Inverse Scattering Problems in Optics

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 Holder 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 x 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.