Abstract

A new class of twisted Schell-model array correlated sources are introduced based on Mercer’s expansion. It turns out that such sources can be expressed as superposition of fully coherent Laguerre-Gaussian modes, and the twistable condition is established. Furthermore, on the basis of a stretched coordinate system and a quadratic approximation, analytical expressions for the mutual coherence function of an anisotropic non-Kolmogorov turbulence and the cross-spectral density of a twisted Gaussian Schell-model array beam are rigorously derived. Due to the presence of the twist phase, the beam spot and the degree of coherence rotate as they propagate, but their rotation centers are different. It is shown that the anisotropy of turbulence causes an anisotropic beam spreading in the horizontal and vertical directions. However, impressing a twist phase on source beams can significantly inhibit this effect. For an anticipated atmospheric channel condition, a comprehensive selection of initial optical signal parameters, receiver aperture size and receiver capability, etc., is necessary. Our work is helpful for exploring new forms of twistable sources, and promotes guidance on optimization of partial coherent beam applications.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The manipulation and characterization of partially coherent beams and their novel spatial behaviors under propagation constitute important fundamentals in many areas of optical science and technology [1–6]. In previous decades, most of the studies concerning the spatial coherence properties have been based on Schell-model (SM) sources, in which the spectral degree of coherences obey Gaussian distributions [1–3]. It is found that spatial coherence of a partially coherent beam apparently influences the evolution behaviors of spectral density, degree of coherence, degree of polarization and state of the polarization [7–11]. Therefore, generation of prescribed structure beams with tailored intensity, polarization and phase by means of correlation functions manipulation, and exploration of nontrivial modulation properties have seen a rapid growth of interest. Based on the nonnegative definiteness criteria of the cross-spectral density (CSD), a sufficient condition for devising genuine correlation functions was proposed, and then quickly extended to vector cases [12–14]. In recent years, inspiring significant advances have been achieved in design and experimental generation of nonconventional correlation sources. It has been shown that light beams with peculiar correlation functions exhibit nontrivial propagation properties such as far-field flat-topped, ring-shaped, self-focusing, self-splitting, optical cage and optical lattices [15–27].

Twist phase is an important modulation dimension of partially coherent sources, which was introduced by Simon and later experimentally generated by Friberg [28–31]. Recently, on the basis of the nonnegative constraint of the CSD, Gori and associates introduced a new method to design twisted sources endowed with circular or rectangular symmetry [32–34]. Due to the existence of the twist phase, twisted partially coherent beams not only have the ability to carry orbital angular momentum, but also have better anti-turbulence self-repairing capabilities, which can find applications in optical tweezers, optical imaging and optical communications [35–38]. By using a modal analysis, it is possible to check whether a general correlation source can be twisted by evaluating the eigenvalues. Unfortunately, in addition to limited number of cases, it is usually not an easy work to find out the eigenvalues of a twisted partially coherent source [12, 33].

On the other hand, the study of the interaction of light beams with turbulence has attracted for a long time attention of the scientific community due to many interesting applications such as astronomy, optical communications, remote sensing, and optical imaging [39–41]. Experimental and theoretical results show that light beams with specific structured amplitude, coherence, phase and polarization can help reduce the signal distortion caused by turbulence [42–48]. For many years, atmospheric turbulence described by Kolmogorov’s power spectral density model has been applied extensively in studies of optical wave propagation in the atmosphere. However, it is shown that the atmosphere’s statistical behaviors in the upper troposphere and stratosphere, or along nonhomogeneous path does not always follow the Kolmogorov power spectrum density model, and a general non-Kolmogorov statistics should be taken into account [49–52]. In addition, experimental results also demonstrated that the turbulence in the free atmosphere can be highly anisotropic at large scales [53–56]. Therefore, it is worthwhile to find new spectrum model in order to describe anisotropic non-Kolmogorov turbulence statistical behaviors [49, 51]. In this paper, we introduce a new class of twisted Schell-model array (TSMA) correlated sources by using Mercer’s theory and the nonnegative definiteness criteria of the CSD, and the twistable condition is established as well. In addition, with the help of a stretched coordinate system and a quadratic approximation, a closed form for the mutual correlation function of a spherical wave in anisotropic non-Kolmogorov turbulence is also derived. It is seen that anisotropy of turbulence leads to an anisotropic beam spreading in the horizontal and vertical directions, and optimizing the source beam parameters such as the coherence parameter and the twist phase can significantly inhibit this effect. Our work is helpful for exploring new classes of twisted partially coherent beams, and promotes helpful guidance on optimization of partial coherent beam turbulence applications.

2. Generation of twisted Schell-model array (TSMA) coherent sources

For a generic partially coherent beam in space-frequency domain, its complete CSD with respect to arbitrary position vectors r1 and r2can be expressed as [2]

W(r1,r2)=S(r1)S(r2)g(r1r2)χ(r1,r2),
whereri(xi,yi)(i=1,2). S(r)represents the optical intensity across the source, which can be modulated by means of an arbitrary (complex) amplitude filter. g() andχ() denote the degree of coherence (DOC) and an additional phase term of the beam, respectively. Here, let’s consider g()to have an M×NSchell-model array (SMA) function [11, 22]
g(r1r2)=1C0nx=NNny=MMexp[(r1r2)22δ02]×exp[iπnx(x1x2)/a]exp[iπny(y1y2)/a],
whereC0=(2N+1)(2M+1) is a normalization constant factor andδ0is a coherence parameter of the source;nxandnyare the beam order parameter anda is a positive constant relates to the array pitch. WhenM=N=0, Eq. (2) reduces to a Gaussian function, and the corresponding CSD reduces to a standard twisted Gaussian Schell-model (TGSM) source.

It is well known that light beams generated from SM sources have the ability to possess a twist phase [28]

χ(r1,r2)=exp(iμkr1×r2),
whereμrepresents the twist factor with dimensions of the inverse of a length; (r1×r2) denotes the orthogonal component of the cross product between two position vectors r1 and r2. Moreover, the possibility of impressing a twist phase on a SM source requires the conditionμ1/kδ02. In the extreme case ofδ0, it shows that one cannot impress a twist phase on to a coherent source. This means the possibility of impressing a twist phase on a SM source depends only on the coherence and does not depend on the intensity. The physical explanation is that the amplitude can be tailored independently by means of an amplitude filter without affecting the coherence and the phase. In other words, whether a light beam can be twisted is only determined by the coherence. Therefore, for the convenience of analysis, we ignore the intensity and the CSD reduces toW¯(r1,r2)=g(|r1r2|)χ(r1,r2).

