Twist phase-induced changes of the statistical properties of a stochastic electromagnetic beam propagating in a uniaxial crystal

Lin Liu; Yahong Chen; Lina Guo; Yangjian Cai

doi:10.1364/OE.23.012454

1. Introduction

In the past decades, partially coherent beams have been studied extensively and found applications in various fields [1–20]. Gaussian Schell-model (GSM) beam is a typical kind of scalar partially coherent beam, whose spectral intensity and degree of coherence satisfy Gaussian distributions [8, 9]. GSM beam can be generated with specially synthesized rough surfaces, spatial light modulators and synthetic acousto-optic holograms [10], and such beam also can be generated with the help of a rotating grounded glass and Gaussian amplitude filter [11, 12]. Through measuring the fourth-order correlation function of partially coherent beam, one can obtain the information of its degree of coherence and coherence width owing to the internal relation between second-order and fourth-order correlation functions [12].

One interesting property of partially coherent beam is that it can carry a twist phase, which can’t exist in a coherent beam. Twisted GSM beam (i.e., GSM beam with twist phase) was introduced by Simon and Mukunda [13]. Experimental demonstration of twisted GSM beam can be found in [14]. Due to the intrinsic chiral property of the twist phase, the beam spot of twisted GSM beam rotates on propagation [13, 15]. Twisted GSM beam can be represented through an incoherent superposition of partially coherent modified Bessel-Gauss beams [16]. Twisted GSM beam carries an orbital angular momentum (OAM) [17], and the analytical expression for its OAM on propagation can be found in [18]. Propagation of a twisted GSM beam can be treated by the Wigner distribution function [19] or the tensor method [3, 20]. The twist phase is useful for free-space optical communication [21–23] and second-harmonic generation [24]. A twist GSM beam can serve as illumination that may produce images with a resolution overcoming the Rayleigh limit [25].

Another interesting property of partially coherent beam is that its polarization and coherence are interrelated, and the degree of polarization of a vector partially coherent beam (e.g., stochastic electromagnetic beam or partially coherent and partially polarized beam) varies on propagation even in free space [26–28]. Since the unified theory of coherence and polarization was formulated by Wolf [29], numerous efforts have been devoted to stochastic electromagnetic beam [2, 7, 30–40]. Electromagnetic Gaussian Schell-model (EGSM) beam was introduced as a natural extension of scalar GSM beam [32, 33]. The realizability conditions for EGSM beam were discussed in [34, 35]. Propagation properties of an EGSM beam in free space, cavity, fiber and turbulent atmosphere were reported in [36–44], and it was found that EGSM beams have an advantage over scalar GSM beams for reducing turbulence-induced scintillation, which is useful for free-space optical communications and laser radar systems. Various methods have been proposed to generate stochastic electromagnetic beam [45, 46], and experimental measurement of the beam parameters of an EGSM beam was reported in [47]. Experimental demonstration of the coupling of an EGSM beam into a single-mode optical fiber was reported in [48]. EGSM beam also is useful for ghost imaging [49, 50]. Twisted EGSM beam (i.e., EGSM beam with twist phase) was introduced in [51], and it is found in [52] that twisted EGSM beam is useful for particle trapping.

The birefringent effect of uniaxial crystal is useful for designing polarizers, compensators and wave plates. Recently, more and more attention is being paid to the propagation of various beams in a uniaxial crystal and it is found that uniaxial crystal is useful for beam shaping [53–60]. Through solving Maxwell’s equation in crystal, one can treat the propagation of laser beams in a uniaxial crystal [53, 54]. Up to now, only few papers were devoted to study the propagation properties of an EGSM beam in a uniaxial crystal [57–60]. To our knowledge no results have been reported up until now on propagation of a twisted EGSM beams in a uniaxial crystal. In this paper, our aim is to treat the paraxial propagation of a twisted EGSM beam in a uniaxial crystal with the help of the tenor method, and explore the twist phase-induced changes of the statistical properties on propagation in a uniaxial crystal. Some interesting and useful results are found.

