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Effect of the atmospheric turbulence on a special correlated radially polarized beam on propagation
Optics Communications 354 (2015) 353–361
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Effect of the atmospheric turbulence on a special correlated radially polarized beam on propagation Yan Cui a, Cun Wei a, Yongtao Zhang a,b, Fei Wang a,n, Yangjian Cai a a College of Physics, Optoelectronics and Energy & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China b Department of Physics Luoyang Normal University, Luoyang 471022, China
art ic l e i nf o
a b s t r a c t
Article history: Received 1 April 2015 Received in revised form 4 June 2015 Accepted 7 June 2015 Available online 10 June 2015
A special correlated radially polarized beam (Phys. Rev. A 89, 013801, 2014) was introduced and demonstrated in experiment recently. In this paper, we investigate the statistical properties of a special correlated radially polarized (SCRP) beam in atmospheric turbulence. Analytical formulas for the average intensity distribution (AID), degree of polarization (DOP) and degree of coherence (DOC) are derived by adopting a beam coherence-polarization (BCP) matrix. With the help of the derived formulas, the evolutions of the AID, DOP and DOC of the SCRP beam in turbulent atmosphere are illustrated through numerical examples in detail, and the results are compared to those of a partially coherent radially polarized (PCRP) beam under equivalent condition. It reveals that the propagation properties of the SCRP beam is much different from those of the PCRP beam in atmosphere, and closely related to the strength of the turbulence and the beam parameters. & 2015 Elsevier B.V. All rights reserved.
Keywords: Partially coherent beam Atmospheric turbulence Radial polarization Propagation
1. Introduction It is well known that the intensity distribution and the polarization properties of light beams on propagation are closely related to the correlation properties of light sources [1,2]. However, in most studies, the correlation functions of the partially coherent beams are restricted on Gaussian function because it is difficult to judge whether other correlation functions are physical realizable or not. Since Gori and co-workers derived the sufficient condition for devising genuine correlation functions for partially coherent scalar and vector beams [3,4], various partially coherent beams with different kinds of correlation functions have been introduced theoretically [5–8]. Meanwhile, several methods were proposed to produce such kinds of partially coherent beams [9–12]. Of particular interest is that such partially coherent beams exhibit some extraordinary propagation characteristics even in free space. For example, a Laguerre–Gaussian correlated Schell-model (LGCSM) beam or a Bessel–Gaussian correlated Schell-model (BGCSM) beam [7] is capable of producing ring-shaped beam profile in the far field; the transverse maximum intensity of a non-uniformly correlated (NUC) beam is shifted during propagation [5]. Not only the intensity distribution, but also the polarization properties can be controlled by modulating the correlation functions [13,14]. n
Corresponding author. E-mail address: [email protected] (F. Wang).
http://dx.doi.org/10.1016/j.optcom.2015.06.017 0030-4018/& 2015 Elsevier B.V. All rights reserved.
Recently, we introduced and experimentally demonstrated a new kind of partially coherent vector beam, named special correlated radially polarized (SCRP) beam [15], and the focusing properties of the SCRP beam passing through a thin lens was investigated both theoretically and experimentally. It was found that the beam profile and the polarization feature of the SCRP beam in the far field become a doughnut shape and radial polarization, respectively, while such beam in the source plane can be considered as an un-polarized beam with intensity distribution being Gaussian profile. These properties are much different from a radially polarized beam with conventional Gaussian Schell-model correlation, i.e., partially coherent radially polarized (PCRP) beam [16,17], whose polarization distribution is radially polarized in the source and de-polarized on propagation due to its limited spatial coherence length. On the other hand, study of the propagation characteristics of laser beams in turbulent atmosphere has received considerable attention for a long time due to its great importance in connection with remote sensing, imaging and free-space optical communication [18]. In general, the atmospheric turbulence will cause the extra spreading beyond the diffraction, wander and scintillation of laser beams, which limits the performance in the previously mentioned applications. Thus, knowledge of the propagation behavior of light beams in atmospheric turbulence is utmost significant [19–34]. It is known that decreasing spatial coherence and modulating polarization distribution of light beams are two effective methods to reduce the turbulence-induced degradation.
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Y. Cui et al. / Optics Communications 354 (2015) 353–361
Very recently, more and more attention has been paid to the behavior of the partially coherent beams with nonconventional correlation functions propagating through atmosphere. Gu and Gbur investigated the NUC beam in atmosphere and found it to have superior scintillation characteristics [35]. The scintillations of multi-Gaussian Schell-model (MGSM) beams were investigated by Yuan et al. theoretically [36] and Korotkova et al. experimentally [37], and also found to have reduced scintillation. Thus, modulating the correlation functions of laser beams provides an optional way to reduce the negative effects in the atmosphere. Meanwhile, the other propagation properties such as average intensity distribution and degree of coherence, of such kinds of partially coherent beams including cosine-Gaussian correlated Schell-model (CGCSM) beam [38–40], NUC beam [41,42], LGCSM beam [43,44] have been extensively studied. However, the statistical properties of the SCRP beam propagation through atmosphere have not been reported yet. In this paper, our aim is to explore the second-order statistics of the SCRP beam in turbulent atmosphere. The analytical expressions for the average intensity distribution (AID), degree of polarization (DOP) and degree of coherence (DOC) of the SCRP beam in atmosphere are derived. The statistical properties of the SCRP beam are comparatively studied with those of the PCRP beam through numerical examples in detail. Some useful and interesting results are presented.
2. Beam coherence-polarization matrix of a special correlated radially polarized beam in atmosphere Let us consider a quasi-monochromatic SCRP beam, propagating close to z axis. Within the validity of the paraxial approximation, the second-order statistical properties of the SCRP beam in the source plane (z ¼0) may be characterized by the 2 2 beam coherence-polarization (BCP) matrix, given by [45]
Fresnel integral formula, and the elements of the SCRP beam in the output plane with the propagation distance z in atmospheric turbulence are expressed as the following integral formula [18] ⎛ k ⎞2 Γαβ (ρ1, ρ 2, z ) = ⎜ ⎟ ⎝ 2πz ⎠
) m d r1d2r2,
(
2
2
+
2⎤ ik(ρ 2 − r2) ⎥ ⎥ 2z ⎥⎦
(α, β = x, y ),
(6)
where ρ1 ¼(ρx1,ρy1) and ρ2 ¼(ρx2,ρy2) are two position vectors in the output plane; k ¼2π/λ is the wave number with λ being the wavelength of the light beam. ψ represents the complex phase fluctuation introduced by the atmospheric turbulence. m denotes the ensemble average over the turbulent media. The ensemble average over the phase fluctuation of the beam induced by the turbulence can be modeled as [46,47] exp⎡⎣ψ*(r1, ρ1) + ψ (r2, ρ 2)⎤⎦
m
⎡ 2 2 2 ⎞ π k z ⎛⎜ = × exp⎢ − (ρ − ρ1) + (ρ2 − ρ1)⋅(r2 − r1) + (r2 − r1)2⎠⎟ ⎢⎣ 3 ⎝ 2 ⎤
×
∫ 0∞ κ 3Φn(κ )dκ ⎥⎥, ⎦
(7)
where Φn(κ ) is the one-dimensional power spectrum of the refractive-index fluctuations of the atmospheric turbulence; κ is the spatial frequency. On substituting Eq. (7) into Eq. (6) and introducing the “sum” and “difference” variables
ρd = ρ2 − ρ1, ρs = (ρ1 + ρ2)/2,
(8)
rd = r2 − r1, rs = (r1 + r2)/2,
(9)
Eq. (6) turns out to be
1 λ z
′ (ρs , ρd , z ) = Γαβ
2 2
∞
∞
∫−∞ ∫−∞ Γαβ(0)(rs − rd/2, rs + rd/2)
⎡ π 2k 2Tz ⎤ r 2d + rd⋅ρd + ρ2d ⎥d2rsd2rd, × exp⎢− 3 ⎣ ⎦
(1)
(0) Γαβ (r1, r2) = Eα*(r1)Eβ(r2) , (α, β = x, y),
(2)
where r1 ¼ (x1,y1) and r2 ¼(x2,y2) denote the position vectors in the source plane perpendicular to z axis. The superscript “(0)” denotes the elements in the source plane. Ex and Ey represent two orthogonal electric fields; the asterisk and the angular brackets stand the complex conjugate and the ensemble average, respectively. For the SCRP beam, the elements of the BCP matrix in the source plane are written as [15] (0) (0) (r2−r1) gxy (r2−r1)⎞ ⎛ r 2 + r 2 ⎞⎛⎜ gxx ↔(0) ⎟ , Γ SC (r1, r2) = exp⎜⎜− 1 2 2 ⎟⎟⎜ 4σ0 ⎠ g (0)(r2−r1) g (0)(r2−r1)⎟ ⎝ yy ⎝ yx ⎠
(
(3)
⎡ (r − r )2 ⎤ ⎛ (α − α )2 ⎞ (0) (r2−r1) = ⎜⎜1 − 2 2 1 ⎟⎟exp⎢− 1 22 ⎥, (α = x, y), gαα 2δ 0 ⎦ δ0 ⎣ ⎝ ⎠ ⎡ (r − r )2 ⎤ exp⎢− 1 22 ⎥, 2δ 0 ⎦ ⎣
′ (ρs , ρd , z ) = Γαβ(ρs − ρd /2, ρs + ρd /2) Γαβ
T=
∫0
∞
κ 3Φn(κ )dκ .
