Generation and propagation of a vector cosine-Gaussian correlated beam with radial polarization

Shijun Zhu; Yahong Chen; Jing Wang; Haiyan Wang; Zhenhua Li; Yangjian Cai

doi:10.1364/OE.23.033099

1. Introduction

It is well known that the spatial correlation function and the spatial coherence length of a partially coherent beam significantly affect its statistical properties on propagation, such as the intensity distribution, the degree of coherence (DOC), the propagation factor, the degree of polarization (DOP) and the state of polarization (SOP) [1–13 ]. It is useful and meaningful to modulate or design the spatial correlation function of a partially coherent beam. Before Gori et al. discussed the sufficient conditions for devising genuine correlation function of a partially coherent beam [14, 15 ], only few partially coherent sources with different correlation functions were introduced and studied, such as Gaussian correlated Schell-model (i.e, Gaussian Schell-model) source, J₀-correlated Schell-model source and Lambertian source [16–18 ]. Recently, more and more attention is being paid to explore the spatial correlation functions of partially coherent beams and the novel properties induced by special spatial correlation function [19–40 ]. It is shown that a nonuniformly correlated beam exhibits self-focusing and lateral shift of the intensity maximum [19], a multi-Gaussian correlated Schell-model beam exhibits a flat-topped or ring-shaped far-field beam profile [20, 21 ], a Laguerre-Gaussian correlated Schell-model beam exhibits ring-shaped far-field beam profile [22, 23 ], a cosine-Gaussian correlated Schell-model (CGCSM) beam of circular [24, 25 ] or rectangular symmetry [26] exhibits ring-shaped or four-beamlets array beam profile in the far-field. Both CGCSM beam of rectangular symmetry [26] and Hermite-Gaussian correlated Schell-model beam [27] display self-splitting properties on propagation in free-space. Furthermore, it is found that modulating the spatial correlation function of a partially coherent beam not only enriches the classical coherence theory, but also has important applications in free-space optical communication [28–31 ], particle trapping [32, 33 ], optical scattering [34] and optical imaging [35].

Polarization is a manifestation of correlations involving components of the fluctuating electric field at a single point. In the past decade, the polarization properties of a light beam also attracted much attention. Unlike uniformly polarized beam (e.g., linearly or elliptically polarized beam), cylindrical vector beam with non-uniform state of polarization (e.g., radially polarized beam, azimuthally polarized beam) exhibits many unique but useful properties [36]. Through focusing a radially polarized beam by a high numerical aperture lens, one can obtain a strong longitudinal electric field and a tightly focused beam spot, which are useful in microscopy, lithography, electron acceleration, proton acceleration, material processing, optical data storage, high-resolution metrology, super-resolution imaging, plasmonic focusing and laser machining [36–43 ]. Radial polarized partially coherent beam with conventional Gaussian correlated Schell-model function (i.e., radial polarized GCSM beam) was studied theoretically and generated experimentally just recently [44–52 ]. It was shown that the degree of polarization (DOP) of a radially polarized GCSM beam changes on propagation in free space, while its state of polarization remains invariant [47]. Spectral changes of a polychromatic radially polarized GCSM beam were studied in [49], and it was found that such changes are quite different from that of a polychromatic linearly polarized partially coherent beam. In [51, 52 ], it was observed in experiment that a radially polarized GCSM beam has advantage over a linearly polarized partially coherent beam for reducing turbulence-induced scintillation, which will be useful in free-space optical communications. Generation of a partially coherent vector beam with special correlation functions was reported in [53]. In this paper, we introduce a new kind of partially coherent vector beam named radially polarized CGCSM beam as a natural extension of scalar CGCSM beam, and discuss the realizability and beam conditions. We analyze the statistical properties of a radially polarized CGCSM beam on propagation in detail, and we report experimental generation of such beam. Some interesting results are found.

2. Theoretical model, realizability and beam conditions of a radially polarized CGCSM beam

Based on the unified theory of coherence and polarization [2], in space-frequency domain, the second-order statistical properties of a partially coherent vector (i.e., electromagnetic) beam can be characterized by the cross-spectral density (CSD) matrix $\overset{\leftrightarrow}{W} (r_{1}, r_{2})$ with elements $W_{α β} (r_{1}, r_{2}) = 〈 E_{α}^{*} (r_{1}) E_{β} (r_{2}) 〉,$ $(α, β = x, y)$ . Here $E_{α} (r)$ is the electric field component along $α$ -axis at position vector $r$ , the asterisk denotes the complex conjugate and the angular brackets denote an ensemble average. As a natural extension of a scalar CGCSM beam of rectangular symmetry [25], we define the elements of the CSD matrix of a radially polarized CGCSM beam in the source plane (z = 0) as follows

W_{α β} (r_{1}, r_{2}) = A_{α} A_{β} α_{1} β_{2} \exp [- \frac{r_{1}^{2} + r_{2}^{2}}{4 w_{0}^{2}}] q_{α β} (r_{1} - r_{2}), (α, β = x, y),

with

q_{α β} (r_{1} - r_{2}) = B_{α β} \cos [\frac{\sqrt{2 π} n (x_{1} - x_{2})}{δ_{α β}}] \cos [\frac{\sqrt{2 π} n (y_{1} - y_{2})}{δ_{α β}}] \exp [- \frac{{(r_{1} - r_{2})}^{2}}{2 δ_{α β}^{2}}],

Here

r_{i} \equiv (x_{i}, y_{i})

is the transverse position vector,

A_{x}

and

A_{y}

are the amplitudes of x and y components of the electric field, respectively, and we assume

A_{x} = A_{y} = 1

.