Recently, based on the nonnegative definiteness criteria of the CSD, Borghi and associates demonstrated a method to determine whether a correlation structure is capable of carrying a twist phase by means of Mercer’s expansion. For any twisted partially coherent source, a Mercer’s expansion of the CSD reads as [29, 33]

W¯(r1,r2)=j=0,1/2,1,...m=jjαj,mϕj,m*(r1)ϕj,m(r2),
where the sum is made over all the possible values of jandm; The asterisk denotes complex conjugate and αj,mis the eigenvalues. It is proven that the coherent modesϕj,m of a twisted SM source in the polar coordinates r(rcosθ,rsinθ) take on Laguerre–Gaussian form
ϕj,m(r)=kuπ[(j|m|)!(j+|m|)!]12(rkμ)2|m|Lj|m|2|m|(kμr2)exp(kμr22)exp(i2mθ),
with j=0,1/2,1,... and m=j,j+1...j. The radial index j|m|and the angular index 2mtake on nonnegative integer values; Lj|m|2|m|(kμr2)are the associated Laguerre polynomials;exp(i2mθ) is the vortex phase, and the signature of mgives the handedness. The closed form of eigenvaluesαj,m is given by
αj,m=2πδ021+kuδ02(1kuδ021+kuδ02)j+m.
Due to the nonnegative definiteness criteria of the CSD, the twistable condition of a twisted SM source requiresμ1/kδ02. Moreover, it is also proven that the Laguerre–Gaussian mode expansion approach is suitable for any more general twisted SM source, and the eigenvalue sequence {αn}n=0is defined by [33]
αn=W¯(r)Ln(kμr2)exp(kμr2/2)d2r,(n=0,1,2,3...),
where we have αj,m=αj+m. Similarly, the necessary and sufficient condition for a more general twistable SM source requires the eigenvalue αnto be nonnegative for any choose ofn.

On submitting from Eq. (2) and (3) into Eq. (7) and integrate overθ, the eigenvalues take on the form

αn=1C0nx=NNny=MM2π0J0(βr)Ln(kμr2)exp[12(δ02+kμ)r2]rdr,
whereβ=πnx2+ny2/a and J0() is a zero-order Bessel function of the first kind. On deriving Eq. (8), we have used the following identity
02πexp(iarcosθ+ibrsinθ)dθ=2πJ0(ra2+b2).
Then, with the help of the formula 7.421.4 of [57], Eq. (8) turns out to be
αn=1C0nx=NNny=MM2πδ021+kuδ02(1kuδ021+kuδ02)nexp[β2δ022(1+kuδ02)]Ln(β2kuδ04k2u2δ041).
It is of interest to find from Eq. (10) that the eigenvalues of a twisted SMA source has the same form of a twisted J0-correlated Gaussian Schell-model source except for the sum item [33]. Under the condition ofM=N=0. Equation (10) reduces to Eq. (6) associated with a twisted SM source. Since the Laguerre polynomials are nonnegative for negative values of their argument, it is easily to find that the nonnegative condition of Eq. (10) is the same as a twisted SM source. Consequently, we have proven that SMA correlated sources have the ability to be twisted and the twistable condition isμ1/kδ02.

3. Propagation of a twisted Gaussian Schell-model array (TGSMA) beam through anisotropic turbulence

In this section, we further extend the work to study the propagation of a TGSMA beam in anisotropic turbulence (Fig. 1). For the sake of convenience, let us consider the source beam has an elliptical Gaussian spectral density and the CSD is expressed as

 figure: Fig. 1

Fig. 1 Propagation of a TGSMA beam in anisotropic turbulence.

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W(r1,r2)=1C0nx=NNny=MMexp(x12+x224wx2y12+y224wy2)exp((r1r2)22δ02)×exp[iπnxa(x1x2)]exp[iπnya(y1y2)]exp(ikur1×r2).

Under the framework of the narrow-angle propagation assumption, the propagation the CSD of a TGSMA beam in weak turbulence can be evaluated by means of the generalized Huygens-Fresnel integral [40]

W(ρ1,ρ2)=(1λz)2W(r1,r2)exp[ik2z(r122ρ1r1+2ρ2r2r22)]×exp[ik2z(ρ12ρ22)]exp[Ψ(r1,ρ1)+Ψ*(r2,ρ2)]d2r1d2r2,
whereρi(ui,vi)(i=1,2) are two arbitrary points at the output plane; perpendicular to z-axis; k=2π/λ(whereλ is the wavelength); the angular brackets denote averaging over the ensemble of turbulent media, and Ψ(r,ρ)is the complex phase perturbation of a spherical wave propagation through turbulence. According to [41], the last term is the mutual correlation function of turbulence having the form
exp[Ψ(r1,ρ1)+Ψ*(r2,ρ2)]=exp(2πk2z01dt0Φn(κ)d3κ{1exp[tρdκ+(1t)rdκ]}),
where ρd=ρ1ρ2andrd=r1r2, and Φn(κ)is the three-dimensional spectral power spectrum of the refractive-index fluctuation in turbulence with κ(κx,κy,κz) being the spatial frequency.

Here, let us consider a generalized non-Kolmogorov anisotropic power spectrum introduced by Andrews [51].

Φn(κx,κy,κz)=εxεyC˜n2A(ξ)(εx2κx2+εy2κy2+κz2)ξ/2,
with
A(ξ)=Γ(ξ1)cos(πξ/2)4π2,3<ξ<4,
whereκ=εx2κx2+εy2κy2+κz2andξis the spectral index or power law; εxand εyare the anisotropic factor in two transverse directions; the symbolΓ()denotes the Gamma function. Whenξ=11/3, andεx=εy=1, the generalized power spectrum reduces to a common isotropic Kolmogorov spectrum. Forεxεy, the orthogonal xy-plane will no longer be circularly symmetric (i.e., isotropic) and this will lead to different statistical values in the horizontal and vertical transverse directions. For further study, we have considered the Markov approximation, which means that the refractive index is supposed to be uncorrelated between two different points along the direction of propagation path. Therefore, the spatial wave number component κzalong z-axis is ignored (i.e., κz=0).

For further analysis of anisotropic atmospheric turbulence, we change the stretched coordinate system for the anisotropic spectrum back to an isotropic one by means of the following substitutions

κx=qxεx=qcosφεx,κy=qyεy=qsinφεy,dκxdκy=dqxdqyεxεy=qdqdφεxεy,q=qx2+qy2,ρd(udεx,vdεy),rd(xdεx,ydεy).
Then, on substituting from Eqs. (14)-(16) into Eq. (13), and integrate overφ, we have
exp[Ψ(r1,ρ1)+Ψ*(r2,ρ2)]=exp(4π2A(ξ)C˜n2k2z01dt0q1ξ{1J0[q|tρd+(1t)rd|]}dq),
with the help of the stretched coordinate system, Eq. (17) has the same form as the mutual correlation function in an isotropic turbulence. Then, by making use of the identity
0x1p[1J0(x|Q|)]dx=2pp|Q|p2Γ(p/2)Γ(p/2),(2<Re(p)<4),
Equation (17) turns out to be
exp[Ψ(r1,ρ1)+Ψ*(r2,ρ2)]=exp[ξΓ(ξ1)Γ(ξ/2)2ξ(ξ1)Γ(ξ/2)cos(πξ2)C˜n2k2z(ρdξ1rdξ1|ρdrd|)].
Under the condition ofξ=11/3, Eq. (19) coincides with Eq. (24) in [41]. Although a complicated numerical calculation is possible for further study, this seems not very fruitful in understanding the propagation characteristics. We shall, therefore, consider a quadratic approximation of Eq. (19), which then can be solved in a closed form. This has been shown to be a good approximation for practical situations [40]. Thus, the mutual correlation function of a spherical wave in anisotropic turbulence can be expressed as
exp[Ψ(r1,ρ1)+Ψ*(r2,ρ2)]exp[1ρ02(z)(ud2+udxd+xd2εx2+vd2+vdyd+yd2εy2)],
with
ρ0(z)=[ξΓ(ξ1)Γ(ξ/2)cos(πξ/2)2ξ(ξ1)Γ(ξ/2)C˜n2k2z]12ξ,
where ρ0(z)is the spatial coherence radius of a spherical wave in anisotropic turbulence. It is important to note that in order to obtain the spatial coherence radius of anisotropic turbulence, a stretched coordinate system have been used.

