Evolution properties of Bessel-Gaussian Schell-model beams in non-Kolmogorov turbulence

Xiaoyang Wang; Mingwu Yao; Zhiliang Qiu; Xiang Yi; Zengji Liu

doi:10.1364/OE.23.012508

1. Introduction

In recent decades, various types of laser beams propagating through turbulent atmosphere are extensively studied, due to their widespread applications in free-space optical (FSO) communications, laser radar systems, remote sensing, optical imaging, semiconductor waveguide, etc [1–7]. It has been found that the propagation of laser beams are inevitably affected by random refractive-index fluctuations of the turbulence and the use of partially coherent beams (PCBs) is an effective way to reduce the deleterious effect of the turbulence under certain circumstances [8–13]. As a result of the constraint of nonnegative definiteness in the choice of the mathematical form of spatial correlation functions for optical fields, most of the papers on propagation of PCBs in the early days are based on the conventional spatial correlation functions, i.e., Gaussian correlated Schell-model functions.

In order to satisfy such a constraint for devising genuine spatial correlation functions of scalar or electromagnetic PCBs, Gori and his collaborators presented a sufficient condition derived from the theory of reproducing kernel Hilbert spaces [14,15]. Since then the PCBs with non-conventional correlation functions, whose spectral degrees of coherence are not Gaussian profiles, have attracted considerable attention. Up to now, a variety of augmented PCBs with non-conventional correlation functions have been introduced, such as beams with locally varying spatial coherence [16], non-uniformly correlated (NUC) beam [17–24], multi-Gaussian Schell-model (MGSM) beam [25–32], Laguerre-Gaussian Schell-model (LGSM) and Bessel-Gaussian Schell-model (BGSM) beam [33–39], cosine-Gaussian Schell-model (CGSM) beam [40–46], sinc Schell-model (SSM) [47,48], and specially correlated radially polarized (SCRP) beam [49]. Previous experiments and theoretical analyses have demonstrated that those partially coherent fields with non-conventional correlation functions have some extraordinary propagation characteristics in free space, such as the peculiar self-focusing effect and a laterally shifted intensity maxima of the NUC beam [17,18,20,22], far fields with flat-topped intensity profiles generated by MGSM beams and the first kind SSM (SSM1) beam [25,29,47], far fields with ring-shaped (i.e., dark hollow) intensity profiles formed by LGSM beams, BGSM beams, CGSM beams and the second kind SSM (SSM2) beam [33,34,43,47], far field with shape-invariant double-layer flat-topped intensity profile produced by the electromagnetic sinc Schell-model (EM SSM) beam [48], etc.

To date, a few papers on propagation of PCBs with non-conventional correlation functions in turbulent atmosphere have been published [19,22–28,38–42,46,48,49]. In [38], the analytical expressions for the average intensity and the beam width of the BGSM beam propagating through a paraxial ABCD optical system in Kolmogorov turbulence were derived, and the evolution properties of the average intensity and the beam width were investigated. However, to the best of the authors’ knowledge, the evolution properties of the spectral degree of coherence (i.e., correlation function), the effective radius of curvature and the propagation factor of the BGSM beam in turbulence have not been known. In this paper, a generalized power spectrum model valid in non-Kolmogorov turbulence is employed, and the explicit analytical expressions for the spectral degree of coherence, the effective radius of curvature and the propagation factor of the BGSM beam in turbulent atmosphere are derived based on the extended Huygens-Fresnel principle and the second-order moments of the Wigner distribution function (WDF). The evolution properties of BGSM beams propagating in non-Kolmogorov turbulence are numerically explored. Some useful observations and conclusions have been proposed.

