Electromagnetic random source for circular optical frame and its statistical properties

Xiayin Liu; Daomu Zhao

doi:10.1364/OE.23.016702

1. Introduction

It is well known in statistic optics that the structure of the source degree of coherence determines the intensity distribution of its far field, i.e., the famous reciprocity relation known to be of Fourier-type law [1]. This law has been used in developing various new model for source correlation function including Bessel correlated source [2, 3 ], Laguarre-Gaussian [4] and cosine-Gaussian [5] that lead to ring-shaped far-fields; two types of sinc Schell-model correlation being capable of producing tunable flat and dark hollow profile in the far field [6]; non-uniformly correlated source leading to a laterally shifted intensity maxima on propagation [7]; Multi-Gaussian Schell-model correlation generating shape-invariant flat-topped circular or rectangular intensity distribution in the far field [8, 9 ]. On these basis, recently, an analysis model whose source degree of coherence is the Fourier transform of two multi-Gaussian functions’ difference is established [10], which essentially consistent with the propositions that the difference between two cross-spectral density is a genuine cross-spectral density [11]. The novel model allows the radiation of circular/elliptical and square/rectangular frames in the far zone of its source, and the linear composition of the degree of coherence lead to the form of the nested frames. Generally, for these augmented models for random sources, the extension from scalar source to full electromagnetic domain is implemented, such as the electromagnetic cosine-Gaussian Schell-model beam [12], the electromagnetic non-uniformly correlated beams [13], the electromagnetic sinc Schell-model beam [14], and electromagnetic Rectangular Gaussian Schell model beam [15] and so on. The statistic properties of such beams including the spectral density, the spectral degree of coherence and polarization properties of these electromagnetic beams are shown to exhibit unique features. Therefore, it is worthwhile to investigate the statistic properties of electromagnetic random source for optical frame.

Since the unified theory of coherence and polarization presented by Wolf, in which it is demonstrated that the changes of the statistic properties of stochastic electromagnetic beams propagating in deterministic or random media can be determined by the $2 \times 2$ cross-spectral density matrix [16, 17 ], some interesting researches have been performed to explore the statistic properties of such beams passing through turbulence atmosphere [18–23 ]. Due to the vectorial nature, the electromagnetic beams are characterized not only by the state of the coherence but also by their polarization properties. It is shown that correlation-induced changes in the degree of polarization in free space and turbulence atmosphere are different [17, 18 ], and the behaviors of the degree of polarization of an isotropic beam and anisotropic electromagnetic Schell-model beams changes differently with the propagation distance in the homogeneous and isotropic turbulence atmosphere [20]. In addition, it is discovered that the degree of polarization and coherence of the source can affect the level of the scintillation index when the beams propagate in turbulence atmosphere and the unpolarized stochastic beams can reduce the scintillation index compared with polarized ones in atmospheric optical communications [24]. Therefore, the studies of the propagation characteristics of stochastic electromagnetic beam, especially the polarization property, are of importance in a turbulent atmosphere.

In this paper, the scalar model proposed in [10] is used as a building for developing electromagnetic source for circular optical frame. The main purpose is to explore the statistic properties for such electromagnetic source propagating in atmosphere turbulence. Unlike the dark hollow far-field produced by elegant Laguarre-Gaussian and dark hollow Gaussian Schell beam [25, 26 ], the thickness of a single circular frame constructed as a difference of two multi-Gaussian distribution can be adjusted by the upper index of the multi-Gaussian function. Thus, the flat-topped circular frame can be formed for larger value of upper index. In addition, for several overlapped frames, the robustness of outer frames can be controlled by the corresponding weights in linear combination of the degrees of coherence, which prevails over the partially coherent stand Laguarre-Gaussian [25]. In view of the unique characteristic of the circular frames and few reports about the polarization properties of the beams producing dark hollow far-field apart from the work [27, 28 ], it will be of interest to explore the statistic properties of electromagnetic source for circular frame. After the scalar source is extended to its electromagnetic domain, the statistics properties including spectral density, the degree of coherence and the degree of polarization for a single frame and two overlapped frames are comparatively analyzed in free-space and in non-Kolmogorov’s atmospheric turbulence with different fractal constant $α$ of the atmospheric spectrum and refractive-index structure constant ${\tilde{C}}_{n}^{2}$ .

