Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence

Jiayi Yu; Yahong Chen; Lin Liu; Xianlong Liu; Yangjian Cai

doi:10.1364/OE.23.013467

Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence

Jiayi Yu, Yahong Chen, Lin Liu, Xianlong Liu, and Yangjian Cai

Optics Express
Vol. 23,
Issue 10,
pp. 13467-13481
(2015)
•https://doi.org/10.1364/OE.23.013467

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Jiayi Yu, Yahong Chen, Lin Liu, Xianlong Liu, and Yangjian Cai, "Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence," Opt. Express 23, 13467-13481 (2015)

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- Atmospheric and Oceanic Optics
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About this Article
History
- Original Manuscript: March 31, 2015
- Revised Manuscript: May 9, 2015
- Manuscript Accepted: May 9, 2015
- Published: May 13, 2015

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Abstract

Elegant Hermite-Gaussian correlated Schell-model (EHGCSM) beam was introduced in theory and generated in experiment just recently [Phys. Rev. A 91, 013823 (2015)]. In this paper, we study the propagation properties of an EHGCSM beam in turbulent atmosphere with the help of the extended Huygens-Fresnel integral. Analytical expressions for the cross-spectral density and the propagation factors of an EHGCSM beam propagating in turbulent atmosphere are derived. The statistical properties, such as the spectral intensity, the spectral degree of coherence and the propagation factors, of an EHGCSM beam in Kolmogorov and non-Kolmogorov turbulence are illustrated numerically. It is found that an EHGCSM beam exhibits splitting and combing properties in turbulent atmosphere, and an EHGCSM beam with large mode orders is less affected by turbulence than an EHGCSM beam with small mode orders or a Gaussian Schell-model beam or a Gaussian beam, which will be useful in free-space optical communications.

1. Introduction

In the past decades, scalar and vector partially coherent beams with conventional correlation functions (i.e., Gaussian correlated Schell-model functions) have been explored in detail both theoretically and experimentally [1–4]. Since the sufficient conditions for devising genuine correlation functions of scalar and vector partially coherent beams were discussed by Gori et al. [5, 6], a great deal of attention has been paid to the correlation functions of partially coherent beams [7–21]. A variety of partially coherent beams with non-conventional correlation functions, such as nonuniformly correlated beam [10], multi-Gaussian correlated Schell-mode beam [11, 12], Laguerre-Gaussian correlated Schell-model (LGSM) beam [13, 14], specially correlated radially polarized beam [15] and elegant Hermite-Gaussian correlated Schell-model (EHGCSM) beam [16], have been introduced theoretically, and such beams display many extraordinary propagation properties, such as self-focusing and a lateral shift of the intensity maximum, far-field flat-topped and ring-shaped beam profile formation, and self-splitting. Experimental generation of various partially coherent beams with non-conventional correlation functions were reported recently [14–20]. A review on generation and propagation of partially coherent beams with nonconventional correlation functions can be found in [21]. EHGCSM beam (called Hermite-Gaussian correlated Schell-model beam in [16]) exhibits self-splitting property on propagation in free space and a focused EHGCSM beam exhibits splitting and combining properties near the focal plane, which may be useful for attacking multiple targets, trapping multiple particles, and guiding atoms.

Due to their important applications in free-space optical communications, remote sensing of atmosphere and target tracking, the propagation properties of various beams in turbulent atmosphere have been studied extensively [22–41]. Most previous literatures are about the propagation of coherent beams or partially coherent beams with conventional correlation functions in turbulent atmosphere, and it has been found that one can use a light beam with special beam profile or phase or polarization or partially coherence to overcome or reduce turbulence-induced degradation [22–35]. Up to now, only few papers were paid to the propagation properties of partially coherent beams with non-conventional correlation functions in turbulent atmosphere [36–41]. In this paper, our aim is to explore the propagation properties of the EHGCSM beam in turbulent atmosphere. It is found that an EHGCSM beam exhibits splitting and combing properties in Kolmogorov and non-Kolmogorov turbulence, and an EHGCSM beam with large mode orders has advantage over an EHGCSM beam with smalle mode orders or a Gaussian Schell-model beam or a Gaussian beam for reducing turbulence-induced degradation. Thus, modulating the correlation function of a partially coherent beam will be useful in free-space optical communications.

2. Cross-spectral density of an EHGCSM beam propagating in turbulent atmosphere

The cross-spectral density (CSD) of an EHGCSM beam in the source plane (z = 0) is expressed as follows [16]

W^{(0)} (r_{1}, r_{2}, 0) = G_{0} \exp (- \frac{r_{1}^{2} + r_{2}^{2}}{4 σ_{0}^{2}}) μ (r_{2} - r_{1}),

where

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

and

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

are two arbitrary transverse position vectors in the source plane,

G_{0}

is a constant,

σ_{0}

denotes the transverse beam width,

μ (r_{2} - r_{1})

represents the spectral degree of coherence given by

μ (r_{2} - r_{1}) = \frac{H_{2 m} [(x_{2} - x_{1}) / \sqrt{2} δ_{0 x}]}{H_{2 m} (0)} \frac{H_{2 n} [(y_{2} - y_{1}) / \sqrt{2} δ_{0 y}]}{H_{2 n} (0)} \exp [- \frac{{(x_{2} - x_{1})}^{2}}{2 δ_{0 x}^{2}} - \frac{{(y_{2} - y_{1})}^{2}}{2 δ_{0 y}^{2}}],

with

δ_{0 x}

and

δ_{0 y}

being the transverse coherence widths along x and y directions, respectively. Here

H_{m}

denotes the Hermite polynomial of order . When

m = n = 0

and

δ_{0 x} = δ_{0 y}

, the EHGCSM beam reduces to the Gaussian correlated Schell-model beam (conventionally called Gaussian Schell-model beam) [1–3]. When

m = n = 0

and

δ_{0 x} \neq δ_{0 y}

, the EHGCSM beam reduces to the elliptical Gaussian correlated Schell-model beam [17]. As shown in [16], the spectral degree of coherence of the EHGCSM beam exhibits array distribution with rectangular symmetry, and the EHGCSM beam exhibits self-splitting properties on propagation in free space (i.e., the initial single beam spot evolves into two or four beam spots in the far field) due to the nonconventional correlation function (i.e., non-Gaussian distribution of the spectral degree of coherence).

Now we study the propagation of an EHGCSM beam in turbulent atmosphere. Paraxial propagation of the CSD of a partially coherent beam in turbulent atmosphere can be treated by the following generalized Huygens-Fresnel integral [22–25, 32]

\begin{array}{l} W (ρ_{1}, ρ_{2}, z) = \frac{1}{λ^{2} z^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} W^{(0)} (r_{1}, r_{2}, 0) \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

〈 ... 〉

denotes ensemble average,

k = 2 π / λ

is wave number with

λ

being the wavelength,

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}

. The ensemble average term in Eq. (3) can be expressed as [22–25, 32]

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

where

Φ_{n} (κ)

is the spatial power spectrum of the refractive-index fluctuations of the turbulent medium. For the simplicity of expression, we set

T = \int_{0}^{\infty} κ^{3} Φ_{n} (κ) d κ .

If the turbulence obeys the non-Kolmogorov statistics and the power spectrum $Φ_{n} (κ)$ has the van Karman form, in which the slope 11/3 is generalized to an arbitrary parameter $α$ , T can be expressed in the following form [32]

T = \frac{A (α)}{2 (α - 2)} {\tilde{C}}_{n}^{2} [β κ_{m}^{2 - α} \exp (κ_{0}^{2} / κ_{m}^{2}) Γ_{1} (2 - α / 2, κ_{0}^{2} / κ_{m}^{2}) - 2 κ_{0}^{4 - α}], 3 < α < 4,

where

Γ_{1}

is the incomplete Gamma function,

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

κ_{0} = 2 π / L_{0}

with

L_{0}

being the outer scale of turbulence,

κ_{m} = c (α) / l_{0}

with

l_{0}

being the inner scale of turbulence, and

A (α) = \frac{1}{4 π^{2}} Γ (α - 1) \cos (\frac{α π}{2}), c (α) = {[\frac{2 π A (α)}{3} Γ (5 - \frac{α}{2})]}^{1 / (α - 5)} .

The term

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

in Eq. (6) is a generalized refractive-index structure parameter with units

m^{3 - α}

, and

Γ (\cdot)

in Eq. (7) represents the Gamma function. Under the condition of

α = 11 / 3

, the power spectrum

Φ_{n} (κ)

reduces to the van Karman spectrum with Kolmogorov statistics.

