Electromagnetic multi-Gaussian Schell-model vortex light sources and their radiation field properties

Zhangrong Mei; Yonghua Mao; Yuyan Wang

doi:10.1364/OE.26.021992

1. Introduction

A multitude of model sources have been proposed in recent years to analyze random sources of any state of coherence and also to elucidate the properties of the fields that the sources generate [1]. Among them, the so-called Schell-model (SM) sources, whose degree of coherence at any two source points depended on the difference in the position vectors of the two points, were introduced in the scalar theory of coherence as a fundamental tool to describe partially coherent light sources [2]. The correlation function of a scalar SM source can be arbitrary, such as Gaussian SM sources [3], multi-Gaussian SM sources [4–6], cosine-Gaussian SM sources [7], sinc SM sources [8] et al, if only a sufficient and necessary condition is satisfied, that is the Fourier transform of their degree of coherence be a nonnegative function [9].

The scalar fields ignore the polarization properties which arise from the vector nature of the electromagnetic field. Actually, the states of coherence and polarization of a light field had been proven to be intimately related and can be treated in a unified theory of polarization and coherence [10]. Recently, great attention has been devoted the generalization of all kinds of scalar sources to the vector case. The electromagnetic Gaussian SM source is the earliest proposed and the simplest analytical model of a stochastic electromagnetic source [11], and has been widely studied on their generation and propagation characteristics [12–22]. Just as the scalar sources, the correlation matrix of the electromagnetic SM sources is also limited by the constraint of non-negative definiteness [23]. Roychowdhury and Korotkova established the conditions for the choice of parameters describing a physically realizable electromagnetic Gaussian SM source [24]. Gori and associates present a sufficient condition satisfying the nonnegative definiteness constraint for the whole class of electromagnetic SM sources [25]. The generalized superposition rule for any type of stochastic electromagnetic source, ensuring that the nonnegative definiteness constraint automatically satisfied, played a significant role in recent studies of partially coherence vector fields [26]. Some novel stochastic electromagnetic sources were devised using the prescriptions of [26], and the statistical characteristics of the beams generated by them on propagation in free space or random medium were explored [27–30]. However, the spatial correlation functions describing these models are expressed as real functions, leaving aside the phase properties of light field. In fact, the far-field statistical characteristics of stochastic beams are closely related to their phase properties of source field [31–36].

In this paper, we employ the generalized superposition rule to determine the mode structure of an electromagnetic multi-Gaussian Schell-model vortex source and of the far field that such a source produces. After some preliminary remarks about the modal theory of stochastic electromagnetic light fields, we give the general expression for the cross-spectral density (CSD) matrix of any electromagnetic Schell-model vortex source. And then, we specialize our analysis to a new class of stochastic electromagnetic vortex source, named electromagnetic multi-Gaussian Schell-model vortex source, and demonstrate how the coherence properties and topological charge of the source field affects the far-field statistical characteristics of the beams generated by this new source.

2. Electromagnetic multi-Gaussian Schell-model vortex light source

Consider a planar, secondary, electromagnetic source, located in the plane z = 0 and radiating into the half-space $z > 0$ . The second-order correlation properties of the source, at points specified by two-dimensional position vectors ${ρ^{'}}_{1}$ and ${ρ^{'}}_{2}$ and angular frequency $ω$ , may be characterized by the electric 2 × 2 cross-spectral density matrix [23]

{\hat{W}}^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}; ω) \equiv [\begin{matrix} W_{x x}^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}; ω) & W_{x y}^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}; ω) \\ W_{y x}^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}; ω) & W_{y y}^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}; ω) \end{matrix}] .

The matrix elements are given by

W_{α β}^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}; ω) = 〈 E_{α}^{*} ({ρ^{'}}_{1}; ω) E_{β} ({ρ^{'}}_{2}; ω) 〉, (α = x, y; β = x, y),

where

E_{α}

and

E_{β}

denote the fluctuating electric field components in two mutually orthogonal x- and y-axis, perpendicular to the z-axis, the asterisk denotes the complex conjugate and the angular brackets denote ensemble average. The polar coordinate and Cartesian coordinate of the vector of

ρ^{'}

are

(ρ^{'}, ϕ^{'})

and

(x^{'}, y^{'})

. From now on, the angular frequency

ω

dependence of all the quantities of interest will be omitted but implied for brevity.

