Vector Hermite-Gaussian correlated Schell-model beam

Yahong Chen; Fang Wang; Jiayi Yu; Lin Liu; Yangjian Cai

doi:10.1364/OE.24.015232

1. Introduction

Polarization is one of the most salient features of a light beam, and it is a manifestation of correlations involving components of the fluctuating electric field at a single point. Based on the state of polarization (SOP), vector beams can be classified as beams with spatially uniform SOP and beams with spatially non-uniform SOP (e.g., radially polarized beam and azimuthally polarized beam) [1]. It is usually assumed that the SOP and degree of polarization (DOP) of a vector beam remain invariant on propagation in free space. In 1994, James first found that the DOP of a partially coherent vector beam may change on propagation in free space [2]. Since then numerous efforts have been paid to partially coherent vector beam [3–22]. In 2003, Wolf developed a unified theory of coherence and polarization for partially coherent vector beam [6], which can be used conveniently to analyze the statistical properties of a partially coherent vector beam on propagation. Partially coherent vector beam with uniform SOP usually is called stochastic (or random) electromagnetic beam, e.g., electromagnetic Gaussian Schell-model beam or partially polarized Gaussian Schell-model beam [5,7]. Realizability and beam conditions of a stochastic electromagnetic beam were discussed in [7–9]. It was found in [10] that the state of polarization of a stochastic electromagnetic beam may change on propagation. Salem and Wolf explored the phenomenon of coherence-induced polarization changes in detail and predicted that the polarization properties of a vector beam can be modulated by varying its spatial coherence width [11]. After then, Vidal et al. proved this prediction in experiment [12]. A review on generation and propagation of a stochastic electromagnetic beam can be found in [13]. Partially coherent vector beam with non-uniform SOP, such as radially or azimuthally polarized partially coherent beam, was introduced in theory and generated in experiment recently [14–16]. Although the DOP of a radially or azimuthally polarized partially coherent beam varies on propagation in free space, its state of polarization remains invariant on propagation [14], which is quite different from that of a stochastic electromagnetic beam. It was shown in [15,16] that modulating the spatial coherence width provides a convenient for shaping the beam profile of a radially or azimuthally polarized partially coherent beam, which is useful for particle trapping. Modulating the coherence and polarization properties of a partially coherent vector beam is useful for ghost imaging [17–19], and can be used to reduce turbulence-induced scintillation [20–22], which is useful in free-space optical communications and laser system.

All above mentioned partially coherent vector beams have conventional correlation functions (i.e., Gaussian correlated Schell-model functions). Recently, more and more attention is being paid to partially coherent beams with nonconventional correlation functions. Gori et al. discussed the sufficient condition for devising a genuine correlation function of a scalar or vector partially coherent beam [23,24], and then partially coherent beams with different correlation functions were introduced and generated [25–40]. Generation and propagation of one kind of partially coherent vector beam with nonconventional correlation function named specially correlated radially polarized beam were reported in [39], and the beam profile of such beam varies on propagation while the SOP of such beam also remains radial polarization in free space. Partially coherent beams with non-conventional correlation functions exhibit many extraordinary properties [25–41] and are useful for particle trapping, free-space optical communications and optical imaging [42–49]. Hermite-Gaussian correlated Schell-model (HGCSM) beam was introduced and generated just recently [40], and such beam exhibits self-splitting properties on propagation in free space, which may be useful for attacking multiple targets, trapping multiple particles, and guiding atoms. Furthermore, it was shown in [48] that the HGCSM beam exhibits interesting splitting and combing properties in turbulent atmosphere and such beam is less affected by turbulence than a conventional Gaussian correlated beam or a Gaussian beam. In this paper, our aim is to introduce a new kind of partially coherent vector beam named vector HGCSM beam as an extension of scalar HGCSM beam, and explore the effect of the structures of correlation functions on the propagation properties of a vector HGCSM beam. Some interesting and useful results are found.

2. Theoretical model for a vector HGCSM beam

In the space-time domain, the statistical properties of a partially coherent vector beam are characterized by the following beam coherence-polarization (BCP) matrix [4]

\hat{Γ} (r_{1}, r_{2}) = (\begin{matrix} Γ_{x x} (r_{1}, r_{2}) & Γ_{x y} (r_{1}, r_{2}) \\ Γ_{y x} (r_{1}, r_{2}) & Γ_{y y} (r_{1}, r_{2}) \end{matrix}),

with elements

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

Here

r_{1}

and

r_{2}

are transverse position vectors in the source plane.

E_{α} (r)

is the fluctuating electric field component along the

α -

axis at point

r

and the angular brackets denote an ensemble average.

In [24], Gori et al. showed that the BCP matrix can be expressed in the following integral form in order to be genuine or physical realizable

Γ_{α β} (r_{1}, r_{2}) = \int p_{α β} (v) ℋ_{α}^{*} (r_{1}, v) ℋ_{β} (r_{2}, v) d^{2} v, (α, β = x, y),

where

ℋ_{x} (r, v)

and

ℋ_{y} (r, v)

are two arbitrary kernels, and

p_{α β} (v)

are the elements of the following weighting matrix,

\hat{p} (v) = (\begin{matrix} p_{x x} (v) & p_{x y} (v) \\ p_{y x} (v) & p_{y y} (v) \end{matrix}),

The elements of the weighting matrix should satisfy the following conditions for any

v

[24],

p_{x x} (v) \geq 0, p_{y y} (v) \geq 0, p_{x x} (v) p_{y y} (v) - {| p_{x y} (v) |}^{2} \geq 0.

Let’s set

\begin{array}{l} p_{α β} (v) = \frac{2^{m_{α β} + n_{α β} + 5} δ_{0 α β}^{2 m_{α β} + 2 n_{α β} + 2}}{H_{2 m_{α β}} (0) H_{2 n_{α β}} (0)} \frac{π}{{(- 1)}^{m_{α β} + n_{α β}}} B_{α β} \\ \times v_{x}^{2 m_{α β}} v_{y}^{2 n_{α β}} \exp (- \frac{δ_{0 α β}^{2}}{2} v_{x}^{2}) \exp (- \frac{δ_{0 α β}^{2}}{2} v_{y}^{2}), \end{array}

ℋ_{α} (r, v) = F_{α} (r) \exp (- 2 π i v \cdot r) .

