## Abstract

An approximate expression of a Bessel-Gaussian beam (BGB) with desired topological charge is introduced using a coherence superposition of decentered Gaussian beams (dGBs). And based on such an expression and the extended Huygens-Fresnel principle, the propagation properties of BGBs traveling in turbulent atmosphere are explored. An analytical expression of the average intensity of a BGB with phase singularity propagating through turbulent atmosphere is obtained and analyzed numerically. It is found that intensity profiles of BGBs experienced successive variations and the phase singularity rapidly fades away during propagating in turbulent atmosphere.

©2008 Optical Society of America

## 1. Introduction

It is well known that the propagation of laser beams through a turbulent atmosphere has many applications in optical communication [1]. The turbulent atmosphere is an inhomogeneous medium in which the refractive index is a function of position and time. When laser beams pass through such a medium, it produces random variations in the amplitude and phase of the laser beams and induces intensity fluctuations and spread of the radiation pattern at the observation plane. In recent years, with the application of laser beams in atmosphere, for example, free space optical links and the propagation of high energy laser beams in atmosphere, turbulent atmosphere has become a very lively area of scientific research and application. The propagation properties of various types of laser beams in turbulent atmosphere have been investigated [2–12]. However, these studies above cited have been mainly restricted to the beams without optical vortices.

Singular optics is a new branch of modern physical optics deals with a wide class of effects associated with phase singularities in wave fields [13]. The concept of phase singularities or optical vortices in wave fields was emphasized by Nye and Berry [14–16]. Higher-order Bessel Gaussian beams (BGBs) are the typical example of singularity beams and have attracted a lot of attention because of their nondiffracting property [17,18]. Although there are a few works to play attention on the propagation of vortex beams in turbulent atmosphere, only the integral expression of the intensity distribution for BGBs in the received plane is provided and the analysis mainly has to be conducted with the numerical evaluations [19–24].

In this paper, we introduce the new combination beam through decentered Gaussian beams(dGBs) with different constant phases. It is found that such new combination beam can be made as a quite well approximate of a BGB with desired topological charge. And based on the approximate BGB expression and the extended Huygens-Fresnel principle, the propagation properties of BGBs traveling in a turbulent atmosphere are explored. Nowadays beam combination has been a subject of current interest for some practical applications, where laser beams with high power or special beam profiles are required. A variety of laser beams, e.g., laser array beams, dark hollow beams, flat-topped beams and general-type beams have been developed through beam combination, and have found application in atomic physics, high-power laser systems, free-space optical communications and inertial confinement fusion [25–30]. However, roughly speaking, in the studies listed above the optical beams entering into the combination beam are assumed to have the same phase.

The paper is organized as follows: Section 2 introduces the dGB combination that can also be expressed a superposition of BGB with change *n*. And the relative weight coefficient of the mode components with the desired topological charge is calculated analytically and numerically. It is found that, carefully adjusting the parameters, a BGB can be represented as the combination of these dGBs. Then, based on the combination beam, the intensity profiles of BGBs with phase singularity propagating through turbulent atmosphere are investigated in detail in Section 3. Some numerical examples are presented to illustrate the influence of atmospheric turbulence and the source beam parameters on the propagating beam properties. Finally, some conclusions are summed in Section 4.

## 2 dGB expansion of BGBs

In this paper we consider the beam combination

where

is a dGB with a constant phase *φ*
* _{m}*=

*nmα*

_{0}(

*m*=0,1,2,…,

*M*-1 and

*α*

_{0}2

*π*/

*M*),

*w*

_{0}is its beam waist size,

*θ*

*=*

_{m}*mα*

_{0}, while

*n*is a integer termed as a spiral number or topological charge. As the combination beam of M Gaussian beams can represent a hollow beam [27–30], the combination beam (1) with an appropriate choice of parameters can mimic an optical vortex beam with desired charge

*n*. Without loss of generality, an unimportant amplitude constant factor has been omitted.

