Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere

Rong Chen; Lin Liu; Shijun Zhu; Gaofeng Wu; Fei Wang; Yangjian Cai

doi:10.1364/OE.22.001871

1. Introduction

In the past decades, partially coherent beams whose degrees of coherence satisfy Gaussian distributions have been studied extensively and have been applied in optical imaging, Fourier optics, laser scanning, inertial confinement fusion, and reduction of noise in photography, free-space optical communications, particle scattering, particle trapping, remote detection and second-harmonic generation [1–12]. Since Gori and collaborators discussed the sufficient condition for devising a genuine correlation function of a scalar or electromagnetic partially coherent beam [13, 14], more and more attention is being paid to partially coherent beams whose degrees of coherence do not satisfy Gaussian distributions, such as nonuniformly correlated Gaussian Schell-model beam [15–18], multi-Gaussian Schell-model beam [19–22], cosine-Gaussian Schell-model beam [23–25], Laguerre-Gaussian Schell-model (LGSM) beam [26–28], and special correlated partially coherent vector beam [29]. Those beams have been found to exhibit some extraordinary propagation properties, such as self-focusing effect and a lateral shift of the intensity maximum, far field flat-topped beam profile formation and ring shaped profile formation. In [27], we established an experimental setup for generating partially coherent beams with different degrees of coherence, and we reported experimental generation of a LGSM beam for the first time. More recently, we reported experimental generation of a cosine-Gaussian-correlated Schell-model beam with rectangular symmetry [25] and a partially coherent vector beam with special correlation functions [29].

Propagation characteristics of different types of beams propagating in the turbulent atmosphere are of interest for free-space optical communications and remote sensing applications [7, 11, 16, 18, 20–22, 24, 28, 30–52]. It has been found that one can overcome or reduce the negative influence of turbulence by use of a laser beam with special beam profile, phase and polarization or a partially coherent beam. Propagation properties of partially coherent beams whose degrees of coherence satisfy Gaussian distributions in turbulent atmosphere have been studied in detail. Up to now, only few papers were devoted to the propagation of partially coherent beams whose degrees of coherence do not satisfy Gaussian distributions [16, 18, 20–22, 24, 28]. In [28], Cang et al. derived the analytical expressions for the average intensity and the beam width of a LGSM beam in turbulent atmosphere and studied the evolution properties of the average intensity and the beam width of such beam in turbulent atmosphere, while they did not derive the analytical expression for the cross-spectral density of the LGSM beam in turbulent atmosphere. To study the statistical properties, such as degree of coherence and the propagation factor, of a LGSM beam in turbulent atmosphere, one should know the expressions for cross-spectral density and the second-order moments of the Wigner distribution function. In this paper, our aim is to derive the analytical expressions for the cross-spectral density and the second-order moments of the Wigner distribution function of a LGSM beam in turbulent atmosphere, and study its statistical properties. Some useful results are found.

2. Cross-spectral density of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere

The cross-spectral density (CSD) of a LGSM beam at z = 0 is defined as [26]

W (r_{1}, r_{2}) = \exp [- \frac{r_{1}^{2} + r_{2}^{2}}{4 σ_{0}^{2}} - \frac{{(r_{1} - r_{2})}^{2}}{2 σ_{g}^{2}}] L_{n}^{0} [\frac{{(r_{1} - r_{2})}^{2}}{2 σ_{g}^{2}}],

where

r_{1} \equiv (x_{1}, y_{1})

and

r_{2} \equiv (x_{2}, y_{2})

are two arbitrary transverse position vectors at z = 0,

σ_{0}

and

σ_{g}

are the transverse beam width and the transverse coherence width of the LGSM beam, respectively,

L_{n}^{0}

denotes the Laguerre polynomial of mode order n and 0. The degree of coherence of the LGSM beam at z = 0 is given as

μ (r_{1}, r_{2}) = \frac{W (r_{1}, r_{2})}{\sqrt{W (r_{1}, r_{1}) W (r_{2}, r_{2})}} = \exp [- \frac{{(r_{1} - r_{2})}^{2}}{2 σ_{g}^{2}}] L_{n}^{0} [\frac{{(r_{1} - r_{2})}^{2}}{2 σ_{g}^{2}}] .