Here

ℋ_{α}

is a Fourier-like structure,

F_{α}

is a possible profile function,

H_{m}

denotes the Hermite polynomial of order m.

Substituting Eqs. (6) and (7) into Eq. (3), we obtain the elements of the BCP matrix as follows

Γ_{α β} (r_{1}, r_{2}) = F_{α}^{*} (r_{1}) F_{β} (r_{2}) μ_{α β} (r_{1}, r_{2}),

where

μ_{α β} (r_{1}, r_{2})

denote the correlation functions of the proposed partially coherent vector beam and are expressed as follows

\begin{array}{l} μ_{α β} (r_{1}, r_{2}) = \frac{B_{α β}}{H_{2 m_{α β}} (0) H_{2 n_{α β}} (0)} \exp (- \frac{{(x_{2} - x_{1})}^{2}}{2 δ_{0 α β}^{2}}) H_{2 m_{α β}} (\frac{(x_{2} - x_{1})}{\sqrt{2} δ_{0 α β}}) \\ \times \exp (- \frac{{(y_{2} - y_{1})}^{2}}{2 δ_{0 α β}^{2}}) H_{2 n_{α β}} (\frac{(y_{2} - y_{1})}{\sqrt{2} δ_{0 α β}}), (α, β = x, y) . \end{array}

Here

B_{α β} = | B_{α β} | \exp (i ϕ_{α β})

is the correlation coefficient between the

E_{α}

and

E_{β}

field components.

H_{2 m_{α β}}

and

H_{2 n_{α β}}

denote the Hermite polynomial of orders

2 m_{α β}

and

2 n_{α β}

, respectively.

σ_{0}

is the width of the spectral density,

δ_{0 x x}

,

δ_{0 y y}

and

δ_{0 x y}

are the widths of the correlation functions

μ_{x x}, μ_{y y}, μ_{x y}

, respectively. We call the partially coherent vector beam whose correlation functions are given by Eq. (9) as a vector HGCSM beam. It can be regarded as a natural extension of a scalar HGCSM beam [40]. The structures of the correlation functions are modulated by varying the mode orders

m_{α β}

and

n_{α β}

as shown in Fig. 1, while the values of

m_{α β}

and

n_{α β}

should be chosen according to certain restrictions as shown in section 3.

Fig. 1 Density plots of the modulus of the correlation functions $| μ_{x x} |, | μ_{y y} |$ and $| μ_{x y} |$ for different values of the mode orders $m_{α β}$ and $n_{α β}$ with $δ_{0 x x} = δ_{0 y y} = δ_{0 x y} = 0.2 mm$ .

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According to [39], vector partially coherent beam with prescribed correlation functions can be formed by an incoherent vector beam with prescribed intensity distribution through propagation. In our paper, Eq. (6) represents the intensity distribution of an incoherent vector beam to form a vector HGCSM beam, and Eq. (7) denotes the response function of the optical path.

The vector HGCSM beam can be classified as a partially coherent vector beam with uniform SOP and a partially coherent vector with non-uniform SOP depending on the detailed expression of the $F_{α}$ . Here we discuss these two cases separately.

(a) Vector HGCSM beam with uniform SOP

If we set $F_{α} (r) = A_{α} \exp [- r^{2} / (4 σ_{0}^{2})], (α = x, y)$ with $A_{α}$ and $σ_{0}$ being a constant and beam width, respectively, the vector HGGCSM beam exhibits uniform SOP, i.e., the state of polarization of any point is the same in the source plane, and we call this beam as electromagnetic HGCSM beam. Under the condition of $m_{α β} = n_{α β} = 0, (α = x, y)$ , the electromagnetic HGCSM beam reduces to the conventional electromagnetic Gaussian Schell-model (GSM) beam [7,8].

(b) Vector HGCSM beam with non-uniform SOP

If we set $F_{α} (r) = α \exp [- r^{2} / (4 σ_{0}^{2})] / (2 σ_{0}), (α = x, y)$ with $σ_{0}$ being the beam width, the vector HGCSM beam exhibits radial polarization in the source plane, and we call this beam as radial polarized HGCSM beam. If we set $F_{x} (r) = y \exp [- r^{2} / (4 σ_{0}^{2})] / (2 σ_{0}),$ and $F_{y} (r) = x \exp [- r^{2} / (4 σ_{0}^{2})] / (2 σ_{0}),$ the vector HGCSM beam exhibits azimuthal polarization in the source plane, and we call this beam as azimuthal polarized HGCSM beam. Under the condition of $m_{α β} = n_{α β} = 0, (α = x, y)$ , the radial or azimuthal polarized HGCSM beam reduces to the conventional radial or azimuthal polarized partially coherent beam with Gaussian correlated Schell-model functions [14–16].

3. Realizability and beam conditions for a vector HGCSM beam

In this section, first, we will derive the general realizability and beam conditions for a vector HGCSM beam with uniform SOP or non-uniform SOP, and then we will derive some additional realizability conditions for a radial polarized HGCSM beam. To be a physical realizable partially coherent vector beam source, the BCP matrix of a vector HGCSM source must be quasi-Hermitian [9], i.e., $Γ_{α β} (r_{1}, r_{2}) = Γ_{β α}^{*} (r_{2}, r_{1})$ , and this requirement can be satisfied if the correlation coefficients $B_{α β}$ and $δ_{0 α β}$ satisfy the following conditions

| B_{α β} | = 1, ϕ_{α β} =0, (α = β), | B_{α β} | \leq 1, (α \neq β), | B_{x y} | = | B_{y x} |, ϕ_{x y} = ϕ_{y x}, δ_{0 x y} = δ_{0 y x} .

