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Propagation of radially polarized multi-cosine Gaussian Schell-model beams in non-Kolmogorov turbulence
Optics Communications 407 (2018) 392–397
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Propagation of radially polarized multi-cosine Gaussian Schell-model beams in non-Kolmogorov turbulence Miaomiao Tang a, *, Daomu Zhao b , Xinzhong Li a , Jingge Wang a a b
School of Physics and Engineering, Henan University of Science and Technology, Luoyang 471023, China Department of Physics, Zhejiang University, Hangzhou 310027, China
a r t i c l e
i n f o
Keywords: Radially polarized multi-cosine Gaussian Shell-model beams Propagation Non-Kolmogorov turbulence Statistical properties
a b s t r a c t Recently, we introduced a new class of radially polarized beams with multi-cosine Gaussian Schellmodel(MCGSM) correlation function based on the partially coherent theory (Tang et al., 2017). In this manuscript, we extend the work to study the statistical properties such as the spectral density, the degree of coherence, the degree of polarization, and the state of polarization of the beam propagating in isotropic turbulence with a non-Kolmogorov power spectrum. Analytical formulas for the cross-spectral density matrix elements of a radially polarized MCGSM beam in non-Kolmogorov turbulence are derived. Numerical results show that lattice-like intensity pattern of the beam, which keeps propagation-invariant in free space, is destroyed by the turbulence when it passes at sufficiently large distances from the source. It is also shown that the polarization properties are mainly affected by the source correlation functions, and change in the turbulent statistics plays a relatively small effect. In addition, the polarization state exhibits self-splitting property and each beamlet evolves into radially polarized structure upon propagation. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Over the past decades, there has been a growing interest in the evolution of partially coherent beams, either in free space or in turbulent atmosphere [1–6]. This interest is motivated by a multitude of potential applications, briefly to free-space optical communications, remote sensing and tracking [7,8]. The random fluctuations in the index of refraction of atmosphere cause spreading of the beam beyond that due to pure diffraction, beam wander, loss of spatial coherence, and random fluctuations in the irradiance and phase. These effects can seriously degrade the signal-to-noise ratio of an optical heterodyne receiver. Therefore, much work has been devoted to a reliable theory for predicting the propagation properties of light beams in turbulent medium. It is demonstrated that the atmosphere exhibits homogeneous and nearly isotropic under the atmospheric boundary layer, which is roughly 1– 2 km above the Earth’s surface. Therefore, the isotropic Kolmogorov power spectrum model of refractive index is generally correct within this inertial sub-range. However, in portions of the troposphere and stratosphere, theoretical and experimental results have shown that the Kolmogorov power spectrum does not properly describe the real turbulence behavior. Consequently, a variety of different power spectrum models and extended turbulence models have been proposed [9–12].
Among them, the most popular one is the non-Kolmogorov spectrum which is proposed by I. Toselli et al. [11]. It was assumed that instead of classic power law11∕3 the power spectrum has a generalized law, defined by parameter 𝛼, in the range 3 < 𝛼 < 5, as the one-dimensional fractal distribution stipulates. Since the atmosphere was shown to be layered in terms of the power spectra at different altitudes, many studies were carried out on the modeling of the non-Kolmogorov spectrum specifically for up/down/slant path propagation. Due to the constraint of non-negative definiteness of the spatial correlation functions, a Gaussian correlated Schell-model (GSM) beam has always been chosen as a typical example of partially coherent beams in the early years. Recently, a new sufficient condition for devising genuine correlation functions of light beams was established by Gori et al. [13,14], a variety of partially coherent beam models with special correlation functions have been proposed in succession. It is shown that these new classes of partially coherent beams exhibit many novel propagation properties. Such as non-uniformly correlated beams lead to self-focus and laterally shifted in their maximum intensity [15,16], multi-Gaussian Schell-model beams and sinc-Schell model beams acquire flat-top profile in the far-field [17,18], cosine-Gaussian Schell beams with circular symmetry possess dark-hollow profile [19].
* Corresponding author.
E-mail address: [email protected] (M. Tang). https://doi.org/10.1016/j.optcom.2017.09.067 Received 13 July 2017; Received in revised form 17 September 2017; Accepted 21 September 2017 0030-4018/© 2017 Elsevier B.V. All rights reserved.
M. Tang et al.
Optics Communications 407 (2018) 392–397
here 𝑘 = 2𝜋∕𝜆 is the wave number with 𝜆 being the wavelength of the light, 𝜓 denotes the complex phase perturbation due to the random medium, and ⟨⋅ ⋯ ⋅⟩𝑚 implies averaging over the ensemble of statistical realizations of the turbulent medium. For points located sufficiently close to the optical axis, the term in the sharp brackets with the subscript 𝑚 in Eq. (2) can be written as ⟨ [ ( ) ( )]⟩ exp 𝜓 ∗ 𝝆𝟏 , 𝝆′1 , 𝑧 + 𝜓 𝝆𝟐 , 𝝆′2 , 𝑧 𝑚 = { 2 2 )( ) ( ) ] ) ( 𝜋 𝑘 𝑧 [( exp − 𝝆𝟏 − 𝝆𝟐 2 + 𝝆𝟏 − 𝝆𝟐 𝝆′𝟏 − 𝝆′2 + 𝝆′𝟏 − 𝝆′𝟐 2 (3) 3 } ∞ × 𝜅 3 𝛷𝑛 (𝜅) 𝑑𝜅 , ∫0
Gaussian Schell-model array beams can radiate desirable lattice-like average intensity in far-field [20]. In addition, the propagation factors of several partially coherent beams have been investigated, one can see that beams with special correlation functions are less affected by turbulence than conventional GSM beams [21–23]. On the other hand, cylindrical vector (CV) beams with non-uniform state of polarization (e.g., radially polarized beams, azimuthally polarized beams) have attracted a wealth of attention due to their unique and interesting properties [24,25], and have widely applications in many research fields, for example, polarization information encryption, optical data storage, and optical tweezers [26]. Partially coherent radially polarized beams with Gaussian Schell-model correlations have been widely investigated both theoretically and experimentally [27–29]. In the past several years, generation and propagation of partially coherent radially polarized beams with peculiar correlations have become a hot topic [30,31]. It has been revealed that under the influence of non-conventional coherence properties, these novel beams exhibit distinctive propagating characteristics. Quite recently, we introduced a new class of partially coherent radially polarized beam with multi-cosine Gaussian Schell-model (MCGSM) correlations, termed as radially polarized MCGSM beams [32]. It is shown that the statistical properties of radially polarized MCGSM beams in the far field can be flexible modulated by varying the source coherence parameters. Moreover, unlike deterministic arrays, once the pattern is formed in the far field it remains structurally invariant on further propagation. Such feature makes the beam of particular importance for certain applications in which a far field with tunable lattice structure must be formed, such as optical trapping, material processing, and free-space and atmospheric optical communications. In this manuscript, we explore the behavior of the statistical properties for radially polarized MGSM beams propagating in atmospheric turbulence by using a non-Kolmogorov power spectrum. The impacts arising from the source correlation functions and the turbulence parameters on the spectral density, the spectral degree of coherence and the polarization properties are emphasized.
