Preface. 1. Diffraction, Fourier Optics and Imaging. 1.1 Introduction. 1.2 Examples of Emerging Applications with Growing Significance. 2. Linear Systems and Transforms. 2.1 Introduction. 2.2 Linear Systems and Shift Invariance. 2.3 Continuous-Space Fourier Transform. 2.4 Existence of Fourier Transform. 2.5 Properties of the Fourier Transform. 2.6 Real Fourier Transform. 2.7 Amplitude and Phase Spectra. 2.8 Hankel Transforms. 3. Fundamentals of Wave Propagation. 3.1 Introduction. 3.2 Waves. 3.3 Electromagnetic Waves. 3.4 Phasor Representation. 3.5 Wave Equations in a Charge-Free Medium. 3.6 Wave Equations in Phasor Representation in a Charge-Free Medium. 3.7 Plane EM Waves. 4. Scalar Diffraction Theory. 4.1 Introduction. 4.2 Helmholtz Equation. 4.3 Angular Spectrum of Plane Waves. 4.4 Fast Fourier Transform (FFT) Implementation of the Angular Spectrum of Plane Waves. 4.5 The Kirchoff Theory of Diffraction. 4.6 The Rayleigh-Sommerfeld Theory of Diffraction. 4.7 Another Derivation of the First Rayleigh-Sommerfeld Diffraction Integral. 4.8 The Rayleigh-Sommerfeld Diffraction Integral For Nonmonochromatic Waves. 5. Fresnel and Fraunhofer Approximations. 5.1 Introduction. 5.2 Diffraction in the Fresnel Region. 5.3 FFT Implementation of Fresnel Diffraction. 5.4 Paraxial Wave Equation. 5.5 Diffraction in the Fraunhofer Region. 5.6 Diffraction Gratings. 5.7 Fraunhofer Diffraction By a Sinusoidal Amplitude Grating. 5.8 Fresnel Diffraction By a Sinusoidal Amplitude Grating. 5.9 Fraunhofer Diffraction with a Sinusoidal Phase Grating. 5.10 Diffraction Gratings Made of Slits. 6. Inverse Diffraction. 6.1 Introduction. 6.2 Inversion of the Fresnel and Fraunhofer Representations. 6.3 Inversion of the Angular Spectrum Representation. 6.4 Analysis. 7. Wide-Angle Near and Far Field Approximations for Scalar Diffraction. 7.1 Introduction. 7.2 A Review of Fresnel and Fraunhofer Approximations. 7.3 The Radial Set of Approximations. 7.4 Higher Order Improvements and Analysis. 7.5 Inverse Diffraction and Iterative Optimization. 7.6 Numerical Examples. 7.7 More Accurate Approximations. 7.8 Conclusions. 8. Geometrical Optics. 8.1 Introduction. 8.2 Propagation of Rays. 8.3 The Ray Equations. 8.4 The Eikonal Equation. 8.5 Local Spatial Frequencies and Rays. 8.6 Matrix Representation of Meridional Rays. 8.7 Thick Lenses. 8.8 Entrance and Exit Pupils of an Optical System. 9. Fourier Transforms and Imaging with Coherent Optical Systems. 9.1 Introduction. 9.2 Phase Transformation With a Thin Lens. 9.3 Fourier Transforms With Lenses. 9.4 Image Formation As 2-D Linear Filtering. 9.5 Phase Contrast Microscopy. 9.6 Scanning Confocal Microscopy. 9.7 Operator Algebra for Complex Optical Systems. 10. Imaging with Quasi-Monochromatic Waves. 10.1 Introduction. 10.2 Hilbert Transform. 10.3 Analytic Signal. 10.4 Analytic Signal Representation of a Nonmonochromatic Wave Field. 10.5 Quasi-Monochromatic, Coherent, and Incoherent Waves. 10.6 Diffraction Effects in a General Imaging System. 10.7 Imaging With Quasi-Monochromatic Waves. 10.8 Frequency Response of a Diffraction-Limited Imaging System. 10.9 Computer Computation of the Optical Transfer Function. 10.10 Aberrations. 11. Optical Devices Based on Wave Modulation. 11.1 Introduction. 11.2 Photographic Films and Plates. 11.3 Transmittance of Light by Film. 11.4 Modulation Transfer Function. 11.5 Bleaching. 11.6 Diffractive Optics, Binary Optics, and Digital Optics. 11.7 E-Beam Lithography. 12. Wave Propagation in Inhomogeneous Media. 12.1 Introduction. 12.4 Beam Propagation Method. 12.5 Wave Propagation in a Directional Coupler. 13. Holography. 13.1 Introduction. 13.2 Coherent Wave Front Recording. 13.3 Types of Holograms. 13.4 Computer Simulation of Holographic Reconstruction. 13.5 Analysis of Holographic Imaging and Magnification. 13.6 Aberrations. 14. Apodization, Superresolution, and Recovery of Missing Information. 14.1 Introduction. 14.2 Apodization. 14.2.1 Discrete-Time Windows. 14.3 Two-Point Resolution and Recovery of Signals. 14.4 Contractions. 14.5 An Iterative Method of Contractions for Signal Recovery. 14.6 Iterative Constrained Deconvolution. 14.7 Method of Projections. 14.8 Method of Projections onto Convex Sets. 14.9 Gerchberg-Papoulis (GP) Algorithm. 14.10 Other POCS Algorithms. 14.11 Restoration From Phase. 14.12 Reconstruction From a Discretized Phase Function by Using the DFT. 14.13 Generalized Projections. 14.14 Restoration From Magnitude. 14.15 Image Recovery By Least Squares and the Generalized Inverse. 14.16 Computation of Hp By Singular Value Decomposition (SVD). 14.17 The Steepest Descent Algorithm. 14.18 The Conjugate Gradient Method. 15. Diffractive Optics I. 15.1 Introduction. 15.2 Lohmann Method. 15.3 Approximations in the Lohmann Method. 15.4 Constant Amplitude Lohmann Method. 15.5 Quantized Lohmann Method. 15.6 Computer Simulations with the Lohmann Method. 15.7 A Fourier Method Based on Hard-Clipping. 15.8 A Simple Algorithm for Construction of 3-D Point Images. 15.9 The Fast Weighted Zero-Crossing Algorithm. 15.10 One-Image-Only Holography. 15.11 Fresnel Zone Plates. 16. Diffractive Optics II. 16.1 Introduction. 16.2 Virtual Holography. 16.3 The Method of POCS for the Design of Binary DOE. 16.4 Iterative Interlacing Technique (IIT). 16.5 Optimal Decimation-in-Frequency Iterative Interlacing Technique (ODIFIIT). 16.5.1 Experiments with ODIFIIT. 16.6 Combined Lohmann-ODIFIIT Method. 17. Computerized Imaging Techniques I: Synthetic Aperture Radar. 17.1 Introduction. 17.2 Synthetic Aperture Radar. 17.3 Range Resolution. 17.4 Choice of Pulse Waveform. 17.5 The Matched Filter. 17.6 Pulse Compression by Matched Filtering. 17.7 Cross-Range Resolution. 17.8 A Simplified Theory of SAR Imaging. 17.9 Image Reconstruction with Fresnel Approximation. 17.10 Algorithms for Digital Image Reconstruction. 18. Computerized Imaging II: Image Reconstruction from Projections. 18.1 Introduction. 18.2 The Radon Transform. 18.3 The Projection Slice Theorem. 18.4 The Inverse Radon Transform. 18.5 Properties of the Radon Transform. 18.6 Reconstruction of a Signal From its Projections. 18.7 The Fourier Reconstruction Method. 18.8 The Filtered-Backprojection Algorithm. 19. Dense Wavelength Division Multiplexing. 19.1 Introduction. 19.2 Array Waveguide Grating. 19.3 Method of Irregularly Sampled Zero-Crossings (MISZC). 19.4 Analysis of MISZC. 19.4.1 Dispersion Analysis. 19.4.2 Finite-Sized Apertures. 19.5 Computer Experiments. 19.6 Implementational Issues. 20. Numerical Methods for Rigorous Diffraction Theory. 20.1 Introduction. 20.2 BPM Based on Finite Differences. 20.3 Wide Angle BPM. 20.4 Finite Differences. 20.5 Finite Difference Time Domain Method. 20.6 Computer Experiments. 20.7 Fourier Modal Methods. Appendix A: The Impulse Function. Appendix B: Linear Vector Spaces. Appendix C: The Discrete-Time Fourier Transform, The Discrete Fourier Transform and The Fast Fourier Transform. References. Index.