Furthermore, the vector HGCSM source should satisfy the nonnegative conditions as shown in Eq. (5). Substituting Eq. (6) into Eq. (5), we find that the first two inequalities always hold and the last one reduces to,

\begin{array}{l} \frac{1}{H_{2 m_{x x}} (0) H_{2 m_{y y}} (0)} \frac{1}{H_{2 n_{x x}} (0) H_{2 n_{y y}} (0)} \frac{2^{m_{x x} + m_{y y} + n_{x x} + n_{y y} + 10} δ_{0 x x}^{2 m_{x x} + 2 n_{x x} + 2} δ_{0 y y}^{2 m_{y y} + 2 n_{y y} + 2}}{{(- 1)}^{m_{x x} + m_{y y} + n_{x x} + n_{y y}}} \\ v_{x}^{2 m_{x x} + 2 m_{y y}} v_{y}^{2 n_{x x} + 2 n_{y y}} \exp (- \frac{δ_{0 x x}^{2} + δ_{0 y y}^{2}}{2} v_{x}^{2}) \exp (- \frac{δ_{0 x x}^{2} + δ_{0 y y}^{2}}{2} v_{y}^{2}) \\ \geq \frac{1}{H_{2 m_{x y}} (0) H_{2 m_{x y}} (0)} \frac{1}{H_{2 n_{x y}} (0) H_{2 n_{x y}} (0)} \frac{2^{2 m_{x y} + 2 n_{x y} + 10} {| B_{x y} |}^{2} δ_{0 x y}^{2 m_{x y} + 2 n_{x y} + 2} δ_{0 x y}^{2 m_{x y} + 2 n_{x y} + 2}}{{(- 1)}^{2 m x y + 2 n x y}} . \\ v_{x}^{4 m_{x y}} v_{y}^{4 n_{x y}} \exp (- δ_{0 x y}^{2} v_{x}^{2}) \exp (- δ_{0 x y}^{2} v_{y}^{2}) \end{array}

Since the function v²^x is a monotonous increasing function versus x when v is a constant and $v^{2} > 1$ , and a monotonous decreasing function versus x when v is a constant and $0 < v^{2} < 1$ , therefore, when the powers of v on both sides of the inequality are different, the inequality will not always hold for arbitrary real value v. Only when the powers of v on both sides of the inequality are the same, the inequality will always hold for arbitrary real values of v, which imply

m_{x x} + m_{y y} = 2 m_{x y}, n_{x x} + n_{y y} = 2 n_{x y} .

Substituting Eq. (12) into Eq. (11), we obtain

\begin{array}{l} \frac{δ_{0 x x}^{2 m_{x x} + 2 n_{x x} + 2}}{H_{2 m_{x x}} (0) H_{2 m_{y y}} (0)} \frac{δ_{0 y y}^{2 m_{y y} + 2 n_{y y} + 2}}{H_{2 n_{x x}} (0) H_{2 n_{y y}} (0)} \exp (- \frac{δ_{0 x x}^{2} + δ_{0 y y}^{2}}{2} v_{x}^{2} - \frac{δ_{0 x x}^{2} + δ_{0 y y}^{2}}{2} v_{y}^{2}) \\ \geq \frac{1}{{[H_{2 m_{x y}} (0)]}^{2}} \frac{1}{{[H_{2 n_{x y}} (0)]}^{2}} {| B_{x y} |}^{2} δ_{0 x y}^{4 m_{x y} + 4 n_{x y} + 4} \exp (- δ_{0 x y}^{2} v_{x}^{2} - δ_{0 x y}^{2} v_{y}^{2}) \end{array}

Since both sides of the inequality equation are monotonic function with v_x and v_y. It is not difficult to derive the following inequality relation

\sqrt{\frac{δ_{0 x x}^{2} + δ_{0 y y}^{2}}{2}} \leq δ_{0 x y} \leq \sqrt[4 m_{x y} + 4 n_{x y} + 4]{\frac{{[H_{2 m_{x y}} (0)]}^{2} δ_{0 x x}^{2 m_{x x} + 2 n_{x x} + 2}}{H_{2 m_{x x}} (0) H_{2 m_{y y}} (0)} \frac{{[H_{2 n_{x y}} (0)]}^{2} δ_{0 y y}^{2 m_{y y} + 2 n_{y y} + 2}}{H_{2 n_{x x}} (0) H_{2 n_{y y}} (0)} \frac{1}{{| B_{x y} |}^{2}}} .

Equations (10), (12) and (14) are the general realizability conditions for a vector HGCSM source with uniform or non-uniform SOP.

Under the condition of $δ_{0 x x} = δ_{0 y y} = δ_{0 x y}$ , Eq. (14) is simplified as follows

\frac{{[H_{2 m_{x y}} (0)]}^{2}}{H_{2 m_{x x}} (0) H_{2 m_{y y}} (0)} \frac{{[H_{2 n_{x y}} (0)]}^{2}}{H_{2 n_{x x}} (0) H_{2 n_{y y}} (0)} \frac{1}{{| B_{x y} |}^{2}} \geq 1.

To generate a beam-like field, the parameters of the vector HGCSM source should satisfy certain conditions. The intensity at a point specified by a position vector $ρ = ρ s$ ( $s$ is a unit vector in its direction) in the far zone is given by the following expression [3]

I^{\infty} (ρ) = {(2 π k / ρ)}^{2} \cos^{2} θ [{\tilde{Γ}}_{x x} (- k s_{⊥}, k s_{⊥}) + {\tilde{Γ}}_{y y} (- k s_{⊥}, k s_{⊥})],

where

k = 2 π / λ

is the wave number of the field with

λ

being the wavelength,

s_{⊥}

is the projection of

s

onto the source plane,

θ

is the angle that the unit vector

s

makes with a positive z direction, and

{\tilde{Γ}}_{α α} (f_{1}, f_{2}) = \frac{1}{{(2 π)}^{4}} \iint Γ_{α α} (r_{1}, r_{2}) \exp [- i (f_{1} \cdot r_{1} + f_{2} \cdot r_{2})] d^{2} r_{1} d^{2} r_{2} .