2. Analytical formula for the cross-spectral density matrix of a twisted EGSM beam propagating in a uniaxial crystal

Based on the unified theory of coherence and polarization, the second-order statistical properties of a stochastic electromagnetic beam can be characterized by the cross-spectral density (CSD) matrix of the electric field in the space-frequency domain, defined by the formula [2, 26–29]

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

where

W_{α β} (r_{1}, r_{2}; ω) = 〈 E_{α}^{*} (r_{1}; ω) E_{β} (r_{2}; ω) 〉, (α, β = x, y),

E_{x} and E_{y}

denote the fluctuating components of the random electric vector, at frequency ω, with respect to two mutually orthogonal x and y directions, perpendicular to the z axis.

r_{1}

and

r_{2}

are position vectors of two arbitrary points in the transverse plane. Here the asterisk denotes the complex conjugate and the angular brackets denote ensemble average.

The elements of the CSD matrix of a twisted EGSM beam are defined as follows [51]

\begin{array}{l} W_{α β} (r_{1}, r_{2}; ω) \\ = A_{α} A_{β} B_{α β} \exp [- \frac{r_{1}^{2}}{4 σ_{α}^{2}} - \frac{r_{2}^{2}}{4 σ_{β}^{2}} - \frac{{(r_{1} - r_{2})}^{2}}{2 δ_{α β}^{2}} - \frac{i k}{2} γ_{α β} {(r_{1} - r_{2})}^{T} J (r_{1} + r_{2})], (α = x, y; β = x, y), \end{array}

where

A_{α}

is the square root of the spectral density of electric field component

E_{α}

,

B_{α β} = | B_{α β} | \exp (i ϕ)

is the correlation coefficient between the

E_{x}

and

E_{y}

field components and satisfy the relation

B_{α β} = B_{β α}^{*}

.

σ_{α β} and δ_{α β}

denote the width of the spectral density and coherence width, respectively.

γ_{α β}

represents the twist factor and is limited by

μ_{α β}^{2} \leq 1 / (k^{2} δ_{α β}^{4})

if α = β due to the non-negativity requirement of

W_{α β} (r_{1}, r_{2}; ω)

.

k = 2 π / λ = ω / c

is the wave number. J is an antisymmetric matrix, i.e.,

J = (\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}) .

Equation (2) can be expressed in the following alternative tensor form [51]

W_{α β} (\tilde{r}; ω) = A_{α} A_{β} B_{α β} \exp (- {\tilde{r}}^{T} {\tilde{M}}_{0 α β}^{- 1} \tilde{r}), (α = x, y; β = x, y),

where

{\tilde{r}}^{T} = (r_{1}^{T}, r_{2}^{T}) = (x_{1}, y_{1}, x_{2}, y_{2})

and

{\tilde{M}}_{0 α β}^{- 1}

is a

4 \times 4

matrix expressed as

{\tilde{M}}_{0 α β}^{- 1} = (\begin{matrix} \frac{1}{2} (\frac{1}{2 σ_{a}^{2}} + \frac{1}{δ_{α β}^{2}}) I & - \frac{1}{2 δ_{α β}^{2}} I + \frac{i k}{2} γ_{α β} J \\ - \frac{1}{2 δ_{α β}^{2}} I + \frac{i k}{2} γ_{α β} J^{T} & \frac{1}{2} (\frac{1}{2 σ_{β}^{2}} + \frac{1}{δ_{α β}^{2}}) I \end{matrix}) .

Here I is

2 \times 2

unit matrix.

Now we study the paraxial propagation of a twisted EGSM beam in a uniaxial crystal orthogonal to the optical axis. The geometry of the propagation of a laser beam in a uniaxial crystal orthogonal to the optical axis is shown in Fig. 1. The optical axis of the crystal coincides with the x -axis, and the dielectric tensor of the crystal can be expressed as

ε = (\begin{matrix} n_{e}^{2} & 0 & 0 \\ 0 & n_{o}^{2} & 0 \\ 0 & 0 & n_{o}^{2} \end{matrix}),

where

n_{o}

and

n_{e}

are the ordinary and extraordinary refractive indices, respectively.

Fig. 1 Geometry of the propagation of a beam in a uniaxial crystal orthogonal to the optical axis.