Substituting Eqs. (1) and (4) into Eq. (3), and by setting

\frac{i k}{z} ρ_{1}^{'} = \frac{i k}{z} ρ_{1} - \frac{π^{2} k^{2} z T}{3} (ρ_{1} - ρ_{2}), \frac{i k}{z} ρ_{2}^{'} = \frac{i k}{z} ρ_{2} - \frac{π^{2} k^{2} z T}{3} (ρ_{1} - ρ_{2}),

A (ρ_{1}, ρ_{2}) = \exp [- \frac{i k}{2 z} (ρ_{1}^{2} - ρ_{2}^{2}) - \frac{π^{2} k^{2} z T}{3} {(ρ_{1} - ρ_{2})}^{2}],

we obtain

\begin{array}{l} W ({ρ^{'}}_{1}, {ρ^{'}}_{2}, z) = \frac{G_{0}}{λ^{2} z^{2}} A (ρ_{1}, ρ_{2}) \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \frac{H_{2 m} [(x_{2} - x_{1}) / \sqrt{2} δ_{0 x}]}{H_{2 m} (0)} \frac{H_{2 n} [(y_{2} - y_{1}) / \sqrt{2} δ_{0 y}]}{H_{2 n} (0)} \\ \times \exp [- \frac{{(x_{2} - x_{1})}^{2}}{2 δ_{0 x}^{2}} - \frac{{(y_{2} - y_{1})}^{2}}{2 δ_{0 y}^{2}}] \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}) - \frac{π^{2} k^{2} z T}{3} {(r_{1} - r_{2})}^{2}] d^{2} r_{1} d^{2} r_{2} . \end{array}

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

\begin{array}{l} r_{s} = \frac{r_{1} + r_{2}}{2}, x_{s} = \frac{x_{1} + x_{2}}{2}, y_{s} = \frac{y_{1} + y_{2}}{2}, \\ r_{d} = r_{1} - r_{2}, x_{d} = x_{1} - x_{2}, y_{d} = y_{1} - y_{2}, \\ {ρ^{'}}_{s} = \frac{{ρ^{'}}_{1} + {ρ^{'}}_{2}}{2}, {ρ^{'}}_{s x} = \frac{{ρ^{'}}_{1 x} + {ρ^{'}}_{2 x}}{2}, {ρ^{'}}_{s y} = \frac{{ρ^{'}}_{1 y} + {ρ^{'}}_{2 y}}{2}, \\ {ρ^{'}}_{d} = {ρ^{'}}_{1} - {ρ^{'}}_{2}, {ρ^{'}}_{d x} = {ρ^{'}}_{1 x} - {ρ^{'}}_{2 x}, {ρ^{'}}_{d y} = {ρ^{'}}_{1 y} - {ρ^{'}}_{2 y}, \end{array}

After tedious integration, Eq. (10) reduces to

\begin{array}{l} W (ρ_{1}, ρ_{2}, z) = \frac{4 G_{0} σ_{0}^{2} π^{2} δ_{0 x} δ_{0 y}}{H_{2 m} (0) H_{2 n} (0) \sqrt{a b} λ^{2} z^{2}} {(1 - \frac{1}{a})}^{m} {(1 - \frac{1}{b})}^{n} H_{2 m} [- \frac{c}{2 a} {(1 - \frac{1}{a})}^{- 1 / 2}] \\ \times H_{2 n} [- \frac{d}{2 b} {(1 - \frac{1}{b})}^{- 1 / 2}] \exp [\frac{c^{2}}{4 a} + \frac{d^{2}}{4 b} - \frac{i k}{z} ({ρ^{'}}_{s x} {ρ^{'}}_{d x} + {ρ^{'}}_{s y} {ρ^{'}}_{d y}) - (\frac{2 π^{2} k^{2} z T}{3} + \frac{σ_{0}^{2} k^{2}}{2 z^{2}}) ({ρ^{'}}_{d x}^{2} + {ρ^{'}}_{d y}^{2})], \end{array}

with

\begin{array}{l} a = 2 δ_{0 x}^{2} (\frac{1}{8 σ_{0}^{2}} + \frac{π^{2} k^{2} z T}{3} + \frac{1}{2 δ_{0 x}^{2}} + \frac{k^{2} σ_{0}^{2}}{2 z^{2}}), b = 2 δ_{0 y}^{2} (\frac{1}{8 σ_{0}^{2}} + \frac{π^{2} k^{2} z T}{3} + \frac{1}{2 δ_{0 y}^{2}} + \frac{k^{2} σ_{0}^{2}}{2 z^{2}}), \\ c = \sqrt{2} δ_{0 x} (\frac{i k {ρ^{'}}_{s x}}{z} + \frac{k^{2} σ_{0}^{2} {ρ^{'}}_{d x}}{z^{2}}), d = \sqrt{2} δ_{0 y} (\frac{i k {ρ^{'}}_{s y}}{z} + \frac{k^{2} σ_{0}^{2} {ρ^{'}}_{d y}}{z^{2}}) . \end{array}

In above derivations, we have used the following integral formulae

δ (s) = \frac{1}{2 π} \int_{- \infty}^{\infty} \exp (- i s x) d x,

\int_{- \infty}^{\infty} f (x) δ^{n} (x) d x = {(- 1)}^{n} f^{(n)} (0), (n = 0, 1, 2),

\int_{- \infty}^{\infty} \exp (- s^{2} x^{2} \pm q x) d x = \frac{\sqrt{π}}{s} \exp (\frac{q^{2}}{4 s^{2}}),

\int_{- \infty}^{+ \infty} \exp [- \frac{{(x - y)}^{2}}{2 m}] H_{n} (x) d x = \sqrt{2 π m} {(1 - 2 m)}^{n / 2} H_{n} [y {(1 - 2 m)}^{- 1 / 2}] .

The spectral intensity of the EHGCSM beam in the output plane is obtained as

S (ρ, z) = W (ρ_{1}, ρ_{2}, z) .

The spectral degree of coherence of the EHGCSM beam in the output plane is obtained as

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

Applying Eqs. (12), (18) and (19), one can study the evolution properties of the spectral intensity and spectral degree of coherence in Kolmogorov or non-Kolmogorov turbulence numerically in a convenient way.

3. Propagation factors of an EHGCSM beam propagating in turbulent atmosphere

In this section, we are going to derive the analytical expressions for the second-order moments of the Wigner distribution function (WDF) of an EHGCSM beam in turbulent atmosphere, and to derive the expressions for the propagation factors of such beam in turbulent atmosphere.

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

ρ_{s} = \frac{ρ_{1} + ρ_{2}}{2}, ρ_{d} = ρ_{1} - ρ_{2},

Equation (3) can be expressed as follows

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

where

W^{(0)} (r_{s}, r_{d}, 0) = W^{(0)} (r_{1}, r_{2}, 0) = W^{(0)} (r_{s} + \frac{r_{d}}{2}, r_{s} - \frac{r_{d}}{2}, 0),

and

H (ρ_{d}, r_{d}, z)

represents the effect of the turbulence defined as

H (ρ_{d}, r_{d}, z) = 4 π^{2} k^{2} z \int_{0}^{1} d ξ \int_{0}^{\infty} [1 - J_{0} (κ | r_{d} ξ + (1 - ξ) ρ_{d} |)] Φ_{n} (κ) κ d κ,

where

κ_{d} \equiv (κ_{d x} κ_{d y})

is the position vector in the spatial-frequency domain,

J_{0}

is Bessel function of zero order.

After some operations as shown in [28], Eq. (21) can be expressed in the following alternative form

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

where

κ_{d} \equiv (κ_{d x} κ_{d y})

is the position vector in the spatial-frequency domain, and

H (ρ_{d}, ρ_{d} + \frac{z}{k} κ_{d}, z) = \frac{π^{2} k^{2} z}{3} (3 ρ_{d}^{2} + 3 \frac{z}{k} ρ_{d} \cdot κ + \frac{z^{2}}{k^{2}} κ_{d}^{2}) T .