2. Electromagnetic source for circular optical frame

The second-order correlation properties of a stationary electromagnetic source at two points $r_{10}$ and $r_{20}$ can be described in terms of the $2 \times 2$ electric cross-spectral density (CSD) matrix ${\hat{W}}^{(0)} (r_{10}, r_{20}; ω)$ , whose element $W_{i j}^{(0)} (r_{10}, r_{20}; ω)$ are given by [1, 16 ]

W_{i j}^{(0)} (r_{10}, r_{20}; ω) = 〈 E_{i}^{*} (r_{10}; ω) E_{j} (r_{20}; ω) 〉; (i, j = x, y),

where the angular brackets mean the ensemble average in the sense of coherence theory in space-frequency domain and asterisk stands for complex conjugate.

E_{i}

is the fluctuating component of the transverse electric vector along the i direction at point

r_{10}

, at the frequency

ω

. In the following analysis, the angular frequency dependence of all the quantities will be omitted but implied.

According to the requirement for non-negative definiteness for the elements of the CSD matrix [29], $W_{i j}^{(0)}$ needs to satisfy the following integral representation

W_{i j}^{(0)} (r_{10}, r_{20}) = \int p_{i j} (v) H_{i}^{*} (r_{10}, v) H_{j} (r_{20}, v) d v,

where

p_{i j} (v)

is a nonnegative, Fourier-transformable function which will define a family of sources with different degree of correlation function and

H_{i}^{*} (r_{10}, v) H_{j} (r_{20}, v)

is an arbitrary kernel, and on setting

H_{i} (r_{10}, v) = A_{i} \exp (- \frac{r_{10}^{2}}{4 σ_{i}^{2}}) \exp (- 2 π i v \cdot r_{10}),

H_{j} (r_{20}, v) = A_{j} \exp (- \frac{r_{20}^{2}}{4 σ_{j}^{2}}) \exp (- 2 π i v \cdot r_{20}),

where

A_{i}

and

A_{j}

are the average amplitudes of the

i

and

j

electric-field components respectively,

σ_{i}

and

σ_{j}

are the rms widths of the

i

and

j

electric-field components. Assuming for simplicity that

σ_{i} = σ_{j} = σ

and on substituting from Eqs. (3) and (4) into Eq. (2), the elements of the CSD matrix can be expressed

W_{i j}^{(0)} (r_{10}, r_{20}) = A_{i} A_{j} \exp [- \frac{r_{10}^{2} + r_{20}^{2}}{4 σ^{2}}] μ_{i j}^{(0)} (r_{10}, r_{20}),

where

μ_{i j}^{(0)} (r_{10}, r_{20})

is source degree of coherence, the Fourier transform of

p {}_{i j}{(v)}

. In order to produce a far field with circular ring profile,

p_{i j} (v)

is supposed to be of the form [10]

p {}_{i j}{(v)} = \frac{B_{i j} A_{0 i j}}{C_{0}} {\sum_{m = 1}^{M} (- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) [\exp (- \frac{m δ_{i j e}^{2} v^{2}}{2}) - \exp (- \frac{m δ_{i j o}^{2} v^{2}}{2})] .

Correspondingly, the

μ_{i j}^{(0)} (r_{10}, r_{20})

has the form

μ_{i j}^{(0)} (r_{10}, r_{20}) = \frac{B_{i j} A_{0 i j}}{C_{0}} \sum_{m = 1}^{M} \frac{{(- 1)}^{m - 1}}{\sqrt{m}} (\begin{matrix} M \\ m \end{matrix}) {\frac{1}{δ_{i j e}} \exp [- \frac{{(r_{20} - r_{10})}^{2}}{2 m δ_{i j e}^{2}}] - \frac{1}{δ_{i j o}} \exp [- \frac{{(r_{20} - r_{10})}^{2}}{2 m δ_{i j o}^{2}}]},

where

δ_{i j e}

and

δ_{i j o}

are outer and inner rms source correlation width, respectively, and

B_{i j}

is the single-point correlation coefficient between the ith and jth electric-field components,

C_{0} = \sum_{m = 1}^{M} \frac{{(- 1)}^{m - 1}}{\sqrt{m}} (\begin{matrix} M \\ m \end{matrix})

the normalization factor. From the condition that a physically genuine field must be quasi-Hermitian [30] it follows that

B_{x x} = B_{y y} = 1, | B_{x y} | = | B_{y x} |, δ_{x y} = δ_{y x} .

