Abstract

The propagation formulae for the propagation factor (known as M2-factor) and beam wander of electromagnetic Gaussian Schell-model (EGSM) array beams in non-Kolmogorov turbulence are derived by using the extended Huygens-Fresnel principle and the second-order moments of the Wigner distribution function. The results indicate that the M2-factor and beam wander depend on the beam parameters and turbulence parameters, and the relative M2-factor has a maximum when the generalized exponent parameter α is equal to 3.1. Otherwise, the changes of the separation distances (x0, y0) have great influence on the relative M2-factor. The relative beam wander increases rapidly when 3<α<3.2; however, it increases slowly when 3.2<α<4. It is also shown that the beam spreading of EGSM array beams is more affected by turbulence than the root mean square beam wander.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Recently, more and more literatures are related to the laser array beams due to the varied applications [14]. 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 [1119]. 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 [11,2022]. 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 [2628]. 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 [28,3135]. 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).

 figure: 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]:

$$\overleftrightarrow W({{{{\boldsymbol{\rho}^{\prime}_1}}},{{{\boldsymbol{\rho}^{\prime}_2}}},0} )= \left[ {\begin{array}{cc} {{W_{xx}}({{{{\boldsymbol{\rho}^{\prime}_1}}},{{{\boldsymbol{\rho}^{\prime}_2}}},0} )} & {{W_{xy}}({{{{\boldsymbol{\rho}^{\prime}_1}}},{{{\boldsymbol{\rho}^{\prime}_2}}},0} )}\\ {{W_{yx}}({{{{\boldsymbol{\rho}^{\prime}_1}}},{{{\boldsymbol{\rho}^{\prime}_2}}},0} )} & {{W_{yy}}({{{{\boldsymbol{\rho}^{\prime}_1}}},{{{\boldsymbol{\rho}^{\prime}_2}}},0} )} \end{array}} \right],$$
where ρ1′= (x1′,y1′) and ρ2′=(x2′,y2′) are arbitrary transverse position vectors at the source plane z = 0. Wpq(ρ1′,ρ2′,0)=〈${E_{p}^\ast}$(ρ1′,0)Eq(ρ2′,0)〉 (p = x,y; q = x,y), Ex and Ey 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]

$$\begin{aligned}{W_{pq}}({{{\rho^{\prime}_1}},{{\rho^{\prime}_2}},0} ) &= {A_p}{A_q}{B_{pq}} \times {\sum\limits_{i = {{ - ({N - 1} )} /2}}^{{(N-1)}/2} {{\sum\limits_{{{j = - ({N - 1} )} / 2}}^{{(N-1)}/2} {\exp \left( { - \frac{{{{({{{x^{\prime}_1}} - i{x_0}} )}^2} + {{({{{y^{\prime}_1}} - j{y_0}} )}^2}}}{{4\sigma_{0p}^2}}} \right)} }}} \\ & {{\times \exp \left( { - \frac{{{{({{{x^{\prime}_2}} - i{x_0}} )}^2} + {{({{{y^{\prime}_2}} - j{y_0}} )}^2}}}{{4\sigma_{0q}^2}}} \right)\exp \left( { - \frac{{\rho_d^2}}{{2\delta_{0pq}^2}}} \right),}}\end{aligned}$$
where Ax and Ay are the amplitudes of the electric field for x and y components. δ0pq (p = q) is the rms width of auto-correlation functions of the x components of the field, or y components of the field, respectively; δ0pq (p≠q) is the rms width of the mutual correlation function of x and y field components. σ0j denotes the rms width of the spectral density along j direction. Bpq denotes the complex correlation coefficient, and Bpq=1 (p = q), Bpq=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]:

$${\overleftrightarrow W_{Tr}}({{{{\boldsymbol{\rho}^{\prime}_1}}},{{{\boldsymbol{\rho}^{\prime}_2}}},0} )= {W_{xx}}({{{{\boldsymbol{\rho}^{\prime}_1}}},{{{\boldsymbol{\rho}^{\prime}_2}}},0} )+ {W_{yy}}({{{{\boldsymbol{\rho}^{\prime}_1}}},{{{\boldsymbol{\rho}^{\prime}_2}}},0} ),$$
where we assume that the electric field Ex and Ey 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],

$${{\boldsymbol{\rho} ^{\prime}_1}} = {\boldsymbol{\rho} ^{\prime}} + {{{{{\boldsymbol{\rho} ^{\prime}_d}}}} {/} 2};{{\boldsymbol{\rho} ^{\prime}_2}} = {\boldsymbol{\rho} ^{\prime}} - {{{{{\boldsymbol{\rho} ^{\prime}_d}}}}/2} ,$$
Therefore, Eq. (3) can be written as:
$$\begin{aligned} {\overleftrightarrow W_{Tr}}({{\boldsymbol{\rho}^{\prime},}{{{\boldsymbol{\rho}^{\prime}_d}}},0} )&= A_x^2{B_{xx}}\sum\limits_{i = {{ - ({N - 1} )} / 2}}^{{{({N - 1} )} {/2}}}\sum\limits_{j = {{- ({N - 1} )} / 2}}^{{{({N - 1} )} {/2}}}{\exp \left({\frac{- 2{{x^{\prime 2}}}- {{{x^{\prime 2}_d}}} /{2 - 2{i^2}}{{x}_{0}^2} + 4i{x}_{0}{x}^{\prime}} {{4{\sigma_{0x}^2}}}}\right)} \\ & \times {\exp \left({\frac{ - 2{{y^{\prime 2}}} - {{{y^{\prime 2}_d}}}/{2 - 2{j^2}}{{y}_{0}^2} + 4j{y}_{0}{y}^{\prime}}{{4{\sigma_{0x}^2}}}} \right)} \times\exp \left( { - \frac{{{\rho_d^2}}}{{2{\delta_{0xx}^2}}}} \right) \\ & + A_y^2{B_{yy}}\sum\limits_{i = {{ - ({N - 1} )} /2}}^{{{({N - 1} )} {/2}}}\sum\limits_{j = {{- ({N - 1} )} /2}}^{{{({N - 1} )} {/ 2}}}{\exp \left( {\frac{ - 2{{x^{\prime 2}}} - {{{x^{\prime 2}_d}}} /{2 - 2{i^2}}{{x}_{0}^2} + 4i{{x}_{0}}{x}^{\prime}}{{4{\sigma_{0y}^2}}}} \right)}\\ & \times {\exp \left( {\frac{ - 2{{y^{\prime 2}}} - {{{y^{\prime 2}_d}}} /{2 - 2{j^2}}{{y}_{0}^2} + 4j{{y}_{0}}{y}^{\prime}}{{4{\sigma_{0y}^2}}}} \right)}\times \exp \left( {\frac{{{ -\rho_d^2}}}{{2{\delta_{0yy}^2}}}} \right).\end{aligned}$$
According to the trace of the CSD, the WDF can be written as [9,23]:
$$h({{\boldsymbol{\rho} },{\boldsymbol{\theta}},0} )= {\left( {\frac{k}{{2\pi }}} \right)^2}\int {{{\overleftrightarrow W}_{Tr}}({{\boldsymbol{\rho} },{{\boldsymbol{\rho} }_d},0} )\exp ({ - ik{\boldsymbol{\theta}} \cdot {{\boldsymbol{\rho} }_d}} ){d^2}{\rho _d}} ,$$
Here, k = 2π/λ, k represents the wave number and λ is the wavelength. θ = (θx, θy).

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

$${{\langle{{x^{{n_1}}}{y^{{n_2}}}\theta_x^{{m_1}}\theta_y^{{m_2}}} \rangle}_0} = \frac{1}{P}\int\!\!\!\int {{x^{{n_1}}}{y^{{n_2}}}\theta _x^{{m_1}}\theta _y^{{m_2}}h({{\boldsymbol{\rho} },{\boldsymbol{\theta}},0} )} {d^2}\rho {d^2}\theta ,$$
where P=∫∫h(ρ,θ,0)d2ρd2θ denotes the total power of beam and we can get the second-order moments 〈x20, 〈y20, 〈x0, 〈y0, 〈θx20, and 〈θy20 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].

