EDITORIAL ADVISORY BOARD
G. S. Agarwal,
Ahmedabad, India
G. Agrawal,
Rochester, USA
T. Asakura,
Sapporo, Japan
A. Aspect,
Orsay, France
M.V Berry,
Bristol, England
A.T. Friberg,
Stockholm, Sweden
V L. Ginzburg,
Moscow, Russia
E Gori,
Rome, Italy
A. Kujawski,
Warsaw, Poland
L.M. Narducci,
Philadelphia, USA
J. Pefina,
Olomouc, Czech Republic
R. M. Sillitto,
Edinburgh, Scotland
H. Walther,
Garching, Germany
Preface This volume presents six review articles devoted to various topics of current interest both in classical and in quantum optics. The first article, by S.Ya. Kilin, entitled Quanta and information, is concerned with a multidisciplinary subject which involves optics, information theory, programming and discrete mathematics. It contains contributions from areas such as computing, teleportation, quantum cryptography and decoherence. The article presents an account of recent results obtained in this relatively new field. The second article. Optical solitons in periodic media with resonant and off-resonant nonlinearities, by G. Kurizki, A.E. Kozhekin, T. Opatrny and B. Malomed, reviews the properties of optical solitons in periodic nonlinear media. The emphasis is on solitons in periodically refractive media (Bragg gratings), incorporating a periodic set of thin layers of two-level systems resonantly interacting with the field. Such media support a variety of bright and dark 'gap solitons' propagating in the band gaps of the Bragg gratings, as well as their multi-dimensional analogs (light bullets). These novel gap solitons differ substantially from their counterparts in periodic media with either cubic or quadratic off-resonant nonlinearities. The article which follows, entitled Quantum Zeno and inverse quantum Zeno effects, by P. Facchi and S. Pascazio, deals with an effect and its inverse which is a manifestation of hindrance and enhancement, respectively, of the evolution of a quantum system by an external agent, such as a detection apparatus. The article includes some examples from quantum optics and quantum electrodynamics. The fourth article, by M.S. Soskin and M.V Vasnetsov, discusses the current status of a relatively new branch of physical optics, sometimes called singular optics. It is concerned with effects associated with phase singularities of wavefields. Wavefronts in the neighborhood of such points exhibit dislocations, optical vortices and other features which are not present in commonly encountered wavefields which have smooth wavefronts. The next article, by G. Jaeger and A.V Sergienko, presents a review of advances in two-photon interferometry and their relation to investigations of the foundations of quantum theory. A recent history of tests of Bell's inequality and the production of entangled photon pairs for testing it is given that illustrates the central role of spontaneous parametric down-conversion in current two-photon
vi
Preface
interferometry. Quantum imaging and quantum teleportation are used to illustrate the current power of entanglement in advanced quantum-optical applications. New, increasingly efficient sources of entangled photon states are described and the manner in which they will assure further progress in multiple-particle interferometry is discussed. Multiple-photon entanglement is shown to provide a new set of phenomena to be investigated in the future by multiple-photon interferometry. The concluding article, by R. Oron, N. Davidson, A.A. Friesem and E. Hasman, is concerned with transverse mode shaping and selection in laser resonators. It presents a review of recent investigations on the shaping and selection of laser modes, by the use of various elements that are inserted into laser resonators. Experimental techniques, as well as basic numerical and analytical methods, are presented. The qualities of the emerging beams, based on different criteria, are discussed, along with various applications for specially designed beams. I wish to use this opportunity to welcome three new members to the Editorial Advisory Board of Progress in Optics, namely Professor G. Agrawal (Rochester), Professor A.T. Friberg (Stockholm) and Professor L.M. Narducci (Philadelphia). Emil Wolf Department of Physics and Astronomy University of Rochester Rochester, New York 14627, USA July 2001
E. Wolf, Progress in Optics 42 © 2001 Elsevier Science B.V. All rights reserved
Chapter 1
Quanta and information by
Sergei Ya. Kilin Quantum Optics Lab, B.I. Stepanov Institute of Physics, National Academy of Sciences of Belarus, Francisc Skarina Avenue, 70, 220072, Minsk, Belarus
Contents
Page Introduction
3
§ 1. The quantum concept
3
§ 2.
Information
16
§ 3. Quantum information
30
§ 4.
The problem of decoherence
74
§ 5.
Conclusions
82
Acknowledgements
84
References
84
Introduction The first volume of the series Progress in Optics, launched and edited by Emil Wolf, contained a chapter by Dennis Gabor [1961] entitled "Light and Information". It was a record of his lecture presented at the University of Edinburgh in 1951. In an effort to answer the question of what information theory can contribute to the physics of light, Gabor suggested a program: "Information theory is of some heuristic use in physics, by asking the right sort of questions''. He also noted that information theory "prepares the mind for quantum theory", and that "in information theory we appear to have the right tool for introducing the quantum point of view in classical physics". These statements, which suggest a potentially useful program, sound very contemporary today. Before implementing the program, however, one important step should be taken, namely the need to realize that quantum objects are radically new objects to information theory, with new information resources, inaccessible for classical objects. In this chapter I will endeavor to show briefly how both "programs" - "Quantum Physics for Information Theory" and "Information Theory for Quantum Physics" - are working together to fiirther progress in understanding the world and to offer new practical applications in technology.
§ 1. The quantum concept The rapid development of quantum optics at the end of the twentieth century, has caused many individuals, not only specialists in quantum physics but also people working far fi-om this field, to appreciate once more the basic statements of quantum theory. Indeed, the abstract basic ideas of quantum physics, to which only a few specialists paid attention not long ago, are now important for almost everyone because of their new applications in technology and, primarily, in optical applications. Quantum computers, quantum teleportation and cryptography, observation and monitoring of single atoms, ions, and molecules, including biological molecules, all belong to the quantum world. This world is extremely difficult to explain in terms of the common classical world of
4
Quanta and information
[1, § 1
macroscopic physics. For its description, it requires a proper definition of quantum mechanics and quantum field theory. Quantum mechanics, which originated in the 1920s by the investigations of Niels Bohr (1885-1962), Erwin Schrodinger (1887-1961) and Werner Heisenberg (1901-1976), provided physicists with the recipes for calculating the energy states of atoms and molecules and the matrix elements of transitions between these states. However, in addition to this aspect which immediately found applications in practical physics, quantum mechanics contained the "ideological", philosophical aspect, which accounts for the odd nature of the quantum world and has remained almost unused until recent years. In the most complete and clear form, sometimes with deliberately paradoxical statements, this part of the quantum theory was presented by Erwin Schrodinger in his famous paper of 1935, which he classified as "a paper or a general confession" {Referat oder Generalbeichte). Using modem terminology, it examines one of the problems of quantum information, namely, what information about the states of quantum objects can we obtain and what happens to the quantum objects while we are obtaining this information? More than half of a century passed before the basic principles formulated by Schrodinger became necessary for an understanding of experiments with practical applications. The present chapter discusses several experiments of this kind. These are, first, experiments on quantum teleportation, quantum cryptography, and, second, quantum computers, which are expected to be extremely beneficial but difficult to construct. Some of the chapter is devoted to single material particles in quantum optics and the methods of their detection. These objects can serve as elements of quantum computers. In conclusion, I examine the problem of decoherence and possible solutions, which is crucial for quantum computation. First, this chapter discusses the language of quantum optics and the statements of quantum theory necessary for fiirther examination.
1.1. Schrodinger and his famous paper of 1935 On November 29, 1935, the journal Die Naturwissenschaften published Erwin Schrodinger's paper "Modem state of quantum mechanics". It was written during his compelled stay in Oxford (fig. 1), after he and Paul Dirac had been awarded the 1933 Nobel prize in physics. As Schrodinger mentioned, his work originated from the discussion started on May 15, 1935 by Albert Einstein, Boris Podolsky and Nathan Rosen in their paper "Can the Quantum Mechanical Description of Reality be Complete?", and continued in Niels Bohr's [1935] paper with the
1, § 1]
The quantum concept
'^w.
Fig. 1. Erwin Schrodinger was bom in Vienna, where he studied at first in a gymnasium, then in the university until graduation in 1910. Schrodinger started working in theoretical physics and soon became a professor in Breslau (now Wroclaw) and then in Zurich, where Einstein had worked earlier. In Zurich, Schrodinger published works that led him to a formulation of the basic equation of nonrelativistic quantum mechanics, the Schrodinger wave equation. For the development of quantum mechanics, Schrodinger together with Dirac was awarded a Nobel prize in 1933. In 1927, he was appointed to the chair of theoretical physics in Berlin, previously headed by Planck. When Hitler attained power, Schrodinger left fascist Germany and accepted an invitation to Oxford. In 1936 he returned to Austria for a short time and held a chair in Graz, but after the Anschluss he had to leave his country again. This time Schrodinger moved to Ireland, to the Institute of Fundamental Research in Dublin. In 1947 he finally returned to his homeland. His health was failing, however, and after a long illness he died in Vienna. [The photograph and biographical note are taken fi-om the anthology Zhizn' Nauki [Science life], edited by S.P. Kapitsa and published by Nauka (Kapitsa [1973]).]
same title. Despite the abstract and complicated style of Schrodinger's paper, its importance was soon realized by Russian scientists, and it was immediately translated into Russian and published in 1936. The English translation did not appear until 1980. In his paper, Schrodinger analyzed difficulties in the quantum-mechanical description of measurement procedures and formulated four basic principles. According to these principles, the states of quantum objects have the following properties: (1) Superposition: A quantum state is described by a linear superposition of the basic states. (2) Interference: The result of measurement depends on the relative phases of the amplitudes in this superposition.
6
Quanta and information
[1» § 1
(3) Entanglement: Complete information about the state of the whole system does not imply complete information about its parts. (4) Nondonability and uncertainty: An unknown quantum state can be neither cloned nor observed without being disturbed. I shall briefly comment on each of these statements. However, let us note first that, until recently, the third and fourth principles were almost unknown to most physicists and were discussed only in connection with the Einstein-PodolskyRosen (EPR) paradox and Bell's inequalities.
1.2. Quantum objects and their states 1.2.1. Superposition and the Schrodinger-cat paradox In contrast to a classical object, a quantum object has statistical origin. However, the probabilistic nature of a quantum object cannot be understood as a classical uncertainty connected, for instance, with incomplete knowledge about the object. For the description of a quantum object, the concept of state is used. By saying that an object is in a quantum state, we mean that there is a list (a catalog, in Schrodinger's language) or, similarly, a wave ftinction, a state vector, or a density matrix containing the information about the possible results of measurement on this object. In the general case, the results of measurement differ fi-om time to time even if the object is prepared in the same quantum state. Hence, the state vector should give statistical information, i.e., distribution functions for the results of measurement. As a simple example, consider the state vector for a system with two orthogonal basic states |1) and |2), e.g. energy states. The state of the object is described by the state vector (wave function) |^) = a|l)+^|2),
(1.1)
where a and /? are complex numbers. In other words, the total state is given by a linear superposition, and the squares of the absolute values of the amplitudes a and (3 are equal to the probabilities of finding the system in the corresponding states (|a|^ + |j8p = 1). As a result of measurement, the coherent superposition (1) is destroyed and reduced to a new state, which is determined by the type of measurement. For instance, an attempt to find the system in state 12) leads to its perturbation by the measurement device. At the moment of measurement, reduction (projection) takes place, |^)^|2)(2|^^)=^|2),
(1.2)
1, § 1]
The quantum concept
7
SO that after the measurement the system is driven into state |2) and the initial state is destroyed ^ A superposition state should be distinguished from a mixed state, which is described by the density matrix Pn,,x = | a | ' | l > ( l | + |i8|'|2)(2|.
(1.3)
In fact, state (1.3) is a classical state, since a system in a mixed state can be found either in state 11) or in state |2), whereas in the superposition state (1.1) the system can be simultaneously found in two states. This principal feature of a superposition state manifests itself in the interference terms of its density matrix p = | W ) ( i I / | = | a | 2 | i ) ( i | + ||3f|2)(2| + a r | l ) ( 2 | + a*/?|2>(l|.
(1.4)
To stress the unusual nature of superposition states, Schrodinger suggests an example disturbing to our common sense. Following Schrodinger, suppose that a steel chamber contains a flask with poison that can be broken by means of some mechanism triggered by the radioactive decay of a single atom. The box also contains a cat (initially alive), which can die as a result of the atom's decay. Similarly to the atom whose state is a superposition of the decayed and nondecayed states, the state of the cat is also given by the superposition of the states of an alive cat, 11), and a dead cat | i ) : | ^^) =L| T ) + ^ | i )• Since quantum superposition states are frequently observed for microscopic systems, such as atoms and molecules, but never observed for macroscopic systems, some effect must be destroying the Schrodinger-cat states for macroscopic systems. This effect, which is called decoherence, is considered below. Note that the problem of pertaining superposition (Schrodinger cat) states for mesoscopic systems is crucial, and its solution will give rise to many applications of quantum information. For further consideration, the superposition state is used to describe a singlephoton beam with a given wave vector, or a single-photon state 11 photon)The state of radiation with a given wave vector can be represented by a
^ Note that measurement, i.e., interaction with a macroscopic measurement device, is an irreversible process in principle. During this process, the state of the measured object changes (reduction takes place). Reduction, like other physical processes, has its own characteristic time scale, specific for each individual measurement. The process of reduction is very short, however, so the question of its internal dynamics, i.e., of the possibility to 'see it with one's own eyes', is usually ignored, although in some measurements, for instance in quantum tomography of ultrashort pulses, it is obviously of interest.
Quanta and information
|2>j|0)^=|W>
|l>l|l>«=|t^>
|i>t|o>«=|J>
[1, § 1
I0>j|2>„=|««>
l«>t|iL=l«> l«>l|0>«=|0>
Fig. 2. A light beam with a fixed wave vector is equivalent to two harmonic oscillators corresponding to two orthogonally polarized modes of the electromagnetic field. A single-photon state of this beam is given by a superposition of two energy-degenerate states of polarized photons | J) and | ^ ) . A twophoton state of this beam is generally a superposition of three energy-degenerate states, two of which represent pairs of photons with equal polarizations | I | ) and | ^ ^ ) , and the third representing a pair of orthogonally polarized photons \l<-^).
set of two quantum-mechanical oscillators, each corresponding to one of two orthogonal polarizations (fig. 2). Denoting the eigenstates of the oscillator with vertical polarization as \n)i and the eigenstates of the oscillator with horizontal polarization as \m)^, two basic vectors, | 1 ) | | 0 ) ^ = | | ) , | 0 ) | | 1 ) ^ = |<-^), are introduced so that any single-photon state can be decomposed as llphoton)
(1.5)
=a\l)-^b\^).
Note that a certain ambiguity exists in the notion of a superposition state. In fact, state (1.5) with a = 13= l/\/2 is a superposition state if considered on the basis of vertical and horizontal polarizations, that is, measured by means of polaroids oriented horizontally and vertically. At the same time, this state (||) + \^)yV2 can be considered as one of the basis states of the pair
I/) = (II) +1-))/v^,
l\) = (II) - MVVi,
(1.6)
which corresponds to a measurement with the polarizers oriented at 45*^ and 135*^. In this basis the state evidently cannot be considered as a superposition state. Therefore, any quantum state is a superposition state, since it is a superposition state in any basis where it is not a basic vector. The choice of basis states implicates the choice of measurements allowed by a quantum object. An important type of measurements arises when a single-mode field with fixed polarization is discriminated by two directions in the phase space (say X and P, fig. 3). In this method of discrimination, which is obviously connected with the type of measurements adopted, one is not interested in how many photons are
1, §1]
The quantum concept
Fig. 3. The stereographical projection from the north pole of the sphere demonstrates the isomorphism between a harmonic oscillator phase plane and the phase space of a two-level system - the Bloch sphere. Two orthogonally squeezed states displaced from the origin by the amplitude a ^ XQ + ipo in the limit of infinite squeezing correspond to two different circles (shown in white) on the sphere. Two states 1^^^^} and \ep^) are defined by the vectors orthogonal to planes crossing the white circles. Therefore, the continuous-variable measurement discriminating between two orthogonally squeezed states is equivalent to a measurement discriminating between two nonorthogonal states \ex^) and \ep^) [{ex^^\ep^) = XQPQ/J(4 + xl){4 + p^)] of a two-state quantum system (e.g., a polarization state of a single photon).
in the object (e.g. in an optical pulse), but rather one wonders how much the light is squeezed in the X or P direction. This method of characterizing quantum objects can be referred to as continuous-variable representation, because of the assignment of the object states to a continuous phase plane. Note that the two-valued continuous variable representation of the harmonic oscillator states is isomorphic to the coherent-state representation of a two-state quantum system (Arecchi, Courtens, Gilmore and Thomas [1972], Pefina [1984]), with a sphere as the phase space (fig. 3) because the stereographical projection provides one-to-one correspondence between sphere and plane. The measurement used for the discrimination is homodyne detection of squeezed light variances, (Ax2) - {(a + fl+)2) - {(a + a^))\ (Ap^) - -{(a - a^f) + {(a - a^))^ (Yuen and Chan [1983]). For details of the isomorphism see § 1.2.2.2. below and the work by Van Enk [1999]. 7.2.2. Entangled states In addition to superposition states, Schrodinger considered the so-called entangled states, which describe the state of a composite system whose parts can be
10
Quanta and information
Initial state
Interaction
[1? § 1
Final state
Fig. 4. As an example of an entangled state, consider the state of a composite system: two-level atom-field. Suppose that the atom is crossing the area of interaction with the field, for instance, a cavity. After a short period of interaction, the atom and the field become spatially separated. However, the state of the whole system remains entangled: the state of the atom is strictly correlated with the state of the field |*^) = |atom)i |field)} + |atom)2 |field)2- Note that the lifetime of such an entangled state may be much larger than the interaction time.
spatially delocalized. States of two systems could serve as examples of an entangled state: the state of a field and the atom emitting it (fig. 4) or a quantum system formed by two single-photon beams with different wave vectors. Each state of such a photon pair can be represented as a superposition of four basic states, 111 +12) = Cii iDi II)2 + c _
| - ) i | - ) 2 + q ^ \l), l - h + c ^ i l - ) i Il>2. (1.7) In the general case, the photonfi*omthe first beam is connected with the photon from the other beam, since the total state vector is not given by the product of the single-photon state vectors. This connection, which is much stronger than in the case of classical correlation, manifests itself in experiments with photon pairs in Bell states. Bell [1964] suggested these states in relation to the EPR paradox. The states l ^ ' ) = (II)ilI)2 + | - - ) i | - - ) 2 ) / V 2 ,
(1.8a)
l<^-)=(II>ill)2-|--)i|--)2)/V2,
(1.8b)
l'^") = (II>i|--)2 + | - ) i l D 2 ) / ^ , l'f'") = ( I I ) i | - - ) 2 - | - ) i l I ) 2 ) / V 2 ,
(1.8c) (1.8d)
form the basis of the Bell states. Each of these entangled states has a remarkable property, namely, as soon as some measurement projects one of the photons onto a state with a definite polarization, the other photon also immediately becomes polarized. For instance, in the case of |y^^) states, if one photon is registered with polarization <-^, the other photon turns out to have the orthogonal polarization!. How can a measurement over one particle have an instantaneous effect on the other, possibly located at a large distance? Einstein, as well as many other outstanding physicists, did not accept this "action of ghosts at a distance", but this behavior of entangled states has been demonstrated in numerous experiments (Clauser and Shimony [1978], Greenberger, Home and Zeilinger [1993]).
1, § 1]
The quantum concept
11
Entangled quantum objects are not absolutely unknown objects in physics. The two electrons of a helium atom in the ground state, for example, have their spins entangled, but this is more a type of unification through unavoidable interactions than entanglement. The entanglement of which Schrodinger spoke is the state of two quantum objects which are noninteracting at the moment of observation. This is an important point in all discussions about entangled states, and it eliminates many questions. Thus, entanglement implies the presence of at least two objects, allowing local manipulations with each. Naturally, it means that at least two possible conclusive results exist for measurements on each part. Only in this case one conclusive measurement leaves an opportunity for a second one to be selective, but not trivial, noninformative. This snag sometimes waits for those who pay attention only to the bipartite aspect of entanglement, for example, considering the two-mode-one-photon state as entangled. Any attempts to leave two opportunities for second measurements after the first conclusive one, for example, by means of nondemolition methods, immediately create a new quantum system, with new outputs (paths) for a photon. As a result, an observer has to describe this new quantum system with an increased number of modes because the two-mode-one-photon system is actually nothing but one quantum system with two possible states. Entangled states have another paradoxical property, which was pointed out by Schrodinger in 1935. According to one of his principles, complete information about the state of the total system still does not provide complete information about the states of its parts. Suppose that we are going to find out the state of a particle in one of the pairs (1.8), for example, in eq. (1.8d). Then we have to average the density matrix of the pure, that is, the most determinate, state
l'^")=(lt),M2-|-)lll)2)/V^ over the states of the second particle. The resulting density matrix of the first particle, p(i) = Tr(|y^-)(«f^-|) = (||), , a | + | ^ > , i ( - | ) / 2 ,
is apparently the density matrix of a mixed state, which is not maximally determinate. In addition, one important optical object possesses the property of entanglement - two orthogonally squeezed modes with fixed polarizations and different wavevectors. The state of this continuous variables object in the limit
12
Quanta and information
[1? § 1
of infinite squeezing is isomorphic to the Bell states of a two-level system discriminated by the measurements in orthogonal basis (1^;^) ± \ep)), composed from nonorthogonal states \ex) and \ep) (see fig. 3). At finite squeezing r a continuous variables entangled state is known from quantum optics as a twomode squeezed state
\^r) = ^,fl('^^'-r\")^\")2-
(1-9)
n=0
Equation (1.9) is written in the Fock basis (Walls and Milbum [1995]). 1.2.2.1. How can one generate entangled states?. Entangled photon pairs can be obtained experimentally by means of cascade decays in atomic systems (Aspect, Grangier and Roger [1981]) or parametric processes involving resonance fluorescence, where two pump photons give birth to a pair of entangled photons (JD\ and 0)2, (OQ + 0)Q -^ W\ -\- (x>2. The quantum correlation for such photons has been predicted by Apanasevich and Kilin [1977, 1979] and observed by Aspect, Roger, Reynaud, Dalibard and Cohen-Tannoudji [1980]. The possibility of obtaining entangled states for massive particles, atoms, was demonstrated experimentally by Hagley, Maitre, Nogues, Wunderlich, Brune, Raimond and Haroche [1997]. At present, the most popular source of entangled photons is spontaneous parametric decay (spontaneous parametric down-conversion) in crystals with quadratic nonlinearity (Zel'dovich and Klyshko [1969], Bumham and Weinberg [1970]). In this process an ultraviolet pump photon decays into a pair of red photons with approximately equal energies, so that the energy and momentum conservation laws are satisfied, ho)^ = hw^ + ho){\ hk^ = hk^ + hki, where hcOj and hkj (j = p, s, i) are the energy and momentum, respectively, of the initial photon (p) and the two output photons, called the signal (s) photon and the idler (i) photon. By using crystals with quadratic nonlinearity and type-II phase matching (fig. 5), one can easily obtain polarization-entangled states in the directions 1 and 2, which are determined by the intersections of phase matching cones for ordinary and extraordinary photons (fig. 5b). In these directions one can observe Bell states of the form (1.8) (Kwiat, Mattle, Weinftirter, Zeilinger, Sergienko and Shih [1995]). Note that recently Atatiire, Sergienko, Saleh and Teich [2000] demonstrated a new technique that can serve as a basic component in the preparation of multiphoton entangled states with the help of femtosecond spontaneous parametric down-conversion. The entangled states of two-mode squeezed optical fields with X-P quantum correlation can be produced by degenerate parametric down-conversion.
The quantum concept
1,§1]
13
Optic axis of the crystal
Type II
Fig. 5. Creation of entangled photon pairs in parametric down-conversion, (a) Momentum conservation inside the crystal, also called "phase matching", is achieved due to the crystal birefringence, which allows the dispersion to be compensated. As a result, the idler and signal photons form a rainbow of colored cones, where conjugated photons are emitted in opposite directions with respect to the pump beam. In the case of type-I phase matching, the signal and idler photons have the same linear polarization, which is orthogonal to the pump polarization, and their cones are concentric with the pump beam. In the case of type-II phase matching, conjugated pairs are formed by a photon with extraordinary polarization and a photon with ordinary polarization. In this case, the cones of signal and idler photons have different axes. For uniaxial negative crystals, like BBO (beta-barium borate), the axis of the cone of extraordinary photons lies between the pump beam and the optic axis, whereas the axis of the cone of ordinary photons is on the opposite side of the pump beam (all the axes and the pump beam are in the same plane), (b) The image of down-converted light emitted by the crystal. Numbers 1 and 2 denote the directions in which polarization-correlated pairs are emitted. In these directions there is no definite polarization; all we know is that the polarization is different for beams 1 and 2.
ICDL ^^ (Oi±Q m2i subthreshold optical parametric oscillator (OPO) (Polzik, Carri and Kimble [1992]). Note that any nonlinear interaction of optical pulses is a possible source for entanglement creation. For example, nonlinear interactions proposed for the nondemolition measurement could be used to create entangled states (Horoshko and Kilin [2000]). The opposite process, disentanglement, when initial entanglement between subsystems is erased while the local properties of the state are preserved, was recently described by Mor and Terno [1999]. 1.2.2.2. How can one measure (project) entangled states?. A Bell state can be distinguished from the other Bell states due to their different symmetry. Among the four states (1.8), the first three have bosonic symmetry, since permutation
14
Quanta and information
[1, § 1
Fig. 6. Scheme for observing intensity interference. Polarization-entangled beams emitted by the crystal are mixed on a 50% beam splitter and registered by two detectors. Photocounts from the detectors are fed to the coincidence circuit. For each photon from any beam, two possibilities exists, to be either reflected or transmitted by the beam splitter. The probability of a photocount is given by the square of the absolute value of the sum of the corresponding quantum amplitudes. The unitary transformation performed by the nonpolarizing beam splitter concerns only the spatial part of the photon wave function. Photons are bosonic particles; therefore, the spatial part of the wave fiinction is symmetrical for the bosonic polarization states | ^ ^ ) , | ^^) and antisymmetrical for the fermionic state | ^ ~ ) . Two-photon interference on a beam splitter demonstrates that two photons are directed by the beam splitter into the same beam in the case of a symmetrical wave fiinction and into different beams in the case of an antisymmetrical wave function. Hence, a coincidence of photocounts from two detectors projects the state of a photon pair onto the fermionic state \^~).
of particles 1 and 2 does not change the signs of their wave functions. The last state (1.8d) is fermionic: permutation of 1 and 2 changes the sign of its wave function. This specific feature of the state \W') = (||)i \^)2 - \^)i 11)2)/A/2 also reveals itself in the intensity interference of beams 1 and 2 (fig. 6). In the figure, both detectors click only if the entangled photon pairs are in the fermionic polarization state | W~). This is a well-known feature of two-photon interference on a beam splitter (Feynman, Leighton and Sands [1964]); in the case of a spatially symmetrical wave function, the beam splitter sends both particles into the same output beam, whereas in the case of a spatially antisymmetrical wave fiinction, the two particles are directed into different output beams. Photons have bosonic statistics; therefore, conservation of the total symmetry requires that the spatial part of the polarization-fermionic wave fiinction | ^ ~ ) is antisymmetrical. A measurement distinguishing the fermionic state among the four states (1.8) is called Bell state measurement (BSM). The problem of complete BSM with the discrimination of all Bell states is not easy to implement experimentally. A simple interferometric setup cannot be of help for the discrimination among the four EPR states. This was reported recently by Liitkenhaus, Calsamiglia and Suominen [1999] and by Vaidman and Yoran [1999], who demonstrated that a complete Bell measurement cannot be performed using only linear elements. Kwiat and Weinfurter [1998] suggested
1, § 1]
The quantum concept
15
a method to overcome this conclusion, but their method requires that the state of interest be embedded in a larger Hilbert space, and therefore can be applied only in the presence of multiple entanglement. To reach this objective, Scully, Englert and Bednar [1999] presented a scheme in which each of the four maximally polarization-entangled two-photon Bell states is uniquely correlated with the counts registered in one of the four detectors. The scheme exploits the nonlinear process - resonant two-photon absorption of atoms, which are suitably prepared in a coherent superposition of hyperfine states. In another proposal, Paris, Plenio, Bose, Jonathan and D'Ariano [1999] described an interferometric setup designed to perform a complete optical Bell measurement. It consisted of a Mach-Zehnder interferometer with the first beam splitter a polarizing one and the second a normal one; inside the interferometer a nondemolition photon number measurement is performed by the Fock filtering technique. The identification of continuous-variables entanglement presents no problems of this kind, because the very notion of entanglement for continuous variables obviously implies the choice of type of measurements in advance: the observer should discriminate between two directions on the phase plane. In optics this identification is realized by homodyne detection including local oscillators and beam splitters. The degree of squeezing of local beams plays a crucial role for the degree of continuous-variables entanglement. 1.2.3. The impossibility of cloning quantum states Since the measurement device destroys the initial quantum state, one can consider a quantum state as a highly sensitive object that all the information about itself keeps secret. The uncertainty principle is a manifestation of this ability. Another typical manifestation is the theorem about the nonclonability of a quantum object state (Wootters and Zurek [1982]). Cloning means creation of an exact copy of an object with the conservation of its initial (and unknown) state. Suppose we have a device for cloning photons. This device reproduces photons with given properties (photons in a given state). If we mean polarization states, the effect is described by the transformation I^I>l I) ^ I^FV)| n),\Rl)\
- > ^ |i?FH)| — ) ,
where \Ri) is the initial state of the cloning device and \Rv\), |^FH) are its final states after cloning photons with vertical and horizontal polarizations. In other words, instead of a single photon with a given polarization, we obtain two photons with the same polarization (fig. 2). However, if we try to clone
16
Quanta and information
[1? § 2
a photon with a polarization that is neither horizontal nor vertical, for instance, a II) +iS |<^), the transformation will be |i?,)ia\ I) +/3| ^ ) ) ^ a|/?Fv)| H) +)3|/?FH)| — ) •
(I-IO)
Even under the condition of equality of |i?Fv) and \RFR), the transformed state does not represent two photons polarized at angle (p = arctanjS/a. Indeed, creation of a single photon polarized at the angle q) = arctan^/a, that is, in the state a ||) + ^ | ^ ) , could be reahzed by applying the creation operator b^ = aciy + ^a^ to vacuum, where ^V'^H ^^^ ^^^ creation operators for the photons polarized vertically and horizontally. They correspond to the right and left harmonic oscillators in fig. 2. Two photons with the same polarization can be obtained from the vacuum by applying this creation operator twice: (b\f
|0) = v/2 (a" III) + ; S 2 1 ^ ^ ) + v/2ay3|I^>) .
(1.11)
For all nonzero a,^ the state (1.11) does not coincide with the field part of state (1.10), that is, a single quantum object cannot be cloned.
§ 2. Information 2.1. Shannon and his classical paper Information theory was bom in a surprisingly rich state in the classical paper of Claude E. Shannon [1948] (fig. 7), 13 years after Schrodinger's paper [1935]. Shannon's paper entitled "The Mathematical Theory of Communication", considered how to describe any communication system without resorting to the concrete type of physical structure of the system (which, of course, can be as diverse as the whole world) and to the meaning of communication messages. In this way he argued two main points of information theory: (1) Information can be measured, and these measures are of practical importance in choosing methods for transmission of messages. (2) Information can be transmittedfaithfully, even if errors are generated during the communication. In his work Shannon discussed almost all classical resources of information. The main points and notions of the work are the following: entropy, relative entropy; information, mutual information; encoding, redundancy of alphabets; typical sequences, data compression; noiseless coding theorems; error-correcting codes; and information of continuous functions, sampling theorem.
1, § 2]
Information
17
Fig. 7. Claude Shannon was bom in Gaylord, Michigan, in april 1916. He studied in Michigan and at the Massachusetts Institute of Technology, and in 1938 published a seminal paper on the application of symbolic logic to relay circuits, which helped transform circuit design from an art into a science. Shannon's famous work, A Mathematical Theory of Communication [1948], outlining what we now know as "information theory", described the measurement of information by binary digits representing "yes-no alternatives", which is the fundamental basis of today's telecommunications. Fortunately, A Mathematical Theory of Communication was written while Shannon was employed by Bell Labs - fortunately, because Shannon was not planning to publish his work, and finally did so at the urging of fellow employees. The paper became the basis for information theory, as we know it today. Shannon's ideas are behind every data-compression algorithm that ever was or will be. Although Shannon's name is not a household name as that of Einstein, his work has contributed just as much or more to this century. Shannon retired at the age of 50 years, although he published papers sporadically over the next 10 years. He died 24 February 2001 in Medford, Massachusetts.
2.2. Discrete objects 2.2.1. Bits and combinatorial interpretation of information The statement that one can count information in any message (object) and therefore compare information in different objects (images, written texts, speech, functions, etc.) is initially rejected by anyone who starts to think about the notions of information. Some illustrative examples, however, will restore this person's ease of mind. One useful example is a historical legend about Bar Kokhba ("the Son of Star"), who was the leader of the Israelite uprising against the Romans in 135 AD: he could obtain important information about the Roman campus from an Israelite scout who had lost his tongue while in Roman captivity. Bar Kokhba just asked questions and the scout answered "yes" or "no" by nodding or shaking his head. This example (Renyi [1976]) shows that, in principle, one can count the
18
Quanta and information
[1, § 2
information of a message by the number of yes-no questions that are necessary to ask an oracle in order to learn the content of a message. In an effort to obtain information from a message that fixes one object in a group of N equivalent ones (a student in a group of students, a letter from an alphabet, a conceived number among a group of numbers, etc.), by asking the sender we can discover how to do this optimally, that is, by a minimal number of questions. The sender (Alice) and receiver (Bob), first, enumerate (encode) all n members of the group by unique numbers; second, they write these numbers in a binary system using the same amount of binary digits for numbering each member of the group (0000, 0001, 0010,..., 1001 for A^= 10); third, Bob asks Alice: "Is the conceived number equal to A^ - 1 written in the binary system?" (for A/^= 10, "Is the number 9= 1001?"), and waits for bit-by-bit answers from Alice; and fourth,ft-omyes-no answers Bob finds the number coding the selected object (for the case of N=\Q and the conceived number 5 = 0101, the series of four answers will be "no, no, yes, yes"). It follows from the solution that the number L of yes-no questions that the receiver should ask to retrieve information is restricted by the inequalities: //o = log2 N ^L<
log2 A^ + 1.
(2.1)
A deep meaning of these inequalities was noted by Hartley [1928], who was interested in the transmission of information via telegraph and telephone. Shannon [1948] went ftirther. He recognized that the inequalities conceal big resources for transmission of messages, one of which is a possible redundancy of the coding scheme. Indeed, using the above method of encoding and decoding, it is possible to communicate not only a message consisting of one selected symbol but also a message consisting of a sequence of M choices from N symbols (objects, letters of alphabet) made at each selection. When symbols are equally likely in messages, then, according to eq. (2.1), the number of bits required to communicate an M-symbol message is bounded below the value of M log2 A^. If Alice and Bob use A^ = 2^ {k integer) symbols for composing the messages, then M log2 N is also an integer, and everything in the communication scheme is uniquely defined. If A^ is not a power of two, however, the value of M log2 N is not an integer, which means that the chosen number of symbols in the alphabet brings Alice and Bob to a coding scheme that admits faithfiil communication even with the use of a longer alphabet. The above "symbol-by-symbol" coding scheme for a 10-letter alphabet requires 4M bits to communicate M symbols, while the limit is Mlog2 10 = 3.33 •••M bits. This is because the chosen code (4 bits per symbol) is more powerfiil than is required for communication
1, § 2]
Information
19
by the 10-letter alphabet (by using a 4 bits per symbol code Alice and Bob could faithfully communicate by messages composed of a 16-symbol alphabet). Another, equivalent explanation of the nonefficiency (redundancy) of the chosen code is in the fact that the amount of noughts is not equal to the amount of ones in the first 10 binary numbers, enumerating 10 symbols of the alphabet, despite the fact that the symbols are equally likely (there are 25 noughts and 15 ones in binary recording of the 10 numbers 0000, 0001, . . . , 1001). Shannon showed that it is possible to minimize the number of bits for communicating one letter in a long message arbitrarily close to log2 N, and he demonstrated that this can be achieved by proper coding. 2.2.1.1. Block coding. One variant of this clever coding is to code blocks of symbols (block coding) instead of single symbols. As an example, when coding pairs of symbols in messages composed of a 10-symbol alphabet, Alice and Bob have to enumerate 100 objects (from 0 to 99). At uniform fixed-length coding, each symbol is encoded by a fixed number of binary digits, and it takes 7 bits per paired symbol (i.e., 3.5 bits per symbol), which is closer to log2 10. Asymptotically the limit log2 10 could be reached by increasing the number of symbols in blocks. Another method is based on the idea that Alice and Bob could use a variable-length code to decrease the number of bits per symbol, but despite this compression, a communicated message has to be left unequally decodable. One such code is the Fano-Shannon code proposed by Fano [1948]. In this encoding Alice and Bob should split all N symbols into two approximately equal groups and assign " 1 " to all members of the first group and, consequently, "0" to all members of the second group as the first bit value in the coding notation of symbols. By continuing these splittings and assignments, a uniquely decodable code is created (fig. 8). The property of unique decodability of a message composed of encoded symbols without gaps in between follows from the fact that any encoded symbol (a string of ones and zeros) a = (ai,a2,...,an) is not a concatenation of any other code symbols, that is, a ^ b\\c (ai,a2,... ,^« ^ bi,b2,-.. ,bk,ci,C2,... ,Cn-k)' In the notation b\\c, b is called a prefix, and therefore another name for decodable code is a prefix code. As follows from the example given in fig. 8, the Fano-Shannon code in the case of N = 10, uses 34 bits overall, that is, 3.4 bits for sending a symbol, which is close to the lower bound Ho = log2 A^ = 3.33 • • •. Huffman [1952] proposed another prefix code (fig. 8b) which is obtained by step-by-step reduction of the initial alphabet {Ai,... ,An} by assigning to each symbol its frequency (pi = l/N for equally likely symbols At) and creating
20
Quanta and information
[1,§2
1
1 1 1
1 0.1 001
0.2 10
0.4 0
2
1 1 0
2 0.1 000
0.1 001
0.4 1 n I 0.4 0
3
1 0 1
3 0.1 Oil
0.1 000
0.2 1(
4
1 0~0^ 1
4 0.1 010
0.1 on
5
1 0 0~~0
5 0.1 1101
0.1 010
6
0 1 1
6 0.1 1100
0.1
1101
7
0 1 0
7 0.1 1111
0.1
1100
8
0 0 1
8 0.1 1110
0.1 -.1111
9
0 0 0 1
9
0.1 JlUO
10
0 0 0 0
10 O.l-'lOO
(a)
o.inioi
^0.6 1
y
(b)
Fig. 8. Prefix code generation schemes for a 10-letter alphabet with equiprobable symbols: (a) FanoShannon; (b) Huffman. The generated codes are printed in bold.
a new, reduced alphabet {^i,.. .An-i,^} with new frequencies p\, ... ,Pn-2, p = Pn-\ -^ Pn- The least likely two symbols of the new alphabet are replaced by one using the same method, and the procedure is continued up to the stage when a two-symbol alphabet is left. Coding the more frequent symbols by 1 and less frequent ones by 0, we start to go back on the chain of reduced alphabets and reconstitute symbols An-\,An by adding to symbol ^ 1 or 0, respectively. As a result, an optimal code is generated. Note that the Huffman code is used in many applications, for example in the JPEG (Joint Photographic Experts Group) format for image file compression. An additional, less known, resource exists in the encoding of messages composed of an equally likely alphabet. The resource exists even in the case oiN = 2^, where a fixed-length code gives an exact lower bound for the number of bits per symbol. This resource is connected with the definition of distance between symbols in the alphabet; it will be discussed later in §2.2.2.3. Thus, from the above consideration it is seen that the lower bound in eq. (2.1), H^ = \og^N,
(2.2)
has a clear combinatorial sense of the minimum number of binary digits (bits) per one symbol in a faithfully communicated message composed of an equally likely alphabet of A^ symbols. To reach this bound, a number of nontrivial encoding methods allowing unique decoding were proposed. By these widely applicable findings, initiated by the works of Hartley and Shannon, one important measure of information, that of combinatorial entropy HQ, was grounded.
1, § 2]
Information
21
Shannon [1948] also recognized the generalization of the measure to the case where the symbols of an alphabet have different frequencies pt in messages, which is the case of any language alphabet. As a result, a generalized measure of information with the same combinatorial sense, namely entropy N
Hi =-J2Pi^''^2Pi ^ Ho,
(2.3)
i= l
was proposed. In comparison with equally likely alphabets, the redundancy of messages becomes their common property, even for N = 2^. The methods of data compression (block coding, variable-length coding) are left the same, but with a clearer interpretation of the coding principle: more frequent symbols are coded by shorter strings or bits. The number of symbols A^ in the alphabet and their frequencies pt are not the only resources for coding and communication messages. An additional peculiarity is connected with the inherent structure of the languages: the conditional frequencies pij of the appearance of symbol Aj in a message after some other symbol Ai do not coincide with the unconditional frequencies Pj. Therefore, if one has guessed the first symbol in a message by using H\ = -J2Pk ^^^iPk yes-no questions, it takes no more than k
H(2\i) = - Y^Pij log2Ay < Hi
(2.4)
J
additional yes-no questions to know the next symbol. Equation (2.4) shows that the amount of information contained in the second symbol, H^2\i), is reduced because of the inequality ptj < p.. The total number of questions to recognize both symbols is H(2\i) = Hi-\- H(2\i).
(2.5)
This example will illustrate the important notions of conditional entropy and mutual information introduced by Shannon. Before doing this, however, note that until now the discussion of information properties has used only the combinatorial interpretation of the latter. Kolmogorov [1965] (see also Cover, Gacs and Gray [1989]), wrote that the future of developing information theory within the combinatorial approach was clouded, because the probabilistic interpretation of information empowers one with a much richer system of notions and relations.
22
Quanta and information
[1? § 2
2.2.2. Probabilistic interpretation of information One of the primary benefits from Shannon's work [1948] was the probabiHstic interpretation of information, embodied in the property of ergodicity. Rather than stating that we can find a clever code in which the number of bits per symbol in a long message will be made as close to Hi as we wish, it is possible to state that we can communicate different messages many times, and in the mean, it takes H\ bits per symbol in faithfial communication. The frequencies in the combinatorial interpretation are replaced by probabilities in the probabilistic approach. In many cases this replacement has practically no ill effects, as proved by the hypothesis of "fast mixing" (Krylov [1950]). In the probabilistic interpretation, communication processes look like the annihilation of initial uncertainty of Bob's knowledge. Each faithfially communicated bit removes a bit of uncertainty, and when the communication of a message is finished, the uncertainty has disappeared completely. Thus, it is natural to apply the notion of information to random processes. If X is a random variable taking N discrete values x, we can reformulate eqs. (2.3)-(2.5). Now Hi = H(X) represents the mean value of yes-no questions in the ensemble of attempts to find one marked value of X. By averaging eq. (2.4) we obtain the conditional entropy of observing the X' value following the observation of X:
H{X'\X) = - Y,Pi T.Py l^&P.-
(2-6)
This relation can be generalized for the conditional entropy of two random variables X and 7, possibly of different dimensions. Therefore, averaging eq. (2.5) leads to an equation for the entropy of joint processes (X, Y): H(X, Y) = H{X)^H{Y\X).
(2.7)
The difference I{X.Y) = H(X) + H(Y)-H(X,
Y) = H(Y)-H(Y\X)
(2.8)
is referred to as mutual information, a measure of the degree to which X and Y contain information of each other. 2.2.2.1. Data compression, typical sequences, and Shannons noiseless coding theorem. Armed with this notion. Shannon investigated transmission of information through noiseless and noisy channels. For the former he showed
1, § 2]
Information
23
that to communicate N values of X, we need only NH\ {X) ^ N bits down the communication channel. This idea is referred to as data compression, also called Shannon's noiseless coding theorem. An illustration of practical importance of this theorem for the equiprobable alphabet is given above. Here we illustrate the proof for the case of a two-letter alphabet with different probabilities for X=\ (p) and X = 0 (l-p). For a sequence of N values, the ones (m) and noughts (N - m) will be distributed according to the binomial distribution: P(m,N-m) - C^p"" (1 -pf-"^ ,
(2.9)
where the binomial coefficient C^ = N\/ (N - m)\ ml is the number of A^-length different sequences with m ones. For the relatively wide region of parameters Np(l -p) > 9 and I < (N + l)p < N, this distribution is approximated by the Gaussian form P(m, N-m)^cxpl
2^^"^ff^^) ] / V^^Np(l -p).
(2.10)
In the limit N, Np -^ oo, the last distribution becomes a 6-function, showing that only those sequences will be presented in communication messages for which m ^ Np. The number of these typical sequences is Nj = C^P ^ p-'^P (1 -pr""^' -P^ = 2^^^W^
(2.11)
where Hi(X) = -p \0g2P - (1 -p) log2 (1 -p). Therefore, Alice and Bob need to encode only Nj equiprobable sequences, which requires log2A^T=A^^i(X)
(2.12)
bits. The idea of data compression is a concept of great practical importance. Today it is used in applications such as conveying television pictures, for data storage in computers, and in modem communications. 2.2.2.2. Error corrections and Shannons noise coding theorem. Transmission of information down a noisy channel can be described by two discrete random variables: X ~ the message sent by Alice, and Y - the message received by Bob and corrupted by the noisy channel. The notion of mutual information (2.8) allows one to define the channel capacity, which is the maximum possible mutual
24
Quanta and information
[1? § 2
information I(X:Y) between the channel input and the output, maximized over all possible input probability distributions {p(x)}: C= max/(X:7).
(2.13)
{P(x)}
As an example, a binary ( - ^ = s _ n r ' ^ ^ l - n f ) symmetrical channel produces noise just by switching randomly two possible output states: when Alice sends x=\, Bob sometimes receives y=\, sometimes y = 0. The conditional probabilities of this noise action are given by the matrix
K-{'-\'_,)-
(2.14)
When the channel is noisefree (e = 0), the matrix P^y is the unity matrix, and C = 1. For the general case (e ^ 0) according to eqs. (2.6) and (2.8), we have I(X:Y) = H(X) + dog2e + (1 - e)log2
(l-s),
and the equally likely input alphabet X gives C = 1 + £log2 ^ + (1 - e) log2(l - e).
(2.15)
Introducing the notion of rate of a source of information RH as a part of an information bit that is conveyed by one bit of message (for fixed-length codes RH = H\/N), Shannon formulated the central result of the classical information theory, namely, the Noise coding theorem, which states: if the rate RH < C, there exists a binary code allowing transmission with an arbitrarily small error probability. 2.2.2.3. Distinguishability, code distance, and error-correcting Hamming code. An example of a code as referred to at the end of the preceding subsection is the error-correcting Hamming code, named after its discoverer (Hamming [1950]). The idea of coding is to make symbols of the alphabet distinguishable to such an extent that adding some noise to them leaves symbols distinguishable. A measure of distinguishability of code symbols is a code distance. The Hamming distance between two «-bit symbols a\ and ai is introduced by the transformation of the binary value of symbols into vectors (e.g., a\ = 1100 is transformed into a = (1,1,0,0), and ^2 = 0110 into a = (0,1,1,0)) and by ^1
^2
1, § 2]
Information
25
the definition of the distance value as the number of nonzero coordinates in the vector difference: PH(«I, «2) = w(_a^ - _a^),
(2.16)
where w stands for weight of vector (the number of nonzero coordinates of vector). In other words, the Hamming distance pn («i, ^2) is the number of noncoinciding binary digits in the symbols (for the above example pii(ai,a2) = 2). The code distance of alphabet (A) is a minimal weight of nonzero code symbols (words): d(A) = mmw(a).
(2.17)
The effect of noise on symbol a can be expressed as a transformation a'=a
+ e,
(2.18)
where the error vector e indicates which bits in a were flipped by the noise (e.g., a = 1101001 -^ a' = 1000001 can be expressed diS a' = a + e, where e = 0101000). If the noise-affected symbol a' does not coincide with any symbol of the alphabet, we can distinguish it and therefore correct. Strictly speaking, code A can correct any combinations of ^bits errors iff (if and only if) its code distance is larger than 2t. The most important error-correcting codes are the linear codes or the paritycheck codes. Such an TV-length code is completely specified by its parity-check matrix H^: each symbol a of the code is a linear independent solution of the equation HN'a=0.
(2.19)
The sum of arbitrary code words a -\- a is also a code word, hence, the name "linear". For example, the parity-check matrix for a Hamming code of length 7 is
Hj=
'0001111 I 0110011 I . 1010101
(2.20)
26
Quanta and information
[1, § 2
Using this code, one can obtain four basic vectors u (/ = 1,2,3,4) and compose a generator matrix
Gi =
/ 1000110 \ 0100101 0010011 \ 0001111/
(2.21)
which is orthogonal to the transposed matrix //f: Gy • H^ = 0. All 2"^ code symbols are generated by the matrix Hj or Gy. As can be seen from (2.21), the code distance for the code is 3; therefore, this code is marked as (7,4,3) Hamming code. The parity-check property (2.19) allows Bob to locate the noise-affected bits in a transmitted symbol. By the operation HN' a^=HN'{a
+ e) = HN' e,
(2.22)
he obtains information about the error e without influencing the communicated symbol a. The vector H^ • e is called the error syndrome. For single-bit errors (like e = (010000)), the error syndrome of a symbol with an error in the /th bit is a column vector of the parity-check matrix. If the parity matrix is written in such a form that each column is a binary representation of this column number (like in eq. 2.20), the error syndrome written in a binary representation coincides with the error bit number! Just by changing the value of the bit. Bob obtains a correct symbol. Error-correction coding is another method from information theory which is of great practical importance to all communication systems. Stemming from Shannon's coding theorem, error-correction theory still faces the notoriously difficult problem of identification of codes with a large rate RH (for {N,k,d) Hamming codes R^ = k/N) and large distance d. This problem has no general solution (MacWilliams and Sloane [1977], Hamming [1986]). 2.3. Information of continuous functions 2.3.1. Sampling theorem It is seen from the foregoing that the information theory has powerful methods to measure and faithfully communicate an item of information, presenting it
1, § 2]
Information
27
in a binary discrete form. It is clear, however, that a fairly large amount of information comes from continuous objects (e.g., images). What is the information content of a picture? The right question to provide the possibility of finding the right answer is: what resources are required to communicate a picture? In an effort to answer this question, we need to represent the continuous function as a discrete and finite set of some basic functions. Generally, this procedure is more or less severely constrained, for both practical and fiindamental reasons. Indeed, without any constraints the information content of a continuous object is infinite. Practically, the constraints posed by the properties of communication channels and by the abilities of both Alice and Bob to recognize an object are described by encoding and decoding fidelities. Kotelnikov [1933] and Shannon [1948] suggested how to find the information content (and, actually, a way of communication) for a continuous fimction f(t) which is however limited in the time {t < T) and frequency {o) < W) domains. The method is based on the Fourier decomposition of such a function (Wiener [1948]) and makes use of the fact that it can be exactly represented by the finite sum \2WT^
^
Jt(2Wt-n)
n=\
By this decomposition (the sampling theorem), the problem of communicati n g / ( ^ is reduced to the problem of communicating N = 2WT values of the fiinction. 2.3.2. Constraints, fidelities, and information of images If we also impose constraints on the accuracy of measurement (observation) of signal/fit) values, say A/, we convert the continuous function communication to the discrete variables communication where all the abovementioned methods of information theory can be applied. For example, with the adopted constraints, the upper bound for the informational resources required to communicate any fiinction/(O limited by (/max,/min), is given by the entropy ^0{/(0}=2^riog2
/max
A/
/m
(2.24)
Note that in the absence of constraints on the numbers of possible values of continuous fiinction, some notions from the discrete case can be generalized to
28
Quanta and information
[1? § 2
the continuous-variables communication; for example, there is a continuous limit for mutual information of two continuous random variables /(Z:r) = / / p ( x , , ) l o g , ( J g L ) d x d , .
(2.25)
As noted by Shannon [1948], however, this generalization can give an unsatisfactory result: "The entropy of a continuous distribution can be negative". The best example of the importance of constraints in the problem of continuous functions information is the information of images. A black and white image is a 2D function of intensity B(x,y). Because of the spatial (x,y ^ L) and spatial-Fourier (Kx,Ky ^ WxW^) limitations, this image will be described by one vector B, with length A^ = W^WyL^/n'^. The human eye, which has 5x 10^ cones and 10^ rod cells composed in a mosaic structure (Miller [2000]), has a fantastic, but limited resolution to sample an image and encode it into a neural image. For example, the human eye cannot distinguish the following two images B' and B" (two A^-length vectors): a uniform background {B') ^'=5,(1,1,...,1),
(2.26)
and a sample with a different intensity and the same background {B"), ^''=5,(1,1,...,1 + ^ , . . . , 1 ) ,
(2.27)
if the contrast A5/5/, is below the differential threshold of the vision recognition (Lebedev and Tsukerman [1965]). Therefore, this limitation allows one to quantize the intensity level. With the help of the Weber-Fehner law (ABJBh = 7 ?^ 0.015-0.02), the quantized levels can be introduced as 5 , = (l + 7')^n.in,
(2.28)
and the number of quantized levels as _ lll(^max/^min)
ln(l + y)
(2.29)
Another approach to the information content of continuous objects such as images is related to the notion of f-entropy and ^-net, introduced by
1, § 2]
Information
29
Kolmogorov and Tikhomirov [1959]. As a result of the limitation, that is, the physical impossibility of distinguishing images whose vectors B' and B" satisfy the condition of indistinguishability (fidelity), expressed in terms of distances R{B^,B"\ in space 5,
^fe'^0 < e,
(2.30)
one can cover the 5-space with the e-regions. The ^-region of the vector B' is a set of vectors B" satisfying condition (2.30). The minimal number m^ of ^-regions covering the whole 5-space defines the ^-entropy of the image H,=N\og^me.
(2.31)
and the centers of e-regions are referred to as the £-net. Some obstacles to the realization of the £-net program for images are present, one of which is the absence of a concise expression for distance (2.30) reflecting the fidelity of images. However, some parts of the program, for example, special transformations in ^-space of images, realized under transformation of spatial Fourier components, are of great importance in holography and optical recognition techniques. 2.4. Algorithmic information; Kolmogorov complexity Kolmogorov [1965] initiated another approach to the information measure. The idea was to extend Shannon's entropy directly to computational problems. For example, it is desirable to quantify the information content of a sequence recorded in the memory of a computer. This suggestion led to an algorithmic approach through the introduction of recursive functions, and eventually to a formal theory of complexity. Briefly, the Kolmogorov complexity, or algorithmic information content K(X) of a string X, is the length S of the shortest program P executed on a universal computer U that will yield X, K(X)=
min S(P).
(2.32)
For example, the problem/* 'givenX, find its square' initially requires L = log2X bits to store the value of X. Then we need to find an algorithm for the calculation on a universal computer. If this universal computer U, usually the Turing
30
Quanta and information
[1, § 3
machine (Turing [1936]), executes the algorithm in S steps, the minimal S among all algorithms gives the complexity of the problem. If an algorithm exists with S given by any polynomial function of Z (e.g. S (x L^ +1), the problem is deemed tractable and is placed in the complexity class P. If S rises exponentially with Z, the problem is hard. If we can verify the found solution of this hard problem spending polynomial time, the problem is referred to as an NP problem. The most interesting and hardest problems belong to the class of A^P-complete problems, for which no known polynomial-time algorithms exist. This is the key class of problems for quantum computations, discussed below in § 3.2.2. In concluding this introductory section on classical information theory, started by Shannon, it is important to note that his work was immediately recognized by mathematicians, engineers and biologists, and, what is more important, it was developed further in various and sometimes unexpected directions. § 3. Quantum information 3.1. Quantum communication Any quantum mechanical experiment can be considered as an object for information theory. Certainly, the primary parts of an experiment, that is, the choice of the basis, preparation of the initial state, interaction, and measurement, can be assigned correspondingly to the choice of a code for a message, encoding, transmission through a channel and decoding with subsequent reconstruction of the communicated message. The conceptual object of quantum theory, the wave function or the density matrix, unavoidably contains the last step of information transmission - a measurement - Schrodinger's "Katalog der Erwartung'\ This section discusses some new resources for communication opened up by the quantum nature of physical objects, beginning with the following setup (fig. 9): at Alice's "station" a source S is emitting quantum obchannel
Alice
Bob
I
^
_^ :-|«.)-|a2)|«i)
\\' I\
u
'
_^ ..\a',)-Hhi)
ancilla 5^
u~^
Fig. 9. Quantum communication channel.
1, § 3]
Quantum information
31
jects (electrons, photons, atoms, etc.), particle by particle in different states |o^i)i, 1^^2)2 5- • • ? WN)N' For certainty, we assume that the pure states |a/) are chosen from a list ofN different states. Alice can send these particles further to Bob without any additional actions or can transform them by unitary operations, discarding some of the particles before communication. Bob, at his station, can reconstruct the state of the received string of particles by adding some ancillas (auxiliary quantum systems) and by unitary operations. After obtaining the reconstructed states 1^1)^; |a2)2 ; • • •; \(^N)N^ ^^^ ^^^ measure them by any projection technique. The communication scheme was analyzed by many authors (Gordon [1964], Holevo [1973, 1979], Levitin [1969], Caves and Drummond [1994]), but it was Schumacher [1995] who clearly recognized that in order to describe all quantum peculiarities in the transmission of a string of quantum particles, one does not have to transform its description into a classical picture with probabilities derived from quantum laws. It is natural to leave, as a minimal resource of quantum information, a quantum object (particle) with two possible states, the quantum bit or qubit, which is the fundamental unit of quantum information. Therefore, for simplicity, but without loss of quantum peculiarities, we will restrict ourselves to 2-state particles. Two general information problems can be considered within the framework of the communication scheme. First, on one hand, Alice can try to convey to Bob the quantum states themselves, but she does not want to send through the communication channel all the quantum particles created in the general initial state of N-qubits string (represented by a vector in Hilbert space of 2^ dimensions)^, but only part of them. How many quantum particles are required to do this, under the condition that Bob can reconstruct the state at his station with good fidelity? Second, on the other hand, Alice can try to convey classical information, that is, information that can be encoded into a sequence of zeros and ones, by means of quantum particles. For example, Alice can encode information as a sequence of coherent field states \a) and \~a) coding |a) as 0 and \-a) as 1 |a)J-a)2|-a)3-.-|a)^.
(3.1)
In this case, another question arises: How many binary digits per particle can Alice convey to Bob faithfiilly? The answers for these two questions are surprisingly similar:
An «-dimensional Hilbert space occupies log2 n qubits.
32
Quanta and information
[1, § 3
(1) For all sufficiently large N, a sequence of A^ particles with certain specified firequencies pt of occurrence of states |«/) may be compressed by a factor of von Neumann entropy S(p) = -Trplog2P,
(3.2)
where p = YliPiMi^il is a single-particle density matrix of input sequences, that is, into NS qubits while retaining arbitrary high fidelity (Jozsa [1994]):
/(p,pO=TrVpi/Vp»/^
(3.3)
where p ' is a single particle density matrix of output sequences, and further compression is impossible in the presence of high fidelity (Jozsa and Schumacher [1994], Schumacher [1995]). (2) If Alice uses certain quantum states |a/) as signals with specified firequencies Pi of occurrence, the number of binary digits per particle she can convey to Bob can be made arbitrarily close to, but not greater than the von Neumann entropy S(p) (3.2) of the ensemble of signals (Holevo [1973, 1979, 1997], Hauslanden, Jozsa, Schumacher, Westmoreland and Wootters [1996]). These two answers (theorems) are of great importance for many aspects of quantum communication. They demonstrate the following: the fundamental role of a qubit as a measure, a ftindamental unit of quantum information resources; a precise information-theoretic interpretation of von Neumann entropy S (p) as the mean number of qubits necessary to encode the state in the ensemble in an ideal coding scheme, or as the number of bits that can be conveyed faithfiilly by one qubit in a long sequence; and a new resource for data compression, quantum data compression. Indeed, for equally likely sequences of length N = 2^, the classical information theory does not see any resources, but in the quantum world it is not the case if the states |a/) chosen for communication are nonorthogonal. 3.1.1. Signaling by two coherent states 3.1.1.1. Nonconclusive measurements, Hamming distances, and quantum data compression. To illustrate the quantum data compression technique, consider some examples of communication by coherent states. In the first example, Alice wants to communicate a message by two coherent states \a) and |-«). To receive the message. Bob should distinguish these two states. He cannot do this perfectly, however, not because of the classical "noise" (letter states \a) and |-a) are transmitted to Bob without alteration in the communication channel), but because
1, § 3]
Quantum information
33
i
/V2a)
t ancilla
|/a)
ancilla
|^^/
Fig. 10. Scheme of an error-free Ivanovic-Peres measurement for unambiguous discrimination between \a) and | - a) coherent states.
of quantum mechanical uncertainty. Trying to distinguish a field in the state \a) fi:om a field in state \-a) by the von Neumann type of measurement, Bob will sometimes receive the answer "yes" on the test and sometimes "no", because of the non-orthogonality of these states x={a\-a)=
exp(-2|ap).
(3.4)
However, another type of measurement allows one to distinguish between the two states without error (Ivanovic [1987], Peres [1988]). An error-free IvanovicPeres measurement for unambiguous discrimination of \a) and \-a) states may be realized by using a 50%: 50% symmetrical beam splitter (Huttner, Imoto, Gisin and Mor [1995], Barnett [1997]), for which the complex transmission and reflection coefficients are 1/V5 and i/\/2, respectively (fig. 10). With the help of the auxiliary field b (the ancilla) prepared in state \ia)b, the mirror transforms the input state \a)a\ia)t into the output state T(jt/4)\a)a\ia)b = \0)c\^V2a)d (T(e) = exp[i0(a+Z? + Z>fl+)]), and \-a)^ \ia)b into |\/2a)^|0)j. Counts at detector D or C determine the state to be |a} or | - a ) , respectively. An inconclusive result occurs, however, when no counts are registered at all. This may happen with probability x. If x is not small, this circumstance creates a big obstacle for communication by weak quantum signals, because quantum noise becomes more pronounced. To avoid this, Alice can simply repeat weak coherent signals Na = j
^
1
-X
(3.5)
34
Quanta and information
[1? § 3
times to let Bob obtain a conclusive result. However, the second of the above statements leads to the following conjecture: Alice can compress the sequence by a factor defined by the eigenvalues of one-particle density matrix of strings (3.1): p = i ( | a ) ( a | + | - a > ( - a | ) = po).
(3-6)
[In eq. (3.6) we assume that \a) and \-a) are equally likely in a string of particles (3.1)]. The diagonalization of the Hermite matrix P(i), Ai) = A+|A+)(A,| + A_|A_){A_|
(3.7)
gives positive eigenvalues A± = (1 ± x)/2 and orthonormalized eigenvectors |A±) = (|a) ± |a))/2v/Al,
(3.8)
which are nothing but even (|A+)) and odd (|A_)) coherent states (Buzek and Knight [1995]). Thus, the number of signals in a long sequence representing matrix (3.6) can be maximally reduced by the factor Saip) = 1 - ^
log2(l + ^ ) - ^
l0g2(l - ^ ) -
(3-9)
How can one achieve the compression? As we know from classical information theory, block coding is a powerful method for data compression and for dealing with noise irrespective of its origin. If we try to use it even in its simplest version of pair coding, we find something distinct from the classical case. The two-particle density matrix that describes the sequences of equally likely pairs of \a) and | - a ) , P(2) = \{\o)x\a)22{a\ i(a| + | - a ) , \-a)^ 2{a\ i(a| + |a)i |-a)2 2 ( - « | i(«| + | - « ) i |«)2 2(«| i ( - « | ) ,
can be rewritten in a more compact form if we use binary numbers to designate pairs of coherent states: |a),|a)2 = |00) = |0), | a ) , | - a ) 2 = |01) = |l), l-a), |a)2 = 110) = |2), l-a), \-a), = |11) = |3).
^'''^
Using these definitions, formula (3.10) is transformed to 1 ^ A2)=22El«)H n=0
(3.12)
1, § 3]
Quantum information
35
Now we can generalize the concept of a distance between the states (3.11) as <.'=W«^"'),
(3.13)
where pu («, n') is the Hamming distance between two 2-bit symbols n and n' (2.16). The diagonalization of the density matrix p(2), that is, diagonalization of the symmetrical matrix of quantum distances dn,n' * /
d=
1 X X X^\ X I X^ X X X^ I X
\x^
X X 1y
gives the following eigenvalues: _(1+Xf A++ —
4
_,2 A, ,
+'
3 _3 _{\+x){l-x)_^ A+_ — A_+ — :
^
^
4
, A+A_,
_(1-J^)'_,2 A__ — A_.
4
(3.14) These are just the products of the eigenvalues of the matrix P(i). The same happens to the orthogonal eigenvectors |A±±) = |A±)i|A±)2,
|A±^) = |A±)i|A^)2,
(3.15)
each being a linear combination of the four states (3.11). In fact, the process of diagonalization of the matrix P(2) may be interpreted as re-encoding with the use of new code states maximally distant from one another. By increasing the number of signals M in one block, we easily find that the M-particle density matrix 2^-1 n=0
has eigenvalues
with the degree of degeneracy C§. Therefore, the probability of generating Xkstrings, composed of K |A+}-states and (M -K) |A_)-states, ;7(A,) = Cj^AfA^-^
(3.17)
coincides with the probability distribution (2.9), thus demonstrating the power of Shannon's noiseless coding theorem. The number of typical sequences of
36
Quanta and information
[1, § 3
z o a: O
if) LU
a: Z)
CO
< LU
MEAN NUMBER OF PHOTONS (|ar)
Fig. 11. The von Neumann entropy Sa, the number N^ of repeated coherent signals \a} and \-a) for receiving a conclusive result, and the number /ii^ of qubits allowing both faithfiil communication and elimination of nonconclusive results for one bit transmission, as functions of the mean number of photons.
qubits required for a faithful communication of M bits of classical information is C^^^ ^ A;^^^ + A:^^ = 2^^^^^>. Thus, instead of sending M bits of classical information encoded in equally likely blocks, one can use for Sa(p) < 1 only NSa(p) qubits in typical sequences and discard the other, untypical sequences (see the example presented by Jozsa and Schumacher [1994]). By combining condition (3.5) and eq. (3.9), we obtain the number of qubits allowing both faithful communication and elimination of nonconclusive results for one-bit transmission: nia = l-x'
(3.18)
Figure 11 demonstrates that a "clever" block coding gives a fair advantage over simple repetition of signals for a small number of photons, smaller than unity. This fact also confirms the idea that communication by means of faint coherent signals has merit. 3.1.1.2. Block coding, nonlinear resources, and entanglement creation. Another lesson from this example is that to make a clever coding, it is necessary to be able to transform initial coherent states into a different basis. In our case this is an orthogonal basis of even |A+) and odd |A_) coherent states, but it
1, § 3]
Quantum information
37
is not a simple task. To perform this transformation by unitary methods, one should use nonlinear resources. A nonlinear mirror with an incredibly high Kerr nonlinearity could realize this transformation; another example of this type of interaction is the nonlinear directional coupler (Pefina [1995]). For the manifestation of the transformation opportunity, let us look at the transformation of field states produced by a nonlinear mirror with Hamiltonian Hx=Xa^ab^b.
(3.19)
If the input and ancillary fields a and b are coherent states, \a) and \I3) respectively, the output states after the transformation at / = >7r (an effective nonlinear interaction length equal to unity) are |a/3)^., = v/A, |A,), m. + V^- |A^). I-/5),.
(3.20)
Therefore, depending on the ancillary field output (|jS)^ or |-/?)^), one can transform (encode) the input signal state |a) into even |A+}^ or odd |A_)^ coherent states. By means of two consecutive transformations realized on two (fig. 12a) or one (fig. 12b) nonlinear mirrors (X\ = X2 = ^ ) , the output of signal fields a and c will be conditioned by the output of the ancillary field:
|A„,>„ |A,,)^ + , y V V |A„_), |A,_>^) 1^), + ( Y Aa^Ay+ \^a-)a \^r+)c + y Aa+Ay- \K+)a \^r-)c 1 \~P)b
= H(l«)Jr). + l-«)J-y>c)l^>. + (Klr).-l-«)J-yUl-^>J. (3.21) These states of two signal fields are in an entangled state analogous to the polarization entangled states (1.8): |0^} when the ancillary field state is \P)b, or |0~) when the ancillary field state is hiS)^. Note that by introducing a phase shift (p = JT, WQ can obtain entangled states analogous to | ^ ^ ) (eq. 1.8c) and \^~) (eq. 1.8d). This example clearly demonstrates how initially independent particles become entangled and that the real price for entanglement is the use of nonlinear resources. Note also that the scheme presented in fig. 12b can generate a chain of entangled states, recently considered by Wootters [2000].
38
[1,§3
Quanta and information
|y)" (a)
(b)
Fig. 12. Scheme of the two consecutive transformations of a string of initially independent coherent field signals |a) • • • |y), realized by (a) two or (b) one nonlinear mirror(s) {X\^ Xi^ ^)- The output of signal fields a and c will be conditioned by the output of the ancillary field b in one of the entangled states (3.21). The round-trip time in the cavity of figure (b) should be equal to the delay time between two input pulses \a) and |y).
3.1.2. Communication of images by coherent states, image recognition and quantum limit of phase space partition 3.1.2.1. Quantum phase space partition theorem. The next example, that is of interest, is the coding and communication of images by coherent signals. The starting point for the coding can be the phase-space-partition-theorem, stating that for the partition of phase space of a harmonic oscillator on cells mn with areas S, the set of coherent states {amn), where Grtjn = mo)\ +no>i, (m,n = 0,1,...)
(3.22)
are the centers of cells with linear independent complex numbers a)\, (JOZ (Im(a;2 ^ i ) ^ 0)» is complete for S = Jt, and the set remains the same if one state is removed from it. Moreover, if S < Jt, the set is overcomplete and it remains the same when the finite number of states are removed. If *S' > ;r, the set is incomplete (Perelomov [1971], Bargmann, Butera, Girardello and Klauder [1971], Bacry, Grossman and Zak [1975]). The theorem conjectures that if Alice wants to encode an image by dividing it into N^ cells (pixels) (for certainty we will consider a square lattice and twograde images) and assigning black pixels to the presence of coherent signals in
1, § 3]
Quantum information
39
the corresponding cells of the phase plane, she can do this without any additional manipulation and resources by choosing a part of the phase plane of area ^ph = JTN^.
(3.23)
Is it possible, however, to squeeze the size ^ph below this value and still retain the ability of image recognition? The phase-space-partition-theorem permits this by discarding some states. Let us see how the problem is solved by means of quantum information methods. 3.1.2.2. Optimal encoding of images. Suppose the images to be encoded belong to the class of equally likely strings of coherent fields defined by the phase space lattice a^ = au = a{k + il)
(|A:|, |/| < L\
(3.24)
without the central pixel (0,0). The single particle density matrix of the lattice is
Ai) = i(El««)<««|-|0)(0|), \k,l = Q
(3-25)
I
where TV is the number of pixels counted {N = AL(L^X) for a square lattice), and |0)(0| = |aoo)(^oo| is the vacuum state. The mean number of photons in the square lattice is {cta)t, = \a\^ ^ . 6
(3.26)
To find the von Neumann entropy of the ensemble of images (3.24), we have to diagonalize the density matrix (3.25); that is, to solve the characteristic equation det
= 0,
(3.27)
where the nonorthogonality of two states is given by (a« I a,,,,) =;,[(*-*'fH/-/')^]/4-.(«'-i'/)/2^ with X = exp (-2a^), where a is real.
(328)
40
Quanta and information
[1, § 3
0.02
CO LU
_l 2 LU
0.01
LU
PIXEL AREA (|a| )
Fig. 13. Eigenstates of the images described by a density matrix of the phase space lattice (see inset), composed of 128 equiprobable coherent states (3.24).
For the simplest case of a cross lattice composed of four coherent states with amplitudes a, - a , ia and - i a , the eigenvalues are Ao,2= (l+xd=x^»^>)/2±jc^i-*V2V4 = x^/2(cosha^i COSa2)/2, (3.29) Ai3= (l-xTx^'^^^/^±x^^-*^/^)/4 = x^/^(sinha2±sina2)/2. The corresponding orthonormalized eigenvectors |Ao,2) = ((\a) + l-a)) ± (\ia) +
\-ia)))/y/X^2,
1^1,3) = ((l«) - |-«)) T (|i«) -
\-ia)))/^X;;,,
(3.30)
are generalized coherent states (Horoshko and Kilin [1997a]), which are also the eigenvalues of the operator exp(ijra+fl/2)a, and which have 4A:, 4A: + 1, 4A: + 2 and 4A: + 3 photons, respectively (k integer), distributed by the Poisson law. Because of the unlikely probabilities of those states, an economic code can be proposed for image transfer (Kilin, Mogilevtsev and Shatokhin [1999]). Note that this method of image encoding should not be confused with investigations on quantum traveling wave imaging (Sokolov, Kolobov and Lugiato [1999]). A much richer structure of the dependence of eigenvalues on pixel size |a|^ arises when the number of pixels is increased. This is illustrated in fig. 13,
1, § 3]
Quantum information
41
where the degeneracy of eigenvalues, that is, their clusterization, is evident for \a\ < 1 and \a\^ > Jt. In the region of \a\^ ^ jr, the eigenvalues become more uniformly distributed, showing in the near vicinity a universal structure ("the sea of information"), which undergoes minor changes with increasing number of pixels A^ in the image. The von Neumann entropy *SV(P(i)) of equally likely distributed images (3.25) per one pixel shows increasing compressibility with decreasing pixel area. An additional resource also appears for communication of images: Alice can send coherent signals without ordering them, but remembering that the distinguishability of signals depends on the distance between them, that is, on their scalar product |(a/ | ay)|. She and Bob can use a protocol by which AHce communicates an image block by block. In the blocks different elements are more distinguishable than the nearest neighbors in the lattice. As the simplest case, Alice can communicate pair-by-pair signals separated by 2a. That is why they become much more distinguishable. The mean value of mutual scalar products I (a/ I aj)\^ over the selected part of the phase plane x = Tr^(p2^)-p(i)/7V)/2
(3.31)
approaches fidelity (3.3) at p = p ' and iV > 1. An estimation of the number of qubits per pixel nia (3.18) with the use of criteria of distinguishability x instead of X points to possible gain for small values of |a| . The quantum data compression discussed in this section optimizes the use of one channel resource, the states of transmitted qubits, but it is possible to transmit an unknown quantum state with perfect fidelity without sending any qubits at all through a communication channel. This process known as quantum teleportation, uses a quantum-mechanical entanglement as a new physical resource. The process is realized by means of local quantum operations over entangled parts shared by Alice and Bob, and an additional classical communication channel between them. The name of this quantum channel is the LOCC-channel. 3.1.3. Quantum teleportation 3.1.3.1. Experimental quantum teleportation. Late in 1997 Anton Zeilinger and his colleagues in Innsbruck (Bouwmeester, Pan, Mattle, Eibl and Zeilinger [1997]) performed an experimental realization of teleportation, the dream of science fiction novelists. The term "teleportation" means that an object disappears at some place and reappears at another place, some distance apart.
42
Quanta and information
[1? § 3
Although the idea of quantum teleportation, that is, of transporting a quantum state from one object to another, had been suggested in 1993 by Charles Bennett and colleagues (Bennett, Brassard, Crepeau, Jozsa, Peres and Wootters [1993]), it was the Innsbruck experiment and other experiments following it (Boschi, Branca, De Martini, Hardy and Popescu [1998], Furusawa, Sorensen, Braunstein, Fuchs, Kimble and Polzik [1998]) that attracted public attention. From the classical viewpoint, teleportation means gaining all possible information about the properties of an object and transposing these properties onto the reconstructed object. This procedure is forbidden in the quantum world, however, because of the above-formulated postulates of projection and destruction of the state during measurement. Another method exists for passing a quantum state from one object to another. In brief, transmission of an unknown quantum state from Alice to Bob is performed as decribed below. Alice has a particle in some unknown quantum state | ^ ) . Teleportation means that Alice destroys the state | W) at her location but some particle at Bob's location is put into the same state ( | ^ ) ) . Neither Bob nor AHce obtains information about the state | W); moreover. Bob does not know that some state was teleported onto his particle. To tell Bob about the teleportation, Alice should use a classical information channel, in which the principal role is played by particles in entangled states. They provide the quantum information channel between Alice and Bob. Suppose that particle 1 (a photon) to be teleported by Alice is initially in the polarization state | ^ ) i = « ||)i +/? | ^ ) i (fig- 14). Alice is connected with Bob by means of photon pairs prepared by an EPR source in an entangled state I '^")23 = (11)2 | - ) 3 - | - ) 2 11)3)/^^-
(3-32)
Photons 2 are sent to Alice and photons 3 are sent to Bob. The joint state of photons 1 and 2 meeting at Alice's station is the product of | ^ ) i and | ^~)23' |^)i|^-)23-|^-)i2(a|-^)3+^||)3)/2+|¥^-)i2(-a|-)3+^II)3)/2 + |^")l2(-^|-)3 + C.|I)3)/2+|0-)i2(^|-)3 + a | I ) 3 ) A (3.33) Consider the wave function (3.33) for three particles, two belonging to Alice and one to Bob. If Alice projects the states of particles 1 and 2 onto the state |y^ )i2, the state of particle 3 at Bob's station is immediately reduced to the state of the first particle, 1^)3 = a l'^)^ + 1^ IDs- In other words, by measuring Bell states formed by mixing photons 1 and 2 on a beam splitter
43
Quantum information
1, § 3 ]
Classical information
->
Bob oo*o
n. in
^" >23-(|:)2l^)3-|^)2l^)3>/^|
= (a\^>,+/^\l>,)/^
Source of EPR photon pairs Fig. 14. Principal scheme of teleportation. Alice is going to transpose the state of particle 1 onto some particle at Bob's station. Alice and Bob obtain photons 2 and 3, which form an EPR pair in the entangled state |^}23- Alice performs the Bell state measurement over particles 1 and 2. This way she also projects the state of particle 3 at Bob's station. In one case of four, detectors Fl and F2 "click" simultaneously, so that Alice knows that the state of particle 3 becomes the same as the initial state of photon 1; that is, that teleportation of the state 1*^)1 occurs. Alice can tell Bob about this through the classical channel. Moreover, if Bob obtains the information through the classical channel and performs an additional unitary transformation over his particle, the state | ^) i will be teleported with 100% probability after each Bell state measurement performed by Alice.
and by registering the coincidences of photocounts from detectors Fl and F2, Alice performs an immediate reduction of photon 3 to the initial state of photon 1, namely, teleportation! Several features of quantum teleportation deserve additional comments. (1) The teleportation procedure does not violate the noncloning theorem for a single quantum object. As soon as Alice performs the Bell state measurement, photon 1 becomes a component of the polarization-entangled pair of photons 1 and 2. Hence, it is no longer an individual particle. Its initial state | ^^) 1 is destroyed. (2) Quantum information can be passed from photon 1 to photon 3 separated by any distance. At present, the largest achieved distance between entangled photons is about 10 kilometers. (3) At the moment of measurement, Alice is aware of the teleportation going on, whereas Bob is not. In fact, teleportation can occur without passing Bob any
(
A J^i^ilpi
Alice
Initial State . ^
•
A^ \ Pump
[1, §3
Quanta and information
44
f
Polarizer
^ fPI^ . ^...•*
R-"
yi
^
e Source of photon pairs
Bob Teleported
P3 ^ / V ?
state
7-V'"""^ii D2 Di
Fig. 15. Scheme of the quantum teleportation experiment (Bouwmeester, Pan, Mattle, Eibl and Zeihnger [1997]). Correlated photons 2 and 3 connecting Alice and Bob were produced by a nonlinear crystal via type-II parametric down-conversion from a UV femtosecond pulsed pump. The reflected pump-generated photon 1, whose state was to be teleported, and photon 4, which was used as a time reference. The Bell state measurement for photons 1 and 2 was performed by mixing them on a beam splitter and then registering by the detectors Fl and F2. The polarization properties of Bob's photon were analyzed by means of a polarizing beam splitter and two detectors Dl and D2.
information about it. Moreover, Alice may not know the state of photon 1 transmitted by her. (4) A classical information channel is required for informing Bob about the teleportation of the unknown state onto photon 3. (5) Suppose that Alice performs a complete Bell state measurement and identifies, in addition to the fermionic state, the three bosonic states, each occurring with a probability of 25%, and sends this information to Bob through the classical channel. Then, by means of an appropriate operation performed over photon 3, Bob can transform its state into the initial state of photon 1 for any result of Alice's measurement. If this procedure is omitted and Alice only projects for the fermionic state, teleportation occurs only in 25% of all trials. This fact has been demonstrated experimentally by Bouwmeester, Pan, Mattle, Eibl and Zeilinger [1997]. Their experimental scheme is shown in fig. 15. Correlated photons 2-3 connecting Alice with Bob were generated by way of type-II parametric down-
1, § 3]
Quantum information
45
conversion in a nonlinear crystal from a UV femtosecond pulsed pump. Photon 1, whose state was to be teleported, was generated from the reflected pump beam. The Bell state measurement fr)r photons 1 and 2 was performed by mixing these photons on a beam splitter and registering coincidences of photocounts from detectors Fl and F2. The polarization properties of Bob's photon were analyzed by means of a polarizing beam splitter and two detectors Dl and D2. Teleportation was experimentally demonstrated by registering coincidences of photocounts from detectors Fl and F2 and one of Bob's detectors (triple coincidences). Suppose that photon 1, which is to be teleported, is polarized at 45° and Bob's polarizing beam splitter is sending -45''-polarized light to detector Dl and +45''-polarized light to detector D2. Then the coincidence of photocounts from Fl and F2 means that photon 3 is polarized at +45*^, that is, a photocount comes from D2 and not from Dl. Hence, if triple coincidence counting rates (D1F1F2) and (D2F1F2) are registered as frmctions of the delay between photons 1 and 2, which is varied by shifting the mirror reflecting the pump, one should expect a gap with complete suppression of coincidences for (D1F1F2) and no dependence for (D2F1F2). Outside the teleportation domain, that is, for delays between photons 1 and 2 so large that these photons hit Fl and F2 independently, the probability of triple coincidences is constant and equal to 50% X 50% = 25% (one 50% is the coincidence probability of photons 1 and 2, the other 50% is the probability that photon 3, which in this case has no definite polarization, hits Dl or D2). The experimental data obtained by Bouwmeester, Pan, Mattle, Eibl and Zeilinger [1997] confirmed these predictions, both for the case of photon 1 polarized at +45"^ (fig. 16a,b) and for the case of photon 1 polarized at -45° (fig. 16c,d). Teleportation was also performed for photons in the superpositions of these polarization states: 0°, 90°, and circularly polarized photons. Note that, despite the relatively low efficiency of teleportation (one out of four attempts), the teleportation fidelity in the first Innsbruck experiment (Bouwmeester, Pan, Mattle, Eibl and Zeilinger [1997]) and in the next experiment (Pan, Bouwmeester, Weinfiirter and Zeilinger [1998]), demonstrating entanglement swapping (Zukowski, Zeilinger, Home and Ekert [1993]), that is, realizing the teleportation of a qubit which itself is still entangled to another one, was relatively high (-0.80) (Bouwmeester, Pan, Weinfiirter and Zeilinger [1999]). At almost the same time as the results of the Innsbruck experiment on teleporation were published, Boschi, Branca, De Martini, Hardy and Popescu [1998] demonstrated their results obtained in the Rome laboratory with a sophisticated scheme of conditioned measurements with pairs of polarizationentangled photons. They managed to realize a complete Bell state measurement
Quanta and information
46
+45°teleportation
-45° teleportation
400
400 c/) o o o
[1, § 3
200
200
CNJ i_
(D Q.
0 O C 0) •g
o c *o o
400
400
_0
200
200
Q.
-100
0
100
Time delay (^m)
-100
0
100
Time delay (|im)
Fig. 16. Triple coincidence counting rates, D1F1F2 (-45°) and D2F1F2 (+45°), as functions of the delay between photons 1 and 2. The delay is varied by moving the mirror reflecting the pump pulses. The teleported photon 1 is polarized at +45° (a,b) and at -45° (c,d).
with the assistance of local operations and a classical communication channel the attributes of a quantum teleportation - but they did not teleport a state of the third unknown object from Alice to Bob. Continuous variables entanglement of the two-mode squeezed state (1.9) with squeezing parameter exp(-2r) = 0.5 was used by Furusawa, Sorensen, Braunstein, Fuchs, Kimble and Polzik [1998] in Pasadena, California, for the first unconditional quantum teleportation of incoming coherent state |vin). The teleportation fidelity for the experiment F = |(Vin | Vin)P was above the classical boundary Fc\ = 0.5. The realization of quantum state teleportation opens up new possibilities for transmitting "fragile" superposition states for large distances without loss of coherence. Solving this problem is crucial for the development of quantum computers, quantum cryptography, and for increasing the communication channel capacity by means of the dense coding method (Bennett and Wiesner [1992]).
1, § 3]
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47
The latter allows Alice to communicate to Bob a two-bit classical message by sending only one qubit through the channel, if they both have shared the entangled particles in advance. In this method, Alice codes the message in one of the Bell states (1.8) by a local unitary operation on her particle of the entangled pair, and sends the particle to Bob. Since Bob possesses both particles, he can distinguish among the four mutually orthogonal states. The method was experimentally demonstrated by Mattle, Weinfurter, Kwiat and Zeilinger [1996]. In addition, quantum teleportation is important in connection with some fundamental problems, such as, for instance, information exchange in complex, spatially separated molecular structures, including biological ones. As a first experimental result in this direction, an attempt at total quantum teleportation of the magnetic states of the hydrogen atom to the states of the chlorine atom within a single trichloroethylene molecule, performed by Nielsen, Knill and Laflamme [1998], should be mentioned. The importance of the paper by Bouwmeester, Pan, Mattle, Eibl and Zeilinger [1997] and subsequent experiments in Pasadena and Rome is clear because, since then, the information aspects of quantum mechanics have been treated not only as leading to "gedanken experiments" but also as "practically important". In addition, the teleportation experiments demonstrated that the classical interpretation of quantum mechanics, which is based on the notions of superposition and reduction and which so far predicted correctly the results of experiments, was confirmed once again. Since any quantum mechanical measurement fixes one of the possible realizations arising from the originally prepared state, Alice's measurements ensure that Bob obtains photon 3 in the original state of photon 1. This is only one of the possibilities that appear from the initial state of the three photons, two of which (2 and 3) are originally in an entangled state generated by a common source. Here, one should not forget that in quantum mechanics, the possibilities for arbitrary initial states are not necessarily described by positive probability distribution functions; that is, their description cannot be reduced to the classical probability theory. Of course, an alternative interpretation based on classical probabilities can be found for certain experiments, measurements, and states. At present, it is unclear, however, if this is interpretation possible in the general case. The state-of-the-art knowledge in this field is given in the review by Klyshko [1998]. It is also necessary to mention some new recent proposals and schemes for quantum teleportation: first, a proposal for teleportation of the wave function of a massive particle using entanglement between motional states of a collection of atoms trapped inside cavities and external propagating fields (Parkins and Kimble [1999]); second, a proposal for teleportation of an internal state of
48
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an atom trapped in a cavity to a second atom trapped in a distant cavity by using the atom cavity entanglement and a projection type of measurement (Bose, Knight, Plenio and Vedral [1999]); note that atom-field entanglement is a source of a variety of unusual effects, such as the quantum instability predicted by Kilin and Shatokhin [1996, 1997a,b]; third, a protocol for swapping of continuous variable entanglement (Van Loock and Braunstein [2000]); fourth, a scheme for dense quantum coding for continuous variables (Braunstein and Kimble [1999]); fifth, a method of conditional photon-number measurement for the improvement of the fidelity of continuous variables teleportation (Opatrny, Kurizki and Welsch [1999]); the same purpose protocol based on a local quantum nondemolition (QND) measurement of the collective excitation number of several continuous variable entangled pairs was proposed by Duan, Griedke, Cirac and Zoller [1999]; and sixth, a new scheme and protocol for continuous variables teleportation based on the entanglement produced by the QND interaction (Horoshko and Kilin [2000]). 3.1.3.2. Quantum teleportation as a new class of physical communication channels and the problem of quantification of entanglement transformations. The successful realizations of quantum teleportation have clearly introduced a new class of physical communication channels: a multipartite quantum system transmitting information by means of entanglement, local quantum operations, and classical communication (LOCC). Here, local operations include any unitary transformations, additions of ancillas, projective measurements, and the discard of parts of the system, each performed by one party on his or her subsystem. Mathematically, the LOCC transformations can be represented as completely positive linear maps that do not increase the trace of the quantum channel density matrix
L{p) = Y,LipLl where the superoperators L/ = ^/ 0 5/ (g) C/ (8) • • • satisfy the relations Y^. L^Li ^ 1. For example, Alice performs a generalized measurement, described by the complete set of operators Ai (^fA'^Ai = 1), and sends the results to Bob, who performs an operation J5,, conditional on the result /. As a consequence of these actions the initial density operator of general system PAB is transformed to the density operator L(pAB) =
'^.BiAiPABAlBt.
The general properties of such LOCC transformations of entangled states have been the subject of extensive work in recent years. The problem was
1, § 3]
Quantum information
49
introduced by three papers (Bennett, Bernstein, Popescu and Schumacher [1996], Bennett, Brassard, Popescu, Schumacher, SmoHn and Wootters [1996], Bennett, DiVincenzo, Smolin and Wootters [1996]). The authors have studied entanglement distillation, solving the problem of transforming some given pure state into (approximate) EPR pairs in the asymptotic limit, where many identical copies of the pure state are initially available. The inverse procedure of entanglement formation solving the problem of transforming EPR pairs into many (approximate) copies of some given pure state, again in asymptotic limit, was also studied. In these investigations, the problem was also generalized to asymptotic and approximate transformations between mixed states and EPR pairs. An important result in this direction was obtained by Nielsen [1999], who proved the theorem {Nielsen s theorem) that any pure state 11/^) of a composite system AB transforms to another pure state \(p) using LOCC transformations if, and only if, the ordered set of the eigenvalues of Alice's initial density matrix PA (t/^) = Tr^ (|V^)(i/^|) is majorized (Marshall and Olkin [1979]) by the same set for the final density matrix p^ (0) = Tr^ (|0)(0|), that is, if for each k k
k
^A,(v;)^5]A,(0), where k\ > A2 > • • •. To prove the theorem, Nielsen used the Schmidt decomposition (Peres [1993]) of the pure state of a composite system
\x) = Yli^i\^^)\^B). where A/ > 0, J ] / ^ / ^ 1? ^"^^ VA) and Iz^) form an orthonormal basis for each subsystem. Jonathan and Plenio [1999a] and Vidal [1999] extended Nielsen's theorem to the case where the transformation for one pure state (say, nonmaximally entangled) to another (say, maximally entangled) need not be deterministic (less than 100%). The same authors (Jonathan and Plenio [1999b]), with the help of Nielsen's theorem, showed that in the case where the target states cannot be reached by LOCC starting from a particular initial state, the assistance of a distributed pair of auxiliary quantum systems can catalyze the transformation, and what is most surprising, these auxiUary quantum systems (catalysts) are left in exactly the same state and remain finally completely uncorrelated to the quantum system of interest. The suggestion was made to call this new class of LOCC transformation "entanglement-assisted local transformations".
