q (r, ω) = Diag adj γ¯ 1/2 , adj γ¯ 1/2 • Q γ¯ 1/2 • r, ω wherein γ¯ 1/2 • γ¯ 1/2 = γ¯ , the problem further simplifies to that for an isotropic dielectric–magnetic medium. Thus, the 3 × 3 DGFs emerge as (Lakhtakia and
3, § 5]
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183
Mackay [2005]) ⎫ 1 • ∇∇ ⎪ ⎪ γ ¯ ⎪ ⎪ ω2 0 μ0 det γ¯ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ × gGAV (r − r , ω) ⎪ ⎪ ⎪ ⎪ ⎪ em −1 • 1/2 ¯ ⎬ G GAV (r − r , ω) = −γ¯ ∇ × I − ω(0 μ0 ) Γ × I , ⎪ • Gee ⎪ (r − r , ω) ⎪ GAV ⎪ ⎪ ⎪ ⎪ em ⎪ ⎪ Gme (r − r , ω) = −G (r − r , ω) ⎪ GAV GAV ⎪ ⎪ ⎪ ⎪ 0 ee ⎪ mm ⎭ G GAV (r − r , ω) = G GAV (r − r , ω) μ0 (5.35)
1/2 • adj γ ¯ (r − r , ω) = iωμ I+ Gee 0 GAV
with the scalar Green function gGAV (r − r , ω) being given by gGAV (R, ω) =
exp{iω(0 μ0 )1/2 [Γ¯ • R + (det γ¯ )1/2 |γ¯ −1/2 • R|]}
5.2.4. Huygens principle
4π|γ¯ −1/2 • R|
.
(5.36)
EBL
The Huygens principle represents a keystone of electromagnetic scattering theory (Brau [2004]). By means of this principle, the electromagnetic fields in a source-free region may be related to the DGF and the tangential field components on a closed surface enclosing the source region. The Huygens principle may be applied, for example, to diffraction from an aperture, wherein the aperture is formally represented as an equivalent source (Chen [1983]); likewise, it can be used to formulate the Ewald–Oseen extinction theorem and the T matrix method (Lakhtakia [1994]). Due to the general scarcity of DGFs in closed form, exact formulations of the Huygens principle are available only for isotropic mediums (Lakhtakia [1994]) and the simplest of anisotropic mediums (Ogg [1971], Lakhtakia, Varadan and Varadan [1989], Tai [1994]).
5.3. Eigenfunction representations 5.3.1. Homogeneous mediums As an alternative to the closed-form representations considered in § 5.2, DGFs may be expressed in terms of expansions of eigenfunction solutions of eq. (5.4).
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Thereby, the spectral representation (Kong [1972]) G(R, ω) =
1 2π
3
ˇ ω) exp(iq • R) d3 q G(q,
(5.37)
q
emerges. Although closed-form representations of DGFs for anisotropic and bianisotropic mediums are relatively scarce, the spatial Fourier transform (Kong [1972]) ˇ ω) = G(R, ω) exp(−iq • R) d3 R G(q, (5.38) R
can always be found. Thus, for the homogeneous bianisotropic medium characterized by the Tellegen 6 × 6 constitutive dyadic K EH (ω), the spatial Fourier transform of eq. (5.4) yields ˇ ω) 1 adj A(q, ˇ G(q, , ω) = ˇ iω det A(q, ω) where ˇ A(q, ω) =
0 −(q/ω) × I
(5.39)
EBL + K
(q/ω) × I 0
EH
(ω).
(5.40)
Whereas the analytic properties of the inverse Fourier transform G(R, ω) =
1 2π
3
1 iω
q
˜ adj[A(q, ω)] exp(iq • R) d3 q ˜ det[A(q, ω)]
(5.41)
have been determined (Cottis and Kondylis [1995]), numerical techniques are generally needed to explicitly evaluate the right side of eq. (5.41). Equation (5.37) is a spectral representation in terms of plane waves. Similar representations of DGFs have been established for other sets of eigenfunctions. Most notably, expansions in terms of cylindrical vector wavefunctions (Chew [1999], Li, Kang and Leong [2001]) have been developed for certain uniaxial bianisotropic mediums (Cheng [1997], Cheng, Ren and Jin [1997], Wu and Yasumoto [1997], Tan and Tan [1998], Li, Leong, Kooi and Yeo [1999], Ren [1999]). Also, formulations are available in terms of spherical harmonics (Monzon [1989]). However, these DGF representations are exceedingly cumbersome to handle and restricted in scope, and generally require numerical implementation.
3, § 5]
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185
5.3.2. Nonhomogeneous mediums DGFs for nonhomogeneous mediums are quite painful to even contemplate. A conspicuous exception is the matrix Green function of a HBM, which may be developed as follows. Let us recall from § 4.6.1 that for a HBM with its helicoidal axis aligned with the Cartesian z axis, as characterized by the Tellegen constitutive relations (3.48), with constitutive dyadics (3.49) and (3.50), propagation in source-free regions is governed by the 4 × 4 matrix differential equation (4.57). In the presence of an Oseen-transformed source term J (z, ω), corresponding to the Oseen-transformed field term F (z, ω), the governing equation (4.57) may be re-expressed as (Lakhtakia and Weiglhofer [1997b]) ∂ F (z, ω) = M (z, ω)F (z, ω) + J (z, ω). ∂z The solution of eq. (5.42) is compactly stated as ˇ F (z, ω) = G
HBM
z
(z, ω)F (0, ω) +
(z HBM
ˇ G
(5.42)
− zs , ω)J (zs , ω) dzs ,
0
(5.43)
EBL
in terms of the spectral matrix Green function −1 ˇ G (z − zs , ω) = Z (z, ω) Z (zs , ω) . HBM
(5.44)
The matrizant Z (z, ω) in eq. (5.44) is the 4×4 matrix that satisfies the differential equation (Yakubovich and Starzhinskii [1975]) ∂ Z (z, ω) = M (z, ω)Z (z, ω), ∂z with boundary condition Z (0, ω) = I.
(5.45)
(5.46)
5.4. Depolarization dyadics Often, it is sufficient to construct approximate solutions of restricted validity. An important example – explored in detail in § 6 – is provided by the homogenization of particulate composite materials (Lakhtakia [1996, 2000a]). Here, the scattering response of an electrically small and homogeneous particle embedded within a homogeneous ambient medium is required. Let K p (ω) and K amb (ω) denote the Tellegen constitutive dyadics for the particle and the ambient mediums, respectively. Then the corresponding Maxwell curl
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Electromagnetic fields in linear bianisotropic mediums
postulates (5.1) may be written in the form L(∇) + iωK amb (ω) • F(r, ω) = Q equiv (r, ω).
[3, § 5
(5.47)
Herein, the particle is represented by an equivalent source current density iω[K amb (ω) − K p (ω)] • F(r, ω), r ∈ V , Q equiv (r, ω) = (5.48) r∈ / V , 0, which is assumed to be uniform within the region V occupied by the particle. As the particle is sufficiently small relative to all relevant electromagnetic wavelengths, the Rayleigh approximation may be implemented to estimate the uniform field F(r, ω) in V as F(r, ω) ≈ Fcf (r, ω) + D(r, ω) • Q equiv (r, ω) (r ∈ V ). The 6 × 6 dyadic D(r, ω) = G(r − r , ω) d3 r
(5.49)
(5.50)
V
is the depolarization dyadic of a region of the same shape, orientation and size as the particle (Lakhtakia [2000a]). The depolarization dyadic owes its existence to the singularity of the DGF of the ambient medium.
EBL
5.4.1. Ellipsoidal shape Consider an ellipsoidal particle with surface parameterized as r e (θ, φ) = ρU • rˆ (θ, φ),
(5.51)
where rˆ (θ, φ) is the radial unit vector specified by the spherical polar coordinates θ and φ, and ρ > 0 is a measure of the linear dimensions. The shape dyadic U is a real-valued 3 × 3 dyadic with positive eigenvalues (Lakhtakia [2000b]), the normalized lengths of the ellipsoid semi-axes being specified by the eigenvalues of U . Suppose that the particle is embedded within a homogeneous ambient medium described by 3 × 3 Tellegen constitutive dyadics amb (ω), ξ (ω), amb ζ (ω) and μ (ω); i.e., the ambient medium is generally bianisotropic. In the amb amb limit ρ → 0, the 6 × 6 depolarization dyadic for the ellipsoidal particle (5.50) may be expressed as (Michel and Weiglhofer [1997]) 2π π D(r, ω) = φ=0 θ=0
ˇ ∞ U −1 • q, ˆ ω sin θ dθ dφ, G
(5.52)
3, § 5]
Dyadic Green functions
187
with qˆ = (sin θ cos φ, sin θ sin φ, cos θ ), and wherein the spectral representation (5.41) is exploited to provide ˇ ˇ ∞ (q, ˆ ω) = lim G(q, ω). G
(5.53)
q→∞
Utilizing eq. (5.39) to expand the integrands in eq. (5.52), we have U/amb D ee (ω) D U/amb (ω) em U/amb (ω) = D , U/amb (ω) (ω) D D U/amb me mm
(5.54)
where U/amb
D λλ and
(ω) = U −1 • d λλ
U/amb
(ω) • U −1
(λλ = ee, em, me, mm), (5.55)
⎫ ⎪ ⎪ ⎪ amb ⎪ ⎪ d U/amb (ω) = q ˆ sin θ dθ dφ q ˆ ⎪ ee ⎪ 4πiωb(θ, φ) ⎪ ⎪ ⎪ φ=0 θ=0 ⎪ ⎪ ⎪ ⎪ π 2π −1 −1 ⎪ • • • • ⎪ ξ (ω) U qˆ qˆ U ⎪ amb U/amb ⎪ d em (ω) = − qˆ qˆ sin θ dθ dφ ⎪ ⎪ ⎪ 4πiωb(θ, φ) ⎪ ⎬ φ=0 θ=0 2π π qˆ • U −1 • μ
(ω) • U −1 • qˆ
EBL(ω)
⎪ ⎪ ⎪ ⎪ amb U/amb − qˆ qˆ sin θ dθ dφ ⎪ d me (ω) = ⎪ ⎪ ⎪ 4πiωb(θ, φ) ⎪ ⎪ ⎪ φ=0 θ=0 ⎪ ⎪ ⎪ ⎪ 2π π qˆ • U −1 • −1 ⎪ • • ⎪ (ω) U q ˆ ⎪ amb U/amb ⎪ d mm (ω) = qˆ qˆ sin θ dθ dφ ⎪ ⎪ ⎪ 4πiωb(θ, φ) ⎭ 2π π
qˆ • U −1 • ζ
•
U −1 • qˆ
.
φ=0 θ=0
(5.56) Herein, b(θ, φ) = qˆ • U −1 • amb (ω) • U −1 • qˆ qˆ • U −1 • μ (ω) • U −1 • qˆ amb − qˆ • U −1 • ξ (ω) • U −1 • qˆ qˆ • U −1 • ζ (ω) • U −1 • qˆ , amb
amb
(5.57) and qˆ = xˆ sin θ cos φ + yˆ sin θ sin φ + zˆ cos θ.
(5.58)
Thus, eqs. (5.56) provide the components of the integrated singularity of the DGF. The depolarization dyadic does not involve the constitutive properties of the particle that occupies V . Depolarization dyadics associated with ellipsoidal particles
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[3, § 5
which are small compared to all relevant wavelengths, but not vanishingly small, may also be derived from the spectral representation (5.41) (Mackay [2004], Cui and Mackay [2007]). Explicit expressions for the integrals in eqs. (5.56) are available for isotropic ambient mediums as well as several classes of anisotropic ambient mediums. Parenthetically, expressions for depolarization dyadics associated with certain cylindrical shapes have also been derived (Cottis, Vazouras and Spyrou [1999], Weiglhofer and Mackay [2002]). 5.4.2. Spherical shape In view of their use in the homogenization formalisms discussed in § 6, let us move on to depolarization dyadics for spherical shapes (i.e., U = I ). 5.4.2.1. Isotropic ambient mediums When the ambient medium is isotropic dielectric–magnetic, i.e., ⎫ η (ω) = η(ω)I (η = , μ)⎬ amb (5.59) , ⎭ ξ (ω) = ζ (ω) = 0 amb
amb
EBL
the corresponding 3 × 3 depolarization dyadics reduce to the well-known forms (Van Bladel [1991]) ⎫ 1 ⎪ I /iso I /iso ⎪ I , D em (ω) = 0 D ee (ω) = ⎬ 3iω(ω) (5.60) . 1 ⎪ /iso /iso ⎭ D Ime (ω) = 0, D Imm (ω) = I⎪ 3iωμ(ω) When the ambient medium is isotropic chiral, i.e., ⎫ (ω) = η(ω)I (η = , μ) ⎬ η amb , ξ (ω) = −ζ (ω) = χ(ω)I ⎭ amb
(5.61)
amb
the 3 × 3 depolarization dyadics have the scalar matrix forms ⎫ μ(ω) χ(ω) ⎪ /iso (ω) = − I , D Iem I⎪ D Iee/iso (ω) = ⎬ 3iωΥ (ω) 3iωΥ (ω) , χ(ω) (ω) ⎪ /iso /iso ⎭ D Ime (ω) = (ω) = I , D Imm I ⎪ 3iωΥ (ω) 3iωΥ (ω)
(5.62)
where Υ (ω) = (ω)μ(ω) + χ 2 (ω).
(5.63)
3, § 5]
Dyadic Green functions
189
5.4.2.2. Anisotropic ambient mediums Let us now turn to the uniaxial dielectric– magnetic scenario wherein the constitutive dyadics of the ambient medium are ⎫ (ω) = η (ω) (η = , μ)⎬ η amb uni (5.64) , ⎭ ξ (ω) = ζ (ω) = 0 amb
amb
with uni (ω) and μ (ω) as specified in eqs. (3.8) and (3.11), respectively. The uni 3 × 3 depolarization dyadics are now given by (Michel [1997]) ⎫ ˆ + Du (ω)uˆ uˆ ⎪ D Iee/uni (ω) = D (ω)(I − uˆ u) ⎪ ⎬ I /uni I /uni D em (ω) = D me (ω) = 0 (5.65) , ⎪ ⎪ ⎭ I /uni μ μ D (ω) = D (ω)(I − uˆ u) ˆ + D (ω)uˆ uˆ u
mm
with ⎫ 1 ⎪ ⎪ D (ω) = ⎪ ⎪ 2iω[η(ω) − ηu (ω)] ⎪ ⎪ ⎪ ⎪ −1 1−γη (ω) 1/2 ⎪ ⎬ γη (ω) sinh γη (ω) × 1− [1 − γη (ω)]1/2 ⎪ ⎪ ⎪ ⎪
⎪ sinh−1 1−γη (ω) 1/2 ⎪ ⎪ 1 γη (ω) ⎪ ⎪ Duη (ω) = − 1 ⎭ 1/2 iω[η(ω) − ηu (ω)] [1 − γη (ω)] η
EBL
(η = , μ),
(5.66) and γη (ω) =
ηu (ω) η(ω)
(η = , μ).
(5.67)
The depolarization dyadics relevant to uniaxial dielectric mediums and uniaxial magnetic mediums follow immediately from eqs. (5.65), upon substituting μ (ω) = μ0 I and uni (ω) = 0 I , respectively. Depolarization dyadics of uni the form (5.65) also arise for a spheroidal shape, with its rotational axis aligned with u, ˆ immersed in an isotropic dielectric–magnetic ambient medium (Mackay and Lakhtakia [2005]). If the ambient medium is either gyrotropic dielectric, i.e., ⎫ ⎪ ⎪ ⎬ ξ (ω) = ζ (ω) = 0 , amb amb ⎪ ⎪ ⎭ μ (ω) = μ I amb (ω) = gyro (ω)
amb
0
(5.68)
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Electromagnetic fields in linear bianisotropic mediums
[3, § 5
or gyrotropic magnetic, i.e., ⎫ ⎪ ⎪ ⎪ ⎬ ξ (ω) = ζ (ω) = 0 , amb amb ⎪ ⎪ ⎪ μ (ω) = μ (ω) ⎭ amb (ω) = 0 I
amb
(5.69)
gyro
(ω) specified in eq. (5.31), the skew-symmetric dyadic with gyro (ω) and μ gyro components make no contribution to the depolarization dyadics.5 Hence, the depolarization dyadics for gyrotropic dielectric mediums and gyrotropic magnetic mediums are the same as those for the corresponding uniaxial dielectric mediums and uniaxial magnetic mediums, respectively. Finally, suppose that the ambient medium is orthorhombic dielectric–magnetic; i.e., η ξ
amb amb
(ω) = ηortho (ω) bi
(ω) = ζ
amb
⎫ (η = , μ)⎬
(ω) = 0
with
⎭
(5.70)
,
EBL
ηortho (ω) = ηx (ω)xˆ xˆ + ηy (ω)yˆ yˆ + ηz (ω)ˆzzˆ bi
(η = , μ).
(5.71)
The corresponding 3 × 3 DGFs for spherical shapes are given as (Weiglhofer [1998b]) D Iee/ortho (ω) =
⎫ 1 ⎪ Dx (ω)xˆ xˆ + Dy (ω)yˆ yˆ + Dz (ω)ˆzzˆ ⎪ ⎪ ⎪ iω0 ⎪ ⎬
/ortho /ortho D Iem = D Ime =0 /ortho D Imm
⎪ ⎪ ⎪ ⎪ 1 μ μ μ ⎭ Dx (ω)xˆ xˆ + Dy (ω)yˆ yˆ + Dz (ω)ˆzzˆ ⎪ = iωμ0
.
(5.72)
5 In fact, it follows from the expressions (5.56) that the skew-symmetric components of any constitutive dyadic vanish from the depolarization dyadics.
3, § 5]
Dyadic Green functions
191
Here, ⎫ 1/2 ηy (ω)[F (λ1 , λ2 ) − E(λ1 , λ2 )] ⎪ ⎪ ⎪ = ⎪ 1/2 ⎪ [ηy (ω) − ηx (ω)][ηz (ω) − ηx (ω)] ⎪ ⎪
1/2 ⎪ ⎪ ⎪ ηx (ω) − ηy (ω) 1 ηz (ω) − ηx (ω) ⎪ η ⎪ ⎪ Dy = − ⎪ ⎪ ηy (ω) − ηx (ω) ηz (ω) − ηy (ω) ηy (ω) ⎪ ⎪
⎬ ηx (ω) , × F (λ1 , λ2 ) ⎪ ηz (ω) − ηx (ω) ⎪ ⎪ ⎪ ⎪ ηy (ω) ⎪ ⎪ ⎪ − E(λ1 , λ2 ) ⎪ ⎪ ηz (ω) − ηy (ω) ⎪ ⎪
⎪
1/2 ⎪ ⎪ ηy (ω) 1 ⎪ η ⎪ ⎭ E(λ1 , λ2 ) 1− Dz = ηz (ω) − ηy (ω) ηz (ω) − ηx (ω) Dxη
(η = , μ),
(5.73)
involve F (λ1 , λ2 ) and E(λ1 , λ2 ) as elliptic integrals of the first and second kinds (Gradshteyn and Ryzhik [1980]), respectively, with arguments ⎫ ηz (ω) − ηx (ω) 1/2 ⎪ ⎪ ⎪ λ1 = tan ⎬ ηx (ω)
1/2 ⎪ ηz (ω)[ηy (ω) − ηx (ω)] ⎪ ⎪ λ2 = ⎭ ηy (ω)[ηz (ω) − ηx (ω)] −1
EBL(η = , μ).
(5.74)
Depolarization dyadics for the corresponding orthorhombic dielectric mediums and orthorhombic magnetic mediums follow immediately from eqs. (5.72), upon (ω) = 0 I , respectively. The form of the substituting μortho (ω) = μ0 I and ortho bi bi depolarization dyadics (5.72) also arises for an ellipsoidal shape, with rotational axes aligned with x, ˆ yˆ and zˆ , in an isotropic dielectric–magnetic ambient medium (Fricke [1953], Weiglhofer [1998b]). 5.4.2.3. Bianisotropic ambient mediums Closed-form representations of depolarization dyadics are not available for a general bianisotropic medium, but analytical progress has been reported for certain uniaxial bianisotropic mediums (Michel and Weiglhofer [1997]). When the ambient medium is characterized by the constitutive relations (4.62), the skew-symmetric components of the magnetoelectric constitutive dyadics do not contribute to the associated depolarization dyadics; therefore, the depolarization dyadics are simply those of the corresponding anisotropic dielectric–magnetic mediums.
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Electromagnetic fields in linear bianisotropic mediums
[3, § 6
§ 6. Homogenization Bianisotropic mediums may be readily conceptualized as homogenized composite mediums (HCMs), arising from constituent mediums which are themselves not bianisotropic (or even anisotropic). The constitutive parameters of HCMs are estimated by homogenization formalisms. A mixture of two or more different mediums may be viewed as being effectively homogeneous provided that wavelengths are much longer than the length scales of the nonhomogeneities (Garland and Tanner [1978], Priou [1992], Lakhtakia [2000a]). In fact, the process of homogenization implicitly underpins our descriptions of anisotropy and bianisotropy, since the constitutive relations relate macroscopic electromagnetic fields which represent the volume-averages of their microscopic counterparts (Van Kranendonk and Sipe [1977], Buchwald [1985]). On a more practical level, homogenization is important in the interpretation of experimental measurements and in material design. This latter topic continues to motivate research. HCMs, and anisotropic and bianisotropic HCMs in particular, furnish prime examples of metamaterials (Walser [2003]). These are artificial composite materials which exhibit properties that are either not exhibited at all or not exhibited to the same extent by their constituent materials. The concept that complex mediums may be realized through the homogenization of relatively simple constituent mediums was promoted initially for isotropic chiral HCMs (Sihvola and Lindell [1990], Lakhtakia [1993]), and subsequently extended to anisotropic and bianisotropic HCMs (Michel [2000]). The origins of formalisms used to estimate the constitutive parameters of HCMs may be traced back to the earliest years of electromagnetic theory. For a historical perspective, see the introduction to an anthology of milestone papers on HCMs (Lakhtakia [1996]). Two of the most widely used formalisms today – the Maxwell Garnett formalism (Maxwell Garnett [1904]) and the Bruggeman formalism (Bruggeman [1935]) – were first developed for isotropic dielectric– magnetic HCMs. However, generalizations of these formalisms appropriate to anisotropic and bianisotropic HCMs emerged much more recently (Weiglhofer, Lakhtakia and Michel [1997]), following the development of expressions for the corresponding depolarization dyadics (Michel and Weiglhofer [1997]). In the conventional approaches to homogenization, as exemplified by the Maxwell Garnett and Bruggeman formalisms, the distributional statistics of the HCM constituents are described solely in terms of their respective volume fractions. A more sophisticated approach is taken in the strong-property-fluctuation theory (SPFT), wherein spatial correlation functions of arbitrarily high order may
EBL
3, § 6]
Homogenization
193
be accommodated (Tsang and Kong [1981]). The development of the SPFT for bianisotropic HCMs began in the 1990s (Zhuck [1994], Michel and Lakhtakia [1995]) and was achieved at a practical level at the beginning of the twenty-first century (Mackay, Lakhtakia and Weiglhofer [2000]). The estimation of constitutive dyadics of linear bianisotropic HCMs is described in the following subsections, using the Maxwell-Garnett, Bruggeman and SPFT formalisms. These approaches are rigorously established, in contrast to extrapolations of isotropic formalisms (Sihvola and Pekonen [1996, 1997], Shanker [1997]).
6.1. Constituent mediums Let us focus upon HCMs arising from two particulate constituent mediums, labelled a and b. Each constituent medium is itself homogeneous. The Tellegen constitutive dyadics of the constituent mediums are denoted by K a (ω) and K b (ω). The constituent particles are taken to be generally ellipsoidal in shape. The ellipsoids of each constituent medium are conformal and have the same orientation, but are randomly distributed. The shapes of the ellipsoids are characterized by the real-symmetric 3 × 3 dyadics U a and U b , as introduced in eq. (5.51). The sizes of the ellipsoids are not specified, but their linear dimensions must be much smaller than the electromagnetic wavelength(s).6 Let V denote the unbounded space occupied by the composite medium. This space is partitioned into the disjoint regions Va and Vb which contain the constituent mediums a and b, respectively. In order to completely fill V with ellipsoids, a fractal-like distribution of the constituent particles is implicit. The distributions of the two constituent mediums throughout V are specified in terms of the characteristic functions 1, r ∈ V ( = a, b). Φ (r) = (6.1) 0, r ∈ / V
EBL
The nth moment of Φ (r) is the ensemble average Φ (r 1 ) · · · Φ (r n ) e , which represents the probability of r 1 , . . . , r n ∈ V ( = a, b). The volume fraction of constituent medium is given by f = Φ (r) e ( = a, b); clearly, fa + fb = 1. 6 We refrain from discussing homogenization formalisms which take into account finite sizes of constituent particles, as these are largely restricted to isotropic dielectric–magnetic (Doyle [1989], Dungey and Bohren [1991]) and isotropic chiral (Shanker and Lakhtakia [1993], Shanker [1996]) HCMs. Exceptions are provided by certain extensions to the Maxwell Garnett formalism (Lakhtakia and Shanker [1993]) and the SPFT (Mackay [2004], Cui and Mackay [2007]).
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Electromagnetic fields in linear bianisotropic mediums
[3, § 6
Only the first moments of Φa,b (r) are used in the Maxwell Garnett and Bruggeman formalisms, whereas arbitrarily high-order moments can be accommodated in the SPFT (Tsang and Kong [1981], Mackay, Lakhtakia and Weiglhofer [2000]). The polarizability density dyadic of an ellipsoid of medium ( = a, b) embedded in a homogeneous ambient medium described by the Tellegen constitutive dyadic K amb (ω), is defined as (Michel [2000]) α /amb (ω) = K (ω) − K amb (ω) • I + iωDU /amb (ω) −1 • K (ω) − K (6.2) (ω) ( = a, b), amb
with DU /amb (ω) being the depolarization dyadic defined in eq. (5.54). The dyadic α /amb (ω) is central to the Maxwell Garnett and Bruggeman formalisms.
6.2. Maxwell Garnett formalism The Maxwell Garnett homogenization formalism has been used extensively, despite its applicability being limited to dilute composites (Lakhtakia [1996]). A rigorous basis for this formalism was established by Faxén [1920] for isotropic dielectric-in-dielectric composite mediums, and its standing was further bolstered by the establishment of the closely related Hashin–Shtrikman bounds (Hashin and Shtrikman [1962]). In the Maxwell Garnett formalism, the mixture of the two constituent mediums may be envisaged as a collection of well-separated particles (of medium a, say) randomly dispersed in a simply connected host medium (medium b, say). The Maxwell Garnett estimate of the Tellegen constitutive dyadic of the HCM is (Weiglhofer, Lakhtakia and Michel [1997]) −1 K MG (ω) = K b (ω) + fa α a/b (ω) • I − iωfa DI /b (ω) • α a/b (ω) . (6.3)
EBL
Herein, the depolarization dyadic DI /b (ω) for a spherical region in constituent medium b indicates the incorporation of a spherical Lorentzian cavity in the Maxwell Garnett formalism. Notice that if DI /b (ω) in eq. (6.3) were to be rea placed by DU /b (ω), then the Bragg–Pippard formalism would arise (Sherwin and Lakhtakia [2002]). The Maxwell Garnett estimate (6.3) is valid for fa 0.3 only. In order to overcome this restriction, the following two refinements of the Maxwell Garnett formalism have been established. In the incremental Maxwell Garnett homogenization formalism, the estimate of the HCM constitutive dyadic is constructed incrementally, by adding the constituent medium a to the constituent medium b not all at once but in a fixed
3, § 6]
Homogenization
195
number N of stages (Lakhtakia [1998b]). After each increment, the composite is homogenized using the Maxwell Garnett formalism. Thus, the iteration scheme 1/N a/n K[n + 1](ω) = K[n](ω) + 1 − fb α (ω) I /n −1 1/N • I − iω 1 − f D (ω) • α a/n (ω) (6.4) b emerges for n = 0, 1, . . . , N − 1, where K[0](ω) = K b (ω). In eq. (6.4), α a/n (ω) is the polarizability density dyadic of a constituent material phase a particle, of shape specified by U a , relative to the homogeneous medium characterized by the Tellegen constitutive dyadic K[n](ω). The incremental Maxwell Garnett estimate of the Tellegen constitutive dyadic of the HCM is delivered as K IMG (ω) = K[N ](ω).
(6.5)
In the limit N → ∞, the incremental Maxwell Garnett formalism gives rise to the differential Maxwell Garnett homogenization formalism (Michel, Lakhtakia, Weiglhofer and Mackay [2001]). That is, the differential Maxwell Garnett estimate K DMG (ω) of the Tellegen constitutive dyadic of the HCM is the solution of the ordinary differential equation 1 ∂ K(ω, s) = α a/s (ω), (6.6) ∂s 1−υ with initial value K(ω, 0) = K b (ω). The continuous variable s represents the volume fraction of constituent medium a added to medium b, and α a/s (ω) is the polarizability density dyadic of a particle made of constituent medium a, and of shape given by U a , immersed in a medium with constitutive dyadic K(ω, s). The particular solution of eq. (6.6) satisfying s = fa is the differential Maxwell Garnett estimate; i.e.,
EBL
K DMG (ω) = K(ω, fa ).
(6.7)
We note that the recursive approach that underpins the incremental and differential Maxwell Garnett formalisms has been applied quite widely within the realm of isotropic dielectric HCMs (Ghosh and Fuchs [1991], Barrera and Fuchs [1995], Fuchs, Barrera and Carrillo [1996], Sosa, Mendoza and Barrera [2001]).
6.3. Bruggeman formalism A central characteristic of the Bruggeman homogenization formalism is that the constituent mediums a and b are treated symmetrically (Milton [1985], Ward
196
Electromagnetic fields in linear bianisotropic mediums
[3, § 6
[2000]). The Bruggeman formalism is thereby applicable for all volume fractions fa ∈ (0, 1), unlike the conventional Maxwell Garnett formalism. The symmetrical treatment of the constituent mediums enables the phenomenon of percolation threshold to be predicted by the Bruggeman formalism, albeit not always correctly (Brouers [1986], Berthier and Peiro [1998], Goncharenko and Venger [2004]). Several variations on the Bruggeman formalism have been developed for isotropic dielectric–magnetic mediums (Hanai [1960], Niklasson and Granqvist [1984], Goncharenko [2003]). Indeed, by a microscopic treatment of the local field effects, the Bruggeman formalism may be shown to arise naturally from the Maxwell Garnett formalism (Aspnes [1982]). The rigorous establishment of the Bruggeman formalism for the most general linear scenarios follows from the SPFT (Mackay, Lakhtakia and Weiglhofer [2000]). The ellipsoidal particles made of the two constituent mediums can have different shapes. The polarizability density dyadics implemented in the Bruggeman formalism, for each constituent material phase, are calculated relative to the HCM itself; i.e., the polarizability density dyadic α /Br (ω) ( = a, b) is defined in terms of the Bruggeman estimate K Br (ω) of the Tellegen constitutive dyadic of the HCM. The assertion that the net polarizability density is zero throughout the HCM underlies the Bruggeman homogenization formalism. Thus, the Bruggeman estimate of the HCM constitutive dyadic is provided implicitly by the nonlinear equation (Weiglhofer, Lakhtakia and Michel [1997])
EBL
fa α a/Br (ω) + fb α b/Br (ω) = 0,
(6.8)
from which K Br (ω) can be extracted by applying the simple Jacobi technique (Michel, Lakhtakia and Weiglhofer [1998]). Hence, the iterative solution K Br [n](ω) = S K Br [n − 1](ω)
(n = 1, 2, . . .)
(6.9)
is developed, with the initial value K Br [0](ω) = K MG (ω). The action of the operator S is defined by −1 a S K Br (ω) = fa K a (ω) • I + iωDU /Br (ω) • K a (ω) − K Br (ω) −1 b + fb K b (ω) • I + iωDU /Br (ω) • K b (ω) − K Br (ω) −1 a • fa I + iωDU /Br (ω) • K (ω) − K (ω) a Br −1 −1 b + fb I + iωDU /Br (ω) • K b (ω) − K Br (ω) . (6.10)
3, § 6]
Homogenization
197
6.4. Strong-property-fluctuation theory 6.4.1. Background The provenance of the strong-property-fluctuation theory7 (SPFT) lies in wavepropagation studies for continuous random mediums (Ryzhov and Tamoikin [1970], Tatarskii and Zavorotnyi [1980]). However, in recent years it has been adapted to estimate the constitutive parameters of HCMs (Tsang and Kong [1981]). In contrast to the conventional Maxwell Garnett and Bruggeman formalisms, the SPFT accommodates a comprehensive description of the distributional statistics of the constituent material phases. Thereby, coherent scattering losses may be accounted for. The SPFT approach to homogenization is based upon the iterative refinement of a comparison medium; i.e., the homogeneous medium specified by the Tellegen constitutive dyadic K comp (ω). The refinement process involves applying a Feynman-diagrammatic technique to ensemble-average a Born series representation of the electromagnetic fields. While the straightforward approach is limited to weak spatial fluctuations in the infinity norm of
EBL a careful consideration of the singularity of the DGF G K a (ω)Φa (r) + K b (ω)Φb (r) − K comp (ω),
(6.11)
(r −r , ω) of the comparison medium allows a reformulation that is applicable for strong fluctuations. When electromagnetic wavelengths are much larger than the length scales of inhomogeneities, the mixture of material phases may viewed as being effectively (ω) homogeneous. In this long-wavelength regime, the nth-order estimate K[n] SPFT of the Tellegen constitutive dyadic of the HCM is delivered by the SPFT as (Mackay, Lakhtakia and Weiglhofer [2000, 2001a]) K[n] (ω) = K comp (ω) − SPFT
comp
−1 1 • Σ [n] (ω), I + Σ [n] (ω) • DU/comp (ω) iω (6.12)
wherein the mass operator term Σ [n] (ω) – which has an infinite series representation – is defined in terms of the DGF for the comparison medium, together with the generalized polarizability density dyadic α U/comp (r, ω) = α a/comp (ω)Φa (r) + α b/comp (ω)Φb (r).
(6.13)
7 It is otherwise known as the strong-permittivity-fluctuation theory in the context of dielectric mediums.
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Electromagnetic fields in linear bianisotropic mediums
[3, § 6
The two populations of ellipsoids are required to have the same shape dyadics; i.e., U a = U b = U . The condition ! α U/comp (r, ω) e = 0 (6.14) is imposed in order to eliminate secular terms from the Born series representation (Frisch [1970]). 6.4.2. SPFT estimates 6.4.2.1. Zeroth and first orders Both the zeroth-order and the first-order mass operator are null-valued; i.e., Σ [0] (ω) = Σ [1] (ω) = 0. The estimates K[0] (ω) = K[1] (ω) = K comp (ω) may be extracted from the condition (6.14). SPFT SPFT In fact, as the condition (6.14) is equivalent to the Bruggeman equation (6.8), the lowest-order SPFT estimates of the Tellegen constitutive dyadic of the HCM are identical to its Bruggeman estimate. 6.4.2.2. Second order The SPFT is most commonly implemented at the secondorder level of approximation – otherwise known as the bilocal approximation (Tsang and Kong [1981], Stogryn [1983], Genchev [1992], Zhuck [1994], Michel and Lakhtakia [1995], Mackay, Lakhtakia and Weiglhofer [2000]). Therein, the distributional statistics of the constituent mediums a and b are characterized by the two-point covariance function ! ! ! Λ(r − r ) = Φa (r)Φa (r ) e − Φa (r) e Φa (r ) e ! ! ! = Φb (r)Φb (r ) e − Φb (r) e Φb (r ) e , (6.15)
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along with its associated correlation length L. Within a region of linear dimensions given by L, and of shape dictated by the covariance function, the correlated responses of scattering centres give rise to an attenuation of the macroscopic coherent field. On the other hand, the responses of scattering centres separated by distances much greater than L are statistically independent. The second-order mass operator is given by (Mackay, Lakhtakia and Weiglhofer [2000]) Σ [2] (ω) = −ω2 α a/comp (ω) − α b/comp (ω) • PΛ (ω) • α a/comp (ω) − α b/comp (ω) , (6.16) where
P (ω) = P
Λ(R)G comp (R, ω) d3 R,
Λ
R
(6.17)
3, § 6]
Homogenization
199
0 with P . . . d3 R denoting principal value integration. The principal value integral (6.17) has been investigated theoretically and numerically for various physically motivated choices of covariance function (Tsang, Kong and Newton [1982], Mackay, Lakhtakia and Weiglhofer [2001b]), and most notably for the step function Λ(r − r ) = Λstep (r − r ) wherein 1 ! f , |U −1 • (r − r )| L Φ (r)Φ (r ) e = (6.18) ( = a, b). f 2 , |U −1 • (r − r )| > L These studies have revealed that the form of the covariance function has only a weak influence upon the SPFT estimate K[2] (ω) (Mackay, Lakhtakia and WeiSPFT glhofer [2001b]). After choosing the covariance function Λstep (r − r ), the principal value integral (6.17) may be expressed in the more tractable form
P
Λstep
f a fb (ω) = 2π 2
2π π ∞ Q(q, ω) sin θ dq dθ dφ,
(6.19)
φ=0 θ=0 q=0
with
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−1 sin qL ˇ0 • q, ω Q(q, ω) = G U − L cos qL , comp q
(6.20)
where q = q qˆ and qˆ = (sin θ cos φ, sin θ sin φ, cos θ ). Furthermore, ˇ ˇ ∞ (q, ˇ 0 (q, ω) = G G (q, ω) − G ˆ ω), comp comp comp
(6.21)
ˇ with G (q, ω) being the spatial Fourier transform of the DGF for the comparcomp ˇ ∞ (q, ˆ ω) in the limit ison medium, as defined in eq. (5.38), which tends to G comp q → ∞, as defined in eq. (5.53). Some analytical progress has been reported towards the simplification of the integral (6.19) on an unbounded domain: for certain anisotropic (Genchev [1992], Zhuck [1994]) and bianisotropic (Mackay, Lakhtakia and Weiglhofer [2000]) comparison mediums, a two-dimensional integral emerges from eq. (6.19), but numerical methods are generally required to evaluate even that integral. 6.4.2.3. Third order To calculate the third-order (i.e., trilocally approximated) mass operator term Σ [3] , a three-point covariance function is required (Miller [1969], Milton and Phan-Thien [1982], Mackay, Lakhtakia and Weiglhofer [2001c]). The two-point covariance function (6.18) incorporated into Λstep (r −r )
200
Electromagnetic fields in linear bianisotropic mediums
[3, § 6
generalizes to yield ⎧ 3 fa , ⎪ ⎪ ⎨ ! f , θa (r)θa (r )θa (r ) = 1 a ⎪ (f + 2fa3 ), ⎪ ⎩ 31 a 3 3 (2fa + fa ),
min{L12 , L13 , L23 } > L, max{L12 , L13 , L23 } L, (6.22) one of L12 , L13 , L23 L, two of L12 , L13 , L23 L,
wherein
⎫ L12 = U −1 • (r − r ) ⎪ ⎪ −1 ⎬ • L13 = U (r − r ) . −1 ⎪ ⎪ ⎭ • (r − r ) L23 = U
(6.23)
Upon using eq. (6.22), the third-order mass operator term emerges as (Mackay, Lakhtakia and Weiglhofer [2001c]) iω3 fa (1 − 2fa ) a/comp α (ω) Σ [3] (ω) = Σ [2] (ω) + 3(1 − fa )2 • V(ω) • α a/comp (ω) • PΛstep (ω) + PΛstep (ω) • α a/comp (ω) • V(ω)
+ PΛstep (ω) • α a/comp (ω) • PΛstep (ω) • α a/comp (ω), in which
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(6.24)
1 −1 (ω) − DU/comp (ω). K (6.25) iω comp Strictly, eq. (6.24) holds for isotropic chiral comparison mediums combined with spherical constituent particles, but it is highly probable that it also holds for weakly bianisotropic mediums with diagonally dominant 3 × 3 constitutive dyadics. By implementing eq. (6.24), SPFT convergence at the level of the bilocal approximation has been demonstrated for certain isotropic chiral HCMs, as well as bianisotropic HCMs of the Faraday chiral type which are both weakly uniaxial and weakly gyrotropic (Mackay, Lakhtakia and Weiglhofer [2001c]). V(ω) =
6.5. Anisotropy and bianisotropy via homogenization Manifestations of anisotropy and bianisotropy may be readily conceptualized through the process of homogenization. HCMs ‘inherit’ the symmetries of their constituent mediums. Therefore, complex HCMs can arise from relatively simple constituent mediums.
3, § 7]
Closing remarks
201
Biaxial HCMs have been comprehensively studied in this regard, in both anisotropic (Mackay and Weiglhofer [2001b, 2000]) and bianisotropic (Mackay and Weiglhofer [2001a]) scenarios. A biaxial HCM generally arises when the constituent mediums present two, noncollinear, distinguished axes of symmetry. These distinguished axes may be either of electromagnetic origin, as in uniaxial constituent mediums for example, or they may derive from the particulate shapes of the constituent mediums, as in spheroidal particles for example. Thus, a biaxial HCM – which is anisotropic or bianisotropic – may develop from constituent mediums a and b wherein: • both constituent mediums are uniaxial and distributed as spherical particles; • both constituent mediums are isotropic and distributed as spheroidal particles; • both constituent mediums are isotropic, with one medium distributed as spherical particles and the other as ellipsoidal particles; or • one of the constituent mediums is uniaxial while the other is isotropic, and one of the constituent mediums is distributed as spheroidal particles while the other as spherical particles. If the distinguished axes of the constituent mediums are orthogonal, then orthorhombic biaxial HCMs generally develop; otherwise monoclinic or triclinic biaxial HCMs arise. A second prime example of bianisotropic HCMs arising from relatively simple constituents is provided by Faraday chiral mediums, as discussed in § 3.3.3. These have been comprehensively investigated using the Maxwell Garnett and Bruggeman formalisms (Weiglhofer, Lakhtakia and Michel [1998], Weiglhofer and Mackay [2000]), as well as the SPFT (Mackay, Lakhtakia and Weiglhofer [2000]).
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§ 7. Closing remarks The most general description of a linear medium in electromagnetics is bianisotropic, which is Lorentz-covariant. But bianisotropy is rarely discussed in even graduate-level treatments of electromagnetics. Bianisotropy arises very naturally when considering descriptions of a linear medium by uniformly moving observers. Furthermore, bianisotropy may be observed in numerous naturally occurring minerals when viewed from a co-moving reference frame. In relation to technological applications, it may well transpire that the most significant manifestation of bianisotropy lies in homogenized composite mediums. Through the process of homogenization, the vast parameter space of bianisotropy may be readily accessed, thereby making possible the realization and exploitation of highly complex electromagnetic behaviour.
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[3
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E. Wolf, Progress in Optics 51 © 2008 Elsevier B.V. All rights reserved
Chapter 4
Ultrafast optical pulses by
Clifford R. Pollock Cornell University, School of Electrical and Computer Engineering, Ithaca, NY, USA
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ISSN: 0079-6638
DOI: 10.1016/S0079-6638(07)51004-8 211
Contents
Page § 1. Overview of ultrashort optical pulses . . . . . . . . . . . . . . . . . .
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§ 2. Fundamental properties of optical pulses . . . . . . . . . . . . . . . .
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§ 3. Ultrashort-pulse generation . . . . . . . . . . . . . . . . . . . . . . .
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§ 4. Ultrafast-pulse characterization . . . . . . . . . . . . . . . . . . . . .
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§ 5. Ultrafast Ti:sapphire lasers and amplifiers . . . . . . . . . . . . . . .
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§ 6. Attosecond pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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§ 7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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§ 1. Overview of ultrashort optical pulses 1.1. Historic developments in short optical pulse development The generation of short-duration optical pulses has been actively pursued by researchers for over 150 years, since Charles Wheatstone used electric discharges to ionize air and generate sparks that would freeze motion. Even before the first lasers were demonstrated, the population dynamics of stimulated emission were predicted to be capable of generating fast pulses, with rise and fall times on a nanosecond time scale (Gould [1959]). At that time, optical pulses of such short duration were almost unheard of. When the laser did arrive two years later, these predictions were soon borne out with many different gain media. The first real modelocking, demonstrated in a HeNe laser in 1964 (Hargrove, Fork and Pollack [1964]), produced 0.5 ns pulses by coherently phase locking all the longitudinal cavity modes together using an intracavity modulator. Theoretical work followed, and the fundamental issues of short-pulse generation using a laser were articulated Crowell [1965]. Reports of many different modelocked systems began to appear, including the generation of 2 ns pulses from the ruby laser (Deutsch [1965]), 20 ns pulses from a passively modelocked CO2 laser (Wood and Schwarz [1968]), and less than 0.5 ns pulses from a Nd:glass laser (DeMaria, Ferrar and Danielson [1966]). Most of these pulses were measured directly with an electro-optic detector and a fast oscilloscope, which together barely had a response time of 0.5 ns. These pulses were faster than anything could measure at the time, hence the term “ultrashort”. “Ultrashort” is a moving target, continuously needing to be redefined as new techniques are discovered for the generation of ever shorter pulses. In the early 1970s, with the demonstration of the passively modelocked dye laser (Ippen, Shank and Dienes [1972]) the generation of a picosecond pulse (10−12 s) was considered to be a significant milestone in any laboratory. As new broadly tunable lasers were discovered, such as the color center laser (Mollenauer and Bloom [1979]) or the Ti:sapphire laser (Moulton [1986]), researchers found creative ways to generate even shorter pulses. Progress in short-pulse generation has advanced since then, until today 50 fs pulses (50 × 10−15 s) are considered commonplace.
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The early pioneering work in ultrashort-pulse generation illustrates some of the common issues that have been a part of ultrashort-pulse generation from its beginnings. These issues include the need for lasers with broad gain bandwidth, the challenge of measuring very short pulses, and the need for mechanisms to couple the modes together. Today, 40 years after much of this work began, these fundamental issues are still at the core of basic research in this area. But much has also been learned in these 40 years. Today researchers are able generate and measure pulses with peak powers exceeding terawatts, push the temporal duration of pulses to the attosecond scale, and generate pulses with wavelengths ranging from the near infrared to the soft X-ray regime.
1.2. Outline of chapter In this review, we will apply the term “ultrafast” to pulses that have a temporal duration less than 50 fs. This is an arbitrary boundary, as there are interesting phenomena that can be observed with longer pulses, and there are many phenomena which require much shorter time frames to be observed (such as the orbital dynamics of a bound electron). This duration is chosen simply because with commonly available lasers, skilled people around the world have been able to generate optical pulses of this range, and are doing many marvellous things with the pulses. Even though such pulses are routine, the temporal resolution they provide is simply astonishing: 30 femtoseconds is to a second what a second is to a million years. The electromagnetic properties of short pulses will first be reviewed. Then the techniques of generating ultrafast pulses will be explored. Since there are many excellent reviews on modelocking solid-state lasers, this chapter will attempt to present the larger picture, and refer interested readers to the more technical monographs where appropriate. With short time domains, even modest pulse energies translate into extreme powers. Using chirped pulse amplifiers, amplification of the energy in an ultrashort pulse by factors of 107 has been reported, with peak powers in the 1010 –1015 W range. Such “tabletop terawatt” systems can be built on a single optical table in a modest laboratory, opening the door for the broad exploration of new states of matter. A major issue that has always accompanied the generation of ultrafast pulses is in measuring these pulses. Usually to measure a time event, one would like an even shorter event to characterize the pulse one wants to measure. The dilemma of measuring ultrafast pulses has been a challenging task, but there are now well-
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established methods. Section 4 will review some of the preferred temporal characterization techniques, including one that provides information on the spectral phase of the pulses. Spectral phase characterization is critical for the development of new sources and the utilization of ultrashort pulses. Finally, intense ultrashort pulses have made it possible to explore highly nonlinear process in atomic, molecular, plasma, and solid-state systems. One application of this work is the generation of even shorter, attosecond pulses (an attosecond is 10−18 s) that generate soft X-rays. These truly ultrafast pulses of soft X-rays can be used to probe short-range dynamics of systems such as the orbiting electron in an atom.
§ 2. Fundamental properties of optical pulses Before considering the details of ultrashort-pulse generation, it is necessary to establish a few basic concepts, such as the relationship between bandwidth, phase, and the ultimate temporal pulsewidth. In this section we review the basic form of the electromagnetic wave, and the Fourier relations between time and frequency. A travelling ultrafast optical pulse is a burst of electromagnetic energy. Formally, the electromagnetic wave is a vector field, E(r, t), that satisfies the electromagnetic wave equation
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∇ 2E +
∂ 2E
= 0, (2.1) ∂t 2 where the magnitude of E describes the amplitude of the electric field, in units of volt/meter. The direction of E indicates which way a positive charge would be accelerated by the field. Here we consider only linearly polarized fields, so only one component of the wave need be considered. This is called the scalar approximation. If polarization is an issue, one would solve this equation for each component of the electric field. Rectilinear boundary conditions lead to easily separable equations, but certain geometries such as propagation in a dielectric cylinder force polarization components to couple so other solution techniques must be used (Pollock [1995]).
2.1. Amplitudes, envelopes, and intensity A general pulse solution to the wave equation (2.1) for a wave travelling in the z direction is given by 1 I (t) exp(iω0 t − ikz) + c.c. E(t, z) = (2.2) 2
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Fig. 1. The electric field of a pulse oscillates with the phase of the carrier frequency. The amplitude of the wave is the envelope of the peak values. The intensity is proportional to the square of the amplitude, and is thus narrower in time.
√ where I (t) represents the scalar amplitude, ω0 is the angular frequency of the wave (in radians/second), and k is the wavevector, which describes the phase accumulated per unit distance as the wave propagates (in radians/meter). The term “c.c.” means the complex conjugate, and is required to make the overall function real. This solution is a function of both time and position. The transverse spatial structure is critical to some aspects of ultrafast pulse propagation, for example Kerr Lens Modelocking (Spence, Kean and Sibbett [1991]), but those issues will be ignored in the present analysis. Spatial effects do not significantly alter any of the discussion below. It is conventional to define the envelope amplitude as the square root of the intensity. This neglects certain constants such as the permittivity and the speed of light, but these constants have no impact on the time-domain solutions that we are concerned about. The time dependence of the intensity describes the envelope of the pulse, and not the fast oscillations due to the carrier frequency. Figure 1 shows a plot of the instantaneous electric field (the dashed line), the pulse envelope, and the intensity of a hyperbolic secant pulse. The intensity of a pulse is shorter in time than the electric field envelope. This fact has proven useful in measuring the temporal pulse shape, discussed in § 4.2. Since the temporal features of the envelope are of primary interest, the fast term that describes the carrier wave, exp(iω0 t), can be separated from the expression for the electric field as shown in eq. (2.2). The carrier wave term is not fundamentally interesting to the pulse dynamics. This separation allows attention to be focussed on the envelope function, described as (2.3) I (t) exp −iφ(t) .
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√ The envelope expression has two components: the amplitude, I (t), which is real; and the phase, φ(t), which can be complex. The amplitude of the envelope is easy to visualize and plot, as shown in fig. 1. The phase φ(t) of the pulse is more subtle. The carrier wave accumulates phase at a steady rate of ω0 rad/s; φ(t) represents deviations from this steady accumulation. Dynamic phase in a pulse is not simple to measure, nor is it simple to visualize. But the impact of phase on the overall pulsewidth can be very strong. As will be shown below, much of the progress in ultrafast-pulse generation has been achieved because researchers have learned how to measure and control the phase of the pulses they were generating. The connection between the amplitude and the phase is described through the Fourier transform that relates the temporal pulse description to a spectral description. The frequency-domain description of the pulse, E(ω), can be found by taking the Fourier transform of the time-domain description of the pulse, E(t), ˜ E(ω) =
∞ E(t) exp(−iωt) dt.
(2.4)
−∞
The spectral form of the field can be converted back into the time domain by using the inverse transform: E(t) =
1 2π
∞
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˜ E(ω) exp(iωt) dω.
(2.5)
−∞
Every feature in the time domain of a pulse will have an impact on the spectral amplitude and phase of the signal in the frequency domain. Reciprocally, every feature in the spectral phase will impact the time domain, possibly impacting the pulse duration. In a manner analogous to the description of the temporal envelope, the spectral ˜ pulse E(ω) is described in terms of its energy spectrum, S(ω), and its spectral phase, φ(ω): ˜ E(ω) = S(ω) exp −iΦ(ω) . (2.6) The energy spectrum S(ω) is easy to visualize – it is what is measured using a spectrometer. The phase Φ(ω) is the spectral phase, not to be confused with the temporal phase φ(t), even though both have units of radians. The spectral phase describes the relative phase between the frequency components that comprise a short pulse. The spectral phase can always be found from the temporal phase through the Fourier transform.
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2.2. Phase, frequency, and group delay The connection between the time domain and the spectral domain is very powerful and useful. Ultrafast-pulse formation requires careful control of the pulse phase. In practice, it is technologically difficult, if not impossible, to implement change in the temporal φ(t) using dynamic adjustments on ultrafast pulses. But it is relatively straightforward to adjust the spectral phase, Φ(ω), of a pulse using dispersive materials and optics. The temporal phase φ(t) relates the carrier frequency to the instantaneous frequency through the familiar relation ω(t) = ω0 − dφ(t)/dt,
(2.7)
where ω0 is the nominal carrier frequency. The temporal phase provides information on how the carrier frequency changes as a function of time. dφ(t)/dt arises in many physical situations, such as in self phase modulation where the index of refraction is a weak function of intensity, n(I ) = n0 + n2 I (t).
(2.8)
A pulse, which naturally has a dI (t)/dt, will directly generate a dφ(t)/dt term, and thus generate new frequencies. The spectral phase provides information on temporal delay. The corresponding relation between time and phase is given by
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τgroup = dΦ(ω)/dω,
(2.9)
where group delay, τgroup , is the time it takes for energy to travel a unit distance. Φ(ω) provides information on relative propagation delays between various frequency components in a pulse. It is useful to describe the spectral phase as a Taylor series, d2 Φ(ω) dΦ(ω) 1 + ···. + (ω − ω0 )2 (2.10) dω 2 dω2 The expansion is useful because there are techniques for adjusting each term in this expansion (see § 5.1). This allows the systematic adjustment of total dispersion in a cavity, or in an optical relay, to achieve the desired phase in a pulse at the target. Typically laser cavities are adjusted for minimal dispersion to generate the shortest pulse duration, whereas there are other applications such as chirped amplifiers where large controlled amounts of dispersion are used to spread a pulse. Both issues are discussed in § 3.2. The zeroth-order phase, Φ0 , is called the absolute phase, and represents the phase of the carrier wave with respect to the envelope function. By convention the Φ(ω) = Φ0 + (ω − ω0 )
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Fig. 2. Simulation of a 30 fs pulse with a quadratic temporal phase of φ2 = −0.05 rad/fs2 . The frequency of the carrier wave almost drops in half due to the huge dφ/dt term.
reference point is chosen to be the peak of the envelope function. The absolute phase is difficult to measure because the group and phase velocities of a pulse are different in media, so Φ0 will vary from pulse to pulse. Except for extremely short pulses which only contain a few cycles, the absolute value of Φ0 has negligible impact on the shape of the envelope. There is, however, increased interest in the absolute phase of the pulse for applications in attosecond-pulse generation (Reider [2004], Dietrich, Kraus and Corkum [2000]). Attosecond-pulse generation is described in § 6. The first-order phase term, dΦ(ω)/dω, can temporally advance or delay a pulse, but has no impact on the shape of the envelope. This first-order dispersion is the cause of the mismatch between phase velocity of a wave and group velocity of its envelope. Generally one is more interested in the shape of the pulse than its arrival time (at least in the laboratory, but perhaps not in a telecommunication system) and so pulse measurements are generally not sensitive to the absolute arrival time of the pulse. If absolute phase control is desired, for example where two modelocked lasers need to be locked together to make a source for frequency metrology using an optical frequency comb, this term can be very important (Ye, Schnatz and Hollberg [2003], Diddam, Bartels, Ramond, Oates, Bize, Curtis, Bergquist and Hollberg [2003]). For short-pulse generation, the interesting effects begin to appear in the secondorder term of the Taylor series expansion. Quadratic variations in Φ(ω) result in a linear ramp of the frequency versus time, which is called a chirp. Figure 2 shows the amplitude of a chirped pulse. In the spectral wavelength region below 1.3 µm, propagation through most materials usually causes a positive chirp, where the frequency of the pulse increases from the leading edge to the trailing edge. The impact of material dispersion can be compensated by adding a section of negative dispersion into a pulse propaga-
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tion path (Fork, Martinez and Gordon [1984]). The effect can also be exploited to temporally expand a pulse for amplification if peak powers in the amplifying medium are a limitation (Pessot, Maine and Mourou [1987]). Once amplified, the pulse can be re-compressed using negative dispersion. The higher-order terms in the Taylor series expansion (2.10) do appear in most materials, so control of third- and fourth-order dispersion is required to achieve the shortest optical pulses. The third-order term dΦ 3 (ω)/dω3 leads to a quadratic chirp. In such a pulse, the central frequencies arrive before the side frequencies arrive. The two side frequencies are slightly different, one being above the central frequency and the other below, so they form a beat (see Trebino [2000] for good examples of this). Compensation of third-order dispersion requires the selection of specific materials for the cavities, and can also be compensated with specially designed mirrors (Szip˝ocs and K˝oházi-Kis [1997]). Experimentally, and from simulations, the shortest pulses have a flat phase profile, whether temporal phase, φ(t), or spectral phase, Φ(ω). Even a pulse that has a relatively strongly modulated spectral profile (lots of bumps and peaks and valleys in the spectrum) will be temporally short if the phase is flat.
2.3. Time–bandwidth product
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A reasonable way to characterize pulses is to calculate the time–bandwidth product. This product is dimensionless, and provides a figure of merit for a pulse. The actual numerical value depends on the shape of the pulse, but for a given shape, the smaller the time–bandwidth product, the better the pulse. A minimal value means that the pulse is not carrying any excess bandwidth, i.e. that the pulse is “transform limited” and cannot be any shorter in time. Using the time–bandwidth product for such comparisons opens the door to the issue of how to define the spectral width and the temporal pulse width. Pulse duration can be defined a number of ways (Trebino [2000]). If the pulse has a smooth shape, the full width at half maximum (FWHM) of the intensity envelope is commonly used. If the pulse is known to be Gaussian, the 1/e halfwidth is often specified. One must be careful to specify which definition is being applied. For a more complex pulse, the root-mean-square (rms) pulsewidth is sometimes used: ! ! 2 τrms (2.11) = t − t 2 = t 2 − t 2 . While the FWHM definition ignores anything below the half-power point, the rms definition emphasizes values that are far from the center of the pulses. If a pulse has long wings, the rms width is a good indicator of this.
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Fig. 3. A full-cycle wave has a net zero area with respect to polarity. The unipolar half-cycle wave has only positive terms.
For very complex pulses the equivalent pulsewidth can be used: τeq =
1 Imax
+∞ I (t) dt.
(2.12)
−∞
This is a useful definition when a pulse has many sub-pulses in its temporal structure. The time–bandwidth product for a large number of pulse shapes has been tabulated by Sala, Kenney-Wallace and Hall [1980]. Depending on the pulse shape, the minimum time–bandwidth product can range from 0.1 to 1. The predicted pulse shape must be specified when a time–bandwidth product is reported.
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2.4. The “zero area” pulse An interesting feature of a travelling electromagnetic wave is that it must have equal amounts of positive and negative field, meaning the shortest optical pulse that could be generated would be one full cycle. It is not possible to generate and transmit a unipolar electromagnetic pulse that consists of only 1/2 cycle. A plot of both cases is shown in fig. 3. This can be easily understood from the Fourier analysis of the propagating wave. The Fourier spectrum of a propagating unipolar pulse (right-hand side of fig. 3) will have a DC term. To launch a unipolar pulse, it would be necessary to transmit a transverse DC potential with the rest of the waves. But it is impossible to establish a transverse DC potential in space, thus it is impossible to create such a pulse. Because no propagating wave can have a DC component, pulses must have equal amounts of positive and negative field. Propagating fields will always have
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an oscillatory structure with at least one cycle. Another way to look at this is to recognize that all physically realizable pulses must have an average area of zero – i.e. there is just as much negative as positive field in a total optical pulse. This fact establishes the limit on how short a pulse can be generated from a given source. The shortest pulse that can be generated from a given laser would consist of one full cycle. For a 1 µm source, where the frequency is 300 THz, the shortest pulse would be 3.3 fs. Going to a shorter wavelength, say 0.5 µm, would reduce the ultimate duration, in this case 1.67 fs. To make increasingly shorter pulses, it is necessary to use shorter wavelengths. We will see in § 6 that attosecond pulses are being generated using nonlinear frequency conversion techniques to convert the available femtosecond pulse to shorter wavelengths.
§ 3. Ultrashort-pulse generation In reviewing the history of ultrashort-pulse generation, one can see that progress was made when new lasers appeared which offered broad spectral gain bandwidth, or new modelocking techniques were developed. In this section we will review a few of the general properties of ultrashort-pulse lasers, then discuss the modelocking processes that are currently being employed.
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3.1. Spectral properties of ultrafast laser materials The dye laser, in particular the Colliding Pulse Modelocked dye laser (Ippen, Shank and Dienes [1972]), was a workhorse for much of the initial modelocking research. The dye laser was the first conveniently operable broadband laser, with a spectral bandwidth extending approximately 50–100 nm in the visible region. Having a very thin dye jet for the gain medium, the laser cavity did not contain a great deal of dispersive material; in fact the beam propagated mostly through air. At its maturity, pulses with durations as short as 27 fs could be obtained when fully pushed to the limit (Valdmanis, Fork and Gordon [1985]). Dye lasers have an annoying limitation, in that the organic dyes used as the gain medium suffer degradation under the intense pumping of the laser. After a few weeks of operation, the output power of a dye laser will be significantly decreased, and the modelocked pulses will lengthen in temporal pulse duration. In the 1980s a number of robust tunable lasers began to appear, based on solidstate materials which did not suffer optically induced degradation. The NaCl color center laser provided broadly tunable power in the 1.4–1.8 µm range (Pinto, Georgiou and Pollock [1986]). The bandwidth of this laser led to direct generation of
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pulses as short as 125 fs (Yakymyshyn, Pinto and Pollock [1989]) and 25 fs with compression (Islam, Sunderman, Soccolich, Bar-Joseph, Sauer, Chang and Miller [1989]). The Cr:forsterite laser (Petri˘cevi´c, Gayen, Alfano, Yamagishi, Anzai and Yamaguchi [1988]) provided broad tuning over the 1.15–1.3 µm range, and was conveniently pumped by a cw Nd:YAG laser. The Cr:YAG laser (Okhrimchuk and Shestakov [1994], Sennaroglu and Pollock [1995]) was similarly pumped, and extended tunable from 1.34 to 1.6 µm (see fig. 6). Both the Cr:forsterite (Chudoba, Fujimoto, Ippen, Haus, Morgner, Kärtner, Scheuer, Angelow and Tschudi [2001]) and Cr:YAG lasers (Ripin, Chudoba, Gopnath, Fujimoto, Ippen, Morgner, Kärtner, Scheuer, Angelow and Tschudi [2002]) have been modelocked to demonstrate ultrafast pulses over these tuning ranges. But the big advance came in 1986 when the Ti:sapphire laser was developed (Moulton [1986]). The Ti:sapphire laser is broadly tunable from 700 to 1100 nm, and can generate watts of cw power when pumped by a green laser. The bandwidth of this laser is sufficient to support pulses as short as 3 fs, and such pulse durations have almost been reached (Binhammer, Rittweger, Morgner, Ell and Kärtner [2006]). In 1991 Spence and colleagues (Spence, Kean and Sibbett [1991]) discovered Kerr Lens Modelocking with the Ti:sapphire laser, and with this discovery the Ti:sapphire laser displaced the dye laser to become the new workhorse for ultrafast-pulse generation. Kerr Lens Modelocking has proven to be very robust and applicable to many systems, and has opened the door to a large number of new developments in ultrafast laser pulse generation. The Ti:sapphire laser combines many ideal properties needed for short-pulse formation. First, it is broadly tunable, providing gain over the 700–1100 nm range. This exceptionally large tuning range provides the necessary bandwidth to create short pulses, as required by the Fourier relation between the frequency and time domains. Second, the nature of the laser active transition enables efficient pumping with almost no competition from higher-lying excited states. Many other broadly tunable systems suffer from excited-state absorption which significantly reduces their conversion efficiency, especially in high-intensity pulsed operation. Finally the solid-state host medium, sapphire, displays self phase modulation that leads to a very fast (femtosecond response time) intensity-dependent lensing in the medium. This latter effect, called a Kerr Lens, can be exploited to form an almost ideal modelocking mechanism (see § 3.4.2). Thus the Ti:sapphire laser has become the foundation of most ultrafast laser pulse generation, either through direct generation of short pulses, or by providing the pump pulse needed to initiate a nonlinear process to generate an ultrafast pulse. Solid-state lasers can be grouped into two broad classes: transition-metal doped lasers (for example, Cr2+ , Cr3+ , Cr4+ , Ti3+ , Co3+ , etc.) and rare-earth doped
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lasers (for example Nd3+ , Er3+ , Yb3+ , etc.). Color center lasers based on defects have also played a role in the generation of ultrashort-pulses (Yakymyshyn, Pinto and Pollock [1989]) but these systems typically require cryogenic operation. Color center lasers have been replaced by room-temperature systems that provide equivalent or even superior performance (for example, Cr:YAG is roughly equivalent to the NaCl:OH− laser in terms of output power, tuning range, and pump requirements, but operates at room temperature instead of 77 K (Sennaroglu and Pollock [1995]). The rare-earth lasers use electronic transitions of electrons in the 4f shell to create optical gain. The 4f electrons are not involved in the bonding of the dopant ion to the surrounding lattice, in fact the 4f shell is shielded from the surrounding lattice. As a result, rare-earth transitions tend to be spectrally sharp, with limited bandwidth available for pulse formation. The gain bandwidth of most rare-earth doped lasers is sufficient to support pulses on the order of a picosecond. Glass hosts, which are much less homogeneous than crystal lattices, can broaden these transitions further. Fiber lasers, which use glass hosts to hold rare-earth ions such as Yb3+ , have been able to generate pulses with around 100 fs pulse durations (Buckley, Wise, Ilday and Sosnowski [2005]). The transition-metal doped solid-state lasers use electrons in the 3d shell to create optical gain. The 3d-shell electrons form the covalent bonds between adjacent ions that hold the lattice together, so the energy levels of the 3d electrons are strongly dependent on the crystal structure, and are modulated by lattice vibrations. This is why the transition-metal ion lasers tend to be broadly tunable. The tuning range of several transition-ion doped lasers are shown in fig. 4. Ti:sapphire spans from 700 to 1100 nm, Cr:forsterite tunes from 1170 to 1350 nm, and Cr:YAG from 1370 to 1600 nm. These broad ranges arise from the phonon broadening of the energy levels associated with the 3d electrons.
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3.2. Modelocking issues Simply having a broadly tunable system with a large spectral bandwidth is not sufficient for making short pulses. A mechanism is required to phase-lock all of the individual modes in the laser cavity to each other, and to do it in a way that minimizes the pulse duration (DiDomenico [1964]). The simplest mechanism that can be implemented is a high-speed shutter that synchronously opens and closes with a pulse circulating around the laser cavity. Such a shutter introduces no loss to a single pulse that is circulating in the cavity so long as the pulse arrives just after the shutter opens, and passes before the shutter closes. All other light
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Fig. 4. The power tuning curves of three popular lasers used today for ultrashort-pulse generation.
created in the cavity, such as spontaneous emission from the gain medium, will be blocked, or at least attenuated by the shutter, and will therefore never reach threshold. In this picture we can see that a shutter can play a role in pulse shaping if it opens and closes quickly, allowing a very short window of time for the pulse to pass, and effectively cutting any leading or lagging tails from the pulse. A more mathematical description of the shutter is to view it as a modulator that couples all the modes circulating within the laser cavity (Kuizenga and Seigman [1970]). If an electromagnetic wave, I0 exp(−iω0 t) + c.c., with intensity I0 and frequency ω0 travels through a modulator that has a time-dependent transmission T (t) = (1 − a) + a exp(−iωc t) + c.c., the transmitted intensity will simply be the product of incident intensity and the modulation function: I (t) = I0 exp(−iω0 t) (1 − a) + a exp(−iωc t) = I0 (1 − a) exp(−iω0 t) + I0 a exp −i(ω0 ± ωc )t . (3.1)
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We see the amplitude modulation reduces the intensity of the single frequency carrier wave by (1 − a), and transfers energy into two sidebands at ±ωc from the carrier wave frequency. Importantly, the frequency of these sidebands has a phase coherence with the carrier, meaning that all three of the waves have the same phase at time t = 0 and then periodically every 2π/ωc seconds. If the modulator frequency is equal to the cavity mode spacing, c/2l, then this energy will be coupled to adjacent cavity modes. If there is sufficient gain bandwidth, these modes will be amplified by the gain medium and will grow in power. These neighboring modes will in turn be modulated and pick up sidebands, coupling to even
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more modes. This process continues until either the laser runs out of active gain bandwidth, or dispersion causes destructive interference of the extreme modes. A key issue of this process is that the modulation frequency must exactly match the cavity mode spacing of the cavity. If passive modulation techniques are used, where the intracavity pulse initiates the modulation, then synchronicity is automatically enforced and mode coupling occurs naturally. If active modulation is applied, much attention must be devoted to ensuring the applied modulating frequency tracks the cavity mode spacing frequency. The transmission modulator performs another task in the modelocked laser, and that is to sweep out any excess light forming in the cavity outside of the envelope of the pulse. Spontaneous emission randomly contributes photons to the cavity, and if not attenuated, these could circulate in the cavity and be built up by the gain until they began to compete with the main pulse. The role of the modulator is to add enough loss in the cavity when the pulse is not passing to suppress this asynchronous build-up of light.
3.3. Active and passive modulation
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As noted above, loss modulation can be active (externally driven) or passive (typically a saturable absorber). The theory of optical modulators for pulse generation has been described thoroughly in many places (Siegman [1986], Yariv [1975], Keller [2004], Haus [1975]). Active modulation can use electro-optic (EO) or acousto-optic (AO) modulation within the cavity to achieve a continuous train of modelocked pulses. This is called cw modelocking. Acousto-optic modulators are more commonly used because they do not require high-voltage RF drivers, and the glass they use tends to be very low loss. Electro-optic cells usually require a much larger voltage to operate. The EO modulator needs to have sufficient aperture to prevent vignetting loss to a intracavity laser beam, but this generally raises the necessary RF voltages to hundreds of volts. AO modulators are simpler, consisting of a high-index glass with an attached piezoelectric transducer. The piezoelectric transducer is driven by an RF current and launches acoustic waves into the material. The acoustic wave creates longitudinal compression and rarefaction in the medium. The change in local density created by the acoustic wave changes the local index of refraction, creating a diffraction grating in the material. Acousto-optic modelockers are designed to form a standing acoustic wave in the material. This creates a standing index grating that oscillates in time at twice the drive frequency of the current. To couple modes within the cavity, the RF drive frequency is set to exactly 1/2 of the
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cavity mode spacing (c/2l for a linear laser of length l, c/d for a ring laser of circumference d) to create the synchronous modulation. The modulator is oriented in the laser cavity to align the grating at the Bragg angle for the laser light, which maximizes the modulation depth. Even when driven strongly, the loss modulation introduced by an AO cell is typically less than 10%. Active modelocking using AO modulation has been used on many systems, especially Nd:YAG lasers that operate with pulsewidths of 50–100 ps. Because the loss is sinusoidally varying in time, the AO modulator does not provide sharp temporal modulation. Using a Taylor series expansion around the moment that the pulse passes through the modulator (call this t = 0), the transmission window has a 1 − ω2 t 2 temporal shape, which forms a relatively gentle modulation on the traversing pulse. The AO modulator does have the virtue of being externally driven, which makes starting the modelocked pulse train simply an issue of turning on the modulation. This fact may seem trivial, but self-stating of modelocking is a serious problem with some modelocking schemes. cw AO modelocking usually can generate a pulse with a few picosecond duration, and is not useful for true ultrafast work. The most effective generation of ultrashort optical pulses has relied on passive modelocking techniques, because these are the only techniques that are fast enough to shape an ultrashort optical pulse. The passive shutter creates a timedependent loss that can be formed through absorption, or through a spatial loss induced by a nonlinear effect such as the Kerr effect. Time-dependent gain can also be used to modelock a system, especially fast systems such as femtosecond Optical Parametric Oscillators (OPO) (Burr, Tang, Arbore and Fejer [1997]). In synchronously pumped OPOs, the parametric gain only exists when the pump pulse is in the nonlinear crystal. The OPO pulse is shaped by the temporal gain, as internal cavity losses will quickly attenuate all light that falls outside of the temporal gain window created by pumping. Saturable absorption modulation, also called Self Absorption Modulation, works by exploiting the dependence of optical transmission on the population dynamics inside an absorber. An absorbing material will have a net transmission
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T = 1 − exp −σ (ω)l N1 (t) − N2 (t)
(3.2)
that is established by the instantaneous populations of the lower, N1 , and upper, N2 , levels of the absorbing transition. In eq. (3.2), σ (ω) is the absorption cross-section of the transition causing the absorption. The cross-section will vary depending on the exact frequencies in the pulse. l is the thickness of the absorber. The total population of absorbers is assumed to be N0 = N1 + N2 /cm3 . If all the
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atoms are in the ground state, N1 , the transmission is minimum, but if N1 = N2 the transmission is 100%. The incident pulse will initially find the system with N2 empty, and will suffer maximum absorption loss. This attenuates the leading edge of the pulse, essentially trimming it back toward the center of the pulse. As more light is absorbed from the pulse and N2 increases, N1 decreases and the strength of the absorption decreases. This absorption saturation can happen very quickly. Ideally the absorption is saturated by the first edge of the pulse, so the rest can pass through without any loss. The challenge in choosing a saturable absorber is to find a system whose absorption matches the wavelength of the laser, choosing the right length to achieve the necessary modulation depth, and selecting the right volume of material in the beam to ensure the cavity pulses can saturate the transition. Keller [2004] has a nice description of these issues. To act as a modelocker, the saturable absorption must completely recover before the next pulse arrives. If the upper state (N2 ) is long-lived, the modulator will simply act like a Q-switch. Assuming the upper state lifetime meets this condition the saturable absorber can be further categorized as being a “fast” or a “slow” modulator. The terms “slow” and “fast” are defined in terms relative to the optical pulse length. If the saturated absorption recovers on a time scale as fast as the pulse duration, the modulator is described as a fast absorber. If the upper state lifetime of the absorber is long compared to the pulse duration, then the modulator will not recover until after the pulse has passed, and it is described as a slow modulator. An example of a slow modulator is a semiconductor absorber used to initiate femtosecond pulses, while an example of a fast modulator is the Kerr Lens effect. The dynamics of the modelocking process are outlined in the next section. Given a choice, a faster absorption will lead to better pulse shaping because it will be able to trim the tail of the pulse as well as the leading edge.
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3.4. Modelocking schemes Passive modelocking occurs in three basic schemes: (i) the laser can use a slow saturable absorber to trim the leading edge of the pulse, and rely on a dynamic change in gain to trim the trailing edge; or (ii) the laser can have a very fast saturable modulation that trims both leading and trailing edges of the pulse; or (iii) the laser can have a slow saturable absorber to initiate the modelocking, but use a balance of Self Phase Modulation and negative cavity dispersion to create a soliton in the cavity.
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The theoretical foundation for each of these models is derived from Haus’ master equation in his theory of modelocking (Haus [1975]). The Haus theory assumes a steady-state situation where the pulse envelope remains unchanged after each round trip in the cavity. Each action in the laser is modelled as a linearized differential operator that acts on the pulse envelope. Haus’ theory has been the starting point for elaborations which explore active mode locking, harmonic mode locking, and the presence of self phase modulation and solitons. A good review of the theory can be found in the original work or in Keller’s [2004] text. We will not describe the theory here, as it is well explained in numerous other texts. Instead we will focus on a phenomenological description of the three modelocking processes with the aim of giving the reader a sound understanding of the issues and differences in each model. 3.4.1. Passive modelocking with a slow saturable absorber and dynamic gain saturation Passive modelocking with a slow saturable absorber and dynamic gain saturation was developed for modelocking dye lasers (Ippen, Shank and Dienes [1972]). The organic dyes used in these lasers have the broad gain bandwidth necessary for short-pulse generation, but they also are characterized by a very large stimulated emission cross-section and consequently a short excited-state lifetime (usually less than 10 ns). The short upper-state lifetime means that there is little energy storage in a dye laser. The gain cross-section is so large that only a small inversion is needed to establish sufficient gain for lasing, and it is rapidly depleted by stimulated emission. Relaxation oscillations, self Q-switching, and other transient behavior that can arise from stored energy in a population inversion are not usually observed in these lasers. The gain in a dye laser is very responsive to the stimulating field, and can be quickly saturated by an incident pulse. Figure 5 shows the time evolution of the gain in the dye as a pulse travels through it (Keller [2004]). Before the pulse arrives, the dye is pumped by an external beam, and the gain linearly increases with time because there is no stimulated emission to depopulate the excited states. When the pulse arrives, the leading edge sees the maximum available gain, and is amplified. Simultaneously the gain begins to saturate, so that by the time the trailing edge gets to the dye, the gain is significantly reduced, thus trimming the tail of the pulse. Since the overall gain that the pulse sees on average will just equal the loss suffered by the pulse in a round trip through the cavity, the average saturated gain in the dye is equal to the threshold value. With each round trip the trailing edge is attenuated simply through gain depletion and the leading edge is sharpened by the excess gain.
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Fig. 5. The gain, saturable loss, and pulse intensity are plotted as a function of time in the gain medium of a saturable gain medium. The loss is quickly saturated by the pulse, as is the saturable gain.
So what does the saturable absorber do? The saturable absorber does pulse sharpening to the leading edge, but importantly, modelocking is initiated by the saturable absorber; it is the starting mechanism. The loss introduced by the saturable absorber must be slightly larger than the peak gain of the dye just before it is saturated by the pulse. This ensures that any weak spontaneous emission in the cavity will not be able to build up in the cavity. The absorber must recover before the next pulse arrives (usually many nanoseconds later), but the absorption loss does not recover fast enough to shape the trailing edge of an ultrafast pulse. Thus it is called a “slow” process.
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3.4.2. Passive modelocking with a fast saturable absorber Solid-state lasers have typically much longer upper-state lifetimes than dye lasers, and hence much smaller stimulated emission cross-sections. To compensate for the smaller cross-section, it is necessary to have a much larger inversion in the solid-state host to establish comparable gain. To illustrate the magnitude of the differences, table 1 compares the gain cross-section, upper-state lifetime, total inversion Ntot , and change in inversion N when a pulse is amplified, for three Table 1 Comparison of dynamic gain saturation in three lasers Rh6Ga NaClb Ti:sapphirec
τ (nsec)
σ (cm2 )
Ntot
N
Remaining %
4 160 3200
1.36 × 10−16 9 × 10−17 4.8 × 10−19
2.34 × 1010 1.25 × 1012 4.27 × 1013
1.51 ×1010 3.52 × 1010 2 × 1010
35 99.7 99.9
a Schäfer [1977]. b Pollock [1987]. c Moulton [1986].
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different laser systems: a dye laser based on Rh6G, the NaCl color center laser, and the Ti:sapphire laser. The data in this table assume that each laser is operating with the same average output power of 500 mW, has a threshold gain of 10%, and that the intracavity intensity in the gain medium is equal to Isat for each respective laser. The only difference between the systems is the central wavelength, which changes the number of excited states that must contribute to the pulse; the longerwavelength pulses will need more photons to create the same pulse energy. The upper-state lifetimes of these three systems span three decades of range (from 4 ns for the dye to 3.2 µs for Ti:sapphire). The gain cross-sections and total inversions also differ by approximately a factor of 1000. Thus, for the same output power, the Ti:sapphire laser has approximately 1000 times as many excited states capable of participating. The final column shows the impact of extracting 5 nJ of energy from the gain. The absolute change in the inversion for each system is approximately the same, scaling inversely with the photon energy of each system. But the change in inversion normalized to the overall inversion is dramatically different for the systems. This is listed in the last column in percentage, and is identical to the percentage change in gain after the pulse passes. The dye laser suffers a dramatic gain change when amplifying the pulse, whereas the Ti:sapphire laser inversion is essentially unchanged by the passage of one pulse. Because the dye laser has such a high gain, the total number of dye molecules needed to boost the power of the pulse by 5 nJ is comparable to the entire inversion contained in the gain volume. The Ti:sapphire laser, on the other hand, under the same conditions of providing a net gain of 10% and boosting a pulse by 5 nJ, only reduces its inversion by less than 1 part in 1000, which is virtually no change. Thus the gain is effectively unchanged by the passing of the pulse. This means there is no pulse shaping that occurs from a dynamic inversion. Because the excited-state lifetime is so much longer in Ti:sapphire than in the dye, the circulating pulse will go through the gain many, many times, before the gain responds. Each ion will have the same chance of participating in the stimulated emission process as the dye molecules do in the their system, so all the excited states will eventually contribute to the lasing field. The slow response time precludes any ability for pulse shaping to occur due to gain dynamics. The color center laser falls in between these systems, but with only a 0.3% gain modulation, the pulse shaping is negligible in this system as well. To modelock a solid-state laser with a long upper-state lifetime, a fast modulator is required, one that will rapidly trim the tail of the pulse as well as the leading edge. Figure 6 shows the plot of the ideal fast modulator (Keller [2004]). The
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Fig. 6. The loss, gain, and optical pulse intensity are plotted as a function of time for the ideal fast modulator. In this case the losses are minimized when the pulse is at maximum intensity, and quickly increase as the intensity falls. Both the leading and trailing edges of the pulse are trimmed by this modulation.
transmission function basically follows the pulse shape. This modulation has a very fast dT /dt function, which will introduce a large amount of frequency modulation to the carrier wave, coupling many modes with spread bandwidth to the pulse. How does one create such a fast modulation? In principle one could hope to find an absorber that relaxed as fast as the pulse, but this would be challenging, as one would have to first match the absorption wavelength to that of the laser, and as the pulse duration changed, the absorber would have to be adjusted or replaced with one that matched the time dynamic of the pulse. This sounds impossible, but in fact several schemes have been demonstrated to provide this fast modulation, and they work quite well. However, instead of absorption which dissipates energy into internal energy of the absorber, these schemes use an intensity-dependent change in the index of refraction, which introduces a spatial or longitudinal phase change that can be translated into a loss with appropriate cavity coupling or alignment. Additive pulse modulation (APM) was one of the first methods to exploit this scheme (Mollenauer and Stolen [1984], Ippen, Haus and Liu [1989]). In APM, a portion of the output is directed into an external cavity where self phase modulation occurs, which both adds bandwidth to the pulse and chirps the pulse. The chirped pulse is then synchronously coupled back into the laser cavity, where it interferes with the circulating pulse. Because the external pulse is chirped, it will destructively interfere with some portions of the pulse, and constructively interfere in other regions. Figure 7 shows how the pulses can recombine to create a shorter pulse. The self phase modulation effect is very fast; in fact its ultimate speed has not been measured. It has proven responsive on time scales down to a few fem-
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Fig. 7. One pulse that has been chirped by self phase modulation is interferometrically added to an unchirped pulse. If the phase is correct, the centers of the pulses constructively add, but the wings destructively interfere, resulting in a shortened pulse.
toseconds. APM was first discovered with color center lasers (Mollenauer and Stolen [1984]), then applied to several other solid-state laser systems, such as Nd:YAG (Liu, Huxley, Ippen and Haus [1990]) and Cr:forsterite (Sennaroglu, Carrig and Pollock [1992]). The requirement of coupled cavities with interferometrically controlled cavity length proved to be a very challenging task for this style of modelocking. In 1991, Spence and colleagues (Spence, Kean and Sibbett [1991]) discovered the Kerr Lens Modelocking technique. KLM is close to the ideal fast saturable absorber shown in fig. 6. The Kerr Lens effect creates a modulation by self-focussing the beam through an aperture in the cavity, which can lead to a loss if the aperture is hard, or can lead to enhanced gain if it improves the spatial overlap of the cavity mode with the pump beam (Piché and Salin [1993]). Either way, the modulation arises from spatial aperturing, and not from absorption losses. Using KLM is not without its challenges. The laser needs to be aligned close to a stability limit, where small changes in focussing power can lead to significant changes in the beam size, at least at some point in the cavity (Lin, Lai and Hsieh [1995]). Once the laser is aligned and operating, however, the pulse train is very stable. A bigger issue with Kerr Lens Modelocking (KLM) is getting the pulse train started. The peak intensity of a 20 fs pulse is about 106 times higher than the intensity of a cw beam. Unlike the saturable absorber, the Kerr effect is almost unnoticeable at low intensities and has an undetectable impact on mode coupling. The Kerr Lens process does not introduce enough modulation at low intensities
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to enable a random noise spike to build up onto a modelocked pulse. Hence KLM lasers typically do not self start. To get KLM operation started, practitioners often just tap the laser. This mechanical shock induces a relaxation oscillation in the solid-state laser, which has a large population inversion ready to contribute to such an event, and the peak power of this transient spike is often sufficient to initiate the KLM effect. Even though there is no absorption loss in KLM, there is a fixed loss from the hard aperture which introduces a fixed loss to the beam at low intensity. When operating with short pulses, the net loss drops to about 1/3 that of the non-pulsing cavity. There is always loss to the wings of the pulse where the intensity is lower than the peak. In this fashion, even the fastest pulse is always shaped, on both the leading and trailing edges. It is not uncommon to see the output power of a KLM laser increase significantly when it modelocks. 3.4.3. Pulse formation with a slow absorber, no gain modulation, but with soliton formation An effective method for creating ultrafast laser pulses uses a saturable absorber coupled with careful dispersion control of the laser cavity. The semiconductor saturable absorber mirror (SESAM) pioneered by Keller (Keller, Miller, Boyd, Chiu, Ferguson and Asom [1992], Keller, Weingarten, Kärtner, Kopf, Braun, Jung, Fluck, Hönninger, Matuschek and Aus der Au [1996]) has been widely used to serve as a saturable absorber for passively modelocking solid-state lasers. The SESAM has been used in various forms, ranging from a semiconductor quantum well grown on top of a Bragg reflector, to a quantum well deposited on top of a silver mirror. With careful material growth, it is possible to deposit materials which have a very fast relaxation time, on the order of a few picoseconds. Detailed guidelines for designing a SESAM for a modelocked laser (and precautions for preventing Q-switched operation) have been given by Keller [1999] and Paschotta and Keller [2003]. Using low-temperature growth techniques, it is possible to create semiconductor structures which have very fast optical recovery times, on the order of a picosecond. This fast response is helpful for pulse shaping, especially with rareearth solid-state lasers that have narrow bandwidths such as Nd:YAG, but even this relaxation time is not sufficiently fast to pulse-shape ultrafast pulses in the 20 fs domain. Nevertheless, modelocking of broadband lasers such as Cr:YAG and Ti:sapphire with SESAMs has produced pulses shorter than were predicted by the simple passive modelocking theory. Pulses 20–30 times shorter than the saturable absorber lifetime have been routinely generated, which is not expected because
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there is no mechanism for pulse-shaping the trailing edge on the ultrafast time scale. The explanation for this ultrafast behavior comes from a combination of effects, primarily the careful balance between self phase modulation of the pulse and the overall cavity dispersion. This balance is called “soliton” modelocking, because of the analogy to the optical soliton in optical fibers. A soliton is a wave (of any type: electromagnetic, water, acoustic, etc.) that does not distort as it propagates over long distances. When a pulse travels through a dispersive medium, it will experience Group Velocity Dispersion (GVD), due to the second-order dispersion term in the Taylor series expansion of spectral phase [eq. (2.10)]. The Group Velocity Dispersion is called positive if the longer-wavelength light travels faster than short-wavelength light. A pulse travelling through a positive-GVD material will acquire a chirp, and begin to spread out in time. GVD can also be negative, where the short-wavelength light travels faster than the long-wavelength light. Again the pulse acquires a chirp and spreads out in time, but this time the chirp goes from high frequency to low frequency. Sapphire displays a positive GVD over the wavelength range of the Ti:sapphire laser, as does fused silica and most optical glasses. For most materials the GVD will reduce as the wavelengths increases, eventually going through zero then reversing sign as the wavelength extends further into the infrared. In fused silica, this happens above 1.3 µm. Dispersion (of either sign) can also be created in physical optics, such as prism sequences, and in dielectric mirrors. We will explore a few examples of those options in § 5.1. To experience the soliton effect, the pulse must experience self phase modulation (SPM). This is the same effect which causes the Kerr Lens effect, except in this case we are only concerned with the longitudinal effect on the instantaneous phase of the pulse. The SPM occurs due to the time dependence of the refractive index,
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n(I ) = n0 + n2 I (t),
(3.3)
where n2 is the Kerr index. The value of n2 is similar in most dielectric hosts for solid-state laser media; it is 3.3 × 10−16 cm2 /W for sapphire, and 3.2 × 10−16 cm2 /W for fused silica. When an ultrafast pulse induces SPM in a material, the frequency of the waves in the leading edge of the pulse experiences a positive dn/dt and is thus downshifted. The trailing edge sees a negative dn/dt, which upshifts the frequency. If this effect occurs in a material with positive GVD, where lower-frequency waves travel faster than higher-frequency waves, the pulse will spread out even faster
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than normal. Not only is the bandwidth increased, which contributes to increased pulse spreading through normal GVD, but the location of the new frequencies exaggerates the impact. Things work out quite differently if the pulse travels through a system with negative GVD. In this case it is possible to balance the two effects, and in fact generate ultrafast pulses. Consider a pulse experiencing SPM on its leading and trailing edges, but travelling in a material with negative GVD. Energy on the leading edge is frequency-downshifted, and as a results begins to travel slower than the pulse envelope. The downshifted light slowly moves toward the back of the pulse where, due to the negative dn/dt, it will be frequency-upshifted. Now it moves faster than the envelope and advances back to the front of the pulse, where the process repeats. In this picture the energy is constantly up- and down-shifted in frequency, and the GVD keeps it bound to the same envelope. A soliton is developed when the GVD exactly compensates the frequency spreading and the pulse maintains its shape over time. In soliton modelocking, the cavity dispersion is adjusted to provide a net negative GVD, just sufficient to compensate the SPM that occurs in the gain medium each time the pulse travels through it. The saturable absorber provides the starting mechanism for this type of modelocking. The saturable absorber also plays a key role in keeping the laser operating with only one pulse. The soliton effect by itself could easily support one, or two, or dozens of pulses in the cavity; there is no mechanism to limit pulse formation. The saturable absorber, because it absorbs energy from the leading edge of the pulse, causes a temporal shift to the pulse each round trip. The pulse envelope actually moves backward with respect to random noise in the cavity. Before this noise can build up into a pulse, it will run into the main pulse and be incorporated into the lone pulse in the cavity. Thus the saturable absorber provides the self-starting mechanism and keeps spontaneous emission noise under control.
