Inverse Source Problems in Optics

H. P. Baltes We begin the introductory chapter with a general definition of the inverse optical problem. Next, we discuss the role of prior knowledge and the questions of uniqueness and stability. We then review the various specific inverse problems in optics as well as the contents of Chapters 2 to 6. Finally, we summarize the notation in co­ herence theory. 1. 1 Direct and Inverse Problems in Optical Physics The "direct" or "normal" problem in optical physics is to :Jredict the emission or propagation of radiation on the basis of a known constitution of sources or scat­ terers. The "inverse" or "indirect" problem is to deduce features of sources or scatterers from the detection of radiation. An intuitive solution of the optical inverse problem is commonplace: we infer the size, shape, surface texture, and ma­ terial of objects from their scattering and absorption of light as detected by our eyes. Intuition has to give way to mathematical reconstruction as soon as we wish to analyze optical data beyond their visual appearance. Examples are the extrapola­ tion and deblurring of optical images, the reconstruction from intuitively inacces­ sible data such as defocused images and interferograms, or the search for information that is "lost" in the detection process such as the phase. Following CHADAN and SABATIER [1. 1], a general definition of inverse optical problems can be attempted as follows. We describe the sources and scatterers by the set (1.

1. Introduction.- 1.1 Direct and Inverse Problems in Optical Physics.- 1.2 Role of Prior Knowledge.- 1.3 Survey of Specific Inverse Problems.- 1.4 Notati on i n Coherence Theory.- References.- 2. The Phase Reconstruction Problem for Wave Amplitudes and Coherence Functions.- 2.1 Phase Reconstruction for Wave Ampl i tudes.- 2.1.1 Relevance of the Phase Problem for Object Structure Determi nation.- 2.1.2 Derivation of the Basic Equations Governing the Phase Probl em.- 2.1.3 General Considerations on the Phase Problem.- 2.1.4 Greenaway’s Proposal for Phase Recovery from a Single Intensity Distribution.- 2.1.5 The Method of Half-Plane Apertures for Semi-Weak Objects.- 2.1.6 The Logarithmic Hilbert Transform: Methods for Circumventing Complications Due to Zeros.- 2.1.7 Phase Retrieval for Strong Objects from Two Defocused Images.- 2.1.8 Phase Retrieval from the Intensity Distributions in Exit Pupil and Image Plane.- 2.1.9 Phase Retrieval from Two Defocused Images for Semi-Weak Objects.- 2.2 Phase Reconstruction for Coherence Functions.- 2.2.1 Phase Determination of Optical Coherence Functions.- 2.2.2 Determination of the Phase of the Spatial Coherence Function with an Incoherent Reference Point Source.- 2.2.3 Determination of the Phase of the Spatial Coherence Function with an Exponential Filter.- 2.2.4 Determination of the Phase of the Spatial Coherence Function from the Intensity in the Fraunhofer Plane.- References.- 3. The Uniqueness of Inverse Problems.- 3.1 Summary of Inverse Problems.- 3.1.1 Inverse Sturm-Liouville Problems.- 3.1.2 Reconstruction Problems.- 3.1.3 Three-Dimensional Reconstruction from Projections.- 3.2 Inverse Diffraction.- 3.2.1 Inverse Diffraction from Far-Field Data.- 3.2.2 Inverse Diffraction from Spherical Surface to Spherical Surface.- 3.2.3 Inverse Diffraction from Plane to Plane.- 3.2.4 Generalization to Arbitrary Surfaces.- 3.2.5 The Determination of the Shape of a Scatterer from Far-Field Data.- 3.3 Non-Radiating Sources.- 3.3.1 Early Results and Special Cases.- 3.3.2 General Theory.- 3.3.3 Integral Equations and Uniqueness by Prior Knowledge.- 3.4 The Determination of an Object from Scattering Data.- 3.4.1 Examples of Nonuniqueness.- 3.4.2 Phase Shift Analysis and the Reconstruction of a Potential.- 3.4.3 The Determination of a Potential or Index of Refraction from the Scattered Fields Generated by a Set of Monochromati c PIane Waves.- 3.4.4 The Unique Determination of an Object from Scattering Data.- 3.4.5 The Analytical Continuation of the Electromagnetic Field from the Exterior to the Interior of a Scatterer and Its Physical Implications.- References.- 4. Spatial Resolution of Subwavelength Sources from Optical Far-Zone Data.- 4.1 Approaches to Superresolution.- 4.1.1 Array of Sources with Known Radiation Pattern.- 4.1.2 Superresolution Using Evanescent Waves.- 4.1.3 x-Locali zed Sources.- 4.2 Partial Waves Associated with Complex Spatial Frequencies.- 4.3 Representations and Expansions of the EM Field.- 4.3.1 Integral Representations.- 4.3.2 Partial-Wave Representation of Exterior Field.- 4.3.3 Multipole Waves.- 4.3.4 Plane Waves.- 4.4 Band-Limiting at Variance with X-Localized Sources.- 4.5 High-Frequency Information in the Far Zone Given a X-Localized Source.- 4.6 X-Localized Sources Reconstructed from Far-Zone Data.- 4.7 Measurement of Phase and Magnitude of the Optical Radiation Pattern.- 4.8 Discussion.- References.- 5. Radiometry and Coherence.- 5.1 The Development of Radiometry.- 5.1.1 The Classical Period.- 5.1.2 The Baroque Period.- 5.1.3 The Modern Period.- 5.2 Coherence of Blackbody Radiation.- 5.2.1 Temporal Coherence.- 5.2.2 Spatial Coherence.- 5.3 First-Order Radiometry.- 5.3.1 Energy Flow in Scalar Fields.- 5.3.2 Coherence Theory and the Radiometrie Quantities.- 5.3.3 The Van Cittert-Zernike Theorem.- 5.3.4 An Example: Quasi stationary Sources.- 5.4 Radiant Intensity and Angular Coherence.- 5.4.1 Source Models.- 5.4.2 Inverse Relations.- 5.4.3 Bessel-Correlated Sources.- 5.4.4 Gauss-Correlated Sources.- 5.4.5 An Application: Coherence of Thermionic Sources.- 5.5 Radiation Efficiency.- 5.5.1 Radiance of Model Sources.- 5.5.2 Emittance and Radiation Efficiency.- 5.5.3 Exampl es.- 5.6 Second-Order Radiometry.- 5.6.1 Radiant Intensity Fluctuation and Autocorrelation.- 5.6.2 Second-Order Radiometric Quantities.- 5.6.3 An Example: Gauss-Correlated Chaotic Source.- References.- 6. Statistical Features of Phase Screens from Scattering Data.- 6.1 Basic Formulation of the Statistical Problem.- 6.1.1 Physical Models.- 6.1.2 Characteristic Functional of the Scattered Light.- 6.1.3 Correlation Functions.- 6.1.4 Gaussian Limit.- 6.2 More General Detection and Coherence Conditions.- 6.2.1 Gaussian Scattered Field.- 6.2.2 Polychromatic Speckle Patterns.- 6.3 Amplitude and Intensity Correlations.- 6.3.1 Information Contained in Amplitude Correlations.- 6.3.2 Information Contained in Intensity Correlations.- 6.3.3 Moving Diffusers.- 6.4 Number-Dependent Effects.- 6.4.1 Moments and Probability Distribution of Intensity.- 6.4.2 Examples.- 6.4.3 Applications.- 6.5 Concluding Remarks.- References.- Additional References with Titles.