Title Page -- Contents -- Introduction -- Chapter 1. Mathematical Prerequisites -- 1.1. Frequently used special functions -- 1.1.1. The "rectangle" function -- 1.1.2. The "sinc" function -- 1.1.3. The "sign" function -- 1.1.4. The "triangle" function -- 1.1.5. The "disk" function -- 1.1.6. The Dirac δ function -- 1.1.6.1. Definition -- 1.1.6.2. Fundamental properties -- 1.1.7. The "comb" function -- 1.2. Two-dimensional Fourier transform -- 1.2.1. Definition and existence conditions -- 1.2.2. Theorems related to the Fourier transform -- 1.2.2.1. Linearity -- 1.2.2.2. Similarity -- 1.2.2.3. Translation -- 1.2.2.4. Parseval's theorem -- 1.2.2.5. The convolution theorem -- 1.2.2.6. The autocorrelation theorem -- 1.2.2.7. The duality theorem -- 1.2.3. Fourier transforms in polar coordinates -- 1.3. Linear systems -- 1.3.1. Definition -- 1.3.2. Impulse response and superposition integrals -- 1.3.3. Definition of a two-dimensional linear shift-invariant system -- 1.3.4. Transfer functions -- 1.4. The sampling theorem -- 1.4.1. Sampling a continuous function -- 1.4.2. Reconstruction of the original function -- 1.4.3. Space-bandwidth product -- Chapter 2. The Scalar Theory of Diffraction -- 2.1. Representation of an optical wave by a complex function -- 2.1.1. Representation of a monochromatic wave -- 2.1.2. Complex amplitude of the optical field in space -- 2.1.2.1. Plane waves -- 2.1.2.2. Spherical waves -- 2.1.3. Complex amplitudes of plane and spherical waves in a front plane -- 2.1.3.1. Complex amplitude of a plane wave in a front plane -- 2.1.3.2. Complex amplitude of a spherical wave in a front plane -- 2.2. Scalar theory of diffraction -- 2.2.1. Wave equation -- 2.2.2. Harmonic plane wave solutions to the wave equation -- 2.2.3. Angular spectrum.

2.2.4. Kirchhoff and Rayleigh-Sommerfeld formulae -- 2.2.5. Fresnel approximation and Fresnel diffraction integral -- 2.2.6. The Fraunhofer approximation -- 2.3. Examples of Fraunhofer diffraction patterns -- 2.3.1. Fraunhofer diffraction pattern from a rectangular aperture -- 2.3.2. Fraunhofer diffraction pattern from a circular aperture -- 2.3.3. Fraunhofer diffraction pattern from a sinusoidal-amplitude grating -- 2.4. Some examples and uses of Fresnel diffraction -- 2.4.1. Fresnel diffraction from a sinusoidal-amplitude grating -- 2.4.2. Fresnel diffraction from a rectangular aperture -- 2.5. Collins' formula -- 2.5.1. Description of an optical system by an ABCD transfer matrix -- 2.5.2. ABCD law and paraxial systems equivalent to a lens -- 2.5.2.1. ABCD law of a spherical wave propagating across an optical system -- 2.5.2.2. System equivalent to a lens -- 2.5.2.3. Properties of the transfer matrix -- 2.5.3. Proof of Collins' formula -- 2.5.3.1. Transmission from a thin lens -- 2.5.3.2. Expression of the ideal image -- 2.5.3.3. Proof of Collins' formula -- 2.5.4. Comparison between Collins' formula and the Fresnel integral -- 2.6. Conclusion -- Chapter 3. Calculating Diffraction by Fast Fourier Transform -- 3.1. Relation between the discrete and analytical Fourier transforms -- 3.1.1. Sampling and periodic expansion of a continuous two-dimensional function -- 3.1.2. The relation between the discrete and continuous Fourier transforms -- 3.2. Calculating the Fresnel diffraction integral by FFT -- 3.2.1. Calculating diffraction by the S-FFT method -- 3.2.2. Numerical calculation and experimental demonstration -- 3.2.3. The D-FFT method -- 3.2.4. Practical sampling conditions due to the energy conservation principle -- 3.2.5. Experimental demonstration of the D-FFT method.

