## Abstract

Recently, we introduced a new class of radially polarized cosine-Gaussian correlated Schell-model (CGCSM) beams of rectangular symmetry based on the partially coherent electromagnetic theory [Opt. Express **23**, 33099 (2015)]. In this paper, we extend the work to study the second-order statistics such as the average intensity, the spectral degree of coherence, the spectral degree of polarization and the state of polarization in anisotropic turbulence based on an extended von Karman power spectrum with a non-Kolmogorov power law $\alpha $ and an effective anisotropic parameter. Analytical formulas for the cross-spectral density matrix elements of a radially polarized CGCSM beam in anisotropic turbulence are derived. It is found that the second-order statistics are greatly affected by the source correlation function, and the change in the turbulent statistics induces relatively small effect. The significant effect of anisotropic turbulence on the beam parameters mainly appears near$\alpha =3.1$, and decreases with the increase of the anisotropic parameter. Furthermore, the polarization state exhibits self-splitting property and each beamlet evolves into a radially polarized structure in the far field. Our work enriches the classical coherence theory and may be important for free-space optical communications.

© 2016 Optical Society of America

## 1. Introduction

For many years, the interaction of light beams with turbulent atmosphere has been attracted much attention due to their important applications in astronomical imaging, free-space optical (FSO) communications, laser radar and remote sensing [1, 2]. The random fluctuations in the index of refraction of atmosphere cause spreading of the beam beyond that due to pure diffraction, beam wander, loss of spatial coherence, and random fluctuations in the irradiance and phase. These effects can seriously degrade the signal-to-noise ratio (SNR) of an optical heterodyne receiver. Therefore, much effort has been devoted to a reliable theory for predicting the propagation properties of light beams in turbulent medium, and a great many achievements have been derived in the past decades [3–19]. It is demonstrated that the atmosphere exhibits homogeneous and nearly isotropic under the atmospheric boundary layer (ABL), which is roughly 1-2km above the Earth’s surface. Therefore, the isotropic Kolmogorov power spectrum model of the refractive index is generally correct within this inertial sub-range. The portion of the atmosphere above the ABL belongs to the free atmosphere (FA) in which the influence of the Earth’s surface friction on the air motion is negligible. However, theoretical and experimental results have shown that the FA can be highly anisotropic at large scales, and the Kolmogorov power spectrum density model does not properly describe the real turbulence behavior. Consequently, a variety of different power spectrum models and extended anisotropic turbulence models have been proposed in the past decades [20–24].

The effects of spatial correlation function of partially coherent beams on the statistical properties in terms of the average intensity, the spectral degree of coherence, the spectral degree of polarization, the orbital angular moment (OAM) and the state of polarization (SOP) have been attracted increasing attention [25–33]. Recently, a new sufficient condition for generation genuine correlation function of light beams was introduced based on the constraint of non-negative definiteness [34, 35]. A variety of new correlation function models have been explored both in theory and in experiment. Partially coherent beams with these special correlation functions exhibit many novel properties, such as self-accelerating, self-focusing and tunable beam profile in the far field such as dark hollow (DH) profile, flat-topped profile and frame profile [36–48]. Among these beams, a class of partially coherent temporal/spatial sources, optical coherence gratings/lattices have found Gaussian intensity profiles and lattice-like spectral degrees of coherence, and can be found applications in metrology, material processing, and optical communications [45, 46]. Moreover, as a particular case of optical coherence lattices, cosine-Gaussian correlated Schell-model (CGCSM) beam of circular [38, 39] or rectangular symmetry [42] exhibits ring-shaped or four-beamlets array beam profile in the far field, and may be useful in particle trapping, optical scattering and optical imaging. It is well known that partially coherent beams with conventional correlation functions (i.e., Gaussian correlated Schell-model functions) are less affected by turbulent atmosphere than fully coherent beams in terms of beam quality, divergence and scintillations. However, propagation properties of partially coherent beams with non-conventional correlation functions in turbulent atmosphere are not clear and need for further study [39, 49–52].