In fact, transforming into the polar coordinates (*ρ*,*θ*) and using the Bessel function expansion $\mathrm{exp}\left(\mathrm{ix}\mathrm{cos}\phi \right)=\sum _{p=-\infty}^{\infty}{i}^{n}{J}_{p}\left(x\right)\mathrm{exp}\left(\mathrm{ip}\phi \right)$ [31], Eq. (1) is reduced to

Considering

as well as performing the sum in Eq. (2) on *m*, finally, Eq. (3) is turned to

The relative weight of the beam component with charge *n*+*Mq* in the combination beam (5) is defined by [16, 31]

$$={I}_{n+\mathrm{Mq}}\left(\frac{{R}^{2}}{4{w}_{0}^{2}}\right)\u2044\sum _{q=-\infty}^{\infty}{I}_{n+\mathrm{Mq}}\left(\frac{{R}^{2}}{4{w}_{0}^{2}}\right)$$

Numerical calculations demonstrate that in the sum (5) only the term with *q*=0 is important when M is sufficient large or R is appropriate small. Therefore, a BGB with charge *n* can be quite well approximated as a combination of M dGBs, that is,

$$\approx \frac{1}{M{i}^{n}}\mathrm{exp}\left(-\frac{{R}^{2}}{4{w}_{0}^{2}}\right)\sum _{m=0}^{M-1}\mathrm{exp}\left[-\frac{{\left(x-\frac{i}{2}R\mathrm{cos}{\theta}_{m}\right)}^{2}+{\left(y-\frac{i}{2}R\mathrm{sin}{\theta}_{m}\right)}^{2}}{{w}_{0}^{2}}+i{\phi}_{m}\right]$$

This is one of the main results obtained in this paper. And with the aid of the approximate expansion (7) of n*th*-order BGB, to obtain an analytical expression for the vortex beam with charge *n* propagating through turbulent atmosphere is possible.

Carefully adjusting the parameters *n*, *M*, and *R*, the results evaluated by Eq. (7) and directly by the BGB may be quite well consistent. For example, for *M*=12, *R*=4*w*
_{0} and *n*=1, 2, 3 and 4, the numerical calculations show that, from the point of view of energy, the relative weight of the component with charge *n* is over 99%, as seen from the Table 1, which means that, in the combination beam, over 99% of the optical energy is converted into a BGB with charge *n*. In Fig. 1 these intensity profiles and corresponding phase distributions are drawn, which clearly represent optical vortex characteristics with desired charge values.

## 3. Propagation of BGBs with topological charges in turbulent atmosphere

In this section we will consider the BGB with change *n* propagating along the z-axis in a turbulent atmosphere using the approximate combination beam (7). Based on the extended Huygens-Fresnel principle, the average intensity at the z-plane can be expressed as

$$\xb7\mathrm{exp}\left(\frac{\mathrm{ik}}{2z}\left[{\left(x-p\right)}^{2}+{\left(y-q\right)}^{2}-{\left(x-\xi \right)}^{2}-{\left(y-\eta \right)}^{2}\right]\right)\mathrm{dpdqd}\xi d\eta $$

here *k*=2*π*/*λ*(*λ* is the wavelength), the asterisk ‘*‘ denotes the complex conjugation and the 〈〉* _{m}* denotes the ensemble average over the medium statistics covering the log-amplitude and phase fluctuations due to the turbulent atmosphere. The ensemble average term is [10]

where *ψ*(*p*,*q*, *x*, *y*) represents the random part of the complex phase of a spherical wave that propagates from the source point to the receiver point, *D*
* _{ψ}*(

*p*-

*ξ*,

*q*-

*η*) is the wave structure function.

