Propagation of multi-Gaussian Schell-model vortex beams in isotropic random media

Miaomiao Tang; Daomu Zhao

doi:10.1364/OE.23.032766

1. Introduction

Over the past decades, the evolution of partially coherent beams in free space and linear isotropic random media has been extensively investigated due to their wide applications in free-space optical communications, remote sensing and tracking [1–10]. In majority of the studies, a Gaussian-Schell model (GSM) beam has been chosen as a typical example of partially coherent beams, whose intensity distribution and degree of coherence has Gaussian shapes [11]. However, since the sufficient condition for devising genuine correlation functions of partially coherence was established by Gori et al. [12, 13], a verity of special model sources have been recently introduced in addition to well-explored GSM source [14–19]. Among them, the multi-Gaussian Schell-model (MGSM) source [15], whose intensity distribution has Gaussian profile while its degree of coherence is modeled by multi-Gaussian distribution, has drawn much attention due to its distinctive characteristics upon propagation.

One the other hand, laser beams with phase singularities have been widely studied due to their important applications in many fields, such as optical trapping, quantum information and optical tweezers [20, 21]. In the past several years, the propagation of various partially coherent vortex beams in random media has become a hot topic [22–28]. Quite recently, a new class of partially coherent vortex beams termed multi-Gaussian Schell-model (MGSM) vortex beams was introduced and the focusing properties are studied [29]. It was found that the focused beam profile can be shaped by varying the initial beam parameters.

In this paper, we explore the behavior of the MGSM vortex beams interact with isotropic random media, for example, the turbulent atmosphere, and do the comparative analysis of effect of the free-space diffraction with the medium-induced diffraction. The dependences of the statistical characteristics on the source correlations and the turbulence properties are also emphasized.

2. Propagation of MGSM vortex beams in linear random media

The field of an optical vortex beam at the source plane $z = 0$ in the Cartesian coordinate system is expressed as [30]

U (ρ^{'}, z = 0) = u (ρ^{'}) {[x^{'} + i sgn (l) y^{'}]}^{l},

where

ρ^{'} \equiv (x^{'}, y^{'})

is the two-dimensional position vector at the

z = 0

plane,

u (ρ^{'})

represents the profile of the background beam envelope,

sgn (\cdot)

specifies the sign function,

l

is the topological charge.

Assume that $u (ρ^{'})$ takes a multi-Gaussian Schell-model form, the cross-spectral density function of MGSM vortex beams at the source plane can be expressed as follows [29]

\begin{array}{l} W^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}, 0) = \frac{1}{C_{0}} {[({x^{'}}_{1} {x^{'}}_{2} + {y^{'}}_{1} {y^{'}}_{2}) + i sgn (l) ({x^{'}}_{1} {y^{'}}_{2} - {x^{'}}_{2} {y^{'}}_{1})]}^{l} \\ \times \exp (- \frac{ρ_{1} {^{'}}^{2} + ρ_{2} {^{'}}^{2}}{w_{0}^{2}}) \sum_{m = 1}^{M} (\begin{matrix} M \\ m \end{matrix}) \frac{{(- 1)}^{m - 1}}{m} \exp [- \frac{{(ρ_{1}^{'} - ρ_{2}^{'})}^{2}}{2 m δ^{2}}], \end{array}

where

{ρ^{'}}_{1}

and

{ρ^{'}}_{2}

are two arbitrary points in the source plane,

C_{0} = \sum_{m = 1}^{M} (\begin{matrix} M \\ m \end{matrix}) \frac{{(- 1)}^{m - 1}}{m}

is the normalization factor,

(\begin{matrix} M \\ m \end{matrix})

stands for binomial coefficients,

w_{0}

is the beam width of the source,

δ

denotes the transverse coherence width, and in the following the topological charge

l

is assumed to be

\pm 1

.

