## Abstract

We present a detailed investigation of the second-order statistics of a twisted Gaussian Schell-model (TGSM) beam propagating in turbulent atmosphere. Based on the extended Huygens-Fresnel integral, analytical expressions for the second-order moments of the Wigner distribution function of a TGSM beam in turbulent atmosphere are derived. Evolution properties of the second-order statistics, such as the propagation factor, the effective radius of curvature (ERC) and the Rayleigh range, of a TGSM beam in turbulent atmosphere are explored in detail. Our results show that a TGSM beam is less affected by the turbulence than a GSM beam without twist phase. In turbulent atmosphere the Rayleigh range doesn’t equal to the distance where the ERC takes a minimum value, which is much different from the result in free space. The second-order statistics are closely determined by the parameters of the turbulent atmosphere and the initial beam parameters. Our results will be useful in long-distance free-space optical communications.

©2010 Optical Society of America

## 1. Introduction

In the past decades, partially coherent beams have been widely investigated and applied in free space optical communication, optical imaging, nonlinear optics, optical trapping, inertial confinement fusion, optical projection and laser scanning [1–6]. Gaussian Schell-model (GSM) beam is a typical and commonly encountered partially coherent beam, whose spectral density and spectral degree of coherence have Gaussian shapes [1,7]. By scattering a coherent laser beam from a rotating grounded glass, then transforming the spectral density distribution of the scattered light into Gaussian profile with a Gaussian amplitude filter, a GSM beam can be generated [8]. GSM beams can also be generated with specially synthesized rough surfaces, spatial light modulators and synthetic acousto-optic holograms (c.f [9].). Propagation properties of a GSM beam have been studied widely [1,10–13]. It has been found that a GSM beam is less affected by the turbulent atmosphere compared to a coherent Gaussian beam, thus have important applications in free space optical communication, remote sensing and radar system [11–13].

A more general partially coherent beam can possess a twist phase, which differs in many respects from the customary quadratic phase factor. In 1993, Simon and Mukunda first introduced the twisted Gaussian Schell-model (TGSM) beam [14]. Unlike the usual phase curvature, the twist phase is bounded in strength due to the fact that the cross-spectral density function must be nonnegative and it is absent in a coherent Gaussian beam. The twist phase has an intrinsic chiral property and is responsible for the rotation of the beam spot on propagation. Friberg et al. first carried out experimental demonstration of TGSM beams [15]. Superposition, coherent-mode decomposition and the analysis of the transfer of radiance of the TGSM beam have been investigated in [16,17]. Dependence of the orbital angular momentum of a partially coherent beam on its twist phase was revealed in Ref [18]. The conventional method for treating the propagation of TGSM beams is the Wigner-distribution function [14]. Lin and Cai have introduced a convenient alternative tensor method for treating the propagation of TGSM beams [19]. With the help of the tensor method, the propagation properties of a TGSM beam through paraxial ABCD optical system, dispersive media and nonlinear media were studied in [20–23]. More recently, Ghost imaging with a TGSM beam was explored in [24]. Zhao et al. studied the radiation force of a TGSM beam on a Rayleigh particle [25]. Twist phase-induced polarization changes in electromagnetic GSM beam were studied in [26].

Investigations of the propagation properties of laser beams in a turbulent atmosphere become more and more important because of their wide applications in e.g. free-space optical communications and remote sensing [2,3,11–13,27–29]. Average intensity and spreading properties of a TGSM beam have been studied in [29]. Recently, more and more attention is being paid to the second-order statistics, such as the propagation factor, the effective radius of curvature (ERC) and the Rayleigh range, of laser beams in turbulent atmosphere [30–34]. To our knowledge no results have been reported up until now on the second-order statistics of a TGSM beam in turbulent atmosphere. The purpose of this paper is to investigate the propagation factor, the ERC and the Rayleigh range of a TGSM beam in turbulent atmosphere, and to explore the advantage of a TGSM beam over a GSM beam for overcoming or reducing the turbulence-induced degradation. Analytical expressions are derived for the second-order moments of the Wigner distribution function of a TGSM beam in turbulent atmosphere, and some useful and interesting results are found.