δ_{x x}

,

δ_{y y}

and

δ_{x y}

are the r.m.s. widths of autocorrelation functions of the x component of the field, of the y component of the field, and of the mutual correlation function of x and y field components, respectively, and

δ_{x y} = δ_{y x}

.

B_{x y} = | B_{x y} | \exp (i ϕ) = B_{y x}^{*}

is the complex correlation coefficient and

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

. n is the beam order parameter, and the radially polarized CGCSM beam reduces to the conventional radially polarized GCSM beam when n = 0 [47, 48 ].

To be a physical realizable radially polarized CGCSM source/beam, the following necessary and sufficient conditions should be satisfied

(1)Realizability conditions for a partially coherent vector source with CGCSM function
(2)Realizablity conditions for a radially polarized CGCSM source
(3)Nonnegative conditions for a radially polarized CGCSM source
(4)Beam condition for radiation generated by a radially polarized CGCSM source

First, we discuss the realizability conditions for a partially coherent vector source with CGCSM function. Equation (1) can be alternatively written as follows

W_{α β} (r_{1}, r_{2}) = \sqrt{W_{α α} (r_{1}, r_{1})} \sqrt{W_{β β} (r_{2}, r_{2})} μ_{α β} (r_{1} - r_{2}), (α, β = x, y),

where

W_{x x}

and

W_{y y}

stand for the diagonal elements of the CSD matrix and they are nonnegative functions. Furthermore, the corresponding correlation functions

μ_{x x}

and

μ_{y y}

, and their Fourier transforms all have to be nonnegative. In fact, due to the nonnegtive definiteness requirement of the whole CSD matrix, any choice of off-diagonal elements of the CSD matrix should satisfy this condition.

For a partially coherent vector source specified by Eq. (3), the nonnegative condition is given by the following quadratic form [54]

ℜ = \sum_{α = x, y} \sum_{β = x, y} \int \int q_{α β} (r_{1}, r_{2}) g_{α}^{*} (r_{1}) g_{β} (r_{2}) d^{2} r_{1} d^{2} r_{2},

where

g_{α}

is an arbitrary function.

μ_{α β}

can be expressed in term of its Fourier transform

{\tilde{q}}_{α β}

as follows

q_{α β} (r_{1} - r_{2}) = \int {\tilde{q}}_{α β} (ξ) \exp [2 π i ξ \cdot (r_{1} - r_{2})] d^{2} ξ .

Since the CSD matrix must be quasi-Hermitian, then we have

q_{x y} (r_{1} - r_{2}) = q_{y x}^{*} (r_{2} - r_{1}), {\tilde{q}}_{x y} (ξ) = {\tilde{q}}_{y x}^{*} (ξ) .

Substituting Eqs. (5) and (6) into Eq. (4), one obtains

ℜ = \int {{| {\tilde{g}}_{x} (ξ) |}^{2} {\tilde{q}}_{x x} (ξ) + {| {\tilde{g}}_{y} (ξ) |}^{2} {\tilde{q}}_{y y} (ξ) + 2 Re [{\tilde{g}}_{x}^{*} (ξ) {\tilde{g}}_{y} (ξ) {\tilde{q}}_{x y} (ξ)]} d^{2} ξ,

where

{\tilde{q}}_{x x}

and

{\tilde{q}}_{y y}

both are nonnegative functions as mentioned above.

In order for $ℜ$ to be nonnegative, one can obtain the following sufficient condition for a partially coherent vector beam [54]

| {\tilde{q}}_{x y} (ξ) | \leq \sqrt{{\tilde{q}}_{x x} (ξ) {\tilde{q}}_{y y} (ξ)},

which meets for any

ξ

.

Applying Eqs. (2), (5) and (8) , it is not difficult to obtain the following realizability condition for a partially coherent vector source with CGCSM function

\begin{array}{l} | B_{x y} | δ_{x y}^{2} \cos h^{2} (2 \sqrt{2} π^{3 / 2} n δ_{x y} ξ) \exp (- 4 π^{2} δ_{x y}^{2} ξ^{2}) \leq \sqrt{δ_{x x}^{2} δ_{y y}^{2} \exp [- 4 π^{2} (δ_{x x}^{2} + δ_{y y}^{2}) ξ^{2}]} \\ \times \sqrt{\cos h^{2} (2 \sqrt{2} π^{3 / 2} n δ_{x x} ξ) \cos h^{2} (2 \sqrt{2} π^{3 / 2} n δ_{y y} ξ)} . \end{array}

Since both sides of the inequality equation (Eq. (9)) are monotonic functions with $ξ$ . It is not difficult to derive the following inequality relations

\sqrt{(δ_{x x}^{2} + δ_{y y}^{2}) / 2} \leq δ_{x y} \leq \sqrt{δ_{x x} δ_{y y} / | B_{x y} |}, | B_{x y} | \leq 2 δ_{x x} δ_{y y} / (δ_{x x}^{2} + δ_{y y}^{2}) \leq 1.