*ρ*

_{0}is the spatial coherence radius of a spherical wave and can be expressed as

*ρ*

_{0}=(0.545

*C*

^{2}

_{n}*k*

^{2}

*z*)

^{-3/5}, where

*C*

^{2}

*is the refractive index structure constant that describes how strong the turbulence is[6]. It is worth pointing out that in order to obtain simpler and viewable analytical results, here we also have employed a quadratic approximation of the authentic Rytov’s phase structure function widely and usually accepted to be valid not only for the case of “weak fluctuations”, but for the case of “strong fluctuations” as well [3,4,32]. From Eqs. (8) and (9), the average intensity at the receiver plane can be represented as*

_{n}$$\xb7\mathrm{exp}\left(\frac{\mathrm{ik}}{2z}\left[{\left(x-p\right)}^{2}+{\left(y-q\right)}^{2}-{\left(x-\xi \right)}^{2}-{\left(y-\eta \right)}^{2}\right]-\frac{1}{{\rho}_{0}^{2}}\left[{\left(p-\xi \right)}^{2}+{\left(q-\eta \right)}^{2}\right]\right)\mathrm{dpdqd}\xi d\eta $$

By inserting Eq. (7) into Eq. (10) and performing the integral[31], the average intensity at the receiver plane can be analytically expressed as

$$\xb7\mathrm{exp}\left\{\frac{{N}_{w}\left(1+i{N}_{w}\right)R}{\Omega {w}_{0}^{2}}\left(x\mathrm{cos}{\theta}_{m}+y\mathrm{sin}{\theta}_{m}\right)+\frac{{N}_{w}\left(1-i{N}_{w}\right)R}{\Omega {w}_{0}^{2}}\left(x\mathrm{cos}{\theta}_{l}+y\mathrm{sin}{\theta}_{l}\right)\right\}$$

where *N*
* _{w}*=

*πw*

^{2}

_{0}/

*λz*is the Fresnel number of dGBs in free space propagation and Ω=1+

*N*

^{2}

_{w}+2

*w*

^{2}

_{0}/

*ρ*

^{2}

_{0}. This is another main result obtained in this paper. Based on the expression, some numerical evaluations can be conducted. In the following calculations

*λ*=632.8nm,

*M*=12,

*R*=2

*w*

_{0}and

*y*=0 are always used.

Figure 2 shows the intensity profile of BGBs during propagations in turbulent atmospheres. For *n*=0, the combination (7) approximates to a 0th order BGB with *n*=0. For the case discussed in this paper, the intensity profile almost retains unchangeable except spreading out for larger distance propagation. However, for the beam with non-zero charge value, the intensity profile changes during the propagation. At different z-planes along propagation direction, the beam profile successively changes from an annular structure to a flattened-top profile and finally to a single-peak profile, which distinguishes from the propagation in free space because the BGB is well-known non-diffracting beams. For example, approximately, for *n*=2 the propagating beam loses its phase singularity at z=0.4*km* and becomes a flattened top profile at z=5*km*. In fact, at z=10*km* all the cases have the extremely similar intensity profiles[4].

The zero intensity located at the center characterizing the phase singularity is missed after a sufficient propagation distance. From Fig. 3(a) it is seen that the propagation distance relies on the charge value of the beam and *C*
^{2}
* _{n}* characterizing the turbulent atmospheric structure. And the larger of the charge

*n*, the farther of the propagation distance before the phase singularity loses. But the bigger of

*C*

^{2}

*, the faster the phase singularity dies away. Additionally, the beam width notably affects the beam spreading during the propagation yet, which can be seen from Fig. 3(b). In particular, the smaller the beam width, the faster the beam spreads.*

_{n}## 4. Conclusions

An approximate expression of a BGB with desired charge is derived with a coherence superposition of dGBs. And based on the expression and the extended Huygens-Fresnel principle, the propagation properties of BGBs traveling in a turbulent atmosphere are explored. An analytical average intensity expression of a BGB with phase singularity propagating through atmospheric turbulences is obtained. The average intensity distributions are studied analytically and numerically. It is found that the intensity profile of BGBs experienced successive changes during the propagation and phase singularity wholly fades out after propagating a small distance. The change speed of the intensity profile and phase singularity sensitively depends on the beam parameters as well as the atmospheric turbulence structure.

## Acknowledgments

One of the authors (X. J. Zheng) thanks the postdoctorial Fond of Central South University.

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