Abstract

A new class of partially coherent radially and azimuthally polarized rotating elliptical Gaussian (PCRPREG and PCAPREG) beams is introduced. The analytical expressions of the PCRPREG and PCAPREG beams propagating through anisotropy oceanic turbulence are derived based on the extended Huygens-Fresnel principle and the spatial power spectrum of oceanic turbulence. The effects of beam waist size w0, coherence width σ0, propagation distance z and oceanic turbulence parameters on the evolution statistics properties of PCRPREG and PCAPREG beams are studied in detail by numerical simulation. Our results indicate that with the increase of the propagation distance in the far field region, the normalized initial profile with a doughnut-like distribution of PCRPREG and PCAPREG beams gradually converts into a flat-topped one, and finally evolves into a Gaussian-like beam profile. We also find that the salinity-induced turbulence fluctuation makes a greater contribution to the decrease of beam quality compared with the temperature-induced turbulence fluctuation. Furthermore, the full width at half maximum becomes wider for the larger propagation distance z and wavelength λ or the smaller dissipation rate ε. Our work will pave the way for the development of underwater optical communication and underwater laser radar in oceanic environment.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The evolutions of laser beams in random media have attracted more and more attentions due to their broad applications spanning from laser radar system [1], remote sensing [2], underwater optical communication [3] and imaging [4] to high-resolution microscopy [5] and single fluorescent molecule [6]. So far, a variety of propagation properties have been developed, such as the power spectrums of oceanic and atmospheric turbulences [7, 8] and the spectrum of turbulent water [9]. It was found both theoretically and experimentally that partially coherent beams are less sensitive to the effects of turbulence than fully coherent beams [10, 11], the propagation of partially coherent laser beams has been a considerable hot subject in recent years, such as partially coherent cylindrical vector beam [12], partially coherent flat-topped laser beam [13], partially coherent radially polarized doughnut beam [14] and partially coherent Airy beam [15].

On the other hand, the evolution behavior of various Gaussian beams through the oceanic turbulence also has been studied, such as multi-Gaussian Schell-model beams [16], asymmetrical Gaussian beam [17], Lorentz Gaussian beam [18], Lommel-Gaussian beam [19]. Moreover, the influences of the turbulent characteristic parameters acting on the normalized intensity distribution [12], the conservation of the topological charge [20], the Strehl ratio [21] and orbital angular momentum [22] have been discussed. In 2000, Nikishov proposed an analytical model for such a power spectrum of oceanic turbulence [7]. Subsequently, the temperature and salinity fluctuations of radially polarized beams in oceanic turbulence have been deeply investigated [23, 24]. Interestingly, although both radial and azimuthal polarizations have identical spectral intensity distribution, their polarization characteristics are actually different [25, 26]. Meanwhile, several methods for generating a laser with radial or azimuthal polarization have been reported [27, 28]. Compared with other beams, radially and azimuthally polarized beams have better tight focusing properties and their focusing characteristics remain unchanged during the propagation process [29, 30].

Recently, rotating elliptical Gaussian beam propagating through the atmosphere turbulent and the oceanic turbulencee has been studied by Zhang et al. [31, 32]. And Ye et al. also analyzed rotating elliptical chirped Gaussian vortex beam through the oceanic turbulence in 2018 [33]. However, to the best of our knowledge, the propagation properties of partially coherent radially and azimuthally polarized rotating elliptical Gaussian (PCRPREG and PCAPREG) beams through the anisotropic oceanic turbulence have not been reported, which have potential application value in underwater wireless optical communication [34] and underwater optical imaging system [35].

Based on the extended Huygens-Fresnel principle and the power spectrum of oceanic turbulence, the analytical expressions for PCRPREG and PCAPREG beams propagating through anisotropy oceanic turbulence are derived. The second-order statistics propagation properties including the normalized spectrum density, the spectral degree of coherence (DOC) and the spectral degree of polarization (DOP) of a PCRPREG and PCAPREG beams in anisotropic oceanic turbulence have been discussed in detail by numerical simulation.

The organization of the paper is as follows. Firstly in Sec. 2, the analytical formula of 2 × 2 cross-spectral density (CSD) matrix of PCRPREG and PCAPREG beams through oceanic turbulence with anisotropy is derived. Then in Sec. 3, we illustrate the evolution behavior of PCRPREG and PCAPREG beams through anisotropic oceanic turbulence numerically. Finally, the main conclusion is drawn in Sec. 4.

2. The CSD matrix of PCRPREG and PCAPREG beams in oceanic turbulence with anisotropy

In the Cartesian coordinate system, the vectorial electric field of radially polarized rotating elliptical Gaussian (RPREG) and azimuthally polarized rotating elliptical Gaussian (APREG) beams can be expressed as the coherent superposition of a TEM01with a polarization direction parallel to the x-axis and a TEM10 with a polarization direction parallel to the y-axis [25, 26, 36, 37]:

(Er(x,y) Eθ(x,y))=exp (x2a2w02y2b2w02ixyc2w02)(xw0yw0  yw0xw0)(ex ey),
where r2=x2+y2, w0 denotes the beam waist size of a rotating elliptical gaussian beam, a, b and c are the arbitrary real constants, ex and ey represent the unit vectors in the x and ydirections, respectively. Figure 1 shows the RPREG (a) and APREG (b) beams at the initial plane.

 

Fig. 1 The RPREG (a) and APREG (b) beams at the initial plane.

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Here, the vertical electric field part can be ignored under the paraxial condition. For a vector partially coherent beam in space-frequency domain, the second-order spatial coherence properties can be characterized by the 2 × 2 CSD matrix of the electric field, defined by the formula [38]

W(x1,y1,x2,y2,0)=(Wxx(x1,y1,x2,y2,0)Wxy(x1,y1,x2,y2,0)Wyx(x1,y1,x2,y2,0)Wyy(x1,y1,x2,y2,0)),
where Wαβ(r1,r2)=Eα*(r1)Eβ(r2), (α,β=x,y), the asterisk denotes the complex conjugate and the angular brackets represent ensemble average.

The elements of the CSD matrix for PCRPREG and PCAPREG beams at source plane (z=0) can be expressed as follows [39]:

Wr(x1,y1,x2,y2,0)=1w02exp (x12+x22a2w02y12+y22b2w02+ix2y2x1y1c2w02)(x1x2x1y2y1x2y1y2)gαβ(r1r2),
Wθ(x1,y1,x2,y2,0)=1w02exp (x12+x22a2w02y12+y22b2w02+ix2y2x1y1c2w02)(y1y2y1x2x1y2x1x2)gαβ(r1r2),
and the spectral DOC is given by
gαβ(r1r2)=exp [(x1x2)2+(y1y2)22σ02],
where r1(x1,y1) and r2(x2,y2) are two arbitary transverse position vectors in the source plane, σ0 represents spatial coherence width.

Here, we investigate the propagation of the PCRPREG and PCAPREG beams in the oceanic turbulence. By applying the well-known extended Huygens-Fresnel integral formula, the element of the CSD matrix Wαβ(r1,r2) at the receiver plane can be expressed as [21, 23, 40]:

Wαβ(ρ1,ρ2,z)=(k2πz)2Wαβ(r1,r2,0)ψ(r1,ρ1)+ψ*(r2,ρ2) ×exp {ik2z[(r1ρ1)2(r2ρ2)2]}d2r1d2r2,
where ρ1(u1,v1) and ρ2(u2,v2) are two arbitary transverse position vectors at receiver plane, k=2πλ is the wave number with λ being the wavelength of the source beam. Under quadratic phase approximations, ψ(r1,ρ1)+ψ*(r2,ρ2) denotes the ensemble average of the turbulent ocean water which can be represented as [41]:
ψ(r1,ρ1)+ψ*(r2,ρ2)=exp [(r1r2)2+(r1r2)(ρ1ρ2)+(ρ1ρ2)2ρocξ2],
where ρocξ is the lateral coherence length of the spherical wave in the anisotropic turbulent ocean, whose expression is as follow [42]:
ρocξ2=π2k2zξ430κ3ψ˜an(κ)dκ,
with 0κ3ψ˜an(κ)dκ and ξ representing the turbulence strength and the anisotropic factor, respectively. We assume that the anisotropy exists only along the propagating direction z of the beam, using an effective anisotropic factor ξ to present the anisotropic power spectrum of oceanic turbulence. This is to say, the smaller ρocξ means the stronger turbulent strength. For the sake of simplicity, we only consider the influence of temperature and salinity fluctuations on the propagation of beams in the ocean turbulence, and ignore the scattering and absorption of the ocean water. In the Markov approximation, the two-dimensional refractive index turbulence spatial spectrum model of the anisotropic oceanic turbulent is written as [43]:
ψ˜an(κ)=0.388×108ε1/3χTξ2(κ)11/3[1+2.35(κη)2/3] ×[exp (ATδ)+ω2exp (ASδ)2ω1exp (ATSδ)],
with AT=1.863×102 , AS=1.9×104 , ATS=9.41×103 , δ=8.284(κη)4/3+12.378(κη)2 and κ=ξκx2+κy2 . Substituting Eq. (9) into Eq. (8), we obtain the coherence length of the spherical wave in anisotropic oceanic turbulence as [21]:
ρocξ=ξ|ω|[1.802×107k2z(εη)1/3χT(0.483ω20.835ω+3.380)]1/2,
where η is the Kolmogorov micro scale, χT is the rate of dissipation of mean square temperature and ranges from 104K2/s to 1010K2/s, ε is the rate of dissipation of turbulent kinetic energy per unit mass of fluid which may vary from 104m2/s3 to 1010m2/s3, ω represents the ratio of the temperature and salinity contribution to the refractive index spectrum, which in the ocean water the value can range from 5 to 0, 5 corresponding to the situation when salinity-driven turbulence prevails, 0 corresponding to the case when temperature-driven turbulence dominates.