As an evident from [23], the correlation function for optical fields cannot be chosen at will because of the non-negative definiteness constraints. According to the superposition rule for CSD matrices [26], this condition is fulfilled if the elements of the CSD matrix can be represented as superposition integral of the form:

W_{α β}^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}) = \int p_{α β} (v) H_{α}^{*} ({ρ^{'}}_{1}, v) H_{β} ({ρ^{'}}_{2}, v) d^{2} v,

where

H_{α} (ρ^{'}, v)

are arbitrary kernels and

p_{α β} (v)

are the elements of the weight matrix

\hat{p} (v)

. A classical class of CSD matrices, leading to the electromagnetic Schell-model sources, is obtained by giving

H_{α} (ρ^{'}, v)

a Fourier-like structure. More explicitly, we set

H_{α} (ρ^{'}, v) = F_{α} (ρ^{'}) \exp (- 2 π i v \cdot ρ^{'}),

where

F_{α} (ρ^{'})

are possible complex profile functions. In previous studies,

F_{α} (ρ^{'})

is usually simply set as a real Gaussian function, and there is no phase factor in the constructed CSD matrices. Here, we assume that the complex profile function in the source plane is a Laguerre-Gaussian mode with separable phase:

F_{α} (ρ^{'}) = A_{α} {(ρ^{'} / σ_{α})}^{l} e x p (- {ρ^{'}}^{2} / σ_{α}^{2}) e x p (- i l ϕ^{'}),

where

A_{α}

and

σ_{α}

are the characteristic amplitude and beam size of the field components in the source plane, and

l

is the topological charge. On substituting from Eq. (5) first into Eq. (4) and then into Eq. (3) we obtain

W_{α β}^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}) = A_{α} A_{β} {(\frac{{ρ^{'}}_{1} {ρ^{'}}_{2}}{σ_{α} σ_{β}})}^{l} \exp [- \frac{{ρ^{'}}_{1}^{2}}{σ_{α}^{2}} - {\frac{{ρ^{'}}_{2}^{2}}{σ_{β}^{2}}}^{2}] F {p_{α β} (v)} (| {ρ^{'}}_{1} - {ρ^{'}}_{2} |) \exp [i l ({ϕ^{'}}_{1} - {ϕ^{'}}_{2}_{1})],

where the symbol

F

indicates the Fourier transform of function

p_{α β} (v)

.

The correlation matrix with the elements (6) characterizes a source which generates the most general electromagnetic Schell-model vortex beam. The choice of $p_{α β} (v)$ defines a family of sources with different correlation functions. Let us suppose $p_{α β} (v)$ to be of the form [37]

\begin{array}{l} p_{α β} (v) = \frac{π B_{α β} δ_{α β}^{2}}{C^{2}} \sum_{m = 1}^{M} (\begin{matrix} M \\ m \end{matrix}) (^{- 1) m - 1} \exp (- m π^{2} δ_{α β}^{2} v_{x}^{2}) \\ \times \sum_{m = 1}^{M} (\begin{matrix} M \\ m \end{matrix}) (^{- 1) m - 1} \exp (- m π^{2} δ_{α β}^{2} v_{y}^{2}), \end{array}

where

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

is the normalization factor,

(\begin{matrix} M \\ m \end{matrix})

stand for binomial coefficients,

B_{α β} = | B_{α β} | e^{i φ_{α β}}

is the single-point correlation coefficient and

δ_{α β}

is the characteristic source correlations.

On inserting from Eq. (7) into Eq. (6) one finds the explicit form of the CSD matrix elements

\begin{array}{l} W_{α β}^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}) = \frac{A_{α} A_{β} B_{α β}}{C^{2}} \exp (- \frac{{x^{'}}_{1}^{2} + {y^{'}}_{1}^{2}}{σ_{α}^{2}}) \exp (- \frac{{x^{'}}_{2}^{2} + {y^{'}}_{2}^{2}}{σ_{β}^{2}}) {(\frac{{x^{'}}_{1} + i {y^{'}}_{1}}{σ_{α}^{2}})}^{l} {(\frac{{x^{'}}_{2} - i {y^{'}}_{2}}{σ_{β}^{2}})}^{l} \\ \times \sum_{m = 1}^{M} \frac{{(- 1)}^{m - 1}}{\sqrt{m}} (\begin{matrix} M \\ m \end{matrix}) \exp [- \frac{{({x^{'}}_{1} - {x^{'}}_{2})}^{2}}{m δ_{α β}^{2}}] \sum_{m = 1}^{M} \frac{{(- 1)}^{m - 1}}{\sqrt{m}} (\begin{matrix} M \\ m \end{matrix}) \exp [- \frac{{({y^{'}}_{1} - {y^{'}}_{2})}^{2}}{m δ_{α β}^{2}}] . \end{array}

We will term such a source the electromagnetic multi-Gaussian Schell-model vortex source.

