Abstract

Recently, we introduced a new class of radially polarized cosine-Gaussian correlated Schell-model (CGCSM) beams of rectangular symmetry based on the partially coherent electromagnetic theory [Opt. Express 23, 33099 (2015)]. In this paper, we extend the work to study the second-order statistics such as the average intensity, the spectral degree of coherence, the spectral degree of polarization and the state of polarization in anisotropic turbulence based on an extended von Karman power spectrum with a non-Kolmogorov power law α and an effective anisotropic parameter. Analytical formulas for the cross-spectral density matrix elements of a radially polarized CGCSM beam in anisotropic turbulence are derived. It is found that the second-order statistics are greatly affected by the source correlation function, and the change in the turbulent statistics induces relatively small effect. The significant effect of anisotropic turbulence on the beam parameters mainly appears nearα=3.1, and decreases with the increase of the anisotropic parameter. Furthermore, the polarization state exhibits self-splitting property and each beamlet evolves into a radially polarized structure in the far field. Our work enriches the classical coherence theory and may be important for free-space optical communications.

© 2016 Optical Society of America

1. Introduction

For many years, the interaction of light beams with turbulent atmosphere has been attracted much attention due to their important applications in astronomical imaging, free-space optical (FSO) communications, laser radar and remote sensing [1, 2]. The random fluctuations in the index of refraction of atmosphere cause spreading of the beam beyond that due to pure diffraction, beam wander, loss of spatial coherence, and random fluctuations in the irradiance and phase. These effects can seriously degrade the signal-to-noise ratio (SNR) of an optical heterodyne receiver. Therefore, much effort has been devoted to a reliable theory for predicting the propagation properties of light beams in turbulent medium, and a great many achievements have been derived in the past decades [3–19]. It is demonstrated that the atmosphere exhibits homogeneous and nearly isotropic under the atmospheric boundary layer (ABL), which is roughly 1-2km above the Earth’s surface. Therefore, the isotropic Kolmogorov power spectrum model of the refractive index is generally correct within this inertial sub-range. The portion of the atmosphere above the ABL belongs to the free atmosphere (FA) in which the influence of the Earth’s surface friction on the air motion is negligible. However, theoretical and experimental results have shown that the FA can be highly anisotropic at large scales, and the Kolmogorov power spectrum density model does not properly describe the real turbulence behavior. Consequently, a variety of different power spectrum models and extended anisotropic turbulence models have been proposed in the past decades [20–24].

The effects of spatial correlation function of partially coherent beams on the statistical properties in terms of the average intensity, the spectral degree of coherence, the spectral degree of polarization, the orbital angular moment (OAM) and the state of polarization (SOP) have been attracted increasing attention [25–33]. Recently, a new sufficient condition for generation genuine correlation function of light beams was introduced based on the constraint of non-negative definiteness [34, 35]. A variety of new correlation function models have been explored both in theory and in experiment. Partially coherent beams with these special correlation functions exhibit many novel properties, such as self-accelerating, self-focusing and tunable beam profile in the far field such as dark hollow (DH) profile, flat-topped profile and frame profile [36–48]. Among these beams, a class of partially coherent temporal/spatial sources, optical coherence gratings/lattices have found Gaussian intensity profiles and lattice-like spectral degrees of coherence, and can be found applications in metrology, material processing, and optical communications [45, 46]. Moreover, as a particular case of optical coherence lattices, cosine-Gaussian correlated Schell-model (CGCSM) beam of circular [38, 39] or rectangular symmetry [42] exhibits ring-shaped or four-beamlets array beam profile in the far field, and may be useful in particle trapping, optical scattering and optical imaging. It is well known that partially coherent beams with conventional correlation functions (i.e., Gaussian correlated Schell-model functions) are less affected by turbulent atmosphere than fully coherent beams in terms of beam quality, divergence and scintillations. However, propagation properties of partially coherent beams with non-conventional correlation functions in turbulent atmosphere are not clear and need for further study [39, 49–52].

Cylindrical vector (CV) beams are solutions of Maxwell equations with cylindrical symmetry both in amplitude and polarization. Typical CV beams such as radially polarization beams exhibit unique focusing properties, which are useful in high-resolution microscopy, plasmonic focusing, and nanoparticle manipulation and high-density storage [53, 54]. Partially coherent radially polarized beams were introduced in theory as an extension of coherent radially polarized beam and generated in experiment [55–57]. It has been found that partially coherent radially polarized beams have advantages over linearly polarized partially coherent beams for reducing turbulence-induced degradation [16, 58–60]. Recently, we introduced a new class of radially polarized CGCSM beams in theory and generated in experiment [48]. Based on the novel vector and coherence properties of a radially polarized CGCSM beam, in this paper, the second-order statistics are studied numerically by using a non-Kolmogorov power spectrum with an anisotropic parameter. Under weak turbulence conditions, analytical formula of 2×2cross-spectral density (CSD) matrix of a radially polarized CGCSM beam propagation through anisotropic turbulence is derived. The effects of the anisotropic parameter, the power law as well as the source correlation function on the average intensity, the spectral degree of coherence and the polarization properties are studied in detail.

2. Cross-spectral density matrix of a radially polarized CGCSM beam in anisotropic non-Kolmogorov turbulence

The vector electric field of a radially polarized beam can be characterized as the coherent superposition of TEM01 Laguerre–Gaussian modes with a polarization direction parallel to the x-axis and a TEM10 with a polarization direction parallel to the y-axis [16, 48, 53, 55–57],

E(r;ω)=Ex(r;ω)ex+Ey(r;ω)ey=xw0exp(r2w02)ex+yw0exp(r2w02)ey,
where r=(x2+y2)1/2is the transversal distance from the beam center and w0is the transverse beam size. For a vector partially coherent beam in space-frequency domain, the second-order spatial coherence properties can be characterized by the 2×2CSD matrix U(r1,r2;ω)with elementsUαβ(r1,r2;ω)=Eα*(r1;ω)Eβ(r2;ω),(α,β=x,y). The asterisk denotes the complex conjugate and the angular brackets represent ensemble average. The elements of the CSD matrix for a partially coherent radially polarized CGCSM beam are described as
Uαβ(r1,r2;ω)=α1β2w02exp(r12+r22w02)gαβ(r1r2;ω),(α=x,y;β=x,y),
and the spectral degree of coherence is given by
gαβ(r1r2;ω)=exp[(r1r2)22σ02]cos[nπ(x1x2)σ0]cos[nπ(y1y2)σ0],
where r1(x1,y1)and r2(x2,y2) are two arbitrary transverse position vectors in the source plane, σ0 represents spatial coherence width and n is the beam order parameter. Whenn=0, a radially polarized CGCSM beam reduces to a conventional radially polarized Schell-model (SM) beam. The realizability conditions for a radially polarized CGCSM source and the beam condition for radiation generated by such source are derived in our recent work [48].

