ADVANCES IN IMAGING AND ELECTRON PHYSICS VOLUME 103
EDITOR-IN-CHIEF
PETER W. HAWKES CEMESIhboratoire d 'Optique Electmnique du Centre National de la Recherche Scientifique Toulouse, France
ASSOCIATE EDITORS
BENJAMIN M A N Xerox Corporation Palo Alto Research Center Palo Alto, California
TOM MULVEY Department of Electronic Engineering and Applied Physics Aston University Birmingham, United Kingdom
Advances in
Imaging and Electron Physics EDITED BY PETER W. HAWKES CEMES/Luboratoired’Optique Electmnique du Centre National de la Recherche Scient@que Toulouse, France
VOLUME 103
ACADEMIC PRESS San Diego London Boston New York Sydney Tokyo Toronto
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ISBN: 0-12-014745-9 Printed in the United States of America 98 99 00 01 BB 9 8 7 6 5 4 3 2
1
CONTENTS CONTRIBUTORS . . . . PREFACE . . . . . .
I. 11. 111.
IV. V. VI.
. . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . .
Space-Time Representation of Ultra Wideband Signals EHUDHEYMANAND n M O R MELAMED Introduction and Outline . . . . . . . . . . . . . . . . Time-Harmonic Radiation from an Aperture . . . . . . . . . Time-Domain Representation of Radiation from an Aperture . . Illustrative Example . . . . . . . . . . . . . . . . . . WavepacketsandhlsedBeamsinaUnifomMedium . . . . . Phase-Space Pulsed Beam Analysis for Time-Dependent Radiation from Extended Apertures . . . . . . . . . . . . . . Appendix: Asymptotic Evaluation of the Beam Field in (125) . . References . . . . . . . . . . . . . . . . . . . . .
vii ix
3 7 15 24 30
44 59
60
The Structure of Relief JANJ. KOENDERINK AND A. J. VAN Doom I. 11. 111. IV. V. VI.
Introduction . . . . . . . . . . . . . The Differential Structure of Images . . . . Global Description of the Relief . . . . . Contours: Envelopes of the Level Curves . . Discrete Representation . . . . . . Conclusion . . . . . . . . . . . . . References . . . . . . . . . . . . .
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. . . .
. . . . . . , . . .
66 83 98 138 140 147 147
Dyadic Green’s Function Microstrip Circulator Theory for Inhomogeneous Ferrite With and Without Penetrable Walls CLIFFORDM. KROWNE I. Overall Introduction . . . . . . . . . . . . . . . . . 152 11. Implicit 3D Dyadic Green’s Function with Vertically Layered External Material Using Mode-Matching . . . . . . . . 153 V
vi
CONTENTS
111. Implicit 3D Dyadic Green’s Function with Simple External Material Using Mode-Matching . . . . . . . . , . . . . . . IV. 2DDyadicGreen’sFunctionforPenetrable Walls . . . . . V. 3D Dyadic Green’s Function for Penetrable Walls . . . . . . VI. Limiting Dyadic Green’s Function Forms for Homogeneous Femte VII. Symmetry Considerations for Hard Magnetic Wall Circulators . VIII. Overall Conclusion . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .
.
196 208 213 228 240 273 274
Charged Particle Optics of Systems with Narrow Gaps: A Perturbation Theory Approach M. I. YAVOR I. Introduction . . . . . . . . . . . . . . . . . . . . 278 II. Applicability of Perturbation Methods in Charged Particle Optics 283 111. Calculation of Weakly Distorted Sector Fields and Their Properties withthe AidofaDirectSubstitutionMethod . . . . . . 295 IV. Transformation of Charged Particle Trajectories in the Narrow Transition Regions Between Electron- and Ion-Optical Elements 318 V. Synthesis of Required Field Characteristics in Sector Energy Analyzers and Wien Filters with the Aid of Terminating Electrodes . . . . . . . . . . . . . . . 336 VI. Calculation of the Elements of Spectrometers for Simultaneous Angular and Energy or Mass Analysis of Charged Particles . . . . . . . . . . . . . . . . . 348 VII. Conclusion . . . . . . . . . . . . . . . . . . . . . 384 Acknowledgments . . . . . . . . . . . . . . . . . . 385 References . . . . . . . . . . . . . . . . . . . . . 385 INDEX .
. . . . . . . . . . . . . . . . . . . . . . . .
389
CONTRIBUTORS Numbers in parentheses indicate the pages on which the author’s contribution begins.
EHUDHEYMAN (l), Department of Electrical Engineering, Physical Electronics, Tel Aviv University, Tel Aviv 69978, Israel JANJ. KOENDERINK (65), Buys Ballot Laboratory, University of Utrecht, Faculty of Physics and Astronomy, Princetonplein 5, P.O. Box 8oo00, 3508 TA Utrecht, The Netherlands CLIFFORD M. KROWNE(151), 3810 Maryland Street, Alexandria, VA 22309-2583 TIMOR MELAMED (l), Boston University, College of Engineering, 110 C u d n g ton Street, Boston, MA 02215
A. J. VAN D o o w (65), Buys Ballot Laboratory, University of Utrecht, Faculty of Physics and Astronomy, Princetonplein 5 , P.O. Box 8000,3508 TA Utrecht, The Netherlands M. I. YAVOR(277), Institute for Analytical Instrumentation RAS,Rizhskij pr. 26, 198103 St. Petersburg, Russia
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The four chapters that make up this volume are drawn from very different areas. In the first contribution, E. Heyman and T. Melamed examine a complicated problem that arises in the study of wide-band signals. The familiar frequency-domain approach is not the most appropriate here and the authors describe and explore at length other analytic techniques. This unified presentation of techniques to which the authors have largely contributed should be very welcome to anyone wishing to become familiar with these ideas. The second contribution is really a short monograph on a vexed and important question: how should relief be described and represented? J. J. Koenderink and A. J. van Doorn, who have made numerous contributions to thinking on this problem, give here a systematic and carefully reasoned study of the various aspects of the topic. The themes are defined in the introductory section, after which the authors describe the differential structure of images, the global description of relief, contours, and discrete representations. Readers familiar with Eberly’s book on Ridges in Image and Data Analysis (Kluwer, 1996) will certainly wish to read this chapter. The third chapter needs less introduction from me, being a further contribution by C. M. Krowne on microstrip circulator theory. The appropriate dyadic Green’s functions are explored thoroughly for several designs. Finally, we have a long account by M. I. Yavor on perturbation methods in electron optics. Several practical systems of great practical importance are examined: sector energy analyzers, Wien filters and several conical designs. The theory is set out fully and the examples are analyzed critically, with the result that designers of such systems should find these studies of direct interest. In conclusion, I thank all the authors most warmly for the time and scholarly effort that they have devoted to their manuscripts. A list of contributions to forthcoming volumes is given below. Peter W. Hawkes
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PREFACE
X
FORTHCOMING CONTRIBUTIONS Mathematical models for natural images Use of the hypermatrix Image processing with signal-dependent noise Near-sensor image processing The Wigner distribution Modem map methods for particle optics Magneto-transport as a probe of electron dynamics in semiconductor quantum dots Distance transforms ODE methods Microwave tubes in space Effect of proton radiation damage on charge-coupled devices Fuzzy morphology Gabor filters and texture analysis Liquid metal ion sources X-ray optics The critical-voltage effect Stack filtering The development of electron microscopy in Spain Number-theoretic transforms and image processing Contrast transfer and crystal images Conservation laws in electromagnetics External optical feedback effects in semiconductor lasers Numerical methods in particle optics Spin-polarized SEM Sideband imaging Computer-aided design using Green’s functions and finite elements Memoir of J. B. Le Poole Well-composed sets Vector transformation
L. Alvarez Leon and J. M. Morel D. Antzoulatos H. H. Arsenault A. Astrom and R. Forchheimer M. J. Bastiaans M. Ben. and colleagues J. Bird G. Borgefors J. C. Butcher J. A. Dayton J. Deen, T. Hardy and R. Murowinski E. R. Dougherty and D. Sinha J. M. H. Du Buf R. G. Forbes E. Forster and F. N. Chukhovsky A. Fox and M. Saunders M. Gabbouj M. I. Herrera and L. Bni A. G. J. Holt and S. Boussakta K. Ishizuka C. Jeffries M. A. Karim and M. F. Alam E. Kasper K. Koike w. Krakow C. M. Krowne van de Laak-Tijssen, E. Coets and T. Mulvey L. J. Latecki W. Li
xi
PREFACE
Complex wavelets Discrete geometry in image processing Electronic tools in parapsychology Z-contrast in the STEM and its applications Phase-space treatment of photon beams Image processing and the scanning electron microscope Representation of image operators Aharonov-Bohm scattering Fractional Fourier transforms HDTV Scattering and recoil imaging and spectrometry The wave-particle dualism Digital analysis of lattice images (DALI) Electron holography X-ray microscopy Accelerator mass spectroscopy Focus-deflection systems and their applications Hexagonal sampling in image processing Confocal microscopy Electron gun system for color cathode-ray tubes Study of complex fluids by transmission electron microscopy New developments in ferroelectrics Organic electroluminescence-materials and devices Electron gun optics Very high resolution electron microscopy Mathematical morphology and scanned probe microscopy Morphology on graphs Representationtheory and invariant neural networks Magnetic force microscopy Structure, fabrication and performance of color CRTs
J.-M. Lina, B .Goulard and P. Turcotte S. Marchand-Maillet R. L. Moms P. D. Nellist and s. J. Pennycook G. Nemes E. Oho B. Olstad M. Omote and S. Sakoda H. M. Ozaktas E. Petajan J. W. Rabalais H. Rauch A. Rosenauer D. Saldin G. Schmahl J. P. F. Sellschop T. Soma R. Staunton E. Stelzer H. Suzuki I. Talmon
J. Toulouse T. Tsutsui and Z. Dechun Y. Uchikawa D. van Dyck J. S. Villarmbia L. Vincent J. Wood C. D. Wright and E. W. Hill E. Yamazaki
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ADVANCES IN IMAGING AND ELECTRON PHYSICS VOLUME 103
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ADVANCES IN (MAGINO AND -ON
PHYSICS. VOL . 103
Space-Time Representation of Ultra Wideband Signals EHUD HEYMAN AND TIMOR MELAMED Department of Electrical Engineering-Physical Electronics Tel-Aviv University. Tel-Aviv 69978. Israel
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . I . Introduction and Outline . . . . . . . . . . . . . . . . . . . . . . II. Time-HarmonicRepresentationsofRadiationfromanAperture . . . . . . . . A . Green's Function Representations . . . . . . . . . . . . . . . . . . B. Plane-Wave Representations . . . . . . . . . . . . . . . . . . . . C. Ray Representation . . . . . . . . . . . . . . . . . . . . . . . D . Radiation Pattern . . . . . . . . . . . . . . . . . . . . . . . In . Time-Domain Representations of Radiation from an Aperture . . . . . . . . . A . Analytic Signals . . . . . . . . . . . . . . . . . . . . . . . . B. Green's Function Representation . . . . . . . . . . . . . . . . . . C . Time-Dependent Plane-Wave Representation . . . . . . . . . . . . . . D. Ray Representation . . . . . . . . . . . . . . . . . . . . . . . E . Time-Dependent Radiation Pattern . . . . . . . . . . . . . . . . . IV. Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . A . The Initial Field . . . . . . . . . . . . . . . . . . . . . . . . B. The Time-Dependent Plane-Wave Spectrum . . . . . . . . . . . . . . C . Radiation Pattern . . . . . . . . . . . . . . . . . . . . . . . D . Special Case: Well-Collimated Condition . . . . . . . . . . . . . . . E. Frequency-Domain Interpretation . . . . . . . . . . . . . . . . . . V. Wavepackets and Pulsed Beams in a Uniform Medium . . . . . . . . . . . . A . General Solution . . . . . . . . . . . . . . . . . . . . . . . . B. Properties and Interpretation . . . . . . . . . . . . . . . . . . . . C. Relation to Complex Source Pulsed Beams (CSPB) . . . . . . . . . . . D . Relation to Time-Harmonic Gaussian Beam . . . . . . . . . . . . . . E. Considerations in UWB Synthesis of Collimated Apertures: Isodifiacting versus Isowidth Apertures . . . . . . . . . . . . . . . . . . . . . . . VI. Phase-Space Pulsed Beam Analysis for Time-Dependent Radiation from Extended Apertures . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Frequency-Domain Formulation: Beam Summation . . . . . . . . . . . B. Time Domain: Pulsed Beam Summation . . . . . . . . . . . . . . . Appendix: AsymptoticEvahationof theBeamFieldin (125) . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .
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ADVANCES IN IMAGING AND ELECTRON PHYSICS Copyright @ 1998 by Academic Rws Inc. AU rights of reproductionin any form reserved. ISSN 1076-5670198$25.00
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LIST OF S m B O L S For the wavefields we use a self-consisted symbol system: Time-dependent fields have no special mark; analytic time dependent fields are identified by +; frequencydomain fields are identified by ; plane-wave fields (in either the time or the frequency domain) are identified by -; functions in the initial (data) plane z = 0 are identified by a subscript 0 .
-
Time-dependent field Analytic time-dependent field; see (21) or (22) Frequency-domain field Time-dependent field in the z = 0 plane Frequency-domain field in the z = 0 plane Time-dependent plane-wave spectrum; see slant-stack transform (33) Analytic time-dependent plane-wave spectrum; see (30) Frequency-domain plane-wave spectrum; see (3) Local (phase-space) spectrum in the frequency domain; see (1 10) Local (phase-space) spectrum in the time domain; see (130) Window function in the frequency domain; see (110) Window function in the time domain; see (131) Phase-space beam propagator in the frequency domain; see (115) Phase-space pulsed-beam propagator in the time domain; see (137) 3D coordinate point Time coordinate Transverse coordinates Transverse coordinates in the z = 0 plane Plane-wave spectral coordinates; see (3) and (30) Direction of the spectral plane wave; see ( 5 ) and (7) Spectral time coordinate; see (30) Phase-space coordinates for frequency-domain fields Phase-space coordinates for time-dependent fields Propagation direction of the phase-space beam; see ( 1 17)
SPACE-TIME REPRESENTATIONOF ULTRA WIDEBAND SIGNALS
I.
3
INTRODUCTION AND OUTLINE
Continuing advances in the development of radiators and receivers for electromagnetic, acoustic, and elastic waves have provided the capability to gather data with a high degree of reliability. The recent trends are toward ever wider signal bandwidths because of the enlarged database with high temporal-spatial resolution provided thereby (see examples in [l-31). To extract from received signals the desired information requires data processing methods that link features (observables) in a signal to features in the environment encounteredby the signal during its travel from source to receiver. Such observables-based parameterization requires a thorough understanding of pulsed-fields synthesis, of propagation and scattering of such fields, and of processing short-pulse scattering data. Because of the broad frequencyband of these fields, the conventional route of inversion of frequency-domain(FD) solutions is often less convenient and physically less transparent than direct treatment in the space-time domain where the fields are well localized. Furthermore, direct time-domain (TD) analysis and processing techniques and “TD thinking” make explicit use of the TD observables and can be used to relate distinct TD observables (events) with instantaneous features of the sources, and vice versa. However, it is desirable, whenever possible, to formulate the TD and the FD techniques within a unified analytical framework, thereby emphasizing the differences introduced by the short-pulse excitation. Choosing the proper processing technique, either in the frequency domain or in the time domain, can then be posed as a trade-off that can be expressed in terms of the problem parameters (e.g., pulselength, space-time observation range, etc.). Good engineering, therefore, requires thinking in both domains. Following this motivation we shall consider several alternative field representations in both the frequency domain and the time domain. They involve Green’s function and plane-wave representations, as well as pulsed beams and local spectrum techniques. Such techniques have been utilized for various applications, including antenna characterization,field analysis, inverse scattering, and so on. (See an overview of the applications in [ 1 4 1 as well in the following discussion and references.) The focus of this paper is thus a review of some analytic techniques and wave solutions that are relevant to the analysis of ultra wideband short-pulse fields and data. To convey the ideas in the simplest format we consider the simple problem of short-pulse radiation from an aperture distribution (Fig. 1). The given field in the initial plane may be the physical time-dependent source (forward problem) or it may be a measured field due to a remote sensing or scattering experiment (inverse problem). In the forward problem one is interested in calculating the radiating field, and in its parameterization and optimization. In the inverse case, the time-dependent data should be processed in order to extract
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FIGURE 1. The-dependent radiation from an aperture in the z = 0 plane. The adopted notations for the Cartesian coordinates are r = (x, e), with x = ( X I , X Z ) . Points in the z = 0 plane are denoted as ro = (m, 0). u(r. 1 ) is the radiated field while the known field distribution in the z = 0 plane may be either the field itself uo(w,t ) = u(r, t)l,=O or its normal derivative U , O ( ~ t, ) = a,u(r, r)l,=o.
the information on the sources (either real or induced sources). ’Qpically, inverse processing involves the same radiation-type integrals used in forward problems. In view with the strategy outlined earlier, we start in Section I1 by reviewing the corresponding FD representations,namely, the Green’s function (or Kirchhoff) integral and the plane-wave integral. The properties of these representations and the physical structure of the field are then discussed by considering explicit approximations, namely, the ray solution and the far-field pattern. Having reviewed the FD formulation, we proceed to discuss the corresponding TD formulations (Section 111). Here, too, we start with the Green’s function and with the plane-wave representations. The first has the well-known form of integration of retardedfields (see, e.g., [ 5 , 61). In the second, the radiating field is expressed as an angular superpositionof time-dependent plane waves. The spectral function (i.e,, the time histories of the plane waves) is found via a Radon transform processing of the space-time data. This operation has the form of a slant stacking of the data; hence, we adopt the notation slant-stack transform (SST). Physically, it extracts the directional information from the space-time data, thereby expressing the field as an angular superposition of time-dependent plane waves. The SST is the TD analog the spatial Fourier transform used in conventional plane-wave representation of time-harmonic fields [5,6]. The TD plane-wave representation has been introduced originally in [7] and [8], and has been extended later on in the spectral theory of transients (STT)in [9] and [lo] as an analytical framework for the analysis of the radiation, propagation and scattering of pulsed fields in various configurations. More recently the SST has been used in the context of near-field scanning of short-pulse antennas [ 111. In [ 121, the TD plane-wave representation has been extended to deal with volume sources, and in [13], this volume formulation has been used as a basis for a theory of TD characterization of antenna systems. The TD plane-wave approach and the SST have also been
SPACE-TIME REPRESENTATIONOF ULTRA WIDEBAND SIGNALS
5
used for the processing of short-pulsescattering data [14,15]. Finally, in [16], this approach has been combined with a Born-type inverse scattering to formulate a TD difiuction tomography (which is the TD analog of the conventional diffraction tomography used for time-harmonic fields [17, 181). In the present paper we demonstrate the time-dependent plane-wave spectrum approach in the context of the problem depicted in Fig. 1. As in the frequency domain, the TD plane-wave spectrumconsists of both propagating and the evanescent spectra; the latter one describes pulsed fields that decay away from the aperture. Using an analytic signal representation, we derive a unified representation that combines both the propagating and the evanescent spectra. However, for the benefit of the practitioner we also consider the separate role of each spectral constituent and present explicit expressions for each of them directly in terms of the real time-dependent data. To clarify the field structure and the numerical properties of the TD plane-wave integral, we also consider approximate field solutions. In the near zone the pulsed field propagates along space-time rays that emerge from the aperture along directions that are determined by the gradient of the delay function (Section 1II.D). We provide explicit expressions for the ray signal, and show that it is Hilbert transformed when the ray passes through a caustic (a ray envelope that forms a focus surface; see Sections II.C and II1.D). In the far zone (Section 1II.E) the field can be approximated by the pulsed radiation pattern, which is directly related to the time-dependent plane-wave spectrum (i.e., the SST) of the source distribution. The distance where the transition from the near-zone local ray representation to the far-zone global plane-wave representation occurs, depends on the spatial width of the aperture and on the pulselength. Borrowing FD terminology, it is termed here the TD Fresnel distance [see (28) and (47)J.Finally, all these TD characteristics are illustrated for an analytic example (Section IV). In Section V we explore the characteristics of well-collimated ultra wideband wavepackets. Such space-time wavepackets are useful in various applications, including modeling of ultra wideband radar or sonar beams, local interrogationof the propagation environment,transmission of localized energy, secured high-rate communication, and so on. Several classes of such localized space-time wavepackets solutions have been introduced recently [ 19-32]. In the eighties, these wavepackets have been studied primarily in the context of possible synthesis of high-energy, nondiffracting, or weakly diffracting wavepackets (i.e., wavepackets that remain localized up to very large distances), but more recently the emphasis has been placed on other applications (see the following discussion). The possible excitation of such nondiffracting waves by physically realizable sources are discussed at the beginning of Section V, based on the time-dependent radiation concepts introduced in Section III. The rest of Section V is devoted to one class of wavepacket solutions, termed pulsed beams (PB) [30-321. These highly localized wave solutions can be
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EHUD HEYMAN AND TIMOR MELAMED
realized by a distributed source in an aperture with jnite support. Unlike other wavepackets, the PBs behave classically in the sense that they remain collimated up to a certain distance (the TD Fresnel distance), and thereafter they diverge along a constant diffraction angle. However, they may be considered as optimal solutions to the collimation problem in the sense that all frequency components remain collimated up to the same distance (the Fresnel distance) [33]. Both the Fresnel distance and the diffraction angle can be controlled by the time-dependent source distribution. The PBs have been used for modeling of wavepackets’ propagation in free space [30,31,33], and for high-resolution probing [34-371. In addition to the foregoing applications, the PBs also furnish a complete basis for general representations of space-time signals that express the field as a phase-space superposition of PBs emerging from all points in the source domain and in all directions. The motivation in these PB summation representations is to obtain local spectral representations that can also be tracked through complicated media. Unlike the more conventional plane-wave representation, each basis function in the PB summation representations accounts only for the local radiation properties of the source near the PB initiation point. Further localization is due to the fact that only those PB basis functions that pass near the space-time observation point need to be considered in the superposition integral. Finally, unlike plane-wave representations, the basis functions in the PB representation can readily be tracked through ambient media, and unlike ray fields, they are insensitive to transition regions (such as caustics or shadow boundaries) [32]. Several alternative schemes for PB summation, which apply for different source configurations have been introduced in [38, 39, 10, 40-421, and will be reviewed in Section VI. As an example for a local PB summation approach, we consider in Section VI the problem of time-dependent radiation from an aperture depicted in Fig. 1. The analysis is based on a local (phase-space) processing of the time-dependent data which extracts the local features of the source and thus expresses the radiated field at a given space-time region only in terms of the relevant local contributions of the source. The processing transform is a local slant-stack transform, which is a windowed version of the SST mentioned earlier and thus extracts the local planewave (directional) information about the window center. Physically, it matches a PB wave propagator that propagates away from the window center along the local radiation direction. This description should be contrasted with the SST that extracts the global directional information of the source distribution, and thus for each direction matches a global time-dependent plane wave to the entire aperture. The local approach is therefore termed a phase-space pulsed beam summation [39, 10, 40, 411. The trade-off in this representation relative to the conventional representationsis an a priori data localization versus more complicated processing integrals and databases. It becomes efficient, therefore, when dealing with very large data objects like those obtained by gathering ultra wideband information over
SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS
7
very large apertures. Specifically,this approach has been utilized in [43] and [44] for local inverse scattering using short-pulse scattering data. This paper is intended to review ideas and techniques associated with spectral representations for time-dependent radiation and propagation, including integral representation, localization techniques, and local wave phenomena. The first half (Sections 11-IV) focuses on “conventional” TD techniques, while the second half (SectionsV-VI) deals with localized waves and local spectrum techniques. Various applications of these techniques have already been mentioned earlier and will be discussed in further detail in the text. This text, however, focuses only on a limited selection of TD formulations and techniques out of those developed recently by these authors and others.
11. TIME-HARMONIC REPRESENTATIONS OF RADIATION FROM AN APERTURE To demonstrate the alternative representationsinvolved in the space-time analysis and processing of time-dependent radiation, we consider the simple model of radiation in a uniform medium with wave speed c (Fig. 1). Specifically we shall be interested in the scalar field u (r, t ) radiated into the half plane z > 0 due a given field distribution in the z = 0 plane. The given field can be either the field itself u o ( q , t ) = u(r, t)lz,O, or its normal derivative U , O ( X O , t ) = a&, t)1,,0, or both. The adopted notations for the Cartesian coordinates are r = (x, z), x = (XI, XZ) while points in the z = 0 are denoted as ro = (xg, 0). This configuration is also relevant for space-time processing wherein uO(r0, t ) is the measured data. We start, however, with the well-known field representations in the frequency domain, which will then be extended to the time domain. Henceforth, we use a circumflex to denote time-harmonic constituents with an assumed exp(-iwt) time dependence (see List of Symbols). In accord with the preceding notations, the known field constituentsin the z = Oplane are either fio(x0) = fi(r)lz=oandor f i z 0 ( x o ) = azfi(r)lz=o. A. Green’s Function Representations
In the Green’s function representation (also termed Kirchhoff or Huygens ’s representation [ 5 ] ) , the known field in the aperture plane z = 0 is replaced by an equivalent source distribution. The radiated field for z > 0 can then be described as a superposition of spherical wave contributions from all points in the aperture (see Fig. 2). In the time-harmonic case, the superposition integral has the form
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EHUD HEYMAN AND TIMOR MELAMED
FIGURE2. Green’s function representation of the radiated field: The radiated field for z z 0 is described as a superposition of spherical wave contributions W from all points xo in the aperture. The signals and the notations correspond to the TD representation (25) with (25a-c). The corresponding time-harmonic representation is given in ( I ) with (1 a x ) .
where $(r, ro) is a spherical wave, emitted from each point ro in the z = 0 plane (Fig. 2). It is related to the data on the z = 0 plane via either one of the alternatives
where &kR
w
G(r, ro) = - k = -, 43rR’
C
R = Ir-rol
is the free-space Green’s function and c is the wave velocity. Its normal derivative with respect to the z = 0 plane is given by
Some words of caution should be added here. The three representations in (1a-c) are equivalent only if the initial field at z = 0 is strictlyforward propagating, that is, if all the sources (real or induced) are located at z -= 0. As implied by (1b, c), the field representations in this case require a knowledge of only l i 0 or of its normal derivative ii,~. Equations (lb, c) cannot be used, however, if ii consists of both forward and backward propagating waves. In this case one should use the representation in (la) wherein the kernel $ extracts the forward propagating part in the initial field iio. The integral (I), therefore, does not describe the total field ii but only a+, that is, the forward propagating part of the field for z > 0 [see (9)].
9
SPACE-TIMEREPRESENTATIONOF ULTRA WIDEBAND SIGNALS
B. Plane- Wave Representations Expression (1) describes the field as a superposition of point source fields, excited by the initial field distribution via the alternative relations in (la-c). The radiated field can be described instead as an angular superposition of plane-wave fields, which are matched to the initial field distribution (or sources). The plane-wave spectral information is recovered via the transverse transform pair
The over-tilde notation for spatial spectrum will also be used for TD spatial spectrum [see (29), (33) and List of Symbols]. Thus, ' will be used for the plane-wave spectrum of time-harmonic fields, obtained via the spatial Fourier transform (3a), whereas will be used for the time-dependent plane-wave spectrum as obtained from the transform in (33). Anticipating extension to the time domain we also use here a frequencynormalized spectral variable (transverse wavenumber) = (61,h).Accordingly, has a frequency-independent geometrical interpretation in terms of the planewave angle [see (7)] that can readily be extended to the time domain. To derive the plane-wave representation of the field we use the spectral representation of 6 [45]
-
<
<
where
c = d1- I ~ I ~Im< , zo
(5)
is the normalized wavenumber in the z-direction. Substituting (4) into (1) with (1a-c), exchanging the order of integrationsand evaluating the xo integrationyields
where (6a-c) Here do(() and dZo(() are found from the data ~O(XO) and i i , o ( x ~ )via (3a).
10
EHUD HEYMAN AND TIMOR MELAMED
RGURE 3. Plane-wave representationof the radiated field: The radiated field for z > 0 is described as an angular superposition of time-dependentplane waves. The signals and the notationscorrespond to the TD representation (30H3I ) (see also Fig. 6): iio(E, 5 ) is a time-dependentplane wave propagating in the direction k = (E, I ' ) in (7). It is found from the data via the slant-stack transform (30) (see Fig. 6).
Equation (6) expresses the field as an angular superposition of plane waves
5 = $O(e> (Fig. 3). For
exp[ik(e x
+ cz)l
161 < 1 each plane wave propagates in the direction
a = (e, 5') = (case,, Cose2,COSO,),
la1 = 1
(7)
where 01,2,z are the angles of k with respect to the corresponding axis. For =- 1 on the other hand, = i hence, the integrand in (6) consists of evanes<1 cent waves that decay in the positive z-direction. The spectral ranges and 161 > 1 are termed, therefore, the propagafing and the evanescenr spectra, respectively. The plane-wave amplitude @O matches the spectrum of plane waves to the initial distribution via either one of the alternatives in (6a-c) which correspond to those in (la-c). As mentioned there, Eqs. (6b,c) are applicable only if iio is strictly forward propagating. Equation (6a), on the other hand, can be applied even if Lio consistsAofboth forward and backward propagating components. In this case the kernel Go in (6a) extracts from the data iio the part corresponding to the forward propagating field, so that the integral for ii yields only of the forward propagating field Li+. To demonstrate this spectral selection we note that the field is described, in general, as a spectrum of forward and backward propagating waves
< d m ,
where z?$ are the spectral amplitudes at z = 0. To express them in terms of the data &andfz0,weapply(8a)atz = 0,givingbo = (hJ+&)andf,o = ikc(i&-bo),
SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS
11
and thus
Thus, if the field cqnsists only of forward propagating waves, then either one of the alternatives for $0 in (6a-c) yields the spectrum hij of the forward propagating wave. If, however, the field consists of both forward and backward propagating waves, then knowing only or cannot recover i$. In view of (8b), however, the spectral kernel (6a) extracts the forward propagating amplitude, giving
50 = st.
(9)
C. Ray Representation In the high-frequencyregime, the Kirchhoff integral (1) and the plane-wave integral (6) are highly oscillatory. Asymptotic approximations may, therefore, be used to simplify the numerical evaluation of these integrals or to derive closed-form approximations for the field. For points in the near zone, the field structure can be explained essentially as a geometrical optics continuation of the aperture field [see (12)]. The integrals here are dominated by stationary point contributions that describe constructive interference of $e integrated wave constituents (i.e., the spherical waves $ or the plane waves $0) along the ray paths. It may be shown that for points near the aperture the stationary point contributions in the Green’s function integral (1) are more localized than in the plane-wave integral (6) ([46]; Appendix A). This can be explained intuitively by noting that many plane waves are needed to synthesize the ray optical “point-to-point” relation (in analogy, many frequencies are needed to synthesize a delta pulse). In the far zone, on the other hand, the radiation pattern exhibits the spectral spread of the entire source distribution;hence, the plane-wave integral (6) is localized around the plane wave that propagates in the observation direction. The Green’s function integral, on the other hand, is distributed over the entire aperture domain (see Section 1I.D). These alternative field descriptors are discussed in this section and the following one. According to (l), each point ro radiates in all directions (Fig. 2). In the highfrequency regime, however, contributions from neighboring points interfere constructively along a preferred direction, describing phase matching of a local plane wave to the field in the aperture plane. To demonstrate this local behavior we express the initial distributions in the form
i i o ( ~w, ) = ~ ( mw)eik@(m) ,
(10)
where 2 and @ are slowly varying functions of ~0 on a wavelength scale. For simplicity it is assumed that @ is real. The local preferred direction of radiation is
12
EHUD HEYMAN AND TIMOR MELAMED
f (PI&) FIGURE 4. Ray representation (contrast Fig. 2). The rays are emitted
from each point in the aperture along the local preferred direction of radiation (1 1). d A is the differential cross section associated with a ray tube. If the phasefront radii of curvature p1,z are negative, the rays converge to a caustic at a distance R = -p1,2 from their initiation point. Beyond the caustic, the signal is filbert transformed [see ( 15) and (43)l.
given via [see (7) and Fig. 41
where VO is the gradient with respect to XO. Equation (11) describes a local matching of the phase at xo to one of the plane waves in (6). Since the plane-wave phase in the z = 0 plane is kf .XO, the local match is determined by f = Vo@(xo). Equation (1 1) then follows by noting from (7)that f = (cos 01, cos 02). The principle of local matching may be established analyticallyby an asymptotic evaluation of the integral representation (1) or (6) (see, e.g., [46]; Appendix A). Indeed, the dominant contributionto the field at a given observation point r comes from a stationary point xg(r), describing a ray that emerges from xg(r) in the preferred direction (1 1) and passes through r (see Fig. 4). The field along this ray is given by [47,48]
i(r) = i(w)d(R)eikR
(12)
where R is the distance along the ray and the amplitude divergence term A(R) is given by
A(R) = [ 1 + [ R 2 d e t @ i j + R ( @ ~ 1 s i n 2 ~ 2 + @ 2 2 s i n 2 8 1 - 2oI2 cosel C O S ~ ~ ) ] / C O S ~ ~ ) - ~ / ~ I ~(13) where aij= a$@ = a2 @ / a x o i a x o j ,6, are defined in (1 1) and cose = Ji - C O S ~el - C O S ~e2.
SPACE-TIMEREPRESENTATION OF ULTRA WIDEBAND SIGNALS
13
Using differential geometry, Eq.(13) may be expressed as
where d A ( r ) is the differential cross section of the ray tube and dAo is the initial value of d A (Fig. 4). In the second expression, the ray tube cross section is expressed in terms of ~ 1 . 2 which , are the principal radii of curvature of the phasefront at xo (Fig. 4). p1,2may readily be found from @ by comparing (13) and (14). Clearly, Eq. (14) for the amplitude divergence coefficient A complies with ray optical and energy conservation considerations [47]. Expression (1 2) is invalid in caustic regions where the differential cross section d A tends to zero. In these regions the ray model has to be augmented by a uniform ray model [48-501. The ray solution can nevertheless be continued through the caustic. Once the ray tube passes through the caustic, d A changes sign; hence, the ray amplitude in (14) can be tracked by using
where the Maslov index M in (15) increasesby an integer as each caustic is passed: The increment is the number of dimensionsthe ray tube loses at the caustic [i.e., M is increased by 1 or 2 if the ray tube cross section reduces to a line (a caustic) or a point (a focus), respectively]. Expression (15) is rather general and may be applied in inhomogeneous media. For the uniform medium configuration consideredhere, the caustic locations along the ray path are determined by R = -p1,2 (Fig. 4) and the radiating ray may touch the corresponding caustic only if p1.2 < 0 (converging ray tube). As an illustrative example, consider the case where @ is a function of xo, only and is independent of xh. From (1 1) cos 81 = &@(XI) and from (13)-(14) dA(r)/dAo = 1 R@Il/cos28so that p1 = @ilcos28. Thus, the rays converge to a caustic only if @11 < 0. Otherwise, pl > 0 and the ray tube diverges. As follows from the preceding discussion, if the spectral spread of the source is large (i.e., the rays emitted from the aperture are not parallel), then the ray representation can be tracked up to the far (or radiation) zone: If p1,z > 0, the ray representation is tracked continuously all the way up to the far zone, whereas if pl,2 < 0, the ray representation breaks at the caustic, but it can be continued beyond it and up to the far zone using (15). For well-collimated apertures, on the other hand, the rays are nearly parallel (i.e., the phase satisfies VO@ 2i const.); hence, the simple ray representation fails in the far zone. In this case, the far-zone field may be expressed effectively by the spectral representation as described later.
+
14
EHUD HEYMAN AND TIMOR MELAMED
D. Radiation Pattern For observation points in the far zone, one may approximate
where cos +(P, Po) = f a PO. We use here the vector notations r 3 Irl, P = r / r and the same notations for the source coordinates ro. When substituted in the exponent e i k ( r - r o ( in (2), the second correction term in (I 6) may be neglected if
where L is a measure of the source dimension in the z = 0 plane. In this case we 'tr may use in (2) 6 2: &eikC'ro and thus obtain from (lc)
where the radiation pattern is
g(P) = -2ik cos8&(i$)lc=x/r.
(19)
Here $0 is the spectral distribution of iio as defined in (3b), 8 is the observation angle relative to the z-axis and x is the transverse coordinates of the observation point (i.e., r = (x, z)). For simplicity we used here only the field representation in (lc): A similar expression for is obtained from (lb) if the data in the z = 0 plane is i i , ~rather than i i ~ . modulated by an From (18), the far field has a form of a spherical wave angular radiation pattern g (note that for large apertures the angular variation due The condition i$ = x/r implies to cos 8 is weak in comparison to that of &l+x,r). that the radiation pattern in the direction P is determined by the spectral plane wave t o that propagates in the direction R = P [see (7)l. We may contrast now the general properties of the alternative integral representations in (1) and (6) as applied for observation points in the far zone. The integral (6) has a highly oscillatory phase exp[ik(i$ x (z)] = exp[ikr(P . A)] where P is the observation direction and R is the spectral propagation direction. Thus, the far-field expression in (19) is obtained as a local stationary point contribution about the pertinent spectral direction A = P. The integrand in (l), on the other as implied by (16). Thus, in general hand, is dominated by a linear phase eikP.ro the integration in (1) must be carried over the entire aperture. The result in (19) is thus a global contribution of the aperture.
+
SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS
15
REPRESENTATIONS OF RADIATION FROM AN APERTURE 111. TIME-DOMAIN
Having reviewed the basic formulations and concepts associated with the timeharmonic problem, we return now to the subject of time-dependent fields. The more efficient way to handle ultra wideband excitations, at least in the shortpulse case, is to formulate the radiation integral directly in the time domain. The alternative approach is to transform the data into the frequency domain, and then calculate the radiation integrals on a frequency-by-frequency basis, and finally transform the results back into the time domain. In particular for large distances, the frequency axis must be sampled very densely. This is due to the fact that typically the time-harmonic field has the form [cf. (12)] ii(w) E,ii,eiurj where ti are propagation delays along the rays. If tj are large, ii is a rapidly oscillatory function of w that has to be sampled densely in frequency when transformed into the time domain. In certain cases, a common term e'"', where t is an average delay, may be extracted, but the procedure for estimating t is quite complicated, in particular under multipath conditions or if ii is specified numerically. Furthermore, this procedure has to be repeated for every observation point. Thus, for short excitation pulses, direct solutions in the time domain where the field is well localized are preferable. With careful interpretation, though, many of the FD concepts and techniques can be translated to the time domain.
-
A. Analytic Signals
The TD field formulations will be derived by transforming the corresponding frequency-domainformulationsinto the time domain via the inverse Fourier transform
However, in order to gain flexibility in the derivation, in particular in those formulations that involve evanescent spectra, it is convenient to use analytic signal representation defined via the one-sided inverse Fourier transform
+
Here f ( t )is the dual analytic signal correspondingto the real signal f (t) with frequency spectrup f ( o )(see List of Symbols). From the integral definition in (20) it follows that f ( t ) is an analytic function in the lower half of the complex t-plane.
16
EHUD HEYMAN AND TIMOR MELAMED
This function may also be defined directly from the real data f ( t ) via
+ In most cases one is interested in the limit of f ( t ) on the real t-axis [see (35) for an examele where one needs the function in the complex domain]. From (22), the limit o f f on the real t-axis is related to the real signal f ( t ) via
+
. k t ) = f ( t > iE f(r),
t real
(23)
where 7-f is the Hilbert transform 7-ff = (-nt)-' 8 f , with 8 denoting a cpnvolution. The real signal for real t can therefore be recovered via f ( t ) = Re f(r).
B. Green's Function Representation We start with the Green's function formulation in (1). Utilizing the TD Green's function [cf. (2)]
we obtain the retarded field integral (Fig. 2) d2xo @(r,ro, t - R / C )
(25)
where R = Ir - rol and the retarded kernels are [cf. (la-c), respectively]
+
( Z / R ) [ R - ~ U ~c-'ub] - u,O
2(z/R)(R-'uo
+C ' U ~ )
(25a-c)
where uo(x0, t ) = u lZ=o and U,O(XO, t ) = 8,u l z are ~ known functions and the prime denotes a time derivative. As discussed in connectionwith (1) and (6) [see also (9)], the three representationsin (25a-c) are equivalent only if the initial field at z = 0 is strictlyforward propagating. As implied by (25b,c), the field representation in this case requires a knowledge of either uo or its normal derivative u , ~ Equations . (25b,c) cannot be used, however, if u consists of both forward and backward propagating waves. In this case one should use the representation in (25a) wherein the kernel 9 extracts the forward propagating part in the initial field U O , so that the integral (25a) describes only theforwardpmpagatingpart of the field for z > 0. Numerical implementation of the integral (25) can be made efficient by utilizing the fact that the radiated field is localized in space-time, in particular if the excitation is a short pulse. For a given observation point r, one is interested in calculating the integrals only within a short time window. Furthermore, taking into
17
SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS
FIGURE 5. Radiation of collimated pulse. An aperture of width L is driven by a pulsed field with pulselength T .
account the delay R / c from each point ro in the aperture to the observationpoint r, one may limit the integration domain to those points that actually contribute at the specified observation time. For observations in the far zone, on the other hand, in particular under well-collimated conditions (see following example), R does not change much as a function of ro. Hence, all points in the aperturecontribute almost simultaneously. Here, it may be preferable to revert to the spectral formulations (Section 1V.B). To illustrate these considerationswe consider a simple example of a collimated pulsed distribution (Fig. 5 ) . It is assumed that the pulse width T is much shorter than the aperture diameter L , namely, L
>> cT
(26)
and that the pulse delay is uniform across the aperture (Fi 5). For simplicity we only consider points along the z-axis. Using R = 21 z &2z, where po denotes the radial coordinate of the integration point Q, we find that the integration domain in (25)is restricted to
d
t - z / c - T < P , ' / ~ c z< t - Z / C
h
+
(27)
The properties of integral (25) are, therefore, determined by what may be termed the TD collimation (or Fresnel) distance F = L2/cT
(28)
In the collimation zone ( z << F) the contributions in (25) come from rings in the
18
EHUD HEYMAN AND TIMOR MELAMED
z = 0 plane as defined by (27). Beyond the collimation distance (z > F), on the other hand, the contributions in the space-time window 0 c t - z/c < T come from the entire aperture. The foregoing discussion provides a TD interpretation to the conventional FD definitions of the near (or Fresnel) zone versus the far (or Fraunhofer) zone. By replacing in (28) T-' w, expression (28) for F reduces to the so-called Fresnel distance k L 2 . Beyond this region [see the Fraunhofercondition (17)], the quadratic phase error in the phase may be neglected, resulting in a global contribution from the aperture in terms of its Fourier transform.
-
C. Time-Dependent Plane- Wave Representation The Green's function representation (25) describes the field as a superposition of spherical wavefronts, which are emitted from each point in the aperture (Fig. 2). In this section we shall consider the dual representation in terms of a spectrum of time-dependent plane waves (see Fig. 3). The waveforms of these plane waves are related to the time-dependent data in the z = 0 plane via what we call the time-dependent spatial spectrum. The time-dependent spatial spectrum can be calculated directly from the timedependent data without recourse to the frequency-domain plane-wave representation (6): The operative relations will be derived by inverting the operations in (3a,b) and (6) to the time domain. To simplify the mathematics, we employ the analytic signal representation, which basically extends the functions into the lower half of the complex t domain where the integral representations converge well. The final formulas will be expressed, of course, for real t . The analytic signal representation also furnishes compact expressions that unify the propagating and the evanescent spectra in a single framework [see discussion after (31)]. We shall use the notational system mentioned after (3a) ( see also List of Symbols): Spatial spectrum constituents are denoted by whereas FD constituents are denoted by a *. Thus, spatial spectrums of time-harmonic and of time-dependent fields are denoted by ' and -,respectively. An over + above any time-dependent field describes its dual analytic signal. With U O ( X O , r) representing the time-dFpendent field in the z = 0 plane, the analytic time-dependent spatial spectrum Go((, t)is defined by
giving, using (3a),
SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS
19
FIGURE 6. The slant-stack transform (SST) of the time dependent data [a Radon transform in the three-dimensional (XO, 1 ) plane]. For a given E, the SST extract from the data a transient plane wave that propagates in the R = (t,5 ) (see Fig. 3).
Note that an important feature of the spectral formulation in (3a) is that the spatial wavenumber is normalized with respect to w and, thus, has afrequencyindependent geometrical meaning in terms of the plane-wave angle [see (7)]. This permits an inversion in the order of integration (legitimate when Im t 5 0 and in the limit of real r ) and a closed-form evaluation of the w-integration, giving
Repeating the same procedure for the field representation in (6) with (6c), one obtains
&r, t ) = -(21rc)-*
s
d26a:Ho[<, t - c-1(<. x
+ {z)].
(31)
Equation (30) is a Radon transform of uo(xg, t) in the three-dimensional (xg, r ) space, consisting of projections of uo(x0,t) along surfaces of linear delay t c-'< . xo = t = const. (Fig. 6). It will, therefore, be termed the slunt-stack transform (SST). Restricting (31) to z = 0, this equation is readily recognized as an inverse transform of (30). The slant-stacking operation in (30) extracts from UO(XO, t) the time-dependent plane-wave signal that propagates in the direction k = (t,{) in (7). Thus, the inverse representation in (31) synthesizes $e field in terms of an angular (i$) superposition of time-dependent plane waves iio[<, t - cP1(< . x {z)] (see Fig. 3). Their propagation properties follow from the delay term c-'(( - x {z): For [
1, where { =ilfl [see (5)], these plane waves decay as z increases away from the aperture. [Thisbehavior follows from the general property of analytic signals which decay as the imaginary parts of their time-argument become more negative; see (21)]. In accord with the FD terminology, the spectral regimes < 1 and 1 are, therefore, termed the propagating and the evanescent spectra, respectively.
+
1e1>
+
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EHUD HEYMAN AND TIMOR MELAMED
The analytic signal representation in (30)-(3 1) incorporates both the propagating and the evanescent spectra in a single analytic framework. The real field is then obtained from the real part of (31). It might be useful, however, to express (30)-(31) directly in terms of the real data UO(XO,t). Expressing u as the sum of the propagating and evanescent spectra, upmp uev, respectively, one immediately finds from (30)-(31) that
+
where fro((, r ) is the real time-dependent plane-wave spectrum as obtained by using uo in (30), that is,
The evanescent spectrum contribution in (3 1) is given by
+
Since ( = i I( I here, this expressionrequires the calculationof LO(<,t) for complex t: t = t - c-l(< x il(1.z). One route to calculate this evanescent spectrum is through the frequency domain: The data UO(XO, r ) are first transformedto i o ( x 0 , w ) for all XO; then the plane-wave spectrum w ) is calculated via (3a) for all o, and finally it is transformed via (29) into GO(<, t ) with complex t. Altem+atively, this calculation may be performed directly in the time domain: Using (22), GO(<, t) for Im t < 0 can be calculated directly from the real data UO(%, t) via
+
to((,
This function should now be substituted in (34). Up to now we assumed that the data on the z = 0 plane are uo. The more general representation is obtained by applying the analytic inverse Fourier transform (29) to (6) with (6a-c). Following the same procedure as described previously, we obtain
where the time-dependent plane-wave kernel
is calculated from the TD data
SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS
21
via either one of the alternatives:
(36a-c) +
+
where go((, t) and fiZo((, r ) are the slant-stack transforms of the data functions uo(x0, t ) and U,O(XO, r). As discussed in connection with (l), (6), and (9), all the alternatives in (36a-c) are equivalent provided that u is strictly forward propagating. In this case it is preferable to use either (36c) or (36b), which only requires the initial field uo or its normal derivative u , ~ ,respectively. If, however, the initial data uo involve both forward and backward propagating waves, one must resort to (36a), which then extracts the forward propagating part in the data, giving [see +
where fid is the spectrum of the forward propagating part of the field. The interpretation and properties of the field representation in (36) are the same as discussed in (29)-(35). 1. On Specrral Localization The plane-wave integrals in (30)-(31) [or (32)] are spectrally distributed. For short-pulse signals, however, dominant contributions are obtained from localized regions in the source domain. These contributions emphasize radiation along the local physical directions. In accord with the time-harmonic field in (lo), we shall assume here that the source distribution has the short-pulse form (Fig. 7) UO(x0, t) = a[%, t
- c-'+o(%)l
(38)
where a(%, t ) is a short temporal pulse and c-'@o(xo) is a delay function, both with slow spatial (XO) variations. It is then found that the dominant contribution to the plane-wave spectrum fro((, t) in (33) comes from the region of the stationary delay point xs((), defined by [see (1 l)]
6 = VO@O(Xs)
(39)
This condition defines the local radiation direction if a pulsed plane wave is locally matched to the source distribution (see Fig. 7), and it therefore generates the dominant contribution to the spectral plane wave 5'. In view of (30) with (38), the corresponding value of T is determined by the so-called Legendre transform
t(O = c-"@o(xs) - 4 . Xsl,
(40)
22
EHUD HEYMAN AND TIMOR MELAMED
FIGURE 7. Asymptotic evaluation of the transient plane-wave integral: The local radiation direction at a given xo is determine by local matching of a pulsed plane wave to the source distribution. For a given 6 this defines the stationary delay point x,(E) in (39) that generates the dominant spectral contribution. The corresponding value of T is determined by the Legendre transform (40).
whose geometrical interpretation is schematized in Fig. 7. Thus, the resulting time-dependent spectral distribution GO(<, r ) is localized in the (<, t ) domain about the surface t(<)of (40). We shall not proceed further with the asymptotic analysis and, therFfore, shall not present here explicit asymptotic expressions for the spectral field GO. Such expressions can be found in [lo] and [46]. Likewise, the time-dependent plane-wave integral (32) can be localized by the stationary delayed evaluation. It can be shown that the dominant contribution for a given observation point r comes from a preferred spectral direction 6. This direction corresponds to the direction of the ray that emerges from the point xo in the z = 0 plane in the preferred direction (1 1) and passes through r [ 10,461.
D. Ray Representation For observation points near the aperture, the radiated field can be described by a geometrical optics continuation of the field in the aperture. In accord with the time-harmonic case (lo), we express the aperture field in the high-frequency form in (38) and it is also assumed that the pulselength of a&, t) is short on the scale of the spatial variations of a and @, that is,
As discussed in connection with (39), and illustrated in Fig. 7, contributions from neighboring points in the aperture add up constructively to generate radiation in a preferred direction (11) (Fig. 4). The waveforms along these rays are found by
SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS
23
applying the analytic Fourier transform (21) to (12). In view of (15), the result is
where 2 (XO, r ) is the analytic excitation pulse. The Maslov index M counts the number of times the ray had touched the caustic as discussed in connection with (15). The real signal is found using (23), giving
u(r, t ) = J m V ' a ( x o , t - R / C )
(43)
where 7f is a Hilbert transform, and l-IM denotes the operator 'H applied M-times, for example, 7f2 = - 1, and so on (recall that M may be at most 2 in the present uniform medium case). Expression (43) fails for observation points near a caustic where d A + 0. A uniform expression for the time-dependent waveform near caustics that utilizes the ray information there has been given in [9] (Section II) and [51] and [52]. As discussed at the end of Section II.C, if the spectral spread of the rays is large, then the ray model can be extended up to the far zone. For well-collimated fields, on the other hand, the field beyond the Fresnel zone [see (28)] should be described by the following spectral radiation pattern.
E. Time-Dependent Radiation Pattern In the far zone the time-dependent field has the form
with the time-dependent radiation pattern
g(?, t ) = 2 8 coseiib((, t)l+nl,
(45)
where iio is calculated from uo(x0, t ) via (33) and 8 is the observation angle relative to the z-axis. Thus, the time-dependent radiation pattern is determined by the time-dependent plane wave iio((, t) that propagates in the spectral direction a = 3. In view of (29), Eqs. (44)-(45) can readily be identified as a Fourier transform into the time domain of the time-harmonic radiation pattern representation in (18)-( 19). Equations (44)-(45) may be derived alternatively from either the time-dependent Green's function integral (25) or the time-dependent plane-wave integral (32), thereby explaining the properties of these integrals for points in the far zone. For example, substituting (16) into (25) with (25c), neglecting the second-order term in (1 6) and also the R-' term in (25c), one obtains
24
EHUD HEYMAN AND TIMOR MELAMED
Noting that ro .f = xo x / r , one readily observes that the xo integration has the form of the SST (33) of ub, thereby reducing to (45). Clearly, the time delay described by the second-order term in (16) can be neglected only if it is much smaller than the pulselength T of U O , that is, if
r>>F=-
L2
(47) cT A similar argument has already been used heuristically in (28). This condition is the TD counterpart of the Fraunhofer condition (17) [see also the TD collimation (Fresnel) distance in (95)l. Alternatively, Eq. (45) can be derived from the time-dependent plane-wave integral (32) via a ‘‘stationary delay” integration, which is the TD analog of the stationary phase principle. Noting that the spectral delay (6 . x [z) in (32) has an extremum (stationary point) in the spectral direction 6 = x / r , it follows that plane waves whose propagation directions are k 2: P arrive to r essentially at the same time and add up constructively to generate the dominant contribution in (45). Plane waves that propagate in other spectral directions arrive at different times, but because of the large gradient of the spectral delay in these directions, they add up destructively. The preceding discussion is a qualitative interpretation of a srationary delay evaluation of Radon transform-typeintegrals, such as the time-dependent plane-wave integral (32), which has been formalized mathematically in [lo]. Thus, while the Green’s function integral in (25) yields in the far zone a global contribution from the entire aperture, the plane-wave integral (32) reduces to a local contribution from the stationary delay point. Both routes, however, yield the same time-dependent radiation pattern in (44)-(45).
+
IV. ILLUSTRATIVE EXAMPLE The preceding considerations are illustrated here for a specific pulsed field distribution U O ( ~t ), for which all operations can be performed analytically. The motivation for this example is to understand the physical content of the field and its spatial spectrum in the time domain.
A. The Initial Field We consider the initial field distribution
25
SPACE-TIME REPRESENTATIONOF ULTRA WIDEBAM) SIGNALS
+
where TO > 0 and a! = (YR i y , with (YR > 0 are given parameters whose role will be discussed in (5 1)-(53). S is the analytic delta function
is the analytic S function with P being Cauchy’s principal value. The lower expression is the distributional limit of the upper one on the real t-axis. To explain the properties of this distribution we separate the real and imaginary parts and rewrite it in a standard form UO(x0, t )
1 =IT
1T 2 P (t - tp)2
+ (fTp)’
where tp(Q)
2 1 = ?l~l’ar/lal C ,
Tp(X0) = To
+ I x ~ I ~ ~ R / ~ ( Y I ~ c(51) .
Thus, for a given Q, U O ( Q , t) is a Lorentzian (or Rayleigh) pulse. It peaks at t , where the peak value is 2 / r T p and the half-magnitude pulselength is T, (see Fig. 8). The pulse is strongest and shortest at Q = 0 where T, = TO and it decays as Tp grows away from the center [see (51) and recall that (YR > 03. Noting that the pulse peak is proportional to T i ’ , the half-amplitude spatial width of the distribution is found by solving T P ( w )= 2T,(O) = 2T0, giving (Fig. 8b)
Do = 2
d
m
.
(52)
Furthermore, the quadratic delay t , in the aperture defines a spherical wavefront M
ct
FIGURE8. The pulsed field distribution (48) in the configuration domain (Q, f) [in view of the cylindrical symmetry, the distribution is shown in the (xo, ,I ) plane]. (a) A 3D view of the initial distribution with a = 1 - 3i and To = The full line below the distribution tracks the pulse maximum at rp(x) of (51). (b) Schematic contour lines of the field distribution identifying the parameters in Eqs. (50)<53).
26
EHUD HEYMAN AND TIMOR MELAMED
whose radius of curvature Ro at the aperture is vt, = lx01~/2Ro.Comparing this expression to (51) yields (Fig. 8b)
Ro = 142/a/
(53)
Thus, the curvature is essentially controlled by a/:If CYI = 0, the distribution has no curvature, whereas if CYI >( 0, the wavefront is converging or diverging, respectively. The distribution is well collimated [see also (66)]if DO >> CTO,
and
F >> RO
(54)
where F is the domain collimation distance as defined in (47). Substituting in (47) L = ;Do,we find that the distribution in (48) is well collimated if the parameters satisfy CYR
>> CTO,
and
(YR
>>a[
(55)
Note that under these conditions we have F 2 C Y R . Indeed, in Section 1V.D the parameter CYR will be identified as the collimation length of the field (see also Section V . 0 .
B. The Time-Dependent Plane- Wave Spectrum The time-dependent spectral distribution iio is found by substituting (48) into the slant-stack transform (33) and evaluating the integral in closed form. The integral is simplified if instead of the real distribution iio we use the analytic distribution LOin (48) (without the real part) and use (30) instead of (33). [The time-dependent spectrum can also be found by inverting the FD spatial spectrum in (68)]. Thus, integrating (30) yields
where
i(-')(t) =
s_b,
dt';(t) = 1
+ (xi)-' In t ,
-n 5 Im In t 5 0
(57)
is the analytic Heaviside function (one may verify that Rei(-')(t) +. H ( t ) as Imt .T 0). Again, to characterizethe spectral distributionwe shall rewrite (56)in a standard form
SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS
27
FIGURE9. Properties of the pulsed field distribution (48)in the spectral domain (& 5 ) . (a) A 3D view of the spectral distribution iio in (56) of the field in Fig. 8(a). For clarity we show the actual plane-wave integral -@o [see (31)-(34)]. The full line describes the spectral peak 7 ( ( ) as found via (b) Schematic the stationary point and the Legendre transform constraint [see discussion after (60)]. contour lines of the spectraldistribution in the (& 7 ) plane, identifyingthe parametersin Eqs. (58x60).
and find for the spectral delay and pulsewidth, respectively, f p ( 0
1 = -~I<12a,/c9
&(
(59)
The spectral width and curvature are found from (59) in a similar manner to (52) and (53), giving (Fig. 9b)
D =2&iqG,
w o = -ff;
1
(60)
The spectral distribution -8$0 is shown in Fig. 9a. In order to further understand this spectral distribution in Fig. 9a we note the distribution in (48) is of the short-pulse form (38) with c-*cDo(x~) = t p ( x 0 ) of (5 1). Thus, the main contribution to iio is obtained from the stationary delay point (39), which gives here = xscx,/la12and thus xs(<) =
+
<
C. Radiation Pattern The time-dependent field is obtain by substituting (56) into the plane-wave integral (31). This integral cannot be evaluated in closed form unless we make some additional assumptions, for example, well collimated [see (66)] or far zone [see (61)l. In the far zone the time-dependent radiation pattern is determined directly
28
EHUD HEYMAN AND TIMOR MELAMED
+
by 60 as in (45). giving
Taking the real part of (61) we obtain for the real field
where 3.1 is the Hilkert transformi This expression involves a sum of two real waveforms: 'a1Re 6 and aRZRe 6. Thus, the far-field waveform (62) is partially Hilbert transformed relative to the real excitation (48) [recall that (.YR > 0 but a1 may be any number; if IYI = 0, then (62) is a pure Hilbert transform of (48)]. In order to understand the waveforms involved in (62), we spall express them explicitly in a standard form. Noting that the argument of the 6 function in (62) has an imaginary part, we shall express it symbolically as 6(r - i c ) where E = TO ; C - ' ~ R sin 8 > 0. Expression (62) involves, therefore, the following standard waveforms:
i +
+ 1 & ( t ) = Re6(t - i c ) = -
1
l7€ ( t / € ) 2
t 1 'Hl&(t)= Im6(t - i c ) = -
l7E
+1
-t/c
(f/€)*
+1
The amplitude of+the radiation pattern (62) is controlled by the imaginary part of the argument of 6: c = ;To ; c - ' ~ Rsin8 > 0. Noting from (45) that the far-field pattern is related to the spectral pattern via 161 + sine, we may express the beamwidth of the radiation pattern in terms of the spectral width of (60)
+
sin@ = D
(64)
Under the well-collimated conditions (54)-(53, we have a narrow spectral spread, that is,
@zD<<1.
(65)
D. Special Case: Well-CollimatedCondition Under the well-collimated conditions (54)-(55), we can evaluate the timedependent plane-wave integral (31) with (56) in closed form. Noting from (65) that D << 1 here, we may approximate here [ = 2: 1 The resulting integral obtains the form of the inverse transform (31) for z = 0 applied
d
w
SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS
+ to Go of (56) with a! + a!
29
+ i z . This immediately yields [cf. (48)]
-ia! + z 1 1x1~ i L(r,t) = -8 t - - - -z - ia! c 2cz -ia! ?TO].
[
This expression describesa pulsed beam field (PB) that propagates along the z-axis. It is a special case of the general PB field (80) [see also (85) with a1 = a2 = a!]. It will be rederived in Section V as a direct solution of the time-dependent wave equation. We therefore defer further discussion of its properties to Section V.B.6. E. Frequency-Domain Interpretation It is instructive to understand the properties of the preceding solution in the frequency domain. The FD counterpart of the initial time-dependent field distribution (48) is
[Indeed, (48) is immediately obtained by substituting (67) in the analytic inverse transform (21)]. The spatial spectrum distribution obtained via (3a) is, therefore,
The spatial and spectral widths of these distributions are
b ( w ) = 21al/&,
D < w >= 2/&
(69)
These widths change with the frequency as w-Il2 (a! is a frequency-independent parameter). The phasefronts radii of curvature of (67) and (68) are w-independent and are given by the corresponding expressions of the time-dependent distributions in (53) and (60). The time-harmonic radiated field is obtained by substituting (68) into the plane-wave integral (6). Under well-collimated conditions k b >> 1, the integral can be evaluated in closed form by using 5' 2: 1 - 1(12 as in (66), giving
This is the well-known expression for a Gaussian beam. Thus, the pulsed beam field in (66) is the TD counterpart of an ultrawideband spectrum of the Gaussian beams (70). These Gaussian beams, however, have the special property that they are characterized by a frequency-independent parameter a!, that is, they have a frequency-independent collimation (Fresnel) distance ( Y R but a frequencydependent width as described by (69). More on these pulsed beams and Gaussian beams will be said in the next section.
30
EHUD HEYMAN AND TIMOR MELAMED
v. WAVEPACKETS AND PULSED BEAMSIN A UNIFORM MEDIUM This section deals with the characteristics of well-collimated, short-pulse (ultra wideband) wavepackets. Such space-time wavepackets are useful in various applications, including modeling of ultra wideband radar or sonar beams, local interrogation of the propagation environment (medical probing and imaging), transmission of localized energy, secured high-rate communication, and so on. During the 1980's, the research emphasized the possible synthesis of high-energy, nondiffracting, or weakly diffracting wavepackets, that is, wavepackets that remain localized up to very large distances, but more recently the emphasis has shifted to other applications. Several classes of localized space-time wavepacket solutions have been introduced to address these applications, in particular in connection with the problem of reducing the far-field diffraction [ 19-32] (see also a critical review paper in [53]). In the case of the focus wave mode (FWM) and its relatives [ 19-23], the nondiffracting mechanism is caused by an interplay of forward and backward propagating fields [24]. If the backward propagating part (which cannot be launched by any physical antenna) is removed, then the collimated wavepacket starts to defocus approximately at the TD collimation (Fresnel) distance mentioned earlier. (To explorethis phenomenon analytically, one should substitute the known analytic expression for the FWM in the Kirchhoff radiation integrals (25c), or equivalently (lc) or (6c) or (35c). As discussed there, the Kirchhoff kernel which involves both uo(x0, t) and U,O(XO,t ) extracts the forward propagating part from the initial field [54]). The weakly diffracting Bessel beams [27, 281 are free-space mode functions (i.e., plane waves in cylindrical coordinates; see [45]) whose JO cross section is concentratednear the propagation axis. Ideally, such beams require infinitely wide source distributions, in which case they remain collimated up to infinity, but for finite apertures, they eventually diffract. Supercollimation is therefore achieved by using apertures (source distribution) that are much wider than the beamwidth (i.e., the width of Jo), in which case the wavepacket stays collimated for longer distances than the collimation distance associated with the beamwidth. On the other hand, this wavepacket defocuses much earlier than the optimal collimation distance that can be achieved with the entire aperture [see (47) and (95)]. Thus, the Bessel beam does not provide an efficient solution for applications involving longrange propagation, yet it can be used as a superresolving wavepacket for near-field applications (e.g., medical interrogation and imaging) where the aperture width can be much wider than the beamwidth. The bullets [25, 261 are wavepacket solutions that behave classically in the sense that they diverge at infinity along diffraction cones and decay like r - ' . The intriguing property of the bullets is that they vanish outside the radiation
SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS
31
cone (the rge). These solutions have been synthesized by using the fact that TD solutions can be expressed directly in terms of the TD radiation pattern. This is done by first associating the TD radiation pattern with the TD plane-wave spectrum [see (44)-(45)], and then synthesizing the exact solution as a plane-wave superposition [see (32)] [55]. (Note, however, that such representation involves only the propagating spectrum and is valid only after the source has been turned off [56, 121.) Thus, the bullets are formally synthesized by specifying a radiation pattern that vanishes outside the given far-field cone. In the near zone, however, these solutions lose their wavepacket structure and in fact require sources with infinite support. The class of wavepacket solutions to be considered here is termedpulsed beam (PB). Unlike other wavepackets, the PBs behave classically in the sense that they remain collimated up to a certain distance (the TD Fresnel distance), and thereafter they diverge along a constant diffraction angle 0 whose width can be reduced by using shorter pulses or wider apertures. Although the PBs diffract in the far zone, they may be considered to be optimal solutions to the collimation problem in the sense that all frequency components remain collimated up to a given distance (the Fresnel distance) (Section V.D) [33]. These PBs are strictly causal and are generated by a finite aperture. Exact solutions for this class of PB can be obtained in an elegant form by extending the source coordinates into the complex space. These globally exact solutions are, therefore, termed complex source pulsed beams (CSPB) [29-3 11. Here, however, we shall only be interested with the solution in the limited domain that brackets the wavepacket. In this domain one obtains approximate expressions which, however, have a simpler analytical form. Such approximate solutions may be found from the exact solutions (see Section V.C), but here we shall derive them directly from the differential wave equation. Via this approach we shall also derive in Section V.A a generalized class of PB solutions that can be extended to a nonhomogeneous medium. These solutions maintain their general structure even through propagation in such a medium or reflections at curved boundaries [32], and thus can be regarded as eigen wavepacket solutions of the time-dependent wave equation. These solutions may, therefore, be used to model local diffraction phenomena and for local interrogation [35-371. Alternatively, it is also possible to derive these PB solutions by synthesizing an appropriate aperture distribution and evaluating the radiation integral [in fact, expression (66), which has been derived as an example for the time-dependent plane-wave synthesis of pulsed aperture distributions, is a special case of the PBs considered later]. An important application of these PBs is their use as basis functions for spectral expansions of general time-dependent fields [38,10,41,42]. This last subject will be considered in Section VI.
32
EHUD HEYMAN AND TIMOR MELAMED
A
x1,2
I1
FIGURE10. Pulsed beam in a uniform medium. The drawing depicts a cross section in the principal plane ( x j , z). T. D j . R j and (YR, denote the pulselength, beamwidth, wavefront curvature, and collimation distance, respectively. Note that D j >> cT [see (96)l. The lines y = const. are the propagation lines [see discussion in connection with (83)l. The thick line in the z = 0 plane represents the rigorous source distribution for the globally exact complex source pulsed beam [30].
A. General Solution We consider pulsed beam (PB) solutions u(r, t ) of the time-dependent wave equation
(a:, + a2: +a:
- c-*a:)u(r, t ) = o
(71)
in a medium with uniform wave velocity c. It is assumed that the PBs propagate along the z-axis in the coordinate frame r = (x,z), x = ( X I , x 2 ) (Fig. 10). From reasons which will be clarified soon [see (76)] we utilize the analytic signal representation. Thus, if A(r, t) is an analytic wave solution, then both UR(r,
t ) = ReA(r, r )
and
ul(r, t ) = ImA(r, t ) = 71uR
(72)
are real wave solutions. Here we shall consider only U R since it also defines U I via (72). Furtheyore, U I or any linear combination of U I and U R may be obtained by multiplying u by an appropriate complex constant and taking the real part. Since the PB is localized in space-time we shall express in a moving coordinate frame
A
+
A(r, t ) = U(r, t),
t =t
- z/c.
(73)
The only approximation in the analysis will be to assume that the shape of the wavepacket changes slowly along the propagation path, that is,
la$l << Ic-'a$l. Thisimplies(a,2-c-*a:)u
= (a,2-2c-'aZa,)U
(74)
2: (-2c-'aza,)UandthusEq.(71)
SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS
33
reduces to what we shall term the wavepacket equation We seek a solution to (75) in the form
+
where A(z) is a slowly varying amplitude function and f is an arbitrary time pulse with a typical length T. I’is a 2 x 2 complex symmetrical matrix and x . r(z) x denotes a quadratic form x:rll xiI‘22 2xlx2r12. To guaranty confinement of U near the z-axis, Im I’must be positive definite. This and other properties of (76) will be discussed in Section V.C. Substituting (76) into (73, one obtains
+
x . (cr2+ I”)xAf”
+
- (A
trace(r}
+ 2c-lA’)f’
=0
(77)
where a prime denotes atderivative with respect to the argument. Thus, (76) is a solution o f (75) for any f if
cr2+ I” = 0
and
A trace(I’)
+ 2c-’A’ = 0.
(78a,b)
The procedure for solving (78a) is well known. Setting I? = Q-’ ,Eq. (78a) yields Q’ = cI and thus Q(z) = Q(0) zcI where I is the unit matrix and finally
+
ryZ)= [r-I(o)
+z ~ ~ ] - l .
(79)
Thus, if the initial conditions matrix I’(0) is positive imaginary definite, then I’ is positive imaginary definite for all z as required in (76). To solve for A we use the general matrix relation 1n’detQ = trace(Q’Q-’}, where In’ f = f ’/f. This relation applies to any Q and in particular here with Q’ = cI. Using it to replace trace(I’) in (78b) by c-’ ln’det Q, we obtain A(z) = A~[detQ(z)]-’/~ where A0 is some constant. Normalizing A(z) = 1 at z = 0, the PB solution becomes 1
i(r, t ) = d d e t I’(z)/det r(0) t - Z / C - ?x.
r(z). x
with r(z)given in (79). B. Properties and Interpretation
The solution in (80) has the characteristics of a pulsed beam (YB).Axial confinement along the beam axis is due to the pulsed behavior of f while transverse confinement is due to the fact that I’is complex and due to the property of analytic signals, which generally decay as the imaginary part of their argument befomes more negative. Since Im I’is positive definite, x . I’.x in the argument of f has a
34
EHUD HEYMAN AND TIMOR MELAMED
positive imaginary part which is quadraticallyproportional to the distance from the axis. The waveform of (80) is therefore strongest on the beam axis and weakens away from the+axis. The beamwidth is therefore determined by Im I? and by the decay rate of f in the lower half of !he complex t-plane. The latter depends, typically, on the frequency content in f ;the higher the frequency content, the faster the decay and the narrower the beam [see an example in (93); other examples can be found in [30], [381, and 13211.
1. The Wavepacket Curvature and the Transverse Amplitude Distribution To quantify tke physical characteristics of the PB in (80) we shall separate the argument o f f into real and imaginary parts and express it as +
h r , t ) = A ( z ) f [ t- ?(x, z) - i y ( x , z)I,
(81) where 5 is the paraxial delay [see (82)], y controls the transverse amplitude [see (83)], and from (80) A is the slowly varying longitudinal amplitude. We shall also separate I' into rR i r,. Each of these matrices is real and symmetrical and, therefore, may be diagonalized by rotating the transverse coordinates x. In general, however, they cannot be diagonalized simultaneously; that is, each matrix has its own principal axes in the transverse plane. The principal axes of rR and of I', change with z and may readily be found via (79). The delay in (80) is given by
+
where the second expression is obtained by rotating the transverse coordinates ( X R ~ X, R J that diagonalizes r R and R I , J ( z )are the reciprocal of theeigenvalues of rR. From (82) R1.2 are identifiedas the wavepacket radii of curvature in the principal directionsXR,,; hence, C r R is the curvature matrix and cI' is termed the complex curvature matrix. Similarly, the imaginary part of the argument of f in (81) is given by x into the principal system
+
where the second expression is obtained by rotating the transverse coordinates into the principal system ( X I , , X I ? ) that diagonalizes I?, and 1:,2(z) > 0 are the reciprocg of the eigenvalues of I', (recall that I', is positive definite). Recalling that f in (81) decays as y increases, it follows that the wavepacket contour lines in the transverse plane are described by y = const. Thus, the wavepacket is elliptical in the transverse plane and its principal semi-axes along ( X I , , x12) y e proportional to I1,2. The wavepacket shape also depends on the decay of f (t - i y ) as a function of y [see a specific example in (93)]. As a function of z, the lines y = const. have the same field magnitude and are, therefore, termed
SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS
35
propagation lines (see Fig. 10). The wavepacket waist in the x ~ ,direction ,~ is, therefore, located at z that minimizes 11.2(z). 2. Special Case: An Isoaxial Astigmatic Pulsed Beam
In the general case, the two real symmetrical matrices r R and I?! cannot be diagonalized simultaneously;hence, Eq.(80) is an astigmatic PB whose amplitude and curvature axes are nonaligned and their orientation also changes with z [3 11. We shall consider here the simpler case of an isoaxiul PB where the principal axes of r R and of I?/ coincide and, thus, their orientation remains constant with z. The resulting PB is still astigmatic but it has a much simpler structure. In this case, the matrix I' has the diagonal form
+
r ( z ) = diag(c-'(z - iaj)-'],
aj = L Y R ~ ia~,( Y R ~> 0
(84)
where the complex constants a j , j = 1,2, are found from the initial value I'(0). Note that (84) complies with (79) and the condition a R j > 0 guarantees positive definiteness of rl. Equation (80) becomes
[Note thyt (66)js a rotationally symmetric special case of ( 8 5 ) with a1 = 4 2 = a, and f(r) = s(t - ;TO). Recall that (66) has been derived from the timedependent plane-wave integral, in the narrow angular-spectrum limit, whereas the present solution is a direct solution of the differential equation (75)]. To parameterize the properties of the PB we note that in this case Rj and Ij of (82)-(83) are found from 1
z-ia,
-
i +R j 1; 1
giving
As discussed in (82), R, are the wavefront radii of curvature, while from (83) 1, controlsthe amplitude decay along XI,. In aplane z = const., the amplitudecontour lines are ellipses with principal axes xi. In the plane (xi, z ) the amplitude contour lines are described by the condition x ~ / I ~ ( = z )const. The waist occurs at z where I, is minimal, that is, at z = - a l j . Near the waist, for Iz ' Y I ~I << C Z R ~1j , 2: f i Rj and the PB stays collimated, whereas for Iz aIjI >> ( Y R ~ Ij , 2: (z a ~ ~ ) / f i ~ ~ ;
+
+
+
36
EHUD HEYMAN AND TIMOR MELAMED
hence, the beam contour line satisfies y = x;/Zf(z) 2 a ~x;/z2 , = const. and the PB opens up along a constant diffraction angle 0,. The foregoing discussion identifies a~~as the collimation length F, of the PB in the (xi,z ) plane. In fact, a key feature in this PB solution is that all its frequency components have the same collimation distance (see discussion in Sections V.E.1 and V.E.2). As has been discussed+after(83), the beamwidth and diffraction angle also depend on the decay rate of f ( r - iy) as y increases. An example is given in (93) later.
3. The Real Field The real PB field is given by taking the real part of (80) [see (71)]. To clarify the structure of the real field we shall express it in terms of the real waveform f y (t) defined via (see (23))
Clearly, f,,( t ) decays as the parameter y increases: On the beam axis y = 0 and fy = f is strongest, but as the distance from the axis increases, y increases and the waveform f,,weakens. Substituting (87) in (81) and taking the real part, using also A(z) = AR i A I , the real PB field has the form
+
where y ( x , z ) is given in (83). Thus, the real PB field is a sum of two real waveforms: f,,and its Hilbert transform 'Flf,. Their relative amplitudes depend on z via A R ( z )and A / ( z ) where A(z) is given in (80). At z = 0, A R = 1 and A/ = 0 and the waveform is f,,,but for z > 0, the waveform is gradually Hilbert transformed as the proportion between AR and A / changes. Note that for z + 00, A(z) z-' as follows from (80). As an example, consider the isoaxial case where A is now given in (85). Here we may obtain explicit expressions for the amplitude functions A R ( z )and A / (z). Considering, for example, the special case of a rotationally symmetric PB with a1 = a2 = a. Here A(z) = - i a / ( z - ia)so that AR = [al(a~ + z ) +cr:]/[(a~ z ) ~ a:] and A / = - ~ R z / [ ( u I z)* a:]. Substituting into (88) we find that the waveforms of U R change from f,,in the z = 0 plane to z-'(a/ ( Y R X ) ~as, , z++ 00.JNote that the astigmatic PB in (66) is a special case of this example with f ( t ) + S(r - TO)]. Another simple example is Q R , # a~~but a/,= a/,= 0 (i.e., the waists in both principal directions are at z = 0). Here the waveforms change from f,,in the z = o plane to z-',/-'Flf, as z + 00.
