Electromagnetic Wave Propagation in Turbulence is devoted to a method for obtaining analytical solutions to problems of electromagnetic wave propagation in turbulence. In a systematic way the monograph presents the Mellin transforms to evaluate analytically integrals that are not in integral tables. Ample examples of application are outlined and solutions for many problems in turbulence theory are given. The method itself relates to asymptotic results that are applicable to a broad class of problems for which many asymptotic methods had to be employed previously.
Electromagnetic Wave Propagation in Turbulence is devoted to a method for obtaining analytical solutions to problems of electromagnetic wave propagation in turbulence. In a systematic way the monograph presents the Mellin transforms to evaluate analytically integrals that are not in integral tables. Ample examples of application are outlined and solutions for many problems in turbulence theory are given. The method itself relates to asymptotic results that are applicable to a broad class of problems for which many asymptotic methods had to be employed previously.
1 Introduction.- 1.1 Book Plan.- 1.2 Introduction to Mellin Transforms.- 1.3 Higher Transcendental Functions ..- 2 Basic Equations for Wave Propagation in Turbulence.- 2.1 Turbulence Spectra.- 2.2 Rytov Approximation.- 2.3 Phase and Log-Amplitude Variances.- 2.4 Power Spectral Density.- 2.5 Beam Shape and Strehl Ratio.- 3 Aperture Filter Functions.- 3.1 Circular Aperture Modes.- 3.2 Piston and Tilt on an Annulus.- 3.3 Finite Apertures and Focal Anisoplanatism.- 3.4 Adaptive-Optics Systems.- 4 Zero-Parameter Problems.- 4.1 Turbulence Models and Moments.- 4.2 Zernike Tilt and Piston for Collimated and Focused Beams.- 4.3 Gradient Tilt.- 4.4 Difference Between Gradient and Zernike Tilt.- 4.5 Beam Movement at a Target.- 4.6 Angle-of-Arrival Jitter.- 4.7 Scintillation for Collimated and Focused Beams.- 4.8 Phase Variance with Finite Servo Bandwidth.- 4.9 Variances for Beams Corrected by Adaptive Optics.- 5 Integral Evaluation with Mellin Transforms.- 5.1 Integral Evaluation with One Parameter.- 5.2 Asymptotic Solutions.- 5.2.1 Alternate Method of Integral Evaluation.- 5.3 Multiple Poles.- 6 Other Applications of Mellin Transforms.- 6.1 Convolutional Transforms.- 6.2 Laplace's Equation in Cylindrical Coordinates.- 6.3 Laplace's Equation in Spherical Coordinates.- 6.4 Time Varying Problems.- 6.5 Integral Equations.- 6.6 Multiple Integrals and Asymptotic Series from Power Series . ..- 6.7 Miscellaneous Additional Applications.- 7 Examples with a Single Positive Parameter.- 7.1 Tilt for the von Kármán Spectrum.- 7.2 Tilt for the Greenwood Spectrum.- 7.3 Tilt with Finite Inner Scale.- 7.4 Piston- and Tilt-Removed Phase Variance on an Annulus .....- 7.5 Effect of Diffraction on Tilt.- 7.6 Tilt Anisoplanatism.- 7.7 Power Spectral Density of Tilt.- 7.8 Scintillation withFinite Apertures and Sources.- 7.9 Scintillation with Finite Inner Scale.- 7.10 Scintillation Anisoplanatism.- 7.11 Focus Anisoplanatism.- 7.12 Focal Anisoplanatism for Point Sources.- 7.13 Focal Anisoplanatism for Distributed Sources.- 7.14 Focal Anisoplanatism for Offset Sources.- 8 Strehl Ratio.- 8.1 Strehl Ratio for Propagation Through Turbulence.- 8.2 Strehl Ratio with Beam Jitter.- 8.3 Strehl Ratio with Anisoplanatism.- 8.4 Strehl Ratio for Various Anisoplanatic Effects.- 9 Mellin Transforms with a Complex Parameter.- 9.1 Mellin-Barnes Integrals with Complex Parameters.- 9.2 Asymptotic Results with a Complex Parameter.- 9.3 The Mellin Transform of an Exponential Times a Bessel Function.- 10 Examples With a Single Complex Parameter.- 10.1 Phase and Log-Amplitude Variances of Beam Waves.- 10.2 Power Spectral Density of Beam Waves.- 10.3 Scintillation on Beam Waves.- 11 Mellin Transforms in N Complex Planes.- 11.1 Convergence of Multi-Parameter Series.- 11.2 Path Closure at Infinity.- 11.3 Integration in Multiple Complex Planes.- 11.4 Asymptotic Solution in Two or More Complex Planes.- 12 Examples with N Parameters.- 12.1 An Integral with Two Bessel Functions and a Sinusoid.- 12.2 An Integral with Three Bessel Functions.- 12.3 Example in Three and N Complex Planes.- 12.4 Effect of Outer Scale on Tilt Anisoplanatism.- 12.5 Tilt with Inner and Outer Scale.- 12.6 Power Spectrum of Tilt with Outer Scale.- 12.7 Structure and Correlation Functions with Inner and Outer Scales.- 13 Beam Shape.- 13.1 General Formula for Beam Shape.- 13.2 Beam Shape for Uncorrected Turbulence.- 13.3 Beam Shape with Tilt Jitter.- 13.4 Beam Shape with Anisoplanatism.- A Additional Mellin Transforms.- B Trancendental Functions.- References.