Next, we investigate the propagation of a radially polarized CGCSM beam in anisotropic non-Kolmogorov turbulence. It is shown that the extended Huygens-Fresnel principle is a simple and effective method for studying the propagation of a light beam in both weak and strong turbulent atmosphere [2]. Within the paraxial approximation, the elements of the CSD at the receiver plane can be expressed as [1–4]

Uαβ(ρ1,ρ2;ω)=(k2πz)2Uαβ(r1,r2;ω)exp[ψ(r1,ρ1;ω)+ψ(r2,ρ2;ω)]×exp[ik2z[(r12r22)2(r1ρ1r2ρ2)+(ρ12ρ22)]]d2r1d2r2,
where ρ1(u1,v1)and ρ2(u2,v2) are two arbitrary transverse position vectors at the receiver plane,k=2π/λ is the wave number and λ is the wavelength, ψ represents the phase distortion of a monochromatic spherical wave in the turbulent atmosphere. Under quadratic phase approximations, the average turbulence phase perturbation is given by [24, 51, 52]
exp[ψ(r1,ρ1;ω)+ψ(r2,ρ2;ω)]=exp[π2k2z3(rΔ2+rΔρΔ+ρΔ2)0κ3Φn(κ,α)dκ],
where rΔr1r2,ρΔρ1ρ2, and Φn(κ,α) is the three-dimensional power spectrum of the refractive-index fluctuations with κbeing the spatial wave number in the spatial-frequency. Here, we use the anisotropic non-Kolmogorov power spectrum introduced in [22, 23]. With the help of a generalized von Karman model [24], the power spectrum of refractive-index fluctuations with an effective anisotropic factorζeff is defined as
Φn(κ,α)=A(α)C˜n2ζeff2(ζeff2κxy2+κz2+κ02)α/2exp(ζeff2κxy2+κz2κH2),
with
κ=ζeff2(κx2+κy2)+κz2=ζeff2κxy2+κz2,A(α)=Γ(α1)4π2cos(πα2),3<α<4,
where α is the power law, κ0=2π/L0withL0being the outer scale of turbulence, κH=C(α)/l0with l0being the inner scale of turbuence, C˜n2=βCn2 is a generalized structure parameter with unit [m3α]andβ is a dimensional constant with unit [m11/3α], C(α)is defined as
C(α)=[πA(α)Γ(32α2)(3α3)]1/(α5),3<α<4,
where Γ(α) is the Gamma function. Whenα=11/3, we find A(11/3)=0.033and the generalized power spectrum reduces to the conventional Kolmogorov spectrum. Also, the generalized structure parameter reduces to the structure parameterCn2 with unit [m2/3]. To point out here, although it is assumed the turbulent cells are anisotropic only along the direction of propagation path (z-axis), they are still isotropic over the planes orthogonal to the z-axis. Based on this assumption, the Kolmogorov theory remains valid because the energy cascade process over the plane orthogonal to the propagation direction is still isotropic, although there is rescaling due to anisotropy for each cell size. For further study, let us consider the Markov approximation, which is commonly used in the theory of wave propagation in random media [2, 20–24]. It is assumed that the refractive index can be expressed as
n(R)=n0+n1(R),
where n0=n(R)1,n1(R)=0.The covariance function—delta correlated in the direction of propagation along the positive z-axis can be expressed as
n1(R1)n1(R2)δ(z1z2)Αn(r1r2).
Equation (10) is often known as Markov approximation. In writing Eq. (10), it has been assumed the covariance is statistically homogeneous so it is a function of only the difference R1R2and Αn(r1r2) is a two-dimensional covariance function. Markov approximation means that the refractive index is supposed to be uncorrelated between two different points along the direction of propagation path. As a consequence, the energy-transfer process in the inertial sub-range develops only over plane orthogonal to the direction of propagation path. Therefore, the spatial wave number componentκz along z-axis is ignored.

In considering the vertical profile, we replace C˜n2 with the average over the path value C˜¯n2, which is defined by

C˜¯n2=1Hh0h0HC˜n2(h)dh,
By means of the typical Hufnagel-Valley (H-V) model, C˜n2(h) is given by [24]:
C˜n2(h)=0.00594(vs27)2(105h)10exp(h1000)+2.7×1016exp(h1500)+C˜n2exp(h100),
where h0andH are the altitudes above the ground where transmitter and receiver are located. The integration in Eq. (5) is derived as follows
Tani=π2k2z30Φn(κ,α)κ3dκ=π2k2zζeff2α6(α2)A(α)C˜¯n2[κ˜H2αβexp(κ02κH2)Γ(2α2,κ02κH2)2κ˜04α],
where κ˜0=κ0/ζeff,κ˜H=κH/ζeff,β=(2κ˜022κ˜H2+ακ˜H2),and Γ(.,.) stands for the incomplete Gamma function. It is seen that anisotropy introduces rescaling of turbulence by the factorζeff2α. On substituting Eqs. (2), (3), and (5) into Eq. (4), the elements of the CSD matrix of a radially polarized CGCSM beam in turbulent atmosphere take on the form
Uxx(ρ1,ρ2;ω)=V(ρ1,ρ2;ω){exp[γv1224M1+Ωv2224Π]+exp[γv1124M1+Ωv2124Π]}×{(Δ+γu11Ωu21+Δ2ΠΩu212)exp[γu1124M1+Ωu2124Π](Δ+γu12Ωu22+Δ2ΠΩu222)exp[γu1224M1+Ωu2224Π]+},
Uyy(ρ1,ρ2;ω)=V(ρ1,ρ2;ω){exp[γu1224M1+Ωu2224Π]+exp[γu1124M1+Ωu2124Π]}×{(Δ+γv11Ωv21+Δ2ΠΩv212)exp[γv1124M1+Ωv2124Π](Δ+γv12Ωv22+Δ2ΠΩv222)exp[γv1224M1+Ωv2224Π]+},
Uxy(ρ1,ρ2;ω)=V(ρ1,ρ2;ω){Ωv22exp[γv1224M1+Ωv2224Π]+Ωv21exp[γv1124M1+Ωv2124Π]}×{(γu11+Δ2ΠΩu21)exp[γu1124M1+Ωu2124Π]+(γu12+Δ2ΠΩu22)exp[γu1224M1+Ωu2224Π]},
Uyx(ρ1,ρ2;ω)=Uxy*(ρ1,ρ2;ω),
where
V(ρ1,ρ2;ω)=k264z2w02M12Π2exp[ik2z(ρ12ρ22)TaniρΔ2],a=n2π/σ0,ξu1=iku1zTaniuΔ,ξu2=iku2zTaniuΔ,ξv1=ikv1zTanivΔ,ξv2=ikv2zTanivΔ,γu11=ξu1+ia,γu12=ξu1ia,γu21=ξu2+ia,γu22=ξu2ia,γv11=ξv1+ia,γv12=ξv1ia,γv21=ξv2+ia,γv22=ξv2ia,Ωu22=Δγu122M1γu22,Ωu21=Δγu112M1γu21,Ωv22=Δγv122M1γv22,Ωv21=Δγv112M1γv21,Δ=1σ02+2Tani,M1=1w02+12σ02+ik2z+Tani,Π=1w02+12σ02ik2zΔ24M1+Tani.
The average intensity and the spectral degree of polarization (DOP) are given by the formulas [25]
I(ρ;ω)=Uxx(ρ,ρ;ω)+Uyy(ρ,ρ;ω),
P(ρ;ω)=14Det[U(ρ,ρ;ω)]{Tr[U(ρ,ρ;ω)]}2,
where Tr and Det denote the trace and the determinant of a matrix. The spectral degree of coherence (DOC) of a vector beam in any transverse plane is defined by [25, 31]

μ(ρ1,ρ2;ω)=TrU(ρ1,ρ2;ω)Tr[U(ρ1,ρ1;ω)]Tr[U(ρ2,ρ2;ω)].