50
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abbreviated to ELOCC. Eisert and Wilkens [1999] went further and studied the catalysis of entanglement manipulation for mixed bipartite states. The problem of entanglement identification for mixed states is of great importance because of the strong pressure of losses and decoherence processes, which transform pure maximally entangled states into partially entangled states of distant modes. As follows fi*om the previous consideration, entanglement arises when the state of a multiparticle system is nonseparable, that is, when it cannot be prepared by acting on the particles individually. Although in recent years important steps have been taken toward the understanding of this quantum resource, we do not know yet in detail how to classify and quantify entanglement for the general multiparticle states, including nonpure states. However, some important results have been reported. Mathematically entangled states are those which do not belong to the class of pure and mixed separable states. A pure state |t/;^^^ > is separable if it can be expressed as a tensor product of states of different parties:
A nonpure mixed state p^^<^ of separable pure states:
is separable if it can be expressed as a mixture
p''^'" = E,^'i^')(v^'i ^ \^')i^'\ ^ \^^)i^' where j9/ > 0, YliPi= 1States of particular importance are the so-called Werner states (Werner [1989]); these are mixtures of a Bell state (say, |0^), eq. 1.8a) and a totally depolarized state, pw=x|0^>(^^| + l ^ i .
(3.34)
These states play an essential role in the understanding of entanglement and distillability properties of two-qubit systems (Wootters [1998]). On one hand, the Werner states are separable when x^j (that is, when fidelity F = (0^ I P ^ I 0+) < ^) and nonseparable when x>\ (F> ^). On the other hand, Bennett, Brassard, Popescu, Schumacher, Smolin and Wootters [1996] showed that one can purify the states with an arbitrarily high fidelity out of many pairs with F > ^ by using LOCC. Furthermore, any arbitrary state can be depolarized to a Werner state without changing the fidelity. The former automatically provides a sufficient criterion for nonseparability, presented by
1, § 3]
Quantum information
51
Peres [1996] and Horodecki [1997] as positive partial transposition, and a distillability criterion presented by M. Horodecki, P. Horodecki and R. Horodecki [1997] as a negative partial transposition property. Diir and Cirac [1999] proposed the generalization of the Werner states to a multiqubit system. For this purpose they used the orthonormal GHZ-basis (Greenberger, Home and Zeilinger [1989]) \Wj^) = (|7)|0) ± |2^-i -j-l)\i))/V2,
(3.35)
where \j) = \j)Ay •••AN-I is the state of the first (N - 1) qubits. For example, for N = 5 andy = 6 this reads \W^) = {\OUO)A, ...AAO)A, ± |1001)^, . . . ^ J l ) ) / A / 2 , since 6 = 0110 in binary notation, and for A^ = 3 and 7 = 0 the state |^^o^) = (|000) + | l l l ) ) / V 2
(3.36)
is the canonical Greenberger-Horne-Zeilinger state. The same authors introduced a family of states to which an arbitrary state p can be depolarized by N local operations while keeping the values of A^ = {W^ \p\ W^) and 2Xj = {W/ \p\ W/) + {Wr \p\ Wr) unchanged. The authors gave a full classification of the states with respect to their separability and distillability properties, which provides sufficient conditions for nonseparability and distillability of arbitrary states. For example, in the case of maximally entangled states of N qubits mixed with a totally depolarized state p(x) = x\W;){W-\ + ^
l
(3.37)
which is a special case of the state pw with Ay = (1 - x ) / 2 ^ , AJ = x + (1 - x ) / 2 ^ , the density matrix p(x) is fiilly nonseparable and distillable to a maximally entangled state if x > 1/(1 + 2 ^ " ^), and fully separable otherwise (see also results by Murao, Plenio, Popescu, Vedral and Knight [1998], Diir, Cirac and Tarrach [1999], Vidal and Tarrach [1999], Braunstein, Caves, Jozsa, Linden, Popescu and Schack [1999], and Smolin [2000]. The progress in understanding the structure of possible transformations of entanglement shows their power for various tasks in quantum information science, one of which is quantum cryptography. 3,1.4. Quantum cryptography One of the most practical aspects of quantum information is quantum cryptography. Its aim is a secret exchange of information between two stations (Alice
52
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and Bob), so that any attempt at eavesdropping messages or breaking the secret code would be unsuccessful. This problem is almost solvable by modem methods of classical cryptography, for instance, in the framework of a "symmetrical" cryptosystem based on a secret code. In this system, Alice and Bob, and nobody else, have a secret code (key), that is, a sequence of random numbers, for instance, decimals, ^ = {12793 41169 42357 • • • } . First, according to some fixed rule, each alphabet letter is put into correspondence with a decimal number. Alice prepares a message where each letter is replaced by a corresponding number. At this step the procedure has no defense and can be broken easily. Now, the obtained message in the form of a sequence of numbers, P = {73997 68279 65867 • • } , is encoded; that is, a digit fi-om the key is added to each digit from the message. As a result, in the lowest order decimal digits form the cryptogram C = {85680 09338 07114 • • } , which can be transmitted through an open channel (telephone, etc.). After receiving the cryptogram, Bob decodes it using the key K and obtains the message P. Note that the above sequences K, P, and C were taken from a real message sent by Che Guevara from Bolivia to Fidel Castro Ruz in Cuba in 1967 (Bennett, Brassard and Ekert [1992]). Shannon [1949], using information theory, proved that such a cryptosystem is absolutely secret if the secret key is truly random, is as long as a message, and is used only once. These statements derive from the requirement of having no redundancy in the sent message P. However, practical realization of this system faces serious difficulties, one of which is the creation and transmission of a large secret key for each message. These difficulties could be avoided by using some physical channel that would be secret because of certain physical principles. This is exactly what quantum physics provides. The possibility of organizing such a secret channel is based, like quantum teleportation, on the impossibility of cloning a single quantum object. If the secret code is transmitted by way of the states of single quantum particles, it cannot be eavesdropped, since any measurement would destroy quantum states. This event can be registered by using a special agreement (protocol) between
1, §3]
Quantum information
53
Quantum channel Alice
\ /
^ \
^.
\ Bob
Classical (open) channel
'jVo
Actions 1 0 0 1 0 0 1 1
1
A^B
2
B measures
3 B=^A: type of measurement A=>B: correct 4 A and B create a code
t
/ ^-^ \ <-^ <-^ X t
+ + X X + X X + t ^-> / \
Secrecy secret Quantum channel open channel secret open channel
/
/ /
/
t
\
z
1
1 0
secret
1
Fig. 17. Procedure of quantum cryptography with polarization encoding.
Alice and Bob. One of the possible protocols can be organized by encoding polarization states of photons in two alternative nonorthogonal bases ^ (Bennett and Brassard [1984], Bennett [1992]). The name of the protocol is BB84. The secret key is transmitted as follows (fig. 17): (1) Alice and Bob first discuss the encoding (for instance, photons with polarizations O'' and 45° correspond to a zero and photons with polarizations 90"^ and 135"" to a one). Alice then randomly changes the polarization of photons sent to Bob through the quantum channel. (2) Bob measures the polarizations of received photons using an analyzer with the orientation randomly changed from 0'',90'' (+) to 45'', 135'' (x).
Each of the digits is encoded by two polarizations to guarantee secrecy. Using only one basis leaves only one quantum channel for transmitting the code from Alice to Bob. In this case, however, even if Alice transmits a random code. Bob has no possibility of checking whether it is correct or broken by eavesdropping attempts.
54
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lAlicel^^
[1. § 3
TvA^ •
>
—
\
iLS2fe,J OR
Fig. 18. Scheme of quantum cryptography with phase modulation and interferometric detection.
(3) Through the open channel, Bob tells Alice which measurement he performed over each photon, and Alice tells him whether the choice was or was not correct. (4) Retaining, of the entire sequence, only measurements chosen correctly, Alice and Bob create a secret key. If an eavesdropper tries to find out the secret key, he or she will cause discrepancies between the codes obtained by Alice and Bob. Alice and Bob can discover this by comparing randomly chosen digits of the code; if the errors exceed the level determined by the detectors, they conclude that an attempt at eavesdropping has been made. Another possible protocol for transmitting a quantum code (Hughes, Aide, Dyer, Luther, Morgan and Schauer [1995]) is provided by phase modulation with interferometric detection based on the interference of a single photon with itself in a setup formed by two Mach-Zehnder interferometers (fig. 18). A photon sent by Alice can hit Bob's detector within one of three separate time intervals, depending on its path. The first time interval corresponds to the case "short arm of interferometer A - short arm of interferometer B". The second interval corresponds to the two indistinguishable cases "short arm of A - long arm of B" and "long arm of A - short arm of B". The third interval corresponds to the case "long arm of A - long arm of B". Due to the indistinguishability of the two paths of photons fitting the second interval, interference must occur, depending on the phase difference between 0A and 0B for the modulators monitored by Alice and Bob. In fact, the probability of detecting a photon within the second interval is PB(XCOS2(1(0A-^)).
Hence, if Alice and Bob use the phases (^A, 0B) = (0, ^JT) for the 0 bits and ( 0 A , ^ ) = (\jt,Jt) for the 1 bits, they obtain an analog of the polarization encoding described above. Ekert [1991] proposed a protocol based on entangled pairs and using Bell's inequality to establish security. Both Alice and Bob receive one particle out of
1, § 3]
Quantum information
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an entangled pair. They perform measurements along at least three different directions on each side, where measurements along parallel axes are used for key generation and oblique angles are used for testing the inequality As Ekert pointed out, eavesdropping inevitably affects the entanglement between the two constituents of a pair and therefore reduces the degree of violation of Bell's inequality. At present, the most convenient medium for the quantum channel is an optical fiber, through which cryptograms can be sent over distances in excess of 100 km. Since an optical fiber has considerable birefringence fluctuations, the "polarization version" of quantum cryptography is connected with certain difficulties, and interferometric detection is preferable. Experimental realization of quantum cryptography is possible with some additional requirements: small losses in the quantum channel (optical fiber has low losses in the IR range for wavelengths 1.3 mm [0.3dB/km] and 1.55 mm); operation of photodetectors in the photon counting regime (for the chosen wavelength of 1.3 mm, existing Ge or InGaAs avalanche photodiodes can be used under certain conditions; Hughes, Aide, Dyer, Luther, Morgan and Schauer [1995]); and no amplifiers introduced into the channel. (From the noncloning theorem, it follows that an amplifier in the quantum channel leads to the same effect as an attempt at eavesdropping.) Quantum cryptography with phase modification was also implemented in a series of experiments (Marand and Townsend [1995], Hughes, Luther, Morgan, Peterson and Simmons [1996], Muller, Herzog, Huttner, Tittel, Zbinden and Gisin [1997]). In an experiment performed by Muller, Zbinden and Gisin [1996, 1997]), the quantum code was transmitted through a standard optical fiber (Swiss Telecom) under Lake Geneva over a distance of 23 km. The length of the code transmitted during the 11 hours of the session was 20 kbit; 1% of errors occurred, mostly caused by the germanium photodiode. As for the polarization encoding, the feasibility of the scheme was demonstrated by Bennett, Bessette, Brassard, Savail and Smolin [1992], Muller, Breguet and Gisin [1993], Franson and Jacobs [1995], and Buttler, Hughes, Kwiat, Lamoreaux, Luther, Morgan, Nordholt, Peterson and Simmons [1998]. The most recent experiment, by Jennewein, Simon, Weihs, Weinfiirter and Zeilinger [1999], realized both the Ekert scheme, which applied Wigner's inequality (Wigner [1970]), and a variant of the BB84 protocol with entangled photons, as proposed by Bennett, Brassard and Mermin [1992]. The system has two completely independent users separated by 360 m, and it generates raw keys at rates of 400 to 800 bits/second with bit error rates around 3%. To implement a smaller bit error rate they used classical error correction over a public channel
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with a result of 0.4%. Finally they were able to transmit securely a 43 200 bit large image by means of the One-Time-Pad. The paper clearly demonstrates the technological ability of entanglement-based cryptography. Some types of protocols for classical and quantum cryptography have been considered above, but many other protocols exist. The most popular cryptosystem with open key transmission is RSA, suggested by Rivest, Shamir and Adelman [1979] (the abbreviation RSA comes from their surnames). In this protocol, two secret codes are used: one for encoding and the other for decoding. In addition, an auxiliary code is transmitted through open channels, which is the product of large prime numbers (containing more than 200 digits). The secrecy is ensured because factoring large numbers is a complicated computational problem, and with modem facilities it cannot be solved in a reasonable polynomial time. It is an NP-complete problem for the existing computers (see § 2.4). Recently, an attempt to solve the mathematical part of this problem led to the suggestion of a fast procedure that could be realized in socalled quantum computers. For these devices to be constructed, a consolidation of efforts in many fields of physics is required, namely, quantum optics, solidstate physics, laser physics, and spectroscopy.
3.2. Quantum computations and computers 3.2. L Reversible and irreversible classical processors Some aspects of the work of ordinary, classical computers will be briefly described in this section before presenting the basic principles of quantum computations and quantum computers, for a more clear demonstration of their peculiarities. Classical computers as devices for calculations must operate with numbers. The simplest device that can represent numbers should have two stable states. For instance, conductors can be in two states: when there is no current, which corresponds to 0, and when a current is present, which corresponds to 1. Such devices can perform operations over numbers written in binary codes. For example, the natural number 9 is written in binary code as 1001 = (1 X 2^) + (0 X 2^) + (0 x 2^) + (1 x 2% and numbers are summed according to the table 0 + 0 = 0, 0 + 1 = 1, 1+0=1, 1 + 1 = 0 (carry 1).
1, § 3]
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57
At present, many devices can perform this operation. As an example, consider the mechanical version of an adder (fig. 19). This device consists of gates and connecting channels, and balls can move along the channels under the action of gravity. When moving, the balls turn the gates into one of two possible positions; the T-state of a gate corresponds to 0, whereas the turned A-state corresponds to 1. Two types of gates are present in the device: gates A and C (gray hatching) fix the state of the channel in which they are located, and gate B ( black color) are logical gates of the summation of two bits. In fact, each gate B has one input channel (to the left) and two output channels (to the right). Among the output channels the lower one is used to carry the bit to the next digit and the upper one, to remove the balls. If the presence of a ball at the input of gate B corresponds to 1 and its absence to 0, this logical gate operates according to table 1. This is equivalent to the operation of two-bit summation if the state of the input B is treated as the first bit and the initial state of gate B as the second. Then the final state of gate B together with the final state of the carrying bit is the result of the summation. Combining such logical gates in a network by connecting the bitcarrying channel to the input of the next-digit logical gate, we obtain a processor for summing arbitrary numbers (fig. 19a-c). The operation of such a processor has an important feature. This processor will perform the summation operation even without gates A and C. In this case, the operation of the adder will be irreversible. In fact, for the transformation "two inputs -^ one output" (initial state of gate B input, initial state of gate B -^ final state of gate B), the information at the output is not sufficient to determine what was at the input (cf. the final states in the second and third rows in table 1) and thus reverse the operation. However, in 1973 Charles Bennett (Bennett [1973]) showed that all logical operations needed to design a computer (only 4^ = 16 for logical gates of the type "two inputs -^ one output") can be made reversible. For the summation operation the initial bit states should be conserved; that is, transformations like (a, b) ^ (a' = a, b' = a + b), where the prime indicates the final states, should be used. In 1980, Toffoli [1980] found how one can describe reversible calculations using the traditional language of Boolean logical gates, such as AND, OR, etc., but having the property of reversibility. One such logical gate, which was shown later to be extremely important for quantum calculations, acts like a controlled NOT (reversible XOR). The bit b (target bit) changes its state if, and only if, the state of the control bit a corresponds to 1; the state of the control bit remains unchanged (fig. 20a). Toffoli also showed that an arbitrary reversible processor can be constructed using only a single logical gate, Toffoli's universal three-bit gate (fig. 20b). In this logical block the state of the target bit (c) changes if, and only if, both variable control bits (a and b)
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EX
:^^v>>^ C4=0
Fig. 19. Mechanical scheme of a summing processor with balls. The device consists of gates and connecting channels. Balls can move along the channels under the action of gravity. When moving, the balls turn the gates in one of two possible positions. A T-state of the gate corresponds to 0, and the turned A-position corresponds to 1. Gates are of two types: gates A and C (gray hatching) record the state of the channel where they are installed, and gates B (black) serve as logical gates for a two-bit addition, (a) The original state of the processor: the states of the gates (B4, B3, B2, Bl) "encode" the number 5 = 0101; the balls at the input "prepare" another number 3 = 0011. (b) The state of the adder after the action of a ball from the first digit: the number 1 is written down in register {Aj}, and the number 0110 (5+1) in register {5/}. (c) The final state of the summing processor after the action of a ball from the second digit: the register contains the initial number 0011. The register {5/} is turned into the state corresponding to the sum 101 + 11 = 1000 (5 + 3 = 8).
1, §3]
59
Quantum information Table 1 Operation of logical gates used in a mechanical adder
Initial state of gate B entrance (1: ball; 0: no ball)
Initial state of gate B ( r = 0, A = 1)
Final state of bottom output channel (bit-carrying channel)
Final state of gate B
a b" c'
1p
^ VJ\) a 0 0 1 1
b 0 1 0 1
a' 0 0 1 1
(a)
b' 0 1 1 0
a 0 0 0 0 1 1 1 1
b 0 0 1 1 0 0 1 1
c 0 1 0 1 0 1 0 1
a' 0 0 0 0 1 1 1 1
b' 0 0 1 1 0 0 1 1
c' 0 1 0 1 0 1 1 0
(b)
Fig. 20. (a) The graphical representation and truth table for an elementary controUed-NOT gate: bit h (the target bit) changes its state if, and only if, the state of the control bit a corresponds to 1, with the control bit state remaining unchanged. Each horizontal line represents the state of a single bit changing in time from left to right. The symbols on two lines connected with the vertical line mean the joint action of two gates on these bits. Clearly, the truth table for this logical gate, which is also called EXCLUSIVE OR (XOR), corresponds to the table of two-bit addition if in the latter the carrying bit is not taken into account, (b) The graphical representation and truth table for Toffoli's three-bit logical gate, which is universal for constructing reversible Boolean logic. Its action reduces to changing the state of target bit c provided that both invariable control bits {a and h) correspond to 1. Each horizontal line represents the state of a single bit, which changes in time from left to right. The symbols near the three lines connected with the vertical line mean joint action of the three gates on these bits.
correspond to 1. Figure 19 also can be used to demonstrate the reversibility of the classical adder. Adding to the consideration the gates A and C, which are used to fix the states of the inputs {Ai ) and carrying bits (C/ ) in each digit, we obtain a reversible adder. In fact, if we rotate this device about the horizontal axis (fig. 19c) and consecutively "let in" the balls that fell into the evacuation
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[1, § 3
Adder in the first digit ^/ Carrying to the second digit
Bi Ci
n
A2 B2 C2 A3 B3
Ca A4 B4 C4
^
I -®^
I
Fig. 21. Logical scheme of a reversible adder with the mechanical scheme shown in fig. 19. The horizontal lines correspond to the states of the bits of two numbers to be summed (A4, A3, A2, Al) and (B4,B3,B2,B1), as well as to the carrying bits (C4,C3,C2,C1). The digit-summing blocks represent an operation that can be performed by a ball falling on the gates Bi in the mechanical scheme: the gate Bj changes its state provided that a ball is present in the channel A^; if the state of the gate Q before the interaction corresponded to 1, the ball is carried into the next digit along the transmission channel changing the state of the gate C, which originally was 0. Carrying blocks perform the controlled-NOT operation by changing the state of gate Bj +1 with a ball that entered the / + 1th summing gate from the carrying bit channel Q.
channels, the final state of the device after the balls have passed coincides with the initial one. A logical network (fig. 21) demonstrating the time evolution of bit states can equivalently describe the operation of such a reversible summation processor. The idea of a quantum processor is a single step ahead of such a logical network of bit states. Feynman [1982, 1985, 1986] made this step when he realized that reversible computation networks can operate, instead of classical bit states, with quantum states of systems governed by reversible Hamiltonian dynamics. This period can be considered to be the beginning of quantum computers history. 3.2.2. Quantum computers 3.2.2.1. Power ofHilbert space, quantum logical operations. Quantum computers are physical devices performing logical operations over quantum states by means of unitary transformations that do not destroy quantum superpositions.
1, § 3]
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61
The work of a quantum computer can be schematically represented as a sequence of three operations: (1) recording (preparation) of the initial state; (2) computation (unitary transformations performed over the initial states); and (3) reading out the result (measurement, or projection, of the final state). 3.2.2.1.1. Recording (preparation) of the initial state. A normal numerical computer operates with bits, Boolean variables that take values 0 and 1. At each stage of calculation, each bit has a definite value that can be measured. At the first stage the initial data should be written into the register, a set of bits, each bit having a definite value (0 or 1). A quantum computer operates with quantum states. The simplest state that plays the role of a bit in the classical computer is a qubit, or a quantum information bit, which is the state of a quantum system with two basic states |0) and |1). The general state of this system is a superposition |^)=co|0)+ci|l), which is somewhat different from a Boolean 0 or 1. A qubit is a quantum superposition of two possibilities, denoted by zero and unity, the possibilities that are fixed and realized only by the choice of measurement! Qubits can be realized in any physical system with two quantum states: photon polarization states, electronic states of isolated atoms or ions, spin states of nuclei, lower states of quantum dots, and so on. The advantage of operating with qubits is apparent even at the first stage of computation. If the initial number is written into a classical register consisting of w bits, w operations are required, since for each bit the values 0 or 1 should be set. As a result, only a single number of length w is written. After w unitary operations are performed over each qubit in the quantum register (a device consisting, for instance, of w quantum dots, see fig. 22, overleaf), a coherent superposition of all Q = 2^ states of the total system, quantum register, is prepared. In this manner, instead of a single number, we obtain 2^ possible readings of the register, a coherent superposition of all possible numbers written in it. In other words, all exponential power of the Hilbert space of w distant, locally separated quantum objects can be reached by only w local operations! It is the quantum mechanical feature that opens up new resources for computations. Naturally, this property can be used for quantum parallel calculations. 3.2.2.1.2. Computation (unitary transformations performed over the initial states). Applying unitary transformations, which play the role of logical operations, to the prepared quantum states, we obtain a quantum processor. The role of connections (wires) is played by qubits, and the role of logical
62
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jo) Classical computations 100 0 Oil... 1 T i XT2 XT3.
register =>
result
computer
jb) Quantum computations
q>i|q>2... |q>N|
|q>=a|0>+p|l> qubit
register
|q>,|q>2....|q>N
Ti XT2 XT3., computer =;
result
MEASUREMENT ! <^*|X|^*>
(c) Quantum register
Q=2 - states |n>
w- qubits Fig. 22. Quantum computers are physical devices performing logical operations on quantum states by means of unitary transformations, so that quantum superpositions are not destroyed during the computation. In contrast to a normal numerical computer (a), which operates with bits, i.e. Boolean variables taking values 0 or 1, a quantum computer (b) operates with qubits, quantum information bits, which are states of a quantum system with two basic states |0) and |1). Physically, qubits can be realized with any systems that have two quantum states. These can be polarization states of photons, electronic states of isolated atoms or ions, spin states of nuclei, lower states in quantum dots, and so on. Schematically, the work of a quantum computer can be represented as a sequence of three procedures: recording (preparation) of the initial state; computation (unitary transformations performed on the initial state); and output of the result (measurement, projecting of the final state). The result of a quantum computation should be treated as some probability distribution and measured in many repeated trials, (c) A quantum register is formed by the states of several qubits. For w qubits in a register, the number of states of the register is ^ = 2^.
blocks (gates) constituting the whole computation process is played by unitary transformations. This concept of quantum processing and quantum gates together with the universal quantum gate (analogous to Toffoli's gate in classical
1, § 3]
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63
computation) was proposed by Deutsch [1989]. Recent investigations showed that one- and two-bit gates are sufficient for obtaining all the necessary set of transformations (DiVincenzo [1995b, Barenco, Deutsch, Ekert and Jozsa [1995], Sleator and Weinfurter [1995]). In particular, these are the negation operation NOT (quantum analog of gates A and C in classical adder on balls), rNOT = |0)(l| + |l)(0|,
(3.38)
acting on a single qubit and transforming its state into rNOT|0) = | l ) ,
rNOT|l) = |0),
and the controlled-NOT, or exclusive-OR (XOR) operation (quantum analog of gates B in classical adder on balls), fxoR = |0)i i(0|/2 + |l)i i(l|r2N0T,
(3.39)
acting on 2 qubits so that the first remains unchanged and the second changes depending on the state of the first one. For instance, rxoR(a|0)i+^|l)i)|0)2 = a|0)i|0)2+iS|l)i|l)2,
(3.40)
that is, the operation TXOR transforms the superposition states into entangled ones and vice versa. A quantum analog of the logical Toffoli gate (controlledcontrolled NOT) (fig. 20b) acts on 3 qubits according to the relation 7^Toffoli(«|0)l + ^ | l ) l ) ( 7 | 0 ) 2 + 5|1)2)(M|0)3 + V|l)3) = [(a|0)i(y|0)2 + (5|l)2) + ^7|l)i|0)2)](M|0)3 + v|l)3)
(3.41)
+ ^(5|1)I|1)2(M|1)3 + V|0)3).
Quantum logical gates combined together and acting on qubit states in a certain order form a quantum network. Using the scheme of the reversible summing processor as an example (see fig. 19) and considering two-level systems instead of states of the gates, with interactions corresponding to unitary transformations (3.39)-(3.41), we obtain (see fig. 21) the simplest quantum network, an adder. 3.2.2.1.3. Reading out the result (measurement, or projection, of the final state). The operation of reading out the result in a classical computer does not differ from any other operation in the course of computation. Computation can be stopped at any stage (with the intermediate result read), and computation is then resumed. In a quantum computer this is different. The final result of a quantum
64
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computation is the state of the quantum register after all unitary transformations. This state is a coherent superposition of all the states possible for this register. Evidently, we cannot obtain all probability amplitudes Cj in the decomposition of this superposition state. According to quantum theory, all that we can obtain from this single quantum object is a set of quadratic forms J2ij Q^f^tj^ given by the measurement of some physical value corresponding to the operator R. It is also clear that the final result of a quantum computation would fluctuate from run to run. Even under such conditions, however, quantum computers can essentially accelerate calculations for some mathematical problems. 3.2.2.2. Quantum computers and mathematical problems. When Feynman first noticed the possibility of constructing a processor based on quantum mechanical principles (Feynman [1982]), it was unclear in what mathematical problems it would have advantages over conventional processors. The first realistic example was found by Shor [1994], who suggested an algorithm for factoring a large ndigit number; the proposed algorithm allowed the calculation time to be reduced fi-om the exponential value exp(«^^^), which is necessary in the case of classical computers, to the polynomial value (n^) required by a quantum computer. Factoring integers belongs to the class of mathematical problems where the solution is sought from among an exponentially large number of candidates. This had actually been a NP-complete problem until Shor suggested using a quantum algorithm for its solution. 3.2.2.2.1. Shors quantum factoring algorithm. The problem of factoring can be reduced to the finding of the period of an auxiliary function. This function is the remainder of dividing a power function a^ by an integer number N to be factored: fyix) = a^ mod A^. For instance, for cf = 11, A^ = 15, the values o f ^ ( x ) at x = 0,1,2,3 are equal to 1,11,1,11, respectively; that is, the period of the function 1V mod 15 equals 2. Furthermore, the procedure of finding prime divisors is reduced to the following operations: 11 ± 1 = 10,12; 15 - 10 = 5; and 15 - 12 = 3. Shor showed that the procedure of finding the period of a periodic ftinction is considerably simplified by using a quantum computation. The following operations should be performed (fig. 23): (1) Preparation of two registers, one for arguments (x-register) and the other for values of the periodic function, for instance, the integer-valued function ^ ( x ) = cf mod A'^ (j-register). Let the number of qubits in the x-register be
1, §3]
65
Quantum information
x-register in the state |l>x+|2)x+- •+|Q--l>x
m
1^1
I
m:-
Fig. 23. Obtaining the period of a periodic function by means of a quantum computation, (a) Preparation of independent states of two registers, x and y. The abscissa corresponds to the numbers n of the basic states \n)x in the x-register, and the ordinate denotes the numbers m of the states \m)y in the ^-register. The hne shows the numbers of the states in two registers that form the initial state, (b) Entangled state of two registers (3.42), with the numbers of connected states of xand >^-registers forming a discrete periodic function m(n) with period r. (c) The same state but with the basis of the x-register changed by the discrete Fourier transform (3.43). Due to the periodicity of the function, only states with numbers k localized in the vicinity of Q/r form the entangled state in the new basis, (d) Because of this localization, several measurements of the x-register state are sufficient for determining the period r.
w, then this register contains Q = 2^ possible states, which will be further denoted by \n) (fig. 23). The number of qubits can be exponentially smaller than the period r of the function fN(x). The y-register contains the same
66
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number of bits. Let us denote its basic states by \m)y. The jc-register, initially in the ground state, is put, after rotating each qubit by 45'', into the state of uniform superposition
(|o).v+ii).v+---+ie-i).v)|o),/ye. Furthermore, by performing an appropriate unitary transformation Uf, the state is transformed into the entangled state of two registers,
(|o).vi/^(o))v+|i),i/^(i)),+...+ie-i).i/^(e-i))>^)/v^. (3.42) This state is schematically shown in fig. 23b as some periodic fiinction, so that each point in the plot corresponds to a pair of integer numbers (n,m = fN(n)), denoting the term \n)x\fN(f^))v ^^ the sum (3.42) with n on the horizontal axis, and m on the vertical. (2) Next, a discrete Fourier transform is performed over the states of the x-register. The corresponding unitary transformation of the basis,
^FT = 7 ^ X ] ^
exp(2jri^«/0|^),,(«|,
(3.43)
applied to the state (3.42), leads to a new state of both registers, Q-l
r
5]^Z^,,,|/:),|/^(m))„
(3.44)
k =0 m=0
where the amplitude of each state in the x-register, ^
^^^2mkrG„,/Q
_ i
4,„, = gexp2-*<'-'")/e = exp2-*"'^e '^P^,^,,^,^ _/, Q^p2jrikr/Q _ J
(3.45)
/=0
has a maximum at k = pQ/r (fig. 23c). Evidently, it is supposed that the total number of states, Q, is not a multiple of the period r. In eq. (3.45), Gm = 7 + ^(Q ^ o ^ ^ ~ ^)l that is, G„j is either equal to the number of periods r in the total number of states Q or exceeds this number by unity, depending on whether or not the remainder on dividing g by r is larger than m.
1, § 3]
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67
(3) The result of measurement (or measurements) of the state of the x-register is approximately given by the ratio Q/r, since the amplitudes Akm are localized near these values (fig. 23c). Hence, we obtain period r. 3.2.2.2.2. Simulations of quantum factorization algorithm and the problem of physical resources. Note that applying a discrete Fourier transform to the problem of finding an unknown period is analogous to measuring the period of some lattice using the diffraction of X-rays or neutrons. If the problem of factoring a 200-digit number were solved by means of diffraction, however, we would need a crystal with period 10^^^ A and size 10^^^ A, and radiation with a wavelength of 1 A. Naturally, this is hardly possible. Many analogous examples show that a quantum gate operation, or factoring and Fourier transformation, could be simulated by means of linear optics (Cerf, Adami and Kwiat [1998], Clauser and Dowling [1996], Howell and Yeazell [2000], Summhammer [1997]). For example, Clauser and Dowling [1996] showed that in a suitably chosen central region on the detection screen behind an A^-slit arrangement, all intensity peaks are equal if, and only if, the quantity n = XR/a^ is a factor of N, where A is the wavelength of the incident radiation, a is the center-to-center distance of the slits, and R is the distance between the slits and the screen. Furthermore, the Fraunhofer limit is assumed (i? > a > A). Different values of n can be tested by adjusting any of the parameters. Despite the bright illustrative character of the above examples, they all should be distinguished from quantum devices that explore the entanglement. All classical or linear quantum mechanical devices using the wave character of quantum objects (like linear transformations of light by means of beam splitters) require much more physical resources (number of loops, beam splitters, or total amount of time needed for factoring N rises polynomially with N rather than just logarithmically). 3.2.2.2.3. Growers search algorithm. Other mathematical problems exist where solutions are sought among an exponentially large number of candidates. Recently, a method of solving another problem was proposed, with the time of calculation reduced polinomially by using quantum computation. This is the NPoracle problem of searching among the elements of a database with each element answering "yes/no" to a query (Grover [1997]). A quantum mechanical analog of the first example in § 2 can be presented as a search of one item, say, the state I7), where 7 is a binary digit among all, say, 2^ states, by acting (asking an oracle) on the joint state of all 2^ qubits. The action pointing to the marked state can be represented by a unitary operator -^oracle
Y.\^ii\-\j){j\-
(3.46)
68
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Starting from the equally weighted superposition 2^-1
l^(^)) = 4 7 E l ^ ' ) '
(^-47)
/=0
one should apply .Soracie, then Fourier-transform, change the sign of all components except |0), and Fourier-transform back. The whole operation is described by the transformation UG\^m = \W(M),
(3.48)
where sin^Ar = 2y/N - \/N. The coefficient of the marked element is now slightly larger than that of all the other elements. The method proceeds simply by applying the operation m times, where m ^ \jZ\fN. As a result, |^(0)) is transformed to UE\ Wm = sin {l.Vl^m)
\J). ^osi^2^-^)
^ |,^,
(3.49)
which means that the quantum state almost precisely equals \j) for large A^. An important further point was proved by Bennett, Bernstein, Brassard and Vazirani [1997], namely, that Grover's algorithm is optimal: no quantum algorithm can do better than 0(VN) times'^. Analogously to Shor's quantum factoring, Grover's algorithm was analyzed to determine the classical or entanglement-free (often optical) simulations of the algorithm. The results of the investigations (Farhi and Gutmann [1998], Spreeuw [1998], Kwiat, Mitchell, Schwindt and White [1999]) was summarized by Lloyd [2000]: Quantum search without entanglement is compared with classical search using waves. Classical devices that rely on waves and interference can also give a ^/N speed-up over classical devices that probe a system using particles alone. However, any search methods without entanglement require exponentially greater resources than the one using entanglement. 3.2.2.2.4. Other quantum algorithms. Abrams and Lloyd [1999] proposed a new polynomial-time quantum algorithm that uses the quantum fast Fourier transform to find eigenvalues and eigenvectors of a local Hamiltonian. This algorithm can be applied in cases (commonly found in ab initio physics
0{VN)
means asymptotically less than a constant times \//V.
1, § 3]
Quantum information
69
and chemistry problems) for which all the known classical algorithms require exponential time. A similar subroutine was previously described by Kitaev [1995] and refined by Cleve, Ekert, Macchiavello and Mosca [1998]. These proposals returned to Feynman's initial idea that a quantum computer might be useful for simulating other quantum systems, the Hilbert space size of which grows exponentially with the number of particles. At the same time, many similar mathematical problems are still waiting for solution. These include, for instance, combinatorial problems and, in particular, the traveling salesman problem (Ore [1962]) to find the shortest path connecting n points with known distances between pairs of points, so that each point is passed only once - a problem that still belongs to the class of NP-complete problems. Other problems of this type are calculations of the optimal way to supply shops with goods and consumers with electricity, to build ring electric communications, and so on. Searching for new algorithms to solve these problems by means of quantum computations is one of the most significant challenges of quantum information theory. Another major challenge is the correction of errors generated in the course of computation. Since quantum states are highly sensitive to external perturbations, the correction of errors in quantum computers is much more important than in classical computers (see § 4.4). 3.2.2.3. Quantum computers and physical problems. At present, the creation of a quantum computer is primarily a physical problem. One of the difficulties is the fast decay of superposition states, with their turning into mixed states. This process is called decoherence, and the analysis of its nature resolves the Schrodinger-cat paradox (see § 4). The effect of decoherence imposes restrictions on the physical elements used in quantum computers: the coherence times of quantum states should exceed the calculation time. Hence, the two possible ways of avoiding coherence decay are to find a quantum system isolated from the surroundings or to increase the coherence time artificially. The possible types of isolated quantum systems are summarized in table 2. Isolation of field quantum systems, modes of an electromagnetic field, is possible in high-g microcavities of optical (Kimble [1994]) and microwave (Raithel, Wagner, Walther, Narducci and Scully [1994]) regions. Such cavities, a few millimeters in size, allow coherence of superposition quantum states to be maintained for times from seconds to microseconds, with the number of photons per mode varying fi-om one to a hundred (Davidovich, Brune, Raimond and Haroche [1996]). Another promising method of field isolation is by using surface modes like the "whispering galleries" mode on microspheres of synthetic silicon (Braginsky, Gorodetsky and Ilchenko [1989]). For these modes
70
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Table 2 Localization of single quantum system Field
Matter
Microcavities
Beams
optical microwave
Ion traps Paul trap end-cap quadrupole ring traps Laser traps
Cavities for "whispering gallery" modes Photonic crystals
Naturally isolated systems molecules in amorphous and polycrystalline matrices impurities in crystals molecules in biological structures (investigation method: single-molecule spectroscopy) Quantum dots Nuclear spins of molecules
it was possible to achieve a quality factor of the order of 10^-10^^, which corresponds to a coherence time of 1-0.1 |is (Collet, Lefevre, Brune, Raimond and Haroche [1993]). A new isolation method is to use three-dimensional periodic dielectric structures, called photonic crystals (John [1987], Yablonovitch [1987]), which perfectly "confine" photons from certain frequency bands. The localization of photons in photonic crystals is so high that a single atom interacting with a photonic crystal should manifest suppression of spontaneous decay and inversion-fi-ee generation of coherent monochromatic sub-Poissonian radiation (Kilin and Mogilevtsev [1992, 1993]). Among several candidates for photonic crystal materials, the most promising at present is synthetic opal (Bogomolov, S.V Gaponenko, Germanenko, Kapitonov, Petrov, N.V Gaponenko, Prokofiev, Ponyavina, Silvanovich and Samoilovich [1997], Romanov, Johnson and De La Rue [1997]). The isolation of single massive particles, such as atoms, molecules, and ions, was historically preceded by one-dimensional isolation in beams (table 2). Other isolation (localization) methods include the following: (1) Quadrupole ion traps, called Paul traps (Fischer [1959]) of various configurations, which can keep a single ion (endcap trap; Schrama, Peik, Smith and Walther [1993]) illuminated by laser beams (Hoffges, Baldauf,
1, § 3]
(2)
(3)
(4)
(5)
Quantum information
71
Lange and Walther [1997]) or several ions (ring quadrupole traps; Birkl, Kassner and Walther [1992]). The last type of trap is also considered as a possible realization of quantum registers (Cirac and Zoller [1995]). A two-bit quantum computer was successfully realized in an experiment with a single cooled beryllium ion (Monroe, Meekhof, King, Itano and Wineland [1995]). Optical traps for neutral atoms (Minogin and Letokhov [1986], Wineland, Wieman and Smith [1994]). Observation of Bose-Einstein condensation (Anderson, Ensher, Matthews, Wieman and Cornell [1995]) suggests that this object also can be useful for quantum computation. Methods of matrix isolation of molecules in polycrystalline and amorphous media (Basche, Moerner, Orrit and Wild [1996]) and gels (Dickson, Norris, Tzeng and Moerner [1996]); impurity centers in crystals (Gruber, Drabenstedt, Tietz, Fleury, Wrachtrup and von Borczyskowski [1997]); and molecules in spatially organized structures such as DNA (Wennmalm, Edman and Rigler [1997]), proteins (Dickson, Cubitt, Tsien and Moerner [1996]), and photosynthetic antenna complexes (Tietz, Chekhlov, Drabenstedt, Schuster and Wrachtrup [1999]). Considerable progress in the study of isolated molecular systems is due to the rapid development of single-molecule experimental and theoretical spectroscopy (Pirotta, Bach, Donley, Renn and Wild [1997], Kilin, Nizovtsev, Berman, von Borczyskowski and Wrachtrup [1997], Kilin, Maevskaya, Nizovtsev, Shatokhin, Berman, von Borczyskowski, Wrachtrup and Fleury [1998]). Quantum dots (Ekert and Jozsa [1996]). A great potential for optical or electronic confinement on a subwavelength size is demonstrated by a unique technique of heterostructures creation in semiconductors (see Ebeling [1996]). As a promising object for quantum computation, one can use spin molecules, which are considerably isolated from the surrounding due to the screening effect. In this case, coherence times can reach several seconds. Kane [1998] proposed to encode information into the nuclear spins of donors (phosphorus, ^^P) in a doped silicon electronic device. Logical operations on individual spins are performed using externally applied electric fields, and spin measurements are made using currents of spin polarized electrons. For this purpose Wrachtrup, Kilin and Nizovtsev [2001] proposed the use of ^^C-nuclear spins located at the nearest-neighbor sites of the NV center in diamond. Individual defect centers in diamond are localized by means of single spectroscopy techniques (Gruber, Drabenstedt, Tietz, Fleury, Wrachtrup and von Borczyskowski [1997]). The center is photostable and has a well-defined structure (Drabenstedt, Fleury, Teitz, Jelesko, Kilin,
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Nizovtsev and Wrachtrup [1999]). As a result of the hyperfine coupling of the ^^C-nuclear spins with the electron spin, the nuclear spins located near the defect center become inequivalent in the presence of an external magnetic field; that is, their individual transitionfi-equenciesbecome distinguishable. Note that a basic difference exists between computations on isolated quantum objects and on quantum objects distributed in some medium. This is analogous to the difference between an experiment with an ensemble of objects and an ensemble of experiments with a single object. It is the NMR bulk-ensemble computation, first proposed by Cory, Fahmy and Havel [1997] and Gershenfeld and Chuang [1997], which tries to work with an ensemble of quantum registers composed of nuclear spins of molecules. Each molecule in the presentation can be considered as a natural elementary quantum computer, but many similar molecules occur in the observation volume. Therefore, the prepared and measured states of the NMR bulk system are mixed states similar to that described by eq. (3.37). As discussed above, the state (3.37) is separable for the usual experimental parameters (x = (hWs/kT) N/2^ ^ 2 • 10"^). Therefore, the present experiments with NMR-bulk systems (see Schack and Caves [1999], for the reference) could be considered as an important implementation of the universe quantum operations, which are necessary for any spinlike quantum processors. For example, Chuang, Vandersypen, Zhou, Leung and Lloyd [1998] were able to perform all unitary quantum operations with nuclear spins of a chloroform molecule that are necessary for implementation of the DeutschJozsa algorithm (Deutsch and Jozsa [1992])^. Another bulk type of systems was proposed for the creation and operation with a macroscopic entangled atomic ensemble. Hald, Sorensen, Schori and Polzik [1999] showed that in the process of propagation of near-resonant light through an optically thick V-type atomic medium, quantum features of light such as squeezing and entanglement can be transferred onto the atomic collective excited spin state. Among other physical problems related to the idea of quantum computations, let us mention the search for physical processes realizing logical operations. The XOR operations can be performed by means of single ions interacting
^ The algorithm can be explained with the help of the well-known game where one player has a coin in his hand and the other player is asked to state whether this coin has two different sides or not. Clearly, having seen both sides of the coin permits a correct answer. But is it possible to give the correct answer having seen only one side of the coin? Modeling such a situation with nuclear spin states of a chloroform molecule, the authors of the paper showed how the answer can be obtained in one run of the quantum processor. They proposed to use the spin states of hydrogen nuclei as an indicator of which side of the coin had been looked at (spin up or down), and the spin states of the carbon nucleus as an indicator of the result of observation.
1, § 3]
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73
with microwave fields in cavities (Kilin and Krinitskaya [1991, 1993], Turchette, Hood, Lange, Mabuchi and Kimble [1995], Turchette, Wood, King, Myatt, Leibfried, Itano, Monroe and Wineland [1998]) or ions oscillating in traps (Cirac and Zoller [1995], Monroe, Meekhof, King, Itano and Wineland [1995]). A promising method of dynamical monitoring of quantum tunneling is by means of laser radiation (Kilin, Berman and Maevskaya [1996]). A recent experiment by Rauschenbeutel, Nogues, Osnaghi, Bertet, Brune, Raimond and Haroche [1999] demonstrated coherent operation of a tunable quantum phase gate acting on qubits carried by a single Rydberg atom and microwave field on a single photon level in a high-Q cavity. The gate operation and nondestructive measurement of a single photon (Nogues, Rauschenbeutel, Osnaghi, Brune, Raimond and Haroche [1999]) require an extremely strong matter-radiation coupling, analogous to the nondemolition measurements in optics based on high values of nonlinearities. If optical fields are not only used for transmission but also for logical operations, it is important to develop methods of measuring their quantum states, which can have forms of complex superpositions. These possibilities are provided by quantum tomography (Smithey, Beck, Raymer and Faridani [1993], Leonhardt, Munroe, Kiss, Richter and Raymer [1996]) and various methods of quantum nondemolition measurements (Braginsky and Khalili [1992]). In devices operating with photon-number field states, such as, for instance, nodes of a quantum information network, one faces a set of problems related to subPoissonian field generation (Rarity and Tapster [1996]). Experimental methods of generating sub-Poissonian fields are based on using either unitary transformations (Kilin and Horoshko [1995]) or nonunitary (projection) transformations (Golubev and Sokolov [1984], Yamamoto, Imoto and Machida [1986], Fofanov [1989], Yamamoto, Machida, Saito, Imoto, Yanagawa, Kitagawa and Bjork [1990], Horoshko and Kilin [1994, 1997a,b], Jann and Ben-Aryeh [1997]). In 1999 two successful realizations of single photon sources were reported. Kim, Benson, Kan and Yamamoto [1999] demonstrated a single-photon turnstile device that generates a regulated stream of photons, where the time of interval between single photons is beyond the Poisson limit. Brunei, Tamarat, Lounis and Orrit [1999] realized a single photon source using the methods of single molecule spectroscopy and a rapid adiabatic passage technique. Sometimes the quantum objects considered as candidates for quantum computation cannot be properly isolated, and the errors caused by their interaction with the environment destroy the quantum coherence. In these cases one can use various methods of quantum error correction, such as increasing the number of information channels with further correction based on some protocol (Shor [1995], Ekert and Macchiavello [1996]), regularization of the interaction
74
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with the environment (Zurek [1981, 1982, 1991], Agarwal [1987], Garraway and Knight [1994a,b], Gerry and Hach [1993], Filho and Vogel [1996], Poyatos, Cirac and Zoller [1996]), feedback methods (Horoshko and Kilin [1997a,b], Kilin and Horoshko [1997, 1998], Vitali, Tombesi and Milbum [1997], Kilin, Horoshko and Shatokhin [1998]), and passive methods (Zanardi and Rasetti [1997]). All these methods solve the decoherence problem, which was illustrated so brightly by the Schrodinger-cat paradox. § 4. The problem of decoherence Some general features of the decoherence process are manifested as a fast transformation of a pure state into a mixed state because of the interaction between the quantum system and the surroundings. As an isolated quantum object, let us consider a harmonic oscillator which can represent a field in a cavity or an ion oscillating in a trap. Let the initial state of the oscillator be a superposition of two coherent states, both corresponding to the same complex amplitude but with opposite phases. We obtain an example of the Schrodinger-cat state (fig. 24). To make this object explicit, one can use Wigner's quasiprobability fiinction W(I3)= ~ Id'^Tr(pcxp(^(^-n-^'^ia-m
(4.1)
Its two-dimensional plot contains complete information about the wave fiinction of the object represented by the harmonic oscillator (a photon, a phonon, or any
(b)
Size of the 'cat' Fig. 24. (a) Wigner function of the superposition formed by two coherent states with the phases differing by jr. The two peaks correspond to the coherent states \a) and | - a). The distance between the peaks determines the size of the Schrodinger cat; that is, it indicates how macroscopic the state is. (b) The projection of the Wigner function. A specific feature of the Wigner function for the state I a) + exp{i6) | - a) is the interference part at the center of the phase plane. Because of the quantum nature of the state, there are points where its Wigner function takes negative values.
1, § 4]
The problem of decoherence
: (a)
i^
Wl9immlS^
75
(c)
'^WSMSXKBS^
(d)
Fig. 25. Evolution of the Wigner function for a quantum harmonic oscillator with damping. The initial state of the oscillator is |a} + exp(iO) | - «). The Wigner function is shown for a = 2 at the following values of/: (a) 0; (b) 1/(16y); (c) 1/(4 y); (d) l/y. Decoherence manifests itself in the fast decay of the interference part of the Wigner function in the course of relaxation.
Other quantum system). For classical states this function is equal to the joint probability distribution function in the variables "coordinate x-momentum p'' (J3 = x + ip). In particular, for the state 11/;+) = N (la) + e'^\ - a))
N'^ = 2 (l + cos 0exp(-2|a|^)),
(4.2)
the Wigner function W(I3) has two maxima localized at points P = ±a and indicating the probabilities of finding the system in the states \a) or \-ot). In addition, an interference structure is at ^ = 0, which in some viewpoints takes negative values. This structure appears due to the quantum interference terms \a) {-a\ exp~^^ + \-a) {a\ exp^^ in the density matrix of the state (4.2). This structure indicates that the state is nonclassical. Relaxation caused, for instance, by the escape of photons from the cavity with the rate y, leads to a specific change in the state of the oscillator: first, the interference part disappears and the superposition state turns into a mixed state, and second, the mixed state gradually becomes the vacuum state (fig. 25). Moreover, the rate t^lcoh "" 2 / |a|^ at which the interference terms decay is higher, the larger the size of the state (4.2) determined by the amplitude a. This feature of relaxation, first pointed out by Zurek [1981], explains why superposition states are easily observed in the microscopic world but are never observed for macroscopic objects, that is, why we never observe the superposition of a dead and an alive cat. However, this fact still offers no solution to the problem of avoiding decoherence. To find the solution, the relaxation process should be investigated in detail. 4.1. Relaxation as a nonunitary evolution of a state; quantum reservoirs engineering Any kind of relaxation is due to the interaction of the system with the reservoir, which is an object with many degrees of freedom and a broad continuous energy
76
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[1, § 4
spectrum. This provides a unidirectional transfer of excitation from the system to the reservoir. A standard model for the reservoir is a large number of harmonic oscillators with distributed eigenfrequencies cOi and creation and annihilation operators b'^^, bf. The states of the system and the reservoir, initially independent, become entangled because of the interaction. As a result, the initial superposition state of the system loses its individuality and becomes a mixed state. The details of this transition depend on the specific form of the interaction between the system and the reservoir. Suppose that the quantum system, represented by an oscillator, interacts with the reservoir by means of the Hamiltonian //.nt = A(a, a^) r +A^(a, a^) T,
(4.3)
where A(a,a^) is a function (nonlinear in the general case) of the creation and annihilation operators for the oscillator, and F = h'^.gibi is a linear function of the reservoir operators. Although Hamiltonian (4.3) is not universal, it still describes numerous physical situations. For linear interaction one has A(a, a^) = a, and the relaxation is described by the equation p= jY {2apa^ -a^ap-pa^a)
(4.4)
for the density matrix of the oscillator a averaged over the initial vacuum state of the reservoir. The constant y = Jtp(a)) Igico)]^ is the energy relaxation rate, and p(co) is the density of states for the reservoir. Solving eq. (4.4), we can obtain the density matrix as a function of time, and predict possible results of measurements performed over the oscillator a. For instance, the initial superposition |i/^+) = N (|a) + exp'^ \-^)) evolves, according to eq. (4.4), as p(t)='-(\ar){ar\ +
\-ar){-a,\)
+ ^ exp-2|«l'^^-^^P^-'^^)>(exp'^ l-a;) (a,| + exp"*^ \at){-at\% This dependence describes a slow decrease in the amplitude at = a exp"^^""^ and a fast transition (with the rate t^^^^^ = 2y|ap) into a mixed state. In the general case of a nonlinear interaction between the oscillator a and the reservoir, relaxation is described by the kinetic equation p=\r{[A,pA^]
+ [Ap,A^]).
(4.6)
From this equation, in combination with the Hamiltonian (4.3), it follows that the relaxation of the oscillator a strongly depends on the form of the interaction A(a, a^). In fact, the eigenstates | W)A of the interaction operator
The problem of decoherence
l.§4]
11
Table 3 Various "system-reservoir" interactions for quantum reservoir engineering Type of interactions
"Pointer basis"
Stationary state
References
A = a + a^ ^ X
Coordinate eigenstates
Vacuum
Zurek [1981, 1982, 1991]
A = a^
Even and odd coherent states
Vacuum
Gerry and Hach [1993], Agarwal [1987]
A = {a + a)(a -a) Even and odd coherent states
Even and odd coherent states
Garraway, Knight [1994a,b], Filho and Vogel [1996], Poyatos, Cirac and ZoUer [1996]
A = a^a
Fock states
Vacuum
Poyatos, Cirac and Zoller [1996]
A = a{a^a - n)
Fock states
Fock states
Poyatos, Cirac and Zoller [1996]
A = e^^«^«fl
Yurke-Stoler superposition state
Vacuum
Horoshko and Kilin [1997a], Kilin, Horoshko and Shatokhin [1998], Kilin and Horoshko [1997]
A(a,a^) remain unperturbed by the interaction with the reservoir, and form the so-called "pointer basis" (Zurek [1991]), which determines the specific form of the relaxational evolution. Hence, by using various forms of the interaction operator A{a, a^), one can create various "pointer bases" and thus vary the relaxation process and, moreover, obtain various stationary states as a result of the relaxation. Several well-known examples of "quantum reservoir engineering" are given in table 3. In recent years, techniques have been realized to generate mesoscopic superpositions of motional states of trapped ions (IVlonroe, IVleekhof, King and Wineland [1996]) and of photon states in the context of the cavity QED (Brune, Hagley, Dreyer, Ivlaitre, IVIaali, Wunderlich, Raimond and Haroche [1996]), where decoherence through coupling to ambient reservoirs and the sensitivity of the rate of decoherence to the size of the superposition were observed. IVlyatt, King, Turchette, Sackett, Klelpinski, Itano, IVlonroe and Wineland [2000] went further, and studied the decoherence into an engineered quantum reservoir, using laser cooling techniques to generate an effectively zero-temperature bath. Note that by replacing the harmonic oscillator by a set of A^ two-level systems representing a quantum register, one can find a subspace of the register states that is completely orthogonal to the states of the reservoir. Such states will not be perturbed by the reservoir. Several special cases have been considered by Zanardi and Rasetti [1997]. Alternatively, a strong correlation exists between the states
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[1? § 4
of a single two-level atom and the reservoir, for instance, radiation. Because of this correlation, the atom dynamics can be varied by changing the state of the field. Thus, if one of the reservoir modes (a resonance mode) is initially in the Yurke-Stoler state, |a) + i | - a ) , and the other reservoir modes are in the vacuum state, the entangled nature of joint states leads to the effect of quantum instability, which manifests itself in the exponential growth of the transition dipole moment of the atom (Kilin and Shatokhin [1996, 1997a]), instead of the usual Rabi oscillations. 4.2. Relaxation as a quantum stochastic process; purity of conditional states Relaxation of the oscillator a also can be considered as a result of averaging quantum stochastic processes of excitation transfer fi*om the oscillator a to the reservoir oscillators. In each process of this kind, such as, for instance, the escape of a photon fi-om a cavity, a quantum is passed from the oscillator a to the reservoir. An instant change results, which is a reduction of the state of the oscillator a. The absence of quantum exchange between the acts of reduction, which occur at random instants of time, does not mean that the state of a remains constant. In fact, the longer one waits for the next quantum to be emitted, the higher the probability that oscillator a will be in the ground state; hence, its amplitude should decrease during such periods. Such a sequence of reductions and intervals of nonunitary evolution is studied by the theory of continuous quantum measurements or quantum jumps (Davies [1976], Holevo [1982], Kilin [1990]). In the case of relaxation with linear interaction, this sequence of random events is described by the conditional state vector of the oscillator a after transmitting exactly n quanta to the reservoir at times ^1,^2? • •» ^n belonging to the interval [0, t) IV^cond(O) = y"S(t,tn)aS{tn,t„-i)a-'aS(tuO)\xl^(0)),
(4.7)
where S(ti,ti_i) = Qxp[-ya^a(ti ~ ti-\)/2] is the nonunitary operator of the evolution between two successive reductions at ^/_i and tf. Emission of quanta at times {ti} results in the reduction of the state. If |V^(0)) = |t/^+), this effect a (|a) ± exp^^ | - a ) ) = a (\a)
=F
exp^^ | - a ) )
(4.8)
increases the relative phase 6 by JT, but the state remains a pure superposition state. The nonunitary evolution S{ti,ti-\) between quantum emissions reduces the amplitude a exponentially, so that the conditional state IV^cond(O) = N(yay (|aexp-^^/2^ + (-l)'^ exp^^ | - aexp-^^^^))
(4.9)
1, § 4]
The problem of decoherence
79
remains pure throughout the evolution period, and its coherence is preserved. Conservation of purity for conditional states in the course of relaxation does not contradict the preceding consideration of the density matrix decoherence: if the conditional density matrix |V^cond(0)(V^cond(Ol is averaged over random realizations of quantum emissions, we immediately obtain result (4.5), which means that the information about the state of the system is partially lost. It is also evident that the first emission event occurring after the average waiting time equal to the decoherence time, t^l^^^ = 2 / |a|^, is sufficient to erase the quantum interference terms.
4.3. Error correction by means of feedback Relaxation considered as a quantum stochastic process also shows that although decoherence is a serious obstacle for quantum information processing, it can still be overcome. To correct errors and uncertainties caused by the interaction of the quantum object with the surroundings, it is not necessary to know the state of the surroundings. It is sufficient to control the times of quantum emissions from the object to the surroundings and to return the system after each reduction to its initial state by means of some unitary transformation (Horoshko and Kilin [1997a,b], VitaH, Tombesi and Milburn [1997] Kilin and Horoshko [1997, 1998], Kilin, Horoshko and Shatokhin [1998]). For the case of Yurke-Stoler coherent states |a) +i | - a ) , this protocol of error correction should be carried out by rotating the phase of the oscillator a by 180'' (Horoshko and Kilin [1997a]). Then the sequence of events in the quantum stochastic process would consist of alternating stages of nonunitary evolution (the absence of emissions), reduction, and phase variation, IV^cond(O) = y^'SiUtn) exp^^-'^aS{tn.tn-x) cxp^^^'^a-"exp^^«'^aS{hMn^^)(4.10) Due to the correcting procedure, which can be realized by the back action on the oscillator a (fig. 26), both the conditional and the unconditional states of the oscillator, obtained by averaging over random realizations of emissions, remain pure superpositions, \\l){t)) = (|aexp-^^/2^ + i |-aexp-^^^2)) / ^
^4^1)
In this case the only sign indicating the existence of relaxation is the exponential amplitude decay (energy relaxation).
80
Quanta and information Modulator
_
[1, § 4
Detector
Fig. 26. Slowing down decoherence by means of an error-correcting feedback. The intracavity field, initially in the Yurke-Stoler state, is continuously detected by a high-efificiency detector. Each photocount is converted into a signal on the phase modulator, which changes the phase of the field by JT. If this procedure is repeated continuously, the superposition state is preserved as long as some photons are in the cavity.
Note that the density matrix of state (4.11) satisfies an equation similar to eq. (4.4):
where the nonlinear interaction operators Ajt = exp'^^ "^ a and A^ = «+ exp~^^^ ^ belong to the class of generalized annihilation/creation operators A(p = exp^^^ ^ a, A'^ = a^ exp"*^'' ^, whose eigenvectors, which are generalized coherent states, have useful quantum properties (Kilin, Horoshko and Shatokhin [1998]). The experimental scheme for the proposed decoherence correction has been demonstrated (Kilin and Horoshko [1997,1998]) (fig. 26). The intercavity field, initially in the Yurke-Stoler state, is continuously registered by a high-efficiency detector. From each photocount of the detector, a signal is fed through a feedback to the phase modulator, which changes the field phase by jr. If this procedure is continued, the superposition state in the cavity is conserved as long as some photons are in the cavity. Suppression of decoherence by means of feedback is a universal method and can be applied to all systems with continuously controllable losses (local nodes of a quantum computer). At present, in addition to the work mentioned above, some other suggestions have been made along these lines (Vitali, Tombesi and Milbum [1997]). Agarwal [2000] demonstrated that considerable slowing down of decoherence can be achieved by fast frequency modulation of the system-heat-bath coupling. If the control of losses is difficult, as in the case of transmission through quantum channels, one should use quantum errorcorrecting methods based on duplicating transmitted qubits (Shor [1995], Ekert and Macchiavello [1996]). 4.4. Hamming code and quantum error corrections From previous examination, it can be concluded that decoherence must be
1, § 4]
The problem of decoherence
81
a unitary process that entangles a system of interest, say a qubit, with the environment. For example, the qubit states |0),|1) and the environment states (not generally orthogonal, not normalized) become entangled due to the interaction: the initial unentangled state |e/)|0) or |^/)|1) transforms like k-)|0) -^ M | 0 ) + |eoi)|l),
|e,)|l) ^ |6io)|0) + |en)|l).
(4.13)
The averaging over environment states transforms a pure initial state of the qubit to a mixture. Classical information theory can successfully both recognize and correct errors of that kind in a string of bits influenced by noise by means of error correcting codes (see § 2.2.2.3). The idea of adapting this method to the quantum situation independently led Shor [1995], Steane [1996a,b] and Calderbank and Shor [1996] to the powerful quantum error correction (QEC) method. The theory of QEC was further advanced by Ekert and Macchiavello [1996], Bennett, DiVincenzo, Smolin and Wootters [1996], and Knill and Laflamme [1997]. The paper by Bennett and colleagues describes the optimal 5-qubit code discovered, also independently, by Laflamme, Miquel, Paz and Zurek [1996]. Gottesman [1996] and Calderbank, Rains, Shor and Sloane [1997] discovered a general group-theoretical framework, introducing the important concept of the stabilizer, which also enabled many more codes to be found (Steane [1996c]). Quantum coding theory reached a further level of maturity with the discovery by Shor and Laflamme [1997] of a quantum analog to the Mac Williams identities of classical coding theory. The idea of QEC is the same as in a classical EC, namely, to encode information in distinguishable strings of bits. For example, instead of 1 qubit, information could be stored in 2 orthogonal superposition states composed of 7 qubits, each state in the superposition being taken from 2^ Hamming code symbols generated by the Gy matrix (eq. 2.21) |0)H = 10000000) + 11010101) + 10110011) + 11100110) + 10001111) + 11011010) + 10111100) + 11101001), |1)H = l l l l l l l l ) + 10101010) + 11001100) + 10011001) + 11110000) + 10100101) + 11000011) + 10010110). If one of any 7 qubits is changed due to the interaction with the environment (and this is the most probable case), then, according to eq. (2.22), one can localize this qubit without affecting the others by the syndrome extraction operation H^ acting on the environmentally influenced state of 7 qubits. The idea of QEC has opened up new possibilities on the way to xQdX fault-tolerant quantum computing (Steane [1998], Preskil [1998]).
82
Quanta and information
[1? § 5
At this point we can allow ourselves to imagine a picture of allegories with a big lake of quantum computational resources blocked by a dump, with the name of "decoherence". Three rivers supply the lake with their own resources: quantum physics provides an understanding of the entanglement and connection with experiments, information theory indicates that we can count information and supplies clever coding, and mathematics formulates the problems made intractable by means of modem computers. Some small holes in the dump give access to the lake resources: quantum error corrections, some proposals combining methods of solid state physics, single molecule spectroscopy, NMR techniques, cavity quantum electrodynamics, and new methods of field confinement. These small sources are transformed into the river which will possibly lead to a fixture low-cost quantum computer.
§ 5. Conclusions Despite the famous names and the long period separating contemporary physicistsfi-omthe basic paper by Schrodinger [1935] and the classical paper by Shannon [1948], the real development of quantum information, with its practical importance for human society, is being started only now. Quantum informatics is developing remarkably rapidly. The scientific race for new achievements in quantum information has involved, joined, and enriched several fields of science, such as discrete mathematics and quantum mechanics, computer science, and quantum optics. Moreover, it has given practical importance to studies that previously seemed to be farfi-ompractical applications, such as the investigation of single quantum objects. All this work stimulates such a rapid development of new approaches, methods, and materials that it is hardly possible to keep abreast of current publications. Some excellent review articles and introductory publications have appeared, reflecting a growing understanding in the field of quantum information (Lloyd [1993], DiVincenzo [1995a], Bennett [1995], Ekert and Jozsa [1996], Steane [1998]). Some special issues of journals are also important milestones in the stream of publications (J. Mod. Opt. 41 (1994) no. 12; 44 (1997) no. 11/12; Philos. Trans. R. Soc. London A 355 (1997) 2215-2416; Proc. R. Soc. London A 454 (1998) 257-482; Phys. Scripta T 76 (1998); Opt. Spectrosc. 87 (1999) no. 4/5). A usefiil source of information is provided by electronic publications and e-preprints available on the Internet earlier than the corresponding hard copies. However, this source of "quantum information" produces new papers incredibly rapidly. For example, the Los Alamos Archive www. l a n l . gov (quant-ph) has been publishing more
1, § 5]
Conclusions
83
than a hundred papers per month for the last two years. To those who are critically assessing this information boom, it may be interesting to read Shannon's warning, published in his short paper "The bandwagon" (Shannon [1956]), in which he notes that there is no single key for all secrets of nature. Even if we understand and adopt these critical warnings, however, we can state that unification of two previously separated fields, namely, quantum physics and classical information theory, has become a reality. It is hard to predict all possible resuhs of the unification. At present, in its initial stage, we can see that once again Nature is giving us a lesson, presenting a new physical resource that has demanded for its description notions and methods that were not available in the huge arsenal of mathematical methods. This resource of quantum entanglement revolutionizes our understanding of the world and opens a window for the unpredictable power of Hilbert space of distant quantum objects. This power promises to be the basis for the next generation of computers and will be able to solve many mathematical problems presently untouchable. In addition, the number theory has been enriched due to the introduction, from physics, of the notion of the qubit - a new measure of quantum information instead of the classical bit. This may possibly be the third revolution in number theory after P3^hagoras' adventure of the irrational numbers and the introduction of complex numbers by Gauss and his contemporaries, a revolution that concurs with Fourier's observation that the investigation of Nature is the richest source of mathematical adventures. To understand the importance of quantum entanglement and describe its potential, some concepts of information theory and the method of reasoning have been used, and an explanation of the wavefixnctionhas again been stressed. The information science language has become an important part of the description of quantum world objects. This kind of thinking and presentation of reality has revived old and fiindamental problems such as symmetries and separability of quantum systems. For example, the Pauli principle forbids using all power of Hilbert space of interacting and, therefore, separately nonperturbable, spins. We understood that entangled systems could serve as a new resource for information storage and handling, but at the same time, the quantum entanglement of all with all creates the primary obstacle to quantum computation, that of decoherence. The solution to the problem of quantum computers creation lies in our ability to find methods for the decoherence harnessing. It seems now that meeting this important challenge will require a major consolidated of effort. In conclusion, many indications have emerged that a new physical view on Nature is being formed. One useful point of view suggests that Nature communicates information encoded by means of a number of "languages", one
84
Quanta and information
[1
of which is quantum mechanics. It is the quantum information that tries to determine, as a branch of science, the structure of this language, which can help us to decode the messages of Nature and to adopt the power of quantum coding for the practical benefits to society.
Acknowledgements The author is grateful to Emil Wolf and Jan Perina for their suggestion to write this review; to PA. Apanasevich, D.B. Horoshko, VN. Shatokhin, A.P Nizovtsev, T.M. Maevskaya, D.S. Mogilevtsev, T.B. Karlovich, and V.A. Zaporozhchenko for their cooperation; and to H. Walther, P Berman, M. Raymer, G. Bjork, C. von Borczyskowski, and J. Wrachtrup for fiiiitful discussions. Gratefial acknowledgment is given for partial financial support fi-om the National Science Foundation, United States (grant NSF9414515 "Spectroscopy of single molecules"); Volkswagen Foundation (grant 1/72171 "Two-level systems in single-molecule spectroscopy"); International Association, European Communityx (grant 96 167 "Generation of single photons and quantum states synthesis"), and the National Research Council, United States (Twinning program "Quantum tomography and other reconstructive measurement methods in quantum optics").
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E. Wolf, Progress in Optics 42 © 2001 Elsevier Science B.V All rights reserved
Chapter 2
Optical solitons in periodic media with resonant and off-resonant nonlinearities by
Gershon Kurizki*, Alexander E. Kozhekin**, Tomas Opatrny*** Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel
and
Boris A. Malomed Department of Interdisciplinary Studies, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
* E-mail:
[email protected] ** Present address: Institute of Physics and Astronomy, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark *** Present address: Friedrich-Schiller-Universitat Jena, Theoretisch-Physikalisches Institut, Max Wien Platz 1, 07743 Jena, Germany and Palacky University, Faculty of Natural Sciences, Svobody 26, 77146 Olomouc, Czech Republic 93
Contents
Page § 1. Introduction
95
§ 2.
Solitons in Bragg gratings with cubic and quadratic nonlinearities
99
§ 3.
Self-induced transparency (SIT) in uniform media and thin
§ 4.
SIT in resonantly absorbing Bragg reflectors (RABR): the model
films
105 109
§ 5. Bright solitons in RABR
120
§ 6.
Dark solitons in RABR
129
§ 7.
Light bullets (spatiotemporal solitons)
136
§ 8.
Experimental prospects and conclusions
139
Abbreviations
142
Acknowledgments
142
References
142
94
§ 1. Introduction The study of light-matter interactions in dielectric structures with periodic modulation of the refractive index has developed into a vast research area. At the heart of this area is the interplay between Bragg reflections, which block the propagation of light in spectral bands known as photonic band gaps (PBGs), and the dynamical modifications of these reflections by nonlinear light-matter interactions (see bibliography compiled by Dowling and Everitt [2000]). Three- or two-dimensional (3D or 2D) PBGs are needed in order to extinguish spontaneous emission in all possible directions of propagation, which requires the nontrivial fabrication of 3D- or 2D-periodic photonic crystals (Yablonovitch [1987, 1993]). For controlling strictly unidirectional propagation, it is sufficient to resort to PBGs in one-dimensional (ID) periodic structures (Bragg reflectors or dielectric multilayer mirrors). Illumination of the periodic dielectric structure at a PBG frequency in the limit of vanishing nonlinearity leads to exponential decay of the incident field amplitude with penetration depth, at the expense of exponential growth of the back-scattered (Bragg-reflected) amplitude. However, this reflection may weaken or cease altogether, rendering the structure transparent, when the illumination intensity and the resulting nonlinearity modify the refractive index so as to shift (or even close down) the PBG. The pulsed mode of propagation in nonlinear periodic structures exhibits a variety of fimdamentally unique and technologically interesting regimes: nonlinear filtering, switching, and distributed-feedback amplification (Scalora, Dowling, Bowden and Bloemer [1994a,b]). Among these regimes, we have chosen here to concentrate on the intriguing solitary waves existing in PBGs, known as gap solitons (GS), and solitons propagating near PBGs. A GS is usually understood as a self-localized moving or standing (quiescent) bright region, where light is confined by Bragg reflections against a dark background. The soliton spectrum is tuned away from the Bragg resonance by the nonlinearity at sufficiently high field intensities. There is also considerable physical interest in finding a dark soliton (DS) in the vicinity of a PBG, i.e., a "hole" of a fixed shape in a continuous-wave (cw) background field of constant intensity (Kivshar and Luther-Davies [1998]). The first type of GS was predicted to exist in a Bragg grating filled with a 95
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Optical solitons in periodic media
[2, § 1
Kerr medium, whose nonlinearity is cubic (Christodoulides and Joseph [1989], Aceves and Wabnitz [1989], Feng and Kneubuhl [1993]). Detailed theoretical studies of these Bragg-grating (Kerr-nonlinear) solitons (see De Sterke and Sipe [1994] for a review) were followed by their experimental observation (Eggleton, Slusher, De Sterke, Krug and Sipe [1996]) in a short (<10cm) piece of optical fiber with a resonant Bragg grating written on it. In theoretical considerations of solitons in a Bragg grating combined with Kerr nonlinearity, a formidable problem is their stability. Current experiments are conducted in short fiber pieces, which do not allow us to test the solitons' stability. Approximate and rigorous treatments of the stability problem (Malomed and Tasgal [1994], Barashenkov, Pelinovsky and Zemlyanaya [1998], de Rossi, Conti and Trillo [1998]) have demonstrated that the Kerr-nonlinear Bragg-grating solitons have instability regions in their fi-equency-amplitude parametric plane, far from the PBG-center, small-amplitude limit. The generation of a slow gap soliton in a Kerr-nonlinear periodic grating via Raman transfer of energy from the pump pulse to the Braggresonant Stokes components was recently proposed by Winfiil and Perlin [2000]. A Bragg grating with quadratic or second-harmonic-generating (SHG) nonlinearity can also give rise to a rich spectrum of solitons (Peschel, Peschel, Lederer and Malomed [1997], He and Drummond [1997], Conti, Trillo and Assanto [1997]). This model has been shown to possess a number of remarkable features. In particular, contrary to Kerr-nonlinear Bragg gratings, in the case of SHG nonlinearity the gap in which solitons may exist is partly empty, only part of it being filled with actually existing soliton solutions (Peschel, Peschel, Lederer and Malomed [1997]). While the above temporal-domain models pertain to light propagation in fibers, distributions of stationary fields in a planar (2D) nonlinear optical waveguide are governed by spatial-domain equations (Stegeman and Segev [1999]). In the planar waveguide, a Bragg grating can be realized as a system of parallel "scratches". The soliton spectrum of this model contains not only the fundamental single-humped solitons but also their two-humped bound states, which, quite unusually, turn out to be dynamically stable (Mak, Malomed and Chu [1998b]). Moreover, it possesses, besides the conventional GSs, also embedded solitons, which are isolated solitary-wave solutions within the continuous spectrum, rather than inside the gap (Champneys and Malomed [2000]). A principally different mechanism of GS formation has been discovered in a periodic array of thin layers of resonant two-level systems (TLS) separated by half-wavelength nonabsorbing dielectric layers, i.e., a resonantly absorbing Bragg reflector (RABR) (Kozhekin and Kurizki [1995], Kozhekin, Kurizki and
2, § 1]
Introduction
97
Malomed [1998], Opatrny, Malomed and Kurizki [1999]). The RABR has been found to allow, for any Bragg reflectivity, a vast family of stable solitons, both standing and moving (Kozhekin and Kurizki [1995], Kozhekin, Kurizki and Malomed [1998]). As opposed to the 2:7r-solitons arising in self-induced transparency, i.e., resonant field-TLS interaction in uniform media (McCall and Hahn [1969, 1970]), gap solitons in RABR may have an arbitrary pulse area (Kozhekin and Kurizki [1995], Kozhekin, Kurizki and Malomed [1998]). As shown below, GS solutions have been consistently obtained only in a RABR with thin active TLS layers. By contrast, an attempt (Akozbek and John [1998]) to obtain such solutions in a periodic structure uniformly filled with active TLS is physically unfounded, and fails for many parameter values, so that such a generalization remains an open problem (see § 4.3 below). An unexpected property of the RABR is that, alongside the stable brightsoliton solutions (Kozhekin and Kurizki [1995], Kozhekin, Kurizki and Malomed [1998]), this system gives rise to a family of dark solitons (DSs), a large part of which are stable (Opatrny, Malomed and Kurizki [1999]). While the existence of stable bright-soliton solutions along with wwstable DSs is a known feature of uniform SHG media (Kivshar and Luther-Davies [1998]), a RABR with thin active layers provides, to the best of our knowledge, the first example of a nonlinear optical medium in which stable bright and dark solitons exist for the same values of the model's parameters (albeit at different frequencies). It is also the first example of the existence of stable bright solitons alongside stable cw (background) solutions. Potential applications of GSs are based on the system's ability to filter out (by means of Bragg reflections) all pulses except for those satisfying the GS dispersion condition, as well as to control the pulses' shape and velocity. It would be clearly desirable to supplement these advantageous properties by immunity to transverse diffiraction of the pulse, i.e., to achieve simultaneous transverse and longitudinal self-localization of light in a RABR. This motivates a quest for multi-dimensional solitons that are localized in both space and time. The concept of optical multidimensional spatio-temporal solitons, alias "light bullets" (LBs), was pioneered by Silberberg [1990], and has since been investigated in various nonlinear optical media, with particular emphasis on the stability of LBs. For a SHG medium, the existence of stable two- and threedimensional (2D and 3D) solitons was predicted as early as 1981 (Kanashov and Rubenchik [1981]), followed by detailed studies of their propagation and stability against collapse (Hayata and Koshiba [1993], Malomed, Drummond, He, Bemtson, Anderson and Lisak [1997], Mihalache, Mazilu, Malomed and Torner [1998], He and Drummond [1998]). Recently, the first experimental
98
Optical solitons in periodic media
[2, § 1
observation of a quasi-2D spatiotemporal soliton in a 3D SHG sample was reported (Liu, Qian and Wise [1999]). The concept of dark LBs was proposed by Chen and Atai [1995]). Stable antidark LBs, i.e., those supported by a finite cw background, were predicted in a generalized nonlinear Schrodinger equation which contains third-order temporal dispersion (Frantzeskakis, Hizanidis, Malomed and Polymilis [1998]). As early as in 1984, simulations indicated that self-focusing of spatiotemporal pulses in a self-induced transparency (SIT) medium could result in the formation of a quasi-stable vibrating object (Drummond [1984]), which hinted at the possible existence of LBs. It has been demonstrated analytically (Blaauboer, Malomed and Kurizki [2000]) that uniform 2D and 3D SIT media can indeed carry stable LBs. This investigation has been extended to the case of RABR, wherein stable, transversely localized SIT solutions combining LB and GS properties are predicted (Blaauboer, Kurizki and Malomed [2000]). RABR with any Bragg reflectivity can support stable LBs, which are closely related to those in uniform SIT media (Blaauboer, Malomed and Kurizki [2000]). Twodimensional LBs supported by a combination of a Bragg reflector with SHG nonlinearity were theoretically investigated earlier (He and Drummond [1998]). Finally, we briefly mention a topic that is outside the scope of this review, namely, quantum bright solitons, which have been a subject of extensive studies in recent years. It has been established that solitons are superpositions of quantum states that correspond to clusters of photons bound together. Most of the initial activity in this area was devoted to Kerr-nonlinear fiber solitons (Lai and Haus [1989a,b], Wright [1991], Yudson [1985], Kartner and Haus [1993], Cheng [1991]). It was shown that optical fiber solitons have quantum analogs which are described by the quantum nonlinear Schrodinger equation (QNLSE). The motivation was to gain understanding of squeezing effects in soliton propagation (Carter, Drummond, Reid and Shelby [1987], Watanabe, Nakano, Honold and Yamamoto [1989], Drummond, Carter and Shelby [1989], Haus and Lai [1990], Rosenbluh and Shelby [1991], see also a review by Sizmann and Leuchs [1999]), two-photon binding effects (Deutsch, Chiao and Garrison [1992, 1993]), and fundamental limits imposed on communication systems employing solitons (Drummond, Shelby, Friberg and Yamamoto [1993], Chiao, Deutsch, Garrison and Wright [1993]). Multidimensional quantum solitons were predicted in Kerr and SHG waveguides (Drummond and He [1997], Kheruntsyan and Drummond [1998a,b, 2000]). The quantization of GSs has attracted considerable attention as well. A Betheansatz solution was given (Cheng and Kurizki [1995]) for quantum GSs consisting of pairwise interacting massive photons (in the effective-mass regime
2, § 2]
Solitons in Bragg gratings with cubic and quadratic nonlinearities
99
of PBGs propagation) in a Kerr-nonlinear ID periodic grating. A mechanism has been found for the creation of two-photon bound states by photons resonantly interacting with identical two-level atoms near PBGs in ID periodic structures (Kurizki, Kofman, Kozhekin and Cheng [1996]). A Bethe-ansatz solution for photons in a PBG material interacting with a single atom was obtained by Rupasov and Singh [1996a,b]. It has later been generalized to an extended many-atom periodic system, where quantum GSs involving pairs of photons and propagating inside a PBG have been found (John and Rupasov [1999]). This subject is of potential interest for quantum communications via entangled twophoton states. This review starts with a brief survey of solitons in Bragg gratings with cubic and quadratic nonlinearities (§ 2) and of self-induced transparency (SIT) in uniform media and thin films (§ 3). It then continues with the derivation of the model for phenomena similar to SIT in resonantly absorbing Bragg reflectors (RABR) (§ 4). Bright and dark solitons in RABR are discussed in § 5 and 6, consecutively. Light bullets in periodic resonantly absorbing media are treated in § 7. Finally, the prospects for experimental progress in this area are summarized in §8.
§ 2. Solitons in Bragg gratings with cubic and quadratic nonlinearities 2.1. Kerr nonlinearity Because the Bragg grating gives rise to very strong effective dispersion, its combination with various optical nonlinearities can create a rich variety of solitons, which, in most cases, are gap solitons (GSs), as their intrinsic frequency must belong to a gap in the spectrum of linear waves in the Bragg grating. Initially, a Bragg grating filled with a Kerr nonlinear medium, whose nonlinearity is cubic, was considered by ChristodouHdes and Joseph [1989], Aceves and Wabnitz [1989] and Feng and Kneubuhl [1993]. The corresponding system of two propagation equations for the right (forward)- and left (backward)-traveling field envelopes E^it, T) and £B{^, ^) with the self- and cross-phase-modulation cubic terms is a generalization of the known Thirring model (in the Thirring model proper, the self-phase modulation terms are omitted): i ^ + i ^ + (I^B|' + i|fFp)fF+fB
=0,
(2.1a)
i^-i^+(|^Fp+i|fBO^B+fF
=0.
(2.1b)
100
Optical solitons in periodic media
[2, § 2
Here the dimensionless time r and length t, are related to the physical time t and length z as t = 4T/(aia)c) and z = 4c^/(flia>c«o), where «o is the mean linear index of refraction, ai is its modulation (see eq. 4.1 below), and cOc is the central frequency of the band gap. Actual values of the electric fields can be obtained from the dimensionless quantities £Y,B upon multiplication by cOc\/a\no/(4SjTX^^^), where x^^^ is the third order susceptibility. In §4.1 we present a general derivation of the Maxwell equations in Bragg gratings. The existence of solitons in a given model is usually closely related to the modulational instability of a continuous-wave (cw) solution in the model (Agrawal [1995]): an array (chain) of bright solitons can be generated from the cw solution by modulational instability. In an optical fiber combining a Bragg grating and Kerr nonlinearity, all the cw states are modulationally unstable, as observed in a direct experiment (which also involved the polarization of light) by Slusher, Spalter, Eggleton, Pereira and Sipe [2000]. However, as was shown by Opatmy, Malomed and Kurizki [1999] (see § 6 below) a RABR gives rise to stable bright solitons coexisting with stable cw states, which is a very unusual property. Unlike the Thirring model proper, its optical version based on eqs. (2.1a) and (2.1b) is not integrable. Moreover, the equations lack any invariance with respect to the reference frame (i.e., the model is neither Galilean nor Lorentz invariant). Nevertheless, exact single-soliton solutions, which contain two independent parameters, viz., an intrinsic frequency co and the soliton's velocity u, were found in the above-mentioned works by Christodoulides and Joseph [1989] and Aceves and Wabnitz [1989], following the pattern of the Thirring-model solitons. At u = 0, the quiescent (standing) solitons completely fill the gap - 1 < co < +1 in the spectrum of the linearized equations (2.1a) and (2.1b), and the velocity takes all the possible values - 1 < u <-\-\ (recall that, in the present notation, the maximum group velocity of light is 1). Solitons with small frequencies have small amplitudes and are close to the classical nonlinear Schrodinger (NLS) solitons, while the ones with values of co^ closer to 1 are strongly different from their NLS counterparts; in particular, they are chirped, i.e., they have a nontrivial intrinsic phase structure. Many properties of these Bragg-grating solitons, as they are frequently called, were reviewed by De Sterke and Sipe [1994] (see also the special issue of Optics Express edited by Brown and Eggleton [1998]). Detailed theoretical studies were finally followed by the experimental observation by Eggleton, Slusher, De Sterke, Krug and Sipe [1996] in a short (<10cm) piece of optical fiber with a resonant Bragg grating written on it. This experiment requires the use of very powerful light beams, with an intensity comparable to the fiber's breakdown threshold.
2]
Solitons in Bragg gratings with cubic and quadratic nonlinearities 1
1
1
1
1
1
1
1
1
1
1
1 1
101
' 1 ' '
5
s o
0
n
i|i
1 1
-
If 1 '^ 5
J
^'
-
f'l
A. n
1
1
1
1
1
100
1
1
1
1
1
1
1
1
200
.
1 .
.
•trAr
300
Time (ps) Fig. 1. Experimental evidence for the existence of solitons in a nonlinear optical fiber with Bragg grating: the broad pulse is far from a Bragg resonance and has passed the fiber without interacting with the grating; the narrow one is a soliton shaped by Bragg-resonant interaction with the grating (see fijrther details in the work by Eggleton, Slusher, De Sterke, Krug and Sipe [1996]).
Despite this difficulty, propagating pulses with soliton-like shapes have been detected in the experiment, see fig. 1. It should be stressed that the above-mentioned exact soliton solutions to eqs. (2.1a) and (2.1b) take the simplest form in the case t; = 0, corresponding to a pulse of standing light. In reality, the soliton observed by Eggleton, Slusher, De Sterke, Krug and Sipe [1996] was moving at a considerable velocity. Generation and detection of zero-velocity solitons in nonlinear Bragg gratings remains an experimental challenge. Theoretical considerations of solitons in Bragg gratings with Kerr nonlinearity face the tough problem of soliton stability. The first approach to the stability problem, developed by Malomed and Tasgal [1994] was based on the variational approximation. Although this approach is not rigorous, it clearly demonstrates that, sufficiently far from the above-mentioned NLS (small-frequency smallamplitude) limit, Bragg-grating solitons can be w«stable. Later, a rigorous analysis of the same stability problem was developed by Barashenkov, Pelinovsky and Zemlyanaya [1998] and by de Rossi, Conti and Trillo [1998]. It has been demonstrated that the Bragg-grating solitons indeed have instability regions in their parametric (co, v) plane, which are close to those originally predicted by means of the variational approximation. The derivation of the standard equations (2.1a) and (2.1b) from the underlying Maxwell equations (see §4.1) neglects all the linear terms with second-order
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Optical solitons in periodic media
[2, § 2
derivatives (which account for the material dispersion and/or diffraction in the medium). As shown by Champneys, Malomed and Friedman [1998], taking the second derivatives into account drastically changes the soliton content of the model (although the propagation distance necessary to observe the change of the solitons' shape may be much larger than that available in current experiments). The relatively simple soliton solutions generated by eqs. (2.1a) and (2.1b) immediately disappear after the inclusion of the second-derivative terms; instead, three completely new branches of soliton solutions emerge, provided that the coefficient in front of the second-derivative terms exceeds a certain minimum value. Very recently, it has been shown by Schollmann and Mayer [2000] that these new branches are, generally, also subject to an instability. Finally, we mention a dual-core nonlinear optical fiber, each core containing a Bragg grating, which is described by four equations (Mak, Malomed and Chu [1998a]), i ^ + i ^
+ (i|fFi|' + |£:Bi|')fFi+fBi+A^F2 = 0,
(2.2a)
i ^ - i ^
+ (i|fBi|' + |fFinfBi+fFi+A£B2 = 0,
(2.2b)
i ^
+ i ^ +
+ \Sm\^) fe +
(2.2c)
i ^
- i ^ + (il^B2|' + |^F2|') ^B2 + f e + A£:BI = 0,
(2.2d)
{\\£Y2?
where the subscripts 1 and 2 stand for the core number, the dimensionless quantities r, ^, fB,F;i,2 have the same meaning as in eq. (2.1), and A is the real linear coupling between the cores. Due to this linear coupling, a soliton in this system necessarily has its components in both cores. As shown by Mak, Malomed and Chu [1998a], an obviously symmetric soliton with equal components in the two cores, £^\ = £^2 and £B\ = ^82, becomes unstable when its energy exceeds a certain threshold. The instability gives rise to a pitchfork bifurcation that generates a pair of stable asymmetric solitons which are mirror images of each other. 2.2. Quadratic nonlinearities A Bragg grating with quadratic (second-harmonic-generating, SHG) nonlinearity gives rise to a vast variety of solitons. This model was introduced
2, § 2]
Solitons in Bragg gratings with cubic and quadratic nonlinearities
103
independently and simultaneously by Peschel, Peschel, Lederer and Malomed [1997], He and Drummond [1997] and Conti, Trillo and Assanto [1997]. The model includes four fields, namely, the fiindamental- and secondharmonic components of the forward- (F-) and backward- (B-) traveling waves. In a normalized notation, the corresponding system takes the form
'"9r ^ '"aF ^ ^^^ ^ ^^^^ + <^B = 0,
(2.3a)
'^—B^^^^x~'Qv + £l + >cg^ = o, V ox i ^
1- ^
(2.3b)
OL, - i ^
- i ^
+ ; ^ ^ B + ^B*eB + fF = 0,
+ X-'QB + f^ + KQV = 0,
(2.3c)
(2.3d)
where £ and Q are the fiindamental- and second-harmonic fields, i; > 0 is the group velocity of the second harmonic relative to the fiandamental harmonic, X is a (generally complex) coupling F-B coefficient in the second harmonic, given that the coupling constant at the fijndamental harmonic is normalized to 1, and the real parameter x determines the phase mismatch of the two harmonics. Analytical and numerical consideration of solitons in this model has revealed a number of nontrivial features. In particular, Peschel, Peschel, Lederer and Malomed [1997] have found that, contrary to the model (2.1) with Kerr nonlinearity, in the system (2.3) the gap in which solitons may exist has voids, so that only a part of it is filled with actually existing soliton solutions. Another noteworthy finding of the same work is that not only single-humped fiindamental solitons, but also their two-humped bound states, which are normally unstable in other versions of the SHG models, are dynamically stable in the four-wave model (2.3). An example of a two-humped soliton which turns out to be fairly stable in simulations of the system (2.3) is shown in fig. 2. A detailed description of gap solitons in the four-wave model combining a Bragg grating and SHG nonlinearity can be found in the review by Etrich, Lederer, Malomed, Peschel and Peschel [2000] on solitons in various SHG media. A different model based on resonant Bragg reflection and SHG nonlinearity can be formulated in the spatial domain [unlike the time-domain models (2.1) and (2.3)], for stationary fields in a planar (2D) nonlinear optical waveguide. In such a waveguide, a ID Bragg grating is realized as a system of parallel
Optical solitons in periodic media
104
[2, §2
intensity
^& forward FF
Fig. 2. An example of a dynamically stable double-humped soliton in the model based on eqs. (2.3a)(2.3d), combining a Bragg grating and second-harmonic-generation nonlinearity. Shown in this figure is the intensity of the forward fundamental-harmonic component |^p| (see Peschel, Peschel, Lederer and Malomed [1997]).
scores ("scratches"). The simplest model of this type involves three waves (Mak, Malomed and Chu [1998b]):
•t-'§*^-^>^-».