3. Far field statistical characteristics of electromagnetic multi-Gaussian Schell-model vortex beams

We now examine the far-field statistical characteristics of a field generated by such a source (9). The elements of the CSD matrix in the far-field plane (i.e., at a propagation distance $z ≫ z_{R}$ , $z_{R} = k σ^{2} / 2$ ) are related to those in the source plane as [23]

W_{α β}^{(\infty)} (ρ_{1}, ρ_{2}) = {(λ z)}^{- 2} \iint \iint W_{α β}^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}) \exp [- i k (ρ_{1} \cdot {ρ^{'}}_{1} - ρ_{2} \cdot {ρ^{'}}_{2}) / z] d^{2} {ρ^{'}}_{1} d^{2} {ρ^{'}}_{2},

where

k = 2 π / λ

is the wave number,

λ

is the wavelength,

ρ_{1}

and

ρ_{2}

are transverse position vectors in the far-field plane. If the off-diagonal elements of the CSD matrix of the beam in the source plane are set at zero, on inserting from Eq. (9) into (10), the Eq. (10) may then be written as

W_{α α}^{(\infty)} (ρ_{1}, ρ_{2}) = A_{α}^{2} {(λ z σ_{α}^{l})}^{- 2} \iint \iint T_{x} T_{y} {({x^{'}}_{1} - i {y^{'}}_{1})}^{l} {({x^{'}}_{2} + i {y^{'}}_{2})}^{l} d^{2} {ρ^{'}}_{1} d^{2} {ρ^{'}}_{2}_{2},

where

T_{t} = \frac{1}{C} \sum_{m = 1}^{M} \frac{{(- 1)}^{m - 1}}{\sqrt{m}} (\begin{matrix} M \\ m \end{matrix}) \exp [- \frac{{t^{'}}_{1}^{2} + {t^{'}}_{2}^{2}}{g^{2}} + \frac{2 {t^{'}}_{1} {t^{'}}_{2}}{m δ_{α β}^{2}} - \frac{i k (t_{1} {t^{'}}_{1} - t_{2} {t^{'}}_{2})}{z}], t = x, y,

\frac{1}{g^{2}} = \frac{1}{σ_{α}^{2}} + \frac{1}{m δ_{α β}^{2}} .

Efficient fast-Fourier-transform algorithms may be used to integrate Eq. (11) numerically. But the four dimensional integrate computation is generally memory intensive. It is possible to expand the binomial terms that allow Eq. (11) to be computed from two-dimensional integrals, that is

W_{α α}^{(\infty)} (ρ_{1}, ρ_{2}) = \frac{A_{α}^{2}}{{(λ z σ_{α}^{l})}^{2}} \sum_{p = 0}^{l} \sum_{q = 0}^{l} (\begin{matrix} p \\ l \end{matrix}) (\begin{matrix} q \\ l \end{matrix}) {(- i)}^{2 l - p - q} \iint T_{x} {x^{'}}_{1}^{p} {x^{'}}_{2}^{q} d {x^{'}}_{1} d {x^{'}}_{2} \iint T_{y} {y^{'}}_{1}^{l - p} {(- {y^{'}}_{2})}^{l - q} d {y^{'}}_{1} d {y^{'}}_{2}_{1} .

The far-field statistical characteristics, including the spectral density $S$ , the degree of polarization $P$ and the degree of coherence μ, can be calculated numerically by the following expression [23]

S (ρ) = W_{x x} (ρ, ρ) + W_{y y} (ρ, ρ),

P (ρ) = | \frac{W_{x x} (ρ, ρ) - W_{y y} (ρ, ρ)}{W_{x x} (ρ, ρ) + W_{y y} (ρ, ρ)} |,

μ (ρ_{1}, ρ_{2}) = \frac{W_{x x} (ρ_{1}, ρ_{2}) + W_{y y} (ρ_{1}, ρ_{2})}{\sqrt{[W_{x x} (ρ_{1}, ρ_{1}) + W_{y y} (ρ_{1}, ρ_{1})] [W_{x x} (ρ_{2}, ρ_{2}) + W_{y y} (ρ_{2}, ρ_{2})]}} .