On substituting from Eqs. (11), (20) and (21) into Eq. (12) and after tedious integration, we finally obtain the following expression for the CSD of a TGSMA beam in the receiver plane

W(ρ1,ρ2)=nx=NNny=MMV(ρ1,ρ2)exp(γu124Χ(+)+γv124ϒ(+)+Ωu24Χ()+Ωv24ϒ()),
with
V(ρ1,ρ2)=k24C0z2Χ(+)ϒ(+)Χ()ϒ()exp[ik2z(ρ12ρ22)ρd2ρ02(z)]γu1=iku1z+iπnxaudεx2ρ02(z),γu2=iku2z+iπnxaudεx2ρ02(z),Δx=12δ02+1εx2ρ02(z),γv1=ikv1z+iπnyavdεy2ρ02(z),γv2=ikv2z+iπnyavdεy2ρ02(z),Δy=12δ02+1εy2ρ02(z),Χ(+)=Δx+14wx2+ik2z,Χ()=Δx+14wx2ik2zΔx2Χ(+)+k2μ24ϒ(+),Ωu=Δxγu1Χ(+)γu2+ikμγv12ϒ(+),ϒ(+)=Δy+14wy2+ik2z,ϒ()=Δy+14wy2ik2zΔy2ϒ(+)+k2μ24Χ(+)+k2μ24Χ()(Δyϒ(+)ΔxΧ(+))2,Ωv=Δyγv1ϒ(+)γv2ikμγu12Χ(+)+ikμΩu2Χ()(Δyϒ(+)ΔxΧ(+)).
Equation (23) provides a convenient analytical expression to study the statistical properties of the TAGSM beam on propagation through anisotropic turbulence such as the spectral density and the DOC

S(ρ)=W(ρ,ρ),
η(ρ1,ρ2)=W(ρ1,ρ2)W(ρ1,ρ1)W(ρ2,ρ2).

4. Numerical analysis

In this section, we numerically analyze the propagation properties of a TGSMA beam propagating through anisotropic non-Kolmogorov turbulence. The source beam parameters the turbulence parameters are set as λ=632.8nm,wx=2wy=4cm,δ0=2cm,a=0.6cm, M=N=1,μ=0.5km1,C˜n2=5×1015m3ξ,εx=1,ξ=11/3,unless different values are specified. Figure 2 depicts the contour plots of the normalized spectral density distribution and the corresponding cross line of a TGSMA beam at different propagation distances. Due to the reciprocal relationship between source coherence and the far-field spectral density distribution, the initial elliptical GSMA beam spot gradually evolves into an array structure in isotropic turbulence [see Figs. 2(a1)-2(a3)]. Meanwhile, Fig. 2(a4) illustrates that an array spectral density will further evolve into a flat-top distribution as the propagation distance further increases due to the turbulence statistics. A comparison of Figs. 2(a1)-2(a4) and Figs. 2(b1)-2(b4) clearly shows that the spectral density distribution within a few kilometers is more sensitive to the source beam parameters than turbulence statistics. While for sufficient distance transmission, the spectral density distribution is determined at most by turbulence statistics, and the anisotropy of turbulence results in an anisotropic beam spreading. Figures 2(c) and 2(d) illustrate the effects of the twist phase and the initial coherence parameter on the spectral density. It is clearly seen that the twist phase not only gives rise to a rotation of beam spot, but also accelerates the beam diffraction, which is similar to that of conventional TGSM beams [35]. As can been seen from Fig. 2(d4) that each far-field elliptical beamlet gradually turns into a circular spot as the initial coherence decreases. This implies that reducing the initial coherence can also help resist the turbulence-induced anisotropy [55,56].

 figure: Fig. 2

Fig. 2 Contour plots of normalized spectral density distributions and the corresponding cross line of a TGSMA beam at several different propagation distances. (a1)-(a4) εy=1,μ=0; (b1)-(b4) εy=3,μ=0; (c1)-(c4)εy=3; (d1)-(d4) εy=3,a=2δ0=12mm.

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In order to examine the effects of turbulence parameters on the far-field spectral density, Fig. 3 illustrates the spectral density distribution of a TGSMA beam at z = 5km for different turbulence parameters. In the absence of the twist phase, the anisotropy of turbulence leads to an anisotropic beam spreading in the horizontal and vertical directions [see Figs. 3(a1)-3(a4)]. For the case ofεx>εy, the long axis of elliptical beamlets is along the y (vertical) direction, while horizontal (long axis is along the x direction) elliptical beamlets correspond toεx<εy. Moreover, due to the inverse square relationship between spectral density and the anisotropy parameter, a smaller anisotropic parameter means a stronger turbulence effect [56]. It is of interest to find from Figs. 3(c1)-3(c4) that the twist phase-induced diffraction increases dramatically with the growth of the twist phase, and plays a major role in determining the spectral density profile. Therefore, a reasonable optimization of the twist phase is beneficial to suppress the effects of turbulence. One possible reason is that the twist phase leads to a more rapid beam diffraction. As a result, a relatively larger beam spot helps to enhance the ability to reduce the turbulence effect [40, 46].

 figure: Fig. 3

Fig. 3 Contour plots of the normalized spectral density distribution and the corresponding cross line of a TGSMA beam at z = 5km. (a1)-(a4)μ=0; (b1)-(b4)μ=0.5km1; (c1)-(c4)μ=1km1.

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Figure 4 illustrates the evolution of the on-axis spectral density as a function of power law ξand anisotropic factor εyfor different values of the twist phaseμ and coherence parameterδ0 at z = 5km. It is clearly seen from Fig. 4(a1) that a GSMA beam without the twist phaseμis more sensitive to the turbulence statistics than a TGSMA beam. Meanwhile, as the increase of the value of the twist phaseμ, the effect of turbulence decreases rapidly. Besides, a significant impact of the anisotropic factor εyon the on-axis spectral density mainly appears nearξ=3.8for a GSMA beam. Similar theoretical simulations for beam width can be found in [49]. The physical interpretation of the maximum value of spectral density appearing nearξ3is that turbulence tends to vanish. On the other extreme ofξ4, the physical explanation is that phase effects dominate in the form of random tilts, which cause beam wander. Compared with Figs. 4(a1)-4(a3) and Figs. 4(b1)-4(b3), one finds that the dependence of the beam on turbulence statistics decreases as the coherence parameter decreases. Therefore, in other words, an appropriate reduction in initial coherence and use of the twist phase can effectively reduce the effects of turbulence on signal beams. However, it is worthwhile to note that it is very difficult to achieve very long-distance transmission of optical signal applications by manipulating the initial beam parameters [42, 43].

 figure: Fig. 4

Fig. 4 On-axis spectral density as a function of ξ and εy at z = 5km, (a1)-(a4)δ0=2cm; (b1)-(b4)δ0=1cm.