## 2. Second-order moments of the Wigner distribution function of a TGSM beam in turbulent atmosphere

A partially coherent beam is generally characterized by the cross-spectral density (CSD) function, and the CSD function of a TGSM beam in the source plane (*z* = 0) is expressed as [14]

*λ*being the wavelength of light field. ${\sigma}_{I0}\text{and}{\sigma}_{g0}$ denote the transverse beam width and spectral coherence width, respectively. ${\mu}_{0}$ is a scalar real-valued twist factor with the dimension of an inverse distance, limited by the double inequality $0\le {\mu}_{0}^{2}\le {\left[{k}^{2}{\sigma}_{g0}^{4}\right]}^{-1}$due to the non-negativity requirement of Eq. (1). In the coherent limit, ${\sigma}_{g0}\to \infty $, the twist factor ${\mu}_{0}$ disappears. In Eq. (1), the symbol

**J**denotes an anti-symmetric matrix given by [14]

Under the condition of${\mu}_{0}=0$, the CSD function in Eq. (1) reduces to the CSD function of a conventional GSM beam without twist phase [7–9]. Due to the existence of the term ${\left({r}_{1}^{\text{'}}-{r}_{2}^{\text{'}}\right)}^{T}J\left({r}_{1}^{\text{'}}+{r}_{2}^{\text{'}}\right)={x}_{1}^{\text{'}}{y}_{2}^{\text{'}}-{x}_{2}^{\text{'}}{y}_{1}^{\text{'}}$ in the right side of Eq. (1), the two-dimensional CSD function cannot be split in a product of two one-dimensional CSD functions.

Within the validity of the paraxial approximation, based on the extended Huygens-Fresnel integral, the CSD function of a partially coherent beam propagating in turbulent atmosphere at *z* is expressed as [11,30–34]

In the derivation of Eq. (3), we have used following sum and difference notations

The term $\mathrm{exp}\left[-H({r}_{d},{r}_{d}^{\text{'}};z)\right]$ in Eq. (3) is the contributions from atmospheric turbulence, and can be written as [11]:

*κ*is the magnitude of the spatial wave-number.

The Wigner distribution function (WDF) of a partially coherent beam on propagation in turbulent atmosphere can be expressed in terms of the CSD function by the formula [30,31]

*z*-direction; $k{\theta}_{x}$and $k{\theta}_{y}$ are the wave vector components along the

*x-*axis and

*y*-axis, respectively.

Substituting Eq. (3) into Eq. (7), we obtain (after some operation) following expression for the WDF of a partially coherent in turbulent atmosphere

According to Ref [30], the moments of order ${n}_{1}+{n}_{2}+{m}_{1}+{m}_{2}$ of the WDF of a laser beam is given by

The second-order statistics of a laser beam, such as the propagation factor, the ERC and the Rayleigh range, are closely related with the second-order moments of the WDF. Substituting Eq. (11) into Eq. (12), we obtain (after tedious integration and operation) following expressions for the second-order moments of the WDF of a TGSM beam propagating in turbulent atmosphere

In Eqs. (14) and (16), the symbols $\u3008r{(z)}^{2}\u3009$ and $\u3008\theta {(z)}^{2}\u3009$ represent the squared beam width and the squared far-field divergence of the TGSM beam in turbulent atmosphere, respectively. The ERC of the TGSM beam is closely determined by $\u3008r(z)\cdot \theta (z)\u3009$ in Eq. (15).

## 3. Propagation factor of a TGSM beam in turbulent atmosphere

The propagation factor (best known as ${M}^{2}$ -factor) proposed by Siegman is a particularly important property of an optical laser beam [1] being regarded as a beam quality factor in many practical applications. Based on the second-order moments of the Wigner distribution function, the ${M}^{2}$-factor of a partially coherent beam is defined as [30–32]

*M*

^{2}-factor of the TGSM beam in free space or in the source plane given by

Under the condition of $T=0$ (without turbulence), Eq. (24) reduces to the expression for the *M*
^{2}-factor of a TGSM beam in free space. Under the condition of ${\mu}_{0}=0$, Eq. (24) reduces to the expression for the *M*
^{2}-factor of a GSM beam without twist phase in turbulent atmosphere. From Eq. (25), it is clear that the *M*
^{2}-factor of a TGSM beam in free space is independent of the propagation distance, and increases with the increase of the absolute value of the twist factor. This phenomenon is caused by the fact that the twist factor cause more rapid spreading of a TGSM beam on propagation.