By using the relation [44]

xnexp (px2+qx)dx=n!exp (q2p)(qp)nπpl=0n/21l!(n2l)!(q24p)l,

Eq. (6) can be analytically integrated as:

Wrxx(ρ1,ρ2,z)=QM1M3{[(N1iN22M2c2w2)N3+(ΔΔ4M1M2c4w4)(1+N322M2)]+[(N1iN22M2c2w2)SiΔN32M2c2w2+(ΔΔ4M1M2c4w4)N3SM3]N42M4+[iΔS2M2c2w2+(ΔΔ4M1M2c4w4)S22M3]12M4(1+N422M4)},
Wrxy(ρ1,ρ2,z)=QM1M4{[N1iN22M2c2w2+(ΔΔ4M1M2c4w4)N32M3]N4 +[iΔ2M2c2w2+(ΔΔ4M1M2c4w4)S2M3](1+N422M4)},
Wryx(ρ1,ρ2,z)=Wrxy*(ρ1,ρ2,z),
Wryy(ρ1,ρ2,z)=QM2M4{(N2iΔN34M1M3c2w2)N4+(ΔiΔS4M1M3c2w2)(1+n422M4)},
Q=k216w2z2M1M2M3M4exp{ik2z(ρ1 2ρ2 2)ρocξ2  (ρ1ρ2)2}exp{N124M1+N224M2+N324M3+N424M4},M1=1a2w02+12σ02+ρocξ2+ik2z,N1=ρocξ2(u1u2)+iku1z,Δ=1σ02+2ρocξ2,M2=14M1c4w04+1b2w02+12σ02+ρocξ2+ik2z,N2=iN12M1c2w02ρocξ2(v1v2)+ikv1z,M3=Δ216M12M1c4w04Δ24M1+1a2w02+12σ02+ρocξ2ik2z,S=iΔ24M1M2c2w02+i1c2w02N3=iN2Δ4M1M2c2w02+N1Δ2M1+ρocξ2(u1u2)iku2z,M4=S24M3Δ24M2+1b2w02+12σ02+ρocξ2ik2z,N4=N3S2M3+N2Δ2M2+ρocξ2(v1v2)ikv2z.

In the similar way, we obtain the following formulas for CSD matrix of a PCAPREG beam propagating in the anisotropy oceanic turbulence:

Wθxx(ρ1,ρ2,z)=Wryy(ρ1,ρ2,z),Wθyy(ρ1,ρ2,z)=Wrxx(ρ1,ρ2,z),Wθxy(ρ1,ρ2,z)=Wθyx(ρ1,ρ2,z)=Wrxy(ρ1,ρ2,z).

The spectral density and the spectral DOC at the output plane are given by the formulas [45]:

I(ρ,z)=TrW(ρ,ρ;z)=Wrxx(ρ,ρ,z)+Wryy(ρ,ρ,z),
μ(ρ1,ρ2,z)=TrW(ρ1,ρ2;z)I(ρ1,z)I(ρ2,z),
where Tr stands for the trace of the matrix.

The spectral DOP of the PCAPREG beam is defined as [45, 46]:

P(ρ,z)=14DetW(ρ,ρ;z)[TrW(ρ,ρ;z)]2=14(WrxxWryyWrxyWryx)(Wrxx+Wryy)2,
where Det denotes the determinant of the matrix. The orientation angle of the polarization ellipse of such beam on propagation can be calculated by the expression [47]:
θ(ρ;z)=12arctan [Re[Wrxy(ρ,ρ,z)]+Re[Wryx(ρ,ρ,z)]Wrxx(ρ,ρ,z)Wryy(ρ,ρ,z)].

Equations (16)-(20) indicate that the PCRPREG and PCAPREG beams has the same evolution properties.

3. Results and discussions

In this section, we will now analyze the behavior of the second-order statistics properties of a PCRPREG beam through anisotropic oceanic turbulence by a set of numerical examples based on the analytical formulas derived in the previous section. Unless other values are specified in caption, the parameters of the source and the oceanic are set as: a=2.0, b=1.8, c=1.7, w0=1.5mm, σ0=2.5mm, λ=416nm, ξ=2.0, ω=2.5, ε=105m2/s3, eta=1mm, χT=109K2/s.

 

Fig. 2 The normalized spectral density I(x,0,z)/Imax(x,0,0) of the PCRPREG beam at several propagation distances in the anisotropic oceanic turbulence for two different values of χT at (a) z=0m, (b) z=10m, (c) z=30m, (d) z=50m, (e) z=80m and (f) z=100m.

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Figure 2 presents normalized spectral density I(x,0,z)/Imax(x,0,0) of a PCRPREG beam at several propagation distances in anisotropic oceanic turbulence for two different values of χT. With the increase of propagation distance z, the doughnut beam profile converts into a flat-topped profile, as appreciably depicted in Figs. 2(a)-2(d). It is observed from Figs. 2(d)-2(f) that the flat-topped profile evolves into a Gaussian-like beam profile as the further increase of z. Besides, the larger the rate of dissipation of mean square temperature χT is, the faster the beam profile is close to the Gaussian-like distribution. The reason for this phenomenon is that as the transmission distance of the doughnut beam increases, the doughnut beam expands due to turbulence and vacuum diffraction, among which the expansion towards the center of the ring causes the spot center to collapse [48].

 

Fig. 3 The evolutions of the spectrum density distributions and the corresponding cross lines of the PCRPREG beam through the anisotropic oceanic turbulence at different propagation distances (a) z=0m, (b) z=10m, (c) z=40m, (d) z=70m, (e) z=100m and (f) z=130m.

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Figure 3 shows the evolutions of the spectrum density distribution and the corresponding cross line of a PCRPREG beam at several different propagation distances through anisotropic oceanic turbulence. Figures 3(a1)-3(f1) are three-dimensional diagrams corresponding to Figs. 3(a)-3(f) under the same conditions. It is significant that the spectrum density pattern has an elliptical annular shape in the initial plane whose major axis is parallel to the x-axis, which gradually rotates clockwise and converts into a quasi-circle annular shape with the increase of propagation distance z. Ultimately, the major axis of the spectrum density pattern is parallel to the y-axis in the far field region. In the oceanic turbulence, the spectrum density of a PCRPREG beam changes from an elliptical dark hollow beam profile to an elliptical flat-topped profile at certain propagation distances. Meanwhile, the elliptical flat-topped profile becomes a Gaussian-like beam profile in the far field region. The above phenomenon can be interpreted as the fact that the stronger oceanic turbulence is, the larger the degradation of the coherence is. One also finds from Fig. 3 that the azimuthal symmetry of the transverse intensity pattern is not affected by the oceanic turbulence although the elliptical dark hollow beam profile disappears due to the anisotropic influence of the oceanic turbulence.

 

Fig. 4 The normalized average spectral density distributions of the PCRPREG beam through anisotropic oceanic turbulence with different variables.

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Figure 4 plots normalized spectral density distribution I/Imax (Imax‒maximum spectral density) of a PCRPREG beam for different values of (a) propagation distance z, (b) wavelength λ, (c) spatial coherence width σ0, (d) transverse beam size w0, (e) rate of dissipation of mean-square temperature χT, (f) anisotropic factor ξ. It is observed from Fig. 4(a) that the normalized average spectral density distribution of a PCRPREG beam through anisotropic oceanic turbulence undergoes several stages of evolution analogous to the propagation of a Guassian Schell-model vortex beam through oceanic turbulence [21]. At the z = 0 plane, the normalized average spectral density distribution in the center part is zero, where the phase becomes singular [20]. With the increase of the propagation distance, a flat-topped spectral density profile occurs, and then evolves into a Gaussian-like beam profile. In Figs. 4(b)-4(e), the evolution behavior of normalized average spectral density distribution of a PCRPREG beam through anisotropic oceanic turbulence is profoundly effected by the anisotropic oceanic turbulence parameters including λ, σ0, w0, and χT. The smaller wavelength λ and w0 are and the larger σ0 and χT are, the faster the change of the beam spectral density distribution is. One can find from Eq. (10) and Fig. 4(e) that the larger χT is, the stronger the oceanic turbulence is. Whereas, the anisotropic factor ξ plays a less definitive role in normalized average spectral density distribution, as depicted in Fig. 4(f). In the meantime, it is shown that the longer propagation distance z is, the larger wavelength λ and χT are and the smaller σ0 and w0 are, the larger the spreading of the beam width is, but the effect caused by anisotropic factor is not significant.

 

Fig. 5 The spectral DOCs of the PCRPREG beam through the anisotropic oceanic turbulence with different variables.