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§ 4. Ultrafast-pulse characterization In order to measure a short time-domain event, it is usually necessary to have something with an even shorter response time. But ultrafast optical pulses currently are the shortest pulses now generated by any means, so this creates an interesting challenge. Also as discussed above, the phase of the pulse plays a major role in determining the overall pulsewidth. So determining both the amplitude and the phase of a pulse is important for the optimization of laser cavities for ultrashort-pulse generation. There are several methods for measuring a pulse, but only a few measure the phase.
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Fig. 8. A collinear autocorrelator: the incident pulse is divided by a beamsplitter, and each pulse is sent to retro-reflectors which translate. The pulses are co-linearly recombined and sent to a SHG crystal. The SHG signal of the superposition is monitored as a function of delay.
Measuring the spectrum is easy to do. One typically uses a monochrometer or interferometer to record the intensity as a function of wavelength of a pulse train. From this one can determine the spectral bandwidth of a pulse, however there is no information about the phase. Without the phase, there are many temporal envelopes E(t) which can fit to a particular spectrum, hence Fourier-transforming the spectrum back to the time domain is inconclusive concerning the temporal characteristics of the pulse. Nevertheless, the most common measurement of ultrafast pulses uses intensity autocorrelation which measures the rms pulsewidth, but provides no information of the spectral phase. The more sophisticated method of frequency-resolved optical gating (FROG), which enables determination of both the temporal and the phase characteristics of a pulse, is outlined in § 4.2.
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4.1. Autocorrelation In intensity autocorrelation, the pulse itself is used as the gate to measure its own temporal shape. The experimental structure for measuring the intensity autocorrelation is shown in fig. 8. The pulse to be measured is split in two by a beamsplitter. These pulses are variably delayed with respect to each other, then recombined on the beamsplitter onto one path. The delayed pulses are then directed onto a nonlinear medium, typically a very thin Second-Harmonic Generating (SHG) crystal, or sometimes a two-photon absorption in a photodiode. The nonlinear process acts like a gate, using one replica of the pulse to gate the transmission of the other pulse. For co-linear recombination, as shown in fig. 8, the pulses travel along the same path and form a superposition of the electric field of the two pulses. The pulses
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Fig. 9. The intensity autocorrelation of a 5 fs pulse. The autocorrelation is symmetric as a function of delay, but does give an excellent indication of the ultrafast speed of this pulse. (Used with permission, from Ell, Morgner, Kärtner, Fujimoto, Ippen, Scheuer, Angelow, Tschudi, Lederer, Boiko and Luther-Davies [2001].)
are delayed with respect to each other by τ , resulting in an electric field as Esum (t, τ ) = E(t) + E(t − τ ).
(4.1)
When sent through a second-harmonic crystal, the resulting second-harmonic intensity is ∞ Ishg (τ ) =
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E(t)2 + 2E(t)E(t − τ ) + E(t − τ )2 2 dt.
(4.2)
−∞
On expansion this expression yields two terms, E(t)2 and E(t − τ )2 , which are independent of the variable delay and represent simple harmonic conversion of the individual pulses. The other terms form an interferogram of the delayed pulses, and provide an 8:1 ratio between the peak and baseline. Figure 9 shows a typical intensity autocorrelation taken from a modelocked laser producing 5 fs pulses. The fringes result from coherent interference between the two pulses as one is delayed with respect to the other. The autocorrelation provides an unambiguous measure of the rms pulsewidth (see § 2.3). This is useful for establishing a maximum value for the pulsewidth, but being an rms measure, it is strongly influenced by any long tails a pulse might have. In general, one is more interested in knowing the FWHM of the pulse, and the pulse shape. Neither of these terms is extractable from the autocorrelation. The intensity autocorrelation is always symmetric, so even a pulse with complex intensity structure will have a smooth symmetric autocorrelation. Certain pulse shapes have been analytically evaluated to predict the relation between the measured autocorrelation and the actual pulsewidth Sala, KenneyWallace and Hall [1980]. A Gaussian pulse will have an autocorrelation FWHM,
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measured in terms of the relative delay τ , that is 1.41 times the actual pulse FWHM. A secant-squared pulse will have an autocorrelation width that is 1.54 times longer. Thus it is necessary to know the actual shape of the pulse envelope in order to predict what the actual pulse duration is. Yet because the phase has not been measured, it is impossible to know with certainty the pulse shape. The autocorrelation gives an approximation of the actual pulse, and until recently was about as good as people could do.
4.2. Frequency-resolved optical gating (FROG) Trebino [2000] in his book very nicely outlines the issues in extracting phase information from an ultrashort-pulse, and the interested reader is encouraged to review that monograph for deeper insight into this problem. The FROG technique is similar to autocorrelation, but one more piece of information is gathered. In this case, the spectrum of the autocorrelation signal is acquired as well as the overall intensity as a function of delay. The optical pulse is again acting as the gate function for probing itself, but unlike the simple autocorrelation, the addition of spectral information enables the phase of the pulse to be estimated. A mathematical description of what is being measured is a spectrogram, which is a two-dimensional plot relating frequency content of a signal as a function of delay: ∞ 2 E (ω, τ ) = E(t)g(t − τ ) exp(−iωt) dt , (4.3) g
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−∞
where g(t − τ ) is the gating function, and the subscript on the summation term indicates the spectrogram uses that particular gate function. For an optical pulse, there are a number of ways to gate the pulse. Secondharmonic generation, where the gate up-converts the input wave, is very similar to the autocorrelator. With sufficient power one can use third-order gates, such as using a field-induced grating or polarization gating that is proportional to the third-order product of electric fields (Trebino [2000, Chapter 3]). Thirdharmonic processes yield shorter gate functions for probing the electric field, which gives more robust results. But third-order processes require higher energy pulses. Given the two-dimensional data, with both intensity and spectrum sampled as a function of delay, it is possible to numerically extract what the pulse actually looks like. With only an intensity autocorrelation, too many temporal pulses can be found that would generate the same spectrum and autocorrelation. But with
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additional connection between the immediate spectrum and autocorrelation as a function of delay, a unique waveform can be found. The FROG algorithm estimates a field, generates a spectrum from that using the Fourier transform, and then compares it to the actual data. From this it generates a correction which gets fed back into the original estimate, and this process repeats. A numeric estimate eventually is found which satisfies both the spectral and temporal data. The process appears to be extremely robust, and has become very useful because it is one of only a few ways to determine what the phase of a pulse is. Once the phase of a pulse is known, it is possible to adjust the cavity if necessary, with the target of creating a pulse with a flat phase profile.
§ 5. Ultrafast Ti:sapphire lasers and amplifiers The Ti:sapphire laser has many desirable properties, especially a broad gain bandwidth spanning 700–1100 nm, high thermal conductivity, and large energy storage capacity, approaching 1 J/cm3 . The large upper state of the system (3.2 µs) at first worried people about this laser’s ultimate ability to generate really short pulses, as discussed in § 3.4.2, but advances in SESAMs, soliton modelocking, and ultimately in Kerr Lens modelocking answered these concerns.
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5.1. Dispersion control The most dramatic advance in Ti:sapphire lasers has been the reduction of pulse duration, to the point where these lasers now hold the record for direct ultrashortpulse generation. This pulse reduction is accomplished primarily by reducing overall cavity dispersion in the laser. This can be done through a combination of methods. Minimizing the amount of optical material in the cavity is a direct way to reduce dispersion. The fastest pulses being currently generated use Ti:sapphire crystals only 2–3 mm long (to list three examples, see Schibli, Kuzucu, Kim, Ippen, Fujimoto, Kärtner, Scheuer and Angelow [2003], Rundquist, Durfee, Chang, Taft, Zek, Backus, Murnane, Kapteyn, Christov and Stoev [1997], and Diddam, Bartels, Ramond, Oates, Bize, Curtis, Bergquist and Hollberg [2003]). But even this short length of material introduces dispersion issues that must be addressed to minimize pulse duration. There are two popular methods for dispersion compensation, either using intracavity prisms, or using specially designed mirrors. The prism method was introduced with the colliding-pulse dye laser (Fork, Martinez and Gordon [1984]), but is still widely used in many ultrafast systems.
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Fig. 10. The prisms introduce controllable GVD through the physical path difference experienced by long-wavelength and short-wavelength light travelling through these pairs. The free space propagation creates negative GVD, while the path through the glass adds a positive GVD. The prisms can be slightly moved in or out to adjust the amount of glass in the cavity.
The basic geometry is shown in fig. 10, where four prisms are oriented at minimum deviation, and the exit face of prism ‘a’ is parallel to the entrance face of prism ‘b’. The first prism disperses the components of an incident beam according to their wavelength, and with this geometry the shorter-wavelength components travel further in the air before entering the second prism. The longer-wave components travel a shorter distance through the air, but then have to travel through more glass than the rays located near the apex. Careful geometrical analysis shows there is a difference in path length for the extreme rays, which results in a negative GVD (i.e. the short wavelengths arrive before the long wavelengths). This negative GVD is proportional to the distance between the prisms and the dispersion of the prisms, so it is possible to adjust the design to compensate arbitrary amounts of positive GVD in a cavity. The glass in the prisms has a positive GVD, and since pushing the prisms toward their apexes does not affect the diffraction of the beams exiting and entering the faces, it is possible to adjust how much glass the beam travels through. Thus by moving the prisms in or out, positive GVD can be controllably added to the cavity, allowing smooth adjustment for fine-tuning the dispersion compensation. The second popular method for controlling dispersion is to design dielectric mirrors with the proper dispersion to balance the effects encountered in the rest of the cavity (Szip˝ocs and K˝oházi-Kis [1997]). Figure 11 shows a typical mirror structure and the paths of a short-wavelength and a long-wavelength beam. By creating a longer path for the longer-wavelength beam, negative GVD can be incorporated in the structure. While this looks simple in concept, the details of these mirrors are daunting. The presence of reflections back into the mirror at the top surface of the mirror where it meets air result can result in strong phase shifts similar to those occurring in a Gires–Tournois Interferometer (Schibli, Kuzucu, Kim, Ippen, Fujimoto, Kärtner, Scheuer and Angelow [2003]). Thus a great deal of effort goes into achieving a good AR coating on the top of these mirror stacks.
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Fig. 11. A chirped mirror is made by adjusting the thickness and fractions of the high-index layers in a controlled fashion. In this example, long wavelengths must travel further before being reflected, so they experience a longer group delay, yielding a negative GVD.
Often the mirrors come in pairs, which individually have strong modulations in their GVD, but which are complementary so that the net effect after bouncing off of both mirrors is smooth. The mirror design allows for higher orders of dispersion to be compensated.
5.2. Ultrashort Ti:sapphire lasers
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Short optical pulses require large spectral bandwidths. To achieve this, the total dispersion in the cavity must be near zero, with small amounts of negative dispersion to compensate any SPM in the gain medium. Recalling eqs. (2.9) and (2.10), the group delay of a pulse can be expressed in terms of higher orders of dispersion. 1 T (ω) = dφ/dω = φ (ω0 ) + φ (ω0 ) ω + φ (ω0 )( ω)2 + · · · . 2
(5.1)
Ideally every frequency in the pulse will experience the same group delay per round trip in the cavity. Thus, the second-, third- and higher-order terms need to be as near zero as possible. The second-order term φ (ω) is the GVD, and is straightforward to control using prisms or a well-designed set of mirrors as described above in § 5.1. GVD can be adjusted to be nearly zero. Correcting for φ (ω) is more challenging. The presence of third-order dispersion becomes significant out in the wings of the pulse, due to the ( ω)2 term. The prism system shown in fig. 10 does not allow for adjustment of third-order dispersion, but it is possible to attack third-order dispersion by carefully choosing the glass used to make the prism. There is also third-order dispersion in the Ti:sapphire crystal, so one of the driving reasons why ultrashort-pulse lasers now use short length (e.g. 2 mm) Ti:sapphire crystals is to minimize this effect. One technique for minimizing third-order dispersion is to choose the operating wavelength of the laser carefully (Rundquist, Durfee, Chang, Taft, Zek, Backus,
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Murnane, Kapteyn, Christov and Stoev [1997]). At a wavelength of 850 nm, the second- and third-order dispersion in Ti:sapphire and fused silica is nearly zero.
5.3. Ti:sapphire amplifiers The output of ultrafast lasers is usually in the 10 nJ range per pulse. When converted to peak power, the numbers are impressive. A 10 nJ pulse of 10 fs duration has a peak power of 1 MW. If this can be tightly focussed to a spot size of 10 µm, the peak intensity reaches ∼1012 W/cm2 . The corresponding electric field √ is E ≈ I × 2Z0 ≈ 3 × 107 V/cm. While this is a formidable electric field, it is much less than the inter-atomic electric field that holds electrons to nuclei, which is on the order of 109 V/m. To generate fields that are comparable to the atomic potentials, more energetic pulses are needed. Without dramatic increases in available pump power, it is unlikely that modelocked lasers will be able to directly generate pulses of substantially higher energy. One can increase the storage time of the cavity by increasing the round-trip time for a single pulse. Dewald has reported 500 nJ pulses of 50 fs duration for a Ti:sapphire laser operating at 6.6 MHz (Dewald, Lang, Schröter, Moshammer, Ullrich, Siegel and Morgner [2006]). Cavity dumping techniques are being pursued which may open the door to 1 µJ pulses, but to date these have yielded long (>100 fs) pulses. See for example (Killi, Steinman, Döring, Morgner, Lederer, Kopf and Fallnich [2005]). It is doubtful powers will ever reach the µJ level with femtosecond pulses without external amplifiers. Pulse amplifiers, on the other hand, are fully capable of boosting the energy of a pulse by more than 106 . The key technique behind this ability is called chirped pulse amplification. The chirped pulse technique used dispersion to first spread the pulse out in time, creating a long, low-amplitude pulse. This long pulse can then be amplified to high total energy, but because it is long, the peak intensity stays safely below the nonlinear limits. Once amplified, the pulse can be re-compressed in time by reversing the dispersion used to stretch the pulse, effectively concentrating the energy back into an ultrashort-pulse. The recompression occurs outside of a dielectric, and thus there is no danger of breakdown or self phase modulation. The Chirped Pulse Amplifier consist of several stages. It starts with an ultrafast Ti:sapphire laser that generates a pulse train of femtosecond pulses. A Pockels cell is used to select an individual pulse from the train for subsequent amplification. The wavelength and pulse duration of this seed laser must be carefully chosen. One issue is the spectral narrowing that occurs when a pulse is amplified by 106 or higher. The spectrum of ultrashort-pulses is quite broad, for example a
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20 fs pulse will have a FWHM spectral bandwidth of approximately 40 nm. The gain bandwidth of Ti:sapphire is approximately 250 nm, but this gain is not spectrally flat. With each pass through the amplifier, the wings will receive a relatively smaller boost. When the pulse emerges from the amplifier, its bandwidth will be narrower, and its pulse duration will necessarily be longer. The short pulse is sent through a stretcher, which consists of a pair of gratings oriented to introduce a positive GVD. The difference in path length travelled by the long- and short-wavelength components determines the GVD, and like the prisms, this GVD is adjustable with separation and with the angular dispersion of the gratings. With light requiring 3.3 ps to travel 1 mm, it is relatively straightforward to stretch a pulse to over 50 ps with only 15 mm path delay. This idea was originally proposed for pulse compression (Treacy [1969]). Power limitations in solid-state laser amplifiers revived this idea, and now pulse expansions of over 1000 are not uncommon (Pessot, Maine and Mourou [1987]). The pulse exiting the grating pair is very highly chirped, but with much lower peak power. The chirped pulse is then multiply passed through a strongly pumped Ti:sapphire crystal, and when it has been amplified to the desired energy level, it is again directed through a grating pair, this time oriented to provide negative GVD of equal magnitude to the initial stretcher. The resulting pulse is extremely powerful, with energies exceeding 1 mJ in pulses shorter than 20 fs. These pulses can then be used for fundamental studies, or for generating even shorter pulses.
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§ 6. Attosecond pulses As a final example of ultrashort-pulses, we will conclude with a description of the shortest pulses yet reported. Looking at the lasers available today, the prospects of ever directly generating sub-femtosecond pulses do not look promising. As discussed in § 2.4, it is necessary to have at least one full cycle of an optical wave if the pulse is going to propagate. Current results in ultrashort-pulse generation are nearly at the single-cycle limit of known lasers (Binhammer, Rittweger, Morgner, Ell and Kärtner [2006]) so shorter pules are going to require UV or even X-ray lasers, and prospects for either of these types of laser are not very promising at the moment. However, the impressive power from the amplified femtosecond pulses is enabling a new way to generate even shorter pulses. This technique was proposed in 1993 (Corkum [1993]) based on the interaction of intense fields with an electron wavepacket on an atom. Today this process is called high-harmonic generation (Levesque and Corkum [2006]). A high-energy (mJ) femtosecond pulse is focused into a thin gas, where the intense electric field extracts an electron from
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Fig. 12. The potential well of a bound atom is strongly tilted in a strong electric field. The electron tunnels out and gains momentum until the field reverses. It returns, increasing momentum, and recollides with the bond state.
an atom or molecule near the crest of the field. The electron is pulled away from its parent ion by the field, and then is driven back when the field reverses. This entire process occurs within a quarter cycle of the applied optical pulse. The electron can then re-collide with its parent ion, and generate soft X-rays. Since the strong time-dependent field of the light pulse directs the electron with subcycle precision, one can control and measure attosecond phenomena. In order to cause tunnelling of an electron from a bound orbit, the applied electric field must be comparable to the inter-atomic fields, which are on the order of 109 V/cm. A 1 mJ, 10 fs pulse has a peak power of about 0.1 terawatt. If focused to a 100 µm diameter the intensity reaches 1015 W/cm2 , which has an electric field in vacuum of about 109 V/cm. So these measurements need pulse energies available only from amplified Ti:sapphire systems. This technique has been able to generate coherent UV and soft X-ray radiation with photon energies up to 1.2 keV (Seres, Seres, Verhoef, Tempea, Streli, Wobrauschek, Yakovlev, Scrinzi, Spielmann and Krausz [2005]). A schematic drawing of the process is shown in fig. 12. When the driving field becomes strong enough, an electron can tunnel from its host atom. The probability of tunnelling is maximal near the peak of the applied wave, but there is a stochastic nature to this process. Once the electron tunnels, it creates a bound wave function and a continuum wavepacket that escapes the ion. One can think of this as an electron interferometer. The tunnelling acts like a beamsplitter for the electron interferometer, and the light pulse creates the probe electron from the atom’s own electrons. So there is a built-in coherence. The free electron is first accelerated away by the field, but when the field reverses, the electron is driven back toward the ion. As it passes through, it interferes with its bound wavepacket. As shown in fig. 12, the tunnelled electron has picked up a lot of momentum so it has a high k-vector. When the two packets super-
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impose, they interfere. The resulting composite wavefunction moves side to side in the direction of the laser field. This rapid oscillating motion continues as long as the re-collision lasts, which is basically the transit time of the electron across the ion wavefunction. The faster the re-collision electron moves, the faster the oscillation of the wavepacket. The wavepacket represents charge moving, so the dipole emits radiation at the frequency of the oscillation. From the polarization of the radiation, one can determine the orientation of the electron as it crosses the atom. This signature provides a window into the electronic structure of the ion, and provides a source of attosecond pulses of coherent light. This system is useful as a light source too. In a real experiment the light pulse interacts with a gas of atoms. As long as the density does not become too high and the interaction region is confined, the emission from each atom in the volume adds coherently. A single-cycle driver will just create one burst of UV and soft X-rays, but a driver pulse with multiple cycles will generate a train of pulses. Due to the temporal coherence of the driving field, the emission will have harmonic features in its energy spectrum. This periodic emission appears as a ladder of harmonics extending to the maximum kinetic energy of the re-collision electron plus the ionization potential of the atom. Attosecond technology has tremendous promise. A classical electron completes a Bohr orbit in about 150 attoseconds. Although quantum mechanics says that the Bohr orbit is not observable, it does allow one to observe something comparable. Using the electron to probe its own atom, the attosecond pulse may enable the observation of the dynamics of electronic wavepackets in atoms and molecules, and to do so with attosecond precision.
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§ 7. Conclusion This chapter has tried to provide an overview of the basic physics and technology behind ultrafast optical pulses. The term “ultrafast” could aptly describe the progress in this field, as it does the product of the research. Each decade dramatic new developments have occurred that enabled new abilities. There is no reason to expect that this pace will slow down. The application of ultrashort-pulse sources seems to have no limit either. On one extreme, ultrashort-pulses are being used to generate higher harmonics which yield even shorter temporal pulses, while on the other extreme, the frequency ladders created by an ultrashort-pulse promise to create the most stable, reliable frequency standards ever known. Development of phase-coherent pulse envelopes, needed for extending the ability to combine different lasers for frequency-ladder
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work, will be immediately helpful to the work on attosecond-pulse generation, where it will allow more precise control of when an electron tunnels from its host ion. There is a tremendous synergy in this field, and undoubtedly many new applications will arise in the next decade that will further exploit these amazing tools.
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E. Wolf, Progress in Optics 51 © 2008 Elsevier B.V. All rights reserved
Chapter 5
Quantum imaging by
Alessandra Gatti, Enrico Brambilla, Luigi Lugiato CNISM, Dipartimento di Fisica e Matematica, Università dell’Insubria, Via Valleggio 11, 22100 Como, Italy
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ISSN: 0079-6638
DOI: 10.1016/S0079-6638(07)51005-X 251
Contents
Page § 1. Introduction to quantum imaging . . . . . . . . . . . . . . . . . . . .
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§ 2. Quantum spatial intensity correlations in optical parametric downconversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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§ 3. Ghost imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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§ 4. Image amplification by parametric down-conversion . . . . . . . . . .
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§ 5. The quantum laser pointer . . . . . . . . . . . . . . . . . . . . . . . .
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§ 6. Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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§ 1. Introduction to quantum imaging With respect to the classical area of imaging, the field of Quantum Imaging (QI) aims to devise novel techniques for optical imaging and parallel information processing at the quantum level, by exploiting the quantum nature of light and the intrinsic parallelism of optical signals. As a matter of fact, the cross-section of a broad-area radiation beam offers the possibility for multichannel configurations, i.e. the availability of several “quantum” channels, each corresponding to a small area in the section. For example, in each of these independent regions the light may be squeezed, or there may be quantum entanglement between pair of channels. Essentially this approach paves the way to parallel quantum information processing, basically different from the sequential configuration which characterizes, for instance, most schemes of quantum cryptography, in which one detects one photon (or one photon pair) at a time. Needless to say, such a parallel configuration allows a remarkable increase of the information capacity of the system. The starting point for QI is provided by the topic of the spatial aspects of quantum-optical fluctuations. Before the late 1980s, the literature on that topic was quite limited (Drummond and Eberly [1982], Heidmann, Reynaud and Cohen-Tannoudji [1984], Grangier, Roger, Aspect, Heidmann and Reynaud [1986], La Porta and Slusher [1991]). The interest toward squeezed states of light relied mainly on the possibility of producing (by optical homodyning) photon beams with a statistics in time more regular than the statistics of random events. Sokolov and Kolobov addressed the issue of generating states of light with a statistical distribution of photons which is regular not only in time but also in space, that is in the cross-section of the beam (Kolobov and Sokolov [1989], Kolobov [1991, 1995], Kolobov and Kumar [1993]). Especially, they introduced the concept of localized squeezing, i.e. squeezing in a small region of the transverse section. Their investigations concerned mainly the squeezed light generated by the down-conversion process in a χ 2 crystal, in the cavityless configuration (optical parametric amplifier, OPA). Systematic investigations on quantum aspects of spatial patterns generated by nonlinear optical cavities, and, conversely, on spatial aspects of non-classical
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states of the radiation field, were activated by our group (Lugiato and Castelli [1992], Lugiato and Gatti [1993], Gatti and Lugiato [1995], Lugiato and Grynberg [1995], Lugiato, Gatti and Wiedemann [1997], Gatti, Wiedemann, Lugiato, Marzoli, Oppo and Barnett [1997], Marzoli, Gatti and Lugiato [1997], Lugiato, Gatti, Ritsch, Marzoli and Oppo [1997], Gatti, Lugiato, Oppo, Martin, Di Trapani and Berzanskis [1997], Lugiato and Grangier [1997], Castelli and Lugiato [1997], Grynberg and Lugiato [1993]). The reason why spatial transverse patterns in nonlinear optical systems display relevant quantum aspects is intrinsic. As a matter of fact, in nonlinear optical processes there is typically the simultaneous creation of two or more photons. As explained e.g. in Lugiato, Brambilla and Gatti [1998], this feature is basic for the understanding of optical pattern formation. On the other hand, the created photons are in a state of quantum entanglement, which is the origin of the quantum nature of spatial correlations of the optical patterns. In particular, we introduced the concept of “Quantum Images” (adopting a name which was suggested to us by Peter Knight), i.e. images which appear in spatial correlation functions rather than in the intensity distribution of the radiation field. Even more important, spatial correlation functions are capable of giving clear spatial evidence of the quantum nature of quantum fluctuations and play a central role in the field of QI. Most of the papers cited above deal with optical parametric down-conversion (PDC). More precisely, they analyze the medium-to-high gain regime, in which the signal/idler field is conveniently described in terms of continuous variables such as amplitude and phase, or quadrature components. On the other hand, a huge literature on PDC deals with the regime of ultra-low gain, in which signal/idler photon pairs are detected one by one, usually in coincidence. Papers relevant for QI are, for example, Hong and Mandel [1985], Klyshko [1988a], Barbosa [1996], Pittman, Strekalov, Klyshko, Rubin, Sergienko and Shih [1996], Burlakov, Chekhova, Klyshko, Kulik, Penin, Shih and Strekalov [1997], Rubin [1996]. A most interesting area of research which was opened by such researches is that of the so-called ghost imaging, which will be discussed later in this chapter. In this introductory section, we will discuss the spatial aspects of squeezing (in particular local squeezing) in §§ 1.1 and 1.2 and of quantum images (§ 1.4), with related aspects such as the interference of signal and idler waves (§ 1.5). In the following sections, we will discuss quantum spatial correlations in spontaneous PDC (§ 2), ghost imaging (§ 3), image amplification by PDC (§ 4), and the detection of small displacements with a precision beyond the standard limit (quantum laser pointer, § 5). In the concluding § 6 we sketch or mention some topics which are not fully included in this review. Previous reviews related to
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Fig. 1. (a) Scheme for parametric down-conversion of type I. (b) Parametric amplification of a plane wave. q is the component of the wavevector in the planes orthogonal to the pump propagation direction.
the field of QI are by Lugiato, Gatti and Wiedemann [1997], Kolobov [1999], Lugiato, Brambilla and Gatti [1998], Lugiato, Gatti and Brambilla [2002]. A volume on QI has been published recently by Kolobov [2006a].
1.1. Optical parametric down-conversion of type I We start by illustrating some models for PDC of type I, both in the single-pass and in the cavity configuration. A model for a single-pass optical parametric amplifier (OPA) of type I, including the transverse degrees of freedom, was derived by Kolobov and Sokolov (Kolobov and Sokolov [1989, 1991], Kolobov [1991, 1995]. They considered a thin slab of a nonlinear type I crystal (Boyd [1992]) of length lc , ideally infinite in the transverse direction, pumped by a coherent plane-wave field of frequency 2ωs (fig. 1a). They took into account nearly degenerate and nearly collinear phase matching conditions into the crystal and neglected the pump depletion along the material. A fraction of the pump photons are down-converted into signal/idler
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photon pairs, with temporal frequencies distributed over a broad band around the degenerate frequency ωs and propagation directions filling a broad cone around the pump propagation direction (z direction). However, each elementary frequency down-conversion process corresponds to the generation inside the crystal of a couple of signal photons, at frequencies ωs + Ω and ωs − Ω, and with transverse wavevectors q and − q , so to preserve energy and transverse momentum. Notice that if, in addition to the pump field, we inject a coherent plane wave with frequency ωs + Ω and transverse wavevector q (fig. 1b), in the output we have a signal wave which corresponds to an amplified version of the input wave, and for this reason the system is called an optical parametric amplifier. Because of the pairwise emission of photons, there is also an idler wave which, close to degeneracy, is symmetrical with respect to the signal wave. On a more formal ground, let us denote by ain ( x , t) the signal/idler complex amplitude envelope operator at the input endface of the crystal slab, and by aout ( x , t) the envelope operator at the output endface, where t indicates time and x ≡ (x, y) is the coordinate vector in the endface planes. We expand ain and aout in Fourier modes in space and time: d q dΩ x , t) = q , Ω)eiq ·x −iΩt , ain ( (1.1) √ ain ( 2π 2π d q dΩ aout ( (1.2) x , t) = q , Ω)eiq ·x −iΩt . √ aout ( 2π 2π By solving the field propagation equations along the crystal slab, Sokolov and Kolobov obtained linear input/output relations in the form of a squeezing transformation:
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† aout ( q , Ω) = U ( q , Ω)ain ( q , Ω) + V ( q , Ω)ain (− q , −Ω),
aout (− q , −Ω) =
(1.3)
† U (− q , −Ω)ain (− q , −Ω) + V (− q , −Ω)ain ( q , Ω).
(1.4) Explicit expressions for the gain coefficients U ( q , Ω) and V ( q , Ω) can be found for example in Kolobov [1999]. They depend on the pump amplitude, on the q , Ω) − parametric gain and on the phase mismatch along z: Δ( q , Ω) = [k1z ( k1z (− q , −Ω) − k0z ]lc , where k1z and k0z are the z-components of the wavevectors for signal and pump, respectively. They obey the unitarity conditions 2 2 U ( q , Ω) + V ( q , Ω) = 1 and U ( q , Ω)V (− q , −Ω) = U (− q , −Ω)V ( q , Ω). Recently, a more general model was formulated by Brambilla, Gatti, Bache and Lugiato [2004], Gatti, Zambrini, San Miguel and Lugiato [2003] for both type I
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and type II phase matching, to include also the finite spatial and temporal profile of the pump field. We return to that model in § 2.3. When the χ (2) material is enclosed in an optical resonator, the device takes the name of optical parametric oscillator (OPO) and presents a characteristic threshold of the pump field intensity above which a macroscopic down-converted field is emitted. The spatial quantum properties of the signal field generated by a degenerate OPO have been analysed in the context of several models (Lugiato and Gatti [1993], Gatti and Lugiato [1995], Lugiato and Marzoli [1995], Lugiato and Grynberg [1995], Lugiato, Gatti and Wiedemann [1997], Gatti, Wiedemann, Lugiato, Marzoli, Oppo and Barnett [1997], Marzoli, Gatti and Lugiato [1997], Lugiato, Gatti, Ritsch, Marzoli and Oppo [1997], Gatti, Lugiato, Oppo, Martin, Di Trapani and Berzanskis [1997], Lugiato and Grangier [1997], Castelli and Lugiato [1997], Grynberg and Lugiato [1993], Petsas, Gatti, Lugiato and Fabre [2003], Lopez, Gigan, Treps, Maître, Fabre and Gatti [2005]). All these models consider cavities with a certain degree of spatial mode degeneracy. The prototype is the OPO with planar mirrors, where transverse translational symmetry is assumed, so that associated to each longitudinal cavity resonance there is a continuum of plane-wave transverse modes of the form ∝eiq ·x . Closer to experimental implementations are cavities with spherical mirrors, which support Gauss–Laguerre modes, labelled by discrete indexes (p, l). Models have considered the quasi-planar case, where the transverse mode separation is much smaller than the cavity linewidth, and the confocal resonator, consisting of two mirrors separated by a distance equal to their radius of curvature, where the cavity resonances consist of whole families or transverse modes with alternatively even or odd spatial parity. The OPO with planar mirror represents the prototype model for describing quantum spatial effects in optics, and has been the object of extensive theoretical analyses. They considered type I, degenerate, collinear phase-matching condition in the nonlinear crystal (Boyd [1992]). The medium is enclosed in a single-port cavity, pumped by a plane-wave field of frequency 2ωs , and it is able to partially convert power from the pump frequency to the signal frequency ωs (fig. 2). The model, which is the generalization of the model of McNeil, Drummond and Walls [1978], Drummond, McNeil and Walls [1980] to include spatial effects, assumes that the cavity is at resonance both with the pump and the signal frequency, and that all the other longitudinal cavity resonances can be neglected. The classical dynamics of such a system has been analysed by Oppo, Brambilla and Lugiato [1994], Oppo, Brambilla, Camesasca, Gatti and Lugiato [1994]. In the quantum description, the intracavity pump and signal fields are two quantum x ), A( x ), associated to the longitudinal resonances closest to 2ωs operators A0 (
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Fig. 2. Degenerate optical parametric oscillator with plane mirrors. A pump field Ain 0 of frequency 2ωs is injected into the cavity. PDC in the χ (2) medium gives rise to the emission of signal photons at frequency ωs (field A). The cavity is doubly resonant both at the pump and at the signal frequencies. Mirror M2 partially transmits the signal beam and totally reflects the pump; mirror M1 is totally reflecting for the signal, while it partially transmits the pump.
and ωs , respectively. ( x is the coordinate in the transverse plane.) As it is appropriate for an open system, the model is formulated in terms of a Master Equation for the reduced density operator of the two intracavity fields (see Gatti and Lugiato [1995], Gatti, Wiedemann, Lugiato, Marzoli, Oppo and Barnett [1997] for more details). The quantum-mechanical interaction Hamiltonian governing the OPO has the typical form g d x A† ( HINT = ih¯ (1.5) x )2 A0 ( x ) − h.c. , 2 describing the annihilation of one pump photon accompanied by the creation of two signal photons and vice versa; g is the coupling constant proportional to the χ (2) coefficient of the material. An interesting approach that permits one to describe the fast dynamics of quantum fluctuations, is that of classical-looking Langevin equations formulated by Gatti, Wiedemann, Lugiato, Marzoli, Oppo and Barnett [1997], Lugiato, Gatti, Ritsch, Marzoli and Oppo [1997], where the Master Equation is turned into a set of stochastic differential equations, for c-number fields in the Wigner representation. By introducing appropriate scaled versions α, α0 of signal and pump fields, the Langevin equations for the OPO with plane mirrors read
∂ 2 α( x , t) + α0 ( x , t)α ∗ ( x , t) α( x , t) = γ − 1 + iθ − i∇⊥ ∂t 1 ξ( x , t) , +√ (1.6) γ0 nth
γ1 2 ∂ x , t) = γ0 − 1 + iθ0 − i ∇⊥ α0 ( x , t) + E0in − α 2 ( x , t) α0 ( ∂t 2γ0
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2 ξ0 ( x , t) . γ0 nth
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(1.7)
In these equations: 2 is the transverse Laplacian, describing diffraction in the paraxial approx– ∇⊥ imation, and transverse coordinates are scaled to the diffraction length ld ∝ √ λL/T , with λ the wavelength, L the cavity round-trip length and T the transmission coefficient of the coupling mirror; – γ , γ0 denote the cavity damping rates of the two fields; – θ = (ωc − ωs )/γ , θ0 = (ω0c − 2ωs )/γ0 are the two cavity detunings of the signal and pump frequencies from the closest longitudinal resonances ωc , ω0c ; – E0in is the scaled amplitude of the input field; x , t) are white-noise terms that model the vacuum fluctuations en– ξ( x , t), ξ0 ( tering the cavity input–output ports. They are Gaussian stationary stochastic processes, with zero average, and non-vanishing correlations: ! ! 1 ξ ∗ ( x , t)ξ( x , t ) = ξ0∗ ( x , t)ξ0 ( x , t ) = δ( x − x )δ(t − t ). 2 2
(1.8)
The parameter nth = γg 2 ld2 represents the number of pump photons in the characteristic area ld2 which are needed to trigger the threshold for signal generation, and it is hence a measure of the system size (Carmichael [1993]). Notice that by dropping the noise terms, or equivalently in the limit nth → ∞, one recovers the classical nonlinear equations derived in Oppo, Brambilla and Lugiato [1994], Oppo, Brambilla, Camesasca, Gatti and Lugiato [1994]. A similar quantum model for a nondegenerate OPO is described in Szwaj, Oppo, Gatti and Lugiato [2000]. Below the threshold for signal generation, the pump depletion can be neglected, and the pump can be approximated as a classical plane wave of amplitude proportional to the input beam, with no quantum fluctuations. In this way a much simpler linear model can be obtained. By solving this model, one obtains for the signal field operator input–output relations that have the same form as eqs. (1.3) and (1.4), with different expressions for the coefficients U and V , which depend on the cavity parameters (Kolobov and Lugiato [1995]). Quantum treatments of the spatial features of OPOs with spherical mirrors have been generally limited to the behavior below threshold (Lugiato and Marzoli [1995], Gatti, Wiedemann, Lugiato, Marzoli, Oppo and Barnett [1997], Marzoli, Gatti and Lugiato [1997], Gatti, Lugiato, Petsas and Marzoli [1999], Gatti, Petsas, Marzoli and Lugiato [2000]). The models are simplified by the assumption of a singly resonant cavity, such that the cavity mirrors are completely transmitting
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Fig. 3. Degenerate optical parametric oscillators with curved mirrors. A plane-wave pump field of frequency 2ωs is injected into the cavity and is completely transmitted by it. PDC in the χ (2) medium gives rise to the emission of signal photons of frequency ωs . The signal field A is only partially transmitted through the outcoupling mirror M2 , the other mirror M1 being totally reflecting at ωs .
for the pump field, while the down-converted field is close to a cavity resonance (fig. 3). If, in addition, a plane-wave pump is assumed, linear input/output relations can be derived for mode operators on Gauss–Laguerre modes. These transformation take the usual form of a squeezing transformation, similarly to eqs. (1.3) and (1.4) (see e.g. Lugiato and Marzoli [1995]). Models for OPOs with confocal resonators have been derived and analysed with various degrees of approximation by Lugiato and Grangier [1997], Petsas, Gatti, Lugiato and Fabre [2003], Lopez, Gigan, Treps, Maître, Fabre and Gatti [2005].
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1.2. Spatially multimode versus single-mode squeezing Almost all literature on squeezing considers single-mode squeezing. A good level of squeezing is detected only when the local oscillator (LO) is matched to the squeezed spatial mode and, in addition, the whole beam is detected. Conversely, if only a portion of the beam is detected the squeezing is immediately degraded, because a portion of a beam necessarily involves higher-order spatial modes, in which squeezing is absent. What we can call local squeezing, i.e. squeezing in small regions of the transverse plane, can be obtained only in the presence of spatially multimode squeezing, i.e. squeezing in a broad band of spatial modes. Kolobov and Sokolov (Kolobov and Sokolov [1989, 1991], Kolobov [1991]) were the first to address the issue of local squeezing. They showed that efficient quantum noise reduction can be observed even in small regions of the output signal beam of a single-pass OPA of type I, provided the size of the detection region is not smaller than the coherence transverse length xcoh , given by the inverse of
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the spatial bandwidth of the amplifier q0 . This hence introduces a lower limit for the dimension of the area over which squeezing can be observed (coherence area). An excellent candidate for generating multimode squeezed light is also parametric down-conversion taking place in a mode-degenerate cavity, e.g. a confocal degenerate OPO below threshold. In a confocal OPO, high levels of squeezing can be observed, independently of the shape of the LO, in small regions of the transverse section of the beam, provided the region is symmetrical with respect to the cavity axis. In the ideal limit of perfect confocality, infinite transverse size of the system, and thin crystal length, a high level of squeezing can be detected in arbitrarily small regions (Lugiato and Grangier [1997]). However, when dropping some of these limiting assumptions a lower limit appears for the size of the region where good squeezing can be detected (Petsas, Gatti, Lugiato and Fabre [2003]). In particular, if the thin-crystal approximation is dropped, as appropriate for describing the typical experimental set-up where the crystal length is on the order of the Rayleigh range of the resonator, it turns out that the size of the region where efficient squeezing can be detected is again limited by the transverse coherence √ length of the crystal xcoh ∝ λlc (Lopez, Gigan, Treps, Maître, Fabre and Gatti [2005]). Refined techniques have been devised by Kim and Kumar [1994], Kim, Li and Kumar [1994], Choi, Li, Kim and Kumar [1997].
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1.3. Spatial structure of squeezed vacuum states in the degenerate optical parametric oscillator below threshold A related question arises, whether it is possible in a measurement to probe the level of squeezing in a single cavity mode (such as a plane-wave mode, or a Gauss–Laguerre mode), rather than in a region of space. To answer this question we have to return to the very definition of spectrum of squeezing, through the homodyne detection as done in Lugiato and Gatti [1993], Gatti and Lugiato [1995], Gatti, Brambilla, Lugiato and Kolobov [1999]. For the sake of definiteness, let us consider the case of a degenerate OPO with plane mirrors, plane pump and below threshold. In a balanced homodyne detection scheme, the field is made to interfere with a strong local oscillator field. When the whole beam is detected, the measured quantity is proportional to the observable:
− 1 2 (+) 2 d x ρL ( d x ρL ( x) x ) Aout ( x , t)e−iφL (x ) EH = + Aout † ( x , t)eiφL (x ) ,
(1.9)
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where we have considered a LO of the form αL ( x ) = ρL ( x ) exp(iφL ( x )), and Aout is the signal field out of the cavity. From eq. (1.9) it is clear that the homodyne detection on the one side selects the quadrature of the field through the phase of the LO, and on the other performs the projection of the field onto the LO spatial x ). Hence the detected level of quantum noise reduction below distribution ρL ( shot noise (i.e. squeezing) depends on the spatial configuration of the LO. We are interested here in the spectrum of fluctuations of the homodyne field (+) (+) around the stationary mean value δEH = EH − EH : ∞ Σ(ω) =
! (+) (+) dt exp−iωt δEH (t)δEH (0)
(1.10)
−∞
= 1 + S(ω),
(1.11)
where we have indicated by S(ω) the normally ordered part of the spectrum Σ(ω). In this way S(ω) = 0 corresponds to the shot-noise level of fluctuations, while S(ω) = −1 indicates complete suppression of quantum noise at frequency ω (that is, perfect squeezing), for the quadrature component φL . Let us consider an orthonormal set of functions in the transverse plane {fl }. By varying the spatial distribution of the LO among the functions of the set one is able to explore the level of fluctuations in the single-mode spatial components of the field, that is, the whole spatial structure of the squeezed states (Lugiato and Gatti [1993], Gatti and Lugiato [1995]). In the case of the OPO with spherical mirrors, the set that appropriately describes the spatial structure of the squeezed vacuum generated below threshold is the Gauss–Laguerre basis. For an OPO with plane mirrors, the appropriate set is that of Fourier modes. x ) ∝ cos ( q · x) corresponds For example, the choice of a LO of the form ρL ( to probing the level of squeezing in two propagation directions, symmetrically q |/kz with respect to the cavity axis, where kz is the longitilted at angle δq ≈ | tudinal component of the wavevector. In fig. 4 the spectrum S, obtained with this choice of LO, changed of sign, is plotted as a function of the frequency ω and the modulus of the transverse component of the wavevector q. For each propagation direction (more precisely, for each value of | q |), the phase φL has been chosen to optimize the level of squeezing. A significant reduction of quantum noise below the shot-noise level is shown for zero frequency by a large region of wavevectors around q = qc , where qc is the critical wavevector where the instability arises at threshold (Lugiato and Gatti [1993]), as will be discussed in the following subsection. As shown by Gatti and Lugiato [1995], Lugiato and Marzoli [1995], closely related to the spectrum of squeezing is the spatial correlation function of a field
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Fig. 4. OPO. The spectrum of squeezing S, with inverted sign, is plotted as a function of the scaled q |, which identifies the frequency ω/γ , and of the modulus of the scaled transverse wavevector ld | angle of emission of the twin photons, at threshold.
quadrature in a plane immediately out of the cavity (near field), defined as ! x , x , t) = :δEH ( x , t)δEH ( x , 0): , Γ φL ( (1.12) (1.13) EBL where : : indicates chronologically and normally ordered expectation value in x , t) = Aout ( x , t)e−iφL + A†out ( x , t)eiφL , EH (
the steady state, and the phase φL is taken constant (i.e. independent of x) for simplicity. For example, let us consider the case of a degenerate OPO below threshold, with plane mirrors and plane-wave pump, so that there is translational invariance and the correlation function Γ ( x , x , t) depends on x and x via x − x . If we quantize the transverse modes by considering a square B of side b in the transverse plane, the spatial correlation function Γ is given by (Gatti and Lugiato [1995]) Γ
φL
+∞
( x − x , t) = −∞
φ Sq L (ω) n
where functions
φ dω iωt 2 cos qn · ( x − x ) Sq L (ω), e 2 n 2π b
(1.14)
n
is the spectrum of squeezing obtained using the orthonormal set of √
2 fqn ( x) = × b
cos( qn · x) sin( qn · x)
(1.15)
with qn = 2π n/b, n ≡ (nx , ny ) nx , ny = 0, 1, 2, . . . . (Notice that the same result φ for Sq L (ω) holds using either the cosine or sine modes.) The explicit expression n
φ
for Sq L (ω) can be found in Lugiato and Gatti [1993], Gatti and Lugiato [1995]. n
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Conversely, Sq L (ω) can be obtained from Γ ( x − x , t) by the inversion formula: φ
n
φ Sq L (ω) n
+∞ −iωt = dt e d x cos qn · ( x − x ) Γ φL ( x − x , t). −∞
(1.16)
B
In the limit b → ∞, 0qn becomes a continuous variable q and, in eq. (1.14), 2 (2/b2 ) n → (2π)−2 d q . Hence, in order to measure the spectrum of squeezing it is not necessary to use a set of local oscillators, each corresponding to a different element of the orthogonal basis. It is enough to measure the spatial correlation function Γ by using just one arbitrary local oscillator, obtaining the spectrum from Γ . In the case of the OPO with plane mirror one uses eq. (1.16). This result in general is shown in Gatti and Lugiato [1995]. We wish to mention also that Kolobov and Sokolov, in analysing the case of an OPA in the cavityless configuration, defined directly the spectrum of squeezing as the spatio-temporal Fourier transform of the correlation function Γ (Kolobov and Sokolov [1991], Kolobov [1999]). They also introduced the idea of antibunching in space–time, referring to the photocurrent density measured by homodyne detection: the system is able to produce a photocurrent distribution more ordered than that of random events (Poisson distribution) not only in time but also in space. See also Belinsky and Rosanov [1991]. The approach to the spectrum of squeezing followed by Lugiato and Gatti [1993], Gatti and Lugiato [1995] was more tailor-made for squeezed light generated in optical cavities, and more operational in the sense of being related to the spatial configuration of the local oscillator. In the case of the OPO with planar mirrors the two definitions of spectrum of squeezing coincide, as one can see from eq. (1.16).
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1.4. Quantum images in the OPO above and below threshold Although we will discuss the case of the degenerate OPO with planar mirrors, some of the following consideration apply to the case of degenerate four-wave mixing as well. 1.4.1. OPO above threshold As is well known, the OPO presents a threshold below which, in the semiclassical theory, the signal field is zero. The threshold value of the cw pump field injected into the cavity can be determined by performing a standard linear stability analysis of the dynamical equations of the model (Oppo, Brambilla and Lugiato [1994], Oppo, Brambilla, Camesasca, Gatti and Lugiato [1994]). As mentioned,
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in the case of OPO with plane mirrors, the classical model equations can be obtained from eqs. (1.6) and (1.7) by dropping the noise terms. These equations are linearized around the trivial stationary solution, in which α( x , t) = 0, and the response of the system to a perturbation of the form eiq ·x is studied. This amounts to introduce a Fourier transformation in the linearized equations. In the case of optical systems the Fourier analysis is not only a mathematical tool since the far field is just the Fourier transform of the near field. The threshold corresponds to the minimum value of the pump intensity such that there is a value qc of | q |, called the critical wavevector, for which the perturbation does not decay. Because of the rotational symmetry, the instability arises over the circle | q | = qc , which can be hence called a critical circle. As shown by Oppo, Brambilla and Lugiato [1994], Oppo, Brambilla, Camesasca, Gatti and Lugiato [1994], the result depends on the sign of the cavity detuning of the signal field θ : – When θ 0, the instability arises in the axial mode q = 0, and the signal field which is emitted above threshold is uniform in the transverse plane. – When θ < 0, qc is positive and the threshold is lower than in the case θ 0. In the latter case, the system spontaneously breaks the translational symmetry, and, because of the pairwise emission of signal photons with conservation of the total transverse momentum, above threshold the signal field has the form of a stripe pattern (fig. 5a) of the form a( x ) = σ exp i(φ+ + qc · x) + σ exp i(φ− − qc · x) , (1.17)
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where qc is an arbitrary wavevector of modulus qc . The intensity distribution 2 φ+ − φ− a( x ) = σ 2 cos2 qc · x + (1.18) 2 corresponds to a stripe pattern with an arbitrary orientation linked to the arbitrary way in which the system breaks the rotational symmetry. The far-field distribution corresponding to eq. (1.18) is shown in fig. 5b; it corresponds to two delta-like peaks at q = qc and q = − qc . One can say that the near field shows the wave aspect of radiation, i.e. the stripe pattern which arises from the interference of the two plane waves tilted in opposite directions, whereas the far field exhibits the particle aspect, i.e. the two spots in the transverse plane generated by the two beams formed by pairs of twin photons emitted in the PDC process. In the quantum version of the model, most interesting phenomena appear. First of all, it turns out that the phase difference φ+ − φ− undergoes large fluctuations, which diverge for zero frequency (Grynberg and Lugiato [1993], Lugiato and Grynberg [1995]). Looking at eq. (1.18), we see that φ+ − φ− controls the position of the stripe pattern in the near field, hence this pattern is washed out
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Fig. 5. (a) Stripe pattern. (b) Far-field configuration of the stripe pattern. | q | = qc is the critical circle.
(note that this is not the case for the far-field pattern). Figure 6 shows the random diffusive motion of the roll pattern in time in a 1D simulation. Therefore, if one averages over the motion (long-time observation, corresponding to zero frequency) the pattern becomes invisible, whereas it can be detected if the observation is fast enough (snapshot). More importantly, even in a long-time observation it is possible to reveal the underlying spatial structure by looking at the spatial correlation function of the signal field [of the form (1.12)] instead of the intensity distribution. The correlation function is not affected by the fluctuations of φ+ − φ− and shows the same spatial dependence of the roll pattern (Gatti and Lugiato [1995], Gatti, Wiedemann, Lugiato, Marzoli, Oppo and Barnett [1997]). We introduced the name quantum image to designate the case of an image encoded in the spatial correlation function instead of the intensity itself. Zambrini, Hoyuelos, Gatti, Colet, Lugiato and San Miguel [2000] discuss the connection between this
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Fig. 6. Random diffusive motion of the roll pattern under the action of quantum fluctuations. The horizontal axis corresponds to space, the vertical axis to time. From Zambrini, Hoyuelos, Gatti, Colet, Lugiato and San Miguel [2000].
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phenomenon and the Goldstone (or neutral) mode which arises in the presence of translational symmetry. Gatti and Mancini [2000] describe similar phenomena arising in the hexagonal pattern that forms in a Kerr medium. Contextually to the presence of large fluctuations in φ+ − φ− , there is the presence of sub-shot-noise fluctuations in the conjugate observable N+ − N− , where N+ and N− are the number of photons in modes qc and − qc , respectively. In particular, the fluctuations of N+ −N− vanish exactly for zero frequency (Lugiato and Castelli [1992]). The twin-photon quantum effect was discovered first in the case of the plane-wave nondegenerate OPO (Reynaud, Fabre and Giacobino [1987]). In our case, we have the same picture related to spatial effects, i.e. the twin beams which form the stripe pattern are quantum-correlated. In Castelli and Lugiato [1997] it is shown also that they exhibit EPR aspects with respect to the variables φ+ , N+ for one beam and φ− , N− for the other, i.e. they are in a state of macroscopic entanglement. The presence of quantum correlation between the beams which form the far field has also been confirmed in the case of more complex structures, such as the hexagonal patterns which form in χ (3) media (Grynberg and Lugiato [1993], Gatti and Mancini [2000]). A general description of these results can be found in Lugiato, Gatti and Wiedemann [1997]. The presence of a roll pattern in the near field and of squeezing in the far field is related to aspects of wave–particle duality, which are discussed in Lugiato and Grynberg [1995] together with the effects of a broken translational invariance. The quantum correlations present for the stripe pattern persist also in the case of a disordered spatial structure to which many modes contribute (frozen chaos, Zambrini, Barnett, Colet and San Miguel [2003]).
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1.4.2. From below to above threshold Figure 7 shows an example of stochastic realization of the system dynamics, based on the numerical simulation of the Langevin equations (1.6), (1.7), describing the fast dynamics of quantum fluctuations in the planar OPO. It is well known that in the Wigner representation the marginals of the quasi-probability distribution in the classical phase space are the correct quantum-mechanical probability distributions. Hence we can interpret a single realization of the stochastic equations as the result of a homodyne measurement, where a single quadrature component of the field is detected. The vertical sequence at left shows snapshots of the near-field distribution of the most amplified quadrature of the signal field, for increasing values of the pump field across the threshold region; the vertical sequence at right displays the corresponding far-field intensity distribution. Parameters are such that above threshold the signal emerges in the form of a stripe pattern.
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EBL Fig. 7. Scan of the threshold region of the OPO. Snapshots of the spatial distribution of the most amplified quadrature component in the near field (left-hand frames) and in the far-field intensity distribution (right-hand frames): (a) 4.2% below threshold, (b) 1% below threshold, (c) 3% above threshold, (d) 6% above threshold.
From a classical viewpoint the signal field below threshold is zero everywhere; in a quantum description, however, even below threshold there are signal photons, and the field vanishes only on average. More importantly, it is able to show a noteworthy level of self-organization, which anticipates the onset of an ordered structure above threshold (Gatti, Wiedemann, Lugiato, Marzoli, Oppo and Barnett [1997], Lugiato, Gatti, Ritsch, Marzoli and Oppo [1997]). Approaching threshold from below (left-hand frames of fig. 7) a speckle-like pattern forms in the near field. The pattern is irregular and the speckles perform a slow random motion in the transverse plane: as a result the time average of the field vanishes and the translational symmetry is recovered. However, in a single snapshot the distribution of speckles is not random, as it would be for traditional (i.e. thermal) speckle patterns. Instead, the probability of finding two spots a distance r apart has maxima when r is an integer multiple of the critical wavelength λc = 2π/qc .
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The far-field distribution is concentrated on the critical circle, indicating the existence of a privileged angle of emission for the photon pair. In a single snapshot bright spots appear always in symmetrical pairs, which provides clear spatial evidence of the emission of correlated twin photons. Below threshold the position of the pairs of spots moves randomly over the critical circle, in such a way that the mean intensity distribution forms an annulus around the critical circle. Due to critical slowing down of fluctuations, this motion, as well as the motion of speckles in the near field, becomes slower and slower as threshold is approached. Above threshold the rotational symmetry is broken by fixing the position of the pair of spots in the far field. This corresponds to selecting an orientation for the stripes in the near field. The sequence of fig. 7 can be found in the form of an animation in Gatti, Lugiato, Oppo, Martin, Di Trapani and Berzanskis [1997]. Stochastic simulations such as those shown in fig. 7 are able to reproduce the dynamics of the quantum image as it would appear under an observation which is fast enough to resolve the dynamics of fluctuations. Under a low-frequency observation, spatial correlation functions represent the main tool for highlighting the spatial properties of a field that on average appears structureless. Let us consider the spatial correlation function of field quadratures given by eq. (1.12), in a plane immediately out of the cavity. For a resonator with plane mirrors and plane-wave input field, analytical calculations can be performed in the framework of the linearized model (see for example Gatti and Lugiato [1995], Gatti, Wiedemann, Lugiato, Marzoli, Oppo and Barnett [1997]). Figure 8 shows examples of the results obtained for zero time delay t = 0, for values of the input field increasing towards the threshold value. As it must be for a system with translational and rotational symmetry, the function Γ φL ( x , x , t) depends only on the distance between the two points. When plotted as a function of this distance, the correlation shows a regular modulation which occurs at the same wavelength λc of the stripe pattern immediately above threshold. The function is exponentially damped, but the correlation length turns out to diverge at threshold (Gatti and Lugiato [1995]). Hence the function is able to reveal the presence of a certain degree of spatial order in the OPO field below threshold, which becomes long-ranged as the critical point is approached, anticipating the onset of a regular spatial structure. For the case of the OPO at threshold, see Zambrini, Barnett, Colet and San Miguel [2003]. It is worth mentioning that these phenomena are still present when an integral over time delay t is performed, that is, if a slow detector is used. Moreover these features of correlation functions are robust toward the injection of a pump beam with finite diameter, instead of a plane-wave beam (Gatti, Wiedemann, Lugiato, Marzoli, Oppo and Barnett [1997]).
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Fig. 8. OPO with plane mirrors. Analytical results for the correlation function of field quadratures. The x , x , 0)/Γ φL ( x , x, 0) is plotted versus the scaled distance r = | x − x |/ ld for φL = 0 ratio Γ φL ( or π/2, θ = −2, and 40% below threshold (solid line), 5% below threshold (dashed line), 1% below threshold (dotted line).
For a resonator with spherical mirror, see Lugiato and Marzoli [1995]. Quantum images in a nondegenerate OPO are discussed in Szwaj, Oppo, Gatti and Lugiato [2000]. All these features of the correlation functions substantiate nicely the traditional analogy between the laser threshold region and second-order phase transitions (Graham and Haken [1970], DeGiorgio and Scully [1970]), by adding the spatial aspect which was absent in the previous treatments. The phenomena here described are “classical”, in the sense that the same functions could describe spatial correlations of classical fluctuations, close to a modulational instability. Signatures of the quantum nature of fluctuations can be found in the phase sensitivity of the correlation function: for the most deamplified field quadrature the correlation (1.12) for t = 0 takes on negative values at small distances. As elucidated in Lugiato, Brambilla and Gatti [1998], this implies that the photocurrent which results from a homodyne measurement on a small region of the near field and over a short time interval may have a statistics more ordered than the Poisson statistics, similarly to what was shown for the case of the OPA by Kolobov and Sokolov [1991], Kolobov [1999]. However, the most robust and evident signatures of the quantum nature of light appear in the far-field correlations; this point will be discussed in § 2.1. A laboratory experiment performed at the University of Insubria in Como (Gatti, Lugiato, Oppo, Martin, Di Trapani and Berzanskis [1997], Beržanskis, Chinaglia, Lugiato, Feller and Di Trapani [1999]) exhibits features closely related to the quantum images described in fig. 7. In this experimental setting, picosecond
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Fig. 9. Experimental results at Como laboratories. (a) Single-shot far-field intensity profile of the signal, (b) corresponding near-field intensity distribution.
pump pulses propagate in a sample of lithium triborate, where non-degenerate frequency down-conversion takes place. Despite the differences from the OPO case described above, the configurations of the signal field in the near field (fig. 9b) and in the far field (fig. 9a) are closely similar to those of figs. 7a, b. This similarity concerns only the classical aspects of quantum images; the quantum nature of these phenomena will be substantiated in §§ 2.3, 2.4. A paper by Santagiustina, Colet, San Miguel and Walgraef [1997] predicted very interesting noise-sustained patterns in a passive optical cavity containing a Kerr medium, when the input beam is slightly tilted with respect to the cavity axis. These patterns arise from a convective instability as a macroscopic manifestation of dynamically amplified quantum noise, with amplification factors up to 105 . The analysis of the model has pointed out states with a non-Gaussian Wigner distribution, an evident manifestation of the nonlinearity (Zambrini, Barnett, Colet and San Miguel [2002]).
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1.5. The interference of signal and idler waves in type I PDC Parametric down-conversion is characterized by quantum EPR correlations between the signal and idler fields (quantum entanglement). A rather natural question is: are there interesting quantum effects if one makes the signal and the idler fields interfere? (of course in the case of type I PDC in which the photons have the same polarization, by “signal” and “idler” waves we mean e.g. two plane waves with opposite transverse moments q and − q ). At first sight, the answer seems rather negative, since PDC lacks just the kind of correlation required for standard interference, based on the measurement of the intensity distribution. The phase difference φ+ − φ− between signal and idler is not at all constant and therefore interference fringes in the intensity distribution can be observed only for times shorter than the (very short) coherence time and are washed out otherwise. In
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turn, this implies that it is necessary to operate in the high-gain regime, i.e. where a large number of photons are emitted in a coherence time. The situation is in part similar to that of the interference of two uncorrelated laser beams emitted above threshold, in which, as is well known, fringes are visible only for times shorter than the coherence time of the lasers (Magyar and Mandel [1963]). In this case the beams are always intense enough to detect the interference fringes. However, also well known (Paul [1986], Kuo, Smithey and Raymer [1991]) is that, in the case of independent lasers, fringes can be well observed even with a long temporal detection or for weak sources, if one measures the normally ordered spatial intensity correlation function E (−) ( x )E (−) ( x )E (+) ( x )E (+) ( x ) (Hanbury-Brown and Twiss [1956], Glauber [1963a, 1963b]). Ghosh and Mandel [1987] reported a fourth-order interference experiment of signal and idler in the very low-gain regime in which one detects coincidences of two photons. They observed fourth interference fringes with visibility definitely larger than 50%, whereas no second order interference fringes were obtained of course. Indeed, the theory (see also Ghosh, Hong, Ou and Mandel [1986]) predicts that the visibility is 1 under ideal conditions. In the case of the interference of two independent laser beams, the fringe visibility in a fourth-order interference experiment cannot exceed 50%. If one calculates the interference fringes in PDC for arbitrary gain by using the same two-mode model (Ghosh and Mandel [1987]), one obtains that the visibility varies from 1 in the low-gain regime to 50% in the high-gain limit. This seems disappointing because in the high-gain limit the result is the same as for the case of two independent lasers, in which no quantum aspects are involved. However, the quantum-image approach suggests how to identify a measurement in which not only the result obtained in the high-gain PDC case differs substantially from that of two uncorrelated lasers, but also one can identify in a clear-cut way the quantum regime (Brambilla, Gatti and Lugiato [2007]). Precisely, the quantity to be considered is not the spatial intensity correlation function, but the spatial correlation function of a field quadrature defined by eq. (1.12). The conceptual scheme is shown in fig. 10. It can apply equally well to single-pass PDC or to the case of a degenerate OPO below threshold with plane mirrors or in a self-imaging configuration. Two pinholes P1 and P2 located in the far field select two symmetrical coherence areas (transverse modes) q and − q which, after reflection on two mirrors M1 and M2 , are led to interfere. In the detection plane the field is mixed with a uniform LO of phase φL , and one measures the spatial correlation function of the selected quadrature component. For the sake of simplicity.
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Fig. 10. Conceptual scheme for an interference experiment in which one detects the spatial correlation function of a field quadrature.
• We assume that the pump is a monochromatic axial plane-wave. • We quantize the transverse modes in a square of size b, with b ≈ pump diameter, consistently with the fact that the pinholes select coherence areas in the far field. x , t) in the detection plane appearing in eq. (1.13) has the form The field Aout ( 1 Aout ( (1.19) x , t) = aqn (t)eiqn ·x , b
EBL
n
where aqn are the photon annihilation operators of the transverse modes and qn is the same as in eq. (1.15), but with nx , ny = 0, ±1, ±2, . . . . All modes lie in the vacuum state with the exception of the two symmetrical modes selected by the two pinholes. Let us now consider the temporal Fourier transform of the correlation function x , x , t), i.e. Γ φL ( Γ
φL
+∞ ( x , x , ω) = dt Γ φL ( x , x , t)e−iωt .
(1.20)
−∞
One finds (Brambilla, Gatti and Lugiato [2007]) that x , x , ω) = Γ φL (
φ 2 cos q · ( x − x ) Sq L (ω), 2 b
(1.21) φ
where, in the case of an OPO with plane mirrors, Sq L (ω) is the same function which appears in eqs. (1.14) and (1.16), evaluated for qn = q. This result is remarkably simple and connected with the result on the spectrum of squeezing illustrated in § 1.3. Several remarks are in order: (a) There are interference fringes, which, as expected, have a wavelength equal to twice that obtained in a fourth-order interference experiment.
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(b) The result (1.21) is substantially different from that of two uncorrelated φ φ lasers, where in eq. (1.21) Sq L (ω) is replaced by the part of Sq L (ω) that is independent of the LO phase φL . φ (c) The quantum regime is clearly identified as that in which Sq L (ω) < 0, i.e. φ Γ L ( x , x, ω) < 0. (d) The interference of the two modes q and − q combines quadratures components of the two modes which vary with the position x. However, thanks to φ the EPR entanglement between the two modes the result, given by Sq L (ω), is the same for all values of ( x − x ). All transverse modes, also those different from q , − q and lying in the vacuum state, contribute to the shot-noise level, which is given by δ( x − x ). There is a basic connection with the results obtained in the near field and discussed in § 1.3. As one can see from eqs. (1.14) and (1.20), the result for Γ˜ φL ( x , x , ω) in the near field just corresponds to the sum of contributions with the form (1.21) over all transverse mode pairs, i.e. it can be interpreted as the interference of all mode pairs. In this superposition fringes disappear, contrary to what happens when two modes q and − q are selected as in fig. 10, which φ leads to eq. (1.21). In this case the factor Sq L (ω), which multiplies the interference fringes in eq. (1.21), can be recovered experimentally by measuring x , x, ω) in the near field and calculating the spatial Fourier component for Γ˜ φL ( qn = q [see eq. (1.16)]. This provides an alternative procedure for the interference fringes, “extracting” them from the near-field correlation function. It is also interesting to remark that, in the light of the discussion above, the oscillations in fig. 8 for the OPO with planar mirrors can be interpreted as fringes arising from the interference of all modes q and − q with | q | = qc . As a matter of fact, when threshold is approached the critical modes become dominant and fringes can appear.
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§ 2. Quantum spatial intensity correlations in optical parametric down-conversion This section is exclusively devoted to the spatial quantum properties of light in PDC, even if some of these phenomena can be found also in the four-wave-mixing configuration (Perez-Arjona and Gatti [2007]). Following the historical development, we will start in § 2.1 with the case of a degenerate OPO with spherical mirrors below threshold (fig. 3). In this case, we will describe the spatial quantum signatures in the far field arising from the (apparent) violation of the Cauchy–Schwartz inequality. This amounts also to the
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spatial analogue of photon antibunching. The experimental observation of spatial correlations in OPOs, and of the spatial distribution of quantum noise in semiconductor laser will also be discussed. In § 2.2 we will present a multimode model for single-pass parametric downconversion. Next, we will focus (§ 2.3) on single-pass PDC of type I, under conditions close to degeneracy. We focus on the high-gain regime in which a large number of signal/idler photons are emitted in each pump pulse. We will illustrate for this case the phenomenon of the near-field/far-field duality. We will then turn (§ 2.4) to the case of single-pass PDC of type II, in which the twin (signal/idler) photons have orthogonal polarization. We will discuss theoretically the simultaneous presence of spatial quantum correlation in the near field as well as in the far field. For the latter case, we will describe and discuss (§ 2.5) the first experimental observation of the quantum nature of the spatial fluctuations in the high-gain regime of PDC and of the spatial analogue of photon antibunching. A theoretical description of the possibility of detecting weak-amplitude objects with a sensitivity beyond the standard quantum limit is also briefly given, in § 2.6. Finally in § 2.7 we discuss theoretically the multimode polarization entanglement in high-gain PDC.
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2.1. Degenerate OPO below threshold, spatial quantum correlation and entanglement Let us first consider the case of degenerate parametric down-conversion taking place in an optical resonator. The fundamental character of the twin photon emission is well known (see for example Hong, Ou and Mandel [1987], Kwiat, Mattle, Weinfurter, Zeilinger, Sergienko and Shih [1995]). In the case of degenerate parametric down-conversion of type I, the two photons of the pair are degenerate both in polarization and in frequency. However, recent analyses (Marzoli, Gatti and Lugiato [1997], Lugiato, Gatti, Ritsch, Marzoli and Oppo [1997]) have shown that the far-field distribution is able to provide a clear spatial evidence of the emission of twin photons, and carries a spatial signature of the quantum nature of the correlation between the photons of the pair. The discussion will be limited to the case of an OPO with spherical mirrors, in the quasi-planar configuration (fig. 3). Similar results hold also for an OPO with plane mirrors (fig. 2). We start by considering the direct detection of photons in a plane at great distance z from the cavity. Our calculations show that the average intensity distribution a out† ( x )a out ( x ) has cylindrical symmetry around the cavity axis, and, well
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Fig. 11. Spatial correlation function of the intensity fluctuations G( x , x , t = 0), plotted as a function of the second point x : (a) near field, (b) far field (z = 200z0 , where z0 is the Rayleigh range), 50% below threshold, and η/γ = 0.1, where γ is the cavity damping rate for the signal field and η is the frequency spacing between adjacent transverse modes of the quasi-planar cavity.
below threshold, corresponds to a broad peak, which does not show any privileged angle of emission for the signal photons. More interesting features are displayed by the correlation function of the intensity fluctuations: ! x , t)δI ( x , 0): G( x , x , t) = :δI ( (2.1) where
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! x , t)a out ( x , t) − a out† ( x , t)a out ( x , t) . δI ( x , t) = a out† (
(2.2)
Figure 11 shows two plots of the function, obtained by fixing point x and varying point x , for zero time delay. In the near field (fig. 11a) the correlation has a single high peak for x = x, much narrower than the intensity distribution. Moving to the far field (fig. 11b) G undergoes a dramatic transformation: the peak at x = x drops, while another peak grows on the other side with respect to the system axis, x . This result gives clear evidence of the emission of twin phothat is for x = − tons, from a point of view of spatial fluctuations. Twin photons are produced with no special angle of emission; but if one of the two is emitted in some direction, its twin will propagate in a direction symmetrical with respect to the system axis, due to conservation of transverse momentum. Propagation at large distances separates the pair, and then in the far field it is natural to find the maximum correlation between two points symmetrically located with respect to the z axis. Even more important, it can be shown that this correlation is of quantum nature, in the sense that no classical field can produce a correlation function G( x , x , 0) such that G( x , − x , 0) > G( x , x, 0). The simple proof, based on the Cauchy– Schwartz inequality, can be found in Lugiato, Gatti, Ritsch, Marzoli and Oppo
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[1997], and it is the spatial counterpart of the proof that a classical field cannot show antibunching in time. A relevant consequence of this shape of the correlation function in the far field is the existence of an observable with sub-shot-noise fluctuations (Marzoli, Gatti and Lugiato [1997]). Let us consider two regions in the cross-section of the beam, say S1 and S2 , symmetrical with respect to the axis of the system. Let us focus on the total intensities detected in these two regions, and on their difference. We calculated the spectrum of fluctuations in the intensity difference, and showed that for zero frequency it is well below the shot-noise level. Provided that the system operates not too close to threshold, and for appropriate choices of the regions S1,2 , we found a quantum noise reduction up to 85% (Marzoli, Gatti and Lugiato [1997]). Again, this result reminds of the behavior of the spectrum of fluctuations in the signal/idler intensity difference (Reynaud, Fabre and Giacobino [1987], Heidmann, Horowicz, Reynaud, Giacobino, Fabre and Camy [1987]) when considering a two-mode non-degenerate OPO. Also in this case the reduction of fluctuations below shot noise indicates the high correlation in the process of parametric down-conversion. Here, however, the correlation does not concern global quantities (i.e. the total signal and idler intensities), but local observables, i.e. the intensity of light collected from symmetrical regions in the transverse plane, and the sub-shot-noise behavior arises for any pair of symmetrical regions. In the literature this phenomenon was given the name of spatial entanglement, even though, properly speaking, a quantum correlation of this kind does not necessarily imply entanglement of the state. As a matter of fact, entanglement in general implies a certain degree of quantum correlation simultaneously for at least two non-commuting observables (Duan, Giedke, Cirac and Zoller [2000]). The name spatial entanglement is in fact appropriate for the case of homodyne detection of the far-field fluctuations (Lugiato, Gatti, Ritsch, Marzoli and Oppo [1997]). Let us consider two pairs of non-commuting observables: X1 , Y1 , corresponding to two orthogonal quadratures of the signal field, detected in a region S1 of the beam cross-section, and X2 , Y2 , corresponding to field quadratures detected in the symmetric and separated region S2 . For a suitable choice of the phase of the local oscillator field which probes the signal, it can be shown that fluctuations in the difference X1 −X2 and in the sum Y1 +Y2 can be reduced below the shot-noise limit. The amount of noise reduction can range up to 80%. This result is not very sensitive either to the size of the detection regions, or to the waist of the Gaussian profile of the LO, provided that the LO phase profile matches the phase profile of the TEM00 mode in the detection plane (Lugiato, Gatti, Ritsch, Marzoli and Oppo [1997], Gatti, Lugiato, Petsas and Marzoli [1999], Gatti, Petsas, Marzoli and Lu-
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giato [2000]). Moreover, it can be shown that the quantum correlation existing between pairs of symmetrical regions in the transverse plane at large distance from the cavity is deeper than that implied by squeezing in the appropriate observables. Actually, variables measured in region S1 are correlated to those measured in S2 also in the sense of Einstein, Podolsky and Rosen [1935] (EPR): from a measurement of X1 , Y1 in region S1 , it is possible to infer the values for variables X2 , Y2 of the spatially separated region S2 with a precision apparently violating the Heisenberg rules (Gatti, Lugiato, Petsas and Marzoli [1999], Gatti, Petsas, Marzoli and Lugiato [2000]). This result is again reminiscent of the EPR aspects demonstrated, theoretically (Reid and Drummond [1988], Reid [1989]) and experimentally (Ou, Pereira and Kimble [1992]), for a non-degenerate OPO below threshold. In that case, the two pairs of conjugate variables were two orthogonal quadratures for a signal and an idler beam, distinguished by their polarization. In our case of a degenerate, spatially extended OPO the “signal” and “idler” beams correspond, respectively, to the field measured in regions S1 and S2 . A first experiment devised to observe quantum spatial correlations in OPOs was realized recently (Martinelli, Treps, Ducci, Gigan, Maître and Fabre [2003]). This followed other experiments of the same group in the classical domain, in which spatial patterns were observed in quasi-degenerate cavities (Vaupel, Maître and Fabre [1999], Ducci, Treps, Maître and Fabre [2001]). The separation between the different transverse modes was experimentally adjusted in order to be much smaller than the cavity damping rate. An iris of variable diameter was inserted at the output of the OPO. The signal and idler beams were separated by a polarizing beamsplitter after the iris, and the noise N− of the intensity difference between the two beams was measured while the diameter of the iris was reduced. In a first experiment executed with a non-degenerate cavity, the level of quantum correlation decreased linearly when the iris diameter was reduced, because the interception of light by the iris is a loss mechanism which destroys the quantum effect like any other kind of loss. Hence in this case the signal and the idler beams are composed of time-correlated photons which are randomly distributed from a spatial viewpoint. With a quasi-confocal cavity, instead, the decrease of N− when the iris is progressively closed is no longer linear. A detailed analysis of the concept of multimode quantum field (Treps, Delaubert, Maître, Courty and Fabre [2005]) demonstrates that this feature is a proof that the generated field is spatially multimode. If the two beams were perfectly correlated from a spatial standpoint, the value of N− would remain constant. In this experiment, instead, it seems that the photons are spatially correlated only in the outer parts of the two beams, a feature which has not yet been theoretically explained. More details can be found in Maitre, Treps and Fabre [2006].
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Some earlier experiments (Poizat, Chang, Ripoll and Grangier [1998], Bramati, Hermier, Khoury, Giacobino, Schnitzer, Michalzik, Ebeling, Poizat and Grangier [1999]) investigated the spatial quantum noise of laser diodes and verticalcavity surface-emitting lasers (VCSELs), respectively, using a movable razor blade to identify precisely the noisy sub-threshold spatial modes and to infer the presence of correlations between the lasing mode and subthreshold higher order modes. Related to these are also investigations on the impact of the Petermann-excess noise factor on the intensity noise squeezing in semiconductor lasers (van der Lee, van Druten, van Exter, Woerdman, Poizat and Grangier [2000]).
2.2. Multimode-model for single-pass parametric down-conversion The following subsections will be devoted to spatial quantum correlation in single-pass PDC; therefore we start by introducing a general model for pulsed PDC. Let us focus for definiteness on type II PDC, in which a pump beam travelling inside a nonlinear χ (2) crystal is partially down-converted into signal and idler photon with orthogonal polarizations. The starting point of the analysis is an equation describing the propagation of the three waves (signal, idler and pump) inside the crystal. We consider a crystal slab of length lc , ideally infinite in the transverse directions, cut for type II quasi-collinear phase-matching. We take the z axis as the laser pump mean propagation direction (fig. 12). In the framework of the slowly-varying envelope approximation the electric field operator is described by means of three quasi-monochromatic wave-packets, of central frequencies ω0 , ω1 and ω2 , associated to the pump, the signal and the idler fields, respectively. These frequencies are taken to satisfy the energy-conservation condition ω1 + ω2 = ω0 . In the following, we shall assume quasi-degenerate phase matching, so that ω1 = ω2 = ω0 /2. We indicate by ai (z, q, Ω) (i = 0, 1, 2) the spatio-temporal Fourier transforms of the slowly-varying envelope operators of the three waves, where q is the transverse component of the wavevector, and Ω is the frequency offset from the carrier frequencies. Furthermore, we assume negligible pump depletion along the crystal (a condition easily satisfied when the pump pulse is not too intense and the crystal length is on the order of millimeters), so that the pump wave can be described by its classical c-number envelope. As derived in more detail by Gatti, Zambrini, San Miguel and Lugiato [2003], the propagation of the three waves along the crystal is then described by the following equation (see also Brambilla, Gatti, Bache and Lugiato [2004] and Scotto and
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Fig. 12. Scheme of type II PDC. The lens used to reach the far-field zone (not shown in the figure) is located at z = lc + f .
San Miguel [2002]): d ai (z, q, Ω) dz dΩ αp ( q + q , Ω + Ω ) = χ d q × aj† (z, q , Ω )e−iΔij (q ,q ;Ω,Ω )z/ lc , i = j = 1, 2,
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(2.3)
where – χ is a parameter proportional to the second-order susceptibility of the medium. – The phase mismatch function along the longitudinal direction is q , q ; Ω, Ω ) Δij ( q , Ω) + kj z ( q , Ω ) − k0z ( q + q , Ω + Ω ) . = lc kiz ( (2.4) q , Ω) = ki ( q , Ω) − q 2 indicates the z-component of the In eq. (2.4), kiz ( q , Ω) being the wavenumber. Note that the wavenumber dewavevectors, ki ( pends on the propagation direction q only for the extraordinary wave(s), a property leading to spatial walk-off. q , Ω) is the Fourier transform of the pump pulse at the input face of the – αp ( crystal z = 0, and the linear propagation of the pump inside the medium has been incorporated into the phase-mismatch function. Equation (2.3) describes all the possible microscopic processes through which a pump photon of frequency ω0 + Ω + Ω , propagating in the direction q + q is annihilated at position z inside the crystal, and gives rise to a signal and an idler photon, with frequencies ω0 /2 + Ω, ω0 /2 + Ω , and transverse wavevectors q, q , with an overall conservation of energy and transverse momentum. The effectiveness of each process is weighted by the phase mismatch function (2.4),
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which accounts for conservation of the longitudinal momentum. In the limit of an infinitely long crystal, where longitudinal light momentum has to be conserved, only those processes are allowed for which Δij = 0. For a finite crystal, however, the phase matching function has finite bandwidths, say q0 in the transverse domain and Ω0 in the frequency domain. Equation (2.3) couples all the signal and idler spatial and temporal frequencies within the angular bandwith of the pump δq ≈ 1/wp , with wp denoting the pump beam waist, and within the pump temporal spectrum δΩ ≈ 1/τp , where τp is the pump pulse duration. In general, no analytical solution is available and one has to resort to numerical methods in order to calculate the quantities of interest, as described in Brambilla, Gatti, Bache and Lugiato [2004]. A limit where analytical results can be obtained is that of sufficiently large pump waist and pump duration that δq q0 , δΩ Ω0 . In this case the pump beam can be approximated by a plane wave αp ( q + q , Ω + Ω ) → αp δ( q+ q )δ(Ω+Ω ). In this limit of plane-wave pump approximation (PWPA) the model equations can be solved in terms of input–output relations in the form of a twomode squeezing transformation: q , Ω) = U1 ( q , Ω)a1in ( q , Ω) + V1 ( q , Ω)a2† in (− q , −Ω), a1out (
(2.5)
a2out ( q , Ω) = U2 ( q , Ω)a2in ( q , Ω) + V2 ( q , Ω)a1† in (− q , −Ω),
(2.6)
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linking only symmetric modes q, Ω and − q , −Ω in the signal and idler fields. Explicit expressions for the gain coefficients Ui , Vi (i = 1, 2) can be found in Gatti, Zambrini, San Miguel and Lugiato [2003] and Brambilla, Gatti, Bache and Lugiato [2004]. It is important to note that the gain functions satisfy the following unitarity conditions: 2 2 Uj ( q , Ω) − Vj ( q , Ω) = 1
(j = 1, 2),
q , Ω)V2 (− q , −Ω) = U2 (− q , −Ω)V1 ( q , Ω), U1 (
(2.7) (2.8)
which guarantee the conservation of the free-field commutation relations after propagation. For type I, quasi-collinear and quasi-degenerate phase matching, similar equations can be been derived (see Brambilla, Gatti, Bache and Lugiato [2004]). In this case, the signal and idler waves cannot be any more distinguished by their polarization, so that a propagation equation similar to eq. (2.3) holds, provided that one put a2 = a1 and all the subscripts 2 → 1. In the limit of a plane-wave pump, the input–output transformations now take the form of the squeezing transformation (1.3), (1.4).
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2.3. Single-pass PDC of type I. Near-field/far-field duality We wish to illustrate the key spatial quantum properties of the field generated by single-pass PDC of type I. A similar behavior arises in the degenerate OPO below threshold. The near-field zone shows the phenomenon of spatially multimode squeezing or local squeezing discussed in § 1.2: a good level of squeezing is found, provided that the detection region has a size not smaller than the coherence length, given by the inverse of the spatial bandwidth of emission q0 . If, on the other hand, one looks at the far-field zone (which can be reached, typically, by using a lens), relevant spatial quantum correlation/spatial entanglement emerge between small regions located symmetrically with respect to the pump direction. If one considers two symmetrical pixels 1 and 2 (fig. 14a) located in the farfield zone, the intensity fluctuations in the two pixels are very well correlated or, equivalently, the fluctuations in the intensity difference between the two pixels are very much below the shot-noise level (Gatti, Brambilla, Lugiato and Kolobov [1999], Brambilla, Gatti, Lugiato and Kolobov [2001], Brambilla, Gatti, Bache and Lugiato [2004]). Precisely, let us consider the numbers of photons N1 and N2 detected in pixels 1 and 2, respectively, and the associated fluctuations
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δNi = Ni − Ni (i = 1, 2). The plane-wave pump model predicts that when the light is collected over a detection time much longer than the PDC coherence time, τcoh ∝ 1/Ω0 , the photon number difference N− = N1 − N2 is fluctuationless, as can be demonstrated by using eqs. (1.3) and (1.4) (Brambilla, Gatti, Lugiato and Kolobov [2001]), that is, ! δN−2 = 0.
(2.9)
This amounts to saying that N1 = N2 , i.e. by measuring N1 the value of N2 can be inferred with perfect precision. This result expresses in the most emphatic way the emission of signal and idler photons in pairs, propagating in symmetrical directions with respect to the pump, so as to preserve to transverse momentum. In the far-field zone, symmetric propagation directions are mapped into symmetric positions, so that perfect correlations are found between any pairs of far-field symmetric positions. Formally one has ! ! ! δN−2 = δN12 + δN22 − 2 δN1 δN2 ,
(2.10)
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and since δN12 = δN22 one finds from eq. (2.9) that the normalized correlation is δN1 δN2 = 1, C≡ (2.11) δN12 δN22 which means perfect correlation. The same effect occurs also for quadrature components, because in the two pixels the fluctuations of the quadrature component X are almost exactly correlated, and those of the quadrature component Y are almost exactly anticorrelated (Navez, Brambilla, Gatti and Lugiato [2001]). In a more realistic model (Brambilla, Gatti, Bache and Lugiato [2004]), the finite waist of the pump profile must be taken into account. This introduces some uncertainty in the relative propagation directions of each photon pair, so that if one photon is emitted in, say, direction q , the propagation direction of its twin will be distributed around − q , with a typical spread determined by δq ≈ 1/wp , with wp the pump waist. As a consequence, perfect spatial correlation in the far field is recovered only when the detection regions have a linear size larger than a resolution length (or coherence length in the far field) xdiff ≈ λf/wp . Conversely, if this condition is not fulfilled the fluctuations of N− just lie below or close to the shot-noise level, i.e. ! δN−2 < N+ = N1 + N2 . (2.12)
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Figure 13 shows an example of this behavior, in the case of a LBO (lithium triborate) 5 mm crystal. This figure was obtained by numerical simulations of the model, based on stochastic equations in the Wigner representation (Brambilla, Gatti, Bache and Lugiato [2004]). The spatial correlation of intensity fluctuations in the far field is quite evident even in single shots (the pump beam is typically pulsed). Figure 14a shows a numerical simulation of the LBO, in a case where the degenerate frequency corresponds to slightly non-collinear phase matching, so that close to this frequency the PDC emission in the far field is ring-shaped. In a single shot, inside the ring one can observe the presence of symmetrical pairs of speckles, corresponding to correlated intensity fluctuations. The speckles become broader and broader as the waist of the pump field is reduced. A similar situation is observed in an experiment performed using a LBO crystal (Lantz and Devaux [2001]) (see fig. 14b). The near-field/far-field duality (i.e. the simultaneous presence of squeezing in the near field and spatial entanglement in the far field) is related to the simple fact that by injecting into a 50/50 beamsplitter two beams squeezed in orthogonal quadrature components one obtains in the output two entangled beams, and vice
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2 / N is plotted as a function Fig. 13. Far-field correlation of type I. The noise reduction factor δN− + of the detector size d, for different values of the pump waist. d is normalized to the typical scale of variation of the far-field intensity distribution x0 .
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Fig. 14. Intensity distribution in the far field for a single shot of the pulsed pump field: (a) Numerical simulations. The waist of the pump beam is 100, 300 and 150 µm in the three frames from top to bottom, respectively. Ni , Xi and Yi (i = 1, 2) denote the photon numbers and the quadrature component measured in the two pixels 1 and 2 respectively. (b) Experimental observation by Devaux and Lantz at University of Besançon (see Lantz and Devaux [2001]).
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versa. This aspect has been discussed by Lugiato, Gatti and Brambilla [2002, 2006]. 2.4. Single-pass PDC of type II. Simultaneous near-field and far-field spatial correlation This subsection is based on Brambilla, Gatti, Bache and Lugiato [2004], and we refer to that article for more details about these results. Figure 15 illustrates the kind of far-field patterns that can be generated by a single pump pulse in type II PDC. They have been obtained by simulating the model equation (2.3) by a stochastic method, based on the Wigner representation. Differently from type I, in type II the walk-off between signal and idler displaces their intensity distributions along the walk-off direction (vertical axis in the figure). Depending on the collinear phase mismatch parameter, the emission is distributed over two intersecting rings (a), two tangent rings (b) or two broad spots (c). As in type I, when the pulse duration is shorter than, or on the order of, the coherence time of PDC light, one can observe speckles in the light distribution, whose size (corresponding to the far-field coherence area) increases as the pump waist is reduced. Each speckle in the signal distribution has a symmetric twin in the idler distribution (the centers of the figures correspond to the pump far-field spot, not shown), which is a clear signature of spatial correlation of intensity fluctuations between symmetric portions of the far field, because of pairwise emission of photons propagating in symmetric directions.
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Fig. 15. Typical far-field patterns of the down-converted light in a type II crystal, observed close to the degenerate frequency. They are obtained by decreasing the collinear phase mismatch parameter. The pump pulse duration is τp = 1.5 ps, the pump beam waist is wp = 664 µm.