(10)
(11)
(12)
On substituting Eqs. (3)–(5) into Eq. (10) and after integrating over rs and rd, we obtain the expressions for the elements of the BCP matrix of the SCRP beam propagating through atmosphere in the plane of z
′ (ρs , ρd , z ) = Γαα
2⎤ ⎡ izραs ⎞ ⎥ 1⎢ 1 ⎛ z2 ⎜ ⎟ 1− Δ ρ − − ⎜ ⎟ 1 Δ2 ⎢⎣ kσ02 ⎠ ⎥⎦ Δ2 k 2σ02δ02 Δ22 δ02 ⎝ αd
⎤ ⎡ ik ⎛ ⎛ Δ ⎞ ρ2 ⎞ × exp⎜⎜− 2s ⎟⎟exp⎢ ⎜1 − 1 ⎟ρs ⋅ρd ⎥ Δ2 ⎠ ⎦ ⎣z⎝ ⎝ 2σ0 Δ2 ⎠ ⎤ ⎡ ⎛ 2 2 k 2σ02 ⎛ Δ12 ⎞⎞ 2 ⎥ π k Tz ⎟⎟ρd , (α = x, y). ⎜ ⎟ 1 × exp⎢−⎜⎜ + − ⎜ ⎟ ⎢⎣ ⎝ 3 Δ2 ⎠⎠ ⎥⎦ 2z 2 ⎝ (13)
(4)
(5)
where s0 and δ0 are the beam width and the spatial coherence length of the SRCP beam. The propagation of the elements of the BCP matrix in turbulent atmosphere can be treated by the well-known extended Huygens–
)
with
with correlation functions being
δ02
2z
⎢⎣
× exp ψ*(r1, ρ1) + ψ (r2, ρ 2)
with elements
(x2 − x1)(y2 − y1)
ik(ρ1 − r1)
⎡ ik ⎤ × exp⎢ (rs⋅rd + ρs ⋅ρd − rs⋅ρd − rd⋅ρs )⎥ ⎣z ⎦
⎛ Γ (0)(r , r ) Γ (0)(r , r )⎞ xx 1 2 xy 1 2 ⎟ ↔(0) , Γ SC (r1, r2) = ⎜ ⎜ Γ (0)(r , r ) Γ (0)(r , r )⎟ ⎝ yx 1 2 yy 1 2 ⎠
(0) (0) gxy (r2−r1) = g yx (r2−r1) = −
⎡
∫ ∫ Γαβ(0)(r1, r2)exp⎢⎢−
′ (ρs , ρd , z ) = − Γxy
1
⎛ ⎜
⎜ρxd Δ12 Δ23 δ02 ⎝
−
⎛ izρys ⎞ iρxs z ⎞⎛ ρ2s ⎞ ⎟ ⎟exp⎜ − ⎟⎜ρ − ⎜ ⎟ 2 ⎟⎜ yd 2 ⎟ 2 kσ0 Δ1 ⎠⎝ ikσ0 Δ1 ⎠ ⎝ 2σ0 Δ2 ⎠
⎡ ⎛ 2 2 2 ⎞ ⎤ ⎡ ik ⎛ ⎤ Δ ⎞ k σ02 ⎛ Δ2 ⎞ π k Tz ⎜1 − 1 ⎟⎟⎟ρ2⎥ , × exp⎢ ⎜1 − 1 ⎟ρs⋅ρd⎥exp⎢ −⎜⎜ + ⎟ d⎥ ⎢ 2 ⎜ ⎢⎣ z ⎝ ⎥⎦ Δ2 ⎠ 3 Δ 2 z 2 ⎠⎠ ⎦ ⎝ ⎣ ⎝
(14)
Y. Cui et al. / Optics Communications 354 (2015) 353–361
′ (ρs , ρd , z ) = ⎡⎣Γxy ′ (ρs , ρd , z )⎤⎦*. Γyx
(15)
Δ1 and Δ2 in Eqs. (13) and (14) are two dimensionless parameters which are
π 2Tz 3 , Δ1 = 1 − 3σ02 Δ2 = 1 +
(16)
1 ⎞ 2π 2Tz 3 z2 ⎛ 1 ⎜ . + 2 ⎟⎟ + 3σ02 δ0 ⎠
⎜ k 2σ02 ⎝ 4σ02
(17)
In derivation of Eqs. (13) and (14), we have applied the following integral formula ∞
∫−∞ xnexp⎡⎣−(x − a)2⎤⎦dx = (2i)−n
π Hn(ia).
(18)
Eqs. (13)–(15) provide us a convenient way to study the second-order statistical properties of the SCRP beam in turbulent atmosphere. In the statistics of interest, what we will consider in follows is the AID, which is obtained by setting ρ ¼ ρ1 ¼ ρ2
↔ I (ρ , z ) = Tr[Γsc (ρ , 0, z )],
(19)
and is the DOP, given by [2]
P (ρ , z ) =
ISC (ρ1, z )ISC (ρ2 , z )
=
⎡ 4P3ρ2 P P P ⎞ρ ρ 1 ⎢ 2ik ⎛ 1 − 3 + 2 sα2 − ⎜1 − 3 1 ⎟ sα dα ⎢ 4P2 P2 z P2 ⎠ P2 ⎝ P ω 2 0 ⎣ 2⎞ 2 ⎤ ⎛ 2 2 P ω 4k ⎛ P ⎞ ρ ⎥ − ⎜ 1 2 + 0 2 ⎜1 − 1 ⎟ ⎟ dα2 ⎥ ⎟ ⎜P P 4z ⎝ 2 ⎠ ⎠ ω0 ⎦ ⎝ 2 2 ⎤ ⎡ ⎛ 2ρ2 ⎞ P 2⎞ ω 2k ⎛ 8π 2Tz 3 × exp⎜⎜ − 2 s ⎟⎟exp⎢ − 0 2 ⎜⎜1 + − 1 ⎟⎟ρd2⎥ 2 ⎢ P2 ⎠ ⎥⎦ 3ω0 ⎣ 8z ⎝ ⎝ ω 0 P2 ⎠
⎡ ik ⎛ ⎤ P ⎞ × exp⎢ ⎜1 − 1 ⎟ρs⋅ρd⎥ , (α = x, y ), ⎢⎣ z ⎝ ⎥⎦ P2 ⎠
(23)
PC Γxy (ρs , ρd , z )
=−
2 ⎡ 2 k ω02 ⎢⎛ P P2⎞ P ⎜P − 2P1 + 3 1 ⎟⎟ρ ρ − 3 16z ρ ρ 2 2 ⎢⎜ 2 P2 ⎠ dx dy P2 ω 4k 2 sx sy 16z P2 ⎣⎝ 0
PP⎞ i 4z ⎛ 8z 2 ρ ρ − ρsy ρdx + ⎜1 − 3 1 ⎟ρs⋅ρd 2 sx dy P2 ⎠ kω02 ⎝ ω04k ⎤ ⎡ ⎛ 2ρ2 ⎞ 2 2⎛ 2 3 ω k P2 ⎞ 8π Tz × exp⎜⎜ − 2 s ⎟⎟exp⎢ − 0 2 ⎜⎜1 + − 1 ⎟⎟ρd2⎥ 2 ⎢ P 3ω0 2⎠ ⎥ ⎦ ⎣ 8z ⎝ ⎝ ω 0 P2 ⎠ ⎡ ik ⎛ ⎤ P1 ⎞ exp⎢ ⎜1 − ⎟ρ ⋅ρ ⎥ , ⎢⎣ z ⎝ P2 ⎠ s d⎥⎦
(
+
with P1 = 1 −
(20)
and the DOC introduced by Wolf [48], i.e.,
↔ TrΓSC (ρ1, ρ2 , z )
PC Γαα (ρs , ρd , z )
)
(24)
* PC PC Γyx (ρs , ρd , z ) = ⎡⎣Γxy (ρs , ρd , z )⎤⎦ ,
↔ 4Det[ΓSC (ρ , ρ , z )] 1− , ⎡ ↔ ⎤2 ⎣TrΓSC (ρ , ρ , z )⎦
μ(ρ1, ρ2 , z ) =
355
4π 2Tz 3 3ω02
, P2 = 1 +
4z 2
(25) ⎛ ⎜
1
ω02k 2 ⎝ ω02
+
1
⎞ ⎟+
δg20 ⎠
8π 2Tz 3 3ω02
, P3 = 1 +
4z 2 ω04k 2
.
Based on the derived formula shown in Eqs. (23)–(25), the statistical properties such as AID, DOP and DOC, of the PCRP beam propagating through atmosphere can be determined.
. (21)
Tr and Det entering in Eqs. (19)–(21) stand the trace and the determinant of the BCP matrix. The quantity μ is connected to the visibility of the electromagnetic interference fringes in the far field formed by two points source located at ρ1 and ρ2 on the beam, and can be determined from the Young's experiment. It is evident from Eq. (21) that μ is independent on the anti-diagonal terms of the BCP matrix of stochastic electromagnetic beams. Before performing the numerical study of the statistical properties of the SCRP beam in atmosphere based on the derived formula, let us briefly review the expression for the BCP matrix of the PCRP beam in atmosphere [26–28]. According to [28], the BCP matrix of the PCRP beam in source plane reads as
⎤ ⎡ ⎛ r2 + r2 ⎞ (0) (r − r )2 ⎛ x1x2 x1y2 ⎞ 1 Γ^PC (r1, r2, 0) = 2 exp⎜⎜− 1 2 2 ⎟⎟exp⎢− 1 2 2 ⎥⎜ ⎟, ⎢⎣ 2δg 0 ⎥⎦⎝ y1x2 y1y2 ⎠ ω0 ω0 ⎠ ⎝
(22)
where ω0 is the beam waist size of fundamental Gaussian beam; δg0 is the spatial coherent length of the beam. It follows from Eq. (22) that the correlation function of the PCRP beam in the source plane is of Gaussian function and the beam profile is of a doughnut shape. When δg0 tends to infinity, Eq. (22) reduces to the BCP matrix of the completely coherent RP beam. In the paraxial region, the propagation of the elements of BCP matrix of the PCRP beam in atmospheric turbulence can also be treated by the extended Huygens–Fresnel integral formula shown in Eq. (6). Following the same procedure of the derivation of the BCP matrix of the SCRP beam, the elements of the BCP matrix of the PCRP beam in atmosphere take the form
3. AID, DOP and DOC of the SCRP beam in turbulent atmosphere: numerical examples In this section, as numerical examples, we investigate the evolution properties of the AID, DOP and DOC of the SCRP beam and the PCRP beam in turbulent atmosphere comparatively. Suppose that the statistics of atmospheric turbulence obeys Kolmogorov spectrum, and the spatial power spectrum Φn(κ ) is adopted by the von Karman spectrum [18]
(
)
Φn(κ ) = 0.033Cn2(κ 2 + κ 02)−11/6exp −κ 2/κm2 ,
(26)
Cn2
where is the structural constant of the refractive-index fluctuations of turbulence; κ0 = 2π /L 0 with L 0 being the outer scale of turbulence and κm = 5.92/l0 with l0 being the inner scale of turbulence. The values of L 0 and l0 are chosen to be 1 m and 0.01 m respectively throughout the paper for numerical calculation. On substituting Eq. (26) into Eq. (12) and integrating over κ , the parameter T yields T = 0.552Cn2. By applying Eqs. (13), (23) and (19), we obtain the AID of the SCRP beam and the PCRP beam propagation through atmosphere at distance z
ISC (ρ , z ) =
IPC (ρ , z ) =
⎤ ⎛ 2⎡ z2 z 2ρ2 ρ2 ⎞ ⎢1 − ⎥exp⎜⎜− ⎟⎟, + 2 2 2 2 2 2 4 Δ2 ⎢⎣ Δ2 k σ0 δ0 2Δ2 k δ0 σ0 ⎥⎦ ⎝ 2σ02Δ2 ⎠ ⎛ ⎛ 2ρ2 ⎞ P ⎞ 1 ⎜ 2z 2 4π 2Tz 3 + + 3 ρ2⎟⎟exp⎜⎜− 2 ⎟⎟, 3 P2 ⎠ ⎝ ω0 P2 ⎠
⎜ ω02P22 ⎝ k 2δg20
(27)
(28)
Fig. 1 illustrates the normalized AID of the SCRP beam at the cross line (ρy ¼ 0) in free space (T ¼0) for several propagation distances. The extension for the 3D-plot of the AID is straightforward
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Y. Cui et al. / Optics Communications 354 (2015) 353–361
Fig. 1. Average intensity distribution of the SCRP beam, the equivalent RP beam (ω0 ¼1.995 mm, δg0-1) and the equivalent PCRP beam (ω0 ¼ 2.261 mm, δg0 ¼ 3 mm) at the cross section (ρy ¼0) in free space (Cn2 = 0) for several propagation distances. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
because the intensity distribution at the beam cross-section is rotational symmetry. The other parameters used in the calculation are s0 ¼10 mm, δ0 ¼2 mm, λ ¼632.8 nm unless different values are specified. In order for comparing the evolution properties of the SCRP beam and PCRP beam in turbulent atmosphere, the equivalent condition between two beams has been applied in the numerical calculation (see Appendix A), i.e.,
1 1 1 1 + = + 2. ω02 δ0 2δg20 8σ02
(29)
The SCRP beam and the PCRP beam satisfying Eq. (29) are called the equivalent beams each other because the spread angle of them both in free space and in turbulent atmosphere keeps same [49]. It can be seen from Eq. (29) that there are different sets of ω0 and δg0 to satisfy Eq. (29) when the beam parameters of the SCRP beam is fixed. The normalized AID of the equivalent RP beam (blue line, for special case of the PCRP beam when δg0-1, ω0 ¼1.995 mm) and the equivalent PCRP beam (green line, ω0 ¼2.261 mm, δg0 ¼ 3 mm) in free space calculated from Eq. (28) are also plotted in Fig. 1. As shown in Fig. 1, the beam profile of the RP beam in free space remains unchanged and keeps doughnut shape during propagation, as expected. Due to its limited spatial coherence length of the equivalent PCRP beam, the beam profile is spatially variant on propagation, and turns from doughnut shape to “hilllike” shape with a little dip in the central region as the propagation
