Scintillation optimization of radial Gaussian beam array propagating through Kolmogorov turbulence

Abstract

The analytical expression for scintillation index of radial Gaussian beam array with coherent combination based on Kolmogorov power-law spectrum in the horizontal path is derived. The influences of the beam number and ring radius on the scintillation index are studied. The results show that the scintillation index can be reduced by increasing beam number and an optimum ring radius is proved to exist. Further, the optimum ring radius greatly depends on the source size and exists only in a certain range of the source size determined by the propagating distance. Additionally, the scintillation index distributions at the receiver greatly depend on the source size.

Appendix: Derivation of Eq. (9)

The incident field on a source plane (z = 0) for the general beam is given by [12]

$$ \begin{aligned} {u^{\rm inc}}\left( {{s_x},{s_y},z = 0} \right)& = \sum\limits_{l = 1}^N {\sum\limits_{\left( {n,m} \right)} {{A_{lnm}}} \exp \left( { - i{\theta _{lnm}}} \right){{\mathop{\rm H}\nolimits} _n}\left( {{\alpha _{xln}}{s_x} + {b_{xln}}} \right)} \\ & \quad\times{{\mathop{\rm H}\nolimits} _m}\left( {{\alpha _{ylm}}{s_y} + {b_{ylm}}} \right)\exp \left[ { - \frac{k}{2}\left( {{\alpha _{xln}}s_x^2 + {\alpha _{yln}}s_y^2} \right)} \right]\\ &\quad \times\exp \left[ { - i\left( {{V_{xln}}{s_x} + {V_{yln}}{s_y}} \right)} \right]. \end{aligned} $$

(10)

where l is the number of the set of multimode content, (n, m) are the specific mode within the l ^th set, A _lnm is the amplitude of the field at the source plane, θ _lnm is the constant phase factor, H _n and H _m are Hermite polynomials of order n and m determining the field distributions in the s _x and s _y directions, α _xln and α _yln are the complex parameters characterizing the width of Hermite polynomials in the s _x and s _y directions, b _xln and b _yln are complex parameters characterizing the displacement of Hermite polynomials in the s _x and s _y directions, V _xln and V _yln are the complex displacement parameters associated with the Gaussian part of the beam in the s _x and s _y directions, respectively.

If n = m = 0, A _lnm  = 1 and $\alpha_{xl}=\alpha_{yl}=\alpha_l={1 / {(kw_0^2)}},$ Eq. (10) can be reduced to Eq. (2) with $V_{xl}={{\left( {i{R_0}\cos {\varphi _n}} \right)} / {w_0^2}}, V_{yl}={{\left( {i{R_0}\sin {\varphi _n}} \right)} / {w_0^2}}$ and $\theta_l=- {{iR_0^2} / {\left( {2w_0^2} \right)}}.$ After using Huygens–Fresnel principle and Rytov method, the scintillation index Eq. (3) under weak fluctuation conditions on the receiving plane can be obtained. Upon substituting Eqs. (4), (5) and (8) into Eq. (3), σ ²_I (p,L) is expressed as

$$ \begin{aligned} \sigma_{\rm I}^2\left( {p,L} \right) &=0.415C_n^2{k^2}{\mathop{\rm Re}\nolimits}\left\{ {{{\left( {\sum\limits_{{n_1} = 1}^N {\sum\limits_{{n_2} =1}^N {{{\rm H}_2}\left( {{n_1},{n_2},p} \right)} } } \right)}^{ - 1}}} \right.\\ &\quad \times \sum\limits_{{n_1} = 1}^N {\sum\limits_{{n_2} = 1}^N {{{\rm H}_2}\left( {{n_1},{n_2},p} \right)} } \int\limits_0^L {{{\rm d}}z\int\limits_0^\infty {{\kappa ^{ - \frac{8}{3}}}\exp \left[ { - {{\rm Q}_2}\left( z \right){\kappa ^2}} \right]{\rm d}\kappa } } \\ &\quad \times\int\limits_0^{2\pi } {{{\rm d}} \theta }\exp \left[ {\left( { - {\rm B}\left( z \right)\cos {\varphi _{{n_1}}} - {{\rm B}^ * }\left( z \right)\cos {\varphi _{{n_2}}} + {\rm D}\left( z \right)p\cos \psi } \right.}\right.\\ &\left.{\quad+{{\rm D}^*} \left( z \right)p\cos \psi } \right)\kappa \cos \theta + \left( { - {\rm B}\left( z \right)\sin {\varphi _{{n_1}}} - {{\rm B}^ * }\left( z \right)\sin {\varphi _{{n_2}}}} \right.\\ & \left. {\left. {\quad+ {\rm D}\left( z \right) p\sin \psi + {{\rm D}^ * }\left( z \right)p\sin \psi } \right)\kappa \sin \theta } \right]\cr& \quad-{\left( {\sum\limits_{{n_1} = 1}^N {\sum\limits_{{n_2} = 1}^N {{{\rm H}_1}\left( {{n_1},{n_2},p} \right)} } } \right)^{ - 1}} \sum\limits_{{n_1} = 1}^N {\sum\limits_{{n_2} = 1}^N {{{\rm H}_1}\left( {{n_1},{n_2},p} \right)} } \\ &\quad \times \int\limits_0^L {{{\rm d}}z}\int\limits_0^\infty {{\kappa ^{ - \frac{8}{3}}}\exp \left[ { - {{\rm Q}_1}\left( z \right) {\kappa ^2}} \right]{{\rm d}}\kappa } \\ &\quad \times\int\limits_0^{2\pi}{{{\rm d}}\theta }\exp\left[ { - {\rm B} \left( z \right)\left( {\cos {\varphi _{{n_1}}} - \cos {\varphi _{{n_2}}}} \right)\kappa \cos \theta } \right.\cr& \left. {\left. {\quad- {\rm B}\left( z \right) \left( {\sin {\varphi _{{n_1}}} - \sin {\varphi _{{n_2}}}} \right)\kappa \sin \theta } \right]} \right\}. \end{aligned} $$

(11)

Recalling the integral formula $\int\nolimits_0^{2\pi } {{\mathop{\rm d}\nolimits} \theta \exp \left( {x\cos\theta + y\sin \theta } \right)} = 2\pi {I_0}\left[ {{{\left( {{x^2} + {y^2}} \right)}^{{1 / 2}}}} \right]$ to perform θ integration, and expanding Bessel function in the integrand in a Maclaurin series, then using the integral formula $\int\nolimits_0^\infty {{\mathop{\rm d}\nolimits} t{t^{\nu - 1}}\exp \left( { - \mu {t^p}} \right)}= {p^{ - 1}}{\mu ^{ - {\nu / p}}}\Upgamma \left( {{\nu / p}} \right), \kappa$ integration can be performed. Finally, Eq. (9) can be obtained and simplified by using the confluent hypergeometric function of first kind ${}_1{{\mathop{\rm F}\nolimits} _1}\left( {a,c,z} \right) .$