-
+
+ +
+
+
37
SPACE-TIME REPRESENTATIONOF ULTRA WIDEBAND SIGNALS
4. The Aperture Field
The PB distribution in the aperture plane is found from (80)
(89) For simplicitylet us assume that Im I'(0) = 0, so that the initial wavefront is planar. ~ , that In this case the PB is isoaxial-astigmatic with real a!j = c ~ R so
Thus, in order to synthesizethe PB, the pulsed sources at the center of the aperture should be strong and short, but as the distance from the center increases, the sources should be weaker with a longer pulse (i.e., with lower-frequency content). Illustrative examples for specific pulse distributions have been given in [30],[38],and [32].Specifically,for the analytic 6 pulse in (93) we obtain the initial distribution in (48) with a! = Q R . From the general properties of analytic signals, the following qualitative characteristics are found: If the real signal f ( t ) is characterized by a pulselength TO, then the pulselength T, and the amplitude A, of the distribution in (90) are
+ h12/2ca!R1-'* (91) Thus, the energy of the off-center pulsed sources also decay like (TO+ 2 y ) - ' . Tp(m)rv TO -k l%12/2ca!R,
Ap(%)
[TO
5 . The Axial Energy It is also instructive to see how the wavepacket energy changes along the propagation axis. Here we shall only consider the axial energy density defined as Ilu(r, f)1121x=o. The off-axis energy distribution as well as the total energy have been calculated in [37, Appendix B ] . In order to evaluate the energy integral it is convenient to use, again, the analytic signal formulation. According to this formu!ation, the+energy llg1I2 = g2(t)dt of areal signal g ( t ) is one halfof the energy llg1I2 = Ig(t)12dr of its dual analytic signal i ( t ) . Applying (85) for x = 0, and assuming for simplicity that the parameters a!j = a ! are ~ real, ~ we obtain
s
s
where llf1I2 is the energy of the real signal f ( r ) . Thus, the energy density remains essentially constant up to the diffraction points a!Rj where it starts to decay like 2-2.
38
EHUD HEYMAN AND TIMOR MELAMED
6. Analytic Delta Pulsed Beam Finally, we consider the characteristics of the PB (80) for a specific excitation pulse. A simple example is provided by the analytic delta pulse [see ( 4 9 ) )
The analytic PB field is given now by using (93)in (80) [or in (85)for the isoaxialastigmatic case]. The real field solutionsare given by (88) where the real signals fy (t) and 'Hf, (t) are given now by S,(r) and 'HSc(t)in (63), with 6 = ;TO y and y is given in (83). From the discussion in (63), the half-amplitude pulsewidth and the peak value of the wavepacket are TO 2 y and n-l (; TO y ) - ' , respectively. Thus, the waveform is shortest and strongest on the beam axis ( y = 0) and it decays as y increases away from the axis. The peak reduces to half of the axial value when To y = TO.Solving for y and substituting in (83) using (86a), we find for the half-amplitude beamwidth in the principal direction xi,
+
+
+
+
D j ( z ) = DO,
d
m
,
DO,
=
2dZZ,
(94)
where Doj is the width at the waist z = -arj. From (94),the far-field diffraction angle is 0, = D0,/aRj = 2 4 c T o / u ~ ~
(944
We therefore observe that the physical characteristics of the wavepacket, namely the waist beamwidth Doj,the collimation length Fj = a~~and the axial pulselength TOare related by what we may call the TD diyraction (or Rayleigh) limit: Fj = Dij/4cTo
(95)
Recall that it is assumed in this section that the PBs are well collimated, implyR70 ~ << 1 [see (55)].Typical parameters in electromagnetics can be ing c T o / ~= 70 2: + One obtains the following relation between pulselength, the beamwidth, and the collimation distance (for simplicity we remove the index j ) with Po
<< 1.
Note that the collimation distance F of the pulsed beams is a frequency-independent parameter [compare (10411. Representative plots for a stigmatic (cylindrical symmetric) PB are given in Fig. 11. These figures have been computed, in fact, for the globally exact cornplex source pulsed beam field (see discussion in Section V.C) but since the field practically vanishes outside the wavepacket region, they also apply for the paraxial
+
FIGURE 11. A pulsed beam (85). Beam parameters: a1 = a2 with a1 = 0 (i.e., waist at z = 0)and collimation distance U R . f is given by (93) with normalized pukelength C T o / ( Y R = t = 5 . w4.Observation times: (a) c t / ( Y R = 1/3; (b) C t / a R = 3. The plots
W
v)
depict axial cross-sectional cuts through the rotationally symmetric 3D-space at specified observatio? times.The wavepacket is localized &Re u (r, t ) . The insets depict the pulse within the space-time window z 2 ct, 1x1 2: 0. In particular, we show the normalized field shape on the beam axis [note that the far-field signal in (b) is a Hilbert transform of the near-field signal (a)].
40
EHUD HEYMAN AND TIMOR MELAMED
PB solutions considered here. The field is computed via (85). The parameters in these figures are (YI =42 with aI= 0 (i.e., waist at z = 0) and collimation distances (YR. The normalized pulselength is To = 0.0005; due to graphical limitations we did not plot a PB with a smaller (more realistic) value of TO. C. Relation to Complex Source Pulsed Beams (CSPB)
Complex source pulsed beams are globally exact wavepacket solutions of the time-dependent wave equation that can be modeled mathematically in terms of radiation from a pulsed source located at a complex coordinate point. It has been shown [30,3 11that physically these solutions are generated by a real pulsed source distribution of finite support (see Fig. 10). Thus, the complex source model may be regarded as a mathematical trick to generate simple field expressions due to these physical distributions. This trick has been introduced originally in order to generate time-harmonic Gaussian beam solutions [57,58], and to explore Gaussian beam interactions with boundaries [59,60]. Detailed discussion of the properties of the CSPB may be found in [30] and [31]. Here we shall only comment that near the axis, the exact CSPB solution has the form of the paraxial PB in ( 8 5 ) [in fact, the CSPB is rotationally symmetric so that it has the paraxial form in (66)]. Although for practical applications we are interested mainly with the PB field in the paraxial beam region, the globally exact complex source model is an important analytical tool for various applications. By analytic continuation of the time-dependent Green's function to complex source coordinates, one may generate analytically the response of the environment to an incident wavepacket. Following a systematic procedure, one may derive exact closed-form expressions for the response in certain canonical configurations. From these expressions one may then extract simplified models for the interaction of the wavepacket with the environment. Because of the local nature of the interaction, these models can be extended to noncanonical configuration. Basic canonical PB scattering problems have already been explored along these lines [34-371. D. Relation to 7ime-Harmonic Gaussian Beam
One may easily verify that the frequency spectrum of the solution in (80) is given by
1.
where f ( w ) is the frequency spectrum of With r ( z ) given by (79), Eq. (97) is the most general expression for a time-harmonic Gaussian beam in a uniform
SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS
41
medium (for Gaussian beams in nonuniform medium, see [61]). The matrices l?R and I?, then describe, respectively, the phasefront curvature and the Gaussian envelope of that beam. In the special case when I? can be diagonalized as in (84), ii is an isoaxialastigmatic Gaussian beam
(98) From (98), the beamwidth in the principal plane ( x j , z ) (defined as twice the value of x j where lii I goes down to e-lI2 of its on-axis value) is given by
The waist is located at z = -al,and the collimationdistanceis a R j . The diffraction angle at infinity is given by 8j = 2/&. The expressions in (97) and (98) are the frequency spectra of the PB fields in (80) and (83, respectively, only if I? and a, are frequency independent. This implies that all frequency components in the PB are Gaussian beams with the same waist planes and collimation distances. At a given z , their phasefront curvatures are frequency independent but their widths are proportional to @-‘I2 as described in (99). Thus, the PB can be synthesized by an ultra wideband source distribution (say, in the z = 0 plane) provided that the width of this distribution is proportional to @-1/2.
E. Considerations in UWB Synthesis of Collimated Apertures: Isodiffracting versus Isowidth Apertures From a FD perspective, synthesis of an ultra wideband aperture adds another dimension to the conventional FD considerations, namely, the frequency variation of the aperture parameters (e.g., the width). One obvious choice, termed here isowidth, is to use the same aperture distribution (and size) at all frequencies. We shall contrast it with the aperture realization of the PB fields considered earlier. As discussed in connection with (99). in this case all the frequency components have the same collimation distance F , so that the size of the aperture must be scaled according to o - 1 / 2 This . wideband aperture will be termed isodi$racting. Here we shall contrast the properties of these realizations by considering the simple examples of rotationally symmetricGaussian distributions. The discussion can nevertheless be extended and phrased in terms of general aperture distributions [33].
42
EHUD HEYMAN AND TIMOR MELAMED
1. Isowidth Distribution A rotationally symmetric time-harmonic Gaussian distribution of width 2 DOhas the form
h(r, w)I,=o
= iio(x0, w ) = f*( w ) e-1m1*/20,2,
(100)
We assume here that the “width” DOisfrequency independent. Thus, the timedependent distribution associated with (100) is uo(x0, t) = f(t)e-Im12/2Di,
(101)
with f ( r ) being the pulse associated with f ( w ) . Assuming that DOis large on a wavelength scale (i.e.. k DO>> 1), the FD expression for the field near the beam axis is given by -i F(w) z - i F ( w ) exp[ik
ci(r, w ) = f ( w )
i1x12
(z
+ z - iF(w)
where F(o)is thefrequency-dependent collimation distance
F(w) = kDi
For a given frequency, (102) is identical to (98) with F replaced by 2 (99) are o-independent whereas here F depends on w ] . however, that ~ 1 . in Expression (102) is a Gaussian beam whose cross-sectional width is Thus, for z < F(w) the beam stays collimated but for z >> F(w) it diverges along the diffraction angle 8 ( w ) = (k DO)-’ and decays like z-’ (Fig. 12a).
(4
(b)
FIGURE 12. Radiation from an ultra wideband isowidth aperture with width DO.(a) Beamwidth in the ( p , z ) plane. (b) Field amplitude along the z-axis. The plots are shown at three frequencies: W I , w = 2wl, and y = 301. (7, is the center frequency. The parameters are: Do = 1 and F ( w l ) = 3 0 0 , so that F(c3) = 4Do and F ( y ) = DO.
SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS
43
On the beam axis, the amplitude in (102) behaves like
where the approximation applies for z >> F(w). As one may observes from (103, the field in the far zone is a time derivative of the near field. This property of the near-to-far field transition has already been noted in (19) and (43, and it applies also off the axis. The analysis in (109, however, demonstrates that the far-field ( - i k ) term is a direct consequence of the frequency dependenceof the collimation distance F ( w ) : As schematized in Fig. 12b, the lower frequencies have a shorter F(w) and thus the field starts its z-' decay earlier. An important outcome of this property is that in order to synthesize an impulselike waveform in the far zone with a flat wideband spectrum, one should compensate at the lower frequencies using [62] f(w)
-
w-'
(106)
This, however, is not an efficientrealization since the lower frequencies are lost by wide-angle diffraction (see Fig. 12a). One may also calculate the TD expressions for the field or the radiation pattern (see [33]), but we shall not pursue this example further. 2. Isodiffracting Distribution As has been explained previously in terms of fundamental wave mechanism, the low-frequency loss in the near-to-far transition (105) is due to the shorter collimation distance at the lower frequencies, so that the low-frequency energy is lost by wide-angle diffraction. It therefore follows that a more efficient transition could be achieved by synthesizing the aperture so that all frequency components will have the same collimation (or diffraction) distance (Fig. 13). As an example, consider the time-harmonic Gaussian distribution in the aperture
which is a special case of (98) with a1,2 = a,applied at z = 0. For simplicity we take a = a~ to be real. As discussed in (98), a~ is the (frequency-independent)collimation (or diffraction) distance. The width of the distribution is given by Do of (99), and thus changes with frequency like w - ' / ~(see Fig. 13). The time-harmonic field due to the distribution in (107) is given by (98) with a1.2 = a ~ Thus, . on the beam axis we find that
44
EHUD HEYMAN AND TIMOR MELAMED
.t P .4
:.z. - .... _____.._....,.... --____ :O -----..-.-.--.--...-...-.-__ - .. ..-... ..
p l ,
.
-2
, ,,,
-.
--4
5
--
- .-
___ _-_. -
'.' .''
L%!-----------;
, . .-.
4
~
~
.-
IF i'p
--
.
________ ________ --...-..__. ~
..
,
__ -
.
15
~
_
2.
FIGURE 13. Radiation from an ultrawideband isodiffracting aperture with a Fresnel distance F . The figure depicts the beamwidth in the ( p . z) plane at three frequencies: 01. (7, = hl, and 0~ = 301. G, is the center frequency of the pulse. The parameters are: F = 9 and DO(@)= F / 9 so that Do(&) = ~ D o ( w zand ) DO(OI) = -~Do(Y).
Thus, here the near-to-far transition is essentiallyfrequency independent whereas in (105) it exhibits a low-frequency loss. Following the interpretation given to (l05), it follows that the flat response in the present case is associated with the fact that here all the frequency components stay collimated up to the same collimation distance OIR (see Fig. 13). The time-dependent field corresponding to (107) is given by ( 8 5 ) with 41,2 = O I R . Thus, on the z = 0 plane the isodiffracting time-dependent distribution is given in (90) with ( ~ 1 . 2= O I R . The parameters of this distributions are discussed in connection with (91) [see also (501. Further discussions and illustrative examples are given in [33].
VI.
PHASE-SPACE PULSED BEAMANALYSIS FOR "ME-DEPENDENT RADIATION FROM EXTENDED APERTURES
As an example for the application of the PBs, we consider here spectral field representations that utilize PBs as basis functions. These representations are the local-spectrum alternatives to the conventional FD and TD representations considered in Sections II and III, which involve Green's functions and plane waves. The conventional representations typically lead to distributed integrals over the entire configuration or spectral domains, but as demonstrated in the previous sections, the actual field may be localized in one of these domains or in both. Mathematically, localization of the distributed integrals is affected by constructive interference (stationary point evaluation). Therefore, it is suggested to use basis functions, which tend to localize the radiation process a priori. Beams (or pulsed beams; we use here the generic term
SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS
45
FIGURE14. Local matching of a beam to an aperture distribution.
beams to describe beam fields in both the frequency and time domains) are good candidates since they may be matched to the source distribution so that each beam accounts collectively for the radiation from a given region in the source domain as schematized in Fig. 14. This leads to a compact spectral representation of the field. In order to gain further insight we shall reconsider the Green’s function and the plane-wave representations. In the former, each point in the aperture radiates (a Green’s function) in all directions (Fig. 2) but eventually, radiation from neighboring points interferes constructively along a preferred direction of radiation (Fig. 4), thereby describing the local ray field emitted from the aperture. Similarly, in the plane-wave representation, each plane wave is generated by the entire aperture (Fig. 3) but practically it is generated by a region in the aperture where the gradient of the source phase (or delay) matches the plane-wave direction [see (1 1) and (39)]. In the phase-space beam representation presented in Fig. 15 the field is described by superposition of beams that emanate from all points in the aperture and in all directions. The excitation amplitudes of these beams are the local spectra of the field distribution. Thus, only those beams that match the local radiation direction are actually excited and should be accounted for in the integral representation. The foregoing discussion implies that the compactizationin the beam representation is due to the fact that each beam has a finite width in both the configuration and the spectral (i.e., directional) domains and, thus, it samples the aperture information in both domains. This intuitive insight will be formalized later within a phase-space format that combines the configuration and the spectral domains [39, 10, 40, 411. The trade-off in this representation relative to the conventional representations is an a priori data localization versus more complicated processing integrals and databases. It becomes efficient, therefore, when dealing with very large data objects, like those obtained by gathering ultrawideband information over very large apertures (TD synthetic aperture radar (TD-SAR) processing and TD inverse scattering [43,44]).
46
EHUD HEYMAN AND TIMOR MELAMED
Diffraction beams
Z
FIGURE15. Phase-space beam (or pulsed beam) summation for radiation from extended source distributions. The arrows represent the beam propagators; large arrows represent strongly excited beams by local features of the source distribution.
A. Frequency-Domain Formulation: Beam Summation 1. Local Spectrum of the Data
Referring to the configuration in Fig. 1 and the problem formulation in the beginning of Section 11, the windowed transform 00of the initial field i2o is defined by
where a(%)represents a spatial window function and the asterisk denotes a complex conjugate. Assuming S(xg) to be localized around xo = (0, 0), one may interpret 00as the local spectral distribution of 60 in the vicinity of %. Referring to (3a) we shall, therefore, refer to (%,t) as phase-space coordinates, and adopt the notation X = (%, Equation (109) may now be written in the abbreviated form
0.
SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS
47
with the kernel W(x0; X ) = 5(q - t)e'kt.xo.Alternatively, the local spectrum 00can be evaluated from the plane-wave spectrum
&(X) = (k/2n)2
J
d2(&(<)W*((; X )
(111)
where t o is the plane-wave spectrum (3a) of the data, while $(<;X ) = I%([ - t ) e-ik(c-l).Twith and I% being the spatial spectra (3a) of W and 5,respectively. Thus, assuming also that I%(<) is localized around = (0, 0), 00may also be considered as the local field corresponding to a sample of $0 around 1. The degree of spatial and spectral localization achieved can be quantified in terms of the spatial and spectral R M S widths of the window defined, respectively, by
<
( 112a,b)
where AxAt 2 1/ k according to the uncertainty principle, and
is the L2 norm of 5. By a straightforward extension of the 1D short-timeFourier transform inversion formula in [63], the inverse phase-space transform may be written formally as [39, 411 iio(x0) = (
J
k / 2 ~ ) ~ N d 4 X 6 0 ( X ) W ( ~X). ;
(1 14)
Note that this representation is not unique as there are many four-variable kernels that can replace 00in (1 14) and still represent iio. Of particular interest is the Gabor kernel [64,65,39]. However, the function 00of (109) yields the minimum energy representation [63].
2. The Radiating Field The phase-space superposition (1 14) of the initial field can be propagated into the region z > 0, giving
*-21
ii(r) = (k/2n)2N
d 4 X & ( X ) b ( r ; X)
(1 15)
48
EHUD HEYMAN AND TIMOR MELAMED
where the phase-space propagator b is the field radiated by each window element W(m;X). b can be expressed in several alternative ways (e.g., by Green’s functions or plane-wave representations). In the present context it is convenient to express fi by the plane-wave representation (6c); that is,
I
B(r; X) = ( k / 2 1 7 ) ~ d2.!jW(I(s; X)eik(e’x+cz)
(1 16)
where 6’ is defined in (1 11). If 12is wide on a wavelength scale, then the spatial and spectral distributions of @ are localized around ~0 = % and = 1,respectively. Consequently, b behaves like a collimated beam whose axis emerges from the t = 0 plane at ~0 = 2 at the direction given in (7)
<
k
= (<, f ) ,
f =4 1-1 11’
(1 17)
Propagating beams occur only for 1 1 1c 1 [or more precisely, for 1 1 1c (1 - At) where At is defined in (1 12b)l. For 151 > 1, b decays exponentially with z . The representation in (1 15)describes the radiated field as a continuous superposition of shifted and tilted beams, centered at % and directed along 1 (see Fig. 15). The phase-space function 00defines the excitation strengths of these beams via local matching to the aperture field ko(x0). An important feature of this representation is the a priori localization around a well-defined region in the X-domain. This localization is affected by both 00and B in (1 15). Since each beam has finite spatial and spectral widths in the aperture plane, it senses via (109)the local radiation properties of ko(m) at %. Accordingly, 00favors a priori the beams that emanate from % along the localpreferred direction (see Fig. 15). If, for example, lio(m) has the high-frequency form in (lo), O0 and thereby the integration domain in (1 15) are limited to the vicinity of the radiation direction constraint
< = V@O(%)
(118)
that defines the local direction of radiation [see (1 l)]. In many situations, however, the form of the constraint may be more complicated. For example, if l i 0 represents the scattered field data due to several localized scatterers, then it consists of several additive terms like (10) and the integration domain in (1 15) involves several constraints like (1 18), with possible overlap. The effective domain of integration in (1 15) is limited further because only those beams that pass near r actually contribute. For a given r this localizes the contributions in (1 15) to the vicinity of a hyperplane in the X-domain, defined by (x - % ) / R = <,
R
= JW.
(1 19)
This observation constraint defines the phase-space beams that pass through r.
SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS
49
Thus, unlike the plane-wave superposition, the phase-spacebeam representation is localized a priori in the X-domain about the skeleton as schematized in Fig. 15. It emphasizes only those beams that emanate along the local radiation direction ( 1 18) and also pass near the observation point as defined in (1 19). 3. Example: A Gaussian Window Gaussian windows have several important properties: (a) they maximize the phasespace localization implied by the uncertainty principle; (b) they generate tractable beam propagators, which are related to the conventional Gaussian beams; (c) they are well suited to performing analytic approximations; and (d) they furnish convenient basis functions for time-dependent representations (see Section VI.B.3). A 2D Gaussian window has the general form $(%) = e-l/25.p-'.5
(120)
where p-' is a symmetrical complex matrix so that % p-' 3 is a quadratic form. For convergence at large x, the matrix Rep-' should be positive definite for w > 0. Henceforth, we shall only consider rotationally symmetric windows where = PI with I being the identity matrix and = /?R with PR > 0 for w > 0. The spatial and spectral window functions now have the form [see (67), (68)l
+ is,
$(%) = e-1/%B-'bdZ
g(e) = 2npk-le-l/2k/JlElZ
(121)
with the norm and width [see (1 12)-( 1 13)]
fi2= (r/k)lSl2/B~,
Ax = ISI/m= AtlSI.
(122)
NotetheuncertaintyprincipleA,Ac = IpI/p~k1 Ilk, withanequalityforp = p ~ Note also that we have kept the frequency parameter k explicit in the exponent of (121). This, of course, has no effect for monochromatic fields, but for wideband solution, it implies that the resulting beam propagators havefrequencyindependent collimation distance and frequency-dependent widths [see ( 127)]. The properties of the resulting TD window and propagators will be considered in Section VI.B.3. The phase-space propagators h(r; X) are calculated by substituting (121) in (1 16). For large k p ~the integral can be evaluated asymptotically as detailed in the appendix. The result is listed next. We utilize the beam axis coordinates ( X b l , X b z , Zb) defined, for a given phase-space point X,by the transformation
[z][ =
cos 6 cos -sin$ sin 6 cos
4 4
cos 6 sin $ cos sin 6 sin $
4
-st [
61 x1
cos 6
;ill
x2 - i
2
( 123)
.
50
EHUD HEYMAN AND TlMOR MELAMED
FIGURE 16. The phase-spacepropagators in the configuration space. The beam field (1 25). (147) is propagating along the beam axis Z b (123). The transverse coordinates Xb = (Xbl, X b 2 ) are rotated such that X b l lies in the ((, k ) plane while Xb2 is orthogonal to Xbl and lies in the q plane.
where (8,4) are the spherical angles that define the beam direction k = (<, <) (Fig. 16). That is, C O S = ~
c,
cos6 = &/IEI,
sin6 = h/I
( 124)
Thus, the zb-axis coincides with the beam axis while the transverse coordinates xb = (xbl, x h l ) are rotated such that xbl lies in the plane (<, 8) (see Fig. 16). Accordingly, the linear phase E . x implied by the window function in the z = 0 plane [see (log)] is operative in the Xbl direction but not in the Xb2 direction. Utilizing the beam coordinates we obtain by saddle point integration (see appendix)
where
Here Xb .I?. x b is a quadratic form, which becomes for the present case of diagonal
r:
kc-l
- ipf2)-I
+ ~ i ~ ( 2 f --I i p p .
The properties of B either as a configuration-space object, where X is kept constant and B is regarded as a function of r, or as a phase-space object, where r is kept constant and X is varied, are discussed next. Regarded as a function of r, B has essentially the form of a Gaussian beam (GB) (97), propagating along the beam axis Zb. In the conventional GB, however, the elements of I? depend only on the location along the beam axis [i.e., on zh; see (79) or (84) wherein z is the beam axis], whereas in (126) they depend on
SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS
51
= Zb - Xbl tan 6. This difference is due to the fact that in the conventional GB, the Gaussian initial conditions are given on a plane normal to the beam axis, whereas here they are defined on a plane of constant z , which is generally inclined with respect to the beam axis. It is important, therefore, to observe that B in (125) conforms smoothly to the specified Gaussian distribution W(x0; X) in z = 0 plane, whereas for large Zb it gradually changes into a conventional GB, described by substituting in (125)-(126) z t - ' = Zb - Xb, tan0 3 Zb. Under this paraxial substitution, the elements of I? in (126) become 2f-I
( z t - ' - i s f 2 ) - ' + ( z b - i s f 2 ) - ' = ( z b - ~1 - iFI)-'
(zt-'
- is)-' + (Zb - i s ) - ' = ( z b - 2 2 - iFZ)-'.
(127)
The parameters Z I ,and ~ F1,2 defined by these relations will be interpreted later. Substituting (127) into (125), we obtain the GB form in (98) for B(r;X) with: --f fit2= FI - iZ1; and 4 2 + fi = F2 - iZ2. Z + 26; X i --f x b j ; The interpretation of this expression follows the discussion after (98). The beam is astigmatic: its waist in the ( 2 , Xb,) plane is at Zb = Zj = -"I,, its collimation distance is F, = ay~~ and its widrh bj is given in (99). This astigmatism is caused by the beam tilt, which reduces the effective initial beam width in the Xb, direction (see Fig. 16). Regarding B as a function of X for a fixed r, one finds that it is strong along the constraint (1 19) and exhibits a Gaussian decay away from it. Keeping constant, one finds that the phase-space widths as X is displaced from this constraint in the Xbl and Xb2 directions are given by
bl
= D ~ l f , D 2 = D2.
(128)
This also determined, via (1 19), the Gaussian widths of the decay as X displaced in the (1 and (2 directions. These conditionsdefine the width of the relevant phasespace integration domain around the constraint ( 1 19) and thereby the numerical efficiency of the representation. Other properties of this representation can be found in [41]. The reader is also referred to [39] for a thorough asymptotic and localization analysis, which is performed, however, in the context of a 2D configuration. B. Time Domain: Pulsed Beam Summation 1. Local Spectrum of the Data
The local spectrum approach will be extended next to analyze time-dependent field solutions directly in the time domain. We refer again to the configuration in Fig. 1 and the problem formulation in the beginning of Section 11.
52
EHUD HEYMAN AND TIMOR MELAMED
The time-dependent local spectral distribution of the initial field U O ( Q , t ) can be defined as an inverse Fourier transform of the Oo(X)in (109)
where 5 denotes the phase-space time variable in the five-dimensional phase-space Y = (X,i) = (a, 1,i). The time shift in the exponent of (129) has been introduced for a convenient interpretation of the results [see discussions following (131) and in (139)]. Next we shall calculate the local spectrum directly from the TD data U O ( Q , t ) . The TD local spectrum transform is obtained by applying (129) to (109) and performing the w integration first, giving
WY)
=Sd2xoSdtuo(xg,t)~(xo,t;~)
( 130)
where
W(Q,t ; Y)= w[xo - X, t
- 7 - c-'E
*
(xo - X)].
(131)
The window function w(xg, t ) is the TD counterpart of the frequency-domain window function I%(%); hence, the phase-space kernel W is the counterpart of the kernel W in (1 10). It is assumed here that w(x0, t ) is localized about the origin: Typical parameters for the space-time support of this window function will be considered in Section VI.B.3. Then, noting the space-time and spectral shifts in the phase-space window W of (131), one identifies (130) as a windowed version of the slant-stack transform (33): It is localized about (xg, r) = (X, i) with the spectral tilt 1 (see Fig. 17 and compare to Fig. 3). This interpretation identifies (130) as a local slant-stack transform (local SST) and UOas the local time-dependent spectrum of uo.
(a) (b) FIGURE17. Local (pulsed beam) spectrum. (a) The local slant-stack transforms of the initial field uo(x0, t)
(cf. Fig. 6). (b) The radiating pulsed beam (cf. Fig. 3).
SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS
53
To gain further insight, it might be helpful to note that in the special case r ) = 6 ( r ) , the integral in (130) reduces to the SST transform, namely,
W(Q,
Uo(Y)+ i i O ( ( , i - c-'E * f)
( 132)
where ii~(& 5 ) is the time-dependent spectrum of uo as obtained from the SST (33). Finally, we note that for proper numerical implementation of the local transform namely in (130) we shall also demand that w is in LCfr,t),
J J d2x
dr Iw(x, r)l = finite.
(133)
The inversion formula corresponding to (130) is found by transforming (1 14) into the time domain, giving UO(x0, r )
=42n4-2
J
dSYUO(Y)W,(Q, t ; Y),
( 134)
where
W,(Q,
t ; Y) = N+W 8 W ( Q ,
t ; Y),
( 135)
with Q3 denoting a convolution and
Note that fi may tend to zero as w + 00, in which case the integral in (136) does not converge. Thus, the integration domain in (136) will be limited to the bandwidth of the excitation signal so that (1 36) is well defined [see discussion after (146)l.
2. The Radiating Field Equation ( 134) can be propagated to z > 0, giving
I
u(r, r ) = - ( 2 ~ c ) - ~ d5YUo(Y)B(r,r ; Y)
(1 37)
where B describes the radiation into the half sphere z > 0 due to field distribution WN(XO,t ; Y)in the z = 0 plane. B can be evaluated from the FD propagator B in ( 1 16) via [see (129)]
Alternatively, B may be calculated directly in the time domain via several alternative formulations [e.g., Kirchhoff integral (25) with (2%) or the plane-wave integral (31)]. Following (1 16), we shall represent it here by the TD plane-wave
54
EHUD HEYMAN AND TIMOR MELAMED t
integral (31), giving B(r, t ; Y) = Re B (r, t ; Y) with
(r, t ; Y) = - ( 2 n ~ ) - ~ J ’ d ~ 4 4 ? W ~t [ ( .c - ’ ( ( . x
+ { z ) ;Y]
(139)
where in (139) i i / ~ ( ( r, ; Y)is the analytic TD plane-waye spectrum (30) of the W ~ ( x 0t ,; Y) in (135). In view o+f(131) it is given by WN(<, r ; Y) = N t ( t ) 18 ii, (( - 1, r -+i c-I( .%)where ii, ((, t)denotes the SS,T (30) of & (XO, t). Since from (131) w is localized about (XO, (, t ) = ( j z , 1,i), B describes a space-time wavepacket (pulsed beam, PB) that emerges from the z = 0 plane at (XO, t ) = ( j z , 7) in the direction I%of (1 17). Its space-time trajectory is described by [see (1 19)]
+
(x - R ) / R = 1, +
t =i
+ R/c.
(140)
Closed-form expressions for B will be considered in Section VI.B.3. The representation in (137) describes the field as a continuous spectrum of shifted, tilted, and delayfd PB (Fig. 15). The PB amplitude is described by the local spectral funFtion UO that matches PB spectrum to the given aperture distribution: Thus, UO senses the local radiation properties of the aperture around 2 and, therefore, excites only those PB propagators that match those properties (Fig. 17). The integral representation (137) is, therefore, localized a priori in the Y domain (without recourse to asymptoticanalysis). This property has been explored analytically in [lo], and will be demonstrated numerically in Section VI.B.3. Assuming, for example, that uo has the short-pulse distribution form (38), then UOenhances PB whose phase-space coordinates Y are in the vicinity of the radiation constraint [see ( 1 18) and Fig. 181
< = VOO(%),
5 = c-lOo(t),
(141)
where the first and second conditions define the local radiation direction and initiation time, respectively. This limits the effective domain of integration in (137) to phase-space regions adjacent to that constraint (Fig. 17). The integration domain is limited further due to the fact that the PB propagators B are confined in space-time, so that nonnegligible contributions are obtained only from PBs that pass near the observation point. For a given (r, t), Eq. (140) can then be considered as an observation constraint that confines the relevant points in the Y domain. Thus, the actual integration domain in (137) is confined a priori to compact domains around the intersection of the constraints in (141) and (140). The degree of localization depends on the choice of the proper window function w , which minimizes the phase-space support of both UOand B for a given class of initial distributions, and for a given observation range. Asymptotic and parametric analysis of this phase-space localization has been considered in [39] and [lo]. This localization will be demonstrated next via a special example.
SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS
55
5
FIGURE 18. The radiation constraint (141) for the short-pulse distribution of the form (38). xe,.? are the end point of the aperture distribution. The local spectrum UOenhances PB whose phase-space coordinates Y are in the vicinity of the radiation constraint (141) as well as diffracting PB emanating from X = x , , , ~in all directions at 7 = @ ~ ( x , , , ~(cf. ) / cFig. 15).
3. Example: A Gaussian &Window The favorable properties of the Gaussian windows (120) or (121) for analysis of time-harmonic problems have already been mentioned at the beginning of Section VI.A.3. In the time domain, a convenient window is obtained by tsansforming (121), with B frequency independent. We note though that convergence of the window in (121) implies BlO<0 = -B*lWso. It is, therefore, convenient to employ only positive frequencies and utilize the analytic signal representation (21). Thus, applying (21) to (121) we obtain
“-1
w(%, t) = Re&(xo, t) = Re6 t
-Ixol’/Bc]
where 8 is the analytic delta function (49). This window has the form of the field distribution in (48) (without the parameter TO):It is localized around (XO, t ) = (0,O). For 1x01 = 0, it is impulsive at t = 0 and decays thereafter like t-’. For 1x01 0 it is a smooth Rayleigh pulse with a peak at t = 0 (for real B) where the lx01’ and peak value and half-amplitude width are given, respectively, by 2v@~/77 Ix0l2//3~u.If, however, B is complex, then the peak occurs at t = lx01’B,/2~1~1* and the window exhibits a quadratic delay. One may readily verify that the decay rate of the window in (142) is not fast enough in t and x and thus it is not in Cfx,f)[see (133)l. In order to obtain a window in Cfx,f) we shall differentiate this window twice. Furthermore, as noted earlier, the window in (142) is impulsive for xo = 0. In numerical processing one may analytically extract the contribution of the &singularity of the local SST integral
+
56
EHUD HEYMAN AND TIMOR MELAMED
(130). In certain cases, however, it might be preferable to use a smooth window. Such window is obtained if an exponential decay e-('/2)wTgis added to zir in (121), where the parameter Tp > 0 is chosen to satisfy
with wmaxdenoting the upper frequency in the data U O ( X O ,t). The reason for this choice will be discussed after (146). Thus, replacing the window in (120) by
and by applying (21) we obtain, instead of (142), the window function
where from (49), &2)(t) = $ " ( i ) = 2/nit3. This window has essentially the same form as (48) (it is raised to the power 3 and a! is replaced by B); hence, its properties are as discussed in (50)-(53). Finally, it should be noted thpt for j?real and very large, the window in (142) behaves like a planar window S " ( t ) so that the local spectrum integral reduces to the SST limit as in (132). The phase-space pulsed-beam propagators B due to the window (145) are obtained by substituting in (139) the spectral window [see (56)]
as obtained by the slant-stacktransform (30). For well-collimatedwindows, closedform expressions for B can be obtained via an asymptotic evaluation of the resulting integral (139) directly in the time domain [cf. the evaluation of (66)]. For simplicity, however, we shall derive these expressions from the asymptotic FD beams B, in (125). Noting that the TD window in (145) corresponds to the FD window in (144), we find that the corresponding FD propagators are obtained by multiplying the propagators B, in (138) by (-iw)2e-(1/2)0Tg.Furthermore, g the using (122) we have for this FD window f i 2 = ( - i ~ ) ~ : g e - ~so~ that integral in (136) for N t diverges like eWTp. However, if TB is chosen according to (143), then it has no effect over the entire frequency band of the data and we may use Tp = 0 in the expression for f l 2 [see also discussion in (136)l. Applying
SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS
57
(138) and, recalling that all the parameters in (125) are w-independent, we obtain B =ReB,
(147) where the beam coordinates ( x b l , nb,Z b ) are defined in (123) and r is given by (126). This expression has the PB form in (80)(subject to a few differences, which will be discussed in the next paragraph). Thus, it is identified as a PB that emanates from the point xo = t, in the z = 0 plane at a time t = i , and propagates in the i% direction alpng the zb-axis. Its pfopyrties have been discussed in (81)-(92), for a general f and in (93)-(96) for f = 8 . As discussed after (126) in connection with the Gaussian beam window, the elements of r in (126) depend only on zf-' = Zb - x b , tan8, whereas in the PB (80) they depend only on the location Zb along the beam axis. Thus, as in (125), expression (147)conforms smoothly with the initial field distribution W(m,t ; Y) at the z = 0 plane. For large Zb, on the other hand, the elements of I? behave as in (127) and, thus, (147) changes gradually into+the conventional PB form (80). Specifically, since in our case I? is diagonal, B in (147) obtains the form i? (8:) with the replacements: z + Z b ; x j +-X b j ; a1 + piz; a2 + p; and f + 8'(t - ~ p as) in (93). The interpretation of this expression is discussed after (93): It is astigmatic PB; in the (z, X b j ) plane the waist is at Zb = -a!,,the collimation distance is F, = c ~ R ~ the beamwidth D, is given in (94),and the diffraction angle at (94a).
4 . Illustrative Example for the Local Spectrum In this example we calculate the time-dependent local spectrum of the initial distribution (48) via the local SST (130), using the Gaussian 8-window (145). The resulting integral can be evaluated in closed form, giving
-2ilsc Uo(Y) = Re a-l + p*-1
(1
i - -To - -Tp i 2
+ c-'(a + /?*I-'
It should be noted that for large and real /? the window in (142) behaves like a plane window 8 ( t ) and the local spectrum reduces to the SST limit in (132) [see discussion in connection with (142)l. Indeed one observes from (148) that if
,
58
EHUD HEYMAN AND TIMOR MELAMED
IBI >> la[,I%l and Ts << TO,this expression reduces into the SST spectrum (56) as in (1 32). By minimizing the imaginary part of the argument in (148) for a given 8, one finds for real B that the waveform in (148) peaks at <(%)= iZa,/lal2,
1
I(%) = -It12a,/clal 2 .
(149) 2 This condition defines the phase-space domain where the data &(Y) actually reside. Using @O(XO) = ct,(xO), with t, of (51) being the time delay in the aperture, one may readily verify that (149) agrees with the radiation constrainr (141). A plot of this constraint in the (i,, $1, i) subdomain with ( i 2 , $2) = 0 is depicted in Fig. 19a.
FIGURE19. The local spectrum (148) for the initial field distribution in Fig. 8a. (a) The phasespace radiation constraint (141) [see also (149)l. Contour plots of U,,(Y) in cross-sectional planes i/T,, = 0. -1/2, -1, -3/2. The contour plots are concentrated about the radiation constraint (149) (full line). The contour levels are 0 , 3 db, 6 db, 9 db, I2 db, and 15 db below the peak level of U,(Y)at (f,(, 7) = (0.0.0). (b)and (c) Snapshots of the spectral waveform in (148). for 7 = 0 and 5 = -Tu. The spectrum is shown in the ( i r , 61) plane for (X2,t 2 )= (0.0). The window parameters are p = 1 and Tp = 0. Note the concentration of the local spectrum in (b, c) near the radiation constraint in (a) for the corresponding values of 5.
59
SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS
o O 0.02
0.04
*
O
3
1
0.01
0 -0.02
-0.01
j
-0.04
-0.021
, ,
-0.06
-1
1
0
-0.03
-1
0
t, 1
x lo"
(4
(b)
FIGURE 20. The local spectrum (148) for the initial field in Fig. 18a, shown in a cross-sectional plane (,Ti,7) whereon is related to ,TI via radiation condition (149). The contour lines at 3 db, 6 db, and 9 db below the peak are shown to be concentrated along the condition i = c-l a,(%)of (149) (dashed line). (a) (Y = 1 - 4i as in the examples in Figs. 18 and 19 and (b) (Y = 1.
<
To demonstrate this phase-space localization, we also show in Fig. 19a contour at several cross-sectionalplanes ?/To = 0, - 1/2, -1, plots of the function Vo(Y) -3/2 in the Y domain. One observes that Vo(Y)is localized about the radiation constraint (149). To clarify the structure of Vo(Y)we also show in Figs. 19b, c snapshots of the distribution of Vo(Y) for two values of i: ? = 0 and i = -T0/2. The distribution is depicted in the (El, 6 , ) plane with (E2, = (0,O). Consequently, we The initial data parameters are a! = 1 - 4i and To = choose a window with 9, = 1 and Tp << To. We observe that indeed the distribution of the local spectrum is concentrated near the radiation constraint (149). Finally, in Fig. 20 we demonstrate the phase-space localizationfrom yet another point of view. The figure shows the contour lines of Uo(Y) (148) sampled on a plane (XI,i) defined by the radiation constraint = Za!i/lU12 [see (14911. l k o different cases of initial conditions are considered: (a) a! = 1 - 4i (as in Figs. 18 and 19), and (b) a! = 1. One observes that the contour plots are concentrated along the constraint i = i l ! i ) 2 a i / v ( ~of1 2(149) (dashed line).
c2)
<
APPENDIX:ASYMPTOTIC EVALUATION OF THE BEAMFIELD IN (125)
In order to analyze B we shall express (1 16) with (121) in the form
60
EHUD HEYMAN AND TIMOR MELAMED
where formal definition of the spectral integrationdomains is given in (6c). Integral (A. 1) has a stationary point i& in the complex 5 domain, defined by
V( q
= ( x - a)
+ is(< - 1)- < z / c = o
at 6 = tS.
('4.2)
This equation has a real solution 5, = 1if and only if X and r are related by < 1. For all other values of X and r, the observation constraint in (1 19) and the solution of (A.2) is complex and cannot be found explicitly. However, for points near the beam axis, an approximate expression can be obtained by a Taylor series expansion of q ( 1 ) about the beam direction 1, which can be cast in the form (A.3) where (A.4a) (A.4b)
(A.4c) From (A.3), the stationary point is given by 5s
=
1- s;'
.Ql
04.5)
and the beam field is evaluated via
B
- B,
=
(is/,
(A.6)
Utilizing the beam coordinates in
where r is given by (126). The final result (125) is obtained by substituting (A.7) into (A.6). From (AS) and (A.7) one readily observes that the displacement of ,$, from the real value 1is proportional to x b . thereby justifying the preceding Taylor analysis for observation points near the beam axis.
SPACE-TIME REPRESENTATION OF ULTRA WIDEBAND SIGNALS
61
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EHLJD HEYMAN AND TIMOR MELAMED
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[48] Kravtsov, Yu. A,, and Orlov, Yu. L. (1991). Geometrical Optics of Inhomogeneous Media, Springer-Verlag. [49] Ludwig, D. (1 966). Uniform asymptotic expansion at a caustic, Comm. Pure and Appl. Math. 19, 2 I 5-260. [SO] Kravtsov, Yu. A. (1964). A modification of the geometrical optics method, Radiojzika (Russian) 7,664-673. [51] Brown, M. G. (1986). The transient wavefield in the vicinity of the cuspoid caustics, J. Acoust. Soc. Am. 79, 1367-1 384. [52J Brown, M. G.(1986). The transient wavefields in the vicinity of the elliptic, hyperbolic and parabolic umbilic caustics, J. Acoust. SOC.Am. 79, 1385-1401. [53] Hafizi, B., and Sprangle, P. (1991). Diffraction effects in directed radiation beams, J. Opt. SOC. Am. A 8,705-717. [54] Heyman, E. (1989). The focus wave mode: A dilemma with causality, IEEE Trans. Antennas Propagai. AP-37, 1604-1608. [55] Moses, H. E., Nagen, R. J., and Sandri, G. v. H. (1992).The general solution of the three-dimension acoustic equation and of Maxwell’s equations in the infinite domain in terms of the asymptotic solution in the wave zone, J. Math. Phys. 33,86101. [56] Devaney, A. J., and Sherman, G. C. (1973). Plane-wave representations for scalar wave fields, SIAM Review 15,765-785. [57] Deschamps, G. A. (197 1). Gaussian beam as a bundle of complex rays, Electron. Lett. 7,684-685. [58] Shin, S. Y., and Felsen, L. B. (1977). Gaussian beam modes by multiples with complex source points, .I. Opt. SOC.Am. 67,699-700. [59] Ra, J. W., Bertoni, H., and Felsen, L. B. (1973). Reflection and transmission of beams at dielectric interfaces, SIAM J. Appl. Math. 24,396-412. [60] Felsen, L. B. ( 1976).Complex-source-point solutions of the field equations and their relation to the propagation and scattering of Gaussian beams, In Symposia Matematica, Vol. XVIII, Academic Press, London, England, pp. 40-56. [61] BabiE, V. M., and Buldyrev, V. S. (1990). Short-Wavelength Diflracrion Theory, Springer-Verlag. 1621 Baum, C. E. (1 989). Radiation of impulse-like transient fields, Sensor Simulation Notes, p. 321. [63] Bastiaans, M. J. (1981). Signal descriptions by means of a local frequency spectrum, In Transformation in Optical Signal Processing, Vol. 373, Proc. SPIE, pp. 49-62. [64] Bastiaans, M. J. (1980). The expansion of an optical signal into a discrete set of Gaussian beams, Optik 57.95-102. [65] Einziger, P. D., Raz, S., and Shapira, M. (1986). Gabor representation and aperture theory, J. Opt. SOC.Am. A 3,508-522.
ACKNOWLEDGMENT This work is based on a report prepared in 1992 for the US. Air Force System Command, Rome Laboratory, under Contract No. F19628-91-C-0113. It has been also supported by the US-Israel Binational Science Foundation, Jerusalem, Israel, under Grant No. 95-00399and by the U.S. Air Force Office of Scientific Research. under Grant No. F49620-93-1-0028.
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ADVANCES IN IMAGING AND ELECTRON PHYSICS VOL. 103
The Structure of Relief JAN J . KOENDERINK AND A . J . VAN DOORN Buys Ballot Laboratory University of Utrecht. Faculty of Physics and Astmnomy Pnncetonplein 5. I? 0. Box 8000.3508 TA Utrecht. The Netherlands I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . A . Scalar Fields in 2D: Images. Reliefs. and Landscapes . . . . . . . . . . B . Intrinsic and Extrinsic Geometry. 3D and (2 I)D Images . . . . . . . . . C. Affine and Arbitrary Monotonic Transformations.Gamma Corrections . . . . D. Images Defined by the Gradient . . . . . . . . . . . . . . . . . . E. The Qualitative Structure of Images . . . . . . . . . . . . . . . . . F. The Nature of Observation. Resolution . . . . . . . . . . . . . . . . G. Local Description of Image Structure . . . . . . . . . . . . . . . . H . Genericity and Structural Stability . . . . . . . . . . . . . . . . . I1. The DifferentialStructure of Images . . . . . . . . . . . . . . . . . . A . Stable Differentiation . . . . . . . . . . . . . . . . . . . . . . B . The Local Jet . . . . . . . . . . . . . .. . . . . . . . . . C. The Congruence of Level Curves and the Creep Field . . . . . . . . . . D. Differential Invariantsof the Second Order . . . . . . . . . . . . . . E. Nature of the Creep Field . . . . . . . . . . . . . . . . . . . . 111. Global Description of the Relief . . . . . . . . . . . . . . . . . . . A . One-DimensionalCase . . . . . . . . . . . . . . . . . . . . . B . Cayley and Maxwell’s “Natural Districts’’ . . . . . . . . . . . . . . C. Integrals of the Creep Equation . . . . . . . . . . . . . . . . . . D. Description in Terms of Intrinsic Properties . . . . . . . . . . . . . . E. The Geometrical Meaning of the Differential Relations . . . . . . . . . . F. The TopographicCurves . . . . . . . . . . . . . . . . . . . . . G. Generic Structure . . . . . . . . . . . . . . . . . . . . . . . H . Transport of Stuff . . . . . . . . . . . . . . . . . . . . . . . I . De Saint-Venant’s Curves . . . . . . . . . . . . . . . . . . . . J . Cliff Curves . . . . . . . . . . . . . . . . . . . . . . . . . K . The ConvedConcave Boundary . . . . . . . . . . . . . . . . . . L. The Loci of Vertices of the Isohypses . . . . . . . . . . . . . . . . IV. Contours: Envelopes of the Level Curves . . . . . . . . . . . . . . . . V. Discrete Representation . . . . . . . . . . . . . . . . . . . . . . A . Triangulations . . . . . . . . . . . . . . . . . . . . . . . . B. Isohypses and Slope Field . . . . . . . . . . . . . . . . . . . . C. Morse Critical Points . . . . . . . . . . . . . . . . . . . . . . D. RidgesandRuts . . . . . . . . . . . . . . . . . . . . . . . . V1. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .
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In many sciences scalar fields on two-dimensional manifolds play an important role. We investigate the structure of such fields (“relief ”). Historically, the study of this problem is related to that of the so-called “topographic curves,” which are the level curves (or contour curves) and fall curves (or curves of steepest descent) of a landscape and its “ridges” and “courses”. The latter entities are important in many applications but have remained the object of considerable controversy in the literature till the present day. We attempt a balanced review of the various definitions and propose novel rational and pragmatic approaches. With modem computer facilities and much improved algorithmic tools (both symbolical and numerical) many calculations of mere theoretical interest in the past have now become viable in everyday scientific work. We provide a short discussion of possibilities.
I. INTRODUCTION A. Scalar Fields in 2 0 : Images, Reliefs, and Landscapes
Scalarfields in two and three dimensions occur with great frequency, both in the sciences and in various other fields of endeavor. In this paper we concentrate on the case of dimension two, though some reference will be made to the case of dimension one and to that of higher dimensions. The two-dimensional case is of special interest because it applies to many immediate applications; moreover, from a more principled perspective, dimension two is also somewhat special in the formal sense. For instance, the Pfaffian differential equation
f (x, Y ) dx + g ( x , Y ) dy = 0
(1)
is guaranteed to possess an integrating factor in dimension two, whereas already in dimension three this is not the case; rather, a special condition has to be satisfied in order for the field of contact elements (line elements in the two-dimensional case, planar elements in the three-dimensionalcase) to possess integral manifolds foliating the space. Important and commonly encountered examples of scalar fields in dimension two are images (think of photographs, video frames, etc.), reliefs (a term referring to the distance function for the surface of some smooth object in three-dimensional space with respect to the vantage point of an observer or a laser ranging device), and geographical landscapes in the sense of low-resolution geomorphology. Indeed, many of the terms used in the description of the structure of scalar fields in two dimensions derive immediately from topography, for example, hill, ridge, and so on. Historically, much of the mathematical framework has been developed in this latter setting (e.g.. the Encyclopea of Mathematical Sciences [44]has a section on “topographiccurves”). Nowadays images have become so important that the latest developments rather derive from the former setting (‘‘image processing”). Relief
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used to be important in the theory of art [32] and in visual science [42]; currently applications abound in orthopedic diagnosis [61], computer-aided design, virtual reality, and so on. These three examples are indeed very similar, though in practice they differ in a number of important ways. For instance, whereas the landscape is generally considered as an approximately smooth,continuous surface of (potentially)infinite extent, images are typically limited in extent and are most often specified discretely (pixels). Image intensity is typically limited from below (nonnegative) and most often also from above (e.g., the range 0 . . .255, or 1 byte is very common). In the case of relief there appears the additional complication that the curves of equal distance possess envelopes and that they cannot be continued after meeting that envelope, the so-called “visual contour.” (Moreover, the envelope may itself possess a number of generic singularities.) In this paper we will largely ignore the complications that arise from the existence of a contour and/or of an arbitrary boundary (edge of the image). Surfaces that have acute slope angles (and thus a single z-value for each location) are sometimes (Miiller [49]) called “graphical surfaces.” In this paper we typically consider graphical surfaces on the entire plane with heights on the entire real axis (thus allowing us to skip various kinds of special cases that would unnecessarily burden the exposition), though at times we will have to restrict the territory or the range of heights. The three examples are useful in the context of this paper since they are equivalent in many important ways, whereas they can also be used to illustrate certain special properties that need not be shared by all scalar fields in two dimensions. Notice that the examples by no means exhaust the potential areas of application of the general theory though. (See Fig. 1.)
FIGURE1. Examples of scalar fields in dimension two. Left: Topographical map showing fall curves. Notice that the gaps in the hatches indicate the isohypses that are the level curves of the height. Middle: Relief of a mannequin obtained with the Moh5 method. Notice that the level curves of the depth approach a common envelope, the visual contour of the object. Right: A posterized image showing the isophotes, which are the level curves of the image intensity.
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In this introductory section we discuss a number of issues that will prove to be importantin the development of the theory but are perhaps not generally recognized in the sciences except for a limited number of specific areas. 1. Cartographic Generalization
A map the size of the landscape would be of little use to the traveler. Maps are useful exactly because they offer a summary description of the landscape and omit most detail. What is and what is not on a map depend on the intended use of the map. Thus, a map is not so much a small copy of the landscape as it is an interpretation of the landscape by a cartographer. The cartographer uses both simplified iconic renderings and conventional symbols to indicate features of the landscape that are of special interest, for example, landmarks or the locations of certain resources. This is known as cartographic generalization, which has remained largely an art, despite the fact that there is also a science of cartography [14]. Features that are often marked in maps and that might interest us in the present setting are descriptors of topographic relief such as isohypses [loci of equal height, also known as contour lines, Schichtenlinien (German), lignes de niveau (French)] and isobaths (loci of equal submarine depth), slope lines orfall curves (integral curves of directions of locally steepest slope), ridges (or divides) and ruts (or water courses), hills and dales. One finds such entities marked on early maps long before formal, mathematical definitions were even attempted. Interestingly though, the exact meaning of an entity such as a ridge has remained elusive even up to today, and this paper is largely concerned with their formal characterization and calculation. In this review we will use level curves as the generic term. In specific instances we will replace this with the appropriate descriptive term such as isohypses, isobaths, isophotes, and so on. In many fields this is necessary to distinguish between various level curves of different types in a single map, for example, in meteorology one might conceivably have isobars (level curves of pressure) and isotherms (level curves of temperature) in a single map.
2. Posterization In image processing the image intensity (we use this term to mean either radiance, irradiance, or whatever the case may be) is typically discretized. For instance, the typical CCD image has about 8 bits per pixel. In many image processing packages one has the choice toposterize the image, that is, to decrease the number of discrete image intensitieseven more. In the extreme case of only two image intensities (binary or black-and-white images) one speaks of thresholding. Posterization vividly shows the curvilinear congruence of loci of equal image intensities or isophotes as the common boundaries of uniform areas.
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Such posterization is also common in topographic maps where one draws isohypses for a given spacing (depending on the terrain and the intended precision of the map), say 5 m. The first such map was drawn by Cruquius [17] in the Netherlands in the early eighteenth century. Cruquius marked the edge of the water level of the river Merwede as it dried out during summer and thus acquired isohypse data empirically. This representation of the terrain is also useful in engineering calculations, for instance, in building or road construction. Indeed, the necessary mathematical formalism was developed in such an engineering setting [49] and represents the earliest theory of scalar fields in two dimensions. The first extensive review is due to Noizet [5 13 in the context of military engineering (fortification).
3. Hildebrand’s Depth Flow The German sculptor Hildebrand [32] wrote an influential book on reliefat the end of the nineteenth century. He stressed the fact that the human observer is hardly sensitive to the depth scale, for example, a sculpture in low relief does not look different from one in the round if both are viewed from some distance. Thus, he stressed the fundamental importance of topological structure of the curvilinear congruence of loci of equal distance and defined relief essentially as the depth field modulo arbitrary depth scalings. Hildebrand also draws attention to the field of integral curves of the local directions of fastest increase of depth, which he calls the “depth flow.” He outlines certain desirable topological properties of the depth flow in order for the sculpture to appear aesthetically pleasing. From a technical point of view Hildebrand proposes that the sculptor starts from the front of the block and works his way inward, removing material so as to reveal depth layer after depth layer, taking care of good composition within each layer of depth and suitable relations between the layers. He ascribes the invention of this method to Michelangelo, probably based upon a famous description by Vasari [63, 161 (which may well be apocryphical): . . . he gradually chisels away the stone till. . .issues forth from the marble, in the same manner that one would lift a wax figure out of a pail of water, evenly and in a horizontal position. First would appear the body, the head, and the knees, the figure gradually revealing itself as it is raised upwards, till there would come to view the relief more than half completed and finally the roundness of the whole.
This method is very similar to modern CAM methods where the computer-controlled tool removes material from a block in essentially the way recommended by Hildebrand.
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4. Scalar Fields
With a scalar field in two dimensions, we intuitively mean a Coofunction @ :R2
+ R,
z = @(x, y),
@:
+ @;
< 00,
where z denotes the height (or depth, or image intensity) and { x ,y } are Cartesian coordinates in the plane. If we regard z as the third coordinate in Cartesian coordinates { x , y, z) of R’ = R2 x R,this defines a surface in terms of a Monge patch parameterization [36]. In general we will want to distinguish sharply between the z and the { x , y } coordinates because (except for the geographical example) they are not commensurable. There are many obvious difficulties with this type of definition. For instance, images are typically discrete, whereas the height in the landscape is an ill-defined probably fractal variable. Certainly, we may not assume anything like continuity or smoothness (e.g., differentiability). Moreover, there exist serious scale problems: Should one count large rocks as hills? And, if so, then how about grains of sand? Similar issues arise in the physics of continuous media since matter is granular on the atomic scale. We will clarify these issues later. B. Intrinsic and Extrinsic Geometry, 3 0 and (2
+ I)D Images
As has been said before, the { x , y} and the z coordinates are typically not commensurable. In the case of the landscape they apparently are, but even here the height is in a quite different category from the location. For instance, when the cartographer draws a cross section of a terrain, he or she typically uses different scales for the vertical and horizontal ranges. When one walks in the mountains, one also notices a distinct difference between such ranges. In case we do not admit any direct comparison of the z and ( x , y ) coordinates, we will speak of the “(2 I)-dimensional” case, otherwise of the “3-dimensional case.” In the (2 I)-dimensional case we can only do extrinsic geometry, with reference to the special (z) direction. In the 3-dimensional case we can also do extrinsicgeometry, of course, but here concepts from intrinsic geometry also make sense. An example would be the (Euclidean) curvature of a hilltop: This is an intrinsic entity that makes no sense in the (2f 1)-dimensionalcase. The distinction is a crucial one and we will have to refer to it often.
+
+
C. ABne and Arbitrary Monotonic Transformations, Gamma Corrections Typically we regard the ( x , y}-plane as an Euclidean plane; that is, geometrical configurations are left invariant by isometries (or even similarities), whereas the
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z-dimension is regarded as an a#ne line; that is, geometrical configurations are left invariant by arbitrary translations and similarities. In that case we don’t work in R3 = R2 x R,but in R3 = R2 x A, where A denotes the affine line. In some cases one might even consider more general transformations such as arbitrary (nonlinear) monotonic transformations of the z-domain. An example of monotonic, nonlinear transformations that is extremely common is that of the gamma corrections in image processing. A gamma correction of the image intensity is a transformation I ’ = a Z Y . Notice, however, that this is nothing but an affinity in the log I-domain. We will consequently consider such transformations as affinities and use log I, rather than image intensity per se, as the z-coordinate in such cases. In summary, we consider arbitrary monotonic transformationsand affine transformations of the z-axis and the three-dimensional case as alternatives in this paper. D. Images Dejined by the Gradient
Often the empirical data are not (samples of) the z-coordinateat all, but rather of its gradient, or even higher-order derivatives. In such cases the image (for instance) is not fully defined, but only by way of the first- (or higher-) order derivatives. In many cases one may even observe less, say only the gradient direction or the orientation of the isophotes. For example, if you use Cruquius’s [ 171 original method (draw the outlines of the water level), or when you have only a level to gauge the landscape, you find a field of isohypse orientations. If the landscape is illuminated by the sun at the zenith, the illuminance of the surface is a nonlinear, monotonic function of the magnitude of the gradient [33]. Psychophysical methods that address the visual perception of relief often obtain samples of the depth gradient [42]. Many additional examples can be identified, many of them important in some branch of the sciences. We thus shall occasionally consider the scalar fields to be specified by (aspects of) their differential structure, rather than by the nominal scalar variable per se. In such cases there will typically remain some ambiguity regarding the value of the scalar variable.
E. The Qualitative Structure of Images In many applications the qualitative structure of images is of much greater importance than their quantitative structure. Examples abound in the consumer industry: Consider a population watching television. No pair of randomly selected television sets will be the same (or even similar). From television set to television set there
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exist rather severe nonlinear (but typically monotonic) differences. (This is even more severe in color television: Most people really haven’t the faintest notion of how to even start tuning their sets to achieve close to veridical color balance.) Yet most of the population is convinced that everybody is watching the same television broadcast. Apparently, folk wisdom implicitly assumes that largely arbitrary monotonic transformations of image intensity are to be considered irrelevant. This means that “the image” is to be understood as the qualitative (topological) structure of the curvilinear congruence of isophotes, not as a quantitative, spatially organized data structure. The qualitative structure of images is much like the qualitative structure of a landscape. When asked to describe it, one is apt to talk in terms of hills (light blobs) and dales (dark blobs), and so forth. Such language describes the structure of the nesting of level curves without reference to the image intensity values identifying these level curves. 1. Morse Critical Points
Topologically the simplest (i.e., the first-order) singularities or critical points of landscape are the summits that are the local maxima of height, the immits that are the local minima of height, and the saddle points. (The term irnmit was coined by Cayley [ 151 and used by Maxwell [46]in his seminal paper “On Hills and Dales.”) Their operational definition is straightforward: Locally the ground is level at a critical point (that is in fact the definition of these locations). Near a summit all paths go downward, near an immit all go upward, but in the neighborhood of a saddle point some paths go upward and some go downward. This clearly exhausts the possibilities.’ In the natural landscapes immits are actually very rare and to a good approximation one encounters only summits and saddles. In other examples immits may be as commonly encountered as summits. In this paper we will proceed as if landscapes contain both immits and summits: Unlike nature, in our discussion an inverted landscape (where the height dimension is inverted) is again a perfectly possible landscape. These singularities are known as the Morse [47] critical poinrs. With critical points one indicates the points at which the gradient of the height vanishes so that the local direction of steepest ascent or descent is undefined. Such points are termed Morse if they are isolated (a small perturbation only pushes them around a little but doesn’t lead to critical points splitting into others) and if all critical
’
One has to be careful though: It is, for instance, not the case that when along any straight line through a point one finds a maximum of height the point thus has to be a summit (although this certainly holds for a true summit)! A famous counterexample is due to Peano [24, 571: take z(x, y) = (2x2 - y)(y - xz). One easily sees that the origin is not an extremum (some descending and some ascending paths are easily found), but along any straight direction through the origin, the origin is an extremum!
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points are of distinct height. In such cases the level curves are very simple. Most are composed of one or more closed loops. (Some may fail to be closed because they either run off to infinity or end on the boundary if there is such an entity.) A number of level curves (of measure zero) is special in that they either consist of isolated points (these are the summits and immits) or are loops that self-intersect. Such self-intersections occur at the saddle points. Another way to characterize the Morse critical points is to say that they are locally like their second-order Taylor approximations; that is, the osculating paraboloids at the Morse critical points characterize the local qualitative structure fully. Morse proved that all critical points may be assumed to be such simple immits, summits, or saddles; exceptions occur with zero probability for the generic landscape. 2. Inloop and Outloop Structure The level curves through the saddle points self-intersectat the saddles. This means that such level curves occur in two distinct flavors: Either they look like a figure eight shape (called “outloop” by Cayley [ 151) or they look like an inverted figure eight shape (called “inloop” by Cayley). The outloop curves consist of two closed loops that meet at the saddle. Both loops encircle either higher or lower areas. Thus, the outloops consist either of a pair of hills or of a pair of dales. Here one should strictly speak of false maxima instead of hills (and similarly for the dales) as Maxwell [46] does, because such a hill might itself contain dales, and so on. The inloop curves consist of a loop and a lune; these are nested closed circuits of opposite orientation. Thus, the inloops define either a hill in a dale or a dale in a hill (of course, the remark on Maxwell’s false extrema applies here as well). (See Fig. 2.) If you consider the level curves between two successive critical values (values of the scalar variable at the critical points), they consist of a (number of) simple closed loop(s) and all curves are mutually homotopic; that is, they are smoothly interpolated by adjacent level curves without going through some singularity. Thus, such curvilinear congruences are equivalence classes of qualitatively similar level
FIGURE 2.
The outloop and inloop structures.
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JAN J. KOENDERLNK AND A. J. VAN DOORN
FIGURE3. Left: Bird-eye’s perspective view of a generic landscape. Right: The same landscape conceived as an image.
curves. If we consider a single element, it typically divides the plane into an interior and an exterior part. If the level increases when we progress into the direction of the interior the curve may be called afalse maximum, otherwise it may be called a false minimum. In Fig. 3 (left) we show a landscape in a bird-eye’s perspective view, in Fig. 4 (left) as a topographic map with equally spaced level curves, in Fig. 3 (right) as an image, and in Fig. 4 (right) as subdivided by the singular level curves through the saddle points. In the remainder of the paper we will have frequent occasion to refer to this paradigmatic landscape2 in order to illustrate various properties. In the sequel we always consider single connected components of the generic level curves. Each of these is a false extremum. The singular level curves through the saddles organize the global nesting of these false extrema. Each singular curve defines a branch in the hierarchy. If we use the notation “[ 1’’ for a false maximum and “( )” for a false minimum, the singular curves define structures like (00) or [[][I], which are the outloops, or (()[I) or [()[]I, which are the inloops. In this way we can write down the structure of the entire landscape as a nested bracket expression, any syntactically correct expression corresponding to a specific qualitative structure. This describes the landscape as “A hill and a dale on a hill . . . ,”and an image as “A light blob and a dark blob in a light blob . . . .” Later we will meet alternative ways to analyze the qualitative structure of landscapes. Each of the various methods has its particular utility. The homotopy of level curves is useful in the context of images when thresholding is a meaningful
*
The landscape was constructed as a random (i.e., whimsically thrown together) wavepacket of trigonometric functions: 0.1 s i n ( 5 . l ~- 3 . 9 ~ ) 0.3 sin(6.91 - 0 . 2 ~ ) 0.7 sin(0.k 0.3y)+ s i n ( 2 . 7 ~+ 0 . 6 ~ ) .Though perhaps not as fully generic an example as might have been desirable, it is good enough for our purposes. The advantage is that the function and its derivatives are fast to evaluate numerically.
+
+
+
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FIGURE 4. Left: A topographic map of the landscape with equally spaced level curves. Right: In this map we have drawn the level curves at the heights of the saddles. In the example landscape the territory turns out to be largely covered with hierarchically nested outloops.
operation and in the context of topography when we are interested in the effects of flooding, and so forth.
3. Topographic Curves Topographiccurves are special curves in the case of the affine z-axis; at least, they Examples are the isohypses are roughly defined in such a way by Liebmann [a]. and their orthotomics (the fall curves), the ridges and ruts, and several others. The isohypses (French: lignes de niveau, German: Horizontallinien, Schichtenlinien) are probably historically the oldest (dating from Cruquius [17] in 1729). The fall curves (French: lignes de plus grande pente, German: Fallinien) were apparently introduced by Monge (according to Ch. Dupin, Essai hist.) and thus are probably next in line. The ridges (French: lignes defaite, German: Kammlinien), ruts (French: ligne de rhalweg, German: Talweg, Rinne linien), cliff and plateau curves (vide infra) all date from the nineteenth century and are still subject of research. Apart from these lines which either form curvilinear congruences (level curves and fall curves) or are isolated curves (ridges, ruts, cliff and plateau curves), there also exist topographic curves, which depart in any direction from any given point. Examples of the latter are the geodesics (in the three-dimensional setting) and the curves of equal inclination (German: Biischungsliniep), which are paths along which the rate of ascent or descent is constant. Such curves have important applications in road engineering since vehicles tend to have a maximum climb rate, so the fastest road up a mountain slope is of constant inclination. Some of these curves even exist in the setting of arbitrary monotonic transformations of the z-axis. For instance, the loci of inflections or vertices of either the level curves or the fall curves are of such a nature. Others are only defined if we
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JAN J. KOENDERINK AND A. J. VAN DOORN
FIGURE 5. Sequences of hills like beads on a string symbolize ridges on early maps. Left: Roman map of the third century (Mediterranean). Right: An eighteenth-century French map of Lapland (the region of Maupertuis’s expedition).
limit the admissible transformations to affinities. An example of the latter are the loci of locally steepest ascent along a fall curve. Many of the topographic curves have special significance in the landscape and were drawn on maps long before their mathematical definition was even attempted. For instance, ruts are drawn as water courses and valley paths [the (in)famous “Thalweg”], ridges as concatenated strings of hilltops. (See Fig. 5 . ) Curves of steepest slope along the fall curves appear as clifs when the slope is serious; thus, we may designate them clif curves. Likewise, curves of shallowest slope along the fall curves appear as plateaus when really shallow; thus, we designate them plateau curves (although in this case the curves are rather ill defined: The plateaus are almost like two-dimensional regions if the slope is very shallow). Loci of steepest (or shallowest) slope along the level curves have been implicated as ridges or ruts and so have the loci of vertices of the level curves (but vide infra). Finally, there exist loci that are only defined in the three-dimensional setting. Examples are the ridges in the conventional differential geometric sense [53]. These are loci of extreme principal curvature along the direction of that principle curvature. Such loci cannot really be termed topographic curves in the proper sense at all because they have no reference whatsoever to the height in the landscape, hence, don’t change if you tilt the whole landscape bodily. Although they have been treated as topographic curves in the literature, we will not do so in this paper and we will indeed ignore them altogether. One should take good notice of the different meanings of the term ridge though. There exists no relation between the ridges in the sense of classical differential geometry of surfaces and the ridges as topographic curves at all. A good introduction to the geometrical nature of the former is Portuous [53].
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E The Nature of Observation,Resolution Consider a simple and common observation, taken from Webster’s [MI book on partial differential equations: For example, if we look at a cumulus cloud from a distance, it appears to have a sharp boundary (the density of the water vapor is discontinuous) but as we approach it we find it impossible to say where the boundary is, and it seems to shade off gradually, while again if we examine it with a microscope we find it is composed of minute drops, each of which we suppose to be composed of millions of atoms, each composed of many smaller parts. We have here an extreme example of the possible variety of points of view.
Webster dismisses the point perhaps too easily. Indeed, this is not an extreme example of the possible variety of points of view at all, but rather this is the generic condition in the natural sciences! It is not essentially different for any images, landscapes, or reliefs. It is remarkable how Webster-who like many others clearly perceived the problem-never attempts to frame a solution. He is satisfied to state: “We shall here adopt the hypothesis of general continuity of properties, . . .”. Apparently he is convinced that the remainder of the book will neverthelessbe of use (the book is addressed to the engineer). Despite the deep cleft between the realm of smooth functions and observed scalarjelds, scientists have not been hesitant to apply the methods of differential calculus to actual data. That this actually works fine is to be considered a tribute to common sense. However, a more principled approach is clearly desirable. Consider a single image printed on different devices. If you study the toning of the page microscopically, you meet with great variety. (See Fig. 6.) Some devices actually put gray tones on the paper. Others (like most laser printers today) simulate gray tones by putting dots of fully black tone on the white paper base. Yet you would have reason for complaint if these images didn’t “look similar” if not identical. They really are the same image though quite different on the microscopic level. This vividly shows that scalar fields can only be defined with respect to someMucial level of resolution. Roughly speaking, we have to specify whether we want to see the leaves, the foliage, the treetop, or the forest. This is not something dictated by the laws of nature; it is the observer’sfree choice. Webster is correct in detecting a problem here with respect to the differential structure of the field: How can you diferentiate the density field (of condensed water) in a cloud? Clearly, you cunnor, though Webster dodges the issue. Such is the common method in the sciences and evidently not without success. In this paper we cannot avoid the issue altogether and we will explain how one may successfully differentiate what is evidently not differentiable. Consider how one might actually measure the density in a laser printer image. Since we know that the image is a gray-tone image, it doesn’t help much to
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FIGURE6. The structure of a halftone(!) print in detail. At its actual size this picture would have the dots spaced at about 50 dpi (dots per inch).
put it under a microscope and notice that we see either the white paper base or a black “fly speck” of carbon deposit (about 1/600 of an inch in diameter for the typical office device of today). Evidently the density is only defined at fairly coarse levels. Thus, we assume a level of resolution d (say) and count the fraction of the paper base covered with pigment within a circular disk of diameter d centered at the fiducial location. This is the density at level of resolution d . Notice that there is no sense in which we could speak of the density as the limit of the observed density for arbitrarily small d : The limit clearly makes no sense. We have to recognize that the only real density is the observed density and that it is a function of both the image and the diameter of the aperture. Thus, we obtain a one-parameter family of images (observed density distributions) with resolution as the parameter. Here we meet with the cartographic generalization again: The images at coarse resolution can be viewed as the generalized version of images at fine resolution. We obtain an atlas with maps of various resolution. As in the atlas we expect to find the cause of a feature in a coarse map in the structure of a finer map, but not vice versa. We would appreciate being able to compute coarse maps from finer maps, though the reverse need not be possible. In doing so it should not matter whether we do the computation in one step (from a very fine map to the desired one) or in several steps (hopping to coarser and coarser maps until we
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arrive at the desired one). This means that the operation of blurring should satisfy the semigroup property. The only way this works is if the blurring is done through convolution with Gaussian kernels, or-what amounts to the same-if the images at various resolutions are simultaneous solutions of the diffusion equation
where s is the scale parameter with the dimension of area. (Roughly speaking, s = d 2 in our example.) Thus, the semigroup property that makes the multiresolution observations into a causal atlas forces us to observe via a Gaussian window instead of a mere circular disk. That the Gaussians are the unique choice [39] follows from the fact that they are the only isotropic kernels of limited support (in the sense that they decrease faster than any power of the distance to the origin) that reproduce their shape (except for size) when convolved with each other. Because of the diffusion equation, constraint blurring does not create spurious resolution in the following sense: Blurring decreases the height of summits and increases the height of immits [39]. Indeed, at an extremum the gradient vanishes and the density can locally be approximatedby aquadratic function. The Laplacean AZ is proportional to the sum of the principal curvatures and is negative for maxima, positive for minima. Because of the diffusion equation we thus have Z, < 0 (we use the convention that subscript s denotes partial differentiation with respect to s) for summits and I, > 0 for immits, which proves the foregoing statement. (See Fig. 7.)
FIGURE 7. An example of spurious resolution. Left: The target image. Right: The image convolved with a circular disk (pill box) aperture. Notice the evident spurious resolution: the inner annulus of 180" out-of-phase sectors. Here the contrast in the image is the exact opposite of that in the object!
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In summary, scalar densities can only be defined relative to a fiducial resolution, and the one-parameterfamily of resolutions forms a causal atlas and avoids spurious resolution if and only if the observations take place with a Gaussian weighing function. This structure is well known today and is generally designated as scale space. The laser printer images exist only within a certain range of scales: If the scale is too fine, you have only white paper base or black pigment; if the scale is too coarse, the whole image becomes a mere point. At the right scale we have a halftone image (which is what we see because the laser printer was designed to deliver black dots on white paper at the correct scale). The scale space setting immediately allows us to define differentiation even for scalar fields that are not differentiableat all [40](Webster’s example). First notice that when we write
we apparently imply that the “real” image I ( x , y ) exists at “infinite” resolution. This is clearly nonsense. The point is that we know nothing about the “real” image, except for the observation, that is, I ( x , y ; s ) . Thus, Q.(4) has to be taken cum gram salis. We will take the equation to mean that I ( x , y ; s) is obtained through an observation (not a computation) via the Gaussian aperture Go(x,y ; s). Then the equation has merely a formal, symbolic meaning. When we write it down we are not committed to the belief that there exists a “real” image of which we possess knowledge besides that which we obtain from the observation itself. In particular we have no knowledge whatsoever concerning its smoothness or differentiability. However, we can rest assured that the observed density is smooth by design. For all we know, there might well exist infinitely many “real” images that would yield the identical observation. We shouldn’t care, since it is irrelevant: They can’t be observed! It is entirely a matter of personal taste whether one wants to believe in the existence of nonobservable entities or not. In any case, we needn’t address the matter here. Notice that we must forget the naive notion of an image as a map
at this point. Rather, we define not images but operarors (such as the Gaussian apertures conceived of as pieces of hardware). Then “images” are what the operators operate upon and all we may ever know about them are the results of the operations, which are samples of values of image intensity at the fiducial level of resolution (which is our free choice to which we are committed once we pick the operators).
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The scale space structure defined here proves to be the correct setting to define derivatives of scalar fields too (vide infra).
G. Local Description of Image Structure We have to distinguish various notions of locality. We will refer to properties such as the height in the landscape as punctal because they are defined at a point. (However, even this has to be taken cum grano salis because of what was said in the previous section.) We will refer to properties such as slope or curvature as local because they depend on the differential structure at a point. The same goes for entities such as fall curves, which are defined by differential equations. Here we will limit the order of the differentials. For instance, we will refer to the “firstorder description.” In this description the slope is a local entity, but curvature is not (because it depends on second-order derivatives). There exists an in-between concept, of multilocality. We refer to a property as being multifocul if it is defined in terms of local differential invariants and a connection. For instance, in the firstorder description, curvature is a bilocal property because we can obtain it from a comparison of directions at infinitesimally close locations (here the “connection” allows us to compare the directions at infinitesimally close but different points). We will refer to properties as global whenever they depend on the manifold as a whole; for instance, a closed curve is such a global concept. [A bitangent line that meets a curve at two distinct (not infinitesimally close) points is a global, not a multilocal entity!] The local structure “up to order n” is the structure revealed by the nth-order Taylor approximation. Here we consider all surfaces with the same nth-order Taylor approximation as equivalent; this equivalence class is the nth-order jet of the surface at that point. The local jet space is the space of all local jets and is spanned by the set of partial derivatives up to (and including) order n. Notice that the jets are geometrical objects (i.e., they don’t depend on the specific coordinate system), whereas the partial derivatives obviously are not.