Cylindrical vector (CV) beams are solutions of Maxwell equations with cylindrical symmetry both in amplitude and polarization. Typical CV beams such as radially polarization beams exhibit unique focusing properties, which are useful in high-resolution microscopy, plasmonic focusing, and nanoparticle manipulation and high-density storage [53, 54]. Partially coherent radially polarized beams were introduced in theory as an extension of coherent radially polarized beam and generated in experiment [55–57]. It has been found that partially coherent radially polarized beams have advantages over linearly polarized partially coherent beams for reducing turbulence-induced degradation [16, 58–60]. Recently, we introduced a new class of radially polarized CGCSM beams in theory and generated in experiment [48]. Based on the novel vector and coherence properties of a radially polarized CGCSM beam, in this paper, the second-order statistics are studied numerically by using a non-Kolmogorov power spectrum with an anisotropic parameter. Under weak turbulence conditions, analytical formula of $2\times 2$cross-spectral density (CSD) matrix of a radially polarized CGCSM beam propagation through anisotropic turbulence is derived. The effects of the anisotropic parameter, the power law as well as the source correlation function on the average intensity, the spectral degree of coherence and the polarization properties are studied in detail.

## 2. Cross-spectral density matrix of a radially polarized CGCSM beam in anisotropic non-Kolmogorov turbulence

The vector electric field of a radially polarized beam can be characterized as the coherent superposition of TEM01 Laguerre–Gaussian modes with a polarization direction parallel to the *x-*axis and a TEM_{10} with a polarization direction parallel to the *y-*axis [16, 48, 53, 55–57],

Next, we investigate the propagation of a radially polarized CGCSM beam in anisotropic non-Kolmogorov turbulence. It is shown that the extended Huygens-Fresnel principle is a simple and effective method for studying the propagation of a light beam in both weak and strong turbulent atmosphere [2]. Within the paraxial approximation, the elements of the CSD at the receiver plane can be expressed as [1–4]

*z*-axis), they are still isotropic over the planes orthogonal to the

*z*-axis. Based on this assumption, the Kolmogorov theory remains valid because the energy cascade process over the plane orthogonal to the propagation direction is still isotropic, although there is rescaling due to anisotropy for each cell size. For further study, let us consider the Markov approximation, which is commonly used in the theory of wave propagation in random media [2, 20–24]. It is assumed that the refractive index can be expressed aswhere ${n}_{0}=\u3008n(R)\u3009\cong 1,\u3008{n}_{1}(R)\u3009=0.$The covariance function—delta correlated in the direction of propagation along the positive

*z*-axis can be expressed as

*z*-axis is ignored.

In considering the vertical profile, we replace ${\tilde{C}}_{n}^{2}$ with the average over the path value ${\overline{\tilde{C}}}_{n}^{2}$, which is defined by