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

$$\begin{aligned} \langle{{\rho^2}} \rangle &= {{\langle{{\rho^2}} \rangle} _0} + 2{{\langle{\rho \cdot \theta } \rangle}_0}z + {{\langle{{\theta^2}} \rangle}_0}{z^2} + \frac{4}{3}{\pi ^2}T{z^3}\\ &= \frac{1}{P}\left[ {A_x^2{B_{xx}}\sum\limits_{i = - {{({N - 1} )}/ 2}}^{(N - 1)/2} {\sum\limits_{j = - {{({N - 1} )}/ 2}}^{{{({N - 1} )}/ 2}} {({2\pi \sigma_{0x}^4 + 2{i^2}x_0^2\sigma_{0x}^2\pi } )} } } \right. + A_y^2{B_{yy}}\\ & \left. { \times \sum\limits_{i = - {{({N - 1} )} / 2}}^{{{({N - 1} )}/ 2}} {\sum\limits_{j = - {{({N - 1} )}/ 2}}^{{{({N - 1} )}/ 2}} {({2\pi \sigma_{0y}^4 + 2{i^2}x_0^2\sigma_{0y}^2\pi } )} } } \right]\\ & + \frac{1}{P}\left[ {A_x^2{B_{xx}}\sum\limits_{i = - {{({N - 1} )}/ 2}}^{{{({N - 1} )}/ 2}} {\sum\limits_{j = - {{({N - 1} )}/ 2}}^{{{({N - 1} )}/ 2}} {({2\pi \sigma_{0x}^4 + 2{j^2}y_0^2\sigma_{0x}^2\pi } )} } } \right.\\ & + \left. {A_y^2{B_{yy}}\sum\limits_{i = - {{({N - 1} )}/ 2}}^{{{({N - 1} )}/ 2}} {\sum\limits_{j = - {{({N - 1} )} / 2}}^{{{({N - 1} )} /2}} {({2\pi \sigma_{0y}^4 + 2{j^2}y_0^2\sigma_{0y}^2\pi } )} } } \right]\\ & + \frac{{2{z^2}}}{{P{k^2}}}\left[ {A_x^2{B_{xx}}\sum\limits_{i = - {{({N - 1} )} / 2}}^{{{({N - 1} )}/ 2}} {\sum\limits_{j = - {{({N - 1} )}/ 2}}^{{{({N - 1} )}/ 2}} {\left( {\frac{1}{{4\sigma_{0x}^2}} + \frac{1}{{\delta_{0xx}^2}}} \right)} } } \right.\\ & + \left. {A_y^2{B_{yy}}\sum\limits_{i = - {{({N - 1} )} / 2}}^{{{({N - 1} )} /2}} {\sum\limits_{j = - {{({N - 1} )} / 2}}^{{{({N - 1} )} / 2}} {\left( {\frac{1}{{4\sigma_{0y}^2}} + \frac{1}{{\delta_{0yy}^2}}} \right)} } } \right] + \frac{4}{3}{\pi ^2}T{z^3}, \end{aligned}$$
$$\begin{aligned} \langle{{\theta^2}} \rangle &= {\langle{\theta_x^2} \rangle _0} + {\langle{\theta_y^2} \rangle _0} + 4{\pi ^2}Tz\\ &= \frac{2}{{P{k^2}}}\left[ {A_x^2{B_{xx}}\sum\limits_{i = - {{({N - 1} )}/ 2}}^{{{({N - 1} )} /2}} {\sum\limits_{j = - {{({N - 1} )} /2}}^{{{({N - 1} )}/ 2}} {\left( {\frac{1}{{4\sigma_{0x}^2}} + \frac{1}{{\delta_{0xx}^2}}} \right)} } } \right. + A_y^2{B_{yy}}\\ & \times \left. {\sum\limits_{i = - {{({N - 1} )} / 2}}^{{{({N - 1} )}/ 2}} {\sum\limits_{j = - {{({N - 1} )}/ 2}}^{{{({N - 1} )}/ 2}} {\left( {\frac{1}{{4\sigma_{0y}^2}} + \frac{1}{{\delta_{0yy}^2}}} \right)} } } \right] + 4{\pi ^2}Tz, \end{aligned}$$
$$\begin{aligned} \langle{\rho \cdot \theta } \rangle &= {{\langle{{\theta^2}} \rangle}_0}\,z + 2{\pi ^2}T{z^2}\\ &= \frac{{2z}}{{P{k^2}}}\left[ {A_x^2{B_{xx}}\sum\limits_{i = - {{({N - 1} )}/ 2}}^{{{({N - 1} )} /2}} {\sum\limits_{j = - {{({N - 1} )} /2}}^{{{({N - 1} )} /2}} {\left( {\frac{1}{{4\sigma_{0x}^2}} + \frac{1}{{\delta_{0xx}^2}}} \right)} } } \right. + A_y^2{B_{yy}}\\ & \times \left. {\sum\limits_{i = - {{({N - 1} )} / 2}}^{{{({N - 1} )}/ 2}} {\sum\limits_{j = - {{({N - 1} )} / 2}}^{{{({N - 1} )} /2}} {\left( {\frac{1}{{4\sigma_{0y}^2}} + \frac{1}{{\delta_{0yy}^2}}} \right)} } } \right] + 2{\pi ^2}T{z^2}, \end{aligned}$$
$$P = A_x^2{B_{xx}}\sum\limits_{i = - {{({N - 1} )}/ 2}}^{{{({N - 1} )} / 2}} {\sum\limits_{j = - {{({N - 1} )}/ 2}}^{{{({N - 1} )}/ 2}} {2\pi \sigma _{0x}^2} } + A_y^2{B_{yy}}\sum\limits_{i = - {{({N - 1} )} / 2}}^{{{({N - 1} )}/ 2}} {\sum\limits_{j = - {{({N - 1} )}/ 2}}^{{{({N - 1} )}/ 2}} {2\pi \sigma _{0y}^2} } .$$
The turbulence quantity T can be given by [28]:
$$T = \int_0^\infty {{\kappa ^3}{\Phi _n}({\kappa ,\alpha } )d\kappa }, $$
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:
$${\Phi _n}({\kappa ,\alpha } )= \frac{{A(\alpha )C_n^2}}{{{{[{{\kappa^2} + {{({{{2\pi }/ {{L_0}}}} )}^2}} ]}^{{\alpha / 2}}}}}\exp \left\{ { - \frac{{{\kappa^2}}}{{{{[{{{c(\alpha )} / {{l_0}}}} ]}^2}}}} \right\}\;,$$
where Cn2 denotes the generalized structure parameter with units m3-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:

$$A(\alpha )= \frac{1}{{4{\pi ^2}}}\Gamma ({\alpha - 1} )\cos \left( {\frac{{\alpha \pi }}{2}} \right),$$
$$c(\alpha )= {\left[ {\frac{{2\pi }}{3}A(\alpha )\Gamma \left( {\frac{{5 - \alpha }}{2}} \right)} \right]^{{1 / {\alpha - 5}}}}\,.$$
where Γ() is the Gamma function.