H. Genericity and Structural Stability In the earlier literature on topographic curves (e.g., the early French literature starting at the end of the eighteenth century, the seminal papers of Cayley [15] and Maxwell [46], etc.) the authors had to deal with large numbers of special cases that essentially prevented them from arriving at general results and led them on numerous tangents. For instance, in treating critical points they had to distinguish higher-order passes (or saddle points). Although many of these authors exercise admirable prudence, these problems often spoil the impact of their efforts.
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Several authors were well aware of the nature of the problem3 and, for instance, Cayley manages to handle problems in ways that ignore nongeneric cases. However, formal methods were wanting and the common sense of the natural sciences (so effectively wielded by Cayley) is formally nothing but mere mathematical handwaving. Effective ways to handle these problems have recently become available [62,2, 111. Since they involve much technical detail, we cannot, deal with them in this paper. We will merely introduce the essential notions of genericity and structural stability and invoke them in the sequel in an informal manner. The notion of genericity is closely related to the conventional notion of general position in geometry. Three points in the plane in general position are not collinear, for if they were the slightest perturbation would destroy this condition. Thus, the configuration of three points in general position is structurally stable in the sense that arbitrary small perturbations will not change the configuration. One says that it is generic for three points in the plane to be in general position (not collinear or coincident) because such special configurations are very rare in the sense that if you generate configurations at random, they usually appear with “probability zero.” In order to make the notion precise, one has to formalize the notion of events occurring “with probability zero.” Clearly this depends on the space (three points on a line will always be collinear) and on the admissible perturbations. Because generic configurations are necessarily equivalence classes of configurations (because small perturbations produce equivalent configurationswith probability one), we can often specify canonical models (that are specific members of the equivalence class with especially convenient properties) that summarize the structure in advantageous ways. We already gave an example: For the Morse critical points we can always find a coordinate system (using rotations in the plane and scalings of the axes) such that they are locally of one of the forms [ 111 (a) z = -x2 - y2 a summit, (b) z = +n2 - y 2 a saddle, (c) z = +x2
+ y2
(7)
(9)
an immit.
Such local models are evidently extremely convenient. When local models are known, we can proceed to analyze the essence of the structurethrough conventional means. The problem of nongenericity has bedeviled the calculus. For instance, there exists a famous example due to Peano [24] of a surface violating Schwarz’s theorem [581 on the equality of mixed partial derivatives. For the function z ( x , y ) = x y ( x 2 - y 2 ) / ( x 2 y 2 ) for (x, y ) (0,0).~(0.0) =0 one has a z / a y a x = -1, whereas a z / a x a y = +1 though the relief seems nice and smooth enough. This is one of Maxwell’s “higher-order passes.” The oddity can be removed through infinitesimal changes that leave the relief essentially unchanged to the human eye; it is a nongeneric example.
+
+
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11.
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DIFFERENTIAL STRUCTURE OF IMAGES A. Stable Diferentiation
Taking Eq. (4)in the formal sense, we notice that the operation of convolution is linear and commutes with differentiation. Thus, we have the (purely formal!) identities
a m Y ; s) = a w y ) 8 c o ( x ,y ; s) = z(x, y ) co aGo(x,Y ; s),
(10)
where a denotes any differential operator, for instance, a/ax. These expressions are purely symbolical and we have to interpret them in order to give them operational significance. The three expressions occumng in Eq.(10) have completely different meanings [40]: The first expression makes at least some formal sense, since Z(x, y ; s) is (ideally) an analytic function, regardless of the “real” image. However, in real life the observations will exist as stored data structures (in which case they are probably available in discrete form) and will typically have suffered some random perturbation (noise) with nonsmooth characteristics. Thus, pragmatically one should never differentiate observed images. (This is an in itself valid point that has given rise to the erroneous belief-common in image processing circles-that one should avoid differentiation like the plague.) The second expression is worse than the first in the sense that one cannot give it any reasonable meaning at all. The “derivative of the real image” is only a figment of the mind. The third expression is the only one that can be given a solid operational meaning. Notice that the derivative of the Gaussian is an analytical function. Like the Gaussian itself, it is a kernel of limited support (in the sense indicated earlier). The third expression symbolizes an observation (not a calculation!), this time with the derivative of the Gaussian as the weighing kernel. Notice that the observation entails an integration (rather than a differentiation) and is evidently a robust and stable process. Thus, we will take the “derivative of the scalar field” in the operational sense symbolized by the third expression in Eq. (10). A kernel such as
will be called “the fuzzy derivative operator of order (n,rn) at level of resolution s.” An observation with this operator yields the (n,m)th-order partial derivative of the scalar field at level of resolution s. By construction such fuzzy derivatives
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behave in every respect like the derivatives from classical differential calculus and can be substituted without the least hesitation when partial derivatives are called for. The main point to be noticed here is that one has to indicate thejducial level of resolufion.This entails an essentially arbitrary choice on the part of the observer. According to the choice, the curvature of outline of a treetop (almost Webster's example) will turn out quite differently: You choose whether to "see" the treetop, the foliage, or the leaves. In many circumstances (for instance, when you can't risk to commit yourself), it makes solid sense to study the scalar field at a range of resolution levels simultaneously. Thus, one should never attempt to differentiate images (indeed, how could one?), but instead observe values of the derivative directly via the fuzzy derivative operators at some (freely chosen) fiducial level of resolution. At the conclusion of this section we note a few relations for which the reader will find frequent application: Gnrn(x9 y ; s)
8 Gkl(x, y ; t ) = Gn+k,rn+l(X, y ; s
+t ) ,
(12)
Thus, concatenation of fuzzy derivatives changes the resolution. The final expression is a simple consequence of the diffusion equation of which the fuzzy derivative operators are solutions.
B. The Local Jet In the previous section we outlined a theory of robust differentiation of scalar fields. Starting from this we can proceed to compute the complete set of partial derivatives up to a given order. From that we obtain the nth-order jet at level of resolution s
where we have omitted the constant term. The space of all local jets is the local nth-order jet space J". In the sequel we will have special occasion to refer to the second-orderjet space J 2 . Then we will use the conventional notation
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Thus, [ p , q } is the gradient of the density (at level of resolution s). We will refer to the function o2 = p 2 q2 as the slope squaredfunction and to its positive root as the slope. We will denote the cubic terms as
+
and the quartic terms as
C. The Congruence of Level Curves and the Creep Field The level curves are characterized by the condition
which is-by construction-an exact direrential. The orthotomics to the level curves are defined by the differential equation
Pdy - qdr = 0,
(20)
which is not (in general) an exact differential. (Only for minimal surfaces with Az = 0 is the equation exact, for then we have
This is not a generic case, rather, the condition Az = 0 will only be satisfied on certain curves.) This latter differential equation defines the directions of the gradient, or-equivalently-the directions of locally extremal slope. We refer to it as the creep equation because the debris of erosion or water seeping through the soil may be expected to make its way down slopes by way of the direction of steepest descent. The creep is the vector field of directions of steepest descent; that is, c = -vz = -p e x - q e y .
(21)
(See Fig. 8.) We use the term creep because it suggests a slow process. Indeed, the creep and its integral curves are of interest as geometrical entities and we are not at all interested in any dynamics, for example, how water runs really downhill. In cases of actual water flow (except for seepage through the soil, but then still other physical factors come into play) one has to reckon with inertial effects (and take the Navier-Stokes equations of hydrodynamics into account), something that is quite beyond the scope (and the intention) of this paper.
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FIGURE8. Creep indicated on a topographic map. Notice that the density of the creep is used by the cartographer to suggest the effect of light and shade due to directional illumination.
The integral curves of the creep are thefall curves or slope lines. These figure in the early literature as lignes de fai'te (French) or Fallinien (German). We will use the terms fall curve, slope line, or creep line interchangeably.
D. Differential Invariants of the Second Order
The individual partial derivatives [ p ,q , r, s, f } that together define the secondorderjet J 2 have no intrinsic meaning for themselves because they are defined with respect to an arbitraryCartesian coordinate frame. The second-orderjet as a whole is a true geometrical object though. Since it is obviously desirable to frame all relations in terms of entities which possess intrinsic meaning, whenever possible we look for differential invariants that might replace the partial derivatives. A differentialinvariant is a function of the partial derivatives that remains unchanged in a rotation of the Cartesian frame except for a possible factor that depends only on the transformation. It is easy enough to find such differential invariants, for instance, the slope squared function is clearly one because the gradient Vz = p ex q ey is a geometrical object. (Thus, a2= Vz .Vz = p 2 + q 2 . ) This also holds for the tilt: The tilt is the direction of the creep and can be specified as the unit vector t = Vz/ 11 Vzll ormore conventionally-by the angle t = arctan(Vz f. Vz x f ) , where arctan(., .) yields a result modulo 2n and f is a fiducial reference direction. This means that both slope and tilt are differential invariants and together they specify the gradient or-equivalently-the local planar contact element. These are the first-order
+
-
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THE STRUCTURE OF RELIEF
+
FIGURE 9. Left: The slope squared function a2 = Vz . Vz = p 2 q2 of the example landscape. Right: The tilt function r = arctan(Vz . f, Vz x f ) for the example landscape. The ragged curves (white-black edges) are at the branch cut of the arctangent function in the definition of the tilt: These “edges” are actually continuous transitions.
differential invariants. For the landscape example, the first-order differential invariants are illustrated in Fig. 9. Second-order invariants are the Hessian determinant H = rr - s2 and the Laplacean L = r r . The sign of the Hessian determinant [see Fig. 10 (left)] reveals whether the surface is (intrinsically) elliptically or (intrinsically) hyperbolically curved: If it is hyperbolically curved, the surface intersects the local planar contact element in two intersecting curves (the tangents to these curves are
+
FIGURE 10. Left: The elliptic and hyperbolic areas of the example landscape (sign of H = r t -s2). The black area is hyperbolic and contains the saddles (marked with circles with x’s). The white area is elliptic and contains the summits (open circles) and immits (closed circles). Right: The zero crossings of the example landscape (sign of L = r 1). Notice the zebra pattern with roughly vertical stripes. Thus, the zero crossings are roughly vertical, equally spaced curves for the example landscape. In early image processing the zero crossings used to be identified with “edges.”
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the asymptotic directions or principal tangents of the surface). If it is elliptically curved, the surface lies fully on one side of the contact element. The principal tangents are invariant directions. The sign of the Laplacean decides on which side the surface lies in the elliptical case: If the Laplacean is positive it lies above, otherwise beneath the tangent plane. (Thus, the Laplacean is negative at summits and positive at immits.) The locus of vanishing Hessian determinant is known as the parabolic curve of the surface; the locus of vanishing Laplacean is often known as a zero crossing (especially in image processing and computer vision one often omits the “of the Laplacean”). At the zero crossings the principal tangents are mutually orthogonal. [See Fig. 10 (right).] The condition r d x Sx
+ s(dx 6 y + 6 x d y ) + t d y 6 y = 0,
(22) defines an invariant relation between the directions of the linear elements ( d x , d y } and ( S x , Sy}. One says that these are (mutually) conjugated directions. The principal tangents are self-conjugated. The conjugated directions of the orientations of the level curves and fall curves are evidently invariant directions. The latter are often used in numerical optimization (summit or immit finding) problems (so-called conjugate gradient following). A further second-order invariant of frequent use is the deformation of the gradient. Notice that the gradient of any vector field can be written as a part depending on the divergence (a uniform expansion), a part depending on the curl (a rigid rotation), and a deformation part. In the case of the gradient the divergence is just the Laplacean and the curl vanishes identically; thus, we have already considered these parts. The remaining part, that is, the deformation, is a symmetric, traceless tensor; in the case of the gradient it is simply
+
thus, we have a scalar invariant (r - t)’ 4s2 and an invariant orientation. It is easy enough to find the invariant expressions for the second-order germ of the surface: First, we notice that a rotation of the coordinate axes
(;) (-sincp
(;)
sincp) coscp changes the partial derivatives in the following manner: =
P = pcoscp-qsincp, Q = psincp+qcoscp, R =r~os~cp-2ssincpcoscp+tsin~cp, S = r sinv coscp
T = r sin2cp
+ s(cos2cp - sin2cp) + t sincp cos cp,
+ 2 sin cp cos cp + t cos2cp.
(28) (29)
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We obtain a canonical representation by selectingthe orientation of the coordinate system in such a way that Q vanishes. Then the first frame vector is directed along the direction of steepest ascent and the second along the level curve. This means that we should choose sinq =
d-,-4 P2 + q2
cosq =
d-
P p2
(30)
+ 42’
Thus. we obtain R=
p2r
+ 2pqs + q 2 t , +
P2 q2 ( P 2 - q2)s- p q ( r - t ) S= P 2 q2 q2r - 2pqs p2t T = P2 q2
+
+
9
+
(33)
7
which are the required mixed first- and second-order differential invariants. The reader easily verifies that the roles of (R, S, TI and (r, s, t } may be reversed in these formulas! Notice that H = rt - s2 = RT - S2 and L = r t = R T (and similar for the deformation of the gradient); thus, these invariants are in no way mutually independent. That the mixed-order differential invariants have rather immediate geometrical significance becomes evident when we express the differential geometrical properties of the level curves and fall curves (i.e., their curvature and torsion) in terms of these quantities. Here we have several options. First, we may find the curvature of the projections of the level curves and fall curves on the xy-plane. We express the results in terms of the slope angle q = arctan n instead of the first-order partial derivatives p , q. The slope angle q is-like the slope squared-a first-order differential invariant. We find (the overbar refers to the projection, suffix f denotes “constant f,”thus suffix z refers to the level curves, suffix w to the fall curves, here w ( x , y) is understood as a complete integral of the creep equation, vide infra):
+
K,
K,
= -T cos q~ curvature of level curve, = S cot q
curvature of fall curve.
+
(34) (35)
Thus, the invariants T and S have a rather immediate significance. Because arbitrary monotonic transformations cannot affect the curvature of the projections of the level curves and fall curves, these invariants also keep their significance in a much more general setting.
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Second, we may compute the curvature of the level curves and fall curves as space curves in the three-dimensional setting. We obtain K~ K,
= -T cos cp
curvature of level curve,
= -R c0s3 q curvature of fall curve.
(36) (37)
The space curvature of the level curves is simply identical to the curvature of their projections since the level curves are planar curves parallel to the plane. Notice that-because cp is an acute angle-all trigonometric functions are positive; thus, the sign is displayed in these expressions. Third, we may find the geodesic curvature and torsion of the level curves and fall curves relative to the surface in the three-dimensional setting. We obtain: g , = - T sin cp tan cp geodesic curvature of level curve,
(38)
tz = -s cos2q
geodesic torsion of level curve,
(39)
g , = S cos2cp cot cp
geodesic curvature of fall curve,
(40)
t , = s COS* cp
geodesic torsion of fall curve.
(41)
That the geodesic torsions of the isohypses and fall curves are identical except for the opposite sign might seem remarkable on first blush. However, this merely expresses the fact that these are orthotomic curvilinear congruences. Notice that only 5 , S, and T are invariants of the relief proper (they remain invariantunder arbitrary monotonic deformationsof the height), whereas I,0 2 ,and R are only Euclidean invariants (thus remain invariant under arbitrary isometries of the xy-plane). The difference is crucial in applications. 1. Complete, Irreducible Sets of Differential Invariants
If it is required to take higher-order structure into account, we need to find the higher-order differential invariants. As a start we simply repeat the procedure followed in the case of the secondorder jet. We study the variation of the coefficients of the cubic and quartic under rotation of the coordinate frame. Then we orient the frame along the direction of steepest ascent and that of the level curve. It is indeed geometrically obvious that we thus obtain invariant expressions for the coefficients of the cubic and quartic part of the fourth-order germ. We omit these rather lengthly expressions here; the reader can easily rederive them (preferably using a symbolic algebra package!). We will denote the invariant cubic terms as 1
-(C30X3 3!
+ 3C2,X2Y + C,2XY2 + C03Y3),
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and the invariant quartic terms as
Next we need to find the differential invariants that are the equivalents of the Hessian determinant and the Laplacean in the case of the second-order germ. This can be done via classical techniques of invariant theory. Most importantly we may construct complete, irreducible sets of differential invariants using these methods. In most cases order four will be amply sufficient and it is fairly simple (thanks to modem symbolic algebra packages) to compute the invariants basis. The simplest way to do this is to write an algorithm for the (symbolical)computation of the rthorder transvectant of any two given homogeneous polynomials in two variables for all of the listed invariants [29] can be expressed in terms of the transvectants. Again, we omit an explicit listing of these (rather complicated) expressions here: The reader can find all the necessary tools to derive them in the classical literature (e.g., reference [29]). The classical differential invariants allow us to find such useful entities as the locus of inflections of the congruences of principal tangents (curves of flecnodal points) and the (isolated) points where these congruences undulate (the biflecnodes) [36].
2. The Local Surface Shapes It is a fairly simple matter to obtain a notion of the possible local surface shapes and to relate them to the values of the differentialinvariants. First of all, we distinguish intrinsic and extrinsic surface curvature. The intrinsic curvature is defined through the relation of the surface to its tangent plane. In the case of elliptical curvature we distinguish convex and concave curvature in reference to the upward direction. The signs of H and L suffice to classify surface curvature intrinsically. The extrinsic curvature is defined with respect to the vertical. This is a notion of curvature that tends to be unfamiliar to most scientists. A slope is called convexly curved if the level curve is curved such as to tend to circumnavigatethe higher part, concavily curved if it tends to circumnavigate the lower part. (In Fig. 11 we show an example of two convex hills. Notice that this notion of convexity might be counterintuitive on first blush.) In this case the sign of the mixed invariant T decides on the issue. Notice that we have two quite distinct notions of convexity (or concavity) here. In order to distinguish these we will use elliprical convex in the intrinsic case whereas convex will always refer to the extrinsicnotion (and similarly for concave). Notice that T = 0 implies H = - S 2 5 0; thus, the division between convex and concave regions runs through the hyperbolic areas. An elliptical area is then either fully convex or fully concave. A summitthus defines an elliptically convex area that is fully convex and an immit an elliptically concave area that is fully concave. (See Fig. 12.) The former may aptly be termed hilltops and the latter valley borroms. The
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FIGURE11. Two convex hills (T c 0). Notice that the hill slopes are intrinsically elliptic and hyperbolic.
hilltops are the largest domes (everywhere convex regions-in both senses!-with interior maximum) we can cut out of the landscape and the valley bottoms are the largest bowls (everywhere concave regions-again in both senses!-with interior minimum). What remains of the territory after cutting out the domes and bowls are transitional regions that are hyperbolical, bounded by parabolic curves, which may contain elliptical areas that lie fully on slopes (contain no internal extremum). At an immit or summit the surface is locally like a parabolic dome or depression with different extremal normal curvatures in two mutually orthogonal directions. The level curves near the critical point (indicatrices of Dupin) are thus concentric ellipses [36]. In practice (probability one) all immits and summits will be of this
FIGURE 12. Left: The domes of the landscape (in white). The convex area is tinted gray (except for the dome areas, that is), whereas the concave area is tinted black. The domes are both convex and elliptic. The critical points are also indicated; notice that only some elliptic and convex (white) areas contain a summit. Only these are truly domes. Right: The bowls of the landscape (in white). In this figure the concave area is tinted gray (except for the bowl areas of course), whereas the convex area is tinted black. The bowls are both concave and elliptic. The critical points are also indicated; notice that only some elliptic and concave (white) areas contain an immit. Only these are truly bowls.
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FIGURE 13. The local surface shape near critical points with the corresponding level curves maps. Left: An immit. From an immit all paths go uphill. Right: A saddle. From the saddle some paths go uphill, some downhill. There are four directions (pairwise colinear) that are initially level.
type. Summits or immits that are isotropic are called umbilical. When slightly perturbed the level curves would become egg shaped (with a pointed and a blunt end, that is) with six vertices of alternately maximum and minimum curvature. However, as said, one is unlikely to encounter such isotropic cases [53] (see Fig. 13) although they will (approximately) turn up in numerical work. At a saddle the indicatrix of Dupin is a pair of hyperbolas [36]. The surface intersects the horizontal plane along two distinct directions (the principle tangents); these are the asymptotes of the indicatrix. Along the bisectrices of the principal tangents we find paths that curve upward in one direction and downward in the orthogonal direction. These paths appear as singular features in the creep. (See Fig. 13.)
3. Local Shape of Level Curves The level curves are generically smooth closed curves. The only local features of interest are points of inflection, undulations, and vertices. It is simple enough to characterize these cases. Inflections are points of vanishing curvature whereas the derivative of the curvature doesn’t vanish. The inflection points of the level curves are the intersections of the level curves with the curve T = 0; this immediately defines the geometrical locus of all level curve inflections. At an undulation of the level curves the curve T = 0 is tangent to the level curve. Such points are interesting because they are implied as likely points for springs (an observation due to Jordan [34]) to occur on the slopes. The inflection points of the fall curves are characterized through S = 0. Such loci have been implied as likely course or ridge curves (a definition due to De Saint-Venant [IS]) and we will treat these in more detail later in the text.
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JAN J. KOENDERINK AND A. J. VAN DOORN
FIGURE14. Example of a slope squared extremum. Left: Curves of equal slope squared. There exists an extremum at the center of the figure. Right: The black curve is the parabolic curve which divides the area into elliptic and hyperbolic regions. The curves R = 0 and S = 0 (drawn in graytone) meet transversely, whereas the curve R = 0 osculates the parabolic curve.
Especially interesting points are those where the curves S = 0 and T = 0 intersect: There both the level curve and the fall curve inflect. At such apoint the invariant RT - S 2 must vanish; thus, the point is also on the parabolic curve. One easily shows that these special points are the critical points of the slope squared function. (Fig. 14; the height function is z = x y 2 / 2 x 3 - 1 . 5 -~ 1 ~ . ~5 ~ 0 ~. 6 ~~" ) The example landscape contains several of such points. Another special point occurs when the curve T = 0 intersects the zero crossing of the Laplacean. One has T = 0 and R T = 0, thus R = 0. Thus the normal curvature of the fall curve has an inflection, which means that the cliff curve also goes through the point. At a vertex the curvature of the level curve reaches an extremum. A simple analysis reveals that such points are characterized by a condition of the third order, namely, C12a- 2S2 = 0. These loci have been implicated as candidate ridges and courses. Indeed, it is intuitively clear that they self-intersect at the saddles and at the extrema, though at the summits and immits one would discard the branch of minimum curvature. These properties are indeed very desirable ones for ridges and courses (vide infru). However, we will demonstrate later that these loci fail on other counts and that more likely candidate ridges and courses are available.
+
+
+
+
E. Nature of the Creep Field The fall curves are the orthotomics of the level curves. This immediately reveals much of the structure of the creep field. Inflections of the fall curves occur when the invariant S vanishes.
THE STRUCTURE OF RELEF
95
Though the level curves are sets of concentric loops and very roughly parallel curves, the fall curves run from the summits to the immits and are often seen to approach asymptotically to certain-apparently special-topographic curves. Such curves have classicically been regarded as the formal equivalents of the intuitive notions of ridges and ruts. The problem has been to define the notion formally, and it is perhaps fair to say that there is still no concensus on this issue in the literature. Because of its importance we consider the problem in some detail here. 1. Slant and Tilt A planar contact element at a point (xo, yo, 20) is a triple { p , q , ZO)such that the points ( x , y, z) that satisfy the relation:
( z - 20)
= p ( x - xo)
+ q(y - Yo),
(44)
are on the planar element. Thus, the differential equation p dx
+ q dy = 0,
(45)
at any point defines a bushel of contact elements that intersect along the direction of the “characteristic” which-in this case-is the direction of the isohypses. As noted before we refer to this direction as the tilt, whereas we refer to the inclination of the surface as its slope D . Both tilt and slope are angularvariables, but the slope is always acute and absolute (i.e., referred to the vertical), whereas the tilt can assume any value (we will typically take it on [-n,n])and is relative (i.e., referred to an arbitrary fiducial direction such as the x-axis or indeed any compass direction). These angular parameters are simply related to the gradient, for (here the tilt is referred to the x-direction) T = -arctan
=
(p, q ) ,
d w .
(46)
(47)
In the latter equation the positive root is understood. Often the slope squared function is more convenient than the slope function itself because the sign convention for the root is not needed. Moreover, at an extremum, say z(e) = e2/2 (e the distance from the origin), the slope squared function is e2whereas the slope is 141: Thus, the slope squared function is smooth whereas the slope itself isn’t. We have also occasion to use the slope angle (p = arctan D . Since (p is an acute angle there can be no ambiguity. The equation p dx q dy = 0 thus specifies the tilt field but yields no information concerning the slope field. If we use a conventional water level we observe the tilt directly; thus, such observations specify the landscape via the equation p dx q dy = 0. This was essentially Cruquius’s method in his empirical study of the shape of the riverbed of the Merwede.
+
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JAN J. KOENDERINK AND A. J. V A N DOORN
2. The Eikonal Equation: Relation with Normal Illumination The Eikonal equation is the partial differential equation p2
5)
+ q2 = (g) + (
= d 2 ( x , y)
9
and specifies the slope field (or rather the slope squared function) but yields no information concerning the tilt. The nonlinear first-order partial differential equation is familiar from classical mechanics [28] (Hamilton-Jacobi equation) and geometrical optics [28] (Eikonal equation). The related Hamilton equations are
x’(t) =
H p = p,
y’(t) = H4 = q
Consequently,
In the terminology of geometricaloptics the characteristics of the Eikonal equation are the rays whereas the level curves of the solution are the wavefronts. For instance, when o2= const, we have the solution z(x, y) = Ax By C, with A2 B2 = u2,i.e., planes. The characteristics [35] (the Monge cones of the Eikonal equation) are circular cones with axes parallel to the z-axis. These cones are themselves also solutions. The general solutions are developable surfaces with generators of fixed inclination. We will have occasion to use such surfaces later; thus, we add a few immediate properties. The fall curves of these surfaces are (obviously) the generators. They are tangent to a certain space curve that is the ridge of the surface (for a cone the curve degenerates into its apex). In the projection on the xy-plane the projections of the fall curves are also tangent to the projection of the ridge. (If we view the projections of the fall curves as light rays-in the geometrical optics interpretation-the projection of the ridge is the caustic.) Thus, the projections of the level curves (the wavefronts) must be the evolvents of the caustic. Consequently, the projections of the level curves are parallel curves that have the caustic as common evolute. [See Fig. 15 (left).] For a larger piece of surface the surface is likely to intersect itself. [See Fig. 15 (right) and 26.1 If we agree to keep only the parts up to the intersection, we obtain
+
+ +
97
THE STRUCTURE OF RELIEF
FIGURE15. Left: A curve with parallel curves and normal rays. Notice how the normal rays envelope a caustic curve. Right: The crest line near a vertex of the curve. This can also be regarded as the topographical map of a surface of constant inclination.
a surface of constant slope with a sharp (finite dihedral angle less than n) crest (German: Grat, French: crtte). The crest ends at a discontinuity (sharp bend, or angle) of the direction of the level curve. Nearby level curves have a vertex (curvature extremum). This is probably the reason that loci of level curve vertices have often been identified (erroneously) as candidate ridges. The crests are indeed the primordial ridges, indeed in this case no one disagrees in the least about the existence or location of the ridge (surely it has to be the crest). In the case of smooth surfaces (no crests) the definition of the ridge is much more involved and opinions disagree (vide infra). In many cases we can observe the slope squared function directly. For instance, when the sun is at the zenith the illumination of the landscape is a monotonic function of the slope squared function (namely, a constant times cos @). Then the isophotes of the illumination are curves of constant slope of the surface. We refer to the congruence of constant slope curves as the isoclines of the surface. The equation for the isoclines is apparently (by direct differentiation)
rpdx + s ( q d x
+ p d y ) + r q d y = 0,
(55)
-
an equation that may (more suggestively) be written as Vz . H dr = 0, where H denotes the Hessian matrix, Vz the gradient, and dr = (dr,dy) an infinitesimal displacement along the isocline. The directions of the isoclines are thus the conjugated directions of the gradient [36]. (See Fig. 16.) One immediate observation is that if the isoclines are tangent to the fall curves it will be along a principal tangent direction; thus, this may only happen in hyperbolic areas (only there the principal tangent directions are real). Generically the isoclines will be transverse to the fall curves.
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JAN J. KOENDERINK AND A. J. V A N DOORN
FIGURE 16. The conjugated directions and the indicatrix of Dupin. The fat black oval is the indicatrix. It can be understood as the intersection of the surface with a plane parallel to a tangent plane: In this drawing the surface is thus locally elliptic. The two diameters drawn as thin black lines are mutually conjugated directions. Notice that the tangents to the indicatrix at the endpoints of a diameter are parallel to the conjugated diameter. If one considers the indicatrix as inscribed in the paralellogram that thus arises, it is geometrically evident that arbitrary affinities will conserve the conjugation relations.
The surface described by the tangents to the fall curves at the isoclines are solutions of the Eikonal equation (48) with u2 = consf. We describe these important surfaces in detail later. 111. GLOBALDESCRIFTIONOF THE RELIEF
Although the local structure of scalar fields is often enough of interest, it is typically the global sfrucrurethat is really sought for. Several of the topographic curves are inherently global geometrical entities (i.e., cannot be defined locally).
A. One-DimensionalCase It is of some interest to consider the one-dimensional case first. Here the global structure is easy enough to grasp intuitively. In one dimension the critical points are either summits or immits (there are no saddles). We may immediately divide the landscape into simple slopes, each simple slope being the interval between two adjacent critical points, necessarily of different types (one an immit, the other a summit). (See Fig. 17.) If we are so inclined we can divide the landscape into hills if we concatenate adjacent simple slopes that share a summit, or we can divide the landscape into dales if we concatenate adjacent simple slopes that share an immit. Notice that these two subdivisionsinto natural districts are complementary: Both the hills and
THE STRUCTUREOF RELIEF
99
A
FIGURE 17. One-dimensional landscape. We label the boundary extrema a and o,the summits a-d and the immits p-r. The slopes between successive extrema are natural districts. We can divide
the entire territory into either (six) hills or (five) dales.
the dales exhaust the territory. According to one’s viewpoint, there exist only hills or only dales. It is somewhat more of a challenge to try to find a hierarchical order in the hills (or dales). One indeed often has the notion that a “small hill rides a larger hill.” There appears to be no principled way to do this though, several approaches have been described in the literature. One judges the relative importance of hills (say) on their height, width, and/or area. Various definitions are possible and it is not obvious that any one represents the more rational choice. A common choice is simply height. Using this choice the example of Fig. 17 can be written as a hierarchy of hills (a((ab)(c(dw)))) or as a hierarchy of dales ((p(qr))(st)). An approach that seems most reasonable to us is to study the landscape in a scale space setting. Through progressive blurring we may remove the critical points in pairs: Each time a summit meets an immit and both annihilate this identifies an elementary slope as the subslope of another one. In this manner we obtain a complete partial order of slopes. This gets rid of most of the uncertainties mentioned previously.
B. Cayley and Maxwells “Natural Districts” The so-called “natural districts” described by Cayley [ 151 and in more detail by Maxwell [46] are an immediate generalization of the one-dimensional case for the plane. (For a modern computer implementation,see [25].) Intuitively natural districts are bounded by ridges (mountain ranges) and/or courses (rivers). In folk psychology it is indeed quite natural to regard the people from across the mountains, or the other side of the river as somehow different. The technical term Thalweg (originally of German origin but adopted by the French mathematicians) indeed derives from the definition of the natural boundary between France and Germany as the course (Thalweg)of the river Mine [26].
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JAN J. KOENDERINK AND A. J. V A N DOORN
FIGURE18. Left: A single slope (natural district). All paths descending from the summit end up at the same immit. The boundary paths eventually anive at the immit by way of the saddles; at the saddles these paths make a sharp turn. Notice that almost all paths (i.e., the full one-parameter manifold of paths with the possible exception of a single one) leave the summit in either of two opposite directions and likewise meet the immit in either of two opposite directions. Thus, the region has either a very narrow (zero!) angle at the immit or summit or a very oblique one (180"!). Right: A natural district with perhaps unexpected structure: All paths descending from the summit end up at rhe same immit! In such a case there is only a single saddle involved and we have a maze with a single edge.
Starting from any summit we may try various descents, always taking the direction of steepest descent (thus, we follow the fall curves). Qpically we will end up at an immit, from which there is no more escape (from the immit all directions ascend). However, not all paths need lead us to the same immit. Thus, we can divide the downward paths into equivalence families of paths that will lead us to particular immits. The (singular) paths that divide these bundles of descents will end up at saddle points or (in the topographical language) passes. From such a path we have the choice of two singular paths that lead us to either one of the two immits that are the endpoints of the paths in the two path bundles that were divided by the singular path. A single path bundle thus covers a (curvilinear) polygonal area that is bounded by singular paths. On each such singular path we find a saddle that is a vertex of the boundary of the polygon. The summit and immit are also vertices. Of course the immits may (in general) also be reached via descents from other summits than the fiducial summit considered here. The polygonal area covered by a single bundle of descents may be termed a single, or elementary slope. It is defined by the fact that from any point on the slope the ascent will lead to the same summit and the descent to the same immit. Thus, the single slopes are characterized by the unique pair (though not every possible pair defines a slope) of a summit and an immit. [See Fig. 18 (left).] The area covered by all descents from the summit defines the hill of which the summit is the highest point. The hill is a (curvilinear) polygonal area of which the vertices are the immits whereas there lies a single saddle on each edge. The territory is exhausted by the union of its hills. If we invert the whole procedure we see that it is equally possible to divide the territory into the union of dales. Each dale contains a single immit as its lowest
THE STRUCTURE OF RELIEF
FIGURE 19. The natural districts
101
for the example landscape.
point. The vertices of its curvilinearpolygonal area are summits, and on each edge there lies an unique saddle. Thus, the landscape can be divided either into hills or into dales: These are dual descriptions. Indeed, if we consider the mesh of singular curves through the saddles we can pick out two dual polygonal meshes in the following sense. One mesh has the hills as its faces; thus, we label its faces with the summits, its edges with the saddles, and its vertices with the immits. The other mesh has the dales as its faces; thus, we label its faces with the immits, its edges with the saddles, and its vertices with the summits. These meshes are dual in the sense that the faces of one correspond to the vertices of the other and vice versa, whereas the edges are self-dual. This division into hills (or dales) is straightforward enough in principle, but one has to be wary of such facts as that it may well happen that a face is bounded by only a single edge. [See Fig. 18 (right).] Possibilitieshave been listed by Nackman [501. From Fig. 19 one sees that the polygonal meshes often have either very acute or very obtuse angles. Indeed, though the angles at the saddle points are finite, those at the extrema generically vanish or are 180". This is indeed geometrically evident since an arbitrary fall curve will always reach an extremum along its direction of least (absolute) normal curvature, but we have never encountered a mention of this observation in the literature. In fact, most often the drawings made to illustrate the concept of natural districts (it is rare enough to find such) fail to reflect this fact. (For instance, the otherwise exemplary paper by Longuet-Higgins [45] contains several such drawings. In all cases the angles of the curvilinear polyhedra are finite; the author makes them look like regular polygons for clarity.) It seems likely that this point escaped the attention of earlier authors.
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JAN J. KOENDERINK AND A. J. VAN DOORN
a
e
’
I
‘
I
FIGURE20. The integration of the various differential equations as an exercise in Cauchy’s problem
+
via the method of characteristics. The case illustrated here is that of the relief z ( x . y ) = ( x 2 y 2 ) / 2 , also illustrated in Fig. 21. Left: Cauchy’s problem for the level curves z ( x , y ) . The height has been specified on a fall curve and is then propagated by way of the characteristics of p d x q d y = 0. By specifying various height distributions we can obtain arbitrary surfaces of revolution. Middle: Cauchy’s problem for the level curves w ( x , y). The label for the fall curves has been specified on an isohypse and then propagated by way of the characteristics of - q d x p d y = 0. In this case we specified the azimuth as a label. Notice that the propagation is by lines through the origin; thus, the resulting solution of the creep equation is simply the azimuth and thus has a singularity at the origin. Right: Cauchy’s problem for the logarithm of the integrating divisor log 9 ( x , y ) . The value has been set a constant on an isohypse and then propagated by way of the characteristics of the differential equation p a log 9 / a x q a log 9 / a y = Az. Notice that there is a singularity at the origin (log9 = -w, thus 9 = 0). This is a (degenerated) Rothe special fall curve.
+
+
+
C. Integrals of the Creep Equation
The creep equation describes the essential structure of the relief. All topographic curves are closely related to the creep equation. In this subsection we consider the nature of the integrals of the creep equation. Notice that the differential condition dz = p d x qdy = 0 can be written as the homogeneous linear first-order partial differential equation [we regard (p, 4) as given functions]
+
which (of course) has the level curves as its characteristics [35,31. Arbitrary transformations of the z-axis leave the congruence of level curves-and thus the equation-invariant. If we take a fall curve and assign an arbitrary height function to its arclength, we can use the method of characteristics and obtain a solution for the surface with this boundary condition [35](Cauchy’s problem). (See Fig. 20.)
THE STRUCTURE OF RELIEF
103
The creep equation is defined by the differential condition q dx - p dy = 0. It can be written as the homogeneous linear first-order partial differential equation [we regard ( p , q ) as given functions]
Its characteristics are the the fall curves. When we assign an arbitrary function to the arclength along a level curve, we can use the method of characteristics to obtain an integral surface w(x, y) (Cauchy’s problem again). (See Fig. 20.) Of course, the integral surfaces cannot necessarily be continued over the full domain. From the general theory of differential equations [35, 5 , 12, 591, we know that if p ( x , y) and q ( x , y) are k-times continuously differentiable and p 2 + q 2 > 0: Every two integrals are dependent, that is, a(w1, w~)/a(x,y) = 0. When w(x, y) is a complete integral (i.e., not constant over any area), then all integrals are the functions with continuous first derivatives that are dependent of w . There exists a complete integral that is k-times continuously differentiable. The integral can be built from the characteristics. If an integral is required to pass through a given curve (Cauchy’s problem), a unique solution exists when the curve is everywhere transverse to the characteristics. If there are isolated singular points and if the characteristics in neigborhoods of the singular points are closed curves, then there exists a complete integral with w: w; > 0 except at the singular points [l, 191.
+
The final item applies to the level curves but not to the creep equation, of course. The creep equation is generically not exact; it fails because p x # -qy, that is, zxx zyy = Az # 0. Thus, the Laplacean of the height measures “how nonexact” the creep equation is. The partial differential equation for an integrating divisor is
+
Itscharacteristicsaretheintegralcurvesofthevectorfield( p ( x , y), q ( x , y), dAz), where p ( x , y), q ( x , y) and Az(x, y) are known functions of ( x , y). Thus, the projection of the characteristics on the xy-plane are the projections of the fall curves, whereas the vertical component is the Laplaceanof the height that measures the degree of nonexactness of the creep equation. Notice that the degree of nonexactness Az equals az/as, that is, the derivative of the height function with respect to its scale. Thus, the integrating divisor d ( x , y)
104
JAN J . KOENDERINK AND A. J. VAN DOORN
has an intimate relation to the structure of the scale space of the height function z(x, y ; s). 1. Nature and Existence of Complete Integrals
The creep equation p dy - q dx = 0 is guaranteed (though not likely globally) to have an integrating divisor 6 ( x , y ) such that O ( x , y ) dw(x, y ) = p d y
- 4 dx,
(59)
where w ( x , y ) denotes a complete integral. Neither the integral nor the integrating divisor are unique, of course. If F ( 6 ) denotes an arbitrary function, then O’(X, Y ) = O ( X 3 y ) F ( w ( x ,Y ) )
(60)
denotes another integrating divisor of the creep equation. Notice that solutions of the creep equation must simultaneously be solutions of
m,y ) F ( w ( x ,Y ) ) = 0.
(61)
We may assume the integrating divisor and the general solution to be independent functions; that is,
We will assume this to be the case throughout the paper. When the integrating divisor and the general solution are dependent, the curves 6 ( x , y ) = c’ would also satisfy w ( x ,y ) = c ; that is, the level curves of the integrating divisor would coincide with the fall curves defined by the general solution.
2. Singular Solutions The curves 6 ( x , y ) = 0 satisfy the creep equation. When 6 and w are independent, these curves cannot be members of the complete integral. Such curves are singular fall cuwes; though they satisfy the creep equation, the complete integral doesn’t capture them [55, 301. We will call the fall curves that can be represented as w ( n , y ) = c as regularfall a w e s .
3. Consistent Labeling of Fall Curves The congruence of level curves has a natural parameterization: We simply use the value of z as a label for the level curve. The only ambiguity that might arise is that the level curves often consist of a number of disconnected components. But at least in a sufficiently limited neighborhood the height serves as a convenient way
THE STRUCTURE OF RELIEF
105
to label the level curves. The classical terminology (German: kofierte Projekfion) stresses especially this aspect. The situation is quite different with the fall curves: On the face of it there appears to be no principled way to label them. One might of course single out a fall curve and give it a label (a,say), but there is no way to go from one to the next in “even steps.” Indeed, in the literature the fall curves are usually left without a system of labeling and one is satisfied to simply name a few ones (a,B, y , . . .) that figure in the discussion. There is a solution to this problem if one is able to specify a complete integral of the creep equation: The complete integral of the creep equation itself provides us with a principled way to parameterize the congruence of fall curves. Because the equation w ( x , y ) = c is the implicit equation for a fall curve, the parameter c - o r the value of w-is a label for the fall curve. Thus, w tells us whichfull curve, just as z tells us which level curve. (See Figs. 21 and 23.) Notice that affine transformations of the z-axis merely relabel the fall curves and level curves but do not change them as mutually orthotomic curvilinear congruences. A simple (but nongeneric!) example is
Here the labeling of the fall curves is uniform on all level curves because it has been taken uniform on one and propagated through the characteristicsof the creep equation (Cauchy’s problem). The integrating divisor has an isolated zero at the origin: This corresponds to the degenerated rut (all water will flow into the origin).
FIGURE 21.
An isotropic bowl. Left: The surface. Right: The integral of the creep equation.
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JAN J. KOENDERINK AND A. J. VAN DOORN
(See Fig. 21.) Notice that the fall curve labeling is simply the azimuth, which is clearly the natural choice given the symmetry of the immit. In general we can assign a smooth labeling to the fall curves on any curve that meets all fall curves to be labeled transversely. (For instance, we may use the arclength along some fiducial level curve.) Then we can propagate the labeling along the fall curves to other regions. The problem here is that at some places the labeling is arbitrarily compressed because the fall curves approach each other asymptotically. This happens exactly at the singular solutions of the creep equation, the curves where the integrating divisor vanishes. At such loci the labeling breaks down. They are the singular solutions of the creep equation; that is, the curves 0 = 0. A simple generic example is the parabolic gutter 2
=x
+ y2/2,
(66)
here a consistent labeling is given by the integral
with integrating divisor
(68)
0 =y.
The various surfaces and the flow are very intuitive (Figs. 22 and 23). The differential equation of the integrating divisor is a0
- +y-
ax
a0 = 0; ay
its characteristics are illustrated in Fig. 24.
Y
I
.U
8
hS
I
1.1
FIGURE22. A parabolic gutter. Left: The surface z(x, y). Right: The level curves z ( x , y) = c and the fall curves w ( x , y) = c.
THE STRUCTUREOF RELIEF
107
1
I
FIGURE 23. A parabolic gutter. Left: The integral surface w ( x , y) of the creep equation. Right: The integrating divisor B(x, y ) .
D. Description in Terms of Intrinsic Properties Since we have consistent labelings for the orthotomic curvilinear congruences of level curves and fall curves, we may free ourselves completely from the arbitrary choice of coordinates (of the plane) and express all relations in terms of the labels. Thus, we arrive at an elegant, intrinsic description of the scalar field.
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JAN J. KOENDERINK AND A. J. VAN DOORN
-1 X
FIGURE 24. The field of directions of the characteristics for the integrating factor. Notice that the plane y = 0 is an asymptote and will enforce a zero of the integrating divisor +(x, y ) at y = 0.
1. Basic Differential Relations By far the fastest way to construct the required framework is to use differential forms [12,60]. A basis of differential forms (or covectors) {wx,my} in the plane are the differentials of the coordinates; that is, wx = d x , oy= dy. Thus, we have the basic contractions
a
w x ( e x )= --dx
ax
a
wx(ey)= --dx
aY
a
w y ( e x )= --dy
ax
a
oy(ey)= --dy aY
= 1,
(70)
= 0, = 0,
(72)
= 1.
(73)
We use the Hodge star operator "*" that rotates the first-order forms over 7r/2 and toggles scalars and two forms: *1 = W,
A
my,
*w, = my,
*ay= -w,, *(wx A w , ) = 1.
It is easy enough to translate expressions in terms of the w , d, A, and in terms of classical differential calculus. For convenience we notice the relations that are most generally useful in the present context: The operator *d *d = A, that is, the Laplacean operator. The scalar product of two gradients such as V f Vg
THE STRUCTURE OF RELIEF
109
appears as *(df A *dg), whereas the exteriorproduct of two gradients Vf x Vg = fxgy - fygx appears as *(df A dg). The defining equation for the level curves is simply dz = 0, whereas the creep equation is the dual expression d z = 0. The functions w and S are defined through the defining relation bdw = *dz. The function w is assumed to be a complete integral of the creep equation whereas 6 is an integrating divisor. We assume w and 6 to be independent; that is, *(dw A do) f 0. Starting from this small set of equations we proceed to apply Cartan's theorem (i.e., d2 = 0) to every differential form in sight until there is nothing left to differentiate. The result of this brute force approach is the following list of basic differential relations
(78)
*dz = 6 dw, *dw = -dz/6, dS A dw = (d6/6)
A
d(6 A dz)/S2 = -(dS/6)
*dz = d * dz, A
*dw = d * dw,
*(dz A dw) = a2/S,
(79) (80) (81) (82)
dw A *dz = dz A *dw = 0, *(dz A *dz) = (T 2 , *(dw A *dw) = a 2 / O 2 . There are quite a few remarkable symmetries and analogies in these relations. For instance, Eqs. (78) and (79) reveal an interesting duality between the parameters w and z. Of course, this is more like window dressing though because these equations merely repeat the defining equations for the fall curves (78) and level curves (79). Eqs. (80) and (81) specify the Laplaceans of w and z and are reminiscent of the Cauchy-Riemann equations from the theory of complex functions. Eq. (80) is a partial differential equation that characterizes the integrating divisor 19. Eq. (82) defines the element of area of the xy-plane in terms of the parameters w and z, likewise Eqs. (84) and (85) define the element of arc in the ny-plane in terms of the intrinsic parameterization. Equation (83) is trivial and merely reflects the fact that the level curves and fall curves are orthotomic curvilinear congruences. Since the reader might find it useful, we list here the conventional expressions for these basic results: BAz = VZ ' VS, S A W= -VW * V6,
vz x v w = (T2/6, v z . v w = 0, IIVZ1l2 = u2, IJVW112= 2 / 0 2 .
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JAN J. KOENDERINK AND A. J. VAN DOORN
These equations have immediate geometrical significance. We will study this in more detail in the next section.
E. The Geometrical Meaning of the Diaerential Relations 1. The Line EEement
The metric in the xy-plane in terms of the intrinsic parameterization (z, w) is given by the classical line element ds2=dx2+dy2=dsi+dsl=
dz2
+ O2 dw2 0 2
The line element has to be understood in the sense ds@ds=dx@dx+dy@dy,
(93)
that is, as a symmetric tensor [60]. In &.92 ds, denotes the arclength along the level curves, whereas ds, denotes the arclength along the fall curves. Apparently the fall curves crowd injnitely close together when the integrating divisor 19 vanishes, that is, along the singular fall curves. Clearly this is one intuitively very important characteristicof the ridges and courses and it will indeed prove to be an important one. Apparently the complete integral w(x, y ) is singular when the integrating divisor 0(x, y) vanishes. Indeed, starting from the relation qdx - p d y = 0dw we may write
thus, w(x, y) fails to be an entire function. This will be the case no matter which complete integral andor integrating divisor we accept. 2. The Gauss-Weingarten Equations of Classical Suface Theory
The material in this section applies primarily to the Euclidean 3-dimensional setting, and not to relief as such (in which we disregard arbitrary monotonic transformations of the z-axis). However, the material will find frequent application and does throw some light on the structure of topographic relief, hence, its inclusion here. In the classical theory of curves and surfacesone expresses all quantities in terms of adaptedframes. In this way the differential invariants appear as by magic. In the theory of space curves one uses the Serret-Frenet frame (tangent, normal and binormal),in the theory of surfacesone uses the Gauss-Weingarten frame (tangents along the parameter curves and the surface normal). The trick is to express the
THE STRUCTURE OF RELIEF
111
movements of the frame as one progresses from one point on the manifold to the next in terms of the adapted frame itself. The second fundamental form of the surfaces in three dimensions is [we use the notation r = { x ( z , w), y ( z , w), z), N denotes the surface normal] Il(dz, dw) = L dz2
+ 2M dz dw + N dw2
- Rdz2 - 2BSdzdw - B2T dw2
a24iT7
(95)
(96)
The coefficients of the connection (the Christoffel symbols; “symbols” because the rfj are not the components of a tensor r, although they certainly look that way) are:
r:,
S
= ---j-p’
These quantities occur in the classical theory of surfaces in the ( z , w}-parameterization. The Gauss equations of classical surface theory express the second-order differential of the position in terms of the Gauss-Weingarten frame; that is,
+ + +
+ + +
rzz = rlllrU rtlrU LN, rzw= rf2ru rf2rU MN, rww= r;2r,, rl2rU NN. The Weingarten equations of classical surface theory express the change of the normal in terms of infinitesimal changes in the parameters. In the present case the normal (which characterizes the attitude of the local contact element or tangent plane) is represented by the slant and tilt; thus, we have to look for expressions that relate the changes in slant and tilt to infinitesimal changes of the values of the
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JAN J. KOENDERINK AND A. J. VAN DOORN
intrinsic parameters z and w . These relations are: dz 6dw Udt = S- - T = Sds, - T d s , , U
U
dz ddw dU = R- - S= Rds, - Sds,. U
U
Evidently the differential invariant S measures the tilt change in the direction of the fall curves and the slope change along the level curves, whereas the differential invariant T measures the tilt change along the level curves, and the differential invariant R measures the slope change along the fall curves. An important invariant relation between directions immediately follows from the Weingarten equations concerning the so-called conjugation relations between directions on the surface. These relations are equally valid for directions in the xy-plane. (We have met them before.) Given two directions in natural coordinates ( d z ,d w ) and (St,Sw), we call them (mutually) conjugated directions when
R d z S z - d S ( d z S w + 6 z d w ) - 8 2 T d w 6 w =0,
R ds, SS,
-S
(ds, SS,
+ ds, SS,)
or,
- T ds, SS, = 0.
(108) (109)
The latter form of this relation is nothing but the definition of the invariants R, S, and T , of course. Notice that d s , = 0, ds, # 0 leads to SSs, TSs, = 0; thus, the conjugated direction of the creep is Ss, : As, = -S : T . Likewise, the conjugated direction of the isohypses is Ss, : Ss, = - R : S . A self-conjugate direction is an asymptotic direction. Thus, the creep direction is self-conjugated at R = 0 (i.e., when the fall curves inflect) and the isohypses are self-conjugated when T = 0 (i.e., when the isohypses inflect). Notice that the curvilinear congruence that is conjugated to the creep is invariant against afinities ofthe z-axis. These are the level curves of the slope squared function, which is itself not an invariant. Closely related is the so-called indicatrix of Dupin, which is defined as the curve
+
Rdz2
+2dSdzdw + d2Tdw2 = f e 2 .
(1 10)
The indicatrix is either an ellipse or a pair of hyperbolae. Geometricallythe indicatrix is obtained as the intersection of the surface with planes parallel to the tangent plane but infinitesimally displaced from the tangent plane. Thus, the indicatrices yield an immediate and very vivid impression of local surface shape. Conjugated directions are simply conjugated diameters of the indicatrix; this explains immediately why the relation is preserved in the projection on the xy-plane. The principal tangents (asymptotic directions) are self-conjugated. (See Fig. 25.) When the Hessian vanishes, the indicatrix degenerates into a pair of parallel lines. This indicates that the surface is locally cylindrical. The principal tangents merge into the direction of the generators of the cylinder. It must, clearly be the
THE STRUCTURE OF RELIEF
113
RGURE 25. The field of indicatrices of Dupin for the example landscape. The graylevel indicates whether the surface lies above or below the cutting plane. When the indicatrix is a hyperbola only the part inside a disk about the fiducial point is drawn. The indicatrices have been sampled for points on a Cartesian lattice. Notice the variation in size (thus curvature) and shape (elliptic and hyperbolic).
case that if a ridge crosses the parabolic curve, it runs along the cylinder axis; thus, these geometrical entities are important features of the topography. Notice that most of the material in this section applies only to the Euclidean 3-dimensionalcase, and not to relief proper. Some conceptscarry over to the case of R2 x A though; for instance, the concepts of conjugacy and asymptotic directions. 3. The Mainardi-CodauiEquations of Sur$ace Consistency The relations discussed in this section also apply to the Euclidean 3-dimensional setting rather than relief proper. The classical conditionsof surfaceintegrity are obtained when we apply Cartan’s operator d to the Weingarten equations and use Cartan’s theorem, that is to say d2 = 0. We obtain (setting d2a = 0 and d2r = 0) Sw
6 + BT, = --(2S2 + T 2+ R T ) ,
(111)
Rw
+ 6 S z = - >fJ6( R + T ) S .
(1 12)
U2
To this we may add the differential equation for the fall curves; that is, (setting d2w = 0) 6z =
6L 7’
Eqs. (111) and (1 12) are the so-called Mainardi-Codazzi equations of classical surface theory. They are satisfied by virtue of the fact that the surface is indeed a surface.
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JAN 1. KOENDERINK AND A. J. VAN DOORN
These equalities are often of some utility if we need to simplify expressions for various differential geometric properties. F: The Topographic Curves
At this point we have developed sufficient theory in order to be able to start the study of the so-called topographic curves [MI. We will start with the simplest examples (i.e., review the level curves and fall curves again) and gradually move to topics where the recent literature has not yet managed to reach a concensus. 1. The Congruence of Fall Tangents and Its Second Caustic Surface
+
Consider the collection [in (2 1)-dimensions] of the tangent planes of the topographic surface @ (say) that possess some fiducial slope, slope angle 01 (say). This one-parameterfamily of planes envelopes a developable surface S (say) that might be called the “slope developable of slope angle (Y.” It is a surface of constant slope, and all its fall curves are straight lines. Its level curves are parallel curves, that is, solutions of the Eikonal equation p 2 q2 = const (wide supra). This developable surface 3 has a certain edge ofregression 6 (say), such that the tangent planes of this curve envelope the developable. The curve 6 must also have constant inclination (Y.The developable S touches the surface @ along a certain curve (say). This is the basic geometry that characterizes a curve of constant inclination (q). (See Fig. 26.) When the topographic surface @ is illuminated by the sun in the zenith the curve appears as a curve of constant illumination or isophote. The generators of the
+
FIGURE26. Left: Surface of constant slope and its crest curve. This is the surface generated earlier, see Fig. 15. Right: The same surface with the generators of the developable surface extended in order to show the edge of regression of the surface. All generators are tangent to the edge of regression. Notice that the surface self-intersectsat the crest curve and is seen to form a swallowtail-likestructure (apparently the edge of regression itself may display singularities, in this case a cusp). It is important to distinguish between the edge of regression 6 and the crest x .
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THE STRUCTURE OF RELIEF
developable 3 are the tangents to the fall curves at the points of JJ; let’s call them the fall tangents of v . When we vary the inclination a,we obtain all slope developables that together envelope the topographic surface 0. At the same time the edges of regression of these slope developables (the curves c ) describe a certain surface W (say). All the fall tangents are also tangent to the surface W; thus, they are bitangents of 0 and q. Thus, we conclude that the original surface 0 and the surface of edges of regression of the fall tangents of a certain inclination are the caustics of the congruence of all fall tangents of the topographic surface. We will denote the original surface theJirst caustic surjace and the surface @ the second caustic surjace of the congruence of fall tangents. If we consider the tangents of any given fall curve, these form a developable surface of which the fall curve is the edge of regression. They touch the second caustic surface along a curve y (say) that is the image of the fall curve on the surface W. In the (singular) case that all fall curves of @ are planar curves the second caustic surface is the envelope of a one-parameter family of planes and thus has to be developable. This simply characterizes all topographic surfaces with planar fall curves, an observation that was first made by Wunderlich [66]. In general the second caustic surface will not be a developable surface, of course. If the topographical surface 0 itself is a developable surface, the second caustic surface degenerates into a curve, for the case of planar fall curves even into a straight line. In the generic case both surfaces @ and W will be nondegenerated and nondevelopable. That the second caustic surface will be a developable surface if all the fall curves of the topographical surface @ are planar curves (not necessarily in vertical planes) is a very special case, which is of historical interest because for some time ridges and courses of generic topographical surfaces were believed to be planar [26]. Wunderlich [67] has extensively researched this issue and has also characterized all surfaces for which the fall curves are especially simple (i.e., quadrics). Such surfaces can often serve as convenient illustration, though one should be aware of their nongenericity (Fig. 27). The notion of the second caustic surface W for any topographic surface 0 is indeed a very useful one and we treat it in some detail in the next subsection. 2. Description of a Topographical Surjace in Terms of Its Support Function
If we regard the topographical surface @ as the envelope of its tangent planes, we can write it as x cos u
+ y sin u + 4V = h(u, v),
(1 14)
where the surface parameters (u, u ) have a simple geometrical significance: The parameter u is the tilt ( 5 ) and the parameter v the tangent of the slope angle (tan q).
FIGURE 27. A surface with confocal hyperbolas and ellipses as level curves and fall curves respectively. The parametric representation of this surface is x = cos u cosh v , y = sin u sinh v , z = sin u. Left The surface. Right: The relief and creep.
THE STRUCTUREOF RELIEF
117
The function h ( u , u ) may be called the supportfunction of the surface: It is the distance from the origin to the line of intersection of the tangent plane at the point { x , y, z} and the xy-plane. This is seen immediately when we note that the surface normal n is given by the expression
n = {cosssin6o,sintsincp,cos6p}= { u c o s u , u s i n u , l ] / ~+l u 2 ,
(115)
thus (d is the distance of the tangent plane at { x , y. z] from the origin)
and we have that h ( u , u ) = d / sin p, which is indeed the distance from the origin of the intersection of tangent plane at the point { x , y, z} with the xy-plane. The support function h(u, u ) describes the surface completely, thus, it can take the place of the height function z(x, y). The height function describes the position of points on the surface whereas the support function describes the position of its tangent planes: These are dual descriptions. An explicit parameterization of the surface is given by the expressions x ( u , u ) = (h
+ uh,)cosu
- h,
sinu,
y(u,u) = (h+uh,)sinu+h,cosu, z(u, u ) = -u 2 h".
(117) (118) (119)
This representation is frequently advantageous since it describes the surface directly in terms of the slant and tilt of its surface elements. We essentially treat the surface as the envelope of its tangent planes. Since we have obtained an explicit parametric expression of the surface (in terms of the support function) we can immediately find all of its differential geometric properties via the standard algorithms. Considerthe developablesurfacesof constantinclinationthat touches the surface 0 along curves: These curves have to be the u-parameter curves. Such a surface will have an edge of regression, which is a curve. When we vary the parameter u these edge curves will sweep out a surface Q (say). The generators of this developable surface are tangent to Q, (they are rays of constant inclination since they are common to two infinitesimallyclose tangent planes of constant inclination) but they are also tangent to the edge curves, thus to Q: Hence, they are the common tangents of our original surface 0 and the surface W. When we consider the line congruence of fall curve tangents, then the original surface Q, and the surface Q are the caustic surjaces of this congruence. The surface Q is the second caustic surface defined by the surface 0. It is not difficultto find an explicitrepresentation of the second caustic surface Q: The developable surface of constant slope E: is characterized by u = const. Here the parameter u is the parameter of the curve arctan u. The edge of regression of
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JAN J. KOENDERINK AND A. J. VAN DOORN
the developable surface thus can immediately be obtained from Eqs. (1 17)-( 1 19). When we then vary both the parameter u (position on an edge of regression) and the parameter u (tells which edge of regression) we obtain the explicit parametric representation of the second caustic surface 9. The parametric representation of the second caustic surface 9 of the fall tangents is: x ( u , u ) = -h, sinu y ( u , u ) = h, cosu z(u, u ) = ( h
- h,, cosu,
+ h,, sinu,
(120)
+h u b .
It is a straightforward exercise to express the differentialgeometry of the surfaces or 9 in terms of the parameters { u , u ) , that is, in terms of the slant and tilt of the local surface. Most importantly, for the surface @ the metric is Q,
ds2 = ( h
+ h,u)2du2 + 2h,,(h + h,,) du du + (4h: + h:,) d u 2 ,
(123)
whereas the second fundamental form is
II(du, du) = L dU2
+ 2M du dv + N du2,
( 124)
with L=
M=
N =
+ h,,,) sin u - (h,, + uh) cos u ) JG-7 u(h,, cos u + (2h, + h,,,) sin u ) diT7 -2h, + u(3h,, cos u + h,,, sin u ) u((h,
dG-7
( 125) ( 126)
(127)
We present an example in Fig. 28.
G. Generic Structure 1. Morse Critical Points At a Morse critical point the second-order germ of the height is Z(X,
1 Y ) = - ( K 1 X 2 -k K 2 Y 2 ) , 2
(128)
where K I , ~2 denote the principal curvatures in the three-dimensional setting. The complete integral is w ( x , y) = x-KZyK',
(129)
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119
FIGURE28. The surface with support function h ( u , u ) = exp(u)u2/2 and its second caustic. A few rays of constant inclination have also been drawn.
and the integrating divisor
Possible ridges and ruts are the coordinate axes x = 0, y = 0. In case the point is elliptic ( ~ 1 ~> 20 ) the level curves are concentric ellipses. The fall curves approach the critical pointfrom a single direction only; thus, only one of the directions x = 0, y = 0 is a candidate ridge or rut. (See Fig. 29.)
FIGURE 29. An anisotropic bowl. Left: The immit surface with level curves. Right: The level curves and creep near the immit.
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JAN J. KOENDERINK AND A. J. VAN DOORN
I
U
e
0s
1
FIGURE30. A nonsymmetric saddle. Left: The saddle surface with level curves. Right: The level curves and creep near the saddle.
In case the point is hyperbolic ( K I ~2 < 0) both the level curves and the fall curves are (mutually orthogonal) hyperbolas. (See Fig. 30.) The special fall curves that meet at the saddle are also curves of steepest ascent or descent. Moreover, the other fall curves approach them asymptotically, reason why these were singled out by Cayley, Maxwell, and Jordan as ridges and ruts. However, these curves may fail certain other intuitively reasonable properties of ridges and ruts (vide infra). These curves are generally termed separatrices because they separate the families of hyperbolic arcs. In general, there seems to be no local way to recognize a separatrix (i.e., except from following it from the saddle point). Thus, Jordan indeed denies that the special fall curves are in any way to be distinguished from the regular ones. This seems to contradict common sense though: One has the impression that it is pretty clear where the courses should run even if we have no occasion to go to the mountains and locate their defining saddles. Even more strongly, it seems counterintuitiveto deny that local courses depend only on local relief, for then local changes high up in the distant mountains could influence their course over here. Such influence should intuitively be limited to the water supply, not to the local course. Indeed, Rothe’s criterium 6 = 0 (vide infra) defines the course locally; there is no need to follow separatrices to the saddles. The Morse critical points of the example landscape are illustrated in Fig. 3 1.
2. Parabolic Curves The parabolic curves are intrinsic geometrical entities; that is, if you tilt the entire landscape, they remain well defined. We may assume that in the generic landscape the parabolic curves are smooth, closed curves (they bound elliptic regions) and that they are typically inclined, that is, that they osculate level curves (or, for that matter, fall curves) at isolated points that are distinct from the critical points. The
THE STRUCTURE OF RELIEF
121
FIGURE31. The Morse critical points of the example landscape. The immits are denoted 0 , the summits 0, and the saddles Q.
parabolic curves contain certain higher-order singularities, namely the Gaussian cusps (where the cylinder axis coincides with the tangent to the parabolic curve, O 0; also termed rufles) and the inflection points. The ruffles are defined by C ~ = we may assume that these typically lie on inclined parts of the curve. In neighborhoods of a parabolic point each point has a mate (at the other side of the parabolic curve) with a parallel tangent plane (same tilt and slope). In neighborhoods of a ruffle we find triples of such points. These properties characterize the parabolic points and ruffles [4]. When the parabolic curve osculates a level curve, this will be a critical point of the slope squared function, which can be either a saddle or an extremum. Such isolated points of locally steepest slope in the landscape are obviously remarkable. If the landscape is illuminated by the sun in the zenith, these points appear as critical points of the illuminance. They have been studied in this context [41] and are of interest in the context of computer vision and human visual psychophysics. (See Fig. 14.)
3. Ridges and Ruts The intuitive notion of a ridge is perhaps most clearly illustrated by the example of the ridge of a conventional western roof the horizontalintersectionof two inclined planes that is the highest part overall. Ridge derives from the Anglo Saxon hrycg or hricg, the Scottish rig or rigg. Principal meanings according to Webster’s are “an animal’s back, a range of hills, or the horizontal line found by the meeting of two sloping surfaces; as in the ridge of a roof.” Ridges are sometimes called “divides” since they act as watersheds. If we invert the height the ridge turns into what may be called a “rut” or a “course” since it is the likely locus for a river. In German the ridge is known as Kummweg (ridge path) or Riickenlinie [back (as an
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JAN J. KOENDERINK AND A. J. VAN DOORN
FIGURE32. Intuitive definition of a ridge as the junction of two roof planes.
animal’s back) line], the rut as Rinnelinie (rut line) or Thalweg (or Talweg, valley path). This latter term (Thalweg) was taken over by the French geometers, who refer to the ridge as the ligne de faite (ridge line). (See Fig. 32.) The definition of these topographic curves has been the subject of long and heated debate [44,551. Even today the issue is not generally been considered settled. One issue is whether there exists a local criterion or whether ridges and courses can only be defined on the global landscape. Jordan [34] flatly denied that there could be such a local criterion. He, as Cayley [ 151and Maxwell [46], defined courses and ridges as the fall curves through the saddles and running into summits and immits to define ridges as strings of summits and courses as strings of immits. This closely approximates the intuitive ideas of the cartographers (Fig. 33). Rather early a concensus arose that the fall curves through the saddles are courses. Jordan grants that courses may also be fall curves that start from the highest point of a concave slope though. If you consider an overall convex hill slope which contains a concave intrusion, the level curves are seen to develop an undulation (where they touch the curve T = 0, the condition is Co3 = 0). At one side of the level of the undulation the level curves are convex throughout; at the other side they have a pair of inflections (where they cut the curve T = 0) with a concave segment. Here we can immediately draw horizontal bitangents to the surface; the undulation is the limiting point of these bitangents. Such points have been implicated by Jordan as points where springs issue forth from the slope: The initial points of courses running down the slope (Fig. 34). In Cayley’sdescription a ridge can be followed from a saddle to a summit; then at the summit we can continue in the same direction (the direction of minimal normal curvature) to end up at another (sometimesthe same!) saddle. (See Fig. 35.) There we can again continue (keeping on a straight course), and so on. Thus, the ridge becomes a string of saddles interspersed with summits and a smooth space curve throughout. The same can be done for the courses of course. Perhaps a slight flaw in this contmction is the fact that though this curve indeed runs through the
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THE STRUCTURE OF RELIEF
FIGURE33. “Caterpillars” in early maps: These signify strings of hills connecting to form mountain ridges.
0
4.5
-1 -1.5 -2
-w -3
-15
-1 -05
0
0.5
1
1.5
2
FIGURE34. A convex surface with concave intrusion. Left: The surface. The black area is elliptically curved, the gray (and also the black) area is concave. The white part of the surface is convex. At the top of the concave intrusion we have the point where a spring might be expected. Right: The structure. The parabolic curve and the convex-concave boundary (T = 0) osculate at their common top.
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JAN J. KOENDERINK AND A. J. VAN DOORN
FIGURE 35. A chain of Cayley ridges. It consists of asymptotic fall curves from the saddles. Since all fall curves generically reach the extrema from one of to opposite directions, the ridges (or courses) N n smoothly (i.e., without change in direction) through the summits (or immits). Near an extremum the (extended) ridge osculates the special fall curve through the extremum, but does not coincide with
it.
summits in the direction of the special fall curve (the one all other fall curves asymptotically approach) it fails to coincide with the special fall curve but only osculates it. Thus, the fall curve congruence slightly deviates from Cayley’s (and Jordan’s, Maxwell’s) ridge at the extrema. (See Fig. 36.) Another choice would have been to follow the fall curves that extend the directions of minimum normal curvature at the summits and immits. However, empirical studies indicate (and a simple analysis verifies this) that these curves are numerically badly behaved. Moreover, the fact that they may entirely miss the saddles counts against them. These curves for the example landscape are illustrated in Fig. 37. Clearly they are not likely candidates for course or ridge-hood. If one considers the relation between a saddle and one of its immits there exist two limiting (nongeneric)cases. In one case an asymptote of the saddle meets the immit along its direction of minimum normal curvature (we imply absolute value here); in the other it meets it along the direction of largest normal curvature. But notice that a slight perturbation will always let it meet the immit along the direction
FIGURE 36. Cayley’s ridges and courses in the example landscape. Left: The ridges. The faces are the dales. Right: The courses. The faces are the hills.
THE STRUCTUREOF RELIEF
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FIGURE37. The fall curves passing through the extrema along the direction of the special direction: Clearly the result is less than fortunate when such curves are thought of as candidates for course or ridge-hood!
of least normal curvature because this is the fall curve to which all others (except the direction of maximum normal curvature) asymptotically approach. The other one has probability zero. In the generic case the asymptote of the saddle will not coincide with the special fall curve from the immit. This latter curve does not meet the saddle at all but proceeds to a summit, which it generically meets along its direction of minimum normal curvature. So what is the course? Empirically(!) perhaps the most reasonable (and numerically stable) definition is to ascend at an immit along the fall curve that connects the immit with the saddle and follow it upstream until the surface fails to be elliptically concave. At that point a course can be said to “spring forth” from the slope (basically Jordan’s insight). A similar (fully symmetric) procedure can be applied near a summit (although it is awkward to speak of a “ridge springing forth,” however, if we invert the landscape we see it as an “inverted” spring). In this picture there will be no courses near a saddle at all, which makes formal sense because there is no true (i.e., only asymptotic) confluence of the creep (vide infra) and also accounts for one’s experience in the actual landscape: Near a saddle one has marshy conditions at most, but no streams. This definition yields very convincing results, but we grant that there is a certain amount of arbitrariness (Fig. 38). However, all definitions we know from the literature suffer from definitely worse defects. That this is not necessarily the case is clear from the example of the (Jordan-type) spring on a slope example. No matter how exactly one defines the ridges and courses, they must necessarily be of Rothe’s type (a = 0), and we fail to understand Rieger [54]’s arguments to the contrary. Rothe’s definition is indeed the only viable one available (since other local definitions like De Saint-Venant’s and others violate Boussinesq’s condition) if the region of interest doesn’t contain (the relevant) critical points.
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JAN J. KOENDERINK AND A. J. VAN DOORN
FIGURE 38. Ridges and courses in the example landscape. The parts of the fall curves connecting the summits and immits with the saddles are drawn only in the dome and bowl areas. Notice that the courses in the landscape run into the immits and end there (the immits act as drainholes) instead of running towards the “ocean” (that is to say: the edge of the picture). Real landscapes don’t contain immits except for some rare cases (lakes).
The first one to suggest a purely local criterion was De Saint-Venant [18]. According to De Saint-Venant a course is the locus of extreme slope along the level curves. (“Linge de pente maxima”.) This is certainly an intuitively reasonable idea. (See Fig. 40.) Thus, the condition for a course is simply (of course it works in the same way for ridges)
(g
= 0)
A
(dz = 0).
This immediately leads to the condition 19s= 0. However, here De Saint-Venant only considers the possibility S = 0 and thus arrives at a differential equation for the courses (named after him):
--(--$) az az
a2z
ax ay
ax2
-
((g)2-($)2)-=0. a2z (132) axay
The alternativepossibility 19 = 0 doesn’t figure at all in De Saint-Venant’s account. According to a novel definition due to Eberly [22,23] ridges should be regarded as Ridges are loci of maximum height along the direction of maximum value of the second order directional derivative. Although this certainly sounds like a novel idea, it nevertheless turns out to lead to De Saint-Venant’s equation again. Thus, perhaps the definition can be said to throw some new light on the meaning of De Saint-Venant’s equation. The De Saint-Venant’s curves for the example landscape are shown in Fig. 39.
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RCURE 39. De Saint-Venant’s curves (the curves S = 0) of the example landscape. Compare these with the Cayley ridges and courses (Fig. 36) and the ridges and courses in our definition (Fig. 38). Notice that there is a superficialsimilaritybut that these curves are differentin most details. Notice also how at many points the De Saint-Venant ridges are tangentto the isohypses,thus violating Boussinesq’s condition that water should run downhill!
A problem with De Saint-Venant’s equationis that the condition S = 0 is highly nongeneric because it is highly unlikely that the condition will coincide with the independent condition dw = 0, that is, that the course is also a special fall curve. This latter condition is equally intuitive and is due to Boussinesq [8,9]: Bluntly stated it means that water tends to run downhill (see Fig. 40). The two conditions (S= dw = 0) taken together imply that d t = 0. This is Breton de Champ’s [lo] memorable observation: The implication is that the courses are planar curves running in vertical planes! De Saint-Venant’s locus only coincides with a fall curve if the surface happens to be a member of a certain class of surfacesthat satisfy a certain third-order differential equation (easy to derive from the representation in terms of the support function) see [66]. Evidently most surfaces are not of this
FIGURE40. Left: De Saint-Venant’s principle: The fall curve a that issues forth from the point P on the isohypse i is steeper than any of the fall curves in its immediate neighborhood. Thus, all the water will tend to collect in a course along this ligne a2 pente murim as Lk Saint-Venant calls it. Right: Boussinesq’s principle: When a is a fall curve, then we don’t expect water to run along the direction p. (Although intuitively obvious most ridge and course finders in common use today don’t recognize this simple observation!)
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JAN J. KOENDERINK AND A. J. VAN DOORN
FIGURE41. A helicoidal gutter surface. Left Piece of the surface. Right: Relief map of a much larger part of this surface. In practice we study only part of this landscape in order to avoid the origin or branch cuts, like the part depicted on the left.
type. This clearly poses a problem: For instance, it forbids rivers to do what they usually do, which is to meander. As a historical curiosity, some topographers actually held the opinion that this is how water runs downhill! These geographers applied this nongeneric solution to geomorphology and drew some weird (rather counterintuitive)conclusions from it (e.g., [26])! That the De Saint-Venant ridges are located at intuitively absurd places can be illustrated rather easily. In Fig. 41 we show a helicoydal gutter surface. Here the De Saint-Venant course is curved, but it is evidently located at the wrong place: This is immediately seen when we numerically integrate the creep equation (Fig. 42). Rieger [54]objects against examples like this that “the heightjumps along
FIGURE 42. Numerically integrated creep field of the helicoYda1 gutter surface. Here the region considered is a hemi-annulus. The circular arc is De Saint-Venant’s course: Notice that the fall curves have inflections on this curve. Evidently the numerically computed fall curves cross this curve and instead asympotically approach a quite different circle, which is the true course. This circle is Rothe’s special fall curve (19 = 0, that is a singular solution of the creep equation). Although some authors still cling to the De Saint-Venant definition of ridges and courses, or at least seriously doubt the validity of Rothe’s explanation of ridges and courses, it is hard to understand why given examples like this.