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Figure 5(a) illustrates the evolution of the transverse DOCs of the PCRPREG beam between two points ρ1=(u,0) and ρ2=(u,0) at z=100m. It can be clearly seen that the μ decreases quickly from 1 to -0.4 and then increases slowly from -0.4 to 0 with the increase of the transverse distance u. As the wavelength decreases, the minimum value appears in a shorter transverse distance. In Figs. 5(b)-5(f), two points are fixed at ρ1=(0.3cm,0.3cm) and ρ2=(0.3cm,0.3cm), the DOCs of the PCRPREG beam are interpreted. For convenience, we choose z=100m for analysis in Fig. 5(c). With the increase of the propagation distance z, μ decreases rapidly from 0 to -0.3, and then increases progressively. The μ changes more significantly with the increase of the wavelength, as shown in Fig. 5(b). It is easy to find from Figs. 5(a) and 5(b) that the choice of wavelength has a great influence on the DOCs of the PCRPREG beam. The effects of the anisotropic oceanic turbulence on the μ are shown in Figs. 5(c)-5(d). Obviously, the μ gradually increases with the increase of ω, and the increasing trend becomes sharp for ω close to 0 value. Namely, the salinity-induced turbulence fluctuation makes a greater contribution to the decrease of beam quality compared with the temperature-induced turbulence fluctuation [21]. On the other hand, the decreasing ε and the increasing z lead to the increasing μ. One can see from Figs. 5(e) and 5(f) that the spectral DOC μ reaches the minimal value and then increases quickly. The larger ξ is and the smaller ω is, the greater the change of the spectral DOC μ is. As the propagation distance increases, theμ decreases from the value of 0 of the initial plane to the minimum value, and then returns to the initial value of 0, which is due to the influence of ocean turbulence as displayed in Figs. 5(b), 5(e) and 5(f). The above phenomenon can be explained by Eq. (10), it is not difficult to find that the smaller ω and the larger ξ are, the stronger the oceanic turbulence is. The spectral DOCs of the PCRPREG beam depend on the oceanic turbulence strength.

 

Fig. 6 The spectral DOPs of the PCRPREG beam and the corresponding cross line (v=0) at different propagation distances.

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Fig. 7 Cross lines (v=0) of the DOPs of the PCRPREG beam for difference values of ω, ε and λ.

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To study about the influence of the oceanic turbulence on the polarization properties of a PCRPREG beam in an anisotropic oceanic turbulence. We calculate the DOP of a PCRPREG beam and the corresponding cross line (v=0) at several propagation distances as shown in Fig. 6. The DOP of a PCRPREG beam equals 1 for all the points across the entire transverse plane when z = 0, which is manifested in Fig. 6. In the anisotropic oceanic turbulence, one can see from Fig. 6 that a dip appears in the distribution of the DOP, where the DOP distribution curve reaches the lowest point when u = 0, but as the propagation distance z increases, the minimum of the spectrum DOP decreases to 0.2 and the distribution is nearly unchanged. We also find that the width of the dip increases with the increasing of propagation distance, which implies that a radial polarization structure of a PCRPREG beam is destroyed during the propagation in an anisotropic oceanic turbulence. Furthermore, one also finds from Fig. 7 that the DOP curve is symmetric about the v axis, and the full width at half maximum is greater for the larger the ratio of the temperature and salinity contributions to the refractive index sprctrum ω (a) and wavelength λ (b) and the smaller the rate of dissipation of turbulent kinetic energy per unit mass of fluid ε (c) .

 

Fig. 8 The orientation angles of the PCRPREG beam in the anisotropic oceanic turbulence at different propagation distances.

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Finally, to learn more about the polarization properties, we calculate the orientation angle of a PCRPREG beam in anisotropic oceanic turbulence at several propagation distances. Figure 8 presents the radial polarization structure is destroyed due to the effects of ocean turbulence. It is interesting to note that the center of polarization structure changes most significantly with the increase of propagation distance.

4. Conclusion

In this paper, the analytical formulae for PCRPREG and PCAPREG beams propagating through anisotropy oceanic turbulence are derived based on the extended Huygens-Fresnel principle and the power spectrum of oceanic turbulence. The propagation evolution properties of PCRPREG and PCAPREG beams in oceanic turbulence have been discussed in detail by numerical simulation. Our results show that the propagation properties of the beam are closely related to beam waist size w0, coherence width σ0,propagation distance z and the oceanic turbulence parameters. The initial profile of PCRPREG and PCAPREG beams gradually disappears with a flat-topped one occurring, and finally the evolves into a Gaussian-like beam profile with the increase of the propagation distance in the far field region. Obviously, the longer propagation distance z is, the larger wavelength λ and the rate of dissipation of mean square temperature χT are and the smaller spatial coherence width σ0 and the beam waist size w0 are, the larger the spreading of the beam width is. In addition, the salinity-induced turbulence fluctuation makes a greater contribution to the decrease of beam quality comparing with the temperature-induced turbulence fluctuation. The larger the ratio of the temperature and salinity contributions to the refractive index sprectrum ω and wavelength λ are or the smaller the rate of dissipation of turbulent kinetic energy per unit mass of fluid ε is, the greater the full width at half maximum is. It is rather remarkable that the radial polarization structure of a PCRPREG beam is destroyed due to the anisotropic influence of the oceanic turbulence. It is generally believed that our work may have potential applications in underwater optical comminication system.

Funding

National Natural Science Foundation of China (NSFC) (11775083, 11374108); Innovation Project of Graduate School of South China Normal University (2018LKXM043).

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27. K. J. Moh, X. C. Yuan, J. Bu, R. E. Burge, and Bruce Z. Gao, “Generating radial or azimuthal polarization by axial sampling of circularly polarized vortex beams,” Appl. Opt. 46(30), 7544–7551 (2007). [CrossRef]  

28. G. Miyaji, N. Miyanaga, K. Tsubakimoto, K. Sueda, and K. Ohbayashi, “Intense longitudinal electric fields generated from transverse electromagnetic waves,” Appl. Phys. Lett. 84(19), 3855 (2004). [CrossRef]  

29. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000). [CrossRef]  

30. C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017). [CrossRef]  

31. J. Zhang, J. Xie, F. Ye, K. Zhou, X. Chen, D. Deng, and X. Yang, “Effects of the turbulent atmosphere and the oceanic turbulence on the propagation of a rotating elliptical Gaussian beam,” Appl. Phys. B 124, 168 (2018). [CrossRef]  

32. J. Zhang, J. Xie, and D. Deng, “Second-order statistics of a partially coherent electromagnetic rotating elliptical Gaussian vortex beam through non-Kolmogorov turbulence,” Opt. Express 26(16), 21249–21257 (2018). [CrossRef]   [PubMed]  

33. F. Ye, J. Zhang, J. Xie, and D. Deng, “Propagation properties of the rotating elliptical chirped Gaussian vortex beam in the oceanic turbulence,” Opt. Commun. 426, 456–462 (2018). [CrossRef]  

34. W. Liu, Z. Xu, and L. Yang, “SIMO detection schemes for underwater optical wireless communication under turbulence,” Photon. Res. 3(3), 48–53 (2015). [CrossRef]  

35. S. Matta, W. Houb, S. Woodsc, W. Goodeb, E. Jarosz, and A. Weidemannb, “A novel platform to study the effect of small-scale turbulent density fluctuations on underwater imaging in the ocean,” Methods in Oceanography 11(5), 39–58 (2014). [CrossRef]  

36. 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]  

37. T. Zhong, J. Zhang, L. Feng, Z. Pang, L. Wang, and D. Deng, “Propagation properties of radially and azimuthally polarized chirped-Airy vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc.Am. B 35(6), 1355–1361 (2018).

38. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5), 263–267 (2003). [CrossRef]  

39. Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19(7), 5979–5992 (2011). [CrossRef]   [PubMed]  

40. M. Tang and D. Zhao, “Regions of spreading of Gaussian array beams propagating through oceanic turbulence,” Appl. Opt. 54(11), 3407–3411 (2015). [CrossRef]  

41. X. Huang, Y. Bai, and X. Fu, “Propagation factors of partially coherent model beams in oceanic turbulence,” IEEE Photonics J. 9(5), 1–11 (2017).

42. Y. Li, Y. Zhang, Y. Zhu, and M. Chen, “Effects of anisotropic turbulence on average polarizability of Gaussian Schell-model quantized beams through ocean link,” Appl. Opt. 55(19), 5234–5239 (2016). [CrossRef]  

43. Y. Wu, Y. Zhang, Y. Li, and Z. Hu, “Beam wander of Gaussian-Schell model beams propagating through oceanic turbulence,” Opt. Commun. 371, 59–66 (2016). [CrossRef]  

44. D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017). [CrossRef]  

45. X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15(25), 16909–16915 (2007). [CrossRef]   [PubMed]  

46. 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]  

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

48. J. Cang and Y. Zhang, “The propagation properties of J0-correlated partially coherent beams in the slant atmosphere,” Acta Physica Sinica 58(4), 2444–2450 (2009).