2. Theory

2.1. Spectral degree of coherence of the BGSM beam in non-Kolmogorov turbulence

Let us consider that a BGSM source generates a beam-like field propagating into the half-space z = 0 filled with turbulent atmosphere nearly parallel to the positive z direction. For brevity, the dependence of all field statistics on the angular frequency ω is implied but omitted in what follows. The cross-spectral density (CSD) of the BGSM beam at the source plane z = 0 is defined by [33]

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

where r₁ ≡ (x₁, y₁) and r₂ ≡ (x₂, y₂) denote two arbitrary transverse position vectors at the source plane, σ is the transverse beam width (i.e., the rms width) of the BGSM beam, and μ(r₁, r₂, 0) is the spectral degree of coherence of the BGSM beam at z = 0 and and characterized by the following form

μ (r_{1}, r_{2}, 0) = exp [- \frac{{| r_{1} - r_{2} |}^{2}}{2 δ^{2}}] J_{0} (β \frac{| r_{1} - r_{2} |}{δ})

where δ is the transverse spatial coherence width of the BGSM width, β is a real constant, and J₀ (·) is the zeroth-order Bessel function of the first kind.

It is clearly seen that the BGSM model represents a family of sources which can reduce to the conventional Gaussian Schell-model (GSM) source when β = 0 and to the J₀-correlated source as β → ∞. The condition for a BGSM source to generate a beam-like field is the same as that for a classical GSM source: 1/4σ² + 1/δ² ≪ 2π²/λ² with λ being the wavelength of the source [33].

Within the validity of the paraxial approximation, the propagation of the CSD of the BGSM beam in turbulence can be studied with the help of the following extended Huygens-Fresnel integral [2,50,51]

\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 (r_{s}, r_{d}, 0) \times exp [\frac{i k}{z} (ρ_{s} - r_{s}) (ρ_{d} - r_{d})] \\ \times exp [- H (ρ_{d}, r_{d}, z)] d^{2} r_{s} d^{2} r_{d} \end{array}

where k = 2π/λ is the wave number with λ being the wavelength of the source. In Eq. (3), the central abscissa coordinate systems are chosen, that is,

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

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

where ρ₁ ≡ (ρ_1x, ρ_1y) and ρ₂ ≡ (ρ_2x, ρ_2y) are two arbitrary transverse position vectors at the receiver plane, perpendicular to the direction of propagation of the beam, and

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

The term exp[−H (ρ_d, r_d; z)] in Eq. (3), represents the contribution of atmospheric turbulence, and H (ρ_d, r_d; z) can be expressed as [25,28,41,42,46,48]

H (ρ_{d}, r_{d}, z) = \frac{π^{2} k^{2} z T}{3} [{r_{d}}^{2} + r_{d} \cdot ρ_{d} + {ρ_{d}}^{2}]

where

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

Φ_n (κ) is the one-dimensional power spectrum of fluctuations in the refractive index of the isotropic turbulent medium, and κ is the scalar spatial wave number (i.e., spatial frequency).

In order to cover a wide scope of atmospheric conditions, our discussion on propagation of the BGSM beam is based on a generalized power spectrum model valid in non-Kolmogorov turbulence [25,28,41,42,46,48,52]

Φ_{n} (κ, α) = A (α) {\tilde{C}}_{n}^{2} \frac{exp [- (κ^{2} / κ_{m}^{2})]}{{(κ^{2} + κ_{0}^{2})}^{α / 2}}, 0 \leq κ < \infty, 3 < α < 4

where the term

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

is a generalized index-of-refraction structure constant (in units of m^3−α), κ₀ = 2π/L₀, κ_m = c(α)/l₀, L₀ and l₀ being the outer and inner scales of the turbulence, respectively, and

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

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

with Γ(·) being the Gamma function. The power spectrum expressed by Eq. (9) reduces to the conventional Kolmogorov spectrum when α = 11/3, A(α) = 0.033,

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

, L₀ → ∞ and l₀ → 0. With the power spectrum in Eq. (9) the integral in Eq. (8) becomes [25,28,41,42,46,48,52]

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

where

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

and Γ(·,·) denotes the incomplete Gamma function.