A_{0 i j}

is set to be

A_{0 i j} = {(\frac{1}{δ_{i j e}} - \frac{1}{δ_{i j o}})}^{- 1}

, which guarantees that the

μ_{i j}^{(0)} (r_{10}, r_{20})

takes on values unity for coinciding point.

Furthermore, according to the required inequalities for the nonnegative definiteness [29]

p_{i j} (v) \geq 0,

p_{x x} (v) p_{y y} (v) - p_{x y} (v) p_{y x} (v) \geq 0,

we find that

δ_{i j e} < δ_{i j o},

\begin{array}{l} B_{x x} A_{0 x x} {{[1 - \exp (- δ_{x x e}^{2} v^{2} / 2)]}^{M} - {[1 - \exp (- δ_{x x o}^{2} v^{2} / 2)]}^{M}} \\ \times B_{y y} A_{0 y y} {{[1 - \exp (- δ_{y y e}^{2} v^{2} / 2)]}^{M} - {[1 - \exp (- δ_{y y o}^{2} v^{2} / 2)]}^{M}}, \\ \geq {| B_{x y} A_{0 x y} |}^{2} {{[1 - \exp (- δ_{x y e}^{2} v^{2} / 2)]}^{M} - {[1 - \exp (- δ_{x y o}^{2} v^{2} / 2)]}^{M}}^{2} \end{array}

which imply

δ_{x y o} - δ_{x y e} \geq \min {δ_{x x o} - δ_{x x e}, δ_{y y o} - δ_{y y e}},

B_{x y} δ_{x y o} δ_{x y e} \leq \min {δ_{x x o} δ_{x x e}, δ_{y y o} δ_{y y e}},

\begin{array}{l} {[1 - \exp (- δ_{x y e}^{2} v^{2} / 2)]}^{M} - {[1 - \exp (- δ_{x y o}^{2} v^{2} / 2)]}^{M} \\ \leq \min {{[1 - \exp (- δ_{x x e}^{2} v^{2} / 2)]}^{M} - {[1 - \exp (- δ_{x x o}^{2} v^{2} / 2)]}^{M}, {[1 - \exp (- δ_{y y e}^{2} v^{2} / 2)]}^{M} - {[1 - \exp (- δ_{y y o}^{2} v^{2} / 2)]}^{M}} . \end{array}

From the inequalities (13)-(15), it can be found that the values of

δ_{x y o}

and

δ_{x y e}

depend on the values of

δ_{x x o}

,

δ_{x x e}

,

δ_{y y o}

,

δ_{y y e}

,

B_{x y}

and

M

. Though it is difficult to obtain a specific analytical formula for the choice of parameters, the values of

δ_{x y o}

and

δ_{x y e}

can be determined for different values of

δ_{x x o}

,

δ_{x x e}

,

δ_{y y o}

,

δ_{y y e}

,

B_{x y}

and

M

by numerical calculation.

3. Electromagnetic source for a single circular frame propagating in non-Kolmogorov’s atmospheric turbulence

Recall that generalized paraxial Huygens-Fresnel principle for the components of the CSD matrix that characterizes any electromagnetic beam propagating in free space or in any random linear medium at two points $r_{1}$ and $r_{2}$ in the transverse plane of the half-space $z > 0$ has form [20–22 ]

W_{i j} (r_{1}, r_{2}, z) = {(\frac{k}{2 π z})}^{2} \iint d^{2} r_{10} \iint d^{2} r_{20} W_{i j}^{(0)} (r_{10}, r_{20}) K (r_{10}, r_{20}, r_{1}, r_{2}),

where

\begin{array}{l} K (r_{10}, r_{20}, r_{1}, r_{2}) = \exp [- i k \frac{{(r_{1} - r_{10})}^{2} - {(r_{2} - r_{20})}^{2}}{2 z}] \\ \times \exp {- \frac{π^{2} k^{2} z}{3} [{(r_{1} - r_{2})}^{2} + (r_{1} - r_{2}) (r_{10} - r_{20}) + {(r_{10} - r_{20})}^{2}] \int_{0}^{\infty} κ^{3} Φ_{n} (κ) d κ}, \end{array}

where

Φ_{n} (κ)

is the spatial power spectrum of the refractive-index fluctuation of the isotropic turbulent medium, i.e [21].

