Far-zone polarization distributionproperties of partially coherent beams with non-uniform source polarization distributions inturbulent atmosphere

Rong Zhang; Xiangzhao Wang; Xin Cheng

doi:10.1364/OE.20.001421

1. Introduction

In recent years, it is an interesting subject for many researchers to study polarization properties of partially coherent electromagnetic beams propagating in free space or in turbulent atmosphere.

James [1] first found that the degree of polarization of an electromagnetic GSM beam can change in free space. Korotkova and Wolf [2] obtained the formulas for the orientation angle and degree of ellipticity of the completely coherent part of a partially coherent electromagnetic beam, and using an electromagnetic GSM beam as an example, showed that the polarization state of the completely coherent part of the field can also change as the beam propagates in free space. Salem and Wolf [3] studied explicitly the effect of source coherence on the polarization properties of an electromagnetic GSM beam propagating in free space.

The evolution properties of the degree of polarization and the state of polarization of electromagnetic GSM beams propagating in turbulent atmosphere were studied by Korotkova and other authors [4–7], and they found that, by changing the source correlation properties, the transverse distributions of the degree of polarization and the state of polarization can all experience different evolution processes, but under the influence of turbulent atmosphere, all the polarization distributions on the source plane will be recovered in the far field. The effect of a source-plane square aperture on the evolution properties of the degree of polarization of an electromagnetic GSM beam propagating in atmospheric turbulence was studied by Ji and Pu [8], and they found that, the size of the aperture will affect the transverse polarization distribution of the beam in the intermediate propagation region, but will not affect that in the far zone, as the beam will regain its source-plane polarization distribution in the far zone. In addition, Ji et al. [9, 10] also studied the polarization properties of partially coherent electromagnetic cosh-Gaussian and Hermite-Gaussian beams in turbulent atmosphere, and found that, besides the source coherence length, other parameters like the cosh-coefficient for cosh-Gaussian beams, the beam order for Hermite-Gaussian beams, can also affect the polarization evolution process, but due to the influence of turbulent atmosphere, in the far field the degree of polarization will always take values same as on the source plane. Kashani et al. [11] studied the on-axis degree of polarization of aberrated partially coherent flat-topped beams, and Zhong et al. [12] studied the on-axis degree of polarization of partially coherent electromagnetic elegant Laguerre-Gaussian beams, in an environment of turbulent atmosphere, and according to their results, these two kinds of beams will also regain their source values of degree of polarization in the far field.

It must be noted that, for all the researches mentioned above, the intensity distribution parameters of the two intensity distributions, which correspond to the two orthogonal components of the source electric field respectively, take completely the same values. Under this condition, the polarization properties are uniform on the source plane. On the other hand, if the two electric components of the source field have different intensity distribution laws, the polarization properties will be non-uniform on the source plane. Du et al. [13] first considered the on-axis polarization properties of electromagnetic GSM beams for which $σ_{0 x} \neq σ_{0 y}$ is satisfied, i.e., the intensity distribution parameters in relation to the two source electric components are different. They found through numerical examples that, the on-axis degree of polarization of this kind of electromagnetic GSM beams in the far field is not equal to that on the source plane, even under the influence of turbulent atmosphere. Du and Zhao [14] also studied the effect of a source-plane slit aperture of variable width on the polarization properties of this kind of beams on propagation in turbulent atmosphere, and found that, by changing the width of the slit aperture, the transverse polarization distribution in the intermediate propagation region and the far-field on-axis degree of polarization can both be affected. One may note that the latter result of Du and Zhao is different from the result of Ji and Pu. Ghafary and Alavinejad [15] researched the on-axis evolution properties of the degree of polarization and the state of polarization of partially coherent flat-topped beams in atmospheric turbulence, and the intensity distribution parameters corresponding to the two source electric components were chosen to be different. They found through numerical illustrations that the beam cannot reacquire its source polarization values in the far field. Lastly, Eyyuboglu et al. [16] investigated the on-axis degree of polarization of partially coherent electromagnetic cosh-Gaussian, cos-Gaussian, Gaussian and annular-Gaussian beams propagating through turbulent atmosphere, and found by numerical examples that, unequal displacement parameters and unequal source sizes with respect to the x and y components of the source electric field can both result in the on-axis degree of polarization at excessive propagation lengths not returning to the original value.

In view of the investigations about the far-field polarization properties of partially coherent electromagnetic beams with non-uniform polarization distributions on the source plane, we think they are incomplete and lack depth, mainly because the researchers studied the far-field polarization properties only for the on-axis points, and they arrived at their conclusions only through numerical examples. On the other hand, while it was frequently shown that certain partially coherent electromagnetic beams can recover their source-plane polarization distributions in the far field when propagating in turbulent atmosphere, due to the lack of a thorough study about the far-field transverse polarization properties of beams with non-uniform polarization distributions on the source plane, it is impossible to come up with a condition to predict whether and how an electromagnetic beam can regain its source polarization distribution in the far zone.

In this paper, using a kind of electromagnetic anisotropic GSM beams, we make a study about the above mentioned problems. We firstly analyze all the possible polarization distribution patterns on the source plane, which can be roughly classified into uniform and non-uniform shapes. Then we derive the far-field mathematical formula for the degree of polarization of this kind of beams propagating in turbulent atmosphere. By using the formula obtained, we analyze the far-zone transverse polarization properties, especially paying attention to those anisotropic electromagnetic GSM beams with non-uniform polarization distributions on the source plane. Based on a complete knowledge of the far-field transverse polarization properties of electromagnetic anisotropic GSM beams with various polarization patterns on the source plane, we put forward a source-plane polarization distribution recovery condition.

