Second-order statistics of a partially coherent electromagnetic rotating elliptical Gaussian vortex beam through non-Kolmogorov turbulence

Jianbin Zhang; Jintao Xie; Dongmei Deng

doi:10.1364/OE.26.021249

1. Introduction

Recently, the evolutions of the vortex in a strain flow [1], the electron vortex beams through the magnetic matter [2] and the orbital-angular-momentum-tunable vortex beam in a dual-off-axis pumped Yb:CALGO laser [3] have been reported, which is potentially relevant to vortex dynamics in atmospheres of the earth. Owing to the unique important property of the scintillation reduction, the propagation of the electromagnetic vortex beams through turbulent atmosphere has attracted extensive attentions [4,5].

Compared with completely coherent beams, partially coherent beams have a significant advantage for reducing the inevitable degradation induced by random refractive-index fluctuations of the turbulence [6,7]. According to the unified theory of coherence and polarization given by Wolf [8], the spectral degree of coherence (DOC), degree of polarization (DOP) of partially coherent electromagnetic beams through the turbulent atmosphere have been studied [8, 9]. Besides, the rotating elliptical Gaussian (REG) beam is regarded as the Gaussian beam with general astigmatism [10] and its symmetry planes are absent due to the rotations of the light spot and the phase front. In practice, the REG beam can be obtained from an ordinary Gaussian beam, which propagates through a system of lenses in the case that principal planes of the astigmatic surfaces are arbitrarily oriented relative to one another [11]. Moreover, the REG beam, which has potential applications in optical tweezers [12], biotechnology [13], and microfluidics [14], is significant to be investigated.

For a long time, Kolmogorov’s power spectrum of refractive-index fluctuations has been widely used to develop the propagation properties of beams in atmospheric turbulence [15]. However, it has been theoretically [16] and experimentally [17] shown that anisotropic turbulence in portions of troposphere and stratosphere deviates from the Kolmogorov model. To address this issue, the non-Kolmogorov power spectrum was introduced by Toselli et al [18]. Based on this theoretical model, the cross-spectral density (CSD) [19], the scintillation index [20], the beam spreading [21, 22], the propagation factor [23] and the second-order moments [24] of optical beams have been explored successfully in atmospheric turbulence. To the best of our knowledge, up to now the partially coherent electromagnetic rotating elliptical Gaussian vortex (PCEREGV) beam has not been proposed and its propagation properties through non-Kolmogorov turbulence are worth investigating, which have potential applications in atmospheric optical trapped [12], optical communication [22], etc. Furthermore, compared with the REG beam, the PCEREGV beam has natural advantages over the reductions of the scintillation and the turbulent fluctuations, which are beneficial for improving the beam quality through the atmospheric turbulence [4–7]. In this paper, we study the second-order statistics including the spectrum density, the spectral DOC and DOP of the PCEREGV beam in non-Kolmogorov turbulence.

The paper is organized as follows. In Sec. 2, the analytical expressions for the CSD matrix elements, the spectral density, the spectral DOC and the spectral DOP of the PCEREGV beam through non-Kolmogorov turbulence are derived. In Sec. 3, the variation properties of the spectral density, the spectral DOC and the spectral DOP for the PCEREGV beam through non-Kolmogorov turbulence are elucidated in detail. Finally, we summarize the main results obtained in this paper in Sec. 4.

2. The theoretical model of a PCEREGV beam in non-Kolmogorov turbulence

In the Cartesian coordinate system, the electric field of the REG vortex beam at the z = 0 source plane is written as