References

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  1. X. Ji, Y. Ma, Z. Pu, and X. Li, “Characteristics of tilted and off-axis partially coherent beams reflected back by a cat-eye optical lens in atmospheric turbulence,” Appl. Phys. B 115(3), 379–390 (2014).
    [Crossref]
  2. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Medium, 2nd ed. (SPIE, Washington, 2005).
  3. S. Arnon and D. Kedar, “Non-line-of-sight underwater optical wireless communication network,” J. Opt. Soc. Am. A 26(3), 530–539 (2009).
    [Crossref]
  4. W. Hou, S. Woods, E. Jarosz, W. Goode, and A. Weidemann, “Optical turbulence on underwater image degradation in natural environments,” Appl. Opt. 51(14), 2678–2686 (2012).
    [Crossref] [PubMed]
  5. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
    [Crossref]
  6. B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
    [Crossref]
  7. V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the seawater refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2010).
    [Crossref]
  8. 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]
  9. R. Hill, “Optical propagation in turbulent water,” J. Opt. Soc. Am. A 68, 1067–1072 (1978).
    [Crossref]
  10. Y. Zhu, M. Chen, Y. Zhang, and Y. Li, “Propagation of the OAM mode carried by partially coherent modified Bessel-Gaussian beams in an anisotropic non-Kolmogorov marine atmosphere,” J. Opt. Soc. Am. A 33(12), 2277–2283 (2016).
    [Crossref]
  11. H. Chen, X. Sheng, F. Zhao, and Y. Zhang, “Orbital angular momentum entanglement states of Gaussian-Schell beam pumping in low-order non-Kolmogorov turbulent aberration channels,” Opt. Laser Technol 49, 332–336 (2013).
    [Crossref]
  12. Y. Dong, L. Guo, C. Liang, F. Wang, and Y. Cai, “Statistical properties of a partially coherent cylindrical vector beam in oceanic turbulence,” J. Opt. Soc. Am. A 32(5), 894–901 (2015).
    [Crossref]
  13. M. Yousefi, S. Golmohammady, A. Mashal, and F. D. Kashani, “Analyzing the propagation behavior of scintillation index and bit error rate of a partially coherent flat-topped laser beam in oceanic turbulence,” J. Opt. Soc. Am. A 32(11), 1982–1992 (2015).
    [Crossref]
  14. W. Fu and H. Zhang, “Propagation properties of partially coherent radially polarized doughnut beam in turbulent ocean,” Opt. Commun. 304, 11–18 (2013).
    [Crossref]
  15. Y. Jin, M. Hu, M. Luo, Y. Luo, X. Mi, C. Zou, L. Zhou, C. Shu, X. Zhu, J. He, S. Ouyang, and W. Wen, “Beam wander of a partially coherent Airy beam in oceanic turbulence,” J. Opt. Soc. Am. A 35(8), 1457–1464 (2018).
    [Crossref]
  16. C. Lu and D. Zhao, “Propagation of electromagnetic multi-Gaussian Schell-model beams with astigmatic aberration in turbulent ocean,” Appl. Opt. 55(29), 8196–8200 (2016).
    [Crossref] [PubMed]
  17. Y. Ata and Y. Baykal, “Effect of anisotropy on bit error rate for an asymmetrical Gaussian beam in a turbulent ocean,” Appl. Opt. 57(9), 2258–2262 (2018).
    [Crossref]
  18. D. Liu, Y. Wang, G. Wang, and H. Yin, “Influences of oceanic turbulence on Lorentz Gaussian beam,” Optik 154, 738–747 (2018).
    [Crossref]
  19. L. Yu and Y. Zhang, “Analysis of modal crosstalk for communication in turbulent ocean using Lommel-Gaussian beam,” Opt. Express 25(19), 22565–22574 (2017).
    [Crossref]
  20. Y. Huang, B. Zhang, Z. Gao, G. Zhao, and Z. Duan, “Evolution behavior of Gaussian Schell-model vortex beams propagating through oceanic turbulence,” Opt. Express 22(15), 17723–17734 (2014).
    [Crossref] [PubMed]
  21. X. Huang, Z. Deng, X. Shi, Y. Bai, and X. Fu, “Average intensity and beam quality of optical coherence lattices in oceanic turbulence with anisotropy,” Opt. Express 26(4), 4786–4797 (2018).
    [Crossref]
  22. Y. Li, L. Yu, and Y. Zhang, “Influence of anisotropic turbulence on the orbital angular momentum modes of Hermite-Gaussian vortex beam in the ocean,” Opt. Express 25(11), 12203–12215 (2017).
    [Crossref]
  23. X. Peng, L. Liu, Y. Cai, and Y. Baykal, “Statistical properties of a radially polarized twisted Gaussian Schell-model beam in an underwater turbulent medium,” J. Opt. Soc. Am. A 34(1), 133–139 (2017).
    [Crossref]
  24. W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A: Pure Appl. Opt. 8, 1052–1058 (2006).
    [Crossref]
  25. H. Wang and X. Li, “Spectral properties of partially coherent azimuthally polarized beam in a turbulent atmosphere,” Optik 122, 2164–2171 (2011).
    [Crossref]
  26. Y. Cai, Q. Lin, H. T. Eyyuboǧlu, 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]
  27. K. J. Moh, X. C. Yuan, J. Bu, R. E. Burge, and Bruce Z. Gao, “Generating radial or azimuthal polarization by axial sampling of circularly polarized vortex beams,” Appl. Opt. 46(30), 7544–7551 (2007).
    [Crossref]
  28. G. Miyaji, N. Miyanaga, K. Tsubakimoto, K. Sueda, and K. Ohbayashi, “Intense longitudinal electric fields generated from transverse electromagnetic waves,” Appl. Phys. Lett. 84(19), 3855 (2004).
    [Crossref]
  29. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000).
    [Crossref]
  30. C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017).
    [Crossref]
  31. J. Zhang, J. Xie, F. Ye, K. Zhou, X. Chen, D. Deng, and X. Yang, “Effects of the turbulent atmosphere and the oceanic turbulence on the propagation of a rotating elliptical Gaussian beam,” Appl. Phys. B 124, 168 (2018).
    [Crossref]
  32. J. Zhang, J. Xie, and D. Deng, “Second-order statistics of a partially coherent electromagnetic rotating elliptical Gaussian vortex beam through non-Kolmogorov turbulence,” Opt. Express 26(16), 21249–21257 (2018).
    [Crossref] [PubMed]
  33. F. Ye, J. Zhang, J. Xie, and D. Deng, “Propagation properties of the rotating elliptical chirped Gaussian vortex beam in the oceanic turbulence,” Opt. Commun. 426, 456–462 (2018).
    [Crossref]
  34. W. Liu, Z. Xu, and L. Yang, “SIMO detection schemes for underwater optical wireless communication under turbulence,” Photon. Res. 3(3), 48–53 (2015).
    [Crossref]
  35. S. Matta, W. Houb, S. Woodsc, W. Goodeb, E. Jarosz, and A. Weidemannb, “A novel platform to study the effect of small-scale turbulent density fluctuations on underwater imaging in the ocean,” Methods in Oceanography 11(5), 39–58 (2014).
    [Crossref]
  36. 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]
  37. T. Zhong, J. Zhang, L. Feng, Z. Pang, L. Wang, and D. Deng, “Propagation properties of radially and azimuthally polarized chirped-Airy vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc.Am. B 35(6), 1355–1361 (2018).
  38. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5), 263–267 (2003).
    [Crossref]
  39. Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19(7), 5979–5992 (2011).
    [Crossref] [PubMed]
  40. M. Tang and D. Zhao, “Regions of spreading of Gaussian array beams propagating through oceanic turbulence,” Appl. Opt. 54(11), 3407–3411 (2015).
    [Crossref]
  41. X. Huang, Y. Bai, and X. Fu, “Propagation factors of partially coherent model beams in oceanic turbulence,” IEEE Photonics J. 9(5), 1–11 (2017).
  42. Y. Li, Y. Zhang, Y. Zhu, and M. Chen, “Effects of anisotropic turbulence on average polarizability of Gaussian Schell-model quantized beams through ocean link,” Appl. Opt. 55(19), 5234–5239 (2016).
    [Crossref]
  43. Y. Wu, Y. Zhang, Y. Li, and Z. Hu, “Beam wander of Gaussian-Schell model beams propagating through oceanic turbulence,” Opt. Commun. 371, 59–66 (2016).
    [Crossref]
  44. D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
    [Crossref]
  45. X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15(25), 16909–16915 (2007).
    [Crossref] [PubMed]
  46. 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]
  47. M. Yao, I. Toselli, and O. Korotkova, “Propagation of electromagnetic stochastic beams in anisotropic turbulence,” Opt. Express 22(26), 31608–31619 (2014).
    [Crossref]
  48. J. Cang and Y. Zhang, “The propagation properties of J0-correlated partially coherent beams in the slant atmosphere,” Acta Physica Sinica 58(4), 2444–2450 (2009).