1 Introduction.- 1.1 Book Plan.- 1.2 Introduction to Mellin Transforms.- 1.3 Higher Transcendental Functions ..- 2 Basic Equations for Wave Propagation in Turbulence.- 2.1 Turbulence Spectra.- 2.2 Rytov Approximation.- 2.3 Phase and Log-Amplitude Variances.- 2.4 Power Spectral Density.- 2.5 Beam Shape and Strehl Ratio.- 3 Aperture Filter Functions.- 3.1 Circular Aperture Modes.- 3.2 Piston and Tilt on an Annulus.- 3.3 Finite Apertures and Focal Anisoplanatism.- 3.4 Adaptive-Optics Systems.- 4 Zero-Parameter Problems.- 4.1 Turbulence Models and Moments.- 4.2 Zernike Tilt and Piston for Collimated and Focused Beams.- 4.3 Gradient Tilt.- 4.4 Difference Between Gradient and Zernike Tilt.- 4.5 Beam Movement at a Target.- 4.6 Angle-of-Arrival Jitter.- 4.7 Scintillation for Collimated and Focused Beams.- 4.8 Phase Variance with Finite Servo Bandwidth.- 4.9 Variances for Beams Corrected by Adaptive Optics.- 5 Integral Evaluation with Mellin Transforms.- 5.1 Integral Evaluation with One Parameter.- 5.2 Asymptotic Solutions.- 5.2.1 Alternate Method of Integral Evaluation.- 5.3 Multiple Poles.- 6 Other Applications of Mellin Transforms.- 6.1 Convolutional Transforms.- 6.2 Laplace's Equation in Cylindrical Coordinates.- 6.3 Laplace's Equation in Spherical Coordinates.- 6.4 Time Varying Problems.- 6.5 Integral Equations.- 6.6 Multiple Integrals and Asymptotic Series from Power Series . ..- 6.7 Miscellaneous Additional Applications.- 7 Examples with a Single Positive Parameter.- 7.1 Tilt for the von Kármán Spectrum.- 7.2 Tilt for the Greenwood Spectrum.- 7.3 Tilt with Finite Inner Scale.- 7.4 Piston- and Tilt-Removed Phase Variance on an Annulus .....- 7.5 Effect of Diffraction on Tilt.- 7.6 Tilt Anisoplanatism.- 7.7 Power Spectral Density of Tilt.- 7.8 Scintillation withFinite Apertures and Sources.- 7.9 Scintillation with Finite Inner Scale.- 7.10 Scintillation Anisoplanatism.- 7.11 Focus Anisoplanatism.- 7.12 Focal Anisoplanatism for Point Sources.- 7.13 Focal Anisoplanatism for Distributed Sources.- 7.14 Focal Anisoplanatism for Offset Sources.- 8 Strehl Ratio.- 8.1 Strehl Ratio for Propagation Through Turbulence.- 8.2 Strehl Ratio with Beam Jitter.- 8.3 Strehl Ratio with Anisoplanatism.- 8.4 Strehl Ratio for Various Anisoplanatic Effects.- 9 Mellin Transforms with a Complex Parameter.- 9.1 Mellin-Barnes Integrals with Complex Parameters.- 9.2 Asymptotic Results with a Complex Parameter.- 9.3 The Mellin Transform of an Exponential Times a Bessel Function.- 10 Examples With a Single Complex Parameter.- 10.1 Phase and Log-Amplitude Variances of Beam Waves.- 10.2 Power Spectral Density of Beam Waves.- 10.3 Scintillation on Beam Waves.- 11 Mellin Transforms in N Complex Planes.- 11.1 Convergence of Multi-Parameter Series.- 11.2 Path Closure at Infinity.- 11.3 Integration in Multiple Complex Planes.- 11.4 Asymptotic Solution in Two or More Complex Planes.- 12 Examples with N Parameters.- 12.1 An Integral with Two Bessel Functions and a Sinusoid.- 12.2 An Integral with Three Bessel Functions.- 12.3 Example in Three and N Complex Planes.- 12.4 Effect of Outer Scale on Tilt Anisoplanatism.- 12.5 Tilt with Inner and Outer Scale.- 12.6 Power Spectrum of Tilt with Outer Scale.- 12.7 Structure and Correlation Functions with Inner and Outer Scales.- 13 Beam Shape.- 13.1 General Formula for Beam Shape.- 13.2 Beam Shape for Uncorrected Turbulence.- 13.3 Beam Shape with Tilt Jitter.- 13.4 Beam Shape with Anisoplanatism.- A Additional Mellin Transforms.- B Trancendental Functions.- References.
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