One finds from Eq. (2) that the degree of coherence of a LGSM beam has a non-Gaussian distribution, which induces unique propagation properties of such beam [26–28]. Under the condition of n = 0, Eq. (1) reduces to the expression for the CSD of a GSM beam.

Within the validity of the paraxial approximation, propagation of the CSD of a partially coherent beam in turbulent atmosphere can be studied with the help of the following extended Huygens-Fresnel integral [7, 31–33, 52]

\begin{array}{l} W (ρ_{1}, ρ_{2}) = \frac{1}{λ^{2} z^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{\infty}^{\infty} \int_{- \infty}^{\infty} W (r_{1}, r_{2}) \times \exp [- \frac{i k}{2 z} {(r_{1} - ρ_{1})}^{2} + \frac{i k}{2 z} {(r_{2} - ρ_{2})}^{2}] \\ \times 〈 \exp [Ψ (r_{1}, ρ_{1}) + Ψ^{*} (r_{2}, ρ_{2})] 〉 d^{2} r_{1} d^{2} r_{2}, \end{array}

where the asterisk denotes the complex conjugate and the angular brackets denote ensemble average,

ρ_{1} \equiv (ρ_{1 x}, ρ_{1 y})

and

ρ_{2} \equiv (ρ_{2 x}, ρ_{2 y})

are two arbitrary transverse position vectors at the receiver plane,

d r_{1} d r_{2} = d x_{1} d y_{1} d x_{2} d y_{2}

,

k = 2 π / λ

is the wave number with

λ

being the wavelength. The expression in the angular brackets in Eq. (3) can be expressed as [7, 31–33, 52]

〈 \exp [Ψ (r_{1}, ρ_{1}) + Ψ^{*} (r_{2}, ρ_{2})] 〉 = \exp [- \frac{{(r_{1} - r_{2})}^{2}}{ρ_{0}^{2}} - \frac{(r_{1} - r_{2}) \cdot (ρ_{1} - ρ_{2})}{ρ_{0}^{2}} - \frac{{(ρ_{1} - ρ_{2})}^{2}}{ρ_{0}^{2}}],

where

ρ_{0} = {(0.545 C_{n}^{2} k^{2} z)}^{- 3 / 5}

is the coherence length of a spherical wave propagating in the turbulent medium with

C_{n}^{2}

being the structure constant. Following [7, 31–33, 52], we have applied the Kolmogorov turbulence spectrum and a quadratic approximation for wave structure function.

Substituting Eqs. (1) and (4) into Eq. (3), we obtain

\begin{array}{l} W (ρ_{1}, ρ_{2}) = \frac{1}{λ^{2} z^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{\infty}^{\infty} \int_{- \infty}^{\infty} \exp [- \frac{i k}{2 z} (ρ_{1}^{2} - ρ_{2}^{2}) - \frac{{(ρ_{1} - ρ_{2})}^{2}}{ρ_{0}^{2}}] \\ \times \exp [(- \frac{i k}{2 z} - \frac{1}{4 σ_{0}^{2}}) r_{1}^{2} + (\frac{i k}{2 z} - \frac{1}{4 σ_{0}^{2}}) r_{2}^{2} + r_{1} \cdot (\frac{i k}{z} ρ_{1} - \frac{ρ_{1} - ρ_{2}}{ρ_{0}^{2}})] \\ \times \exp [- r_{2} \cdot (\frac{i k}{z} ρ_{2} - \frac{ρ_{1} - ρ_{2}}{ρ_{0}^{2}}) - (\frac{1}{2 σ_{g}^{2}} + \frac{1}{ρ_{0}^{2}}) {(r_{1} - r_{2})}^{2}] L_{n}^{0} [\frac{{(r_{1} - r_{2})}^{2}}{2 σ_{g}^{2}}] d^{2} r_{1} d^{2} r_{2} . \end{array}

If we set

\frac{i k}{z} ρ_{1}^{'} = \frac{i k}{z} ρ_{1} - \frac{ρ_{1} - ρ_{2}}{ρ_{0}^{2}}, \frac{i k}{z} ρ_{2}^{'} = \frac{i k}{z} ρ_{2} - \frac{ρ_{1} - ρ_{2}}{ρ_{0}^{2}},