Applying Eqs. (3), (6) and (7), and calculating the integral with respect to r_s by using the following formula [53]

\int_{- \infty}^{\infty} exp (- p^{2} x^{2} \pm q x) d x = \frac{\sqrt{π}}{p} exp (\frac{q^{2}}{4 p^{2}}), (p > 0)

we obtain the expression for the CSD of the BGSM beam in the receiver plane

W (ρ_{s}, ρ_{d}, z) = Q \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} exp [- a r_{d}^{2} - b ρ_{s} \cdot r_{d} + c ρ_{d} \cdot r_{d}] J_{0} (β \frac{| r_{d} |}{δ}) d^{2} r_{d}

where

Q = \frac{σ^{2} k^{2}}{2 π z^{2}} exp [\frac{i k}{z} ρ_{s} \cdot ρ_{d} - (\frac{π^{2} k^{2} z T}{3} + \frac{σ^{2} k^{2}}{2 z^{2}}) ρ_{d}^{2}]

a = \frac{1}{8 σ^{2}} + \frac{1}{2 δ^{2}} + \frac{σ^{2} k^{2}}{2 z^{2}} + \frac{π^{2} k^{2} T z}{3}, b = \frac{i k}{z}, c = \frac{σ^{2} k^{2}}{z^{2}} - \frac{π^{2} k^{2} T z}{3}

With the application of the coordinate transformation, Eq. (14) can be rewritten as

W (ρ_{s}, ρ_{d}, z) = Q \int_{0}^{\infty} \int_{0}^{2 π} exp [- a r_{d}^{2} - | b ρ_{s} - c ρ_{d} | r_{d} cos φ] J_{0} (β \frac{r_{d}}{δ}) r_{d} d r_{d} d φ

where r_d is the scalar magnitude of r_d, φ is the angle between bρ_s − cρ_d and r_d. By using the following formulas [53]

\int_{0}^{2 π} exp (\pm x cos φ) d φ = 2 π I_{0} (x)

\int_{0}^{\infty} x exp (- γ x^{2}) I_{ν} (p x) J_{ν} (q x) d x = \frac{1}{2 γ} exp (\frac{p^{2} - q^{2}}{4 γ}) J_{ν} (\frac{p q}{2 γ}), [Re γ > 0, Re ν > - 1]

where J_ν (·) is the ν-order Bessel function of the first kind and I_ν (·) is the ν-order modified Bessel function of the first kind, respectively. Eq. (17) is simplified to

\begin{array}{l} W (ρ_{s}, ρ_{d}, z) = & \frac{σ^{2} k^{2}}{2 a z^{2}} exp [\frac{{| b ρ_{s} - c ρ_{d} |}^{2}}{4 a} + \frac{i k}{z} ρ_{s} \cdot ρ_{d} - (\frac{π k^{2} z T}{3} + \frac{σ^{2} k^{2}}{2 z^{2}}) ρ_{d}^{2}] \\ \times exp (- \frac{β^{2}}{4 a δ^{2}}) J_{0} (β \frac{| b ρ_{s} - c ρ_{d} |}{2 a δ}) \end{array}

Accordingly, the spectral density of the BGSM beam in the receiver plane is obtained as S (ρ, z) = W (ρ, 0, z), where ρ ≡ (ρ_x, ρ_y) denotes an arbitrary transverse position vector at the receiver plane. Furthermore, the spectral degree of coherence of the BGSM beam in the receiver plane is acquired by

μ (ρ_{1}, ρ_{2}, z) = \frac{W (\frac{ρ_{1} + ρ_{2}}{2}, ρ_{1} - ρ_{2}, z)}{\sqrt{S (ρ_{1}, z) S (ρ_{2}, z)}}

2.2. Second-order moments of the BGSM beam in non-Kolmogorov turbulence

On the basis of inverse Fourier transform of the Dirac delta function and its property of even function [54], and after some operations as shown in [50], the CSD of the BGSM beam in the receiver plane can be rewritten as