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Exactly as in type I, the plane-wave pump model predicts the existence of a perfect correlation between the photon numbers detected between any two symmetric portion of the far-field PDC emission: δN−2 = 0, for whatever size of the detection pixels. In a more realistic model, where the shape of the pump pulse is taken into account, one has to consider detection of photons over two symmetric regions of finite area. If, for example, the detectors are the pixels of a CCD camera, they measure the total number of incoming photons down-converted in a single pump shot and the measurement time can be identified with the pump pulse duration. A good level of correlation is recovered only when the pixel size is larger than the far-field coherence length (the size of the speckles in fig. 15) xdiff ≈ λf/wp . Let us now turn to the near-field correlation. Spatial correlations in the nearfield zone arise because twin photons are generated at the same position inside the crystal: if this is not too long, there is a high probability to find them close one to each other at the exit face of the crystal. In general, during propagation along the crystal photons are diffracted, so that their relative positions at the crystal √ endface have a characteristic spread proportional to the diffraction length λlc . In the type II case, the signal and idler near-field distributions can be physically separated by means of a polarizing beamsplitter, since the fields are orthogonally polarized. Hence one can locate two pixel detectors at the same corresponding positions in the signal/idler beam and detect the photon number difference. It has been shown (Brambilla, Gatti, Bache and Lugiato [2004]) that perfect correlation (i.e. δN−2 = 0) holds only asymptotically in the limit of vanishing crystal length, or, more physically, when the transverse area of the detector pixel is larger than 2 , where the coherence area xcoh
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xcoh ≡
1 ∝ λlc . q0
(2.13)
See Brambilla, Gatti, Bache and Lugiato [2004] for detailed results of both nearand far-field correlation from numerical simulations. A key point that has to be noticed is the simultaneous presence of quantum spatial correlations, ideally perfect, in both the far- and the near-field zone of the PDC emission. This feature represents a definite signature of the entanglement of the PDC state, which cannot be shown by any separable (i.e. non-entangled state) (Gatti and Cirac [2007]). From the microscopic point of view, the simultaneous presence of near- and far-field correlations corresponds to the simultaneous correlation of both momentum and position of the signal/idler photon pairs. This is very much reminiscent of the original EPR paradox for position and momen-
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tum, and recently an inseparability criterium based on position and momentum correlation of a photon pair has been formulated and experimentally tested with PDC photons (Howell, Bennink, Bentley and Boyd [2004], D’Angelo, Kim, Kulik and Shih [2004]). In order to understand the spatial entanglement properties of the signal/idler fields, it is convenient to turn to the state (Schrödinger-like) picture instead of the operator (Heisenberg-like) picture we used up to now. For simplicity, we ignore the time and frequency variables t and Ω in the remainder of this section. The state at the input face of the slab is the vacuum state for all modes, i.e. 3 |ψ in = (2.14) |0, q 1 |0, q 2 , q
where we indicate by |n, q i the Fock state with n photons in mode q , i = 1(2) indicating the signal (idler) as usually. On the other hand, as derived in detail by Gatti, Zambrini, San Miguel and Lugiato [2003], in the PWPA the state of the signal/idler field at the output face of the slab is 1 ∞ 3 |ψ out = (2.15) cn ( q )|n, q 1 |n, − q 2 , q
n=0
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q ) = {U1 ( q )V2 (− q )}n |U1 ( q )|−(2n+1) . Equation (2.15) is a superposiwhere cn ( tion of states with the same number of photons in mode q for the signal field and mode (− q ) for the idler field. This expresses in a very emphatic way the momentum entanglement of the signal and idler photons which, in the far field, gives rise to the spatial correlation between symmetric pixels. Equation (2.15) is an eigenstate of N− = N1 − N2 with eigenvalue zero, which explains why N− is fluctuationless. If only one of the two fields is detected, e.g. the signal field, while the idler is disregarded, the output state of the signal field, obtained by tracing away the degrees of freedom of the idler, is described by the density matrix 1 ∞ 3 2 cn ( %1out = (2.16) q ) |n, q 11 n, q | . q
n=0
It can be verified that 2 cn ( q ) =
n( q ) n , [1 + n( q ) ]n+1
(2.17)
where n( q ) is the average number of photons in mode q, so that the photon statistics of the signal field is thermal for all modes q . The same is true for the idler.
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The position entanglement of signal and idler photons, which gives rise to the near-field spatial correlation, can be retrieved in the limit of a short crystal, where diffraction and walk-off along the crystal are negligible. The coefficients Ui ( q) q ) in eqs. (2.5) and (2.6) become practically constant with respect to q and Vi ( and can be replaced with their values for q = 0. Back-transforming to the real space x, we obtain † x ) = U1 ( q = 0)a1in ( x ) + V1 ( q = 0)a2in ( x ), a1out ( † x ) = U2 ( q = 0)a2in ( x ) + V2 ( q = 0)a1in ( x ). a2out (
(2.18)
The input–output relations (2.18) are local in the position x in the crystal output plane (“near field”), and the corresponding output state reads 1 ∞ 3 cn ( q = 0)|n, x 1 |n, x 2 , |ψ = (2.19) x
n=0
where |n, x is the Fock state with n photons at point x. In this limit, there is ideally a perfect correlation in the number of signal/idler photons at the same near-field position (position entanglement).
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2.5. Detection of sub-shot-noise spatial correlation in the high gain regime of type II PDC. Spatial analogue of photon antibunching In this subsection we describe the first experimental observation of spatial correlation at the quantum level in the far field of PDC. The experiment uses as a detector a high-efficiency scientific CCD camera, whose pixels have an area on 2 . The results of this secthe order of the characteristic far-field coherence area xdiff tion are based on the work of Jedrkiewicz, Jiang, Brambilla, Gatti, Lugiato and Di Trapani [2004, 2006]. There is by now a large literature on spatial effects in the low-gain regime, where photon pairs are detected via coincidence counting. Genuine spatial quantum effects have been shown by Nogueira, Walborn, Pádua and Monken [2001], demonstrating spatial antibunching, and by Howell, Bennink, Bentley and Boyd [2004], who reported on the realization of an EPR paradox regarding the position– momentum uncertainty relation for photons. The theoretical investigations reported in §§ 2.2 and 2.3 predict that in any gain regime, symmetric portions of the signal and idler emission cones corresponding to phase-conjugate modes are quantum-correlated. This far-field correlation
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amounts to the multimode version of the well-known twin-beam effect, a subshot-noise correlation between the whole signal and idler beams which was evidenced for example by Aytür and Kumar [1990] in the medium-gain regime of PDC. The spatial correlation has also been investigated in the low-gain regime by Jost, Sergienko, Abouraddy, Saleh and Teich [1998], Oemrawsingh, van Drunen, Eliel and Woerdman [2002] with a high-sensitivity CCD camera, although it was not possible to determine its quantum nature quantitatively. More recently, the measurement of small displacements beyond the Rayleigh limit (Treps, Andersen, Buchler, Lam, Maître, Bachor and Fabre [2002]) and the realization of noiseless amplification of optical images (Mosset, Devaux and Lantz [2005]) made evident the application potentials of such multimode quantum correlations in the regime of large photon numbers. The experiment of Jedrkiewicz, Jiang, Brambilla, Gatti, Lugiato and Di Trapani [2004, 2006] aimed at demonstrating the predicted quantum character of the far-field correlation, by characterizing the detection of the far field of the PDC radiation emitted by a β-barium borate (BBO) nonlinear crystal pumped by a low-repetition-rate (2 Hz) pulsed high-power laser (1 GW–1 ps). Thanks to the huge number of radiation transverse modes, it is possible to select a portion of the parametric fluorescence close to the collinear direction and within a narrow frequency bandwidth around degeneracy. This portion still contains a large (>1000) number of pairs of signal/idler correlated phase-conjugate modes. In the far field, where the measurement is performed, the couples of modes correspond to pairs of symmetrical spots, which can be considered as independent and equivalent spatial replicas of the same quantum system. Thanks to the very large number of these, the statistical ensemble averaging necessary for the quantum measurement can be done solely over the spatial replicas for each, single, pump-laser pulse. Thus, differently from the experiment of Aytür and Kumar [1990], where the statistics was performed over different temporal replicas of the system, here no temporal averages over successive laser shots are considered. A simple far-field detection set-up was initially mounted as shown in fig. 16a. A deep-depletion back-illuminated charge-coupled device (CCD) camera with quantum efficiency η ≈ 89% triggered by a pulse from the laser system, was placed in the focal plane of a single large-diameter lens (f = 5 cm), which collects at a distance f the far-field PDC radiation emitted by the BBO. The CCD detection array has 1340×400 pixels, with a pixel size of 20 µm×20 µm. By means of an interferential filter (IF), centered at 704 nm, it is possible to visualize the signal and idler far-field beams emitted close to degeneracy. Typical far-field images recorded at degeneracy in a single shot (for 1 ps pump pulse) are shown in
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Fig. 16. (a) Scheme of the diagnostics for the far-field detection of the signal/idler ring-type patterns close to degeneracy. (b) Detailed scheme of the experimental set-up used for the spatial correlation measurements. The third harmonic of the Nd:Glass laser, at 352 nm, is used to pump the BBO crystal which is cut for degeneracy at 704 nm (θ = 49.05◦ , φ = 0).
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figs. 17a, b, for two different values of the pump intensity and spot sizes. The shape of the rings is similar to the numerical fig. 15b. The existence of spatial correlation is already apparent from the symmetrical properties of the signal and idler patterns shown in fig. 17. In order to reveal the quantum character of the correlation the experimental set-up illustrated in fig. 16b is employed where, with respect to fig. 16a, a different diagnostics configuration is adopted. A portion of fluorescence around the collinear direction is selected, and the radiation is transmitted through a polarizing beamsplitter (PBS) that separates the signal and idler beams. The far-field zones of the two beams are imaged onto two separate regions of the CCD, which is placed in the common focal plane of the two lenses (f = 10 cm). In this set-up the correlation measurements are performed without using any narrow-band IFs, since these unavoidably introduce relevant transmission losses reducing the visibility of sub-shot-noise correlations. The estimated quantum efficiency of each detection line, which accounts for both the transmission losses and the detector efficiency, is ηtot 75%. Figure 18a shows a typical far-field image recorded in a single shot in the experimental configuration of fig. 16b, where a fairly broad-band radiation (i.e., the one transmit-
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Fig. 17. Experimental single-shot far-field images of the signal (left-hand ring) and idler (right-hand ring) beams close to degeneracy, recorded in the scheme of fig. 16a, with pump intensity (a) I ≈ 30 GW/cm2 , and (b) I ≈ 50 GW/cm2 , and FWHM pump beam size of (a) 1 mm and (b) 0.4 mm, respectively.
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Fig. 18. (a) Single-shot far-field image recorded by the CCD in the scheme of fig. 16b, for a pump intensity I 30 GW/cm2 . The spatial regions for statistics are indicated by the black boxes, and were selected within the frequency-degenerate emission by temporarily inserting a 10 nm IF, as shown in (b). (c) Zoom of two symmetrical areas of the signal and idler far-fields.
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ted by the rectangular aperture) is acquired in the signal (left) and idler (right) branches. The selection of the desired temporal and angular bandwidth around degeneracy is made by temporarily inserting in front of the CCD a 10-nm wide IF around 704 nm, allowing one to locate the collinear degeneracy point (see fig. 18b). The data analysis is restricted within two rectangular boxes (indicated in black in fig. 18a). Each of the selected regions contain 4000 pixels, over which the mean number of photons is approximately uniform, so that spatial averages are performed over identical replicas. A zoom of the selected areas is presented in fig. 18c, where the rather spectacular symmetry of the intensity distribution in the signal and idler branches shows the twin-beam character of the phase-conjugate modes. We are interested in the symmetrical pixel-pair correlation, which is evaluated 2 of the PDC photoelectrons (pe) experimentally by measuring the variance σs−i difference ns − ni of the signal/idler pixel pair versus the mean total number of down-converted pe of the pixel pair. This variance is ! 2 σs−i (2.20) = (ns − ni )2 − ns − ni 2 . In the experiment the averages are evaluated as spatial averages performed over the set of equivalent symmetrical pixel pairs contained in the chosen sample regions. Each single shot of the laser provides a different ensemble, characterized by its pixel-pair average pe number ns + ni , in turn related to the parametric gain. In the experiment, ensembles corresponding to different gains are obtained by varying the pump-pulse energy. Figure 19 shows the experimental results, where each point is associated to a different laser shot. The data are normalized to the shot-noise level. Although the noise in the individual signal and idler pixels is found to be very high and much greater than their shot-noise-level (= ns and ni respectively), we observe an evident sub-shot-noise pixel-pair correlation up to gains characterized by ns + ni ≈ 15–20. The maximum level of noise reduction observed experimentally agrees with the theoretical limit (dotted line in fig. 19) determined by the total losses of the system (∼ 1 − ηtot ). Further relevant remarks are the following. Let us start from the inequality characterizing the sub-shot-noise character of the signal/idler photon number difference, i.e. ! (ns − ni )2 − ns − ni 2 < ns + ni . (2.21)
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If we assume (as is the case) that ns = ni ≡ n ,
! ! δn2s = δn2i ≡ σ 2 ,
(2.22)
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2 normalized to the shot-noise level n + n . Each point Fig. 19. Intensity difference variance σs−i s i (open circle) corresponds to a pump pulse where the spatial ensemble statistics has been performed over a 100 × 40-pixel region. The triangles (each one obtained by averaging the experimental points corresponding to a certain gain) and their linear fit illustrate the trend of the data in the region between ns + ni = 8 and 20.
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where δns,i = ns,i − ns,i , and introduce the normally ordered expectation values: :δns δni : = δns δni , ! ! :δn2s : = :δn2i : = σ 2 − n , we obtain easily from (2.21) the inequality ! ! !1/2 :δns δni : :δn2s,i : = :δn2s : :δn2i : ,
(2.23) (2.24)
(2.25)
which states that the cross-correlation between signal and idler is larger than the (normally ordered) self-correlation. This corresponds to an apparent violation of the Cauchy–Schwartz inequality (see also § 2.1). This effect was predicted by Marzoli, Gatti and Lugiato [1997], Lugiato, Gatti, Ritsch, Marzoli and Oppo [1997], Szwaj, Oppo, Gatti and Lugiato [2000] for the case of the optical parametric oscillator and then generalized by Brambilla, Gatti, Lugiato and Kolobov [2001] to the case of the traveling-wave type I optical parametric amplifier. It represents a spatial analogue of the phenomenon of photon antibunching in time and was experimentally demonstrated in the coincidence regime by Nogueira, Walborn, Pádua and Monken [2001]. Our experiment provides the first evidence of this phenomenon in the high-gain regime.
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The analogy with the usual photon antibunching in time can be seen more clearly on the basis of the following arguments. In the experiment described above, the signal and idler photons are spatially separated with the help of a polarizing beamsplitter. If, instead, we do not separate them, in the same pixel there will be both signal and idler photons and we can consider the total number of x ) = ns ( x ) + ni ( x ), where x indicates the position of the pixel. Let photons Nt ( us now consider two symmetrical pixels labeled by x and − x . One can prove that x ) = Nt ( x ) − Nt (− x ) are below the shot-noise level, i.e. the fluctuations of N− ( ! 2 ! !2 δN−2 ( x ) = Nt ( x ) − Nt (− x ) − Nt ( x ) − Nt (− x) ! < Nt ( x ) + Nt (− x) ,
(2.26)
exactly as in type I materials [see eq. (2.12)]. Next, following the same steps of eqs. (2.20)–(2.24) with ns and ni replaced by x ) and Nt (− x ), we arrive at the inequality Nt ( ! ! :δNt ( x )δNt (− x ): > :δNt ( x )δNt ( x ): .
(2.27)
Let us now consider photon antibunching in time, assuming for simplicity a single-mode field. The condition which defines antibunching is
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g (2) (t, t ) > g (2) (0, 0),
t > t ,
(2.28)
where (Glauber [1963a, 1963b]) g (2) (t, t ) =
E (−) (t )E (−) (t)E (+) (t)E (+) (t ) :n(t)n(t ): = n 2 n 2
(2.29)
and : : denotes normal and time ordering. Condition (2.28) amounts to ! ! :n(t)n(t ): > :n(0)n(0): ! ! ⇒ :δn(t)δn(t ): > :δn(0)δn(0): ,
(2.30)
where the last passage is obtained by subtracting n2 on both sides. Clearly, inequality (2.27) is an analogue in space of eq. (2.30). It states that the joint probability of finding a photon at point x and a second photon at point x is larger than the probability of detecting two photons at the same point in space. Similarly, photon antibunching in time corresponds to the case that the joint probability of detecting two photons at different times is larger than the probability of finding them at the same time.
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Fig. 20. Detection of a weak amplitude object by measuring the intensity difference I1 −I2 . The object is located in the far field.
2.6. Detection of weak amplitude objects beyond the standard quantum limit The experimental results shown in the previous section pave the way to the following opportunity. Let us consider the case of a weak-amplitude object which is located, say, in the signal branch of the PDC far field (fig. 20). Both signal and idler are very noisy with respect to intensity fluctuations; we have shown in eqs. (2.16) and (2.17) that their photon number statistics is thermal. Therefore, in the high-gain regime the signal-to-noise ratio (SNR) characterizing the detection of a weak object in the signal field can be quite low. However, because of the spatial quantum correlation, fluctuations in the intensity difference between signal and idler are small. Hence if we detect the object by performing the signal/idler intensity difference, the signal-to-noise ratio becomes much better. This scheme is the generalization to the spatially multimode configuration of a single-mode scheme utilized to detect a weak spectroscopic signal (Ribeiro, Schwob, Maitre and Fabre [1997]). We have recently shown that, when the absorption is small enough, and the level of correlation is good enough, it is possible to increase the SNR beyond the standard quantum limit (Brambilla, Caspani, Gatti and Lugiato [2007]). The numerical simulation in fig. 21 shows an example of the SNR improvement (Brambilla, Caspani, Gatti and Lugiato [2007]) when the object is a rectangular mask. The opportunity of increasing the SNR can be important especially in the case of biological samples which cannot be strongly illuminated. The possibility of detecting a weak-phase object beyond the standard quantum limit using spatially multimode squeezed light was predicted in Kolobov and Kumar [1993].
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2.7. Multimode polarization entanglement in high-gain PDC In this subsection we turn our attention to the mixing of polarization, spatial and temporal degrees of freedom of PDC beams. The results reported here are based
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Fig. 21. Numerical simulation of the improvement of the SNR if, instead of detecting the intensity distribution of the signal beam alone (a), one detects the signal/idler intensity difference (b).
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on Gatti, Zambrini, San Miguel and Lugiato [2003], where more details can be found. Similar results can be demonstrated for the OPO configuration, as discussed by Zambrini, Gatti, Lugiato and San Miguel [2003]. Parametric down-conversion in a type II crystal represents probably the most well-known source of polarization entangled photons. The interesting configuration is that of fig. 15a, where the far-field distribution of signal and idler close to degeneracy has the form of two intersecting rings. In the two regions at the ring intersections the polarization of a photon is completely undetermined. However, once the polarization of one photon has been measured, the polarization of the twin photon, which propagates at the symmetric position, is exactly determined. In other words, when considering photodetection from these two regions, the two-photon state can be described as the ideal polarization-entangled state (Kwiat, Mattle, Weinfurter, Zeilinger, Sergienko and Shih [1995]). The question addressed by Gatti, Zambrini, San Miguel and Lugiato [2003] is whether this microscopic photon polarization entanglement leaves any trace in the regime of high parametric down-conversion efficiency, where a rather large number of photons is produced, and in which form. Quantum-optical polarization properties of light beams are conveniently described within the formalism of Stokes operators, which are the quantum counterparts of the Stokes vectors of classical optics. These operators obey angularmomentum-like commutation rules, and the associated observables are in general
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non-compatible. Gatti, Zambrini, San Miguel and Lugiato [2003] performed a detailed study of the quantum correlation between Stokes operators measured from symmetric portions of the far-field beam cross-section. Denoting by aH , aV the field operators of the horizontally and vertically polarized field components (say, signal and idler), and by x the position in the far-field zone of PDC, “local” versions of Stokes operators are defined as † sˆ0 ( x , t) = aH ( x , t)aH ( x , t) + aV† ( x , t)aV ( x , t),
(2.31)
† sˆ1 ( x , t) = aH ( x , t)aH ( x , t) − aV† ( x , t)aV ( x , t),
(2.32)
sˆ2 ( x , t) = sˆ3 ( x , t) =
† aH ( x , t)aV ( x , t) + aV† ( x , t)aH ( x , t), † −i aH ( x , t)aV ( x , t) − aV† ( x , t)aH ( x , t) .
(2.33) (2.34)
The first two operators represent the sum of and the difference between the number of horizontal and vertical photons. In the limit of a detection area larger than 2 , the investigations reported in § 2.3 the characteristic far-field coherence area xdiff show that the number of H and V photons collected from any two symmetric portions of the far-field plane are perfectly correlated observables. This implies an ideally perfect correlation, both between sˆ0 ( x ) and sˆ0 (− x ), and between sˆ1 ( x ) and −ˆs1 (− x ), for any choice of the position x in the far field (notice that sˆ0 ( x ) commutes with sˆ1 ( x )). This is a direct consequence of pairwise emission of photons with vertical and horizontal polarizations propagating in symmetric directions, as required by transverse momentum conservation. Quite different is the situation with respect to the other two Stokes operators, which involve measurements of the photon number in the oblique and circular polarization bases. Figure 22b shows a typical result for the noise in the difference between Stokes operators sˆ2 measured from small (but larger than xdiff ) symmetric portions of the far field. Results for sˆ3 are identical. For comparison, fig. 22a shows the mean intensity distribution. Parameters are those of a 2 mm-long BBO crystal, cut at 49.6 degrees for degenerate type II phase matching at 702 nm, and the emission is observed in a bandwidth 8 nm wide around degeneracy. Plot (b) clearly shows two large zones, in correspondence with the intersections of the rings, where the Stokes operator correlation is almost perfect. Out of these regions, basically no spatial correlation at the quantum level exists for Stokes operators sˆ2 and sˆ3 . Figure 22 was obtained by using a relatively narrow frequency filter. Remarkably, when a broader frequency filter is employed, the regions where Stokes parameters are correlated stretch to form a ring-shaped region around the pump direction. As explained in more detail by Gatti, Zambrini, San Miguel and Lu-
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Fig. 22. (a) Far-field intensity distribution of the down-converted field. (b) Distribution of the noise in the difference between sˆ2 measured from symmetric portions of the beam cross-section, scaled to the shot-noise level. The distribution for sˆ3 is identical.
giato [2003] this shape can be understood by considering the geometry of the down-conversion cones emitted at the various frequencies by a BBO crystal. We can hence conclude that the polarization entanglement of photon pairs emitted in parametric down-conversion survives in high-gain regimes. It takes the form of non-classical spatial correlations of all Stokes operators associated with polarization degrees of freedom. In the regions where the two rings intersect (in a ring-shaped region around the pump direction when a broad frequency filter is employed) all the Stokes operators are highly correlated at a quantum level, realizing in this way a macroscopic polarization entanglement. Although Stokes parameters are extremely noisy (it can be demonstrated that the state is unpolarized), the measurement of a Stokes parameter in any polarization basis in one far-field region determines the Stokes parameter collected from the symmetric region, within an uncertainty far below the standard quantum limit. Spatial correlations in optical angular momentum, relevant for quantum imaging, have been discussed by Altman, Köprülü, Corndorf, Kumar and Barbosa [2005].
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§ 3. Ghost imaging The topic of “Ghost Imaging” (GI) has attracted considerable attention in recent years. Invented by Klyshko many years ago (Klyshko [1988a, 1988b], Belinsky and Klyshko [1994]) with the idea of exploiting the quantum entanglement in photon pairs generated by PDC, this technique was also called entangled (two-photon) imaging until recently. In the mid-1990s some pioneering experiments were performed (Ribeiro, Pádua, Machado da Silva and Barbosa [1994],
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Fig. 23. Ghost imaging with entangled photon pairs. The object is located in the test arm 1. The functions h1 and h2 are the impulse response functions which take into account the light propagation, the presence of optical elements (e.g. lenses, apertures) and the object. (a) Case of extremely low gain, in which the information is obtained from the coincidence counts of the two photons, by varying the position x2 of the point-like detector D2 in the transverse plane of the reference arm, while D1 is fixed. (b) Case of high gain. The information is obtained from the intensity correlation function x1 )I2 ( x2 ) as a function of x2 by keeping D1 fixed and using an array of point-like detectors in I1 ( the reference arm.
Strekalov, Sergienko, Klyshko and Shih [1995], Pittman, Shih, Strekalov and Sergienko [1995]). It is by now clear that appropriate classically correlated beams can also be used to implement such a technique. In a standard imaging configuration one has a source which illuminates the object, an imaging system and a detection system. In GI, instead, one exploits the correlation between two beams to retrieve information about an unknown object. Let us describe this technique for the case of entangled photon pairs as originally conceived by Klyshko and later systematized by Saleh, Abouraddy, Sergienko and Teich [2000], Abouraddy, Saleh and Sergienko [2001a, 2002]. The photons of a pair are spatially separated and each propagates through a distinct imaging system, usually called the test and the reference arms (fig. 23a). Information is not obtained by direct measurement of photon 1, because, e.g., detector D1 is point-like and is held fixed, or D1 is a “bucket” detector which measures the total intensity of beam 1 and is therefore unable to reveal the transverse position of photon 1. Information is retrieved, instead, from the coincidences of signal/idler photon pairs as a function of the transverse position of photon 2, since detector 2 is point-like and its position scans the transverse plane. The name “ghost imaging” originates exactly from the fact that the result is obtained by scanning the position of the photon which never passed through the object. By changing the optical elements in the two arms, one can obtain different kinds of information about the object, e.g., the image (the intensity distribution) of the object (ghost image) or the diffraction pattern (the modulus square of the Fourier transform) of the object (ghost diffraction).
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The rationale behind the GI technique is that for some reasons it is not easy to act on the test arm and/or to locate in that arm an array of point-like detectors (e.g. when we wish to keep the detection hidden), hence we wish to keep the configuration of the optical elements and of the detection system in the test arm as simple as possible and fixed once and forever, whereas we have full freedom of action in the reference arm (e.g. to scan detector D2 or, equivalently, to locate an array of point-like detectors in arm 2, or to modify arm 2 so as to pass from a ghost-image to a ghost-diffraction configuration). Such a two-arm configuration provides more flexibility than standard imaging procedures. For example, there is the possibility of illuminating the object at a given light frequency in the test arm and of performing a spatially resolved detection in the other arm with a different light frequency, or of processing the information from the object by only operating on the imaging system of the reference arm (Bache, Brambilla, Gatti and Lugiato [2004a]). In addition, it opens the possibility for performing coherent imaging by using, in a sense, spatially incoherent light, since each of the two down-converted beams taken separately is described by a thermal-like mixture and only the two-beam state is pure [see eqs. (2.15) and (2.16)]. In the high-gain regime of PDC, where a large number of photon pairs are produced and detected together, this technique has been generalized to the measurement of the signal/idler correlation function of the intensity fluctuations (Gatti, Brambilla and Lugiato [2003, 2005], Bache, Brambilla, Gatti and Lugiato [2004a, 2004b]) as a function of the position x2 in the reference arm (fig. 23b). In § 3.1 we discuss the theory of GI with entangled beams, which is illustrated in § 3.2 for two paradigmatic imaging schemes. In § 3.3 we show the benefits introduced by an appropriate spatial averaging technique and discuss a homodyne detection scheme. In § 3.4 we illustrate the debate on whether entanglement is necessary or not in GI. The GI technique using splitted thermal-like beams is discussed in § 3.5 (theory) and § 3.7 (experiment). Section 3.6 is devoted to resolution, correlation, and visibility aspects, and § 3.8 shows the complementarity between the GI technique and the classical Hanbury-Brown–Twiss correlation technique.
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3.1. General theory of ghost imaging with entangled beams In the following analytical treatment we consider for simplicity only spatial variables and ignore the time argument. In addition, we assume translational invariance in the transverse plane, which amounts to requiring that the cross-section of the source is much larger than the object and all the optical elements.
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Our starting point are the input–output relations (2.18). The two outgoing beams each travel through a distinct imaging system, described by the impulse response functions h1 ( x1 , x1 ) and h2 ( x2 , x2 ), respectively (see fig. 23). The fields at the detection planes are given by ci ( (3.1) xi ) = d xi hi ( xi , xi )aiout ( xi ) + Li ( xi ), i = 1, 2, where L1 , L2 account for possible losses in the imaging systems, and depend on vacuum field operators uncorrelated with aiout . Information about the object is extracted by measuring the spatial correlation function of the intensities detected by D1 and D2 , as a function of the position x2 of the pixel of D2 : ! ! I1 ( (3.2) x1 )I2 ( x2 ) = c1† ( x1 )c1 ( x1 )c2† ( x2 )c2 ( x2 ) . All of the object information is concentrated in the correlation function of intensity fluctuations: ! ! ! G( x1 , x2 ) = I1 ( (3.3) x1 )I2 ( x2 ) − I1 ( x1 ) I2 ( x2 ) , where Ii ( xi ) = ci† ( xi )ci ( xi ) is the mean intensity of the ith beam. When using a bucket detector in arm 1, the measured quantity corresponds to the integral over x1 of both sides of eq. (3.3). Since c1 and c2† commute, all the terms in eqs. (3.2) and (3.3) are normally ordered and L1 , L2 can be neglected, thus obtaining x1 d x2 d x2 x1 d G( x1 , x2 ) = d
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× h∗1 ( x1 , x1 )h1 ( x1 , x1 )h∗2 ( x2 , x2 )h2 ( x2 , x2 ) † ! † × a1out ( x1 )a1out ( x1 )a2out ( x2 )a2out ( x2 ) ! † ! † ( x1 )a1out ( x1 ) a2out ( x2 )a2out ( x2 ) . − a1out
(3.4)
The four-point correlation function in eq. (3.4) has special Gaussian factorization properties. As can be obtained from eqs. (2.5) and (2.6): ! † † a1out ( x1 )a1out ( x1 )a2out ( x2 )a2out ( x2 ) ! † ! † ( x1 )a1out ( x1 ) a2out ( x2 )a2out ( x2 ) = a1out ! ! † † ( x1 )a2out ( x2 ) a1out ( x1 )a2out ( x2 ) . + a1out (3.5) By inserting this result into eq. (3.4) one obtains GPDC ( x2 h1 ( x1 , x2 ) = d x1 d x1 , x1 ) x2 , x2 ) × h2 (
! 2 ,
a1out ( x1 )a2out ( x2 )
(3.6)
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where, by using relations (2.5) and (2.6) and taking into account that the fields ai,in are in the vacuum state, ! d q iq ·(x −x ) 1 2 U1 ( x1 )a2out ( x2 ) = e q )V2 (− q ). a1out ( (3.7) (2π)2 The two-field correlation has hence a correlation length, or transverse coherence length xcoh , determined by the inverse of the bandwidth q0 of the function U1 ( q )V2 (− q ), which in turn is the phase-matching bandwidth of type II PDC √ q0 ∝ 1/ λlc . An essential feature is that in eq. (3.6) the modulus is outside the integral, which ensures the possibility of coherent imaging via correlation measurement. A 3D version of ghost imaging with PDC beams called quantum holography has been formulated by Abouraddy, Saleh, Sergienko and Teich [2001a].
3.2. Two paradigmatic imaging schemes Let us now analyse two paradigmatic examples of imaging systems as sketched in fig. 24. In both examples the set-up of arm 1 is fixed, consisting of an object, described by a complex transmission function T ( x ), and a lens located at a focal distance f from the object and from the detection plane. Hence, i 2πi h1 ( (3.8) x1 , x1 ) = − exp − x1 · x1 T ( x1 ), λf λf
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In arm 2 a single lens is placed at a distance z both from the source and from the detection plane 2; for simplicity we take the two lenses identical. (a) Ghost diffraction scheme. We assume z = f so that i 2πi h2 ( x2 , x2 ) = − exp − x2 · x2 . λf λf By inserting this propagator into eq. (3.6), and taking into account eq. (3.8), we obtain
2π 2π ˜ 2π 2 GPDC ( x1 , x2 ) ∝ U1 − , V2 x2 T ( x2 + x1 ) x2 λf λf λf (3.9) 0 dx −iq ·x ˜ e T ( x ) is the amplitude of the diffraction pattern where T ( q) = 2π
from the object. Note that a shift of the position x1 of detector D1 produces a translation of the pattern. The whole diffraction pattern of the object can be reconstructed via the correlation function, provided that the spatial
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Fig. 24. Imaging scheme. L denotes two identical lenses of focal length f . D1 is a point-like detector. The distance z is either z = f or z = 2f .
bandwidth q0 is larger than the maximal transverse wavenumber qmax in the diffraction pattern, or equivalently, provided that xcoh < lo , where lo is the smallest scale of variation of the object spatial distribution. Notice that, when xcoh < lo , we are in the limit of incoherent illumination, and no information about the diffraction pattern of the object can be obtained by direct detection of the intensity distribution in arm 1 without making use of the correlations with the reference arm. In fact, one can easily get:
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! I1 ( x1 ) ∝
2 2 2π d q ˜ V1 ( q ) . − q T x 1 2 λf (λf )
(3.10)
q )|2 is practically constant in the region For xcoh < lo , i.e. q0 > qmax , |V1 ( where T˜ does not vanish, and can be taken out of the integral so that the resulting expression does not depend on x1 any more. We notice that the reference arm detector is placed in the far field with respect to the PDC source; hence detection of a reference photon at some position x2 corresponds to the detection of its transverse momentum. Since the transverse momenta of the twin photons are correlated (§ 2.3), this measurement conditions the momentum of the test photon before the object. In order to form the diffraction pattern of the object, the scheme thus exploits the far-field spatial correlation of the PDC source. (b) Ghost image scheme. In the second example, we set z = 2f , so that h2 ( x2 , x2 )
=
δ( x2 + x2 ) exp
π −i| x2 | . λf 2
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Inserting this into eq. (3.6), and taking into account (3.8), we get ! ∗ i 2π x ·x1 2 1 λf GPDC ( x1 , x2 ) ∝ d x1 a1 ( x1 )a(− x2 ) T ( x1 )e 2 2π x1 2π x1 2 ≈ U1 T (− x2 ) , V2 − λf λf
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(3.11) (3.12)
where in the second line the limit xcoh lo was assumed. Since the correlation function a1 ( x1 )a(− x2 ) , which depends on x1 + x2 [see eq. (3.7)], x2 , this condition ensures is non-zero in a region of size xcoh around x1 = − that T ( x1 ) is roughly constant in this region and it can be taken out of the integral in eq. (3.11), thus obtaining eq. (3.12). In this example the intensity correlation function provides information about the image of the object. In the general case (3.11), the image reconstructed via the correlation function is a convolution of the object image with the second-order correlation function (3.7); therefore the coherence length xcoh fixes the resolution of the imaging scheme. In this scheme the reference arm detector is placed in the image plane with respect to the PDC near field, so that detection of a reference photon at position x2 corresponds to detecting its position at the crystal exit face. Because of position correlation of twin photons (§ 2.3), this measurement conditions the position of the test photon before the object. We see that in this case, in order to form the image of the object, the scheme exploits the near-field spatial correlation of the PDC source. These examples show that it is possible to pass from the diffraction pattern to the image of a given object by simply changing the optics of the reference beam (the one that never passed through the object) and leaving untouched both the source and the test arm. This possibility relies on the simultaneous presence of a spatial correlation in both the far field and the near field of the PDC source, as discussed in § 2.3. The above are basic examples; actually ghost imaging can be performed in several configurations. For example the point-like detector in arm 1 can be replaced by a “bucket” detector, 0 which collects all the radiation in arm 1; in this case the measured quantity is d x1 GPDC ( x1 , x2 ) and, in the same limit of incoherent illumination xcoh < lo , one gets: 2 x1 , x2 ) ∝ T (− x2 ) d x1 GPDC ( (3.13)
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which, again, provides the image of the object. An advantage of the bucket detector is that the lens in the test arm can be avoided and the relative position of the detection plane and the object in the test arm becomes immaterial, provided the
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detection plane is beyond the object. (Notice that in the ghost diffraction scheme the bucket detector cannot be used because it washes out the diffraction pattern.) Moreover, in the ghost image scheme, under proper limit, one can vary at will the positions of the object in the test arm and the lens in the reference arm; the image of the object is reconstructed provided that the distance p1 between the object and the lens in the reference arm (calculated as the sum of the distance between the object and the χ (2) slab along the test arm and the distance between the slab and the lens along the reference arm) and the distance p2 between the lens and the detection plane in the reference arm obey the thin-lens law (Pittman, Shih, Strekalov and Sergienko [1995]) 1 1 1 + = . p1 p2 f
(3.14)
Zeilinger [1999] provides an extensive analysis of a scheme identical to that of fig. 24, but with the detectors in arms 1 and 2 interchanged, and points out several relevant wave–particle aspects. Gatti, Brambilla and Lugiato [2003] showed that the results for imaging and the wave–particle duality features, which have been demonstrated in the microscopic case (coincidence regime), persist in the macroscopic case (high-gain regime of PDC).
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3.3. Spatial average in ghost diffraction: Increase of spatial bandwidth and of speed in retrieval. Homodyne detection scheme This subsection is based on Bache, Brambilla, Gatti and Lugiato [2004a]. From eq. (3.9) we see that the correlation provides information about the diffraction pattern of the object if we fix x1 and scan x2 , but since the gain also depends on x2 there is a limit to the information we can extract. Precisely, the gain x2 2π x2 2π U1 (− λf )V2 ( λf ) introduces a cut-off of the reproduced spatial Fourier frequencies at the imaging bandwidth of the PDC source q0 . It can be proven that an average over the position x1 of the point-like detector in arm 1 circumvents this limitation and, in addition, notably increases the speed of retrieval of the ghost diffraction pattern. This spatial averaging technique works basically in the same way in the case of ghost imaging with thermal-like beams, discussed later in this chapter. Another possibility is to exchange the detectors in the two arms, but this is not in the spirit of ghost imaging and, in addition, does not increase the speed of retrieval. We also showed that, in a ghost image scheme, by using a bucket detector instead of a point-like detector in the test arm one increases the speed of retrieval of
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the image, but no increase of image resolution is achieved. In addition, it is necessary to use a narrow-band interference filter, otherwise a significant degrading in resolution is observed. In Bache, Brambilla, Gatti and Lugiato [2004b] we analyzed the scheme of fig. 24 with the important difference that, in both the test and the reference arm, instead of a direct intensity detection, a homodyne detection is performed by introducing 50/50 beamsplitters and appropriate local oscillators which select the phases of the observed quadrature components. In the case of ghost image detection, in the reference arm one must use a telescopic two-lens configuration instead of the single-lens z = 2f configuration shown in fig. 24, whereas in the case of ghost diffraction the configuration remains as in fig. 24 with z = f . An initial motivation for using a homodyne scheme for ghost imaging came from the need to circumvent the problems related to information visibility in the macroscopic regime. Specifically, when intensity detection is performed the measured quanx1 )I2 ( x2 ) includes the homogeneous background term I1 ( x1 ) I2 ( x2 ) . tity I1 ( This term, which can be rather large, does not contain any information about the object and lowers the image visibility. Instead, by using homodyne detection the signal/idler correlation becomes second-order instead of fourth-order in the fields, and hence this background term is absent. Another advantage of homodyne detection is that arbitrary quadrature components of the test and reference beams can be measured, which means that the homodyne detection scheme allows for both amplitude and phase measurements of the object. Another very interesting point about homodyne detection is that, by measuring both quadratures in the far-field distribution, we may reconstruct the complete near-field object distribution from this information by using the inverse Fourier transform. In fig. 25 we used two different objects: (a) an amplitude transmission mask with the letters “INFM”, and (b) a more complicated amplitude transmission mask showing a picture of a wolf. We show the ghost images obtained by inverse Fourier transforms for different numbers of shots, and evidently the simple mask (a) converges faster than the more complicated mask (b). Nevertheless in both cases a good, sharp image is obtained after few shot repetitions, implying that the corresponding far-field diffraction patterns converge very fast and with a very large bandwidth. After additional averaging over shots (using 500 shots, as shown in the last frames) the irregularities gradually disappear. Recently two papers (Borghi, Gori and Santarsiero [2006], Baleine, Dogariu and Agarwal [2006]) presented schemes, different from homodyne detection, to retrieve both phase and amplitude information about the diffraction pattern. This was done by using pseudo-thermal beams.
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Fig. 25. Using the spatial average technique in an f-f set-up with two different objects (a) and (b) to obtain the ghost image using the inverse Fourier transform. The correlations are calculated from a full 3 + 1D simulation, and averaging additionally over the number of repeated pump shots shown on the right.
3.4. Debate: Is quantum entanglement really necessary for ghost imaging? This question was addressed rather early by Pittman, Shih, Strekalov and Sergienko [1995]. A more recent theoretical analysis (Abouraddy, Saleh, Sergienko and Teich [2001b]) gave arguments that the ghost imaging scheme truly requires entanglement. The topic became hot after the ghost image experiment of Pittman, Shih, Strekalov and Sergienko [1995] was successfully reproduced using classically correlated beams (Bennink, Bentley and Boyd [2002]). In this experiment a classical source produced pairs of single-mode angularly correlated
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pulses that served as classical analogs of momentum-correlated pairs of photons produced by PDC. In the accompanying theoretical discussion, Bennink, Bentley and Boyd [2002] presented arguments that although the results of any single experiment in quantum imaging could be reproduced by classical sources with proper statistical correlation, a given classical source cannot emulate the behavior of a quantum-entangled source for any arbitrary test and reference systems. In Gatti, Brambilla and Lugiato [2003] we addressed this question starting from the consideration that a key feature of the entangled state produced by PDC is the simultaneous presence of (ideally) perfect spatial correlation in the near field and the far field of the signal/idler beams (see § 2.3). The important point is that only the pure EPR state displays perfect signal/idler spatial correlation both in the near field and in the far field. On the basis of this and of more precise arguments, we argued that only in the presence of quantum entanglement is it possible to produce both the image and the diffraction pattern of an object by using a single source and by solely operating on the reference arm. Thus we pointed out the importance of performing in combination the two experiments with z = f and z = 2f in fig. 24. This interpretation was received rather well in the quantum imaging community and was generally viewed as a possibility to discriminate between the presence of quantum entanglement and classical correlation in the source. In particular, Bennink, Bentley, Boyd and Howell [2004] and D’Angelo, Kim, Kulik and Shih [2004] successfully carried out combined experiments for ghost diffraction and ghost image with entangled beams. However, in later works (Gatti, Brambilla, Bache and Lugiato [2003, 2004a, 2004b]), we found a basic counterexample, which partially contradicts the previous picture. Namely, we predicted that the classical correlation between the two beams obtained by dividing a thermal-like beam on a beamsplitter, allows one to perform in combination both experiments with z = f and z = 2f . We showed that there is a deep analogy between the use of entangled beams from PDC and of a splitted thermal-like beam in ghost imaging. In the latter case, the correlation between the two beams is not perfect either in the near or in the far field, but it is enough to perform ghost imaging very efficiently. Other papers dealing with this kind of ghost imaging are by Cheng and Han [2004], Cao, Xiong and Wang [2005], Scarcelli, Valencia and Shih [2004a]. This concludes the debate in the sense that quantum entanglement is not necessary for ghost imaging, even if it bears an important advantage in special situations, as will discussed below.
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Fig. 26. Ghost imaging with incoherent thermal light. The thermal-like beam a is divided by the beamsplitter BS into beams b1 and b2 that travel through a test and a reference arm, respectively. The rest of the set-up is identical to that in fig. 23b.
3.5. Ghost imaging by splitted thermal-like beams: Theory As we mentioned already, ghost imaging offers the possibility of performing coherent imaging using incoherent beams, since both the signal and the idler beams, taken separately, are incoherent. In this section we show that ghost imaging can be implemented using truly incoherent light, such as the radiation produced by a thermal (or thermal-like) source. We consider a scheme in which a thermal beam is divided by a beamsplitter (BS) and the two outgoing beams are handled in the same way as the PDC beams in entangled imaging (fig. 26). In this case, we start from the input– output relations of a beamsplitter,
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b1 ( x ) = ra( x ) + tv( x ),
b2 ( x ) = ta( x ) + rv( x ),
(3.15)
where t and r are the complex transmission and reflection coefficients of the mirror, a is a thermal-like field and v is a vacuum field uncorrelated with a. We assume that the state of beam a is characterized by Gaussian field statistics, in which any correlation function of arbitrary order is expressed via the second-order correlation function (Mandel and Wolf [1995]) ! d q −iq ·(x −x ) Γ ( x , x ) = a † ( (3.16) x )a( x ) = e n( q ) th . (2π)2 Here n( q ) th denotes the expectation value of the photon number in mode q in the thermal state. In the second step of this equation, we assumed for simplicity the hypothesis of translational invariance of the source, under which
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Γ ( x , x ) = Γ ( x − x ) (this is equivalent to the plane-wave pump approximation in the PDC case). In particular, the following factorization property holds: ! x )a( x )a † ( x )a( x ): :a † ( ! ! ! ! = a † ( (3.17) x )a( x ) a † ( x )a( x ) + a † ( x )a( x ) a † ( x )a( x ) . By using eqs. (3.1)–(3.4), with the PDC beams aiout replaced by the splitted thermal fields bi , by taking into account the transformation (3.15) and that v is in the vacuum state, and by using eq. (3.17), one can write the final result !2 2 ∗ † x2 h1 ( Gth ( x1 , x2 ) = |tr| d x1 , x1 )h2 ( x2 , x2 ) a ( x1 )a( x2 ) , x1 d (3.18) a † ( x1 )a( x2 )
where is given by eq. (3.16). At this point the analogy between the two cases, quantum entangled beams and splitted thermal beams, clearly emerges. Apart from the trivial numerical factor |tr|2 and the presence of h∗1 instead of h1 , the thermal second-order correlation x )a( x ) in eq. (3.18) plays the same role as the PDC signal/idler correlation a † ( x )a2out ( x ) in eq. (3.6). Consequently, from eqs. (3.16) and (3.7), function a1out ( q )V2 (− q) the thermal mean photon number n( q ) th plays the same role as U1 ( † x )a( x ) governs the properties in the PDC case. The correlation function a ( of spatial coherence of the thermal source (Gatti, Brambilla, Bache and Lugiato [2004a], Mandel and Wolf [1995], Svelto and Hanna [1998]). Its correlation length, or transverse coherence length, is the inverse of the bandwidth q0 of the x )a2out ( x ) function n( q ) th . The same comments hold for the correlation a1out ( q )V2 (− x ) in the entangled case. and the function U1 ( On the basis of this precise analogy, referring to the imaging scheme of fig. 24, we can expect that all the results for the detection of the diffraction pattern of the object [eqs. (3.9) and (3.10)], as well as the result for the ghost image case [eq. (3.12)] still hold, provided we replace U1 V2 with n th . This is actually true, except for the feature that in the diffraction pattern result (3.9) the argument of T˜
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x1 + x2 )2π/λf as a consequence of the fact that is ( x1 − x2 )2π/λf instead of ( † x1 )a( x2 ) in eq. (3.17) replaces a1out ( x1 )a2out ( x2 ) in eq. (3.6). In both cases a ( of PDC beams and thermal-like beams, the resolution in the ghost image retrieval is determined by the transverse coherence length. In addition, the replacement of a1out in eq. (3.6) by a † in eq. (3.18) brings another difference: in the thin-lens law (3.14), in the thermal case, p1 is calculated as the difference, instead of the sum, of the distance between the beamsplitter and the lens along the reference arm and the distance of the object and the beamsplitter along the reference arm (Cao, Xiong and Wang [2005]).
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3.6. Resolution aspects, correlation aspects, visibility aspects As we noted in the previous section, the resolution in the near field (ghost image retrieval) is determined by the transverse coherence length (i.e. the speckle size) at the object plane, say Δxn , exactly as in the case of PDC beams. Hence the more incoherent the thermal beam is, the better is the resolution. An example of “thermal” light whose coherence properties can be engineered is offered by, e.g., chaotic radiation obtained by scattering laser light through random media (Martienssen and Spiller [1964]). If the limit of translational invariance of the source illuminating the object is dropped, the thermal field in the far field with respect to the object acquires a finite transverse coherence length (the speckle size), say Δxf ∝ λf/ws , where ws is the transverse size of the beam illuminating the object. It turns out that the spatial resolution of the ghost diffraction scheme is determined by this length (Ferri, Magatti, Gatti, Bache, Brambilla and Lugiato [2005]). In the PDC case we have a similar situation when the plane-wave pump approximation is dropped, with ws substituted by the transverse size of the pump. Performing the ghost image and ghost diffraction experiments in combination as in the scheme of fig. 24 requires the simultaneous presence of a very good level of spatial correlation between the two arms in both the near and the far field of the object. To prove that this condition holds also in the thermal case, let us consider the number of photons detected in two small identical portions S (“pixels”) thermal beams in the near field immediately before the object, 0 of the Ni = S d x bi† ( x )bi ( x ), i = 1, 2, and the difference N− = N1 − N2 . Making use of the transformation (3.15), for |r|2 = |t|2 = 1/2 it can be proven that the variance δN−2 = N−2 − N− 2 is given by ! δN−2 = N1 + N2 , (3.19)
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which corresponds exactly to the shot-noise level. Remarkably, eq. (3.19) holds regardless of the statistical properties of the input beam a, provided that in the other input port there is the vacuum state. On the other hand, by using the identity (2.10), and taking into account that δN12 = δN22 for |r|2 = |t|2 , the degree of spatial correlation is given by C=
δN1 δN2 N1 =1− . δN12 δN12 δN22
(3.20)
For any state one has 0 |C| 1, where the upper bound is imposed by the Cauchy–Schwarz inequality. The lower bound corresponds to the coherent state, for which δN12 = N1 . For the thermal state, there is always some excess noise
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with respect to the coherent state δN12 > N1 , so that the correlation (3.20) never vanishes. Remarkably, a high degree of spatial correlation between beams b1 and b2 is ensured by the presence of a high level of excess noise in the input beam. As shown in detail in the Appendix of Gatti, Brambilla, Bache and Lugiato [2004b], for thermal systems with a large number of photons, provided that the beam is intense enough and the pixel size is not too small with respect to Δxn , one has that N1 / δN12 1, and C can be made close to its maximum value. Even more important, in the absence of losses it is not difficult to show that eqs. (3.19) and (3.20) hold in any plane linked to the near-field plane by a Fresnel transformation (Gatti, Brambilla, Bache and Lugiato [2004b]). Moreover, the field after propagation is still described by thermal statistics, since the Gaussian statistics and the factorization property (3.17) of the fourth-order correlation function are preserved by linear unitary transformations. Therefore, also in the far field the correlation can be very good, provided that the size of the detection region is not too small with respect to Δxf and the thermal beam is intense enough. A key issue is the visibility of the information in the PDC and thermal regimes. The information about the object is retrieved by subtracting the background term x1 ) I2 ( x2 ) as indicated in eq. (3.3). A measure of this visibility is given by I1 ( evaluating the following quantity in relevant positions:
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G( x1 , x2 ) G( x1 , x2 ) = , (3.21) I1 ( x1 )I2 ( x2 ) I1 ( x1 ) I2 ( x2 ) + G( x1 , x2 ) with 0 V 1. A first remark concerns the presence of n( q ) th in eqs. (3.18) and (3.16) in place of U1 ( q )V2 (−q) in eqs. (3.6) and (3.7). As a consequence, in the thermal x1 , x2 ) scales as n( q ) 2th . In the entangled case, GPDC ( x1 , x2 ) scales as case Gth ( 2 2 q )V2 (−q)| = n( q ) PDC + n( q ) PDC , where n( q ) PDC = |V2 (− q )|2 = |U1 ( 2 q )| is the mean number of photons per mode in the PDC beams, and |V1 ( q )|2 = 1 + |V1 ( q )|2 . The difference between the two cases is immaterial |U1 ( when the mean photon number is large, while it emerges clearly in the smallphoton-number regime ( n( q ) 1). Actually, in the thermal case the visibility x1 , x2 ) does not exceed the value 1/2, whatever the value of n( q ) th , since Gth ( scales in the same way as the background term. On the contrary, in the PDC case the visibility can approach the value 1 in the small-photon-number regime, x1 , x2 ) is n( q ) PDC and this term since in this case the leading scale of GPDC ( x1 ) I2 ( x2 ) ∝ n( q ) 2PDC . becomes dominant with respect to the background I1 ( Hence, in the regime of single photon-pair detection the entangled case presents a much better visibility of the information than with classically correlated thermal beams (see also Belinsky and Klyshko [1992], Mandel and Wolf [1995], Saleh, Abouraddy, Sergienko and Teich [2000]). V=
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In addition, very important for the visibility are the detection time and the size of the detected area. Standard calculations (Mandel and Wolf [1995], Saleh, Abouraddy, Sergienko and Teich [2000]) show that the visibility scales as the ratio of the coherence time τcoh of the source to the detection time. This implies that conventional thermal sources, with very small coherence times, are not suitable for the schemes studied here. A suitable source should present a relatively long coherence time, such as for example the chaotic light produced by scattering a laser beam through a random medium (Martienssen and Spiller [1964]). Similarly, the visibility scales as the ratio of the transverse coherence area to the pixel area (Gatti, Brambilla, Bache and Lugiato [2004b]), and this feature points to the same conclusions as inferred from the temporal scaling. There is a tradeoff between the degree of correlation and the visibility for what concerns the pixel size. In the same way, there is also a trade-off between resolution and visibility (Cai and Zhu [2004], Gatti, Bache, Magatti, Brambilla, Ferri and Lugiato [2006]). Pseudo-thermal sources (Martienssen and Spiller [1964]) appear as the best candidates to overcome major visibility problems, because of the possibility of engineering the speckle size independently in the near and in the far field. A quite interesting aspect of ghost imaging, both with and without quantum entanglement, is that the visibility (3.21) is independent of the level of illumination. As one can show that the visibility basically coincides with the S/N ratio (Jing and Yi-Jing [2005]), this is quite different from the case of coherent illumination.
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3.7. Ghost imaging with splitted thermal beams: Experiment In this subsection we illustrate some of the experimental results reported by Ferri, Magatti, Gatti, Bache, Brambilla and Lugiato [2005], Gatti, Bache, Magatti, Brambilla, Ferri and Lugiato [2006], with a combined ghost image/ghost diffraction scheme. A ghost image experiment, also utilizing pseudo-thermal light but without a ghost diffraction counterpart, has been done by Valencia, Scarcelli, D’Angelo and Shih [2005]. A ghost image experiment utilizing true thermal light has been reported bu Zhai, Chen, Zhang and Wu [2005]. The experimental set-up is sketched in fig. 27. The source of pseudo-thermal light is provided by a scattering medium illuminated by a laser beam. The medium is a slowly rotating ground glass placed in front of a scattering cell containing a turbid solution. When this is illuminated with a large collimated He–Ne laser beam, the stochastic interference of the waves emerging from the source produces at large enough distances a time-dependent speckle pattern, characterized
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Fig. 27. Scheme of experimental set-up (see text for details).
by a chaotic statistics and by a correlation time τcoh on the order of 0.1 s. At a distance of 400 mm from the thermal source, a 3 mm diaphragm selects an angular portion of the speckle pattern, allowing the formation of an almost collimated speckle beam characterized by a huge number (on the order of 104 ) of speckles of transverse size Δxn ≈ 25 µm. The speckle beam is separated by the beamsplitter (BS) into two “twin” speckle beams that exhibit a high (although classical) level of spatial correlation. The two beams emerging from the BS illuminate two different non-overlapping portions of the CCD camera. The data are acquired with an exposure time (1–3 ms) much shorter than τcoh , allowing the recording of high-contrast speckle patterns. The experiment demonstrates a reconstruction of both the image (fig. 28) and the diffraction pattern (fig. 29) of the object by using a single classical source, and by operating only on the optical set-up of the reference arm. The optical set-up of the object arm 1 is fixed. An object, consisting of a thin needle inside a rectangular aperture, is placed in this arm at a distance d1 from the BS. The object plane is taken as the reference plane, and referred to as the near-field plane (this is not to be confused with the source near-field, as the object plane is in the far zone with respect to the pseudo-thermal source). A single lens is placed after the object, at a distance p1 from the object and q1 = F (F is the focal length) from the CCD. Hence the CCD images the far-field plane with respect to the object. However, since the light is spatially incoherent, the diffraction pattern of the object is not visible on the CCD, as shown in fig. 29a. We consider two different set-ups for the reference arm 2. In the first, an additional lens of focal length F is inserted in arm 2 immediately before lens F . This allows us to locate the plane conjugate to the CCD plane, by temporarily inserting the object in arm 2 and determining the position that produces a well-focused image on the CCD with laser illumination (fig. 28b). The object is then translated in the object arm. The distances in the reference arm approximately obey a thin-lens
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Fig. 28. Reconstruction of the object image. (a) Correlation of intensity fluctuations. (b) Image observed using laser light. (c) Averages of 500 horizontal data sections from (a) and (b). (d) Same as (a) but with 30,000 data acquisitions instead of 5000. When the average is performed over 30,000 data acquisitions, the contrast increases significantly.
equation of the form 1/(p2 − d1 ) + 1/q2 ≈ 1/F2 with 1/F2 = 1/F + 1/F . The data of the intensity distribution of the reference arm are acquired, and each pixel is correlated with the total photon counts of arm 1, which corresponds to having a “bucket” detector there. Averages performed over 5000 data acquisitions show a well-resolved image of the needle (fig. 28a) that can be compared with the image obtained with laser illumination (fig. 28b). Figure 28c compares the corresponding horizontal sections. The spatial resolution shown by correlated imaging with incoherent light is comparable with that obtained via coherent illumination. In the second set-up the lens F is simply removed from the scheme of fig. 27, so that the CCD is in the focal plane of lens F also in arm 2. The spatial crosscorrelation of the intensities is calculated as a function of the displacement x2 − x1 between the pixel positions in the two arms, by making an additional average over pixel positions at each fixed x2 − x1 . Thus, averages over only 500 independent frames are enough to show a sharp reproduction of the diffraction pattern of the object. This is comparable with the diffraction pattern obtained by laser illu-
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Fig. 29. Reconstruction of the diffraction pattern. (a) Single-shot intensity distribution in arm 1. (b) Intensity fluctuation correlation G( x2 − x1 ). (c) Object diffraction pattern observed using laser light. (d) Horizontal cut of (b) and (c).
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mination (fig. 29c). Horizontal sections of the two patterns display a very good agreement (figs. 29b, d). We measured the fourth-order autocorrelation function in arm 2 in both the near-field and far-field planes. In this way we could determine the speckle sizes (i.e. the coherence lengths Δxn , Δxf ), which govern the ghost image and ghost diffraction resolution. We found that the near-field resolution Δxn (the ghost image resolution) was Δxn ≈ 34.3±0.6 µm, while the far-field resolution (the ghost diffraction resolution) was Δxf ≈ 15.6 ± 0.6 µm. The latter in turn corresponds to a spread in transverse wavevectors Δq = (2π/λF )Δxf = (1.94 ± 0.07) × 10−3 µm−1 , so that we find for our classical beams Δxn Δq = 0.06 ± 0.003. This has to be compared with the bound Δxn Δq > 1, which in earlier literature (Bennink, Bentley, Boyd and Howell [2004], D’Angelo, Kim, Kulik and Shih [2004]) was believed to limit the product of resolutions of ghost imaging with classical (non-entangled) beams. In addition, our result is roughly four times smaller than the results reported by Bennink, Bentley, Boyd and Howell [2004], D’Angelo, Kim, Kulik and Shih [2004] where entangled photons were used. Notice that, however, our result for Δxn Δq is not violating any EPR bound, contrary to the experiments carried out with single photon pairs Bennink, Bentley, Boyd and Howell [2004], D’Angelo, Kim, Kulik and Shih [2004]. In fact, in any plane, the probability of detecting a photon at position x2 in beam 2 conditioned to the
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detection of a photon at x1 in beam 1 is given by [see eq. (3.3)]: !4 ! P ( x2 | x1 ) ∝ I2 ( x2 )I1 ( x1 ) I1 ( x1 ) ! 4 ! x2 ) + G( x1 , x2 ) I1 ( x1 ) . = I2 (
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In our thermal case, the two terms have roughly the same height, but the first one, originating from the background, is much broader than the second, because the beam diameter is much larger than the coherence length. Hence in good approximation the conditional variance in position is the beam spot size. The product of the variances in the near and far zones is bounded by a Fourier limit, and hence does not violate any EPR bound. The crucial point is that the conditional variance and the resolution of ghost imaging coincide only in the special case where the background is negligible, as e.g. in the coincidence count regime of PDC. Only in this case, which in principle corresponds to 100% visibility, the Fourier bound holds also for resolutions. We finally observe that in the object plane (near field) the speckle size is much smaller than the spatial scale which characterizes the object, hence the illumination is incoherent, as it is clear also from the absence of the interference pattern in fig. 29a. In addition we note that the illumination is incoherent also from a temporal viewpoint, even if the acquisition time is smaller than the coherence time. As a matter of fact, one retrieves the ghost diffraction pattern on averages over thousands of data acquisitions, that is, over a time interval much longer than the coherence time. In conclusion, the only advantage of entanglement with respect to classical correlation lies in the better visibility of information. This implies a better signalto-noise ratio, which is important in high-sensitivity measurements or in quantum information schemes (where for instance the information needs to be hidden for a third party), but it does not give any practical advantage in processing information from a macroscopic classical object.
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3.8. Complementarity between “thermal” ghost imaging and the classic Hanbury-Brown–Twiss (HBT) correlation technique, with respect to spatial coherence The approach of “thermal” ghost diffraction is reminiscent of the HanburyBrown–Twiss interferometric method for determining the stellar diameter (Hanbury-Brown and Twiss [1956], Mandel and Wolf [1995]). However, in a standard HBT scheme the object is placed in the thermal beam before the beamsplitter, contrary to the ghost imaging approach where it is placed in the test arm after the
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beamsplitter. This feature introduces a basic difference: in the HBT configuration the intensity correlation function provides the Fourier transform of the modulus square of the object transmission function instead of the Fourier transform of the object, thus losing any phase information. In the ghost diffraction scheme, where the object is located in only one arm of the two, phase information about the object can be extracted and, e.g., the diffraction pattern from a pure phase object can be reconstructed. x ). Therefore the result of In the case of a double slit, |T ( x )|2 coincides with T ( the HBT scheme, which measures the autocorrelation function of the light transmitted by the object, is the same as the result of the ghost diffraction scheme, which measures the cross-correlation function between arm 1 and arm 2. The difference arises when the object introduces also a phase modulation. The most interesting case is that of a pure phase modulation, in which the result of the HBT auto-correlation measurement in the far field corresponds to a “delta-function” x )| constant), whereas the ghost pattern (the Fourier transform of |T ( x )|2 with |T ( diffraction experiment provides (modulus square of) the Fourier transform of the object. Hence only ghost imaging truly corresponds to coherent imaging with incoherent light (Bache, Magatti, Ferri, Gatti, Brambilla and Lugiato [2006]). A ghost diffraction experiment with a pure phase object, using entangled photon, has been reported by Abouraddy, Stone, Sergienko, Saleh and Teich [2004]. We recently reported (Bache, Magatti, Ferri, Gatti, Brambilla and Lugiato [2006]) about an experimental observation of the ghost diffraction of a pure phase object, utilizing pseudo-thermal light. As the features of object were very small (on the order of a few µm), in order to realise a spatially incoherent illumination of the object it was necessary to engineer a speckle beam with ∼2 µm speckles, which provides the best resolution ever attained in ghost imaging experiments, with entangled or classical light. The results of the experiment fully confirmed the picture outlined above: the diffraction pattern of the object can be efficiently reconstructed from in the ghost diffraction scheme, whereas information about the object is completely absent in the HBT scheme. We also analysed the transition from spatially incoherent to coherent illumination, by gradually increasing the speckle size in the near field (object plane). The result is that the diffraction pattern in the ghost diffraction experiment (crosscorrelation function) gradually disappears, whereas it gradually appears in the HBT auto-correlation function as well as in the direct intensity detection in the object arm. This shows not only that coherent imaging can be performed with spatially incoherent light, but also that the spatial incoherence of the source is an essential ingredient to perform ghost imaging (see also Gatti, Bache, Magatti, Brambilla, Ferri and Lugiato [2006]). Moreover, the result reveals the presence
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of a remarkable complementarity between ghost diffraction and the HBT scheme with respect to spatial coherence, when phase information is involved. Our results also provide an experimental confirmation of the increase of spatial bandwidth introduced by the spatial average, as predicted by Bache, Brambilla, Gatti and Lugiato [2004b].
§ 4. Image amplification by parametric down-conversion In this section we discuss the topic of twin quantum entangled images (§ 4.1) and that of noiseless amplification of images. The latter is introduced in § 4.2, discussed theoretically in § 4.3 and experimentally in § 4.4 for the pumped regime and in § 4.5 for the cw regime.
4.1. Twin (quantum entangled) images Let us return to the configuration of fig. 1b. We now assume that, instead of a coherent plane wave at frequency close to ωs , we inject a coherent monochromatic image (fig. 30) of frequency ωs [which corresponds to taking Ω = 0 in eqs. (1.3) and (1.4)]. Parametric image amplification has been studied extensively from a classical viewpoint (see, for instance, Gavrielides, Peterson and Cardimona [1987], Laferriere, Wetterer, Schelonka and Kramer [1989], Devaux and Lantz [1995a, 1995b, 1995c], Lantz and Devaux [2001]). A basic point in fig. 30 is that, if the image is injected off-axis, one obtains in the output a signal image that represents an amplified version of the input image, and also a symmetrical idler image. An interesting situation arises if one has, in addition to the amplifier, a pair of lenses located at focal distances with respect to the object plane, to the amplifier and to the image plane (fig. 31). As was shown by our group (Gatti, Brambilla, Lugiato and Kolobov [1999, 2000], Navez, Brambilla, Gatti
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Fig. 30. Off-axis injection of an image and generation of twin entangled images.
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Fig. 31. A possible scheme of the parametric optical image amplifier.
and Lugiato [2001]), in the limit of large amplification the two output images can be considered twins, even from a quantum-mechanical viewpoint. As a matter of fact, they do not only display the same intensity distribution but also the same local quantum fluctuations. Precisely, let us consider two symmetrical pixels in the two images (fig. 32). It turns out that the intensity fluctuations in the two pixels are identical, i.e. exactly correlated/synchronized. On the other hand, the phase fluctuations are exactly anticorrelated. So in this way, from one image one obtains twin images in a state of spatial entanglement which involves also the quadrature components X and Y , as was already described for the case of parametric fluorescence without any signal injection in § 2.2. The twin-images phenomenon is basically the spatially multimode generalization of the “quantum cloning amplifier” of Levenson, Abram, Rivera, Fayolle, Garreau and Grangier [1993]. As a matter of fact, the off-axis injection of an image gives rise to an amplified image on the one hand, and to a symmetrical “clone” idler image on the other. In the case of type II materials (as shown by Levenson, Abram, Rivera, Fayolle, Garreau and Grangier [1993]) twin images can be ob-
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Fig. 32. The spatial entanglement between the two output images concerns intensity and phase fluctuations, and also the fluctuations of quadrature components.
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tained also by injecting an arbitrary image (even symmetrical with respect to the axis) in the field polarized at 45◦ , and detecting the polarized component of the output image at 0◦ and 90◦ . For an experimental observation of twin images, see § 4.5.
4.2. Noiseless amplification of images It is well known that a phase-insensitive amplifier adds at least 3 dB of quantum noise in the output, i.e. the signal-to-noise ratio (SNR) in the output is smaller by at least a factor 2 than the SNR in the input, whereas by using a phase-sensitive amplifier the amplification may be “noiseless”, i.e. the SNR in the output may be equal to the SNR in the input (Caves [1982]). Let us focus now on the case of PDC and, more precisely, on the type I configuration of fig. 1b. The input–output relations (1.3), (1.4) are identical to those of a nondegenerate parametric amplifier (Lugiato, Gatti and Brambilla [2006]), and therefore by injecting only a coherent plane wave corresponding to q , ωs + Ω one has phase-insensitive amplification. The same is true for the off-axis injection of an image of frequency ωs in figs. 30 and 31. Hence in this case the amplification process degrades the SNR. The idea to obtain noiseless amplification is suggested by the following argument: let us re-express the two-mode field
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a( q , Ω = 0)eiq ·x + a(− q , Ω = 0)e−iq ·x ,
(4.1)
in terms of sin and cos modes as follows: q ) cos q · x + b2 ( q ) sin q · x, b1 (
(4.2)
with q ) = a( q , Ω = 0) + a(− q , Ω = 0), b1 ( q ) = i a( q , Ω = 0) − a(− q , Ω = 0) . b2 ( The input–output relations (1.3), (1.4) in terms of modes b1 and b2 read † q ) = U ( q )b1in ( q ) + V ( q )b1in ( q ), b1out (
(4.3)
† U ( q )b2in ( q ) + V ( q )b2in ( q ),
(4.4)
q) = b2out (
where we have set U ( q ) = U ( q , Ω = 0) = U (− q , Ω = 0) and similarly for V ( q ). Such relations are uncoupled and have the form of the input–output relations for a degenerate parametric amplifier (Lugiato, Gatti and Brambilla [2006]) which produces phase-sensitive amplification. Hence if one injects symmetrically two plane waves a( q , Ω = 0) and a(− q , Ω = 0) in the same coherent state, this
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Fig. 33. Scheme for parametric optical image amplifier, but with a symmetrical injection.
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Fig. 34. Symmetrical injection of an image.
corresponds to the injection of the b1 mode in a coherent state, and the amplification can be noiseless if the phase of the coherent state is chosen appropriately. This idea can be exploited to attain noiseless amplification of images with the scheme of fig. 31, in which the amplification occurs in the far field with respect to the object. Precisely, as shown by figs. 33 and 34, one injects a symmetrical image into the amplifier. In this case one has in the input two identical but uncorrelated images and in the output two amplified images in a state of spatial quantum entanglement. An aperture of diameter a is introduced into the model to keep the average intensity distribution in the output finite (Kolobov and Lugiato [1995]). The arguments above apply to type I materials. In the case of type II materials, phase-insensitive amplification can be obtained by injecting an arbitrary input image polarized at 0◦ (and detecting its polarized component at 0◦ ) and phasesensitive amplification by injecting an arbitrary input image polarized at 45◦ , and detecting (in the input and in the output) its polarized components at 0◦ and 45◦ .
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Fig. 35. Alternative scheme for a parametric amplifier.
An alternative amplification scheme is shown in fig. 35 (Wang, Yang, Gatti and Lugiato [2003]). In this case, before and after the amplifying crystal there is a telescope. The first telescope carries the object to the input face of the crystal, hence in this case the amplification occurs in the near field of the object. The first detailed theory for noiseless image amplification was formulated for the scheme of fig. 33 (Kolobov and Lugiato [1995], Sokolov, Kolobov and Lugiato [1999]), while the experiments described later utilise the scheme of fig. 35. The model corresponding to the scheme of fig. 35 does not require the introduction of an aperture and, in addition, also allows the amplification of asymmetrical images. A last scheme that has been analysed theoretically for noiseless image amplification is based on the use of a confocal ring resonator (Mancini, Gatti and Lugiato [2000]); it is shown in fig. 36. An aperture P is introduced for the same reasons as in Kolobov and Lugiato [1995], Sokolov, Kolobov and Lugiato [1999]. Another analogy with the scheme of fig. 33 is that it amplifies symmetrical images.
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Fig. 36. Optical scheme of the parametric image amplifier with confocal cavity. The latter consists of a ring resonator containing two equal lenses whose focal length f is one fourth of the cavity round trip length. Mirror M1 has non-zero transmittivity, while the others are totally reflecting at the signal wavelength. All the distances O − L1 , L1 − R, R − L2 , L2 − C, C − L1 , C − L3 , L3 − P , P − L4 , L4 − I are equal to f .
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4.3. Theory of noiseless amplification of optical images Let us consider first the scheme of fig. 33. The image resolution in this case is determined by the “diffraction area” Sdiff =
(λf )2 , Sp
(4.5)
where λ is the wavelength, f is the focal length, and Sp = (π/4)a 2 is the area of the aperture (Kolobov and Lugiato [1995], Sokolov, Kolobov and Lugiato [1999]). Let us now indicate by S( x ) the complex, even, stationary input image, with S( x ) = eiφ(x ) S( (4.6) x ). This scheme is capable of amplifying in an approximately uniform way a central portion of the object of area Samp = (λf q0 )2 , or an annular portion, according to the phase-matching conditions which govern the variation of the functions 2π 2π u( x ) ≡ U q = x, Ω = 0 , v( x ) ≡ V q = x, Ω = 0 , λf λf with respect to x. In particular, the phase φ( x ) of the object must be correlated with the phases of u( x ) and v( x ) (Sokolov, Kolobov and Lugiato [1999]). In order to analyse the issue of noiseless amplification, one considers the noise figure
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F =
(SNR)in ; (SNR)out
(4.7)
the amplification is noiseless when F = 1. Note that in the definition of (SNR)in one must assume that it corresponds to ideal (i.e. equal to unity) quantum efficiency in the photodetection. As explained by Kolobov and Lugiato [1995], this is necessary to ensure that the noise figure characterises the noise added by the amplifier but not the noise due to the imperfection in the pre-amplification (like nonideal photodetection in the object plane, see Levenson, Abram, Rivera, Fayolle, Garreau and Grangier [1993], Levenson, Abram, Rivera and Grangier [1993]). Without such a prescription, i.e. allowing for non-ideal photodetection in the object plane, the noise figure may be lower than unity. Assuming that the detection time Td is much larger than the characteristic time of the amplifier Tamp = 2π/Ω0 , where Ω0 is, as usual, the temporal bandwidth of PDC, a necessary condition for attaining noiseless amplification is that the object is intense enough, precisely 2 1 S( x ) Sdiff Tamp . 4
(4.8)
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If we denote by S0 the object area amplified in a uniform way, by Sel the area which characterises the smallest elements of the object, and by Sd the area of the pixels in the detection array, the optimum choice for noiseless amplification with sufficient spatial resolution is S0 = Samp Sel Sd , Sdiff .
(4.9)
Under such conditions, the number of object elements resolved by the amplifier is Nel ≈
Samp SP = = SP q02 , Sdiff Scoh
(4.10)
2 = 1/q 2 . where Scoh = xcoh 0 It must be added that noiseless amplification can be attained both for Sd Sdiff and for Sd Sdiff . In the latter case the noise figure may be smaller than unity, as in the case of non-ideal photodetection in the object plane (Sokolov, Kolobov and Lugiato [1999], Kolobov and Lantz [2006]). More details can be found in these references. In the case of the scheme of fig. 35, the scenario is basically the same, if one replaces the area Sdiff with the coherence area Scoh which governs the resolution in the near field and Samp , SP by the pumped area in eqs. (4.9) and (4.10). The image resolution is controlled by the coherence area Scoh , and the number of object elements resolved by the amplifier is given again by eq. (4.10). In the broadband limit q0 → ∞, the functions U ( q , Ω = 0) and V ( q , Ω = 0) become practically constant. Finally, let us consider the scheme of fig. 36 based on the use of a confocal resonator. The peculiarity of a perfectly confocal resonator is that, if one considers only symmetrical input images, it is diffractionless (Mancini, Gatti and Lugiato [2000]). In the ideal case that the pump is a plane wave, the coefficients of the input–output relations are constant. In this configuration any symmetrical input image is amplified in a perfectly uniform way. The condition (4.8) for noiseless amplification holds also for this scheme. In Mancini, Gatti and Lugiato [2000] one finds also the description of the effects which arise when the confocality is imperfect. The issue of noiseless amplification of optical images has been discussed theoretically also for the case of intracavity second-harmonic generation (Scotto and San Miguel [2002], Scotto [2003]). These researches in the quantum domain were accompanied by works of the same group at a classical level, concerning the possibility of all-optical image processing using second-harmonic generation (Scotto, Colet and San Miguel [2003], Jacobo, Colet, Scotto and San
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Miguel [2005]). While the classical investigations were carried out for intracavity second-harmonic generation, the studies in the quantum domain were done for the traveling-wave configuration. The issue of noiseless amplification was discussed for a scheme basically identical to that of fig. 33, both for type I (Scotto and San Miguel [2002], Scotto [2003]) and for type II (Scotto, Colet, Jacobo and San Miguel [2006]) second-harmonic generation. Details can be found in Scotto, Colet, Jacobo and San Miguel [2006].
4.4. Noiseless amplification of optical images: Experiments in the pumped regime Noiseless amplification of temporal optical signals was demonstrated experimentally by Ou, Pereira and Kimble [1993] for cw and by Levenson, Abram, Rivera, Fayolle, Garreau and Grangier [1993] for pulsed signals. Quantum fluctuations in parametric image amplification have been studied experimentally by Choi, Vasilyev and Kumar [1999] and by Mosset, Devaux and Lantz [2005]. The first experiment reported the observation of noiseless image amplification for the first time, investigating the temporal behavior of quantum fluctuations recorded by a photodiode while this was scanned over an image. The second experiment, instead, addressed directly the issue of spatial distribution of quantum fluctuations in the image by means of spatial averages, similar to those of Jedrkiewicz, Jiang, Brambilla, Gatti, Bache, Lugiato and Di Trapani [2004], demonstrating spatially noiseless amplification for the first time. Let us now discuss these two experiments in order. The experimental set-up of Choi, Vasilyev and Kumar [1999] is shown in fig. 37. The signal and pump pulses are provided respectively by the fundamental (1064 nm) and the second harmonic of a Q-switched mode-locked Nd:YAG laser at a repetition rate of 1 kHz. The amplification is performed in a KTP crystal. The input image is polarized at 45◦ of the crystal neutral axis, in order to obtain phase-sensitive amplification. The phase of the pump beam relative to that of the signal beam is locked to maximize the parametric gain. The object is a resolution chart. By taking into account the measured overall detection efficiency, the average value for the noise figure turns out to be slightly larger than unity for phasesensitive amplification, and is clearly lower than the quantum limit of an ideal phase-insensitive amplifier of the same gain. The experimental set-up of Mosset, Devaux and Lantz [2005] is shown in fig. 38. The signal and pump pulses are provided respectively by the second har-
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Fig. 37. Scheme of the experimental set-up to measure the noise figure of an image amplified by a traveling-wave OPA. A feedback signal to the PZT is used to lock for maximum amplification in the phase-sensitive configuration. From Choi, Vasilyev and Kumar [1999].
monic (1.2 ps duration FWHM at 527.5 nm) and the fourth harmonic of a Qswitched mode-locked Nd:Glass laser at a repetition rate of 33 Hz. The type I amplification is performed by a BBO crystal. Collinear interaction is phase-sensitive, while phase-insensitive amplification is obtained by a slight angular shift between the pump and signal beams introduced by appropriately shifting the spatial filter in fig. 38. The object is, again, a resolution chart. The measurement of (SNR)out is performed by using single-shot images, in order to take into account the strong variation of gain from one shot to another in the phase-sensitive scheme because of the non-controlled variation of the relative phase between the signal and the pump. Good results have been obtained by
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Fig. 38. Experimental set-up for spatially noiseless amplification. From Kolobov and Lantz [2006].
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Fig. 39. Example of amplified image in the PSA scheme. Area of 3266 pixels. From Mosset, Devaux and Lantz [2005].
performing difference of images, in order to eliminate the spatial defects that are reproducible from one shot to another. The (SNR)out is measured by selecting an area with approximately constant mean intensity, over which the spatial average can be operated. The intensity of pure PDC (i.e. without input image) is subtracted from the amplified images. The noise figure has been evaluated for both phase-insensitive and phasesensitive operation. In the latter case, because the relative phase is not controlled, it is more difficult to find pairs of images that correspond both to the same gain and the same phase. Figure 39 shows an example of a selected image. The noise figure is measured for different groupings of the pixels of the CCD camera (achieved by software) in order to consider effective detector areas Sd smaller or greater than Scoh . The results turn out to be in good quantitative agreement with the theoretical predictions for both phase-insensitive and phase-sensitive conditions. More details can be found in Mosset, Devaux and Lantz [2005]; see also Fabre [2005].
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4.5. Noiseless amplification of optical images: Experiment in the cw regime. Experimental observation of twin images Very recently we learnt (Lopez, Treps, Fabre and Maitre [2007]) about the first experimental observation of noiseless amplification in the cw regime. This was obtained using OPOs below threshold (that is, in the amplification regime) with a degenerate cavity, pumped by a cw laser. The configuration of the experimental set-up is of the kind of fig. 36, with the relevant difference that the cavity has separate input and output mirrors. This configuration maximizes the spatial quantum correlations, but introduces an extra source of vacuum noise that degrades
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the signal-to-noise ratio. Hence this experiment does not correspond to noiseless amplification strictu sensu, but it is undoubtedly image amplification in the quantum regime. This can be seen by measuring an “on–off” noise figure F00 , defined as the relative variation of the SNR measured at the output when the amplifier pump field varies from its actual value “on” to zero (“off”). The experiment has been done on the one hand with a type I LiNbO3 crystal in a doubly resonant confocal cavity, on the other with a type II KTP crystal in a triply resonant, dual (Gigan, Lopez, Delaubert, Treps, Fabre and Maitre [2006]) half-confocal cavity. In a half-confocal cavity, only half of the even modes are simultaneously resonant with the cavity, and the injected image gets mixed at the output with its spatial Fourier transform (Gigan, Lopez, Treps, Maître and Fabre [2005]). The relative phase between the pump and the input image is controlled by a piezo-electric transducer. The input image is created via a USAF resolution chart. In both type I and type II materials, the on–off noise figure for the phasesensitive configuration turns out to be close to the theoretical curve and substantially smaller than in the phase-insensitive case. In the type II case, the output image was decomposed into its 0◦ and 90◦ components, and it is observed that the noise in the difference between the intensities in corresponding pixels of the CCD camera is 30% smaller than the shot-noise level: this amounts to the first experimental observation of the twin-image phenomenon described in § 4.1.
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§ 5. The quantum laser pointer Precision optical imaging using CCD cameras or photodetector arrays is necessary in many sciences, for instance in astronomy or biology. The ultimate limit to the performances of optical imaging technology is imposed by quantum mechanics. Of special importance from an applicative standpoint is the measurement of image displacement, for example the displacement of a laser beam. This problem has been recently studied from a quantum-mechanical viewpoint by Fabre, Fouet and Maitre [2000]. According to the Rayleigh diffraction limit, the position of the laser beam can be measured with an error on the order of the beam section. However one can use, rather, a split detector which measures the intensities i1 and i2 from the two halves of the beam cross-section, as shown in fig. 40. This technique is used, for instance, in atomic force microscopy (Putman, De Grooth, Van Hulst and Greve [2006]), in the measurement of very small absorption coefficients via the mirage effect (Boccara, Fournier and Badoz [1980]), and in the observation of the motion of a single molecule (Kojima, Muto, Higuchi and Yanagida [1997]). If
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Fig. 40. Measurement of very small displacements in the transverse plane. (Courtesy C. Fabre.)
one gradually displaces the beam with respect to the detector and plots the intensity difference, one obtains a curve such as that shown in fig. 40 and the precision of the measurement of the displacement is limited only by noise. With a classical, shot-noise-limited laser source, the smallest displacement that can be measured (with a SNR of one) is given by Fabre, Fouet and Maitre [2000] π w0 dSQL = (5.1) √ , 8 N where w0 is the waist of a Gaussian TEM00 mode and N is the total number of photons recorded by the two detectors during the measurement time. In this way one can measure shifts in the sub-nanometric range. However, in the case of the microscopic observation of biological samples one cannot raise the photon number too much to avoid damaging the sample. Similarly, in atomic force microscopy excessive laser power leads to radiation pressure noise (Smith [1995]). Therefore it is important to have the possibility of going beyond the classical limit by reducing the noise of i1 − i2 below the shot-noise level. To achieve this, one possibility is to use a beam in a state of spatial entanglement between its upper and its lower part, because in this way the numbers of photons in the two parts are basically identical and the fluctuations in the intensity difference are very small. For example if, in an experiment like that of Jedrkiewicz, Jiang, Brambilla, Gatti, Bache, Lugiato and Di Trapani [2004] one does not separate the signal and idler fields using a polarizing beamsplitter, the beam has the same total (signal + idler) number of photons in symmetrical pixels or, more precisely, the photon number difference between the two pixels is below shot noise [see eq. (2.26)]. Alternatively, one can use a type I material, with the same result. Fabre, Fouet and Maitre [2000] devised a different procedure to reduce the quantum noise in the difference (i1 − i2 ). It consists in engineering a multimode
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Fig. 41. Electric field profile of the two constituents used to form the non-classical multimode beam. From Treps, Andersen, Buchler, Lam, Maître, Bachor and Fabre [2002].
non-classical beam which displays the wanted spatial correlations. Let us now illustrate this approach in the simplest configuration, in which one wants to measure displacements in one direction only. Figure 41 shows a Gaussian TEM00 mode u0 (x) and a flipped mode u1 (x), obtained by flipping the left part of the Gaussian mode. Such two modes can be viewed as the first two elements of a complete orthonormal basis. The key point is that, as one can show, only these two modes must be considered in calculating the quantum noise, and a beam obtained by superimposing them has the correlation property required to reduce (i1 − i2 ) below shot noise (Fabre, Fouet and Maitre [2000]).
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5.1. 1D experiment In the experimental realization of the scheme described in the preceding subsection (Treps, Andersen, Buchler, Lam, Maître, Bachor and Fabre [2002]) the special non-classical mode which is synthesized arises from the superposition of a Gaussian mode in a squeezed vacuum state and a flipped mode which lies in a coherent state. The flipped mode is obtained from a Gaussian beam by sending it through a specially designed phase plate, consisting of two birefringent half-wave plates, one rotated by 90◦ with respect to the other. The squeezed output from an OPA is superimposed onto the squeezed flipped mode using a beamsplitter which reflects 92% of the squeezed state and transmits 8% of the coherent state. The reflected output is then sent to a quadrant detector with quantum efficiency larger than 90%; only two of the four quadrants are used in the experiment. The noise measured in the difference (i1 − i2 ) turns out to be at 2.34 ± 0.05 dB below the shot-noise level. This result suggests that the beam is made of two strongly correlated parts. This spatial noise correlation has been used to improve the precision of displacement measurements in the image plane. In order to pro-
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Fig. 42. Noise spectrum of the photocurrent difference in the presence of an oscillating displacement of amplitude 2.9 Å and frequency 4.5 MHz (resolution bandwidth: 100 kHz): (a) using a coherent state of light, (b) using the two-mode non-classical state of light. This curve is obtained by averaging the signal over 10 successive traces. From Treps, Andersen, Buchler, Lam, Maître, Bachor and Fabre [2002].
duce a small controllable beam displacement in the frequency range of the measurements, a modulation at 4.5 MHz as signal for the displacement measurement has been introduced; this can easily be distinguished from the low-frequency beam displacements induced by mechanical or acoustic vibrations. Figure 42 shows the differential signal monitored by a spectrum analyzer when the light beam undergoes a displacement modulation with an amplitude of 2.9 Å. The setup records a modulation peak in the Fourier spectrum. Plot (a) of the figure shows the trace when vacuum instead of the squeezed vacuum is used in mode u0 (x); this noise floor gives the standard quantum limit in such a displacement measurement. The SNR of this measurement is 0.68. When the two-mode non-classical beam is utilized in the measurement (plot (b) in fig. 42), a SNR of 1.20 is obtained, which shows an improvement of the displacement measurement sensitivity by a factor of 1.7. These results demonstrate that the noise floor of displacement measurements can actually be reduced to below the standard quantum limit.
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5.2. 2D quantum laser pointer This experiment (Treps, Grosse, Bowen, Fabre, Bachor and Lam [2003]) demonstrated the possibility of measuring both coordinates of the beam’s center with a precision beyond the standard quantum limit, using a quadrant detector (fig. 43).
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Fig. 43. Measurement of laser beam direction. A laser beam is incident on a quadrant detector. Simple arithmetic operations are performed on the four photocurrents to produce signals Ix and Iy , which are proportional to the displacement in the horizontal and vertical axes, respectively. An example signal for Ix is plotted. The standard deviation of the signal σI defines the quantum-noise limited displacement dQNL . From Treps, Grosse, Bowen, Fabre, Bachor and Lam [2003].
