Propagation factor and beam wander of electromagnetic Gaussian Schell-model array beams in non-Kolmogorov turbulence

Biling Zhang; Yonggen Xu; Xiaoyan Wang; Youquan Dan

doi:10.1364/OSAC.2.000162

1. Introduction

Recently, more and more literatures are related to the laser array beams due to the varied applications [1–4]. Furthermore, laser arrays can provide higher system output powers [5]. The correlated radial stochastic electromagnetic array beam has been introduced by use of tensor method [5]. Besides, the characteristics of the angular spread of Gaussian Schell-model array beams propagating through atmospheric turbulence have been studied [6]. In Ref. [7], it is shown that the resulting beam of off-axis partially coherent beams for the superposition of the cross-spectral density function is more affected by the turbulence than that for the superposition of the intensity. Cai et.al studied the off-axis GSM beam and partially coherent laser array beam in a turbulent atmosphere [8]. In addition, some theoretical analyses have indicated that the laser beams with high-order and low coherence are less susceptible to turbulence [9,10]. Therefore, it is important to study the propagation properties of the partially coherent electromagnetic Gaussian Schell-model (EGSM) array beams through turbulent atmosphere. The EGSM beam as a typical stochastic electromagnetic beam has been studied in the past several years, because of its importance in the theories of coherence and polarization of light [11–19]. What’s more, the M²-factor, which is defined based on the second–order moments of the Wigner distribution function (WDF), has been adopted widely for characterizing laser beams [11,20–22]. In some reports [9,17,23,24], the M²-factor of a laser beam propagating in the turbulent atmosphere has been studied.

Beam wander may appear when the beam propagates through the atmospheric turbulence [25], which is characterized by the random displacement of the instantaneous center of laser beam as it propagates in the atmospheric turbulence [26–28]. Moreover, beam wander as an important characteristic of laser beams determines their utility for practical applications [29,30]. In the past years, the beam wander of different types of beams have been studied widely [28,31–35]. In Ref. [31], Wen et al. reported the beam wander of partially Airy beams, and provided an effective way to control the beam wander of partially airy beams. Yu et al. discussed the influence of the phase curvature and polarization on the beam wander in 2012 [35]. In Ref. [36], Wu et al. studied the beam wander of random EGSM vortex beams propagating through Kolmogorov turbulence, which found that the beams with smaller coherent length, smaller wavelength, and larger topological charge can effectively reduce beam wander in free-space optical communication. Therefore, it is necessary to study the beam wander of EGSM array beams.

In summary, the M²-factor of EGSM beam through inhomogeneous atmospheric turbulence has been discussed [17] and the beam wander of EGSM beam has also been studied [35]. Besides, the beam wander of EGSM vortex beam has been studied in detail [36]. However, M²-factor and beam wander of EGSM array beams in non-Kolmogorov turbulence have not been reported. Consequently, according to the extended Huygens-Fresnel principle and the second-order moments of the WDF, the purpose of this paper is to investigate the M²-factor and beam wander of EGSM array beams propagating through the non-Kolmogorov turbulence. In order to illustrate our theoretical results, the numerical examples are given. In addition, the influences of the beam parameters and turbulence parameters on the M²-factor and beam wander of EGSM array beams have been discussed in detail.

2. Theoretical model

For the sake of convenience, the laser array beams studied in this paper take square array as an example (Fig. 1).

Fig. 1. Schematic drawing of EGSM array beams.

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According to the 2×2 cross spectral density matrix (CSDM), the second-order statistical properties of electromagnetic beam in the source plane z = 0 are characterized as [14,17]:

(1)$$\overleftrightarrow W({{{{\boldsymbol{\rho}^{\prime}_1}}},{{{\boldsymbol{\rho}^{\prime}_2}}},0} )= \left[ {\begin{array}{cc} {{W_{xx}}({{{{\boldsymbol{\rho}^{\prime}_1}}},{{{\boldsymbol{\rho}^{\prime}_2}}},0} )} & {{W_{xy}}({{{{\boldsymbol{\rho}^{\prime}_1}}},{{{\boldsymbol{\rho}^{\prime}_2}}},0} )}\\ {{W_{yx}}({{{{\boldsymbol{\rho}^{\prime}_1}}},{{{\boldsymbol{\rho}^{\prime}_2}}},0} )} & {{W_{yy}}({{{{\boldsymbol{\rho}^{\prime}_1}}},{{{\boldsymbol{\rho}^{\prime}_2}}},0} )} \end{array}} \right],$$

where ρ₁′= (x₁′,y₁′) and ρ₂′=(x₂′,y₂′) are arbitrary transverse position vectors at the source plane z = 0. W_pq(ρ₁′,ρ₂′,0)=〈${E_{p}^\ast}$(ρ₁′,0)E_q(ρ₂′,0)〉 (p = x,y; q = x,y), E_x and E_y denote the fluctuating components of the random electric vector for x and y direction. Here, the asterisk represents the complex conjugate and the angle brackets denote the ensemble average.