\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 (r^{'}, ρ_{d} + \frac{z}{k} κ_{d}, 0) \\ \times exp [- i ρ_{s} \cdot κ_{d} + i κ_{d} \cdot r^{'} - H (ρ_{d}, ρ_{d} + \frac{z}{k} κ_{d}, z)] d^{2} r^{'} d^{2} κ_{d} \end{array}

where κ_d ≡ (κ_dx, κ_dy) is the position vector in the spatial-frequency domain. The term

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

is obtained by inserting Eqs. (1) and (2) into Eq. (6), as follows

W (r^{'}, ρ_{d} + \frac{z}{k} κ_{d}, 0) = exp [- \frac{1}{2 σ^{2}} {r^{'}}^{2} - (\frac{1}{8 σ^{2}} + \frac{1}{2 δ^{2}}) {(ρ_{d} + \frac{z}{k} κ_{d})}^{2}] \times J_{0} (\frac{β}{δ} | ρ_{d} + \frac{z}{k} κ_{d} |)

and the term

H (ρ_{d}, ρ_{d} + \frac{z}{k} κ_{d}, z)

is acquired from Eq. (7), that is

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

As is well known, the WDF being closely related to the second-order statistical properties of PCBs on propagation in turbulent atmosphere, is especially suitable for the treatment of PCBs, and the WDF can be expressed in terms of the CSD by the following formula [26,39,50,51]

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 ≡ (θ_x, θ_y) denotes an angle which the vector of interest makes with the z-direction, kθ_x and kθ_y are the wave vector components along the x-axis and y-axis, respectively.

On substituting from Eqs. (22)–(24) into Eq. (25) and calculating the integral with respect to r′ by using Eq. (13), the WDF of the BGSM beam in non-Kolmogorov turbulence is obtained as

\begin{array}{l} h (ρ_{s}, θ, z) = & \frac{k^{2} σ^{2}}{8 π^{3}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} exp [- η_{1} ρ_{d}^{2} - η_{2} κ_{d}^{2} - η_{3} ρ_{d} \cdot κ_{d} - i ρ_{s} \cdot κ_{d} - i k θ \cdot ρ_{d}] \\ \times exp [- H (ρ_{d}, ρ_{d} + \frac{z}{k} κ_{d}, z)] J_{0} (\frac{β}{δ} | ρ_{d} + \frac{z}{k} κ_{d} |) d^{2} ρ_{d} d^{2} κ_{d} \end{array}

where

η_{1} = \frac{1}{8 σ^{2}} + \frac{1}{2 δ^{2}}, η_{2} = (\frac{1}{8 σ^{2}} + \frac{1}{2 δ^{2}}) \frac{z^{2}}{k^{2}} + \frac{1}{2} σ^{2}, η_{3} = (\frac{1}{4 σ^{2}} + \frac{1}{δ^{2}}) \frac{z}{k}

The moments of order n₁ + n₂ + m₁ + m₂ of the WDF of a laser beam are defined by [39,42,50,51]

〈 ρ_{s x}^{n_{1}} ρ_{s 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} ρ_{s x}^{n_{1}} ρ_{s y}^{n_{2}} θ_{x}^{m_{1}} θ_{y}^{m_{2}} h (ρ_{s}, θ, z) d^{2} ρ_{s} d^{2} θ

where ρ_sx and ρ_sy are the coordinates of of the transverse position vector ρ_s at the receiver plane,

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

is the total irradiance of the beam, and P = 2πσ² for the case of BGSM beams.

Substituting Eq. (26) into Eq. (28), we obtain (after tedious integration) the following expressions for the second-order moments of the WDF of the BGSM beam in non-Kolmogorov turbulence

〈 ρ_{s}^{2} 〉 = 〈 ρ_{s x}^{2} 〉 + 〈 ρ_{s y}^{2} 〉 = 2 σ^{2} + (\frac{1}{2 σ^{2}} + \frac{2 + β^{2}}{δ^{2}}) \frac{z^{2}}{k^{2}} + \frac{4}{3} π^{2} T z^{3}