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Assume that the twisted EGSM source is located in the plane $z = 0$ and radiates into the half-space $z > 0$ . The paraxial propagation of the elements of the CSD matrix of a twisted EGSM beam in a uniaxial crystal orthogonal to the optical axis can be treated by the following integral formulae [57–60]

\begin{array}{l} W_{x x} (ρ_{x 1}, ρ_{y 1}, ρ_{x 2}, ρ_{y 2}; ω) = \frac{k^{2} n_{o}^{2}}{4 π^{2} z^{2}} \iint \iint W_{x x} (x_{1}, y_{1}, x_{2}, y_{2}; ω) d x_{1} d y_{1} d x_{2} d y_{2} \\ \times \exp {\frac{k}{2 i z n_{e}} [n_{o}^{2} {(ρ_{x 1} - x_{1})}^{2} + n_{e}^{2} {(ρ_{y} - y_{1})}^{2}] - \frac{k}{2 i z n_{e}} [n_{o}^{2} {(ρ_{x 2} - x_{2})}^{2} + n_{e}^{2} {(ρ_{y 2} - y_{2})}^{2}]}, \end{array}

\begin{array}{l} W_{y y} (ρ_{x 1}, ρ_{y 1}, ρ_{x 2}, ρ_{y 2}; ω) = \frac{k^{2} n_{o}^{2}}{4 π^{2} z^{2}} \iint \iint W_{y y} (x_{1}, y_{1}, x_{2}, y_{2}; ω) d x_{1} d y_{1} d x_{2} d y_{2} \\ \times \exp {\frac{k n_{o}}{2 i z} [{(ρ_{x 1} - x_{1})}^{2} + {(ρ_{y} - y_{1})}^{2}] - \frac{k n_{o}}{2 i z} [{(ρ_{x 2} - x_{2})}^{2} + {(ρ_{y 2} - y_{2})}^{2}]}, \end{array}

\begin{array}{l} W_{x y} (ρ_{x 1}, ρ_{y 1}, ρ_{x 2}, ρ_{y 2}; ω) = \frac{k^{2} n_{o}^{2}}{4 π^{2} z^{2}} \exp [i k z (n_{o} - n_{e})] \iint \iint W_{x y} (x_{1}, y_{1}, x_{2}, y_{2}; ω) d x_{1} d y_{1} d x_{2} d y_{2} \\ \times \exp {\frac{k}{2 i z n_{e}} [n_{o}^{2} {(ρ_{x 1} - x_{1})}^{2} + n_{e}^{2} {(ρ_{y} - y_{1})}^{2}] - \frac{k n_{o}}{2 i z} [{(ρ_{x 2} - x_{2})}^{2} + {(ρ_{y 2} - y_{2})}^{2}]}, \end{array}

W_{y x} (ρ_{x 1}, ρ_{y 1}, ρ_{x 2}, ρ_{y 2}; ω) = W_{x y}^{*} (ρ_{x 2}, ρ_{y 2}, ρ_{x 1}, ρ_{y 1}; ω),

Equations (7)-(9) can be alternatively expressed in the following tenor form

W_{x x} (\tilde{ρ}; ω) = \frac{1}{π^{2} [\det {({\tilde{B}}_{x x})}^{1 / 2}]} \iint \iint W_{x x} (\tilde{r}; ω) \exp (- {\tilde{r}}^{T} {\tilde{B}}_{x x}^{- 1} \tilde{r} - {\tilde{ρ}}^{T} {\tilde{B}}_{x x}^{- 1} \tilde{ρ} + 2 {\tilde{r}}^{T} {\tilde{B}}_{x x}^{- 1} \tilde{ρ}) d \tilde{r},

W_{y y} (\tilde{ρ}; ω) = \frac{1}{π^{2} [\det {({\tilde{B}}_{y y})}^{1 / 2}]} \iint \iint W_{y y} (\tilde{r}; ω) \exp (- {\tilde{r}}^{T} {\tilde{B}}_{y y}^{- 1} \tilde{r} - {\tilde{ρ}}^{T} {\tilde{B}}_{y y}^{- 1} \tilde{ρ} + 2 {\tilde{r}}^{T} {\tilde{B}}_{y y}^{- 1} \tilde{ρ}) d \tilde{r},

