Statistics properties of a cylindrical vector partially coherent beam

Yiming Dong; Yangjian Cai; Chengliang Zhao; Min Yao

doi:10.1364/OE.19.005979

Optics Express
Vol. 19,
Issue 7,
pp. 5979-5992
(2011)
•https://doi.org/10.1364/OE.19.005979

Statistics properties of a cylindrical vector partially coherent beam

Yiming Dong, Yangjian Cai, Chengliang Zhao, and Min Yao

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Yiming Dong, Yangjian Cai, Chengliang Zhao, and Min Yao, "Statistics properties of a cylindrical vector partially coherent beam," Opt. Express 19, 5979-5992 (2011)

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Table of Contents Category
- Coherence and Statistical Optics
Optics & Photonics Topics
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The topics in this list come from the Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: February 11, 2011
- Revised Manuscript: March 9, 2011
- Manuscript Accepted: March 10, 2011
- Published: March 16, 2011

Abstract

Cylindrical vector partially coherent beam is introduced as a natural extension of cylindrical vector coherent beam based on the unified theory of coherence and polarization. Analytical propagation formula for the cross-spectral density matrix of a cylindrical vector partially coherent beam passing through paraxial ABCD optical system is derived based on the generalized Collins integral formula. As an application example, the statistics properties, such as the average intensity, spreading and the degree of polarization, of a cylindrical vector partially coherent beam propagating in free space are studied in detail. It is found that the statistics properties of a cylindrical vector partially coherent beam are much different from a cylindrical vector coherent beam. Our results may find applications in connection with laser beam shaping and optical trapping.

1. Introduction

In the past several years, laser beams with cylindrical symmetry in polarization (i.e., cylindrical vector beams) have been widely investigated and applied in optical trapping, laser making, dark field imaging, free-space optical communications, singular optics, data storage, 3D tailoring of the focus shape, optical inspection and metrology [1–20]. Different methods have been developed to generate cylindrical vector beams [1,8–13]. Different theoretical models for cylindrical vector beams have been proposed, and the tight focusing properties, propagation properties and second-harmonic generation of cylindrical vector beams have been explored in detail [1,14–21].

In the past decades the two important properties of light waves: coherence and polarization were studied separately [22,23]. After the unified theory of coherence and polarization was formulated [24] it became evident that these properties are interrelated [25]. Recently, more and more attention is being paid to vector partially coherent beams (also named stochastic electromagnetic beams) owing to their importance in theories of coherence and polarization of light [25–45]. As a typical vector partially coherent beam, electromagnetic Gaussian Schell-model (GSM) beams were introduced as a natural extension of the scalar GSM beams [26–28]. Generation and propagation of EGSM beams in free space and through turbulent atmosphere, human tissues, paraxial optical systems, and resonators have been then widely investigated [29–41]. It was found that the EGSM beams may have reduced levels of intensity fluctuations compared to the scalar GSM beams [32], which makes them attractive for free-space optical communications and active laser radar systems [33,34]. We can modulate or control the spectral, coherence, propagation factor, polarization properties of a stochastic beam by a Gaussian cavity by choosing suitable cavity parameters and the parameters of the source beam [37–41]. Ghost imaging with an EGSM beam was examined in [42], and it was revealed that EGSM beam is useful in novel optical imaging. The radiation force of EGSM beams on a Rayleigh dielectric sphere is explored in [43], and it was found that we can increase trapping ranges by choosing suitable source polarization and coherence, which makes them useful in optical trapping. Degree of paraxiality of an EGSM beam was examined in [44], and it was found that the nonparaxial properties of a stochastic electromagnetic beam were related with not only coherence and also polarization. Focusing of stochastic electromagnetic GSM beams through a high numerical aperture objective was explored recently [45].

In this paper, we extend the cylindrical vector beam to the partially coherent case based on the unified theory of coherence and polarization. Analytical propagation formula for the cross-spectral density matrix of a cylindrical vector partially coherent beam passing through paraxial ABCD optical system is derived. The statistics properties of a cylindrical vector partially coherent beam on propagation in free space are illustrated numerically. Some interesting phenomena are found.