(2.4a)
,d£2_.d£2 i^-i^+^i+^3fr=0, d^ ^ dx
(2.4b)
2 l - ^ - q£2> + ^ - ^ 2 " + ^1^2 = 0.
(2.4c)
Here t and x are the propagation and transverse coordinates, respectively; the fields £\ and £2 are two components of the fundamental harmonic that are transformed into each other by resonant reflections on the ID Bragg grating, £3 is the second-harmonic component, and D is an effective diffraction coefficient for the second harmonic. The wave vectors ^1,2,3 of the three waves are related by the resonance condition, k\+k2 = k^,, the real parameter q accounts for a residual phase-mismatch. The configuration corresponding to this model assumes that the second harmonic propagates parallel to the Bragg grating (which has the form of the above-mentioned scores). It is therefore necessary to take into account the diffraction of this component, while for the two fundamental harmonics the effective diffraction induced by resonant Bragg scattering is much stronger than normal diffraction, which is neglected here. The soliton spectrum of this model is fairly rich. It contains not only fiindamental single-humped solitons but also their two-humped bound states.
2, § 3]
Self-induced transparency (SIT) in uniform media and thin films
105
some of which, as in the case of the four-wave model (2.3), may be dynamically stable (Mak, Malomed and Chu [1998b]). A rigorous stability analysis for various solitons in the model (2.4), based on computation of eigenvalues of the corresponding linearized equations, was performed by Schollmann and Mayer [2000]. This analysis has shown that some of these solitons, although quite stable in direct dynamical simulations, are subject to a very weak oscillatory instability, whereas other solitons in this model are stable in the rigorous sense. The three-wave model (2.4) possesses, besides the traditional GSs, numerous branches of embedded solitons: isolated solitary-wave solutions existing within the continuous spectrum, rather than inside the gap (Champneys and Malomed [2000]). Solutions of this kind appear also when the second-derivative terms are added to the generalized Thirring model (2.1). Finally, the four-wave model (2.3) with quadratic nonlinearity can be extended to the two- and three-dimensional cases, by adding transverse diffraction terms to each equation of the system. Physically, this generalization corresponds to spatiotemporal evolution of the fields in a two- or three-dimensional layered medium. Because, as is well known, quadratic nonlinearity does not give rise to wave collapse in any number of physical dimensions, the latter model can support stable spatiotemporal solitons, frequently called light bullets. Direct numerical simulations reported by He and Drummond [1998] have confirmed the existence of stable "bullets" in a multidimensional SHG medium embedded in a Bragg grating.
§ 3. Self-induced transparency (SIT) in uniform media and thin films 3.1. SIT in uniform media Self-induced transparency(SIT) is the solitary propagation of electromagnetic (EM) pulses in near-resonant atomic media, irrespective of the carrierfrequency detuning from resonance. This striking effect, which is of paramount importance in nonlinear optics, was discovered by McCall and Hahn [1969, 1970]. If the pulse duration is much shorter than the transition (spontaneousdecay) lifetime {Ti) and dephasing time {T2), then the leading edge of the pulse is absorbed, inverting the atomic population, while the remainder of the pulse causes atoms to emit stimulated light and thus return the energy to the field. When conditions for the process are met, it is found that a steady-state pulse envelope is established and then propagates without attenuation at a velocity that may be considerably less than the phase velocity of light in the medium.
106
Optical solitons in periodic media
[2, § 3
We start with the Hamiltonian for a single atom in the field, H=^^^E.d,
(3.1)
where w^\e){e\-\g)(g\
(3.2)
is the atomic inversion operator, COQ is the atomic transition firequency, \g) and \e) denote the atomic ground and excited states, respectively, E is the electric field vector, and d is the atomic dipole-moment operator. We take the projection on the field direction, so that E d = Ed, where d^^(p
+ P^),
(3.3)
pi being the dipole moment matrix element (chosen real) and P = 2\g){e\
(3.4)
is the atomic polarization operator. We express the electric field at a given point by means of the Rabifi-equencyQ as E=^{QQ"''^' + Q'e''^').
(3.5)
The Heisenberg equations of motion dA/dt = \/{\h) A,H , for the atomic polarization and inversion operators (3.4) and (3.2), yield the Bloch equations for their expectation values (c-numbers) P and w, respectively dtP{z, t) = w(z, t)Q -'\{ojo- con)P,
(3.6a)
S,w(z,0 = -J [ P * ( z , 0 ^ + c.c] .
(3.6b)
The Maxwell equations (Newell and Moloney [1992]) reduce in the rotatingwave and slow-varying approximations to
^ | 4 ) ^ = ro-P, nodz
(3.7)
at J
where
ro = " ^ J ^ ,
(3.8)
is the cooperative resonant absorption time, Qo being the TLS density (averaged over z), and no is the refraction index of the host media.
2, § 3]
Self-induced transparency (SIT) in uniform media and thin films
107
In the simplest case, when the driving field is in resonance with the atomic transition, a^o = ^c, the Bloch equations (3.6) can be easily integrated and the Maxwell equation (3.7) then reduces to the sine-Gordon equation FP-9
dm
sin a
(3.9)
for the "rotation angle", 0= /
QAt',
(3.10)
J-C
in terms of the dimensionless variables x = (t- noz/c)/ro and C = mz/cXo, This sine-Gordon equation is known to have solitary-wave solutions, for which the total area under the pulse is conserved and equal to In - the so-called pulsearea theorem by McCall and Hahn [1969, 1970]: 0(C,r) = (ro)-Uosech[i3(£-i;f)] ,
(3.11)
where the pulse width /? is an arbitrary real parameter uniquely defining the amplitude ^o = 2/^ and group velocity v = \/0^ of the soliton. Since its inception, SIT has become an active research area with many practical applications, for which we refer readers to excellent reviews by Lamb Jr [1971], Poluektov, Popov and Roitberg [1975], Maimistov, Basharov and Elyutin [1990] and references therein. In this section we will only briefly discuss results which are pertinent to the present review, such as SIT in thin films and collisions of counterpropagating SIT solitons. 3.2. SIT in th in films The interaction of light with a thin film of a nonlinear resonant medium located at the interface between two linear media has been described by Rupasov and Yudson [1982, 1987], who have shown that a nonlinear thin film of TLS can be a nearly ideal mirror for weak pulses, but transparent for pulses of sufficient intensity. The problem of light pulse transmission through the nonlinear medium boundary has been studied under conditions of coherent interaction with the matter. The system can be described by a set of nonlinear Maxwell-Bloch-like equations which effectively take the presence of the reflected wave into account by imposing boundary conditions on the electromagnetic fields at the interface. It has been shown (Rupasov and Yudson [1987]) that these equations are exactly
108
Optical solitons in periodic media
[2, § 3
integrable by the inverse scattering method, and 2jr-soHton-pulse transmission through the film has been studied. If the atomic density is such that on average there is more than one atom per cubic resonant wavelength, then near-dipole-dipole (NDD) interactions, or local-field effects, can no longer be ignored, contrary to the case of more dilute media. NDD effects necessitate a correction to the field that couples to an atom in terms of the incident field and volume polarization (Bowden, Postan and Inguva [1991], Scalora and Bowden [1995]). This effect can give rise to bistable optical transmission of ultrashort light pulses through a thin layer consisting of two-level atoms (Basharov [1988], Benedict, Malyshev, Trifonov and Zaitsev [1991]): the local-field correction leads to an inversion-dependent resonance frequency, and generates a new mechanism of nonlinear transparency. When the excitation frequency is somewhat larger than the original resonant frequency, the transmission of the layer exhibits a transient bistable behavior on the time scale of superradiance (Basharov [1988], Benedict, Malyshev, Trifonov and Zaitsev [1991]). It was shown that if an ultrashort pulse is allowed to interact with a thin film of optically dense two-level systems, the medium response is characterized by a rapid switching effect (Crenshaw, Scalora and Bowden [1992], Crenshaw and Bowden [1992]). This behavior is more remarkable than the response of conventional two-level systems, because the medium can only be found in one of two states: either fiilly inverted or in the ground state, depending (quasiperiodically) on the ratio between the peak field-strength and the NDD coupling strength. This feature was found to be impervious to changes in pulse shape, and to be independent of the pulse area (Crenshaw, Scalora and Bowden [1992]). Passage of light through a system of two thin TLS films of two-level atoms has been considered by Logvin and Samson [1992] and Logvin and Loiko [2000] who have shown that if the distance between the films is an integer multiple of the wavelength, then the system is bistable. Self-pulsations, i.e., periodically generated output, arise if an odd number of half-wavelengths can be fitted between the films and absorption in the medium is insignificant. In general, the dynamics admit both regular and chaotic regimes.
3.3. Collisions of counterpropagating SIT solitons Situations in which it is necessary to consider the interaction of incident (forward) and reflected (backward) light waves include: intrinsic optical bistability (Inguva and Bowden [1990]), dynamics of excitations in a cavity (Shaw and Shore [1990]) and collisions of counterpropagating SIT solitons
2, § 4]
SIT in RABR: model
109
(Afanas'ev, Volkov, Dritz and Samson [1990], Shaw and Shore [1991]). The field in such problems is represented as a superposition of forward- and backward-traveling waves. The atomic response to this field is determined by solving the Bloch equations (3.6) in the rotating-wave approximation. The population inversion w(z, t) and polarization P may be represented by a quasiFourier expansion over a succession of spatial harmonic carriers and slow varying envelopes, entangled in a fashion which leads to an infinite hierarchy of equations. The truncation of this hierarchy can only be justified by phenomenological arguments, such as atom movement in an active atomic gas. When the forward (F-) and backward (B-) wave pulses overlap in space and time, the resulting interference pattern of nodes modifies the atomic excitation pattern. The spatial quasi-Fourier expansion provides an efficient way of treating the spatial inhomogeneities of the response in those regions where the F- and Bpulses overlap, each successive Rabi cycle increasing the number of terms that contribute to the expansion (Shaw and Shore [1990]). Collisions of optical solitons produce observable effects on both the atoms and the pulses. The overlap of two counterpropagating pulses can produce an appreciable spatially localized inversion of the atomic population, thus causing optical solitons to lose energy. It was found that, whereas large-energy solitons passed freely through each other, solitons whose initial energy fell below a critical value were destroyed by collisions. In addition, the residual atomic dipole, created by the excitation, acts as a fiirther source of radiation. This radiation appears as an oscillating tail on the postcoUisional pulses and, over longer time scales, as fluorescence (Afanas'ev, Volkov, Dritz and Samson [1990], Shaw and Shore [1991]). § 4. SIT in resonantly absorbing Bragg reflectors (RABR): the model 4.1. Maxwell equations Let us assume (Kozhekin and Kurizki [1995], Kozhekin, Kurizki and Malomed [1998], Opatmy, Malomed and Kurizki [1999]) a one-dimensional (ID) periodic modulation of the linear refractive index n{z) along the z direction of the electromagnetic wave propagation (see fig. 3). The modulation can be written as the Fourier series n^(z) = nl[\+a\ cos(2A:cZ) + ^2 cos(4A:cz)+ • • •],
(4.1)
where n^, aj and kc are constants, and the medium is assumed to be infinite and homogeneous in the x and y directions.
Optical solitons in periodic media
no
[2. §4
Fig. 3. Schematic description of the periodic RABR and of the decomposition of the electric field into modes 2"+ and Z_. The shading represents regions with different index of refi-action; the darker the shading the larger n is. The black regions correspond to the TLS layers. The upper solid curve represents the electric field, the lower solid curves correspond to the components Re(2'+)cos^cZ and -\m{l_)smkcz; the dashed curves are the envelopes Re(2V) and -Im(2'_). The vertical dotted lines denote the positions of the TLS.
The periodic grating gives rise to photonic band gaps (PBGs) in the system's Unear spectrum, i.e., the medium is totally reflective for waves whose frequency is inside the gaps. The central frequency of the fundamental gap is 0)^ = kcc/no, c being the vacuum speed of light, and the gap edges are located at the frequencies 0)1,2 = CO, (1 ±
\a),
(4.2)
where a\ is the modulation depth from eq. (4.1). We further assume that very thin TLS layers (much thinner than l/A:,), whose resonance frequency o^o is close to the gap center H)^, are placed at the maxima of the modulated refraction index. In other words, the thin active layers are placed at the points Ziayer such that COS(A:c^layer) = ± 1 .
We shall study the propagation of electromagnetic waves with frequencies close to CL^C through the described medium. Let us write the Maxwell equation for one component of the field vector propagating in the z direction as
-n\z)
(4.3)
with the refraction index n modulated as in eq. (4.1), E being the electric field component and P„\ the nonlinear polarization. We use the substitution E= [£-F(z,0e'^»^ + £-B(2,0e-'''^]
(4.4)
SIT in RABR: model
4]
111
with 0)c satisfying the dispersion relation HQWC = Kc and £^ and £B denoting the forward and backward-propagating field components. We work in the slowly varying envelope approximation d^£ B,F d^£,B,F
df
d£ B,F dz
<
KQ
<
COr
(4.5a)
d£B,¥ dt
(4.5b)
Substituting eq. (4.4) into eq. (4.3), using eqs. (4.5a) and (4.5b), multiplying by e^^^^'^^^^'^^^ and averaging over the wavelength A = 2jt/kc and the period T = Ijt/cOc we get c d£^
d£^
\a\0)c
no dz
dt
4
c d£B no dz
d£B dt
fR + £*F +
IIXTI
Ipirl
(4.6a) (4.6b)
where we define ipLxl I d^P,^ dt^ hWcnl
i(±kcz+(i)ct)
(4.7) l,T
with IJ, the dipole moment (§ 3), and TQ a constant chosen here to be the medium absorption time (see eq. 3.8). The averaging is defined as
^^.^-Ar/X ^T JUT
• • dt dz.
(4.8)
We express the field components £^^B by means of the dimensionless quantities I ' i , -F,B
h (^+±^-), Aixxo
(4.9)
so that the electric field E = E(z, t) is E(z, 0 = hiiiXoT^ (Re[i:+(z, t)Q~''^'^] cos kcz - lm[l^(z,t)Q-''^'^'] sink^z).
(4.10)
To obtain the equations of motion in the most compact form, it is convenient to introduce the dimensionless time r, coordinate ^ and detuning <5 as no_ Z, 6 = (COo - COC)T^0, (4.11) cto where To is the characteristic absorption time of the field by the TLS medium as defined by eq. (3.8). Substituting eq. (4.9) into eqs. (4.6a) and (4.6b), using ^0
112
Optical solitons in periodic media
[2, § 4
eqs. (4.11) and (3.8), then differentiating the equations again with respect to C and r, we arrive, after algebra, at the form
-g^ - ^
= -rf^.
+ iriiP, +P-}+Q^ (P. +P-)+Q^
(P^ - ^ - ) '
(4.12a)
Here the dimensionless modulation strength rj is the ratio of the TLS absorption distance /abs = TQCMQ to the Bragg reflection distance /refi = 4c/{a\(DcnQ), which can be expressed as ^ = ^ =^li^.
(4.13)
The equations for the electric field components Il± can be solved once we know the averaged polarization P±. To find P±, we express the polarization Pni at a given point as the dipole moment density Pni = ^JtQ{d) = IJiQ^ UP) + ( M ) ) = -IJTiQpi (Pe-^^^^ -P*e^^'^^),
(4.14)
where Q is the number of the two-level atoms in a unit volume. Neglecting the time derivatives of P with respect to those of Q~^^^\ we can write for the polarization derivative
— / = iTticolQii (Pe-^^^^ -P*e^^^^),
(4.15)
so that P± on the right-hand side of eqs. (4.6a) and (4.6b) read as
To determine the evolution of P±, we have to make certain approximations. Let us first consider the situation when the TLSs are confined to infinitely thin layers. We will subsequently study the influence of finite width of the layers.
2, § 4]
SIT in RABR: model
113
4.2. Two-level systems (TLS) in infinitely thin layers Let us now assume that the atomic density Q is concentrated in zero-width layers located at Zj such that
i.e., it is described by p=^^6(z-z,),
(4.18)
where QQ is the bulk density averaged over the whole wavelength. We also assume that the spatial dependence of P is to a good approximation anti-periodic with respect to A/2, i.e., P{z + A/2) ^ -P{z). This is in agreement with the approximate anti-periodicity of the electric field with A/2. Denoting by PQ the value of P in even-numbered layers (the value in the odd-numbered layer being -P^), we get the spatial average in eq. (4.16) as 2
P± =
:"/""Po.
(4.19)
finf.
Note that P+ = P- is the consequence of the zero width of the layers. Due to the choice of TQ as in eq. (3.8), we obtain the simple relation P±=Po.
(4.20)
Using this expression in eqs. (4.12a) and (4.12b), we obtain the following evolution equations for I± # ^ _ | ^ = ^ ^ ^ . 2 i , P . , | ,
where we have omitted the index 0 of PQ, foi* simplicity.
(4.21a)
114
Optical solitons in periodic media
[2, § 4
The equations for the polarization P and inversion w in the even-numbered layers can be obtained from eqs. (3.6) by substituting for Q [combining eqs. (4.10) and (3.5)] Q = To ^ (i:+ cos k^z + '\I_ sin k^z)
{A21)
and applying eq. (4.17) at the positions of these layers. Expressing the detuning as in eq. (4.11), we obtain the equations ^
=-idP + I^w,
(4.23a)
^ or
= -Re (^^P*).
(4.23b)
The set of equations (4.21) and (4.23) was first derived by Mantsyzov and Kuz'min [1984, 1986] for the particular case of a periodic array of thin TLS layers without modulation of the linear index of refi*action, i.e., r/ = 0. In this case, these equations can be reduced to the sine-Gordon equation (3.9) for the area of the "forward" wave W(z,t)=
/
I4z,t')dt'.
(4.24)
J -o
It follows from eq. (4.24) that the lit solitary wave (SIT), associated with the sum of the forward- and backward-propagating waves, is an exact solution to the coupled-mode Maxwell-Bloch equations for a periodic stack of resonant thin films satisfying the Bragg condition. These solitary wave solutions are referred to as "two-wave solitons" (Mantsyzov and Kuz'min [1986]). If the Bragg condition is not exactly satisfied, then the system can exhibit a rich, multi-stable behavior, which has been studied numerically (Mantsyzov [1992], Lakoba and Mantsyzov [1992], Lakoba [1994]). In the case of a slight violation of the Bragg condition at the exact two-level resonance, an analytical solution for the 2jt gap soliton with a phase modulation has been obtained (Mantsyzov [1995]). These studies have also revealed the existence of an oscillating pulse, whose amplitude and velocity sign change periodically. Combining eqs. (4.23a) and (4.23b), one can eliminate the TLS population inversion:
Vi-i^p
(4.25)
2, § 4]
SIT in RABR: model
115
Without the field-induced polarization, the TLS population is not inverted (w = -1), hence the lower sign must be chosen in eq. (4.25). Thus, the remaining equations for 2V and P form a closed system, = ~Y]^I^ + 2i(r] - 8)P - 2 y ^ l - | P | 2 I^,
9r
= -i6P-^l-\P\^I^,
(4.26a) (4.26b)
and I-, the field component driven by dP/d^, can then be found from eq. (4.21b).
4.3. Finite width of TLS layers So far, we have assumed that the TLS layers are infinitesimally thin. We now proceed to estimate effects of non-zero width of the active layers, which represent more realistic physical situations. We still assume the width of the active layers to be small in comparison to the wavelength. This allows us to expand the polarization as a Taylor series in the position within the layer, and consider only terms up to the second order. Averaging the polarization over the entire wavelength yields the source term of the Maxwell equations (4.6). Let us assume that the TLS density is given, instead of eq. (4.18), by
e = ^m,
(4.27)
where QQ is the density averaged over the whole wavelength and the function/(z) has the properties /(z)>0, pZj + X/4
J'
/ ( z + A/2)=/(z), nZj + X/A
nZj + k/A
/(z)dz=l, / z/(z)dz = 0, / z^f{z)dz = (4.28) Jzj-X/A Jzj-l/4 where yz,-A/4
116
[2, §4
Optical solitons in periodic media
To calculate (gPe"^^^^^)^ of eq. (4.16), we expand the spatial dependence of P and Qxp(±ikcz) in a Taylor series, />Z2j + A/4
(gP,.±i^cz\
— ^±iA:<
/(^) Jzjj
-
A/4
PO + P'o(z - Z2j) + -P^(Z - Z2jf + • •
(4.30)
1 zb ik,(z - Z2j) - y ( ^ - ^2jf + • dz, where PQ, ^O ^^^ ^o ^^^^^ to the values of P and its spatial first and second derivatives at the positions of the even-numbered layers. Neglecting higher than quadratic terms and using eqs. (4.28), we arrive at <()Pe±'^'--),«po Po + y^
Pl.}Po_Po 2k^ k
(4.31)
2
so that
"--""-'^(i^^-f
(4.32)
Using these values in eq. (4.16), substituting them into eqs. (4.12a) and (4.12b), and defining ^0
^PiZ2j),
(4.33)
PB
^ - - — k dz
(4.34)
p£-'^
(4.35)
we obtain the field equations of motion in the form 8x2
d^2
dr^
8^2
=-riI+
+ 2it]Po + 2di
= -v'l-~2^
+^^'^ r
. „
dPc
dPc
dPs
+ f -ir/Pfi 8t
dr
OPB
(4.36a) (4.36b)
To obtain the equations of motion for the atomic parameters, we use eq. (4.22) for Q in eq. (3.6) and rewrite the Bloch equations at a point z as = -idP + [cos(^cZ)2'+ + i sin(^cz)-^-] w. dr 8w ^ = -\ [cos(kcz)I+ + i sin(A:cz)2L] P* + c.c.
ar
(4.37a) (4.37b)
2, §4]
SIT in RABR: model
117
Considering that the spatial derivatives of ^± are much smaller than those of sin(A:cz) and cos(A:cz), and calculating the first and second derivatives at the positions of the even-numbered layers, we get 9Po dr
(4.38)
= -idPo + I+WQ,
dn
(4.39)
dr
dPl = -idPQ - k^I+wo + 2i^c^-Wo + dr
I+WQ,
(4.40)
and dwo = -i^^Po*+c.c., ~dx dr
(4.41)
ife-^_Po*-^^+^o*+c.c.,
i^c^_P^*-i^+i^r + c.c. dr Using definitions (4.33)-(4.35) and defining Wo = Wi
(4.43)
(4.44)
w(Z2j),
i dw k dz
(4.42)
(4.45) Z = Z2j
(4.46)
W2 = Z2j
we then arrive at the equations of motion dPo = -i<5Po + wo2+. dr dPB = -ISPB + 2woS- - 2wi2V, dr dPc = -idPc + W22+ + 2w\Z-, dr
(4.47) (4.48) (4.49)
= -i^+Po*+c.c.,
(4.50)
dwi ~dr
= 2 ^ - ^ 0 ~ 4 ^ + ^ B ~ ^•^•'
(4.51)
^
= X , ( P o * - i P a ) - i ^ - / ' S + c.c.
(4.52)
"a7
Equations (4.36a), (4.36b) and (4.47)-(4.49) with their complex conjugates and eqs. (4.50)-(4.52) form a closed set of 13 equations for the variables 2'+, 21,
118
Optical solitons in periodic media
[2, § 4
PQ, PB, PC and their complex conjugates and WQ, wi and W2, i.e., 13 real variables together. The equations are parametrized by three real parameters rj, 6 and y. Note that for y = 0 eqs. (4.36a), (4.36b), (4.47) and (4.50) are identical to eqs. (4.21a), (4.21b), (4.23a) and (4.23b), respectively, with PQ standing for P. Even though the number of equations and variables has now increased, they are still relatively easy to solve numerically, and can realistically express the properties of the system over rather long times. The case of a uniform active medium embedded in a Bragg grating calls either for a solution of the ^ / / second-order Maxwell equation, without the spatial slow varying approximation, or for a soluton of an infinite set of coupled equations for Pi and wi as in the case discussed in § 3.3. This makes the present analysis principally at variance with that of Akozbek and John [1998], where the slow varying approximation for atomic inversion and polarization is assumed, thus arbitrarily truncating the infinite hierarchy of equations to its first two orders.
4.4. Energy densities To reveal the physical meaning of the quantities 2'± and P, we express the energy density of the electromagnetic field as W,-\hw,po{\IA^ + \^A^),
(4.53)
that of the TLS excitations (considering the limit of infinitely thin TLS layers) as ^A = \hcD,p, U - ^1 - \PA ,
(4.54)
and the energy density of the TLS-field interaction as W, = \hp^^'\m{I^P^).
(4.55)
From eq. (4.53) we conclude that |2VP and \Il-\^ are proportional to the number of photons per TLS (atom) in the standing-wave symmetric and anti-symmetric modes, whose anti-nodes and nodes, respectively, coincide with the active layers (seefig.3). Since the interaction time TQ (see eq. 3.8) is usually much larger than the optical period 2jr/(Oc, the interaction energy is negligible in comparison with the energies of the field and atomic excitations.
2, §4]
119
SIT in RABR: model
Fig. 4. The RABR dispersion curves (dimensionless frequency x versus dimensionless wavevector A:) at r; = 0.5 and 6 = -0.2. The solid lines show the dispersion branches corresponding to the 'bare' (noninteracting) grating, while the dashed and dash-dotted lines stand for the dispersion branches of the grating 'dressed' by the active medium. The frequency bands that support the standing dark and bright solitons are shaded. The arrow indicates a complete gap, where no field propagation of any kind can take place.
4.5. Linearized spectrum To reach general understanding of the dynamics of the model, the first necessary step is to consider the spectrum produced by the linearized version of eqs. (4.21b), (4.26a) and (4.26b), which describe weak fields in the limit of infinitely thin TLS layers. Setting w = - 1 , and (4.56a) (4.56b) (4.56c) we obtain from the linearized equation (4.26b) that C = i(d-xT^^- Substituting this into eqs. (4.21b) and (4.26a), we arrive at the dispersion relation for the wavenumber K and frequency x ^^ the form
(x'-K'-rj')(X-d)x{(x-d)[x'-K'-(2
+ r]')]+2(rj-d)}=0.
(4.57) Different branches of the dispersion relation generated by eq. (4.57) are shown in fig. 4. The roots x =" i ^ / c ^ + yji (corresponding to the solid lines in fig. 4) originate from the driven equation (4.21b) and represent the dispersion relation of the Bragg reflector with the gap \x\ < rj (cf eq. 4.2), that does not feel the
120
Optical solitons in periodic media
[2, § 5
interaction with the active layers. The degenerate root x = <5 is trivial, as it corresponds to the eigenmode (4.56c) with ^ = 5 = 0. Important roots are those given by the curly brackets in eq. (4.57) (shown by the dashed and dash-dotted lines in fig. 3), since they give rise to nontrivial spectral features. They will be shown below to correspond to bright or dark solitons in the indicated (shaded) bands. The fi-equencies corresponding to A: = 0 are X^ = r] and Xo,± --\{r]-d)±
^2^\{r]
+ d)\
(4.58)
while at A:^ -^ oo the asymptotic expressions for different branches of the dispersion relation are x ^ ^k and ^ ^ <5 + 2{r]~ S)k~'^. Thus, the linearized spectrum always splits into two gaps, separated by an allowed band, except for the special case, r] = r/o = | 6 + \ / l + ^(3^, when the upper gap closes down. The upper and lower band edges are those of the periodic structure, shifted by the induced TLS polarization in the limit of a strong reflection. They approach the SIT spectral gap for forward- and backward-propagating waves (Mantsyzov [1995]) in the limit of weak reflection. The allowed middle band corresponds to a polaritonic (collective atomic polarization) excitation in the periodic structure. It is different fi-om single-photon hopping in a PBG via resonant dipole-dipole interactions (John and Quang [1995]).
§ 5. Bright solitons in RABR 5.1. Standing (quiescent) self-localized pulses Stationary solutions of eqs. (4.26a) and (4.26b) corresponding to bright solitons have been found by Kozhekin, Kurizki and Malomed [1998]. Such solutions for the symmetric-mode field 2'+ and polarization P are sought in the form 2^ = e-^^5(e),
P = ie-^^^P(e),
(5.1)
with real V and S. Substituting this into eq. (4.26b), we eliminate V in favor of
^_sign(X-c3).5
2, §5]
121
Bright solitons in RABR
Fig. 5. The potentials U{S) of eq. (5.5) leading to the bright {x = - 3 2 ) and dark (x = -1.6) standing solitons. The system parameters are 6 = -2, and rj = 2.5.
and obtain an equation for 5(C),
S'' =
>(^-Z)sign(;^-(5)
(t-X^)S-2S'-
(5.3)
^(x-sy^s^ '
where the prime stands for d/dt- Equation (5.3) can be cast into the form of Newton's equation of motion for a particle with the coordinate S(^) moving in a potential U(S): (5.4) where
U(S) = -^-(rj^-x')S' + 2(rj-x) • sign(x-6W(x-df+
S^.
(5.5)
The potential gives rise to bright solitons (Newell and Moloney [1992]), provided it has two symmetric minima (see fig. 5). As follows from eq. (5.5), the latter condition implies that the quadratic part of the potential is concave, i.e., \x\ > T], and the second (asymptotically linear) part of the expression (5.5) is convex, so that X < ^' Moreover, two minima separated by a local maximum in <S = 0 appear if W(0) < 0. From this inequality it follows that bright solitons can appear in two frequency bands x, the lower band Xi <X<min{;^2,-^,5},
(5.6)
and the upper band max{xi,//,(5} <X <X2,
(5.7)
122
Optical solitons in periodic media
[2,
where the boundary frequencies X\,2 are given by (5.8)
X\,2 = \ ^ - ^ T \ / ( r 7 + (5)2 + 8
The lower band exists for all values r/ > 0 and (5, while the upper one only exists for (5.9)
l/rj.
6 > Tj-
which follows from the requirement Xi > V (see eq. (5.7). On comparing these expressions with the spectrum shown in fig. 4, we conclude that part of the lower gap is always empty from solitons, while the upper gap is completely filled with stationary solitons in the weak-reflectivity case (5.9), and completely empty in the opposite limit. It is relevant to mention that a partly empty gap has also been found in a Bragg grating with second-harmonic generation (Peschel, Peschel, Lederer and Malomed [1997]), see §2. The bright soliton corresponds to the solution of the Newton equation (5.4) with the "particle" sitting at time -oo on the local maximum S = 0, then swinging to one side and finally returning to «S = 0 at time +CXD. Such solutions have been found in an implicit form by Kozhekin, Kurizki and Malomed [1998]:
SiO = 2\x-d\ni^){\-n\C)y
(5.10)
with
ia = i 2 x-s
(l-7^o)
x-n
+ (2710)-'In
tan
d-Y-j^
7^o + ^Jnl-n? n
(5.11)
and
nl = i
\(X^rj)(x-6)\
(5.12)
(note that 1Zl is positive under the conditions 5.6-5.9). It can be checked that this zero-velocity (ZV) gap soliton is always single-humped. Its amplitude can be found from eq. (5.11), 47^o *->max
r
-'
vU + ^l The polarization amplitude V is determined by S via eq. (5.2).
(5.13)
§5] 16 141
123
Bright solitons in RABR
(a)
x10"
^2\10
6
4l 2
C
50
100
150
-150
-100
-50
50
100
150
Fig. 6. Zero-velocity (RABR) solitons |2'+(f)p: (a) 5 = 0, ry = 0.9, x = -0.901 (divergent width and amplitude); (b) idem, but for / = 0.901 (divergent width and finite amplitude).
To calculate the electric field in the antisymmetric I- mode, we substitute I^ = iQ-'^^A(0
(5.14)
into eq. (4.21b) and obtain (5.15) which can be easily solved by the Fourier transform, once V(C) is known. An example of bright solitons is depicted in fig. 6. Note that, depending on the parameters rj, 6 and x, the main part of the soliton energy can be carried either by the 1+ or the 2^ mode. The most drastic difference of these new solitons firom the well-known SIT pulses is that the area of the ZV soliton is not restricted to 2jt, but, instead, may take an arbitrary value. As mentioned above, this basic new result shows that the Bragg reflector can enhance (by multiple reflections) the field coupling to the TLS, so as to make the pulse area effectively equivalent to In. In the limit of the small-amplitude and small-area solitons, T^Q
(5.16)
In the opposite limit, I - Til ~^ ^' i-^' ^^^ vanishingly small \x + r]\, thQ soliton's amplitude (5.13) becomes very large, and fiirther analysis reveals that, in this case, the soliton is characterized by a broad central part with a width ~(l -IZl) (fig. 6a). Another special limit is / - ry -> 0. It can be
124
Optical solitons in periodic media
[2, § 5
checked that in this Hmit, the ampHtude (5.13) remains finite, but the soliton width diverges as \x - r]\~^^'^ (fig. 6b). Thus, although the ZV soliton has a single hump, its shape is, in general, strongly different firom that of the traditional nonlinear-Schrodinger (NLS) sech pulse. 5.1.1. Stability The stability of the ZV gap solitons was tested numerically, by means of direct simulations of the ftill system (4.26), the initial condition taken as the exact soliton with a small perturbation added to it. Simulations at randomly chosen values of the parameters, have invariably shown that the ZV GS are apparently stable. However, the possibility of their dynamical and structural instability needs be further investigated, as has been done in the case of GS in a Kerr-nonlinear fiber with a grating by Barashenkov, Pelinovsky and Zemlyanaya [1998] and Schollmann and Mayer [2000]. 5.2. Moving solitons Although the system of eqs. (4.26) is not explicitly Galilean- or Lorentzinvariant, translational invariance is expected on physical grounds. Hence, a full family of soliton solutions should have velocity as one of its parameters. This can be explicitly demonstrated in the limit of the small-amplitude large-width solitons (cf eq. 5.16). We search for the corresponding solutions in the form ^ 4 C , r ) = 5(C,r)exp(-ixor), P(?,r) = i7^(?,r)exp(-ixor) (cf. eqs. 5.1), where Xo is the frequency corresponding to A: = 0 on any of the three branches of the dispersion relation (4.58) (see fig. 4), and the functions 5(C, t) and P ( t , T) are assumed to be slowly varying in comparison with exp(-ixo7)- Under these assumptions, we arrive at the following asymptotic equation for <S(?, r):
(5.17)
Since this equation is of the NLS form, it has the full two-parameter family of soliton solutions, including the moving ones (Newell and Moloney [1992]). In order to check the existence and stability of the moving solitons numerically, the following procedure has been used by Kozhekin, Kurizki and Malomed [1998]: eqs. (4.26) were simulated for an initial configuration in the form of
2, §5]
125
Bright solitons in RABR
(b)
(a)
Fig. 7. Pulses obtained as a result of "pushing" a zero-velocity RABR soliton (dashed lines): (a) push, characterized by the initial multiplier exp(-zpj) after a sufficiently long evolution (r = 400) (solid lines), 5 = 0, r/ = 4, x = -4.4, and /? = 0.1; (b) idem, but for/? = 0.5.
the ZV soliton multiplied by exp(i/>^) with some wavenumber p, in order to "push" the soliton. The results demonstrate that, at sufficiently small p, the "push" indeed produces a moving stable soliton (fig. 7a). However, if p is large enough, the multiplication by exp(i/^t) turns out to be a more violent perturbation, splitting the initial pulse into two solitons, one quiescent and one moving (fig. 7b). Another one-parameter subfamily of moving GS was found in the exact form of a phase-modulated 2:/r-soliton by Kozhekin and Kurizki [1995]: (5.18)
^+ = ^0 exp [i (/re - xr)] sech [jS (C - vr)],
where x is the detuning from the gap center, AQ is the amplitude of the solitary pulse, j8 its width and v its group velocity. Substituting djP from eq. (4.23a) into eq. (4.21a), we may express P in terms of 2V and the population inversion w. Then, upon eliminating P and using ansatz eq. (5.18), we can integrate eq. (4.23b) for the population inversion w, obtaining -1-
Alix-K/v) 2{d-r])
1 cosh^[iS(e-i;r)]'
(5.19)
Using these explicit expressions for P and w in eqs. (4.21a) and (4.23a), we reduce our system to a set of algebraic equations for the coefficients K, X that determine the spatial and temporal phase modulation, and the pulse width ^ as fiinctions of the velocity v\ 2{X~K/v)-(\-\/v'){8-r])
= (),
(5.20a)
{X-S){ P^v^ -l3^ + K^-x^ + ri^ + 2) + ip^v^ix - K/V) + 2{d -^) = 0,
(5.20b)
(/3V - ^ 2 + K^-X" + ri' + 2)-2{x
(5.20c)
+ S){x-^/v)
= 0.
126
Optical solitons in periodic media
[2, § 5
The soliton amplitude is then found to satisfy \AQ\ = 2^v, exactly as in the case of usual SIT (see § 3). This implies, by means of eq. (5.18), that the area under the 2V envelope is 2jt. Let us consider the most illustrative case, when the atomic resonance is exactly at the center of the optical gap, 6 = 0. Then the solutions for the above parameters are -
-
^1-3^^^
2u 1 -1;2 '
(5.21a)
2 I -u^
In the frame moving with the group velocity of the pulse, ^' = ^ - or, the temporal phase modulation will be (KU-X) ^, which is found from eq. (5.21) to be equal to -rjT. Since t] is the (dimensionless) "bare" gap width (see § 4), this means that the frequency is detuned in the moving frame exactly to the band-gap edge. The band-gap edge corresponds (by definition) to a standing wave, whence this result demonstrates that such a pulse is indeed a soliton, which does not disperse in its group-velocity frame. The allowed range of the solitary group velocities may be determined from eq. (5.21c) through the condition (3^ > 0 for a given rj. The same condition implies |?7| < rj^ax, where ,8.^(1-.^) '/max
(1+^2)2
'
^ •
^
It follows from eq. (5.22) that the condition for Ijt SIT gap soliton (5.18) is |ry| < 1, r/max = 1 corresponding to t; = l / \ / 3 . This condition means that the cooperative absorption length CTOMQ should be shorter than the reflection (attenuation) length in the gap Ac/{a\(ji)cnQ), i.e., that the incident light should be absorbed by the TLS before it is reflected by the Bragg structure. In addition, both these lengths should be much longer than the light wavelength for the weakreflection and slow-varying approximation to be valid. From eq. (4.21b) we find 1- = I+/u. The envelopes of both waves (forward and backward) propagate in the same direction; therefore the group-velocity of the backward wave is in the direction opposite to its phase-velocity! This is analogous to climbing a descending escalator. Analogously to Kerr-nonlinear gap solitons (§ 2), the real part of the nonlinear polarization ReP creates a traveling "defect" in the periodic Bragg reflector
2, § 5 ]
Bright solitons in RABR
127
Fig. 8. Dependence of the solitary pulse velocity (solid line) and amplitude (dashed line) in RABR on frequency detuning from the gap center for rj = OJ. The "bare" gap edge marked by dotted line.
structure which allows the propagation at band-gap frequencies. The real part of the nonlinear polarization is governed by the frequency detuning from the TLS resonance. Exactly on resonance (which we here take to coincide with the gap center) x = ^ = 0, RQP = 0, and our solutions (5.21) yield imaginary values of the velocity v and modulation coefficient K. The forward field envelope then decays with the same exponent as in the absence of TLS in the structure. Because of this mechanism, SIT exists only on one side of the band-gap center, depending on whether the TLS are in the region of the higher or the lower linear refractive index. This result may be understood as the addition of a nearresonant non-linear "refiractive index" to the modulated index of refraction of the gap structure. When this addition compensates the linear modulation, soliton propagation becomes possible (see fig. 3). On the "wrong" side of the band-gap center, soliton propagation is forbidden even in the allowed zone, because then the nonlinear polarization cannot compensate even for a very weak loss of the forward field due to reflection. The soliton amplitude and velocity dependence on frequency detuning fi:om the gap center (which coincides with atomic resonance) are illustrated in fig. 8. They demonstrate that forward soliton propagation is allowed well within the gap, for X satisfying (1 - ^J\ - r]^)/r] < x < (1 + \ / l - rj^)/rj. In addition to frequency detuning from resonance, the near-resonant GS possesses another unique feature: spatial self-phase modulation Kt, of both the forward and backward field components.
128
Optical solitons in periodic media
[2, §5
5.3. Numerical simulations To check the stability of the analytical solution eq. (5.18), as well as the possibility to launch a moving GS by the incident light field, numerical simulations of eqs. (4.6) were performed by Kozhekin and Kurizki [1995]. As the launching condition, the incident wave was taken in the form SY = A exp[ix(^-^)] /cosh|jS(r-/o)/ro] without a backward wave {£B = 0) at the boundary of the sample z = 0. By varying the detuning x and amplitude A we investigate the field evolution inside the structure. When these parameters are close to those allowed by eqs. (5.21) and (5.22), we observe the formation and lossless propagation of both forward and backward soliton-like pulses with amplitude ratios predicted by our solutions (fig. 9). By contrast, exponential decay of the forward pulse in the gap is numerically obtained in the absence ofTLS(fig. 10).
Fig. 9. Numerical simulations of the intensities of (a) "forward" and (b) "backward" waves in the RABR gap, when eqs. (5.21) and (5.22) are obeyed {r] = 0.7, group velocity v ~ 0.3).
Fig. 10. Numerical simulations of the intensities of "forward" waves in the RABR gap without TLS (same rj and incident pulse as in fig. 9).
2, §6]
129
Dark solitons in RABR
The analysis surveyed in §§ 5.1-5.3 strongly suggests, but does not rigorously prove, that the solution subfamily (5.18) belongs to a far more general two-parameter family, whose other particular representatives are the exact ZV solitons (5.11) and the approximate small-amplitude solitons determined by eq. (5.17). 5.4. Collisions between gap solitons An issue of obvious interest is that of collisions between GSs moving at different velocities in RABR. In the asymptotic small-amplitude limit reducing to the NLS equation (5.17), the collision must be elastic. To get a more general insight, we simulated collisions between two solitons given by (5.18). The conclusion is that the collision is always inelastic, directly attesting to the nonintegrability of the model. Typical results are displayed in fig. 11, which demonstrates that the inelasticity may be strong, depending on the parameters.
T=5
^ ^
w
lU 5 n
' 7^: A
T=10
^
^ ]
'JK
T=0/^
:_
'-'^ A
:
!
"
A-25
: 1 ^ :
, - ^ - ^
^
^ 10
15
20
- ^ 25
30
Fig. 11. Typical example of inelastic collisions between the RABR solitons eq. (5.18) at (5 = 0 and r/ = 0.5, with the velocities (normalized to c) u\ = 0.6, U2 = -0.75.
§ 6. Dark solitons in RABR 6.1. Existence conditions and the form of the soliton Dark solitons (DSs) in RABR have been studied by Opatrny, Malomed and Kurizki [1999]. They are obtained similarly to the bright ones, by solving
130
Optical solitons in periodic media
[2, § 6
eq. (5.4) with the potential (5.5). The potential will give rise to DS's provided that it has two symmetric maxima (seefig.5). In this case the quadratic part of the potential is convex, i.e., \x\ < T], and the second (asymptotically linear) part of the expression (5.5) is concave, so that X > ^- From these two inequalities, a simple necessary restriction on the model's parameters follows, d
(6.1)
The condition for the existence of the symmetric maxima determines the following frequency interval x (recall that rj is defined to be positive): max{(5,-rj} <x < min{x2, rj},
(6.2)
Making use of eq. (5.8), one can easily check that, once the condition (6.1) is satisfied, the DS-supporting band (6.2) always exists. The DS frequency range defined by eq. (6.2) is marked by shading (to the right from zero) infig.4. The maxima of the potential are located at the points SM = ± \ / 4 ( / 7 + X ) - ' - ( / - 6 ) 2 ,
(6.3)
which correspond to the polarization values
Vu = T^Jl-\{X-^m + rJy•
(6.4)
Integrating eq. (5.4) by means of energy conservation in the formal mechanical problem, we obtain S(t,) in an implicit form,
£
-I' - ± -V2J0 /
with
dS, ^UM-U{SX)
^•^'
(6.5)
V2J0 a=\{n^-x\fi
= 2(r]-x),y = X-^-
(6.6)
The solution (6.5) corresponds to a trajectory beginning at the potential maximum ±«SM at "time" ^ = -CXD and arriving at the other maximum, T*^M at "time" ^ = 00 (seefig.5). In terms of the 2"+ mode of the electric field, this is exactly a quiescent (zero-velocity) DS with the background cw amplitude <SM.
6]
Dark solitons in RABR
131
Fig. 12. A typical example of a dark soliton, presented in terms of the variables S, V and A. The parameters are ry = 0.6, b = - 2 , and x = 0.25.
The integral (6.5) can be formally expressed in terms of incomplete elliptic integrals, but, practically, it is more helpful to evaluate it numerically. As in § 5, the polarization amplitude V is determined by S via eq. (5.2) and the amplitude of the IL mode is obtained by solving eq. (5.15). An example of the amplitude S for DS in the 2V mode, together with the corresponding quantities V and A, are plotted in fig. 12. The energy density of the |2VP field mode always has the shape of a hole in the background (see fig. 13). The energy density of \l-\^ has a hump, which is the counterpart of the hole in the 2V mode. The net electromagnetic energy density may have either a hole (which never drops to zero) or a hump, depending on the system parameters rj, 6, and the soliton frequency X6.2. Background stability An obvious necessary condition for the stability of DS is the stability of its cw background. To tackle this problem, we use eq. (4.26b) to eliminate 2"+ in favor of P,
^^ = -iP,-xbP){\-\P\X
1/2
,
(6.7)
and insert it into eq. (4.26a). The resulting equation for P is linearized around the stationary value VM (see eq. 6.4), substituting P = PMe-'^^[l+a(S,r) + iKe,T)],
(6.8)
132
Optical solitons in periodic media
[2, § 6
Fig. 13. Field energy densities as a function of the coordinate t, inside the dark soliton shown in fig. 12: |2Vp (dashed line), |lLp (dash-dotted line), and their sum (continuous line).
where a and b are small real perturbations. We can look for its general solution in the form [cf. eq. (4.56c)], a(^, T) = «oe'<''f-'''>, bit, r) = Z>oe'<'^f-"^>,
(6.9)
ivhich leads to a dispersion relation for Q and K, that consists of two parts:
(6.10) and
(6.11) As checked numerically by Opatmy, Malomed and Kurizki [1999] for many values of r], d and x that support DSs according to the results obtained in the previous section, all the roots of eqs. (6.10) and (6.11) are real for any real K. This implies the stability of the background for these values of rj, 6 and XEquations (6.10) and (6.11) represent new dispersion relations, which are valid under the condition of strong background field Su and replace the zero-field dispersion relations of eq. (4.57). This kind of optical bistability can be compared
2, §6]
133
Dark solitons in RABR
to the distributed feedback bistability with Kerr nonHnearity studied by Winful, Marburger and Garmire [1979]. 6.3. Direct numerical stability tests Even though there is no evidence of background instabihty, it is necessary to simulate the full system of the partial differential equations, in order to directly test the DS stability. Equations (4.26a) and (4.26b) have been integrated numerically by Opatmy, Malomed and Kurizki [1999], with initial conditions differing from the exact DS solution by a small added perturbation. The results strongly depend on the parameters 77, d and X- for some values, an explosion of the initial perturbation occurs, leading to a completely irregular pattern, whereas for others the DS shape remained virtually undisturbed. The dependence of the stability on the parameters Y] and x with fixed 8 is shown in fig. 14. The darkest area of the DS parameter region corresponds to the stable regime where no instability has occurred during the entire simulation time (typically, r '^ 500). In the rest of the parameter region, DSs are unstable: in the lightest part of the DS region, the instability develops very quickly (at r < 50), whereas in the intermediate part, the instability builds up relatively slowly. As can be seen, the unstable behavior occurs closer to the boundaries of the existence region, X = ^r] and ;(: = c5, whereas along the boundary X "" Xi (corresponding to the
1
0.5
0.5
JC2__
-1
unstable
--.^
1.5
6 2.5 -3
~~Yi
3.5 -A
Fig. 14. Parameter regions {r] vs. x) for dark (eq. 6.2) and bright (eqs. 5.6 and 5.7) solitons at 6 = - 2 . The boundaries X\ and Xi are given by eq. (5.8). In the DS region, the darkest area corresponds to stable behavior whereas in the remaining part the (numerical) solutions are unstable: in the lightest area (of DSs) instability develops very quickly, while in the intermediate area the DS exists for a much longer time before the onset of the instability.
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Optical solitons in periodic media
[2, § 6
DS-supporting background which degenerates into the trivial zero solution), DSs are stable. 6.4. Coexistence of the dark and bright solitons A very interesting question is whether the system can support bright and dark solitons at the same values of the parameters. As mentioned above, DSs always exist in the frequency interval (6.2), once the inequality (6.1) is satisfied. On the other hand, bright solitons are found in two frequency bands X-> given in eqs. (5.6) and (5.7). From the discussion in § 5 it follows that the DS frequency band always coexists with one or two bands supporting the bright solitons. The special case when there are two bright-soliton bands coexisting with the DS band is singled out by the condition T]
< 6 < ri.
(6.12)
rj
One can readily check that the coexisting frequency bands supporting bright and dark solitons never overlap, i.e., quite naturally, the bright and dark solitons cannot have the same frequency. 6.5. Moving dark solitons Thus far we have considered only the quiescent DSs. A challenging question is whether they also have their moving counterparts. Adding the velocity parameter to the exact DS solution is not trivial, as the underlying equations (4.26a) and (4.26b) have no Galilean or Lorentzian invariance. The physical reason for this is the existence of the special (laboratory) reference frame, in which the Bragg grating is at rest. In principle it is possible, in analogy to the stationary solutions and eq. (5.1), to substitute functions of the argument (^ - vr) into the set (4.26a) and (4.26b), so as to obtain an ordinary differential equation. However, this would be a complicated complex nonlinear equation of the third order, containing all the lower-order derivatives, so that we would not be able to take advantage of the Newton-like structure, as in §§ 5 and 6. Though it is possible to solve such an equation numerically, it is more suitable to deal with the original set of partial differential equations, in order to better understand the nature of the evolution. In contrast to the case of bright solitons where the moving solutions can be found by multiplying, in the initial conditions, the quiescent DS by a factor
•6]
Dark solitons in RABR
135
Fig. 15. Moving dark soliton: the values of the parameters are the same as in fig. 12, the background phase-jump parameter (j) (see eqs. (6.13) and (6.14)) is 0 = - | j r . Dashed line: r = 0, continuous line: r = 600.
proportional to exp(i/ct) (see Kozhekin, Kurizki and Malomed [1998] and § 5), it has proven possible to generate stable moving DSs from the quiescent ones in a different way (Opatrny, Malomed and Kurizki [1999]). To this end, recall that a DS corresponds to a transition between two different values of the background cw field. The background field generally takes complex values (note the real values in the expressions (6.3) and (6.4) above are only our choices adopted for convenience). The quiescent DS corresponds to a transition between two background values with phases differing by Jt. A principal difference of the DSs in the present model from those in the NLS equation (Kivshar and Luther-Davies [1998]) is that here a moving DS is generated by introducing SL phase jump ^ n across the DS. Thus, one can take the initial condition for the system of equations (4.26a) and (4.26b) as X4C,0) =cos(^0)5q(t) + isin(|0)5M,
(6.13)
P = cos(^0)j9q(C) + isin(^0)j9M,
(6.14)
where <Sq and Pq are the (real) fimctions corresponding to the quiescent DS, «SM and Vu are given by eqs. (6.3) and (6.4), and (j) is the deviation from Jt of the background phase jump across DS. A typical result obtained by means of this modification of the initial state is displayed in fig. 15: the DS moves at a velocity that is proportional to (j). The resulting form of the moving DS is slightly different from that of the quiescent soliton. The moving DS appears to be stable over the entire simulation time.
136
Optical solitons in periodic media
[2, § 7
§ 7. Light bullets (spatiotemporal solitons) A promising direction is the study of solitons in resonantly absorbing multidimensional (2D and 3D) media, in which the quasi-ID periodic structures can be realized as thin homogeneous layers set perpendicular to the direction of light propagation. In such media, spatiotemporal solitons, i.e., those localized in all dimensions, both transverse (spatial proper) and longitudinal (effectively, temporal), may exist. Spatiotemporal optical solitons or "light bullets" (LBs) in various nonlinear media are surveyed in § 1. Here we are concerned with LBs in RABRs that consist of thin TLS layers embedded in a 2D- or 3D-periodic dielectric medium. We will follow a recent analysis by Blaauboer, Kurizki and Malomed [2000], which extends an earlier prediction of stable LBs in uniform 2D and 3D SIT media (Blaauboer, Malomed and Kurizki [2000]). We start by considering a 2D SIT medium with a refractive index «(z,x) periodically modulated in the propagation direction z, which represents the quasione-dimensional Bragg grating. Light propagation in the medium is described by the lossless Maxwell-Bloch equations (Newell and Moloney [1992]):
dP ^-£w
=0,
(7.1b)
=0.
(7.1c)
OT
^-^^(£*P
+ P'S)
Here (as in § 4.1) £^ and P are the slowly varying amplitudes of the electric field and medium's polarization, w is the population inversion, C and x are longitudinal and transverse coordinates (measured in units of the resonant-absorption length), and r is time (measured in units of the input pulse duration). The Fresnel number, which governs the transverse diffraction in the 2D and 3D propagation, was incorporated into jc, and the detuning of the carrier frequency COQ from the central atomic-resonance fi-equency was absorbed into £ and P. To neglect the polarization dephasing and inversion decay, we assume pulse durations that are short on the time scale of the relaxation processes. Equations (7.1) are then compatible with the local constraint |Pp + w^ = 1, which represents the so-called Bloch-vector conservation. In a ID case, i.e., in the absence of the x-dependence and for «(z,x) = 1, eq. (7.1a) reduces to the sine-Gordon (SG) equation, which has a commonly known soliton solution, see eq. (3.11) and § 3. To search for LBs in a 2D medium subject to a resonant periodic longitudinal modulation, one may assume a periodic modulation of the refi-active index as
2, § 7]
Light bullets (spatiotemporal solitons)
137
per eq. (4.1). The RABR is then constructed by placing very thin layers (much thinner than X/k^) of two-level atoms, whose resonance frequency is close to (D^, at maxima of this modulated refractive index. The objective is to consider the propagation of an electromagnetic wave with a frequency close to o)^ through a 2D RABR. Due to the Bragg reflections, the electric field £ gets decomposed into forward- and backward-propagating components 8^ and £Q, which satisfy equations that are a straightforward generalization of the ID equations derived by Kozhekin and Kurizki [1995], Kozhekin, Kurizki and Malomed [1998], and Opatrny, Malomed and Kurizki [1999] (see also eqs. 4.21 and 4.23 in this review): ,d^l^
.d^I^ .d^I-
d^I^ .d^I+
d^I+ d^I-
d^I+ d^I-
.
.dP
,. „
^
,,,,
d^I-
2^
.dP
^
,^^^,
^ + {dP- 2^w = 0, ox ^ + UiiP or ^
+ i:+p*) = 0.
(7.2c) (7.2d)
Here I± is defined by eq. (4.9), and ?/ is a ratio of the resonant-absorption length in the two-level medium to the Bragg-reflection length, which was defined above by eq. (4.13). To construct an analytical approximation to the LB solutions, the starting point adopted by Blaauboer, Kurizki and Malomed [2000] is a subfamily of the exact ID soliton solutions to eqs. (7.2), which was found by Kozhekin and Kurizki [1995] (see also §5.2 in this review) and is given by eqs. (5.18) and (5.19). These solutions were taken with parameters satisfying eqs. (5.20):
K = -^8^
- r]\
x = 6.
(7.3b)
These solutions were chosen as a pattern to construct an approximate solution for LBs because the shape of the fields 1+ and I- in the solutions is similar to that of the SG soliton in the ID uniform SIT medium (see §3.1). Inspired by this analogy and by the fact that there exist LBs in the uniform 2D SIT medium which reduce to the SG solitons in the ID limit (Blaauboer, Malomed and Kurizki [2000]), one can search for an approximate LB solution to the 2D
138
Optical solitons in periodic media
[2, § 7
equations (7.2), which also reduces to the exact soliton in ID. To this end, the following approximation was assumed: I^ = Aoy/sQcheiSQche2c'^''^-^'^^V'',
(7.4a)
^_ = I+/v,
(7.4b)
P = y/sQchO\ sech6)2 X |(tanh6)i +tanhe2) 6-rj ,r . -C^[(tanhei-tanh6)2) 4r/ 2(sech^6)i+sech^6)2)] ^ j1 w=
,1/2
[i-|/^IT '
(7.4c) e^i(K-t-xr)+iv
(^-^d)
with 6)i(r, C) = i3(C - i;r) + 00 + Cx, e2(^, C) = iS(C - ^^) + 6)o - Cc, the phase v and coefficients OQ and C being real constants, while the other parameters are defined by eqs. (7.3). The ansatz (7.4) satisfies eqs. (7.2a) and (7.2b) exactly, while eqs. (7.2d) are satisfied to order y/S/t]- IC^, which requires that y/d/rj- \C^ <^ 1. The ansatz applies for arbitrary r/, admitting both weak (rj < 1) and strong (rj > 1) reflectivities of the Bragg grating, provided that the detuning remains small with respect to the gap fi*equency. Comparison with numerical simulations of eqs. (7.2), using eq. (7.4) as an initial configuration (a finite-difference method, with Fourier transform scheme, described by Drummond [1983], was used), tests this analytical approximation and shows that it is indeed fairly close to a numerically exact solution; in particular, the shape of the bullet remains within 98% of its originally presumed shape after having propagated a large distance, as is shown in fig. 16. Three-dimensional LB solutions with axial symmetry have also been constructed in an approximate analytical form and succesfiilly tested in direct simulations, following a similar approach (Blaauboer, Kurizki and Malomed [2000]). Generally, they are not drastically different from their 2D counterparts described above. A challenging problem which remains to be considered is the construction of spinning light bullets in the 3D case (doughnut-shaped solitons, with a hole in the center, carrying an intrinsic angular momentum). Recently, spinning bullets were found by means of a sophisticated version of the variational approximation in a simpler 3D model, viz., the nonlinear Schrodinger equation with self-focusing cubic and self-defocusing quintic nonlinearities, by Desyatnikov, Maimistov
2, § 8]
Experimental prospects and conclusions
139
X
Fig. 16. The forward-propagating electric field of the two-dimensional "light bullet" in the Bragg reflector, \Sp\, vs. time r and transverse coordinate x, after having propagated the distance z = 1000. The parameters are rj = OA, 6 = 0.2, C = 0.1 and 0Q = -1000. The field is scaled by the constant /2/4ro/i«o-
and Malomed [2000]. Further direct simulations have demonstrated that these spinning bullets (unlike their zero-spin counterparts) are always subject to an azimuthal instability, that eventually splits them into a few moving zero-spin solitons, although the instability can sometimes be very weak (Mihalache, Mazilu, Crasovan, Malomed and Lederer [2000]). At present, it is not known whether spinning LBs can be completely stable in any 3D model. It is relevant to stress that two- (and three-) dimensional LB solutions of the variable-separated form, 2"+ ~ 2'_ ~ / ( r , £) • g(x), do not exist in the RABR model. Indeed, the substitution of this into eqs. (7.2a) and (7.2b) yields only a plane-wave solution of the form I± ^ exp (iAr -\- iBx), with constant A mdB.
§ 8. Experimental prospects and conclusions This review has focused on properties of solitons in RABR, combining a periodic refractive-index (Bragg) grating and a periodic set of thin active layers (consisting of two-level systems resonantly interacting with the field). It has been demonstrated that the RABR supports a vast family of bright gap solitons, whose properties differ substantially from their counterparts in periodic structures with either cubic or quadratic off-resonant nonlinearies reviewed in § 2. The same RABR can support, depending on the initial conditions, either dark or bright stable solitons, without any changes of the system parameters, which is a unique feature for nonlinear optical media (§§5, 6). Zero-velocity dark solitons can be found in an analytical form, as well as traveling dark
140
Optical solitons in periodic media
[2, § 8
solitons with a constant phase difference (^ :/r) of the background amplitudes across the soUton (§6). The latter property is a major difference with respect to dark solitons of the NLS equation, whose motion is supported by giving the background a nonzero wavenumber. Depending on the values of the parameters, the frequency band of the quiescent dark solitons coexists with one or two bands of the stable bright ones, without an overlap. Direct numerical simulations demonstrate that some darksoliton solutions are stable against arbitrary small perturbations, whereas others are unstable when they are close to the "dangerous" boundaries of their existence domain. A multidimensional version of the RABR model, corresponding to a periodic set of thin active layers placed at the maxima of the refractive index, which is modulated along the propagation direction of light has been considered too. It has been found to support stable propagation of spatiotemporal solitons in the form of two- and three-dimensional "light bullets" (LBs). The best prospect of realizing a RABR which is adequate for observing the solitons and light bullets discussed in §§5-7 is to use thin layers of rareearth ions (Greiner, Boggs, Loftus, Wang and Mossberg [1999]) embedded in a spatially periodic semiconductor structure (Khitrova, Gibbs, Jahnke, Kira and Koch [1999]). The two-level atoms in the layers should be rare-earth ions with the density of 10^^-10^^ cm~^, and large transition dipole moments. The parameter r/ can vary from 0 to 100 and the detuning is -10^^-10^^ s"^ Cryogenic conditions in such structures can strongly extend the dephasing time T2 and thus the soliton's or LB's lifetime, well into the \is range (Greiner, Boggs, Loftus, Wang and Mossberg [1999]), which would greatly facilitate the experiment. The construction of suitable structures constitutes a feasible experimental challenge. In a RABR with the transverse size of 10|im, LBs can be envisaged to be localized on the time and transverse-length scales, respectively, ^10"^^ s and 1 j^m. The incident pulse has uniform transverse intensity and the transverse diffraction is strong enough. One needs d^/hh^Xo < 1, where /abs, AQ and d are the resonant-absorption length, carrier wavelength, and the pulse diameter, respectively (Slusher [1974]). For /abs ^ 10"^ m and AQ ~ 10""^ m, one thus requires d < 10~^m, which implies that the transverse medium size L^ must be a few (uim. Effects of TLS dephasing and deexcitation in RABRs can be studied by substituting the values -\d - FiT^ for the frequency term -id in eqs. (4.47)(4.49) and the loss terms -riro(wo + 1) in eq. (4.50), -FxWi in eq. (4.51) and -r\(w2 - 2) in eq. (4.52). We have checked that these modifications
2, § 8]
Experimental prospects and conclusions
141
do not influence the qualitative behavior of the solutions on the time scale r < To < l/ri,2. Let us now discuss the experimental conditions for the realization of RABR solitons using quantum wells embedded in a semiconductor structure with periodically alternating linear index of refraction (Khitrova, Gibbs, Jahnke, Kira and Koch [1999]). We can assume the following values: the average refraction index is n^ ^ 3.6, the wavelength (in the medium) A ^ 232 nm, which corresponds to the angular frequency (Oc ^ 2.26 x 10^^ s"^ Excitons in quantum wells can, under certain conditions (such as low densities and proximity of the operating frequency to an excitonic resonance, see Khitrova, Gibbs, Jahnke, Kira and Koch [1999]) may be regarded as effective two-level systems (TLS's). We consider their surface density to be ^ 10^^-10^^ cm"^, which corresponds to a bulk density po ^ 10^^-10^^ cm~^. If we assume that the excitons are formed by electrons and holes displaced by ^ 1-10 nm, then the characteristic absorption time TQ defined in eq. (3.8) is TQ ^ 10"^^-10~^^ s, and the corresponding absorption length is CTQ/WO ^ 10-100 |im. The dephasing time for excitons discussed by Khitrova, Gibbs, Jahnke, Kira and Koch [1999] is l/r2 ^ 10~^^ s, which seems to be the chief limitation of the soliton lifetime for this system. The structures shown in figs. 6, 12, 13 and 15, occupying regions of approximately 100 absorption lengths, would require a device with a total width of approximately 1 mm to 1 cm, which corresponds to ?=^ 10^ to 10^ unit cells. The modulation of the refraction index can be as high as «i ?^ 0.3, so that the parameter rj (see eq. 4.13) can vary from 0 to 100. The unit of the dimensionless detuning 8 would represent a 10~^-10~^ fraction of the carrier frequency. The intensities of the applied laser field corresponding to 2"^- ^ 1 are then of the order 10^-10^ W/cm^. In the work by Khitrova, Gibbs, Jahnke, Kira and Koch [1999], the width of the active layers (quantum wells) is considered to be 5-20 nm, which corresponds to the parameter y^ (see eq. 4.29) in the range 10^^-2 x 10"^. In the simulations discussed by Opatrny, Malomed and Kurizki [1999], taking the largest of these values and the parameters as in fig. 12, i.e., t] = 0.6, 8 = - 2 , and x ^ 0.25, we have observed the time evolution of the system (4.36a)-(4.52). As the initial condition, both the DS solution corresponding to zero width of the active layers and the DS solution including the finite width correction, have been taken. In both cases, the evolution was quite regular over the observed time r ?^ 50, and the zero-width solution (with the quantities <S, V, A as given by eqs. 6.5, 5.2 and 5.15) started to change after r ^ 10. We can now sum up the discussion of experimental perspectives for the realization of RABR solitons: (a) The prospects appear to be good for
142
Optical solitons in periodic media
[2
gratings incorporating thin layers of rare-earth ions under cryogenic conditions, (b) The reahzation of these sohtons in excitonic superlattices would require much longer dephasing times than those currently achievable in such structures.
Abbreviations DS
dark soliton
GS
gap soliton
LB
light bullet
NDD
near-dipole-dipole
NLS
nonlinear Schrodinger
PBG
photonic band gap
QNLSE
quantum nonlinear Schrodinger equation
RABR
resonantly absorbing Bragg reflector
SHG
second-harmonic-generating
SIT
self-induced transparency
TLS
two-level system
ZV
zero-velocity
Acknowledgments We are grateful to M. Blaauboer for discussions and help. G.K. acknowledges the support of the EU (ATESIT) and US-Israel BSE TO. thanks the Deutsche Forschungsgemeinschaft for support. A.K. acknowledges support of the Thomas B. Thriges Center for Quantum Information.
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E. Wolf, Progress in Optics 42 © 2001 Elsevier Science B. V All rights reserved
Chapter 3
Quantum Zeno and inverse quantum Zeno effects by
Paolo Facchi and Saverio Pascazio Dipartimento di Fisica, Universitd di Bari and Istituto Nazionale di Fisica Nucleare, Sezione di Bari, 70126 Bari, Italy
147
Contents
Page § 1. Introduction
149
§ 2.
152
Two-level systems and Bloch vector
§ 3. Pulsed observation
155
§ 4.
Dynamical quantum Zeno effect
163
§ 5.
Continuous observation
167
§6.
Novel definition of quantum Zeno effect
173
§7.
Zeno effects in down-conversion processes
175
§ 8.
Genuine unstable systems and Zeno effects
191
§ 9.
Three-level system in a laser
field
199
§ 10. Concluding remarks
213
Acknowledgments
214
References
214
148
§ 1. Introduction Zeno and his master Parmenides lived about 2500 years ago in Elea, a small Italian town not far from Naples, in the Mediterranean region called "Magna Graecia". Parmenides, a profound and innovative philosopher, believed that senses are deceptive and our perception of reality in continuous change is an illusion. In his conception, there is a unique, indivisible Truth ("being") that does not undergo any change ("becoming") and cannot be decomposed into smaller entities. Zeno was Parmenides' most brilliant disciple and in order to support his master's ideas he would challenge the most "obvious" conclusions of common sense by putting forward many paradoxical examples. His captious arguments are ingenious and very famous. For example, he argued that Achilles, who starts running from point A, cannot reach a turtle that at the same time starts moving from point B, because when the former reaches B the latter has moved to C, and so on ad infinitum. Against the very idea of dividing an object into parts, he claimed that if a finite segment is made up of an infinite number of points then one runs into a contradictory conclusion: indeed, if a single point has a finite size, then the size of the segment is infinite; if, on the other hand, a single point has no size, their sum (the segment itself) cannot have a nonvanishing size. Zeno lacked the concept of infinitesimal. Modem infinitesimal calculus resolves the first paradox by introducing the concept of velocity as the derivative of position with respect to time: Achilles will reach the turtle in 2i finite time because it takes a very small time to cover a very small distance, where "very small" times and distances are infinitesimal of the same order. The solution of the second paradox is even subtler. An uncountable ensemble of points can have a nonvanishing size. At any rate, it is undeniable that his provocative arguments foreran very subtle concepts of infinitesimal calculus, such as derivatives, Riemann and Lebesgue measures. More to this, one should not forget that Zeno aimed at challenging the "obviousness" of common sense, in order to support the philosophy of his master Parmenides and bring to light the difficulties inherent in the very idea of "becoming". One of Zeno's paradoxes will be the object of the present investigation: A sped arrow never reaches its target, because at every instant of time, if we look at the arrow, we see that it occupies a portion of space equal to its own size. At any 149
150
Quantum Zeno effects
[3, § 1
given moment the arrow is therefore immobile, and by summing up many such "immobihties" it is clearly impossible, according to Zeno, to obtain motion. It is amusing that some quantum-mechanical states, under particular conditions, behave in a way that is reminiscent of this paradox. In this chapter we shall review the main features of the so-called quantum Zeno effect (von Neumann [1932], Beskow and Nilsson [1967], Khalfin [1968], Misra and Sudarshan [1977]). In very few words, the evolution of a quantummechanical state can be slowed down (or even halted in some limit) when very frequent measurements are performed on the system, in order to check whether it is still in its initial state: Zeno's quantum arrow (the wave function) does not move, if it is continuously observed. The interest in the quantum Zeno effect (QZE) has been revived, during the last decade, mainly because of some interesting proposals that made it liable to experimental investigation. Unlike previous studies, confined to a purely academic level, the investigation of the last few years has focused on practical experiments, possible applications, as well as theoretical implications and interpretative issues. In this chapter we shall first review the main features of the QZE and discuss some simple examples. We shall then concentrate our attention on more interesting physical situations and emphasize the occurrence of new physical phenomena, such as the "inverse" quantum Zeno effect. The quantum Zeno effect has been mainly investigated for oscillating systems (Cook [1988], Itano, Heinzen, Bollinger and Wineland [1990], Pascazio, Namiki, Badurek and Rauch [1993], Kwiat, Weinfurter, Herzog, Zeilinger and Kasevich [1995], Luis and Pefina [1996]), whose Poincare time is finite. However, the discussion cannot be limited to oscillating systems. New and somewhat unexpected phenomena are disclosed when one considers unstable systems, whose Poincare time is infinite (Bemardini, Maiani and Testa [1993], Facchi and Pascazio [1998], Maiani and Testa [1998], Joichi, Matsumoto and Yoshimura [1998], Alvarez-Estrada and Sanchez-Gomez [1999]). Unfortunately, in this case the analysis becomes more complicated and requires a quantum field theoretical framework. The QZE has recently become such a wide subject of investigation, that it is difficult to discuss all its multiple facets. In this chapter we shall therefore discuss only some of its aspects, by focusing our attention on quantum optics and quantum electrodynamics. As a general philosophy, we shall always start by considering simple physical systems and then extend our analysis to more complicated cases. As already emphasized, it would be wrong to limit the analysis to elementary situations (such as oscillating systems), because in doing so one would overlook a great deal of interesting physical effects. We will
3, § 1]
Introduction
151
therefore try to follow Einstein's precept: Things should be made as simple as possible, but not simpler. In this chapter we will often work in natural units {h = c= 1), but will put the physical constants back in the final formulas whenever it will be helpfiil to get a feeling for the numbers. After setting up the notation in § 2, we introduce in § 3 the fundamentals of the quantum Zeno and "inverse" quantum Zeno effect (IZE), by making use of elementary quantum-mechanical techniques. We shall first use the seminal formulation of QZE in terms of projection operators: This is the usual approach and makes use of what we might call a "pulsed" observation of the quantum state (Mihokova, Pascazio and Schulman [1997], Schulman [1998]). We then explain in § 4 that it is not necessary to use projection operators and nonunitary dynamics. A fiilly dynamical explanation of the QZE is possible, involving Hamiltonians and no projectors (Pascazio andNamiki [1994], Petrosky, Tasaki and Prigogine [1990]). In §5 we introduce the notion of "continuous" observation of the quantum state, e.g., performed by means of an intense field. Although this idea has been revived only recently (Mihokova, Pascazio and Schulman [1997], Schulman [1998]), it is contained, in embryo, in earlier papers (Kraus [1981], Peres [1980], Plenio, Knight and Thompson [1996]). This idea will lead us to a novel definition of QZE in § 6. We then discuss, in § 7, an interesting example of QZE in quantum optics, both with pulsed and continuous measurements. We look at a down-conversion process in a nonlinear crystal as a "decay" of a pump photon into a pair of signal and idler photons of lower frequency and study how the "decay" is modified by a measurement process of some sort. Interestingly, this system discloses the presence of an inverse Zeno effect. We shall see that by increasing the strength of the observation, the "decay" is sometimes accelerated rather than hindered. In § 8 we consider the QZE and IZE for bona fide unstable systems. This is a more complicated problem, because it requires the use of quantum field theoretical techniques. The study of a solvable (but significant) example enables us to understand the role played by the Weisskopf-Wigner approximation (Gamow [1928], Weisskopf and Wigner [1930a,b], Breit and Wigner [1936]) and the Fermi "golden rule" (Fermi [1932, 1950, I960]). Moreover, we shall see that for an unstable system, the form factors of the interaction play a fundamental role and determine the occurrence of a Zeno or an inverse Zeno regime, depending on the physical parameters describing the system. We finally investigate, in § 9, the intriguing possibility that the lifetime of an unstable quantum system be modified by the presence of a very intense electromagnetic field. We shall look at the temporal behavior of a three-level system (such as an atom or a molecule) illuminated by an intense laser field (Pascazio and Facchi [1999], Facchi and Pascazio [2000]) and see
152
Quantum Zeno effects
[3, § 2
that, for physically sensible values of the intensity of the laser, the decay can be enhanced. This will be interpreted as an inverse quantum Zeno effect. The choice of the subjects that appear in this review article reflects our own point of view on the quantum Zeno problem. This was inevitable and requires an apology. Some important aspects of this problem were either left out or only briefly mentioned. We regret, in particular, that some very important features of the temporal evolution, arisingfi-oma genuine quantum field theoretical analysis, are not even mentioned. Moreover, we will not explore other very important topics, such as neutron physics, irreversibility and deviationsfi*omMarkovianity. On the other hand, we intentionally did not discuss some academic issues of no practical interest. We shall not look in detail at the characteristics of the quantum-mechanical evolution law at short times (Beskow and Nilsson [1967], Khalfin [1968], Wilkinson, Bharucha, Fischer, Madison, Morrow, Niu, Sundaram and Raizen [1997]) and long times (Mandelstam and Tamm [1945], Fock and Krylov [1947], Hellund [1953], Namiki and Mugibayashi [1953], Khalfin [1957, 1958]). These are summarized in Nakazato, Namiki and Pascazio [1996] and will often be taken for granted. An excellent account of the most recent results on the QZE can be found in Home and Whitaker [1997] and Whitaker [2000]. Our attention will mainly be focused on quantum optics and quantum electrodynamics. Specific examples will be given particular importance and will therefore play a fiindamental role. We will not attempt to generalize, unless necessary. The leitmotif of this chapter is that the quantum Zeno effect is a dynamical phenomenon, that can be explained in terms of the Schrodinger equation, without making use of projection operators. We will implicitly assume, throughout our discussion, that a projection operator is just a shorthand notation that summarizes the effects of a much more complicated underlying dynamical process, involving a huge number of elementary quantummechanical systems (Namiki, Pascazio and Nakazato [1997]). This idea will constitute the "backbone" of our work. Only if this concept is fiilly elaborated and completely digested, can one realize that a broader definition of Zeno effect is required, that takes into account the very concept of continuous measurement, performed for example by a quantum field.
§ 2. Two-level systems and Bloch vector We start by considering a two-level system undergoing Rabi oscillations. This is the simplest nontrivial quantum-mechanical example, for it involves 2x2
3, § 2]
Two-level systems and Block vector
153
matrices and very simple algebra. One can think of an atom illuminated by a laser field whose frequency resonates with one of the atomic transitions, or a neutron spin in a magnetic field. We shall neglect the energy difference between the two states |ib). The (interaction) Hamiltonian reads /f, = D ( T , = 0 ( | + ) H + | - ) ( + | ) = ( ^ ^
^ ) ,
(2.1)
where ^ is a real number, Oj (j = 1,2,3) the Pauli matrices and
;.
H=;
(2.2)
are eigenstates of 03. We will use the above notation interchangeably. Let the initial state be
|V%) = I + ) = ( J ) ,
(2.3)
SO that the evolution yields \^l>t) = e-^^'\^Po) = cos(f20|+)-isin(i30|-) = ( _ f s i n ^ ^ ) •
^^.4)
In the following, we shall often make use of the rotating coordinates, introduced by Bloch [1946] and Rabi, Ramsey and Schwinger [1954], and of well-known computational techniques due to Feynman, Vernon and Hellwarth [1957]. In terms of the polarization (Bloch) vector R(t)={ilJt\o\ilJt)
= (RuR2,R3)',
(2.5)
where ^ denotes the transposed matrix, the Schrodinger equation reads ^ ( 0 = 2Qx R(tl
(2.6)
where Q = (Q,0,Of. The norm of the Bloch vector is preserved: \\R(t)\\ = 1,V^. See fig. 1.
(2.7)
Quantum Zeno effects
154
[3, §2
Fig. 1. The Poincare sphere and the Bloch vector.
The density matrix of a two-level system is expressed in terms of the Bloch vector according to the formula P=
P-+
P-
= ^(.l+R-a),
(2.8)
so that (2.9) where P± = p±± is the probability that the system is in level ± . Notice that Tr p = P+ + P. = I (normalization) and Tr{ pa) = R. Vice versa, the Bloch vector is readily expressed in terms of the density matrix:
/?,
=p^+p-^,
R2 = i ( p + - - p - + ) , R3 =p+^-p
(2.10)
^=P+-P-.
The level configuration and the dynamics of the oscillations are shown in fig. 2. Observe that the probability returns to its initial value after a time Tp = JT/Q. This is a very simple instance of Poincare recurrence time.
3, § 3 ]
Pulsed observation
l+>
155
0.5
Vtt/Tl
l-> Fig. 2. Rabi oscillations in a two-level system.
§ 3. Pulsed observation Let us introduce the fundamental features of the quantum Zeno effect. We shall follow the "historical" approach (von Neumann [1932], Beskow and Nilsson [1967], Khalfin [1968], Misra and Sudarshan [1977]), by considering "pulsed" measurements. The alternative notion of continuous measurement will be discussed in § 5. For the sake of simplicity we shall often refer to two-level systems. This will make our analysis more transparent. It goes without saying that more general and formal approaches are also possible (Nakazato, Namiki and Pascazio [1996], Home and Whitaker [1997]). 3.1. Survival probability under pulsed measurements We define the survival amplitude At) = {MH>t) = (V^ole-^^IV^o)
(3.1)
and the survival probability P{t)=\A(t)\^
= \{rPo\c-^'\tPo)\\
(3.2)
where H is the total Hamiltonian of the system. These quantities represent the amplitude and probability that a quantum system, initially prepared in state | V^o), is still in the same state at time t. An elementary expansion shows that the behavior of the survival probability at short times is quadratic P(t) = 1 -t^/rl + . . . ,
r~^ = {%\H^\%) - {^o\H\il^of.
(3.3)
For instance, with the Hamiltonian (2.1) one finds .4(0 = cos Qt,
(3.4)
156
Quantum Zeno effects
[3, § 3
P{t) = cos^ Qt,
(3.5)
rz
(3.6)
= Q-'.
The quantity iz is the "Zeno time" and seemingly yields a quantitative estimate of the short-time behavior. As we shall see in this chapter, this is a misleading estimate in many situations. Strictly speaking, Tz is simply the convexity of P(t) in the origin. Note that if one writes H = Ho+Hu
with Ho\m) = Wo\il^o), (V^ol^ilV^o) = 0,
(3.7)
the Zeno time reads T^'= {iPo\H^\%)-
(3.8)
Therefore iz depends only on the square of the off-diagonal part of the Hamiltonian. Let us now perform N measurements at time intervals r, in order to check whether the system is still in its initial state. The survival probability after the measurements reads
/'no=/'(rf=p(^y
N/ large large
-
^
//
*t
\ A'-^oo ^
)
(3.9) where ^ = A^r is the total duration of the experiment. The N ^^ oo limit was originally named limit of "continuous observation" and regarded as a paradoxical result (Misra and Sudarshan [1977]): Infinitely fi-equent measurements halt the quantum-mechanical evolution and "freeze" the system in its initial state. Zeno's quantum-mechanical arrow (the wave fiinction), sped by the Hamiltonian, does not move, if it is continuously observed. The investigation of the last few years has shown that the QZE is not paradoxical. Although the A/^ ^ oo limit must be considered as a mathematical abstraction (Ghirardi, Omero, Weber and Rimini [1979], Nakazato, Namiki, Pascazio and Ranch [1995], Venugopalan and Ghosh [1995], Pati [1996], Hradil, Nakazato, Namiki, Pascazio and Ranch [1998]), the evolution of a quantum system can indeed be slowed down for sufficiently large A^ (Itano, Heinzen, Bollinger and Wineland [1990], Petrosky, Tasaki and Prigogine [1990, 1991], Peres and Ron [1990], Ballentine [1991], Itano, Heinzen, Bollinger and Wineland [1991], Frerichs and Schenzle [1992], Inagaki, Namiki and Tajiri [1992], Home and Whitaker [1992, 1993], Pascazio, Namiki, Badurek and Ranch [1993], Blanchard and Jadczyk [1993], Altenmuller and Schenzle
3, §3]
Pulsed observation
157
Fig. 3. Evolution with frequent "pulsed" measurements: quantum Zeno effect. The dashed (solid) line is the survival probability without (with) measurements.
0(5x^)
Fig. 4. Short-time evolution of phase and probability.
[1994], Pascazio and Namiki [1994], Schulman, Ranfagni and Mugnai [1994], Berry [1995], Beige and Hegerfeldt [1996], Kofman and Kurizki [1996], Schulman [1997], Thun and Pefina [1998]). The Zeno evolution is shown in fig. 3. In a few words, the QZE is ascribable to the following mathematical properties of the Schrodinger equation. In a short time 8r ^ 1//V, the phase of the wave function evolves like 0((5r), while the probability changes by 0((5r^), so that
pW(Oc:^
•2x1 ^^ [\-0{\/N^)]
N^oo
(3.10)
This is sketched in fig. 4; it is a very general feature of the Schrodinger equation. 3.2. Quantum Zeno and Inverse quantum Zeno effects It is convenient to rewrite eq. (3.9) in the following way {t = Nr) P^''\t) = P(rf
= exp(7VlogP(r)) = exp(-7eff(r)0,
(3.11)
158
Quantum Zeno effects
[3, §3
0 n T2 Fig. 5. (a) Determination of r*. The solid line is the survival probability, the dashed line is the exponential e~'^^ and the dotted line is the asymptotic exponential Ze~^^ in eq. (3.17). (b) Quantum Zeno vs. inverse Zeno ("Heraclitus") effect. The dashed line represents a typical behavior of the survival probability P{t) when no measurement is performed: the short-time Zeno region is followed by an approximately exponential decay with a natural decay rate y. When measurements are performed at time intervals r, we get the effective decay rate YeffiT). The solid lines represent the survival probabilities, and the dotted lines their exponential interpolations, according to eq. (3.11). For Ti < r* < 12 the effective decay rate YeffirO [7eff(^2)] is smaller (QZE) [larger (IZE)] than the "natural" decay rate y. When r = r* one recovers the natural lifetime, according to eq. (3,15).
where we introduced an effective decay rate (3.12)
yeff(r) = - - l o g P ( r ) .
For instance, for times r such that P(T) :^ exp(-r^/r|) with good approximation, one easily checks that yeff is a linear function of r yeffCT") ^
for
0.
(3.13)
Notice that 7eff(^) in eq. (3.12) represents the effective decay rate of a system that evolves freely up to time r and is measured at time r. One expects to recover the "natural" decay rate y (if it exists), in agreement with the Fermi "golden" rule, for sufficiently long times, i.e., after the initial quadratic region is over yeff(r) •''2^' ' Y-
(3-14)
The quantitative meaning of the expression "long" in the above equation represents an interesting conceptual problem and will be tackled in § 8. Suffice it to say, at this stage, that TZ is not the right time scale. We now concentrate our attention on a truly unstable system, with decay rate y. We ask whether it is possible to find a finite time r* such that yeff(r*) = y.
(3.15)
If such a time exists, then by performing measurements at time intervals r* the system decays according to its "natural" lifetime, as if no measurements were performed. By eqs. (3.15) and (3.12) one gets P(T*) = e-'^^*,
(3.16)
i.e., r* is the intersection between the curves P(t) and e~^^ Figure 5 illustrates
3, § 3 ]
Pulsed observation
159
Fig. 6. Study of the case Z > \. The soHd Hne is the survival probability, the dashed line is the renormalized exponential e"^^ and the dotted line is the asymptotic exponential ^e~^^ (a) If P(0 and e"^^ do not intersect, then no finite solution r* exists, (b) If P{t) and e~^^ intersect, then a finite solution r* exists. (In this case there are always at least two intersections.)
an example in which such a time r* exists. By looking at this figure, it is evident that if r = Ti < r* one obtains a QZE. Vice versa, if r = r2 > r*, one obtains an inverse Zeno effect (IZE). In this sense, r* can be viewed as a transition time fi*om a quantum Zeno to an inverse Zeno effect. Paraphrasing Misra and Sudarshan (Misra and Sudarshan [1977]) we can say that r* determines the transition from Zeno (who argued that a sped arrow, if observed, does not move) to Heraclitus (who replied that ever5^hing flows). We shall see that in general it is not always possible to determine r*: eq. (3.15) may have no finite solutions. This will be thoroughly discussed in the following, but it is interesting to anticipate some general conclusions. As we shall see in §§ 7 and 8, for an unstable system and for sufficiently "long" times (the definition of "long" times will be sharpened later) the survival probability reads with very good approximation P{t) = \A(t)\^ c^ ZQ-
(3.17)
where Z, the intersection of the asymptotic exponential with the ^ = 0 axis, is the wave fixnction renormalization and is given by the square modulus of the residue of the pole of the propagator. We claim that a sufficient condition for the existence of a solution r* of eq. (3.15) is that Z < I. This is easily proved by graphical inspection. The case Z < I is shown in fig. 5a: P(t) and e~^^ must intersect, since according to (3.17) P(t) ^ ZQ~^^ for large t, and a finite solution r* can always be found. The other case, Z > 1, is shown in fig. 6. A solution may or may not exist, depending on the features of the model investigated. We shall come back to the Zeno-Heraclitus transition in §§ 7 and 8. The occurrence of an inverse Zeno effect has been discussed by several authors, in different contexts (Pascazio [1996], Schulman [1997], Pascazio and Facchi [1999], Kofinan and Kurizki [1999, 2000], Facchi and Pascazio [2000], Facchi, Nakazato and Pascazio [2001]). There are situations (e.g., oscillatory systems, whose Poincare time is finite) where y and Z cannot be defined. As we shall see, these cases require a different
160
Quantum Zeno effects
[3, § 3
treatment, for the very definition of Zeno effect becomes somewhat delicate. This will be discussed in §§ 6-8. 3.3. Pitfalls: "repopulation" and conceptual difficulties The quantum Zeno effect has become very popular during the last decade, mainly because of an interesting idea due to Cook (Cook [1988]), who proposed to test the QZE with a two-level system, and the subsequent experiment performed by Itano and collaborators (Itano, Heinzen, Bollinger and Wineland [1990]). This experiment provoked a very lively debate and was discussed by many authors (Petrosky, Tasaki and Prigogine [1990, 1991], Peres and Ron [1990], Ballentine [1991], Itano, Heinzen, Bollinger and Wineland [1991], Frerichs and Schenzle [1992], Inagaki, Namiki and Tajiri [1992], Home and Whitaker [1992, 1993], Blanchard and Jadczyk [1993], Pascazio, Namiki, Badurek and Ranch [1993], AltenmuUer and Schenzle [1994], Pascazio and Namiki [1994], Schulman, Ranfagni and Mugnai [1994], Berry [1995], Beige and Hegerfeldt [1996], Schulman [1997], Thun and Pefina [1998]). However, we shall follow here a different route: rather than analyzing Cook's proposal and the related experiment, we shall consider a physically equivalent situation that better suits our discussion and can be easily compared to the analysis of the following sections. The central mathematical quantity considered by Misra and Sudarshan (Misra and Sudarshan [1977]) is "the probability V(0,T;po) that no decay is found throughout the interval A = [0, T] when the initial state of the system was known to be po." (Italics in the original. Some symbols have been changed.) In the notation of § 3.1, this reads V(0,T;po)=\im P^^\Ty
(3.18)
N—^oc
Notice that the above-mentioned "survival probability" is the probability of finding the system in its initial state po at every measurement, during the interval A. This is a subtle point, as we shall see. Consider a three-level (atomic) system, shined by an rf field offi-equencyD, that provokes Rabi oscillations between levels |+) and | - ) . The equations of motion (2.6)-(2.7), with initial condition (in this section we omit the symbol ^ of vector Transposition) R(0) ~ (0,0,1) (only level |+) is initially populated), yield R(t) = (0, sin 2Qt, cos 2Qt).