The paraxial form of the Huygens-Fresnel principle which describes the interaction of waves with linear random media implies that the cross-spectral density function at two points $(ρ_{1}, z)$ and $(ρ_{2}, z)$ in the same transverse plane of the half-space $z \geq 0$ is written as [4]

\begin{array}{l} W (ρ_{1}, ρ_{2}, z) = {(\frac{k}{2 π z})}^{2} \iint \iint W^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}, 0) \exp [- i k \frac{{(ρ_{1} - {ρ^{'}}_{1})}^{2} - (ρ_{2} - {ρ^{'}}_{2})}{2 z}] \\ \times {〈 \exp [ψ^{*} (ρ_{1}, {ρ^{'}}_{1}, z) + ψ (ρ_{2}, {ρ^{'}}_{2}, z)] 〉}_{R} d^{2} {ρ^{'}}_{1} d^{2} {ρ^{'}}_{2}, \end{array}

were

k = 2 π / λ

is the wave number with

λ

being the wavelength of the light,

ψ

denotes the complex phase perturbation due to the random medium, and

{〈 ... 〉}_{R}

implies averaging over the ensemble of statistical realizations of the turbulent medium. For points located sufficiently close to the optical axis, the term in the sharp brackets with the subscript

R

in Eq. (3) can be written as [4]

\begin{array}{l} {〈 \exp [ψ^{*} (ρ_{1}, ρ_{1}^{'}, z) + ψ (ρ_{2}, ρ_{2}^{'}, z)] 〉}_{R} = \\ \exp {- \frac{π^{2} k^{2} z}{3} [{(ρ_{1} - ρ_{2})}^{2} + (ρ_{1} - ρ_{2}) ({ρ^{'}}_{1} - {ρ^{'}}_{2}) + {({ρ^{'}}_{1} - {ρ^{'}}_{2})}^{2}] \int_{0}^{\infty} κ^{3} Φ_{n} (κ) d κ}, \end{array}

where

Φ_{n} (κ)

is the one-dimensional spatial power spectrum of the refractive-index fluctuations of random medium,

κ

being spatial frequency.

For instance, the non-Kolmogorov turbulence model characterizing atmospheric fluctuations at various altitudes is known to have form [31]

Φ_{n} (κ) = A (α) {\tilde{C}}_{n}^{2} \frac{\exp [- κ^{2} / κ_{m}^{2}]}{{(κ^{2} + κ_{0}^{2})}^{α / 2}}, 0 \leq κ < \infty,

where

3 < α < 4

and the term

{\tilde{C}}_{n}^{2}

is a generalized refractive-index structure parameter with units

m^{3 - α}

,

κ_{0} = \frac{2 π}{L_{0}}, κ_{m} = \frac{c (α)}{l_{0}},

c (α) = {[\frac{2 π}{3} Γ (5 - \frac{α}{2}) A (α)]}^{\frac{1}{α - 5}},

A (α) = \frac{1}{4 π^{2}} Γ (α - 1) \cos (\frac{α π}{2}),

L_{0}

and

l_{0}

are the outer and inner scales of turbulence, respectively, and

Γ (\cdot)

is the Gamma function. With the power spectrum in Eq. (5), the integral in Eq. (4) becomes

I = \int_{0}^{\infty} κ^{3} Φ_{n} (κ) d κ = \frac{A (α)}{2 (α - 2)} {\tilde{C}}_{n}^{2} κ_{m}^{2 - α} β \exp (\frac{κ_{0}^{2}}{κ_{m}^{2}}) Γ (2 - \frac{α}{2}, \frac{κ_{0}^{2}}{κ_{m}^{2}}) - 2 κ_{0}^{4 - α},

where

β = 2 κ_{0}^{2} - 2 κ_{m}^{2} + α κ_{m}^{2}

and

Γ (\cdot, \cdot)

denotes the incomplete Gamma function.

On introducing new variables $u = \frac{{ρ^{'}}_{1} + {ρ^{'}}_{2}}{2}$ , $v = {ρ^{'}}_{1} - {ρ^{'}}_{2}$ and substituting from Eqs. (4)-(9) into (3), one finds that

