## Abstract

A class of random, wide-sense stationary optical beams with uniform correlations, named the partially coherent quasi-rectangular beam, is introduced theoretically. Based on the extended Huygens-Fresnel principle, the analytical expressions for the cross-spectral density (CSD), effective radius of curvature, and beam wander of the beam in the non-Kolmogorov turbulence are derived. It is found that the position of maximum intensity of the partially coherent quasi-rectangular beams shifts farther from the axis at intermediate distance, the shift in the turbulence is depressed compared to that in free space. As the effective radius of curvature decreases from infinity to a constant with the increase of the coherence length, it always takes a higher value than that in free-space propagation when the other parameters are fixed. In addition, the beam wander can be reduced by picking a relative small initial beam width, short coherence length, or long wavelength. These results are of importance for optical systems operating through long-range turbulent channels in which a beam must have a range-dependent tilt, e.g. on travelling around an obstacle on the axis.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The source degree of coherence [1] of the partially coherent beams (PCBs) with special correlation structure is modulated by a well-behaved correlation function, the beams exhibit self-splitting [2], self-focusing [3], self-drifting [4], self-shaping [5] and the other unique physical phenomena during the propagating process [6]. The typical sources contain non-uniformly correlated partially coherent light [7,8], cosine-Gaussian correlated Schell-model (CGCSM) beam [9,10], Hermite-Gaussian correlated Schell-model beam [11], electromagnetic multi-Gaussian Schell-model beam [12], and various PCBs with special correlation structure. The laser beam propagating in turbulence has promising applications such as atmosphere optical communications [13], remote sensing [14] and the laser radar [15], which effectively promotes the development of optics in interdisciplinary fields.

Compared with generalized coherent laser beams, the partially coherent beams have advantages of effectively reducing turbulence induced extra intensity scintillation, beam spreading and beam wander [16]. In 1972, Kon and Tatarski [17] considered the mutual coherence function for a partially coherent light source in a turbulent medium, and investigated the behavior of the effective coherence radius which determines the angular width. Based on the mutual coherence function, Belen’kii et al. [18] derived a solution of the problem of turbulence-induced spreading of the image of a partly coherent laser source formed in the focal plane of the receiver lens. Salem et al. [19] theoretically studied the intensity distribution of the PCBs and fully coherent beams in modified von Karman spectrum atmospheric turbulence after long-distance propagation based on the cross-spectral density (CSD) functions of this two kinds of beams. Li and Cai [20] derived an approximate analytical formula for a circular flattened Gaussian beam (FGB) propagating through an apertured misaligned ABCD optical system, the mode order and the coherence length have a significant effect on its propagation properties. Wang et al. [21] introduced theoretically and generated experimentally a partially coherent crescent-like beam, the beam is shown to be capable of forming an off-axis intensity maximum at a specific distance from the source in von Karman power spectrum. Based on the standard Huygens-Fresnel integral and Wigner distribution, Gbur [22] investigated the propagation properties of the PCBs in the Kolmogorov spectrum, considered turbulence effects related to the scintillation properties of the field. Optical communication is limited by atmospheric turbulence that has been described for many years by various turbulent refractive index power, and the Kolmogorov’s power density model has become a convenient choice for its simplicity. After that, to meet the demand of describing the portions of the troposphere and stratosphere of the atmosphere more accurate, Toselli et al. [23] presented a non-Kolmogorov power spectrum, the non-Kolmogorov power spectrum is now accepted by many scholars.

Over the years, there have been many reports on propagation properties of PCBs in turbulent atmosphere, the beams hold much promise in multifarious application for their resistance to the deleterious effects of turbulence [24,25]. Benefit from the pioneering work of Gori and his collaborators on the correlation function of PCBs [9], in this paper, we establish a novel source degree of coherence of a non-uniformly partially coherent beam. We will then study the propagation properties of the partially coherent quasi-rectangular beam in anisotropic turbulence with the help of extended Huygens-Fresnel principle [26]. The non-Kolmogorov refraction power spectrum [27] would be used to derive the expressions of intensity distribution, effective radius of curvature and beam wander, some interesting and useful results will be discussed.

## 2. Theoretical analysis

#### 2.1 Intensity distribution

A partially coherent beam can be characterized by the mutual intensity in the space-time domain or the CSD function in the space-frequency domain [28,29]. The CSD of the field in the source plane can be presented in a two-point correlation function in the form of [25]

*x*and

*y*directions that can have quite arbitrary dependence on the wavelength,

*M*is beam order,

*s*is called the decentered beam parameter [30] which describes the decentered degree and the non-uniformity of the radiation field. Equation (3) reduces to the field of rectangular beams with uniformity field when

*s*approaches to infinity.