E (r^{'}; 0) = A_{0} exp (- \frac{x^{' 2}}{a^{2} w^{2}} - \frac{y^{' 2}}{b^{2} w^{2}} - \frac{i x^{'} y^{'}}{c^{2} w^{2}}) (\frac{x^{'}}{a w} + \frac{i y^{'}}{b w}),

where r′ ≡ (x′, y′) is the original position vector; A₀ is the optical amplitude; a, b and c are the elliptical parameters; w denotes the initial beam waist. The second-order statistics of the PCEREGV beam in the source plane can be characterized by the 2 × 2 CSD matrix W⃡(r′₁, r′₂; 0) with elements described as [9]

\begin{array}{l} W_{i j} (r_{1}^{'}, r_{2}^{'}; 0) & = A_{0}^{2} A_{i} A_{j} B_{i j} [(\frac{x_{1}^{'} x_{2}^{'}}{a^{2} w^{2}} + \frac{y_{1}^{'} y_{2}^{'}}{b^{2} w^{2}}) + i (\frac{x_{1}^{'} y_{2}^{'} - x_{2}^{'} y_{1}^{'}}{a b w^{2}})] exp (- \frac{x_{1}^{' 2} + x_{2}^{' 2}}{a^{2} w^{2}} \\ - \frac{y_{1}^{' 2} + y_{2}^{' 2}}{b^{2} w^{2}} + i \frac{x_{2}^{'} y_{2}^{'} - x_{1}^{'} y_{1}^{'}}{c^{2} w^{2}}) exp [- \frac{{(x_{1}^{'} - x_{2}^{'})}^{2} + {(y_{1}^{'} - y_{2}^{'})}^{2}}{2 δ_{i j}^{2}}], (i, j = x, y) \end{array}

where r′₁ ≡ (x′₁, y′₁) and r′₂ ≡ (x′₂, y′₂) are positions of two points at the z = 0 plane; A_i, A_j and B_ij are cofficients; δ_ij is a positive constant characterizing the correlation length, in which δ_xx and δ_yy are auto-correlated while δ_xy and δ_yx are cross-correlated. Importantly, the following conditions of the CSD matrix should be satisfied by some parameters [25]:

\frac{A_{x}^{2} δ_{x x}^{2}}{δ_{x x}^{2} + 4 w^{2}} - \frac{2 A_{x} A_{y} | B_{x y} | δ_{x y}^{2}}{δ_{x y}^{2} + 4 w^{2}} + \frac{A_{y}^{2} δ_{y y}^{2}}{δ_{y y}^{2} + 4 w^{2}} \geq 0

,

\frac{δ_{x x}^{2}}{δ_{x x}^{2} + 4 w^{2}} - \frac{2 δ_{x y}^{2}}{δ_{x y}^{2} + 4 w^{2}} + \frac{δ_{y y}^{2}}{δ_{y y}^{2} + 4 w^{2}} \leq 0

,

\frac{1}{4 w^{2}} + \frac{1}{δ_{x x}^{2}} ≪ \frac{2 π}{λ^{2}}

,

\frac{1}{4 w^{2}} + \frac{1}{δ_{y y}^{2}} ≪ \frac{2 π}{λ^{2}}

,

max {δ_{x x}, δ_{y y}} \leq δ_{x y} \leq min {\frac{δ_{x x}}{\sqrt{B_{x y}}}, \frac{δ_{y y}}{\sqrt{B_{x y}}}}

.

Based on the extended Huygens-Fresnel integral, the elements of the CSD matrix for the PCEREGV beam propagating through non-Kolmogorov turbulence are given by [9,21]

\begin{array}{l} W_{i j} (r_{1}, r_{2}; z) & = \frac{k^{2}}{4 π^{2} z^{2}} \int \int d^{2} r_{1}^{'} \int \int d^{2} r_{2}^{'} W_{i j} (r_{1}^{'}, r_{2}^{'}; 0) \\ \times exp {- \frac{i k}{2 z} [{(r_{1} - r_{1}^{'})}^{2} - {(r_{2} - r_{2}^{'})}^{2}]} \\ \times < exp [ψ^{*} (r_{1}, r_{1}^{'}) + ψ (r_{2}, r_{2}^{'})] >, \end{array}

where r₁ ≡ (x₁, y₁) and r₂ ≡ (x₂, y₂) are two arbitrary transverse position vectors at the receiver plane;