The beam used in this experiment is the superposition of three modes: one is a Gaussian TEM00 mode, the second is obtained from the Gaussian mode by flipping it with respect to the horizontal, and the third is obtained from the Gaussian mode by flipping it with respect to the horizontal and to the vertical. The simply flipped mode is in an intense coherent state and the other two modes are in a squeezed-vacuum state (about 4 dB) produced by two OPAs driven by one laser. The mixing of the two squeezed modes was obtained by a ring optical cavity, using a novel technique. Finally, the resulting beam was mixed with the coherent beam on a 95%/5% beamsplitter. In this way, they achieved simultaneously 3.05 ± 0.1 dB of noise reduction in the horizontal direction and 2.0 ± 0.1 dB in the vertical direction. Instead of the average position of the beam, which is difficult to measure because of low-frequency vibrations and air-index fluctuations, the authors measured oscillations of the beam position with ultrasmall amplitude dmod at high temporal frequencies Ω (Ω > 1 MHz). They first measured the quantum noise n(Ω) with no modulation, using standard quantum optics techniques (Bachor [1998]), and then added a very small modulation of the beam position d(Ω). To determine the oscillation amplitude at frequency Ω they used a spectrum analyzer, which demodulates the signal and measures the power spectral density. It displays the value for n(Ω)2 + d(Ω)2 on a logarithmic dB scale (fig. 44A). The detection time was Δt = 10 µs. To perform the oscillation, a mirror mounted on a piezo-electric transducer was used. The modulation amplitude dmod was slowly
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EBL Fig. 44. (A) Measurement of the horizontal displacement signal ramped up in time, with (i) coherent beams and (ii) spatially squeezed beams. Both radio-frequency spectrum analyzer traces are the averages of 20 runs. The observed noise reduction of the squeezed beams measurement is 3.3 ± 0.2 dB. (B) Data from (A) processed to show signal-to-noise improvement (left vertical axis) plotted against the inferred displacement. Traces (iii) and (iv) show the results from data (i) and (ii), respectively. The squeezing translates into an increase in displacement sensitivity. Choosing a 99% confidence level (right vertical axis), the smallest displacement detectable improved from 2.3 to 1.6 Å. From Treps, Grosse, Bowen, Fabre, Bachor and Lam [2003].
increased over time. The upper curve corresponds to the measurement performed with a coherent state, the lower curve corresponds to the same measurement but with the non-classical beam. The signal differs from the background noise for a modulation amplitude, which is greater for the coherent beam than for the spatially squeezed beam. In fig. 44B, the same data are normalized to the respective noise levels. The vertical axis is now the difference between the measured signal and the noise with no displacement, and the traces for the coherent and the spatially squeezed beams are superimposed. In the case of the spatially squeezed
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light, the average of the signal trace crosses the threshold of confidence, set here at 3 standard deviations, for a smaller oscillation amplitude. A corresponding oscillation amplitude of 1.6 Å was found; this corresponds to an improvement by a factor of 1.5 compared with the standard quantum noise limit. This improvement was achieved for two simultaneous measurements performed on the beam, namely the oscillation amplitudes in two orthogonal directions. Thus the experiment of Treps, Grosse, Bowen, Fabre, Bachor and Lam [2003] utilized a multimode squeezed beam with squeezing in more than one mode. The main point in the approach followed by these two experiments (in 1D and 2D) is to identify the modes which introduce noise and reduce the noise level below shot noise by setting such modes in a state of squeezed vacuum. Such an approach is general, and has been followed not only by Fabre, Fouet and Maitre [2000] but also in the quantum study of Kolobov and Fabre [2000]. This analysis has been extended to an arbitrary number of independent measurements, each being a channel from the viewpoint of information theory. To improve an n-channel measurement beyond the standard quantum limit one has to use n squeezed states in appropriate modes (Treps, Bachor and Fabre [2006]). The configuration of the experiment in 2D is scalable, and it is possible to increase the number of independent modes with a slight variation of the layout. From a quantum-information point of view, each transverse mode is an independent communication channel, and the usual quantum communication proposals can be extended to the multichannel configuration (Sokolov, Kolobov, Gatti and Lugiato [2001]). Hence this approach and these experiments are interesting not only for quantum imaging but, more in general, for quantum information, and pave the way to the use of multimode light in the parallel processing of quantum information. Such issues are illustrated in general in an article by Treps, Bachor and Fabre [2006] that formulates a general theory for a modal synthesis quantum imaging, and also includes a discussion of the optical read-out of a Gaussian beam after interaction with the surface of a CD using an array detector, and the description of a spatial homodyne technique to measure a small displacement in an optimal way.
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§ 6. Miscellaneous In this article we selected, somewhat arbitrarily, according to our personal interest and taste, some topics in the field of quantum imaging and we described them in some details in the previous sections. However, there are several other relevant topics which deserve a lot of attention, and we try to review them in this section, some of them with a concise description, some basically with a list of references.
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Fig. 45. Scheme of imaging system. (Courtesy M.I. Kolobov.)
6.1. Object reconstruction Kolobov and Fabre [2000] have studied the quantum limits in the process of image reconstruction. The very well-known scheme they analysed is shown in fig. 45. The object is contained in a finite region of size X. The imaging system is composed of two lenses and an aperture. Because of the transverse finiteness of the imaging system, diffraction comes into play, with the consequence that some details of the object are blurred. In order to better reconstruct the object, one can proceed as follows. One considers the linear operator H which transforms the object into the image, its eigenvalues and its eigenfunctions, which are called prolate spheroidal functions:
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Hfn ( x ) = μn fn ( x ).
(6.1)
If the imaging were perfect, H would be the identity operator and all eigenvalues would be equal to unity. An imperfect imaging introduces a sort of loss so that μn 1. Now one expands the image e( x ) on the basis of the eigenfunctions: e( x) = (6.2) cn fn ( x) n
and obtains the coefficients cn . The object a( x ) is reconstructed by the following expression: cn a( x) = (6.3) x ), √ fn ( μn n √ where the coefficients cn have been replaced by cn / μn . In principle the reconstruction is perfect, but a problem is introduced by the circumstance that when the index n is increased the eigenvalues μn quickly approach zero. As shown by Kolobov and Fabre [2000], this feature makes the reconstruction of fine details very sensitive to noise, so that again quantum noise represents the ultimate limitation. The idea proposed by Kolobov and Fabre [2000] is
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to illuminate the object by bright-squeezed light instead of coherent light, and to replace the vacuum state in the part of the object plane outside the object itself by squeezed-vacuum radiation. In this way one can obtain a definite improvement of the resolution in the reconstruction, beyond the standard quantum limit. This quantum theory of super-resolution has been developed further in Kolobov, Fabre, Scotto, Colet and San Miguel [2003] and Beskrovnyy and Kolobov [2005]; especially, a review of the state of the art in this problem is given in Kolobov [2006b]. 6.2. Entangled two-photon microscopy In recent years there has been a lively interest in entangled photon microscopy (Teich and Saleh [1997]). The surge in the development of fluorescence microscopy based on two-photon excitation using laser light has been driven by the principal advantages of this technique over single-photon excitation: a pair of lowenergy photons can deposit as much energy as a single ultraviolet photon, thereby exciting a fluorescent molecule within a sample with greater penetration depth, better resolution and less risk of damage upon absorption along the optical path. However, in order to obtain two-photon absorption with a classical light source such as a laser, a very large photon-flux density is necessary to place two photons within a small enough volume and time window that the fluorescent molecule can absorb them. In this latter case, a femtosecond-pulsed high-power laser is used directly as the source of light, which can produce undesired photodamage of the specimen. On the other hand, entangled photons generated by the process of spontaneous parametric down-conversion in a nonlinear crystal comprise intrinsically paired photons within a small volume and short time window. In principle, therefore, far smaller photon-flux densities can be used to affect absorption so that the risk of photodamage to the specimen is reduced. Yet other possible advantages arise from the direct proportionality of absorption to photon-flux density (rather than the quadratic relation that holds for ordinary two-photon absorption) and the fact that the sum of the photon energies of each entangled pair is a constant and equal to the energy of the down-converted pump photon (whereas the sum energy is far broader for photons from a femtosecond laser). However, one of the principal challenges in implementing this form of microscopy is obtaining an entangled-photon flux that is sufficient to excite two-photon transitions, which have a limited cross-section (Fei, Jost, Popescu, Saleh and Teich [1997]).
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Further developments are described by Saleh, Jost, Fei and Teich [1998], Nasr, Abouraddy, Booth, Saleh, Sergienko, Teich, Kempe and Wolleschensky [2002], Lissandrin, Saleh, Sergienko and Teich [2004].
6.3. Quantum-optical coherence tomography This technique is devised for carrying out tomographic measurements with dispersion-cancelled resolution (Abouraddy, Nasr, Saleh, Sergienko and Teich [2002]). It bears a resemblance to classical optical coherence tomography, but it utilizes an entangled two-photon light source that permits measurements to be made at depths greater than those accessible to standard optical coherence tomography, which is affected by the deterioration effects arising from sample dispersion. Quantum-optical coherence tomography also offers higher sensitivity as well as an enhancement of resolution by a factor of two for the same source bandwidth. Experimental demonstrations of this technique to carry out axial optical sectioning, including the factor-two enhancement in resolution, were given by Nasr, Saleh, Sergienko and Teich [2003, 2004]. A polarization-sensitive version was formulated by Booth, Di Giuseppe, Saleh, Sergienko and Teich [2004]. The possibility of enhancing the axial resolution by chirped quasi-phase matching was theoretically shown by Carrasco, Torres, Torner, Sergienko, Saleh and Teich [2004].
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6.4. Quantum ellipsometry A subject related to quantum imaging is quantum ellipsometry, an interferometric technique for performing ellipsometric measurements which also relies on the use of a non-classical source, namely, the polarization-entangled twin photons generated by spontaneous PDC, in conjunction with a coincidence-detection scheme (Abouraddy, Toussaint, Sergienko, Saleh and Teich [2001, 2002]). Measurements acquired by this technique are absolute, since they do not require source and detector calibration. An experimental demonstration has been given by Toussaint, Di Giuseppe, Bycenski, Sergienko, Saleh and Teich [2004]. 6.5. Transverse distribution of quantum fluctuations in free-space spatial solitons With the lively interest in the subject of spatial solitons and the spectacular development of the field of quantum information, a natural question arises: can solitons be of some interest as potential supports not only of classical digital information,
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but also of quantum information? Hence it is highly interesting to study the quantum properties of spatial solitons. A Green-function approach has been used in this investigation (Treps and Fabre [2000], Lantz, Sylvestre, Maillotte, Treps and Fabre [2004]). The main results concern the spatial distribution in 1D solitons. Because of diffraction, as the soliton propagates strong quantum anticorrelation develops between symmetrical points inside the scalar soliton spot, and between the polarization components for the χ (2) soliton. Because its formation is not due to phase effects, squeezing in a χ (2) soliton can be observed in the amplitude quadrature, without homodyne detection. In view of applications to quantum information, an exploration of the quantum properties of soliton arrays has been started (Lantz, Cambournac and Maillotte [2002]). More details can be found in Lantz, Treps and Fabre [2006]. 6.6. Quantum fluctuations in cavity solitons In this case one considers spatial solitons which are generated in optical cavities, or cavity solitons (Lugiato [2003]). Hence these are dissipative solitons, whereas standard spatial solitons arise in conservative contexts. Recent studies have analysed the quantum features of cavity solitons arising in degenerate OPOs from the locking of domain walls. It turns out that quantum fluctuations can be responsible for the growth of arrays of cavity solitons (Rabbiosi, Scroggie and Oppo [2002, 2003]) and for the appearance of remarkable quantum correlations in the near and the far field (Zambrini, Barnett, Colet and San Miguel [2003]). A detailed account of these results can be found in Oppo and Jeffers [2006].
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6.7. Quantum holographic teleportation and dense coding of optical images This subsection is more directly connected with the general field of quantum information. Two well-known protocols of quantum information are quantum teleportation and quantum dense coding; both are rooted in the phenomenon of quantum entanglement. Here we discuss the extension of such protocols, in their continuous-variable version, from the usual single-mode regime to the multimode configuration of optical images. In the case of quantum teleportation of optical images (Sokolov, Kolobov, Gatti and Lugiato [2001], Gatti, Sokolov, Kolobov and Lugiato [2004]), we consider an OPA in the output of which we have two twin fields which are very noisy but still in a state of spatial entanglement; one is sent to Alice’s station and the
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other to Bob’s station. The teleportation scheme generalizes the one formulated by Vaidman [1994], Braunstein and Kimble [1998] for quantum teleportation in terms of continuous variables. The input image to be teleported lies in an arbitrary quantum state, and using a 50/50 beamsplitter it is combined with one of the two noisy twin images, so that its quantum state is completely corrupted at this stage. However, the system as a whole works as a quantum fax machine, which is capable of reproducing in the output image not only the average properties of the input image, but also the details of its quantum state. This is due to the fact that, thanks to the perfect correlation between the two noisy entangled twin images, all the noise introduced at the initial state is exactly cancelled in the final stage. This process is called “holographic quantum teleportation” because it shows striking analogies with standard holography, apart from the fact that, of course, it works on a quantum level. This generalized protocol opens new potential applications of teleportation as quantum interface in two-dimensional parallel quantum computing, in parallel quantum communication, quantum memory, error correction, etc. The continuous-variable quantum dense coding, elaborated by Braunstein and Kimble [2000], has been generalized to the spatially multimode configuration in Golubeva, Golubev, Sokolov and Kolobov [2006]. More details can be found in Sokolov [2006]. More generally it can be said that these investigations, together with those described in §§ 6.1, 6.2 and 6.3, pave the way to parallel quantum information processing. The multimode quantum entanglement produced by PDC at the single photon-pair level is preserved in the high-gain regime where entangled light beams containing a macroscopic number of photons are produced. It leads in the latter case to strong correlations between continuous optical variables, such as intensity, or field quadratures, which can be used to drastically reduce the detrimental effect of quantum fluctuations in optical data processing. Such a regime is characterized, rather than by entangled photon pairs, by pairs of modes linked by a continuous variable entanglement. Since a huge number of entangled mode pairs is simultaneously available, this configuration provides a parallel operation, which is alternative to the usual sequential operation where single photon pairs are detected one after the other. In experiments such as those described in Jedrkiewicz, Jiang, Brambilla, Gatti, Bache, Lugiato and Di Trapani [2004], Mosset, Devaux and Lantz [2005] one has a very large number of replicas of the same system (pairs of entangled spatial modes) simultaneously available in a single pump pulse. In this connection, see also O’Sullivan-Hale, Ali Khan, Boyd and Howell [2005].
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Fig. 46. Scheme for quantum-optical lithography. From Agarwal, Boyd, Nagasako and Bentley [2001].
6.8. Quantum-optical lithography This topic was pioneered by Rathe and Scully [1995] and has become a focus of attention after the theoretical paper by Boto, Kok, Abrams, Braunstein, Williams and Dowling [2000]. In this work there are two key points (fig. 46). First, the entangled photon pairs emitted by an OPA in a regime of pure parametric downconversion are conveyed to a 50/50 beamsplitter, and second the lithographic interference pattern is obtained by two-photon absorption. The effect is that the wavelength of the interference modulation is halved with respect to the standard one-photon absorption pattern. This is due to the fact that, according to a wellknown experiment by Hong and Mandel [1985], the entangled photons emerge from the beamsplitter either as a pair on the upper side or as a pair on the lower side of the beamsplitter itself, and never with one photon on each side, so that no coincidences are measured. It is just the quantum interference between the two possible two-photon paths which produces the halved interference pattern. An experiment by D’Angelo, Chekhova and Shih [2001] seemed to provide a first principle demonstration of the phenomenon, but it turned out later (Gatti, Brambilla, Bache and Lugiato [2004b]) that the observed halving effect was purely a consequence of the detection procedure, rather than a quantum effect. As a matter of fact, the results of the experiment by D’Angelo, Chekhova and Shih [2001] were reproduced using a split pseudo-thermal beam (Scarcelli, Valencia and Shih [2004b], Xiong, Cao, Huang, Li, Sun and Wang [2005]). Agarwal, Boyd, Nagasako and Bentley [2001] considered, instead of the regime of single photon-pair detection, the case in which the OPA produces a large number of photon pairs. They showed that in this limit the halving effect is well preserved, even if the fringe visibility is reduced by a factor of 5. A simple interpretation of the halving effect in this case is obtained by observing that the two entangled beams emitted by the OPA are transformed by the beamsplitter into a pair of squeezed beams, and therefore one detects the two-photon absorption
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Fig. 47. Interpretation of quantum lithography as interference between two squeezed vacuum beams.
of the interference of two squeezed vacuum fields. In fig. 47 we see the squeezing ellipses of the two beams. One of the two ellipses rotates with respect to the other, corresponding to the variation of the phase χ (see fig. 47) which represents the phase difference between the two paths. Clearly it is not necessary to perform a rotation of 2π to return to the initial configuration. A rotation of π is enough, and this produces the halving of the wavelength. This consideration suggests an alternative procedure for attaining an experimental demonstration of the principle of optical lithography: the interference of two vacuum-squeezed beams produced by two optical parametric oscillators below threshold which share the same pump beam. As is well known, a minimum uncertainty squeezed vacuum state has the form (Schleich [2001], Scully and Zubairy [1997], Barnett and Radmore [1997]) exp ζ a 2 − ζ ∗ (a † )2 |0 , (6.4)
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where ζ is a c-number and |0 is the vacuum state. One can generalize the argument above by considering a state of the form exp ζ a N − ζ ∗ (a † )N |0 , (6.5) which can be obtained, for example, by a material with a χ N nonlinearity, in which a single pump photon is down-converted into N entangled photons. One can consider the interference of two beams, one of which is in state (6.5), while the other lies in the state N N exp ζ aeiχ − ζ ∗ a † e−iχ (6.6) |0 , obtained from eq. (6.5) by a rotation of the angle χ which corresponds to the phase difference between the two paths plus, possibly, a constant. Clearly, a rotation of 2π/N is enough to return to the initial configuration, and as a consequence the wavelength of the interference pattern, detected by N -photon absorption, is reduced by a factor N . Of course, this argument does not predict the visibility of
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E. Wolf, Progress in Optics 51 © 2008 Elsevier B.V. All rights reserved
Chapter 6
Assessment of optical systems by means of point-spread functions by
Joseph J.M. Braat, Sven van Haver Optics Research Group, Technical University Delft, Lorentzweg 1, 2628 CJ Delft, The Netherlands e-mail:
[email protected];
[email protected] url: http://www.nijboerzernike.nl
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Augustus J.E.M. Janssen, Peter Dirksen Philips Research Europe, HTC 36/4, 5600 JA Eindhoven, The Netherlands e-mail:
[email protected];
[email protected]
ISSN: 0079-6638
DOI: 10.1016/S0079-6638(07)51006-1 349
Contents
Page § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
351
§ 2. Theory of point-spread function formation . . . . . . . . . . . . . . .
355
§ 3. Energy density and power flow in the focal region . . . . . . . . . . .
391
§ 4. Quality assessment by inverse problem solution . . . . . . . . . . . .
409
§ 5. Quality assessment using the Extended Nijboer–Zernike diffraction theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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§ 6. Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . . . .
454
Acknowledgements
455
Appendix A: Derivation of Weyl’s plane wave expansion of a spherical wave 456 Appendix B: The Debye integral in the presence of aberrations . . . . . .
457
Appendix C: Series expansion of the diffraction integral at large defocus
458
m (r, f ) Vn,j
. . . .
459
Appendix E: The predictor–corrector procedure . . . . . . . . . . . . . . .
463
Appendix D: Series expansion for the diffraction integral
Appendix F: Zernike coefficients for circularly symmetric polarization states 465 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
350
466
§ 1. Introduction The subject of this chapter is the computation of the point-spread function of optical imaging systems and the characterization of these systems by means of the measured three-dimensional structure of the point-spread function. The pointspread function, accessible in the optical domain only in terms of the energy density or the energy flow, is a nonlinear function of the basic electromagnetic field components in the focal region. That is why the reconstruction of the amplitude and phase of the optical far-field distribution that produced a particular intensity point-spread function is a nonlinear procedure that does not necessarily have a unique solution. For a long time, a detailed measurement of the point-spread function was not possible because of the lack of adequate intensity recording media. The eye of a human subject, although close to perfection over the typical diameter of its iris, is not capable of appreciating the small imperfections that may be present in high-quality instruments for optical observation. In the 17th and 18th centuries, telescopes and microscopes were still manufactured in a craftsmanship way, without feedback from reliable and objective optical measurement. Generally speaking, one could say that the modern epoch of high-quality instrument making started with the pioneering work by Joseph von Fraunhofer who combined his gifts in optical design with a professional approach to optical measurement technology and manufacturing. In a few decades, as of 1850, the trial-anderror methods from the past were ruled out and scientific instrument making was gradually introduced. Today, the perfection of optical instruments has reached a level that was thought to be impossible until recently, as is apparent from the statement in Conrady [1929]: “it is no use to acquire a microscope objective with a numerical aperture beyond 0.80 because a still larger cone of light will only contribute to light gathering and not to improved imaging”. Especially since the 1970s, the quality of optical imaging systems (telescopes, microscope objectives, high-quality projection lenses for optical lithography, space observation cameras) has been pushed to the extreme limits. At this level of perfection, a detailed analysis of the optical point-spread function is necessary to understand the image formation by these instruments, especially when they operate at high
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numerical aperture. In terms of imaging defects, we are allowed to suppose that the wavefront aberration of such instruments is not substantially larger than the wavelength λ of the light. In most cases, the aberration even has to be reduced to a minute fraction of the wavelength of the light to satisfy the extreme specifications of these imaging systems. In the following subsections we briefly present the past work on point-spread function analysis and its application to the assessment of imaging systems. We conclude this introduction with a brief outline of the further contents of the chapter.
1.1. The optical point-spread function A very comprehensive overview of the early history of point-spread function analysis can be found in a review paper by Wolf [1951]; in this subsection we mention the most important steps in the remote past that have led to our present knowledge and then sketch in some more detail the recent developments since the 1950s. The early point-spread function analysis was based on ray optics, and it focused on the influence of spherical aberration (see early work by Christiaan Huygens, reported in 1941). Because of the increasing quality of optical components at the beginning of the 19th century and the refinement of, for instance, astronomical observations, a more sophisticated analysis of the optical point-spread function in focus was required. This led to the expression given in Airy [1835] that is based on the wave theory of light and takes into account the diffraction of light on its passage through an aperture with limited extent. Point-spread function interaction when imaging incoherent sources was studied by Rayleigh [1879] leading to his still frequently used criterion for minimum star separation in astronomy. An important step forward in the analysis of the point-spread function was by Lommel [1885] who derived analytic expressions for the out-of-focus region, thus for the first time systematically adding the axial dimension in the analysis of diffraction images. An interesting criterion in quality assessment of optical systems was introduced by Strehl [1894]. He defined the ratio of the maximum on-axis intensity of the point-spread function of an actual imaging system and its theoretical value in the absence of aberrations, given by Airy’s expression. This quantity was given the name ‘Definitionshelligkeit’, later called Strehl definition or Strehl ratio in the English literature. Since then, various authors have focused on numerical evaluations (Conrady [1919]) and analytic expressions for the diffraction image or its Strehl ratio in the presence of certain typical aberrations like spherical aberration (Steward [1925]) and coma or astigmatism, see Picht [1925].
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During this period, a continuous subject of research was the optimum distribution or ‘balancing’ of aberrations of various orders and types to optimize the quality of the point-spread function; see Richter [1925] for a discussion of this topic. This subject continues to be of great practical importance for optical system designers, who need useful rules-of-thumb in their laborious optimization activities. A breakthrough in point-spread function analysis and the study of aberrations was the introduction of the circle polynomials in optical diffraction problems, see Zernike [1934]. They were applied to the study of weakly or moderately aberrated point-spread functions by Nijboer [1942] and Zernike and Nijboer [1949]. The orthogonality of the Zernike circle polynomials provided the optical system design community with a general solution to the ‘balancing’ problem of residual aberrations in well-corrected imaging systems. The circle polynomials also proved their usefulness when studying the allowable amount of aberration of an optical system that is to attain a certain minimum on-axis intensity (Strehl intensity). According to a result derived by Maréchal [1947], the deviation from unit Strehl ratio for small aberrations is given by Var(Φ), the variance of the phase departure Φ of the focusing wave over the exit pupil of the optical system. Applying the circle polynomials to expand the phase function Φ leads to an expression for Var(Φ) that is a simple weighted sum of squares of the Zernike expansion coefficients, see Born and Wolf [2002]. An important new development in the study of the point-spread function of an imaging system is related to the extension of light propagation from the common scalar to the more intricate vector model. The complete set of electric and magnetic field vectors has to be calculated in the focal region of the optical system and, from these, the relevant electromagnetic quantities like energy density and the flow components related to energy, impulse and angular momentum can be obtained. A first series of publications by Ignatowsky [1919] on the vector field in focus passed relatively unnoticed by the community. Some qualitative considerations on the vector aspects of the field in focus were put forward by Hopkins [1943]. It finally was a set of two papers, Wolf [1959] and Richards and Wolf [1959], that triggered the interest for the rigorous study of high-quality imaging systems with a numerical aperture higher than, say, 0.60. Today, the vector diffraction theory proposed in these papers is widely used, together with alternative representations that will be equally discussed in this chapter. Fields of application are high-resolution three-dimensional microscopy, high-density optical data storage, and high-resolution optical lithography.
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1.2. Quality assessment by inverse problem solving It was mentioned above that assessing the quality of a highly specified optical system has to be done in the wavefront domain down to a fraction of the wavelength of the light. Interferometric methods are mostly used for this purpose. Although the achievable precision is very high, these methods need refined and delicate optical set-ups and, in practice, special laser sources to achieve sufficient signal-tonoise ratio. When a measurement at a specific wavelength is needed for which an adequate source is not available, interpolation from measurements at other wavelengths would be required and the measurement accuracy can become a problem. For that reason, a direct measurement of the point-spread function (or intensity impulse response) can be of great practical interest if it is possible to derive from such an intensity distribution the relevant quality data of the optical system, in particular the wavefront aberration. The strongly nonlinear relationship between the phase departure in the exit pupil of the optical system and the detected intensity in the focal plane leads to an ill-posed inversion problem. The first publications on this type of inversion problems go back to Gerchberg and Saxton [1971, 1972] and Frieden [1972]. To improve the stability of the inversion process, extra information from, e.g., the pupil-intensity distribution (optical far-field intensity) or from several image planes in the focal region is incorporated, as in the ‘phase diversity’ method proposed by Gonsalves [1982], or the multiple images phase retrieval method in electron microscopy by VanDijck and Coene [1987]. An early ‘phase retrieval’ method is found in Fienup [1982]; later developments can be found in Barakat and Sandler [1992], Frieden and Oh [1992], Fienup, Marron, Schultz and Seldin [1993], Iglesias [1998] and Fienup [1999]. The focus in this chapter will be on the assessment of optical systems using the optical point-spread function, especially in the case of systems with very high values of the numerical aperture in image space. The reconstruction of complete objects, a much broader subject, is outside the scope of this chapter. We will pay special attention to methods for representing the pupil function of the optical system and the analytic or numerical steps that are needed to obtain the point-spread function in the focal region. The stability of the various pupil-function representations in the inversion process is studied and the range of wavefront aberration that can be retrieved is tested. The organization of the chapter is as follows. We first present in § 2 the calculation method in the forward direction, to arrive from the complex amplitude in the exit pupil of an optical system to the amplitude in the focal region in image space. Various levels of approximation in solving the pertinent diffraction integral are addressed, leading to the Rayleigh and Debye integral expressions and
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the so-called paraxial approximation; both the scalar and the vector diffraction formalism are discussed. One important subject in this section is the efficient and stable representation of the exit-pupil distribution or far field by means of Zernike polynomials. Section 3 uses the results of § 2 to develop analytic expressions for the energy density and the Poynting vector components in the focal region, this in the presence of a general exit-pupil function characterized by its complex Zernike expansion coefficients. In § 4 we address the general inverse problem in optical imaging and the various solution methods that have been devised so far. In § 5, the emphasis is on the application of the Extended Nijboer–Zernike diffraction theory to the optical inverse problem. By using the information from through-focus point-source images we describe a method to assess the quality of the optical system regarding its optical aberrations, transmission defects and birefringence. In this section, both the scalar and vector diffraction theory will be applied to the solution of the optical inverse problem. The final, short section presents the conclusions and an outlook to further research in this field. Several appendices give detailed derivations of results that were used in the main text.
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§ 2. Theory of point-spread function formation In this section we describe the optical model used for calculating the point-spread function of optical systems that suffer from relatively small wavefront aberrations. Analytic or semi-analytic expressions for both the in-focus and the out-of-focus point-spread function are given, based on the work by Lommel [1885] and Nijboer [1942]. Recent extensions deal with the description of through-focus point-spread functions in the presence of non-negligible aberrations, as these are crucial for solving the inverse problem. As the basis for our calculation of the point-spread function we will use the Debye diffraction integral. Its derivation from more general diffraction integrals and its limits of applicability are discussed in some detail. The optical model is first based on the common scalar approximation and is then extended to include vector diffraction effects.
2.1. Field representations and the diffraction integral In representing a field distribution on a surface and its propagated and/or diffracted version elsewhere, it is possible to use either the basic principle of Huygens’ spherical wavelets or the more recently developed plane-wave expansion and the associated concept of Fourier transformation. In the latter case, the field
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Assessment of optical systems by means of point-spread functions
[6, § 2
˜ ; kx , ky ) Fig. 2.1. The field distribution E(r ) and its two-dimensional spatial Fourier transform E(z are given in the plane z = z . The field has to be calculated in an arbitrary point P , given by the general position vector r.
distribution is described in terms of the complex spectrum of spatial frequencies, each spatial frequency set kx , ky , kz corresponding to a plane wave with wavevector k = (kx , ky , kz ). The time dependence of the monochromatic field components is given by exp{−iωt} and will generally be omitted when using the complex representation of time-harmonic fields. The result of the dispersion relation at frequency ω yields the relationship kx2 + ky2 + kz2 = n2 k02 = k 2 , with n the (complex) refractive index of the medium. We now define the two-dimensional forward and inverse Fourier transforms of the complex field E(r) according to (see fig. 2.1 for the geometry of the problem)
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˜ ; kx , ky ) = E(z
+∞
E(x , y , z ) exp −i[kx x + ky y ] dx dy ,
(2.1)
−∞
1 E(x , y , z ) = (2π)2
+∞
˜ ; kx , ky ) exp i[kx x + ky y ] dkx dky . (2.2) E(z
−∞
˜ ; kx , ky ) in the plane z = z , the field in a general Using the Fourier transform E(z point P with position vector r is given by 1 E(x, y, z; z ) = (2π)2
+∞ −∞
˜ ; kx , ky ) E(z
6, § 2]
Theory of point-spread function formation
357
× exp i kx x + ky y + kz (z − z ) dkx dky . (2.3) The value of kz equals k 2 − kx2 − ky2 for kx2 + ky2 k 2 and +i kx2 + ky2 − k 2 for kx2 + ky2 > k 2 . The relationship between the propagation method using a Fourierbased plane-wave expansion and the physically more intuitive Huygens’ spherical wavelet model can be established by using Weyl’s [1919] result for the plane-wave expansion of a spherical wave, i exp(ikr) = r 2π
+∞ −∞
exp{i[kx x + ky y + kz z]} dkx dky , kz
(2.4)
where r = + z2 )1/2 . A proof of Weyl’s result is given in Appendix A. The dual approach to wave propagation has been described more systematically in well-known textbooks like those by Born and Wolf [2002] and Stamnes [1986], especially in the context of focused fields. To illustrate the connection between both approaches we follow the arguments in Stamnes [1986] and study the propagated field in the case that not the field itself is given in the plane z = z , but its derivative with respect to z, ∂E(x, y, z; z )/∂z. Using eq. (2.3), taking the z-derivative and then putting z = z , we find (x 2
+ y2
1 ∂E(x, y, z ; z ) = ∂z (2π)2
EBL ˜ ;k ,k ) ik E(z
+∞
z
x
y
−∞
× exp i[kx x + ky y] dkx dky .
(2.5)
After taking the Fourier transform of this quantity and some manipulation we find the following relationship:
∂E(x, y, z ; z ) ˜ d (z ; kx , ky ) = ikz E(z ˜ ; kx , ky ), =E FT (2.6) ∂z where the subscript d indicates that we have taken the Fourier transform of the z-derivative of the field. The propagated field according to eq. (2.3) is now alternatively written as 1 E(x, y, z; z ) = (2π)2
×
+∞
E˜ d (z ; kx , ky )
−∞
exp{i[kx x + ky y + kz (z − z )]} dkx dky . ikz
(2.7)
Following Sherman [1967], the above expression is interpreted as the Fourier transform of the product of two functions that can be put equal to the convolu-
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[6, § 2
tion of their transforms 1 f (x, y) = (2π)2 +∞ =
+∞
F1 (kx , ky )F2 (kx , ky ) exp i[kx x + ky y] dkx dky
−∞
f1 (x , y )f2 (x − x , y − y ) dx dy ,
(2.8)
−∞
where the lower-case functions are the inverse Fourier transforms of the corresponding capital functions. ˜ d (z ; kx , ky ) and F2 = exp{ikz (z − z )}/(ikz ) and using the By putting F1 = E result of eq. (2.4) we find after some arrangement the expression Ed (x, y, z; z ) +∞ ∂E(x, y, z ; z ) −1 = 2π ∂z −∞
×
exp{ik[(x − x )2 + (y − y )2 + (z − z )2 ]1/2 } dx dy . [(x − x )2 + (y − y )2 + (z − z )2 ]1/2
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(2.9)
Like before, the subscript d indicates that the field has been obtained using the z-derivative values in the plane z = z as single-sided boundary conditions, which means that we neglect any counter-propagating wave components. It can be shown similarly that a comparable expression can be obtained using the field values in the plane z = z as starting condition, and this leads to the expression Ef (x, y, z; z ) −1 = 2π
+∞
E(x , y , z ; z )
−∞
∂ exp{ik[(x − x )2 + (y − y )2 + (z − z )2 ]1/2 } × dx dy . ∂z [(x − x )2 + (y − y )2 + (z − z )2 ]1/2 (2.10) Ef (x, y, z; z ) and Ed (x, y, z; z ) are generally referred to as, respectively, the Rayleigh-I and Rayleigh-II diffraction integrals, based on the propagation of spherical waves or their z-derivatives. An equally weighted sum of both solutions leads to a third integral expression, the well-known Kirchhoff diffraction formula (Stamnes [1986]). This relationship between the Rayleigh and Kirchhoff integrals
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Fig. 2.2. The field propagated to a general point P in the presence of an obstructing aperture A in the plane z = z where the incident field is given. A portion of a secondary spherical wave, emanating from a general point Q in the aperture, has been indicated schematically.
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is valid only if the assumption holds that there are no counter-propagating wave components. These three equivalent representations of the propagated field remain directly applicable when the effective source area is limited by an aperture A, see fig. 2.2. The effect of this aperture is either included in the integration range or it is accounted for by adding a multiplying ‘aperture’ function in the integrand of the diffraction integrals. If necessary, this aperture function is complex to account for possible phase changes introduced on the passage of the radiation through the aperture.
2.2. The Debye integral for focused fields When calculating a point-spread function, the field in the aperture is basically a spherical wave converging to the focal point F . Especially for high-numericalaperture focused beams, it is customary to use the plane-wave-expansion based integral of eq. (2.3) to calculate the focal field distribution. The focusing incident wave passes through the diaphragm A and produces a diffraction image in the focal region near F , see fig. 2.3. We now temporarily restrict ourselves to a scalar wave phenomenon, characterized by a single quantity E to describe the field. We
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Fig. 2.3. The incident field is a spherical wave focused at the point F (xf , yf , zf ) with the incident field in a general point Q given by eq. (2.11). The diffracted field in a point P is calculated by means of an integration over the solid angle Ω that is determined by the lines joining the rim of the aperture A and the focal point F .
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suppose that the field in the aperture is given by EA (x , y , z ) = E0 (x , y )
exp{−ikRQF } , RQF
(2.11)
with RQF given by the distance from a general point Q in the aperture with coordinates (x , y , z ) to the focal point F (xf , yf , zf ). The function E0 (x , y ) (dimension field strength times meter) accounts for any perturbations of the incident spherical wave in amplitude or phase; for a perfectly spherical wave we have E0 (x , y ) ≡ 1. The minus sign in the exponential for a converging wave stems from the choice of the phase reference point: commonly the focal point F . The angular spectrum of the field in the aperture is given by exp{−ikRQF } ˜ E0 (x , y ) E(z ; kx , ky ) = RQF A
× exp −i[kx x + ky y ] dx dy ,
(2.12)
where possible aberrations or aperture transmission variations can be incorporated in the function E0 (x , y ). The field limitation by the aperture boundary is geometrically ‘sharp’, not taking into account possibly more smooth electromagnetic boundary conditions. This so-called ‘hard’ Kirchhoff boundary condition
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is adequate when the typical dimension of the aperture is on the order of many wavelengths, a condition satisfied in most practical optical imaging systems. ˜ ; kx , ky ) basically extends to infinity, The angular spectrum of the function E(z among others because of the hard Kirchhoff boundary condition. This is a serious complication when carrying out the integration of eq. (2.3). A frequently used ˜ ; kx , ky ), originally proposed by Debye [1909], is approximation for E(z ⎧ ky kx 2π ⎪ ⎪ ik E0 {xf − kz (zf − z ), yf − kz (zf − z )} ⎪ ⎨ z × exp{−i[kx xf + ky yf + kz (zf − z )]}, ˜ ; kx , ky ) = E(z (2.13) ⎪ ⎪ inside Ω, ⎪ ⎩ 0, outside Ω, where rf = (xf , yf , zf ) is the position vector of the focal point F , and Ω denotes the solid angle that the aperture subtends at F . The solid angle Ω equals the solid angle of the cone of light created by the incident spherical wave after truncation by the aperture following the laws of geometrical optics. The expression (2.13) can be obtained by asymptotic expansion of eq. (2.12) for the aberration-free case by finding the stationary points of the phase func˜ ; kx , ky ) tion (Stamnes [1986]). In Appendix B we give the expression for E(z in the presence of an aberrated incident wave. The diffraction integral (2.3) now becomes
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−i E(x, y, z; z ) = 2π
E {x − 0 f
kx kz (zf
− z ), yf −
kz × exp i kx (x − xf ) + ky (y − yf ) + kz (z − zf ) dkx dky .
ky kz (zf
− z )}
Ω
(2.14)
In most cases the coordinate z will be that of the center of the aperture plane and then equals zero. The Debye approximation is thus equivalent to the introduction of a sharp boundary in the plane-wave spectrum following from geometrical-optics arguments. It has been shown by Stamnes [1986] that the Debye approximation is equivalent to an asymptotic value of the integral (2.3) where only the interior stationary point has been kept. The conditions of applicability of the Debye approximation have been examined by Wolf and Li [1981]. The result of their analysis is that the Debye integral is a sufficient approximation to the field values in the focal region if the condition zf − z π/{k sin2 (θ/2)} is fulfilled, with sin θ the numerical aperture of the focusing wave divided by the refractive index of the medium.
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2.3. The Rayleigh-I integral for focused fields In this subsection, we will focus on the first version of the Rayleigh diffraction integrals, the so-called Rayleigh-I integral. For an incident focused field, this integral is obtained by substituting eq. (2.11) in eq. (2.10) and, including the aberration phase Φ(x , y ) introduced in Appendix B, we get z − z −i Ef (x, y, z; z ) = E (x , y ) exp iΦ(x , y ) 0 2 R λ RQP QF A × exp ik(RQP − RQF ) dx dy , (2.15) where Q(x , y , z ) again is the general point in the diffracting aperture A, and (z − z )/RQP can be recognized as an obliquity factor for the strength of the emitted secondary waves. The integral expression above neglects the diffracted near-field contribution but, for kRQP 1, it is sufficiently accurate if the Kirchhoff boundary conditions apply. A direct comparison of the Rayleigh and Debye integral expressions can be carried out by transforming the Debye integral (2.14) from an integration over the (kx , ky )-domain back to the (x , y )-domain in the planar diffracting aperture A. With the focal point F located on the z-axis, the relation between the coordinates (x , y ) and the wavevector components (kx , ky ) is given by (see fig. 2.4)
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kx =
kz k x = − x, zf − z RQF
ky =
kz k y = − y, zf − z RQF
(2.16)
2 = x 2 + y 2 + (z − z )2 . The Jacobian of the transformation yields with RQF f
dkx dky = dx dy
k 2 (zf − z )2 4 RQF
and, with kz = k cos αQ = k(zf − z )/RQF and after some rearrangement, we find the transformed Debye integral according to zf − z −i E0 (x , y ) exp iΦ(x , y ) E(x, y, z; z ) = 3 λ RQF A × exp ik · (rQP − rQF ) dx dy , (2.17) with the aberration function Φ of the incident wave explicitly included in the integral, and the components of the vector k defined by eq. (2.16). Discrepancies between the Rayleigh-I and Debye integrals are found in the amplitude or obliquity factor where the difference between RQP and RQF and the difference between z and zf is neglected in the Debye expression. Another important difference is found in the pathlength exponential. The pathlength difference
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Fig. 2.4. Schematic drawing of the aperture A limiting the incident wave with its focus in the axial point F . The possible amplitude and phase variation over the beam cross-section in A are preferably measured or calculated on the exit-pupil sphere with radius R, centered in F and intersecting the z-axis in the point Q 0 . In the figure the aperture cross-section is chosen to be circular, but a more general shape can also be accommodated.
RQP − RQF of the Rayleigh-I integral is approximated by the scalar product s · (rQP − rQF ), with s the unit vector in the propagation direction. Like for the obliquity factor above, the expressions are sufficiently accurate when P and F are close and R is very large with respect to λ. The pathlength expression in the Debye integral is exact if R → ∞; it then corresponds to the pathlength definition along a geometrical ray given by Hamilton in the framework of his eikonal functions (Born and Wolf [2002]). The evaluation of the function E0 (x , y ) exp{iΦ(x , y )} can be carried out in the plane of the aperture A by measuring in A the amplitude and phase differences between the actual wave and the ideal spherical wave. The function E0 exp(iΦ) carries the information about the amplitude and phase of the bundles of rays that have been traced through the optical system. These quantities are preferably defined on the exit-pupil sphere of the optical system, the sphere with radius R, centered on F in fig. 2.4 and truncated by the physical aperture A.
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2.4. Comparison of the various diffraction integrals A comparison of the various diffraction integrals for focused fields leads to the following order in terms of accuracy and degree of approximation: • Rayleigh-I integral: The Rayleigh-I integral according to eq. (2.15) is the most accurate one, within the framework of scalar diffraction theory. The integral is related to the amplitude distribution in a plane. A comparably accurate integral can be obtained from eq. (2.9); this is the so-called Rayleigh-II integral. • Debye integral: The Debye integral, eq. (2.14), yields accurate results once the distance from pupil to focal point is large (Q 0 F = R → ∞) and the aperture of the cone of plane waves is sufficiently large. The angular spectrum is truncated according to the geometrical-optics approximation, but this truncation has less and less influence when R increases (see Wolf and Li [1981] for the residual error of this integral). The function E0 , see eq. (2.14), accounts for a nonuniform (complex) amplitude of the incident spherical wave. A numerical comparison of the axial intensity in the focal region according to the Rayleigh-I and the Debye integral is given in fig. 2.5. The upper, middle and lower panels apply to increasing aperture diameters of 10λ, 100λ and 105 λ, respectively. The graphs in the left-hand column are for a numerical aperture of the focused beam s0 = 0.25, while the right-hand column is for s0 = 0.50. The plotted intensity patterns, in arbitrary units, have been normalized with respect to the most accurate result, following from the Rayleigh-I integral (solid lines). The dotted curves follow from the Debye approximation of the diffraction integral. The variable plotted along the horizontal axis is the defocusing (z − zf ) in units of λ. The graphs in the upper panel show that for a very small aperture diameter the difference between the Rayleigh-I and Debye integral is large. The Rayleigh-I integral leads to a strong asymmetry with respect to the nominal focus position, and the highest intensity is at an axial position closer to the aperture than the nominal focal point F . These effects are relaxed by an increase of the numerical aperture as shown by the upper right graph. The strong intensity oscillations at the negative defocus values −20 z − zf −10 in the upper left graph correspond to axial points that are very close to the diffracting aperture itself. They can be explained by the interference effect between the wave diffracted from the circular rim of the aperture and the undiffracted focused wave, both having comparable amplitudes close to the aperture. The axial range beyond the intensity maximum and the focal point F does not show these deep oscillations because in this region the direct undiffracted spherical wave has, by far, the largest amplitude on axis. The effect of a higher numerical aperture is a less pronounced focus offset of the Rayleigh-I integral; one also observes an increased fidelity of the Debye integral
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Fig. 2.5. The axial intensity in the focal region calculated according to the Rayleigh-I (solid lines) and Debye integral expressions (dotted lines). The aperture diameter 2a is 10λ in the upper row, has increased to 100λ in the middle row, and is 105 λ in the bottom row. In the left-hand graphs the numerical aperture s0 of the focusing beam is 0.25, in the right-hand graphs it is 0.50. The intensity in arbitrary units has been normalized to the result of the Rayleigh-I integral. The defocusing z − zf has been plotted along the horizontal axis in units of the wavelength λ of the light.
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regarding maximum intensity. An increase of the aperture diameter to 100λ makes the focus offset almost disappear, especially in the graph on the right with s0 = 0.50. The asymmetry around focus in the position of the relative maxima is still visible, but the Debye approximation has strongly improved relative to the upper row of graphs, including its prediction of maximum intensity. The correspondence between both representations gets increasingly better, and beyond the value 2a = 500λ hardly any difference is noticeable. This is illustrated in the lower panel, which applies to the very large apertures encountered in practical optical systems, for instance 2a = 5 mm with λ = 0.5 µm. For clarity the Debye approximation results are represented by the heavy dots, and these coincide extremely well with the Rayleigh-I integral results in the range of numerical apertures of interest for high-resolution applications. It is for imaging systems in this domain that the quality assessment using point-spread functions will be carried out. The graphs in the lower panel show that in this case it is fully justified to resort to the analytically more accessible Debye integral. 2.4.1. Paraxial approximation of the Debye integral The paraxial approximation to the Debye integral is allowed if the aperture shape is such that kz2 (kx2 + ky2 ) within the cone of integration Ω, see eq. (2.14). The kz -factor in the nominator of the integrand is put equal to k. The variables (kx , ky ) are transformed according to kx = −k(xs /R) and ky = −k(ys /R), with (xs , ys ) Cartesian coordinates on the exit-pupil sphere through Q 0 with radius R that has its center in the focal point F , located on the z-axis, see fig. 2.4. After some manipulation, and expanding the square root for kz in the pathlength exponential up to the first power, we obtain
−i (x 2 + y 2 ) Ef (x, y, z; z ) ≈ exp ik(z − zf ) exp ik 2R λR 2
2 (xs + ys 2 ) exp −ik(z − zf ) Es (xs , ys ) × 2R 2
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A
xxs + yys × exp iΦs (xs , ys ) exp −ik dxs dys . R (2.18) Es (xs , ys )
exp{iΦs (xs , ys )},
and phase function describThe amplitude function ing the departure of the complex amplitude of the focusing wave from that of a uniform spherical wave, are now defined on the exit-pupil sphere where they can easily be calculated or measured.
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The paraxial approximation (2.18) is often modified to allow the use of dimensionless coordinates. The aperture coordinates are normalized with respect to the lateral dimension a of the aperture. The lateral field coordinates of the image point P are normalized with respect to the quantity λR/a or λ/s0 , the diffraction unit in the focal region (s0 = sin αmax = a/R is the numerical aperture of the focusing beam). The axial coordinate z is normalized with respect to the axial diffraction unit, λ/(πs02 ). With these transformations we find Ef (xn , yn , zn ) ≈
−is02 i2(zn − zn,f ) πλ 2 2 x + y exp exp i n n λ s02 Rs02 × exp −i(zn − zn,f ) xn 2 + yn 2 E(xn , yn ) An
× exp iΦ(xn , yn ) exp −i2π(xn xn + yn yn ) dxn dyn . (2.19) Using normalized polar coordinates (ρ, θ ) in the aperture and cylindrical coordinates (r, φ, f ) in the focal region (origin of the normalized axial coordinate f is in F ) yields the expression
−is02 i2f πλr 2 exp 2 exp i Ef (r, φ, f ) ≈ λ s0 Rs02 × exp −ifρ 2 E(ρ, θ ) exp iΦ(ρ, θ)
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An
× exp −i2πrρ cos(θ − φ) ρ dρ dθ,
(2.20)
where the amplitude and aberration functions in Cartesian coordinates have now been replaced by their analogs in polar coordinates.
2.5. The amplitude of the point-spread function produced by an optical system The intensity distribution in the point-spread function is strongly dependent on the departure of the incident focusing wave from its reference shape, that of a spherical wave with a uniform amplitude. In this subsection we discuss, especially for the high-numerical-aperture case, the various factors that influence the complex amplitude distribution of the focusing wave, measured in the exit pupil of the imaging system. We also discuss the various methods for representing the wavefront aberration on the exit-pupil sphere.
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Fig. 2.6. An incident wave is described by its complex amplitude on the entrance-pupil sphere S0 (flat in this picture with the object point at infinity) and propagates from the entrance pupil through the optical system towards the exit-pupil sphere S1 and to the focal region with its center in F . The coordinates in object and image space are referred to by (x0 , y0 , z0 ) and (x, y, z), respectively, with respect to the origins in object and image space. The general point Q on the exit-pupil sphere is defined by means of its polar coordinates (ρ, θ) with respect to the z-axis. The aperture of the imaging pencil (diameter 2a) is given by s0 = sin αmax . The distance from Q 0 to F is denoted by R, with the origin for the z -coordinate on the exit-pupil sphere chosen in Q 0 .
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2.5.1. Amplitude distribution in the exit pupil For the calculation of the amplitude in the focal region of an optical imaging system we need the complex amplitude distribution on the exit-pupil sphere of the system. In most practical cases we are able to specify the complex amplitude distribution on the entrance-pupil sphere (or the entrance-pupil plane in the frequently occurring case that the object conjugate of the system is at infinity). The transfer of complex amplitude from entrance to exit pupil depends on numerous factors like diaphragm shape, reflection losses at the intermediate optical surfaces, light absorption in the lens materials, etc. These effects, particular for each optical system, can be accounted for in the complex transmission function E(ρ, θ ) exp{iΦ(ρ, θ)}. A more general aspect is the pupil imaging telling us how the complex amplitude distribution in object space is mapped to the exit-pupil sphere in image space. In fig. 2.6 we show the geometry that is relevant for this process of mapping from object to image space. Several options may occur in practical systems. To study these options, we consider the intensities in an annular region of the entrance pupil and the corresponding annulus on the exit-pupil
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sphere. The more general situation with a finite object distance and a spherical entrance-pupil surface does not basically change the result. Supposing loss-free light propagation, the relation between the power flow through the annular regions of entrance pupil and exit pupil is given by 2πI0 r0 dr0 = 2πp(α)I1 R 2 sin α dα, (2.21) where r0 = x02 + y02 is a function of α that determines the mapping effect and the ratio I1 /I0 = (fL /R)2 follows from the paraxial magnification between the exit pupil and entrance pupil (fL is the focal distance of the imaging system). The function p(α) is unity on axis but can deviate from this value for α = 0 to account for non-paraxial behavior of finite rays in the imaging system. Integration of eq. (2.21) from 0 to a general aperture value given by sin α yields α 2 2 r0 = fL p(α) sin α dα. (2.22) 0
We consider two options: • p(α) = 1, yielding r0 = 2fL sin(α/2) • p(α) = cos α, yielding r0 = fL sin α
(Herschel’s condition);
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(Abbe’s sine-condition).
(2.23)
(2.24)
The two pupil imaging conditions applying to non-paraxial rays have already been proposed in the 19th century. They emerge as special cases in the framework of general isoplanatic imaging conditions, see Welford [1986]. Outside the paraxial imaging regime, the Herschel condition favors the imaging of axial points in front of and beyond F ; the Abbe sine-condition has been designed to guarantee good imaging for image points in the focal image plane through F . The vast majority of optical systems obeys the Abbe sine condition, and for this reason, in the following, we will adhere to the condition p(α) = cos(α). The function p(α) pertains to the ratio of intensities. In the case of a uniform amplitude distribution in the entrance pupil, we will thus apply the rule, with cos2 α = (1 − s02 ρ 2 ), that the amplitude function on the exit-pupil sphere, E(ρ, θ ), contains a factor (1 − s02 ρ 2 )1/4 . This amplitude factor is often referred to as the radiometric effect. As defined before, ρ is the normalized radial coordinate on the exit-pupil sphere. 2.5.2. Phase distribution in the exit pupil The aberration function Φ(ρ, θ) originates from the possible aberration that is already present in the incident beam and from the aberration imparted to the beam
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on its traversal of the optical system. It is common practice to effectively project back the effect on the aberration of all optical surfaces and media in the system onto the exit-pupil sphere, yielding the global aberration function Φ(ρ, θ) of the system. For sufficiently small aberrations, typically Φ 2π, this method is allowed. The representation of the aberration function has its particular history. From the start of modern aberration theory by Seidel, see Welford [1986], based on a power series expansion of optical pathlength differences in an optical system, it was common practice to represent Φ as Φ(ρ, θ) = akl ρ k cosl (θ ). (2.25) Certain combinations of k and l yield a characteristic aberration. This is more or less true for the lowest-order aberration types that occur in an optical system with rotational symmetry when k + l = 4. But for higher-order aberration terms, the expression (2.25) becomes rather confusing because of the non-orthogonality of the expansion in both ρ and θ . A breakthrough in aberration theory was the advent of the Zernike circle polynomials (see Zernike [1934] and Nijboer [1942]), and the expansion for Φ now reads m m Φ(ρ, θ) = (2.26) Rnm (ρ) αn,c cos mθ + αn,s sin mθ , nm
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where Rnm (ρ) is the radial Zernike polynomial of radial order n and azimuthal order m with n, m 0 and n − m even. An alternative representation is Φ(ρ, θ) = (2.27) αnm Rn|m| (ρ) exp{imθ }, nm
where n 0, with (n − |m|) even, and m now also assumes negative values. This latter expansion will be used and, more generally, we will also allow complex coefficients αnm so that a complex function Φ(ρ, θ) can be expanded. The relam tionship between the now equally complex coefficients αn,c/s and the αnm is then given by m αn,c = αnm + αn−m , m αn,c = αnm + αn−m , m = − αnm − αn−m , αn,s m αn,s (2.28) = αnm − αn−m . Other representations of the complex amplitude on the exit-pupil sphere have been proposed. We mention the expansion of the far field using ‘multipole waves’, see Sheppard and Török [1997]. As a function of the azimuthal and elevation angles
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on the exit-pupil sphere, the far field is described with the aid of spherical harmonics. The coefficients that yield the optimum far-field match are then used to propagate the multipole waves towards and beyond the focal point. The propagation as a function of the distance r is described in terms of well-behaving spherical Bessel functions of various orders. So far, the analysis has been restricted to circularly symmetric geometries with amplitude (transmission) variation on the exit pupil and to an infinitely distant exit pupil. But the method can be extended to more general geometries and to aberrated waves. The method is applicable not only to scalar diffraction problems but equally well to high-numerical-aperture systems requiring a vector diffraction treatment, and can be extended to birefringent media, see Stallinga [2004a]. Another representation uses the so-called Gauss–Laguerre polynomials that are orthogonal on the interval [−∞ < r < +∞] and emerge as eigenfunctions of the solution of the paraxial wave equation according to Siegman [1986]. They have been further studied by Barnett and Allen [1994] to make them suitable for the non-paraxial case. We will not further consider this type of amplitude and aberration representation because its Gaussian shape is not well suited for the hard-limited aperture functions that are mostly encountered in optical imaging systems. But the Gauss–Laguerre elementary solutions with azimuthal order number m = 0 are well suited to represent a phase departure of the pupil function that shows a so-called helical phase profile with a phase jump of 2mπ. This is interesting when discussing optical beams with orbital angular momentum, see for instance Beijersbergen, Coerwinkel, Kristensen and Woerdman [1994]. However, using the Zernike polynomial representation of eq. (2.27) with the exponential azimuthal dependence, it is equally well possible to represent helical phase profiles by selecting a single non-zero αnm -coefficient instead of an automatic combination of αnm and αn−m . Using the appropriate expressions for the amplitude and aberration function on the exit-pupil sphere we are now able to evaluate eq. (2.20) and to obtain the amplitude of the scalar point-spread function in the paraxial approximation. It is possible to extend the scalar integral (2.20) beyond the paraxial domain by incorporating the defocus exponential (2.14) according to 1 kx2 + ky2 (z − zf ) . exp ikz (z − zf ) = exp ik 1 − (2.29) k2
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With the same coordinate transformation as used in deriving eq. (2.18) and switching to normalized polar coordinates (ρ, θ ) on the exit-pupil sphere, we obtain 5 6 exp ikz (z − zf ) = exp ik 1 − s02 ρ 2 (z − zf ) . (2.30)
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The axial coordinate z − zf is normalized in the high-numerical-aperture case according to f , (2.31) ku0 with u0 = 1 − 1 − s02 , and one then finds the final expression for the highnumerical-aperture defocus exponential, viz.
8 f f 7 2 2 exp ikz (z − zf ) = exp −i (2.32) 1 − 1 − s0 ρ . exp i u0 u0 z − zf = −
The scalar integral for high numerical aperture then reads
−is02 −if πλr 2 Ef (r, φ, f ) ≈ exp exp i λ u0 Rs02 1 % & 1 − 1 − s02 ρ 2 × exp if u0 An
× E(ρ, θ ) exp iΦ(ρ, θ) × exp −i2πrρ cos(θ − φ) ρ dρ dθ,
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(2.33)
is again evaluated on the exit-pupil sphere using the data from ray tracing or other propagation methods of the wave through the optical system. In several instances in the literature, the minus sign in the exponential with the factor cos(θ − φ) in eq. (2.33) has also been suppressed. This means that the results apply to an azimuth shift of π for the axis φ = 0 in image space. In § 3 and further of this chapter, we will adhere to this latter sign convention. The integral above is an improvement with respect to the paraxial approximation, beyond an aperture of 0.60, on the condition that polarization effects are not dominating. In practice, this might be the case when the optical system is illuminated with effectively unpolarized or ‘natural’ light. 2.5.3. The high-numerical-aperture vector point-spread function The extension of the point-spread function analysis to the vector components of the electrical and magnetic fields was first carried out by Ignatowsky [1919]. In a series of three papers he analyzed the electromagnetic field in focus of a parabolic mirror and in focus of a general imaging system. He also analyzed the amplitude conversion from entrance to exit pupil following from the various pupil imaging conditions, see eqs. (2.23)–(2.24). The subject was reformulated and cast in
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ˆ in the entrance and exit pupil that are Fig. 2.7. Definition of the orthogonal unit vector sets (ν s , ν p , k) used to describe the components of the electromagnetic field vectors in image space. The azimuthal plane defined by the angle θ is the plane of incidence. The origin of the exit-pupil coordinates is chosen in Q 0 , with Q a general point on the exit-pupil sphere.
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the form of a generalized Debye integral by Wolf [1959] and Richards and Wolf [1959], applied to an optical system that is illuminated by a parallel beam from infinity with the optimum focus of the point-spread function in the geometrical focal point F . The Debye integral is used to solve the diffraction problem for each Cartesian vector component of the fields. The vector components of the fields on the exit-pupil sphere are obtained using the condition of Abbe for the mapping of the field components from the entrance pupil to the exit pupil (aplanatic imaging). From the geometry of the problem, see fig. 2.7, one easily derives the required unit vectors in image space that are associated with the s- and p-polarization components and the unit propagation vector: ⎛
⎛ ⎞ ⎞ cos θ cos α sin θ ν p = ⎝ sin θ cos α ⎠ , ν s = ⎝ − cos θ ⎠ , sin α 0 ⎛ ⎞ − cos θ sin α kˆ = ⎝ − sin θ sin α ⎠ . cos α
(2.34)
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The incident field is specified in terms of the linearly polarized electric components along the x0 - and y0 -axis in the entrance pupil according to E = a0 xˆ + b0 yˆ , with a0 and b0 complex numbers to allow for an arbitrary state of polarization of the incident beam. The p- and s-polarization components of the field on the exit-pupil sphere are now given by ⎛ ⎞ cos θ cos α Ep ∝ {a0 cos θ + b0 sin θ} ⎝ sin θ cos α ⎠ , sin α ⎛ ⎞ sin θ Es ∝ {a0 sin θ − b0 cos θ} ⎝ − cos θ ⎠ . (2.35) 0 The x-, y- and z-components of the field on the exit-pupil sphere are obtained by evaluating the scalar products with the Cartesian unit vectors, and this yields Es,x
1/2 fL kz kz kz = − cos 2θ 1 − a0 1 + k k 2Rk 1/2 kz , − b0 sin 2θ 1 − k 1/2
fL kz kz = sin 2θ 1 − −a 0 k 2Rk 1/2 kz kz + cos 2θ 1 − , + b0 1 + k k
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Es,y
1/2
f L kr k z (a0 cos θ + b0 sin θ ), (2.36) Rk 3/2 with kr = k 1 − kz2 /k 2 , and where we have included the amplitude mapping factor from the entrance to the exit pupil (see § 2.5.1). The unit vector that points in the direction of the magnetic field is given by hˆ = kˆ × eˆ , yielding hˆ p = (cos α cos θ, cos α sin θ, sin α) and hˆ s = (− sin θ, cos θ, 0). The Cartesian components of the magnetic induction are then given by Es,z =
Bs,x
1/2
nr fL kz kz = −a0 sin 2θ 1 − k 2cRk 1/2 kz kz − cos 2θ 1 − , − b0 1 + k k
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Bs,y =
375
1/2 nr fL kz kz kz + cos 2θ 1 − a 1 + 0 k k 2cRk 1/2 kz + b0 sin 2θ 1 − , k 1/2
n r f L kr k z (a0 sin θ − b0 cos θ ), (2.37) cRk 3/2 with nr the refractive index of the image space. With the expressions for the electric and magnetic field components in terms of the wavevector components kx and ky , the Debye integral (2.14), with xf = yf = 0, yields for the field components in the focal region near F Es (−kx , −ky ) −i E(x, y, z) = 2π kz Ω × exp i kx x + ky y + kz (z − zf ) dkx dky , Bs (−kx , −ky ) −i B(x, y, z) = 2π kz Ω × exp i kx x + ky y + kz (z − zf ) dkx dky , (2.38) Bs,z =
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with P (x, y, z) the coordinates of the considered point in the focal region. Using the more appropriate normalized cylindrical coordinates (ρ, θ ) on the exit-pupil sphere and (r, φ) in the focal region, we obtain −is02 Es (ρ, θ + π) −if exp E(r, φ, f ) = λ u0 (1 − s02 ρ 2 )1/2 C
1/2 if 1 − 1 − s02 ρ 2 × exp u0 × exp i2πrρ cos(θ − φ) ρ dρ dθ, (2.39) with C the scaled integration area on the exit-pupil sphere (in a standard situation this is equal to the unit circle); a comparable expression holds for the B-field components. In arriving at eq. (2.39), we used eq. (2.31) and the following coordinate transformations and normalizations: kx = kr cos θ = ρkr,max cos θ,
kr,max = ks0 ,
ky = ρkr,max sin θ, 1/2 1/2 kz = k 2 − kx2 − ky2 = k 1 − s02 ρ 2 , 1/2 ks0 2 r= , x + y2 2π
(2.40)
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with the field strength function Es , originally defined as a function of the wavevector components (kx , ky ), now to be measured as a function of the normalized radial aperture coordinate ρ on the exit-pupil sphere and the azimuthal coordinate θ + π. The position on the exit-pupil sphere is obtained from eq. (2.13) using x = xf − (kx /k)R, y = yf − (ky /k)R, leading to real space coordinates of (x = −ρa cos θ, y = −ρa sin θ ), in normalized polar coordinates (ρ, θ + π); note that this latter phase offset of π is missing in Wolf [1959].
2.6. Analytic expressions for the point-spread function in the focal region (scalar case) The first analytic solution of the aberration-free point-spread function integral (2.20) goes back to Lommel [1885] and is treated in detail in Born and Wolf [2002]. The solution for the aberrated case has been studied by various authors, see Conrady [1919], Steward [1925], Picht [1925], Richter [1925]. A more systematic analysis of the influence of aberrations on the point-spread function became possible after the introduction of the Zernike polynomials to describe the wavefront aberration, see Zernike [1934], Nijboer [1942] and Zernike and Nijboer [1949]. Considering the integral (2.20) for a circular aperture (unit circle), we first substitute the Zernike expansion for the aberration function and use the approximation exp(iΦ) ≈ 1 + iΦ for small values of Φ, typically Φ 1. We thus obtain
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2π 1 Ef (r, φ, f ) ≈ 0
exp ifρ 2 E(ρ, θ ) 1 + i αnm Rn|m| (ρ) exp(imθ ) nm
0
× exp i2πrρ cos(θ − φ) ρ dρ dθ.
(2.41)
Carrying out the integration over θ and using the property 2π
exp(imθ ) exp i2πrρ cos(θ − φ) dθ
0
= 2πim Jm (2πrρ) exp(imφ), we get the expression originally derived by Nijboer [1942] in his thesis, 1 Ef (r, φ, f ) ≈ 2πi 0
exp ifρ 2 E(ρ, θ )
(2.42)
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377
1 + α00 J0 (2πrρ)
im+1 αnm Rn|m| (ρ)Jm (2πrρ) exp(imφ) ρ dρ,
(2.43)
nm
where the summation now has to be carried out over all possible (n, m)-values with the exception of m = n = 0. As usual, the function Jm (x) denotes the Bessel function of the first kind of order m. We remark here that, instead of expanding the function Φ itself using the α-coefficients, it is also possible to expand the complete pupil function E(ρ, θ ) exp{iΦ(ρ, θ)} in terms of Zernike polynomials, as was first proposed by Kintner and Sillitto [1976]. A basic result from aberration theory, initially derived by Nijboer, is the following: 1
Rn|m| (ρ)Jm (2πrρ)ρ dρ = (−1)
n−|m| 2
Jn+1 (2πr) . 2πr
(2.44)
0
In the perfectly focused case and for E(ρ, θ ) ≡ 1, the integral above is sufficient to analytically calculate the amplitude Ef with zn = 0 in eq. (2.43). However, in the defocused case, the analysis becomes more complicated and Bauer’s formula has been used by Nijboer [1942] to obtain a workable expression: ∞ f f 0 (2n + 1)in jn (ρ). R2n exp ifρ 2 = exp i (2.45) 2 2
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n=0
The spherical Bessel function jk (x) of the first kind is defined by π J 1 (x), k = 0, 1, . . . , jk (x) = (2.46) 2x k+ 2 see Born and Wolf [2002, Chapter 9], and Abramowitz [1970, Chapter 10]. The extra radial Zernike polynomials that result from the expansion of the quadratic defocus exponential can be treated by formulae that express the product of two Zernike polynomials in a series of Zernike polynomials with differing upper or lower indices. This approach has been discussed by Nijboer. In practice, his solution allows to solve the problem of the defocused aberrated point-spread function for modest defocus values, for instance |f | < π/2. When trying to reconstruct aberrations from defocused intensity distributions, numerically reliable expressions for the intensity of the out-of-focus point-spread function are required over a larger range of f -values. The work described in Nijboer’s thesis did not yet provide such results. It should be added that, even if these results would have been available, the lack of advanced computational means would have prohibited any further activity in that direction at the time.
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2.6.1. Analytic solution for the general defocused case A semi-analytic solution of the aberrated diffraction integral in the defocused case was presented by Janssen [2002], and its application and convergence domain was studied by Braat, Dirksen and Janssen [2002]. The basic integral occurring in, for instance, eq. (2.43) reads 1 Vnm (r, f )
=
exp ifρ 2 Rn|m| (ρ)Jm (2πrρ)ρ dρ,
(2.47)
0
and its solution is found to be an infinite Bessel function series according to Vnm (r, f ) = m exp[if ]
p ∞ J|m|+l+2j (2πr) (−2if )l−1 vlj . l(2πr)l l=1
(2.48)
j =0
In eq. (2.48) we have to choose m = −1 for odd m < 0 and m = 1 otherwise. The function Vnm (r, f ) provides us with the analytic solution of the integrals that occur in the general expression of Ef in eq. (2.43) and they are associated with a typical Zernike aberration of radial order n and azimuthal order m. To obtain the total expression for Ef , it is just required to insert the appropriate azimuthal dependence. Denoting
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n + |m| n − |m| , q= , p= (2.49) 2 2 the coefficients vlj in eq. (2.48) are given as vlj = (−1)p |m| + l + 2j 9 |m| + j + l − 1 j + l − 1 l−1 q +l+j × , (2.50) l−1 l−1 p−j l for l = 1, 2, . . . , j = 0, 1, . . . , p. The binomial coefficients are defined by n n(n − 1) · · · (n − m + 1) = (2.51) m m! with the remark that any binomial with n < m is put equal to zero. To illustrate the accuracy of the series expansion, it can be shown that an absolute accuracy of 10−6 requires a number lmax of terms in the summation that is given by lmax = |3f | + 5. With this number of terms and a range |f | 2π, the amplitude in the focal region of interest of well-corrected optical imaging systems can be calculated with ample precision. Some special cases for the scalar amplitude Ef of eq. (2.43) can be derived directly using the results of eqs. (2.48)–(2.51).
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• Nijboer’s in-focus result (2.44) is obtained for the special case of (2.48) with f = 0, where the summation over l is now restricted to the term with l = 1 and the coefficient v1j is identical (−1)(n−|m|)/2 , regardless the value of j . • The special case with m = n = 0 corresponds to the aberration-free situation that should yield the result originally obtained by Lommel. Referring to eq. (2.47), Lommel’s solution reads, see Born and Wolf [2002], 1 V00 (r, f )
=
exp ifρ 2 J0 (2πrρ)ρ dρ
0
C(r, 2f ) + iS(r, 2f ) . 2 The functions C and S are given by =
(2.52)
sin(f/2) cos(f/2) U1 (r, f ) + U2 (r, f ), f/2 f/2 sin(f/2) cos(f/2) S(r, f ) = U1 (r, f ) − U2 (r, f ), f/2 f/2 C(r, f ) =
(2.53)
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with the general Lommel-function Un (r, f ) defined as ∞ f 2s+n 2s Un (r, f ) = (−i) J2s+n (2πr). 2πr
(2.54)
s=0
The substitution of Lommel’s results in eq. (2.52) leads, after some rearrangement, to the expression V00 (r, f ) = exp(if )
∞ Jl (2πr) (−2if )l−1 . (2πr)l
(2.55)
l=1
It is seen that this compact expression of Lommel’s result is equivalent to the special case with n = m = 0 of eq. (2.48) once we have substituted the value vl0 = l. • The on-axis amplitude distribution is obtained from eq. (2.43) with r = 0. If we limit ourselves to circularly symmetric aberrations with m = 0 and use Bauer’s formula of (2.45) for the defocus exponential, the integral over ρ is easily evaluated with the aid of the properties of the inner products of the radial Zernike polynomials and we find 1 ∞ n 0 Ef (0, 0, f ) ≈ iπ exp(if/2) j0 (f/2) + (2.56) i α2n jn (f/2) . n=0
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In the aberration-free case, the axial dependence is given by the spherical Bessel function of zero order. With the identity j0 (x) = sin(x)/x we find the Lommel result above. Another analytic result from Janssen [2002] is related to diffraction integrals of the type 1 Tnm (r, f ) =
exp ifρ 2 ρ n Jm (2πrρ)ρ dρ.
(2.57)
0
These integrals with a ρ-monomial in the integrand can be considered to be the building blocks for more general integrals containing a polynomial like a Zernike polynomial. Of course, they are also useful in the context of the aberration representation according to Seidel. The Bessel series solution of this type of integral is given by Tnm (r, f ) = m exp[if ]
p ∞ J|m|+l+2j (2πr) (−2if )l−1 slj . (2πr)l l=1
(2.58)
j =0
In eq. (2.58) we again choose m = −1 for odd m < 0 and m = 1 otherwise. The coefficients slj are given by 9 |m| + j + l − 1 q +l+j j |m| + l + 2j p slj = (−1) , (2.59) q +1 j l−1 q +1
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for the same ranges as in the case of vlj : l = 1, 2, . . . , j = 0, 1, . . . , p. For the case of circularly symmetric aberrations a different expansion of the diffraction integral has been proposed by Cao [2003]; this expansion produces 0 (r, f ) defined above. The exponential analytic expressions for the functions T2p 2 factor exp(ifρ ) is written as a Taylor series in f and this gives rise to the appearance of the so-called Jinc-functions with index n according to 1 Jincn (r) = (2πr)2n+2
2πr x 2n+1 J0 (x) dx,
(2.60)
0
for which Bessel series expansions are given. A convergence problem is present with respect to the power series expansion in f , and the condition |f | 15 should be respected to obtain an accuracy of 10−3 in amplitude, 10−6 in intensity. The analytic expressions for the general functions Vnm (r, f ) and Tnm (r, f ) are composed of a Bessel function expansion with the argument 2πr and a power series expansion with respect to f . The latter series expansion, especially for the V -functions, gives rise to numerical convergence problems once the value of |f | is larger than, say, 5π, and an accuracy of 10−8 in amplitude cannot be achieved
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Fig. 2.8. Axial cross-section of a defocused intensity distribution (f = 23) caused by the presence of a Fresnel zone plate in the plane z = 0. The beam shows some circularly symmetric aberration that becomes visible in the figure on the right in which the axial intensity has been plotted. The numerical aperture is 0.60, the wavelength amounts to λ = 248 nm.
for larger f -values. A drastic improvement in accuracy is obtained once the power series expansion in f can somehow be replaced by a more stable expression. To this goal, we use Bauer’s expansion and write the exponential exp(ifρ 2 ) according to eq. (2.45). Using this in the expression for Vnm (r, f ), we find ∞ 1 f m k Vn (r, f ) = exp if (2k + 1)i jk 2 2
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k=0
1 ×
0 R2k (ρ)Rn|m| (ρ)Jm (2πrρ)ρ dρ.
(2.61)
0
To proceed further, a general expression is needed that writes the product of a 0 (ρ) and a general polynomial R m (ρ) radially symmetric Zernike polynomial R2k n as a series of Zernike polynomials according to |m| 0 |m| R|m|+2p = wkl R|m|+2l . R2k (2.62) l
Janssen, Braat and Dirksen [2004] have given explicit expressions for the coefficients wkl and the range of the summation index l (see also Appendix C). A numerical implementation of these results has shown that |f |-values as large as 1000 can be dealt with. To illustrate the kind of intensity distributions that can be handled numerically by the analysis according to eqs. (2.61)–(2.62), we show in fig. 2.8 cross-sections of strongly defocused intensity distributions.
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Fig. 2.9. Measured intensity distribution in a strongly defocused image plane. The value of f is approximately 75. Note that the axial intensity corresponds to a minimum due to the presence of an even number of Fresnel zones in the aperture as seen from the defocused position.
In the left-hand figure, a contour map is shown of an axial cross-section of a focal intensity distribution that is offset by approximately 15 focal depths from its nominal focal setting. The intensity distribution has been produced by means of a Fresnel zone lens and is affected by spherical aberration. In the right-hand figure we show the axial intensity distribution that shows an asymmetry around focus due to this residual aberration of the focusing beam. Figure 2.9 produces a picture of the measured intensity distribution in a strongly defocused image plane (f ≈ 75). A typical Fresnel diffraction pattern is observed. Some spurious structure is visible due to light scattering at imperfections on the optical surfaces in the experimental set-up. The number of luminous rings NF in the Fresnel diffraction pattern is approximately given by NF = (f − π)/2π. In fig. 2.10 we show a radial cross-section of such a defocused intensity distribution, from the central position on axis up to the geometrical shadow region. It is interesting to note that part of the fine structure in the fringes, predicted by the calculations, is also visible in the measured distribution, despite the high sensitivity of the measured intensity to spurious coherent light scattered in the set-up. The calculated distribution was fit to the measured one by varying the f -parameter; the optimum value for these measured data was f = 75. When comparing the semi-analytic methods based on the out-of-focus extension of the Nijboer–Zernike diffraction theory and the numerical methods of the Fast Fourier Transform (FFT) type for solving the various diffraction integrals, we conclude the following. The analytic method has a clear advantage regarding the accuracy and monitored convergence of the solutions. In practice, the numer-
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Fig. 2.10. Comparison of a measured intensity distribution and the best-fit calculated intensity distribution. In this case the value of f is 75 radians. The measured intensity shows a less pronounced modulation than the calculated distribution which may be attributed to some scattered background light and to a diaphragm rim that is not perfectly spherical.
ical effort is a linear function of the defocus parameter and machine precision is the limiting factor when ultimate accuracy is sought for. Typical aliasing problems like in an FFT calculation are of no concern. The aberration value can basically be increased to large values with respect to the wavelength of the light, although the number of Zernike aberration terms might become unwieldy. The calculation of the Zernike coefficients of the aberrated wavefront is a numerical investment that has to be carried out beforehand. Once the coefficients have been obtained, the actual calculation of the scalar amplitude or the field components in the focal region is extremely fast. The analytic decomposition in (r, f )- and φ-dependent functions produces an important economic advantage regarding computation time. At this moment we can state that the numerical effort to produce the amplitude in a single image plane using the analytic approach is comparable to that needed in the Fast Fourier Transform case. However, the equal numerical effort situation for both methods is limited to relatively coarse calculations. As soon as an accuracy better than, say, 10−4 is required, the FFT calculation time has to be strongly in-
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creased. The main reason for this is the intrinsically discontinuous nature of the pupil function which asks for high sample rates. For the calculation of the amplitude distribution in a large number of defocused planes, the semi-analytic method offers a very distinct advantage because the incremental numerical effort is small when going from one defocused plane to another, the only difference being the calculation of the V (r, f )-functions at another f -setting. 2.7. Analytic expressions for the point-spread function in the vector diffraction case In the case of a non-uniform and aberrated amplitude distribution at the exit of the optical system, we have to further specify the expressions for the field components in eqs. (2.36)–(2.37) by inserting a complex amplitude transmission function. Because of the vector nature of the diffraction problem, the transmission function possibly depends on the state of polarization. A general field distribution that can be encountered in the entrance or exit pupil of an optical system was described by Stallinga [2001]. The general coherent field is written as the superposition of two orthogonal polarization states. We take the linear polarization states along, respectively, the x- and the y-axis as basic orthogonal states. A general elliptical state of polarization is obtained via a linear superposition of the two basic linear states with relative amplitude weights and a certain phase difference, according to the complex numbers a0 and b0 defined in § 2.5.3. The complex transmission functions for the two orthogonal linear polarizations in the entrance pupil are written tx (ρ, θ ) = Ax (ρ, θ ) exp iΦx (ρ, θ ) , ty (ρ, θ ) = Ay (ρ, θ ) exp iΦy (ρ, θ ) , (2.63)
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where Ax and Ay are real-valued functions and describe the transmission factors in the x- and y-direction, and Φx (ρ, θ ) and Φy (ρ, θ ) are also real-valued and describe the phase distortion on the exit-pupil sphere due to the wavefront aberration W (ρ, θ ) = Φ(ρ, θ)/k introduced by the optical system. The phase difference (ρ, θ ) = Φy (ρ, θ ) − Φx (ρ, θ ) is the result, after traversal of the optical system, of a spatially varying birefringence due to medium transitions, optical coatings, internal material stresses, etc. In well-corrected optical systems, the maximum value of the birefringence func-
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385
tion (ρ, θ ) should be small and certainly restricted to the range [−π, +π]. Other orthogonal polarization states, for instance radial and azimuthal polarization distributions (see Quabis, Dorn, Eberler, Glöckl and Leuchs [2000]), can also be accounted for by appropriately chosen functions tx (ρ, θ ) and ty (ρ, θ ). The x- and y-dependent transmission functions of the optical system, given the incident linear state of polarization, are expanded with the aid of Zernike polynomials. We choose to apply a general expansion with complex Zernike coefficients βx and βy so that a complex transmission function can be represented, see Kintner and Sillitto [1976]. From now on, we will use the letter β for such an expansion to distinguish it from an expansion of the phase aberration function Φ only with coefficients α according to eq. (2.27). We thus have m βn,x Rn|m| (ρ) exp{imθ }, tx (ρ, θ ) = nm
ty (ρ, θ ) =
m βn,y Rn|m| (ρ) exp{imθ }.
(2.64)
nm
In the forward direction, the expansion is unique. In the reverse direction, obtaining the functions A and Φ from the β-coefficients, the uniqueness of such an expansion can be questioned because of the contribution to both the real and imaginary part by the exponential factor in eq. (2.63). But phase-unwrapping techniques can be used, like in optical interferometry (Ettl and Creath [1996]), to recover the functions A and Φ independently if they show a smooth behavior. The transmission functions tx,y are now inserted in eq. (2.39) to obtain the following expressions for the electric and magnetic field components in the focal region:
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E(r, φ, f )
−iπfL s02 if im exp[imφ] = exp λ u0 n,m ⎡ ⎛ m s2 m exp[2iφ] + V m exp[−2iφ]) ⎞ Vn,0 + 20 (Vn,2 n,−2 ⎢ ⎟ 2 m ⎜ × ⎣ax βn,x ⎝ i s0 (−V m exp[2iφ] + V m exp[−2iφ]) ⎠ n,2 n,−2 2 ⎛
m exp[iφ] + V m exp[−iφ]) is0 (−Vn,1 n,−1 s02
m exp[2iφ] + V m exp[−2iφ]) i 2 (−Vn,2 n,−2
⎞⎤
⎟⎥ 2 m ⎜ + ay βn,y ⎝ V m − s0 (V m exp[2iφ] + V m exp[−2iφ]) ⎠⎦ , (2.65) n,0 n,2 n,−2 2 m exp[iφ] + V m exp[−iφ]) −s0 (Vn,1 n,−1 and
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B(r, φ, f )
−iπnr fL s02 if = im exp[imφ] exp cλ u0 n,m ⎡ ⎛ s2 m exp[2iφ] + V m exp[−2iφ]) ⎞ i 20 (−Vn,2 n,−2 ⎢ ⎟ 2 m ⎜ × ⎣ax βn,x ⎝ V m − s0 (V m exp[2iφ] + V m exp[−2iφ]) ⎠ n,0 n,2 n,−2 2 m exp[iφ] + V m exp[−iφ]) −s0 (Vn,1 n,−1 2 ⎛ s0 m m m exp[−2iφ]) ⎞⎤ −Vn,0 − 2 (Vn,2 exp[2iφ] + Vn,−2 ⎟⎥ 2 m ⎜ + ay βn,y ⎝ +i s0 (V m exp[2iφ] − V m exp[−2iφ]) ⎠⎦ , 2
n,2
n,−2
m exp[iφ] − V m exp[−iφ]) is0 (Vn,1 n,−1
(2.66) with ax and ay the complex amplitude factors that determine the total power in the incident beam and allow to specify the state of global polarization of the light incident on the entrance pupil. The quantity nr is the refractive index of m that depend on the image space medium. The newly introduced functions Vn,j the normalized radial coordinate r and the defocus parameter f are given by (j = −2, −1, 0, 1, 2) : ;−|j |+1
: 1 ; 1 + 1 − s02 ρ 2 if m 2ρ2 (r, f ) = ρ |j | exp 1 − s 1 − Vn,j 0 u0 (1 − s02 ρ 2 )1/4
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0
× Rn|m| (ρ)Jm+j (2πrρ)ρ dρ.
(2.67)
m (r, f ) we have incorporated the so-called radiometric In the expression for Vn,j effect in imaging the entrance-pupil distribution onto the exit-pupil sphere, which leads to the factor (1−s02 ρ 2 )−1/4 in the integrand; this was discussed in § 2.5.1 and follows from Abbe’s sine condition for the imaging of the pupils, see eq. (2.24). A series expansion can be devised to quickly obtain accurate values of the integral above. The functions that can be used in the expansions and the values of the corresponding expansion coefficients are given in Appendix D. To illustrate these results, we reproduce in figs. 2.11–2.13 some grayscale graphs from Braat, Dirksen, Janssen and van de Nes [2003] obtained with our high-accuracy analytic formulae. The series expansions for the vector components of the electric field were truncated so as to have the inaccuracy in the field components below an absolute value of 10−6 , and the corresponding inaccuracy in intensity below 10−12 . Finally, in fig. 2.14 we have produced cross-sections of the energy density in the optimum focal plane (z = f = 0). The ‘Airy disc’
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Fig. 2.11. Graph of the absolute values of the Ex -field component, and its phase in the focal volume for a high-numerical-aperture focused beam (s0 = 0.95, linearly polarized incident light along the x-axis). Both the radial coordinate r and the axial coordinate z have been expressed in units of λ/s0 . Phase singularities are observed at the locations of zero amplitude.
approximation (solid curve) at high numerical aperture (NA) has been obtained from the analytic expression 4J 2 (2πr)/(2πr)2 ; its theoretical minima have been plotted explicitly. The dashed and dotted curves have been sampled using our semi-analytic series expansions for the field components in the focal plane. Due to this discrete sampling, the zero values that are theoretically possible in the crosssection perpendicular to the incident polarization have not been represented exactly (dashed curve). Figure 2.14 clearly shows that in the cross-section parallel to the incident polarization no explicit zeros are observed. The different full widths at half maximum (FWHM) in the two main cross-sections, already demonstrated by Richards and Wolf [1959], are also clearly observed. The narrower FWHM for the cross-section perpendicular to the plane of polarization of the incident light can be explained by the relative amplitude enhancement as a function of the normalized exit-pupil coordinate ρ at the pupil rim. This phenomenon is proper to an optical system satisfying Abbe’s sine condition. Its behavior resembles that of a system with an annular aperture yielding a smaller value of the FWHM of the intensity profile and showing increased side-lobe intensity.
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Fig. 2.12. Same as fig. 2.11, now for the y-component of the electric field.
Fig. 2.13. Same as fig. 2.11, now for the z-component of the electric field.
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Fig. 2.14. Cross-sections of the normalized energy density (∝ |E|2 ) in the optimum focal plane of a high-numerical-aperture focused beam (s0 = 0.95, linearly polarized incident light) as a function of the radial normalized coordinate r in units of λ/s0 . Solid line: hypothetic Airy disc profile, extrapolated to this high-NA value from the low-aperture scalar model. Dashed line: cross-section of the normalized energy density, in a plane perpendicular to the incident linear polarization (y-cross-section). Dotted line: same, now in the plane of the incident linear polarization parallel to the x-axis.
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2.8. The point-spread function in a stratified medium So far, we have supposed that the image space was homogeneous. In practice, when using point-spread functions for assessing the quality of an optical system, the detection of the free-space focused beam is carried out in a relatively thin detection medium, for instance photo-resist on a Si substrate or a photosensitive thin slice of doped crystalline silicon. The final transition to the detecting medium, within the focal volume, induces angle-dependent amplitude and phase changes that produce a difference between the free-space intensity distribution and the effectively detected distribution in the medium. The effect of a single transition has been studied by Ling and Lee [1984] in the framework of focused microwaves, and by Török, Varga, Laczik and Booker [1995] for optical microscopy when a mismatched aberrating cover glass is used. A general analysis of highnumerical-aperture imaging has been described by Mansuripur [1986, 1989]. The basic mathematical tool in the analysis of these references is a Fourier transformation, to propagate through free space, but equally well through optical components
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of various nature (gratings, birefringent plates, thin-film stacks), both for pointsource imaging and for the imaging of extended objects. Through its very general nature and flexibility, this approach is less convenient for numerical treatment of a specific problem like the formation of a point-spread function issued from a single tiny source point. The specific problem of detecting a point-spread function in a thin recording layer at very high numerical aperture was discussed by Flagello and Milster [1992], Török, Sheppard and Varga [1996] and Flagello, Milster and Rosenbluth [1996]. The analysis applies the matrix technique devised for reflection and transmission of light through thin films, see Macleod [1989]. The plane wave components in the Debye integral expressions that are incident at a certain angle on the detection layer are given by the typical complex Fresnel coefficients belonging to the multiple reflection and transmission phenomena in the thin-film stack at that specific angle. A modified version of the approach by Flagello and colleagues can be found in van de Nes, Billy, Pereira and Braat [2004]. Here, the analysis has been simplified by using polar coordinates throughout, thus exploiting the special behavior of the electric field components in the plane of incidence and perpendicular to it, the p- and s-components, respectively. For point-spread functions in a narrow field close to the optical axis, this approach is permitted; it is not applicable when studying oblique beams and strongly off-axis pointspread functions. The superposition of the forward- and backward-propagating plane waves in the layered medium with their angle-dependent amplitude and phase can be taken into account in modified Zernike expansions of the pupil function. One modified expansion applies to the forward-propagating waves, the other to the backward-propagating waves, creating a second ‘virtual’ exit-pupil function. With these modified pupil functions, two point-spread functions pertaining to the two propagation directions can be calculated in the corresponding layer of the stratified medium. The coherent superposition of the field components of the two point-spread functions allows the calculation of the local electric energy density that shows the typical standing-wave phenomena in such a thin-film stack. The first noticeable effect of the penetration of the focused beam in the recording layer is a circularly symmetric phase defect to be classified as spherical aberration, see Visser and Wiersma [1991]. At high numerical aperture, amplitude and phase deviations without circular symmetry become visible in the case of, for instance, linearly polarized light in the entrance pupil. Projected back to the exit pupil of the imaging system, this is equivalent to the introduction of apparent astigmatism in the focusing beam. The further treatment of point-spread function formation in layered media is outside the scope of this chapter.
EBL
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§ 3. Energy density and power flow in the focal region In the preceding section we have derived expressions for the complex disturbance (scalar approximation) or the complex field vectors (high-numerical-aperture vector case) in the focal region. These basic quantities are needed for evaluating the optically measurable quantities like energy density, intensity, linear impulse and angular momentum that are built up in and transported through the focal region. Although these quantities, time-averaged at optical frequencies, can be obtained in each point from the electromagnetic field vectors only, we will carry out, in this section, a detailed analysis of the typical distributions of energy density and intensity that are associated with particular aberrations. We will need these expressions when we address the main subject of this chapter, the assessment of optical systems using the point-spread function intensity in the focal region. As the starting point, we will use the expressions (2.65) and (2.66) we had derived for the field vectors. The model does not only incorporate wavefront aberration, but also a non-uniform amplitude distribution on the exit-pupil sphere, as this is, in any case, to be expected for high-numerical-aperture imaging. In this section we will express the quadratic quantities in the focal region (energy density, Poynting m (r, f ). Regardvector components) in terms of the basic diffraction integral Vn,j ing the energy density, we will restrict ourselves to the electric energy density because this is the quantity that is relevant for detectors or recording media at optical frequencies. The energy flow, described e.g. by the Cartesian or cylindrical components of the Poynting vector, will be discussed relatively briefly. The direction and magnitude of the vector yield the flow of the total electromagnetic energy, including the magnetic part. The divergence of the Poynting vector yields the energy outflow per unit volume. If the power loss in the detection medium is due to its finite conductance only, the loss can be attributed to ohmic dissipation and is proportional to the electric energy density in the medium. In this case, the electric energy density can be used directly; the Poynting-vector data can serve as a check on the calculations based on the electric energy density. Finally, the electric energy density distribution in the focal region according to the scalar approximation will emerge as the low-aperture limit of the result for an arbitrary opening angle of the focused beam.