For an EHGCSM beam, using Eq. (1), we can express the CSD

W (r_{s}^{'}, ρ_{d} + \frac{z}{k} κ_{d}, 0)

as follows

\begin{array}{l} W ({r^{'}}_{s}, ρ_{d} + \frac{z}{k} κ_{d}, 0) = \frac{G_{0}}{H_{2 m} (0) H_{2 n} (0)} \exp [- \frac{{r^{'}}_{s}^{2}}{2 σ_{0}^{2}} - \frac{1}{8 σ_{0}^{2}} {(ρ_{d} + \frac{z κ_{d}}{k})}^{2}] \\ \times H_{2 m} [- \frac{1}{\sqrt{2} δ_{0 x}} (ρ_{d x} + \frac{z}{k} κ_{d x})] H_{2 n} [- \frac{1}{\sqrt{2} δ_{0 y}} (ρ_{d y} + \frac{z}{k} κ_{d y})] \\ \times \exp [- \frac{1}{2 δ_{0 x}^{2}} {(ρ_{d x} + \frac{z}{k} κ_{d x})}^{2}] \exp [- \frac{1}{2 δ_{0 y}^{2}} {(ρ_{d y} + \frac{z}{k} κ_{d y})}^{2}] . \end{array}

The Wigner distribution of a partially coherent beam can be expressed in terms of the CSD by the following formula [28]

h (ρ_{s}, θ, z) = {(\frac{k}{2 π})}^{2} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} W (ρ_{s}, ρ_{d}, z) \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. (24)-(27), we obtain the following expression for the WDF of the EHGCSM beam in turbulent atmosphere

h (ρ_{s}, θ, z) = h (ρ_{s x}, θ_{x}, z) h (ρ_{s y}, θ_{y}, z),

with

\begin{array}{l} h (ρ_{s j}, θ_{j}, z) = \frac{\sqrt{2 π G_{0} σ_{0}^{2}} k}{4 π^{2} H_{2 l} (0)} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} H_{2 l} [- \frac{1}{\sqrt{2} δ_{0 x}} (ρ_{d x} + \frac{z}{k} κ_{d x})] \\ \times \exp (- a_{x} ρ_{d x}^{2} - b_{x} κ_{d x}^{2} - c_{x} ρ_{d x} κ_{d x} - i ρ_{s x} κ_{d x} - i k θ_{x} ρ_{d x}) d κ_{d x} d ρ_{d x}, (j = x, y l = m, n), \end{array}

where

\begin{array}{l} a_{j} = \frac{1}{2 δ_{0 j}^{2}} + \frac{1}{8 σ_{0}^{2}} + π^{2} k^{2} z T, b_{j} = \frac{z^{2}}{2 k^{2} δ_{0 j}^{2}} + \frac{z^{2}}{8 k^{2} σ_{0}^{2}} + \frac{σ_{0}^{2}}{2} + \frac{π^{2} z^{3} T}{3}, \\ c_{j} = \frac{z}{k δ_{0 j}^{2}} + \frac{z}{4 k σ_{0}^{2}} + π^{2} z^{2} k T, (j = x, y) . \end{array}

The moments of order $n_{1} + n_{2} + m_{1} + m_{2}$ of the WDF of a beam is defined as [28, 33]

〈 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_{s}, θ, z) d^{2} ρ_{s} d^{2} θ,

where

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

Substituting Eq. (28) into Eqs. (31) and (32), we obtain (after tedious integration) the following expressions for the second-order moments of WDF of the EHGCSM beam in a turbulent atmosphere

〈 ρ_{j}^{2} 〉 = \frac{\sqrt{2 π G_{0} σ_{0}^{2}}}{P_{j} H_{2 l} (0)} [\frac{2 z^{2} {(- 1)}^{l - 1} (2 l)!}{δ_{0 j}^{2} k^{2} (l - 1)!} - 2 b_{j} \frac{{(- 1)}^{l} (2 l)!}{l!}],

〈 θ_{j}^{2} 〉 = \frac{\sqrt{2 π G_{0} σ_{0}^{2}}}{P_{j} H_{2 l} (0)} [\frac{2 {(- 1)}^{l - 1} (2 l)!}{δ_{0 j}^{2} k^{2} (l - 1)!} - 2 \frac{a_{j}}{k^{2}} \frac{{(- 1)}^{l} (2 l)!}{l!}],

〈 ρ_{j} θ_{j} 〉 = \frac{\sqrt{2 π G_{0} σ_{0}^{2}}}{P_{j} H_{2 l} (0)} [\frac{2 z {(- 1)}^{l - 1} (2 l)!}{δ_{0 j}^{2} k^{2} (l - 1)!} - \frac{c_{j}}{k} \frac{{(- 1)}^{l} (2 l)!}{l!}],

with

P_{j} = \frac{{(- 1)}^{l} (2 l)! \sqrt{2 π G_{0} σ_{0}^{2}}}{l! H_{2 l} (0)}, (j = x, y, l = m, n) .

The propagation factor (also named M² factor) introduced by Siegman is an important property of a beam being regarded as a beam quality factor in many practical applications [42]. Gori et al. introduced the definition of the propagation factor of a partially coherent beam [43, 44]. The propagation factor of a partially coherent beam in turbulent atmosphere was introduced in [28], and is related with the second-order moments of the WDF by the following formula

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

Because the EHGCSM beam is of rectangular symmetry, its propagation factors are related with the second-order moments of the WDF by the following formulae [45]

M_{x}^{2} (z) = 2 k {(〈 ρ_{x}^{2} 〉 〈 θ_{x}^{2} 〉 - {〈 ρ_{x} θ_{x} 〉}^{2})}^{1 / 2}, M_{y}^{2} (z) = 2 k {(〈 ρ_{y}^{2} 〉 〈 θ_{y}^{2} 〉 - {〈 ρ_{y} θ_{y} 〉}^{2})}^{1 / 2},

where

M_{x}^{2} (z)

and

M_{y}^{2} (z)

are the propagation factors along x and y directions, respectively.

Substituting Eqs. (33)-(36) into Eq. (38), we obtain the following explicit expressions for the propagation factors of the EHGCSM beam in turbulent atmosphere

\begin{array}{l} M_{j}^{2} (z) = 2 {[\frac{z^{2}}{δ_{0 j}^{2} k^{2}} (2 l + 1) + \frac{z^{2}}{4 k^{2} σ_{0}^{2}} + σ_{0}^{2} + \frac{2 π^{2} z^{3} T}{3}] [\frac{1}{δ_{0 j}^{2}} (2 l + 1) + \frac{1}{4 σ_{0}^{2}} + 2 π^{2} k^{2} z T] \\ {- {[\frac{z}{k δ_{0 j}^{2}} (2 l + 1) + \frac{z}{4 k σ_{0}^{2}} + π^{2} z^{2} k T]}^{2}}}^{1 / 2}, (j = x, y, l = m, n) . \end{array}

Under the condition of $Φ_{n} (κ) = 0$ (without turbulence), Eq. (39) reduces to the expressions for the propagation factors of the EHGCSM beam in free space

M_{j}^{2} (z) = {[4 \frac{σ_{0}^{2}}{δ_{0 j}^{2}} (2 l + 1) + 1]}^{1 / 2} = {\frac{4 σ_{0}^{2}}{δ_{0 j}^{2}} [1 - \frac{4 l (2 l - 1) H_{2 l - 2} (0)}{H_{2 l} (0)}] + 1}^{1 / 2}, (j = x, y, l = m, n) .

Equation (40) is consistent with results reported in [16].

Under the condition of m = n = 0, Eqs. (39) and (40) reduce to the expressions for the propagation factors of a Gaussian Schell-model beam in turbulent atmosphere and in free space, respectively. Under the condition of m = n = 0 and $δ_{0 x} = δ_{0 y} = \infty$ , Eqs. (39) and (40) reduce to the expressions for the propagation factors of a coherent Gaussian beam in turbulent atmosphere and in free space, respectively.

4. Statistical properties of an EHGCSM beam propagating in turbulent atmosphere

Now we study the statistical properties of an EHGCSM beam propagating in turbulent atmosphere by using the formulae derived in above sections. In the following numerical examples, the parameters of the beam and the turbulence are set as $λ = 632.8 n m$ , $σ_{0} = 10 m m$ , $δ_{0 x} = δ_{0 y} = 4 m m$ , $L_{0} = 1 m$ , $l_{0} = 1 m m$ , ${\tilde{C}}_{n}^{2} = 5 \times 10^{- 15} m^{3 - α}$ .

We calculate in Fig. 1 the 3D-normalized spectral intensity distribution of an EHGCSM beam at several propagation distances in Kolmogorov turbulence with m = n = 5 and $α = 11 / 3$ , and in Fig. 2 the 3D-normalized spectral intensity distribution of an EHGCSM beam at several propagation distances in non-Kolmogorov turbulence with m = n = 5 and $α = 3.1$ . For the convenience of comparison, the 3D-normalized spectral intensity distribution of an EHGCSM beam at several propagation distances in free space with m = n = 5 and $α = 11 / 3$ is shown in Fig. 3. One finds from Figs. 1 and 2 that the EHGCSM beam has a Gaussian beam profile in the source plane, and in Kolmogorov or non-Kolmogorov turbulence, the EHGCSM beam exhibits splitting properties at short propagation distance (i.e., the initial single beam spot evolves into four beam spots on propagation), which is similar to its propagation properties in free space (see Fig. 3), while at long propagation distance, the EHGCSM beam exhibits combing properties (i.e., the four beam spots evolves into one beam spot on propagation). One can explain this phenomenon by the fact that the influence of turbulence can be neglected and the free-space diffraction plays a dominant role at short propagation distance, thus the propagation properties EHGCSM beam in turbulence is similar to those in free space. With the further increase of the propagation distance, the influence of turbulence accumulates and plays a dominant role gradually, and the four beam spots evolves into one beam spot again at long propagation distance due to the isotropic influence of turbulence.