k = \frac{2 π}{λ}

is the wave number with λ being the wavelength of the PCEREGV beam; <> respresents ensemble averaging over the turbulent atmosphere; ψ denotes the phase distortion of a monochromatic spherical wave in the atmospheric turbulence. For the case of quadratic phase approximations, the average turbulence phase perturbation can be expressed as [9]

\begin{array}{l} < exp [ψ^{*} (r_{1}, r_{1}^{'}) + ψ (r_{2}, r_{2}^{'})] > = exp {- \frac{π^{2} k^{2} z}{3} \int_{0}^{\infty} κ^{3} Φ_{n} (κ, α) \\ \times [{(r_{1} - r_{2})}^{2} + (r_{1} - r_{2}) \cdot (r_{1}^{'} - r_{2}^{'}) + {(r_{1}^{'} - r_{2}^{'})}^{2}] d κ}, \end{array}

where Φ_n(κ, α) respresents the three-dimensional power spectrum of the refractive-index fluctuations with κ being the spatial wave number in the spatial-frequency. With the help of the non-Kolmogorov power spectrum model given in [16], the integral expression in Eq. (4) is obtained as

\begin{array}{l} T = - \frac{π^{2} k^{2} z}{3} \int_{0}^{\infty} κ^{3} Φ_{n} (κ, α) d κ = - \frac{π^{2} k^{2} z A (α) \tilde{C_{n}^{2}}}{6 (α - 2)} [κ_{m}^{2 - α} β \\ \times exp (\frac{κ_{0}^{2}}{κ_{m}^{2}}) Γ (2 - \frac{α}{2}, \frac{κ_{0}^{2}}{κ_{m}^{2}}) - 2 κ_{0}^{4 - α}], 0 \leq κ < \infty, 3 < α < 4 \end{array}

with

κ_{m} = \frac{c_{1} (α)}{l_{0}}

;

κ_{0} = \frac{2 π}{L_{0}}

;

c_{1} (α) = {[\frac{2 π}{3} A (α) Γ (5 - \frac{α}{2})]}^{\frac{1}{α - 5}}

;

A (α) = \frac{1}{4 π^{2}} Γ (α - 1) cos (\frac{α π}{2})

;

β = 2 κ_{0}^{2} + (α - 2) κ_{m}^{2}

, where l₀ and L₀ are the inner and the outer scales of atmospheric turbulence, respectively; Γ(.) is Gamma function; α is the generalized exponent parameter;

\tilde{C_{n}^{2}}

is the generalized refractive-index structure parameter with units m^3−α.

Substituting Eqs. (2) and (4) into Eq. (3), we perform the integral by using the formulae in [26], and the analytical expressions for the CSD matrix elements of the PCEREGV beam through non-Kolmogorov turbulence are obtained as

\begin{array}{l} W_{i j} (r_{1}, r_{2}, z) & = \frac{k^{2} A_{0}^{2} A_{i} A_{j} B_{i j}}{4 z^{2} \sqrt{M_{1 i j} M_{2 i j} M_{3 i j} M_{4 i j}}} exp [- \frac{i k}{2 z} (x_{1}^{2} - x_{2}^{2} + y_{1}^{2} - y_{2}^{2}) + T {(x_{1} - x_{2})}^{2} \\ + T {(y_{1} - y_{2})}^{2} + \frac{N_{1}^{2}}{4 M_{1 i j}} + \frac{N_{2 i j}^{2}}{4 M_{2 i j}} + \frac{N_{3 i j}^{2}}{4 M_{3 i j}} + \frac{N_{4 i j}^{2}}{4 M_{4 i j}}] \times {\frac{1}{a^{2} w^{2}} [\frac{N_{1} M_{2 i j} f_{i j}}{N_{2 i j} M_{1 i j}} \\ - \frac{i (f_{i j} + g_{i j} + h_{i j})}{2 c^{2} w^{2} M_{1 i j}} + 2 i c^{2} w^{2} M_{2 i j} h_{i j}] + \frac{q_{i j}}{b^{2} w^{2}} + \frac{i}{a b w^{2}} [\frac{N_{1} N_{4 i j}}{4 M_{1 i j} M_{4 i j}} \\ - \frac{i q_{i j}}{2 c^{2} w^{2} M_{1 i j}} + \frac{u_{i j} p_{i j}}{M_{1 i j}} - f_{i j} - g_{i j} - h_{i j}]}, \end{array}