3. Numerical examples of a radially polarized CGCSM beam in anisotropic non-Kolmogorov turbulence

In this section, we give out some numerical results for studying the second-order statistics of a radially polarized CGCSM beam in anisotropic non-Kolmogorov turbulence. The parameters are set as: w0=4cm,σ0=2cm,λ=632.8nm,C˜n2=1.7×1014m3α,vs=21m/s,l0=1mm,L0=10m, h0=0,H=30km,n=1,α=3.5,ζeff=3(unless stated otherwise). On submitting from Eq. (12) into Eq. (11), one finds the average refractive index structure constantC˜¯n2=7.45×1017m3α. Figure 1 shows the normalized average intensity distribution and the corresponding cross line of a radially polarized CGCSM beam at several different propagation distances for different values ofz=5km. For the case ofn=0, Figs. 1(a1)-1(a4) show that a doughnut shaped radially polarized SM beam gradually evolves into a Gaussian shape as expected [16]. For the case ofn0, a circularly symmetric radially polarized CGCSM beam gradually decomposes into a four-beamlets array distribution in the far field, and the distance between each beamlet becomes larger as the beam orderz=5km increases. Figure 2 plots the normalized average intensity distribution, composition components Uxx(ρ,ρ),Uyy(ρ,ρ) and the corresponding cross lines of a radially polarized CGCSM beam at several different propagation distances withn=2. One finds that the source radially polarized beam spot decomposes into four radially polarized beamlets in the far field due to the source correlation function, which significantly differs from those conventional radially polarized SM beams. Moreover, Figs. 1 and 2 demonstrate that the behavior of partially coherent beams in turbulent atmosphere is greatly affected by source correlation function, even at large propagation distances. The physical interpretation is that the disturbance of turbulence is homogeneous, which causes a reduction of spatialcoherence of light beam. Therefore, the average intensity distribution is mainly affected by the source correlation function, and the turbulence leads to a spatial broadening of the beam spot. Our results is consistent with the conclusion derived from conventional SM beams propagation in isotropic turbulence [6–11]. Figure 3 illustrates the evolution of the on-axis average intensity for different values ofz=5km. As is seen from Figs. 3(a1)-3(a3), the on-axis average intensity generally increases first causing by the source correlation, and then decreases due to the diffraction as well as the turbulence-induced spatial broadening of the light beam. Furthermore, the distance corresponding to the peak value gradually shorten with the growth of valuez=5km. Figures 3(b1)-3(b3) illustrate that the significant impact of the effective anisotropic parameterζeff on the on-axis average intensity mainly appears nearα=3.1. However, with the increase ofζeff, the average intensity gradually grows forn=0, while it deceases forn0. This implies that the behavior of radially polarized CGCSM beams is more sensitive to the source correlations, rather than the turbulence parameters. Similar results have been found both in isotropic turbulence and other class of beams [24].

 figure: Fig. 1

Fig. 1 Normalized average intensity distribution and the corresponding cross line of a radially polarized CGCSM beam at several different propagation distances for different values ofn.

Download Full Size | PPT Slide | PDF

 figure: Fig. 2

Fig. 2 Normalized average intensity distribution, composition componentsUxx(ρ,ρ), Uyy(ρ,ρ) and the corresponding cross lines of a radially polarized CGCSM beam at several different propagation distances withn=2.

Download Full Size | PPT Slide | PDF

 figure: Fig. 3

Fig. 3 On-axis average intensity of a radially polarized CGCSM beam for different values ofn. (a1)-(a3) as a function of distance z andζeff, (b1)-(b3) as a function of α andζeffat z=10km.

Download Full Size | PPT Slide | PDF

Figure 4 shows the modulus of the spectral DOC and the corresponding cross line of a radially polarized CGCSM beam at different propagation distances for different values ofz=5km. One finds that the spectral DOC of a conventional radially polarized SM beam [see Figs. 4(a1)-4(a4)] is a typical circular symmetry distribution as expected [16]. For the case ofn0, the spectral DOC exhibits array distribution with rectangular symmetry, and both the shape and the symmetry change during transmission. Due to the influence of turbulence, the spectral DOC gradually evolves into a Gaussian shape after propagating a sufficient distance, which quite differs from its free space behavior [48]. In addition, because of the diffraction, the coherence width significantly increased. Figure 5 illustrates the modulus of the spectral DOC μ(0cm,0cm,5cm,0cm) of a radially polarized CGCSM beam for different values ofz=5km. As is shown in Figs. 5(a1)-5(a3), ζeff plays a crucial role in determining the correlation property, and the spectral DOC generally increases as the growth ofζeff. However, for a high value of z=5km such asn=2 [see Fig. 5(a3)], the effect of ζeffon the spectral DOC is weakened. Unlike Figs. 3(b1)-3(b3), and Fig. 5(b1)-5(b3) demonstrate that the spectral DOCs for different values ofz=5km are similar and the effect of ζeff on the spectral DOC is gradually reduced as the valuez=5km increases. As a result, for lower effective anisotropic parameter valueζeff, the effects induced by source correlation on the spectral DOC become dominant. Similar results have been reported for conventional electromagnetic SM beams in anisotropic turbulence [24].

 figure: Fig. 4

Fig. 4 The modulus of the spectral DOC μ(u,v,0cm,0cm)distribution and the corresponding cross line of a radially polarized CGCSM beam at different propagation distances for different values ofn.

Download Full Size | PPT Slide | PDF

 figure: Fig. 5

Fig. 5 The modulus of the spectral DOC μ(0cm,0cm,5cm,0cm)of a radially polarized CGCSM beam for different values ofn. (a1)-(a3) as a function of distance z andζeff, (b1)-(b3) as a function of α andζeffat z=10km.

Download Full Size | PPT Slide | PDF

Figure 6 shows the spectral DOP and the corresponding cross line of a radially polarized CGCSM beam at different propagation distances for different values ofz=5km. It is clearly seen that only the on-axis spectral DOP remains invariant during propagation, actually equals to zero, which implies that the on-axis polarization of a radially polarized CGCSM beam is not destroyed on propagation even in anisotropic turbulence. However, it has been predicted that the spectral DOP of a coherent radially polarized beam is always equal to one in free space for entire transverse light field [16]. This implies that the behavior of polarization structure of a radially polarized CGCSM beam determined mostly by the source correlation function, rather than by the turbulence parameters. Furthermore, the distribution of the spectral DOP of a radially polarized CGCSM beam in turbulent atmosphere is quite different from those of conventional radially polarized SM beams and is closely determined by the beam orderz=5km. It is clearly seen that the distributions of spectral DOP are greatly affected by the CGCSM correlation functions, even at large propagation distances. This behavior is in agreement with the result derived in detail previously for conventional SM beams propagation in isotropic or anisotropic turbulence [7–10]. Figure 7 shows the spectral DOP of a radially polarized CGCSM beam for different values ofz=5km at (5cm, 0cm). An obvious difference is found in Figs. 7(a1)-7(a3) that the spectral DOP generally decreases for a conventional radially polarized SM beam, while it reduces first and then grows to a certain value and finally deceases for a radially polarized CGCSM beam due to the modulation of the source cosine correlation function. Similar to the on-axis average intensity, a significant effect of ζeff on the spectral DOP mainly appears nearα=3.1, and decreases with the increase ofζeff.

 figure: Fig. 6

Fig. 6 The spectral DOP distribution and the corresponding cross line of a radially polarized CGCSM beam at different propagation distances for different values ofn.

Download Full Size | PPT Slide | PDF

 figure: Fig. 7

Fig. 7 The spectral DOP of a radially polarized CGCSM beam for different values ofn at (5cm, 0cm). (a1)-(a3) as a function of distance z andζeff, (b1)-(b3) as a function of α andζeffat z=10km.

Download Full Size | PPT Slide | PDF

Finally, to learn more about the polarization property, Fig. 8 shows the SOP of a radially polarized CGCSM beam at different propagation distances for different values ofz=5km. It is seen that the SOP of a radially polarized SM beam remains invariant in the far field. However, the SOP of a radially polarized CGCSM beam is closely determined by the beam orderz=5km, and the radial polarization structure is destroyed in the far field due to the non-conventional source correlation function. This means that the SOP is greatly affected by the source correlation function, similarly to its free space behavior, and the change in the turbulent statistics induces relatively small effect. In addition, although the overall SOP of a radially polarized CGCSM beam is broken, the SOP of each beamlet evolves into a radial polarization structure in the far field. Here, we obtain an important conclusion that we can generate various class of vector beams in the far field with designated polarization structures such as polarization array structure by modulating the source correlation function. Our results may be found potential applications in optical trapping, high-resolution microscopy, FSO communications, and remote polarization sensing.

 figure: Fig. 8

Fig. 8 The SOP of a radially polarized CGCSM beam at different propagation distances for different values ofn.