Hence, the element of the cross spectral density (CSD) for EGSM array beams at the plane z = 0 can be shown to be [12,13]

(2)$$\begin{aligned}{W_{pq}}({{{\rho^{\prime}_1}},{{\rho^{\prime}_2}},0} ) &= {A_p}{A_q}{B_{pq}} \times {\sum\limits_{i = {{ - ({N - 1} )} /2}}^{{(N-1)}/2} {{\sum\limits_{{{j = - ({N - 1} )} / 2}}^{{(N-1)}/2} {\exp \left( { - \frac{{{{({{{x^{\prime}_1}} - i{x_0}} )}^2} + {{({{{y^{\prime}_1}} - j{y_0}} )}^2}}}{{4\sigma_{0p}^2}}} \right)} }}} \\ & {{\times \exp \left( { - \frac{{{{({{{x^{\prime}_2}} - i{x_0}} )}^2} + {{({{{y^{\prime}_2}} - j{y_0}} )}^2}}}{{4\sigma_{0q}^2}}} \right)\exp \left( { - \frac{{\rho_d^2}}{{2\delta_{0pq}^2}}} \right),}}\end{aligned}$$

where A_x and A_y are the amplitudes of the electric field for x and y components. δ_0pq (p = q) is the rms width of auto-correlation functions of the x components of the field, or y components of the field, respectively; δ_0pq (p≠q) is the rms width of the mutual correlation function of x and y field components. σ_0j denotes the rms width of the spectral density along j direction. B_pq denotes the complex correlation coefficient, and B_pq=1 (p = q), B_pq=0 (p≠q); N represents the beam array number, x₀ and y₀ represent the beam separation distance in the x and y directions.

In order to calculate conveniently, the trace of the CSDM of EGSM array beams in the source plane z = 0 is expressed as follows [12,14,15]:

(3)$${\overleftrightarrow W_{Tr}}({{{{\boldsymbol{\rho}^{\prime}_1}}},{{{\boldsymbol{\rho}^{\prime}_2}}},0} )= {W_{xx}}({{{{\boldsymbol{\rho}^{\prime}_1}}},{{{\boldsymbol{\rho}^{\prime}_2}}},0} )+ {W_{yy}}({{{{\boldsymbol{\rho}^{\prime}_1}}},{{{\boldsymbol{\rho}^{\prime}_2}}},0} ),$$

where we assume that the electric field E_x and E_y are uncorrelated; and Tr is the trace of CSDM of an EGSM array beam.

For the sake of convenience, we have used the central abscissa coordinate systems, i.e [9],

(4)$${{\boldsymbol{\rho} ^{\prime}_1}} = {\boldsymbol{\rho} ^{\prime}} + {{{{{\boldsymbol{\rho} ^{\prime}_d}}}} {/} 2};{{\boldsymbol{\rho} ^{\prime}_2}} = {\boldsymbol{\rho} ^{\prime}} - {{{{{\boldsymbol{\rho} ^{\prime}_d}}}}/2} ,$$

Therefore, Eq. (3) can be written as:

(5)$$\begin{aligned} {\overleftrightarrow W_{Tr}}({{\boldsymbol{\rho}^{\prime},}{{{\boldsymbol{\rho}^{\prime}_d}}},0} )&= A_x^2{B_{xx}}\sum\limits_{i = {{ - ({N - 1} )} / 2}}^{{{({N - 1} )} {/2}}}\sum\limits_{j = {{- ({N - 1} )} / 2}}^{{{({N - 1} )} {/2}}}{\exp \left({\frac{- 2{{x^{\prime 2}}}- {{{x^{\prime 2}_d}}} /{2 - 2{i^2}}{{x}_{0}^2} + 4i{x}_{0}{x}^{\prime}} {{4{\sigma_{0x}^2}}}}\right)} \\ & \times {\exp \left({\frac{ - 2{{y^{\prime 2}}} - {{{y^{\prime 2}_d}}}/{2 - 2{j^2}}{{y}_{0}^2} + 4j{y}_{0}{y}^{\prime}}{{4{\sigma_{0x}^2}}}} \right)} \times\exp \left( { - \frac{{{\rho_d^2}}}{{2{\delta_{0xx}^2}}}} \right) \\ & + A_y^2{B_{yy}}\sum\limits_{i = {{ - ({N - 1} )} /2}}^{{{({N - 1} )} {/2}}}\sum\limits_{j = {{- ({N - 1} )} /2}}^{{{({N - 1} )} {/ 2}}}{\exp \left( {\frac{ - 2{{x^{\prime 2}}} - {{{x^{\prime 2}_d}}} /{2 - 2{i^2}}{{x}_{0}^2} + 4i{{x}_{0}}{x}^{\prime}}{{4{\sigma_{0y}^2}}}} \right)}\\ & \times {\exp \left( {\frac{ - 2{{y^{\prime 2}}} - {{{y^{\prime 2}_d}}} /{2 - 2{j^2}}{{y}_{0}^2} + 4j{{y}_{0}}{y}^{\prime}}{{4{\sigma_{0y}^2}}}} \right)}\times \exp \left( {\frac{{{ -\rho_d^2}}}{{2{\delta_{0yy}^2}}}} \right).\end{aligned}$$