Equation (10) is the realizability (i.e., necessary and sufficient) conditions for a partially coherent vector source with CGCSM function.

Second, we discuss the realizability conditions for a radially polarized CGCSM source. Besides the general condition (Eq. (10)) should be satisfied, the following two additional conditions for a radially polarized CGCSM source should be satisfied

(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 state of polarization of the completely polarized portion can be studied with the help of a polarization ellipse as well as the degree of ellipticity $ε$ and the orientation angle $θ$ [7]

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

ε (r, r) = A_{-} (r, r) / A_{+} (r, r), 0 \leq ε \leq 1.

where

A_{+}

and

A_{-}

stand for the major and minor semi-axes of the polarization ellipse and take the following form

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

The additional condition (a) implies $ε = 0$ . With the requirements of additional conditions (a) and (b), and applying Eqs. (1) and (11)-(14) , we obtain the following realizability conditions for a radially polarized CGCSM source

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

δ_{x x} = δ_{y y} = δ_{x y} = δ_{y x} = δ_{0} .

Third, we discuss the nonnegative conditions for a radially polarized CGCSM source. According to [15], to be a genuine CSD matrix, for a radially polarized CGCSM source, the following nonnegative conditions should be satisfied

Λ_{x x} (u) \geq 0, Λ_{y y} (u) \geq 0, Λ_{x x} (u) Λ_{y y} (u) - {| Λ_{x y} (u) |}^{2} \geq 0,

where

Λ_{α β} (u) = \cos h (\frac{2 n \sqrt{2 π} u_{x}}{σ_{α β}}) \cos h (\frac{2 n \sqrt{2 π} u_{y}}{σ_{α β}}) \exp [- \frac{2 (u_{x}^{2} + u_{y}^{2})}{σ_{α β}^{2}}], (α, β = x, y) .

Here

u = (u_{x}, u_{y})

,

σ_{α β}

is a positive constant determined by

δ_{α β}

, thus

σ_{α α} = σ_{β β} = σ_{α β} = σ_{0}

. Since

\cosh (x)

and Gaussian function are nonnegative functions, it is easy to find Eq. (17) is indeed satisfied for a radially polarized CGCSM source

Finally, we discuss the beam condition for radiation generated by a radially polarized CGCSM source. To generate a beam-like field from a radially polarized CGCSM source, some beam condition should be satisfied. The spectral density at point $ρ = ρ \vec{s}$ ( $\vec{s}$ is unit vector) in the far field has the following form

S^{\infty} (ρ) = 4 π^{2} k^{2} ρ^{- 2} \cos^{2} φ [{\tilde{W}}_{x x} (- k {\vec{s}}_{⊥}, k {\vec{s}}_{⊥}) + {\tilde{W}}_{y y} (- k {\vec{s}}_{⊥}, k {\vec{s}}_{⊥})],

with

{\tilde{W}}_{α α} (f_{1}, f_{2}) = \frac{1}{{(2 π)}^{4}} \iint W_{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},

where

k = 2 π / λ

is the wave number with

λ

being the wavelength,

f_{1} = - k {\vec{s}}_{1 ⊥}, f_{2} = k {\vec{s}}_{2 ⊥},

\cos φ = {\vec{s}}_{z}

,

{\vec{s}}_{⊥}

is the projection of

\vec{s}

onto the source plane,

φ

is the angle that the unit vector

\vec{s}

makes with a positive z direction. Substituting Eqs. (1), (2), (15) and (16) into Eq. (20), one obtains

\begin{array}{l} {\tilde{W}}_{x x} (- k {\vec{s}}_{⊥}, k {\vec{s}}_{⊥}) = \frac{k^{2} w_{0}^{2} \cos^{2} φ}{16 ρ^{2} M_{1}^{2} M_{2}} {e x p [- \frac{{(k {\vec{s}}_{y} - a)}^{2}}{2 M_{2}}] + e x p [- \frac{{(k {\vec{s}}_{y} + a)}^{2}}{2 M_{2}}]} \\ \times {[δ_{0}^{- 2} + \frac{{(k {\vec{s}}_{x} + a)}^{2}}{4 w_{0}^{2} M_{2}}] \exp (- \frac{{(k {\vec{s}}_{x} + a)}^{2}}{2 M_{2}}) + [δ_{0}^{- 2} + \frac{{(k {\vec{s}}_{x} - a)}^{2}}{4 w_{0}^{2} M_{2}}] \exp (- \frac{{(k {\vec{s}}_{x} - a)}^{2}}{2 M_{2}})}, \end{array}