A (ρ_{1}, ρ_{2}) = \exp [- \frac{i k}{2 z} (ρ_{1}^{2} - ρ_{2}^{2}) - \frac{{(ρ_{1} - ρ_{2})}^{2}}{ρ_{0}^{2}}],

Equation (5) is simplified as

\begin{array}{l} W (ρ_{1}^{'}, ρ_{2}^{'}) = \frac{A (ρ_{1}, ρ_{2})}{λ^{2} z^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{\infty}^{\infty} \int_{- \infty}^{\infty} \exp [- (\frac{1}{2 σ_{g}^{2}} + \frac{1}{ρ_{0}^{2}}) {(r_{1} - r_{2})}^{2}] L_{n}^{0} [\frac{{(r_{1} - r_{2})}^{2}}{2 σ_{g}^{2}}] \\ \times \exp [(- \frac{i k}{2 z} - \frac{1}{4 σ_{0}^{2}}) r_{1}^{2} + (\frac{i k}{2 z} - \frac{1}{4 σ_{0}^{2}}) r_{2}^{2} + \frac{i k}{z} (r_{1} \cdot ρ_{1}^{'} - r_{2} \cdot ρ_{2}^{'})] d^{2} r_{1} d^{2} r_{2} . \end{array}

For the convenience of integration, we introduce the following “sum” and “difference” coordinates

r = \frac{r_{1} + r_{2}}{2}, r_{d} = r_{1} - r_{2}, ρ^{'} = \frac{ρ_{1}^{'} + ρ_{2}^{'}}{2}, ρ_{d}^{'} = ρ_{1}^{'} - ρ_{2}^{'},

then Eq. (8) can be expressed as

\begin{array}{l} W (ρ^{'}, ρ_{d}^{'}) = \frac{A (ρ_{1}, ρ_{2})}{λ^{2} z^{2}} \int_{\infty}^{\infty} \int_{- \infty}^{\infty} \exp [- (\frac{1}{2 σ_{g}^{2}} + \frac{1}{ρ_{0}^{2}} + \frac{1}{8 σ_{0}^{2}}) r_{d}^{2} + \frac{i k}{z} r_{d} \cdot ρ^{'}] L_{n}^{0} (\frac{r_{d}^{2}}{2 σ_{g}^{2}}) d^{2} r_{d} \\ \times \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \exp [- \frac{1}{2 σ_{0}^{2}} r^{2} + r \cdot (- \frac{i k}{z} r_{d} + \frac{i k}{z} ρ_{d}^{'})] d^{2} r . \end{array}

After integration over r, Eq. (10) reduces to

\begin{array}{l} W (ρ^{'}, ρ_{d}^{'}) = \frac{A (ρ_{1}, ρ_{2})}{λ^{2} z^{2}} 2 π σ_{0}^{2} \exp (- \frac{k^{2} σ_{0}^{2}}{2 z^{2}} ρ_{d}^{' 2}) \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} L_{n}^{0} (\frac{r_{d}^{2}}{2 σ_{g}^{2}}) \\ \times \exp [- (\frac{1}{2 σ_{g}^{2}} + \frac{1}{ρ_{0}^{2}} + \frac{1}{8 σ_{0}^{2}} + \frac{k^{2} σ_{0}^{2}}{2 z^{2}}) r_{d}^{2} + r_{d} \cdot (\frac{i k}{z} ρ^{'} + \frac{k^{2} σ_{0}^{2}}{z^{2}} ρ_{d}^{'})] d^{2} r_{d} . \end{array}

If we set

a = \frac{1}{2 σ_{g}^{2}} + \frac{1}{ρ_{0}^{2}} + \frac{1}{8 σ_{0}^{2}} + \frac{k^{2} σ_{0}^{2}}{2 z^{2}}, b = \frac{i k}{z}, c = \frac{k^{2} σ_{0}^{2}}{z^{2}},