Φ_{n} (κ) = A (α) {\tilde{C}}_{n}^{2} \exp [- (κ^{2} / κ_{m}^{2})] / {(κ^{2} + κ_{0}^{2})}^{α / 2}, 0 \leq κ \leq \infty, 3 < α < 4,

where

{\tilde{C}}_{n}^{2}

is a generalized refractive-index structure parameter with units

m^{3 - α}

,

κ_{0} = 2 π / L_{0}

and

κ_{m} = c (α) / l_{0}

,

L_{0}

and

l_{0}

being the outer and the inner scale of turbulence, and

c (α) = {[Γ (5 - α / 2) A (α) 2 π / 3]}^{1 / (α - 5)}

,

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

, with

Γ (x)

being the Gamma function. With the power spectrum in Eq. (18) the integral in Eq. (17) results in the expression

I = \int_{0}^{\infty} κ^{3} Φ_{n} (κ) d κ = \frac{A (α)}{2 (α - 2)} {\tilde{C}}_{n}^{2} [κ_{m}^{2 - α} β \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}],

where

β = 2 κ_{0}^{2} - 2 κ_{m}^{2} + a κ_{m}^{2}

, and

Γ (\cdot, \cdot)

is the incomplete Gamma function.

On substituting from Eqs. (17) and (19) into Eq. (16), the elements of the CSD matrix for the circular optical frame at distance $z$ are calculated to be

W_{i j} (r_{1}, r_{2}, z) = W_{i j e} (r_{1}, r_{2}, z) - W_{i j o} (r_{1}, r_{2}, z),

\begin{array}{l} W_{i j a} (r_{1}, r_{2}, z) = \frac{A_{i} A_{j} B_{i j} A_{0 i j}}{C_{0} δ_{i j a} Δ_{i j a} (z)} \sum_{m = 1}^{M} \frac{{(- 1)}^{m - 1}}{\sqrt{m}} (\begin{matrix} M \\ m \end{matrix}) \exp [- \frac{{(r_{1} + r_{2})}^{2}}{8 σ^{2} Δ_{i j a}^{2} (z)}] \exp [\frac{i k (r_{2}^{2} - r_{1}^{2})}{2 R_{i j a} (z)}] \\ \times \exp {- [\frac{1}{2 Δ_{i j a}^{2} (z) Ω_{i j a}^{2}} + \frac{1}{3} π^{2} k^{2} z I (1 + \frac{2}{Δ_{i j a}^{2} (z)}) - \frac{π^{4} k^{2} z^{4} I^{2}}{18 σ^{2} Δ_{i j a}^{2} (z)}] {(r_{1} - r_{2})}^{2}}, \\ (a = e, o), \end{array}

with

\frac{1}{Ω_{i j a}^{2}} = \frac{1}{4 σ^{2}} + \frac{1}{m δ_{i j a}^{2}}, Δ_{i j a}^{2} (z) = 1 + \frac{z^{2}}{k^{2} σ^{2} Ω_{i j a}^{2}} + \frac{2 π^{2} z^{3} I}{3 σ^{2}}, R_{i j a} (z) = \frac{σ^{2} Δ_{i j a}^{2} (z) z}{σ^{2} Δ_{i j a}^{2} (z) + \frac{1}{3} π^{2} z^{3} I - σ^{2}} .