Here

{\tilde{Γ}}_{α α} (f_{1}, f_{2})

is the four-dimensional Fourier transform of

Γ_{α α} (r_{1}, r_{2})

.

For a beam-like field, its far-field intensity must be negligible except for directions within a narrow solid angle about the z axis. Substituting Eqs. (8) and (9) into Eqs. (16) and (17), after some operations, we obtain the following beam conditions for a vector HGCSM source

\frac{1}{4 σ_{0}^{2}} + \frac{1}{δ_{0 x x}^{2}} ≪ \frac{2 π^{2}}{λ^{2}}, \frac{1}{4 σ_{0}^{2}} + \frac{1}{δ_{0 y y}^{2}} ≪ \frac{2 π^{2}}{λ^{2}} .

Besides the above general realizability and beam conditions for a vector HGCSM beam, the following two additional conditions for a radial polarized HGCSM beam should be satisfied [14],

(a) Any point in the source plane is linearly polarized
(b) The orientation angle of the polarization at any point in the source plane should satisfies $θ (x, y) = arc \tan (y / x) .$

It is known that a partially coherent vector beam can be decomposed into a superimposition of completely polarized portion and completely unpolarized portion. The SOP of the completely polarized portion can be studied with the help of a polarization ellipse, whose degree of ellipticity $ε$ and orientation angle $θ$ are expressed as follows [10]

θ (r) = \frac{1}{2} arc \tan [\frac{2 Re [Γ_{x y} (r, r)]}{Γ_{x x} (r, r) - Γ_{y y} (r, r)}],

ε (r) = \frac{A_{2} (r)}{A_{1} (r)},

where

A_{1} (r)

and

A_{2} (r)

stand for the major and minor semi-axes of the polarization ellipse given by

\begin{array}{l} A_{1} (r) = \frac{1}{\sqrt{2}} [\sqrt{{[Γ_{x x} (r, r) - Γ_{y y} (r, r)]}^{2} + 4 {| Γ_{x y} (r, r) |}^{2}} \\ {+ \sqrt{{[Γ_{x x} (r, r) - Γ_{y y} (r, r)]}^{2} + 4 {[Re Γ_{x y} (r, r)]}^{2}}]}^{1 / 2}, \end{array}

\begin{array}{l} A_{2} (r) = \frac{1}{\sqrt{2}} [\sqrt{{[Γ_{x x} (r, r) - Γ_{y y} (r, r)]}^{2} + 4 {| Γ_{x y} (r, r) |}^{2}} \\ {- \sqrt{{[Γ_{x x} (r, r) - Γ_{y y} (r, r)]}^{2} + 4 {[Re Γ_{x y} (r, r)]}^{2}}]}^{1 / 2} . \end{array}

With the requirements of additional conditions (a) and (b), and applying Eqs. (8), (9), (11) and (20)-(23), we obtain the following additional realizability conditions for a radially polarized HGCSM beam,

B_{x y} = B_{y x} = 1, δ_{0 x x} = δ_{0 y y} = δ_{0 x y} = δ_{0}, m_{x x} = m_{y y} = m_{x y} = m, n_{x x} = n_{y y} = n_{x y} = n .

4. Analytical formulae for a vector HGCSM beam propagating in free space

Within the validity of the paraxial approximation, paraxial propagation of the elements of the BCP matrix of a partially coherent vector beam in free space can be studied with the help of the following generalized Huygens-Fresnel integral

\begin{array}{l} Γ_{α β} (ρ_{1}, ρ_{2}) = \frac{1}{{(λ z)}^{2}} \exp [- \frac{i k}{2 z} (ρ_{1}^{2} - ρ_{2}^{2})] \\ \times \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} Γ_{α β} (r_{1}, r_{2}) \exp [- \frac{i k}{2 z} (r_{1}^{2} - r_{2}^{2})] \exp [\frac{i k}{z} (r_{1} \cdot ρ_{1} - r_{2} \cdot ρ_{2})] d^{2} r_{1} d^{2} r_{2}, \end{array}

where

ρ_{1} \equiv (ρ_{1 x}, ρ_{1 y})

and

ρ_{2} \equiv (ρ_{2 x}, ρ_{2 y})

are two arbitrary transverse position vectors in the receiver plane.

Substituting the elements $Γ_{α β} (r_{1}, r_{2})$ of the BCP matrix of an electromagnetic HGCSM beam or a radially polarized HGCSM beam in the source plane into Eq. (25), after integration and operation, we obtain the analytical formulae for the elements of the BCP matrix with $ρ_{1} = ρ_{2}$ on propagation in free space (see Appendix A).

The average intensity and the degree of polarization (DOP) of a vector HGCSM beam in the receiver plane are obtained as

I (ρ) = Γ_{x x} (ρ, ρ) + Γ_{y y} (ρ, ρ) = I_{x x} (ρ) + I_{y y} (ρ),

P (ρ) = \sqrt{1 - \frac{4 Det \hat{Γ} (ρ, ρ)}{{[Tr \hat{Γ} (ρ, ρ)]}^{2}}} .

The SOP of a vector HGCSM beams can be calculated by Eqs. (20)-(23) just by replacing the elements of the BCP matrix in the source plane by those in the receiver plane.