Now we study the evolution properties of the *M*
^{2}-factor of a TGSM beam in turbulent atmosphere. In the following numerical examples, we adopt the Tatarskii spectrum for the spectral density of the index-of-refraction fluctuations, which is expressed as [11]

Substituting Eq. (27) into Eq. (24), we can calculate the${M}^{2}$-factor of a TGSM beam in turbulent atmosphere numerically.

For the convenience of comparison, we now study the normalized *M*
^{2}-factor of a TGSM beam defined as${M}^{2}(z)/{M}^{2}(0)$on propagation in turbulent atmosphere. Figure 1
shows the normalized *M*
^{2}-factor of a TGSM beam on propagation in turbulent atmosphere for different values of the structure constant${C}_{n}^{2}$. As illustrated by Fig. 1, the normalized *M*
^{2}-factor of a TGSM beam in turbulent atmosphere increases on propagation, which is much different from its propagation-invariant properties in free space (${C}_{n}^{2}=0$). As the value of the structure constant${C}_{n}^{2}$ increases (i.e., turbulence becomes strong) or the value of the inner scale ${l}_{0}$ decreases, the normalized *M*
^{2}-factor increases more rapid on propagation. Figure 2
shows the normalized *M*
^{2}-factor of a TGSM beam on propagation in turbulent atmosphere for different values of ${\sigma}_{g0}^{}$ and ${\mu}_{0}$with ${l}_{0}=0.01\text{m}$. One finds from Fig. 2(a) that the normalized *M*
^{2}-factor of a TGSM beam increases slower on propagation as its initial coherence width ${\sigma}_{g0}^{}$ decreases, which means that a TGSM beam with lower coherence is less affected by turbulent atmosphere as expected [30–32]. One finds from Fig. 2(b) that the normalized *M*
^{2}-factor of a TGSM beam increases slower than that of a GSM beam without twist phase (${\mu}_{0}=0$) on propagation in turbulent atmosphere, which means that a TGSM beam is less affected by atmospheric turbulence than a GSM beam. Furthermore, as shown by Fig. 2(b), the TGSM beam with larger absolute value of ${\mu}_{0}$is less affected by the turbulence than that with smaller absolute value of ${\mu}_{0}$.

In order to show the advantage of a TGSM beam over a GSM beam in turbulent atmosphere quantitatively, we introduce a parameter $\Delta {M}^{2}(z)$ named the deviation percentage of the normalized *M*
^{2}-factor to show the difference between the normalized *M*
^{2}-factor of a TGSM beam and that of a GSM beam. The deviation percentage of the normalized *M*
^{2}-factor is defined as

The advantage of a TGSM beam over a GSM bam increases with the increase of the deviation percentage$\Delta {M}^{2}(z)$.

We calculate in Fig. 3
the deviation percentage of the normalized *M*
^{2}-factor versus the propagation distance z for different values of ${\mu}_{0}$ with${\sigma}_{I0}=10mm$, ${\sigma}_{g0}=10mm$, ${l}_{0}=0.01\text{m}$ and ${C}_{n}^{2}={10}^{-14}{\text{m}}^{-\text{2/3}}$. As shown in Fig. 3, the parameter $\Delta {M}^{2}(z)$ increases on propagation, and it approaches to a constant value in the far field. The constant value increases as the absolute value of ${\mu}_{0}$ increases. For the case of $\left|{\mu}_{0}\right|=1.5{\text{km}}^{-1}$, the parameter$\Delta {M}^{2}(z)$approaches to 5%, which is quite significant. In practical experiment, we can convert a GSM beam into a TGSM beam with a six-element astigmatic lens system as shown in [15], and control the twist phase by controlling the astigmatic lens. A GSM beam can be generated with the help of a rotating grounded glass and a Gaussian amplitude filter conveniently [8]. Thus it is economic and realizable to generate a TGSM beam for application in free-space optical communications.