According to the trace of the CSD, the WDF can be written as [9,23]:

(6)$$h({{\boldsymbol{\rho} },{\boldsymbol{\theta}},0} )= {\left( {\frac{k}{{2\pi }}} \right)^2}\int {{{\overleftrightarrow W}_{Tr}}({{\boldsymbol{\rho} },{{\boldsymbol{\rho} }_d},0} )\exp ({ - ik{\boldsymbol{\theta}} \cdot {{\boldsymbol{\rho} }_d}} ){d^2}{\rho _d}} ,$$

Here, k = 2π/λ, k represents the wave number and λ is the wavelength. θ = (θ_x, θ_y).

From Eq. (6), the moments of the order n₁+n₂+m₁+m₂ of WDF can be shown as [9,23]:

(7)$${{\langle{{x^{{n_1}}}{y^{{n_2}}}\theta_x^{{m_1}}\theta_y^{{m_2}}} \rangle}_0} = \frac{1}{P}\int\!\!\!\int {{x^{{n_1}}}{y^{{n_2}}}\theta _x^{{m_1}}\theta _y^{{m_2}}h({{\boldsymbol{\rho} },{\boldsymbol{\theta}},0} )} {d^2}\rho {d^2}\theta ,$$

where P=∫∫h(ρ,θ,0)d²ρd²θ denotes the total power of beam and we can get the second-order moments 〈x²〉₀, 〈y²〉₀, 〈xθ_x〉₀, 〈yθ_y〉₀, 〈θ_x²〉₀, and 〈θ_y²〉₀ from Eq. (7).

Substituting Eq. (5)–Eq. (6) into Eq. (7), the new Eq. (7) can be given , and using the Eqs. (8)–(10) to simplify the new Eq. (7) [9].

(8)$$\delta (s )= \frac{1}{{2\pi }}\int {\exp ({ - isx} )} dx,$$

(9)$${\delta ^{(n )}}(s )= \frac{1}{{2\pi }}\int {{{({ - ix} )}^n}\exp ({ - isx} )} dx\quad ({n = 0,\mbox{ }1,\mbox{ }2} ),$$

(10)$$\int {f(x )} {\delta ^{(n )}}(x )dx = {({ - 1} )^n}{f^{(n )}}(0 )\quad ({n = 0,\mbox{ }1,\mbox{ }2} ),$$

where δ(x) is the Dirac delta function and δ⁽ⁿ⁾(x) is the nth derivatives of δ(x); f(x) is the arbitrary function and f⁽ⁿ⁾(x) is the nth derivatives of f(x).

Finally, some second-order moments of EGSM array beams in the non-Kolmogorov turbulence turn out to be [9,23]:

(11)$$\begin{aligned} \langle{{\rho^2}} \rangle &= {{\langle{{\rho^2}} \rangle} _0} + 2{{\langle{\rho \cdot \theta } \rangle}_0}z + {{\langle{{\theta^2}} \rangle}_0}{z^2} + \frac{4}{3}{\pi ^2}T{z^3}\\ &= \frac{1}{P}\left[ {A_x^2{B_{xx}}\sum\limits_{i = - {{({N - 1} )}/ 2}}^{(N - 1)/2} {\sum\limits_{j = - {{({N - 1} )}/ 2}}^{{{({N - 1} )}/ 2}} {({2\pi \sigma_{0x}^4 + 2{i^2}x_0^2\sigma_{0x}^2\pi } )} } } \right. + A_y^2{B_{yy}}\\ & \left. { \times \sum\limits_{i = - {{({N - 1} )} / 2}}^{{{({N - 1} )}/ 2}} {\sum\limits_{j = - {{({N - 1} )}/ 2}}^{{{({N - 1} )}/ 2}} {({2\pi \sigma_{0y}^4 + 2{i^2}x_0^2\sigma_{0y}^2\pi } )} } } \right]\\ & + \frac{1}{P}\left[ {A_x^2{B_{xx}}\sum\limits_{i = - {{({N - 1} )}/ 2}}^{{{({N - 1} )}/ 2}} {\sum\limits_{j = - {{({N - 1} )}/ 2}}^{{{({N - 1} )}/ 2}} {({2\pi \sigma_{0x}^4 + 2{j^2}y_0^2\sigma_{0x}^2\pi } )} } } \right.\\ & + \left. {A_y^2{B_{yy}}\sum\limits_{i = - {{({N - 1} )}/ 2}}^{{{({N - 1} )}/ 2}} {\sum\limits_{j = - {{({N - 1} )} / 2}}^{{{({N - 1} )} /2}} {({2\pi \sigma_{0y}^4 + 2{j^2}y_0^2\sigma_{0y}^2\pi } )} } } \right]\\ & + \frac{{2{z^2}}}{{P{k^2}}}\left[ {A_x^2{B_{xx}}\sum\limits_{i = - {{({N - 1} )} / 2}}^{{{({N - 1} )}/ 2}} {\sum\limits_{j = - {{({N - 1} )}/ 2}}^{{{({N - 1} )}/ 2}} {\left( {\frac{1}{{4\sigma_{0x}^2}} + \frac{1}{{\delta_{0xx}^2}}} \right)} } } \right.\\ & + \left. {A_y^2{B_{yy}}\sum\limits_{i = - {{({N - 1} )} / 2}}^{{{({N - 1} )} /2}} {\sum\limits_{j = - {{({N - 1} )} / 2}}^{{{({N - 1} )} / 2}} {\left( {\frac{1}{{4\sigma_{0y}^2}} + \frac{1}{{\delta_{0yy}^2}}} \right)} } } \right] + \frac{4}{3}{\pi ^2}T{z^3}, \end{aligned}$$

