Statistical properties of a partially coherent ... - OSA Publishing

894

Research Article

Vol. 32, No. 5 / May 2015 / Journal of the Optical Society of America A

Statistical properties of a partially coherent cylindrical vector beam in oceanic turbulence YIMING DONG,1,* LINA GUO,2 CHUNHAO LIANG,2 FEI WANG,2

AND

YANGJIAN CAI2,3

1

Department of Physics, Shaoxing University, Shaoxing 312000, China School of Physical Science and Technology, Soochow University, Suzhou 215006, China 3 e-mail: [email protected] *Corresponding author: [email protected] 2

Received 5 February 2015; revised 12 March 2015; accepted 17 March 2015; posted 17 March 2015 (Doc. ID 233935); published 27 April 2015

Propagation of a partially coherent cylindrical vector Laguerre–Gaussian (PCCVLG) beam passing through oceanic turbulence is studied with the help of the extended Huygens–Fresnel integral formula and unified theory of coherence and polarization of light. Analytical formula for the cross-spectral density matrix of a PCCVLG beam propagating in oceanic turbulence is derived, and the statistical properties, such as intensity distribution and degree of polarization, of a PCCVLG beam on propagation in oceanic turbulence are illustrated in detail. It is found that the statistical properties of a PCCVLG beam in oceanic turbulence vary as the sea-related parameters, initial coherence length, and mode orders vary, and such beam is depolarized on propagation. Our results will be useful in optical underwater communications, imaging, and sensing. © 2015 Optical Society of America OCIS codes: (030.1640) Coherence; (030.7060) Turbulence; (350.5500) Propagation. http://dx.doi.org/10.1364/JOSAA.32.000894

1. INTRODUCTION According to polarization property, a light beam can be classified as a uniformly or nonuniformly polarized beam. As a typical nonuniformly polarized beam, the cylindrical vector (CV) beam, such as a radially polarized beam or azimuthally polarized beam, has attracted increasing attention in recent years due to its unique tightly focusing properties and potential applications. A strong longitudinal electric field and a much smaller focal spot can be produced by a tightly focused radially polarized beam [1,2]. A tightly focused azimuthally polarized beam has a dark hollow profile near the focal plane [1,2]. The CV beam has been used in many research fields: focus shaping technique, optical trapping, dark field imaging, optical communication, optical data storage, etc. [1,3–8]. Different theoretical models, including Laguerre–Gaussian, Bessel–Gaussian, and modified Bessel–Gaussian, for CV beams have been proposed [1,9,10]. Different methods, including active and passive techniques, have been developed to generate CV beams [11–16]. The propagation properties, tightly focused properties, and second-harmonic generation of CV beams have been investigated in detail [2,8,17–19]. More recently, a partially coherent vector beam with a spatially nonuniform state of polarization, called the partially coherent cylindrical vector Laguerre–Gaussian (PCCVLG) beam, was introduced as a natural extension of a coherent cylindrical vector Laguerre–Gaussian (CCVLG) beam [20]. A PCCVLG beam can be used to describe partially coherent radially or 1084-7529/15/050894-08$15/0$15.00 © 2015 Optical Society of America

azimuthally polarized beams under certain conditions. Paraxial and nonparaxial propagation properties of a PCCVLG beam in free space have been investigated in [20,21]. Experimental generation of a partially coherent radially or azimuthally polarized beam can be found in [22–24], and it is realized that a focused beam spot with a doughnut, fat-top, or Gaussian profile can be obtained by varying the initial spatial coherence of a partially coherent radially or azimuthally polarized beam passing through a thin lens, which is useful for trapping particles. Theoretical and experimental studies of the spectral changes of a polychromatic partially coherent radially polarized beam were reported in [25]. The statistical properties in Young’s interference pattern formed with a partially coherent radially polarized beam were revealed in [26]. On the other hand, propagation of a laser beam in random medium has attracted much attention due to its important applications in communication and remote sensing. Two typical examples of an optical random medium are the atmosphere and ocean. However, more attention has been paid to propagation of a laser beam in atmospheric turbulence than in oceanic turbulence [27–40]. It has been found that a partially coherent beam has advantages over a fully coherent one for reducing atmospheric turbulence-induced degradation [33–43]. Propagation properties of a PCCVLG beam in turbulent atmosphere were revealed in [44], and experimental study of the turbulence-induced scintillation of a partially coherent radially polarized beam was reported in [45]. It is interesting