EBL
3.1. Expression for the electric energy density The time-averaged value of the electric field energy density we is 0 we = n2r |E|2 , 2
(3.1)
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with n2r = the relative dielectric constant. The electric field components in the presence of aberrations in a high-numerical-aperture system, as given by eq. (2.65), are needed to compute the scalar product E∗ · E. The direct evaluation of we leads to a quadruple sum over the indices n, m, n and m that appear in the products of the electric field components. In our approach to quality assessment of optical systems, we focus on the typical azimuthal dependencies that appear in the through-focus intensity distribution in the presence of aberrations. For that reason, a more systematic approach is required in the evaluation of the expression for the energy density. We combine terms with equal azimuthal dependence exp(imφ), and these collected terms will play an important role in the quality assessment problem that we address in the second part of this chapter. A general term in the expression of eq. (3.1) is written m m im exp[imφ]βn,1 Vn,k (r, f ) exp[ikφ] Gkl (β1 , β2 ) = n,m
×
n ,m
=
∗
∗
i−m exp[−im φ]βnm ,2 Vnm ,l (r, f ) exp[−ilφ]
exp i(m − m )π/2 exp i(m − m + k − l)φ
n,m,n ,m ∗ m m ∗ m βn ,2 Vn,k (r, f )Vnm ,l (r, f ), × βn,1
EBL has the sets of Zernike coefficients β
(3.2)
m n,1
m βn,2
and as variwhere the function Gkl ables, written in shorthand notation as β1 and β2 in the argument of Gkl . The m exp[i(j φ)]-functions that indices {kl} stem from the various combinations of Vn,j occur in the summation. The azimuthal dependence of the energy density terms 2 is made explicit by writing the general quadruple series n,m,n ,m an,m,n ,m according to a diagonal summation scheme as 1 am,m ;n,n = am,m;n,n m,m ,n,n
n
+ + +
m μ max
(am,m+μ;n,n μ=1 m 1 ν max
+ am+μ,m;n,n )
(am,m;n,n+ν + am,m;n+ν,n )
ν=1 μ max μ=1
n
m
n
m
(am,m+μ;n,n+ν + am+μ,m;n,n+ν
+ am,m+μ;n+ν,n + am+μ,m;n+ν,n ) .
(3.3)
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The summation ranges for the indices m, n, μmax and νmax are derived from the transformation from a rectangular summation scheme to a summation scheme along the central diagonal and the off-axis diagonals. The following expression for Gkl results: m m∗ m m∗ βn,1 βn,2 Vn,k Vn,l Gkl (β1 , β2 ) = exp i(k − l)φ +
n
μ max
m
exp[−iμπ/2] exp[−iμφ]
μ=1
×
m
m+μ∗
m βn,1 βn,2
n
+ exp[iμπ/2] exp[iμφ] +
ν max
m+μ∗
m Vn,k Vn,l
m
m+μ m∗ m+μ m∗ βn,1 βn,2 Vn,k Vn,l
n
m m∗ m m∗ βn,1 βn+ν,2 Vn,k Vn+ν,l
n m ν=1 m m∗ m m∗ + βn+ν,1 βn,2 Vn+ν,k Vn,l μ max
EBL
+
exp[−iμπ/2] exp[−iμφ]
μ=1
×
m+μ∗
m+μ∗
m m βn,1 βn+ν,2 Vn,k Vn+ν,l
n
m
m+μ∗ m m+μ∗ m + βn+ν,1 βn,2 Vn+ν,k Vn,l
+ exp[iμπ/2] exp[+iμφ] m+μ m∗ m+μ m∗ × βn,1 βn+ν,2 Vn,k Vn+ν,l n
m
,
m+μ m∗ m+μ m∗ + βn+ν,1 βn,2 Vn+ν,k Vn,l
(3.4)
where we have suppressed in the notation the (r, f )-dependence of the V -functions. With the G-function notation above, the electric energy density is readily written as
! 0 n2r k02 fL2 s04 we (r, φ, f ) = |ax |2 G0,0 (βx , βx ) 8 + s02 G0,2 (ax βx , ax βx − iay βy )
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+ G0,−2 (ax βx , ax βx + iay βy ) s04 G2,2 (ax βx − iay βy , ax βx − iay βy ) 4 + G−2,−2 (ax βx + iay βy , ax βx + iay βy ) +
s04 G2,−2 (ax βx − iay βy , ax βx + iay βy ) 2 + |ay |2 G0,0 (βy , βy ) − s02 G0,2 (ay βy , iax βx + ay βy ) + G0,−2 (ay βy , −iax βx + ay βy )
+
s04 G2,2 (iax βx + ay βy , iax βx + ay βy ) 4 + G−2,−2 (−iax βx + ay βy , −iax βx + ay βy )
+
s04 G2,−2 (iax βx + ay βy , −iax βx + ay βy ) 2 + s02 G1,1 (iax βx + ay βy , iax βx + ay βy ) + G−1,−1 (−iax βx + ay βy , −iax βx + ay βy ) 2 + 2s0 G1,−1 (iax βx + ay βy , −iax βx + ay βy ) , (3.5) +
EBL
with k0 the vacuum wave number and fL the focal length of the imaging system. The amplitude factors ax,y apply to the strength of the incident x- and y-polarization components, and the indices x, y of β in the arguments of the Gkl functions refer to the sets of Zernike coefficients to be used. They correspond to m and β m , respeceither x- or y-linearly polarized light and are denoted by βn,x n,y tively. A further rearrangement of the expression for we (r, φ, f ) can be made using the properties Gkl (β1 + β2 , β3 + β4 ) = Gkl (β1 , β3 ) + Gkl (β1 , β4 ) + Gkl (β2 , β3 ) + Gkl (β2 , β4 ), Gkl (β1 , β2 ) =
G∗lk (β2 , β1 ).
This leads to the expression
! 0 n2r k02 fL2 s04 we (r, φ, f ) = |ax |2 G0,0 (βx , βx ) + |ay |2 G0,0 (βy , βy ) 8 + s02 |ax |2 G0,2 (βx , βx ) + iax ay∗ G0,2 (βx , βy ) + iax∗ ay G0,2 (βy , βx ) − |ay |2 G0,2 (βy , βy )
(3.6) (3.7)
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+ s02 |ax |2 G0,−2 (βx , βx ) − iax ay∗ G0,−2 (βx , βy ) − iax∗ ay G0,−2 (βy , βx ) − |ay |2 G0,−2 (βy , βy ) s04 |ax |2 G2,2 (βx , βx ) + iax ay∗ G2,2 (βx , βy ) 2 − iax∗ ay G2,2 (βy , βx ) + |ay |2 G2,2 (βy , βy )
+
s04 |ax |2 G−2,−2 (βx , βx ) − iax ay∗ G−2,−2 (βx , βy ) 2 + iax∗ ay G−2,−2 (βy , βx ) + |ay |2 G−2,−2 (βy , βy ) + s02 |ax |2 G1,1 (βx , βx ) + iax ay∗ G1,1 (βx , βy ) − iax∗ ay G1,1 (βy , βx ) + |ay |2 G1,1 (βy , βy ) + s02 |ax |2 G−1,−1 (βx , βx ) − iax ay∗ G−1,−1 (βx , βy ) + iax∗ ay G−1,−1 (βy , βx ) + |ay |2 G−1,−1 (βy , βy ) + 2s02 −|ax |2 G1,−1 (βx , βx ) + iax ay∗ G1,−1 (βx , βy ) + iax∗ ay G1,−1 (βy , βx ) + |ay |2 G1,−1 (βy , βy ) . (3.8) +
EBL
This is, within the framework of the approximations related to the Debye integral, a rigorous expression for the energy density in the focal region for a general incident state of polarization with the optical system suffering from transmission defects, wavefront aberration and birefringence that is spatially varying over the pupil cross-section. Once we eliminate all optical defects (βx = βy = β with β00 = 1 and all other βnm ≡ 0) and limit ourselves to linear polarization along the x-axis (ay = 0), the expression is greatly simplified to
! 0 n2r k02 fL2 s04 |ax |2 we (r, φ, f ) = G0,0 (β, β) 8 + s02 G0,2 (β, β) + G0,−2 (β, β) − 2G1,−1 (β, β) s4 + 0 G2,2 (β, β) + G−2−2 (β, β) , (3.9) 2 and, on substituting the expression for the functions Gkl in the case of a perfect system, we find ! 0 n2r k02 fL2 s04 |ax |2 0 2 V + 2s 2 V 0 2 we (r, φ, f ) = 0 0,0 0,1 8 0 2 0 2 + V 0 V 0∗ . + s04 V0,2 + cos 2φ V0,1 0,0 0,2
(3.10)
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This special case corresponds to the original result on high-numerical-aperture focusing by a perfect system treated by Richards and Wolf [1959]. When we strongly reduce the numerical aperture, only the leading term with s04 remains, leading to Lommel’s result. Finally, in the nominal focal plane (f = 0), and 0 with s → 0, we obtain the expression using the analytic expression for V0,0 0 corresponding to Airy’s result:
! 0 n2r |ax |2 π 2 fL2 s04 2J1 (2πr) 2 we (r, φ, f ) = (3.11) , 2 2πr λ2 where the factor in square brackets can be identified as the energy ‘concentration’ factor in optimum focus. If we limit ourselves to optical systems that are free of birefringence, we can study characteristic energy density patterns in the focal region corresponding to special cases of incident polarization. A general state of polarization in the entrance pupil of the system is determined by the field coefficients ax = px A and ay = py A, with px and py complex numbers satisfying |px |2 + |py |2 = 1 for the purpose of normalization. The absence of birefringence allows us to write for m = β m = β m . Using the result of eq. (3.8), we each aberration coefficient βn,x n,y n find
! 0 n2r k02 fL2 s04 |A|2 we (r, φ, f ) = G0,0 (β, β) 8 + s02 |px |2 − |py |2 G0,2 (β, β) − 2 (px py∗ ) G0,2 (β, β) + s02 |px |2 − |py |2 G0,−2 (β, β) + 2 (px py∗ ) G0,−2 (β, β)
EBL
s04 1 − 2 (px py∗ ) G2,2 (β, β) 2 + 1 + 2 (px py∗ ) G−2,−2 (β, β) + s02 1 − 2 (px py∗ ) G1,1 (β, β) + 1 + 2 (px py∗ ) G−1,−1 (β, β) − 2s02 |px |2 − |py |2 G+1,−1 (β, β) ∗ + 2 (px py ) G+1,−1 (β, β) . +
(3.12)
Alternatively, the various terms in the expression above can be arranged according to their contribution to the four Stokes vector components, as shown by Stallinga
6, § 3]
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[2001]. Some special cases can be distinguished regarding the incident state of polarization. 3.1.1. Linear polarization • x-direction, (px , py ) = (1, 0). The energy density reduces to ! wex (r, φ, f ) 0 ∝ G0,0 (β, β) + s02 G0,2 (β, β) + G0,−2 (β, β) s04 G2,2 (β, β) + G−2,−2 (β, β) 2 + s02 G1,1 (β, β) + G−1,−1 (β, β) − 2s02 G+1,−1 (β, β) . +
(3.13)
• y-direction, (px , py ) = (0, 1). The energy density is now proportional to ! y we (r, φ, f ) 0 ∝ G0,0 (β, β) − s02 G0,2 (β, β) + G0,−2 (β, β) s04 G2,2 (β, β) + G−2,−2 (β, β) 2 + s02 G1,1 (β, β) + G−1,−1 (β, β) + 2s02 G+1,−1 (β, β) . +
(3.14) EBL Figure 3.1 presents grayscale plots of the intensity distribution in the focal volume in the presence of typical lower-order aberrations. In all graphs the polarization state is linear, with the plane of polarization parallel to the x-axis. The defocus range, going from left to right, extends over four focal depths (|f | = π/2) of z = λ/(4u0 ); this quantity, defined in eq. (2.31), is used in the case of high-numerical-aperture values, to be compared with z = λ/(2s02 ) for low values of s0 . The upper row shows the energy density in the aberration-free case. The well-known broadening of the point-spread function in the cross-section parallel to the plane of polarization in the entrance pupil is clearly visible, leading to an overall elliptical profile of the point-spread function. In the second row, the coef0 , describing the phase departure in the entrance pupil due to spherical ficient α4,c √ aberration, has been put equal to 2. For this aberration value the classical Strehl intensity, see Born and Wolf [2002], would become 0.60. This means that the aberration pushes the quality of the imaging system beyond the ‘just-diffractionlimited’ level corresponding to a Strehl ratio of 0.80. The columns on the right, corresponding to the defocus region beyond the nominal focal point F , start to show a narrower ring-shaped pattern; closer to the exit pupil, the central lobe of the point-spread function is broadened. This asymmetry with respect to ‘best’ focus is characteristic for spherical aberration. In the low-aperture scalar case, it
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EBL
Fig. 3.1. Intensity of the point-spread function of a high-numerical-aperture imaging system (s0 = 0.95). The columns pertain to the defocus parameter with f = π, π/2, 0, −π/2, −π , from left to right, going away from the exit pupil according to the sign convention. Radial size: |r| 3. 0 = 1.41; row 3: α 1 = 1.79; row 4: α 1 = 1.79; row 5: Upper row: aberration-free case; row 2: α4,c 3,c 3,s 2 = 1.55; row 6: α 2 = 1.55; row 7: α 0 = 0.81, α 1 = 1.03, α 2 = 0.89. The grayscale graphs α2,c 2,s 3,c 2,c 4,c have been normalized in each row to the top intensity in the focal plane with f = 0. The maximum intensity of the aberrated graphs in rows 2 to 7 is approximately 0.60 (Strehl intensity).
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leads to sharp and narrow diffraction rings beyond focus, and a more blurred pat0 is positive as in fig. 3.1. tern at the short side of F , at least when the sign of α4,c In the high-numerical-aperture case, the typical spherical aberration effects are somewhat blurred by the intrinsic ‘unsharpness’ that arises from the vector diffraction effects. The third and fourth rows are point-spread functions of an imag1 = 1.79 in the ing system suffering from comatic aberration, with a coefficient α3,c 1 in the fourth row. The characteristic asymthird row and the same value for α3,s metric pattern of comatic aberration is present in both cases. In the third row, the comatic blur is superimposed on the intrinsic blur in the x-direction due to the linear state of polarization. It also becomes apparent from the graphs that comatic aberration does not perturb the axial symmetry with respect to the nominal focal plane in F . In the fifth row we have introduced lowest-order astigmatism with 2 = 1.55. The vector diffraction effects influence the widths of the coefficient α2,c the focal lines in the x- and y-directions in the defocused state. The focal line in the y-direction, farther away from focus, has suffered a strong broadening due to 2 , the two outer fothe polarization effect. Note that with the chosen value of α2,c cus settings approximately correspond to the sharp focal line positions predicted by geometrical optics. The sixth row shows, with the astigmatic aberration rotated by 45◦ , the typical interaction between the vector ‘blur’ direction along the x-axis and the astigmatic effect that is diagonal. The net visual effect is a small extra rotation of the point-spread function towards the y-axis. Finally, the graphs in the seventh row correspond to a mixture of the three basic aberrations above, so that the total rms phase departure is the same and the Strehl intensity is again close to 0.60. Certain azimuthal dependencies in the energy density patterns can be made visible by the subtraction of ‘orthogonal’ states of polarization. In this case, the subtraction of the two orthogonal linearly polarized patterns at 0◦ and 90◦ with the x-axis yields ! ! y wl,0 = wex (r, φ, f ) π/2 − we (r, φ, f ) 0 = 2s02 G0,2 (β, β) + G0,−2 (β, β) − 2G+1,−1 (β, β) . (3.15)
EBL
The subtraction of two energy patterns belonging to diagonal linear polarization states gives rise to ! ! y wl,π/4 = wex (r, φ, f ) 3π/4 − we (r, φ, f ) π/4 = 2s02 G0,2 (β, β) − G0,−2 (β, β) + 2G+1,−1 (β, β) . (3.16) The G-functions above, in the absence of aberrations, contain the basic 2φazimuthal dependence in the energy density, in general a mixture of the cos 2φ-
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and sin 2φ-function from their real and imaginary parts, depending on the orientation of the linear state of polarization. The addition of aberrations introduces higher- and lower-order φ-dependencies in the energy density function. This phenomenon plays an important role when studying the quality of an optical system with the aid of its through-focus point-spread functions. 3.1.2. Circular polarization (LC and RC) If we substitute py = −ipx and py = ipx , we define right- (RC) and left-handed (LC) circularly polarized light, respectively. This definition of polarization handedness follows from our convention exp{i(kz − ωt)} for an outgoing plane wave in the positive z-direction. The energy density is proportional to ! weRC (r, φ, f ) 0 ∝ G0,0 (β, β) + s04 G2,2 (β, β) + 2s02 G1,1 (β, β)
(3.17)
and, in a corresponding way, ! weLC (r, φ, f ) 0 ∝ G0,0 (β, β) + s04 G−2,−2 (β, β) + 2s02 G−1,−1 (β, β).
EBL
(3.18)
The difference between right- and left-handed polarization density distributions thus equals wC,0 = s04 G2,2 (β, β) − G−2,−2 (β, β) + 2s02 G1,1 (β, β) − G−1,−1 (β, β) . (3.19) In the aberration-free case, this difference is identically zero. However, in the presence of aberrations the non-zero difference function with its specific azimuthal dependencies carries information on the aberration function of the optical system. Like for the linear state of polarization, this fact will be exploited below when dealing with the assessment of optical systems regarding their imaging quality. 3.1.3. Unpolarized or ‘natural’ light The energy density in the case of incoherent (‘natural’) light is obtained by adding the energy density patterns corresponding to two orthogonal polarization states. We find ! weN (r, φ, f ) = G0,0 (β, β) + s02 G1,1 (β, β) + G−1,−1 (β, β) +
s04 G2,2 (β, β) + G−2,−2 (β, β) . 2
(3.20)
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In the aberration-free case, the energy densities for natural light and circularly polarized light are equal. This is generally not the case in the presence of aberrations. The case of partially polarized light can be accounted for by defining a total energy density that is a weighted sum of a fully polarized pattern and an unpolarized pattern, the weight being determined by the degree of polarization. 3.1.4. Circularly symmetric states of polarization (radial and tangential) Other states of polarization with a specific structure have been studied in the context of the angular momentum of light beams, in particular Gauss–Laguerre beam modes (Allen, Beijersbergen, Spreeuw and Woerdman [1992], He, Friese, Heckenberg and Rubinsztein-Dunlop [1995]). These special beams have a polarization state that varies spatially over the beam cross-section, and they cannot be represented by a combination of two complex numbers specifying their constant x- and y-components. The Gauss–Laguerre beam modes apply to paraxial beams with a field strength that gradually decreases as a function of radial position. To incorporate finite-size beams with a sharp aperture limitation as is common in imaging systems, we give below the Zernike expansion of radially or tangentially polarized beams, or, more generally, of polarized beams with circular symmetry. The expansion to be used is that of eq. (2.64) in which the linear phase terms, carrying angular momentum, are present as exp(imθ ) with m also assuming negative values to allow for the positive or negative sign of angular momentum. Two sets of expansion coefficients are required to represent the x- and y-components of the field in the entrance pupil. We write the state of polarization of the incident field in the entrance pupil as, see Braat, Dirksen, Janssen and van de Nes [2003],
EBL
Ex = A0 cos(θ + θ0 ), Ey = A0 sin(θ + θ0 ) exp(ixy ).
(3.21)
With xy ≡ 0, we find as limiting cases radial polarization for θ0 = 0 and tangential or azimuthal polarization for θ0 = π/2. The constant phase xy allows a gradual change from linearly polarized light along the x- and y-axis to elliptically polarized light with maximum eccentricity along the diagonal sections of the aperture. m Limiting ourselves to xy = 0, we can calculate the general coefficients βn,x m of the Zernike expansion by forming the inner products with the field and βn,y components of eq. (3.21). The details of the derivation are given in Appendix F; the result is +1 β2n+1,x = fc (n) exp(iθ0 ), −1 β2n+1,x = fc (n) exp(−iθ0 ),
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EBL
Fig. 3.2. Intensity of the point-spread function for various states of polarization in the entrance pupil of an aberration-free imaging system (s0 = 0.95). The columns pertain to the defocus parameter with f = π, π/2, 0, −π/2, −π , from left to right. Radial size: |r| 3. Upper row: incident light is unpolarized; row 2: circular polarization; row 3: radial polarization; row 4: tangential polarization.
+1 β2n+1,y = fc (n) exp i(θ0 − π/2) , −1 β2n+1,y = fc (n) exp −i(θ0 − π/2) , with
fc (n) = (−1)n
2n + 2 , (2n + 1)(2n + 3)
(3.22)
and with n = 0, 1, . . . , nmax , where the value of nmax depends on the convergence of the expansion. Some special polarization states are depicted in fig. 3.2. In all cases, the phase departure of the focusing wave in the exit pupil is zero. With the through-focus planes of each column chosen as in fig. 3.1, we first present the point-spread function intensity resulting from imaging with unpolarized light, obtained by adding with equal weights the intensity distributions belonging to two orthogonal states of polarization. The second row pertains to circularly polarized light, with no difference between right- or left-handed circular polarization in the aberration-free case. Moreover, as is also suggested by fig. 3.2, the point-spread functions in rows 1 and 2 are identical in the aberration-free case,
6, § 3]
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see eqs. (3.17), (3.18) and (3.20). In the third and fourth rows we show throughfocus point-spread functions corresponding to, respectively, radial and tangential polarization in the entrance pupil. The wavefront deformation in the exit pupil is ±1 ±1 and β2n+1,y are now still zero, but non-zero complex Zernike coefficients β2n+1,x needed to represent the continuously varying state of polarization in the entrance pupil. In the case of radial polarization (row 3), a strong z-component of the electric field is present in the optimum focal point F . It is seen that the radial polarization leads to an overall blurring effect when the total energy density is considered. Particular components of the electric field, in this case the z-component, give rise to a contribution to the energy density distribution that is significantly narrower than the typical width of the point-spread function. If a detecting material can be devised that is preferentially sensitive to the z-component, this narrowing of the point-spread function can be exploited in practice. In the fourth row, with tangential polarization, we observe a zero on-axis, surrounded by a strong ring in the plane of best focus. This singular point in the energy density distribution can be exploited, for instance, when pointing accuracy in metrology is important. As a final remark we note that, apart from dimensional issues, the geometry of the electric field distribution for radial polarization is identical to that of the magnetic field in the case of tangential polarization, and vice versa.
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3.2. Expression for the Poynting vector In this subsection we briefly discuss the behavior of the components of the Poynting vector in a high-numerical-aperture system with general aberration and state of polarization in the entrance pupil. The energy flow in the focal volume is determined by the time-averaged values of the Cartesian components of the Poynting vector S. In a passive system, the quantity | ∇ · (E × B) |/μ0 , integrated over a certain volume, determines the total power loss in that volume. If the loss is ohmic only, this quantity can also be written as σ |E|2 /2, with σ the local electric conductivity in units of Sm−1 . The exposure in a photo-resist layer or the total charge collected by a detector thus depends on either the electric energy density or the divergence of the Poynting vector. In practice, it will be easier to calculate exposure or detected charge by using the directly available electric energy density. However, despite the evident advantage of using |E|2 , we present the expressions for the Poynting vector components to enable a comparison and numerical check of both approaches. For harmonic fields, the time average of the Poynting vector reads S =
0 c 2 E × B∗ . 2
(3.23)
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We write the Poynting vector in a medium (refractive index nr ) as 0 nr c 2 2 4 2 (3.24) k f s |A| (Px , Py , Pz ), 8 0 L 0 where the incident electric field is represented by ax = px A, ay = py A, like we did before in the case of an optical system that is free of birefringence. Using eqs. (2.65) and (2.66) for the electromagnetic field vectors, we obtain, after some lengthy manipulation, the following expressions for the components Px and Py : Px = s0 −2ipx∗ py (G0,1 ) + (G0,−1 ) + i |px |2 [G∗0,1 − G∗0,−1 ] − |py |2 [G0,1 − G0,−1 ] − s03 −2 (px py∗ ) + 1 (G2,1 ) − 2 (px py∗ ) + 1 (G−2,−1 ) + 2i (px py∗ ) (G1,−2 ) − (G2,−1 ) + i |px |2 − |py |2 (G1,−2 ) − (G2,−1 ) , (3.25) ∗ Py = s0 2ipx py (G0,1 ) − (G0,−1 ) + |px |2 [G0,1 + G0,−1 ] + |py |2 [G∗0,1 + G∗0,−1 ] + s03 −2 (px py∗ ) + 1 (G2,1 ) + 2 (px py∗ ) + 1 (G−2,−1 ) + 2i (px py∗ ) (G1,−2 ) − (G2,−1 ) − i |px |2 − |py |2 (G1,−2 ) − (G2,−1 ) . (3.26) S = (Sx , Sy , Sz ) =
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The z-component of the Poynting vector equals Pz = G0,0 − s02 i |px |2 − |py |2 (G0,2 ) + 2i (px py∗ ) (G0,2 ) + i |px |2 − |py |2 (G0,−2 ) − 2i (px py∗ ) (G0,−2 ) s04 1 − 2 (px py∗ ) G2,2 + 1 + 2 (px py∗ ) G−2,−2 . (3.27) 2 The real parts of the above expressions are needed to obtain the time-averaged energy flow and to eliminate the reactive part of the Poynting vector, yielding the adapted Poynting vector components Px,r = s0 −2 (px py∗ ) (G0,1 + G0,−1 ) + (G0,1 − G0,−1 ) + s03 +2 (px py∗ ) (G2,1 + G−2,−1 ) − (G2,1 − G−2,−1 ) , (3.28) ∗ Py,r = s0 −2 (px py ) (G0,1 − G0,−1 ) + (G0,1 + G0,−1 ) + s03 −2 (px py∗ ) (G2,1 − G−2,−1 ) + (G2,1 + G−2,−1 ) , (3.29) −
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s04 (G2,2 + G−2,−2 ) 2 − 2 (px py∗ )(G2,2 − G−2,−2 ) .
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Pz,r = G0,0 −
(3.30)
In fig. 3.3 we have plotted these Poynting vector components in three crosssections of the focal volume with f = π, 0, −π. The top row (no aberrations) shows a uniformly directed flow on both sides of best focus, towards the focal point F in front of F and outgoing beyond F . In the plane with f = 0, the only non-zero flow component is that along the z-axis. Despite the non-circularly symmetric energy density in the high-numerical-aperture case, the flow pattern of the Poynting vector is circularly symmetric. This is because of the complementary field distributions of the E- and B-vectors, yielding a circularly symmetric function regarding their product. At a very fine scale, not visible in the figure, vortices in the flow pattern are possible, close to the regions in space of zero energy density where the z-component of the Poynting vector changes sign, see Richards and Wolf [1959]. In the second row of fig. 3.3, in the presence of spherical aberration, the global flow pattern of the aberration-free case is still there, but finite radial components are now visible in the plane f = 0 because of the ‘distributed’ axial focus in the beam. The asymmetry around focus, already present in the energy density function, is also visible in the Poynting-vector flow pattern. In the third row, the leftmost graph shows the enhanced energy flow in the x-direction towards the asymmetrical coma distribution in the plane f = 0. On the other side of the best focal plane, the outflow in the x-direction is also stronger to get back to the balanced flow pattern at large defocus value in the far field. Finally, in the fourth row, we observe in the energy flow pattern the effect of the splitting of the ideal focal point into two focal lines, due to the astigmatism at 45◦ . The slight change in orientation of the ‘focal lines’ towards the y-axis is due to the initial state of linear polarization that was chosen along the x-axis.
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3.2.1. The aberration-free system as a special case It is interesting to reduce the general expressions for the Poynting vector components to their simpler form in the case of an aberration-free imaging system. In that case, we put βnm = 0 with the exception of β00 ≡ 1. The G-functions reduce to 0 Gkl (β) = exp i(k − l)φ Ψ0;k,l (3.31) (r, f ), 0 (r, f ) = V 0 (r, f )V 0∗ (r, f ). After some mawith the shorthand notation Ψ0;k,l 0,k 0,l nipulation, the Cartesian Poynting vector components are found to be
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Fig. 3.3. Flow lines of the electromagnetic energy in the focal region of a high-numerical-aperture beam (s0 = 0.95, linear polarization in the x-direction), going from a defocused plane closer to the exit pupil, via the best focus plane in F to a defocused plane beyond F (the defocus distance equals two focal depths). Lateral dimensions in the x- and y-direction are −1.11 x, y +1.11 in dimensionless units of λ/s0 . The x- and y-components of the Poynting vector are represented by arrows; the amplitude of the z-component is represented by the gray shading in each plane z = constant. Top 0 = 1.41; row 3: α 1 = 1.79; row 4: α 2 = 1.55. row: aberration-free case; row 2: α4,c 3,c 2,s
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0 Px,r = 2s0 2 (px py∗ ) Ψ0;0,1 (r, f ) sin φ 0 (r, f ) cos φ + Ψ0;0,1 0 + 2s03 2 (px py∗ ) Ψ0;2,1 (r, f ) sin φ 0 (r, f ) cos φ , − Ψ0;2,1 0 Py,r = 2s0 −2 (px py∗ ) Ψ0;0,1 (r, f ) cos φ 0 (r, f ) sin φ + Ψ0;0,1 0 + 2s03 −2 (px py∗ ) Ψ0;2,1 (r, f ) cos φ 0 (r, f ) sin φ , − Ψ0;2,1 0 0 Pz,r = Ψ0;0,0 (r, f ) − s04 Ψ0;2,2 (r, f ).
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(3.32)
(3.33) (3.34)
Because of the basic circular symmetry of many diffraction problems, it is appropriate to express the Poynting vector components in cylindrical coordinates, and we find for the polar components 0 0 (r, f ) − s02 Ψ0;2,1 (r, f ) , Pr = 2s0 Ψ0;0,1 0 0 Pφ = −4s0 (px py∗ ) Ψ0;0,1 (3.35) (r, f ) − s02 Ψ0;2,1 (r, f ) .
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Some conclusions can be drawn from the above expressions. Using the property 0 (r, −f ) = Ψ 0∗ (r, f ), we can state that Ψ0;k,l 0;k,l • On-axis (r = 0), the power flow is directed along the axis of the beam. • The radial component changes sign through focus. • The azimuthal component is only present if the incident radiation is not linearly polarized (arg(px ) = arg(py )), see Boivin, Dow and Wolf [1967] and Stallinga [2001]. This corresponds to the presence of angular momentum in the focused beam. • The azimuthal component is maximum in the case of circularly polarized light. • The azimuthal component maintains its sign on both sides of the optimum focus. • For large s0 , all components can locally change sign with respect to their average values if the (r, f )-dependent function becomes zero. This can give rise to regions where the energy flow is in the negative z-direction and also to the appearance of vortices in the energy flow pattern. • Regardless of the incident state of polarization, the z-component possesses circular symmetry. This is because the Poynting vector describes the energy flow of both electric and magnetic energy. Any lack of circular symmetry in the energy densities of these contributions is cancelled in their sum. • In the low-aperture case, the z-component is the only one surviving.
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EBL Fig. 3.4. Flow lines of the Poynting vector in the focal region of an ideal imaging system (s0 = 0.95) for various states of polarization in the entrance pupil. Left column: f = π ; central column: f = 0; right column: f = −π . Lateral dimensions in the x- and y-direction are −1.11r x, y +1.11. Upper row: linear polarization along the x-axis; middle row: right-handed circularly polarized light (px = 1, py = −i); bottom row: radial polarization.
The graphs in fig. 3.4 illustrate the observations made above for the aberrationfree case. With the upper row (linear polarization in the x-direction) as a reference, we see in the second row the influence of circular polarization in the entrance pupil on the flow pattern in the focal region. The state of polarization is right-handed, the flow circulation in focus becomes left-handed. Everywhere on the optical axis the azimuthal component is zero. The beam possesses a certain amount of angular momentum. With the positive z-direction as the viewing direction, we observe that the rotation sense of the electric field vector is lefthanded for RC-polarized light and this rotation sense is in accordance with the
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sign of the angular momentum of the electric field distribution in the focal region. In the bottom row we have plotted the power flow in the case of radial polarization. Because of the interchangeability of the E- and B-vectors between radial and azimuthal polarization, the flow pattern would have been the same for azimuthal polarization. It is important to remark that the z-component of the Poynting vector is zero everywhere on-axis, despite the fact that the energy density is finite on-axis in the case of radial polarization. The same holds for the magnetic energy density in the case of azimuthal polarization. However, because of the vanishing transversal components of either the electric (radial polarization) or magnetic (azimuthal) field components on the z-axis, no power flow is possible in the z-direction on the axis itself. § 4. Quality assessment by inverse problem solution In the preceding section, a thorough description has been presented of the energy density and the intensity in the focal region. Various methods are available for calculating these quantities. The propagation from the exit-pupil sphere to the image region is the crucial and most labor-intensive step for obtaining the above quantities. Numerical methods are frequently used, in most cases based on the Fast Fourier Transform. These methods can handle a large variety of practical situations; with the correct sampling density quite arbitrary geometries are tractable. Analytic methods generally are less versatile, but they have the advantage of providing an immediate solution of the diffraction integral, thus saving much computational effort. Considerations of this nature are important when addressing the so-called ‘inverse problem’, how to obtain reliable information on the properties of the optical imaging system to assess its quality. Although an excellent method like interferometry is available to evaluate the aberrations of an optical system, this method cannot always be implemented easily in a practical situation. Special sources, preferably lasers, are required at the specific design wavelengths of the optical system. Perfect optics are needed in the reference branch of an amplitudesplitting interferometer or a wavefront splitting device in a shearing interferometer. Coherence-length requirements have to be satisfied and the necessary mechanical and environmental stability is difficult to obtain. For these reasons, the reconstruction or retrieval of system information from images in the focal region is interesting because this method avoids most of the subtle experimental conditions encountered in interferometry. The images to be studied for ‘retrieval’ of system parameters can be of a general nature or they can be of the simplest form, the image of a point source, the so-called point-spread function. After a brief discussion of the more general methods for phase retrieval, we will focus in this
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section on the point-source or quasi-point-source option as it is, experimentally, most easily realized. The quasi-point-source is the practical compromise between the smallest possible object dimension and light efficiency in the measurement set-up. The analyses in this section will be limited to the scalar approximation of image formation, valid for relatively low-numerical-aperture imaging systems. The extension of the inverse problem to high numerical aperture will be treated in the following section. 4.1. Intensity measurements and phase retrieval The most important quantity impairing the performance of an optical instrument is its phase aberration. It may be due to environmental perturbations like atmospheric turbulence in astronomical imaging, it may be caused by the design of the system, or induced by fabrication and mounting errors of the system components, such as the improper figuring of an aspheric optical surface. In this subsection we briefly discuss the model for retrieval of phase information from intensity measurements based on an approximate solution of the scalar wave equation for quasi-parallel or low-aperture beams. The starting point of the analysis is the so-called paraxial or parabolic Helmholtz wave equation (Kogelnik [1965]), ∂Q ∇t2 Q + 2ik (4.1) + k 2 Q = 0, ∂z with Q(x, y; z) the slowly varying amplitude in the z-direction of the scalar solution U for the wave U (x, y, z; t) = Q(x, y; z) exp i(kz − ωt) . (4.2)
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The circular wavenumber of the wave propagating in the z-direction is again given by k, the circular frequency is ω. The ∇t2 -sign means that only the second derivatives in the transversal direction need to be taken. The complex amplitude is more explicitly written as Q(x, y; z) = A(x, y; z) exp{iφ(x, y; z)} where, again, the parameter-wise notation of the z-coordinate indicates that the functions vary only slowly as a function of this variable. The intensity of the wave phenomenon is given by |U |2 = QQ∗ = I , and an equation can be derived (Teague [1983]) that relates the measured intensity I to the phase function φ(x, y; z) of the wave: the so-called Intensity Transport Equation (ITE). Multiplying eq. (4.1) by Q∗ and the complex conjugate expression of eq. (4.1) by Q itself and subtracting the results, we obtain after some rearrangement
2 ∂ 2φ ∂A ∂φ ∂A ∂φ ∂ φ ∂A2 + + 4iA − + 2iA2 = 0, 2ik (4.3) ∂z ∂x ∂x ∂y ∂y ∂x 2 ∂y 2
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Fig. 4.1. Schematic drawing of a quasi-parallel beam and the variation of its intensity in the axial and in the transverse beam direction due to, for instance, wavefront curvature. The lower intensity regions have been shaded. The energy propagation direction is given by the arrows, the dashed curve indicates the curved wavefront. All effects have been strongly exaggerated in the figure.
which is commonly written as the ITE in compact notation ∂I (4.4) = ∇t · (I ∇t φ) = I ∇t2 φ + ∇t I · ∇t φ. ∂z If the intensity across the beam is locally constant, the second term on the righthand side of eq. (4.4) is zero and the axial derivative of the intensity is proportional to the wavefront curvature. In regions where the intensity across the beam varies, the gradient or slope of the wavefront determines the transfer of intensity across the beam (see fig. 4.1). Methods to solve the intensity transport equation using Green’s functions have been presented by Woods and Greenaway [2003]. The experimental evaluation of the z-derivative of the intensity is generally done by measuring the intensity distribution in two axially shifted planes. The ITE-based phase retrieval method is limited to paraxial beams. In this chapter the interest is focused on the characterization of high-quality imaging systems where, in most cases, imaging pencils with a medium to large opening angle are encountered. The characterization of such systems is treated in the following subsections. −k
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4.2. The optical inverse problem for finite-aperture imaging systems The aberration of an optical imaging systems influences its complex ‘lens’ transmission function. Measurements only give access to the intensity, the modulus squared of the complex amplitude. In this subsection we discuss inverse schemes
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Fig. 4.2. Schematic drawing of the exit-pupil domain (far field, coordinates Q(u, v)) and the image domain (coordinates P (x, y, z)). The propagation forth and back between the two domains is governed by the scalar diffraction integral. In a different set-up, the light is focused first on the object in a plane z = z0 ; the far field then becomes available on the detection sphere D. The angular extent αmax of the imaging pencil has been exaggerated in the figure.
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that go back to the complex lens function in a generally unambiguous way using appropriate intensity measurements. However, a unique way back to the lens function is not guaranteed in all cases, among others given the problem of phase retrieval from data that extend beyond the basic retrieval interval for phase data of [−π, +π] (phase-unwrapping problem). The first scheme for phase retrieval or pupil function reconstruction goes back to Gerchberg and Saxton [1971]. The method is based on the frequent back and forth propagation of the complex amplitude between the image plane and the exit pupil, see fig. 4.2. A Fast Fourier Transform (FFT) is executed and the Fraunhofer or Fresnel approximation is used for calculating the pathlength from a general point Q in the exit pupil to a point P in the image plane. After each forward and backward propagation, the calculated results are updated to better match them with the measurements and the physical boundary condition (non-negativity of the intensity). Denoting the image-plane complex amplitude in a certain plane z = z0 by f (x, y, z0 ), we have the basic Fourier relationship +∞ +∞ f (x, y, z0 ) exp −i(kx x + ky y) dx dy, F (kx , ky ; z0 ) = −∞ −∞
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+∞ +∞ 1 f (x, y, z0 ) = F (kx , ky ; z0 ) (2π)2 −∞ −∞ × exp +i(kx x + ky y) dkx dky ,
413
(4.5)
where kx and ky are the optical ‘far-field’ coordinates and where we have neglected the exact radiation factors that should precede the integrals. Both F (kx , ky ; z0 ) and f (x, y, z0 ) are complex-valued according to F (kx , ky ; z0 ) = F (kx , ky ; z0 ) exp iφF (kx , ky ; z0 ) , f (x, y, z0 ) = f (x, y, z0 ) exp iφf (x, y, z0 ) . (4.6) In the most comfortable situation, both the optical far field |F | and near field |f | are accessible to measurements via their squared moduli. In other circumstances, for instance when measuring the far-field distribution scattered from a microscopic periodic transmitting object, |F | is available and we want to get back to the object transmission function f , including its phase φf . In the case of a single intensity measurement, a supplementary property of the object needs to be specified. This could be, for example, the frequently occurring boundary condition that the object is partly obscuring the light, yielding a transmission function f (x, y; z0 ) that should be real and positive. The Gerchberg–Saxton algorithm then proceeds as follows: (1) produce an estimate of the object and calculate its Fourier transform to obtain the far field, (2) replace the calculated modulus of the far field by its measured value to obtain a better estimate of the far field, (3) apply an inverse Fourier transform to this function to obtain an image estimate, (4) replace the modulus of the image by the measured modulus to further improve the image. Following the analysis by Fienup [1982], the equations governing the four steps in the nth cycle are, starting with, for instance, the nth estimate fn of the object function Fn (kx , ky ; z0 ) = Fn (kx , ky ; z0 ) exp iφF,n (kx , ky ; z0 ) = F fn (x, y, z0 ) , (4.7) Fn (kx , ky ; z0 ) = F (kx , ky ; z0 ) exp iφF,n (kx , ky ; z0 ) , (4.8) fn (x, y, z0 ) = fn (x, y, z0 ) exp iφf,n (x, y, z0 ) = F −1 Fn (kx , ky ; z0 ) , (4.9)
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fn+1 (x, y, z0 ) = f (x, y, z0 ) exp iφf,n+1 (x, y, z0 ) = f (x, y, z0 ) exp iφ (x, y, z0 ) . f,n
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(4.10)
Here, F and F −1 stand for the forward and backward Fourier transform operations; the primed quantity Fn has been obtained by using the measurement result for |F |. In a comparable way, the non-primed quantity fn+1 in the fourth equation has been obtained from the primed fn by substituting the measurement result of |f |. If a complete measurement of F or f is not available, partial adjustments can be made in steps (2) and (4). These are related to possible negative intensity values or to the spreading of F or f outside a fixed domain for these functions whose limits are known a priori. The whole cycle is repeated until no further improvement is obtained, which means that either the optimum image or pupil function has been found or that stagnation has occurred. The improvement is monitored by calculating a merit or cost function at each cycle according to fn+1 (x, y, z0 ) − f (x, y, z0 )2 dx dy, Mn,f = (4.11) n 1 F (kx , ky ; z0 ) − Fn (kx , ky ; z0 )2 dkx dky , Mn,F = (4.12) n 2 (2π) where either the image plane or the far-field data are monitored. Convergence of the merit function can be proved in the following way. Applying Parseval’s theorem first to Mn,F we find 1 F (kx , ky ; z0 ) − Fn (kx , ky ; z0 )2 dkx dky Mn,F = n 2 (2π) f (x, y, z0 ) − fn (x, y, z0 )2 dx dy. = (4.13) n
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Inspection of eqs. (4.7)–(4.10) yields the result |fn − fn | |fn | − |f | = |fn − fn+1 |,
(4.14) fn
and fn are different in since the argument values of the complex quantities general. This allows us to write 1 F (kx , ky ; z0 ) − Fn (kx , ky ; z0 )2 dkx dky n 2 (2π) fn+1 (x, y, z0 ) − f (x, y, z0 )2 dx dy, n or: Mn,F Mn,f .
(4.15)
In an analogous manner, Parseval’s theorem applied to Mn,f yields the result Mn,f Mn+1,F , which leads to the conclusion that each iteration cycle will produce smaller (or equal) values of the merit function, defined either in the far field
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Fig. 4.3. A simple set-up to simultaneously produce two intensity patterns on the detectors D1 and D2 of an object. The required phase diversity between the two detected images is obtained by the defocusing (heavily exaggerated in the figure) of the point-spread function on D2 . The object is a point source S, the exit pupil of the optical system O has been denoted by P .
or in the image domain. The original method by Gerchberg and Saxton can be classified in the more general framework of optimization methods; it then becomes a special case of the method of steepest descent.
4.3. Solving the optical inverse problem using phase diversity
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The initial solution to optical inverse problems required the measurement of one intensity pattern, subject to certain constraints in the corresponding Fourier domain, or the measurement of two intensity patterns in both Fourier domains. The success of a solution method strongly depends on a fast propagation algorithm to switch between the two domains; the preferred tool to carry out this propagation is the Fast Fourier Transform. If it is not possible to have access to both domains, or if the measurements in the second domain add virtually no information, the inverse problem can be extended to more than one measurement in a single domain. The first suggestion to do so goes back to Gonsalves [1982]; in the case of sufficiently small aberration, the procedure for unique phase retrieval using two intensity measurements is explained as follows (Gonsalves [2001]). In fig. 4.3 we have depicted a system producing phase diversity by means of the beam splitter that delivers a defocused image on detector D2 . In what follows, both the aberrations of the system O itself and the defocusing on the second detector are small with respect to the wavelength λ of the light. The pupil function of the imaging system O is now given by P = A exp[iΦ] ≈ A[1 + iΦ] = A + i[AΦ1 + AΦ2 ] = A + i[Ψ1 + Ψ2 ],
(4.16)
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where we have suppressed, for ease of notation, the dependence of the functions on the pupil coordinates (u, v). Moreover, we have split the phase function Φ in two: an even part Φ1 and an odd part Φ2 . The total phase departure Φ should remain small, typically π/2. The amplitude function A is supposed to be real and positive (0 < A 1). If the detector D1 is positioned in the nominal imaging plane, the amplitude point-spread function is given by the Fourier transform of the pupil function P and is written p = a + i(ψ1 + ψ2 ) = a + ψ3 + iψ1 ,
(4.17)
with the lower-case symbols indicating the Fourier transforms of the corresponding functions in the pupil domain. The Fourier transform ψ1 of the even function Ψ1 is real and even. Because Ψ2 is odd, its transform ψ2 is purely imaginary and odd; for that reason it has been replaced by ψ3 = iψ2 with ψ3 real and odd to have only real functions in eq. (4.17). The measured intensity on D1 is proportional to pp ∗ , yielding SD1 = S11 + S12 = a 2 + ψ12 + ψ32 + 2aψ3 , (4.18) where we have split the detector signal into an even part S11 and an odd part S12 . The function ψ2 = −iψ3 = −iS12 /(2a) is purely imaginary. After applying a backward or inverse Fourier transform we get the result AΦ2 = {Ψ2 } for the odd part of the pupil function. Having obtained the odd part of the pupil function, eq. (4.18) allows to find the function ψ 1 using S11 , the even part of the detector signal, and retrieve AΦ1
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using ψ1 = S11 − a 2 − ψ32 and applying an inverse Fourier transform to this function. The problem in doing this is the uncertainty about the sign when taking the square root. Phase diversity is a useful tool to eliminate this problem. The defocusing in the second measurement branch corresponds to a modified pupil function P = A exp i(Φ + Φd ≈ A + i[Ψ1 + Ψ2 + Ψd ], (4.19) with Ψd = AΦd and Φd the quadratic phase function due to defocusing, again small in amplitude. Using the same arguments as above, the signal on the second defocused detector is now given by 2 SD2 = S21 + S22 = a 2 + ψ3 + (ψ1 + ψd )2 , (4.20) where ψd is the Fourier transform of the defocus function AΦd . Combining the two detector measurements we write SD2 = SD1 + 2ψ1 ψd + ψd2 ,
(4.21)
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with the solution ψ1 =
SD2 − SD1 − ψd2 . 2ψd
(4.22)
Finally, the value of AΦ1 is obtained after applying the backward Fourier transform to ψ1 and taking the real part of it. Possible divisions by zero in obtaining ψ2 above and ψ1 in eq. (4.22) can be avoided numerically by replacing a division by a by the expression a/(a 2 + ) where the value of decides about the allowed range of function values of the ψ1 - and ψ2 -functions. The basic approach of phase diversity as described above has been extended to problems of a more general nature (Fienup [1999]), for instance in the context of the Hubble space telescope recovery action, or the analysis of microscope objectives (Wesner, Heil and Sure [2002]). The ‘diversity’ in images to be treated can be extended to the wavelength domain (broadband imaging), to varying aberration settings and to changes in pupil geometry. The extension to broadband imaging is especially interesting when dealing with astronomical images of extremely weak objects (photon-starved regime); a restriction to quasi-monochromatic detection would give rise to a prohibitively low signal-to-noise ratio of the corresponding images.
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§ 5. Quality assessment using the Extended Nijboer–Zernike diffraction theory In this section we focus on the retrieval of the pupil function using an expansion in terms of Zernike polynomials. The coefficients of the expansion are obtained by means of an inversion or retrieval process that uses as input the data from various intensity patterns collected in the focal region of the optical system. To be able to treat these defocused intensity patterns, an analytic extension of the Nijboer– Zernike theory was recently devised by Janssen [2002] and Braat, Dirksen and Janssen [2002]. The out-of-focus extension of the classical Nijboer–Zernike diffraction theory allows for a fast and numerically stable forward propagation from the exit pupil to the focal ‘volume’ of the imaging system. This can be done independently for each of the amplitude and intensity patterns belonging to a typical Zernike aberration. In this way, Fast Fourier Transform methods can be avoided. Although the latter are reputedly fast with respect to the classical numerical evaluation of a diffraction integral, they still constitute the major numerical burden in an inversion process. Because of the discontinuous behavior of the pupil function, high sample rates are needed. A semi-analytic approach as made possible by using the Extended Nijboer–Zernike theory offers very substantial advantages
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in the speed with which an inversion can be carried out. Moreover, due to the analytic decomposition of the inversion problem, certain computational tasks are independent of the parameter values and can be carried out beforehand, making the effective inversion operation extremely fast. Both the amplitude and phase parts of the pupil function can be retrieved by using the complex Zernike coefficient expansion that was given in eq. (2.27) for the scalar diffraction case and in eq. (2.64) for the vector diffraction case. Due to space limitations, most considerations in this section are of a global nature; analytic and computational details can be found in the references. We first treat the retrieval process for systems with relatively low numerical apertures, using the scalar diffraction model. This is a good approximation up to aperture values of 0.60 in air, or equivalently, up to a full cone convergence angle of the focused pencil below 75◦ , independently of the refractive index of the image-space medium. An extension to higher aperture values first needs an adaptation of the defocus factor in the diffraction integral from the quadratic form to the exact expression at high numerical aperture. With this extension, one can still adequately describe imaging up to a numerical aperture value of, say, 0.85 if the imaging is carried out with natural light so that polarization effects in the focal region are not yet pronounced, see Dirksen, Braat, Janssen and Leeuwestein [2005] and van der Avoort, Braat, Dirksen and Janssen [2005]. But in the case of fully polarized illumination in the entrance pupil and imaging with high numerical aperture, one should use the full vector diffraction integral and the complete parametric description of the pupil function according to eq. (2.64). With this expansion one can detect possible birefringence introduced by the optical system, a non-negligible factor of quality deterioration in imaging systems, especially at high numerical aperture. The limits of applicability of the Extended Nijboer–Zernike diffraction model are given in practice by the restrictions on the illumination. The illumination of the entrance pupil should basically be done with a point source, or, in practice, a source that is much smaller than the diffraction unit λ/sE in object space, with sE the object-side numerical aperture of the optical system. Because of the limited throughput of such a tiny source, it is important to be able to extend the model to sources with a finite lateral extent. This can be done by including the analytically known non-uniform far-field pattern of a finite source in the effective pupil function of the optical system (van der Avoort, Braat, Dirksen and Janssen [2005]). In this way, the model is made applicable to practical experimental circumstances where exposure time or detector sensitivity are critical factors.
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5.1. Scalar retrieval process using the Extended Nijboer–Zernike theory The starting point for the retrieval process is the analytic expression for the complex amplitude in the focal region that was obtained in § 2.6, using the result for the diffraction integral of eq. (2.47). We describe the complex amplitude distribution on the exit-pupil sphere, in analogy with eq. (2.27), by m |m| P (ρ, θ ) = E(ρ, θ ) exp iΦ(ρ, θ) = (5.1) βn Rn (ρ) exp{imθ }, nm
βnm
where we have used the symbol for the complex expansion coefficients for the complete pupil function to distinguish them from the (real) expansion coefficients αnm that are related to the expansion of the phase function Φ only. In the case of sufficiently smooth functions E and Φ, it is possible to go back from the complex βnm to the separate amplitude and phase parts of the pupil function. Using the above expansion for the exit-pupil function in eq. (2.20), we readily obtain the complex amplitude in the focal region with the aid of eq. (2.47): Ef (r, φ, f ) ∝ (5.2) 2im βnm Vnm (r, f ) exp{imφ}, nm
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where the summation over the index m has to be carried out over both positive and negative values. The function Vnm (r, f ) and its series expansion have been given in eqs. (2.47)–(2.50). The intensity I in the focal region is obtained by taking the squared modulus of Ef . In the case of relatively small aberrations and transmission non-uniformity, the intensity is approximated by 2 2 I (r, φ, f ) ≈ 4 β00 V00 (r, f ) + 8β00
+∞ ∞
i−m βnm∗ V00 (r, f )Vnm∗ (r, f )
n=0 m=−∞
× exp[−imφ] ,
(5.3)
where the term with m = n = 0 has to be omitted in the summation. The coefficient β00 is chosen to be real and > 0; any non-zero phase of β00 can be accounted for by a phase offset in the coefficients βnm with n = m = 0, as the absolute phase of the wavefront in the exit pupil is of no concern when measuring the image intensity. The approximation in eq. (5.3) is justified when β00 is the dominant coefficient in the Zernike expansion of the pupil function, thus allowing the deletion of any terms quadratic in β that do not contain a β00 coefficient. In eq. (5.3) the leading term is the aberration-free intensity pattern, which for f = 0 is equal to the Airy disc intensity.
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Although the parametric representation of the pupil function with the aid of the β-coefficients is mathematically efficient and compact, the relationship between the amplitude and phase parts of the pupil function and the structure of the intensity distribution in the focal volume tends to become obscure. For that reason, we temporarily resort to the more common harmonic representation with possibly complex βc - and βs -coefficients for the cosine and sine terms and the following relationship with the general β-coefficients m m m m βn,c βn,c βn,s βn,s (5.4) −i , βn−m = +i . 2 2 2 2 When the pupil function shows only weak phase aberration, one uses the approximate representation m m Rnm (ρ) αn,c cos θ + αn,s sin θ . (5.5) P (ρ, θ ) ≈ 1 + iΦ(ρ, θ) = 1 + i
βnm =
nm
The prime on the summation sign indicates that the term with n = m = 0 should be excluded from the summation. The corresponding β-coefficients are given by m m m m αn,c αn,c αn,s αn,s + , βn−m = i − , 2 2 2 2 m m m = iαn,c , βn,s = iαn,s , β00 = 1,
βnm = i m βn,c
m αn,c
and
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(5.6)
m αn,s
and now specifically linked to the phase part with the real coefficients of the pupil function. For this latter, frequently occurring practical case, eq. (5.3) can be written out in detail to expose the separate φ- and (r, f )-dependencies in the intensity distribution in the focal region. These dependencies will be exploited later in the retrieval scheme. After some manipulation, eq. (5.3) is rewritten as m m I (r, φ, f ) ≈ Ψ00 + Ψnm (r, f ) βn,c cos mφ + βn,s sin mφ n,m
m m cos mφ + βn,s sin mφ , + Ψnm (r, f ) βn,c
(5.7)
where we have introduced the function Ψnm (r, f ) = γm i−m V00 (r, f )Vnm∗ (r, f )
(5.8)
with γ0 = 4 and γm = 8 for m = 1, 2, . . . . With the phase-only pupil function and the coefficients according to eq. (5.6) we find m m Ψnm (r, f ) αn,c cos mφ + αn,s sin mφ . I (r, φ, f ) ≈ Ψ00 + (5.9) n,m m and This expression represents a set of equations in the unknown coefficients αn,c m αn,s , of which a certain set is needed to represent in a sufficiently accurate way the
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phase part of the pupil function Φ. Various methods can be applied to solve for the unknown coefficients, like point-matching or a least-squares solution. Here, we present an approach that is based on the formation of a set of equations using inner products, see Dirksen, Braat, Janssen and Juffermans [2003], van der Avoort, Braat, Dirksen and Janssen [2005]. 5.1.1. Solution method using inner products To solve for the unknown aberration coefficients we first carry out a harmonic analysis of the intensity data in the focal volume by evaluating Ξcm (r, f )
1 = 2π
2π I (r, φ, f ) cos mφ dφ,
(5.10)
0
and the corresponding sine-transform Ξsm (r, f ). Carrying out the same harmonic analysis on the right-hand side of eq. (5.9) we obtain the equations 1 m m Ξcm (r, f ) = δmm Ψ00 (r, f ) + α Ψn (r, f ) , 2 n n,c 1 m m Ξsm (r, f ) = (5.11) αn,s Ψn (r, f ) , 2 n
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with δmm the Kronecker symbol. We define the (normalized) inner product of two functions Ψnm by Ψ1 (r, f ), Ψ2 (r, f ) =
1 2πR 2 F
R F
Ψ1 (r, f )Ψ2∗ (r, f )r dr df,
(5.12)
0 −F
where the integration extends over the focal volume up to certain limits within which reliable data have been made available by measurement. We now apply the inner-product operation to both sides of eq. (5.12) and obtain 1 m m m m αn,c Ψn (r, f ) , Ψnm , Ξc , Ψn = δmm Ψ00 , Ψnm + 2 n m m 1 Ξs , Ψn = (5.13) α m Ψnm (r, f ) , Ψnm . 2 n n,s By choosing a sufficiently large number of n values, a preferably square system of equations is constructed for each specific value of the harmonic component m. The inner products on the right-hand side of eq. (5.13) are calculated beforehand, a single time. The inner products on the left-hand side depend on the measurement
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Fig. 5.1. Measurement set-up for quality assessment of a high-resolution projection lens. The intermediate image of the projection lens is captured by a measurement objective that produces a magnified image on the image sensor. The picture inserted at the right is a typical defocused image.
data and have to be calculated anew for each system measurement. The solution of the system of equations is further facilitated by the fact that the matrix system of linear equations, built via the inner product method, is sparse with an almost perfect diagonal structure; this is because of the ‘almost’ orthogonality properties of the elementary functions Ψnm (r, f ) over the two-dimensional (r, f )-integration domain in the focal region.
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5.1.2. Experimental results of aberration retrieval in the scalar approximation In fig. 5.1 we have sketched an experimental set-up for measuring the lens quality of a high-resolution projection lens meant for optical lithography. The lens is illuminated via a tiny pinhole at the mask or reticle location using the standard illumination system of the lens. The pinhole image is captured in the high-numericalaperture image space with the aid of a measuring objective that produces a magnified image on a high-density image sensor. A typical image produced by such a sensor is shown in fig. 5.1 at the right. Through-focus images are successively captured by a defocusing of the measuring objective. A typical number of images to be treated in the total measurement series is 2N + 1, preferably symmetrically arranged around the optimum focal plane (see the upper row of fig. 5.2). The total excursion on either side of optimum focus is of the order of 3 to 4 focal depths (−2π f +2π). The total amount of data captured from the 2N + 1 through-focus images is used to construct an interpolated 3D picture of the intensity distribution in the focal volume. The data are submitted to the harmonic decomposition of eq. (5.10), yielding the cosine and sine components of the intensity variations over a circle of radius r at axial position f .
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Fig. 5.2. A through-focus collection of point-spread functions captured by the image sensor. In this case seven images, symmetrically arranged around best focus, have been recorded for aberration retrieval (upper row). The calculated result of the retrieval process is shown in the lower row of pictures where, for an easy comparison, the same pixel structure as during recording has been kept.
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Fig. 5.3. A lens quality map obtained with the aid of the Extended Nijboer–Zernike method in terms of measured Zernike coefficients of the most frequently occurring aberrations. The measurement data are from the upper row of fig. 5.2. The coefficients are expressed in units of milli-wavelength (mλ) of the light. The vertical scales of each order m has been adapted to the size of the aberration coefficients. Horizontal axis: Zernike polynomial index according to the Fringe Convention.
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The solution of the system of linear equations then yields the Zernike coefficients. Figure 5.3 shows the set of measured coefficients that result from the point-spread function analysis of the upper row of fig. 5.2. The Zernike aberration coefficients have been arranged according to their azimuthal order mθ , ranging from 0 to 5. The numbering along the horizontal axis follows the Fringe Convention ordering of Zernike polynomials (Mahajan [1998]). It is seen that the lowest-order aberrations (m 3) are dominating. The stability of the retrieval process has been tested by adding numerical noise to the captured images. The amplitude of the noise was 10% of the top intensity, measured in the best-focus region. An example of such a series of perturbed images and their originals is given in fig. 5.4. The retrieval process using the noisy images gave only minor deviations from the original result. Maximum coefficient deviation remained below 10 mλ, resulting in an rms wavefront deviation of less than 3 mλ. This value would be too large for reliable quality assessment of high-resolution projection lenses for microlithography, but, fortunately, the image detection conditions are favorable there. The noisy pictures on the right are typical for photon-starved detection conditions. These are encountered in astronomical observation where aberration detection with 10 mλ precision is generally more than sufficient. Up to now, the through-focus images were experimentally obtained using an intermediate magnifying objective with its possible own optical defects. Although a separate assessment of such an objective is feasible, it is preferable to have a direct measurement of the point-spread function in the high-numerical-aperture image space. One method to obtain the intensity profile in the high-NA focal region is to print the point-source images in a thin layer of photo-resist. To this goal, a point source is imitated by means of a tiny hole in the chromium layer of the object mask of the projection lens, with a diameter of typically 0.5λ/s0 , the diffraction unit at the mask side (see fig. 5.5). For a certain exposure, the printed resist image will be developed up to a certain contour of equal intensity in the point-source image. By varying the exposure between successive displaced images in the resist layer, a whole range of equalintensity contours of the point-source images is obtained. By analyzing the contour plot of each resist image for the complete series of defocused resist images at varying exposure, we can go back to the three-dimensional intensity distribution in the focal volume. Here we should note that the contour plots carry with them intrinsically both the effect of using a finite hole size and the effect of image blur due to finite exposure. Fortunately, the Extended Nijboer–Zernike formalism is flexible enough to account for such effects (see, for instance, Dirksen, Braat and Janssen [2006]). Below we show how we account for the effects of finite hole size,
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Fig. 5.4. A selection from a through-focus set of seventeen point-source images (top set) and the same pictures with noise artificially added to them (bottom set of pictures). The retrieval process applied to the noisy images shows only minor differences with respect to the original result.
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Fig. 5.5. A finite-size hole in the object mask chromium layer for a high-resolution projection lens (left-hand picture) and its developed resist image in the focal region, obtained with a certain exposure dose (electron microscope images). The geometric measurement of a large number of such defocused and over- or under-exposed images yields the required three-dimensional intensity distribution in the focal region.
while § 5.1.3 deals with blur effects. For a circular object hole, illuminated by a plane wave, the normalized far-field amplitude distribution is given by (scalar approximation), 2J1 (πD sin αf /λ) In (sin αf ) = (πD sin αf /λ)
π 2D2 π 2D2 2 2 sin αf ≈ exp − sin αf , ≈1− (5.14) 4λ2 4λ2
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with αf the polar far-field angle and J1 (x) the first-order Bessel function of the first kind (van der Avoort, Braat, Dirksen and Janssen [2005]). In eq. (5.14) we have also used, for sufficiently small values of the argument x, the approximation 2J1 (x)/x ≈ 1 − x 2 /4 ≈ exp{−x 2 /4}. The quadratically decreasing amplitude on the entrance pupil is accounted for in the diffraction integral of eq. (2.20) by allowing the defocus parameter f to become complex according to f˜ = f − ifd . If the numerical aperture at the object side is given by sE , the matching of the far-field amplitude function at the rim of the entrance pupil (ρ = 1) yields 2 2 2 π D sE 2 exp −if˜ρ 2 = exp −ifρ 2 exp − (5.15) ρ , 4λ2 with the appropriate value of the amplitude factor fd . Dirksen, Braat, Janssen, Juffermans and Leeuwestein [2003] have discussed calibration of the retrieval method using resist images. A lithographic projection objective (NA = 0.63, λ = 193 nm) was analyzed in its nominal setting and in some ‘detuned settings’. The detuning operation allows to introduce well-defined aberration increments by axial displacement, decentering or tilt of individual lens elements of the objective.
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Fig. 5.6. Measurement results for a microlithographic projection objective (NA = 0.63, λ = 193 nm). The original aberrational data (lowest orders) of the objective are represented by the black bars (absolute values). The set of white bars represents the measurement results for a detuned aberration setting 2 , respectively. of the objective with an increment of 50 mλ for the astigmatic coefficient α2,c
Figure 5.6 shows retrieved lens data for the nominal lens (black bars) and a particular detuned state of the objective. The objective has been detuned so that 2 for x-oriented astigmatism should have been increased by an the coefficient α2,c amount of 50 mλ. The retrieved aberration coefficients of the detuned objective are represented by the white bars, showing the approximate increase by 50 mλ for 2 -coefficient. The measurement spread α m the α2,c n,c/s in the other fixed aberration coefficients is of the order of ±10 mλ and has to be imputed to small drifts in the objective itself and to the accuracy of the retrieval method. The short-term repeatability of the retrieved coefficients is of the order of a few mλ. An analysis of the convergence and the robustness of the through-focus retrieval method using the Extended Nijboer–Zernike theory has been carried out by van der Avoort, Braat, Dirksen and Janssen [2005]. In this analysis, an approximated scalar retrieval scheme at high-numerical-aperture values was also given. It was found, as a rule of thumb, that the values of the wavefront coefficients |αnm | should
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Table 5.1 Simulation of detection noise (zero mean) and its influence on the values of some lower-order retrieved aberration coefficients Indices
Input values
n
m
2 1 3 2
0 1 1 2
0.3000 0.1000 0.4000 0.2000
Retrieved values SNR = ∞
SNR = 100
SNR = 10
0.3000 0.1000 0.4000 0.2000
0.2996 0.1003 0.4002 0.1991
0.2989 0.1027 0.4082 0.2087
not collectively give rise to a variance of the phase aberration larger than 0.4, equivalent to an rms wavefront aberration of less than λ/10. This corresponds to a Strehl ratio of 0.60, well outside the ‘just’-diffraction-limited regime. To achieve good retrieval results for these larger aberration coefficients, it is necessary to extend the linearized retrieval scheme with a predictor–corrector iteration method (van der Avoort, Braat, Dirksen and Janssen [2005]). We will discuss this iteration method in the next section on the vector diffraction case. Van der Avoort, Braat, Dirksen and Janssen [2005] also simulated the addition of zero-average detection noise in the captured images, and compared the resulting coefficients with the original values for defocus, wavefront tilt, coma and astigmatism. These data are reproduced in table 5.1; it can be concluded that a signal-to-noise ratio as small as 10 is still capable of yielding retrieval errors not larger than 5%. It has been observed that the detection noise should have zero mean. If this not the case, serious offsets in the retrieved coefficients can occur.
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5.1.3. Refinements of the scalar retrieval model The influence of the finite object pinhole size is an example of how to extend the basic imaging and retrieval model to real experimental conditions. In this subsection we present some refinements of the basic model that allow a broader range of application of the retrieval method. With these refinements one can treat images affected by focus or in-plane noise during acquisition, by blurring due to chromatic aberration of an imaging system, or by lateral blurring due to image sensor digitization or photo-resist diffusion effects. These are typical ‘blurring’ effects that arise in the treatment of point-source images produced by high-resolution projection lenses. For these lenses, the quality assessment has to be carried out with utmost accuracy and such a refined model is mandatory. As a side effect, such a retrieval process also allows to establish the magnitude of the parameters that cause the blurring, for instance the photo-resist diffusion parameters in the case of image capture via developed photo-resist images in a high-NA space.
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The exposure in the focal region, affected by in-plane or axial position noise, depends on the statistics of the movements. Following Dirksen, Braat and Janssen [2006], it is customary to adopt Gaussian statistics for the movements, with a probability density function given by 1 2 (t) x (t) = √ (5.16) exp − x 2 , 2σx 2πσx with σx the standard deviation of the movement in the x-direction. Comparable expressions hold for the statistical excursions in the y- and f -direction with standard deviations of σy and σf , respectively. The incremental exposure in a resist layer or the time-integrated intensity recorded by a sufficiently small detector pixel is given by dEr (x, y, f ) = I x − x (t), y − y (t), f − f (t) dt. (5.17) In the case of a sufficiently small standard deviation value σr with respect to the full width at half maximum (FWHM) of the lateral intensity profile, and a sufficiently small value of σf of the axial movements with respect to, for instance, the focal depth of the imaging system, a Taylor expansion of the intensity distribution can be applied. The total exposure, integrated over sufficiently long time, is then approximated, at least up to second order, by +∞
Er (r, f ) ≈
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I (r, f ) − x (t) −∞
∂I ∂I ∂I − y (t) − f (t) ∂x ∂y ∂f
1 ∂ 2I 1 ∂ 2I 1 ∂ 2I + 2x (t) 2 + 2y (t) 2 + 2f (t) 2 2 2 2 ∂x ∂y ∂f 2 2 ∂ I ∂ I + x (t) y (t) + x (t) f (t) ∂x∂y ∂x∂f ∂ 2I + · · · dt. + y (t) f (t) ∂y∂f
(5.18)
In what follows, the expressions are simplified by considering the case that the intensity distribution possesses radial symmetry and that equally large statistical xand y-movements are present (isotropic case) each with a standard deviation σm . This limitation to the radially symmetric case is adequate in most practical cases. In the radially symmetric case, one can easily introduce a further effect that is very important in resist imaging, that is, the chemical diffusion of the exposure profile in the resist layer. This chemical blurring of the exposure profile occurs partly during the exposure and development process (latent image formation) but it is most pronounced during the so-called post-exposure bake. It can be adequately
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incorporated in a total radial standard deviation given by σr2 = σm2 + 2Dtb ,
(5.19)
where 2Dtb is the Fickian diffusion length, the product of the acid diffusion coefficient D and the baking time tb (Lavery, Vogt, Prabhu, Lin and Wu [2006]). The integration of eq. (5.18), formally over the total time of resist exposure up to the end of the post-exposure bake, gives rise to
tE σr2 ∂ 2 1 ∂ + I (r, f ) Er (r, f ) = E0 (r, f ) + 2 r ∂r ∂r 2 +
tE σf2 ∂ 2 I (r, f ) , 2 ∂f 2
(5.20)
all other terms yielding zero in the radially symmetric case due to statistical averaging. The leading term E0 (r, f ) = I (r, f )tE is called the static exposure term, with the exposure time tE sufficiently long for statistical averaging to effectively have taken place. For the evaluation of the first and second derivative of the function I (r, f ) it is sufficient to use the approximated expression of eq. (5.3) with m = 0 in the summation term. Dirksen, Braat and Janssen [2006] have given the derivatives of the 0∗ (r, f ). To cover still larger blurring effects, an expansion products V00 (r, f )V2n term of the fourth order has been included in eq. (5.18). This fourth-order Taylor term, in general contributing ∂2 2 tE σr4 ∂ 2 + I (r, f ) (5.21) 8 ∂x 2 ∂y 2
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to the exposure of eq. (5.20), is then restricted in this reference to the leading term of I (r, f ), the aberration-free term |V00 (r, f )|2 . Figure 5.7 presents three physical causes for blurring of the point-spread function, all three for a radially symmetric case. The contour plots apply to a pointspread function produced by an objective with an NA = 0.63, operating at a wavelength of 193 nm (deep UV illumination). The scales are in absolute units, although different along the radial and axial axes. In the left graph, an (r, f ) contour plot (dashed) is given for a point-spread function suffering from spherical aberration. The Zernike coefficient α40 of the wavefront aberration equals 75 mλ, giving rise to an rms wavefront aberration of 33 mλ, well within the diffraction limit of 71 mλ. The solid curves apply to the aberration-free case. The typical asymmetric behavior of the point-spread function with respect to the plane f = 0 is well visible in the figure. A blurring of comparable magnitude, but now with focus symmetry, is presented in the middle graph (dashed contours). In this case,
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Fig. 5.7. Contour plots for typical blur effects (dashed lines) in a point-spread function (radial symmetry). The ideal point-spread function contours have been represented with solid lines. The left-hand graph has been obtained by introducing spherical aberration in the imaging system (α40 = 75 mλ). In the central graph, a radial blur (chemical diffusion in the resist pattern) has been applied, σr = 30 nm. The graph on the right presents the effect of an axial blur due to focus noise (σf = 150 nm).
the exposure profile in the recording resist layer has introduced a radial diffusion of the exposure pattern of 30 nm rms value (σr ); the axial blur is very small. Finally, in the rightmost figure, the point-spread function was subjected to a focus blurring with an rms value of 150 nm, to be compared with the approximate focal depth of 240 nm. In the case of focus noise, the symmetry around focus is also preserved, the blurring being mainly in the axial direction. Further refinements in the scalar retrieval method are possible to account for other experimental conditions. Instead of mechanical focus noise, an axial blur can also be produced by axial chromatism of the imaging system. As was mentioned earlier, in some applications broadband light has to be used. This leads to a blurring of the incoherently superimposed monochromatic point-spread functions by the scaling of the diffraction unit λ/NA.
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5.2. Pupil function retrieval for high-NA imaging systems The foregoing section has extensively discussed the retrieval mechanism for the complex pupil function using the scalar imaging model. In this section we replace the scalar through-focus intensity distribution by its vector equivalent and then apply this new expression to basically the same retrieval procedure. The complications that arise in the vector-imaging model stem from the influence of the polarization state in the exit pupil on the intensity distribution in the focal region, see § 3. In most cases, the state of polarization in the exit pupil is uniquely determined by the state of polarization in the entrance pupil. This direct relationship disappears when the optical system introduces birefringence. The vector-imaging
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model is capable to keep track of this birefringence and, using more than one set of through-focus images with different states of polarization in the entrance pupil, it is even possible to also retrieve the birefringence properties of the system. We first present a linearized expression for the intensity distribution in the focal volume in the absence of birefringence and use this expression in a vector diffraction retrieval scheme. The extension to retrieval in the presence of birefringence is addressed further on in this section but will not be worked out in full detail. 5.2.1. Approximated linearized intensity distribution in the focal volume In this section we use the expressions for the Cartesian electric field vectors in the focal region of a high-NA imaging system in the case of a point-source object. The field in the entrance pupil is described by a coherent superposition of two orthogonally polarized linear states of polarization according to E = (px , py , 0)A0 , where px and py generally are complex numbers and A0 is the constant amplitude factor. The influence of the non-perfect high-numerical-aperture imaging system is the introduction of wavefront deformation and transmission changes, on top of an intrinsic amplitude distribution on the exit-pupil sphere that is different for each Cartesian field component and that has been described by Wolf [1959] and Richards and Wolf [1959]. The field in the focal region is obtained by calculating three basic integrals. In what follows, the behavior of an aberrated optical system is described by an expansion of the complex lens-transmission function in terms of Zernike polynomials with complex coefficients that are supposed to be identical for each polarization state. We suppose that each vector component of the electric field in the exit pupil has to be multiplied by the complex pupil-transmission m = β m = β m yielding a function of eq. (2.63) with the Zernike coefficients βn,x n,y n pupil function βnm Rn|m| (ρ) exp (imθ ). P (ρ, θ ) = (5.22)
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n,m
As usual, we suppose that β00 is the leading term; this will be the case for optical systems that are close to the diffraction limit. Using the Extended Nijboer–Zernike theory, the complex field vectors in the focal region are now calculated using m , V m , V m , V m , V m , where eq. (2.64) with the modified integrals Vn,−2 n,−1 n,0 n,+1 n,+2 the indices (n, m) pertain to the Zernike polynomial expansion on the exit-pupil function. These integrals can be evaluated using the analytic schemes given in Appendix D. The general expression for the energy density has been given in eq. (3.8). An important reduction in complexity is obtained for an isotropic imaging system
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with the pupil function defined by eq. (5.22); the resulting expression is eq. (3.12). The expression for the energy density that will be used in the retrieval scheme allows a further approximation that was also used in the scalar case by exploiting the fact that β00 is the leading term in the pupil function expansion. The expression for a general term Gk,l of eq. (3.12) is then given by Gk,l (β, β) =
β00 exp i(k − l)φ 2 μ∗ × (2 − δνμ ) βνμ∗ Ψν;k,l (r, f ) exp(−iμφ) ν μ μ μ + βν Ψν;l,k (r, f ) exp(+iμφ) .
(5.23)
In this expression, μ assumes both positive and negative values and δνμ stands for the Kronecker symbol. We also introduced the shorthand notation μ
μ
0∗ (r, f )Vν,l (r, f ). Ψν;k,l (r, f ) = (+i)μ V0,k
(5.24)
5.2.2. Aberration retrieval scheme for the vector diffraction case Like in the scalar case, the retrieval scheme for obtaining the complex pupil function is based on a Fourier analysis of the measured and the analytically proposed intensity data. In contrast with the analysis of § 5.1 where we dealt with separate expressions for the cosine and sine harmonic components, we will use from now on the complex exponentials exp(imθ ) and the complex β-coefficients with mindices running from −∞ to +∞. The harmonic dependence in the focal region is represented by complex exponentials exp(imφ). The Fourier decomposition is carried out with respect to the harmonics in the through-focus intensity distribution. To this goal we evaluate
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m (r, f ) Ψan
1 = 2π
+π
! we (r, φ, f ) exp(imφ) dφ,
(5.25)
−π
with the subscript “an” referring to the analytically calculated intensity distribution. A comparable operation is performed on the measured intensity data, yielding functions Ψ m (r, f ). m in comThe analytic energy density distribution (3.12) is used to calculate Ψan bination with the linearized Gkl -functions, and after some lengthy manipulation we obtain (see Appendix A of Braat, Dirksen, Janssen, van de Nes and van Haver [2005] and van Haver, Braat, Dirksen and Janssen [2006])
m∗ β 0 m∗ m m∗ m∗ (r, f ) ≈ 0 + s02 Ψν;1,1 + Ψν;−1,−1 βν (2 − δν,m ) Ψν;0,0 Ψan 2 ν
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m∗ s02 m∗ m∗ ∗ m∗ + − Ψν;−1,−1 Ψν;2,2 + Ψν;−2,−2 − 2 (px py ) Ψν;1,1 2 s02 m∗ m∗ Ψν;2,2 − Ψν;−2,−2 + 2
−m −m −m + βν−m (2 − δν,m ) Ψν;0,0 + s02 Ψν;1,1 + Ψν;−1,−1
−m s02 −m −m −m ∗ + Ψν;−2,−2 − 2 (px py ) Ψν;1,1 − Ψν;−1,−1 Ψ + 2 ν;2,2 s 2 −m −m − Ψν;−2,−2 + 0 Ψν;2,2 2 + βν(m−2)∗ |px |2 − |py |2 + 2i (px py∗ ) (2 − δν,m−2 ) s02 (m−2)∗ (m−2)∗ (m−2)∗ Ψν;−2,0 + Ψν;0,2 − 2Ψν;−1,+1 2 + βν(m+2)∗ |px |2 − |py |2 − 2i (px py∗ ) (2 − δν,m+2 ) ×
s02 (m+2)∗ (m+2)∗ (m+2)∗ Ψν;2,0 + Ψν;0,−2 − 2Ψν;+1,−1 2 + βν(−m−2) |px |2 − |py |2 − 2i (px py∗ ) (2 − δν,−m−2 ) ×
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s02 (−m−2) (−m−2) (−m−2) Ψν;−2,0 + Ψν;0,2 − 2Ψν;−1,+1 2 + βν(−m+2) |px |2 − |py |2 + 2i (px py∗ ) (2 − δν,−m+2 )
s02 (−m+2) (−m+2) (−m+2) + Ψν;0,−2 − 2Ψν;+1,−1 . × (5.26) Ψ 2 ν;2,0 ×
We recall that the expression above is not exact but applies to the linearized approximation of the Gk,l -functions. The equations to be solved now read m Ψ m (r, f ) ≈ Ψan (r, f ),
(5.27)
to be solved for each separate m-value. In practice, we merge these equations into one large system of linearized equations. Like in the scalar case, the practical solution procedure consists of taking inner products on both sides with the functions m (r, f ) and to solve this new system of equations. This more global method Ψn;k,l replaces the direct approach that would try to find the solution with the optimum match for each point in the (r, f )-cross-section of the focal volume. The inner products are defined by eq. (5.12), in line with the scalar retrieval method. In the m (r, f ) and Ψ m (r, f ) are now vector retrieval case, the functions Ψ m (r, f ), Ψan n;k,l all complex; the integration limits are determined by the axial and lateral range of the collected intensity data.
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5.3. Retrieval examples for high-NA systems In this subsection we present an example that shows the inadequacy of a retrieval method based on scalar diffraction theory when applied to a high-NA imaging system. Secondly, we show the ranges of aberration and transmission defects that can be handled by the linearized system of equations based on eq. (5.27). Subsequently, we evaluate the error that is introduced in the retrieval process when system parameters, such as the numerical aperture and incident polarization, are not exactly known. We then present a simulated retrieval example performed in such a way that it closely resembles the treatment of experimental data. Experimental intensity data are analyzed for a high numerical aperture projection lens illuminated by a source with a partial degree of polarization equal to zero. 5.3.1. Retrieval of a high-NA system assuming scalar conditions To underline the necessity of applying the full vectorial case when assessing imaging systems with a high numerical aperture, we analyze the following retrieval operation on simulated data. A through-focus intensity distribution is constructed from five axially displaced through-focus images with focus parameter values f = −2, −1, 0, 1, 2, in dimensionless focal units according to the definition in eq. (2.31). The intensity distribution is calculated for an aberration-free optical system with a numerical aperture of 0.95, whose entrance pupil is illuminated by a linearly polarized wave with the plane of polarization parallel to the x-axis. We apply the forward-calculation scheme using the expressions for the electric field components of eq. (2.65). The high-NA data set acquired through this operation is then analyzed using the scalar version of the retrieval scheme. This scheme neglects the vector character of the optical field but takes into account the exact phase departure of a defocused wave at high numerical aperture as is done in eq. (2.33). This gives rise to the set of retrieved β-coefficients presented in the rightmost column of table 5.2. The fit imposed by this limited set of β-coefficients is remarkably good as can be seen from fig. 5.8; the maximum intensity deviations are of the order of 5%. Nevertheless, on comparing the β-coefficients obtained through the scalar retrieval operation with the actual β’s (aberration-free case) used for the simulation, we observe a very poor correspondence, see fig. 5.9. Although at first sight the scalar model with high-NA focus adaptation is able to fit a high-NA intensity distribution, the retrieved β-coefficients do not have a physical relevance. We observe that the β20 -, β22 - and β2−2 -coefficients are all real and thus affect the transmission part of the pupil function. This amplitude modulation, composed of a circularly symmetric part and a contribution with a cos 2θ dependence, is necessary to describe the strongly elliptical energy density distribution
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Table 5.2 Comparison between the initial and the retrieved β-coefficients when retrieving a simulated aberration-free intensity distribution (NA = 0.95, x-polarization) using the scalar retrieval scheme Input β00 β20 β2−2 β22
Retrieved
1.000
1.038
0.000
−0.593
0.000
−0.466
0.000
−0.466
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Fig. 5.8. In-focus (f = 0) intensities as a function of the positive x- and y-coordinate (upper left graph, dashed and solid line, respectively). Right upper graph: grayscale plot in the (x, y)-plane of the difference between the aberration-free intensity distribution (NA = 0.95, x-polarization) and its retrieved version based on a scalar model. In the second row the same information is presented for an out-of-focus position with f = −2.
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Fig. 5.9. The strongly deformed pupil function resulting from a scalar retrieval method applied to an imaging system with NA = 0.95. Left-hand graph: false-color plot of the modulus of the retrieved pupil function. Right-hand graph: phase of the pupil function (black region with zero phase, white region with a phase of π ). The figure shows that the outer part of the pupil function in the x-direction assumes negative values.
in the focal region at high values of the numerical aperture. Note that the strong amplitude deformation even includes a region in the pupil where the amplitude is negative, see fig. 5.9. This effectively means that this region is subjected to a phase shift of π. This example with simulated data shows that the β-coefficients found by applying scalar theory to intensity distributions governed by the vectorial model do not have direct physical relevance and no longer give a correct description of the system under consideration. Physically relevant β-coefficients for a high-NA optical system can only be expected if the full vectorial case is applied at the retrieval stage. The high-NA retrieval formalism was introduced above and it will now be further examined using both simulated data and a set of experimentally obtained data.
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5.3.2. Accuracy of the high-numerical-aperture retrieval scheme When applying the retrieval scheme for the high-NA case, the retrieved βcoefficients are generally not exact because of our approximated version of the m -functions according to eq. (5.26). An exception is the aberration-free system Ψan where the retrieved β-coefficients are exact, giving the trivial solution β00 = 1 and all other βnm equal to zero. When aberrations are present in the system, described by additional β-coefficients with n or m = 0, the retrieved βnm -coefficients will show residual errors originating from the linearization applied in eq. (5.26). An impression of the magnitude of the errors in the retrieved β-coefficients is obtained by performing the following simulations. Starting from a perfect
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Fig. 5.10. Plot of the maximum error present in the retrieved β -coefficients as a function of the magnitude of the β22 -coefficient that served as input. The imaging system was free of any other defect; incident light x-polarized, NA = 0.95.
(aberration-free) system, represented by a single β-coefficient, β00 = 1, we introduce one additional non-zero β-coefficient, for instance β22 = 0. Next, this pair of β-coefficients is used to simulate a through-focus intensity distribution that serves as input for the retrieval operation. The retrieval process will now generate an estimate for this pair of β-coefficients describing the system. These estimates, which we shall denote as β , are not exact and include a certain error. The above process having been repeated at ever increasing size of β22 leads to the error behavior in the set of retrieved β shown in fig. 5.10. In the figure, the maximum error in the set of retrieved β-coefficients is plotted as a function of the magnitude of the β22 -coefficient that was used as an input to the simulation. Figure 5.10 indicates a quadratic relation between the maximum observed error in the retrieved β -coefficients and the input value β22 . This was to be expected, as we omit ex actly the cross-terms depending on βnm βnm in the linearized expression for Gk,l , see eq. (5.23). The linearized retrieval method is very accurate for well-corrected optical systems (β 10−1 ). On the other hand, if we have a system influenced by larger aberrations (β of the order of 1), the errors present in the retrieved βcoefficients will be of the same order, which means that the quality of the retrieval method is poor. Fortunately, iterative schemes can be applied to the solution of linearized systems of equations, updating the system at each iteration step with an estimate of the ignored quadratic term. Such a correction scheme can be applied to the retrieval method and it produces strongly improved retrieval results
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Fig. 5.11. A plot of the residual errors in the retrieved β-values versus the number of iterative steps taken in the predictor–corrector procedure. The colors pertain to various aberration terms that were either initially present or that were erroneously detected at the start of the iterative retrieval process. The end value is determined by machine precision. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this chapter.)
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to the extent that the retrieved β-coefficients converge to their correct values. The correction scheme of the “predictor–corrector” type is discussed in more detail below. 5.3.3. Predictor–corrector method for improved system assessment A wider range of aberrated pupil functions can be handled once we use the so-called predictor–corrector extension when solving the linearized equations. It leads to a better assessment of the quality of imaging systems over a range that substantially exceeds the classical diffraction limit (Strehl ratio 0.80). Although the final quality of high-resolution imaging systems should be well within this limit, it is well known that during the quality-tuning stage in the manufacturing process these systems can show appreciable deviations from the diffraction limit. The medium-to-large aberrations that can be expected during this stage should be accessible for an assessment method of more general use. A so-called predictor– corrector iteration scheme for improving scalar retrieval results has already been proposed and thoroughly tested, see van der Avoort, Braat, Dirksen and Janssen [2005]. An equivalent iterative procedure has been devised for the high-NA case. It is discussed in detail in Appendix E. Figure 5.11 presents the results of high-NA retrieval using the predictor– corrector procedure. The optical system suffers from astigmatism in the x-
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Table 5.3 Set of β-coefficients used for the description of the field on the exit-pupil sphere of the imaging system under consideration (NA = 0.95) Input values β-coefficients β00
β20
β3−1
β31
β2−2
β22
1.0000
0.0200i
0.0500i
0.0500i
0.0100i
0.0100i
direction (β00 = 1, β22 = β2−2 = 0.5i). The numerical aperture is 0.95, the incident light is polarized in the x-direction. One observes a steady decrease of the error in the retrieval result with the number of iterations. For the synthetic data used in this example, the error eventually attains the typical machine precision of the calculation software, equal to ≈10−15 in the case of fig. 5.11. Note that accuracies customary for practical applications, typically of the order of a few times 10−3 , are reached in less than 10 cycles. For real experimental data, when numerous inaccuracies and perturbations such as noise are inevitable, the attainable precision will be limited. Still, in that case, the residual errors in the retrieved β-values obtained through the predictor– corrector procedure will be small and of the same (or lower) order of magnitude as the noise present in the data. We have observed such a performance for aberration values that can be as large as twice the diffraction limit, up to an rms wavefront deviation of 0.15λ. The influence of noise will be discussed later in this section; we now first investigate the effect on the retrieval quality of systematic errors in the parameters describing the optical imaging system.
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5.3.4. Systematic errors and their influence on the quality assessment One of the possible complications encountered when going from simulated data to experimental data obtained from a real optical system, is that certain system parameters are not exactly known. Investigations on the effect of these uncertainties on the retrieval quality have been carried out by van Haver, Braat, Dirksen and Janssen [2006] for the assumed cases of incorrect values of the numerical aperture and incorrect values for the azimuth of the linear state of polarization. To this end, a through-focus intensity distribution according to the β-values of table 5.3 was synthesized. Next, the system defined in table 5.3 was subjected to the pupil-function retrieval procedure based on the vector diffraction model. A range of values for the numerical aperture was assumed from very low to the extreme value of unity. The error in the assessment of the imaging system was monitored by recording several β-coefficients; the results are presented in fig. 5.12. One observes that the residual error in all β-coefficients is minimal for the cor-
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Fig. 5.12. The error present in some relevant β-coefficients when retrieving the system defined in table 5.3 while assuming different values for the numerical aperture NA. The colors point to typical aberration coefficients; the residual errors for all aberrations are smallest at the correct value of the numerical aperture s0 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this chapter.)