\begin{array}{l} {\tilde{W}}_{y y} (- k {\vec{s}}_{⊥}, k {\vec{s}}_{⊥}) = \frac{k^{2} w_{0}^{2} \cos^{2} φ}{16 ρ^{2} M_{1}^{2} M_{2}} {e x p [- \frac{{(k {\vec{s}}_{x} - a)}^{2}}{2 M_{2}}] + e x p [- \frac{{(k {\vec{s}}_{x} + a)}^{2}}{2 M_{2}}]} \\ \times {[δ_{0}^{- 2} + \frac{{(k {\vec{s}}_{y} + a)}^{2}}{4 w_{0}^{2} M_{2}}] \exp (- \frac{{(k {\vec{s}}_{y} + a)}^{2}}{2 M_{2}}) + [δ_{0}^{- 2} + \frac{{(k {\vec{s}}_{y} - a)}^{2}}{4 w_{0}^{2} M_{2}}] \exp (- \frac{{(k {\vec{s}}_{y} - a)}^{2}}{2 M_{2}})}, \end{array}

with

a = n \sqrt{2 π} / δ_{0}, M_{1} = 1 / 4 w_{0}^{2} + 1 / 2 δ_{0}^{2}, M_{2} = 1 / 4 w_{0}^{2} + 1 / δ_{0}^{2} .

For a beam-like field, its far-field spectral density must be negligible except for directions within a narrow solid angle about the z axis. Then one can derive a general beam condition for a radially polarized CGCSM beam as follows

e x p (- k^{2} {\vec{s}}_{x}^{2} / 2 M_{2}) \approx 0, e x p (- k^{2} {\vec{s}}_{y}^{2} / 2 M_{2}) \approx 0.

Therefore, the beam condition for radiation generated by a radially polarized CGCSM source is given as

1 / 4 w_{0}^{2} + δ_{0}^{- 2} ≪ 2 π^{2} / λ^{2} .

3. Statistical properties of a radially polarized CGCSM beam on propagation

In this section, we are going to derive the analytical formulae for a radially polarized CGCSM beam propagating through a stigmatic ABCD optical system and analyze the statistical properties of such beam focused by a thin lens.

Within the validity of the paraxial approximation, propagation of the elements of the CSD matrix of a partially coherent vector beam through a stigmatic ABCD optical system can be studied with the help of the following extended Collins formula [55, 56 ]

\begin{array}{l} W_{α β} (ρ_{1}, ρ_{2}) = \frac{k^{2}}{4 π^{2} B^{2}} \iint W_{α β} (r_{1}, r_{2}) \exp [\frac{i k}{2 B} (A r_{2}^{2} - 2 r_{2} \cdot ρ_{2} + D ρ_{2}^{2})] \\ \times \exp [- \frac{i k}{2 B} (A r_{1}^{2} - 2 r_{1} \cdot ρ_{1} + D ρ_{1}^{2})] d^{2} r_{1} d^{2} r_{2}, \end{array}

where

ρ_{j} \equiv (u_{j}, v_{j})

is the transverse position vector in the receiver plane,

A, B, C,

and

D

are elements of the transfer matrix of the optical system.

Substituting Eqs. (1) and (2) into Eq. (26), one obtain (after tedious integration and operation) the following expressions for the elements of the CSD matrix of a radially polarized CGCSM beam in the receive plane

\begin{array}{l} W_{x x} (ρ_{1}, ρ_{2}) = \frac{k^{2} Γ (ρ_{1}, ρ_{2})}{64 B^{2} N_{1}^{2} Π^{2}} [e x p (- \frac{ζ_{v 12}^{2}}{4 N_{1}} - \frac{Ω_{v 22}^{2}}{4 Π}) + e x p (- \frac{ζ_{v 11}^{2}}{4 N_{1}} - \frac{Ω_{v 21}^{2}}{4 Π})] \\ \times {(δ_{0}^{- 2} - ζ_{u 11} Ω_{u 21} - \frac{Ω_{u 21}^{2}}{2 δ_{0}^{2} Π}) \exp (- \frac{ζ_{u 11}^{2}}{4 N_{1}} - \frac{Ω_{u 21}^{2}}{4 Π}) \\ + (δ_{0}^{- 2} - ζ_{u 12} Ω_{u 22} - \frac{Ω_{u 22}^{2}}{2 δ_{0}^{2} Π}) \exp (- \frac{ζ_{u 12}^{2}}{4 N_{1}} - \frac{Ω_{u 22}^{2}}{4 Π})}, \end{array}