2 Realizable conditions of an electromagnetic anisotropic GSM source and its polarization distribution patterns

Consider a planar, secondary, partially coherent, electromagnetic source being placed in the x-y plane, and the beam generated by it propagates along the z axis. In the space-frequency domain, the second-order correlation properties of this source are usually described by an $2 \times 2$ electric cross-spectral density matrix [17, 18]

{\overset{\leftrightarrow}{W}}^{(0)} (s_{1}, s_{2}, ω) = (\begin{matrix} W_{x x}^{(0)} (s_{1}, s_{2}, ω) & W_{x y}^{(0)} (s_{1}, s_{2}, ω) \\ W_{y x}^{(0)} (s_{1}, s_{2}, ω) & W_{y y}^{(0)} (s_{1}, s_{2}, ω) \end{matrix}),

where

W_{i j}^{(0)} (s_{1}, s_{2}, ω) = 〈 E_{i}^{*} (s_{1}, 0, ω) E_{j} (s_{2}, 0, ω) 〉

,

i

and

j

each stand for the subscripts x and y,

E_{x}

and

E_{y}

each represent the two orthogonal components of the random electric field,

s_{1}

and

s_{2}

denote the position vectors of two points on the source plane,

ω

denotes the angular frequency of the light, and the angle brackets represent taking ensemble average of the random electric field.

The elements of the cross-spectral density matrix of a Schell-type partially coherent electromagnetic source share the following general structure [19, 20]:

W_{i j}^{(0)} (s_{1}, s_{2}, ω) = \sqrt{S_{i} (s_{1}, ω)} \sqrt{S_{j} (s_{2}, ω)} μ_{i j} (s_{1} - s_{2}, ω),

where

S_{i} (s, ω)

is the average intensity distribution function of the

i

component of the electric field, and

μ_{i j} (s_{1} - s_{2}, ω)

is the spatial coherence distribution function between the

i

and

j

components of the electric field. When the mathematical expressions for

S_{i}

and

μ_{i j}

(

i, j = x, y

) are all circularly Gaussian, the electromagnetic GSM sources mentioned in the introduction are obtained. If these quantities have expressions as

\begin{array}{l} S_{i} (s, ω) = A_{i}^{2} \exp (- \frac{1}{2} s^{T} {\tilde{σ}}_{i} s), \\ μ_{i j} (s_{1} - s_{2}, ω) = B_{i j} \exp [- \frac{1}{2} {(s_{1} - s_{2})}^{T} {\tilde{δ}}_{i j} (s_{1} - s_{2})], \end{array}

the most general electromagnetic anisotropic GSM source proposed by Wang et al. [21] is obtained. Here,

A_{x}^{2}

and

A_{y}^{2}

denote the average intensities with respect to the x and y electric components at the central point of the source plane, respectively;

B_{i j}

represents the correlation between the

i

and

j

electric components at a coincident source point and

B_{x x} = B_{y y} = 1

,

B_{x y} = B_{y x}^{*}

, and

| B_{x y} | \leq 1

should be obeyed;

{\tilde{σ}}_{i}

and

{\tilde{δ}}_{i j}

, being real, symmetric, and positive definite

2 \times 2

matrices, describe the intensity distribution shapes and the spatial coherence distribution shapes, respectively. In this paper, to avoid complicated mathematical calculations and meanwhile take into account the anisotropy feature of the intensity distributions and the spatial coherence distributions, we adopt the way in which Li and Wolf defined the scalar anisotropic GSM beam [22] and make simplifications to the intensity distribution matrices and the spatial coherence distribution matrices as follows:

\begin{array}{l} {\tilde{σ}}_{x} = [\begin{matrix} {[σ_{0 x}^{(x)}]}^{- 2} & 0 \\ 0 & {[σ_{0 y}^{(x)}]}^{- 2} \end{matrix}], \\ {\tilde{σ}}_{y} = [\begin{matrix} {[σ_{0 x}^{(y)}]}^{- 2} & 0 \\ 0 & {[σ_{0 y}^{(y)}]}^{- 2} \end{matrix}], \\ {\tilde{δ}}_{x x} = [\begin{matrix} {[δ_{0 x}^{(x)}]}^{- 2} & 0 \\ 0 & {[δ_{0 y}^{(x)}]}^{- 2} \end{matrix}], \\ {\tilde{δ}}_{y y} = [\begin{matrix} {[δ_{0 x}^{(y)}]}^{- 2} & 0 \\ 0 & {[δ_{0 y}^{(y)}]}^{- 2} \end{matrix}], \\ {\tilde{δ}}_{y y} = [\begin{matrix} {[δ_{0 x}^{(y)}]}^{- 2} & 0 \\ 0 & {[δ_{0 y}^{(y)}]}^{- 2} \end{matrix}], \\ {\tilde{δ}}_{x y} = [\begin{matrix} {[δ_{0 x}^{(n)}]}^{- 2} & 0 \\ 0 & {[δ_{0 y}^{(n)}]}^{- 2} \end{matrix}] . \end{array}

Considering these simplifications in Eq. (3), we are led to the following expressions for the elements of the cross-spectral density matrix of our electromagnetic anisotropic GSM source:

W_{x x}^{(0)} (s_{1}, s_{2}, ω) = A_{x}^{2} \exp [- \frac{ξ_{1}^{2} + ξ_{2}^{2}}{4 σ_{0 x}^{(x) 2}} - \frac{η_{1}^{2} + η_{2}^{2}}{4 σ_{0 y}^{(x) 2}} - \frac{{(ξ_{1} - ξ_{2})}^{2}}{2 δ_{0 x}^{(x) 2}} - \frac{{(η_{1} - η_{2})}^{2}}{2 δ_{0 y}^{(x) 2}}],

