Optics

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Optics (encyc) - Optics entry in the encyclopedia.

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)
B_{x}(\tau )=\int _{0}^{\infty }S_{x}(\omega )\cos(\omega \tau )\,d\omega \,

The power spectrum is defined as
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 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 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

\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 D_{n}(R)=C_{n}^{2}R^{{2/3}}\, and the integral formula \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:

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

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