〈 θ^{2} 〉 = 〈 θ_{x}^{2} 〉 + 〈 θ_{x}^{2} 〉 = (\frac{1}{2 σ^{2}} + \frac{2 + β^{2}}{δ^{2}}) \frac{1}{k^{2}} + 4 π^{2} T z

〈 ρ_{s} \cdot θ 〉 = 〈 ρ_{s x} θ_{x} 〉 + 〈 ρ_{s y} θ_{y} 〉 = (\frac{1}{2 σ^{2}} + \frac{2 + β^{2}}{δ^{2}}) \frac{z}{k^{2}} + 2 π^{2} T z^{2}

In the above derivations, we have used the series representation of the Bessel function of the first kind [53]

J_{ν} (x) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} {(x / 2)}^{2 n + ν}}{n! Γ (n + ν + 1)}, | x | < \infty

and the following formulas [54]

δ (p x) = \frac{1}{| p |} δ (x)

δ^{(n)} (s) = \frac{1}{2 π} \int {(- i x)}^{n} exp (- i s x) d x, (n = 0, 1, 2)

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

In accordance with the definition of the effective radius of curvature of a laser beam [26,55,56], we obtain the explicit expression for the effective radius of curvature of the BGSM beam in non-Kolmogorov turbulence with the help of Eqs. (29) and (31)

R (z) = \frac{〈 ρ_{s}^{2} 〉}{〈 ρ_{s} \cdot θ 〉} = \frac{2 σ^{2} + (\frac{1}{2 σ^{2}} + \frac{2 + β^{2}}{δ^{2}}) \frac{z^{2}}{k^{2}} + \frac{4}{3} π^{2} T z^{3}}{(\frac{1}{2 σ^{2}} + \frac{2 + β^{2}}{δ^{2}}) \frac{z}{k^{2}} + 2 π^{2} T z^{2}}

Based on the second-order moments of the WDF, the propagation factor of a partially coherent beam in turbulence is defined by [39,42,50,51]

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

Applying Eqs. (29)–(31) and (37), we obtain the propagation factor of the BGSM beam in non-Kolmogorov turbulence

M^{2} (z) = {[2 σ^{2} (\frac{1}{2 σ^{2}} + \frac{2 + β^{2}}{δ^{2}}) + 8 π^{2} k^{2} σ^{2} T z + \frac{4}{3} π^{2} (\frac{1}{2 σ^{2}} + \frac{2 + β^{2}}{δ^{2}}) T z^{3} + \frac{4}{3} π^{4} k^{2} T^{2} z^{4}]}^{\frac{1}{2}}

Eq. (38) indicates that the propagation factor of the BGSM beam in non-Kolmogorov turbulence increases with the propagation distance z, which means the beam quality is in decline on propagation of the beam as a result of the presence of atmospheric turbulence.

Under the condition of Φ_n (κ) = 0, Eq. (38) reduces to the expression for the propagation factor of the BGSM beam in free space, that is M² (z) = [1 + 2σ² (2 + β²)/δ²]^1/2. One can find that the propagation factor of the BGSM beam in free space is independent of the propagation distance z, and only determined by the beam width σ, the spatial coherence width δ and the parameter β of the BGSM source. When Φ_n (κ) = 0 and β = 0, Eq. (38) reduces to the expression for the propagation factor of a GSM beam in free space, i.e., M² (z) = [1 + 4σ²/δ²]^1/2, which is consistent with the previous result reported in [57].