5. Paraxial propagation properties of a vector HGCSM beam in free space

In this section, with the formulae derived in above sections, we will study the paraxial propagation properties of an electromagnetic HGCSM beam and a radially polarized HGCSM beam, respectively. In the following numerical examples, we set $A_{x} = A_{y} = 1, λ = 632.8 nm,$ $B_{x y} = 0.1 \exp (i π / 3),$ $σ_{0} = 1 mm, δ_{0 x x} = δ_{0 y y} = δ_{0 x y} = δ_{0} = 0.2 mm .$

First, we analyze the propagation properties of an electromagnetic HGCSM beam in free space. We calculate in Fig. 2 the density plot of the normalized intensity distribution of an electromagnetic HGCSM beam with $m_{x x} = m_{y y} = n_{x x} = n_{y y} =5$ at several propagation distances in free space, and in Fig. 3 the density plot of the normalized intensity distribution of an electromagnetic HGCSM beam at z = 5m for different values of $m_{x x}, m_{y y},$ $n_{x x}, n_{y y}$ . One finds from Figs. 2 and 3 that the electromagnetic HGCSM beam exhibits self-splitting properties on propagation in free space, and under the condition of $δ_{0 x x} = δ_{0 y y}$ and $m_{x x} = m_{y y} = n_{x x} = n_{y y} \neq 0$ , the splitting properties on propagation are similar to those of a scalar HGCSM beam, e.g., the initial single beam spot evolves into four beam spots in the far field. While by varying the values of the beam parameters $m_{x x}, m_{y y}, n_{x x}, n_{y y}$ (i. e., varying the structures of the correlation functions $μ_{x x}$ , $μ_{y y}$ ), the electromagnetic HGCSM beam displays more flexible self-splitting properties, e.g., the initial single beam spot evolves into more complicated beam pattern in the far field (see Fig. 3), which are caused by the fact that both I_xx and I_yy exhibit self-splitting properties and evolve in different ways on propagation. The behaviors of the electromagnetic HGCSM beam on propagation are quite different from that of a conventional electromagnetic GSM beam, whose beam spot is always of circular symmetry and don’t exhibit self-splitting properties. Thus, modulating the structures of the correlation functions of a partially coherent vector beam provides a novel way for controlling the self-splitting properties and may be useful for particle trapping.

Fig. 2 Density plot of the normalized intensity distribution of an electromagnetic HGCSM beam with m_xx = m_yy = n_xx = n_yy = 5 at several propagation distances in free space.

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Fig. 3 Density plot of the normalized intensity distribution of an electromagnetic HGCSM beam at z = 5m for different values of m_xx, m_yy, n_xx, n_yy.

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To learn about the properties of the SOP and the DOP of an electromagnetic HGCSM beam on propagation in free space, we calculate in Figs. 4-6 the density plot of the normalized intensity distribution and the distribution of the SOP of an of an electromagnetic HGCSM beam for different values of m_xx, m_yy, m_xy, n_xx, n_yy, n_xy at several propagation distances in free space, and in Fig. 7 the density plot of the DOP of an electromagnetic HGCSM beam at several propagation distances in free space for different values of m_xx, m_yy, m_xy, n_xx, n_yy, n_xy. The parameters used in the calculation are chosen to satisfy the realizability and beam conditions derived in section 2. One finds from Fig. 4 that the electromagnetic HGCSM beam has a uniform SOP in the source plane as expected, while the SOP becomes non-uniform on propagation due to the influence of the initial correlation functions. This phenomenon can be explained in the following way. The unequal values of the beam orders m_xx, m_yy and m_xy lead to the asymmetric distribution of correlation functions ( $μ_{x x}$ , $μ_{y y}$ , $μ_{x y}$ ), which lead to the asymmetric distribution of the elements of the BCP matrix, i.e., $Γ_{x x} (ρ, ρ) \neq Γ_{y y} (ρ, ρ) \neq Γ_{x y} (ρ, ρ)$ . Therefore, the uniform SOP of the electromagnetic HGCSM beam is destroyed under the conditions of unequal values of beam orders m_xx, m_yy and m_xy. From Fig. 5, we find that the SOP of an electromagnetic HGCSM beam in the far field displays different distributions as the values of m_xx, m_yy, m_xy, n_xx, n_yy, n_xy vary. Furthermore, we also find that the SOP of an HGCSM beam remains invariant on propagation in free space under the condition of $δ_{0 x x} = δ_{0 y y} = δ_{0 x y}$ and m_xx = m_yy = m_xy = n_xx = n_yy = n_xy (see Fig. 6). In Fig. 6, the orientation angle of the polarization ellipses is equal to $π / 4$ , the degree of ellipticity is equal to $\sqrt{3} / 3$ and the polarization ellipses are right-handed, and their properties remain invariant on propagation. From Fig. 7, one sees that the DOP of the electromagnetic HGCSM beam also displays uniform distribution across the source plane, while it becomes non-uniform on propagation and its value varies on propagation in free space and is modulated by m_xx, m_yy, m_xy, n_xx, n_yy, n_xy. Under the condition of $δ_{0 x x} = δ_{0 y y} = δ_{0 x y}$ and m_xx = m_yy = m_xy = n_xx = n_yy = n_xy, the DOP remains invariant on propagation. Thus, one comes to the conclusion that modulating the structures of the correlation functions of a partially coherent vector beam provides a novel way for modulating its SOP and DOP on propagation.

Fig. 4 Density plot of the normalized intensity distribution and the distribution of the SOP of an electromagnetic HGCSM beam with m_xx = n_xx = 1, m_yy = n_yy = 5, m_xy = n_xy = 3 at several propagation distances in free space.

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Fig. 5 Density plot of the normalized intensity distribution and the distribution of the SOP of an electromagnetic HGCSM beam at z = 5m in free space for different values of m_xx, m_yy, m_xy, n_xx, n_yy, n_xy.

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Fig. 6 Density plot of the normalized intensity distribution and the distribution of the SOP of an electromagnetic HGCSM beam with m_xx = m_yy = m_xy = n_xx = n_yy = n_xy = 5 at several propagation distances in free space.

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Fig. 7 Density plot of the DOP of an electromagnetic HGCSM beam at several propagation distances in free space for different values of m_xx, m_yy, m_xy, n_xx, n_yy, n_xy.