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the course” (he eventually discards the Rothe-type solution 6 = 0 altogether), but he is evidently mistaken since we may simply define a nice region of interest (e.g., an open disk) such as to avoid this trivial problem. (Another simple “solution” is to solve the problem on the infinitely covered plane, taking the origin as a branch point. The surface then becomes a helicoydal gutter with infinitely many turns.) The possibility ignored (or rather: missed) by De Saint-Venant is 6 = 0. Clearly this is the generic case though! These curves also satisfy the creep equation; thus, they are the singularfall curves. It was Rothe’s [55] seminal insight that De Saint-Venant’s and Boussinesq’s conditions are not at odds at all, but serve to single out the singular fall curves as candidate courses and ridges. In our opinion Rothe’s solution definitively solves the problem and also very nicely shows why De Saint-Venant’s reasoning-which is certainly very reasonable-typically leads to nonsensical results. Here we shall take the singular solutions of the creep equation as Candidate ridges and courses: Some additional requirements have to be fulfilled for them to qualify (vide infra). Notice that the creep equation is exact whenever qy = - p x , that is, whenever L = R T = 0, which again is the case for the so-called minimal surfaces. In such cases we don’t need an integrating divisor; 6 can be taken a constant. Thus, minimal surfaces can have no ruts or ridges in Rothe’s sense. An example is the saddle z(x, y) = ( x 2 - y 2 ) / 2 . However, the generic hyperbolic quadric is
+
1 2
z(x, y > = - ( a x 2
- by2),
(133)
and the complete integral of the creep equation is w ( x , y) = xby-‘, the integrating divisor O(x, y) = xb-’y-’-’. Thus, the symmetric saddle is a very special case. Since it is arbitrarily close to saddles with the x - and y-axes as singular solutions, it makes sense to simply define these as singular solution for the symmetric case. Whether a singular fall curve will be a candidate ridge or course can be very simply decided: It depends solely on concavity or convexity of the surface and can thus be decided purely locally. The sign of the differential invariant T is decisive. For the saddles two of the branches are (candidate)courses and two are (candidate) ridges. In case the zero locus of T transversely meets the singularfall curve, we have the changeover from a candidate course to a candidate ridge. This is in fact Jordan’s observation that a course can start on a slope where the surface turns concave. A singular fall curve can indeed be ridge in some parts and course in another. In very special (nongeneric)cases the singular fall curve may actually be both ridge and course [ 5 5 ] ! Needless to say that one needn’t bother with such cases. (see Fig. 34.)
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H. Transport of Stuff In the real landscape the rivers transport all water raining down on them toward the ocean. If we assume a constant, uniform rainfall, a certain amount of stuff per area element has to be carried along by the creep. After a sufficiently long period an equilibrium state will have been reached in which all streams have reached a given size such that the stuff is transported to the ocean without local accumulations or depletions. In the true landscape there are hardly immits; in general the water will tend to flow into the immits (rather than the ocean) of course. In the landscape the immits will fill and spill over, thus forming lakes with an outlet. We will ignore such effect here and consider immits as drain holes where the inflow simply vanishes. Clearly, conceptually it makes but little difference to the concept whether the flow is into the ocean (infinity) or into the immits. The equilibrium size of the streams and immits is obviously an interesting geometrical entity that depends on the structure of the landscape and lets us judge from the size of the streams how extensive their basin at any point along their course is. This measure might well prove important in image processing where it will yield (global) a measure of the relative importance (river or spring) of the courses. We know of no attempt to compute such data, nor any formal attempts to characterize the information contained in these types of data. It remains an important open problem. A numerical calculation via a simulation is easy enough, of course: One simply “rains” points (from a uniform distribution) on the regions and follows the flow through numerical integration. Density can be monitored by counting how many flow lines traverse each pixel. 1. Confluence
In his article on topographic curves in the Encyklopudie der mathemarischen Wissenschaften mit EinschluJ ihrer Anwendungen Liebmann [44] remarks that “in general the fall curves approach the courses,” the kind of insipid statement that appears strangely out of place in the Encyclopedia of Mathematical Sciences. The observation is quite apt though. Rothe [55] correctly notices that in order to form a river the river bed has to be in a position to collect water from an extended region (the so-called basin of the river). Thus, we need an additional condition of confluence: Somehow fall curves should approach and merge the course (and similarly for ridges, of course). In the topographical parlance one might say that the singular fall curve needs tributaries, which actually merge it at junctions before we will consider to call it a course. The singular fall curves are in a good position here because we have already seen that the fall curves tend to crowd closely together near a singular fall curve. The candidate course is given by B(n, y ) = 0, with a ( w , B ) / a ( x , y ) f 0. The direction of a regular fall curve is given by q / p = --w,/wy. In order for the
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FIGURE 43. Left: A “false course” z = x + y 2 / 2 - x 3 / 3 . Though there exists a candidate course that runs from the saddle to the irnrnit, there fails to be confluence;thus, this is in no way a true course. It can be seen that the water actually runs into the irnrnit from a direction orthogonal to the course. Right: The relief for this case.
regular fall curve to meet the singular curve, one thus has to check the condition O x P y
= wxlwy = - q / P .
( 134)
Because of the condition a(w,O)/a(x, y) f 0 the issue can certainly be decided. The junction can either be at some definite point on the candidate course (e.g., an extremum) or it can be at infinity (e.g., at the asymptote from a saddle). This condition immediately discards many candidate courses as viable (see Fig. 43). (A small perturbation completely changes the qualitative structure; see Fig. 44.)In the case of the saddles thejunction is ar injinity. Thus, there will be no true course near the saddle, and whether the candidate course is an actual one depends on the shape
.I
u
.
u
I
u
-1.5
-I
u
0
0.5
I
IS
FIGURE 44. Left: The previous case being nongeneric we here show the effect of a not too large perturbation z = x + y 2 / 2 - x 3 / 3 + x y / 2 . Now the water suddenly runs into the immit from the orthogonal direction (according to the perturbation from either side)! This really doesn’t change the final conclusion that this candidate course should be discarded. Right: The relief for this case.
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FIGURE 45. Left: Cylindrical gutter with circular isohypses. Right: Isohypses and flow lines for the gutter. The isohypses are circular arcs; thus, the vertex locus is ill defined: Yet there evidently exists a well-defined course.
of the landscape far away from the saddle: In many cases the saddle is essentially irrelevant in determining the course, in contradistinction with Jordan, Cayley, and Maxwell’s course definition. The most reasonable definition of courses seems to be the special fall curves that connect the saddles with the immits, taken to the points where the surface changes from concave to convex. Such courses are thus curvilinear line segments. An analog definition can be framed for the ridges. In cases where there are no saddles (e.g., in image processing with small images, or in our “gutter” examples) the only way to proceed is to consider Rothe’s special fall curves. Other definitions (e.g., De Saint-Venant’s or the loci of level curve vertices) need not even be considered seriously because they (generically) violate Boussinesq’s condition. That the locus of isohypse vertices is not a particularly fortunate ridge definition is clear from such simple obervations as that a gutter with circular isohy ses (every point is a vertex point! See Fig. 45, the height function z =x - 1 - y2) has a well-defined course and that a gutter with elliptical isohypses has a course that fails to coincide with the vertex locus (Fig. 46, height function z = x - 3y/5 - 2d-15).
e
\
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I
\
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\
\
\
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\
\
\
\
-
-
\
FIGURE 46. Left: The isohypses of this gutter are elliptical. The upper line is the vertex locus, the lower line is Rothe’s special fall curve (8= 0). Right: Numerically integrated flow. Clearly the flow converges on a course that does not coincide with the vertex locus, which is indicated by the horizontal line.
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1
15
2
25
3
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133
4
FIGURE 47. Left: An infinitely extended river bed like the FloridaEverglades. The height function is z = x + y2/2 - x 3 / 3 . The flow lines are asymptotically horizontal, but at different location. Each fall curve has its own asymptote (which is parallel to the singular fall curve) thus, it will never have a junction with the course, not even at infinity. There is no confluence. Right: The same flow with a perturbation term x y / 2 . Notice that the perturbation doesn’t really change much. Small (but finite!) perturbations really don’t affect the conclusion that one doesn’t have a well-defined river bed here.
In cases where it remains undecided whether the singular fall curve would be a ridge or a course ( T = 0) additional complications occur. Of course this case is nongeneric since the condition T = 0 typically defines a curve. However, in practice it may happen that the value of T is small over an extended area and that due to experimental uncertainties the sign of T must remain in doubt. In such areas both the level curves and the fall curves must be straight fines, and since they are orthotomic both must be congruences of parallel lines. In such cases there can be no confluence, hence no course, but the river bed assumes a large area like the Everglades in Florida. (See Fig. 47.) This case is, of course, not generic. However, slight perturbations really don’t destroy this state of affairs, thus, one may easily encounter it in practice. An interesting problem concerns thejunction of two courses. It is easy to build simple models for such a case, an example is Z(X,
y ) = x4/4
+yx2 +
EX.
(135)
For E = 0 we have a simple, symmetrical situation. If one computes the flow, it is seen that the course splits into two subcourses, the junction being of the pitchfork variety. Indeed, the configuration of ridges and courses must be like a trident, the junction being at an osculation of the concave/convexboundary with a level curve. If one slightly “tilts the landscape” (by letting E # 0), we have a “flip-flop” where the water either takes one subcourse or the other, the minor course will run dry. This apparently simulates the behavior of actual streams quite well. (See Figs. 48 and 49.)
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RGURE 48. Left: The symmetrical junction relief. Right: The symmetrical junction Row. The ridgekourse configuration has a trident structure.
If one follows a river upstream, one meets with countless junctions of the preceding type. The tributaries grow smaller and smaller. At some distance upstream it becomes rather arbitrary to distinguish between the main stream and the tributaries (one has a fractal behavior). According to the resolution, the branches become arbitrarily small. Eventually they end at points where they spring forth from the slopes. Higher upstream the flow is as seepage through the soil. We end this section with the discussionof the most remarkable generic properties of the loci considered in this paper. We will not prove these relations in exrenso. A simple way to derive all these relations is to write the height as a truncated Taylor expansion (a fourth-order binary polynomial suffices) and then to use a symbolic package like Mathematica to compute all relations of interest (be sure not to display these relations since they become very complicated!). Then study the various specializations at the origin: These are simple expressions and the results follow without any hardship. In the cases treated here order four is sufficient. In
FIGURE 49. Left: The perturbed junction relief. Right: The perturbed junction Row. This is the generic case. All the water takes one of the branches, the other runs dry. The configuration of ridges and courses now has two branches.
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general one will pick an order such that the final result is still general; that is, at all stages in the computation one should carry superfluous terms. By specializing the expressions to the origin, one rids oneself from this fluff automatically and arrives at the general expression.
I. De Saint-Venant’s Curves Though the condition of De Saint-Venant fails as a local criterion for ridges or courses, these curves are of some interest in their own right. When a fall curve intersects such a curve it has an inflection of the horizontal curvature: That is essentially De Saint-Venant’s condition. The curves may intersect the concavekonvexboundary ( T = 0) transversely. At such points the Hessian ( R T - S 2 ) must also vanish; thus, we have an intersection of De Saint-Venant’s curve, the convexkoncaveboundary, and the parabolic curve. Both the level curves and the fall curves inflect. (See Fig. 14.) The curves have transverse self-intersectionsat the Morse critical points. At the extrema one branch is locally steepest, the other locally shallowest. At points where the curve is tangent to the fall curves it changes from locally steepest to locally shallowest. J. CliffCurves
The cliff and plateau curves are characterized through the condition R = 0. Cliffs are often marked on topographic maps since they are evidently important for human traffic. Their properties simply follow from the defining condition. First of all, since H = RT - S2 we have H = -S2 5 0 on these curves: They are apparently restricted to hyperbolical areas. The curves S = 0 are loci of inflection of the fall curves in the horizontal plane. When a cliff curve meets such a curve, it has to be on a parabolic point since H necessarily vanishes. The cliff curves have osculations with the parabolic curve. Near such a point they run completely in the hyperbolic area. The fall curve intersects both the cliff curve and the parabolic curve transversely at such a point (see Figs. 50 and 5 1). The example illustrates the height function z(x, y) = x y2/2 x2y/2 xy2/2 y3/6. Cliff curves change into plateau curves when they are tangent to a fall curve. If you are only interested in cliffs, you may say that the cliff curves end at such points. At a saddle the cliff curves self-intersect transversely. We have four halfbranches of cliff and plateau curves. The cliff curves for the example landscape are illustrated in Fig. 52. Cliff curves often occur in working drawings of sculptors since they vividly reveal the relief as seen from a given vantage point. The cliff curves for a sculptor’s drawing of a male face en projil are illustrated in Fig. 53.
+
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JAN J. KOENDERINK AND A. J. VAN DOORN 1 0.7s
0.S
O U 0
-
.
a= 4.1
4.75 I
a75
as
QZ(
o
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u
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FIGURE50. The level curves and fall curves for a case of osculation of a cliff curve with the parabolic curve (in the center of the figure).
FIGURE51. The case of osculation of a cliff curve with the parabolic curve. Left: The local surface habitus. Right: The cliff curve, fall curve, and parabolic cuwe (the black curve).
FIGURE 52. The cliff curves of the example landscape. Here the area has been tinted according to the sign of the invariant R; the cliff curves are thus the boundaries of the tinted areas.
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FIGURE 53. The cliff curves in a technical sculptor’s drawing. Cliff curves are quite similarly indicated in cartography.
K. The ConvedConcave Boundary The convedconcave boundary runs fully in the hyperbolic area. The level curves inflect on this boundary. These curves are very important because they decide on the ridginess/courseness of the candidate ridges/courses (19= 0 loci). When the convedconcave boundary meets a cliff curve the Laplacean must vanish; thus, these three curves meet transversely at special points. Both the level curve and the fall curve inflect there. When a ridge or course curve meets the convedconcave boundary transversely, the type toggles; that is, ridge becomes course and vice versa. Hence, such points have been implicated (by Jordan) as the likely origins of springs and we may well call them spring points. The convedconcave boundary of the example landscape is illustrated in Fig. 54.
FIGURE 54. The convedconcave boundary of the example landscape. The critical points are also indicated: Notice the relation to the saddles.
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L. The Loci of Vertices of the Isohypses As a short calculation will reveal, the vertices of the level curves are characterized through the (cubic) condition
2s2- C12a = 0.
(136)
These curves pass both through the extrema and through the saddles, and they have a self-intersection at both. It appears evident that the vertex loci run all the way from the extrema to the saddle points, though they can certainly bifurcate and reunite again. The generic pattern has (to the best of our knowledge) not been studied yet.
OF THE LEVELCURVES Iv. CONTOURS: ENVELOPES
In the natural landscape overhangs essentially don’t occur since gravity would soon remove them. Thus, the level curves will simply foliate the plane and have no envelopes. In images we can only have one image intensity per pixel; thus, envelopes likewise don’t occur. In the case of relief envelopes are very common though. The reason is simply that most reliefs are due to the surfaces of objects of limited extent. From any vantage point one sees only the front of the object, that is, only part of its surface. The level curves of the distance have singularities at the locations where the lines of sight are tangent to the object: In such cases the projection of the tangent plane degenerates into a line (Figs. 55 and 56). These curves are evident in the topographic rendering as envelopes of the curvilinear congruence of level curves. At an envelope the level curve is generically tangent to the envelope and one branch (the part of the level curve at one side of the point
FIGURE55. Generation of the contour. The bean-shaped object is projected upon the plane n from the point P.The grazing visual rays form a cone C that meets the plane ll in the curve C. The preimage of C on the object (the curve p, called the rim) is smooth, whereas the curve C has a cusp due to the fact that the rim may be tangent to the visual ray at isolated points.
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FIGURE 56. Generic behavior of level curves at the contour. Left: A generic contour or fold. Middle: A near point on the contour. Right: A far point on the contour.
of tangency) will be visible, whereas the other branch will not be visible due to the fact that it belongs to points on the backside of the object. Since the sign of the normal curvature of the surface at a point of the preimage of the contour is well determined (because the object has to lie at one side of the contour), the sign of the curvature of the contour in the plane of projection is simply the sign of the Gaussian curvature of the surface. Thus, the contour is concave when the surface is (locally hyperbolic) and convex if it is (locally) elliptic (Fig. 57). The envelopes will typically be smooth curves, except for self-intersections(socalled T-junctions) and so-called cusp points. At a cusp the (visible) branch of the envelope ends. Cusps occur when the line of sight runs along the preimage of the envelope on the surface of the object (it can be shown that the line of sight then touches the surface along a principal tangent or asymptotic direction, hence the surface will locally be hyperbolic). In that case the projection of the preimage degenerates into a point (Fig. 58). The fall curves cusp at the envelope, although only one of the branches will be visible. Indeed, when the fall curve meets the envelope the line of sight has to be tangent to the preimage of the fall curve. At isolated points of the contour the level curves may meet the contour in such a way that either only an isolated point, or both branches of the level curve are visible. This happens when the line of
FIGURE 57. Convex and concave contour segments. The surface part marked H is hyperbolic, that marked E is elliptic. The parabolic curve meets the contour at P and Q. Thus, we obtain the concave segments 02, u3, and the convex segments 01and u4.
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FIGURE58. Generic singularities of the contour. P is the typical case: The point is not on a contour. Q denotes the fold: The point is on the contour. R denotes two transversely intersecting folds: The point is a 7'-junction. S is the cusp: The point is the endpoint of a contour.
sight touches the surface along a principal direction, the preimage of the envelope then runs along the other principal direction. At such points the preimage of the contour has a local extremum of depth. These extrema have the character of a depth maximum or a depth saddle. (See Fig. 59.) These special points complement the Morse critical points in the interior of the contour.
V. DISCRETEREPRESENTATION In applications we rarely meet with scalar fields as such: We typically have access to observations of samples at a (possibly very large but certainly finite) number of sample points. This introduces a number of important and difficult problems. d
d
FIGURE59. Near and far points of the contour. Here o denotes the contour and (p the fall curve through the singular point. Notice that point P is like a saddle, whereas point Q is like a depth maximum.
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A. Triangulations
In order to deal with surface properties we certainly need more than just a collection of sample points. Some data structure is needed that allows us to glue these completely distinct items together. The way to handle this is to introduce a triangulation of which the sample points are the vertices. Since we have samples of the values on the vertices we then have a polyhedral approximation of the surface. In order to define the triangulation we have to define its edges andfaces. The edges will be ordered pairs of distinct vertices such that any pair of vertices defines at most a single edge. Not all possible pairs will be used to define edges because we will certainly require that no two edges will intersect each other. We need a sufficient number of edges to ensure that they divide the territory into a number of triangular faces. Aface is an ordered number of vertices such that when (vl , u2, ug) is a facet, then (u1, 1121, (u2, ug), and (2)3,01) are edges, possibly in the wrong order (e.g., (u2, vl] is the edge ( u ~u2} , in reverse order). We require that the faces tesselate a simply connected area. This still leaves an enormous amount of ambiguity in the choice of triangulation. We describe some common choices next. The required data structures are then first of all a list of sampled values ( z l , . . . , z ~ )where , V denotes the number of sample points. The order in this list (index {vl , . . . , u v ) = ( 1, . . . , V}) is used as a label for the vertex. Then we need a listofedges((ui,u j ) ,...1, wheretheorderinthelist((e1, ...,eE) = (1, ...,E)) is used as a label for the edge. Finally, we need a list of faces ( ( u i , vj, u k } , . . .), where the order in the list ((f~, . . . , f ~ =) (1, . . . , F } ) is used as a label for the face. This type of data structure is the most basic one and turns out to be very convenient in most algorithms when implemented in mathematics packages such as Mathematica or Maple. For many algorithms one desires more explicit relations between the vertices, edges, and faces (although in principle everything can be found from the elementary data structures). It usually pays off to precompute a number of additional data structures, depending on the task. These data structures mainly serve to be able to find neighbors or boundary elements and to proceed from one item to the next. Examples are the edges that form the boundaries of a face, the neighbor of a face for a given edge, the ordered list of edges that meet a given vertex, and so on. 1. Regular Lattices
Regular lattices are either Cartesian grids with added diagonals or lattices with hexagonal symmetry. In practice the differences tend to be slight, either in quality of the results or the complexity of the algorithms. This is the case because the Cartesian lattices with additional diagonals are simply hexagonal lattices, at least, if all diagonals are drawn in the same direction. The hexagonal lattices have the
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FIGURE60. Left: Cartesian lattice. The edges list is ((1.2). (2,3), (4,5), (5.6). (7.8). (8.9). (1,4), (4,7), (2.5). (5,8), (3.6). (6.9). (I.5), (2.61, (4,8), (5,911 and the faces list is ((1,2,5),(1,5,4), (2,3,6), (2,6,5),(4.53). (43.7). (5,6,9), (5,9,8)). Right: Hexagonal lattice. The edges list is ((1.2). (1,3), (l,4), (1,5). (1,6). (1,7). (2,3), (3,4), (4.5). (5.6). (6.7). ((7,211, and the faces list is ((1.2.3). (1,3,4), (1,4,5), (lS.6). (1.6.7). (1.72)).
theoretical advantage because they are more nearly isotropic; the Cartesian lattices are typically more convenient to handle. The choice is essentially arbitrary. (See Fig. 60.) 2. Delaunay Triangulations
When the sample points are irregular (or actually random) the Delaunay triangulation is the optimal choice. This type of triangulation leads to faces that are as closely equilateral as can be. This is very important because very “thin” triangles lead to numerically unstable behavior. Nowadays computational geometry has provided us with optimal algorithms for actually computing Delaunay triangulations and-although costly-they typically are a viable option. (See Fig. 61.) Mathematics packages such as Mathematica or Maple readily perform the task.
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3. TriangulationEditing In most cases of interest the triangulation will be a very large data structure and it will be next to impossible to make changes manually without assistencefrom some smart program. A triangulation editor lets one graphically pick vertices, edges, or faces and allows one to, for example, delete an element (the editor should automatically “mend” the hole and update the data structure), add an element (the editor should automatically “splice it in” and update the data structure), barycentrically subdivide a face (the editor should automatically subdivide neighboring faces so as to add up with a valid triangulation and update the data structure), and so on. It should be possible to indicate an element and retrieve various properties of it, or to obtain pertinent data concerning the triangulation as a whole.
B. Isohypses and Slope Field The level curves and fall curves on the triangulated surface degenerate into piecewise linear polygonal arcs. This is because we have only toconsider planar surfaces: Any point on the triangulation is generically in a face, and thus part of a unique planar surface. The level curves and fall curves for planes are simply straight lines which are easy to compute. In fact, most math packages such as Mathematica or Maple support contour maps, which simply plot the level curves for a Cartesian-basedtriangulation. (See Fig. 62.) The fall curves are more problematic, not because they are particularly
FIGURE62. Level curves in the discrete case. Left: The triangulation. Right: The contour curves. Over the interior of the faces the level curves are straight-line segments.
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difficult to compute, but because there is no principled way to space them. In order to do that one would need a complete integral of the creep equation. In practice one has to be satisfied with an essentially arbitrary selection of drawn fall curves. One useful method is to pick a fiducial level curve. Then the fall curves may be launched from points of the fiducial level curve spaced at equal arclength intervals.
C. Morse Critical Points To find Morse extrema (summits and immits) on the triangulated surface is an easy task. A vertex is a summit if all neighboring vertices are below it. Thus, one simply has to find all neighbors of a vertex and check them. The neighbor vertices are the vertices that are part of edges that also contain the fiducial vertex. For the immits the procedure is similar. In order to find all summits and immits we simply visit all vertices and check them for the required properties. The only problem that might occur is that some vertices might turn out to be at the same height. This is actually a rather likely event if the samples have been discretized as is most often the case in image processing. The simplest way to avoid the problem is toforce genericity. One sorts the samples and finds the equal samples. Then one perturbs the heights by small amounts (less than a discretization step) in order to make all heights distinct. This introduces some arbitrariness, but this is irrelevant as the perturbations are in the noise level anyway. (In practice one simply replaces the heights with the rank order, where the order is assigned arbitrarily in the case of ties.) More principled methods tend to be (much) more complicated because a great number of possible exceptions has to be handled and the chunks of data that appear as single entities can become arbitrarily large (when all values turn out to be equal). Such methods are often more of a pain in the neck than a real boon. It is much more difficult to find the saddles. In order to find saddles we need to order the neighbors of a vertex something that is quite unnecessary if we look only for extrema. When we visit all neighbors in order and notice whether the sampled values are higher or lower than the fiducial values we end up with the following possibilities (of course one should mind the fact that the order is periodic): All values are lower: Then the fiducial vertex is a summit. All values are higher: Then the fiducial value is an immit. There is a run of lower and a run of higher values: Then the fiducial value is a generic slope point. In all other cases the fiducial vertex is a saddle. There are severalcomplicationsthat might occur here. First of all, we have assumed that the vertex is not on the boundary of the triangulation. These boundary vertices have to be handled separately, we will not go into that here. Second, the saddle may be either of the Morse type in which case we should have an alternation of runs
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of lower, higher, lower and again higher neighbor vertices, or it may be a “higherorder” saddle. For instance, for the (most common) vertex with six neighbors we may find the sequence - which indicates a “monkey saddle.” There are several ways to handle this complication. Since we can find the maximum number of neighbors for any given vertex we can figure out beforehand what types of higher-order saddles are at all possible and simply include these in our classification. Another method is to introduce dummy vertices and edges and split the higher-order saddle into generic Morse ones. The first possibility is typically better if one uses a Cartesian-based triangulation, since there exist no reasonable ways to handle the messed-up lattice (which counts as a disadvantage). The latter method is to be preferred if it is not prohibitive to complicate the triangulation slightly because then we have the advantage of Morse critical points throughout without (awkward) exceptions.
+ + +-,
D. Ridges and Ruts
Finding ridges and ruts on the triangulated surface turns out to be a simple task. In fact, one might say that ridges and ruts are more easily defined on triangulated surfaces than on smooth ones. At least everyone seems to agree on the definition in the discrete case whereas there is as yet no concensus on the definition in the continuous case! Ridges and ruts will definitely have to be sought among edge progressions because there can be no such entities on the faces since these are planar. The surface at an edge is simply the butterfly structure of the two faces that meet at the edge (so-called winged edge representation). When we consider the height gradient on both faces we meet with the following possibilities (Fig. 63): There are paths of steepest descent that run from one face, over the edge, onto the other face. If this is the case the edge is certainly not part of a rut or ridge but part of a generic slope region.
FIGURE 63. The creep field at discrete ridges and ruts. Left: A regular edge: The flow simply runs over the edge from one face to another. Middle: An edge that is a course: From both adjacent faces the flow runs into the edge. The flow can only continue its descent via the edge, which acts as a true course. Right: An edge that is a ridge: In both adjacent faces the flow runs from the edge into the face. The flow can only start its descent at the edge which acts as a true watershed.
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FIGURE 64. Morse critical points, ridges and ruts in the discrete case.
There exist descending paths on each of the two faces that meet at a point of the edge. Then the edge must be a rut since the creep will run from the faces to the edge and continue by way of the edge. There exist ascending paths on each of the two faces that meet at a point of the edge. Then the edge must be a ridge since the edge will act as a watershed and the creep will run from the edge into each of the faces. We can simply check these conditions and do an exhaustive search for ridge and rut elements. Afterward we can concatenate the individually found elements into edge progressions of maximum length. Slight complications arise here because bifurcations do occur and ruts and ridges may meet, but in principle the procedure is indeed straightforward. (See Fig. 64.)We have used such methods routinely for triangulations of up to about a thousand faces. For larger triangulations one should search for optimum algorithms. As far as we know such work is not available at present though. In order to find the ridges and ruts as indicated one really needs the heights at the vertices and the layout in the xy-plane, the mere height order and triangulation topology doesn’t suffice (as it does for the computation of the Morse critical points). To see what the problem is consider the situation depicted schematicallyin Fig. 63. Suppose that B is above A , and Q above P. Then if Q is above both A and B and P below both A and B we certainly have a regular edge. If the depth order of the points is APQB we cannot decide, for consider any situation (regular, ridge,
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or course), then you can always place points P, Q on the two faces in the given order, thus-conversely-the order is consisent with any of these interpretations.
VI. CONCLUSION We have extensively discussed the theory of topographic maps (structure of scalar fields in two dimensions). This is what is availabletoday. It will be evident from our exposition that there are still quite a few “white areas on the map.” Indeed, though the subject has a venerable history (serious mathematics started [20] at about the beginning of the nineteenth century with much activity continuing to the present day) it cannot be regarded as a closed subject. Such important and intriguing topics as the proper definition of ridges and courses still remains a matter of hot debate. A major lack of knowledge is in the area of generic properties of the topographic features. Most of the early work focused on nongeneric, specific examples, which where often generalized wrongly. Since many of the interesting properties are of a rather high order it will be a nontrivial task to fill in the required understanding. If we-as is almost mandatory-study the topography with scale (or resolution) as a parameter the task becomes even more daunting. However, even on the level of fairly simplistic descriptions there are still surprises in store; in this paper we have several illustrations of generic facts that have apparently completely escaped earlier authors. An almost completely novel area is that of the discrete representation. Certainly real observations were used in the past (most of the developments were done with very practical issues in mind), but one typically used techniques of descriptive geometry [49, 561, which are quite alien to modem computer analysis. With the advent of modem symbolic algebra and novel geometrical algorithms,possibilities have been created that were completely out of reach even a decade ago. In the area of the structure of scalar field there are still extensive opportunities for progress in these fields.
REFERENCES [I] Alden, H. H. (1934). Solution o f f ( x , y ) g + g(x, y)% = 0 in a Neighborhood o f a Singulur Point, Am. J. Math. LVI, pp. 593-612. [2] Arnold, V. I. (1984). Catastrophe Theory, Springer, Berlin. [3] Arnold, V. I. (1987). Ordinary Differential Equurions, transl. R. A. Silverman, MIT Press, Cambridge, MA. [4] Banchof, T., Gaffney, T., and McCrory, C. (1982). Cusps of Gauss Muppings, Pitman, Boston, MA. [5] Birkhoff, G. and Rota, G.-C. (1962). Ordinary Differenrial Equations, Blaisdell Publishing Company, Waltham, MA.
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[6] Brown, L. A. (1949). The Story of Maps, Dover Publications, Inc., New York. [7] Boscovich, R. J. (1966). A Theory of Natural Philosophy, The MIT Press, Cambridge, MA, first edition Vienna 1758. [8] Boussinesq, J. ( 187 1). Sur unepmpriiti remarquable despoints oli les lignes deplus grandepente d’une surface ont leur plans osculateurs verricaux, et sur la diffirence qui existe giniralement, a la surface de terre, entre les lignes de faite et de thalweg et celles le long desquelles la pente du sol est un minimum, C.R. 73, pp. 1368. [9] Boussinesq, J. (1887). Cours d’analyse infinitisimule I , Vol. 2, pp. 229-243, Paris. [ 101 Breton de Champ, M. (1877). Mdmoire sur les lignes de faite et de thalweg que 1 ’onest conduit a considirer en topographie, J. de Math. pure et appl. 30, pp. 99-1 14. [I 11 Bruce, J. W. B. and Giblin, P. J. (1984). Curves and Singulariries, Cambridge University Press, Cambridge. [I21 Cartan, H. (1983). Differential Calculus, Hermann, Paris. [I31 Cartan, H. (1983). Direrential Forms, Hermann, Paris. [ 141 Greenwood, D. (1944). Mapping, The University of Chicago Press, Chicago and London. [ 151 Cayley, A. (1970). On contour andslope lines, The London, Edinburgh and Dublin Philosophical Magazine and J. of Science, Vol. 18, No. 120, pp. 264-268. [I61 Cellini, B. (1967). Technical annex to the “Vita”, in: The treatises of Benvenuto Cellini on goldsmithing and sculpture, transl. C. R. Ashbee, Dover, New York. [17] Cruquius, N. S. (1729). Published 1733. See: J. L. LiEka, Zur Geschichre der Horizontalhien oder Isohypsen, Z. f. Vermessungswesen 9, Stuttgart, 1880. [ 181 De Saint-Venant. (1 852).Surfacesriplusgrandepenteconstituiessurdes lignescourbes,Bulletin de la sc.philomath. de Paris, March 6. [ 191 Digel, E. (1936). uberdie Existenz vonlntegralen derpartiellen Diferenrialgleichung f (x, y ) g(x, y ) $ in der Umgebung eines singularen Punktes, Math. Z. 42, pp. 231-237. [20] Dupuis-Torcy and Brissot, B. (1808). Sur l’arrdepmjeter les canaux de navigation, J. k. Polyt., t. VII, 14, pp. 262-288. [21] Eckhart, L. (1922). uber Flachen vierter Onlnung, deren Fallinien Kegelschnitte sind, Sitz. ber. Akad. Wiss. Wien, 131, pp. 417427. [22] Eberly, D., Gardner, R., Morse, B., Pizer, S., and Scharlach, C. (1993). Ridgesforlmage Analysis, Manuscript UNC Dept. Computer Science, Februari. [23] Eberly, D. (1996). Ridges in Image and Data Analysis, Kluwer, Dordrecht. [24] Genocchi, A. (1899). Calcolo diferenziale pubblicato con aggiunte dal G.feano, Torino, 1884. German Translation: A. Genocchi, Differentialrechnung und Grundziige der Integralrechnung. herausgegeben von G. Peano, Teubner, Leipzig. [25] Griffinn, L. D. (1992). Scale and segmentation of grey-scale images using maximum gradient paths, Image and Vision Computing 10, pp. 389402. [26] Giinter. S. (1 902). uber gewisse hydmlogisch-topographische Gnmdbegriffe, Sitzungsber. d. math.-phys. Klasse der kgl. bayerischen Akademie der Wissenschaften, Vol. 32, pp. 17-38. [27] Gurevich, G. B. (1964). Foundations of the theory of algebraic invariants, Transl. J. R. M. Radok and A. J. M. Spencer, P. Noordhoff Ltd, Groningen. [28] Frank, Ph. and von Mises, R. ( 196 1). Die Differential- undlntegralgleichungen der Mechanik und fhysik, 11. Physikalischer Teil, Friedrich Vieweg and Son, Braunschweig, and Dover Publication, New York. [29] Grace, J. H. and Young, A. (1903). The Algebra of Invariants. Chelsea Publ. Company, New York. [30] Hamburger, M. (1893). uber die singularen L2isungen der gewohnlichen Direrentialgleichungen erster Ordnung, J. f. Math. 112, pp. 205-246. [31] Hartman. Ph. (1964). Ordinary Differential Equations, Baltimore, 1973, original edition John Wiley and Sons, Inc.
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[32] Hildebrand, A. ( I 893). Das Problem der Form in der bildenden Kunst, Strassburg. [33] Horn, B. K. P. and Brooks, M. J. (1989). Shape from shading, The MIT Press, Cambridge, MA. [34] Jordan, C. ( 1872).Sur les lignes defaite etde Thalweg, Paris, CR 74, pp. 1459-1475; 75, pp. 625627, pp. 1023-1025. [35] Kamke, E. ( 1965).Differentialgleichungen, Usungsmethoden und Liisungen. II: Parfielie Differentialgleichungen erster Ordnungfur eine gesuchre Funktion, Akademische Verlagsgesellschaft Geest und Portig K.-G., Leipzig. [36] Koenderink, J. J. (1991). Solid shape, The MIT Press, Cambridge, MA. [37] Koenderink, J. J. and van Doom, A. J. (1993). Local features of smooth shapes: Ridges and courses, in: Geometrical methods in computer vision II, ed. B. C. Vemuri, SPIE Vol. 2031. [38] Koenderink,J. J. and van Doom, A. J. (1994). Two-Plus-One-DimensionalDi$erentialGeometry, Pattern Recognition Letters 15, pp. 439-443. [39] Koenderink, J. J. and van Doom, A. J . (1984). The Structure of Images, Biol.Cybern. 50, pp. 363-370. [40] Koenderink, J. J. and van Doom, A. J. (1992). Generic Neighborhood Operators, IEEE Trans. PAMI 14, pp. 597-605. [41] Koenderink, J. J . and van Doom, A. J. (1980). Photometric Invariants Related to Solid Shape, Optica Acta 27, pp. 981-996. [42] Koenderink, J. J. and van Doom, A. J. (1995). Relief: pictorial and otherwise, Image and Vision Computing 13, pp. 321-334. [43] Lefschetz, S. (1957). Differential equations: Geometric theory, Interscience Publ., Inc., New York. [44] Liebmann, H. ( 1902-1927). Geometrische Theorie der Differentialgleichungen:Die topographischen Kurven, in: W. Fr. Meyer and H. Mohrmann, Eds., Encyklopadie der Mathematischen Wissenschajlen mit Einschluss ihrer Anwendungen, Vol. III,3rd part, B. G. Teubner, Leipzig. [45] Longuet-Higgins, M. S. (1960). Reflection and Refraction at a Random Moving Surface. J. Opt. SOC.Am 50, pp. 838-844. 1461 Maxwell, J. C. (1859). On Hills and Dales, The London, Edinburgh and Dublin Philosophical Magazine and J. of Science, Vol. 40, No. 269, pp. 421-425. [47] Morse, M. and Cairns, S. S. ( 1969). Critical point theory in global analysis and differential topology, Academic Press, New York and London. [48] Miiller, E. (191 I). Eine Abbildung krummer Flachen auf eine Ebene und ihre Verwertung zur konstrirktiven Behandlung der Schraub- urid Schieb-jachen, Sitz. ber. Akad. Wiss. Wien, 120, pp. 1763-1810. [49] Miiller, E. and Kruppa, E. (1936). khrbuch der Darstellenden Geometrie, B. G. Teubner, Leipzig and Berlin. [SO] Nackman, L. R. ( 1984). Two-Dimensional Crirical Point Conjguration Graphs, IEEE Trans. PAMI 6, pp. 442-449. 15I] Noizet. F. ( 1823). Mimoire sur la Giomitrie Appliquie au Dessin de la Fortification, MCmorial de I’Officier du GCnie, Nr.6, Paris. 1521 Petrovski, I. G. (1966). Ordinary diferential equations, Transl. R. A. Silverman, Dover Publ., Inc., New York. [53] Poneous. 1. R. (1994). Geometric differentiation for the intelligence of curves and surfaces, Cambridge University Press, Cambridge. (541 Rieger, J. H. (1997). Topograpliicalpropertiesofgeneric images, Int. J . Comp. Vis. 23, pp. 79-92. [55] Rothe. R. (1915). Zuni Problem des Talwegs, Sitz. ber. d. Berliner Math. Gesellschaft, 14, pp. 5 1-69, [56] Rothe, R. (19 14). Darstellende Geometrie des Gelandes, Leipzig. (571 Scheeffer,L. ( 1890). Theorie von Maxima urid Minima einer Funktion von zwei Variahelen,Math. Ann. 35, pp. 541-576.
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[58] Schwarz, H. A. (1980). ober ein vollstandiges System voneinander unabhiingiger Voraussetzungen zum Beweise des Satzes ( = & , Ges. math. Abhandlungen. Bd. 11, Springer, Berlin. [59] Sneddon, I. N. (1957). Elements ofpartial differential equations, McGraw-Hill Book Company, Inc., New York. [60] Spivak, M. (1975). A comprehensive introduction to differential geometry, Vol. 111, Publish or Perish, Houston, Texas. [61] Takasaki, H. (1970). Moire‘topography, Applied Optics 9, pp. 1457-1472. 1621 Thorn, R. (1972). Stabilitt! structurelle et rnorphoge‘ntse, Benjamin, New York. 1631 Vasari, G.(1960). Technical introduction to the “Vita”, Vasari on technique, transl. L. S . Maclehose, Dover, New York. 1641 Webster, A. G.(1955). Partial Direrential Equations of Mathematical Physics, Dover, New York, (first edition 1927), p. 3. [65] Weitzenbock. R. ( 1923). lnvarianten Theorie, P. Noordhoff, Groningen. [66] Wunderlich, W. (1961). Fluchen mit ebenen Fallinien, Monatschr. f . Math. LXV,pp. 291-300. 1671 Wunderlich, W. (1961). Flachen mit Kegelschnirte als Fallinien, J. f . d. reine u. angew. Math. 208, pp. 204-220. 1681 Wunderlich, W. (1938). Darstellende Geometrie der Spiralflichen, Monatschr. Math. Phys., 46, pp. 248-265.
6y )
ADVANCES IN IMAGING AND ELECTRON PHYSICS. VOL. 103
Dyadic Green’s Function Microstrip Circulator Theory for Inhomogeneous Ferrite with and without Penetrable Walls CLIFFORD M. KROWNE Microwave Technology Branch. Electronics Science and Technology Division. Naval Research Laboratory. Washington. DC 230375
I . Overall Introduction . . . . . . . . . . . . . . . . . . . . . . . I1. Implicit 3D Dyadic Green’s Function with Vertically Layered External Material Using Mode-Matching . . . . . . . . . . . . . . . . . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . B . Self-Adjoint Operators for Vertically Layered External Material . . . . . . C . Implicit Dyadic Green’s Function Construction . . . . . . . . . . . . D. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . I11. Implicit 3D Dyadic Green’s Function with Simple External Material Using ModeMatching . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . B . Fields. Mode-Matching Technique, Nonsource and Source Equations . . . . . C. Implicit Dyadic Green’s Function . . . . . . . . . . . . . . . . . D. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . IV. 2D Dyadic Green’s Function for Penetrable Walls . . . . . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . B . Formulation . . . . . . . . . . . . . . . . . . . . . . . . . C . Dyadic Green’s Function . . . . . . . . . . . . . . . . . . . . D . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . V. 3D Dyadic Green’s Function for Penetrable Walls . . . . . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . B . Formulation . . . . . . . . . . . . . . . . . . . . . . . . . C . Dyadic Green’s Function . . . . . . . . . . . . . . . . . . . . D . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . VI . Limiting Dyadic Green’s Function Forms for Homogeneous Ferrite . . . . . . A. 2D Dyadic Green’s Function . . . . . . . . . . . . . . . . . . . B . 3D Dyadic Green’s Function . . . . . . . . . . . . . . . . . . . VII . Symmetry Considerations for Hard Magnetic Wall Circulators . . . . . . . . A . 2D Dyadic Green’s Function . . . . . . . . . . . . . . . . . . . B . 3D Dyadic Green’s Function . . . . . . . . . . . . . . . . . . . VIII . Overall Conclusion . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .
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ADVANCES IN IMAGING AND ELECTRON PHYSICS Copyright @ 1998 by Academic Ress Inc. All rights of reproduction in any form reserved. ISSN 1076-5670/98$25.00
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I. OVERALLINTRODUCTION
Our focus here is on canonically shaped circulators, namely, those with circular symmetry. Great advantage may be taken of mathematical physics tools when restricting the problems examined to canonically shaped objects, such as those with circular symmetry. For electromagnetic circulators, many actual devices possess some degree of circular symmetry, so symmetry restrictions are not unrealistic. As one moves further and further away from simple objects which are symmetrical, like a circle in a two-dimensional model or a circular cylinder in a threedimensionalmodel, the ability to derive explicit dyadic Green’s functions becomes more difficult. Finally, when the complexity is increased to the extent when the circulator puck (composed of ferrite anisotropic material) is surrounded by layers of substrate material and layers of superstrate material, with radially changing composition, all that is reasonable to seek is an implicit Green’s function, which retains some of the features originally employedto find the simpler explicit Green’s function, and new aspects found in the mode-matching method. Adding in the mode-matching technique (Sections I1 and 111) with its variable number of matching modes seems to irrevocably eliminate the ability to solve for explicit dyadic Green’s function elements. Systems of equations, or equivalently, matrices in matrix equations contain the properties of the environment external to the puck and the manner in which the outlying areas interface with the puck itself. But what is lost in acquiring the field solutions through explicit dyadic Green’s functions is gained in treating much more complicated surrounding environments, which still retain circular symmetry. The radially changing composition mentioned above may be thought of as zones of specific radial width, reaching out toward infinite radial size. There may be a finite number of zones, and the last zone can be either terminated at infinity or at an electric or magnetic wall. Another area where an attempt to obtain explicit dyadic Green’s functions may shed more light on the electromagnetics is that of field extension beyond the circulator perimeter (Sections IV-VI). It is often a reasonable approximation, substantiated by comparison to experiment, to model the contour between microstrip ports as an impenetrable magnetic wall. But it might be very instructive to obtain expressions giving the dependence on the external material outside of the inhomogeneous puck. This may be done under specific assumptions for both two- and three-dimensional approaches regarding the field behavior and coupling between the internal puck fields and the outside fields on either side of the nonport boundaries. Significant simplifications occur when the inhomogeneity of the ferrite puck is taken to limit to a single uniform disk (two-dimensionalmodel) or cylinder (three-dimensionalmodel) of material (Section VI). Although the magnetic sources (driving functions) which represent the microstrip feeding lines can be placed at arbitrary azimuthal angular locations along
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the circulator perimeter, for both two- and three-dimensional models, tremendous simplifications occur in the various dyadic Green’s functions used to obtain the s-parameters and electromagneticfields when symmetricplacement of the ports is imposed (Section VII). For threefold, fourfold, or sixfold symmetry of the ports, where the mathematics is still manageable, drastic reduction of the required number of Green’s function elements used in building up s-parameter solutions occurs in the two-dimensional treatment.
11. IMPLICIT 3D DYADICGREEN’SFUNCTION WITHVERTICALLYLAYERED EXTERNAL MATERIAL USING MODE-MATCHING A . Introduction
Developing Green’s function approaches for canonical structures can be particularly advantageous when solving inhomogeneousboundary-valued problems, as is the case for planar circulator problems of the microstrip variety. The driving force occurs on the r = R surface at the point 4 = 4’ and z = z’ or on a strip at 4 = 4’. Obtaining explicit dyadic Green’s function expressions is known to be very convenient and allows extremely rapid numerical computation of electromagneticfields and s-parameters [ 1, 21. In that work, the circulator was a circular femte puck, but with completely arbitrary radial variation of the descriptive parameters of the problem. The puck itself was made up of a number of annular rings, each with different widths, and with different material properties for the magnetization M, and demagnetization factor N,,. The magnetic biasing field Happwas also allowed to vary in an arbitrary radial manner. Two- and three-dimensional dyadic Green’s functions were obtained, which depended upon recursive relations to find final expressions. Although these expressions are compact and explicit, the recursive nature of the development necessarily contains embedded information, making the actual algebraic dyadic Green’s functions immensely complicated. Therefore, computer techniques are essential in studying the behavior of the dyadic Green’s functions. But because of the canonical nature of the structure geometry, and the theoretical techniques employed in the derivations, these dyadic Green’s functions lead to field evaluationsbetween 1,000 and 10,000 times faster than intensive numerical techniques like finite difference or finite element methods. What we are desirous of doing in this section is dropping this complex inhomogeneous puck into a medium which consists of radial zones beyond the circulator puck perimeter. Each zone is made up of an arbitrary number of horizontal layers, stacked vertically in the z-direction. This arrangement outside of the puck will constitute yet another inhomogeneous problem, in addition to that of the puck
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itself. In principle, the region above the puck, bounded on the lower side by an electric wall formed by a microstrip conductor, and on the top by a metal limiting wall, can also be viewed as a zone. Such a structure as outlined above can be used to treat the case where a circulator ferrite puck is dropped into a hole in a substrate, and possibly covered by a superstrate. Both the substrate and superstrate may be broken up into layers. All of the material external to the puck will be considered to be isotropic, but with the possibility that the cylinder above the puck and the layers in the radial zones can have permittivity properties, permeability properties (unbiased), or both simultaneous permittivity and permeability properties. Each radial zone, stretching vertically from the lower ground plane to the top horizontal wall, made up of many different layered regions, is viewed as a waveguide section, with a collective radial waveguide propagation constant. At a cylindrical wall r = r , , mode-matching is applied. The j index increases in value from j = 1 at the the puck-external medium interface rl = R, to jN at the last interface. The last interface may be chosen as open in which case a radiation condition could be applied or as an electric or magnetic wall requiring explicit but simple mode-matching conditions for the last zone’s vertically stacked regions. Here we will treat a specific case of the general situation outlined in the last paragraph. The puck will be placed inside a substrate like that found in microwave monolithic integrated circuits (MMICs), with a ground plane bounding it from below. An electric wall, representing microstrip metal, will constrain the fields within the femte puck material from above, and this electric wall will be flush with the substrate surface. Immediately above the puck will be isotropic material, not necessarily the same as that for the medium beyond the circulator puck perimeter. One zone exists beyond the perimeter, and it consists of the substrate on the bottom and another material region on top, not necessarily the same as the inner zone above the puck. The top layer, consisting of an inner and outer radial ordered set, constitutes the superstrate, which could be chosen by default in the simplest situation to be air. For the substrate being part of an MMIC, it could be one of a number of semiconductor materials like Si, GaAs, or even heterostructure material. For the case where a more hybrid-like circuit is used, it could be an unbiased magnetic material, even the same or related to that used for the puck itself. Furthermore, depending upon how the biasing magnetic field is obtained for the puck, the electric wall above the puck may be a microstrip-keeper metal combination to allow self-biasing of the femte material in the puck. With the use of a conventional biasing magnet, the origin of this field is considered to come from outside the whole structure shown schematically in Fig. 1. Allowing for a magnet to be placed in a layered arrangement above the puck (as in Fig. 2 with no space shown here, for this particular diagram, between the magnet and the microstrip disk) constitutes a greater complication to the problem which won’t be addressed in this paper, although
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multl-layered vertical zone
Top Cover
/ / / / / / / / /I / / / / / / / / / /
OT EdT P d T source wall
OB Ed0 P d 0
I
hl
c e.w.
OT hO
TI
rL
C
hc
lnhomo Ferrite
OB 7
Ground Plane R
RGURE1. The ferrite circulator structure including the regions above and surrounding the device puck. This figure is formed by taking a cut plane at d, = const (in 3D).
the theoretical principles for accomplishing such inclusion will be treated in this work. Self-adjoint operators are found for the differential equations describing the z-dependent field variation in the medium zone external to the circulator puck. The external medium is in general inhomogeneously layered, consisting of media with permittivity properties, magnetic properties, or both. For the simplest case in which each zone has regions of only one trait (i.e., not mixed), and that trait is
R
FIGURE2. The femte circulator sttllcture including the regions above and surrounding the device puck with the presence of an external biasing magnet. This figure is formed by taking a cut plane at d, = const (in 3D).
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dielectric, information is available on the TE axisymmetriceigenvalue equation [3], eigenvector forms [4,5], or scalar potential governing equations and the eigenvalues and eigenvectors using self-adjointness properties [6]. Eigenvalue equations characterizingthe radially sectioned medium outside the puck are found, as are the eigenvectors. When the z-dependent parts are multiplied with the radial and azimuthal dependences,the complete field expressions are determined. Source constraint equations driving the circulator are then combined with the mode-matching technique to obtain in direct space implicit dyadic Green’s function elements. Mode orthogonality is employed to encourage sparsity in matrix system development where appropriate or convenient. The self-adjoint operators lead to testing functions which may be used to test field continuity equations, thereby reducing some infinite summations to single-term contributions. The implicit Green’s function is particularly useful because field information and s-paramaters may be found in real space, completely avoiding typical inverse transformations. Consideration of field extension into the surrounding medium, beyond the circulator perimeter, including fringing such that fields may extend out and then above the height of the circulator nonreciprocal puck, is an essentially physical motivation for this theoretical work. The approach is a good approximation to a very complicated geometric and inhomogeneous problem, given the irregular effects arising from application of the dc biasing magnetic field and the actual finite-width microstrip input and output lines. For narrow microstrip lines the expectation that the fields extend beyond the device perimeter, with azimuthal symmetry, is very good, and essential to this canonical treatment. When some of the microstrip lines attain widths which are a noticeable fraction of the puck radius, the error introduced by the symmetry assumption for r > R for the fields will be directly related to the fraction of the circumference occluded by the presence of the line itself. B. Self-Adjoint Operatorsfor Vertically Layered External Material 1. Introduction
Consider the situation where the electromagnetic field occupies three areas (see Fig. 1). The first area C is that filled by the femte nonreciprocal material for r < R and 0 p z p hc. The second area or zone 0 has two regions for r > R, a bottom region O B with0 Iz p hc andatopregion OT withhc 5 z Iho. Andthethird region I has r < R and hc Iz Iho. Conducting walls are assumed at z = 0, h (all r ) and z = hc ( r < R ) , and the radiation condition in effect as r + 00. Fig. 3 shows a cross section through a z = const plane, with microstrip lines coming into (or out of) the ports located at $1, &, . . . ,4i, . . . ,4~~on the r = R puck surface. In order to maintain the same field structure formulas and parallel construction techniques inside and outside of the puck, governing equations are developed in the zone r > R with the stacked 0 B and 0 T regions utilizing field components
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Port - femte region
7/ /
interface
,
Port 1
[E
Inhomo. Wall ( non - port)
\\
; (UPo, K)I
Multi-layered external vertical zone [Ed,
dl
Port i
FIGURE3. This sketch represents a cut plane at z = const (in 3D) for the femte. circulator structure.
from the puck in the isotropic limit. Specifically, TM, (e mode) and TE, (h mode) equations are sought. Maxwell’s sourceless equations assuming eiordependence are
V x E = -iwpH
V x H = ioEE
( 1ab)
2. TM, Operator Properties Take the curl of (la), noting that both p and E depend upon spatial location, namely, that p = p(r) = p ( r , 4, z ) and E = E(r) = ~ ( r4,, z ) [71.
V x V x E = -iwV x (pH)
+
= o2p&E vp(r) x V x E LL
using both curl expressions (1) to remove any H field dependence. Using an identity to eliminate the curlcurl term,
Realizing that the divergence of a curl in the left-hand side of (lb) is zero, the divergence of the electric field in (3) may be replaced by
V .E =
--a
Vdr) E E
Equation (4)has been obtained from the general spatial variation being reduced to that in only the z-direction because of regional changes within a zone (i.e., for
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CLIFFORD M.KROWNE
a two-region zone as being considered here, E ( Z ) = &fB for 0 5 z 5 hc and for hc 5 z 5 ho with a discontinuity at z = hc). E(Z) = Inserting (4) into (3), the vector electric field governing equation is found:
k2 = W'PE = 02p(r)E(r)= W * P ( Z ) E ( Z )
(6)
If the TM to z is selected as the basic mode electric type (out of the two required for a complete field description) with only the electric field existing perpendicular to the transverse plane, this being precisely the same directional field setup inside the puck, then (5) vastly simplifies to +k2Ez = O
(7)
+k2E, = 0
(8)
Since all of the other field components (transverse) depend on formulas written in terms of E,, only its expansion need be considered first. m
h
n=-m
k,=b
bo
n=-mm,=l
Here it is seen that the electric field solution is the sum over the radial variation, z directed harmonic, and azimuthal harmonic products. The A coefficient provides
THEORY FOR INHOMOGENEOUS FERRITE
159
the term by term weighting. The infinite sum over k, propagation constants in forward ( f ) and backward (b) directions may be changed into only a backward wave summation,but now requiring explicit forward and backward A coefficients. added to the A coefficients store that wave direction Superscripts "-"and information, letting the subscript propagation index being simplified to merely k, = b. This process is collected together in the next line. Recognizing that in each zone only the radial propagation constant n is definable (and therefore capable of being indexed over the entire zone), but that k, varies from region to region within a zone, the fifth line is obtained. Finally, the n solutions to be determined later, can be assigned for the solutions, index numbers ordered as me = 1 , 2 , . . . , 00. The associations for Z i e( z ) in (9) can be summarized as
"+"
acting as the separation constant,
Select the first two derivative terms of (13), plus the first of the last bracketed sum, as the inhomogeneous linear operator L T M ,
Invoking ( 14) enables ( 13) to be recast as
L TM z ; < ( z ) = -A;< Zi,(Z)
(15)
Operator equation (1 5) is in eigenvalue form, the eigenvalue operator on the righthand side of the equation merely the eigenvalue constant. Requiring the radiation condition to hold as r += 00 makes
R ; p ) = Kl (0:r)
(16)
Next let us find the adjoint of L to better understand the behavior of the inhomogeneously loaded waveguide zone. For ease of mathematical argument and brevity, abbreviate two different Z i e( z ) solutions as u and u and define the inner
160
CLIFFORD M.KROWNE
product on the interval (0,h o ) as
with weight w(z). To find the adjoint, study
1
ho
(u, Lu) =
w(z)u(Lu}d z
Sometimes it is convenient to explicitly place the weight in the bracketed expression when we wish to be reminded of its presence, as in ( w u , Lu). Anyway, we seek to convert (18) into the form (L'u, u) by repeated application of integration by parts and thereby identify the adjoint form La of the operator 15. The weight we choose here is
Therefore,
THEORY FOR INHOMOGENEOUS FERRITE
=-loe$du
161
+ (qu,u )
From (20) the adjoint form of the operator can be identified as L$M = L T M
(21)
Thus, it is seen that the operator is self-adjoint [8,9,10]. Equation (21) was obtained by using a number of boundary conditions, which will be covered in this section briefly. Electric wall conditions at z = 0, h~ require transverse field components E, and E4 to be zero. From the Ed field component expression [ 111, for example,
one observes that pure Neumann conditions hold at the ends of the inner product domain. This conclusion follows because (22) originates when exponential (harmonic, plane waves) are chosen for the z-functionalbehavior. Thus, the coefficient p,, from the second piece of E4 generating the TM mode, indexed for j z-directed
162
CLIFFORD M. KROWNE
modes, contains a ik, factor, implying the presence of a partial derivative a/az operator.
The same conditions, of course, hold for u. Because within a region of a zone E(Z) is stipulated to vary continuously, and for the zone to vary, at most, piecewise continuously, because the discontinuitiesonly occur at the region interfaces, ds/dz will be well behaved at the domain ends. Furthermore, requiring E ( Z ) to be constant within a region makes a Neumann condition also hold for it. Thus,
Neumann conditions (24) on d&(z)/dz are not required for obtaining the selfadjoint relation (20). This is also the case for TE modes, as will be demonstrated later, It is emphasized here that although L T M is self-adjoint, it is not representing lossless media. The media can be dielectrically lossy, magnetically lossy, or lossy in both regards simultaneously. Thus, the eigenvalue in (15) may be complex. In fact, for ordinary media, we expect it to be complex. Now let us review in light of this fact, the short derivation of orthogonality as implied by self-adjointness. Consider that any self-adjoint operator L obeys ( L u , u ) = (u,L u )
(25)
Let
L T M U= -AhCu; L T M U= -A',ume
(26ab)
associating u and u with, respectively, the eigenvalue indices me and m:. Placing (26) into (25) yields
(-A;/,
u) = (U,-A;p)
or
(A;; - A;,) (u,u ) = 0 or
(u,u ) = 0
(27)
Orthogonality relation (27) holds precisely because the z-eigenfunctionsare associated with different eigenvalues. In our case, the eigenvalue difference is between different complex eigenvalues. Relation (27) says that the z-eigenfunctions are orthogonal no matter how many different regions are stacked in a zone, and this
THEORY FOR INHOMOGENEOUS FERRITE
163
is true regardless whether we use only dielectric regions, magnetic regions (unbiased), or intermix these two types of regions, or even if we further complicate the situation by using regions with both dielectric and magnetic characteristics. As a final point of discussion, proof of L T M operator self-adjointness can be found in a more general manner than by (20) by associating L T M with a general second-order derivative operator, showing the connection between this secondorder derivative operator and another standard operator form Ls, and then finally demonstrating that L S is indeed self-adjoint. A general second-order derivative operator can be expressed as
d2 d L, = 4 z ) - b(z)dz2 dz Next define the L S operator as
+
1 d w(z)dz In L S make the following associations:
Substituting ~
( z )p ,( z ) , and q ( z ) into L s ,
1
= L,
+ c(z)
164
CLIFFORD M.KROWNE
We can readily show that L S is self-adjoint by studying
THEORY FOR INHOMOGENEOUS FERRITE
165
Self-adjointness of L S was assured because evaluation of d i n i t e integrals n two steps of the derivation relied upon either Dirichlet or Neumann boundary conditions holding, and this is known to be true for many classes of operators acting on variables, as it is for true for the LTM operator. In fact, for the LTM operator ( 14)Neumann conditionsare valid for the variables u and u of the problem. Now let us associate the LTM operator with a general second-order derivative operator by finding the coefficients a ( z ) ,b ( z ) , and c ( z ) . Expand the second operator term in (14),
{
-l d_s } + k 2 L T M = -d+2 - - -I+ d& -d dz2 s d z d z dz s d z Comparison to L , allows the identifications 1d s = 1; b ( z ) = --; E dz C ( Z ) may also be written as U(Z)
C(Z)
=k
At last, the coefficients in the L S operator can be determined.
+
q(z)= c(z)=w2s(z)p(z)
3. TE, Operator Properties Following the philosophy of the TM, development in Section B.2, take the curl of (lb).
V
x
V
x H = iwV x
= u2psH
(EE)
+VE(r) x V x H &
166
CLIFFORD M.KROWNE
using both curl expressions (1) to remove any E field dependence. Using an identity to eliminate the curlcurl term,
+
V2H- V ( V H) W'PEH
+V E ( r )x V x H = 0 E
(29)
Realizing that the divergence of a curl in the left-hand side of (la) is zero, the divergence of the magnetic field in (29) may be replaced by
Equation (30) has been obtained from the general spatial variation being reduced to that in only the z-direction because of regional changes within a zone (i.e., for a two-region zone as being considered here, p(z) = pdOB for 0 5 z 5 hc and p(z) = p y T for hc Iz 5 ho with a discontinuity at z = h c ) . Inserting (30) into (29), the vector magnetic field governing equation is found:
':)A
a:(
. ( ( l a -H- -z - q ) r a4
a2)i}=0
(31)
If the TE to z is selected as the basic mode magnetic type with only the magnetic field existing perpendicular to the transverse plane, then (3 1) greatly simplifies to
or
Since all of the other field components (transverse) depend on formulas written in terms of H,, only its expansion need be found. c
o
b
167
THEORY FOR INHOMOGENEOUS FERRITE
=
1
Rnhk,(r)[e
ik z
hO- + e-ikzzAhO+ Ank, nk,
Ie'
in@
n=-ca k,=b
n=-camh=l
Here it is seen that the magnetic field solution is the sum over the radial variation, z-directed harmonic, and azimuthal harmonic products. The a solutions to be determined later, more correctly denoted as am,,are assigned for the solutions are different than urn< index numbers ordered as m h = 1,2, . , 00. Solutions amh for the T M , case. The associations for Zkh(I)in (34) can be summarized as
..
h Ahozh mh (z) = z:(z)= z k z ( z ) = eikzz nk,
+ e-ik,zAhO+
nk,
= eik,zAhO
nk2
+ e-ik,zAhO
n,-k,
(35)
Now inserting (34) into (33) and applying separation of variables, with 2
= -(amok>
(36)
acting as the separation constant,
Select the first two derivative terms of (38), plus the first of the last bracketed sum, as the inhomogeneous linear operator L T E ,
(39) Utilizing (39) enables (38) to be recast as
Requiring that the radiation condition to hold again as r + 00 makes
168
CLIFFORD M.KROWNE
Next let us find the adjoint of L to better understand the behavior of the inhomogeneously loaded waveguide zone. For ease of mathematical argument and brevity, again abbreviate two different ZLh(z)solutions as u and u and use the inner product definitions (17) and (18) on the interval (0, h o ) . We seek to convert ( 1 8) into the form ( L a u ,u ) in order to identify the adjoint form L" of the operator L = L T E . The weight we choose here is w(z) = W ( Z )
Therefore,
(42)
THEORY FOR IMIOMOGENEOUS FERRITE
= ( L $ ~ v u, )
169
(43)
From (43) the adjoint form of the operator can be identified as L$E = L T E
(44)
Thus, it is seen that the operator is self-adjoint. Equation (44)was obtained by using a number of boundary conditions, which will be covered in this section briefly. Electric wall conditions at z = 0, ho require transverse field components E, and E$ to be zero. From the E$ field component expression in (22), one observes that pure Dirichlet conditionshold at the ends of the inner product domain. Coefficient s,, from the first piece of E+ generating the TE mode, indexed for j z-directed modes, has no ik, factor, implying u ( 0 ) = 0; u(ho) = 0
(45ab)
The same conditionshold for u. Because within aregion of a zone p(z)is stipulated to vary continuously, and for the zone to vary, at most, piecewise continuously, because the discontinuitiesonly occur at the region interfaces, dp/dz will be well behaved at the domain ends. Furthermore, requiring p(z) to be constant within a region makes a Neumann condition also hold for it. Thus,
Although (46) holds, conditions on d p ( z ) / d z are not needed for obtaining the self-adjoint relation (43). This is like the case for the TM mode, where (24) was not required.
170
CLIFFORD M.KROWNE
Finishing the presentation of TE, operator properties, again something may be said about the Ls operator as was done for the TM, operator properties. L s may be found for the TE modes merely by referring to the previous TM presentation and replacing E by p everywhere and noting that Dirichlet boundary conditions replace Neumann boundary conditions. 4. Eigenvalues for TM, Modes Eigenvalues umeof the TM, modes, and the consequent values of A&, can be found by applying the boundary conditions on the electric field component
E;'(r, 4, z ) = 0; z = ho
(47)
Ezo(r, 4, z ) = 0; z = 0
(48)
and continuityconditionson the azimuthalcomponents of the electric and magnetic fields
E;o(r, 49 z)lhc+ = E;o(r, 4, z)lhc-
(49)
H;o(r, 4, z)lh,+ = H;o(r, 47 z)lhc-
(50)
EZo can be written using the third line form of (9), and (22).
n=-w
k,=b
ikzj
.
'
where P'- - k i - kZj'
i W d
s.---
ki
- kzj
.
, u'--
-
iWEd
ki
- kZj
(52abc)
The comparable H: expression to (22) is [ 111
Electric wall boundary conditions (47) and (48) yield with the help of (5 1) eikPThoAeOT- - e-ikPThoAeOT+ -0 nm, eikpn.O
e O B - - e-ik;n.O
nm,
eOB+
=0
(55)
(56) Anmc Anmc These conditionshave resulted from the global nature of the radial om,propagation constant or me index, allowing the radial function and factors depending only on this index to drop out of the equations. Superscripts T and B denote, respectively, the top and bottom regions in the outside zone. It should be realized here that kZ has an implicit dependence on me so that when we see the perpendicular propagation
THEORY FOR INHOMOGENEOUS FERRITE
171
constant, it is understood that it constitutes an abbreviation for :k (for the top region, for example). Continuity conditions (49) and (50) become, using respectively (51) and (54), [ e i k p r h c A rnm, O T - - e-ik;ThcAeOT+ nm,
3 k z o T - [eik;Bhc A enm,O B - - e-ikpBhc A:::']
kpB (57)
Adding and subtracting these two equations from each other, and utilizing (56) gives the top region amplitude coefficients.
It is also helpful to define a ratio of the two amplitudes, AeOT+
so that
Similarly, refemng back to (56),
Insert amplitude relations (59) and (60) into the electric wall condition (55) to find the characteristic eigenvalue equation:
172
CLIFFORD M. KROWNE
+
(a),
Keeping the bracketed groupings in identifying 81 = k,OThl kPBhc and 82 = kfTh, - kpBhc, the exponential transcendental eigenvalue equation can be converted to a trigonometric form.
+
I-
sin[kfThl k,OBhC sin[kpThl - kpBhc] -
5-g d
+
$ d
Here h , = ho - hc. Note that for the limiting case where the top and bottom regions become identical, the numerator becomes zero and (65) reduces to the familiar form sin[kpho] = 0. Another useful form may be obtained by grouping according to the ratios kpB/k,OT and & f B / & f T . Again, the exponential transcendental eigenvalue equation is changed into a trigonometric expression, which can be set down in two ways. E fB k: O T tan[k,O T h,] = --
koT
kp koB
‘d
Ed
‘d
tan[k,OBhC]
5 tan[kPTh,] = -+tan[kpBhc] The first form of this tangent relationship is written in terms of the propagation constant and dielectric ratios. The second form relates the top region quantities to the bottom region quantities. Equation (66b) can be shown to have the same form as indicated for a stacked inhomogeneous dielectric zone [6]. But we must be aware that now the material of the regions can be dielectric, magnetic, or of mixed permittivity and permeability character. The transcendental eigenvalue equation for the TE, will be seen later to differ explicitly from the pure dielectric form even though the TM, has not. That (66), or (65), constitute eigenvalue equations for , ; a can be understood if the separation equations are found from the differential governing equation ( 13)for Z i Cwith the help of (10) providing the exponential Z& form. Inserting (10) into (13) for the top and bottom regions in the zone, realizing that the inhomogeneous dielectric constant term drops out inside each region, yields
Invoking (1 1) for the separation constant, and taking the positive branches in (67), gives
THEORY FOR INHOMOGENEOUS FERRITE
173
Inserting (68) into (66b) gives a single transcendental equation in terms of the unknown Material region propagation constants are delineatable once (6) is examined.
02.
OT 2 (kd
)
OT O T
2 =@
pd
Ed
;
OB 2 (kd
)
2 OB O B
= w pd
Ed
(69)
Return to the amplitude ratio R f T . Following the same reasoning in finding (66) using grouping by k,OB/k,OTand
Eliminating the bottom region information in (70) by using the eigenvalue equation form (66b), the compact formula
results. Using ratio grouping and (66), the backward and forward amplitude coefficients in (59) and (60) for the top region are now
5 . Eigenvalues f o r TE, Modes Eigenvalues a,,,,,of the TE, modes, and the consequent values of A;*, can be found by applying the boundary conditions on the electric field component E j o ( r , 4, z) = 0; E;'(r,
4, z) = 0;
z = ho
(74)
t =0
(75)
and continuity conditionson the azimuthal components of the electric and magnetic fields E;O(r,
4 7
Z)Ihc+
= E j 0 ( r 9 4, Z ) l h c -
H;O(r, #, z)lhc+ = ~
i O ( r4, , z)lhc-
E j o can be written using the third line form of (34), and (22).
(76) (77)
174 H:'
CLIFFORD M. KROWNE
can be expressed, using the first part of (53), as
Electric wall boundary conditions (74) and (75) yield with the help of (78) eikp'hoAhOTnm, eikpB.O
hOBAnmt
+e-ikpThoAhOT+
=0
nm, + e-ikpB.O
hOB+ Anme
=0
(80) (81)
These conditions have resulted from the global nature of the radial am,propagation constant or m h index, allowing the radial function and factors depending only on this index to drop out of the equations. It should be realized here that k, has an implicit dependence on m h so that when we see the perpendicular propagation constant, it is understood that it constitutes an abbreviation for kg: (for the top region, for example). Continuity conditions (76) and (77) become, using respectively (78) and (79).
Adding and subtracting these two equations from each other, and utilizing (81) gives the top region amplitude coefficients.
Define a ratio of the backward and forward wave amplitudes,
175
THEORY FOR INHOMOGENEOUS FERRITE
so that
Similarly, referring back to (81),
Insert amplitude relations (84) and (85) into the electric wall condition (80) to find the characteristiceigenvalue equation:
(89)
+
Keeping the bracketed groupings in (89), identifying 81 = k,OThr k,OBhCand 0, = k P T h l - kPBhc ,the exponential transcendental eigenvalue equation can be converted to a trigonometric form.
For the limiting case where the top and bottom regions become identical, the numerator becomes zero and (90) reduces to the familiar form sin[k?ho] = 0. Another useful form may be obtained by grouping according to the perpendicular propagation constant and permeability ratios k,OB/k,OTand pfB/pdOT.Again, the exponential transcendental eigenvalue equation is changed into a trigonometric expression, which can be set down in two ways. OT
5tan[kfThr] = Pd
k: tan[kfBhC] k,OB
PJoT
PdOB
--
= -- tan[kfBhC] k,O k,OB The first form of this tangent relationship is written in terms of the propagation constant and dielectric ratios. The second form relates the top region quantities to the bottom region quantities. Equation (91b) shows that the magnetic differences between regions in a zone appear explicitly in the eigenvalue equation for the TE, -tan[kfThl]
176
CLIFFORD M. KROWNE
modes, whereas (66b) demonstratedthat the dielectric differences between regions in a zone appear explicitly in the eigenvalue equation for the T M , modes. ~ , be understood if That (91), or (90), constitute eigenvalue equations for C T ~ can the separation equations are found from the differential governing equation (38) for ZLh with the help of (35) providing the exponential Zkh form. Inserting (35) into (38) for the top and bottom regions in the zone, realizing that the inhomogeneous permeability term drops out inside each region, yields
Invoking (36) for the separation constant, and taking the positive branches in (92), gives
Inserting (93) into (91b) gives a single transcendental equation in terms of the unknown ath. Looking at the amplitude ratio R f T in (87) again, using the same procedure in finding (91) using grouping by k p B / k p T and p f B / p f T ,
Eliminating the bottom region informationin (94) by using the eigenvalue equation form (91b), the compact formula
results. Using ratio grouping and (91), the backward and forward amplitude coefficients in (84) and (85) for the top region are now
THEORY FOR INHOMOGENEOUS FERRITE
177
6. Eigenvectors of the TM, Modes Return to (lo), extracting out the third exponential wave form. Factor out the backward wave,
Writing (98) in terms of the different regions (two here), using the definitions (61) and (63,
The fourth and sixth forms of (99) created the scaled formulas for the eigenfunctions, separated from the unknown coefficients found in ZZJz). Known ratios
available from (59), enable the unknown fields to be written in terms of one unknown amplitude for each zone (one external zone beyond the circulator in the current problem). The scaled forms are related to one another by
178
CLIFFORD M.KROWNE
Retrieving (71) and (72), the first scaled eigenfunction (the generator for the eigenvector) form in the top region can be evaluated, completing its explicit formula:
(2cos(kgz);
Oszihc
One can easily show that (102) corresponds directly to a continuity condition at the z = hc interface if it is multiplied by E(z), thereby satisfying the normal component of the displacement field continuity. D,e0(C 4, Z ) l h C + = D,e% 49 Z ) I h c -
(103)
Enlisting (99), the orthogonal property for Zip(z) given by (27) carries over to TmP(z).That is,
(z:* (z), 2;; ( z ) ) = (z:< (z),
z; (2))
=0
( 104)
It is instructive to verify that (102) indeed satisfies (104). Proceeding,
cos (k,O,h, ) cos(k;! h / )
Referring to a compilation of integrals [ 121, the first definite integral in (105) is evaluated to be %hCc0s(kgz)c o s ( k 2 z ) dz
Referencing the integration variable in the second integral to the z = h o top plane, and making appropriate constant changes, allows the use of (106) again. The difference between perpendicular propagation constant eigenvaluesin the different regions, which act as a prefactors in both integrals, are found to be equal by (67)
179
THEORY FOR INHOMOGENEOUS FERRITE
to
( 0 2 )-~(cJ:)~, reducing the inner product expression in (105) to
(Qz)7
.Qz))
invoking the eigenvalue equation (66). Thus, we have verified (104)directly. The orthogonality relation may now be stated as
( Z i e(z), T m L (z)) = C:,Sm,,m:
(107)
Let us borrow the right-hand side of (103, and particularize to me = mi,to obtain : , . the square of the normalization constant, which is equal to C
where [ 12J
1 lc
"
hr
11, =
cos2(k&fz)dz = - - sin(2kghc) k%: 4
ho
12e
=
2
OB
cos (kZm,[ho- z])dz =
+
*] 2
sin(2kghl)
+ k'Th']2
( 109a)
(109b)
7. Eigenvectors of the TE, Modes
Go back to ( 3 3 , extracting out the third exponential wave form. Factor out the backward wave,
180
CLIFFORD M. KROWNE
Writing (1 10) in terms of the different regions (two here), using the definitions (86) and (88), [eikpTZ + e - i k p ' z R O T h +,-ikpnzR0B h e ~ k P T z+ e - i k p T z R O T h
3
AhOT-
;
]
AhOB-
; OPzshc
nmh
nmh
OT-. ] A hnmh
2i sin(kpBz ) A:::-; [eikpTz + e - i k p ' z ~ f T ] R T B mh
9
hcIZ5ho
hc 5 Z l h 0 OPzshc
; hc P z 5 h o
}
OSzihc - Z -
h
AhOBmh
=
{
nmh
+
[eikpTz e-ikp'zRfT]; hc 5 z 5 h o 2i sin(kpBz)R:T-; 0 5 z 5 hc
- zh A h O T mh
nmh
(111)
The fourth and sixth forms of (1 11) created the scaled formulas for the eigenfunctions, separated from the unknown coefficients found in ZLh( z ) . Known ratios
available from (84), enable the unknown fields to be written in terms of one unknown amplitude for each zone. The scaled forms are related to one another by
Retrieving (95) and (96), the first scaled eigenfunction (the generator for the eigenvector) form in the top region can be evaluated, completing its explicit formula:
Equation (1 14) can be shown to correspond directly to a continuity condition at the z = hc interface if it is multiplied by p ( z ) , thereby satisfying the normal component of the B field continuity.
181
THEORY FOR INHOMOGENEOUS FERRITE
Utilizing (1 1 l), the orthogonal property for Zkh(z) given by (27) carries over to (z) when using the proper weight function in the inner product construction. That is,
zkh
z:h(z), 2;:
(Zkh(z), z;
(z)) = (
(z)) = 0
(1 16)
It is also informative to verify that (1 14) does satisfy (1 16). Proceeding,
6"
sin(kg[ho - zl) sin(kz![ho
- z1)dz
(117)
Refemng to a compilation of integrals [12], the first definite integral in (1 17) is evaluated to be
Referencing the integration variable in the second integral to the z = ho top plane, and making appropriate constant changes, allows the use of (1 18) again. The difference between perpendicular propagation constant eigenvalues in the different regions, which act as prefactors in both integrals, are found to be equal by (92) to (a? )2 reducing the inner product expression in (1 17) to mh
invoking the eigenvalue equation (91). Thus, we have verified (1 16) directly.
182
CLIFFORD M. KROWNE
The orthogonality relation may now be stated as
( Z : h ( ~ )Z:;C~)> ' = CLh6mn,rn;
(1 19)
Let us borrow the right-hand side of (1 17), and particularize to mh = mi,to obtain the square of the normalization constant, which is equal to Clh. CLh= -4pfB
1
hc
sin2 (kzm,z) O B dz - 4
[s]= OB
where [ 121
1
hc
Ilh
=
sin2(k:2z)dz
=
k%?
