Propagation factors of multi-sinc Schell-model beams in non-Kolmogorov turbulence

Zhenzhen Song; Zhengjun Liu; Keya Zhou; Qiongge Sun; Shutian Liu

doi:10.1364/OE.24.001804

1. Introduction

The evolution properties of laser beams on propagation in turbulent atmosphere have been extensively investigated for their significant applications in free-space optical communications and remote sensing [1–9 ]. The random turbulence can inevitably degrade the beam quality. Studies show that partially coherent beams propagating in turbulent atmosphere have substantial advantage over completely coherent beams for reducing turbulence-induced degradation [4–9 ]. Since Gori and his collaborators established a sufficient condition for devising genuine spatial correlation functions [10,11], various partially coherent light sources with non-conventional correlation functions have been proposed and investigated both theoretically and experimentally [12–26 ]. The beam-like fields generated by those sources demonstrate extraordinary propagation characteristics, such as self-focusing and laterally shifted intensity maxima of the non-uniformly correlated (NUC) beams [12–15 ]; tunable flat far-fields of multi-Gaussian Schell-model (MGSM) beams [16, 17] and the first kind sinc Schell-model (SSM1) beams [18], dark-hollow far-fields of Laguerre-Gaussian Schell-model (LGSM) beams [19,20], Bessel-Gaussian Schell-model (BGSM) beams [19], Cosine-Gaussian Schell-model (CGSM) beams [21, 22], and the second kind sinc Schell-model (SSM2) beams [18]; self-splitting properties of Hermite-Gaussian correlated Schell-model (HGCSM) beams [23] and CGSM beams with rectangular symmetry [24, 25], double-layer flat-topped far-fields of electromagnetic sinc Schell-model (EMSSM) beam [26], and so on. The propagation factor (M ²-factor), which can be calculated utilizing the Wigner distribution function (WDF) [27, 28], is regarded as a beam quality factor in many practical applications. Recently, the M ²-factor of several partially coherent beams with nonconventional correlation functions in turbulent atmosphere have been studied [29, 30]. For the modulation of special functions to the degree of coherence, the affection of the turbulence to these beams is less than the conventional GSM beams.

A novel class of partially coherent beams of Schell type named multi-sinc Schell-model (MSSM) beam was recently proposed based on the superposition technique [31]. It demonstrates adjustable multi-rings far-fields in free space. The propagation of MSSM beams in turbulent atmosphere has also been studied [32]. However, to the best of our knowledge, the root-mean-square (rms) angular width and the M ²-factor of MSSM beams propagating in turbulence have not been reported. The main purpose of this paper is to investigate the root-mean-square (rms) angular width and the M ²-factor of MSSM beams on propagation in non-Kolmogorov turbulence. We derive the explicit analytical expressions on propagation based upon the extended Huygens-Fresnel principle and second-order moments of the WDF. The effect of MSSM beams parameters and non-Kolmogorov turbulence on the rms angular width and the M ²-factor in the turbulence are also analyzed by numerical examples. At last, the propagation behaviors of the MSSM beams are compared with that of the conventional GSM beam.

2. Second-order moments of MSSM beams in non-Kolmogorov turbulence

Let us consider a statistically stationary optical field generated by the MSSM source. The cross-spectral density (CSD) function at the source plane z = 0 is expressed as [31, 32]

W_{0} (r_{1}, r_{2}, 0) = \frac{1}{C} exp (- \frac{r_{1}^{2} + r_{2}^{2}}{2 σ^{2}}) \sum_{n = 1}^{N} \frac{{(- 1)}^{n - 1}}{B_{m n}} sinc (\frac{r_{1} - r_{2}}{B_{m n} δ}),

where r ₁ and r ₂ denotes two points at the input plane, σ represents the beam waist width, δ is the correlation width, B_mn = {(2N − 1)/[2^m (2N − 2n + 1)]}^1/m is the modified parameter, and the index m is an arbitrary positive real number,

C = \sum_{n = 1}^{N} {(- 1)}^{n} / B_{m n}

is the normalization factor. When N = 1, Eq. (1) reduces to the CSD of the SSM1 beams. When N = 2, Eq. (1) reduces to the CSD of the SSM2 beams.