2018 (8)

Y. Jin, M. Hu, M. Luo, Y. Luo, X. Mi, C. Zou, L. Zhou, C. Shu, X. Zhu, J. He, S. Ouyang, and W. Wen, “Beam wander of a partially coherent Airy beam in oceanic turbulence,” J. Opt. Soc. Am. A 35(8), 1457–1464 (2018).
[Crossref]

Y. Ata and Y. Baykal, “Effect of anisotropy on bit error rate for an asymmetrical Gaussian beam in a turbulent ocean,” Appl. Opt. 57(9), 2258–2262 (2018).
[Crossref]

D. Liu, Y. Wang, G. Wang, and H. Yin, “Influences of oceanic turbulence on Lorentz Gaussian beam,” Optik 154, 738–747 (2018).
[Crossref]

X. Huang, Z. Deng, X. Shi, Y. Bai, and X. Fu, “Average intensity and beam quality of optical coherence lattices in oceanic turbulence with anisotropy,” Opt. Express 26(4), 4786–4797 (2018).
[Crossref]

J. Zhang, J. Xie, F. Ye, K. Zhou, X. Chen, D. Deng, and X. Yang, “Effects of the turbulent atmosphere and the oceanic turbulence on the propagation of a rotating elliptical Gaussian beam,” Appl. Phys. B 124, 168 (2018).
[Crossref]

J. Zhang, J. Xie, and D. Deng, “Second-order statistics of a partially coherent electromagnetic rotating elliptical Gaussian vortex beam through non-Kolmogorov turbulence,” Opt. Express 26(16), 21249–21257 (2018).
[Crossref] [PubMed]

F. Ye, J. Zhang, J. Xie, and D. Deng, “Propagation properties of the rotating elliptical chirped Gaussian vortex beam in the oceanic turbulence,” Opt. Commun. 426, 456–462 (2018).
[Crossref]

T. Zhong, J. Zhang, L. Feng, Z. Pang, L. Wang, and D. Deng, “Propagation properties of radially and azimuthally polarized chirped-Airy vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc.Am. B 35(6), 1355–1361 (2018).

2017 (6)

2016 (5)

2015 (6)

2014 (4)

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

S. Matta, W. Houb, S. Woodsc, W. Goodeb, E. Jarosz, and A. Weidemannb, “A novel platform to study the effect of small-scale turbulent density fluctuations on underwater imaging in the ocean,” Methods in Oceanography 11(5), 39–58 (2014).
[Crossref]

Y. Huang, B. Zhang, Z. Gao, G. Zhao, and Z. Duan, “Evolution behavior of Gaussian Schell-model vortex beams propagating through oceanic turbulence,” Opt. Express 22(15), 17723–17734 (2014).
[Crossref] [PubMed]

X. Ji, Y. Ma, Z. Pu, and X. Li, “Characteristics of tilted and off-axis partially coherent beams reflected back by a cat-eye optical lens in atmospheric turbulence,” Appl. Phys. B 115(3), 379–390 (2014).
[Crossref]

2013 (2)

H. Chen, X. Sheng, F. Zhao, and Y. Zhang, “Orbital angular momentum entanglement states of Gaussian-Schell beam pumping in low-order non-Kolmogorov turbulent aberration channels,” Opt. Laser Technol 49, 332–336 (2013).
[Crossref]

W. Fu and H. Zhang, “Propagation properties of partially coherent radially polarized doughnut beam in turbulent ocean,” Opt. Commun. 304, 11–18 (2013).
[Crossref]

2012 (1)

2011 (2)

H. Wang and X. Li, “Spectral properties of partially coherent azimuthally polarized beam in a turbulent atmosphere,” Optik 122, 2164–2171 (2011).
[Crossref]

Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19(7), 5979–5992 (2011).
[Crossref] [PubMed]

2010 (1)

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the seawater refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2010).
[Crossref]

2009 (2)

S. Arnon and D. Kedar, “Non-line-of-sight underwater optical wireless communication network,” J. Opt. Soc. Am. A 26(3), 530–539 (2009).
[Crossref]

J. Cang and Y. Zhang, “The propagation properties of J0-correlated partially coherent beams in the slant atmosphere,” Acta Physica Sinica 58(4), 2444–2450 (2009).

2008 (1)

2007 (2)

2006 (1)

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A: Pure Appl. Opt. 8, 1052–1058 (2006).
[Crossref]

2004 (1)

G. Miyaji, N. Miyanaga, K. Tsubakimoto, K. Sueda, and K. Ohbayashi, “Intense longitudinal electric fields generated from transverse electromagnetic waves,” Appl. Phys. Lett. 84(19), 3855 (2004).
[Crossref]

2003 (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5), 263–267 (2003).
[Crossref]

2001 (1)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

2000 (2)

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[Crossref]

K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000).
[Crossref]

1978 (1)

R. Hill, “Optical propagation in turbulent water,” J. Opt. Soc. Am. A 68, 1067–1072 (1978).
[Crossref]

Andrews, L. C.

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

Arnon, S.

Ata, Y.

Bai, Y.

X. Huang, Z. Deng, X. Shi, Y. Bai, and X. Fu, “Average intensity and beam quality of optical coherence lattices in oceanic turbulence with anisotropy,” Opt. Express 26(4), 4786–4797 (2018).
[Crossref]

X. Huang, Y. Bai, and X. Fu, “Propagation factors of partially coherent model beams in oceanic turbulence,” IEEE Photonics J. 9(5), 1–11 (2017).

Baykal, Y.

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

Brown, T.

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

Bu, J.

Burge, R. E.

Cai, Y.

X. Peng, L. Liu, Y. Cai, and Y. Baykal, “Statistical properties of a radially polarized twisted Gaussian Schell-model beam in an underwater turbulent medium,” J. Opt. Soc. Am. A 34(1), 133–139 (2017).
[Crossref]

C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017).
[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]

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]

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. Dong, L. Guo, C. Liang, F. Wang, and Y. Cai, “Statistical properties of a partially coherent cylindrical vector beam in oceanic turbulence,” J. Opt. Soc. Am. A 32(5), 894–901 (2015).
[Crossref]

Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19(7), 5979–5992 (2011).
[Crossref] [PubMed]

Y. Cai, Q. Lin, H. T. Eyyuboǧlu, 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]

Cang, J.

J. Cang and Y. Zhang, “The propagation properties of J0-correlated partially coherent beams in the slant atmosphere,” Acta Physica Sinica 58(4), 2444–2450 (2009).

Chen, H.

H. Chen, X. Sheng, F. Zhao, and Y. Zhang, “Orbital angular momentum entanglement states of Gaussian-Schell beam pumping in low-order non-Kolmogorov turbulent aberration channels,” Opt. Laser Technol 49, 332–336 (2013).
[Crossref]

Chen, M.

Chen, X.

J. Zhang, J. Xie, F. Ye, K. Zhou, X. Chen, D. Deng, and X. Yang, “Effects of the turbulent atmosphere and the oceanic turbulence on the propagation of a rotating elliptical Gaussian beam,” Appl. Phys. B 124, 168 (2018).
[Crossref]

Chen, Y.

Deng, D.

J. Zhang, J. Xie, F. Ye, K. Zhou, X. Chen, D. Deng, and X. Yang, “Effects of the turbulent atmosphere and the oceanic turbulence on the propagation of a rotating elliptical Gaussian beam,” Appl. Phys. B 124, 168 (2018).
[Crossref]

J. Zhang, J. Xie, and D. Deng, “Second-order statistics of a partially coherent electromagnetic rotating elliptical Gaussian vortex beam through non-Kolmogorov turbulence,” Opt. Express 26(16), 21249–21257 (2018).
[Crossref] [PubMed]

F. Ye, J. Zhang, J. Xie, and D. Deng, “Propagation properties of the rotating elliptical chirped Gaussian vortex beam in the oceanic turbulence,” Opt. Commun. 426, 456–462 (2018).
[Crossref]

T. Zhong, J. Zhang, L. Feng, Z. Pang, L. Wang, and D. Deng, “Propagation properties of radially and azimuthally polarized chirped-Airy vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc.Am. B 35(6), 1355–1361 (2018).

Deng, Z.

Dong, Y.

Du, X.

Duan, Z.

Eyyuboglu, H. T.

Feng, L.

T. Zhong, J. Zhang, L. Feng, Z. Pang, L. Wang, and D. Deng, “Propagation properties of radially and azimuthally polarized chirped-Airy vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc.Am. B 35(6), 1355–1361 (2018).

Fu, W.

W. Fu and H. Zhang, “Propagation properties of partially coherent radially polarized doughnut beam in turbulent ocean,” Opt. Commun. 304, 11–18 (2013).
[Crossref]

Fu, X.

X. Huang, Z. Deng, X. Shi, Y. Bai, and X. Fu, “Average intensity and beam quality of optical coherence lattices in oceanic turbulence with anisotropy,” Opt. Express 26(4), 4786–4797 (2018).
[Crossref]

X. Huang, Y. Bai, and X. Fu, “Propagation factors of partially coherent model beams in oceanic turbulence,” IEEE Photonics J. 9(5), 1–11 (2017).

Gao, Bruce Z.

Gao, Z.

Golmohammady, S.

Goode, W.

Goodeb, W.

S. Matta, W. Houb, S. Woodsc, W. Goodeb, E. Jarosz, and A. Weidemannb, “A novel platform to study the effect of small-scale turbulent density fluctuations on underwater imaging in the ocean,” Methods in Oceanography 11(5), 39–58 (2014).
[Crossref]

Guo, L.

He, J.

Hecht, B.

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[Crossref]

Hill, R.

R. Hill, “Optical propagation in turbulent water,” J. Opt. Soc. Am. A 68, 1067–1072 (1978).
[Crossref]

Hou, W.

Houb, W.