4
sin(2kghc)
1
k: +hC:2
(121a)
(121b)
C. Implicit Dyadic Green's Function Construction 1. Fieldsfor Puck, Interiol; and External Zones
Electromagnetic fields to be used in the following continuity conditions employing the mode-matching method are summarized here from the previous section in the 0 zone, plus those found also in the I and C regions:
m=l n=-m
THEORY FOR INHOMOGENEOUS FERRITE
183
m=O n = - w
w
w
m,=l n = - w 0 0 0 0
mh=l n = - w
w
w
m=l n = - w c a m
m=O n=-w
Field expressions in (130)-( 133) for the circulator puck region are available in [2] and [l I].
184
CLIFFORD M. KROWNE
2. Mode-Matching Theory at Intelfaces
Use of tangential field continuity at the r = R interface will be used to connect the various regions with z as the variable. That is, {Ezv E,, Hz, H#)lr=R- = {Ez, Ed, Hz, H&)lr=R+
(134)
This expression must be applied at the CO interface and at the I 0 interface. It will not hold at the CO interface for the magnetic field components because the r = R surface contains singular forcing functions (delta functions). In the spirit of applying rigorous mode-matching theory by projecting testing functions on the continuity equations and integrating, we find for the I 0 interface
and for the CO interface,
Finally, for the O(C
+ I ) interface,
Here $r>,m are z-dependent testing functions of the particular region i = I, 0,C as applied to the f type of field continuity equation for component j type, with the mode index m. Projection of these functions on the field continuity equations in the fashion of ( 1 3 9 4 142) allows their orthogonal properties to be used, with proper attention paid to weighting functions, thereby reducing the complexity of the eventual system of equations describing the problem. Wherever orthogonality is exploited, matrix sparsity is enhanced. For the exterior zone problem, the work required to determine the orthogonal properties of the scalar generating potentials is balanced by their convenience in use and sparse matrix behavior. We will use such potential properties here, noting that it is possible to use an external unloaded
THEORY FOR INHOMOGENEOUSFERRITE
185
zone cavity testing function (with no orthogonal features, but easy to identify). These testing functions are
+Lzm = +&,, = sin(k:,[z
- hc]);
I mn k,, = -, hl
m = 1,2,. . . , Mf"
m = 0, 1, . . . , M C
$5,= sin(kFmZ);
(144)
(145)
m = 1,2, ..., MC
Notice that the infinite summations contained in the field representationshave been truncated to Mi, i = I, 0, C in the respective regions, with mode-type differentiation in the external zones. Mi must be chosen with some care. M Iand MC may be chosen relative to M o (ignoring, for the moment, mode-type differentiation) as
There is some flexibility in the specific ratio converting M o into the other summation limits [ 131. These issues are referred to as the relative convergence behavior. How large to choose M o is an issue of absolute convergence, specifically how accurate an answer we desire for the problem solution.
3. Nonsource Governing Equations Selection of the subset (if a full set is not needed) of continuity equations is not unique [ 141, the choice being dependent upon individual inclination, sometimes numerical advantage, and the requirement that the number of equations equal the number of unknowns when the missing source equations are included (to be addressed later). Thus, we choose (133, (136), (141), and (142), which become, once the testing function projections and integrations are performed,
186
CLIFFORD M.KROWNE
m=l
m=l
m=l
mh = 1,2,.
. . , M:
(153)
In these previous relationships, for the internal I region,
(ki)2 = ( D : ) ~
+ (k:m)2
Also, the normalized S:h constant used in (153) is
The overlap integrals found in (150)-(153) are given by [ 121
(154)
THEORY FOR INHOMOGENEOUS FERRITE
187
188
CLIFFORD M. KROWNE
(165) Orthogonality relationships relevant to these equations are found in (104) and (1 16) for respectively and [used in (152) and (153)], and below for (150) and (151).
ce
zkh
m,m'> 0 where Em
{
112; m = O = 1; m>O
Also
was used in (158), and a similar formula exists for the h mode.
(167)
189
THEORY FOR INHOMOGENEOUS FERRITE
+
+
+
+
Four system equation set (150H153) constitute Mf M: M$ M: 1 individual equations [2(M, M o ) 1 if the e and h mode index limits are the same]. But the unknowns are
+
+
m anmO; 1
= 0, 1, . . . ,Mc
( 170a)
1,2,. . . ,MC
( 170b)
2
anmO; m =
,
me
= 1,2, . . . , M ;
(170c)
AhoT-; nmh
mh
= 1 , 2 , . . . , M:
( 170d)
A:;;
m = 0, 1 , . . . , MI"
( 170e)
Ail;
m = 1 , 2 , . . . , M:
(170f)
+
+
AeOT-.
+
+
+
+
making a total of M ; M: M; M: ~ ( M C 1) of them MI MO+ MC 1) if the e and h mode index limits are the same]. Notice that the m = 0 case has been left off of the second radial mode coefficient index listing (170b) because it corresponds to the zero perpendicular thickness situation and we expect the first radial mode to completely dominate. Therefore, we see that exactly 2Mc 1 more equations are required to describe the structure.
+
+
4. Source Governing Equations The missing equations come from two source equations governing the exchange of energy between the magnetic delta forcing functions acting on the r = R surface through a point aperture, and the puck structure. The components of the surface magnetic field are chosen to couple the external environment to the circulator puck structure, thereby defining a Green's function construction. The two source equations are H@A(#, Z) = H$'S(Z - z')S(#
- #')A#'
H z ~ ( #Z> . = H;'S(Z - z%(# - #')A#'+
+ H t ( @ ,Z)
(171)
H,O(#,
(172)
Z)
Both H ~ and A H z may ~ be expressed by a Fourier series constructed from the same expansion functions as used to represent the circulator puck field components on the interval (0, hc) by simply using the extended field on (-hc, h c ) ~ 5 1 .
190
CLIFFORD M.KROWNE
H$Am
(177)
= HiAm =0
Here ex indicates extended field and (177) results from the cos or sin nature of the field component variation within the circulator puck. It is convenient to retain the exponential Fourier series on (-n,n) for an additional representation of the # dependent coefficients found in the z-expansions (173) and (174). Thus,
n=-co m
n=-w
With the foregoing information, namely (171)-( 18 l), the proper constraints on the sources, can be imposed. It is done through the requirements that the tangential magnetic field components to the r = R surface be continuous in a limiting process just to the inside and outside of the device perimeter. Hi
Ir=~-
= H @ AIr=R+
Hf
Ir=~-
= H z A Ir=R+
(182ab)
Use of the azimuthal orthogonality property and recognizing the similarity of the perpendicular coordinate expansions in both the puck and aperture (source) surface, Hcir @mn
= HiAmn
cir
Hzmn
- HfArnn
(183ab)
where we have written out cir to emphasize the difference between the puck expansion coefficients and the aperture mnth coefficient, which happens to be a cosine type. Placing (175) into (180) gives, considering the left-hand side of
THEORY FOR INHOMOGENEOUS FERRITE
191
(183a) first, H;Amn
=Em f x H m , ( a ,
nhc
r ) c o s (m F n) ez - i n 4 d + d z
0
(184)
Next insert the source equation (171) into (184),
-
(
rnnz )H$' A+'e-i"@' + 0 integral 5 cos F nhC
Retrieving H: in (129), the second integral in (185) can be evaluated for the 0B and 0 T regions being identical, demonstrating the reduction of one infinite summation:
At this stage, constraint (183a) can be imposed, invoking (133) for the righthand side, using the previous (185) and (186) results. The fifth system equation, containing MC 1 subequations, can now be stated as
+
192
CLIFFORD M. KROWNE
Overlap integrals on the right-hand side of (187) are determined to be
The sixth system equation containing Mc subequations is found by treating (182b). Place (176) into (18 I), obtaining
Inserting the aperture source expression (172) for the perpendicular magnetic field component into (190) gives
(F )
1 mnz HzCAmn= -sin Hfr' Aq5ie-in@ 0 integral XhC
+
(191)
Using (127), the outside integral second term is expressed as .
b
o
o
0
With the help of (132) for the puck field on the left-hand side of (182b), and employing (191) and (192), the final perpendicular source equation is written as
m = l , 2 , ...,Mc (193)
THEORY FOR INHOMOGENEOUSFERRITE
193
The overlap integral in (193) is given by
1
hc
I$.mh =
sin(kFmz)zt,(z) dz = 2i
sin (kFmz) s i n ( k g z ) dz
Orthogonality relationships implicitly contained in (187) and (193) for the puck test eigenfunctions are
1
hc
LcI,b&,I,b&,dz =
hC m, m' > 0 sin(kFmz)sin(kFm,z)dz= -L3mm~; 2 (195)
5. Complete System Equation and Dyadic Green 's Function
The six system equations, four sourceless as seen in (150)-(153), and two with magnetic sources in (187) and (193), may be stated in compact form in a single
A representativenmth element for each class of unknowns is shown in the left-hand side column vector. The matrix entries are as follows: zero for no contribution of the subscript type of unknown, one for a single entry contribution, and X for a sum of all that particular class of unknowns indicated by the subscript. On the right-hand side of the equation are the source forcing terms. It is possible to reduce
194
CLIFFORD M.KROWNE
the size by analytical effort of the system matrix (197) due to the appearance of null and unity entries, and solve a smaller inhomogeneous linear matrix problem, albeit with fewer unknowns determined initially. The remaining unknowns are captured by solving subsidiary matrix equations of generally smaller size than the reduced system matrix. Setting H:'A#' = 1 and H$'A# = 0 (or the reverse) allows for the solution of the suite of unknown coefficients in the column vector. When these solutions (or the reverse) are placed in the electromagnetic field formulas, the dyadic Green's function elements are generated. This may be put down symbolically as
S represents system (197) and S( 1,O) corresponds to the azimuthal magnetic source turned off. F is the field equation operator and produces the correct component (first superscript) for the desired field (electric or magnetic field, indicated by the first subscript). It is clear that if we had only one term for each summation, then the compressed form in (197) would represent a 6 x 6 system, and it would be possible but extremely tedious to pull the forcing terms through the determinant solution for each unknown and obtain explicit real space dyadic Green's function elements. But for our problem here with incommensurateheights (or other geometrical dimensions), the problem is in practice impossible to solve for an explicit dyadic Green's function. One would look naturally to reciprocal space to obtain by analytical means compact explicit dypdic Green's function expressions. But there is a tremendous advantage in not going to reciprocal space, and that is that the implicit dyadic Green's function can be used to obtain the field behavior directly without any transformations. Furthermore, these Green's functions can be used to obtain the s-parameters for the circulator structure too. Mention is made here that the diagonal nature of the relation between different region amplitudes in the outer zone allows one amplitude (per mode class) to characterize the entire zone (the T amplitude was chosen). Once they are determined by (197), the subsidiary diagonal equations (100) and (1 12), stated explicitly below, may be employed to capture the B amplitudes.
RBT-
0
...
0
0
R2BT-
...
0
lo
0
. .. R L i -
THEORY FOR INHOMOGENEOUS FERRITE
195
D. Conclusion Source constraint equations have been combined with the mode-matching technique to obtain in direct space implicit dyadic Green’s function elements for a very general canonical circulator geometry. The approach allows the inclusion of layered surrounding material beyond the radius of the ferrite puck, as well as a covering material above the puck, enabling a more realistic or complete description of the circulator structure. New self-adjoint operators are found consistent with the surroundingmaterial having dielectric, magnetic, or simultaneouscharacter. Inclusion of substrate and superstrate effects are possible with this treatment. Assessment of vertical field fringing is a natural consequence of this analysis approach. Obtaining the dyadic Green’s function as described in the paper is particularly useful because field information and s-paramaters may be found in real space, completely avoiding typical inverse transformations. In this approach are included the inhomogeneous properties of the circulator puck due to chosen radial variation of the ferrite material parameters, nonuniform applied magnetic dc biasing field, or finite puck geometric effects on the bias field. All of these inhomogeneousproperties in the puck region are naturallyincorporated into the dyadic Green’s function. Numerical evaluation should be efficient and only limited by the well-understood features of the mode-matching technique. The geometry considered correspondsto that found when one of two self-biasing configurations are employed using hexagonal ferrite films [16-181: (1). The puck itself is a hexagonal material with the exterior zone a layered combination of materials or (2) the puck is spinel or garnet material and the exterior material a hexagonal material (it may be anisotropic, a situation not treated here). In the first structure the electric wall condition above the puck is maintained by microstrip metal. For the second structure, it is maintained by a combination of the microstrip metal and low-coercivity“keeper” plate (or cover) permalloy. Hexagonalmaterials include Ba, Pb, or Sr, iron oxide compounds. The garnets are the usual Y, iron oxide compounds, and the spinels are the Ni, Li, or MgMn, iron oxide compounds. If we had not insisted on self-biasing,but used an external magnet lying immediately above the circulator puck, configuration Fig. 2 would have resulted. (This structure was not be addressed here.) The methods covered in this section still apply, with extra allowance made for the magnet excluding fields in the I region if
196
CLIFFORD M. KROWNE
it is viewed as a perfect conductor forming electric walls at its boundaries. Field exclusion occurs for a volume encompassing the magnet’s thickness and diameter 2R. Operator methods developed in this section are also still required. One may wonder what happens if the external radial zones can be constructed out of simple uniform material for each zone. For such a situation, the operator work here may be avoided and considerableease is found in Section I11 in obtaining a formulation compared to the labor required in this section.
111. IMPLICIT 3D DYADICGREEN’S F U ” I O N WITH SIMPLE EXTERNAL MATERIAL USING MODE-MATCHING A. Introduction Recently, consideration of field extension into the surrounding medium, beyond the circulator perimeter, has been examined with the idea of constructing a more descriptive dyadic Green’s function, which can allow assessment of external dielectric effects (actually both isotropic permittivity and permeability). This has been done for both 2D [ 191 and 3D models [20]. But neither model has the ability to enable field extension and fringing such that fields may extend out and then above the height of the circulator nonreciprocal puck. It is the intent here to show that a 3D approach may correct this deficiency by combining source constraint equations with a rigorous mode-matching treatment where the outlying radial zones are single cylindrical regions with one material throughout each zone. Restriction of the structure radial zones vastly simplifies the analysis compared to Section I1 where considerable work was required in an operator theory development. None of that is needed here. Mode orthogonality is utilized to encourage sparsity in matrix system development, eventually resulting in implicit dyadic Green’s function elements. When the Green’s function elements are obtained this way, they are in direct space, and consequently field information and s-parameters arise immediately, completely avoiding typical inverse transformations. The approach allows the inclusion of surrounding dielectric material as well as a cover plate above the ferrite puck. The approach is a good approximation to a very complicated geometric and inhomogeneous problem, given the irregular effects arising from application of the dc biasing magnetic field and the actual finite-width microstrip input and output lines. For narrow microstrip lines the expectation that the fields extend beyond the device perimeter, with azimuthal symmetry, is very good, and the dyadic Green’s function elements will be nearly exact, only limited by numerical precision. When some of the microstrip lines attain widths which are a noticeable fraction of the puck radius, the error introduced by the symmetry assumption for r > R for the
THEORY FOR INHOMOGENEOUSFERRITE
197
R
FIGURE 4. The ferrite circulator structurewith a uniform radial zone and a region above the device puck. This figure is formed by taking a cut plane at I$= const (in 3D).
fields will be directly related to the fraction of the circumference occluded by the presence of the line itself.
B. Fields, Mode-Matching Technique,Nonsource and Source Equations Consider the situation where the electromagneticfield occupies three regions (see Fig. 4). The first region C is that filled by the ferrite nonreciprocal material for r < R andO 5 z 5 h c . The second region 0 has r > R andO 5 z 5 h o . And the third region I has r < R and hc 5 z 5 h o . Conducting walls are assumed at z = 0, ho and the radiation condition in effect as r + 00. Such a geometry could result from magnetless, self-biased circulators.
m=O n=-m m
m
m = l n=-m m
m
L
m = l n=-m
J
- uLA:!,,airnJ,' (uimr)eing
198
CLIFFORD M.KROWNE m o o
m=On=-m o o m
m=l n=-m
+ Li pRoAn ;
L
: K, (afmr)
c a m
m=O n=-m
M
M
Field expressions in (210)-(211) for the circulator puck region are available in [2, 111. Use of tangential field continuity at the r = R interface will be used to connect the various regions at the appropriate z value. That is,
This expression must be applied at the CO interface and at the I0 interface. It will not hold at the CO interface for the magnetic field components because the r = R surface contains the singular forcing function (a delta function). In the spirit of applying rigorous mode-matching theory by projecting testing functions on the
THEORY FOR INHOMOGENEOUS FERRITE
199
continuity equations and integrating, we find for the I0 interface
and for the CO interface,
Lastly for the 0 (I
+ C) interface,
+ij,
are z-dependent eigenfunction factors of the particular region i = Here I, 0, C as applied to the f type of field continuity equation for component j type. Projection of these functions on the field continuity equations in the fashion of (215)-(219) allows their orthogonal properties to be used, thereby reducing the complexity of the eventual system of equations describing the problem. These testing functions are
+,:
=
= cos(kim[z- h c l ) ;
+Lzm = $i#,= ~ i n ( k ; ~ [-z h c l ) ;
,+:
= cos(k;,z);
+$,
= sin(kF,z);
+zm= cos(kgz); +&,
mlr
k,,c - -, hC mlt , :k = -, hC 0 mlt k,, = -, h0
I mlt kzm = -, hl I
mn
k,, = -, hI
m = 0, 1, ..., M I (220) m = 1 , 2 , . . . , M I (221)
m = 0, 1 , . . . , M C
(222)
m = 1 , 2 , . .. , M c
(223)
m = 0, 1 , . .. , M o
(224)
m = 1,2,. .. , Mo k,,0 = -, m= (225) h0 Notice that the infinite summations contained in the field representations have i = I, 0, C in the respective regions. Mi must be chosen been truncated to M i , = sin(kgz);
200
CLIFFORD M. KROWNE
carefully. M Iand Mc may be chosen relative to M o as
There is some flexibility in the specific ratio converting M o into the other summation limits, and discussions regarding this limit management is found elsewhere [ 131. These issues are referred to as the relative convergence behavior. How large to choose M o is an issue of absolute convergence, specifically how accurate an answer we desire for the problem solution. Selection of the subset (if a full set is not needed) of continuity equations is not unique [ 141, the choice being dependent upon individual inclination, sometimes numerical advantage, and the requirement that the number of equations equal the number of unknowns when the missing source equations are included (to be addressed later). Thus, we choose (215a), (215b). (218), and (219). The orthogonality and projection relationships relevant to these equations are [12]
m, m' > 0
(229)
THEORY FOR INHOMOGENEOUS FERRITE
m,m’ > 0
=ilosin(Ez
) sin (T-:[ z
20 1
(234)
)
- h] dz = Ismtm I0 (235)
where
Placing (227)-(236) into (215a), (215b), (218), and (219), where all summations in the field components in the inner products [i.e., as in (215)] are switched to m’ indexing, yields the four system equations
202
CLIFFORD M.KROWNE
m'=O
m'=O
m = 0, I , . . . , M o
(240)
m'= 1
m = 1 , 2 , . ..,M o These four system equations constitute the unknowns are
(241)
MI + M o + 1) individual equations. But
1 anmO; m = 0, 1,. . . , MC
2 anmO; m = 1 , 2 , ...,M c
A;:;
m = 0 , 1 , ..., M O
hO Anm; m = l , 2 , ..., M o
A$;
m = 0, 1 , . . . , M I
hl Anm; m = 1 , 2 , .. . , M I
+
+
+
making a total of 2 ( M , M o Mc) 3 of them. Notice that the m = 0 case has been left off of the second radial mode coefficient index listing because it corresponds to the zero perpendicular thickness situation and we expect the first
203
THEORY FOR INHOMOGENEOUS F E R R m
+
radial mode to completely dominate. Therefore, we see that exactly 2Mc 1 more equations are required to describe the structure. The missing equations come from two source equations governing the exchange of energy between the magnetic delta forcing functions acting on the r = R surface through a point aperture and the structure. The componentsof the surface magnetic field are chosen to couple the external environment to the circulator structure, thereby defining a Green's function construction. The two source equations are H @ A ( Z~) , = H$'S(Z - z')S(# - #')A#' H z ~ ( 4Z), = H;'S(Z - z ' ) S ( ~ - #')A#
+ H:(#,
+ HP(#,
Z)
(248)
Z)
(249)
Both H+A and H z may ~ be expressed by a Fourier series constructed from the same expansion functions as used to represent the circulator puck field components on the interval (0, h c ) by simply using the extended field on ( - h c , h c ) [15].
(254)
H$Am = H;Am = 0
Here ex indicates extended field and (252) results from the cos or sin nature of the field component variation within the circulator puck. It is convenient to retain the exponential Fourier series on (-n,n) for an additional representation of the 4 dependent coefficients found in the z-expansions (250) and (25 1). Thus,
n=-w 60
n=-w
204
CLIFFORD M.KROWNE
With the foregoing information, namely (248)-(258), the proper constraints on the sources can be imposed. It is done through the requirements that the tangential magnetic field componentsto the r = R surface be continuous in a limiting process just to the inside and outside of the device perimeter. Hilr=R-= H & A I r = R +
Hflr=R-
= HzAIr=R+
(259ab)
Use of the azimuthal orthogonality property and recognizing the similarity of the perpendicular coordinate expansions in both the puck and aperture (source) surface, Hcir
6mn = H i A m n
cir Hzmn
- HfAmn
(260ab)
where we have written out cir to emphasize the difference between the puck expansion coefficients and the aperture mnth coefficient which happens to be a cosine type. Placing (252) into (257) gives, considering the left-hand side of (260a) first,
HiAmn= 5 J h x H m , ( ~z)cos( ,
nhc 0 Next insert the source equation (248) into (261),
Retrieving
m )zez- i n 6 d 4 d z F
H t in (209), the second integral in (262) can be evaluated:
(261)
THEORY FOR INHOMOGENEOUSF E m
205
At this stage, constraint (260a) can be imposed, invoking (213) for the righthand side, using the previous (262) and (263) results. The fifth system equation, containing Mc 1 subequations, can now be stated as
+
:AAmNaLmO
+ :A2mNanm02
- U 0, , eOA ~0 ~ , ~ ~0 ~ , K ~ ( ~ m~ =~0,, R1, ..., ) ] IMC ~ ? ~ ;
(264)
The sixth system equation containing MC subequations is found by treating (219b). Place (253) into (258). obtaining
Inserting the aperture source expression for the perpendicular magnetic field component into (265) gives
Using (207), the outside integral second term is expressed as
With the help of (212) for the puck field on the left-hand side of (259b), and employing (266) and (267), the final perpendicular source equation is written as I
i [2AArnNanrnO
+ hz ~ n2m N a n 2 m OHZCA,. I m = 1 , 2 , . .. , MC (268)
C. Implicit Dyadic Green's Function The six system equations, four sourceless as seen in (238)-(241), and two with magnetic sources in (262) and (264), may be stated in compact form in a single
206
CLIFFORD M.KROWNE
matrix formula: o h0 ChO
Oh 0 IhO
ChO ChO
A representative nmth element for each class of unknowns is shown in the lefthand-side column vector. The matrix entries are as follows: zero for no contribution of the subscript type of unknown, one for a single entry contribution, and C for a sum of all that particular class of unknowns indicated by the subscript. On the right-hand side of the equation are the source forcing terms. It is possible to reduce the size by analytical effort of the system matrix (269) due to the appearance of null and unity entries, and solve a smaller inhomogeneous linear matrix problem, albeit with fewer unknowns determined initially. The remaining unknowns are captured by solving subsidiary matrix equations of generally smaller size than the reduced system matrix. Setting H:rA@' = 1 and H$A@ = 0 (or the reverse) allows for the solution of the suite of unknown coefficients in the column vector. When these solutions (or the reverse) are placed in the electromagnetic field formulas, the dyadic Green's function elements are generated. This may be put down symbolically as Gr3@,z;z E,H;H(r,
4, z) = Fi,Y[S(I?0)l G'&3;H(rr @ z.@ 4, z ) = Fi,Y[S(O,111
(270) (27 1)
S representssystem (269) and S( 1,O) correspondsto the azimuthal magnetic source turned off. F is the field equation operator and produces the correct component (first superscript) for the desired field (electric or magnetic field, indicated by the first subscript). It is clear that if we had only one term for each summation, then the compressed form in (269) would represent a 6 x 6 system, and it would be possible but extremely tedious to pull the forcing terms through the determinant solution for each unknown
THEORY FOR INHOMOGENEOUS FERRITE
207
and obtain explicit real space dyadic Green’s function elements. But for our problem here with incommensurate height (or other geometrical dimensions), the problem is in practice impossible to solve for an explicit dyadic Green’s function. One would look naturally to reciprocal space to obtain by analytical means compact explicit dyadic Green’s function expressions. But there is a tremendous advantage in not going to reciprocal space, and that is that the implicit dyadic Green’s function can be used to obtain the field behavior directly without any transformations. Furthermore, these Green’s functions can be used to obtain the s-parameters for the circulator structure too.
D. Conclusion Here we have combined source constraint equations with the mode-matching technique to obtain in direct space implicit dyadic Green’s function elements for a more general circulator geometry. The approach allows the inclusion of surrounding dielectric material as well as a cover plate above the ferrite puck, enabling a more realistic or complete description of the circulator structure. An assessment of vertical field fringing is possible with this treatment. Obtaining the dyadic Green’s function in this manner is particularly useful because field information and s-paramatersmay be found in real space, completelyavoiding typical inverse transformations. But the theory avoids many of the complicationsinherent in the more complicated radial zones seen in the geometries considered in Section 11. In this approach are included the inhomogeneous properties of the circulator puck due to chosen radial variation of the ferrite material parameters, nonuniform applied magnetic dc biasing field, or finite puck geometric effects on the bias field. All of these inhomogeneous properties in the puck region are naturally incorporated into the dyadic Green’s function. Numerical evaluation should be efficient and only limited by the well-understood features of the mode-matching technique. The geometry considered correspondsto that found when one of two self-biasing configurations are employed using hexagonal ferrite films [16-181: (1) the puck itself is a hexagonal material with the exterior material a dielectric or (2) the puck is spinel or garnet material and the exterior material is a hexagonal material. In the first structure the electric wall condition above the puck is maintained by microstrip metal. For the second structure, it is maintained by a combination of the microstrip metal and low-coercivity “keeper” plate (or cover) permalloy. Hexagonal materials include Ba, Pb, or Sr, iron oxide compounds. The garnets are the usual Y, iron oxide compounds, and the spinels are the Ni, Li, or MgMn, iron oxide compounds. If we had not insisted on self-biasing but used an external magnet lying immediately above the circulator puck, configuration Fig. 5 would have resulted. Again
208
CLIFFORD M. KROWNE / / /// / / / / / / / / / / / / / / /
I t
I
,
i R
FIGURE 5. The femte circulator structure with a uniform radial zone and a region above the device
puck with the presence of an external biasing magnet. This figure is formed by taking a cut plane at
4 = const (in 3D).
the methods covered in this section apply, with extra allowance made for the magnet excluding fields in the I region if it is viewed as a perfect conductor forming electric walls at its boundaries. Field exclusion occurs for a volume encompassing the magnet’s thickness and diameter 2R. This structure will not be addressed here.
N.2D DYADICGREEN’SFUNCTION
FOR PENETRABLE WALLS
A. Introduction In the previous inhomogeneous 2D model for microstrip circulators [l], it is assumed that the non port boundaries are magnetic walls, confining energy exchange to only the ports. However, in a theoretical treatment which goes into great lengths to correctly model the detailed internal circulator field behavior, and allow extraction of the device s-parameters, by viewing the circulator itself as made up of a sufficient number of annuli, it may be both reasonable and perhaps necessary to understand the effects of electromagnetic field extension into the non-ferrite region. This extension is accomplished by still having the ports interface with the external circuit, but now replacing all magnetic walls on contour lines between ports with penetrable boundaries obeying the proper electromagnetic interfacial boundary conditions. We develop here a new dyadic Green’s function [21], which allows us to find out the actual effect in switching from a hard impenetrable magnetic wall to an external dielectric material with permittivity &d and permeability pd (see Fig. 6). The three common cases of air as the external medium, a dielectric as the external
THEORY FOR INHOMOGENEOUS FERRITE
Port - femte region
a
& -\
; (CL, PO,
DIRECT
209
ISOLATED
Inhomo. Wall ( non - port)
b
FIGURE6. Cross section through a z = const circulator plane parallel to the ground plane. The diagram shows the special case of only three ports, labelled counterclockwise. External material exists beyond the r = R puck radius.
medium, and an unmagnetized ferrite as the external medium are all easy to treat with the new dyadic Green's function. Obtaining an explicit dyadic Green's function is a result of reducing the problem down from three dimensions examined in Sections Il and III to two dimensions here.
B. Formulation Applying the radiation condition as r + 00, leads to the selection of the modified Bessel function of the second kind K,,(kdr) for use in the external field construction, r > R . It is assumed that the same field modes are maintained in the device for radii exceeding the circulator radius so that a consistent 2D modeling procedure holds internal and external to the device. Additionally, the microstrip edge effect and fringing field contribution provide the correct field available from the circulator puck for coupling to the external environment when multiplied by the factor f. With these constraints, the internal TM, persists for r > R , = [ l / ( i W p d ) ] a E , d / a rand ,
Hi
Requiring continuity of the perpendicular electric field at r = R , f E f ( R ,4) = E,d(R9 $ 1 9
210
CLIFFORD M. KROWNE
Factor f is not a simple quantity, but one multiplicative component making it up may be thought of as the ratio of the difference between the total integrated field over the entire open surface to that contained in the puck, divided by that in the puck. It may be computed by capacitance calculations, and is estimated to fall between a few percent and 10% for typical device pucks. Factor f in an approximate way must both allow for a consistent fringing in the 2D model and for the fact that the 2D model is only approximately reflective of the 3D nature of the actual physical device. More discussion on these points is found in Section VI covering limiting dyadic Green's function forms for homogeneous ferrite devices. The forcing function for the Green's function is applied at (r', 4'). r = R , through the equality
H p ( R , 4) = H y ~ a ( 4 4'1A4'
+ H $ ( R , 9 # 4')
(274)
The perimeter azimuthal magnetic field can be represented by a ID Fourier expansion,
Azimuthal magnetic field in the circulator, as r + R from the inside within the last annulus i = N, is given by 00
H i ( R , 4) =
an0 [anN(reCuf)C:haN(r)+b.~(recur)C:~~,,,(r)]e'"@
(276)
n=-co
Here the coefficients anN(recur) and b,N(recur) come from a recursive transfer matrix process. Equating the fields in (275) and (276),
H i ( R , 4) = H F ( R , 4) an0 [anN(recur)c$,N(r)
(277)
+ bniv(recur)~$bN(r)]
Inserting (272) into (2781, r + R from the outside, utilizing azimuthal orthogonality, yields ano{ [w(recur)C:haN(r) +bnN(reC~r)~:hbN(r)~
THEORY FOR INHOMOGENEOUS FERRITE
21 1
C. Dyudic Green's Function With the solution of u,o from (279) in terms of the forcing field H ~ and A using the development in [ 1 11 as a guide, the elements of the dyadic Green's function in the i th annulus may be derived as l b O ani (recu)Cneai ( r ) GLdH;(r,4; R , 4') = 2n n = - w ynN- fkd
+ bni (recu>cnebi( r )e;n(,#,-,#,') yzc
i o p d K , &:R)
KA (kd R ,
(280a)
Dyadic Green's function element subscripts are interpreted as follows. The first refers to the resulting field type, the second to the forcing field type, and the third to the annulus under consideration. For the superscripts,the first is the resulting field component and the second the forcing field component. It is evident from (280) that the new dyadic Green's function elements are those of a circulator device with hard walls (namely, magnetic walls), but with a modification to the form of the denominator term. This modification is in the form of a subtraction from , depends on the properties of the external the original circulator divisor y n ~and medium, on the internal circulator field behavior through y i i , and on the factor f . The elements of the dyadic Green's function external to the circulator, in the nonport regions, are
(281a)
(281b)
(281c)
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CLIFFORD M.KROWNE
These r > R dyadic Green’s function elements are completely new and not only contain the denominatorcorrection term but also functionalforms which assure that any fields constructedfrom them will properly decay outside of the device. Results in both (280) and (281) correspond to a Green’s function which requires a radian azimuthal contour integration. That is, the differential element is d 4 . To make the Green’s function correspond to a line integral of differential element length ds, each dyadic element like G$Hd(r,4; R, 4’) should be multiplied by l/R. Besides being able to treat the case of an external dielectric with Ed and the case of the external medium being air is accomplished by setting &d = and pd = po, the free-space values. Finally, the case of an unmagnetized ferrite is done by setting &d = &f and pd = pf (the unmagnetized 2D diagonal value) which are available. In order to see that coupling the external medium to the circulator puck affects the field behavior, it is merely necessary to show that the Green’s function (GF) elements are modified. This may be done by examining the lower-order leading azimuthal terms. All GF elements have their denominators modified by the multiplier (1 - me;)where the ratio part is
Now from running both 2D Green’s function and FEM simulationsof a MnMg ferrite puck device of radius R = 0.270 cm and thickness h = 0.0675 cm [ h / ( 2 R )= h/D =0.125], we find the center frequency to be about 9.0 GHz [22]. For this device w,,, = -yMs = (2.8MHz/Oe) x (2300G) = 6.44 GHz (magnetization frequency). The off-diagonal relative permeability tensor element K M w m / w = 0.716 when wo x 0 (internal dc bias field small) and a,,,M 0 (losses neglected), and the diagonal relative permeability tensor element p x 1. The effective permeability will be pe = 0.488. The relative dielectric constant of the ferrite material is & f r = 13.3. The propagation constant in a surrounding air environment kd = 1.89 m = 4.80 radianskm. Electrical length radianskm and in the puck k, = w angles are defined as 6d = kdR = 0.509 rad and 0, = k,R = 1.30 rad. In terms of electrical wavelengths, D/Ad = 0.162 and D/Ae = 0.413. To simplify matters, let us assume enough uniformity to use the limiting form of (282) as the required number of annuli plus the disk N’ + 1 (N’ = N 1). Then we can write for the ratio
+
Evaluating (283) for the axisymmetric mode n = 0 and the first few asymmetric modes with nonreciprocal contributions n = f l and n = f2,we find that they are all a few tenths or smaller. The negative asymmetric modes are about ten times
THEORY FOR INHOMOGENEOUS FERRITE
213
smaller than the positive asymmetric modes. And for n = f 1 0 , the value is down by another factor of a thousand. Thus, we see from this example that the inclusion of the external medium modestly affects the Green’s function, and so we expect both modest changes in the field behavior and s-parameters.
D. Conclusion Dyadic Green’s function applicable to the case where the inhomogeneous ferrite circulator walls are no longer perfect magnetic conductors, a good but imperfect assumption at best, has been derived. The 2D dyadic Green’s function elements are seen to contain a modification from the impenetrable wall case, which is easy to recognize. The modification may easily be turned off, returning the device to the original hard wall situation. External permittivity and permeability effects on device electromagnetic fields and s-parameters may be found from the new dyadic Green’s function. This Green’s function is expected to be particularly relevant to new work proceeding on developing microstrip thin film ferrite circulators [16]. We note in passing here that the present approach will yield numerical results in a few secondsper frequency point versus times exceeding a thousandfold increase using intensive numerical 2D and 3D finite element method (FEM) or finite difference time domain (FDTD)method. The next section to come follows the philosophy in this section in that an explicit dyadic Green’s function is sought, but in three dimensions. A number of assumptions and approximations must be made to accomplish that, and these are covered in Section V.
V. 3D DYADIC GREEN’SFUNCTION FOR PENETRABLE WALLS A. Introduction In the previous inhomogeneous3D microstripmodel [2], it is assumed that the nonport boundaries are magnetic walls, confining energy flow to only the ports. This may be a reasonable assumption in view of the thin dimensions in the z-direction compared to the extent on the lateral surface. But by its very nature, a 3D model could allow accurate description of field extension beyond the circulator perimeter and fringing fields, depending upon how general and complex the formulation. Here we will develop a treatment which extends the perfect electric wall at the ground plane beyond the perimeter and assumes that the fields exit the device with no vertical fringing, essentially maintaining a distributionin the 0 < z < h region. This approach will enable a manageable derivation of an explicit dyadic Green’s function and the ability to model the field extension into the outlying dielectric
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CLIFFORD M. KROWNE
region while also maintaining the radial mode splitting inside the puck due to finite device thickness of the planar structure. Issues such as the real fringing near the microstrip line-circulatorinterface or the field fringing above the circulator height can’t be addressed. There are procedures to address the field fringing above the circulator height and they will be partly treated in B and C in this section, but for more general structures the procedures entail substantial complications because of the different electric wall heights beyond the perimeter, and this height may even vary depending upon the environment external to the actual device region. Some attempt to address these complications was already presented in Sections I1 and 111. Field extension is accomplished by replacing all magnetic walls on surfaces (in a cross section, a contour line) between ports with penetrable boundaries obeying the proper electromagnetic interfacial boundary conditions. A new 3D dyadic Green’s function is found, which allows one to find out the actual effect in switching from a hard impenetrable magnetic wall to an external dielectric material with permittivity &d and permeability pd. The three common cases of air as the external medium, a dielectric as the external medium, and an unmagnetized ferrite as the external medium are all easy to treat with the new dyadic Green’s function.
B. Formulation Figure 7 shows a three-dimensional sketch of the circulator structure. It is highly exaggerated in the axial direction z so as to make viewing easier. The circulator thickness is h. There are a finite number of ports placed at azimuthal locations 4; with angular widths A& and having their microstrip lines with physical widths w ; where i = 1,2, . . . , N,. For r < R, there is a solid cylinder of magnetized ferrite material, which is radially inhomogeneous. For r =- R, there is a homogeneous filling material, except of course where the ports exit and enter the circulator puck. At r = R is located the ferrite-external region inhomogeneous boundary, which constitutes an imperfect wall, that is, where field coupling is allowed. The ground plane is at z = 0, and the microstrip circulator metal restricting the puck fields is at z = h and r < R. Conductive metal is also found on the incoming and exiting microstrip lines at z = h. Microstrip line widths are required to be in total angular width A ~ =T X ( i = 1,2, . . ., N , ) A4; << 2n,assuring minimal microstrip line effect on the external fields. Figure 8 shows a cross section of the structure in Fig. 7 for the three-port case with ports labeled counterclockwise. Port a is the input port, port b the direct or coupled port and port c the isolated or terminated port. The ferrite material is characterized by permittivity E = &(r,4, z ) = E(r), and tensor permeability , po. On-diagonal elements p = p ( r , 4 , z ) = p ( r ) , K = K ( r , 4 , z) = ~ ( r )and
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215
Port Np
Z
a1
Port 1
FIGURE 7. Three-dimensional sketch of the circulator structure. Thickness h is exaggerated to make viewing easy. The inhomogeneous ferrite puck is contained within the r 5 R region.
permeability elements are p and po and the off-diagonal is K . External material is characterized by permittivity &d and permeability pd. This material completely surrounds the puck except for the small space caused by the exclusion due the microstrip lines, required to be a small angular proportion of the outside space.
Port - ferrite region interface
C
ISOLATED
Inhomo. Wall ( non - port)
a
DIRECT FIGURE 8. Cross section through a z = const circulator plane in Fig. 7. The diagram shows the special case of only three ports, labeled counterclockwise. External material exists beyond the r = R puck radius.
216
CLIFFORD M. KROWNE
None of the ports are ideal and in fact may have imperfect reflection, coupling, or isolation, for the respective a, b, and c ports. It is on the microstrip-puck interface surface given by coordinates (4, z ) at r = R where the forcing function for the Green’s function approach is applied with the particular distributions of the forms g(z)S(# - 6’)and h ( z ) 6 ( 4 - +’), to be discussed later in more detail in regard to magnetic field source components in the 6-and z-directions. g(z) and h(z) may be chosen to approximate that found for actual microstrip lines, or selected to be S(z - z’) with the required extra perpendicular integration. Either way is equivalent, and actual substrate thickness may dictate the best course of action. The magnetized femte material is analyzed by breaking up the cylindrical puck into one inner disk and N annuli, with the annuli being able to have different radial thicknesses. In general, each annulus has anisotropic and nonreciprocally acting material. However, it is possible for some annuli to be intentionally selected as dielectric and reciprocal material, thereby constructing ring like devices. Field structure for the inside of the puck is understood and so, other than brief explanations provided here with a summary of basic formulas, the reader is referred to the appropriate literature for more information. It is the intent here to connect the complicated field structure inside the inhomogeneous puck with that external to the puck, while maintaining as much of the earlier dyadic Green’s function format as possible, because it is known that numerical implementations are exceedingly efficient [23]. Coupled Helmholtz equations which govem the electromagnetic field behavior inside the circulator puck are given by [11,24l
V:E, + a E , -tbH, = 0
(284a)
+ c H , + dE, = 0
(284b)
V:H,
where the field coefficients including the coefficients of coupling b and d are
a = kz - k:
(285a)
b = -iwpok,c = -Po (k P
2
K
P
(285b)
- k z2)
K
d = iwEkz-
P
(28%) (285d)
The permeability tensor elements appearing in (285) come from the tensor
THEORY FOR INHOMOGENEOUS FERRITE
217
due to dc bias magnetic field in the z-direction inside the ferrite circulator puck. Relevant squared propagation constant definitions to the identifications in (285) are k2 = w 2& p
(287a)
k i = w2&po kz = w2&pL.,
(287b) (287c)
with the effective in-plane permeability pe ( x y or r# plane) given as Ile
=
p2 - K 2 ~
I-L
Transverse field components can be determined once (284) is solved for the perpendicular field components E, and Hz using the expressions SaH, aH, q a E , aE, E ---+?-++-+pr r a# ar r a# ar E @ = -r-a- H s -,+ - - - qaH, paE, aE, ar r a# ar r a# qaHz aH, u a E , aE, H - -- +p-+-+Ir - r a# ar r a# ar t aE, aE, paH, aH, 4 = -r a# - 4 ar + ra# - U- ar
(289a) (289b) (289c) (289d)
Here the constant coefficients in the derivative terms are defined as s = iW(pek2- p k : ) / D ,
(290a)
r = wKk:/D,
(290b)
K
q = -kzk2/D,
(290c)
Il
p = ik,(kl - k 2 ) / D S
(290d)
-k2)/Ds
(29oe)
u = iws(kl
K 2 t = -w&-k /Ds LL
D, = - ( k 2 - K:)2
(2900
+( w ~ E K ) ~
(291)
With the summary of the electromagneticfield solution within the circulator puck complete, we return to the problem of interest, which utilizes the field formulas inside of the puck and the properly obtained fields external to the puck.
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CLIFFORD M.KROWNE
Applying the radiation condition as r --f 00 leads to the selection of the modified Bessel function of the second kind K,,(adjr) for use in the external field construction, r > R. It is assumed that field conditions are maintained external to the puck in such a manner for radii exceeding the circulator radius that the same z-indexing modal set can be used internal and external to the device. With this assumption,
j=O n = - w w o o
He(r, 4, z) =
iufhj Sin(k,jZ)Kn(adjr)ein9
(293)
j=1 n=--w
Here the characteristic equation for the radial separation constant a d j is given in the outside region by adj
=
dm,kd
=W
m
(294ab)
where the perpendicular indexing for the discrete spectrum of allowed values is done according to
jn k,j = -; h
j = (0 or l), 2,
...
(295)
with the first j index choice determined by the first nontrivial field component. The azimuthal magnetic field component H$j (only the transverse part) may be made to retain a form congruent with the puck field construction by setting the coefficient factors q and t of the partial differential operators a / & and 8 / 8 4 equal to zero in (289d) (see [ 111 where exponential form is associated with the perpendicular propagation index j ) : (296) with (297ab) Similarly, the azimuthal electric field component E,di (only the transverse part) may be made to retain a form congruent with the puck field construction by setting the coefficient factors F and q of the partial differential operators a/a# and a / a r equal to zero in (289b):
THEORY FOR INHOMOGENEOUS FERRITE
219
with iwuA
(299)
Requiring continuity of the perpendicular electric field at r = R ,
f q 4)~= E,, ~ ( R4),
(300)
Factor f is chosen to correct for the actual vertical fringing seen in a 3D circulator and not treated by the model adopted here for simplicity reasons. The approach here does account for external field extension by assuming to first approximation that the jth perpendicular mode spectrum inside the puck holds outside also. f used here modifies the inhomogeneous formulas found in [24], and enables the effect of the external environment to be turned off (set f = 0), or left on (f # 0). Null f value must be done with some thought in the formulas to follow, but it can be demonstrated that the limit exactly returns the problem to the hard wall 3D case. Circulator field ES in the i = N t h ring at its most extreme position r = R is expressed as M
M
Here the modal coefficients aANj and b f N j ,i = 1 or 2, of the radial functions C,$,jNR, g = a or b , can be related by the recursive nature of the problem to the modal coefficients in the disk (i = 0; containing an implicit singularity at r = 0 which has been removed)
The "recur" indicates that the 4 x 2 matrix is determinedrecursively. Putting (292) and (301) into (300), and using longitudinal and azimuthal orthogonality, yields
f [:AAj&j
+ :A$&,]
making the coefficient in the external region
= a,djKn@djR)
(303)
220
CLIFFORD M.KROWNE
with e 1 zAnj
- II - ‘nNj
eA2. z nJ = 2I.
czln e a j N R + bt%jCiibjNR
n N J. c ; l z : a j ~ ~
+
b:ijci;bjNR
+aikjCr$ajNR
+ b%jCi2bjNR
(305)
+
+
(306)
a n22N j C i 2 a j N R
b:’,iCi;bjbjNR
Requiring continuity of the azimuthal electric field at r = R, (307) Circulator field EG in the i = Nth ring at its most extreme position r = R is expressed as
+
aiNjc::ajNR
+
bi~jcf:bjNR]
ein@
(308)
The dielectric region field E$ as r + R from the outside is obtained by (298) with the help of (292) and (293).
(309) Inserting (308) and (309) into (307), and using longitudinal and azimuthal orthogonality, yields
THEORY FOR INHOMOGENEOUS FERRITE
22 1
Here g(z) is the functional behavior of the forcing azimuthal magnetic field in the z-direction. The perimeter azimuthal magnetic field can be represented by a 1D Fourier expansion, 00
H,P"'(R,#) =
1 A,,e'"@, n=-m
A,, = - / r H F ( R , $)e-'"@d+ 2rc
Azimuthal magnetic field in the circulator, as r last annulus i = N, is given by
(315ab)
R from the inside within the
3
(3 18) with h 1 h
2
11
@1
+ b!thjctibjNR +a i i j c % j N R +b;fijc:ibjNR
(319)
12
@1
12 $1 + bnNjCnhbjNR + a%jc!tajNR
(320)
= 'nNjcnhajNR = 'nNj'nhajNR
+b;fijc:lbjNR
and H @ t ~ being j the overlap integral specifying the amount of the jth mode behavior contained in the forcing azimuthal magnetic field. It is
(321) H$j is found from m o o
$(r, 9 , Z) =
1 C cos(kzjz)[
h d 1 1 +Anja,oj
j=O
n=--60
where (296) has been used to find
+ ~Anjanoi]e'"@ h d 2 2
(322)
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CLIFFORD M. KROWNE
Inserting (322) into (3 18), letting r + R from the outside, and utilizing azimuthal orthogonality, yields
The perpendicular magnetic field forcing function for the Green's function is applied at (r', 4'), r = R , through the equality H p ( R , 4) = H ~ A ~ ( z )-S 4')A4' (~
+ H e ( R , 4 # 4')
(326)
Here h(z) is the functional behavior of the forcing perpendicular magnetic field in the z-direction. The perimeter perpendicularmagnetic field can also be represented by a 1D Fourier expansion,
Perpendicular magnetic field in the circulator, as r + R from the inside within the last annulus i = N, is given by
Equating the fields in (327) and (328),
gives
with
j the overlap integral specifying the amount of the jth mode behavior and H z ~being contained in the forcing perpendicular magnetic field. It is
1
h 2 HzAj = h H z ~ sin(kzjz)h(z)d z
(333)
THEORY FOR INHOMOGENEOUSFERRITE
223
H i is found from (293) and (3 10). Inserting (293) into (330), letting r -+ R from the outside, and utilizing azimuthal orthogonality, yields
C. Dyadic Green's Function
Let us introduce some new definitions in order to make transparent the forms of the disk modal coefficients in (325) and (334).
Introducingthese formulas into the two equations forming a simultaneous solution set for the disk modal coefficientsin terms of the driving field magnitudes,
A'.a' . + A2.a2 . = -HqAjA4'e-'"@', 1 nJ no/
nJ n0J
2rr (337ab)
-I
+
-2
2
1
Bnja;oj BnjanOj= -H ,A#re-in@' 2rr zAJ
enables a solution to be found
These equation forms are identical to those derived for the 3D circulator problem with hard walls between the ports. The right-hand sides are the driving terms. From (338) and (339) the dyadic Green's function elements for the ith ring are
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CLIFFORD M. KROWNE
THEORY FOR INHOMOGENEOUS FERRITE
225
Here z-directed weights K: and K $ are given by
.:=;I
h
21
h
COs(kzjz)g(z) d z ,
Kij =
sin(k,jz)h(z) dz
(353ab)
by invoking (321) and (333). It should be noted that the T ( r ) radial indexed functions are related when r = R to the definitions in (305), (306), (31 l), (312), (319), (320), (331), and (332), and their specific forms may be found in [lo]. The dyadic elements for r > R are to be discussed next. Consider (293), (310), (338), and (339) to obtain
Employing (292), (304), (338), and (339), one obtains
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CLIFFORD M.KROWNE
Generalizing (322) for r # R (r > R ) , and using (338) and (339) yields
ein(@-4') x [h$Aij(r)A:j- hd@Anj(r)AAi] 2
(359)
where the radial functions are
(361) Setting coefficient factors r: = 0 and q = 0 of the partial differential operators a/ar and 8/84 equal to zero in (289),
From (362). the next two dyadic Green's function elements are identified as
(363ab) remembering the definition of dyadic elements [25], and writing it for this situation as
Thus, using (354)-(357), and (363),
227
THEORY FOR INHOMOGENEOUSFERRITE
Enlisting (298), with the help of (364),
(367ab) Thus, the corresponding dyadic Green's function elements are
Setting coefficient factors q = 0 and r = 0 of the partial differential operators 8 / 3 4 and a/& equal to zero in (289c),
The two associated dyadic Green's function elements are identified as G$Hdj = pj-
ac$Hj u j
+--r
ar
a4
9
aGzHj GzHdj = p j - ar
u j acgHj + -r a+
(37 1ab)
Therefore, using (354)-(357), and (371), ,
0
0
0
0
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CLIFFORD M. KROWNE
D. Conclusion For the case where the inhomogeneousferrite circulator walls are no longer perfect magnetic conductors, a revised and extended 3D dyadic Green’s function can be obtained, which describes both the fields internal to (within the circulator perimeter) and external to the nonreciprocal medium. It is seen that updated disk modal coefficients are found enabling a transparent extension of the dyadic Green’s function within the inhomogeneous annuli regions. Dyadic Green’s function element expressions external to the puck are also provided, and of course they are entirely new in relation to the old perfect wall case. External permittivity and permeability effects on device electromagnetic fields and s-parameters may be found from the new 3D circulator dyadic Green’s function. The next section, Section VI, will treat the much simpler case of a homogeneous circulator puck. Insights gained in this section and in the previous one on 2D aspects of the dyadic Green’s function should be helpful.
VI. LIMITING DYADICGREEN’SFUNCTION FORMSFOR HOMOGENEOUS FERRITE A. 2 0 Dyadic Green’s Function 1. Introduction
Ferrite planar circulators are often built with circular symmetry, and they are electrically thin enough to warrant a model based upon a 2D approach [ 191. Furthermore, the use of a canonical structure can provide guidance on design for structures with noncanonical geometry. It has already been shown that s-parameter and field results may be obtained numerically with great efficiency using a 2D microstrip dyadic Green’s function, which is based upon judicious treatment of the source point singularity and mode-matching [23]. Such an approach avoids explicit use of the completenesstheorem for the homogeneous part of the problem, a tremendous advantage since the final dyadic Green’s function has one less infinite summation.
THEORY FOR INHOMOGENEOUS FERRITE
229
But in the previous 2D model [l], it is assumed that the nonport boundaries are magnetic walls, confining energy exchange to only the ports. This seems like a reasonable assumption, given that the actual device may only have magnetized material in the circular region, and that the surface current perpendicular to the perimeter on the microstrip goes to zero at the boundary. (This particle current being zero does not mean displacement current is null, thereby allowing a finite H$ value.) Convincing experimental evidence for this supposition is certainly found in the literature, and this is also indicated by recent theoretical and experimental data reviewed [22]. Nevertheless, in order to find out the actual effect in switching from a hard impenetrable magnetic wall to an external dielectric material with dielectric constant ~ d and , also thereby achieving the capability of varying the permittivity of the external dielectric, we develop here a new dyadic Green’s function satisfactory to accomplish the task. The permeability pd is also allowed to differ from the free space value. We restrict ourselves to the 2D homogeneous circulator case because it approximates many actual operating devices, readily allows explicitly developed compact Green’s functions, and enables the modificationsto the hard wall device to become evident. Thus, more transparent formulas will arise in this section compared to Section IV.
2. Fields and Constraints Applying the radiation condition as r +. 00 leads to the selection of the modified Bessel function of the secondkind K,, (kdr)for use in the external field construction, r > R: M
(374a) (374b) Requiring continuity of the perpendicular electric field at r = R,
we find (376) where
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CLIFFORD M. KROWNE
f attempts in an approximate way to allow for a consistent fringing in the 2D model, which has some inherent degree of 3D nature. The forcing function for the Green's function is applied at (r', 4'), r = R , through the equality
H r ( R , 4) = H @ A J (-~ #')A$'
+ H $ ( R , 4 # 4')
(378)
The perimeter azimuthal magnetic field can be represented by a 1D Fourier expansion, W
H,p"(R, 4) =
A,ein@
(379)
n=-W
Azimuthal magnetic field in the circulator disk, as r + R from the inside, is given by
Equating the fields in (379) and (381),
or
Inserting (374) into (380), r + R from the outside, utilizing azimuthal orthogonality, yields
3. Dyadic Green's Function With the solution of ano from (384) in terms of the forcing field H ~ Athe , elements of the dyadic Green's function for r < R (within the circulator puck) may be written
THEORY FOR INHOMOGENEOUS FERRITE
23 1
down as
4. Factor f Factor f is estimated as f = f w f,. f w weights the parameter dependence expressed in f,. Closed-form formulas, based upon self-consistent static solutions, exist for microstrip capacitive (electric) end effect [26]. Stretching the microstrip end so as to connect one corner to the other constructs the circulator perimeter, and allows us to roughly obtain f,. f
CT - C - _ Cf h - -cf c c AEd
,-
(39 1)
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CLIFFORD M. KROWNE
Assign A = nR2 and W = 2nR, and place them in (391) and in the equivalent additional radial length Alf expression, which relates to the fringing capacitance
c,:
c is the speed of light in vacuum, h the substrate thickness, Z, the microstrip impedance based upon &d dielectric loading causing an effective dielectric constant &de. We replace &de by &d under the W / h > > 1 limit. The left-hand side of (391) is given by [27,28]:
Using (392) and (393) in (391), the final formula for f is (noting that the r subscript denotes relative value) 0.824h &,d fp
=
&,d
+ 0.300 R/h + 0.042 - 0.258 R/h + 0.127 0.2217 +0.1061n
Cover location h’ >> h in deriving (394). One would expect the f w prefactor to contain information on the azimuthal mode structure, and this will be displayed as a dependence on the azimuthal mode index n. The prefactor f w may be very complicated, and the best that one can do is to obtain some reasonable degree of approximation. But that will not be done here.
B. 3 0 Dyadic Green’s Function 1. Introduction Although the theoretical work has been completed for a 3D model of the ferrite circulator, it has been done for the relatively general case of an inhomogeneous device puck. Here we would like to present the considerably simplified case for only a disk which is homogeneous [20]. The motivation for this is twofold. One is a historical parallel with the developments for the original planar work, which in the earlier years found sufficient accuracy using a uniform ferrite region in a 2D model. The other is the more streamlined formulas and more transparent formulation. The other part of what we want to do here is to allow for either perfect magnetic walls as before, containing the fields within the nonreciprocal region, or for
THEORY FOR INHOMOGENEOUS FERRITE
233
imperfect walls leading to field extension into the area external to the device ferrite material. In the previous inhomogeneous 3D microstrip model [2], it is assumed that the nonport boundaries are magnetic walls, confining energy flow to only the ports. This may be a reasonable assumption in view of the thin dimensions in the z-direction compared to the extent on the lateral surface. But by its very nature, a 3D model could allow accurate description of field extension beyond the circulator perimeter and fringing fields, depending upon how general and complex the formulation. Here we will develop a treatment which extends the perfect electric wall at the ground plane beyond the perimeter, and the microstrip conductor covering the ferrite puck into the region r > R but excluding all port areas where microstrip lines enter and exit the device.
2. Fields and Constraints Applying the radiation condition as r + 00 leads to the selection of the modified Bessel function of the second kind K,(adjr) for use in the external field construction, r > R. It is assumed that the electric wall conditions are maintained in the manner described previously for radii exceeding the circulator radius so that the same z-indexing modal set can be used internal and external to the device. With this assumption,
Here the characteristic equation for the radial separation constant udj is given in the outside region by
where the perpendicular indexing for the discrete spectrum of allowed values is done according to jn kzj = -; h
j = (0 or l), 2,. . .
(398)
with the first j index choice determined by the first nontrivial field component. Azimuthal magnetic field component H$ (only the transverse part) may be made to retain a form congruent with the puck field constructionby setting the coefficient factors q and t of the partial differential operators a/ar and a/a# equal to zero
234
CLIFFORD M. KROWNE
(see [l 11):
with
p=-
ikzj
ki - kZj
, u=-
iW&d k i - kZj
Similarly, the azimuthal electric field component E$ (only the transverse part) may be made to retain a form congruent with the puck field construction by setting the and a/ar equal coefficient factors F and q of the partial differential operators to zero:
with
Requiring continuity of the perpendicular electric field at r = R ,
f q R , 4) = E,d(R,4)
(403)
Circulatorfield ES (in the i = Nthring, N = 0) at its most extreme position r = R is expressed as 0 3 0 3
EE =
C C c o s ( k ~ j Z ) [ a ~ o j J n ( a l j+a~OjJn(azjR)]e'"@ R) j=O
(404)
n=-m
Putting (395) and (402) into (401), and using longitudinal and azimuthal orthogonality, yields
f [Jn(aljR)anoj 1 Jn(a2jR)a;oj] = a,djKn(adjR)
(405)
making the coefficient in the external region
Requiring continuity of the azimuthal electric field at r = R, (407) f E ; ( R 4 ) = E,d(R. 4 ) Circulatorfield E; (in the i = Nth ring, N = 0) at its most extreme position r = R is
235
THEORY FOR INHOMOGENEOUSFERRITE
expressed as
Q2j + -(iwpo + sj)cl,)J;(a2jR)
(408)
bj
The dielectric region field E$ as r -+ R from the outside is obtained by (401) with the help of (395) and (396). 0 0 0 0
$!I
=
C
j=l n=-m
i sin(kzjz)
iPjn
d
4--an,Kn(ad,R) R
1
e'"'
(409) Inserting (408) and (409) into (407), and using longitudinal and azimuthal orthogonality, yields
The azimuthal magnetic field forcing function for the Green's function is applied at (r', #), r = R, through the equality
H p R , 4) = Ht$'Ag(z)a($
- #')A#' 4-H:(R, # # 9')
(413)
Here g(z) is the functional behavior of the forcing azimuthal magnetic field in the z-direction. The perimeter azimuthal magnetic field can be represented by a 1D
236
CLIFFORD M. KROWNE
Fourier expansion, 00
Hp(R,#) =
1 A,e'"$, n=-m
An =
2n
H r ( R , #)e-'"@d#
(414)
--11
Azimuthal magnetic field in the circulator, as I + R from the inside within the last annulus i = N = 0, is given by
Equating the fields in (414) and (413),
H;(R, 4 ) = H r ( R , 9 ) gives
and H@fAjbeing the overlap integral specifying the amount of the jth mode behavior contained in the forcing azimuthal magnetic field. It is
Htj is found from
;=O n=-w
where (398) has been used to find
237
THEORY FOR INHOMOGENEOUS FERRITE
where 2AAj and
are given by
Inserting (419) into (4 17), letting r -+ R from the outside, and utilizing azimuthal orthogonality, yields
= - 1H t ~ j A 4 I e -in@'
2n
The perpendicular magnetic field forcing function for the Green's function is applied at (r', + I > , r = R, through the equality
Hp"'(Rt4) = H z A ~ ( Z ) S (&A# ~
-k H,d(R, 4
# 4')
(425)
Here h(z) is the functional behavior of the forcing perpendicular magnetic field in the z-direction. The perimeter perpendicular magnetic field can alsobe represented by a 1D Fourier expansion,
c oi)
Hp"'(R,#) =
n=-w
Bnein@,
Bn = 2n
/" -"
H:a(R, #)e-'"@d 4
(426)
Perpendicular magnetic field in the circulator, as r + R from the inside within the last annulus i = N = 0, is given by
Equating the fields in (426) and (427),
H;(R, 4) = H p ( R , 4 )
(428)
238
CLIFFORD M.KROWNE
gives
and H , A ~being the overlapintegral specifyingthe amount of the jth mode behavior contained in the forcing perpendicular magnetic field. It is
H S is found from (396) and (410). Inserting (396) into (429), letting r + R from the outside, and utilizing azimuthal orthogonality, yields
3. Dyadic Green's Function Let us introduce some new definitions in order to make transparent the forms of the disk modal coefficients in (428) and (437).
-1 c.-)12j B n1.=I Jn(a2jR)- f h:Aij, bj
-2 cj - A l j Jn(a2jR) - f ':A:j Bnj = bj (434ab) Introducingthese formulas into the two equations forming a simultaneous solution set for the disk modal coefficients in terms of the driving field magnitudes,
1 + Aijui0 = H ~ I AA4~e 2n I
Atja;,
whose solution is
-in@'
,
B,!,ja;,
1
+ Bijaio = Hz~jA@ e 2n I
-in@'
(435ab)
THEORY FOR INHOMOGENEOUS FERRITE
239
These equation forms are identical to those derived for the 3D circulator problem with hard walls between the ports. The right-hand sides are the driving terms. From (436) and (437) the dyadic Green’s function elements are determined to be (c subscript on the left-hand side of the equations denotes circulator disk since the ith ring degenerates to i = 0)
(439)
c
1 ” 0 3 1 i K$ sin(kzjz)[L?ijCif,jO(r) G z H c= 2n j = I n=-w DABj
240
CLIFFORD M. KROWNE
. " "
(447)
(449) Here z-directed weights K$ and Kij are given by (450)
by invoking (419) and (431). The dyadic elements for r > R possess the same form as previously provided in Section V and so will not be repeated here.
VII. SYMMETRY CONSIDERATIONS FOR HARD MAGNETIC WALLCIRCULATORS A. 2 0 Dyadic Green$ Function 1. Introducrion
Green's function methods are widely employed for analyzing circulators. Most circulators are designed with symmetry [29], and this fact may be used to significantly reduce the number of Green's function elements which must be calculated when determining the s-parameters and electric field. Although we have
THEORY FOR INHOMOGENEOUS FERRITE
24 1
$b = 2n/3
b
Port - ferrite region
-
( non port)
C $C =
- 2d3
FIGURE9. Two-dimensional (2D) circulator diagram showing the special case of only three ports, labeled counterclockwise. Port interfaces act as source driving functions, with the rest of the perimeter acting as a magnetic wall. Inside the perimeter located at r = R , an inhomogeneous ferrite material exists, which is broken up into N annuli, each annulus being considered uniform in material.
demonstrated how such simplified elements may be obtained from the general expressions which are valid for arbitrarily placed ports for the case of a three-port inhomogeneous device [2], here derivations for this device as well as the fourport and six-port cases will be given to provide a complete picture for multiport symmetric inhomogeneous circulators. The value of the Green’s function approach, besides its great mathematical elegance at times, is that it allows fast calculation for canonical structures such as the circular circulator. Thus, the utilization of the formulas provided here greatly assists the effort in simulating the s-parameters and z-component of the electric field E,. Focus is on finding the s-parameters because from them inhomogeneous circulator performance can be assessed. However, discussion is given on getting E,(r, 4) so that field plots are available. Presentation to follow will use a counterclockwiselabeling of the ports, shown in Fig. 9 for the case of a symmetric three port device. Note that the circulator puck is composed of concentric rings with differing ferrite parameters, creating inhomogeneous internal loading. 2. Three-Port Symmetric Circulator
There are a tremendous amount of simplifications which result by constraining the inhomogeneous circulator to a symmetric disposition of the port locations. Ea
= GaaHa
GabHb
+ GacHc
(45 la)
242
CLIFFORD M.KROWNE
= G b a H a -k
-k
GbcHc
Ec = G c a H a -k G c b H b 4-
GccHc
Eb
GbbHb
(45 1b) (45 1c)
In the general case for the 2D model, (45 1) holds for an inhomogeneousthree-port device. Examination of the EH dyadic Green's function element for n = N, the last ring, allows full advantage to be taken of the inherent threefold symmetry [lo]. .
m
Let us make this expression more transparent by defining the source azimuthal location to be 4j = 4: and the field location to be 4 = $i . Furthermore, abbreviate
G"EQ;IN (R,4; R,4:)
I
= G(4iT 4j)
(453)
@: =@ j
On the right-hand side of (452), collect the azimuthal exponents into one factor, while noting that the radial variation, here with r = R, is stored in the prefactor 7;; = 7;; (R). Then (451) becomes
(454) Right away, we notice from (454) that the Green's function only depends upon the difference between the source and field locations. That is,
G(4iv 4 j ) = G(4i - 4 j )
(455)
This does not mean, however, that G(@i,4,) = G($,, 4i). In fact, because the material is a nonreciprocatingmedium, we know this can't hold. Certainly, circulating action would cease if this type of Green's function dyadic element symmetry existed. Another clue that this type of symmetry doesn't exist is seen by reexamining (454) again. Let us find the G($,, 4i) dyadic element from (454) by switching azimuthal angles. . o o
Now define a new summation index as n' = -n. Substitutingthis into (455) yields
.
-02
(457)
But because notation is arbitrary, change the n' into n for the index, and recognize that the order of summation doesn't matter, especially as the same integers are
THEORY FOR INHOMOGENEOUS FERRITE
used as in the original G(&,
243
Green’s function. .
w
Comparing this to the expression for G(qji, 4j) in (454) enables us to see that except for the prefactor, the dyadic Green’s function formulas are identical. It is this prefactor, nevertheless, which is all important here and maintains the device nonreciprocity! pZNis not equal to p;i, or
(459) Because of the all important relationships in (459), G ( @ j 4i) , # G(4ii,@ j )
(460)
is true and we are assured of our nonreciprocity. However, property (455) and the symmetric angular distribution of the port locations will reduce the number of actual Green’s function elements required to be calculated. Note that the notation Gij
= G(4iV 4j)
(461)
is used in (45 1). Thus, by evaluating the Green’s dyadic for @i = # j , the self-terms, it is found that .
w
All self-term Green’s function dyadic elements are equal, and we denote this fact by assigning
Equation (462) obviously also means that G(4i, 4i) = G ( + j , 9j)
(464)
For the off-diagonal Green’s function dyadic elements, it is found that the number of unique terms is less than the total number of off-diagonal elements. Denote the total number of radially located ports as N n p . Then the total number of diagonal elements is NT,diag
= Nm
(465)
but the number of unique diagonal elements is one by the argumentsin the previous
244
CLIFFORD M. KROWNE
paragraph. The total number of off-diagonal elements is
but the number of unique off-diagonal elements is considerably less,
for an even number of ports and
for an odd number of ports. A device with three ports would have Nu,+d = 2. This is a much smaller number than the result in (466), giving NT,~-,,= 6. The savings for larger Nnp rapidly goes up because of the quadratic term in (466). The determinant of the system to solve for the radial azimuthal &-fields also displays all the individual Green's function dyadic elements.
Let us define
- 4 .J
A#.. '1- 4 .I
(470)
Recognizing that a right-handed system with (r, 4) or ( x , y) in the plane of the paper requires counterclockwise labeled ports (i.e., i = a, b, c or i = 1, 2, 3) to have progressively more positive valued azimuthal angles, if the input port angle is set to 0 radians, then $a
= 0;
4b
= 2x13; 4~= 4x13
(47 1)
In order to see that there are only two unique dyadic Green's function elements requiring calculation,it is best to begin to evaluate the particular off-diagonalterms in (469). It will become apparent what the trend is once this examination process is started. By (454), (461), and (469),
Moving down the first column, excluding the diagonal, the ij = 21 or ba element is .
w
(473)
THEORY FOR INHOMOGENEOUSFERRITE
245
Invoking (47 l), %a
= #b - #a = 2x13
(474)
and inserting this into (474) yields one of the unique dyadic Green’s function elements.
(475) with Gba = G+
(476)
Moving down to the next element in the system matrix, the 31 or ca element, we find the angular argument
and put it into
obtaining the other unique dyadic Green’s function element.
with Gca = G -
(480)
The dyadic Green’s function elements in the first row in (469), are found from those already determined by noting that the azimuthal angle differences have their signs reversed. Finally, the cb (and bc) element is determined once it is noted that A#& = #c - #b = 4x13 - 2x13 = 2 ~ / 3
(481)
By (472), this implies that the cb element is G - . Therefore, in summary we have found that
(482a) (482b)
(482c)
246
CLIFFORD M. KROWNE
Placing (482) into (469) gives
Now the three azimuthal H-fields can be simplified as follows:
(484a)
(484b)
(484c)
+
Note that the and - indices on the Green’s functions are mislabeled in [2] for the Hb and Hc results (they have been reversed inadvertently). The s-parameters are found in the usual manner, (485a) (485b) (48%)
247
THEORY FOR INHOMOGENEOUSFERRITE
3. Four-Port Symmetric Circulator Equation (451) comes from the more general construction to be covered here. The z-component of the electric field at the circulator perimeter, r = R, is given by
where Hq = H$. Evaluating this at 4 = 4q,p = 1,2, 3 , 4 for
=4 gives
or written out explicitly for all p indices, dropping the understood component index on the electric field,
In matrix form this reads as
[ [zi: E4
G12 G13 = cZ2G23 G31 G32 G33 G41 G42 G43
[21
G14 G ~ ~ ] G34 G44 H4
Making port 1 the input port,
The remaining three ports have the impedance conditions
or put in a more compact form,
(489)
248
CLIFFORD M. KROWNE
Placing (490) and (492) into (489) produces the matrix equation for finding the azimuthal H-fields.
The H-field solutions are
(494a)
(494b)
(494c)
H4
1 =D4x4
2 - -D4x4
G2I
(G22
+ <2)
G31
G32
G4l
G42
G23 (G33
+ 1'3)
(343
(494d)
249
THEORY FOR INHOMOGENEOUS FERRITE
where the determinant of (493) is (GI1 D4x4
=
+ (1)
G2I (331
G4I
Gl2
GI3
G 14
(G22 (2) G32 G42
G23
G24
+
(G33
+ (3)
G43
G34 (G44 (4)
(495)
+
The s-parameters are given as an extension from the three-port case. Sll = 1 - flH1
(496) (497a) (497b) (497c)
For the symmetric situation, evaluating (470), A+i,i+I
= -n/2
(498a)
A+i+l,i
=~ 1 2
(498b)
In contrast to (498), the three-port circulator had A@i,i+l= Wi+I,i
-21~13
= 2x13
(499a) (499b)
Evaluate the followingdyadic Green's function elements creating the definitionsof the four unique elements. Note that 41 = 0, $Q = n/2,43 = n,and 44 = 3x12. G(A412) = G(O - n/2) = G(--7t/2) = G-I
(50w
G(A+l3) = G(0 - n) = G ( - n ) = G-2
(5oob)
G(A414) = G(0 - 3x12) = G(-3~/2)
= G(-2n
+ ~ / 2 =) G(n/2) = G I
G(A+,l) = G(JT- 0 ) = C ( X )= G2
(50oC)
(5cw
The four unique off-diagonal dyadic Green's function elements are (501a) .
d
o
(501b)
250
CLIFFORD M.KROWNE
(501c) (501d) We note that the self-term dyadic Green's function elements are still given by (463). In fact, this is the case for any number of ports. The system matrix (495), using (498) and (500), is
G -2
The H-fields are found from (494). (503a)
(503b)
(503c)
(503d) From these expressions,the s-parameters may be derived utilizing (496) and (497). 4. Six-Port Symmetric Circulator Equation (486) may be written down for the N T =~ 6 six-port ~ case.
c 6(4p 6
E: =
4pq
q=l
Written out explicitly for all p indices, dropping the understood component index on the electric field,
+ + + G14H4 + GlsHs + G1,jHs E2 = G21 HI + G22H2 + G23H3 + G24 H4 + G25 H5 + G26 H6
El = G I I H I G I z H ~ G13H3
(505a) (505b)
THEORY FOR INHOMOGENEOUS FERRITE
25 1
In matrix form this reads as GI1
GI2
GI3
GI4
GI5
GI6
G21
G22
G23
G24
G25
G3I G41
G32
(333
G34
G35
G26 G36
G42
G43
G44
G45
G46
G5l
G52
G53
G54
G56
-G61
G62
G63
G64
G55 G65
G66
Making port 1 the input port,
The remaining five ports have the impedance conditions
or put in a more compact form,
Placing (507) and (508) into (506) produces the matrix equation for finding the azimuthal H-fields.
252
CLIFFORD M. KROWNE
The H-field solutions are
1
(G22
+ 5'2)
G 24
G23 (G33
G32
+ 5'3)
G 25
G34 (G44
+
G42
G43
G52
G53
G54
G62
G63
G64
5'4)
G 26
(335
G36
G45
G46
(G55
+ 5'5)
G56
(G66
G65
+ 5'6) (51la)
(321
G3I
G23 (G33
+ 5'3)
G 24
G 25
G26
G34
G35
G36
G45
G46
(G44
+ 5'4)
G41
G43
G5l
G53
(354
G61
G63
G64
(G55
+ 5'5)
G56 (G66
G56
+
5'6)
(5 1 lb) G26
H3
G36
2
=-
G46
D6x6
G 56 I
61
G64
G62
(5 1lc) G21 H4
= --
2
D6x6
(G22
+ (2)
G23
+ (3)
G3I
G32
G4I
G42
G43
G5l
G52
G53
G6I
G62
G63
(G33
G 25
G26
G35
G36
G45 (G55
+
G46 G56
5'5)
G65
(G66
+ 5'6) (511d)
H5
2
=D6x6
G 24
G 26
G34
G36
G2I
(G22 -k c2)
G31
G32
G41
G42
G43
G5l
G52
(353
G54
G6I
G62
G 63
G64
G23
(G33
+ 5'3)
(G44
+ (4)
G46 G56
(G66
+ 5'6) (511e)
G2l
(G22
+ (2)
G23
(G33
+ 53)
G 24
G25
G34
G35
G31
G32
G4l
G42
G43
G5l
G52
(353
G54
G6l
G62
G63
G64
(G44
+ 5'4)
G45
(G55
+ 5'5)
G 65 (5110
THEORY FOR INHOMOGENEOUS FERRITE
253
where the determinant of (5 10) is
D6x6 =
The s-parameters are given as an extension from the four-port case.
(514a)
(5 14d)
In compact form, for the other ports besides the input,
For the symmetric situation, evaluating (470), A4i,i+l = - ~ / 3
(516a)
A4i+l,i = ~ / 3
(5 16b)
Evaluate the following dyadic Green's function elements creating the definitions of the six unique elements. Note that 4, = 0, & = x / 3 , 43 = 2x13, 4 4 = R, 45 = 4x13, and 4 6 = 5x13. Gl2 = G(A412) = G(0 - x / 3 ) = G ( - n / 3 ) = G-1
(517a)
G13 = G(A413) = G ( 0 - 2x13) = G ( - 2 ~ / 3 )= G-2
(517b)
GI4
= G(A414) = G(O - x ) = G (-n ) = G-3
(517c)
254
CLIFFORD M. KROWNE
(518a) .
a
(518b) (5 18c) (518d)
r
"
The self-term dyadic Green's function elements are still given by (463). The system matrix (512), using (516) and (517), is
The H-fields are found from (511). 2 HI = D6x6
THEORY FOR INHOMOGENEOUS FERRITE
255
(520b)
H3
2 =-
(520c)
D6x6
(520d)
From these expressions, the s-parameters may be derived utilizing (506)and (508).