(3.19)
3, § 3]
Pulsed observation
161
If the transition between the two levels is driven by an on-resonant Jt/2 pulse, of duration T=^,
(3.20)
one gets R{T) = (0,0,-1), so that only level |-) is populated at time T. Perform a measurement at time r = T/N = Jt/INQ, by shining on the system a very short "measurement" pulse, that provokes transitions from level |-) to a third level |M), followed by the rapid spontaneous emission of a photon. The measurement pulse "projects" the atom onto level |-) or |+) and "kills" the offdiagonal terms p ± ^ of the density matrix, while leaving unaltered its diagonal terms p±±, so that, from eq. (2.10),
Then the evolution restarts, always governed by eq. (2.6), but with the new initial condition R^^\ After N measurements, at time T = Nr = Jt/2Q, R(T) = (o, 0, cos^ - ) = R^"^^
(3.22)
and the probabilities that the atom is in level |+) or |-) read (see eq. 2.9) Vi^\T)
= 1 ( l + 7 ? f ) = 1 ( l + cos^ I ) ,
r(N)^T) = \ (l -Rf)
= \(\-
cos^ I ) ,
(3.23) (3.24)
respectively Since V^+\T) ^ 1 and V(_^\T) -> 0 as A^ -^ oo, this looks like a quantum Zeno effect. However, it is not the quantum Zeno effect a la Misra and Sudarshan: eq. (3.23) [(3.24)] expresses only the probability that the atom is in level |+) [|-)] at time T, after N measurements, independently of its past history. In particular, eqs. (3.23)-(3.24) take into account ihQpossibility that one level gets repopulated after the atom has made transitions to the other level. In order to shed light on this rather subtle point, let us look explicitly at the first two measurements. After the first measurement, by eq. (3.21), R
- sin^ ^
= P<') -V<}\
(3.25)
162
Quantum Zeno effects
[3,
:^(l) where VJ^ is the occupation probabiHty of level |±) at time r = JT/INQ, the first measurement pulse. After the second measurement, one obtains
(2)
R'
Jt
after
(3.26)
^os''-^=V['^-V['\
where the occupation probabilities at time 2r = Ji/NQ read JZ
p(2) ^
•pm =
.
7
Jl
Jt -)
(3.27)
Jl
(3.28)
'^^" 2A^^^^ IN'
It is then obvious that V^+\ in eq. (3.27), is not the survival probability of level |+), according to definition (3.18). It is just the probability that level |+) is populated at time t = Jt/NQ, including the possibility that the transition |+) -^ |-) -^ |+) took place, with probability sm^(ji/2N) ' sin^(Jt/2N) = siYi^(Jt/2N). By contrast, the survival probability, namely the probability that the atom is found in level |+) both at the first and second measurements, is given by P|^'^^ = cos^(jr/27V) • cos^(jt/2N) = cos^(jt/2N). Figure 7 shows what happens during the first two measurements. After N measurements, the probability that level |+) is populated at time T, independently of its "history", is given by eq. (3.23), and includes the possibility that transitions to level |-) took place. As a matter of fact, it is not difficult to realize that eqs. (3.23)-(3.24) conceal a binomial distribution:
E
^2«^2(A^-„)
„2N
E
(s/cy2«
Ti:[(:;)(r^(:)<-'"(r ^2N
n=0 L \
/
\
/
[{l^(s/cff^(l-(s/cff] = ^ [l+cos^(jr/iV)]
= Pf Vr), (3.29) where ^ ^ ^^^^ is a sum over all even values of « between 0 and N,s = sin{jt/2N) and c = cos(Ji/2N). Clearly, eq. (3.23) includes all possible transitions between levels 1+) and |-) and is conceptually very different from Misra and Sudarshan's
163
Dynamical quantum Zeno effect
3, § 4 ]
2
v+ = o
c \
T'«=c2
c'
7>f = c4 + s4
c
V^^^ = 2s^c^
2
V- = 0
j,m = ,2
t=0
t = n/N
2/
t=2n/N
Fig. 7. Transition probabilities after the first two measurements for an oscillating system [s = sm(jt/2N) and c = cos(jt/2N)].
survival probability (3.18). The correct formula for the survival probability, in the present case, is obtained by considering only the « = 0 term in eq. (3.29):
PT\T) = cos^
27V'
(3.30)
This is a bona fide "survival probability", namely the probability that level |+) is populated at every measurement, at times nx = nT/N (n= 1,..., A^). The conclusions drawn in this section are always valid when the temporal behavior of the system under investigation is of the oscillatory type and no precautions are taken in order to prevent repopulation of the initial state (Nakazato, Namiki, Pascazio and Ranch [1996]). For instance, this problematic feature is present in the interesting proposal by Cook [1988] and the beautiful experiment by Itano, Heinzen, Bollinger and Wineland [1990]. On the other hand, no repopulation of the initial state takes place in other experiments involving neutron spin (Pascazio, Namiki, Badurek and Ranch [1993]) or photon polarization (Kwiat, Weinfurter, Herzog, Zeilinger and Kasevich [1995]). We have seen that P^f\T), in eq. (3.30), is a bona fide survival probability, but Vf^\T\ in eq. (3.23) is not (at least not according to Misra and Sudarshan's definition). However, both quantities tend to the same limiting value 1 as A/^ ^ oo and for large N the evolution is, in fact, hindered. We are therefore led to wonder whether it would not be meaningfixl to extend the notion of QZE beyond Misra and Sudarshan's definition of survival probability. This will be the subject of §6.
§ 4. Dynamical quantum Zeno effect In the usual formulation of QZE the measurement process is schematized by making use of projection operators a la von Neumann (Copenhagen
164
Quantum Zeno effects
[3, § 4
c+l+>®|l^)
(c+|+) + c_|-))®|l,)
7 C_|->®|1^ Fig. 8. The generalized spectral decomposition.
interpretation), without endeavoring to shed Hght on the underlying dynamics. However, a quantum-mechanical measurement is a very complicated physical process, taking place in a finite time and involving complex (macroscopic) physical systems. It is possible to give a dynamical explanation of the Zeno effect (Pascazio and Namiki [1994], Pascazio [1997]), that involves only the Schrodinger equation and makes no use of projection operators. Let us briefly sketch how this is accomplished by introducing the notion of "generalized spectral decomposition" (GSD). Consider again a two level system, prepared in a superposed state. A GSD is a d5mamical (Hamiltonian) process by which different states of the system become associated (entangled) with different external "channels" (e.g., different degrees of fi-eedom of a larger system). See fig. 8. One can think, for example, of a two-level atomic system getting entangled with different photon states of the electromagnetic field. The notion of "spectral decomposition" was introduced by Wigner [1963], who considered the Stem-Gerlach decomposition of an initial spin state, where each component of the spin becomes associated with a different wave packet. It is worth observing that the external channels the system gets entangled with need not be "external": for example, different wave packets of the system itself can act as "external" degrees of fi-eedom. A GSD is realized by the following Hamiltonian:
//GSD(0
= g{t) [|+)(+|c7^ + \-){-\Oy] Oa = g(t)H\
f \(t) dt = \jt, Jo
(4.1) where the interaction is switched on during the time interval [0, to], g is a. real
3, § 4]
Dynamical quantum Zeno effect
165
function, a^ = o^ (the index ^ = a,P,y labels the channel infig.8) and the effect of (7^ is defined by o,\0,) = \l,),
a^|l,.) = |0^>,
(4.2)
SO that if there is a "particle" in channel fi the operator a^ destroys it, while if there is no particle, a^ creates one. The effect of a^ (V/i) is therefore identical to that of the first Pauli matrix. We set [o^,Oy]=0.
(4.3)
The action of the Hamiltonian //QSD is ^GSD(c+|+)+c_|-))0|la,O^,Oy)a (c+|+>0|O«,l^,Oy>+c_|-)0|O«,O^,ly» (4.4) and consists in sending the |+) (|-)) state of the system in the upper (lower) channel infig.8, thus performing a GSD. In general, the only effect of a GSD is to set up a perfect correlation between the two states of the system and different external channels (namely, a univocal and unambiguous correspondence between different states of the system and different external channels). This is easily accomplished: the evolution engendered by i/osD can be explicitly calculated (Pascazio and Namiki [1994]) and the result is exp - i
['HGSD{t')dt' ( c ^ | + ) + c _ | - ) ) 0 | l « )
^^^^
Jo
= - i (c+|+) (8) |1^) +c_|-) (g) |ly))
(t > to),
where we suppressed all Os for simplicity. A projection operator represents an instantaneous measurement. This is clearly a very idealized situation that cannot correspond to a real physical process, taking place at a microscopic level. The problem is therefore to understand how we can simulate such an instantaneous and unphysical process in our analysis, that makes use only of unitary evolutions. We observe that, in general, a GSD must take place in a very short time. Obviously, the term "very short time" must be understood at a macroscopic level of description, because the time microscopically required to efficaciously perform a GSD can be very long. Therefore, if we restrict our analysis to a macroscopic level of description, we can describe an (almost) instantaneous GSD by means of the so-called impulse approximation to^O\ (4.6) / ''g(t)dt='.Jt, . , Jo which roughly amounts to setting g(t) -^ ^Jtb(t) as to -^ 0, where 6 is the Dirac fiinction/J'8(0 = 1. This is our alternative description of a von Neumann-like
166
Quantum Zeno effects
[3, § 4
instantaneous projection. It is a good approximation of the physical situation whenever ^ is much shorter than the characteristic time of the free evolution of the system under observation. By making repeated use of GSDs it is very simple to get quantum Zeno dynamics. A general proof is given by Pascazio and Namiki [1994] (a somewhat simpler version can be found in Pascazio [1997]), but here let us only sketch the main idea by looking at the example (2.1). The initial state (2.3), that we rewrite by including the external channel (wave packet) in the description, |^o) = | + ) 0 | l « ) ,
(4.7)
evolves after a short time r into state (2.4): 1^,) = e-'^'H^o) = [cos(Or)|+) -isin(r2T)|-)] 0 |1«).
(4.8)
The GSD then yields (for ^ < ^2"^) l^r + ro)=exp -i r HG^^{t')dt' \^r) Jo /o oc cos(Dr)|+) (g) \lp) -ism(QT)\-)
(4.9) (g) |ly).
apart from a phase factor. Observe that the quantum coherence is perfectly preserved, during this evolution. At the next "step" of the evolution, channels (3 and y become new incoming channels and the system evolves again under the action of//i for a time r and ^GSD for a time to. After N steps the final wave function reads ^^(r..o)) = n
| e x p [ - i 2 V S D ( ^ 0 c i / ' ] exp[-i//ir]| \Wo)
oc cos^(Or)|+) (g
'ft
(4.10)
\lf^)+0(N-^),
An)
where //QSQ is the Hamiltonian that performs a generalized spectral decomposition at the «th step and | iL^^) (all Os were suppressed) represent the wave packet traveling in channel j8 at step N. Note that N(T + to) is kept finite. The contribution of all the other channels is 0(A^"^): a QZE is obtained because the particle, initially in state (4.7), ends up with probability [1-0(1/A^^)]^ ~ 1-0(1/7V)
(4.11)
in state |+) 0 |1^ ^). The "external" degrees of freedom are irrelevant and can be traced out (or recombined with the initial one).
3, § 5]
Continuous observation
167
We would like to emphasize that the very dynamical mechanism leading to QZE is curious: QZE is obtained via repeated use of generalized spectral decompositions HGSD% ^ven though the interaction Hamiltonian Hi "attempts" to drive |+) into |-) for a finite time Ni. This is probably the reason why QZE is often considered a counterintuitive phenomenon.
§ 5. Continuous observation A projection a la von Neumann (von Neumann [1932]) is a handy way to "summarize" the complicated physical processes that take place during a quantum measurement. A measurement process is performed by an external (macroscopic) apparatus and involves dissipative effects, that imply an exchange of energy with and often a flow of probability towards the environment. The external system performing the observation need not be a bona fide detection system, namely a system that "clicks" or is endowed with a pointer. It is enough that the information on the state of the observed system be encoded in the state of the apparatus. For instance, a spontaneous emission process is often a very effective measurement process, for it is irreversible and leads to an entanglement of the state of the system (the emitting atom or molecule) with the state of the apparatus (the electromagnetic field). The von Neumann rules arise when one traces away the photonic state and is left with an incoherent superposition of atomic states. We shall now introduce several alternative descriptions of a measurement process and discuss the notion of continuous measurement. This is to be contrasted with the idea of pulsed measurements, discussed in §3. Both formulations lead to QZE. 5.1. Mimicking the projection with a non-Hermitian Hamiltonian It is useful for our discussion on the QZE and probably interesting on general grounds to see how the action of an external apparatus can be mimicked by a non-Hermitian Hamiltonian. Let us consider the following Hamiltonian:
^^^(S -i?r)^"'^^^^'''' ^ = (^'0,iK)\
(5.1)
that yields Rabi oscillations of frequency Q, but at the same time absorbs away the |-) component of the Hilbert space, performing in this way a "measurement".
168
Quantum Zeno effects
[3, §5
Pit) 1
, • • • ' • • " " " '
_i;;^= 10Q \
N.
/
l+>\\ Q.
0.5
/ ,/
\
Y=AV
0.5
Q^/TT
Fig. 9. Survival probability for a system undergoing Rabi oscillations in the presence of absorption {V = 0.4,2, \0Q). The gray line is the undisturbed evolution (V = 0).
Due to the non-Hermitian features of this description, probabihties are not conserved: we are concentrating our attention only on the |+) component. An elementary SU(2) manipulation yields the following evolution operator: h • iJ
^-m = ^-vt cosh(/zO - i ^ — sinh(/zO
(5.2)
where h = VV^ - Q^ and we supposed V > Q. Let the system be initially prepared in the state (2.3): the survival amplitude reads
=e
sinh(VF2-^2^)
cos h(VF2-r22/) + y/V^-Q^ -{V-\/V^-Q^)t
^1 1+
(5.3)
2
V
-{VWV^-Q^)t
The above results are exact and display some interesting and very general aspects of the quantum Zeno dynamics. The survival probability P{t) = \A(t)\^ is shown infig.9 for F = 0.4,2,10^. As expected, probability is (exponentially) absorbed away as ^ —» oo. However, as V increases, by using eq. (5.3), the survival probability reads
/'(0~(l + g)exp(-^r),
(5.4)
and the effective decay rate 7eff(F) = ^^/V becomes smaller, eventually halting the "decay" (absorption) of the initial state and yielding an interesting example
3, § 5]
Continuous observation
169
of QZE: a larger V entails a more "effective" measurement of the initial state. We emphasize that the expansion (5.4) becomes valid very quickly, on a time scale of order V'^. Notice that this example is not affected by the repopulation drawback described in § 3.3 (once the probability is absorbed away, it does not flow back to the initial state). 5.2. Coupling with aflat continuum We now show that the non-Hermitian Hamiltonian (5.1) can be obtained by considering the evolution engendered by a Hermitian Hamiltonian acting on a larger Hilbert space and then restricting the attention to the subspace spanned by {|+), | - ) } . Consider the Hamiltonian i / = 0(|+)(H + |-)(+|) + | d a ; H ^ ) H + y ^ / d a > ( | - ) H + |w)(-|), (5.5) which describes a two-level system coupled to the photon field in the rotatingwave approximation. The state of the system at time t can be written as \xl)t) =x(t)\-^)+y(t)\-)
+ Jdajz{(o,t)\a)),
(5.6)
and the Schrodinger equation reads ix(t) = Qy{tX iy(t) = Qx{t) + \hr-
/ dco z(ft), t),
(5.7)
i z(a;, t) = o)z{w, 0 + y ^ >'(^)By using the initial condition x(0) = 1 and ^(0) = Z{(D, 0) = 0 one obtains z(ca, t) = -i\hf- I dr e-^^('-"V(^) V In Jo
(5.8)
iy{t) = ^ ^ ( 0 - i ^ fdcofdr
(5.9)
and e-^"(^-^V(r) = Qx(t)-i^y(ty
Therefore z(a),t) disappears from the equations and we get two first-order differential equations for x and y. The only effect of the continuum is the
170
Quantum Zeno effects
[3, § 5
appearance of the imaginary frequency -iF/l. Incidentally, this is ascribable to the "flatness" of the continuum [there is no form factor or frequency cutoff in the last term of eq. (5.5)], which yields a purely exponential (Markovian) decay of>^(0. In conclusion, the dynamic in the subspace spanned by |+) and |-) reads ijc(0 = Qy{t\
{y{t) = -i^y-\-Qx(t).
(5.10)
Of course, this dynamic is not unitary, for probability flows out of the subspace, and is generated by the non-Hermitian Hamiltonian / / = f2(|+)(-| + | - > ( + | ) - i | | - ) ( - | .
(5.11)
This Hamiltonian is the same as (5.1) when one sets F = 4V. QZE is obtained by increasing F: a larger coupling to the environment leads to a more effective "continuous" observation on the system (quicker response of the apparatus), and as a consequence to a slower decay (QZE). The processes described in this section and the previous one can therefore be viewed as "continuous" measurements performed on the initial state. The non-Hermitian term - 2 i F is proportional to the decay rate F of state | - ) , quantitatively F = 4V. Therefore, state |-) is continuously monitored with a response time l/F: as soon as it becomes populated, it is detected within a time 1/r. The "strength" T = 4F of the observation can be compared to the frequency T~^ = (t/Ny^ of measurements in the "pulsed" formulation. Indeed, for large values of F one gets from eq. (5.4) 4Q^ 4 7eff(r)--- = ^ - , i
r^i
for
r-.oc,
(5.12)
which, compared with eq. (3.13), yields an interesting relation between continuous and pulsed measurements (Schulman [1998]) r - -
4 T
4N =_ .
(5.13)
t
5.3. Continuous Rabi observation The two previous examples might lead the reader to think that absorption and/or probability leakage to the environment (or in general to other degrees
3, § 5]
Continuous observation
171
of freedom) are fundamental requisites to obtain QZE. This expectation would be incorrect. Let us analyze a somewhat different situation by coupling one of the two levels of the system to a third one, which will play the role of a measuring apparatus. The (Hermitian) Hamiltonian is /fi = 0 ( | + ) H + |-)(+|) + ^ ( | - ) ( M | + |M>(-|)=
/O O \0
Q 0 ^
0\ K], 0/
(5.14)
where AT G R is the strength of the coupling to the new level M, and (+1 = (1,0,0),
(-1 = (0,1,0),
{M\ = (0,0,1).
(5.15)
This is probably the simplest way to include an "external" apparatus in our description: as soon as the system is in | - ) , it undergoes Rabi oscillations to \M). Similar examples were considered by Peres [1980] and Kraus [1981]. We expect level \M) to perform better as a measuring apparatus when the strength K of the coupling becomes larger. The above Hamiltonian is easily diagonalized. Its eigenvalues and eigenvectors are Ao = 0,
\uo) =
, ^^
^ 1 ^ 1 '
V2(K^^m
\
K
J
Let the initial state be
The evolution is easily computed:
(5.18) and the survival probability reads p(t)=
1
2
K^ + Q^ cos(y/K^ + Q^ t)
This is shown in fig. 10 forK=
1,3,9Q.
(5.19)
172
Quantum Zeno effects
[3, § 5
p(t) 1
^;-5!^-S^>^
^^^''~''^,<-"~
^^XT^'^^^^'^^ IM>-
\ 0.5
^^
/
\
/^-^
/
\ir}
^
^
x^^^—
0
Qt/lT
0.5
Fig. 10. Survival probability for a continuous Rabi "measurement" with K= 1,3,9Q\ quantum Zeno effect.
We notice that for large K the state of the system does not change much: as K is increased, level \M) performs a better "observation" of the state of the system, hindering transitions from |+) to | - ) . This can be viewed as a QZE due to a "continuous", yet Hermitian observation performed by level \M). This simple example triggers also another remarkable observation. The Zeno time is easily computed and turns out to be much longer than the Poincare time (we are assuming K :$> Q)
rz = Q-^ >rp = o(A:-').
(5.20)
As a matter of fact, the Zeno time yields only the convexity of the survival probability in the origin: it does not give any relevant information about the duration of the short-time quadratic region. This contradicts many erroneous claims in the literature of the last few years. We shall come back to this point in § 8. A few more comments are necessary. First of all, the example considered in this section is not free from repopulation effects like those considered in § 3.3. As a matter of fact, the situation here is even worse: unlike the case studied in § 3.3, where there was a probability repopulation of the initial state, in the present case there is a (coherent) amplitude repopulation phenomenon. However, even if these cases are at variance with Misra and Sudarshan's definition [see (3.18) and the paragraph preceding it], they call, in our opinion, for a broader formulation of QZE. This will be proposed in the following section. Let us see how "effective" the Rabi "measurement" is, compared to the case of pulsed measurements. Notice that by performing pulsed observations on system (2.1) one gets from eq. (3.5)
^-«-(^)"=(cosf)"-(.
#2
1-
N
(5.21)
3, § 6]
Novel definition of quantum Zeno effect
173
for large values of N. On the other hand, in the present case of continuous observation, for large values of ^ , eq. (5.19) reads P'^)(0~ ( l - 2 ^ )
(l+2gcos(^^^772^o)
whence, by taking the average over a very short time of order l/K, . O ,2
P(K)(t)~l-2r^.
(5.23)
By comparing eq. (5.21) and eq. (5.23), one sees that the evolution is hindered (QZE) for ^ - ^ ( « l ) ,
(5.24)
namely K^^-f.
(5.25)
This relation is similar to (5.13): strong coupling is equivalent to frequent measurements. A final comment is in order. All the situations analyzed in § 5 lead to QZE but never to IZE. The reason for this is profound and lies in the absence of the form factors of the interactions. The importance of form factors and the role they play in this context will be discussed later.
§ 6. Novel definition of quantum Zeno effect The diverse examples and particular cases considered in the previous sections motivate us to look for a broader definition of Zeno effect, that includes "continuous" observations as well as somewhat delicate situations in which repopulation effects (in amplitude or probability) take place. Let us consider a quantum system whose evolution is described by a Hamiltonian H. Let the initial state be po (not necessarily a pure state) and the
174
Quantum Zeno effects
[3, § 6
survival probability P{t). Consider the evolution of the system under the effect of an additional interaction, so that the total Hamiltonian reads HK-H
(6.1)
+ H^UK\
where K'lsdi set of parameters and Hmeas(K = 0) = 0. This Hamiltonian includes also as a particular case the GSD described in § 4; moreover, since a GSD is basically equivalent to a bona fide measurement, the above Hamiltonian includes, for all practical purposes, the usual formulation of quantum Zeno effect in terms of projection operators. Notice that H is not necessarily the free Hamiltonian; rather, one should think of ^ as a full Hamiltonian, containing interaction terms, and //meas(^) should be viewed as an "additional" interaction Hamiltonian performing the "measurement". We have considered many examples in this chapter: all of them fit in the scheme (6.1). We shall say that the system displays a QZE if there exists an interval / W = [/^^),4^)] such that P^^\t) > P{t\
\/t e I^^\
(6.2)
where P^^\t) and P{t) are the survival probabilities under the action of the Hamiltonians HK and H, respectively. We shall say that the system displays an inverse QZE if there exists an interval /^^^ such that P^^\t) < P{t\
\/t e I^^K
(6.3)
The time interval I^^^ must be evaluated case by case. However, tf^ ^ Tp,
(6.4)
where 7p is the Poincare time. Obviously, for the definition (6.2)-(6.3) to be meaningful from a physical point of view, the length of the interval I^^^ must be of order Tp. The above one is a very broad definition, for it includes a huge class of systems (even trivial cases like time translations P(t) -^ P{t - to) are included). We have not succeeded in finding a more restrictive definition and we do not think it would be meaningful. This is in line with our general philosophy: the Zeno effects are very common phenomena. In order to elucidate the meaning of the above definition, let us look at some particular cases considered in this chapter. The situations considered in figs. 9 and 10 are both QZEs; according to this definition one has ^^ = 0 and tf^ ^ Tp = Jt/Q [and (rf^ - rj^^) = 0(rp)]. The case outlined in fig. 3 is also a QZE, with tf^ = 0 and 4^^ ^ Tp (notice that Tp may even be infinite).
3, § 7]
Zeno effects in down-conversion processes
175
If we deal with an unstable system, the definition of Zeno effect can be made more stringent, by simply generalizing the results of § 3.2 to a broader class of measurements. Indeed, in such a case, one need not refer to an interval /^^^ and can consider the global behavior of the survival probability. By introducing the lifetime y, one can define the occurrence of a QZE or an IZE if YMK)
S
7,
(6.5)
respectively, where yeff(^) is the new (effective) lifetime under the action of HK. Notice that this case is in agreement with the definitions (6.2)-(6.3). Moreover, 4 —> oo for IZE, while tf'^ ^ ^pow for QZE, where ^pow is the time at which a transition from an exponential to a power law takes place. [Such a time is of order log (coupling constant), at least for renormalizable quantum field theories (Facchi and Pascazio [1999]).] The definition (6.5) includes all the cases considered in §3.2. See for example fig. 5b. Notice that when a Zeno effect is obtained by repeated use of projection operators (at equal time intervals r), one always gets an exponential behavior with a well defined yeff(^) (see § 3.2). A problem arises with oscillating systems (or in general with systems whose Poincare time is finite) because of the impossibility of defining the "natural" decay rate y (see, for instance, fig. 9). From this perspective, we cannot help feeling that the very concept of QZE is somewhat less meaningful for purely oscillating systems, exhibiting no bona fide instability. We shall adopt these new definitions of Zeno effects in the following. They are novel and have not been proposed before. They work in all the cases considered in this chapter and also comprise, in a more general framework, all the examples of Zeno effects considered in the literature. These definitions should be kept in mind while considering the examples proposed in the following sections.
§ 7. Zeno effects in down-conversion processes The differences and analogies between pulsed and continuous observations analyzed in the previous sections will now be discussed by considering a quantum optical example. A down-conversion process in a nonlinear crystal can be thought of as the decay of a pump photon into a pair of signal and idler photons of lower frequency. If the pumping is sufficiently strong and there is phase matching, the energy of the spontaneously down-converted light monotonously increases and that of the pump beam monotonously decreases. In this sense, the down-conversion process may be looked at as the decay process of an unstable system.
176
Quantum Zeno effects
[3, § 7
Let us first discuss the case of "pulsed" observation. There are similar ideas in the literature (Pascazio, Namiki, Badurek and Ranch [1993], Kwiat, Weinfurter, Herzog, Zeilinger and Kasevich [1995], Facchi, Klein, Pascazio and Schulman [1999]), but here we shall discuss an interesting example first proposed by Luis and Pefina [1996]. A pump beam illuminates a nonlinear crystal that is transversely cut in N pieces, which are then carefiilly aligned so that the signal and pump photons leave a given slice that becomes the input signal and pump photons for the next slice of the crystal, while the idler photons are taken out at each step (see fig. 12 in the following). By increasing the number N of slices, the probability of emission of the down-converted pair decreases: this is QZE. However, if the phase matching condition is not fulfilled in the process of down-conversion (Luis and Sanchez-Soto [1998], Thun and Pefina [1998]), the observation may, on the contrary, enhance the emission for a properly chosen N\ this is an IZE. We shall see that such a behaviour occurs also when, instead of cutting the crystal into N pieces, the idler beam is coupled to an auxiliary mode (see fig. 13 in the following). The continuous interaction with the auxiliary mode is a sort of "steady gaze" at the system and performs a continuous observation. The Zeno-inverse Zeno interplay that takes place in this model can be easily understood in the light of the techniques outlined in § 3.2 and will be the object study of this section. 7.1. The system Consider a nonlinear crystal through which three modes, pump /?, signal s and idler /, propagate in the same direction. The nonlinear waveguide is filled with a second-order nonlinear medium in which ultraviolet pump photons are downconverted into signal and idler photons of lower fi-equency. We will assume that all modes are monochromatic and their frequencies fixed, e.g., by placing narrow interference filters in front of the detectors. Provided the amplitudes of the fields vary little during an optical period (SVEA approximation), the effective Hamiltonian reads {h= \) H = WpOJjap + 0)salas -\- o)ia]ai 4-apala]e^'-^alasaiC-'^' F
(7.1)
where cOa is the frequency of mode a, A = (kp -ks- ki)z is the nonlinear phase mismatch and the propagation variable z has been replaced with the evolution parameter /. The nonlinear coupling constant F is proportional to the second order nonlinear susceptibility x(2) (Hong and Mandel [1985]). We suppose that
3, § 7]
Zeno effects in down-conversion processes
177
the incident pump field is intense and that the pump mode Up can be treated classically, as a field of complex amplitude ap = ^ Qxp(-i(Opt), where ^ and cOp denote the complex amplitude and the fi-equency of the classical pump wave, respectively. In this approximation the Hamiltonian (7.1) has only two quantized field modes and reads H = (Ogalas + (jOiaJat +•^Ute-i(a^,-^X F
+ ^
e^(a;,-^^^
(7.2)
where the amplitude ^ has been absorbed in the coupling constant F (taken real for simplicity). Notice that the strong pump wave approximation will cease to be valid once appreciable depletion of the pump field occurs. Therefore the solution of eq. (7.2) properly describes the process of parametric downconversion under the restriction {nsj(t))
which obey the same commutation rules as the a% the Heisenberg equations of motions take the form a', = -iW,,H%
a[ = -i[a[,H'l
(7.4)
with the time-independent Hamiltonian
H'=iaU:+ia%UF
o!}d} + a>;
(7.5)
where the frequency matching condition co^ = a;^ + o^i was used. The state of the field at time ^ = 0 is taken to be the vacuum for the signal and idler modes |t/^o) = |0„0,).
(7.6)
Under the action of the Hamiltonian (7.5) this state is unstable and spontaneously decays, continuously generating photon pairs. For example, when z\ = 0, the average number of signal and idler photons originating in the crystal of length t, (at(Oa.(0) = (V^ol4(0«.(OlV^o> = (t/^ol4(O^KOlV^o) = sinh'(rO,
(7.7)
is an exponentially increasing function of ^ In eq. (7.7) and henceforth all (slowly varying) operators are written without primes to simplify the notation.
178
Quantum Zeno effects
[3, § 7
Our interest is focused on the survival amplitude of the vacuum state under the action of the Hamiltonian (7.5). It is somewhat more convenient to consider the evolution of the following linear combinations a = - ^ ,
b = - ^ .
(7.8)
In terms of these two modes the Hamiltonian (7.5) reads H = | a t « + L [«t2 + ^2] ^ l^tft _ £: [^t2 + ^2]
(79)
and the modes a and ^ are completely decoupled. It is now straightforward to evaluate the time evolution of |i/^o) by considering the properties of the generalized two-photon coherent states (Mandel and Wolf [1995]). Remember that by letting U(jj.,v) be a unitary transformation that generates the pseudo-annihilation operator A{iu, V) = U{^, V) a U\ii, V) = ^a + vaK
with
\fi\^ -1 v|^ = 1,
(7.10)
for two complex numbers /i, v, the generalized two-photon coherent state |[^, y;u]) is defined by operating U(fi, v) on the coherent state |t;), i.e., \[lx,v;u]) = U(ii,v)\u).
(7.11)
The scalar product between this state and a coherent state has the following form {o\[^,y;w]) = - ^ e x p
111?
II
i2
1 ^
*2
^
*
1 ^
2
(7.12) and the most general unitary transformation that has the property (7.10) is U(ti, V) = exp[-i(^a^fl + A«^ + A:*a+^)],
(7.13)
where K is real and k, a constant. The complex numbers // and v are related to the parameters K and k by the relation rM = cosh(A^)4-i^si^^^^^ \ V = i H sinh(A/:),
with
where we considered, for simplicity, k real.
Ak = V4k^^K-\
(7.14)
179
Zeno effects in down-conversion processes
3, § 7 ]
Let us look at the survival amplitude of the vacuum state under the action of the quadratic Hamiltonian (7.9), corresponding to a two-photon interaction with a classical pump for the two independent modes a and b. By using eq. (7.12) this reads A{t) = (0,, 0,|e-^^|0„ 0,) = (0,, Ob\[^a(tl Vait); 0,], [^,(0, v^(0; Ob]) 1 (7.15) y/fla(t)!^b{ty In our case A l^a{t) = libit) = cosh(ArO + i^T-TT; 2Ar sinh(ArO,
with
/ A^ A r = \ F^ 4 ' (7.16)
whence one gets the survival probability P{t)=\A{t)\
2 _
1
lM.(OP
cosh^ArOH-
4Ar2
sinh'(ArO
(7.17)
We can now look at the features of this system. At short times p{t) -1 - ^^^f
---f
= \-
(7.18)
r^t\
4 so that the Zeno time reads (7.19)
T£2 = (H^) = — (0„,0i|(aV2 + 62fet2)|0„06) = r\
The long-time behavior depends on the value of Ar^ in (7.16): if A < IF, at long times, P(t)-
1 LQ^AH^
A^ ^
i6Ar2
4Ar^ 2Art
-2Art
(7.20)
so that .
P{t)-ZQ-^\
where
y^\/Ar^-A^
4 r 2 _ A2
and
Z=
-.—.
(7.21)
This formula, as already emphasized in §3.2 (see eq. 3.17), is an excellent approximation at long times and enables us to discuss the Zeno-inverse Zeno
180
[3, §7
Quantum Zeno ejfects
P{t)
Pit)
Fig. 11. Survival probability of the initial (vacuum) state, (a) A = \/2r (< VSF) and Z = 2; the solid line is the survival probability (7.17), the dotted line is the asymptotic exponential (7.21), and the dashed line is the renonnalized exponential exp(-70- ^ ( 0 ^nd exp(-yO do not intersect, r* does not exist, and only a QZE is possible, (b) A = VlTir (V^F < A < IF) and Z = 0.5; the solid line is the survival probability (7.17), the dotted line is the asymptotic exponential (7.21), and the dashed line is the renormalized exponential exp(-yO. Pit) and exp(-yO intersect, r* exists, and a Zeno-inverse Zeno transition is possible. The gray line is the survival probability (7.22) for A = 3r {> iry. m this case one gets an oscillatory behavior and Z cannot be defined.
transition. The condition Z <\ reads A > VSF. In fig. 11 the vacuum survival probability (7.17) is shown for different values of the parameters. We shall see in § 8 that, in general, Z is due to the renormalization of the wave function. As we have seen, when A < IF the survival probability decreases exponentially; on the other hand, when A > IF the behavior is oscillatory
p(o = i A o r -
1
l^.(OP
cos2(|Ar|0 +
4|Ar|^
• sin^(|Ar|0
(7.22)
with 4|Arp/Z\^ < P(t) < 1. In this case a decay rate y and, as a consequence, Z cannot be defined. We note that the vacuum state never decays completely. We can now discuss pulsed and continuous observation. 7.2. Pulsed observation Let us first consider pulsed observations performed at time intervals r = t/N (Luis and Pefina [1996]). The nonlinear crystal is divided into N equal parts of length L/N, corresponding to an interaction time r = t/N, as shown in fig. 12. Assume that the signal beams at each slice are perfectly superimposed and aligned and that reflection at each step is made negligible, for instance by embedding the A^ pieces in a linear medium with exactly the same refi-active index. On the other hand, the idler path is interrupted after each slice, for instance by means of mirrors. At each step the output idler beam is completely removed and replaced by a new input idler beam in the vacuum state. With this
181
3, § 7 ]
Fig. 12. Outline of a "sliced" parametric down-conversion scheme. The down-converter is cut into N crystals of length L/N. After each slice, the output idler beams a^- ^ are removed by means of mirrors inserted in the idler path and replaced by different input idler fields aik in vacuimi.
modification it is possible to detect the emission of the idler photons, for instance, by means of N photodetectors. By using the definition (7.8) and the evolution law (7.10) or, alternatively, by directly solving the Heisenberg equations (7.4) for the Hamiltonian (7.5), one gets for a single slice f as(r) = li{T)as + v(r)aj 1 ai{r) = ii{T)ai + v{r)a\ '
(7.23)
where pi{t) = fiaiO is defined by eq. (7.16) and v(t) = Va(t) = i r sinh(ArO/Ar. Remember that unitarity requires |^(0P - k(OP "" 1We study how the survival probability of the vacuum state is modified by frequent interruptions of the idler path. To this end we will look at the modified evolution of the signal mode, following Luis and Pefina [1996], Luis and Sanchez-Soto [1998] and Thun and Pefina [1998]. By using (7.23) we can express the signal annihilation operator after the Nth slice as in terms of the annihilation (creation) operator of the signal (idler) mode before it W = Ai(r)af-i> + v(r)a^
(7.24)
a
where we used the fact that a different vacuum mode atk, k = I,... ,N, is at the idler input of each of the N crystals. By iterating eq. (7.24) we obtain
«f^ = MV+v^M^-*4.
(7.25)
k=l
The mean value of the number of signal photons reads
(af)t«f)) = |vpEH 2iN-k) k=i
= V
2i/^r-i_ IMP
= iMr-i,
(7.26)
182
Quantum Zeno effects
[3, § 7
where the unitarity condition was used. For large values of A^, by making use of eq. (7.17) we get ( a f ^^flf ^) = \Kt/N)\^^ - 1 = Pit/N)-'' - 1 - 1 -P{t/Nf
= 1 -P^^HO (7.27) and the mean number of photons coincides with the probability of emitting one signal photon (for the probability of emission of more than one photon is negligible), i.e., with the (modified) decay probability of the vacuum state. By using the short-times expansion of the survival probability (7.18) we get P^^HO-exp(-r2rO,
(7.28)
i.e., an effective decay rate yeff(r) = r 2 r ,
(7.29)
which is in accord with eq. (3.13), because iz = l/F (as shown by eq. 7.19). In the TV ^ oo limit the effective decay rate approaches zero and the decay is completely frozen, i.e., no photons are emitted (QZE). If, in (7.21), Z>\^A<
V3r,
(7.30)
we are in the situation outlined in figs. 6a and 11a, and according to the analysis of § 3.2 only a QZE can occur. On the other hand, if Z<\,
(7.31)
then, according to the analysis of §3.2 (see figs. 5 and 6), a transition time r* exists and by decreasing the frequency of measurements one observes the transition fi-om a Zeno to an inverse Zeno (Heraclitus) regime. By using eq. (7.21), the condition (7.31) reads A > Vsr.
(7.32)
In particular, if the phase mismatch A is close to the value A = 2 r , the linear approximation (7.29) is valid up to r*, because y in eq. (7.21) approaches zero, and we get r* =
'- = J-1?
=_ =
whence for A/^* < F^t/IAF the photon production is enhanced (IZE).
(7 33)
3, § 7]
Zeno effects in down-conversion processes
183
So far, we supposed A
2r (see eq. 7.22). Indeed, in this case the phase mismatch is so large that the downconversion process is no longer exponential, but has an oscillatory behavior
(at(fK(0) = mt |Arp
-1 = cos^dArio + - ^ sin^dArio -1 ' ' (7.34) sin^(|Ar|0,
which is bounded by (al(?K(0>MAX = T ^ .
(7.35)
On the other hand, by cutting the crystal and removing the idler path, one gets ( a f )t«f)) = (^1 + ^
y
_ 1 . expCr^rO- 1,
(7.36)
and an explosive exponential behavior is recovered (IZE). It is easy to check that r* = 0, a quite remarkable situation. Independently of the frequency of measurements N, one always obtains an IZE, see fig. l i b (a QZE is recovered only in the iV ^ oo limit). Remember also that Z cannot be defined, in this case. We will rigorously justify this interpretation in § 8, by considering complete and incomplete Rabi oscillations as the limiting case of a truly unstable system with a finite-width form factor. We notice that the process described is always unitary and it actually makes no difference whether any measurements on the idler modes are actually carried out or not. It is sufficient that such measurements could in principle be made, as stressed in § 4. We also emphasize that the situation just analyzed is affected by (probability) repopulation effects like those described in § 3.3. 7.3. The nonlinear coupler: continuous observation We now modify the system considered in the previous subsections and discuss continuous observation. Consider a nonlinear coupler made up of two waveguides, through which four modes, pump /?, signal s, idler / and auxiliary mode b propagate in the same direction, see fig. 13. The nonlinear waveguide is again filled with a second-order nonlinear medium in which ultraviolet pump photons are down-converted to signal and idler photons of lower frequency.
184
[3, §7
Quantum Zeno effects signal
pump p
i
1-1
nonlinear waveguide
1 1 \
t
idler
linear waveguide
Fig. 13. Outline of the nonlinear coupler.
but in addition, the idler mode is allowed to exchange energy, e.g., by means of evanescent waves, with the auxiliary mode b propagating through a linear medium (Rehacek, Pefina, Facchi, Pascazio and Mista [2000]). We assume again the validity of the SVEA approximation and we consider the linear coupling weak enough so that it can be described by the coupled modes theory (Bom approximation) (Stich and Bass [1985], Yariv and Yeh [1984], Saleh and Teich [1991]). With the help of the strong pump wave approximation the Hamiltonian of our problem is simplified as follows: H^=H + Hm,as(i^l
with
HmcasW = ic(a^ib + aib'^) + a)bb'^b,
(131)
where H is the down-converter Hamiltonian (7.2) and K the linear coupling constant. By introducing the slowly varying operators (with a slightly different choice for a'^ and a-, which is somewhat more convenient for the following discussion)
a: = e^'^'as
a[ = e^'^'-^^'at.
b' = e^'^'-^^'b,
(7.38)
we get, instead of (7.5), the new time-independent Hamiltonian //x- = Aa]ai + r{ala] + a^a/) + Ab^b + K{a]b + atb^).
(7.39)
where we assumed again that thefi"equencymatching conditions, cOp-Ws- cOi = 0 and cOb = cOj, hold and we suppressed all primes. The dynamics of the nonlinear coupler (7.39) reduces to the dynamics of the spontaneous down-conversion process (7.5) provided that /c = 0. In the case of phase matching A = 0, the average number of signal and idler photons originating in the crystal of length t, eq. (7.7), is an exponentially increasing function of ^ The behavior of the down-conversion process changes completely when one of the two down-converted modes (say, the idler mode) is coupled to an auxiliary
3, § 7]
185
Zeno effects in down-conversion processes
mode via the linear interaction, performing the "continuous observation". The Hamiltonian (7.39) yields, when ^ = 0 (phase matching), Qs = -if a], at = -iral - iKb,
{A = 0)
(7.40)
b = -IK at,
and we are interested in the regime of weak nonlinearity, expressed by the condition K > F. Notice that two opposite tendencies compete in eqs. (7.40): an elliptic structure, leading to oscillatory behavior, governed by the coupling parameter /f, 'di = -K^Qi,
b = -K^b
(7.41)
and a hyperbolic structure, yielding exponential behavior, governed by the nonlinear parameter T, hs = r^as,
di = r^at.
(7.42)
The threshold between these two regimes occurs for F ^ K. The system of equations (7.40) is easily solved and the number of output signal photons, which is the same as the number of pump photons decays, reads (al(0«.(0> = ^ sin^ xt + ^ ( 1 - cosxtf, A
(7.43)
A
where x == V/c*^ ~ F^. Unlike the case of phase matched down-conversion (7.7), the exchange of energy between all modes now becomes periodical when K > F. As the linear coupling becomes stronger, the period of the oscillations gets shorter and the amplitude of the oscillations decreases as /c"^, namely {al(t)as(t))
F^ F^ T sin^ Kt+—r(l/C^
K^
4F^ Kt cos Ktf = —^ sin^ — /f^
2
( / c > F).
(7.44) For strong coupling the down-conversion process is completely frozen, the medium becomes effectively linear and the pump photons propagate through it without "decay". [In the regime of very large /c, however, the coupled modes theory breaks down and some other experimental realization of the Hamiltonian (7.39) should be found.] Notice that in this situation, even if t is increased, the number of down-converted photons is bounded [compare with the opposite
186
Quantum Zeno effects
[3, § 7
case (7.7)]. This is QZE in the following sense: by increasing the coupling with the auxiliary mode, a better "observation" of the idler mode (and therefore of the decay of the pump) is performed and the evolution is hindered. There is an intuitive explanation of this behavior: since the linear coupling changes the phases of the amplitudes of the interacting modes, the constructive interference yielding exponential increase of the converted energy (7.7) is destroyed and down-conversion is frozen (see § 7.5 in the following). In agreement with the final part of §5.3, by comparing eq. (7.44) with eqs. (7.27)-(7.29), we find that the linear coupling is effective as the square root of the number of pulsed measurements, namely
.-^-f.
,7.45,
Consider now the Hamiltonian (7.39) when K = 0, describing down-conversion with phase mismatch A. It is apparent that the coupling and the phase mismatch influence the down-conversion process in the same way. Indeed for large values of the phase mismatch A it is easy to find from eq. (7.34) that («t(?K(0) ~ ^
sin^ y
(A » n,
(7.46)
which is to be compared with eq. (7.44). The interesting interplay between coupling K and mismatch A will be investigated in the following subsection. 7.4. Competition between the coupling and the mismatch In the previous section we saw that the nonlinear interaction was affected by both linear coupling and phase mismatch in the same way. The effectiveness of the nonlinear process drops down under their action. In this section we show that when both disturbing elements are present in the dynamics of the downconversion process, the linear coupling can compensate for the phase mismatch and vice versa, so that the probability of emission of the signal and idler photons can almost return back to its undisturbed value. We start from the equations of motion generated by the full interaction Hamiltonian (7.39) Us = -iFa], at = -lAai - ira] - iKb,
(A^O,K^
0).
(7.47)
b = -iAb - iKai Although it is easy to write down the explicit solution of the system (7.47), we shall provide only a qualitative discussion of the solution. The main features are
3, § 7]
Zeno effects in down-conversion processes
187
then best demonstrated with the help of a figure. Eliminating idler and auxiliary mode variables fi-om eq. (7.47) we get a differential equation of the third order for the annihilation operator of the signal mode. Its characteristic polynomial (upon substitution as{t)=^ exp(-iAO) A^ + lAX^ + {A^ -K^ + r^)X + r^A,
K^O,
(IAS)
is a cubic polynomial in A with real coefficients. An oscillatory behavior of the signal mode occurs only provided the polynomial (7.48) has three real roots (casus irreducibilis), i.e., if its determinant D obeys the condition Z) < 0. Expanding the determinant in the small nonlinear coupling parameter F and keeping terms up to the second order in F we obtain D--—
[(K^ - A^f - (5A^ + 3K^)F^] ,
F<^A,K.
(7.49)
It is seen that a mismatched down-conversion behaves in either an oscillatory or a hyperbolic way, depending on the strength of the coupling with the auxiliary mode. The values of K lying at the boundary between these two types of dynamics are determined by solving the equation Z) = 0. The only two nontrivial solutions are
/c-1,2 = \JA^ +^-F^± VlAF.
(7.50)
The case Z\ > T is of main interest here. Hence we can, eventually, drop F^ in eq. (7.50). The resulting intervals are hyperbohc behavior: oscillatory behavior:
K ^ {A- \[lF,A + \/lF), /r G (0, Zi - v ^ r ) U (z\ + \ / 2 r , oo).
n ^\\
The behavior of the mismatched down-conversion process is shown in fig. 14a for a particular choice of A. In absence of linear coupling the down-converted light shows oscillations and the overall effectiveness of the nonlinear process is small due to the presence of phase mismatch A. However, as we switch on the coupling between the idler and auxiliary mode, the situation changes. By increasing the strength K of the coupling the period of the oscillations gets longer and their amplitude larger. When K becomes larger than A - VlF, the oscillations are no longer seen and the intensity of the signal beam starts to grow monotonously. We can say that in this regime the initial nonlinear mismatch has been compensated by the coupling.
[3, §7
Quantum Zeno effects
(b)
(a)
Fig. 14. (a) Mean number of signal photons («s) behind the nonlinear medium as a function of interaction length t and strength K of the linear coupling. The nonlinear mismatch is Z\=10r. (b) Interplay between linear coupling and phase mismatch. The mean number of signal photons {«s) behind the nonlinear medium of length Ft = 1.5 is shown versus the strength K of the linear coupling and the nonlinear mismatch A. A significant production of signal photons, viewed as a "decay" of the initial state (vacuum), is a clear manifestation of an inverse Zeno effect.
The interplay between nonlinear mismatch and linear coupling is illustrated in fig. 14b. A significant production of signal photons is a clear manifestation of IZE. In accord with the observations of Luis and Sanchez-Soto [1998] and Thun and Pefina [1998], such an IZE occurs only if a substantial phase mismatch is introduced in the process of down-conversion. This is the condition (7.32) for having Z < 1 in the decay of the vacuum state. It is worth comparing the interesting behavior seen in fig. 14b with the Zeno and inverse Zeno effects in a sliced nonlinear crystal discussed in § 7.3. The coupling parameter K here plays a role similar to the number of slices A^, so that one can state again that K ~ \/N in the sense of § 5.
7.5. Dressed modes We now look for the modes dressed by the interaction K. This will provide an alternative interpretation and a more rigorous explanation of the result obtained above. Let us diagonalize the Hamiltonian (7.39) with respect to the linear coupling. It is easy to see that in terms of the dressed modes ^ (at + b)/V2,
d = {at - b)/V2,
(7.52)
3, § 7 ]
189
Zeno ejfects in down-conversion processes C
A +K
A - K
d
r~ Fig. 15. Energy scheme of a mismatched down-conversion process subject to linear coupling. The bottom solid lines denote a resonant process.
the Hamiltonian (7.39) reads
r
r
HK = 0)cC^c + o)dd^d+—i={alc^+asC)+ —=(ald^-\-asd), v2 v2 where the dressed energies are a)c = A + K,
cOd =
A-K.
(7.53)
(7.54)
The coupling of the idler mode at with the auxiliary mode b yields two dressed modes c and d that the pump photon can decay to. They are completely decoupled and due to their energy shift (7.54), exhibit a phase mismatch A±K, Since the phase mismatch effectively shortens the time during which a fixedphase relation holds between the interacting beams, the amount of converted energy is smaller than in the ideal case of perfectly phase-matched interaction, A = 0. A strong linear coupling then makes the subsequent emissions of converted photons interfere destructively and the nonlinear interaction is frozen. In this respect the disturbances caused by the coupling and by frequently repeated measurements are similar and we can interpret the phenomenon as a QZE. The energy scheme implied by the Hamiltonian (7.53) is shown in fig. 15. Under the influence of the coupling with the auxiliary mode b the mismatched downconversion splits into two dressed energy-shifted interactions. It is apparent that when K = ±A, one of the two interactions becomes resonant. The other one is "counter-rotating" and acquires a phase mismatch 2A, yielding oscillations. Also, the amplitude of such oscillations decreases as A~^ and the mode output becomes negligible compared to the other one. The use of the rotating wave
190
Quantum Zeno effects
[3, § 7
approximation in eq. (7.53) is fiilly justified in this case and the system is easily solved. The output signal intensity reads {al{t)as{t)) = sinh^ ( ^ M
(^ = =^^' ^^ > 1)
C^-^^)
(compare with eq. 7.7). The linear coupling to an auxiliary mode compensates for the phase mismatch up to a change in the effective nonlinear coupling strength
r -^ r/Vi. As a matter of fact, the condition K = ±A can also be interpreted as a condition for achieving the so-called quasi-phase-matching in the nonlinear process. A quasi-phase-matched regime of generation (Armstrong, Bloembergen, Ducuing and Pershan [1962], Fejer, Magel, Jundt and Byer [1992], Chirkin and Volkov [1998]) is usually forced by creating an artificial lattice inside a nonlinear medium, e.g., by periodic modulation of the nonlinear coupling coefficient. A periodic change of sign of F (rectangular modulation) yields the effective coupling strength F -^ 2F/jt, where, as before, F is the coupling strength of the phase-matched interaction. Thus the continuous "observation" of the idler mode even gives a slightly better enhancement of the decay rate than the most common quasi-phase-matching technique. To summarize, the statement "the down-conversion process is mismatched" means that the nonlinear process is out of resonance in the sense that the momentum of the decay products (signal and idler photons) differs from the momentum carried by the pump photon before the decay took place. When the linear interaction is switched on, the system gets dressed and the energy spectrum changes. A careful adjustment of the coupling strength K makes it possible to tune the nonlinear interaction back to resonance. In this way the probability of pump photon decay can be greatly enhanced. This occurs when K ~ ±A and explains why the inverse Zeno effect takes place along the lines K = ±A in fig. 14b. In some sense, on very general grounds, the Zeno effect is a consequence of the new dynamical features introduced by the coupling with an external agent that (through its interaction) "looks closely" at the system. When this interaction can be effectively described as a projection operator a la von Neumann, we obtain the usual formulation of the quantum Zeno effect in the limit of very frequent measurements. In general, the description in terms of projection operators may not apply, but the dynamics can be modified in such a way that an interpretation in terms of Zeno or inverse Zeno effect is appealing and intuitive. This is the main reason why we think that examples of the type analyzed in this chapter call for a broader definition of Zeno effects.
3, § 8]
Genuine unstable systems and Zeno effects
191
§ 8. Genuine unstable systems and Zeno effects We will now study the Zeno-inverse Zeno transition in greater detail, by making use of a quantum field theoreticalfi-amework,and discuss the primary role played by the form factors of the interaction. As usual, rather than analyzing the general case, we shall focus on simple examples. We generalize the two-level Hamiltonian (2.1) to N states \j) (J = 1,..., A/^) Qi 0
/ 0 Qi 7=1
0
0
0 /
0
0 (Oi
0
\ 0
0
(ON
\QN
(8.1)
and introduce different energies /COQ
' 7=1
(8.2) J
In order to obtain a truly unstable system we need a continuous spectrum, so we will consider the continuum limit of these Hamiltonians H = Ho-\-Hi = a;o|+)(+|+ / dw co\a)){a)\+ / dcog(a))(\+){(jD\-\-\a)){+\). (8.3) The transition to a quantum field theoretical framework is an important component of our analysis, as we shall see. As before, we take as the initial state IV^o) = |+). The interaction of this normalizable state with the continuum of states \co) is responsible for its decay and depends on the form factor g((o). We reobtain the physics of two-level systems in the limit g^((o) = Q^b(a)). The Fourier-Laplace transform of the survival amplitude for this model can be given a convenient analytic expression. Notice that the transform of the survival amplitude is the expectation value of the resolvent
A{E) = Jdt
c^'Ait) = (+1 j
At e^'^-'"'\+) = (+l;^r^l+). (8.4)
and is defined for ImE > 0. By using twice the operator identity 1 E—H
1 E — HQ
E
1 .. 1 -M — HQ E — H
(8.5)
192
Quantum Zeno effects
[3, § 8
one obtains
' ^ E^^^H,^ ^-A-H,-A-H,- ' — HQ E — HQ E — HQ E — HQ E — H
A{E) = (+1
E — Ho
E -COQ
+ E-^— /dw '^''"""" ME)- COQ J E -co
(8.6) In the above derivation we used the fact that Hi is completely off-diagonal in the eigenbasis of HQ, {|+), |a;)}, which is a resolution of the identity |+)(+|+ fd(jo\a)){co\ = l.
(8.7)
The algebraic equation (8.6) can be solved and gives
E-
(OQ- Z(E)
where the self-energy function ^{E) is related to the form factor g{coi) by a simple integration
Z ( £ ) = / d a , K l » ! = / d a , ^ . J E-w J E-o)
(8.9)
By inverting eq. (8.4) we finally get r Af
\
C
^(^) = / ^ ^'''AE) =^UE 7B 2 ^
271 J^
e~*^^
^ ^^ ^.„., E-COQ-
(8.10)
1(E)
the Bromwich path B being a horizontal line ImE = constant > 0 in the half plane of convergence of the Fourier-Laplace transform (upper-half plane). We consider now a particular case. Let the form factor be Lorentzian
g(co)=^J^^^.
(8.11)
This describes, for instance, an atom-field coupling in a cavity with high finesse mirrors (Lang, Scully and Lamb [1973], Ley and Loudon [1987], GeaBanacloche, Lu, Pedrotti, Prasad, Scully and Wodkiewicz [1990]). (Notice that
193
Genuine unstable systems and Zeno effects
3, § 8 ]
%
i-/A 0
cjo
2A
Fig. 16. (a) Form factor g^(co) and position of the initial state energy COQ. (b) Poles of the propagator in the complex £'-plane.
the Hamiltonian in this case is not lower bounded and we expect no deviations from exponential behavior at very large times.) In this case one easily obtains ^(^) = ^
,
(8.12)
whence the propagator iC^" + iA) (E - cooXE + iA) - Q^'
A{E)
(8.13)
has two poles in the lower-half energy plane (see fig. 16). Their values are El = (JOo+ A - i | ,
(8.14)
^2 = - A - i ( A - | ) ,
where ^^4+4^2^2+^2
A = — M _L M ^ 2 "^ 2
2a)i
with ^V
{y=^
g^ = cof) + 4Q^-A^
(8.15)
+4ft;2A2-g2
2
(Notice that g^ can be negative.) The survival amplitude reads ,, , A(t)
^1 + iA = -^
El
:p ,
— Q '
-E2
£"2 + i A - TT
EI-E2
:z7 ,
— Q '
a>o + A + i(A - 7/2) g
i(coo+A)t^-yt/2
(8.16)
0^0 + 2 A + i ( A - y)
A -17/2 _QiAt^-{A~y/2)t a>o + 2A + i(A - 7) = ( 1 - 7^) Q-^(^o+A)t^-yt/2
^
^Q^At^-{A-Y/2)t^
194
Quantum Zeno effects
[3, § 8
where I V wi i _ 2 ' ' ( £ , ) a)o + 2A + iiA-Y) is the residue of the pole E\ = a>o + A- iy/2 of the propagator. The survival probability reads P(t) = \Ait)\^ = Zexp(-yO + 2Re[7l*(l - TZ)e-*"*"'^^^"]exp(-AO +
\n\'cxp[-i2A-y)tl
(8.18) where Z = 11 - T^p is the wave function renormahzation. We now focus on the Zeno-inverse Zeno transition and the conditions for it to take place. The reader should refer to the discussion of § 3.2. For the sake of simplicity, we consider the weak coupling limit Q
coo-co
y = 2jTg\coo) + O(Q^) = 2A
coUA^ f
^^^^^
, + O(O^).
(OQ + A ^
Notice that the latter formula is the Fermi "golden" rule. The second exponential in eq. (8.16) is damped very quickly, on a time scale A~^ much faster than y"^ whence, after a short initial quadratic (Zeno) region of duration A ^ the decay becomes purely exponential with decay rate y. Notice that the corrections are of order Q^ ^ = - ^ ^ ^ + 0 ( f 3 ^ ) (8.20) coj + A^ coo + lA and the Zeno time is iz = Q~^ > A~\ i.e., the initial quadratic (Zeno) region is much shorter than the Zeno time. In general, the Zeno time does not yield a correct estimate of the duration of the Zeno region. The approximation P(t) ^ 1 - f/rl (see eq. 3.3) holds for times f < A~^
COQ-IA
02
(Oo + iA +o(Q^)=i-2 ; '
r/)2 - /V2
^^»
t+oi^')
ft>5 + A^O)^Q + A^
(8.21)
namely (ol>A^
+ O(O^).
(8.22)
This is a condition of asymmetry of the initial energy COQ with respect to the form factor (see fig. 16). Notice that in atomic or molecular systems coupled to
Genuine unstable systems and Zeno effects
195
7eff(T)/7
Fig. 17, Effective decay rate yeff(r) for the model (8.11), for A = 0.1 and different values of the ratio O)Q/A (indicated). The horizontal line shows the "natural" decay rate y: its intersection with yeff(^) yields the solution r* of eq. (3.15). The asjmiptotic value of all curves is y, as expected. A Zeno (inverse Zeno) effect is obtained for r < r* (t > T*). Notice the presence of a linear region for small values of r and observevfcat T* does not belong to such linear region as the ratio (DQ/A decreases. Above a certain threshold, given by eq. (8.22) in the weak coupling limit of the model (and in general by the condition £^ = 1), eq. (3.15) has no finite solutions: only a Zeno effect is realizable in such a case.
the free space photon vacuum this condition is always extremely well satisfied (Facchi and Pascazio [1998]). In such a case, namely, if COQ > A^, the transition time is well approximated by eq. (3.13), which yields (remember eq. 3.15)
Y^l
lA
^t^'
+ 0(0^).
(8.23)
When o^o gets closer to the peak of the form factor, the effective decay rates yQ^{T) in eq. (3.12) are not linear in r anymore and the solution r* of the equation yeff(7*) = 7 becomes larger than the approximation (8.23), eventually going to infinity when the condition (8.22) is no longer valid. In such a case, only a QZE is possible. This situation is outlined in fig. 17, where the effective decay rate 7eff(^) is displayed for different values of the ratio COQ/A. The conclusions drawn for the Lorentzian model (8.11) are of general validity (Facchi, Nakazato and Pascazio [2001]). The occurrence of a Zeno-inverse Zeno transition depends on the form factor of the coupling and on the ratio (WQ - (JOg)/A, where cOg is the energy of the ground state. We will not go into additional details here. It is interesting to consider some limits of the model investigated. In the limit
196
Quantum Zeno effects
[3, § 8
of large bandwidth A > (DQ, Q, from eq. (8.15) one gets y = 2Q^/A + 0(A"^) and in order to have a nontrivial result with a finite decay rate, we let Q^
A ^ oo,
Q -^ oo,
with
y
- r == x = const
(8.24)
In this limit the continuum has a flat band, g(co) = y/y/2jt =const, and we expect to recover the results of § 5.2. Indeed, in this case one gets 1Z = 0 and A = 0, whence A(E) = ^——, ^ ^ E-coo + iy/2' so that the survival amplitude and probability read ^ ( 0 = exp (-icoot - ^A
and P{t) = Qxp(-yt).
(8.25) ^ ^ (8.26)
In this case the propagator (8.25) has only a simple pole and the survival probability (8.26) is purely exponential. Therefore measurements cannot modify the free behavior. Indeed, the conditions for occurrence of a Zeno effect are always ascribable to the presence of an initial non-exponential behavior of the survival probability, which is caused by a propagator exhibiting a richer structure than a simple pole in the complex energy plane. In the limit of narrow bandwidth A
(8.27)
and the continuum is "concentrated" in G; = 0. Therefore the continuum behaves as a second discrete level and one obtains Rabi oscillations. In fact one gets
y = 0, A = - ^ + r2,,
n=Ul-^),
(8.28)
where Qx = \lQ^ + ^
(8.29)
is the usual Rabi frequency of a two-level system with energy difference (JOQ and coupling Q. By (8.28) the survival amplitude and probability read
Ao=Hi + ^)e-'"-'^'"+Hi-2iti^
(8.30)
In this case, if o^o = 0 (initial state energy at the center of the form factor), the survival probability (8.30) oscillates between 1 and 0 and only a QZE is
3, § 8]
Genuine unstable systems and Zeno effects
197
possible. On the other hand, if o^o ^ 0 (initial state energy strongly asymmetric with respect to the form factor of "width" A = 0) the initial state never decays completely. By measuring the system, the survival probability will vanish exponentially, independently of the strength of observation, whence only an IZE is possible. This is what we found for a down-conversion process with a sufficiently large phase mismatch at the end of § 7.2. Incidentally, notice that the Zeno time is still Xz = ^"^ and now yields a good estimate of the duration of the Zeno region. This is, so to say, a "coincidence" due to the oscillatory features of the system. Another interesting case is that of strong coupling, O c^ A. Roughly speaking, it yields the situation outlined in fig. 6b. This is a typical case in which the strong coupling provokes violent oscillations before the system reaches the asymptotic regime. In the limit O > A, a>o, we get 4 = i3-^+0(r3-'),
y = -4+0(i2->),
7 ^ = i - ^
+ 0(i2-3), (8.31)
whence the survival amplitude reads
AO-exp(-i^;-y?
1
WQ + iA\
:„,
/1
(»o + iA ^ jat
(8.32) which yields fast oscillations of frequency Q damped at a rate A
I dcodcj' i\(o)((o,(o'\ + \(o,a}'){co\)+ fdo}' \co'){co'\.
(8.33) As soon as a photon is emitted, it is coupled to another boson of frequency (o' (notice that the coupling has no form factor). By following a reasoning identical to that of § 5.2, one can show that the dynamics of the Hamiltonian (8.3)-(8.33), in the relevant subspace, is generated by //== Wo|+)(+|+
do) (CO-ir/2)\co){co\+ / d(wg((w)(|+>(w| + |(o)(+|),
(8.34) and an effective continuous observation on the system is obtained by increasing r . Indeed it is easy to see that the only effect due to F in eq. (8.34) is the substitution of I{E) with I(E + i r / 2 ) in eq. (8.8), namely, E - COQ- Z(E +11/2)
198
Quantum Zeno effects
[3, § 8
For large values of T, i.e., for very quick response of the "apparatus", the selfenergy function (8.9) has the asymptotic behavior ^{E
+ i^--ij
j dcDg\(D) = - i ^ ,
for
r-.oc,
(8.36)
where the definition of g{a)) was used, jdcog\co)
= jdw
|(+|//i|a;)|' = (+|//2|+) = - i .
(8.37)
[Notice that T -^ oo in (8.36) means T > A, the fi-equency cutoff of the form factor.] In this case the propagator (8.35) reads
E-U)o + iyeff(r)/2 and the survival probability decays with the effective exponential rate (valid for r>A) YMn=:^-
(8.39)
The effective rate (8.39) is the same result (5.12) we found for the particular model considered in § 5.2. We see that it is a general result. The equivalence (5.13) is therefore of general validity. Additionally, we have here a scale for the validity of the linear approximation for /eff. The linear term in the asymptotic expansion (8.36) approximates well the self-energy fiinction only for values of T that are larger than the bandwidth A. For smaller values of F one has to take into account the nonlinearities arising from the successive terms in the expansion. Note that the flat-band case (8.25), yielding a purely exponential decay, is also unaffected by the continuous measurement. Indeed in that case 1(E) = -iy/2 is a constant independent of E, whence I(E + iF/l) = 1(E) is independent of F. The same thing happens if one considers the Weisskopf-Wigner approximation. In this case one neglects the whole structure of the propagator in the complex energy plane and retains only the dominant pole near the real axis. This yields, as we have seen, a self-energy fiinction which does not depend on energy and a purely exponential decay (without any deviations), that cannot be modified by any observations. Notice that the very existence of a QZE is related to the existence of an initial quadratic behavior of the survival probability, i.e., to a finite value of tz.
3, § 9]
Three-level system in a laser
field
199
As eq. (8.37) shows, this is related to the convergence of the integral of the form factor. In general, in a quantum field theoretical framework, the Zeno time Tz (the inverse of the second moment of the Hamiltonian) cannot be defined, because it vanishes for pointlike particles. It becomes necessary to introduce form factors and cutoffs and use more sophisticated techniques. These problems will not be discussed here. See Bemardini, Maiani and Testa [1993], Facchi and Pascazio [1998], Joichi, Matsumoto and Yoshimura [1998], Maiani and Testa [1998], Alvarez-Estrada and Sanchez-Gomez [1999].
§ 9. Three-level system in a laser field We shall now investigate a realistic situation in which a continuous observation performed by a laser field leads to an inverse Zeno effect, in a way very similar to that outlined in § 5. We shall look at the temporal behavior of a three-level system (such as an atom or a molecule), where level 11) is the ground state and levels |2), 13) are two excited states. (See fig. 18.) The system is initially prepared in level 12) and if it follows its natural evolution, it will decay to level |1). The decay will be (approximately) exponential and characterized by a certain lifetime, that can be calculated from the Fermi golden rule. But if one shines an intense laser field on the system, tuned at the transition frequency 3-1, the evolution can be different. This problem was investigated by Mihokova, Pascazio and Schulman [1997], who found that the lifetime of the initial state depends on the intensity of the laser field. We shall see that for physically sensible values of the intensity of the laser, the decay is enhanced. This can be viewed as an inverse quantum Zeno effect (Facchi and Pascazio [2000], Pascazio and Facchi [1999]). An important role will be played by the form factor of the interaction Hamiltonian. I3>-
I2>-
ll>Fig. 18. Level configuration.
200
Quantum Zeno effects
[3, ^
9.1. The system We consider the Hamiltonian (h = c = 1) H = Ho-\- Hint
(9.1) where the first two terms are the free Hamiltonian of the 3-level atom (whose states |/) (/ = 1,2,3) have energies E\ =0, coo= E2-E1 > 0, Qo= E^-Ei > 0), the third term is the free Hamiltonian of the EM field and the last two terms describe the 1 ^ 2 and 1 ^ 3 transitions in the rotating wave approximation, respectively. (See fig. 18.) States |2) and |3) are chosen so that no transition between them is possible (e.g., because of selection rules). The matrix elements of the interaction Hamiltonian read
*" " T C F S / " ' " * ' ' " ^•'*>
(9.2)
kX *" = ^!5FS/"'""''"--•"<'>•
where -e is the electron charge, 60 the vacuum permittivity, V the volume of the box, (o = \k\, Ekx thQ photon polarization andyfi the transition current of the radiating system. For example, in the case of an electron in an external field, we haveyfi = ipjaipi where i/^i and xpf are the wavefiinctions of the initial and final state, respectively, and a is the vector of Dirac matrices. For the sake of generality we are using relativistic matrix elements, but our analysis can also be performed with nonrelativistic ones y'fi = 'H)^p\l)\/me, where p/nie is the electron velocity. We shall concentrate on a 3-level system bathed in a continuous laser beam, whose photons have momentum k^ (|^o| = -^o) and polarization AQ, and assume, throughout this chapter, that 0itoAo=O,
(9.3)
i.e., the laser does not interact with state |2). The laser is in a coherent state |ao) with a very large average number NQ = |aoP of ^0-photons in volume V
3, § 9]
Three-level system in a laser
field
201
[we will eventually consider the thermodynamical limit; see eq. (9.21)]. In the picture defined by the unitary operator nt) = exp (ao*e^^«^a,,Ao - ao^'^'^^^oAo) '
(^-4)
the UQ mode is initially in the vacuum state (Mollow [1975], Cohen-Tannoudji, Dupont-Roc and Grynberg [1998]) and the Hamiltonian becomes (A^o > 1) / / ^ (^o|2)(2| + Oo|3){3| + ^ k,X
a>,4,a,, + ^ ' ( 0 , , 4 J 1 ) ( 2 | + 0^^^^ k,?i
+ (^,„Ao<e'^'|l)(3| + $;,„aoe-«"|3)(l|), (9.5) where a prime means that the summation does not include (A:o,Ao) (due to hypothesis (9.3). In the above equations and henceforth, the vector |/;/2u) represents a state in which the atom is in state \i) and the electromagnetic field in a state with W^^A (k, A)-photons. We shall analyze the behavior of the system under the action of a continuous laser beam of high intensity. Under these conditions, level configurations similar to that of fig. 18 give rise to the phenomenon of induced transparency (Tewari and Agarwal [1986], Harris, Field and Imamoglu [1990], Boiler, Imamoglu and Harris [1991], Field, Hahn and Harris [1991], Zhu, Narducci and Scully [1995], Zhu and Scully [1996], Huang, Zhu, Zubairy and Scully [1996]), for laser beams of sufficiently high intensities. Our interest, however, will be focused on unstable initial states. We shall study the temporal behavior of level |2) when the system is shined by a continuous laser of intensity comparable to those used to obtain induced transparency. The operator
M-\2){2\+J2'ala,,,
(9.6)
k,X
satisfies [H,M] = 0,
(9.7)
which implies the conservation of the total number of photons plus the atomic excitation [Tamm-Dancoff approximation (Tamm [1945], Dancoff [1950])]. The Hilbert space splits therefore into sectors that are invariant under the action of the Hamiltonian. In our case, the system evolves in the subspace labeled by the eigenvalue J\f = I and the analysis can be restricted to this sector (Radmore and Knight [1982], Knight and Lauder [1990]).
202
Quantum Zeno effects
[3, § 9
9.2. Schrodinger equation and temporal evolution We will study the temporal evolution by solving the time-dependent Schrodinger equation i^|t/'(0>=i/(Ol'/'(0>,
(9.8)
where the states of the total system in the sector M = 1 read \xp(t))=x{t)\2-0) + Y^yuxm;
hx) + Y^z,,{t)^''"^'\3;
k,X
Ux)
(9.9)
k,X
and are normalized:
{w)m)) = \x{t)?+Y^\yutt+Y^\zutt = 1 (vo-
(9.10)
By inserting (9.9) in (9.8) one obtains the equations of motion \x(t) = Wox(t) + Yl
^xykxiO,
*"^ (9.11) ^hxit) = 0kxx(t) + Wkykxit) + aQ0koXo^kx(tX izkxit) = OoO^^ykxit) + cOkZkxit), where a dot denotes time derivative. At time ^ = 0 we prepare our system in the state \m))
= |2;0)
^
x(0) = 1, ykxiO) = 0, Zkx(0) = 0.
(9.12)
By Fourier-Laplace transforming the system of differential equations (9.11) and incorporating the initial conditions (9.12) the solution reads E-
COQ- I(B,
E)
with
^iB,E) = Y.\<^.x\\^^-;'_^,
(9.16)
k,X
and where B' = ^o|4'*„Aol' (9.17) is proportional to the intensity of the laser field and can be viewed as the "strength" of the observation performed by the laser beam on level |2) (in the
3, § 9]
Three-level system in a laser
field
203
sense of § 5). Note that the coupHng B is related to the Rabi frequency by the simple relation B = ORabi/2. In the continuum limit ( F —> oo), the matrix elements scale as follows }^:^^^,Y.j^^\t>i'^\^^s'(^ox\o)),
(9-18)
A
where Q is the solid angle. The (dimensionless) function x{^) and coupling constant g have the following general properties (Facchi and Pascazio [2000]) X {(O) ^ { g^
a
'.
^^ A
(9.19)
= a{coQ/AfJ^^^\
(9.20)
where j is the total angular momentum of the photon emitted in the 2 ^ 1 transition, =F represent electric and magnetic transitions, respectively, ^(> 1) is a constant, a the fine structure constant and A a natural cutoff (of the order of the inverse size of the emitting system, e.g., the Bohr radius for an atom), that can be explicitly evaluated and determines the range of the atomic or molecular form factor (Berestetskii, Lifshits and Pitaevskii [1982], Moses [1972a,b, 1973], Seke [1994a,b]). In order to scale the quantity B, we take the limit of a very large cavity, by keeping the density of Oo-photons in the cavity constant: K ^ cxo,
iVo ^ oo,
with
TVo — = WQ = const.,
(9.21)
and obtain from (9.17) B^ = noV\0koXo\' = {2jtyno\(pUko)\\
(9.22)
where q) = 0V^^^/(2jty^^ is the scaled matrix element of the 1-3 transition. If the 1-3 transition is of the dipole type, the above formula reads B^ = 2jtaQo\el,^'Xu\W
(9.23)
where xu is the dipole matrix element. In terms of laser power P and laser spot area A, eq. (9.23) reads P
X^
PX^
where P is expressed in watt, XL (laser wavelength) in |im, A in ^m^ and hPii, in eV In eq. (9.24) the quantity B is expressed in suitable units and can be
204
Quantum Zeno effects
[3, § 9
easily compared to a^o (the ratio B/w^ being the relevant quantity, as we shall see). For laser intensities that are routinely used in the study of electromagnetic induced transparency, the inverse quantum Zeno effect should be experimentally observable. For a quick comparison remember that B is just half the Rabi frequency of the resonant transition 1-3. 9.3. Laser off Let us first look at the case B = 0. The laser is off and we expect to recover the well-known physics of the spontaneous emission of a two-level system prepared in an excited state and coupled to the vacuum of the radiation field. In this case the self-energy function 1(0, E) reads, in the continuum limit (see eq. 8.9), 1(E) = g'cooq(E) = g'wo r do; 1 ^ , (9.25) Jo 1^ - (o where x is defined in (9.18). The function x(E) in eq. (9.13) (with ^ = 0) has a logarithmic branch cut, extending from 0 to +CXD, and no singularities on the first Riemann sheet (physical sheet) (Facchi and Pascazio [1998]). On the other hand, it has a simple pole on the second Riemann sheet, that is the solution of the equation E~coo-g^cooqu(E)
= 0,
(9.26)
where qu(E) = ^(£e-2^^0 = q(E) - 2Jiix\E)
(9.27)
is the determination oiq(E) on the second Riemann sheet. We note that g^q(E) is 0(g^), so that the pole can be found perturbatively. By expanding q\i(E) around 0)Q we get a power series, whose radius of convergence is Re = COQ because of the branch point at the origin. The circle of convergence lies half on the first Riemann sheet and half on the second sheet (fig. 19). The pole is well inside the convergence circle, because l^pde - (Oo\ '^ g^ojo "^ Re, and we can write ^poie = (Oo+g^cooqn((JOo-iO^)-^0(g'^) = a)o-^g^cooq(coo^iO^) +
0(g\ (9.28) because qu(E) is the analytic continuation of q(E) below the branch cut. By setting E^,,, = (Oo + A-i^,
(9.29)
3, § 9 ]
205
Three-level system in a laser field
,
I
\E
\
COo ^~-~.
I
n
X Z7
"A
\.
•vil \ Fig. 19. Cut and pole in the £'-plane {B = 0) and convergence circle for the expansion of ^{E) around E = COQ.I and II are the first and second Riemann sheets, respectively. The pole is on the second Riemann sheet, at a distance 0{g^)firomCL>O-
one obtains from eq. (9.25)
7 = 2jtg'(Da\(o^) + 0{g%
A = g'cooP r JQ
doj ^ ^ ^ + 0{g\ (OQ-CO
(9.30) which are the Fermi "golden rule" and the second-order correction to the energy of level 12) (see eqs. 8.19). The Weisskopf-Wigner approximation consists in neglecting all branch cut contributions and approximating the self-energy function with a constant (its value in the pole), that is (9.31)
x(E) = E-COQ-
1(E)
E~COO-
2'ii(£'pole)
E - £'pole
where in the last equality we used the pole equation (9.26). This yields a purely exponential behavior, x(t) = exp(-L£'poie05 without short-time (and long-time) corrections. As is well known, the latter are all contained in the neglected branch cut contribution. 9.4. Laser on We now turn our attention to the situation with the laser switched on (B ^ 0) and tuned at the 1-3 transition frequency QQ. The self-energy fiinction I(B,E)
206
Quantum Zeno effects
[3, § 9
in (9.16) depends on B and can be written in terms of the self-energy function 1{E) in absence of the laser field (eq. 9.25), by making use of the following remarkable property:
^(i..«=ii:fcp(^-j-^.^-^) ,,,
X - -^
-
-
/
(9.32)
= \{1{E-B) + I{E + B)\. Notice, incidentally, that in the continuum limit {V ^ oo), due to the above formula, I(B,E) scales just like 1(E). The position of the pole £'poie (and as a consequence the lifetime IE = y~^ = -l/2Im£'poie) depends on the value of ^. There are now two branch cuts in the complex E plane, due to the two terms in eq. (9.32). They lie over the real axis, along [-B, +oo) and [+^, +oo). The pole satisfies the equation E-(Oo-I(B,E) = 0,
(9.33)
where I(B,E) is of order g^, as before, and can again be expanded in a power series around E = (OQ, in order to find the pole perturbatively. However, this time one has to choose the right determination of the fiinction I(B,E). Two cases are possible: (a) The branch point +B is situated at the lefi; of COQ, SO that (Oo lies on both cuts; see fig. 20a. (b) The branch point +B is situated at the right of c^o, so that WQ lies only on the upper branch cut; seefig.20b. We notice that in the latter case (B > COQ) a number of additional effects should be considered. Multi-photon processes would take place, the other atomic levels would start to play an important role and our approach (3-level atom in the rotating wave approximation) would no longer be completely justified. Notice also that our approximation still applies for values ofB that are of the same order of magnitude as those utilized in the electromagnetic-induced transparency. In this case the influence of the other atomic levels can be taken into account and does not modify the main conclusions (Facchi and Pascazio [2000]). In case (a), i.e., for B < COQ, the pole is on the third Riemann sheet (under both cuts) and the power series converges in a circle lying half on thefirstand half on the third Riemann sheet, within a convergence radius Re = COQ - B, which decreases as B increases [fig. 20a]. On the other hand, in case (b), i.e., for B > (OQ, the pole is on the second Riemann sheet (under the upper cut only) and the power series converges in a circle lying half on the first and half on the second Riemann sheet, within a convergence radius Re = B - CJOQ, which increases with B (fig. 20b).
3, § 9 ]
207
Three-level system in a laser field
0)„ -\-B
|H)1L-
\+B [II
ZZ'l-nilc
III
(a)
(b)
Fig. 20. Cuts and pole in the £^-plane (B ^ 0) and convergence circle for the expansion of I(B, E) around E = (DQ.1,11 and III are the first, second and third Riemann sheets, respectively, (a) B < COQ. (b) B > coQ.ln both cases, the pole is at a distance 0(g^)fi*omCOQ.
In either cases we obtain, for \Epo\Q - a>o| < ^c = |^ - ^o|, ^poie = (Oo+\ [^((JOo +B + /0+) + I{m
-B + /0+)] + 0(g4)
(9.34) = 0^0+ \g^(JOo [q{0)Q + ^ + /0+) + q{(OQ -B + /0+)] + 0{g^). We write, as in eq. (9.29), .reff(5) ^pole = ^ 0 + A{B) - i
(9.35)
Substituting (9.25) into (9.34) and taking the imaginary part, one obtains the following expression for the decay rate yeff(5) = V a > o [x\oj^ +B) + x\o)^ -B)d(coo - B)] + 0(g'),
(9.36)
which yields, by (9.30), _ ^^ x\coo +B) + x\o)o - B)d{wo - B) + 0(g4). (9.37) =y 2x\coo) Equation (9.37) expresses the "new" lifetime yeff (^)~^ when the system is bathed in an intense laser field B, in terms of the "ordinary" lifetime y"^ when there is no laser field. By taking into account the general behavior (9.19) of the matrix elements X^(^) and substituting into (9.37), one gets to 0(g'^) YMB)
y.ff{B)
i+—
+
1
B \^^'^^
e{a)Q-B)
(B < A),
COoJ
(9.38) where =F refers to 1-2 transitions of electric and magnetic type, respectively. Observe that, since A ^ inverse Bohr radius, the case B < (OQ <^ A is the
208
[3, § 9
Quantum Zeno effects
physically most relevant one. The decay rate is profoundly modified by the presence of the laser field. Its behavior is shown in fig. 21 for a few values ofy. In general, fory > 1 (1-2 transitions of electric quadrupole, magnetic dipole or higher), the decay rate /eff (^) increases with B, so that the lifetime yeff(^)"^ decreases as B is increased. Since B is the strength of the observation performed by the laser beam on level |2), this is an IZE, for decay is enhanced by observation. 7eff(5)/7
leAB)/j 4
eff(i^)/7 2\
/
3 1
nl 0
^/^'o 0.4
0.8
0
/
8
2 1
12
_
^ B/uo
0
3 = 1
/
4
0.4 0.8 J = 2
0-"""^
0
0.4
B/uJo 0.8
J = 3
Fig. 21. The decay rate yeffW versus B, for electric transitions withy = 1,2,3; XeffW is in units y and 5 is in units COQ. Notice the different scales on the vertical axis.
As already emphasized, eq. (9.38) is valid for B <^ A. In the opposite case 5 > A, by (9.19) and (9.37), one gets to 0(g^)
7eff ( ^ )
^
7
X\B)
2 x\m)
oc (B/A)-
(B^A)
(9.39)
This result is similar to that obtained by Mihokova, Pascazio and Schulman [1997]. If such high values ofB were experimentally obtainable, the decay would be considerably hindered (QZE). A final remark is now in order. If one would use the Weisskopf-Wigner approximation (9.31) in eq. (9.32), in order to evaluate the new lifetime, by setting 1(E) = 2'(£'poie) = const, one would obtain I{B,E) = 1(E) = 2'(£'poie), i.e., no ^-dependence. Therefore, the effect we are discussing is ultimately due to the nonexponential contributions arising from the cut. In particular, viewed from the perspective of the time domain, this effect is ascribable to the quadratic short-time behavior of the |2) -^ |1) decay.
Three-level system in a laser field
9]
209
9.5. Photon spectrum, dressed states and induced transparency It is interesting to look at the spectrum of the emitted photons. It is easy to check that, in the Weisskopf-Wigner approximation, the survival probability \x{t)\^ decreases exponentially with time. In this approximation, for any value of 5, the spectrum of the emitted photons is Lorentzian. The proof is straightforward and is given in Facchi and Pascazio [2000]. One finds that, for B = 0, the probability to emit a photon in the range (a>, O) + do;) reads (9.40) where WQ = WQ + ^{E) and 1
(9.41)
/ L ( a > ; 7 ) = 0)2 + y 2 / 4 -
On the other hand, when B ^0 one gets: dPB = g^o)oX^{o))\ [Mco -oJo-B;
yeff(^)) +fL((o -(bo+B; y^B))] do;.
(9.42) The emission probability is given by the sum of two Lorentzians, centered in WQ ± B. We see that the emission probability of a photon of frequency a)o + ^ (d)o - B) increases (decreases) with B (fig. 22). The linewidths are modified according to eq. (9.38). When B reaches the "threshold" value COQ, only the photon of higher fi-equency (a)o +B) is emitted (with increasing probability vs. B). dP/duj
dPslduj
0.5
1.5
2
(a) Fig. 22. The spectrum (9.42) of the emitted photons. The height of the Lorentzians is proportional to the matrix element X^i^J^) (dashed line). We chose an electric quadmpole transition, withy = 2 and Y = 10~^a)o, and used arbitrary units on the vertical axis, (a) 5 = 0; (b) 5 = COQ/S; note that, from eq. (9.38), 7eff (^) = (28/25)y.
210
Quantum Zeno effects
[3, § 9
Photons of different frequencies are therefore emitted at different rates. In order to understand better the features of the emission, let us look at the dressed states of the system. For simplicity, since the average number TVO of ^O-photons in the total volume V can be considered very large, we consider number (rather than coherent) states of the electromagnetic field. Henceforth, the vector \i\nkx,M{)) represents an atom in state |/), with riux (A:, A)-photons and MQ laser photons. The Hamiltonian (9.1) becomes H ~ (yo|2)(2| +Do|3)(3| + ^ a > , 4 , a u + $ ] ' (^hxa\,\\){2\ + k,X
r,,a,x\2){\\)
k,X
+ ( ^ ^ o A o < A o | l ) ( 3 | + ^;oAo^AoAo|3)(l|),
(9.43) where a prime means that the summation does not include (A:o,Ao) (due to hypothesis 9.3). Besides (9.6), there is now another conserved quantity: indeed the operator •A/'o = |3)(3|+aUa*oAo
(9-44)
satisfies [H,Mo] = [^fo,^f]-0.
(9.45)
In this case, the system evolves in the subspace labeled by the two eigenvalues J\f = 1 and J\fo= No, whose states read |i/;(0)=x(0|2;0,iVo) + ^ W ( 0 | l ; U A , ^ o ) + ^ W ) | 3 ; U A , A ^ o - l ) . k^
k,X
(9.46) By using the Hamiltonian (9.43) and the states (9.46) and identifying A/Q with ^^0 = I aoP of §9.1, the Schrodinger equation yields again the equations of motion (9.11), obtained by assuming a coherent state for the laser mode. Our analysis is therefore independent of the statistics of the driving field, provided it is sufficiently intense, and the (convenient) use of number states is completely justified. Energy conservation implies that if there are two emitted photons with different energies (see eq. 9.42), there are two levels of different energies to which the atom can decay. This can be seen by considering the laser-dressed (Fano) atomic states (Fano [1961], Cohen-Tannoudji and Reynaud [1977a-c], Yoo and Eberly [1985]). The shift of the dressed states can be obtained directly
3, § 9]
Three-level system in a laser
field
211
from the Hamiltonian (9.43). In the sector A/Q = NQ, the operator A/Q is proportional to the unit operator, the constant of proportionality being its eigenvalue. Hence one can write the Hamiltonian in the following form H = H- QoJVo + QoNo,
(9.47)
which, by the setting Ei -\-NoQo = 0, reads H = Ho-\- ifint = a;o|2)(2| + Y^'cokal^aki + ^ ' (^^k?.al^\l){2\ + (l>l^akx\2){l\) k,l
k,l
+ (^*«Ao«Lll>(3| +
HMMQ
(^Mo
(9.48)
with M = 1 and Afo = No, the last term
(^MO\/A^|1)(3| + ^ M O V ^ | 3 ) ( 1 | ) .
(9.49) This operator is easily diagonalized in terms of the (orthonormal) noninteracting states
|±) = -J=(|1) + |3))
(9.50)
[this is the simplest choice (Facchi and Pascazio [2000])]. A simple manipulation yields H = Hl,+Hi„
(9.51)
where the primed free and interaction Hamiltonians read, respectively. Hi, = a;o|2)(2| +5|+)(+| - ^ | - ) ( - | + Y^o^kal^aux, k,l
"(- ' E ' [(^""l+X^I + §«.AI2)(+I)
(9.52)
and B^ = \0k^Xo\^No. The dressed states |+) and |-) have energies -{-B and -B and interact with state |2) with a coupling ^kx/\[l. Since IB = ^Rabi this is
212
Quantum Zeno effects
I2>
r
l+> ii>
'"o
^AV
l->
^'
+fi 0
|2>—
ii> "
-fi
(a)
(h)
Fig. 23. Shift of the dressed states |+) and |-) vs. B. (a) For B < (JOQ there are two decay channels, with y_ > y^. (b) For 5 > o^o, level |+) is above level |2) and only the y_ decay channel remains.
the well-known Autler-Townes doublet (Autler and Townes [1955], Townes and Schawlow [1975]). Therefore, by applying the Fermi golden rule, the decay rates into the dressed states read
r.-2^,W^^^,
y--2.,W^^^,
(9.53)
and the total decay rate of state |2) is given by their sum 7eff(^)=y^ + 7-,
(9.54)
which yields (9.36). One sees why there is a threshold at 5 = WQ- For B < COQ, the energies of both dressed states |±) are lower than that of the initial state |2) (fig. 23a). The decay rate y_ increases with B, whereas y+ decreases with B; their sum y increases with B. These two decays (and their lifetimes) could be easily distinguished by selecting the frequencies of the emitted photons, e.g. by means of filters. On the other hand, when B > COQ, the energy of the dressed state |+) is larger than that of state |2) and this decay channel disappears (fig. 23b). Finally, let us emphasize that if state |2) were below state 11), our system would become a three-level system in a ladder configuration, and the shift of the dressed states would give rise to electromagnetically induced transparency (Tewari and Agarwal [1986], Harris, Field and Imamoglu [1990], Boiler, Imamoglu and Harris [1991], Field, Hahn and Harris [1991]). The situation we consider and the laser power required to bring these effects to light are therefore similar to those used in induced transparency.