The CSD function of any physically realizable random source should satisfy the condition of non-negative definiteness, and it can be written as [31]

*p*is a non-negative function for any value of argument $({v_x},{v_y})$, asterisk represents complex conjugate,

*H*has the form of Fourier-like kernel [31]

*σ*

_{0}denotes the beam width.

The general cross spectral density takes the form of [25]

Here, the tilde denotes one-dimensional Fourier transform. The choice of $p({v_x},{v_y})$ function determines the correlation function of the source and intensity distribution of the far field. In order to obtain a quasi-rectangular beam, the weight function $p({v_x},{v_y})$ can be expressed as

From Eqs. (2) and (8) we obtain for the cross-spectral density (CSD) (1) the formula

According to the extended Huygens-Fresnel integral and approximation, the spectral density (SD) of a beam after propagating through a turbulent atmosphere in the receiver plane is given by [21]

*z*axis, ${\theta _2}$ is the angle between the vector ${{\textbf r}_2}$ and the

*z*axis, $\sin {\theta _1} = {{\textbf S}_{1 \bot }}$, ${{\textbf S}_{1z}}$ is the projection of ${{\textbf S}_1}$ along the direction of the

*z*axis, ${{\textbf S}_{1 \bot }}$ is the projection of ${{\textbf S}_1}$ along the direction of perpendicular to the

*z*axis. Meanwhile, $\cos {\theta _2} = {S}_{2z}$,$\sin {\theta _2} = {S}_{2 \bot}$(see Fig. 1).

The angular brackets with subscript in Eq. (10) denote the ensemble average of the medium realizations. $\Psi ({{\mathbf \rho },{\textbf r},z} )$ denotes the complex phase perturbation induced by the random distribution of the medium’s refractive index. If the medium fluctuations are anisotropic, the phase correlation function is related to the power spectrum *T* by formula [32]

*x*and

*y*transverse directions,${\mu _z}$ is the anisotropic factor in direction of propagation, 3<

*α<*4 is the non-Kolmogorov slope. When ${\mu _x} = {\mu _y} = {\mu _z} = 1$ and $\alpha = 11/3$, the anisotropic non-Kolmogorov spectrum reduces to isotropic Kolmogorov spectrum [33]. And the power spectrum of the anisotropic turbulence is chosen as [26]

Upon substituting from Eqs. (9) and (11)—(14) into Eq. (10), by setting ${\textbf r} = {{\textbf r}_1} = {{\textbf r}_2}$, and using the following formula [34]

Upon substituting *y* with *x* in the Eq. (17), one arrives at the parameters ${\Delta _y}(z )$.

#### 2.2 Effective radius of curvature

A finite laser beam will experience random deflections as it propagating in the turbulence, causing further spreading of the spot by large scale inhomogeneities of the atmosphere. The effective radius of curvature of the beam along the *x* or *y* direction is deduced as

In Eq. (18), the last term under the square root on the right side indicates the initial beam width. The first and second terms indicate that the diffraction of the beam induced by beam width and the coherence length respectively as it propagation in free space, they are all proportional to the square of the propagation distance. The third term represents the diffraction caused by the non-Kolmogorov turbulence, it is inversely proportional to the square of the anisotropic factor [33]. The diffraction caused by the anisotropic turbulence of the beam can be ignored when the propagation distance of the beam is not too far. As the propagation distance increases, the value of the third term in the formula is proportional to the ${z^3}$, and the values of the first and second terms are proportional to the ${z^2}$, so the diffraction induced by anisotropic turbulence plays a domain role gradually, and more details would be discussed in Fig. 5.

#### 2.3 Beam wander

Movement of the short-term beam instantaneous center or hot spot is commonly called beam wander. The beam wander can be characterized statistically by the variance of hot spot displacement along an axis or by the variance of the magnitude of the hot spot displacement. According to the model of the beam wander given by Andrews and Phillips [35]

By means of geometrical optics approximation [12], the last term simplified as

Upon substituting from Eqs. (12), (13) and (20)–(22) into Eq. (19), the beam wander of the partially coherent quasi-rectangular beam in the turbulent environment is

Equations (2), (16), (18) and (23) are the main analytical results of this paper. One can study the propagation properties of the beam in the turbulent atmosphere by those derived equations.

## 3. Numerical results and analysis

With the analytical expressions for the partially coherent quasi-rectangular beam propagating in non-Kolmogorov turbulence, the propagation properties is investigated by illustrating numerical examples, including the normalized intensity distribution, the off-axis distance of the maximum intensity, effective radius of curvature and the beam wander of a quasi-rectangular beam propagating through turbulence atmosphere. The initial parameters are set as *λ*=632.8 *nm*, *M*=20, *α=*11/3, *δ*_{0}*=δ _{x}*(

*λ*)

*=δ*(

_{y}*λ*)

*=*5

*mm*,

*σ*

_{0}=5.5

*mm*,

*s*=0.02,

*ɛ*=0.01, $\textrm{ }C_n^2$

*=*3×10

^{−14}

*m*

^{-2/3},

*l*

_{0}

*=*0.01

*m*and

*L*

_{0}

*=*1

*m*throughout the paper unless different values are specified.