5 . Reverse S-Parameters Interest in the reverse s-parameters sji occurs because it is useful to establish for the inhomogeneous circulator that reverse propagation is reduced significantly in those directions where such characteristics have been designed. For example, if the forward coupled s21 s-parameter has a significant throughput, making the magnitude Is211 sizable for a three-port circulator, then we expect Is121 to be small in comparison. That is, we expect Is21I >> Is12 1. It is easy to establish by symmetry argumentsthat knowledge of the direct s j ; s-parameters ( j > i ) leads to knowledge of the reverse sij s-parameters (i < j ) . Consider the inhomogeneous three-port device first. View the voltage wave arriving at each port j as due to the excitation at port i . This leads to the definition
256
CLIFFORD M.KROWNE
1
4 3 =
@1=
0
- 2d3
10. A 2D, three-port, inhomogeneous circulator as in Fig. 9, but with the port labeling changed to numeric integers. The s-parameter port-to-port relationships are shown in terms of clockwise and counterclockwise waves for s12 and s3I. FIGURE
of the notation and construction of the s-parameter s i j . Looking down on the circulator cross-sectional r+-plane (z is normal to this plane), recognize that the measured voltage is due to a clockwise circulating wave (denoted by superscript c), and a counterclockwisecirculating wave (denote by a superscript cc). (It is not necessary to worry about the radial propagation behavior as this is common for both types of +-circulating waves.) Thus, we can write two sets of equations, one for the direct s-parameters, and a second for the indirect or reverse (see Fig. 10 for a diagram):
for the direct, and
for the indirect. Looking at (521) and (522) leads us to conclude = Is311
(523a)
Is131= ls21I
(523b)
ls12I
Next, let us address the four-port inhomogeneous circulator. Instead of 2n/3 being the basic azimuthal increment angle, n / 2 is. Therefore, the two sets of
THEORY FOR INHOMOGENEOUS FERRITE
257
equations equivalent to (71) and (72) are (524a) (524b) (5244
+
$12
= ~ : 2 ( ~ / 2 )~ f ; ( 3 ~ / 2 )
(525a)
$13
= sf3(n) -k Sfg(n>
(525b)
= Sy4(3n/2)
+ S:i(15/2)
(52%) for the indirect. In a similar fashion to (523), combining (524) and (525) leads to $14
(526a) (526b) (526) Lastly, consider the six-port inhomogeneous circulator. The basic azimuthal increment angle is now n/3. Thus, the direct and indirect sets of equations are
+ S;f(lr/3)
$21
= $1 (577/3)
$31
= 3$1(4n/3) sSf(2n/3)
+
(527a) (527b)
= $l(n)
(527c)
$51
+ Cf(n) = ~5c1(2n/3)+ ~ , ' f ( 4 ~ / 3 )
$61
=
(527d) (527e)
$41
(n/3) -k S:f(5lr/3)
and
(529a) (529b) (529c) (529d) (529e)
25 8
CLIFFORD M. KROWNE
6. Electric Field Within the Annuli Convenient expression for E , within an annulus (see Fig. 9) is desirable since it can allow contour field plotting, and this is known to be helpful [30] in understanding how the port load terminations, construction of the annuli, and frequency behavior contribute to the inhomogeneouscirculator performance. What we need is the dyadic Green's function element for an arbitrary radial location. Using the generalization of (452), where the ring subscript N has been replaced by i [ 1 11, GZEHi(r94; R ,
4;)
n=-m
Here the Cieai(r)and Ciebi(r)radial functions are constructed from Bessel functions, y: is defined as
+
Y$ = a,,v(recur)CzhaN(r) bnN(recur)C,hbN(') @
(531)
and the recursion coefficients a,i (recur) and bni(recur) relate successive annuli [ 11, and are equivalent to a matrix cascaded process. If we define
+
yi:(r) = a,i (recur)C,Zeai( r ) b,i (recur>C,Zebi (r)
(532)
and
then (530) may be written in compressed form as .
M
(534) The field within the inhomogeneous circulator, in the ith ring, is given by a form similar to (45 la), which was done only for the perimeter, and an idealized circulator with an incident wave at port a (as well as a reflected wave), but only exiting waves at ports b and c. For the same port conditions, Ezi(r, 4) = G(r, 4: # a ) &
-k G(r, 4; 4 b ) H b
+ G(r, 4; 4 c ) H c
(535)
Magnetic fields Ha, H b , H, are determined by (484) and G(r,4; I&), p = a, b, c , by (534). Equation (535) is based upon an arbitrariness in setting the incident wave at the input port a. It may be thought of as unity for the purposes of statement (535). However, for the case of an imperfect circulator with finite reflections at the output port b and isolated port c, the total field is no longer given by (535). For this situation, the subscripts on the port angles and magnetic fields must be changed to in, out, and is0 and hntaken as an independent variable, indicative of rotating the
259
THEORY FOR INHOMOGENEOUS FERRITE
-
+b = 2 r / 3
$b = 2 t / 3
- -
gin ga
$1n = $b = +a 2 n / 3 $Out $C = @ + 2 d 3 $is0 = $a 2n/3 +
Hiso
out
isolated @c
--
2d3
=
-
2n/3
@in= gc = @a - 2x13 $out =@a = @b - 2 r / 3 $is0 = @b = +c 2 d 3
isolated
-
(
Hout
)=It
@a= 0
FIGURE1 1. The drivinfload constraints for the three configurations building up the complete EL out of the Green's functions for a 2D, three-port inhomogeneous circulator. Loads are equal on all ports. The three configurations in (a), (b), and (c) are related to one another by successive 2n/3 rotations (following 4, making the sense counterclockwise).
drivinglloading conditions of the device by 2rrm/3, where m = 0, 1, 2. Angles and hs0may be viewed as functionally dependent on +in9 that is, +out(&) and $iiso(&,). Figure 11 shows the resulting three drivingfloading conditions in pictorial form. Also, because the port fields are a result of unity driving incident field in the idealized device, the magnetic field must be replaced by the products Hp(+in,(l)~i"c(hn), p =in, out, iso. $out
260
CLIFFORD M. KROWNE
Examining the perimeter Green's functions constituting the H p , p = a , b, c, we realize that they are solely determined by relative positional information. For a threefold symmetric device, they must be invariant as & varies, &n = #b, &. Therefore, Hin(1) = H a ; and
Hout(hn)(l)= Hout(1) = Hb; Hiso(h,)(l)= Hiso(1) = Hc (537) &
~ f ; ' ( r ,4) =
C { ~ ( r4;, hnIHin(1) + ~ ( r4;, 4out(4in)>Hout(l) hi.=&
+ G (r, 4; 4iso ( 4 i n ) ) Hiso( 1 ) ) Einc(#in)
(538) Numerical results based upon (538), with a theoretical technique employed to derelative to Ei"C(4a), has been done for specific circulator parameters rive EinC(+in) ~311.
7. Conclusion Sets of Green's function elements pertinent to the perimeter of the inhomogeneous circulator have been determined for various multiport symmetric circulators. It has been demonstrated how these Green's function elements are employed for finding the device s-parameters. Utilization of these formulas leads to an economy of effort spent in numerical evaluations. Also, derivation of an expression suitable for electric field contour plotting within the inhomogeneouscirculator boundary has been provided. B. 3 0 Dyadic Green's Function 1. Introduction In part A of this section, we have presented the symmetric considerations for 2D circulators, and have found a great reduction in the number of specific Green's fucntions which must be evaluated to determine the s-parameters. And it was also seen that only one type of Green's function is required to act as a generator to construct the rest of the Green's functions needed to completely represent the internal electric field. Here we present the 3D dyadic Green's function results.
2. Three-Port Symmetric Circulator There are a tremendous amount of simplifications which result by constraining the circulator to a symmetric disposition of the port locations. For a three-port circulator [ 1 11 Ea = TiaHa Eb = T;a Ha
+ T:b Hb + TfCHc + T;b Hb + TlC Hc
(539a) (539b)
26 1 (539c) (540a) (540b) (540c) (541a) (541b) (541c) (542a) (542b) (542c) (543a) (543b) (543c) (544a) (544b) (544c)
(545a) (545b) Gfh
(cc)MCc
(54%) (546a)
(546b)
(546c)
(547a)
(547b)
262
THEORY FOR INHOMOGENEOUS FERRITE
263
The dyadic Green's functions elements at the circulator perimeter are given by the following formulas, which are then used to obtain the azimuthal dyadic Green's functions elements employed in the preceding equations.
(556a)
(556b)
(556~)
(556d)
(556e)
(5560
(557a)
264
CLIFFORD M.KROWNE
x [AAjT'ijN(R) - AijTL~jN(R)]e-i"4e'"0
7".( R )
x [AAj
- A i j T'),
( R ) ]e-i"@': ein@
(557d)
(5570
To get the analysis started for the 3D case, we must try to compact these complicated expressions. With this desire in view, consider how factorizable non-4dependent parts of the 2D Green's function elements were created earlier. The same thing may be done here by setting the z-coordinate to z = zs, and defining
Psy = Kzj+ COS(kzNj+Zs)-DABj 1 [Bn,TnejN(R)- B,!jTi;N(R)] 2
p:$jh
1
= Kzj+ COS(kzNj+Zs)-
DABj
= iKzj+ sin(k,Nj+z,>-
1
DABj
Pnr$jh = iKzj+ Sin(kZ,vj+zS)-
1
DAEj
zl
[AL,T,Z,:.N(R)- AijTi:jN(R)]
(558b)
[BijT,',:.,(R) - B,!jTLiN(R)] ( 5 5 8 ~ ) [AAjT':,(R)
- AijTL:jN(R)] (558d)
[BnjTnejN(R) 2 4J1 - BjjT2,(R)] DABj @ezh 1 pnNj = iKd+ sin(kzNj+zs)[ A A j T S N ( R) Aij7"jN(R)] DABj = iKzj+ sin(kZNj+zs)-
(558a)
(558e) (5580
THEORY FOR INHOMOGENEOUS FERRITE
265
The Green’s function elements may then be written in the much abbreviated forms (560a) .
w
w
(560b)
(560d)
(561a)
266
CLIFFORD M.JSROWNE .
w
w
(561b) .
w
o
o
(561c) .
w
w
(561d)
(561e) .
w
w
By constructingazimuthal summationsover the new coefficients j&iS ‘sf where ,f2 are electric or magnetic fields, removing the extra j-index arising from the z-variation, allows the 3D Green’s function elements to take on forms identical to the earlier 2D forms studied. The modified coefficients are defined as
(,,tSare cylindricalcoordinatesand fl
w
(562a) j-0 w
(562b) j=O
(562c) j=O
(562d) j=O
(562e) m
(563a) j=O
THEORY FOR INHOMOGENEOUS FERRITE
267 (563b)
(563c) j=O w
(563d)
(563e)
j=O
The Green's function element expressions can now be set down using these f p They are modified ~ ~ ~ " scoefficients.
.
w
(565a) (565b)
268
CLIFFORD M.KROWNE
(565c) .
w
(565d) (565e)
Because our interest is in only the z-directed component of the electric field E,, and a subsidiary z-component magnetic field H,is employed, the first superscripts may be removed since they are understood and the compressed notation used for the only four Green’s function elements required.
(567a) Gih(i?j ) = Gih($iy 4,) = G i h ( R , $ i ; R , 4 j ) = %
C ;7; W
zhein(@,-4,)
n=-w
(567b) Following the logic in the 2D theory for a three-port device, unique azimuthal symmetry dyadic Green’s function elements are defined for the four basic types of dyadic Green’sfunctionelements. Each basic dyadic Green’s function element will have three unique azimuthal symmetry dyadic Green’s function elements. Thus, a total of twelve are needed and provided here. For the E, H4 symmetry elements *
w
(568a)
THEORY FOR INHOMOGENEOUS FERRITE .
269
w
(568b)
(568c) For the Ez Hzsymmetry elements (569a) , 1
‘?,eh
=
c w
= ze. zh 2inlr/3 YnN
(569b)
n=-w
(569c) For the HzHb symmetry elements (570a) .
W
(570b) .
m
(570c) And finally for the HzHzsymmetry elements .
W
(571a) I G$,hh
=
c
= zh, zh 2 i n n / 3
YnN
(571b)
n=-w
(571c) The simplification of the azimuthal dyadic Green’s function elements includes the following rules for the EzHb symmetry elementsbased upon counterclockwise
270
CLIFFORD M.KROWNE
port labeling
For the H, Hd symmetry elements
And finally for the H,H, symmetry elements
THEORY FOR INHOMOGENEOUS FERRITE
27 1 (579a) (579b) (579c) (580a) (580b) (58oc)
(581a) (581b) (581c)
(582a)
(582b)
(582c)
(583a)
(583b)
272
CLIFFORD M.KROWNE
=
['f,
hh
-
'1
2
- 'k,
hh':,
(584c)
hh
The Mi, expressions are related to one another because of the symmetry simplifications.
-
Maa
= Mbb = Mcc =
Mab
= Mbc = GI, hh ['f, hh = -Mcb =
('5,
[G:,hh
2 hh)
3'
2
-'t.hhGt3hh
'3 - ('t, - 't, :,[' hh
(585b)
hh) hh
-
(585a)
'1
(585c)
The T: relationships may be put into matrix form again for compactness and ease of viewing their properties.
,!'
eh
,!'
eh
'$,
eh
1
THEORY FOR INHOMOGENEOUS FERRITE
273
Once all the preceding symmetry relations are found, the electromagneticfields and, as a consequence, the s-parameters, can be found.
3. Conclusion Sets of Green’s function elements pertinent to the perimeter of the inhomogeneous 3D circulator have been determined for a single multiport symmetric circulator case, namely, the most common three-port case. It has been demonstrated how these Green’s function elements are employed for finding the device s-parameters. Compared to the 2D situation, these formulas are much more involved. But a similar economy in the number of Green’s function elements occurs here and is passed down throughout the expressions required to find the s-parameters. The same process may be undertaken for four or six symmetrically displaced ports, with the provision, however, that the algebra involved in working with the Green’s functions increases tremendously, as we would expect.
VIII. OVERALLCONCLUSION Electromagnetic circulators, which are commonly utilized at several hundred MHz, at a few GHz, in the microwave regime where the free space wavelength is on the order of centimeters, to the millimeter wave region which exists beyond 30 GHz up to 300 GHz, are a key component of high-frequency electronics and will probably be around in the foreseeable future. Circulators are employed in very high-power applications as well as in moderate- and low-power electronics. The lower-power applications involve hybrid and monolithic circuits, which are integrated with solid state devices, and it is for these higher-frequency, lowerpower applications where much of the interest and utility of microstrip circulators arises.
274
CLIFFORD M. KROWNE
We have tried in this contribution to bring together several approximation possibilities or economizing techniques which may be applicable toward fast computeraided design (CAD) using rigorous mathematical physical methods for microstrip circulators, and to address or handle real issues encountered by devices in circuit environments, which have not appeared in the literature or are of very recent origin. Consequently, the effect of placing the circulator puck in an external medium affecting its behavior has been examined from several aspects. The puck has been placed in a vertically layered, radially zoned medium (Sections I1 and 111). The puck has been placed in a much simpler medium where only a single outlying radial region exists (Sections IV through VI). Finally, ways to employ symmetry to reduce significantly the number of Green’s function elements and vastly speed up numerical computation have been presented (Section VII). Our focus has been on techniques that are more important for their mathematical physics content and potential for user-friendly engineering design capability than for their ability to obtain nearly exact solutionswith numerically intense simulators, which use extensive computer code, memory, and time.
REFERENCES [I] Krowne, C. M., and Neidert, R. E. (1996). Theory and numerical calculations for radially inhomogeneous circular ferrite circulators, IEEE Trans. Microwave Theory & Tech. 44(3), 419-431, March. [2] Krowne, C. M. (1996). 3D dyadic green’s function for radially inhomogeneous circular ferrite circulator, IEEE Microwave Theory Tech. Symposium Digest, San Francisco, CA, June 18, pp. 121-124. [3] Maystre, D., Vincent, P., and Mage, J. C. (1983). Theoretical and experimental study of the resonant frequency of a cylindrical dielectric resonator, IEEE Trans. Microwave Theory & Tech. 31(10), 844-848, October. [4] Hui, W. K., and Wolff, I. (1991). Dielectric ring-gap resonator for application in MMIC’s, IEEE Trans. Microwave Theory & Tech. 39(12), 2061-2068, December. [5] Hui, W. K.,and Wolff, I. (1994). A multicomposite, multilayered cylindrical dielectric resonator for application in MMIC’s, IEEE Trans. Microwave Theory & Tech. 42(3). 415423, March. [6] Michalski, K. A. (1986). Rigorous analysis methods, In Dielectric Resonators (D. Kajfez and P. Guillon Eds.). Dedham, MA: Artech House, pp. 185-258. [7] Krowne, C.M. (1998). Implicit 3D dyadic green’s function using self-adjoint operators for inhomogeneous planar ferrite circular with vertically layered external material employing modematching, IEEE Trans. Microwave Theory Tech. M I T 46(4), April. [8] Morse, P. M., and Feshbach, H. (1953). Purr I, Methods of Theoretical Physics, New York McGraw-Hill. [9] Friedman, B.(1956). Principles and Techniques of Applied Mathematics, New York: Wiley. [lo] Courant, R. (1962). Partial differential equations, in Purr 11, Methods of Mathematical Physics, by R. Courant and D. Hilbert. New York: Interscience. [ 1 I] Krowne, C. M. (1996).Theory of the recursive dyadic green’s function for inhomogeneous ferrite canonically shaped microstrip circulators, in Advances in Zmuging and Electron Physics, Vol. 98 (P. W. Hawkes, Ed.). Academic Press,pp. 77-321.
THEORY FOR INHOMOGENEOUS FERRlTE
275
[12] Gradshteyn, I. S., and Ryzhik, I. M. (1980). Table oflntegrals, Series, and Products, Academic Press, San Diego. [13] Wang, I. J. H. (1996). Generalized Moment Methods in Electromagnetics. Wiley, New York. [I41 Chu, T.S., Itoh, T.,and Shh, Y.-C. (1985). Comparative study of modematching formulations for microstrip discontinuityproblems, IEEE Trans. Microwave Theory Tech. MlT 33(10), 10181023, October. [ 151 Mautz, J. R. (1995). On the electromagneticfield in a cavity fed by a tangential electric field in an aperture in its wall, IEEE Trans. Microwave Theory Tech. M l T 43(3), 620-626, March. [I61 Carosella, C. A., Chrisey, D. B., Lubitz, P.,Honvitz, J. S., Dorsey, P., Seed, R., and Vittoria, C. (1992). Pulsed laser deposition of epitaxial BaFelzO19 thin films, J. Appl. Physics 71(10), 5107-51 10, May 15. [I71 Karim, R., Ball, S. D., Truedson, J. R., and Patton, C. E. (1993). Frequency dependence of the ferromagnetic resonance linewidth and effective linewidth in manganese substitutedsingle crystal barium ferrite, J. Appl. Physics 73(9), 45 1 2 4 515, 1 May. [ 181 Morisako, A., Nakanishi, H., Matsumoto, M., and Naoe. M. (1994). Low temperature deposition of hexagonal ferrite films by sputtering,J. Appl. Physics 75(10), 5969-5971, May 15. [ 191 Krowne, C. M. (1997). 2D dyadic green’s function for homogeneousferrite microstrip circulator with soft walls, The 22nd Intern. Conf. Infrared MillimeterWaves Dig., pp. 168-169, Wintergreen Resort, VA, July 19-25. [20] Krowne, C. M. (1997). Homogeneous ferrite microstrip circulator 3D dyadic green’s function with and without perimeter interfacial walls, The 22nd Intern. Conf. Infrared Millimeter Waves Dig., pp. 170-171, Wintergreen Resort, VA, July 19-25. [21] Krowne, C. M. (1998). Inhomogeneous ferrite microstrip circulator 2D dyadic green’s function for penetrable walls. Intern. J. Electronics 84(6), June. [22] Newman, H. S., Webb, D. C., and Krowne, C. M. (1996). Design and realization of millimeterwave microstrip circulators, Intern. Con$ Millimeter Submillimeter Waves Appl. Ill Dig., SPIE Proceedings 2842, 181-191. [23] Krowne, C. M., and Neidert, R. E. ( 1995).Inhomogeneous ferrite microstrip circulator:Theory and numerical calculationsusing a recursive green’s function, 25th European Microwave Conference Dig., pp. 414420, September 4-7, Bologna, Italy. [24] Krowne, C. M. (1997). Femte microstrip circulator 3D dyadic green’s function with perimeter interfacial walls and internal inhomogeneity, Microwave Optical Tech. Lens. 15(4), 235-242, July. [25] Tai, C.-T. (1993).Dyadic Green’s Functions in Electromagnetic Theory, IEEEPress, Piscataway, NJ. [26] Gupta, K. C., Garg, R., and Bahl, I. J. (1979). Microstrip Lines and Slotlines, Artech House: Dedham, MA. [27] Hammerstad, E. 0.. and Bekkadal, F. (1975).Microstrip Handbook, ELAB report STF44 A74169, The University of Trondheim, The Norwegian Institute of Technology. [28] Garg, R., and Bahl, I. J. (1978). Microstripdiscontinuities,Intern. J. Electronics 45.81-87, July. [29] Krowne, C. M. (1997). Symmetry considerations based upon 2D EH dyadic green’s functions for inhomogeneous microstrip ferrite circulators, Microwave Optical Technology Letts. 16(5), 176-186, October 20. (301 Newman, H. S., Neidert, R. E., Krowne, C. M., Vaugh, J. T., and Popelka, D. J. (1996). CAD tools for planar ferrite Circulators:Development, verification, and utilization, Femte CAD Workshop, IEEE Microwave Th.Tech. Symposium, San Francisco, CA, June 17. [31] Newman, H. S., and Krowne, C. M. (1998). Analysis of ferrite circulators by 2D finite element and recursive green’s function techniques, IEEE T m s . Microwave Theory Tech. M7T 46(2), 167-177, February.
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ADVANCES IN IMAGING AND ELECTRON PHYSICS,VOL. 103
Charged Particle Optics of Systems with Narrow Gaps: A Perturbation Theory Approach M. I. YAVOR Institute for Analytical Instrumentation RAS, Rizhskij p r 26, 198103 St. Petersburg, Russia Tel. +7 (812) 251-86-63. Fax +7 (812) 251-70-38, Email: [email protected]
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . II. Applicability of Perturbation Methods in Charged Particle Optics . . . . . . . A. Electromagnetic Field Structures Suitable for Application of Perturbation Methods B. How to Apply Perturbation Methods Correctly . . . . . . . . . . . . . 111. Calculation of Weakly Distorted Sector Fields and Their Propertieswith the Aid of a Direct Substitution Method . . . . . . . . . . . . . . . . . . . . . . . A. Electrostatic Field and Charged Particle Trajectories in an Imperfectly Manufactured Sector Energy Analyzer . . . . . . . . . . . . . . . . . . . . . B. Magnetostatic Field and Charged Particle Trajectoriesin an Imperfectly Manufactured Sector Magnet . . . . . . . . . . . . . . . . . . . . . . . . C. Electromagnetic Field and Charged Particle Trajectories in an Imperfectly Manufactured Wien Filter . . . . . . . . . . . . . . . . . . . . . . . D. Parasitic Beam Distortions in Charged Particle Analyzers Based on Sector Fields and Wien Filters and Their Correction . . . . . . . . . . . . . . . . N. Transformation of Charged Particle Trajectoriesin the Narrow Transition Regionsbetween Electron- and Ion-Optical Elements . . . . . . . . . . . . . . . . . . A. Charged Particle Beam Transport through the Gaps of Multiple Magnetic Prisms B. Transformation of Charged Particle Trajectories in the Gaps between Lenses of Closely Packed Quadrupole Multiplets . . . . . . . . . . . . . . . . V. Synthesis of Required Field Characteristicsin Sector Energy Analyzers and Wien Filters with the Aid of Terminating Electrodes . . . . . . . . . . . , . . . . A. Electrostatic Field in the Gap between Two CurvilinearElectrodes Terminated by Split Shielding Plates . . . . . . . . . . . . . . . . . . . . . . . B. Influence of Split Matsuda Plates on the Field and Optical Propertiesof Sector Energy Analyzers and Wien Filters . . . . . . . . . . . . . . . . . C. Synthesis of a Required Field Distribution in a Vicinity of the Beam Main Path in a Sector Analyzer with the Aid of Split Shielding Plates . . . . . . . . . . VI. Calculation of the Elements of Spectrometers for Simultaneous Angular and Energy or Mass Analysis of Charged Particles . . . . , . . . . . . . . . . . . . A. Electrostatic Field of a Poloidal Analyzer . . . . . . . . . . . . B. Calculation of Particle Trajectories in a Poloidal Analyzer . . . . . . . . C. Focusing Properties of a Poloidal Analyzer . . . . . . . . . . . . . . D. Electrostatic Field of a Conical Mirror . . . . . . . . . . . . . . . E. Electrostatic Field of a Conical Lens with Longitudinal Electrodes . . . . . . F. Electrostatic Field of a Slit Conical Lens . . . . . . . . . , . . . . . G. Elimination of a Beam Deflection in Conical Lenses . . . . . . . . . . H. Focusing of Hollow Charged Particle Beams by Conical Lenses . . . . . . VII. Conclusion . . . . . . . . . , . . . . . . . . . . . . . . . .
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211 Volume 103 ISBN 0-12-014745-9
ADVANCES IN IMAGING AND ELECTRON PHYSICS Copyright@ 1998 by Academic Press. Inc. All righu of repduction in any form reserved. ISSN l076-5670/98525.00
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385
Acknowledgments References
I. INTRODUCTION
A design of any charged particle optical system (electron or ion microscope, energy or mass analyzer, mass separator,particle accelerator,etc.) is always based on simulation of electron or ion trajectories through an electromagnetic field created by a certain set of electrodes, currents, or magnetic pole faces. At the early stage of development of charged particle optics, simple model field distributions were used for this simulation such as bell-shaped lens fields or sector and multipole fields in a sharp-cutoff approximation; these distributions allowed analytical representation that made trajectory analysis relatively simple. The simulation usually resulted in analytical formulas, which clearly revealed physical properties of the system under investigation and thus enabled the achievement of optimal solutions easily. For practical purposes, however, a framework of analytical field models appeared to be too narrow. Even a small deviation from an analytical model could be treated only with the aid of a numerical solution of the partial differential equations for electromagnetic field distributions followed by either a subsequent numerical ray tracing of trajectories or, at best, by a calculation of aberration integrals. A progress in computer engineering has led during the last decade to development of a variety of sophisticated modem numerical methods for electromagnetic field calculation (a survey of these methods can be found, for example, in [ 11) and corresponding software for charged particle optical simulation [2]. However, practical optimization of electron- and ion-optical devices with a help of such software usually is a complicatedproblem, and the reason is not only that numerical computations are time-consuming and wasteful of computer resources. With a numerical simulation, the main advantage of analytical models is lost: a clearness and visuality of the physics involved. Instead of general physical properties of the system and trends in its behavior with changes of parameters, a designer sees only a result of a particular calculation for a given set of the system parameters, and this result as a rule does not tell much about a way or even a possibility to optimize a system in a desired direction. Thus, a question arises: How would it be possible to enlarge the area of application of analytical calculation methods in charged particle optics? A positive answer to this question may be a development of approximate analytical methods, provided that their accuracy in practical cases would be generally comparable to the accuracy of conventional numerical simulation.
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
279
In many areas of physical knowledge where complex processes are studied (such as hydrodynamics, propagation and diffraction of electromagnetic or acoustical waves), a perturbation theory techniqueis widely used, which enables us to enlarge considerably the area of application of analytical calculation methods with all their advantages being preserved. The perturbation theory methods may be applied to a problem if one or several small physical parameters are inherent in it; this allows one to obtain a mathematical solution of the problem by the successive approximation method in a form of power serii in these small parameters. The goal of the present chapter is to demonstrate that perturbation methods can be successfully and fruitfully used also in the optics of charged particles. It should be clearly emphasized that the perturbation methods just mentioned have nothing to do with the approach of aberration expansions that has remained the base of the charged particle optics for a long time. We imply perturbation theory expansions to be the serii in powers of small parameters that are characteristic for a physical system and thus are present in coefficients of the differential equations describing this system. In optics of charged particles the corresponding parameters characterize specifics of the electromagneticfield distribution (for example, a small length of the interval of the field inhomogeniety, like that observed in a sector magnet fringing field). In the theory of charged particle beam aberrations the situation is different: Studied here are not the expansions of the particle trajectory coordinates with respect to the small parameters inherent in the electromagnetic field structures (usually it is not assumed at all that such parameters are present in the system) but the expansions in power serii with respect to small initial conditions for the charged particles (these conditions can be defined either explicitly or through the beam aperture parameters). Actually the aberration coefficients are just the Taylor serii coefficients, and successiveapproximationmethod is only a tool for obtaining these coefficients in case where explicit solutions of the trajectory equations are unknown. Note that aberration Coefficientscan be also calculated without using any successive approximation methods, for example, by differential algebraic methods [3]. A presence in electromagnetic field structures of charged particle optic systems of features described by small parameters is very rarely used for calculation of both the fields themselves and the particle trajectories in these fields. Actually, in a trajectory simulation the only example of the application of a perturbation method is a development of the so-called fringing field integral method to describe a charged particle beam transformation in the fringing fields of sector electrostatic and magnetic analyzers as well as of quadrupole lenses and multipoles (the essence of the method is summarized in [4]; see Section II.A.2 for a more detailed history of the problem). Among the electromagnetic field calculation methods, the only application of a perturbation approach is an investigation of field deviations caused by weak distortionsof the electrodes,magnetic pole pieces, or coils; however, even this investigation usually requires a numerical simulation of the field deviation
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distribution, if performed with the aid of the general perturbation methods by F. Bertain, M. A. Monastyrskij,and other authors, which are surveyed in [ 5 ] . There were only few attempts to obtain analyticalrepresentations of electromagneticfield deviations, which are mentioned in Section 1I.A. 1. It should be noted that even these rare attempts to apply perturbation methods in charged particle optics in many cases finished with incorrect results, which can be found in the publications devoted to both the development of the fringing field integral method and the investigation of electromagnetic field deviations due to the electrode or magnetic pole distortions. The reason for the corresponding mistakes is due to neglecting an important feature of the perturbation theory. As a matter of fact, from the mathemathical point of view, perturbation expansions are specific cases of the so-called formal asymptotic expansions. A characteristic feature of the asymptotic serii is that these serii are, as a rule, diverging. One may only state with a certain confidence that, provided that the expansion parameter is small and that there are no any other “side” large values (coefficients or functions) in the equation under consideration, the main contribution to the solution of the problem is represented by the first term of the asymptotic expansion while the succeeding terms describe smaller corrections. However, a presence of the side large parameters is typical for many physical problems, including the problems of charged particle optics. For example, a small parameter used (explicitly or implicitly) in the fringing field theory considerations is usually a small length of the fringing field region as compared with the length of the main field. But at the same time, the smaller the fringing field length is the larger the field gradient is in the fringing field region. If this fact is not taken into account properly, the fringing field transfer matrix may be calculated partially or completely incorrectly. Generally, an arbitrary specification of an “evident” small parameter without a thorough study of a problem to be solved may lead to confusion. A very good and widely known example is deducing a formula for the focal length of a thin round electrostatic lens. It is well known [6] that, being obtained in two different ways (starting with an ordinary paraxial trajectory equation and with an equation relative to a transformed coordinate function), the corresponding formulas are different! It is not easy to understand directly which of the resulting expressions is correct and which one is wrong; an explanation given in [6] seems to be rather unclear. A real difference between the two ways discussed is a different specification of the small parameter in the problem. Working with the ordinary paraxial trajectory equation, one implicitly assumes that a small parameter in the problem is a small deviation of the charged particle coordinate inside the lens from its initial value; however, an explicit attempt to build the asymptotic expansion based on this small parameter fails. Following the second way, a small parameter is assumed to a small deviation of the ratio of the particle coordinate to the fourth root of the electrostatic field potential from the initial value of this ratio. Such a definition of
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the small parameter leads to a correct final analytical expression for the thin lens focal length. Thus, a development of the perturbation methods in charged particle optics requires, first, a specification of the circle of the problems where application of these methods may be promising and, second, an attentive and thorough analysis of such problems in order to provide a correct use of perturbation algorithms. In the present chapter we will mainly concentrate on the optics of charged particle spectrometers. The reason is that the relations inherent in geometrical dimensions of the elements of such spectrometersvery often allow one to introduce some typical small parameters. For example, energy or mass dispersive elements like electrostatic or magnetic sectors or electrostatic cylindrical or spherical mirrors usually have gaps between the electrodesor pole faces that are small compared to their curvature radii. Typically lengths of the particle trajectories inside these elements as well as inside quadrupole lenses or multipoles are considerably larger than the sizes of the fringing fields. In some types of electrostatic lenses often used in electron and ion spectrometers, like so-calledtransaxial and conical lenses, the electrostatic field is concentrated in the regions whose sizes are small as compared with the curvature radii of the electodes. Generally, all these elements may be called optical elements with narrow gaps, though explicit mathematical expressions for the corresponding small parameters may be different. In spite of the similar nature of all these small parameters, they allow the perturbation methods to be applied to very different kinds of problems in charged particle optics. The three main applications, which will be considered in the present chapter, are calculation of effects of weak distortions of the electrode and magnetic pole surfaces; investigation of a transformation of charged particle beams in the fringing fields and short transition regions between adjacent elements; and, finally, calculation of electromagneticfields and electron optical properties of the elements with narrow gaps. The state of the development of the corresponding problems and possible directions of the use of perturbation methods to their solution are discussed in Section 11, which contains also some general rules for a correct application of perturbation expansions. The subsequent sections are devoted to specific problems in charged particle optics solved by perturbation techniques. Section I11 presents a way of analytical calculation of the electromagnetic field disturbance and its influence on charged particle beam distortions in sector field analyzers and Wien filters. A method used in this section is based on the idea of the substitution of the Taylor expansion for the field potential written on the optic axis to the boundary conditions defined at distorted electrode or magnet pole surfaces. Section IV describes an extension of the fringing field integral method for sector fields and quadrupole lenses to the case of closely packed arrays of such elements where the fringing fields of the adjacent elements overlap. Section V demonstrates an original and elegant analytical method for calculation and synthesis of the electrostaticfield distribution
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1
Systems with narrow gaps: (air gap)/(curvaiure radiw) << 1
/
methods I
Direct substitution of
\ Reduction of the
boundary conditions
RGURE 1. A scheme of possible applications of pertubation methods to electron and ion optics of systems with narrow gaps.
with the aid of the multiply divided terminating electrodes in sector analyzers and Wien filters. Finally, Section VI is devoted to calculation of electrostatic fields and optical properties of the elements of electron and ion spectrometers for angle resolved energy or mass analysis. Generally, the scheme proposed for application of perturbation methods in the present chapter is shown in Fig. 1. The scheme illustrates how the general idea of using a small parameter characteristic for electron- and ion-optical systems with narrow gaps may be applied, through different mathematical methods, to the investigation of various properties of these systems. As mentioned earlier, since the perturbation methods for calculation of electromagnetic fields and charged particle trajectories result in asymptotic expansions of the field strength or trajectory coordinates with respect to some small parameters, these expansions do not contain any internal estimations of the accuracy given by their several first terms used in practical calculations. Such an estimation may be performed by a comparison of the results obtained by a perturbation method and by a numerical simulation with the aid of any high-precision numerical simulation. For the fringing field integral method this estimation, which demonstrated a high accuracy of the perturbation algorithms, was performed recently by B. Hartmann
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[7] who used a differential algebraic approach for accuracy control. In the present chapter we present a comparison of the results obtained by the perturbation methods, proposed for calculation of electrostatic fields in Sections V and VI, with the results of a numerical simulation or available experimental data. Generally this comparison also shows very high accuracy of the perturbation methods even in cases where the used dimensionless small parameter values are rather large (about 0.5). 11. APPLICABILITY OF PERTURBATION METHODS IN CHARGED
PARTICLE Omcs A. Electromagnetic Field Structures Suitablefor Application of PerturbationMethods
As mentioned in Section I, a perturbation method may be applied to the solution of an equation for an electromagnetic field distribution or for charged particle trajectories, if the coefficients of these equations contain one or several small parameters; that is, the equations may be considered "disturbed" by such parameters. The small parameters should be dimensionlessin order to exclude any uncertainties and changes of these parameters with resizing a whole system. In other words, the small parameters should be ratios of geometrical dimensions characteristic for the system (they may also include characteristic field strengths and particle energies, masses, and charges). A typical small parameter which arises in studyingtolerance problemsin charged particle optics is a weak (as compared with the system dimensions) distortion of the electrodes, pole faces, or coils forming the electromagnetic field. From a mathematical point of view the tolerance problems are specific, since disturbed here are not the coefficients of the differential equations describing the field but the boundaries of the regions where the solutions of these equations are to be obtained. However, such a specific is not essential because in principle a problem of the solution of a nondisturbed equation in a disturbed region may be reduced by the coordinate frame variation to a problem of the solution of a disturbed equation in a nondisturbed region, though technically this way is often inconvenient. A progress of the current state of the methods of tolerance theory is surveyed in [ 5 ] . Some basic references are found in Section 1I.A.1. An existence of another small parameter typical for charged particle spectrometers is connected with the fact that in many elements of such spectrometersone can conventionally consider separately the effects of the main andfringing fields, the length of the particle trajectory in the main field being usually large as compared with its length in the fringing field. The corresponding small parameter allows one to apply the so-called fringing field integral method, which actually gives
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the expansion of the aberration integrals of the fringing fields in powers of this small parameter. The present state of the method and the possible directions of the development of the fringing field integral method are discussed in Section II.A.2. However, the most promising from the point of view of the application of perturbation methods is, to our opinion, a class of systems that can be characterized as systems whose geometry is similar in some sense to two-dimensional ones. Indeed, it is well known that electron- and ion-optical systems possessing planar symmetry (i.e., systems where the electromagnetic field is independent of one of the Cartesian coordinates) enable most wide possibilities for the exact analytical calculation of the field distribution. This fact is both due to a relative simplicity of the application of the coordinate separation method and due to the presence of a powerful conformal mapping tool for solution of the two-dimensional Laplace equation. The systems that we call similar to two-dimensionalones are the systems where the curvatures of the equipotential (or flux density) lines in the direction of one of the Cartesian coordinates is small as compared with their curvatures in perpendicular directions. In this case the field of such a system may be considered as a perturbed field of the two-dimensional system, so that there is a chance to obtain an approximate analytical description of this field. The examples of the systems in question are given in Section II.A.3. Section 1I.B. contains some simple advice, which can help to avoid incorrect realization of perturbation methods. 1, Charged Particle Optical Systems with Weakly Distorted
Electrode and Pole &$aces
Investigation of effects of deviations of electromagnetic fields in charged particle optical systems due to imperfect machining and assembling of their electrodes, magnet poles, or coils is an important part of the tolerance calculations. Besides, the deviations just mentioned may be introduced deliberately for optimization of electron- or ion-optical properties of a system [8]. A numerical simulation of the field disturbances is usually performed in the framework of perturbation methods because a direct numerical calculation does not give an acceptable accuracy due to a considerable difference between the characteristic dimensions of the system and the scale of the electrode or magnet pole distortions. A perturbation method allows one to separate the evaluation of the nondisturbed field and of its deviation, which increases the accuracy of the calculation and simplifies a physical analysis of the consequences of various field disturbances. A most well-known method for simulation of disturbances of electrostatic and magnetostatic fields of electron- and ion-optical systems due to small distortions of their electrodes, magnet poles, or coils is the Bertein method. This method was proposed for investigation of effects of deformations of the electrodes of electrostatic round lenses in [9] and then applied by Sturrock [ 101 and Archard [ 1 11 for similar
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studies of round magnetic lenses. The essence of the method is a reduction of the solution of the Laplace equation in a weakly distorted region to a sum of two terms: The first one describes the field in the nondistorted system and the second one approximately gives the field deviation as the solution of the Laplace equation with some specifically defined boundary conditions at the same nondistorted system’s boundaries. In spite of certain limitations of the Bertein method, connected with severe restrictions concerning the smoothness of the boundaries, their distortions, and boundary conditions, this approach appeared to be so simple and convenient in practice, that it is still used very widely. It should be noted that most studies based on the Bertein method are performed in the framework of a numerical solution of the equation for the field deviation. For example, in [ 12, 131 a field distrubance in round lenses was calculated with the aid of the finite difference method. To solve a similar problem, in [14,151 a finite element method was used. The same numerical approaches were applied for an estimation with the Bertein method of field disturbances in electrostatic deflectors [ 16,171. The method under considerationallows one to obtain analytical solutions only for some simple electrodeor magnet pole geometries,where an exact analytical representation exists for a nondisturbed field, and only for some model types of the system’s boundary distortions [ 18,5]. Looking for ways to overcome the Bertein method limitations, M.A. Monastyrsky and S. R. Kolesnikov have formulated an integral equation method “in variations” [ 191. In this approach an electromagneticfield disturbance is found as a solution of an integral equation in the nondisturbed region. This method was applied successfully for an investigation of sophisticated systems [20]. However, like the Bertein method, the integral equation algorithm is not intended for obtaining analytical solutions of the perturbation problems. There exists one more general approach for investigation of tolerance problems: a method of the coordinateframe variation. According to this method, a description of the distorted system’s boundaries is changed back to the nondistorted form in some new coordinates, and the disturbance “passes” to the coefficients of the differential equation for the electromagnetic field. The ideology of this method can be found in [21]. As the methods just mentioned, this approach implies an application of numerical algorithms for determining the field deviation. Two methods intended for obtaining analytical formulas for electromagnetic field disturbances caused by manufacturing imperfectionsare known. The first one was proposed by Doynikov for investigation [22] and correction [23] of defects of the field distribution in magnetic quadrupole lenses; it was based on ideas of approximate conformal mappings [24]. In principle this approach seems to be promising, though the particular problem considered by Doynikov can be solved also in the framework of the Bertein method [ 181. Another analytical method, suitable for systems with narrow air gaps, is to represent an electrostatic or magnetostatic field in the system under consideration
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as a Taylor expansion in the vicinity of a (generally curvilinear) beam main path. Then, since the electrode or magnet pole surface are located close by this path, this Taylor expansion may be substituted directly in the boundary conditions specified at these surfaces. Such a procedure allows one to define the initially unknown coefficients of the Taylor expansion in an analytical form. A basic idea of a direct substitution was first proposed by A. J. H. Boerboom for calculation of the field distribution in conical sector magnets [25] and toroidal electrostatic sector analyzers [26]. However, in the latter paper a form of the asymptotic expansion of the Taylor coefficients was chosen not quite correctly, so that the results appeared to be correct only for the toroidal condensers with small vertical curvatures of the electrodes. Later the same author applied the method of a direct substitution to the problem of calculation of electrostatic field disturbances in imperfectly manufactured cylindrical [27] and toroidal[28] electrostatic sector analyzers. Again, in the latter paper the same inaccuracy was made as in [26]. Only after a thorough analysis of the problem was performed [29], a correct result was obtained for the toroidal electrostatic analyzer [30,31]. A similar technique was also developed for inhomogeneous sector magnets [32] and Wien filters [33]. The method of the direct substitution of the Taylor expansion to the boundary conditions and its application to the investigation of the beam distortions in static sector field analyzers is discussed in Section III. Since for weakly distorted electrostatic and magnetic sector fields this method was recently surveyed in [5], it will be not reproduced here in all details; only some basic ideas, principal points, and results will be discussed. Instead, we will point out a possibility to extend the results obtained for sector fields to inhomogeneous Wien filters, and pay attention to investigation, based on the obtained analytical expressions, of parasitic aberrations in multistage static sector field spectrometers. 2. Narrow Inhomogeneous Field Regions From the point of view of calculation of charged particle optical systems, most simple are devices based on a sequence of some standard elements in which a field distribution (or even trajectory coordinates) may be evaluated analytically. In the theory of static mass spectrometers, separators, and particle accelerators, these elements are sector electrostatic and magnetic analyzers, crossed field analyzers as well as quadrupole lenses and multipoles. A possibility to represent the field distribution in an analytical form also considerably simplifies calculation of some other types of charged particle analyzers, for example, mirror electrostatic analyzers and wedge magnets. Analytical algorithms for calculation of electron- and ion-optical devices allow one to design multistage systems with the aid of efficient gradient [34,35] or random search [361 optimization procedures. A calculation of the systems just mentioned is, however, still difficult due to a presence of inhomogeneous fringing fields in the vicinity of the edges of
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the optical elements. In these fields an exact analytical trajectory description is not only impossible but also in some cases even an analytical representation of the electromagnetic field distribution is impossible. To overcome this difficulty, Herzog [37,38] introduced the concept of an efecfivefield boundary and laid the basis of the fringing field integral method. The idea of this method is that, because of a relatively small length of the fringing field, charged particles are influenced only by some averaged characteristics of this field, expressed in terms of the integrals over the fringing field region of the electromagneticfield distribution and, besides, in terms of the parameters independent of this distribution. As a result, a transformation of the coordinates and slopes of charged particle trajectories by the entrance fringing field of an optical element may be described accurately as a composition of three transformations [4]: a transformationin a field-free space up to some virtual surface called an effective field boundary; a discontinuous transformation at this surface, expressed in terms of fringing field integrals; and, finally, a transformation in the ideal (“main”) field of the element, beginning just from the effective boundary. An analogous representation also holds, of course, for the exit fringing field. Such a description is valid if a researcher is interested not in the details of the trajectory behavior inside the fringing field region but only in the overall transformation from some point located outside the element to a point located inside it (or vice versa). The field integral method for calculation of fringing field effects in sector field analyzers and multipoles appeared to be so convenientthat it is still developingup to now and remains most widely used in calculations of static mass analyzers. Much publication is devoted to the investigationof fringing field effectswith the aid of this method; we will mention here only some of them. Such an investigation, initially performed in the second-order approximation for electrostatic and magnetic fields [39], was then extended to the third aberration order, mainly due to the efforts of H. Matsuda and H. Wollnik (electrostatictoroidal sector analyzerswere considered in [40], sector magnets in [41,42], and quadrupole lenses in [43]); similar studies were also performed in [44,45]. Recently an interest appeared in an investigation of the fifth-order effects in the fringing fields of electrostatic analyzers [46] and quadrupole lenses [47,48]. The fringing field effects in magnets with a complex boundary shape [49] and solenoids [SO] were considered. A high accuracy of the fringing field method was verified by a comparison of the results of charge particle trajectory simulationperformed with the aid of this method and by direct numerical integration using a differential algebra approach [5 11. It should be noted that application of the fringing field integral method, as a particular case of perturbation methods, requires an attention to the statement of the problem, scaling of variables, definition of a small parameter, and performing the expansion [52]. The absence of such an attention led to more or less serious inaccuracies in the results of many publications, including some of the works by Matsuda and Wollnik and even more recent papers (inaccuracies are contained,
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FIGURE2. A multiple magnetic prism for reflection low-energy electron microscopes that consists of two homogeneous bending magnets. This prism can be made stigmatically focusing by a proper choice of the ratio of the magnet excitations. Fringing fields of the outer and inner magnets overlap in the narrow gap between these magnets.
for example, in [53],which can be revealed by a comparison of the results listed in this paper with more accurate results of [54]. In spite of a large number of publications devoted to the development of the fringing field method, its resources seem to be far from being exhausted. Up to recently this approach was used to treat only single fringing fields, that is, to describe particle trajectory transformation between a field-free space and the main field of an electron- or ion-optical element. In Section IV we will present an extension of the fringing field integral method to a situation where several optical elements are placed close by each other so that the fringing fields of the adjacent elements overlap. An example of such a situation may be a quadrupole multiplet where lenses are closely packed in order to achieve higher lens optical power and reduce aberrations by increasing the rod lengths whereas a moderate total multiplet length is preserved [%I. Another example is a multiple magnetic prism (see Fig. 2) that consists of several differently excited homogeneous deflecting magnets placed close by each other. Such a device was proposed in [56]and is used for separation of illuminating and deflected electron beams in reflection low-energy electron microscopes [57, 581. Obviously, an action of the overlapping fringing fields of adjacent optical elements is not the same as the two successive actions of the exit fringing field of the first element and the entrance fringing field of the second element. Indeed, consider two homogeneous magnets placed close by each other. It is well known that both entrance and exit fringing fields of sector magnets diverge a charged particle beam in the vertical direction in case of magnet boundaries being normal to the beam. If the fringing field actions were considered separately, the entrance
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fringing field of the succeeding magnet should have been added to the fringing field action of the preceding one. However, in a limiting case of equal excitations of both magnets and a negligibly small gap between them the summary action of the fringing fields vanishes! This means that a concept of the effective boundary of a single fringing field is not valid for overlapping fringing fields. Nevertheless, the fringing field integral technique can be extended to the situation where two optical elements are closely packed, if the concept of a single effective boundary between rwo elements is introduced instead of effective field boundaries for each element separately. Then a transformation of charged particle trajectories in the narrow gap between adjacent electron- or ion-optical elements may be described as a composition of a transformation in the ideal (main) field of the first element up to the effective boundary between the elements, a discontinuoustransformation at this boundary, expressed in terms of fringing field integrals, and a transformation in the ideal field of the second element, beginningjust from the effective boundary. This way was proposed for multiple magnetic prisms in [59] and for quadrupole multiplets in [60]. It should be noted that for an investigation of fringing field effects it seems attractive to use perturbation methods not only for calculation of charged particle trajectory transformations but also for evaluation of the fringing fields themselves. Actually, since Herzog proposed a model of a plane condenser with a shielding aperture [611 for analytical calculation of fringing field distributionsin electrostatic sector analyzers and sector magnets, there were only few attempts to treat more complicated geometries. In particular, a fringing field distribution in a clamped cylindrical analyzer was obtained using a conformal mapping method in [62]. In more complicated cases these distributions were determined either by numerical simulation [63,64] or using some rough analytical models that contained arbitrary assumptions [65]. Only recently a perturbation method was successfully applied for analytical calculation of the fringing field of a toroidal electrostatic analyzer. The results were shown to be of importance for correct evaluation of a third-order transfer matrix of this fringing field [66]. However, in the present chapter we will not reproduce the corresponding fringing field calculation. The idea of this calculation is very similar to the methods used in Sections V and VI; an interested reader can find a description of the method for a toroidal analyzer fringing field calculation in [67].
3. Field Structures Similar to Two-Dimensional Ones Sector Field Analyzers of Charged Particles As noted in Section I, perturbation methods can be advantageously used for calculation of electromagnetic field distributions in systems whose geometries are in some sense similar to twodimensional ones, that is, where the curvatures of equipotential surfaces are small
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RGURE 3. An electrostatic cylindrical deflector whose interelectrodespace is terminated by Matsuda plates split into two parts. By changing the plate potentials V1 and V2 with respectto the cylindrical electrode potentials &VO,one can vary electron- or ion-optical properties of the deRector.
along one of the Cartesian coordinates. From this point of view, a field distribution in a sector electrostatic or magnetic analyzer practically always can be considered as being close to a two-dimensional one, since as a rule a gap between electrodes or pole faces in such an analyzer is much smaller than azimuthal curvature radii of equipotential surfaces; that is, a presence of this curvature leads to only a minor distortion of a field distribution in a plane containing the symmetry axis.' This fact was implicitly used for calculation of fields in a conical sector magnet [25] and in a toroidal sector analyzer [26]. The property just mentioned of sector fields is applied in Section V for a solution of a problem of an analytical electrostatic field representation in an electrostatic sector analyzer whose interelectrode space is terminated in the vertical (axial) direction by conducting plates located at the top and bottom sides of the condenser. Such plates were first proposed in [68], though the terminating plates are conventionally called Matsudu plates (Matsuda studied properties of a cylindrical analyzer supplied with these plates in [69]). Initially the additional terminating electrodes were intended to change first-order focusing properties of an analyzer; however, it was then found that large third-order aberrations are induced by such a change. To enlarge possibilitiesof a synthesis of a required field distribution in the analyzer, a more flexible design of multiply divided Matsuda plates was proposed in [70] (a similar idea was earlier discussed in [7 11). ShieldingMatsuda plates split into several parts, as shown in Fig. 3, are efficiently applied for different purposes, most often to adjust the field distribution or shield it from the influence of other equipotential surfaces (in particular, magnet pole faces in a Wien filter). Though in a particular case of a cylindrical deflector with single (nonsplit) terminating plates, provided that a gap between these plates and deflector electrodes is negligibly small, a field potential distribution can be obtained in an exact I
This minor distortion may, however, considerably change optical properties of a sector field.
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analytical form [72, 731; in a more complicated situation of multiply divided plates a calculation and optimization of the field distribution were performed up to recently only by a numerical simulation [70,74]. This way of a synthesis of a field with required electron-optical properties is very laborious, since a designer needs to vary several parameters (the potentials at the split terminating electrodes). At the same time such a synthesis is of major interest because only split Matsuda plates allow one to avoid large geometrical aberrations. In Section V an original perturbation method is described, which enables analytical calculation with a high accuracy of an electrostatic field distribution in a sector analyzer or Wien filter supplied with multiply divided Matsuda plates. This method was originally proposed for a cylindrical energy analyzer [67,75] and then extended to spherical and toroidal electrostatic analyzers as well as homogeneous and inhomogeneous Wien filters [76,77]. A most valuable feature of the proposed approach is that it allows an analytical solution of not only a direct problem (i.e., a field calculation for given Matsuda plate potentials) but also of an inverse problem of calculation of potentials at the split terminating electrodes, which provides a required electrostatic field distribution in the analyzer. Angle-Resolved Analyzers of Wide Charged Particle Beams Besides sector field analyzers, there are other types of the systems for focusing and analysis of charged particle beams, which can be investigated using the same ideology of perturbation methods. We mean electrostatic systems possessing axial symmetry and intended for a simultaneous angular and energy analysis or mass analysis of wide hollow electron or ion beams [78]. Since charged particles in optical elements of these systems as a rule move far enough from the symmetry axis, curvature radii of equipotential surfaces in the azimuthal direction are large as compared with a characteristicradial size of the region where the field is concentrated. Thus, a distribution of the field in a plane containing the axis of symmetry (meridianal plane) is similar to a two-dimensional field distribution. This fact is used in Section VI for calculation of electrostatic fields and investigation of optical properties of the elements applied in spectrometers under consideration. A popular system for an angle-resolved analysis of wide charged particle beams is a so-called polar-toroidal (or, simply, poloidal) analyzer. It consists of two toroidal surfaces between which particles move in a meridianal direction, as shown in Fig. 4. This direction of motion is different from the azimuthal direction used in conventional toroidal sector electrostatic deflectors. Focusing properties of an electrostatic field of a poloidal type were first investigated in [79]. Most often poloidal condensers are used for simultaneous angular and energy analysis of ions scattered on solid or gaseous targets [80,81] or of photoelectrons [82, 831. Poloidal analyzers also appeared to be attractive for investigation of a space plasma. This plasma is characterized by low density, a wide energy range, an anisotropy, and a rapid change of properties, though it consists of only a limited
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FIGURE4. A 90" poloidal deflector for an angle-resolved energy analysis of charged particles. Particles leaving asource, located at the axis of symmetry, perpendicular to this axis, are then focused on the ring-shapedimage at the position-sensitive detector plane, so that each point of the ring corresponds to a certain initial azimuthal angle. The energy dispersion takes place in the radial direction.
number of components. For this reason studying space plasma properties is advantageously performed by analyzers which possess a moderate resolving power but can register an energy, velocity, or mass spectra in a full solid angle within several seconds. From this point of view poloidal analyzers are very convenient because they allow a simultaneousanalysis of ions in a full range of polar angles from 0 to 360 degrees with a rather high angular acceptance (up to 10 degrees). This is why these analyzers have been designed and applied for a space plasma investigation in various configurations: as a single-stage analyzer [84], as a preliminary stage for a time-of-flight mass analyzer [85], and in combination with a toroidal magnet as a static mass analyzer [86]. Devices based on poloidal condensers can also operate as time-of-flight analyzers. It is well known that a conventional electrostatic sector field may be used for an energy-isochronoustime-of-flight mass analysis of ion beams [87]. Generally difficulties in manufacturing and low acceptance of sector-type time-of-flight spectrometers do not allow them to compete with widely used mirror-type reflectrons [88], though time-of-flight mass analyzers based on a toroidal geometry are still being designed [89, 901. A development of poloidal analyzers enabled timeof-flight properties of a toroidal field to be combined with a possibility of the angle-resolvedanalysis of ion beams. In [91] a scheme of a two-stage angular and velocity ion spectrometer was proposed; a device for a simultaneous angular and time-of-flight mass analysis of a space plasma is described in [92].
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Since there is no exact analytical representation of the electrostatic field distribution in a poloidal condenser, the field calculation and optimization of electronoptical properties of a system based on a poloidal geometry was as a rule performed by numerical simulation [93, 941. This made difficult a solution of actual problems for the design of angle-resolved analyzers: increasing the acceptance and the angular resolution, reducing aberrations, and improving the linearity of the energy or mass spectra. An application of perturbation methods allows one to obtain an approximate analytical representation of the electrostatic field potential in a polar-toroidal analyzer [95], accurate enough to perform third-order calculation of charged particle trajectories. This calculation can be based on a direct ray tracing or evaluation of aberration integrals. The latter approach was used in [96] for designing an ionoptical scheme of a time-of-flight mass analyzer with a high angular resolution. For an angle-resolved analysis of low-energy charged particle beams another type of analyzer can be used: a conical mirror. It acts similarly to a conventional cylindrical mirror analyzer [97,98];however, the reflecting fieldof aconical mirror is formed by a pair of coaxial conical electrodes,either parallel to each other [99] or with a common apex [ 1001. Application of a conical mirror instead of a cylindrical one enables analysis of hollow conical beams with a large cone opening angles or of disklike beams. These mirrors can be also used for a deflection of a conical beam [ 1011; in case this beam enters the conical field through the outer electrode. An approximate analytical electrostatic field representation in a mirror analyzer formed by two coaxial cones with parallel surfaces was first proposed based on a perturbation technique in [102]. In Section V1.D a more accurate perturbation approach will be proposed. In the present chapter we will not, however, go into details of trajectory calculations and only briefly mention general properties of a conical mirror analyzer; an interested reader can find further details in [102]. As a rule, a device for a simultaneous angular and energy analysis or mass analysis should provide for several types of focusing: spatial focusing in radial and azimuthal directions, spatial and time-of-flight energy focusing in time-of-flight mass analyzers, and so on. Besides, retarding charged particles in the analyzer may be desirable. Fulfilling all such conditions and technical constraints may become impossible without using additional lenses. Electrostatic lenses for focusing hollow conical beams usually have a conical shape of electrodes (see Fig. 5 ) and are called conical lenses. There are two types of conical lenses applied in angle-resolved analyzers: lenses with longitudinal electrodes [82] and slit lenses [83]. Up to recently optical properties of such lenses were not studied systematically; only a paraxial trajectory equation in a radial direction in a conical lens was obtained [ 1031. A perturbation method allows one to obtain an approximate analytical representation of the electrostatic field distribution in both lenses with longitudinal
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FIGURE5 . An electrostatic two-electrode conical lens with longitudinal electrodes. Charged particles forming a hollow conical beam move between two coaxial conical surfaces.
electrodes and slit conical lenses, using as a zero-order approximation a field in two-dimensional lenses of a similar geometry. For conical lenses with longitudinal electrodes such a method was developed in 1671 and for slit lenses in [104]. The proposed approach considerably simplifies an investigation of properties of conical lenses [104, 1051. B. How to Apply Perturbation Methods Correctly As mentioned in Section I, in order to avoid obtaining inaccurate results while
using perturbation expansions, attention should be paid to correct specification of small parameters and the form of expansions that represent a solution of the problem in question. Here we list several simple general rules which help avert the danger of incorrect application of a perturbation approach. 1. A small parameter (or several parameters), with respect to which perturbation expansions are constructed, should be explicitly separated out in the equations to be solved. One should avoid working with variables or parameters which are implicity assumed to be small or large. 2. A small parameter (or parameters) should be dimensionless. 3. Independent variables, parameters, and functions, present in equations to be solved, should be scaled to a dimensionless form. Care should be taken that in the final representation of the equations there are no large variables, parameters, or functions, as well as functions with large first- or higher-order derivatives. From this point of view, it is advantageous to scale the independent variables (coordinates) so that the intervals, where all the
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
295
functions are defined, are of the unit length (or a length that is of the order of unity). 4. A form of the expansion for the solutions of the equations is to be chosen in accordance with the expected behavior of the solution. For example, if a solution is expected to change rapidly in some narrow region, using so-called boundary layer expansions is preferrable (such type of expansions is sometimes used while considering emission electron-optical systems [20]; in the present chapter, however, all the expansions used are simple power serii with respect to a small parameter).
SECTOR FIELDS AND THEIR 111. CALCULATION OF WEAKLY DISTORTED PROPERTIES WITH THE AID OF A DIRECT SUBSTITlJTION METHOD
In this section an influence on the fields and charged particle trajectories of the smooth distortions of electrode and magnet pole surfaces in sector electrostatic and magnetic analyzers as well as Wien filters is investigated, neglecting fringing field effects. A. Electrostatic Field and Charged Particle Trajectories in an Imperfectly Manufactured Sector Energy Analyzer We consider here a general case of a toroidal electrostatic sector analyzer. It conists of two electrodes possessing axial symmetry with respect to the z-axis of a cylindrical coordinate frame (r,4, z } . Sections by a plane containingthe symmetry axis through the surfaces of the electrodes are circle arcs whose curvature centers lie in the median plane z = 0 (see Fig. 6) and generally do not coincide. The radii Rl and R2 of the curvatures of these arcs are assumed to be not too different from each other, that is, 1 R2 - R I1 << min( Rl , R2). The electrode surfaces intersect with the median plane z = 0 at the points with radial coordinates rl = ro - be and r2 = ro be,ro being a radius of the main beam path (optical axis) in the analyzer. The circular arcs of an “ideal” (nondistorted) toroidal analyzer are described by the equations
+
r - ro = G(z) -2
+ b e g j ( z ) , j = 1,2, -4
+
where Ro = ( R I R2)/2, a subscript j = 1 corresponds to the inner electrode and j = 2 to the outer one. The functions g j ( z ) depend on the position of the curvature centers of circular arcs. In particular, in a most common case of
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M. I. YAVOR
FIGURE6. A section through the electrodes of a toroidal electrostatic sector analyzer and a sector magnet by a meridianal plane containing the symmetry axis z. This figure also represents a section through the electrodes and magnetic poles of an inhomogeneous Wien filter by a plane x y perpendicular to these surfaces. In the latter case the electromagnetic field is assumed to possess planar symmetry but not rotational symmetry.
coinciding curvature centers R I = Ro - be, R2 = Ro
+ be these functions read
be
&(z)
=1
-+-.I
+ [1 - be 2 R i 2R03
22
+.
Distorted toroidal condenser electrode surfaces can be generally represented by the equations
+
+
r - ro = G ( z ) be[Ej(z) Sgj(Z, @)I,
(3)
where the functions 6gj characterize distortions. We assume the distortions to be small (16gj I) << max 1 gj I) and smooth (the latter means that the derivatives of the functions Sgj are small too). Now we proceed to dimensionless coordinates. These coordinates can be introduced in two different ways: by scaling cylindrical coordinates r and z with respect either to half the interelectrode gap be or to the mean beam path radius curvature ro. Formally, from the point of view of application of perturbation methods, the former way might be advantageous, since normalized, half the radial distance between the electrodes in the median plane becomes equal to one (see Section 1I.B).
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
297
However, in this section we will use the lamer way of scaling and introduce the dimensionless coordinates r] = ( r - ro)/ro and c = z/ro, since these coordinates are conventionally used for representing the equations of charged particle motion in sector fields. We also define a small dimensionless parameter E
= b,/ro
which characterizes the analyzer as a system with a narrow (as compared with the electrode curvature radii) interelectrode gap. In the new coordinates Eq. (1) reads
v = ~ ( c +) E J j ( O 9 where
and C(0)
= ro/Ro;
8
the functions in case of coinciding centers of curvature of electrode sections by a meridianal plane 4 = const read
Then for distorted electrode surfaces Eq. (3) yields
v = F ( < )+
+ Sfj(cp
911,
(4)
where Sfj(c, 4) = Sg,(ro(, 4 ) are small and smooth functions. Note that the derivatives of all the functions in Eq. (4) are not large if the value Ro is not too small as compared with ro. This fact will allow us to obtain analytical expressions for the field potential distribution in the toroidal condenser in a form of asymptotic power serii with respect to the small parameter E. As examples of surface distortions, we can consider two simple cases (see Fig. 7): 1. A variation of the interelectrode gap. Consider the inner and outer
electrodes to be displaced in the radial direction so that the radial distances between their real and nominal positions are A, (9);then Sfj (4) = Sg, (4) = A, ( 4 ) / b ethat ; is, the functions fj represent the ratio of the electrode surface displacements to half of the interelectrode gap.
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M.I. YAVOR
FIGURE7. A section by a meridianal plane through an imperfectly manufactured cylindrical electrostatic analyzer. The inner electrode of the analyzer is shifted with respect to its nominal position by A in the radial direction and inclined by the angle 0 in the vertical direction.
2. A mutual vertical inclination of the electrodes. Consider the electrodes to be manufactured inaccurately so that their surfaces are not normal to the median plane at the points of their intersection with this plane; let O j ( 4 )~b((a6gj/aZ>I,=o = E(a6fj/a<)(,=o be small vertical angles between the electrode surfaces and the z-axis, measured at the median plane. Then the derivatives of the functions 6fj with respect to represent the ratio of the angles of the vertical inclinations of the electrodes to the small parameter 6. Note that these angles are thus assumed to be small enough as compared with 6.
<
We consider the electrostatic field potential u in the toroidal analyzer in the form of an expansion in the vicinity of the main beam path: w
.
+
The coefficients H i k can be represented as the sums H i & ( + ) = h i k x i k ( 4 ) where the constants hik are the expansion coefficients for the nondistorted analyzer and the terms X i k ( 4 ) describe a small deviation of the potential caused by electrode distortions. To obtain analytical formulas for h i k and x i k , we represent them as asymptotic power serii
n=O
n=O
and substitute Eq. ( 5 ) together with expansions of Eq. (6) into a Laplace equation
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
299
for the electrostatic potential
and boundary conditions at the distorted electrode surfaces
+ ~{4([) + Sfj(C9
u[F([)
911, 91 = V j , j C q
= 1929
(8)
where V I and V2 are electrode potentials. As a result of such a substitution, we obtain a set of algebraic equations for coefficients hj,"' and xi(kn). The procedure of deriving these equations is described in detail in [5] and, therefore, we do not reproduce it here. After straightforwardbut lengthy calculations, one can extract from the correspondingequations analytical formulas for the coefficients with arbitrary numbers i, k, and n. We list next the coefficientshik for i k 5 3 and X i k for i k 5 2, assuming the electrode potentials V I and V2 to be chosen so that hw = 0 and h10 = 1:
+
hoi = h03 = hi1 = h21 = 0,
+
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M. I. YAVOR
Here the subscripts5‘ and 4 denote the derivatives with respect to the corresponding variables in the plane 5 = 0. In the coefficients x i k retained are only the terms linear with respect to the functions Sf,; these functions are also assumed to be calculated at = 0. The analytical expressions for the coefficients hik are of an interest by themselves because they allow one to calculate the field distribution in an ideal toroidal analyzer in a more general situation and with a better accuracy than the expressions obtained in earlier papers [26, 1061. In particular, the expression for the coefficient h02 directly represents a so-called toroidal factor c, which determines electronoptical properties of the analyzer to the first-order (linear) approximation [ 1061. In the most common situation of the coinciding centers of the electrode vertical curvatures, we have
In case the interelectrode gap of the toroidal analyzer is not very narrow, Eq. (16) allows one to use for calculation a more precise value of the toroidal factor than the approximation c = c(0)conventionally used in practice. Most important is, however, the analytical representation of the coefficients x i k that characterize a disturbance of the toroidal electrostatic sector field caused by manufacturing inaccuracies. Suppose that the nominal kinetic energy KO of the charged particles in the beam is such that a particle possessing this energy moves along a circular path of the radius ro inside the ideal analyzer (where the electrode potentials are chosen so that hm = 0 and hlo = l), and a = ( K / q Ko/qo)/(Ko/qo) is a relative deviation of the energy-to-chargeratio for an arbitrary charged particle with respect to a reference particle possessing the nominal kinetic energy. Then the trajectory equations in the toroidal analyzer take the following form: It 2vi2 vi2 (1 v12 q ----]-v = 1 + a 2 u I+v
+ rr2+ +
A substitution of the potential distribution in the imperfectly manufactured analyzer into Eqs. (10) and (1 l ) leads to the equations that describe beam distortions caused by inaccuracies of the electrode surface shapes. It is convenientto represent the particle coordinates as sums
v(4) = ij(4) + 41 ( 4 ) + 42(4)> (‘(4)= MI+ h<4>+ t2(4I9
(12) (13)
where i j and f describethe trajectoriesin the nondisturbed field of the ideal toroidal condenser, 41 and f~ are small displacements of the beam as a whole in the radial and vertical directions, and 4 2 and f 2 describe other trajectory distortions. In the
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
301
first aberration order the corresponding trajectory equations read
+ Afj = C , f ” + BS = 0,
(14)
~ =I bo,
(16)
= co,
(17)
fj”
4;’ + A
l; + BtI
(15)
+ = blfj + b2fj’+ b3f + b4c’ + b 5 +~ bby, t; + B f 2 = clfj + + cgf + c4f’ + c5a + c6y.
6;’
A42
c2fj’
(18) (19)
Here the parameter y = ( m / q - mo/qO)/(mo/qo)denotes a relative deviation of the mass-to-charge ratio for an arbitrary particle with respect to the reference particle possessing a nominal mass. This parameter evidently does not contribute to the equations for charged particle trajectories in the toroidal analyzer (i.e., b6 = C6 = 0); it is introduced in 9 s . (18) and (19) to unify the forms of these equations with the forms of equations in cases of a sector magnet and a Wien filter, considered in Sections 1II.B and 1II.C. The coefficients in Eqs. (14)-( 19) are A
=3+h20,
B=h02,
C=U,
bo = -2xOO - XI07 co = bl = - ( l O h 2 0 h30 IS>% - 2 h 2 0 x 0 0 - 12x00- 6x10- x 2 0 , b2 = 24; x&, b3 = -(%2 h 1 2 1 f 1 - 2x01- X I I , 64 = -2fi, -KO19
+
b5 = (hzo CI
=-
c2
= 2i;,
+ +
+
+ 6)41 + 4x00+ XIO, +
~ o 2 h12)l1
- 4x01- x11.
+ + c5 = h 0 2 h + XOl,
= -(4ho2 h i d 4 1 - 2ho2XOO - X02, c4 = 24; XLO, C3
In these coefficientsretained are only the terms linear with respect to the parameters characterizing the field disturbances. Since Eqs. (14) and (15) can be easily solved analytically, the solutions of Eqs. (16)-( 19), which describe beam distortions, can be expressed in a form of abrerration integrals. For some particular types of manufacturing defects, as, for example, electrode shifts or ellipticity, these integralscan be evaluated analytically. The beam distortions are represented by Eqs. (16H19) in the first aberration order. However, the equations for higher-order parasitic aberrations caused by manufacturing imperfections can be also obtained using the procedure just described.
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M. I. YAVOR
B. Magnetostatic Field and Charged Particle Trajectories in an Imperfectly Manufactured Sector Magnet
In this section we consider a general case of an inhomogeneous sector magnet, whose pole surfaces are rotationally symmetric with respect to the z-axis of a cylindricalcoordinateframe {r,4, z) and also symmetricwith respect to the median plane z = 0 (see Fig. 6). Generally an equation describing the pole surfaces of such a magnet is
where the subscript j = 1 corresponds to a lower pole, j = 2 to the upper one, ro is a main beam path radius inside the magnet, and b, is half a gap between the pole faces, measured at the radial distance r = ro from the sector symmetry axis z. The function Q can be represented as a power series
Here the coefficients pm are defined by the shapes of the pole surfaces. For a homogeneous magent P m = 0 for all m z 1 ;in the case of a conical magnet with half a cone opening angle (Y (see Fig. 6) pm = 0 form z 2 and p1 = (rotan a)/b,; finally, for a toroidal magnet with a radius of the pole face curvature R one has p2 = r;/(2bmR),p3 = 0, p4 = r$/(8bmR3),and so on. Note that we assume all the coefficients pm not to be large; in particular, this means that for a toroidal magnet a condition R >> ro should hold. In the presence of inaccuracies in machining or assembling the magnet poles, Eq.(20) transforms to
where the functions 6qj describe small smooth distortions of the magnet pole surfaces. As in Section IILA, we proceed now to the dimensionless coordinates q = (r ro)/ro and = z/ro and introduce a small dimensionless parameter E
= bm/ro.
Then Eq. (22) takes the following form:
< = (-l)’e[g(~) + Sqj(q9 411.
(23)
As examples of manufacturing defects, we can consider the following two simple cases:
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
303
1. A variation of the air gap in the magnet. Consider the lower and upper pole faces to be displaced in the vertical direction so that the vertical distances between their real and nominal positions are A j (4);then Sqj (4) = (-l)jAj(4)/bm; that is, the functions q j represent the ratio of the magnet pole face displacements to half of the air gap (note the opposite signs adopted for displacements in the same direction of the lower and upper poles). 2. A mutual inclination of the magnet pole faces in the radial direction. Consider the pole faces of a homogeneous magnet to be manufactured inaccurrately so that they are not parallel to the median plane z = 0 but slightly inclined as in case of a conical magnet; let O j (4) = (-l)jc(t36qj/t3q)(4=0 be small angles in the radial direction between the pole surfaces and the median plane, measured at the radial distance r = ro. Then the derivatives of the functions Sqj with respect to q represent the ratio of the angles of the inclinations of the magnet pole surfaces in the radial direction to the small parameter e . Note that these angles are thus assumed to be small enough as compared with 6 . Note also that the opposite signs are adopted for inclinations in the same direction of the lower and upper poles. Similar to what was done previously in Section III.A, we describe the magnetic field in the inhomogeneous magnet by its scalar magnetic potential w; we represent the latter in the form of an expansion in the vicinity of the main beam path: M
.
The coefficients A i k can be represented as the sums A i k ( 4 ) = aik 4- ( Y i k ( 4 ) where the constants aik are the expansion coefficientsfor the nondistorted magnet and the terms cxik ( 4 )describe a small deviation of the scalar magnetic potential caused by pole face distortions. To obtain analytical formulas for a i k and (Yik, we represent them as asymptotic power serii 00
00
n=O
n=O
and substitute Eq. (24) together with expansions of Eq.(25) into a Laplace equation L(w)= 0 for the scalar magnetic potential, where the operator L is expressed by Eq. (7), as well as into the boundary conditions at the pole faces (we assume here an infinite magnetic permeability of the poles, so that the scalar potential values at the pole surfaces are constants fWO):
~ [ v (-1)jc{q(~) , + Sqj(q9
41= ( - 1 ) j ~ o .
(26)
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M.I. YAVOR
As a result of such a substitution, we obtain a set of algebraic equations for coefficients a::) and a!:’. The procedure of deriving these equations, which is similar to the procedure used in the case of an electrostatic sector field, is described in detail in [5] and is not reproduced here. From the corresponding equations, analytical formulas for the coefficients with arbitrary numbers i , k, and n can be extracted. We list next the coefficients a i k for i k 5 3 and a i k for i k 5 2, assuming the magnetic scalar potential WO to be chosen so that the condition aol = I is satisfied:
+
+
= a10 = a12 = a20 = a02 = a30 = 0,
+ E 2 [ . .I, + PI + 2P2 + e2[’
a21 = 2(P: - P 2 ) a03
= -2p: €
am(@>= --(Sq2
2
€
a10(4J)= p
q
*
*
* I 7
- 6%) +€3[*-1,
+ P l ( S q 2 - 6ql)l+ 63[*-1, - h),, + PI(Sq2 - w,
2 - Sqd,
E
azo(@) = $-(&I2
- 2 ( P ? -P2) ( 8 q 2 - 6 q l ) l + ~ 3 [ . * * 1 . 1 UOl(@) = --(h +Sq2) + € 2[...l, 2 1 all(@) = Pl(6ql 6q2) - -(6q1 8 q 2 ) , E 2 [ * *I, 2
+
E
402(@) = p
% 2
- Sql)dd + (&I2
+
+ - b),, + (1 - PI) (Sqz - Sql),
+ (2P: - p2 - P I ) (6q2 - h ) l + E 3 [ . . .I. Here the subscripts r] and 4 denote the derivatives with respect to the corresponding variables, calculated at = 0. In the coefficients aik retained are only the terms linear with respect to the functions S q j ; these functions are also assumed to be calculated at r] = 0. The formulas obtained for the coefficients a i k of the magnetic field distribution in an ideal inhomogeneous magnet generalize to the case of a toroidal magnet earlier results reported in [25] for a conical magnet. Most interesting, however, are the analytical expressionsfor the coefficients ajk which characterize a magnetic field disturbance. Suppose that, similarly to the case of a toroidal electrostatic analyzer, the nominal kinetic energy KOand the nominal mass rno of the charged particles in the beam
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
305
is such that a particle possessing this energy and mass moves along a circular path of the radius ro inside the ideal magnet (where a magnetic flux density is chosen so that the coefficient aol = 1). Then the trajectory equations in the inhomogeneous sector magnet take the following form:
where u and y are relative deviations of the energy-to-charge and mass-to-charge ratios for an arbitrary charged particle with respect to a reference particle possessing the nominal kinetic energy and mass. A substitution of the field distribution in the imperfectly manufactured magnet into Eqs. (27) and (28) leads to the equations that describe beam distortions caused by inaccuracies of the pole surface shapes. If we represent the particle coordinates as sums of Eqs. (12) and (13), then in the first aberration order the resulting equations for the terms of these sums take the form of Eqs. (14)-( 19), where the coefficients are
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M.I. YAVOR
c3 = (2all Al c4 = 91,
+ azdii1 + a l l ,
l A 1 c5 = c(j = --all
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
307
Now taking a required limit in the formulas obtained previously in Sections 1II.A and 1II.B is straightforward. In this limit a coordinate frame ( x , y , s}becomes a Cartesian one. In a resulting Cartesian coordinate frame the equations describing the electrode surfaces of an electrostatic cylindrical condenser can be given in the following form: x =X(y)+xj(y)
+ S j ( y , s),
j = 172.