3. Numerical calculation results and analysis

In this section, we will examine the evolution properties of BGSM beams on propagation in non-Kolmogorov turbulence by a set of numerical examples based on the analytical formulas derived in the previous section. Due to the unit of the generalized structure constant ${\tilde{C}}_{n}^{2}$ depending on the spectral index α, analysis of the beam propagation in non-Kolmogorov turbulence involving ${\tilde{C}}_{n}^{2}$ for various α may easily cause misleading interpretations. Charnotskii has discussed this issue and suggested to use a dimensional argument of the same dimension or some dimensionless argument to eliminate the unit choice inconsistency, such as coherence radius, Fresnel number and Rytov variance [58,59]. In this paper, the generalized Rytov variance for a plane wave propagating in weak generalized atmospheric turbulence is chosen, which can be expressed by $σ_{R}^{2} = - 2 Γ (α - 1) Γ (1 - α / 2) α^{- 1} cos (α π / 2) cos [(α - 2) / 4] {\tilde{C}}_{n}^{2} k^{3 - α / 2} z^{α / 2}$ [60]. Without loss of generality, the parameters of the source and turbulent atmosphere are chosen to be ${\tilde{C}}_{n}^{2} = 10^{- 14} m^{3 - α}$ , α = 3.8, L₀ = 1 m, l₀ = 0.001 m, σ = 0.02 m, δ = 0.01 m, λ = 632.8 nm unless other parameters are specified in each figure caption.

Figure 1 illustrates the evolution properties of the modulus of the transverse spectral degree of coherence of the BGSM beam at several propagation distances in non-Kolmogorov turbulence for different values of the parameter β. For comparison, the spectral degree of coherence of a GSM beam (β = 0) is given under the same conditions, which maintains Gaussian profile for all different propagation distances. From Fig. 1(a), one can find that the spectral degree of coherence of the BGSM beam with β > 0 is non-Gaussian profile, which has several side lobes around the main lobe. The larger the parameter β, the more the side lobes in the spectral degree of coherence. For relatively short propagation distances, the side lobes gradually disappear and the spectral degree of coherence converts to a single Gaussian profile for all β at a distance of approximately 2.6 km (for the given BGSM source and atmospheric parameters) [Figs. 1(b)–1(d)], Moreover, the spectral degree of coherence goes through another change for z > 2.6 km and the side lobes reconstruct successively depending on the values of the parameter β [Figs. 1(e)–1(g)]. In the end, the spectral degree of coherence of the BGSM beam evolves into Gaussian profile again in the far field [Fig. 1(h)]. This interesting phenomenon can be attributed to the combined effect of the source correlation and atmospheric turbulence. To be precise, the effect of the source correlation plays a dominant role at short propagation distances and the influence of atmospheric turbulence at long propagation distances.

Fig. 1 Modulus of the transverse spectral degree of coherence of the BGSM beam as a function of ρ_1x at several propagation distances in non-Kolmogorov turbulence for different values of parameter β.

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Figure 2 shows the behaviors of the modulus of the transverse spectral degree of coherence of the BGSM beam at propagation distance z = 2 km in non-Kolmogorov turbulence for different values of spectral index α and generalized Rytov variance $σ_{R}^{2}$ . From Fig. 2(a), one can find that the spectral degree of coherence of the BGSM beam is non-Gaussian profile for different values of the spectral index α. As the fluctuations of atmospheric turbulence increase, the spectral degree of coherence of the BGSM beam eventually converges to Gaussian profile, and more rapidly for smaller spectral index α. It can be concluded that the BGSM beam is less affected by the generalized atmospheric turbulence with larger spectral index α.

Fig. 2 Modulus of the transverse spectral degree of coherence of the BGSM beam with β = 5 as a function of ρ_1x at propagation distance z = 2 km in non-Kolmogorov turbulence for different values of spectral index α and generalized Rytov variance $σ_{R}^{2}$ .

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Moreover, numerical calculations for the normalized propagation factor of the BGSM beam in non-Kolmogorov turbulence were performed on the basis of Eq. (38). Some typical examples are compiled in Figs. 3–8.

Fig. 3 Normalized propagation factor of the BGSM beam for different values of parameter β versus propagation distance z in non-Kolmogorov turbulence with different values of generalized refractive-index structure constant ${\tilde{C}}_{n}^{2}$ .

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Fig. 4 Normalized propagation factor of the BGSM beam versus propagation distance z in non-Kolmogorov turbulence for different values of parameter β and beam width σ.