Fig. 1 3D-normalized spectral intensity distribution $S (ρ, z) / S {(ρ, z)}_{\max}$ of an EHGCSM beam at several propagation distances in Kolmogorov turbulence with m = n = 5 and $α = 11 / 3$ .

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Fig. 2 3D-normalized spectral intensity distribution $S (ρ, z) / S {(ρ, z)}_{\max}$ of an EHGCSM beam at several propagation distances in non-Kolmogorov turbulence with m = n = 5 and $α = 3.1$ .

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Fig. 3 3D-normalized spectral intensity distribution $S (ρ, z) / S {(ρ, z)}_{\max}$ of an EHGCSM beam at several propagation distances in free space with m = n = 5 and $α = 11 / 3$ .

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Figure 4 shows the ratio of the spectral intensity in the optical axis ( $ρ = 0$ ) to the maximum spectral intensity in the transverse plane of an EHGCSM beam versus the propagation distance in Kolmogorov ( $α = 11 / 3$ ) or non-Kolmogorov turbulence ( $α = 3.1$ ) for different values of the mode orders $m$ and $n$ . One finds from Fig. 4 that the ratio of the spectral intensity in the optical axis ( $ρ = 0$ ) to the maximum spectral intensity in the transverse plane equals to one when the propagation distance is very short, and the ratio decreases gradually on propagation, which means the beam spot gradually splits into four beam spots. The ratio recovers to one at long propagation distance, which means the four beam spots combine to one beam spot again. Furthermore, the ratio of the spectral intensities of an EHGCSM beam with large mode orders m and n recovers slower than that of an EHGCSM beam with small mode orders $m$ and $n$ both in Kolmogorov and non-Kolmogorov turbulence, which means the EHGCSM beam with large mode orders m and n is less affected by turbulence.

Fig. 4 Ratio of the spectral intensity in the optical axis ( $ρ = 0$ ) to the maximum intensity in the transverse plane of an EHGCSM beam versus the propagation distance z in Kolmogorov ( $α = 11 / 3$ ) or non-Kolmogorov turbulence ( $α = 3.1$ ) for different values of the mode orders $m$ and $n$ .

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Figure 5 shows the density plot of the modulus of the spectral degree of coherence and the corresponding cross line of an EHGCSM beam at several propagation distances in free space ( ${\tilde{C}}_{n}^{2} = 0$ ) for different values of the mode orders m and n. Figures 6 and 7 show the density plot of the modulus of the spectral degree of coherence and the corresponding cross line of an EHGCSM beam at several propagation distances in Kolmogorov ( $α = 11 / 3$ ) and non-Kolmogorov turbulence ( $α = 3.1$ ) for different values of the mode orders m and n. One finds that the spectral degree of coherence of the EHGCSM beam exhibits array distribution in the source plane, and the array distribution gradually disappears on propagation in free space and finally evolves into diamond distribution (see Fig. 5). In Kolmogorov or non-Kolmogorov turbulence, the evolution properties of the spectral degree of coherence are similar to those in free space at short propagation distance, i.e., the array distribution evolves into diamond distribution. Due to the influence of the turbulence, the diamond distribution evolves into array distribution again at intermediate propagation distance, and the spectral degree of coherence finally becomes of Gaussian distribution in the far field (see Figs. 6 and 7). Furthermore, the evolution properties of the spectral degree of coherence are closely related to the mode orders m and n. We find from Figs. 6 and 7 that the conversion from the array distribution to Gaussian distribution becomes slower as the beam orders $m$ and $n$ increases, which means that an EHGCSM beam with large $m$ and $n$ is less affected by turbulence both in Kolmogorov and non-Kolmogorov turbulence from the aspect of the spectral degree of coherence.

Fig. 5 Density plot of the modulus of the spectral degree of coherence and the corresponding cross line of an EHGCSM beam at several propagation distances in free space ( ${\tilde{C}}_{n}^{2} = 0$ ) for different values of the mode orders m and n.

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Fig. 6 Density plot of the modulus of the spectral degree of coherence and the corresponding cross line of an EHGCSM beam at several propagation distances in Kolmogorov turbulence ( $α = 11 / 3$ ) for different values of the mode orders m and n.

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Fig. 7 Density plot of the modulus of the spectral degree of coherence and the corresponding cross line of an EHGCSM beam at several propagation distances in non-Kolmogorov turbulence ( $α = 3.1$ ) for different values of the mode orders m and n.

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To learn about the evolution properties of the propagation factors of an EHGCSM beam in turbulence, we calculate in Fig. 8 the normalized propagation factors of an EHGCSM beam versus the propagation distance z in Kolmogorov ( $α = 11 / 3$ ) or non-Kolmogorov turbulence ( $α = 3.1$ ) for different values of the mode orders $m$ and $n$ with $λ = 632.8 n m$ , $σ_{0} = 10 m m$ , $δ_{0 x} = δ_{0 y} = 4 m m$ , $L_{0} = 1 m$ , $l_{0} = 1 m m$ , ${\tilde{C}}_{n}^{2} = 5 \times 10^{- 15} m^{3 - α}$ . For the convenience of comparison, the corresponding result (dark line) of a Gaussian beam (m = n = 0 and $δ_{0 x} = δ_{0 y} = \infty$ ) is also shown in Fig. 8. The propagation factor is a parameter denoting the quality of a beam, and it is invariant on propagation in free space. One finds from Fig. 8 that the propagation factors change on propagation in Kolmogorov or non-Kolmogorov turbulence, which means the quality of the beam is degraded by the turbulence and the beam widths diverge more rapidly in turbulent atmosphere than in free space. Furthermore, we note that the normalized propagation factors of an EHGCSM beam with large $m$ and $n$ increase slower than an EHGCSM beam with small $m$ and $n$ or a Gaussian Schell-model beam ( $m = n = 0$ ) or a Gaussian beam on propagation, which means that the EHGCSM beam with large $m$ and $n$ is less affected by turbulence from the aspect of the propagation factor. Kolmogorov model and non-Kolmogorov model represent homogeneous and non-homogenous turbulence in three dimensions, respectively. Our results clearly show that EHGCSM has advantage in both Kolmogorov and non-Kolmogorov turbulence, which will be useful in free-space optical communications.

Fig. 8 Normalized propagation factors of an EHGCSM beam versus the propagation distance z in Kolmogorov ( $α = 11 / 3$ ) or non-Kolmogorov turbulence ( $α = 3.1$ ) for different values of the mode orders $m$ and $n$ . The dark line denotes the corresponding result of a Gaussian beam.

Download Full Size | PPT Slide | PDF

5. Summary

We have studied the paraxial propagation of an EHGCSM beam in Kolmogorov turbulence or non-Kolmogorov turbulence. Analytical expressions for the CSD and the propagation factors of an EHGCSM beam in turbulence have been derived and the evolution properties of the spectral intensity, the spectral degree of coherence and the propagation factors of such beam have illustrated numerically. We have found that an EHGCSM beam exhibits splitting and combing properties in Kolmogorov and non-Kolmogorov turbulence, and an EHGCSM beam with large mode orders is less affected by turbulence than an EHGCSM beam with small mode orders or a Gaussian Schell-model beam or a Gaussian beam. Our results show that modulating the correlation function of a partially coherent beam will be useful in free-space optical communications.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grants Nos. 11474213 and 11404234; the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions; the Innovation Plan for Graduate Students in the Universities of Jiangsu Province under Grant Nos. KYLX-1218 and KYZZ_0334; the Key Lab Foundation of The Modern Optical Technology of Jiangsu Province, Soochow University, under Grant No. KJS1301.

References and links

1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

2. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

3. Y. Cai and F. Wang, “Tensor method for treating the propagation of scalar and electromagnetic Gaussian Schell-model beams: a review,” Open Opt. J. 4(1), 1–20 (2010). [CrossRef]

4. Y. Cai, F. Wang, C. Zhao, S. Zhu, G. Wu, and Y. Dong, “Partially coherent vector beams: from theory to experiment,” in Vectorial Optical Fields: Fundamentals and Applications, Q. Zhen, ed. (World Scientific, 2013), Chap. 7, pp. 221–273.

5. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007). [CrossRef] [PubMed]

6. F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009). [CrossRef]

7. R. Martínez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34(9), 1399–1401 (2009). [CrossRef] [PubMed]

8. M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014). [CrossRef] [PubMed]

9. F. Gori and M. Santarsiero, “Difference of two Gaussian Schell-model cross-spectral densities,” Opt. Lett. 39(9), 2731–2734 (2014). [CrossRef] [PubMed]

10. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011). [CrossRef] [PubMed]

11. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012). [CrossRef] [PubMed]

12. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012). [CrossRef] [PubMed]

13. Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013). [CrossRef] [PubMed]

14. Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014). [CrossRef] [PubMed]

15. Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014). [CrossRef]

16. Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015). [CrossRef]

17. F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013). [CrossRef] [PubMed]

18. Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014). [CrossRef] [PubMed]

19. Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014). [CrossRef] [PubMed]

20. F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014). [CrossRef] [PubMed]

21. Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review [Invited],” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014). [CrossRef] [PubMed]

22. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Washington, 2001).

23. Y. Cai, “Generation of various partially coherent beams and their propagation properties in turbulent atmosphere: a review,” Proc. SPIE 7924, 792402 (2011). [CrossRef]

24. G. Gbur, “Partially coherent beam propagation in atmospheric turbulence [Invited],” J. Opt. Soc. Am. A 31(9), 2038–2045 (2014). [PubMed]

25. F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited review),” Prog. Electromagnetics Res. 150, 123–143 (2015). [CrossRef]

26. X. Liu, F. Wang, C. Wei, and Y. Cai, “Experimental study of turbulence-induced beam wander and deformation of a partially coherent beam,” Opt. Lett. 39(11), 3336–3339 (2014). [CrossRef] [PubMed]

27. R. J. Noriega-Manez and J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007). [CrossRef] [PubMed]

28. 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] [PubMed]

29. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008). [CrossRef]

30. W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009). [CrossRef] [PubMed]

31. P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35(7), 1043–1045 (2010). [CrossRef] [PubMed]

32. E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express 18(10), 10650–10658 (2010). [CrossRef] [PubMed]

33. 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] [PubMed]

34. F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013). [CrossRef]

35. L. Lu, X. Ji, and Y. Baykal, “Wave structure function and spatial coherence radius of plane and spherical waves propagating through oceanic turbulence,” Opt. Express 22(22), 27112–27122 (2014). [CrossRef] [PubMed]

36. Z. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012). [CrossRef] [PubMed]

37. Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013). [CrossRef] [PubMed]

38. Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013). [CrossRef]

39. Z. Mei, E. Shchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013). [CrossRef] [PubMed]

40. S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013). [CrossRef]

41. O. Korotkova, S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, Y. Gu, and G. Gbur, “Scintillation reduction in multi-Gaussian Schell-model beams propagating in atmospheric turbulence,” Proc. SPIE 9224, 92240M (2014). [CrossRef]

42. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2C14 (1990).

43. F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82(3-4), 197–203 (1991). [CrossRef]

44. M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16(1), 106–112 (1999). [CrossRef]

45. R. Martínez -Herrero, P. M. Mejías, and H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993). [CrossRef]

References

View by:
Article Order
|
Year
|
Author
|
Publication

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
Y. Cai and F. Wang, “Tensor method for treating the propagation of scalar and electromagnetic Gaussian Schell-model beams: a review,” Open Opt. J. 4(1), 1–20 (2010).
[Crossref]
Y. Cai, F. Wang, C. Zhao, S. Zhu, G. Wu, and Y. Dong, “Partially coherent vector beams: from theory to experiment,” in Vectorial Optical Fields: Fundamentals and Applications, Q. Zhen, ed. (World Scientific, 2013), Chap. 7, pp. 221–273.
F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]
F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]
R. Martínez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34(9), 1399–1401 (2009).
[Crossref] [PubMed]
M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014).
[Crossref] [PubMed]
F. Gori and M. Santarsiero, “Difference of two Gaussian Schell-model cross-spectral densities,” Opt. Lett. 39(9), 2731–2734 (2014).
[Crossref] [PubMed]
H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[Crossref] [PubMed]
S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[Crossref] [PubMed]
O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
[Crossref] [PubMed]
Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[Crossref] [PubMed]
Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref] [PubMed]
Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[Crossref]
Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]
F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]
Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
[Crossref] [PubMed]
Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]
F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
[Crossref] [PubMed]
Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review [Invited],” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
[Crossref] [PubMed]
L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Washington, 2001).
Y. Cai, “Generation of various partially coherent beams and their propagation properties in turbulent atmosphere: a review,” Proc. SPIE 7924, 792402 (2011).
[Crossref]
G. Gbur, “Partially coherent beam propagation in atmospheric turbulence [Invited],” J. Opt. Soc. Am. A 31(9), 2038–2045 (2014).
[PubMed]
F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited review),” Prog. Electromagnetics Res. 150, 123–143 (2015).
[Crossref]
X. Liu, F. Wang, C. Wei, and Y. Cai, “Experimental study of turbulence-induced beam wander and deformation of a partially coherent beam,” Opt. Lett. 39(11), 3336–3339 (2014).
[Crossref] [PubMed]
R. J. Noriega-Manez and J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007).
[Crossref] [PubMed]
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] [PubMed]
O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008).
[Crossref]
W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009).
[Crossref] [PubMed]
P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35(7), 1043–1045 (2010).
[Crossref] [PubMed]
E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express 18(10), 10650–10658 (2010).
[Crossref] [PubMed]
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] [PubMed]
F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).
[Crossref]
L. Lu, X. Ji, and Y. Baykal, “Wave structure function and spatial coherence radius of plane and spherical waves propagating through oceanic turbulence,” Opt. Express 22(22), 27112–27122 (2014).
[Crossref] [PubMed]
Z. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
[Crossref] [PubMed]
Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[Crossref] [PubMed]
Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]
Z. Mei, E. Shchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013).
[Crossref] [PubMed]
S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[Crossref]
O. Korotkova, S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, Y. Gu, and G. Gbur, “Scintillation reduction in multi-Gaussian Schell-model beams propagating in atmospheric turbulence,” Proc. SPIE 9224, 92240M (2014).
[Crossref]
A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2C14 (1990).
F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82(3-4), 197–203 (1991).
[Crossref]
M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16(1), 106–112 (1999).
[Crossref]
R. Martínez -Herrero, P. M. Mejías, and H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993).
[Crossref]

2015 (2)

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited review),” Prog. Electromagnetics Res. 150, 123–143 (2015).
[Crossref]

2014 (12)

X. Liu, F. Wang, C. Wei, and Y. Cai, “Experimental study of turbulence-induced beam wander and deformation of a partially coherent beam,” Opt. Lett. 39(11), 3336–3339 (2014).
[Crossref] [PubMed]

G. Gbur, “Partially coherent beam propagation in atmospheric turbulence [Invited],” J. Opt. Soc. Am. A 31(9), 2038–2045 (2014).
[PubMed]

L. Lu, X. Ji, and Y. Baykal, “Wave structure function and spatial coherence radius of plane and spherical waves propagating through oceanic turbulence,” Opt. Express 22(22), 27112–27122 (2014).
[Crossref] [PubMed]

Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref] [PubMed]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[Crossref]

Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
[Crossref] [PubMed]

Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
[Crossref] [PubMed]

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review [Invited],” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
[Crossref] [PubMed]

M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014).
[Crossref] [PubMed]

F. Gori and M. Santarsiero, “Difference of two Gaussian Schell-model cross-spectral densities,” Opt. Lett. 39(9), 2731–2734 (2014).
[Crossref] [PubMed]

O. Korotkova, S. Avramov-Zamurovic, C. Nelson, R. Malek-Madani, Y. Gu, and G. Gbur, “Scintillation reduction in multi-Gaussian Schell-model beams propagating in atmospheric turbulence,” Proc. SPIE 9224, 92240M (2014).
[Crossref]

2013 (7)

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).
[Crossref]

Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[Crossref] [PubMed]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[Crossref]

Z. Mei, E. Shchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013).
[Crossref] [PubMed]

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[Crossref]

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[Crossref] [PubMed]

2012 (3)

Z. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
[Crossref] [PubMed]

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[Crossref] [PubMed]

O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
[Crossref] [PubMed]

2011 (2)

H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[Crossref] [PubMed]

Y. Cai, “Generation of various partially coherent beams and their propagation properties in turbulent atmosphere: a review,” Proc. SPIE 7924, 792402 (2011).
[Crossref]

2010 (4)

Y. Cai and F. Wang, “Tensor method for treating the propagation of scalar and electromagnetic Gaussian Schell-model beams: a review,” Open Opt. J. 4(1), 1–20 (2010).
[Crossref]

P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35(7), 1043–1045 (2010).
[Crossref] [PubMed]

E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express 18(10), 10650–10658 (2010).
[Crossref] [PubMed]

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] [PubMed]

2009 (3)

W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009).
[Crossref] [PubMed]

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

R. Martínez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34(9), 1399–1401 (2009).
[Crossref] [PubMed]

2008 (2)

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] [PubMed]

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008).
[Crossref]

2007 (2)

R. J. Noriega-Manez and J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007).
[Crossref] [PubMed]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]

1999 (1)

M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16(1), 106–112 (1999).
[Crossref]

1993 (1)

R. Martínez -Herrero, P. M. Mejías, and H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993).
[Crossref]

1991 (1)

F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82(3-4), 197–203 (1991).
[Crossref]

Avramov-Zamurovic, S.