\begin{array}{l} W_{y y} (ρ_{1}, ρ_{2}) = \frac{k^{2} Γ (ρ_{1}, ρ_{2})}{64 B^{2} N_{1}^{2} Π^{2}} [e x p (- \frac{ζ_{u 12}^{2}}{4 N_{1}} - \frac{Ω_{u 22}^{2}}{4 Π}) + e x p (- \frac{ζ_{u 11}^{2}}{4 N_{1}} - \frac{Ω_{u 21}^{2}}{4 Π})] \\ \times {(δ_{0}^{- 2} - ζ_{v 11} Ω_{v 21} - \frac{Ω_{v 21}^{2}}{2 δ_{0}^{2} Π}) \exp (- \frac{ζ_{v 11}^{2}}{4 N_{1}} - \frac{Ω_{v 21}^{2}}{4 Π}) \\ + (δ_{0}^{- 2} - ζ_{v 12} Ω_{v 22} - \frac{Ω_{v 22}^{2}}{2 δ_{0}^{2} Π}) \exp (- \frac{ζ_{v 12}^{2}}{4 N_{1}} - \frac{Ω_{v 22}^{2}}{4 Π})}, \end{array}

\begin{array}{l} W_{x y} (ρ_{1}, ρ_{2}) = \frac{k^{2} Γ (ρ_{1}, ρ_{2})}{64 B^{2} N_{1}^{2} Π^{2}} {Ω_{v 22} e x p [- \frac{ζ_{v 12}^{2}}{4 N_{1}} - \frac{Ω_{v 22}^{2}}{4 Π}] + Ω_{v 21} e x p [- \frac{ζ_{v 11}^{2}}{4 N_{1}} - \frac{Ω_{v 21}^{2}}{4 Π}]} \\ \times {(- ζ_{u 11} - \frac{Ω_{u 21}}{2 δ_{0}^{2} Π}) \exp [- \frac{ζ_{u 11}^{2}}{4 N_{1}} - \frac{Ω_{u 21}^{2}}{4 Π}] + (- ζ_{u 12} - \frac{Ω_{u 22}}{2 δ_{0}^{2} Π}) \exp [- \frac{ζ_{u 12}^{2}}{4 N_{1}} - \frac{Ω_{u 22}^{2}}{4 Π}]}, \end{array}

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

with

\begin{array}{l} ζ_{u 11} = k u_{1} / B + a, ζ_{u 12} = k u_{1} / B - a, ζ_{u 21} = k u_{2} / B + a, ζ_{u 22} = k u_{2} / B - a, \\ ζ_{v 11} = k v_{1} / B + a, ζ_{v 12} = k v_{1} / B - a, ζ_{v 21} = k v_{2} / B + a, ζ_{v 22} = k v_{2} / B - a, \\ Ω_{u 22} = ζ_{u 12} / 2 δ_{0}^{2} N_{1} - ζ_{u 22}, Ω_{u 21} = ζ_{u 11} / 2 δ_{0}^{2} N_{1} - ζ_{u 21}, Ω_{v 22} = ζ_{v 12} / 2 δ_{0}^{2} N_{1} - ζ_{v 22}, \\ Ω_{v 21} = ζ_{v 11} / 2 δ_{0}^{2} N_{1} - ζ_{v 21}, N_{1} = M_{1} + i k A / 2 B, Π = M_{1} - i k A / 2 B - 1 / 4 δ_{0}^{4} N_{1}, \\ Γ (ρ_{1}, ρ_{2}) = \exp [- i k D (ρ_{1}^{2} - ρ_{2}^{2}) / 2 B] . \end{array}

There are two definitions of the DOC for a paraxial partially coherent vector beam [2, 57 ]. Here, we use the definition introduced by Tervo et al., which is convenient for experimental measurements. The DOC between two arbitrary points $ρ_{1}$ and $ρ_{2}$ is defined as [57]

{\vec{q}}^{2} (ρ_{1}, ρ_{2}) = \frac{T r [{\overset{\leftrightarrow}{W}}^{*} (ρ_{1}, ρ_{2}) \overset{\leftrightarrow}{W} (ρ_{1}, ρ_{2})]}{T r [\overset{\leftrightarrow}{W} (ρ_{1}, ρ_{1})] T r [\overset{\leftrightarrow}{W} (ρ_{2}, ρ_{2})]},

where Det and Tr denote the determinant and the trace of the matrix. The average intensity and the DOP at point

ρ

in the receiver plane are given as

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

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

The information of the SOP of the completely polarized portion of the partially coherent vector beam in the receiver plane can be studied by Eqs. (12)-(14) just by replacing

r

with

ρ

. Applying Eqs. (27)-(34) , we can study the statistical properties of a radially polarized CGCSM beam on propagation conveniently.

As an example, we study the focusing properties of a radially polarized CGCSM beam. Assume that the beam passes through a thin lens with focal length f and then arrives at the receiver plane $ρ$ . The distances from the source to the thin lens and from the thin lens to the receiver plane $ρ$ are f and z, respectively. The transfer matrix between the source plane and the plane $ρ$ reads as

(\begin{matrix} A & B \\ C & D \end{matrix}) = (\begin{matrix} 1 & z \\ 0 & 1 \end{matrix}) (\begin{matrix} 1 & 0 \\ - 1 / f & 1 \end{matrix}) (\begin{matrix} 1 & f \\ 0 & 1 \end{matrix}) = (\begin{matrix} 1 - z / f & f \\ - 1 / f & 0 \end{matrix}) .