W_{y y}^{(0)} (s_{1}, s_{2}, ω) = A_{y}^{2} \exp [- \frac{ξ_{1}^{2} + ξ_{2}^{2}}{4 σ_{0 x}^{(y) 2}} - \frac{η_{1}^{2} + η_{2}^{2}}{4 σ_{0 y}^{(y) 2}} - \frac{{(ξ_{1} - ξ_{2})}^{2}}{2 δ_{0 x}^{(y) 2}} - \frac{{(η_{1} - η_{2})}^{2}}{2 δ_{0 y}^{(y) 2}}],

\begin{array}{l} W_{x y}^{(0)} (s_{1}, s_{2}, ω) = A_{x} A_{y} B_{x y} \exp [- \frac{ξ_{1}^{2}}{4 σ_{0 x}^{(x) 2}} - \frac{η_{1}^{2}}{4 σ_{0 y}^{(x) 2}}] \exp [- \frac{ξ_{2}^{2}}{4 σ_{0 x}^{(y) 2}} - \frac{η_{2}^{2}}{4 σ_{0 y}^{(y) 2}}] \\ \times \exp [- \frac{{(ξ_{1} - ξ_{2})}^{2}}{2 δ_{0 x}^{(n) 2}}] \exp [- \frac{{(η_{1} - η_{2})}^{2}}{2 δ_{0 y}^{(n) 2}}], \end{array}

W_{y x}^{(0)} (s_{1}, s_{2}, ω) = W_{x y}^{(0) *} (s_{2}, s_{1}, ω) .

It should be noted that $A_{x}$ , $A_{y}$ , $B_{x y}$ , the intensity distribution parameters ( $σ_{0 x}^{(x)}$ , $σ_{0 y}^{(x)}$ , $σ_{0 x}^{(y)}$ , $σ_{0 y}^{(y)}$ ), and the spatial coherence distribution parameters ( $δ_{0 x}^{(x)}$ , $δ_{0 y}^{(x)}$ , $δ_{0 x}^{(y)}$ , $δ_{0 y}^{(y)}$ , $δ_{0 x}^{(n)}$ , $δ_{0 y}^{(n)}$ ) are all independent of position but, in general, may depend on the frequency.

The cross-spectral density matrix of a realizable partially coherent electromagnetic source should satisfy a non-negative definiteness requirement [20]

\sum_{i = x, y} \sum_{j = x, y} \iint f_{i}^{*} (s_{1}) f_{j} (s_{2}) W_{i j}^{(0)} (s_{1}, s_{2}, ω) d s_{1} d s_{2} \geq 0,

where

f_{x} (s)

and

f_{y} (s)

are two arbitrary, well-behaved functions. Gori provided a sufficient and necessary condition to determine whether the cross-spectral density matrix of a Schell-type partially coherent electromagnetic source satisfies the non-negative definiteness requirement [23]. According to Gori’s method, suppose that

{\tilde{μ}}_{i j} (η)

represents the Fourier transformation of

μ_{i j} (ρ)

, i.e.

{\tilde{μ}}_{i j} (η) = \int μ_{i j} (ρ) \exp (- i 2 π η \cdot ρ) d^{2} ρ, (ρ \equiv s_{1} - s_{2}),

then, for the cross-spectral density matrix to be non-negatively definite, the following three inequalities must be met for arbitrary values of

η_{x}

and

η_{y}

:

{\begin{cases} {\tilde{μ}}_{x x} (η_{x}, η_{y}) \geq 0 \\ {\tilde{μ}}_{y y} (η_{x}, η_{y}) \geq 0 \\ | {\tilde{μ}}_{x y} (η_{x}, η_{y}) | \leq \sqrt{{\tilde{μ}}_{x x} (η_{x}, η_{y}) {\tilde{μ}}_{y y} (η_{x}, η_{y})} \end{cases} .

Substituting the expressions for $μ_{x x}$ , $μ_{y y}$ , and $μ_{x y}$ which can be obtained from Eqs. (5)-(7) into Eq. (10) to get the corresponding Fourier transformations, and then applying these Fourier transformations in Eq. (11), one can find that, as long as the coherence parameters $δ_{0 x}^{(x)}$ , $δ_{0 y}^{(x)}$ , $δ_{0 x}^{(y)}$ , and $δ_{0 y}^{(y)}$ take real positive values, the first two inequalities will be satisfied spontaneously. For the third inequality to be satisfied, the following inequality should be further met:

\begin{array}{l} | B_{x y} | δ_{0 x}^{(n)} δ_{0 y}^{(n)} \exp [- 2 π^{2} (δ_{0 x}^{(n) 2} η_{x}^{2} + δ_{0 y}^{(n) 2} η_{y}^{2})] \\ \leq \sqrt{δ_{0 x}^{(x)} δ_{0 y}^{(x)} δ_{0 x}^{(y)} δ_{0 y}^{(y)}} \exp {- π^{2} [(δ_{0 x}^{(x) 2} + δ_{0 x}^{(y) 2}) η_{x}^{2} + (δ_{0 y}^{(x) 2} + δ_{0 y}^{(y) 2}) η_{y}^{2}]} . \end{array}

It can be easily seen that, for the inequality in Eq. (12) to be sound for arbitrary values of $η_{x}$ and $η_{y}$ , the following three inequalities should be satisfied:

{\begin{cases} | B_{x y} | δ_{0 x}^{(n)} δ_{0 y}^{(n)} \leq \sqrt{δ_{0 x}^{(x)} δ_{0 y}^{(x)} δ_{0 x}^{(y)} δ_{0 y}^{(y)}} \\ 2 δ_{0 x}^{(n) 2} \geq (δ_{0 x}^{(x) 2} + δ_{0 x}^{(y) 2}) \\ 2 δ_{0 y}^{(n) 2} \geq (δ_{0 y}^{(x) 2} + δ_{0 y}^{(y) 2}) \end{cases} .

In summary, as the coherence parameters would definitely be positive, provided that the three inequalities in Eq. (13) are satisfied, the cross-spectral density matrix given by Eqs. (5)-(8) would surely satisfy the non-negative definiteness requirement, thus the source described by it can be realized physically.