3.3. Calculation of the classical diffraction formulae using FFT -- 3.3.1. Kirchhoff and Rayleigh-Sommerfeld formulae in convolution form -- 3.3.2. Unitary representation of the classical diffraction formulae -- 3.3.3. Study of the sampling conditions of the classical formulae -- 3.3.4. Example of calculations of the classical diffraction formulae -- 3.3.5. Calculation of diffraction by convolution: summary -- 3.4. Numerical calculation of Collins' formula -- 3.4.1. Collin's direct and inverse formulae -- 3.4.2. Calculating Collins' formula by S-FFT -- 3.4.3. Calculating the inverse Collins' formula by S-FFT -- 3.4.4. Calculating Collins' formula by D-FFT -- 3.4.5. Calculating the inverse Collins' formula by D-FFT -- 3.4.6. Numerical calculation and experimental demonstration -- 3.4.6.1. Demonstration of the S-FFT and S-IFFT methods -- 3.4.6.2. Demonstration of the D-FFT method -- 3.5. Conclusion -- Chapter 4. Fundamentals of Holography -- 4.1. Basics of holography -- 4.1.1. Leith-Upatnieks holograms -- 4.1.1.1. Illumination in the propagation direction of the original reference wave -- 4.1.1.2. Illumination with a wave propagating along the z-axis -- 4.1.2. Condition for the separation of the twin images and the zero order -- 4.1.2.1. Case where the reference wave is planar -- 4.1.2.2. The case where the reference wave is no longer planar -- 4.2. Partially coherent light and its use in holography -- 4.2.1. Analytic signal describing a non-monochromatic wave -- 4.2.1.1. Analytic signal describing a monochromatic wave -- 4.2.1.2. Analytic signal describing a non-monochromatic wave -- 4.2.1.3. Analytic signal and spectrum of a laser wave -- 4.2.2. Recording a hologram with non-monochromatic light -- 4.2.3. Total coherence approximation conditions -- 4.2.3.1. Identical wave train model.

4.2.3.2. Temporal coherence of a source emitting identical wave trains -- 4.2.4. Recording a Fresnel hologram -- 4.3. Study of the Fresnel hologram of point source -- 4.3.1. Reconstructing the hologram of a point source -- 4.3.1.1. Hologram illuminated by a spherical wave with the same wavelength -- 4.3.1.2. Hologram illuminated by a spherical wave with a different wavelength -- 4.3.1.3. Case where the reference and reconstruction waves are plane waves -- 4.3.2. Magnifications -- 4.3.2.1. Transverse magnification of the reconstructed image -- 4.3.2.2. Longitudinal magnification -- 4.3.3. Resolution of the reconstructed image -- 4.3.3.1. Influence of the size of the illuminating source -- 4.3.3.2. Influence of the spectral width of the light source -- 4.4. Different types of hologram -- 4.4.1. The Fraunhofer hologram -- 4.4.2. The Fourier hologram -- 4.4.2.1. Component Cũ1(xi, yi) -- 4.4.2.2. Component Cũ2(xi, yi) -- 4.4.2.3. Component Cũ3(xi, yi) -- 4.4.2.4. Component Cũ4(xi, yi) -- 4.4.3. The lensless Fourier hologram -- 4.4.4. The image hologram -- 4.4.5. The phase hologram -- 4.5. Conclusion -- Chapter 5. Digital Off-Axis Fresnel Holography -- 5.1. Digital off-axis holography and wavefront reconstruction by S-FFT -- 5.1.1. Characteristics of the diffraction from a digital hologram impacted by a spherical wave -- 5.1.1.1. Virtual object -- 5.1.1.2. Conjugate object -- 5.1.1.3. Zero order -- 5.1.2. Optimization of the experimental parameters -- 5.1.3. Experimental reconstruction by S-FFT -- 5.1.4. Quality of the reconstructed image -- 5.2. Elimination of parasitic orders with the S-FFT method -- 5.2.1. Diffraction efficiency of a digital hologram -- 5.2.2. Methods of direct elimination -- 5.2.2.1. Method directly eliminating the object and reference waves.

5.2.2.2. Method with an arbitrary phase shift of the reference wave -- 5.2.3. Method of extracting the complex amplitude of the object wave -- 5.3. Wavefront reconstruction with an adjustable magnification -- 5.3.1. Convolution with adjustable magnification -- 5.3.2. Experiment with adjustable magnification -- 5.3.3. Elimination of the perturbation due to the zero order -- 5.3.3.1. Spectral distribution and determination of the center of the object wave -- 5.3.3.2. Spectral position of the object wave -- 5.3.4. Method eliminating the perturbation due to the zero order -- 5.4. Filtering in the image and reconstruction planes by the FIMG4FFT method -- 5.4.1. Adjustable magnification reconstruction by the FIMG4FFT method -- 5.4.1.1. Filtering the image plane -- 5.4.1.2. Experimental results -- 5.4.1.3. Local reconstruction by the FIMG4FFT method -- 5.5. DBFT method and the use of filtering in the image plane -- 5.5.1. DBFT method -- 5.5.2. Sampling of the DBFT algorithm -- 5.5.3. Improvement of the DBFT method -- 5.5.4. Experimental demonstration of the DDBFT method -- 5.6. Digital color holography -- 5.6.1. Recording a digital color hologram -- 5.6.2. Standardization of the physical scale of reconstructed monochromatic images -- 5.6.3. Fresnel transform with wavelength-dependant zero-padding -- 5.6.4. Experimental study of the different methods of reconstructing color images -- 5.6.4.1. Two-color image calculated by the zero-padding method -- 5.6.4.2. Reconstruction by the FIMG4FFT and the DDBFT with adjustable magnification -- 5.6.4.3. Three-color digital holography -- 5.7. Digital phase hologram -- 5.7.1. Formation of a digital phase hologram and reconstruction by S-FFT -- 5.7.2. Experimental demonstration -- 5.8. Depth of focus of the reconstructed image -- 5.8.1. Theoretical analysis.

5.8.2. Comparison with a digital holographic simulation.