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Now, let’s turn our attention to evaluate the evolution properties of the DOC of a TGSMA beam in anisotropic turbulence. Figure 5 shows the DOC and the corresponding cross line of a TGSMA beam at several different propagation distances. It is clearly seen that for relatively short distance turbulence transmission, the initial coherence and the twist phase play a significant role in determining the DOC, while the anisotropy parameter has little effects on the DOC. However, it is worth noting that for sufficiently long distance transmission, the spatial coherence property are mainly determined by turbulence statistics [40, 42]. In addition, it is of interest to find that although the spectral density and the DOC rotate during transmission, their rotation center are different. The rotation center of the spectral density splits as the beam splits during transmission, while the rotation center of the DOC remains invariant.

 figure: Fig. 5

Fig. 5 The modulus of the DOCη(u,v,0,0) and the corresponding cross line of a TGSMA beam at several different propagation distances. (a1)-(a4) εy=1,μ=0; (b1)-(b4) εy=3,μ=0; (c1)-(c4) εy=3; (d1)-(d4) εy=3,a=2δ0=12mm.

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Figure 6 illustrates the DOC and the corresponding cross line of a TGSMA beam at several different propagation distances. It is interesting to find that the DOC rotates clockwise in the case ofεx>εy, while it rotates counterclockwise forεx<εy . In addition, as the increase of the twist phase, the array structure of the DOC gradually appears, which is completely opposite to the evolution of spectral density. In other words, the ability of a TGSMA beam to resist anisotropic turbulence is enhanced as the twist phase increases. In Fig. 7, we display the behavior of the DOC η(u,0,0,0) at z = 5km as a function of vertical anisotropic factor εy or power lawξ for different twist phaseμ. In the absence of the twist phase [see Fig. 7(a1)] for a conventional Kolmogorov turbulence, the DOC is insensitive to the vertical anisotropic parameter εyand maintains Gaussian distribution. While for a TGSMA beam, Figs. 7(a2) and 7(a3) show that the DOC has been a significant change aroundεy=1. Meanwhile, it is found that the DOC array structure alongudirection significantly enhanced with the increase of anisotropic parameterεy, which means that the influence of turbulence is weakened. In addition, a remarkable change in the DOC near the boundary value of power lawξindicates that the turbulence strength changes sharply. Furthermore, a comparison of Figs. 7(b2) and 7(b3) shows the array structure of the DOC grows apparently with the increase of the twist phase. This implies that the use of a twisted partially coherent beam is beneficial for suppressing the effect of turbulence.

 figure: Fig. 6

Fig. 6 The modulus of the DOC η(u,v,0,0)and the corresponding cross line of a TGSMA beam at z = 5km for different turbulence statistics and initial coherence. (a1)-(a4) μ=0; (b1)-(b4) μ=0.5km1; (c1)-(c4) μ=1km1.

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 figure: Fig. 7

Fig. 7 The modulus of the DOC η(u,0,0,0) for different values of power law ξ and anisotropic factorεy at z = 5km. (a1)-(a3)ξ=11/3; (b1)-(b3)εy=3.

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5. Conclusion

In this work, we have introduced a new class of TGSMA correlated sources based on the nonnegative definiteness criteria of the CSD and a Mercer’s expansion. It turns out that such sources can be expressed as the superposition of fully coherent Laguerre-Gaussian modes, and the twistable condition is established. Furthermore, we extend our work to evaluate the statistical properties of a TGSMA beam in an anisotropic non-Kolmogorov turbulence. It has been seen that the twist phase not only gives rise to a rotation of beam spot, but also accelerates the beam diffraction. One also finds that the rotation center of the spectral density splits as the beam splits during transmission, while the rotation center of the DOC remains invariant. Moreover, the DOC distribution rotates clockwise in the case ofεx>εy, while a counterclockwise rotation corresponds toεx<εy. The anisotropy of turbulence causes an anisotropic beam diffraction in the horizontal and vertical directions. However, using a twisted partially coherent beam can significantly inhibit this effect. It is worth noting that optimizing the initial beam parameters such as the coherence and the twist phase can help enhance turbulent resistance. However, this results in a larger beam spot and a reduced power incident on the receiver. Therefore, a comprehensive selection of initial beam parameters, receiver aperture size and receiver capability, etc., for the anticipated atmospheric channel conditions is necessary. Our work is helpful for exploring new classes of twisted partially coherent beams, and promotes helpful guidance on optimization of partial coherent beam turbulence applications.

Funding

National Natural Science Foundation of China (NSFC) (11504172, 91750201); National Natural Science Fund for Distinguished Young Scholar (11525418); Natural Science Foundation of Jiangsu Province (BK20150763); China Postdoctoral Science Foundation (7131701018); Postdoctoral Science Foundation of Jiangsu Province (7131707317); Fundamental Research Funds for the Central Universities (30917011336, 30916014112-001); Jiangsu Overseas Visiting Scholar Program for University Prominent Young & Middle-aged Teachers and Presidents.

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19. Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review,” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014). [CrossRef]   [PubMed]  

20. O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014). [CrossRef]   [PubMed]  

21. L. Ma and S. A. Ponomarenko, “Optical coherence gratings and lattices,” Opt. Lett. 39(23), 6656–6659 (2014). [CrossRef]   [PubMed]  

22. Z. Mei, D. Zhao, O. Korotkova, and Y. Mao, “Gaussian Schell-model arrays,” Opt. Lett. 40(23), 5662–5665 (2015). [CrossRef]   [PubMed]  

23. L. Wan and D. Zhao, “Optical coherence grids and their propagation characteristics,” Opt. Express 26(2), 2168–2180 (2018). [CrossRef]   [PubMed]  

24. S. Zhu, Y. Chen, J. Wang, H. Wang, Z. Li, and Y. Cai, “Generation and propagation of a vector cosine-Gaussian correlated beam with radial polarization,” Opt. Express 23(26), 33099–33115 (2015). [CrossRef]   [PubMed]  

25. M. W. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl. 6(6), 064030 (2016). [CrossRef]  

26. M. Santarsiero, R. Martínez-Herrero, D. Maluenda, J. C. G. de Sande, G. Piquero, and F. Gori, “Partially coherent sources with circular coherence,” Opt. Lett. 42(8), 1512–1515 (2017). [CrossRef]   [PubMed]  

27. C. Ding, M. Koivurova, J. Turunen, and L. Pan, “Self-focusing of a partially coherent beam with circular coherence,” J. Opt. Soc. Am. A 34(8), 1441–1447 (2017). [CrossRef]   [PubMed]  

28. R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993). [CrossRef]  

29. R. Simon, N. Mukunda, and K. Sundar, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10(9), 2008 (1993). [CrossRef]  

30. R. Simon and G. S. Agarwal, “Wigner representation of Laguerre--Gaussian beams,” Opt. Lett. 25(18), 1313–1315 (2000). [CrossRef]   [PubMed]  

31. A. T. Friberg, B. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11(6), 1818–1826 (1994). [CrossRef]  

32. F. Gori and M. Santarsiero, “Devising genuine twisted cross-spectral densities,” Opt. Lett. 43(3), 595–598 (2018). [CrossRef]   [PubMed]  

33. R. Borghi, F. Gori, G. Guattari, and M. Santarsiero, “Twisted Schell-model beams with axial symmetry,” Opt. Lett. 40(19), 4504–4507 (2015). [CrossRef]   [PubMed]  

34. R. Borghi, “Twisting partially coherent light,” Opt. Lett. 43(8), 1627–1630 (2018). [CrossRef]   [PubMed]  

35. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006). [CrossRef]  

36. F. Wang and Y. Cai, “Second-order statistics of a twisted Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 18(24), 24661–24672 (2010). [CrossRef]   [PubMed]  

37. Y. Cai and S. Zhu, “Orbital angular moment of a partially coherent beam propagating through an astigmatic ABCD optical system with loss or gain,” Opt. Lett. 39(7), 1968–1971 (2014). [CrossRef]   [PubMed]  

38. Z. Mei and O. Korotkova, “Random sources for rotating spectral densities,” Opt. Lett. 42(2), 255–258 (2017). [CrossRef]   [PubMed]  

39. A. Ishimaru, Wave propagation and scattering in random media (Academic, 1978).

40. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Medium, 2nd ed. (SPIE, 2005).

41. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11(6), 1399–1406 (1972). [CrossRef]   [PubMed]  

42. S. A. Ponomarenko and E. Wolf, “Solution to the inverse scattering problem for strongly fluctuating media using partially coherent light,” Opt. Lett. 27(20), 1770–1772 (2002). [CrossRef]   [PubMed]  

43. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19(8), 1592–1598 (2002). [CrossRef]   [PubMed]  

44. M. Tang and D. Zhao, “Propagation of multi-Gaussian Schell-model vortex beams in isotropic random media,” Opt. Express 23(25), 32766–32776 (2015). [CrossRef]   [PubMed]  

45. G. R. Salla, C. Perumangattu, S. Prabhakar, A. Anwar, and R. P. Singh, “Recovering the vorticity of a light beam after scattering,” Appl. Phys. Lett. 107(2), 021104 (2016). [CrossRef]  

46. S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010). [CrossRef]   [PubMed]  

47. J. Wang, S. Zhu, H. Wang, Y. Cai, and Z. Li, “Second-order statistics of a radially polarized cosine-Gaussian correlated Schell-model beam in anisotropic turbulence,” Opt. Express 24(11), 11626–11639 (2016). [CrossRef]   [PubMed]  

48. X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, “Propagation of optical coherence lattices in the turbulent atmosphere,” Opt. Lett. 41(18), 4182–4185 (2016). [CrossRef]   [PubMed]  

49. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008). [CrossRef]  

50. I. Toselli, O. Korotkova, X. Xiao, and D. G. Voelz, “SLM-based laboratory simulations of Kolmogorov and non-Kolmogorov anisotropic turbulence,” Appl. Opt. 54(15), 4740–4744 (2015). [CrossRef]   [PubMed]  

51. L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014). [CrossRef]  

52. J. Wang, S. Zhu, and Z. Li, “Vector properties of a tunable random electromagnetic beam in non-Kolmogrov turbulence,” Chin. Opt. Lett. 14(8), 80101–80105 (2016). [CrossRef]  

53. A. I. Kon, “Qualitative theory of amplitude and phase fluctuations in a medium with anisotropic turbulent irregularity,” Waves Random Complex Media 4(3), 297–306 (1994). [CrossRef]  

54. Y. Baykal, Y. Luo, and X. Ji, “Scintillations of higher order laser beams in anisotropic atmospheric turbulence,” Appl. Opt. 55(33), 9422–9426 (2016). [CrossRef]   [PubMed]  

55. F. Wang, I. Toselli, J. Li, and O. Korotkova, “Measuring anisotropy ellipse of atmospheric turbulence by intensity correlations of laser light,” Opt. Lett. 42(6), 1129–1132 (2017). [CrossRef]   [PubMed]  

56. J. Wang, H. Wang, S. Zhu, and Z. Li, “Second-order moments of a twisted Gaussian Schell-model beam in anisotropic turbulence,” J. Opt. Soc. Am. A 35(7), 1173–1179 (2018). [CrossRef]   [PubMed]  

57. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

References

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  16. F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
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  17. S. G. Reddy, A. Kumar, S. Prabhakar, and R. P. Singh, “Experimental generation of ring-shaped beams with random sources,” Opt. Lett. 38(21), 4441–4444 (2013).
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  18. Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91, 013832 (2014).
  19. Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review,” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
    [Crossref] [PubMed]
  20. O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
    [Crossref] [PubMed]
  21. L. Ma and S. A. Ponomarenko, “Optical coherence gratings and lattices,” Opt. Lett. 39(23), 6656–6659 (2014).
    [Crossref] [PubMed]
  22. Z. Mei, D. Zhao, O. Korotkova, and Y. Mao, “Gaussian Schell-model arrays,” Opt. Lett. 40(23), 5662–5665 (2015).
    [Crossref] [PubMed]
  23. L. Wan and D. Zhao, “Optical coherence grids and their propagation characteristics,” Opt. Express 26(2), 2168–2180 (2018).
    [Crossref] [PubMed]
  24. S. Zhu, Y. Chen, J. Wang, H. Wang, Z. Li, and Y. Cai, “Generation and propagation of a vector cosine-Gaussian correlated beam with radial polarization,” Opt. Express 23(26), 33099–33115 (2015).
    [Crossref] [PubMed]
  25. M. W. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl. 6(6), 064030 (2016).
    [Crossref]
  26. M. Santarsiero, R. Martínez-Herrero, D. Maluenda, J. C. G. de Sande, G. Piquero, and F. Gori, “Partially coherent sources with circular coherence,” Opt. Lett. 42(8), 1512–1515 (2017).
    [Crossref] [PubMed]
  27. C. Ding, M. Koivurova, J. Turunen, and L. Pan, “Self-focusing of a partially coherent beam with circular coherence,” J. Opt. Soc. Am. A 34(8), 1441–1447 (2017).
    [Crossref] [PubMed]
  28. R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993).
    [Crossref]
  29. R. Simon, N. Mukunda, and K. Sundar, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10(9), 2008 (1993).
    [Crossref]
  30. R. Simon and G. S. Agarwal, “Wigner representation of Laguerre--Gaussian beams,” Opt. Lett. 25(18), 1313–1315 (2000).
    [Crossref] [PubMed]
  31. A. T. Friberg, B. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11(6), 1818–1826 (1994).
    [Crossref]
  32. F. Gori and M. Santarsiero, “Devising genuine twisted cross-spectral densities,” Opt. Lett. 43(3), 595–598 (2018).
    [Crossref] [PubMed]
  33. R. Borghi, F. Gori, G. Guattari, and M. Santarsiero, “Twisted Schell-model beams with axial symmetry,” Opt. Lett. 40(19), 4504–4507 (2015).
    [Crossref] [PubMed]
  34. R. Borghi, “Twisting partially coherent light,” Opt. Lett. 43(8), 1627–1630 (2018).
    [Crossref] [PubMed]
  35. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
    [Crossref]
  36. F. Wang and Y. Cai, “Second-order statistics of a twisted Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 18(24), 24661–24672 (2010).
    [Crossref] [PubMed]
  37. Y. Cai and S. Zhu, “Orbital angular moment of a partially coherent beam propagating through an astigmatic ABCD optical system with loss or gain,” Opt. Lett. 39(7), 1968–1971 (2014).
    [Crossref] [PubMed]
  38. Z. Mei and O. Korotkova, “Random sources for rotating spectral densities,” Opt. Lett. 42(2), 255–258 (2017).
    [Crossref] [PubMed]
  39. A. Ishimaru, Wave propagation and scattering in random media (Academic, 1978).
  40. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Medium, 2nd ed. (SPIE, 2005).
  41. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11(6), 1399–1406 (1972).
    [Crossref] [PubMed]
  42. S. A. Ponomarenko and E. Wolf, “Solution to the inverse scattering problem for strongly fluctuating media using partially coherent light,” Opt. Lett. 27(20), 1770–1772 (2002).
    [Crossref] [PubMed]
  43. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19(8), 1592–1598 (2002).
    [Crossref] [PubMed]
  44. M. Tang and D. Zhao, “Propagation of multi-Gaussian Schell-model vortex beams in isotropic random media,” Opt. Express 23(25), 32766–32776 (2015).
    [Crossref] [PubMed]
  45. G. R. Salla, C. Perumangattu, S. Prabhakar, A. Anwar, and R. P. Singh, “Recovering the vorticity of a light beam after scattering,” Appl. Phys. Lett. 107(2), 021104 (2016).
    [Crossref]
  46. S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010).
    [Crossref] [PubMed]
  47. J. Wang, S. Zhu, H. Wang, Y. Cai, and Z. Li, “Second-order statistics of a radially polarized cosine-Gaussian correlated Schell-model beam in anisotropic turbulence,” Opt. Express 24(11), 11626–11639 (2016).
    [Crossref] [PubMed]
  48. X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, “Propagation of optical coherence lattices in the turbulent atmosphere,” Opt. Lett. 41(18), 4182–4185 (2016).
    [Crossref] [PubMed]
  49. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
    [Crossref]
  50. I. Toselli, O. Korotkova, X. Xiao, and D. G. Voelz, “SLM-based laboratory simulations of Kolmogorov and non-Kolmogorov anisotropic turbulence,” Appl. Opt. 54(15), 4740–4744 (2015).
    [Crossref] [PubMed]
  51. L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
    [Crossref]
  52. J. Wang, S. Zhu, and Z. Li, “Vector properties of a tunable random electromagnetic beam in non-Kolmogrov turbulence,” Chin. Opt. Lett. 14(8), 80101–80105 (2016).
    [Crossref]
  53. A. I. Kon, “Qualitative theory of amplitude and phase fluctuations in a medium with anisotropic turbulent irregularity,” Waves Random Complex Media 4(3), 297–306 (1994).
    [Crossref]
  54. Y. Baykal, Y. Luo, and X. Ji, “Scintillations of higher order laser beams in anisotropic atmospheric turbulence,” Appl. Opt. 55(33), 9422–9426 (2016).
    [Crossref] [PubMed]
  55. F. Wang, I. Toselli, J. Li, and O. Korotkova, “Measuring anisotropy ellipse of atmospheric turbulence by intensity correlations of laser light,” Opt. Lett. 42(6), 1129–1132 (2017).
    [Crossref] [PubMed]
  56. J. Wang, H. Wang, S. Zhu, and Z. Li, “Second-order moments of a twisted Gaussian Schell-model beam in anisotropic turbulence,” J. Opt. Soc. Am. A 35(7), 1173–1179 (2018).
    [Crossref] [PubMed]
  57. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