The paraxial propagation of the CSD of a partially coherent beam through turbulent atmosphere can be treated by the extend Huygens-Fresnel principle. Supposing ρ ₁ and ρ ₂ are two spatial points at an arbitrary transverse plane of the half-space z = const > 0, the propagation equation of the CSD is described as [33]

\begin{array}{l} W (ρ_{1}, ρ_{2}, z) & = & {(\frac{k}{2 π z})}^{2} \iint W_{0} (r_{1}, r_{2}, 0) exp [- i k \frac{{(ρ_{1} - r_{1})}^{2} - {(ρ_{2} - r_{2})}^{2}}{2 z}] \\ \times & exp {- \frac{k^{2} z T}{2} [{(ρ_{1} - ρ_{2})}^{2} + (ρ_{1} - ρ_{2}) (r_{1} - r_{2}) + {(r_{1} - r_{2})}^{2}]} d^{2} r_{1} d^{2} r_{2}, \end{array}

with

T = \frac{2 π^{2}}{3} \int_{0}^{\infty} κ^{3} Φ (κ, α) d κ,

where k = 2π/λ is the wave number, the term Φ(κ, α) is the spatial power spectrum of the refractive index fluctuations of the turbulent medium, κ is the magnitude of two-dimensional spatial frequency, and α is the power-law exponent. If T = 0, Eq. (2) represents the propagation in free space, and the spectral intensity of MSSM beams at any point (ρ, z) can be obtained as [31]

S (ρ, z) = W (ρ, ρ, z) = \frac{δ σ^{2}}{C ω^{2} (z)} \int exp [- \frac{{(ρ + 2 π z υ / k)}^{2}}{ω^{2} (z)}] \sum_{n = 1}^{N} {(- 1)}^{n - 1} rect (B_{m n} δ υ) d^{2} υ,

where

ω^{2} (z) = σ^{2} + z^{2} / (k^{2} σ^{2}) .

Figure 1 shows the cross line (ρ_y = 0) of the spectral intensity of MSSM beams with different parameters N and m at propagation distance z = 10km in free space. The other parameters of MSSM beams are chosen to be σ = 0.05m, δ = 0.02m, λ = 632.8nm. It can be easily seen that the far field of MSSM beams with different parameter N demonstrates different profile, such as flat-top profile with N = 1, dark-hollow profile with N = 2, and multi-rings profile with N = 6. Moreover, the dark-hollow size increases with increasing parameter m.

Fig. 1 The cross line (ρ_y = 0) of the spectral intensity of MSSM beams for several values of parameters N and m at propagation distance z = 10km in free space.

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For convenience of the subsequent calculations, the sum and difference vector notation r = (r ₁ + r ₂)/2, r _d = r ₁ − r ₂, ρ = (ρ ₁ + ρ ₂)/2, ρ _d = ρ ₁ − ρ ₂ are used to rewrite Eq. (2) as follows [34–36 ]

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

Utilizing the inverse Fourier transform of the Dirac delta function and its property of even function [37], and variable substitution r=r′ and r _d = ρ _d + z/k ·κ _d, the CSD of the MSSM beams at the output plane is expressed as

\begin{array}{l} W (ρ, ρ_{d}, z) & = & {(\frac{1}{2 π})}^{2} \iint W_{0} (r^{'}, ρ_{d} + \frac{z}{k} κ_{d}, 0) exp i r^{'} κ_{d} - i ρ κ_{d}) \\ \times & exp [- \frac{k^{2} z T}{2} (3 ρ_{d}^{2} + 3 \frac{z}{k} ρ_{d} \cdot κ_{d} + \frac{z^{2}}{k^{2}} κ_{d}^{2})] d^{2} r^{'} d^{2} κ_{d}, \end{array}

where κ _d is the position vector in the spatial-frequency domain, and

\begin{array}{l} W_{0} (r^{'}, ρ_{d} + \frac{z}{k} κ_{d}, 0) & = & \frac{1}{C} exp [- \frac{1}{σ^{2}} r^{' 2} - \frac{1}{4 σ^{2}} {(ρ_{d} + \frac{z}{k} κ_{d})}^{2}] \\ \times & \sum_{n = 1}^{N} \frac{{(- 1)}^{n - 1}}{B_{m n}} sinc [\frac{1}{B_{m n} δ} (ρ_{d} + \frac{z}{k} κ_{d})] . \end{array}

As is well known, the WDF is very suitable for the treatment of partially coherent beams on propagation in turbulent atmosphere, and it can be expressed as a two-dimensional spatial Fourier transform of the CSD by the following formula [35, 36, 38, 39]

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

where vector 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.