In terms of the equal point ( $s_{1} = s_{2} = s$ ) cross-spectral density matrix, the degree of polarization at a point $s$ on the source plane is given by [1, 17, 20]

P (s) = \sqrt{1 - \frac{4 \det W (s, s, ω)}{{[T r W (s, s, ω)]}^{2}}},

where

\det W

and

T r W

are the determinant and the trace of the matrix

W (s, s, ω)

, respectively. Letting

s_{1} = s_{2} = s

in Eqs. (5)-(8), substituting the resulted expressions into Eq. (14), and after a further arrangement, we find the degree of polarization of our anisotropic electromagnetic GSM source is given by the formula

P (s) = \sqrt{1 - \frac{4 (A_{x}^{2} / A_{y}^{2}) (1 - {| B_{x y} |}^{2})}{{\begin{cases} (A_{x}^{2} / A_{y}^{2}) \exp [\frac{1}{4} (\frac{1}{σ_{0 x}^{(y) 2}} - \frac{1}{σ_{0 x}^{(x) 2}}) ξ^{2} + \frac{1}{4} (\frac{1}{σ_{0 y}^{(y) 2}} - \frac{1}{σ_{0 y}^{(x) 2}}) η^{2}] \\ + \exp [- \frac{1}{4} (\frac{1}{σ_{0 x}^{(y) 2}} - \frac{1}{σ_{0 x}^{(x) 2}}) ξ^{2} - \frac{1}{4} (\frac{1}{σ_{0 y}^{(y) 2}} - \frac{1}{σ_{0 y}^{(x) 2}}) η^{2}] \end{cases}}^{2}}} .

From Eq. (15) one can see that, by choosing the values of the factors $(1 / σ_{0 x}^{(y) 2} - 1 / σ_{0 x}^{(x) 2})$ and $(1 / σ_{0 y}^{(y) 2} - 1 / σ_{0 y}^{(x) 2})$ appropriately, an anisotropic electromagnetic GSM source can have various polarization distribution patterns formed by points with equal degree of polarization. These polarization distribution patterns generally fall into four categories:

(I) Ellipse-shaped patterns. For a polarization distribution pattern to be ellipse-shaped, the intensity distribution parameters should meet one of the following two conditions: ${\begin{cases} \frac{1}{σ_{0 x}^{(y) 2}} - \frac{1}{σ_{0 x}^{(x) 2}} > 0 \\ \frac{1}{σ_{0 y}^{(y) 2}} - \frac{1}{σ_{0 y}^{(x) 2}} > 0 \\ \frac{1}{σ_{0 x}^{(y) 2}} - \frac{1}{σ_{0 x}^{(x) 2}} \neq \frac{1}{σ_{0 y}^{(y) 2}} - \frac{1}{σ_{0 y}^{(x) 2}} \end{cases},$
or
${\begin{cases} \frac{1}{σ_{0 x}^{(y) 2}} - \frac{1}{σ_{0 x}^{(x) 2}} < 0 \\ \frac{1}{σ_{0 y}^{(y) 2}} - \frac{1}{σ_{0 y}^{(x) 2}} < 0 \\ \frac{1}{σ_{0 x}^{(y) 2}} - \frac{1}{σ_{0 x}^{(x) 2}} \neq \frac{1}{σ_{0 y}^{(y) 2}} - \frac{1}{σ_{0 y}^{(x) 2}} \end{cases} .$
(II) Circle-shaped patterns. For a polarization distribution pattern to be circle-shaped, the intensity distribution parameters should satisfy the following condition: $\frac{1}{σ_{0 x}^{(y) 2}} - \frac{1}{σ_{0 x}^{(x) 2}} = \frac{1}{σ_{0 y}^{(y) 2}} - \frac{1}{σ_{0 y}^{(x) 2}} \neq 0.$
(III) Hyperbolic-curve-shaped patterns. For a polarization distribution pattern to be hyperbolic-curve-shaped, one of the following two conditions should be satisfied by the intensity distribution parameters: ${\begin{cases} \frac{1}{σ_{0 x}^{(y) 2}} - \frac{1}{σ_{0 x}^{(x) 2}} > 0 \\ \frac{1}{σ_{0 y}^{(y) 2}} - \frac{1}{σ_{0 y}^{(x) 2}} < 0 \end{cases},$
or
${\begin{cases} \frac{1}{σ_{0 x}^{(y) 2}} - \frac{1}{σ_{0 x}^{(x) 2}} < 0 \\ \frac{1}{σ_{0 y}^{(y) 2}} - \frac{1}{σ_{0 y}^{(x) 2}} > 0 \end{cases} .$
(IV) Homogeneous distribution patterns. Lastly, for a polarization distribution pattern to be uniform on the source plane, the intensity distribution parameters should meet the following condition: ${\begin{cases} σ_{0 x}^{(x)} = σ_{0 x}^{(y)} \\ σ_{0 y}^{(x)} = σ_{0 y}^{(y)} \end{cases} .$

3 The far-field polarization properties of an anisotropic electromagnetic GSM beam propagating through turbulent atmosphere

According to the extended Huygens-Fresnel method [24, 25], the elements of the equal-point cross-spectral density matrix of a partially coherent electromagnetic beam at propagation distance $z$ in turbulent atmosphere can be calculated by

\begin{array}{l} W_{i j} (r, r, z, ω) = {(\frac{k}{2 π z})}^{2} \iint W_{i j}^{(0)} (s_{1}, s_{2}, ω) \exp {\frac{- i k}{2 z} [{(r - s_{1})}^{2} - {(r - s_{2})}^{2}]} \\ \times 〈 exp[ψ (r, s_{1}, z) + ψ^{*} (r, s_{2}, z)] 〉 d s_{1} d s_{2} . \end{array}