3, § 10]
Concluding remarks
213
For physically sensible values of the intensity of the laser field, the decay of level |2) is faster when the laser is present. Equations (9.37)-(9.38) (valid to 4th order in the coupling constant) express the new lifetime as a function of the "natural" one and other parameters characterizing the physical system. The initial state decays to the laser-dressed states with different lifetimes, yielding an IZE.
§ 10. Concluding remarks The usual formulation of quantum Zeno effect in terms of repeated ("pulsed") measurements a la von Neumann is a very effective one. It motivated quite a few theoretical proposals and experiments and provoked very interesting discussions on their physical meaning. In general, the quantum Zeno effect is a straightforward consequence of the new dynamical features introduced by a series of measurement processes. In turn, a measurement process is due to the coupling with an external apparatus that, after interacting with the system, gets entangled with it. It is then very natural to think that a quantum Zeno effect can also be obtained if the (Hamiltonian) dynamics is such that the interaction takes a sort of "close look" at the system. When such an interaction can be effectively described as a projection operator a la von Neumann, we obtain the usual formulation of the quantum Zeno effect in the limit of very frequent measurements. Otherwise, if the description in terms of projection operators does not apply, but one can still properly think in terms of a "continuous gaze" at the system, an explanation in terms of Zeno can still be very appealing and intuitive. These considerations and the diverse examples analyzed in this chapter motivated us to interpret several physical phenomena as quantum Zeno or inverse quantum Zeno effects. We believe that this approach is prolific. Not only does it often yield a simple intuitive picture of the dynamical features of the system, it also enables one to look at these dynamical features from a different, new perspective. The very concept of inverse Zeno effect is a good example. Other examples are the phenomena discussed in §§7 and 9. The underlying idea is that coupling the system to an "observer" (like a laser) can sometimes enhance the evolution. This is close to Heraclitus' viewpoint, who used to argue (against Zeno and Parmenides) that ever3^hing flows. The physical features of the dynamical evolution laws have profound implications (Prigogine [1980]) and always provide matter for thoughts. In this way, one even finds links with instability (Facchi, Nakazato, Pascazio, Pefina and Rehacek [2001]), chaos (Facchi, Pascazio and
214
Quantum Zeno effects
[3
Scardicchio [1999], Kaulakys and Gontis [1997]) and geometrical phases (Berry and Klein [1996], Facchi, Klein, Pascazio and Schulman [1999]). The very fact that these links may not always be obvious is in itself a motivation to pursue the investigation in this direction.
Acknowledgments It is a pleasure to thank the many colleagues who have collaborated with us during the last few years on the topics discussed in this chapter. We would like to mention in particular G. Badurek, Z. Hradil, A.G. Klein, H. Nakazato, M. Namiki, J. Pefina, H. Ranch, J. Rehacek, A. Scardicchio and L.S. Schulman. We owe them much of our own comprehension of the diverse phenomena known as Zeno effects.
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E. Wolf, Progress in Optics 42 © 2001 Elsevier Science B.V All rights reserved
Chapter 4
Singular optics by
M.S. Soskin and M.V Vasnetsov Institute of Physics, National Academy of Sciences of Ukraine, Prospect Nauki 46, Kiev, 03650 Ukraine
219
Contents
Page § 1.
1. Introduction
221
§ 2. Anticipations of singular optics
223
§ 3. Wave-front dislocations - phase defects
226
§ 4.
Circular and linear edge dislocations
231
§ 5.
Screw wave-front dislocation - axial OV
239
§ 6.
Reflection, refraction, interference and diffraction of OVs . . .
244
§ 7.
Topology of wave fronts and vortex trajectories
249
§ 8.
Gouy phase shift in singular optics
258
§9.
Statistics of phase dislocations
261
§ 10. Optical vortices in frequency conversion processes
263
§ 11. AppUcations
268
§ 12. Conclusions
271
References
272
220
§ 1. 1. Introduction What is singular optics? This new branch of modern physical optics deals with a wide class of effects associated with phase singularities in wave fields, as well as with the topology of wave fronts. Although phase is an auxiliary fiinction in electromagnetic field description, it is very usefiil because it gives a visual perception of wave propagation and transformation along its path. An important relevant conception is a wave front, or a surface of equal phase, usually associated with a crest of a wave, where the field strength attains its highest value. The wave fronts follow each other with spatial separation of one wavelength, and between two neighboring wave fronts there are two surfaces where field strength becomes zero, and one surface where the field reaches minimum (negative) value (trough). This perfect regular motion, being true for a plane wave, can sometimes be violated for real waves. In brief, phase of a wave can experience a ";r-jump", corresponding to a step on half of a wavelength in a wave train, producing a phase defect of a wave front along a continuous line in space. For instance, some physical reasons can be responsible for local retardation or acceleration of the phase velocity across a wave front. Resulting wave-front bending can lead to a tear of the wave front, and the phase becomes indeterminate, or singular along the tear. The necessary condition for a phase singularity to appear is the vanishing of the field amplitude. M. Berry has recently emphasized that the study of wave singularities in optics and physics in general started in "the miraculous 1830s" (Berry [2000]). In 1838 Airy showed that the rainbow is the caustic where light rays are gathered. He recognized also that the infinite brightness of caustic predicted by ray optics is softened to finite value because of the wave origin of light. Whewell in 1833 discovered the phase singularities in the tide waves (for more details see Berry [1981]). Hamilton in 1832 discovered the third type of singularities, namely polarization light singularities, as the "effect of conical refi-action" (see Born and Wolf [1999]). The general types of polarization singularities were analyzed later by Nye (see Nye [1999] and references therein). More recently the results of optical singularities investigations are treated in the Singular Optics, a branch of modern optics studying new important features of light, which are absent in the traditional optics of waves with smooth wave 221
222
Singular optics
[4, § 1
fronts (Soskin and Vasnetsov [1999a,b]). More precisely, there are three levels of optical singularities: (i) ray singularities (caustics) considered as optical catastrophes, (ii) singularities of plane polarized waves (scalar fields), (iii) polarization singularities of vector light fields. An important step to understanding light singularities was made when it was recognized that light flow could create vortices (Braunbek and Laukien [1952], Boivin, Dow and Wolf [1967], Nye and Berry [1974], Gullet, Gil and Rocca [1989]). In general, vortices are inherent to any wave phenomena, including even complex probability wave ftinction in quantum mechanics, e.g. encountered in connection with the Dirac monopole (Dirac [1931]) and the Aharonov-Bohm effect (Aharonov and Bohm [1959], Berry, Ghambers, Lange, Upstill and Walmsley [1980]). Vortices exist in various physical systems from superconductors (de Gennes [1989]) and superfluids (Donnely [1991]) up to cosmic strings (Vilenkin and Shellard [1994]). In the case of phase singularities in light, much of the revived interest has come from the ability of lasers to generate and easily manipulate a great variety of optical fields. The aim of this report is to collect a "harvest" of the results. By the end of the 20-century a few review papers and monographs appeared. Namely, the linear "catastrophe" optics was developed in a comprehensive way by Wright [1979], Berry and Upstill [1980] and summarized by Nye [1999]. The nonlinear optical catastrophe was observed recently during self-action of an elliptical Gaussian beam focused into a Kerr-like medium (Deykoon, Soskin and Swartzlander [1999]). Some important topics of singular optics where considered in a monograph "Optical Vortices" (Vasnetsov and Staliunas [1999]). The optical vortex solitons in nonlinear media where considered by Kivshar and LutherDavies [1998] and Akhmediev and Ankiewicz [1997], and vortices in nonlinear systems by Pismen [1999]. The presence of orbital angular momentum, one of the most specific features of optical vortices, was reviewed recently by Allen, Padgett and Babiker [1999]. Due to the large scope of research in the area, we shall restrict our review of singular optics to systematic exposition of its physical background and to the new features absent in light fields with a smooth wave front. Propagation of a singular laser beam through free space and/or nonlinear media will be considered. The rapidly developing field of nonlinear optics is accompanied by studies of transverse pattern formation in wide-aperture nonlinear optical systems (Rosanov [1996]). Full overview of pattern formation and competition in active and passive optical systems was given recently by Arecchi, Boccaletti and Ramazza [1999], where phase singularities were also briefly considered. Some aspects of phase
4, § 2]
Anticipations of singular optics
223
singularities in laser systems were considered by Weiss and Vilaseca [1991] and in an overview "Solitons and vortices in lasers" by Weiss, Vaupel, Staliunas, Slekys and Taranenko [1999].
§ 2. Anticipations of singular optics The history of optics embraces geometrical optics, according to which light propagates along straight lines (rays) and wave optics, which explains interference of light beams and diffraction, elucidating light penetration into a shadow area. Wave optics teaches us that light energyflow,associated with the Poynting vector, does not follow a straight line. In any case, the flow in a free space should be laminar, as it is in inviscid fluid. The question is, under which circumstances does turbulence in a light flow appear, or is it possible to create at least one isolated "optical vortex"? At first glance, the answer is negative. However, the nature of light brings more surprises than one might expect. It appears that the possibility of a local backward light flow has been first noted by Ignatowskii [1919] who studied a field structure in a focal plane of a focusing lens. This fact was analyzed later in detail by Richards and Wolf [1959], who computed the Poynting vector distribution in the focal plane of an aplanatic system. The energy flow was found to have vortices around certain lines in the focal plane (Boivin, Dow and Wolf [1967]) (fig. 1). The important feature of the light vortex is that the field amplitude vanishes on the axis of the vortex. Another way which has led to the discovery of an "optical vortex" was shown by Braunbek and Laukien [1952] (fig. 2). The vortices were found in an interference field produced by an incident plane wave and its reflection from a semi-infinite perfectly reflecting half-plane screen. Due to the presence of the screen edge, the standing-wave field in fi-ont of the screen was slightly modulated. Instead of the zero-amplitude nodal interference planes only lines parallel to the edge appeared where the field amplitude vanished. Again, the lines of zero amplitude were the axes of vortices. The two examples illustrated in figures 1 and 2 reveal a possibility of generation of vortices in a monochromatic light wave. As was pointed out by Sommerfeld [1950], a deviation from monochromaticity violates the periodic wave front sequence in a traveling light wave. Phase defects appear at the points where the beating of waves with slight differences in frequencies gives a zero value for the field amplitude. Around zero-amplitude point the crest of the wave transforms to the trough, and the phase becomes undetermined, or singular. An important feature of a phase defect is that its center, i.e. zero-amplitude point.
224
[4, §2
Singular optics
'15
-i.Q
iM m
ifi
^.i
'OS
Fig. 1. Flow lines of the time-averaged Poynting vector, showing vortex behavior around the phase singularity at the focal plane (Boivin, Dow and Wolf [1967], fig. 1).
••m'i
i
71 / '
It
/ / ; M
I "lip'' /
///;,
/ / / /11 k
///I I
'illi,.,
ll'll'l 'iiiiin'i'.iil
Fig. 2. Flow lines of the time-averaged Poynting vector at the vicinity of a perfectly conducting halfplane, illuminated by a plane wave (Braunbek and Laukien [1952], fig. 3).
4, § 2]
Anticipations of singular optics
225
belongs to a continuous line in space, where the amplitude vanishes. The phase circulates around the line, creating a vortex. However, Sommerfeld concluded: "due to zero value of amplitude, the influence of these points is no stronger than other points of variable amplitude". The essential role of phase singularities has been recognised only after pubHcation of the seminal paper of Nye and Berry [1974], who introduced a new concept into wave theory based on phase singularities in a wave field as a new class of objects in optics, and more generally in electromagnetic waves. In analogy with defects in crystals, wave-front dislocations were introduced. Furthermore, Berry [1981] concluded that phase singularities are the most remarkable features of wave fronts. To our knowledge, Bryngdahl [1973], Bryngdahl and Lee [1974] and Lee [1978] first reported the idea of artificial introduction of phase singularities into a smooth wave-front beam in connection with the radial-fringe interferogram formation. The phase structure of Laguerre-Gaussian (LG) "doughnut" modes of laser emission, which possess zero value of intensity on the axis, was the point of investigation of Vaughan and Willetts [1979]. The phase singularities have been considered as exotic objects until Zel'dovich with co-workers found phase dislocations in speckle fields (Baranova, Zel'dovich, Mamaev, Pilipetskii and Shkunov [1981]). The speckle structure of coherent light scattered by a rough surface was observed still in 1962 immediately after the development of the first lasers, namely a cw He-Ne laser (Rigden and Gordon [1962]). These speckles, or local intensity maxima, arise from constructive interference of partial waves with random phases. It was shown that there are also points of fiilly destructive interference with zero amplitude, and consequently with appearance of phase singularities in the form of screw wave-front dislocations (Baranova, Zel'dovich, Mamaev, Pilipetskii and Shkunov [1982]). They noted a "sea" of optical vortices and also established the important features of screw wave-front dislocations in a speckle field, in which amplitude distribution is a smooth random function of coordinates with the following properties: (i) the average number of "positive" and "negative" screw dislocations per unit area of a beam cross-section is equal; (ii) continuous zero-amplitude lines possess snake-like structures and extend along the beam propagation axis (Baranova, Mamaev, Pilipetskii, Shkunov and Zel'dovich [1983]). An intriguing term "optical vortex" (OV) was introduced by Gullet, Gil and Rocca [1989], stimulating a wave of new researches. Sometimes the terms "phase dislocations", "phase singularity lines" and "optical vortices" are used as equivalent (Berry and Dennis [2000b]). In our
226
Singular optics
[4, § 3
opinion, they are not all totally identical but rather complement each other. Indeed, a phase dislocation is the loci of the zero amplitude that can be as a surface, as a line. The case of a line of zero field amplitude is a phase singularity line. The optical vortex has a complicated structure with a dark core (zero-amplitude axis) with phase circulation around it. The screw wave-front dislocation appears as a helicoidal wave-front structure around the dislocation line. By the end of the 1980s, a "critical mass" in the field of phase singularities of wave fronts for singular optics was attained.
§ 3. Wave-front dislocations - phase defects Nye and Berry [1974] introduced the term "wave-front dislocations" as close analog of those found in crystals. Let us examine this analogy deeper. According to how the Burgers vector is arranged relative to the direction of the dislocation line, dislocations may be of edge, screw or mixed edge-screw type (Nye and Berry [1974]). Figure 3 gives a comparison between two main types of defects in a periodic structure in a crystal and in a wave-front sequence. Burgers [1939] showed that a contour formed from the main translation vectors of a lattice embracing any point is closed in an ideal crystal, but in a defect crystal the contour embracing a dislocation is torn. The additional vector connecting the end point to the start point is called the Burgers vector. In the case of edge dislocation (fig. 3a) the Burgers vector is perpendicular to the dislocation line, in the case of screw dislocation (fig. 3b) it is parallel to the dislocation line. Let us determine how the phase grows progressively with the optical path, say along z-axis. For a plane wave the phase 0, which depends both on time t and distance z, appears as 0(z,t) = kz-(ot,
(3.1)
where k is the wavenumber, co is the light frequency, and co/k = c is the speed of light. This choice is rather arbitrary, and we can determine the phase to grow progressively with time, but we prefer expression (3.1) for convenience. For more complicated beams, the phase should be also dependent on transverse coordinates. Fortunately, often it is possible to separate terms responsible for the wave-front shape ("transversal" phase) and for propagation along the beam axis ("longitudinal" phase). A family of lines of equal phase (contour phase map) infig.3c is a momentary cross-section of the phase of a wave propagating along the z-axis, with phase
4, § 3 ]
Wave-front dislocations - phase defects
111
Fig. 3. Comparison between edge and screw dislocations in a crystal (a,b) and in a wave front (c, d). Edge dislocation axis, shown in (a) as a cross, is perpendicular to the picture plane. Burgers contour ABCDEF embracing a screw dislocation is shown (b). Burgers vector FA connecting points F and A is parallel to the dislocation line, the sign of dislocation is positive, (c) Edge dislocation of a wave front, (d) Helical wave front with an axial OV possessing unity charge. The vector FA is the analog of the Burgers vector, the sign is positive.
interval Ji/A, Wave fronts (crests) are shown by a thick line, trough contour is indicated by a dashed line. We can attribute for selected continuous wave-front surface (crest of a wave) zero phase value, as shown in fig. 3c. Phase grows along the z-axis, with a correspondence Ijt in phase -^ one wavelength in space. Very similar to the edge dislocation in a crystal, a wave-front dislocation can appear as an edge of an extra sheet between neighbor wave fronts, as fig. 3c shows. The end point of this extra sheet in the XZ cross-section is depicted as O. Here the crest of the wave gradually transforms to the trough, and the field amplitude necessary attains zero value at this point. Looking on the left side of the wave-front sequence, we detect one full wavelength, while on the right side two wavelengths could be found.
228
Singular optics
[4, § 3
At a saddle point S, two troughs are shown to meet together. In a half period of time oscillation, two crests (which are normally separated by one wavelength in a regular wave) will meet here. To understand the physical reason for this rather strange behavior, let us have a look at a separation between equiphase lines in fig. 3c. It is seen with the naked eye that the lines are compressed on the right side of the edge dislocation, where an extra-sheet of the waveft-ontappears. This is evidence that the phase velocity is somewhat slower here, and higher on the other side of the dislocation. An analogue of screw dislocation in a crystal (fig. 3b), wave-fi'ont screw dislocation can also exist. Figure 3d is a sketch of helicoid-shape wave front with a "Burgers contour" ABCDEF, where the "Burgers vector" length is a wavelength (Nye [1997]). Of course, we can choose any other plane to make a section of a running wave and obtain a contour phase map. However, the picture which occurs in any section perpendicular to the z-axis will show nothing but parallel lines, the same for any section perpendicular to the x-axis. These pictures are not very informative. The best choice is to use a plane, which is transpierced by the dislocation line. In this plane, we can easily recognize the phase singularity manifesting itself as a point radiating equiphase lines (or, equivalently, equiphase lines terminate there), see e.g. a detail view of the equiphase structure around the dislocation line (point O in fig. 3c). One round-trip around point O by any closed contour will change the phase on 2jr. A wave dislocation can be defined in terms of an integral around a circuit that contains within an isolated dislocation line (Nye and Berry [1974]):
/
d 0 = 2mjz,
(3.2)
where the integer m, which may be positive or negative, is the winding number, or the charge of a dislocation. For a monochromatic wave a dislocation is stationary in space, forming an isolated interference fringe. However, in contrast to the stationary dislocations in crystals, wave-fi-ont dislocations are dynamic objects due to the continuous motion of light and therefore phase variation. We shall see further how the phase circulates around a dislocation line, producing an "optical vortex". To begin analysis, we have to check how the used field description fits our task. First, we shall restrict ourselves within a frame of scalar field, i.e., linear field polarization. The oscillations of the electric field are assumed to occur in a plane, which contains the 7-axis, as is shown in fig. 4. Three optical rays are shown, OA, OB and OC: OA is of arbitrary direction, OB is in YZ plane, OC is
Wave-front dislocations - phase defects
4, § 3 ]
229
,'AEy
Fig. 4. Electric field of a spherical wave emitted by a point source located at the coordinate origin O. All three polarization components Ey, E^ and E^ are present in observation point A for a wave with the plane polarization.
in xz-plane. The electric field component in the transversal electromagnetic wave is perpendicular to the direction of propagation, therefore we can consider E{C) to have the only Ey component, E(B) will have both Ey and E^ components, and finally E{A) has all three E^, Ey and Ez components. The amplitude Ey amounts to E cos Yy, Ez=E sin y^ cos YX, Ex=E sin Yy sin Yx- The paraxial approximation eliminates both Ex and E^ components, assuming the angles Yy ^^^ Yx to be small. The only Ey=E component is used for the field description. We start with the scalar wave equation written in cylindrical coordinates p, q), z and time t to describe a light wave propagating along the z-direction: dE\
1 d^E
d^E _ 1 d^E
pdp where E(p,q),z,t) the frequency co:
(3.3)
is the strength of the electric field, oscillating in time with
E{p, (p,z, 0 = E(p, (p,z) Qxpi-icot).
(3.4)
The wave oscillation in space can be represented by introducing the wave number k, resembling propagation of a plane wave (eq. 3.1): E(p, cp,z) = E{p, (p,z)exp(i^z).
(3.5)
where the amplitude E(p, cp,z) at the right-hand side of eq. (3.5) is a "slowly varying" function of p, q) and z. The essence of the paraxial approximation
230
Singular optics
[4, § 3
consists in neglecting the second derivative of the slowly varying amplitude, and results in the final equation \ d ( dE\
1 d^E
^.BE
^
^_^^
In a physical sense, the paraxial approximation is based on an assumption that the wave amplitude varies very little over a distance of the order of a wavelength. In the further analysis we shall use the solution without any azimuthal dependence, therefore the second term in eq. (3.6) vanishes. The merit of eq. (3.6) is the analytical solution in a form of the Gaussian beam, £'(p,z) = £G — exp (
r ) ^^P f i^7^ ~ i ^^^tan — ) ,
(3.7)
where EQ is the amplitude parameter, k = 2jt/X is the wavenumber, A is the wavelength, WQ is the beam waist parameter, with associated Rayleigh range, ZR = kwl/2, where the transversal beam dimension w = wo(l +Z^/ZR)^^^ enlarges in ^/2 times with respect to the waist. The radius of the wave-front curvature is R = z(\ -^z\/z^). The phase term includes the longitudinal phase fe, transversal phase kp^/lR and the Gouy phase shift (Siegman [1986]) appears as 0 = kz+^-
arctan (—\
2R
.
(3.8)
\z^J
Let us examine solution (3.7). First, the presence of the Gouy phase shift arctan(z/zR) influences the phase velocity of the Gaussian beam (Siegman [1986]). On its way from the waist (z = 0) to the far field (z^oo) the beam experiences an additional phase shift -nil with respect to a plane wave, and the corresponding shift amounts to -;r/4 on a distance equal to the Rayleigh range. This phase shift can be interpreted as a small acceleration of the beam in the region near its waist, where its transversal dimension is compressed. The effect of the Gouy phase shift is considered in § 8. Second, the beam has a variable curvature of the wave front, being a function of z. At the waist, the wave front is plane ( 0 = 0) due to the fact that 7^(0)—>oc. Near the waist (z
— + -^] ZR
2Z|
.
(3.9)
4, § 4]
Circular and linear edge dislocations
231
Far enough from the waist, the spherical approximation of the wave-front shape is adequate. We can also determine at the waist the phase velocity distribution across the beam:
On a circle with radius WQ the phase velocity exactly equals c, it exceeds c within the circle, and is less than c outside it.
§ 4. Circular and linear edge dislocations It is well known that the so-called Airy pattern appears in the focal plane of a uniformly illuminated lens (Born and Wolf [1999]). The bright central spot is surrounded by concentric dark rings, along which the field amplitude vanishes. The origin of the dark rings is boundary diffraction and destructive interference of waves. The phase of the field changes by Jt, while crossing any dark ring. These dark rings have been recognized as phase singularities (Boivin, Dow and Wolf [1967], Basistiy, Soskin and Vasnetsov [1995] and Karman, Beijersbergen, van Duijl and Woerdman [1997]). On the dark ring the phase is undefined, as its gradient diverges. Therefore a natural defect of a wave field appears similar to defects in crystal lattice. In the case of an axial symmetry, those dark rings form a family of "circular edge dislocations" of a wave front. A gradual variation of lens illumination conditions can lead to the drastic transformation of the dark rings structure, including annihilation of the circular edge dislocations and birth of new ones (Karman, Beijersbergen, van Duijl and Woerdman [1997]). These transformations are called topological reactions, because topological objects, namely phase saddles, extrema {minima and maxima), as well as vortices, are involved in these reactions. We shall pay more attention to the topological reactions in § 7. To study the structure of the circular edge dislocation more thoroughly, we need to create an isolated dark (zero-amplitude) ring within a light beam, with an analytical description of the field. Interference of two co-axial Gaussian beams can be chosen for this purpose (Vasnetsov, Gorshkov, Marienko and Soskin [2000]). This choice provides an analytical description of the amplitude and the phase of the wave superposition in space, and therefore we can examine in detail the wave-front shape in the vicinity of the circular phase dislocation. To obtain the circular edge dislocation in this simple way, let us consider the interference of two Gaussian beams with phase difference of jr. Because both
232
Singular optics
[4, § 4
beams have a plane wave front in the shared waist at z = 0, on superposition one has the ampHtude distribution E{p,z = 0) = Ei Qxp(-^j
exp(i0i) + ^ 2 e x p f - ^ j exp(i02),
(4.1)
where E\, Ei are the amplitude parameters of the beams (£"1 > E2), w\ and W2 are their waist sizes {w\ < W2), and the phases at the waist are 0 i ( p , z = 0) = ^{p,z = 0) ± JT. The zero-amplitude circle is located at the waist plane with the radius Po = A H - ^ - ^ l n ( £ i / £ 2 ) -
(4.2)
To find how the phase behaves in the vicinity of the dark line, we must calculate the interference field as a sum of two complex amplitudes E(p,z) = E\(p,z)Qxp[i
tg
(4.3)
Re[£(p,z)] where the wave amplitudes £'i,2(p,^) and the phases 0\2(p,z) are defined in eqs. (3.7) and (3.8). After some trigonometric transformations, we obtain the equation in implicit form for the equiphase line 0 = const.: / kp^ z \ £'i(p,z)sin 01 - 0~\-kz-\- ^rzr-— -arctan — V 2Ri(z) ZR, y / kp^ z \ = E2(p,z) sin a>i - 0 + ^z + :—-- - arctan — V
2R2(Z)
(4.4)
ZR2 /
Figure 5 shows the result of superposing the beams, as the amplitude distribution at the waist (fig. 5a) and the equiphase lines at the plane p,z (fig. 5b) as a numerical solution of eq. (4.4) with parameters E\ = 1, £"2 = 0.5, A: = 1, wi = 10, W2 = 100, 01 = 0.5, 0 = 0, ±2jr. The chosen equiphase lines are sections of the surfaces of constant phase (crests of the wave), and by varying parameter 0i one can simulate the wave-fi-ont motion through the plane z = 0. The dark circle within the combined beam at p = po is localized at the plane z = 0. Even a small displacement from the plane z = 0 violates the
4]
233
Circular and linear edge dislocations
iilimil
5
10
15 20 25
30
35
40
I
45 50
(b)
(a)
Fig. 5. (a) Modulus of the amplitude of two Gaussian components (solid lines) and the resulting combined beam (dashed line) at the waist. The destructive interference of the components produces a zero-amplitude circle at p = Po- (b) The wave-front sequence cross-section in the plane p,z (0 = 0, ±2JT). One wave front is lost within the circle.
condition of the field vanishing, because of the phase mismatch caused by the terms responsible for the wave-front curvature and the Gouy phase shift. The Gouy phase shift plays a very important role in this situation. We shall see below, how under some circumstances two new edge dislocations can appear near the waist plane in a topological reaction of saddle-vortex collision. At first, the wave front before the waist (z < 0) in fig. 5b looks bent and has a ledge toward the direction of the beam propagation. After the waist the ledge is inverted. The lack of one wave front results in a small wavelength difference, or phase velocity variation, within the circle of the dislocation and outside it. The phase velocity is greater within the dislocation circle: the physical reason is the Gouy effect (Gouy [1890]). A spatial compression of a light beam results in the increase of phase velocity on the axis (Hariharan and Robinson [1996]). To calculate the phase velocity v of the Gaussian beam in the waist, eq. (3.8) can be applied and one finds that 1 v=c
(4.5)
There is a distance p where v = c\ pi^ = WQ. When p < Pu, the phase velocity exceeds c, and vice versa. Due to this phase velocity variation, the wave front of the Gaussian beam being plane at the waist gradually transforms to a spherical one. Arriving at the waist, the ledge of the wave front joins with the next wave front, which has a "hole" at this moment (fig. 6a). Then the "holed" wave front
[4, §4
Singular optics
(a)
(b)
(c)
Fig. 6. (a) The wave front comes close to the plane of dislocation ((P\ = 0); (b) two neighboring wave fronts join; (c) the wave front becomes holed, and the next one restores continuity. Black dot, position of the dislocation line cross-section; open circle, the saddle point.
takes the top of the ledge and restores a continuous surface (fig. 6b). Without the missed part, the preceding wave front becomes holed (fig. 6c), and the process repeats. In other words, the phase biftircation occurs at the circular dislocation. Figure 7 exhibits in more detail the equiphase lines with spacing 6 = Ji/4 around the dislocation line. The crest of the wave (0 = 0) is now at z = 0 within the circular edge dislocation and transforms to the trough {0 = Ji) outside the circle p= Po- The familiar OV structure (phase star) about the core is evidently seen. Also, there is the phase saddle, where the bifurcation occurs. The phase gradient is consistent with the distance z: the larger is z, the bigger is the phase. (This is a result of our choice of the sign of the phase in eq. 3.1.) However, this consistence is violated at the vicinity of the phase dislocation. As is seen, the sequence of equiphase lines between vortex and saddle corresponds to their rotational motion, which becomes possible due to the bifurcation shown in fig. 6. The specific motion of the equiphase lines around the OV core is a reflection of the energy flow at this region. If we draw the direction of the local Poynting vector as arrows which are perpendicular to the equiphase line at any point (taking into account the general motion of the wave along the z-axis), we find that there is a closed area, where light rotates around the dislocation line. The separatrix loop is a boundary of this area and outside it the energy flow is directed in an ordinary manner, toward the beam propagation. Saddle point is a stagnation point, or standing-wave point. The dimension of the rotation area (roughly the distance fi-om the saddle to OV core) in the example above is about 10~^ of the wavelength along the transversal coordinate and about 10~^ of the wavelength along the z-axis. The light is "captured" within the torous. These dimensions are much less than the narrow Gaussian beam waist size (about 1.5 wavelengths). Vasnetsov, Gorshkov, Marienko and Soskin [2000] discussed the validity of the paraxial approximation in this case. This fascinating phenomenon of the torous light rotation around the circular
4, §4]
235
Circular and linear edge dislocations
kz 2
\
\
\
\
1 0 -1
I /1
\
^—.\ ^
/
-2^
(b) 1
8.2 8.3 8.4 8.5 Transverse coordinate P
8.6
8
/ 1
1
1
1
r—
8.2 8.3 8.4 8.5 Transverse coordinate P
1
8.6
Fig. 7. (a) Detailed view of the equiphase lines around the dislocation line (the black dot indicates the point where it pierces the plane p,z). The open circle denotes the saddle point. The crest of the wave transforms to the trough at the dislocation point, (b) The same as (a) with arrows directed along the local energy flow. The continuous line, which passes through the saddle point and makes a loop around the dislocation, is the separatrix.
edge dislocation justifies the adaptation of the term "optical vortex" to the fi-ee propagation of a light wave. The effect of the energy rotation around the OV core is probably the most important feature of OV. Analysis shows that there are two possibilities for the saddle point to exist, inside or outside the circle of the edge dislocation (Vasnetsov, Gorshkov, Marienko and Soskin [2000]). The location of the saddle was found to be determined by the distribution of the phase velocity around the dislocation line: vortex winds toward the area of higher velocity. This result can explain the physical sense of the Airy ring pattern structure, which exhibits a very similar picture of the Poynting vector flow lines, see fig. 1 (Boivin, Dow and Wolf [1967]). In a more in-depth approach, non-paraxial beams should be considered. Surprisingly, circular edge dislocations are shown to exist even for a single nonparaxial Gaussian beam (Carter [1973], Berry [1998a]). Let us turn now to the case of linear edge dislocations in a two-dimensional (2D) scalar wave field, similar to the structure shown in fig. 2. As was pointed out by Nye, Hajnal and Hannay [1988], linear edge dislocations can be created in pairs with opposite direction of the energy rotation around them, or a pair can meet and destroy one another. For a monochromatic wave there is the topological necessity that two phase saddles should participate in this event. It was established also that there are no phase maxima or minima in the contour phase map for 2D waves, but only saddles and vortices. We note that we did
236
Singular optics
[4, § 4
not encounter phase extrema in the above consideration of two interfering 3D Gaussian beams. Very similar to the above analyzed circular edge dislocation, a pair of linear edge dislocations can be constructed as interference nulls in the interference field of two 2D Gaussian beams (Pas'ko, Soskin and Vasnetsov [2001]). This configuration can be regarded as a degeneracy of a circular case, but the reduction to 2D waves permits us to avoid any consideration of polarization components except the one parallel to the dislocation line. Of course, this makes the Poynting vector calculations much easier. In paraxial approximation, the 2D Gaussian beam, which satisfies the wave equation, has the following representation: E{X,Z) = EG
( ^ ) ' ' " e x p ( - ^ j exp i (kz-
kx^ 2R{z)
1 z - 2 - arctanZR,
(4.6) where EQ is the amplitude parameter, the other parameters are defined in eq. (3.7). We can arrange two zero-amplitude lines parallel to the j-axis by superposition of two 2D Gaussian beams at the waist, similar to the superposition of 3D Gaussian beams which creates a circular edge dislocation (eq. 4.1): E{x,z = 0) = £, e x p f - ^ j exp(i0i) + £2 e x p f - ^ j exp(ia>2),
(4.7)
where E2, wi < W2. The edge dislocation position is given by
xo = ±J^^^\n(E,/E2l
(4.8)
where the ratio E2/E1 = rj will be used below as the governing parameter. The equation for determining the saddle position Xs can be obtained as a coordinate of the branching point for the equiphase line 0 = 0:
^+^_J_ = :^Wffc+J^__LV 2z^i
2ZRI
Ei(x) \
(4.9)
2z^2 2zR2y
The position of the saddle point can be inside the distance XQ, (XS < XQ), or outside it {Xs > XQ). First, we shall analyze the case Xs < XQ, which is similar to the circular edge dislocation structure described above. The phase map looks
4, § 4]
Circular and linear edge dislocations
231
identical, showing the vortices around zero-amplitude lines, and now we can look for some particular details. With increasing of r], XQ diminishes according to eq. (4.8), and two dislocation lines approach each other as well as saddles. The saddles, which are located between the vortices, will meet at x = 0 point when rj = ZR2(2^ZRI - 1)/ZRI(2^ZR2 - 1). This is the first stage of the collapse (fig. 8a). Then saddles appear again separately, but are located now on the z-axis symmetrically to the x-axis (fig. 8b). The maximum distance between the saddle points will be reached for rj ^ 0.995 and amounts approximately to unity (fig. 8b). At this point the distance between vortices becomes equal to the distance between the saddles. With the further increase of rj the saddles as well as the vortices come closer. Finally, vortices and saddles collide and they annihilate altogether at the point x = 0 (fig. 8c). With decreasing of rj, the saddle point moves toward the vortex and meets 2_ 2
it when rj = exp 2/^2^^\^ (this condition corresponds to the equality of phase velocities of the interfering waves on the dislocation line). At this collision, the saddle-vortex point appears, which is a "pathological" topological object. After the crossing event, the saddle appears outside the interval between the dislocation lines, and thus the light flow circulation around the dislocation reverses direction. This reaction of saddle-vortex collision is accompanied by the creation of two similar objects, asfig.9 shows. Let us examine how new edge dislocations can "gemmate" from the saddlevortex line ("unfolding" of an edge dislocation, according to Nye [1998]). First, we have to find the conditions for the edge dislocation to exist out of the waist plane. There are two necessary conditions, one for destructive interference, another for the equality of the amplitudes of the interfering waves. The phase condition is ^i(x,z) -
I
z
- -2 arctan ZRi
kx^
—-2R2{z)
^
,, ^r^^
+ -2 arctan — =0. ZR2
I
z
(4.10)
There is a trivial solution z = 0 (the waist plane), where two initial edge dislocations are located. The surface of destructive interference, on which new edge dislocations can appear, is given by ^ 1 (z^^ziA (Z^+ZIJ) ( Z Z\ x^ = -r T^^ ^ arctan arctan — . kz 42~4i V ^Ri ^R2/
(4.11)
The cross-section of this surface is shown infig.9 by a solid line. The line crosses the x-axis exactly at the point that corresponds to the equality of phase velocities of the interfering waves. This coincidence is hardly accidental, and
238
Singular optics
[4, §4
Fig. 8. Collapse of edge dislocations, (a) Equiphase lines around the dislocations, which are close to the beam center, saddle points have joined, (b) Vortices approach closer; saddles are split along the z-axis. (c) All topological objects collide and annihilate at rj = 1.
4, §5]
Screw wave-front dislocation - axial OV
239
0.613 0.612
Fig. 9. Creation of two new edge dislocations in a saddle-vortex collision. Solid line, cross-section of the surface of destructive interference {ic phase difference between interfering beams) for x > 0; dashed lines, cross-section of the surface of the amplitude equality. Positions of edge dislocations are shown by dots. For rj > TJQ there is only one possible crossing point and therefore one zeroamplitude line (the edge dislocation). For rj < TJQ there are three points of crossing, and therefore three edge dislocations exist.
gives a hint to understanding the vortex behavior. It can be concluded that the vortex rotation is directed toward the area of higher phase velocity (Soskin, Vasnetsov and Pas'ko [2001]). § 5. Screw wave-front dislocation - axial OV A screw wave-front dislocation has an interesting feature: the spatial structure of the wave front has the form of a helicoid around the dislocation axis. Evidently, this helicoid can be left-handed or right-handed, therefore its "topological charge" can be positive (conventionally, for right helicoid) or negative for left helicoid, as we have determined using the Bruster vector. The examples of waves containing optical vortices are the Laguerre-Gaussian modes LG^ (Kogelnik and Li [1966]), which are solutions of the wave equation in the paraxial approximation for a cavity with circular symmetry (cylindrical coordinates p, cp, z are used): E(LG') = Ei^Gexp 1 feH-
\/2p
\i\
exp
W2]H
W2
(5.1)
+ l(p- Q arctan — 2R{z) where ^LG is the amphtude parameter, ZR is the Rayleigh range, R(z) is the radius of the wave-front curvature in the cross-section along the z-axis. Ly{-^)
240
Singular optics
[4, § 5
is the Laguerre polynomial with the integer index / equal to the value of the phase change in a closed loop around the circumference of the beam axis (in 2Jt modulo), and the number of nodes along the radial distribution of wave amplitude equals /? if / = 0, and /? + 1 if / ^ 0 (axial zero in amplitude distribution appears, corresponding to the axial OV). The index Q is equal to 2/? + |/| + 1. Modes with equal Q belong to the same mode family, propagating with equal phase velocity [note that Q governs the small difference in phase velocity determined by the Gouy phase shift arctan(z/z/?)]. Below we consider only the case of singleringed (p = 0) modes. The intensity distribution in a transverse cross-section of the beam now has the form of an annulus: at the beam axis ( p = 0) amplitude vanishes and reaches the maximum value on the circle p = Pmax? which is connected with the transversal beam dimension w by the relation Pm..=w{\l\/2f\
(5.2)
The value of the amplitude maximum at the waist is related to the amplitude parameter £LG as Em..=E^G(\l\/2ef'\
(5.3)
We can also determine the radius of the central crater pc using the criterion d^ \E(p)\^ /dp^ = 0 to find the inflection point of the intensity distribution: p2.^2miizVmil,
(5.4)
o
The topological charge of the OV associated with the LG^ mode is m = I. The helicoidal wave-fi-ont shape is determined by the phase term in eq. (5.1): Icp + kz^ljin,
« = 0,±1,±2...
(5.5)
(here we neglect the small influence of the wave-fi-ont spherical curvature and the Gouy phase shift on the wave-front shape). With / = i l , any wave front (number n) is a part of a helical surface with a pitch equal to the wavelength A. The next wave fi-ont (number « + 1) joins the preceding one continuously. With |/| > 1, the following wave fi-ont at the distance of one wavelength is enclosed within the preceding one, which has pitch |/|A. As a result, a |/|-start helicoid is built, with the distance between neighboring 2«;r-phase surfaces equal to the wavelength with small deviations caused by the change of the curvature and the Gouy phase shift.
4, §5]
241
Screw wave-front dislocation — axial OV -;=r (-111
>
{i
__ -• /=0
/ = T/8
/ = T/4
/ = 3T/8
r = T/2
Fig. 10. Contour lines of a momentary electric field strength distribution across the beam crosssection.
Transverse coordinate p Fig. 11. Amplitude and intensity distribution of OV beam (LGi mode) (a); intensity distribution in a gray scale (b); wave-front shape (c).
Why is the hehcoidal wave-front beam a vortex? Let us examine how the field ampHtude varies with the time in any given cross-section of the beam (z = const). Figure 10 shows the distribution of the electric field strength in a cross-section of the beam (LGQ mode) at different instants of time, with an interval of 1/8 time period T of the wave oscillation. At half of the period, the field distribution makes a half-turn around the beam axis. The time averaging over a period results in a "doughnut" with zero field value on the axis (fig. lla,b). Rotation of the field around the axis produces the vortex, and the combination of the field circulation and of the longitudinal propagation of the wave results in the helicoidal wave-front structure (fig. lie). Another consequence of these phenomena is the existence of the orbital angular momentum (OAM) in the beam. The origin of the orbital angular momentum for a beam with an axial OV can be explained by a simple consideration. As the wave front has a helicoidal shape, the Poynting vector 5 ( p , ^),z), which is perpendicular to the wave-front surface, has a nonzero, tangential component S(p{p, q),z) at each point (fig. 1 Ic). The value of this component can be found in the paraxial approximation from analysis of the wave-front geometry. Figure 12 schematically shows a narrow
242
Singular optics
[4, § 5
f Z
Pd
sector dcp of di helicoidal wave front. The slope of the surface with respect to the xy-plane on a distance p from the heHcoid axis is a = arctan(dz/pd(p).
(5.6)
The displacement dz along the z-axis can be found from the equation of the helicoid surface (5.5), /d(p = -A:dz,
(5.7)
which determines the slope (far enough from the axis) a « -l/kp.
(5.8)
Assuming that the Poynting vector is perpendicular to the wave-front surface (Bom and Wolf [1999]), we find the tangential component of the Poynting vector to be given by the expression S^ = aS.
(5.9)
Because of the proportionality of the modulus of the Poynting vector to the intensity of the wave \E{p,q),z)\^, the resulting tangential component will be proportional to the local intensity: Scf(x-l\E(p,(p,z)f/kp.
(5.10)
The density of the distribution of the OAM M^ oc pS^p will therefore be M,(x-l\E(p,(p,z)\^.
(5.11)
4, § 5]
Screw wave-front dislocation - axial OV
243
In a more rigorous approach, we shall consider the vector product of electric and magnetic components of the electromagnetic wave (linearly polarized and propagating in free space) to determine the 0AM density M as M = rxS/c\
(5.12)
Here r is the radius-vector, c is the speed of light. With the known expression for 5, viz. S = E X H, and using the relation c^ = eo/io, B = ^H, we obtain for M the expression M = eQrx{ExB).
(5.13)
Assuming that the field is stationary and that the light is monochromatic, E = (E(x,y,z)Qxp(-i(Ot)-\-c.c.y2,
B = (B(x,y,z)exp(-ia)t)
+ c.c.)/2, (5.14) the expression for the magnetic field component B can be derived fi-om a Maxwell equation as BR - — = Vx^,
(5.15)
B=^\/xE.
(5.16)
10)
The resulting expression for the 0AM of a beam becomes M = e^r X
fE + E"" B + B* —-— X
= T^rx
\(E* xV
x V x JE-*)! , (5.17) where the first term in the round brackets does not depend on time and the second oscillates with the frequency 2o) and vanishes on averaging. The detailed derivation of the expression for the OAM distribution can be found in a recent survey (Allen, Padgett and Babiker [1999]). The expression is .^dE
xE-ExV
_dE*\
2(0
xE*) + (E X V xE-E""
f.dE
^dE^
(5.18)
In the case of an axial OV (not necessarily LG mode), the expression for the OAM distribution is MAx,y,z) = - ^ (O
\E{x,y,zf,
(5.19)
244
Singular optics
[4, § 6
which agrees with the expression (5.11). Therefore, for a beam with the axial vortex the distribution of the 0AM coincides the intensity distribution. The total 0AM value L^ is the integral of M^ over the beam cross-section:
4=
/
fu.dxdy,
(5.20)
which results in the expression L, = -mW/w,
(5.21)
where W is the beam energy. The expression which we obtained does not depend on a radial function of the intensity distribution (e.g. LG mode or Bessel-Gauss beam) with axial symmetry. If the beam energy is expressed in terms of the number of photons N, W = Nhco, the result obtained is L, = -mm.
(5.22)
In other words, each photon in the beam carries an 0AM mh, independent of the frequency. This result, obtained within a classical description of a field, can be a coincidence, reflecting the natural property of a photon to be able to carry quantized orbital angular momentum (Tiwari [1999]).
§ 6. Reflection, refraction, interference and diffraction of OVs As we have just seen, OV beams possess some unusual properties in freespace propagation. It is worthwhile to examine how they behave on reflection, refraction and diffraction. Reflection of an OV by an ordinary flat mirror changes the OVs topological charge, because a right helicoid becomes a left helicoid, and vice versa. Second, OV reflection possesses some specific feature: in contrast with the linear momentum transfer, there is no 0AM transfer to a flat mirror at normal incidence. However, at oblique incidence, the transfer of the OAM does occur, as fig. 13 shows. We note also the existence of some subtle effects caused by the helical wave-front structure of the OV beam (Fedoseyev [2001]). The same effect of the angular momentum transfer appears in a symmetrical refraction by a prism, as shown in fig. 14. In general, OV beam refraction is also accomplished with the transfer of the orbital angular momentum, but includes the transversal beam deformation. This feature is, of course, important for any
4, § 6]
Reflection, refi-action, interference and diffraction of OVs Left helicoid \
/Right
245
helicoid
Fig. 13. Transfer of the 0AM from a light beam to a mirror in an oblique incidence. Gray arrows indicate vectors of the 0AM of the light beams and the angular momentum transferred to the mirror.
Left helicoid
Left helicoid
Fig. 14. Transfer of the 0AM from a hght beam to a prism: due to the change of the direction of the light beam, a residual 0AM appears producing a torque on the prism.
optical system transporting an OV beam, and causes the light-induced rotation for small refracting particles (Heckenberg, Friese, Nieminen and RubinszteinDunlop [1999]). There is an interesting discussion of mechanical properties of light waves with phase singularities (Bekshaev [1999, 2000]). There are also some interesting peculiarities in the focusing of OV beams. The use of an ordinary spherical lens does not change the properties of LG modes carrying OVs, but focusing with an astigmatic lens (cylindrical for example) will exhibit drastic spatial and topological transformations of the OV, as has been shown by Bekshaev, Vasnetsov, Denisenko and Soskin [2001]. At a distance of the focal length, an axially symmetric "doughnut" divides into two bright spots, with a zero-amplitude line between them, as fig. 15 shows. The wave front appears to be cut into two parts, shifted by half of the wavelength. This is an example of the edge dislocation, familiar from § 4. However, OV exists within the focused beam in front of the focal plane, as well as behind it. Therefore, there is a crossing between "longitudinal" OV and the "transversal" edge dislocation. Figure 15 shows the set of calculated distributions of intensity and the wavefront shape of the initial beam and the focused beam before the focus, in the
246
Singular optics
[4, § 6
focal plane and behind it respectively. After the dislocations crossing, the beam returns to the "doughnut" shape (which is now stretched) with an axial zero, but the sign of the wave-front helicity is reversed (fig. 15). This is an example of how the dislocation crossing changes the topological charge of the OV Of course, phase circulation around the edge dislocation is also reverted at the point of the crossing. The beam transformation behind the focal plane can be "frozen" if we place an identical cylindrical lens on a distance of two focal lengths: this is a configuration of a JT mode converter (Allen, Padgett and Babiker [1999]). The Till mode converter can also be realized which transforms the "doughnut" LGQ mode into the Hermite-Gaussian HGoi mode. Padgett, Arlt, Simpson and Allen [1996] described in a tutorial paper how a range of LG modes can be produced from the corresponding HG modes using the mode converter. There is one important question: if the sign of the vortex is reversed after the crossing of dislocation lines, how does the 0AM associated with the vortex change? In our opinion, the 0AM, which is a conserving quantity on propagation in free space, will be redistributed between the "vortex" part and the complicated phase-amplitude structure of the beam (Poynting vector distribution in the beam cross-section). The interference of light beams carrying OVs demonstrates fascinating pictures. First, off-axis interference between an ordinary (Gaussian) beam and a "singular" beam produces "forks" in the interference pattern, the orientation being dependent on the angle between a singular beam and reference wave (Basistiy, Soskin and Vasnetsov [1995]). For co-axial interference, the picture appears like fringes spiraling fi-om the center (Bazhenov, Soskin and Vasnetsov [1992]), the number of fringes is equal to the absolute value of the topological charge of the OV in a singular beam. The number of vortices in a beam, which is a co-axial superposition of the Gaussian beam, and the singular beam with m-charged OV can vary along the propagation path, because new OVs can enter or leave the beam and amount to |m|, 2\m\ or zero depending on the conditions. The total topological charge is m or zero (Soskin, Gorshkov, Vasnetsov, Malos and Heckenberg [1997]). The orbital angular momentum of the beam is conserved regardless of the variations of the number of vortices, and amounts to the value of the orbital angular momentum of the singular component alone. Diffraction of light beams carrying optical vortices is of particular interest from the point of view of optical information transmission. The first attempt of the study of OV beam diffraction by an edge of nontransparent screen has recently been performed (Vasnetsov, Marienko and Soskin [2000], Masajada
4, §6]
Reflection, refi-action, interference and diffraction of OVs
247
'mi,::.
Fig. 15. First row: intensity distribution and wave front of a singular beam just after the focusing cylindrical lens. Second row: the same before the focus. Third row: the same exactly in the focal plane. There is a dark line crossing the beam intensity distribution and a jr-step across the wave front. Last row: after the focal plane the vortex reverses the sign.
248
Singular optics
[4, §6
z=0
z = 0.2ZR
Z = ZR
Z = 2ZR
Z = 5ZR
Fig. 16. Results of calculations of intensity distributions (left column) in the transverse section of a beam with an axial OV with the topological charge m = - 1 at different distances (in units of Rayleigh range) behind the screen, which cuts off the dashed part of the beam at z = 0. Central column: the corresponding phase distributions. Right column: patterns of interference with a plane reference wave. The transverse size of an unperturbed beam at the same distance is indicated by a dashed circle.
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[2000]). Slit diffraction in the near field also was analyzed (Abramochkin and Volostnikov [1993]). It was shown that the spatial truncation of an OV beam does not change its structure drastically, if the OV core is not cut off by an aperture. However, even when the vortex core is cut off with the edge of nontransparent screen, a pair of vortices with opposite charges nucleates at some distance behind the screen. One vortex then leaves the beam and the other, possessing the same charge as the initial one, remains within the beam up to the far field. This is a kind of the "regeneration" of an OV (Vasnetsov, Marienko and Soskin [2000]). Figure 16 shows the numerical simulation of the diffraction process. The truncated beam at z = 0 plane is represented by
Jo, 7 < 70 ^^^'^^ " \ V(^^ + / ) exp [- {x' ^yywl
+ icp] , y>y,.
^^'^^
The beam, apertured by a half-plane y ^ y^, contains only a peripheral part, which involves no phase singularity, when j^o > 0. Intensity and phase distributions of the truncated OV beam were calculated for different distances after the screen cut off more than half of the beam, as well as the patterns of interference with a tilted reference wave. As the beam propagates behind the screen, over distances less than the Rayleigh range for the unperturbed beam, diffraction bands are formed and the light penetrates asymmetrically into the area of geometric shadow. The latter effect was described as the rotation of a light spot around the z-axis (Vasnetsov, Basistiy, Kreminskaya, Marienko and Soskin [1998]). The direction of this rotation is determined by the sign of the OV charge in an initial beam. Within the interval from one to two Rayleigh ranges, the formation of a phase step was detected, which subsequently generates a pair of vortices. Then, one of them quickly disappears from the beam. The remaining vortex with the initial sign is located within the beam up to the far field, as fig. 16 shows. The vortex regeneration effect does not depend on the degree of the beam screening, as has been established experimentally (Vasnetsov, Marienko and Soskin [2000]). A detailed theory of a vortex beam diffraction on a half-plane under strong truncation, which cuts the vortex core, was elaborated recently by Gorshkov, Soskin and Kononenko [2001].
§ 7. Topology of wave fronts and vortex trajectories By definition, a wave front is a surface of equal phase. Both hills and valleys may be located on this surface, as well as saddles. (Their positions, of course.
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Fig. 17. Schematic views of wave-front fragments: (a) spherical; (b) charge minus-one helicoid; (c) charge minus-two helicoid; (d) OV dipole.
will depend on the choice of the reference plane, in contrast to vortices, as Berry [1998b] has emphasized). We shall speak about topological objects on the wave front, because of the visual perception of them, and we always take the reference plane to be z = const. At any given cross-section of a beam (z = const), we can make a "topographic map" of the wave front. For instance, a spherical wave front will be projected as a set of concentric circles (fig. 17a). Helicoidal wave front has the phase map as a "star" (fig. 17b). Being a continuous surface, a wave front carrying topological objects (extrema, saddles and vortices) satisfies the general topological laws. The topological index is attributed to each of these objects: extremum (maximum or minimum) and vortex (irrespective of the value of the topological charge) possess index +1, and a saddle possesses index - 1 (see, for example Freund [1995]). The index theorem asserts that the total topological index, which is the
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sum of the indices of all the objects, remains constant on wave propagation in free space. Now we shall deal with the more complicated wave field than a simple onaxis OV-like LGQ mode. A new important object is the vortex trajectory, which is a line of zero field amplitude in a 3D space. Generally, this type of phase dislocation can be referred to as a mixed edge-screw dislocation. A vortex trajectory can be a closed line, or start and end on infinity, z—> ± oo, or p—>oc (Soskin, Gorshkov, Vasnetsov, Malos and Heckenberg [1997], Freund [2000]). Again, we shall demonstrate some features of a vortex trajectory on simple examples. As the Laguerre-Gaussian beams are self-similar in the propagation from the waist to the far field, no interesting spatial dynamics is expected for them. However, coherent addition of even two LG modes, which belong to different mode families (LGQ and LGQ), immediately demonstrates the effect of vortex "rotation" around the beam axis. Three modes create a "dipole" of vortices (Indebetouw [1993]). Coherent superposition of four modes was shown to stimulate collision of vortices of opposite charge, therefore produce a closed dislocation line (Indebetouw [1993]). There is currently much discussion about the interpretation of a vortex trajectory and related definitions. Berry [1998b] has emphasised that actually there is not any annihilation of opposite-charged vortices in a collision, because the whole zero-amplitude line is a core of a vortex. When the line becomes at some point perpendicular to the direction of beam propagation, there is only transformation of a mixed edge-screw dislocation to pure edge dislocation, happening at the cusp of the dislocation line. We have considered previously one type of closed OV trajectory, namely circular edge dislocation in § 4. As we have seen, the wave-front dislocations are zero-amplitude lines, or "threads of darkness" (Berry [1998b]). Their general properties, especially spatial orientation and statistics, were considered in a set of papers during recent years. We will consider the above mentioned speckle fields of scattered coherent light. Speckles appear as a result of coherent interference of many partial waves with random phases and were first analyzed from the point of view of phase singularities by Zel'dovich and coworkers (Baranova, Mamaev, Pilipetskii, Shkunov and Zel'dovich [1983] and references therein). They also considered the "snake structure" of dislocation lines in space. Freund in numerous papers (Freund [1999, 2000] and references therein) performed the detailed investigation of topological properties of monochromatic 2D speckle fields with random phases (Gaussian statistics). We briefly note here the main results. A random (Gaussian) speckle pattern is completely determined by the
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Structure of its dislocation network. Random phase fields represent themselves as the system offully correlated critical points (singularities, saddles and extrema). It was shown theoretically that an "explosion" of a high-order vortex could often be controlled by an external field parameter (Freund [1999]). The next point is connected with the characterization of the dislocation lines. They are in general closed curved lines in space. Their direction in each point s is defined by the tangential unit vector n{s). Then the strength ^S of a dislocation is the (positive or negative) integer defined as (Berry [1998a]) S = sgnlmVi/;* xV\p'n,
(7.1)
where \p is the complex scalar wave fianction. There are two approaches to the choice of the sign of the unit vector n along the dislocation line. Berry [1998b] proposed one of them. It is based on next assumptions: (i) the choice of the sign of n is arbitrary, (ii) it is convenient to make the choice of the sign of n(s) in some dislocation point s and then define n{s) elsewhere along the line in continuity. In this approach a "collision" and "annihilation" of dislocations at some point in space are artefacts, because the location of the annihilation event will depend on the choice of a family of spanning surfaces I. We will give an overview of the main results, important to the understanding of singular optics. Let us start with Berry's general consideration of the problem (Berry [1998b]). He stressed at first that the exact sense of the time evolution for a stationary field is the description of its structure in a given observation plane when the observation plane is moved gradually. For example, OVs' "nucleation" or their "annihilation" in pairs means only that they appear (disappear) in the observation plane. However, we have to emphasize that the mathematical virtual observation plane is not physical, because it should be transparent for the propagating light: a real observer will see nothing on the screen. Any absorption necessary for the field detection on the screen will change the field structure in the critical points of the vortices nucleation and annihilation, where an infinitesimal backward light current exists (see § 4). Berry's approach was used for a general topological analysis of possible dislocation trajectories (Freund [2000]). It is based on the so-called "sign rule", which requires that vortex signs must alternate while looking along lines of zero of the real or imaginary parts of the wave fiinction in every planar section of a 3D field (Shvartsman and Freund [1994], Freund and Shvartsman [1994]). Specific rules were formulated that constrain the relationships between OVs trajectories. Possible configurations were considered for these trajectories and
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manifolds of various phase surfaces. For example, "planes" and "tubes" appear to be generic manifolds within a host Gaussian laser beam. It was shown also that if the orientation of any one trajectory on a manifold is known, this automatically determines the orientation of the other trajectories on the manifold (Freund [2000]). However, the way to realize most key trajectories that were considered remains open. The second approach to the dislocation structure description is based on the assumption that their main properties are defined by the direction of the Poynting vector in the vicinity of the dislocation line. This approach is more natural from the physical point of view. It can be used when monochromatic OV beam possesses a well-defined propagation direction (Basistiy, Bazhenov, Soskin and Vasnetsov [1993], Basistiy, Soskin and Vasnetsov [1995]). The designation of a topological charge for a vortex is legitimate in such situations (Berry [1998b]). In this case the topological charge can be positive or negative, depending on the helicity of the wavefi*ontabout the vortex axis. Therefore even for a closed dislocation line the vortex around it can have an opposite sign of topological charge, associated with the direction of the host beam propagation. In these terms, the dislocation "nucleation" and "annihilation" by pairs is a real experimentally observable event, which obey the topological charge conservation law. What is known so far about OVs trajectories that were realized? They can exist within a beam up to infinity, i.e., to the far field, for instance in a "pure" LGQ mode or mode superposition. Dislocation lines can be created outside a laser cavity by use of a computer-generated hologram (Bazhenov, Vasnetsov and Soskin [1990], Bazhenov, Soskin and Vasnetsov [1992], Heckenberg, McDuff, Smith, Rubinsztein-Dunlop and Wegener [1992]) or mode converter (Tamm and Weiss [1990], Allen, Courtial and Padgett [1999], O'Neil and Courtial [2000] and references therein). Another source of initial or additional phase singularities is the diffraction of the usual or OV beam on some obstacle (Vasnetsov, Marienko and Soskin [2000]). A well-known case of a closed dislocation line is the Airy ring (see also § 4). It was shown in § 4 that a circular edge dislocation can be created in the common waist plane due to the destructive interference of two coaxial Gaussian beams. A closed dislocation line and the vortices "annihilation" were experimentally achieved with the aid of computer-synthesized hologram producing OV dipole (Basistiy, Bazhenov, Soskin and Vasnetsov [1993]). Combined light beams carrying optical vortices possess many nontrivial features (Soskin, Gorshkov, Vasnetsov, Malos and Heckenberg [1997]). First it was demonstrated that additional vortices could appear from the far periphery
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of a beam or disappear at a beam periphery. It was shown that a superposition of a m-charged axial-vortex beam and a Gaussian beam may change the number of vortices. They can vary between 0, |m| and 2\m\ over free space propagation. At the same time, the total topological charge can be m or 0. The total 0AM is conserved in all cases of free space propagation. These results are in full agreement with experiments. It was shown recently that the superposition of noncoaxial OV light beams creates light patterns with a richer OV content than given by the arithmetic summation of the topological charges of the individual beams (Molina-Terriza, Recolons and Tomer [2000]). Is the nucleation of OVs possible in a single, say, Gaussian beam due to its topological transformations? As was stated by Berry [1981], the universal mechanism of phase dislocation appearance in some wave fields at a given space-time point is the fully destructive interference of the different partial wave contributions to this point. Such situations are realized in speckle fields. More generally, each event of a phase singularity's creation or "annihilation" is a result of a broken symmetry of the system (Pismen [1999]). In singular optics, the notion "system" includes both incident light beam and a linear or nonlinear medium through which this beam propagates. We shall see that symmetry breaking of each system component can result in the appearance of phase singularities. It was shown both theoretically (Kreminskaya, Soskin and Khizhnyak [1998]) and experimentally (see below) that such nucleation of singularity arises due to the self-action of a laser beam in a nonlinear medium. Two conditions should be fulfilled simultaneously: (i) the nonlinear refractive-index modulation creates a lens-like structure with coexisting focusing and defocusing parts in contrast to a usual spherical lens, (ii) the optical strength of the induced lens reaches some definite threshold value. For the laser beam with Gaussian amplitude distribution, the self-induced thin "Gaussian-like" lens is created with the complex transmission fiinction T(x,y) = To exp {ikDAn(x,y)} = To exp {iOexp (-2 [x^ + ( / / « ' ) ] /rl)} .
(7.2)
Here To is the coefficient of amplitude transmission, D is the thickness of a medium, 6 is the nonlinear phase shift on the beam axis (optical strength of Gaussian lens), a is the coefficient of astigmatism connected with possible anisotropy of the medium, ro is the radius of the Gaussian lens equal to the size of the waist of the laser beam.
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The distribution of the complex amplitude of the beam at some cross-section on a distance z after the Gaussian lens located at incident beam waist z = 0 can be calculated by use of the Kirchhoff-Fresnel integral
E(xuyuz)-
dxdyE(x,y)T(x,y)Qxph—
^(x-xif+
(y-yif^j
,
(7.3) where ZR is the Rayleigh range of the incident beam. As we have seen, the search for the singularity's location is reduced to detection of the location of zero-amplitude points, where real and imaginary parts of the field simultaneously equal to zero. The analysis of the Gaussian beam evolution behind the Gaussian lens can be performed by the method of "complex rays" (Kravtsov, Forbes and Asatryan [1999]). A necessary condition for the formation of an optical singularity in this approach is the appearance of ray caustics behind the lens. The optical singularities exist only at the points where the complex amplitude vanishes. Outside the caustic rays can not intersect and optical singularities are absent. Inside the beak three rays intersect at every point. The location of the phase singularities can be found by solving the appropriate equations (Kreminskaya, Soskin and Khizhnyak [1999]). Circular edge dislocation, which is located in a plane perpendicular to the direction of beam propagation, can be stretched to produce a line in 3D space (tennis-ball seam shape). In any cross-section of the dislocation line we shall detect four OVs with altering charges, so called OV quadruple (Kreminskaya, Soskin and Khizhnyak [1998]). Along the direction of the beam propagation, one can find appearance of the plane of the closed dislocation line and the plane of its cancellation (see fig. 18). The angle Y(Z) between a vortex trajectory and the z-axis at any cross-section Zn ^ z ^ Zan is the mcasurc of the edge-screw dislocation character in this cross-section (Berry [1998b]). From this point of view, the evolution of the stretched "tennis-ball seam" trajectory (fig. 18) is clear enough. Starting angle Y{zn) = TC/1 corresponds to the pure edge dislocation in two trajectory points where dislocation appears to the observer. The edge-screw dislocation is realized for all intermediate distances z„
Singular optics
256
[4, §7
045
Fig. 18. OV trajectories in the case of nonlinear self-induced defocusing Gaussian lens. Stigmatic lens produces circular edge dislocation, astigmatic lens gives rise to the stretched OV trajectory: dislocation line exists only in the interval z„ ^ z ^ z^n-
symmetry broken modes whose amplitudes and phases are determined by the astigmatic lens. This model gives the same qualitative results. The experiments with various types of nonlinear media are in good agreement with predictions of the theory. The circular edge dislocation nucleation was observed only in a case of a laser beam self-action in an axial symmetric dye-doped liquid crystal (LC) cell, which was located in the waist plane of the focused Ar^ laser beam (A = 488 nm) (Reznikov, Soskin, Slussarenko, Pishnyak, Fedorenko and Vasnetsov [1998]). The axial Gaussian-like lens was induced at low intensity (3-0.3 W/cm^) due to thermal nonlinearity, which is essential under these conditions (homeotropic LC cell orientation). The system of dark rings in a laser beam transmitted through an LC cell is well-known (Zolot'ko, Kitaeva and Terskov [1994]). These rings are actually "gray" as a result of partial destructive interference. Only one of them can create phase dislocation due to fully destructive interference at a definite distance depending on light intensity, according to theoretical prediction (Kreminskaya, Soskin and Khizhnyak [1998]). Thus, a circular edge dislocation is generated (fig. 19). The extra large refractive index variation up to A« ^ 0.3 appears at rather high intensities (~ 120-3200 W/cm^) and the induced Gaussian lens in a LC planar cell becomes astigmatic due to the orientational nonlinearity. As a result, the circular edge dislocation transforms immediately to the stretched closed line with
7]
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Fig. 19. Intensity distribution (a) and corresponding interferogram (b) of the laser beam observed in a screen after passing through the self-induced Gaussian-like lens in a LC cell. The circular edge dislocation is shown by arrows.
the shape of tennis-ball seam, which is seen by an observer as a quadruple of optical vortices in any cross-section. The spontaneous optical vortices nucleation producing the OV quadruple was observed in a laser beam passed through photorefractive LiNbOs crystal (Vasnetsov, Ilyenkov and Soskin [1994], Ilyenkov, Khizhnyak, Kreminskaya, Soskin and Vasnetsov [1996]). The complex nonlinear (mainly defocusing) lens was self-induced due to the effect of so-called "optical damage" (Ashkin, Boyd, Dziedzik, Smith, Ballman and Nassau [1966]). Two and even three quadruples of optical vortices with the same sign distribution were observed at higher exposures. The formation of dislocations even inside a crystal was obtained when the lens was strong enough. The symmetry breaking of a system "light + nonlinear medium" was demonstrated later by Ackemann, Kriege and Lange [1995]. The OV quadruple was obtained as inside as outside of a cell with high-density sodium vapor by self-focusing of a slightly astigmatic (up to 6%) laser beam from a cw dye laser tuned close to the Di resonance line. Another mechanism of Gaussian-like lens creation is realized in the SEN crystal (Sr;tBai_xNb206:Fe) (Ilyenkov, Kreminskaya, Soskin and Vasnetsov [1997]). The non-focused beam linearly polarized along crystal C-axis with power up to lOOmW was directed on the crystal input face. When an Ar+ laser beam (A = 488nm) passes through the SBN crystal it induces a thermal nonlinear (mostly focusing) lens in the crystal due to the absorption of the radiation in the green spectral range. The optical vortices' quadruple "nucleation" takes place at some "threshold" input beam power at the given distance z„ behind the crystal. The distribution of OV signs was reversed in the case of the LiNbOs crystal according to prediction of the theory. Opposite-sign vortices approach each other along the C-axis. Then
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they "annihilate" in pairs with the increase of the distance from the crystal to some final value Zan and never appear at z > Zan-
§ 8. Gouy phase shift in singular optics It was found more than a century ago (Gouy [1890]) that focused light exhibits a 71 shift of its optical path as it crosses the focal point. This effect is known now as "Gouy phase shift" or "phase anomaly" (Bom and Wolf [1999]). Gouy has shown also that this phase shift exists for waves of any nature, including sound waves that pass through a focus. It was shown later (Simon and Mukunda [1993], Subbarao [1995]) that the Gouy phase shift possesses deep physical sense and is a special case of the additional topological Berry phase (Berry [1984]) of electromagnetic field associated with the light beam. Gouy phase shift can be treated as the geometrical quantum effect caused by the changing of transverse dimensions of a light beam (Hariharan and Robinson [1996]). Therefore, it is distributed continuously in space along the axis of a beam propagating with variable cross-section dimensions. The actual elements for singular optics are LG modes, which possess caustic (waist) and change their cross-section size during free-space propagation. For example, the Gouy phase shift A0G for LG^ modes is A0G = - (2/7 + |/| + 1) arctan (Z/ZR) = -Q arctan (Z/ZR) ,
(8.1)
where Q is the index of the mode family. A0G attains minimum absolute value for the Gaussian beam with/? = / = 0, and grows with the increase of Q. It is natural that most rapid phase changes occur near the waist, where a beam is mostly compressed. For the same reason, the presence of the Gouy phase shift has been discovered in the focal region for usual light beams. The increase of the Gouy phase shift with radial index/? is also natural because the subsection of a beam by circular nodal surfaces makes its structure sectioned on pieces with smaller dimensions comparable to wo, while the total transverse beam dimension increases. We have noted in § 4 the importance of the Gouy phase shift for phase velocity variation in Gaussian beams, necessary for "transversal OV" to appear (Pas'ko, Soskin and Vasnetsov [2001]). Fascinating manifestation of the Gouy phase shift in the beams with singleringed (/? = 0) LG modes has been demonstrated. The variation of the longitudinal phase enlarges the distance between neighboring surfaces of a
4, § 8]
Gouy phase shift in singular optics
259
helical wave front according to eq. (5.1). Therefore this additional phase shift A0G affects the azimuth phase as given by the expression cp' = cp- sgn(/) arctan (Z/ZR) ,
(8.2)
where sgn(/) = 1 for / > 0, 0 for / = 0 and - 1 for / < 0. This new effective azimuth dependence demonstrates the influence of the Gouy phase shift as the axial rotation of light beams with helical wave fronts first considered by Abramochkin and Volostnikov [1993]. The "angular velocity" in the case of the Gaussian envelope of a singular beam with axial optical vortex (/ 0) is
dz
ZR
1 + (Z/ZR)^
It follows from eq. (8.3) that the sign of Q is determined by the sign of the topological charge: the right-handed helical wave rotates in space counterclockwise, and clockwise for the left-handed wave. The whole beam turn amounts to Jt/2 from the waist to the far field, independently on the index / (/ ^ 0). For / = 0, no rotation occurs. The axial symmetry of the intensity distribution does not allow direct observation of the rotation. Indirectly, we can judge the circulation of the light flow by means of diffraction experiments (§ 6). The generalization of optical vortices rotation to an OV array nested in a smooth host (Gaussian) beam was given by Indebetouw [1993]. It was shown that a system of single-charged vortices with the same sign rotates rigidly in one direction and expands or contracts together with the host beam. Vortices of opposite charge, in contrast, rotate in opposite directions and "attract" each other. The condition for their junction was analyzed by Indebetouw [1993] and demonstrated experimentally by Basistiy, Bazhenov, Soskin and Vasnetsov [1993]. The off-axis vortices' rotation effect was first used to our knowledge for direct measurement of the Gouy phase shift and its spatial dependence (Basistiy, Bazhenov, Soskin and Vasnetsov [1993]). The dark centers of a pair of singlecharged vortices produced by a computer-generated hologram within a read-out Gaussian beam were used as the marks on the helicoidal wave front. Due to the Gouy phase shift they exhibit rotation during the host beam propagation from beam waist to the far field (fig. 20). The measured dependence of the Gouy phase shift on distance agrees with theory. The important parameter of the realized scheme is a ratio between a read-out beam waist size and an actual dimension of the singularity core registered on a hologram. When the beam waist is much larger than approximately two periods
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angle of rotaiion 6, dag
5 L, meter* Fig. 20. (a) Schematic view of a grating with two closely located dislocations of equal strength 1. (b) Rotation of an OV pair during free-space propagation, (c) Experimental results of the angle of rotation 6 vs. distance L (squares) and the theoretical dependence (solid line).
of the grating, the vortices will appear as point-like vortices within a wide host beam (Rozas, Sacks and Swartzlander [1997]). Again, the Gouy phase shift of the corresponding modes responsible for these point-like OVs produces their rotation around the beam axis. Propagation dynamics of OVs was described in terms of hydrodynamic paradigm both for linear and nonlinear media (Rozas, Law and Swartzlander [1997], Rozas, Sacks and Swartzlander [1997]). It was shown both numerically and experimentally that two off-axis point-like OVs of the same charge orbit one another during free-space propagation at rates that are inversely proportional to the squared distance of their separation. This approach is valid for short enough propagation distances, until the vortices' cores, broadened due to the strong diffraction, do not overlap. In general, we must keep in mind that vortices in linear optics "do not have the dynamics of their own" (Pismen [1999]). The unwanted effect of point-like vortices' diffraction spread-out can be substantially suppressed in nonlinear self-defocusing media due to the formation of genuine vortex filaments where real OVs dynamics can be realized (Rozas and Swartzlander [2000]). Simulation shows that the large rotation angles may be
4, §9]
Statistics of phase dislocations
261
Fig. 21. Experimental set-up. Gaussian beam from Argon laser 1 is directed to the hologram 2, and the first diffracted order carrying an off'-axis single-charged OV is launched into the separator.
achieved simultaneously with stable orbits due to nonlinear interaction between the rotating vortex solitons. This prediction was justified by an example of a pair of point-like closely spaced vortices propagating through a long glass cell (290 mm) filled with slightly dyed methanol. When the power of a host Ar+ laser beam increases, the vortex core contracts and a pair of vortex filaments is formed. The fluid-like rotation of these filaments was 3.5 times larger than in linear media and the total rotation angle exceeded the Jt/2 limit of the Gouy phase shift. The difference of the Gouy phase shift for LG modes with different modal indices Q opens the nontrivial possibility to separate modes in space from a beam which is a modal superposition. According to eq. (8.1), a confocal twolens telescope adds to the propagating LG modes the Gouy phase shift equal to -Qjt. Therefore modes with even and odd indices Q will attain even and odd quantities of half-wavelength, or it, correspondingly. The occurring phase difference can be used for the mode separation in a double-beam interferometric arrangement. This possibility was proved in two-beam interferometer (fig. 21). The laser beam with single-charged off-axis OV was synthesized by the computergenerated hologram (Bazhenov, Vasnetsov and Soskin [1990]) with a lateral shift of readout Gaussian beam from the hologram center (fig. 22a). The results of precise separation of an axial OV and smooth-wave-front component from the combined beam carrying OV are seen in fig. 22 (Vasnetsov, Slyusar and Soskin [2001]).
§ 9. Statistics of pliase dislocations Berry and Dennis [2000a,b] have analyzed the 3D statistics for most general cases of isotropic random scalar and vector fields. Both monochromatic and
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Fig. 22. (a) Schematic view of a computer-synthesized hologram and the light spot of a read-out Gaussian beam on it. (b) Near field of the first-order diffracted beam which contains an off-axis OV (c, d) Output beams from channels A and B, and their interferograms (e, f).
blackbody radiation were considered. The average values of main characteristics of such fields were found. The dislocation density for the isotropic random monochromatic scalar field in space equals d = 4jr/3A^ (length of dislocation lines in a volume, length/volume) and d = 2jr/3A^ for the plane section of the same wave (dislocation point density in the plane) with mean spacing 0.69A. Therefore, on average there are nearly two dislocation points per A^ square. It is natural that dislocation lines, which appear as the result of isotropic 3D random
4, § 10]
Optical vortices in frequency conversion processes
263
waves interference, are very strongly curved. It was shown that the average radius of the curvature is only 0.218A, indicating that dislocation lines are sharply curved in the subwavelength scale. The same is valid, of course, for an average dimension of 3D speckles (local "grains" of intensity maximum). The dislocation lines are typically screwed rather than closed flat rings. Another consequence of dislocation lines' snake structure is the elliptical shape of amplitude cores around dislocation lines (anisotropic vortices). It was found that the eccentricity of these ellipses is large, about 0.83. A full set of average quantities was also found for blackbody radiation. For instance, the dislocation line density (in volume) i^ d = 40/63jtkj, where kr = k^T/hc (A:B is Boltzmann's constant). For a plane section of the same wave the density of dislocation points d = 20/63Jtkj^. A measure of the radius of curvature is 0.026Ar (Ar = 2jt/kT is the "thermal wavelength"), i.e., dislocation lines are even more sharply curved than in the monochromatic case due to the interference of waves with various frequencies. The next much more complicated case is the statistics of polarization singularities in an isotropic random vector field (Berry and Dennis [2000b]). Optical monochromatic vector fields possess an elliptical polarization with variable parameters from point to point. Therefore, the optical vortices of scalar fields considered above are not realized in vector waves. Nevertheless, they possess their own specific polarization singularities. As was shown by Nye and Hajnal [1987] there are two types of polarization singularities in optical vector fields in space: C-lines and L-lines, which are loci of pure circular polarization (C-line index di^) and pure linear polarization (L-line index ±1), respectively. They were reinterpreted as loci of photon spin 1 (C-lines) and 0 (L-lines), which generalizes the familiar relations between the photon spin and polarization of light fields. The isotropic random vector field possesses plane-wave components randomly oriented in space with random phases and polarization. The polarization is uniformly distributed on the Poincare sphere. It was found that for monochromatic waves, the density of C-lines is equal to dc = 0.21096A:^, and for L-lines d^ = 0.21360A:^, i.e., they are nearly but not entirely equal. The density of C-lines for blackbody radiation is much higher due to its broad spectrum: dc = 156.51/X^. In general, the density of polarization singularities is much higher than in random scalar fields. § 10. Optical vortices in frequency conversion processes Both frequency up- and down-conversion processes possess essential peculiarities for light beams with OVs. Let us start from the well-known frequency up-
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Fig. 23. Schematic view of wave-front transformation in the process of coUinear second-harmonic generation: (a) an initial helical wave front of a singular pump wave (m = -I) with the pitch X^^\ and (b) the wave front corresponding to the second-harmonic wave (m = -2) with the same pitch, but now a two-started helicoid. The resulting distance between equiphase surfaces along the z-axis becomes A^^^^).
conversion effects. Basistiy, Bazhenov, Soskin and Vasnetsov [1993] first realized the collinear second-harmonic generation (SHG) with an OV pump beam. The authors started fi-om two facts: the amplitude of converted signal is defined locally by the amplitude of the pump beam (frequency co); and the highest conversion efficiency is realized at phase-matching conditions (Bloembergen [1965], Boyd [1992], Dmitriev, Gurzadyan and Nikogosyan [1994]). Both crests and troughs of the ftindamental wave generate crests of the double frequency (Ico) wave. Then, the resulting wave-front helicoid at double frequency wave has the same sign of chirality as the fiindamental OV beam, but with doubled pitch ("helical phase matching conditions"). Therefore, all frequency harmonics in collinear (type I) scheme have to be OVs due to zero amplitude on the pump beam axis and helicoidal structure of the wave fi-ont. Figure 23 illustrates the corresponding wave-fi*ont structure (Soskin and Vasnetsov [1998]). This leads automatically to the doubling of the OV topological charge for second harmonics: ^i2co) ^
2m^''\
(10.1)
This prediction was confirmed by collinear SHG of a moderate-power unfocused beam generated by pulse Nd:YAG laser (A^^'^^ = 1.06/im) with an axial singlecharged OV (Basistiy, Bazhenov, Soskin and Vasnetsov [1993]). As we know, "all roads lead to Rome". Therefore, it is not surprising that Dholakia, Simpson, Padgett and Allen [1996] came to the same result three years later starting fi*om the conservation law of total orbital angular momentum (OAM) in the process of SHG for both noncritical (type I) and critical (type II) phase-matching schemes (Bloembergen [1965]). The family
4, § 10]
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of high-order LG^ modes up to / = 7 from NdiYAG laser (A^^^ = 1.06 ^m) with a linear polarization was used for SHG to achieve double frequency waves with azimuth phase term e^^^^. The number of 0AM equals Ih per photon for LG modes and does not depend on the light frequency but on the photon number only. Assume that N^^"^ photons on the frindamental frequency are converted to j^iio,) = \_^((^) ^-^j^ Qj^y^ 2lh per photon. Therefore, A^^^^ • Ih = |7V^^^ • 2lh, i.e., net 0AM conserves for any efficiency of SHG. The same is valid of course for generation of higher-order harmonics. Courtial, Dholakia, Allen and Padgett [1997] showed that second harmonics of LGQ mode is LGQ mode. In contrast, the second harmonic of LGJ, modes with/? > 0 is no longer a single LG mode. As we see, the topological charge of the SHG beam is dictated by the charge of the input fundamental harmonic. The same result was obtained for sum and difference frequency mixing (Berzanskis, Matijosius, Piskarskas, Smilgevisius and Stabinis [1997, 1998]). It was shown that total topological charge of pumping beams conserved during any parametric scattering. The beam walk-off essentially changes the dynamics of OVs' interaction during sum-frequency mixing. As a result, the higher-order vortex decays into single-charged vortices with the same sign of topological charge aligned perpendicular to the walk-off direction. The three-wave mixing of OV beams with moderate power in an x^^"^ medium is also defined at negligible depletion by their topological charges only. Much more interesting is the evolution of OV beams during SHG when an input OV beam is mixed with a Gaussian second-harmonic seed beam (Petrov and Torner [1998], Petrov, Molina-Terriza and Torner [1999]). It was shown that the qualitative behavior of the combined beam formed by the mutual coherent seeded and generated second-harmonic beams during propagation inside the quadratic medium. The two pairs of the second-harmonic OVs that appeared, have had zero net charge consistent with predictions of the theory. It is remarkable that the OVs did not form any type of solitary waves. Nontrivial dynamics of the multicharged OV was predicted for SHG with a seed beam (Molina-Terriza and Torner [2000]). The vortex streets exist under the combined effects of diffiraction and Poynting vector walk-off in SHG (Molina-Terriza, Torner and Petrov [1999]). SHG of intense OV beams is accompanied by new nonlinear phenomena due to the appearance of /^^^ nonlinearity in a quadratic medium. It was shown by numerical simulations that second-harmonic waves with higher-order OVs are parametrically azimuthal unstable and decay into set of stable bright solitary waves both in types I and II SHG schemes (Torner and Petrov [1997], Torres, Soto-Crespo, Tomer and Petrov [1998a,b]). This opens the possibility of producing a new class of optical devices that can potentially process
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information by mixing of topological charges and to form certain patterns of bright soliton spots. What is the form of the 0AM conservation law during transformation of OV beams to bright solitons? The answer was given by Firth and Skryabin [1997] and Skryabin and Firth [1998] for the case of axial OV transformations in self-focusing saturable Kerr-like and quadratic nonlinear media. It was shown that when the OV is broken into a family of bright filaments due to azimuthal modulation instability they move off along straight-line trajectories tangentially to the initial ring structure without any spiraling due to orbital angular momentum conservation. To our knowledge, this important conclusion is not yet proven. Self-action of laser beams in nonlinear media (Askaryan [1962]) leads to formation of solitons (Boardman and Xie [1998]) including OV ("black") solitons in self-focusing media (Snyder, Poladian and Mitchell [1992]). OV solitons were realized first in the self-focusing Kerr-like media (Swartzlander, Andersen, Regan, Yin and Kaplan [1991]) and then in the photorefi-active crystal SBN (Duree, Morin, Salamo, Segev, Crosignani, Di Porto, Sharp and Yariv [1995]). OV solitons in this crystal were investigated in detail (see review by Mamaev and Zozulya [1999] and references therein). Nonlinear transformation of an OV beam into a soliton was observed in a LiNbOs crystal (Chen, Segev, Wilson, Miller and Maker [1997]). OV solitons were obtained and investigated thoroughly in saturable atomic rubidium vapor (Kivshar, Nepomnyashchy, Tikhonenko, Christou and Luther-Davies [2000] and references therein). The breakup of vortex beams in self-focusing nonlinear media to bright solitons is quite complicated. Actually, it is known that the vortex can disappear in only two possible processes: annihilation in collision with an opposite-sign vortex with the same absolute value of topological charge, or disappearance on the border of the beam. Therefore, a fi-eely propagated axial OV beam has to conserve the topological charge and zero-amplitude trajectory even while breaking up into several bright solitons. The unusual structure of the vortex wave front in this case has not yet been considered. The reconfigurable self-induced waveguides are very promising for photonics applications. It was shown that they can be realized due to "cascade nonlinearities" first considered by Karamzin and Sukhorukov [1976] and revisited in the middle of the ninety's (Torruellas, Wang, Hagan, VanStryland and Stegeman [1995]). It was demonstrated that an intense (above some threshold value) smooth wave-front pump beam focused into a quadratic medium results in the formation of two-dimensional spatial solitary waves. It happens due to the mutual trapping of fiindamental and generated second-harmonic fields. This
4, § 10]
Optical vortices in frequency conversion processes
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Strong cascading nonlinear coupling counteracts both diffraction and walk-off of the created beams. Dark OV solitons in quadratic media without exact phase matching are stabilized by incoherent coupling between the harmonics due to the selfdefocusing Kerr effect for both harmonics (Alexander, Buryak and Kivshar [1998]). This interplay between diffraction and parametric coupling of the harmonics field carrying OVs leads to formation of a new class of solitons parametric vortex solitons. A general approach (Alexander, Kivshar, Buryak and Sammur [2000]) predicts two novel types of vortex solitons: (i) a "ring-vortex'' soliton, which is a vortex in a harmonic field that guides a ring-like localized mode of the fi^ndamental frequency field, and (ii) a "halo-vortex'\ consisting of a two-component vortex surrounded by a bright ring of its harmonic field, which appears as a result of a third-harmonic generation in a medium with defocusing Kerr nonlinearity. Quite nontrivial are the parametric down-conversion processes with OV beams. It is known (Bloembergen [1965], Boyd [1992]) that they are the result of threewave nonlinear coupling of pump, signal and idler waves with frequencies cO/, wave vectors hi and phases 0/. These interactions obey the energy conservation law 0^1 + 0^2 = CO3 and phase-matching conditions k\+k2 = k^, or (Pi + ^2 ^ ^3These conditions have to be supplemented with the conservation law of 0AM l\h + hh = l^h, which leads automatically to conservation of the OV topological charge l\ -V h = h- At last, the phase terms Of contain the azimuthal phase term //(p. The clearest case is the situation when both pump and signal beams exist at the input of a nonlinear medium. The OV properties in three-wave nonlinear coupling were first investigated theoretically for the degenerate case when signal and idler waves are identical (Staliunas [1992]). The first experiments with a singular signal beam and an ordinary pump beam were performed by Berzanskis, Matijosius, Piskarskas, Smilgevisius and Stabinis [1997]. The "spontaneous parametric fluorescence", or parametric scattering (Klyshko [1988]), when only one pump beam is launched into a nonlinear medium, is quite nontrivial. In this case, both signal and idler waves build up from quantum noise. Di Trapani, Berzanskis, Mirandi, Sapone and Chinaglia [1998] have shown both experimentally and numerically that Bessel-like JQ beams and Bessel-like vortices are generated in a traveling-wave optical parametric amplifier with a ringshape gain angular spectrum. These results are of fundamental importance and show that the vortex structure is characteristic of vacuum quantum noise as well. The situation of only one OV pump beam launched into nonlinear medium was investigated for a cw laser beam (wavelength 532 nm) transmitted through
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[4, § 11
the lithium triborat crystal with types I and II critical schemes (Arlt, Dholakia, Allen and Padgett [1999]). It was observed that the down-converted signal and idler beams are spatially incoherent and that the OAM is not conserved "within the classical wave fields". This important result finds a natural explanation in spatial uncorrelated phases
§ 11. Applications It is natural that the unique properties of OV laser beams open a variety of new possibilities for singular optics applications. At the same time, they are rather delicate and suitable for only a few special problems. There are some up-to-date examples described below. The "hollow" OV beam can be a "potential tube" for trapped cold atoms, what was first mentioned by Bazhenov, Soskin and Vasnetsov [1992]. Several constructions of such magneto-optical traps were proposed or experimentally realized (Snadden, Bell, Clarke and Riis [1997], Clifford, Arlt, Courtial and Dholakia [1998], Kuga, Torii, Shiokawa and Hirano [1997]). The dark OV core became the physical instrument to create optical vortex tweezers (Heckenberg, Friese, Nieminen and Rubinsztein-Dunlop [1999], and references therein). The best equilibrium position for a trapped particle is just on the OV axis at a short distance above a microscope focal plane. The OV angular momentum can be transferred to trapped particles, initiating their rotation {optical vortex spanners), (Simpson, Allen and Padgett [1996]). The OV tweezers were realized and used successfiilly (He, Heckenberg and RubinszteinDunlop [1995], Simpson, Dholakia, Allen and Padgett [1997], Friese, Nieminen, Heckenberg and Rubinsztein-Dunlop [1998]). The calculation shows that the size of the rotating particle should not exceed few microns (Courtial and Padgett [2000]). The use of the combined beams carrying OVs with varying relative phases of the component modes (Soskin, Gorshkov, Vasnetsov, Malos and Heckenberg [1997], Tikhonenko and Akhmediev [1996], Freund [1999]) would open additional opportunities for developing the multi-channel optical vortex tweezers based on high-charged OVs. Singular beam stability on propagation in free space up to the far field within the scaling factor and the axial rotation opens the opportunity for a steady
4, § 11]
Applications
269
transmission of optical information through optical communication systems and for optical processing. Abramochkin and Volostnikov [1993, 2001] have performed the most essential steps in this direction. They have worked out the effective technique of synthesis of a "spiral-type" singular beams with the predetermined intensity distribution. The elaborated algorithm allows synthesis of figures, characters and closed polygonal lines, etc., which can be transported through free-space connectors or by optical fibers without distortions. It is known that the speckle-field structure with a multitude of vortices appears in a laser beam propagating through the turbulent atmosphere. An interesting approach to define the structure of such a wave field was proposed by Aksenov, Banakh and Tikhomirova [1998] and Banakh and Faltis [2001]. The motion of the scattering objects leads to the fluctuation of speckle fields in space and time. The statistics of OVs behavior was used successfully to obtain information about the scattering medium, which is very useful for many applications (Harris [1995]). Integer topological charge of OV is in principle a new tool for topological arithmetic, optical computing and processing (Basistiy, Bazhenov, Soskin and Vasnetsov [1993], Freund [1999]). Parametric frequency converters are very promising for these purposes (Berzanskis, Matijosius, Piskarskas, Smilgevisius and Stabinis [1997, 1998]). The phase singularities are promising for high-resolution metrology. The second half of the 20th century was marked by the origin and rapid successful development of near-field optical microscopy with lateral resolution, which essentially exceeds the Rayleigh resolution criterion in classical optics (Born and Wolf [1999]). The scanning near-field microscopy with a sharp tip brought to the vicinity of an illuminated sample is the most effective. A scattered evanescent field is detected then in the far field (Adam, Bijeon, Viardot and Royer [2000]). We have seen that the trajectories of phase dislocations are mathematical lines and appear as dark points in a cross-section. Therefore, it seems that wavefront singularities provide an opportunity to develop a new type of the superresolution metrology. Indeed, an arbitrary small lateral shift of the OV center can be obtained in a combined-beam configuration when the weak coherent background smoothly changes the amplitude and phase of the combined beam. The development of opto- and nano-electronics demands creation and testing of surface gratings and other regular structures with sub wavelength dimensions. According to the previous exposition, it is natural to expect the appearance of edge-type phase singularities during diffraction of smooth wave-front light fields by step-like structures. These optical singularities became the subject of numerous attempts to realize new types of super-resolution optical microscopy
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on this basis. Indeed, Tychinsky [1991] first observed some phase structures produced by the gratings with the subwavelength period and the deep grooves. The technique of lateral super-resolution microscopy was proposed on this basis. A rigorous analysis (Totzeck and Tiziani [1997a,b]) has shown that attempts to develop real sub-micron optical metrology using phase singularities can be the source of overrated image interpretation. The same conclusion that the term "super-resolution" is not suitable for phase step-like structures, was given after careful theoretical and experimental investigations by Blattner [1999]. The general case of arbitrary incident 3D field on 2D gratings was investigated by rigorous solution of Maxwell's equations (Urbach and Merkx [1992], Urbach [2000]). It was also shown that the resolution could be only slightly better than the Rayleigh criterion. On the other hand, it was shown recently by Dandliker, Rockstuhl, Herzig and Blattner [2001], that a real super-resolution can be achieved during rigorous inspection of periodical phase elements with preliminary known structure. The surface step-like gratings with sub-wavelength structures cause the specific "edge birefiingence". They work as polarization selective optical devices (Fujimoto, Okuno and Matsuda [1999]). This phenomenon became the physical background of the phase-shifting polarization interferometry for microstructures inspection (Totzeck, Jacobsen and Tiziani [1999]). The liquidcrystal phase shifter was included in the imaging section of a microscope. Subwavelength structures down to 0.3 A were measured. It was found that the images depend strongly on the position of the focus, etc., as in the usual twobeam interferometry. Summarizing, the phase-singularities based on interference microscopy of step-like structures give, up to now, a resolution only slightly better than the Rayleigh criterion. Nevertheless, it seems that real topometry of subwavelength objects can be achievable if their structure is approximately known. For example, it would be useful for the proof of errors for a given specimen during massive production of nano-scale structures. OVs in optical fibers are of considerable importance due to wide applications of fibers in optical communications, optoelectronics, remote sensors, etc. OVs are usually observed in the speckle output of a multimode fiber due to mode interference (Bazhenov, Vasnetsov and Soskin [1990]). It was shown both theoretically and experimentally that the single OV is created during the propagation of a beam with circular polarization through the multimode fiber (Darsht, Zel'dovich, Kataevskaya and Kundikova [1995]). The behavior of laser beams in a multimode fiber is rather complicated due to the "optical Magnus effect", spin-orbit interaction of a photon, conservation of the circular
4, § 12]
Conclusions
271
polarization in an axially symmetric step-like profile multimode fiber, etc. (Zel'dovich and Liberman [1990], Doogin, Kundikova, Liberman and Zel'dovich [1992], Savchenko and Zel'dovich [1992]). Low-mode fibers are most appropriate for OV creation (Volyar and Fadeeva [1996a,b], Alexeyev, Fadeyeva, Volyar and Soskin [1998]). It was shown recently that a two-beam interferometer with a low-mode fiber in the detecting arm allows one to achieve high sensitivity. It is connected with the spiral structure of an OV coaxial interference fringe with a Gaussian reference beam, for example (Bazhenov, Soskin and Vasnetsov [1992]). As a result, rotation of this fringe can be measured with nearly one angular degree preciseness.