(12)$$\begin{aligned} \langle{{\theta^2}} \rangle &= {\langle{\theta_x^2} \rangle _0} + {\langle{\theta_y^2} \rangle _0} + 4{\pi ^2}Tz\\ &= \frac{2}{{P{k^2}}}\left[ {A_x^2{B_{xx}}\sum\limits_{i = - {{({N - 1} )}/ 2}}^{{{({N - 1} )} /2}} {\sum\limits_{j = - {{({N - 1} )} /2}}^{{{({N - 1} )}/ 2}} {\left( {\frac{1}{{4\sigma_{0x}^2}} + \frac{1}{{\delta_{0xx}^2}}} \right)} } } \right. + A_y^2{B_{yy}}\\ & \times \left. {\sum\limits_{i = - {{({N - 1} )} / 2}}^{{{({N - 1} )}/ 2}} {\sum\limits_{j = - {{({N - 1} )}/ 2}}^{{{({N - 1} )}/ 2}} {\left( {\frac{1}{{4\sigma_{0y}^2}} + \frac{1}{{\delta_{0yy}^2}}} \right)} } } \right] + 4{\pi ^2}Tz, \end{aligned}$$

(13)$$\begin{aligned} \langle{\rho \cdot \theta } \rangle &= {{\langle{{\theta^2}} \rangle}_0}\,z + 2{\pi ^2}T{z^2}\\ &= \frac{{2z}}{{P{k^2}}}\left[ {A_x^2{B_{xx}}\sum\limits_{i = - {{({N - 1} )}/ 2}}^{{{({N - 1} )} /2}} {\sum\limits_{j = - {{({N - 1} )} /2}}^{{{({N - 1} )} /2}} {\left( {\frac{1}{{4\sigma_{0x}^2}} + \frac{1}{{\delta_{0xx}^2}}} \right)} } } \right. + A_y^2{B_{yy}}\\ & \times \left. {\sum\limits_{i = - {{({N - 1} )} / 2}}^{{{({N - 1} )}/ 2}} {\sum\limits_{j = - {{({N - 1} )} / 2}}^{{{({N - 1} )} /2}} {\left( {\frac{1}{{4\sigma_{0y}^2}} + \frac{1}{{\delta_{0yy}^2}}} \right)} } } \right] + 2{\pi ^2}T{z^2}, \end{aligned}$$

(14)$$P = A_x^2{B_{xx}}\sum\limits_{i = - {{({N - 1} )}/ 2}}^{{{({N - 1} )} / 2}} {\sum\limits_{j = - {{({N - 1} )}/ 2}}^{{{({N - 1} )}/ 2}} {2\pi \sigma _{0x}^2} } + A_y^2{B_{yy}}\sum\limits_{i = - {{({N - 1} )} / 2}}^{{{({N - 1} )}/ 2}} {\sum\limits_{j = - {{({N - 1} )}/ 2}}^{{{({N - 1} )}/ 2}} {2\pi \sigma _{0y}^2} } .$$

The turbulence quantity T can be given by [28]:

(15)$$T = \int_0^\infty {{\kappa ^3}{\Phi _n}({\kappa ,\alpha } )d\kappa }, $$

where Φn (κ,α) denotes the spatial power spectrum of the refractive-index fluctuations of the turbulent medium. For the non-Kolmogorov case, Φn (κ,α) can be shown as:

(16)$${\Phi _n}({\kappa ,\alpha } )= \frac{{A(\alpha )C_n^2}}{{{{[{{\kappa^2} + {{({{{2\pi }/ {{L_0}}}} )}^2}} ]}^{{\alpha / 2}}}}}\exp \left\{ { - \frac{{{\kappa^2}}}{{{{[{{{c(\alpha )} / {{l_0}}}} ]}^2}}}} \right\}\;,$$

where C_n² denotes the generalized structure parameter with units m^3-α, α (3<α<4) is the generalized exponent parameters. L₀ and l₀ represent outer scale of turbulence and inner scale of turbulence, respectively. κ (0≤κ<∞) is the magnitude of the spatial wavenumber.