with

u_{i j} = \frac{1}{2 δ_{i j}^{2}} - T

,

M_{1 i j} = \frac{1}{a^{2} w^{2}} + \frac{i k}{2 z} + u_{i j}

,

M_{2 i j} = \frac{1}{b^{2} w^{2}} + \frac{i k}{2 z} + u_{i j} + \frac{1}{4 c^{4} w^{4} M_{1 i j}}

,

v_{i j} = 1 - \frac{u_{i j}^{2}}{M_{1 i j} M_{2 i j}}

,

M_{3 i j} = \frac{1}{a^{2} w^{2}} - \frac{i k}{2 z} + u_{i j} - \frac{u_{i j}^{2}}{M_{1 i j}} + \frac{u_{i j}^{2}}{4 c^{4} w^{4} M_{1 i j}^{2} M_{2 i j}}

,

M_{4 i j} = \frac{1}{b^{2} w^{2}} - \frac{i k}{2 z} + u_{i j} - \frac{u_{i j}^{2}}{M_{2 i j}} + \frac{v_{i j}^{2}}{4 c^{4} w^{4} M_{3 i j}}

,

N_{1} = \frac{i k x_{1}}{z} + T x_{1} - T x_{2}

,

N_{2 i j} = \frac{i k y_{1}}{z} + T y_{1} - T y_{2} - \frac{i N_{1}}{2 c^{2} w^{2} M_{1 i j}}

,

N_{3 i j} = - \frac{i k x_{2}}{z} - T x_{1} + T x_{2} + \frac{u_{i j} N_{1}}{M_{1 i j}} - \frac{i u_{i j} N_{2 i j}}{2 c^{2} w^{2} M_{1 i j} M_{2 i j}}

,

N_{4 i j} = - \frac{i k y_{2}}{z} - T y_{1} + T y_{2} + \frac{u_{i j} N_{2 i j}}{M_{2 i j}} + \frac{i N_{3 i j} v_{i j}}{2 c^{2} w^{2} M_{3 i j}}

,

f_{i j} = \frac{N_{2 i j}}{2 M_{2 i j}} (\frac{N_{3 i j}}{2 M_{3 i j}} + \frac{i v_{i j} N_{4 i j}}{4 c^{2} w^{2} M_{3 i j} M_{4 i j}})

,

p_{i j} = \frac{N_{3 i j} N_{4 i j}}{4 M_{3 i j} M_{4 i j}} + \frac{i v_{i j} (1 + \frac{N_{4 i j}^{2}}{2 M_{4 i j}})}{4 c^{2} w^{2} M_{3 i j} M_{4 i j}}

,

g_{i j} = \frac{u_{i j} p_{i j}}{M_{2 i j}}

,

q_{i j} = \frac{N_{2 i j} N_{4 i j}}{4 M_{2 i j} M_{4 i j}} + \frac{u_{i j} (1 + \frac{N_{4 i j}^{2}}{2 M_{4 i j}})}{2 M_{2 i j} M_{4 i j}} - \frac{i g_{i j}}{2 c^{2} w^{2} M_{1 i j}}

,

h_{i j} = - \frac{i u_{i j}}{4 c^{2} w^{2} M_{1 i j} M_{2 i j} M_{3 i j}} [1 + \frac{N_{3 i j}^{2}}{2 M_{3 i j}} - \frac{v_{i j}^{2} (1 + \frac{N_{4 i j}^{2}}{2 M_{4 i j}})}{4 c^{4} w^{4} M_{3 i j} M_{4 i j}} + \frac{i N_{3 i j} N_{4 i j} v_{i j}}{2 c^{2} w^{2} M_{3 i j} M_{4 i j}}]

.

