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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