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rect value of the numerical aperture. This feature not only enables a tuning of the retrieval process to the exact value of the numerical aperture, but also suggests a procedure to accurately determine the numerical aperture of an unknown system. A comparable simulation was presented for an erroneous input of the azimuth of the linear state of polarization in the entrance pupil of the optical system; the numerical aperture was fixed in this case at a value of 0.95. The synthetic data representing the through-focus intensity distribution are obtained by assuming an exact linearly polarized illumination of the entrance pupil along the x-axis. The linear state of polarization is then varied by adapting the coefficients px and py while having px and py both real and maintaining (|px |2 +|py |2 ) = 1 for normalization purposes. This leads to a rotation of the plane of polarization towards a certain finite angle with respect to the x-axis. This presumed linear state of polarization, different from the actual state of the system, leads to an error in the retrieved set of β-coefficients. The results of these simulations are given in fig. 5.13, where the error in the set of retrieved βnm -coefficients is plotted versus the angle between the supposed and the actual orientation of the plane of polarization. From fig. 5.13 one observes that it is also important to have accurate knowledge of the polarization state of the system under consideration in order to obtain good retrieval quality. An angular deviation of one degree is a practical limit when it is needed to assess very high-quality imaging systems with sufficient accuracy.
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Fig. 5.13. The error present in some selected retrieved β-coefficients (solid lines) as a function of the offset angle between the chosen plane of polarization at the construction stage of the synthetic intensity data (x-polarization) and the angle of the polarization plane at the retrieval stage. The dotted 2 line is a plot of the total rms error of the retrieved β-coefficients, defined by ( nm | βnm |2 )1/2 , with βnm the error in each coefficient.
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The above example illustrates an interesting relation between the possible inaccuracy in the polarization state and the error in the retrieved β-coefficient. For an optical system of which the polarization state is approximately known, an equivalent operation as used for the generation of fig. 5.13 can be put into place to determine the polarization state with great accuracy. For the general case that one has no knowledge whatsoever about the state of polarization, the above procedure is no longer applicable. This is caused by the fact that the predictor–corrector procedure is not applicable once the deviations from the real state of polarization are very large. In this case, a basic retrieval operation can be used to obtain the approximate polarization state, after which the predictor–corrector method can be applied to determine the state of polarization with great accuracy. 5.3.5. Quality assessment of a high-NA optical system in the presence of noise Like in the scalar approach, a good estimate of the influence of intensity noise on the retrieved β-coefficients is required. Zero-mean-intensity noise can be added to a synthetic data set to perform a numerical noise experiment. Van Haver, Braat, Dirksen and Janssen [2006] constructed a through-focus intensity distri-
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Table 5.4 Set of β-coefficients needed to describe the field in the exit pupil of the optical system (NA = 0.95) subjected to a numerical test with noise added to the through-focus intensity distribution SNR = ∞
Input coeff.
Lin. retr. β00
β1−1 β11 β3−1 β31 β20 β2−2 β22 β3−3 β33
SNR = 10 Pr.-Corr.
Lin. retr.
Pr.-Corr.
1.0
1.1294
1.0000
1.1291
1.0004
0.0 +i0.5
0.1002 +i0.4278
0.0000 +i0.5000
0.0934 +i0.4277
0.0050 +i0.4933
0.0 +i0.5
0.0997 +i0.4576
0.0000 +i0.5000
0.0973 +i0.4598
−0.0124 +i0.5068
0.5
0.4545
0.5000
0.4401
0.4688
+i0.0
+i0.0028
+i0.0000
+i0.0144
+i0.0099
−0.5
−0.4330
−0.5000
−0.4339
−0.5041
+i0.0
−i0.0008
+i0.0000
−i0.0174
−i0.0385
0.0 +i0.0
0.0382 +i0.0000
0.0000 +i0.0000
0.0220 −i0.0176
−0.0264 −i0.0270
0.0 +i0.5
0.1138 +i0.5813
0.0000 +i0.5000
0.1276 +i0.5306
0.0112 +i0.4327
0.0 +i0.5
0.1113 +i0.3039
0.0000 +i0.5000
0.1122 +i0.3095
0.0060 +i0.5137
−0.5
−0.3269
+i0.0
−i0.0843
0.5 +i0.0
0.5534 +i0.0869
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−0.5000
−0.3821
−0.5468
+i0.0000
−i0.0641
+i0.0253
0.5000 +i0.0000
0.5631 +i0.0535
0.5139 −i0.0296
The input data were a noise-free intensity distribution and a distribution with a highest signal-to-noise ratio of 10 at best-focus; from the pictures in the second row of fig. 5.14 it is clear that the SNR value is much lower for the out-of-focus intensity distributions. Both distributions are formed from synthetic data obtained by a forward calculation. The retrieved coefficients have been obtained by linearized retrieval and by repeated application of the predictor–corrector scheme.
bution corresponding to the β-coefficients listed in the first two columns of table 5.4. The resulting through-focus intensity distribution is sampled in 2N + 1 axial planes, several of which are shown in the top row of fig. 5.14. Noise is added, yielding the corresponding pictures in the second row. The noisy distribution (SNR = 10 with respect to the highest measured intensity in the best focal plane) is used for a retrieval operation in which all other parameters are assumed to be known exactly. The retrieval results are shown in the third and fourth rows. The pictures in the third row are obtained after a single retrieval step, while the fourth row gives the results after convergence using the predictor– corrector method. Although visual inspection hardly shows any difference be-
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EBL Fig. 5.14. Through-focus intensity distributions pertaining to the system defined by the β-coefficients of table 5.4 (NA = 0.95, x-polarization). The upper row is the synthesized actual distribution, the second row is the modified one after adding noise with a SNR of 10. The third row gives the distribution defined by the first β-estimates and the last row is the distribution resulting after the predictor–corrector procedure. Note that all images have been scaled according to their maximum value in order to show maximum detail. The SNR-value of 10 applies to the highest intensity of the in-focus distribution.
tween the original and the retrieved intensity distributions of fig. 5.14, the differences are made clear by inspection of table 5.4. With the chosen values of the β-coefficients, a predictor–corrector step is essential for obtaining the correct input values. It is striking that the presence of the noise does not change the first estimate after linear retrieval. It is the convergence towards the final correct values with the predictor–corrector method that is obstructed by the noise. With the signal-to-noise ratio of 10, the final coefficients show deviations of the order of 10 to 15%.
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Fig. 5.15. Radial cross-section of the normalized in-focus intensity in the case of a high-numerical-aperture aberration-free optical system (s0 = 0.85) that is illuminated with effectively unpolarized light. The radial coordinate is v = 2π r with r expressed in the diffraction unit λ/s0 .
5.3.6. Quality assessment of a high-NA system using experimental data In this subsection we discuss the quality assessment of a lithographic projection lens with a numerical aperture of 0.85 (Dirksen, Braat, Janssen and Leeuwestein [2005]). The lithographic lens is illuminated by a special illuminator using an excimer laser emitting radiation at λ = 193 nm. The radiation of the multimode laser source is effectively unpolarized after integration over the exposure time that is needed to create a developable latent image in the photo-resist layer. The point-spread function created by such an effectively unpolarized focused beam is given by eq. (3.20). In the aberration-free case, it possesses circular symmetry with respect to the optical axis, and a cross-section of the theoretical radial crosssection in the optimum focal plane is shown in fig. 5.15. The salient features are the broadening of the central width of the profile with respect to the scalar prediction by approximately 10% and the absence of dark rings (see graph on the right). The broadening effect for unpolarized illumination is predicted by the shape of the orthogonal intensity cross-sections in fig. 2.14 for linearly polarized illumination. The broadening effect of the unpolarized point-spread function is likely to reduce the resolution capability of the imaging system although it does not necessarily affect the spatial bandwidth of the imaging system. The assessment of the imaging system with an object point source is carried out in reality with a circular hole in a chromium mask with finite diameter. In the retrieval procedure based on the Extended Nijboer–Zernike diffraction theory, the effect of the finite source diameter is accounted for by a complex defocus
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parameter, see eq. (5.15). Including such a finite hole size in the point-spread calculations gives rise to the through-focus cross-sections of fig. 5.16. In the top graph, a comparison with the theoretical point-source response would show only minor differences because of the 12.5 nm geometrical size at the image side. The bottom graph shows an appreciable broadening, the scaled source size becoming comparable to the halfwidth of the theoretical point-spread function. The retrieval procedure using the inner-product method exploits the (nonperfect) orthogonality of the through-focus intensity patterns corresponding to a specific Zernike coefficient of the pupil function. As examples of two such patterns, we first produce in fig. 5.17 the aberration-free pattern that is basically associated with the coefficient |β00 |2 |V00 |2 with β00 = 1 (top graph). The bottom graph is the through-focus pattern of a system that suffers from lowest-order spherical aberration and is, in a first order, proportional to the sum of a contribution associated with (β00 )2 and contributions stemming from the product of β00 β40∗ and the corresponding product of V -functions. The bottom graph has been calculated with a spherical aberration coefficient of β40 ≈ 0.25i or a value Z9 = 40 mλ (Fringe Convention). In the case of fig. 5.17, the partial orthogonality between the two patterns stems from the asymmetry through focus of the point-spread function with spherical aberration. As was pointed out in § 5.1.3, the detected point-spread function is blurred by various effects like mechanical movements in the axial direction or in the plane of the image, or by chemical diffusion of the latent resist image, mainly during the post-exposure bake of the developed resist layer. These effects can be mapped on a radial and axial blurring of the image, the magnitude of which is represented by the standard deviations σr and σf . The bottom graph in fig. 5.18 shows that a radial diffusion with a typical value of σr = 40 nm has a very large impact on the radial scale of the detected point-spread function. With the aberration-free full width at half maximum of the point-spread function in the radial direction given by 65 nm, we observe an increase to 85 nm in the bottom graph, showing that the diffusion effect in resist imaging should not be neglected. With the extensions described above, a retrieval operation can be carried out on experimental data, including lateral diffusion and axial blurring. The throughfocus point-spread function data were obtained with the resist printing method sketched in fig. 5.5. The contour lines of the experimental through-focus pointspread function are shown as the solid curves in fig. 5.19. They apply to a lithographic projection objective with a numerical aperture of 0.85, used in water immersion. The refractive index of water is 1.4367 at the deep UV wavelength of 193 nm. It is important to note that the contour lines cannot be constructed experimentally at relatively high intensity levels where the contour diameter becomes
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Fig. 5.16. Through-focus cross-sections of the radially symmetric point-spread function in the presence of a finite diameter of the source. The diameter of the circular source is 50 nm (top figure) and 200 nm (bottom figure) and has to be scaled by the magnification (1/4) of the imaging system (s0 = 0.85, λ = 193 nm). The solid lines are obtained using a lithographic simulation package, SOLID-C, the dashed lines have been calculated using the Extended Nijboer–Zernike formalism. The vertical and horizontal coordinates, z and r respectively, apply to the focal position and the radial image coordinate, both expressed in real space units of µm.
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Fig. 5.17. Through-focus cross-section of a radially symmetric point-spread function for the aberration-free case, top figure, and in the case where spherical aberration of lowest order is present (β40 ≈ 0.25i, bottom figure). The solid lines are obtained using the lithographic simulation package SOLID-C, the dashed lines have been calculated using the Extended Nijboer–Zernike formalism. Vertical and horizontal coordinates in µm; s0 = 0.85, λ = 193 nm.
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Fig. 5.18. Through-focus cross-section of the aberration-free point-spread function (top graph) and the point-spread function contours of the recorded latent image after a diffusion process in the resist layer (bottom figure) with σr = 40 nm. The solid lines are again obtained using the SOLID-C package, the dashed lines have been calculated with the Nijboer–Zernike formalism. Vertical and horizontal coordinates in µm; s0 = 0.85, λ = 193 nm.
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Fig. 5.19. An experimentally obtained through-focus contour plot (solid lines) of a point-spread function in a thin photo-resist layer. The dashed lines have been obtained after Zernike coefficient retrieval using the Extended Nijboer–Zernike formalism. Vertical and horizontal coordinates in µm; λ = 193 nm, water immersion with n = 1.4367, s0 = 0.85/1.4367.
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small. The corresponding resist images are beyond the recording capabilities of current photo-resist materials. This fact has not proven to be detrimental in the retrieval process because the point-spread-function deformation by aberrations is most visible at larger values of the radial coordinate and there where the diffraction ring structure is found. The retrieval procedure carried out on the experimental data gave rise to a very low aberration level with |β40 | 0.05. The dominating factor for the contour-line broadening was due to the diffusion and blurring in the lateral and axial directions. The standard deviations of these blurring phenomena could also be retrieved by using them as free parameters in the retrieval method. In line with Dirksen, Braat and Janssen [2006], a merit function can be defined according to 2 M(σr , σf ) =
0 (σ ,σ )−β 0 ]}2 { [β2n r f 2n,o 2n+1 0 (σ ,σ )− [β 0 ]|2 2 |β2n r f 2n,o n 2n+1
n=0
,
(5.28)
0 ] with n 2 represent the deviation from unity of the where the real parts [β2n,o transmission function of the imaging system, in most cases available by a priori knowledge about the system. The blurring due to lateral diffusion and defocus could become visible in the retrieved coefficients by a change in their real parts
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that affect the transmission function of the imaging system. A reduced resolution because of blurring can be enforced by a reduction in the amplitude transmission function at the pupil rim to effectively reduce the numerical aperture of the system. Given the original circularly symmetric transmission function of 0 ], the quantity the imaging system, well represented by the coefficients [β2n,o M(σr , σf ) weighs the change in these coefficients with respect to the total power in the retrieved complex β-coefficients. The combination of σr - and σf -values that minimizes M is the most likely solution for the retrieval problem at hand. When the retrieval procedure using minimization of M was applied to the data set of fig. 5.19, the values σr = 16 nm and σr = 100 nm were obtained. 5.3.7. Birefringence assessment in a high-NA imaging system The initial state of polarization of the incident wave in the entrance pupil of an optical system is modified when it propagates through the optical system. The change in state of polarization because of a change in beam convergence has been extensively described for a high-NA system in § 2.5.3. Another factor that influences the state of polarization on the exit-pupil sphere is associated with the polarization-dependent amplitude and phase changes on transmission through the (coated) air–lens and lens–air interfaces of an imaging system or the reflections at mirror surfaces; these are especially appreciable in imaging systems with a high numerical aperture where the ray incidence angles at intermediate surfaces can be large. We also mention possible anisotropy of the lens materials or the reflective coatings. This anisotropy can be induced by structural properties but also by residual stresses in the coatings or lens materials. The result is a gradual or abrupt change in the state of polarization on propagation of the radiation through the imaging system. In Braat, Dirksen, Janssen, van de Nes and van Haver [2005] it is indicated how these polarization-sensitive effects can be dealt with in a highnumerical-aperture system. By a number of retrieval steps with different states of polarization in the entrance pupil of the system, one obtains several sets of complex Zernike coefficients corresponding to these incident polarization states. The combination of these retrieved sets of coefficients leads to the value of the geometric wavefront deformation and to the amplitude and axis orientation of the cumulative system birefringence in each point on the exit-pupil sphere. The exact procedure is briefly pointed out below.
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I. Obtaining the required field components Ex and Ey in the exit pupil The cumulative effect of birefringence in the optical system is represented in each point on the exit-pupil sphere by a certain value of the retardation due to the bire-
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fringence, b , and the azimuths of the orthogonal principal axes, for instance, by defining the angle αb of the ‘slow’ axis of the birefringence. For most practical cases it is allowed to simplify the analysis by neglecting the dichroism introduced by the optical system. This means that we trust that the amplitude anisotropy, for instance due to modulus changes in the transmission or reflection coefficients at surfaces, will have much less influence on the system performance than the corresponding phase anisotropy (Van Wiggeren and Roy [1999]). This is a reasonable assumption as long as we do not employ resonant structures like multi-layer coatings in an optical system. Thus, in the modified expressions for the E-field components in the exit pupil due to birefringence, only phase ‘retardation’ will be considered. The state of polarization in the exit pupil is now analyzed using the Jones matrix analysis. The matrix relation between the x- and y-components of the input and output electric fields belonging to a plane wave that has traversed a birefringent optical element is given by (Jones [1941], McGuire and Chipman [1990]) Ex,j px,j J11 J12 px,j (5.29) = Aj = J (ρ, θ ) Aj , Ey,j J21 J22 py,j py,j where the complex amplitudes of the x- and y-components of the incident electric field at the j th exposure or detection are given by (px,j Aj , py,j Aj ), with Aj the modulus of the amplitude of the incident fields. The field components (Ex,j , Ey,j ) depend on the position (ρ, θ ) in the entrance pupil because, in general, the J -matrix is spatially varying. The functions describing the locally varying complex amplitudes can be expanded with the aid of Zernike polynomials m and β m . The field components (E , E ), affected and their coefficients, βn,x x,j y,j n,y by the birefringence of the optical system, will now formally replace the original components (px,j , py,j )Aj in the entrance pupil. The Zernike expansions corresponding to (Ex,j , Ey,j ) are then used to determine the vector components of the field on the exit-pupil sphere, and these are used to evaluate the field in the focal region, see eq. (2.65).
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II. Linearized energy density in the presence of birefringence Basically, we need to evaluate the four complex matrix elements Jij for each sample point in the exit pupil, leading to eight independent quantities to be determined. But since we have excluded dichroism, the above matrix has a special structure (Lu and Chipman [1994], Stallinga [2004b]) and can be written as J11 J12 J = (5.30) , ∗ ∗ −J12 J11
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with the property |J11 |2 + |J12 |2 = 1. The eigenstates of this matrix are elliptical in general. Once the eigenvalues and eigenstates have been found, the orientation α of the slow and fast axes and the value b of the phase birefringence are known. The retrieval procedure starts with eq. (3.8) where we use the still unknown m and β m to construct the energy density in the focal volume. The coefficients βn,x n,y effect of birefringence has to be accounted for in each basic function Gkl (βx , βy ), 0 or see eq. (3.4). Linearization of Gkl (βx , βy ), keeping only products with β0,x 0 , gives contributions from lines 1 and 7–12 in eq. (3.4). Ordering these β0,y terms leads to the following general expression, comparable to eq. (5.23) for the isotropic case: Gk,l (β1 , β2 ) = exp i(k − l)φ μ∗ μ∗ 0 β0,1 βν,2 Ψν;k,l (r, f ) exp(−iμφ) × ν0 μ=0
μ 0∗ μ + (1 − δνμ )β0,2 βν,1 Ψν;l,k (r, f ) exp(+iμφ) .
(5.31)
0 and β 0 is due to the average bireThe difference between the coefficients β0,1 0,2 fringence over the exit pupil. Because of the neglect of dichroism, these two coefficients have equal moduli but show a phase difference 12 defined by 0 = β 0 exp{i /2} β0,1 12 0 0 0 = β0,1 exp{−i 12 }. (5.32) or, β0,2 0 = β 0 exp{−i /2} β0,2 12 0
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With this definition of the (0, 0)-coefficients, we obtain Gk,l (β1 , β2 ) = exp i(k − l)φ β00 exp{i 12 /2} μ∗ μ∗ × βν,2 Ψν;k,l (r, f ) exp(−iμφ) ν0 μ=0
μ μ + (1 − δνμ )βν,1 Ψν;l,k (r, f ) exp(+iμφ) .
(5.33)
Inspection of this expression shows that using the property 12 = − 21 the relationship Gkl (β1 , β2 ) = G∗lk (β2 , β1 ) remains valid. Because of the special structure of the unitary matrix J , three independent quantities need to be determined in addition to the geometrical wavefront aberration and transmission defects of the system. We thus need four retrieval operations to determine the complex quantities J11 and J12 plus the polarizationindependent geometrical defects of the system. Preferred polarization states (px,j , py,j ) are two orthogonal √ linear polarization states, e.g. (1, 0) and (0,√1) and the circular ones, viz. (1, i)/ 2 for left-circularly polarized and (1, −i)/ 2
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for right-circularly polarized light. The four exposures or detection steps with the preferred polarization states lead, after retrieval, to four different sets of βx - and βy -coefficients. The four sets of β-coefficients are now used to obtain the complex amplitude in any point of the exit pupil for four different polarization states. This is basically sufficient to uniquely determine the size and the orientation of the cumulative birefringence of the optical system in that specific point of the exit pupil. In addition, we obtain the geometrical defects of the system that are independent of the state of polarization in the entrance pupil of the optical system. § 6. Conclusion and outlook The assessment of optical systems regarding their imaging capability focuses on the measurement of the exit-pupil function. Especially the phase part of the exitpupil function is important as it determines the imaging aberrations and possible birefringence of the system. An accurate knowledge of the exit-pupil function, both in the design and in the manufacturing stage, allows for the correction of aberrations or birefringence in the fine-tuning manufacturing step. Precision interferometry is the most direct method to measure the exit-pupil function. In practice, its applicability is somewhat limited because of the requirement of special sources and extremely well-specified reference surfaces. In this chapter we have discussed the measurement of the point-spread function of an imaging system as a means to obtain accurate information about the defects of the wave in the exit-pupil, not only regarding the aberrations of the system under test but also including its transmission and birefringence defects. A prerequisite for solving the so-called inverse problem is the accurate knowledge of the point-spread function in the focal volume as a function of the aberrations and other defects of the exit-pupil function. For systems with a high numerical aperture, the state of polarization of the incident field has to be included and the vector diffraction integral of the aberrated system has to be solved. Based on the use of Zernike polynomials, we have discussed semi-analytic expressions for the complex amplitude and the energy density in the focal region. They first apply to the scalar diffraction case at low numerical aperture, then to an extended scalar regime for apertures up to values of 0.85, and finally to the vector diffraction case at very high aperture close to unity regarding the sine of the opening angle of the focusing beams. The availability of analytic expressions offers an interesting short-cut to the more laborious propagation methods based on a numerical solution of the diffraction integrals involved in the propagation from the exit-pupil to the focal region. Inverse problem solution in optical imaging has been addressed by several authors, based on the matching of measured intensity distributions in the image
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6]
Acknowledgements
455
plane and the intensity of the exit-pupil function. Certain physical constraints are applied like the non-negativity of optical intensity or the limited extent of a pointspread or pupil function to assure convergence towards a most likely solution. Like in optical interferometry, the method of phase diversity has been introduced in the solution of optical inverse problems to avoid the phase ambiguity that is possible when intensity measurements are used to retrieve the complex amplitude of the exit-pupil function. We have focused in the last part of this chapter on a special application of the phase diversity principle, the collection of through-focus pointspread functions to reconstruct the complex amplitude of the exit pupil. With our analytic expression for the energy density or, equivalently, for the power flow in the focal region, we are capable to address the inverse problem of how to retrieve the coefficients of the Zernike expansion of the complex pupil function that created the measured through-focus point-spread function. Using a linearized version of the energy density function for weak system defects, a system of equations was developed to solve for the unknown complex Zernike coefficients. At larger defect values, a predictor–corrector method is applied to improve the solution. The total range of aberration and other defects that can be covered in this way is well in excess of what is needed for good-quality imaging systems. The quality assessment method takes into account the particularities of the detecting medium. For very high-resolution systems, a photo-resist layer is needed to record the extremely fine features of the through-focus point-spread functions. The chemical diffusion in a resist layer is accounted for and a matching procedure allows the precise estimate of the magnitude of this diffusion effect. The same matching principle can be applied to other blurring factors like in-plane and axial vibrations during the point-spread-function exposure time. It is equally possible to obtain an accurate estimate of the value of the numerical aperture of the system under test and of the state of polarization of the light incident on the entrance pupil. Future work in this direction will address the assessment of systems with a more general pupil configurations, especially those with a central obstruction. An extension of the analysis to systems with broadband illumination would be of great practical importance because this would bring within reach applications in the fields of, for instance, astronomical observation and ophthalmology.
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Acknowledgements One of the authors (J.B.) thanks the Technical University of Delft for the opportunity of a sabbatical leave and Profs. Mario Bertolotti and Concita Sibilia of University ‘La Sapienza’, Rome, for their generous hospitality during this sabbatical leave.
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Appendix A: Derivation of Weyl’s plane wave expansion of a spherical wave The time-independent Helmholtz equation is given by 2 ∇ + k 2 f (r, r ) = −δ(r − r ),
(A.1)
where the point source is located at r = r . The (normalized) solution for the three-dimensional geometry is given by the expression for a scalar spherical wave f3 (x, y, z; x , y , z ) =
exp(ik|r − r |) , 4π|r − r |
(A.2)
with |r − r | given by {(x − x )2 + (y − y )2 + (z − z )2 }1/2 . A solution for the onedimensional geometry where a uniform coherent source covers the entire plane z = z is obtained from the Helmholtz equation 2 ∇ + kz2 f (x, y, z; kx , ky , z ) = −δ(z − z ), (A.3) where the value of kz is given by kz2 = k 2 − kx2 − ky2 . The individual plane-wave solutions of this equation are given by f1 (x, y, z; kx , ky , z ) =
i exp ikz |z − z | exp i(kx x + ky y) . 2kz
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(A.4)
The validity of this plane-wave solution is easily checked by substituting the solution in eq. (A.3) and by verifying, using Gauss’ theorem, that ∇f1 · n dS = (A.5) −δ(z − z ) − kz2 f dV , where the surface S is that of a thin box at z = z that infinitely extends in the xand y-directions and has a vanishing thickness in the z-direction, n is the outward normal to S and the volume V is the interior of the box. The complete solution for the one-dimensional geometry is obtained by an integration over all (kx , ky ). The one-dimensional solution can equally well be constructed from the threedimensional solution by a 2D-integration over the source plane at z = z . This leads to the alternative expression f1 (x, y, z; kx , ky , z ) +∞ = f3 (x, y, z; x , y , z ) exp i kx (x − x ) + ky (y − y ) dx dy −∞
= exp i[kx x + ky y]
+∞ −∞
exp{ik|r − r |} exp −i[kx x + ky y ] dx dy . 4π|r − r | (A.6)
6, Appendix B]
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457
When comparing eqs. (A.4) and (A.6), it can be concluded that the second integral in eq. (A.6), the Fourier transform of f3 (x, y, z; x , y , z ), should equal (i/2kz ) exp{ikz (|z − z |)}. By taking the inverse Fourier transform of both quantities we obtain
exp{ik|r − r |} FT−1 FT 4π|r − r | +∞ exp{i[kz |z − z | + kx x + ky y]} i = (A.7) dkx dky , 2kz (2π)2 −∞
or, in its most elementary notation, i exp(ik|r|) = |r| 2π
+∞ −∞
exp{i[kz z + kx x + ky y]} dkx dky . kz
(A.8)
Appendix B: The Debye integral in the presence of aberrations
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The angular spectrum of the field distribution in the aperture A is given by eq. (2.12). In the aberrated case the field E0 (x , y ) is written as A0 (x , y ) × exp{iΦ(x , y )}. The function Φ(x , y ) = kW (x , y ) is the phase deviation resulting from the aberration W (x , y ) from the spherical shape of the incident wavefront. Our sign convention implies that a positive wavefront aberration corresponds to a position of the wavefront further away from the focal point F than ˜ ; kx , ky ), the stain the aberration-free case. To obtain an asymptotic value of E(z tionary points of the integrand in eq. (2.12) with respect to x and y have to be found. Writing the integrand as A0 (x , y ) exp −i kRQF + kx x + ky y − Φ(x , y ) RQF = g(x , y ) exp ih(x , y ) ,
(B.1)
we find the stationary points by putting ∂h/∂x = 0 and ∂h/∂y = 0, and this yields the equations k(x − xf ) ∂Φ = −kx + , RQF ∂x k(y − yf ) ∂Φ = −ky + . RQF ∂y
(B.2)
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These equations can be solved by squaring both equations, next summing them 2 to the result, finally yielding and adding (z − zf )2 /RQF
(z − zf )2 1 ∂Φ 2 ∂Φ 2 1= (B.3) + − + k − k . x y 2 ∂x ∂y k2 RQF Neglecting the second-order terms in ∂Φ/∂x and ∂Φ/∂y we find RQF = k kz 1 +
zf − z
2 kz2
∂Φ ∂Φ kx ∂x + ky ∂y
1/2 .
(B.4)
For small derivative values of Φ, the stationary points are approximated by 2
k − ky2 ∂Φ kx (zf − z ) ky ∂Φ − xs ≈ xf − 1− , kz ∂x kx kz2 kz2 ∂y 2
ky (zf − z ) k − kx2 ∂Φ kx ∂Φ ys ≈ yf − (B.5) 1− 2 − . kz ∂y kz ∂x ky kz2 In the aberration-free case, we find the stationary points that already appeared in the argument of E0 in eq. (2.13). It is common practice to neglect the influence of aberrations on the position of the stationary points in the diffracting aperture. If the aberration is small and the aperture size is many wavelengths large, the derivatives of Φ will remain relatively small and the factors in braces in eq. (B.5) can be put equal to unity. A more careful examination is needed at high numerical-aperture values when the various pre-factors containing kx , ky and kz can become large; high-order aberrations also yield higher derivative values in eq. (B.5).
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Appendix C: Series expansion of the diffraction integral at large defocus The diffraction integral corresponding to Vnm (r, f ) is written using Bauer’s formula according to eq. (2.45) yielding ∞ if (2k + 1)ik jk (f/2) Vnm (r, f ) = exp 2 k=0
1 ×
0 R2k (ρ)Rnm (ρ)Jm (2πrρ)ρ dρ.
(C.1)
0
The polynomial product in the integrand of eq. (C.1) is written as 0 m m R2k (ρ)Rm+2p (ρ) = wkl Rm+2l (ρ). l
(C.2)
m (r, f ) Series expansion for the diffraction integral Vn,j
6, Appendix D]
459
Janssen, Braat and Dirksen [2004] have shown that the coefficients wkl can be represented by a double series involving the product of three other coefficients according to wkl =
p
min(k,s)
s=0
t=0
m m fps gk+s−2t,l bkst ,
(C.3)
with the coefficients f , g and b given by m+p−s−1m+p+s m−1 m p−s 2s + 1 , s = 0, . . . , p, fps = (−1) p+s s p+s+1 s m u+l m + 2l + 1 u−l l m gul = , u = l, . . . , l + m, m + u + l + 1 m+l+u
(C.4) (C.5)
m+l
2s1 + 2s2 − 4t + 1 As1 −t At As2 −t , 2s1 + 2s2 − 2t + 1 As1 +s2 −t t = 0, . . . , min(s1 , s2 ),
bs1 s2 t =
2k
(C.6)
and the coefficients Ak given by the binomial coefficient k . We have wkl = 0 only when l between max(0, k − p − m, p − k) and k + p. Also wkl 0 in all cases. A special case arises when m = 0 and the expressions of eqs. (C.4) and (C.5) are now defined as
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0 fps = δps ,
0 gul = δul ,
(C.7)
with δab equal to Kronecker’s delta symbol. Here, as before, n, m are integers 0 with n − m 0 and even, and the definitions for p and q are, respectively, p = (n − m)/2 and q = (n + m)/2. Appendix D: Series expansion for the diffraction integral V m n,j (r, f ) In this appendix we present two methods for obtaining a series expansion of the m (r, f ) given by integral Vn,j :
1 m Vn,j (r, f )
=
ρ 0
|j |
1+
;−|j |+1 1 − s02 ρ 2 (1 − s02 ρ 2 )1/4
× Rn|m| (ρ)J|m+j | (2πrρ)ρ dρ,
: ; if 2 2 exp 1 − 1 − s0 ρ u0 (D.1)
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with m, n, j integers and n − |m| 0. We have used the absolute value of the order number (m + j ) of the Bessel function. The relation J−n (x) = (−1)n Jn (x) assures that we can also accommodate negative values of the order index (m + j ). The first method, discussed by Braat, Dirksen, Janssen and van de Nes [2003], applies a series expansion that is based on the scalar diffraction integral Vnm (r, f ) according to eqs. (2.47). The second method uses an expansion with the functions Tnm (r, f ) of eq. (2.57) as basis functions. Both functions, V (r, f ) and T (r, f ), have their own Bessel series expansion and expansion coefficients given by eqs. (2.48)–(2.50) and eqs. (2.58)–(2.59), respectively.
D.1: Expansion using the functions Vnm (r, f ) The following steps are required to transform the integral (D.1) into a tractable form: • We write
: ;−|j |+1 ; : if exp 1 − 1 − s02 ρ 2 1 + 1 − s02 ρ 2 u0 ∞ 0 = exp gj + ifj ρ 2 (D.2) hkj R2k (ρ),
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k=0
and define the coefficients gj and fj by requiring the best fit for the constant and the quadratic term in ρ in the expression. The series of Zernike polynomials with coefficients hkj will normally be limited to a constant term h0j close to unity, and a relatively small higher-order term h2j . If the value of s0 , the geometrical numerical aperture, approaches a value of, say, 0.90, or the defocus parameter exceeds the value of 2π, higher-order coefficients hkj are needed. • To reduce the integral Vnm,j (r, f ) to the analytically known result Vnm (r, f ), the upper index of the Zernike polynomial and the order of the Bessel function should be identical. Recursion formulae, already presented by Nijboer [1942], can be used and the following general relationship can be established ρ |j | Rn|m| (ρ) =
|j |
|m+j |
cn|m|j s Rn+|j |−2s (ρ).
(D.3)
s=0
• Having determined the two or three new Zernike polynomials that we denote by |m+j | Rn+|j |−2s (ρ), we need to evaluate products of these Zernike polynomials with 0 (ρ) that appeared in the first step. It can be shown that a general polynomial R2k
m (r, f ) Series expansion for the diffraction integral Vn,j
6, Appendix D]
461
the following general relationship exists: |m+j |
0 R2k (ρ)Rn+|j |−2s (ρ) =
∞
|m+j |
dn|m|j skt Rn+|j |−2s+2t (ρ),
(D.4)
t=0
and the number of terms t in the summation is normally limited to three. Note that the problem of finding the coefficients dn|m|j skt was already solved in Appendix C where an explicit expression for the coefficients wkl was given via eqs. (C.4)–(C.6). m as a linear When combining the above steps we have succeeded in writing Vn,j |m+j | combination of a modest number of terms of the form Vn+|j |−2s+2t (r, fj ) exp (gj ). A detailed derivation of the expressions for the coefficients gj and fj , for the coefficient cn|m|j s with running index s and the coefficient dn|m|j skt with running index t can be found in Braat, Dirksen, Janssen and van de Nes [2003].
D.2: Expansion using the functions Tnm (r, f ) The expression according to the second method is more straightforward than the first method and can be applied when m and n are not too large (say, 20). We start with the slightly more general integral
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1 I (r, f ) =
ρ
k
: ;−p+1 1 + 1 − s02 ρ 2 (1 − s02 ρ 2 )1/4
0
: ; if 2 2 exp 1 − 1 − s0 ρ u0
× Rnu (ρ)Jl (2πrρ)ρ dρ,
(D.5)
where k, p, u, n, l 0 and both n − u and k + u − l are 0 and even. We write n−u
Rnu (ρ)
=ρ
u
2
n−u
Cs = n−u
Cs ρ 2s ,
2
s=0
(−1) 2 −s , − s !s!(n − s)!
(D.6)
and :
1+
;−p+1 1 − s02 ρ 2
(1 − s02 ρ 2 )1/4
: ; if exp 1 − 1 − s02 ρ 2 u0
∞ = exp g + if ρ 2 Bt ρ 2t . t=0
(D.7)
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Assessment of optical systems by means of point-spread functions
[6, Appendix D
The coefficients Bt are obtained from a computation scheme that is given below. Using these coefficients we write the integral I (r, f ) as n−u
I (r, f ) =
∞ 2
l Cs Bt Tk+u+2s+2t (r, f ),
(D.8)
s=0 t=0
with the Bessel series expansion for T (r, f ) itself given by eq. (2.58); there further holds k + u + 2s + 2t − l 0 and even. The computation scheme for the coefficients Bt runs as follows. As a first step, we bring the fraction on the left-hand side of eq. (D.7) into the exponential function and split off the optimal quadratic part g + if ρ 2 . The Taylor expansion of 2 2t the remaining part in the exponential is written as ∞ t =0 At ρ . The coefficients Bt are defined by % ∞ & ∞ 2t 2t Bt ρ = exp At ρ . (D.9) t=0
t =0
Using u0 = 1 − s02 and d0 = (u0 /s0 )2 as auxiliary quantities, the detailed scheme now looks as follows: ⎧ b0 = 12 d0 + ln ud00 , a0 = 12 − 16 d0 , ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎨ c0 = − 1 + 1−s2 0 ln(1 − s 2 ) , 0 s0 (D.10) ⎪ ⎪ a1 = 12 1 − 15 d02 , b1 = − 14 d0 (2 − d0 ), ⎪ ⎪ ⎪ ⎪ 2 s02m ⎩ c1 = −3 ∞ m=1 (m+1)(m+2) , 1 g = − 14 (c0 − c1 ) + (−p + 1)(b0 − b1 ) + if (a0 − a1 ), (D.11) f = 12 ic1 + 2f a1 − 2i(−p + 1)b1 , ⎧ ⎨ A0 = (−p + 1) ln 2 − g , A1 = 14 s02 p + 2if u0 − if , (D.12) ⎩ A = (p − 1)− 12 (−1)t + 1 − if − 12 (−1)t s02t , t = 2, 3, . . . , t 2 u0 t −1 2t t
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with the binomial coefficients again given by eq. (2.51). The values of the Bcoefficients in eq. (D.9) are then given by B0 = exp(A0 ), Bt+1 =
t t +1−j At+1−j Bj , t +1 j =0
t = 0, 1, . . . .
(D.13)
6, Appendix E]
The predictor–corrector procedure
463
The number of coefficients to be used can be judged from the convergence of the series expansion for I (r, f ). Like in the first case, even for numerical aperture values s0 as high as 0.95, a limited number of coefficients is generally needed to attain an accuracy of, say, 10−4 in complex amplitude, or, equivalently, 10−8 in intensity. Such an accuracy is largely sufficient in practical applications where the measured intensity data, to be used in a retrieval process, are affected by noise at a much higher level.
Appendix E: The predictor–corrector procedure The predictor–corrector procedure has been described and tested in simulations by van der Avoort, Braat, Dirksen, Janssen [2005, Section 4], for the case of relatively low-numerical-aperture systems that allow a scalar treatment of the image formation. The extension to the high-numerical-aperture vectorial case is rather straightforward. The basic principles being identical, we only present a brief outline. The starting point of the predictor–corrector method is the availability of a measured through-focus intensity distribution I . From this distribution, by Fourier analysis, we form the various parts with azimuthal dependence exp(imφ) that are represented in the form 2 0,0 ∗ m,m m,0 Im = β00 χ0,0 (E.1) +2 β00 βnm∗ χn,0 + βnm βnm χn,n .
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n,m
n,m n ,m
0,0 m,0 Here we use the symbols χ0,0 and χn,0 to identify the dominant aberration-free self-interference term and the dominant cross-terms, respectively, that arise in accordance with eqs. (3.12) and (5.24). The third term on the right-hand side of m,m m m ∗ eq. (E.1), χn,n , is an elaborate term that involves products Vn;j Vn ;j , pertaining to relatively small cross-terms. The primes in the summations in eq. (E.1) indicate that the terms with n = m = 0 and n = m = 0 should be deleted. In the basic linearized retrieval scheme, we choose the β’s in the small cross-term-deleted version 0 2 0,0 m,0 β0 χ0,0 + 2 (E.2) β00 βnm∗ χn,0 n,m
of eq. (E.1) such that the match between eq. (E.2) and Im is maximal; this is done in accordance with eqs. (5.26) and (5.27). The resulting β’s are denoted by βnm (1) and form a first estimate of the β’s in eq. (E.1) that serve to represent Im .
464
Assessment of optical systems by means of point-spread functions
[6, Appendix E
In the linearized retrieval scheme, matching was done with the small crossterms deleted. Now that an estimate βnm (1) has been found, the small cross-term expression in eq. (E.1) can be estimated as n,m n ,m
∗
m,m βnm (1)βnm (1)χn,n
(E.3)
in which the unknown βnm are replaced by their first estimates βnm (1). A dim,m are rect computation of eq. (E.3) is, however, quite involved since the χn,n rather complicated, so we proceed in a different manner. We compute, using the forward scheme for computing the field components Ei in eq. (2.65), the through-focus intensity point-spread function I (1) = |E(1)|2 of the optical system with pupil function P (1) of eq. (5.22) where we have set βnm = βnm (1) throughout. Then, in accordance with eq. (E.1), the quantity (E.3) is given by 2 0,0 m,0 −2 β00 (1)βnm∗ (1)χn,0 , I (1) − β00 (1) χ0,0
(E.4)
n,m
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and its computation is now much easier and feasible in a strongly reduced time. Having available now the double summation cross-term of (E.3), we perform basic retrieval with the Im replaced by Im −
n,m n ,m
∗
m,m βnm (1)βnm (1)χn,n .
(E.5)
Hence, the β’s in eq. (E.2) now maximize the match between eqs. (E.2) and (E.5) to yield a new collection of coefficients βnm (2). This whole process of adjusting Im is repeated until convergence is reached. When convergence is reached we have obtained coefficients βnm (∞) that satisfy Im −
n,m n ,m
∗
m,m βnm (∞)βnm (∞)χn,n
2 0,0 m,0 = β00 (∞) χ0,0 −2 β00 (∞)βnm∗ (∞)χn,0 .
(E.6)
n,m
By bringing the double summation in the left-hand side of eq. (E.6) to the right, we see that we have managed to represent Im in the form (E.1) using βnm = βnm (∞).
6, Appendix F]
Zernike coefficients for circularly symmetric polarization states
465
Appendix F: Zernike coefficients for circularly symmetric polarization states The incident field in the entrance pupil is given by eq. (3.21), and using the expansion of eq. (2.64) with m βn,x Rn|m| (ρ) exp(imθ ), Ex = m Ey = (F.1) βn,y Rn|m| (ρ) exp(imθ ), |m |
we form the inner products with a general Zernike polynomial Rn (ρ) exp(im θ ). For the Ex -component we obtain 1 cos(θ + θ0 )Rn|m | (ρ) exp(−im θ )ρ dρ dθ π A
2π 1 1 m βn,x Rn|m| (ρ) exp(+imθ )Rn|m | (ρ) exp(−im θ )ρ dρ dθ = π n,m 0
=
m 2βn,x
1
0
2 Rn|m | (ρ) ρ dρ
m βn,x = , n+1
(F.2)
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0
where we have used the properties of the inner products of the radial Zernike poly|m| nomials Rn , see Born and Wolf [2002]. Evaluating the first integral in eq. (F.2) then yields m βn,x
n+1 = π
1 2π
0
exp{i(θ + θ0 )} + exp{−i(θ + θ0 )} 2
0
1
× exp(−im θ ) dθ
Rn|m | (ρ)ρ dρ
0
= (n + 1) exp(iθ0 )δm ,+1 + exp(−iθ0 )δm ,−1
1
Rn|m | (ρ)ρ dρ, (F.3)
0
with δm,m the Kronecker symbol. Using the properties of the Jacobi polynomials, see Braat, Dirksen, Janssen and van de Nes [2003], an analytic solution of the integral over ρ can be found m 1 1 2 p m p Rm+2p (ρ)ρ dρ = (−1) m+2 2 2 0
p+1
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Assessment of optical systems by means of point-spread functions
with the Pochhammer symbol (m)p given by (m)p = m(m + 1) · · · (m + p − 1), (m)0 = 1.
[6
(F.4)
±1 For our special case |m | = 1 we then find the only non-zero coefficients β2n+1,x ±1 and β2n+1,y of eq. (3.22).
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Huygens, C., 1941, in: Korteweg, D. (Ed.), Complete Works, Royal Dutch Academy of Arts and Sciences, Amsterdam. Iglesias, I., 1998, Appl. Opt. 37, 5427. Ignatowsky, V.S., 1919, Tr. Opt. Inst. 1 (4), 1. Janssen, A.J.E.M., 2002, J. Opt. Soc. Am. A 19, 849. Janssen, A.J.E.M., Braat, J.J.M., Dirksen, P., 2004, J. Mod. Opt. 51, 687. Jones, R.C., 1941, J. Opt. Soc. Am. 31, 488. Kintner, E.C., Sillitto, R.M., 1976, Opt. Act. 23, 607. Kogelnik, H., 1965, Appl. Opt. 4, 1562. Lavery, K.A., Vogt, B.D., Prabhu, V.M., Lin, E.K., Wu, W.-L., 2006, J. Vac. Sci. Techn. B 24, 3044. Ling, H., Lee, S., 1984, J. Opt. Soc. Am. A 1, 965. Lommel, E., 1885, Abh. Bayer. Akad. 15 (2), 233; Abh. Bayer. Akad. 15 (3) (1886) 531. Lu, S.-Y., Chipman, R.A., 1994, J. Opt. Soc. Am. A 11, 766. Macleod, H.A., 1989, Thin Film Optical Filters, McGraw–Hill, New York, USA. Mahajan, V.N., 1998, Optical Imaging and Aberrations. Part I: Ray Geometrical Optics, SPIE, Bellingham, Wash. Mansuripur, M., 1986, J. Opt. Soc. Am. A 3, 2086. Mansuripur, M., 1989, J. Opt. Soc. Am. A 6, 786. Maréchal, A., 1947, Rev. Opt. (théor. instrum.) 26, 257; J. Opt. Soc. Am. 37 (1947) 982. McGuire, J.P., Chipman, R.A., 1990, J. Opt. Soc. Am. A 7, 1614. Nijboer, B.R.A., 1942, Thesis, University of Groningen, Groningen. An electronic version can be downloaded from http://www.nijboerzernike.nl. Picht, J., 1925, Ann. Phys. Lpz. 77, 685; Ann. Phys. Lpz. 80 (1926) 491. Quabis, S., Dorn, R., Eberler, M., Glöckl, O., Leuchs, G., 2000, Opt. Commun. 179, 1. Rayleigh, 1879, Phil. Mag. 8, 403. Richards, B., Wolf, E., 1959, Proc. R. Soc. London Ser. A 253, 358. Richter, R., 1925, Z. Instrumentkunde 45, 1. Sheppard, C.J.R., Török, P., 1997, J. Mod. Optics 44, 803. Sherman, G.C., 1967, J. Opt. Soc. Am. 57, 546. Siegman, A.E., 1986, Lasers, University Science Books, Mill Valley, CA, USA. SOLID-C, 2004, A software product (release 6.3.0) of SIGMA-C GmbH, Thomas-Dehlerstrasze 9, D-81737, Munich, Germany. Stallinga, S., 2001, J. Opt. Soc. Am. A 18, 2846. Stallinga, S., 2004a, J. Opt. Soc. Am. A 21, 1785. Stallinga, S., 2004b, J. Opt. Soc. Am. A 21, 2406. Stamnes, J.J., 1986, Waves in Focal Regions, Adam Hilger, Bristol. Steward, G.C., 1925, Phil. Trans. Roy. Soc. A 225, 131; Trans. Camb. Phil. Soc. 23 (1926) 235. Strehl, K., 1894, Theorie des Fernrohrs, Barth, Leipzig. Teague, M.R., 1983, J. Opt. Soc. Am. 73, 1434. Török, P., Sheppard, C.J.R., Varga, P., 1996, J. Mod. Optics 43, 1167. Török, P., Varga, P., Laczik, Z., Booker, G.R., 1995, J. Opt. Soc. Am. A 12, 325. van de Nes, A.S., Billy, L., Pereira, S.F., Braat, J.J.M., 2004, Opt. Express 12, 1281. van der Avoort, C., Braat, J.J.M., Dirksen, P., Janssen, A.J.E.M., 2005, J. Mod. Optics 52, 1695. van Haver, S., Braat, J.J.M., Dirksen, P., Janssen, A.J.E.M., 2006, J. Eur. Opt. Soc. -RP 1, 06004, 1. Van Wiggeren, G.D., Roy, R., 1999, Appl. Opt. 38, 3888. VanDijck, D., Coene, W., 1987, Optik 77, 125. Visser, T.D., Wiersma, S.H., 1991, J. Opt. Soc. Am. A 8, 1404. Welford, W.T., 1986, Aberrations of Optical Systems, Adam Hilger, Bristol. Wesner, J., Heil, J., Sure, T., 2002, Proceedings of SPIE 4767, 32. Weyl, H., 1919, Ann. Phys. (Leipzig) 60 (22), 481.
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E. Wolf, Progress in Optics 51 © 2008 Elsevier B.V. All rights reserved
Chapter 7
The discrete Wigner function by
Gunnar Björk School of Information and Communication Technology, Royal Institute of Technology (KTH), Electrum 229, SE-164 40 Kista, Sweden
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Andrei B. Klimov Departamento de Física, Universidad de Guadalajara, 44420 Guadalajara, Jalisco, Mexico
Luis L. Sánchez-Soto Departamento de Óptica, Facultad de Física, Universidad Complutense, 28040 Madrid, Spain
ISSN: 0079-6638
DOI: 10.1016/S0079-6638(07)51007-3 469
Contents
Page § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
471
§ 2. Continuous Wigner function . . . . . . . . . . . . . . . . . . . . . . .
476
§ 3. Discrete finite space and finite fields . . . . . . . . . . . . . . . . . .
477
§ 4. The generalized Pauli group . . . . . . . . . . . . . . . . . . . . . . .
480
§ 5. Mutually unbiased bases . . . . . . . . . . . . . . . . . . . . . . . . .
485
§ 6. The discrete Wigner function . . . . . . . . . . . . . . . . . . . . . .
488
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§ 7. Reconstruction of the density operator from the discrete Wigner function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
498
§ 8. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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§ 9. Discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . .
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Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
513
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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§ 1. Introduction Quantum mechanics describes physical systems through a mathematical object, the state vector (or, more generally, the density operator ρ), ˆ that contains all the relevant information. For systems with continuous degrees of freedom, this density operator lives in an infinite-dimensional complex Hilbert space and the relations between ρˆ and the physical properties of the system are far from trivial. To avoid this abstract construction, Weyl [1928] and Wigner [1932], soon after the discovery of quantum mechanics, developed an alternative description based on the so-called quasidistributions. These functions are defined on the phase space, which results in a striking formal similarity with classical statistical mechanics (Schleich [2001]). There are, however, important differences with respect to the classical description. They come from the non-commuting nature of conjugate quantities (like position and momentum), which precludes their simultaneous precise measurement and, therefore, imposes a fundamental limit on the accuracy with which we can determine a point in phase space. As a distinctive consequence of this, there is no unique rule by which we can associate a classical phase-space variable to a quantum operator. Therefore, depending on the operator ordering, various quasidistributions can be defined, of which the best known are the Glauber–Sudarshan P function (Glauber [1963], Sudarshan [1963]), the Husimi Q function (Husimi [1940], Kano [1965]), and the Wigner function (Wigner [1932]), corresponding to normal, antinormal, and symmetric order, respectively, in the associated characteristic functions. Although all these distributions, and other generalizations such as the s-ordered quasidistributions (Cahill and Glauber [1969]), have their own pros and cons, it seems indisputable that the Wigner function plays an exceptional role among all of them (Balazs and Jennings [1984], Hillery, O’Connell, Scully and Wigner [1984], Lee [1995]). There are three main reasons for that: First, when integrated along any direction in phase space it yields a non-negative function with unit area, i.e., the corresponding marginal distribution; second, it is covariant under linear transformations, which guarantees that it behaves exactly in the same way as a classical probability distribution; third, it is self-dual, which gives a particularly simple re-
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The discrete Wigner function
[7, § 1
lation for expressing the overlap between states or operators. Nevertheless, the quantumness imposes that, in contrast to the classical distributions, it can also take negative values, and for this reason it is known as a quasidistribution. Until quite recently Wigner functions were used mostly for infinite-dimensional Hilbert spaces. However, many quantum systems can be appropriately described in a finite-dimensional Hilbert space. These include, among other, spin systems, multi-level atoms, optical fields with a fixed number of photons, electrons occupying a finite number of sites, etc. Such systems were studied originally by Schwinger [1960a, 1970] and, in a mathematical context, by Weil [1964]. Today they are again being studied very intensively, especially in connection with quantum information (Nielsen and Chuang [2000]). The problem of generalizing the Wigner function to finite systems has a long history. Much of the older work focused on spin, trying to represent spin states by continuous functions of angle variables. This idea was initiated by Stratonovich [1957], Berezin [1975], and Agarwal [1981]. The resulting Wigner function, naturally related to the SU(2) dynamical group, was studied by a number of authors (Scully [1983], O’Conell and Wigner [1984], Cohen and Scully [1986], Várilly and Gracia-Bondía [1989], Wolf [1996], Ali, Atakishiyev, Chumakov and Wolf [2004], Heiss and Weigert [2000]), was applied to some problems in quantum optics (Dowling, Agarwal and Schleich [1994], Agarwal, Puri and Singh [1997], Benedict and Czirjak [1999]), and was extended to more general groups (Brif and Mann [1998, 1999]). These Wigner functions, although describing discrete systems, are not defined in a discrete phase space (Vourdas [2004]). A possible approach for doing that was taken by Hannay and Berry [1980], who considered a phase space constrained to admit only periodic probability distributions, which implies that it is effectively a 2d × 2d-dimensional torus, d being the dimension of the system. By adapting the definition of the continuous Wigner function and, especially, by demanding covariance under linear canonical transformations, they arrive at a compact Wigner function. Another discrete Wigner function using a 2d × 2d grid was proposed by Leonhardt [1995, 1996], who suggested a scheme for reconstructing it from the marginal distributions, bringing this discrete formalism even closer to physical measurements. Note that this method in discrete phase space has both integer and half-odd coordinates. This redundancy of information makes the associated Wigner function to involve ‘ghost variables’ with vanishing probability. This approach has been used to deal with different aspects of quantum computing (Miquel, Paz and Saraceno [2002], Miquel, Paz, Saraceno, Knill, Laflamme and Negrevergne [2002], Paz [2002]).
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Other important generalizations of the Wigner function apply when the phase space is pictured as a d × d lattice. This line was started by Buot [1974], who introduced a discrete Weyl transform that generates a Wigner function on the toroidal lattice Zd (the integers modulo d, with d odd). This is in the same vein as previous mathematical discoveries of Schwinger [1960a], who clearly recognized that the expansion of arbitrary operators in terms of a certain operator basis was the crucial concept in setting a proper phase-space description. In fact, he identified the finite counterpart of the Weyl–Heisenberg group, which describes canonical conjugacy and can be used to define a d × d phase space. More recently, these ideas have been rediscovered and developed further by other authors (Wootters [1987], Galetti and De Toledo Piza [1988, 1993] Cohendet, Combe, Sirugue and Sirugue-Collin [1988], Kasperkovitz and Peev [1994], Opatrný, Miranowicz and Bajer [1996], Galetti and Marchiolli [1996], Opatrný, Welsch and Buzˇek [1996], Rivas and Ozorio de Almeida [1999], Mukunda, Chaturvedi and Simon [2004a, 2004b], Chaturvedi, Ercolessi, Marmo, Morandi, Mukunda and Simon [2005]). A salient feature of these proposals is that one can sum the Wigner function along different axes (including skew ones) to obtain correct marginal probability distributions for observables associated with those axes. Other discrete Wigner functions have been considered that do not fulfill that (Luis and Peˇrina [1998], Takami, Hashimoto, Horibe and Hayashi [2001]), although there is a wide consensus that such a tomographic property is central. For the standard continuous Wigner function, the axes are associated with two complementary observables, namely position and momentum (this continuous theory can be extended to conjugate variables other than Cartesian ones, such as action and phase (Vaccaro [1995a, 1995b], Opatrný, Miranowicz and Bajer [1996]) or angle and angular momentum (Mukunda [1979], Bizarro [1994], Galetti and De Toledo Piza [1995]). In the discrete case, each ‘axis’ can again be associated with a specific observable, and although the ‘axis observables’ cannot be complementary in the usual sense (their commutator cannot be proportional to the identity), they will have a closely related property: every eigenstate of either one of them is a state of maximum uncertainty with respect to the other. This makes a deep connection with the notion of mutually unbiased bases (Delsarte, Goethals and Seidel [1975], Wootters [1986]), which were introduced as a central tool for quantum state reconstruction (Wootters and Fields [1989]). They also play a relevant role in a proper understanding of complementarity (Kraus [1987], Lawrence, Brukner and Zeilinger [2002], Chaturvedi [2002], Wootters [2006]) and in cryptographic protocols (Bechmann-Pasquinucci and Peres [2000], Asplund, Björk and Bourennane [2001]), due to the complete uncer-
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tainty about the outcome of a measurement in some basis after the preparation of the system in another, if the bases are mutually unbiased. Mutually unbiased bases are important for quantum error correction codes (Gottesman [1996], Calderbank, Rains, Shor and Sloane [1988, 1997]) and recently they have also found application in quantum game theory, particularly to provide a solution to the socalled mean king problem (Aharonov and Englert [2001], Englert and Aharonov [2001], Aravind [2003], Hayashi, Horibe and Hashimoto [2005], Paz, Roncaglia and Saraceno [2005], Durt [2006], Kimura, Tanaka and Ozawa [2006]). It has been shown that the maximum number of mutually unbiased bases can be at most d + 1 (Ivanovi´c [1981]). Actually, it is known that if d is prime or a power of a prime, this bound can be achieved (Calderbank, Cameron, Kantor and Seidel [1997]). The construction of mutually unbiased bases is closely related to the possibility of finding d + 1 disjoint classes, each one having d − 1 commuting operators, so that the corresponding eigenstates form sets of mutually unbiased bases (Bandyopadhyay, Boykin, Roychowdhury and Vatan [2002]). Different explicit constructions of mutually unbiased bases in prime power dimensions have been suggested in a number of recent articles (Klappenecker and Rötteler [2004], Lawrence [2004], Parthasarathy [2004], Pittenger and Rubin [2005], Durt [2005], Planat and Rosu [2005], Klimov, Sánchez-Soto and de Guise [2005]). Remarkably though, there is no known answer for any other values of d, although there have been some attempts to find a solution to this problem in simple cases, such as d = 6 or when d is a non-prime integer squared (Grassl [2004], Archer [2005], Wocjan and Beth [2005]). Recent works have suggested that the answer to this question may well be related with the non-existence of finite projective planes of certain orders (Saniga, Planat and Rosu [2004], Bengtsson and Ericsson [2005]) or with the problem of mutually orthogonal Latin squares in combinatorics (Zauner [1999], Wootters [2006]). Coming back to the Wigner function, when the Hilbert space dimensionality is a power of a prime, d = p N , one can label the points in the d × d grid with elements of the finite Galois field GF(p N ). At first sight, the use of elements of GF(p N ) as coordinates could be seen as an unnecessary complication, but it turns out to be an essential step: only by doing so can we endow the phase-space grid with the same geometric properties as the ordinary plane (where, for example, two lines that are not parallel have only one intersection). Once the phase-space arena is built in this way, one can, as in the continuous case, impose a quantum structure by assigning a quantum state to each line. Any good assignment of quantum states to lines is called a ‘quantum net’, and can be used to define a discrete Wigner function. In fact, there is no unique quantum net. One can manage to construct
EBL
7, § 1]
Introduction
475
lines and striations (sets of parallel lines) in this phase space: after an arbitrary choice that does not lead to anything fundamentally new, it turns out that the orthogonal bases associated with each striation are mutually unbiased. Alternatively, one can proceed in just the opposite way and start by considering the geometrical structures in phase space that are compatible with the notion of unbiasedness (Björk, Romero, Klimov and Sánchez-Soto [2007], Klimov, Romero, Björk and Sánchez-Soto [2007]). These admissible structures can be classified into lines (or rays) and Abelian curves (and the former also in exceptional and regular, depending on whether they are self-intersecting or not). To each bundle of curves we can associate a set of mutually unbiased bases and a Wigner function, and these bases are related by local transformations that do not change the corresponding entanglement properties (Romero, Björk, Klimov and Sánchez-Soto [2005]). The approach of Gibbons, Hoffman and Wootters [2004] is recovered for the special case of straight lines. Although the restriction to powers of primes seems to rule out many quantum systems, we stress that this approach is well suited to study quantum-information problems: quantum computers made out of N qubits have a Hilbert space whose dimension d = 2N is precisely a power of a prime. In addition, operators made out of tensor products of Pauli operators acting on each qubit play an important role for quantum computers and have a central role in phase-space methods, since they represent phase-space translation operators. Finally, we address the question of what states have non-negative discrete Wigner function. Quite recently it has been shown (Galvão [2005], Cormick, Galvão, Gottesman, Paz and Pittenger [2006]) that they are precisely stabilizer states, i.e., simultaneous eigenstates of generalized Pauli operators. Moreover, the group of unitaries that preserve the non-negativity of the Wigner function constitutes a subgroup of the Clifford group. This means that such states and unitaries are ‘classical’ in the sense of allowing for an efficient classical simulation scheme using the stabilizer formalism. When one considers the coherent superpositions of ‘classical’ states, the discrete Wigner function behaves in a way that differs drastically from its continuous counterpart: the interference spreads over all of phase space, affecting even the regions where the original states are localized. As a consequence, the presence of interference may become hard to identify when using this class of Wigner functions. However, given two orthogonal stabilizer ‘classical’ states it is possible to define a Wigner function (Cormick and Paz [2006]) such that all coherent superpositions of those states have a phase-space representation in which the quantum interference is localized.
EBL
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It is still unclear if the use of phase-space methods will help in expanding our understanding of some of the problems of quantum information. However, we think that it is highly likely that the various definitions of the Wigner function will make it possible to analyze in a very elegant and compact way a number of different and appealing problems.
§ 2. Continuous Wigner function The continuous Wigner function (Wigner [1932]) was proposed as a tool for finding a quantum-mechanical equivalent of the classical Gibbs–Boltzmann formula for systems in thermal equilibrium. In the original paper, Wigner referred to the function as a “probability function”, but with time the term Wigner function has been adopted both for the original function and for its various extensions. Wigner was of course keenly aware of the fact that the function was not a ‘proper’ probability distribution, as such a function would assign a probability for the system to simultaneously be in a near eigenstate of the position x and the momentum p, clearly a contradiction. The continuous Wigner function corresponding to a density matrix ρˆ for the canonical observables pˆ and q, ˆ usually taken to be the momentum and the position, is defined as
EBL
1 Wρˆ (q, p) = π
+∞ exp(2ipx) q − x|ρ|q ˆ + x dx.
(2.1)
−∞
Here, and in the following, we set h¯ = 1. The Wigner function has three properties of particular importance for what follows: I. The function is real. II. When integrated along any direction in phase space it yields a non-negative function with unit area, i.e., a marginal probability distribution. That is, +∞ P (qϑ ) = W (q cos ϑ − p sin ϑ, q sin ϑ + p cos ϑ) dp
(2.2)
−∞
is the marginal probability distribution of the observable qˆϑ = qˆ cos ϑ + pˆ sin ϑ.
(2.3)
III. The Wigner function is translationally covariant. This property can be stated mathematically as follows: Let ρˆ be a state obtained from the state ρˆ
7, § 3]
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477
Fig. 1. Schematic illustration of a covariant translation of a continuous Wigner function.
by a displacement (q0 , p0 ) in phase space, viz. ˆ ρˆ exp −i(q0 pˆ − p0 q) ˆ . ρˆ = exp i(q0 pˆ − p0 q)
(2.4)
Then the Wigner function W corresponding to ρˆ is obtained from W via the transformation (2.5) EBL In other words, when the density matrix is translated, the Wigner function W (q, p) = W (q − q0 , p − p0 ).
follows along rigidly as illustrated in fig. 1. Replacing the density operator ρˆ in eq. (2.1) with any operator Oˆ k one gets the Wigner function of the operator, which we shall denote by WOˆ k (q, p). The following important overlap relation holds: +∞ +∞ WOˆ k (q, p)WOˆ l (q, p) dq dp. Tr(Oˆ k Oˆ l ) = 2π
(2.6)
−∞ −∞
This equation is particularly convenient for calculating expectation values Tr(ρˆ Oˆ l ).
§ 3. Discrete finite space and finite fields Before making the transition from the continuous Wigner function to its discrete counterpart it is worthwhile to lay a basic foundation of the algebra of discrete space. While in the continuous case it is possible to translate a state by an infinite distance, this is clearly not possible if the space is finite. To ‘prevent’ a state from
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[7, § 3
‘escaping’ the finite phase space it is natural and convenient to use finite fields in the representation of the states. A set L is a commutative ring if two binary operations: addition and multiplication (both commutative and associative) are defined. The simplest example of a ring is a set of natural numbers {0, 1, . . . , N − 1} = ZN , where the algebraic operations are taken modulus N . A field F is a commutative ring with division, i.e., for any α ∈ F there exists −1 α ∈ F so that α −1 α = αα −1 = 1 (excluding the zero element). Elements of a field form commutative (Abelian) groups with respect to addition in F and multiplication in F ∗ = F − {0}, where the notation indicates that the zero is excluded. We will use the standard symbol + to indicate summation. The characteristic of a finite field is the smallest integer p so that p · 1 = 1< + 1 + =>· · · + 1? = 0, p times
and this is always a prime number. Any finite field contains a prime subfield Zp and has d = pN elements, where N is a natural number. Moreover, the finite field containing p N elements is unique and is usually called a Galois field, GF(p N ). GF(p N ) is an extension of degree N of Zp , i.e., elements of GF(p N ) can be obtained with Zp and one of the roots of an irreducible polynomial of order N (that is, one which cannot be factorized in Zp ) with coefficients in Zp . N The multiplicative group of GF(p N )∗ = GF(p N ) − {0} is cyclic: θ p = θ , θ ∈ GF(p N ). The generators of this group are called primitive elements, and every primitive element θ of GF(p N ) is a root of the irreducible polynomial. This polynomial is called a minimal polynomial. As a simple example of non-prime fields let us consider the roots of the following irreducible polynomial on Z2 : x 2 + x + 1 = 0; then, if θ is a root of this polynomial, then the elements {0, 1, θ, θ 2 = θ + 1 = θ −1 } satisfy the summation and multiplication rules listed in table 1 and form the finite field GF(22 ). The element θ is clearly the generator of the multiplicative group GF(22 )∗ . Let us point out that one needs to be careful with the factorization of polynomials over finite fields. For example, the polynomial x 2 + 1 is not irreducible over Z2 : x 2 + 1 = x 2 + 2x + 1 = (x + 1)(x + 1) since 2x = 0 modulus 2. The map α → α p , where α ∈ GF(p N ), is a linear automorphism of GF(p N ). (α + β)N = α N + β N , (αβ)N = α N β N , which is called a Frobenius automork phism: Sk (α) = α p . The elements of the prime field are invariant under action of the Frobenius automorphism.
EBL
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479
Table 1 GF(22 ) summation and multiplication tables Summation 0 0 1 θ θ −1
0 1 θ θ −1
Multiplication 1 1 0 θ −1 θ
θ
θ −1
0
1
θ
θ −1
θ θ −1 0 1
θ −1
0 0 0 0
0 1 θ θ −1
0 θ θ −1 1
0 θ −1 1 θ
θ 1 0
The trace operation tr(α) = α + α 2 + · · · + α p
N−1
=
N−1
Sk (α)
(3.1)
k=0
maps any field element into an element of the prime field, tr(α) GF p N −→ Zp , and thus leaves the elements of the prime field invariant. It satisfies the property (3.2) ⇒ tr(−α) = −tr(α). EBL Do not confuse the field-element trace denoted ‘tr’ with the standard Hilberttr(α1 + α2 ) = tr(α1 ) + tr(α2 )
space trace that is denoted ‘Tr’. Also remember that −α implies the element that satisfies α + (−α) = 0. In the above example of GF(22 ), in particular, we obtain tr(θ ) = θ + θ 2 = θ + θ + 1 = 1, tr θ 2 = θ 2 + θ 4 = (θ + 1) + θ = 1, 3 = 0, tr θ = tr(1) = 1 + 1 tr(0) = 0 + 0 = 0. The additive characters are defined as (Lidl and Niederreiter [1986])
2πi χ(α) = exp tr(α) . p
(3.3)
Note that they will have the role of the integration kernel in eq. (2.1) and they possess two important properties: χ(α1 + α2 ) = χ(α1 )χ(α2 ) and χ(α1 α2 ) = p N δ0,α2 . (3.4) α1 ∈GF(p N )
Any finite field GF(p N ) can be also considered as an N -dimensional linear vector space, and there is a basis {θk , k = 1, . . . , N } in this vector space so that
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2 any α ∈ GF(p N ), α = N k=1 ak θk with ak ∈ Zp . Then, for any function f (α) one has f (α) = f (a1 θ1 + · · · + aN θN ). (3.5) α∈GF(p N )
a1 ,...,aN
There are several natural bases, one of which is the polynomial basis {1, θ, θ 2 , . . . , θ N−1 }, where θ is a primitive element of GF(p N ). Another basis is the norN−1 mal basis {θ, θ p , . . . , θ p }, so one can choose an appropriate basis according to the specific problem. Two bases {α1 , . . . , αN } and {β1 , . . . , βN } in the same field are dual if tr(αk βl ) = δk,l . A basis which is dual to itself is called a self-dual basis, tr(αk αl ) = δk,l . In the case of GF(22 ) the elements {θ, θ 2 } are both roots of the primitive polynomial. The polynomial basis is {1, θ }, whose dual basis is {θ 2 , 1}: tr(11) = 0, tr 1θ 2 = 1, 2 tr(θ1) = 1. tr θ θ = 0, The normal basis {θ, θ 2 } is self-dual: tr(θ θ ) = 1, tr θ θ 2 = 0, tr θ 2 θ 2 = 1. tr θ 2 θ = 0,
EBL
The self-dual basis cannot always be found, and the following important theorem (Jungnickel [1993]) holds: For every prime power d = p N , there exists an almost self-dual basis of GF(p N ) over Zp . Moreover, it has a self-dual basis if and only if either p is even or both N and p are odd. The almost self-dual basis satisfies the properties tr(θk θl ) = 0 when k = l and tr(θk2 ) = 1, with one possible exception. For instance, in the case of GF(32 ) a self-dual basis does not exist and two elements {θ 2 , θ 4 }, θ being a root of the irreducible polynomial x 2 + x + 2 = 0, form an almost self-dual basis, i.e. tr θ 4 θ 4 = 2, tr θ 2 θ 4 = tr θ 4 θ 2 = 0. tr θ 2 θ 2 = 1,
§ 4. The generalized Pauli group 4.1. Prime-dimensional spaces We can now use the field algebra to define physical operators that will help us to construct the discrete Wigner function in a general manner. Let us start with
7, § 4]
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481
the prime-dimensional case, so that d = p. Hence, GF(p) Zp is a field with a prime number of elements. Assume that {|n , n = 0, 1, . . . , p−1} form an orthonormal basis in a p-dimensional Hilbert space Hp . Then, the following operators (Schwinger [1960b, 1960c]), usually referred to as the momentum and position operators, form the Pauli group Zˆ ≡
p−1
Xˆ ≡
ω |n n|, n
n=0
p−1
|n + 1 n|.
(4.1)
n=0
ˆ The monomials They satisfy Xˆ p = Zˆ p = 1. Zˆ k Xˆ l
(4.2)
with k, l = 0, 1, . . . , p − 1 form an operator basis in the space of all the operators acting in Hp . From the definitions it follows that ˆ Z|n = ωn |n = χ(n)|n , ˆ Zˆ Xˆ = ωXˆ Zˆ = χ(1)Xˆ Z,
ˆ X|n = |n + 1 , (4.3)
where ω = exp(2πi/p) is a root of unity and all the algebraic operations are taken modulus p. The following relations hold for the roots: ω
k+l
=ω ω, k l
p−1
EBL
ω = pδl,0 , kl
(4.4)
k=0
similar to eq. (3.4). One can also introduce the Fourier transform operator, which has the familiar form p−1 1 mn ˆ ω |m n|. F =√ p
(4.5)
m,n=0
ˆ Fˆ Xˆ Fˆ † = Zˆ and Fˆ † Zˆ Fˆ = X. ˆ The Fourier operator transforms between Xˆ and Z:
4.2. Power-of-a-prime-dimensional spaces When the space dimension is a power of a prime, d = pN , we have to use elements of the finite field GF(p N ) to label states of the system and operators acting on the corresponding Hilbert space. In particular, we will denote as |α , α ∈ GF(p N ) (in the case of Zp we have used Latin characters, instead), an orthonormal basis in the Hilbert space of the quantum system, α|β = δα,β . Operationally, the elements of the basis can be labeled by powers of the primitive
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[7, § 4
elements. These vectors will be considered as eigenvectors of the generalized position operator which belong to the generalized Pauli group. The generators of this group, the generalized momentum and position operators, are defined as follows: Xˆ β |α ≡ |α + β
Zˆ β |α ≡ χ(αβ)|α ,
(4.6)
so that Zˆ β† = Zˆ −β ,
Xˆ β† = Xˆ −β ,
(4.7)
and Zˆ α Xˆ β = χ(αβ)Xˆ β Zˆ α ,
(4.8)
where χ(θ) is the additive character defined in eq. (3.3). The operators (4.6) are related through the finite Fourier transform operator, defined in a similar way as eq. (4.5) (Klimov, Sánchez-Soto and de Guise [2005]): 1 Fˆ = pN
(4.9)
χ(αβ)|α β|,
α,β∈GF(p N )
so that Fˆ Fˆ † = Fˆ † Fˆ = 1ˆ and Fˆ Xˆ α Fˆ † = Zˆ α .
EBL
(4.10)
= 1ˆ for d = (where p = 2), and = 1ˆ for d = 2N . In addition, The Fourier transform offers us the possibility of introducing the conjugate basis, which is related to the basis |α as follows: Fˆ 4
|α ˜ = Fˆ |α ,
Fˆ 2
pN
Zˆ β |α ˜ = |α + β ,
Xˆ β |α ˜ = χ ∗ (αβ)|α . ˜
(4.11)
Hence, the vectors of the conjugate basis are eigenvectors of the Xˆ β operators. In applications, we have to establish a relation between abstract states, labeled with elements of the field, and states of a given physical system. Such interrelation depends strongly on the system. For instance, the system can be a single ‘particle’ with p N energy levels or it can consist of N ‘particles’ (degrees of freedom) with p ‘levels’. In the latter case the mapping Hd ↔ Hp ⊗ Hp ⊗ · · · ⊗ Hp from the abstract Hilbert space to the N -particle vector space can be achieved by expanding an element of the field in a convenient basis {θ1 , . . . , θN }: α = a1 θ1 +· · ·+aN θN , ak ∈ Zp , so that |α → |a1 1 ⊗ · · · ⊗ |aN N ≡ |a1 , . . . , aN , and the coefficients ak play the role of quantum numbers √ of each particle. For example, in the case of two qubits, the state (|0 + |θ 3 )/ 2 can be mapped onto
7, § 4]
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483
Table 2 Two mappings between physical and abstract states in GF(22 ) State
Polynomial basis
Self-dual basis
00 01 10 11
0+0=0 0+θ =θ 1 + 0 = θ2 + θ = θ3 1 + θ = θ2
0+0=0 0 + θ2 = θ2 θ +0=θ θ + θ2 = θ3
√ the physical state (|00 + |10 )/ 2 in the polynomial basis √ (1, θ ), whereas in the self-dual basis (θ, θ 2 ) it is associated with (|00 + |11 )/ 2, see table 2. Observe that while the first state is factorizable, the other one is entangled. The mapping onto an N -particle vector space implies that every operator Zˆ β is factorized into a product of single-particle Zˆ operators (4.3). That is, Zˆ β = θ , {θ , . . . , θ } is the basis dual to Zˆ b1 ⊗ · · · ⊗ Zˆ bN , where β = b1 θ1 + · · · + bN n N 1 {θ1 , . . . , θN }, and bl ∈ Zp . To obtain a better understanding of this aspect, let us recall the definition of the operator Zˆ α given in eq. (4.7) above and choose a basis {θ1 , . . . , θN } to expand β. Then, taking α = θk as an element of the dual basis, we obtain Zˆ θk =
3 EBL 2πi exp tr(θ θ ) |b b |,
χ(θk β)|β β| =
β∈GF(p N )
N ⊗ p−1
l=1
bl =0
p
k l
l
l
@ where ⊗ l denotes the tensor product over the index l that indicates the lth particle. If k = l the duality implies that tr(θk θl ) = 0, and thus p−1 bl =0
p−1 2πbl i exp |bl bl | = 1ˆ l . tr(θk θl ) |bl bl | = p bl =0
If, on the other hand, k = l we have tr(θk θk ) = 1 and then, p−1 bk =0
p−1 2πbk i 2πbk i exp exp tr(θk θk ) |bk bk | = |bk bk | = Zˆ k . p p bk =0
Finally, we obtain the factorization Zˆ θk = 1ˆ 1 ⊗ · · · ⊗ 1ˆ k−1 ⊗ Zˆ k ⊗ 1ˆ k+1 ⊗ · · · ⊗ 1ˆ n . That is, one Zˆ operator operating on Hilbert space Hk (on the kth ‘particle’) and all the other factors are identity operators.
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The discrete Wigner function
[7, § 4
In particular, making use of an almost self-dual basis, β=
N
bk θ k ,
tr(θk θl ) = ql δk,l ,
bk , qk ∈ Zp ,
(4.12)
k=1
where qk = 1 only for a single basis element (say qk = q = 1), we have Zˆ β =
N ⊗ 3
Zˆ bl ql = Zˆ b1 ⊗ · · · ⊗ Zˆ bk−1 ⊗ Zˆ qbk ⊗ · · · ⊗ Zˆ bN .
(4.13)
l=1
In any almost self-dual basis the Xˆ β operator is a factorized product of single particle Xˆ operators (4.3) in an obvious way |β + θk β| = 1ˆ 1 ⊗ · · · ⊗ 1ˆ k−1 ⊗ Xˆ k ⊗ 1ˆ k+1 ⊗ · · · ⊗ 1ˆ N , Xˆ θk = β∈GF(p N )
where we use the expansion (4.12) for β, thus Xˆ β =
N ⊗ 3
Xˆ bk .
(4.14)
EBL
k=1
The above equation is a result of the relation (4.10) and the factorization of the finite Fourier transform operator (4.9) in the almost self-dual basis into N -particle Fourier operators (4.5), Fˆ =
N−1 3⊗
Fˆk ⊗ F˜N ,
(4.15)
k=1
where p−1 1 qmn ω |m n|. F˜N = √ p
(4.16)
m,n=0
For instance, in the case of GF(32 ) (when the self-dual basis does not exist) the Fourier operator is factorized into Fˆ = Fˆ1 ⊗ Fˆ2† in the almost self-dual basis {θ 2 , θ 4 }, where θ is a root of the irreducible polynomial x 2 + x + 2 = 0. In a self-dual basis, Fˆ is factorized into a product of single particle Fourier operators, Fˆ =
N ⊗ 3 k=1
Fˆk .
7, § 5]
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485
§ 5. Mutually unbiased bases The continuous observables qˆ and pˆ that parameterize the continuous Wigner function obey the commutation relation [q, ˆ p] ˆ =i
(5.1)
(still with h¯ = 1). These operators are canonical (and hence complementary). The natural question to ask is if there are similar observables in a discrete space, and if so, what are they? A related problem was studied already in the early years of quantum theory by Weyl [1928]. He considered a phase φ and two unitary operators Dˆ and Dˆ in a Hilbert space of dimension d such that ˆ Dˆ Dˆ = exp(iφ)Dˆ D.
(5.2)
He found that when φ = 2π/d, or more generally, when exp(idφ) = 1, there is a solution. If the (orthonormal) eigenvectors |n of Dˆ are taken as the standard basis, then the eigenvectors of Dˆ are d−1 1 exp(−i2πmn/d)|n , |m ˜ =√ d n=0
EBL
(5.3)
where 0 m d − 1, and where the tilde sign has been used to distinguish the conjugate basis from the standard one. We see that the two bases are related through the discrete Fourier transformation (4.5) and therefore 2 1 n|m ∀m, n. ˜ = d An interpretation of this equation is that if the state |m ˜ is measured by an obˆ servable having the same eigenvectors as D, every outcome is equiprobable, with probability 1/d. Such bases were named mutually unbiased in Wootters [1986] and were studied in the context of state reconstruction by Ivanovi´c [1981]. Both sets of eigenvectors discussed above constitute complete bases on the vector space. One may ask, given the first set, is the second set unique? The answer is no, and since the choice of the standard basis between Dˆ and Dˆ was arbitrary, any third basis should therefore be mutually unbiased with respect to both of them. It was shown by Ivanovi´c [1981] that at most there can be d + 1 bases where all are mutually unbiased. Ivanovi´c also provided an explicit construction of such bases when d is a prime. In matrix notation the basis generating unitary operators he derived are: 1 [Dˆ k ]uv = √ exp 2iπk(u + v − 1)2 /d , 2 k d − 1, d
486
The discrete Wigner function
[7, § 5
[Dˆ 0 ]uv = δu,v , 1 [Dˆ d ]uv = √ exp(2iπuv/d), (5.4) d where we have enumerated the standard basis 0 as usual. We note that the basis {|d, m } ≡ {|m } ˜ is the basis we have denoted conjugate basis by eq. (4.11) and above. We will henceforth use an extended set of unitary operators and denote ˆ them D(k, l), where the index k corresponds to the subscript of Ivanovi´c’s opˆ erators, eq. (5.4). The eigenvectors of D(k, l) will be denoted as |k, m . That ˆ ˆ is, D(k, l) and D(k, l ) where l = l have the same set of eigenvectors. The standard basis should be denoted |0, n in this notation, but for this basis we will suppress the basis index. The bra associated to the ket |k, m will be written k, m|. A few years later, Wootters and Fields [1989] gave an explicit construction of d + 1 mutually unbiased bases when d was an integer power of a prime. When d = p N , where p is an odd prime and N is a positive integer, an explicit state construction is 1 exp (2πi/d)tr κν 2 + μν |ν , |κ, μ = √ (5.5) d ν∈GF(d)
EBL
where κ, μ, ν ∈ GF(d) (there are d of them) and {|ν } is the standard basis. When d = 2N things become a little more complicated. Let the field elements {αu } be a basis in GF(2N ). Then any product of field elements can be written αu αv =
d−1
(w) auv αw ,
(5.6)
w=0 (w) where auv ∈ Z2 = {0, 1}. We can look at a (w) as a symmetric d × d matrix and we can also define κ, μ, ν as N -component vectors with elements in Z2 . Hence, for each of the three indices there are d vectors that can be ordered. With these prerequisites the explicit construction of the d bases mutually unbiased and all unbiased to the standard basis {|ν } becomes (Wootters and Fields [1989]): d−1 1 ν T (κ·a)ν ¯ |κ, μ = √ i (−1)μ·ν |ν , d ν=0
(5.7)
where a¯ is the d-component vector of a (w) matrices and T denotes transpose. When d is a composite number, but not an integer power of a prime, then it is presently not known how many unbiased bases exist. Many attempts have been made to answer this question (Grassl [2004], Archer [2005], Wocjan and Beth
7, § 5]
Mutually unbiased bases
487
[2005]), and the indicative answer is that it is not possible to find d + 1 bases, but the question is still open. However, for any dimension d there are at least three mutually unbiased bases. The fact that there are no two unique operators (or equivalently, no unique two sets of complete, mutually unbiased bases) that will correspond to qˆ and pˆ is perhaps surprising and disappointing. As we shall see below, several workers in the field have chosen to select two bases and try, to the largest extent possible, to retain the correspondence between these two bases in the discrete-space case to qˆ and pˆ in the continuous case. This is perfectly legitimate, but it feels to be somewhat in contradiction to the fact that mutual unbiasedness is a ‘symmetric’ property that does not single out any basis. To further motivate this section’s digression about mutually unbiased bases, a topic that will return eventually, we note that they have mainly been motivated by considering optimal strategies for state reconstruction. A general state in a d-dimensional Hilbert space can be described by a density operator ρ. ˆ Expressed in matrix form, the operator contains d 2 − 1 independent real numbers (since Tr ρˆ = 1 and ρˆ = ρˆ † ). State tomography aims to estimate these numbers by making clever measurements on an ensemble of identically prepared states. Obviously one could project the state onto the d basis vectors of d + 1 mutually unbiased bases. Hence, one would obtain (d + 1)d = d 2 + d probabilities, of which d − 1 in each basis would be linearly independent. One would get (d − 1)(d + 1) = d 2 − 1 linearly independent probabilities, and from these probabilities one could reconstruct the density matrix. However, as the obtained probabilities would not be exact, but only measured estimates, one would be well advised to make the projectors ‘as linearly independent as possible’ to minimize the estimation errors. This is precisely what the mutually unbiased bases accomplish. As an alternative to representing the state with a density matrix, or the marginal probability distributions of d + 1 mutually unbiased bases, one could instead use a discrete Wigner function that is a real function defined on a d × d grid of phase-space points. The normalization condition requires that the sum of all these numbers equals unity, so only one of them would not be linearly independent. From the arguments above it follows that a normalized Wigner function represents d 2 − 1 independent real numbers and therefore it can represent an arbitrary state. We shall see below that there is, indeed, a very natural connection between the marginal probability distributions of the mutually unbiased states and the discrete Wigner-function coefficients.
EBL
488
The discrete Wigner function
[7, § 6
§ 6. The discrete Wigner function As we have hinted in the previous sections, the dimensionality of the system plays an important role when defining a discrete Wigner function. Note the indefinite article in the previous sentence, for there is no unique way of defining a discrete Wigner function, and several different constructions have been proposed, as discussed in § 1. Several dividing lines between different constructions can be identified. The simplest case is when the space dimension d is prime. Another case is when d is odd but composite with at least two different factors. A third case is when d is even but d = 2. Finally, a fourth and fifth case are when d = p N , where p is a prime, and here the construction depends on whether p = 2 or p is an odd prime. Another dividing line, when d is a composite number, is whether or not to decompose the space into factor spaces. In some cases, in particular if the system is composed of ‘natural’ subsystems (e.g., N ‘particles’), it seems reasonable to do so. In other cases, e.g., for a spin 5/2-system (having d = 6 = 2 · 3) it is questionable what could be gained from such a factorization, except for mathematical simplicity.
EBL
6.1. Wigner function in prime-dimensional spaces In this space, due to the existence of finite groups for prime-dimensional spaces, there is a universally accepted construction algorithm. However, we shall derive (or define) this function in two different ways, one closely connected to physical operators and their actions, and one more formal and abstract, using field algebra. 6.1.1. A derivation from physical states and operators Choose any orthonormal state-vector set as the standard basis labeled {|n }. A mutually unbiased basis can be constructed from the operators in eq. (5.4). If we choose the dth operator (the conjugate basis that we sometimes denote with a tilde), we get d−1 1 exp(−2iπmn/d)|n , |d, m = √ d n=0
(6.1)
where we have followed the phase-sign convention in Leonhardt [1995] rather than Ivanovi´c’s in eq. (5.4) above.
7, § 6]
The discrete Wigner function 5
4
2
3
1
5
3
4
1
2
5
2
1
4
3
5
1
3
2
4
o
0
0
0
0
489
Fig. 2. The six possible rays (all crossing the origin) in the phase-space mapping the five-dimensional Hilbert space.
A discrete Wigner function Wρˆ (a, b) of the density matrix ρˆ can then be defined as 1 4πian exp − b − n|ρ|b ˆ + n Wρˆ (a, b) = d n d 4πibm 1 exp = (6.2) d, b − m|ρ|d, ˆ b + m . d m d More generally, by replacing ρˆ with an arbitrary operator the Wigner function of that operator is obtained. The definition (6.2) implies that for a standard basis state |n 1 W|n n| (a, b) = δb,n . d
EBL
(6.3)
In analogy with the continuous Wigner function, the discrete Wigner function of the ‘conjugate’ mutually unbiased state |d, m will consist of vertical ‘lines’, i.e., W|d,m d,m| (a, b) =
1 δa,m . d
(6.4)
We note that the lines corresponding to the standard basis and the ‘conjugate’ mutually unbiased basis have slopes zero and infinity, respectively. For each slope, there are d parallel lines. Each such set of d parallel lines was called a striation by Gibbons, Hoffman and Wootters [2004]. As the two striations described by eqs. (6.3) and (6.4) have different slopes, any two lines, one line from each striation, will have one, and only one, crossing point, corresponding to their unbiasedness. It is not hard to guess that the d − 1 additional unbiased bases correspond to striations in phase space with slopes k, where k = 1, . . . , d − 1. These striations can be parameterized (a, b) = (a, ka + l), where a = 0, . . . , d − 1, 0 k d denotes the striation, and 1 l d denote the line in the striation. Figure 2 illustrates the six possible rays in the 5 × 5 phase-space. The horizontal ray 0 corresponds to the standard basis state |0 , while the vertical ray 5 corresponds to
490
The discrete Wigner function
[7, § 6
the conjugate state |5, 0 . All rays cross at the phase-space origin which has been labeled ‘o’. The discrete Wigner function (6.2) is real-valued for (Hermitian) density matrices. It obeys a similar relation for marginal probability, here called P (k, m), where k denotes the basis and m the basis state, as in the continuous case, namely P (0, n) ≡ n|ρ|n ˆ =
d
W|n n| (a, n)
(6.5)
a=1
and P (d, m) ≡ d, m|ρ|d, ˆ m =
d
W|d,m d,m| (m, b).
(6.6)
b=1
Let us now construct a Hermitian operator from the standard basis as follows: Hˆ 0 =
d−1 2πn n=0
d
|n n|.