Therefore,

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

$$\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 Mf2 indicates the M2-factor of EGSM beams through free space, i.e., T = 0. And the relative M2-factor is introduced as follows:
$$M_r^2(z )= \frac{{{M^2}(z )}}{{M_f^2(0 )}}.$$
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]:
$$\langle{r_c^2} \rangle = 4{\pi ^2}{k^2}W_{\textrm{FS}}^2\int_0^L {\int_0^\infty {\kappa {\Phi _n}(\kappa )\exp ({ - {\kappa^2}W_{\textrm{LT}}^2} )\left[ {1 - \exp \left( { - \frac{{2{L^2}{\kappa^2}{{({1 - {z / L}} )}^2}}}{{{k^2}W_{\textrm{FS}}^2}}} \right)} \right]} } d\kappa dz\;.$$
where WFS represents the beam widths in the presence of turbulence, and WLT 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:

$$\begin{aligned} \langle{r_c^2} \rangle &= \frac{{4{\pi ^2}C_n^2A(\alpha ){L^2}}}{{\alpha - 2}}\kappa _0^{ - \alpha }{\int_0^L {\left( {1 - \frac{z}{L}} \right)} ^2}\left\{\vphantom{{{\left( {\frac{{{\kappa_0}}}{{{\kappa_m}}}} \right)}^2}}{ - 2\kappa_0^4} + \kappa _0^\alpha \kappa _m^2{\left({W_{LT}^2 + \kappa_m^{ - 2}} \right)^{{\alpha / 2}}}[{2\kappa_0^2\left({1 + \kappa_m^2W_{LT}^2} \right)} \right.\\ &\left. { + ({\alpha - 2} )\kappa_m^2} ]\times {({1 + \kappa_m^2W_{LT}^2} )^{ - 2}}\exp \left[ {{{\left( {\frac{{{\kappa_0}}}{{{\kappa_m}}}} \right)}^2} + \kappa_0^2W_{LT}^2} \right] {\Gamma \left[ {2 - \frac{\alpha }{2},{{\left( {\frac{{{\kappa_0}}}{{{\kappa_m}}}} \right)}^2} + \kappa_0^2W_{LT}^2} \right]} \right\}dz. \end{aligned}$$
The beam width WLT is shown as [28]:
$$W_{LT}^2 = \langle{{\rho^2}} \rangle = {{\langle{{\rho^2}} \rangle}_0} + 2{{\langle{\rho \cdot \theta } \rangle}_0}z + {{\langle{{\theta^2}} \rangle}_0}{z^2} + \frac{4}{3}T{\pi ^2}{z^3}.$$
Thus, the beam wander Bw and the relative beam wander Bwr of EGSM array beams in non-Kolmogorov can be shown as:
$${B_w} = {\left[{\langle{r_c^2} \rangle } \right]^{{1 /2}}},$$
$${B_{wr}} = {\left[{{{\langle{r_c^2} \rangle }/ {W_{LT}^2}}} \right]^{{1 /2}}}.$$
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
$${P_0}({{\boldsymbol{\rho}^{\prime}};0} )= \sqrt {1 - \frac{{4Det\mathord{\buildrel{{\leftrightarrow}} \over W} ({\boldsymbol{\rho} ^{\prime}},{\boldsymbol{\rho} ^{\prime}};0)}}{{{{\left[ {Tr\mathord{\buildrel{{\leftrightarrow}} \over W} ({{\boldsymbol{\rho}^{\prime}},{\boldsymbol{\rho}^{\prime}};0} )} \right]}^2}}}} ,$$
where we assumed Ay2 ≤Ax2, so we can obtain 0 ≤ P0≤1.

3. Numerical examples

The numerical calculation results are given in Figs. 25 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 σ0x 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).

 figure: Fig. 2.

Fig. 2. Relative M2-factor vs. propagation distance z, λ=632.8 nm, δ0xx=5 mm, δ0yy=3 mm, (a) N = 3, l0=20 mm, L0=50 m, Cn2=10−14m3-α, x0=y0=10 mm, α=11/3, P0=0.5. (b) σ0x=10 mm, σ0y=5 mm, l0=20 mm, L0=50 m, Cn2=10−14m3-α, x0=y0=10 mm, α=11/3, P0=0.5. (c) σ0x=10 mm, σ0y=5 mm, N = 3, L0=50 m, Cn2=10−14m3-α, x0=y0=10 mm, α=11/3, P0=0.5. (d) σ0x=10 mm, σ0y=5 mm, N = 3, l0=20 mm, Cn2=10−14m3-α, x0=y0=10 mm, α=11/3, P0=0.5. (e) σ0x=10 mm, σ0y=5 mm, N = 3, l0=20 mm, L0=50 m, x0=y0=10 mm, α=11/3, P0=0.5. (f) σ0x=10 mm, σ0y=5 mm, N = 3, l0=20 mm, L0=50 m, Cn2=10−14m3-α, α=11/3, P0=0.5. (g) σ0x=10 mm, σ0y=5 mm, N = 3, l0=20 mm, L0=50 m, Cn2=10−14m3-α, x0=y0=10 mm, P0=0.5. (h) σ0x=10 mm, σ0y=5 mm, N = 3, l0=20 mm, L0=50 m, Cn2=10−14m3-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.

 figure: Fig. 3.

Fig. 3. Relative M2-factor vs. the generalized exponent parameter α, λ=632.8 nm, N = 3, δ0xx=5 mm, δ0yy=3 mm, σ0x=10 mm, σ0y=5 mm, z = 10 km, (a) P0=0.5, x0=y0=10 mm, L0=50 m, Cn2=10−14m3-α. (b) P0=0.5, x0=y0=10 mm, l0=20 mm, Cn2=10−14m3-α. (c) P0=0.5, Cn2=10−14m3-α, l0=20 mm, L0=50 m. (d) Cn2=10−14m3-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 Cn2. 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 Bwr<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).

 figure: Fig. 4.

Fig. 4. The rms beam wander Bw and the relative beam wander Bwr vs. the generalized exponent parameter α, λ=632.8 nm, N = 3, δ0xx=5 mm, δ0yy=3 mm, σ0x=10 mm, σ0y=5 mm, l0=20 mm, L0=50 m, L = 10 km (a) P0=0.5, x0=y0=10 mm. (b) x0=y0=10 mm, Cn2=10−14m3-α. (c) and (d) P0=0.5, Cn2=10−14m3-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 Cn2 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.

 figure: Fig. 5.

Fig. 5. The beam wander and the relative rms beam wander Bwr vs. propagation distance L λ=632.8 nm, N = 3, δ0xx=5 mm, δ0yy=3 mm, l0=20 mm, L0=50 m. (a) and (c) Cn2=10−14m3-α, σ0x=10 mm, σ0y=5 mm, P0=0.5, x0=y0=10 mm. (b) α=11/3, σ0x=10 mm, σ0y=5 mm, P0=0.5, x0=y0=10 mm, (d) α=11/3, Cn2=10−14m3-α, P0=0.5, x0=y0=10 mm. (e) α=11/3, Cn2=10−14m3-α, σ0x=10 mm, σ0y=5 mm, x0=y0=10 mm. (f) α=11/3, Cn2=10−14m3-α, σ0x=10 mm, σ0y=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 (Cn2), and with decreasing of beams order (N), the initial degree of polarization (P0), initial beam widths (σ0x, σ0y), 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, σ0x, σ0y, Cn2, 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.

References

1. Y. Ai and Y. Dan, “Range of turbulence-negligible propagation of Gaussian Schell-model array beams,” Opt. Commun. 284(13), 3216–3220 (2011). [CrossRef]  

2. X. Li, X. Ji, and F. Yang, “Beam quality of radial Gaussian Schell-model array beams,” Opt. Laser Technol. 42(4), 604–609 (2010). [CrossRef]  

3. X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010). [CrossRef]  

4. Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16(22), 18437–18442 (2008). [CrossRef]  

5. X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009). [CrossRef]  

6. X. Ji and Z. Pu, “Angular spread of Gaussian Schell-model array beams propagating through atmospheric turbulence,” Appl. Phys. B 93(4), 915–923 (2008). [CrossRef]  

7. X. Ji, E. Zhang, and B. Lü, “Superimposed partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. B 25(5), 825–833 (2008). [CrossRef]  

8. Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278(1), 157–167 (2007). [CrossRef]  

9. Y. Dan and B. Zhang, “Beam Propagation Factor of Partially Coherent Flat-topped Beams in a Turbulent Atmosphere,” Opt. Express 16(20), 15563–15575 (2008). [CrossRef]  

10. F. Wang and Y. Cai, “Second-order statistics of a twisted Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 18(24), 24661–24672 (2010). [CrossRef]  

