## Abstract

The propagation of a partially coherent hollow vortex Gaussian beam through a paraxial *ABCD* optical system in turbulent atmosphere has been investigated. The analytical expressions for the average intensity and the degree of the polarization of a partially coherent hollow vortex Gaussian beam through a paraxial *ABCD* optical system are derived in turbulent atmosphere, respectively. The average intensity distribution and the degree of the polarization of a partially coherent hollow vortex Gaussian beam in turbulent atmosphere are numerically demonstrated. The influences of the beam parameters, the topological charge, the transverse coherent lengths, and the structure constant of the atmospheric turbulence on the propagation of a partially coherent hollow vortex Gaussian beam in turbulent atmosphere are also examined in detail. This research is beneficial to the practical applications in free-space optical communications and the remote sensing of the dark hollow beams.

©2012 Optical Society of America

## 1. Introduction

Due to the important applications in atom optics, dark hollow beams have received considerable interest in the past decades [1–8]. Different theoretical beam models, e.g. the $TE{M}_{01}^{\ast}$doughnut beam [1], the superposition of off-axis Gaussian beams [9], the higher order Bessel-Gaussian beam [10], the hollow Gaussian beam [11], and the controllable dark hollow beam [12] have been proposed to describe dark hollow laser beams, respectively. In experiment, dark hollow beams can be generated by using different methods. A dark hollow beam can be generated from a multimode fiber, and the dependence of the output beam profile on the incident angle of laser beam has been analyzed [13]. Hollow Gaussian beams can be created by spatial ðltering in the Fourier domain with spatial ðlters that consist of binomial combinations of even-order Hermite polynomials [14]. By coupling a partially coherent beam into a multimode ðber with a suitable incident angle, a high-quality partially coherent dark hollow beam can be experimentally generated [15]. The dark hollow beam can be produced via coherent combination based on adaptive optics [16]. Also, the dark hollow femtosecond pulsed beam has been generated by means of phase-only liquid crystal spatial light modulator [17]. By means of the different beam models, the properties of dark hollow beams propagating in free space and the turbulent atmosphere have been extensively theoretically investigated [18–25]. However, the researches denote that the existing beam models have good propagation stability only in the region close to the source [11, 21]. With the increase of the propagation distance, the dark region decreases until it finally disappears. This defect seriously affects the atom manipulation with dark hollow beams. To overcome the above defect, a hollow vortex Gaussian beam is introduced, whose dark region always exists under arbitrary conditions. If the hollow Gaussian beam goes through a spiral phase plate, it becomes a hollow vortex Gaussian beam. The spiral phase plate can modulate the wave-front phase of the hollow Gaussian beam. As the initial vortex phase is introduced, the hollow vortex Gaussian beam can no longer be regarded as the superposition of a series of Laguerre-Gaussian beams. The propagation of a hollow vortex Gaussian beam in free space indicates that it has the remarkable propagation stability. In practical optical systems, however, laser beams are almost partially coherent [26], which denotes that fully coherent laser sources are only the ideal cases. Therefore, we now consider the partially coherent hollow vortex Gaussian beam. The research in the propagation of laser beams in a turbulent atmosphere is vital to the applications in free-space optical communications and the remote sensing. The propagation of various kinds of laser beams in a turbulent atmosphere has been extensively investigated [27–36]. As to the practical applications of a laser beam in turbulent atmosphere, a series of optics systems is often used to direct or redirect a laser beam to a distant target plane. Accordingly, the analysis of propagation of a laser beam through an *ABCD* optical system in turbulent atmosphere is indispensable. In the remainder of this paper, therefore, the propagation of a partially coherent hollow vortex Gaussian beam through a paraxial *ABCD* optical system in turbulent atmosphere is investigated. Analytical expressions of the average intensity and the degree of the polarization are derived and illustrated by numerical examples.

## 2. Propagation of a partially coherent hollow vortex Gaussian beam through a paraxial *ABCD* optical system in turbulent atmosphere

In the Cartesian coordinate system, the *z*-axis is taken to be the propagation axis. The second order coherence and polarization properties of a partially coherent hollow vortex Gaussian beam in the source plane *z* = 0 is characterized by the following 2 × 2 cross spectral density matrix [37]

*E*

_{j}_{0}is the characteristics amplitude. (

*ρ*

_{01},

*θ*

_{01}) and (

*ρ*

_{02},

*θ*

_{02}) are the radial and the azimuthal coordinates of two points in the source plane, respectively.