\begin{array}{l} W_{x y} (\tilde{ρ}; ω) = \frac{A_{x} A_{y} B_{x y}}{π^{2} [\det {({\tilde{B}}_{x y})}^{1 / 2}]} \exp [i k z (n_{o} - n_{e})] \\ \times \iint \iint W_{x y} (\tilde{r}; ω) \exp (- {\tilde{r}}^{T} {\tilde{B}}_{x y}^{- 1} \tilde{r} - {\tilde{ρ}}^{T} {\tilde{B}}_{x y}^{- 1} \tilde{ρ} + 2 {\tilde{r}}^{T} {\tilde{B}}_{x y}^{- 1} \tilde{ρ}) d \tilde{r}, \end{array}

where

{\tilde{B}}_{x x} = [\begin{matrix} - \frac{2 i z n_{e}}{k n_{o}^{2}} & 0 & 0 & 0 \\ 0 & - \frac{2 i z}{k n_{e}} & 0 & 0 \\ 0 & 0 & \frac{2 i z n_{e}}{k n_{o}^{2}} & 0 \\ 0 & 0 & 0 & \frac{2 i z}{k n_{e}} \end{matrix}], {\tilde{B}}_{y y} = [\begin{matrix} - \frac{2 i z}{k n_{o}} & 0 & 0 & 0 \\ 0 & - \frac{2 i z}{k n_{o}} & 0 & 0 \\ 0 & 0 & \frac{2 i z}{k n_{o}} & 0 \\ 0 & 0 & 0 & \frac{2 i z}{k n_{o}} \end{matrix}], {\tilde{B}}_{x y} = [\begin{matrix} - \frac{2 i z n_{e}}{k n_{o}^{2}} & 0 & 0 & 0 \\ 0 & - \frac{2 i z}{k n_{e}} & 0 & 0 \\ 0 & 0 & \frac{2 i z}{k n_{o}} & 0 \\ 0 & 0 & 0 & \frac{2 i z}{k n_{o}} \end{matrix}] .

Substituting Eq. (4) into Eqs. (11)-(13), we obtain (after some vector integration and tensor operation) the following expressions for the elements of the CSD matrix of a twisted EGSM beam in the output plane

W_{x x} (\tilde{ρ}; ω) = \frac{A_{x}^{2} B_{x x}}{{[\det (\tilde{I} + {\tilde{B}}_{x x} {\tilde{M}}_{0 x x}^{- 1})]}^{1 / 2}} \exp (- {\tilde{ρ}}^{T} {\tilde{M}}_{1 x x}^{- 1} \tilde{ρ}),

W_{y y} (\tilde{ρ}; ω) = \frac{A_{y}^{2} B_{y y}}{{[\det (\tilde{I} + {\tilde{B}}_{y y} {\tilde{M}}_{0 y y}^{- 1})]}^{1 / 2}} \exp (- {\tilde{ρ}}^{T} {\tilde{M}}_{1 y y}^{- 1} \tilde{ρ}),

W_{x y} (\tilde{ρ}; ω) = \frac{A_{x} A_{y} B_{x y} \exp [i k z (n_{o} - n_{e})]}{{[\det (\tilde{I} + {\tilde{B}}_{x y} {\tilde{M}}_{0 x y}^{- 1})]}^{1 / 2}} \exp (- ρ^{T} {\tilde{M}}_{1 x y}^{- 1} ρ),

where

\tilde{I}

is

4 \times 4

unit matrix and

{\tilde{M}}_{1 x x}^{- 1} = {({\tilde{M}}_{0 x x} + {\tilde{B}}_{x x})}^{- 1}, {\tilde{M}}_{1 y y}^{- 1} = {({\tilde{M}}_{0 y y} + {\tilde{B}}_{y y})}^{- 1}, {\tilde{M}}_{1 x y}^{- 1} = {({\tilde{M}}_{0 x y} + {\tilde{B}}_{x y})}^{- 1} .

The spectral density of a stochastic electromagnetic beam at point $ρ$ in the output plane is defined as follows [2, 27]

I (ρ; ω) = Tr \overset{\leftrightarrow}{W} (ρ, ρ; ω) = W_{x x} (ρ, ρ; ω) + W_{y y} (ρ, ρ; ω) = I_{x} (ρ; ω) + I_{y} (ρ; ω) .