11. S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010). [CrossRef]  

12. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A: Pure Appl. Opt. 3(1), 1–9 (2001). [CrossRef]  

13. O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004). [CrossRef]  

14. E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, Cambridge, 2007).

15. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998). [CrossRef]  

16. G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17(11), 2019–2023 (2000). [CrossRef]  

17. B. Zhang, Y. Xu, Y. Dan, X. Deng, and Z. Zhao, “Beam spreading and M2-factor of electromagnetic Gaussian Schell-model beam propagating in inhomogeneous atmospheric turbulence,” Optik 149, 398–408 (2017). [CrossRef]  

18. B. Kanseri and H. C. Kandpal, “Experimental determination of electric cross-spectral density matrix and generalized Stokes parameters for a laser beam,” Opt. Lett. 33(20), 2410–2412 (2008). [CrossRef]  

19. H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32(15), 2215–2217 (2007). [CrossRef]  

20. B. Zhang, X. Chu, and Q. Li, “Generalized beam-propagation factor of partially coherent beams propagating through hard-edged apertures,” J. Opt. Soc. Am. A 19(7), 1370–1375 (2002). [CrossRef]  

21. X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A: Pure Appl. Opt. 11(10), 105705 (2009). [CrossRef]  

22. G. Zhou, “Generalzied M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. A: Pure Appl. Opt. 12(1), 015701 (2010). [CrossRef]  

23. Y. Dan and B. Zhang, “Second moments of partially coherent beams in atmospheric turbulence,” Opt. Lett. 34(5), 563–565 (2009). [CrossRef]  

24. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyubolu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009). [CrossRef]  

25. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed.; SPIE: Bellingham, 2005.

26. J. Recolons, L. C. Andrews, and R. L. Phillips, “Analysis of beam wander effects for a horizontal-path propagating Gaussian-beam wave: focused beam case,” Opt. Eng. 46(8), 086002 (2007). [CrossRef]  

27. Y. Huang, A. Zeng, Z. Gao, and B. Zhang, “Beam Wander of Partially Coherent Array Beams through Non-Kolmogorov Turbulence,” Opt. Lett. 40(8), 1619–1622 (2015). [CrossRef]  

28. Y. Xu, H. Tian, Y. Dan, H. Feng, and S. Wang, “Beam wander and M2-factor of partially coherent electromagnetic hollow Gaussian beam propagating through non-Kolmogorov turbulence,” J. Mod. Opt. 64(8), 844–854 (2017). [CrossRef]  

29. D. Y. Vasylyev, A. A. Semenov, and W. Vogel, “Toward Global Quantum Communication: Beam Wandering Preserves Nonclassicality,” Phys. Rev. Lett. 108(22), 220501 (2012). [CrossRef]  

30. W. Wen and X. Chu, “Beam wande of partially coherent Airy beams,” J. Mod. Opt. 61(5), 379–384 (2014). [CrossRef]  

31. G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: The effect of partial coherence,” Phys. Rev. E 76(5), 056606 (2007). [CrossRef]  

32. H. T. Eyyuboğlu and C. Z. Cil, “Beam wander of dark hollow, flat-topped and annular beams,” Appl. Phys. B 93(2-3), 595–604 (2008). [CrossRef]  

33. C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95(4), 763–771 (2009). [CrossRef]  

34. C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J(0)- and I(0)-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98(1), 195–202 (2010). [CrossRef]  

35. S. Yu, Z. Chen, T. Wang, G. Wu, H. Guo, and W. Gu, “Beam wander of electromagnetic Gaussian-Schell model beams propagating in atmospheric turbulence,” Appl. Opt. 51(31), 7581–7585 (2012). [CrossRef]  

36. G. Wu, W. Dai, H. Tang, and H. Guo, “Beam wander of random EGSM vortex beams propagating through a Kolmogorov turbulence,” Opt. Commun. 336, 55–58 (2015). [CrossRef]  

References

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  • |
  • |

  1. Y. Ai and Y. Dan, “Range of turbulence-negligible propagation of Gaussian Schell-model array beams,” Opt. Commun. 284(13), 3216–3220 (2011).
    [Crossref]
  2. X. Li, X. Ji, and F. Yang, “Beam quality of radial Gaussian Schell-model array beams,” Opt. Laser Technol. 42(4), 604–609 (2010).
    [Crossref]
  3. X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
    [Crossref]
  4. Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16(22), 18437–18442 (2008).
    [Crossref]
  5. X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009).
    [Crossref]
  6. X. Ji and Z. Pu, “Angular spread of Gaussian Schell-model array beams propagating through atmospheric turbulence,” Appl. Phys. B 93(4), 915–923 (2008).
    [Crossref]
  7. X. Ji, E. Zhang, and B. Lü, “Superimposed partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. B 25(5), 825–833 (2008).
    [Crossref]
  8. Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278(1), 157–167 (2007).
    [Crossref]
  9. Y. Dan and B. Zhang, “Beam Propagation Factor of Partially Coherent Flat-topped Beams in a Turbulent Atmosphere,” Opt. Express 16(20), 15563–15575 (2008).
    [Crossref]
  10. F. Wang and Y. Cai, “Second-order statistics of a twisted Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 18(24), 24661–24672 (2010).
    [Crossref]
  11. S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010).
    [Crossref]
  12. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A: Pure Appl. Opt. 3(1), 1–9 (2001).
    [Crossref]
  13. O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004).
    [Crossref]
  14. E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, Cambridge, 2007).
  15. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998).
    [Crossref]
  16. G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17(11), 2019–2023 (2000).
    [Crossref]
  17. B. Zhang, Y. Xu, Y. Dan, X. Deng, and Z. Zhao, “Beam spreading and M2-factor of electromagnetic Gaussian Schell-model beam propagating in inhomogeneous atmospheric turbulence,” Optik 149, 398–408 (2017).
    [Crossref]
  18. B. Kanseri and H. C. Kandpal, “Experimental determination of electric cross-spectral density matrix and generalized Stokes parameters for a laser beam,” Opt. Lett. 33(20), 2410–2412 (2008).
    [Crossref]
  19. H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32(15), 2215–2217 (2007).
    [Crossref]
  20. B. Zhang, X. Chu, and Q. Li, “Generalized beam-propagation factor of partially coherent beams propagating through hard-edged apertures,” J. Opt. Soc. Am. A 19(7), 1370–1375 (2002).
    [Crossref]
  21. X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A: Pure Appl. Opt. 11(10), 105705 (2009).
    [Crossref]
  22. G. Zhou, “Generalzied M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. A: Pure Appl. Opt. 12(1), 015701 (2010).
    [Crossref]
  23. Y. Dan and B. Zhang, “Second moments of partially coherent beams in atmospheric turbulence,” Opt. Lett. 34(5), 563–565 (2009).
    [Crossref]
  24. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyubolu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009).
    [Crossref]
  25. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed.; SPIE: Bellingham, 2005.
  26. J. Recolons, L. C. Andrews, and R. L. Phillips, “Analysis of beam wander effects for a horizontal-path propagating Gaussian-beam wave: focused beam case,” Opt. Eng. 46(8), 086002 (2007).
    [Crossref]
  27. Y. Huang, A. Zeng, Z. Gao, and B. Zhang, “Beam Wander of Partially Coherent Array Beams through Non-Kolmogorov Turbulence,” Opt. Lett. 40(8), 1619–1622 (2015).
    [Crossref]
  28. Y. Xu, H. Tian, Y. Dan, H. Feng, and S. Wang, “Beam wander and M2-factor of partially coherent electromagnetic hollow Gaussian beam propagating through non-Kolmogorov turbulence,” J. Mod. Opt. 64(8), 844–854 (2017).
    [Crossref]
  29. D. Y. Vasylyev, A. A. Semenov, and W. Vogel, “Toward Global Quantum Communication: Beam Wandering Preserves Nonclassicality,” Phys. Rev. Lett. 108(22), 220501 (2012).
    [Crossref]
  30. W. Wen and X. Chu, “Beam wande of partially coherent Airy beams,” J. Mod. Opt. 61(5), 379–384 (2014).
    [Crossref]
  31. G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: The effect of partial coherence,” Phys. Rev. E 76(5), 056606 (2007).
    [Crossref]
  32. H. T. Eyyuboğlu and C. Z. Cil, “Beam wander of dark hollow, flat-topped and annular beams,” Appl. Phys. B 93(2-3), 595–604 (2008).
    [Crossref]
  33. C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95(4), 763–771 (2009).
    [Crossref]
  34. C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J(0)- and I(0)-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98(1), 195–202 (2010).
    [Crossref]
  35. S. Yu, Z. Chen, T. Wang, G. Wu, H. Guo, and W. Gu, “Beam wander of electromagnetic Gaussian-Schell model beams propagating in atmospheric turbulence,” Appl. Opt. 51(31), 7581–7585 (2012).
    [Crossref]
  36. G. Wu, W. Dai, H. Tang, and H. Guo, “Beam wander of random EGSM vortex beams propagating through a Kolmogorov turbulence,” Opt. Commun. 336, 55–58 (2015).
    [Crossref]