Research Article


to find that a PCCVLG or partially coherent radially polarized beam has advantage over a linearly polarized beam or scalar beam for reducing atmospheric turbulence-induced degradation [44,45], which will be useful in free-space optical communication. Compared with the atmospheric turbulence, there is much less investigation on light propagation in oceanic turbulence. It is well known that the propagating properties in oceanic turbulence are generally influenced by temperature and salinity fluctuations. The influence of temperature and salinity fluctuations on light propagation in oceanic turbulence has been studied separately for a long time, an analytical model combining salinity and temperature fluctuation has been developed only recently [46,47]. Since then, some work has been carried out on the laser beams propagating in the turbulent ocean [48–52]. However, to the best of our knowledge, the propagation properties of a partially coherent vector beam with a spatially nonuniform state of polarization in oceanic turbulence have not yet been reported. Since a PCCVLG beam is regarded as a convenient and generalized model for describing a partially coherent vector beam with spatially nonuniform state of polarization, in this paper, our aim is to study the statistical properties of a PCCVLG beam in oceanic turbulence based on the extended Huygens–Fresnel formula and the unified theory of coherence and polarization of light. For simplicity, we ignore the absorption and scattering and only consider the effects of the fluctuating refractive index on light’s propagation.

2. THEORY In cylindrical coordinates, the electric field of a CCVLG beam in the source plane (z 0) can be expressed as [9] 2 2 2 n1∕2 r 2r n1 2r L Er; z 0 exp − 2 p w0 w20 w20 cosnϕeϕ ∓ sinnϕer ; (1) × sinnϕeϕ cosnϕer where r ≡ r; ϕ is the position vector in the source plane. r and ϕ are the radial and azimuthal (angle) coordinates. Lpn1 denotes the Laguerre polynomial with mode orders p and n 1. w0 is the beam waist of the fundamental Gaussian mode. When p n 0, Eq. (1) degenerates to the electric field of a radially or azimuthally polarized Gaussian beam. By using er cos ϕex sin ϕey and eϕ − sin ϕex cos ϕey , Eq. (1) can be expressed as follows: 2 2 2 n1∕2 r 2r n1 2r Lp Er; 0 exp − 2 w0 w20 w20 ∓ sinnϕ ϕex cosnϕ ϕey ; (2) × cosnϕ ϕex sinnϕ ϕey where ex and ey denote the unit vectors along the x and y directions in the Cartesian coordinate, respectively. From Eqs. (1) and (2), it is found that there are four expressions for the CCVLG beam. For simplicity and without loss of generality, we only choose one of the four expressions, which can be written as

895

2 2 n1∕2 2 r 2r n1 2r Er; 0 exp − 2 L p w0 w20 w20 × cosn 1ϕex sinn 1ϕey :

(3)

Based on the unified theory of coherence and polarization, the CSD matrix of the PCCVLG beam in the source plane, can be expressed as follows [53]: E x r1 ; 0 ↔ E x r1 ; 0; E y r2 ; 0 W r1 ; r2 ; 0 E y r2 ; 0 W xx r1 ; r2 ; 0W xy r1 ; r2 ; 0 ; (4) W yx r1 ; r2 ; 0W yy r1 ; r2 ; 0 with W αβ r1 ; r2 ; 0 hE α r1 ; 0E β r2 ; 0i;

α; β x; y; (5)

where the angular brackets denote the ensemble average and the asterisk denotes the complex conjugate. If we assume that the PCCVLG beam is generated by a Schell-model source, Eq. (5) can be expressed as [20] W αβ r1 ; r2 ; 0 E α r1 ; 0E β r2 ; 0g αβ r1 − r2 ; 0; α; β x; y; with g αβ r1 − r2 ; 0 B αβ