distance increases (see Fig. 1(b) and (c)). If the spatial coherent length deteriorates further, our numerical results show that the beam profile of the equivalent PCRP beam (ω0 ¼2.814 mm, δg0 ¼2 mm, not shown here) becomes a Gaussian profile at z¼0.5 km or z ¼5 km, while the evolution of beam profile of the SCRP beam is rather different due to its special correlation structure. It changes from Gaussian shape to doughnut shape gradually with the increase of propagation distance. When the propagation distance is sufficient large (z 45 km), the beam profiles of the SCRP beam and the equivalent RP beam are nearly the same. In the presence of atmospheric turbulence, the evolution properties of the normalized AID of three equivalent beams at several propagation distances are shown in Fig. 2. One finds that the effect of the turbulence is to degenerate the beam profile of three equivalent beams to a Gaussian-like shape, irrespective of the intensity distribution and the structure of the correlation function of the beam in the source plane. As a result, the beam profiles among three equivalent beams become almost same at the propagation distance z ¼10 km. This phenomenon can be explained by the fact that the SCRP beam propagating in atmosphere is affected by the free-space diffraction and the atmospheric turbulence together. The effect of the former factor is to turn the beam profile to doughnut shape due to the special coherence structure of the SCRP beam, while the latter factor is to degenerate the coherence structure of the SCRP beam to Gaussian profile (see Eq. (3)) and its effect accumulates on propagation.
Fig. 2. Average intensity distribution of the SCRP beam, the equivalent RP beam (ω0 ¼1.995 mm, δg0-1) and the equivalent PCRP beam (ω0 ¼2.261 mm, δg0 ¼3 mm) at the cross-line (ρy ¼ 0) in turbulent atmosphere (Cn2 = 10−13m−2/3) for several propagation distances.
Y. Cui et al. / Optics Communications 354 (2015) 353–361
After propagating a long distance, the effect of turbulence overpasses the effect of free-space diffraction. Thus, the AID of the SCRP beam will become a Gaussian shape after a long propagation distance in turbulence. To learn about the change of the beam shape of three equivalent beams propagating in free space or in turbulence, a parameter named dark hollow rate (DHR) is introduced to describe the difference of evolution properties of the AID among three beams, defined as
H (z ) = I (0, z )/[I (ρ , z )]max
(30)
The quantity H(z) is the ratio of the average intensity at the central point to the maximum intensity in the corresponding cross-section at the propagation distance z. When H(z)¼ 0 the beam profile is a standard dark hollow shape, no dip in the central region of the beam profile will be observed if H(z)¼1. Fig. 3 shows the dependence of the DHR of three equivalent beams on the propagation distance with different spatial coherence length and strength of the turbulence. In free space (Cn2 = 0), the DHR of the equivalent RP beam keeps the value 0 on propagation, which is consistent with the analysis in Fig. 1, implying that the beam profile is a standard doughnut shape with zero intensity at the onaxis point. For the case of the equivalent PCRP beam, the value of DHR increases dramatically as the propagation distance increases, and then tends to a stable value for further propagation. Notice that this stable value is closely related to the initial spatial coherence length of the equivalent PCRP beam. The increase of the initial spatial coherence length leads to the decrease of the stable value (see Fig. 3(a) and (d)). For the equivalent SCRP beam, the DHR in free space first keeps unit, and then starts to decrease as the propagation distance increases, at last remains a value almost unchanged when the propagation distance is long enough. The distance keeps the DHR of the equivalent SCRP beam unit and the constant value at the long propagation distance is also determined by the spatial coherence length δ0. The larger the spatial coherence length is, the longer the distance keeping unit is and the larger the constant value at the long propagation distance is. In the atmospheric turbulence, the DHR of the equivalent RP beam and the equivalent PCRP beam increases monotonously with the increase of the propagation distance, and the slope becomes large as the strength of the turbulence enhances, which implies that the
357
ability for maintaining the dark hollow shape fades in turbulence. The situation of the equivalent SCRP beam in turbulence is similar to that of the equivalent RP beam or PCRP beam, which is that the dark hollow profile appearing on propagation is also degenerated, i.e., The DHR of the SCRP beam increases on propagation when the propagation distance is long enough (see Fig. 3(c) and (f)). Note that the value of DHR of the equivalent SCRP beam (δ0 ¼5 mm) is always 1 in the strong turbulence, indicating that no dark hollow beam shape will be observed during propagation under that circumstance. Let us turn to study the behavior of the DOP of the SCRP beam and the equivalent RP or PCRP beam during propagation comparatively. By applying Eq. (20), the DOP of the SCRP beam and the PCRP beam in atmospheric turbulence take the following form
PSC (ρ , z ) =
PPC (ρ , z ) =
ρ2
2Δ22 k 2δ02σ04/z 2 − 2Δ2 σ02 + ρ2 ρ2 2
ρ +
ω02P2(P2
− P3)/2P3
,
,
(31)
(32)
It can be seen from Eqs. (31) and (32) that the distributions of the DOP of the SCRP beam and the PCRP beam across the source plane (z ¼0) are all uniform, while the value of the DOP equals 1 for the PCRP beam and 0 for the SCRP beam. We plot in Fig. 4 the variance of the DOP of the SCRP beam, the equivalent RP beam and the equivalent PCRP beam as a function of ρx(ρy ¼0) both in free space and in turbulence. As expected, the DOP of the equivalent RP beam keeps unit invariant across the transverse plane during propagation, while the DOP of the SCRP beam and the equivalent PCRP beam on propagation forms a dip pattern with the value of the onaxis point keeping zero. In turbulent atmosphere, the distributions of the DOP of the three beams on propagation are similar to each other, which all form the dip pattern. However, the width of the dip among them in the short propagation distance is different (see Fig. 4(d)–(e), (g)–(h)). With the increase of the propagation distance or enhancing the strength of the turbulence, the distribution of the DOP between the SCRP beam and the equivalent RP beam becomes almost same (see Fig. 4(f) and (i)). To trace the variance of the width of the formed dip on
Fig. 3. Variance of dark hollow rate of the SCRP beam, the equivalent RP beam and the equivalent PCRP beam with propagation distance in free space and turbulent atmosphere for two different values of spatial coherence length. The beam parameters in (a)–(c) are same with those in Fig. 2. The beam parameters in (d)–(f) are ω0 ¼4.924 mm, δg0-1 for equivalent RP beam and ω0 ¼ 5.250 mm, δg0 ¼10 mm for equivalent PCRP beam.
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Fig. 4. Evolution of the DOP of the SCRP beam, the equivalent RP beam and the equivalent PCRP beam in free space and turbulent atmosphere on propagation.
propagation, a parameter ΔP (z ) is introduced to quantity the width of the dip, given by
ΔP (z ) = (ρx2 )P (ρx2 , z )= 0.5 − (ρx1)P (ρx1, z )= 0.5 .
(33)
According to Eq. (33), the width of the dip of the DOP of the SCRP and the PCRP beam deduced from Eqs. (31) and (32) are
ΔPSC (z ) = 2σ0 2Δ22 k 2δ02σ02/z 2 − 2Δ2 ,
(34)
ΔPPC (z ) = 2ω0 P2(P2 − P3)/2P3 ,
(35)
Fig. 5 shows the dependence of ΔPSC (z ) of the SCRP beam on the propagation distance in turbulence for different values of spatial coherence length. The corresponding variance ΔPPC (z ) of the
equivalent RP beam (δg0-1) is also plotted in Fig. 5 (dashed line) for convenient comparison. The width of dip of the equivalent RP beam increases monotonously with the increase of propagation distance, while the dip width of the SCRP beam first decreases dramatically, reaches a minimum value, and then increases during propagation. It is worth noting that the difference of the width of dip becomes large in the far field (z ¼5 km) with the increase of the spatial coherence length of the SCRP beam (see Fig. 5(a)–(c)). This phenomenon can be explained by the fact that if the propagation distance is sufficient large, Eq. (34) and Eq. (35) could simplify to 4 2 π 2kTz 2δ0/3 and 4 2 π 2kTz 2ω0/3 respectively, to a good approximation. So, the dip width is proportional to δ0 for the SCRP beam and to ω0 for the equivalent RP beam. Under the equivalent condition, the SCRP beam with s0 ¼ 10 mm and δ0 ¼ 2 mm corresponds to the equivalent RP beam with
Fig. 5. Dependence of the dip width of the SCRP and the equivalent RP beam on the propagation distance for different spatial coherence length in atmosphere.