Besides, A(α) and c(α) are shown as:

(17)$$A(\alpha )= \frac{1}{{4{\pi ^2}}}\Gamma ({\alpha - 1} )\cos \left( {\frac{{\alpha \pi }}{2}} \right),$$

(18)$$c(\alpha )= {\left[ {\frac{{2\pi }}{3}A(\alpha )\Gamma \left( {\frac{{5 - \alpha }}{2}} \right)} \right]^{{1 / {\alpha - 5}}}}\,.$$

where Γ() is the Gamma function.

Therefore,

(19)$$T(\alpha )= \frac{{A(\alpha )\tilde{C}_n^2}}{{2({\alpha - 2} )}}\left\{ {[{2\kappa_0^2 + ({\alpha - 2} )\kappa_m^2} ]\kappa_m^{2 - \alpha }\exp \left( {\frac{{\kappa_0^2}}{{\kappa_m^2}}} \right)\Gamma \left( {2 - \frac{\alpha }{2},\frac{{\kappa_0^2}}{{\kappa_m^2}}} \right) - 2\kappa_0^{4 - \alpha }} \right\},$$

κ_m=c(α)/l₀, κ₀=2π/L₀, and Γ(.,.) is the incomplete Gamma function.

On the basis of the definition of the M²-factor, it is listed as [9]:

(20)$$\begin{aligned} {M^2}(z )&= k{\left[{\langle{{\rho^2}} \rangle \langle{{\theta^2}} \rangle - {{({\langle{\rho \cdot \theta } \rangle } )}^2}} \right]^{{1 / 2}}}\\ &= k\left\{ \left\{ \frac{1}{P}\left[ {A_x^2{B_{xx}}\sum\limits_{i = - {{({N - 1} )} / 2}}^{{{({N - 1} )}/ 2}} {\sum\limits_{j = - {{({N - 1} )}/ 2}}^{{{({N - 1} )}/ 2}} {\left({2\pi \sigma_{0x}^4 + 2{i^2}x_0^2\sigma_{0x}^2\pi } \right)} } } \right.\right.\right.\\ &\left.\left.\left. + A_y^2{B_{yy}} { \times \sum\limits_{i = - {{({N - 1} )} / 2}}^{{{({N - 1} )}/ 2}} {\sum\limits_{j = - {{({N - 1} )} / 2}}^{{{({N - 1} )} / 2}} {({2\pi \sigma_{0y}^4 + 2{i^2}x_0^2\sigma_{0y}^2\pi } )} } } \right] \right.\right.\\ &\left.\left. + \frac{1}{P}\left[ A_x^2{B_{xx}}\sum\limits_{i = - {{({N - 1} )}/ 2}}^{{{({N - 1} )} /2}} \sum\limits_{j = - {{({N - 1} )} / 2}}^{{{({N - 1} )} / 2}} \left({2\pi \sigma_{0x}^4 + 2{j^2}y_0^2\sigma_{0x}^2\pi } \right) \right.\right.\right.\\ &\left.\left.\left.+ {A_y^2{B_{yy}}\sum\limits_{i = - {{({N - 1} )} / 2}}^{{{({N - 1} )} / 2}} {\sum\limits_{j = - {{({N - 1} )} / 2}}^{{{({N - 1} )}/ 2}} {\left({2\pi \sigma_{0y}^4 + 2{j^2}y_0^2\sigma_{0y}^2\pi } \right)} } } \right]\right.\right.\\ &\left.\left. + \frac{{2{z^2}}}{{P{k^2}}}\left[ A_x^2{B_{xx}}\sum\limits_{i = - {{({N - 1} )} / 2}}^{{{({N - 1} )}/ 2}} {\sum\limits_{j = - {{({N - 1} )} / 2}}^{{{({N - 1} )}/ 2}} {\left( {\frac{1}{{4\sigma_{0x}^2}} + \frac{1}{{\delta_{0xx}^2}}} \right)} } \right.\right.\right.\\ &\left.\left.\left.+ {A_y^2{B_{yy}}\sum\limits_{i = - {{({N - 1} )}/ 2}}^{{{({N - 1} )} / 2}} {\sum\limits_{j = - {{({N - 1} )} / 2}}^{{{({N - 1} )} / 2}} {\left( {\frac{1}{{4\sigma_{0y}^2}} + \frac{1}{{\delta_{0yy}^2}}} \right)} } } \right] + {\frac{4}{3}{\pi^2}T{z^3}} \right\}\right.\\ &\left.\times \left\{ {\frac{2}{{P{k^2}}}} \left[ A_x^2{B_{xx}}\sum\limits_{i = - {{({N - 1} )} / 2}}^{{{({N - 1} )} / 2}} {\sum\limits_{j = - {{({N - 1} )}/ 2}}^{{{({N - 1} )}/ 2}} {\left( {\frac{1}{{4\sigma_{0x}^2}} + \frac{1}{{\delta_{0xx}^2}}} \right)} } \right.\right.\right.\\ &\left.\left.\left. + A_y^2{B_{yy}} {\sum\limits_{i = - {{({N - 1} )}/ 2}}^{{{({N - 1} )}/ 2}} {\sum\limits_{j = - {{({N - 1} )} / 2}}^{{{({N - 1} )} / 2}} {\left( {\frac{1}{{4\sigma_{0y}^2}} + \frac{1}{{\delta_{0yy}^2}}} \right)} } } \right] + 4{\pi^2}Tz \right\}\right.\\ &\left. - \left\{ {\frac{{2z}}{{P{k^2}}}} \left[ A_x^2{B_{xx}}\sum\limits_{i = - {{({N - 1} )}/ 2}}^{{{({N - 1} )}/ 2}} {\sum\limits_{j = - {{({N - 1} )} / 2}}^{{{({N - 1} )}/ 2}} {\left( {\frac{1}{{4\sigma_{0x}^2}} + \frac{1}{{\delta_{0xx}^2}}} \right)} } \right.\right.\right.\\ &\left.\left.\left.+ A_y^2{B_{yy}}{\sum\limits_{i = - {{({N - 1} )}/ 2}}^{{{({N - 1} )}/ 2}} {\sum\limits_{j = - {{({N - 1} )}/ 2}}^{{{({N - 1} )}/ 2}} {\left( {\frac{1}{{4\sigma_{0y}^2}} + \frac{1}{{\delta_{0yy}^2}}} \right)} } } \right] + 2{\pi^2}T{z^2} \right\}^2\right\}^{1/2}. \end{aligned}$$