(6)

r1 − r2 2 exp − ; 2σ 2αβ

α; β x; y; (7)

where σ αβ denotes the initial coherence length of the PCCVLG beam. B αβ is the complex correlation coefficients between E α and E β field components. As shown in [38], the realizability condition of a PCCVLG beam is B αβ 1, σ xx σ xy σ yy σ 0 . We can obtain the relation: g xx r1 − r2 ; 0 g xy r1 − r2 ; 0 g yx r1 − r2 ; 0 g yy r1 − r2 ; 0 gr1 − r2 ; 0. When a PCCVLG beam propagates in oceanic turbulence, the elements of the CSD matrix can be derived by using the following extended Huygens–Fresnel formula [48–52]: 2 ZZ k W αβ ρ1 ; ρ2 ; z W αβ r1 ; r2 ; 0 2πz r1 − ρ1 2 − r2 − ρ2 2 × exp −ik 2z −ρ1 − ρ2 2 − ρ1 − ρ2 r1 − r2 − r1 − r2 2 dr1 dr2 ; × exp ρ20 z (8) where ρ1 and ρ2 denote the position vector of two points in the receiver plane. k 2π∕λ is the wavenumber. λ is the wavelength of the light. ρ0 is the coherence length of a spherical wave propagating in oceanic turbulence, expressed as 1∕2 3 R : (9) ρ0 π 2 k2 z 0∞ κ 3 Φn κdκ For this model, we assume that the turbulence is isotropic and homogeneous, and the power spectrum has the form

896


Research Article Z k 4 ku1 ˜ ku1 Qβ Q˜ α 2πz 2πz 2πz k u ρ du1 ; × R˜ − 2πz 1

Φn κ 0.388 × 10−8 ε−1∕3 κ −11∕3 1 2.35κη2∕3 f κ; w; χ T ;

W αβ ρ; z

(10) where ε is the dissipation rate of turbulent kinetic energy per unit mass of fluid. It can be varied from 10−4 to 10−10 m2 ∕s3 . η 10−3 m is the Kolmogorov microscale (inner scale) and χ (11) f κ; w; χ T T2 w2 e −AT δ e −AS δ − 2we −AT S δ ; w where χ T is the dissipation rate of mean-square temperature, AT 1.863 × 10−2 , AS 1.9 × 10−4 , AT S 9.41 × 10−3 , and δ 8.284κη4∕3 12.978κη2 . w is the relative strength of temperature and salinity fluctuations, which can be varied from −5 to 0, with −5 and 0 corresponding to dominating salinity-induced and temperature-induced optical turbulence, respectively. On substituting Eqs. (6) and (7) into Eq. (8), we obtain 2 k Δρ2 exp − 2 W αβ ρS ; Δρ; z ρ0 z 2πz ZZ ikρS · Δρ drS dΔr × exp z × Q α rS − Δr∕2Q β rS Δr∕2gΔr; 0 ik −Δρ · Δr Δr2 ik − Δr · ρS ; × exp − rS · Δρ exp − ρ20 z z z

where

Δr2 RΔr gΔr; 0 exp − 2 ; ρ0 z

rS r1 r2 ∕2;

Δr r2 − r1 ;

(13)

ρS ρ1 ρ2 ∕2;

Δρ ρ2 − ρ1 ;

(14)

Q j r E j r; 0 exp

ik 2 r ; 2z

j α; β.

(15)

Q α rS − Δr∕2 and Q β rS Δr∕2 can be expressed in the following forms: ZZ k2 ku1 ˜ Qα Q α rS − Δr∕2 2 2 4π z 2πz ik r − Δr∕2 · u1 du1 ; (16) × exp z S ZZ k2 ku2 ˜ Qβ Q β rS Δr∕2 2 2 4π z 2πz ik × exp − rS Δr∕2 · u2 du2 ; z

where a1 2σ1 2 ρ21z . 0

−p−n−2

n1 2−n3∕2 a a2 − 1p un1 π n2 wn3 0 i 2 2π 2 w20 u2 n1 π 2 w20 u2 Lp C φ; (21) × exp − 4a2 2a2 1 − a2 α