Here $r$ is the position vector of a point on the observation plane, $k$ is the vacuum wave number of the light, $ψ (r, s, z)$ is the random complex phase imposed by the turbulent atmosphere on a spherical wave propagating from the point $(s, 0)$ to the point $(r, z)$ , and $〈〉$ denotes taking ensemble average of this random complex phase. The ensemble average term can be explicitly expressed as

\begin{array}{l} 〈 exp[ψ (r, s_{1}, z) + ψ^{*} (r, s_{2}, z)] 〉 \\ = \exp {- 4 π^{2} k^{2} z \int_{0}^{1} \int_{0}^{\infty} κ Φ_{n} (κ) [1 - J_{0} (κ ξ | Q |)] d κ d ξ} \\ = \exp [- \frac{1}{2} D_{s p} (| Q |, z)], \end{array}

where

Φ_{n} (κ)

is the spatial power spectral density of the refractive-index fluctuations of the turbulent atmosphere,

J_{0} (•)

is the Bessel function of first kind and zero order,

D_{s p} (| Q |, z)

is structure function of the random complex phase, and

Q = s_{1} - s_{2}

. Assuming a Kolmogorov turbulent spectrum [25], it follows from Eq.(23) that

D_{s p} (| Q |, z) = 1.09 C_{n}^{2} k^{2} z Q^{5 / 3},

where

C_{n}^{2}

is the refractive-index structure parameter, and can be considered constant for a horizontal propagation path, with its value as a measure of the strength of the refractive index fluctuations. However, it is impossible to obtain an analytical result of Eq. (22) if Eqs. (23) and (24) are used directly in it. In order to obtain an analytical result of Eq. (22), we adopt the quadratic approximation proposed by Leader [26] in Eq. (24), i.e., approximating the exponent 5/3 in Eq. (24) by the exponent 2, and find that

〈 exp[ψ (r, s_{1}, z) + ψ^{*} (r, s_{2}, z)] 〉 = \exp (- \frac{Q^{2}}{ρ_{0}^{2}}),

where

ρ_{0} = {(0.545 C_{n}^{2} k^{2} z)}^{- 3 / 5}

is the transverse coherence length of a spherical wave propagating in atmospheric turbulence. Because the quadratic approximation can ensure the acquisition of analytic results in most cases, it is widely used by researchers to study the propagation properties of various beams in turbulent atmosphere [8-10, 12-16, 27-43]. By employing partially coherent fundamental, hyperbolic, sinusoidal, and annular Gaussian beams, Eyyuboglu [44] once showed that no substantially different receiver intensity profiles will be obtained via the use of the quadratic approximation and the use of the exact five-thirds power law.

Substituting Eqs. (5)-(7) and Eq. (25) into Eq. (22), changing the variables of integration from $s_{1}$ and $s_{2}$ to ${s^{'}}_{1}$ and ${s^{'}}_{2}$ by making

{\begin{cases} s_{2} + s_{1} = 2 {s^{'}}_{1} \\ s_{2} - s_{1} = {s^{'}}_{2} \end{cases},

and also making use of the integral formula

\int_{- \infty}^{\infty} \exp (- z_{1} t^{2}) \exp (- i z_{2} t) d t = \sqrt{\frac{π}{z_{1}}} \exp (- \frac{z_{2}^{2}}{4 z_{1}}), (Re z_{1} > 0),

after time-consuming calculations and arrangements, we obtain

W_{x x} (x, y, z, ω) = \frac{A_{x}^{2}}{M_{x}^{(x)} M_{y}^{(x)}} \exp (- \frac{x^{2}}{2 σ_{0 x}^{(x) 2} M_{x}^{(x) 2}}) \exp (- \frac{y^{2}}{2 σ_{0 y}^{(x) 2} M_{y}^{(x) 2}}),

with

\begin{array}{l} [^{M_{x}^{(x)} (z)] 2} = 1 + \frac{z^{2}}{k^{2} σ_{0 x}^{(x) 2}} (\frac{1}{4 σ_{0 x}^{(x) 2}} + \frac{1}{δ_{0 x}^{(x) 2}} + \frac{2}{ρ_{0}^{2}}), \\ [^{M_{y}^{(x)} (z)] 2} = 1 + \frac{z^{2}}{k^{2} σ_{0 y}^{(x) 2}} (\frac{1}{4 σ_{0 y}^{(x) 2}} + \frac{1}{δ_{0 y}^{(x) 2}} + \frac{2}{ρ_{0}^{2}}); \\ W_{y y} (x, y, z, ω) = \frac{A_{y}^{2}}{M_{x}^{(y)} M_{y}^{(y)}} \exp (- \frac{x^{2}}{2 σ_{0 x}^{(y) 2} M_{x}^{(y) 2}}) \exp (- \frac{y^{2}}{2 σ_{0 y}^{(y) 2} M_{y}^{(y) 2}}), \end{array}

with

{[M_{x}^{(y)} (z)]}^{2} = 1 + \frac{z^{2}}{k^{2} σ_{0 x}^{(y) 2}} (\frac{1}{4 σ_{0 x}^{(y) 2}} + \frac{1}{δ_{0 x}^{(y) 2}} + \frac{2}{ρ_{0}^{2}}),

{[M_{y}^{(y)} (z)]}^{2} = 1 + \frac{z^{2}}{k^{2} σ_{0 y}^{(y) 2}} (\frac{1}{4 σ_{0 y}^{(y) 2}} + \frac{1}{δ_{0 y}^{(y) 2}} + \frac{2}{ρ_{0}^{2}});

and

\begin{array}{l} W_{x y} (x, y, z, ω) = W_{y x}^{*} (x, y, z, ω) \\ = \frac{A_{x} A_{y} B_{x y}}{M_{x}^{(n)} M_{y}^{(n)}} \exp [- \frac{x^{2}}{2 σ_{x}^{(n) 2} M_{x}^{(n)}^{2}}] \exp [- \frac{y^{2}}{2 σ_{y}^{(n) 2} M_{y}^{(n)}^{2}}], \end{array}

with

σ_{x}^{(n) 2} = \frac{2 σ_{0 x}^{(x) 2} σ_{0 x}^{(y) 2}}{σ_{0 x}^{(x) 2} + σ_{0 x}^{(y) 2}},