§ 12. Conclusions Modern optics shows continual progress in many directions, which is clearly seen from the content of 40 volumes of Progress in Optics. One of the new branches of optics is Singular Optics, which studies a wide class of effects associated with phase singularities of wave fronts as well as topology of wave fronts possessing singular points. It was shown that optical singularities (optical vortices, etc.) exhibit some new fundamental features absent in the "usual" fields with smooth wave fronts. Namely, optical vortices possess orbital angular momentum (Allen, Padgett and Babiker [1999]), topological charge for helical wave front of beams with well-defined direction of propagation, e.g., Laguerre-Gaussian modes, or dislocation strength in general cases (Berry [1998b]). As a result, an interesting spatial evolution can be generated such as optical vortices "nucleation" and "annihilation" by pairs with participation of phase saddles, often called "optical chemistry" (Freund [2000]). Moreover, it was shown recently in nonlinear fi-equency down-conversion experiments with a parametric amplifier without seed light signal that "vortex" structure is characteristic of vacuum quantum noise (Di Trapani, Berzanskis, Mirandi, Sapone and Chinaglia [1998]). A review of the physics of singular beams reveals the significance of optical vortices and the orbital angular momentum at a fundamental level (Tiwari [1999]). A short but successfiil history of singular optics shows clearly that the majority of well-known optical phenomena possess new features for fields with phase singularities. Nye and Berry [1974] have shown in their seminal paper that phase singularities, or wave-front dislocations, appear as a result of partial spatial overlapping of two quasi-monochromatic pulses. They predicted the gliding and climbing of wave front dislocations on the pulse envelope. The level of quantum electronic development makes it possible to produce femtosecond pulses with
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[4
only a few field vibrations. Therefore, it is time for the appearance of singular optics of ultrashort light pulses. The theory of such ultrashort pulses propagating in nonlinear media can be considered as a first step in this direction (Eisenberg and Silberberg [1999]). The same is valid for the study of OVs in stimulated Raman and Brillouin scatterings (Drampyan [2001], Starikov and Kochemasov [2001]), and the investigation of polarization singularities of speckle fields (Angelsky, Mokhun, Mokhun and Soskin [2001]).
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E. Wolf, Progress in Optics 42 © 2001 Elsevier Science B.V. All rights reserved
Chapter 5
Multi-photon quantum interferometry by
Gregg Jaeger and Alexander V Sergienko Dept. of Electrical and Computer Engineering, Boston University, 8 St. Marys St., Boston, MA 02215, USA
277
Contents
Page § 1. Introduction: practical optical tests of Bell inequalities § 2.
. . . .
Two-photon interferometry with type-I phase-matched SPDC . .
279 283
§ 3. Two-photon interferometry with type-II SPDC
290
§4.
311
Higher multiple-photon entanglement
References
320
278
§ 1. Introduction: practical optical tests of Bell inequalities The often deeply counterintuitive predictions of quantum mechanics have been the focus of intensive discussions and debates among physicists since its introduction. In the early 1930s, the first explicit quantum theory of measurement was presented in the work of Landau and Peierls [1931], von Neumann [1932], Fock [1932] and Pauli [1933]. Since these early days, the entangled states (quantum states of multiple particles that cannot be represented as products of independent single-particle states: W ^ ip\ 0 "(pi ^ •" ^ ^^n, where ^pi are subsystem states, for n particles) were prepared in order to better differentiate quantum behavior from classical behavior. As Schrodinger [1935], who defined entanglement, put it, entanglement is "the characteristic trait of quantum mechanics". After the theorem of von Neumann concerning hidden-variables theories, of which quantum mechanics might be the statistical counterpart, and the argument of Einstein, Podolsky and Rosen (EPR) [1935], that quantum mechanics is an incomplete theory of physical objects, practical tests of nonclassical behavior were carried out by Wu and Shaknov [1950] with states of spin, as discussed by David Bohm [1951] and analyzed by Bohm and Aharonov [1957] in the 1950s ^. Next the way was prepared for the systematic study of quantum-scale behavior that could not be explained by the class of local, deterministic hidden variables theories. In 1964, John Bell derived a general inequality that introduced a clear empirical borderline between local, classically explicable behavior and less intuitive forms of behavior (such as nonlocality, contextuality and stochasticity), which he called "nonlocality" (Bell [1964])^. The atomic cascade decay was at the time the best optical source of entangled states. An experiment using such a source was made soon after by Kocher and Commins [1967], to distinguish between quantum mechanics and local hidden-variables theories. Clauser, Home,
In this experiment, polarization measurements were made of two high-energy photons produced by spin-zero positronium annihilation (Wu and Shaknov [1950]). ^ The premise resulting from Bell's heuristics motivated by his "locality" thesis was later shown (Jarrett [1984]) to be decomposable into the conjunction of two independent conditions, known as parameter and outcome independence (Shimony [1990]). 279
280
Multi-photon quantum interferometry
[5, § 1
Shimony and Holt (CHSH) then modified Bell's treatment to fit any experimental arrangement similar to that of a two-spin atomic system, deriving the inequality |E(0i,02) + E(0i,0^) + E(0(,02)-E(0;,0O|<2, where the E's are the expectation values of the products of measurement outcomes for measurement parameter values 0j,0[,02 ^^^ ^2 (Clauser, Home, Shimony and Holt [1969]). This resuh set the stage for more definitive experimental tests of quantum theory that involved a new generation of quantum optical sources of entangled photon pairs, i.e., of two-photons. Radiative atomic cascade two-photon sources were used by groups at Berkeley, Harvard and Texas A&M in the early 1970s (see Clauser and Shimony [1978]). However, one particular problem with Bell inequality tests arose during this period: only single-channel polarizers consisting of a stack of glass plates at the Brewster angle were available, giving access to only positive measurement outcomes (see Aspect [1999]). In 1976, John Clauser performed one such experiment with results that suggested that such Bell-type inequalities might in fact hold, so that a local hidden-variable theory might be valid at the atomic scale. However, advances in laser physics and optics allowed for a new generation of elegant experiments by the group of Aspect at Orsay in the early 1980s, based on the use of photon pairs produced by nonlinear laser excitations of an atomic radiative cascade and the use of two-channel polarizers. These experiments paved the way for future quantum-interferometric experiments involving entangled photons, finally producing an unambiguous violation of a Bell-type inequality by tens of standard deviations and strong agreement with quantum-mechanical predictions (Aspect, Grangier and Roger [1981]). Though producing photon pairs using atomic cascade decays had allowed for the initial demonstration of the significance of entangled states, the method was obviously far from optimal considering the sophisticated tests already proposed (Fry and Thompson [1976]). Meanwhile, a new, more powerfiil source of entangled photons, the optical parametric oscillator (OPO, fig. 1), was being developed independent of tests of basic principles of quantum mechanics. OPOs were operational in major nonlinear optics research groups around the world almost immediately following the development of the laser (for more details, see Bloembergen [1982]). It was seven years after the first OPOs were introduced that an international collection of experimental groups [Byer at Stanford University (Harris, Oshman and Byer [1967]), Magde and Mahr at Cornell University (Magde and Mahr [1967]), and Klyshko et al. at Moscow State University (Klyshko [1967])] independently discovered the spontaneous emission of polarized photons in an optical parametric amplifier.
Introduction:
5, § 1 ]
practical
optical tests of Bell
inequalities
281
Parametric osctHator Focusing lens
Optic axis o f crystal c j | , coj, a>3
«*
il--
w,
Nonlinear /?, =* 100% crystal /?a^lOO% ^3=0%
S
w, -
K, (high but < 1 ) Rj (high but < 1 ) R,= 0%
Fig. 1. Optical parametric oscillator (OPO).
This very weak OPO spontaneous noise occupied a very broad spectral range, from near the blue pump frequency through to the infrared absorption band. A corresponding spatial distribution of different frequencies followed the wellknown and simple phase-matching conditions of nonlinear optical systems (see §2). This effect had several names at the time: "parametric fluorescence", "parametric luminescence", "spontaneous parametric scattering", and "splitting". The existence of such a process follows from the quantum consideration of a parametric amplifier developed by Louisell, Yariv and Siegman [1961]. The Hamiltonian for the down-conversion process is given by H, = \ j dvP.Ep{rj)=\ j dvxflE,{rJ)E2{rj)Ep,
(1.1)
where P is the nonlinear polarization induced in the medium by the pump field E. The polarization is defined in terms of the second-order dielectric susceptibility of the medium Xup^ coupling the pump field to the two output fields E\ and E2. The field annihilation operators for photons at two output frequencies a)\ and 0)2 can be written as ai(t)
=
e^-ift^iV '^'\axQ coshg^ + ie '^4o sinhgO.
(1.2)
a2(t) = e ''^\a2o coshgt + ie '^ajo sinhgO. where g is a parametric amplification coefficient proportional to the secondorder susceptibility, the crystal length and the pump field amplitude, ato and AJQ are the initial operator values, and (/) is determined by the pump wave phase. Accordingly, the average number of photons per mode in the outputfieldsni(t) and ^2(0 is n\(t)= (a\(t)a\(t)\
= n \ocosh^ gt-\-(I+n2o)smh^
gt, (1.3)
«2(0 = ( 4 ( 0 ^ 2 ( 0 / = n2o cosh^ gt + (I +/7io)sinh^g^
282
Multi-photon quantum interferometty
[5, § 1
where «io and «2o are the inputs into the n\{t) and «2(0 fields, respectively. This describes a two-component gain feature having the odd property that the " 1 " in the second terms means that there is nonzero output, even when both input fields are zero. This "extra" one photon per mode - due to vacuum fluctuations - can be viewed as stimulating spontaneous down-conversion. The practical theory describing the generation of such radiation was analyzed in detail in 1967-1968 by Mollow and Glauber [1967], Giallorenzi and Tang [1968, 1969] and Klyshko [1967]. The next step in the theory was to analyze the statistics of photons appearing in such spontaneous conversion of one photon into a pair. This was done by Zel'dovich and Klyshko [1969], and Mollow in 1969 (and later treated in detail by Mollow [1973] and Kleinman [1968]), demonstrating the existence of very strong correlations between these photons in space, time and fi-equency. Bumham and Weinberg [1970] first demonstrated the unique and explicitly nonclassical features of states of two-photons generated in the spontaneous regime fi-om the parametric amplifier. Quantum correlations involving twophotons were exploited again 10 years later in experimental work by Malygin, Penin and Sergienko [1981a,b]. Because of a very active research program at the University of Rochester led by Mandel, and the work of Alley at the University of Maryland, the use of highly correlated pairs of photons for the explicit demonstration of Bell inequality violations has become popular and convenient since the mid-1980s. The contemporary name for the process of generating these states, "spontaneous parametric down-conversion" (SPDC), has become widely accepted in the research community, and new, high-intensity sources of SPDC have been developed (see § 3). A number of excellent reviews on the topic of two-photon quantum interference exist. Among the most comprehensive recent reviews covering the topic of entangled-photon interference are Quantum Optics and the Fundamentals of Physics by Perina, Hradil and Jurco [1994], Optical Coherence and Quantum Optics by Mandel and Wolf [1995], Hariharan and Sanders [1996], Quantum Optics by Scully and Zubairy [1997], and The Physics of Quantum Information by Bouwmeester, Ekert and Zeilinger [2000]. Though quantum optics has always kept the attention of the physics community, these reviews have mainly covered the subjects of quantum coherence, squeezed states, quantum non-demolition measurement and, most recently, quantum information. The main goal of this review is to exhibit several different contemporary trends in the development of entangled-photon interferometry using SPDC. We shall concentrate mainly on developments in the area of experimental two-photon interferometry, which has received a significant boost recently due to the importance of the properties
5, § 2]
Two-photon interferometry with type-Iphase-matched SPDC
283
of quantum entanglement in such exciting, but still relatively young areas as quantum teleportation, quantum cryptography and quantum computing (see also § 3 of ch. 1 in this volume).
§ 2. Two-photon interferometry with type-I phase-matched SPDC Spontaneous parametric down-conversion (SPDC) of one photon into a pair is said to be of one two types, type I or type II, depending on whether the two photons of the down-conversion pair have the same polarization or orthogonal polarizations. The two photons of a pair can also leave the down-converting medium either in the same direction or in different directions, the collinear and noncollinear cases respectively. A medium is required for down-conversion, as conservation laws exclude the decay of one photon into a pair in vacuum. The medium is usually some sort of birefringent crystal, such as potassium dihydrogen phosphate (KDP), having a x^^^ optical nonlinearity Upon striking such a nonlinear crystal there is a small probability (on the order of 10~^) that an incident pump photon will be down-converted into a two-photon (see fig. 2). If down-conversion occurs, these conserved quantities are carried into that of the resulting photon pair under the constraints of their respective conservation laws, with the result that the phases of the corresponding wavefunctions match, in accordance with the relations 0)1 + (D2 = (Dp,
ki+k2=kp,
(2.1)
known as the "phase-matching" conditions, where the kt and (Ot are momenta and firequencies for the three waves involved. The individual photons (here labeled / = 1,2) are often arbitrarily called "signal" and "idler", for historical reasons. When the two photons of a pair have different momenta or energies, entanglement will arise in SPDC, provided that the alternatives are in principle experimentally indistinguishable. The two-photon state produced in type-I down-conversion can be written \^)=
da;i (picoi, COQ - coi) \a)i) \(0o - (Oi),
(2.2)
Jo where (l)(a)i,(Oo - (Oi) is the frequency density and the two photons leave the nonlinear medium with the same polarization, orthogonal to the polarization of the pump beam photons. Down-conversion photons are thus produced in two
284
Multi-photon quantum interferometry
[5, §2
Pump bdam
Fig. 2. Spontaneous parametric down-conversion (Saleh [1998]).
thick spectral cones, one for each photon, within which two-photons appear each as a pair of photons on opposite sides of the pump-beam direction (see fig. 2). In a pioneering experiment in the mid-1980s, Hong, Ou and Mandel [1987] created noncollinear, type-I phase-matched SPDC photon pairs in KDP crystal using an ultraviolet continuous-wave (cw) laser pump beam (see fig. 3). These photon pairs were directed to a movable beamsplitter by two mirrors, so that the two resulting spatially superposed beams impinged on two photodetectors Di and D2. Filters placed in the apparatus determined the fi-equency spread of the down-converted photons. This experiment empirically demonstrated the strong temporal correlation of the two-photons. The correlation fiinction for two-photons is g(t) = G(tyG(0). In the experiment G{t) = Jdt(t>[(coo/2) + (jOi,(a)o/2) - a)\], where the down-converted light was frequency-degenerate, so that 0 peaked at a){ = JCDQ = (O2, with COQ = 351.1 nm; g was nearly Gaussian in o) with a bandwidth Aco. The probability of joint detection of the two photons of the pair at Di and D2, at times t and t-\-T respectively, in such an experiment is given by Pl2(0 = ^ E[-\t)E^{^(t-\-T)E^^\t^r)E\'^
(O)
= K\G(0)\' {T'\g(T)\' + R'\g(2AT - T)p - R T \ g \ x ) g { 2 A r - r) + c.c.]} , (2.3) where the £, are the electric fields at detectors D/, and ^ is a constant characterizing the detectors. In the Hong-Ou-Mandel (HOM) experiment, the coincidence rate for photon joint detection at D\ and D2 was studied as the beamsplitter (BS) was translated vertically from its central location by small distances c 5r, giving rise to optical path differences for the two outgoing beams. With R/T = 0.95, the corresponding joint count rate A^^ exhibited a sharp dip.
5, §2]
Two-photon interferometry
with type-I phase-matched
285
SPDC
1 Amp. 1 T*" Counter j Disc. 1 V
U-
Coincidence Counter
—¥-
PDF 11/23+ A
Amp.
& [
Counter
\ Disc. 1
Fig. 3. Hong-Ou-Mandel interferometer (Hong, Ou and Mandel [1987]).
^
260
2B0
300
320
340
380
Position of beam splitter (yum) Fig. 4. Hong-Ou-Mandel dip (Hong, Ou and Mandel [1987]).
near the time difference 8r, having a width determined by the length of the wavepacket (or, equivalently, the coherence time) of the two-photons. This nonclassical coincidence dip was seen to fall to a few percent from the maximum value (see fig. 4), whereas classical optics predicts a visibility that cannot exceed 0.5 (IVlandel [1983]) and Bell-type inequality violations can be obtained once coincidence visibilities exceed 71% (see, for example Tittel, Brendel, Gisin and Zbinden [1999]). Such a dip - hereafter referred to as the "Hong-Ou-]VIandel dip" - also provided for an empirical measure of the time intervals between the two photon arrivals with sub-picosecond precision. Unlike methods requiring the observation of second-order (i.e., single-photon) interference, this technique does not require keeping path differences stable to within a fraction of a wavelength. In 1988, a similar arrangement and light frequency was used by Ou and IVIandel [1988a,b] to demonstrate the violation of Bell's inequality by six standard deviations, in addition to disagreement with classical optical
286
Multi-photon quantum interferometry j/
[5, §2
/ 3 - B a B 2 0 4 gnd
PRISM 2
PRISM I 4"^
150 PS 7 0 P P S / NO-YAG LASER )
M - MIRROR N D - N O FILTERS L - LENS P - PIN H O L E B - 5 0 - 5 0 BEAM S P L I T T E R A - GLAN-THOMPSON POLARIZATION ANALYSER F - NARROW BAND SPECTRAL FILTER D - DETECTOR
,e3 Fig. 5. Shih-Alley experiment (Shih and Alley [1988]).
predictions. In that experiment, the idler photon was rotated by 90 degrees in one beam before reaching the beamsphtter, and polarizers were placed before D\ and D2 at angles 0\ and 62, respectively, to obtain count rates corresponding to the joint probabilities of the left-hand side of Bell's inequality. Taking into account alignment imperfections, the observed joint probabilities were found in this experiment to be in agreement with the quantum mechanical predictions and in violation of the CHSH inequality. According to quantum theory, the choices 0i=jr/8, 02 = ^ / 4 , 0 / = 3jr/8, 02^ = 0, for example, yield 5" = \K{y/2-\)>Q, where 5" < 0 is the Clauser-Home variant of the inequality (Clauser and Home [1974]). The corresponding two-photon interference visibility was empirically found to be F = 0.76. That same year, Shih and Alley [1988] used a similar experimental arrangement, but replaced the cw pump laser with a pulsed laser operating at 266 nm (fig. 5), to demonstrate a three-standard-deviation Bell-type inequality violation. In particular, it was found that d = \ [R^{\JI)-R^iljt)]/RQ\ = 0.34 ± 0.03 > \, where 5 ^ ^ is the Freedman-Clauser variant of the inequality (Freedman and Clauser [1972]). Furthermore, the results were in good agreement with the quantum-mechanical prediction of 6 = ^\/2 = 0.35. In a variation on the same experimental arrangement. Rarity and Tapster obtained a coincidence dip by translating right-angle prisms, instead of fixed mirrors, placed in the beam paths before the beamsplitter. They next explored the frequency non-degenerate case of SPDC to obtain an interferogram exhibiting additional oscillations (Rarity and Tapster [1990a]). The time resolution was improved to approximately 40 fs and the observed visibility reached V = 0.84.
2]
287
Two-photon interferometry with type-I phase-matched SPDC
Fig. 6. Rarity-Tapster experiment (Rarity and Tapster [1990b]).
During the same period, after proposals by Home, Pykacz, Shimony, Zeilinger and Zukowski (Home and Zeilinger [1985], Zukowski and Pykacz [1988], Home, Shimony and Zeilinger [1989]), Rarity and Tapster [1990b] used a modified arrangement involving two beamsplitters and two balanced MachZehnder interferometers to test Bell's inequality. In this case, the variable of the state entanglement was momentum-direction and phase-shifting elements were placed in space-like separated locations (see fig. 6). The measured value for the left-hand side of the CHSH inequality using this arrangement was found to reach S = 22\ at an interference visibility oiV = 0.78, amounting to an inequality violation by 10 standard deviations. SPDC had also previously been used for similar experiments by Ou and Mandel [1988b] using polarization variables. A different interferometric arrangement having two spatially separated, unbalanced Mach-Zehnder interferometers, each involving a phase shift 0/ (/ = 1,2) between the long and short beam paths, was also proposed by Franson [1989], in order to test a Bell-type inequality for position and energy without the involvement of polarization variables or polarizers. This latter sort of experiment was carried out by Franson [1991] (see fig. 7). The interferometer was pumped by a cw laser that produced energy-degenerate two-photons by SPDC. Brendel, Mohler and Martienssen [1992], Kwiat, Steinberg and Chiao [1993] and Shih, Sergienko and Rubin [1993] carried out similar experiments, though with somewhat different arrangements. The initial two-photon state for such experiments can be written \n>) = \ (|5)i \S), - e'<^'-^^> |£), \L), + e'^^ \S), \L), + e'^' \L),
\S),), (2.4)
288
[5, §2
Multi-photon quantum interferometry
-i^-S^
^l^i2^ >2
- W Atom j-4 F2
»
Mi
L2
Fig. 7. Franson interferometer (Franson [1991]).
where S and L refer to the temporal position corresponding to short and long optical path lengths, respectively. By using a sufficiently large path-length difference between long and short options, the last two terms may be neglected an entangled two-photon state results. In the above experiments, the difference of optical paths in the two interferometers, AL, satisfies the requirement cT^oh <^ AL, where TJoh is the coherence time of the down-converted photons, so as to exclude single-photon (i.e., secondorder) interference and to allow only two-photon (i.e., fourth-order) interference to occur. Since a cw laser was used as a pump, the photon emission times were unknowable (i.e., experimentally indistinguishable in principle), allowing no way of determining which of the two photons detected takes the short path and which takes the long path, while preserving the energy correlations between photons. The resulting coincidence counting rate was R^ = \cos^ ((j)[ +02). This ostensibly allowed the demonstration of nonlocal effects due to quantum theory and the testing of local realistic theories because of their in-principle capability of providing 100% two-photon interference visibility while no singlephoton interference would arise when the individual phases 0- (0^ = 0i/2, 02 = 02/2 + 0o) were varied. However, it should be noted that it recently has been shown that such experiments cannot provide tests of local realism, since there exists a local hidden-variable model that reproduces quantum predictions for joint measurements using this apparatus (Larsson, Aerts and Zukowski [1998]). A time-frequency Bell inequality test has also been proposed, though not realized (Davis [1989]). The idea is to measure the detection time of one of two-photons produced by down-conversion of a bandwidth-limited, pulsed pump beam, with respect to the center of each down-converted pump pulse and the spectral frequency of the other. The detection time of the first photon provides information regarding the arrival time of the second photon fi-om the same two-photon. Thus the arrival time and the spectral fi-equency of the second photon become known precisely enough to violate the time-frequency bandwidth
5, §2]
Two-photon interferometry with type-I phase-matched SPDC I
289
Pump laser
Fig. 8. Apparatus for testing local realism with a Bell inequality (Torgerson, Branning, Monken and Mandel [1995]).
product. Using time-dependent-physical-spectrum (TDPS) measurements, using a single-photon detector behind a Fabry-Perot etalon, the photon arrival time is to be measured to a precision limited only by the reciprocal of the etalon frequency resolution. The quality factor of the etalon then determines which combination of the time and frequency FQ, is observed. Positive or negative values are to be attributed based on whether or not the photon is transmitted through the etalon within a given time window. The necessary correlations between such measurements on space-like separated photons will arise due to their simultaneous production under phase-matching constraints. Considering three sets of coincidence count rates under the assumptions of counterfactual definiteness of photon properties and the Einstein locality condition, given the above direct correlations allows a Bell-type inequality to be derived for these variables: Pr[F^i^ = +, f^2> = - ] ^ Vx[F^'^ = +, r^^^ = - ] + Pr[f^^) = +, f^^^ = - ] ,
(2.5)
where F and T refer to the special cases of pure frequency and pure time measurements, and the superscripts each refer to one of the two photons, arbitrarily labeled 1 and 2. There followed a truly remarkable violation of local realism by roughly 40 standard deviations. This result was achieved in an experiment by Torgerson, Branning, Monken and Mandel [1995] (see fig. 8). Motivated by the ambiguous results of Bell-type inequality tests in which two photons pass through QWPs before reaching polarization analyzers, these workers obtained tremendous inequality violations that removed any lingering questions about nature's ability to violate such inequalities.
290
Multi-photon quantum interferometry
[5, § 3
In late 1994, an argument was presented to the effect that many experiments involving SPDC cannot be used to properly test Bell-type inequalities because the states they utilized are in fact product states (De Caro and Garuccio [1994]). That is, only by post-selecting from the full ensemble of down-conversion events that one-half of events in which joint-detections occur, can Bell tests be simulated. In particular, such a method appears invalid because the intrinsic efficiency of detection required for loophole-free tests is 67% (Kwiat [1995], Kwiat, Eberhard, Steinberg and Chiao [1994], Eberhard [1993]). However, it was subsequently pointed out that even type-I phase-matched down-conversion sources can be configured so as to produce genuine entanglement without the need for post-selection (Kwiat [1995]). This concern can be completely avoided by using type-II phase-matched down-conversion sources that produce a state truly entangled in regard to polarization. Furthermore, the CHSH inequality may be slightly modified so as to allow the use of the full ensemble in a valid Belltype inequality test. § 3. Two-photon interferometry with type-II SPDC In the case of type-II spontaneous parametric down-conversion (SPDC), the two photons of each down-conversion pair have orthogonal rather than identical polarizations. This allows the entanglement of their states to involve polarization in addition to those other quantities potentially involved in the type-I case. This sort of entanglement, including multiple degrees of freedom, has been referred to as "hyper-entanglement" (Kwiat [1997]). In the type-II case, if the two photons of a pair leave the down-converting medium in different directions, i.e., noncollinearly, their entanglement will involve both directions - as it is not possible to identify which photon went in each direction - and polarizations. Moreover, for a nearly monochromatic, continuous-wave laser pump any sort of down-conversion pair entanglement will involve energy, yielding hyperentanglement with three relevant quantities. Such states are generally given by H^) = \ I Jo
do)(l)(co, COQ - (o) \co) \a)o - co)
• (I*,) 1*0 + exp[i0] 1*2) 1*2)) (Ie) |o) + |o) |e)), where the orthogonal polarizations of the down-conversion photons are labeled "e" and "o", according to their orientation relative to the polarizations associated with the extraordinary and ordinary axes of the nonlinear crystal used for downconversion. Unlike the case of type-I phase-matched down-conversion, the two
5, §3]
Two-photon interferometry with type-II SPDC
/ ^
291
Entangled-state emission directions
ordinary Fig. 9. SPDC under type-II phase-matching conditions (Kwiat [1997]).
V—B
351.1 nm
BBO Type li
^
Ar laser
No I Fig. 10. Bell inequality tests using type-II phase-matched two-photons (Kiess, Shih, Sergienko and Alley [1993]).
down-conversion light cones are not concentric about the direction of the pump beam (see fig. 9, and contrast with fig. 2). The new ingredient in the type-II case (eq. 3.1), compared with the type-I case (eq. 2.2), is the involvement of polarization in the entanglement. Entangled states of this kind were used by Kiess, Shih, Sergienko and Alley [1993] to find CHSH inequality violation by 22 standard deviations. In that experiment, a 351.1 nm cw laser pump was used to produce two-photons in BBO crystal at 702.2 nm. These collinear-photon pairs were deflected by a nonpolarizing beamsplitter to two Glan-Thompson polarization analyzers followed by photodetectors, and the resulting coincidence detections were studied (see fig. 10).
292
Multi-photon quantum interferometry
[5, §3
2500
a tn
2000
R
§ S3: 3 (O
1500 1000
5
33
o
?
P -100
-50
0
SO
Optical Delay Al -15
too
150
(|xm)
-10 -5 0 5 10 Number of Quartz Plates Inserted
15
Fig. 11. Polarization two-photon coincidences varying optical delay (Rubin, Klyshko, Shih and Sergienko [1994]).
Shortly thereafter, a comprehensive theoretical treatment of these type-II phase-matched two-photons was given by Rubin, Klyshko, Shih and Sergienko [1994]. A review of several experiments done at the University of MarylandBaltimore County verifying this treatment was presented therein. Quantum beating between polarizations was also observed as absolute polarizations were varied while relative polarization was kept orthogonal (see fig. 11). A similar experimental arrangement was then used to demonstrate the violation of two Bell-type inequalities, one for polarization and one for spacetime, in a single experimental arrangement (Pittman, Shih, Sergienko and Rubin [1995]). In order to test the latter, EPR states were produced by probabilityamplitude cancellation. The experimental arrangement was similar to that of fig. 10, but included also a large quartz polarization delay line and a number of thinner reorientable birefiingent quartz plates placed before the predetector polarization analyzers. Two optical paths to each detector were thus created, so that a two-photon state of the form W=A{XuX2)-A{Y,,Y2) was created, where 1 and 2 label the fast-axis path and the slow-axis path respectively, analogously to the short and long paths of the Franson interferometer, and X and Y indicate two orthogonal linear polarizations. Notably different from the Franson interferometer, however, is that the entangled state here arises from probability-amplitude cancellation rather than from the use of a short coincidence counting time window. In the position test, by activating two spacelike separated Pockels cells, a coincidence counting
5, § 3]
Two-photon interferometry with type-II SPDC
293
Fig. 12. High-intensity two-photon source (Kwiat, Mattle, Weinfurter, Zeihnger, Sergienko and Shih [1995]).
rate R^ = i^o [1-cos(ft>iZ\i - 0)2^2)] was found, where the A are the total optical delay between the optical paths of the two detectors, and (J)\ and CO2 are the signal and idler frequencies. An inequality violation of more than 14 standard deviations was achieved. Similarly, a test in polarization was made by rotating polarization analyzers behind each Pockels cell with coincidence counting rate Rc{(p), where (j) is the difference in polarization analyzer angles at counters 1 and 2, such that 8 = \ [i?c(^^)-^c(|^)]/^o| = 0.309 zb 0.009 > \. A violation of the constraints of local hidden variables theory by more than six standard deviations was observed. In 1995, a new high-intensity, type-II phase-matched SPDC two-photon source was developed in order to take full advantage of two-photon entanglement involving polarization. Two-photons were produced noncollinearly and directly, i.e., without the use of extra beamsplitters or mirrors previously required to emulate entanglement post-selectively (see fig. 12) (Kwiat, Mattle, Weinfurter, Zeilinger, Sergienko and Shih [1995]). This source allowed the observation of CHSH inequality violations by more than 100 standard deviations in less than 5 minutes. Furthermore, all four polarization Bell-states |if'±> = i ( | H , V ) ± | V , H ) ) ,
|0±> = i(|H,H>±|V,V)),
(3.2)
were readily produced. The use of a half-wave plate (HWP) allowed for polarization flipping between ordinary and extraordinary, that is H and V, states. It thus allowed for the exchange of states |*^~) and | ^ " ) , and states | ^ ^ ) and |0+). Similarly, a birefi-ingent phase-shifter allowed for a sign change between two-photon joint amplitudes, so that an exchange between two-photon states |^+) and | ^ ~ ) , and between |(P+) and |cp"), was also accompHshed. Bell-type inequalities were tested using all four Bell states, with significant violations in each case. In addition to the problem of creating high-intensity sources two-photons with entanglement involving polarization, there have been other difficulties associated
294
Multi-photon quantum interferometry
[5, § 3
with entangled optical states. First, long crystals capable of producing entangled states with two polarizations give rise to nontrivial walk-offs. This problem can arise in the form of spatial walk-off: a photon of one polarization moves more quickly through the crystal than the other (yielding longitudinal walk-off) and, though they will leave the crystal collinearly, they can move in different directions while within the crystal (transverse walk-off). For sufficiently short crystals, one can completely compensate for the walk-off, as interference occurs pairwise between processes where the photon pair is created at equal distances but on opposite sides of the crystal central axis. This is accomplished by the introduction in each of the two photon paths of a similar crystal half as long (or in one path and of identical length) after polarization rotation of the photons. This makes the polarization that was previously fast the slow polarization, and vice versa (Rubin, Klyshko, Shih and Sergienko [1994], Kwiat, Mattle, Weinfurter, Zeilinger, Sergienko and Shih [1995]). Similarly, optimal transverse walk-off compensation is accomplished. However, for a sufficiently long crystal, the o and e rays may separate by more than the coherence length of the pump photons, making complete compensation impossible. After the Hong-Ou-Mandel (fig. 3) and Shih-Alley (fig. 5) experiments, it was often intuitively believed that the two-photon interference could be understood in terms of the simultaneous arrival - and hence possible interaction of the two photons of each pair at the common beamsplitter. This is incidental, however. The essential requirement is the equality of optical path length to within the coherence length of the photons, resulting in in-principle indistinguishability. Type-II phase-matched two-photons provided an opportunity to demonstrate this. Pittman, Strekalov, Migdall, Rubin, Sergienko and Shih used collinear type-II phase-matched SPDC in a similar arrangement to observe two-photon interference, where the two photons of each pair were made to reach the common beamsplitter at times greater than the coherence length of their 702.2 nm photons yet still yield two-photon interference (Pittman, Strekalov, Migdall, Rubin, Sergienko and Shih [1996]) (see fig. 13). This provided a counterexample to the intuitive, local picture of some local influence at a common beamsplitter "telling them" which way to travel afterward. First, a phase shifter (r^^) was placed in the path of the signal photon. Then, since that alone could eliminate the indistinguishability of the two-photon alternatives necessary for coincidence interference, "postponed compensation" was used, the leading photon was delayed for Ti^ = 2r«v after the beamsplitter. Thus the arrival of the photons at the two detectors was accomplished in exactly the same order and time difference, successfiilly restoring indistinguishability of detection events, as can be clearly seen in a space-time portrayal of alternative events (see
5, § 3 ]
Two-photon interferometry with type-II SPDC
295
Fig. 13. Schematic of postponed compensation demonstration (Pittman, Strekalov, Migdall, Rubin, Sergienko and Shih [1996]). time
tirfie
space space (a) Balanced HOM/SA Interferometer time
^tim.e
/El
Ek
i^x /
1 ^ \ / « space spac (b) PostpK)ned Compensation Experiment Fig. 14. Space-time diagram of restored indistinguishability (Pittman, Strekalov, Migdall, Rubin, Sergienko and Shih [1996]).
fig. 14). In fig. 13, the delay, state and path labels are identical, allowing for direct comparison with those of fig. 14, if one reorients the apparatus schematic so that the down-conversion crystal is placed at the bottom with its output directed upward. Such an apparatus later proved useful for high-precision polarization mode dispersion measurements (see below).
296
[5, § 3
Multi-photon quantum interferometry 351 nm
detector 1
Pockels cell
collection lens
zr—=" BBO
quartz compensator
quartz rod --r-^-L ^
analyzer
fjiter detector 2
Fig. 15. Apparatus for postselection-free Bell-inequality test energy (Strekalov, Pittman, Sergienko, Shih and Kwiat [1996]).
In 1996, another step exhibiting the nonlocal character of two-photon quantum interference was taken when the new high-intensity type-II phase-matched SPDC source was used by some of the same investigators to make a post-selectionfree test of a Bell inequality with entanglement involving energy (Strekalov, Pittman, Sergienko, Shih and Kwiat [1996]). Unlike the experiment of Franson (fig. 7), where a short-duration time window was used to post-select the coincidence alternatives of interest, this experimental arrangement avoided unwanted alternatives by design: the short-long and long-short alternatives were engineered out. Noncollinear beams of 702.2 nm-photon pairs were created in a symmetrical configuration and passed through a quartz compensator, quartz compensator rods, Pockels cells and polarization analyzers (see fig. 15). The quantum state of two-photons that emerged fi-om the birefiingent rods along the two propagation directions was |
[|s),|s)2e'<°^^'-|f),|f)2]sin(0i+02) (3.3) |s),|f)2e'«-|f),|s)2e''*]cos(,+^2),
where a and ji are the phase shifts introduced by each of the two Pockels cells. The resulting interference visibility, 95%, was found to exceed the limit set by Bell locality by 17 standard deviations. It is noteworthy that the pair of quartz rods, rather than the usual pair of spatial paths as in the Franson apparatus, provide the interfering quantum alternatives in this experiment. In 1998, Kwiat and Weinfiirter showed how higher-dimensional Hilbert spaces could be used to distinguish Bell states (eq. 3.2), in what they called "embedded
5, § 3]
Two-photon interferometry with type-II SPDC
297
b45
0
50-50 beam splitter
/
Polarizing beam splitters (aloi^g 45")
Fig. 16. Bell-state analyzer (Kwiat and Weinflirter [1998]).
Bell-State analysis" (Kwiat and Weinfurter [1998]). Previously, Vaidman and Luetkenhaus had shown that the Bell states cannot be fully distinguished using only linear optical elements, if multiple entanglements are not involved. The main idea of Kwiat and Weinfurter was to use entanglements including variables beyond those of the Bell states, for example energies in the case of a polarizationstate analysis, and the additional interferometric measurements they allow, to fully distinguish the four Bell states. In the chosen example, first photon pairs in Bell states are sent to a 50-50 beamsplitter and the emerging beams sent to polarizing beamsplitters (see fig. 16). The scheme works here because only state | W) can give rise to one photon in each beamsplitter output beam, allowing the state to be readily identified by coincidence measurements in detector pairs a and b. Then, by looking at only one of the wings corresponding to one beam, one is able to distinguish between the remaining three states. Birefringent material, with axes oriented to correspond to the H-V basis states, was introduced for this purpose. That gives rise to temporal shifts between H and V polarization components, distinguishing | ^ ^ ) from 10"^). IW') was distinguished by detecting two photons in one wing separated in time, making sure that this time difference is still less than the coherence time of the pump photons. Finally, in the dz45'^ basis, |0=^) are distinguished fi-om one another by the polarizing beamsplitter: for \
298
[5, § 3
Multi-photon quantum interferometry H-polarizcd """^L V-polarized (iVoni #2)
Fig. 17. Two-crystal ultra-bright two-photon source (Kwiat, Waks, White, Appelbaum and Eberhard [1999]).
accordingly observed a far higher violation of a CHSH Bell-type inequality, as much as 242 standard deviations within three minutes. Considerations of source symmetry suggest that an increase in intensity of 10000 times was achieved in the entire output. Two relatively thin (0.59 mm) BBO crystals were identically cut and oriented so that their optic axes lay in perpendicular planes. They were then pumped by a 351.1 nm laser line polarized 45 degrees to both the e and o crystal axes (see fig. 17). The result was the production of two-photons in which all pairs of a given color were entangled, as the amplitudes for the orthogonal modes were non-zero in only one of the two and in alternate crystals. That is, two-photons will be described by the polarization state |t/;) = ^ ( | H ) | H ) + e'^|V)|V)).
(3.4)
The phase 0 is adjustable by changing crystal tilt, by phase shifting one of the output beams or adjusting the phase relation between horizontal and vertical polarization components of the pump state. A two-photon interference visibility of V = 0.996 was thus achieved. In another significant step forward, Zeilinger et al. tested a Bell-type inequality under strict locality conditions, in order to close one loophole of previous Bell tests, since it is conceivable that polarizers might somehow communicate their settings to one another before two-photons reach them (Weihs, Jennewein, Simon, Weinfiarter and Zeilinger [1998]). In this they surpassed the previous attempt of Aspect, Dalibard and Roger [1982], whose polarizer orientations had only been rapidly, periodically changed during the photons' flight from their source to polarizers separated by a distance of 12 m, which accordingly suffered from what has come to be known as the "periodicity loophole" (Shimony [1990], Weihs, Weinfurter and Zeilinger [1997]).
5, § 3]
Two-photon interferometry with type-II SPDC
299
Zeilinger et al. separated their polarizers by 400 m, thereby allowing them a full I3jiis to make ultrafast physically random (as opposed to physically deterministic pseudorandom) polarizer orientations, as well as to independently register measurement results from each wing of the apparatus. At the end of each cycle of the experiment, the two sets of data were brought together to view the correlations. 97% coincidence interference visibility was obtained, and the CHSH inequality was violated by 30 standard deviations. There remains only the "detection loophole", to such tests after this work. This final loophole can be blocked when photodetection efficiencies can be improved beyond 0.841 (Shimony [1990]). As a practical application of two-photon interferometry, two-photon interference patterns similar to the HOM dip (see fig. 4) have proven usefiil for measuring polarization mode dispersion (PMD), the difference in propagation rate between two polarization modes in a birefringent medium (see, for example, Dauler, Jaeger, Muller, Migdall and Sergienko [1999]). PMD is important, among other reasons, in understanding propagation of polarized light in optical fibers, such as has been proposed for the purposes of quantum cryptography (see below). The extreme constraint on the simultaneity of the creation of the two photons of a down-conversion pair allows for the high resolution achieved using such a method. The PMD can be determined with sub-femtosecond resolution by studying the effect of dispersive media on this interference feature. An important advantage of this technique, relative to some non-white light interferometric methods, is that it determines the optical delay absolutely, as opposed to simply measuring the delay modulo a wavelength. The PMD is directly determined from the temporal shifi; of the HOM-type interference feature produced by the insertion of a birefringent sample into the interferometer (see fig. 18). Two ways of producing a coincidence event are arranged so that they cannot be distinguished (even in principle). A differential delay line is used to delay one polarization relative to the other. The coincidence rate from spatially separated detectors is recorded as this delay line is varied. When the two photons are separated at the beamsplitter by more than their coherence time the two coincidence events can be distinguished, so no interference is possible; the total coincidence rate is simply the sum of the two individual rates. When the two photons reach the beamsplitter to within their coherence time, however, destructive interference occurs, as the detector polarizers are oriented at 45 degrees and 135 degrees. The two types of coincidence, the first photon produced in the e polarization and the second in the o polarization, or vice versa, become indistinguishable. The temporal correlations are limited by the length
300
[5, § 3
Multi-photon quantum interferometry
\>>
d Easer
BBO Crystal ^•^rrr 351 nm mirror 2 photons (702 nm)
Quartz wedgei
y r 3 0 fs/mm
AA filter EI" beam < splitter
Coinc,
lAT" Optical Delay
/ ^Coinc.
Coincidence circuit Fig. 18. Apparatus for high-precision PMD measurements (Dauler, Jaeger, Muller, Migdall and Sergienko [1999]).
of the down-conversion crystal, and a triangular-shaped interference feature is seen. This occurs because the effect is to convolve two rectangular two-photon wavefiinctions. The shift of the center of this interference feature is identical to the PMD of the sample. The uncertainty limit of the method was determined by how well the center of that feature was determinable. This was found to be as low as 0.15fs. Most recently, the two-photon interferometer has been modified to produce a modified interferogram, with additional "internal fiinging" (see fig. 19) (Branning, Migdall and Sergienko [2000]). This feature of "fiinging in the HOM dip" is introduced by moving the additional variable delay line of the first arrangement for PMD measurement after the first beamsplitter (see fig. 20). Using this improved technique allows one to measure the PMD with a precision of 8 attoseconds. In the quantum-informational context, decoherence-fi-ee subspaces within multiple-photon Hilbert spaces have been a subject of interest. They could be
5, §3]
ffl.
Two-photon interferometry with type-II SPDC
©Ulfl
i3
301
302 Multi-photon quantum interferometry
O O
m §
TS
5
[5, § 3
5, § 3]
Two-photon interferometry with type-II SPDC
303
Birefringent Crystals Down-Conversion Crystal (BBO) Pump{UV)
\
-^
^
* ^ Phase Tuner Correlated Pairs Detectors
Fig. 21. Preserving a two-photon decoherence-free polarization subspace (Berglimd [2000]).
useful, for example, in the redundant coding of quantum information (Ekert, Palma and Suominen [2000]). One could, for example, encode the states |6) , 11) as follows: |6) = |0,1),
|1) = |1,0),
(3.5)
where information-bearing states built from the double eigenstate are chosen not to be susceptible to decoherence in a way that those constructed using the single eigenstate might be. The decoherence-free (DF) subspaces of the Hilbert space of hyperentangled polarization states have recently begun to be studied (Berglund [2000] and Kwiat, Berglund, Altepeter and White [2000]). In particular, it has been shown that, while the energy correlations required by down-conversion phase-matching conditions can render two-photons susceptible to decoherence under the influence of an environment where frequencypolarization coupling is present, the DF subspace can be readily preserved (fig. 21). By appropriately symmetrizing the induced phase errors for the antisymmetric polarization state | V^~), its decoherence-free character can be demonstrated (Zanardi [1997]). The Bell states |t/;±) = ^ ( | H , V ) ± | y H ) ) ,
|^±) = ^ ( | H , H ) ± | V , V ) ) ,
(3.6)
are initially considered, with identical birefringent crystals placed across both paths L and R; denoting the thickness of the nonlinear crystal as C, the phase difference between the two arms will be co^^. The off-diagonal elements of the density matrix for It/;"^) approach zero as the crystal thickness surpasses the coherence length of the down-conversion photons, which is proportional to c/bcD, with b(D the width of the frequency spectrum. In this configuration, \(j)^) do not undergo decoherence, while |i/;=^) do.
304
Multi-photon quantum interferometry
[5, § 3
In particular, at the analyzer, |0=^) is transformed to
which has no effect on the magnitude of off-diagonal density matrix elements. However, these states do not generate a DFS, since in the left/right-circular polarization basis they are written |0±) = i ( | L , R ) ± | R , L ) ) ,
(3.8)
and thus can lose phase information in crystals with eigenmodes |L),|R). By contrast, the antisymmetric state is rotationally invariant, so there is hope for recovering its DFS. By rotating the nonlinear crystal in one arm by 90 degrees, the states 10"^) and |i/;^) will be seen to decohere, while the state |i/^~) will not: its state at the analyzers will be |t/;'-) = ^ ( | H , V ) - e x p [ i ^ ^ ] |V,H)).
(3.9)
Quantifying the fidelity of the transmission process, F = Tr(pinPout) or, for a mixed input state, F = [Tr(y^y^pout v0in)] » decoherence-free subspaces will have F = 1, which is the case for |t/;~). The currently most advanced form of quantum information experimentation is taking place in quantum cryptography - more precisely, quantum key distribution (QKD). QKD is the distribution of a secret key (bit sequence) between two interested parties, usually called Alice and Bob. This key can be used to encrypt and decrypt secret messages using the safe one-time pad method of encryption. The security of QKD is not based on complexity, but on quantum mechanics, since it is generally not possible to measure an unknown quantum system without altering it. Any eavesdropping introduces physical errors in the transmitted data (see also §3.1.4 of ch. 1 in this volume). The basic QKD protocols are the BB84 scheme (Bennett and Brassard [1984]) and the Ekert scheme (Ekert [1991]). BB84 uses single photons transmitted from sender (Alice) to receiver (Bob), which are prepared at random in four partly orthogonal polarization states: 0, 45, 90 and 135 degrees. When an eavesdropper. Eve, tries to obtain information about the polarization, she introduces observable bit errors, which Alice and Bob can detect by comparing a random subset of the generated keys. The Ekert protocol uses entangled pairs and a Belltype inequality. In that scheme, both Alice and Bob receive one particle of
5, § 3]
Two-photon interferometry with type-II SPDC
305
the entangled pair. They perform measurements along at least three different directions on each side, where measurements along parallel axes are used for key generation, and those along oblique angles are used for security verification. Several innovative experiments have been made using entangled photon pairs to implement quantum cryptography in the recent period, 1999-2000 (Sergienko, Atatiire, Walton, Jaeger, Saleh and Teich [1999], Jennewein, Simon, Weihs, Weinfurter and Zeilinger [2000] and Tittel, Brendel, Zbinden and Gisin [2000]). Quantum cryptography experiments have had two principal implementations: weak coherent state realizations of QKD and those using two-photons. The latter approach made use of the nonlocal character of polarization Bell states generated by spontaneous parametric down-conversion. The strong correlation of photon pairs, entangled in both energy-time and momentum-space, eliminates the problem of excess photons faced by the coherent-state approach, where the exact number of photons actually injected is uncertain. In the entangled-photon technique, one of the pair of entangled photons is measured by the sender, confirming for the sender that the state is the appropriate one. It has thus become the favored experimental technique. The first of the recent innovative experiments using SPDC demonstrated a more flexible and robust method of quantum secure key distribution with type-II phase-matched two-photons, in an improved configuration (Sergienko, Atatiire, Walton, Jaeger, Saleh and Teich [1999]). The high contrast and stability of the fourth-order quantum interference, along with the available knowledge of the exact number of photons present in the quantum communication channel, clearly show the performance of EPR-state-based quantum key distribution to be superior to the coherent-state-based technique. The entangled-photon technique had previously used type-I phase-matched pairs and, as a result, suffered from low visibility (only up to 85%) and poor stability of the intensity interferometer. This has primarily been due to the need in previous experiments for the synchronous manipulation of interferometers well separated in space. The intervention of any classical measurement apparatus (eavesdropping) will cause an immediate reduction of the visibility to 70.7%, so high visibility is required to ensure key security. Only an undisturbed EPR state can produce 100% interference visibility. Previous attempts to demonstrate the feasibility of quantum key distribution using EPR photons had failed to attain the high-visibility coincidences. A double, strongly unbalanced, distributed polarization intensity interferometer was used to avoid the simultaneous spatial manipulation that compromised previous attempts. A frequency-doubled femtosecond Ti:sapphire laser was used to generate 80-fs pulses at 541.5 nm that were sent through a 0.1-mm-thick BBO crystal, oriented so as to yield collinearly propagating type-II phase-matched EPR pairs. The
306
Multi-photon quantum interferometry
I
[5, §3 Alice
M*
Ar* Laser
ThSapphire A. = 830 nm Variable polarization delay line BBO crystal Type-il
Bs
i\\
' ri
APD Detector 1
Analyzer-modulator 450(-45<*)
•^r
Analyzer-modulator 45° (-45O) APD Detector 2
Bob
|v|^ ^ ^
Coincidence Counter
Fig. 22. Two-photon entangled-state QKD scheme 2000 (Sergienko, Atatiire, Walton, Jaeger, Saleh and Teich [1999]).
photons entered two spatially separated interferometer arms via a polarizationinsensitive 50-50 beamsplitter BS, which allowed photons of both ordinary and extraordinary polarization to be reflected and transmitted with equal probability. One output port led to a controllable polarization-dependent optical delay - the e-ray/o-ray loop - then to detector 1. The other led, through an optical channel, to detector 2. Polarization analyzers were placed in front of each photon-counting detector and were oriented at 45'' or 315°. Coincidence counts between the two detectors were monitored as a function of the optical delay between the orthogonally polarized photons. In this quantum key-distribution arrangement, the first
5, §3]
Two-photon interferometry with type-II SPDC
307
0° relative phase shift (45° and 45° )
^
1000
130
135
140
145 150 155 160 Optical Delay (fsec)
165
170
90° relative phase shift (45° and -45° )
130
135
140
145 150 155 160 Optical Delay (fsec)
165
170
Fig. 23. Two-photon quantum cryptographic signals (Sergienko, Atatiire, Walton, Jaeger, Saleh and Teich [1999]).
beamsplitter is located with the quantum key sender (Alice), while one of the output beamsplitters is located at a distance with the receiver (Bob), as in fig. 22. The high-fi-equency carrier that resides under the HOM-type interference feature reflects the period of the UV pump wavelength rather than that of individual waves, and arises from the nonlocal entanglement of the twin beams. As shown in fig. 23a, a 90^^ phase shift of one of the analyzers modifies the quantum interference pattern so that the central fringe is constructive rather than destructive. The contrast is very high, 98%, as is evident firom fig. 23b. This demonstrates that cryptographic key qubits - one value corresponding to each of the two sorts of interference - can be sent with a high degree of fidelity using this
308
Multi-photon quantum interferometry
;K O
<
\sUt)^ ; \IUS)A |s)p.|s>.
f
[5, §3
|S>P.|/)B;|/)PJS>B
\Uf>A
\I)P.\I)B
LU UL
^ / Alice
Bob
Fig. 24. Time-energy entanglement quantum key distribution scheme (Tittel, Brendel, Zbinden and Gisin [2000]).
apparatus. Key distribution works as follows. The polarizations of the photons are randomly modulated by switching each analyzer-modulator in the rectilinear basis (45'' and SIS*'), providing O*' or 90'' relative phase shift between them. In order to fiilly complete the procedure of quantum key distribution, it would also be necessary to randomly switch the polarization parameters of the two-photon entangled state between two nonorthogonal polarization bases, such as rectilinear and circular polarization. This could be accomplished using fast Pockels-cell polarization rotators. These sets of randomly selected angles force the mutual measurements by Alice and Bob to be destructive (a binary "0") or constructive (a binary "1") with a 50-50 probability, depending on the mutual orientation of the modulators on both sides. Communications between Alice and Bob, which give the set of polarizer orientations selected during each measurement but not the measurement outcomes themselves, are then to be sent over a public classical communication channel. Other protocols may be devised to endow this configuration with the full security that has been added to other configurations. A second experiment uses a scheme that combines using photon pairs and energy-time entanglement (Tittel, Brendel, Zbinden and Gisin [2000]). This scheme realizes the initial concept of using photon-pair correlations (Ekert [1991]) for QKD (fig. 24). However, it implements Bell states, and the robustness of energy-time entanglement allows the information produced using this second method to be preserved over long distances. In this scheme, a light pulse sent at time to enters an initial interferometer imposing a large path length difference
5, § 3]
Two-photon interferometry with type-II SPDC
309
relative to the pulse length. The pulse is split in two, so that the subpulses leave time-separated but with a definite phase difference. The 655 nm, 80 MHz pulses entered a down-converting (KnbOa) crystal creating two-photons described by |t/<) = ^ ( | s ) p | s ) p + e'^|^)p|£)p),
(3.10)
where |s)p and \t)^ are photon states created by a pump photon that traveled via the short or the long arm of the initial fiber interferometer, respectively. The fiill set of Bell states are thus achievable by the appropriate choosing of and/or interchange of a short or long photon state for one of the photons. The photons were separated and sent, one to the Alice wing and one to the Bob wing. Each photon then traveled through another fiber interferometer, introducing the same path difference through one or the other arm as the initial, pre-crystal interferometer. In order to use this system for QKD, Alice and Bob are to carry out the following procedure. The photons arrive at Alice's detectors in one of three time slots relative to ^o- The time-of-arrival to Alice does not give a fijll description of the two-photon state, since the path of the photon traveling to Bob is unknown to Alice. To obtain the quantum key, Alice and Bob then use a classical channel to find those events where both detect a photon in a side peak, without revealing the detector. The other half of the events are discarded, leaving them with correlated detection times. Finally, they assign bit value "0" to the short cases and " 1 " to the long cases. When both find the photon in the central temporal position, by choosing appropriate phase settings, Alice and Bob will always find perfect correlations in the output ports. Either both detect the photons in their "+" detector (bit value "0"), or both in their " - " detector ("1"). Bit rates of roughly 33 bps and bit error rates of around 4% were achieved. A third recent QKD experiment (fig. 25) used |t/;~)-state two-photons created in BBO to approximate a single photon source (Jennewein, Simon, Weihs, Weinfiirter and Zeilinger [2000]). It implemented a novel key distribution scheme using the Wigner-version Bell-type inequality (Wigner [1970]) to test the security of the quantum channel, as well as a variant of the BB84 protocol (Bennett and Brassard [1984]). To use the Wigner inequality analogous to the CHSH inequality in the Ekert protocol, observer Alice chose between two polarization measurements along the axes x and w, corresponding to angles x ^^^ V^? with the possible results 1 and - 1 , on photon A; Bob chose between measurements along u and V, corresponding to angles i/> and (O, on photon B. When the polarization was parallel to the analyzer axis the result was 1; with polarization orthogonal
310
Multi-photon quantum interferometry
Source
Alice Detectors
-1
Electro Optic Modulator
Photdn>^
Clock
Photon a
Random Number Generator
-r-\-^... T
Bob Electro Optic Modulator
Optical Flber^
Polarizer. &Po\a^
[5,
Classical Communication
Detectors
«\ arizer^ [ Polarizer Random Number . ^ j | - - | R b | Generator | Clock
......... ^r^
Fig. 25. Realization of two-photon QKD over a long distance (Jennewein, Simon, Weihs, Weinfiirter and Zeilinger [2000]).
to the analyzer axis the result was - 1 . By assuming premeasurement values for properties along x, u and v and perfect anticorrelation of measurements along parallel axes, the probabilities for obtaining 1 on both sides obey the inequality P++( X, ^) -^P++(^, 0)) -/7++( X, (o) ^ 0.
(3.11)
The quantum-mechanical prediction for arbitrary analyzer settings a with Alice and P with Bob given the linear polarization singlet Bell state W' is QM
(a,l3)=\sm\a-P).
(3.12)
Maximum violation of the inequality is thus obtained for x ^ -30'', i/; = O"", w = 30'', when the l.h.s. reaches -1/8. To send the quantum key Alice and Bob randomly change their analyzer settings: Alice between -30'' and O'', Bob between 0" and 30". Four combinations of analyzer settings can thus occur: the three oblique settings allow a test of Wigner's inequality, the remaining combination of parallel settings allows key generation using perfect anticorrelations. When the probabilities violated Wigner's inequality, then the generated key was taken to be secured. The second QKD realization of this experiment implemented a variant of the BB84 protocol with entangled photons, with the same |V^")-state polarizationentangled photon pairs approximating the single-photon realization of BB84. Alice and Bob randomly changed their polarizer settings between 0" and 45°. They observed perfect anticorrelations whenever their analyzers were parallel. They obtained identical keys by simply inverting all resulting bit values. Whenever Alice made a measurement on photon A, photon B was projected into the orthogonal state that was analyzed by Bob, or vice versa. After the initial bit distribution, key security could be checked by classically comparing a small subset of their keys to check the security via the error rate.
5, § 4]
Higher multiple-photon entanglement
311
The nonlinear crystal used was again BBO, producing polarization-entangled photon pairs at a wavelength of 702 nm from cw pump light of 351 nm at a power of 350 mW. The photons were each coupled into 500 m long optical fibers and transmitted to "Alice" and "Bob", respectively, who were separated by 360 m. Wollaston polarizing beamsplitters were used as polarization analyzers. The users generated raw keys at rates of 400-800 bps, with bit error rates of approximately 3%.
§ 4. Higher multiple-photon entanglement The entanglement of three or more photons has been a subject of great interest since the proposals in 1989 and 1990 of Greenberger, Home, Zeilinger and Shimony to test locality, reality and completeness assumptions of EPR using entangled three-particle states (Greenberger, Home and Zeilinger [1989], Greenberger, Home, Shimony and Zeilinger [1990], Bernstein, Greenberger, Home and Zeilinger [1993], Klyshko [1993], Aravind [1997]). Bell's inequality had provided a test of these assumptions using statistical correlations, with the most striking results involving Bell states. The GHZ theorem provided a test involving perfect correlations without the use of inequalities, through the use of "GHZ states". The GHZ states can be written as | 0 ± ) = -^(|H)|H)|H)±|V)|V>|V)), |
V
(4.1)
|f^±) = - ^ ( | H ) | V ) | H ) ± | V ) | H ) | V ) ) , |'f'3^) = ^ ( | H ) | H ) | V ) ± | V ) | V > | H » , in the three-polarization case. Among other reasons, such states are interesting because, as in the Bell states (3.6), the polarization of each photon is indeterminate while the three particles are certainly perfectly correlated in polarization. A similar, beam-entangled set of states has also been introduced (Greenberger, Home, Shimony and Zeilinger [1990]). These states allow entanglement to be studied in a less trivial context than that of the traditional two particles. Interferometric studies subsequently sought to exhibit these correlations and to use them for various means.
312
Multi-photon quantum interferometty
[5, § 4
Fig. 26. Three-photon beam-entanglement source (Zeilinger, Home, Weinfiirter and Zukowski [1997]).
In 1997, Zeilinger, Home, Weinfiirter and Zukowski proposed a scheme for generating GHZ states using the concept of quantum erasure, following an earlier direction of investigation initiated by Yurke and Stoler [1992] that they developed further (Zukowski, Zeilinger, Home and Ekert [1993], Zukowski, Zeilinger and Weinfurter [1995], Zeilinger, Home, Weinfiirter and Zukowski [1997]). This approach allows one to achieve entanglement while avoiding the problematic need for particle interaction, as had previously been used for this end; its use was explicated for both polarization and beam entanglement. This scheme begins with two independent sources of two-photons, followed by the "erasing" of source information of one of the four photons at a beamsplitter (see, for example, figs. 26 and 28). This was first done using a pair of laser pulses. An illustration of this principle is shown in fig. 26 for the case involving beam entanglement. The states of the initial down-conversion pairs can be written -^(|a)|d)±|a')|c')),
-^(|d')|b')±|c)|b)).
(4.2)
After one of the four photons triggers a detector in the source, three-photon states arise for the remaining three particles, yielding the entangled state of beamdirection eigenvectors -^(|a)|b)|c)+e'^|a')|b')|c')).
(4.3)
The possibilities represented in this state then can be made to interfere in an apparatus such as that shown schematically in fig. 27, with triple-incidences, for example, at detectors DA, DB, DC, that vary sinusoidally in 0a + 0b + 0cIn general, when three-particle interference visibilities surpass 50%, a violation of a Bell locality can be demonstrated (Mermin [1990b]). A polarization
5, §4]
Higher multiple-photon entanglement
313
Fig. 27. Three-photon beam-entanglement interferometer (Zeilinger, Home, Weinfurter and Zukowski [1997]).
Fig. 28. Three-photon polarization-entanglement source (Pan and Zeilinger [1998]).
GHZ-state source analogous to that for beams in fig. 26 is shown in fig. 28. A GHZ-state analyzer (see fig. 29) can be constructed, just as a Bell-state analyzer (see fig. 16) can, in a manner that can also be extended to construct an w-particle entangled state analyzer. For three photons in the modes A, B and C, the scheme of fig. 29 will give rise to triple-incident detections when the photons are in GHZ states | ^ ^ ) . These two states can then be distinguished because, after the half-wave plates (HWPs), \0'^) results in one or three photons with polarization H and zero or two photons with polarization V, while \0') results in just the opposite situation. The first experimental proof of entanglement of more than two spatially separated particles was only recently produced (Bouwmeester, Pan, Daniell, Weinfiarter and Zeilinger [1999]). In this demonstration, the first two photon pairs
314
[5, § 4
Multi-photon quantum interferometry
PBSl
-J^
.-p'
PBS2
A"
Dv3
HWPM
HWPI
mode 1
mode 21
PBSA
ffl
mode BC
mode 3
PBS„,
I—fr-32" HWP
PBS3
Fig. 29. GHZ-state analyzer (Pan and Zeilinger [1998]).
t
°j
POL BS
Fig. 30. Three-photon polarization-entanglement source (Bouwmeester, Pan, Weinfiirter and Zeilinger [1998]).
were generated from a single PDC source (BBO) using a 394 nm pulsed laser pump (see fig. 30) to create the state -^(|H)a|V)b-|V)a|H)b).
(4.4)
Narrow-bandwidth filters made the coherence time of the photons (500 fs) more than twice as long as that of the initial UV pulse (200 fs). In arm a, a polarizing beamsplitter reflected only vertical polarizations, which were subsequently rotated 45"" by an HWP; in arm b, an ordinary 50-50 beamsplitter reflected both polarizations with equal likelihood. The two arms were arranged so as to meet at a polarizing beamsplitter from opposite
5, § 4]
Higher multiple-photon entanglement
315
sides. The events of interest were those in which two photon pairs were created simultaneously. The GHZ state in those cases, |^3^) = ^ ( | H ) | H ) | V ) + |V)|V)|H)),
(4.5)
was then post-selected by considering only events where four simultaneous detections were made at detectors Di, D2, D3 and the trigger detector T. Though the detection efficiency for such joint events was only 10"^^/pulse, the pulse rate was nearly 10^/s and the observed coincidence detection rate was nearly one detection every 2.5 minutes. The ratio of desired versus undesired detections (by polarization state) for the joint detections was 12:1. Coherent superposition, as opposed to an undesirable mixture, of the two desired states was verified by measuring the first photon in the 45'' polarization state and finding the second and third photons to be entangled by virtue of their polarizations being seen to be identically polarized along the 45° direction, as predicted for "entangled entanglement" (Krenn and Zeilinger [1996]). With the arrangement of fig. 30, only the triple coincidences (of particles 1, 2 and 3) predicted by quantum mechanics were observed and none of those predicted by local realism were found, within experimental uncertainties. Entangled three-particle states were created with a purity of 71%. An interference visibility of 75% was obtained. The optimal Bell-type inequality for three particles was derived by Mermin [1990a] to be I {^yy) + {y^y) + {yy^) - {xxx) \ ^ 2,
(4.6)
withx the outcome for measurements in the basis {\x±) = ^TjdH) ib |V))}, and j that for a measurement in {|y±) = 7f (1^^) =^ i|^))}- This inequality requires only a visibility >50% to be violated. With this arrangement, the l.h.s. of eq. (4.6) was found to reach 2.83±0.09 (Bouwmeester, Pan, Daniell, Weinfiirter and Zeilinger [2000]), in clear violation of local realism. Another recent experimental discovery was that two particles, each initially entangled with one other's partner particle, can be placed in an entangled state by making a Bell measurement, giving rise to "entanglement swapping" (Bennett, Brassard, Crepeau, Jozsa, Peres and Wootters [1993], Zukowski, Zeilinger, Home and Ekert [1993] and Bose, Vedral and Knight [1998]). In a recent experiment, a single nonlinear crystal was used as the source of a pair of two-photons to be used in entanglement swapping (Pan, Bouwmeester, Weinfijrter and Zeilinger [1998]). In that experiment, calling the photons from one source (I) 1 and 2, and those from the other source (II) 3 and 4, a Bell-state measurement was made of two
Multi-photon quantum interferometry
316
[5, §4
Beil MeaswemeiJt
^
Polarizing Beam Splitter
Fig. 31. Experimental arrangement for entanglement swapping (Bouwmeester, Pan, Weinfurter and Zeilinger [1998]).
photons from different sources, say photons 2 and 3 (see fig. 31). The result was that the state of the other two photons, 1 and 4, was projected onto an entangled state. This can be seen as follows. The initial state of the two photon pairs is \E) = 1 (|H)|V) - |V)|H)),2 (|H)|V) - |V)|H))34
(4.7)
^ 1 (m)l^2^3) + 1*^1-4)1^23) + I^H)I^2"3) + l^r4)l^2-3))2
Photons 2 and 3 are then measured, projecting their state onto one of the Bell states, |^±)^3 = ^ ( | H ) | V ) ± | V ) | H ) ) , 1 |0±)^3 = — ( | H ) | H ) ± | V ) | V ) ) .
(4.8)
As a resuh, the pair of photons 1 and 4 are in the state that was found in measurement, as can be seen from the expansion of l^"). Similarly, in the so-called quantum polarization teleportation process, a laser pump-pulse was used to provide the opportunity to create two pairs of photons: on the path from left to right the pulse creates an entangled pair, the so-called "ancillary" pair of photons 2 and 3 (see fig. 32). One of these photons is passed on to Alice and the other one to Bob, who receives the polarization state. On the return path from the mirror the pulse again creates a pair of photons (photons 1 and 4). The second photon of that pair (photon 4) is sent to a trigger detector, p, that is used to reject all events in which this second pair was not created. In the experiment the entangled photons, photons 2 and 3, were produced in the
5, §4]
Higher multiple-photon entanglement
317
ALICE
Classical Information "Coincidence"
initial state
K.
UV-pulse .l^'**€>
a EPR source
Analysis Fig. 32. Experimental arrangement for polarization teleportation (Bouwmeester, Pan, Weinflirter and Zeilinger [2000]).
antisymmetric state 1^2-^). Alice subjects both the photon to be teleported and her ancillary photon to a partial Bell-state measurement using a beamsplitter. Observation of a coincidence at the Bell-state analyzer detectors, f 1 and f2, then informs Alice that her two photons were projected into the antisymmetric state [^12). This then implies that Bob's photon is projected by Alice's Bell-state measurement onto the original state completing the process. Some work has also been done to assess the quality of quantum state teleportation (Bouwmeester, Pan, Mattle, Eibl, Weinfiirter and Zeilinger [1997]). In particular, it has been pointed out that teleportation fidelity and teleportation efficiency are distinct. The quality of a quantum teleportation procedure can be evaluated on the basis of three properties: (i) How well any arbitrary quantum state that it was designed to transfer can be teleported (fidelity of teleportation); (ii) How often it succeeds when given an input it was designed to teleport (efficiency of teleportation); (iii) How well it rejects a state it was not designed to teleport (cross-talk rejection efficiency). The aim of the above experiment was to teleport with high fidelity a quantum bit of information, i.e., a two-dimensional quantum state, given by the polarization state of a single photon. When the teleportation system does not output a single photon carrying the desired qubit it is similar to an absorption process in a
318
Multi-photon quantum interferometry
[5, § 4
communication channel; after renormalization of a two-dimensional state, the original state, the qubit, is obtained again without any influence on fidelity. In the above experiment, any incoming UV pump pulse had two chances to create pairs of photons. So those cases where only one pair resulted are rejected, since only those situations are accepted in which the trigger detector p fires together with both Bell-state analyzer detectors f 1 and f2. Similarly, any cases where more than two pairs are created is ignored in the experiment since the likelihood of creating one pair per pulse in the modes detected corresponds to less than one event per day. Further, a three-fold coincidence p-f l-f2 has only two explanations. First, teleportation of the initial qubit could be properly encoded in photon 1 as was demonstrated for the 5 polarizer settings H, V, +45°, -45° and R (circular). These bases involve quite different directions on the Poincare sphere, proving that teleportation works for arbitrary superpositions. Second, both photon pairs could be created by the pulse on its return trip; in that case no teleported photon arrives at Bob's station and teleportation does not happen, but Alice still records a coincidence count at her Bell-state detector. This leads to a high intrinsic cross-talk rejection efficiency. Nonetheless, only one of the four Bell states was identified, i.e. the protocol works in only one out of every four possible situations. However, this only reduces the efficiency of the scheme, not the fidelity of the teleported qubit. Another method of teleportation avoids the problem of performing a joint Bell measurement on two particles, following an initial proposal of Popescu [1995]. This is done (see fig. 33) by encoding the two quantum states to be measured by Alice on two degrees of freedom of one particle (Boschi, Branca, De Martini, Hardy and Popescu [1998]). The price that is paid for this ability is that the preparer must select a pure quantum state (here a polarization state), rather than an arbitrary state, and give it directly to the EPR photon of Alice. Both a linear polarization state and an elliptical polarization state were teleported using this method and a 200mWcw pump laser of 351 nm, with interference visibilities exceeding 80%. Another subject in fundamental quantum theory of multiple photons that can be investigated using SPDC is attempted quantum cloning. Ideally, a quantum cloning machine could be constructed that creates an arbitrary number of highfidelity copies of an arbitrary quantum state of a given quantum system. While it has long been known that such a device cannot be constructed as a matter of principle (Wootters and Zurek [1982], Dieks [1982]) - it would allow one to send signals faster than light (Herbert [1982]) - a device can be constructed that makes imperfect copies (Buzek and Hillery [1996, 1998], Bruss, DiVincenzo, Ekert, Macchiavello and Smolin [1998], Gisin and Massar [1997], Bruss, Ekert and
5, §4]
319
Higher multiple-photon entanglement
ALICE
EPR SOURCE
Fig. 33. Experimental arrangement for polarization teleportation (Boschi, Branca, De Martini, Hardy and Popescu [1998]).
Clones BS
Pump
.^
Pubxiel
V ''11 \ Anti-Ctenes
\>
Mirror
^1
Trigger
Fig. 34. Schematic for quantum cloning using down-conversion (Simon, Weihs and Zeilinger [2000]).
IMacchiavello [1998]). Recently, Simon et al. investigated the question of such universal cloning via parametric down-conversion (Simon, Weihs and Zeilinger [2000]). They considered type-II phase-matched parametric down-conversion with pulsed light input for polarization-entangled two-photon singlet-state output (see fig. 34). Utilizing quasi-collinear outputs and cloning one photon of an entangled pair, they entangled all three output photons. By considering cloning beginning with TV identical photons, i.e., the initial state \\^i) =
(4i) A^o'^^
|0), the portion of the output state containing a
fixed number of photons in mode 1 is proportional to M-N
.
EH)'( /=o
^
M-l N
1/2
\M-l)^,\l)^,\l)^,\M-N-l)
/H2 :
(4.9)
320
Multi-photon quantum interferometry
[5
which is the output of an optimal cloning machine for the initial state. These workers also investigated the practicality of creating such an apparatus. In the laboratory, pair-production probabilities of 0.004 were achieved using a 76 MHz pulse rate at a UV power of 0.3 W and a 1 mm BBO crystal (a situation designed for maximum overlap of photons from different down-conversion pairs). Assuming a realistic detection of 10%, this would allow for a twopair detection every few seconds. By changing to a 300 kHz laser system an improvement in detection rate of more than an order of magnitude could be expected.
References Aravind, P.K., 1997, Borromean entanglement of the GHZ state, in: Potentiality, Entanglement and Passion-at-a-Distance: Quantum-mechanical Studies for Abner Shimony, Vol. 2, eds R.S. Cohen et al. (Kluwer, Dordrecht) p. 53. Aspect, A., 1999, Bell's inequality test: more ideal than ever. Nature 398, 189. Aspect, A., J. Dalibard and G. Roger, 1982, Experimental test of Bell's inequalities using time-varying analyzers, Phys. Rev Lett. 49, 1804. Aspect, A., P. Grangier and G. Roger, 1981, Experimental tests of realistic local theories via Bell's theorem, Phys. Rev Lett. 47, 460. Bell, J.S., 1964, On the Einstein Podolsky Rosen paradox. Physics 1, 195. Bennett, C.H., and G. Brassard, 1984, Quantimi cryptography: public key-distribution and cointossing, in: Proc. IEEE Conf on Computers, Systems and Signal Processing, Bangalore, India (IEEE, New York), p. 175. Bennett, C.H., G. Brassard, C. Crepeau, R. Jozsa, A. Peres and W.K. Wootters, 1993, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett. 70, 1895. Berglund, A.J., 2000, Quantum coherence and control in one-photon and two-photon optical systems, LANL Archives quant-ph/0010001 v2 (3 October 2000). Bernstein, H.J., D.M. Greenberger, M.A. Home and A. Zeilinger, 1993, Bell's theorem without inequalities, Phys. Rev. A 47, 78. Bloembergen, N., 1982, Nonlinear optics and spectroscopy. Rev. Mod. Phys. 54, 685. Bohm, D., 1951, Quantum Theory (Prentice Hall, Englewood Chffs, NJ) p. 614. Bohm, D., and Y. Aharonov, 1957, Discussion of experimental proof for the paradox of Einstein, Rosen and Podolsky, Phys. Rev 108, 1070. Boschi, D., S. Branca, R De Martini, L. Hardy and S. Popescu, 1998, Experimental realization of teleporting an unknown pure quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev Lett. 80, 1121. Bose, S., V Vedral and PL. Knight, 1998, Multiparticle generalization of entanglement swapping, Phys. Rev A 57, 822. Bouwmeester, D., A.K. Ekert and A. Zeilinger, eds, 2000, The Physics of Quantum Information (Springer, Berlin). Bouwmeester, D., J.-W. Pan, M. Daniell, H. Weinfurter and A. Zeilinger, 1999, Observation of three-photon Greenberger-Home-Zeilinger entanglement, Phys. Rev. Lett. 82, 1345.
5]
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E. Wolf, Progress in Optics 42 © 2001 Elsevier Science B. V All rights reserved
Chapter 6
Transverse mode shaping and selection in laser resonators by
Ram Oron, Nir Davidson, Asher A. Friesem Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel
and
Erez Hasman Optical Engineering Laboratory, Faculty of Mechanical Engineering, Technion - Israel Institute of Technology, Haifa 32000, Israel
325
Contents
Page § 1. Introduction
327
§ 2.
328
Transverse modes
§ 3. Intra-cavity elements and resonator configurations
334
§ 4.
Properties of the laser output beams
366
§ 5.
Concluding remarks
383
Acknowledgements
383
References
383
326
§ 1. Introduction One of the important properties of a laser is the transverse field distribution of the beam emerging from the laser resonator. This field distribution determines the divergence of the emerging beam and how well this beam can be focused. Due to diffraction, the field distribution is generally not uniform, but is a combination of discrete transverse patterns. Each pattern of the field distribution is related to a specific mode or a combination of modes that propagates inside the laser resonator. The transverse shape of the field distribution is maintained along the propagation path inside the resonator and after emerging from the laser. Laser modes can be selected, controlled, and modified by inserting specially designed elements inside the resonator so as to obtain a desired laser output beam. These output beams from the laser could be further manipulated and shaped outside the resonator. The intra-cavity elements can tailor the field distribution of specific transverse modes, resulting in a field distribution which changes along the path inside the resonator, but returns to the original distribution after a round-trip of the resonator. Also, selection of specific transverse modes can be obtained by elements that introduce low losses to a specific desired mode, but high losses to other modes. In recent years, new fabrication technologies, most of which emerged from the semiconductor industry, enabled the realization and even mass production of intra-cavity elements with small feature size. This allowed the exploitation of new mode discrimination and mode shaping methods. In this article, the basics of laser transverse modes are reviewed, along with numerical methods to calculate them. Various mode shaping and mode selection techniques are presented in detail, along with experimental data. Also, the output beam properties, as well as applications of specially designed beams, are discussed. Reviews of laser modes can be found in textbooks (see for example Siegman [1986] and Hodgson and Weber [1997]). Section 2 describes transverse modes in stable and unstable resonators, as well as numerical and analytical methods to determine the field distributions of the transverse modes. Section 3 describes various methods to select specific transverse modes in laser resonators, along with techniques for fabricating the needed intra-cavity elements. The anal3^ical tools for describing the properties 327
328
Transverse mode shaping and selection
[6, § 2
of the laser output beams, along with selected applications are presented in § 4. Finally, § 5 presents some concluding remarks.
§ 2. Transverse modes Laser resonators are generally categorized as either stable or unstable. In a stable resonator, a ray launched inside the resonator parallel to the optical axis remains inside it, whereas in an unstable resonator, the ray may bounce off the resonator after a few round-trips. For example, a resonator of length L and mirror curvatures R\ and R2 is stable if 0 <{\ - LIR\){\ - LIR2) <\, and unstable otherwise. Another important property of the resonator is the Fresnel number, given by N^^c^l'kL, where a is the resonator radius, and A the operating wavelength. Hodgson and Weber [1997] introduced an equivalent Fresnel number, which is also dependent on the curvature of the mirrors. In such laser resonators, the transverse modes, each having a specific field distribution, should reproduce themselves after each round-trip. Such self-consistent field distributions could be determined by solving the round-trip wave propagation equation of the resonator. In this section, an introduction to transverse modes of both stable and unstable resonators is given, along with analytical methods of calculating them. 2.1. Transverse modes in laser resonators The round-trip wave-beam propagation equations were solved in either circular or rectangular symmetry to yield the Laguerre-Gaussian and the HermiteGaussian transverse modes respectively (see Kogelnik and Li [1966] and Siegman [1986]). For cylindrical coordinates, TEM^/ Laguerre-Gaussian modes are characterized by p radial nodes and / angular nodes, whereas in Cartesian coordinates, TEM^„ Hermite-Gaussian modes are characterized by m and n modes in the horizontal and vertical directions respectively. With circular symmetry, the field distribution E(r,d) of a nondegenerate Laguerre-Gaussian TEM^ / mode inside a laser resonator is expressed by £^,/(r, 0) = ^opl'"'ll/l(p) exp(-p/2) exp(i/0),
(1)
where r and Q are the cylindrical coordinates, £"0 the magnitude of the field, p = lr'^lw^ with w the spot size of the Gaussian beam (see Hodgson and Weber [1997] for a detailed discussion), and Lj, are the generalized Laguerre
6, § 2]
Transverse modes
329
polynomials of order p and index / (note that for nondegenerate modes, / may be positive or negative). Some specific values of the Laguerre polynomials are 4(p)=l,
L[{p) =
l+\-p,
L[{p) = \p^ - (/ + 2)p + \{l + 1)(/ + 2),
(2)
4(p) = (-iy5Wz^^/(p). Now, modes with the same radial index p and opposite angular index / have the same radial distribution (albeit with opposite helical phases). They are usually degenerate, and appear simultaneously, leading to a TEM^/ (degenerate Laguerre-Gaussian) mode with Epi{r, 6) = Eop'^%(p) exp(-p/2) cos(/0).
(3)
For clarity, a sign for the / value (e.g., TEMi, + 2) will indicate the nondegenerate modes, and the degenerate modes (where / is always nonnegative), will appear without a sign (e.g., TEM12). Note, in general, the intensity distribution of the TEM^, ±1 modes will have a circularly symmetric annular shape, whereas those of the TEMpi will have lobes, except for the fundamental, Gaussian-shaped TEMoo mode. Also, adjacent lobes will have opposite phases (Jt phase shift). The normalization factor for the TEM^,±/ mode is -t/ . ^l\i\)\^ whereas for the TEMpi mode it is J n^^ w( +/)p with SQI = 1 when / = 0 and (5o/ = 0 otherwise. Similarly, with rectangular symmetry, we obtain the TEM^„ HermiteGaussian modes, given by Emn(x,y)
= EoCXp[-(f
+ Xl)^)/2]Hm{mnm.
(4)
where x and y are the Cartesian coordinates, § = A/2X/>V, \l) = \/2ylw, normalization factor is ^/l/jt and the Hermite polynomials H are //o(x) = 1,
Hx{x) = 2x,
H3(x) = Sx^ - 12x,
H2{x) = 4x^ - 2,
H4(x) = lex"^ - 48x^ + 12.
Note, the Hermite-Gaussian (HG) modes with m,n^l Gaussian (LG) modes, namely, TEMSP^TEM^G,
TEMfo^ = TEM^G^),
the
TEM^i^ =
are also Laguerre-
TEM^G^),
TEM«G = TEM^f,
where the subscripts (x) and (y) denote the axis connecting the two lobes of the TEMoi mode, and correspond to cosine or sine functions in eq. (3), respectively.
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[6, § 2
Generally, in stable resonators, the number of modes existing in a resonator increases with the Fresnel number, as the width of the resonator increases and the losses of all modes decrease. The Fresnel number can be controlled either by changing the length of the resonator, or, most commonly, by inserting an aperture into the resonator and adjusting its radius. When the Fresnel number is small the laser operates with a single fundamental mode of Gaussian shape and lowest losses; when the Fresnel number is high, it operates with multi-transverse modes. Typically, a multimode laser operation results in a relatively poor beam quality. When a high-quality laser beam is required, one usually operates the laser with the fundamental mode. However, this concomitantly results in a significant decrease in output power, so there is a trade-off between output power and beam quality. The unstable resonators are divided into two types, namely negative- and positive-branch, depending on the product {\-LIR\){\-LIR2), which is either negative or greater than unity, respectively. Negative-branch unstable resonators exhibit a focal spot in the resonator, at which there is an undesired high intracavity intensity, so they are less common. As for stable resonators, the modes of unstable resonators are determined by solving the round-trip propagation equation. However, in contrast to stable resonators, the beam propagation can be readily approximated by geometrical optics. Indeed, unstable resonators can be characterized, in the geometrical limit, by two spherical waves that reproduce themselves after each round-trip (see Siegman [1974]). In such resonators the round-trip waves diverge, resulting in magnification M. This leads to a roundtrip mode loss of 1 -(1/M)^. A more precise determination of the mode losses and patterns requires solving the Kirchhoff integral directly. Siegman [1974] presented such solutions for various values of the Fresnel number, where he showed that the lowest loss mode depends on the Fresnel number. This is in contrast to stable resonators where the fiindamental Gaussian mode has the lowest losses for every Fresnel number. Moreover, the power loss per pass is lower than expected fi-om the geometrical approximation. This is attributed to diffraction effects that act to reshape the exact eigenmode in such a way as to reduce the losses below the geometrical value. Overall, unstable resonators with relatively large Fresnel numbers can operate with a single mode. Thus, they can have relatively high single mode output powers. 2.2. Methods of analysis and design In this subsection, we describe analytical methods to determine the modes of
6, § 2]
Transverse modes
331
given resonator and intra-cavity elements that are needed to control or shape specific mode patterns. We begin with the round-trip propagation equation, given by KUn = YnUn.
(7)
where the eigenvectors Un represent the field distribution of the resonator modes, the power loss per pass 1 - |7„p is obtained fi*om the eigenvalues y„ and K represents the round-trip Kirchhoff propagation kernel. The fi*ee-space propagation is represented by the Kirchhoff-Fresnel integral (see Hodgson and Weber [1997] for details), whereas other intra-cavity elements, such as mirrors, lenses and phase elements are represented by appropriate operators, from which the round-trip operator K is obtained. Solving eq. (7) is equivalent to the diagonalization of a matrix operator K. However, when the matrix K is large, there could be a practical problem of diagonalizing it directly. In the following, several methods for solving eq. (7), as well as tailoring resonators for specific desired modes (eigenvectors Z7„), are presented. 2.2.1. Fox-Li Probably the most commonly used method is the iterative Fox-Li method (Fox and Li [1962]). Here, one starts from an arbitrary field distribution (initial vector U = VQ) and propagates it back and forth in the resonator by applying repeatedly the round-trip propagation kernel K, to obtain a sequence of vectors Vm + \=KVm' If the initial vector VQ is represented by a linear combination of the eigenvectors, namely, Vo = J2n(^nUn, then consequently, the series of vectors Vm is given by K^ = ^ „ a „ y ^ r / „ . If the eigenvalues are ordered so that 7i ^ 72 ^ 73 ^ • • • ^ yiv, then for large m values, the eigenvector of the fundamental mode U\ will be dominant, namely, V^ ^ «i y^U\, and y\ could be obtained hy yi^Vm + i/Vm- Thus, after typically tens or hundreds of iterations, only the lowest loss fundamental mode is obtained. A lower number of iterations is needed if the initial vector VQ is close to Ui. However, for a multimode resonator, where 7i ~ 72 ~ 73 • •, a combination of all the modes, along with their common eigenvalue, is obtained. When the modes Un are orthogonal, the Fox-Li method can be extended to obtain the other modes, after finding the fundamental mode U\. This could be done by selecting a new initial vector VQ, which is orthogonal to U\ (the component along Ui could also be removed from other vectors in the series F^). In such a way, the second lowest-order mode is determined. This procedure is
332
Transverse mode shaping and selection
[6, §2
then repeated for higher-order modes. Another method for finding several lowestlosses modes is the Prony method, described in detail by Siegman and Miller [1970]. 2.2.2. Gerchberg-Saxton The Gerchberg-Saxton method involves an iterative algorithm (Gerchberg and Saxton [1972]), with which one can calculate the field distribution at one location in the resonator based on the intensity or phase constraints in two locations. For example, for a given intensity distribution at the output and a certain phase constraint at the back mirror, one can determine the field distribution at any location in the resonator. This method is particularly usefiil for designing mode shaping elements that are inserted into the resonator (see for example Makki and Leger [1999]). A schematic diagram of the algorithm in a laser resonator configuration is presented in fig. 1. One starts with a field having a desired
Phase constraint ^ c
llntensity Ij I phase
Propagation
^if^ > llntensity / j phase (jp^r. Propagation Back mirror
phase (Pi
Desired intensity ^
llntensity /^
Output coupler
Fig. 1. Laser resonator configuration including the Gerchberg-Saxton algorithm with a desired intensity / ^ at the output coupler and a phase constraint cp2C ^^ ^^^ ^^^^ mirror.
intensity distribution IQ and random phase at the output coupler. This field propagates to the back mirror, to obtain v^exp(i(p2)- Then, a phase constraint is imposed so that the reflected field is y/h^^piWic)The phase constraint may directly depend on the incoming wave, such as for phase conjugation (see §§3.1 and 3.2), or perform a transformation from q)2 to a uniform or spherical phase. This new reflected field distribution propagates back to the output coupler, resulting in y/T^exp(iq)i), to which the intensity constraint ID is applied again. This procedure is repeated, typically hundreds of times, until I\ converges to ID2.2.3. Propagation-matrix diagonalization The propagation-matrix diagonalization method is based on directly solving the round-trip propagation equation (see eq. 7). This is done by first representing each of the intra-cavity elements (such as mirrors, apertures.