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Fig. 5 Normalized propagation factor of the BGSM beam with β = 2.5 as a function of propagation distance z in non-Kolmogorov turbulence for different values of (a) outer scale L₀ and (b) inner scale l₀.

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Fig. 6 Normalized propagation factor of the BGSM beam versus parameter β at propagation distance z = 10 km in non-Kolmogorov turbulence for different values of (a) spatial coherence width δ and (b) wavelength λ.

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Fig. 7 Normalized propagation factor of the BGSM beam versus spatial coherence width δ at propagation distance z = 10 km in non-Kolmogorov turbulence for different values of parameter β.

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Fig. 8 Normalized propagation factor of the BGSM beam versus spectral index α at propagation distance z = 1 km in non-Kolmogorov turbulence with $σ_{R}^{2} = 0.2$ for different values of parameter β.

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Figure 3 plots the normalized propagation factor of the BGSM beam versus propagation distance z in non-Kolmogorov turbulence for different values of parameter β and generalized refractive-index structure constant ${\tilde{C}}_{n}^{2}$ . Figure 4 shows the normalized propagation factor of the BGSM beam versus propagation distance z in non-Kolmogorov turbulence for different values of parameter β and beam width σ. From ^{Fig. 3–4}, it can be clearly seen that the normalized propagation factor of the BGSM beam in turbulence increases with the propagation distance z, which is totally different from that in free space being independent of the propagation distance z. The normalized propagation factor of the BGSM beam with larger parameter β increases more slowly than that with smaller parameter β or a GSM beam, especially in the cases of larger structure constant ${\tilde{C}}_{n}^{2}$ and larger beam width σ. That implies the influence of atmospheric turbulence can be significantly reduced by using the BGSM beam with larger parameter β.

Figure 5 depicts the normalized propagation factor of the BGSM beam as a function of propagation distance z in non-Kolmogorov turbulence for different values of outer scale L₀ and inner scale l₀, respectively. From Fig. 5, it can be readily seen that the normalized propagation factor of the BGSM beam increases with outer scale L₀ for a fixed propagation distance z while decreasing with inner scale l₀, and more sensitive to the change of inner scale l₀. The outer scale L₀ forms the upper limit of the inertial range and increases with the strength of atmospheric turbulence. The inner scale l₀, which forms the lower limit of the inertial range, has a larger value in weak turbulence and a smaller value in strong turbulence. The increasing of outer scale L₀ or the decreasing of inner scale l₀ is equivalent to increasing the strength of the turbulence. In these cases, the beam will meet more turbulence eddies along its propagation paths. As a result, the beam experiences more spreading and has larger propagation factor [2,52].

Figure 6 plots the normalized propagation factor of the BGSM beam versus parameter β at propagation distance z = 10 km in non-Kolmogorov turbulence for different values of spatial coherence width δ and wavelength λ, respectively. As indicated by Fig. 6, the normalized propagation factor of the BGSM beam decreases monotonically with the parameter β, more apparently for larger spatial coherence width δ and smaller wavelength λ. However, the benefits gained by increasing the parameter β continue to diminish and can be ignored when the parameter β is larger than 15. For smaller values of spatial coherence width δ, the normalized propagation factor of the BGSM beam is always smaller as shown in Fig. 6(a). The reason is that the turbulence-induced spread of a weaker coherent beam is less than that of a stronger coherent beam. In addition, Fig. 6(b) implies that the BGSM beam with larger wavelength has advantage over the BGSM beam with smaller wavelength in reducing the turbulence-induced degradation.

Figure 7 presents the normalized propagation factor of the BGSM beam versus spatial coherence width δ at propagation distance z = 10 km in non-Kolmogorov turbulence for different values of parameter β. As shown in Fig. 7, the normalized propagation factor of the BGSM beam can be reduced by increasing the value of the parameter β when the value of the spatial coherence width δ is fixed, which is in line with the depiction of Fig. 6(a). The normalized propagation factor of the BGSM beam converges to a constant when the spatial coherence width δ is large enough or small enough, whatever the value of the parameter β is. That means the beam quality cannot get improved by changing the value of the parameter while the source is a fully coherent wave or an incoherent wave.