with a2 12 −

ikw20 4z

,

C α φ

cosl φ α x : sinl φ α y

On substituting Eqs. (20) and (21) into Eq. (18), after tedious integration, we obtain the elements of the CSD matrix of a PCCVLG beam propagating in oceanic turbulence as follows: kw0 2n6 1 ja2 j−2p−2n−4 ja2 − 1j2p W xx ρ; z 2z 2n4 a1 X p X p pl pl a 4 a 5 ρ2 st × × exp − −1 a4 a5 p−s p−t s0 t0 s t 2 2 a4 ρ a a Γs t n 2 0 − L × 3 3 s!t! a4 a5 stn2 stn1 a4 a5 2l a24 ρ2 n1 a4 ρ Γs t 1 2n2 − L −1 a4 a 5 a4 a5 st2n3 st × cos2n 1θ ; (22)

where Q˜ j · is the Fourier transform of Q j ·. In this paper, we focus on the evolution properties of the average intensity and degree of polarization of a PCCVLG beam in oceanic turbulence. We only consider the case of ρ1 ρ2 ρ. On substituting Eqs. (16) and (17) into Eq. (12), we can obtain

0

For a PCCVLG beam, we can obtain the analytical expression for Q˜ α u in the cylindrical coordinate system: Z ˜ Q α u Q α r exp2πir · udr

kw0 2z

2n6

1 ja2 j−2p−2n−4 ja2 − 1j2p 2n4 a1 p p pl pl a a ρ2 X X × exp − 4 5 −1stn1 a4 a5 s0 t0 p−s p−t 2 2 s t 2n2 a ρ a a a ρ Γs t 1 2n2 Lst − 4 × 3 3 × 4 st2n3 s!t! a4 a 5 a4 a5

W xy ρ; z (17)

(19)

˜ is the Fourier transform of RΔr, with R· Z k ik ˜ u ρ RΔr exp − Δr · u1 ρ dΔr R − 2πz 1 z 2 π k 2 (20) exp − 2 u1 ρ ; 4z a1 a1

(12) where

(18)

× sin2n 1θ; W yx ρ; z W xy ρ; z ;

(23) (24)


Research Article

kw0 2z

2n6

1 ja2 j−2p−2n−4 ja2 − 1j2p 2n4 a1 X p X p pl pl a a ρ2 × exp − 4 5 −1st × a 4 a5 p−s p−t s0 t0 s t 2 2 a3 a3 Γs t n 2 0 a4 ρ − L × s!t! a4 a5 stn2 stn1 a4 a5 a24 ρ2 a ρ2l Γs t 1 2n2 − L − −1n1 4 a4 a5 a4 a5 st2n3 st × cos2n 1θ ; (25)

W yy ρ; z

where ρ and θ are the radial and azimuthal coordinates in the k 2 w20 , 2 1−a 2

output plane, respectively. a3 8z 2 a k 2 w20 1 8z 2 a 2

a4 4zk2 a , and 2

1

a5 In above derivations, we have used the following formulae: Z 2π expinϕ expix cosϕ − φ 2πi n J n x expil φ; 1 a2 .

0

Z 0

(26) ∞

x v1 e −βx Lvn αx 2 J v xydx 2−v−1 β−v−n−1 β − αn y v 2 y αy2 Lvn : (27) × exp − 4β 4βα − β 2

The average intensity and degree of polarization at the output plane are defined as [53] ↔

hI ρ; zi Tr W ρ; z W xx ρ; z W yy ρ; z;

(28)

and

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ↔ u 4Det W ρ; z t Pρ; z 1 − ; ↔ 2 Tr W ρ; z

897

(29)

respectively. Tr and Det stand for the trace and determinant of the CSD matrix, respectively.

3. NUMERICAL RESULTS Now we will investigate the statistical properties of a PCCVLG beam in oceanic turbulence numerically with the help of the formulae derived in the above section. In the following calculations, the parameters λ, w0 , and η are assumed as λ 632.8 nm, w0 2 cm, and η 10−3 m. We calculate in Fig. 1 the normalized intensity distribution (cross line) of a PCCVLG beam at several propagation distances in oceanic turbulence for different values of χ T . Other parameters are chosen as σ 0 2 cm, ε 10−7 m2 ∕s3 , w −2.5, p 1, and n 0. As shown in Fig. 1, the intensity distribution will change from a doughnut beam spot to a Gaussian beam spot in turbulent ocean. This transition is closely affected by χ T . The beam profile approaches the Gaussian profile more rapidly with the increase of the rate of dissipation of meansquare temperature χ T (i.e., the strength of turbulence increases). The evolution properties of the intensity distribution of PCCVLG beam in oceanic turbulence are similar to those of the PCCVLG beam with low coherence in free space [20]. This phenomenon can be explained by the fact that the oceanic turbulence decreases the beam coherence when it propagates in turbulent ocean.