Y. Cui et al. / Optics Communications 354 (2015) 353–361
ω0 ¼1.995 mm. Therefore, the dip width between two equivalent beams is almost same in the far field (see Fig. 5(a)). However, the SCRP beam with s0 ¼10 mm and δ0 ¼20 mm corresponds to the equivalent RP beam with ω0 ¼16.3 mm. This leads to the difference of the width of dip between two beams in the far field (see Fig. 5(c)). Let us now explore the behavior of the DOC of the SCRP beam and the equivalent PCRP beam both in the free space and in the turbulent atmosphere. On substituting Eqs. (13) and (23) into Eq. (21), the expressions for the DOC of two beams at a pair of points −ρ and ρ are N1 − 2Δ12 ρ2/Δ22 δ02
μSC ( − ρ, ρ, z ) =
2
N1 + z 2ρ2/2Δ22 k σ04δ02 ⎡ ⎛ 2 2 2 ⎞ ⎤ 2k σ02 ⎛ Δ2 ⎞ 4π k Tz 1 ⎟ 2⎥ ⎜⎜1 − 1 ⎟⎟ − × exp⎢ −⎜⎜ + ρ ⎢ 3 Δ2 ⎠ 2Δ2 σ02 ⎟⎠ ⎥⎦ z2 ⎝ ⎣ ⎝
with N1 = 1 − z
2
(36)
/Δ2 k 2σ02δ02 .
μPC ( − ρ , ρ , z ) =
⎛ P2 − P3 − 2⎜P12 + ⎝
ω04k 2 4z 2
2⎞
(P2 − P1) ⎟⎠ρ2/P2ω02
359
dislocation [50] could be found in the DOC of the SCRP beam in the source plane, i.e., the trace of the zero value of the DOC forms a circle. The value of the DOC inner and outer of the ring is positive and negative, respectively. Thus, a π phase jump occurs in the inner and outer of the ring. When the two equivalent beams propagate from the source plane to the plane z4 0 in free space, two side lobes in the DOC of the equivalent PCRP beam can be observed clearly, whereas the distribution of the DOC of the SCRP beam gradually turns to Gaussian-like shape (see in Fig. 6(b) and (c)). In the turbulent atmosphere, both the distribution of the SCRP beam and the equivalent PCRP beam degenerates to Gaussian-like profile and the difference between them gets smaller with the increase of propagation distance (see Fig. 6(e) and (f)). This interesting result may be explained that in the far field, the turbulence effect plays the dominant role to determine the coherence properties of the PCRP beam or SCRP beam. Therefore, the patterns of DOC of two equivalent beams become similar when the propagation distance is long enough. It is worth noting that although the ring dislocation is hardly observed at z¼ 2 km (See Fig. 6(f)), in fact it exists. From Eqs. (36) and (37), if we set μ( − ρ, ρ, z ) = 0, the equation of the radius of the ring of two equivalent beams are
RSC (z ) = δ0 Δ22 N1/2Δ12 ,
P2 − P3 + 2P3ρ2 /P2ω02
(38)
(
2
)
RPC (z ) = ω0 P2(P2 − P3)/2 P12 + ω04k 2(P2 − P1) /4z 2 , ⎤ ⎡ ⎛ 2 2 P2 ⎞ ω 2k 2 ⎛ 4π k Tz 2 ⎞⎟ 2⎥ × exp⎢−⎜⎜ + 0 2 ⎜⎜1 − 1 ⎟⎟ − ρ ⎟ ⎢⎣ ⎝ 3 P2 ⎠ P2ω02 ⎠ ⎥⎦ 2z ⎝
(37)
It is obvious from Eqs. (36) and (37) that the DOC of two beams are rotational symmetry in the beam cross section. Fig. 6 gives the cross-line (ρy ¼0) of the DOC of the SCRP beam and the equivalent PCRP beam at two points ρx and ρx both in free space and in turbulent atmosphere at several propagation distances. One finds that the patterns of the DOC between two equivalent beams are much different in the source plane. The distribution of the DOC of the equivalent PCRP beam are of Gaussian shape in the source plane, while the distribution of the DOC of the SCRP beam has two side lobes besides the central peak. In the 3D-plot case, a ring
(39)
Fig. 7 illustrates the location of the ring in the DOC of two equivalent beams at z¼ 0, 1 km, 3 km in turbulent atmosphere (Fig. 7(a) and (c)) and the dependence of the radius of the ring on the propagation distance (Fig. 7(b) and (d)). The red, green and pink pair points represent the typical coherence singularities where the value of the DOC at the pair of point ( ρ, ρ) equals zero at z¼ 0, z ¼1 km and z ¼3 km, respectively. Note that no coherence singularities in the DOC of the equivalent PCRP beam could be found in the source plane because the distribution of DOC is a Gaussian function. Both of the radius of the ring RSC(z) and RPC(z) in the free space increase monotonously with the increase of propagation distance since the distribution of the DOC spreads in the transverse plane due to the diffraction
Fig. 6. Variance of DOC of the SCRP beam and the equivalent PCRP beam as a function of ρx(ρy ¼ 0) at a pair points ρx and ρx for three different propagation distances. The beam parameters of the SCRP beam and equivalent PCRP beam are s0 ¼ 2.814 mm, δ0 ¼2 mm and ω0 ¼ 2.814 mm, δg0 ¼ 2 mm, respectively.
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Fig. 7. (a,c) Circles where the ring dislocations locate for different propagation distances, the read, green and pink pair points are typical coherence singularities pairs at z¼ 0, 1 km and 3 km, respectively. (b,d) The evolution of the radius of ring against the propagation distance in free space and in atmosphere. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
effect of the beam on propagation. However, in the atmospheric turbulence, the evolution properties of the radius of the ring are rather different between two equivalent beams. For the case of RSC(z), it increases and drop dramatically near the critical propagation distance. The reason is that the coefficient Δ1 in Eq. (38) tends to zero at a certain propagation distance in turbulence, while this will not happen in free space. The critical propagation distance where RSC(z) tends to infinity in turbulence can be obtained by setting Δ1 = 0, which is zc = 3 3σ02/π 2T . The larger the strength of the turbulence is, the shorter the critical propagation is. For the case of RPC(z), it first increases as the propagation distance increases, reaches a maximum value, and then decrease with the increase of the propagation distance further due to the de-coherence effect of the turbulence, and no discontinuity occurs in RPC(z) on propagation through atmosphere.
4. Conclusion In summary, we have derived the analytical formulas for the AID, DOP and DOC of the SCRP beam propagating through turbulent atmosphere with the help of the extended Huygens–Fresnel integration. Based on the derived formulas, we present the numerical study of the evolution properties of the AID, DOP and DOC of the SCRP beam in atmospheric turbulence on propagation. The corresponding results of the equivalent PCRP beam are also illustrated for comparison. It is found that the AID of the SCRP beam degenerates to Gaussian-like profile in atmosphere for the propagation distance long enough, and the evolution properties are closely determined by the strength of turbulence and the spatial coherence length. The behavior of the DOP of the SCRP beam in turbulence is similar to that of the equivalent PCRP beam on propagation. The distributions of the DOP of two equivalent beams form the “Inverse-Gaussian” shape during propagation. More
interestingly, the evolutions of the ring dislocation in the DOC of the SCRP beam and the equivalent PCRP beam are much different in turbulent atmosphere. The ring radius in the DOC of the SCRP beam first increases dramatically with the increase of the propagation distance, tends to infinity at the critical propagations, and then decreases as the propagation distances increases, while no such critical propagation distance where the ring radius tends to infinity occurs in the DOC of the equivalent PCRP beam.
Acknowledgment This work is supported by the National Natural Science Foundation of China under Grant nos. 11474213, 11274005 and 11474143. The Project is Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars (To Dr. Cai), State Education Ministry.
Appendix A. : Derivation of the equivalent condition of the SCRP beam and the PCRP beam The variance of the mean-squared width of light beams with the propagation distance in turbulent atmosphere can be written as [51]
ρ2 (z ) =
∫ ρ2I (ρ, z )d2ρ , ∫ I (ρ, z )d2ρ
(A1)
where I (ρ, z ) denotes the average intensity distribution in the plane of z. The denominator in Eq. (A1) stands the total energy carried by the light beams and remain unchanged propagating
Y. Cui et al. / Optics Communications 354 (2015) 353–361
through the atmospheric turbulence. For the SCRP beam, On substituting from Eq. (27) into Eq. (A1), after tedious integration and operation, we obtain 2 ρSC (z ) = 2σ02 +
4 ⎞ 4π 2Tz 3 z2 ⎛ 1 ⎜⎜ . + 2 ⎟⎟ + 3 δ0 ⎠ k 2 ⎝ 2σ02
(A2)
The second term on the right side of Eq. (A2) denote the beam spreading induced by the diffractive effect in free space, while the third term represents the spreading due to atmospheric turbulence. From Eq. (A2), the angular spread of the SCRP beam in atmosphere turbulence reads as
θSC
(ρ = lim
2 (z ) SCRP
z →∞
z
1/2
)
=
1⎛ 1 4 ⎞ 4π 2Tz ⎜⎜ . + 2 ⎟⎟ + 3 k 2 ⎝ 2σ02 δ0 ⎠
(A3)
(A4)
Compared to Eq. (A2), one finds that the beam spreading induced by turbulence between two beams is same. From Eq. (A4), we obtain the expression for the angular spread of the PCRP beam in turbulence, i.e.,
θPC =
⎛ ⎞ 2 ⎜ 4 + 2 ⎟ 1 + 4π Tz . ⎜ 2 2 ⎟ 2 3 δg 0 ⎠ k ⎝ ω0
[10] [11] [12] [13] [14] [15] [16] [17] [18]
Let us turn to derive the angular spread of the PCRP beam. On substituting Eq. (28) into Eq. (A1), after integrating, the meansquared width of the PCRP beam in turbulence yields
⎛ ⎞ 4 2 z2 4Tπ 2z 3 2 . ρPC (z ) = ω02 + ⎜⎜ 2 + 2 ⎟⎟ 2 + 3 δg 0 ⎠ k ⎝ ω0
[3] [4] [5] [6] [7] [8] [9]
[19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
(A5) [37]
If sources generate the same angular spread both in free space and in turbulent atmosphere, such sources are called equivalent beams each other. Thus, the equivalent condition for the SCRP beam and PCRP beam deduced from Eqs. (A3) and (A5) is
1 1 1 1 + = + 2. ω02 δ0 2δg20 8σ02
(A6)
Reference [1] L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, Cambridge University press, Cambridge, 1995. [2] E. Wolf, Introduction to the Theory of Coherence and Polarization of Light, Cambridge University press, Cambridge, 2007.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Effect of the atmospheric turbulence on a special correlated radially polarized beam on propagation Yan Cui a, Cun Wei a, Yongtao Zhang a,b, Fei Wang a,n, Yangjian Cai a a College of Physics, Optoelectronics and Energy & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China b Department of Physics Luoyang Normal University, Luoyang 471022, China
art ic l e i nf o
a b s t r a c t
Article history: Received 1 April 2015 Received in revised form 4 June 2015 Accepted 7 June 2015 Available online 10 June 2015
A special correlated radially polarized beam (Phys. Rev. A 89, 013801, 2014) was introduced and demonstrated in experiment recently. In this paper, we investigate the statistical properties of a special correlated radially polarized (SCRP) beam in atmospheric turbulence. Analytical formulas for the average intensity distribution (AID), degree of polarization (DOP) and degree of coherence (DOC) are derived by adopting a beam coherence-polarization (BCP) matrix. With the help of the derived formulas, the evolutions of the AID, DOP and DOC of the SCRP beam in turbulent atmosphere are illustrated through numerical examples in detail, and the results are compared to those of a partially coherent radially polarized (PCRP) beam under equivalent condition. It reveals that the propagation properties of the SCRP beam is much different from those of the PCRP beam in atmosphere, and closely related to the strength of the turbulence and the beam parameters. & 2015 Elsevier B.V. All rights reserved.