For the sake of comparison, the M_f² indicates the M²-factor of EGSM beams through free space, i.e., T = 0. And the relative M²-factor is introduced as follows:

(21)$$M_r^2(z )= \frac{{{M^2}(z )}}{{M_f^2(0 )}}.$$

The variance of this displacement of the instantaneous center of the beam can be described by beam wander, and the model of beam wander can be expressed by [25]:

(22)$$\langle{r_c^2} \rangle = 4{\pi ^2}{k^2}W_{\textrm{FS}}^2\int_0^L {\int_0^\infty {\kappa {\Phi _n}(\kappa )\exp ({ - {\kappa^2}W_{\textrm{LT}}^2} )\left[ {1 - \exp \left( { - \frac{{2{L^2}{\kappa^2}{{({1 - {z / L}} )}^2}}}{{{k^2}W_{\textrm{FS}}^2}}} \right)} \right]} } d\kappa dz\;.$$

where W_FS represents the beam widths in the presence of turbulence, and W_LT denotes the beam widths in the absence of turbulence. z is the distance of an intercept point from the input plane at z = 0, and L is the total propagation distance.

Substituting Eq. (16)–Eq. (18) into Eq. (22), and we can simplify Eq. (22) as follows:

(23)$$\begin{aligned} \langle{r_c^2} \rangle &= \frac{{4{\pi ^2}C_n^2A(\alpha ){L^2}}}{{\alpha - 2}}\kappa _0^{ - \alpha }{\int_0^L {\left( {1 - \frac{z}{L}} \right)} ^2}\left\{\vphantom{{{\left( {\frac{{{\kappa_0}}}{{{\kappa_m}}}} \right)}^2}}{ - 2\kappa_0^4} + \kappa _0^\alpha \kappa _m^2{\left({W_{LT}^2 + \kappa_m^{ - 2}} \right)^{{\alpha / 2}}}[{2\kappa_0^2\left({1 + \kappa_m^2W_{LT}^2} \right)} \right.\\ &\left. { + ({\alpha - 2} )\kappa_m^2} ]\times {({1 + \kappa_m^2W_{LT}^2} )^{ - 2}}\exp \left[ {{{\left( {\frac{{{\kappa_0}}}{{{\kappa_m}}}} \right)}^2} + \kappa_0^2W_{LT}^2} \right] {\Gamma \left[ {2 - \frac{\alpha }{2},{{\left( {\frac{{{\kappa_0}}}{{{\kappa_m}}}} \right)}^2} + \kappa_0^2W_{LT}^2} \right]} \right\}dz. \end{aligned}$$

The beam width W_LT is shown as [28]:

(24)$$W_{LT}^2 = \langle{{\rho^2}} \rangle = {{\langle{{\rho^2}} \rangle}_0} + 2{{\langle{\rho \cdot \theta } \rangle}_0}z + {{\langle{{\theta^2}} \rangle}_0}{z^2} + \frac{4}{3}T{\pi ^2}{z^3}.$$

Thus, the beam wander B_w and the relative beam wander B_wr of EGSM array beams in non-Kolmogorov can be shown as:

(25)$${B_w} = {\left[{\langle{r_c^2} \rangle } \right]^{{1 /2}}},$$

(26)$${B_{wr}} = {\left[{{{\langle{r_c^2} \rangle }/ {W_{LT}^2}}} \right]^{{1 /2}}}.$$

In addition, the effect of the initial degree of polarization on the beam propagation is discussed in this paper. And the initial degree of polarization of the initial source beam at point ρ′ can be expressed as

(27)$${P_0}({{\boldsymbol{\rho}^{\prime}};0} )= \sqrt {1 - \frac{{4Det\mathord{\buildrel{{\leftrightarrow}} \over W} ({\boldsymbol{\rho} ^{\prime}},{\boldsymbol{\rho} ^{\prime}};0)}}{{{{\left[ {Tr\mathord{\buildrel{{\leftrightarrow}} \over W} ({{\boldsymbol{\rho}^{\prime}},{\boldsymbol{\rho}^{\prime}};0} )} \right]}^2}}}} ,$$

where we assumed A_y² ≤A_x², so we can obtain 0 ≤ P₀≤1.