2017 (2)

B. Zhang, Y. Xu, Y. Dan, X. Deng, and Z. Zhao, “Beam spreading and M2-factor of electromagnetic Gaussian Schell-model beam propagating in inhomogeneous atmospheric turbulence,” Optik 149, 398–408 (2017).
[Crossref]

Y. Xu, H. Tian, Y. Dan, H. Feng, and S. Wang, “Beam wander and M2-factor of partially coherent electromagnetic hollow Gaussian beam propagating through non-Kolmogorov turbulence,” J. Mod. Opt. 64(8), 844–854 (2017).
[Crossref]

2015 (2)

Y. Huang, A. Zeng, Z. Gao, and B. Zhang, “Beam Wander of Partially Coherent Array Beams through Non-Kolmogorov Turbulence,” Opt. Lett. 40(8), 1619–1622 (2015).
[Crossref]

G. Wu, W. Dai, H. Tang, and H. Guo, “Beam wander of random EGSM vortex beams propagating through a Kolmogorov turbulence,” Opt. Commun. 336, 55–58 (2015).
[Crossref]

2014 (1)

W. Wen and X. Chu, “Beam wande of partially coherent Airy beams,” J. Mod. Opt. 61(5), 379–384 (2014).
[Crossref]

2012 (2)

D. Y. Vasylyev, A. A. Semenov, and W. Vogel, “Toward Global Quantum Communication: Beam Wandering Preserves Nonclassicality,” Phys. Rev. Lett. 108(22), 220501 (2012).
[Crossref]

S. Yu, Z. Chen, T. Wang, G. Wu, H. Guo, and W. Gu, “Beam wander of electromagnetic Gaussian-Schell model beams propagating in atmospheric turbulence,” Appl. Opt. 51(31), 7581–7585 (2012).
[Crossref]

2011 (1)

Y. Ai and Y. Dan, “Range of turbulence-negligible propagation of Gaussian Schell-model array beams,” Opt. Commun. 284(13), 3216–3220 (2011).
[Crossref]

2010 (6)

X. Li, X. Ji, and F. Yang, “Beam quality of radial Gaussian Schell-model array beams,” Opt. Laser Technol. 42(4), 604–609 (2010).
[Crossref]

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
[Crossref]

F. Wang and Y. Cai, “Second-order statistics of a twisted Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 18(24), 24661–24672 (2010).
[Crossref]

S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010).
[Crossref]

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J(0)- and I(0)-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98(1), 195–202 (2010).
[Crossref]

G. Zhou, “Generalzied M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. A: Pure Appl. Opt. 12(1), 015701 (2010).
[Crossref]

2009 (5)

Y. Dan and B. Zhang, “Second moments of partially coherent beams in atmospheric turbulence,” Opt. Lett. 34(5), 563–565 (2009).
[Crossref]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyubolu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009).
[Crossref]

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A: Pure Appl. Opt. 11(10), 105705 (2009).
[Crossref]

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95(4), 763–771 (2009).
[Crossref]

X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009).
[Crossref]

2008 (6)

2007 (4)

J. Recolons, L. C. Andrews, and R. L. Phillips, “Analysis of beam wander effects for a horizontal-path propagating Gaussian-beam wave: focused beam case,” Opt. Eng. 46(8), 086002 (2007).
[Crossref]

G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: The effect of partial coherence,” Phys. Rev. E 76(5), 056606 (2007).
[Crossref]

H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32(15), 2215–2217 (2007).
[Crossref]

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278(1), 157–167 (2007).
[Crossref]

2004 (1)

2002 (1)

2001 (1)

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A: Pure Appl. Opt. 3(1), 1–9 (2001).
[Crossref]

2000 (1)

1998 (1)

Agrawal, G. P.

Ai, Y.

Y. Ai and Y. Dan, “Range of turbulence-negligible propagation of Gaussian Schell-model array beams,” Opt. Commun. 284(13), 3216–3220 (2011).
[Crossref]

Andrews, L. C.

J. Recolons, L. C. Andrews, and R. L. Phillips, “Analysis of beam wander effects for a horizontal-path propagating Gaussian-beam wave: focused beam case,” Opt. Eng. 46(8), 086002 (2007).
[Crossref]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed.; SPIE: Bellingham, 2005.

Baykal, Y.

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J(0)- and I(0)-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98(1), 195–202 (2010).
[Crossref]

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
[Crossref]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyubolu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009).
[Crossref]

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95(4), 763–771 (2009).
[Crossref]

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278(1), 157–167 (2007).
[Crossref]

Berman, G. P.

G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: The effect of partial coherence,” Phys. Rev. E 76(5), 056606 (2007).
[Crossref]

Borghi, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A: Pure Appl. Opt. 3(1), 1–9 (2001).
[Crossref]

Cai, Y.

F. Wang and Y. Cai, “Second-order statistics of a twisted Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 18(24), 24661–24672 (2010).
[Crossref]

S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010).
[Crossref]

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J(0)- and I(0)-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98(1), 195–202 (2010).
[Crossref]

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95(4), 763–771 (2009).
[Crossref]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyubolu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009).
[Crossref]

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278(1), 157–167 (2007).
[Crossref]

Chen, Z.

Chu, X.

Chumak, A. A.

G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: The effect of partial coherence,” Phys. Rev. E 76(5), 056606 (2007).
[Crossref]

Cil, C. Z.

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J(0)- and I(0)-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98(1), 195–202 (2010).
[Crossref]

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95(4), 763–771 (2009).
[Crossref]

H. T. Eyyuboğlu and C. Z. Cil, “Beam wander of dark hollow, flat-topped and annular beams,” Appl. Phys. B 93(2-3), 595–604 (2008).
[Crossref]

Dai, W.

G. Wu, W. Dai, H. Tang, and H. Guo, “Beam wander of random EGSM vortex beams propagating through a Kolmogorov turbulence,” Opt. Commun. 336, 55–58 (2015).
[Crossref]

Dan, Y.

Y. Xu, H. Tian, Y. Dan, H. Feng, and S. Wang, “Beam wander and M2-factor of partially coherent electromagnetic hollow Gaussian beam propagating through non-Kolmogorov turbulence,” J. Mod. Opt. 64(8), 844–854 (2017).
[Crossref]

B. Zhang, Y. Xu, Y. Dan, X. Deng, and Z. Zhao, “Beam spreading and M2-factor of electromagnetic Gaussian Schell-model beam propagating in inhomogeneous atmospheric turbulence,” Optik 149, 398–408 (2017).
[Crossref]

Y. Ai and Y. Dan, “Range of turbulence-negligible propagation of Gaussian Schell-model array beams,” Opt. Commun. 284(13), 3216–3220 (2011).
[Crossref]

Y. Dan and B. Zhang, “Second moments of partially coherent beams in atmospheric turbulence,” Opt. Lett. 34(5), 563–565 (2009).
[Crossref]

Y. Dan and B. Zhang, “Beam Propagation Factor of Partially Coherent Flat-topped Beams in a Turbulent Atmosphere,” Opt. Express 16(20), 15563–15575 (2008).
[Crossref]

Deng, X.