Keywords: Partially coherent beam Atmospheric turbulence Radial polarization Propagation
1. Introduction It is well known that the intensity distribution and the polarization properties of light beams on propagation are closely related to the correlation properties of light sources [1,2]. However, in most studies, the correlation functions of the partially coherent beams are restricted on Gaussian function because it is difficult to judge whether other correlation functions are physical realizable or not. Since Gori and co-workers derived the sufficient condition for devising genuine correlation functions for partially coherent scalar and vector beams [3,4], various partially coherent beams with different kinds of correlation functions have been introduced theoretically [5–8]. Meanwhile, several methods were proposed to produce such kinds of partially coherent beams [9–12]. Of particular interest is that such partially coherent beams exhibit some extraordinary propagation characteristics even in free space. For example, a Laguerre–Gaussian correlated Schell-model (LGCSM) beam or a Bessel–Gaussian correlated Schell-model (BGCSM) beam [7] is capable of producing ring-shaped beam profile in the far field; the transverse maximum intensity of a non-uniformly correlated (NUC) beam is shifted during propagation [5]. Not only the intensity distribution, but also the polarization properties can be controlled by modulating the correlation functions [13,14]. n
Corresponding author. E-mail address: [email protected] (F. Wang).
http://dx.doi.org/10.1016/j.optcom.2015.06.017 0030-4018/& 2015 Elsevier B.V. All rights reserved.
Recently, we introduced and experimentally demonstrated a new kind of partially coherent vector beam, named special correlated radially polarized (SCRP) beam [15], and the focusing properties of the SCRP beam passing through a thin lens was investigated both theoretically and experimentally. It was found that the beam profile and the polarization feature of the SCRP beam in the far field become a doughnut shape and radial polarization, respectively, while such beam in the source plane can be considered as an un-polarized beam with intensity distribution being Gaussian profile. These properties are much different from a radially polarized beam with conventional Gaussian Schell-model correlation, i.e., partially coherent radially polarized (PCRP) beam [16,17], whose polarization distribution is radially polarized in the source and de-polarized on propagation due to its limited spatial coherence length. On the other hand, study of the propagation characteristics of laser beams in turbulent atmosphere has received considerable attention for a long time due to its great importance in connection with remote sensing, imaging and free-space optical communication [18]. In general, the atmospheric turbulence will cause the extra spreading beyond the diffraction, wander and scintillation of laser beams, which limits the performance in the previously mentioned applications. Thus, knowledge of the propagation behavior of light beams in atmospheric turbulence is utmost significant [19–34]. It is known that decreasing spatial coherence and modulating polarization distribution of light beams are two effective methods to reduce the turbulence-induced degradation.
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Very recently, more and more attention has been paid to the behavior of the partially coherent beams with nonconventional correlation functions propagating through atmosphere. Gu and Gbur investigated the NUC beam in atmosphere and found it to have superior scintillation characteristics [35]. The scintillations of multi-Gaussian Schell-model (MGSM) beams were investigated by Yuan et al. theoretically [36] and Korotkova et al. experimentally [37], and also found to have reduced scintillation. Thus, modulating the correlation functions of laser beams provides an optional way to reduce the negative effects in the atmosphere. Meanwhile, the other propagation properties such as average intensity distribution and degree of coherence, of such kinds of partially coherent beams including cosine-Gaussian correlated Schell-model (CGCSM) beam [38–40], NUC beam [41,42], LGCSM beam [43,44] have been extensively studied. However, the statistical properties of the SCRP beam propagation through atmosphere have not been reported yet. In this paper, our aim is to explore the second-order statistics of the SCRP beam in turbulent atmosphere. The analytical expressions for the average intensity distribution (AID), degree of polarization (DOP) and degree of coherence (DOC) of the SCRP beam in atmosphere are derived. The statistical properties of the SCRP beam are comparatively studied with those of the PCRP beam through numerical examples in detail. Some useful and interesting results are presented.
2. Beam coherence-polarization matrix of a special correlated radially polarized beam in atmosphere Let us consider a quasi-monochromatic SCRP beam, propagating close to z axis. Within the validity of the paraxial approximation, the second-order statistical properties of the SCRP beam in the source plane (z ¼0) may be characterized by the 2 2 beam coherence-polarization (BCP) matrix, given by [45]
Fresnel integral formula, and the elements of the SCRP beam in the output plane with the propagation distance z in atmospheric turbulence are expressed as the following integral formula [18] ⎛ k ⎞2 Γαβ (ρ1, ρ 2, z ) = ⎜ ⎟ ⎝ 2πz ⎠
) m d r1d2r2,
(
2
2
+
2⎤ ik(ρ 2 − r2) ⎥ ⎥ 2z ⎥⎦
(α, β = x, y ),
(6)
where ρ1 ¼(ρx1,ρy1) and ρ2 ¼(ρx2,ρy2) are two position vectors in the output plane; k ¼2π/λ is the wave number with λ being the wavelength of the light beam. ψ represents the complex phase fluctuation introduced by the atmospheric turbulence. m denotes the ensemble average over the turbulent media. The ensemble average over the phase fluctuation of the beam induced by the turbulence can be modeled as [46,47] exp⎡⎣ψ*(r1, ρ1) + ψ (r2, ρ 2)⎤⎦
m
⎡ 2 2 2 ⎞ π k z ⎛⎜ = × exp⎢ − (ρ − ρ1) + (ρ2 − ρ1)⋅(r2 − r1) + (r2 − r1)2⎠⎟ ⎢⎣ 3 ⎝ 2 ⎤
×
∫ 0∞ κ 3Φn(κ )dκ ⎥⎥, ⎦
(7)
where Φn(κ ) is the one-dimensional power spectrum of the refractive-index fluctuations of the atmospheric turbulence; κ is the spatial frequency. On substituting Eq. (7) into Eq. (6) and introducing the “sum” and “difference” variables
ρd = ρ2 − ρ1, ρs = (ρ1 + ρ2)/2,
(8)
rd = r2 − r1, rs = (r1 + r2)/2,
(9)
Eq. (6) turns out to be
1 λ z
′ (ρs , ρd , z ) = Γαβ
2 2
∞
∞
∫−∞ ∫−∞ Γαβ(0)(rs − rd/2, rs + rd/2)
⎡ π 2k 2Tz ⎤ r 2d + rd⋅ρd + ρ2d ⎥d2rsd2rd, × exp⎢− 3 ⎣ ⎦
(1)
(0) Γαβ (r1, r2) = Eα*(r1)Eβ(r2) , (α, β = x, y),
(2)
where r1 ¼ (x1,y1) and r2 ¼(x2,y2) denote the position vectors in the source plane perpendicular to z axis. The superscript “(0)” denotes the elements in the source plane. Ex and Ey represent two orthogonal electric fields; the asterisk and the angular brackets stand the complex conjugate and the ensemble average, respectively. For the SCRP beam, the elements of the BCP matrix in the source plane are written as [15] (0) (0) (r2−r1) gxy (r2−r1)⎞ ⎛ r 2 + r 2 ⎞⎛⎜ gxx ↔(0) ⎟ , Γ SC (r1, r2) = exp⎜⎜− 1 2 2 ⎟⎟⎜ 4σ0 ⎠ g (0)(r2−r1) g (0)(r2−r1)⎟ ⎝ yy ⎝ yx ⎠
(
(3)
⎡ (r − r )2 ⎤ ⎛ (α − α )2 ⎞ (0) (r2−r1) = ⎜⎜1 − 2 2 1 ⎟⎟exp⎢− 1 22 ⎥, (α = x, y), gαα 2δ 0 ⎦ δ0 ⎣ ⎝ ⎠ ⎡ (r − r )2 ⎤ exp⎢− 1 22 ⎥, 2δ 0 ⎦ ⎣
′ (ρs , ρd , z ) = Γαβ(ρs − ρd /2, ρs + ρd /2) Γαβ
T=
∫0
∞
κ 3Φn(κ )dκ .