Download Full Size | PPT Slide | PDF

4. Conclusion

In this work, we investigated the statistics of a radially polarized CGCSM beam in anisotropic turbulence based on an extended von Karman power spectrum with a non-Kolmogorov power lawα and an effective anisotropic parameter. With the help of Markov approximation, effective anisotropic parameter describes anisotropy along the vertical direction. Analytical formulas for the cross-spectral density matrix elements of a radially polarized CGCSM beam in turbulent atmosphere are derived. One finds that the average intensity, the spectral DOC, the spectral DOP and the SOP of a radially polarized CGCSM beam evolve into four-beamlets array. The second-order statistics are determined mostly by the source correlation function, and the change in the turbulent statistics induces relatively small effect. The significant effect of anisotropic parameter on the beam parameters mainly appears nearα=3.1, and decreases with the growth of anisotropic parameter. Furthermore, the initial radial polarization structure splits together with the source beam spot, and the SOP of each beamlet evolves into a radial polarization structure in the far field. This provides us with a new method to generate various vector beams such as tunable radially polarized beams, azimuthally polarized beams as well as cylindrical vector beam array. Our work is helpful in understanding the correlation modulation theory, and may be useful in optical trapping, high-resolution microscopy, polarization communications and remote polarization sensing.

Acknowledgments

The author’s research is supported by the National Natural Science Foundation of China under Grant Nos. 11504172 and 11274005, the National Natural Science Fund for Distinguished Young Scholar under Grant No. 11525418, the Natural Science Foundation of Jiangsu Province under Grant No. BK20150763, and the project of the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions.

References and links

1. A. Ishimaru, Wave Propagation and Scattering in Random Media, vol. 2 (Academic, 1978).

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

3. 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]  

4. A. I. Kon and V. I. Tatarskii, “On the theory of the propagation of partially coherent light beams in a turbulent atmosphere,” Radiophys. Quantum Electron. 15(10), 1187–1192 (1972). [CrossRef]  

5. 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]  

6. A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28(1), 10–12 (2003). [CrossRef]   [PubMed]  

7. H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005). [CrossRef]  

8. M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14(4), 513–523 (2004). [CrossRef]  

9. O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004). [CrossRef]  

10. G. Gbur and E. Wolf, “Partially coherent beam propagation in atmospheric turbulence [invited],” J. Opt. Soc. Am. A 31(9), 2038–2045 (2014). [CrossRef]   [PubMed]  

11. G. Zhou and X. Chu, “Average intensity and spreading of a Lorentz-Gauss beam in turbulent atmosphere,” Opt. Express 18(2), 726–731 (2010). [CrossRef]   [PubMed]  

12. P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35(7), 1043–1045 (2010). [CrossRef]   [PubMed]  

13. Y. Baykal, H. T. Eyyuboğlu, and Y. Cai, “Scintillations of partially coherent multiple Gaussian beams in turbulence,” Appl. Opt. 48(10), 1943–1954 (2009). [CrossRef]   [PubMed]  

14. H. T. Eyyuboğlu, Y. Baykal, E. Sermutlu, O. Korotkova, and Y. Cai, “Scintillation index of modified Bessel-Gaussian beams propagating in turbulent media,” J. Opt. Soc. Am. A 26(2), 387–394 (2009). [CrossRef]   [PubMed]  

15. X. Ji, H. T. Eyyuboğlu, G. Ji, and X. Jia, “Propagation of an Airy beam through the atmosphere,” Opt. Express 21(2), 2154–2164 (2013). [CrossRef]   [PubMed]  

16. Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008). [CrossRef]   [PubMed]  

17. 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]  

18. S. Zhu and Y. Cai, “M2-factor of a truncated electromagnetic Gaussian Schell-model beam,” Appl. Phys. B 103(4), 971–984 (2011). [CrossRef]  

19. F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited review),” Prog. Electromagnetics Res. 150, 123–143 (2015). [CrossRef]  

20. 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]  

21. M. Charnotskii, “Common omissions and misconceptions of wave propagation in turbulence: discussion,” J. Opt. Soc. Am. A 29(5), 711–721 (2012). [CrossRef]   [PubMed]  

22. I. Toselli, B. Agrawal, and S. Restaino, “Light propagation through anisotropic turbulence,” J. Opt. Soc. Am. A 28(3), 483–488 (2011). [CrossRef]   [PubMed]  

23. I. Toselli, “Introducing the concept of anisotropy at different scales for modeling optical turbulence,” J. Opt. Soc. Am. A 31(8), 1868–1875 (2014). [CrossRef]   [PubMed]  

24. M. Yao, I. Toselli, and O. Korotkova, “Propagation of electromagnetic stochastic beams in anisotropic turbulence,” Opt. Express 22(26), 31608–31619 (2014). [CrossRef]   [PubMed]  

25. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

26. J. Turunen, A. Vasara, and A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A 8(2), 282–289 (1991). [CrossRef]  

27. S. A. Ponomarenko, W. Huang, and M. Cada, “Dark and antidark diffraction-free beams,” Opt. Lett. 32(17), 2508–2510 (2007). [CrossRef]   [PubMed]  

28. S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18(1), 150–156 (2001). [CrossRef]   [PubMed]  

29. G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28(11), 878–880 (2003). [CrossRef]   [PubMed]  

30. G. Gbur and T. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1-6), 117–125 (2003). [CrossRef]  

31. J. Tervo, T. Setälä, and A. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003). [CrossRef]   [PubMed]  

32. 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]  

33. S. Zhu and Z. Li, “Theoretical and experimental studies of the spectral changes of a focused polychromatic partially coherent flat-topped beam,” Appl. Phys. B 118(3), 481–487 (2015). [CrossRef]  

34. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007). [CrossRef]   [PubMed]  

35. F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A 11(8), 085706 (2009). [CrossRef]  

36. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011). [CrossRef]   [PubMed]  

37. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012). [CrossRef]   [PubMed]  

38. Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013). [CrossRef]   [PubMed]  

39. Z. Mei, E. Schchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013). [CrossRef]   [PubMed]  

40. Y. Zhang and D. Zhao, “Scattering of multi-Gaussian Schell-model beams on a random medium,” Opt. Express 21(21), 24781–24792 (2013). [CrossRef]   [PubMed]  

41. L. Pan, C. Ding, and H. Wang, “Diffraction of cosine-Gaussian-correlated Schell-model beams,” Opt. Express 22(10), 11670–11679 (2014). [CrossRef]   [PubMed]  

42. C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014). [CrossRef]   [PubMed]  

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

44. F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014). [CrossRef]   [PubMed]  

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

46. L. Ma and S. A. Ponomarenko, “Free-space propagation of optical coherence lattices and periodicity reciprocity,” Opt. Express 23(2), 1848–1856 (2015). [CrossRef]   [PubMed]  

47. Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015). [CrossRef]  

48. 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]  

49. Z. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012). [CrossRef]   [PubMed]  

50. Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013). [CrossRef]   [PubMed]  

51. R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014). [CrossRef]   [PubMed]  

52. J. Yu, Y. Chen, L. Liu, X. Liu, and Y. Cai, “Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 23(10), 13467–13481 (2015). [CrossRef]   [PubMed]  

53. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009). [CrossRef]  

54. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef]   [PubMed]  

55. F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012). [CrossRef]  

56. S. Zhu, X. Zhu, L. Liu, F. Wang, and Y. Cai, “Theoretical and experimental studies of the spectral changes of a polychromatic partially coherent radially polarized beam,” Opt. Express 21(23), 27682–27696 (2013). [CrossRef]   [PubMed]  

57. S. Zhu, F. Wang, Y. Chen, Z. Li, and Y. Cai, “Statistical properties in Young’s interference pattern formed with a radially polarized beam with controllable spatial coherence,” Opt. Express 22(23), 28697–28710 (2014). [CrossRef]   [PubMed]  

58. Y. Gu, O. Korotkova, and G. Gbur, “Scintillation of nonuniformly polarized beams in atmospheric turbulence,” Opt. Lett. 34(15), 2261–2263 (2009). [CrossRef]   [PubMed]  

59. R. Martínez-Herrero and F. Prado, “Polarization evolution of radially polarized partially coherent vortex fields: role of Gouy phase of Laguerre-Gauss beams,” Opt. Express 23(4), 5043–5051 (2015). [CrossRef]   [PubMed]  