where 𝛷𝑛 (𝜅) is the one-dimensional spatial power spectrum of the refractive-index fluctuations of random medium, 𝜅 being spatial frequency. For the non-Kolmogorov case, the spatial power spectrum of the refractive index fluctuations of the turbulent atmosphere is known to have form [33] [ ] exp −𝜅 2 ∕𝜅𝑚2 2 ̃ 𝛷𝑛 (𝜅) = 𝐴 (𝛼) 𝐶𝑛 ( , 0 ≤ 𝜅 < ∞, (4) ) 𝜅 2 + 𝜅02 𝛼∕2 where 3 < 𝛼 < 4 and the term 𝐶̃𝑛2 is a generalized refractive-index structure parameter with units m3−𝛼 , 𝑐(𝛼) 2𝜋 , 𝜅𝑚 = 𝐿𝑜 𝑙𝑜 1 [ ( ) ] 𝛼−5 2𝜋 𝛼 𝑐 (𝛼) = 𝛤 5− 𝐴 (𝛼) , 3 2 ( ) 1 𝛼𝜋 , 𝐴 (𝛼) = 𝛤 (𝛼 − 1) cos 2 4𝜋 2 𝜅0 =
The elements of the cross-spectral density (CSD) matrix of a radially polarized MCGSM beam in the source plane are described as [32] ( ′2 [ ( ′ ) ] ′ ) 𝝆1 + 𝝆22 𝑥1 − 𝑥′2 2 ) 𝛼1′ 𝛽2′ (0) ( ′ ′ exp − exp − 𝑊𝛼𝛽 𝝆1 , 𝝆2 , 0 = 𝑁2 4𝜎 2 2𝛿02 [ ( ′ ] ) ] [ (𝑁−1)∕2 ′ 2 ∑ 𝑦 − 𝑦2 ) 2𝜋𝑛 ( ′ × exp − 1 × 𝑥1 − 𝑥′2 cos 𝛿0 2𝛿 2 𝑛=−(𝑁−1)∕2 0
×
𝑛=−(𝑁−1)∕2
[
cos
] ) 2𝜋𝑛 ( ′ 𝑦1 − 𝑦′2 , 𝛿0
(6) (7)
𝐿0 and 𝑙0 are the outer and inner scales of turbulence, respectively, and 𝛤 (⋅) is the Gamma function. With the power spectrum in Eq. (4), the integral in Eq. (3) becomes ( 2) ∞ 𝜅0 𝐴 (𝛼) ̃2 2−𝛼 𝐼= 𝜅 3 𝛷𝑛 (𝜅) 𝑑𝜅 = 𝐶𝑛 𝜅𝑚 𝛽 exp ∫0 2 (𝛼 − 2) 𝜅𝑚2 ( ) 2 𝛼 𝜅 × 𝛤 2 − , 0 − 2𝜅04−𝛼 , (8) 2 𝜅𝑚2
2. Analytic solutions for radially polarized MCGSM beams in nonKolmogorov turbulence
(𝑁−1)∕2 ∑
(5)
where 𝛽 = 2𝜅02 − 2𝜅𝑚2 + 𝛼𝜅𝑚2 and 𝛤 (⋅, ⋅) denotes the incomplete Gamma function. Substituting Eqs. (1) and (3) into Eq. (2) and calculating the resulting integral we arrive at the formulas ( ) ( ) 𝑊𝑥𝑥 𝝆1 , 𝝆2 , 𝑧 = 𝛤 𝝆1 , 𝝆2 ( 2 [ ( 2 2 ) 2 )] (𝑁−1)∕2 𝛾𝑦1+ 𝛺𝑦22 𝛾𝑦1− 𝛺𝑦21 ∑ + + exp + × exp 4𝑀 4𝛱 4𝑀 4𝛱 𝑛=−(𝑁−1)∕2 {( ) ( 2 ) (𝑁−1)∕2 2 2 (9) 𝛾𝑥1+ 𝛺𝑥22 𝛥𝛺𝑥22 ∑ × 𝛥 + 𝛾𝑥1+ 𝛺𝑥22 + exp + 2𝛱 4𝑀 4𝛱 𝑛=−(𝑁−1)∕2 ( ) ( 2 )} 2 2 𝛥𝛺𝑥21 𝛾𝑥12 𝛺𝑥21 + 𝛥 + 𝛾𝑥1− 𝛺𝑥21 + exp + , 2𝛱 4𝑀 4𝛱
(1)
where (𝛼, 𝛽 = 𝑥, 𝑦), 𝝆′1 and 𝝆′2 are two-dimensional position vectors in the source plane, 𝜎 denotes its source width , 𝛿0 represents spatial coherence width, and 𝑁 is the positive integer. When 𝑁 = 1, the radially polarized MCGSM beam reduces to a conventional radially polarized Gaussian Schell-model beam. The realizability conditions for a radially polarized MCGSM source and corresponding beam conditions are derived in our recent work [32]. The paraxial form of the extended Huygens–Fresnel principle which describes the interaction of waves with random implies ( medium ) ( the) elements of the CSD matrix at two points 𝐫𝟏 = 𝝆𝟏 , 𝑧 and 𝐫2 = 𝝆2 , 𝑧 in the same transverse plane of the half-space 𝑧 > 0 are related to those in the source plane as [33] ) ( ) ( 𝑘 )2 (0) ( ′ 𝑊𝛼𝛽 𝝆1 , 𝝆′2 , 0 𝑊𝛼𝛽 𝝆𝟏 , 𝝆2 , 𝑧 = ⨌ 2𝜋𝑧 [ ( ) ( )] 𝝆𝟏 − 𝝆′1 2 − 𝝆2 − 𝝆′2 × exp −𝑖𝑘 2𝑧 ⟨ [ ∗( ) ( )]⟩ ′ × exp 𝜓 𝝆𝟏 , 𝝆1 , 𝑧 + 𝜓 𝝆𝟐 , 𝝆′2 , 𝑧 𝑚 d2 𝝆′1 d2 𝝆′2 , (2)
( ) ( ) 𝑊𝑦𝑦 𝝆1 , 𝝆2 , 𝑧 = 𝛤 𝝆1 , 𝝆2 , [ ( ) ( 2 )] (𝑁−1)∕2 2 2 2 𝛾𝑥1+ 𝛺𝑥22 𝛾𝑥1− 𝛺𝑥21 ∑ × exp + + exp + 4𝑀 4𝛱 4𝑀 4𝛱 𝑛=−(𝑁−1)∕2 {( ) ( 2 2 2 ) (𝑁−1)∕2 𝛥𝛺𝑦22 𝛾𝑦1+ 𝛺𝑦22 ∑ × 𝛥 + 𝛾𝑦1+ 𝛺𝑦22 + exp + 2𝛱 4𝑀 4𝛱 𝑛=−(𝑁−1)∕2 ( ( 2 2 ) 2 )} 𝛥𝛺𝑦21 𝛾𝑦12 𝛺𝑦21 exp , + 𝛥 + 𝛾𝑦1− 𝛺𝑦21 + + 2𝛱 4𝑀 4𝛱 393
(10)
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Optics Communications 407 (2018) 392–397
Fig. 1. Normalized spectral intensity of a radially polarized MCGSM beam with 𝑁 = 5 at several different propagation distances in non-Kolmogorov turbulence.