σ_{y}^{(n) 2} = \frac{2 σ_{0 y}^{(x) 2} σ_{0 y}^{(y) 2}}{σ_{0 y}^{(x) 2} + σ_{0 y}^{(y) 2}},

{[M_{x}^{(n)} (z)]}^{2} = 1 + \frac{z^{2}}{k^{2} σ_{x}^{(n) 2}} [\frac{1}{2 (σ_{0 x}^{(x) 2} + σ_{0 x}^{(y) 2})} + \frac{1}{δ_{0 x}^{(n) 2}} + \frac{2}{ρ_{0}^{2}}] - \frac{i z}{k} \frac{σ_{0 x}^{(y) 2} - σ_{0 x}^{(x) 2}}{2 σ_{0 x}^{(x) 2} σ_{0 x}^{(y) 2}},

{[M_{y}^{(n)} (z)]}^{2} = 1 + \frac{z^{2}}{k^{2} σ_{y}^{(n) 2}} [\frac{1}{2 (σ_{0 y}^{(x) 2} + σ_{0 y}^{(y) 2})} + \frac{1}{δ_{0 y}^{(n) 2}} + \frac{2}{ρ_{0}^{2}}] - \frac{i z}{k} \frac{σ_{0 y}^{(y) 2} - σ_{0 y}^{(x) 2}}{2 σ_{0 y}^{(x) 2} σ_{0 y}^{(y) 2}} .

When the propagation distance is so far that the approximations

{[M_{x}^{(x)} (z)]}^{2} \approx 2 z^{2} ρ_{0}^{- 2} / k^{2} σ_{0 x}^{(x) 2}

,

{[M_{y}^{(x)} (z)]}^{2} \approx 2 z^{2} ρ_{0}^{- 2} / k^{2} σ_{0 y}^{(x) 2}

,

{[M_{x}^{(y)} (z)]}^{2} \approx 2 z^{2} ρ_{0}^{- 2} / k^{2} σ_{0 x}^{(y) 2}

,

{[M_{y}^{(y)} (z)]}^{2} \approx 2 z^{2} ρ_{0}^{- 2} / k^{2} σ_{0 y}^{(y) 2}

,

{[M_{x}^{(n)} (z)]}^{2} \approx 2 z^{2} ρ_{0}^{- 2} / k^{2} σ_{x}^{(n) 2}

, and

{[M_{y}^{(n)} (z)]}^{2} \approx 2 z^{2} ρ_{0}^{- 2} / k^{2} σ_{y}^{(n) 2}

can be made, we think that the beam enters the far field of its polarization distribution. In the far-field region, we have

W_{x x}^{(f f)} (r, z) = \frac{A_{x}^{2} k^{2} σ_{0 x}^{(x)} σ_{0 y}^{(x)}}{2 z^{2} ρ_{0}^{- 2}} \exp (- \frac{k^{2} r^{2}}{4 z^{2} ρ_{0}^{- 2}}),

W_{y y}^{(f f)} (r, z) = \frac{A_{y}^{2} k^{2} σ_{0 x}^{(y)} σ_{0 y}^{(y)}}{2 z^{2} ρ_{0}^{- 2}} \exp (- \frac{k^{2} r^{2}}{4 z^{2} ρ_{0}^{- 2}}),

W_{x y}^{(f f)} (r, z) = \frac{A_{x} A_{y} B_{x y} k^{2} σ_{x}^{(n)} σ_{y}^{(n)}}{2 z^{2} ρ_{0}^{- 2}} \exp (- \frac{k^{2} r^{2}}{4 z^{2} ρ_{0}^{- 2}}),

W_{y x}^{(f f)} (r, z) = \frac{A_{x} A_{y} B_{x y}^{*} k^{2} σ_{x}^{(n)} σ_{y}^{(n)}}{2 z^{2} ρ_{0}^{- 2}} \exp (- \frac{k^{2} r^{2}}{4 z^{2} ρ_{0}^{- 2}}) .

Here the superscript letter “ $f f$ ” indicates the far-field region. Applying Eqs. (31)-(34) in the mathematical formula for the degree of polarization given by Eq. (14), we obtain the far-field degree of polarization of an anisotropic electromagnetic GSM beam in turbulent atmosphere as below:

P^{(f f) 2} = 1 - \frac{4 (A_{x}^{2} / A_{y}^{2}) (σ_{0 x}^{(x)} σ_{0 y}^{(x)} σ_{0 x}^{(y)} σ_{0 y}^{(y)} - {| B_{x y} |}^{2} σ_{x}^{(n) 2} σ_{y}^{(n) 2})}{{[(A_{x}^{2} / A_{y}^{2}) σ_{0 x}^{(x)} σ_{0 y}^{(x)} + σ_{0 x}^{(y)} σ_{0 y}^{(y)}]}^{2}} .

In what follows, making use of Eq. (35), we analyze separately the far-field polarization distribution properties of anisotropic electromagnetic GSM beams with non-uniform as well as uniform source polarization distributions in turbulent atmosphere.