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Now we analyze the propagation properties of a radially polarized HGCSM beam in free space. We calculate in Fig. 8 the density plot of the normalized intensity distribution and the distribution of the SOP of a radially polarized HGCSM beam at several propagation distances in free space for different values of m and n. For the convenience of comparison, the corresponding results of a conventional radially polarized GSM beam (m = n = 0) are also shown in Fig. 8. One finds that the intensity distribution of a radially polarized HGCSM beam also displays self-splitting properties on propagation in free space, e.g., the initial single beam spot evolves into two or four beam spots in the far field, which is similar to that of a scalar HGCSM beam, but is quite different from that of a conventional radially polarized GSM beam, which doesn’t show self-splitting properties although its beam shape also varies on propagation. The distances between spots in the far field are controlled by the values of m and n.

Fig. 8 Density plot of the normalized intensity distribution and the distribution of the SOP of a radially polarized HGCSM beam at several propagation distances in free space for different values of m and n.

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One also finds from Fig. 8 that for a conventional radially polarized GSM beam (m = n = 0), it displays radial polarization in the source plane and the SOP remains invariant on propagation in free space as expected [14], and there always exists one polarization singularity in the beam center. For a radially polarized HGCSM beam, although it also displays radial polarization in the source plane, but the SOP varies and more polarization singularities appears on propagation in free space. In Fig. 8, the polarization singularity in fact is a vector point singularity (V-point), where the orientation angle $θ$ of the polarization ellipse is undefined and both the major and minor semi-axes of the polarization ellipse are equal to zero (i.e., the Stokes parameters S₁, S₂, S₃ are equal to zero) [50]. From Eqs. (20), (22) and (23), the V-point can be obtained under the conditions of $Re [Γ_{x y} (r, r)] = 0$ and $Γ_{x x} (r, r) - Γ_{y y} (r, r) = 0.$ To show the polarization singularities more clearly, based on the definition of V-point, we calculate in Fig. 9 the distribution of the polarization singularities of a radially polarized HGCSM beam at several propagation distances in free space for different values of m and n. One finds from Fig. 9 that the single V-point of a radially polarized GSM beam indeed keeps invariant on propagation, while the single V-point of a radially polarized HGCSM beam in the source plane is split into more V-points, and the number of the V-points are determined by the values of m and n. Figure 10 shows the distribution of the DOP of a radially polarized HGCSM beam at several propagation distances in free space for different values of m and n. One finds that the distribution of the DOP of a conventional radially polarized GSM beam always displays circular symmetry although its value varies on propagation, while the distribution of the DOP of a radially polarized HGCSM beam gradually evolves from circular symmetry into non-circular symmetry, both its shape and value are closely determined by m and n. Thus, modulating the structures of the correlation functions provides a novel way for modulating the polarization properties of a partially coherent radially polarized beam.

Fig. 9 Distribution of the polarization singularities of a radially polarized HGCSM beam at several propagation distances in free space for different values of m and n.

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Fig. 10 Distribution of the DOP of a radially polarized HGCSM beam at several propagation distances in free space for different values of m and n.

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6. Experimental generation of a radially polarized HGCSM beam

In principle, vector HGCSM beam can be generated with the help of the spatial light modulators and the rotating ground-disk [41,51]. In this section, we report experimental generation of a radially polarized HGCSM beam through conversion of a linearly polarized HGCSM beam by a radial polarization converter (RPC).

Figure 11 shows our experimental setup for generating a radially polarized HGCSM beam. The procedure for generating a scalar or linearly polarized HGCSM beam is the same with that reported in [40]. A full coherent He-Ne laser beam transmits through a beam expander (BE) and is reflected by a reflecting mirror (RM), then the reflected beam passes through a spatial light modulator (SLM, Holoeye LC2002), and the diffraction patterns appear. Here the SLM acts as a phase grating designed by the method of computer-generated holograms. The grating pattern of the hologram loaded on the SLM is calculated by the interference of a beam whose intensity is given by Eq. (6) and a plane wave, and the phase grating for the beam whose intensity is given by Eq. (6) with m = n = 5 or m = 5, n = 0 is shown as inset in Fig. 11. The first-order diffraction pattern is regarded as the beam whose intensity is given by Eq. (6) and selected out by a circular aperture (CA). The transmitted beam from the CA illuminates a rotating ground glass (RGGD) producing an incoherent beam whose intensity is given by Eq. (6). The output beam from the RGGD passes through free space distance f₁ = 150 mm, a thin lens L₁ with focal length f₁ = 150 mm, a Gaussian amplitude filter (GAF), and a linear polarizer LP₁, then becomes a linearly polarized HGCSM beam. After passing through the RPC, the linearly polarized HGCSM beam is converted into a radially polarized HGCSM beam. The generated radially polarized HGCSM beam passes through a thin lens L₂ with focal length f₂ = 400 mm and goes towards a charge-coupled device (CCD), which is used to measure the intensity distribution $I (ρ)$ of the focused radially polarized HGCSM beam. The distance between L₂ and CCD is z. The intensity components $I_{x x} (ρ)$ and $I_{y y} (ρ)$ can be measured by adding linear polarizer LP₂ between L₂ and CCD.

Fig. 11 Experimental setup for generating a radially polarized HGCSM beam and measuring its focused intensity. BE, beam expander; RM, reflecting mirror; SLM, spatial light modulator; CA, circular aperture; RGGD, rotating ground-disk; L₁ and L₂, thin lenses; GAF, Gaussian amplitude filter; RPC, radial polarization converter; LP₁ and LP₂, linear polarizers; CCD, charge-coupled device; PC₁ and PC₂, personal computers.

