Inverse Scattering Problems in Optics

When, in the spring of 1979, H.P. Baltes presented me with the precursor of this vo 1 ume, the book on "Inverse Source Problems in Opti cs", I expressed my gratitude in a short note, 11hich in translation, reads: "Dear Dr. Ba ltes, the mere titl e of your unexpected gift evokes memori es of a period, which, in the terminology of your own contribution, would be described as the Stone Age of the Inverse Problem. Those were pleasant times. Walter Kohn and I lived in a cave by ourselves, drew pictures on the walls, and nobody seemed to care. Now, however, Inversion has become an Industry, which I contemplate with as much bewilderment as a surviving Tasmanian aborigine gazing at a modern oil refinery with its towers, its fl ares, and the confus i ng maze of its tubes." The present volume makes me feel even more aboriginal - impossible for me to fathom its content. What I can point out, however, is one of the forgotten origins of the Inverse Scattering Problem of Quantum Mechanics: Werner Heisenberg's "S-Matrix Theory" of 1943. This grandiose scheme had the purpose of eliminating the notion of the Hamiltonian in favour of the scattering operator. If Successful, it would have done away once and for all with any kind of inverse problem.

1. Progress in Inverse Optical Problems.- 1.1 Inverse Problems in Optics and Elsewhere.- 1.2 Survey of Recent Results.- 1.2.1 Phase, Uniqueness, and Estimation.- 1.2.2 Radiometry and Coherence.- 1.2.3 A Moment Problem.- 1.3 Construction of Lambertian Scatterers.- 1.3.1 Lambertian Source Correlation.- 1.3.2 Random Scatterer Models.- 1.4 Organization of this Volume.- References.- 2. The Inverse Scattering Problem in Structural Determinations.- 2.1 Philosophical Background.- 2.2 The Direct Scattering Problem.- 2.2.1 Description of the Medium.- 2.2.2 The Scattered Fields.- 2.2.3 Expression for the Intensity.- 2.3 Analytic Description and Properties of Scattered Fields.- 2.3.1 Entire Functions of the Exponential Type.- 2.3.2 Distributions of Zeros for Functions of Class E.- 2.3.3 Encoding of Information by Zeros.- 2.4 The Deterministic Problem.- 2.4.1 Limitations of Measurements.- 2.4.2 The Phase Problem.- 2.4.3 Solutions to the Zero Problem.- 2.4.4 Zero Location.- 2.4.5 Extensions of the Method.- 2.5 The Statistical Problem.- 2.5.1 Overall Characterization of the Medium.- 2.5.2 Analytical Properties of Overall Descriptors.- 2.5.3 Determination of Overall Descriptors from Finite Records.- 2.6 Conclusions.- References.- 3. Photon-Counting Statistics of Optical Scintillation.- 3.1 Introductory Remarks.- 3.2 Photon-Counting Statistics.- 3.2.1 Single-Interval Statistics.- 3.2.2 Photon-Correlation Spectroscopy.- 3.2.3 Instrumental Effects.- 3.2.4 Noise and Statistical Accuracy.- 3.3 Scattering Theory.- 3.3.1 Mechanisms and Theories for Strong Scattering.- 3.3.2 The “Discrete-Scatterer” Model.- 3.3.3 K Distributions.- 3.3.4 Correlation Functions.- 3.4 Limit Distributions in the Random Walk Problem.- 3.4.1 The Gaussian Limit.- 3.4.2 Negative Binomial Number Fluctuations.- 3.4.3 A Population Model.- 3.5 Experiments.- 3.5.1 Dynamic Scattering by Nematic Liquid Crystals.- 3.5.2 Hot-Air Phase Screen.- 3.5.3 Extended Atmospheric Turbulence.- 3.5.4 Other Experiments.- 3.6 Concluding Remarks.- References.- 4. Microscopic Models of Photodetection.- 4.1 Photoelectron and Photon Statistics.- 4.1.1 Definition of the Problem.- 4.1.2 Ideal and Real Detection.- 4.2 Models for Ideal Detection ― a Review.- 4.2.1 Mandel’s Formula.- 4.2.2 Perturbation Approach.- 4.2.3 Field Attenuation.- 4.2.4 Inversion Problem.- 4.3 Open-System Detection Scheme.- 4.3.1 Detector Model.- 4.3.2 Relation Between Atomic and Field Dynamics.- Field Dynamics.- Dynamics of the Atomic Moments.- 4.3.3 Photocounting Probability.- 4.4 Disturbing Effects.- 4.4.1 Dark Currents and Noise.- Photodetectors.- Noise in Photoconductive Detectors.- Noise in Photomultipliers.- PMT Statistics.- 4.4.2 Dead Time Effects.- 4.4.3 Coherence and Sampling Effects.- Time Effects.- Spatial Effects.- Sampling Effects.- Other Counting Experiments.- 4.5 Temperature Effects in Photodetection.- 4.5.1 Langevin Equations of Motion.- The Field Equation.- Connection Between Atomic and Field Dynamics.- 4.5.2 Photocounting Probability.- 4.5.3 Applications.- Numerical Examples and Discussion.- 4.6 Summary of Statistical Methods.- 4.6.1 Random Variables.- Examples.- 4.6.2 Stochastic Processes.