S. Matta, W. Houb, S. Woodsc, W. Goodeb, E. Jarosz, and A. Weidemannb, “A novel platform to study the effect of small-scale turbulent density fluctuations on underwater imaging in the ocean,” Methods in Oceanography 11(5), 39–58 (2014).
[Crossref]

Hu, M.

Hu, Z.

Y. Wu, Y. Zhang, Y. Li, and Z. Hu, “Beam wander of Gaussian-Schell model beams propagating through oceanic turbulence,” Opt. Commun. 371, 59–66 (2016).
[Crossref]

Huang, X.

X. Huang, Z. Deng, X. Shi, Y. Bai, and X. Fu, “Average intensity and beam quality of optical coherence lattices in oceanic turbulence with anisotropy,” Opt. Express 26(4), 4786–4797 (2018).
[Crossref]

X. Huang, Y. Bai, and X. Fu, “Propagation factors of partially coherent model beams in oceanic turbulence,” IEEE Photonics J. 9(5), 1–11 (2017).

Huang, Y.

Jarosz, E.

S. Matta, W. Houb, S. Woodsc, W. Goodeb, E. Jarosz, and A. Weidemannb, “A novel platform to study the effect of small-scale turbulent density fluctuations on underwater imaging in the ocean,” Methods in Oceanography 11(5), 39–58 (2014).
[Crossref]

W. Hou, S. Woods, E. Jarosz, W. Goode, and A. Weidemann, “Optical turbulence on underwater image degradation in natural environments,” Appl. Opt. 51(14), 2678–2686 (2012).
[Crossref] [PubMed]

Ji, X.

X. Ji, Y. Ma, Z. Pu, and X. Li, “Characteristics of tilted and off-axis partially coherent beams reflected back by a cat-eye optical lens in atmospheric turbulence,” Appl. Phys. B 115(3), 379–390 (2014).
[Crossref]

Jin, Y.

Kashani, F. D.

Kedar, D.

Korotkova, O.

Li, X.

X. Ji, Y. Ma, Z. Pu, and X. Li, “Characteristics of tilted and off-axis partially coherent beams reflected back by a cat-eye optical lens in atmospheric turbulence,” Appl. Phys. B 115(3), 379–390 (2014).
[Crossref]

H. Wang and X. Li, “Spectral properties of partially coherent azimuthally polarized beam in a turbulent atmosphere,” Optik 122, 2164–2171 (2011).
[Crossref]

Li, Y.

Li, Z.

Liang, C.

Lin, Q.

Liu, D.

D. Liu, Y. Wang, G. Wang, and H. Yin, “Influences of oceanic turbulence on Lorentz Gaussian beam,” Optik 154, 738–747 (2018).
[Crossref]

D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
[Crossref]

Liu, L.

X. Peng, L. Liu, Y. Cai, and Y. Baykal, “Statistical properties of a radially polarized twisted Gaussian Schell-model beam in an underwater turbulent medium,” J. Opt. Soc. Am. A 34(1), 133–139 (2017).
[Crossref]

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A: Pure Appl. Opt. 8, 1052–1058 (2006).
[Crossref]

Liu, W.

Liu, X.

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]

Lu, C.

Lu, W.

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A: Pure Appl. Opt. 8, 1052–1058 (2006).
[Crossref]

Luo, M.

Luo, X.

D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
[Crossref]

Luo, Y.

Ma, Y.

X. Ji, Y. Ma, Z. Pu, and X. Li, “Characteristics of tilted and off-axis partially coherent beams reflected back by a cat-eye optical lens in atmospheric turbulence,” Appl. Phys. B 115(3), 379–390 (2014).
[Crossref]

Mashal, A.

Matta, S.

S. Matta, W. Houb, S. Woodsc, W. Goodeb, E. Jarosz, and A. Weidemannb, “A novel platform to study the effect of small-scale turbulent density fluctuations on underwater imaging in the ocean,” Methods in Oceanography 11(5), 39–58 (2014).
[Crossref]

Mi, X.

Miyaji, G.

G. Miyaji, N. Miyanaga, K. Tsubakimoto, K. Sueda, and K. Ohbayashi, “Intense longitudinal electric fields generated from transverse electromagnetic waves,” Appl. Phys. Lett. 84(19), 3855 (2004).
[Crossref]

Miyanaga, N.

G. Miyaji, N. Miyanaga, K. Tsubakimoto, K. Sueda, and K. Ohbayashi, “Intense longitudinal electric fields generated from transverse electromagnetic waves,” Appl. Phys. Lett. 84(19), 3855 (2004).
[Crossref]

Moh, K. J.

Nikishov, V. I.

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the seawater refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2010).
[Crossref]

Nikishov, V. V.

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the seawater refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2010).
[Crossref]

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[Crossref]

Ohbayashi, K.

G. Miyaji, N. Miyanaga, K. Tsubakimoto, K. Sueda, and K. Ohbayashi, “Intense longitudinal electric fields generated from transverse electromagnetic waves,” Appl. Phys. Lett. 84(19), 3855 (2004).
[Crossref]

Ouyang, S.

Pang, Z.

T. Zhong, J. Zhang, L. Feng, Z. Pang, L. Wang, and D. Deng, “Propagation properties of radially and azimuthally polarized chirped-Airy vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc.Am. B 35(6), 1355–1361 (2018).

Peng, X.

Phillips, R. L.

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

Ping, C.

Pu, Z.

X. Ji, Y. Ma, Z. Pu, and X. Li, “Characteristics of tilted and off-axis partially coherent beams reflected back by a cat-eye optical lens in atmospheric turbulence,” Appl. Phys. B 115(3), 379–390 (2014).
[Crossref]

Sheng, X.

H. Chen, X. Sheng, F. Zhao, and Y. Zhang, “Orbital angular momentum entanglement states of Gaussian-Schell beam pumping in low-order non-Kolmogorov turbulent aberration channels,” Opt. Laser Technol 49, 332–336 (2013).
[Crossref]

Shi, X.

Shu, C.

Sick, B.

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[Crossref]

Sueda, K.

G. Miyaji, N. Miyanaga, K. Tsubakimoto, K. Sueda, and K. Ohbayashi, “Intense longitudinal electric fields generated from transverse electromagnetic waves,” Appl. Phys. Lett. 84(19), 3855 (2004).
[Crossref]

Sun, J.

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A: Pure Appl. Opt. 8, 1052–1058 (2006).
[Crossref]

Tang, M.

Toselli, I.

Tsubakimoto, K.

G. Miyaji, N. Miyanaga, K. Tsubakimoto, K. Sueda, and K. Ohbayashi, “Intense longitudinal electric fields generated from transverse electromagnetic waves,” Appl. Phys. Lett. 84(19), 3855 (2004).
[Crossref]

Wang, F.

Wang, G.

D. Liu, Y. Wang, G. Wang, and H. Yin, “Influences of oceanic turbulence on Lorentz Gaussian beam,” Optik 154, 738–747 (2018).
[Crossref]

D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
[Crossref]

Wang, H.

Wang, J.

Wang, L.

T. Zhong, J. Zhang, L. Feng, Z. Pang, L. Wang, and D. Deng, “Propagation properties of radially and azimuthally polarized chirped-Airy vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc.Am. B 35(6), 1355–1361 (2018).

Wang, Y.

D. Liu, Y. Wang, G. Wang, and H. Yin, “Influences of oceanic turbulence on Lorentz Gaussian beam,” Optik 154, 738–747 (2018).
[Crossref]

D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
[Crossref]

Weidemann, A.

Weidemannb, A.

S. Matta, W. Houb, S. Woodsc, W. Goodeb, E. Jarosz, and A. Weidemannb, “A novel platform to study the effect of small-scale turbulent density fluctuations on underwater imaging in the ocean,” Methods in Oceanography 11(5), 39–58 (2014).
[Crossref]

Wen, W.

Wolf, E.

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5), 263–267 (2003).
[Crossref]

Woods, S.

Woodsc, S.

S. Matta, W. Houb, S. Woodsc, W. Goodeb, E. Jarosz, and A. Weidemannb, “A novel platform to study the effect of small-scale turbulent density fluctuations on underwater imaging in the ocean,” Methods in Oceanography 11(5), 39–58 (2014).
[Crossref]

Wu, Y.

Y. Wu, Y. Zhang, Y. Li, and Z. Hu, “Beam wander of Gaussian-Schell model beams propagating through oceanic turbulence,” Opt. Commun. 371, 59–66 (2016).
[Crossref]

Xie, J.

J. Zhang, J. Xie, and D. Deng, “Second-order statistics of a partially coherent electromagnetic rotating elliptical Gaussian vortex beam through non-Kolmogorov turbulence,” Opt. Express 26(16), 21249–21257 (2018).
[Crossref] [PubMed]

F. Ye, J. Zhang, J. Xie, and D. Deng, “Propagation properties of the rotating elliptical chirped Gaussian vortex beam in the oceanic turbulence,” Opt. Commun. 426, 456–462 (2018).
[Crossref]

J. Zhang, J. Xie, F. Ye, K. Zhou, X. Chen, D. Deng, and X. Yang, “Effects of the turbulent atmosphere and the oceanic turbulence on the propagation of a rotating elliptical Gaussian beam,” Appl. Phys. B 124, 168 (2018).
[Crossref]

Xu, Z.

Yang, L.

Yang, X.