*w*

_{0}is the waist size of the Gaussian part.

*n*denotes the beam order, and

*m*represents the topological charge.

*δ*is the transverse coherent length in the

_{jj}*j*-direction. The complex degree of spatial coherence in Eq. (2) is generated by a Schell-model source. The off-diagonal elements of the cross spectral density matrix for a partially coherent hollow vortex Gaussian beam in the source plane ${W}_{xy}^{}({\rho}_{01},{\rho}_{02},{\theta}_{01},{\theta}_{02},0)$and ${W}_{yx}^{}({\rho}_{01},{\rho}_{02},{\theta}_{01},{\theta}_{02},0)$ are set to be zero, which denotes that the two mutually orthogonal components

*E*and

_{x}*E*of the electric vector are uncorrelated at each point of the source [38–40]. This class of sources is a special case, which results in the simple outcome. The average intensity and the degree of the polarization for the partially coherent hollow vortex Gaussian beam propagating through a paraxial

_{y}*ABCD*optical system in turbulent atmosphere are given by [37]

*ρ*,

*θ*) are the radial and the azimuthal coordinates in the observation

*z*-plane, respectively. Tr denotes the trace, and det stands for the determinant. $\overleftrightarrow{W}(\rho ,\rho ,\theta ,\theta ,z)$ is the cross spectral density matrix of the partially coherent hollow vortex Gaussian beam propagating through a paraxial

*ABCD*optical system in turbulent atmosphere.

*A*,

*B*,

*C*, and

*D*are the transfer matrix elements of the paraxial optical system between the source and the observation planes. Apparently,

*W*(

_{xy}*ρ*,

*ρ*,

*θ*,

*θ*,

*z*) and

*W*(

_{yx}*ρ*,

*ρ*,

*θ*,

*θ*,

*z*) are equal to zero. Based on the extended Huygens-Fresnel diffraction integral, the cross spectral density of the partially coherent hollow vortex Gaussian beam propagating through a paraxial

*ABCD*optical system in turbulent atmosphere can be obtained by [41]

*k*= 2

*π*/

*λ*with

*λ*being the optical wavelength.

*ψ*(

*ρ*

_{01},

*ρ*) and

*ψ*(

*ρ*

_{02},

*ρ*) are the solutions to the Rytov method that represents the random part of the complex phase. The angle brackets denote the ensemble average over the medium statistics covering the log-amplitude and phase fluctuations due to the turbulent atmosphere. The asterisk means the complex conjugation. The ensemble average term is given by [42]

*σ*

_{0}is the spherical-wave lateral coherence radius due to the turbulence of the entire optical system and is defined as [42]

*C*

_{n}^{2}is the constant of refraction index structure and describes the turbulence level.

*b*(

*z*) corresponds to the approximate matrix element for a ray propagating backwards through the system.

*L*is the axial distance between the source and the observation planes. Substituting Eqs. (2) and (7) into Eq. (6), the cross spectral density of the partially coherent hollow vortex Gaussian beam in the observation plane reads as

*ξ*,

*α*

_{1},

*α*

_{2}, and

*α*

_{3}being given by

*J*(⋅)is the

_{l}*l*-th order Bessel function of the first kind.

*I*(⋅)is the

_{m}*m*-th order modified Bessel function of the first kind. The cross spectral density of the partially coherent hollow vortex Gaussian beam in the observation plane turns out to be

*p*

_{1}and

*p*

_{2}are given byAs a result, the average intensity of the partially coherent hollow vortex Gaussian beam in the observation plane is given by