(6.7)
We can subsequently write a corresponding unitary displacement operator d−1 −2iπkn ˆ D(k, 0) = exp(−iHˆ 0 k) = (6.8) exp |n n|. d
EBL
n=0
Now let us see what happens if we apply such an operator to one of the mutually unbiased basis states. In the standard basis these states have the expansion d−1 1 iφk,m,n |k, m = √ e |n , d n=0
(6.9)
where φk,m,n are real-valued phases. All these phases are roots of the equation eidφk,m,n = 1. For example, when k = 0 we get d−1 † 1 2iπn(s−l)/d k, m| Dˆ l (1, 0) Dˆ s (1, 0)|k, m = e d n=0 1 if s − l = 0, d, 2d, . . . , = 0 otherwise,
(6.10)
where s and l are integers. The interpretation of this result, since it is valid for any ˆ m, s, l, and k = 0, is that the operator D(1, 0) permutes each of the vectors |k, m into another vector in the same basis. The (shortest) cycle of this permutation is d, ˆ Hence we can generate a whole basis-vector set of any of the so Dˆ d (1, 0) = 1. ˆ bases (except the standard basis) by applying, sequentially, D(1, 0) to any state in the basis. Specifically, in the conjugate basis d, given explicitly by eq. (6.1), we
7, § 6]
The discrete Wigner function 4
1
3
0
2
3
0
2
4
1
2
4
1
3
0
1
3
0
2
4
0
2
4
1
3
491
Fig. 3. The five lines of striation 3 in fig. 2. The phase-space points belonging to the ray are denoted 0.
find that ˆ D(1, 0) =
d
|d, m + 1 d, m|
⇔
ˆ Dˆ k (1, 0) = D(k, 0),
(6.11)
m=1
where the summation should be interpreted cyclicly, so that d +1 = 1. From this it ˆ becomes clear that D(k, 0) is a discrete displacement operator along the horizontal axis (direction) in the d ×d-dimensional phase-space spanned by eq. (6.2). It is also clear that this operator will uphold translational covariance as we can express any density operator in the basis {|d, m }, and every vector in this basis will be ˆ translated k phase-space points horizontally by the operator D(k, 0) (respecting the modulus d algebra). We can also define the translation operator
EBL
ˆ D(0, l) =
d−1
|n + l n|.
(6.12)
n=0
This operator translates the standard basis states cyclicly l ‘places’, and permutes all other bases, mutually unbiased to basis d, in a d-cyclic fashion. In discrete phase space it is a vertical displacement operator (translating all phase-space points l points upwards), thus upholding translational covariance. From fig. 3 we ˆ can see that applying D(0, 1) sequentially to the state |3, 0 one will generate, in turn, the states |3, 1 , |3, 2 , |3, 3 and |3, 4 (and the corresponding, vertically ˆ translated lines 1–4). If, instead, we apply the operator D(1, 0) to the state |3, 0 (ray 3 in fig. 2) the five orthogonal states |3, 2 , |3, 4 , |3, 1 and |3, 3 (nonintersecting lines 2, 4, 1, and 3) are generated, in that order. The discrete version of the overlap relation (2.6) also holds. That is, if the discrete Wigner function of an operator Oˆ k is denoted WOˆ k (a, b), then Tr(Oˆ k Oˆ l ) = d
d d
WOˆ k (a, b)WOˆ l (a, b).
a=1 b=1
Again, the relation is useful for calculating ensemble averages.
(6.13)
492
The discrete Wigner function
[7, § 6
We have stressed above that the bases 0 and d have no particular physical significance, but by defining the Wigner function according to eq. (6.2) we have chosen these bases to represent the horizontal and vertical axes (corresponding to qˆ and pˆ in the continuous case). Of course we can define unitary displacement operators Dˆ k in the same manner as in eqs. (6.8) and (6.12) from the basis states of any basis. They will act as displacement operators such that Dˆ k will d-cyclicly permute the vectors of any basis, mutually unbiased to basis k. The simplest example of such a set of translation operators is when d = 2. In this case the operators Dˆ k will have real eigenvalues, so they can be taken to be the Pauli operators σˆ x , σˆ y , σˆ z . If a particle is spinning along the positive z-axis (i.e., the particle is in an eigenstate of σˆ z ), a rotation of ±π around either the x-axis or the y-axis (accomplished by the operators σˆ x and σˆ y , respectively) will make the particle spin along the negative z-axis, i.e., be transformed into the orthogonal state. By simply relabeling the coordinate system, one understands that the relations are identical between any of the three operators σˆ x , σˆ y and σˆ z , reflecting their mutually unbiasedness. 6.1.2. A derivation from abstract states and operators Recall from eq. (4.2) that the operators k, l = 0, 1, . . . , d − 1, (6.14) EBL where φ(k, l) is an arbitrary phase, form a complete (displacement) operator basis Dˆ φ (k, l) = e−iφ(k,l) Zˆ k Xˆ l ,
in the prime-dimensional space Hd (Schwinger [1960b, 1960c]). However, the unitary condition (here we assume that d = 2) Dˆ φ† (k, l) = Dˆ φ−1 (k, l) = Dˆ φ (−k, −l)
(6.15)
and the periodicity condition Dˆ φ (k, l) = Dˆ φ (k + d, l + d) restrict the possible choices of the phase φ. Note that Dˆ φ (k, l) is a generalization of eq. (6.8) and therefore will be a cyclic operator as indicated by eq. (6.10). Different choices of the phases have been analyzed in Vourdas [2004], Wootters [1987], Luis and Peˇrina [1998], Hakioglu [1998], Galetti and Ruzzi [1999, 2000], Klimov and Muñoz [2005]. To get a simple relation with the case when the space dimension is the power of a prime, d = p N , we will consider the following phase choice (Vourdas [2004], Wootters [1987]): 2π −1 φ= (6.16) 2 kl , d where 2−1 is the multiplicative inverse of 2 in Zp , and only the factors inside the brackets are field elements (where the multiplications are to be taken modulus d),
7, § 6]
The discrete Wigner function
493
so that −1 ˆ D(k, l) = ω−2 kl Zˆ k Xˆ l ,
(6.17)
and satisfy the composition relation ˆ 1 , l1 )D(k ˆ 1 + k2 , l1 + l2 ), ˆ 2 , l2 ) = ω2−1 k1 l2 −2−1 k2 l1 D(k D(k and the relation 1 ˆ ˆ D(k, l) = P. d
(6.18)
k,l
Pˆ is the parity operator ˆ P|n = |−n ,
(6.19)
where n + (−n) = 0 modulus d. Physically, this is the basis for translational covariance as subsequent translations by the unitary operators add up in a natural manner. The discrete Weyl symbol WOˆ of an operator Oˆ is defined as an invertible mapping WOˆ (k, l) = so that Oˆ =
EBL
1 ˆ ˆ l) , Tr O Δ(k, d
ˆ l), WOˆ (k, l)Δ(k,
(6.20)
(6.21)
k,l
where the operational kernel is defined as (Vourdas [2004]) 1 kn−lm ˆ ˆ l) = ω Δ(k, D(m, n) = Δˆ † (k, l). d m,n
(6.22)
ˆ l) can also be It is worth noting here that the condition of Hermiticity of Δ(k, used for fixing the phase φ in (6.14) instead of the unitary condition (6.15). An important point we stress is that the kernel (6.22) satisfies the so-called Stratonovich–Weyl postulates (Stratonovich [1956], Várilly and Gracia-Bondía [1989]), i.e., it is Hermitian and normalized ˆ ˆ l) = d 1; (6.23) Δ(k, k,l
covariant under transformations of the Pauli group, ˆ ˆ + m, l + n); ˆ l)Dˆ † (m, n) = Δ(k D(m, n)Δ(k,
(6.24)
494
The discrete Wigner function
and admits the orthogonality relation ˆ l)Δˆ † (k , l ) = dδk,k δl,l . Tr Δ(k,
[7, § 6
(6.25)
As intended, the discrete Weyl symbol WOˆ (k, l) is defined in the phase space Zd ⊗ Zd , a set of points that form a finite geometry (Lidl and Niederreiter [1986]) when d is a prime. In this case one can define a concept of lines, discussed above, and other geometric structures that will be discussed briefly below. The kernel (6.22) can conveniently be represented as follows ˆ ˆ l) = D(k, ˆ 0)Dˆ † (k, l), Δ(k, l)Δ(0,
(6.26)
where ˆ 0) = Δ(0,
1 ˆ D(k, l) d k,l
for any choice of the phase φ, although in the case (6.16) it acquires a form similar ˆ where Pˆ is the parity ˆ 0) = P, to the continuous flat situation (Royer [1977]), Δ(0, operator defined in eq. (6.19). The Weyl symbol of the density operator is the discrete Wigner function in the Zp ⊗ Zp space. Hence, replacing Oˆ by ρˆ in eq. (6.20) one has expressed eq. (6.2) in an abstract form. The symbols (6.20) are also covariant under transformations from the generalized Pauli group and satisfy the overlap relation (6.13). Expanding an arbitrary operator in the basis (6.17), ˆ Ok,l D(k, l), Oˆ = (6.27)
EBL
k,l
we can easily obtain the following useful representation for the Wigner function of the operator Oˆ in terms of its coefficients Ok,l : 1 WOˆ (k, l) = (6.28) Om,n ωlm−kn . d m,n Let us now consider some typical examples: The Wigner functions corresponding to the eigenstates of the Zˆ and the Xˆ operators are given by eqs. (6.3) and (6.4), respectively. Furthermore, the Wigner functions of the Zˆ and Xˆ operators are WZˆ (k, l) = ωl and WXˆ (k, l) = ω−k . There are two important observations to retain from this section. The first is that there is a relation between unitary transformations in Hilbert space and translations in phase space. This connection will be used extensively below. The second is that the definition of the Wigner function usually assigns special significance to one pair of mutually unbiased operators, a significance that is not reflected in the physical theory.
7, § 6]
The discrete Wigner function
495
6.2. Wigner function in composite-dimensional spaces As mentioned above, even- and odd-dimensional Hilbert spaces may have to be treated differently. The reasons for this are subtle, but roughly speaking one reason is that one root to the equation eidφk,m,n = 1 is −1 for any even number d, and this root is real, which is not the case for any of the non-trivial roots when d is odd. 6.2.1. Odd dimensionality For odd composite dimensions, the Wigner function defined for odd primedimensions in § 6.1 works well. However, it is important to realize that while we can always find two mutually unbiased bases (even three, and sometimes more), there is no known algorithm to construct d +1 mutually unbiased bases. In fact, as mentioned in § 5, how many bases exist in such a space is still an open question. 6.2.2. Even dimensionality When the composite space dimension d is even (but larger than two), the definition given in eq. (6.2) does no longer represent the state properly (and the assumption in § 6.1.2, that d is a prime no longer holds). This can be seen, e.g., from the marginal distribution d b=1
Wρˆ (n, b) =
1 d
d b=1 m
EBL
exp
4πibm d, b − m|ρ|d, ˆ b + m d
= n|ρ|n ˆ + n + d/2|ρ|n ˆ + d/2 = P (0, n).
(6.29)
One way to solve this problem is to increase the phase-space grid to 2d × 2d points. We also let the grid indices and summation indices a, b, m and n take on half odd and integer values between −d/2 + 1/2 and d/2. One can subsequently define a discrete Wigner function (Leonhardt [1996]) 1 4πian exp − b − n|ρ|b ˆ + n Wρˆ (a, b) = 2d n d 1 4πibm = (6.30) exp d, b − m|ρ|d, ˆ b + m . 2d m d In order for the definitions to make sense a further convention is needed, namely that matrix elements with half-integer values vanish. The definition makes the graphical representation of states a bit more difficult. For example, the representation of the states |n (or |d, m ), that were visually straight horizontal (vertical) lines in the odd-dimensional Hilbert space, now becomes two horizontal (vertical)
496
The discrete Wigner function
[7, § 6
lines, one at b = n and another at b = n + d/2. The Wigner-function coefficients along the latter have the alternating values (2d)−1 and −(2d)−1 , so the sum of the Wigner-function coefficients over this line is zero. Such lines were called ‘ghost lines’ by Leonhardt [1996]. 6.2.3. Factorized Wigner function An alternative to the two Wigner-function definitions above, for composite odd and composite even dimensions, respectively, is to factor the space into its prime factors and then define the Wigner function as the Cartesian product of the Wigner functions in each prime space. This view was advocated by Wootters [1987] and Asplund and Björk [2001]. If the system has a natural physical division that coincides with the Hilbert-space factorization it seems natural to make such a factorization. If the system is not in an entangled state, then the total-system Wigner function can be written as a convex sum over the Wigner functions of the individual spaces. In general, however, this is not possible. The approach has the advantage of treating all spaces on equal footing, and not to introduce redundancy. A price to be paid is that the inversion algorithm, obtaining states or operators from the Wigner function, becomes somewhat involved (Asplund and Björk [2001]). Obtaining the Wigner function, or the density operator, from the marginal probability distributions also becomes less efficient in the power-of-a-primedimensional case. If one factors the Wigner function into N functions, each defined on a p × p phase space, one needs, for each such phase space, p − 1 independent probabilities for each of p +1 measurement bases. In all, (p 2 −1)N probabilities are needed to reconstruct the state in this manner. If we instead use the construction discussed in the next subsection, that uses the p N + 1 mutually unbiased bases in this space, we need (pN + 1)(p N − 1) = p 2N − 1 measurement probabilities, clearly fewer than in the preceding case.
EBL
6.3. Wigner function for pN -dimensional space In a power-of-a-prime-dimensional space the operational basis is formed by the displacement operators having a form similar to the prime-dimensional case (6.14): ˆ D(α, β) = φ(α, β)Zˆ α Xˆ β ,
(6.31)
where the phase factor φ(α, β) is such that the unitary condition (6.15) is satisfied, implying that φ(α, β)φ ∗ (α, β) = 1.
(6.32)
7, § 6]
The discrete Wigner function
497
Also, we impose the conditions φ(α, 0) = φ(0, β) = 1, which means that the displacements along the horizontal axes α and vertical axes β, corresponding to the generalized momentum and position operators, are not associated with any phase. The operator basis (6.31) becomes orthogonal, ˆ 1 , β1 )D(α ˆ 2 , β2 ) = dδ−α1 ,α2 δ−β1 ,β2 , Tr D(α (6.33) ˆ −β), if the phase φ(α, β) satisfies the followor equivalently Dˆ † (α, β) = D(−α, ing condition: φ(α, β)φ(−α, −β) = χ(−αβ),
(6.34)
and for the particular case of the fields of even characteristic, d = 2N , which is the case for two or more qubits, it implies that φ 2 (α, β) = χ(αβ),
(6.35)
ˆ ˆ which is tantamount to Dˆ † (α, β) = D(α, β), that is, D(α, β) is Hermitian. In general, the displacement operators are non-Hermitian and thus cannot be used for mapping Hermitian operators onto real phase-space functions. Nevertheless, a desirable Hermitian kernel can be defined (Klimov and Muñoz [2005], Vourdas [2005], Klimov, Muñoz and Romero [2006]) in the same manner as in the prime-dimensional case (6.22): 1 ˆ ˆ β) = Δ(α, (6.36) χ(αλ − βκ)D(κ, λ). d
EBL
κ,λ∈GF(d)
In this equation, the kernel indices are elements of the field GF(p N ) instead of the numbers 1, . . . , p as in the prime-dimensional case. It is important to stress that the Stratonovich–Weyl postulates (6.23), (6.24) and (6.25) are still satisfied for the kernel. Therefore, the Wigner function of an operator Oˆ can be defined in the analogous way as in the prime-dimensional case (6.20), (6.21): 1 ˆ ˆ β) . Tr O Δ(α, d The inversion relation is ˆ β). WOˆ (α, β)Δ(α, Oˆ = WOˆ (α, β) =
α,β∈GF(d)
The overlap relation has the standard form WOˆ k (α, β)WOˆ l (α, β), Tr(Oˆ k Oˆ l ) = d α,β∈GF(d)
(6.37)
(6.38)
498
The discrete Wigner function
[7, § 7
and, as a particular case, the average value of the operator Oˆ is calculated as ˆ =d Wρˆ (α, β)WOˆ (α, β), Tr(ρˆ O) (6.39) α,β∈GF(d)
as expected. In terms of the expansion coefficients of the operator Oˆ [which are identical in form to those in eq. (6.27)] in the operational basis (6.31), the Weyl symbol of Oˆ has the form 1 WOˆ (α, β) = (6.40) Oκ,λ χ ∗ (αλ − βκ), d κ,λ∈GF(d)
which again is essentially the same expression as (6.28). To give some simple examples, we find that the Wigner functions of the operators Zˆ κ and Xˆ λ are 1 1 χ(βκ), WXˆ λ (α, β) = χ(−αλ), (6.41) d d which, in the particular case of prime dimension, reduce to the above-mentioned symbols of eq. (4.3). In the same way, the Wigner functions of the basis states |ν and |μ ˜ are WZˆ κ (α, β) =
EBL
1 1 W|ν ν| (α, β) = δβ,ν , W|μ ˜ μ| ˜ (α, β) = δα,μ , d d in full similarity with eqs. (6.3) and (6.4). The phase appearing in the definition of the displacement operators can be determined by imposing an additional geometric condition on the Wigner function WOˆ (α, β), introduced by Wootters [1987] and intimately related to the desired invariance properties.
§ 7. Reconstruction of the density operator from the discrete Wigner function 7.1. Lines and rays In the discrete space GF(d) × GF(d) the concept of a line can be introduced in a similar way as in the Euclidean plane. A set of points (α, β) ∈ GF(d) × GF(d) satisfying the relation κα + λβ = θ,
(7.1)
7, § 7]
Reconstruction of the density operator from the discrete Wigner function
499
where κ, λ and θ are some fixed elements of GF(d) is called a line. Two lines κ α + λ β = θ ,
κα + λβ = θ,
(7.2)
are parallel if they have no common points, which implies that λκ = κλ . If the lines (7.2) are not parallel they cross each other at a single point. A line that passes through the origin is called a ray and its equation has the form α = 0,
or
β = κα,
(7.3)
so that α = 0 and β = 0 are the vertical and horizontal axes respectively. Each ray is characterized by the value of the ‘slope’ κ. We denote by Λκ the ray corresponding to β = κα, and by Λ˜ the ray corresponding to the vertical axis. There are d − 1 lines parallel to each of d + 1 rays, parameterized β = κα + μ, so that the total number of lines is d(d + 1). The collection of d parallel lines is the striation. The displacement operators, labeled with points of the phase space, belonging to the same ray commute (here we omit the phase factor): Zˆ α1 Xˆ β1 =κα1 Zˆ α2 Xˆ β2 =κα2 = Zˆ α2 Xˆ β2 =κα2 Zˆ α1 Xˆ β1 =κα1 ,
(7.4)
EBL
and thus, they have a common system of eigenvectors {|κ, μ , κ, μ ∈ GF(d)}: Zˆ α Xˆ κα |κ, μ = exp(iξκ,μ )|κ, μ ,
(7.5)
where κ is fixed and exp(iξκ,μ ) is the corresponding eigenvalue, so that |0, ν ≡ |ν are eigenstates of the Zˆ α operators (displacement operators labeled with the points of the ray Λ0 , i.e., the horizontal axis) and |˜ν = Fˆ |ν are the eigenstates ˜ of the Xˆ β operators (displacement operators labeled with the points of the ray Λ, the vertical axis). Let us suppose that |κ, 0 is a state associated to a ray in the κth striation, Zˆ α Xˆ κα |κ, 0 = exp(iξκ,0 )|κ, 0 . Then all the other states associated to lines of this striation can be obtained by applying the operator Xˆ μ to the state |κ, 0 : |κ, μ = Xˆ μ |κ, 0 ,
(7.6)
and exp(iξκ,μ ) = exp(iξκ,0 )χ(αμ), which allows us to associate the state |κ, μ with the line β = κα + μ. Of course, we can also obtain any of the lines (except for striation κ = 0) by operating with Zˆ α on the associated ray. In fig. 4 the five rays generated in this manner in the 4 × 4-dimensional phase space are shown. The rays are labeled with the ‘slope’ κ = 0, 1, θ , and θ −1 . ˜ The multipliThe vertical ray, corresponding to the conjugate basis, is labeled 4. cation rules for the field elements are given in the multiplication table 1 in § 3.
500
The discrete Wigner function 4˜
θ −1 θ
4˜
θ
1
4˜
1
θ −1 θ
o
0
0
[7, § 7
1 θ −1
0
Fig. 4. The five rays in the phase-space mapping the four-dimensional Hilbert space.
7.2. Marginal probability density and the density operator The discrete Wigner function can be constructed using projective measurements leading to the marginal probability distributions P (κ, μ), associated with a summation over the lines (Wootters [1987], Asplund and Björk [2001]), and corresponding to the integration along a certain direction in the continuous case as indicated in eq. (2.2). Let us start with the relation between the marginal distribution probability P (κ, μ) and the Wigner function: Wρˆ (α, β)δβ,κα+μ = Wρˆ (α, κα + μ) P (κ, μ) = (7.7) α,β∈GF(d)
and for the conjugate basis P (d, μ) = μ|ρ| ˜ μ ˜ =
EBL
α∈GF(d)
Wρˆ (μ, β).
(7.8)
β∈GF(d)
The full set of marginal probability densities can be used to express the density operator (Ivanovi´c [1981], Asplund, Björk and Bourennane [2001]): ˆ P (κ, μ)|κ, μ κ, μ| + P (d, μ)|μ ˜ μ| ˜ − 1. ρˆ = (7.9) κ,μ∈GF(d)
μ∈GF(d)
We can expand the marginal probability densities in terms of the components of the density matrix in the basis of the displacement operators (6.17). The expansion is obtained by taking into account the relations (3.2), (3.4), (6.40), (7.7) and (7.8): %λ,κλ χ(λμ), P (κ, μ) = λ∈GF(d)
P (d, μ) =
%0,λ χ ∗ (λμ).
(7.10)
λ∈GF(d)
In eq. (7.10) we have used %κ,λ to denote the density-matrix expansion coefficients ˆ in the basis D(κ, λ) so that ˆ %κ,λ D(κ, λ), ρˆ = (7.11) κ,λ∈GF(d)
7, § 7]
Reconstruction of the density operator from the discrete Wigner function
501
in analogy with eq. (6.27). Equation (7.10) can immediately be inverted using eq. (3.4): %λ,κλ =
1 d
P (κ, μ)χ(−λμ),
μ∈GF(d)
or, changing the indices, %κ,μ =
1 d
P λ−1 μ, λ χ(−κλ),
(7.12)
λ∈GF(d)
where κ = 0. To reconstruct the matrix element %0,ν we have to use the results of measurements in the conjugate basis (7.8), which leads to %0,ν =
1 d
P (d, μ)χ(μν)
⇒
μ∈GF(d)
%0,0 =
1 . d
(7.13)
Finally, using eqs. (6.33) and (7.11) we also establish the relation %κ,λ =
1 ˆ† Tr ρˆ D (κ, λ) . d
7.3. Tomographic reconstruction
(7.14)
EBL
To express the Wigner function in terms of the marginal probabilities, we first note that this function is normalized, which follows from eq. (7.7) and the fact that any probability distribution is normalized: P (κ, μ) = 1 = Wρˆ (α, κα + μ) = Wρˆ (α, β). μ∈GF(d)
μ,α∈GF(d)
α,β∈GF(d)
(7.15) Above, we have associated the line (α, κα + μ) to the state |κ, μ . To express Wρˆ (α, β) in terms of P (κ, μ), that is, to tomographically construct the Wigner function, we write
1 Wρˆ (α, β) = (7.16) P κ, β + (−κα) + P (d, α) − 1 , d κ∈GF(d)
where (−κα) indicates the field element such that κα + (−κα) = 0. That is, we add the marginal probabilities of all the projections onto mutually unbiased states whose associated lines cross the point (α, β). The sum in eq. (7.16) sums one marginal distribution probability from each striation. Every marginal, in turn, is the sum of the Wigner function along the line associated with the state |κ, μ . Since
502
The discrete Wigner function
[7, § 7
the lines in the sum correspond to unbiased states, the lines only cross at (α, β). Hence, the Wigner function at this point is summed d + 1 times, while every other point is summed once. Using eq. (7.15), we arrive at eq. (7.16). With the alternative density operator representation (7.11), we can use eqs. (7.7) and (7.12) to write 1 Wρˆ α, κ −1 μα + λ χ(−κλ) %κ,μ = (7.17) d α,λ∈GF(d)
and %0,μ =
1 d
Wρˆ (λ, β)χ(μλ).
(7.18)
β,λ∈GF(d)
7.4. Rotation operators The ‘rotation’ operators Vˆκ , which transform eigenstates of the operators associated with the ray β = κα ˆ Zˆ α Xˆ κα , Zˆ α Xˆ κα , . . .} {1, 1 1 2 2
EBL
(7.19)
into eigenstates of the operators labeled with points of the ray β = (κ + κ )α ˆ Zˆ α Xˆ (κ+κ )α , Zˆ α Xˆ (κ+κ )α , . . .}, {1, 1 2 1 2 are defined through the relations (Klimov, Muñoz and Romero [2006]) ˆ Vˆκ Zˆ α Vˆκ† = exp iϕ(α, κ) Zˆ α Xˆ κα , [Vˆκ , Xˆ μ ] = 0, Vˆ0 = 1,
(7.20)
(7.21)
for all κ, μ ∈ GF(d). Thus, if |κ, μ (7.5) is a state assigned to the ray β = κα, then the state Vˆκ |κ, μ is an eigenstate of the set (7.20). That is: Zˆ α Xˆ (κ+κ )α Vˆκ |κ, μ = exp i ξκ,μ − ϕ(α, κ ) Vˆκ |κ, μ . (7.22) This means that we can interpret the action of the operator Vˆκ on the ray Λκ as a ‘rotation’ in the discrete phase space: Vˆκ
Λκ −→ Λκ+κ ,
(7.23)
although care should be taken in the case of GF(2N ), as we will discuss below. Note that one cannot reach the vertical axis Λ˜ by applying Vˆκ to any other ray. Equation (7.22) together with eq. (7.6) suggest a form for the discrete phasespace construction: we arbitrarily associate the eigenstate |0, 0 ≡ |0 of the set of Zˆ α operators, with the horizontal axis, β = 0 (and fix in such way a class of
7, § 7]
Reconstruction of the density operator from the discrete Wigner function
503
quantum nets, see Wootters [2004] and Gibbons, Hoffman and Wootters [2004]). It is suitable to choose the state |0 as the eigenstate with all eigenvalues equal to unity (this state is unique). All the other states of the ‘first’ striation are obtained by applying the displacement operator Xˆ ν to |0 , so that the state |ν = Xˆ ν |0
(7.24)
is associated with the horizontal line that crosses the vertical axis at the point (0, ν), i.e., with the line β = ν. All the other striations are constructed as follows: first we apply the rotation operator Vˆκ to the state |0 and obtain the first state |κ, 0 = Vˆκ |0 . It is associated with the ray β = κα. The state |κ, 0 is an eigenstate of the set {Zˆ α Xˆ κα } according to eq. (7.22). All the other states of the κth striation are obtained by applying the operator Xˆ μ to the state |κ, 0 : |κ, μ = Xˆ μ |κ, 0 .
(7.25)
The states |κ, μ are associated with the lines β = κα + μ. The phase exp[−iϕ(α, κ)] in the above equation is defined in eq. (7.21). The last striation, corresponding to the set of vertical lines with an obvious association |˜ν = Fˆ |ν , corresponds to the line crossing the horizontal axis at the point (ν, 0). The states constructed according to eqs. (7.24) and (7.25) form mutually unbiased bases. The rotation operator Vˆκ is diagonal in the conjugate basis (4.11),
EBL
Vˆκ =
cμ,κ |μ ˜ μ|, ˜
c0,κ = 1,
(7.26)
μ∈GF(d)
where the coefficients cκ satisfy the following condition: ∗ = exp iϕ(α, μ) χ(−μακ). cκ+α,μ cκ,μ In particular, for κ = 0 we obtain ∗ = cα,μ , exp iϕ(α, μ) = cα,μ c0,μ
(7.27)
that is, ∗ = cα,μ χ(−μακ), cκ+α,μ cκ,μ
(7.28)
and substituting α = 0 we get |cκ,μ |2 = 1, which also follows from the unitary ˆ condition Vˆμ Vˆμ† = Vˆμ† Vˆμ = 1.
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The discrete Wigner function
[7, § 7
7.4.1. Fields of odd characteristic In the case of fields of odd characteristic, we impose an additional restriction on the rotation operators: we demand that Vˆκ form an Abelian group Vˆκ Vˆκ = Vˆκ+κ ,
(7.29)
∗ which implies that cκ,μ cκ,μ = cκ,μ+μ and, in particular, cκ,μ = cκ,−μ , leading to
Vˆκ† = Vˆ−κ . In this case the relation (7.23) is well defined, i.e., the operator Vˆκ transforms a state associated with the ray Λκ into a state associated with the ray Λκ+κ . It can be shown that the condition (7.29) cannot be satisfied for fields of even characteristic, so that this case should be considered separately. For fields of odd characteristic a solution of eq. (7.28) can be found as cκ,μ = χ −2−1 κ 2 μ , (7.30) so that
Vˆκ =
χ −2−1 μ2 κ |μ ˜ μ|. ˜
EBL
μ∈GF(d)
(7.31)
In the prime field case, GF(p), p = 2, the whole set of rotation operators is produced by taking powers of a single operator (Vourdas [2004]) Vˆ =
d−1
−1 m2
ω−2
|m ˜ m|. ˜
(7.32)
m=0 In particular, the state Vˆ k |k, 0 is associated with the ray b = (k + k )a, so that
Vˆ
Λκ −→ Λκ+1 ,
Vˆ 2
Λκ −→ Λκ+2 ,
etc.
7.4.2. Fields of even characteristic The situation is more complicated when d = 2N . In fact, it follows from eq. (7.28) that 2 = χ κ 2μ , cκ,μ (7.33) whose solution is not unique, and thus there is an ambiguity in solving eq. (7.28). One consequence of this ambiguity is that operators of the form (7.26), where cκ,μ is a particular solution [for a fixed value of μ and κ ∈ GF(2N )] of eq. (7.28), do not form a group. The operator Vˆκ2 is, hence, not the identity operator. In fact,
7, § 7]
Reconstruction of the density operator from the discrete Wigner function
using eq. (7.33) we have 2 Vˆκ2 = cκ,μ |μ ˜ μ| ˜ = μ∈GF(2N )
χ μ2 κ |μ ˜ μ|, ˜
505
(7.34)
μ∈GF(2N )
which due to the property tr θ = tr θ 2 , θ ∈ GF(2N ), can be transformed into N−1 |μ ˜ μ| ˜ = Xˆ κ 2N−1 , χ μκ 2 Vˆκ2 = (7.35) μ∈GF(2N ) N−1
where the relation μ2 κ = (μκ 2 )2 has been used. It also follows from eq. (7.35) that in the set {Vˆκ , κ ∈ GF(2N )} an inverse operator to a given Vˆκ from this set does not exist. To find the inverse operator to some Vˆκ we have to extend the set {Vˆκ , κ ∈ GF(2N )} to the whole collection of rotation operators. To do that we note that eq. (7.28) is automatically satisfied after the substitution ν = cκ,μ χ(−κμ), cκ,μ → cα,μ
which means that different sets of operators Vˆκ have the form (7.36) EBL is constructed using an arbitrary solution of eq. (7.28). Then, one can
Vˆκ,λ = Vˆκ Xˆ λ ,
where Vˆκ conclude from eq. (7.35) that
(Vˆκ,ν )−1 = Vˆκ,κ 2N−1 +ν , which implies the following relation between cκ,μ : N−1 ∗ cκ,μ . = χ κμ2 cκ,μ
(7.37)
We will fix the operator Vˆκ,μ=0 in such a way that the coefficients cκ,κ , corresponding to the basis elements of the field, κ = θ1 , . . . , θn , are chosen positive, so that we have % N−1 & N N 3 kl θ l kj θ j χ ks2 θs2 μ , cκ,μ = χ μ (7.38) l=1
κ=
N
kl θ l ,
j =l+1
kl ∈ Z2 ,
s=1
(7.39)
l=1
where the principal branch of the square root in eq. (7.38) is chosen. All the other possible rotation operators can be obtained according to eq. (7.36).
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The discrete Wigner function
[7, § 7
To illustrate how this procedure works let us apply it to the case of GF(22 ). In the self-dual basis {θ, θ 2 } the solution of eq. (7.33) for, say, κ = θ 3 = 1, is c1,0 = 1,
c1,θ = ±i,
c1,θ 2 = ±i.
Then, the last coefficient is given by c1,θ 3 = c1,θ+θ 2 = c1,θ c1,θ 2 χ(1) = (±i)(±i)(1), and one can see that there exist 4 different possible operators Vˆ1,θ 3 . A similar calculation can be made for the operators Vˆ1,θ and Vˆ1,θ 2 . It is convenient to fix positive signs of the coefficients cκ,μ corresponding to the elements of the field basis, i.e., cal σl ,μ = χ al2 θl2 μ , l = 1, . . . , N, and form the ‘first’ set of the rotation operators Vˆκ with these coefficients. Then, all the other sets of Vˆκ,μ can be obtained according to eq. (7.36). Once we have the coefficients cκ,μ we can easily obtain the corresponding phase factors for the displacement operator (7.43). In the following we will choose the ‘first’ set of rotation operators in the above example as Vˆθ = diag(1, 1, i, −i), Vˆθ 3 = diag(1, i, i, −1),
EBL
Vˆθ 2 = diag(1, i, 1, −i), (7.40)
where ‘diag’ means a diagonal matrix in the conjugate basis with the listed coefficients along the diagonal. The whole set of operators {Vˆκ,μ } form a group, Vˆκ,μ Vˆκ ,μ = Vˆκ+κ ,μ+μ +f (κ,κ ) , where the real function f (κ, κ ) = f (κ , κ) is a solution of the equations cκ,μ cκ,μ = χ κf (μ, μ ) cκ,μ+μ . (7.41) In the particular case of GF(22 ) using the self-dual basis (θ, θ 2 ) and fixing the rotation operators as in eq. (7.40), one obtains f (θ, θ ) = θ 2 , f θ, θ 2 = 0, f θ, θ 3 = θ 2 , f θ 2 , θ 2 = θ, f θ 2 , θ 3 = θ, f θ 3, θ 3 = θ 3.
(7.42)
7, § 7]
Reconstruction of the density operator from the discrete Wigner function
507
The four non-standard mutually unbiased bases are generated by applying the rotation operators to the elements of the standard basis |0 = (1, 0, 0, 0)T , |θ = (0, 1, 0, 0)T , |θ 2 = (0, 0, 1, 0)T , |θ 3 = (0, 0, 0, 1)T , were θ 2 + θ + 1 = 0. In the standard basis the rotation operators and the Fourier transformation operator in GF(22 ) have the form ⎡ ⎡ ⎤ ⎤ 1 i 1 −i 1 1 i −i 1⎢ i 1⎢ 1 1 −i 1 ⎥ 1 −i i ⎥ ⎥, ⎥, Vˆθ = ⎢ Vˆθ 2 = ⎢ ⎣ ⎣ ⎦ i i −i 1 1 ⎦ 2 1 −i 1 2 −i 1 i 1 −i i 1 1 ⎡ ⎡ ⎤ ⎤ i 1 1 −i 1 1 1 1 1⎢ 1 1 ⎢ 1 −1 1 −1 ⎥ i −i 1 ⎥ ⎥, ⎥. Fˆ = ⎢ Vˆθ 3 = ⎢ ⎣ ⎦ 1 2 1 −i i 2 ⎣ 1 1 −1 −1 ⎦ −i 1 1 i 1 −1 −1 1 Since Vˆθ |0 (Vˆθ |θ k ) yields the vector that is the first (kth) column of Vˆθ , etc., the four bases’ vectors can be read off from the columns of the four operators above. It is obvious that they are all unbiased with respect to the standard basis, and one can easily check that they are all mutually unbiased.
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7.5. The phase of the displacement operator
The phase ϕ(α, κ) appearing in eq. (7.21) is intimately related to the phase φ(α, β) of the displacement operator (6.31). Let us impose the natural condition that the sum of the Wigner function along the line β = κα + ν be equal to the average value of the density matrix over the state |κ, ν associated with that line as expressed by eq. (7.7) (Wootters [2004], Gibbons, Hoffman and Wootters [2004]). Using the phase-space construction described above, we obtain the following relation (Klimov, Muñoz and Romero [2006]): φ(α, β) = cα,α −1 β .
(7.43)
In GF(22 ) we thus obtain the following phase factors: φ θ, θ 3 = i, φ(θ, θ) = i, φ θ, θ 2 = 1, φ θ 2 , θ 3 = i, φ θ 2 , θ 2 = i, φ θ 2 , θ = 1, φ θ 3 , θ 3 = −1. φ θ 3 , θ 2 = −i, φ θ 3 , θ = −i,
(7.44)
Equation (7.43) immediately implies that for fields of odd characteristic φ(α, β) = χ −2−1 αβ ,
(7.45)
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The discrete Wigner function
[7, § 7
and the condition (6.34) is automatically satisfied. The displacement operator in this case has the form ˆ D(α, β) = χ −2−1 αβ Zˆ α Xˆ β , (7.46) ˆ so that the unitarity condition Dˆ † (α, β) = D(−α, −β) is satisfied and the kernel (6.36) can be represented in the familiar form ˆ ˆ β) = D(α, Δ(α, β)Pˆ Dˆ † (α, β), where 1 Pˆ = d
ˆ D(α, β)
(7.47)
α,β∈GF(d)
is the parity operator defined for prime spaces in eq. (6.19). Nevertheless, for fields of even characteristic, d = 2N , there is a freedom in the election of the phase φ, which takes values ±1, ±i, related to different possibilities of choosing the rotation operators Vˆκ,ν = Vˆκ Xˆ ν for the phase-space construction. Once the set {Vˆκ,ν } of rotation operators is fixed we can find the phase φ from eq. (7.43), and thus construct the Wigner function (7.49). In general, for a density matrix defined in the standard basis |ν as ρμ,ν |μ ν|, ρˆ = (7.48)
EBL
μ,ν∈GF(d)
the Wigner function takes the form 1 χ κ(ν − β) + α(ν − μ) Wρˆ (α, β) = d μ,ν,κ∈GF(d)
× φ(κ, ν − μ)ρμ,ν ,
(7.49)
which still can be simplified (summed over κ) for fields of odd characteristic taking into account the explicit expression (7.45) for the phase φ. In fig. 5 we have illustrated the interrelations between the four state representations ρ, ˆ %k,m , P (k, m) and Wρˆ (α, β). For p = 2 the kernel operator (6.36) acquires the form 1 ˆ β) = χ αλ − βκ − 2−1 κλ Zˆ κ Xˆ λ , Δ(α, (7.50) d κ,λ∈GF(d)
and thus can easily be factorized into one-particle operators in any almost selfdual basis. In fact, due to eqs. (4.13) and (4.14) and the factorization of the generalized character, we have: χ(αβ) =
N 3 k=1
ωαk βk qk ,
7, § 8]
Applications
509
(7.14)
%κ,μ
ρm,n (7.11)
(7.17)
(6.6) (6.38)
(6.37)
(7.12)
(7.10)
(7.9) (6.40)
(7.7)
W (α, β)
P (κ, μ) (7.16)
Fig. 5. The interconnecting equations between the four representations of a state.
where αk , βk ∈ Zp are coefficients of expansion of α and β, respectively, in an almost self-dual basis α=
p−1
ak θ k ,
β=
k=0
p−1
(7.51)
bk θ k ,
k=0
EBL
and tr(θk2 ) = qk = (q − 1)βkN + 1. Taking into account eq. (3.5), we have ˆ β) = Δ(α,
N ⊗
3 1 k=1
=
N−1 3⊗
p
χ αk λk − βk κk − 2
−1
κk λk qk Zˆ κk qk Xˆ λk
κk ,λk ∈GF(p N )
ˆ ˆ k , βk ) ⊗ Δ(qα Δ(α N , βN ).
k=1
ˆ β) = Thus, in any self-dual basis Δ(α,
@⊗ N
(7.52)
ˆ
k=1 Δ(αk , βk ).
§ 8. Applications In this section we consider two relevant physical problems to give a flavor of the kind of problems one expects this phase-space formalism to be a useful tool for. We start by considering the so-called mean king problem (Vaidman, Aharonov and Albert [1987]). In modern terms, Alice has to prepare a quantum bit in some state. Then she should give the qubit to the mean king. The king performs a projective measurement with respect to one of the Pauli spin observables σˆ x , σˆ y or σˆ z and then hands the qubit back to Alice. She can then perform any measurement on the state. When Alice’s measurement has been performed, the king reveals which
510
The discrete Wigner function
[7, § 8
observable he measured, and then Alice has to state the outcome of the king’s measurement to save her life. A solution of the mean king problem is only possible if Alice entangles the particle to be sent to the king with another identical particle she keeps in her own laboratory. We will analyze this problem using phase-space methods following Paz, Roncaglia and Saraceno [2005]. Let us consider the initial state prepared by √ Alice to be the Bell state |Φ = (|00 + |11 )/ 2 (written in ordinary qubit notation). When the king measures one of the three components of the spin of the first particle he can obtain one of two values, ±1/2, and the state collapses to one of the ˜ |θ˜2 , depending on the measurement outcome states |0 , |θ 2 , |1, 0 , |1, θ 2 , |0 , and the king’s choice of measurement basis, σˆ z , σˆ y or σˆ x , respectively. In writing these states we have used the polynomial basis notation in table 2. These are all ‘line states’: vertical lines are associated with the measurement of σˆ x , horizontal lines are associated with the measurement of σˆ z , and the striation containing the main diagonal corresponds to the measurement of σˆ y . The possible states that the king could give us according to his measurement constitute a net of six states represented in fig. 6, where the black lines are associated with the eigenvalue 1/2 and the gray lines with the eigenvalue −1/2. Once the king has made his measurement, Alice has to devise a measurement scheme that would enable her to retrodict the result of the king’s measurement once she is informed about the measured observable. The solution is to measure a collective observable of the two-particle system with four distinct eigenvalues. Each of the four eigenstates of this observable must have vanishing overlap with one and only one of the two states generated by each of the king’s three possible measurements. When the king announces what was the measured observable, Alice will hence always be able to infer what was the measured result. For example, one such state, which we denote as |ϕ1 , should be orthogonal to the gray-line states. Once we obtain this function we can find the Wigner functions associated to the other three states that complete the basis of the Hilbert space by implementing simple translations. We note that the translation Zˆ θ 2 , corresponding to a translation of the Wigner function by W (α, β) → W (α + θ 2 , β), will interchange the black and the gray vertical and diagonal lines while leaving the horizontal lines invariant. Since the state this new Wigner function represents now has vanishing overlap with the states |1, θ 2 and |θ˜2 , it can be made orthogonal to |ϕ1 . Hence, we require
EBL
ϕ1 |Zˆ θ 2 |ϕ1 = ϕ1 |ϕ2 = 0. Using the same line of reasoning with the translations Xˆ θ 2 and Zˆ θ 2 Xˆ θ 2 one can derive Alice’s measurement basis once |ϕ1 is found.
7, § 8]
Applications
θ2 θ 1 0
511
−1/16 3/16 3/16 3/16
−1/16 1/16 1/16 3/16
−1/16 1/16 1/16 3/16
3/16 −1/16 −1/16 −1/16
0
1
θ
θ2
Fig. 6. Wigner functions of the king’s six possible measurement outcomes, top. The Wigner function of the state |ϕ1 , bottom.
To find the Wigner function of the state |ϕ1 we need to determine its value in all the 4 × 4 points of the phase-space grid. By imposing all physical conditions, e.g., that along the three gray lines the function adds up to zero (orthogonality), that ϕ1 |Zˆ θ 2 |ϕ1 = 0 and similarly for Xˆ θ 2 and Zˆ θ 2 Xˆ θ 2 , that the function is normalized, that it represents a pure state (i.e., the sum over the square of all its coefficients equals 1/16), etc., one arrives at the Wigner function depicted at the bottom in fig. 6. The Wigner functions of the remaining three states in the measurement basis then follow from translation of this function as described in the previous paragraph. So, if Alice’s measurement projects the two-qubit state to, say, |ϕ2 (that has vanishing overlap with the gray horizontal line, and the black vertical and diagonal lines), then the king’s outcome must have been 1/2 if he measured σˆ z and −1/2 if he measured σˆ x or σˆ y . If, instead, Alice’s measurement projects the state onto |ϕ1 (having vanishing overlap with all grey lines), the king must have obtained the measurement result 1/2, regardless of his choice of measurement. We next discuss the phase-space representation of a quantum error-correction code. A detailed introduction to this topic is outside the scope of this review. Following Paz, Roncaglia and Saraceno [2005], let us consider the simplest case, in which we encode one qubit of quantum information using N physical qubits. The space of encoded states is a two-dimensional subspace of the total Hilbert space formed by the set of common eigenstates of the tensor products of Pauli operators acting on each qubit. The stabilizer is chosen in such a way that the code corrects a set of errors Ei , which are also translation operators. The code will correct against errors Ei (i = 1, . . . , 2N−1 − 1) if the encoded states |φL
EBL
512
The discrete Wigner function
[7, § 9
are mapped by the errors Ei onto subspaces which are mutually orthogonal for different values of i. As encoded states are eigenstates of translation operators, their corresponding Wigner functions must be invariant under the same translations. Moreover, as errors are also translation operators, when translating the Wigner function of an encoded state by a correctable error one should obtain a Wigner function which is orthogonal to the original one. The task of finding the Wigner function of encoded states could be accomplished using the symmetries of the state in the first place. In Paz, Roncaglia and Saraceno [2005], the reader can see the discussion of the simplest stabilizer quantum error-correcting code, which encodes one qubit of quantum information using three physical qubits and corrects against errors of the ˆ 1, ˆ 1⊗ ˆ Z⊗ ˆ 1⊗ ˆ Z; ˆ 1⊗ ˆ 1ˆ and 1⊗ ˆ i.e., phase errors. The stabilizer of the code form Z⊗ ˆ Xˆ ⊗ X. ˆ Since is defined by the translation operators Sˆ1 = Xˆ ⊗ Xˆ ⊗ 1ˆ and Sˆ2 = 1⊗ each error maps eigenstates of Sˆi onto eigenstates of the same operators with different eigenvalues, the code can correct against all errors. Indeed, if one defines the code space as the set of all states with eigenvalues +1 for both Sˆ1 and Sˆ2 , the action of errors on encoded states turns out to be the following: Zˆ ⊗ 1ˆ ⊗ 1ˆ errors map encoded states onto eigenstates of Sˆ1 and Sˆ2 with eigenvalues −1 and +1, while for 1ˆ ⊗ Zˆ ⊗ 1ˆ and 1ˆ ⊗ 1ˆ ⊗ Zˆ errors the corresponding eigenvalues are (−1, −1) and (+1, −1), respectively. The Wigner function of encoded states must be symmetric under the translations Sˆ1 and Sˆ2 . Each of these symmetries reduces by a half the total number of independent parameters defining the Wigner function. So, invariance under Sˆ1 and Sˆ2 reduces the number of parameters defining the Wigner function of encoded states from 8×8 to 2×2. An explicit expression for the Wigner function of general encoded states can be provided using further symmetry arguments.
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§ 9. Discussion and outlook Compared to the continuous Wigner function, the discrete Wigner function is an adolescent formulation, slowly developing into adult maturity. As we have shown, discrete phase space imposes several new challenges, some connected to the fact that unitary transformations are expected to lead to covariant translations. In a finite space this leads to an intricate mapping of the Wigner function phase-space coefficients. As we have argued above, such a mapping is most conveniently undertaken by resorting to mathematical field notation. However, finite fields can only be found for certain dimensions, namely if the dimension is a prime or the power of a prime. In these cases the construction method is
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relatively straightforward, but as we have seen, even and odd dimensions will have to be dealt with in slightly different ways. Moreover, several choices must be made along the way. In this chapter we have only considered striations corresponding to lines, but for a system consisting of a number of identical subsystems it has recently been shown that it is also possible to construct striations consisting of curves, i.e., a parameterization of the striations that is non-linear in the field elements (Klimov, Romero, Björk and Sánchez-Soto [2007]). Such constructions correspond to different structures with respect to the separability of the bases (Romero, Björk, Klimov and Sánchez-Soto [2005], Björk, Romero, Klimov and Sánchez-Soto [2007]). The problem at hand should dictate the construction method adopted. So far, the number of problems where the Wigner function representation facilitates the solution has been limited. This may be due to the fact that the discrete Wigner-function formalisms are relatively new. In addition, the absence of a natural ordering of field elements complicates the situation as the mapping between the phase-space formalism and the physical states becomes a matter of choice. Finally we must stress that this chapter only gives a brief overview over the vast field of discrete Wigner function formulations and applications. An INSPEC search on ‘discrete AND Wigner AND (function OR distribution)’ yields more than 400 hits. In a single chapter it is impossible to do justice to all the workers who have contributed to this branch of science. We have tried to provide a mathematical derivation and justification for a certain construction, based on the desire to ‘import’ some of the continuous-function features to the discrete counterpart. Many other starting points are possible and reasonable, and they will likewise provide a tool for computation and visualization, but a somewhat different tool than the one we have presented.
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Acknowledgements We would like to acknowledge discussions with, and help from, Mr. R. Asplund, Dr. I. Ghiu, Prof. J. Gracia-Bondía, Prof. H. de Guise, Dr. J.L. Romero, Dr. J. Söderholm, and Prof. A. Vourdas. This work was supported by the Swedish Foundation for International Cooperation in Research and Higher Education (STINT), the Swedish Research Council (VR), the Swedish Foundation for Strategic Research (SSF), the Grant 45704 of Consejo Nacional de Ciencia y Tecnologia (CONACyT), Mexico, and the Spanish Research Directorate (DGI), Grant FIS2005-0671.
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Author index for Volume 51 Ashcroft, N.W., 148 Asom, M.T., 234 Aspect, A., 253 Asplund, R., 473, 496, 500 Aspnes, D.E., 196 Atakishiyev, N.M., 472 Atkinson, R., 85, 86 Aus der Au, J., 234 Aydin, K., 15 Aydina, K., 15 Aytür, O., 289
A Abouraddy, A., 289, 299, 302, 307, 312, 313, 318, 338 Abram, I., 320, 324, 326 Abramowitz, M., 169, 377 Abrams, D., 341, 343 Adler, E., 95 Agarwal, G., 306, 341 Agarwal, G.S., 472 Agranovich, V.M., 3, 6, 8, 15, 18, 19, 29, 42, 168 Aharonov, Y., 474, 509 Ahmad, M.M., 103 Airy, G.B., 352 Akhmediev, N.N., 102, 103, 106 Akozbek, N., 7, 28, 29 Aktsipetrov, O.A., 102, 103, 106 Albert, D.Z., 509 Alekseyev, L., 17 Alekseyev, L.V., 17, 23, 24 Alfano, R.R., 223 Ali, S.T., 472 Ali Khan, I., 340 Alkalsi, M.M., 23 Allen, L., 371, 401 Altman, A., 298 Altman, C., 139, 140 Alu, A., 7, 15, 19, 20, 55, 57 An, Y.Q., 101 Anand, S., 15, 16 Andersen, U., 289, 331, 332 Andersson, S.K., 74, 81 Angelow, G., 223, 238, 240, 241 Anzai, H., 223 Aravind, P.K., 474 Arbore, M.A., 227 Archer, C., 474, 486 Arfken, G.B., 96, 127 Asahara, Y., 31
B Bache, M., 256, 279, 281–283, 285, 286, 288, 289, 300, 305, 306, 308, 310–313, 318, 319, 326, 330, 340, 341 Bachor, H., 289, 331–335 Bachor, H.-A., 335 Backus, S., 240, 242, 243 Badoz, J., 329 Bajer, J., 473 Balazs, N.L., 471 Baleine, E., 306 Bandyopadhyay, S., 474 Bar-Joseph, I., 223 Barakat, R., 354 Baranova, I.M., 102, 103, 106 Barasch, M.L., 139 Barbosa, G., 254, 298 Barnett, R., 342 Barnett, S., 254, 257–259, 266–269, 339 Barnett, S.M., 271, 371 Barrera, R.G., 195 Barron, L.D., 90, 92 Bartels, A., 219, 240 Basharov, A.M., 35 Baughman, R.H., 6, 15, 18, 19, 29, 42 Becchi, M., 127 Bechmann-Pasquinucci, H., 473 Befort, O., 80, 103
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518
Author index for Volume 51
Beijersbergen, M.W., 371, 401 Belinsky, A., 264, 298, 312 Belkin, M.A., 72, 90, 93 Belov, P.A., 17 Benedict, M.G., 472 Bengtsson, I., 474 Bennink, R., 287, 288, 307, 308, 316 Bentley, S., 287, 288, 307, 308, 316, 341 Berezin, F.A., 472 Bergquist, J.C., 219, 240 Berkovic, G., 79, 81, 89 Berrier, A., 15, 16 Berry, M.V., 168, 472 Berthier, S., 196 Bertolotti, M., 7, 28 Berzanskis, A., 254, 257, 269, 270 Beržanskis, A., 270 Beskrovnyy, V., 337 Beth, T., 474, 486, 487 Bhattacharyya, K., 79 Billy, L., 390 Binhammer, T., 223, 244 Bizarro, J.P., 473 Bize, S., 219, 240 Björk, G., 473, 475, 496, 500, 513 Blaikie, R., 25 Blaikie, R.J., 23, 25 Bloembergen, N., 71, 72, 79, 89, 93, 94, 97, 98, 101, 102, 109 Bloemer, M.J., 7, 28, 29, 34–36, 42, 44 Bloom, D.M., 213 Boardman, A.D., 3, 36, 44–48 Boccara, A.C., 329 Bock, J., 73 Bohren, C.F., 138, 193 Boiko, A., 238 Boivin, A., 407 Booker, G.R., 389 Booth, M., 338 Borghi, R., 306 Born, M., 108, 145, 146, 157, 353, 357, 363, 376, 377, 379, 397, 465 Bossard, J., 15, 19 Bosshard, Ch., 100 Boto, A., 341, 343 Bourennane, M., 473, 500 Bowen, W., 332–335 Bower, J.R., 73, 94, 103 Boyd, G.D., 234
Boyd, G.T., 79 Boyd, R., 255, 257, 287, 288, 307, 308, 316, 341 Boyd, R.W., 40, 71, 75, 79, 96, 114, 340 Boykin, P.O., 474 Braat, J., 421, 424, 426, 429, 430, 450 Braat, J.J.M., 378, 381, 386, 390, 401, 417, 418, 421, 426–428, 433, 439, 440, 442, 445, 451, 459–461, 463, 465 Braginsky, L., 3 Bramati, A., 279 Brambilla, E., 255, 256, 261, 272, 273, 279, 281–283, 285, 286, 288, 289, 293, 295, 300, 305, 306, 308, 310–313, 318–321, 326, 330, 340, 341 Brambilla, M., 254, 255, 257, 259, 264, 265, 270 Bratz, A., 80, 102, 103 Brau, C.A., 183 Braun, B., 234 Braunstein, S., 340, 341, 343 Brewer, R., 5 Brif, C., 472 Brock, J.B., 21 Brouers, F., 196 Brown, F., 73, 94 Brueck, S.R.J., 9–13, 55 Bruggeman, D.A.G., 192 ˇ 473 Brukner, C., Buchler, B., 289, 331, 332 Buchwald, J.Z., 192 Buck, M., 103 Buckingham, A.D., 90 Buckley, J., 224 Buin, A.K., 31 Bulua, I., 15 Buot, F.A., 473 Burke, B.J., 91 Burlakov, A.V., 254 Burr, K.C., 227 Burstein, E., 5 Button, K.J., 151, 152 Buzˇek, V., 473 Bycenski, K., 338 Byers, J.D., 86, 91, 92
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C Cade, N.A., 103, 106 Caglayana, H., 15
Author index for Volume 51 Cahill, K.E., 471 Cai, W., 7–11, 13, 25, 55, 58 Cai, Y., 313 Caiazzo, M., 7 Calderbank, A.R., 474 Callen, H.B., 139 Cambournac, C., 339 Cameron, P.J., 474 Camesasca, D., 257, 259, 264, 265 Campley, R.E., 4, 5 Camy, G., 277 Cao, D., 308, 310 Cao, D.-Zh., 341 Cao, Q., 380 Cardimona, D., 319 Carmichael, H., 259 Carrasco, S., 338 Carrig, T.J., 233 Carrillo, J.L., 195 Casimir, H.B.G., 139 Caspani, L., 295 Castelli, F., 254, 257, 267 Cattaneo, S., 72, 73, 81–84, 87, 89, 90, 93, 96, 99, 100, 112–115 Caves, C., 321 Censor, D., 151 Centini, M., 7, 28 Chambers, Ll.G., 161 Champagne, B., 90 Chan, C.T., 34 Chandrasekhar, S., 145, 153, 154 Chang, R.K., 72, 79, 89, 93, 94, 97, 98 Chang, T., 279 Chang, T.Y., 223 Chang, Z., 240, 242, 243 Chaturvedi, S., 473 Chekhova, M., 341 Chekhova, M.V., 254 Chen, C.K., 73, 94, 101, 106 Chen, G., 50 Chen, H., 7 Chen, H.C., 127, 132, 141, 146, 150, 152, 157, 162, 164, 165, 169, 179, 183 Chen, J., 79, 165 Chen, J.M., 73, 94, 103 Chen, K., 15, 19 Chen, S.-Y., 4
519
Chen, W., 34, 35, 71–73, 89, 94, 97, 98 Chen, X.-H., 313 Cheng, D., 184 Cheng, J., 308 Chettiar, U., 9 Chettiar, U.K., 7–13, 55, 58 Chew, W.C., 184 Chi, L.F., 74 Chinaglia, W., 270 Chipman, R.A., 452 Chiu, T.H., 234 Choi, S., 326, 327 Choi, S.-K, 261 Christov, I., 240, 242, 243 Chuang, I.L., 21, 472 Chudoba, C., 223 Chumakov, S.M., 472 Cirac, I., 286 Cirac, J.I., 277 Cnossen, G., 103 Coene, W., 354 Coerwinkel, R.P.C., 371 Cohen, L., 472 Cohen-Tannoudji, C., 253 Cohendet, O., 473 Colet, P., 266, 267, 269, 271, 325, 326, 337, 339 Collin, R.E., 152 Combe, Ph., 473 Conboy, J.C., 92, 93 Conrady, A., 352, 376 Conrady, A.E., 351 Corkum, P., 219 Corkum, P.B., 244 Cormick, C., 475 Corndorf, E., 298 Côté, C.-Y., 82 Cottis, P.G., 184, 188 Courty, J., 278 Crawford, M.J., 91 Creath, K., 385 Cross, G.H., 106 Crowell, J.E., 94 Crowell, M.H., 213 Cubukcu, E., 15 Cui, J., 188, 193 Cummer, S.A., 7, 58, 59 Curtis, E.A., 219, 240 Czirjak, A., 472
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D Dadap, J.I., 101 D’Aguanno, G., 7, 28, 29, 34–36, 42, 44 D’Angelo, M., 287, 308, 313, 316, 341 Danielson, G.E., 213 Dannenberger, O., 103 Darmanyan, S.A., 44 Davis, C.C., 24 de Castro, A.R.B., 94 de Gennes, P.G., 145, 154 De Grooth, B., 329 de Guise, H., 474, 482 de Lange, O.L., 138 de Sterke, C.M., 15, 34 De Toledo Piza, A.F.R., 473 Debye, P., 361 DeGiorgio, V., 270 Delaubert, V., 278, 329 Delsarte, P., 473 DeMaria, A.J., 213 Denariez-Roberge, M.-M., 82 Depine, R.A., 11, 139 Deutsch, T., 213 Devaux, F., 283, 284, 289, 319, 326, 328, 340 Dewald, S., 243 Di Giuseppe, G., 338 Di Trapani, P., 254, 257, 269, 270, 288, 289, 326, 330, 340 Diaz, A., 15, 19 Dick, B., 103, 106 Diddam, S.A., 219, 240 DiDomenico, M., 224 Dienes, A., 213, 222, 229 Dietrich, P., 219 Ding, J., 165 d’Inverno, R., 155 Dirksen, P., 378, 381, 386, 401, 417, 418, 421, 424, 426–430, 433, 439, 440, 442, 445, 450, 451, 459–461, 463, 465 Dogariu, A., 306 Dolling, G., 9, 12, 13, 174 Döring, J., 243 Dorn, R., 385 Dorsch, D., 73 Dow, J., 407 Dowling, J., 341, 343 Dowling, J.P., 472 Downer, M.C., 72, 101
Doyle, W.T., 193 Drabe, K.E., 103 Drachev, P., 9 Drachev, V.P., 3, 9–14, 31, 55 Drummond, P., 257, 278 Drummond, P.D., 253 Duan, L.M., 277 Dubrovskii, A.V., 90 Ducci, S., 278 Ducuing, J., 79, 94 Dungey, C.E., 193 Durant, S., 24 Durfee, C., 240, 242, 243 Durt, T., 474 E Earls, J.D., 106 Ebeling, K., 279 Eberler, M., 385 Eberly, J.H., 253 Economou, E.N., 15 Efros, A.L., 25 Egan, P., 36, 44–47 Eggleton, B.J., 15, 34 Eidenschnik, R., 73 Einstein, A., 278 Eisenthal, B., 79 Eisenthal, K.B., 101 Eisert, F., 103 Eliel, E., 289 Ell, R., 223, 238, 244 Elser, J., 18 Engheta, N., 7, 15, 19, 20, 24, 53–55, 57, 152 Englert, B.-G., 474 Enkrich, C., 7, 9, 12, 13, 27 Enoch, S., 15 Ercolessi, E., 473 Ericsson, A., 474 Ettl, P., 385 Everly, R.M., 85–87
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F Fabre, C., 257, 260, 261, 267, 277, 278, 289, 295, 328–337, 339 Fainman, Y., 12 Fallnich, C., 243 Fan, W., 9–13, 55 Fan, Y.-X., 165
Author index for Volume 51 Fang, N., 24, 25 Faxén, H., 194 Fayolle, P., 320, 324, 326 Feast, W.J., 103 Fei, H., 337 Fei, H.-B., 338 Feise, M.W., 7, 36 Fejer, M.M., 227 Felderhof, B.U., 80, 81, 102, 103 Feller, K.-H., 270 Feller, M.B., 73 Felsen, L.B., 149, 177 Ferguson, J.F., 234 Ferrar, C.M., 213 Ferri, F., 311, 313, 318 Fields, B.D., 473, 486 Fienup, J.R., 354, 413, 417 Figliozzi, P., 72, 101 Firsov, A.A., 9 Fischer, P., 72, 90, 93 Flagello, D.G., 390 Flörsheimer, M., 100 Fluck, R., 234 Flytzanis, C., 81, 91 Fork, R.L., 213, 220, 222, 240 Foteinopoulou, S., 15 Fouet, J., 329–331, 335 Fournier, D., 329 Frey, J.G., 91 Fricke, H., 191 Frieden, B.R., 354 Friese, M.E.J., 401 Frisch, U., 198 Fuchs, R., 195 Fujimoto, J.G., 223, 238, 240, 241 G Gabitov, I.R., 7, 28, 30, 32, 33, 36, 42, 144 Galetti, D., 473, 492 Galvão, E.F., 475 Garcia, N., 23 Garland, J.C., 192 Garmire, E., 34, 35 Garreau, J., 320, 324, 326 Gartstein, Yu.N., 3, 15 Gatti, A., 254–270, 272–279, 281–283, 285–289, 293, 295–298, 300, 305, 306, 308, 310–313, 318–321, 323, 325, 326, 330, 335, 339–341
521
Gavrielides, A., 319 Gayen, S.K., 223 Geiger, F., 81 Geim, A.K., 9 Genchev, Z.D., 198, 199 Genov, D.A., 25 Georgiou, E., 222 Gerardin, J., 168 Gerchberg, R.W., 354, 412 Gersten, J.I., 149 Ghosh, K., 195 Ghosh, R., 272 Giacobino, E., 267, 277, 279 Gibbons, K.S., 475, 489, 503, 507 Gibbs, H.M., 35 Giedke, G., 277 Gierulski, A., 73, 103, 106 Gigan, S., 257, 260, 261, 278, 329 Ginzburg, V.L., 8, 168 Giordmaine, J.A., 90 Girling, I.R., 103, 106 Glauber, R., 272, 294 Glauber, R.J., 471 Gleeson, H.F., 9 Glöckl, O., 385 Goethals, J.M., 473 Golubev, Y., 340 Golubeva, T., 340 Goncharenko, A.V., 196 Gonsalves, R., 354, 415 Gopnath, J.T., 223 Gordon, J.P., 220, 222, 240 Gori, F., 306 Gorkunov, M., 27 Gottesman, D., 474, 475 Gould, G., 213 Gracia-Bondía, J.M., 472, 493 Gradshteyn, I.S., 191 Graham, R., 270 Gralak, B., 15 Grangier, P., 253, 254, 257, 260, 261, 279, 320, 324, 326 Granqvist, C.G., 196 Grassl, M., 474, 486 Graugnard, E., 15, 19 Greenaway, A.H., 411 Greene, R.F., 139 Greve, J., 329
EBL
522
Author index for Volume 51
Grigorenko, A.N., 9, 10 Grosse, N., 332–335 Grubb, S., 73 Gruzdkov, Y.A., 91 Grynberg, G., 254, 257, 265, 267 Günter, P., 100 Guvena, K., 15 Guyot-Sionnest, P., 71–73, 89, 94, 95, 97–99, 102, 114 H Hache, F., 72, 74, 81, 91, 93 Hafner, Ch., 3 Haken, H., 270 Hakioglu, T., 492 Hall, G.E., 221, 238 Han, S., 308 Han, S.H., 90 Hanai, T., 196 Hanbury-Brown, R., 272, 317 Hanna, D., 310 Hannay, J.H., 472 Hargrove, L.E., 213 Hartstein, A., 5 Hashimoto, T., 473, 474 Hashin, Z., 194 Haslam, S., 91 Hata, C., 31 Haus, H., 223, 226, 229, 232 Haus, H.A., 233 Hayashi, A., 473, 474 Hayden, L.M., 103 Hayman, E., 34 He, H., 401 Hecht, L., 92 Heckenberg, N.R., 401 Hegde, R.S., 7, 34, 36 Hehl, F.W., 138, 139, 143 Heidmann, A., 253, 277 Heil, J., 417 Heinz, T.F., 71, 73, 74, 77–80, 91, 94–102, 104, 106, 109, 112–114 Heiss, S., 472 Held, H., 97–100 Hellemans, L., 81, 87, 93 Hermier, J., 279 Hicks, J.M., 72, 79, 86, 91–93 Hietala, V.M., 15 Higuchi, H., 329
Hilgevoord, J., 138 Hillenbrand, R., 23 Hillery, M., 471 Hoffman, J.D., 172 Hoffman, M.J., 475, 489, 503, 507 Holden, A., 8, 9 Hollberg, L., 219, 240 Hollberg, L.W., 219 Hong, C., 254, 272, 275, 341 Hong, S.-C., 97, 99, 100 Hönninger, C., 234 Hopkins, H.H., 353 Horibe, M., 473, 474 Horowicz, R.J., 277 Hoshi, H., 81 Houck, A.A., 21 Howell, J., 287, 288, 308, 316 Howell, J.C., 340 Hoyuelos, M., 266 Hsieh, W.-F., 233 Hsiung, H., 73 Hu, L., 175 Huang, F., 341 Huffman, D.R., 138 Hulliger, J., 100 Hung, Y.J., 24 Husimi, K., 471 Hutter, J., 74 Huxley, J.M., 233 Huygens, C., 352
EBL I
Iglesias, I., 354 Ignatowsky, V.S., 353, 372 Ikonen, M., 80 Ikushima, A.J., 31 Ilday, F.O., 224 Indik, R.A., 7, 28, 30, 32, 33 Ippen, E.P., 213, 222, 223, 229, 232, 233, 238, 240, 241 Ishikawa, K., 81 Islam, M.N., 223 Ivanovi´c, I.D., 474, 485, 500 J Jackson, J.D., 77, 107, 125, 133 Jackson, J.L., 139 Jacob, Z., 23, 24
Author index for Volume 51 Jacobo, A., 325, 326 Jaggard, D.L., 152 Janner, A.M., 98 Janssen, A.J.E.M., 378, 380, 381, 386, 401, 417, 418, 421, 424, 426–430, 433, 439, 440, 442, 445, 450, 451, 459–461, 463, 465 Jedrkiewicz, O., 288, 289, 326, 330, 340 Jeffers, J., 339 Jennings, B.K., 471 Jha, S.S., 72, 89, 93, 94, 97, 98 Jiang, H., 7 Jiang, Y., 72, 101 Jiang, Y.-K., 288, 289, 326, 330, 340 Jin, Y.-Q., 184 Jing, C., 313 Joannopoulos, J.D., 15, 16, 33, 34 John, S., 15 Johnson, S.G., 15, 16 Jones, E.D., 15 Jones, R.C., 452 Jones, R.J., 103 Jost, B., 289, 337 Jost, B.M., 338 Juffermans, C., 421, 426 Jung, I.D., 234 Jungnickel, D., 480 Justice, B.J., 7, 58, 59 K Kaatz, P., 100 Kalinin, V.A., 13 Kaneko, S., 31 Kang, X.-K., 184 Kano, Y., 471 Kantor, W.M., 474 Kapteyn, H.C., 240, 242, 243 Kärtner, F.X., 223, 234, 238, 240, 241, 244 Kasperkovitz, P., 473 Katz, T.J., 81, 87, 93 Kauranen, M., 71–73, 75, 76, 80–84, 87, 89–93, 95, 99, 100, 102, 104, 109–111, 113–115 Kawakami, S., 15 Kawashima, T., 15 Kazantseva, E.V., 7, 42 Kean, P.N., 216, 223, 233 Keller, R.A., 82 Keller, U., 226, 228, 229, 231, 234
523
Kemnitz, K., 79 Kempe, M., 338 Kenney-Wallace, G.A., 221, 238 Khoo, I.C., 15, 19 Khoury, A., 279 Khrushchev, I.Y., 9 Kildishev, A.V., 3, 7–14, 55, 58 Killi, A., 243 Kim, C., 261 Kim, D., 102, 103 Kim, H., 44 Kim, J.-W., 240, 241 Kim, M.W., 73 Kim, Y., 287, 308, 316 Kimble, H., 278, 326, 340 Kimura, G., 474 King, F.W., 138 King, N., 3, 36, 44, 48 Kintner, E.C., 377, 385 Kissel, V.N., 23 Kivshar, Y., 7, 28, 38 Kivshar, Y.S., 7, 28, 36, 38, 39, 44–47 Kivshar, Yu.S., 6, 25–27, 33, 39, 42, 49–51 Klappenecker, A., 474 Klar, T.A., 3, 11, 12, 14 Klein, M.W., 7, 27 Klimeck, G., 9 Klimov, A.B., 474, 475, 482, 492, 497, 502, 507, 513 Klyshko, D., 254, 298, 299, 312 Klyshko, D.N., 254 Knill, E., 472 Kockaert, P., 28 Kogelnik, H., 410 K˝oházi-Kis, A., 220, 241 Kojima, H., 329 Kok, P., 341, 343 Kolinsky, P.V., 103, 106 Kolobov, M., 253, 255, 256, 259–261, 264, 270, 282, 293, 295, 319, 322–325, 327, 335–337, 339, 340 Kondylis, G.D., 184 Kong, J.A., 130, 131, 140, 141, 157, 184, 193, 194, 197–199 Kooi, P.-S., 184 Koopmans, B., 98 Kopf, D., 234, 243 Köprülü, K., 298 Korobkin, D., 23
EBL
524
Author index for Volume 51
Koroteev, N.I., 90 Kosaka, H., 15 Koschny, T., 9, 12 Kourakis, I., 7, 28 Kowarz, M.W., 152 Kozyrev, A.B., 44 Kramer, M., 319 Kraus, F., 219 Kraus, K., 473 Krausz, F., 245 Kriech, M.A., 92, 93 Kristensen, M., 371 Krowne, C.M., 140 Krug, P.A., 15, 34 Kubrakov, N.F., 85, 86 Kuhta, N.A., 24 Kuizenga, D.J., 225 Kulik, S., 287, 308, 316 Kulik, S.P., 254 Kumar, P., 253, 261, 289, 295, 298, 326, 327 Kuo, S., 272 Kurkela, J., 80 Kuzucu, O., 240, 241 Kwiat, P., 275, 296 L
Lazarides, N., 7, 28 Leblanc, R., 82 Lederer, F., 35, 36 Lederer, M.J., 238, 243 Lee, C.H., 72, 73, 89, 93, 94, 97, 98, 103 Lee, H., 25 Lee, H.W., 471 Lee, J.-B., 15, 16 Lee, S., 389 Leeuwestein, A., 418, 426, 445 Lemmetyinen, H., 73, 80–84, 87, 89, 90, 93, 114, 115 Leong, M.-S., 184 Leonhardt, U., 7, 55, 56, 472, 488, 495, 496 Leuchs, G., 385 Levenson, J., 320, 324, 326 Levesque, J., 244 Li, H.-G., 341 Li, J., 34 Li, L.-W., 184 Li, R.-D., 261 Li, Y., 361, 364 Lidl, R., 479, 494 Lifshitz, E.M., 4, 155 Lin, E.K., 430 Lin, K.-H., 233 Lin, S.Y., 15 Lin, Z., 175 Lindell, I.V., 178, 182, 192 Linden, S., 7, 9, 12, 13, 27, 174 Ling, H., 389 Lissandrin, F., 338 Litchinitser, N.M., 7, 15, 28, 30, 32, 33, 36, 144 Litvak, A.G., 46 Liu, L.Y., 232, 233 Liu, Z., 24, 25 Lomakin, V., 12 Lommel, E., 352, 355, 376 Lopez, L., 257, 260, 261, 328, 329 Loy, M.M.T., 74 Lu, S.-Y., 452 Lu, W.T., 15 Lugiato, L., 254–270, 272, 273, 275–279, 281–283, 285, 287–289, 293, 295–298, 300, 305, 306, 308, 310, 312, 313, 318–326, 330, 335, 339–341 Lugiato, L.A., 256, 257, 259, 264, 265, 270, 277–279, 281–283, 285, 286, 311, 313
EBL
La Porta, A., 253 Laczik, Z., 389 Laferriere, P., 319 Laflamme, R., 472 Lagarkov, A.N., 10, 23 Lai, Y., 233 Lakhtakia, A., 11, 126, 133, 134, 138, 139, 141, 144, 146, 151–155, 161, 163, 166, 168, 170–172, 174, 175, 179, 182, 183, 185, 186, 189, 192–201 Lam, P., 289, 331–335 Lamb, H., 8 Landau, L.D., 4, 155 Lang, T., 243 Langbein, U., 35, 36 Langeveld-Voss, B.M.W., 81 Lantz, E., 283, 284, 289, 319, 325–328, 339, 340 Lapine, M., 27 Larkin, I.A., 23 Lavery, K.A., 430 Lawrence, J., 473, 474 Lax, B., 151, 152
Author index for Volume 51 Luis, A., 473, 492 Luo, C., 15, 16 Lüpke, G., 74 Luther-Davies, B., 238 Lvovsky, A.I., 97–100 M Machado da Silva, J., 298 Machida, S., 79 Maci, S., 7 Mackay, T.G., 147, 152, 153, 155, 166, 170, 172, 174, 175, 182, 183, 188, 189, 193–201 Macleod, H.A., 390 Magatti, D., 311, 313, 318 Magyar, G., 272 Mahajan, V.N., 424 Maillotte, H., 339 Maimistov, A.I., 7, 28, 30, 32, 33, 36, 42, 144 Maine, P., 220, 244 Maitre, A., 278, 295, 328, 329–331, 335 Maître, A., 257, 260, 261, 278, 289, 329, 331, 332 Makhnovskiy, D.P., 10 Maki, J.J., 71, 75, 80, 81, 87, 91, 92, 104, 109–111 Maksimchuk, A., 4 Malloy, K.J., 9–13, 55 Mancini, S., 267, 323, 325 Mandel, L., 254, 272, 275, 309, 310, 312, 313, 317, 341 Mandel, P., 15, 20 Mandelshtam, L.I., 8 Mann, A., 472 Mansuripur, M., 149, 389 Maradudin, A.A., 5 Marburger, J.H., 34, 35 Marchiolli, M.A., 473 Marcuse, D., 48, 50 Marcuvitz, N., 149, 177 Maréchal, A., 353 Marklund, M., 7, 28 Marmo, G., 473 Marowsky, G., 73, 74, 79–81, 89, 102, 103, 106 Marron, J.C., 354 Martienssen, W., 311, 313 Martin, R., 254, 257, 269, 270
Martinelli, M., 278 Martinez, O.E., 220, 240 Marzoli, I., 254, 257–260, 262, 266, 268–270, 275–278, 293 Mate, C.M., 94 Matlis, N., 72, 101 Matsuoka, M., 73, 94 Mattern, B., 72, 101 Mattiucci, N., 7, 28, 29, 34–36, 42, 44 Mattle, K., 275, 296 Matuschek, N., 234 Maxwell Garnett, J.C., 192 McAloney, R.A., 88 McCall, M.W., 144, 174, 175 McGuire, J.P., 452 McHugh, D., 85, 86 McNeil, K., 257 McPeake, D., 27 McPhedran, R.C., 57 Meade, R.D., 33, 34 Meijer, E.W., 81 Melville, D., 25 Melville, D.O.S., 23, 25 Mendoza, B.S., 72, 101 Mendoza, C.I., 195 Merlin, R., 23 Mermin, N.D., 148 Messier, R., 155, 171, 172 Michalzik, R., 279 Michel, B., 152, 163, 186, 189, 191–196, 198, 201 Miettinen, K., 73, 115 Miller, B.I., 223 Miller, D.A.B., 234 Miller, M.N., 199 Mills, D.L., 4, 5, 34, 35 Milster, T., 390 Milton, G.W., 7, 24, 55, 57, 58, 195, 199 Ming, N.-B., 165 Miquel, C., 472 Miranda, P.B., 102, 103 Miranowicz, A., 473 Mironov, V.A., 46 Mishina, E.D., 102, 103, 106 Mitchell, S.A., 81, 88 Mizrahi, V., 96–99, 102, 106 Moad, A.J., 85–87, 91 Möbius, D., 74, 80, 103 Mochan, W.L., 72, 101
EBL
525
526
Author index for Volume 51
Mock, J.J., 7, 58, 59 Mollenauer, L.F., 213, 232, 233 Monken, C.H., 288, 293 Monzon, J.C., 180, 184 Morandi, G., 473 Morgner, U., 223, 238, 243, 244 Moshammer, R., 243 Moss, D.J., 98 Mosset, A., 289, 326, 328, 340 Moulton, P.F., 213, 223, 230 Mourou, G., 220, 244 Muenchausen, R.E., 82 Mukunda, N., 473 Mulot, M., 15, 16 Munger, G., 82 Muñoz, M., 492, 497, 502, 507 Murnane, M.M., 240, 242, 243 Müstecaplıoglu, Ö.E., 15, 20 Muto, E., 329
O
N Nagasako, E., 341 Nakamura, A., 31 Nakotte, H., 31 Narimanov, E., 17, 23, 24 Narimanov, E.E., 3, 10, 13, 15, 17, 18, 23, 24 Nasr, M., 338 Navez, P., 283, 319, 320 Neal, D.B., 103 Negrevergne, C., 472 Nemat-Nasser, S.C., 8 Neviere, M., 44 Newton, R.W., 199 Nicorovici, N.-A.P., 7, 55, 57, 58 Nicorovici, N.A., 57 Niederreiter, H., 479, 494 Nielsen, M.A., 472 Nieto-Vesperinas, M., 23 Nijboer, B.R.A., 353, 355, 370, 376, 377, 460 Niklasson, G.A., 196 Nogar, N.S., 82 Nogueira, W., 288, 293 Nolte, R.J.M., 92 Notomi, M., 15, 16 Novak, V.R., 102, 103, 106 Nuckolls, C., 81, 87, 93 Nye, J.F., 145, 147, 157
Oates, C.W., 219, 240 O’Brien, S., 27 Obukhov, Y.N., 138 Obukhov, Yu.N., 139, 143 O’Conell, R.F., 472 O’Connell, R.F., 471 O’Dell, T.H., 123 Oemrawsingh, S., 289 Ogg, N.R., 183 Oh, C., 354 Okhrimchuk, A.G., 223 Oktel, M.Ö., 15, 20 Oldano, C., 127 Olyslager, F., 178, 182 Omi, S., 31 Onsager, L., 139 Opatrný, T., 473 Oppo, G., 259, 270, 293 Oppo, G.-L., 254, 257–259, 264–266, 268–270, 275–277, 293, 339 Oseen, C.W., 170 Osgood, R.M., 9–13, 55 O’Sullivan-Hale, M., 340 Ou, Z., 272, 275, 278, 326 Ozawa, M., 474 Ozbay, E., 15 Ozorio de Almeida, A.M., 473
EBL P
Padilla, W.J., 8 Pádua, S., 288, 293, 298 Pafomov, V.E., 8 Palenberg, M., 81 Pancharatnam, S., 166 Panina, L.V., 10 Panoiu, N.C., 9–13, 55 Parimi, P.V., 15 Park, B., 81 Park, W., 15, 16 Parks, R.E., 94 Parthasarathy, K.R., 474 Paschotta, R., 234 Patterson, D.B., 15 Paul, H., 272 Payrastre, C., 81, 91 Paz, J.P., 472, 474, 475, 510–512 Peev, M., 473
Author index for Volume 51 Peiro, J., 196 Pekonen, O.P.M., 193 Pendry, J.B., 3, 7–9, 13–16, 21, 23, 24, 27, 37, 55–59, 174 Penin, A.N., 254 Pereira, S., 278, 326 Pereira, S.F., 390 Peres, A., 473 Perez-Arjona, I., 274 Peˇrina, J., 473, 492 Pershan, P.S., 71, 95, 101, 102, 109 Persoons, A., 71, 72, 75, 76, 80, 81, 87, 91–93, 95, 102, 104, 109–111 Pessot, M., 220, 244 Peterson, I.R., 103, 106 Peterson, P., 319 Petralli-Mallow, T., 86, 91 Petri˘cevi´c, V., 223 Petrovic, J., 9 Petsas, K., 257, 259–261, 277, 278 Petsas, K.I., 259, 277, 278 Petty, M.C., 103 Phan-Thien, N., 199 Philbin, T.G., 7 Phillips, K.E.S., 81, 87, 93 Piché, M., 233 Picht, J., 352, 376 Pinnow, M., 103 Pinto, G.R., 79 Pinto, J.F., 222–224 Pitaevskii, L.P., 4 Pittenger, A.O., 474, 475 Pittman, T., 254, 299, 305, 307 Planat, M., 474 Platzman, P.M., 23 Plébanski, J., 155 Plocinik, R.M., 85–87 Pocklington, H.C., 8 Podolskiy, V.A., 10, 11, 13, 15, 17, 18, 23, 24 Podolsky, B., 278 Poizat, J., 279 Poizat, J.-P., 279 Pokrovsky, A.L., 25 Poliakov, E.Y., 7, 28, 29 Polizzi, M.A., 86, 91 Pollack, M.A., 213 Pollock, C.R., 215, 222–224, 230, 233 Ponti, S., 127 Popescu, S., 337
527
Popov, A.K., 7, 14, 39, 40, 43, 44 Post, E.J., 131, 138 Prabhu, V.M., 430 Prêtre, Ph., 100 Priou, A., 192 Probert, J.M., 91 Prost, J.A., 145, 154 Provencher, P., 82 Puri, R.R., 472 Putman, C., 329 Q Qing, De-Kui, 50 Qiu, M., 15, 16 Quabis, S., 385 R Raab, R.E., 138 Rabbiosi, I., 339 Radmore, P., 342 Rains, E.M., 474 Ramakrishna, S., 13, 14 Ramakrishna, S.A., 3, 27, 37, 174 Ramond, T.M., 219, 240 Rapoport, Y., 36, 44, 48 Rasing, Th., 73, 98 Rathe, U., 341 Rautian, S.G., 31 Rayleigh„ 352 Raymer, M., 272 Reid, M., 278 Reider, G.A., 73, 78, 91, 95, 97, 98, 219 Ren, W., 184 Rentzepis, P.M., 90 Reynaud, S., 253, 267, 277 Ribeiro, P., 295, 298 Ricard, D., 73, 94, 101, 106 Richards, B., 353, 373, 387, 396, 405, 432 Richter, R., 353, 376 Rieger, B., 73 Righini, R., 90 Ringhofer, K.H., 13, 27 Ripin, D.J., 223 Ripoll, O., 279 Ritsch, H., 254, 257, 258, 268, 275–277, 293 Rittweger, E., 223, 244 Rivas, A.M.F., 473 Rivera, J.-P., 138
EBL
528
Author index for Volume 51
Rivera, T., 320, 324, 326 Robbins, D.J., 8, 9 Roberts, G.G., 103 Roders, O., 80, 102, 103 Roger, G., 253 Romero, J.L., 475, 497, 502, 507, 513 Roncaglia, A.J., 474, 510–512 Rosanov, D., 264 Rosen, N., 278 Rosenbluth, A.E., 390 Rosu, H., 474 Rötteler, M., 474 Rowlen, K.L., 81 Roy, A., 81, 91 Roy, R., 452 Roychowdhury, V., 474 Royer, A., 494 Rubin, M., 254 Rubin, M.H., 474 Rubinsztein-Dunlop, H., 401 Rudnick, J., 73, 94 Rundquist, A., 240, 242, 243 Ruppin, R., 44, 45, 49 Russell, P.St.J., 15 Ruzzi, M., 492 Ryzhik, I.M., 191 Ryzhov, Yu.A., 197
Scalora, M., 7, 28, 29, 34–36, 42, 44 Scarcelli, G., 308, 313, 341 Schäfer, F.P., 230 Schanne-Klein, M.C., 74, 81, 91 Schelonka, L., 319 Scheuer, V., 223, 238, 240, 241 Schibli, T.R., 240, 241 Schleich, W., 155, 342 Schleich, W.P., 471, 472 Schmid, H., 123, 138 Schnatz, H., 219 Schnitzer, P., 279 Schonbrun, E., 15, 16 Schouten, A.J., 92 Schröter, C.D., 243 Schultz, S., 8, 21, 174 Schultz, T.J., 354 Schurig, D., 7, 55–59 Schuster, A., 8 Schwarz, S.E., 213 Schwarzberg, E., 79, 81 Schwinger, J., 472, 473, 481, 492 Schwob, C., 295 Scotto, P., 279, 280, 325, 326, 337 Scrinzi, A., 245 Scroggie, A., 339 Scully, M., 341, 342 Scully, M.O., 155, 270, 471, 472 Seidel, J.J., 473, 474 Seldin, J.H., 354 Sennaroglu, A., 223, 224, 233 Seres, E., 245 Seres, J., 245 Sergienko, A., 254, 275, 289, 296, 299, 302, 305, 307, 312, 313, 318, 338 Sergienko, A.V., 307 Setiawan, S., 172, 174, 175 Shadrivov, I., 7, 28, 38 Shadrivov, I.V., 6, 7, 25–28, 33, 36, 38, 39, 42, 44–47, 49–51, 53 Shalaev, V.M., 3, 7–14, 25, 28, 30–33, 36, 39, 40, 43, 44, 55, 58, 144, 174 Shamonina, E., 13 Shan, J., 101 Shank, C.V., 213, 222, 229 Shanker, B., 193 Shelby, R.A., 8, 21, 174 Shen, J.T., 23 Shen, N.-H., 165
EBL
S Sakai, T., 81 Sala, K.L., 221, 238 Salandrino, A., 15, 19, 20, 24 Saleh, B., 289, 299, 302, 307, 312, 313, 318, 337, 338 Salin, F., 233 San Miguel, M., 256, 266, 267, 269, 271, 279–281, 287, 296–298, 325, 326, 337, 339 Sánchez-Soto, L.L., 474, 475, 482, 513 Sandler, B.H., 354 Saniga, M., 474 Santagiustina, M., 271 Santarsiero, M., 306 Saraceno, M., 472, 474, 510–512 Sarychev, A.K., 9–13, 55 Sato, T., 15 Sauer, N., 223 Sawatzky, G.A., 98 Saxton, W.O., 354, 412
Author index for Volume 51 Shen, Y.R., 6, 15, 18, 19, 29, 38, 40, 42, 47, 71–75, 78, 79, 82, 89, 90, 93–103, 106, 107, 114 Sheng, P., 34 Sheppard, C.J.R., 370, 390 Sherman, G.C., 357 Sherwin, J.A., 194 Shestakov, A.V., 223 Shih, Y., 254, 275, 287, 296, 299, 305, 307, 308, 313, 316, 341 Shih, Y.H., 254 Shklover, V., 3 Shkurinov, A.P., 90 Shor, P.W., 474 Shtrikman, S., 194 Shukla, P.K., 7, 28 Shvets, G., 12, 18, 23 Sibbett, W., 216, 223, 233 Sibilia, C., 7, 28 Siegel, M., 243 Siegman, A.E., 225, 226, 371 Sieverdes, F., 80, 102, 103 Sihvola, A.H., 192, 193 Sillitto, R.M., 377, 385 Siltanen, M., 81, 87, 90, 93 Simon, R., 473 Simpson, G.J., 81, 85–87, 91 Singh, R.P., 472 Sioncke, S., 72, 81, 93 Sipe, J.E., 15, 34, 96–99, 102, 104–110, 192 Sirugue, M., 473 Sirugue-Collin, M., 473 Sivukhin, D.V., 8 Skrotskii, G.V., 155 Slabko, V.V., 7, 39 Sleeper, A.M., 94 Sloane, N.J.A., 474 Slusher, R., 253 Slusher, R.E., 15, 34 Smith, D., 330 Smith, D.R., 3, 7, 8, 21, 55–59, 174 Smith, F.W., 149 Smithey, D., 272 Smolyaninov, I.I., 24 Snauwaert, J., 81, 87, 93 Soccolich, C.E., 223 Sokolov, I., 253, 255, 260, 264, 270, 323–325, 335, 339, 340 Solymar, L., 13
529
Sommerfeld, A., 44 Somorjai, G.A., 94 Soneson, J.E., 7, 28, 30, 32, 33 Sosa, I.O., 195 Sosnowski, T., 224 Soukoulis, C.M., 9, 12, 13, 15, 174 Spence, D.E., 216, 223, 233 Spielmann, C., 245 Spiller, E., 311, 313 Spreeuw, R.J.C., 401 Spyrou, C., 188 Sridhar, S., 15 Stallinga, S., 371, 384, 396, 397, 407, 452 Stamnes, J.J., 357, 358, 361 Starr, A.F., 7, 58, 59 Starzhinskii, V.M., 185 Stegeman, G.I., 96–99, 106 Stegun, I.A., 169, 377 Steinhoff, R., 73, 74, 89 Steinman, A., 243 Stenflo, L., 7, 28 Stern, E.A., 73, 94 Steward, G.C., 352, 376 Stewart, W.J., 8, 9, 13 Stockman, M.I., 23 Stoev, V., 240, 242, 243 Stogryn, A., 198 Stolen, R.H., 232, 233 Stolle, R., 79, 81 Stone, P., 318 Stratonovich, R.L., 472, 493 Strehl, K., 352 Strekalov, D., 254, 299, 305, 307 Strekalov, D.V., 254 Streli, C., 245 Suau, A., 82 Suchy, K., 139, 140 Sudarshan, E.C.G., 471 Sukhorukov, A.A., 44–47, 49–51 Summers, C.J., 15, 19 Sun, C., 25 Sun, L., 72, 101 Sun, X.-J., 341 Sunderman, E.R., 223 Sure, T., 417 Sutter, K., 100 Svelto, O., 310 Swillo, M., 15, 16 Sylvestre, T., 339
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Author index for Volume 51
Syrchin, M., 7, 28, 29 Szip˝ocs, R., 220, 241 Szwaj, C., 259, 270, 293
Tsang, L., 193, 194, 197–199 Tschudi, T., 223, 238 Tsironis, G.P., 7, 28 Tuttle, G., 12 Twiss, R., 272, 317
T ’t Hooft, G.W., 23 Taft, G., 240, 242, 243 Tai, C.T., 175, 183 Takami, A., 473 Takezoe, H., 81 Talneau, A., 15, 16 Tamamura, T., 15 Tamir, T., 50 Tamoikin, V.V., 197 Tan, E.L., 184 Tan, S.Y., 184 Tanaka, H., 474 Tang, C.L., 227 Tanji, H., 31 Tanner, D.B., 192 Tassin, P., 28 Tatarskii, V.I., 197 Taubner, T., 23 Tayeb, G., 15 Teague, M.R., 410 Teerenstra, M.N., 92 Teich, M., 289, 299, 302, 307, 312, 313, 318, 337, 338 Tempea, G., 245 Thommen, Q., 15, 20 Thompson, W.A., 74 Thylén, L., 15, 16 Tian, K., 82 Tian, Y., 165 Tinker, M., 15, 16 Tlidi, M., 28 Tokizaki, T., 31 Tom, H.W.K., 71, 73, 74, 79, 94, 102, 114 Tomita, A., 15 Torner, L., 338 Török, P., 370, 389, 390 Torres, J., 338 Toussaint Jr., K., 338 Treacy, E.B., 244 Trebino, R., 220, 239 Treps, N., 257, 260, 261, 278, 289, 328, 329, 331–335, 339 Tretyakov, S.A., 14 Trutschel, U., 35, 36
U Uchida, K., 31 Ullrich, J., 243 Umstadter, D., 4 Urzhumov, Y., 12, 23 V Vaccaro, J., 473 Vaidman, L., 340, 509 Valdmanis, J.A., 222 Valencia, A., 308, 313, 341 Valint Jr., P., 73 Van Bladel, J., 179, 188 van de Nes, A.S., 386, 390, 401, 433, 451, 460, 461, 465 van der Avoort, C., 418, 421, 426–428, 439, 463 van der Lee, A., 279 Van der Sande, G., 28 van der Weide, D.W., 44 van der Woude, F., 98 van Driel, H.M., 98 van Drunen, W., 289 van Druten, N., 279 Van Elshocht, S., 71, 75, 76, 80, 81, 87, 91, 93 van Exter, M., 279 van Haver, S., 433, 440, 442, 451 Van Hulst, N., 329 Van Kranendonk, J., 192 Van Rompaey, Y., 81, 93, 102 Van Wiggeren, G.D., 452 VanDijck, D., 354 Varadan, V.K., 163, 183 Varadan, V.V., 163, 183 Varga, P., 389, 390 Várilly, J.C., 472, 493 Vasilyev, M., 326, 327 Vatan, V., 474 Vaupel, M., 278 Vazouras, C.N., 188 Velasco, L., 3, 36, 44, 48
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Author index for Volume 51 Venger, E.F., 196 Venugopal, V.C., 154 Verbiest, T., 71, 72, 75, 76, 80, 81, 87, 91–93, 95, 102 Veretennicoff, I., 28 Veretenov, N., 28 Verhoef, A.J., 245 Veselago, V., 3 Veselago, V.G., 3, 7, 8, 11, 21, 22 Vier, D.C., 8 Visser, T.D., 390 Vodo, P., 15 Vogt, B.D., 430 Voigt, W., 166 Von Laue, M., 8 Vourdas, A., 472, 492, 493, 497, 504 Vuorimaa, E., 73, 81–84, 87, 89, 90, 93, 114, 115 W Wagnière, G., 74 Walborn, S., 288, 293 Walgraef, D., 271 Wallis, R.F., 5 Walls, D., 257 Walser, R.M., 123, 192 Wang, C.C., 98 Wang, C.S., 73, 94, 103 Wang, F.X., 90 Wang, H.-T., 165 Wang, K., 308, 310, 323, 341 Wang, L., 15 Wangberg, R., 18 Ward, A.J., 55, 56 Ward, L., 195, 196 Weber, H.J., 96, 127 Wecht, K.W., 90 Wegener, M., 4, 7, 9, 12, 13, 27, 174 Wei, X., 90, 97–100 Weigert, S., 472 Weiglhofer, W.S., 126, 128, 138, 141, 143, 144, 146, 147, 149, 151–153, 155, 171, 174, 175, 177, 178, 181, 182, 185, 186, 188, 190–201 Weil, A., 472 Weinfurter, H., 275, 296 Weingarten, K.J., 234
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Welford, W.T., 369, 370 Welsch, D.-G., 473 Werner, D., 15, 19 Wesner, J., 417 Wetterer, C., 319 Weyl, H., 357, 471, 485 White, C.W., 72, 101 Wiedemann, H., 254, 255, 257–259, 266–269 Wierenga, H.A., 98 Wiersma, D.A., 103 Wiersma, D.S., 90 Wiersma, S.H., 390 Wigner, E.P., 471, 472, 476 Williams, C., 341, 343 Williams, J.M., 23 Williams, Y., 15, 19 Wiltshire, M.C.K., 3, 13 Win, Q., 165 Winful, H., 7, 34, 36 Winful, H.G., 34, 35 Winn, J.N., 33, 34 Wise, F.W., 224 Withrow, S.P., 72, 101 Wobrauschek, P., 245 Wocjan, P., 474, 486, 487 Woerdman, J., 279, 289 Woerdman, J.P., 371, 401 Wolf, C.R., 25 Wolf, E., 108, 145, 146, 157, 309, 310, 312, 313, 317, 352, 353, 357, 361, 363, 364, 373, 376, 377, 379, 387, 396, 397, 405, 407, 432, 465 Wolf, K.B., 472 Wolleschensky, R., 338 Wong, G.K., 102, 103 Wong, T.M., 91 Wood, O.R., 213 Woods, S.C., 411 Wootters, W.K., 473–475, 485, 486, 489, 492, 496, 498, 500, 503, 507 Wu, L.-A., 313 Wu, W.-L., 430 Wu, X.B., 184
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X Xiao, S., 13 Xiong, J., 308, 310, 341
532
Author index for Volume 51
Y Yablonovitch, E., 15 Yakovlev, V., 245 Yakubovich, V.A., 185 Yakymyshyn, C.P., 223, 224 Yamagishi, K., 223 Yamaguchi, Y., 223 Yamanoto, Y., 79 Yanagida, T., 329 Yang, G., 323 Yariv, A., 226 Yasumoto, K., 184 Ye, J., 219 Ye, P.X., 103 Ye, Z., 23 Yee, H.I., 86, 91, 92 Yen, T.-J., 25 Yeo, T.-S., 184 Yi-Jing, Y., 313 Yoo, J.-G., 81 Yuan, H., 9–11, 13, 55 Yuan, H.-K., 10, 13 Yuan, H.K., 9 Z
Zauner, G., 474 Zavorotnyi, V.U., 197 Zeilinger, A., 275, 296, 305, 473 Zek, E., 240, 242, 243 Zernike, F., 353, 370, 376 Zhai, Y.-H., 313 Zhang, C.H., 102, 103 Zhang, D., 313 Zhang, L., 12 Zhang, S., 9–13, 55 Zhang, T.G., 102, 103 Zhang, X., 24, 25 Zhang, Y., 9 Zharov, A., 7, 28, 38 Zharov, A.A., 6, 7, 25–28, 33, 39, 42, 44–47, 50 Zharova, N., 7, 28, 38 Zharova, N.A., 39, 42 Zheltikov, A.M., 7, 28, 29 Zhou, J., 9, 12 Zhou, L., 34 Zhu, S., 7, 313 Zhu, X.D., 94 Zhuang, X., 102, 103 Zhuck, N.P., 193, 198, 199 Ziolkowski, R.W., 7, 34, 55 Zoller, P., 277 Zubairy, M., 342
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Zakhidov, A.A., 6, 15, 18, 19, 29, 42, 44 Zambrini, R., 256, 266, 267, 269, 271, 279, 281, 287, 296–298, 339
Subject index for volume 51 dense coding 339, 340 density operator 471, 477, 487, 494, 496, 498–501, 503, 505, 507 detection – plane 272, 273, 277, 301, 302, 304, 305 – system 299, 300 detectors 288, 299, 300, 303–305, 330, 391, 415 dielectric – host 18 – permittivity 3, 4, 25, 29 diffraction – images 352, 359 – integrals 355, 359, 364, 380, 382, 454 – pattern 299, 302–306, 308, 310, 314–316, 318 Dirac delta function 136, 176 displacement operators 496–500, 503, 507, 508 dyadic Green’s functions 175 dye laser 222, 223, 229–231 dynamic gain saturation 229, 230
aberration function 362, 369–371, 376, 400 anisotropic mediums 123, 124, 144, 145, 166–168, 183 Bessel function 377, 460 bianisotropic mediums 123, 124, 141, 142, 151, 152, 157, 168, 172, 175, 176, 182, 184, 191, 192, 200 – – , homogeneous 152, 158, 182, 184 – – , linear 121, 123, 151 Bragg gap 33, 34 Bruggeman formalism 192, 195–197, 201
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cavity solitons 339 chiral – invariance 130, 132, 135 – sculptured thin films 154, 155, 170 chirality 81, 90, 92, 93 circular symmetry 390, 401, 407 classical optics 296 cloaking 55, 57, 58 coherence time 272, 285, 313, 317 coherent – imaging 318 – state 311, 312, 322, 331, 332, 334 complete theoretical model 116 composite materials 10, 185 correlation – function 264, 266, 270, 273, 276, 277, 301, 302, 304, 309, 310 – – , second-order 304, 309 – length 302, 310 cost function 414 covariance function 199 cross-correlation function 318 crystal optics 145, 157, 164
eigenfunctions 184, 336 eigenstates 287, 453, 473, 492, 494, 502, 503, 510, 512 eigenvectors 167, 482, 485, 499 electric field envelope function 30 electromagnetic theory 127, 192 electron tunnelling 245, 247 ellipsometry 86, 115 emission 246, 269, 276, 282, 285 – , stimulated 213, 229 – of twin photons 275, 276 energy – density 351, 353, 355, 391, 395, 397, 400, 401, 407, 409, 432, 433, 453, 454 – – distribution 403 – – function 455 – – patterns 399, 400
Debye approximation 361, 364 degeneracy 256, 275, 289–292, 296 533
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Subject index for volume 51
– flow 40, 51, 351, 391, 403, 407 – levels 224, 244 enhanced nonlinearity 25 entangled – beams 283, 300, 308, 310, 341 – two-photon microscopy 337 envelope function 219 excitation 10, 11, 34 excited states 229, 231 exciton 18, 19 exit-pupil function 355, 432, 454, 455 Extended Nijboer–Zernike theory 417, 419, 427, 432 factorization 478, 483, 484, 488, 496, 508 far-field – correlations 270, 284, 286, 288, 289 – distribution 265, 269, 275, 296, 306 – zone 24, 280, 282, 290, 297 Faraday chiral mediums 152, 159, 168, 169, 175, 201 Fast Fourier transform 382, 409, 412, 415 Fourier domains 415 frequency-resolved optical gating 237, 239 function – analysis 352, 353, 372, 424 – of delay 237–240 fundamental – beams 74, 76, 80–83, 85, 87, 91, 95, 99, 100, 105–108 – properties of optical pulses 215
image amplification 254, 319, 321, 323, 325, 327, 329 – – , noiseless 323, 326 impulse response functions 299, 301 intensity – correlation function 299, 304, 318 – fluctuations 276, 282, 295, 300, 301, 315, 320 – patterns 415, 417 interferometry 409, 455 inverse – Fourier transform 184, 306, 307, 413, 457 – problem 355, 409, 410, 415, 454, 455 inversion 96, 131, 229–231, 417, 418 isotropic – chiral mediums 144, 152, 153, 161, 179 – materials 17, 72, 73, 96, 98, 99 – mediums 123, 124, 142–145, 157, 176, 178, 183 Kerr lens modelocking technique 233, 234 Kronig relations 136, 138
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gap solitons 33, 34 Gaussian mode 331, 333 generalized – nonlinear Schrödinger equation 28 – Pauli group 480, 482, 494 ghost – diffraction scheme 302, 305, 318 – image retrieval 310, 311 – imaging 254, 298–301, 303–309, 311, 313, 315, 317, 318 Green’s functions 124, 175, 180, 181, 185, 411 gyrotropy 148, 149, 151 Herglotz functions 138 high-resolution projection lens 422, 426, 428 homogenization 152, 153, 175, 192, 197, 200, 201 Huygens principle 183
lasers – , doped 223, 224 – , pulsed high-power 289, 337 – , semiconductor 275, 279 – , solid-state 223, 230, 231, 234 – , Ti:sapphire 213, 223, 231, 235, 240, 242, 243 – , ultrashort-pulse 222 – , YAG 227, 326 lens – , conventional 22, 23 – function 412 – materials 368, 451 linear – bianisotropic mediums 121, 123, 151 – mediums 138, 142, 143, 145, 147, 149, 151, 153 – properties 3, 25, 30, 48, 102–104, 114 liquid crystals 20, 73, 145, 153 Lorentz – covariance 132, 138 – reciprocity 139, 142 magnetic – field components 4, 6, 25, 39, 375, 385 – permeability 3, 4, 6, 25, 26, 29, 57
Subject index for volume 51 marginal probability distributions 476, 487, 496, 500 Maxwell curl postulates 126, 128, 129, 140, 141, 157, 161, 170, 180 Maxwell postulates 124, 125, 127, 129–135, 137, 139, 141, 175 Maxwell-Garnett estimate 194 Maxwell’s equations 31, 55, 56 metamaterials 3, 6, 7, 9, 25, 28–30, 55–58, 174, 192 microscopy 329, 330, 337 modelocking 224, 228–230, 233, 234, 236 – solid-state lasers 214, 234 modified pupil functions 390 multimode configuration 295, 339, 340 nano-resonators 53 nanoparticles 19, 20, 31, 32 nanostructures 12, 30, 31, 33 – , plasmonic 30 negative – phase velocity 170, 174 – refraction 3, 7, 15–18, 20, 21, 23, 144, 174 negative-index metamaterials 3, 5–9, 11, 12, 14, 23–25, 28, 29, 33, 34, 36, 38–44, 46–49, 59, 60 Nijboer–Zernike diffraction theory 355, 382, 417, 445 nonhomogeneous mediums 153, 157, 168, 170, 185 nonlinear – material 35, 90, 101, 103 – negative-index metamaterials 47, 48 – optics 38, 116 – polarization 33, 39, 77, 78, 95, 102, 108 – properties 3, 6, 26 – response 6, 25, 36, 71, 75, 78, 80, 82, 87, 93–96, 98, 99 nonlinearity 7, 27, 34, 36, 72, 73, 94, 106, 271 numerical aperture, high 372, 387, 390, 410, 435, 451, 454
535
– fibers 235 – imaging 253, 329, 355, 454 – instruments 351, 410 – inverse problem 355, 411, 415, 455 – lithography 351, 353, 422 – materials 3, 240 – memory 36, 55 – metamaterials 3, 6, 55, 60 – negative index metamaterials 3, 8, 13, 14, 36, 59 – parametric – – amplification 42, 43 – – down-conversion 254, 255, 274 – properties 7, 31, 154 – – , linear 73–75, 78, 99, 103, 109, 110, 113–115 – solitons 235 Pauli group, generalized 480–483, 494 periodic structures 6, 25, 26, 34 permittivity 24, 58, 156, 162, 175, 180, 182, 216 photon antibunching 275, 288, 293, 294 photonic crystals 15, 16 plane-wave propagation 134, 156, 157, 162, 164–166, 168–170, 174 point-spread functions 351, 353, 355, 366, 389, 390, 399, 400, 402, 431, 455 polariton waves 18, 19 polarization – effects 73, 75, 77, 79, 81, 83, 85, 87, 89 – techniques 69, 71–74, 76, 78, 80, 82, 84, 86, 90–100, 114–116 – – , applications of 90, 91, 93, 95, 97, 99 positive-index metamaterials 5, 33, 34, 37, 39–44, 54 – – , linear 36, 47, 48 Poynting vector 17, 29, 41, 391, 403, 404, 407–409 – – components 355, 391, 404, 405, 407 probability density function 429 pupil function 354, 384, 390, 414, 416–418, 420, 421, 432, 433, 435, 437, 446, 455 – – , complex 431, 433, 455 – – , retrieval 431, 437
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object transmission function 318 Onsager relations 139, 141 optical – activity 81 – bistability 7, 27, 35, 36 – cavity 253, 271, 333, 339 – domains 6, 351 – effects 102
quantum – correlation 267, 277, 278, 297, 339 – ellipsometry 338 – imaging 251, 253–274, 276, 278, 280, 282, 284, 298, 308, 334–336, 338
536
Subject index for volume 51 – polaritons 19, 44–49, 51, 53 – waves 10, 44, 46, 48, 51
– laser pointer 254, 329, 331–333 – mechanics 27, 246, 329, 471 – optics 472 – teleportation 339, 340 quantum-optical – coherence tomography 338 – lithography 341 refractive index 9, 11, 14, 16, 20–23, 32, 72, 73, 76, 90, 94, 102–104 – – , negative 6, 10, 17, 29, 31 resonators 44, 45, 47, 49, 51, 53, 54, 261, 269, 270, 323 retrieval 24, 305, 409, 410, 417, 432, 435, 454 scalar – approximation 215, 391, 410, 422, 426 – function 136 – retrieval 419, 427, 428, 431, 435–437, 439 second-harmonic light 87, 89, 91 semiconductor lasers 275, 279 semiconductors 16, 234 simulations, numerical 267, 283, 284, 286, 295, 296 solid-state lasers 223, 230, 231, 234 soliton – effect 235, 236 – modelocking 235, 236, 240 solitons 27, 30, 34, 42, 228, 229, 235, 236, 338, 339 spatial – correlation functions 192, 254, 262–264, 266, 269, 272, 273, 276, 301 – quantum correlation 275, 279, 295, 328 – solitons 7, 36, 338, 339 spherical – aberration 352, 382, 390, 430, 431, 448 – Bessel functions 371, 380 spontaneous emission 225, 226 Stokes – operators 296–298 – parameters 297, 298 strong-property-fluctuation theory 192, 197 superlens 13, 20–23, 44, 57 surface – nonlinear optics 69, 71–74, 76, 78, 80, 82, 84, 86, 98, 116
theoretical models 74, 82, 87, 101, 102, 114, 116 thermal beams 309, 311, 317 through-focus images 422, 424, 432 Ti:sapphire – amplifiers 243 – lasers 213, 223, 231, 235, 240, 242, 243 translation operators 491, 511, 512 translational symmetry 267, 268 transmission – function 232, 384, 411, 450, 451 – , optical 227 twin photons 265, 275, 276, 286, 296, 303, 304 two-photon absorption 237, 337, 341 ultrafast – pulses 214, 215, 223, 230, 235, 236 – Ti:sapphire lasers 240, 243 ultrashort-pulse generation 214, 215, 222, 225, 236, 244 uniaxial – dielectric mediums 146, 161, 162, 164, 168, 181, 189 – magnetic mediums 162–164, 181
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Voigt waves 166, 168 wavefront aberration 352, 354, 367, 376, 391, 395 waveguides 7, 17, 44, 45, 47, 49–51, 53 wavevector 10, 17–19, 41, 158, 161, 255, 265, 279, 280, 356 Wigner function 471–477, 488, 489, 492, 494–498, 501, 508, 511, 512 – – , continuous 472, 473, 476, 477, 489, 512 – – , discrete 469, 472–475, 480, 487–491, 494, 495, 498–500, 512 – – , factorized 496 YAG laser 227, 326 Zernike polynomials 355, 376, 377, 380, 385, 417, 432, 454, 460, 465
Contents of previous volumes*
VOLUME 1 (1961) 1 2 3 4 5 6 7 8
The modern development of Hamiltonian optics, R.J. Pegis Wave optics and geometrical optics in optical design, K. Miyamoto The intensity distribution and total illumination of aberration-free diffraction images, R. Barakat Light and information, D. Gabor On basic analogies and principal differences between optical and electronic information, H. Wolter Interference color, H. Kubota Dynamic characteristics of visual processes, A. Fiorentini Modern alignment devices, A.C.S. Van Heel
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1– 29 31– 66 67–108 109–153 155–210 211–251 253–288 289–329
VOLUME 2 (1963) 1 2 3 4 5 6
Ruling, testing and use of optical gratings for high-resolution spectroscopy, G.W. Stroke The metrological applications of diffraction gratings, J.M. Burch Diffusion through non-uniform media, R.G. Giovanelli Correction of optical images by compensation of aberrations and by spatial frequency filtering, J. Tsujiuchi Fluctuations of light beams, L. Mandel Methods for determining optical parameters of thin films, F. Abelès
1– 72 73–108 109–129 131–180 181–248 249–288
VOLUME 3 (1964) 1 2 3
The elements of radiative transfer, F. Kottler Apodisation, P. Jacquinot, B. Roizen-Dossier Matrix treatment of partial coherence, H. Gamo
1 2 3 4
Higher order aberration theory, J. Focke Applications of shearing interferometry, O. Bryngdahl Surface deterioration of optical glasses, K. Kinosita Optical constants of thin films, P. Rouard, P. Bousquet
1– 28 29–186 187–332
VOLUME 4 (1965)
* Volumes I–XL were previously distinguished by roman rather than by arabic numerals.