The parameters used in the following calculations are set as

λ = 632.8 nm,

w_{0} = 1 mm,

δ_{x x} = δ_{y y} = δ_{x y} = δ_{0} = 0.6 mm,

f_{3} =150mm

.

We calculate in Fig. 1 the density plots of the square of the DOC ${\vec{q}}^{2} (x, y, 1 m m, 1 m m)$ , ${\vec{q}}_{x x}^{2} (x, y, 1 m m, 1 m m)$ , ${\vec{q}}_{x y}^{2} (x, y, 1 m m, 1 m m)$ , ${\vec{q}}_{y y}^{2} (x, y, 1 m m, 1 m m)$ of a radially polarized CGCSM beam in the source plane for different values of $n$ , and in Fig. 2 the spectral intensity distribution and the corresponding cross line of a radially polarized CGCSM beam in the source plane for different values of $n$ . For the case of n = 0, the radially polarized CGCSM beam reduces to the conventional radially polarized GCSM beam, whose square of the DOC (see Figs. 1(a)-1(a3)) and spectral intensity (see Fig. 2(a)) all have circular symmetries. For the case of $n \neq 0$ , the square of the DOC of a radially polarized CGCSM beam exhibits array distribution with rectangular symmetry and the number of the beamlets increases with the increase of the beam order $n$ (see Figs. 1(b)-1(b3) and Figs. 1(c)-1(c3)), which is similar to that of a scalar CGCSM beam, while its spectral intensity is the same with the case n = 0. Due to the special distribution of the DOC, the radially polarized CGCSM beam exhibits interesting propagation properties, which are quite different from those of the radially polarized GCSM beam as shown later.

Fig. 1 Density plots of the square of the DOC ${\vec{q}}^{2} (x, y, 1 m m, 1 m m)$ , ${\vec{q}}_{x x}^{2} (x, y, 1 m m, 1 m m)$ ${\vec{q}}_{x y}^{2} (x, y, 1 m m, 1 m m)$ , ${\vec{q}}_{y y}^{2} (x, y, 1 m m, 1 m m)$ of a radially polarized CGCSM beam in the source plane for different values of $n$ .

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Fig. 2 Spectral intensity distribution and the corresponding cross line of a radially polarized CGCSM beam in the source plane for different values of $n$ .

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We calculate in Fig. 3 the spectral intensity distribution $I (ρ)$ and the corresponding cross line of a radially polarized CGCSM beam focused by a thin lens at several propagation distances for different values of n, and in Fig. 4 the spectral intensity distribution $I (ρ)$ , its composition components $W_{x x} (ρ, ρ)$ , $W_{y y} (ρ, ρ)$ , and the corresponding cross lines of a radially polarized CGCSM beam focused by a thin lens at several propagation distances with n = 1. For the case of n = 0, the spectral intensity distribution of the conventional radially polarized GCSM beam always has circular symmetry on propagation as expected (see Figs. 3(a1)-3(e1)) [47, 48 ]. For the case of $n \neq 0$ , the radially polarized CGCSM beam exhibits quite different propagation properties, i.e., the initial dark hollow beam profile gradually disappears on propagation and evolves into a four-beamlets array distribution in the focal plane, and the distance between each beamlets is closely related with the beam order n (see Figs. 3(a2)-3(e2) and (a3)-(e3)). Furthermore, from Fig. 4, one sees that the composition components $W_{x x} (ρ, ρ)$ and $W_{y y} (ρ, ρ)$ of the spectral intensity distribution of a radially polarized CGCSM beam also evolve into four-beamlets array distributions in the focal plane, which are also quite different from those of the conventional radially polarized GCSM beam.

Fig. 3 Spectral intensity distribution $I (ρ)$ and the corresponding cross line of a radially polarized CGCSM beam focused by a thin lens at several propagation distances for different values of $n$ .

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Fig. 4 Spectral intensity distribution $I (ρ)$ , its composition components $W_{x x} (ρ, ρ)$ , $W_{y y} (ρ, ρ)$ , and the corresponding cross lines of a radially polarized CGCSM beam focused by a thin lens at several propagation distances with n = 1.