## 3. Numerical examples of a radially polarized CGCSM beam in anisotropic non-Kolmogorov turbulence

In this section, we give out some numerical results for studying the second-order statistics of a radially polarized CGCSM beam in anisotropic non-Kolmogorov turbulence. The parameters are set as: ${w}_{0}\text{=4}cm\text{,}{\sigma}_{0}\text{=2}cm\text{,}\lambda \text{=632}\text{.8}nm\text{,}{\tilde{C}}_{n}^{2}=1.7\times {10}^{-14}{m}^{3-\alpha}\text{,}{v}_{s}=21m/s,{l}_{0}=1mm,{L}_{0}=10m,$ ${h}_{0}=0,H=30km,n=1,\alpha =3.5,{\zeta}_{eff}=3$(unless stated otherwise). On submitting from Eq. (12) into Eq. (11), one finds the average refractive index structure constant${\overline{\tilde{C}}}_{n}^{2}=7.45\times {10}^{-17}{m}^{3-\alpha}$. Figure 1 shows the normalized average intensity distribution and the corresponding cross line of a radially polarized CGCSM beam at several different propagation distances for different values of$z=5km$. For the case of$n=0$, Figs. 1(a1)-1(a4) show that a doughnut shaped radially polarized SM beam gradually evolves into a Gaussian shape as expected [16]. For the case of$n\ne 0$, a circularly symmetric radially polarized CGCSM beam gradually decomposes into a four-beamlets array distribution in the far field, and the distance between each beamlet becomes larger as the beam order$z=5km$ increases. Figure 2 plots the normalized average intensity distribution, composition components ${U}_{xx}(\rho ,\rho ),{U}_{yy}(\rho ,\rho )$ and the corresponding cross lines of a radially polarized CGCSM beam at several different propagation distances with$n=2$. One finds that the source radially polarized beam spot decomposes into four radially polarized beamlets in the far field due to the source correlation function, which significantly differs from those conventional radially polarized SM beams. Moreover, Figs. 1 and 2 demonstrate that the behavior of partially coherent beams in turbulent atmosphere is greatly affected by source correlation function, even at large propagation distances. The physical interpretation is that the disturbance of turbulence is homogeneous, which causes a reduction of spatialcoherence of light beam. Therefore, the average intensity distribution is mainly affected by the source correlation function, and the turbulence leads to a spatial broadening of the beam spot. Our results is consistent with the conclusion derived from conventional SM beams propagation in isotropic turbulence [6–11]. Figure 3 illustrates the evolution of the on-axis average intensity for different values of$z=5km$. As is seen from Figs. 3(a1)-3(a3), the on-axis average intensity generally increases first causing by the source correlation, and then decreases due to the diffraction as well as the turbulence-induced spatial broadening of the light beam. Furthermore, the distance corresponding to the peak value gradually shorten with the growth of value$z=5km$. Figures 3(b1)-3(b3) illustrate that the significant impact of the effective anisotropic parameter${\zeta}_{eff}$ on the on-axis average intensity mainly appears near$\alpha =3.1$. However, with the increase of${\zeta}_{eff}$, the average intensity gradually grows for$n=0$, while it deceases for$n\ne 0$. This implies that the behavior of radially polarized CGCSM beams is more sensitive to the source correlations, rather than the turbulence parameters. Similar results have been found both in isotropic turbulence and other class of beams [24].

Figure 4 shows the modulus of the spectral DOC and the corresponding cross line of a radially polarized CGCSM beam at different propagation distances for different values of$z=5km$. One finds that the spectral DOC of a conventional radially polarized SM beam [see Figs. 4(a1)-4(a4)] is a typical circular symmetry distribution as expected [16]. For the case of$n\ne 0$, the spectral DOC exhibits array distribution with rectangular symmetry, and both the shape and the symmetry change during transmission. Due to the influence of turbulence, the spectral DOC gradually evolves into a Gaussian shape after propagating a sufficient distance, which quite differs from its free space behavior [48]. In addition, because of the diffraction, the coherence width significantly increased. Figure 5 illustrates the modulus of the spectral DOC $\mu \left(0cm,0cm,5cm,0cm\right)$ of a radially polarized CGCSM beam for different values of$z=5km$. As is shown in Figs. 5(a1)-5(a3), ${\zeta}_{eff}$ plays a crucial role in determining the correlation property, and the spectral DOC generally increases as the growth of${\zeta}_{eff}$. However, for a high value of $z=5km$ such as$n=2$ [see Fig. 5(a3)], the effect of ${\zeta}_{eff}$on the spectral DOC is weakened. Unlike Figs. 3(b1)-3(b3), and Fig. 5(b1)-5(b3) demonstrate that the spectral DOCs for different values of$z=5km$ are similar and the effect of ${\zeta}_{eff}$ on the spectral DOC is gradually reduced as the value$z=5km$ increases. As a result, for lower effective anisotropic parameter value${\zeta}_{eff}$, the effects induced by source correlation on the spectral DOC become dominant. Similar results have been reported for conventional electromagnetic SM beams in anisotropic turbulence [24].