3. Numerical examples

The numerical calculation results are given in Figs. 2–5 to show the propagation factor and beam wander of EGSM array beams in non-Kolmogorov turbulence, where λ=632.8 nm is kept fixed.

Figure 2 shows that the relative M²-factor of EGSM array beams versus the propagation distance z in non-Kolmogorov turbulence. From Fig. 2(a) it can be found that the relative M²-factor with larger σ_0x is less affected by turbulence. It could be found that the relative M²-factor increase with decreasing of N from Fig. 2(b). Besides, the relative M²-factor increases fast when the propagation distance z is more than 2 km. It can be seen from Fig. 2(d) that the relative M²-factor increases slowly as the outer scale of turbulence is more than 50 m. It indicates that the relative M²-factor with smaller L₀ is less affected by turbulence. Figs. 2(c), 2(e)–2(h) show that the relative M²-factor increases obviously with larger the generalized structure parameter, and with smaller the initial degree of polarization, inner scale of turbulence, the separation distances (x₀, y₀), and the generalized exponent parameter. It should be noticed that the separation distances have a great influence on the relative M²-factor from Fig. 2(f).

Fig. 2. Relative M²-factor vs. propagation distance z, λ=632.8 nm, δ_0xx=5 mm, δ_0yy=3 mm, (a) N = 3, l₀=20 mm, L₀=50 m, C_n²=10⁻¹⁴m^3-α, x₀=y₀=10 mm, α=11/3, P₀=0.5. (b) σ_0x=10 mm, σ_0y=5 mm, l₀=20 mm, L₀=50 m, C_n²=10⁻¹⁴m^3-α, x₀=y₀=10 mm, α=11/3, P₀=0.5. (c) σ_0x=10 mm, σ_0y=5 mm, N = 3, L₀=50 m, C_n²=10⁻¹⁴m^3-α, x₀=y₀=10 mm, α=11/3, P₀=0.5. (d) σ_0x=10 mm, σ_0y=5 mm, N = 3, l₀=20 mm, C_n²=10⁻¹⁴m^3-α, x₀=y₀=10 mm, α=11/3, P₀=0.5. (e) σ_0x=10 mm, σ_0y=5 mm, N = 3, l₀=20 mm, L₀=50 m, x₀=y₀=10 mm, α=11/3, P₀=0.5. (f) σ_0x=10 mm, σ_0y=5 mm, N = 3, l₀=20 mm, L₀=50 m, C_n²=10⁻¹⁴m^3-α, α=11/3, P₀=0.5. (g) σ_0x=10 mm, σ_0y=5 mm, N = 3, l₀=20 mm, L₀=50 m, C_n²=10⁻¹⁴m^3-α, x₀=y₀=10 mm, P₀=0.5. (h) σ_0x=10 mm, σ_0y=5 mm, N = 3, l₀=20 mm, L₀=50 m, C_n²=10⁻¹⁴m^3-α, x₀=y₀=10 mm, α=11/3.

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Figure 3 shows the relative M²-factor of EGSM array beams versus the generalized exponent parameter α propagating through the non-Kolmogorov turbulence. From Fig. 3(a)–3(d) it could be found that the relative M²-factor increases rapidly as α increases from 3 to 3.1, and it is also interesting to find that the relative M²-factor decreases as α increases from 3.1 to 4. It is clear that the relative M²-factor has a maximum when α is about 3.1. When α >3.6, it can be easily seen that the relative M²-factor is less sensitive to the change of l₀ [from Fig. 3(a)]. However, it is more sensitive to the change of L₀ [from Fig. 3(b)]. Moreover, it is clearly seen from Figs. 3(c)–3(d) that as α increases the relative M²-factor increases with the decreasing of separation distances (x₀, y₀) and the initial degree of polarization.

Fig. 3. Relative M²-factor vs. the generalized exponent parameter α, λ=632.8 nm, N = 3, δ_0xx=5 mm, δ_0yy=3 mm, σ_0x=10 mm, σ_0y=5 mm, z = 10 km, (a) P₀=0.5, x₀=y₀=10 mm, L₀=50 m, C_n²=10⁻¹⁴m^3-α. (b) P₀=0.5, x₀=y₀=10 mm, l₀=20 mm, C_n²=10⁻¹⁴m^3-α. (c) P₀=0.5, C_n²=10⁻¹⁴m^3-α, l₀=20 mm, L₀=50 m. (d) C_n²=10⁻¹⁴m^3-α, x₀=y₀=10 mm, l₀=20 mm, L₀=50 m.