(10)
(11)
(12)
On substituting Eqs. (3)–(5) into Eq. (10) and after integrating over rs and rd, we obtain the expressions for the elements of the BCP matrix of the SCRP beam propagating through atmosphere in the plane of z
′ (ρs , ρd , z ) = Γαα
2⎤ ⎡ izραs ⎞ ⎥ 1⎢ 1 ⎛ z2 ⎜ ⎟ 1− Δ ρ − − ⎜ ⎟ 1 Δ2 ⎢⎣ kσ02 ⎠ ⎥⎦ Δ2 k 2σ02δ02 Δ22 δ02 ⎝ αd
⎤ ⎡ ik ⎛ ⎛ Δ ⎞ ρ2 ⎞ × exp⎜⎜− 2s ⎟⎟exp⎢ ⎜1 − 1 ⎟ρs ⋅ρd ⎥ Δ2 ⎠ ⎦ ⎣z⎝ ⎝ 2σ0 Δ2 ⎠ ⎤ ⎡ ⎛ 2 2 k 2σ02 ⎛ Δ12 ⎞⎞ 2 ⎥ π k Tz ⎟⎟ρd , (α = x, y). ⎜ ⎟ 1 × exp⎢−⎜⎜ + − ⎜ ⎟ ⎢⎣ ⎝ 3 Δ2 ⎠⎠ ⎥⎦ 2z 2 ⎝ (13)
(4)
(5)
where s0 and δ0 are the beam width and the spatial coherence length of the SRCP beam. The propagation of the elements of the BCP matrix in turbulent atmosphere can be treated by the well-known extended Huygens–
)
with
with correlation functions being
δ02
2z
⎢⎣
× exp ψ*(r1, ρ1) + ψ (r2, ρ 2)
with elements
(x2 − x1)(y2 − y1)
ik(ρ1 − r1)
⎡ ik ⎤ × exp⎢ (rs⋅rd + ρs ⋅ρd − rs⋅ρd − rd⋅ρs )⎥ ⎣z ⎦
⎛ Γ (0)(r , r ) Γ (0)(r , r )⎞ xx 1 2 xy 1 2 ⎟ ↔(0) , Γ SC (r1, r2) = ⎜ ⎜ Γ (0)(r , r ) Γ (0)(r , r )⎟ ⎝ yx 1 2 yy 1 2 ⎠
(0) (0) gxy (r2−r1) = g yx (r2−r1) = −
⎡
∫ ∫ Γαβ(0)(r1, r2)exp⎢⎢−
′ (ρs , ρd , z ) = − Γxy
1
⎛ ⎜
⎜ρxd Δ12 Δ23 δ02 ⎝
−
⎛ izρys ⎞ iρxs z ⎞⎛ ρ2s ⎞ ⎟ ⎟exp⎜ − ⎟⎜ρ − ⎜ ⎟ 2 ⎟⎜ yd 2 ⎟ 2 kσ0 Δ1 ⎠⎝ ikσ0 Δ1 ⎠ ⎝ 2σ0 Δ2 ⎠
⎡ ⎛ 2 2 2 ⎞ ⎤ ⎡ ik ⎛ ⎤ Δ ⎞ k σ02 ⎛ Δ2 ⎞ π k Tz ⎜1 − 1 ⎟⎟⎟ρ2⎥ , × exp⎢ ⎜1 − 1 ⎟ρs⋅ρd⎥exp⎢ −⎜⎜ + ⎟ d⎥ ⎢ 2 ⎜ ⎢⎣ z ⎝ ⎥⎦ Δ2 ⎠ 3 Δ 2 z 2 ⎠⎠ ⎦ ⎝ ⎣ ⎝
(14)
Y. Cui et al. / Optics Communications 354 (2015) 353–361
′ (ρs , ρd , z ) = ⎡⎣Γxy ′ (ρs , ρd , z )⎤⎦*. Γyx
(15)
Δ1 and Δ2 in Eqs. (13) and (14) are two dimensionless parameters which are
π 2Tz 3 , Δ1 = 1 − 3σ02 Δ2 = 1 +
(16)
1 ⎞ 2π 2Tz 3 z2 ⎛ 1 ⎜ . + 2 ⎟⎟ + 3σ02 δ0 ⎠
⎜ k 2σ02 ⎝ 4σ02
(17)
In derivation of Eqs. (13) and (14), we have applied the following integral formula ∞
∫−∞ xnexp⎡⎣−(x − a)2⎤⎦dx = (2i)−n
π Hn(ia).
(18)
Eqs. (13)–(15) provide us a convenient way to study the second-order statistical properties of the SCRP beam in turbulent atmosphere. In the statistics of interest, what we will consider in follows is the AID, which is obtained by setting ρ ¼ ρ1 ¼ ρ2
↔ I (ρ , z ) = Tr[Γsc (ρ , 0, z )],
(19)
and is the DOP, given by [2]
P (ρ , z ) =
ISC (ρ1, z )ISC (ρ2 , z )
=
⎡ 4P3ρ2 P P P ⎞ρ ρ 1 ⎢ 2ik ⎛ 1 − 3 + 2 sα2 − ⎜1 − 3 1 ⎟ sα dα ⎢ 4P2 P2 z P2 ⎠ P2 ⎝ P ω 2 0 ⎣ 2⎞ 2 ⎤ ⎛ 2 2 P ω 4k ⎛ P ⎞ ρ ⎥ − ⎜ 1 2 + 0 2 ⎜1 − 1 ⎟ ⎟ dα2 ⎥ ⎟ ⎜P P 4z ⎝ 2 ⎠ ⎠ ω0 ⎦ ⎝ 2 2 ⎤ ⎡ ⎛ 2ρ2 ⎞ P 2⎞ ω 2k ⎛ 8π 2Tz 3 × exp⎜⎜ − 2 s ⎟⎟exp⎢ − 0 2 ⎜⎜1 + − 1 ⎟⎟ρd2⎥ 2 ⎢ P2 ⎠ ⎥⎦ 3ω0 ⎣ 8z ⎝ ⎝ ω 0 P2 ⎠
⎡ ik ⎛ ⎤ P ⎞ × exp⎢ ⎜1 − 1 ⎟ρs⋅ρd⎥ , (α = x, y ), ⎢⎣ z ⎝ ⎥⎦ P2 ⎠
(23)
PC Γxy (ρs , ρd , z )
=−
2 ⎡ 2 k ω02 ⎢⎛ P P2⎞ P ⎜P − 2P1 + 3 1 ⎟⎟ρ ρ − 3 16z ρ ρ 2 2 ⎢⎜ 2 P2 ⎠ dx dy P2 ω 4k 2 sx sy 16z P2 ⎣⎝ 0
PP⎞ i 4z ⎛ 8z 2 ρ ρ − ρsy ρdx + ⎜1 − 3 1 ⎟ρs⋅ρd 2 sx dy P2 ⎠ kω02 ⎝ ω04k ⎤ ⎡ ⎛ 2ρ2 ⎞ 2 2⎛ 2 3 ω k P2 ⎞ 8π Tz × exp⎜⎜ − 2 s ⎟⎟exp⎢ − 0 2 ⎜⎜1 + − 1 ⎟⎟ρd2⎥ 2 ⎢ P 3ω0 2⎠ ⎥ ⎦ ⎣ 8z ⎝ ⎝ ω 0 P2 ⎠ ⎡ ik ⎛ ⎤ P1 ⎞ exp⎢ ⎜1 − ⎟ρ ⋅ρ ⎥ , ⎢⎣ z ⎝ P2 ⎠ s d⎥⎦
(
+
with P1 = 1 −
(20)
and the DOC introduced by Wolf [48], i.e.,
↔ TrΓSC (ρ1, ρ2 , z )
PC Γαα (ρs , ρd , z )
)
(24)
* PC PC Γyx (ρs , ρd , z ) = ⎡⎣Γxy (ρs , ρd , z )⎤⎦ ,
↔ 4Det[ΓSC (ρ , ρ , z )] 1− , ⎡ ↔ ⎤2 ⎣TrΓSC (ρ , ρ , z )⎦
μ(ρ1, ρ2 , z ) =
355
4π 2Tz 3 3ω02
, P2 = 1 +
4z 2
(25) ⎛ ⎜
1
ω02k 2 ⎝ ω02
+
1
⎞ ⎟+
δg20 ⎠
8π 2Tz 3 3ω02
, P3 = 1 +
4z 2 ω04k 2
.
Based on the derived formula shown in Eqs. (23)–(25), the statistical properties such as AID, DOP and DOC, of the PCRP beam propagating through atmosphere can be determined.
. (21)
Tr and Det entering in Eqs. (19)–(21) stand the trace and the determinant of the BCP matrix. The quantity μ is connected to the visibility of the electromagnetic interference fringes in the far field formed by two points source located at ρ1 and ρ2 on the beam, and can be determined from the Young's experiment. It is evident from Eq. (21) that μ is independent on the anti-diagonal terms of the BCP matrix of stochastic electromagnetic beams. Before performing the numerical study of the statistical properties of the SCRP beam in atmosphere based on the derived formula, let us briefly review the expression for the BCP matrix of the PCRP beam in atmosphere [26–28]. According to [28], the BCP matrix of the PCRP beam in source plane reads as
⎤ ⎡ ⎛ r2 + r2 ⎞ (0) (r − r )2 ⎛ x1x2 x1y2 ⎞ 1 Γ^PC (r1, r2, 0) = 2 exp⎜⎜− 1 2 2 ⎟⎟exp⎢− 1 2 2 ⎥⎜ ⎟, ⎢⎣ 2δg 0 ⎥⎦⎝ y1x2 y1y2 ⎠ ω0 ω0 ⎠ ⎝
(22)
where ω0 is the beam waist size of fundamental Gaussian beam; δg0 is the spatial coherent length of the beam. It follows from Eq. (22) that the correlation function of the PCRP beam in the source plane is of Gaussian function and the beam profile is of a doughnut shape. When δg0 tends to infinity, Eq. (22) reduces to the BCP matrix of the completely coherent RP beam. In the paraxial region, the propagation of the elements of BCP matrix of the PCRP beam in atmospheric turbulence can also be treated by the extended Huygens–Fresnel integral formula shown in Eq. (6). Following the same procedure of the derivation of the BCP matrix of the SCRP beam, the elements of the BCP matrix of the PCRP beam in atmosphere take the form
3. AID, DOP and DOC of the SCRP beam in turbulent atmosphere: numerical examples In this section, as numerical examples, we investigate the evolution properties of the AID, DOP and DOC of the SCRP beam and the PCRP beam in turbulent atmosphere comparatively. Suppose that the statistics of atmospheric turbulence obeys Kolmogorov spectrum, and the spatial power spectrum Φn(κ ) is adopted by the von Karman spectrum [18]
(
)
Φn(κ ) = 0.033Cn2(κ 2 + κ 02)−11/6exp −κ 2/κm2 ,
(26)
Cn2
where is the structural constant of the refractive-index fluctuations of turbulence; κ0 = 2π /L 0 with L 0 being the outer scale of turbulence and κm = 5.92/l0 with l0 being the inner scale of turbulence. The values of L 0 and l0 are chosen to be 1 m and 0.01 m respectively throughout the paper for numerical calculation. On substituting Eq. (26) into Eq. (12) and integrating over κ , the parameter T yields T = 0.552Cn2. By applying Eqs. (13), (23) and (19), we obtain the AID of the SCRP beam and the PCRP beam propagation through atmosphere at distance z
ISC (ρ , z ) =
IPC (ρ , z ) =
⎤ ⎛ 2⎡ z2 z 2ρ2 ρ2 ⎞ ⎢1 − ⎥exp⎜⎜− ⎟⎟, + 2 2 2 2 2 2 4 Δ2 ⎢⎣ Δ2 k σ0 δ0 2Δ2 k δ0 σ0 ⎥⎦ ⎝ 2σ02Δ2 ⎠ ⎛ ⎛ 2ρ2 ⎞ P ⎞ 1 ⎜ 2z 2 4π 2Tz 3 + + 3 ρ2⎟⎟exp⎜⎜− 2 ⎟⎟, 3 P2 ⎠ ⎝ ω0 P2 ⎠
⎜ ω02P22 ⎝ k 2δg20
(27)
(28)
Fig. 1 illustrates the normalized AID of the SCRP beam at the cross line (ρy ¼ 0) in free space (T ¼0) for several propagation distances. The extension for the 3D-plot of the AID is straightforward
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Y. Cui et al. / Optics Communications 354 (2015) 353–361
Fig. 1. Average intensity distribution of the SCRP beam, the equivalent RP beam (ω0 ¼1.995 mm, δg0-1) and the equivalent PCRP beam (ω0 ¼ 2.261 mm, δg0 ¼ 3 mm) at the cross section (ρy ¼0) in free space (Cn2 = 0) for several propagation distances. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
because the intensity distribution at the beam cross-section is rotational symmetry. The other parameters used in the calculation are s0 ¼10 mm, δ0 ¼2 mm, λ ¼632.8 nm unless different values are specified. In order for comparing the evolution properties of the SCRP beam and PCRP beam in turbulent atmosphere, the equivalent condition between two beams has been applied in the numerical calculation (see Appendix A), i.e.,
1 1 1 1 + = + 2. ω02 δ0 2δg20 8σ02
(29)
The SCRP beam and the PCRP beam satisfying Eq. (29) are called the equivalent beams each other because the spread angle of them both in free space and in turbulent atmosphere keeps same [49]. It can be seen from Eq. (29) that there are different sets of ω0 and δg0 to satisfy Eq. (29) when the beam parameters of the SCRP beam is fixed. The normalized AID of the equivalent RP beam (blue line, for special case of the PCRP beam when δg0-1, ω0 ¼1.995 mm) and the equivalent PCRP beam (green line, ω0 ¼2.261 mm, δg0 ¼ 3 mm) in free space calculated from Eq. (28) are also plotted in Fig. 1. As shown in Fig. 1, the beam profile of the RP beam in free space remains unchanged and keeps doughnut shape during propagation, as expected. Due to its limited spatial coherence length of the equivalent PCRP beam, the beam profile is spatially variant on propagation, and turns from doughnut shape to “hilllike” shape with a little dip in the central region as the propagation