6, § 2]
Transverse modes
333
lenses and free-space propagation) with an appropriate matrix, then multiplying all matrices in the appropriate order to obtain the round-trip matrix. For example, for a simple resonator configuration with only two mirrors, M(round-trip) = M(free-space) * ^ ( m i r r o r l ) * ^(free-space) * M(mirror2). T h e n , d i a g O U a l i z e t h e
matrix M(round-trip) to find the eigenvectors U, which represent the modes, and the eigenvalues y, and thereby the round-trip losses. Unlike the Fox-Li method, the propagation-matrix diagonalization method provides more detailed information on the number and shape of the individual modes in the resonator, and allows for a better physical insight. However, the propagation-matrix diagonalization method requires lengthy computations for the diagonalization, since the round-trip matrix is typically large. Some optimization of the computation is possible. For example, Sanderson and Streifer [1969] suggested an optimal procedure, which is based on a combination of two diagonalization techniques, yielding round-trip matrices of smaller size. Also, they compared between different methods of mode calculation, mainly, the FoxLi method and several matrix diagonalization methods. Abrams and Chester [1974] optimized the diagonalization method for waveguide lasers. In both cases, the optimization is mainly based on the appropriate selection of a basis for describing the modes. Oron, Danziger, Davidson, Friesem and Hasman [1999b] reduced the complexity of the analysis by exploiting the symmetry of the resonators. They selected a more general basis and considered a radially symmetric laser operating with Laguerre-Gaussian modes, so the analysis was reduced to one-dimensional (similar simplification was performed for Hermite-Gaussian modes by treating each of the Cartesian coordinates separately). Here, the free-space propagation was based on the Bessel-Fourier transformation, and could include various azimuthal field distributions, depending on the azimuthal index / (see also Ehrlichmann, Habich and Plum [1993]) to yield Ui{r2.L) = i^^^kL-^Qxp(-ikL) X f Ui(ruO)Ji(krir2/L)Qxp[-ik(rl + rly(2L)]n dn,
^^^
where k = 2jt/X. For every value of /, the integral in eq. (8) could also be represented by a matrix operator M/(free-space), namely, f//(r,I)=M/(free-space)'^/(^,0). In a similar manner, each of the intra-cavity elements, such as mirrors, lenses or phase elements, can be represented by an appropriate matrix M. Some representative results for a radially symmetric resonator are shown in fig. 2. Figures 2a-c show three sets of intensity distribution for the lowest order modes with azimuthal indices of / = 0, /= 1 or / = 2. Figure 2d shows the loss per pass
334
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Transverse mode shaping and selection
20
00^
0
0.4
0.8
1.2
Fresnel number Np (d)
Fig. 2. Intensity distributions cross sections in the radial direction for Laguerre-Gaussian modes and their loss per pass as a function of N^, (a) 1 = 0, (b) /= 1, (c) 1 = 2, and (d) calculated mode losses as a fimction of N^.
as a function of the Fresnel number N^ for these modes, which enables the prediction of the transverse mode content for a given Fresnel number and laser gain.
§ 3. Intra-cavity elements and resonator configurations Intra-cavity elements can be incorporated into laser resonators in order to shape specific transverse mode patterns and to discriminate and select a specific single mode out of the many modes that exist in the resonator. Most mode-shaping methods modify the transverse field distribution of the lowest-order mode, from a Gaussian into another desired distribution such as a super-Gaussian or a nearly flat-top. Since the modified field distribution is no longer a mode of free space (that does not change as it propagates), the desired transverse field distribution is obtained only at a certain location in the resonator (for example, at the output coupler), whereas in other locations, the field distribution is different. Most mode selection methods involve discrimination and selection of a specific "natural" (Laguerre-Gaussian or Hermite-Gaussian) single mode by introducing relatively high losses to all modes except the specific desired mode. While doing so, the desired mode will maintain the same transverse field distribution at all locations within the resonator. The simplest mode selection method involves the insertion of a circular aperture into the resonator in order to obtain the ftindamental Gaussian mode. Other methods discriminate and select a specific high-order mode, introducing losses to all other modes including the
6, §3]
335
Intra-cavity elements and resonator configurations
fundamental Gaussian mode. An early example of such a method was presented by Rigrod [1963], who isolated Laguerre-Gaussian modes in a laser resonator, by inserting wires into the resonator and adjusting the intra-cavity aperture diameter. In this section, both mode shaping and mode selection techniques are described along with experimental results. These are based on various types of intra-cavity elements, including specially designed mirrors, diffractive elements, phase elements, and polarizing elements, that can be incorporated into both stable and unstable resonator configurations, to produce controlled output beams.
3.1. Graded phase mirrors Graded phase mirrors (GPMs) have a nonspherical phase profile. The incorporation of such mirrors into laser resonators was proposed and analyzed by Belanger and Pare [1991] (and Pare and Belanger [1992]) in order to obtain a shaped (non-Gaussian) fiindamental mode, with a predetermined field distribution at the output coupler. A resonator configuration with a GPM is presented in fig. 3. The output coupler is a plane mirror, whereas the back mirror is a GPM whose phase profile deviates from that of a spherical mirror of radius ro = 1 by a phase shift of A()9. By controlling this phase shift, one can shape the field distribution of the mode. In order to determine the phase profile of the GPM, one begins with a desired intensity distribution at the output coupler (the phase distribution at this point should be uniform). Then, this desired field propagates a distance of L/2, and the resulting phase distribution is found using the Kirchhoff-Fresnel diffraction integral. The phase profile of the GPM is then set as the conjugate of this phase, namely, a passive phase conjugate mirror for the desired mode. Aperture 1 Output I coupler
Aperture 2 GPM
i
Gain medium
Output beam
C>
K(p -LI2
Fig. 3. Resonator configuration with a graded phase mirror (GPM). A(j9 is the phase difference between the GPM profile and a spherical mirror (dashed line) of radius r^^L. (From Pare and Belanger [1992].)
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Transverse mode shaping and selection
[6, §3
1001
2 4 6 Fresnel number Np2
Fig. 4. Loss per pass (1 - | y p ) as a function of the Fresnel number (A/^F2) ^ ^ the first four modes. Sohd Unes depict a 6th-order super-Gaussian resonator with a GPM; dashed Hnes depict a reference Gaussian resonator. (From Pare and Belanger [1992].)
In order to obtain significant mode discrimination between the shaped fundamental mode and the higher order modes, the length of the laser resonator, 1/2, should equal approximately the Rayleigh distance, namely JZMP-IX. With shorter resonators, the phase distribution at the back mirror is flat in the center and steep near the edges, leading to a complex GPM design and high alignment sensitivity. On the other hand, with longer resonators, the phase distribution is nearly spherical, so the phase shift A()9 is very small, leading to decreased shaping ability. A desired output intensity distribution for a GPM resonator could be a superGaussian of order «, given by U{r) = exp[-(r/wr].
(9)
A high-order super-Gaussian mode has a nearly flat-top intensity distribution, which better fills the gain medium than a Gaussian mode, leading to higher output powers. Moreover, a laser operating with a super-Gaussian mode has a low loss for the fundamental mode and high losses for the higher order modes. These losses were calculated using the Prony method (see § 2.2), and the results for Gaussian and super-Gaussian modes are shown in fig. 4. As evident, with the GPM resonator, even for relatively high Fresnel numbers at aperture 2, there is significant mode discrimination between the fundamental and high-order modes, while maintaining low losses for the fundamental mode. Belanger, Lachance and Pare [1992] and Van Neste, Pare, Lachance and Belanger [1994] performed experiments with a CO2 laser in which GPMs were inserted in order to select super-Gaussian modes of orders 4 and 6, and compared the results to reference Gaussian modes. Figure 5 shows representative results of the laser output power as a function of aperture diameter, for a 6th-order superGaussian. Evidently, the output power first increases with aperture diameter
6, §3]
Intra-cavity elements and resonator configurations
337
16 J
6
8
10
12
Aperture diameter [mm]
Fig. 5. Output power as a function of aperture diameter for a 6th-order super-Gaussian resonator with a GPM. (From Van Neste, Pare, Lachance and Belanger [1994],)
to reach a plateau of approximately 5.1 W for a single-mode operation at an aperture diameter of 8 to 11 mm. The corresponding Gaussian output power was only 4.3 W, indicating a power increase of approximately 20% with the super-Gaussian. For larger aperture diameters, a multimode operation with higher output powers is obtained. Angelow, Laeri and Tschudi [1996] showed that by applying a simulatedannealing optimization algorithm, it is possible to design GPMs with higher modal selectivity than obtained with the phase conjugation approach. However, this concomitantly implies somewhat higher fundamental mode losses. Paakkonen and Turunen [1998] designed and analyzed resonator configurations with graded phase (aspheric) mirrors, which operate with Bessel-Gauss modes, and compared their mode losses with those of other Bessel-Gauss resonators.
3.2. Diffractive elements Diffractive optical elements can transform one wavefi*ont into another, so they are potentially useful for mode shaping. Leger, Chen and Wang [1994] replaced the back mirror in a laser resonator with a diffractive mirror in order to shape the mode intensity distribution. This mirror, designed in a similar manner to passive phase conjugate GPMs, is nearly flat and has a maximal phase variation of In, so the phase profile includes many In steps. The diffractive mirror was designed to form a square 20th-order super-Gaussian mode, and was incorporated into an Nd:YAG laser resonator. The experimental output profile results are shown in fig. 6. A nearly flat-top square was obtained, as expected for a super-Gaussian profile.
338
Transverse mode shaping and selection
43
-0.8
[6, §3
-^M
-eJS
0
&a
0L4
<S.6
OS
Distance [mm] Fig. 6. Measured output intensity profile fi-om an Nd:YAG laser with a dififractive mirror. (From Leger, Chen and Wang [1994].) Mode selecting
Aperture? ^^^^^..^^
Aperture 1 O " * " ' • coupler
element Gain medium
|_^ Output beam
:>
r Fig. 7. Resonator configuration with a flat output coupler, a diffractive back mirror, and an additional intra-cavity diffractive element. (From Leger, Chen and Dai [1994].)
As for the laser resonator with a GPM, the length of the resonator with a diffractive mirror should be comparable to the Rayleigh distance. Thus, relatively long resonators are required. Leger, Chen and Dai [1994] proposed and demonstrated the insertion of an additional intra-cavity diffractive grating, which allows for high modal discrimination, yet with shorter resonator lengths. A typical resonator configuration is shown in fig. 7. The output mirror is simply flat, the back mirror is a diffractive mirror, and the internal diffractive phase grating is placed approximately in the middle of the resonator. The diameters of two apertures in the resonator are so chosen that negligible loss is introduced to the fundamental mode, but high losses to other modes. Typically, the internal diffractive element is a sinusoidal phase grating of the form exp[iwsin(2:/r/gx], where m is the modulation index and/g is the spatial fi-equency of the grating. Figure 8 shows the calculated modal threshold gain (given by l/|y|^) for the second-order mode in a specific resonator geometry, whereas for the fiandamental mode it is nearly unity; for higher-order modes the threshold gain would of course be higher, so it need not be considered. As evident, the modal
6, §3]
Intra-cavity elements and resonator configurations
0
10
20
339
30
Grating frequency [1/mm]
Fig. 8. Calculated modal threshold gain for the second order mode, for a laser with an internal diflfractive siriusoidal grating. (From Leger, Chen and Dai [1994].)
threshold gain, which indicates the modal discrimination, is low for very low and very high grating frequencies, but reaches a maximum of approximately 2.5 at a grating frequency/g ?^ 6 mm"^. This could be understood by considering the field at the diffractive mirror. This field consists of a multiplicity of near-field patterns resulting from several different orders of the internal grating that are separated by Az2/g. For very low grating frequencies (typically/g < 3 mm~^), the patterns greatly overlap, leading to relatively low mode discrimination. For very high grating frequencies (typically/g > 10 mm"^), there is little, if any, overlap, leading to relatively low mode overlap between the diffraction patterns, again leading to very low mode discrimination. For moderate grating frequencies (typically 3mm~^
340
Transverse mode shaping and selection
Distance [mm]
Distance [mm]
(a)
(b)
[6, §3
Frequency [mm-^] (c)
Fig. 9. Intensity distribution and cross sections for a laser operating with a wire to select the second order mode (denoted TEMQI): (a) experimental near-field intensity distribution; (b) calculated nearfield cross section; (c) experimental far-field cross section. (From Chen, Wang and Leger [1995].)
1.0
1.2
1.4
1.6
1.8
2.0
Aperture size [mm]
Fig. 10. Measured modal threshold gain of the ftmdamental (denoted TEMoo) and two higher-order modes (denoted TEMQI and TEMn), as a function of output aperture diameter. Note, these modes are not the Hermite-Gaussian modes, but specific to the super-Gaussian resonator. (From Chen, Wang and Leger [1995].)
the two higher-order modes as flinctions of the aperture diameter. Evidently, high modal discrimination can be obtained. Leger, Chen and Mowry [1995] also analyzed a pseudorandom phase plate internal diffractive element. The pseudorandom phase plate was designed with the Gerchberg-Saxton method to have a high bandwidth, so as to obtain higher mode discrimination than with the sinusoidal phase grating. Such pseudorandom phase plates are generally difficult to realize and more sensitive to misalignment. Napartovich, Elkin, Troschieva, Vysotsky and Leger [1999] suggested an internal diffi-active element having the form of a simple phase step. Calculations showed that high modal discrimination with low fundamental-mode losses could be maintained. Lin and Wang [2000] analyzed laser resonators in which one of the mirrors was flat in the center, and had sinusoidal phase grating at the edges. By properly choosing the size of the central region and the spatial frequency of
6, § 3]
Intra-cavity elements and resonator configurations
(a)
341
(b)
Fig. 11. Near- and far-field experimental results for a laser designed to produce a "X" intensity distribution at the far field. (From Zeitner, Wyrowski and Zellmer [2000].)
the phase grating, a nearly flat-top output beam with uniform phase could be obtained. Zeitner, Wyrowski and Zellmer [2000] investigated a different approach in order to obtain a desired laser output intensity distribution at the far field. They replaced the back mirror of the laser resonator with a diffractive mirror, in order to obtain an intensity distribution which is the Fourier transformation of the desired distribution. In order to introduce the appropriate phase to the emerging output beam (which has a uniform phase), another external phase element was placed adjacent to the output coupler. The far-field pattern was then simply obtained by a lens. The experimental results for the near and far fields are shown in fig. 11. A A^-shaped far-field pattern is obtained, albeit with some blurring effects attributed to inhomogeneous pumping and nonlinear effects.
3.3. Binary phase elements Phase elements can change the phase distribution of an incident beam, just as diffractive elements can. The feature sizes in phase elements are significantly larger than those in diffractive elements, thereby alleviating fabrication concerns. In this subsection, mode-selecting binary phase elements, which consist of only two phase levels, are described along with experimental results in different resonator configurations. The simplest method to selectively attenuate certain regions of the intensity distribution of a mode and thereby discriminate it from other modes is to insert some absorbing elements such as wire grids into the laser resonator. Such elements introduce losses by absorption, and they heat up, so they are relatively inefficient. Better efficiency can be obtained with non-absorbing phase elements that introduce the desired losses by diffraction and interference. Hence, we consider how binary phase elements can be exploited to introduce losses to
342
[6, §3
Transverse mode shaping and selection Aperture
OutP"* coupler DPE,
Back mirror
Output beam
Gain medium
^
^ dm Fig. 12. Laser resonator configuration with a DPE inserted next to the output coupler.
specific modes, to discriminate between many modes, and to select one that will exist in the laser resonator. Kol'chenko, Nikitenko and Troitskii [1980] replaced the absorbing wires by phase-shifting masks in order to select high-order transverse modes. The phaseshifting masks were designed to have a nearly n phase shift along narrow lines instead of wires, while no phase shift was introduced to other areas. These phase masks were ft)und to introduce relatively low losses to the desired mode, while introducing high losses to other modes. Moreover, the mode discrimination was higher compared to wires. Different and more general binary phase elements, useful for selecting only one desired high-order transverse mode, were developed by Oron, Danziger, Davidson, Friesem and Hasman [1999a]. They inserted binary discontinuous phase elements (DPEs), designed to match the phase distribution and selectively reverse the phases of the desired mode. Because DPEs have a specific phase distribution with sharp phase changes, the insertion of DPEs into the laser resonator results in minimal losses for a desired transverse mode but high losses to others. Typically, the DPE is inserted near one of the resonator mirrors, preferably near the output coupler, as shown in fig. 12. The DPE is designed to ensure that discontinuous phase changes of either 0 or JT occur at the interfaces between adjacent parts of a desired mode distribution, where the intensity is very low. Specifically, the DPE designed to select the azimuthal index / introduces an angular-dependent phase shift, of the form 0
3jr/2 + Ijim > IB > K/1 + Inm, 5K/2 + 2jTm > 16 > 3jt/2 + Ijzm.
m integer.
(10)
Note that for every (positive) value of /, a singular point appears in the origin of the DPE, which corresponds to the zero intensity in the origin for a TEMpi mode with / ^ 0. DPEs could also be designed to select the radial index/?. Some
6, §3]
Intra-cavity elements and resonator configurations
343
Fig. 13. Representative DPEs: (a) designed to select TEMQI; (b) designed to select TEM02; (c) designed to select TEM20.
representative DPEs, for selecting the TEMQI, TEM02 or TEM20 modes, are shown in fig. 13. As a result of passing through the DPE, adjacent spots and adjacent rings of the field distribution in the desired mode, which normally have opposite phases (jr phase shift), will now have the same phase. The output mirror then reflects the modified mode distribution so it passes once more through the DPE, and all parts of the mode field distribution revert back to their original values. This is due to the fact that the total phase change introduced by the DPE is 0 or In. If the distance d between the DPE and the mirror is sufficiently short compared to the resonator length L (i.e., d<^L), then the overall distribution of the desired mode does not change as it passes through the DPE twice in a round-trip. Specifically, the phase change introduced by the first passage is canceled by the return passage. However, all other modes, whose intensity distributions are different, suffer a sharp phase change at locations where their intensity is typically strong. Since ( i > A , this leads to a strong divergence, where the phase change introduced by the first passage through the DPE is no longer canceled by the phase change introduced by the return passage. As a result, all modes except the desired mode suffer a loss, and most, if not all, are suppressed. Some modes, which are higher harmonics of the desired mode, may be unaffected by the DPE, but could easily be suppressed by a simple aperture. Finally, by placing the DPE next to the output coupler (as in fig. 12), all parts of the desired mode distribution are in phase, so that the far field of the output beam intensity has a high central peak, with some side-lobes. Several theoretical and experimental representative results are presented in figs. 14-17. Figure 14 shows results with an Nd:YAG laser in which a DPE designed to select the TEM02 mode (such as shown in fig. 13b) was inserted next
Transverse mode shaping and selection
344
mm mm
A ilk
(a)
(b)
[6, §3
mm
Fig. 14. Theoretical and experimental intensity distributions that emerge from an Nd:YAG laser in which a DPE was incorporated to obtain the TEM02 mode: (a) theoretical near field; (b) experimental near field; (c) theoretical far field; (d) experimental far field; and (e) theoretical (solid line) and experimental (dashed) cross sections along the center of the far fields. (From Oron, Danziger, Davidson, Friesem and Hasman [1999a].)
to the output coupler. The DPEs were formed by means of photoHthographic and etching processes on fused silica, and antireflection (AR) coated. The length of the resonator was 60 cm. Figure 14a shows the theoretical near-field intensity distribution as calculated by eq. (3). The corresponding experimental nearfield intensity distribution detected with a CCD camera is shown in fig. 14b. Both show the expected four lobes. Figure 14c shows the theoretical far-field intensity distribution obtained after applying a Fourier transformation acting on the absolute value of the near-field distribution (i.e., all four lobes are in phase). Figure 14d shows the corresponding experimental far-field intensity distribution. Finally, fig. 14e shows the corresponding central cross-sections of the far-field intensity distributions. As evident, there is a strong central peak, indicating that all four lobes of the near-field distribution are in phase. Using the DPE to make the laser operate with the TEM02 mode, the output power was 3.5 W, with an internal aperture set at a = 1.1 mm (i.e., N^ = 1.9). This was higher than the output power fi-om the laser operating with the single fundamental mode (TEMoo), which was obtained by reducing the initial aperture to a = 0.7 mm (i.e., 7VF = 0.77). In a similar manner, the Laguerre-Gaussian TEM03 and TEM04 modes were selected. The near- and far-field intensity distributions fi-om a laser operating with these modes are shown in fig. 15. Here again, we note the many lobes in the near field and a high central peak in the far field. Single high-order modes
6, §3]
Intra-cavity elements and resonator configurations
345
Fig. 15. Experimental intensity distributions that emerge from an Nd:YAG laser: (a) TEMQS near field; (b) TEMQS far field; (c) TEM04 near field; (d) TEM04 far field.
•'Wf!ifS:'''^Kt"
(b)
Fig. 16. Experimental intensity distributions that emerge from a pulsed (q-switched) Nd:YAG laser in which a DPE was incorporated to obtain the TEM02 niode: (a) near field; (b) far field.
were also obtained with a lamp-pumped NdiYAG laser which was pulsed by applying electro-optical g-switching. Results for the TEM02 mode operation, with an output energy of 15mJ per pulse, are shown in fig. 16. In comparison with the fundamental Gaussian mode operation, the output energy was less than lOmJ per pulse. Figure 17 shows results fi*om a CO2 laser with a DPE inserted to select the TEMoi mode (such as shown in fig. 13a). Figure 17a shows the theoretical near-field intensity distribution, which was calculated by eq. (3). Figure 17b shows the corresponding experimental intensity distributions obtained with a pyroelectric camera. Both results show the expected two lobes. Figures 17c,d show the theoretical and experimental far-field intensity distributions, with a high central peak and two low side-lobes. Finally, fig. 17e shows the corresponding central cross-sections of the far-field intensity distributions. Inserting an internal aperture set at a = 4mm (i.e., A^F = 2.5), an output power of 3.7 W was obtained, with 80% of the power concentrated in the central lobe. This was higher by more
346
Transverse mode shaping and selection
[6, § 3
Fig. 17. Theoretical and experimental intensity distributions that emerge from a CO2 laser in which a DPE was incorporated to obtain the TEMQI mode: (a) theoretical near field; (b) experimental near field; (c) theoretical far field; (d) experimental farfield; and (e) theoretical (solid line) and experimental (dashed) cross sections along the center of the far fields. (From Oron, Danziger, Davidson, Friesem and Hasman [1999a].)
than 50% than the output power of the single fundamental mode (TEMQO), which was obtained by reducing the aperture to a = 3.4mm (i.e., A^F = 1-8). The performance of the DPEs was compared to that of absorbing wires of various thicknesses. The maximal single-mode TEMQI power of the CO2 laser, obtained for a wire diameter of 75 |im, was 20% lower than with the DPE, confirming the advantage of the phase elements.
3.4. Spiral phase elements Spiral phase elements (SPEs) introduce a phase shift of exp(i7V0), where 6 is the azimuthal angle, to the beam passing through them. Representative examples of SPEs are shown in fig. 18. Figure 18a shows an SPE with N=\, and
Fig. 18. Spiral phase elements (SPEs) with (a) A^=l and (b) 7V = 2. The height discontinuities represent In phase shifi:s.
6, § 3]
Intra-cavity elements and resonator configurations
347
fig. 18b shows an SPE with A^ = 2, where N represents the number of In phase discontinuities. The height discontinuities needed to obtain a Ijt phase shift per passage {An phase shift in reflection) are A for a reflective element and X/{n - 1) for a transmittive element {n is the refractive index). As evident, phase singularities appear in the origin of the SPEs. Beijersbergen, Coerwinkel, Kristiansen and Woerdman [1994] exploited SPEs in order to transform Gaussian beams into helical beams outside the laser resonator. This transformation generally results in a combination of helical beams of different helicity. For example, a Gaussian beam incident on an SPE with N=\ results in 78% of the power in a TEMo,+i beam, and the rest of the power in other high-order helical beams. The degradation in efficiency is attributed to different intensity distributions of the original and transformed beams, so in general, the efficiency is higher as the intensity distribution of the original and transformed beams are closer, and it could reach 100% for beams having the same intensity distribution (such as the TEMo,+i and TEMo,-i). A helical mode can be formed inside a laser resonator, so a helical beam would emerge directly from the laser. Harris, Hill, Tapster and Vaughan [1994] exploited a helical laser mode operation based on coherent intra-cavity coupling and summation of two nonhelical modes. Sherstobitov and Rodionov [2000] proposed to apply SPEs in order to select helical modes in unstable resonators. In this case the SPE replaces the output coupler mirror, and a curved roof reflector serves as the back mirror. Calculations predicted that the emerging beam is indeed nearly helical. Oron, Danziger, Davidson, Friesem and Hasman [1999b] applied SPEs inside laser resonators in order to discriminate and select high-order helical LaguerreGaussian modes. In this approach, based on azimuthal mode discrimination, the SPEs are essentially lossless for the desired high-order mode, but introduce high losses to all other modes, especially to the fundamental Gaussian mode. The mode discrimination removes the degeneracy and separates helical modes with opposite / (angular momentum). The SPEs, in essence, change the phase of a wavefront passing through them, in accordance to either exp(+L/V0) or exp(-i7V0). Three laser resonator configurations with SPEs are shown schematically in fig. 19. Figure 19a shows the basic laser resonator configuration with the two SPEs adjacent to the resonator mirrors. Alternatively, reflective SPEs can replace the mirrors. The first SPE changes the angular mode index of the mode passing through it twice by -IN, i.e., the mode of angular index / changes to I-IN and the second SPE changes it back to /. With these SPEs the modes with an angular index of l = -\-N wiU be changed to those with l = -N. As a result.
348
Transverse mode shaping and selection
[6, §3
Reflective SPE
/-27V Output coupler
(Back mirror)
Cylindrical sJens
Output coupler
l-N i-l+M) Back mirror
Reflective SPE
/
Cylindrical lens^
Cylindrical lens^
-1+2N (0|
(-/+2A0
1-2. (Back mirror)
Output coupler (c)
Fig. 19. Laser resonator configurations with SPEs: (a) a configuration with two SPEs, each placed adjacent to a laser mirror; (b) a configuration with a single SPE, placed next to the output coupler, and a cylindrical lens (which reverses the angular phase) focused on the back mirror; (c) a configuration with an added external cylindrical lens, to form helical beams. The angular indices are indicated along the round-trip (second trip in parentheses).
since modes of opposite / have the same intensity distribution, only these modes will maintain the same distribution before and after passing through the SPEs. However, the radial distribution for the other modes will have the form of a combination of Laguerre-Gaussian modes, each with a different angular index /, so they will be wider, and the losses for all modes having I ^N will be higher. Thus, in accordance to the design of the SPEs, it is possible to select a specific mode that will propagate inside the laser resonator. Moreover, it is possible to separate two TEM^, ±i modes with opposite angular indices, so as to lead to a pure TEM^, +/ distribution. Specifically, the mode will be linearly polarized with a field distribution of the form given by eq. (1), i.e., doughnut-shaped, rather
6, § 3]
Intra-cavity elements and resonator configurations
349
than that of TEM^/ of the form in eq. (3), i.e., a distribution with distinct and separate lobes. Figure 19b shows a laser resonator configuration in which one of the SPEs is replaced by an element that reverses the angular phase (such as a cylindrical lens focused on the resonator mirror, or a Porro prism). In this configuration, each passage through the SPE changes the angular index of the mode passing through it by -N, and the angular phase-reversing element simply reverses the sign of the angular index. Here again, the Laguerre-Gaussian radial distribution is maintained after each round-trip only for those modes having an angular index l=N. For other modes, however, the radial distribution is wider, resulting in significantly higher losses. As evident from figs. 19a,b, the emerging laser beam passes through the SPE only once. As a result, the phase distribution of the output beam is constant. Thus, it converges to a relatively small single lobe in the far field, with no need for any external element. This concomitantly leads to an improvement in M^, as will be discussed in sect. 4. Alternatively, by replacing the back mirror with an output coupler and adding an external cylindrical lens (fig. 19c), it is possible to obtain the internal mode pattern outside the laser. Such configuration was applied by Oron, Davidson, Friesem and Hasman [2000b] to form pure helical beams. This could also be obtained by inserting a beam splitter inside the resonator so it will serve as an output coupler for getting the internal beam out of the resonator. Here, the back mirror is replaced with a reflective SPE, which changes the phase of the wavefront upon reflection by exp(+2iA^0) for a desired helical mode with l=N. Thus, a helical mode with phase exp(-i/0) is converted into exp(+i/0) after reflection by the SPE. The cylindrical lens, which is located inside the resonator and focused on the output coupler, inverts the helicity of the mode back to exp(-i/0) after a round-trip, to ensure self-consistency of the desired helical mode. The beam emerging fi-om the resonator is collimated by another external cylindrical lens, so its distribution will have the same form as the intra-cavity helical mode pattern. Note that the helicity of the SPE determines the helicity of the helical mode in the laser resonator. Consequently, by designing the SPE with a specific helicity, it is possible to control the helicity of the helical beam emerging from the laser. In all three configurations, the insertion of an aperture inside the resonator ensures that the laser operates with the lowest-order helical mode, i.e., TEMo,+/. In order to determine the aperture diameter, the mode discrimination for such resonator configurations was calculated by the matrix-diagonalization method. Due to the circular symmetry of the resonator, the usually complex matrices, based on Bessel-Fourier transformation, become simpler (see sect. 2.2.3). The
350
Transverse mode shaping and selection
0.5 1 1.5 2 Fresnel number
%
0.5 1 1.5 2 Fresnel number
[6, §3
2.5
Fig. 20. Diffraction losses in a round-trip for various Laguerre-Gaussian transverse modes as a function of the Fresnel number NY=a^lXL. The bold curves represent the TEMo,+1 mode: (a) laser configuration with no SPEs; (b) laser configuration with a single SPE oiN=\ and a phase reverting element, inserted into the resonator. (From Oron, Danziger, Davidson, Friesem and Hasman [1999b].)
calculated results are presented in fig. 20. The power loss per round-trip as a fiinction of the Fresnel number A'^p is shown for the different modes in the laser resonator. Figure 20a shows the losses for a laser resonator configuration without any SPEs, whereas fig. 20b shows the losses for a configuration with one SPE of A^ = 1, and a phase-reversing element. Evidently, the losses of the modes with / = 1 are not affected by the SPE, while all other modes (including the ftindamental TEMQO mode) suffer very high losses. Thus, laser operation with a single high-order mode can be obtained with a Fresnel number of approximately 2, which is significantly larger than that of a laser operating with the fijndamental Gaussian mode. An SPE was incorporated into a CO2 laser resonator configuration with a cylindrical lens, such as shown in fig. 19b. The laser was a discharge-pumped CO2 laser whose length was 60cm, the SPE was formed fox N=\ on GaAs substrates by means of a 16-level photolithographic process and the cylindrical lens of ZnSe had a focal length of 7.5 cm. The SPE and the cylindrical lens had AR coating for A= 10.6 [xm. A variable aperture, inside the laser resonator, was adjusted until the emerging beam contained only one mode, leading to a Fresnel number of 2. The results are shown in fig. 21, along with the corresponding calculated results from eq. (1). Figure 21a depicts the calculated laser output near-
6, §3]
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351
Fig. 21. Theoretical and experimental intensity distributions that emerge from a CO2 laser in which a SOE with A^= 1 was incorporated to obtain the TEMo^+1 mode: (a) theoretical near field; (b,c) experimental near field; (d) theoretical far field; (e,f) experimental far field. (From Oron, Danziger, Davidson, Friesem and Hasman [1999b].)
field intensity distribution corresponding to the TEMo,+i mode. Figures 21b,c present the corresponding experimental near-field pattern and contour plot. The expected doughnut-shape distribution is clearly evident in both calculated and experimental results. The nonuniformities in the experimental distribution arise fi-om the 16-level discontinuities, caused by the Jtl^ phase steps. Figure 2Id depicts the theoretical far-field laser output intensity distribution, which is calculated by the Fourier transformation of the near-field pattern with a uniform phase. Figures 2 le,f depict the corresponding experimental far-field pattern and contour plot. The excellent agreement between the theoretical and experimental far-field patterns is evident, indicating that all parts of the near-field pattern are in phase. Also, opening the aperture to allow for higher-order transverse mode operation, and carefiilly adjusting the resonator length L so as to control the longitudinal modes, lead to laser operation with high-order modes. This is attributed to the coupling between longitudinal and transverse modes in such a CO2 laser. The laser output near-field intensity distributions corresponding to these higher modes are shown in fig. 22. As evident, the helical phase causes spiral-like patterns, and the higher the mode order, the higher number of lobes in the laser output near-field intensity distributions. SPEs were also incorporated into an NdiYAG laser resonator. The laser resonator configuration included two SPEs, as shown in fig. 19a. The laser was a flashlamp-pumped NdiYAG laser whose length was 60 cm, and both
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Fig. 22. Near-field intensity distributions corresponding to higher-order modes in a resonator with a SPE: (a) second-order mode with four lobes; (b) third-order mode with six lobes.
Fig. 23. Experimental intensity distributions that emerge from an Nd:YAG laser with SPEs that operated with the TEMQ, +1 mode and without SPEs that operated with the fimdamental TEMQO mode: (a) TEMo,+ i near field; (b) TEMQO near field; (c) TEMo,+1 far field; and (d) TEMQO far field. (From Oron, Danziger, Davidson, Friesem and Hasman [1999b].)
SPEs were formed for A/^= 1, on fiised silica substrates in a single-stage etching process using a gray-scale mask, and AR coated for A= 1.06 |im. Here again, an internal aperture was varied in order to achieve single-mode operation. The experimental results with the NdiYAG laser are shown in fig. 23. Figure 23a presents the laser output near-field intensity distribution, corresponding to the TEMo,+i mode, with the expected doughnut-shaped distribution. Figure 23b depicts the near-field intensity distribution, corresponding to the fimdamental TEMoo mode pattern (with no SPEs). Figure 23c depicts the corresponding farfield intensity distribution, where the single main lobe is evident, again indicating that all parts of the near-field pattern are in phase. Figure 23d depicts the farfield pattern fi-om a laser operating with the fimdamental TEMQO mode, whose cross-section area is, as expected, similar to that from a laser operating with the higher-order TEMo,+i mode. As evident, there is some asymmetry in the
6, §3]
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353
(b)
•Hi \ •Vu \ '1/ vn \ jj
W (d)
\
!
Fig. 24. Near- and far-field intensity distributions of a helical beam: (a) experimental near field; (b) experimental far field; (c) calculated and experimental near-field cross sections; dashed lines, experimental results for the x and y axes; solid lines, calculated results; (d) calculated and experimental far-field cross sections. (From Oron, Davidson, Friesem and Hasman [2000b].)
intensity distributions, which is attributed to the nonuniformity of the SPEs. An output power of 5.2 W was obtained when the laser operated with the TEMo,+i mode, which was higher by up to 50% with respect to that obtained from the same laser operating with the TEMQO mode and no SPEs. The resonator configuration shown in fig. 19c was experimentally tested with a linearly polarized discharge-pumped CO2 laser, operating with a single longitudinal mode. The reflective SPE was fabricated on a silicon substrate in a multistage etching process, to form 32 phase levels with a combined depth of A, which corresponds to A/^= 1. The depth accuracy in the fabrication process was less than 3% and the RMS surface quality was better than 20 nm. Its reflectivity was better than 98%, adequate to serve as a laser reflector mirror. The diameter of the laser tube was 11 mm, and the length of the laser was 65 cm. The intracavity cylindrical lens ( / = 12.5 cm) was focused on the concave (r = 3 m) output coupler, while an identical lens was positioned outside the cavity to collimate the output beam. Figure 24 shows the near- and far-field intensity distributions of a helical beam that emerges from the laser after passing through the external cylindrical lens. The near-field distribution, shown in fig. 24a, had the expected doughnut shape, albeit with some distortions, due mainly to imperfections in the fabrication process of the SPE. The output power was 1.2 W. The corresponding farfield intensity distribution, shown in fig. 24b, was obtained by focusing the output beam with a spherical lens ( / = 50cm). Here again the beam was
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[6, § 3
doughnut shaped. The x and y cross-sections of the near- and far-field intensity distributions, compared to the calculated cross-sections derived from eq. (1), are given in figs. 24c and 24d, respectively. As evident, there was good agreement between the predicted and experimental results, including the low intensity at the center. As shown, there was some asymmetry between the x and y cross-sections of the intensity distributions, which is due to some astigmatism caused by the intra-cavity cylindrical lens. The intensity distribution at other planes maintained the same shape. By replacing the SPE with a reflective mirror in the setup of fig. 19c and adjusting the internal aperture, the laser operated with the fundamental TEMQO mode, where the output power was 0.9 W. This is significantly lower than the 1.2 W obtained when the laser operated with the TEMo,+i helical mode.
3.5. Self-imaging and Fourier resonators It is also possible to discriminate and select a single high-order mode by resorting to specialized resonator configurations. In this subsection we describe two types of resonators: resonator designs based on the Fourier transformation, and selfimaging resonators based on the Talbot effect (see Talbot [1936]). Generally, in Fourier resonators a single spatial filter is not sufficient to select a highorder mode, so two different spatial filters are used. Self-imaging resonators exploit the Talbot effect, where periodic structures, such as high-order modes, are reproduced after propagating certain distances (namely Talbot lengths). Kermene, Saviot, Vampouille, Colombeau, Froehly and Dohnalik [1992] presented a Fourier resonator with two binary amplitude spatial filters, each placed next to a laser mirror. In this resonator, an intra-cavity focusing lens performed the Fourier transformation between the two planes where the laser mirrors were placed. The spatial filter next to the output coupler had a similar pattern to that of the desired output beam, whereas the spatial filter next to the back mirror was designed to produce the desired output beam with a uniform phase. Specifically, this back filter was designed to have high absorption in regions where the Fourier transformation of the desired intensity distribution has low intensity. For example, for a desired square intensity distribution, a square spatial filter was placed next to the output coupler and a spatial filter with absorbing lines along the zeros of a two-dimensional sine function (which is the Fourier transformation of a square) was placed next to the back mirror. A similar design, for obtaining a uniform-phase circular intensity distribution at the laser output, was demonstrated by Saviot, Mottay, Vampouille and Colombeau
6, § 3]
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355
N'»f*T»f»H^I W 4 ^ l>:|"-l'!'i";;".l";.'g*l
(b)
Fig. 25. Near- and far-field intensity distributions emerging fi-om a slab waveguide CO2 laser, operating with the 12th-order mode, with an intra-cavity wire grid. (From Morley, Yelden, Baker and Hall [1995].)
[1993]. Here the spatial filter next to the output coupler was simply a circular aperture, whereas the spatial filter next to the back mirror had concentric rings, corresponding to the zeros of the Airy pattern. Experimental results with a pulsed Nd:YAG laser, operating with such square and circular intensity distributions with output energies of 200 mJ, were obtained. Bourliaguet, Mugnier, Kermene, Barthelemy and Froehly [1999] showed that the performance of a pulsed optical parametric oscillator (OPO) could be improved by intra-cavity spatial filtering. Specifically, a five-fold increase of brightness with respect to the multimode operation was demonstrated when applying an intracavity two-dimensional wire grid designed to form a few lobes in the far field. Le Gall and Bourdet [1994] also investigated a Fourier resonator configuration in which an internal spatial filter coupled the phases of an array of CO2 waveguide lasers. Fourier resonators were also investigated by Wolff, Messerschmidt and Fouckhardt [1999] for selecting high-order modes in broad area lasers. Abramski, Baker, Colley and Hall [1992] exploited a one-dimensional wire grid in a slab waveguide CO2 laser in order to select a single high-order mode. The wire grid spacing d was designed to match the periodicity of the desired mode. In principle, this by itself could lead to high modal discrimination, but in practice the alignment tolerances cannot be met, so excessive losses are introduced. The losses can be significantly reduced by resorting to a resonator with intra-cavity coherent self-imaging, based on the Talbot effect. Specifically, the resonator length L was chosen to match the Talbot length, namely, L = \p(flX, where ;? is a small integer that corresponds to the number of imaging planes in a round-trip. Such a self-imaging Talbot effect is particularly advantageous in a waveguide laser where the boundaries reflect the light, leading to a "kaleidoscope" effect, in which a much larger periodic structure is more efficiently self-imaged. The modal properties of such a slab waveguide CO2 laser were experimentally investigated by Morley, Yelden, Baker and Hall [1995], and their results are presented in fig. 25. It shows the near- and far-field intensity
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distributions cross-sections, when the laser operated with the 12th-order mode. As evident, the 12 lobes in the near-field pattern transformed into two main lobes (of opposite phase) in the far field, indicating that the laser operates with a single mode. The wires were 75 |Jim thick, and output powers of up to 65 W were obtained with the single high-order mode, compared to 90 W power for the multimode operation, in a resonator length of 25.4 cm. Self-imaging resonators based on the Talbot effect were also applied for mode selection in waveguide lasers (Baneiji, Davies and Jenkins [1997]), to coherently lock arrays of diode lasers (e.g., Jansen, Yang, Ou, Botez, Wilcox and Mawst [1989]), and for phase matching the modes in waveguide CO2 lasers with intracavity binary phase elements (Glova, Elkin, Lysikov and Napartovich [1996]). Similarly, Tang, Xin and Ochkin [1998] replaced the back mirror in a CO2 laser with a reflective binary phase element to obtain high-output powers. In these cases, either in-phase or anti-phase operation resulted in either one high central lobe or two main lobes in the far field.
3.6. Polarization-selective resonators The light emerging ft-om most resonators is either linearly polarized or unpolarized. Linear-polarization operation is typically obtained by either inserting into the resonator a Brewster window or other polarization selective elements (such as birefringent crystals, polarizers or polarizing beam splitters), or by a polarizationsensitive pumping system (such as RF-excited slab lasers). Unpolarized light is simply obtained where there are no polarization-sensitive elements in the resonator. Also, circularly polarized light can be obtained by inserting a quarterwave (A/4) plate into the resonator (see for example Trobs, Balmer and Graf [2000]). In all the above, the light polarization is uniform across the entire laser output beam. In this subsection, we present laser resonator configurations in which the polarization in different parts of the output beam can be varied, namely a laser output beam with space-variant polarization. Space-variant polarization, such as azimuthal and radial polarizations, results in completely symmetric laser beams that can be exploited in various applications. Such polarizations have been obtained, outside the laser resonator, either by transmitting a linearly polarized laser beam through a twisted nematic liquid crystal (Stadler and Schadt [1996]) or by combining two linearly polarized laser output beams interferometrically (Tidwell, Kim and Kimura [1993]). Azimuthal and radial polarizations have also been obtained by inserting polarization-selective elements into the laser resonator. Pohl [1972] inserted a
6, § 3]
357
Intra-cavity elements and resonator configurations Output coupler
Aperture + stop Calcite
Gain medium (Ruby)
Back mirror
Output beam
C>
U
-Telescope>J -
Q-^switch
Fig. 26. Resonator configuration for selecting an azimuthally polarized mode. (From Pohl [1972].)
birefringent calcite crystal, in which the principal axis was along the z-axis (z cut), into a pulsed ruby laser in order to discriminate between azimuthal and radial polarizations. The resonator configuration is shown in fig. 26. The calcite crystal was inserted inside a two-lens telescope arrangement, so as to increase the divergence of the mode inside the crystal and thereby also the polarization discrimination. Specifically, due to different angles of refraction, the diameter of the azimuthally polarized mode differed from that of the radially polarized beam; so discrimination and selection of an azimuthally polarized mode were obtained by inserting an aperture and a stop with the appropriate diameters. Wynne [1974] generalized this method and showed experimentally, with a wavelength-tunable dye laser, that it is possible to select either the azimuthally or the radially polarized mode. This was achieved by controlling the telescope length and location, so in a certain range of telescope lengths and locations, the azimuthally polarized mode is stable whereas the radially polarized mode is unstable or vice versa. Mushiake, Matsumura and Nakajima [1972] used a conical intra-cavity element to select a radially polarized mode. The conical element introduced low reflection losses to the radially polarized mode but high reflection losses to the azimuthally polarized mode. This method is somewhat similar to applying a Brewster window for obtaining a linear polarization. Similarly, Tovar [1998] suggested using complex Brewster-like windows, of either conical or helical shape, to select radially or azimuthally polarized modes. Nesterov, Niziev and Yakunin [1999] replaced one of the mirrors of a highpower CO2 laser by a sub-wavelength diffractive element. This element consisted of either concentric circles (for selecting azimuthal polarization) or straight lines through a central spot (for selecting radial polarization) to obtain different reflectivities for the azimuthal and radial polarizations. Experimentally, high output power of 1.8 kW was obtained, but the polarization purity was relatively low, with mixed transverse mode operation. Liu, Gu and Yang [1999] analyzed a
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[6, § 3
00^® (b)
Fig. 27. Coherent superposition of two orthogonally polarized TEMQI modes to form azimuthally and radially polarized modes: (a) azimuthally {0) polarized doughnut mode; (b) radially (r) polarized doughnut mode.
resonator configuration, into which two sub-wavelength diffractive elements were incorporated, to obtain a different fiindamental mode pattern for two different polarizations. Oron, Blit, Davidson, Friesem, Bomzon and Hasman [2000] presented a method for efficiently obtaining an essentially pure either azimuthally or radially polarized beam directly from a laser. It is based on the selection and coherent summation of two linearly polarized transverse modes that exist inside the laser resonator; specifically, two orthogonally polarized TEMQI modes. The coherent summation of TEMoi(.r) and TEMoi(v) Laguerre-Gaussian modes (or TEMio and TEMoi Hermite-Gaussian modes), having orthogonally linear polarizations, leads to the formation of either an azimuthally or radially polarized mode, whose vectorial field distributions have the form Azimuthal:
£(r, 6) = yEon,)(r, d)-xEoiiy)(r, 6) = 0Eop2exp(-p/2%
Radial:
E(r, 6) = xEo^.^ir, 6) + j^oiCv)(^, 8) = r^op2 exp(-p/2),
where 0 and r are unit vectors in the azimuthal and radial directions, respectively. This coherent summation is illustrated in fig. 27. Figure 27a depicts an azimuthally polarized mode, obtained by a coherent summation of a >^-polarized TEMoi(x) mode and an x-polarized TEMoi(j;) mode, whereas fig. 27b depicts a radially polarized mode, obtained by a coherent summation of an x-polarized TEMoi(jc) mode and a >^-polarized TEMoi(v) mode. The laser resonator configuration in which specific transverse modes are selected and coherently summed is schematically shown in fig. 28. Here, the
6, §3]
Intra-cauity elements and resonator configurations
359
X polarization Combined DPE
0 «» T^
•
y polarization
•
Birefringent beam displacer
I Back mirror
\
Gain medium
Aperture Alignment plate
Output coupler
Fig. 28. Laser resonator configuration with a discontinuous phase element (DPE) for forming azimuthally or radially polarized beam. (From Oron, Blit, Davidson, Friesem, Bomzon and Hasman [2000].)
light propagating inside the laser is split and displaced by means of a birefringent beam displacer to obtain two separate paths with orthogonally polarized light. A differently oriented discontinuous phase element (DPE) is inserted in each path, adjacent to the back mirror, to select the TEMoi mode. Specifically, one of these modes is TEMoi(x), and the other is TEMoi(3;). In practice, the two DPEs can be fabricated on the same substrate. In order to add the two modes coherently with the appropriate phase between them, an additional aligning plate is inserted into one of the paths (in the region after separation), so as to control the optical path by slightly tilting the window. Note that exact phase locking between the two orthogonal modes is obtained by a small coupling between them; the alignment plate brings the two modes close enough to allow this locking to occur. At the back mirror, two spatially separated TEMoi modes evolve, each with a different linear polarization. However, as a result of the coherent summation of these two modes, a circularly symmetric doughnut-shaped beam emerges from the output coupler. This approach was verified experimentally with a continuous-wave lamppumped Nd:YAG laser into which were inserted a calcite crystal as the birefringent beam displacer, two DPEs for selecting the orthogonally polarized TEMoi modes, and an alignment plate to adjust the phase between the two orthogonally polarized TEMQI modes. The calcite crystal was 4 cm long, so the two orthogonally polarized light paths were displaced 4 mm apart. The phase elements were aligned to obtain two orthogonal TEMQI modes. The alignment plate was simply a flat-fused silica window with antireflection layers on both faces. To ensure that the beam emerging from the laser is indeed azimuthally or
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Fig. 29. Experimental intensity distributions of an azimuthally polarized beam that emerges from an NdiYAG laser: (a) directly from the laser with no external elements; (b) after passing a horizontal A/4 plate and a polarizer oriented at 45 degrees; (c) after passing a polarizer oriented in the horizontal direction; (d) after passing a polarizer oriented at 45 degrees; (e) after passing a polarizer oriented in the vertical direction. (From Oron, Blit, Davidson, Friesem, Bomzon and Hasman [2000].)
radially polarized, it was passed through a linear polarizer at 45 degrees. Then, the alignment plate was tilted until the intensity distribution after the polarizer had two lobes perpendicular (for azimuthally polarized) or parallel (for radially polarized) to the polarization direction. This indicated that the orthogonal TEMQI modes add coherently. Some results for an azimuthally polarized beam are shown in figs. 29 and 30. Figure 29 shows the intensity distributions, detected with a CCD camera, that emerge fi"om an NdiYAG laser, which emits an azimuthally polarized beam. Figure 29a shows the near-field intensity distribution of the azimuthally polarized beam, emerging directly fi-om the laser. Here the doughnut shape is evident. In order to determine the polarization of the output beam, four additional intensity distributions were detected. These are shown in figs. 29b-e. Figure 29b shows the intensity distribution of the emerging beam after passing through a quarter wave plate, whose main axis was oriented in the horizontal direction, and a polarizer oriented at 45 degrees. Here, the nearly doughnut-shaped intensity distribution (with approximately half the power) indicates that the polarization of the original beam is linear at each point. Figures 29c-e show the intensity distributions of the beam emerging fi-om the laser, after passing a single linear polarizer oriented at different orientations. Figure 29c shows the intensity distribution with the polarizer oriented in the horizontal direction, fig. 29d that in the diagonal (45 degrees) direction and fig. 29e in the vertical direction. At these three
3]
361
Intra-cavity elements and resonator configurations
•v
t
'^ f ^ '^ >^ A
^
f ^
^
^ <. <. <-
•^ ^ * \ 1^ 1^
^ ^
A^ ^
^ ^
(b) Fig. 30. Experimental plot of the space variant polarization directions of the emerging (a) azimuthally and (b) radially polarized beams. (From Oron, Blit, Davidson, Friesem, Bomzon and Hasman [2000].)
orientations, the intensity distributions have two lobes, along a line perpendicular to the polarization direction, as expected for an azimuthally polarized beam. By measuring the intensities at each point of the distributions in figs. 29b-e, the Stokes parameters S^-S^ were calculated at each point of the beam, from which the polarization ellipse parameters were deduced at each point. These results, for the experimental polarization orientations for an azimuthally polarized beam, are presented in fig. 30, where the arrows indicate the direction of the main axis of the local polarization ellipse (azimuthal angle i/^), calculated by arctan(*S'2/*S'i). The deviation from the desired direction of polarization and the average ellipticity angle x ^ arcsin(*S'3/*S'o) were also calculated. The calculated deviation from the desired polarization orientation was 10 degrees, and the average ellipticity angle was found to be 8 degrees. The overall polarization purity (percentage of power that is azimuthally polarized) was determined to be 95%. As expected with a TEMoi-mode operation that exploits more of the laser gain medium than the fijndamental Gaussian mode, the output power of our laser with azimuthal polarization was 5.2 W, that is 50% higher than with the fiindamental Gaussian mode. Similar results were obtained for a radially polarized beam, as shown in fig. 30b. Note that other polarization states can be obtained by applying higher-order modes. These include high-order rotational polarization by applying TEMQ/ modes where / ^ 2, and azimuthally or radially polarized beams having a few concentric rings with TEM^/ modes, where/> ^ 1. In a similar resonator configuration, it is possible to simultaneously select two different orthogonally polarized transverse modes, in order to more efficiently
362
[6, §3
Transverse mode shaping and selection
Output Coupler 2
/
/ Output beam s-polarization
Back Mirror Output beam p-polarization
IJOutput Aperture 1
Coupler 1
Fig. 31. Resonator configuration in order to obtain two orthogonal polarized TEMQO and TEM02 modes. (From Oron, Shimshi, Blit, Davidson, Friesem and Hasman [2001].)
exploit the gain medium. Oron, Shimshi, BHt, Davidson, Friesem and Hasman [2001] investigated laser operation with orthogonally polarized TEMQO and TEM02 modes using the configuration shown in fig. 31. The total output power with the two modes was 5.7 W, compared to 4.7 W with the single TEM02 mode and 3.2 W with the single TEMQO mode; thus, an improvement of the output power by approximately 25% over that of a laser operating with the TEM02 mode was obtained. Furthermore, since the two modes are orthogonally polarized, each could be manipulated separately and then combined to obtain high-output beam quality as well. 3.7. Unstable resonators In this subsection, we present various mode-shaping and mode-selection techniques, which are unique to unstable resonator configurations. Several other methods involve the use of a graded-reflectivity mirror (GRM). GRMs have a continuous, nonuniform reflectivity, where typically the reflectivity is higher in the center and lower near the edges of the mirror. Generally, in a laser with a GRM output coupler, there is a dip in the center of the near-field pattern, since the reflection of the GRM is higher in the center. The analysis of GRM unstable resonators could either be based on geometrical optics, which is accurate only in a certain range of resonator parameters (see for example Bowers [1992]), or be based on diffraction analysis, which is more general (see for example Morin [1997]). Other mode-shaping and mode-selection techniques involve either the exploitation of intra-cavity apertures with various shapes, or replacing the back mirror by a mirror with a phase step.
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363
Several methods were suggested for shaping the internal modes to obtain a specified laser output profile. Belanger and Pare [1994] replaced the output coupler by a GRM in order to obtain a certain output beam profile with lower diffraction than with a conventional mirror. The method usually results in some decrease in the output power. Massudi and Piche [1997] showed theoretically that nearly flat-top output beams could be obtained by inserting an aperture into certain negative-branch unstable resonators. In practice however, such output beams are difficult to realize, because of the existence of a focal spot in the negative-branch resonators. Makki and Leger [2001] controlled the profile of the output beam from an unstable resonator by replacing the output coupler with a GRM and replacing the back mirror with a reflecting phase element. The phase element allowed additional shaping of the intra-cavity mode pattern so as to eliminate the dip in the center of the output beam. By proper design of the phase element, a nearly flat-top output beam was obtained. Piche and Cantin [1991] demonstrated that by introducing a nearly Jt phase step in the center of a mirror, one can significantly lower the round-trip losses of the unstable resonator. This is attributed to focusing of the laser mode near the optical axis, caused by the phase step, so the output coupler reflects a larger portion of the beam. Calculations based on the Prony method were supported by experimental results with a CO2 laser. Specifically, lasers with a phase step can operate with higher magnification than without a phase step. Van Eijkelenborg, Lindberg, Thijssen and Woerdman [1998] isolated higherorder modes by inserting a strip aperture, a square aperture or a circular aperture into the unstable laser resonator. The mode selection was performed indirectly, by controlling the longitudinal mode of a HeXe laser, operating in the infrared (A = 3.51fim). Different high-order modes were obtained for each aperture shape, in good agreement with calculated predictions. McDonald, Karman, New and Woerdman [2000] calculated the shapes of high-order modes in unstable resonators in which differently shaped two-dimensional apertures were inserted. They found that the high-order modes have kaleidoscope-like patterns.
3.8. Alternative methods In this subsection, we present mode-selection and mode-shaping techniques, based on inserting into the resonator either specialized mirrors or specialized prisms, or elements that replace the commonly used hard-edged (binary) aperture. Rioux, Belanger and Cormier [1977] replaced one of the laser mirrors
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by a conical (axicon) mirror, to obtain high-order annular modes. In this case, either single-mode or multimode operation was obtained, depending on the resonator length and intra-cavity aperture diameter. Uehara and Kikuchi [1989] exploited an annular back mirror in order to obtain nearly Bessel-Gaussian output beams. The experimental intensity distributions were measured in good agreement with expected calculated results. Abramochkin, Losevsky and Volostnikov [1997] obtained different spiral-type output beams from a ring laser resonator in which an intra-cavity prism that rotated the internal beam was inserted. The output beams resulted from unusual combinations of Laguerre-Gaussian beams, and they generally contained a few azimuthal lobes and exhibited rotational symmetry. Zhou, Fu, Lu, Li and Yu [1991] replaced the intra-cavity aperture by a capillary tube, and showed both theoretically and experimentally, that increased modal discrimination between the frindamental and high-order modes is obtained. Ait-Ameur [1993] proposed to replace the hard-edged aperture by a super-Gaussian aperture. Calculations revealed that with a super-Gaussian aperture of order 5 to 10, both high transverse-mode discrimination and low ftindamental-mode losses could be obtained. Tovar and Casperson [1998] considered an apodized aperture with a specified amplitude transmission, in order to obtain a laser operating with hyperbolic sine Gaussian modes. 3.9. Fabrication of intra-cavity elements In general, great care must be taken when fabricating elements that are inserted into the laser resonator, because any losses are greatly magnified. This is particularly true for mode-selecting and mode-shaping elements, whose light efficiency must be very high. Thus, various specialized techniques have been developed for fabricating these intra-cavity elements. These involve advanced electron beam recording, photolithographic and etching thin film deposition, and diamond turning technologies. The fabrication of some representative intracavity elements will be briefly described. Binary diffractive or phase elements are generally fabricated with a one-stage etching process, using a single binary mask that is generally recorded with an electron beam, whose information is transferred to photo-resist layers with conventional photolithographic technologies. Continuous diffractive or phase elements (such as spiral phase elements) are fabricated with a more complex, multistage, etching process (see for example Hasman, Davidson and Friesem [1991]). In this fabrication method, a number N^n of binary masks are first individually recorded, and their information is transferred to photo-resist layers
6, § 3]
Intra-cavity elements and resonator configurations
365
that are individually etched to form the nearly continuous phase element with 2^"^ phase levels. Generally, when using such elements outside the resonator (for example, for beam shaping), a high efficiency of 98.6% is obtained for an element with 16 levels (A^m = 4). For an intra-cavity element one might typically need even higher-level resolution due to two main factors. First, an intra-cavity element is often placed next to a mirror or replaces it, so light passes through it twice in a short distance, requiring twice the level depth resolution. Second, losses are magnified inside the laser resonator, so higher efficiencies, and thereby lower losses, are required from the element. Thus, a larger number of masks A^m may be required. This implies that the multistage process has a basic drawback in fabrication complexity. Alternatively, a one-stage etching process with a gray-scale mask can be exploited. Suleski and O'Shea [1995] recorded such a gray-scale mask on lowcontrast films using visible illuminators and photo-reduction techniques. For their final blazed grating, diffraction efficiencies up to 85% were measured. Daschner, Stein, Long, Wu and Lee [1996] recorded the gray-scale mask on a specially designed high-energy beam sensitive glass, using a computer-controlled high-energy electron beam. Borek and Brown [1999] also exploited a one-stage etching process, with a gray-scale mask. Such one-stage etching processes lead to elements with a desirable continuous depth profile. However, the process requires very exact calibration procedures. Bourderionnet, Huot, Brignon and Huignard [2000] showed that holographic optical elements can also serve as intra-cavity elements. The holographic elements were recorded in a thick photo-polymer material, using a computercontrolled spatial light modulator to control the phase of the signal beam. The holographic elements were designed to obtain a super-Gaussian output from an Nd:YAG laser {X= 1.06[im). Unfortunately, highest diffraction efficiency of the holographic element was only 95%, which resulted in a laser output power approximately 40%) lower than that from a laser with no holographic elements. Thin-film technology could also be used to form a binary phase element (see for example Kol'chenko, Nikitenko and Troitskii [1980] or Piche and Cantin [1991]). Here, instead of etching into the substrate, a relatively thick layer is deposited on the substrate through a mask with the desired shape, which results in a binary phase element. Some elements are fabricated with a diamond-turning technique. This technique is generally exploited for forming aspherical mirrors or phase elements, such as conical elements of GPMs (see for example Van Neste, Pare, Lachance and Belanger [1994]), where there is a circular symmetry. Due to the relatively low attainable resolution, this method is mostly used for longer wavelengths, namely, CO2 lasers.
366
[6, §4
Transverse mode shaping and selection Reflective coating Focus electrode e^ Copper Piezoceramic actuator disks
Common ground electrode
Fig. 32. Adaptive mirror. The phase profile is controlled by the three electrodes el-e3. (From Cherezova, Chesnokov, Kaptsov and Kudryashov [1998a].)
Cherezova, Chesnokov, Kaptsov and Kudryashov [1998a,b] showed that it is possible to replace the fixed back mirror of a laser resonator with an adaptive mirror. An adaptive mirror fabricated with three electrodes can control piezoceramic actuators to form a radially symmetric phase pattern. The shape of the adaptive mirror can be electrically controlled both to obtain a predetermined shape and to correct possible aberrations in the resonator. A schematic diagram of the adaptive mirror is shown in fig. 32. This mirror was applied as a GPM in order to select super-Gaussian output beams of various orders or a doughnutshaped output beam. Finally, most if not all intra-cavity elements require high-quality reflective or antireflective coatings in order to minimize losses. Also, high surface quality (low roughness) is required to reduce scattering losses, and high parallelism is required fi'om transmittive elements in order to maintain the optical path after insertion.
§ 4. Properties of the laser output beams In this section, we consider certain properties of the laser output beams. These include beam quality, output power and field distributions. Several specific laser output beams are presented.
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Properties of the laser output beams
367
4.1. Beam quality There are various criteria for evaluating the quality of laser output beams. Such criteria are not universal, but depend on the specific application. The most widely used criteria for beam quality are based on focusability, or a product of beam width and beam divergence. However, there are different definitions of width and divergence. Other criteria are based on phase space analysis, on coherence, or on thermodynamic limits (entropy). Here, several beam quality criteria are considered. 4.1.1. Second-order moments (M^) The quality of the beam emerging from a laser is most commonly defined by its focusability, determined by the product of beam waist and beam divergence. The measure for the beam quality is the second-order moment beam propagation factor introduced by Siegman [1990], namely, the M^ value, given by the ratio between the space-bandwidth products of the beam and that of a Gaussian beam, as
M^ =
^1
,
(12)
^Gaussian "Gaussian
where w = ({w^))2 and 6 = ({6^))2. Consequently, the optimal beam has a Gaussian shape, with a minimal waist-divergence product, which is limited only by diffraction. Such an optimal beam, with M^ = l, can be obtained from a laser operating with a single fimdamental TEMQO mode. Modifications of the beam-quality criteria are based on different definitions of w and 6, namely, the angle or waist in which a certain percentage of the energy is contained. These criteria agree with the M^ value for Gaussian beams. However, for other types of beams, such as high-order modes or nearly flat-top beams, the criteria may deviate significantly. For any beam, the M^ value is left unchanged by propagation of the beam or by simple optical elements such as spherical mirrors and lenses, a combination of those, or any optical system that can be represented by an ABCD matrix (see Yariv [1991] for details). Moreover, Siegman [1993] showed that binary (jt) phase plates cannot improve the M^ value. Later, Zhao [1999] also showed that other step-phase plates cannot improve the M^ value. Also, aberrations in an optical system may be related to the deterioration of the M^ value for a beam passing through it. M^ values for various types of beams have been calculated (see for example Siegman [1990] or Saghafi and Sheppard [1998]). Specifically, for a beam
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[6, § 4
derived from a laser operating with the Hermite-Gaussian modes, the M^ value is given by 1 + « + m, while a beam derived from a laser operating with the Laguerre-Gaussian modes has \+2p+\l\. For a multimode laser, the M^ value is mostly determined by that of the highest-order mode, although the total number of modes, Nj = }^M'^{M'^+ \), could be very large. So, the mode content of a laser cannot generally be determined from the M^ value. Another disadvantage of this criterion is that certain types of beams that are desired for various applications, such as a flat-top beam, have very high M^ values. For beams whose field distribution cross-section differs between the x and y axes, it is useful to define a more general value, namely M^=Ml-Ml. A similar definition could be applied for radially symmetric beams. M^ values have been calculated for beams derived from lasers operating with the HermiteGaussian and the Laguerre-Gaussian modes, yielding M'^ = (l +2m)(l +2«) and M 4 = (1 +2jr7+ |/|)2-/2, respectively (see Murphy [1999]). 4.1.2. Wigner distribution function (WDF) The Wigner distribution frinction (WDF) was introduced by Wigner [1932] and has been applied in many branches of physics and optics (see Dragoman [1997]). One of these applications relates to beam quality. The WDF of a beam is given by
1
W{r, p) = X~' / d'Ar exp(-2;ripAr/A) \l){r + ^Ar) \l)\r - ^Ar),
(13)
where r = (jc,j;) is the spatial coordinate,/? = (/?;c,;?,v) is the frequency coordinate, and \l) denotes the field distribution. Gase [1995] provided a WDF representation of Laguerre-Gaussian modes. Simon and Agarwal [2000] simplified this representation, and obtained simple relations between the WDF of Laguerre-Gaussian modes and the WDF of Hermite-Gaussian modes. The WDF of Laguerre-Gaussian modes is ^^,/(r,p) = (2/A)(-l)2^^l'lL^,|/|(4[eo ± Q2\)\{mo
T e2])exp(-4eo),
(14) where go = \{r'^lw^ + 7i^\\?-p^l}}\ Q2 = JtX~\xpy -ypx), and Lp are the Laguerre polynomials of order p. For the TEMQO ftmdamental mode, a four-dimensional Gaussian is obtained. For all other modes, a more complicated fiinction, but with a larger Gaussian envelope, is obtained. A method to establish the WDF of a beam experimentally was suggested by Hodgson, Haase, Kostka and Weber [1992]. Various beam-quality parameters
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Properties of the laser output beams
369
can be obtained from the WDF. Specifically, either the M^ or the M^ values can be derived from the second-order moments of the WDF (see Dragoman [1997]), which corresponds to the fr)ur-dimensional volume surrounded by the envelope of the WDF (or to the phase-space area, given by Murphy [1999]). Also, the total coherence function K was derived by Eppich, Johansson, Laabs and Weber [2000], as K = X f W(r,pfdrdp
= XTr(mV%
(15)
which is maximal for a coherent (or a single-mode) beam. The total coherence function of eq. (15) is inversely proportional to the actual (net) volume of the WDF. This volume may be only a part of the envelope volume. For lasers operating with the fundamental Gaussian mode or multimodes, the actual and envelope volumes coincide, whereby the envelope is completely frill. Moreover, for lasers operating with any single high-order mode, the total coherence frinction K is unity, just as for a laser operating with the fundamental Gaussian mode. This implies that the actual volume of the WDF is minimal. For lasers operating with a few modes (each of equal power), the actual volume of the WDF is the number of modes. 4.1.3. Coherence and entropy Wolf and Agarwal [1984] exploited coherence theory (see for example Born and Wolf [1965]) to show that if there is no degeneracy, each mode is completely spatially coherent. Moreover, complete spatial coherence can occur only in a laser operating with a single mode (or in some combinations of degenerate modes) and cannot occur in a multimode laser. Visser, Friberg and Wolf [2001] introduced the phase-space product, which can be exploited to obtain a measure of the beam quality for partially coherent beams. This product expresses the effective coherence area of the source multiplied by the effective angular spread of the beam, and coincides with the M^ value for fully coherent beams. A minimal value of unity was obtained for a class of laser beams, which also included the lowest-order Gaussian mode. Another approach, which differentiates between single-mode and multimode laser operation, was introduced by Graf and Balmer [1996]. The analysis of the beam quality is based on the second law of thermodynamics, namely, on entropy. In order to study the limits of beam shaping, the entropy of a laser operating with various transverse modes was calculated to yield *S'l ^ ^
^ln(«mode)
for
Wmode >
1,
(16)
370
Transverse mode shaping and selection !
1
[6, §4
"
S
j .
1/
W^PSx^ \
1 \//
MDFv 1
\f"^^% f
1
1
1
„,...
i.„ 1
0.5
0
Fig. 33. Calculation of entropy, MDF and ln(A/^), for a superposition of the two lowest-order modes, as a function of the portion of the lowest-order mode CQ. The entropy was calculated for a total number of 100 photons. (From Graf and Balmer [1996].)
where «mode is the number of photons in the resonator mode. For a constant number of photons «tot = X]mode«mode, the maximal entropy is obtained when the photons are equally distributed among all the modes, whereas the minimal entropy is obtained when the laser operates with a single mode. Moreover, the second law of thermodynamics (dS ^ 0) implies that photons can only be transferred from mode 1 to mode 2 if «i >«2, so a transformation from multimode operation to single-mode operation is not possible. Also, the entropy of a single high-order mode is equal to that of a single Gaussian mode. Thus, it is possible thermodynamically to transform a high-order mode into a Gaussian beam without losses. The entropy in eq. (16) depends on the total number of photons. Similarly, it is possible to define the mode distribution frmction (MDF) that does not depend on the number of photons. This MDF was referred to as the information entropy by Bastiaans [1986], namely ^^^
/ ^ ^mode ^^ ^mode? mode
^mode
^mode
(17)
«tot
The entropy and MDF were compared to the M^ value for a laser operating with the two lowest-order modes. The results, as a function of the relative number of photons in each mode, are presented in fig. 33. Here, both the entropy and the MDF reach a maximum (poorest beam quality) when the photons are equally divided between the two modes, namely e{) = e\ = \, whereas the M^ value decreases monotonically with EQ. Thermodynamically, it is possible to reduce the M^ value from that shown by point 1 to that shown by point 2 since the entropy of these two states is the same.
6, § 4]
Properties of the laser output beams
371
It should be noted that the coherence properties, the MDF and the entropy depend significantly on the modal structure of the beam. This modal structure can be evaluated from intensity distribution cross-section measurements (see for example Cutolo, Isernia, Izzo, Pierri and Zeni [1995] or Santarsiero, Gori, Borghi and Guattari [1999]) or coherence measurements (Warnky, Anderson and Klein [2000]). Thus, the MDF and entropy can be measured experimentally. Also, the entropy is related to the possible brightness improvement of a beam. The brightness is inversely proportional to the M^ value, thus, a brightness improvement is concomitantly obtained with the reduction of the M^ value. Note that there are two possible orthogonal polarization states, and the above discussion is valid for each of them.
4.2. Intensity and phase distributions In this subsection, we consider the field distributions of beams that emerge from laser resonators. Properties of such field distributions along with methods to distinguish between them are presented. Moreover, methods to improve the focusability of beams having specified field distributions are demonstrated both theoretically and experimentally. 4.2.1. Uniform phase distribution Beams with a uniform phase distribution can be shaped or transformed using various techniques (see for example Bryngdahl [1974] and Davidson, Friesem and Hasman [1992]). The most widespread laser output beam with uniform phase is the Gaussian beam, in which the transverse intensity distribution is maintained while propagating, leading to simple propagation properties. Such Gaussian beams can be readily obtained fi:om lasers operating with only the ftindamental TEMQO mode. For other beams with uniform phase, the transverse intensity distribution is changed during propagation, and their M^ value is greater than unity. Their propagation properties depend on their intensity distributions in the near field. For example, the propagation properties of super-Gaussian beams, which can be obtained from laser resonators with intra-cavity diffractive elements or GPMs, were analyzed by Parent, Morin and Lavigne [1992]. 4.2.2. Binary phase distribution Beams emerging from a laser operating with a high-order Hermite-Gaussian mode or a high-order degenerate (non-helical) Laguerre-Gaussian mode have
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Transverse mode shaping and selection
[6, § 4
binary phase distributions, which consist of lobes and rings, where neighboring lobes or rings have opposite phases {n phase shift). Casperson [1976] compensated for the phase differences between neighboring lobes or rings, by letting the output beam pass through a binary phase element with Jt phase shifts in proper locations. This increased the peak intensity and power in the main lobe of the farfield intensity distribution, implying a better beam quality. Yet, Siegman [1993] calculated that the beam quality, in terms of M^ is not improved but remains the same. Indeed, he concluded that binary phase plates cannot improve the M^ value. The contradiction between the two approaches results from different criteria for beam quality. The beam quality in accordance to percentage of power in the main lobe criterion is hardly affected by low-power side-lobes, whereas in accordance to the M^ criterion the side-lobes contribute significantly to the improvement in the M^ value. Optimized binary phase-compensating elements were tested experimentally by Casperson [1977] and Casperson, Kincheloe and Stafsudd [1977] with HeNe and CO2 lasers, yielding improvement in the peak power and percentage of power in the main lobe, in agreement with predictions. Lescroart and Bourdet [1995] analyzed binary phase-compensating elements for improving the far-field characteristics of an array of waveguide lasers and determined the trade-off between a main lobe with high peak but with side-lobes to that of main lobes of lower power concomitant with very low side-lobes. Lapucci and Ciofini [1999] optimized the design of a binary phase-compensating element for narrow annular laser sources. Note that, in a laser configuration in which a DPE is inserted next to the output coupler, as shown in fig. 12 (see sect. 3.3), there is no need for a phase-compensating element, since the mode-selecting DPE acts also as a phasecompensating element. Baker, Hall, Hornby, Morley, Taghizadeh and Yelden [1996] showed that by introducing a binary phase-compensating element, the beam emerging from a waveguide laser operating with a high-order antisymmetric mode is transformed so that the far-field distribution consists of a high-intensity main lobe and lowintensity side-lobes. Moreover, by resorting to spatial filtering in the far-field, the side-lobes were eliminated, thereby significantly improving the M^ value with a relatively small decrease in power. Specifically, an original beam with M J =21.7 was transformed into a beam with M^ close to unity with an efficiency of 59%. 4.23. Helical phase distribution In general, the intensity distribution of a helical laser beam is the same as that of a doughnut-shaped laser beam, but their field distributions are distinctly different.
6, § 4]
Properties of the laser output beams
373
Fig. 34. Experimental interference fringe patterns: (a) Gaussian beam; (b) lowest-order helical beam; (c) lowest-order helical beam of opposite helicity. (From Harris, Hill, Tapster and Vaughan [1994].)
Specifically, the doughnut-shaped laser beams are composed of an incoherent superposition of two TEMQ/ modes. For example, when the two field distributions of the TEMoi(x) and TEMQ 1(3;) modes in eq. (6) are added incoherently, they form a hybrid mode whose intensity distribution is doughnut-shaped. On the other hand, when they are added coherently with the appropriate phase, they form a pure helical mode. Several techniques were developed to distinguish between helical and doughnut beams, and between helical beams of opposite helicity. In one technique, the determination whether a beam is helical, having a phase of exp(i0), is done by examining the interference of the beam with its mirror image or with a reference beam (see Indebetouw [1993], Harris, Hill and Vaughan [1994], or Harris, Hill, Tapster and Vaughan [1994]). Examples of such interference patterns are shown in fig. 34. As evident, the helicity can be easily obtained from the fringe pattern. Alternatively, one can let the emerging beam pass through another SPE. An SPE having a phase of exp(-i/0) will focus the helical beam to obtain a main lobe with a high central peak intensity, whereas one having a phase of exp(+i/0) will diverge it fiirther away from the center. This property is unique to the helical beams formed by the TEMo,+/ modes. For the beams formed by the hybrid mode, either one of these two SPEs will focus the hybrid beam to a main lobe with a high central peak intensity, since all parts of the beam are approximately in phase. Experimental results for the helical beams formed by the TEMo,+i mode are shown in fig. 35, along with those predicted for hybrid and helical beams. Figure 35a shows the cross-sections of the far-field intensity distributions with the first phase-correcting SPE having a phase of exp(-i0). As evident, there is a main lobe with a high central peak intensity and very low side-lobes, in agreement with those predicted for a helical beam, while the incoherent hybrid beam has more power spreading. Figure 35b shows the corresponding far-field cross-sections of the intensity distributions with the second SPE of exp(i0). Here the energy spreads out from the center to form an annular shape, as expected for a helical beam. However, for a hybrid beam, no spreading should occur, and there
[6, § 4
Transverse mode shaping and selection
374
Fig. 35. Experimental and calculated far-field intensity distribution cross sections with an additional transmittive SPE: (a) SPE of exp(-i0); (b) SPE of exp(+i0) (dashed lines, experimental results; sohd lines, calculated results for the coherent-helical; dotted lines, calculated results for the incoherenthybrid). (From Oron, Davidson, Friesem and Hasman [2000b].)
Still is one main central lobe. These results clearly indicate that the emerging beam is indeed helical. An interesting property of helical beams is that their M^ value can be significantly improved. Oron, Davidson, Friesem and Hasman [2000a] showed that continuous-spiral phase elements can improve the M^ value of helical beams. Specifically, a single high-order helical beam was transformed into a nearly Gaussian beam. An arrangement for transforming the helical output beam into a nearly Gaussian beam is shown schematically in fig. 36. A helical TEMo,+/ beam, with a field distribution given by eq. (1), emerges from the laser in which a reflective SPE is inserted. The beam is collimated by a cylindrical lens, and its M^ value is 1 + /. In the optical mode converter, the collimated beam first passes through a transmissive SPE, which introduces a phase of exp(-i/0), thereby modifying the helical-phase distribution into a uniform distribution yielding £opl'l''L]/'(p)exp(-p/2). Laser resonator
Optical mode converter Transmissive SPE
U
-^u—>\
Spatial filter (Back mirror)
Fig. 36. Basic configuration of a laser resonator that yields a high-order helical mode and an optical mode converter that yields a nearly Gaussian mode. (From Oron, Davidson, Friesem and Hasman [2000a].)
6, § 4]
Properties of the laser output beams
375
Table 1 Initial and final M^ values and transformation efficiency r/, for a laser operating with either the fiindamental mode or high-order helical modes Mode
Initial M^
Final M^
Transformation efficiency r]
TEMoo
1
1
TEMo, + i
2
1.036
94%
TEMo, + 2
3
1.06
87%
TEMo, + 3
4
1.07
80%
TEM0. + 4
5
1.07
74%
100%
Analysis based on Fourier transformation of the near field and the secondorder moments reveal that the phase modification with the external SPE reduces the M^ value significantly, fi-om 1+/ to (1+/)^^^. This result is in contrast with that obtained for a laser operating with degenerate modes, where a correcting binary-phase plate can improve the peak power of the far-field intensity distribution, but not the M^ value. Moreover, the phase modification significantly changes the far-field intensity distribution, yielding a high central lobe and low ring-shaped side-lobes that contain only a small portion of the total power (e.g., 6% for a laser operating with the TEMo,±i modes). Thus, by exploiting a simple spatial filter (e.g., a circular aperture), it is possible to obtain a further significant improvement in the M^ value. Specifically, a nearly Gaussian beam, withM^ near 1 (theoretically 1.036 for the TEMo,+i mode), with only a small decrease in output power is obtained. Table 1 shows the calculated initial and final (after spatial filtering) M^ values, as well as the transformation efficiency ry, denoting the percentage of power in the main lobe, for a laser operating with either the fundamental mode or in high-order helical modes. Note that the transformation efficiency decreases as the order of the mode increases. The configuration shown in fig. 36 was tested with a linearly polarized CO2 laser in which a reflective SPE replaced the usual back mirror. The SPE was designed to ensure that the laser operated with the helical TEMo,+i mode, as described in sect. 3.4. The optical mode converter contained a transmissive SPE formed on zinc selenide substrate, a telescope configuration of two lenses the first (/i =50 cm) placed 50 cm from the SPE and the second (/2 = 25cm) 75 cm from the first - and a spatial filter in the form of a circular aperture. The intensity distributions were detected at the spatial filter plane and the output plane with a pyroelectric camera. The results are presented in figs. 37 and 38. Figure 37 shows the detected intensity distributions, along
376
Transverse mode shaping and selection
[6,
Fig. 37. Detected intensity distributions and experimental and calculated intensity cross sections at the spatial-filter plane: (a) without SPEs; and (b) with a transmissive SPE. (Solid lines, calculated; dashed lines, experimental). (From Oron, Davidson, Friesem and Hasman [2000a].)
with calculated and experimental intensity cross-sections at the spatial filter plane. Figure 37a shows the intensity distribution and cross-sections without the transmissive SPE. Thus, the usual nearly doughnut-shaped distribution of a helical beam whose phase was not compensated by the transmissive SPE is obtained. Figure 37b shows the intensity distribution and cross-sections when the transmissive SPE was inserted. As is evident, there is a high central peak with low side-lobes that are removed by spatial filtering, yielding a nearly Gaussian beam. Moreover, the detected intensity distribution is narrower than that obtained with no SPE, indicating the improvement of M^. Figure 38 shows photographs of the detected intensity distributions along with calculated and experimental intensity cross-sections at the output of the optical mode converter. Here, the calculated results were obtained by Fourier transformation of the field distribution in the spatial filter plane. Figure 38a shows the intensity distribution and cross-sections at the output plane, when the mode converter includes the SPE but no spatial filter. This is simply an image of the doughnut-shaped helical beam from the laser, whose intensity distribution results from a TEMo,+i mode. The SPE in this case does not affect the intensity distribution at the output plane but only its phase. Figure 38b shows the detected intensity distribution and cross-sections at the output plane with both the SPE and the spatial filter in the mode converter. As predicted, the intensity distribution has a Gaussian shape. In this case the efficiency r] was 85%, which is somewhat lower than the calculated limit of 94%. The M^ value of this beam was measured to be better than 1.1, as expected.
6, §4]
Properties of the laser output beams
?>11
Fig. 38. Detected intensity distributions and calculated and experimental intensity cross sections at the output of the optical mode converter: (a) without a spatial filter; and (b) with a spatial filter (solid lines, calculated; dashed lines, experimental). (From Oron, Davidson, Friesem and Hasman [2000a].)
4.2.4. Several transverse modes When the laser operates with multiple modes, i.e., fundamental and higher order modes, the emerging beam quality is relatively poor and is mainly determined by the highest-order mode. In such lasers the phase distribution of the output beam is random, and little, if anything, can be done to improve the quality of the beam. When the laser operates with a single high-order mode, the emerging beam quality is still inferior to that from a laser operating with the fundamental mode, because the intensity distribution and the divergence of the beam are relatively large. Yet, a beam which originates from a laser operating with a single highorder mode has well-defined amplitude and phase distributions, so in accordance to entropy, it is allowed thermodynamically to efficiently transform it into a nearly Gaussian beam (see sect. 4.1.3). A laser may also operate with only a few modes, where most of the modes between the fundamental and the highest-order modes are not present. Here again the phase distribution is undefined at any point of the beam emerging from the laser. Yet, Oron, Davidson, Friesem and Hasman [2001] demonstrated that beam quality could be improved in a laser operating with a limited number of modes N, much smaller than A/^T, namely. N
<^NT
= \[M^{M^
+ 1)] ^ |max(l +2;?+ |/|)l
(18)
In accordance with the total coherence parameter K of eq. (15), the beam quality depends only on the number of modes and their relative powers. Thus,
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Transverse mode shaping and selection
[6, § 4
according to the Wigner algebra, it is possible to reduce the WDF envelope volume of a beam from a laser operating with few modes, where the condition in eq. (18) is valid, towards that of the actual volume, thereby allowing the desired transformation that will improve the quality of the laser output beam. Such an improvement will be illustrated for a laser operating with only two incoherent transverse modes TEMi ±1, whose M^ value is 4. For comparison, the number of modes in a multimode laser with the same M^ would be 10 (see eq. 18). An Nd: YAG laser in which an annular DPE, consisting of a single discontinuity ring with a specified radius was inserted into the resonator, as shown in fig. 12, to simultaneously select both helical TEMi, ±1 modes. These two selected modes have the same radial dependence, but a different azimuthal dependence; thus, even an incoherent combination of these modes can be manipulated together in the radial direction. The optical beam converter contained a telescope configuration of two lenses with a spatial filter in the form of a circular aperture. The intensity distributions next to the output coupler, at the spatial filter plane and at the output plane, were detected with a CCD camera. Since the DPE is placed next to the output coupler, the emerging field distribution does not have the usual n phase shift between the two rings, so the far-field intensity distribution will be significantly affected. It now has a central ring, which contains most of the output power, and ring-shaped side-lobes that contain only a small portion of the power (e.g., 20.4% for a laser operating with the TEMi, ±1 modes). Thus, by exploiting the additional converter with a simple spatial filter (e.g., a circular aperture), it was possible to obtain a significant improvement in the M^ value. Specifically, a nearly doughnut-shaped beam, having M^ near 2 (theoretically 2.04 for the TEMi ±1 modes), was obtained with a relatively small decrease (20.4%) in output power. Such a reduction in M^ leads to a significant increase in the brightness of the beam, proportional to PIM^, where P is the power, leading to an improvement by a factor of 2.9. The WDF of the TEMi,+i mode has been calculated, and some representative results are shown in fig. 39. Since the Wigner distribution is four-dimensional, only a subspace which includes the origin and where r is parallel to p is presented. Figure 39a shows the WDF of a TEMi,+i mode, which consists of a central negative distribution, indicating a central phase singularity (vortex), and surrounding concentric rings. Figure 39b shows the WDF after passing through a DPE which eliminates the n phase shift between the two rings in the field distribution of the TEMi,±i mode. As evident, the shapes of the rings are not whole, leading to a high (and larger) central negative distribution, and more spread-out side-lobes. This is the first stage in contracting the WDF. Figure 39c shows the far-field WDF, which is obtained after Fourier transformation. This
6, § 4]
Properties of the laser output beams
(a)
(b)
(c)
(d)
379
Fig. 39. Wigner distribution function of helical TEMi+i mode at various locations: (a) WDF of original TEMi+i mode; (b) WDF of TEMi+i mode after passing an annular DPE; (c) far-field WDF of TEMi +1 mode after passing an annular DPE; (d) WDF of TEMi +1 mode after passing an annular DPE and spatial filtering. All distributions are identical to those of the TEMi _i mode and to an incoherent summation of the two modes. (From Oron, Davidson, Friesem and Hasman [2001].)
Fourier transformation practically rotates the WDF by 90 degrees, to exchange the r and/; axes. The last stage consists of spatial filtering, which cleans the WDF from most of the side-lobes, yielding a WDF similar to that of a TEMo,+i mode, still with a central negative distribution, but only a single outer ring. This WDF is shown in fig. 39d. Since the two modes TEMi,+i and TEMi,_i have similar radial dependence (though different azimuthal dependence), the above subspace of the WDF is the same for both modes, so figs. 39a-d actually show the WDF of either one of the two TEMi ±i modes or of an incoherent superposition of the two (since the WDF of an incoherent summation of two modes is the sum of the WDFs of these modes). Note that for only one of these modes is it possible to further improve the beam quality with an SPE (see sect. 4.2.3). The experimental results are presented in figs. 40 and 41. Figure 40 shows the detected intensity distributions, along with calculated and experimental intensity cross-sections in the near and far fields respectively. Figure 40a shows the intensity distribution and cross-sections at the output from the laser. The intensity distribution shows the two rings of the TEMi ±i modes. Figure 40b shows the corresponding far-field intensity distribution and cross-sections. There is a central doughnut-shaped pattern with some low side-lobes, that are later removed by the spatial filtering, to obtain a nearly doughnut-shaped beam. Figure 41 shows the near- and far-field intensity distributions along with
380
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[6, § 4
Fig. 40. Detected intensity distributions and calculated and experimental intensity cross sections at near and far fields: (a) at the output of the laser; and (b) at the spatial filter plane. (Solid lines, calculated; dashed lines, experimental). (From Oron, Davidson, Friesem and Hasman [2001].)
Fig. 41. Detected intensity distributions and calculated and experimental intensity cross-sections at the near and far fields after the optical mode converter: (a) near field; (b) far field. (Solid lines, calculated cross sections; dashed lines, experimental cross-sections). (From Oron, Davidson, Friesem and Hasman [2001].)
calculated and experimental intensity cross-sections of the beam at the output of the optical beam converter. As predicted, the intensity distribution is doughnut-shaped in both the near and far fields. In this case the measured efficiency was 76%, in agreement with the calculated limit of 79.6%. The M^ value of this beam was nearly 2, as expected, leading to significant brightness improvement, in agreement with prediction.
6, § 4]
Properties of the laser output beams
381
The output power was 9.5 W directly from the laser, and 7.2 W after the mode converter. For a similar laser, with no phase element, the output power was only 5.5 W, obtained by simply opening the aperture to allow TEMo,±i-inode operation; however, in this case, the laser operation included the fundamental mode, which was not suppressed. Overall, a high-quality doughnut beam was obtained, with a significantly higher power than would normally be obtained. Note, the transformation efficiency in this case was 79.6%, but it could be increased to 100% by resorting to SPEs, rather than DPEs, and selecting highorder helical modes having different p but the same /, such as TEMQ, +2 and TEMi,+2 modes for laser operation.