## 3. The numerical results and analyses

Now, the average intensity and the degree of the polarization of a partially coherent hollow vortex Gaussian beam in turbulent atmosphere are calculated by using the formulae derived above. For simplicity, no other optical element is placed in turbulent atmosphere, which denotes that the matrix elements are *A* = 1, *B* = *z*, *C* = 0, and *D* = 1. In this case, ${\sigma}_{0}={(0.545{C}_{n}^{2}{k}^{2}z)}^{-3/5}$. *λ* is set to be 632.8nm. Figures 1
and 2
represent the normalized intensity distributions of the partially coherent hollow vortex Gaussian beams with different beam parameters in the reference planes of turbulent atmosphere *z* = 0, 0.5km, 1km, and 1.5 km. In Figs. 1 and 2, *δ _{xx}* = 0.02m,

*δ*= 0.04m,

_{yy}*P*

^{(0)}= 0.5, and

*C*

_{n}^{2}= 10

^{−14}m

^{-2/3}. With the increase of the propagation distance in turbulent atmosphere, the on-axis normalized intensity gradually increases from zero and finally becomes the maximum, which is caused by the isotropic influence of the atmosphere turbulence. When the propagation distance is large enough, the dark region disappears and the intensity distribution of a partially coherent hollow vortex Gaussian beam in turbulent atmosphere tends to a Gaussian-like distribution. With increasing the waist size of the Gaussian part

*w*

_{0}, the on-axis normalized intensity in the fixed reference plane decreases and the propagation distance within which the dark region will preserve increases. The effect of the increase of the beam order

*n*is similar to that of the increase of the waist size of the Gaussian part

*w*

_{0}. If one wants to keep the propagation distance within which the dark region will preserve longer, the increase of the beam order

*n*is inferior to the increase of the waist size of the Gaussian part

*w*

_{0}. The normalized average intensity distributions of a partially coherent hollow vortex Gaussian beam with different topological charge in the reference planes of the turbulent atmosphere are shown in Fig. 3 . The reference planes are same as those in Figs. 1 and 2. Besides the topological charge

*m*, other parameters are kept unvaried. The sign of the topological charge

*m*doesn’t affect the average intensity distribution, which is omitted to save space. The effect of the increase of the absolute value of the topological charge

*m*is slight superior to that of the increase of the waist size of the Gaussian part

*w*

_{0}. Figure 4 represents the normalized average intensity distributions of a partially coherent hollow vortex Gaussian beam with different transverse coherent lengths in the reference planes of the turbulent atmosphere. Except the transverse coherent lengths, the rest of the parameters are kept unvaried. With the increase of the transverse coherent lengths, the propagation distance within which the dark region will preserve also increases. However, the effect of the increase of the transverse coherent lengths is inferior to that of the increase of the beam order

*n*. Figure 5 denotes the normalized average intensity distribution of a partially coherent hollow vortex Gaussian beam in the reference planes of the atmosphere with different turbulent level. Except

*C*

_{n}^{2}, other parameters are unvaried. When propagating in the atmosphere of the weak turbulence, the propagation distance within which the dark region will preserve will be lengthened. The effect of the decrease of the structure constant of the atmospheric turbulence is approximately equivalent to that of the increase of the transverse coherent lengths. Moreover, the effect of the decrease of the structure constant of the atmospheric turbulence is also inferior to that of the increase of the beam order

*n*. If only the degree of the polarization in the source plane is changed, the normalized intensity distributions of the partially coherent hollow vortex Gaussian beams in the fixed reference plane of turbulent atmosphere will not changed, the corresponding figures are omitted to save space.

Figures 6
-10
represent the degree of the polarization of a partially coherent hollow vortex Gaussian beam in the different reference planes of the turbulent atmosphere. Figures 6-10 correspond to Figs. 1-5, respectively. Although the degree of the polarization in the everywhere of the source plane is identical, the degree of the polarization of a partially coherent hollow vortex Gaussian beam propagating in turbulent atmosphere is generally varied. Only under the condition of *δ _{xx}* =

*δ*, the degree of the polarization of a partially coherent hollow vortex Gaussian beam propagating in turbulent atmosphere is kept unvaried and is equal to the degree of the polarization in the source plane. Except the fluctuation in the edge, the degrees of the polarization in the different reference planes have the nearly same variational law. In a general way, the degrees of the polarization will first decrease, then increase, and finally fluctuate with increasing the value of the radial coordinate