537
1– 36 37– 83 85–143 145–197
538 5 6 7
Contents of previous volumes The Miyamoto–Wolf diffraction wave, A. Rubinowicz Aberration theory of gratings and grating mountings, W.T. Welford Diffraction at a black screen, Part I: Kirchhoff’s theory, F. Kottler
199–240 241–280 281–314
VOLUME 5 (1966) 1 2 3 4 5 6
Optical pumping, C. Cohen-Tannoudji, A. Kastler Non-linear optics, P.S. Pershan Two-beam interferometry, W.H. Steel Instruments for the measuring of optical transfer functions, K. Murata Light reflection from films of continuously varying refractive index, R. Jacobsson X-ray crystal-structure determination as a branch of physical optics, H. Lipson, C.A. Taylor 7 The wave of a moving classical electron, J. Picht
1– 81 83–144 145–197 199–245 247–286 287–350 351–370
VOLUME 6 (1967) 1 2 3 4 5 6 7 8
Recent advances in holography, E.N. Leith, J. Upatnieks Scattering of light by rough surfaces, P. Beckmann Measurement of the second order degree of coherence, M. Françon, S. Mallick Design of zoom lenses, K. Yamaji Some applications of lasers to interferometry, D.R. Herriot Experimental studies of intensity fluctuations in lasers, J.A. Armstrong, A.W. Smith Fourier spectroscopy, G.A. Vanasse, H. Sakai Diffraction at a black screen, Part II: electromagnetic theory, F. Kottler
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1– 52 53– 69 71–104 105–170 171–209 211–257 259–330 331–377
VOLUME 7 (1969) 1 2 3 4 5 6 7
Multiple-beam interference and natural modes in open resonators, G. Koppelman Methods of synthesis for dielectric multilayer filters, E. Delano, R.J. Pegis Echoes at optical frequencies, I.D. Abella Image formation with partially coherent light, B.J. Thompson Quasi-classical theory of laser radiation, A.L. Mikaelian, M.L. Ter-Mikaelian The photographic image, S. Ooue Interaction of very intense light with free electrons, J.H. Eberly
1– 66 67–137 139–168 169–230 231–297 299–358 359–415
VOLUME 8 (1970) 1 2 3 4 5 6
Synthetic-aperture optics, J.W. Goodman The optical performance of the human eye, G.A. Fry Light beating spectroscopy, H.Z. Cummins, H.L. Swinney Multilayer antireflection coatings, A. Musset, A. Thelen Statistical properties of laser light, H. Risken Coherence theory of source-size compensation in interference microscopy, T. Yamamoto 7 Vision in communication, L. Levi 8 Theory of photoelectron counting, C.L. Mehta
1– 50 51–131 133–200 201–237 239–294 295–341 343–372 373–440
Contents of previous volumes
539
VOLUME 9 (1971) 1 2 3 4 5 6 7
Gas lasers and their application to precise length measurements, A.L. Bloom Picosecond laser pulses, A.J. Demaria Optical propagation through the turbulent atmosphere, J.W. Strohbehn Synthesis of optical birefringent networks, E.O. Ammann Mode locking in gas lasers, L. Allen, D.G.C. Jones Crystal optics with spatial dispersion, V.M. Agranovich, V.L. Ginzburg Applications of optical methods in the diffraction theory of elastic waves, K. Gniadek, J. Petykiewicz 8 Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions, B.R. Frieden
1– 30 31– 71 73–122 123–177 179–234 235–280 281–310 311–407
VOLUME 10 (1972) 1 2 3 4 5 6 7
Bandwidth compression of optical images, T.S. Huang The use of image tubes as shutters, R.W. Smith Tools of theoretical quantum optics, M.O. Scully, K.G. Whitney Field correctors for astronomical telescopes, C.G. Wynne Optical absorption strength of defects in insulators, D.Y. Smith, D.L. Dexter Elastooptic light modulation and deflection, E.K. Sittig Quantum detection theory, C.W. Helstrom
1– 44 45– 87 89–135 137–164 165–228 229–288 289–369
VOLUME 11 (1973) 1 2 3 4 5 6 7
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Master equation methods in quantum optics, G.S. Agarwal Recent developments in far infrared spectroscopic techniques, H. Yoshinaga Interaction of light and acoustic surface waves, E.G. Lean Evanescent waves in optical imaging, O. Bryngdahl Production of electron probes using a field emission source, A.V. Crewe Hamiltonian theory of beam mode propagation, J.A. Arnaud Gradient index lenses, E.W. Marchand
1– 76 77–122 123–166 167–221 223–246 247–304 305–337
VOLUME 12 (1974) 1 2 3 4 5 6
Self-focusing, self-trapping, and self-phase modulation of laser beams, O. Svelto Self-induced transparency, R.E. Slusher Modulation techniques in spectrometry, M. Harwit, J.A. Decker Jr Interaction of light with monomolecular dye layers, K.H. Drexhage The phase transition concept and coherence in atomic emission, R. Graham Beam-foil spectroscopy, S. Bashkin
1– 51 53–100 101–162 163–232 233–286 287–344
VOLUME 13 (1976) 1
On the validity of Kirchhoff’s law of heat radiation for a body in a nonequilibrium environment, H.P. Baltes 2 The case for and against semiclassical radiation theory, L. Mandel 3 Objective and subjective spherical aberration measurements of the human eye, W.M. Rosenblum, J.L. Christensen 4 Interferometric testing of smooth surfaces, G. Schulz, J. Schwider
1– 25 27– 68 69– 91 93–167
540
Contents of previous volumes
5
Self-focusing of laser beams in plasmas and semiconductors, M.S. Sodha, A.K. Ghatak, V.K. Tripathi 6 Aplanatism and isoplanatism, W.T. Welford
169–265 267–292
VOLUME 14 (1976) 1 2 3 4 5 6 7
The statistics of speckle patterns, J.C. Dainty High-resolution techniques in optical astronomy, A. Labeyrie Relaxation phenomena in rare-earth luminescence, L.A. Riseberg, M.J. Weber The ultrafast optical Kerr shutter, M.A. Duguay Holographic diffraction gratings, G. Schmahl, D. Rudolph Photoemission, P.J. Vernier Optical fibre waveguides – a review, P.J.B. Clarricoats
1– 46 47– 87 89–159 161–193 195–244 245–325 327–402
VOLUME 15 (1977) 1 2 3 4 5
Theory of optical parametric amplification and oscillation, W. Brunner, H. Paul Optical properties of thin metal films, P. Rouard, A. Meessen Projection-type holography, T. Okoshi Quasi-optical techniques of radio astronomy, T.W. Cole Foundations of the macroscopic electromagnetic theory of dielectric media, J. Van Kranendonk, J.E. Sipe
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1– 75 77–137 139–185 187–244 245–350
VOLUME 16 (1978) 1 2 3 4 5
Laser selective photophysics and photochemistry, V.S. Letokhov Recent advances in phase profiles generation, J.J. Clair, C.I. Abitbol Computer-generated holograms: techniques and applications, W.-H. Lee Speckle interferometry, A.E. Ennos Deformation invariant, space-variant optical pattern recognition, D. Casasent, D. Psaltis 6 Light emission from high-current surface-spark discharges, R.E. Beverly III 7 Semiclassical radiation theory within a quantum-mechanical framework, I.R. Senitzky
1– 69 71–117 119–232 233–288 289–356 357–411 413–448
VOLUME 17 (1980) 1 2 3
Heterodyne holographic interferometry, R. Dändliker Doppler-free multiphoton spectroscopy, E. Giacobino, B. Cagnac The mutual dependence between coherence properties of light and nonlinear optical processes, M. Schubert, B. Wilhelmi 4 Michelson stellar interferometry, W.J. Tango, R.Q. Twiss 5 Self-focusing media with variable index of refraction, A.L. Mikaelian
1– 84 85–161 163–238 239–277 279–345
VOLUME 18 (1980) 1 2
Graded index optical waveguides: a review, A. Ghatak, K. Thyagarajan Photocount statistics of radiation propagating through random and nonlinear media, J. Pe˘rina
1–126 127–203
Contents of previous volumes Strong fluctuations in light propagation in a randomly inhomogeneous medium, V.I. Tatarskii, V.U. Zavorotnyi 4 Catastrophe optics: morphologies of caustics and their diffraction patterns, M.V. Berry, C. Upstill
541
3
204–256 257–346
VOLUME 19 (1981) 1 2 3 4 5
Theory of intensity dependent resonance light scattering and resonance fluorescence, B.R. Mollow Surface and size effects on the light scattering spectra of solids, D.L. Mills, K.R. Subbaswamy Light scattering spectroscopy of surface electromagnetic waves in solids, S. Ushioda Principles of optical data-processing, H.J. Butterweck The effects of atmospheric turbulence in optical astronomy, F. Roddier
1– 43 45–137 139–210 211–280 281–376
VOLUME 20 (1983) 1 2 3 4 5
Some new optical designs for ultra-violet bidimensional detection of astronomical objects, G. Courtès, P. Cruvellier, M. Detaille Shaping and analysis of picosecond light pulses, C. Froehly, B. Colombeau, M. Vampouille Multi-photon scattering molecular spectroscopy, S. Kielich Colour holography, P. Hariharan Generation of tunable coherent vacuum-ultraviolet radiation, W. Jamroz, B.P. Stoicheff
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1– 61 63–153 155–261 263–324 325–380
VOLUME 21 (1984) 1 2 3 4 5
Rigorous vector theories of diffraction gratings, D. Maystre Theory of optical bistability, L.A. Lugiato The Radon transform and its applications, H.H. Barrett Zone plate coded imaging: theory and applications, N.M. Ceglio, D.W. Sweeney Fluctuations, instabilities and chaos in the laser-driven nonlinear ring cavity, J.C. Englund, R.R. Snapp, W.C. Schieve
1– 67 69–216 217–286 287–354 355–428
VOLUME 22 (1985) 1 2 3 4 5
Optical and electronic processing of medical images, D. Malacara Quantum fluctuations in vision, M.A. Bouman, W.A. Van De Grind, P. Zuidema Spectral and temporal fluctuations of broad-band laser radiation, A.V. Masalov Holographic methods of plasma diagnostics, G.V. Ostrovskaya, Yu.I. Ostrovsky Fringe formations in deformation and vibration measurements using laser light, I. Yamaguchi 6 Wave propagation in random media: a systems approach, R.L. Fante
1– 76 77–144 145–196 197–270 271–340 341–398
VOLUME 23 (1986) 1
Analytical techniques for multiple scattering from rough surfaces, J.A. DeSanto, G.S. Brown 2 Paraxial theory in optical design in terms of Gaussian brackets, K. Tanaka 3 Optical films produced by ion-based techniques, P.J. Martin, R.P. Netterfield
1– 62 63–111 113–182
542 4 5
Contents of previous volumes Electron holography, A. Tonomura Principles of optical processing with partially coherent light, F.T.S. Yu
183–220 221–275
VOLUME 24 (1987) 1 2 3 4 5
Micro Fresnel lenses, H. Nishihara, T. Suhara Dephasing-induced coherent phenomena, L. Rothberg Interferometry with lasers, P. Hariharan Unstable resonator modes, K.E. Oughstun Information processing with spatially incoherent light, I. Glaser
1– 37 39–101 103–164 165–387 389–509
VOLUME 25 (1988) 1
Dynamical instabilities and pulsations in lasers, N.B. Abraham, P. Mandel, L.M. Narducci 2 Coherence in semiconductor lasers, M. Ohtsu, T. Tako 3 Principles and design of optical arrays, Wang Shaomin, L. Ronchi 4 Aspheric surfaces, G. Schulz
1–190 191–278 279–348 349–415
VOLUME 26 (1988) 1 2 3 4 5
Photon bunching and antibunching, M.C. Teich, B.E.A. Saleh Nonlinear optics of liquid crystals, I.C. Khoo Single-longitudinal-mode semiconductor lasers, G.P. Agrawal Rays and caustics as physical objects, Yu.A. Kravtsov Phase-measurement interferometry techniques, K. Creath
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1–104 105–161 163–225 227–348 349–393
VOLUME 27 (1989) 1 2 3 4
The self-imaging phenomenon and its applications, K. Patorski Axicons and meso-optical imaging devices, L.M. Soroko Nonimaging optics for flux concentration, I.M. Bassett, W.T. Welford, R. Winston Nonlinear wave propagation in planar structures, D. Mihalache, M. Bertolotti, C. Sibilia 5 Generalized holography with application to inverse scattering and inverse source problems, R.P. Porter
1–108 109–160 161–226 227–313 315–397
VOLUME 28 (1990) 1 2
Digital holography – computer-generated holograms, O. Bryngdahl, F. Wyrowski Quantum mechanical limit in optical precision measurement and communication, Y. Yamamoto, S. Machida, S. Saito, N. Imoto, T. Yanagawa, M. Kitagawa 3 The quantum coherence properties of stimulated Raman scattering, M.G. Raymer, I.A. Walmsley 4 Advanced evaluation techniques in interferometry, J. Schwider 5 Quantum jumps, R.J. Cook
1– 86 87–179 181–270 271–359 361–416
Contents of previous volumes
543
VOLUME 29 (1991) 1 2
Optical waveguide diffraction gratings: coupling between guided modes, D.G. Hall Enhanced backscattering in optics, Yu.N. Barabanenkov, Yu.A. Kravtsov, V.D. Ozrin, A.I. Saichev 3 Generation and propagation of ultrashort optical pulses, I.P. Christov 4 Triple-correlation imaging in optical astronomy, G. Weigelt 5 Nonlinear optics in composite materials. 1. Semiconductor and metal crystallites in dielectrics, C. Flytzanis, F. Hache, M.C. Klein, D. Ricard, Ph. Roussignol
1– 63 65–197 199–291 293–319 321–411
VOLUME 30 (1992) 1
4 5
Quantum fluctuations in optical systems, S. Reynaud, A. Heidmann, E. Giacobino, C. Fabre Correlation holographic and speckle interferometry, Yu.I. Ostrovsky, V.P. Shchepinov Localization of waves in media with one-dimensional disorder, V.D. Freilikher, S.A. Gredeskul Theoretical foundation of optical-soliton concept in fibers, Y. Kodama, A. Hasegawa Cavity quantum optics and the quantum measurement process, P. Meystre
1 2 3 4 5 6
Atoms in strong fields: photoionization and chaos, P.W. Milonni, B. Sundaram Light diffraction by relief gratings: a macroscopic and microscopic view, E. Popov Optical amplifiers, N.K. Dutta, J.R. Simpson Adaptive multilayer optical networks, D. Psaltis, Y. Qiao Optical atoms, R.J.C. Spreeuw, J.P. Woerdman Theory of Compton free electron lasers, G. Dattoli, L. Giannessi, A. Renieri, A. Torre
2 3
1– 85 87–135 137–203 205–259 261–355
VOLUME 31 (1993)
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VOLUME 32 (1993) 1 2 3 4
Guided-wave optics on silicon: physics, technology and status, B.P. Pal Optical neural networks: architecture, design and models, F.T.S. Yu The theory of optimal methods for localization of objects in pictures, L.P. Yaroslavsky Wave propagation theories in random media based on the path-integral approach, M.I. Charnotskii, J. Gozani, V.I. Tatarskii, V.U. Zavorotny 5 Radiation by uniformly moving sources. Vavilov–Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena, V.L. Ginzburg 6 Nonlinear processes in atoms and in weakly relativistic plasmas, G. Mainfray, C. Manus
1– 59 61–144 145–201 203–266 267–312 313–361
VOLUME 33 (1994) 1 2 3 4
The imbedding method in statistical boundary-value wave problems, V.I. Klyatskin Quantum statistics of dissipative nonlinear oscillators, V. Peˇrinová, A. Lukš Gap solitons, C.M. De Sterke, J.E. Sipe Direct spatial reconstruction of optical phase from phase-modulated images, V.I. Vlad, D. Malacara 5 Imaging through turbulence in the atmosphere, M.J. Beran, J. Oz-Vogt 6 Digital halftoning: synthesis of binary images, O. Bryngdahl, T. Scheermesser, F. Wyrowski
1–127 129–202 203–260 261–317 319–388 389–463
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Contents of previous volumes VOLUME 34 (1995)
1 2 3 4 5
Quantum interference, superposition states of light, and nonclassical effects, V. Bužek, P.L. Knight Wave propagation in inhomogeneous media: phase-shift approach, L.P. Presnyakov The statistics of dynamic speckles, T. Okamoto, T. Asakura Scattering of light from multilayer systems with rough boundaries, I. Ohlídal, K. Navrátil, M. Ohlídal Random walk and diffusion-like models of photon migration in turbid media, A.H. Gandjbakhche, G.H. Weiss
1–158 159–181 183–248 249–331 333–402
VOLUME 35 (1996) 1 2 3 4 5 6
Transverse patterns in wide-aperture nonlinear optical systems, N.N. Rosanov Optical spectroscopy of single molecules in solids, M. Orrit, J. Bernard, R. Brown, B. Lounis Interferometric multispectral imaging, K. Itoh Interferometric methods for artwork diagnostics, D. Paoletti, G. Schirripa Spagnolo Coherent population trapping in laser spectroscopy, E. Arimondo Quantum phase properties of nonlinear optical phenomena, R. Tana´s, A. Miranowicz, Ts. Gantsog
1– 60 61–144 145–196 197–255 257–354 355–446
VOLUME 36 (1996) 1 2 3 4 5
Nonlinear propagation of strong laser pulses in chalcogenide glass films, V. Chumash, I. Cojocaru, E. Fazio, F. Michelotti, M. Bertolotti Quantum phenomena in optical interferometry, P. Hariharan, B.C. Sanders Super-resolution by data inversion, M. Bertero, C. De Mol Radiative transfer: new aspects of the old theory, Yu.A. Kravtsov, L.A. Apresyan Photon wave function, I. Bialynicki-Birula
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1– 47 49–128 129–178 179–244 245–294
VOLUME 37 (1997) 1 2 3 4 5 6
The Wigner distribution function in optics and optoelectronics, D. Dragoman Dispersion relations and phase retrieval in optical spectroscopy, K.-E. Peiponen, E.M. Vartiainen, T. Asakura Spectra of molecular scattering of light, I.L. Fabelinskii Soliton communication systems, R.-J. Essiambre, G.P. Agrawal Local fields in linear and nonlinear optics of mesoscopic systems, O. Keller Tunneling times and superluminality, R.Y. Chiao, A.M. Steinberg
1– 56 57– 94 95–184 185–256 257–343 345–405
VOLUME 38 (1998) 1 2 3
Nonlinear optics of stratified media, S. Dutta Gupta Optical aspects of interferometric gravitational-wave detectors, P. Hello Thermal properties of vertical-cavity surface-emitting semiconductor lasers, W. Nakwaski, M. Osi´nski 4 Fractional transformations in optics, A.W. Lohmann, D. Mendlovic, Z. Zalevsky 5 Pattern recognition with nonlinear techniques in the Fourier domain, B. Javidi, J.L. Horner 6 Free-space optical digital computing and interconnection, J. Jahns
1– 84 85–164 165–262 263–342 343–418 419–513
Contents of previous volumes
545
VOLUME 39 (1999) 1 2
Theory and applications of complex rays, Yu.A. Kravtsov, G.W. Forbes, A.A. Asatryan Homodyne detection and quantum-state reconstruction, D.-G. Welsch, W. Vogel, T. Opatrný 3 Scattering of light in the eikonal approximation, S.K. Sharma, D.J. Somerford 4 The orbital angular momentum of light, L. Allen, M.J. Padgett, M. Babiker 5 The optical Kerr effect and quantum optics in fibers, A. Sizmann, G. Leuchs
1– 62 63–211 213–290 291–372 373–469
VOLUME 40 (2000) 1 2 3 4
Polarimetric optical fibers and sensors, T.R. Woli´nski Digital optical computing, J. Tanida, Y. Ichioka Continuous measurements in quantum optics, V. Peˇrinová, A. Lukš Optical systems with improved resolving power, Z. Zalevsky, D. Mendlovic, A.W. Lohmann 5 Diffractive optics: electromagnetic approach, J. Turunen, M. Kuittinen, F. Wyrowski 6 Spectroscopy in polychromatic fields, Z. Ficek, H.S. Freedhoff
1– 75 77–114 115–269 271–341 343–388 389–441
VOLUME 41 (2000) 1 2 3 4 5 6 7
Nonlinear optics in microspheres, M.H. Fields, J. Popp, R.K. Chang Principles of optical disk data storage, J. Carriere, R. Narayan, W.-H. Yeh, C. Peng, P. Khulbe, L. Li, R. Anderson, J. Choi, M. Mansuripur Ellipsometry of thin film systems, I. Ohlídal, D. Franta Optical true-time delay control systems for wideband phased array antennas, R.T. Chen, Z. Fu Quantum statistics of nonlinear optical couplers, J. Peˇrina Jr, J. Peˇrina Quantum phase difference, phase measurements and Stokes operators, A. Luis, L.L. Sánchez-Soto Optical solitons in media with a quadratic nonlinearity, C. Etrich, F. Lederer, B.A. Malomed, T. Peschel, U. Peschel
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1– 95 97–179 181–282 283–358 359–417 419–479 483–567
VOLUME 42 (2001) 1 2 3 4 5 6
Quanta and information, S.Ya. Kilin Optical solitons in periodic media with resonant and off-resonant nonlinearities, G. Kurizki, A.E. Kozhekin, T. Opatrný, B.A. Malomed Quantum Zeno and inverse quantum Zeno effects, P. Facchi, S. Pascazio Singular optics, M.S. Soskin, M.V. Vasnetsov Multi-photon quantum interferometry, G. Jaeger, A.V. Sergienko Transverse mode shaping and selection in laser resonators, R. Oron, N. Davidson, A.A. Friesem, E. Hasman
1– 91 93–146 147–217 219–276 277–324 325–386
VOLUME 43 (2002) 1 2 3
Active optics in modern large optical telescopes, L. Noethe Variational methods in nonlinear fiber optics and related fields, B.A. Malomed Optical works of L.V. Lorenz, O. Keller
1– 69 71–193 195–294
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4
Canonical quantum description of light propagation in dielectric media, A. Lukš, V. Peˇrinová 5 Phase space correspondence between classical optics and quantum mechanics, D. Dragoman 6 “Slow” and “fast” light, R.W. Boyd, D.J. Gauthier 7 The fractional Fourier transform and some of its applications to optics, A. Torre
295–431 433–496 497–530 531–596
VOLUME 44 (2002) 1 2 3
Chaotic dynamics in semiconductor lasers with optical feedback, J. Ohtsubo Femtosecond pulses in optical fibers, F.G. Omenetto Instantaneous optics of ultrashort broadband pulses and rapidly varying media, A.B. Shvartsburg, G. Petite 4 Optical coherence tomography, A.F. Fercher, C.K. Hitzenberger 5 Modulational instability of electromagnetic waves in inhomogeneous and in discrete media, F.Kh. Abdullaev, S.A. Darmanyan, J. Garnier
1– 84 85–141 143–214 215–301 303–366
VOLUME 45 (2003) 1 2 3 4 5 6
Anamorphic beam shaping for laser and diffuse light, N. Davidson, N. Bokor Ultra-fast all-optical switching in optical networks, I. Glesk, B.C. Wang, L. Xu, V. Baby, P.R. Prucnal Generation of dark hollow beams and their applications, J. Yin, W. Gao, Y. Zhu Two-photon lasers, D.J. Gauthier Nonradiating sources and other “invisible” objects, G. Gbur Lasing in disordered media, H. Cao
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1– 51 53–117 119–204 205–272 273–315 317–370
VOLUME 46 (2004) 1 2
Ultrafast solid-state lasers, U. Keller Multiple scattering of light from randomly rough surfaces, A.V. Shchegrov, A.A. Maradudin, E.R. Méndez 3 Laser-diode interferometry, Y. Ishii 4 Optical realizations of quantum teleportation, J. Gea-Banacloche 5 Intensity-field correlations of non-classical light, H.J. Carmichael, G.T. Foster, L.A. Orozco, J.E. Reiner, P.R. Rice
1–115 117–241 243–309 311–353 355–404
VOLUME 47 (2005) 1 2 3 4 5 6
Multistep parametric processes in nonlinear optics, S.M. Saltiel, A.A. Sukhorukov, Y.S. Kivshar Modes of wave-chaotic dielectric resonators, H.E. Türeci, H.G.L. Schwefel, Ph. Jacquod, A.D. Stone Nonlinear and quantum optics of atomic and molecular fields, C.P. Search, P. Meystre Space-variant polarization manipulation, E. Hasman, G. Biener, A. Niv, V. Kleiner Optical vortices and vortex solitons, A.S. Desyatnikov, Y.S. Kivshar, L.L. Torner Phase imaging and refractive index tomography for X-rays and visible rays, K. Iwata
1– 73 75–137 139–214 215–289 291–391 393–432
Contents of previous volumes
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VOLUME 48 (2005) 1 2 3 4 5
Laboratory post-engineering of microstructured optical fibers, B.J. Eggleton, P. Domachuk, C. Grillet, E.C. Mägi, H.C. Nguyen, P. Steinvurzel, M.J. Steel Optical solitons in random media, F. Abdullaev, J. Garnier Curved diffractive optical elements: Design and applications, N. Bokor, N. Davidson The geometric phase, P. Hariharan Synchronization and communication with chaotic laser systems, A. Uchida, F. Rogister, J. García-Ojalvo, R. Roy
1– 34 35–106 107–148 149–201 203–341
VOLUME 49 (2006) 1 2 3 4 5 6
Gaussian apodization and beam propagation, V.N. Mahajan Controlling nonlinear optical processes in multi-level atomic systems, A. Joshi, M. Xiao Photonic crystals, H. Benisty, C. Weisbuch Symmetry properties and polarization descriptors for an arbitrary electromagnetic wavefield, C. Brosseau, A. Dogariu Quantum cryptography, M. Dušek, N. Lütkenhaus, M. Hendrych Optical quantum cloning, N.J. Cerf, J. Fiurášek
1– 96 97–175 177–313 315–380 381–454 455–545
VOLUME 50 (2007) 1 2 3 4 5 6 7 8
From millisecond to attosecond laser pulses, N. Bloembergen Conical diffraction: Hamilton’s diabolical point at the heart of crystal optics, M.V. Berry, M.R. Jeffrey Historical papers on the particle concept of light, O. Keller Field quantization in optics, P.W. Milonni The history of near-field optics, L. Novotny Light tunneling, H.M. Nussenzveig The influence of Young’s interference experiment on the development of statistical optics, E. Wolf Planck, photon statistics, and Bose–Einstein condensation, D.M. Greenberger, N. Erez, M.O. Scully, A.A. Svidzinsky, M.S. Zubairy
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Cumulative index – Volumes 1–51* Abdullaev, F. and J. Garnier: Optical solitons in random media Abdullaev, F.Kh., S.A. Darmanyan and J. Garnier: Modulational instability of electromagnetic waves in inhomogeneous and in discrete media Abelès, F.: Methods for determining optical parameters of thin films Abella, I.D.: Echoes at optical frequencies Abitbol, C.I., see Clair, J.J. Abraham, N.B., P. Mandel and L.M. Narducci: Dynamical instabilities and pulsations in lasers Agarwal, G.S.: Master equation methods in quantum optics Agranovich, V.M. and V.L. Ginzburg: Crystal optics with spatial dispersion Agrawal, G.P.: Single-longitudinal-mode semiconductor lasers Agrawal, G.P., see Essiambre, R.-J. Allen, L. and D.G.C. Jones: Mode locking in gas lasers Allen, L., M.J. Padgett and M. Babiker: The orbital angular momentum of light Ammann, E.O.: Synthesis of optical birefringent networks Anderson, R., see Carriere, J. Apresyan, L.A., see Kravtsov, Yu.A. Arimondo, E.: Coherent population trapping in laser spectroscopy Armstrong, J.A. and A.W. Smith: Experimental studies of intensity fluctuations in lasers Arnaud, J.A.: Hamiltonian theory of beam mode propagation Asakura, T., see Okamoto, T. Asakura, T., see Peiponen, K.-E. Asatryan, A.A., see Kravtsov, Yu.A.
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Babiker, M., see Allen, L. Baby, V., see Glesk, I. Baltes, H.P.: On the validity of Kirchhoff’s law of heat radiation for a body in a nonequilibrium environment Barabanenkov, Yu.N., Yu.A. Kravtsov, V.D. Ozrin and A.I. Saichev: Enhanced backscattering in optics Barakat, R.: The intensity distribution and total illumination of aberration-free diffraction images Barrett, H.H.: The Radon transform and its applications Bashkin, S.: Beam-foil spectroscopy * Volumes I–XL were previously distinguished by roman rather than by arabic numerals.
549
48, 35 44, 303 2, 249 7, 139 16, 71 25, 1 11, 1 9, 235 26, 163 37, 185 9, 179 39, 291 9, 123 41, 97 36, 179 35, 257 6, 211 11, 247 34, 183 37, 57 39, 1 39, 291 45, 53 13,
1
29, 65 1, 67 21, 217 12, 287
550
Cumulative index – Volumes 1–51
Bassett, I.M., W.T. Welford and R. Winston: Nonimaging optics for flux concentration Beckmann, P.: Scattering of light by rough surfaces Benisty, H. and C. Weisbuch: Photonic crystals Beran, M.J. and J. Oz-Vogt: Imaging through turbulence in the atmosphere Bernard, J., see Orrit, M. Berry, M.V. and C. Upstill: Catastrophe optics: morphologies of caustics and their diffraction patterns Bertero, M. and C. De Mol: Super-resolution by data inversion Bertolotti, M., see Chumash, V. Bertolotti, M., see Mihalache, D. Beverly III, R.E.: Light emission from high-current surface-spark discharges Bialynicki-Birula, I.: Photon wave function Biener, G., see Hasman, E. Björk, G., A.B. Klimov and L.L. Sánchez-Soto: The discrete Wigner function Bloembergen, N.: From millisecond to attosecond laser pulses Bloom, A.L.: Gas lasers and their application to precise length measurements Bokor, N. and N. Davidson: Curved diffractive optical elements: Design and applications Bokor, N., see Davidson, N. Bouman, M.A., W.A. Van De Grind and P. Zuidema: Quantum fluctuations in vision Bousquet, P., see Rouard, P. Boyd, R.W. and D.J. Gauthier: “Slow” and “fast” light Braat, J.J.M., S. van Haver, A.J.E.M. Janssen and P. Dirksen: Assessment of optical systems by means of point-spread functions Brambilla, E., see Gatti, A. Brosseau, C. and A. Dogariu: Symmetry properties and polarization descriptors for an arbitrary electromagnetic wavefield Brown, G.S., see DeSanto, J.A. Brown, R., see Orrit, M. Brunner, W. and H. Paul: Theory of optical parametric amplification and oscillation Bryngdahl, O.: Applications of shearing interferometry Bryngdahl, O.: Evanescent waves in optical imaging Bryngdahl, O., T. Scheermesser and F. Wyrowski: Digital halftoning: synthesis of binary images Bryngdahl, O. and F. Wyrowski: Digital holography – computer-generated holograms Burch, J.M.: The metrological applications of diffraction gratings Butterweck, H.J.: Principles of optical data-processing Bužek, V. and P.L. Knight: Quantum interference, superposition states of light, and nonclassical effects
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Cagnac, B., see Giacobino, E. Cao, H.: Lasing in disordered media Carmichael, H.J., G.T. Foster, L.A. Orozco, J.E. Reiner and P.R. Rice: Intensity-field correlations of non-classical light Carriere, J., R. Narayan, W.-H. Yeh, C. Peng, P. Khulbe, L. Li, R. Anderson, J. Choi and M. Mansuripur: Principles of optical disk data storage
27, 161 6, 53 49, 177 33, 319 35, 61 18, 257 36, 129 36, 1 27, 227 16, 357 36, 245 47, 215 51, 469 50, 1 9, 1 48, 107 45, 1 22, 77 4, 145 43, 497 51, 349 51, 251 49, 315 23, 1 35, 61 15, 1 4, 37 11, 167 33, 389 28, 1 2, 73 19, 211 34,
1
17, 85 45, 317 46, 355 41, 97
Cumulative index – Volumes 1–51 Casasent, D. and D. Psaltis: Deformation invariant, space-variant optical pattern recognition Cattaneo, S., see Kauranen, M. Ceglio, N.M. and D.W. Sweeney: Zone plate coded imaging: theory and applications Cerf, N.J. and J. Fiurášek: Optical quantum cloning Chang, R.K., see Fields, M.H. Charnotskii, M.I., J. Gozani, V.I. Tatarskii and V.U. Zavorotny: Wave propagation theories in random media based on the path-integral approach Chen, R.T. and Z. Fu: Optical true-time delay control systems for wideband phased array antennas Chiao, R.Y. and A.M. Steinberg: Tunneling times and superluminality Choi, J., see Carriere, J. Christensen, J.L., see Rosenblum, W.M. Christov, I.P.: Generation and propagation of ultrashort optical pulses Chumash, V., I. Cojocaru, E. Fazio, F. Michelotti and M. Bertolotti: Nonlinear propagation of strong laser pulses in chalcogenide glass films Clair, J.J. and C.I. Abitbol: Recent advances in phase profiles generation Clarricoats, P.J.B.: Optical fibre waveguides – a review Cohen-Tannoudji, C. and A. Kastler: Optical pumping Cojocaru, I., see Chumash, V. Cole, T.W.: Quasi-optical techniques of radio astronomy Colombeau, B., see Froehly, C. Cook, R.J.: Quantum jumps Courtès, G., P. Cruvellier and M. Detaille: Some new optical designs for ultra-violet bidimensional detection of astronomical objects Creath, K.: Phase-measurement interferometry techniques Crewe, A.V.: Production of electron probes using a field emission source Cruvellier, P., see Courtès, G. Cummins, H.Z. and H.L. Swinney: Light beating spectroscopy
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Dainty, J.C.: The statistics of speckle patterns Dändliker, R.: Heterodyne holographic interferometry Darmanyan, S.A., see Abdullaev, F.Kh. Dattoli, G., L. Giannessi, A. Renieri and A. Torre: Theory of Compton free electron lasers Davidson, N. and N. Bokor: Anamorphic beam shaping for laser and diffuse light Davidson, N., see Bokor, N. Davidson, N., see Oron, R. De Mol, C., see Bertero, M. De Sterke, C.M. and J.E. Sipe: Gap solitons Decker Jr, J.A., see Harwit, M. Delano, E. and R.J. Pegis: Methods of synthesis for dielectric multilayer filters Demaria, A.J.: Picosecond laser pulses DeSanto, J.A. and G.S. Brown: Analytical techniques for multiple scattering from rough surfaces Desyatnikov, A.S., Y.S. Kivshar and L.L. Torner: Optical vortices and vortex solitons
551
16, 289 51, 69 21, 287 49, 455 41, 1 32, 203 41, 283 37, 345 41, 97 13, 69 29, 199 36, 1 16, 71 14, 327 5, 1 36, 1 15, 187 20, 63 28, 361 20, 1 26, 349 11, 223 20, 1 8, 133 14, 1 17, 1 44, 303 31, 321 45, 1 48, 107 42, 325 36, 129 33, 203 12, 101 7, 67 9, 31 23, 1 47, 291
552
Cumulative index – Volumes 1–51
Detaille, M., see Courtès, G. Dexter, D.L., see Smith, D.Y. Dirksen, P., see Braat, J.J.M. Dogariu, A., see Brosseau, C. Domachuk, P., see Eggleton, B.J. Dragoman, D.: The Wigner distribution function in optics and optoelectronics Dragoman, D.: Phase space correspondence between classical optics and quantum mechanics Drexhage, K.H.: Interaction of light with monomolecular dye layers Duguay, M.A.: The ultrafast optical Kerr shutter Dušek, M., N. Lütkenhaus and M. Hendrych: Quantum cryptography Dutta, N.K. and J.R. Simpson: Optical amplifiers Dutta Gupta, S.: Nonlinear optics of stratified media
20, 1 10, 165 51, 349 49, 315 48, 1 37, 1
Eberly, J.H.: Interaction of very intense light with free electrons Eggleton, B.J., P. Domachuk, C. Grillet, E.C. Mägi, H.C. Nguyen, P. Steinvurzel and M.J. Steel: Laboratory post-engineering of microstructured optical fibers Englund, J.C., R.R. Snapp and W.C. Schieve: Fluctuations, instabilities and chaos in the laser-driven nonlinear ring cavity Ennos, A.E.: Speckle interferometry Erez, N., see Greenberger, D.M. Essiambre, R.-J. and G.P. Agrawal: Soliton communication systems Etrich, C., F. Lederer, B.A. Malomed, T. Peschel and U. Peschel: Optical solitons in media with a quadratic nonlinearity
7, 359
EBL
Fabelinskii, I.L.: Spectra of molecular scattering of light Fabre, C., see Reynaud, S. Facchi, P. and S. Pascazio: Quantum Zeno and inverse quantum Zeno effects Fante, R.L.: Wave propagation in random media: a systems approach Fazio, E., see Chumash, V. Fercher, A.F. and C.K. Hitzenberger: Optical coherence tomography Ficek, Z. and H.S. Freedhoff: Spectroscopy in polychromatic fields Fields, M.H., J. Popp and R.K. Chang: Nonlinear optics in microspheres Fiorentini, A.: Dynamic characteristics of visual processes Fiurášek, J., see Cerf, N.J. Flytzanis, C., F. Hache, M.C. Klein, D. Ricard and Ph. Roussignol: Nonlinear optics in composite materials. 1. Semiconductor and metal crystallites in dielectrics Focke, J.: Higher order aberration theory Forbes, G.W., see Kravtsov, Yu.A. Foster, G.T., see Carmichael, H.J. Françon, M. and S. Mallick: Measurement of the second order degree of coherence Franta, D., see Ohlídal, I. Freedhoff, H.S., see Ficek, Z. Freilikher, V.D. and S.A. Gredeskul: Localization of waves in media with one-dimensional disorder
43, 433 12, 163 14, 161 49, 381 31, 189 38, 1
48,
1
21, 355 16, 233 50, 275 37, 185 41, 483 37, 95 30, 1 42, 147 22, 341 36, 1 44, 215 40, 389 41, 1 1, 253 49, 455 29, 321 4, 1 39, 1 46, 355 6, 71 41, 181 40, 389 30, 137
Cumulative index – Volumes 1–51 Frieden, B.R.: Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions Friesem, A.A., see Oron, R. Froehly, C., B. Colombeau and M. Vampouille: Shaping and analysis of picosecond light pulses Fry, G.A.: The optical performance of the human eye Fu, Z., see Chen, R.T. Gabitov, I.R., see Litchinitser, N.M. Gabor, D.: Light and information Gamo, H.: Matrix treatment of partial coherence Gandjbakhche, A.H. and G.H. Weiss: Random walk and diffusion-like models of photon migration in turbid media Gantsog, Ts., see Tana´s, R. Gao, W., see Yin, J. García-Ojalvo, J., see Uchida, A. Garnier, J., see Abdullaev, F. Garnier, J., see Abdullaev, F.Kh. Gatti, A., E. Brambilla and L. Lugiato: Quantum imaging Gauthier, D.J.: Two-photon lasers Gauthier, D.J., see Boyd, R.W. Gbur, G.: Nonradiating sources and other “invisible” objects Gea-Banacloche, J.: Optical realizations of quantum teleportation Ghatak, A. and K. Thyagarajan: Graded index optical waveguides: a review Ghatak, A.K., see Sodha, M.S. Giacobino, E. and B. Cagnac: Doppler-free multiphoton spectroscopy Giacobino, E., see Reynaud, S. Giannessi, L., see Dattoli, G. Ginzburg, V.L.: Radiation by uniformly moving sources. Vavilov–Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena Ginzburg, V.L., see Agranovich, V.M. Giovanelli, R.G.: Diffusion through non-uniform media Glaser, I.: Information processing with spatially incoherent light Glesk, I., B.C. Wang, L. Xu, V. Baby and P.R. Prucnal: Ultra-fast all-optical switching in optical networks Gniadek, K. and J. Petykiewicz: Applications of optical methods in the diffraction theory of elastic waves Goodman, J.W.: Synthetic-aperture optics Gozani, J., see Charnotskii, M.I. Graham, R.: The phase transition concept and coherence in atomic emission Gredeskul, S.A., see Freilikher, V.D. Greenberger, D.M., N. Erez, M.O. Scully, A.A. Svidzinsky and M.S. Zubairy: Planck, photon statistics, and Bose–Einstein condensation Grillet, C., see Eggleton, B.J.
EBL
553
9, 311 42, 325 20, 63 8, 51 41, 283 51, 1 1, 109 3, 187 34, 333 35, 355 45, 119 48, 203 48, 35 44, 303 51, 251 45, 205 43, 497 45, 273 46, 311 18, 1 13, 169 17, 85 30, 1 31, 321 32, 267 9, 235 2, 109 24, 389 45, 53 9, 281 8, 1 32, 203 12, 233 30, 137 50, 275 48, 1
554
Cumulative index – Volumes 1–51
Hache, F., see Flytzanis, C. Hall, D.G.: Optical waveguide diffraction gratings: coupling between guided modes Hariharan, P.: Colour holography Hariharan, P.: Interferometry with lasers Hariharan, P.: The geometric phase Hariharan, P. and B.C. Sanders: Quantum phenomena in optical interferometry Harwit, M. and J.A. Decker Jr: Modulation techniques in spectrometry Hasegawa, A., see Kodama, Y. Hasman, E., G. Biener, A. Niv and V. Kleiner: Space-variant polarization manipulation Hasman, E., see Oron, R. Heidmann, A., see Reynaud, S. Hello, P.: Optical aspects of interferometric gravitational-wave detectors Helstrom, C.W.: Quantum detection theory Hendrych, M., see Dušek, M. Herriot, D.R.: Some applications of lasers to interferometry Hitzenberger, C.K., see Fercher, A.F. Horner, J.L., see Javidi, B. Huang, T.S.: Bandwidth compression of optical images
29, 321 29, 1 20, 263 24, 103 48, 149 36, 49 12, 101 30, 205 47, 215 42, 325 30, 1 38, 85 10, 289 49, 381 6, 171 44, 215 38, 343 10, 1
Ichioka, Y., see Tanida, J. Imoto, N., see Yamamoto, Y. Ishii, Y.: Laser-diode interferometry Itoh, K.: Interferometric multispectral imaging Iwata, K.: Phase imaging and refractive index tomography for X-rays and visible rays
40, 77 28, 87 46, 243 35, 145 47, 393
Jacobsson, R.: Light reflection from films of continuously varying refractive index Jacquinot, P. and B. Roizen-Dossier: Apodisation Jacquod, Ph., see Türeci, H.E. Jaeger, G. and A.V. Sergienko: Multi-photon quantum interferometry Jahns, J.: Free-space optical digital computing and interconnection Jamroz, W. and B.P. Stoicheff: Generation of tunable coherent vacuum-ultraviolet radiation Janssen, A.J.E.M., see Braat, J.J.M. Javidi, B. and J.L. Horner: Pattern recognition with nonlinear techniques in the Fourier domain Jones, D.G.C., see Allen, L. Joshi, A. and M. Xiao: Controlling nonlinear optical processes in multi-level atomic systems
5, 247 3, 29 47, 75 42, 277 38, 419
Kastler, A., see Cohen-Tannoudji, C. Kauranen, M. and S. Cattaneo: Polarization techniques for surface nonlinear optics Keller, O.: Local fields in linear and nonlinear optics of mesoscopic systems Keller, O.: Optical works of L.V. Lorenz Keller, O.: Historical papers on the particle concept of light
5, 1 51, 69 37, 257 43, 195 50, 51
EBL
20, 325 51, 349 38, 343 9, 179 49, 97
Cumulative index – Volumes 1–51 Keller, U.: Ultrafast solid-state lasers Khoo, I.C.: Nonlinear optics of liquid crystals Khulbe, P., see Carriere, J. Kielich, S.: Multi-photon scattering molecular spectroscopy Kilin, S.Ya.: Quanta and information Kinosita, K.: Surface deterioration of optical glasses Kitagawa, M., see Yamamoto, Y. Kivshar, Y.S., see Desyatnikov, A.S. Kivshar, Y.S., see Saltiel, S.M. Klein, M.C., see Flytzanis, C. Kleiner, V., see Hasman, E. Klimov, A.B., see Björk, G. Klyatskin, V.I.: The imbedding method in statistical boundary-value wave problems Knight, P.L., see Bužek, V. Kodama, Y. and A. Hasegawa: Theoretical foundation of optical-soliton concept in fibers Koppelman, G.: Multiple-beam interference and natural modes in open resonators Kottler, F.: The elements of radiative transfer Kottler, F.: Diffraction at a black screen, Part I: Kirchhoff’s theory Kottler, F.: Diffraction at a black screen, Part II: electromagnetic theory Kozhekin, A.E., see Kurizki, G. Kravtsov, Yu.A.: Rays and caustics as physical objects Kravtsov, Yu.A. and L.A. Apresyan: Radiative transfer: new aspects of the old theory Kravtsov, Yu.A., G.W. Forbes and A.A. Asatryan: Theory and applications of complex rays Kravtsov, Yu.A., see Barabanenkov, Yu.N. Kubota, H.: Interference color Kuittinen, M., see Turunen, J. Kurizki, G., A.E. Kozhekin, T. Opatrný and B.A. Malomed: Optical solitons in periodic media with resonant and off-resonant nonlinearities
EBL
Labeyrie, A.: High-resolution techniques in optical astronomy Lakhtakia, A., see Mackay, T.G. Lean, E.G.: Interaction of light and acoustic surface waves Lederer, F., see Etrich, C. Lee, W.-H.: Computer-generated holograms: techniques and applications Leith, E.N. and J. Upatnieks: Recent advances in holography Letokhov, V.S.: Laser selective photophysics and photochemistry Leuchs, G., see Sizmann, A. Levi, L.: Vision in communication Li, L., see Carriere, J. Lipson, H. and C.A. Taylor: X-ray crystal-structure determination as a branch of physical optics Litchinitser, N.M., I.R. Gabitov, A.I. Maimistov and V.M. Shalaev: Negative refractive index metamaterials in optics Lohmann, A.W., D. Mendlovic and Z. Zalevsky: Fractional transformations in optics
555 46, 1 26, 105 41, 97 20, 155 42, 1 4, 85 28, 87 47, 291 47, 1 29, 321 47, 215 51, 469 33, 1 34, 1 30, 205 7, 1 3, 1 4, 281 6, 331 42, 93 26, 227 36, 179 39, 1 29, 65 1, 211 40, 343 42, 93 14, 47 51, 121 11, 123 41, 483 16, 119 6, 1 16, 1 39, 373 8, 343 41, 97 5, 287 51, 1 38, 263
556
Cumulative index – Volumes 1–51
Lohmann, A.W., see Zalevsky, Z. Lounis, B., see Orrit, M. Lugiato, L., see Gatti, A. Lugiato, L.A.: Theory of optical bistability Luis, A. and L.L. Sánchez-Soto: Quantum phase difference, phase measurements and Stokes operators Lukš, A. and V. Peˇrinová: Canonical quantum description of light propagation in dielectric media Lukš, A., see Peˇrinová, V. Lukš, A., see Peˇrinová, V. Lütkenhaus, N., see Dušek, M.
40, 271 35, 61 51, 251 21, 69
Machida, S., see Yamamoto, Y. Mackay, T.G. and A. Lakhtakia: Electromagnetic fields in linear bianisotropic mediums Mägi, E.C., see Eggleton, B.J. Mahajan, V.N.: Gaussian apodization and beam propagation Maimistov, A.I., see Litchinitser, N.M. Mainfray, G. and C. Manus: Nonlinear processes in atoms and in weakly relativistic plasmas Malacara, D.: Optical and electronic processing of medical images Malacara, D., see Vlad, V.I. Mallick, S., see Françon, M. Malomed, B.A.: Variational methods in nonlinear fiber optics and related fields Malomed, B.A., see Etrich, C. Malomed, B.A., see Kurizki, G. Mandel, L.: Fluctuations of light beams Mandel, L.: The case for and against semiclassical radiation theory Mandel, P., see Abraham, N.B. Mansuripur, M., see Carriere, J. Manus, C., see Mainfray, G. Maradudin, A.A., see Shchegrov, A.V. Marchand, E.W.: Gradient index lenses Martin, P.J. and R.P. Netterfield: Optical films produced by ion-based techniques Masalov, A.V.: Spectral and temporal fluctuations of broad-band laser radiation Maystre, D.: Rigorous vector theories of diffraction gratings Meessen, A., see Rouard, P. Mehta, C.L.: Theory of photoelectron counting Méndez, E.R., see Shchegrov, A.V. Mendlovic, D., see Lohmann, A.W. Mendlovic, D., see Zalevsky, Z. Meystre, P.: Cavity quantum optics and the quantum measurement process Meystre, P., see Search, C.P. Michelotti, F., see Chumash, V. Mihalache, D., M. Bertolotti and C. Sibilia: Nonlinear wave propagation in planar structures
28, 87 51, 121 48, 1 49, 1 51, 1
EBL
41, 419 43, 295 33, 129 40, 115 49, 381
32, 313 22, 1 33, 261 6, 71 43, 71 41, 483 42, 93 2, 181 13, 27 25, 1 41, 97 32, 313 46, 117 11, 305 23, 113 22, 145 21, 1 15, 77 8, 373 46, 117 38, 263 40, 271 30, 261 47, 139 36, 1 27, 227
Cumulative index – Volumes 1–51 Mikaelian, A.L.: Self-focusing media with variable index of refraction Mikaelian, A.L. and M.L. Ter-Mikaelian: Quasi-classical theory of laser radiation Mills, D.L. and K.R. Subbaswamy: Surface and size effects on the light scattering spectra of solids Milonni, P.W.: Field quantization in optics Milonni, P.W. and B. Sundaram: Atoms in strong fields: photoionization and chaos Miranowicz, A., see Tana´s, R. Miyamoto, K.: Wave optics and geometrical optics in optical design Mollow, B.R.: Theory of intensity dependent resonance light scattering and resonance fluorescence Murata, K.: Instruments for the measuring of optical transfer functions Musset, A. and A. Thelen: Multilayer antireflection coatings Nakwaski, W. and M. Osi´nski: Thermal properties of vertical-cavity surface-emitting semiconductor lasers Narayan, R., see Carriere, J. Narducci, L.M., see Abraham, N.B. Navrátil, K., see Ohlídal, I. Netterfield, R.P., see Martin, P.J. Nguyen, H.C., see Eggleton, B.J. Nishihara, H. and T. Suhara: Micro Fresnel lenses Niv, A., see Hasman, E. Noethe, L.: Active optics in modern large optical telescopes Novotny, L.: The history of near-field optics Nussenzveig, H.M.: Light tunneling
EBL
Ohlídal, I. and D. Franta: Ellipsometry of thin film systems Ohlídal, I., K. Navrátil and M. Ohlídal: Scattering of light from multilayer systems with rough boundaries Ohlídal, M., see Ohlídal, I. Ohtsu, M. and T. Tako: Coherence in semiconductor lasers Ohtsubo, J.: Chaotic dynamics in semiconductor lasers with optical feedback Okamoto, T. and T. Asakura: The statistics of dynamic speckles Okoshi, T.: Projection-type holography Omenetto, F.G.: Femtosecond pulses in optical fibers Ooue, S.: The photographic image Opatrný, T., see Kurizki, G. Opatrný, T., see Welsch, D.-G. Oron, R., N. Davidson, A.A. Friesem and E. Hasman: Transverse mode shaping and selection in laser resonators Orozco, L.A., see Carmichael, H.J. Orrit, M., J. Bernard, R. Brown and B. Lounis: Optical spectroscopy of single molecules in solids Osi´nski, M., see Nakwaski, W. Ostrovskaya, G.V. and Yu.I. Ostrovsky: Holographic methods of plasma diagnostics
557 17, 279 7, 231 19, 45 50, 97 31, 1 35, 355 1, 31 19, 1 5, 199 8, 201
38, 165 41, 97 25, 1 34, 249 23, 113 48, 1 24, 1 47, 215 43, 1 50, 137 50, 185 41, 181 34, 249 34, 249 25, 191 44, 1 34, 183 15, 139 44, 85 7, 299 42, 93 39, 63 42, 325 46, 355 35, 61 38, 165 22, 197
558
Cumulative index – Volumes 1–51
Ostrovsky, Yu.I. and V.P. Shchepinov: Correlation holographic and speckle interferometry Ostrovsky, Yu.I., see Ostrovskaya, G.V. Oughstun, K.E.: Unstable resonator modes Oz-Vogt, J., see Beran, M.J. Ozrin, V.D., see Barabanenkov, Yu.N.
30, 87 22, 197 24, 165 33, 319 29, 65
Padgett, M.J., see Allen, L. Pal, B.P.: Guided-wave optics on silicon: physics, technology and status Paoletti, D. and G. Schirripa Spagnolo: Interferometric methods for artwork diagnostics Pascazio, S., see Facchi, P. Patorski, K.: The self-imaging phenomenon and its applications Paul, H., see Brunner, W. Pegis, R.J.: The modern development of Hamiltonian optics Pegis, R.J., see Delano, E. Peiponen, K.-E., E.M. Vartiainen and T. Asakura: Dispersion relations and phase retrieval in optical spectroscopy Peng, C., see Carriere, J. Peˇrina Jr, J. and J. Peˇrina: Quantum statistics of nonlinear optical couplers Pe˘rina, J.: Photocount statistics of radiation propagating through random and nonlinear media Peˇrina, J., see Peˇrina Jr, J. Peˇrinová, V. and A. Lukš: Quantum statistics of dissipative nonlinear oscillators Peˇrinová, V. and A. Lukš: Continuous measurements in quantum optics Peˇrinová, V., see Lukš, A. Pershan, P.S.: Non-linear optics Peschel, T., see Etrich, C. Peschel, U., see Etrich, C. Petite, G., see Shvartsburg, A.B. Petykiewicz, J., see Gniadek, K. Picht, J.: The wave of a moving classical electron Pollock, C.R.: Ultrafast optical pulses Popov, E.: Light diffraction by relief gratings: a macroscopic and microscopic view Popp, J., see Fields, M.H. Porter, R.P.: Generalized holography with application to inverse scattering and inverse source problems Presnyakov, L.P.: Wave propagation in inhomogeneous media: phase-shift approach Prucnal, P.R., see Glesk, I. Psaltis, D. and Y. Qiao: Adaptive multilayer optical networks Psaltis, D., see Casasent, D.
39, 291 32, 1 35, 197 42, 147 27, 1 15, 1 1, 1 7, 67
EBL
37, 57 41, 97 41, 359 18, 127 41, 359 33, 129 40, 115 43, 295 5, 83 41, 483 41, 483 44, 143 9, 281 5, 351 51, 211 31, 139 41, 1 27, 315 34, 159 45, 53 31, 227 16, 289
Qiao, Y., see Psaltis, D.
31, 227
Raymer, M.G. and I.A. Walmsley: The quantum coherence properties of stimulated Raman scattering Reiner, J.E., see Carmichael, H.J.
28, 181 46, 355
Cumulative index – Volumes 1–51 Renieri, A., see Dattoli, G. Reynaud, S., A. Heidmann, E. Giacobino and C. Fabre: Quantum fluctuations in optical systems Ricard, D., see Flytzanis, C. Rice, P.R., see Carmichael, H.J. Riseberg, L.A. and M.J. Weber: Relaxation phenomena in rare-earth luminescence Risken, H.: Statistical properties of laser light Roddier, F.: The effects of atmospheric turbulence in optical astronomy Rogister, F., see Uchida, A. Roizen-Dossier, B., see Jacquinot, P. Ronchi, L., see Wang Shaomin, Rosanov, N.N.: Transverse patterns in wide-aperture nonlinear optical systems Rosenblum, W.M. and J.L. Christensen: Objective and subjective spherical aberration measurements of the human eye Rothberg, L.: Dephasing-induced coherent phenomena Rouard, P. and P. Bousquet: Optical constants of thin films Rouard, P. and A. Meessen: Optical properties of thin metal films Roussignol, Ph., see Flytzanis, C. Roy, R., see Uchida, A. Rubinowicz, A.: The Miyamoto–Wolf diffraction wave Rudolph, D., see Schmahl, G.
EBL
Saichev, A.I., see Barabanenkov, Yu.N. Saito, S., see Yamamoto, Y. Sakai, H., see Vanasse, G.A. Saleh, B.E.A., see Teich, M.C. Saltiel, S.M., A.A. Sukhorukov and Y.S. Kivshar: Multistep parametric processes in nonlinear optics Sánchez-Soto, L.L., see Björk, G. Sánchez-Soto, L.L., see Luis, A. Sanders, B.C., see Hariharan, P. Scheermesser, T., see Bryngdahl, O. Schieve, W.C., see Englund, J.C. Schirripa Spagnolo, G., see Paoletti, D. Schmahl, G. and D. Rudolph: Holographic diffraction gratings Schubert, M. and B. Wilhelmi: The mutual dependence between coherence properties of light and nonlinear optical processes Schulz, G.: Aspheric surfaces Schulz, G. and J. Schwider: Interferometric testing of smooth surfaces Schwefel, H.G.L., see Türeci, H.E. Schwider, J.: Advanced evaluation techniques in interferometry Schwider, J., see Schulz, G. Scully, M.O. and K.G. Whitney: Tools of theoretical quantum optics Scully, M.O., see Greenberger, D.M. Search, C.P. and P. Meystre: Nonlinear and quantum optics of atomic and molecular fields
559 31, 321 30, 1 29, 321 46, 355 14, 89 8, 239 19, 281 48, 203 3, 29 25, 279 35, 1 13, 69 24, 39 4, 145 15, 77 29, 321 48, 203 4, 199 14, 195 29, 65 28, 87 6, 259 26, 1 47, 1 51, 469 41, 419 36, 49 33, 389 21, 355 35, 197 14, 195 17, 163 25, 349 13, 93 47, 75 28, 271 13, 93 10, 89 50, 275 47, 139
560
Cumulative index – Volumes 1–51
Senitzky, I.R.: Semiclassical radiation theory within a quantum-mechanical framework Sergienko, A.V., see Jaeger, G. Shalaev, V.M., see Litchinitser, N.M. Sharma, S.K. and D.J. Somerford: Scattering of light in the eikonal approximation Shchegrov, A.V., A.A. Maradudin and E.R. Méndez: Multiple scattering of light from randomly rough surfaces Shchepinov, V.P., see Ostrovsky, Yu.I. Shvartsburg, A.B. and G. Petite: Instantaneous optics of ultrashort broadband pulses and rapidly varying media Sibilia, C., see Mihalache, D. Simpson, J.R., see Dutta, N.K. Sipe, J.E., see De Sterke, C.M. Sipe, J.E., see Van Kranendonk, J. Sittig, E.K.: Elastooptic light modulation and deflection Sizmann, A. and G. Leuchs: The optical Kerr effect and quantum optics in fibers Slusher, R.E.: Self-induced transparency Smith, A.W., see Armstrong, J.A. Smith, D.Y. and D.L. Dexter: Optical absorption strength of defects in insulators Smith, R.W.: The use of image tubes as shutters Snapp, R.R., see Englund, J.C. Sodha, M.S., A.K. Ghatak and V.K. Tripathi: Self-focusing of laser beams in plasmas and semiconductors Somerford, D.J., see Sharma, S.K. Soroko, L.M.: Axicons and meso-optical imaging devices Soskin, M.S. and M.V. Vasnetsov: Singular optics Spreeuw, R.J.C. and J.P. Woerdman: Optical atoms Steel, M.J., see Eggleton, B.J. Steel, W.H.: Two-beam interferometry Steinberg, A.M., see Chiao, R.Y. Steinvurzel, P., see Eggleton, B.J. Stoicheff, B.P., see Jamroz, W. Stone, A.D., see Türeci, H.E. Strohbehn, J.W.: Optical propagation through the turbulent atmosphere Stroke, G.W.: Ruling, testing and use of optical gratings for high-resolution spectroscopy Subbaswamy, K.R., see Mills, D.L. Suhara, T., see Nishihara, H. Sukhorukov, A.A., see Saltiel, S.M. Sundaram, B., see Milonni, P.W. Svelto, O.: Self-focusing, self-trapping, and self-phase modulation of laser beams Svidzinsky, A.A., see Greenberger, D.M. Sweeney, D.W., see Ceglio, N.M. Swinney, H.L., see Cummins, H.Z.
13, 169 39, 213 27, 109 42, 219 31, 263 48, 1 5, 145 37, 345 48, 1 20, 325 47, 75 9, 73 2, 1 19, 45 24, 1 47, 1 31, 1 12, 1 50, 275 21, 287 8, 133
Tako, T., see Ohtsu, M. Tanaka, K.: Paraxial theory in optical design in terms of Gaussian brackets
25, 191 23, 63
EBL
16, 413 42, 277 51, 1 39, 213 46, 117 30, 87 44, 143 27, 227 31, 189 33, 203 15, 245 10, 229 39, 373 12, 53 6, 211 10, 165 10, 45 21, 355
Cumulative index – Volumes 1–51
561
Tana´s, R., A. Miranowicz and Ts. Gantsog: Quantum phase properties of nonlinear optical phenomena Tango, W.J. and R.Q. Twiss: Michelson stellar interferometry Tanida, J. and Y. Ichioka: Digital optical computing Tatarskii, V.I. and V.U. Zavorotnyi: Strong fluctuations in light propagation in a randomly inhomogeneous medium Tatarskii, V.I., see Charnotskii, M.I. Taylor, C.A., see Lipson, H. Teich, M.C. and B.E.A. Saleh: Photon bunching and antibunching Ter-Mikaelian, M.L., see Mikaelian, A.L. Thelen, A., see Musset, A. Thompson, B.J.: Image formation with partially coherent light Thyagarajan, K., see Ghatak, A. Tonomura, A.: Electron holography Torner, L.L., see Desyatnikov, A.S. Torre, A.: The fractional Fourier transform and some of its applications to optics Torre, A., see Dattoli, G. Tripathi, V.K., see Sodha, M.S. Tsujiuchi, J.: Correction of optical images by compensation of aberrations and by spatial frequency filtering Türeci, H.E., H.G.L. Schwefel, Ph. Jacquod and A.D. Stone: Modes of wave-chaotic dielectric resonators Turunen, J., M. Kuittinen and F. Wyrowski: Diffractive optics: electromagnetic approach Twiss, R.Q., see Tango, W.J.
47, 75 40, 343 17, 239
Uchida, A., F. Rogister, J. García-Ojalvo and R. Roy: Synchronization and communication with chaotic laser systems Upatnieks, J., see Leith, E.N. Upstill, C., see Berry, M.V. Ushioda, S.: Light scattering spectroscopy of surface electromagnetic waves in solids
48, 203 6, 1 18, 257 19, 139
EBL
Vampouille, M., see Froehly, C. Van De Grind, W.A., see Bouman, M.A. van Haver, S., see Braat, J.J.M. Van Heel, A.C.S.: Modern alignment devices Van Kranendonk, J. and J.E. Sipe: Foundations of the macroscopic electromagnetic theory of dielectric media Vanasse, G.A. and H. Sakai: Fourier spectroscopy Vartiainen, E.M., see Peiponen, K.-E. Vasnetsov, M.V., see Soskin, M.S. Vernier, P.J.: Photoemission Vlad, V.I. and D. Malacara: Direct spatial reconstruction of optical phase from phase-modulated images Vogel, W., see Welsch, D.-G.
35, 355 17, 239 40, 77 18, 204 32, 203 5, 287 26, 1 7, 231 8, 201 7, 169 18, 1 23, 183 47, 291 43, 531 31, 321 13, 169 2, 131
20, 63 22, 77 51, 349 1, 289 15, 245 6, 259 37, 57 42, 219 14, 245 33, 261 39, 63
562
Cumulative index – Volumes 1–51
Walmsley, I.A., see Raymer, M.G. Wang Shaomin, and L. Ronchi: Principles and design of optical arrays Wang, B.C., see Glesk, I. Weber, M.J., see Riseberg, L.A. Weigelt, G.: Triple-correlation imaging in optical astronomy Weisbuch, C., see Benisty, H. Weiss, G.H., see Gandjbakhche, A.H. Welford, W.T.: Aberration theory of gratings and grating mountings Welford, W.T.: Aplanatism and isoplanatism Welford, W.T., see Bassett, I.M. Welsch, D.-G., W. Vogel and T. Opatrný: Homodyne detection and quantum-state reconstruction Whitney, K.G., see Scully, M.O. Wilhelmi, B., see Schubert, M. Winston, R., see Bassett, I.M. Woerdman, J.P., see Spreeuw, R.J.C. Wolf, E.: The influence of Young’s interference experiment on the development of statistical optics Woli´nski, T.R.: Polarimetric optical fibers and sensors Wolter, H.: On basic analogies and principal differences between optical and electronic information Wynne, C.G.: Field correctors for astronomical telescopes Wyrowski, F., see Bryngdahl, O. Wyrowski, F., see Bryngdahl, O. Wyrowski, F., see Turunen, J.
EBL
Xiao, M., see Joshi, A. Xu, L., see Glesk, I.
28, 181 25, 279 45, 53 14, 89 29, 293 49, 177 34, 333 4, 241 13, 267 27, 161 39, 63 10, 89 17, 163 27, 161 31, 263 50, 251 40, 1 1, 155 10, 137 28, 1 33, 389 40, 343 49, 97 45, 53
Yamaguchi, I.: Fringe formations in deformation and vibration measurements using laser light Yamaji, K.: Design of zoom lenses Yamamoto, T.: Coherence theory of source-size compensation in interference microscopy Yamamoto, Y., S. Machida, S. Saito, N. Imoto, T. Yanagawa and M. Kitagawa: Quantum mechanical limit in optical precision measurement and communication Yanagawa, T., see Yamamoto, Y. Yaroslavsky, L.P.: The theory of optimal methods for localization of objects in pictures Yeh, W.-H., see Carriere, J. Yin, J., W. Gao and Y. Zhu: Generation of dark hollow beams and their applications Yoshinaga, H.: Recent developments in far infrared spectroscopic techniques Yu, F.T.S.: Principles of optical processing with partially coherent light Yu, F.T.S.: Optical neural networks: architecture, design and models
28, 87 28, 87 32, 145 41, 97 45, 119 11, 77 23, 221 32, 61
Zalevsky, Z., D. Mendlovic and A.W. Lohmann: Optical systems with improved resolving power
40, 271
22, 271 6, 105 8, 295
Cumulative index – Volumes 1–51 Zalevsky, Z., see Lohmann, A.W. Zavorotny, V.U., see Charnotskii, M.I. Zavorotnyi, V.U., see Tatarskii, V.I. Zhu, Y., see Yin, J. Zubairy, M.S., see Greenberger, D.M. Zuidema, P., see Bouman, M.A.
563 38, 263 32, 203 18, 204 45, 119 50, 275 22, 77
EBL
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EBL