Fig. 1. Normalized intensity distribution (cross line) of a PCCVLG beam at several propagation distances in oceanic turbulence for different values of χ T .

898


Research Article

Fig. 2. Normalized intensity distribution (cross line) of a PCCVLG beam for different values of (a) w and (b) ε at z 400 m.

In order to show the influence of the parameters w (relative strength of temperature and salinity fluctuations) and ε (rate of dissipation of turbulent kinetic energy per unit mass of fluid) on the intensity distribution of a PCCVLG beam in turbulent ocean, we calculate in Fig. 2 the normalized intensity distribution (cross line) at z 400 m for different values of the parameters w and ε with χ T 5 × 10−10 . Other parameters are the same with those used in Fig. 1. It is clear from Fig. 2(a) that, for a larger w, the beam profile will become the Gaussian profile more rapidly. It is known that, for the ocean water, w can vary in the interval −5; 0, with −5 and 0 corresponding to dominating salinity-induced and temperature-induced optical turbulence, respectively. One finds from Fig. 2(a) that the normalized intensity distribution is more affected by temperature fluctuations than salinity fluctuations. From Fig. 2(b), we find that the beam profile evolves into a Gaussian spot more rapidly with the decreasing of the parameter ε (rate of dissipation of turbulent kinetic energy per unit mass of fluid). Figure 3 shows the normalized intensity distribution (cross line) of a PCCVLG beam at several propagation distances for different values of initial coherence length σ 0 with χ T 10−9 . Other parameters are the same with those used in Fig. 1. It is found that the beam profile approaches Gaussian shape more quickly with the decrease of the initial coherence length, which is similar to the evolution properties when the strength of turbulence increases (i.e., turbulence-induced decrease of the beam’s coherence increases) (see Fig. 1). Figure 4 shows the normalized intensity (cross line) of a PCCVLG beam at z 600 m for different values of mode

orders p and n with χ T 10−9 . Other parameters are the same as those in Fig. 1. For the convenience of comparison, the corresponding result of a partially coherent radially polarized beam is shown in Fig. 4(a). It can be seen in Fig. 4 that, when the normalized intensity distribution of the partially coherent radially polarized beam or PCCVLG beam with lower-mode orders becomes a Gaussian beam profile [see Figs. 4(a) and 4(e)], the PCCVLG beam with higher-mode orders still remains a non-Gaussian beam profile, which means that the PCCVLG beam with higher-mode orders is less affected by oceanic turbulence than that with lower-mode orders (i.e., PCCVLG beam with higher-mode orders can endure the oceanic turbulence in comparison with lower-mode orders). The reason is that the mode structure of a PCCVLG beam with higher-order modes places more energy away from the beam axis; thus, it requires longer propagation distance to become a circular Gaussian beam in oceanic turbulence. Now we will study the properties of the degree of polarization of a PCCVLG beam propagating in oceanic turbulence. Substituting Eqs. (22)–(25) into Eq. (29), we calculate in Fig. 5 the degree of polarization (cross line) of the PCCVLG beam versus the propagation distance in oceanic turbulence for different values of ρ, p, and n with χ T 10−9 . The degree of polarization in the source plane equals 1, except at the point ρ 0, where the intensity is zero. One finds from Fig. 5 that the PCCVLG beam is depolarized on propagation in oceanic turbulence (i.e., the PCCVLG beam varies from a fully polarized beam to a partially polarized one on propagation in oceanic turbulence). Furthermore, the PCCVLG beam is

Fig. 3. Normalized intensity distribution (cross line) of a PCCVLG beam at several propagation distances in oceanic turbulence for different values of initial coherence length σ 0 .