By taking some special cases in Eq. (6): a = 1, b = 1, c → ∞ and ignoring the effect of the vortex, we obtain the expressions the same as those in [27], which verify the validity of the analytical expressions for the CSD matrix elements. Accordingly, the spectral density S(r; z), the spectral DOC η(r₁, r₂; z) and DOP P(r; z) of the PCEREGV beam in the receiver plane are given by the formulas as follows [8,9]

S (r; z) = Tr \overset{\leftrightarrow}{W} (r, r; z) = W_{x x} (r, r; z) + W_{y y} (r, r; z),

η (r_{1}, r_{2}; z) = \frac{Tr \overset{\leftrightarrow}{W} (r_{1}, r_{2}; z)}{\sqrt{S (r_{1}; z) S (r_{2}; z)}},

P (r; z) = \sqrt{1 - \frac{4 Det \overset{\leftrightarrow}{W} (r, r; z)}{{[Tr \overset{\leftrightarrow}{W} (r, r; z)]}^{2}}},

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

3. The numerical results and discussions

Unless specified in captions, the parameters used in calculations are set as follows: a = 2, b = 0.5, c = 1.2, w = 2mm, λ = 0.8μm, A_x = A_y = 2, B_xx = B_yy = 1, $B_{x y} = 0.2 exp (\frac{i π}{6})$ , $B_{y x} = 0.2 exp (- \frac{i π}{6})$ , δ_xx = 0.003, δ_yy = 0.002, δ_xy = δ_yx = 0.004, α = 3.10, l₀ = 0.01m, L₀ = 10m, $\tilde{C_{n}^{2}} = 10^{- 14} m^{3 - α}$ , r = |r₁ − r₂| = 0.001m with $r_{1} = \frac{r}{2}$ and $r_{2} = - \frac{r}{2}$ .

For the sake of verifying the theoretical analytical results, we first perfrom the numerical calculations of a PCEREGV beam with different values of a/b at the initial input plane in Fig. 1. Apparently, the analytical results of the spectrum density patterns can be constructed by varying the ratio of a/b, as revealed in Figs. 1(a1)–1(c1). In our numerical calculations, the holograms in Figs. 1(a3)–1(c3) are acquired by calculating the interference intensity patterns of the PCEREGV beam at the z = 0 plane and a tilted plane wave in Figs. 1(a2)–1(c2). Subsequently, we simulate the reflection from holograms and numerically reconstruct the information of the encoded beam via a spatial filtering 4f system. At last, the numerical calculation results of the spectrum density distributions are exhibited in Figs. 1(a4)–1(c4), which are accurate and in accordance well with analytical results.

Fig. 1 Numerical calculations of a PCEREGV beam at the initial input plane with (a) a = b = 2, (b) a = 2, b = 1, (c) a = 2, b = 0.5. (a1)–(c1) Analytical results of the spectrum density distributions; (a2)–(c2) interference intensity of the initial generated beam and a plane wave; (a3)–(c3) computer-generated holograms; (a4)–(c4) transverse spectrum density distributions obtained from the numerical calculation.

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Figure 2 plots the transverse spectrum density and the normalized cross line of the PCEREGV beam through non-Kolmogorov turbulence. From Fig. 2(a), it is obvious that the spectrum density pattern has an elliptical annular shape in the initial plane whose major axis is parallel to the x-axis. Subsequently, the hollow portion of the shape enlarges gradually with the propagation distance increasing, which induces the spectrum density pattern splitting into two light spots. Due to the existence of vortex, the peak spectrum density of the PCEREGV beam increases within a certain propagation distance (2m). With the PCEREGV beam further propagating, the peak spectrum density decreases because the effect of diffraction is stronger than that of vortex. Besides, the spectrum density pattern spins anticlockwise until the major axis of the elliptical shape parallels to the y-axis at about 500m. Afterwards, the pattern no longer spins but broadens along the y-axis, which can be interpreted by the spinning angle θ = arctan(z/Z_R), where Z_R = kw²/2 is the beam Rayleigh range [28]. It is worth mentioning that the rotation phenomena is caused by the inhomogeneity of the transverse energy flow [28]. With propagating, the angular momentum and the angular velocity of the PCEREGV beam decrease to zero gradually, which induces that the intensity pattern becomes non-rotated [29,30]. As is shown in Fig. 2(d), the angle between the major axis of the pattern and the x-axis is 45° at the distance of 16m (about 1Z_R). As the PCEREGV beam rotates during the propagation, the beam width in the x-direction increases more slowly than that in the y-direction so that the former is much smaller than the latter in the far field. Additionally, the spectrum density versus x has two peak values when z < 50m and keeps a unimodal distribution when z > 50m, which is totally different from the case versus y.