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Diffraction, Fourier Optics and Imaging

METHOD AND APPARATUS FOR ULTRASONIC IMAGING USING A LINE SOURCE AND A LINEAR RECEIVER ARRAY ABSTRACT Method and apparatus for ultrasonically imaging objects of interest. The apparatus includes an elongate source of coherent acoustic waves which are brought to a line focus and propagated in an object of interest. The acoustic waves are detected by an elongate linear array of sub-stantially point receivers which are located parallel to the elongate source and substantially at the line focus. The elongate source and the linear receiver array are translated together relative to the object of interest so that an area is swept out across the object of interest. The amplitude and the phase of the acoustic waves scattered by the object of interest are detected by the receiver array, digitized and back wave reconstructed to yield a digital image of the object of interest.

Method and apparatus for ultrasonic imaging using a line source and a linear receiver array

This paper describes a technique that numerically reconstructs the complex acoustic amplitude (i.e. the acoustic amplitude and phase) of a compression acoustic wave in the interior volume of a specimen from a set of full-field optical measurements of the instantaneous displacement of the surface. The volume of a thick specimen is probed in transmission mode by short bursts of narrowband compression acoustic waves generated at one of its faces. The temporal evolution of the displacement field induced by the bursts emerging at the opposite surface is measured by pulsed digital holographic interferometry (pulsed TV holography). A spatio-temporal 3D Fourier transform processing of the measured data yields the complex acoustic amplitude at the plane of the surface as a sequence of 2D complex-valued maps. Finally, a numerical implementation of the Rayleigh–Sommerfeld diffraction formula is employed to reconstruct the complex acoustic amplitude at other planes in the interior volume of the specimen. The whole procedure can be regarded as a combination of optical digital holography and acoustical holography methods. The technique was successfully tested on aluminium specimens with and without an internal artificial defect and sample results are presented. In particular, information about the shape and position of the defect was retrieved in the experiment performed on the flawed specimen, which indicates the potential applicability of the technique for the nondestructive testing of materials.

https://iopscience.iop.org/article/10.1088/0957-0233/22/2/025109/pdf

Pulsed TV holography measurement and digital reconstruction of compression acoustic wave fields: application to nondestructive testing of thick metallic samples

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Earth and Planetary Sciences, 1979.

Experimental study of seismic scattering by a penny-shaped crack

The image of the acoustic field near a driven, submerged plate has been obtained experimentally using underwater nearfield holography. In addition to a discussion of the results, this paper develops theoretical explanation, for mechanical and acoustical processes, and describes aspects of the work which may be of interest to others attempting similiar experiments.

Van Rooy

Papers

Digital ultrasonic wavefront reconstruction in the near field