( ) ( ) 𝑊𝑥𝑦 𝝆1 , 𝝆2 , 𝑧 = 𝛤 𝝆1 , 𝝆2 ( 2 ( 2 [ 2 ) 2 )] (𝑁−1)∕2 𝛾𝑦1+ 𝛺𝑦22 𝛾𝑦1− 𝛺𝑦21 ∑ + + 𝛺𝑦22 × exp + 𝛺𝑦21 exp × 4𝑀 4𝛱 4𝑀 4𝛱 𝑛=−(𝑁−1)∕2 {( ( 2 ) ) 2 (𝑁−1)∕2 𝛾𝑦1+ 𝛺𝑦22 ∑ 𝛥𝛺𝑥22 × 𝛾𝑥1+ + exp + 2𝛱 4𝑀 4𝛱 𝑛=−(𝑁−1)∕2 ( 2 )} ( ) 2 𝛾𝑦12 𝛺𝑦21 𝛥𝛺𝑥21 + 𝛾𝑥1− + exp + , (11) 2𝛱 4𝑀 4𝛱 ( ) ( ) ∗ 𝝆1 , 𝝆2 , 𝑧 , 𝑊𝑦𝑥 𝝆1 , 𝝆2 , 𝑧 = 𝑊𝑥𝑦
Fig. 2. Normalized spectral intensity of a radially polarized MCGSM beam at propagation distance 𝑧 = 1 km for different values of 𝑁.
(12)
with ( ) 𝜇 𝝆1 , 𝝆2 , 𝑧 = √
( ) ( ) 𝑖𝑘𝑥2 𝑖𝑘𝑥1 − 𝐼 𝑥1 − 𝑥2 , 𝜍𝑥2 = − 𝐼 𝑥1 − 𝑥2 , 𝜍𝑥1 = 𝑧 𝑧 ( ) ( ) 𝑖𝑘𝑦1 𝑖𝑘𝑦2 2𝜋𝑛 , 𝜍𝑦1 = − 𝐼 𝑦1 − 𝑦2 , 𝜍𝑦2 = − 𝐼 𝑦1 − 𝑦2 , 𝑎= 𝛿 𝑧 𝑧
, ( ) ↔( ) Tr𝑊 𝝆1 , 𝝆1 , 𝑧 Tr𝑊 𝝆2 , 𝝆2 , 𝑧 [ [ ] ] 2𝑅𝑒 𝑊𝑥𝑦 (𝝆, 𝑧) 1 , 𝜃 (𝝆, 𝑧) = arc tan 2 𝑊𝑥𝑥 (𝝆, 𝑧) − 𝑊𝑦𝑦 (𝝆, 𝑧)
𝛾𝑥1± = 𝜍𝑥1 ± 𝑖𝑎, 𝛾𝑥2± = 𝜍𝑥2 ± 𝑖𝑎,
𝛥𝛾 𝛺𝑥22 = 𝑥1− − 𝛾𝑥2− , 2𝑀 𝛥𝛾𝑦1− 𝛺𝑦22 = − 𝛾𝑦2− , 2𝑀3
1 1 𝑖𝑘 + + + 𝐼, 4𝜎 2 2𝛿 2 2𝑧 0
3. Numerical examples In this section, we will now illustrate the second-order statistical properties of a radially polarized MCGSM beam propagating through non-Kolmogorov turbulence by a set of numerical examples. Unless specified in captions, the source and the medium parameters are chosen as follows: 𝜆 = 632 nm, 𝜎 = 1 cm, 𝛿0 = 0.5 cm, 𝑁 = 5, 𝐿0 = 1 m, 𝑙0 = 1 mm, 𝛼 = 3.1 and 𝐶̃𝑛2 = 10−13 m3−𝛼 . In Fig. 1 the evolution of the normalized spectral intensity of a radially polarized MCGSM beam with 𝑁 = 5 on propagation in nonKolmogorov turbulence is shown at several selected distances. As is seen from Figs. 1(a)–1(c), the beam profile gradually transforms from darkhollow at the initial plane to a 5 × 5 Gaussian arrays pattern at a certain distance. However, unlike in the case of free space propagation, where the lattice pattern remains for any large distance [32], in the atmospheric turbulence, the lattice pattern is gradually destroyed along with further propagation. What is more, after passing through the turbulence at sufficiently large distances, the beam shape eventually converges into Gaussian profile. This phenomenon can be explained by
𝛱=
Based on the elements of the CSD matrix, the spectral density, the spectral degree of coherence, the spectral degree of polarization, and the orientation angle of the polarization ellipse of such beam on propagation can be calculated by the expressions [34,35] ↔
𝑆 (𝝆, 𝑧) = Tr𝑊 (𝝆, 𝝆, 𝑧) , √ [↔ ] √ √ √ 4Det 𝑊 𝝆, 𝑧) (𝝆, √ √ 𝑃 (𝝆, 𝑧) = √1 − , ↔ Tr𝑊 (𝝆, 𝝆, 𝑧)
(17)
(13)
1 1 𝑖𝑘 + − + 𝐼, 4𝜎 2 2𝛿 2 2𝑧 0 [ ] ( ) −𝑖𝑘 𝝆21 − 𝝆22 ( ) ( ) 𝑘2 𝛤 𝝆1 , 𝝆2 = exp − 𝐼 𝝆1 − 𝝆2 . 2𝑧 64𝑧2 4𝜎 2 𝑀 2 𝛱 2 𝑀=
(16)
↔
where Det and Tr stand for determinant and trace of the matrix, respectively.
𝛾𝑦1± = 𝜍𝑦1 ± 𝑖𝑎, 𝛾𝑦2± = 𝜍𝑦2 ± 𝑖𝑎, 𝛥𝛾𝑥1+ 𝛺𝑥21 = − 𝛾𝑥2+ , 2𝑀 𝛥𝛾𝑦1+ − 𝛾𝑦2+ , 𝛺𝑦21 = 2𝑀3
↔ ( ) Tr𝑊 𝝆1 , 𝝆2 , 𝑧
(14)
(15)
394
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Optics Communications 407 (2018) 392–397
Fig. 3. Spectral degree of polarization and the corresponding cross line of a radially polarized MCGSM beam at propagation distance 𝑧 = 1 km for different values of 𝑁.
Fig. 4. Spectral degree of polarization of a radially polarized MCGSM beam with 𝑁 = 5 at several different propagation distances, (a) 𝑧 = 0; (b) 𝑧 = 0.5 km; (c) 𝑧 = 1 km; (d) 𝑧 = 4.5 km.
Fig. 5. Modulus of the spectral of coherence of a radially polarized MCGSM beam with 𝑁 = 5 at several different propagation distances, (a) 𝑧 = 0.2 km; (b) 𝑧 = 0.5 km; (c) 𝑧 = 1 km; (d) 𝑧 = 8 km.
the fact that the evolution behavior of the beam is determined by the source correlations and atmospheric turbulence. At small propagation distance, the beam is mostly modified by the source correlations, as the beam passes to sufficiently large distance through turbulence, the effect of turbulence plays a dominate role for determining the beam profile. Fig. 2 displays the intensity distribution at the propagation distance 𝑧 = 1 km for 𝑁 = 3 and 𝑁 = 6, respectively. One sees that the periodicity of the array intensity can be adjusted by the choice of the summation index 𝑁. Fig. 3 reveals the behavior of the spectral degree of polarization of such beam at the propagation distance 𝑧 = 1 km for different values of 𝑁 in 2D and 3D versions. One finds that the polarization contours exhibit periodic oscillations near the axis, and the periodicity of the lattice-like distribution is completely depended on the index 𝑁. Fig. 4 exhibits the evolution of the degree of the polarization at several propagation distances. It is clearly seen that the oscillations on the polarization distribution persevere for any large distance, and gradually suppressed due to the uniformly correlated turbulence. This implies that the behavior of the polarization distribution of a radially polarized
Fig. 6. The influences of the turbulence parameter 𝐶̃𝑛2 on the properties of a radially polarized MCGSM beam with 𝑁 = 5 in the plane 𝑧 = 1 km, (a) Degree of polarization (b) Modulus of the spectral degree of coherence.
beam is mostly determined by the source correlations, rather than by the medium parameters. Fig. 5 shows the modulus of the spectral degree of coherence of a radially polarized MCGSM beam with 𝑁 = 5 at different propagation distances. One sees that at relatively small distances, the coherence 395
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Optics Communications 407 (2018) 392–397
Fig. 7. The orientation angle of a radially polarized MCGSM beam in the source plane 𝑧 = 0 for different values of 𝑁.