Here

I_{x}

and

I_{y}

denote the spectral densities of the x and y components of the beam.

The degree of polarization of a stochastic electromagnetic beam at point $ρ$ in the output plane is defined as follows [2, 27]

P (ρ; ω) = \sqrt{1 - \frac{4 D e t \overset{\leftrightarrow}{W} (ρ, ρ; ω)}{{[T r \overset{\leftrightarrow}{W} (ρ, ρ; ω)]}^{2}}},

The degree of coherence of a stochastic electromagnetic beam at a pair of transverse points with position vectors $ρ_{1} and ρ_{2}$ is defined by the formula [2, 27]

μ (ρ_{1}, ρ_{2}; ω) = \frac{Tr \overset{\leftrightarrow}{W} (ρ_{1}, ρ_{2}; ω)}{\sqrt{Tr \overset{\leftrightarrow}{W} (ρ_{1}, ρ_{1}; ω) Tr \overset{\leftrightarrow}{W} (ρ_{2}, ρ_{2}; ω)}} .

Equations (15)-(18) are the main analytical results of this paper. Applying Eqs. (15)-(21), one can study the statistical properties of a twisted EGSM beam propagating in a uniaxial crystal orthogonal to the optical axis conveniently.

3. Statistical properties of a twisted EGSM beam propagating in a uniaxial crystal orthogonal to the optical axis

In this section, we study the statistical properties of a twisted EGSM beam propagating in a uniaxial crystal orthogonal to the optical axis, and mainly discuss the twist phase-induced changes of the statistical properties.

In the following text, we only discuss the propagation of a twisted EGSM beam whose cross-spectral density matrices are diagonal. If the off-diagonal elements are to be included in the calculation, the realizability conditions of the source should relate the on- and off-diagonal twist factors. But these conditions are not known so far. Thus, for the convenience of analysis, the off-diagonal elements are not included in the following numerical examples. The parameters $λ, n_{o}, σ_{x}, σ_{y}$ are set as $λ = 1.064 μ m,$ $n_{o} = 2.616,$ $σ_{x} = σ_{y} = 100 μ m$ , $A_{x}^{2} = A_{y}^{2} = 0.5$ , $δ_{x x} = δ_{y y} = 3 μ m$ .

We calculate in Fig. 2 the normalized spectral densities $I_{x} / I_{x \max}, I_{y} / I_{y \max}, I / I_{\max}$ of an EGSM beam without twist phase at several propagation distances in isotropic medium ( $n_{e} = n_{o}$ ) and in Fig. 3 the normalized spectral densities $I_{x} / I_{x \max}, I_{y} / I_{y \max}, I / I_{\max}$ of a twisted EGSM beam at $z = 3000 μ m$ in isotropic medium for different values of the twist factors $γ_{x x}$ , $γ_{y y}$ . One finds from Fig. 2 that the distributions of the spectral densities $I_{x} / I_{x \max}, I_{y} / I_{y \max}, I / I_{\max}$ of an EGSM beam without twist phase on propagation in isotropic medium all have Gaussian beam profiles and exhibit circular symmetries, which are similar to their distributions in free space. From Fig. 3, we find that the twist factors only affect the widths of the spectral densities in isotropic medium, and don’t affect the symmetries of the spectral densities. Thus, it is hard for us to determine whether an EGSM beam carries twist phase or not from the distributions of the spectral densities in isotropic medium or in free space.

Fig. 2 Normalized spectral densities $I_{x} / I_{x \max}, I_{y} / I_{y \max}, I / I_{\max}$ of an EGSM beam without twist phase at several propagation distances in isotropic medium ( $n_{e} = n_{o}$ ).

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Fig. 3 Normalized spectral densities $I_{x} / I_{x \max}, I_{y} / I_{y \max}, I / I_{\max}$ of a twisted EGSM beam at $z = 3000 μ m$ in isotropic medium ( $n_{e} = n_{o}$ ) for different values of the twist factors $γ_{x x}$ , $γ_{y y}$ .