- 4.6.3 The Statistical Description of the Radiation Field.- 4.7 The Statistical Description of Open Systems.- 4.7.1 Equation of Motion of the Reduced Density Matrix.- 4.7.2 Langevin Equations.- References.- 5. The Stability of Inverse Problems.- 5.1 Ill-Posedness in Inverse Problems.- 5.1.1 Well-Posed and Ill-Posed Problems.- 5.1.2 Ill-Posedness and Numerical Instability.- 5.1.3 General Formulation of Linear Inverse Problems.- 5.1.4 Prior Knowledge as a Remedy to Ill-Posedness.- 5.1.5 Hölder and Logarithmic Continuity.- 5.2 Regularization Theory.- 5.2.1 An Outline of Miller’s Theory.- 5.2.2 Eigenfunction Expansions and Numerical Filtering.- 5.2.3 Tikhonov Regularization Method.- 5.2.4 Stability Estimates.- 5.3 Optimum Filtering.- 5.3.1 Random Variables in a Hilbert Space.- 5.3.2 Best Linear Estimates.- 5.3.3 Mean-Square Errors.- 5.3.4 Comparison with Miller’s Regularization Method.- 5.4 Linear Inverse Problems in Optics.- 5.4.1 Inverse Problems in Fourier Optics.- Prolate Spheroidal Wave Functions (PSWF).- Perfect Lowpass Filter.- Bandwidth Extrapolation.- 5.4.2 Inverse Diffraction.- Inverse Diffraction from Plane to Plane.- Inverse Diffraction for Cylindrical Waves.- Inverse Diffraction from Far-Field Data.- 5.4.3 An Inverse Scattering Problem for Perfectly Conducting Bodies.- 5.4.4 Inverse Scattering Problems in the Born Approximation.- 5.4.5 Object Reconstruction from Projections and Abel Equation.- 5.4.6 Concluding Remarks and Open Problems.- References.- 6. Combustion Diagnostics by Multiangular Absorption.- 6.1 Absorption in Homogeneous Media.- 6.2 Multiangular Scanning.- 6.2.1 Basic Equation.- 6.2.2 Two-Dimensional Fourier Transform.- 6.2.3 Linear Superposition Techniques.- 6.2.4 Algebraic Reconstruction Techniques (ART).- 6.2.5 Applications and Results.- 6.3 The Reconstruction Procedure.- 6.3.1 Reconstruction Errors.- 6.3.2 An Observation of the Oversampling Requirement of Reconstruction.- 6.3.3 Number of Measurements M × N in Combustion Application.- 6.3.4 The Convolution Algorithm.- 6.3.5 Simulated Test Functions and Results.- 6.3.6 Algebraic Reconstruction.- 6.3.7 Benefits of Additional Digital Signal Processing.- 6.3.8 Conclusion.- 6.4 Experimental Aspects.- References.- 7. Polarization Utilization in Electromagnetic Inverse Scattering.- 7.1 Scope.- 7.1.1 Definitions of the Electromagnetic Inverse Problem.- 7.1.2 Definitions of Exact, Unique, and Approximate Methods.- 7.1.3 Incompleteness and A Priori Knowledge, Data Limitedness and Self-Consistency.- 7.2 The Vector Diffraction Integral, Its Far-Field Approximations, and Some Tauberian Relations.- 7.2.1 Basic Scattering Phenomena, Nomenclature, and Radar Definitions.- 7.2.2 The Stratton-Chu Vector Diffraction Integral and the Vector-Current Integral Equations.- 7.2.3 Far Scattered Fields in the Physical Optics Limit and Their Vector Corrections.- 7.2.4 Time-Domain Target Modeling: Utilization of Some Tauberian Theorems.- 7.3 The Radar Scattering and Target Polarization Matrices.- 7.3.1 Basic Electromagnetic Polarization Descriptors.- 7.3.2 Radar Scattering Matrices and Radar Measurables.- 7.3.3 Kennaugh’s Optimum Polarization Pairs.- 7.3.4 Radar Target and Clutter Characteristic Operators.- Single Radar Target Classification.- The Time-Varying Distributed Target.- Synthetic Aperture Imagery.- 7.4 Inverse Scattering Theories in Various Electromagnetic Frequency Regimes.- 7.4.1 The Low Frequency Regime: Rayleigh-Gans Theory.- 7.4.2 The Resonant Frequency Regime: Natural Frequency Expansion.- 7.4.3 Physical Optics Far-Field Inverse Scattering Theories: Broad-Band Approach.- Fourier Transform Method of Physical Optics.- POFFIS in Time, Frequency, and Projection Domain.- The Limited Aperture Problem.- Polarizational Correction.- 7.4.4 Geometrical Optics Inverse Scattering Asymptotic Theories.- GOIS and the Minkowski Problem.- Vector Extension of GO Equivalent Curvature Inverse Method.- Scattering Center Discrimination: Kell’s Monostatic-Bistatic Equivalence Theorem.- 7.5 Vector Holography and Polarization Utilization.- 7.5.1 Vector Wavefront Reconstruction and Interferometry.- 7.5.2 Polarization Dependence in Millimeter and Microwave Holography.- 7.5.3 The Postulate of Inverse Boundary Conditions.- 7.5.4 Near-Field Approach to Vector Inverse Scattering.- 7.6 Conclusions.- 7.6.1 Summary.- 7.6.2 Unresolved Vector Inverse Problems.- 7.6.3 Limitations and Omissions.- 7.6.4 Recommendations.- References.- Additional References with Titles.