Fig. 8. As in Fig. 7 but for 𝑧 = 1 km.
distribution is mainly modulated by multi cosine function Gaussian distribution, However, as the propagation distance grows, the oscillatory trend on the coherence curves is suppressed by medium fluctuations. Moreover, after passing through the turbulence at sufficiently large distances, the turbulence modifies the coherence curve to Gaussian distribution while narrowing the degree of coherence. Fig. 6(a) exhibits the profiles of the degree of polarization for selected turbulence parameters. It turns out that, as the atmospheric fluctuations become more severe (𝐶̃𝑛2 grows), the oscillations on the polarization curve become less evident. It can also be seen form Fig. 6(b) that the degree of coherence is significantly affected by the strength of the atmosphere turbulence, the larger the value of 𝐶̃𝑛2 is, the more efficiently the coherence profile approaches to a Gaussian distribution. Finally, to learn more about the polarization properties, we calculate in Figs. 7 and 8 the orientation angle of a radially polarized MCGSM beam for different values of 𝑁 in source plane and in turbulence propagation, respectively. It turns out that the state of polarization with different values of 𝑁 all present radial polarization in the source plane as expected. While after passing through the atmospheric turbulence as shown in Fig. 8, the radial polarization structure is destroyed due to the non-conventional source correlation function, and the state of polarization displays a more complex structure, similarly to its free space behavior [32]. This means that the state of polarization is greatly affected by the source correlation, and change in the turbulent statistics plays a relatively small effect. It is also interesting to note that the although the overall state of polarization is broken, the state of polarization of each beamlet evolves into a radially polarization structure on propagation.
state exhibits self-splitting property and each beamlet evolves into radially polarized structure upon propagation. Moreover the degree of coherence first broadens due to the source correlations and then narrows down, when the effect of turbulence starts to dominate the effect of source coherence. As we all known, spatially periodic arrays of light beams are of particular interest for applications in many fields. Moreover, vector partially coherent arrays have significant advantages in comparison with conventional scalar array beams. Our results obtained in this work is helpful in understanding the correlation modulation theory, and may be useful in optical trapping, high-resolution microscopy, and polarization communications. Acknowledgment This work is supported by the National Natural Science Foundation of China under Grant Nos. 11704098, 11504091, and 61775052. References [1] F. Gori, Opt. Commun. 31 (1983) 4. [2] L.C. Andrews, R.L. Phillips, Laser Beam Propagation through Random Media, Bellingham, 1998. [3] J. Wu, J. Modern Opt. 37 (1990) 671. [4] T. Shirai, A. Dogariu, E. Wolf, J. Opt. Soc. Am. A 20 (2003) 1094. [5] G. Wu, H. Guo, S. Yu, B. Luo, Opt. Lett. 35 (2010) 715. [6] T. Shirai, A. Dogariu, E. Wolf, Opt. Lett. 28 (2003) 610. [7] A. Dogariu, S. Amarande, Opt. Lett 28 (2003) 10. [8] J. Ricklin, F. Davidson, J. Opt. Soc. Am. A 20 (2003) 856. [9] A.I. Kon, Waves Random Complex Media 4 (1994) 297. [10] M. Charnotskii, J. Opt. Soc. Am. A 29 (2012) 711. [11] I. Toselli, B. Agrawal, S. Restaino, J. Opt. Soc. Am. A 28 (2011) 483. [12] I. Toselli, J. Opt. Soc. Am. A 31 (2014) 1868. [13] F. Gori, M. Santarsiero, Opt. Lett. 32 (2007) 3531. [14] F. Gori, M. Santarsiero, R. Borghi, V.R. Sanchez, J. Opt. Soc. Am. A 25 (2008) 1016. [15] H. Lajunen, T. Saastamoinen, Opt. Lett. 36 (2011) 4104. [16] Z. Tong, O. Korotkova, J. Opt. Soc. Am. A 29 (2012) 2154. [17] S. Sahin, O. Korotkova, Opt. Lett. 37 (2012) 2970. [18] Z. Mei, Opt. Lett. 39 (2014) 4188. [19] Z. Mei, O. Korotkova, Opt. Lett. 38 (2013) 2578. [20] Z. Mei, D. Zhao, O. Korotkova, Y. Mao, Opt. Lett. 40 (2015) 5662. [21] H. Xu, Z. Zhang, J. Qu, W. Huang, Opt. Express 22 (2014) 22479. [22] X. Wang, M. Yao, Z. Qiu, X. Yi, Z. Liu, Opt. Express 23 (2015) 12508. [23] Z. Song, Z. Liu, K. Zhou, Q. Sun, S. Liu, Opt. Express 24 (2016) 1804. [24] K.S. Youngworth, T.G. Brown, Opt. Express 7 (2000) 77. [25] R. Dorn, S. Quabis, G. Leuchs, Phys. Rev. Lett. 91 (2003) 233901.
4. Concluding remarks In this manuscript, we have investigated the statistical properties of radially polarized MCSM beams in atmospheric turbulence with a non-Kolmogorov power law. The analytical propagation formulas for the second-order characteristics have been derived based on the extended Huygens–Fresnel principle. Via numerical examples we have illustrated the fact that the lattice-like intensity patterns of the beam on propagation in free space is destroyed when it passes at sufficiently large distances form the source through the non-Kolmogorov turbulence. It is also shown that the oscillatory trend of the degree of polarization is gradually suppressed by medium fluctuation. Besides, the polarization 396
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[26] P. Wróbel, J. Pniewski, T. Antosiewicz, T. Szoplik, Phys. Rev. Lett. 102 (2009) 183902. [27] F. Wang, Y. Cai, Y. Dong, O. Korotkova, Appl. Phys. Lett. 100 (2012) 051108. [28] G. Wu, F. Wang, Y. Cai, Opt. Express 20 (2012) 28301. [29] S. Zhu, X. Zhu, L. Liu, F. Wang, Y. Cai, Opt. Express 21 (2013) 27682. [30] S. Zhu, Y. Chen, J. Wang, H. Wang, Z. Li, Y. Cai, Opt. Express 23 (2015) 33099.