Figure 2 gives the modulus of the SDOC of the partially coherent quasi-rectangular beams for several selected sets of parameters, the first column is *M*=1, *δ*_{0}=*δ _{x}*(

*λ*)

*=δ*(

_{y}*λ*)

*=*0.56

*mm*; the second column is

*M*=40,

*δ*(

_{x}*λ*)

*=δ*(

_{y}*λ*)

*=*0.56

*mm*; the third column is

*M*=40,

*δ*(

_{x}*λ*) = 1.6

*mm*,

*δ*(

_{y}*λ*) = 0.56

*mm*. The contour of the SDOC is a circular Gaussian profile when

*M*= l. As shown in the first row, when

*s*=0.02, the circular Gaussian profile gradually evolves into a rectangular profile with sidelobes. Different values of the coherence length lead to different evolution speeds of the SDOC along the

*x*axis and the

*y*axis. As shown in the second row, when

*s*=0.002, the circular Gaussian profile gradually evolves into a quasi-rectangular profile with sidelobes. It should be pointed out that the SDOC is asymmetric here, the sidelobes on the left side to become almost invisible, while the sidelobes on the right side are obvious as shown in Fig. 2(f).

To better understand the evolution of the intensity distribution of the beam propagating through the anisotropic turbulence. We now compare the theoretical models in anisotropic turbulence with that in free space. The normalized intensity distribution of the partially coherent quasi-rectangular beam in free space and anisotropic turbulence is shown in Fig. 2 and Fig. 3, respectively. By Eq. (16), we can obtain the intensity expression for the beams propagation in free space by setting *T=*0. Furthermore, the generalized structure parameter of refractive index is chosen as $\textrm{ }C_n^2$*=*3×10^{−14}*m*^{-2/3} in non-Kolmogorov spectrum.

In Fig. 3(a), the normalized intensity distribution of the beam at the source plane is a circular Gaussian profile. However, the beam’s profile gradually evolves into a quasi-rectangular asymmetric beam profile, and retains its shape during the further propagation in free space while its size enlarges due to diffraction [see Fig. 3(b)-Fig. 3(f)]. It is worth noting that the hot pot of the beam shifts away from the axis, and it shifts further with the increase of the propagation distance. Figure 4 shows that the evolution of beam’s profiles are similar to that in free space when the distance smaller than 0.3km [see Fig. 3(a)-Fig. 3(c) and Fig. 4(a)-Fig. 4(c)]. This is because the beam’s profiles are determined by the specially adjusted correlation function of the source when the propagation distance is not too far. After that, the quasi-rectangular shape gradually becoming a quasi-rectangular asymmetric shape with the further increase of propagation distance because the turbulence-induced diffraction gradually plays the dominant role in determining the beam’s profiles. Nevertheless, by comparing Fig. 3(f) with Fig. 4(f), under the condition of the anisotropic turbulence, the intensity distribution of the beam converts to the quasi-rectangular asymmetric shape faster than that case in free space. Which indicates the turbulence atmosphere brings influence on the beam’s propagation progress indeed.

Figure 5 presents the normalized intensity distribution of a partially coherent quasi-rectangular beam under the turbulence atmosphere at propagation distance *z*=2 *km*. In Fig. 5(a), the off-axis distance is affected by the increasing of the initial beam width. In turbulence atmosphere, when the value of the initial beam width *σ*_{0} is 4.5 *mm*, 5.5 *mm* and 6.5 *mm*, the off-axis distance belonging to the position of maximum intensity is 0.48 *mm*, 0.61 *mm* and 0.74 *mm*, respectively. The off-axis distance increases with the increasing of the beam width. Figure 5(b) shows the off-axis distance increases with the increasing of the beam order. In Fig. 5(c), when the value of the coherence length ${\delta _0}$ is 2.5 *mm*, 5.0 *mm* and 10.0 *mm*, the off-axis distance belonging to the position of maximum intensity *S _{x}* is 0.19

*mm*, 0.61

*mm*and 1.34

*mm*, respectively. The off-axis distance decreases with the increasing of coherence length, which shows the opposite trend with the increasing of the initial beam width and the beam order. That is to say one may control the transverse acceleration of the intensity by modulating the initial beam parameters. Figure 5(d) presents the off-axis distance of maximum intensity is also affected by the change of refractive-index structure parameter. The normalized intensity distribution is affected obviously by the structure parameter when it changes from 3×10

^{−14}

*m*

^{-2/3}to 3×10

^{−13}

*m*

^{-2/3}, while the normalized intensity distribution is minimally affected when it changes from 3×10

^{−14}

*m*

^{-2/3}to 3×10

^{−15}

*m*

^{-2/3}. Implying the normalized intensity distribution is strongly affected by the turbulence change from strong to moderate and it is little change with the change of structure parameter within the moderate or weak turbulence. In addition, the strong turbulence contributes to smaller off-axis distance. The lateral shift in the turbulence is depressed compared to that in free space because the shift of the intensity maximum is simultaneously affected by the free-space diffraction and the turbulence.