## 4. Effective radius of curvature of a TGSM beam in turbulent atmosphere

According to [33,34], the ERC of a laser beam at *z* is defined in terms of the ratio of $\u3008r{(z)}^{2}\u3009$ to $\u3008r(z)\cdot \theta (z)\u3009$ as follows

Substituting Eqs. (14) and (15) into Eq. (29), we obtain following expression for the ERC of a TGSM beam in turbulent atmosphere

From Eq. (30), one finds that the ERC of a TGSM beam on propagation are determined by the beam parameters (i.e., beam width ${\sigma}_{I0}$, the coherence width ${\sigma}_{g0}$, the wavelength*λ*and the twisted factor ${\mu}_{0}$) and the parameters of the turbulent atmosphere (i.e., structure constant ${C}_{n}^{2}$ and the inner scale ${l}_{0}$) together. Under the condition of $T=0$ and ${\mu}_{0}=0$, Eq. (30) reduces to the ERC of a GSM beam in free space. Equation (30) provides a convenient way for studying the evolution properties of a GSM beam with or without twist phase in turbulent atmosphere.

We calculate in Fig. 4 the ERC of a TGSM beam in turbulent atmosphere versus the propagation distance for different values of the structure constant${C}_{n}^{2}$ and the inner scale ${l}_{0}$with ${\sigma}_{I0}=10mm$ and ${\sigma}_{g0}=10mm$. One finds from Fig. 4 that the ERC of a TGSM beam on propagation in free space (${C}_{n}^{2}=0$) or in turbulent atmosphere will initially display a downward trend in the near field, but after reaching a dip, will star to increase. The value of the ERC on propagation decreases as the structure constant${C}_{n}^{2}$ increases or the inner scale ${l}_{0}$ decreases especially in the far field.

Figure 5 shows the ERC of a TGSM beam in turbulent atmosphere versus the propagation distance for different values of ${\mu}_{0}$ and ${\sigma}_{g0}^{}$ with ${\sigma}_{I0}=10mm$. One finds from Fig. 5 (a) that the difference between the ERC of a TGSM beam in free space and that in turbulence is smaller than the difference between the ERC of a GSM beam in free space and that in turbulent atmosphere, which means a TGSM beam is less affected by the turbulent atmosphere than a GSM beam from the aspect of ERC. From Fig. 5 (b), it is also clear that the TGSM beam with lower coherence is less affected by the turbulence than that with higher coherence. To show the advantage of a TGSM beam over a GSM beam quantitatively, we introduce a parameter $\Delta R(z)$ named the deviation percentage of the ERC to show the difference between the ERC of a TGSM or GSM beam in turbulent atmosphere and that of a TGSM or GSM beam in free space. The deviation percentage of the ERC is defined as

The advantage of a TGSM beam over a GSM beam increases with the increase of the deviation between the$\Delta R(z)$of a TGSM beam and that of a GSM beam. We calculate in Fig. 6
the deviation percentage of the ERC of a TGSM or GSM beam versus the propagation distance *z* with${\sigma}_{I0}=10mm$, ${\sigma}_{g0}=10mm$, ${l}_{0}=0.01\text{m}$. One finds from Fig. 6 that the deviation between the$\Delta R(z)$of a TGSM beam with $\left|{\mu}_{0}\right|=1.5k{m}^{-1}$ and that of a GSM beam increases on propagation, and it approaches to a constant value (about 5%) in far field. This result agrees well with the result shown in Fig. 3. One also finds from Fig. 5 that evolution of the ERC of a TGSM beam is little different from that in free space. In free space, the value of the ERC of a TGSM beam with larger absolute value of${\mu}_{0}$ (or smaller ${\sigma}_{g0}^{}$) on propagation is always smaller than that with smaller absolute value of${\mu}_{0}$ (or larger ${\sigma}_{g0}^{}$) in the near field or in the intermediate propagation distance. With the increase of propagation distance, the difference between the ERC of TGSM beams with different ${\mu}_{0}$ or ${\sigma}_{g0}^{}$ becomes smaller, and in the far field, the ERC tends to $R(z)\to z$. In turbulent atmosphere, there exists a critical propagation length ${z}_{c}$ where the TGSM beams with different ${\mu}_{0}$ or ${\sigma}_{g0}^{}$ have the same value of ERC. For the case of $z<{z}_{c}$, the value of the ERC of the TGSM beam with larger absolute value of ${\mu}_{0}$ (or smaller ${\sigma}_{g0}^{}$) is smaller that that with smaller absolute value of ${\mu}_{0}$ (or larger ${\sigma}_{g0}^{}$). For the case of $z>{z}_{c}$, the reverse situation occurs. From Eq. (30), we obtain following expression for the critical propagation length