_{yy}*ρ*. However, there are also exceptions. When

*δ*= 0.03m and

_{xx}*δ*= 0.04m, the degree of the polarization in the reference plane

_{yy}*z*= 1.5km gradually increases with increasing the value of

*ρ*. This phenomenon may be interpreted as follow. Equations (10) and (11) denote that the smallest one of

*w*

_{0},

*σ*

_{0},

*δ*or

_{xx}*δ*plays the dominant role on the intensity distribution. With the increase of the axial propagation distance

_{yy}*z*, the spherical-wave lateral coherence radius

*σ*

_{0}decreases. When

*z*= 1.5km,

*σ*

_{0}= 0.018m. Therefore,

*w*

_{0}and

*σ*

_{0}both dominate the intensity distribution. Secondly,

*δ*is close to

_{xx}*δ*. The combination of the above two situations results in the strange behavior of the degree of the polarization. When

_{yy}*δ*= 0.02m and

_{xx}*δ*= 0.01m, the degree of the polarization in the different reference planes mainly experiences first increases, then decreases, and finally increases with increasing the value of

_{yy}*ρ*, which is caused by

*δ*being far smaller than

_{yy}*w*

_{0},

*σ*

_{0}, and

*δ*. In this case, the degree of the polarization even reaches to zero, which means that

_{xx}*W*(

_{xx}*ρ*,

*ρ*,

*θ*,

*θ*,

*z*) is equivalent to

*W*(

_{yy}*ρ*,

*ρ*,

*θ*,

*θ*,

*z*). The fluctuation of the degree of the polarization at larger value of the radial coordinate

*ρ*decreases with increasing the propagation distance

*z*. The degree of the polarization is related to the divergence of the beam. When the propagation distance

*z*is small, the beam spot is also small. The local change of the degree of the polarization is relatively drastic in the edge of the small beam spot. The beam spot augments with increasing the propagation distance

*z*. The local change of the degree of the polarization is relatively slow in the edge of the large beam spot.

Upon propagation in the turbulent atmosphere, the dark region of the partially coherent hollow vortex Gaussian beam will diminish until it disappears. Now, we consider the variation of the degree of the polarization in the axis upon propagation, which is shown in Fig. 11
. The degree of the polarization in the axis first fluctuates, then slowly decreases, and finally tends to *P*^{(0)}. The fluctuation of the degree of the polarization in the axis at small *z* values is also caused by the small beam spot in this case.

## 4. Conclusions

The propagation of a partially coherent hollow vortex Gaussian beam through a paraxial *ABCD* optical system in turbulent atmosphere is investigated. By means of the mathematical techniques, the analytical expressions of the average intensity and the degree of the polarization of a partially coherent hollow vortex Gaussian beam in turbulent atmosphere are derived without any approximations. The average intensity distribution and the degree of the polarization of a partially coherent hollow vortex Gaussian beam are numerically examined. The influences of the beam parameters, the topological charge, the transverse coherent lengths, and the structure constant of the atmospheric turbulence on the propagation of a partially coherent hollow vortex Gaussian beam in turbulent atmosphere are also discussed, respectively. With the increase of the propagation distance in turbulent atmosphere, the dark region gradually diminishes until it completely disappears. If one wants to keep the propagation distance within which the dark region will preserve longer, the optimal method is the increase of the topological charge *m*, and the second preferential method is the increase of the waist size of the Gaussian part *w*_{0}, and the third preferential method is the increase of the beam order *n*. Of course, the increase of the transverse coherent lengths or the decrease of the structure constant of the atmospheric turbulence will also elongate the propagation distance within which the dark region will preserve. When *δ _{xx}*≠

*δ*, the degree of the polarization of a partially coherent hollow vortex Gaussian beam propagating in turbulent atmosphere is varied. In a general way, the degree of the polarization will first decreases, then increases, and finally fluctuates with increasing the value of the radial coordinate

_{yy}*ρ*. However, there are also exceptions. This research is beneficial to the practical applications in free-space optical communications and remote sensing involving in the dark hollow beams.

## Acknowledgments

Guoquan Zhou acknowledges the support by the National Natural Science Foundation of China under Grant Nos. 10974179 and 61178016. 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 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. The authors are indebted to the reviewers for valuable comments.

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