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Figure 5 shows the density plot of the square of the DOC ${\vec{q}}^{2} (ρ_{1}, ρ_{2} = 0)$ and the corresponding cross line of a radially polarized CGCSM beam focused by a thin lens at different propagation distances for different values of n. One finds from Figs. 5(a1)-5(e1) that the square of the DOC of the conventional radially polarized GCSM beam (n = 0) always has circular symmetry on propagation although its shape changes on propagation as expected [48]. From Figs. 5(a2)-5(e2) and Figs. 5(a3)-5(e3), we see that the square of the DOC of a radially polarized CGCSM beam ( $n \neq 0$ ) exhibits array distribution with rectangular symmetry near the source, and both the shape and the symmetry change on propagation, and the square of the DOC evolves into a rhombus profile, which are quite different from those of the radially polarized GCSM beam. Figure 6 shows the density plot of the DOP and the corresponding cross line of a radially polarized CGCSM beam focused by a thin lens at different propagation distances for different values of n. One sees that the distribution of the DOP of the conventional radially polarized GCSM beam (n = 0) always has circular symmetry and the value of the on-axis points remains zero on propagation as expected [48]. The evolution properties of the distribution of the DOP of the radially polarized CGCSM beam ( $n \neq 0$ ) are quite different from those of the radially polarized GCSM beam and are closely determined by the beam order n. To learn about the properties of the SOP, we calculate in Fig. 7 the SOP of a radially polarized CGCSM beam in the source plane and in the focal plane for different values of n. The SOP of a radially polarized CGCSM beam is independent of the beam order n. The SOP of a radially polarized GCSM beam (n = 0) remains invariant on propagation and still displays radial polarization in the focal plane. The SOP of a radially polarized CGCSM beam changes on propagation and the radial polarization structure is destroyed due to different source correlation function, and the SOP displays a complex structure in the focal plane. From above discussions, one comes to the conclusion that one can modulate the statistical properties of a partially coherent vector beam through modulating the structure of its correlation function.

Fig. 5 Density plot of the square of the DOC ${\vec{q}}^{2} (ρ_{1}, ρ_{2} = 0)$ and the corresponding cross line of a radially polarized CGCSM beam focused by a thin lens at different propagation distances for different values of n.

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Fig. 6 Density plot of the DOP and the corresponding cross line of a radially polarized CGCSM beam focused by a thin lens at different propagation distances for different values of n.

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Fig. 7 SOP of a radially polarized CGCSM beam in the source plane and in the focal plane for different values of n.

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4. Experimental generation of a radially polarized CGCSM beam

Part I in Fig. 8 shows our experimental setup for generating a radially polarized CGCSM beam. A laser beam emitted from a He-Ne laser ( $λ = 632.8 nm$ ) first passes through a beam expander (BE), then is reflected by a Mirror (M). The reflected beam transmits through a spatial light modulator (SLM) which is used to generate a cosh-Gaussian beam. A circular aperture (CA) is used to select out the first-order diffraction pattern, and the LP is used to converts the selected beam into a linearly polarized beam. The transmitted beam from the RGGD can be regarded as a spatially incoherent beam if the diameter of the beam spot on the RGGD is much larger than the inhomogeneity scale of the ground glass. After transmitting a distance $f_{1}$ , thin lens L₁ and the GAF, the incoherent cosh-Gaussian field becomes a linearly polarized CGCSM beam. The RPC converts the linearly polarized CGCSM beam into a radially polarized CGCSM beam. Part II in Fig. 8 shows our experimental setup for measuring the square of the modulus of its DOC, and the focused intensity of the generated radially polarized CGCSM beam. The generated beam is split into two beams by a beam splitter (BS). The transmitted beam is focused by L₃, with focal length $f_{3}$ , and then arrives at a beam profile analyzer (BPA), which is used to measure the intensity distribution. The reflected beam passes through an imaging lens L₂ with focal length $f_{2}$ , and arrives at a CCD, which is used to measure the square of the modulus of the DOC of the generated beam. Detailed principle and process for measuring the square of the modulus of the DOC can be found in [13].

Fig. 8 Experimental setup for generating a radially polarized CGCSM beam, measuring the square of the modulus of its DOC and its focused intensity. Laser, He-Ne laser; BE, beam expander; M, mirror; SLM, spatial light modulator; CA, circular aperture; LP, linear polarizer; RGGD, rotating ground-disk; L₁, L₂, L₃, thin lenses; GAF, Gaussian amplitude filter; RPC, radial polarization converter; BS, beam splitter; CCD, charge-coupled device; BPA, beam profile analyzer; PC₁, PC₂; personal computers.

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Figure 9 shows our experimental results of the intensity distribution and the corresponding cross line of the generated radially polarized CGCSM beam with n = 1 in the output plane of RPC, as well as the composition components $W_{x x} (ρ, ρ)$ and $W_{y y} (ρ, ρ)$ . One finds from Fig. 9 that generated source beam is indeed radially polarized as expected. Through theoretical fit of the experimental data, we obtain $w_{0} = 0.96 mm$ . Figure 10 shows our experimental results of the square of the DOC and the corresponding cross line of the generated radially polarized CGCSM beam in the output plane of RPC with n = 1. One finds from Fig. 10 that the distributions of the square of the DOC ${\vec{q}}^{2}$ , ${\vec{q}}_{x x}^{2}$ , ${\vec{q}}_{x y}^{2}$ , ${\vec{q}}_{y y}^{2}$ all have array distributions with rectangular symmetries as expected by Fig. 1. Through theoretical fit of the experimental data, we obtain $δ_{0} = 0.12 mm$ .

Fig. 9 Experimental results of the intensity distribution and the corresponding cross line of the generated radially polarized CGCSM beam with n = 1 in the output plane of RPC, as well as the composition components $W_{x x} (ρ, ρ)$ and $W_{y y} (ρ, ρ)$ . The solid curve denotes the theoretical fit of the experimental data.