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Figure 4 shows the normalized spectral densities $I_{x} / I_{x \max}, I_{y} / I_{y \max}, I / I_{\max}$ of an EGSM beam without twist phase at several propagation distances in a uniaxial crystal ( $n_{e} = 1.5 n_{o}$ ). From Fig. 4, one finds that the distribution of the spectral density $I_{x} / I_{x \max}$ gradually becomes of elliptical symmetry on propagation in uniaxial crystal due to the anisotropic effect of the crystal, which is much different from its distribution in isotropic medium or in free space. The distribution of the spectral density $I_{y} / I_{y \max}$ remains circular symmetry on propagation in a uniaxial crystal. The distribution of the total spectral density of the EGSM beam without twist phase exhibits elliptical symmetry on propagation in uniaxial crystal, which is also different from its distribution in isotropic medium or in free space. The long-axis of the elliptical beam spot of $I_{x} / I_{x \max}$ or $I / I_{\max}$ is along x direction. Figures 5-7 show the normalized spectral densities $I_{x} / I_{x \max}, I_{y} / I_{y \max}, I / I_{\max}$ of a twisted EGSM beam at $z = 3000 μ m$ in a uniaxial crystal ( $n_{e} = 1.5 n_{o}$ ) for different values of the twist factors $γ_{x x}$ , $γ_{y y}$ . From Figs. 5 and 6, we find that the twist phase affects the distributions of the spectral densities of the EGSM beam. The twist factor $γ_{x x}$ affects the widths and orientation of the elliptical beam spot of the spectral density $I_{x} / I_{x \max}$ , and the twist factor $γ_{y y}$ affects the widths of the Gaussian beam spot of the spectral density $I_{y} / I_{y \max}$ . The orientation of the beam spot of the total spectral density $I / I_{\max}$ is only determined by $γ_{x x}$ , while the beam widths of the total spectral density $I / I_{\max}$ are affected by $γ_{x x}$ and $γ_{y y}$ together. Thus one may determine whether an EGSM beam carries twist factor $γ_{x x}$ or not with the help of a uniaxial crystal through measuring the orientation angle of the beam spot in the output plane.

Fig. 4 Normalized spectral densities $I_{x} / I_{x \max}, I_{y} / I_{y \max}, I / I_{\max}$ of an EGSM beam without twist phase at several propagation distances in a uniaxial crystal ( $n_{e} = 1.5 n_{o}$ ).

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Fig. 5 Normalized spectral densities $I_{x} / I_{x \max}, I_{y} / I_{y \max}, I / I_{\max}$ of a twisted EGSM beam at $z = 3000 μ m$ in a uniaxial crystal ( $n_{e} = 1.5 n_{o}$ ) for different values of the twist factors $γ_{x x}$ with $γ_{y y} = 0$ .

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Fig. 6 Normalized spectral densities $I_{x} / I_{x \max}, I_{y} / I_{y \max}, I / I_{\max}$ of a twisted EGSM beam at $z = 3000 μ m$ in a uniaxial crystal ( $n_{e} = 1.5 n_{o}$ ) for different values of the twist factors $γ_{y y}$ with $γ_{x x} = 0$ .

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Fig. 7 Normalized spectral densities $I_{x} / I_{x \max}, I_{y} / I_{y \max}, I / I_{\max}$ of a twisted EGSM beam at $z = 3000 μ m$ in a uniaxial crystal ( $n_{e} = 1.5 n_{o}$ ) for different values of the twist factors $γ_{x x}$ , $γ_{y y}$ .

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We calculate in Fig. 8 the distribution of the degree of polarization of a twisted EGSM beam at $z = 3000 μ m$ in isotropic medium ( $n_{e} = n_{o}$ ) for different values of the twist factors $γ_{x x}$ , $γ_{y y}$ and in Fig. 9 the distribution of the degree of polarization of a twisted EGSM beam at $z = 3000 μ m$ in a uniaxial crystal ( $n_{e} = 1.5 n_{o}$ ) for different values of the twist factors $γ_{x x}$ , $γ_{y y}$ . One finds from Fig. 8 that the distribution of the degree of polarization of a twisted EGSM beam after propagation in isotropic medium exhibits circular symmetry and the twist factors only affect the widths of the distribution. From Fig. 9, one finds that the distribution of the degree of polarization of a twisted EGSM beam after propagation in a uniaxial crystal exhibits non-circular symmetry, and the orientation angel of the distribution of the degree of polarization is controlled by the twisted factor $γ_{x x}$ , while its widths are affected by $γ_{x x}$ and $γ_{y y}$ together. Thus one may determine whether an EGSM beam carries twist factor $γ_{x x}$ or not with the help of a uniaxial crystal through measuring the orientation angle of the distribution of the degree of polarization in the output plane.