Substituting Eqs. (7) and (8) into Eq. (9), and calculating the integral with respect to r′ by using the following formula [40]

\int exp (- a x^{2} \pm b x) d x = \sqrt{\frac{π}{a}} exp (\frac{b^{2}}{4 a}), (a > 0),

the WDF of MSSM beams propagating in turbulent atmosphere is obtained as

\begin{array}{l} h (ρ, θ, z) & = & \frac{k^{2} σ^{2}}{16 π^{3} C} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} exp [- \frac{σ^{2} κ_{d}^{2}}{4} - \frac{1}{4 σ^{2}} {(ρ_{d} + \frac{z}{k} κ_{d})}^{2}] \\ \times & \sum_{n = 1}^{N} \frac{{(- 1)}^{n - 1}}{B_{m n}} sinc [\frac{1}{B_{m n} δ} (ρ_{d} + \frac{z}{k} κ_{d})] exp (- i ρ κ_{d} - i k θ \cdot ρ_{d}) \\ \times & exp [- \frac{k^{2} z T}{2} (3 ρ_{d}^{2} + 3 \frac{z}{k} ρ_{d} \cdot κ_{d} + \frac{z^{2}}{k^{2}} κ_{d}^{2})] d^{2} κ_{d} d^{2} ρ_{d} . \end{array}

The moments of the order of the WDF for three-dimensional beams is defined as [35, 36, 39]

〈 ρ_{x}^{n_{1}} ρ_{y}^{n_{2}} θ_{x}^{m_{1}} θ_{y}^{m_{2}} 〉 = \frac{1}{P} \iint ρ_{x}^{n_{1}} ρ_{y}^{n_{2}} θ_{x}^{m_{1}} θ_{y}^{m_{2}} h (ρ, θ, z) d^{2} ρ d^{2} θ,

where P = ∬h(ρ, θ, z)d ² ρ d ² θ is the total power of the beams. For a partially coherent Schell beam, the general formulas of the second-order moments of WDF on propagation in turbulent atmosphere can be easily derived as [41, 42]

〈 ρ^{2} 〉 = {〈 ρ^{2} 〉}_{0} + {〈 θ^{2} 〉}_{0} z^{2} + 2 T z^{3},

〈 θ^{2} 〉 = {〈 θ^{2} 〉}_{0} + 6 T z,

〈 ρ \cdot θ 〉 = {〈 θ^{2} 〉}_{0} z + 3 T z^{2},

where

{〈 θ^{2} 〉}_{0} = {〈 θ_{x}^{2} 〉}_{0} + {〈 θ_{y}^{2} 〉}_{0}

and

{〈 ρ^{2} 〉}_{0} = {〈 ρ_{x}^{2} 〉}_{0} + {〈 ρ_{y}^{2} 〉}_{0}

are the second-order moments of WDF at the input plane. Equations (13)–(15) imply that the second-order moments of the beams propagating in turbulent atmosphere only relates to the initial second moments, the propagation distance, and the spatial power spectrum of the refractive index fluctuations.

Substituting Eq. (11) into Eq. (12), the second-order moments of WDF of MSSM beams in non-Kolmogorov turbulence at the source plane z = 0 can be obtained as

{〈 ρ^{2} 〉}_{0} = σ^{2},

{〈 θ^{2} 〉}_{0} = \frac{2}{k^{2}} \frac{1 / (2 σ^{2}) + \sum_{n = 1}^{N} [{(- 1)}^{n - 1} π^{2}] / (B_{m n}^{3} δ^{2})}{\sum_{n = 1}^{N} {(- 1)}^{n - 1} / B_{m n}} .

The rms angular width and the M ²-factor are important parameters characterizing light beams. According to the definition of the rms angular width and the M ²-factor of laser beams [28], the analytical expressions of the rms angular width and the M ²-factor of MSSM beams is described as

\begin{array}{l} θ_{N} (z) & \equiv & {(〈 {| θ - 〈 θ 〉 |}^{2} 〉)}^{1 / 2} = {(〈 θ^{2} 〉)}^{1 / 2} \\ = & {\frac{2}{k^{2}} \frac{1 / (2 σ^{2}) + \sum_{n = 1}^{N} [{(- 1)}^{n - 1} π^{2}] / (B_{m n}^{3} δ^{2})}{\sum_{n = 1}^{N} {(- 1)}^{n - 1} / B_{m n}} + 6 T z}^{1 / 2} \end{array},

\begin{array}{l} M^{2} (z) & = & k {(〈 ρ^{2} 〉 〈 θ^{2} 〉 - {〈 ρ \cdot θ 〉}^{2})}^{1 / 2} \\ = & {(2 σ^{2} + 4 T z^{3}) \frac{1 / (2 σ^{2}) + \sum_{n = 1}^{N} [{(- 1)}^{n - 1} π^{2}] / (B_{m n}^{3} δ^{2})}{\sum_{n = 1}^{N} {(- 1)}^{n - 1} / B_{m n}} + 6 k^{2} σ^{2} T z + 3 k^{2} T^{2} z^{4}}^{1 / 2} . \end{array}

Equations (18) and (19) are used to analyze the statistical properties of MSSM beams on propagation in non-Kolmogorov turbulence. As comparison, a conventional GSM beam is considered in our analysis, and the M ²-factor is given as [35]

M_{G}^{2} (z) = {[(2 σ^{2} + 4 T z^{3}) (\frac{1}{2 σ^{2}} + \frac{2}{δ^{2}}) + 12 k^{2} σ^{2} T z]}^{1 / 2} .