Figure 1 shows the spectral density of electromagnetic multi-Gaussian Schell-model vortex beam at the plane z = 20z_R for different values of the coherence length and topological charge. Without loss of generality the other parameter values of the source are chosen to be A_x = 1, A_y = 0.5, λ = 632.8nm, σ_x = σ_y = 1mm, M = 10. From left to right, it illustrates the typical variation of far-field spectral density with coherence length, $δ_{x x} = 10 mm$ , $δ_{y y} = 9 m m$ in the left column, $δ_{x x} = 5 mm$ , $δ_{y y} = 3 m m$ in the middle column, $δ_{x x} = 0.5 mm$ , $δ_{y y} = 0.3 m m$ in the right column. From top to bottom, the topological charge $l$ is increased from 1 to 4. As can be seen, the far-field distributions of spectral density depend on the coherence length and the topological charge. The high coherence cases produce a central dark vortex core with a minimum intensity that is close to zero. The area of dark core increases with the increase of the topological charge $l$ , the central intensity reach zero value when $l > 1$ . With the decrease of the coherence length, the central intensity increase gradually. In the low coherence cases, the core is filled with the shape of a rectangular flat top, the area of flat top decreases with the increase of the topological charge $l$ .

Fig. 1 Far-field spectral densities radiated by a random source defined by Eq. (9) for various coherence lengths and topological charge.

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The influence of coherence length and topological charge on the far-field distributions of the degree of polarization are shown in Fig. 2. In the high coherence case, the degree of polarization has a circularly symmetric profile with a central depression. The value of the center gradually decreases with the increase of topological charge l, and eventually reaches zero. With the decrease of coherence length, the center gradually rises, the profile gradually converts from a circular symmetry to a rectangular distribution and forms a plane with protuberant and uniform polarization distribution in the central region.

Fig. 2 Far-field distributions of the degree of polarization corresponding to Fig. 1.

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Figures 3 and 4 show the modulus and the phase of the degree of coherence corresponding to Fig. 1. One clearly sees that in general the coherence vortex core is generated in the high coherence case. With the decrease of coherence length, the value of the outer circle gradually decrease, the degree of coherence eventually possess Laguerre-Gaussian profiles for the low coherence case. Some striking dark ring, which persists as the coherence length is varied, is evident in profile of the degree of coherence. The number of dark ring dislocations is just equal to topological charge of a given stochastic electromagnetic vortex beam. Each dark ring exhibits a phase dislocation that is characterized by a π phase jump across the circular boundary, as shown in Fig. 4. Therefore, it can be used to as a method of measuring the azimuthal index l according to the number of dark rings or the times of phase jumps. The radius of the rings increases as the coherence length of the source field decreases.

Fig. 3 The modulus of the degree of coherence corresponding to Fig. 1.

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Fig. 4 The phase of the degree of coherence corresponding to Fig. 1.

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4. Concluding remarks

In this paper, we have introduced a new class of stochastic electromagnetic sources with multi-Gaussian Schell-model coherence function and helicoidal phase. The far-field statistical properties of the beams generated by such sources have been studied in detail by numerical examples. It is shown that these properties are closely related to the initial coherent length and topological charge. For the high coherence cases, the profiles of statistical properties, including the far-field spectral density, the degree of coherence and the degree of polarization, all possess a vortex structure. In addition, the degree of coherence exhibit ring dislocations that is in a one-to-one correspondence with the topological charge. There is a π phase jump across the each circular boundary. With the weakening of the coherence of the source field, these vortex structures gradually disappear. The spectral density and the degree of polarization of such beams in the low coherence cases are similar to those of the usual electromagnetic multi-Gaussian Schell-model beams without vortex phase. But the degree of coherence becomes the Laguerre-Gaussian profile and maintains the number of ring dislocations. This remarkable property can be used as a method to measure the topological charge by the number of ring dislocations whether for the cases of high coherence or low coherence. We remark that our results presented here may provide a feasible method for further explorations to new electromagnetic vortex beams and can be of interest in the realm of current research of coherence and polarization theory.

Funding

Zhejiang Provincial Natural Science Foundation of China (LY16F050007).

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Electromagnetic multi-Gaussian Schell-model vortex light sources and their radiation field properties

Abstract

1. Introduction

2. Electromagnetic multi-Gaussian Schell-model vortex light source

3. Far field statistical characteristics of electromagnetic multi-Gaussian Schell-model vortex beams

4. Concluding remarks

Funding

References

Cited By

Figures (4)

Equations (17)

Optics Express