Research Article


899

Fig. 4. Normalized intensity distribution (cross line) of a PCCVLG beam at z 600 m in oceanic turbulence for different values of mode orders p and n.

polarized more slowly, as the mode order n decreases or p increases. Figure 6 shows the degree of polarization of a PCCVLG beam at z 400 m for different values of the parameters χ T , w, ε, and σ 0 . One finds from Fig. 6 that the depolarization of the PCCVLG beam is also affected by the parameters χ T , w, ε, and σ 0 . The on-axis degree of polarization is always equal to

zero, and a dip appears near the z axis. The width of the dip on propagation varies as the parameters χ T , w, ε, and σ 0 vary. From the above discussion, one comes to the conclusion that the PCCVLG beam is depolarized on propagation in oceanic turbulence, and the depolarization is affected by the parameters (χ T , w, ε) of the turbulence and the parameters (σ 0 , n, p) of the initial beam.

Fig. 5. Degree of polarization of a PCCVLG beam versus the propagation distance z in oceanic turbulence for different values of ρ, p, and n.

900


Research Article

Fig. 6. Degree of polarization (cross line) of a PCCVLG beam for different values of (a) χ T , (b) w, (c) ε, and (d) σ 0 at z 400 m.

4. CONCLUSION In conclusion, by using the extended Huygens–Fresnel integral formula and the unified theory of coherence and polarization of light, we have derived the analytical formulae for the CSD matrix of a PCCVLG beam propagating in oceanic turbulence. The statistical properties, such as the intensity distribution and degree of polarization, have been discussed in detail with the help of numerical calculations. It is shown that the intensity distribution will change from a doughnut beam spot to a Gaussian beam spot in turbulent ocean. The transition happens more rapidly with the increase of the dissipation rate of the mean-square temperature, the relative strength of temperature and salinity fluctuations, or the decrease of the dissipation rate of turbulent kinetic energy per unit mass of fluid, the mode orders, and the initial coherence length of beam. Furthermore, the distribution of the degree of polarization generates a dip on propagation in oceanic turbulence, which means a PCCVLG beam is depolarized on propagation. The depolarization is affected by the sea-related parameters, initial coherence length, and mode orders. Our results will be useful in optical underwater communications, imaging, and sensing. National Natural Science Foundation of China (11474213, 11404067, 11304204, 11274005); National Natural Science Foundation of Zhejiang Province of China (LQ13A040003). REFERENCES 1. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009). 2. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000). 3. W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265, 411–417 (2006).

4. B. Roxworthy and K. Toussaint, Jr., “Optical trapping with pi-phase cylindrical vector beams,” New J. Phys. 12, 073012 (2010). 5. Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18, 10828– 10833 (2010). 6. D. Biss, K. Youngworth, and T. Brown, “Dark-field imaging with cylindrical-vector beams,” Appl. Opt. 45, 470–479 (2006). 7. W. Cheng, J. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17, 17829– 17836 (2009). 8. Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10, 324–331 (2002). 9. A. Tovar, “Production and propagation of cylindrically polarized Laguerre–Gaussian laser beams,” J. Opt. Soc. Am. A 15, 2705–2711 (1998). 10. C. Li, “Integral transformation solution of free-space cylindrical vector beams and prediction of modified Bessel–Gaussian vector beams,” Opt. Lett. 32, 3543–3545 (2007). 11. I. Moshe, S. Jackel, and A. Meir, “Production of radially or azimuthally polarized beams in solid-state lasers and the elimination of thermally induced birefringence effects,” Opt. Lett. 28, 807–809 (2003). 12. R. Zheng, C. Gu, A. Wang, L. Xu, and H. Ming, “An all-fiber laser generating cylindrical vector beam,” Opt. Express 18, 10834–10838 (2010). 13. K. Moh, X. Yuan, J. Bu, D. Low, and R. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett. 89, 251114 (2006). 14. R. Kurti, K. Halterman, R. Shori, and M. Wardlaw, “Discrete cylindrical vector beam generation from an array of optical fibers,” Opt. Express 17, 13982–13988 (2009). 15. G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre-Gaussian beams,” Opt. Commun. 237, 89–95 (2004). 16. G. He, J. Guo, B. Wang, and Z. Jiao, “Generation of radially polarized beams based on thermal analysis of a working cavity,” Opt. Express 19, 18302–18309 (2011).