B. Zhang, Y. Xu, Y. Dan, X. Deng, and Z. Zhao, “Beam spreading and M2-factor of electromagnetic Gaussian Schell-model beam propagating in inhomogeneous atmospheric turbulence,” Optik 149, 398–408 (2017).
[Crossref]

Du, X.

X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009).
[Crossref]

Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16(22), 18437–18442 (2008).
[Crossref]

Eyyuboglu, H. T.

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
[Crossref]

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J(0)- and I(0)-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98(1), 195–202 (2010).
[Crossref]

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95(4), 763–771 (2009).
[Crossref]

H. T. Eyyuboğlu and C. Z. Cil, “Beam wander of dark hollow, flat-topped and annular beams,” Appl. Phys. B 93(2-3), 595–604 (2008).
[Crossref]

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278(1), 157–167 (2007).
[Crossref]

Eyyubolu, H. T.

Feng, H.

Y. Xu, H. Tian, Y. Dan, H. Feng, and S. Wang, “Beam wander and M2-factor of partially coherent electromagnetic hollow Gaussian beam propagating through non-Kolmogorov turbulence,” J. Mod. Opt. 64(8), 844–854 (2017).
[Crossref]

Gao, Z.

Gori, F.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A: Pure Appl. Opt. 3(1), 1–9 (2001).
[Crossref]

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998).
[Crossref]

Gorshkov, V. N.

G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: The effect of partial coherence,” Phys. Rev. E 76(5), 056606 (2007).
[Crossref]

Gu, W.

Guo, H.

G. Wu, W. Dai, H. Tang, and H. Guo, “Beam wander of random EGSM vortex beams propagating through a Kolmogorov turbulence,” Opt. Commun. 336, 55–58 (2015).
[Crossref]

S. Yu, Z. Chen, T. Wang, G. Wu, H. Guo, and W. Gu, “Beam wander of electromagnetic Gaussian-Schell model beams propagating in atmospheric turbulence,” Appl. Opt. 51(31), 7581–7585 (2012).
[Crossref]

Huang, Y.

Ji, X.

X. Li, X. Ji, and F. Yang, “Beam quality of radial Gaussian Schell-model array beams,” Opt. Laser Technol. 42(4), 604–609 (2010).
[Crossref]

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
[Crossref]

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A: Pure Appl. Opt. 11(10), 105705 (2009).
[Crossref]

X. Ji and Z. Pu, “Angular spread of Gaussian Schell-model array beams propagating through atmospheric turbulence,” Appl. Phys. B 93(4), 915–923 (2008).
[Crossref]

X. Ji, E. Zhang, and B. Lü, “Superimposed partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. B 25(5), 825–833 (2008).
[Crossref]

Jia, X.

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A: Pure Appl. Opt. 11(10), 105705 (2009).
[Crossref]

Kandpal, H. C.

Kanseri, B.

Korotkova, O.

Li, Q.

Li, X.

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
[Crossref]

X. Li, X. Ji, and F. Yang, “Beam quality of radial Gaussian Schell-model array beams,” Opt. Laser Technol. 42(4), 604–609 (2010).
[Crossref]

Lin, Q.

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278(1), 157–167 (2007).
[Crossref]

Lü, B.

Mondello, A.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A: Pure Appl. Opt. 3(1), 1–9 (2001).
[Crossref]

Phillips, R. L.

J. Recolons, L. C. Andrews, and R. L. Phillips, “Analysis of beam wander effects for a horizontal-path propagating Gaussian-beam wave: focused beam case,” Opt. Eng. 46(8), 086002 (2007).
[Crossref]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed.; SPIE: Bellingham, 2005.

Piquero, G.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A: Pure Appl. Opt. 3(1), 1–9 (2001).
[Crossref]

Pu, Z.

X. Ji and Z. Pu, “Angular spread of Gaussian Schell-model array beams propagating through atmospheric turbulence,” Appl. Phys. B 93(4), 915–923 (2008).
[Crossref]

Qu, J.

Recolons, J.

J. Recolons, L. C. Andrews, and R. L. Phillips, “Analysis of beam wander effects for a horizontal-path propagating Gaussian-beam wave: focused beam case,” Opt. Eng. 46(8), 086002 (2007).
[Crossref]

Salem, M.

Santarsiero, M.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A: Pure Appl. Opt. 3(1), 1–9 (2001).
[Crossref]

Semenov, A. A.

D. Y. Vasylyev, A. A. Semenov, and W. Vogel, “Toward Global Quantum Communication: Beam Wandering Preserves Nonclassicality,” Phys. Rev. Lett. 108(22), 220501 (2012).
[Crossref]

Simon, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A: Pure Appl. Opt. 3(1), 1–9 (2001).
[Crossref]

Tang, H.

G. Wu, W. Dai, H. Tang, and H. Guo, “Beam wander of random EGSM vortex beams propagating through a Kolmogorov turbulence,” Opt. Commun. 336, 55–58 (2015).
[Crossref]

Tian, H.

Y. Xu, H. Tian, Y. Dan, H. Feng, and S. Wang, “Beam wander and M2-factor of partially coherent electromagnetic hollow Gaussian beam propagating through non-Kolmogorov turbulence,” J. Mod. Opt. 64(8), 844–854 (2017).
[Crossref]

Vasylyev, D. Y.

D. Y. Vasylyev, A. A. Semenov, and W. Vogel, “Toward Global Quantum Communication: Beam Wandering Preserves Nonclassicality,” Phys. Rev. Lett. 108(22), 220501 (2012).
[Crossref]

Vogel, W.

D. Y. Vasylyev, A. A. Semenov, and W. Vogel, “Toward Global Quantum Communication: Beam Wandering Preserves Nonclassicality,” Phys. Rev. Lett. 108(22), 220501 (2012).
[Crossref]

Wang, F.

Wang, H.

Wang, S.

Y. Xu, H. Tian, Y. Dan, H. Feng, and S. Wang, “Beam wander and M2-factor of partially coherent electromagnetic hollow Gaussian beam propagating through non-Kolmogorov turbulence,” J. Mod. Opt. 64(8), 844–854 (2017).
[Crossref]

Wang, T.

Wang, X.

Wen, W.

W. Wen and X. Chu, “Beam wande of partially coherent Airy beams,” J. Mod. Opt. 61(5), 379–384 (2014).
[Crossref]

Wolf, E.

Wu, G.

G. Wu, W. Dai, H. Tang, and H. Guo, “Beam wander of random EGSM vortex beams propagating through a Kolmogorov turbulence,” Opt. Commun. 336, 55–58 (2015).
[Crossref]

S. Yu, Z. Chen, T. Wang, G. Wu, H. Guo, and W. Gu, “Beam wander of electromagnetic Gaussian-Schell model beams propagating in atmospheric turbulence,” Appl. Opt. 51(31), 7581–7585 (2012).
[Crossref]

Xu, Y.

Y. Xu, H. Tian, Y. Dan, H. Feng, and S. Wang, “Beam wander and M2-factor of partially coherent electromagnetic hollow Gaussian beam propagating through non-Kolmogorov turbulence,” J. Mod. Opt. 64(8), 844–854 (2017).
[Crossref]

B. Zhang, Y. Xu, Y. Dan, X. Deng, and Z. Zhao, “Beam spreading and M2-factor of electromagnetic Gaussian Schell-model beam propagating in inhomogeneous atmospheric turbulence,” Optik 149, 398–408 (2017).
[Crossref]

Yang, F.

X. Li, X. Ji, and F. Yang, “Beam quality of radial Gaussian Schell-model array beams,” Opt. Laser Technol. 42(4), 604–609 (2010).
[Crossref]

Yang, K.

Yu, S.

Yuan, Y.

Zeng, A.

Zhang, B.

Zhang, E.