Equation (11) reduces to

\begin{array}{l} W (ρ^{'}, ρ_{d}^{'}) = \frac{A (ρ_{1}, ρ_{2})}{λ^{2} z^{2}} 2 π σ_{0}^{2} \exp (- \frac{k^{2} σ_{0}^{2}}{2 z^{2}} ρ_{d}^{' 2}) \\ \times \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \exp [- a r_{d}^{2} + r_{d} \cdot (b ρ^{'} + c ρ_{d}^{'})] L_{n}^{0} (\frac{r_{d}^{2}}{2 σ_{g}^{2}}) d^{2} r_{d} . \end{array}

By using the following expansion formulae [53]

L_{n}^{0} (x) = \sum_{p = 0}^{n} (\begin{array}{l} n \\ p \end{array}) \frac{{(- 1)}^{p}}{p!} x^{p},

{(x^{2} + y^{2})}^{p} = \sum_{m = 0}^{p} (\begin{matrix} p \\ m \end{matrix}) x^{2 (p - m)} y^{2 m},

Equation (13) can be expressed in the following alternative form

\begin{array}{l} W (ρ_{x}^{'}, ρ_{y}^{'}, ρ_{d x}^{'}, ρ_{d y}^{'}) = \frac{2 π σ_{0}^{2}}{λ^{2} z^{2}} A (ρ_{1}, ρ_{2}) \exp [- \frac{k^{2} σ_{0}^{2}}{2 z^{2}} (ρ_{d x}^{' 2} + ρ_{d y}^{' 2})] \sum_{p = 0}^{n} \sum_{m = 0}^{p} (\begin{array}{l} n \\ p \end{array}) (\begin{matrix} p \\ m \end{matrix}) \frac{{(- 1)}^{p}}{2^{p} p! σ_{g}^{2 p}} \\ \times \int_{\infty}^{\infty} \int_{- \infty}^{\infty} \exp [- a r_{d x}^{2} + r_{d x} (b ρ_{x}^{'} + c ρ_{d x}^{'}) - a r_{d y}^{2} + r_{d y} (b ρ_{y}^{'} + c ρ_{d y}^{'})] r_{d x}^{2 (p - m)} r_{d y}^{2 m} d r_{d x} d r_{d y} . \end{array}

With the help of the following integral formulae [54]

\int_{- \infty}^{\infty} x^{n} \exp [- {(x - β)}^{2}] d x = {(2 i)}^{- n} \sqrt{π} H_{n} (i β),

where

H_{n}

denotes the Hermite polynomial of mode order n, after integration over

r_{d x}

and

r_{d y}

, we obtain the following expression for the CSD of a LGSM beam in the output plane

\begin{array}{l} W (ρ_{1 x}, ρ_{1 y}, ρ_{2 x}, ρ_{2 y}) = \sum_{p = 0}^{n} \sum_{m = 0}^{p} (\begin{array}{l} n \\ p \end{array}) (\begin{array}{l} p \\ m \end{array}) \frac{1}{2^{3 p} p! σ_{g}^{2 p} a^{p + 1}} \frac{2 π^{2} σ_{0}^{2}}{λ^{2} z^{2}} \exp {- \frac{k^{2} σ_{0}^{2}}{2 z^{2}} [{(ρ_{1 x}^{'} - ρ_{2 x}^{'})}^{2} + {(ρ_{1 y}^{'} - ρ_{2 y}^{'})}^{2}]} \\ \times \exp [- \frac{i k}{2 z} (ρ_{1 x}^{2} + ρ_{1 y}^{2} - ρ_{2 x}^{2} - ρ_{2 y}^{2}) - \frac{{(ρ_{1 x} - ρ_{2 x})}^{2} + {(ρ_{1 y} - ρ_{2 y})}^{2}}{ρ_{0}^{2}}] \\ \times \exp {\frac{1}{4 a} {[b (\frac{ρ_{1 x}^{'} + ρ_{2 x}^{'}}{2}) + c (ρ_{1 x}^{'} - ρ_{2 x}^{'})]}^{2}} H_{2 (p - m)} [i \frac{b (\frac{ρ_{1 x}^{'} + ρ_{2 x}^{'}}{2}) + c (ρ_{1 x}^{'} - ρ_{2 x}^{'})}{2 \sqrt{a}}] \\ \times \exp {\frac{1}{4 a} {[b (\frac{ρ_{1 y}^{'} + ρ_{2 y}^{'}}{2}) + c (ρ_{1 y}^{'} - ρ_{2 y}^{'})]}^{2}} H_{2 m} [i \frac{b (\frac{ρ_{1 y}^{'} + ρ_{2 y}^{'}}{2}) + c (ρ_{1 y}^{'} - ρ_{2 y}^{'})}{2 \sqrt{a}}], \end{array}