Research Article


17. Y. Cai, Q. Lin, H. Eyyubog˘ lu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16, 7665–7673 (2008). 18. D. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B 23, 1228–1234 (2006). 19. A. Ohtsu, Y. Kozawa, and S. Sato, “Calculation of second-harmonic wave pattern generated by focused cylindrical vector beams,” Appl. Phys. B 98, 851–855 (2010). 20. Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19, 5979–5992 (2011). 21. Y. Dong, F. Feng, Y. Chen, C. Zhao, and Y. Cai, “Statistical properties of a nonparaxial cylindrical vector partially coherent field in free space,” Opt. Express 20, 15908–15927 (2012). 22. F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100, 051108 (2012). 23. Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86, 013840 (2012). 24. G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express 20, 28301–28318 (2012). 25. 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, 27682–27696 (2013). 26. 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, 28697– 28710 (2014). 27. C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002). 28. H. T. Eyyubog˘ lu and Y. Baykal, “Reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Express 12, 4659–4674 (2004). 29. Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14, 1353–1367 (2006). 30. Y. Cai and S. He, “Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. 31, 568–570 (2006). 31. Y. Cai, Y. Chen, H. T. Eyyubog˘ lu, and Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32, 2405–2407 (2007). 32. X. Chu, “Evolution of an Airy beam in turbulence,” Opt. Lett. 36, 2701–2703 (2011). 33. T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094–1102 (2003). 34. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006). 35. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342– 2348 (2008).

901

36. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyubog˘ lu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17, 17344– 17356 (2009). 37. F. Wang and Y. Cai, “Second-order statistics of a twisted Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 18, 24661–24672 (2010). 38. F. Wang, Y. Cai, H. Eyyuboglu, and Y. Baykal, “Average intensity and spreading of partially coherent standard and elegant LaguerreGaussian beams in turbulent atmosphere,” Prog. Electromagn. Res. 103, 33–56 (2010). 39. F. Wang, Y. Cai, H. T. Eyyubog˘ lu, and Y. Baykal, “Twist phaseinduced reduction in scintillation of a partially coherent beam in turbulent atmosphere,” Opt. Lett. 37, 184–186 (2012). 40. X. Liu, Y. Shen, L. Liu, F. Wang, and Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38, 5323–5326 (2013). 41. X. Liu, F. Wang, C. Wei, and Y. Cai, “Experimental study of turbulence-induced beam wander and deformation of a partially coherent beam,” Opt. Lett. 39, 3336–3339 (2014). 42. Y. Cai, “Generation of various partially coherent beams and their propagation properties in turbulent atmosphere: a review,” Proc. SPIE 7924, 792402 (2011). 43. F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited review),” Prog. Electromagn. Res. 150, 123–143 (2015). 44. R. Chen, Y. Dong, F. Wang, and Y. Cai, “Statistical properties of a cylindrical vector partially coherent beam in turbulent atmosphere,” Appl. Phys. B 112, 247–259 (2013). 45. F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103, 091102 (2013). 46. V. Nikishov and V. Nikishov, “Spectrum of turbulent fluctuation of the sea-water refraction index,” Int. J. Fluid Mech. Res. 27, 82–98 (2000). 47. O. Korotkova and N. Farwell, “Effect of oceanic turbulence on polarization of stochastic beams,” Opt. Commun. 284, 1740–1746 (2011). 48. E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulent ocean,” Appl. Phys. B 105, 415–420 (2011). 49. O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22, 260–266 (2012). 50. Y. Zhou, K. Huang, and D. Zhao, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams propagating through the oceanic turbulence,” Appl. Phys. B 109, 289–294 (2012). 51. Y. Zhou, Q. Chen, and D. Zhao, “Propagation of astigmatic stochastic electromagnetic beams in oceanic turbulence,” Appl. Phys. B 114, 475–482 (2014). 52. J. Xu, M. Tang, and D. Zhao, “Propagation of electromagnetic non-uniformly correlated beams in the oceanic turbulence,” Opt. Commun. 331, 1–5 (2014). 53. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).