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The beam width and the coherence width of the generated radially polarized HGCSM beam just beside the RPC are measured to be 1mm and 0.2mm, the principle and procedure for measuring such parameters can be found in [40,41]. Figures 12 and 13 show our experimental results of the intensity distribution of generated radially polarized HGCSM beam with m = n = 5 and m = 5, n = 0 focused by thin lens L₂ with focal length f₂ = 400 mm and its corresponding components $I_{x x} (ρ)$ and $I_{y y} (ρ)$ at different propagation distance z. One finds from our experiment results that the focused radially polarized HGCSM beam indeed displays splitting properties on propagation as predicted by our theoretical results, e.g., the initial vector beam with single beam spot evolves into a vector beam with two or four beam spots on propagation. Furthermore, from the experimental results of the intensity components $I_{x x} (ρ)$ and $I_{y y} (ρ)$ , one finds that the vector beam with more polarization singularities appear on propagation as expected.

Fig. 12 Experimental results of intensity distribution of the generated radially polarized HGCSM beam with m = n = 5 focused by thin lens L₂ with focal length f₂ = 400 mm and its corresponding components $I_{x x} (ρ)$ and $I_{y y} (ρ)$ at different propagation distance z.

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Fig. 13 Experimental results of intensity distribution of the generated radially polarized HGCSM beam with m = 5, n = 0 focused by thin lens L₂ with focal length f₂ = 400 mm and its corresponding components $I_{x x} (ρ)$ and $I_{y y} (ρ)$ at different propagation distance z.

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7. Summary

In this paper, vector HGCSM beam was introduced as a natural extension of recently introduced scalar HGCSM beam. We have derived theoretical models, realizability and beam conditions, and propagation formulae for a vector HGCSM beam with uniform SOP (i.e., electromagnetic HGCSM beam) and a vector HGCSM beam with non-uniform SOP (radially polarized HGCSM beam), respectively. The propagation properties of an electromagnetic HGCSM beam and a radially polarized HGCSM beam have been illustrated numerically and analyzed in detail. Furthermore, we have reported experimental generation of a radially polarized HGCSM beam for the first time. We have found that modulating the structures of the correlation functions provides a novel way for modulating the intensity distribution and polarization properties of a partially coherent vector beam, which will be useful for in many applications, where light beam with prescribed beam shape and polarization properties are required.

Appendix A Propagation formulae for the elements of the BCP matrix with $ρ_{1} = ρ_{2}$ on propagation in free space

(a) The elements of the BCP matrix for an electromagnetic HGCSM beam on propagation

\begin{array}{l} Γ_{α β} (ρ, ρ) = A_{α} A_{β} B_{α β} \frac{k^{2} σ_{0}^{2}}{2 z^{2}} \frac{1}{H_{2 m_{α β}} (0) H_{2 n_{α β}} (0)} \frac{1}{a_{α β}} \exp [- {(\frac{k}{z})}^{2} \frac{(ρ_{x}^{2} + ρ_{y}^{2})}{4 a_{α β}}] \\ \times {(1 - \frac{1}{2 a_{α β} δ_{0 α β}^{2}})}^{m_{α β}} H_{2 m_{α β}} [\frac{i k}{z} \frac{ρ_{x}}{2 \sqrt{a_{α β}} {(2 a_{α β} δ_{0 α β}^{2} - 1)}^{1 / 2}}] \\ \times {(1 - \frac{1}{2 a_{α β} δ_{0 α β}^{2}})}^{n_{α β}} H_{2 n_{α β}} [\frac{i k}{z} \frac{ρ_{y}}{2 \sqrt{a_{α β}} {(2 a_{α β} δ_{0 α β}^{2} - 1)}^{1 / 2}}], (α, β = x, y), \end{array}

with

a_{α β} = \frac{1}{8 σ_{0}^{2}} + \frac{1}{2 δ_{0 α β}^{2}} + \frac{σ_{0}^{2}}{2} {(\frac{k}{z})}^{2} .

(b) The elements of the BCP matrix for a radially polarized HGCSM beam on propagation

\begin{array}{l} Γ_{x x} (ρ, ρ) = \frac{1}{4} \frac{π^{2}}{{(λ z)}^{2}} \frac{1}{H_{2 m_{x x}} (0) H_{2 n_{x x}} (0)} \frac{1}{a} \\ \times \exp [- {(\frac{k}{z})}^{2} \frac{(ρ_{x}^{2} + ρ_{y}^{2})}{4 a}] H_{2 n_{x x}} [\frac{i k}{z} \frac{ρ_{y}}{2 \sqrt{a} {(2 a δ_{0}^{2} - 1)}^{1 / 2}}] {(1 - \frac{1}{2 a δ_{0}^{2}})}^{n_{x x}} \\ \times {2 σ_{0}^{2} {{(1 - \frac{1}{2 a δ_{0}^{2}})}^{m_{x x}} H_{2 m_{x x}} [\frac{i k}{z} \frac{ρ_{x}}{2 \sqrt{a} {(2 a δ_{0}^{2} - 1)}^{1 / 2}}] - σ_{0}^{2} δ_{0}^{2} {(\frac{k}{z})}^{2} Ω_{x x}} - \frac{1}{2} δ_{0}^{2} Ω_{x x}}, \end{array}

\begin{array}{l} Γ_{y y} (ρ, ρ) = \frac{1}{4} \frac{π^{2}}{{(λ z)}^{2}} \frac{1}{H_{2 m_{y y}} (0) H_{2 n_{y y}} (0)} \frac{1}{a} \\ \times \exp [- {(\frac{k}{z})}^{2} \frac{(ρ_{x}^{2} + ρ_{y}^{2})}{4 a}] H_{2 m_{y y}} [\frac{i k}{z} \frac{ρ_{x}}{2 \sqrt{a} {(2 a δ_{0}^{2} - 1)}^{1 / 2}}] {(1 - \frac{1}{2 a δ_{0}^{2}})}^{m_{y y}} \\ \times {2 σ_{0}^{2} {{(1 - \frac{1}{2 a δ_{0}^{2}})}^{n_{y y}} H_{2 n_{y y}} [\frac{i k}{z} \frac{ρ_{y}}{2 \sqrt{a} {(2 a δ_{0}^{2} - 1)}^{1 / 2}}] - σ_{0}^{2} δ_{0}^{2} {(\frac{k}{z})}^{2} Ω_{y y}} - \frac{1}{2} δ_{0}^{2} Ω_{y y}}, \end{array}