where

ρ_{1 x}^{'} = (1 - \frac{z}{i k ρ_{0}^{2}}) ρ_{1 x} + \frac{z}{i k ρ_{0}^{2}} ρ_{2 x}, ρ_{1 y}^{'} = (1 - \frac{z}{i k ρ_{0}^{2}}) ρ_{1 y} + \frac{z}{i k ρ_{0}^{2}} ρ_{2 y},

ρ_{2 x}^{'} = (1 + \frac{z}{i k ρ_{0}^{2}}) ρ_{2 x} - \frac{z}{i k ρ_{0}^{2}} ρ_{1 x}, ρ_{2 y}^{'} = (1 + \frac{z}{i k ρ_{0}^{2}}) ρ_{2 y} - \frac{z}{i k ρ_{0}^{2}} ρ_{1 y} .

The average intensity of the LGSM beam in the output plane is obtained as $I (ρ_{x}, ρ_{y}) = W (ρ_{x}, ρ_{y}, ρ_{x}, ρ_{y})$ , and the degree of coherence of the LGSM beam in the output plane is obtained as

μ (ρ_{1 x}, ρ_{1 y}, ρ_{2 x}, ρ_{2 y}) = \frac{W (ρ_{1 x}, ρ_{1 y}, ρ_{2 x}, ρ_{2 y})}{\sqrt{W (ρ_{1 x}, ρ_{1 y}, ρ_{1 x}, ρ_{1 y}) W (ρ_{2 x}, ρ_{2 y}, ρ_{2 x}, ρ_{2 y})}} .

Applying Eqs. (18) and (21), one can study the evolution properties of the degree of coherence of a LGSM beam in turbulent atmosphere numerically.

3. Second-order moments of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere

The second-order statistical properties, such as the propagation factor, the effective radius of curvature and the Rayleigh range, of a partially coherent beam in turbulent atmosphere are closely related with the second-order moments of the Wigner distribution function [39, 40, 47], in this section, we derive the analytical expressions for the second-order moments of the Wigner distribution function of a LGSM beam in turbulent atmosphere, and then derive the expression for the propagation factor of such beam.

Applying the following “sum” and “difference” coordinates,

r = \frac{r_{1} + r_{2}}{2}, r_{d} = r_{1} - r_{2}, ρ = \frac{ρ_{1} + ρ_{2}}{2}, ρ_{d} = ρ_{1} - ρ_{2},

Equation (3) can be expressed as

\begin{array}{l} W (ρ, ρ_{d}) = {(\frac{k}{2 π z})}^{2} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} W (r, r_{d}) \exp [- \frac{i k}{z} (ρ - r) \cdot (ρ_{d} - r_{d})] \\ \times \exp (- \frac{r_{d}^{2}}{ρ_{0}^{2}} - \frac{r_{d} \cdot ρ_{d}}{ρ_{0}^{2}} - \frac{ρ_{d}^{2}}{ρ_{0}^{2}}) d^{2} r d^{2} r_{d}, \end{array}

where

W (r, r_{d}) = W (r_{1}, r_{2}) = W (r + \frac{r_{d}}{2}, r - \frac{r_{d}}{2}) .

After some operations as shown in [39], i.e., by expressing $W (r, r_{d})$ in term of its Fourier transform and applying the properties of the Dirac delta function, Eq. (23) can be expressed in the following alternative form

\begin{array}{l} W (ρ, ρ_{d}) = {(\frac{1}{2 π})}^{2} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} W (r^{'}, ρ_{d} + \frac{z}{k} κ_{d}) d^{2} r^{'} d^{2} κ_{d} \\ \times \exp (i ρ \cdot κ_{d} - i r^{'} \cdot κ_{d} - \frac{3}{ρ_{0}^{2}} ρ_{d}^{2} - \frac{z^{2}}{ρ_{0}^{2} k^{2}} κ_{d}^{2} - \frac{3 z}{ρ_{0}^{2} k} ρ_{d} \cdot κ_{d}), \end{array}

where

κ_{d} \equiv (\begin{matrix} κ_{d x} & κ_{d y} \end{matrix})

is the position vector in the spatial-frequency domain.