In this paper, we adopt a generalized power spectrum model, which is valid in non-Kolmogorov turbulence, and its expression is [43, 44]

Φ (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 A(α) = (2π)⁻²Γ(α − 1)cos(απ/2) with Γ(·) is the gamma function,

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

is a index-of-refraction structure constant with units m³⁻ ^α, κ ₀ = 2π/L ₀, κ_m = c(α)/l ₀ with c(α) = [2πΓ(5 − α/2) A(α)/3]^1/(α−5), L ₀ and l ₀ are the outer and inner scales of turbulence, respectively. If α = 11/3, L ₀ = ∞ and l ₀ = 0, Eq. (21) is simplified as follows

Φ (κ, 11 / 3) = 0.033 C_{n}^{2} κ^{- 11 / 3},

which represents the ideal Kolmogorov power spectrum. The analytical expression of the strength of non-Kolmogorov turbulence can be obtained by substituting Eq. (21) to Eq. (3), as follows

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

where

β = (α - 2) κ_{m}^{2} + 2 κ_{0}^{2}

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

3. Numerical calculation results and analysis

We now investigate the statistical properties of MSSM beams on propagation in non-Kolmogorov turbulence by numerical simulation. In the following analysis, the calculation parameters of MSSM beams are the same with that used in Fig. 1, and of non-Kolmogorov turbulence are ${\tilde{C}}_{n}^{2} = 10^{- 15} m^{3 - α}$ , α = 3.1, L ₀ = 1m, l ₀ = 1mm unless other variable parameters are specified in calculation.

Figure 2 shows the normalized rms angular width of MSSM beams propagating through non-Kolmogorov turbulence for several values of parameters N and m. It can be seen that the normalized rms angular width increases monotonically with the propagation distance. Figure 2(a) indicates that the normalized rms angular width of MSSM beams with the parameter N = 2 is smaller than other values of N when m = 1. While in Fig. 2(b), the normalized rms angular width of MSSM beams decreases with increasing the parameter m when N = 2. It means that the beam with dark-hollow far-fields (N = 2) in free space has stronger resistance to the turbulence than that with flat-topped far-fields (N = 1) and multi-rings far-fields (N = 6), and the resistance increases with the increase of dark-hollow size. The MSSM beams with the parameter N = 2 and larger m are less affected by non-Kolmogorov turbulence from the aspect of the normalized rms angular width.

Fig. 2 Normalized rms angular width of MSSM beams propagating through non-Kolmogorov turbulence for several values of (a) parameter N and (b) parameter m, respectively.

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For a partially coherent beam propagating in turbulent atmosphere, the beam quality will be improved with lower spatial coherence. The effect of wavelength λ and beam width σ of MSSM beams on the normalized M ²-factor is considered as follows. Figure 3 shows the dependence of the normalized M ²-factor of the MSSM beams for several values of parameters N and m at propagation distance z = 10km in non-Kolmogorov turbulence on wavelength λ and beam width σ, respectively. One finds that the normalized M ²-factor decreases with increasing wavelength λ or beam width σ. It can be concluded that MSSM beams with longer wavelength or larger beam width are less affected by non-Kolmogorov turbulence.

Fig. 3 Normalized M ²-factor of MSSM beams at propagation distance z = 10km in non-Kolmogorov turbulence versus (a) wavelength λ and (b) beam width σ, respectively.

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We will now analyze the influence of non-Kolmogorov turbulence on the propagation factor of MSSM beams. Figure 4 shows the dependence of the normalized M ²-factor of MSSM beams for several values of parameters N and m on the power-law exponent α at propagation distance z = 10km in non-Kolmogorov turbulence. According to Fig. 4, one finds that the normalized M ²-factor of MSSM beams with different parameters N and m reaches the maximum point at the same power-law exponent α = 3.07. When α < 3.07, it increases quickly with the increase of α. When α > 3.07, it decreases gradually with the further increase of α. This result is similar with the dependence of the strength of the non-Kolmogorov turbulence on the power-law exponent α. Therefore, it can be explained that the beam quality is degraded the most at the maximum strength of the turbulence, and improved under the weaker turbulence.