Baykal, Y.

Borghi, R.

Cai, Y.

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited review),” Prog. Electromagnetics Res. 150, 123–143 (2015).
[Crossref]

F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
[Crossref] [PubMed]

Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
[Crossref] [PubMed]

Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref] [PubMed]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[Crossref]

Y. Cai, “Generation of various partially coherent beams and their propagation properties in turbulent atmosphere: a review,” Proc. SPIE 7924, 792402 (2011).
[Crossref]

Y. Cai and F. Wang, “Tensor method for treating the propagation of scalar and electromagnetic Gaussian Schell-model beams: a review,” Open Opt. J. 4(1), 1–20 (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] [PubMed]

Chen, Y.

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
[Crossref] [PubMed]

Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref] [PubMed]

Cheng, W.

W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009).
[Crossref] [PubMed]

Cincotti, G.

Dan, Y.

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] [PubMed]

de Sande, J. C. G.

M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014).
[Crossref] [PubMed]

Du, S.

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[Crossref]

Eyyuboglu, H. T.

Gbur, G.

G. Gbur, “Partially coherent beam propagation in atmospheric turbulence [Invited],” J. Opt. Soc. Am. A 31(9), 2038–2045 (2014).
[PubMed]

Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[Crossref] [PubMed]

Gori, F.

M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014).
[Crossref] [PubMed]

F. Gori and M. Santarsiero, “Difference of two Gaussian Schell-model cross-spectral densities,” Opt. Lett. 39(9), 2731–2734 (2014).
[Crossref] [PubMed]

R. Martínez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34(9), 1399–1401 (2009).
[Crossref] [PubMed]

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]

F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82(3-4), 197–203 (1991).
[Crossref]

Gu, J.

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

Ji, X.

Korotkova, O.

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[Crossref] [PubMed]

Z. Mei, E. Shchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013).
[Crossref] [PubMed]

Z. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
[Crossref] [PubMed]

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[Crossref] [PubMed]

O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
[Crossref] [PubMed]

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008).
[Crossref]

Lajunen, H.

H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[Crossref] [PubMed]

Liang, C.

F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
[Crossref] [PubMed]

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[Crossref]

Liu, L.

Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref] [PubMed]

Liu, X.

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited review),” Prog. Electromagnetics Res. 150, 123–143 (2015).
[Crossref]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

Liu, Z.

P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35(7), 1043–1045 (2010).
[Crossref] [PubMed]

Lu, L.

Ma, Y.

P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35(7), 1043–1045 (2010).
[Crossref] [PubMed]

Malek-Madani, R.

Martínez -Herrero, R.

R. Martínez -Herrero, P. M. Mejías, and H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993).
[Crossref]

Martínez-Herrero, R.

R. Martínez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34(9), 1399–1401 (2009).
[Crossref] [PubMed]

Mei, Z.

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[Crossref] [PubMed]

Z. Mei, E. Shchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013).
[Crossref] [PubMed]

Mejías, P. M.

R. Martínez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34(9), 1399–1401 (2009).
[Crossref] [PubMed]

R. Martínez -Herrero, P. M. Mejías, and H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993).
[Crossref]

Nelson, C.

Noriega-Manez, R. J.

R. J. Noriega-Manez and J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007).
[Crossref] [PubMed]

Piquero, G.

M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014).
[Crossref] [PubMed]

Qu, J.

Ramírez-Sánchez, V.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Saastamoinen, T.

H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[Crossref] [PubMed]

Sahin, S.

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[Crossref] [PubMed]

O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
[Crossref] [PubMed]

Santarsiero, M.

M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014).
[Crossref] [PubMed]

F. Gori and M. Santarsiero, “Difference of two Gaussian Schell-model cross-spectral densities,” Opt. Lett. 39(9), 2731–2734 (2014).
[Crossref] [PubMed]

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]

F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82(3-4), 197–203 (1991).
[Crossref]

Shchepakina, E.

Z. Mei, E. Shchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013).
[Crossref] [PubMed]

O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
[Crossref] [PubMed]

Shirai, T.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Sona, A.

F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82(3-4), 197–203 (1991).
[Crossref]

Tong, Z.

Z. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
[Crossref] [PubMed]

Vahimaa, P.

Wang, F.

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited review),” Prog. Electromagnetics Res. 150, 123–143 (2015).
[Crossref]

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
[Crossref] [PubMed]

Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref] [PubMed]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

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] [PubMed]

Y. Cai and F. Wang, “Tensor method for treating the propagation of scalar and electromagnetic Gaussian Schell-model beams: a review,” Open Opt. J. 4(1), 1–20 (2010).
[Crossref]

Wang, X.

P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35(7), 1043–1045 (2010).
[Crossref] [PubMed]

Weber, H.

R. Martínez -Herrero, P. M. Mejías, and H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993).
[Crossref]

Wei, C.

Yuan, Y.

F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
[Crossref] [PubMed]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[Crossref]

Zhan, Q.

W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009).
[Crossref] [PubMed]

Zhang, B.

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] [PubMed]

Zhao, C.

Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref] [PubMed]

Zhao, H.

P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35(7), 1043–1045 (2010).
[Crossref] [PubMed]

Zhou, P.

P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35(7), 1043–1045 (2010).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

J. Opt. A, Pure Appl. Opt. (1)

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

J. Opt. Soc. Am. A (4)

O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
[Crossref] [PubMed]

G. Gbur, “Partially coherent beam propagation in atmospheric turbulence [Invited],” J. Opt. Soc. Am. A 31(9), 2038–2045 (2014).
[PubMed]

Open Opt. J. (1)

Y. Cai and F. Wang, “Tensor method for treating the propagation of scalar and electromagnetic Gaussian Schell-model beams: a review,” Open Opt. J. 4(1), 1–20 (2010).
[Crossref]

Opt. Commun. (3)

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008).
[Crossref]

F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82(3-4), 197–203 (1991).
[Crossref]

Opt. Express (10)

Z. Mei, E. Shchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013).
[Crossref] [PubMed]

W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009).
[Crossref] [PubMed]

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] [PubMed]

R. J. Noriega-Manez and J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007).
[Crossref] [PubMed]

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] [PubMed]

Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref] [PubMed]

Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
[Crossref] [PubMed]

Opt. Laser Technol. (1)

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[Crossref]

Opt. Lett. (13)

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
[Crossref] [PubMed]

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[Crossref] [PubMed]

R. Martínez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34(9), 1399–1401 (2009).
[Crossref] [PubMed]

M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014).
[Crossref] [PubMed]

F. Gori and M. Santarsiero, “Difference of two Gaussian Schell-model cross-spectral densities,” Opt. Lett. 39(9), 2731–2734 (2014).
[Crossref] [PubMed]

H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[Crossref] [PubMed]

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[Crossref] [PubMed]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]

Z. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
[Crossref] [PubMed]

Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[Crossref] [PubMed]

P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35(7), 1043–1045 (2010).
[Crossref] [PubMed]

Opt. Quantum Electron. (1)

R. Martínez -Herrero, P. M. Mejías, and H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993).
[Crossref]

Phys. Rev. A (2)

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

Proc. SPIE (2)

Y. Cai, “Generation of various partially coherent beams and their propagation properties in turbulent atmosphere: a review,” Proc. SPIE 7924, 792402 (2011).
[Crossref]

Prog. Electromagnetics Res. (1)

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited review),” Prog. Electromagnetics Res. 150, 123–143 (2015).
[Crossref]

Other (5)

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Washington, 2001).

Y. Cai, F. Wang, C. Zhao, S. Zhu, G. Wu, and Y. Dong, “Partially coherent vector beams: from theory to experiment,” in Vectorial Optical Fields: Fundamentals and Applications, Q. Zhen, ed. (World Scientific, 2013), Chap. 7, pp. 221–273.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2C14 (1990).