\begin{array}{l} Γ_{x y} (ρ, ρ) = - \frac{1}{4} [σ_{0}^{4} {(\frac{k}{z})}^{2} + \frac{1}{4}] \frac{δ_{0}^{2} π^{2}}{{(λ z)}^{2}} \frac{1}{H_{2 m_{x y}} (0) H_{2 n_{x y}} (0)} \frac{1}{a} \exp [- {(\frac{k}{z})}^{2} \frac{(ρ_{x}^{2} + ρ_{y}^{2})}{4 a}] \\ \times \sum_{p = 0}^{\min (2 m_{x y}, 1)} 2^{p} p! (\begin{matrix} 2 m_{x y} \\ p \end{matrix}) (\begin{matrix} 1 \\ p \end{matrix}) {(1 - \frac{1}{2 a δ_{0}^{2}})}^{m_{x y} + 0.5 - p} H_{2 m_{x y} + 1 - 2 p} [\frac{i k}{z} \frac{ρ_{x}}{2 \sqrt{a} {(2 a δ_{0}^{2} - 1)}^{1 / 2}}] \\ \times \sum_{q = 0}^{\min (2 n_{x y}, 1)} 2^{q} q! (\begin{matrix} 2 n_{x y} \\ q \end{matrix}) (\begin{matrix} 1 \\ q \end{matrix}) {(1 - \frac{1}{2 a δ_{0}^{2}})}^{n_{x y} + 0.5 - q} H_{2 n_{x y} + 1 - 2 q} [\frac{i k}{z} \frac{ρ_{y}}{2 \sqrt{a} {(2 a δ_{0}^{2} - 1)}^{1 / 2}}], \end{array}

Γ_{y x} (ρ, ρ) = Γ_{x y}^{*} (ρ, ρ),

with

a = \frac{1}{8 σ_{0}^{2}} + \frac{1}{2 δ_{0}^{2}} + \frac{σ_{0}^{2}}{2} {(\frac{k}{z})}^{2},

\begin{array}{l} Ω_{x x} = {(1 - \frac{1}{2 a δ_{0}^{2}})}^{m_{x x}} H_{2 m_{x x}} [\frac{i k}{z} \frac{ρ_{x}}{2 \sqrt{a} {(2 a δ_{0}^{2} - 1)}^{1 / 2}}] + \frac{1}{2} \sum_{p = 0}^{\min (2 m_{x x}, 2)} 2^{p} p! (\begin{matrix} 2 m_{x x} \\ p \end{matrix}) \\ \times (\begin{matrix} 2 \\ p \end{matrix}) {(1 - \frac{1}{2 a δ_{0}^{2}})}^{m_{x x} + 1 - p} H_{2 m_{x x} + 2 - 2 p} [\frac{i k}{z} \frac{ρ_{x}}{2 \sqrt{a} {(2 a δ_{0}^{2} - 1)}^{1 / 2}}], \end{array}

\begin{array}{l} Ω_{y y} = {(1 - \frac{1}{2 a δ_{0}^{2}})}^{n_{y y}} H_{2 n_{y y}} [\frac{i k}{z} \frac{ρ_{y}}{2 \sqrt{a} {[2 a δ_{0}^{2} - 1]}^{1 / 2}}] + \frac{1}{2} \sum_{q = 0}^{\min (2 n_{y y}, 2)} 2^{q} q! (\begin{matrix} 2 n_{y y} \\ q \end{matrix}) \\ \times (\begin{matrix} 2 \\ q \end{matrix}) {(1 - \frac{1}{2 a δ_{0}^{2}})}^{n_{y y} + 1 - q} H_{2 n_{y y} + 2 - 2 q} [\frac{i k}{z} \frac{ρ_{y}}{2 \sqrt{a} {(2 a δ_{0}^{2} - 1)}^{1 / 2}}] . \end{array}

Acknowledgments

National Natural Science Fund for Distinguished Young Scholar (11525418); National Natural Science Foundation of China (NSFC) (11274005, 11404234); Project of the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions.

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Vector Hermite-Gaussian correlated Schell-model beam

Abstract

1. Introduction

2. Theoretical model for a vector HGCSM beam

(a) Vector HGCSM beam with uniform SOP

(b) Vector HGCSM beam with non-uniform SOP

3. Realizability and beam conditions for a vector HGCSM beam

4. Analytical formulae for a vector HGCSM beam propagating in free space

5. Paraxial propagation properties of a vector HGCSM beam in free space

6. Experimental generation of a radially polarized HGCSM beam

7. Summary

Appendix A Propagation formulae for the elements of the BCP matrix with $ρ_{1} = ρ_{2}$ on propagation in free space

(a) The elements of the BCP matrix for an electromagnetic HGCSM beam on propagation

(b) The elements of the BCP matrix for a radially polarized HGCSM beam on propagation

Acknowledgments

References and links

Cited By

Figures (13)

Equations (36)

Optics Express

Abstract

1. Introduction

2. Theoretical model for a vector HGCSM beam

(a) Vector HGCSM beam with uniform SOP

(b) Vector HGCSM beam with non-uniform SOP

3. Realizability and beam conditions for a vector HGCSM beam

4. Analytical formulae for a vector HGCSM beam propagating in free space

5. Paraxial propagation properties of a vector HGCSM beam in free space

6. Experimental generation of a radially polarized HGCSM beam

7. Summary

Appendix A Propagation formulae for the elements of the BCP matrix with ρ1=ρ2on propagation in free space

(a) The elements of the BCP matrix for an electromagnetic HGCSM beam on propagation

(b) The elements of the BCP matrix for a radially polarized HGCSM beam on propagation

Acknowledgments

References and links

Cited By

Figures (13)

Equations (36)

Optics Express

Appendix A Propagation formulae for the elements of the BCP matrix with $ρ_{1} = ρ_{2}$ on propagation in free space