2. Theory

In cylindrical coordinates, the electric field of coherent cylindrically polarized Laguerre-Gaussian (LG) beam at z = 0 is expressed as follows [14]

\vec{E} (r, ϕ, z) = \exp (- \frac{r^{2}}{w_{0}^{2}}) {(\frac{2 r^{2}}{w_{0}^{2}})}^{(n \pm 1) / 2} L_{p}^{n \pm 1} (\frac{2 r^{2}}{w_{0}^{2}}) {\begin{cases} \cos (n ϕ) {\vec{e}}_{ϕ} \mp \sin (n ϕ) {\vec{e}}_{r} \\ \pm \sin (n ϕ) {\vec{e}}_{ϕ} + \cos (n ϕ) {\vec{e}}_{r} \end{cases}},

where r and ϕ are the radial and azimuthal (angle) coordinates,

L_{p}^{n \pm 1}

denotes the Laguerre polynomial with mode orders p and

n \pm 1

, is the beam width of the fundamental Gaussian mode. When

n = 0

, Eq. (1) reduces to the electric field for the well-known radially or azimuthally polarized LG beam. When

p = 0

and

n = 0

, Eq. (1) degenerates to the electric field of a radially or azimuthally polarized Gaussian beams.

By use of the formulae ${\vec{e}}_{r} = \cos ϕ {\vec{e}}_{x} + \sin ϕ {\vec{e}}_{y}$ and ${\vec{e}}_{ϕ} = - \sin ϕ {\vec{e}}_{x} + \cos ϕ {\vec{e}}_{y}$ , Eq. (1) can be expressed in the following alternative form

\begin{array}{l} \vec{E} (r, ϕ, 0) = \exp (- \frac{r^{2}}{w_{0}^{2}}) {[\frac{2 r^{2}}{w_{0}^{2}}]}^{(n \pm 1) / 2} L_{p}^{n \pm 1} [\frac{2 r^{2}}{w_{0}^{2}}] {\begin{cases} \sin [(n \pm 1) ϕ] {\vec{e}}_{x} + \cos [(n \pm 1) ϕ] {\vec{e}}_{y} \\ \cos [(n \pm 1) ϕ] {\vec{e}}_{x} + \sin [(n \pm 1) ϕ] {\vec{e}}_{y} \end{cases}} = \exp (- \frac{r^{2}}{w_{0}^{2}}) \\ \times {[\frac{2 r^{2}}{w_{0}^{2}}]}^{(n \pm 1) / 2} L_{p}^{n \pm 1} [\frac{2 r^{2}}{w_{0}^{2}}] {\begin{cases} \frac{\exp [i (n \pm 1) ϕ] - \exp [- i (n \pm 1) ϕ]}{2 i} {\vec{e}}_{x} + \frac{\exp [i (n \pm 1) ϕ] + \exp [- i (n \pm 1) ϕ]}{2} {\vec{e}}_{y} \\ \frac{\exp [i (n \pm 1) ϕ] + \exp [- i (n \pm 1) ϕ]}{2} {\vec{e}}_{x} + \frac{\exp [i (n \pm 1) ϕ] - \exp [- i (n \pm 1) ϕ]}{2 i} {\vec{e}}_{y} \end{cases}}, \end{array}

By use of the following relation between an LG mode and an Hermite-Gaussian (HG) mode

{[\frac{2 r^{2}}{w_{0}^{2}}]}^{(n \pm 1) / 2} L_{p}^{n \pm 1} [\frac{2 r^{2}}{w_{0}^{2}}] \exp [i (n \pm 1) ϕ] = \frac{{(- 1)}^{p}}{2^{2 p + n \pm 1} p!} \sum_{m = 0}^{p} \sum_{s = 0}^{n \pm 1} i^{s} (\begin{array}{l} p \\ m \end{array}) (\begin{array}{l} n \pm 1 \\ s \end{array}) H_{2 m + n \pm 1 - s} (\frac{\sqrt{2} x}{w_{0}}) H_{2 p - 2 m + s} (\frac{\sqrt{2} y}{w_{0}}),

with

H (x)

being the Hermite polynomial,

(\begin{array}{l} p \\ m \end{array})

and

(\begin{array}{l} n \pm 1 \\ s \end{array})

being binomial coefficients, Eq. (2) can be expressed in the following alternative form in Cartesian coordinates