For a LGSM beam, we can express its CSD $W (r^{'}, ρ_{d} + \frac{z}{k} κ_{d})$ as follows

W (r^{'}, ρ_{d} + \frac{z}{k} κ_{d}) = \exp [- \frac{1}{2 σ_{0}^{2}} r^{'}^{2} - (\frac{1}{8 σ_{0}^{2}} + \frac{1}{2 σ_{g}^{2}}) {(ρ_{d} + \frac{z}{k} κ_{d})}^{2}] L_{n}^{0} [\frac{1}{2 σ_{g}^{2}} {(ρ_{d} + \frac{z}{k} κ_{d})}^{2}] .

The Wigner distribution of a partially coherent beam can be expressed in terms of the CSD by the formula [39]

h (ρ, θ) = {(\frac{k}{2 π})}^{2} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} W (ρ, ρ_{d}) \exp (- i k θ \cdot ρ_{d}) d^{2} ρ_{d},

where

θ \equiv (θ_{x}, θ_{y})

denotes an angle which the vector of interest makes with the z-direction,

k θ_{x}

and

k θ_{y}

are the wave vector components along the x-axis and y-axis, respectively.

Applying Eqs. (25)-(27), we obtain the following expression for the Wigner distribution of a LGSM beam in turbulent atmosphere

\begin{array}{l} h (ρ, θ) = \frac{k^{2}}{16 π^{4}} 2 π σ_{0}^{2} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} L_{n}^{0} [\frac{1}{2 σ_{g}^{2}} {(ρ_{d} + \frac{z}{k} κ_{d})}^{2}] \\ \times \exp (- a ρ_{d}^{2} - b κ_{d}^{2} - c ρ_{d} \cdot κ_{d} - i k θ \cdot ρ_{d} + i ρ \cdot κ_{d}) d^{2} κ_{d} d^{2} ρ_{d} . \end{array}

where

a = \frac{1}{8 σ_{0}^{2}} + \frac{1}{2 σ_{g}^{2}} + \frac{3}{ρ_{0}^{2}}, b = \frac{z^{2}}{8 k^{2} σ_{0}^{2}} + \frac{z^{2}}{2 k^{2} σ_{g}^{2}} + \frac{σ_{0}^{2}}{2} + \frac{z^{2}}{k^{2} ρ_{0}^{2}}, c = \frac{z}{4 k σ_{0}^{2}} + \frac{z}{k σ_{g}^{2}} + \frac{3 z}{k ρ_{0}^{2}} .

The moments of order $n_{1} + n_{2} + m_{1} + m_{2}$ of the Wigner distribution function of a beam is defined as

< x^{n_{1}} y^{n_{2}} θ_{x}^{m_{1}} θ_{y}^{m_{2}} > = \frac{1}{P} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} x^{n_{1}} y^{n_{2}} θ_{x}^{m_{1}} θ_{y}^{m_{2}} h (ρ, θ) d^{2} ρ d^{2} θ,

where

P = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} h (ρ, θ, z) d^{2} ρ d^{2} θ .

Substituting Eq. (28) into Eqs. (30) and (31), we obtain (after integration) the following expressions for the second-order moments of the Wigner distribution function of a LGSM beam in a turbulent atmosphere

〈 ρ^{2} 〉 = \frac{z^{2}}{k^{2}} [\frac{1}{2 σ_{0}^{2}} + \frac{2}{σ_{g}^{2}} (1 + n)] + 2 σ_{0}^{2} + \frac{4 z^{2}}{k^{2} ρ_{0}^{2}},

〈 θ^{2} 〉 = \frac{1}{k^{2}} [\frac{1}{2 σ_{0}^{2}} + \frac{2}{σ_{g}^{2}} (1 + n)] + \frac{12}{k^{2} ρ_{0}^{2}},

〈 ρ \cdot θ 〉 = - \frac{z}{k^{2}} [\frac{1}{2 σ_{0}^{2}} + \frac{2}{σ_{g}^{2}} (1 + n)] - \frac{6 z}{k^{2} ρ_{0}^{2}} .