When the source polarization distribution is non-uniform, the intensity distribution parameters will satisfy one of the conditions given by Eqs. (16)-(20). No matter which condition is satisfied, Eq. (35) cannot be further simplified. In other words, Eq. (35) is the ultimate expression for the degree of polarization of the beam in the far field. We can see from Eq. (35) that, when an anisotropic electromagnetic GSM beam with a non-uniform polarization distribution on the source plane propagates in turbulent atmosphere, its far-field polarization distribution has two characteristics:

(I) The far-field degree of polarization is independent of the transverse coordinate, i.e., the far-field degree of polarization is uniformly distributed. Thus generally speaking, a partially coherent electromagnetic beam, having an inhomogeneous distribution of the degree of polarization on the source plane, cannot recover the source-plane polarization distribution in the far field when propagating through turbulent atmosphere.
(II) The far-field degree of polarization is determined by $A_{x}^{2} / A_{y}^{2}$ , $| B_{x y} |$ , and the source intensity parameters ( $σ_{0 x}^{(x)}$ , $σ_{0 y}^{(x)}$ , $σ_{0 x}^{(y)}$ , $σ_{0 y}^{(y)}$ ), but is independent of the source spatial coherence parameters ( $δ_{0 x}^{(x)}$ , $δ_{0 y}^{(x)}$ , $δ_{0 x}^{(y)}$ , $δ_{0 y}^{(y)}$ , $δ_{0 x}^{(n)}$ , $δ_{0 y}^{(n)}$ ) and the propagation distance $z$ . It should be emphasized here that, although the source spatial coherence properties have no influence on the far-field degree of polarization, because of the appearance of the source spatial coherence parameters in the elements of the equal-point cross-spectral density matrix at an intermediate propagation distance, the source spatial coherence properties will generally affect the transverse polarization distribution in the intermediate propagation region.

When the source polarization distribution is uniform, the intensity parameters satisfy the condition given by Eq. (21). Substituting Eq. (21) into Eq. (15), it can be further simplified as

P^{(0)} = \sqrt{1 - \frac{4 A_{x}^{2} / A_{y}^{2} (1 - {| B_{x y} |}^{2})}{{(1 + A_{x}^{2} / A_{y}^{2})}^{2}}};

upon substituting Eq. (21) into Eq. (35), we find the far-field degree of polarization is also given by the right hand side of Eq. (36). This indicates that, the source-plane polarization distribution is regained in the far field.

We would like to make a short summary about the far-zone polarization properties of anisotropic electromagnetic GSM beams we have found so far. For an anisotropic electromagnetic GSM beam propagating in turbulent atmosphere, we find, if its source intensity parameters satisfy one of the conditions given in Eqs.(16)-(20), the degree of polarization is inhomogeneous on the source plane while it is homogenous on a transverse plane in the far field. On the other hand, if the source intensity parameters satisfy the condition given by Eq.(21), the distribution of the degree of polarization is homogeneous on the source plane and it can be recovered in the far field. Here we would like to interpret that Eq.(21) means that the two intensity distributions corresponding to the two components of the source electric field are the same, or are different only for a constant parameter. Based on a complete knowledge of the far-field polarization properties of a kind of more general electromagnetic GSM beams, i.e., anisotropic electromagnetic GSM beams, we can arrive at a source-plane polarization distribution recovery condition for electromagnetic GSM beams propagating in turbulent atmosphere. More specifically, electromagnetic GSM beams can recover their source-plane polarization distributions in the far field when propagating in turbulent atmosphere if and only if the two intensity distributions corresponding to the two components of the source electric field are the same, or are different only for a constant parameter. This condition is valid no matter whether the two kinds of source intensity distributions are isotropic or anisotropic, and also no matter whether the three kinds of source spatial coherence distributions are same or different from one another, isotropic or anisotropic.

4 Evaluation of the validity of the far-field polarization properties obtained by using the quadratic approximation through numerical calculation of the exact quadruple integrations

All our conclusions about the far-field polarization properties of anisotropic electromagnetic GSM beams on propagation in turbulent atmosphere are based on the quadratic approximation. In this section, we evaluate the validity of our conclusions through making a comparison of the polarization distributions at excessively long distances obtained from the analytic formulas and obtained by numerically calculating the quadruple integrations in Eq. (22).

Choosing three different propagation distances, we calculate polarization curves in the x direction ( $y = 0$ ) as well as in the y direction ( $x = 0$ ) for four different anisotropic electromagnetic GSM beams in Figs. 1 and 2 . The solid curves represent the results calculated by the analytic formulas, and the dotted curves represent the results calculated by numerically evaluating the exact quadruple integrations. In our calculations, $A_{x}^{2} / A_{y}^{2} = 1$ , $| B_{x y} | = 0.5,$ $k = 10^{7} m^{- 1}$ , $C_{n}^{2} = 10^{- 14} m^{- 2 / 3}$ , and all these values keep unchanged. The two beams in Fig. 1(a) and Fig. 1(b) have completely same values of source intensity parameters and different values of source spatial coherence parameters, and similarly, the two beams in Fig. 2(a) and Fig. 2(b) have completely same values of source intensity parameters and different values of source spatial coherence parameters. For the two beams in Fig. 1, we choose their values of source intensity parameters to satisfy Eq.(20), thus the polarization distributions on the source plane are inhomogeneous, and have a hyperbolic-curve shape. Substituting the explicit values of related parameters given in the caption of Fig. 1 into Eq. (35), one can find that the far-field degree of polarization predicted by the approximate theory is 0.208. For the two beams in Fig. 2, we choose their values of source intensity parameters to satisfy Eq. (21), thus the polarization distributions on the source plane are homogeneous. Using Eq. (36), one can find that the degree of polarization on the source plane is 0.5. And according to the approximate theory, the degree of polarization in the far field in turbulent atmosphere is also 0.5.

