The effects of atmospheric turbulence on the spectral changes of diffracted pulsed hollow higher-order cosh-Gaussian beam

Abstract

In this paper, we derive by using the Fourier transform and the extended Huygens-Fresnel, the analytical formulae for a truncated and non-truncated diffracted pulsed Hollow higher-order Cosh-Gaussian beam propagating in a turbulent atmosphere. Numerical examples are presented to illustrate the behavior of the spectral intensity of the propagated beam under the change of the initial beam parameters and the structure constant of the atmospheric turbulence. It is shown that the on-axis spectrum is blue-shifted, and the spectrum becomes red-shifted as the transverse distance grows. Also, some conclusions are presented to explain the effect of the considered medium and beam parameters on spectral shifts at different observation positions upon the propagation. Several studies could be derived from our principal result as special cases. It is to be noted that the obtained results in the present work would be helpful for the optical communications and remote sensing.

Appendix A

In this Appendix, we will demonstrate the integral used in the theoretical formulation

$$H = \int\limits_{0}^{R} {e^{{ - p\rho^{2} + 2q\rho }} \rho^{\varepsilon } d\rho } = e^{{{{q^{2} } \mathord{\left/ {\vphantom {{q^{2} } p}} \right. \kern-0pt} p}}} \int\limits_{0}^{R} {e^{{ - p\left( {\rho - \delta } \right)^{2} }} \rho^{\varepsilon } d\rho } ,$$

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where $\delta = {q \mathord{\left/ {\vphantom {q p}} \right. \kern-0pt} p}$ and ${\text{Re}} \,p > 0.$

With the help of the following change of variables $t = \sqrt p \left( {\rho - \delta } \right)$ and the identities (Gradshteyn et al. 2007)

$$\left( {t + \sqrt p \delta } \right)^{\varepsilon } = \sum\limits_{r = 0}^{\varepsilon } {\left( {\begin{array}{*{20}c} \varepsilon \\ r \\ \end{array} } \right)\left( {\sqrt p \,\,\delta } \right)^{\varepsilon - r} t^{r} } ,$$

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$$\int\limits_{0}^{\,\delta } {e^{{ - \mu t^{2} }} t^{2v - 1} dt\, = \,\,} \frac{1}{{2\mu^{v} }}\frac{{\left( {\mu \varepsilon^{2} } \right)^{v} }}{v}{}_{1}F_{1} \left( {v;\nu + 1; - \mu \varepsilon^{2} } \right),$$

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the expression of integral H becomes

$$H = \frac{{e^{{{{q^{2} } \mathord{\left/ {\vphantom {{q^{2} } p}} \right. \kern-0pt} p}}} }}{{p^{{\frac{\varepsilon + 1}{2}}} }}\sum\limits_{r = 0,\varepsilon } {\left( {\begin{array}{*{20}c} \varepsilon \\ r \\ \end{array} } \right)} \left( {\sqrt p \,\,\delta } \right)^{\varepsilon - r} H_{r} ,$$

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where

$$H_{r} = \left( { - 1} \right)^{r} H_{\delta } + H_{{\left( {R - \delta } \right)}} ,$$

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with

$$H_{\xi } = \int\limits_{0}^{{\sqrt {p\,\,} \,\xi }} {e^{{ - t^{2} }} t^{r} dt\quad with\quad \left( {\xi = \delta \quad or\quad \left( {R - \delta } \right)} \right)} .$$

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So, $H_{\delta }$ and $H_{{\left( {R - \delta } \right)}}$ can be written as

$$H_{\delta } = \frac{{\left( {p\delta^{2} } \right)^{{\frac{r + 1}{2}}} }}{{\left( {r + 1} \right)}}{}_{1}F_{1} \left( {\frac{r + 1}{2};\frac{r + 3}{2}; - p\delta^{2} } \right),$$

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and

$$H_{{\left( {R - \delta } \right)}} = \frac{{\left[ {p\left( {R - \delta } \right)^{2} } \right]^{{\frac{r + 1}{2}}} }}{{\left( {r + 1} \right)}}{}_{1}F_{1} \left( {\frac{r + 1}{2};\frac{r + 3}{2}; - p\left( {R - \delta } \right)^{2} } \right).$$

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The final expression of our integral can be rearranged as

$$\begin{aligned} \int\limits_{0}^{R} {\rho^{\varepsilon } e^{{ - p\rho^{2} + 2q\rho }} } d\rho = & \frac{{e^{{\frac{{q^{2} }}{p}}} }}{{p^{{\frac{\varepsilon + 1}{2}}} }}\sum\limits_{r = 0}^{\varepsilon } {\left( {\begin{array}{*{20}c} \varepsilon \\ r \\ \end{array} } \right)} \left( {\frac{q}{\sqrt p }} \right)^{\varepsilon - r} \\ & \quad \times \left[ {\left( { - 1} \right)^{r} \frac{{\left( {\frac{{q^{2} }}{p}} \right)^{{\frac{r + 1}{2}}} }}{r + 1}{}_{1}F_{1} \left( {\frac{r + 1}{2};\frac{r + 3}{2}; - \frac{{q^{2} }}{p}} \right) + \frac{{\left[ {p\left( {R - \frac{q}{p}} \right)^{2} } \right]^{{\frac{r + 1}{2}}} }}{r + 1}{}_{1}F_{1} \left( {\frac{r + 1}{2};\frac{r + 3}{2}; - p\left( {R - \frac{q}{p}} \right)^{2} } \right)} \right]. \\ \end{aligned}$$

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