## 5. Rayleigh range of a TGSM beam in turbulent atmosphere

The Rayleigh range is an important beam parameter for characterizing the distance within which the laser beam can be considered effectively non-spreading. The Rayleigh range is defined as the distance ${z}_{R}$ along the propagation direction of a beam from the beam waist to the place where the area of the cross section is doubled (i.e., the diameter of the spot size increases by a factor $\sqrt{2}$ compared to the spot size at the beam waist) [36]. The range of the minimum effective radius of curvature is defined as the distance ${z}_{m}$along the propagation direction of a beam from the beam waist to the place where the ERC of the beam takes the minimum vale. In free space, the Rayleigh range ${z}_{R}$ equals to the range of the minimum effective radius of curvature ${z}_{m}$. What will happen in turbulent atmosphere? Now let’s study the properties of the Rayleigh range ${z}_{R}$ and the range of the minimum effective radius of curvature ${z}_{m}$ in turbulent atmosphere. Based on the definition of ${z}_{R}$ and ${z}_{m}$ [36], they can be obtained by solving following equations

Substituting Eqs. (14) and (30) into Eqs. (33) and (34) respectively, we obtainUnder the condition of *T* = 0 (free space), Eqs. (35) and (36) reduce to the same quadratic equation. After some calculation, we obtain following analytical expression for${z}_{R}$ and ${z}_{m}$ of a TGSM beam in free space

We calculate in Fig. 7 ${z}_{R}$ and ${z}_{m}$ of a TGSM beam in turbulent atmosphere different values of twist factor ${\mu}_{0}$and coherence width ${\sigma}_{g0}$with ${\sigma}_{I0}=10mm$and ${C}_{n}^{2}={10}^{-14}{\text{m}}^{\text{-2/3}}$. For the convenience of comparison, the corresponding results in free space are also shown. As shown in Fig. 7, ${z}_{R}$ and ${z}_{m}$ don’t coincide with each other due to the influence of turbulence. ${z}_{m}$ in turbulent atmosphere is always larger than that in free space, and ${z}_{R}$ in turbulent atmosphere is always smaller than that in free space. As the absolute value of twist factor ${\mu}_{0}$increases or the coherence width ${\sigma}_{g0}$decreases, the difference between ${z}_{R}$ and ${z}_{m}$ becomes smaller, which means that a TGSM beam with larger absolute value twist factor or lower coherence is less affected by the turbulence.

## 6. Conclusion

In conclusion, we have derived the analytical expressions for the second-order moments of the WDF of a TGSM beam in turbulent atmosphere based on the extended Huygens-Fresnel integral. The second-order statistics, such as the propagation factor, the ERC and the Rayleigh range, of a TGSM beam propagating in turbulent atmosphere have been studied and compared with the results in free space. Our numerical results show that a TGSM beam is less affected by the turbulence than a GSM beam, and a TGSM beam with larger absolute value of twist factor or lower coherence is less affected by the turbulence than that with smaller twist factor or higher coherence. Our results will be useful in long-distance free-space optical communications.

## Acknowledgments

Yangjian Cai acknowledges the support by the National Natural Science Foundation of China under Grant No. 10904102, the Foundation for the Author of National Excellent Doctoral Dissertation of PR China under Grant No. 200928, the Natural Science of Jiangsu Province under Grant No. BK2009114, the Huo Ying Dong Education Foundation of China under Grant No. 121009 and the Key Project of Chinese Ministry of Education under Grant No. 210081.

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