\begin{array}{l} W (ρ_{1}, ρ_{2}, z) = \frac{1}{C_{0}} {(\frac{k}{2 π z})}^{2} \sum_{m = 1}^{M} (\begin{matrix} M \\ m \end{matrix}) \frac{{(- 1)}^{m - 1}}{m} \exp [- \frac{i k}{2 z} (ρ_{1}^{2} - ρ_{2}^{2})] \exp [- \frac{π^{2} k^{2} z I}{3} {(ρ_{1} - ρ_{2})}^{2}] \\ \times \iint d^{2} u \iint d^{2} v [(u^{2} - \frac{v^{2}}{4}) \mp i (u_{x} v_{y} - u_{y} v_{x})] \exp (- \frac{2}{w_{0}^{2}} u^{2}) \exp [\frac{i k}{z} (ρ_{1} - ρ_{2}) u] \\ \times \exp (- A v^{2} - \frac{i k}{z} u v) \exp {v [\frac{i k}{2 z} (ρ_{1} + ρ_{2}) - \frac{π^{2} k^{2} z I}{3} (ρ_{1} - ρ_{2})]}, \end{array}

where

A = \frac{1}{2 w_{0}^{2}} + \frac{1}{2 m δ^{2}} + \frac{π^{2} k^{2} z I}{3},

u_{x} = \frac{{x^{'}}_{1} + {x^{'}}_{2}}{2}

,

u_{y} = \frac{{y^{'}}_{1} + {y^{'}}_{2}}{2}

,

v_{x} = {x^{'}}_{1} - {x^{'}}_{2}

,

v_{y} = {y^{'}}_{1} - {y^{'}}_{2}

, the symbol

‘ \mp ’

corresponds to

l = \pm 1

at the source plane.

Recalling the integral formulae

\int \exp (- p x^{2} + 2 q x) d x = \exp (\frac{q^{2}}{p}) \sqrt{\frac{π}{p}},

\int x \exp (- p x^{2} + 2 q x) d x = \exp (\frac{q^{2}}{p}) \sqrt{\frac{π}{p}} (\frac{q}{p}),

\int x^{2} \exp (- p x^{2} + 2 q x) d x = \frac{1}{2 p} \exp (\frac{q^{2}}{p}) \sqrt{\frac{π}{p}} (1 + \frac{2 q^{2}}{p}),

after tedious but straightforward integral calculations, we obtain

\begin{array}{l} W (ρ_{1}, ρ_{2}, z) = \frac{1}{C_{0}} {(\frac{k}{2 π z})}^{2} \sum_{m = 1}^{M} (\begin{matrix} M \\ m \end{matrix}) \frac{{(- 1)}^{m - 1}}{m} \exp [- \frac{i k}{2 z} (ρ_{1}^{2} - ρ_{2}^{2})] \\ \times \exp {- \frac{π^{2} k^{2} z I}{3} [{(x_{1} - x_{2})}^{2} + {(y_{1} - y_{2})}^{2}]} \\ \times [(N_{1} - N_{2}) \mp (N_{3} - N_{4})], \end{array}

where

N_{1} = \frac{π^{2}}{A C} B_{x} B_{y} \exp (\frac{D_{x}^{2} + D_{y}^{2}}{C}) (\frac{D_{x}^{2} + D_{y}^{2}}{C^{2}} + \frac{1}{C}),

N_{2} = \frac{π^{2} w_{0}^{2}}{8 F} \exp (- \frac{k^{2} w_{0}^{2}}{8 z^{2}} {| ρ_{1} - ρ_{2} |}^{2}) \exp (\frac{G_{x}^{2} + G_{y}^{2}}{F}) (\frac{G_{x}^{2} + G_{y}^{2}}{F^{2}} + \frac{1}{C}),

N_{3} = \frac{i π^{2} w_{0}}{\sqrt{2 A C F}} B_{y} \frac{D_{x} G_{y}}{C F} \exp [- \frac{k^{2} w_{0}^{2}}{8 z^{2}} {(y_{1} - y_{2})}^{2}] \exp (\frac{D_{x}^{2}}{C} + \frac{G_{y}^{2}}{F}),

N_{4} = \frac{i π^{2} w_{0}}{\sqrt{2 A C F}} B_{y} \frac{D_{y} G_{x}}{C F} \exp [- \frac{k^{2} w_{0}^{2}}{8 z^{2}} {(x_{1} - x_{2})}^{2}] \exp (\frac{D_{y}^{2}}{C} + \frac{G_{x}^{2}}{F}),

B_{x} = \exp [- \frac{k^{2}}{16 A z^{2}} {(x_{1} + x_{2})}^{2}] \exp [\frac{π^{4} k^{4} z^{2} I^{2}}{36 A} {(x_{1} - x_{2})}^{2}] \exp [- \frac{i π^{2} k^{3} I}{12 A} (x_{1}^{2} - x_{2}^{2})],