Fig. 2 The evolutions of the spectrum density and the corresponding normalized cross line at the y = 0 plane (a)–(d) and x = 0 plane (e)–(h) of a PCEREGV beam at several propagation distances through non-Kolmogorov turbulence.

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Figure 3 presents the curves of the normalized spectrum density versus x for a PCEREGV beam through non-Kolmogorov turbulence by controlling different variables. From Fig. 3(a), it is seen that the normalized spectrum density is initially y-axis symmetry with a minimum at the center and two peaks at both sides, but as the propagation distance further increases, it evolves into a unimodal distribution with a peak at the center. During the evolution in Fig. 3(a), the distance between the initial two peaks is almost invariable but the minimum rises to 1 with the increase of the propagation distance. By changing the initial beam width, we find out that the normalized spectrum density distribution is the most concentrated and the minimum is the biggest when w = 2mm, as shown in Fig. 3(b). The physics behind the phenomenon is that the rotation effect causes the effective beam width in x-direction of the PCEREGV beam with a larger initial beam width to increase more slowly within a certain propagation distance. At z = 50m, the PCEREGV beam with w = 2mm has the smallest effective beam width in x-direction compared with other values. Moreover, the normalized spectrum density is hardly affected by the inner scale l₀ while greatly influenced by the wavelength on the minimum, as depicted in Figs. 3(c) and 3(d). With a bigger wavelength, the normalized spectrum density distributes more dispersedly and its minimum becomes larger. In general, we discover that the effect of the turbulence is much smaller than that of the beam’s intrinsical parameters within a short propagation range, implying that the PCEREGV beam has a better capacity of anti-interference for turbulent disturbance. This is because the rotation of the PCEREGV beam dominates the evolution of the spectrum density.

Fig. 3 Normalized spectrum density of a PCEREGV beam as a function of x through non-Kolmogorov turbulence, (a) with different z, (b) with different w at z = 50m, (c) with different l₀ at z = 50m, (d) with different λ at z = 50m.

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To learn more about the coherence property of the PCEREGV beam through non-Kolmogorov turbulence, the variation of the spectral DOC is displayed in Fig. 4. One can see from Fig. 4(a) that when the propagation distance z < 1m, the spectral DOC η is equal to 0.82; when 1m < z < 12m, η first increases slightly and then decreases steeply until it reaches the minimal value (η_min = 0); when 12m < z < 1000m, η increases to the maximal value (η_max = 1), and finally decreases to 0 gradually for further propagation. In particular, the spectral DOC of the PCEREGV beam with α = 3.10 decreases fastest and that with α = 3.99 decreases slowest when z > 1000m. Owing to the influence of the vortex, the spectrum DOC has a zero point at z = 12m, where the phase of the PCEREGV beam is uncertain. As manifested in Fig. 4(b), with the increase of the exponent parameter α, η first decreases to a minimum when α is about 3.10 and then increases gradually. We discover that T in Eq. (5) decreases with the increasing α for the range 3 < α < 3.10 and then increases when 3.10 < α < 4, indicating that under the condition of α = 3.10 the disturbance of turbulence is the largest so that the spectral DOC will be reduced to the minimum. Meanwhile, the PCEREGV beam with a smaller r has a bigger and stabler η. From Fig. 4(c), it can be clearly seen that η decreases quickly from 1 to 0.1 and slowly from 0.1 to 0 with the increase of the parameter r. In addition, the smaller the wavelength is, the faster the spectral DOC decreases to the minimal value. As the turbulent inner scale l₀ increases, η rises gradually with a weakened acceleration shown in Fig. 4(d). Moreover, the smaller the wavelength is, the smaller the spectral DOC is.