Zhang, T.

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A: Pure Appl. Opt. 11(10), 105705 (2009).
[Crossref]

Zhao, D.

X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009).
[Crossref]

Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16(22), 18437–18442 (2008).
[Crossref]

Zhao, Z.

B. Zhang, Y. Xu, Y. Dan, X. Deng, and Z. Zhao, “Beam spreading and M2-factor of electromagnetic Gaussian Schell-model beam propagating in inhomogeneous atmospheric turbulence,” Optik 149, 398–408 (2017).
[Crossref]

Zhou, G.

G. Zhou, “Generalzied M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. A: Pure Appl. Opt. 12(1), 015701 (2010).
[Crossref]

Zhu, S.

Zhu, Y.

Appl. Opt. (1)

Appl. Phys. B (5)

H. T. Eyyuboğlu and C. Z. Cil, “Beam wander of dark hollow, flat-topped and annular beams,” Appl. Phys. B 93(2-3), 595–604 (2008).
[Crossref]

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95(4), 763–771 (2009).
[Crossref]

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J(0)- and I(0)-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98(1), 195–202 (2010).
[Crossref]

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
[Crossref]

X. Ji and Z. Pu, “Angular spread of Gaussian Schell-model array beams propagating through atmospheric turbulence,” Appl. Phys. B 93(4), 915–923 (2008).
[Crossref]

J. Mod. Opt. (2)

W. Wen and X. Chu, “Beam wande of partially coherent Airy beams,” J. Mod. Opt. 61(5), 379–384 (2014).
[Crossref]

Y. Xu, H. Tian, Y. Dan, H. Feng, and S. Wang, “Beam wander and M2-factor of partially coherent electromagnetic hollow Gaussian beam propagating through non-Kolmogorov turbulence,” J. Mod. Opt. 64(8), 844–854 (2017).
[Crossref]

J. Opt. A: Pure Appl. Opt. (3)

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A: Pure Appl. Opt. 11(10), 105705 (2009).
[Crossref]

G. Zhou, “Generalzied M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. A: Pure Appl. Opt. 12(1), 015701 (2010).
[Crossref]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A: Pure Appl. Opt. 3(1), 1–9 (2001).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Opt. Commun. (4)

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278(1), 157–167 (2007).
[Crossref]

Y. Ai and Y. Dan, “Range of turbulence-negligible propagation of Gaussian Schell-model array beams,” Opt. Commun. 284(13), 3216–3220 (2011).
[Crossref]

X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009).
[Crossref]

G. Wu, W. Dai, H. Tang, and H. Guo, “Beam wander of random EGSM vortex beams propagating through a Kolmogorov turbulence,” Opt. Commun. 336, 55–58 (2015).
[Crossref]

Opt. Eng. (1)

J. Recolons, L. C. Andrews, and R. L. Phillips, “Analysis of beam wander effects for a horizontal-path propagating Gaussian-beam wave: focused beam case,” Opt. Eng. 46(8), 086002 (2007).
[Crossref]

Opt. Express (5)

Opt. Laser Technol. (1)

X. Li, X. Ji, and F. Yang, “Beam quality of radial Gaussian Schell-model array beams,” Opt. Laser Technol. 42(4), 604–609 (2010).
[Crossref]

Opt. Lett. (6)

Optik (1)

B. Zhang, Y. Xu, Y. Dan, X. Deng, and Z. Zhao, “Beam spreading and M2-factor of electromagnetic Gaussian Schell-model beam propagating in inhomogeneous atmospheric turbulence,” Optik 149, 398–408 (2017).
[Crossref]

Phys. Rev. E (1)

G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: The effect of partial coherence,” Phys. Rev. E 76(5), 056606 (2007).
[Crossref]

Phys. Rev. Lett. (1)

D. Y. Vasylyev, A. A. Semenov, and W. Vogel, “Toward Global Quantum Communication: Beam Wandering Preserves Nonclassicality,” Phys. Rev. Lett. 108(22), 220501 (2012).
[Crossref]

Other (2)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed.; SPIE: Bellingham, 2005.

E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, Cambridge, 2007).

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Figures (5)

Fig. 1.
Fig. 1. Schematic drawing of EGSM array beams.
Fig. 2.
Fig. 2. Relative M2-factor vs. propagation distance z, λ=632.8 nm, δ0xx=5 mm, δ0yy=3 mm, (a) N = 3, l0=20 mm, L0=50 m, Cn2=10−14m3-α, x0=y0=10 mm, α=11/3, P0=0.5. (b) σ0x=10 mm, σ0y=5 mm, l0=20 mm, L0=50 m, Cn2=10−14m3-α, x0=y0=10 mm, α=11/3, P0=0.5. (c) σ0x=10 mm, σ0y=5 mm, N = 3, L0=50 m, Cn2=10−14m3-α, x0=y0=10 mm, α=11/3, P0=0.5. (d) σ0x=10 mm, σ0y=5 mm, N = 3, l0=20 mm, Cn2=10−14m3-α, x0=y0=10 mm, α=11/3, P0=0.5. (e) σ0x=10 mm, σ0y=5 mm, N = 3, l0=20 mm, L0=50 m, x0=y0=10 mm, α=11/3, P0=0.5. (f) σ0x=10 mm, σ0y=5 mm, N = 3, l0=20 mm, L0=50 m, Cn2=10−14m3-α, α=11/3, P0=0.5. (g) σ0x=10 mm, σ0y=5 mm, N = 3, l0=20 mm, L0=50 m, Cn2=10−14m3-α, x0=y0=10 mm, P0=0.5. (h) σ0x=10 mm, σ0y=5 mm, N = 3, l0=20 mm, L0=50 m, Cn2=10−14m3-α, x0=y0=10 mm, α=11/3.
Fig. 3.
Fig. 3. Relative M2-factor vs. the generalized exponent parameter α, λ=632.8 nm, N = 3, δ0xx=5 mm, δ0yy=3 mm, σ0x=10 mm, σ0y=5 mm, z = 10 km, (a) P0=0.5, x0=y0=10 mm, L0=50 m, Cn2=10−14m3-α. (b) P0=0.5, x0=y0=10 mm, l0=20 mm, Cn2=10−14m3-α. (c) P0=0.5, Cn2=10−14m3-α, l0=20 mm, L0=50 m. (d) Cn2=10−14m3-α, x0=y0=10 mm, l0=20 mm, L0=50 m.
Fig. 4.
Fig. 4. The rms beam wander Bw and the relative beam wander Bwr vs. the generalized exponent parameter α, λ=632.8 nm, N = 3, δ0xx=5 mm, δ0yy=3 mm, σ0x=10 mm, σ0y=5 mm, l0=20 mm, L0=50 m, L = 10 km (a) P0=0.5, x0=y0=10 mm. (b) x0=y0=10 mm, Cn2=10−14m3-α. (c) and (d) P0=0.5, Cn2=10−14m3-α.
Fig. 5.
Fig. 5. The beam wander and the relative rms beam wander Bwr vs. propagation distance L λ=632.8 nm, N = 3, δ0xx=5 mm, δ0yy=3 mm, l0=20 mm, L0=50 m. (a) and (c) Cn2=10−14m3-α, σ0x=10 mm, σ0y=5 mm, P0=0.5, x0=y0=10 mm. (b) α=11/3, σ0x=10 mm, σ0y=5 mm, P0=0.5, x0=y0=10 mm, (d) α=11/3, Cn2=10−14m3-α, P0=0.5, x0=y0=10 mm. (e) α=11/3, Cn2=10−14m3-α, σ0x=10 mm, σ0y=5 mm, x0=y0=10 mm. (f) α=11/3, Cn2=10−14m3-α, σ0x=10 mm, σ0y=5 mm, P0=0.5.