D_{x} = \frac{1}{2} [\frac{i k}{z} (x_{1} - x_{2}) + \frac{k^{2}}{4 A z^{2}} (x_{1} + x_{2}) + \frac{i π^{2} k^{3} I}{6 A} (x_{1} - x_{2})],

G_{x} = \frac{1}{2} [\frac{i k}{z} (x_{1} + x_{2}) - \frac{π^{2} k^{2} z I}{3} (x_{1} - x_{2}) + \frac{k^{2} w_{0}^{2}}{4 z^{2}} (x_{1} - x_{2})],

C = \frac{2}{w_{0}^{2}} + \frac{k^{2}}{4 A z^{2}},

F = A + \frac{k^{2} w_{0}^{2}}{8 z^{2}} .

Due to the symmetry, $B_{y}$ , $D_{y}$ and $G_{y}$ can be obtained from $B_{x}$ , $D_{y}$ and $G_{x}$ by replacing $x_{1}$ and $x_{2}$ with $y_{1}$ and $y_{2}$ , respectively.

Substituting $ρ_{1} = ρ_{2} = ρ$ into Eq. (15), the spectral density at any point $(ρ, z)$ can be obtained

\begin{array}{l} S (ρ, z) = W (ρ, ρ, z) = \frac{1}{C_{0}} {(\frac{k}{2 π z})}^{2} \sum_{m = 1}^{M} (\begin{matrix} M \\ m \end{matrix}) \frac{{(- 1)}^{m - 1}}{m} {\frac{π^{2}}{A C} \exp [(\frac{H^{2}}{C} - H) ρ^{2}] \\ \times (\frac{H^{2} ρ^{2}}{C^{2}} + \frac{1}{C}) - \frac{π^{2} w_{0}^{2}}{8 F} \exp (\frac{J^{2} ρ^{2}}{F}) (\frac{J^{2} ρ^{2}}{F^{2}} + \frac{1}{F})}, \end{array}

with

H = \frac{k^{2}}{4 A z^{2}}, J = \frac{i k}{2 z} .

The spectral degree of coherence is defined as [11]

μ (ρ_{1}, ρ_{2}, z) = \frac{W (ρ_{1}, ρ_{2}, z)}{{[S (ρ_{1}, z) S (ρ_{2}, z)]}^{\frac{1}{2}}} .

The spreading of beam induced by turbulence can be examined based on the mean-squared (r. m. s) beam width, which is defined as [7]

w (z) = \sqrt{\frac{\int ρ^{2} S (ρ, z) d^{2} ρ}{\int S (ρ, z) d^{2} ρ}} .

Making use of the integral formula

\int x^{2 p} \exp (- β x^{2 q}) d x = Γ (\frac{2 p + 1}{2 q}) / q β^{\frac{2 p + 1}{2 q}},

and substituting Eq. (25) into Eq. (28), the r. m. s beam width of MGSM vortex beams is given by

w (z) = \frac{\sum_{m = 1}^{M} (\begin{matrix} M \\ m \end{matrix}) \frac{{(- 1)}^{m - 1}}{m} {\frac{π^{2}}{A C} {(H - \frac{H^{2}}{C})}^{- 3} [\frac{2 H^{2}}{C^{2}} + \frac{1}{C} (H - \frac{H^{2}}{C})] + \frac{π^{2} w_{0}^{2} J^{- 4}}{8}}}{\sum_{m = 1}^{M} (\begin{matrix} M \\ m \end{matrix}) \frac{{(- 1)}^{m - 1}}{m} {\frac{π^{2}}{A C} {(H - \frac{H^{2}}{C})}^{- 2} [\frac{H^{2}}{C^{2}} + \frac{1}{C} (H - \frac{H^{2}}{C})]}} .

