Comparative analysis of some Schell-model beams propagating through turbulent atmosphere

In this paper, we investigate a comparative analysis of some Generalized Laguerre–Gaussian Schell-model beams through a paraxial ABCD optical system in a turbulent atmosphere. Based on the extended Huygens–Fresnel diffraction integral, analytical expressions for the spectral density in the receiver plane of the studied beams are derived in detail. By studying the effects of the source coherence parameters and the atmospheric turbulence strength, the numerical results indicate that the profile of these Schell-model beams takes different shapes during their propagation.


Introduction
In recent years, the propagation of laser beams in a turbulent atmosphere has attracted much attention from researchers (Cai 2006;Cai and He 2006a;Chu 2007;Cai et al. 2007aCai et al. , 2008Qu et al. 2010;Zhou 2011;Kinani et al. 2011;Liu et al. 2016;Boufalah et al. 2016Boufalah et al. , 2018Belafhal et al. 2011;Ez-zariy et al. 2016;Saad and Belafhal 2017;Yaalou et al. 2019) due to their large applications such as in free space communications (Navidpour et al. 2007), active optical imaging (Hajjarian et al. 2010), and remote sensing (Andrews et al. 2001). Furthermore, the specific nature of partially coherent laser beams has led to their extensive study in turbulent atmosphere by many authors (Wolf 2007;Korotkova 2004;Cai and He 2006b;Gbur and Korotkova 2007;Cai et al. 2007b;Lu et al. 2007;Yang et al. 2008Yang et al. , 2009Yuan et al. 2009;Zhou and Chu 2009;Wang et al. 2010;Zhao and Cai 2011;Zhong et al. 2011;Zhou et al. 2012;Chib et al. 2020). Lu et al. (2007) have interested to the change of degree of coherence of partially coherent Gaussian Schell-model beam during its propagation in a turbulent atmosphere. Zhou and Chu (2009) have examined the comportment of partially coherent cosine-Gaussian beam in a turbulent atmosphere through the study of the average intensity. The effect of the atmospheric turbulence and the input beam parameters on the beam width and the average intensity of partially coherent Hermite-cosh-Gaussian beams traveling through atmospheric turbulence has been studied by Yang et al. (2009), and recently, Chib et al. (2020) have investigated the propagation characteristics of partially coherent Generalized flattened Hermite-cosh-Gaussian beam in a turbulent atmosphere.
On the other hand, Gori and Santarsiero (2007) have presented the sufficient condition of the cross-spectral density (CSD) to be physically realizable. Thereby, considerable analytical models for CSD function of random sources generating flat and ring-shaped intensity profiles in the far field have been introduced such as: multi-Gaussian Schell-model (MGSM) sources, Laguerre-Gaussian Schell-model (LGSM) and Bessel-Gaussian Schellmodel (BGSM) sources (Sahin and Korotkova 2012;Mei and Korotkova 2013). These models have been the subject of different studies; for example, Cang et al. (2013a) have examined the evolution of the average spectral density, the spectral degree of coherence and the beam width of MGSM beams traveling through atmospheric turbulence. In another work, Cang et al. (2013b) have reported the propagation characteristics of LGSM and BGSM beams in a turbulent atmosphere. In 2014, Chen et al. (2014) have derived the analytical expression for the CSD and the second-order moments of the Wigner distribution function of LGSM beam in a turbulent atmosphere in order to study the statistical properties of this laser beam.
The main objective of the current paper is to study the comportment of the Standard and Elegant Generalized Laguerre-Gaussian Schell-model (GLGSM) beams propagating through a turbulent atmosphere by developing analytically the spectral density, which to our best knowledge, has not evaluated previously. The remainder of the paper is organized as follows: Sect. 2 presents a general theoretical formulation for the spectral density of Schell-model beams through a paraxial ABCD optical system in atmospheric turbulence. In Sect. 3, we derive the analytical expressions of the spectral density for the Gaussian Schell-model, Standard and Elegant Generalized Laguerre-Gaussian Schell-model beams in atmospheric turbulence. Discussions and comparative analysis for the spectral density of the studied Shell-model beams in the considered medium are exposed detailly in Sect. 4. A last Section outlines a general conclusion.