distance increases (see Fig. 1(b) and (c)). If the spatial coherent length deteriorates further, our numerical results show that the beam profile of the equivalent PCRP beam (ω0 ¼2.814 mm, δg0 ¼2 mm, not shown here) becomes a Gaussian profile at z¼0.5 km or z ¼5 km, while the evolution of beam profile of the SCRP beam is rather different due to its special correlation structure. It changes from Gaussian shape to doughnut shape gradually with the increase of propagation distance. When the propagation distance is sufficient large (z 45 km), the beam profiles of the SCRP beam and the equivalent RP beam are nearly the same. In the presence of atmospheric turbulence, the evolution properties of the normalized AID of three equivalent beams at several propagation distances are shown in Fig. 2. One finds that the effect of the turbulence is to degenerate the beam profile of three equivalent beams to a Gaussian-like shape, irrespective of the intensity distribution and the structure of the correlation function of the beam in the source plane. As a result, the beam profiles among three equivalent beams become almost same at the propagation distance z ¼10 km. This phenomenon can be explained by the fact that the SCRP beam propagating in atmosphere is affected by the free-space diffraction and the atmospheric turbulence together. The effect of the former factor is to turn the beam profile to doughnut shape due to the special coherence structure of the SCRP beam, while the latter factor is to degenerate the coherence structure of the SCRP beam to Gaussian profile (see Eq. (3)) and its effect accumulates on propagation.
Fig. 2. Average intensity distribution of the SCRP beam, the equivalent RP beam (ω0 ¼1.995 mm, δg0-1) and the equivalent PCRP beam (ω0 ¼2.261 mm, δg0 ¼3 mm) at the cross-line (ρy ¼ 0) in turbulent atmosphere (Cn2 = 10−13m−2/3) for several propagation distances.
Y. Cui et al. / Optics Communications 354 (2015) 353–361
After propagating a long distance, the effect of turbulence overpasses the effect of free-space diffraction. Thus, the AID of the SCRP beam will become a Gaussian shape after a long propagation distance in turbulence. To learn about the change of the beam shape of three equivalent beams propagating in free space or in turbulence, a parameter named dark hollow rate (DHR) is introduced to describe the difference of evolution properties of the AID among three beams, defined as
H (z ) = I (0, z )/[I (ρ , z )]max
(30)
The quantity H(z) is the ratio of the average intensity at the central point to the maximum intensity in the corresponding cross-section at the propagation distance z. When H(z)¼ 0 the beam profile is a standard dark hollow shape, no dip in the central region of the beam profile will be observed if H(z)¼1. Fig. 3 shows the dependence of the DHR of three equivalent beams on the propagation distance with different spatial coherence length and strength of the turbulence. In free space (Cn2 = 0), the DHR of the equivalent RP beam keeps the value 0 on propagation, which is consistent with the analysis in Fig. 1, implying that the beam profile is a standard doughnut shape with zero intensity at the onaxis point. For the case of the equivalent PCRP beam, the value of DHR increases dramatically as the propagation distance increases, and then tends to a stable value for further propagation. Notice that this stable value is closely related to the initial spatial coherence length of the equivalent PCRP beam. The increase of the initial spatial coherence length leads to the decrease of the stable value (see Fig. 3(a) and (d)). For the equivalent SCRP beam, the DHR in free space first keeps unit, and then starts to decrease as the propagation distance increases, at last remains a value almost unchanged when the propagation distance is long enough. The distance keeps the DHR of the equivalent SCRP beam unit and the constant value at the long propagation distance is also determined by the spatial coherence length δ0. The larger the spatial coherence length is, the longer the distance keeping unit is and the larger the constant value at the long propagation distance is. In the atmospheric turbulence, the DHR of the equivalent RP beam and the equivalent PCRP beam increases monotonously with the increase of the propagation distance, and the slope becomes large as the strength of the turbulence enhances, which implies that the
357
ability for maintaining the dark hollow shape fades in turbulence. The situation of the equivalent SCRP beam in turbulence is similar to that of the equivalent RP beam or PCRP beam, which is that the dark hollow profile appearing on propagation is also degenerated, i.e., The DHR of the SCRP beam increases on propagation when the propagation distance is long enough (see Fig. 3(c) and (f)). Note that the value of DHR of the equivalent SCRP beam (δ0 ¼5 mm) is always 1 in the strong turbulence, indicating that no dark hollow beam shape will be observed during propagation under that circumstance. Let us turn to study the behavior of the DOP of the SCRP beam and the equivalent RP or PCRP beam during propagation comparatively. By applying Eq. (20), the DOP of the SCRP beam and the PCRP beam in atmospheric turbulence take the following form
PSC (ρ , z ) =
PPC (ρ , z ) =
ρ2
2Δ22 k 2δ02σ04/z 2 − 2Δ2 σ02 + ρ2 ρ2 2
ρ +
ω02P2(P2
− P3)/2P3
,
,
(31)
(32)
It can be seen from Eqs. (31) and (32) that the distributions of the DOP of the SCRP beam and the PCRP beam across the source plane (z ¼0) are all uniform, while the value of the DOP equals 1 for the PCRP beam and 0 for the SCRP beam. We plot in Fig. 4 the variance of the DOP of the SCRP beam, the equivalent RP beam and the equivalent PCRP beam as a function of ρx(ρy ¼0) both in free space and in turbulence. As expected, the DOP of the equivalent RP beam keeps unit invariant across the transverse plane during propagation, while the DOP of the SCRP beam and the equivalent PCRP beam on propagation forms a dip pattern with the value of the onaxis point keeping zero. In turbulent atmosphere, the distributions of the DOP of the three beams on propagation are similar to each other, which all form the dip pattern. However, the width of the dip among them in the short propagation distance is different (see Fig. 4(d)–(e), (g)–(h)). With the increase of the propagation distance or enhancing the strength of the turbulence, the distribution of the DOP between the SCRP beam and the equivalent RP beam becomes almost same (see Fig. 4(f) and (i)). To trace the variance of the width of the formed dip on
Fig. 3. Variance of dark hollow rate of the SCRP beam, the equivalent RP beam and the equivalent PCRP beam with propagation distance in free space and turbulent atmosphere for two different values of spatial coherence length. The beam parameters in (a)–(c) are same with those in Fig. 2. The beam parameters in (d)–(f) are ω0 ¼4.924 mm, δg0-1 for equivalent RP beam and ω0 ¼ 5.250 mm, δg0 ¼10 mm for equivalent PCRP beam.
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Fig. 4. Evolution of the DOP of the SCRP beam, the equivalent RP beam and the equivalent PCRP beam in free space and turbulent atmosphere on propagation.
propagation, a parameter ΔP (z ) is introduced to quantity the width of the dip, given by
ΔP (z ) = (ρx2 )P (ρx2 , z )= 0.5 − (ρx1)P (ρx1, z )= 0.5 .
(33)
According to Eq. (33), the width of the dip of the DOP of the SCRP and the PCRP beam deduced from Eqs. (31) and (32) are
ΔPSC (z ) = 2σ0 2Δ22 k 2δ02σ02/z 2 − 2Δ2 ,
(34)
ΔPPC (z ) = 2ω0 P2(P2 − P3)/2P3 ,
(35)
Fig. 5 shows the dependence of ΔPSC (z ) of the SCRP beam on the propagation distance in turbulence for different values of spatial coherence length. The corresponding variance ΔPPC (z ) of the
equivalent RP beam (δg0-1) is also plotted in Fig. 5 (dashed line) for convenient comparison. The width of dip of the equivalent RP beam increases monotonously with the increase of propagation distance, while the dip width of the SCRP beam first decreases dramatically, reaches a minimum value, and then increases during propagation. It is worth noting that the difference of the width of dip becomes large in the far field (z ¼5 km) with the increase of the spatial coherence length of the SCRP beam (see Fig. 5(a)–(c)). This phenomenon can be explained by the fact that if the propagation distance is sufficient large, Eq. (34) and Eq. (35) could simplify to 4 2 π 2kTz 2δ0/3 and 4 2 π 2kTz 2ω0/3 respectively, to a good approximation. So, the dip width is proportional to δ0 for the SCRP beam and to ω0 for the equivalent RP beam. Under the equivalent condition, the SCRP beam with s0 ¼ 10 mm and δ0 ¼ 2 mm corresponds to the equivalent RP beam with
Fig. 5. Dependence of the dip width of the SCRP and the equivalent RP beam on the propagation distance for different spatial coherence length in atmosphere.