4. Generalization to the circular frame combinations

In [10] it was pointed out that the linear superposition of the degree of coherence may lead to optical fields with the nested intensity combination of individual frames. In this case, the degree of coherence of the electromagnetic source for several frames is set to be a linear combination of those in Eq. (7)

μ_{i j} = B \sum_{n = 1}^{N} a_{n} μ_{n i j}, B = {(\sum_{n = 1}^{N} a_{n})}^{- 1},

where factor

B

guarantees that maximum value of the degree of coherence is unity.

a_{n}

is the weights which will manipulate the relative intensities of individual frame. N determines the number of the frames.

Due to the effect of linear superposition of the degree of coherence, the initial element of the CSD matrix of the source becomes the linear combination of the components defined in Eq. (5),

W_{i j}^{(0)} = B \sum_{n = 1}^{N} a_{n} W_{n i j}^{(0)}

where

\begin{array}{l} W_{n i j}^{(0)} (r_{10}, r_{20}) = A_{i} A_{j} \exp [\frac{r_{10}^{2} + r_{20}^{2}}{4 σ^{2}}] \frac{B_{n i j} A_{n 0 i j}}{C_{0}} \\ \times \sum_{m = 1}^{M} \frac{{(- 1)}^{m - 1}}{\sqrt{m}} (\begin{matrix} M \\ m \end{matrix}) {\frac{1}{δ_{n i j e}} \exp [- \frac{{(r_{20} - r_{10})}^{2}}{2 m δ_{n i j e}^{2}}] - \frac{1}{δ_{n i j o}} \exp [- \frac{{(r_{20} - r_{10})}^{2}}{2 m δ_{n i j o}^{2}}]} \end{array}

In order for the elements of nth CSD matrix to be a genuine one, the choice of the source parameters

B_{n i j}

,

δ_{n i j e}

and

δ_{n i j o}

must satisfy the inequalities (11) and (13)-(15).

The propagation law, for the components in Eq. (24), is the same with Eq. (16), so the specific process of derivation is omitted. It can be found that the elements of the CSD matrix of the source for the circular frame combinations propagating in turbulent atmosphere at the distance z are expressed

W_{i j} = B \sum_{n = 1}^{N} a_{n} W_{n i j},

W_{n i j} (r_{1}, r_{2}, z) = W_{n i j e} (r_{1}, r_{2}, z) - W_{n i j o} (r_{1}, r_{2}, z),

\begin{array}{l} W_{n i j a} (r_{1}, r_{2}, z) = \frac{A_{i} A_{j} B_{n i j} A_{n 0 i j}}{C_{0} δ_{n i j a} Δ_{n i j a} (z)} \sum_{m = 1}^{M} \frac{{(- 1)}^{m - 1}}{\sqrt{m}} (\begin{matrix} M \\ m \end{matrix}) \exp [- \frac{{(r_{1} + r_{2})}^{2}}{8 σ^{2} Δ_{n i j a}^{2} (z)}] \exp [\frac{i k (r_{2}^{2} - r_{1}^{2})}{2 R_{n i j a} (z)}] \\ \times \exp {- [\frac{1}{2 Δ_{n i j a}^{2} (z) Ω_{n i j a}^{2}} + \frac{1}{3} π^{2} k^{2} z I (1 + \frac{2}{Δ_{n i j a}^{2} (z)}) - \frac{π^{4} k^{2} z^{4} I^{2}}{18 σ^{2} Δ_{n i j a}^{2} (z)}] {(r_{1} - r_{2})}^{2}}, \end{array}

where

\begin{array}{l} A_{n 0 i j} = {(\frac{1}{δ_{n i j e}} - \frac{1}{δ_{n i j o}})}^{- 1}, \frac{1}{Ω_{n i j a}^{2}} = \frac{1}{4 σ^{2}} + \frac{1}{m δ_{n i j a}^{2}}, \\ Δ_{n i j a}^{2} (z) = 1 + \frac{z^{2}}{k^{2} σ^{2} Ω_{n i j a}^{2}} + \frac{2 π^{2} z^{3} I}{3 σ^{2}}, R_{n i j a} (z) = \frac{σ^{2} Δ_{n i j a}^{2} (z) z}{σ^{2} Δ_{n i j a}^{2} (z) + \frac{1}{3} π^{2} z^{3} I - σ^{2}} . \end{array}

When

N = 1

, it is just the case for a single circular frame as Eqs. (20) and (21) express. In the following section, utilizing the Eqs. (25)-(27) , we can directly study the changes in the spectral density, the degree of polarization and the degree of coherence of the electromagnetic source for circular frame combinations propagating through free space and turbulent atmosphere. The spectral density and the degree of polarization at the point

(r, z)

and the degree of coherence at a pairs of points

(r_{1}, z)

and

(r_{2}, z)

are given by the formulas [1, 16 ]

S (r, z) = Tr \overset{\leftrightarrow}{W} (r, r, z),

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

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

where Det and Tr denote the determinant and the trace of the CSD matrix with components (25).