(29)
The first term in the right-hand side of Eq. (29) represents some reference surface located between the ideal electrodes:
Y 4 - ... X ( y ) = --Y 2 - 2Ro 8 R i where Ro = (R1+ R2)/2, R I and R2 are the electrode curvature radii (see Fig. 6). The second term in the right-hand side of Eq. (29) describes deviations of the shapes of ideal (nondistorted)electrodes from the shape of this reference surface (a subscript j = 1 corresponds to the inner electrode and j = 2 to the outer one). The last term in Eq. (29) gives the electrode distortions. Note that in principle we do not confine ourselves with a case of coinciding centers of the electrode curvatures and the cylindrical shape of the electrodes. We only require that the electrodes be symmetric with respect to the median plane y = 0. If they are not cylindrical, the values R1 and R2 can represent local curvature radii of the electrodesat the points of their intersectionwith the median plane. Anyway, we suppose the following conditionsto be satisfied: XI (0) = -be* x2(0) = be and x j ( - y ) = x , ( y ) ,where be is half an interelectrode gap measured in the median plane. For cylindrical electrodes with the equal curvature radii R I = R2 = Ro one has X I = - be,x2 = be, for the coaxial cylindrical electrodes
Now to unify the representation of the results of this and the two previous sections, we introduce again the scaled coordinates, which we denote as in Section III.A, as ( r ] , {, #}; these coordinates,however, now are the Cartesian ones: r] = pox, ( = poy, # = pos. We also introduce a small dimensionless parameter 6,
= Pobe
which is a ratio of half the interelectrode gap to the curvature radius of charged particle trajectories in the pure magnetic field with a flux density equal to the flux density of the magnetic field at the optical axis of the Wien filter. In the coordinate
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M. I. YAVOR
framejust describedEq.( 2 9 )becomes identical to Eq.(4),if we suppose in the latter F ( c ) = P X ( C / P O ) ,f j C ~= xj(C/fi)/beg 4) = S j ( C / h , # / P O ) / b e . Similarly as it was for a toroidal analyzer considered in Section III.A, in case of a displacement of an electrode in the x-direction by a small distance A , the function 6 f j equals A j / b e ;in case of a small vertical inclination of the electrode by an angle 0 , the derivative ( 6 f j ) < w Oj/Ee. The electrostatic potential ii of the ideal condenser field can be represented as afj(C9
where
and the disturbance 6u of the potential reads
(31) where the expressions for the coefficients XOO, x l o , X O I , X I 1 , x 0 2 coincide with the , x20 = - xO2 e,(Sfi 6f 2 ) @ @ / 2 formulas given in Section 1II.A with E = E ~ and . .I. Here the subscripts denote the derivatives with respect to the corresponding variables; all the functions and their derivatives are calculated at the optical axis q = 5 = 0. Analogously, the equations which describe distorted pole faces of an inhomogeneous magnet in a Wien filter, read
+
+
+€:[a
Y = (-1)j[y(x)
+ yj(x9
~ 1 1 7
(32)
where the term Y ( x ) defines the shape of the ideal (nondistorted)magnet pole faces, and the term yj small distortions of this shape (a subscript j = 1 corresponds to the lower pole and j = 2 to the upper one). The ideal magnet pole faces are assumed to be symmetric with respect to the median plane y = 0. The function Y ( x ) can be represented in a form of an expansion W
Y ( x ) = bm
+ C Pnxn
9
(33)
n= I
where b, is half an air gap between the poles, measured at x = 0, the coefficients P,,depend on the shape of the pole faces. For a homogeneous magnet P,, = 0 for
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
309
all n 2 1. In case of a wedge magnet Pl = tana, where a is half a wedge angle, and P,, = 0 with n 1 2. For a magnet with cylindrical pole faces P2 = 1/(2R) where R is a pole face curvature radius. In the dimensionless coordinates [ q , (,41 Eq. (32) takes the form of Eq. (23), whereq(r~)=Y(t7/~0)/bm,aqj(q, 4) = Y~(V/PO, 4/fi)/brn, Pn = Pn/brn. Inparticular, in case of a small vertical displacement A j of a magnet pole face the function Sqj equals (-l)jA,/bm; in case where a pole face of a homogeneous magnet is inclined with respect to the median plane y = 0 by a small angle Oj, the derivative (6qj)q x (-1)jOj/em, where we use a notation ern = Pobm.
A scalar potential ib of the field of the ideal magnet can be represented in the following form:
(35) where the expressions for the coefficients am, ale, a01,all, a02, and a20 coincide with the formulas given in Section III. B with 6 = em. Here, as in case of an electrostatic field, the subscripts denote the derivatives with respect to the corresponding variables; all the functions and their derivatives are calculated at the optical axis q={=O. Suppose that the nominal kinetic energy KO and the nominal mass rno of the charged particles in the beam are such that a particle possessing this energy and mass moves along a straight optical axis of the ideal Wien filter, where the distributions of the electrostatic and magnetic fields are given by Eqs. (30) and (34). Then, taking into account the adopted normalization of the field potentials, the trajectory equations in the inhomogeneous Wien filter take the following form:
$’ = - 1 + ql* + t i 2 a u 1+a-2u
,au
[a,- 17- a,]
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M.I. YAVOR
where, as in the previous sections, (T and y are relative deviations of the energyto-charge and mass-to-charge ratios for an arbitrary charged particle with respect to a reference particle possessing the nominal kinetic energy and mass. A substitution of the field distribution in the imperfectly manufactured Wien filter into Eqs. (36) and (37) leads to the equations that describe beam distortions caused by inaccuracies of the electrode and pole surface shapes. If we represent the particle coordinates as sums of Eqs. (12 ) and (13), then in the first aberration order the resulting equations for the terms of these sums take the form of Eqs. (14)-(19), where the coefficients are
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
31 1
In these coefficientsthe only terms retained are linear with respect to the parameters characterizing the distortions of the electrode and magnet pole surfaces.
D. Parasitic Beam Distortions in Charged Particle Analyzers Based on Sector Fields and Wien Filters and Their Correction We describe a trajectory of a charged particle at the profile plane of the entrance slit of a static energy or mass analyzer by its initial coordinates xo and yo, the angles a0 and POof an inclination of the x- and y-projections of the particle trajectories with respect to the optical axis z, and by relative deviations cr and y of the particle energy-to-charge and mass-to-charge ratios with respect to their nominal values. For the analyzers possessing a symmetry plane y = 0, the values of the lateral coordinates x and y in an arbitrary plane z = const can be represented in a form of aberration expansions
+
x = (x la>ao (x I x h o
+ (x I c-JT)cr + (x I Y)Y
+ (x I aa).,” + (x I acr)aocr +
Y = (Y
* * *
,
I BIB0 + (Y I Y>Yo + (Y I aB)aoSo + -
* *
(38) (39)
with respect to the powers of the initial parameters ao, xo, SO,yo, cr, and y. A position of a final x-image is characterizedby the condition (x I a) = 0. In doublefocusing mass spectrometers the condition (x I 6) = 0 also holds. As a rule, optical schemes of high-performance analyzers are optimized so that some higher-order aberrations are eliminated in the final image plane, for example, (x I aa) = 0. In the presence of machining or assembling inaccuracies in the analyzer, a dependence of the final trajectory coordinates on the initial parameters becomes more complicated: x = AXm
+ [(x I a)+ Axalao + [(x I x) + Axx1x0
+ A x u b + [(x I v>+ AX,IY + AxpBo + AXyYO + [(x I aa)+ Axaalai + [(x I + A - ~ ~ ~ l a o ~ + [(x
ID)
(YO)
+ AxapaoBo + Y = AYm
* * *
,
+ [(YI B ) + A Y ~ I B +o [(Y I Y) + A Y ~ I Y+O
(40) * *
(41)
where the coefficients Ax and Ay with various subscripts are small values depending on the type of defects. In an image plane each of these coefficients correspond to a certain type of parasitic aberrations and beam distortions. The terms AXm and Aymare responsible for a lateral shift of a main beam path and the image as a whole. The term Axa is responsible for displacement of a Gaussian image plane along the optical axis and, thus, for an image defocusing at the ideal position of the exit analyzer slit.
3 12
M. I. YAVOR
To the same effect leads a displacement along the optical axis of an energy focus (achromatic plane) in double-focusing mass analyzers in case of a nonvanishing coefficient Axo. The coefficient Axx is responsible for a change of a linear magnification of the image and the coefficient Ax,, for a change of the mass dispersion in a mass analyzer. The aberrationsjust listed are caused by defects that do not violate a symmetry of the system with respect to its median plane. If this symmetry is violated as a result of some manufacturing defects, two more parasitic aberrations appear, which are characterized by the coefficients Ax, and A x y . Consider these aberrations in more detail. Assume for the moment that a charged particle analyzer is triple focusing [ ( x I a)= ( x I 6) = (y I /?) = 01 and its entrance slit is infinitely narrow. Then particles leaving one point of this slit are focused to the first order in a point with the zero x-coordinate in the Gaussian image plane, whereas the image of all the slit is a vertical line (whose height Hi = (y I y)&, where HOis the entrance slit height). If only the parasitic aberration An, is nonvanishing in Eq. (40), then charged particles leaving one point of the entrance slit will still be focused in one point, but the x-coordinate of this point will depend on the initial y-coordinate of the trajectory. This means that the image of the whole entrance slit is still a thin line; it is, however, no longer vertical but inclined by some angle with respect to the y-axis. If, as it is most often the case, the parasitic aberration AxB is also nonvanishing, the particles leaving one point of the entrance slit under different vertical angles form a linear image parallel to the x-axis; the length of the line is proportional to the coefficient Ax, and to the vertical angular spread in the beam. Thus, the image of the whole entrance slit is defocused and has a parallelogram-like shape. In case the analyzer is not focusing in the y-direction, the shape of the image in the situation just considered generally remains (in the approximation of the first aberration order) a parallelogram, whose upper and lower sides are, however, not parallel to the x-axis. An example of the parallelogram-like defocusing of the image, simulated by a computer program ISIOS [ 1071, is shown in Fig. 8. This type of defocusing can be also observed experimentally 11081. The described effects exhaust the types of the first-order parasitic beam distortions in the x-direction, observed in the image plane. Thus, these types are 1. lateral image shift, 2. shift of the Gaussian image plane along the optical axis and the corresponding image defocusing in the nominal Gaussian plane, 3. change of the linear magnification of the image, 4. change of the energy dispersion or a longitudinal shift of the energy focus (achromatic plane) and a corresponding image defocusing in the Gaussian plane of a double-focusingmass analyzer, 5 . change of the mass dispersion,
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
313
RGURE 8. Computer simulation of images, produced by the mass analyzer of [108], which are created by the ions of three masses 999.96, 1o00, and 1ooO.04 a m u emerging from the 0.008-mm-wide and 4-mm-high entrance slit with a 1.6" horizontal angular spread and a 0.4" vertical angular spread: in a perfectly manufactured analyzer (a), in the case of an imperfectly manufactured inner electrode of a cylindrical electrostatic deflector, which has a conical shape with a cone opening angle of 0.5 mrad (b), and in the case where the effect of this imperfection is reduced by an excitation of a rotated quadrupole corrector that eliminates the parasitic aberration Axa (c).
6. tilt of the image, 7. parallelogram-like image defocusing. First-order parasitic aberrations in the vertical direction y, except the vertical shift of the main beam path and the image as a whole mentioned above, are not of a practical interest for charged particle analyzers because they do not influence their energy or mass resolution and can only slightly decrease the transmission of the analyzer. Note that the quality of the image in charged particle analyzersis deterioratednot only by the first-order parasitic effects but also because of higher-order parasitic beam distortions, such as, for example, a parasitic second-order angular aberration Axaa (which can be important in the systems where the angular aberration (x I aa) is eliminated) or a mixed angular aberration AX,^, which appears if the symmetry of a system with respect to its median plane is violated. The higher-order parasitic aberrations can be estimated based on the same approach that was used in Sections 1II.A-1II.C for deducing equations which describe first-order parasitic effects. Actually, the second-order parasitic beam distortions in practice determine tolerances for the optical elements of charged particle analyzers, since the first-order parasitic aberrations can be more or less successfully compensated for by correcting elements. Consider now what types of the electrode or magnet pole distortions are responsible for different kinds of parasitic beam distortions. A shift of the optical axis in the x-direction is described by Eq. (16). In an electrostatic sector analyzer this shift is caused by nonvanishing values of the coefficients x~ and XIO of the electrostatic field expansion, that is, by changes of the field potential and strength on the ideal circular main beam path. The formulas
3 14
M.I. YAVOR
obtained for these coefficients in Section 1II.A show that such changes are mainly contributed by the variation of the interelectrode gap, characterized by the difference ( S f 2 - Sfl), and the variation of the position of the middle of the gap relative to the ideal circular optical axis, characterized by the sum (Sf:! Sfi). The latter variation, which is the contribution to the beam axis shift of the pure potential change (i.e., of the variation of the beam kinetic energy), is considerably smaller than the former variation, which is the contributionof the pure field strength change (i.e., of the variation of the beam deflecting force), for comparable values of the distortions Sfj: indeed, in the formulas for the coefficients xw and x l o the sum (Sf2 Sfi) is multiplied by the small parameter E . In a sector magnet the shift of the optical axis in the x-direction is caused by a nonvanishing value of the coefficient sol, that is, by a change of the magnetic flux density on the ideal optical axis. This coefficient (see Section 1II.B) is proportional to the variation (Sql 6q2) of the air gap between the pole surfaces. In a Wien filter the shift of the optical axis in the x-direction, as follows from the results obtained in Section III.C, is caused by the summary action of all the listed factors. A shift of the optical axis in the y-direction is described by Eq. (17). In an electrostatic sector analyzer this deflection is caused by a nonvanishing value of the coefficient x o l , that is, by a sum ~ ( S f l Sf.)(. characterizing the summary inclination of the electrodes with respect to the analyzer median plane. In a sector magnet this shift is determined by the coefficient ( ~ 1 0 . This coefficient (see Section 1II.B) is proportional to the summary inclination of the magnet poles with respect to the median plane; in case of a conical magnet the value a10 is contributed also by a deviation of the middle of the air gap from this plane. In a Wien filter the shift under consideration is caused by the action of all the listed factors. Anyway, a sensitivity of the vertical shift of the optical axis to the electrode and magnet pole face distortions is smaller than the sensitivity of the horizontal shift of the optical axis, since both coefficients xol and a10are proportional to the small parameter E . The effects of the image defocusing in the x-direction are described by Eq. (18). In particular, a longitudinal shift along the optical axis of a Gaussian image plane and of the achromatic plane in the double-focusing mass spectrometer is determined by the coefficients bl ,b2 and b5 in this equation. In an electrostatic sector analyzer (considered by itself or as a stage of a mass analyzer) these effects are caused by the same factors as the shift of the optical axis in the x-direction and depend on the coefficient x20 which, in turn, is dependent on the sums (Sf1 Sf2)cc and (Sfl+ Sf2)## characterizing the deviation of the electrode curvatures from their nominal values. In a sector magnet the effects in question, apart from the factors which cause the horizontal shift of the optical axis, are determined also by the coefficient all which, in turn, is dependent on the sum (Sgl 8 q ~ )that ~ , is, on the rate of the mutual inclination of the homogeneous magnet pole faces (or
+
+
+
+
+
+
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
3 15
on the inaccuracy of the mutual inclination of the conical magnet pole faces). In a Wien filter parasitic effects under consideration are influenced by all the listed factors. Finally, a defocusing of the x-image caused by a violation of the symmetry of the system’s geometry with respect to the median plane is determined by the coefficients b3 and b4 of Eq. (18). In an electrostatic sector analyzer this defocusing is influenced by the same factor as a vertical shift of the optical axis and by a nonvanishing value of the coefficient ~ 1 1 . This coefficient, in turn, depends on the rate of a mutual inclination of the electrodes in the vertical direction, characterized by the expression (Sf2 - Sfl)(. In a sector magnet the parallelogram-like defocusing is also caused by the same factor as a vertical shift of the optical axis and by a nonvanishing value of the coefficient 4 0 2 and a longitudinal variation of the coefficient 400,that is, as shown by the analysis of the corresponding formulas given in Section III.B, by parasitic pole face curvatures and inclinations of the pole faces with respect to the median plane. Note, however, that both these coefficients are proportional to the small parameter E . Thus, the parallelogramlike defocusing in a sector magnet is considerably smaller than in an electrostatic sector analyzer for comparable magnitudes of the electrode and magnet pole face distortions. In a Wien filter the defocusing in question is caused by all the listed factors. The conclusions made in the previous paragraph are important for a practical design of charged particle analyzers based on sector fields. They mean that a most significant source of the parallelogram-like defocusing (which is most difficult to correct) in sector field mass analyzers is an electrostatic sector analyzer. Calculations show (see [109]) that a typical mutual inclination of the electodes of a cylindrical sector analyzer of 1 mrad (such an inclination can be caused not only by inaccurate assembling of the electrodes but also by occasional wavelike distortions of the electrodes in the vertical direction) can considerably decrease the resolving power of a double-focusing mass spectrometer with compensated second-order aberrations. Moreover, such an inclination creates not only large first-order parasitic beam distortions but also considerable second-order parasitic aberrations like Ax,p, which prevent restoring a nominal mass resolving power of the analyzer by aberration correctors. Thus, much attention should be given to an accurate machining and assembling of the electrostatic sector condenser electrodes. In some cases, where a precise manufacturing is difficult (in particular, in mass separators with large electrode sizes), it can be advantageous to avoid using not only toroidal electrodes that are difficult to assemble properly but electrostatic sector analyzers as well, replacing them by achromatic magnetic systems whose stages are floated at different electrostatic potentials [ 1101. Consider possibilities of compensation for first-order parasitic effects. A shift of the optical axis in the x-direction can be compensated for by a change of the
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electrostatic field strength in a sector condenser and Wien filter or of the magnetic flux density in a sector magnet and Wien filter. Since this compensation provides only a spatial correction in a given profile plane but not an angular compensation, in multistage systems it is advantageous to place additional deflecting elements along the optical axis. A vertical shift of the optical axis cannot be compensated for by a change of the excitations of sector field elements or a Wien filter. In this case different ways of correction are possible. In a sector magnet an efficient means of correcting a vertical beam deflection can be a small inclination of the magnet as a whole in the radial direction. In an electrostatic analyzer or Wien filter supplied by Matsuda plates, compensation may be achieved by applying additional electrostatic potentials of the opposite signs to the upper and lower Matsuda plates. Similar to the case of the horizontal deflection, it is advantageous to place additional deflecting elements in the system. A longitudinal shift of the image plane can also be easily compensated for in different ways: by changing the lens excitations, if any lenses are present in the optical scheme, by applying voltages to the Matsuda plates in electrostatic sector analyzers or Wien filters, or just by mechanical shift of the entrance or exit slits. Similarly, one can correct the position of the achromatic plane in a double-focusingmass spectrometer;however, to restore nominal positions of both shifted Gaussian image plane and achromatic plane require combination of these methods. Most difficultis to compensatefor a parallelogram-likeimage defocusing caused by a violation of the system's symmetry with respect to the median plane. Obviously such compensation requires that additional correcting elements be present in a system that can induce fields not symmetrical to the median plane. A most efficient element of this type is a quadrupole corrector, which is a quadrupole lens (electrostatic or magnetic) rotated about its axis to 45". Note that though rotated quadrupoles have been routinely used as stigmators in electron microscopes for many decades, in mass spectrometersthey are still not used commonly. Instead of a rotated quadrupole corrector one can employ electrostatic hexapole or octopole elements in the system to which additional voltages are applied in a special manner [ 1111. One can also apply voltages to electrically isolated poles of magnetic quadrupole lenses [ 1121. Nevertheless, the correction of the parallelogram-likedefocusing is rather complicated. The problem is that there are two first-order parasitic aberrations, Axp due to the vertical aperture angle and Ax,, due to the vertical size of the entrance slit, in a system whose symmetry with respect to the median plane is violated. One rotated quadrupole corrector can eliminate only one of these aberrations; both aberrations can be compensated for only by two rotated quadrupoles. Sometimes it is possible to use one corrector and additionally tilt the exit slit for the full correction (see Fig. 8) but this way is not suitable for multicollection devices.
CHARGED PARTICLE OpIlCS OF SYSTEMS WITH NARROW GAPS
317
The effectiveness of the correction is rather sensitive to the position of the rotated quadrupole corrector with respect to the imperfect stage of the system because an improperly located corrector may increase one of the parasitic aberrations Axp or Axy while correcting another [ 1071. Besides Matsuda plates, whose action is consideredin detail in Section V, we can mention two more special correctors of parasitic beam distortions. A widely used device for tuning magnetic mass spectrometers and especially mass separators is the correction coils [ 1131. The coils are mainly produced as printed circuit boards [114, 1151 though they can be also made of explicit wires or produced by cutting sheets of copper with thicknesses of several millimeters. Correction coils are very effective in compensating for a shift of the image plane (nl-coils) and the correction of the second-order angular aberration (nz-coils), which can become nonvanishing in the Gaussian image plane after its shift [1161. In principle n 1 -coils can also compensate for a shift of the energy focus plane; however, it is difficult to achieve coincidence of the Gaussian image and energy focus planes by using only one correcting element, since it shifts both these planes. Thus, at least two correctors are required. The lateral displacementof the beam and longitudinal shifts of the image and energy focus planes can be achieved by conventional quadrupolecorrectors. Though the quadrupole lenses used for the beam transport and focusing can be in principle applied for these purposes, it is adviseable to include in the system more flexible adjustable multipole elements [117]. At least one such element should be placed before the dispersive elements to allow the image plane position to be shifted independently of the energy focus plane. It should be noted here that such multipoles can be also used for correction of the parallelogram-like image defocusing, since they can provide quadrupole field components rotated by 45" with respect to the field of conventional quadrupole lenses. An example of the use of adjustable multipole correctors can be a set of three such elements implemented in the ISOLDE-3 mass separator [ 1181, which allows considerable improvement in its mass resolving power. We emphasize again that all the listed methods and elements are efficient for correction of the first-order parasitic effects. It should be noted that in case of large enough manufacturing inaccuracies such a correction can lead to an increase of second-order parasitic effects. For example, a correction of a longitudinal shift of the Gaussian image plane in analyzers with eliminated second-order angular aberration inevitably leads to a violation of the condition of elimination, which prevents achievement of a nominal performance of the analyzer. Secondorder parasitic aberrations present in imperfectly manufactured systems or induced by correction elements restrict a possibility of compensation for parasitic beam distortions and determine finally technological machining and assembling tolerances.
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M. I. YAVOR
Iv.
TFtANSFORMATION OF CHARGED PARTICLE 'rkAJECTORIES IN THE NARROW TRANSITION REGIONS BETWEEN ELECTRONAND ION-OFTICAL ELEMENTS
This section presents an extension of a fringing field integral method to a situation where effects of overlapped fringing fields between two adjacent electron-optical elements, placed close by each other, are considered. However, a technique of scaling with explicit separating out a small parameter, proposed in this section, is a useful tool which provides a correct and efficient application of a fringing field integral method to various problems, including investigation of effects of single fringing fields. A. Charged Particle Beam Transport through the Gaps of Multiple Magnetic Prisms
In this subsection we study a transformation of charged particle trajectories in a narrow transition region between two homogeneousdeflecting magnets, characterized by different magnetic flux densities in their air gaps. Such a region is formed, for example, in multiple magnetic prisms used for separation of illuminating and reflected electron beams as shown in Fig. 2. First of all, we will obtain a power series expansion for the magnetic field in the transition region based on the magnetic field distribution in the direction normal to the magnet boundaries; a magnetic field is characterized by its scalar magnetic potential W and we suppose the magnetic permeability of the poles to be infinite. We consider a general case where two homogeneous magnets with the same median plane Y = O are separated by a gap whose width G is comparable with air gaps 2Kl and 2K2 between the poles of these magnets. Generally a gap between the magnets may be circular with the radius of curvature R and inclined by an angle h with respect to the beam optical axis T (see Fig. 9). To be precise, R is the radius of some circle arc whose center coincides with the centers of the magnet boundary curvaturesand which intersects the beam optical axis at the same point as a so-called effective boundary between the magnets; the concept of the effective boundary will be specified later. The curvature of the gap is assumed to be positive if the second magnet boundary is concave. We introduce a Cartesian coordinate frame (X, Y, Z) whose origin C lies at the point of the intersection of the beam optical axis with the effective boundary between the magnets. The Z-axis is tangent to the beam optical axis; the XY-plane is the effective boundary plane. A condition specifying the position of the point C will is given later by E q . (59). We also introduce another Cartesian coordinate frame Y, 2) with the same origin C, which is tilted with respect to the coordinate frame ( X ,Y, Z) by the angle h about the Y-axis.
{x,
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
319
FIGURE9. A particle trajectory T passing through a narrow gap of a width G and a curvature radius R between two magnets. Shown are coordinate systems used in calculation.
We make now a first important step: namely, we proceed to dimensionless coordinates x = X / K 2 , y = Y / K 2 , z = Z / K z , t = x / K z , = z / K 2 and introduce a dimensionless cylindrical coordinate p = (r - R ) / K z (the axis of the cylindrical coordinate frame passes through the center of the curvatures of the magnet boundaries). Thus, in the new coordinates the air gap of the second magnet is formed by the surfaces described by equations y = f 1, and the dimensionless width G / K2 of the transition region between the magnets in the new coordinate frames is comparable to 1. A scalarmagnetic potential W in the gap between the magnets can be represented in a form of an expansion
<
We substitute the expansion of Eq. (42) to the Laplace equation, which in the cylindrical coordinate frame (p, y ) reads
a2w a2w +--- p aw -+- 0, ay2 ap2 1+PP ap where the parameter p = K 2 / R is assumed to be small. Having expanded the denominator of the last term in the Laplace equation by powers of this parameter, we come to the following sequence of relations between the coefficients wi(p):
wi+2 = -wI’ - (p - p 2 p
+ p 3p 2 - * * . > w ; ,
where prime denotes a derivative with respect to p.
(43)
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M. I. YAVOR
If we assume the distribution of the flux density B ( p ) = -K;’aW/ayl,,o in the median plane Y = O to be known from calculations or measurements, then all the coefficients wi can be evaluated from Eq. (43). Let us suppose wo(p) = W(0, p ) = 0. Then w2 = w4 = 0, w3 = K 2 [ B ” ( p ) ( p - p2p .)B’ ( p ) ] ,205 = Kz[-B””(p) - 2pB”’(p) . .]. With small 6 we obtain the following relations between the coordinates p , 6 , and 2‘:
+.
+
+
a
Using these relations, we can finally represent the magnetic field distribution in the vicinity of the <-axis as
K T ’ W ( 6 , Y , C) = -B(<)Y
1 + g1B ” ( 5 ) y 3- 75jB’”’(C)y5
(44) Note that the magnetic flux density B ( < )changes from some initial constant value B1 inside the first magnet to a final constant value B2 inside the second magnet at an interval of the coordinate whose length is of an order of unity. In the Cartesian coordinate frame {t,y , C } the equations for the charged particle trajectories are
where prime denotes a derivative with respect to C,
Po =
40K2 4r9 mo@o
6Q + y‘2 + 1 F = J (1 + + Y)’
mo,00, and qo are a mass, kinetic energy, and charge of a reference particle moving along the optical axis, and a and y are relative deviations of the energy-to-charge and mass-to-charge ratios for an arbitrary particle with respect to the reference particle. A condition of motion of the reference particle along the circle arc of a
CHARGED PARTICLE OF'TICS OF SYSTEMS WITH NARROW GAPS
32 1
radius 1 2 in the second magnet is
In Eq. (47) we introduced a dimensionless parameter E , which is assumed to be small. A presence of such a small parameter, which also characterizes the ratio of the width of the transition region between the magnets to the radius of the reference particle trajectory in the second magnet, allows one to apply a perturbation approach for calculation of a charged particle beam transformation by a narrow gap between two magnets placed close by each other. A next step is to represent the functions{(() and y ( ( ) as well as their derivatives with respect to ( in a form of expansions
6 = t o + €61 + r2t2 + E 3 t 3 + ...,
y' = y;,
+ Ey; + E2y; + r3y; + . * * *
Since the right-hand sides of Eqs. (45) and (46) are proportional to POand consequently to the small parameter E , the zero-order approximations and yo satisfy simple equations
6''0 -- 0, y;; = 0.
(49)
Now we will specify initial conditions for the functions &, yk and their derivatives. First of all, we define a concept of the so-called egective trajectory. This is a curved line consisting of two parts as shown in Fig. 10, which are two particle trajectories calculated in the ideal homogeneous fields of the first and second magnets up to the effective boundary (not taking into account the inhomogeniety of the field in the transition region) and that coincide with a real trajectory inside both magnets far from the transition region. At the effective boundary the effective trajectory is discontinuous, and our goal is to calculate a relation between
FIGURE10. Real and effective trajectories in a transition region between two magnets. The effective trajectory consists of two circle arcs of the radii rl and rz that are the deflection radii in the homogeneous fields of the magnets. Note that the effective trajectory is discontinuous at the effective boundary plane.
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M. I. YAVOR
the coordinates and slopes of the first and second parts of the effective trajectory at the effective boundary plane t = 0. This relation will completely describe a transformation of the corresponding real trajectory by the inhomogeneous field in the gap between the magnets, if we are not interested in a precise behavior of this trajectory in this field and are only interested in the behavior of trajectories far from the transition region. The latter will be calculated by an analytical transformation of the simple effective trajectories at the effective boundary instead of integration of real trajectories in the inhomogeneous field. Therefore, we define the coordinates R = xI,=o, 1 = yl,=o and their derivatives R‘ = dx/dtl,=o and J’ = dy/dzl,=o for a virtual particle that moves in the ideal homogeneous field of the first magnet up to the effective boundary and whose trajectory coincides with a real particle trajectory far from the gap between the magnets. We assume the values f and jj to be of an order of unity, and the derivatives R’ and J’ to be small values of the order of the parameter r ; then we introduce parameters a!
= XI/€, p = Y‘/€,
(50)
that are of the order of unity. Technically it is most convenient to solve the trajectory equations in the coordinate frame {t,y, 5). For this reason we define initial values of the coordinates and derivatives in this tilted coordinate frame. The initial coordinates { and 6 for the effective trajectory are
-
( = -sR,
-
6
=cf.
(51)
Here and later we use notations s = sin A , c = cos A, t = tan A. The relations between the derivatives in the tilted and untilted coordinate frames are
With Eq. (50) we can rewrite Eqs. (52) as follows:
Eqs. (53) define the initial conditions for the coefficients of the expansions of Eq. (48) for the derivatives 6’ and y’. The solutions of Eqs. (49) with the initial conditions, which are specified by the main terms (not containing e ) of Eqs. (51) and (53), are
6; = t ,
60
= cx
+ t (( + s f ) ,
y;, = 0, yo = J .
(54)
323
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
Substituting Eqs. (48) and (54) into the formula for the coefficient F of Eqs. (45) and (46), we can rewrite this coefficient in the following form:
F=
1
+
( r + E . $ I + E 2 ~ ~ + E 3 ~ ~ + . . . ) 2 + (Ey; + E 2 y ; + € 3 y ; + - - * ) 2
(1
+ ES)(1 +
9
EV)
(55) where we used notations 6 = cr/e and v = y / E . Since the parameters cr and y are assumed to be small, we suppose that the values S and v are of an order of unity. Finally, expanding Eq. (55) by powers of the small parameter c, substituting this expansion as well as expansions of Eq.(48) and the relation Po = E/B2 [see Eq. (47)] into Eqs. (45) and (46) and separating the terms of the different orders of magnitude (i.e., with different powers of E ) , we come to equations for the functions .$k and yk with k = 1 , 2 , .. . . With k = 1 these equations read
t y;’ = --b’(()jj c
t + -b”’(()J3, 6c
(57)
where b ( ( ) = B ( ( ) / B z . Equation (56) allows one to determine a position of the effective boundary. To obtain the transfer matrix that gives a transformation of an effective trajectory at the effective boundary in the third aberration order, one also has to deduce analogously equations for the functions .$k and yk with k = 2,3. This procedure is straightforward and, therefore, the corresponding lengthy equations are not reproduced here. Note that the separation of equations that describe effects of different orders of magnitude is very easy in the framework of the proposed approach, since scaling of the coordinates and functions is performed in such a way that the only small value in Eqs. (45) and (46) after all the substitutionsjust listed is a parameter E ; all the other variables, functions, and their derivatives are of the order of unity, as well as the interval of the coordinate ( where the equations are to be solved. A transformation of a real trajectory by the inhomogeneousfield in the narrow gap between two magnets may be represented by the relation between two sets of parameters (coordinate and slopes) which characterize this trajectory. As a first set we choose the coordinates and slopes at the effective boundary of the first part of the corresponding effective trajectory, that is, of a trajectory coinciding with the real one well inside the first magnet but passing up to the effective boundary in the homogeneous field of this magnet. The second set contains the coordinates and slopes at the effective boundary of the second part of the effective trajectory, that is, of the trajectory which coincides with the real one well inside the second magnet but passes directly from the effective boundary in the homogeneous
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M.I. YAVOR
field of this magnet. To calculate the relations between these sets, one obviously should
1. integrate the trajectory equations with the initial conditions given at the left side of the effective boundary (which is the plane Z = 0), that is, at the point f at the (-axis, back in the ideal field of the first magnet to some plane ( = M where the point ( = M is situated well inside this magnet, 2. integrate trajectory equations in the real inhomogeneous field for the trajectory coinciding with the trajectory resulting from the first step at the plane ( = M ,from this plane to some plane ( = N where the point ( = N is situated well inside the second magnet, and 3. integrate the trajectory equations for the trajectory resulting from the previous step at the plane ( = N , back from this plane in the ideal homogeneous field of the second magnet to some point at the plane Z = 0. The coordinate ( of the final point, which we denote as i , is A
( =-si,
where 2 is a final x-coordinate of the effective trajectory at the effective boundary. Since this coordinate is subject to evaluation, the value f is also unknown in advance and has to be calculated. To determine a position of the effective boundary, one should integrate Eq. (56) as it was just described. The result of the integration is a first-order approximation for the relation between the derivatives 2’ and X’, which characterize the slopes of the effective trajectory with respect to the Z-axis. This relation reads €
A
P ’ = X ’ + t ( l - 6 ) ~- -(( - f ) C
1
b(()d( +EM - N , C
(58)
where 6 = B I / B ~u , = E X . Equation ( 5 8 ) shows that the main term of the expansion of the trajectory coordinate 6 is independent of the field distribution in the transition region between the magnets, and thus - f = -t(g - i$) = - s ( i - 2 ) is a small value of the order of E . Since all the terms in Eq. ( 5 8 ) are proportional to E, besides the term e ( t - f ) / c that is proportional to E * , the latter term can be neglected in the first-order approximation. Then for the effective trajectory to experience no refraction at the effective boundary to the first order in case of the magnet boundaries normal to the beam (t = 0), the following condition is to be fulfilled:
l
B ( ( ) d ( = B2N - B I M .
(59)
This condition determines the position of the effective boundary between two magnets. Its mathematical sense is similar to the mathematical sense of the single
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
325
RGURE1 1 . A position of the effective boundary between two magnets is chosen so that the areas of the two marked segments SIand S2,bounded by the effective and real magnetic flux density distributions at the <-axis,coincide.
magnet effective boundary [4]:The segments S1 and S2 in Fig. 11 between the line representing the effective magnetic flux density distribution (Beff = B1 with { c 0 and Be# = B2 with { > 0) and a curve representing a real magnetic flux density distribution have equal areas. Thus, the concept just introduced of the effective boundary between two magnets generalizes the well-known concept of the effective boundary of a single magnet. Evidently the condition of Fq.(59) can be satisfied only with B1# B2. In case the magnetic flux densities are equal in both magnets (B1 = B2), a similar method may be used for calculation of the transformation of charged particle trajectories in the gap between the magnets, but the concept just introduced of the effective boundary loses its sense and the final results will have a different structure. However, in the present chapter we confine ourselves only to the practically most interesting case of different magnetic flux densities in the air gaps of the first and second magnets. Integration of the trajectory equations in the second- and higher-order approximations is performed similarly to the integration of the first-order equations. It results in an analytical formula for the transformation of the effective trajectory at the effective boundary between the magnets. It is most convenient to represent this transformation in the coordinates u = E X and Y = ey. We also define the derivatives u’ = du/(e dz) = dx/dz and u’ =~ u / ( Edz) = dy/dz. We denote by a hat the parameters of the effective trajectory at its side turned to the second magnet and omit the bar denoting the corresponding parameters defined at the effective boundary side turned to the first magnet. We also introduce the notations q = r z / R
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M.1. YAVOR
and A = (1 - 6) = (B2 - B I ) / & . Then in the second-order approximation we obtain the following transformations (the corresponding transformations in the third aberration order are listed in [59]):
where
t2 aUu= - - A , 2 1 a,, = - A 2c2
b, = l - c 2 {
b,, = t 2 A
+
+
t(62 5) 54, 2c
+E
-E
[
q t ( 3 r 2 + 2 ) + (1 C
t(2r2 C
+ 1 ) 54, J 1 + c 3 [ - ( 3q2t2 ~ + * ) J z + ( g + $ ) ] J 3 ,
CO=--E~($+;)
C2
c, = t A + r 2 { ‘
t(
3t2 + 2 )
q(3t2+ 1 )
C2
+ +
c3
6
[$
+
+
qt2(3t2 C 2 ) ] }J I ,
327
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
3
c,,
=-[g+bfi-]A, t
c,, = - - A , 2 cvv
1
q t r3 = -+-+-(2-6) 2
2c3
+E
[
2
qr(6;z2+ 5)
culv= - t 2 A - E
+
+
+
r2(10t2 7) - &t2(2r2 1) 2c C
+ 1) 34
t(2t2
A
]
J4r
9
C
dui = 1
+ c2
dv = -tA --6
+2{ 1 dulu= --A
JI
C
2t2
[-
+1
C
Js} ,
54
qZt(3t2 C2
-E
+
+ 2) + q(3t4 + t 2 - 1) + 6 (qt2(3t2 + 2) - f)]
+
J1
c
C
C2
t (6t2 5)
C2
C
54,
C
duiu= - t 2 A - 6
+ 1) 54
t(2t2
9
C
t
dvr = - A + € 2
2t2
+ 154.
C
The coefficientsin Eqs. (60)-(63) contain sixfield integrals J k , k = 1 - 6 of the magnetix flux density distribution along the <-axis, that is, in the direction normal to the magnet boundaries. These integrals are
// N
JI =
M
b(<)d< d< -
51"' + 6 M 2 ) + ~
M N ,
328
M. I. YAVOR
( J b ( o d ( ) 2 d{ + 2 6 M J 1 + 6 M N ( N - 6 M ) - -1( N 3 - J 2 M 3 ) , 3
J~ = N
54 =
b2(<)d { - N
+b2M,
their values are independent on the exact positions of the points M and N. In the second aberration order the details of the magnetic field distribution in the transition region, described by the integrals J k , contribute only to small correction to the transformation coefficients, since all the integrals are multiplied by some powers of the small parameter E. Note that all the integrals are of an order of unity, since the function b(c) changes from the constant value 6 inside the first magnet to the constant value 1 inside the second magnet at the interval of the coordinate whose length is of an order of unity. The tables of the field integrals can be found in [59]. In the sharp-cutoff approximation, where the contribution of all the field integrals are neglected, the transformation of Eqs. (60)-(63) coincides with the product of the transformations given by the exit fringing field of the single first magnet and by the entrance fringing field of the single second magnet. However, this is not the case if the contributions just mentioned are taken into account. The main contribution of the field integrals is a defocusing action of the gap between the magnets in the Y-direction, represented by the term proportional to €54 in the coefficient d,. Equations (60) and (61) show some parallel shift at the effective boundary of all the effective trajectories, given by the coefficient QO, and a small deflection of all the trajectories, given by the coefficient CO. These effects are similar to those observed in the fringing field of a single magnet [4 11. Equations (60)-(63) contain as a limiting case the trajectory transformations produced by the entrance fringing field of a single magnet. This case corresponds to the zero magnetic field inside the first magnet: 6 = 0 and A = 1. The corresponding transformations coincide with the results given in [54] (taking into account some differences in the representation of the results in the paper just mentioned and in the present section, namely, different coordinate frames and opposite signs of the magnet boundary curvature).
<
329
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
B. Transformation of Charged Particle Trajectories in the Gaps between Lenses of Closely Packed Quadrupole Multiplets
In this subsection we study a transport of a charged particle beam through a narrow gap between two electrostaticor magnetic quadrupole lenses placed close by each other in a quadrupole multiplet. We assume that the lens apertures are considerably smaller than their lengths, so that the field inhomogeniety in the transition region between the lenses does not influence an ideal field distribution well inside the lenses, where the electrostatic or scalar magnetic potential is a quadratic function of the coordinates in a profile plane normal to the lens axis. Consider first a case of an electrostatic lens. A distribution of the electrostatic field potential U in the transition region between two lenses can be represented in the form of an expansion
by powers of the Cartesian coordinatesx and y in a profile plane orthogonal to the lens axis z . The trajectory equations in the transition region read
where @O and qo are a kinetic energy and charge of some reference particle, a is a relative deviation of the energy-to-charge ratio for an arbitrary particle with respect to a reference one, prime denotes a derivative with respect to z . Now we introduce dimensionless coordinates X = x / r 2 , Y = y/r2, and 2 = z/r2. where r2 is the aperture radius of the second lens. We rewrite the function k ( z ) in a form k ( z ) = 2V2 f ( Z ) / r : , where V2 is a potential at the electrodes of the second lens, crossing the x-axis. The function f (2)tends to 1 well inside the second lens and to the value h = V1ri/(V2r:)inside the first lens (here V1 is a potential at the electrodes of the first lens, crossing the x-axis). Note that, with the width of the gap between the lenses being comparable with the lens aperture radii rl and 1-2, the length of the transition region in the dimensionlesscoordinates is of an order of unity and thus the derivatives of the function f (2)with respect to its argument Z are also of an order of unity (i.e., these derivatives are not large). In the new coordinate frame the normalized potential u = qoU/@o reads
f ( Z ) ( X 2- Y 2 )- f " o ( X 4 12 where
E
= 4-
- Y 4 )+
..-I
(67)
is a dimensionless parameter, which we assume to be
330
M. I. YAVOR
small. The latter condition always holds in practical situations for a lens whose aperture radius is considerably smaller than its length (otherwise beam focusing inside the lens in one direction and a strong beam divergence in the other direction would be observed, that is usually not the case in the systems where quadrupole multiplets are used). Since for a correct application of a perturbation method it is essential that the length of the transition region in terms of the dimensionless coordinate Z be of the order of unity, a small value of E also means a small length of the transition region as compared with the lengths of the quadrupole lenses. Note that a linear change of a second lens aperture radius r:!requires a quadratic change of the potential V2 at the electrodes of this lens in order to preserve an optical power of the lens; for this reason we choose the parameter E to be quadratically dependent on the potential. Now to apply the field integral method one needs to rewrite Eqs. (65) and (66) in the coordinate frame { X, Y , Z ) , using the representation of Eq. (67) for the electrostatic field potential. Substituting into the resulting trajectory equations the coordinate functions in the form of their expansions with respect to E:
x = xo + E X ,
+2x2
+ 2x3 +. . . ,
Y = YO+EY1+€2Y2+€3Y3+...
(68)
and assuming in the trajectory equations the parameter 0 to be of the order of magnitude of the parameter E , we finally come to a sequence of equations for the coefficients of Eqs. (68). In the zero-order approximation these equations read x“0 -- y“0 --0 , (69) which means that
X o ( 2 ) = s, YO(Z) = P,
(70)
where bars denote initial coordinates of an effective trajectory (i.e., of the trajectory which coincides with a real trajectory well inside the first lens and passes up to the effective boundary in the main (ideal) field of this lens, then passes after some discontinuous transformation at the effective boundary in the main field of the second lens and coincides with the real trajectory well inside this lens) at the left side of the effectiveboundary. For the functions X 1 and Y Ithe same equations hold: XN - Y r = 0. For the functions X2 and YZ we obtain the following equations:
,
1
X ’N--6 - f ” ( Z ) S 3 - f ( Z ) X ,
Integratingthese equations back from the effective boundary to some plane 2 = M in the main field of the first lens, then from this plane to some plane Z = N in the real field and then back from the latter plane to the effective boundary in the main
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
33 1
FIGURE12. A position of the effective boundary between two quadrupole lenses is chosen so that the areas of the two marked segments SIand SZ, bounded by the effective and real field gradient distributions at the optical axis, coincide.
field of the second lens, we come to the conditions relating the parameters of the effective trajectory at the both sides of the effective boundary. From these relations one can obtain that in order for the effective trajectory to experienceno refraction at the effective boundary in the second-order approximation,the following condition must hold rN
where f, = f ( M ) and Z* is the coordinate of the effective boundary. Equation (73) determines the position of the effective boundary. Its mathematical sense is analogous to the sense of Eq. (59) that defined the position of the effective boundary between two magnets: namely, the areas of two segments S1 and S, in Fig. 12, bounded by the real and effective field distributions, are equal. Hereafter we will assume that the origin of the coordinate frame {X,Y, Z} is chosen so that Z* = 0 and thus the XY plane is the plane of the effective boundary. In case the first lens is absent (fi = 0), Eq. (73) gives a position of the effective boundary of a single quadrupole lens as defined in [43]. Note that the effective boundary position can be calculated based on Eq. (73) only in the case where f i # 1, that is, where the first and second lenses have different excitations. However, the case of equally excited adjacent quadrupole lenses is not interesting from the practical point of view. Integration of the equations for higher-order coefficients gives the relations between the effective trajectory coordinatesand slopes at both sides of the effective
332
M. I. YAVOR
boundary. Similar to the case of two magnets considered in Section IV.A, these relations completely describe the transformation of the charged particle beam by the inhomogeneous field in the transition region between two quadrupole lenses; they allow one to replace a laborious integration of trajectories in the real field in the gap between the lenses by a simple integration of the effective trajectories up to the effective boundary and application of the analytical transformation of the effective trajectories at this boundary. Following we give the relations between the effective trajectory coordinates at different sides of the effective boundary in the third-order approximation. In these relations we come back to the initial coordinate frame [x, y, z] assuming that the xy-plane is the plane of the effectiveboundary. We denote X I = r 2 z and yl = r2 P the effective trajectory coordinates at the left side of the effective boundary (turned = arctan bl the angles of inclination of to the first lens), a1 = arctan a1 and the x- and y- projections of the effective trajectory at this side; let x2, y2, a2 = arctan a2 and /?2 = arctan bz be the corresponding coordinates and angles at the right side of the effective boundary (turned to the second lens). We also introduce a notation KO2 = 40ko/(2@0).Then the relations in question read
Here A = (k2 - k l ) / k 2 ,where kl and k2 are the constant values of the function k ( z ) in the first and second lenses, respectively, Z j ( j = 1,2,3,4) are the field integrals
+ - (1z ' , - - h z L ) , 6
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
333
- -1 ( z i - h 2 z M3 ) , 3
’ /“
I4 = -
k;
k ( z ) 2d z - Z N
+ h2z,u,
ZM
and Z N are some points at the z-axis located well inside the first and the second lens, respectively. The field integral values are independent of the exact positions of these points. In case the first lens is absent (kl = 0), Eqs. (74)-(77) give the transformation of charged particle trajectories at the effective boundary of a single electrostatic quadrupole lens [43]. We proceed now to magnetic quadrupole lenses. The distribution of the scalar magnetic potential in a narrow transition region between two such lenses can be represented as ZM
W(X,y , z )
k(Z) 2
= --xy
k”(z) + -xy 12
(x2
+y2)+
* * *
,
(78)
and the trajectory equations in this region read
where mo is a mass of a reference particle and y is a relative deviation of the mass-to-charge ratio for an arbitrary particle with respect to the reference one. A way of calculation of the transformation of effective trajectoriesat the effective boundary between the magnetic lenses is completely identical to the case of electrostaticlenses, ifwedefine the function f ( Z ) bytheformulak(z) = 2 B 2 f ( Z ) / r : , where B2 is a magnetic flux density value at the second lens pole tip at the point closest to the lens axis, and the small parameter by the relation c2 = qoB2/,/=. We introduce a notation KO2 = qoko/po, where po = m0u0 is a momentum of the particle with the nominal mass and velocity UO, t = (p - po)/po is a relative momentum deviation for an arbitrary particle with respect to the reference one, h = kl/k2. The effective boundary position is determined, as in case of electrostatic lenses, by Eq. (73). The transformation of an effective trajectory at the effective boundary
334
M. I. YAVOR
reads
where the integrals I, ( j = 1 , 2 , 3 , 4 ) are defined by the same formulas as in case of electrostatic lenses. If the first lens is absent (kl = O ) , 4 s . (81)-(84) give the transformation of charged particle trajectories at the effective boundary of a single magnetic quadrupole lens [43]. An influence on a charged particle beam transformation of the inhomogeneous field between the quadrupole lenses can be illustrated by an example of an electrostatic quadrupole multiplet consisting of two identical symmetric quadrupole triplets. We consider each triplet to be formed by a mirror pair of doublets; a doublet focuses particles, emerging from a point, into a parallel beam in both x- and y-directions. We choose the effective lengths of the first and the third lenses to be 15 mm with the potentials at their electrodes ~ 3 1 . 6V (these lenses defocus the beam in the x-direction). The effective length of the middle lens of the triplet is chosen to be 40 mm with the potentials at the electrodes of this lens being f14.55 V. The distance between the object plane and the effective entrance boundary of the first lens is 30 mm and the distance between the triplets is 60 mm. The aperture radii of the lenses are 3 mm and the effective distances between the middle and outer lenses in the triplet are 5.4 mm (it is supposed that the effective boundary of a single quadrupole lens is 3.3 mm away from the mechanical lens boundary, so that the mechanical length of a lens is 6.6 mm smaller than its effective length; this means that the distance between the mechanical boundaries of the middle and outer lenses of the triplet is 12 mm). In Fig. 13a paraxial trajectories of the 1 KeV ions are shown emerging from a point source located at the optical axis in the object plane. These trajectories were calculated in a sharp-cutoff approximation; that is, the lengths of the lenses were assumed to be their own effective lengths and effects of overlapped fringing fields
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
335
FIGURE13. Shown are paraxial charged particle trajectories in a pair of quadrupole triplets, calculated in a sharp-cutoff approximation (a) and taking into accounta transformation of the trajectories in the narrow gaps between the lenses of the triplets (b). In the latter case the fields of the adjacent lenses are formally extended up to the effective boundaries between these lenses, and a transformation of the trajectories is calculated at these boundaries.
were not taken into account. In this approximation the behavior of the trajectories in the first and second triplets are identical. Figure 13b shows the result of a calculation of the trajectories that takes into account their transformation in the gaps between the middle and outer lenses of the triplets. According to the method just presented, these lenses were formally extended up to the effective boundaries between the lenses, and the transformation of Eqs. (74)-(77) was performed at these boundaries. The figure demonstrates a noticeable difference of the behavior of the trajectories calculated in the two approximations considered. In particular, the system that was designed to be stigmatically focused in the sharp-cutoff approximation appears to be astigmatic in reality. Note that in case a quadrupole multiplet is designed without taking into account the effects of the transition regions between closely situated lenses, a correction of its first-order properties can be in principle performed by changing lens excitations. However, in practice such a correction may become difficult since it requires an
336
M. I. YAVOR
empirical matching of several potentials simultaneously to achieve a good-quality image, which can be usually controlled only at the final image plane of a beam transport channel. Thus, an initially correct calculation of quadrupole multiplets is of great importance. The effects of the trajectory transformation in overlapped fringing fields of quadrupole lenses considerably contribute, as in the case of the fringing fields of single lenses, not only to the first-order focusing properties of multiplets but to their higher-order aberrationstoo. For this reason these effects should be also taken into account in a design of high-performance transporting channels and particle spectrometers. The perturbation method described in the present section allows one to perform calculation of aberrations in the framework of the field integral approach.
v.
SYNTHESIS OF REQUIRED m L D CHARACTERISTICS IN SECTOR ENERGY ANALYZERS AND W E N FILTERS WITH THE AID OF TERMINATING ELECTRODES
This section is devoted to an investigation of the effects of a set of electrodes placed at the top and bottom sides of an electrostatic sector condenser or a Wien filter. These electrodes known as split Matsuda plates can correct aberrations or protect the electrostatic field distribution in the interelectrode gap from external influences. The present study neglects the fringing field effects produced by the terminating electrodes at the entrance and the exit of the analyzer. A. Electrostatic Field in the Gap between Two Curvilinear Electrodes Terminated by Split Shielding Plates
Consider an electrostatic sector energy analyzer with a beam main path radius ro, where an electrostatic field is formed by two electrodes (generally toroidal) with the azimuthal curvature radii r1 = ro - b and r.2 = ro b and the curvature radii in the vertical direction R1 = Ro - b and R2 = Ro b, as well as by a set of electrodes that terminate the interelectrode space from the top and bottom sides of the analyzer; the potentials VI and V2 are applied to the inner and outer toroidal electrodes, respectively. We introduce a toroidal coordinate system [R, 8,4] as shown in Fig. 14a. At the moment we do not specify a geometry of the terminating electrodes; we only assume that these electrodes create some potential distribution V (R) at the conical surfaces 8 = f O symmetric with respect to the median plane 8 = 0 of the analyzer. This distribution is an arbitrary (may be discontinuous) function of the coordinate R. The electrostatic field inside the toroidal condenser is symmetric with respect to 8 and independent of 4.
+ +
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
337
FIGURE 14. The coordinate frames used for calculation of effects of terminating electrodes in a toroidal (a) and a cylindrical (b) electrostatic sector analyzer.
We introduce dimensionless coordinates 1 R z = -e x = -1nL’ L Ro-b’ where L = In((& b)/(Ro - b)). In the coordinate frame {x, z , 4) the Laplace equation for the electrostatic field potential U reads au = o. a2u a2u COS(LZ)%- sin(Lz)= -+-+L (85) ax2 az2 COS(LZ) exp(-Lx)
+
Taking into account that x = 0 with R = R1 and x = 1 with R = R2, we obtain the following boundary conditions for the potential U : U ( x = 0, z) = VI , U (x = 1, z) = V2 and U ( x , z = -+Z) = V ( R ( x ) ) , where Z = O / L . Note that in the new coordinate frame the dimensionless width of the gap between the toroidal electrodes is 1. Now we use the fact that in most practical situations the interelectrode gap in an analyzer is considerably smaller than the radius ro of the beam main path; thus, we assume that the parameter c = b/ro
is small. Then the left-hand side of Eq. (85) can be expanded with respect to the powers of this small parameter. We introduce a normalized field potential U ( X , z) = ( U ( x ,z) - V1)/(V2 - V I ) and represent it in the form of an expansion
u = Uo
+
€241
+
€2112
+E3U3 + . * * *
(86)
338
M. 1. YAVOR
Substituting Eq. (86) into the left-hand side of Eq. (85) and collecting the terms with equal powers of E , we finally come to the following sequence of equations for the functions uj with j = 1,2, . . . :
a2uo -+ax2
a2uo a22
= 0,
a2ul + a 2 u l = -2-, auo
-
ax2
ax
a22
and so on, where co = ro/Ro is approximately equal to a toroidal factor of the analyzer [see Eq. (9)]. The boundary conditions for the function uo are uo(0, z ) = 0, uo(1, z) = 1, uo(x, * 2 ) = u(x), (90) where u(x) = (V(R(x)) - Vl)/(V, - VI). The functions U I , 1.42, . . . satisfy zero boundary conditions. A solution of Eq. (87) with the boundary conditions of Eq. (90) can be repW O ,functions UO and w o being the Fourier serii resented as a sum uo = (lo
+
UO(X, z ) = x
+ -2 TI
c b3
n=l
(-l), cosh(nnz) sin(rrnx), n cosh(nnZ)
-
(91)
here the constant parameters a, are the Fourier coefficients of the function u(x): a, = 2
I'
u(x) sin(nnx) d x .
(93)
Since as a rule the analyzer electrode height is not smaller than its interelectrode gap, we will suppose that the distribution u(x) is specified at the sections 8 = f O , for which the arc length ROOis not smaller than b. With L x 2cor this condition gives 2 2 1/2. In this case a parameter p = exp(-nZ) is small, so that we can expand cosh-I ( n n 2 ) = 2p" (1 - p2, p4"-. .). Denoting p = E u and expanding the denominators in the right-hand sides of Eqs. (91) and (92), we come to the following expression for the function uo in the median plane 0 = 0 of the analyzer:
+
+ 2rdlu sin(nx) + 2~*d2u'sin(2nx) + 2c3[d3sin(3nx) - dl sin(nx)]u3+ . . ., where dl = a1 - 2/n, d2 = a2 + l / n and d3 = a3 - 2/(3n). UO(X, 0) = x
(94)
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
339
A solution of Eq. (88) with the zero boundary conditions can be represented as - xwo w1, where
+
U I = (1 - x)Uo
00
cosh(nnz) sin(nnx), cosh(nn Z)
w1(x, z) = Cb, II=
I
(95)
rl
b, = 2
x v ( x ) sin(nnx) d x .
Performing the transformations analogous to those applied to the function U O ,we obtain U I ( X ,0) = x(l
[: +
- x ) + E --(1
1
- x ) - 2alx + 261 usin(lrx)
+ e2 [z(1- x ) - 2a2x
1
+
2b2 u2 sin(2nx) . . . . (96) n Handling Eq. (89) is performed in a similar way, though it requires more laborious calculations. As a result the following representation can be obtained for the function u2 at the median plane of the analyzer:
2 - co
u2(x,O) = x ( l - x)(l - 2 x )
3
+
€
4(2Co - 1) lr3
where
Now we can calculate the field potential distribution in the median plane. This distribution can be represented as an expansion with respect to the powers of the dimensionless coordinate p = (R - Ro)/ro:
Here E O = L ~ U / L ~ ~ ~ , , = O is , ~a , =normalized O nominal field strength at the optical axis of an “infinitely high” analyzer, that is, of an analyzer whose interelectrode space is not restricted in the vertical direction by additional terminating
340
M.I. YAVOR
electrodes and whose electrode height is so large that the fringing field effects do not contribute noticeably to the field in the vicinity of the optical axis. In Eq. (99) a1 = ( E - Eo)/Eo is a relative deviation of the field strength at the circular optical axis with respect to the nominal value in presence of terminating electrodes; Eoao is a field potential deviation at the optical axis, caused by terminating electrodes (it is supposed that the electrode potentials V I and V2 are chosen so that the field potential at the optical axis of the infinitely high analyzer equals zero). Finally, A2, A3 and A4 are the higher-order coefficients of the field potential expansion in the infinitely high analyzer and a2,a3, and a4 are their deviations caused by the presence of the terminating electrodes. Note that up to now we did not make any particular assumptions concerning the geometry of the latter electrodes and the potential distribution at these electrodes; we only supposed that in presence of the terminating electrodes the potential distribution at the surfaces 8 = k0 is some known function V (R). With Eqs. (86), (94),(96), and (97), we come to the following analytical expressions for the coefficients a0 and 41,which characterize the potential and field strength deviations at the beam optical axis with respect to their nominal values in the infinitely high condenser, caused by the action of terminating electrodes:
+ r 3 -4(b2I7 + l)u2 + 2gl u + 2d2u2n + -2lg , U I 7 2 C o l + . . . ,
[
(101)
+
where gl = d1/2 - bl 2/17. Obviously the nominal .values of the potential and field strength at the optical axis can be restored by changing the potentials V1 and V2. Since the coefficients a0 and a1 are small, it is easy to obtain from Eq. (99) that the required potential correction is
Generally Eq. (99) can be rewritten in the following form:
1 + -(A3 +8 6
+ 241 +
3 ) ~ -(A4 ~
+
)
. ,
8 4 ) ~ * .~
(103)
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
341
+ +
where hj stands for ( A j +a,)/( 1 al)- A j . It is well known that the parameter 82 determines first-order focusing properties of an electrostatic sector analyzer, the parameter a30 = A3 6 3 determines its second-order aberrations and a40 = A4 +h4 its third-order aberrations. From Eqs. (86), (94), (96), and (97) one can obtain the following formulas for the parameters E j that are responsible for the deviations of the optical properties of the analyzer supplied with terminating electrodes, with respect to the infinitely high analyzer: a20 = A2
+
+COV
[
(-dl
(q+ t) + 9) +
+ v27r2[d: (-2
2) -
-
(f + $)]
Codln2
2d2nco]
1
+ v3n2[9d3+ dl (1 - 4nd2)] + c3(* *
a),
- d2(24
+2 ~ ' )
1 + v3n4[ - y d 3 - -dl (1 - 4nd2) + c(. . .). 4
1
A detailed analysis of Eqs. (104)-(106) will be given in Sections V.B and V.C.
342
M.1. YAVOR
Here we only note that these equations show that with the terminating electrodes creating some potential distribution V (R) at the surfaces 0 = f O , the coefficients of Eq. (103)are mainly influenced by the first, second, and to some extent by the third Fourier harmonics a, of Eq. (93)and (to a smaller degree) by the corresponding harmonics of Eqs. (95) and (98). This circumstance considerably simplifies the calculation of the coefficientsjust mentioned and the analysis of effects of the terminating electrodes. Up to this point we considered an arbitrary toroidal condenser. In case of a cylindrical analyzer (see Fig. 3), the toroidal coordinates R and 0 lose their sense since Ro = 00, and all the considerations should be performed in the cylindrical coordinates {r. z, 4) as shown in Fig. 14b. Nevertheless, the final results retain the form of Eqs. (99)-(106), if we assume p to be a dimensionless cylindrical coordinate p = (r - ro)/ro, the coordinate x to be replaced by x = (r - r l ) / b , where b is half an interelectrodegap of the analyzer, the potential distribution V ( x ) to be created by the terminating electrodes at the surfaces z = f z o , the parameter 2 to be equal Z = zo/(2b), and, finally, set co = 0. Analogous considerationscan be also performed for the electrostaticfield formed by the electrodes (generally curvilinear)of a Wien filter with a straight optical axis. This situation corresponds to the limiting case ro + 00. Then the coordinates R , 0, x , and z, introduced for a toroidal analyzer, still have a sense, but the parameter E tends to zero and co to infinity. Therefore, in this case it is necessary to redefine the parameter E as E = b/Ro and the variable p as p = ( R - Ro)/Ro. Then the electrostatic field distribution in the median plane of a Wien filter is described by Eq. (103),where
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
343
TABLE 1 COMPARISON OF RESULTS OBTAINED BY CALCULATION OF A mRODIAL FACTOR C AND ITS DERIVATIVES C’ AND C” WITH THE AID OF THE PERTURBATION APPROACH, WITH THEIR EXACTVALUES, IN CASE OF A CYLINDRICAL ANALYZER WITH THE OPTICAL AXIS RADIUS110 mm, THE INTERELECTRODE GAP20 mm,AND THE ELEC~RODE HEIGHT 40 mm FOR DIFFERENT VALUES OF THE POTENTIALS AT THE SOLID (NONSPLIT)
WINATING ELXTRODES.
c u
0.5 0.6 0.7 0.8 0.9 1.0
Perturb.
Exact
Perturb.
Exact
Perturb.
-0.13576 0.46418 1.0653 1.6676 2.2712 2.8759
-8.9906 -8.4222 -7.1319 -5.1157 -2.3694 1.1111
-8.8963 -8.3217 -7.0223 -4.9938 -2.2319 1.2678
33.651 -152.85 -336.36 -514.24 -683.85 -842.49
32.368 -153.44 -336.45 -513.74 -682.64 -840.45
Exact -0.13632 0.46316 1.0638 1.6655 2.2684 2.8724
C”
CI
Though formally these expressions as well as Eq. (103) lose their sense in case of planar electrodes (Ro = oo),this sense can be easily restored by the substitution of Eqs. (107)-( 111) into Eq. (103) with u = P / E and p = E ( R- Ro)/b, after which one should set E = 0. The formulas obtained previously with the aid of the perturbation method for the coefficients of the field distribution are accurate enough for practical calculations. To illustrate this accuracy, we give a comparison of the results of calculation of the field characteristics,performed by the perturbation method for a simple model of a cylindrical analyzer terminated by solid (nonsplit) terminating electrodes, with the exact numbers resulting from exact analytical formulas available for this model [72, 731. This comparison is presented in Table 1 for different values of the normalized potential u = (V - V , ) / ( V 2- Vl) at the terminating electrodes. Given in the table are not the coefficients aio (i = 2,3,4) of the expansion of the field potential but the related values of a toroidal factor c and its derivatives c’ c” (see [ 1061): c = -1 -a20,
+ 2~ + c2 - ~ 3 0 , C” = -6 + 3 ~ ’1(+ C) - ~ ( + 6 3~ + c’) C’ = 2
-~ 4 0 .
It is seen that for a characteristic value of the small parameter E being 0.1, the accuracy of the calculation of the toroidal factor c is about 0.1 %, and the accuracy of the calculation of its derivatives is within 2%.
344
M. 1. YAVOR
B. Influence of Split Matsuda Plates on the Field and Optical Properties of Sector Energy Analyzers and Wien Filters We assume now that some known field potential distribution V ( R ) at the surfaces 8 = f O in a toroidal sector analyzer or a Wien filter (or a distribution V ( x )at the surfaces z = f z o in a cylindrical analyzer) is formed by electrodes whose surfaces are described by the equations B = fa,R I < R < R2 (or z = f ZO,rl < r < r2 for a cylindrical analyzer). In practice such an assumption is acceptable for a Wien filter or a cylindrical analyzer, if we suppose the gaps between the terminating electrodes and the main analyzer electrodes to be negligibly small, but looks somewhat artificial for a toroidal analyzer, since in the latter case the terminating electrodes are as a rule designed as planar rings whose surfaces are parallel to the median plane of the analyzer (as it is in case of a cylindrical analyzer) and not conical rings described by the equations just mentioned. Nevertheless, in case of a toroidal analyzer an acceptable accuracy of results can be expected, if one approximates the real planar terminating electrodes by slightly inclined conical ones, since the angle 0 is always small and, besides, as it was noted in the previous section, the electrostatic field distribution in the vicinity of the optical axis of a charged particle beam is determined not by a detailed field potential distribution at the surfaces 8 = f O but only by its several first Fourier harmonics. Next we consider in detail the case of a toroidal sector analyzer, since, because of similar forms of the field potential representationobtained in Section V.A, all the conclusions made for a toroidal sector analyzer will also hold for the electrostatic field of a cylindrical condenser and a Wien filter as well as for the influence of this field on optical properties of these devices. Suppose the terminating electrodes to be divided into N ring-shaped parts with negligibly narrow gaps between them, so that these split electrodes create a piecewise-constant potential distribution along the lines 8 = f 0 of the form V ( R )= W1 with R1 < R < R ( ' ) , V ( R ) = W2 with R(') < R < R(2),. . , , V ( R ) = W , with R(,-') < R < R2. Then the Fourier coefficients a,, in Eq. (93) as well as the coefficients b, in Eq. (95) and CI in Eq. (98) can be represented as linear combinations of the parameters W1,. . . , W,. Consider an influence of the potential distribution V ( R ) ,created by split terminating electrodes, on the coefficients &, d3, and E4 described by Eqs. (104)-( 106). These equations can be rewritten as
a 3 = v n 2 ( a I - ~ ) [ ~ + c 0 ( ~ + 3 +) 4] ( a 2 + f ) n 3 u 2 + y 3 ,
(113)
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
h4=-
l ( 4€2
u1--
i ) * Y 3 , ( 32,) nv--vn
u3--
+y4,
345 (1 14)
where thenotations y2, y3, and y4 stand fortheremaining termsinEqs. (104)-(106) that are small with respect to the terms explicitly given in Eqs. (1 12)-( 114). Equation (1 12) shows that the first-order optical properties of the analyzer are mainly determined by the first Fourier harmonic a1 of the potential distribution V (R)at the terminating electrodes. The influence of higher-order harmonics on these properties is smaller (generally, the higher the number of the harmonic, the smaller is its effect). One can see from Eq. (1 13) that the second-order aberrations are determined mainly by the first Fourier harmonic a1 and the second harmonic u2. Finally, the third-order aberrations are determined by the first and, to a smaller extent, by the third harmonic, as is seen from Eq. (1 14). Equation (1 14) allows one to make one more important conclusion. Namely, if we leave in Eqs. (1 12) and (1 14) only the main terms with respect to E [i.e., the terms proportional to (a1 - 2/n)l, we come to the relation a4 - x ~ / ( ~ E ~ ) & . Note that the factor n2/(4c2) for a typical value of E being about 0.025 is a large value of an order of lo3. This means that a variation of the first-order focusing properties of the analyzer relative to the properties of the infinitely high analyzer, caused by a change of the first Fourier harmonic u l , necessarily leads to a strong increase of the coefficient h4 and thus to an increase of the third-order aberrations of the analyzer. This effect is always observed in analyzers with nonsplit terminating electrodes. From the physical point of view this effect is caused by the fact that, while the analyzer focusing properties for paraxial particles, moving in the vicinity of the optical axis, are determined by local curvatures of the equipotential surfaces influenced by terminating electrodes, the analyzer properties for the particles moving closer to the main electrodes are determined by the curvatures of equipotential surfaces located close by these electrodes. Such curvatures evidently are to a greater extent dependent on the mechanical curvatures of the main electrodes than on the influence of the terminating electrodes. In other words, the terminating electrodes mainly change the field distribution in the vicinity of the optical axis and not in the vicinity of the analyzer electrodes. (We do not consider here a complicated field structure in the vicinity of the terminating electrodes, since we assume that charged particles do not move too close by the upper and lower edges of the analyzer.) The effect just described obviously takes place independently of the potential distribution at the terminating electrodes, including multiply divided ones. All one can do for a correction of this undesirable effect is to enlarge the region around the optical axis where the local curvature of equipotential surfaces is not changed considerably, that is, to reduce the coefficient a4 which controls the third-order aberrations. Since the harmonic a1 can be considered fixed in order to provide for required first-order properties of the analyzer, and the harmonic a2 should be
346
M.I. YAVOR
chosen to provide a small enough value of the coefficient zY3 (in order not to create large second-order aberrations of the analyzer), the only way to reduce the thirdorder aberrations is to create such a distribution of the potential at the terminating electrodes that has a large third Fourier harmonic a3. This way requires using the terminating electrodes split at least into three parts, with the potential at the middle part being as a rule considerably lower (for a positive particle charge) than the potentials at the outer parts. Besides, in order to allow the third Fourier harmonic of the potential distribution at the terminating electrodes to affect the field characteristics in the vicinity of the optical axis efficiently, the ratio of the analyzer electrode height to the interelectrode gap should not exceed 2 to 2.5. We emphasize that the conclusion concerning the necessity to apply the terminating electrodes split into three parts holds only in case one wants to design an analyzer with a variable focal length. Otherwise, if the terminating electrodes are required only for protection of the field distribution inside the analyzer from a penetration of the parasitic external fields into a wide (as compared with the analyzer height) interelectrodegap, one needs to provide for the zero values of the coefficients a, with j = 2 , 3 , 4 . In this case it is sufficient to use the terminating electrodes split into two parts. Such electrodes consisting of only two parts may be advantageous to use, in particular, in Wien filters to protect the field distribution between the electrodes from the influence of equipotential surfaces of the magnet poles. In principle one can require a possibility to tune slightly the first-order properties of the analyzer with high enough electrodes. In this case it is sufficient to use nonsplit electrodes. The optimal ratio of the electrode height to the interelectrode gap in this case is 5 to 6; a smaller ratio leads to an undesirable change of higher-order aberrations and a larger ratio leads to a necessity to apply overly high voltages to terminating electrodes. For the ratio just proposed of the electrode height to the interelectrode gap, application of the split terminating electrodes is senseless, since the field distribution in the vicinity of the optical axis appears to be practically insensitive to the values of the second- and higher-order Fourier harmonics of the potential distribution at the terminating electrodes.
C. Synthesis of a Required Field Distribution in a Ecinity of the Beam Main Path in a Sector Analyzer with the Aid of Split Shielding Plates To solve a problem of shielding the interelectrode space of an analyzer from penetration of outer fields or to design an analyzer with a variable focal length, one needs to determine the potentials, applied to the terminating electrodes, that create a required field distribution in the vicinity of the optical axis of the analyzer. This problem is an inverse one with respect to a problem solved in Section V.A. A remarkable property of Eqs. (112)-( 114) is that they enable a simple analytical solution of such an inverse problem.
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
347
Specifying some required optical properties of the analyzeruniquely determines the field distribution in the vicinity of the optical axis, that is, the coefficients 52,53,and 5 4 . A number of the coefficients that can be changed independently with the aid of variation of potentials at the terminating electrodes coincides with the number N of the ring-shaped parts of these electrodes. Consider, for example, the case N = 3. Then a solution of the problem of the synthesis of the field with some given characteristics looks as follows. As a first step the zero-order approximations for the Fourier coefficients a ] ,u2, and a3 are calculated for given values of 52,83,and 54, based on the system of linear algebraic equations that result from Eqs. ( 1 12)-(114), where the terms y2, y3, and y4 are set to zero. Then the potentials W1,W2,and W3 are easily evaluated in this approximation as the solutions of Eqs. (93) with n = 1 , 2 , 3 . Based on the calculated potentials, the terms n,y3, and y4 are evaluated and substituted into Eqs.(l12)-( 114). From the resulting system of inhomogeneous linear algebraic equations, the first-order approximation is calculated for the coefficients u l , u2, and u3. The process of successive approximations is repeated until the required values of the potentials W I, W2,and W3 are obtained with a sufficient accuracy. As a rule such a process converges in a few steps. As an illustration we consider an application of the proposed method to the calculation of a cylindricalcondenser with terminating electrodes, where the spherical electrostatic field distribution is to be created. For this example we have deliberately chosen a geometry of the condenser with a relatively large interelectrode gap, which allows to demonstrate the specifics of the mode of operation of terminating electrodes as well as the possibility to apply the perturbation method successfully in case the parameter E would be not very small. Namely, we suppose the beam main path radius in the analyzer to be ro = 45 mm, whereas the interelectrode gap is 2b = 30 mm; that is, the small parameter value is E = 1/3. We choose the ratio of the condenser height (i.e., the distance between the upper and lower terminating electrodes) to the interelectrode gap to be 2 and the potentials of the inner and outer electrodes to be 0 and 1 V, respectively. In this case the calculation of the potentials at the terminating electrodes, which give the values of the coefficients 52,53,and 5 4 specific for the spherical field, leads to the following results: 1. In the case of nonsplit terminating electrodes, the required potential at these electrodes is 1.155 V. 2. In the case of terminating electrodes split into two equally wide parts, the required potentials at the inner and outer parts of the terminating electrodes are 1.99 V and 0.379 V, respectively. 3. In the case of terminating electrodes split into three equally wide parts, the required potentials at the inner, middle, and outer parts of the terminating electrodes are 8.153 V, -4.125 V, and 5.245 V, respectively.
348
M. I. YAVOR
RGURE 15. Equipotentials(solid lines) imitating a spherical potential distribution(dashed lines) in a cylindrical analyzer with solid (nonsplit) terminating electrodes (a) as well as terminating electrodes split into two (b) and three (c) parts.
Evidently in the first case the spherical field distribution in the vicinity of the optical axis is created only in the second-order approximation by the powers of the deviation of the radial coordinate with respect to this axis. In the second case this distribution is created in the more accurate third-order approximation, and in the third case it is created in the most accurate fourth-order approximation. Note a typically low value of the potential at the middle part of the terminating electrodes. Three equipotential lines in the condenser under consideration are shown in Fig. 15. In accordancewith the predictions of the theory developed in Sections V.A and V.B, nonsplit terminating electrodes (Fig. 15a) and electrodes split into two parts (Fig. 15b) do not provide a similarity of the equipotential shapes to the spherical ones in a large enough interval A. A good coincidence of these shapes is observed only when the electrodes split into three parts are used, as shown in Fig. 15c.
VI. CALCULATION OF THE ELEMENTSOF SPECTROMETERS FOR SIMULTANEOUS ANGULAR AND ENERGY OR MASS ANALYSIS OF CHARGED PARTICLES In this section different types of electron-opticalelements for separation and focusing of wide hollow beams are considered: poloidal analyzers, conical mirrors, and conical electrostatic lenses. For all these elements a perturbation approach allows one to obtain analytical representation of electrostatic fields , which considerably simplifies analysis of charged particle trajectories.