60. Z. Chen, S. Cui, L. Zhang, C. Sun, M. Xiong, and J. Pu, “Measuring the intensity fluctuation of partially coherent radially polarized beams in atmospheric turbulence,” Opt. Express 22(15), 18278–18283 (2014). [CrossRef]   [PubMed]  

References

  • View by:
  • |
  • |
  • |

  1. A. Ishimaru, Wave Propagation and Scattering in Random Media, vol. 2 (Academic, 1978).
  2. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Medium, 2nd ed. (SPIE, 2005).
  3. 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]
  4. A. I. Kon and V. I. Tatarskii, “On the theory of the propagation of partially coherent light beams in a turbulent atmosphere,” Radiophys. Quantum Electron. 15(10), 1187–1192 (1972).
    [Crossref]
  5. 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]
  6. A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28(1), 10–12 (2003).
    [Crossref] [PubMed]
  7. H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005).
    [Crossref]
  8. M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14(4), 513–523 (2004).
    [Crossref]
  9. O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
    [Crossref]
  10. G. Gbur and E. Wolf, “Partially coherent beam propagation in atmospheric turbulence [invited],” J. Opt. Soc. Am. A 31(9), 2038–2045 (2014).
    [Crossref] [PubMed]
  11. G. Zhou and X. Chu, “Average intensity and spreading of a Lorentz-Gauss beam in turbulent atmosphere,” Opt. Express 18(2), 726–731 (2010).
    [Crossref] [PubMed]
  12. P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35(7), 1043–1045 (2010).
    [Crossref] [PubMed]
  13. Y. Baykal, H. T. Eyyuboğlu, and Y. Cai, “Scintillations of partially coherent multiple Gaussian beams in turbulence,” Appl. Opt. 48(10), 1943–1954 (2009).
    [Crossref] [PubMed]
  14. H. T. Eyyuboğlu, Y. Baykal, E. Sermutlu, O. Korotkova, and Y. Cai, “Scintillation index of modified Bessel-Gaussian beams propagating in turbulent media,” J. Opt. Soc. Am. A 26(2), 387–394 (2009).
    [Crossref] [PubMed]
  15. X. Ji, H. T. Eyyuboğlu, G. Ji, and X. Jia, “Propagation of an Airy beam through the atmosphere,” Opt. Express 21(2), 2154–2164 (2013).
    [Crossref] [PubMed]
  16. Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
    [Crossref] [PubMed]
  17. 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]
  18. S. Zhu and Y. Cai, “M2-factor of a truncated electromagnetic Gaussian Schell-model beam,” Appl. Phys. B 103(4), 971–984 (2011).
    [Crossref]
  19. F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited review),” Prog. Electromagnetics Res. 150, 123–143 (2015).
    [Crossref]
  20. 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]
  21. M. Charnotskii, “Common omissions and misconceptions of wave propagation in turbulence: discussion,” J. Opt. Soc. Am. A 29(5), 711–721 (2012).
    [Crossref] [PubMed]
  22. I. Toselli, B. Agrawal, and S. Restaino, “Light propagation through anisotropic turbulence,” J. Opt. Soc. Am. A 28(3), 483–488 (2011).
    [Crossref] [PubMed]
  23. I. Toselli, “Introducing the concept of anisotropy at different scales for modeling optical turbulence,” J. Opt. Soc. Am. A 31(8), 1868–1875 (2014).
    [Crossref] [PubMed]
  24. M. Yao, I. Toselli, and O. Korotkova, “Propagation of electromagnetic stochastic beams in anisotropic turbulence,” Opt. Express 22(26), 31608–31619 (2014).
    [Crossref] [PubMed]
  25. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  26. J. Turunen, A. Vasara, and A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A 8(2), 282–289 (1991).
    [Crossref]
  27. S. A. Ponomarenko, W. Huang, and M. Cada, “Dark and antidark diffraction-free beams,” Opt. Lett. 32(17), 2508–2510 (2007).
    [Crossref] [PubMed]
  28. S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18(1), 150–156 (2001).
    [Crossref] [PubMed]
  29. G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28(11), 878–880 (2003).
    [Crossref] [PubMed]
  30. G. Gbur and T. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1-6), 117–125 (2003).
    [Crossref]
  31. J. Tervo, T. Setälä, and A. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003).
    [Crossref] [PubMed]
  32. 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]
  33. S. Zhu and Z. Li, “Theoretical and experimental studies of the spectral changes of a focused polychromatic partially coherent flat-topped beam,” Appl. Phys. B 118(3), 481–487 (2015).
    [Crossref]
  34. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
    [Crossref] [PubMed]
  35. F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A 11(8), 085706 (2009).
    [Crossref]
  36. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
    [Crossref] [PubMed]
  37. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
    [Crossref] [PubMed]
  38. Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
    [Crossref] [PubMed]
  39. Z. Mei, E. Schchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013).
    [Crossref] [PubMed]
  40. Y. Zhang and D. Zhao, “Scattering of multi-Gaussian Schell-model beams on a random medium,” Opt. Express 21(21), 24781–24792 (2013).
    [Crossref] [PubMed]
  41. L. Pan, C. Ding, and H. Wang, “Diffraction of cosine-Gaussian-correlated Schell-model beams,” Opt. Express 22(10), 11670–11679 (2014).
    [Crossref] [PubMed]
  42. C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
    [Crossref] [PubMed]
  43. Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review [invited],” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
    [Crossref] [PubMed]
  44. F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
    [Crossref] [PubMed]
  45. L. Ma and S. A. Ponomarenko, “Optical coherence gratings and lattices,” Opt. Lett. 39(23), 6656–6659 (2014).
    [Crossref] [PubMed]
  46. L. Ma and S. A. Ponomarenko, “Free-space propagation of optical coherence lattices and periodicity reciprocity,” Opt. Express 23(2), 1848–1856 (2015).
    [Crossref] [PubMed]
  47. Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
    [Crossref]
  48. 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]
  49. Z. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
    [Crossref] [PubMed]
  50. Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
    [Crossref] [PubMed]
  51. R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
    [Crossref] [PubMed]
  52. J. Yu, Y. Chen, L. Liu, X. Liu, and Y. Cai, “Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 23(10), 13467–13481 (2015).
    [Crossref] [PubMed]
  53. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
    [Crossref]
  54. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
    [Crossref] [PubMed]
  55. F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
    [Crossref]
  56. S. Zhu, X. Zhu, L. Liu, F. Wang, and Y. Cai, “Theoretical and experimental studies of the spectral changes of a polychromatic partially coherent radially polarized beam,” Opt. Express 21(23), 27682–27696 (2013).
    [Crossref] [PubMed]
  57. S. Zhu, F. Wang, Y. Chen, Z. Li, and Y. Cai, “Statistical properties in Young’s interference pattern formed with a radially polarized beam with controllable spatial coherence,” Opt. Express 22(23), 28697–28710 (2014).
    [Crossref] [PubMed]
  58. Y. Gu, O. Korotkova, and G. Gbur, “Scintillation of nonuniformly polarized beams in atmospheric turbulence,” Opt. Lett. 34(15), 2261–2263 (2009).
    [Crossref] [PubMed]
  59. R. Martínez-Herrero and F. Prado, “Polarization evolution of radially polarized partially coherent vortex fields: role of Gouy phase of Laguerre-Gauss beams,” Opt. Express 23(4), 5043–5051 (2015).
    [Crossref] [PubMed]
  60. Z. Chen, S. Cui, L. Zhang, C. Sun, M. Xiong, and J. Pu, “Measuring the intensity fluctuation of partially coherent radially polarized beams in atmospheric turbulence,” Opt. Express 22(15), 18278–18283 (2014).
    [Crossref] [PubMed]

2015 (7)

2014 (12)

Z. Chen, S. Cui, L. Zhang, C. Sun, M. Xiong, and J. Pu, “Measuring the intensity fluctuation of partially coherent radially polarized beams in atmospheric turbulence,” Opt. Express 22(15), 18278–18283 (2014).
[Crossref] [PubMed]

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
[Crossref] [PubMed]