5. Numerical examples

We will now illustrate the behaviors of the spectral density, degree of polarization and degree of coherence of the electromagnetic source for a single circular frame and two superposed circular frames ( $N = 2$ ) through the free space and atmospheric turbulence in greater detail. A comparative analysis between the cases of $N = 1$ and $N = 2$ is made. Considering that quite a few source parameters are needed to be set, here, we mainly analyze the statistics properties of the source with uncorrelated field components, i.e. with zero off-diagonal elements of the CSD matrix. The source and the medium parameters are set as follows: $k = 10^{7} m^{- 1}$ , $L_{0} = 1 m$ , $l_{0} = 1 mm$ , $σ = 10 mm$ , $λ = 632.8 nm$ , $δ_{1 x x e} = 1.5 mm$ , $δ_{1 x x o} = 10 mm$ , $δ_{1 y y e} = 2 mm$ , $δ_{1 y y o} = 8 mm$ , $δ_{2 x x e} = 0.4 mm$ , $δ_{2 x x o} = 0.8 mm$ , $δ_{2 y y e} = 0.1 mm$ , $δ_{2 y y o} = 0.6 mm$ , $A_{x} = 1.5$ , $A_{y} = 1.0$ , $a_{1} = 1$ and $a_{2} = 5$ .

Figure 1 shows the evolution of the spectral density in the axial direction (left) and the transverse distribution of the spectral density at the distance $z = 1 km$ (middle and right) in the turbulence atmosphere. One can see that with the transmission distance increasing Gaussian profile of the source field gradually self-splits into far fields with a single circular frame for $N = 1$ and two overlapped frames for $N = 2$ . It is shown that the dark center of the single frame or the inner one of two nested frames is gradually filled as the values of ${\tilde{C}}_{n}^{2}$ increase, i.e. the atmospheric turbulence starts to dominate. But the shape of the outer frame is slightly affected by the turbulent, the property being agree with the situation that the inner square frame resembles a circle while the shape of the outer one remains the same at certain distances [31].

Fig. 1 Evolution of the spectral density of the single circular optical frame (the top row) and the two overlapped circular frames (the bottom row) propagating in free space and turbulent atmosphere. (a) and (d) the longitudinal distribution, ${\tilde{C}}_{n}^{2} = 10^{- 13} m^{3 - α}$ , $α = 3.667$ and $M = 10$ . (b) and (e) the 2-Dimensional distribution at the plane $z = 1 km$ ; (c) and (f) the transverse distribution for different values of parameter ${\tilde{C}}_{n}^{2}$ .

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Figure 2 exhibits the transverse distribution of the spectral density of the circular optical frame at the propagation distance $z = 5 km$ for different values of parameters ${\tilde{C}}_{n}^{2}$ , $α$ and $M$ . Figure 2(a) shows that at larger distances the circular flat-topped frame can be formed in free space ( ${\tilde{C}}_{n}^{2} = 0$ ) for larger values of $M$ , but converts to be the circular frame in the atmosphere. By comparing Figs. 2(a) and 2(d) with Figs. 2(b) and 2(e), it can be found that the effect of the non-Kolmogorov turbulence ( $α = 3.001$ and $α = 3.1$ ) with local refractive-index fluctuation ${\tilde{C}}_{n}^{2} = 10^{- 13} m^{3 - α}$ on the frame is much stronger than that of the Kolmogorov turbulence ( $α = 3.667$ ). Especially, the single circular frame is completely reshaped to be Gaussian-like when $α = 3.1$ , which is shown to be an important value for other beams [14, 21 ]. Figures 2(c) and 2(f) show that the circular frame is less susceptible to the effect of atmospheric turbulence for larger values of $M$ , being similar with that of rectangular multi-Gaussian Schell model beam in atmospheric turbulent [32].