2018 (4)

2017 (5)

2016 (7)

S. Zhu, J. Wang, X. Liu, Y. Cai, and Z. Li, “Generation of arbitrary radially polarized array beams by manipulating correlation structure,” Appl. Phys. Lett. 109(16), 161904 (2016).
[Crossref]

M. W. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl. 6(6), 064030 (2016).
[Crossref]

Y. Baykal, Y. Luo, and X. Ji, “Scintillations of higher order laser beams in anisotropic atmospheric turbulence,” Appl. Opt. 55(33), 9422–9426 (2016).
[Crossref] [PubMed]

J. Wang, S. Zhu, and Z. Li, “Vector properties of a tunable random electromagnetic beam in non-Kolmogrov turbulence,” Chin. Opt. Lett. 14(8), 80101–80105 (2016).
[Crossref]

J. Wang, S. Zhu, H. Wang, Y. Cai, and Z. Li, “Second-order statistics of a radially polarized cosine-Gaussian correlated Schell-model beam in anisotropic turbulence,” Opt. Express 24(11), 11626–11639 (2016).
[Crossref] [PubMed]

X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, “Propagation of optical coherence lattices in the turbulent atmosphere,” Opt. Lett. 41(18), 4182–4185 (2016).
[Crossref] [PubMed]

G. R. Salla, C. Perumangattu, S. Prabhakar, A. Anwar, and R. P. Singh, “Recovering the vorticity of a light beam after scattering,” Appl. Phys. Lett. 107(2), 021104 (2016).
[Crossref]

2015 (5)

2014 (6)

2013 (2)

2012 (1)

2010 (3)

2009 (3)

2008 (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

2007 (1)

2006 (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
[Crossref]

2002 (2)

2001 (1)

2000 (1)

1996 (1)

E. Wolf and D. F. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59(6), 771–818 (1996).
[Crossref]

1994 (2)

A. T. Friberg, B. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11(6), 1818–1826 (1994).
[Crossref]

A. I. Kon, “Qualitative theory of amplitude and phase fluctuations in a medium with anisotropic turbulent irregularity,” Waves Random Complex Media 4(3), 297–306 (1994).
[Crossref]

1993 (2)

1991 (1)

1972 (1)

Agarwal, G. S.

Andrews, L. C.

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

Anwar, A.

G. R. Salla, C. Perumangattu, S. Prabhakar, A. Anwar, and R. P. Singh, “Recovering the vorticity of a light beam after scattering,” Appl. Phys. Lett. 107(2), 021104 (2016).
[Crossref]

Baykal, Y.

Borghi, R.

Bose-Pillai, S.

M. W. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl. 6(6), 064030 (2016).
[Crossref]

Cai, Y.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

S. Zhu, J. Wang, X. Liu, Y. Cai, and Z. Li, “Generation of arbitrary radially polarized array beams by manipulating correlation structure,” Appl. Phys. Lett. 109(16), 161904 (2016).
[Crossref]

J. Wang, S. Zhu, H. Wang, Y. Cai, and Z. Li, “Second-order statistics of a radially polarized cosine-Gaussian correlated Schell-model beam in anisotropic turbulence,” Opt. Express 24(11), 11626–11639 (2016).
[Crossref] [PubMed]

X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, “Propagation of optical coherence lattices in the turbulent atmosphere,” Opt. Lett. 41(18), 4182–4185 (2016).
[Crossref] [PubMed]

S. Zhu, Y. Chen, J. Wang, H. Wang, Z. Li, and Y. Cai, “Generation and propagation of a vector cosine-Gaussian correlated beam with radial polarization,” Opt. Express 23(26), 33099–33115 (2015).
[Crossref] [PubMed]

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91, 013832 (2014).

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review,” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
[Crossref] [PubMed]

Y. Cai and S. Zhu, “Orbital angular moment of a partially coherent beam propagating through an astigmatic ABCD optical system with loss or gain,” Opt. Lett. 39(7), 1968–1971 (2014).
[Crossref] [PubMed]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

F. Wang and Y. Cai, “Second-order statistics of a twisted Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 18(24), 24661–24672 (2010).
[Crossref] [PubMed]

S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010).
[Crossref] [PubMed]

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
[Crossref]

Chen, Y.