3. Numerical examples and analysis

In this section, we will now illustrate the behavior of the spectral density, the degree of coherence and the beam spreading of a MGSM vortex beam with the topological charge $l = + 1$ propagating through non-Kolmogorov turbulence by a set of numerical examples, and for comparison the corresponding results of the beam on propagation in free space are compiled together. Unless specified in captions, the source and the medium parameters are chosen as follows: $λ = 632 nm$ , $w_{0} = 2 cm$ , $δ = 0.5 cm$ , $l = + 1$ , $M = 5$ , $L_{0} = 1 m$ , $l_{0} = 1 mm$ , $α = 3.8$ and ${\tilde{C}}_{n}^{2} = 10^{- 14} m^{3 - α}$ .

In Fig. 1 the evolution of the normalized spectral distribution $S (x, 0, z) / S {(x, 0, z)}_{\max}$ of a MGSM vortex beam $(M = 5)$ and a GSM vortex beam $(M = 1)$ is shown at several selected distances $z$ from the source plane. Free-space Fig. 1(a) and atmospheric Fig. 1(b) clearly show the difference in the beam behavior. As can be seen, the MGSM vortex source presents a dark core at the center in the source plane $z = 0$ , and the phase becomes singular. While on propagation in free space, unlike the classic GSM vortex beam converts its initial hollow beam profile into a Gaussian distribution, for the case of MGSM vortex beam, within relatively short ranges from the source $(z = 0.8 km)$ , the normalized spectral density changes its form to a quasi-Gaussian distribution, but it exhibits a flat-top distribution at sufficiently larger distance. This modification can be attributed to the source correlations. However, on propagation in optical turbulence as shown in Fig. 1(b), the most important atmospheric effect is destroying the far zone flatted intensity profile, and the beam shape eventually converges into Gaussian profile. Similar effect of turbulence on Gaussian vortex beams and stochastic electromagnetic vortex beams were early reported in [26, 28].

Fig. 1 Evolution of normalized spectral intensity of a GSM vortex beam ( $M = 1$ ) and a MGSM vortex beam with ( $M = 5$ ) propagating in (a) free space and (b) non-Kolmogorov atmospheric turbulence.

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The influence of the source correlation width $δ$ on the far-zone intensity profile on propagation in free space and non-Kolmogorov turbulence is given in Fig. 2(a) and 2(b), respectively. One finds from Fig. 2(a) that in free space propagation, the far-zone intensity distribution is completely determined by the source parameters, thus the far zone beam shape can be adjusted by the choice of the source correlation width, which is in agreement with results previously reported (see [29],). However, under the presence of the isotropic turbulent atmosphere, all curves with different $δ$ preserve Gaussian distribution. This phenomenon can be explained by the fact that the evolution behavior of the beam is determined by three factors: the source multi-Gaussian correlations, the vortex phase and atmospheric turbulence. At small propagation distance the beam is mostly modified by the source correlations and the vortex phase, as the beam passes to sufficiently large distance through turbulence, the effect of turbulence plays a dominate role for determining the beam profile.

Fig. 2 (a) The normalized spectral distribution of the MGSM vortex beam with $M = 5$ at propagation distance $z = 10 km$ with different $δ$ in (a) free space and (b) non-Kolmogorov atmospheric turbulence.

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Figure 3(a) exhibits the profiles of the spectral distribution at certain transverse plane on propagation for several different values of $M$ . It turns out that the larger the index $M$ is, the smaller its height and the faster the on-axis intensity increases. It can be seen from Fig. 3(b) that the spectral distribution is significantly affected by the strength of the atmosphere turbulence, as the atmospheric fluctuations become more severe ( ${\tilde{C}}_{n}^{2}$ grows), the more efficiently the beam profile approaches to a Gaussian distribution.

Fig. 3 The normalized spectral distribution of a MGSM vortex beam propagation in atmospheric turbulence at propagation distance $z = 500 m$ , (a) for different $M$ with ${\tilde{C}}_{n}^{2} = 10^{- 14} m^{3 - α}$ ; (b) for different ${\tilde{C}}_{n}^{2}$ with $M = 5$ .

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We now turn our attention to the changes in the spectral degree of coherence calculated with the help of Eq. (27). Figure 4 shows the free-space evolution of the absolute value of the degree of coherence as a function of separation distance $x_{d} = | x_{1} - x_{2} |$ where the two points are chosen at locations symmetric, i.e. $ρ_{1} (x, 0)$ and $ρ_{2} (- x, 0)$ . As can be seen the coherence curves possess a turning point due to intrinsic vortex phase, and the multi-Gaussian function modulation makes such trend even more complex. However, as the propagation distance grows, the dependence on the values of $M$ gradually disappears and all the curves evolve in the same way.