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Figures 4(a)–4(b) give the rms beam wander and the relative beam wander for different the parameters versus the generalized exponent parameter α in non-Kolmogorov turbulence. From Fig. 4(a) it could be found that the relative beam wander increases obviously with the larger C_n². Figure 4(b) shows that the relative beam wander increases rapidly with 3<α<3.2, and then increases slowly with 3.2<α<4. Besides, Fig. 4(b) implies that the relative beam wander B_wr<1 which means the influence of turbulence on the rms beam wander is less than that on beam spreading. It can be found from Fig. 4(d) that the rms beam wander is very obvious as the increase of α when L = 10 km. However, Fig. 4(c) indicates that the beam wander is much smaller than the beam spreading. When the 3<α<3.6, the beam wander increases obviously as the separation distances (x₀, y₀) decrease in Fig. 4(d).

Fig. 4. The rms beam wander B_w and the relative beam wander B_wr vs. the generalized exponent parameter α, λ=632.8 nm, N = 3, δ_0xx=5 mm, δ_0yy=3 mm, σ_0x=10 mm, σ_0y=5 mm, l₀=20 mm, L₀=50 m, L = 10 km (a) P₀=0.5, x₀=y₀=10 mm. (b) x₀=y₀=10 mm, C_n²=10⁻¹⁴m^3-α. (c) and (d) P₀=0.5, C_n²=10⁻¹⁴m^3-α.

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Figures 5(a)–5(f) reveal the rms beam wander and the relative beam wander vs. L for different parameters. It can be found that the rms beam wander increases with the increase of α from Fig. 5(a). Besides, the generalized structure parameter C_n² is larger, and the relative beam wander is larger from Fig. 5(b). And it is also can be seen from Fig. 5(c) that the relative beam wander increases rapidly when L is about less than 2 km, and then increases slowly when L is larger than 2 km less than 15 km. It is indicated that the relative beam wander of the long distance transmission tends to be stable. Figure 5(a) and Fig. 5(c) imply that the influence of α on the rms beam wander in comparison with the relative beam wander is obvious. Figure 5 (d) – Fig. 5(f) show that the relative beam wander increases with the increasing of initial beam widths, the initial degree of polarization and the decreasing of the separation distances (x₀, y₀). And it is also can be seen from Fig. 5(f) that the effect of the separation distances (x₀, y₀) on the relative beam wander is small.

Fig. 5. The beam wander and the relative rms beam wander B_wr vs. propagation distance L λ=632.8 nm, N = 3, δ_0xx=5 mm, δ_0yy=3 mm, l₀=20 mm, L₀=50 m. (a) and (c) C_n²=10⁻¹⁴m^3-α, σ_0x=10 mm, σ_0y=5 mm, P₀=0.5, x₀=y₀=10 mm. (b) α=11/3, σ_0x=10 mm, σ_0y=5 mm, P₀=0.5, x₀=y₀=10 mm, (d) α=11/3, C_n²=10⁻¹⁴m^3-α, P₀=0.5, x₀=y₀=10 mm. (e) α=11/3, C_n²=10⁻¹⁴m^3-α, σ_0x=10 mm, σ_0y=5 mm, x₀=y₀=10 mm. (f) α=11/3, C_n²=10⁻¹⁴m^3-α, σ_0x=10 mm, σ_0y=5 mm, P₀=0.5.

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4. Conclusions

In conclusion, the M²-factor and beam wander of the EGSM array beams in non-Kolmogorov have been investigated. The results indicate that the relative M²-factor increases with increasing of outer scale of turbulence (L₀), the generalized structure parameter (C_n²), and with decreasing of beams order (N), the initial degree of polarization (P₀), initial beam widths (σ_0x, σ_0y), inner scale of turbulence (l₀), the separation distances (x₀, y₀), the generalized exponent parameter (α). And the relative beam wander increases rapidly with lager α, σ_0x, σ_0y, C_n², P₀, with smaller x₀, y₀. It also is shown that the relative M²-factor first increases rapidly as α increases from 3 to 3.1, and then decreases as α increases from 3.1 to 4. Otherwise, the relative beam wander increases rapidly with 3<α<3.2, and then increases slowly with 3.2<α<4. And the influence of α on the beam wander in comparison with the relative beam wander is obvious. Besides, It could be found that beam spreading is more effected by turbulence than the rms beam wander.

Funding

Young scholar for reserve talents of Xihua University (0220170303); Education Department of Sichuan Province (16ZA0160); National Natural Science Foundation of China (NSFC) (U1433127); Civil Aviation Administration of China (CAAC) (U1433127); Civil Aviation University of China (CAUC) (JG2016-17, JG2017-17, JG2018-17).

Acknowledgments

Thanks to Xihua University for supporting the paper.

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Propagation factor and beam wander of electromagnetic Gaussian Schell-model array beams in non-Kolmogorov turbulence

Abstract

1. Introduction

2. Theoretical model

3. Numerical examples

4. Conclusions

Funding

Acknowledgments

References

Cited By

Figures (5)

Equations (27)

OSA Continuum