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Fig. 1 3D-normalized spectral intensity distribution

S (ρ, z) / S {(ρ, z)}_{\max}

of an EHGCSM beam at several propagation distances in Kolmogorov turbulence with m = n = 5 and

α = 11 / 3

View in Article | Download Full Size | PPT Slide | PDF

Fig. 2 3D-normalized spectral intensity distribution

S (ρ, z) / S {(ρ, z)}_{\max}

of an EHGCSM beam at several propagation distances in non-Kolmogorov turbulence with m = n = 5 and

α = 3.1

View in Article | Download Full Size | PPT Slide | PDF

Fig. 3 3D-normalized spectral intensity distribution

S (ρ, z) / S {(ρ, z)}_{\max}

of an EHGCSM beam at several propagation distances in free space with m = n = 5 and

α = 11 / 3

View in Article | Download Full Size | PPT Slide | PDF

Fig. 4 Ratio of the spectral intensity in the optical axis (

ρ = 0

) to the maximum intensity in the transverse plane of an EHGCSM beam versus the propagation distance z in Kolmogorov (

α = 11 / 3

) or non-Kolmogorov turbulence (

α = 3.1

) for different values of the mode orders

m

and

n

View in Article | Download Full Size | PPT Slide | PDF

Fig. 5 Density plot of the modulus of the spectral degree of coherence and the corresponding cross line of an EHGCSM beam at several propagation distances in free space (

{\tilde{C}}_{n}^{2} = 0

) for different values of the mode orders m and n.

View in Article | Download Full Size | PPT Slide | PDF

Fig. 6 Density plot of the modulus of the spectral degree of coherence and the corresponding cross line of an EHGCSM beam at several propagation distances in Kolmogorov turbulence (

α = 11 / 3

) for different values of the mode orders m and n.

View in Article | Download Full Size | PPT Slide | PDF

Fig. 7 Density plot of the modulus of the spectral degree of coherence and the corresponding cross line of an EHGCSM beam at several propagation distances in non-Kolmogorov turbulence (

α = 3.1

) for different values of the mode orders m and n.

View in Article | Download Full Size | PPT Slide | PDF

Fig. 8 Normalized propagation factors of an EHGCSM beam versus the propagation distance z in Kolmogorov (

α = 11 / 3

) or non-Kolmogorov turbulence (

α = 3.1

) for different values of the mode orders

m

and

n

. The dark line denotes the corresponding result of a Gaussian beam.

View in Article | Download Full Size | PPT Slide | PDF

Equations (40)

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W^{(0)} (r_{1}, r_{2}, 0) = G_{0} \exp (- \frac{r_{1}^{2} + r_{2}^{2}}{4 σ_{0}^{2}}) μ (r_{2} - r_{1}),

μ (r_{2} - r_{1}) = \frac{H_{2 m} [(x_{2} - x_{1}) / \sqrt{2} δ_{0 x}]}{H_{2 m} (0)} \frac{H_{2 n} [(y_{2} - y_{1}) / \sqrt{2} δ_{0 y}]}{H_{2 n} (0)} \exp [- \frac{{(x_{2} - x_{1})}^{2}}{2 δ_{0 x}^{2}} - \frac{{(y_{2} - y_{1})}^{2}}{2 δ_{0 y}^{2}}],

\begin{array}{l} W (ρ_{1}, ρ_{2}, z) = \frac{1}{λ^{2} z^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} W^{(0)} (r_{1}, r_{2}, 0) \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}

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

T = \int_{0}^{\infty} κ^{3} Φ_{n} (κ) d κ .

T = \frac{A (α)}{2 (α - 2)} {\tilde{C}}_{n}^{2} [β κ_{m}^{2 - α} \exp (κ_{0}^{2} / κ_{m}^{2}) Γ_{1} (2 - α / 2, κ_{0}^{2} / κ_{m}^{2}) - 2 κ_{0}^{4 - α}], 3 < α < 4,

A (α) = \frac{1}{4 π^{2}} Γ (α - 1) \cos (\frac{α π}{2}), c (α) = {[\frac{2 π A (α)}{3} Γ (5 - \frac{α}{2})]}^{1 / (α - 5)} .

\frac{i k}{z} ρ_{1}^{'} = \frac{i k}{z} ρ_{1} - \frac{π^{2} k^{2} z T}{3} (ρ_{1} - ρ_{2}), \frac{i k}{z} ρ_{2}^{'} = \frac{i k}{z} ρ_{2} - \frac{π^{2} k^{2} z T}{3} (ρ_{1} - ρ_{2}),

A (ρ_{1}, ρ_{2}) = \exp [- \frac{i k}{2 z} (ρ_{1}^{2} - ρ_{2}^{2}) - \frac{π^{2} k^{2} z T}{3} {(ρ_{1} - ρ_{2})}^{2}],

\begin{array}{l} W ({ρ^{'}}_{1}, {ρ^{'}}_{2}, z) = \frac{G_{0}}{λ^{2} z^{2}} A (ρ_{1}, ρ_{2}) \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \frac{H_{2 m} [(x_{2} - x_{1}) / \sqrt{2} δ_{0 x}]}{H_{2 m} (0)} \frac{H_{2 n} [(y_{2} - y_{1}) / \sqrt{2} δ_{0 y}]}{H_{2 n} (0)} \\ \times \exp [- \frac{{(x_{2} - x_{1})}^{2}}{2 δ_{0 x}^{2}} - \frac{{(y_{2} - y_{1})}^{2}}{2 δ_{0 y}^{2}}] \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}) - \frac{π^{2} k^{2} z T}{3} {(r_{1} - r_{2})}^{2}] d^{2} r_{1} d^{2} r_{2} . \end{array}

\begin{array}{l} r_{s} = \frac{r_{1} + r_{2}}{2}, x_{s} = \frac{x_{1} + x_{2}}{2}, y_{s} = \frac{y_{1} + y_{2}}{2}, \\ r_{d} = r_{1} - r_{2}, x_{d} = x_{1} - x_{2}, y_{d} = y_{1} - y_{2}, \\ {ρ^{'}}_{s} = \frac{{ρ^{'}}_{1} + {ρ^{'}}_{2}}{2}, {ρ^{'}}_{s x} = \frac{{ρ^{'}}_{1 x} + {ρ^{'}}_{2 x}}{2}, {ρ^{'}}_{s y} = \frac{{ρ^{'}}_{1 y} + {ρ^{'}}_{2 y}}{2}, \\ {ρ^{'}}_{d} = {ρ^{'}}_{1} - {ρ^{'}}_{2}, {ρ^{'}}_{d x} = {ρ^{'}}_{1 x} - {ρ^{'}}_{2 x}, {ρ^{'}}_{d y} = {ρ^{'}}_{1 y} - {ρ^{'}}_{2 y}, \end{array}

\begin{array}{l} W (ρ_{1}, ρ_{2}, z) = \frac{4 G_{0} σ_{0}^{2} π^{2} δ_{0 x} δ_{0 y}}{H_{2 m} (0) H_{2 n} (0) \sqrt{a b} λ^{2} z^{2}} {(1 - \frac{1}{a})}^{m} {(1 - \frac{1}{b})}^{n} H_{2 m} [- \frac{c}{2 a} {(1 - \frac{1}{a})}^{- 1 / 2}] \\ \times H_{2 n} [- \frac{d}{2 b} {(1 - \frac{1}{b})}^{- 1 / 2}] \exp [\frac{c^{2}}{4 a} + \frac{d^{2}}{4 b} - \frac{i k}{z} ({ρ^{'}}_{s x} {ρ^{'}}_{d x} + {ρ^{'}}_{s y} {ρ^{'}}_{d y}) - (\frac{2 π^{2} k^{2} z T}{3} + \frac{σ_{0}^{2} k^{2}}{2 z^{2}}) ({ρ^{'}}_{d x}^{2} + {ρ^{'}}_{d y}^{2})], \end{array}

\begin{array}{l} a = 2 δ_{0 x}^{2} (\frac{1}{8 σ_{0}^{2}} + \frac{π^{2} k^{2} z T}{3} + \frac{1}{2 δ_{0 x}^{2}} + \frac{k^{2} σ_{0}^{2}}{2 z^{2}}), b = 2 δ_{0 y}^{2} (\frac{1}{8 σ_{0}^{2}} + \frac{π^{2} k^{2} z T}{3} + \frac{1}{2 δ_{0 y}^{2}} + \frac{k^{2} σ_{0}^{2}}{2 z^{2}}), \\ c = \sqrt{2} δ_{0 x} (\frac{i k {ρ^{'}}_{s x}}{z} + \frac{k^{2} σ_{0}^{2} {ρ^{'}}_{d x}}{z^{2}}), d = \sqrt{2} δ_{0 y} (\frac{i k {ρ^{'}}_{s y}}{z} + \frac{k^{2} σ_{0}^{2} {ρ^{'}}_{d y}}{z^{2}}) . \end{array}

δ (s) = \frac{1}{2 π} \int_{- \infty}^{\infty} \exp (- i s x) d x,

\int_{- \infty}^{\infty} f (x) δ^{n} (x) d x = {(- 1)}^{n} f^{(n)} (0), (n = 0, 1, 2),

\int_{- \infty}^{\infty} \exp (- s^{2} x^{2} \pm q x) d x = \frac{\sqrt{π}}{s} \exp (\frac{q^{2}}{4 s^{2}}),

\int_{- \infty}^{+ \infty} \exp [- \frac{{(x - y)}^{2}}{2 m}] H_{n} (x) d x = \sqrt{2 π m} {(1 - 2 m)}^{n / 2} H_{n} [y {(1 - 2 m)}^{- 1 / 2}] .