Fig. 2 The transverse distribution of the circular optical frame’s spectral density at the plane $z = 5 km$ for different parameters ${\tilde{C}}_{n}^{2}$ , $α$ and $M$ . (a) and (d) for different ${\tilde{C}}_{n}^{2}$ with $α = 3.667$ , $M = 20$ ; (b) and (e) for different $α$ with ${\tilde{C}}_{n}^{2} = 10^{- 13} m^{3 - α}$ , $M = 20$ ; (c) and (f) for different $M$ with ${\tilde{C}}_{n}^{2} = 10^{- 13} m^{3 - α}$ , $α = 3.667$ . The top row corresponding to the single circular frame and the bottom row corresponding to the two combined circular frame.

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The evolutions of on-axial degree of polarization of the circular frames in free space and atmospheric turbulence for difference values of parameters are demonstrated in Fig. 3 . It is found that the degree of polarization of the circular frame in free space increases to a certain value with the propagation distance increasing and keeps invariant, but returns to its initial value in the source plane after propagating a long distance in turbulent atmosphere. This behavior is in agreement with the cases of partially coherent electromagnetic beam and the electromagnetic sinc Schell model beam [14, 18–20 ]. It is observed that the degree of polarization experiences a different change of fluctuation, which essentially is attributed to the unique coherence property of the source. Figure 3(b) and 3(e) show that the fluctuation is suppressed and the humps disappear in the intermediate propagation range when $α = 3.1$ , and the degree of polarization returns to its initial value at the relatively shorter propagation distance. The degree of polarization of two overlapped circular frames is characterized by more humps during propagation, and it finally resumes the initial value at a much longer distance compared with the case of a single circular frame.

Fig. 3 The change in the spectral degree of polarization of the single circular frame (the top row) and the two superimposed ones (the bottom row) along z-axis for different parameters ${\tilde{C}}_{n}^{2}$ , $α$ and $M$ . (a) $α = 3.667$ , $M = 10$ ; (b) ${\tilde{C}}_{n}^{2} = 10^{- 13} m^{3 - α}$ , $M = 10$ ; (c) ${\tilde{C}}_{n}^{2} = 10^{- 13} m^{3 - α}$ , $α = 3.667$ .

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Figure 4 shows the transverse distribution of the degree of polarization at $z = 5 km$ for different values of parameters ${\tilde{C}}_{n}^{2}$ , $α$ and $M$ . It can be found that for different values of such parameters the change of the degree of polarization of a single circular frame versus the radial almost follows the same trend: first drop then rise gradually towards a saturated value of unity. Apart from some special points where the degree of polarization is independent on the choice of parameter ${\tilde{C}}_{n}^{2}$ and $α$ , the nearer the axial the off-axial point is, the more obvious the change of the degree of polarization is. Like the case of the distribution of spectral density, the behavior of the degree of polarization is less resistant to the affluence of non-Kolmogorov turbulence as Figs. (b) and (e) shows: when $α = 3.1$ , the central prominent part disappears. In addition, for the two superposed frames, the unpolarized off-axial points exist, which are related to the values of the parameters ${\tilde{C}}_{n}^{2}$ , $α$ and $M$ .

Fig. 4 The transverse distribution of the spectral degree of polarization of the single circular frame (the top row) and the two overlapped frame (the bottom row) at the plane $z = 5 km$ . (b) and (c) ${\tilde{C}}_{n}^{2} = 10^{- 14} m^{3 - α}$ , the other parameters set as Fig. 2.