Figure 8 gives the normalized propagation factor of the BGSM beam versus spectral index α at propagation distance z = 1 km in non-Kolmogorov turbulence with $σ_{R}^{2} = 0.2$ for different values of parameter β. One can find from Fig. 8 that the relationship between the normalized propagation factor of the BGSM beam and the spectral index α is monotonic. The normalized propagation factor of the BGSM beam decreases with the spectral index α, and the reason is that the influence of atmospheric turbulence becomes weaker with the increasing of the spectral index α. At the same time, we can deduce that the BGSM beam with larger parameter β is less affected by the turbulence than that with smaller parameter β.

Figure 9 represents the effective radius of curvature of the BGSM beam propagating in non-Kolmogorov turbulence for different values of parameter β and generalized refractive-index structure constant ${\tilde{C}}_{n}^{2}$ . As manifested in Fig. 9, the effective radius of curvature of the BGSM beam displays a downward trend in the near field and then turns to increase after reaching a dip, as a result of the diffraction of a collimated beam in atmospheric turbulence. There exists a minimum R_min(z) as the propagation distance z increases, and the position of R_min(z) is further close to the source plane for the BGSM beam with larger parameter β than that with smaller parameter β. The difference between the effective radius of curvature of the BGSM beam in free space and that in non-Kolmogorov turbulence is smaller than that of a GSM beam, and the larger the parameter β the smaller the difference. It is also found that the BGSM with larger parameter β is less affected by the turbulence than that with smaller parameter β from the aspect of the evolution properties of the effective radius of curvature.

Fig. 9 Effective radius of curvature of the BGSM beam propagating in non-Kolmogorov turbulence for different values of parameter β and generalized refractive-index structure constant ${\tilde{C}}_{n}^{2}$ .

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4. Concluding remarks

In this paper, we have derived the analytical expressions for the spectral degree of coherence, the effective radius of curvature and the propagation factor of the BGSM beam in non-Kolmogorov turbulence with the help of the extended Huygens-Fresnel principle and the second-order moments of the WDF, which can be simplified to those of a GSM beam when β =0. We have examined the variation of three main statistical properties of the BGSM beam: the spectral degree of coherence, the effective radius of curvature and the normalized propagation factor, on propagation in non-Kolmogorov turbulence by some numerical examples. It is found that the BGSM beam is less affected by the turbulence than a GSM beam, and this advantage is enhanced for larger parameter β, larger beam width σ or in the case of stronger fluctuations of the turbulence. The beam quality of the BGSM beam can further be improved by decreasing of the spatial coherence width δ or increasing the wavelength λ. Furthermore, the effects of outer scale L₀, inner scale l₀, and the spectral index α of the turbulence on the propagation factor are also explored at great length. The BGSM beam has greater spreading in the generalized atmospheric turbulence with smaller inner scale l₀, larger outer scale L₀ or smaller spectral index α. Our theoretical results will be useful in the long-distance free-space optical communications. In the future work, we will try to carry out the experiment and find whether the experiment results agree with the theoretical predictions. The experimental generation of partially coherent beams with different complex degrees of coherence was recently reported by Wang et al. [34]. It is possible to generate the BGSM beam with the similar method and realize the experiment of the BGSM beam propagating in non-Kolmogorov turbulence.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant 61001129 and the Fundamental Research Funds for the Central Universities.

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Evolution properties of Bessel-Gaussian Schell-model beams in non-Kolmogorov turbulence

Abstract

1. Introduction

2. Theory

2.1. Spectral degree of coherence of the BGSM beam in non-Kolmogorov turbulence

2.2. Second-order moments of the BGSM beam in non-Kolmogorov turbulence

3. Numerical calculation results and analysis

4. Concluding remarks

Acknowledgments

References and links

Cited By

Figures (9)

Equations (38)

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