397
S. Zhu, F. Wang, Z. Li, Appl. Phys. Lett. 109 (2016) 161904. M. Tang, D. Zhao, . L, . Li, . H, . Li, Opt. Commun. 396 (2017) 249. E. Shchepakina, O. Korotkova, Opt. Express 18 (2010) 10650. E. Wolf, Phys. Lett. A 32 (2003) 263. E. Wolf, Introduction to the Theories of Coherence and Polarization of Light, Cambridge University, 2007.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Propagation of radially polarized multi-cosine Gaussian Schell-model beams in non-Kolmogorov turbulence Miaomiao Tang a, *, Daomu Zhao b , Xinzhong Li a , Jingge Wang a a b
School of Physics and Engineering, Henan University of Science and Technology, Luoyang 471023, China Department of Physics, Zhejiang University, Hangzhou 310027, China
a r t i c l e
i n f o
Keywords: Radially polarized multi-cosine Gaussian Shell-model beams Propagation Non-Kolmogorov turbulence Statistical properties
a b s t r a c t Recently, we introduced a new class of radially polarized beams with multi-cosine Gaussian Schellmodel(MCGSM) correlation function based on the partially coherent theory (Tang et al., 2017). In this manuscript, we extend the work to study the statistical properties such as the spectral density, the degree of coherence, the degree of polarization, and the state of polarization of the beam propagating in isotropic turbulence with a non-Kolmogorov power spectrum. Analytical formulas for the cross-spectral density matrix elements of a radially polarized MCGSM beam in non-Kolmogorov turbulence are derived. Numerical results show that lattice-like intensity pattern of the beam, which keeps propagation-invariant in free space, is destroyed by the turbulence when it passes at sufficiently large distances from the source. It is also shown that the polarization properties are mainly affected by the source correlation functions, and change in the turbulent statistics plays a relatively small effect. In addition, the polarization state exhibits self-splitting property and each beamlet evolves into radially polarized structure upon propagation. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Over the past decades, there has been a growing interest in the evolution of partially coherent beams, either in free space or in turbulent atmosphere [1–6]. This interest is motivated by a multitude of potential applications, briefly to free-space optical communications, remote sensing and tracking [7,8]. The random fluctuations in the index of refraction of atmosphere cause spreading of the beam beyond that due to pure diffraction, beam wander, loss of spatial coherence, and random fluctuations in the irradiance and phase. These effects can seriously degrade the signal-to-noise ratio of an optical heterodyne receiver. Therefore, much work has been devoted to a reliable theory for predicting the propagation properties of light beams in turbulent medium. It is demonstrated that the atmosphere exhibits homogeneous and nearly isotropic under the atmospheric boundary layer, which is roughly 1– 2 km above the Earth’s surface. Therefore, the isotropic Kolmogorov power spectrum model of refractive index is generally correct within this inertial sub-range. However, in portions of the troposphere and stratosphere, theoretical and experimental results have shown that the Kolmogorov power spectrum does not properly describe the real turbulence behavior. Consequently, a variety of different power spectrum models and extended turbulence models have been proposed [9–12].
Among them, the most popular one is the non-Kolmogorov spectrum which is proposed by I. Toselli et al. [11]. It was assumed that instead of classic power law11∕3 the power spectrum has a generalized law, defined by parameter 𝛼, in the range 3 < 𝛼 < 5, as the one-dimensional fractal distribution stipulates. Since the atmosphere was shown to be layered in terms of the power spectra at different altitudes, many studies were carried out on the modeling of the non-Kolmogorov spectrum specifically for up/down/slant path propagation. Due to the constraint of non-negative definiteness of the spatial correlation functions, a Gaussian correlated Schell-model (GSM) beam has always been chosen as a typical example of partially coherent beams in the early years. Recently, a new sufficient condition for devising genuine correlation functions of light beams was established by Gori et al. [13,14], a variety of partially coherent beam models with special correlation functions have been proposed in succession. It is shown that these new classes of partially coherent beams exhibit many novel propagation properties. Such as non-uniformly correlated beams lead to self-focus and laterally shifted in their maximum intensity [15,16], multi-Gaussian Schell-model beams and sinc-Schell model beams acquire flat-top profile in the far-field [17,18], cosine-Gaussian Schell beams with circular symmetry possess dark-hollow profile [19].
* Corresponding author.
E-mail address: [email protected] (M. Tang). https://doi.org/10.1016/j.optcom.2017.09.067 Received 13 July 2017; Received in revised form 17 September 2017; Accepted 21 September 2017 0030-4018/© 2017 Elsevier B.V. All rights reserved.
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Optics Communications 407 (2018) 392–397
here 𝑘 = 2𝜋∕𝜆 is the wave number with 𝜆 being the wavelength of the light, 𝜓 denotes the complex phase perturbation due to the random medium, and ⟨⋅ ⋯ ⋅⟩𝑚 implies averaging over the ensemble of statistical realizations of the turbulent medium. For points located sufficiently close to the optical axis, the term in the sharp brackets with the subscript 𝑚 in Eq. (2) can be written as ⟨ [ ( ) ( )]⟩ exp 𝜓 ∗ 𝝆𝟏 , 𝝆′1 , 𝑧 + 𝜓 𝝆𝟐 , 𝝆′2 , 𝑧 𝑚 = { 2 2 )( ) ( ) ] ) ( 𝜋 𝑘 𝑧 [( exp − 𝝆𝟏 − 𝝆𝟐 2 + 𝝆𝟏 − 𝝆𝟐 𝝆′𝟏 − 𝝆′2 + 𝝆′𝟏 − 𝝆′𝟐 2 (3) 3 } ∞ × 𝜅 3 𝛷𝑛 (𝜅) 𝑑𝜅 , ∫0
Gaussian Schell-model array beams can radiate desirable lattice-like average intensity in far-field [20]. In addition, the propagation factors of several partially coherent beams have been investigated, one can see that beams with special correlation functions are less affected by turbulence than conventional GSM beams [21–23]. On the other hand, cylindrical vector (CV) beams with non-uniform state of polarization (e.g., radially polarized beams, azimuthally polarized beams) have attracted a wealth of attention due to their unique and interesting properties [24,25], and have widely applications in many research fields, for example, polarization information encryption, optical data storage, and optical tweezers [26]. Partially coherent radially polarized beams with Gaussian Schell-model correlations have been widely investigated both theoretically and experimentally [27–29]. In the past several years, generation and propagation of partially coherent radially polarized beams with peculiar correlations have become a hot topic [30,31]. It has been revealed that under the influence of non-conventional coherence properties, these novel beams exhibit distinctive propagating characteristics. Quite recently, we introduced a new class of partially coherent radially polarized beam with multi-cosine Gaussian Schell-model (MCGSM) correlations, termed as radially polarized MCGSM beams [32]. It is shown that the statistical properties of radially polarized MCGSM beams in the far field can be flexible modulated by varying the source coherence parameters. Moreover, unlike deterministic arrays, once the pattern is formed in the far field it remains structurally invariant on further propagation. Such feature makes the beam of particular importance for certain applications in which a far field with tunable lattice structure must be formed, such as optical trapping, material processing, and free-space and atmospheric optical communications. In this manuscript, we explore the behavior of the statistical properties for radially polarized MGSM beams propagating in atmospheric turbulence by using a non-Kolmogorov power spectrum. The impacts arising from the source correlation functions and the turbulence parameters on the spectral density, the spectral degree of coherence and the polarization properties are emphasized.