The effects of anisotropic turbulence on the effective radius of curvature $w(z)$ can be computed according to Eq. (18) and the results are presented in Fig. 6. Figure 6(a) compares the effective radius of curvature of the beam along the *x* and *y* direction. The anisotropic factors in *x* and *y* direction are chosen to be ${\mu _x} = 1$ and ${\mu _y} = 3$, the maximum of ${w_y}(z)$ is the three times of the maximum of ${w_x}(z)$. The $w(z)$ increased first, but it gradually decreased after it reaches max value with increasing of $\alpha$. From the raw data of software Origin, the ${w_x}(z)$ and ${w_y}(z)$ reach max value when *α=*3.13. Earlier in the paper, the *α=*11/3 had been mentioned, and the anisotropic spectrum would reduce to isotropic spectrum. The off-axis of the maximum intensity may account for the *α* is 3.13 but not 11/3. The position of effective radius of curvature also shifts away from the axis. In Fig. 6(b), the $w(z)$ increases with the increase of the propagation distance, an increase in the initial beam width exacerbates such increasing. In view of ${w_y}(z)$ shows the same evolution law as the ${w_x}(z)$, ${w_x}(z)$ would be discussed more in the below. From the Fig. 6(c), the effective radius of curvature along the *x* direction decreases from infinity to a constant with the increase of the coherence length, the ${w_x}(z)$ always takes a higher value than that for the free space. In the presence of turbulence, the smaller refractive-index structure parameter can mitigate the influence of the turbulence on the effective radius of curvature. As shown in Fig. 6(d) that, a quasi-rectangular beam in the smaller inter scale of the turbulence has larger ${w_x}(z)$.

According to Eq. (23) and Eq. (24), the beam wander of the partially coherent quasi-rectangular beam is shown in Fig. 7. Physically the turbulence acts like many lenses of different size that random change the effective optical path of the beam. The beam wander is used as one of the main parameters to evaluate the intensity profile along the path. In Fig. 7(a), when the other parameters are fixed, the beam wander along the *x* direction always takes higher value than that along the *y* direction, which means that anisotropic turbulence causes about more beam wander in the one direction under the case ${\mu _x} \ne {\mu _y}$. Also in Fig. 7(a), the beam wander reaches max value when $\alpha = 3.30$, this $\alpha$ is more closer to 11/3 compare with 3.13, the beam wander is influenced by the non-Kolmogorov slope as well. From the Fig. 7(b)-Fig. 7(d), the increase in the initial beam width, refractive-index structure parameter, the outer scale of the turbulence or the coherence length exacerbates increasing of the beam wander, while the increase in the wavelength alleviates the increasing of the beam wander.

## 4. Conclusion

In conclusion, a class of random, wide-sense stationary optical beams with uniform correlations which produce far field a quasi-rectangular intensity distribution with the maximum at an off-axis position is introduced theoretically. Based on the extended Huygens-Fresnel principle, the expressions of intensity distribution, effective radius of curvature, and beam wander of the partially coherent quasi-rectangular beam in non-Kolmogorov turbulence are derived. Numerical results have shown that the off-axis distance of maximum intensity of the quasi-rectangular beam increases with the decrease of the coherence length and the refractive-index structure parameter in non-Kolmogorov turbulence. This is similar to the corresponding result for the partially coherent crescent-like optical beams [21]. The effective radius of curvature is increases with the increase of the beam width, refractive-index structure parameter, the propagation distance, the outer scale of the turbulence or the decrease of the inter scale of the turbulence. The partially coherent quasi-rectangular beam has larger beam wander in the turbulence with large refractive-index structure parameter, large initial beam width, large outer scales or short wavelength. The off-axis distance can be precise controlled by the appropriate selection of the beam’s initial parameters. Considering that the maximum intensity traces a curved trajectory on propagation, the beam can be applied in the situations where the optical signal must travel around an obstacle on the axis embedded in turbulent medium.

## Funding

Natural Science Foundation of Anhui Province (1808085QA10); National Natural Science Foundation of China (11974219, 12074005).

## Acknowledgments

In appreciate of Laser Control and Transmission Lab of Anhui Normal University for providing a research environment. Dedicated to the help of Shandong Normal University and the cooperation of Anhui University of Science and Technology.

## Disclosures

The authors declare no conflicts of interest.

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