J. Zhang, J. Xie, F. Ye, K. Zhou, X. Chen, D. Deng, and X. Yang, “Effects of the turbulent atmosphere and the oceanic turbulence on the propagation of a rotating elliptical Gaussian beam,” Appl. Phys. B 124, 168 (2018).
[Crossref]

Yao, M.

Ye, F.

J. Zhang, J. Xie, F. Ye, K. Zhou, X. Chen, D. Deng, and X. Yang, “Effects of the turbulent atmosphere and the oceanic turbulence on the propagation of a rotating elliptical Gaussian beam,” Appl. Phys. B 124, 168 (2018).
[Crossref]

F. Ye, J. Zhang, J. Xie, and D. Deng, “Propagation properties of the rotating elliptical chirped Gaussian vortex beam in the oceanic turbulence,” Opt. Commun. 426, 456–462 (2018).
[Crossref]

Yin, H.

D. Liu, Y. Wang, G. Wang, and H. Yin, “Influences of oceanic turbulence on Lorentz Gaussian beam,” Optik 154, 738–747 (2018).
[Crossref]

D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
[Crossref]

Youngworth, K.

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

Yousefi, M.

Yu, L.

Yuan, X. C.

Zhang, B.

Zhang, H.

W. Fu and H. Zhang, “Propagation properties of partially coherent radially polarized doughnut beam in turbulent ocean,” Opt. Commun. 304, 11–18 (2013).
[Crossref]

Zhang, J.

F. Ye, J. Zhang, J. Xie, and D. Deng, “Propagation properties of the rotating elliptical chirped Gaussian vortex beam in the oceanic turbulence,” Opt. Commun. 426, 456–462 (2018).
[Crossref]

J. Zhang, J. Xie, and D. Deng, “Second-order statistics of a partially coherent electromagnetic rotating elliptical Gaussian vortex beam through non-Kolmogorov turbulence,” Opt. Express 26(16), 21249–21257 (2018).
[Crossref] [PubMed]

J. Zhang, J. Xie, F. Ye, K. Zhou, X. Chen, D. Deng, and X. Yang, “Effects of the turbulent atmosphere and the oceanic turbulence on the propagation of a rotating elliptical Gaussian beam,” Appl. Phys. B 124, 168 (2018).
[Crossref]

T. Zhong, J. Zhang, L. Feng, Z. Pang, L. Wang, and D. Deng, “Propagation properties of radially and azimuthally polarized chirped-Airy vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc.Am. B 35(6), 1355–1361 (2018).

Zhang, Y.

Zhao, C.

Zhao, D.

Zhao, F.

H. Chen, X. Sheng, F. Zhao, and Y. Zhang, “Orbital angular momentum entanglement states of Gaussian-Schell beam pumping in low-order non-Kolmogorov turbulent aberration channels,” Opt. Laser Technol 49, 332–336 (2013).
[Crossref]

Zhao, G.

Zhong, T.

T. Zhong, J. Zhang, L. Feng, Z. Pang, L. Wang, and D. Deng, “Propagation properties of radially and azimuthally polarized chirped-Airy vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc.Am. B 35(6), 1355–1361 (2018).

Zhou, K.

J. Zhang, J. Xie, F. Ye, K. Zhou, X. Chen, D. Deng, and X. Yang, “Effects of the turbulent atmosphere and the oceanic turbulence on the propagation of a rotating elliptical Gaussian beam,” Appl. Phys. B 124, 168 (2018).
[Crossref]

Zhou, L.

Zhu, S.

Zhu, X.

Zhu, Y.

Zou, C.

Acta Physica Sinica (1)

J. Cang and Y. Zhang, “The propagation properties of J0-correlated partially coherent beams in the slant atmosphere,” Acta Physica Sinica 58(4), 2444–2450 (2009).

Appl. Opt. (6)

Appl. Phys. B (2)

J. Zhang, J. Xie, F. Ye, K. Zhou, X. Chen, D. Deng, and X. Yang, “Effects of the turbulent atmosphere and the oceanic turbulence on the propagation of a rotating elliptical Gaussian beam,” Appl. Phys. B 124, 168 (2018).
[Crossref]

X. Ji, Y. Ma, Z. Pu, and X. Li, “Characteristics of tilted and off-axis partially coherent beams reflected back by a cat-eye optical lens in atmospheric turbulence,” Appl. Phys. B 115(3), 379–390 (2014).
[Crossref]

Appl. Phys. Lett. (1)

G. Miyaji, N. Miyanaga, K. Tsubakimoto, K. Sueda, and K. Ohbayashi, “Intense longitudinal electric fields generated from transverse electromagnetic waves,” Appl. Phys. Lett. 84(19), 3855 (2004).
[Crossref]

IEEE Photonics J. (1)

X. Huang, Y. Bai, and X. Fu, “Propagation factors of partially coherent model beams in oceanic turbulence,” IEEE Photonics J. 9(5), 1–11 (2017).

Int. J. Fluid Mech. Res. (1)

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the seawater refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2010).
[Crossref]

J. Opt. A: Pure Appl. Opt. (1)

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A: Pure Appl. Opt. 8, 1052–1058 (2006).
[Crossref]

J. Opt. Soc. Am. A (7)

J. Opt. Soc.Am. B (1)

T. Zhong, J. Zhang, L. Feng, Z. Pang, L. Wang, and D. Deng, “Propagation properties of radially and azimuthally polarized chirped-Airy vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc.Am. B 35(6), 1355–1361 (2018).

Laser Phys. (1)

D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
[Crossref]

Methods in Oceanography (1)

S. Matta, W. Houb, S. Woodsc, W. Goodeb, E. Jarosz, and A. Weidemannb, “A novel platform to study the effect of small-scale turbulent density fluctuations on underwater imaging in the ocean,” Methods in Oceanography 11(5), 39–58 (2014).
[Crossref]

Opt. Commun. (3)

F. Ye, J. Zhang, J. Xie, and D. Deng, “Propagation properties of the rotating elliptical chirped Gaussian vortex beam in the oceanic turbulence,” Opt. Commun. 426, 456–462 (2018).
[Crossref]

W. Fu and H. Zhang, “Propagation properties of partially coherent radially polarized doughnut beam in turbulent ocean,” Opt. Commun. 304, 11–18 (2013).
[Crossref]

Y. Wu, Y. Zhang, Y. Li, and Z. Hu, “Beam wander of Gaussian-Schell model beams propagating through oceanic turbulence,” Opt. Commun. 371, 59–66 (2016).
[Crossref]

Opt. Express (13)

Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19(7), 5979–5992 (2011).
[Crossref] [PubMed]

X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15(25), 16909–16915 (2007).
[Crossref] [PubMed]

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]

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

L. Yu and Y. Zhang, “Analysis of modal crosstalk for communication in turbulent ocean using Lommel-Gaussian beam,” Opt. Express 25(19), 22565–22574 (2017).
[Crossref]

Y. Huang, B. Zhang, Z. Gao, G. Zhao, and Z. Duan, “Evolution behavior of Gaussian Schell-model vortex beams propagating through oceanic turbulence,” Opt. Express 22(15), 17723–17734 (2014).
[Crossref] [PubMed]

X. Huang, Z. Deng, X. Shi, Y. Bai, and X. Fu, “Average intensity and beam quality of optical coherence lattices in oceanic turbulence with anisotropy,” Opt. Express 26(4), 4786–4797 (2018).
[Crossref]

Y. Li, L. Yu, and Y. Zhang, “Influence of anisotropic turbulence on the orbital angular momentum modes of Hermite-Gaussian vortex beam in the ocean,” Opt. Express 25(11), 12203–12215 (2017).
[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]

Y. Cai, Q. Lin, H. T. Eyyuboǧlu, 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]

J. Zhang, J. Xie, and D. Deng, “Second-order statistics of a partially coherent electromagnetic rotating elliptical Gaussian vortex beam through non-Kolmogorov turbulence,” Opt. Express 26(16), 21249–21257 (2018).
[Crossref] [PubMed]

K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000).
[Crossref]

C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017).
[Crossref]

Opt. Laser Technol (1)

H. Chen, X. Sheng, F. Zhao, and Y. Zhang, “Orbital angular momentum entanglement states of Gaussian-Schell beam pumping in low-order non-Kolmogorov turbulent aberration channels,” Opt. Laser Technol 49, 332–336 (2013).
[Crossref]

Optik (2)

H. Wang and X. Li, “Spectral properties of partially coherent azimuthally polarized beam in a turbulent atmosphere,” Optik 122, 2164–2171 (2011).
[Crossref]

D. Liu, Y. Wang, G. Wang, and H. Yin, “Influences of oceanic turbulence on Lorentz Gaussian beam,” Optik 154, 738–747 (2018).
[Crossref]

Photon. Res. (1)

Phys. Lett. A (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5), 263–267 (2003).
[Crossref]

Phys. Rev. Lett. (2)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[Crossref]

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]

Other (1)

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

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Figures (8)

Fig. 1
Fig. 1 The RPREG (a) and APREG (b) beams at the initial plane.
Fig. 2
Fig. 2 The normalized spectral density I ( x , 0 , z ) / I m a x ( x , 0 , 0 ) of the PCRPREG beam at several propagation distances in the anisotropic oceanic turbulence for two different values of χT at (a) z=0m, (b) z=10m, (c) z=30m, (d) z=50m, (e) z=80m and (f) z=100m.
Fig. 3
Fig. 3 The evolutions of the spectrum density distributions and the corresponding cross lines of the PCRPREG beam through the anisotropic oceanic turbulence at different propagation distances (a) z=0m, (b) z=10m, (c) z=40m, (d) z=70m, (e) z=100m and (f) z=130m.
Fig. 4
Fig. 4 The normalized average spectral density distributions of the PCRPREG beam through anisotropic oceanic turbulence with different variables.
Fig. 5
Fig. 5 The spectral DOCs of the PCRPREG beam through the anisotropic oceanic turbulence with different variables.
Fig. 6
Fig. 6 The spectral DOPs of the PCRPREG beam and the corresponding cross line ( v = 0 ) at different propagation distances.
Fig. 7
Fig. 7 Cross lines ( v = 0 ) of the DOPs of the PCRPREG beam for difference values of ω, ε and λ.
Fig. 8
Fig. 8 The orientation angles of the PCRPREG beam in the anisotropic oceanic turbulence at different propagation distances.