S. Zhu, F. Wang, Y. Chen, Z. Li, and Y. Cai, “Statistical properties in Young’s interference pattern formed with a radially polarized beam with controllable spatial coherence,” Opt. Express 22(23), 28697–28710 (2014).
[Crossref] [PubMed]

L. Pan, C. Ding, and H. Wang, “Diffraction of cosine-Gaussian-correlated Schell-model beams,” Opt. Express 22(10), 11670–11679 (2014).
[Crossref] [PubMed]

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[Crossref] [PubMed]

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

F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
[Crossref] [PubMed]

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

I. Toselli, “Introducing the concept of anisotropy at different scales for modeling optical turbulence,” J. Opt. Soc. Am. A 31(8), 1868–1875 (2014).
[Crossref] [PubMed]

M. Yao, I. Toselli, and O. Korotkova, “Propagation of electromagnetic stochastic beams in anisotropic turbulence,” Opt. Express 22(26), 31608–31619 (2014).
[Crossref] [PubMed]

G. Gbur and E. Wolf, “Partially coherent beam propagation in atmospheric turbulence [invited],” J. Opt. Soc. Am. A 31(9), 2038–2045 (2014).
[Crossref] [PubMed]

2013 (6)

2012 (4)

2011 (3)

2010 (3)

2009 (5)

2008 (1)

2007 (2)

2005 (1)

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005).
[Crossref]

2004 (2)

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14(4), 513–523 (2004).
[Crossref]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[Crossref]

2003 (5)

2002 (1)

2001 (1)

1994 (1)

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]

1991 (1)

1972 (2)

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]

A. I. Kon and V. I. Tatarskii, “On the theory of the propagation of partially coherent light beams in a turbulent atmosphere,” Radiophys. Quantum Electron. 15(10), 1187–1192 (1972).
[Crossref]

Agrawal, B.

Amarande, S.

Baykal, Y.

Bogatyryova, G. V.

Cada, M.

Cai, Y.

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited review),” Prog. Electromagnetics Res. 150, 123–143 (2015).
[Crossref]

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

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]

J. Yu, Y. Chen, L. Liu, X. Liu, and Y. Cai, “Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 23(10), 13467–13481 (2015).
[Crossref] [PubMed]

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
[Crossref] [PubMed]

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[Crossref] [PubMed]

F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (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]

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

S. Zhu, F. Wang, Y. Chen, Z. Li, and Y. Cai, “Statistical properties in Young’s interference pattern formed with a radially polarized beam with controllable spatial coherence,” Opt. Express 22(23), 28697–28710 (2014).
[Crossref] [PubMed]

S. Zhu, X. Zhu, L. Liu, F. Wang, and Y. Cai, “Theoretical and experimental studies of the spectral changes of a polychromatic partially coherent radially polarized beam,” Opt. Express 21(23), 27682–27696 (2013).
[Crossref] [PubMed]

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
[Crossref]

S. Zhu and Y. Cai, “M2-factor of a truncated electromagnetic Gaussian Schell-model beam,” Appl. Phys. B 103(4), 971–984 (2011).
[Crossref]

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. Baykal, H. T. Eyyuboğlu, and Y. Cai, “Scintillations of partially coherent multiple Gaussian beams in turbulence,” Appl. Opt. 48(10), 1943–1954 (2009).
[Crossref] [PubMed]

H. T. Eyyuboğlu, Y. Baykal, E. Sermutlu, O. Korotkova, and Y. Cai, “Scintillation index of modified Bessel-Gaussian beams propagating in turbulent media,” J. Opt. Soc. Am. A 26(2), 387–394 (2009).
[Crossref] [PubMed]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
[Crossref] [PubMed]

Charnotskii, M.

Chen, R.

Chen, Y.

Chen, Z.

Chu, X.

Cui, S.

Ding, C.

Dogariu, A.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14(4), 513–523 (2004).
[Crossref]

A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28(1), 10–12 (2003).
[Crossref] [PubMed]

Dong, Y.

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
[Crossref]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Eyyuboglu, H. T.

Fel’de, C. V.

Friberg, A.

Friberg, A. T.

Gbur, G.

Gori, F.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A 11(8), 085706 (2009).
[Crossref]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]

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(1), 013823 (2015).
[Crossref]

Gu, Y.

Huang, W.

Ji, G.

Ji, X.

Jia, X.

Kon, A. I.

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]

A. I. Kon and V. I. Tatarskii, “On the theory of the propagation of partially coherent light beams in a turbulent atmosphere,” Radiophys. Quantum Electron. 15(10), 1187–1192 (1972).
[Crossref]

Korotkova, O.

M. Yao, I. Toselli, and O. Korotkova, “Propagation of electromagnetic stochastic beams in anisotropic turbulence,” Opt. Express 22(26), 31608–31619 (2014).
[Crossref] [PubMed]

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[Crossref] [PubMed]

Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[Crossref] [PubMed]

Z. Mei, E. Schchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013).
[Crossref] [PubMed]

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[Crossref] [PubMed]

Z. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
[Crossref] [PubMed]

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
[Crossref]

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]

H. T. Eyyuboğlu, Y. Baykal, E. Sermutlu, O. Korotkova, and Y. Cai, “Scintillation index of modified Bessel-Gaussian beams propagating in turbulent media,” J. Opt. Soc. Am. A 26(2), 387–394 (2009).
[Crossref] [PubMed]

Y. Gu, O. Korotkova, and G. Gbur, “Scintillation of nonuniformly polarized beams in atmospheric turbulence,” Opt. Lett. 34(15), 2261–2263 (2009).
[Crossref] [PubMed]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[Crossref]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14(4), 513–523 (2004).
[Crossref]

Lajunen, H.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Li, Z.

Liang, C.

Lin, Q.

Liu, L.

Liu, X.

Liu, Z.

Ma, L.

Ma, Y.

Martínez-Herrero, R.

Mei, Z.

Pan, L.

Polyanskii, P. V.

Ponomarenko, S. A.

Prado, F.

Pu, J.

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Ramírez-Sánchez, V.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A 11(8), 085706 (2009).
[Crossref]

Restaino, S.

Roychowdhury, H.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005).
[Crossref]

Saastamoinen, T.

Sahin, S.

Salem, M.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14(4), 513–523 (2004).
[Crossref]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[Crossref]

Santarsiero, M.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A 11(8), 085706 (2009).
[Crossref]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]

Schchepakina, E.

Sermutlu, E.

Setälä, T.

Shirai, T.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A 11(8), 085706 (2009).
[Crossref]

Soskin, M. S.

Sun, C.

Tatarskii, V. I.

A. I. Kon and V. I. Tatarskii, “On the theory of the propagation of partially coherent light beams in a turbulent atmosphere,” Radiophys. Quantum Electron. 15(10), 1187–1192 (1972).
[Crossref]

Tervo, J.

Tong, Z.

Toselli, I.

Turunen, J.

Vasara, A.

Visser, T.

G. Gbur and T. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1-6), 117–125 (2003).
[Crossref]

Wang, F.

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited review),” Prog. Electromagnetics Res. 150, 123–143 (2015).
[Crossref]

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
[Crossref] [PubMed]

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[Crossref] [PubMed]

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

F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
[Crossref] [PubMed]

S. Zhu, F. Wang, Y. Chen, Z. Li, and Y. Cai, “Statistical properties in Young’s interference pattern formed with a radially polarized beam with controllable spatial coherence,” Opt. Express 22(23), 28697–28710 (2014).
[Crossref] [PubMed]

S. Zhu, X. Zhu, L. Liu, F. Wang, and Y. Cai, “Theoretical and experimental studies of the spectral changes of a polychromatic partially coherent radially polarized beam,” Opt. Express 21(23), 27682–27696 (2013).
[Crossref] [PubMed]

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
[Crossref]

Wang, H.

Wang, J.

Wang, X.

Wolf, E.