Crabbs, R.

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

de Sande, J. C. G.

Ding, C.

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

Friberg, A. T.

Gbur, G.

G. Gbur and T. D. Visser, “The structure of partially coherent fields,” Prog. Opt. 55, 285–341 (2010).
[Crossref]

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19(8), 1592–1598 (2002).
[Crossref] [PubMed]

Gori, F.

Gu, J.

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91, 013832 (2014).

Guattari, G.

He, S.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
[Crossref]

Hyde, M. W.

M. W. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl. 6(6), 064030 (2016).
[Crossref]

James, D. F.

E. Wolf and D. F. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59(6), 771–818 (1996).
[Crossref]

Ji, X.

Koivurova, M.

Kon, A. I.

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S. Zhu, J. Wang, X. Liu, Y. Cai, and Z. Li, “Generation of arbitrary radially polarized array beams by manipulating correlation structure,” Appl. Phys. Lett. 109(16), 161904 (2016).
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X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, “Propagation of optical coherence lattices in the turbulent atmosphere,” Opt. Lett. 41(18), 4182–4185 (2016).
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F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
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Ma, L.

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Pan, L.

Perumangattu, C.

G. R. Salla, C. Perumangattu, S. Prabhakar, A. Anwar, and R. P. Singh, “Recovering the vorticity of a light beam after scattering,” Appl. Phys. Lett. 107(2), 021104 (2016).
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Ponomarenko, S. A.

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G. R. Salla, C. Perumangattu, S. Prabhakar, A. Anwar, and R. P. Singh, “Recovering the vorticity of a light beam after scattering,” Appl. Phys. Lett. 107(2), 021104 (2016).
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G. R. Salla, C. Perumangattu, S. Prabhakar, A. Anwar, and R. P. Singh, “Recovering the vorticity of a light beam after scattering,” Appl. Phys. Lett. 107(2), 021104 (2016).
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Setälä, T.

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G. R. Salla, C. Perumangattu, S. Prabhakar, A. Anwar, and R. P. Singh, “Recovering the vorticity of a light beam after scattering,” Appl. Phys. Lett. 107(2), 021104 (2016).
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G. Gbur and T. D. Visser, “The structure of partially coherent fields,” Prog. Opt. 55, 285–341 (2010).
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M. W. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl. 6(6), 064030 (2016).
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M. W. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl. 6(6), 064030 (2016).
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I. Toselli, O. Korotkova, X. Xiao, and D. G. Voelz, “SLM-based laboratory simulations of Kolmogorov and non-Kolmogorov anisotropic turbulence,” Appl. Opt. 54(15), 4740–4744 (2015).
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Appl. Phys. Lett. (3)

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[Crossref]

S. Zhu, J. Wang, X. Liu, Y. Cai, and Z. Li, “Generation of arbitrary radially polarized array beams by manipulating correlation structure,” Appl. Phys. Lett. 109(16), 161904 (2016).
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Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
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Opt. Eng. (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

Opt. Express (6)

Opt. Lett. (19)

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F. Gori and M. Santarsiero, “Devising genuine twisted cross-spectral densities,” Opt. Lett. 43(3), 595–598 (2018).
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R. Borghi, F. Gori, G. Guattari, and M. Santarsiero, “Twisted Schell-model beams with axial symmetry,” Opt. Lett. 40(19), 4504–4507 (2015).
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F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
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S. G. Reddy, A. Kumar, S. Prabhakar, and R. P. Singh, “Experimental generation of ring-shaped beams with random sources,” Opt. Lett. 38(21), 4441–4444 (2013).
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S. A. Ponomarenko and E. Wolf, “Solution to the inverse scattering problem for strongly fluctuating media using partially coherent light,” Opt. Lett. 27(20), 1770–1772 (2002).
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X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, “Propagation of optical coherence lattices in the turbulent atmosphere,” Opt. Lett. 41(18), 4182–4185 (2016).
[Crossref] [PubMed]

M. Santarsiero, R. Martínez-Herrero, D. Maluenda, J. C. G. de Sande, G. Piquero, and F. Gori, “Partially coherent sources with circular coherence,” Opt. Lett. 42(8), 1512–1515 (2017).
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Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91, 013832 (2014).

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M. W. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl. 6(6), 064030 (2016).
[Crossref]

Proc. SPIE (1)

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

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G. Gbur and T. D. Visser, “The structure of partially coherent fields,” Prog. Opt. 55, 285–341 (2010).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 Propagation of a TGSMA beam in anisotropic turbulence.
Fig. 2
Fig. 2 Contour plots of normalized spectral density distributions and the corresponding cross line of a TGSMA beam at several different propagation distances. (a1)-(a4) ε y = 1 , μ = 0 ; (b1)-(b4) ε y = 3 , μ = 0 ; (c1)-(c4) ε y = 3 ; (d1)-(d4) ε y = 3 , a = 2 δ 0 = 12 mm .
Fig. 3
Fig. 3 Contour plots of the normalized spectral density distribution and the corresponding cross line of a TGSMA beam at z = 5km. (a1)-(a4) μ = 0 ; (b1)-(b4) μ = 0.5 km 1 ; (c1)-(c4) μ = 1 km 1 .
Fig. 4
Fig. 4 On-axis spectral density as a function of ξ and ε y at z = 5km, (a1)-(a4) δ 0 = 2 cm ; (b1)-(b4) δ 0 = 1 cm .
Fig. 5
Fig. 5 The modulus of the DOC η ( u , v , 0 , 0 ) and the corresponding cross line of a TGSMA beam at several different propagation distances. (a1)-(a4) ε y = 1 , μ = 0 ; (b1)-(b4) ε y = 3 , μ = 0 ; (c1)-(c4) ε y = 3 ; (d1)-(d4) ε y = 3 , a = 2 δ 0 = 12 mm .
Fig. 6
Fig. 6 The modulus of the DOC η ( u , v , 0 , 0 ) and the corresponding cross line of a TGSMA beam at z = 5km for different turbulence statistics and initial coherence. (a1)-(a4) μ = 0 ; (b1)-(b4) μ = 0.5 km 1 ; (c1)-(c4) μ = 1 km 1 .
Fig. 7
Fig. 7 The modulus of the DOC η ( u , 0 , 0 , 0 ) for different values of power law ξ and anisotropic factor ε y at z = 5km. (a1)-(a3) ξ = 11 / 3 ; (b1)-(b3) ε y = 3 .