Fig. 8 Distribution of the degree of polarization of a twisted EGSM beam at $z = 3000 μ m$ in isotropic medium ( $n_{e} = n_{o}$ ) for different values of the twist factors $γ_{x x}$ , $γ_{y y}$ .

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Fig. 9 Distribution of the degree of polarization of a twisted EGSM beam at $z = 3000 μ m$ in a uniaxial crystal ( $n_{e} = 1.5 n_{o}$ ) for different values of the twist factors $γ_{x x}$ , $γ_{y y}$ .

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We calculate in Fig. 10 the distribution of the degree of coherence of a twisted EGSM beam at $z = 3000 μ m$ in isotropic medium ( $n_{e} = n_{o}$ ) for different values of the twist factors $γ_{x x}$ , $γ_{y y}$ and Fig. 11 the distribution of the degree of coherence of a twisted EGSM beam at $z = 3000 μ m$ in a uniaxial crystal ( $n_{e} = 1.5 n_{o}$ ) for different values of the twist factors $γ_{x x}$ , $γ_{y y}$ . After propagation in isotropic medium, the distribution of the degree of coherence of a twisted EGSM beam exhibits circular symmetry and the twist factors only affect the widths of the distribution (see Fig. 10). From Fig. 11, one finds that the distribution of the degree of coherence of a twisted EGSM beam after propagation in a uniaxial crystal exhibits elliptical symmetry, and the orientation angel of the distribution of the degree of coherence is controlled by the twist factor $γ_{x x}$ , and its widths are affected by $γ_{x x}$ and $γ_{y y}$ together. Thus one may determine whether an EGSM beam carries twist factor $γ_{x x}$ or not with the help of a uniaxial crystal through measuring the orientation angle of the distribution of the degree of coherence in the output plane.

Fig. 10 Distribution of the degree of coherence of a twisted EGSM beam at $z = 3000 μ m$ in isotropic medium ( $n_{e} = n_{o}$ ) for different values of the twist factors $γ_{x x}$ , $γ_{y y}$ .

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Fig. 11 Distribution of the degree of coherence of a twisted EGSM beam at $z = 3000 μ m$ in a uniaxial crystal ( $n_{e} = 1.5 n_{o}$ ) for different values of the twist factors $γ_{x x}$ , $γ_{y y}$ .

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4. Summary

We have derived the analytical formula for the cross-spectral density matrix of a twisted EGSM beam propagating in a uniaxial crystal orthogonal to the optical axis, and studied the effect of twist phase on the statistical properties of an EGSM beam propagating in a uniaxial crystal. We have found that the spectral density, the degree of polarization and the degree of coherence of a twisted EGSM beam all exhibits non-circular symmetries and their orientation angles are determined by the twist phase, which are quite different from those of a twisted EGSM beam in isotropic medium or in free space. Thus one may determine whether an EGSM beam carries twist phase or not with the help of a uniaxial crystal through measuring the statistical properties in the output plane. The uniaxial crystal provides a convenient way for modulating the statistical properties of a stochastic electromagnetic beam, which may be useful in some applications, where partially coherent beam with prescribed properties are required.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant Nos.11274005, 11404067 and 11404234, the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, the Innovation Plan for Graduate Students in the Universities of Jiangsu Province under Grant No. KYLX-1218, the Key Lab Foundation of The Modern Optical Technology of Jiangsu Province, Soochow University, under Grant No. KJS1301, and the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

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Twist phase-induced changes of the statistical properties of a stochastic electromagnetic beam propagating in a uniaxial crystal

Abstract

1. Introduction

2. Analytical formula for the cross-spectral density matrix of a twisted EGSM beam propagating in a uniaxial crystal

3. Statistical properties of a twisted EGSM beam propagating in a uniaxial crystal orthogonal to the optical axis

4. Summary

Acknowledgments

References and links

Cited By

Figures (11)

Equations (21)

Optics Express