G. Gbur and E. Wolf, “Partially coherent beam propagation in atmospheric turbulence [invited],” J. Opt. Soc. Am. A 31(9), 2038–2045 (2014).
[Crossref] [PubMed]

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005).
[Crossref]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[Crossref]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14(4), 513–523 (2004).
[Crossref]

G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28(11), 878–880 (2003).
[Crossref] [PubMed]

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]

Wu, G.

Xiong, M.

Yao, M.

Yu, J.

Yuan, Y.

Yura, H. T.

Zhan, Q.

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

Zhang, L.

Zhang, Y.

Zhao, D.

Zhao, H.

Zhou, G.

Zhou, P.

Zhu, S.

S. Zhu and Z. Li, “Theoretical and experimental studies of the spectral changes of a focused polychromatic partially coherent flat-topped beam,” Appl. Phys. B 118(3), 481–487 (2015).
[Crossref]

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]

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (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]

S. Zhu, F. Wang, Y. Chen, Z. Li, and Y. Cai, “Statistical properties in Young’s interference pattern formed with a radially polarized beam with controllable spatial coherence,” Opt. Express 22(23), 28697–28710 (2014).
[Crossref] [PubMed]

S. Zhu, X. Zhu, L. Liu, F. Wang, and Y. Cai, “Theoretical and experimental studies of the spectral changes of a polychromatic partially coherent radially polarized beam,” Opt. Express 21(23), 27682–27696 (2013).
[Crossref] [PubMed]

S. Zhu and Y. Cai, “M2-factor of a truncated electromagnetic Gaussian Schell-model beam,” Appl. Phys. B 103(4), 971–984 (2011).
[Crossref]

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]

Zhu, X.

Adv. Opt. Photonics (1)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

Appl. Opt. (2)

Appl. Phys. B (2)

S. Zhu and Y. Cai, “M2-factor of a truncated electromagnetic Gaussian Schell-model beam,” Appl. Phys. B 103(4), 971–984 (2011).
[Crossref]

S. Zhu and Z. Li, “Theoretical and experimental studies of the spectral changes of a focused polychromatic partially coherent flat-topped beam,” Appl. Phys. B 118(3), 481–487 (2015).
[Crossref]

Appl. Phys. Lett. (1)

F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012).
[Crossref]

J. Mod. Opt. (1)

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52(11), 1611–1618 (2005).
[Crossref]

J. Opt. A (1)

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A 11(8), 085706 (2009).
[Crossref]

J. Opt. Soc. Am. A (8)

Opt. Commun. (2)

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[Crossref]

G. Gbur and T. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1-6), 117–125 (2003).
[Crossref]

Opt. Express (18)

J. Tervo, T. Setälä, and A. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003).
[Crossref] [PubMed]

M. Yao, I. Toselli, and O. Korotkova, “Propagation of electromagnetic stochastic beams in anisotropic turbulence,” Opt. Express 22(26), 31608–31619 (2014).
[Crossref] [PubMed]

X. Ji, H. T. Eyyuboğlu, G. Ji, and X. Jia, “Propagation of an Airy beam through the atmosphere,” Opt. Express 21(2), 2154–2164 (2013).
[Crossref] [PubMed]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
[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]

G. Zhou and X. Chu, “Average intensity and spreading of a Lorentz-Gauss beam in turbulent atmosphere,” Opt. Express 18(2), 726–731 (2010).
[Crossref] [PubMed]

R. Martínez-Herrero and F. Prado, “Polarization evolution of radially polarized partially coherent vortex fields: role of Gouy phase of Laguerre-Gauss beams,” Opt. Express 23(4), 5043–5051 (2015).
[Crossref] [PubMed]

Z. Chen, S. Cui, L. Zhang, C. Sun, M. Xiong, and J. Pu, “Measuring the intensity fluctuation of partially coherent radially polarized beams in atmospheric turbulence,” Opt. Express 22(15), 18278–18283 (2014).
[Crossref] [PubMed]

F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
[Crossref] [PubMed]

L. Ma and S. A. Ponomarenko, “Free-space propagation of optical coherence lattices and periodicity reciprocity,” Opt. Express 23(2), 1848–1856 (2015).
[Crossref] [PubMed]

Z. Mei, E. Schchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013).
[Crossref] [PubMed]

Y. Zhang and D. Zhao, “Scattering of multi-Gaussian Schell-model beams on a random medium,” Opt. Express 21(21), 24781–24792 (2013).
[Crossref] [PubMed]

L. Pan, C. Ding, and H. Wang, “Diffraction of cosine-Gaussian-correlated Schell-model beams,” Opt. Express 22(10), 11670–11679 (2014).
[Crossref] [PubMed]

S. Zhu, X. Zhu, L. Liu, F. Wang, and Y. Cai, “Theoretical and experimental studies of the spectral changes of a polychromatic partially coherent radially polarized beam,” Opt. Express 21(23), 27682–27696 (2013).
[Crossref] [PubMed]

S. Zhu, F. Wang, Y. Chen, Z. Li, and Y. Cai, “Statistical properties in Young’s interference pattern formed with a radially polarized beam with controllable spatial coherence,” Opt. Express 22(23), 28697–28710 (2014).
[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]

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
[Crossref] [PubMed]

J. Yu, Y. Chen, L. Liu, X. Liu, and Y. Cai, “Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 23(10), 13467–13481 (2015).
[Crossref] [PubMed]

Opt. Lett. (15)

Z. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
[Crossref] [PubMed]

Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[Crossref] [PubMed]

Y. Gu, O. Korotkova, and G. Gbur, “Scintillation of nonuniformly polarized beams in atmospheric turbulence,” Opt. Lett. 34(15), 2261–2263 (2009).
[Crossref] [PubMed]

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[Crossref] [PubMed]

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

P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35(7), 1043–1045 (2010).
[Crossref] [PubMed]

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]

A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28(1), 10–12 (2003).
[Crossref] [PubMed]

G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28(11), 878–880 (2003).
[Crossref] [PubMed]

S. A. Ponomarenko, W. Huang, and M. Cada, “Dark and antidark diffraction-free beams,” Opt. Lett. 32(17), 2508–2510 (2007).
[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. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]

H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[Crossref] [PubMed]

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[Crossref] [PubMed]

Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[Crossref] [PubMed]

Phys. Rev. A (1)

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

Phys. Rev. Lett. (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Prog. Electromagnetics Res. (1)

F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited review),” Prog. Electromagnetics Res. 150, 123–143 (2015).
[Crossref]

Radiophys. Quantum Electron. (1)

A. I. Kon and V. I. Tatarskii, “On the theory of the propagation of partially coherent light beams in a turbulent atmosphere,” Radiophys. Quantum Electron. 15(10), 1187–1192 (1972).
[Crossref]

Waves Random Complex Media (1)

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]

Waves Random Media (1)

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14(4), 513–523 (2004).
[Crossref]

Other (3)

A. Ishimaru, Wave Propagation and Scattering in Random Media, vol. 2 (Academic, 1978).

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

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1 Normalized average intensity distribution and the corresponding cross line of a radially polarized CGCSM beam at several different propagation distances for different values of n .
Fig. 2
Fig. 2 Normalized average intensity distribution, composition components U x x ( ρ , ρ ) , U y y ( ρ , ρ ) and the corresponding cross lines of a radially polarized CGCSM beam at several different propagation distances with n = 2 .
Fig. 3
Fig. 3 On-axis average intensity of a radially polarized CGCSM beam for different values of n . (a1)-(a3) as a function of distance z and ζ e f f , (b1)-(b3) as a function of α and ζ e f f at z = 10 k m .
Fig. 4
Fig. 4 The modulus of the spectral DOC μ ( u , v , 0 c m , 0 c m ) distribution and the corresponding cross line of a radially polarized CGCSM beam at different propagation distances for different values of n .
Fig. 5
Fig. 5 The modulus of the spectral DOC μ ( 0 c m , 0 c m , 5 c m , 0 c m ) of a radially polarized CGCSM beam for different values of n . (a1)-(a3) as a function of distance z and ζ e f f , (b1)-(b3) as a function of α and ζ e f f at z = 10 k m .
Fig. 6
Fig. 6 The spectral DOP distribution and the corresponding cross line of a radially polarized CGCSM beam at different propagation distances for different values of n .
Fig. 7
Fig. 7 The spectral DOP of a radially polarized CGCSM beam for different values of n at (5cm, 0cm). (a1)-(a3) as a function of distance z and ζ e f f , (b1)-(b3) as a function of α and ζ e f f at z = 10 k m .
Fig. 8
Fig. 8 The SOP of a radially polarized CGCSM beam at different propagation distances for different values of n .