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It is discovered that the linear combination of the source degree of coherence has no effect on the whole changing trend for the degree of coherence of the electromagnetic source propagating in free space and in the atmospheric turbulence. Therefore, we will only consider the evolutions of the degree of coherence of a single frame for different values of parameters ${\tilde{C}}_{n}^{2}$ , $α$ and $M$ . Figure 5 demonstrates the change of the modulus of the degree of coherence at two points (0, 0, z) and ( $r_{1} = 5 mm, r_{2} = 5 mm,$ z) along the z-axis. It can be found that the degree of coherence at the two field points in free space tends to unity in the far-zone, on the contrary, this quantity tends to zero at a certain distance in turbulent atmosphere. This phenomenon implies that the degree of coherence is also destroyed by the atmospheric turbulence, and the stronger turbulence causes the degree of coherence to tend to zero at shorter distances from the source. In addition, the degree of coherence takes on a fluctuation and attains minimum value (zero) at a special distance both in free space and in the atmosphere, which is well understood based on the fact that the element of the CSD matrix of the circular frame during propagation can be expressed to be the difference of two other ones as Eq. (20) shows.

Fig. 5 Change in the spectral degree of coherence of a single circular frame along the z-axis for different parameters as in Fig. 3.

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Figure 6 displays the transverse degree of coherence for the single circular frame in the plane $z = 1 km$ . It is shown that the profile of the degree of coherence of the single circular frame in free space is analogy to the intensity distribution of circular dark hollow beam at $z = 1.5 km$ in free space, i.e. these is a small bright ring around the brightest center [26]. This phenomenon can be elucidated by one of the reciprocity that the far-zone spectral degree of coherence is proportional to the spatial Fourier transform of the spectral intensity distribution across the source [1]. Meanwhile, it is found that the form of the spatial Fourier transform of the spectral intensity distribution across the source determined by Eq. (6) is similar to the expression of the electric field of the dark hollow beam [26]. However, in the atmosphere, the degree of coherence profile converts to be Gaussian shape, and the width of the Gaussian profile become smaller for a larger structure constant and is smallest when the slope $α = 3.1$ . Figure 6(c) shows that the degree of coherence is almost independent on the values of the summation index $M$ , which can be explained by the well-known reciprocity relation involving the Fourier-transform pairs between source and far-zone.

Fig. 6 Evolution of the transverse degree of coherence as a function of $r_{2}$ at the plane $z = 1 km$ for different parameters as in Fig. 5.

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6. Conclusions

In this article, we generalize the scalar source for circular optical frame to electromagnetic domain. The analytical formula for the electromagnetic source for circular frames combination on propagation in turbulent atmosphere is derived and specially used to explore the evolution of the statistics properties of a single circular frame and two superimposed ones. By numerical calculation, it is of interest to find that the degree of polarization experiences a change of oscillation, in particular that of the two nested frames is characterized by more humps. There is off-axis point and on-axis points where the degree of polarization and the degree of coherence is zero separately. In addition, it is illustrated that the characteristic parameters of the media and the source, including the refractive-index structure constant ${\tilde{C}}_{n}^{2}$ , the fractal constant $α$ of the atmospheric spectrum and the upper index in the source degree of coherence, have differently affected the spectral density, the degree of polarization and the degree of coherence of a single circular frame and two nested ones. For sufficiently large values of ${\tilde{C}}_{n}^{2}$ and for $α = 3.1$ , the influence of the atmosphere on such statistics properties is the strongest: the single flat-topped circular frame and the inner one of the two superimposed frames are shaped to be Gaussian-like, the humps characterizing the degree of polarization are suppressed and the degree of polarization returns to its initial value at the source plane at a earlier propagation distance, and the degree of coherence also tends to zero at a shorter distances from the source. However, for larger values of $M$ , such statistic characteristics are less susceptible to the effect of the atmospheric turbulence.

The results presented here indicate that the novel electromagnetic source for circular optical frames carries potential for practical application involving remote sensing, material surface processing, especially the polarization properties can be utilized to influence the behavior of performance parameter in an atmosphere optical communications link.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) (11274273 and 11474253) and the Fundamental Research Funds for the Central Universities (2015FZA3002).

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Electromagnetic random source for circular optical frame and its statistical properties

Abstract

1. Introduction

2. Electromagnetic source for circular optical frame

3. Electromagnetic source for a single circular frame propagating in non-Kolmogorov’s atmospheric turbulence

4. Generalization to the circular frame combinations

5. Numerical examples

6. Conclusions

Acknowledgments

References and links

Cited By

Figures (6)

Equations (32)

Optics Express