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Fig. 10 Experimental results of the square of the DOC and the corresponding cross line of the generated radially polarized CGCSM beam in the output plane of RPC with n = 1. (a) ${\vec{q}}^{2} (x, 0, 0.2 mm, 0.2 mm)$ , (b) ${\vec{q}}^{2} (x, y, 0.2 mm, 0.2 m m)$ , (c) ${\vec{q}}_{x x}^{2} (x, y, 0.2 mm, 0.2 m m)$ , (d) ${\vec{q}}_{y y}^{2} (x, y, 0.2 mm, 0.2 m m)$ . The solid curve denotes the theoretical fit of the experimental data.

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Figure 11 shows our experimental results of intensity distribution of the generated radially polarized CGCSM beam with n = 1 focused by a thin lens with f ₃ = 150mm and its corresponding composition components $W_{x x} (ρ, ρ)$ and $W_{y y} (ρ, ρ)$ at different propagation distances with n = 1. The solid curve in Fig. 11 denotes the theoretical result. It is found from Fig. 11 that the evolution properties of the intensity distribution of the focused radially polarized CGCSM beam is similar to that of a scalar CGSM beam [26] but are quite different from those of a radially polarized GCSM beam [47, 48 ], and both the intensity distribution and its composition components are consistent with the theoretical results. Figure 12 shows the experimental results of the square of the DOC ${\vec{q}}^{2}$ and the corresponding cross line of the generated radially polarized CGCSM beam with n = 1 focused by a thin lens with f ₃ = 150mm at different propagation distances. As expected, the array distribution of the DOC gradually disappears and evolves into a rhombus profile in the focal plane. Figure 13 shows our experimental results of the DOP (cross line $v = 0$ ) of the generated radially polarized CGCSM beam with n = 1 focused by a thin lens with f ₃ = 150mm at different propagation distances. The solid curve in Fig. 13 denotes the theoretical result. The detailed principle and process for measuring the DOP can be found in [47]. One finds from Fig. 13 that our experimental results agree reasonably well with the theoretical result, only slight difference between experimental result and numerical result exists, which may be caused by the fluctuation of the source beam, the non-ideal optical elements, and the resolution limits of the RPC, CCD and BPA.

Fig. 11 Experimental results of intensity distribution of the generated radially polarized CGCSM beam with n = 1 focused by a thin lens with f ₃ = 150mm and its corresponding composition components $W_{x x} (ρ, ρ)$ and $W_{y y} (ρ, ρ)$ at different propagation distances with $n = 1$ .

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Fig. 12 Experimental results of the square of the DOC ${\vec{q}}^{2}$ and the corresponding cross line of the generated radially polarized CGCSM beam with n = 1 focused by a thin lens with f ₃ = 150mm at different propagation distances. The solid curve denotes the theoretical result.

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Fig. 13 Experimental results of the DOP (cross line v = 0) of the generated radially polarized CGCSM beam with $n = 1$ focused by a thin lens with f ₃ = 150mm at different propagation distances. The solid curve denotes the theoretical result.

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5. Summary

As a summary, we have introduced a new kind of partially coherent vector beam with radial polarization and special correlation function named radially polarized CGCSM beam and discussed the realizability and beam conditions. Analytical formulae for the elements of the CSD matrix of a radially polarized CGCSM beam propagating through a stigmatic ABCD optical system have been derived, and the statistical properties of a radially polarized CGCSM beam focused by a thin lens have been studied numerically and compared with that of a conventional radially polarized GCSM beam. We have found that the structure of correlation function closely determined the statistical properties of a radially polarized CGCSM beam, e.g., the spectral intensity in the focal plane displays four-beamlet array distribution, the DOC and DOP all display non-circular symmetries, which all are quite different from that of a radially polarized GCSM beam. We also have found that SOP of a radially polarized CGCSM beam in the focal plane significantly differs from that in the source plane, which means the polarization structure is broken due the special distribution of the source correlation function. Furthermore, we have reported experimental generation of a radially polarized CGCSM beam for the first time, and measured its focusing properties, which are consistent with our theoretical results. Modulating the correlation function provides a novel for modulating the statistical properties of partially coherent vector beam, and may be useful in some applications, where prescribed beam properties are required.

Acknowledgments

This work is supported by the National Natural Science Fund for Distinguished Young Scholar under Grant No. 11525418, the National Natural Science Foundation of China under Grant Nos. 11504172&61371167&11274005, the Natural Science Foundation of Jiangsu Province under Grant No. BK20150763, the project of the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions, and the Innovation Plan for Graduate Students in the Universities of Jiangsu Province under Grant No. KYLX-1218.

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Generation and propagation of a vector cosine-Gaussian correlated beam with radial polarization

Abstract

1. Introduction

2. Theoretical model, realizability and beam conditions of a radially polarized CGCSM beam

3. Statistical properties of a radially polarized CGCSM beam on propagation

4. Experimental generation of a radially polarized CGCSM beam

5. Summary

Acknowledgments

References and links

Cited By

Figures (13)

Equations (35)

Optics Express