Fig. 4 The spectral DOC of a PCEREGV beam through non-Kolmogorov turbulence, (a) versus z on a log scale with different α, (b) versus α with different r at z = 10km, (c) versus r with different λ at z = 10km, (d) versus l₀ with different λ at z = 10km.

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The spectral DOP of the PCEREGV beam with different parameters through non-Kolmogorov turbulence is investigated in Fig. 5. Similar to the variation of the spectral DOC, the value of the spectral DOP is almost equal to 0.2 when the propagation distance z < 1m is shown in Fig. 5(a). The spectral DOP first increases to the maximal value at the distance of about 8m and then decreases drastically from z = 8m to z = 100m. After that, it increases gradually to a certain extent at the distance of about 100km and lastly decreases to 0.2 slowly with further propagation. Interestingly, the spectral DOP will be influenced by the parameter α only for the case of z > 100m, where the spectral DOP with α = 3.10 increases fastest but decreases slowest and that with α = 3.99 increases slowest but decreases fastest. With the propagation distance increasing, the spectral DOP of the PCEREGV beam with w = 1mm increases to the maximal value at first and then decreases steadily but that with w = 2mm increases monotonically in Fig. 5(b). Additionally, the spectral DOP of the PCEREGV beam with w > 2mm first decreases to the minimal value and then increases gradually. For the sufficiently long distance propagation (18km), the bigger the initial beam width is, the larger the spectral DOP is. As Fig. 5(c) displays, it is pronounced that the smaller the propagation distance is, the larger the fluctuation of the spectral DOP is with the increase of the parameter r. For instance, when z = 5m, P_min = 0.2 and P_max = 1 but when z = 500m, P is almost constant. In Fig. 5(d), the spectral DOP of the PCEREGV beam at z = 5m increases but that at z > 50m decreases with wavelength increasing. The variation curves of the spectrum DOC and DOP manifest the interaction between the inherent property of the PCEREGV beam and the turbulence.

Fig. 5 The spectral DOP of a PCEREGV beam through non-Kolmogorov turbulence, (a) versus z on a log scale with different α, (b) versus z on a basic scale with different w, (c) versus r with different z, (d) versus λ with different z.

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4. Conclusion

In conclusion, we have derived the analytical formulas for the CSD matrix elements of a PCEREGV beam in non-turbulence by using the extended Huygens-Fresnel integral. Meanwhile, the numerical calculations of the PCEREGV beam are carried out and confirm the validity of the analytical expressions. Results show that the transverse spectrum density pattern with an elliptical annular shape in the initial plane spins anticlockwise until its major axis parallels to the y-axis at z = 500m. Furthermore, it evolves into two light spots during the propagation. The PECREGV beam with w = 2mm and a smaller wavelength has a more concentrated distribution of the normalized spectrum density. With respect to the spectral DOC, we can reduce it easily by choosing the appropriate parameters α, r, λ and l₀ during the propagation. For instance, the spectral DOC declines with r, λ and l₀ decreasing, and it is smallest when α = 3.10 in the far field. Meanwhile, the influences of the beam and turbulent parameters expect α on the spectral DOP are great and obvious in the near field but weak in the far field. By contrast, the initial beam parameters have a great impact on the second-order statistics of the PCEREGV beam through non-Kolmogorov turbulence. Our work is helpful for atmospheric optical communication, optical processing and other fields.

Funding

National Natural Science Foundation of China (11775083 and 11374108); National Training Program of Innovation and Entrepreneurship for Undergraduates.

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Second-order statistics of a partially coherent electromagnetic rotating elliptical Gaussian vortex beam through non-Kolmogorov turbulence

Abstract

1. Introduction

2. The theoretical model of a PCEREGV beam in non-Kolmogorov turbulence

3. The numerical results and discussions

4. Conclusion

Funding

References and links

Cited By

Figures (5)

Equations (9)

Optics Express