Fig. 4 Modulus of the spectral degree of coherence of a MGSM vortex beam with several values of $M$ as a function of separation distance $x_{d}$ at different propagation distances in free space, (a) $z = 0.5 km$ ; (b) $z = 1 km$ ; (c) $z = 2 km$ ; (d) $z = 5 km$ .

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In Fig. 5 the evolution of the modulus of the degree of coherence, as a function of separation distance $x_{d}$ , in the atmosphere for different values of $M$ is shown. One sees that at relatively small distances, the degree of coherence goes through the similar processes of evolution as in free space, However, as the propagation distance grows, the oscillatory trend on the coherence curves is suppressed gradually by medium fluctuations. What is more, after passing through the turbulence at sufficiently large distances, the medium modifies the coherence curves to Gaussian distribution while narrowing the degree of coherence.

Fig. 5 As in Fig. 4 but on propagation in non-Kolmogorov turbulence.

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Figure 6 illustrates the behavior of the degree of coherence for selected sources and turbulence parameters. One sees that although the degree of coherence varies slightly with the source correlation width $δ$ , it alters significantly with the change in ${\tilde{C}}_{n}^{2}$ for non-Kolmogorov turbulence, and the larger the value of ${\tilde{C}}_{n}^{2}$ is, the faster the degree of coherence decays.

Fig. 6 Modulus of the spectral degree of coherence of the MGSM vortex beam propagating in atmospheric turbulence as a function of separation distance $x_{d}$ at propagation distance $z = 2 km$ , (a) for different ${\tilde{C}}_{n}^{2}$ with $δ = 0.5 c m$ ; (b) for different $δ$ with ${\tilde{C}}_{n}^{2} = 10^{- 14} m^{3 - α}$ .

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The relative width $w {(z)}_{turb} / w {(z)}_{free}$ of the MGSM vortex beams propagating in turbulent atmosphere is depicted in Fig. 7. From Fig. 7(a) and 7(b) it is seen that the MGSM vortex beam is less affected turbulence with a lager value of $M$ or a small value of $δ$ . It can also be seen from Fig. 7(c) that the exponent $α$ of the power spectrum plays a crucial role in beam spreading.

Fig. 7 Relative width of the MGSM vortex beams propagating through atmospheric turbulence, (a) for different $M$ with $δ = 0.5 cm$ as a function of $z$ ; (b) for different $δ$ with $M = 5$ as a function of $z$ ; (c) at several propagation distances as a function of $α$ .

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4. Concluding remarks

In this paper, we have explored the behavior of MGSM vortex beams interacting with isotropic and homogeneous turbulent media. The analytic expressions for the cross-spectral function of MGSM vortex beams with $l = \pm 1$ propagating in atmospheric turbulence with general non-Kolmogorov power spectrum are derived and used to explore the evolution of the second-order characteristics in various atmospheric conditions. Via numerical examples we have illustrated that while the MGSM vortex beam can shape its beam profile by adjusting the source parameters in the far zone in vacuum, such modification is suppressed when it passes at sufficiently large distance through the atmospheric turbulence. It is also shown that for free-space propagation the oscillatory trend of the degree of coherence is preserved for any propagation distances, but is destroyed by the turbulence. Moreover the degree of coherence first broadens due to the source correlations and then narrows down, when the effect of turbulence starts to dominate the effect of source coherence. Besides, the numerical results demonstrate that the statistical properties are closely related to the beam index $M$ , the correlation width of the source, and the parameters of the turbulence. The results obtained in this paper would be useful for potential application of vortex beam communication.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) (11274273 and 11474253), the Fundamental Research Funds for the Central Universities (2015FZA3002) and SUTD-ZJU/PILOT/01/2014.

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Propagation of multi-Gaussian Schell-model vortex beams in isotropic random media

Abstract

1. Introduction

2. Propagation of MGSM vortex beams in linear random media

3. Numerical examples and analysis

4. Concluding remarks

Acknowledgments

References and links

Cited By

Figures (7)

Equations (30)

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