Theoretical formulation
Based on the extended Huygens-Fresnel integral, the CSD at z plane of some Schell-model beams in atmospheric turbulence will be investigated. The considered beams in this study are: Gaussian Schell-model (GSM) and GLGSM beams.
Within the paraxial approximation, the corresponding CSD of the collimated beam can be written as In the above expression, and are two-dimensional position vectors, is the angular frequency, s is the rms width of the source and (0) is the spectral degree of coherence of the source, which takes a specific form for each beams family. By applying the extended Huygens-Fresnel integral, the propagation of partially coherent beams through a paraxial ABCD optical system in a turbulent atmosphere can be written as (Zhou and Chu 2009;Hanson 1987, 1989) where = 1x , 1y , 2 = 2x , 2y are two arbitrary transverse position vectors at the receive plane, A, B, and D are the transfer matrix elements and d 2 d 2 = dr 1x dr 1y dr 2x dr 2y .
By setting = = in Eq. (2a) we obtain the expression for the spectral density (i.e. the optical intensity at frequency ) of the beam field as follows where z is the propagation distance, k = 2 ∕ is the wave number with is the wavelength and F 0 is the wave-front radius of curvature of the input field. D w is the generalized wavestructure function given, in the Kolmogorov turbulence spectrum, by Yura and Hanson (1987) with 0 = 0.545C 2 n k 2 z −3∕5 is the spherical wave lateral coherence length of the optical system and C 2 n represents the refraction index structure constant. By substituting Eqs. (1) and (3) into Eq. (2b), and by introducing the change of variables: = ( + ) 2 and = − , the following expression for the spectral density of the beam, at a particular frequency , is obtained After some algebraic calculations and integral operation, Eq. (4) can be written as where Note that from Eq. (3), the generalized wave-structure function is a power somewhat less than two, and that the exact value 5∕3 is not crucial. Therefore, Wandzura (1981) has proposed a quadratic approximation of D w by expanding the 5∕3 power as with = 1 3 . Applying this last equation to the factor exp − To evaluate the spectral density S( , z; ) at a particular frequency, the two integrals I( , z; ) and J( , z; ) for each beam will be derived in the below Section. Note that Eq. (9) is our first general main result in the present work.

Spectral density of some Schell-model beams
In the following, we will develop the analytical expression of the spectral density for the Gaussian Schell-model beams and GLGSM beams.

Gaussian Schell-model beams
The spectral degree of coherence of the source for the Gaussian Schell-model (GSM) beams is given as where is a coherence width. By substituting this last expression in Eq. (9), the following integrals are yielded and with and The evaluation of Eq. (11) is realizable with the help of the identity (Gradshteyn et al. 2007) with Re( ) > 0, Re(s + ) > −1 , J is the Bessel function of the first kind and order , Γ is the gamma function and 1 F 1 is the Kummer's series of the confluent hypergeometric series.
So, one obtains To evaluate Eq. (12), we recall the following identities (Gradshteyn et al. 2007) and (10) Γ(x) . By using the identity (Chu and Liu 2010) where chi and shi are the hyperbolic cosine and sine integral functions, with z = 2 4 , Eq. (17) can be written as Finally, Eq. (9) can be rearranged, in the case of GSM beams, as The result obtained in Eq. (20) is our second main finding of the spectral density of GSM beams in a turbulent atmosphere. In our best knowledge, this closed form is not published yet in the literature.
For the axial spectral density (i.e. = � ⃗ 0 ), Eq. (19) becomes with is the Euler's constant, and Eq. (20) can be written, in this case, as Equation (22) describes the approximation expression of the axial spectral density of GSM beams in a turbulent atmosphere.

Generalized Laguerre-Gaussian Schell-model beams
The spectral degree of coherence of the source for the Generalized Laguerre-Gaussian Schell-model (GLGSM) beams is expressed by where L ( ) l is the generalized Laguerre polynomial, is a coherence width and q is a parameter, which defines the different types of Laguerre-Gaussian beams. For q = √ 2, Eq. (23) presents the spectral degree of coherence of the source for the Standard GLGSM beams and for q = 1 it reduces to the spectral degree of coherence of the source for the Elegant GLGSM beams.
For GLGSM beams, Eqs. (9a and 9b) take the following forms and where One notes that these last quantities given by Eqs. (24) and (25) depend also on the parameter of the generalized Laguerre polynomial.
The below integral, established recently , is used to compute the quantity I (l, ) GLGSM ( , z; ) Then, Eq. (24) can be written as and if = 0 , this last equation becomes = � + 1 2 , withis given by Eq. (9-c).  where I (l,0) LGSM ( , z; ) and J (l,0) LGSM ( , z; ) are given by Eqs. (29) and (33), respectively. From the above equations, it is interesting to deduce the expressions of the spectral density propagating through a paraxial ABCD optical system in free space without turbulence which are given as.