The propagation factor of a partially coherent beam in a turbulent atmosphere is defined in terms of the second-order moments as follows [39]

M^{2} (z) = k {(〈 ρ^{2} 〉 〈 θ^{2} 〉 - {〈 ρ \cdot θ 〉}^{2})}^{1 / 2} .

Substituting Eqs. (32)-(34) into Eq. (35), we obtain the following expression for the propagation factor of a LGSM beam in turbulent atmosphere

\begin{array}{l} M^{2} (z) = {[\frac{z^{2}}{2 k^{2} σ_{0}^{2}} + \frac{2 z^{2}}{k^{2} σ_{g}^{2}} (1 + n) + 2 σ_{0}^{2} + \frac{4 z^{2}}{k^{2} ρ_{0}^{2}}] [\frac{1}{2 σ_{0}^{2}} + \frac{2}{σ_{g}^{2}} (1 + n) + \frac{12}{ρ_{0}^{2}}] \\ - {[\frac{z}{2 σ_{0}^{2}} + \frac{2 z}{σ_{g}^{2}} (1 + n) + \frac{6 z}{ρ_{0}^{2}}]}^{2}} \end{array}

Under the condition of $ρ_{0} = \infty$ , Eq. (36) reduces to the following expression for the propagation factor of a LGSM beam in free space

M^{2} = {[1 + \frac{4 σ_{0}^{2}}{σ_{g}^{2}} (1 + n)]}^{1 / 2} .

From Eq. (37), we see that the propagation factor of a LGSM beam in free space is independent of the propagation distance as expected, and its value increases as the beam order increases.

Under the condition of $ρ_{0} = \infty$ and n = 0, Eq. (37) reduces to the following expression for the propagation factor of a GSM beam in free space [55]

M^{2} = {(1 + \frac{4 σ_{0}^{2}}{σ_{g}^{2}})}^{1 / 2} .

4. Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere

In this section, we study the statistical properties of a LGSM beam in turbulent atmosphere numerically by applying the formulae derived in above sections.

Figure 1 shows the modulus of the degree of coherence of a LGSM beam at several propagation distances in free space with $λ = 632.8 nm, σ_{g} = 5 mm,$ $σ_{0} = 10 mm$ and n = 2. One finds from Fig. 1 that the degree of coherence of the LGSM beam in the source plane has non-Gaussian distribution, and there are side robes around the main peak. The side robes in the degree of coherence disappear gradually on propagation in free space. In the far field, only the main peak exists which also has non-Gaussian distribution. Figure 2 shows the modulus of the degree of coherence of a LGSM beam at several propagation distances in turbulent atmosphere for different values of the structure constant $C_{n}^{2}$ with $λ = 632.8 nm,$ $σ_{g} = 5 mm,$ $σ_{0} = 10 mm$ and n = 1. From Fig. 2, one finds that the evolution properties of the degree of coherence at short propagation distance in turbulent atmosphere are similar to the corresponding evolution properties in free space, i.e., the side robes disappear gradually on propagation. While at long propagation distance, the distribution of the degree of coherence in turbulent atmosphere is much different from that in free space. In turbulent atmosphere, the degree of coherence becomes of Gaussian distribution in the far field. We may explain this phenomenon by the fact that at the short propagation distance, the influence of the turbulence can be neglected and the role of free-space diffraction plays a dominant role. At long propagation distance, the influence turbulence plays a dominant role, and the degree of coherence takes a Gaussian distribution due to the isotropic influence of the turbulence. Figure 3 shows modulus of the degree of coherence of a LGSM beam at several propagation distances in turbulent atmosphere for different values of the mode order n. One finds from Fig. 3 that the evolution properties of the degree of coherence of the LGSM beam in turbulent atmosphere are also affected by the mode order n. The conversion from the non-Gaussian distribution to Gaussian distribution becomes slower as the mode order n increases, which means that a LGSM beam with larger n is less affected by turbulence.