\begin{array}{l} \vec{E} (x, y, 0) = {\begin{matrix} E_{x} (x, y, 0) {\vec{e}}_{x} + E_{y} (x, y, 0) {\vec{e}}_{y} \\ E_{y} (x, y, 0) {\vec{e}}_{x} + E_{x} (x, y, 0) {\vec{e}}_{y} \end{matrix}} \\ = \exp (- \frac{x^{2} + y^{2}}{w_{0}^{2}}) \frac{{(- 1)}^{p}}{2^{2 p + n \pm 1} p!} {\begin{matrix} \frac{1}{2 i} \sum_{m = 0}^{p} \sum_{s = 0}^{n \pm 1} i^{s} [1 - {(- 1)}^{s}] (\begin{array}{l} p \\ m \end{array}) (\begin{array}{l} n \pm 1 \\ s \end{array}) H_{2 m + n \pm 1 - s} (\frac{\sqrt{2} x}{w_{0}}) H_{2 p - 2 m + s} (\frac{\sqrt{2} y}{w_{0}}) {\vec{e}}_{x} \\ \frac{1}{2} \sum_{m = 0}^{p} \sum_{s = 0}^{n \pm 1} i^{s} [1 + {(- 1)}^{s}] (\begin{array}{l} p \\ m \end{array}) (\begin{array}{l} n \pm 1 \\ s \end{array}) H_{2 m + n \pm 1 - s} (\frac{\sqrt{2} x}{w_{0}}) H_{2 p - 2 m + s} (\frac{\sqrt{2} y}{w_{0}}) {\vec{e}}_{x} \end{matrix} \\ \begin{matrix} + \frac{1}{2} \sum_{m = 0}^{p} \sum_{s = 0}^{n \pm 1} i^{s} [1 + {(- 1)}^{s}] (\begin{array}{l} p \\ m \end{array}) (\begin{array}{l} n \pm 1 \\ s \end{array}) H_{2 m + n \pm 1 - s} (\frac{\sqrt{2} x}{w_{0}}) H_{2 p - 2 m + s} (\frac{\sqrt{2} y}{w_{0}}) {\vec{e}}_{y} \\ + \frac{1}{2 i} \sum_{m = 0}^{p} \sum_{s = 0}^{n \pm 1} i^{s} [1 - {(- 1)}^{s}] (\begin{array}{l} p \\ m \end{array}) (\begin{array}{l} n \pm 1 \\ s \end{array}) H_{2 m + n \pm 1 - s} (\frac{\sqrt{2} x}{w_{0}}) H_{2 p - 2 m + s} (\frac{\sqrt{2} y}{w_{0}}) {\vec{e}}_{y} \end{matrix}} . \end{array}

Now we extend the cylindrically polarized Laguerre-Gaussian (LG) beam to the partially coherent case. Let us consider a planar, secondary, vector (i.e., electromagnetic) partially coherent source located in the plane $z = 0$ and radiating into the half-space $z > 0$ . Based on the unified theory of coherence and polarization, the second-order correlation properties of the source, in space–frequency domain, can be characterized by the cross-spectral density matrix of the electric field, defined by the formula [24]

\hat{W} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = (\begin{array}{l} W_{x x} (x_{1}, y_{1}, x_{2}, y_{2}, 0) W_{x y} (x_{1}, y_{1}, x_{2}, y_{2}, 0) \\ W_{y x} (x_{1}, y_{1}, x_{2}, y_{2}, 0) W_{y y} (x_{1}, y_{1}, x_{2}, y_{2}, 0) \end{array}),

with elements

W_{α β} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = 〈 E_{α} (x_{1}, y_{1}, 0) E_{β}^{*} (x_{2}, y_{2}, 0) 〉, (α = x, y; β = x, y),

where

E_{x}

and

E_{y}

denote the components of the random electric vector, along two mutually orthogonal x and y directions perpendicular to the z-axis. Here the asterisk denotes the complex conjugate and the angular brackets denote ensemble average.