where 𝛷𝑛 (𝜅) is the one-dimensional spatial power spectrum of the refractive-index fluctuations of random medium, 𝜅 being spatial frequency. For the non-Kolmogorov case, the spatial power spectrum of the refractive index fluctuations of the turbulent atmosphere is known to have form [33] [ ] exp −𝜅 2 ∕𝜅𝑚2 2 ̃ 𝛷𝑛 (𝜅) = 𝐴 (𝛼) 𝐶𝑛 ( , 0 ≤ 𝜅 < ∞, (4) ) 𝜅 2 + 𝜅02 𝛼∕2 where 3 < 𝛼 < 4 and the term 𝐶̃𝑛2 is a generalized refractive-index structure parameter with units m3−𝛼 , 𝑐(𝛼) 2𝜋 , 𝜅𝑚 = 𝐿𝑜 𝑙𝑜 1 [ ( ) ] 𝛼−5 2𝜋 𝛼 𝑐 (𝛼) = 𝛤 5− 𝐴 (𝛼) , 3 2 ( ) 1 𝛼𝜋 , 𝐴 (𝛼) = 𝛤 (𝛼 − 1) cos 2 4𝜋 2 𝜅0 =
The elements of the cross-spectral density (CSD) matrix of a radially polarized MCGSM beam in the source plane are described as [32] ( ′2 [ ( ′ ) ] ′ ) 𝝆1 + 𝝆22 𝑥1 − 𝑥′2 2 ) 𝛼1′ 𝛽2′ (0) ( ′ ′ exp − exp − 𝑊𝛼𝛽 𝝆1 , 𝝆2 , 0 = 𝑁2 4𝜎 2 2𝛿02 [ ( ′ ] ) ] [ (𝑁−1)∕2 ′ 2 ∑ 𝑦 − 𝑦2 ) 2𝜋𝑛 ( ′ × exp − 1 × 𝑥1 − 𝑥′2 cos 𝛿0 2𝛿 2 𝑛=−(𝑁−1)∕2 0
×
𝑛=−(𝑁−1)∕2
[
cos
] ) 2𝜋𝑛 ( ′ 𝑦1 − 𝑦′2 , 𝛿0
(6) (7)
𝐿0 and 𝑙0 are the outer and inner scales of turbulence, respectively, and 𝛤 (⋅) is the Gamma function. With the power spectrum in Eq. (4), the integral in Eq. (3) becomes ( 2) ∞ 𝜅0 𝐴 (𝛼) ̃2 2−𝛼 𝐼= 𝜅 3 𝛷𝑛 (𝜅) 𝑑𝜅 = 𝐶𝑛 𝜅𝑚 𝛽 exp ∫0 2 (𝛼 − 2) 𝜅𝑚2 ( ) 2 𝛼 𝜅 × 𝛤 2 − , 0 − 2𝜅04−𝛼 , (8) 2 𝜅𝑚2
2. Analytic solutions for radially polarized MCGSM beams in nonKolmogorov turbulence
(𝑁−1)∕2 ∑
(5)
where 𝛽 = 2𝜅02 − 2𝜅𝑚2 + 𝛼𝜅𝑚2 and 𝛤 (⋅, ⋅) denotes the incomplete Gamma function. Substituting Eqs. (1) and (3) into Eq. (2) and calculating the resulting integral we arrive at the formulas ( ) ( ) 𝑊𝑥𝑥 𝝆1 , 𝝆2 , 𝑧 = 𝛤 𝝆1 , 𝝆2 ( 2 [ ( 2 2 ) 2 )] (𝑁−1)∕2 𝛾𝑦1+ 𝛺𝑦22 𝛾𝑦1− 𝛺𝑦21 ∑ + + exp + × exp 4𝑀 4𝛱 4𝑀 4𝛱 𝑛=−(𝑁−1)∕2 {( ) ( 2 ) (𝑁−1)∕2 2 2 (9) 𝛾𝑥1+ 𝛺𝑥22 𝛥𝛺𝑥22 ∑ × 𝛥 + 𝛾𝑥1+ 𝛺𝑥22 + exp + 2𝛱 4𝑀 4𝛱 𝑛=−(𝑁−1)∕2 ( ) ( 2 )} 2 2 𝛥𝛺𝑥21 𝛾𝑥12 𝛺𝑥21 + 𝛥 + 𝛾𝑥1− 𝛺𝑥21 + exp + , 2𝛱 4𝑀 4𝛱
(1)
where (𝛼, 𝛽 = 𝑥, 𝑦), 𝝆′1 and 𝝆′2 are two-dimensional position vectors in the source plane, 𝜎 denotes its source width , 𝛿0 represents spatial coherence width, and 𝑁 is the positive integer. When 𝑁 = 1, the radially polarized MCGSM beam reduces to a conventional radially polarized Gaussian Schell-model beam. The realizability conditions for a radially polarized MCGSM source and corresponding beam conditions are derived in our recent work [32]. The paraxial form of the extended Huygens–Fresnel principle which describes the interaction of waves with random implies ( medium ) ( the) elements of the CSD matrix at two points 𝐫𝟏 = 𝝆𝟏 , 𝑧 and 𝐫2 = 𝝆2 , 𝑧 in the same transverse plane of the half-space 𝑧 > 0 are related to those in the source plane as [33] ) ( ) ( 𝑘 )2 (0) ( ′ 𝑊𝛼𝛽 𝝆1 , 𝝆′2 , 0 𝑊𝛼𝛽 𝝆𝟏 , 𝝆2 , 𝑧 = ⨌ 2𝜋𝑧 [ ( ) ( )] 𝝆𝟏 − 𝝆′1 2 − 𝝆2 − 𝝆′2 × exp −𝑖𝑘 2𝑧 ⟨ [ ∗( ) ( )]⟩ ′ × exp 𝜓 𝝆𝟏 , 𝝆1 , 𝑧 + 𝜓 𝝆𝟐 , 𝝆′2 , 𝑧 𝑚 d2 𝝆′1 d2 𝝆′2 , (2)
( ) ( ) 𝑊𝑦𝑦 𝝆1 , 𝝆2 , 𝑧 = 𝛤 𝝆1 , 𝝆2 , [ ( ) ( 2 )] (𝑁−1)∕2 2 2 2 𝛾𝑥1+ 𝛺𝑥22 𝛾𝑥1− 𝛺𝑥21 ∑ × exp + + exp + 4𝑀 4𝛱 4𝑀 4𝛱 𝑛=−(𝑁−1)∕2 {( ) ( 2 2 2 ) (𝑁−1)∕2 𝛥𝛺𝑦22 𝛾𝑦1+ 𝛺𝑦22 ∑ × 𝛥 + 𝛾𝑦1+ 𝛺𝑦22 + exp + 2𝛱 4𝑀 4𝛱 𝑛=−(𝑁−1)∕2 ( ( 2 2 ) 2 )} 𝛥𝛺𝑦21 𝛾𝑦12 𝛺𝑦21 exp , + 𝛥 + 𝛾𝑦1− 𝛺𝑦21 + + 2𝛱 4𝑀 4𝛱 393
(10)
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Fig. 1. Normalized spectral intensity of a radially polarized MCGSM beam with 𝑁 = 5 at several different propagation distances in non-Kolmogorov turbulence.