Equations (21)

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

( E r ( x , y )   E θ ( x , y ) ) = exp  ( x 2 a 2 w 0 2 y 2 b 2 w 0 2 i x y c 2 w 0 2 ) ( x w 0 y w 0     y w 0 x w 0 ) ( e x   e y ) ,
W ( x 1 , y 1 , x 2 , y 2 , 0 ) = ( W x x ( x 1 , y 1 , x 2 , y 2 , 0 ) W x y ( x 1 , y 1 , x 2 , y 2 , 0 ) W y x ( x 1 , y 1 , x 2 , y 2 , 0 ) W y y ( x 1 , y 1 , x 2 , y 2 , 0 ) ) ,
W r ( x 1 , y 1 , x 2 , y 2 , 0 ) = 1 w 0 2 exp  ( x 1 2 + x 2 2 a 2 w 0 2 y 1 2 + y 2 2 b 2 w 0 2 + i x 2 y 2 x 1 y 1 c 2 w 0 2 ) ( x 1 x 2 x 1 y 2 y 1 x 2 y 1 y 2 ) g α β ( r 1 r 2 ) ,
W θ ( x 1 , y 1 , x 2 , y 2 , 0 ) = 1 w 0 2 exp  ( x 1 2 + x 2 2 a 2 w 0 2 y 1 2 + y 2 2 b 2 w 0 2 + i x 2 y 2 x 1 y 1 c 2 w 0 2 ) ( y 1 y 2 y 1 x 2 x 1 y 2 x 1 x 2 ) g α β ( r 1 r 2 ) ,
g α β ( r 1 r 2 ) = exp  [ ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 2 σ 0 2 ] ,
W α β ( ρ 1 , ρ 2 , z ) = ( k 2 π z ) 2 W α β ( r 1 , r 2 , 0 ) ψ ( r 1 , ρ 1 ) + ψ * ( r 2 , ρ 2 )   × exp  { i k 2 z [ ( r 1 ρ 1 ) 2 ( r 2 ρ 2 ) 2 ] } d 2 r 1 d 2 r 2 ,
ψ ( r 1 , ρ 1 ) + ψ * ( r 2 , ρ 2 ) = exp  [ ( r 1 r 2 ) 2 + ( r 1 r 2 ) ( ρ 1 ρ 2 ) + ( ρ 1 ρ 2 ) 2 ρ o c ξ 2 ] ,
ρ o c ξ 2 = π 2 k 2 z ξ 4 3 0 κ 3 ψ ˜ a n ( κ ) d κ ,
ψ ˜ a n ( κ ) = 0.388 × 10 8 ε 1 / 3 χ T ξ 2 ( κ ) 11 / 3 [ 1 + 2.35 ( κ η ) 2 / 3 ]   × [ exp  ( A T δ ) + ω 2 exp  ( A S δ ) 2 ω 1 exp  ( A T S δ ) ] ,
ρ o c ξ = ξ | ω | [ 1.802 × 10 7 k 2 z ( ε η ) 1 / 3 χ T ( 0.483 ω 2 0.835 ω + 3.380 ) ] 1 / 2 ,
x n exp  ( p x 2 + q x ) d x = n ! exp  ( q 2 p ) ( q p ) n π p l = 0 n / 2 1 l ! ( n 2 l ) ! ( q 2 4 p ) l ,
W r x x ( ρ 1 , ρ 2 , z ) = Q M 1 M 3 { [ ( N 1 i N 2 2 M 2 c 2 w 2 ) N 3 + ( Δ Δ 4 M 1 M 2 c 4 w 4 ) ( 1 + N 3 2 2 M 2 ) ] + [ ( N 1 i N 2 2 M 2 c 2 w 2 ) S i Δ N 3 2 M 2 c 2 w 2 + ( Δ Δ 4 M 1 M 2 c 4 w 4 ) N 3 S M 3 ] N 4 2 M 4 + [ i Δ S 2 M 2 c 2 w 2 + ( Δ Δ 4 M 1 M 2 c 4 w 4 ) S 2 2 M 3 ] 1 2 M 4 ( 1 + N 4 2 2 M 4 ) } ,
W r x y ( ρ 1 , ρ 2 , z ) = Q M 1 M 4 { [ N 1 i N 2 2 M 2 c 2 w 2 + ( Δ Δ 4 M 1 M 2 c 4 w 4 ) N 3 2 M 3 ] N 4   + [ i Δ 2 M 2 c 2 w 2 + ( Δ Δ 4 M 1 M 2 c 4 w 4 ) S 2 M 3 ] ( 1 + N 4 2 2 M 4 ) } ,
W r y x ( ρ 1 , ρ 2 , z ) = W r x y * ( ρ 1 , ρ 2 , z ) ,
W r y y ( ρ 1 , ρ 2 , z ) = Q M 2 M 4 { ( N 2 i Δ N 3 4 M 1 M 3 c 2 w 2 ) N 4 + ( Δ i Δ S 4 M 1 M 3 c 2 w 2 ) ( 1 + n 4 2 2 M 4 ) } ,
Q = k 2 16 w 2 z 2 M 1 M 2 M 3 M 4 exp { i k 2 z ( ρ 1   2 ρ 2   2 ) ρ o c ξ 2     ( ρ 1 ρ 2 ) 2 } exp { N 1 2 4 M 1 + N 2 2 4 M 2 + N 3 2 4 M 3 + N 4 2 4 M 4 } , M 1 = 1 a 2 w 0 2 + 1 2 σ 0 2 + ρ o c ξ 2 + i k 2 z , N 1 = ρ o c ξ 2 ( u 1 u 2 ) + i k u 1 z , Δ = 1 σ 0 2 + 2 ρ o c ξ 2 , M 2 = 1 4 M 1 c 4 w 0 4 + 1 b 2 w 0 2 + 1 2 σ 0 2 + ρ o c ξ 2 + i k 2 z , N 2 = i N 1 2 M 1 c 2 w 0 2 ρ o c ξ 2 ( v 1 v 2 ) + i k v 1 z , M 3 = Δ 2 16 M 1 2 M 1 c 4 w 0 4 Δ 2 4 M 1 + 1 a 2 w 0 2 + 1 2 σ 0 2 + ρ o c ξ 2 i k 2 z , S = i Δ 2 4 M 1 M 2 c 2 w 0 2 + i 1 c 2 w 0 2 N 3 = i N 2 Δ 4 M 1 M 2 c 2 w 0 2 + N 1 Δ 2 M 1 + ρ o c ξ 2 ( u 1 u 2 ) i k u 2 z , M 4 = S 2 4 M 3 Δ 2 4 M 2 + 1 b 2 w 0 2 + 1 2 σ 0 2 + ρ o c ξ 2 i k 2 z , N 4 = N 3 S 2 M 3 + N 2 Δ 2 M 2 + ρ o c ξ 2 ( v 1 v 2 ) i k v 2 z .
W θ x x ( ρ 1 , ρ 2 , z ) = W r y y ( ρ 1 , ρ 2 , z ) , W θ y y ( ρ 1 , ρ 2 , z ) = W r x x ( ρ 1 , ρ 2 , z ) , W θ x y ( ρ 1 , ρ 2 , z ) = W θ y x ( ρ 1 , ρ 2 , z ) = W r x y ( ρ 1 , ρ 2 , z ) .
I ( ρ , z ) = T r W ( ρ , ρ ; z ) = W r x x ( ρ , ρ , z ) + W r y y ( ρ , ρ , z ) ,
μ ( ρ 1 , ρ 2 , z ) = T r W ( ρ 1 , ρ 2 ; z ) I ( ρ 1 , z ) I ( ρ 2 , z ) ,
P ( ρ , z ) = 1 4 D e t W ( ρ , ρ ; z ) [ T r W ( ρ , ρ ; z ) ] 2 = 1 4 ( W r x x W r y y W r x y W r y x ) ( W r x x + W r y y ) 2 ,
θ ( ρ ; z ) = 1 2 arctan  [ R e [ W r x y ( ρ , ρ , z ) ] + R e [ W r y x ( ρ , ρ , z ) ] W r x x ( ρ , ρ , z ) W r y y ( ρ , ρ , z ) ] .

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