4.3. Selected applications In this subsection, selected applications that require special laser beam properties are briefly described, along with the possible mode-shaping or mode-selecting methods. Some applications require that the laser beam have special transverse intensity distributions, such as doughnut shape, which are particularly useful for trapping particles and atoms. Specifically, Sato, Harada and Waseda [1994] exploited doughnut-shaped beams in order to trap small metallic particles. Kuga, Torii, Shiokawa, Hirano, Shimizu and Sasada [1997] exploited doughnut-shaped Laguerre-Gaussian beams to trap laser-cooled rubidium atoms in the dark region of the beams. Doughnut-shaped beams can have additional interaction with trapped particles. He, Friese, Heckenberg and Rubinsztein-Dunlop [1995] showed that if the trapping beam is helical, the angular momentum from the laser beam is transferred to absorptive particles. One can control the angular momentum transfer by changing the helicity of the beam. Moreover, a radially polarized laser beam can be applied for accelerating electrons (Liu, Cline and He [1999]). Doughnut-shaped beams were also used by Charters, Luther-Davies and Ladouceur [1999] for directly exposing photosensitive materials to form waveguides. They showed that with doughnut-shaped beams the uniformity of the waveguide refraction index is better than with conventional beams with Gaussian profile. Other applications require that the laser beams have special phase distribution such as a helical beam with a phase singularity. The properties of such helical beams, namely optical vortices, were analyzed by Soskin, Gorshkov, Vasnetsov, Malos and Heckenberg [1997], who developed various rules regarding the topological charge and angular momentum that characterize the propagation of such beams or combinations of such beams, and showed annihilation and
382
Transverse mode shaping and selection
(a)
[6, § 4
(b)
Fig. 42. Ray-tracing geometry and polarized light orientation for focusing coUimated beams: (a) linearly polarized beam; and (b) radially polarized beam. (From Quabis, Dom, Eberler, Glockl and Leuchs [2000].)
appearance of new vortices. Courtial, Robertson, Dholakia, Allen and Padgett [1998] studied the rotational frequency shift of light beams, which is proportional to the photon angular momentum. Experimental measurements were performed with helical Laguerre-Gaussian beams and radially polarized (or other specially polarized) beams, in the millimeter wavelength range. Still other applications require that the laser beams have special polarization properties. Specifically, Marhic and Garmire [1981] exploited azimuthally polarized CO2 laser beams for transmission through hollow metallic waveguides. The azimuthally polarized beam has a significantly higher reflection, and consequently lower transmission losses through the hollow waveguides. In laser material processing applications, the proper choice of polarization can result in desirable lower reflectivity and higher absorption. Niziev and Nesterov [1999] analyzed the influence of beam polarization on laser cutting efficiency, and showed that beams with radial polarization are more effective in cutting and drilling than linearly polarized beams, whereas azimuthally polarized beams are very poor. This is because laser beams with azimuthal polarization are readily reflected whereas those with radial polarization are highly absorbed leading to more efficient cutting. Also, Quabis, Dom, Eberler, Glockl and Leuchs [2000] showed that radially polarized beams could be focused into smaller spots than linearly polarized beams (or azimuthally polarized beams). This is due to vectorial effects, and can be explained with the aid of fig. 42. With the linear polarization in fig. 42a, a partial cancellation of the field (destructive interference) between different parts of the beam occurs, whereas with the radial polarization shown in fig. 42b, all parts of the beam focus constructively to obtain a field which is essentially parallel to the optical axis. Youngworth and Brown [2000] performed some experiments on focusing of such beams, and the results agree with predictions.
6]
References
383
§ 5. Concluding remarks An important property in many applications is that the power of the laser output beams be as high as possible. In many cases, high powers can be obtained by resorting to lasers operating with a high-order mode, whose intensity distribution has a larger cross-section than the fundamental Gaussian mode, so it could more efficiently exploit the gain medium. Indeed, high powers were demonstrated either with lasers in which GRMs, DPEs or SPEs were inserted into the resonator, or with self-imaging resonators. Moreover, because spatial coherence of the beams emerging from such lasers is high, they could be efficiently transformed into a nearly Gaussian beam (see sect. 4.2). The use of intra-cavity mode shaping and single high-order mode selection, rather than external mode shaping, has two advantages. First, the laser output power is relatively high since a larger volume of the gain medium is exploited. Second, there is no need for external beam shaping, which introduces both additional losses and some distortions to the output intensity distributions. When choosing between the different mode-selecting and mode-shaping methods, one should note that several methods are susceptible to small changes in the resonator, such as its length, or lensing properties of the gain medium (which can be caused by operating the laser with different pump powers). These include GPMs, diffractive elements and self-imaging or Fourier resonators, which are designed only for a specific set of parameters. Also, several methods are limited to specific resonator configurations. For example, selfimaging (Talbot) resonators are practical mainly for waveguide or slab-laser configurations and are significantly less efficient for other types of resonators; Fourier resonators require the existence of Fourier planes in the resonator; and some methods are specific to unstable resonators. Other methods are more general, and could be exploited in various resonator configurations and parameters. Acknowledgements The authors would like to thank the Pamot Venture Capital Fund and the Israeli Ministry of Science for their support. References Abramochkin, E.G., N. Losevsky and VV Volostnikov, 1997, Opt. Commun. 141, 59. Abrams, R.L., and A.N. Chester, 1974, Appl. Opt. 13, 2117.
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Author index for volume 42
Aravind, PK. 311 Arecchi, ET. 9, 222 Arlt, J. 246, 268 Armstrong, L.A. 190 Asatryan, A.A. 255 Ashkin, A. 257 Askaryan, G.A. 266 Aspect, A. 12, 280, 298 Assanto, G. 96, 103 Atai, J. 98 Atatiire, M. 12, 305-307 Autler, S.H. 212
Abramochkin, E.G. 249, 259, 269, 364 Abrams, D.S. 68 Abrams, R.L. 333 Abramski, K.M. 355 Aceves, A.B. 96, 99, 100 Ackemann, T. 255, 257 Adam, P.M. 269 Adami, C. 67 Adelman, L. 56 Aerts, S. 288 Afanas'ev, A.A. 109 Agarwal, G.S. 74, 77, 80, 201, 212, 368, 369 Agrawal, G.P. 100 Aharonov, Y. 222, 279 Ait-Ameur, K. 364 Akhmediev, N.N. 222, 268 Akozbek, N. 97,118 Aksenov, V 269 Aide, D.M. 54, 55 Alexander, T.J. 267 Alexeyev, A.N. 271 Allen, A. 268 Allen, L. 222, 243, 246, 253, 264, 265, 268, 271, 382 Alley, CO. 286, 291 AltenmuUer, T.P. 156, 160 Altepeter, J. 303 Alvarez-Estrada, R.F. 150, 199 Andersen, D.R. 266 Anderson, B.L. 371 Anderson, D. 97 Anderson, M.A. 71 Angelow, G. 337 Angelsky, O.V 272 Ankiewicz, A. 222 Apanasevich, PA. 12 Appelbaum, I. 297, 298
B Babiker, M. 222, 243, 246, 271 Bach, H. 71 Bacry, H. 38 Badurek, G. 150, 156, 160, 163, 176 Baker, H.J. 355, 372 Baldauf, H.W. 70 Ballentine, L.E. 156, 160 Ballman, A.A. 257 Balmer, J.E. 356, 369, 370 Banakh, V 269 Banakh, VA. 269 Banerji, J. 356 Baranova, N.B. 225, 251 Barashenkov, I.V 96, 101, 124 Barenco, A. 63 Bargmann, V 38 Barnett, S.M. 33 Barthelemy, A. 355 Basche, T. 71 Basharov, A.M. 107, 108 Basistiy, I.V 231, 246, 249, 253, 259, 264, 269 Bass, M. 184 387
388
Author index for volume 42
Bastiaans, MJ. 370 Bazhenov, VYu. 246, 253, 259, 261, 264, 268-271 Beck, M. 73 Bednar, C.J. 15 Beige, A. 157, 160 Beijersbergen, M.V 231 Beijersbergen, M.W. 347 Bekshaev, A.Ya. 245 Belanger, P.A. 335-337, 363, 365 Bell, A.S. 268 Bell, J.S. 10, 279 Ben-Aryeh, Y.J. 73 Benedict, M.G. 108 Bennett, C.H. 42, 46, 49, 50, 52, 53, 55, 57, 68, 81, 82, 304, 309, 315 Benson, O. 73 Berestetskii, VB. 203 Berglund, A.J. 303 Berman, P.R. 71, 73 Bemardini, C. 150, 199 Bernstein, E. 68 Bernstein, H.J. 49, 311 Bemtson, A. 97 Berry, M.V 157, 160, 214, 221, 222, 225, 226, 228, 235, 250-255, 258, 261, 263, 271 Bertet, P. 73 Berzanskis, A. 265, 267-269, 271 Beskow, A. 150, 152, 155 Bessette, F. 55 Bharucha, C.F. 152 Bijeon, J.P 269 Birkl, G. 71 Bjork, G. 73 Blaauboer, M. 98, 136-138 Blanchard, Ph. 156, 160 Blattner, P 270 Blit, S. 358-362 Bloch, F. 153 Bloembergen, N. 190, 264, 267, 280 Bloemer, M.J. 95 Boardman, A.D. 266 Boccaletti, S. 222 Boggs, B. 140 Bogomolov, VN. 70 Bohm, D. 222, 279 Bohr, N. 4 Boivin, A. 222-224, 231, 235 Boiler, K.J. 201, 212 Bollinger, J.J. 150, 156, 160, 163
Bomzon, Z. 358-361 Borek, G.T. 365 Borghi, R. 371 Bom, M. 221, 231, 242, 258, 269, 369 Boschi, D. 42,45, 318, 319 Bose, S. 15,48,315 Botez, D. 356 Bourderionnet, J. 365 Bourdet, G. 372 Bourdet, G.L. 355 Bourliaguet, B. 355 Bouwmeester, D. 41, 44, 45, 47, 282, 313317 Bowden, CM. 95, 108 Bowers, M.S. 362 Boyd, G.D. 257 Boyd, R.W. 264, 267 Braginsky, VB. 69, 73 Branca, S. 42,45,318, 319 Branning, D. 289, 300-302 Brassard, G. 42, 52, 53, 55, 68, 304, 309, 315 Brassard, J. 49, 50 Braunbek, W. 222-224 Braunstein, S.L. 42, 46, 48, 51 Breguet, J. 55 Breit, G. 151 Brendel, J. 287, 305, 308 Brendel, N. 285 Brignon, A. 365 Brown, D.R. 365 Brown, T.G. 100, 382 Brune, M. 12, 69, 70, 73, 77 Brunei, Ch. 73 Bruss, D. 318 Bryngdahl, O. 225, 371 Burgers, J.M. 226 Bumham, D.C. 12, 282 Buryak, A.V 267 Butera, P 38 Buttler, W.T. 55 Buzek, V 34, 318 Byer, R.L. 190, 280
Calderbank, A.R. 81 Calsamiglia, J. 14 Cantin, D. 363, 365 Carri, J. 13
Author index for volume 42 Carter, SJ. 98 Carter, W.H. 235 Casperson, L.W. 364, 372 Caves, CM. 31, 51, 72 Cerf, N.X 67 Chambers, R.G. 222 Champneys, A.R. 96, 102, 105 Chan, VW.S. 9 Charters, R.B. 381 Chekhlov, O. 71 Chen, D. 337-340 Chen, YJ. 98 Chen, Z. 266 Cheng, Z. 98, 99 Cherezova, T.Y. 366 Chesnokov, S.S. 366 Chester, A.N. 333 Chiao, R. 287, 290 Chiao, R.Y. 98 Chinaglia, W. 267, 268, 271 Chirkin, A.S. 190 Christodoulides, D.N. 96, 99, 100 Christou, J. 266 Chu, P.L. 96, 102, 104, 105 Chuang, I.L. 72 Ciofini, M. 372 Cirac, J.I. 48, 51, 71, 73, 74, 77 Clarke, R.B.M. 268 Clauser, J.F. 10, 67, 280, 286 Cleve, R. 69 CUlford, M.A. 268 Cline, D. 381 Coerwinkel, R.RC. 347 Cohen-Tannoudji, C. 12, 201, 210 Collet, L. 70 Colley, A.D. 355 Colombeau, B. 354 Commins, E.D. 279 Conti, C. 96, 101, 103 Cook, R.J. 150, 160, 163 Cormier, M. 363 Cornell, E.A. 71 Cory, D.G. 72 Courtens, E. 9 Courtial, J. 253, 265, 268, 382 Cover, T.M. 21 Crasovan, L.-C. 139 Crenshaw, M.E. 108 Crepeau, C. 42, 315
Crosignani, B. 266
389
Cubitt, A.B. 71 Cullet, P. 222, 225 Cutolo, A. 371
D Dai, K. 338, 339 Dalibard, J. 12, 298 Dancoff, S. 201 Dandliker, R. 270 Daniell, M. 313,315 Danziger, Y 333, 342, 344, 346, 347, 350352 D'Ariano, G.M. 15 Darsht, M.Ya. 270 Daschner, W. 365 Dauler, E. 299, 300 Davidovich, L. 69 Davidson, N. 333, 342, 344, 346, 347, 349353, 358-362, 364, 371, 374, 376, 377, 379, 380 Davies, A.R. 356 Davies, E.B. 78 Davis, L. 288 De Caro, L. 290 de Gennes, P.G. 222 De La Rue, D.M. 70 De Martini, E 42, 45, 318, 319 de Rossi, A. 96, 101 De Sterke, CM. 96, 100, 101 Denisenko, VG. 245 Dennis, M. 225, 261, 263 Desyatnikov, A. 138 Deutsch, D. 63, 72 Deutsch, I.H. 98 Deykoon, A.M. 222 Dholakia, K. 264, 265, 268, 382 Di Porto, P 266 Di Trapani, P 267, 268, 271 Dickson, R.M. 71 Dieks, D. 318 Dirac, PA.M. 222 DiVincenzo, D.P 49, 63, 81, 82, 318 Dmitriev, VG. 264 Dohnalik, T. 354 Donley, E. 71 Donnely, R.J. 222 Doogin, A.V 271 Dom, R. 382 Dow, J. 222-224,231,235
390
Author index for volume 42
Dowling, IP. 67, 95 Drabenstedt, A. 71 Dragoman, D. 368, 369 Drampyan, R.Kh. 272 Dreyer, X 77 Dritz, VV 109 Drummond, P.D. 31, 96-98, 103, 105, 138 Duan, L.-N. 48 Ducuing, J. 190 Dupont-Roc, J. 201 Dur, W. 51 Duree, G. 266 Dyer, P. 54, 55 Dziedzik, J.M. 257
Ebeling, K.J. 71 Eberhard, PH. 290, 297, 298 Eberler, M. 382 Eberly J.H. 210 Edman, L. 71 Eggleton, B.J. 96, 100, 101 Ehrlichmann, D. 333 Eibl, M. 41,44,45,47, 317 Einstein, A. 4, 279 Eisenberg, H.S. 272 Eisert, J. 50 Ekert, A. 304, 308 Ekert, A.K. 45, 52, 54, 63, 69, 71, 73, 80-82, 282, 303,312, 315, 318 Elkin, N.N. 340, 356 Elyutin, S. 107 Englert, B.-G. 15 Ensher, J.R. 71 Eppich, B. 369 Etrich, C. 103 Everitt, H.O. 95
F Facchi, P 150, 151, 159, 175, 176, 184, 195, 199, 203, 204, 206,,209,,211.,213,,214 Fadeeva, T.A. 271 Fadeyeva, T.A. 271 Fahmy, A.F 72 Faltis, A.V 269 Fano, R.M. 19 Fano, U. 210 Farhi, E. 68
Faridani, A. 73 Fedorenko, D.V 256 Fedoseyev, VG. 244 Fejer, M.M. 190 Feng, J. 96, 99 Fermi, E. 151 Feynman, R.P 14, 60, 64, 153 Field, J.E. 201, 212 Filho, M.R.L. 74, 77 Firth, W.J. 266 Fischer, E.Z. 70 Fischer, M.C. 152 Fleury, L. 71 Fock, VA. 152, 279 Fofanov, Ya.A. 73 Forbes, G.W. 255 Fouckhardt, H. 355 Fox, A.G. 331 Franson, J.D. 55, 287, 288 Frantzeskakis, D. 98 Freedman, S.J. 286 Frerichs, V 156, 160 Freund, I. 250-253, 268, 269, 271 Friberg, A.T. 369 Friberg, S.R. 98 Friedman, M.J. 102 Friese, M.E.J. 245, 268, 381 Friesem, A.A. 333, 342, 344, 346, 347, 349353, 358-362, 364, 371, 374, 376, 377, 379, 380 Froehly, C. 354, 355 Fry, E.S. 280 Fu, C.H. 364 Fuchs, C.A. 42, 46 Fujimoto, M. 270 Furusawa, A. 42, 46
G Gabor, D. 3 Gacs, P 21 Gamow, G. 151 Gaponenko, N.V 70 Gaponenko, S.V 70 Garmire, E. 133, 382 Garraway, B.M. 74, 77 Garrison, J.C. 98 Garuccio, A. 290 Gase, R. 368 Gea-Banacloche, J. 192
Author index for volume 42 Gerchberg, R.W. 332 Germanenko, I.N. 70 Gerry, C. 74, 77 Gershenfeld, N.A. 72 Ghirardi, G.C. 156 Ghosh, R. 156 Giallorenzi, T.G. 282 Gibbs, H.M. 140, 141 Gil, L. 222, 225 Gilmore, R. 9 Girardello, L. 38 Gisin, N. 33, 55, 285, 305, 308, 318 Glauber, R.I. 282 Glockl, O. 382 Glova, A.F. 356 Golubev, Yu.M. 73 Gontis, V 214 Gordon, E.I. 225 Gordon, J.P. 31 Gori, F. 371 Gorodetsky, M.L. 69 Gorshkov, VN. 231, 234, 235, 246, 249, 251, 253, 268, 381 Gottesman, D. 81 Gouy, M. 233, 258 Graf, T. 356, 369, 370 Grangier, R 12, 280 Gray, R.M. 21 Greenberger, D.M. 10,51,311 Greiner, C. 140 Griedke, G. 48 Grossman, A. 38 Grover, L.K. 67 Gruber, A. 71 Grynberg, G. 201 Gu, B.Y. 357 Guattari, G. 371 Gurzadyan, G.G. 264 Gutmann, S. 68
H Haase, T. 368 Habich, U. 333 Hach, E.E. 74,77 Hagan, D.J. 266 Hagley, E. 12, 77 Hahn, E.L. 97, 105, 107 Hahn, K.H. 201,212 Hajnal, J.V 235, 263
391
Hald, J. 72 Hall, D.R. 355, 372 Hamming, R.W. 24, 26 Hannay, J.H. 235 Harada, Y. 381 Hardy, L. 42,45,318,319 Hariharan, R 233, 258, 282 Haroche, S. 12, 69, 70, 73, 77 Harris, M. 269, 347, 373 Harris, S.E. 201, 212, 280 Hartley, R.VL. 18 Hasman, E. 333, 342, 344, 346, 347, 349353, 358-362, 364, 371, 374, 376, 377, 379, 380 Haus, H.A. 98 Hauslanden, R 32 Havel, T.R 72 Hayata, K. 97 He, H. 96-98, 103, 105, 268, 381 He, P. 381 Heckenberg, N.R. 245, 246, 251, 253, 268, 381 Hegerfeldt, G. 157, 160 Heinzen, D.J. 150, 156, 160, 163 Hellund, E.J. 152 Hellwarth, R.W. 153 Herbert, N. 318 Herzig, H.P. 270 Herzog, T. 55 Herzog, T.J. 150, 163, 176 Hill, C.A. 347, 373 Hillery, M. 318 Hirano, T. 268, 381 Hizanidis, K. 98 Hodgson, N. 327, 328, 331, 368 Hoffges, XT. 70 Holevo, A.S. 31, 32,78 Holt, R.A. 280 Home, D. 152, 155, 156, 160 Hong, C.K. 176, 284, 285 Honold, A. 98 Hood, C.J. 73 Hornby, A.M. 372 Home, M.A. 10, 45, 51, 280, 286, 287, 311-313, 315 Horodecki, M. 51 Horodecki, P. 51 Horodecki, R. 51 Horoshko, D.B. 13, 40, 48, 73, 74, 77, 79, 80 Howell, J.C. 67
392
Author index for volume 42
Hradil, Z. 156, 282 Huang, H. 201 Huffman, D.A. 19 Hughes, RJ. 54, 55 Huignard, J.-P. 365 Huot, N. 365 Huttner, B. 33, 55
I Ignatowskii, VS. 223 Ilchenko, VS. 69 Ilyenkov, A. 257 Ilyenkov, A.V 257 Imamoglu, A. 201, 212 Imoto, N. 33, 73 Inagaki, S. 156, 160 Indebetouw, G. 251, 259, 373 Inguva, R. 108 Isemia, T. 371 Itano, W.H. 150, 156, 160, 163 Itano, W.M. 71, 73, 77 Ivanovic, I.D. 33 Izzo, I. 371
Jacobs, B.C. 55 Jacobsen, H. 270 Jadczyk, A. 156, 160 Jaeger, G. 299, 300, 305-307 Jahnke, F. 140, 141 Jann, A. 73 Jansen, M. 356 Jarrett, J. 279 Jelesko, F. 71 Jenkins, R.M. 356 Jennewein, T. 55, 298, 305, 309, 310 Johansson, S. 369 John, S. 70,97,99, 118, 120 Johnson, N.P. 70 Joichi, I. 150, 199 Jonathan, D. 15, 49 Joseph, R.I. 96, 99, 100 Jozsa, R. 32, 36,42, 51, 63,71, 72, 82,315 Jundt, D.H. 190 Jurco, B. 282 K Kan, H. 73 Kanashov, A.
97
Kane, B.E. 71 Kapitonov, A.M. 70 Kapitsa, S.P. 5 Kaplan, A.E. 266 Kaptsov, L.N. 366 Karamzin, Y.N. 266 Karnian, G.P 231,363 Kartner, EX. 98 Kasevich, M. 150, 163, 176 Kassner, S. 71 Kataevskaya, I.V 270 Kaulakys, B. 214 Kermene, V 354, 355 Khalfin, L.A. 150, 152, 155 Khalih, EYa. 73 Kheruntsyan, K.V 98 Khitrova, G. 140, 141 Khizhnyak, A.I. 254-257 Kiess, T.E. 291 Kikuchi, H. 364 Kilin, S.Ya. 12, 13, 40, 48, 70, 71, 73, 74, 77-80 Kim, G.H. 356 Kim, J. 73 Kimble, H.J. 13, 42, 46-^8, 69, 73 Kimura, W.D. 356 Kincheloe, N.K. 372 King, B.E. 71, 73, 77 Kira, M. 140, 141 Kiss, T. 73 Kitaev, A.Yu. 69 Kitaeva, VE 256 Kitagawa, M. 73 Kivshar, Yu.S. 95, 97, 135, 222, 266, 267 Klauder, J.R. 38 Klein, A.G. 176, 214 Klein, C.A. 371 Klein, S. 214 Kleinman, D.A 282 Klelpinski, D. 77 Klyshko, D.N. 12, 47, 267, 280, 282, 292, 294, 311 Kneubuhl, EK. 96, 99 Knight, PL. 34, 48, 51, 74, 77, 151, 201, 315 Knill, E. 47, 81 Koch, S.W. 140, 141 Kochemasov, G. 272 Kocher, C.A. 279 Koftnan, A.G. 99, 157, 159 Kogelnik, H. 239, 328
Author index for volume 42 Kol'chenko, A.P. 342, 365 Kolmogorov, A.N. 21, 29 Kolobov, M.I. 40 Kononenko, A.N. 249 Koshiba, M. 97 Kostka, R. 368 Kotelnikov, VA. 27 Kozhekin, A.E. 96, 97, 99, 109, 120, 122, 124, 125, 128, 135, 137 Kraus, K. 151, 171 Kravtsov, Yu.A. 255 Kreminskaya, L.V 249, 254-257 Krenn, G. 315 Kriege, E. 255, 257 Krinitskaya, T.B. 73 Kristiansen, M. 347 Krug, P.A. 96, 100, 101 Krylov, N.S. 22, 152 Kudryashov, A.V 366 Kuga, T. 268, 381 Kimdikova, N.D. 270, 271 Kurizki, G. 48, 96-100, 109, 120, 122, 124, 125, 128, 129, 132, 133, 135-138, 141, 157, 159 Kuz'min, R.N. 114 Kwiat, P.G. 12, 14, 47, 55, 67, 68, 150, 163, 176, 287, 290, 291, 293, 294, 296-298, 303
L Laabs, H. 369 Lachance, R.L. 336, 337, 365 Ladouceur, F. 381 Laeri, E 337 Laflamme, R. 47, 81 Lai, Y. 98 Lakoba, T.L 114 Lamb Jr, G.L. 107 Lamb Jr, W.E. 192 Lamoreaux, S.K. 55 Landau, L.D. 279 Lang, R. 192 Lange, M.D. 222 Lange, W. 70, 73, 255, 257 Lapucci, A. 372 Larsson, J.-A. 288 Lauder, M.A. 201 Laukien, G. 222-224 Lavigne, P. 371 Law, C.T. 260
393
Le Gall, J. 355 Lebedev, D.C. 28 Lederer, E 96, 103, 104, 122, 139 Lee, S.H. 365 Lee, W.-H. 225 Lefevre, L. 70 Leger, XR. 332, 337-340, 363 Leibfried, D. 73 Leighton, R.B. 14 Leonhardt, U. 73 Lescroart, G. 372 Letokhov, VS. 71 Leuchs, G. 98, 382 Leung, D.W. 72 Levitin, L.B. 31 Ley, M. 192 Li, Q.X. 364 Li, T. 239, 328, 331 Liberman, VS. 271 Lifshits, E.M. 203 Lin, Q. 340 Lindberg, A.M. 363 Linden, N. 51 Lisak, M. 97 Liu, J. 357 Liu, X. 98 Liu, Y. 381 Lloyd, S. 68, 72, 82 Loftus, T. 140 Logvin, Yu.A. 108 Loiko, N.A. 108 Long, P 365 Losevsky, N. 364 Loudon, R. 192 Louisell, W.H. 281 Lounis, B. 73 Lu, N. 192 Lu, Z.G. 364 Lugiato, L.A. 40 Luis, A. 150, 176, 180, 181, 188 Luther, G.G. 54, 55 Luther^Davies, B. 95, 97, 135, 222, 266, 381 Liitkenhaus, N. 14 Lysikov, A.Y 356
M Maali, A. 77 Mabuchi, H. 73 Macchiavello, C. 69, 73, 80, 81, 318
394
Author index for volume 42
Machida, S. 73 Mac Williams, F.J. 26 Madison, K.W. 152 Maevskaya, T.M. 71, 73 Magde, O. 280 Magel, G.A. 190 Mahr, H. 280 Maiani, L. 150, 199 Maimistov, A. 107, 138 Mair, A. 268 Maitre, X. 12 Mak, W.C.K. 96, 102, 104, 105 Maker, P.D. 266 Makki, S. 332, 363 Malomed, B.A. 96-98, 100-105, 109, 120, 122, 124, 129, 132, 133, 135-139, 141 Malos, J.T. 246, 251, 253, 268, 381 Malygin, A.A. 282 Malyshev, VA. 108 Mamaev, A.V 225,251,266 Mandel, L. 176, 178, 282, 284, 285, 287, 289 Mandelstam, L. 152 Mantsyzov, B.I. 114, 120 Marand, C. 55 Marburger, J.H. 133 Marhic, M.E. 382 Marienko, I.G. 231, 234, 235, 246, 249, 253 Marshall, A.V 49 Martienssen, W. 287 Masajada, J. 246 Massar, S. 318 Massudi, R. 363 Matijosius, A. 265, 267, 269 Maitre, X. 77 Matsuda, T. 270 Matsumoto, Sh. 150, 199 Matsumura, K. 357 Matthews, M.R. 71 Mattle, J.-W. 317 Mattle, K. 12, 41, 44, 45, 47, 293, 294 Mawst, L. 356 Mayer, A.R 102, 105, 124 Mazilu, D. 97, 139 McCall, S.L. 97, 105, 107 McDonald, G.S. 363 McDuff, R. 253 Meekhof, D.M. 71, 73, 77 Merkx, R.T.M. 270 Mermin, N.D. 55, 312, 315 Messerschmidt, D. 355
Migdall, A. 294, 295, 299-302 Mihalache, D. 97, 139 Mihokova, E. 151, 199, 208 Milbum, G.J. 12, 74, 79, 80 Miller, D.T. 28 Miller, H.Y. 332 Miller, R.E. 266 Minogin, VG. 71 Miquel, C. 81 Mirandi, S. 267, 268, 271 Misra, B. 150, 155, 156, 159, 160 Mista, L. 184 Mitchell, D.J. 266 Mitchell, J.R. 68 Moemer, W.E. 71 Mogilevtsev, D.S. 40, 70 Mohler, E. 287 Mokhun, A.I. 272 Mokhim, I.I. 272 Molina-Terriza, G. 254, 265 Mollow, B.R. 201, 282 Moloney, J.V 106, 121, 124, 136 Monken, C.H. 289 Monroe, C. 71, 73, 77 Mor, T. 13, 33 Morgan, G.L. 54, 55 Morin, M. 266, 362, 371 Morley, R.J. 355, 372 Morrow, RR. 152 Mosca, M. 69 Moses, H.E. 203 Mossberg, T.W. 140 Mottay, E. 354 Mowry, G. 340 Mugibayashi, N. 152 Mugnai, D. 157, 160 Mugnier, A. 355 Mukunda, N. 258 Muller, A. 55, 299, 300 Munroe, M. 73 Murao, M. 51 Murphy, J.B. 368, 369 Mushiake, Y. 357 Myatt, C.J. 73, 77 N Nakajima, N. 357 Nakano, H. 98 Nakazato, H. 152, 155, 156, 159, 163, 195, 213
Author index for volume 42 Namiki, M. 150-152, 155-157, 160, 163166, 176 Napartovich, A.A. 340 Napartovich, A.P. 356 Narducci, L.M. 69, 201 Nassan, K. 257 Nepomnyashchy, A. 266 Nesterov, A.V 357, 382 New, G.H.C. 363 Newell, A.C. 106, 121, 124, 136 Nielsen, M.A. 47, 49 Nieminen, T.A. 245, 268 Nikitenko, A.G. 342, 365 Nikogosyan, D.N. 264 Nilsson, J. 150, 152, 155 Niu, Q. 152 Niziev, VG. 357, 382 Nizovtsev, A.P. 71 Nogues, G. 12, 73 Nordholt, J.E. 55 Norris, D.J. 71 Nye, IF. 221, 222, 225, 226, 228, 235, 237, 263, 271
O Ochkin, VN. 356 Okuno, Y. 270 Olkin, I. 49 Omero, C. 156 O'Neil, A.T. 253 Opatmy, T. 48, 97, 100, 109, 129, 132, 133, 135, 137, 141 Ore, O. 69 Oron, R. 333, 342, 344, 346, 347, 349-353, 358-362, 374, 376, 377, 379, 380 Orrit, M. 71, 73 O'Shea, DC. 365 Oshman, M.K. 280 Osnaghi, S. 73 Ou, S.S. 356 Ou, Z.Y. 284, 285, 287
Paakkonen, R 337 Padgett, M.J. 222, 243, 246, 253, 264, 265, 268, 271, 382 Palma, G.M. 303 Pan, J.-W. 41, 44, 45, 47, 313-317
395
Pare, C. 335-337, 363, 365 Parent, A. 371 Paris, M.G.A. 15 Parkins, A.S. 47 Pascazio, S. 150-152, 155-157, 159, 160, 163-166, 175, 176, 184, 195, 199, 203, 204, 206,208,209, 211,213, 214 Pas'ko, VA. 236, 239, 258 Pati, A. 156 Pauli, W. 279 Paz, J.P 81 Pedrotti, L.M. 192 Peieris, R. 279 Peik, E. 70 Pelinovsky, D.M. 96, 101, 124 Penin, A.N. 282 Pereira, S. 100 Perelomov, A.M. 38 Peres, A. 33, 42, 49, 51, 151, 156, 160, 171, 315 Perina, J. 282 Perina, J. 9, 37, 150, 157, 160, 176, 180, 181, 184, 188, 213 Periin, V 96 Pershan, PS. 190 Peschel, T. 96, 103, 104, 122 Peschel, U. 96, 103, 104, 122 Peterson, C.G. 55 Petrosky, T. 151, 156, 160 Petrov, D. 265 Petrov, D.V 265 Petrov, E.P 70 Piche, M. 363, 365 Pierri, R. 371 Pilipetskii, N.E 225,251 Pirotta, M. 71 Pishnyak, O.P 256 Piskarskas, A. 265, 267, 269 Pismen, L.M. 222, 254, 260 Pitaevskii, L.R 203 Pittman, T. 296 Pittman, T.B. 292, 294, 295 Plenio, M.B. 15, 48, 49, 51, 151 Plum, H.D. 333 Podolsky, B. 4, 279 Pohl, D 356, 357 Poladian, L. 266 Poluektov, LA. 107 Polymilis, C. 98 Polzik, E.S. 13, 42, 46, 72
396
Author index for volume 42
Ponyavina, A.N. 70 Popescu, S. 42, 45, 49-51, 318, 319 Popov, Y.M. 107 Postan, A. 108 Poyatos, J.F. 74, 77 Prasad, S. 192 Preskil, J. 81 Prigogine, I. 151, 156, 160, 213 Prokofiev, A.V 70 Pykacz, J. 287 Q Qian, L.J. 98 Quabis, S. 382 Quang, T. 120
Rabi, I.I. 153 Radmore, RM. 201 Raimond, J.M. 12, 69, 70, 73, 77 Rains, E.M. 81 Raithel, G. 69 Raizen, M.G. 152 Ramazza, PL. 222 Ramsey, N.F. 153 Ranfagni, A. 157, 160 Rarity, J.G. 73, 286, 287 Rasetti, M. 74, 77 Rauch, H. 150, 156, 160, 163, 176 Rauschenbeutel, A. 73 Raymer, M. 73 Recolons, J. 254 Regan, J.J. 266 Rehacek, J. 184,213 Reid, M.D. 98 Renn, A. 71 Renyi, A. 17 Reynaud, S. 12, 210 Reznikov, Yu.A. 256 Richards, B. 223 Richter, Th. 73 Rigden, J.D. 225 Rigler, R. 71 Rigrod, WW. 335 Riis, E. 268 Rimini, A. 156 Rioux, M. 363 Rivest, R. 56 Robertson, D.A. 382 Robinson, PA. 233, 258
Rocca, F. 222,225 Rockstuhl, C. 270 Rodionov, A.Y. 347 Roger, G. 12, 280, 298 Roitberg, VS. 107 Romanov, S.G. 70 Ron, A. 156, 160 Rosanov, N. 222 Rosen, N. 4, 279 Rosenbluh, M. 98 Royer, P 269 Rozas, D. 260 Rubenchik, A. 97 Rubin, M.H. 287, 292, 294, 295 Rubinsztein-Dunlop, H. 245, 253, 268, 381 Rupasov, VI. 99, 107
Sackett, C.A. 77 Sacks, Z.S. 260 Saghafi, S. 367 Saito, S. 73 Salamo, G. 266 Saleh, B.E.A. 12, 184, 284, 305-307 Sammur, R.A. 267 Samoilovich, S.M. 70 Samson, A.M. 108 Samson, B.A. 109 Sanchez-Gomez, J.L. 150, 199 Sanchez-Soto, L.L. 176, 181, 188 Sanders, B.C. 282 Sanderson, R.L. 333 Sands, M. 14 Santarsiero, M. 371 Sapone, S. 267, 268, 271 Sasada, H. 381 Sato, S. 381 Savail, L. 55 Savchenko, A.Yu. 271 Saviot, A. 354 Saviot, F 354 Saxton, WO. 332 Scalora, M. 95, 108 Scardicchio, A. 213 Schack, R. 51, 72 Schadt, M. 356 Schauer, M. 54, 55 Schawlow, A.L. 212 Schenzle, A. 156, 160
Author index for volume 42 SchoUmann, J. 102, 105, 124 Schorl, C. 72 Schrama, C.A. 70 Schrodinger, E. 4, 16, 82, 279 Schulman, L.S. 151, 157, 159, 160, 170, 176, 199, 208, 214 Schumacher, B. 31, 32, 36, 49, 50 Schuster, J. 71 Schwindt, P.D.D. 68 Schwinger, J. 153 Scully, M.O. 15, 69, 192, 201, 282 Segev, M. 96, 266 Seke, J. 203 Sergienko, A.V 12, 282, 287, 291-296, 299302, 305-307 Shaknov, I. 279 Shamir, A. 56 Shannon, C.E. 16-18, 21, 22, 27, 28, 52, 82, 83 Sharp, E. 266 Shatokhin, VN. 40, 48, 71, 74, 77-80 Shaw, M.I. 108, 109 Shelby, R.M. 98 Shellard, E.P.S. 222 Sheppard, C.J.R. 367 Sherstobitov, VE. 347 Shih, Y.H. 12, 286, 287, 291-296 Shimizu, Y. 381 Shimony, A. 10, 279, 280, 287, 298, 299, 311 Shimshi, L. 362 Shiokawa, N. 268, 381 Shkunov, VV 225, 251 Shor, P.W. 64, 73, 80, 81 Shore, B.W. 108, 109 Shvartsman, N. 252 Siegman, A.E. 230, 281, 327, 328, 330, 332, 367, 372 Silberberg, Y. 97, 272 Silvanovich, N.I. 70 Simmons, CM. 55 Simon, C. 55, 298, 305, 309, 310, 319 Simon, R. 258, 368 Simpson, N. 246 Simpson, N.B. 264, 268 Singh, M. 99 Sipe, IE. 96, 100, 101 Sizmann, A. 98 Skryabin, D.V 266 Sleator, T. 63 Slekys, G. 223
397
Sloane, N.J.A. 26, 81 Slusher, R.E. 96, 100, 101, 140 Slussarenko, S.S. 256 Slyusar, VA. 261 Smilgevisius, V 265, 267, 269 Smith, C.R 253 Smith, R.J. 257 Smith, S.J. 71 Smith, W.W. 70 Smithey, D.T. 73 Smolin, J.A. 49-51, 55, 81, 318 Snadden, M.J. 268 Snyder, A.W. 266 Sokolov, I.V 40, 73 Sommerfeld, A. 223 Sorensen, XL. 42, 46, 72 Soskin, M.S. 222, 231, 234-236, 239, 245, 246, 249, 251, 253-259, 261, 264, 268-272, 381 Soto-Crespo, J.M. 265 Spalter, S. 100 Spreeuw, R.J.C. 68 Stabinis, A. 265, 267, 269 Stadler, M. 356 Stafsudd, O.M. 372 Staliunas, K. 222, 223, 267 Starikov, F. 272 Steane, A.M. 81, 82 Stegeman, G.I. 96, 266 Stein, R. 365 Steinberg, A. 287, 290 Stich, M.L. 184 Stoler, D. 312 Streifer, W. 333 Strekalov, D.V 294^296 Subbarao, D. 258 Sudarshan, E.C.G. 150, 155, 156, 159, 160 Sukhorukov, A.R 266 Suleski, T.J. 365 Summhammer, J. 67 Sundaram, B. 152 Suominen, K.A. 303 Suominen, K.-A. 14 Swartzlander, G.A. 222, 260, 266 Swartzlander Jr, G.A. 260
Taghizadeh, M.R. 372 Tajiri, T. 156, 160
398
Author index for volume 42
Talbot, H.E 354 Tamarat, Ph. 73 Tamm, C. 253 Tamm, I. 152, 201 Tang, C.L. 282 Tang, X.T 356 Tapster, PR. 73, 286, 287, 347, 373 Taranenko, VB. 223 Tarrach, R. 51 Tasaki, S. 151, 156, 160 Tasgal, R.S. 96, 101 Teich, M.C. 12, 184, 305-307 Teitz, C. 71 Temo, D.R. 13 Terskov, D.B. 256 Testa, M. 150, 199 Tewari, S.P 201, 212 Thijssen, M.S. 363 Thomas, H. 9 Thompson, R.C. 151, 280 Thun, K. 157, 160, 176, 181, 188 Tidwell, S.C. 356 Tietz, C. 71 Tikhomirov, VM. 29 Tikhomirova, O. 269 Tikhonenko, V 266, 268 Tittel, W. 55, 285, 305, 308 Tiwari, S.C. 244, 271 Tiziani, H.J. 270 ToffoH, T 57 Tombesi, P 74, 79, 80 Torgerson, J.R. 289 Torii, Y. 268, 381 Tomer, L. 97, 254, 265 Torres, J.P 265 Torruellas, W.E. 266 Totzeck, M. 270 Tovar, A.A. 357, 364 Townes, C.H. 212 Townsend, PD. 55 Trifonov, E.D. 108 Trillo, S. 96, 101, 103 Trobs, M. 356 Troitskii, Y.K. 342, 365 Troschieva, VN. 340 Tschudi, T 337 Tsien, R.Y. 71 Tsukerman, I.I. 28 Turchette, Q.A. 73, 77 Turing, A.M. 30
Turunen, J. 337 Tychinsky, V 270 Tzeng, Y.-L. 71
U Uehara, K. 364 Upstill, C. 222 Urbach, H.P 270
Vaidman, L. 14 Vampouille, M. 354 van Duijl, A. 231 Van Eijkelenborg, M.A. 363 Van Enk, S.J. 9 Van Loock, P 48 Van Neste, R. 336, 337, 365 Vandersypen, L.M.K. 72 VanStryland, E.W. 266 Vasnetsov, M.V 222, 231, 234-236, 239, 245, 246, 249, 251, 253, 256-259, 261, 264, 268-271, 381 Vaughan, J.M. 225, 347, 373 Vaupel, M. 223 Vazirani, U. 68 Vedral, V 48, 51, 315 Venugopalan, A. 156 Vernon Jr, EL. 153 Viardot, G. 269 Vidal, G. 49, 51 Vilaseca, R. 223 Vilenkin, A. 222 Visser, TD. 369 Vitali, D. 74, 79, 80 Vogel, W. 74, 77 Volkov, VM. 109 Volkov, VV 190 Volostnikov, VG. 249, 259, 269 Volostnikov, VV 364 Volyar, A.V 271 von Borczyskowski, C. 71 von Neumann, J. 150, 155, 167, 279 Vysotsky, D.V 340
W Wabnitz, S. 96, 99, 100 Wagner, C. 69
Author index for volume 42 Waks, E. 297, 298 Walls, D.F. 12 Walmsley, J.C. 222 Walther, H. 69-71 Walton, Z. 305-307 Wang, L. 340 Wang, T. 140 Wang, Z. 266, 337-340 Wamky, CM. 371 Waseda, Y. 381 Watanabe, K. 98 Weber, H. 327, 328, 331, 368, 369 Weber, T. 156 Wegener, M.X 253 Weihs, G. 55,298,305,309,310,319 Weinberg, D.L. 12, 282 Weinfurter, H. 12, 14, 45, 47, 55, 63, 150, 163, 176, 293, 294, 297, 298, 305, 309, 310, 312-317 Weiss, CO. 223, 253 Weisskopf, V 151 Welsch, D.-G. 48 Wennmalm, S. 71 Werner, R.F. 50 Westmoreland, M. 32 Whitaker, M.A.B. 152, 155, 156, 160 White, A.G. 68 White, A.J. 297, 298, 303 Wieman, CE. 71 Wiener, N. 27 Wiesner, S.J. 46 Wigner, E. 309, 368 Wigner, E.P. 55, 151, 164 Wilcox, J. 356 Wild, UP. 71 Wilkens, M. 50 Wilkinson, S.R. 152 Willetts, D.V 225 Wilson, D.W 266 Wineland, D.J. 71, 73, 77, 150, 156, 160, 163 Winful, H.G. 96, 133 Wise, F.W. 98 Wodkiewicz, K. 192 Woerdman, J.P. 231, 347, 363 Wolf, E. 178, 221-224, 231, 235, 242, 258, 269, 282, 369 Wolff, S. 355 Wood, CS. 73 Wootters, WK. 15, 32, 37, 42, 49, 50, 81, 315,318
399
Wrachtrup, J. 71 Wright, E.M. 98 Wright, E.W 98 Wright, EJ. 222 Wu, C 365 Wu, CS. 279 Wunderlich, C . 12, 77 Wynne, J.J. 357 Wyrowski, F. 341
X Xie, K. 266 Xin, J. 356
Yablonovitch, E. 70, 95 Yakunin, VP 357 Yamamoto, Y 73, 98 Yanagawa, T. 73 Yang, G.Z. 357 Yang, J.J. 356 Yariv, A. 184, 266, 281, 367 Yeazell, J.A. 67 Yeh, P 184 Yelden, E.F 355, 372 Yin, H. 266 Yoo, H.-I. 210 Yoran, N. 14 Yoshimura, M. 150, 199 Youngworth, K. 382 Yu, Z.X. 364 Yudson, VI. 98, 107 Yuen, H.P 9 Yurke, B. 312
Z Zaitsev, A.I. 108 Zak, J. 38 Zanardi, P 74, 77, 303 Zbinden, H. 55, 285, 305, 308 Zeilinger 317 Zeilinger, A. 10, 12, 41, 44, 45, 47, 51, 55, 150, 163, 176, 268, 282, 287, 293, 294, 298, 305, 309-317, 319 Zeitner, U.D. 341 Zel'dovich, B.Ya. 12, 225, 251, 270, 271, 282 Zellmer, H. 341
400 Zemlyanaya, E.V Zeni, L. 371 Zhao, D. 367 Zhou, J.Y. 364 Zhou, X. 72 Zhu, S.-Y. 201
Author index for volume 42 96, 101, 124
Zoller, P. 48, Zolot'ko, A.S. Zozulya, A.A. Zubairy, M.S. Zukowski, M. Zurek, W.H.
71, 73, 74, 77 256 266 201,282 45,287,288,312,313,315 15,74,75,77,81,318
Subject index for volume 42
dipole-dipole interactions Dirac matrices 200 - monopole 222 discrete Fourier transform
action at a distance 10 Aharonov-Bohm effect 222 Airy pattern 231,355 Autler-Townes doublet 212
66
E Einstein-Podolsky-Rosen (EPR) paradox entangled state 9-15, 37, 42, 291 entanglement 6, 50, 83 -, multiple-photon 311 - based cryptography 56 - distillation 49 - formation 49 - swapping 48 entropy, conditional 22 error-correcting code 16, 24
B BB84 protocol 53, 304, 309 Bell state 10, 12-15, 42, 45, 50, 297, 310, 311 - -, optical tests of 279 - - measurement 45, 315, 317, 318 Bell's inequality 6, 55, 280, 284-292, 298, 304, 309, 315 Berry phase 258 Bessel-Fourier transformation 349 Bessel-Gauss beam 244 - - mode 337 - resonator 337 Bloch equations 106,116 - vector 152, 153 Bragg condition 114 - grating 95, 103, 105, 118, 122, 138 - reflection 95, 103, 123, 137 - distance 112 - reflector, resonantly absorbing 96 - resonance 95 - scattering 104 Burgers vector 226-228
cavity quantum electrodynamics channel capacity 23 coherent-state representation 9
120
6
Fano-Shannon code 19 Fermi golden rule 199 Fox-Li method 331, 333 Franson interferometer 292 Fresnel number 330, 336
Gerchberg-Saxton method 332 Gouy effect 233 - phase shift 230, 240, 258, 259, 261 graded phase mirror 335 Greenberger-Horne-Zeilinger (GHZ) basis 51 state 51, 313, 315 theorem 311 Grover's search algorithm 67, 68
82
D data compression 16, 22 dense coding method 46, 48
H Hamming code 24, 25, 80, 81 - distance 24, 32, 35 401
402
Subject index for volume 42
Heisenberg equation 106 Hermite matrix 34 Hermite-Gaussian mode 246, 255, 328, 334, 339, 368, 371 hidden variables 279 Hilbert space 60, 61, 83 Huffman code 20
information theory
3
K Kerr-nonlinear fiber soliton 98 — grating 99 - nonlinearity 37, 96, 99, 101, 124, 267 Kirchhoff'-Fresnel integral 331, 335 Kolmogorov complexity 29
Laguerre-Gaussian beam 251,271, 364, 381, 382 - distribution 349 - - mode 225, 239, 240, 333-335, 339, 344, 368, 371 Lebesgue measure 149 M Mach-Zehnder interferometer 54, 287 Mac Williams identities 81 magneto-optical trap 268 Maxwell equations 100, 101, 109, 110, 115, 118,270 Maxwell-Bloch equations 136 mode selection 327 - shaping 327 N Nielsen's theorem
-matching 185,190 - space product 369 photonic band gap 95,110 photorefi^active crystal 257 Pockels cell 292, 308 Poincare sphere 263 Poynting vector 223, 236, 241, 242, 246 pulse-area theorem 107
49
O optical parametric oscillator
13
parametric down-conversion 12, 13, 44, 267 , spontaneous 282, 283 -fluorescence 281 paraxial approximation 241 Paul trap 70 Pauli principle 83 phase anomaly 258
quantum cloning 15, 52 - communication 30-56 - computer 3, 4, 46, 56-74, 283 - cryptography 3, 4, 46, 51-56, 283 - data compression 32 - d o t 71 - encoding of image 3 9 ^ 1 - entanglement 283 - error correction 73, 80-82 - fast Fourier transform 68 - information 30-74, 282, 303 - key distribution 304-308 - logical operations 60 - nondemolition measurement 73, 282 - phase space partition theorem 38 - processor 61 - teleportation 3, 4, 41-52, 283, 316, 317 - fidelity 46 - tomography 7, 73 - well 141 qubit 61
Rabi fi-equency 203 - oscillation 152, 160, 167, 171, 196 Rayleigh distance 338 - range 239, 249 refi-active index, periodic modulation of resonance fluorescence 12 Riemann measure 149 RSA cryptosystem 56 Rydberg atom 73
sampling theorem 16, 26 Schmidt decomposition 49 Schrodinger-cat paradox 6, 74 - - state 7, 74 - equation, nonlinear 98, 138 second-harmonic generation 122
95
403
Subject index for volume 42 self-focusing 138 - induced transparency 105 in resonantly absorbing Bragg reflector 109-120 thin films 107 Shannon's entropy 29 - noise coding theorem 23 - noiseless coding theorem 35 Shor's quantum factoring algorithm 64-67 sine-Gordon equation 107,114,136 singular optics 221, 223, 271 soliton, Bragg-grating 96, 99-101 -, dark 95, 97, 129, 131 -, gap 95, 99, 126, 129 -, quantum bright 98 squeezed state, two-mode 12 superposition 5
T Talbot effect 354-356 -length 355 Tamm-Dancoff approximation 201 Thirring model 99, 100 Toffoli's universal gate 57, 62 two-photon interferometer 300
U unstable resonators
362, 363
von Neumann entropy 32,41 vortex traj ectory 251 W Weber-Fehner law 28 Weisskopf-Wigner approximation Werner state 50 Wigner distribution 378, 379 - fiinction 74, 75 -inequality 310
Yurke-Stoler coherent state
205, 208
79, 80
Zeno effect, dynamical quantum 163-167 — , in down-conversion processes 175-190 — , inverse quantum 150-152, 157, 159, 199, 204, 213 - -, quantum 150-175, 190, 191, 196, 213 - time 156, 197 Zeno-Heraclitus transition 159 Zeno's paradox 149
Contents of previous volumes*
VOLUME 1 (1961) 1 The modem development of Hamiltonian optics, R.J. Pegis 2 Wave optics and geometrical optics in optical design, K. Miyamoto 3 The intensity distribution and total illumination of aberration-free diffraction images, R. Barakat 4 Light and information, D. Gabor 5 On basic analogies and principal differences between optical and electronic information, H. Wolter 6 Interference color, H. Kubota 7 Dynamic characteristics of visual processes, A. Fiorentini 8 Modem alignment devices, A.C.S. Van Heel
1-29 31-66 67-108 109-153 155-210 211-251 253-288 289-329
VOLUME 2 (1963) 1
Ruling, testing and use of optical gratings for high-resolution spectroscopy, aw Stroke 2 The metrological applications of diffraction gratings, J.M. Burch 3 Diffusion through non-uniform media, i?. G. G/ot;fl«e/// 4 Correction of optical images by compensation of aberrations and by spatial frequency filtering, J. Tsujiuchi 5 Fluctuations of light beams, L. Mandel 6 Methods for determining optical parameters of thin films, F. Abeles
1-72 73-108 109-129 131-180 181-248 249-288
VOLUME 3 (1964) 1 The elements of radiative transfer, F Kottler 2 Apodisation, P. Jacquinot, B. Roizen-Dossier 3 Matrix treatment of partial coherence, H. Gamo
1- 28 29-186 187-332
VOLUME 4 (1965) 1 Higher order aberration theory, J. Focke 2 Applications of shearing interferometry, O. Bryngdahl 3 Surface deterioration of optical glasses, K. Kinosita 4 Optical constants of thin films, P. Rouard, P. Bousquet
* Volumes I-XL were previously distinguished by roman rather than by arable numerals. 405
1- 36 37- 83 85-143 145-197
406 5 6 7
Contents of previous volumes The Miyamoto-Wolf diffraction wave, A. Rubinowicz Aberration theory of gratings and grating mountings, WT. 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. Frangon, S. Mallick Design of zoom lenses, A^. Kzwfly/ 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 IL electromagnetic theory, F. Kottler
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, JH. 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. M^/i/fl
1- 50 51-131 133-200 201-237 239-294 295-341 343-372 373-440
Contents of previous volumes
407
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, VM. Agranovich, V.L. Ginzburg Applications of optical methods in the diffraction theory of elastic waves, K. Gniadek, JPetykiewicz 8 Evaluation, design and extrapolation methods for optical signals, based on use of the prolate fiinctions, 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 4 5 - 87 89-135 137-164 165-228 229-288 289-369
VOLUME 11 (1973) 1 2 3 4 5 6 1
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, JA. Arnaud Gradient index lenses, E.W Marchand
1- 76 77-122 123-166 167-221 223-246 1A1-?>0A 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, JA. 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, WM. Rosenblum, J.L. Christensen 4 Interferometric testing of smooth surfaces, G Schulz, J. Schwider
1-- 25 27- 68 69- 91 93--167
408
Contents of previous volumes
Self-focusing of laser beams in plasmas and semiconductors, M.S. Sodha, A.K. Ghatak, V.K. Tripathi 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, RJ. Vernier Optical fibre waveguides - a review, P.J.B. Clarricoats
1- 46 47- 87 89-159 161-193 195-244 245-325 321^02
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
1- 75 71-131 139-185 187-244 245-350
VOLUME 16 (1978) 1 2 3 4 5
Laser selective photophysics and photochemistry, VS. 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 fi-om high-current surface-spark discharges, R.E. Beverly III 7 Semiclassical radiation theory within a quantum-mechanicalfi^amework,I.R. Senitzky
1- 69 71-117 119-232 233-288 289-356 357-411 413^48
VOLUME 17 (1980) 1 Heterodyne holographic interferometry, R. Ddndliker 2 Doppler-fi-ee multiphoton spectroscopy, E. Giacobino, B. Cagnac 3 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 reft-action, A.L. Mikaelian
1-84 85-161 163-238 239-277 279-345
VOLUME 18 (1980) 1 Graded index optical waveguides: a review, A. Ghatak, K. Thyagarajan 2 Photocount statistics of radiation propagating through random and nonlinear media, J Pefina
1-126 127-203
Contents of previous volumes 3 4
409
Strong fluctuations in light propagation in a randomly inhomogeneous medium, V.I. Tatarskii, VU. Zavorotnyi 204-256 Catastrophe optics: morphologies of caustics and their diffraction patterns, M. V. Berry, C. Upstill 257-346
VOLUME 19 (1981) 1 2 3 4 5
Theory of intensity dependent resonance light scattering and resonance fluorescence, B.R. Mollow 1- 43 Surface and size effects on the light scattering spectra of solids, D.L. Mills, K.R. Subbaswamy 45-137 Light scattering spectroscopy of surface electromagnetic waves in solids, ^. Ushioda 139-210 Principles of optical data-processing,//.J ^M/^erwecA: 211-280 The effects of atmospheric turbulence in optical astronomy, F. Roddier 281-376 VOLUME 20 (1983)
1 2 3 4 5
Some new optical designs for ultra-violet bidimensional detection of astronomical objects, G. Courtis, P. Cruvellier, M. Detaille, M. Saisse 1-61 Shaping and analysis of picosecond light pulses, C Froehly, B. Colombeau, M. Vampouille 63—153 Multi-photon scattering molecular spectroscopy, S. Kielich 155-261 Colour holography, P. Hariharan 263-324 Generation of tunable coherent vacuum-ultraviolet radiation, W. Jamroz, B.P. Stoicheff 325-380
VOLUME 21 (1984) 1 2 3 4 5
Rigorous vector theories of diffraction gratings, D. Maystre Theory of optical bistability, Z.^. Lt/g/a^o 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 Optical and electronic processing of medical images, D. Malacara 2 Quantum fluctuations in vision, M.A. Bouman, W.A. Van De Grind, P. Zuidema 3 Spectral and temporal fluctuations of broad-band laser radiation, A. V. Masalov 4 Holographic methods of plasma diagnostics, G.V Ostrovskaya, Yu.I. Ostrovsky 5 Fringe formations in deformation and vibration measurements using laser light, /. 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) Analytical techniques for multiple scattering from rough surfaces, J.A. DeSanto, G.S. Brown Paraxial theory in optical design in terms of Gaussian brackets, K. Tanaka Optical films produced by ion-based techniques, P.J. Martin, R.P Netterfield
1- 62 63-111 113-182
410
Contents of previous volumes
4 Electron holography, A. Tonomura 5 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. Harihamn Unstable resonator modes, K.E. Oughstun Information processing with spatially incoherent light, /. Glaser
1- 37 39-101 103-164 165-387 389-509
VOLUME 25 (1988) Dynamical instabilities and pulsations in lasers, N.B. Abraham, P. Mandel, L.M. Narducci Coherence in semiconductor lasers, M Ohtsu, T. Tako Principles and design of optical arrays, Wang Shaomin, L. Ronchi 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, /. 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
1-104 105-161 163-225 227-348 349-393
VOLUME 27 (1989) 1 The self-imaging phenomenon and its applications, K. Patorski 2 Axicons and meso-optical imaging devices, L.M. Soroko 3 Nonimaging optics for flux concentration, I.M. Bassett, W.T. Welford, R. Winston 4 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 Digital holography - computer-generated holograms, O. Bryngdahl, F. Wyrowski 2 Quantum mechanical limit in optical precision measurement and communication, Y. Yamamoto, S. Machida, S. Saito, N. Imoto, T. Yanagawa, M. Kitagawa, G. Bjork 3 The quantum coherence properties of stimulated Raman scattering, M.G. Raymer, LA. Walmsley 4 Advanced evaluation techniques in interferometry, J. Schwider 5 Quantum jumps, i?./CooA:
1- 86 87-179 181-270 271-359 361-416
Contents of previous volumes
411
VOLUME 29 (1991) 1 Optical waveguide diflfraction gratings: coupling between guided modes, D.G. Hall 1-63 2 Enhanced backscattering in optics, YuM. Bambanenkov, Yu.A. Kravtsov, V.D. Ozrin, A.L Saichev 65-197 3 Generation and propagation of ultrashort optical pulses, LP. Christov 199-291 4 Triple-correlation imaging in optical astronomy, G. Weigelt 293-319 5 Nonlinear optics in composite materials. 1. Semiconductor and metal crystallites in dielectrics, C. Flytzanis, E Hache, M.C. Klein, D. Ricard, Ph. Roussignol 321-411 VOLUME 30 (1992) 1 2 3 4 5
Quantum fluctuations in optical systems, S. Reynaud, A. Heidmann, E. Giacobino, C. Eabre 1- 85 Correlation holographic and speckle interferometry, Yu.I. Ostrovsky, V.P Shchepinov 87-135 Localization of waves in media with one-dimensional disorder, V.D. Ereilikher, S.A. Gredeskul 137-203 Theoretical foundation of optical-soliton concept in fibers, Y Kodama, A. Hasegawa 205-259 Cavity quantum optics and the quantum measurement process, P Meystre 261-355 VOLUME 31 (1993)
1 2 3 4 5 6
Atoms in strong fields: photoionization and chaos, PW. 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
1-137 139-187 189-226 227-261 263-319 321^12
VOLUME 32 (1993) 1 Guided-wave optics on silicon: physics, technology and status, B.P Pal 1- 59 2 Optical neural networks: architecture, design and models, ET.S. Yu 61-144 3 The theory of optimal methods for localization of objects in pictures, L.P Yaroslavsky 145-201 4 Wave propagation theories in random media based on the path-integral approach, M.I. Charnotskii, J. Gozani, V.I. Tatarskii, VU. Zavorotny 203-266 5 Radiation by uniformly moving sources. Vavilov-Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena, V.L. Ginzburg 267-312 6 Nonlinear processes in atoms and in weakly relativistic plasmas, G. Mainfray, C. Manus 313-361 VOLUME 33 (1994) 1 The imbedding method in statistical boundary-value wave problems, V.I. Klyatskin 2 Quantum statistics of dissipative nonlinear oscillators, V Pefinovd, A. Luks 3 Gap solitons, CM. De Sterke, J.E. Sipe 4 Direct spatial reconstruction of optical phase from phase-modulated images, VI 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, E Wyrowski
1-127 129-202 203-260 261-317 319-388 389^63
412
Contents of previous volumes VOLUME 34 (1995)
1 2 3 4 5
Quantum interference, superposition states of light, and nonclassical effects, V Buzek, P.L. Knight 1-158 Wave propagation in inhomogeneous media: phase-shift approach, L.P. Presnyakov 159-181 The statistics of dynamic speckles, T. Okamoto, T. Asakura 183-248 Scattering of light from multilayer systems with rough boundaries, /. Ohlidal, K. Navrdtil, M. Ohlidal 249-331 Random walk and diffusion-like models of photon migration in turbid media, A.H. Gandjbakhche, G.H. Weiss 333^02 VOLUME 35 (1996)
1 Transverse patterns in wide-aperture nonlinear optical systems, N.N. Rosanov 2 Optical spectroscopy of single molecules in solids, M. Orrit, J. Bernard, R. Brown, B. Lounis 3 Interferometric multispectral imaging, K. Itoh 4 Interferometric methods for artwork diagnostics, D. Paoletti, G. Schirripa Spagnolo 5 Coherent population trapping in laser spectroscopy, E. Arimondo 6 Quantum phase properties of nonlinear optical phenomena, R. Tanas, A. Miranowicz, Ts. Gantsog
1-60 61-144 145-196 197-255 257-354 355^46
VOLUME 36 (1996) 1 Nonlinear propagation of strong laser pulses in chalcogenide glass films, V. Chumash, I. Cojocaru, E. Fazio, E Michelotti, M. Bertolotti 2 Quantum phenomena in optical interferometry, P. Hariharan, B.C. Sanders 3 Super-resolution by data inversion, M Bertero, C. De Mol 4 Radiative transfer: new aspects of the old theory, Yu.A. Kravtsov, L.A. Apresyan 5 Photon wave function, /. Bialynicki-Birula
\- ATI 49-128 129-178 179-244 245-294
VOLUME 37 (1997) 1 The Wigner distribution fimction in optics and optoelectronics, D. Dragoman 2 Dispersion relations and phase retrieval in optical spectroscopy, K.-E. Peiponen, E.M. Vartiainen, T. Asakura 3 Spectra of molecular scattering of light, I.L. Fabelinskii 4 Soliton communication systems, R.-J. Essiambre, G.P Agrawal 5 Local fields in linear and nonlinear optics of mesoscopic systems, O. Keller 6 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 Nonlinear optics of stratified media, S. Dutta Gupta 2 Optical aspects of interferometric gravitational-wave detectors, P. Hello 3 Thermal properties of vertical-cavity surface-emitting semiconductor lasers, W.Nakwaski, M. Osinski 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-^18 419-513
Contents of previous volumes
413
VOLUME 39 (1999) 1 Theory and applications of complex rays, Yu.A. Kravtsov, G. W. Forbes, A.A. Asatryan 1- 62 2 Homodyne detection and quantum-state reconstruction, D.-G. Welsch, W. Vogel, T.Opatrny 63-211 3 Scattering of light in the eikonal approximation, S.K. Sharma, D.J. Somerford 213-290 4 The orbital angular momentum of light, L. Allen, M.J. Padgett, M. Babiker 291-372 5 The optical Kerr effect and quantum optics in fibers, A. Sizmann, G. Leuchs 373-469 VOLUME 40 (2000) 1 Polarimetric optical fibers and sensors, T.R. Wolinski 2 Digital optical computing, J TflmV/fl, Z/c/zzoA^fl 3 Continuous measurements in quantum optics, V. Pefinovd, A. Luks 4 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 and H.S. Freedhoff
1- 75 11-114 115-269 271-341 343-388 389-441
VOLUME 41 (2000) 1 Nonlinear optics in microspheres, M.H. Fields, J. Popp, R.K. Chang 2 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 3 EUipsometry of thin film systems, /. Ohlidal, D. Franta 4 Optical true-time delay control systems for wideband phased array antennas, R.T. Chen, Z. Fu 5 Quantum statistics of nonlinear optical couplers, J. Pefina Jr, J. Pefina 6 Quantum phase difference, phase measurements and Stokes operators, A. Luis, L.L. Sdnchez-Soto 7 Optical solitons in media with a quadratic nonlinearity, C. Etrich, F Lederer, B.A. Malomed, T. Peschel, U. Peschel
1- 95 97-179 181-282 283-358 359-417 419-479 483-567
Cumulative index - Volumes 1-42*
Abeles, E: Methods for determining optical parameters of thin films Abella, I.D.: Echoes at optical fi-equencies Abitbol, C.I., see Clair, J.J. Abraham, N.B., R Mandel, L.M. Narducci: Dynamical instabilities and pulsations in lasers Agarwal, G.S.: Master equation methods in quantum optics Agranovich, VM., VL. Ginzburg: Crystal optics with spatial dispersion Agrawal, G.P.: Single-longitudinal-mode semiconductor lasers Agrawal, G.R, see Essiambre, R.-J. Allen, L., D.G.C. Jones: Mode locking in gas lasers Allen, L., M.J. Padgett, M. Babiker: The orbital angular momentum of light Ammann, E.O.: Synthesis of optical birefi^ingent networks Anderson, R., see Carriere, J. Apresyan, L.A., see Kravtsov, Yu.A. Arimondo, E.: Coherent population trapping in laser spectroscopy Armstrong, J.A., A.W. Smith: Experimental studies of intensity fluctuations in lasers Amaud, 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. Babiker, M., see Allen, L. 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, YD. Ozrin, A.I. Saichev: Enhanced backscattering in optics Barakat, R.: The intensity distribution and total illumination of aberration-fi:ee diffraction images Barrett, H.H.: The Radon transform and its applications Bashkin, S.: Beam-foil spectroscopy Bassett, I.M., W.T Welford, R. Winston: Nonimaging optics for flux concentration Beckmatm, P.: Scattering of light by rough surfaces Beran, M.J., J. Oz-Vogt: Imaging through turbulence in the atmosphere Bernard, J., see Orrit, M.
^ Volumes I-XL were previously distinguished by roman rather than by arable numerals. 415
2, 249 7, 139 16, 71 25, 11, 9, 26, 37, 9, 39, 9, 41, 36, 35, 6, 11, 34, 37, 39,
1 1 235 163 185 179 291 123 97 179 257 211 247 183 57 1
39, 291 13,
1
29, 65 1, 21, 12, 27, 6, 33, 35,
67 217 287 161 53 319 61
416
Cumulative index - Volumes 1-42
Berry, M.V, C. Upstill: Catastrophe optics: morphologies of caustics and their diffraction patterns Bertero, M., C. De Mol: Super-resolution by data inversion Bertolotti, M., see Mihalache, D. Bertolotti, M., see Chumash, V Beverly III, R.E.: Light emission from high-current surface-spark discharges Bialynicki-Birula, I.: Photon wave function Bjork, G., see Yamamoto, Y. Bloom, A.L.: Gas lasers and their application to precise length measurements Bouman, M.A., W.A. Van De Grind, P. Zuidema: Quantum fluctuations in vision Bousquet, P., see Rouard, P. Brown, G.S., see DeSanto, J.A. Brown, R., see Orrit, M. Brunner, W., H. Paul: Theory of optical parametric amplification and oscillation Bryngdahl, O.: AppHcations of shearing interferometry Bryngdahl, O.: Evanescent waves in optical imaging Bryngdahl, O., F. Wyrowski: Digital holography - computer-generated holograms Bryngdahl, O., T. Scheermesser, F. Wyrowski: Digital halftoning: synthesis of binary images Burch, J.M.: The metrological applications of diffraction gratings Butterweck, H.J.: Principles of optical data-processing Buzek, V, PL. Knight: Quantum interference, superposition states of light, and nonclassical effects Cagnac, B., see Giacobino, E. Carriere, J., R. Narayan, W.-H. Yeh, C. Peng, P Khulbe, L. Li, R. Anderson, J. Choi, M. Mansuripur: Principles of optical disk data storage Casasent, D , D. Psaltis: Deformation invariant, space-variant optical pattern recognition Ceglio, N.M., D.W. Sweeney: Zone plate coded imaging: theory and applications Chang, R.K., see Fields, M.H. Chamotskii, M.I., J. Gozani, VI. Tatarskii, VU. Zavorotny: Wave propagation theories in random media based on the path-integral approach Chen, R.T, Z. Fu: Optical true-time delay control systems for wideband phased array antennas Chiao, R.Y, A.M. Steinberg: Tunneling times and superluminality Choi, J., see Carriere, J. Christensen, J.L., see Rosenblum, WM. Christov, LP: Generation and propagation of ultrashort optical pulses Chumash, V, I. Cojocaru, E. Fazio, F. Michelotti, M. Bertolotti: Nonlinear propagation of strong laser pulses in chalcogenide glass films Clair, J.L, C.I. Abitbol: Recent advances in phase profiles generation Clarricoats, P.LB.: Optical fibre waveguides - a review Cohen-Tannoudji, C , A. Kastler: Optical pumping Cojocaru, \., see Chumash, V Cole, T.W: Quasi-optical techniques of radio astronomy Colombeau, B., see Froehly, C. Cook, R.J.: Quantum jumps Courtes, G., P. Cruvellier, M. Detaille, M. Saisse: Some new optical designs for ultraviolet bidimensional detection of astronomical objects Creath, K.: Phase-measurement interferometry techniques Crewe, A.V: Production of electron probes using a field emission source
18, 257 36, 129 27, 227 36, 1 16, 357 36, 245 28, 87 9, 1 22, 77 4, 145 23, 1 35, 61 15, 1 4, 37 11, 167 28, 1 33, 389 2, 73 19, 211 34,
1
17, 85 41, 97 16, 289 21, 287 41, 1 32, 203 41, 37, 41, 13, 29,
283 345 97 69 199
36, 16, 14, 5, 36, 15, 20, 28,
1 71 327 1 1 187 63 361
20, 1 26, 349 11, 223
All
Cumulative index - Volumes 1-42 Cruvellier, P., see Courtes, G. Cummins, H.Z., H.L. Swimiey: Light beating spectroscopy
20, 1 8, 133
Dainty, J.C: The statistics of speckle patterns Dandliker, R.: Heterodyne holographic interferometry DattoH, G., L. Giannessi, A. Renieri, A. Torre: Theory of Compton free electron lasers Davidson, N., see Oron, R. De Mol, C , see Bertero, M. De Sterke, CM., J.E. Sipe: Gap solitons Decker Jr, J.A., see Harwit, M. Delano, E., R.J. Pegis: Methods of synthesis for dielectric multilayer filters Demaria, A.J.: Picosecond laser pulses DeSanto, J.A., G.S. Brown: Analytical techniques for multiple scattering from rough surfaces Detaille, M., see Courtes, G. Dexter, D.L., see Smith, D.Y. Dragoman, D.: The Wigner distribution fimction in optics and optoelectronics Drexhage, K.H.: Interaction of light with monomolecular dye layers Duguay, M.A.: The ultrafast optical Kerr shutter Dutta, N.K., J.R. Simpson: Optical amplifiers Dutta Gupta, S.: Nonlinear optics of stratified media
14, 17, 31, 42, 36, 33, 12, 7, 9,
1 1 321 325 129 203 101 67 31
23, 20, 10, 37, 12, 14, 31, 38,
1 1 165 1 163 161 189 1
Eberly, J.H.: Interaction of very intense light with free electrons Englund, J.C, R.R. Snapp, W.C. Schieve: Fluctuations, instabilities and chaos in the laser-driven nonlinear ring cavity Ennos, A.E.: Speckle interferometry Essiambre, R.-J., G.P. Agrawal: Soliton communication systems Etrich, C , F. Lederer, B.A. Malomed, T. Peschel, U. Peschel: Optical solitons in media with a quadratic nonlinearity Fabelinskii, I.L.: Spectra of molecular scattering of light Fabre, C , see Reynaud, S. Facchi, P., 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 Ficek, Z. and H.S. Freedhoff: Spectroscopy in polychromatic fields Fields, M.H., J. Popp, R.K. Chang: Nonlinear optics in microspheres Fiorentini, A.: Dynamic characteristics of visual processes Flytzanis, C , F. Hache, M.C. Klein, D. Ricard, 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. Frangon, M., S. Mallick: Measurement of the second order degree of coherence Franta, D., see Ohlidal, I. Freedhoff, H.S., see Ficek, Z. Freilikher, VD., S.A. Gredeskul: Localization of waves in media with one-dimensional disorder Frieden, B.R.: Evaluation, design and extrapolation methods for optical signals, based on use of the prolate fiinctions Friesem, A.A., see Oron, R.
7, 359 21, 355 16, 233 37, 185 41, 483 37, 30, 41, 22, 36, 40, 41, 1,
95 1 \A1 341 1 389 1 253
29, 4, 39, 6, 41, 40,
321 1 1 71 181 389
30, 137 9, 311 42, 325
418
Cumulative index - Volumes 1-42
Froehly, C , B. Colombeau, 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. Gabor, D.: Light and information Gamo, H.: Matrix treatment of partial coherence Gandjbakhche, A.H., G.H. Weiss: Random walk and diffusion-like models of photon migration in turbid media Gantsog, Ts., see Tanas, R. Ghatak, A., K. Thyagarajan: Graded index optical waveguides: a review Ghatak, A.K., see Sodha, M.S. Giacobino, E., B. Cagnac: Doppler-free multiphoton spectroscopy Giacobino, E., see Reynaud, S. Giannessi, L., see Dattoli, G. Ginzburg, VL., see Agranovich, VM. Ginzburg, VL.: Radiation by uniformly moving sources. Vavilov-Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena Giovanelli, R.G.: Diffusion through non-uniform media Glaser, L: Information processing with spatially incoherent light Gniadek, K., J. Petykiewicz: Applications of optical methods in the diffraction theory of elastic waves Goodman, J.W.: Synthetic-aperture optics Gozani, J., see Chamotskii, M.I. Graham, R.: The phase transition concept and coherence in atomic emission Gredeskul, S.A., see Freilikher, YD.
20, 63 8, 51 41, 283 1, 109 3, 187 34, 35, 18, 13, 17, 30, 31, 9,
333 355 1 169 85 1 321 235
32, 267 2, 109 24, 389 9, 8, 32, 12, 30,
281 1 203 233 137
Hache, R, 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., B.C. Sanders: Quantum phenomena in optical interferometry Harwit, M., J.A. Decker Jr: Modulation techniques in spectrometry Hasegawa, A., see Kodama, Y. 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 Herriot, D.R.: Some applications of lasers to interferometry Homer, J.L., see Javidi, B. Huang, T.S.: Bandwidth compression of optical images
29, 29, 20, 24, 36, 12, 30, 42, 30, 38, 10, 6, 38, 10,
321 1 263 103 49 101 205 325 1 85 289 171 343 1
Ichioka, Y., see Tanida, J. Imoto, N., see Yamamoto, Y Itoh, K.: Interferometric multispectral imaging
40, 77 28, 87 35, 145
Jacobsson, R.: Light reflection ft-om films of continuously varying refractive index Jacquinot, P., B. Roizen-Dossier: Apodisation Jaeger, G., A.V Sergienko: Multi-photon quantum interferometry Jahns, X: Free-space optical digital computing and interconnection Jamroz, W., B.P. Stoicheff: Generation of tunable coherent vacuum-ultraviolet radiation
5, 3, 42, 38, 20,
247 29 277 419 325
419
Cumulative index - Volumes 1-42 Javidi, B., XL. Horner: Pattern recognition with nonlinear techniques in the Fourier domain Jones, D.G.C., see Allen, L.
38, 343
9, 179
Kastler, A., see Cohen-Tannoudji, C. Keller, O.: Local fields in linear and nonlinear optics of mesoscopic systems Khoo, I.e.: 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. Klein, M.C., see Flytzanis, C. Klyatskin, VI.: The imbedding method in statistical boundary-value wave problems Knight, PL., see Buzek, V Kodama, Y, A. Hasegawa: Theoretical foundation of optical-soliton concept in fibers Koppelman, G.: Multiple-beam interference and natural modes in open resonators Kottler, R: The elements of radiative transfer Kottler, R: Diffraction at a black screen, Part I: Kirchhoff's theory Kottler, R: 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., see Barabanenkov, Yu.N. Kravtsov, Yu.A., L.A. Apresyan: Radiative transfer: new aspects of the old theory Kravtsov, Yu.A., G.W. Rorbes, A.A. Asatryan: Theory and applications of complex rays Kubota, H.: Interference color Kuittinen, M., see Turunen, J. Kurizki, G., A.E. Kozhekin, T. Opatrny, B.A. Malomed: Optical solitons in periodic media with resonant and off-resonant nonlinearities
5, 1 37 257 26, 105 41, 97 20, 155 42 1 4, 85 28, 87 29, 321 33 1 34 1 30 205 7, 1 3, 1 4 281 6 331 42 93 26, 227 29 65 36 179 39, 1 1, 211 40 343
Labeyrie, A.: High-resolution techniques in optical astronomy Lean, E.G.: Interaction of light and acoustic surface waves Lederer, R, see Etrich, C. Lee, W.-H.: Computer-generated holograms: techniques and applications Leith, E.N., J. Upatnieks: Recent advances in holography Letokhov, VS.: Laser selective photophysics and photochemistry Leuchs, G., see Sizmann, A. Levi, L.: Vision in communication Li, L., see Carriere, J. Lipson, H., C.A. Taylor: X-ray crystal-structure determination as a branch of physical optics Lohmann, A.W., D. Mendlovic, Z. Zalevsky: Rractional transformations in optics Lohmann, A.W., see Zalevsky, Z. Lounis, B., see Orrit, M. Lugiato, L.A.: Theory of optical bistability Luis, A., L.L. Sanchez-Soto: Quantum phase difference, phase measurements and Stokes operators Luks, A., see Pefinova, V Luks, A., see Pefinova, V
14 11 41 16 6 16 39 8 41
47 123 483 119 1 1 373 343 97
5 38 40 35 21
287 263 271 61 69
Machida, S., see Yamamoto, Y Mainfray, G., C. Manus: Nonlinear processes in atoms and in weakly relativistic plasmas
28 87 32 313
42 93
41 419 33 129 40 115
420
Cumulative index - Volumes 1-42
Malacara, D.: Optical and electronic processing of medical images Malacara, D., see Vlad, VI. Mallick, S., see Frangon, M. 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. Marchand, E.W.: Gradient index lenses Martin, P.J., 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 Mendlovic, D., see Lohmann, A.W. Mendlovic, D., see Zalevsky, Z. Meystre, P.: Cavity quantum optics and the quantum measurement process Michelotti, F, see Chumash, V Mihalache, D., M. Bertolotti, C. Sibilia: Nonlinear wave propagation in planar structures Mikaelian, A.L., M.L. Ter-Mikaelian: Quasi-classical theory of laser radiation Mikaelian, A.L.: Self-focusing media with variable index of refraction Mills, D.L., K.R. Subbaswamy: Surface and size effects on the light scattering spectra of solids Milonni, PW., B. Sundaram: Atoms in strong fields: photoionization and chaos Miranowicz, A., see Tanas, R. Miyamoto, K.: Wave optics and geometrical optics in optical design MoUow, B.R.: Theory of intensity dependent resonance light scattering and resonance fluorescence Murata, K.: Instruments for the measuring of optical transfer functions Musset, A., A. Thelen: Multilayer antireflection coatings
22, 1 33, 261 6, 71 41, 483 42, 93 2, 181 13, 27 25, 1 41, 97 32, 313 11, 305 23, 113 22, 145 21, 1 15, 77 8, 373 38, 263 40, 271 30, 261
36,
1
27, 227 7, 231 17, 279 19, 45 31, 1 35, 355 1, 31 19, 1 5, 199 8, 201
Nakwaski, W., M. Osihski: Thermal properties of vertical-cavity surface-emitting semiconductor lasers Narayan, R., see Carriere, J. Narducci, L.M., see Abraham, N.B. Navratil, K., see Ohlidal, I. Netterfield, R.P, see Martin, PJ. Nishihara, H., T. Suhara: Micro Fresnel lenses
38, 41, 25, 34, 23, 24,
165 97 1 249 113 1
Ohlidal, I., K. Navratil, M. Ohlidal: Scattering of light from multilayer systems with rough boundaries Ohlidal, I., D. Franta: Ellipsometry of thin film systems Ohlidal, M., see Ohlidal, I. Ohtsu, M., T Tako: Coherence in semiconductor lasers Okamoto, T, T. Asakura: The statistics of dynamic speckles Okoshi, T: Projection-type holography Ooue, S.: The photographic image Opatmy, T, see Welsch, D.-G. Opatrny, T, see Kurizki, G.
34, 41, 34, 25, 34, 15, 7, 39, 42,
249 181 249 191 183 139 299 63 93
421
Cumulative index - Volumes 1-42 Oron, R., N. Davidson, A.A. Friesem, E. Hasman: Transverse mode shaping and selection in laser resonators Orrit, M., J. Bernard, R. Brown, B. Loimis: Optical spectroscopy of single molecules in solids Osinski, M., see Nakwaski, W. Ostrovskaya, G.V, Yu.I. Ostrovsky: Holographic methods of plasma diagnostics Ostrovsky, Yu.L, see Ostrovskaya, G.V. Ostrovsky, Yu.L, VP. Shchepinov: Correlation holographic and speckle interferometry Oughstun, K.E.: Unstable resonator modes Oz-Vogt, J., see Beran, M.J. Ozrin, VD., see Barabanenkov, Yu.N.
42, 325
35, 61 38, 165 22, 197 22 197 30 87 24 165 33, 319 29, 65 291 1 197 147 1 1 1 67
Padgett, M.J., see Allen, L. Pal, B.P.: Guided-wave optics on silicon: physics, technology and status Paoletti, D., G. Schirripa Spagnolo: Interferometric methods for artwork diagnostics Pascazio, ^., 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, T. Asakura: Dispersion relations and phase retrieval in optical spectroscopy Peng, C , see Carriere, J. Pefina, X: Photocount statistics of radiation propagating through random and nonlinear media Pefina, J., see Pefina Jr, J. Pefina Jr, J., J. Pefina: Quantum statistics of nonlinear optical couplers Pehnova, V, A. Luks: Quantum statistics of dissipative nonlinear oscillators Pefinova, V, A. Luks: Continuous measurements in quantum optics Pershan, PS.: Non-linear optics Peschel, T., see Etrich, C. Peschel, U., see Etrich, C. Petykiewicz, J., see Gniadek, K. Picht, J.: The wave of a moving classical electron 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 Psaltis, D., see Casasent, D. Psaltis, D., Y Qiao: Adaptive multilayer optical networks
39 32 35 42 27 15 1 7
Qiao, Y, see Psaltis, D.
31, 227
Raymer, M.G., LA. Walmsley: The quantum coherence properties of stimulated Raman scattering Renieri, A., see Dattoli, G. Reynaud, S., A. Heidmann, E. Giacobino, C. Fabre: Quantum fluctuations in optical systems Ricard, D., see Flytzanis, C. Riseberg, L.A., M.J. Weber: Relaxation phenomena in rare-earth luminescence
37 57 41 97 18 41 41 33 40 5 41 41 9 5 31 41
127 359 359 129 115 83 483 483 281 351 139 1
27 34 16 31
315 159 289 227
28, 181 31, 321 30, 1 29, 321 14, 89
422
Cumulative index - Volumes 1-42
Risken, H.: Statistical properties of laser light Roddier, E: The effects of atmospheric turbulence in optical astronomy 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., XL. Christensen: Objective and subjective spherical aberration measurements of the human eye Rothberg, L.: Dephasing-induced coherent phenomena Rouard, P., P. Bousquet: Optical constants of thin films Rouard, P., A. Meessen: Optical properties of thin metal films Roussignol, Ph., see Flytzanis, C. Rubinowicz, A.: The Miyamoto-Wolf diffraction wave Rudolph, D., see Schmahl, G.
8, 239 19, 281 3, 29 25, 279 35, 1
Saichev, A.I., see Barabanenkov, Yu.N. Saisse, M., see Courtes, G. Saito, S., see Yamamoto, Y. Sakai, H., see Vanasse, G.A. Saleh, B.E.A., see Teich, M.C. Sanchez-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., D. Rudolph: Holographic diffraction gratings Schubert, M., B. Wilhelmi: The mutual dependence between coherence properties of light and nonlinear optical processes Schulz, G., J. Schwider: Interferometric testing of smooth surfaces Schulz, G.: Aspheric surfaces Schwider, J., see Schulz, G. Schwider, J.: Advanced evaluation techniques in interferometry Scully, M.O., K.G. Whitney: Tools of theoretical quantum optics Senitzky, LR.: Semiclassical radiation theory within a quantum-mechanical fi-amework Sergienko, A.V, see Jaeger, G. Sharma, S.K., D.J. Somerford: Scattering of light in the eikonal approximation Shchepinov, VP, see Ostrovsky, Yu.I. Sibilia, C , see Mihalache, D. Simpson, J.R., see Dutta, N.K. Sipe, J.E., see Van Kranendonk, J. Sipe, J.E., see De Sterke, CM. Sittig, E.K.: Elastooptic light modulation and deflection Sizmann, A., 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, 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, VK. 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
29, 65 20, 1 28, 87 6, 259 26, 1 41, 419 36, 49 33, 389 21, 355 35, 197 14, 195
13, 69 24, 39 4, 145 15, 77 29, 321 4, 199 14, 195
17, 163 13, 93 25, 349 13, 93 28, 271 10, 89 16, 413 42, 277 39, 213 30, 87 27, 227 31, 189 15, 245 33, 203 10, 229 39, 373 12, 53 6, 211 10, 165 10, 45 21, 355 13, 169 39, 213 27, 109
Cumulative index - Volumes 1-42
423
Soskin, M.S., M.V Vasnetsov: Singular optics Spreeuw, R.J.C., J.P. Woerdman: Optical atoms Steel, W.H.: Two-beam interferometry Steinberg, A.M., see Chiao, R.Y. Stoicheff, B.P., see Jamroz, W. 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. Sundaram, B., see Milonni, RW. Svelto, O.: Self-focusing, self-trapping, and self-phase modulation of laser beams Sweeney, D.W, see Ceglio, N.M. Swinney, H.L., see Cummins, H.Z,
42, 31, 5, 37, 20, 9,
219 263 145 345 325 73
Tako, T, see Ohtsu, M. Tanaka, K.: Paraxial theory in optical design in terms of Gaussian brackets Tanas, R., A. Miranowicz, Ts. Gantsog: Quantum phase properties of nonlinear optical phenomena Tango, W.J., R.Q. Twiss: Michelson stellar interferometry Tanida, J., Y. Ichioka: Digital optical computing Tatarskii, VI., VU. Zavorotnyi: Strong fluctuations in light propagation in a randomly inhomogeneous medium Tatarskii, VI., see Chamotskii, M.I. Taylor, C.A., see Lipson, H. Teich, M.C., B.E.A. Saleh: Photon bunching and antibunching Ter-MikaeHan, M.L., see Mikaelian, A.L. Thelen, A., see Musset, A. Thompson, B.J.: Image formation with partially coherent hght Thyagarajan, K., see Ghatak, A. Tonomura, A.: Electron holography Torre, A., see Dattoli, G. Tripathi, VK., see Sodha, M.S. Tsujiuchi, I : Correction of optical images by compensation of aberrations and by spatial frequency filtering Turunen, J., M. Kuittinen, R Wyrowski: Diffractive optics: electromagnetic approach Twiss, R.Q., see Tango, W.J.
25, 191 23, 63
2, 1 19, 45 24, 1 31, 1 12, 1 21, 287 8, 133
35, 355 17, 239 40, 77 18, 32, 5, 26, 7, 8, 7, 18, 23, 31, 13,
204 203 287 1 231 201 169 1 183 321 169
2, 131 40, 343 17, 239
Upatnieks, X, see Leith, E.N. Upstill, C , see Berry, M.V Ushioda, S.: Light scattering spectroscopy of surface electromagnetic waves in solids
6, 1 18, 257 19, 139
Vampouille, M., see Froehly, C. Van De Grind, W.A., see Bouman, M.A. Van Heel, A.C.S.: Modem alignment devices Van Kranendonk, X, XE. Sipe: Foundations of the macroscopic electromagnetic theory of dielectric media Vanasse, G.A., H. Sakai: Fourier spectroscopy Vartiainen, E.M., see Peiponen, K.-E. Vasnetsov, M.V, see Soskin, M.S. Vernier, P.X: Photoemission
20, 63 22, 77 1, 289 15, 6, 37, 42, 14,
245 259 57 219 245
424
Cumulative index - Volumes 1-42
Vlad, VI., D. Malacara: Direct spatial reconstruction of optical phase from phasemodulated images Vogel, W., 5e^ Welsch, D.-G. Walmsley, LA., see Raymer, M.G. Wang Shaomin, L. Ronchi: Principles and design of optical arrays Weber, M.J., see Riseberg, L.A. Weigelt, G.: Triple-correlation imaging in optical astronomy Weiss, G.H., see Gandjbakhche, A.H. Welford, WT: Aberration theory of gratings and grating mountings Welford, W.T.: Aplanatism and isoplanatism Welford, WT, see Bassett, I.M. Welsch, D.-G., W Vogel, T. Opatmy: 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.R, see Spreeuw, R.J.C. Wolihski, 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. 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, M. Kitagawa, G. Bjork: 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. Yoshinaga, H.: Recent developments in far infrared spectroscopic techniques Yu, F.TS.: Principles of optical processing with partially coherent light Yu, F.TS.: Optical neural networks: architecture, design and models Zalevsky, Z., see Lohmann, A.W ^ Zalevsky, Z., D. Mendlovic, A.W. Lohmann: Optical systems with improved resolving power Zavorotny, VU., see Chamotskii, M.I. Zavorotnyi, VU., see Tatarskii, V.I. Zuidema, P., see Bouman, M.A.
33, 261 39, 63 28, 25, 14, 29, 34, 4, 13, 27,
181 279 89 293 333 241 267 161
39, 10, 17, 27, 31, 40,
63 89 163 161 263 1
1, 10, 28, 33, 40,
155 137 1 389 343
22, 271 6, 105 8, 295 28, 28, 32, 41, 11, 23, 32,
87 87 145 97 77 221 61
^^^ ^^^ 40, 32, 18, 22,
271 203 204 77