Y. Cui et al. / Optics Communications 354 (2015) 353–361
ω0 ¼1.995 mm. Therefore, the dip width between two equivalent beams is almost same in the far field (see Fig. 5(a)). However, the SCRP beam with s0 ¼10 mm and δ0 ¼20 mm corresponds to the equivalent RP beam with ω0 ¼16.3 mm. This leads to the difference of the width of dip between two beams in the far field (see Fig. 5(c)). Let us now explore the behavior of the DOC of the SCRP beam and the equivalent PCRP beam both in the free space and in the turbulent atmosphere. On substituting Eqs. (13) and (23) into Eq. (21), the expressions for the DOC of two beams at a pair of points −ρ and ρ are N1 − 2Δ12 ρ2/Δ22 δ02
μSC ( − ρ, ρ, z ) =
2
N1 + z 2ρ2/2Δ22 k σ04δ02 ⎡ ⎛ 2 2 2 ⎞ ⎤ 2k σ02 ⎛ Δ2 ⎞ 4π k Tz 1 ⎟ 2⎥ ⎜⎜1 − 1 ⎟⎟ − × exp⎢ −⎜⎜ + ρ ⎢ 3 Δ2 ⎠ 2Δ2 σ02 ⎟⎠ ⎥⎦ z2 ⎝ ⎣ ⎝
with N1 = 1 − z
2
(36)
/Δ2 k 2σ02δ02 .
μPC ( − ρ , ρ , z ) =
⎛ P2 − P3 − 2⎜P12 + ⎝
ω04k 2 4z 2
2⎞
(P2 − P1) ⎟⎠ρ2/P2ω02
359
dislocation [50] could be found in the DOC of the SCRP beam in the source plane, i.e., the trace of the zero value of the DOC forms a circle. The value of the DOC inner and outer of the ring is positive and negative, respectively. Thus, a π phase jump occurs in the inner and outer of the ring. When the two equivalent beams propagate from the source plane to the plane z4 0 in free space, two side lobes in the DOC of the equivalent PCRP beam can be observed clearly, whereas the distribution of the DOC of the SCRP beam gradually turns to Gaussian-like shape (see in Fig. 6(b) and (c)). In the turbulent atmosphere, both the distribution of the SCRP beam and the equivalent PCRP beam degenerates to Gaussian-like profile and the difference between them gets smaller with the increase of propagation distance (see Fig. 6(e) and (f)). This interesting result may be explained that in the far field, the turbulence effect plays the dominant role to determine the coherence properties of the PCRP beam or SCRP beam. Therefore, the patterns of DOC of two equivalent beams become similar when the propagation distance is long enough. It is worth noting that although the ring dislocation is hardly observed at z¼ 2 km (See Fig. 6(f)), in fact it exists. From Eqs. (36) and (37), if we set μ( − ρ, ρ, z ) = 0, the equation of the radius of the ring of two equivalent beams are
RSC (z ) = δ0 Δ22 N1/2Δ12 ,
P2 − P3 + 2P3ρ2 /P2ω02
(38)
(
2
)
RPC (z ) = ω0 P2(P2 − P3)/2 P12 + ω04k 2(P2 − P1) /4z 2 , ⎤ ⎡ ⎛ 2 2 P2 ⎞ ω 2k 2 ⎛ 4π k Tz 2 ⎞⎟ 2⎥ × exp⎢−⎜⎜ + 0 2 ⎜⎜1 − 1 ⎟⎟ − ρ ⎟ ⎢⎣ ⎝ 3 P2 ⎠ P2ω02 ⎠ ⎥⎦ 2z ⎝
(37)
It is obvious from Eqs. (36) and (37) that the DOC of two beams are rotational symmetry in the beam cross section. Fig. 6 gives the cross-line (ρy ¼0) of the DOC of the SCRP beam and the equivalent PCRP beam at two points ρx and ρx both in free space and in turbulent atmosphere at several propagation distances. One finds that the patterns of the DOC between two equivalent beams are much different in the source plane. The distribution of the DOC of the equivalent PCRP beam are of Gaussian shape in the source plane, while the distribution of the DOC of the SCRP beam has two side lobes besides the central peak. In the 3D-plot case, a ring
(39)
Fig. 7 illustrates the location of the ring in the DOC of two equivalent beams at z¼ 0, 1 km, 3 km in turbulent atmosphere (Fig. 7(a) and (c)) and the dependence of the radius of the ring on the propagation distance (Fig. 7(b) and (d)). The red, green and pink pair points represent the typical coherence singularities where the value of the DOC at the pair of point ( ρ, ρ) equals zero at z¼ 0, z ¼1 km and z ¼3 km, respectively. Note that no coherence singularities in the DOC of the equivalent PCRP beam could be found in the source plane because the distribution of DOC is a Gaussian function. Both of the radius of the ring RSC(z) and RPC(z) in the free space increase monotonously with the increase of propagation distance since the distribution of the DOC spreads in the transverse plane due to the diffraction
Fig. 6. Variance of DOC of the SCRP beam and the equivalent PCRP beam as a function of ρx(ρy ¼ 0) at a pair points ρx and ρx for three different propagation distances. The beam parameters of the SCRP beam and equivalent PCRP beam are s0 ¼ 2.814 mm, δ0 ¼2 mm and ω0 ¼ 2.814 mm, δg0 ¼ 2 mm, respectively.
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Fig. 7. (a,c) Circles where the ring dislocations locate for different propagation distances, the read, green and pink pair points are typical coherence singularities pairs at z¼ 0, 1 km and 3 km, respectively. (b,d) The evolution of the radius of ring against the propagation distance in free space and in atmosphere. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
effect of the beam on propagation. However, in the atmospheric turbulence, the evolution properties of the radius of the ring are rather different between two equivalent beams. For the case of RSC(z), it increases and drop dramatically near the critical propagation distance. The reason is that the coefficient Δ1 in Eq. (38) tends to zero at a certain propagation distance in turbulence, while this will not happen in free space. The critical propagation distance where RSC(z) tends to infinity in turbulence can be obtained by setting Δ1 = 0, which is zc = 3 3σ02/π 2T . The larger the strength of the turbulence is, the shorter the critical propagation is. For the case of RPC(z), it first increases as the propagation distance increases, reaches a maximum value, and then decrease with the increase of the propagation distance further due to the de-coherence effect of the turbulence, and no discontinuity occurs in RPC(z) on propagation through atmosphere.
4. Conclusion In summary, we have derived the analytical formulas for the AID, DOP and DOC of the SCRP beam propagating through turbulent atmosphere with the help of the extended Huygens–Fresnel integration. Based on the derived formulas, we present the numerical study of the evolution properties of the AID, DOP and DOC of the SCRP beam in atmospheric turbulence on propagation. The corresponding results of the equivalent PCRP beam are also illustrated for comparison. It is found that the AID of the SCRP beam degenerates to Gaussian-like profile in atmosphere for the propagation distance long enough, and the evolution properties are closely determined by the strength of turbulence and the spatial coherence length. The behavior of the DOP of the SCRP beam in turbulence is similar to that of the equivalent PCRP beam on propagation. The distributions of the DOP of two equivalent beams form the “Inverse-Gaussian” shape during propagation. More
interestingly, the evolutions of the ring dislocation in the DOC of the SCRP beam and the equivalent PCRP beam are much different in turbulent atmosphere. The ring radius in the DOC of the SCRP beam first increases dramatically with the increase of the propagation distance, tends to infinity at the critical propagations, and then decreases as the propagation distances increases, while no such critical propagation distance where the ring radius tends to infinity occurs in the DOC of the equivalent PCRP beam.
Acknowledgment This work is supported by the National Natural Science Foundation of China under Grant nos. 11474213, 11274005 and 11474143. The Project is Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars (To Dr. Cai), State Education Ministry.
Appendix A. : Derivation of the equivalent condition of the SCRP beam and the PCRP beam The variance of the mean-squared width of light beams with the propagation distance in turbulent atmosphere can be written as [51]
ρ2 (z ) =
∫ ρ2I (ρ, z )d2ρ , ∫ I (ρ, z )d2ρ
(A1)
where I (ρ, z ) denotes the average intensity distribution in the plane of z. The denominator in Eq. (A1) stands the total energy carried by the light beams and remain unchanged propagating
Y. Cui et al. / Optics Communications 354 (2015) 353–361
through the atmospheric turbulence. For the SCRP beam, On substituting from Eq. (27) into Eq. (A1), after tedious integration and operation, we obtain 2 ρSC (z ) = 2σ02 +
4 ⎞ 4π 2Tz 3 z2 ⎛ 1 ⎜⎜ . + 2 ⎟⎟ + 3 δ0 ⎠ k 2 ⎝ 2σ02
(A2)
The second term on the right side of Eq. (A2) denote the beam spreading induced by the diffractive effect in free space, while the third term represents the spreading due to atmospheric turbulence. From Eq. (A2), the angular spread of the SCRP beam in atmosphere turbulence reads as
θSC
(ρ = lim
2 (z ) SCRP
z →∞
z
1/2
)
=
1⎛ 1 4 ⎞ 4π 2Tz ⎜⎜ . + 2 ⎟⎟ + 3 k 2 ⎝ 2σ02 δ0 ⎠
(A3)
(A4)
Compared to Eq. (A2), one finds that the beam spreading induced by turbulence between two beams is same. From Eq. (A4), we obtain the expression for the angular spread of the PCRP beam in turbulence, i.e.,
θPC =
⎛ ⎞ 2 ⎜ 4 + 2 ⎟ 1 + 4π Tz . ⎜ 2 2 ⎟ 2 3 δg 0 ⎠ k ⎝ ω0
[10] [11] [12] [13] [14] [15] [16] [17] [18]
Let us turn to derive the angular spread of the PCRP beam. On substituting Eq. (28) into Eq. (A1), after integrating, the meansquared width of the PCRP beam in turbulence yields
⎛ ⎞ 4 2 z2 4Tπ 2z 3 2 . ρPC (z ) = ω02 + ⎜⎜ 2 + 2 ⎟⎟ 2 + 3 δg 0 ⎠ k ⎝ ω0
[3] [4] [5] [6] [7] [8] [9]
[19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
(A5) [37]
If sources generate the same angular spread both in free space and in turbulent atmosphere, such sources are called equivalent beams each other. Thus, the equivalent condition for the SCRP beam and PCRP beam deduced from Eqs. (A3) and (A5) is
1 1 1 1 + = + 2. ω02 δ0 2δg20 8σ02
(A6)
Reference [1] L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, Cambridge University press, Cambridge, 1995. [2] E. Wolf, Introduction to the Theory of Coherence and Polarization of Light, Cambridge University press, Cambridge, 2007.
[38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51]
361
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