We assume the cylindrical vector partially coherent LG beam is radiated from a Schell-model source whose spectral degree of correlation satisfy Gaussian distribution [24–28], namely each of the four elements of the cross-spectral density matrix of such a source has the form

W_{α β} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = \sqrt{E_{α} (x_{1}, y_{1}, 0)} \sqrt{E_{β} (x_{2}, y_{2}, 0)} μ_{α β} (x_{1} - x_{2}, y_{1} - y_{2}, 0), (α = x, y; β = x, y),

where

μ_{α β} (x_{1} - x_{2}, y_{1} - y_{2}, 0) = B_{α β} \exp [- \frac{{(x_{1} - x_{2})}^{2} + {(y_{1} - y_{2})}^{2}}{2 σ_{α β}^{2}}], (α = x, y; β = x, y),

here is the correlation coefficient between the

E_{x}

and

E_{y}

field components and satisfies the relation

B_{α β} = B_{β α}^{*}

, ϕ is the phase difference between the x-and y-components of the field and is removable in most case,

σ_{α β}

denotes the width of the spectral degree of correlation. The realizability conditions for the parameters of an electromagnetic Schell-model source can be found in [27] and [28].

By applying Eqs. (4)–(8), for the case of $\vec{E} (x, y, 0) = E_{x} (x, y, 0) {\vec{e}}_{x} + E_{y} (x, y, 0) {\vec{e}}_{y}$ , we can express the elements of the cross-spectral density matrix of a cylindrical vector partially coherent LG beam as follows

\begin{array}{l} W_{x x} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = \frac{1}{4} \exp [- \frac{x_{1}^{2} + x_{2}^{2} + y_{1}^{2} + y_{2}^{2}}{w_{0}^{2}} - \frac{{(x_{1} - x_{2})}^{2} + {(y_{1} - y_{2})}^{2}}{2 σ_{x x}^{2}}] \\ \times \frac{1}{2^{4 p + 2 n \pm 2} {(p!)}^{2}} \sum_{m = 0}^{p} \sum_{s = 0}^{n \pm 1} \sum_{l = 0}^{p} \sum_{h = 0}^{n \pm 1} i^{s} {(- i)}^{h} [1 - {(- 1)}^{s}] [1 - {(- 1)}^{h}] (\begin{array}{l} p \\ m \end{array}) (\begin{array}{l} p \\ l \end{array}) (\begin{array}{l} n \pm 1 \\ s \end{array}) (\begin{array}{l} n \pm 1 \\ h \end{array}) H_{2 m + n \pm 1 - s} (\frac{\sqrt{2} x_{1}}{w_{0}}) \\ \times H_{2 l + n \pm 1 - h} (\frac{\sqrt{2} x_{2}}{w_{0}}) H_{2 p - 2 m + s} (\frac{\sqrt{2} y_{1}}{w_{0}}) H_{2 p - 2 l + h} (\frac{\sqrt{2} y_{2}}{w_{0}}), \end{array}

\begin{array}{l} W_{x y} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = \frac{B_{x y}}{4 i} \exp [- \frac{x_{1}^{2} + x_{2}^{2} + y_{1}^{2} + y_{2}^{2}}{w_{0}^{2}} - \frac{{(x_{1} - x_{2})}^{2} + {(y_{1} - y_{2})}^{2}}{2 σ_{x y}^{2}}] \\ \times \frac{1}{2^{4 p + 2 n \pm 2} {(p!)}^{2}} \sum_{m = 0}^{p} \sum_{s = 0}^{n \pm 1} \sum_{l = 0}^{p} \sum_{h = 0}^{n \pm 1} i^{s} {(- i)}^{h} [1 - {(- 1)}^{s}] [1 + {(- 1)}^{h}] (\begin{array}{l} p \\ m \end{array}) (\begin{array}{l} p \\ l \end{array}) (\begin{array}{l} n \pm 1 \\ s \end{array}) (\begin{array}{l} n \pm 1 \\ h \end{array}) H_{2 m + n \pm 1 - s} (\frac{\sqrt{2} x_{1}}{w_{0}}) \\ \times H_{2 l + n \pm 1 - h} (\frac{\sqrt{2} x_{2}}{w_{0}}) H_{2 p - 2 m + s} (\frac{\sqrt{2} y_{1}}{w_{0}}) H_{2 p - 2 l + h} (\frac{\sqrt{2} y_{2}}{w_{0}}), \end{array}