S (ρ, z) = W (ρ_{1}, ρ_{2}, z) .

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

ρ_{s} = \frac{ρ_{1} + ρ_{2}}{2}, ρ_{d} = ρ_{1} - ρ_{2},

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

W^{(0)} (r_{s}, r_{d}, 0) = W^{(0)} (r_{1}, r_{2}, 0) = W^{(0)} (r_{s} + \frac{r_{d}}{2}, r_{s} - \frac{r_{d}}{2}, 0),

H (ρ_{d}, r_{d}, z) = 4 π^{2} k^{2} z \int_{0}^{1} d ξ \int_{0}^{\infty} [1 - J_{0} (κ | r_{d} ξ + (1 - ξ) ρ_{d} |)] Φ_{n} (κ) κ d κ,

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

H (ρ_{d}, ρ_{d} + \frac{z}{k} κ_{d}, z) = \frac{π^{2} k^{2} z}{3} (3 ρ_{d}^{2} + 3 \frac{z}{k} ρ_{d} \cdot κ + \frac{z^{2}}{k^{2}} κ_{d}^{2}) T .

\begin{array}{l} W ({r^{'}}_{s}, ρ_{d} + \frac{z}{k} κ_{d}, 0) = \frac{G_{0}}{H_{2 m} (0) H_{2 n} (0)} \exp [- \frac{{r^{'}}_{s}^{2}}{2 σ_{0}^{2}} - \frac{1}{8 σ_{0}^{2}} {(ρ_{d} + \frac{z κ_{d}}{k})}^{2}] \\ \times H_{2 m} [- \frac{1}{\sqrt{2} δ_{0 x}} (ρ_{d x} + \frac{z}{k} κ_{d x})] H_{2 n} [- \frac{1}{\sqrt{2} δ_{0 y}} (ρ_{d y} + \frac{z}{k} κ_{d y})] \\ \times \exp [- \frac{1}{2 δ_{0 x}^{2}} {(ρ_{d x} + \frac{z}{k} κ_{d x})}^{2}] \exp [- \frac{1}{2 δ_{0 y}^{2}} {(ρ_{d y} + \frac{z}{k} κ_{d y})}^{2}] . \end{array}

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

h (ρ_{s}, θ, z) = h (ρ_{s x}, θ_{x}, z) h (ρ_{s y}, θ_{y}, z),

\begin{array}{l} h (ρ_{s j}, θ_{j}, z) = \frac{\sqrt{2 π G_{0} σ_{0}^{2}} k}{4 π^{2} H_{2 l} (0)} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} H_{2 l} [- \frac{1}{\sqrt{2} δ_{0 x}} (ρ_{d x} + \frac{z}{k} κ_{d x})] \\ \times \exp (- a_{x} ρ_{d x}^{2} - b_{x} κ_{d x}^{2} - c_{x} ρ_{d x} κ_{d x} - i ρ_{s x} κ_{d x} - i k θ_{x} ρ_{d x}) d κ_{d x} d ρ_{d x}, (j = x, y l = m, n), \end{array}

\begin{array}{l} a_{j} = \frac{1}{2 δ_{0 j}^{2}} + \frac{1}{8 σ_{0}^{2}} + π^{2} k^{2} z T, b_{j} = \frac{z^{2}}{2 k^{2} δ_{0 j}^{2}} + \frac{z^{2}}{8 k^{2} σ_{0}^{2}} + \frac{σ_{0}^{2}}{2} + \frac{π^{2} z^{3} T}{3}, \\ c_{j} = \frac{z}{k δ_{0 j}^{2}} + \frac{z}{4 k σ_{0}^{2}} + π^{2} z^{2} k T, (j = x, y) . \end{array}

〈 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_{s}, θ, z) d^{2} ρ_{s} d^{2} θ,

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

〈 ρ_{j}^{2} 〉 = \frac{\sqrt{2 π G_{0} σ_{0}^{2}}}{P_{j} H_{2 l} (0)} [\frac{2 z^{2} {(- 1)}^{l - 1} (2 l)!}{δ_{0 j}^{2} k^{2} (l - 1)!} - 2 b_{j} \frac{{(- 1)}^{l} (2 l)!}{l!}],

〈 θ_{j}^{2} 〉 = \frac{\sqrt{2 π G_{0} σ_{0}^{2}}}{P_{j} H_{2 l} (0)} [\frac{2 {(- 1)}^{l - 1} (2 l)!}{δ_{0 j}^{2} k^{2} (l - 1)!} - 2 \frac{a_{j}}{k^{2}} \frac{{(- 1)}^{l} (2 l)!}{l!}],

〈 ρ_{j} θ_{j} 〉 = \frac{\sqrt{2 π G_{0} σ_{0}^{2}}}{P_{j} H_{2 l} (0)} [\frac{2 z {(- 1)}^{l - 1} (2 l)!}{δ_{0 j}^{2} k^{2} (l - 1)!} - \frac{c_{j}}{k} \frac{{(- 1)}^{l} (2 l)!}{l!}],

P_{j} = \frac{{(- 1)}^{l} (2 l)! \sqrt{2 π G_{0} σ_{0}^{2}}}{l! H_{2 l} (0)}, (j = x, y, l = m, n) .

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

M_{x}^{2} (z) = 2 k {(〈 ρ_{x}^{2} 〉 〈 θ_{x}^{2} 〉 - {〈 ρ_{x} θ_{x} 〉}^{2})}^{1 / 2}, M_{y}^{2} (z) = 2 k {(〈 ρ_{y}^{2} 〉 〈 θ_{y}^{2} 〉 - {〈 ρ_{y} θ_{y} 〉}^{2})}^{1 / 2},

\begin{array}{l} M_{j}^{2} (z) = 2 {[\frac{z^{2}}{δ_{0 j}^{2} k^{2}} (2 l + 1) + \frac{z^{2}}{4 k^{2} σ_{0}^{2}} + σ_{0}^{2} + \frac{2 π^{2} z^{3} T}{3}] [\frac{1}{δ_{0 j}^{2}} (2 l + 1) + \frac{1}{4 σ_{0}^{2}} + 2 π^{2} k^{2} z T] \\ {- {[\frac{z}{k δ_{0 j}^{2}} (2 l + 1) + \frac{z}{4 k σ_{0}^{2}} + π^{2} z^{2} k T]}^{2}}}^{1 / 2}, (j = x, y, l = m, n) . \end{array}

M_{j}^{2} (z) = {[4 \frac{σ_{0}^{2}}{δ_{0 j}^{2}} (2 l + 1) + 1]}^{1 / 2} = {\frac{4 σ_{0}^{2}}{δ_{0 j}^{2}} [1 - \frac{4 l (2 l - 1) H_{2 l - 2} (0)}{H_{2 l} (0)}] + 1}^{1 / 2}, (j = x, y, l = m, n) .

Abstract

1. Introduction

2. Cross-spectral density of an EHGCSM beam propagating in turbulent atmosphere

3. Propagation factors of an EHGCSM beam propagating in turbulent atmosphere

4. Statistical properties of an EHGCSM beam propagating in turbulent atmosphere

5. Summary

Acknowledgments

References and links

References

2015 (2)

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2011 (2)

2010 (4)

2009 (3)

2008 (2)

2007 (2)

1999 (1)

1993 (1)

1991 (1)

Avramov-Zamurovic, S.

Baykal, Y.

Borghi, R.

Cai, Y.

Chen, Y.

Cheng, W.

Cincotti, G.

Dan, Y.

de Sande, J. C. G.

Du, S.

Eyyuboglu, H. T.

Gbur, G.

Gori, F.

Gu, J.

Gu, Y.

Gutiérrez-Vega, J. C.

Haus, J. W.

Ji, X.

Korotkova, O.

Lajunen, H.

Liang, C.

Liu, L.

Liu, X.

Liu, Z.

Lu, L.

Ma, Y.

Malek-Madani, R.

Martínez -Herrero, R.

Martínez-Herrero, R.

Mei, Z.

Mejías, P. M.

Nelson, C.

Noriega-Manez, R. J.

Piquero, G.

Qu, J.

Ramírez-Sánchez, V.

Saastamoinen, T.

Sahin, S.

Santarsiero, M.

Shchepakina, E.

Shirai, T.

Sona, A.

Tong, Z.

Vahimaa, P.

Wang, F.

Wang, X.

Weber, H.

Wei, C.

Yuan, Y.

Zhan, Q.

Zhang, B.

Zhao, C.

Zhao, H.

Zhou, P.

Appl. Phys. Lett. (1)

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