Fig. 1 Comparison of the polarization curves calculated by the analytic formulas and by numerically evaluating the exact quadruple integrations, in $x$ ( $y = 0$ ) and $y$ ( $x = 0$ ) directions at three different propagation distances, of an anisotropic electromagnetic GSM beam propagating in turbulent atmosphere. The solid curves represent the results calculated by the analytic formulas, and the dotted curves represent the results calculated by exact numerical evaluations. For both beams in (a) and (b), the source intensity parameters are chosen as $σ_{0 x}^{(x)} = 0.02 m$ , $σ_{0 y}^{(x)} = 0.03 m$ , $σ_{0 x}^{(y)} = 0.09 m$ , $σ_{0 y}^{(y)} = 0.006 m$ . For the beam in (a), the source spatial coherence parameters are chosen as $δ_{0 x}^{(x)} = 0.002 m$ , $δ_{0 y}^{(x)} = 0.004 m$ , $δ_{0 x}^{(y)} = 0.006 m$ , $δ_{0 y}^{(y)} = 0.001 m$ , $δ_{0 x}^{(n)} = 0.0045 m$ , $δ_{0 y}^{(n)} = 0.003 m$ , and for the beam in (b), the source spatial coherence parameters are chosen as $δ_{0 x}^{(x)} = 0.003 m$ , $δ_{0 y}^{(x)} = 0.004 m$ , $δ_{0 x}^{(y)} = 0.001 m$ , $δ_{0 y}^{(y)} = 0.002 m$ , $δ_{0 x}^{(n)} = 0.0025 m$ , $δ_{0 y}^{(n)} = 0.0035 m$ .

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Fig. 2 Same as in Fig. 1, except that the source intensity parameters are chosen as $σ_{0 x}^{(x)} = σ_{0 x}^{(y)} = 0.04 m$ , $σ_{0 y}^{(x)} = σ_{0 y}^{(y)} = 0.02 m$ .

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We can see from Fig. 1 and Fig. 2 that, for both the polarization curves in the x direction and in the y direction, as the propagation distance gets longer, the difference between the approximate and the accurate curves gets smaller, and the transverse variations of the approximate and the accurate curves also become smaller. Especially, we can see that at the extremely long propagation distance ( $z = 5 * 10^{6} m$ ), the approximate and the accurate curves in both the x and y directions are distributed very near the value of 0.208 in Fig. 1, and they are distributed very near the value of 0.5 in Fig. 2. These features indicate that, all the far-field polarization properties obtained by the approximate theory, including the far-field distribution properties of the degree of polarization, the effects of source intensity parameters and source spatial coherence parameters on the far-field degree of polarization, can be observed from the accurate curves. Therefore, for the research aims of our paper, the quadratic approximation is feasible. In addition, compared with the accurate numerical calculation, we think the approximate theory is superior in two aspects: it is more time-saving and it can give the far-field polarization properties and the corresponding mathematical formulas clearly and directly.

Some additional explanations are provided here. When we chose the values of the source spatial coherence parameters for the beams in Figs. 1 and 2, we required that they satisfy the realizable conditions given by Eq.(13). Thus all the beams in Figs. 1 and 2 can be realized in practice. Although in Figs. 1 and 2 the beams need to propagate very long distances to enter the far field of their polarization distributions, through varying the values of source parameters as well as the value of the refractive-index structure parameter, we think there exist situations in which such long propagation distance is not necessary. In this sense, our research may be useful for practical applications.

5 Conclusions

In this paper, we investigated the far-zone polarization distribution properties of partially coherent electromagnetic beams with non-uniform polarization distributions on the source plane and propagating in turbulent atmosphere. And we proposed a condition for a partially coherent electromagnetic beam to regain its source polarization distribution in the far field in turbulent atmosphere.

For these purposes, we put forward a type of electromagnetic anisotropic GSM sources which can have non-uniform polarization distributions. We analyzed all the possible polarization distribution patterns of such sources and the corresponding condition for the appearance of each pattern. Then, by using the quadratic approximation, we derived the analytic formula for the far-field degree of polarization of an electromagnetic anisotropic GSM beam propagating in turbulent atmosphere. With this formula, we found that, those anisotropic electromagnetic GSM beams with non-uniform source polarization distributions will have uniform far-field polarization distributions, and the far-field degree of polarization is influenced by the source intensity parameters, but not by the source spatial coherence parameters. On the other hand, we found that, as many other beams, those anisotropic electromagnetic GSM beams with uniform source polarization distributions will regain these distributions in the far field. Combining the conclusions about the far-field polarization distribution properties of anisotropic electromagnetic GSM beams with uniform as well as non-uniform polarization distributions on the source plane, we arrived at a source-plane polarization distribution recovery condition. This condition states that, an anisotropic electromagnetic GSM beam can recover its source-plane polarization distribution in the far field if and only if the two intensity distributions corresponding to the two field components on the source plane are the same, or are different only for a constant parameter. The validity of this condition is unaffected by the intensity distribution and the spatial coherence distribution profiles of anisotropic electromagnetic GSM sources.

In order to evaluate the validity of all our conclusions obtained on the basis of the quadratic approximation, we also made a comparison of the polarization curves at very long propagation distances calculated by the analytic formulas and by numerically solving the quadruple integrations. The illustration plots clearly show that as the propagation distance becomes extremely long, the difference between the approximate results and the accurate results becomes negligibly small, indicating that all our conclusions derived from the approximate theory are reliable.

Acknowledgments

This work was supported by a grant from the National Natural Science Foundation of China under no.60938003.

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Far-zone polarization distribution properties of partially coherent beams with non-uniform source polarization distributions in turbulent atmosphere

Abstract

1. Introduction

2 Realizable conditions of an electromagnetic anisotropic GSM source and its polarization distribution patterns

3 The far-field polarization properties of an anisotropic electromagnetic GSM beam propagating through turbulent atmosphere

4 Evaluation of the validity of the far-field polarization properties obtained by using the quadratic approximation through numerical calculation of the exact quadruple integrations

5 Conclusions

Acknowledgments

References and links

Cited By

Figures (2)

Equations (42)

Optics Express