Numerical simulations and discussion
In this Section, the propagation properties of GSM and GLGSM beams in a turbulent atmosphere are numerically illustrated by using the analytical expressions of the spectral density developed in Sect. 3. For the simulations, the wave-front radius curvature of the beams is considered to tend toward infinity and the rms width of the source is taken as The influences of the atmospheric turbulence and the incident beams parameters on the spectral density at the receiver plane of the studied beams family are represented in the below figures. In Fig. 1, the normalized spectral density of GSM beams versus is plotted for various values of the turbulent strength C 2 n at z = 6 km and 8 km with = 1064 nm and = 4.24 cm.
It is noted from the figure that the maximum of the spectral density of GSM beams weakens when the environment is more turbulent as the propagation distance increases.  Figure 2 shows the effects of the coherence width and the wavelength on the behavior of GSM beams upon their propagation through an atmospheric turbulence. The other parameters are set to be z = 6 km and C 2 n = 5 × 10 −15 m −2∕3 . From this figure, it can be observed that as the coherence width is large as the maximum of the spectral density increases (see Fig. 2a). The same comportment is observed by increasing the value of the wavelength of the GSM beams for the values lower than = 1064 nm , while the maximum of the spectral density starts to decrease when > 1064 nm (see Fig. 2b).
We illustrate in Figs. 3 and 4 the normalized spectral density given by Eqs. (34) and (36) versus for various values of the turbulent strength C 2 n and the beam order l at two values of the propagation distance, with = 1064 nm , = 0.25 and = 4.24 cm . From Figs. 3A-B, we note that the profiles of Standard GLGSM and Elegant GLGSM beams traveling through atmospheric turbulence and free space display different stages of evolution. It is also observed that the maximum of the spectral density distribution for the two considered beams decreases when the atmosphere is very turbulent. Figure 3A shows that the beam profile of the Standard GLGSM beams evolves from flat-top distribution into Gaussian distribution as the C 2 n increases for l = 1. However, when l is equal to two a ring-shaped beam profile appears in free space without turbulence and transforms into Gaussian profile with the increase of the  takes ring-shaped form and remains almost unchanged while the hollowness of the ringshaped beam profile is affected by the variation of the turbulent strength C 2 n . The behavior evolution of the Elegant GLGSM beams passing through atmospheric turbulence and free space without turbulence differs slightly compared to that one of the Standard GLGSM beams previously treated. From Fig. 3B, one sees that the beam profile of the Elegant GLGSM beams remains Gaussian distribution for small value of l . When l is greater than or equal to five, the Elegant GLGSM beams exhibit another type of profile, gradually passing from a Gaussian profile to a ring-shaped profile. We can also note that the widen of the ring-shaped beam increases with the beam order.
The Standard GLGSM beams distribution becomes a ring-shaped beam profile in free space without turbulence when the propagation distance increases (Fig. 4A). Moreover, the depth of the dip in the spectral density profile increases with the source coherence parameter l . While the Elegant GLGSM beams take the same comportment by changing its profile from Gaussian shape, then flat-top shape, and finally turn into ringshaped when l increases. We can clearly observed in the presence of the turbulence that, with a small value of l , the profiles of the Standard GLGSM and the Elegant GLGSM beams keep a Gaussian shape unchanged with the increase of the turbulent strength (see Fig. 4A and B). When l = 2 , the evolution of the comportment for the Standard GLGSM and the Elegant GLGSM beams propagating in a turbulent atmosphere is similar to that of Fig. 3, with the exception of the depth of the dip is extremely important. From this investigation, we can conclude that this beams family has interesting characteristics, which give them the advantage of being better using in certain applications like trapping and transport of particles.
The effect of the incident beam parameter on the evolution of the normalized spectral density of GLGSM beams is plotted in Fig. 5. The calculation parameters are: l = 10, C 2 n = 5 × 10 −15 m −2∕3 , = 1064 nm , = 4.24 cm and z = 6 km. One sees that the beam profile of Standard GLGSM beams gradually transforms from a central-dip beam shape to a flat-topped beam shape and finally to a Gaussian beam shape when the incident beam parameter gradually increases (see Fig. 5A). For the Elegant GLGSM beams, its profile undergoes the same process as that of the Standard GLGSM beams while the conversion from a central-dip profile into a Gaussian beam occurs more rapidly. We can note that the spectral density of GLGSM beams given by Eq. (34) is reduced to the spectral density of GSM beams for = 0 and l = 0 . As it is difficult to prove it analytically, we proceeded to a numerical illustration (see Fig. 6) to confirm our results.
From the plot of this figure, it can be clearly seen that the obtained result from the GLGSM beams is in a good agreement with that in the case of the GSM beams, given by Eq. (20). In our numerical analysis, the following parameters are used: = 0 , l = 0 , C 2 n = 5 × 10 −15 m −2∕3 , z = 6 km , = 1064 nm and = 4.24 cm. By using Eqs. (35) and (37), we illustrate in Fig. 7 the normalized spectral density of ELGSM beams propagating in turbulent atmosphere at two different receiver planes. The simulation parameters are taken the same as those of Ref. (Cang et al. 2013b). From the plots of Fig. 7, we can note that the obtained results are in good agreement with those of the reference (Cang et al. 2013b) (see Fig. 1 of this reference). In addition, the results of Ref. (Cang et al. 2013b) is considered as special case of the present work; which make our study more generalized. In this work, we have derived the analytical expression of the spectral density for GSM and GLGSM beams based on the extended Huygens-Fresnel integral formula in the paraxial approximation. The evolution of the propagation for GSM, Standard and Elegant GLGSM beams through a paraxial ABCD optical system through a turbulent atmosphere is illustrated numerically by varying the input beam parameters and the turbulent strength. From the obtained results, different shapes of beam profile are observed upon the propagation of GLGSM beams through a turbulent atmosphere, which explains that the profile of the studied beams is influenced by the source coherence parameters and the atmospheric turbulence of the environment. These characteristic properties of beams can be useful in practice, such as for optical projection and communications systems.