Fig. 1 Modulus of the degree of coherence of a LGSM beam at several propagation distances in free space.

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Fig. 2 Modulus of the degree of coherence of a LGSM beam at several propagation distances in turbulent atmosphere for different values of the structure constant $C_{n}^{2}$ .

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Fig. 3 Modulus of the degree of coherence of a LGSM beam at several propagation distances in turbulent atmosphere for different values of the mode order n.

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Figure 4 shows the normalized propagation factor of a LGSM beam versus the propagation distance in turbulent atmosphere for different values of the mode order n and the structure constant $C_{n}^{2}$ with $λ = 632.8 nm, σ_{0} = 10 mm, σ_{g} = 10 mm .$ Fig. 5 shows the normalized propagation factor of a LGSM beam versus the propagation distance in turbulent atmosphere for different values of the mode order n and the coherence width $σ_{g}$ with $λ = 632.8 nm, σ_{0} = 10 mm, C_{n}^{2} = 5 \times 10^{- 15} m^{- 2 / 3}$ . Figure 6 shows the normalized propagation factor of a LGSM beam versus the propagation distance in turbulent atmosphere for different values of the mode order n and the wavelength $λ$ with $C_{n}^{2} = 5 \times 10^{- 15} m^{- 2 / 3},$ $σ_{0} = 10 m m, σ_{g} = 10 m m .$ . One finds from Figs. 4-6 that the normalized propagation factor of a LGSM beam increases on propagation in turbulent atmosphere, which is much different from its properties in free space, where the propagation factor is independent of the propagation distance z. Thus, the turbulence degrades the beam quality of the LGSM beam on propagation. We find that the normalized propagation factor of a LGSM beam with larger n increases slower than a LGSM beam with smaller n or a GSM beam (n = 0) on propagation, which means that the LGSM beam with larger n is less affected by turbulence. Furthermore, we note that the advantage of a LGSM beam with larger n over a LGSM beam with smaller n or a GSM beam is enhanced for larger structure constant $C_{n}^{2}$ , larger coherence width $σ_{g}$ and smaller wavelength $λ$ . Thus, it is necessary for us to take these parameters into consideration in practical applications.

Fig. 4 Normalized propagation factor of a LGSM beam versus the propagation distance in turbulent atmosphere for different values of the mode order n and the structure constant $C_{n}^{2}$ .

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Fig. 5 Normalized propagation factor of a LGSM beam versus the propagation distance in turbulent atmosphere for different values of the mode order n and the coherence width $σ_{g}$ .

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Fig. 6 Normalized propagation factor of a LGSM beam versus the propagation distance in turbulent atmosphere for different values of the mode order n and the wavelength $λ$ .

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5. Summary

We have derived the analytical expressions for the CSD and the second-order moments of a LGSM beam in turbulent atmosphere, and we have studied the statistical properties, such as the degree of coherence and the propagation factor, of a LGSM beam in turbulent atmosphere with the help of the derived formulae. We have found that a LGSM beam with larger mode order n is less affected by turbulence than a LGSM beam with smaller mode order n or a GSM beam by choosing suitable beam parameters. In [7], it is shown that a GSM beam has advantage over a coherent Gaussian beam for reducing the turbulence-induced degradation, thus it is useful in free-space optical communication. The results in our manuscript have shown that a LGSM beam has advantage over a GSM beam for reducing the turbulence-induced degradation, thus we can expect that the LGSM beam will be useful in free-space optical communication.

Acknowledgments

This research is supported by the National Natural Science Foundation of China under Grant Nos. 11274005 &11104195, the Huo Ying Dong Education Foundation of China under Grant No. 121009, the Key Project of Chinese Ministry of Education under Grant No. 210081, the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, the Universities Natural Science Research Project of Jiangsu Province under Grant No. 11KJB140007, the Innovation Plan for Graduate Students in the Universities of Jiangsu Province under Grant Nos. CXLX12_0780 & CXZZ13_07, and the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

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Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere

Abstract

1. Introduction

2. Cross-spectral density of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere

3. Second-order moments of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere

4. Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere

5. Summary

Acknowledgments

References and links

Cited By

Figures (6)

Equations (38)

Optics Express