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

E ( r ; ω ) = E x ( r ; ω ) e x + E y ( r ; ω ) e y = x w 0 exp ( r 2 w 0 2 ) e x + y w 0 exp ( r 2 w 0 2 ) e y ,
U α β ( r 1 , r 2 ; ω ) = α 1 β 2 w 0 2 exp ( r 1 2 + r 2 2 w 0 2 ) g α β ( r 1 r 2 ; ω ) , ( α = x , y ; β = x , y ) ,
g α β ( r 1 r 2 ; ω ) = exp [ ( r 1 r 2 ) 2 2 σ 0 2 ] cos [ n π ( x 1 x 2 ) σ 0 ] cos [ n π ( y 1 y 2 ) σ 0 ] ,
U α β ( ρ 1 , ρ 2 ; ω ) = ( k 2 π z ) 2 U α β ( r 1 , r 2 ; ω ) exp [ ψ ( r 1 , ρ 1 ; ω ) + ψ ( r 2 , ρ 2 ; ω ) ] × exp [ i k 2 z [ ( r 1 2 r 2 2 ) 2 ( r 1 ρ 1 r 2 ρ 2 ) + ( ρ 1 2 ρ 2 2 ) ] ] d 2 r 1 d 2 r 2 ,
exp [ ψ ( r 1 , ρ 1 ; ω ) + ψ ( r 2 , ρ 2 ; ω ) ] = exp [ π 2 k 2 z 3 ( r Δ 2 + r Δ ρ Δ + ρ Δ 2 ) 0 κ 3 Φ n ( κ , α ) d κ ] ,
Φ n ( κ , α ) = A ( α ) C ˜ n 2 ζ e f f 2 ( ζ e f f 2 κ x y 2 + κ z 2 + κ 0 2 ) α / 2 exp ( ζ e f f 2 κ x y 2 + κ z 2 κ H 2 ) ,
κ = ζ e f f 2 ( κ x 2 + κ y 2 ) + κ z 2 = ζ e f f 2 κ x y 2 + κ z 2 , A ( α ) = Γ ( α 1 ) 4 π 2 cos ( π α 2 ) , 3 < α < 4 ,
C ( α ) = [ π A ( α ) Γ ( 3 2 α 2 ) ( 3 α 3 ) ] 1 / ( α 5 ) , 3 < α < 4 ,
n ( R ) = n 0 + n 1 ( R ) ,
n 1 ( R 1 ) n 1 ( R 2 ) δ ( z 1 z 2 ) Α n ( r 1 r 2 ) .
C ˜ ¯ n 2 = 1 H h 0 h 0 H C ˜ n 2 ( h ) d h ,
C ˜ n 2 ( h ) = 0.00594 ( v s 27 ) 2 ( 10 5 h ) 10 exp ( h 1000 ) + 2.7 × 10 16 exp ( h 1500 ) + C ˜ n 2 exp ( h 100 ) ,
T a n i = π 2 k 2 z 3 0 Φ n ( κ , α ) κ 3 d κ = π 2 k 2 z ζ e f f 2 α 6 ( α 2 ) A ( α ) C ˜ ¯ n 2 [ κ ˜ H 2 α β exp ( κ 0 2 κ H 2 ) Γ ( 2 α 2 , κ 0 2 κ H 2 ) 2 κ ˜ 0 4 α ] ,
U x x ( ρ 1 , ρ 2 ; ω ) = V ( ρ 1 , ρ 2 ; ω ) { e x p [ γ v 12 2 4 M 1 + Ω v 22 2 4 Π ] + e x p [ γ v 11 2 4 M 1 + Ω v 21 2 4 Π ] } × { ( Δ + γ u 11 Ω u 21 + Δ 2 Π Ω u 21 2 ) exp [ γ u 11 2 4 M 1 + Ω u 21 2 4 Π ] ( Δ + γ u 12 Ω u 22 + Δ 2 Π Ω u 22 2 ) exp [ γ u 12 2 4 M 1 + Ω u 22 2 4 Π ] + } ,
U y y ( ρ 1 , ρ 2 ; ω ) = V ( ρ 1 , ρ 2 ; ω ) { e x p [ γ u 12 2 4 M 1 + Ω u 22 2 4 Π ] + e x p [ γ u 11 2 4 M 1 + Ω u 21 2 4 Π ] } × { ( Δ + γ v 11 Ω v 21 + Δ 2 Π Ω v 21 2 ) exp [ γ v 11 2 4 M 1 + Ω v 21 2 4 Π ] ( Δ + γ v 12 Ω v 22 + Δ 2 Π Ω v 22 2 ) exp [ γ v 12 2 4 M 1 + Ω v 22 2 4 Π ] + } ,
U x y ( ρ 1 , ρ 2 ; ω ) = V ( ρ 1 , ρ 2 ; ω ) { Ω v 22 e x p [ γ v 12 2 4 M 1 + Ω v 22 2 4 Π ] + Ω v 21 e x p [ γ v 11 2 4 M 1 + Ω v 21 2 4 Π ] } × { ( γ u 11 + Δ 2 Π Ω u 21 ) exp [ γ u 11 2 4 M 1 + Ω u 21 2 4 Π ] + ( γ u 12 + Δ 2 Π Ω u 22 ) exp [ γ u 12 2 4 M 1 + Ω u 22 2 4 Π ] } ,
U y x ( ρ 1 , ρ 2 ; ω ) = U x y * ( ρ 1 , ρ 2 ; ω ) ,
V ( ρ 1 , ρ 2 ; ω ) = k 2 64 z 2 w 0 2 M 1 2 Π 2 exp [ i k 2 z ( ρ 1 2 ρ 2 2 ) T a n i ρ Δ 2 ] , a = n 2 π / σ 0 , ξ u 1 = i k u 1 z T a n i u Δ , ξ u 2 = i k u 2 z T a n i u Δ , ξ v 1 = i k v 1 z T a n i v Δ , ξ v 2 = i k v 2 z T a n i v Δ , γ u 11 = ξ u 1 + i a , γ u 12 = ξ u 1 i a , γ u 21 = ξ u 2 + i a , γ u 22 = ξ u 2 i a , γ v 11 = ξ v 1 + i a , γ v 12 = ξ v 1 i a , γ v 21 = ξ v 2 + i a , γ v 22 = ξ v 2 i a , Ω u 22 = Δ γ u 12 2 M 1 γ u 22 , Ω u 21 = Δ γ u 11 2 M 1 γ u 21 , Ω v 22 = Δ γ v 12 2 M 1 γ v 22 , Ω v 21 = Δ γ v 11 2 M 1 γ v 21 , Δ = 1 σ 0 2 + 2 T a n i , M 1 = 1 w 0 2 + 1 2 σ 0 2 + i k 2 z + T a n i , Π = 1 w 0 2 + 1 2 σ 0 2 i k 2 z Δ 2 4 M 1 + T a n i .
I ( ρ ; ω ) = U x x ( ρ , ρ ; ω ) + U y y ( ρ , ρ ; ω ) ,
P ( ρ ; ω ) = 1 4 D e t [ U ( ρ , ρ ; ω ) ] { T r [ U ( ρ , ρ ; ω ) ] } 2 ,
μ ( ρ 1 , ρ 2 ; ω ) = T r U ( ρ 1 , ρ 2 ; ω ) T r [ U ( ρ 1 , ρ 1 ; ω ) ] T r [ U ( ρ 2 , ρ 2 ; ω ) ] .

Metrics