A. Electrostatic Field of a Poloidal Analyzer A poloidal analyzer is a toroidal condenser where charged particles move in the meridianal direction as shown in Fig. 4 (unlike a conventional toroidal analyzer
349
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
RGURE 16. A section through the electrodes of a toroidal condenser by a meridianal plane.
where particles move in the azimuthal direction). Therefore, to calculate particle trajectories in a poloidal analyzer one needs to know the potential distribution for an arbitrary polar angle. We calculate a field distribution in a poloidal analyzer in a toroidal coordinate frame [r,0 ,@ } as shown in Fig. 16, where @ is the azimuthal angle (i.e.. the angle of rotation about the symmetry axis Z), r and 0 are a polar radial coordinate and a polar angle in a meridianal plane, respectively;the origin of the polar coordinate frame is a point 0 whose offset from the Z-axis is a (the point 0 is a center of the electrode curvatures in a meridianal direction). In this toroidal coordinate frame the electrode surfaces of the poloidal condenser are described by equations r1,2 = ro 7 b, where 2b is an interelectrode gap. The electrostatic field potential q ( r , 0)in the space between the condenser electrodes satisfies the Laplace equation cos 0 a+rcos 1 a2q +--r2 a 0 2
sin 0
r(a
1T a\rr
aq
+ r cos 0)a@ = 0.
(1 15)
Consider the potential distribution in a vicinity of some fixed point [ro, OO}; we introduce dimensionless coordinates u = ( r - ro)/b with rl Ir Ir2 (i.e., IuI 5 1) and u = (0 - Oo)ro/b with IuI 5 1. We represent the Laplace operator as an expansion b2AQ = Ao(u,U ; 0 o ) W
+ EAi(U, U ;
00)"
+ E ' A ~ ( uU,;
00)"
+.
, ( 1 16)
E
= b/ro being the dimensionless parameter which we assume to be small. In
350
M. I. YAVOR
Eq. (1 16)A, are some linear differential operators; in particular, A0 = a 2 / a u 2
+
a2/au2. We represent the potential \I, in the form of an expansion
c bo
W ( u ,u ; 0 0 ) =
Wj(U, u ; 0 0 ) d (1 17) j=O Substituting Eq. (1 17) into Eq. (1 15) and taking into account Eq. (1 16) gives the following sequence of equations for the coefficients W j of Eq. (1 17):
(118) (119)
AoWo = 0, AoWl = -AlWo,
...
9
AoWj = -AiWj-i - ... - AjWo
(120)
with boundary conditions Wo(-l, u ; 00) = VI, Wo(1, u ; 0 0 )= V2, where V , and V2 are the electrode potentials, and zero boundary conditions Wj (fl,u ; 0")= 0 with j = 1,2, . ... Equations (1 18)-(120) with the corresponding boundary conditions have exact analytical solutions for an arbitrary number j . These solutions are valid only in the vicinity u 5 l of the angular direction 0 = 00. However, since 00is an arbitrary angle, we can set in the final solution u = 0; then the poloidal condenser potential in an arbitrary point [ u , 0 = 001can be represented as the expansion of Eq. (1 17) with u = 0. Several first coefficients in this expansion are
Wo(u,0; 0 ) = w
+ vu,
V W I ( U , 0; 0 ) = --(1 2 V W 2 ( u ,0; 0 ) = -(1 3
+ C)(u2 - l), + c + C 2 ) ( U 3 - u),
"
1
1 + C)(1 + C2) - -(1 - C)(C + (u4 - 1) 24 + 5""3(1+ C)(1 + c + C2) + 2(11 - C ) ( C + S2) (u2 - l ) , 19 + 9 c 2 + -c3 + 12c4 + 4s2 W4(u, 0; 0 ) = 60 2 W 3 ( u , 0; 0 ) =
v
$1
S2)
1
- 3c2s2- cs2 (u5 - u ) +
+
1
19 18
Here W = (VI V2)/2, V = (VZ - V1)/2, C = cosO/F, S = s i n @ / F with F = d cos 0 and d = a/ro.
+
CHARGED PARTICLE OPTICS OF SYSTEMS WlTH NARROW GAPS 0 5b
0.0002
0 54
0.0001
351
0
I):,2
-0.0001
05
-0.0002
na
-0.on03 0 4G
-u.ouu4
0 44
-0.0005
u 42
-0.0006
11 4
-0.no07
0
lrfz
?r
3x12
2n
(a)
FIGURE17. (a) A potential W distribution along an optical axis r = ro = 100 mm in a poloidal condenser with t = 0.25 and the difference A W between this distribution calculated by a perturbation method and by an accurate numerical simulation; (b) acorresponding field strength distribution En and the difference AE,, between this distribution calculated by a perturbation method and by an accurate numerical simulation.
Since Eq.(1 17) gives an asymptotic expansion of the potential distribution with respect to the small parameter E , its accuracy improves as the interelectrode gap in the condenser narrows. However, even when the interelectrode gap 2b is comparable with the main beam path radius ro, the accuracy of the proposed potential representation remains very good. The accuracy of the perturbation method can be illustrated by a comparison of a result of an analytical calculation with a result of a calculation performed by a high-accuracy numerical boundary element method (for numerical calculations we used the computer program POISSON-2 by V. Ivanov [2]). Figure 17 shows the results of the calculation of the potential @
352
M. I. YAVOR
at the circular beam main path r = ro and of the component of the field strength En normal to this path. We chose the following geometrical parameters: a = 150 mm, ro = 100 mm, and b = 25 mm, so that the value of the small parameter E is 0.25. In this case the accuracy of the perturbation method is about 0.1% for both the potential and the field strength. The analogous calculation for a smaller parameter E = 0.1 reveals the corresponding accuracy to be about 0.00 1% ,which is already the numerical computation noise level. Thus, for practical situations where the value of 6 is usually about 0.1 to 0.3, the perturbation method is very accurate. The perturbation method preserves a good accuracy even in more extremal situations. Figure 18 shows the results of the calculation of the potential and the field strength at the circular orbit r = 117.5 mm for a poloidal condenser with a = 150 mm, ro = 95 mm, and b =45 111111, which corresponds to E = 0.474. Note that in this case the minimal raduis of curvature of the outer electrode is a - ro - b= 10 rnm, which is much smaller than the interelectrode gap. The accuracy of the potential and the field strength calculation by the perturbation method is about 1% for all 0 except the vicinity of the point 0 = 180" closest to the axis of rotational symmetry where this accuracy is about 3%.
B. Calculation of Particle Trajectories in a Poloidal Analyzer We will represent trajectory equations for charged particles in a poloidal analyzer in a cylindrical coordinate frame, since initial conditions for these trajectories are more conveniently defined in linear rather than angular coordinates. We introduce a cylindricalcoordinate frame { p , 8, J } with the axis perpendicular to some meridianal plane 8 = 0 and located at the distance a from the system's symmetry axis. In such a meridianal plane we have 7 = 0, p = r , and 8 = 0 ;general relations between the cylindrical and toroidal coordinates read
p sin 8
tan 0 = JJ~
tan 4 =
+ (a + p cos 812 - a'
I a+pcos8'
We proceed now to dimensionless coordinates x = ( p - ro)/ro and y = J / r o . The toroidal coordinates u and 0 can be expanded by powers of coordinates x and y . Accurate to the fourth order these expansions are =x
1
2
1
2 + y y Y Y + -u,yyxy
2
+ -41u x x y y x y + -u241 2 2
4
YYYYY
9
SdVf) M O W N HUM SNXLSAS
ES€
mvl
I
1
I
l
I
I
I
200'0
I
I
I
I
I
A,."
I
swag-
-
1000'0-
-
- E1000'0- 2000'0- szooo'o@ I
21°C
'
r;
x
I
'
I
- mnnSE000'0-
21"
0
with ro = 95 mm and E = 0.474 and the difference A @ between this distribution calculated by a perturbation method and by an accurate numerical simulation; (b) a corresponding field strength distribution En and the difference A En between this distribution calculated by a perturbation method and by an accurate numerical simulation.
"7.
I
,,
FIGURE18. (a) A potential \I, distribution along a circle r = 117.5 mm in a poloidal condenser
354
M. I. YAVOR
~ X X Y Y=
uyyyy
2 C O S e~ - 9
f3
3d
3 cos28
f3
f2
= - - -,
oyy= --,sin 8 f
@xyy
=
sinB(d
+ 2cos 0 ) , f2
%yy
=-
@YYYY
=
2sine(3fcose + d 2 ) 7
f3
3sin8(1 + 2 f cost)) f =d
f3
f
+ Cose.
We introduce a concept of a reference particle, which is a particle that in the limiting case of 6 0 moves in a condenser along the circular trajectory of the radius ro. Let KObe akinetic energy of this particle (which is related to the potential difference V by the formula V = 2rKo/40), mo its mass, and 40 its charge, which we call nominal ones. Then exact particle trajectory equations in the poloidal condenser take the following form: (1
+ x)2 + x R + y‘2 ,
(126)
+ x)2 + + y’2 ,
(127)
1+6-2\I,
(1
XR
1+6-2Q
where 6 is a relative deviation of the particle energy-to-charge ratio with respect to its nominal value, and prime denotes a derivative with respect to the angle 8. The equation for the normalized flight time
of an arbitrary particle through the sector with the angle is a nominal time of flight of the reference particle along the circle arc of the radius ro through the same sector, reads 7’+1=
d
(1
+ +X R + XI2
1+6-2\I,
f2
(1
+ Y),
where y is a relative mass-to-charge ratio deviation for a particle with respect to its nominal value.
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
355
Having substituted the field potential, rewritten in the cylindrical coordinate frame, into the trajectory equations, we finally come to the following equations for the first-order terms [XI, y1, q],second-order terms [x2,y2, t 2 ] , and third-order ] describe x- and y-coordinates of charged particle trajectories terms [x3,y3, t 3that in a poloidal analyzer as well as the flight time t:
- -63 5 16
+ -62y 3 + -6y 1 2 + - 1y 3 . 16 16 16
(137)
356
M. I. YAVOR
The coefficients in these equations are sd ax, = UY' = -ax's = - Q y q = -6 2 -, 2f C
29 36
--c
31 c2 180 f 2
+--
1 2 c ay = -c + € 2 - -c3 - -c2 - - c + - - 6 3 2f2 2f2
(
= --1 - -3€ 2 ( c + I ) , 2 4 1 1 cy = - - 2 ( c l), 2 4 1 co = - r 2 ( C 1) €4 2 Cs
+
+
+ +
ayy = c - - c + e 2
+
11 c 12 f 2
1 2
1 c3d 4 f
-c4+-c3+----c2-
2(:
1cd 4 f3
1 3f2
+--
4 f3
1 4
1 c2d - -7_ c2 --
4 f
12f2
-c
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
UYtY’ =
iC- i),
+ €2 (+
-1
1 2
= 2c2 - 3c + € 2
10 --c+ 3
-c4
17 c 6 f2
1 c2d 7 c2 2 1 c3d 5 + -c3 + - - - -c2 - - - - - 3 2 f 6 2 f 6f2 1 cd 5 +-2 f 3 3f2 2 f 3
1
1 1 l), yy-2 4 3 15 C&& = -2(c l) , 8 16 1 3 cay = -- - - 2 ( c l), 4 8 1 1 ‘ Y Y = -8 - -16 €2(c 11,
Cx!,f
=c
1
- - +-2(c
+
+
+
+
+
1 2 6f2 6f2 sd = ay’xx= -ax’yy= -ayyy = -, 7 , 6
a x x x = - c -5c
a,’,,
+ - c19- - + - - -c
2f
+ 14, 5 3 9 2 C 1 axyy= - - c + -c - 3c - -+ -, 2 2 2f2 2f2 axx8= c2 - 7~
axrxcx= c - 3,
ay’y’x =c
- 1,
23 3 ’
357
358
M.I. YAVOR
axss = c
- 7,
3 + -c, 2 5 13 axxy= --c3 + -c 2 2 axys= -2c2 + 5c, ayyA = -c2
1
- 5c - -+ -, 2f2 2f2 C
d 2f3 artxty = a y ~ y =~ ayss y = -byS = -c, 1 3
aYYY= -c 2
- -,
+
b,, = -2(c2 - 4~ 5 ) , by, = 2c2 - C,
b,s = -C
+ 5,
bxy= 2c2 - 3c, 1 5 ,,c , = -c2 - -c 3 3
+ 3-,7
1 c,,, = 1 - -c,
4
3 + -c, 2
cxyy= -c2
3 cyys= --c, cyyy
4 1 = -c, 4
d,, = -C dyy
+ 4,
= c,
where c = cos elf and s = sin e/f. Since no analytical solutions of the homogeneous equations with the left-hand sides of Eqs. (129) and (130) are known, their solutions are to be found by a numerical integration. However, after this integration is performed, the inhomogeneous Eqs. (129)and (1 3 1)-( 137) are easily solved by the method of variation of constants, which leads to a set of aberration integrals for second- and third-order aberrations of the analyzer. Note that Eq. ( 1 29) is inhomogeneous even for the reference particle that possesses the nominal kinetic energy [6 = 01 due to the presence of the coefficient ao. This means that the reference particle slightly deviates from the circular optical
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
359
axis of the analyzer. For this reason exact values of the coefficients of aberrations of some fixed order are determined not only by the solutions of equations corresponding to this order, but also by the solutions of higher-order equations. In particular, the values of the third-order aberration coefficients, obtained as solutions of Eqs. (135)-(137), are not quite accurate if contributions of higher-order equations are not taken into account. For a typical value of the small parameter E = 0.05 the accuracy of calculation of the first- and second-order aberration coefficients is within 0.1%, whereas the accuracy of the third-order aberration coefficients (in case the contributions of the fourth- and higher-order equations are not taken into account) is within 10%. The accuracy of calculation of optical properties of poloidal analyzers with the aid of the approach just proposed can be illustrated also by a comparison of the results obtained by this method and by a numerical simulation, for some wellknown geometry of the poloidal analyzer. As an example we consider an analyzer for investigation of a space plasma whose geometry is shown in Fig. 19. Such a two-stage scheme was proposed in [91]; its modifications can be also found in [92, 951. The original scheme of [9l] included two poloidal condensers with the optical radius of the first condenser being rl = 43 mm and in the second one 1-2 = 60 mm; the corresponding deflection angles are 81 = 127.6’ and 82 = 85”, the offsets of the centers of the meridianal curvature of the electrodes with respect to the system’s symmetry axis are u1= 119 nunand a2 = 100.2 mm. The entrance angle of the particle beam with respect to the symmetry axis is 00 = 73”, the distance between the entrance slit and the entrance boundary of the first condenser
FIGURE 19. A typical geometry of a two-stage poloidal analyzer for an investigation of properties of a space plasma [a configuration is reprinted from the Int. J. Mass Spectrorn. Ion Proc., vol. 130, M. I. Yavor, B. Hartmann, and H. Wollnik, “A new time-of-flight mass analyzer of poloidal geometry,” pp. 223-226 (1994) with kind permission of Elsevier Scientx-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands].
360
M.I. YAVOR TABLE 2 TRANSFER MATRIXCOEFFICIENTS OF THE m 0 - S T A G E ANALYZER, CALCULATED BY THE PERTURBATION METHOD, WTH THE AID OF THE APPROXIMATE METHODBY GHIELMETIIAND SHELLEY, AND BY A NUMERICAL RAY TRACING.
Perturbation
Ghielmetti
Numerical
0.875 1.124 0.819 -0.690 - 1.246 2.000 1.966
0.867 1.047 0.827 -0.656 -1.138 1.880 I .954
0.870 1.110 0.821 -0.680 - 1.234 1.991 1.967
~
( x Ix )
(aIx) (alff) (a 16) ( 5 1x1
(?la) (sly)
11 = 3 mm, between the condensers 12 = 11 mm, from the exit boundary of the
second condenser to the detector (image) plane 13 = 37.5 mm. Note that the analyzer under investigation possesses the angular focusing in the radial direction (i.e., the transfer matrix coefficient ( x I a) = 0, where a is the angle of inclination of a trajectory with respect to the circular optical axis in the radial direction), as well as spatial energy focusing ( x I S) = 0. In Table 2 a comparison is given of the first-order transfer matrix coefficients of the analyzer, calculated with the aid of the perturbation method (taking into account contributions of the second- and third-order equations), with the aid of an approximate method proposed by Ghielmetti and Shelley [91] and by a numerical ray tracing. The results obtained by the latter two methods are taken from [91]. The coefficients are given in the dimensionless coordinates adopted in the latter paper, where the initial coordinate is normalized to the optical axis radius in the first stage and the final coordinate is normalized to the optical axis radius in the second stage (that is why, for example, the product of the linear ( x I x ) and angular (a1 a) magnifications as given in the table is not 1); the angular coordinates are expressed in tangents of the angles, energy and mass deviations in parts of their nominal values, the time variable in the relative deviation of the flight time with respect to the nominal one, multiplied by the total path length and normalized to the optical axis radius in the second stage. The table demonstrates a good accuracy of the perturbation method, being about 1%. This accuracy is not so high as was declared earlier due to the fact that fringing field effects were neglected in calculations with the aid of the perturbation method. In Table 3 a comparison is given of several experimentally measured firstorder transfer matrix coefficients with their values calculated by the perturbation method, for a two-stage system, whose geometry was described in [86]. This geometry is similar to the geometry of [91]; however, the dimensions are slightly
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
361
TABLE 3 COMPARISON OF CALCULATED AND MEASURED TRANSFER MATRIX COFFICIENTS OF A "O-STAGE POLOIDAL ANALYZER FOR DIFFERENT POSITIONS OF THE OBSERVA'flON PLANE.
Measurement
Calculation
(x~x) (X
la)
(XIS)
I
U
III
I
U
m
-1.31 -0.78 1.35
0.89 0.00 -0.04
1.38 0.34 -0.38
-1.30 -0.86 1.33
0.90 -0.08 -0.17
1.40 0.38 -0.37
different. Namely, for the system under consideration f l = 6 mm, rl = 44 mm, a1 = 104 mm, 60 = 76", 81 = 122.5", f 2 = 19 mm, r2 = 51 nun, a2 = 85.4 mm, 62 = 86.3'. The experimental data were found in Table 2 of [86]. Column I of Table 3 corresponds to the coefficients measured at the distance of 40 mm downstream the first stage (in the absence of the second one), column I1 to the coefficients measured at a distance of 24 mm downstream the second stage, and column I11 at a distance of 42 mm downstream the second stage. The comparison shows a good agreement of the results. Thus, the proposed perturbation method possesses a high accuracy and allows one to optimize efficiently various systems based on poloidal analyzers. Some examples of two-stage poloidal analyzers, calculated with the aid of this approach, can be found in [95,96]. C. Focusing Properties of a Poloidal Analyzer If we neglect the small terms proportional to c2 in Egs. (129) and (130), these equations take the following form:
+
2d cos 8 x=s "'+ d + c o s 6 cos e YN y = 0, d+cod where the subscript 1 is omitted in the functions X I and y l . This pair of equations locally (i.e., in the vicinity of some fixed value of the polar angle 0) describes a paraxial charged particle motion in a toroidal sector condenserwith a toroidal factor
+
COS C =
e
d+cosO' Thus, a poloidal analyzer can be regarded as a sector condenser with a toroidal factor that varies along the beam path. Consider first a case where both entrance and exit polar angles of the poloidal analyzer electrodes satisfy the condition -n/2 < 8 < n/2. Then we have 0 c c c 1
362
M.I. YAVOR
and, thus, the focal lengths of such an analyzer are somewhere between the focal lengths of the cylindrical and spherical analyzer. This poloidal analyzer focuses particles in both the radial (x) and the azimuthal (y) directions, but its focusing power in the azimuthal direction is always weaker than in the radial direction. The larger is the offset a, the closer are the focusing properties of the analyzer to the properties of the cylindrical sector deflector. Thus, the analyzer in question cannot be made stigmatically focusing. However, for the systems that use poloidal analyzers another relation between the radial and azimuthal focusing can be of interest. Namely, first of all, as in any analyzer that separates the particles according to their energy-to-chargeratio in the x-direction, it is necessary that such a system possess a point-to-point focusing in the radial direction (in other words, that the final x-coordinate be independent of the initial angle a! of the trajectory inclination in the radial direction); in the notations adopted in the transfer matrix representation,this condition reads (x I a!) = 0. At the same time, to achieve a good angular resolution in the azimuthal direction, a bundle of particles starting from the object parallel to the azimuthal direction must be focused to a point; the set of all these points corresponding to different starting azimuthal angles forms a ring-shaped image as shown in Fig. 4. In other words, the final y-coordinate of the particle must not depend on its initial y-coordinate; in the notations adopted in the transfer matrix representation, this condition reads (Y I Y ) = 0. Both conditionsof the point-to-point focusing in the radial direction and parallelto-point focusing in the azimuthal direction can be satisfied in a single-stage poloidal analyzer by a proper choice of the lengths of the drift spaces in front and behind the poloidal condenser. For the initial polar angle 8 = -n/2 of the condenser electrodes, the deflection angle of 90 degrees and the object situated at the axis of rotational symmetry (this case is shown in Fig. 4), a geometry satisfying the two conditions just mentioned was found in [80]; its parameters are d = a/ro = 0.48 and l / r o = 0.75, where 1 is the distance between the exit face of the poloidal condenser and the final image plane. The same conditions can be also satisfied in more complicatedenergy analyzers containing two poloidal condensers as shown in Fig. 20. This design can help to decrease a radius of the ring-shaped image while preserving a normal incidence of the beam to the detector plane [95]. Considernow a poloidal condenser where a part of the electrodes lies in the range of the polar angles 7r/2 < 8 < 3n/2. In this case c < 0 and thus in this range of the angles the beam is focused in the x-direction and defocused in the y-direction. Note that in the interval of the angles 8 where the sum d cos 8 is small (this can happen if d I1 or d is larger but close to 1) x-focusing as well as y-defocusing becomes very strong; it is better to avoid such a situation, since higher-order aberrations, induced by the corresponding part of the condenser, also become large. Anyway, usually it is difficult to achieve parallel-to-point focusing in the azimuthal direction in the analyzers under consideration. For example, calculation
+
CHARGED PARTICLE OFTICS OF SYSTEMS WITH NARROW GAPS
363
FIGURE 20. A possible geometry of the two-stage device for an angle-resolvedenergy analysis [a configuration is reprinted from Nucl. Instrum Merh. A, vol. 311, M. I. Yavor, H. Wollnik, M. Nappi and B. Hartmann “Image aberrations of poloidal toroid electrostatic analyzers:’ pp. 448-152 (1992) with kind permission of Elsevier Science-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands].
shows that two-stage analyzers whose geometry is shown in Fig. 19 (the time-offlight velocity and mass analyzers of this type were studied in [91, 92, 951) do not possess focusing in the azimuthal direction because the exit part of the second poloidal condenser is defocusing the charged particle beam in the y-direction. To avoid this disadvantage, a two-stage system can be arranged differently, with the beam intersecting the axis of rotational symmetry between the stages as shown in Fig. 21 (this geometry was proposed in [96]). D. Electrostatic Field of a Conical Mirror A conical mirror is a simple analyzer which consists of two coaxial conical electrodes. A charged particle beam is injected into the conical mirror through a window in the inner electrode, is deflected by the electrostatic field between two electrodes and after passing again through another window in the inner electrode is focused at a detector plane (see Fig. 22). Like a conventional cylindrical mirror, a conical mirror allows one to analyze wide hollow particle beams. The advantage of the conical mirror analyzer as compared with the cylindrical one is that charged particles can enter the analyzer in a direction perpendicular to the symmetry axis, so that not only conical but also disklike beams can be analyzed.
364
M. I. YAVOR
FIGURE 21. A geometry of the two-stage poloidal time-of-flight analyzer, which possesses a parallel-to-point focusing in the azimuthal direction [a configuration is reprinted from the Int. J. Mass Spectmm. Ion Pmc., vol. 130, M. I. Yavor, B. Hartmann and H. Wollnik “A new time-of-flight mass analyzer of poloidal geometry,” pp. 223-226 (1994) with kind permission of Elsevier Science-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands].
A disadvantage of the conical mirror analyzer is that there is no exact analytical electrostatic field representation in it. However, similarly to the case of a poloidal analyzer, a perturbation approach allows one to obtain a simple and very accurate approximate analytical formula for the electrostatic field potential. We represent the potential q in the conical coordinate frame ( X ,Z, $1 shown in Fig. 22, where $ is the azimuthal angle with respect to the axis of the rotational symmetry of the conical electrodes and [X, Z] the Cartesian coordinate frame in the meridianal plane C#J = const; its origin 0 lies at an arbitrary point of the Z-axis directing along the line which is equally far from both the electrode surfaces. Since the potential does not depend on 4, the Laplace equation written in conical coordinates takes for this potential the following form a2q a 2 q 1 -+-+ ax2 az2 r o + c x +(cg+Sg) ~~ =o.
(138)
Here ro is the distance of the origin 0 of the conical coordinate frame from the symmetry axis and C = cos8, S = sin 8, where 8 is half the cone opening angle of the electrodes.
CHARGED PARTICLEOPTICS OF SYSTEMS WITH NARROW GAPS
FIGURE
365
22. A geometry of a conical mirror analyzer; shown is the conical coordinate frame
(X,z, $1.
Now we adopt an important assumption that the interelectrode gap b of the analyzer is sufficiently smaller than the distance ro (this assumption holds in most practical applications). Then, proceeding in Eq. (138) to dimensionless coordinates tf = X/b and 5 = Z / b , we obtain the following equation
where E = b/ro is a small parameter. Substituting in Eq.(139)the potential in the form of the expansion q = q ( 0 ) + &')
+ &(2)
+ E3*(3) + .. . ,
( 140)
expanding the denominator in Eq. (139) in powers of E and equating terms with different powers of E in the resulting equation, we finally come to a recursive set of partial differential equations for the functions W(j). The first four equations with j = 1-4 read
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M.I. YAVOR
The function q ( O ) satisfies the boundary conditions q(O)(-l, 5 ) = V , , W(O)(l,c ) = V2, where Vl and V2 are the electrode potentials; the functions Q ( j ) with j 2 1 satisfy zero boundary conditions at t = f1. All the differential equations for the functions q ( j ) with the boundary conditions just listed can be solved analytically; their solutions are polynomials in .$ and 5‘. However, since the important assumptionof the perturbation method is that there are no large variables in the corresponding solutions, the potential representation of Eq. (140) is valid only in a small vicinity of the X-axis, that is, with { being of the order of unity and, thus, Z being of the order of E . Therefore, we first calculate the potential Q ( t , O ) at the X-axis; the solution of Eqs. (141)-(144) yields q ( . $ , o )=
w + vt - €-((2vc 2
- 1)
vc2 + E2-(.53 3 12
+
- 6) 24
(6’
-
1)
+ c4(. . .)
(145)
where W = ( V I V 2 ) / 2 and V = (V2 - V1)/2. Now we recall that the position of the origin 0 of the conical coordinate frame was chosen arbitrarily at the Z-axis; each particular position of the point 0 uniquely corresponds to a certain value of the distance ro (except for the limiting case 8 = 0 of a cylindrical mirror). Thus, Eq. (145) gives the field potential distribution along the X-coordinate line for an arbitrary choice of the position of this line, which is fixed by the value of the parameter E = b/ro;this parameter plays the role of the second coordinate in Eq. (145). In the limiting case of a cylindrical mirror, the parameter E in Eq. (145) is constant in the Z-direction, and the field depends only on the .$-coordinate;Eq. (145) then gives a polynomial expansion of the logarithmic field potential.
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
361
The comparison of calculationsperformed for the conical mirror field potential with the aid of the perturbation method and by a numerical integration shows that the accuracy of Eq. (145) for a typical value of the small parameter E =0.25 is within 0.2%; this accuracy can be further improved by taking into account higherorder terms (proportional to e4, etc.) in the potential expansion. Particle trajectories in the field of the conical mirror generally can be calculated by a numerical integration; the analytical representation of Eq. (145) helps to avoid time-consuming numerical field evaluation during this integration. Only in a rough approximation where the field does not vary along the Z-axis can particle trajectories in the meridianal plane be integrated analytically [102]. Optical properties of a conical mirror were shown in [ 1021 to be similar to the properties of a cylindrical or planar mirror. In particular, the relative dispersion of the analyzer is close to the relative dispersion 2 cos-* A of a two-dimensional field, where h is the angle of the beam entrance into the analyzer with respect to the Z-axis. There exists a mode with second-orderfocusing in the conical mirror; the corresponding parameters of the analyzer depend on the cone opening angle 8. Some more details concerning the modes of operation of the conical mirror analyzer can be found in [ 1021.
E. Electrostatic Field of a Conical Lens with Longitudinal Electrodes Consider a two-electrode immersion lens whose electrode surfaces lie at two coaxial cones (see Fig. 5 ) with the angle between the axis of rotational symmetry and the cone generating lines being 8. We call an optical axis of the lens in a meridianal plane 4 = const a line that passes between the electrodes at an equal distance b between their surfaces. We introduce in this meridianal plane a Cartesian coordinate frame [ X,Z} as shown in Fig. 23, the coordinate axis Z directing along the lens optical axis; let the distance between the origin of the frame and the symmetry axis be ro. In this section we assume that the gap between the pair of the electrodes, whose surfaces lie at one cone, is negligibly small and its 2-coordinate is zero. Under this assumption the field of a three-electrode lens is just a superposition of the fields of two two-electrode lenses; for this reason we will obtain the analytical representation only for a two-electrode lens. Let the potentials at its electrodes be 0 for Z < 0 and V for Z > 0. Now we proceed to the dimensionless coordinates ij = X/b and { = Z / b ; in these new coordinates, which are identical to the coordinates introduced in Section VI.D, the Laplace equation for the field potential takes the form of Eq. (139) with C = cos 8 , S = sine, and E = b/ro. The latter parameter is assumed to be small; in other words, we suppose that the region where the lens field is concentrated is far enough (as compared with the size of this region) from the axis of the lens rotational symmetry.
368
M. I. YAVOR
FIGURE23. A section through a two-electrodeimmersion conical lens with longitudinal electrodes by a plane @ = const.
The substitution of the potential Q(6, f), represented in the form of Eq. (140), leads to a sequence of equations for the coefficients Q ( j ) of Eq. (140). The first four equations (for j = 1-4) are Eqs. (141)-(144). The function *(')satisfies the boundaryconditionsQ(')(fl,{) =Owithf O; the functions Q ( j ) with j >_ 1 satisfy zero boundary conditions at the electrode surfaces. Unlike the case of the conical mirror, only three equations [Eqs. (141)-(143)] can be solved analytically. However, the accuracy of the result represented by the first three terms in Q. (140) is shown later to be high enough for a practical use. The main contribution to the lens field potential can be described in terms of the field of a two-dimensionallens whose potential satisfies Eq.(141). We call this lens a reference one. Note that such a contribution is not exactly a two-dimensional field, since ,$ and f are conical coordinates. The curvature of the electrodes is additionally taken into account by terms satisfying Eqs. (142) and (143). The function *(') can be represented in a simple analytical form: 9(0)(,$, {)
v +v arctan sinh(n 5 /2)
=2
n
COS(?r{/2)
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
369
The functions q(I)and Q(2) can be also expressed analytically. Namely, we represent the function q(')as a sum
SubstitutingEq. (147) into Eq. (142) and taking into account boundary conditions for the functions and \ Y ( l ) , we obtain that the function w1 satisfies the Laplace equation
with the boundary conditions
A solution of Eq. (148) with the boundary conditions given by Eq. (149) is the function
where
and
We proceed now to calculation of the function \Y('). following form: 3
1
q(2)(t7 0 = p"'(t, O ( G + so2- p
( t , c ) ( a
We represent it in the
+ s o + w2@, el. (151)
Substituting Eq.(151) into Eq.(143) and taking into account Eqs. (141) and (148), we come to the Poisson equation for the function w2:
370
M.I. YAVOR
with the boundary conditions
Equation (152) with the boundary conditions of Eq. (153) has the following exact analytical solution:
and
Thus, in a quadratic approximation with respect to the small parameter E , the electrostatic field potential of a conical lens with longitudinal electrodes is expressed through elementary functions and fast-converging serii. Note that the field strength in a conical lens contains a component normal to the optical axis 6 = O and, thus, deflecting a charged particle beam from the initial conical surface. Though such a parasitic effect is comparably small (since the deflecting field component is proportional to E ) , it can make using a conical lens difficult in case it would be located in front of another stage, say, a poloidal analyzer, and its mode of operation requires a change of the excitation. A way of eliminatingthe beam deflecting field component will be discussed in Section V1.G. To estimatethe accuracy of the proposed perturbation method, we give a comparison of the results of a calculation performed by the foregoing analytical formulas and by a high-precision numerical integration. Figure 24 shows the distributions
37 1
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS 0.0001
1
09
\Y'
'
I
'
'
'
I
0
0.2
0.4
0.6
0.8
0
u.n 07
-0.0001
00:
-0.0002
OR
-0.0003
04
u3
-0.0004
02
-0.0005
01
-0.0006
U
~
-08 -0.6 -0 4 -0.2
0
0.2
0.4
~~~
0.6 0.8
-0.8 -0.6 -0.4 -0.2
(4
0.01
0
\
0 -0.001
0 IlCi 4 07
'
- u s -06
'
-0.002 -
VI '
- 0 4 -02
0
-
-0.003 . 02
04
06
'
'
'
08
RGURE 24. (a) A potential \I, distribution along the optical axis of a conical immersion lens with longitudinal electrodes in the case of 0 = 30" and e =0.25 (it is assumed that V = 1 and the linear dimensions are referred to TO), and the difference A Y between this distribution calculated by a perturbation method and by an accurate numerical simulation; (b) a corresponding field strength distribution En and the difference AE, between this distribution calculated by a perturbation method and by an accurate numerical simulation.
of the potential UJ and the field strength component En normal to the lens optical axis, along this axis in a conical lens with 6 = 30" for a typical value of the small parameter 6 = 0.25. It is seen that the accuracy of the potential calculated by the perturbation method is about 0.2% in this case and the accuracy of the calculation of En is about 3%. The poorer accuracy of the field strength representationas compared with the potential distribution is due to the fact that a small normal component of the field strength is contributed only by two terms Q(') and Q(*) of Eq. (140). However, no better accuracy is usually needed for evaluation of this parasitic component, which is responsible for a weak beam deflection from the optical axis.
372
M.I. YAVOR
FIGURE25. A section through a three-electrode conical slit lens by a plane 4 = const.
E Electrostatic Field of a Slit Conical Lens In this section we consider a lens whose electrodes are parts of conical surfaces, with their sections by a meridianal plane C#J = const being perpendicular to the lens optical axis (see Fig. 25). A charged particle beam in such a lens passes along the Z-axis through the slits in the electrodes, and the lens is called a slit conical lens. Let b be a characteristic size of the region where the lens field is concentrated, for example, half a gap of one of the slits. The equations for the coefficients of the field potential expansion with respect to the small parameter 6 have, such as in the case of a lens with longitudinal electrodes, the form of Eqs. (141)-( 144); however, the solutions of these equations take a form different from the one obtained in Section V.E. We will obtain later the solutions of two equations (141) and (142); that is, we will describe a potential in the linear approximation with respect to the small parameter 6. Note that for the adopted model of the slit lens configuration more accurate calculation of higher-order terms is of no practical sense. The reason is that even the field of the two-dimensional reference slit lens is only roughly described by the model that does not take into account a real lens environment, since long tails of the field outside the outer electrodes, which noticeably contribute optical properties of the lens, usually are not predicted by such a solution with a good accuracy.
373
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
Consider a three-electrode einzel slit lens and a two-electrode immersion slit lens. In the former case we assume the {-coordinate of the middle electrode to be zero and b to be half a slit width of this electrode. Then the boundary > t j , { j ) = 0,j = 1 , 2 for conditions for the function @(') take a form @(')(1(1 the outer electrodes and @(')(I.$I > 1,O) = V for the middle electrode. For the two-electrode lens we assume that one of the electrodes has the zero potential and the other one, with half the slit width being b, has the potential V; we suppose that the {-coordinate of the latter electrode is zero. Then the boundary conditions at the electrodes are @(')( 16 I > (2, (2) = 0 and @(')( It I > 1,O) = V. In both cases the function @ ( I ) satisfies the zero boundary conditions at the electrodes. The solution of Eq. (141) can be obtained with the aid of the Schwarz-Christoffel transformation that gives a conformal mapping of the plane z = { it to the upper half-plane of the variable w = u iu. This conformal mapping can be found in [119, 1201. Having obtained this solution, the solution of Eq. (142) is represented in the following form:
+
+
where the functions $1 and +2 satisfy the equations
Besides, both these functions satisfy the zero boundary conditions at the electrodes and are to be finite at the infinity. The function +I that satisfies Eq.(156) and the boundary conditionsjust listed is
where x = 0 for a three-electrode einzel lens. For a two-electrode immersion lens this function is x ( t , { )= u ( t , {) with u being the imaginary part u = 3 w ( z ) of the Schwarz-Christoffeltransformation mentioned previously. The function $2 can be represented as
A substitution of Eq. (159) into Eq. (157) shows that the function h ( t , {) satisfies the two-dimensional Laplace equation and the following boundary conditions at
314
M. I. YAVOR
the electrodes:
h(( > ( j ,
Tj)
= H j l h ( t c - t j , < j > = -Hj$
(160)
with j = 0, 1 , 2 for a three-electrode lens, j = 0 , 2 for a two-electrode lens, and the constant parameters H , being
The constants H j for some electrode geometries are tabulated in [ 1041. The function h ( 6 , C) is easily calculated with the aid of the Schwarz-Christoffel transformation discussed earlier. Note that this function is antisymmetric with respect to the (-coordinate and that
To estimate the accuracy of the perturbation method, we give a comparison of the results of a calculation performed by the proposed analytical formulas and by a high-precision numerical integration. Figure 26 shows the distributions of the potential q and the field strength component En normal to the lens optical axis, along this axis in an einzel three-electrode conical slit lens with f3= 30” and a typical value of the small parameter E =0.25. Calculations were performed for the lens geometry where all the slit gaps were the same and the distances between the central and outer electrodes equal b. The constants H , in this case are H1 = H2 = -0.097, HO = 0.244. It is seen that the accuracy of the potential calculated by the perturbation method inside the lens is about 2% and the accuracy of the calculation of En is about 15%; this accuracy becomes somewhat lower at the tails of the field. G. Elimination of a Beam Deflection in Conical Lenses The electrostatic field component En considered in Sections V1.E and V1.F causes a deflection of the charged particle beam from the lens optical axis. This effect is due to a “shift” of the potential distribution in a conical lens in the direction of the axis of rotational symmetry, as compared with the potential distribution in a two-dimensional reference lens. Such a deflection is usually small but still can achieve several degrees in case of strong lenses. Thus, in case a variation of the lens excitation is required, it is desirable to reduce the field strength component normal to the optical axis. Consideralens with longitudinalelectrodes. A field strength component En(6, C) = -aQ((, <)/at along a line parallel to the lens optical axis with a small offset
375
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
0.15 0.1
-0.007 0.05
-
-0.008 -
- '
'
'
-1.5 - 1
-0.5
0
0.5
1
1.5
2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-1.5
-0.5
0
0.5
1
1.5
2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-0.009 -2
0.08 0.06 0 04
0.02 0
-n 02 -0.04
-0.06 -2
-1
FIGURE 26. (a) A potential W distribution along the optical axis of an einzel conical slit lens in the case of 8 = 30" and c = 0.25 (it is assumed that V = 1 and the linear dimensions are referred to ro). and the difference A W between this distributioncalculated by a perturbationmethod and by an accurate numerical simulation: (b) a corresponding field strength distribution En and the difference between A & between this distribution calculated by a perturbation method and by an accurate numerical simulation.
of 6 from this axis can be roughly [retaining only the terms linear with respect to 6 and E in Eqs. (147) and (151)l expressed as
A behavior of this function is approximately the same as the behavior of the exact distribution of En as shown in Fig. 24b. Thus, we can try to find the offset top, where the amplitude of the normal field strength componentis minimal, by setting
376
M. I. YAVOR 0.002
0 -0.002 -0.004 -0.006 -0.008 -0.01 -0.012 -0.014 -0.016 -0.018
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
FIGURE 27. The deflecting field component En for the immersion two-electrode conical lens with longitudinal electrodes in the case of 0 = 30" and 6 = 0.25, calculated at the distance 6 = -0.03 away from the lens optical axis.
the derivative of the right-hand side of Eq. (163) with respect to ( at ( = 0 to zero. A calculation yield
topf= -0.116~C. With 8 = 30" and c = 0.25 we have to,,, = -0.025. More accurate numerical calculation shows that the minimum amplitude of E, is achieved in this case at top,= -0.03. The distribution of En(top,,() is shown in Fig. 27. The amplitude is four times smaller than at of the normal component of the field strength at top, 6 = 0. Thus, a beam deflection in a conical lens with longitudinal electrodes may be considerably reduced by passing the beam in the lens not along its optical axis but parallel to it with the small offset of top,.Calculation shows that in conical slit lenses this way of reducing of En is not so efficient: for example, a numerical simulation of the distribution of the field strength component, normal to the beam optical axis, along the lines parallel to this axis in case of the threeelectrode einzel slit conical lens with 8 = 30°, E =0.25, equal slit widths of 2b and distances between the electrodes of b, demonstrates that the minimum amplitude = -0.058 (see Fig. 28), is only 1.5 times smaller than the of E n , achieved at topf amplitude of E, along the optical axis, shown in Fig. 26b. However, in a slit lens the deflecting field component at the lens optical axis may be more effectively reduced by changing the electrode potentials. Indeed, Eqs. (159) and (162) yield
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS 0.03
I
I
I
I
377
I
_En
0.02
-
0.01 0 -0.01
-
-0.02
-
-003
-
-0.04
-
-0 05
..
-2
I
I
I
-1.5
-1
-0.5
0
1
I
0.5
1
, 2 1.5
2
FIGURE28. The deflecting field component En for the einzel three-electrode conical slit lens in the case of 0 = 30" and E = 0.25, calculated at the distance 6 = -0.058 away from the lens optical axis.
Thus, the normal component En of the field strength in a conical slit lens is approximately proportional (with the factor - V E C ) to the normal field strength component of a field created by setting the electrode potentials to f H j that are the boundary conditions for the function h ( . $ , { )[see Eq. (160)l.In other words, to reduce the deflecting field component at the lens optical axis, one needs to add to the electrode potentials small additional components F V E C H ~(the upper sign corresponds to the electrodes located at .$ > 0 and the lower sign to the electrodes located at .$ < 0). Figure 29 shows the field component En calculated numerically in cases where such additional potentials have been applied to the electrodes of the three-electrode einzel slit lens with 8 = 30" and E = 0.25. Acomparison of Figs. 29 0.015
0.01
0.005 0 -0.005 -0.01
-0.015 -0.02
-0.025 -0.03
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
FIGURE 29. The deflecting field component En at the optical axis of the einzel three-electrode slit conical lens with 0 = 30" and t = 0.25 in the case where small additional potentials are applied to the lens electrodes.
378
M.I. YAVOR
and 26b shows that changing the electrode potentials reduced the deflecting field component about 2.5 times.
H. Focusing of Hollow Charged Particle Beams by Conical Lenses Consider first a slit conical lens. We assume that the field strength component normal to the beam optical axis is compensated for as proposed in Section V1.G and, consequently, the field potential is completely defined by its distribution at the optical axis. We fix now a meridianal plane (say, 4 = 0) and introduce a Cartesian coordinate frame ( X , Y, Z ) so that the X - and Z-axes coincide with the corresponding axes of the conical coordinateframe introduced in Section V1.D and used in Sections V1.E and V1.E We also consider dimensionless Cartesian coordinates ( x , y , z ) , where x = X / b , y = Y / b and z = Z / b . In the linear approximation with respect to the small parameter E these coordinates are related to the conical coordinates 6 and 5 in a vicinity of the chosen meridianal plane by the following formulas:
Then the potential distribution in the same linear approximation in the Cartesian coordinate frame takes the form
U(x,
Y7
z ) = * ( W ,y , z ) , T ( x , y , 7.)) = * ( O ) ( X , z) *(‘)(x, Z )
C + -W;’)(x, 2
z)y2
S + -@:‘)(x, 2
1
z)y2 ,
(165)
where subscripts denote the derivatives with respect to the correspondingvariables. Since the functions @(’) and * ( I ) satisfy Eqs. (141) and (142), respectively, it is easy to show that their expansions in the vicinity of the beam optical axis are
C + -*‘‘’“x3
6
1 + - [*(I)”” 24
+ 2s*(o’’”]x4 +
. .. .
(167)
Here and everywhere following the functions with omitted arguments denote their values at the beam optical axis: W ( k )= @ ) ( O , z), and prime denotes the derivative with respect to z. Substituting Eqs. (166) and (167) into Eq. (165) and denoting by U with omitted arguments the sum U = @(‘I E * ( ’ ) , we finally come to the following
+
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
379
TABLE 4 FOCALLENGTHS OF A THREE-ELECTRODE EINZELSLIT CONICAL LENS wrm = 0.1. e = 30”. AND EQUAL HALF-SLIT WIDTHSAND INTERELECTRODE GAPS (61 = (2 = 51 = -52 = 1). AS WELL AS FOR A REFERENCE~WO-DIMENSIONAL LENS;FOCALLENGTHSARE NORMALIZEDTO THE HALF-SLITWIDTH OF THE MIDDLE ELECI’RODE [REPRINTEDFROM NUCLINSTRUM.M ~ HA .VOL. 363, M. I. YAVOR AND E. V. STRIGOVA, “FIELDDISTRIBUTION AND ELECTRON O ~ C APROPERTIES L OF ELECTROSTATIC CONICAL SLIT LENSES:’ pp. 445-450 (1995) WlTH KIND PERMISSION OF ELSEVIERSCIENCE-m, SARA BURGERHARTSTRAAT 25,1055 KV AMSTERDAM, THE NETHERLANDS].
V l *a,,
-1.0 -0.5 0.5 1 .o 1.5 2.0 3.0 5.0 7.0 9.0
fx (reference)
fx (conical)
fy (conical)
9.62 55.6 104.8 33.8 19.0 13.3 8.78 6.17 5.53 5.47
9.57 55.1 105.3 33.8 19.0 13.3 8.75 6.14 5.48 5.42
16310 23050 7710 3160 1840 1260 740 390 256 193
expression: 24
cc +6 u
I1
(x
3
- 3xy ) + 12 U”(X4 - 3 x 2 y 2 ) + . . . . 2
6s
The first group of terms in the brackets represents a potential of a two-dimensional lens with a potential distribution at the optical axis, which differs from the potential distribution in the reference lens by the contribution of the function W(’).A term proportional to (y2 - x2) describes a quadrupole field component. This component is small for small angles 8 and is maximal in a limiting case of the so-called transaxial lens [121] with 8 = 90”. Besides, an influence of this component on the focusing properties of the lens is comparably small for einzel lenses, where the derivative U’of the potential changes its sign along the beam path, and is larger in two-electrode immersion lenses where it does not. In the latter case the action (focusing or defocusing) of this component on the lens properties in the y-direction is different for accelerating and retarding lenses, provided that in both cases the particles move in the positive direction of the Z-axis.
380
M. I. YAVOR TABLE 5 FOCALLENGTHSOF A ’~WO-ELECTRODE IMMERSION SLIT CONICAL LENS WITHE = 0.1, e = 30°, AND EQUALHALF-SLITWIDTHS AND INTERELECTRODE GAP ((2 = -12 = I ) , AS WELL AS FOR A REFERENCE ’WO-DIMENSIONAL LENS; FOCAL LENGTHS ARE NORMALIZED TO HALF-SLIT WIDTHS OF THE EXIT ELECTRODE [REPRINTED FROM NUCL INSTRUM.M ~ HA .VOL. 363, M. I. YAVORAND E . V. STRIGOVA, “FIELD DISTRIBUTION AND ELECTRON OPTICAL PROPERTIESOF ELECTROSTATIC CONICAL SLIT LENSES,” pp. 445450 (1995) WITH KIND PERMISSION OF ELSEVIERSCIENCE-m, SARA BURGERHARTSTRAAT 25, 1055 KV AMSTERDAM, THE NETHERLANDS]. V l *a,,
fi.(reference)
fx (conical)
fy (conical)
-0.9 -0.7 -0.5 -0.3 0.5 1.o 1.5 2.0 3.0 5.0 7.0 9.0
1.64 5.64 17.3 68.6 64.8 24.5 15.3 11.6 8.35 6.24 5.51 5.19
1.77 6.74 25.7 193.2 41.6 18.7 12.6 9.84 7.44 5.83 5.29 5.07
10.4 25.7 50.2 105.3 -109.6 -68.1 -53.9 -46.7 -39. I -32.6 -29.6 -27.7
~~
A term proportional to ( x 3 - 3 x y 2 ) describes a hexapole field component; its presence leads to nonvanishing second-order geometrical aberrations. The remaining term describes a higher-order field component. Both these components do not affect too much the lens properties because the second and third derivatives of the field potential change their signs along the beam optical axis. A substitutionof Eq.(168) into the Lorentz equation, which describes a charged particle motion, leads after some simple transformations to trajectory equations in a conical lens. We give these equations, retaining in them only linear terms and the terms responsible for the second-order geometrical aberrations. Denoting by u the field potential normalized to the ratio of the kinetic energy K of the particle in the field-free object space to the particle charge Q :u = - Q U /K ,we obtain
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
38 1
TABLE 6 FOCUSINGAND DEFLECTINGPROPERTIES OF A WO-ELECTRODE IMMERSION CONICAL LENSWITHLONGITUDINALELFCTRODES.
-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 0.2 0.4 0.6 0.8 1.o 1.2 1.4 1.6 2.0 2.5 3.0 5.0 9.0
1.08 3.24 5.04 10.68 25.68 93.88 -393.12 - 150.44 61.92 27.84 17.68 13.00 10.36 8.68 7.56 6.76 5.64
4.84 4.32 3.36 2.92
3.36 5.48 9.28 17.00 36.48 121.68 -47 1.20 -168.52 56.36 23.36 13.80 9.48 7.12 5.64 4.64 3.92 3.00 2.28 1.84 0.96 0.40
-0.84 -2.28 -4.96 - 10.60 -25.52 -93.52 39 1.96 150.12 -62.08 -28.00 -17.80 -13.12 - 10.48 -8.80 -7.68 -6.84 -5.32 -4.88 -4.36 -3.32 -2.64
3.16 5.96 20.52 -20.72 -7.80 -5.12 -3.96 -3.36 -2.32 -2.12 -2.00 -1.88 -1.84 - 1.76 -1.74 -1.72 - 1.69 -1.68 -1.69 -1.72 -1.92
-0.024 -0.007 -0.001 0.001 0.002 0.002 0.002 0.001 -0.001 -0.003 -0.004 -0.005 -0.006 -0.007 -0.008 -0.009 -0.010 -0.012 -0.013 -0.016 -0.018
Note that the potential is assumed to be zero in the object space outside the lens. The last terms in the right-hand sides of a s . (169) and (170) describe a contribution of the quadrupole field component. Focusing properties of slit conical lenses are illustrated by Tables 4 and 5 of the focal lengths in the image space (behind the lens in the positive direction of the Zaxis) of two lenses, einzel and immersion;for acomparison the correspondingfocal lengths of the reference two-dimensional lens are also given. The focal lengths in the tables are normalized to the half-slit width b, and the potential V of the electrode is normalized with respect to the accelerating potential Qacc = -K/Q, so that a positive value of V / Qacc correspondsto an accelerationof the beam inside the lens. Note that, according to the definition of the potential, which equals zero in the object space in front of the lens, the ratio V / Qacc is different from the ratio V*/VI used for electron lenses in the tables by Harting and Read [ 1221(the relation 1). Note also that the numerical between these two ratios is V2/ V , = V / Qacc results given in the tables are somewhat dependent on the behavior of the field tails outside the lens. In our calculation these tails were cut off at the distances f20b away from the lens.
+
382
M.1. YAVOR TABLE 7 FOCUSINGAND DEFLECTING PROPERTIES OF A THREE-ELECTRODE EINZELCONICAL LENSWITH LONGITUDINAL ELECTRODES WITH 1 = b V l ~ a c c
-1.2 -1.1
-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 0.2 0.4 0.6 0.8 1 .O 1.2 1.4 1.6 1.8 2.0 2.5 3.0 4.0
fi
Z(Fi)
Z(Fo)
zc
tan (I
2.72 3.56 4.76 6.48 9.08 13.08 19.68 31.60 55.88 116.60 344.20 192.76 63.68 33.76 2 1.96 16.00 12.52 10.32 8.80 7.72 6.92 5.64 4.92 4.24
3.16 4.04 5.28 7.08 9.12 13.84 20.64 32.80 57.52 119.08 348.96 190.16 61.92 32.32 20.64 14.76 11.28 9.08 7.52 6.40 5.56 4.08 3.16 1.96
-2.80 -3.60 -4.72 -6.36 -8.80 -12.64 - 19.08 -30.68 -54.52 - 1 14.32 -339.60 -195.32 -65.28 -34.92 -22.88 -16.72 -13.12 -10.76 -9.12 -7.96 -7.04 -5.52 -4.56 -3.36
1.24 1.48 1.84 2.36 3.16 4.52 7.36 16.52 - 182.04 - 16.00 -9.00 -4.16 -3.68 -3.40 -3.32 -3.28 -3.32 -3.40 -3.52 -3.68 -3.88 -4.44 -5.28 -7.88
-0.01 I -0.008 -0.006 -0.004 -0.002 -0.001 -0.001 0.000 0.000 0.000 0.000 0.000
-0.001 -0.002 -0.002 -0.003 -0.004 -0.004 -0.005 -0.005 -0.006 -0.006 -0.007 -0.006
Table 4 illustrates focal properties of a three-electrode einzel slit conical lens (a difference between the focal lengths of a reference lens, given in Tables 4 and 5 , as compared with the data reported in [ 1221, is due to an absence in our model of the lens of any outer field shields and thus a more diffuse distribution of the field potential along the lens optical axis). In accordance with theoretical predictions, a quadrupole field component practically does not influence the focusing properties of the lens; that is, focal lengths in the x-direction are very close to the corresponding lengths for the reference lens, and there is only a very weak focusing in the y -direction. A situation is, however, different in the case of a two-electrode immersion lens, whose focusing properties are illustrated by Table 5. It is seen that for an accelerating lens a quadrupole field component leads to a noticeable additional
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
383
TABLE 8 FOCUSING AND DEFLECTING PROPERTIES OF A THREE-ELECTRODE EINZELCONICAL LENSWITH LONGITUDINAL ELECTRODESWITH 1 = 2b.
-1.2 -1.1
-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 0.2 0.4 0.6 0.8 1 .o
I .2 1.4 1.6 1.8 2.0 2.5 3.0 4.0
0.64 0.84 1.20 1.76 2.60 3.96 6.32 10.6 19.56 42.12 124.80 92.60 32.12 18.00 12.36 9.48 7.84 6.76 6.08 5.60 5.28 4.84 4.84 5.68
0.80 1.20 1.64 2.24 3.16 4.60 7.08 11.56 20.84 44.04 128.40 90.08 30.28 16.32 10.72 7.80 6.04 4.84 4.00 3.28 2.72 1.60 0.60 -1.68
-0.92 -1.16 -1.52 - 1.96 -2.72 -3.96 -6.12 - 10.20 -18.76 -40.56 -121.56 -95.00 -33.60 -19.04 -13.08 -9.48 -8.12 -6.84 -5.92 -5.24 -4.68 -3.64 -2.84 -1.24
0.48 0.72 0.92 1.20 1.56 2.12 3.16 5.40 13.40 -61.24 -1 1.28 -4.24 -3.92 -3.92 -4.08 -4.36 -4.80 -5.32 -6.00 -6.80 -7.84 -12.00 -21.8 80.72
-0.067 -0.047 -0.031 -0.020 -0.013 -0.007 -0.004 -0.002 -0.001 0.OOO 0.OOO -0.001 -0.002 -0.003 -0.004 -0.005 -0.006 -0.006 -0.006 -0.006 -0.006 -0.005 -0.004 0.001
focusing in the x-direction and to a defocusing in the y-direction. In the case of a retarding lens, the situation is just the opposite. Consider now a conical lens with longitudinal electrodes. All the conclusions just made for slit conical lenses, regarding their focusing properties as well as a difference between these properties and the properties of two-dimensional lenses, also hold for lenses with longitudinal electrodes. For this reason we do not give here a comparison of corresponding numerical values. Instead, based on a sample geometry of a lens with longitudinal electrodes, we illustrate an action of the field strength component normal to the beam optical axis, which deflects a beam from the initial conical surface. To calculate this action, we used paraxial trajectory equations obtained in [ 1231. In Tables 6 8 are shown the results of a calculation of cardinal elements, a center of the beam deflection, and the angle of the beam deflection for some sample lenses
3 84
M. I. YAVOR
with 8 = 30" and c = 0.25. In Table 6 parameters are given of a two-electrode immersion lens. In Tables 7 and 8 parameters of three-electrode einzel lenses are given with the length 1 of a middle electrode that equals 1 = b = 0.25r0 and 1 = 2b = 0.5r0,respectively. In the tables f is a focal length, Z ( F) is a coordinate of the focal point, the subscript i correspondsto the image space and o to the object space, Z , is a position of the center of deflection, a! is the beam deflection angle (negative 01 corresponds to a deflection toward the axis of rotational symmetry of the lens). All the linear dimensions are measured from the geometrical center of the lens, that is, from the point Z = 0 for the two-electrode lens and from the point corresponding to the center of the middle electrode for the three-electrode lens. These dimensions are referred to half the interelectrode gap b. A potential V is the potential at the exit electrode for the two-electrode lens (the potential of the entrance electrode a5 well as the potential of the object space are assumed to be zero), or the potential of the middle electrode for the three-electrode lens (the potentials of the outer electrodes are assumed to be zero). The particles were supposed to move in the positive direction of the Z-axis. The angle of the beam deflection is rather small in all the calculated ranges of the lens excitations. As a rule it is negative; that is, the beam is deflected toward the axis of rotational symmetry of the lens.
VII. CONCLUSION In this chapter we considered a wide variety of problems of electron- and ionoptical systems with narrow gaps, based on a perturbation theory approach. The main advantage of this approach was shown to be the possibility of obtaining simple analyticalformulas that describe electromagneticfield distributionsor charged particle beam transformations in these systems. This possibility considerably simplifies an analysis of optical properties of such systems, and, finally, a design of corresponding devices. The author is sure that the potential of perturbation methods is far from being exhausted by the considerations given in the present chapter. An extension of these methods in charged particle optics can be developed both in the directions mentioned in the chapter (investigation of tolerance problems, application of the fringing field integral method, and especially calculation of various rotationally symmetric systems for an angle-resolvedanalysisof hollow beams) and in different fields like investigation of devices based on dynamic fields. The author hopes that the methods and results described in the previous sections will be useful not only from a direct practical point of view but also as examples that encourage specialists in charged particle optics to more widely apply the perturbation approach in their studies.
CHARGED PARTICLE OPTICS OF SYSTEMS WITH NARROW GAPS
385
ACKNOWLEDGMENTS
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[ 1111 [112] [113] [ I 141
INDEX A
calculation of particle trajectories in a poloidal analyzer, 352-361 conical lenses, 293-294 conical mirror analyzers, 293 correct applications, 294-295 electromagnetic field structures suitable for, 283-294 electrostatic field in sector field analyzers with split shielding plates, 336-343 electrostatic field of conical lens with longitudinal electrodes, 367-37 1 electrostatic field of conical mirror analyzers, 363-367 electrostatic field of poloidal analyzers, 348-352 electrostatic field of slit conical lenses, 372-374 elimination of beam deflection in conical lenses, 374-378 focusing of hollow beams by conical lenses, 378-384 focusing properties of a poloidal analyzer, 361-363 fringing field integral method, 286-289 magnetic analyzers, 302-306 Matsuda plates, 290 Matsuda plates and effects on sector field analyzers and Wien filters, 344346 parasitic beam distortions in, 31 1-317 poloidal analyzers, 291-293 sector field analyzers, 289-29 1, 295-301
Adapted frames, 110-1 11 Affine monotonic transformations, 70-7 1 Analytic delta pulsed beam, 38-40 Analytic signal representation, 15-16 Analytic time-dependent spatial spectrum, 18 Angle-resolved analyzers, 29 1-294 Aperture field, 37 Arbitrary monotonic transformations, 70-7 1 Axial energy, 37
B Bertein method, 284-285 Bessel beams, 30 Blumng, 79 Bullets, 30-3 1
C Cartan’s theorem, 113 Cartographic generalization, 68 Cayley’s natural districts, 99-102 Charged particle analyzers, parasitic beam: distortions in, 311-3 17 Charged particle optics, perturbation theory and: angle-resolved analyzers, 29 1-294 background information, 278-283 beam transport through gaps of multiple magnetic prisms, 318-328 beam transport through gaps of quadrupole multiplets, 329-336 389
390
INDEX
Charged particle optics (cont.) shielding sector field analyzer with split shieldingplates, 346-348 Wien filters, 306-3 11 Cliff curves, 75,76, 135-137 Collimation distance, 5, 17-18, 24 frequency-independent, 29 well-collimated condition, 28-29 Complex source pulsed beams (CSPB), 3 1.40 Conical lenses, 293-294 electrostatic field of, with longitudinal electrodes, 367-37 1 electrostatic field of slit, 372-374 elimination of beam deflection, 374-378 focusing of hollow charged particle beams by, 378-384 Conical mirror analyzers, 293 electrostatic field o f , 363-367 Conjugated directions, 88,97, 112 Contours, visual, 67, 138-140 Convedconcave boundary, 137 Correction coils, 3 17 Correctors, quadrupole, 3 17 Creep equation, 85, 102-106 Creep field: congruence of level curves and, 85-86 Eikonal equation, 96-98 nature of, 94-98 slope and tilt, 95 Creep line, 86 Critical points, Morse, 72-73 Curvature (wavepacket), transverse amplitude distribution and, 34-35 Curvature matrix, 34 Curves: See also Fall curves; Level curves; Topographic curves
cliff, 75,76, 135-137 De Saint-Venant’s, 125-129, 135 parabolic, 88, 120-121 Cusp points, 139
D Dales, 100-101 De Saint-Venant’s curves, 125-129, 135 Deformation of the gradient, 88 Delaunay triangulations, 142 Delta function, 25 Differential invariants: complete irreducible sets of, 90-91 of first order, 86-87 local shape of level curves, 93-94 local surface shapes, 91-93 of second order, 87-94 Differential structure of relief. See Relief, differential structure of Dyadic Green’s function (2D), for hard magnetic wall circulators: background information, 240-24 1 electric field within the annuli, 258-260 four-port symmetric circulator, 247-250 reverse s-parameters, 255-257 six-port symmetric circulator, 250-255 three-port symmetric circulator, 241-246 Dyadic Green’s function (2D), for homogeneous ferrite: background information, 228-229 construction of, 230-23 1 electromagnetic fields and constraints, 229-230 factor f ,231-232
INDEX
Dyadic Green’s function (2D), for penetrable walls: background information, 208-209 elements of, derived, 21 1-213 formulation, 209-2 10 Dyadic Green’s function (3D), for hard magnetic wall circulators: background information, 260 three-port symmetric circulator, 260-273 Dyadic Green’s function (3D), for homogeneous ferrite: background information, 232-233 construction of, 238-240 electromagnetic fields and constraints, 233-238 Dyadic Green’s function (3D), mode-matching for simple material: background information, 196197 electromagneticfields, mode-matching theory, and nonsourcelsource equations, 197-205 implicit construction, 205-207 Dyadic Green’s function (3D), mode-matching for vertically layered material: background information, 153-156 eigenvalues for TE, modes, 173-176 eigenvalues for TM, modes, 170-173 eigenvectors for TE, modes, 179-182 eigenvectors for TM, modes, 177-179 electromagnetic fields for, 182-1 83 equations combined to form, 193-195
39 1
implicit construction, 182-1 95 mode-matching theory at interfaces, 184-185 nonsource governing equations, 185-1 89 self-adjoint operators, 156-182 source governing equations, 189-1 93 TE, operator properties, 165-170 TM, operator properties, 157-165 Dyadic Green’s function (3D), for penetrable walls: background information, 213-2 14 explicit construction, 223-228 formulation, 214-223
E Edges, 141 Effective field boundary, 287 beam transport through gaps of multiple magnetic prisms, 318,321-325 beam transport through gaps of quadrupole multiplets, 330-336 Effective trajectory, 321-324 beam transport through gaps of multiple magnetic prisms, 321-324 beam transport through gaps of quadrupole multiplets, 330-336 Eigen wavepacket solutions, 3 1 Eigenvalues: for TE, modes, 173-176 for TM, modes, 170-173 Eigenvectors: for TE, modes, 179-182 for TM, modes, 177-179 Eikonal equation, 96-98
392 Electromagnetic field structures, perturbation methods and suitable: angle-resolved analyzers, 29 1-294 Bertein method, 284-285 conical lenses, 293-294 conical mirror analyzers, 293 coordinate frame variation method, 285 direct substitution of Taylor expansion, 286 fringing field integral method, 286-289 integral equation method, 285 main versus fringing fields, 283-284 Matsuda plates, 290 poloidal analyzers, 29 1-293 sector field analyzers, 289-291 weakly distorted electrode and pole: surfaces, 284-286 Wien filters, 306-3 11 Electrostatic field: of conical lens with longitudinal electrodes, 367-37 1 of conical mirror analyzers, 363-367 of poloidal analyzers, 348-352 in sector field analyzers with split shielding plates, 336-343 of slit conical lenses, 372-374 Envelopes, 67, 138-140 Evanescent spectra, 10, 19-20 Extrinsic geometry, 70
INDEX
False maximum curve, 74 False minimum curve, 74 Far-field diffraction, 30,38 Field integrals, 327-328 Focus wave mode (FWM), 30 Focusing of hollow charged particle beams: by conical lenses, 378-384 Focusing properties of a poloidal analyzer, 361-363 Fourier transform: analytic, 23 inverse, 15,20,52 one-sided inverse, 15 Fraunhofer zone, 18,24 Frequency-domain (FD): evanescent spectrum and, 20 interpretation, 29 isodiffracting versus isowidth apertures, 4 1-44 phase-space pulsed beams and, 46-5 1 problems with using, 3 Fresnel distance, 5, 17-18,24 frequency-independent, 29 Fringing field integral method, 286-289 beam transport through gaps of multiple magnetic prisms, 318-328 beam transport through gaps of quadrupole multiplets, 329-3 36 field integrals, 327-328 Fuzzy derivatives, 83-84
F Faces, 141 Fall curves, 75,86 consistent labeling of, 104-106 regular, 104 singular, 104, 110, 129
G Gamma corrections, 7 1 Garbor kernel, 47 Gaussian beams, 29 Gaussian window example:
INDEX
frequency-domain formulation, 49-5 1 time domain formulation, 55-57 Gauss-Weingartenequations of classical surface theory, 110-1 13 Genericity, 82 Geometrical meaning of differential relations, 110-1 14 Global property, 8 1 Global structure of relief. See Relief, global structure of Graphical surfaces, 67 Green’s function representation: See also Dyadic Green’s function, 2D; Dyadic Green’s function, 3D: time-domain representation of radiation, 16-18,45,48 time-harmonic radiation, 3,4, 7-8
H Hamilton-Jacobi equation, 96 Heaviside function, 26 Helmholtz equations, 2 16 Hessian determinant, 87-88 Hilbert transform, 16,23,28,36 Hildebrand’s depth flow, 69 Hills, 66, 100 Hilltops, 91-92 Huygens’s representation. See Green’s function representation
I Image intensity, 68 Images: See also Relief, differential structure of defined by gradient, 7 1 local structure of, 81
393
posterization, 68-69 scalar fields in 2D, 66-70 use of term, 66,67 Images, qualitative structure of: applications, 7 1-72 inloop and outloop structure, 73-75 Morse critical points, 72-73 topographic curves, 75-76 Immits, 72,92-93 Indicatrices of Dupin, 92-93, 112 Inflections, 93-94 Initial field distribution, 24-26 Inloop structure, 73-75 Integral equation method, 285 Intrinsic description of scalar field, 107-1 10 Intrinsic geometry, 70 Isoaxial astigmatic pulsed beam, 35-36 Isobaths, 68 Isoclines, 97 Isodiffracting versus isowidth apertures, 41-44 Isohypses, 68,75, 143-144 Isophotes, 68
K Kirchhoff representation: See Green’s function representation
L Landscapes, geographical: cartographic generalization, 68 inverted, 72 Morse critical points, 72-73 scalar fields in 2D, 66-70 use of term, 66,67 Laplacean determinant, 87,88 Lattices, regular, 141-142 Legendre transform, 21-22,27
394 Level curves, 68 creep field and congruence of, 85-86 envelopes of, 138-140 inloop and outloop structure, 73-75 local shape of, 93-94 nesting of, 72 Line element, 110 Local jet space, 8 1 , 8 4 8 5 Local spectral distribution: example, 57-59 frequency-domain formulation, 46-47 time domain formulation, 51-53 Local structure, 81 Local surface shapes, differential invariants and, 91-93 Longitudinal amplitude, 34 LorentziadRayleigh pulse, 25 Lune, 73
M Magnet analyzers, magnetostatic field and charged particle trajectories in sector, 302-306 Magnetic prisms, beam transport through gaps of multiple, 3 18-328 Mainardi-Codazzi equations of classical surface theory, 113-1 14 Maslov index, 13,23 Matsuda plates, 290 See also Split shielding plates on sector field analyzers and Wien filters, 344346 Maxwell’s natural districts, 99-102 Monotonic transformations, 70-7 1 Morse critical points, 72-73, 118-120,144145 Multilocality, 8 1
lNDEX
Multiplets, beam transport through gaps of quadrupole, 329-336
N Natural districts, 99-102 Nonsource governing equations, 185-1 89, 197-205
0 Observation constraint, 54 Observation, nature of, 77-8 1 Outloop structure, 73-75 Overhangs, 138
P Parabolic curves, 88, 120-121 Parasitic beam distortions in charged particle analyzers, 3 1 1 design considerations to correct, 3 15-3 17 types of electrodehagnet pole distortions, 3 13-3 I5 types of first-order, 3 12-3 13 Paraxial delay, 34 Perturbation theory. See Charged particle optics, perturbation theory and Pfaffian differential equation, 66 Phase-space coordinates, 46 Phase-space pulsed beams. See Pulsed beams, phase-space Plane-wave representations, 3,9-11, 45,48 spectral localization, 2 1-22 time-dependent, 18-22 Plateau curves, 75,76 Poloidal analyzers, 29 1-293 calculation of particle trajectories in, 352-361 electrostatic field of, 348-352 focusing properties of, 361-363 Posterization, 68-69
INDEX
Propagating and evanescent spectra, 10,19-20 Propagation lines, 35 Pulsed beams (PBs): analytic delta, 38-40 aperture field, 37 applications, 5-6,3 1 axial energy, 37 complex source (CSPB), 31,40 derived from differential wave equation, 3 1 globally exact, 40 isoaxial astigmatic, 35-36 properties and interpretation, 33-40 real field, 36 relation to time-harmonic Gaussian beams, 40-41 solution of time-dependent wave equation, 32-33 wavepacket curvature and transverse amplitude distribution, 34-35 Pulsed beams, phase-space, 29 applications, 6-7 frequency-domain formulation, 46-5 1 Gaussian window example, 49-5 1, 55-57 local spectrum approach, 46-47, 5 1-53,57-59 radiating field, 47-49,53-54 time-dependent radiation from extended apertures and, 44-60 time domain formulation, 5 1-59
Q
Quadrupole correctors, 3 17 Quadrupole multiplets, beam transport through gaps of, 329-336
395
Qualitative structure of images. See Images, qualitative structure of
R Radiating field: frequency-domain formulation, 47-49 time domain formulation, 53-54 Radiation constraint, 54,58 Ray representation: time-domain representation of radiation, 22-23 time-harmonic radiation, 11-13 Rayleigh limit, 38 Real field, 36 Reference particle, 354,358-359 Regular fall curves, 104 Relief: applications, 66-67 Hildebrand’s depth flow, 69 use of term, 66 Relief, differential structure of creep field and congruence of level curves, 85-86 creep field, nature of, 94-98 differential invariants of second order, 86-94 local jet, 8 4 8 5 stable, 83-84 Relief, global structure of: cliff curves, 135-137 convexlconcave boundary, 137 creep equation, 102- 106 De Saint-Venant’scurves, 125-129,135 Gauss-Weingarten equations of classical surface theory, 110-1 13 geometrical meaning of differential relations, 110-1 14
396 Relief, global structure of (con?.) intrinsic description, 107-1 10 line element, 110 Mainardi-Codazzi equations of classical surface theory, 113-1 14 Morse critical points, 118-120 natural districts, 99-102 one-dimensional, 98-99 parabolic curves, 120-121 ridges and ruts, 121-129 river transport, 130-135 topographic curves, 1 14-1 18 vertex loci, 138 Relief, structure of: contours, 138-140 discrete representation, 140-147 genericity and structural stability, 81-82 images defined by gradient, 7 1 intrinsic and extrinsic geometry, 70 local structure, 81 monotonic transformations, 70-7 1 Morse critical points, 72-73, 118-120, 144-145 nature of observationhesolution, 77-8 1 qualitative structure of images, 7 1-76 ridges and ruts, 66,75,76, 121-129, 145-147 scalar fields in 2D, 66-70 triangulations, 141-144 Resolution, nature of, 77-8 1 Retarded fields, 4, 16 Ridges, 66,75,76, 121-129, 145-147 Ruffles, 121 Ruts, 75,76,121-129, 145-147
S Saddle point integration, 50,59-60 Saddle points, 72, 144-145
INDEX
Scalar fields in 2D: applications, 66,67 cartographic generalization, 68 defined, 70 Hildebrand’s depth flow, 69 posterization, 68-69 Scale space, 80 Schwarz-Christoffel transformation, 373-374 Sector field analyzers, 289-291 dimensionless coordinates by scaling, 296-297 electrostatic field and charged particle trajectories in toroidal, 295-301 electrostatic field in, with split shielding plates, 336-343 magnetostatic field and charged particle trajectories in magnet, 302-306 Matsuda plates and effects on, 344-346 shielding, with split shielding plates, 346-348 Self-adjoint operators: eigenvalues for TE, modes, 173-176 eigenvalues for TM, modes, 170-173 eigenvectors for TE, modes, 179-182 eigenvectors for TM, modes, 177-179 TE, operator properties, 165-170 TM, operator properties, 157-165 Self-conjugated, 88, 112 Separatrices, 120 Serret-Frenet frame, 110 Singular fall curves, 104,110, 129 Slant-stack transform (SST), 4, 19 local, 6,52-53 Slope angle, 95 Slope line, 86
INDEX
Slope squared function, 85,86,87, 97,112 Source governing equations, 189-193,197-205 Spatial spectrum, time-dependent, 18 Spectral formulations, 17 Spectral theory of transients (STT), 4 Split shielding plates: See also Matsuda plates electrostatic field in sector field analyzers with, 336-343 shielding sector field analyzer with, 346-348 Spring points, 137 Stationary delay point, 21,24,27,59, 60 Structural stability, 82 Summits, 72,92-93 Support function, 115, 117-1 18
T TE, operator properties, 165-170 eigenvalues for, 173-176 eigenvectors for, 179-182 Thresholding, 68 Tilt, 86-87, 95 Time-dependent plane-wave: representation, 18-22 spectrum example, 26-27 well-collimated condition, 28-29 Time-dependent radiation: patterns, 23-24, 27-28 phase-space pulsed beam analysis for, 44-60 Time-dependent spatial spectrum, 18 Time-dependent wave equation, pulsed beam solutions of, 32-33 Time-domain (TD): diffraction limit, 38 diffraction tomography, 5 Fresnekollimation distance, 5, 17-1 8,24
397
phase-space pulsed beams and, 5 1-59 problems with using, 3 Time-domain representation of radiation, 15-24 analytic signal representation, 15-16 Green’s function representation, 16-18 radiation patterns, 23-24 Ray representation, 22-23 time-dependent plane-wave representation, 18-22 Time-harmonic Gaussian beams, 40-41 Time-harmonic radiation: Green’s function representation, 7-8 plane-wave representation, 9-1 1 propagating and evanescent spectra, 10,19-20 radiation patterns, 14 Ray representation, 11-13 Time-of-flight analyzers, 292 T-junctions, 139 TM,operator properties, 157-165 eigenvalues for, 170-173 eigenvectors for, 177-179 Topographic curves, 75-76 congruence of fall tangents and second caustic surface, 114-1 15 global relief and, 114-1 18 support function, 115, 117-1 18 Toroidal analyzers. See Sector field analyzers Transverse amplitude distribution, wavepacket curvature and, 34-35 Triangulations, 141-144
398
INDEX
U
W
Ultra wideband signals, space-time representation of frequency-domain interpretation, 29 initial field distribution, 24-26 overview, 3-7 phase-space pulsed beam analysis for time-dependent radiation, 44-60 radiation patterns, 27-28 time-dependent plane-wave spectrum example, 2 6 2 7 time-domain representation of radiation, 15-24 time-harmonic radiation, 7-14 wavepackets and pulsed beams, 30-44 well-collimated condition, 28-29 Umbilical, 93 Undulations, 93
V
Wavepackets: See also Pulsed beams (PBs) applications, 30 Bessel beams, 30 bullets, 30-3 1 curvature and transverse amplitude distribution, 34-35 eigen solutions, 3 1 equation, 33 forward versus backward propagating fields, 30 isodiffracting versus isowidth apertures, 41-44 Weingarten equations of classical surface theory, 1 10-1 13 Well-collimated condition, 28-29 Wien filters: electromagnetic field and charged particle trajectories for, 306-3 1 1 Matsuda plates and effects on, 344-346 Winged edge representation, 145
Valley bottoms, 9 1-92 Vertices, 93,94 Visual contour, 67
Zero crossing, 88
Z
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