( ) ( ) 𝑊𝑥𝑦 𝝆1 , 𝝆2 , 𝑧 = 𝛤 𝝆1 , 𝝆2 ( 2 ( 2 [ 2 ) 2 )] (𝑁−1)∕2 𝛾𝑦1+ 𝛺𝑦22 𝛾𝑦1− 𝛺𝑦21 ∑ + + 𝛺𝑦22 × exp + 𝛺𝑦21 exp × 4𝑀 4𝛱 4𝑀 4𝛱 𝑛=−(𝑁−1)∕2 {( ( 2 ) ) 2 (𝑁−1)∕2 𝛾𝑦1+ 𝛺𝑦22 ∑ 𝛥𝛺𝑥22 × 𝛾𝑥1+ + exp + 2𝛱 4𝑀 4𝛱 𝑛=−(𝑁−1)∕2 ( 2 )} ( ) 2 𝛾𝑦12 𝛺𝑦21 𝛥𝛺𝑥21 + 𝛾𝑥1− + exp + , (11) 2𝛱 4𝑀 4𝛱 ( ) ( ) ∗ 𝝆1 , 𝝆2 , 𝑧 , 𝑊𝑦𝑥 𝝆1 , 𝝆2 , 𝑧 = 𝑊𝑥𝑦
Fig. 2. Normalized spectral intensity of a radially polarized MCGSM beam at propagation distance 𝑧 = 1 km for different values of 𝑁.
(12)
with ( ) 𝜇 𝝆1 , 𝝆2 , 𝑧 = √
( ) ( ) 𝑖𝑘𝑥2 𝑖𝑘𝑥1 − 𝐼 𝑥1 − 𝑥2 , 𝜍𝑥2 = − 𝐼 𝑥1 − 𝑥2 , 𝜍𝑥1 = 𝑧 𝑧 ( ) ( ) 𝑖𝑘𝑦1 𝑖𝑘𝑦2 2𝜋𝑛 , 𝜍𝑦1 = − 𝐼 𝑦1 − 𝑦2 , 𝜍𝑦2 = − 𝐼 𝑦1 − 𝑦2 , 𝑎= 𝛿 𝑧 𝑧
, ( ) ↔( ) Tr𝑊 𝝆1 , 𝝆1 , 𝑧 Tr𝑊 𝝆2 , 𝝆2 , 𝑧 [ [ ] ] 2𝑅𝑒 𝑊𝑥𝑦 (𝝆, 𝑧) 1 , 𝜃 (𝝆, 𝑧) = arc tan 2 𝑊𝑥𝑥 (𝝆, 𝑧) − 𝑊𝑦𝑦 (𝝆, 𝑧)
𝛾𝑥1± = 𝜍𝑥1 ± 𝑖𝑎, 𝛾𝑥2± = 𝜍𝑥2 ± 𝑖𝑎,
𝛥𝛾 𝛺𝑥22 = 𝑥1− − 𝛾𝑥2− , 2𝑀 𝛥𝛾𝑦1− 𝛺𝑦22 = − 𝛾𝑦2− , 2𝑀3
1 1 𝑖𝑘 + + + 𝐼, 4𝜎 2 2𝛿 2 2𝑧 0
3. Numerical examples In this section, we will now illustrate the second-order statistical properties of a radially polarized MCGSM beam propagating through non-Kolmogorov turbulence by a set of numerical examples. Unless specified in captions, the source and the medium parameters are chosen as follows: 𝜆 = 632 nm, 𝜎 = 1 cm, 𝛿0 = 0.5 cm, 𝑁 = 5, 𝐿0 = 1 m, 𝑙0 = 1 mm, 𝛼 = 3.1 and 𝐶̃𝑛2 = 10−13 m3−𝛼 . In Fig. 1 the evolution of the normalized spectral intensity of a radially polarized MCGSM beam with 𝑁 = 5 on propagation in nonKolmogorov turbulence is shown at several selected distances. As is seen from Figs. 1(a)–1(c), the beam profile gradually transforms from darkhollow at the initial plane to a 5 × 5 Gaussian arrays pattern at a certain distance. However, unlike in the case of free space propagation, where the lattice pattern remains for any large distance [32], in the atmospheric turbulence, the lattice pattern is gradually destroyed along with further propagation. What is more, after passing through the turbulence at sufficiently large distances, the beam shape eventually converges into Gaussian profile. This phenomenon can be explained by
𝛱=
Based on the elements of the CSD matrix, the spectral density, the spectral degree of coherence, the spectral degree of polarization, and the orientation angle of the polarization ellipse of such beam on propagation can be calculated by the expressions [34,35] ↔
𝑆 (𝝆, 𝑧) = Tr𝑊 (𝝆, 𝝆, 𝑧) , √ [↔ ] √ √ √ 4Det 𝑊 𝝆, 𝑧) (𝝆, √ √ 𝑃 (𝝆, 𝑧) = √1 − , ↔ Tr𝑊 (𝝆, 𝝆, 𝑧)
(17)
(13)
1 1 𝑖𝑘 + − + 𝐼, 4𝜎 2 2𝛿 2 2𝑧 0 [ ] ( ) −𝑖𝑘 𝝆21 − 𝝆22 ( ) ( ) 𝑘2 𝛤 𝝆1 , 𝝆2 = exp − 𝐼 𝝆1 − 𝝆2 . 2𝑧 64𝑧2 4𝜎 2 𝑀 2 𝛱 2 𝑀=
(16)
↔
where Det and Tr stand for determinant and trace of the matrix, respectively.
𝛾𝑦1± = 𝜍𝑦1 ± 𝑖𝑎, 𝛾𝑦2± = 𝜍𝑦2 ± 𝑖𝑎, 𝛥𝛾𝑥1+ 𝛺𝑥21 = − 𝛾𝑥2+ , 2𝑀 𝛥𝛾𝑦1+ − 𝛾𝑦2+ , 𝛺𝑦21 = 2𝑀3
↔ ( ) Tr𝑊 𝝆1 , 𝝆2 , 𝑧
(14)
(15)
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Fig. 3. Spectral degree of polarization and the corresponding cross line of a radially polarized MCGSM beam at propagation distance 𝑧 = 1 km for different values of 𝑁.
Fig. 4. Spectral degree of polarization of a radially polarized MCGSM beam with 𝑁 = 5 at several different propagation distances, (a) 𝑧 = 0; (b) 𝑧 = 0.5 km; (c) 𝑧 = 1 km; (d) 𝑧 = 4.5 km.
Fig. 5. Modulus of the spectral of coherence of a radially polarized MCGSM beam with 𝑁 = 5 at several different propagation distances, (a) 𝑧 = 0.2 km; (b) 𝑧 = 0.5 km; (c) 𝑧 = 1 km; (d) 𝑧 = 8 km.
the fact that the evolution behavior of the beam is determined by the source correlations and atmospheric turbulence. At small propagation distance, the beam is mostly modified by the source correlations, as the beam passes to sufficiently large distance through turbulence, the effect of turbulence plays a dominate role for determining the beam profile. Fig. 2 displays the intensity distribution at the propagation distance 𝑧 = 1 km for 𝑁 = 3 and 𝑁 = 6, respectively. One sees that the periodicity of the array intensity can be adjusted by the choice of the summation index 𝑁. Fig. 3 reveals the behavior of the spectral degree of polarization of such beam at the propagation distance 𝑧 = 1 km for different values of 𝑁 in 2D and 3D versions. One finds that the polarization contours exhibit periodic oscillations near the axis, and the periodicity of the lattice-like distribution is completely depended on the index 𝑁. Fig. 4 exhibits the evolution of the degree of the polarization at several propagation distances. It is clearly seen that the oscillations on the polarization distribution persevere for any large distance, and gradually suppressed due to the uniformly correlated turbulence. This implies that the behavior of the polarization distribution of a radially polarized
Fig. 6. The influences of the turbulence parameter 𝐶̃𝑛2 on the properties of a radially polarized MCGSM beam with 𝑁 = 5 in the plane 𝑧 = 1 km, (a) Degree of polarization (b) Modulus of the spectral degree of coherence.
beam is mostly determined by the source correlations, rather than by the medium parameters. Fig. 5 shows the modulus of the spectral degree of coherence of a radially polarized MCGSM beam with 𝑁 = 5 at different propagation distances. One sees that at relatively small distances, the coherence 395
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Fig. 7. The orientation angle of a radially polarized MCGSM beam in the source plane 𝑧 = 0 for different values of 𝑁.
