# Optics

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These problems and the information needed to solve them come from these two books:

Andrews, Larry C. Laser Beam Propagation through Random Media. 1998. ISBN: 0-8194-2787-X.
Andrews, Larry C. Special Functions of Mathematics for Engineers. 1998. ISBN: 0-8194-2616-4.

The covariance function is defined as (among other things)
${\displaystyle B_{x}(\tau )=\int _{0}^{\infty }S_{x}(\omega )\cos(\omega \tau )\,d\omega \,}$

The power spectrum is defined as
${\displaystyle S_{x}(\omega )={\frac {1}{\pi }}\int _{0}^{\infty }B_{x}(\tau )\cos(\omega \tau )\,d\tau \,}$

1. solution Find the associated power spectrum given the covariance function ${\displaystyle B_{x}(\tau )=\exp \left(-\left|{\frac {\tau }{\tau _{0}}}\right|\right),\tau _{0}>0\,}$

2. solution Find the associated power spectrum given the covariance function ${\displaystyle B_{x}(\tau )=\exp \left[-a^{2}\left({\frac {\tau }{\tau _{0}}}\right)^{2}\right],\tau _{0}>0,a>0\,}$

3. solution Given that the power spectrum and structure function are related by

${\displaystyle \phi _{n}(\kappa )={\frac {1}{4\pi ^{2}\kappa ^{2}}}\int _{0}^{\infty }{\frac {\sin(\kappa R)}{\kappa R}}{\frac {\partial }{\partial R}}\left[R^{2}{\frac {\partial D_{n}(R)}{\partial R}}\right]\,dR\,}$,

use the structure function ${\displaystyle D_{n}(R)=C_{n}^{2}R^{2/3}\,}$ and the integral formula ${\displaystyle \int _{0}^{\infty }x^{\alpha }\sin(x)\,dx=2^{\alpha }{\sqrt {\pi }}\,{\frac {\Gamma ({\frac {\alpha }{2}}+1)}{\Gamma ({\frac {1}{2}}-{\frac {\alpha }{2}})}}\,}$ to obtain the Kolmogorov power law spectrum:

${\displaystyle \phi _{n}(\kappa )=0.033C_{n}^{2}\kappa ^{-11/3},\,\,\,{\frac {1}{L_{0}}}<\!\!<\kappa <\!\!<{\frac {1}{l_{0}}}\,}$