Equations (27)

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W ( ρ 1 , ρ 2 , 0 ) = [ W x x ( ρ 1 , ρ 2 , 0 ) W x y ( ρ 1 , ρ 2 , 0 ) W y x ( ρ 1 , ρ 2 , 0 ) W y y ( ρ 1 , ρ 2 , 0 ) ] ,
W p q ( ρ 1 , ρ 2 , 0 ) = A p A q B p q × i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 exp ( ( x 1 i x 0 ) 2 + ( y 1 j y 0 ) 2 4 σ 0 p 2 ) × exp ( ( x 2 i x 0 ) 2 + ( y 2 j y 0 ) 2 4 σ 0 q 2 ) exp ( ρ d 2 2 δ 0 p q 2 ) ,
W T r ( ρ 1 , ρ 2 , 0 ) = W x x ( ρ 1 , ρ 2 , 0 ) + W y y ( ρ 1 , ρ 2 , 0 ) ,
ρ 1 = ρ + ρ d / 2 ; ρ 2 = ρ ρ d / 2 ,
W T r ( ρ , ρ d , 0 ) = A x 2 B x x i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 exp ( 2 x 2 x d 2 / 2 2 i 2 x 0 2 + 4 i x 0 x 4 σ 0 x 2 ) × exp ( 2 y 2 y d 2 / 2 2 j 2 y 0 2 + 4 j y 0 y 4 σ 0 x 2 ) × exp ( ρ d 2 2 δ 0 x x 2 ) + A y 2 B y y i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 exp ( 2 x 2 x d 2 / 2 2 i 2 x 0 2 + 4 i x 0 x 4 σ 0 y 2 ) × exp ( 2 y 2 y d 2 / 2 2 j 2 y 0 2 + 4 j y 0 y 4 σ 0 y 2 ) × exp ( ρ d 2 2 δ 0 y y 2 ) .
h ( ρ , θ , 0 ) = ( k 2 π ) 2 W T r ( ρ , ρ d , 0 ) exp ( i k θ ρ d ) d 2 ρ d ,
x n 1 y n 2 θ x m 1 θ y m 2 0 = 1 P x n 1 y n 2 θ x m 1 θ y m 2 h ( ρ , θ , 0 ) d 2 ρ d 2 θ ,
δ ( s ) = 1 2 π exp ( i s x ) d x ,
δ ( n ) ( s ) = 1 2 π ( i x ) n exp ( i s x ) d x ( n = 0 ,   1 ,   2 ) ,
f ( x ) δ ( n ) ( x ) d x = ( 1 ) n f ( n ) ( 0 ) ( n = 0 ,   1 ,   2 ) ,
ρ 2 = ρ 2 0 + 2 ρ θ 0 z + θ 2 0 z 2 + 4 3 π 2 T z 3 = 1 P [ A x 2 B x x i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 2 π σ 0 x 4 + 2 i 2 x 0 2 σ 0 x 2 π ) + A y 2 B y y × i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 2 π σ 0 y 4 + 2 i 2 x 0 2 σ 0 y 2 π ) ] + 1 P [ A x 2 B x x i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 2 π σ 0 x 4 + 2 j 2 y 0 2 σ 0 x 2 π ) + A y 2 B y y i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 2 π σ 0 y 4 + 2 j 2 y 0 2 σ 0 y 2 π ) ] + 2 z 2 P k 2 [ A x 2 B x x i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 1 4 σ 0 x 2 + 1 δ 0 x x 2 ) + A y 2 B y y i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 1 4 σ 0 y 2 + 1 δ 0 y y 2 ) ] + 4 3 π 2 T z 3 ,
θ 2 = θ x 2 0 + θ y 2 0 + 4 π 2 T z = 2 P k 2 [ A x 2 B x x i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 1 4 σ 0 x 2 + 1 δ 0 x x 2 ) + A y 2 B y y × i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 1 4 σ 0 y 2 + 1 δ 0 y y 2 ) ] + 4 π 2 T z ,
ρ θ = θ 2 0 z + 2 π 2 T z 2 = 2 z P k 2 [ A x 2 B x x i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 1 4 σ 0 x 2 + 1 δ 0 x x 2 ) + A y 2 B y y × i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 1 4 σ 0 y 2 + 1 δ 0 y y 2 ) ] + 2 π 2 T z 2 ,
P = A x 2 B x x i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 2 π σ 0 x 2 + A y 2 B y y i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 2 π σ 0 y 2 .
T = 0 κ 3 Φ n ( κ , α ) d κ ,
Φ n ( κ , α ) = A ( α ) C n 2 [ κ 2 + ( 2 π / L 0 ) 2 ] α / 2 exp { κ 2 [ c ( α ) / l 0 ] 2 } ,
A ( α ) = 1 4 π 2 Γ ( α 1 ) cos ( α π 2 ) ,
c ( α ) = [ 2 π 3 A ( α ) Γ ( 5 α 2 ) ] 1 / α 5 .
T ( α ) = A ( α ) C ~ n 2 2 ( α 2 ) { [ 2 κ 0 2 + ( α 2 ) κ m 2 ] κ m 2 α exp ( κ 0 2 κ m 2 ) Γ ( 2 α 2 , κ 0 2 κ m 2 ) 2 κ 0 4 α } ,
M 2 ( z ) = k [ ρ 2 θ 2 ( ρ θ ) 2 ] 1 / 2 = k { { 1 P [ A x 2 B x x i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 2 π σ 0 x 4 + 2 i 2 x 0 2 σ 0 x 2 π ) + A y 2 B y y × i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 2 π σ 0 y 4 + 2 i 2 x 0 2 σ 0 y 2 π ) ] + 1 P [ A x 2 B x x i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 2 π σ 0 x 4 + 2 j 2 y 0 2 σ 0 x 2 π ) + A y 2 B y y i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 2 π σ 0 y 4 + 2 j 2 y 0 2 σ 0 y 2 π ) ] + 2 z 2 P k 2 [ A x 2 B x x i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 1 4 σ 0 x 2 + 1 δ 0 x x 2 ) + A y 2 B y y i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 1 4 σ 0 y 2 + 1 δ 0 y y 2 ) ] + 4 3 π 2 T z 3 } × { 2 P k 2 [ A x 2 B x x i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 1 4 σ 0 x 2 + 1 δ 0 x x 2 ) + A y 2 B y y i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 1 4 σ 0 y 2 + 1 δ 0 y y 2 ) ] + 4 π 2 T z } { 2 z P k 2 [ A x 2 B x x i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 1 4 σ 0 x 2 + 1 δ 0 x x 2 ) + A y 2 B y y i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 1 4 σ 0 y 2 + 1 δ 0 y y 2 ) ] + 2 π 2 T z 2 } 2 } 1 / 2 .
M r 2 ( z ) = M 2 ( z ) M f 2 ( 0 ) .
r c 2 = 4 π 2 k 2 W FS 2 0 L 0 κ Φ n ( κ ) exp ( κ 2 W LT 2 ) [ 1 exp ( 2 L 2 κ 2 ( 1 z / L ) 2 k 2 W FS 2 ) ] d κ d z .
r c 2 = 4 π 2 C n 2 A ( α ) L 2 α 2 κ 0 α 0 L ( 1 z L ) 2 { ( κ 0 κ m ) 2 2 κ 0 4 + κ 0 α κ m 2 ( W L T 2 + κ m 2 ) α / 2 [ 2 κ 0 2 ( 1 + κ m 2 W L T 2 ) + ( α 2 ) κ m 2 ] × ( 1 + κ m 2 W L T 2 ) 2 exp [ ( κ 0 κ m ) 2 + κ 0 2 W L T 2 ] Γ [ 2 α 2 , ( κ 0 κ m ) 2 + κ 0 2 W L T 2 ] } d z .
W L T 2 = ρ 2 = ρ 2 0 + 2 ρ θ 0 z + θ 2 0 z 2 + 4 3 T π 2 z 3 .
B w = [ r c 2 ] 1 / 2 ,
B w r = [ r c 2 / W L T 2 ] 1 / 2 .
P 0 ( ρ ; 0 ) = 1 4 D e t W ( ρ , ρ ; 0 ) [ T r W ( ρ , ρ ; 0 ) ] 2 ,

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