Equations (25)

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W ( r 1 , r 2 ) = S ( r 1 ) S ( r 2 ) g ( r 1 r 2 ) χ ( r 1 , r 2 ) ,
g ( r 1 r 2 ) = 1 C 0 n x = N N n y = M M exp [ ( r 1 r 2 ) 2 2 δ 0 2 ] × exp [ i π n x ( x 1 x 2 ) / a ] exp [ i π n y ( y 1 y 2 ) / a ] ,
χ ( r 1 , r 2 ) = exp ( i μ k r 1 × r 2 ) ,
W ¯ ( r 1 , r 2 ) = j = 0 , 1 / 2 , 1 , ... m = j j α j , m ϕ j , m * ( r 1 ) ϕ j , m ( r 2 ) ,
ϕ j , m ( r ) = k u π [ ( j | m | ) ! ( j + | m | ) ! ] 1 2 ( r k μ ) 2 | m | L j | m | 2 | m | ( k μ r 2 ) exp ( k μ r 2 2 ) exp ( i 2 m θ ) ,
α j , m = 2 π δ 0 2 1 + k u δ 0 2 ( 1 k u δ 0 2 1 + k u δ 0 2 ) j + m .
α n = W ¯ ( r ) L n ( k μ r 2 ) exp ( k μ r 2 / 2 ) d 2 r , ( n = 0 , 1 , 2 , 3... ) ,
α n = 1 C 0 n x = N N n y = M M 2 π 0 J 0 ( β r ) L n ( k μ r 2 ) exp [ 1 2 ( δ 0 2 + k μ ) r 2 ] r d r ,
0 2 π exp ( i a r cos θ + i b r sin θ ) d θ = 2 π J 0 ( r a 2 + b 2 ) .
α n = 1 C 0 n x = N N n y = M M 2 π δ 0 2 1 + k u δ 0 2 ( 1 k u δ 0 2 1 + k u δ 0 2 ) n exp [ β 2 δ 0 2 2 ( 1 + k u δ 0 2 ) ] L n ( β 2 k u δ 0 4 k 2 u 2 δ 0 4 1 ) .
W ( r 1 , r 2 ) = 1 C 0 n x = N N n y = M M exp ( x 1 2 + x 2 2 4 w x 2 y 1 2 + y 2 2 4 w y 2 ) exp ( ( r 1 r 2 ) 2 2 δ 0 2 ) × exp [ i π n x a ( x 1 x 2 ) ] exp [ i π n y a ( y 1 y 2 ) ] exp ( i k u r 1 × r 2 ) .
W ( ρ 1 , ρ 2 ) = ( 1 λ z ) 2 W ( r 1 , r 2 ) exp [ i k 2 z ( r 1 2 2 ρ 1 r 1 + 2 ρ 2 r 2 r 2 2 ) ] × exp [ i k 2 z ( ρ 1 2 ρ 2 2 ) ] exp [ Ψ ( r 1 , ρ 1 ) + Ψ * ( r 2 , ρ 2 ) ] d 2 r 1 d 2 r 2 ,
exp [ Ψ ( r 1 , ρ 1 ) + Ψ * ( r 2 , ρ 2 ) ] = exp ( 2 π k 2 z 0 1 d t 0 Φ n ( κ ) d 3 κ { 1 exp [ t ρ d κ + ( 1 t ) r d κ ] } ) ,
Φ n ( κ x , κ y , κ z ) = ε x ε y C ˜ n 2 A ( ξ ) ( ε x 2 κ x 2 + ε y 2 κ y 2 + κ z 2 ) ξ / 2 ,
A ( ξ ) = Γ ( ξ 1 ) cos ( π ξ / 2 ) 4 π 2 , 3 < ξ < 4 ,
κ x = q x ε x = q cos φ ε x , κ y = q y ε y = q sin φ ε y , d κ x d κ y = d q x d q y ε x ε y = q d q d φ ε x ε y , q = q x 2 + q y 2 , ρ d ( u d ε x , v d ε y ) , r d ( x d ε x , y d ε y ) .
exp [ Ψ ( r 1 , ρ 1 ) + Ψ * ( r 2 , ρ 2 ) ] = exp ( 4 π 2 A ( ξ ) C ˜ n 2 k 2 z 0 1 d t 0 q 1 ξ { 1 J 0 [ q | t ρ d + ( 1 t ) r d | ] } d q ) ,
0 x 1 p [ 1 J 0 ( x | Q | ) ] d x = 2 p p | Q | p 2 Γ ( p / 2 ) Γ ( p / 2 ) , ( 2< Re( p ) < 4 ) ,
exp [ Ψ ( r 1 , ρ 1 ) + Ψ * ( r 2 , ρ 2 ) ] = exp [ ξ Γ ( ξ 1 ) Γ ( ξ / 2 ) 2 ξ ( ξ 1 ) Γ ( ξ / 2 ) cos ( π ξ 2 ) C ˜ n 2 k 2 z ( ρ d ξ 1 r d ξ 1 | ρ d r d | ) ] .
exp [ Ψ ( r 1 , ρ 1 ) + Ψ * ( r 2 , ρ 2 ) ] exp [ 1 ρ 0 2 ( z ) ( u d 2 + u d x d + x d 2 ε x 2 + v d 2 + v d y d + y d 2 ε y 2 ) ] ,
ρ 0 ( z ) = [ ξ Γ ( ξ 1 ) Γ ( ξ / 2 ) cos ( π ξ / 2 ) 2 ξ ( ξ 1 ) Γ ( ξ / 2 ) C ˜ n 2 k 2 z ] 1 2 ξ ,
W ( ρ 1 , ρ 2 ) = n x = N N n y = M M V ( ρ 1 , ρ 2 ) exp ( γ u 1 2 4 Χ ( + ) + γ v 1 2 4 ϒ ( + ) + Ω u 2 4 Χ ( ) + Ω v 2 4 ϒ ( ) ) ,
V ( ρ 1 , ρ 2 ) = k 2 4 C 0 z 2 Χ ( + ) ϒ ( + ) Χ ( ) ϒ ( ) exp [ i k 2 z ( ρ 1 2 ρ 2 2 ) ρ d 2 ρ 0 2 ( z ) ] γ u 1 = i k u 1 z + i π n x a u d ε x 2 ρ 0 2 ( z ) , γ u 2 = i k u 2 z + i π n x a u d ε x 2 ρ 0 2 ( z ) , Δ x = 1 2 δ 0 2 + 1 ε x 2 ρ 0 2 ( z ) , γ v 1 = i k v 1 z + i π n y a v d ε y 2 ρ 0 2 ( z ) , γ v 2 = i k v 2 z + i π n y a v d ε y 2 ρ 0 2 ( z ) , Δ y = 1 2 δ 0 2 + 1 ε y 2 ρ 0 2 ( z ) , Χ ( + ) = Δ x + 1 4 w x 2 + i k 2 z , Χ ( ) = Δ x + 1 4 w x 2 i k 2 z Δ x 2 Χ ( + ) + k 2 μ 2 4 ϒ ( + ) , Ω u = Δ x γ u 1 Χ ( + ) γ u 2 + i k μ γ v 1 2 ϒ ( + ) , ϒ ( + ) = Δ y + 1 4 w y 2 + i k 2 z , ϒ ( ) = Δ y + 1 4 w y 2 i k 2 z Δ y 2 ϒ ( + ) + k 2 μ 2 4 Χ ( + ) + k 2 μ 2 4 Χ ( ) ( Δ y ϒ ( + ) Δ x Χ ( + ) ) 2 , Ω v = Δ y γ v 1 ϒ ( + ) γ v 2 i k μ γ u 1 2 Χ ( + ) + i k μ Ω u 2 Χ ( ) ( Δ y ϒ ( + ) Δ x Χ ( + ) ) .
S ( ρ ) = W ( ρ , ρ ) ,
η ( ρ 1 , ρ 2 ) = W ( ρ 1 , ρ 2 ) W ( ρ 1 , ρ 1 ) W ( ρ 2 , ρ 2 ) .

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