Figure 6 shows the spectral DOP and the corresponding cross line of a radially polarized CGCSM beam at different propagation distances for different values of$z=5km$. It is clearly seen that only the on-axis spectral DOP remains invariant during propagation, actually equals to zero, which implies that the on-axis polarization of a radially polarized CGCSM beam is not destroyed on propagation even in anisotropic turbulence. However, it has been predicted that the spectral DOP of a coherent radially polarized beam is always equal to one in free space for entire transverse light field [16]. This implies that the behavior of polarization structure of a radially polarized CGCSM beam determined mostly by the source correlation function, rather than by the turbulence parameters. Furthermore, the distribution of the spectral DOP of a radially polarized CGCSM beam in turbulent atmosphere is quite different from those of conventional radially polarized SM beams and is closely determined by the beam order$z=5km$. It is clearly seen that the distributions of spectral DOP are greatly affected by the CGCSM correlation functions, even at large propagation distances. This behavior is in agreement with the result derived in detail previously for conventional SM beams propagation in isotropic or anisotropic turbulence [7–10]. Figure 7 shows the spectral DOP of a radially polarized CGCSM beam for different values of$z=5km$ at (5*cm*, 0*cm*). An obvious difference is found in Figs. 7(a1)-7(a3) that the spectral DOP generally decreases for a conventional radially polarized SM beam, while it reduces first and then grows to a certain value and finally deceases for a radially polarized CGCSM beam due to the modulation of the source cosine correlation function. Similar to the on-axis average intensity, a significant effect of ${\zeta}_{eff}$ on the spectral DOP mainly appears near$\alpha =3.1$, and decreases with the increase of${\zeta}_{eff}$.

Finally, to learn more about the polarization property, Fig. 8 shows the SOP of a radially polarized CGCSM beam at different propagation distances for different values of$z=5km$. It is seen that the SOP of a radially polarized SM beam remains invariant in the far field. However, the SOP of a radially polarized CGCSM beam is closely determined by the beam order$z=5km$, and the radial polarization structure is destroyed in the far field due to the non-conventional source correlation function. This means that the SOP is greatly affected by the source correlation function, similarly to its free space behavior, and the change in the turbulent statistics induces relatively small effect. In addition, although the overall SOP of a radially polarized CGCSM beam is broken, the SOP of each beamlet evolves into a radial polarization structure in the far field. Here, we obtain an important conclusion that we can generate various class of vector beams in the far field with designated polarization structures such as polarization array structure by modulating the source correlation function. Our results may be found potential applications in optical trapping, high-resolution microscopy, FSO communications, and remote polarization sensing.

## 4. Conclusion

In this work, we investigated the statistics of a radially polarized CGCSM beam in anisotropic turbulence based on an extended von Karman power spectrum with a non-Kolmogorov power law$\alpha $ and an effective anisotropic parameter. With the help of Markov approximation, effective anisotropic parameter describes anisotropy along the vertical direction. Analytical formulas for the cross-spectral density matrix elements of a radially polarized CGCSM beam in turbulent atmosphere are derived. One finds that the average intensity, the spectral DOC, the spectral DOP and the SOP of a radially polarized CGCSM beam evolve into four-beamlets array. The second-order statistics are determined mostly by the source correlation function, and the change in the turbulent statistics induces relatively small effect. The significant effect of anisotropic parameter on the beam parameters mainly appears near$\alpha =3.1$, and decreases with the growth of anisotropic parameter. Furthermore, the initial radial polarization structure splits together with the source beam spot, and the SOP of each beamlet evolves into a radial polarization structure in the far field. This provides us with a new method to generate various vector beams such as tunable radially polarized beams, azimuthally polarized beams as well as cylindrical vector beam array. Our work is helpful in understanding the correlation modulation theory, and may be useful in optical trapping, high-resolution microscopy, polarization communications and remote polarization sensing.

## Acknowledgments

The author’s research is supported by the National Natural Science Foundation of China under Grant Nos. 11504172 and 11274005, the National Natural Science Fund for Distinguished Young Scholar under Grant No. 11525418, the Natural Science Foundation of Jiangsu Province under Grant No. BK20150763, and the project of the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions.

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