Fig. 4 Normalized M ²-factor of MSSM beams versus the power-law exponent α at propagation distance z = 10km in non-Kolmogorov turbulence.

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Figure 5 plots the normalized M ²-factor of MSSM beams for several values of parameters N and m versus the index-of-refraction structure constant ${\tilde{C}}_{n}^{2}$ at propagation distance z = 10km in non-Kolmogorov turbulence. According to Eq. (23), the strength of the turbulence increases with increasing ${\tilde{C}}_{n}^{2}$ . So it can be seen from Fig. 5 that there is almost no distinct difference for different parameters N and m for weak turbulence. As the strength of the turbulence increases, the normalized M ²-factor of MSSM beams with dark-hollow far-fields (N = 2) in free space is obviously smaller than the one with flat-topped far-fields (N = 1) or multi-rings far-fields (N = 6), and the difference becomes bigger when only increase the parameter m. It can be concluded that a MSSM beam with dark-hollow far-fields in free space has advantage over the one with flat-topped or multi-rings far-fields for reducing the turbulence-induced degradation, and this advantage will become more obvious with larger dark-hollow size.

Fig. 5 Normalized M ²-factor of MSSM beams for several values of parameters N and m versus the index-of-refraction structure constant ${\tilde{C}}_{n}^{2}$ at propagation distance z = 10km in non-Kolmogorov turbulence.

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Figure 6 shows the normalized M ²-factor of the MSSM beams with N = 2 and m = 5 on propagation in non-Kolmogorov turbulence for different power-law exponent α, outer scale L ₀, and inner scale l ₀. The turbulence with α = 3.1 is stronger than the case α = 3.67. According to Fig. 6, it can be clearly seen that the normalized M ²-factor of MSSM beams is not sensitive to the change of outer scale L ₀, especially in stronger turbulence. And at a fixed propagation distance z, MSSM beams are less affected by the turbulence with smaller outer scale L ₀ or larger inner scale l ₀.

Fig. 6 Normalized M ²-factor of the MSSM beams with N = 2 and m = 5 on propagation in non-Kolmogorov turbulence for different power-law exponent α, outer scale L ₀, and inner scale l ₀.

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Figure 7 shows the difference between the normalized M ²-factor of MSSM beams and a conventional GSM beam on propagation in non-Kolmogorov turbulence. One finds that the normalized M ²-factor of the conventional GSM beam is obviously larger than that of MSSM beams. That is to say, the affection of non-Kolmogorov turbulence to MSSM beams compared to the conventional GSM beam is significantly less. The results presented in this paper might be useful in free-space optical communications.

Fig. 7 Normalized M ²-factor of the MSSM beams for several values of parameters N and m on propagation in non-Kolmogorov turbulence compared with the case of GSM beam.

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4. Conclusion

We have derived the analytical expressions of the rms angular width and the M ²-factor of MSSM beams in non-Kolmogorov turbulence by utilizing the extended Huygens-Fresnel principle and second-order moments of the WDF. The effect of beams parameters and non-Kolmogorov turbulence on the normalized rms angular width and the normalized M ²-factor have been investigated in detail. Numerical results have been shown that a MSSM beam with dark-hollow far-fields in free space has advantage over the one with flat-topped or multi-rings far-fields for reducing the turbulence-induced degradation. Such an advantage will become more obvious with larger dark-hollow size. Beam quality of MSSM beams can be further improved with longer wavelength and larger beam width. The effect of non-Kolmogorov turbulence on the normalized M ²-factor of MSSM beams has also been studied. The MSSM beams will be less affected under the condition of weaker turbulence, such as larger power-law exponent α and outer scale L ₀ or smaller index-of-refraction structure constant ${\tilde{C}}_{n}^{2}$ and inner scale l ₀. In addition, compared with the conventional GSM beam, the effect of non-Kolmogorov turbulence on MSSM beams is significantly less. Our results will have potential application in free-space optical communications.

Acknowledgments

This work was supported by the National Basic Research Program of China under Grant No. 2013CBA01702, and the National Natural Science Foundation of China under Grant NOs. 11104049, 10974039, 61377016, 61308017, 61575053 and 61575055, and the Fundamental Research Funds for the Central Universities (No. HIT.BRETIII.201406) and the Program for New Century Excellent Talents in University (NCET-12-0148).

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Propagation factors of multi-sinc Schell-model beams in non-Kolmogorov turbulence

Abstract

1. Introduction

2. Second-order moments of MSSM beams in non-Kolmogorov turbulence

3. Numerical calculation results and analysis

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Equations (23)

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