Fig. 8. As in Fig. 7 but for 𝑧 = 1 km.
distribution is mainly modulated by multi cosine function Gaussian distribution, However, as the propagation distance grows, the oscillatory trend on the coherence curves is suppressed by medium fluctuations. Moreover, after passing through the turbulence at sufficiently large distances, the turbulence modifies the coherence curve to Gaussian distribution while narrowing the degree of coherence. Fig. 6(a) exhibits the profiles of the degree of polarization for selected turbulence parameters. It turns out that, as the atmospheric fluctuations become more severe (𝐶̃𝑛2 grows), the oscillations on the polarization curve become less evident. It can also be seen form Fig. 6(b) that the degree of coherence is significantly affected by the strength of the atmosphere turbulence, the larger the value of 𝐶̃𝑛2 is, the more efficiently the coherence profile approaches to a Gaussian distribution. Finally, to learn more about the polarization properties, we calculate in Figs. 7 and 8 the orientation angle of a radially polarized MCGSM beam for different values of 𝑁 in source plane and in turbulence propagation, respectively. It turns out that the state of polarization with different values of 𝑁 all present radial polarization in the source plane as expected. While after passing through the atmospheric turbulence as shown in Fig. 8, the radial polarization structure is destroyed due to the non-conventional source correlation function, and the state of polarization displays a more complex structure, similarly to its free space behavior [32]. This means that the state of polarization is greatly affected by the source correlation, and change in the turbulent statistics plays a relatively small effect. It is also interesting to note that the although the overall state of polarization is broken, the state of polarization of each beamlet evolves into a radially polarization structure on propagation.
state exhibits self-splitting property and each beamlet evolves into radially polarized structure upon propagation. Moreover the degree of coherence first broadens due to the source correlations and then narrows down, when the effect of turbulence starts to dominate the effect of source coherence. As we all known, spatially periodic arrays of light beams are of particular interest for applications in many fields. Moreover, vector partially coherent arrays have significant advantages in comparison with conventional scalar array beams. Our results obtained in this work is helpful in understanding the correlation modulation theory, and may be useful in optical trapping, high-resolution microscopy, and polarization communications. Acknowledgment This work is supported by the National Natural Science Foundation of China under Grant Nos. 11704098, 11504091, and 61775052. References [1] F. Gori, Opt. Commun. 31 (1983) 4. [2] L.C. Andrews, R.L. Phillips, Laser Beam Propagation through Random Media, Bellingham, 1998. [3] J. Wu, J. Modern Opt. 37 (1990) 671. [4] T. Shirai, A. Dogariu, E. Wolf, J. Opt. Soc. Am. A 20 (2003) 1094. [5] G. Wu, H. Guo, S. Yu, B. Luo, Opt. Lett. 35 (2010) 715. [6] T. Shirai, A. Dogariu, E. Wolf, Opt. Lett. 28 (2003) 610. [7] A. Dogariu, S. Amarande, Opt. Lett 28 (2003) 10. [8] J. Ricklin, F. Davidson, J. Opt. Soc. Am. A 20 (2003) 856. [9] A.I. Kon, Waves Random Complex Media 4 (1994) 297. [10] M. Charnotskii, J. Opt. Soc. Am. A 29 (2012) 711. [11] I. Toselli, B. Agrawal, S. Restaino, J. Opt. Soc. Am. A 28 (2011) 483. [12] I. Toselli, J. Opt. Soc. Am. A 31 (2014) 1868. [13] F. Gori, M. Santarsiero, Opt. Lett. 32 (2007) 3531. [14] F. Gori, M. Santarsiero, R. Borghi, V.R. Sanchez, J. Opt. Soc. Am. A 25 (2008) 1016. [15] H. Lajunen, T. Saastamoinen, Opt. Lett. 36 (2011) 4104. [16] Z. Tong, O. Korotkova, J. Opt. Soc. Am. A 29 (2012) 2154. [17] S. Sahin, O. Korotkova, Opt. Lett. 37 (2012) 2970. [18] Z. Mei, Opt. Lett. 39 (2014) 4188. [19] Z. Mei, O. Korotkova, Opt. Lett. 38 (2013) 2578. [20] Z. Mei, D. Zhao, O. Korotkova, Y. Mao, Opt. Lett. 40 (2015) 5662. [21] H. Xu, Z. Zhang, J. Qu, W. Huang, Opt. Express 22 (2014) 22479. [22] X. Wang, M. Yao, Z. Qiu, X. Yi, Z. Liu, Opt. Express 23 (2015) 12508. [23] Z. Song, Z. Liu, K. Zhou, Q. Sun, S. Liu, Opt. Express 24 (2016) 1804. [24] K.S. Youngworth, T.G. Brown, Opt. Express 7 (2000) 77. [25] R. Dorn, S. Quabis, G. Leuchs, Phys. Rev. Lett. 91 (2003) 233901.
4. Concluding remarks In this manuscript, we have investigated the statistical properties of radially polarized MCSM beams in atmospheric turbulence with a non-Kolmogorov power law. The analytical propagation formulas for the second-order characteristics have been derived based on the extended Huygens–Fresnel principle. Via numerical examples we have illustrated the fact that the lattice-like intensity patterns of the beam on propagation in free space is destroyed when it passes at sufficiently large distances form the source through the non-Kolmogorov turbulence. It is also shown that the oscillatory trend of the degree of polarization is gradually suppressed by medium fluctuation. Besides, the polarization 396
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[26] P. Wróbel, J. Pniewski, T. Antosiewicz, T. Szoplik, Phys. Rev. Lett. 102 (2009) 183902. [27] F. Wang, Y. Cai, Y. Dong, O. Korotkova, Appl. Phys. Lett. 100 (2012) 051108. [28] G. Wu, F. Wang, Y. Cai, Opt. Express 20 (2012) 28301. [29] S. Zhu, X. Zhu, L. Liu, F. Wang, Y. Cai, Opt. Express 21 (2013) 27682. [30] S. Zhu, Y. Chen, J. Wang, H. Wang, Z. Li, Y. Cai, Opt. Express 23 (2015) 33099.
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S. Zhu, F. Wang, Z. Li, Appl. Phys. Lett. 109 (2016) 161904. M. Tang, D. Zhao, . L, . Li, . H, . Li, Opt. Commun. 396 (2017) 249. E. Shchepakina, O. Korotkova, Opt. Express 18 (2010) 10650. E. Wolf, Phys. Lett. A 32 (2003) 263. E. Wolf, Introduction to the Theories of Coherence and Polarization of Light, Cambridge University, 2007.