W_{y x} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = {[W_{x y} (x_{1}, y_{1}, x_{2}, y_{2}, 0)]}^{*},

\begin{array}{l} W_{y y} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = \frac{1}{4} \exp [- \frac{x_{1}^{2} + x_{2}^{2} + y_{1}^{2} + y_{2}^{2}}{w_{0}^{2}} - \frac{{(x_{1} - x_{2})}^{2} + {(y_{1} - y_{2})}^{2}}{2 σ_{y y}^{2}}] \\ \times \frac{1}{2^{4 p + 2 n \pm 2} {(p!)}^{2}} \sum_{m = 0}^{p} \sum_{s = 0}^{n \pm 1} \sum_{l = 0}^{p} \sum_{h = 0}^{n \pm 1} i^{s} {(- i)}^{h} [1 + {(- 1)}^{s}] [1 + {(- 1)}^{h}] (\begin{array}{l} p \\ m \end{array}) (\begin{array}{l} p \\ l \end{array}) (\begin{array}{l} n \pm 1 \\ s \end{array}) (\begin{array}{l} n \pm 1 \\ h \end{array}) H_{2 m + n \pm 1 - s} (\frac{\sqrt{2} x_{1}}{w_{0}}) \\ \times H_{2 l + n \pm 1 - h} (\frac{\sqrt{2} x_{2}}{w_{0}}) H_{2 p - 2 m + s} (\frac{\sqrt{2} y_{1}}{w_{0}}) H_{2 p - 2 l + h} (\frac{\sqrt{2} y_{2}}{w_{0}}) . \end{array}

In a similar way, for the case of $\vec{E} (x, y, 0) = E_{y} (x, y, 0) {\vec{e}}_{x} + E_{x} (x, y, 0) {\vec{e}}_{y}$ , we can express the elements of the cross-spectral density matrix of a cylindrical vector partially coherent LG beam as follows

\begin{array}{l} W_{1 x x} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = W_{y y} (x_{1}, y_{1}, x_{2}, y_{2}, 0), W_{1 y y} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = W_{x x} (x_{1}, y_{1}, x_{2}, y_{2}, 0), \\ W_{1 x y} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = W_{y x} (x_{1}, y_{1}, x_{2}, y_{2}, 0), W_{1 x y} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = W_{x y} (x_{1}, y_{1}, x_{2}, y_{2}, 0) . \end{array}

where

W_{x x}, W_{x y}, W_{y x}, W_{y y}

are given by Eqs. (9)–(12).

Now we study the propagation of a cylindrical vector partially coherent LG beam through paraxial ABCD optical system. Within the validity of the paraxial approximation, the propagation of the elements of cross-spectral density matrix of a cylindrical vector partially coherent LG beam through paraxial ABCD optical system in free space can be studied with the help of the following generalized Collins formula [46,47]

\begin{array}{l} W_{α β} (u_{1}, v_{1}, u_{2}, v_{2}, z) = {(\frac{1}{λ | B |})}^{2} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} W_{α β} (x_{1}, y_{1}, x_{2}, y_{2}, 0) \\ \times \exp [\frac{i k}{2 B} (A x_{1}^{2} + A y_{1}^{2} - 2 x_{1} u_{1} - 2 y_{1} v_{1} + D u_{1}^{2} + D v_{1}^{2})] \\ \times \exp [- \frac{i k}{2 B^{*}} (A^{*} x_{2}^{2} + A^{*} y_{2}^{2} - 2 x_{2} u_{2} - 2 y_{2} v_{2} + D^{*} u_{2}^{2} + D^{*} v_{2}^{2})] d x_{1} d x_{2} d y_{1} d y_{2}, \end{array}

where

x_{i}

y_{i}

and

u_{i}

v_{i}

are the position coordinates in the input and output planes, A, B, C, and D are the transfer matrix elements of optical system and * denotes the complex conjugate being required for a general optical system with loss or gain, although it does not appear in Eq. (13) of [46],

k = 2 π / λ

is the wave number with λ being the wavelength.

Substituting from Eqs. (9)–(12) into Eq. (14), we obtain (after tedious integration over $x_{1}, x_{2}, y_{1}, y_{2}$ ) the following expressions for the elements of cross-spectral density matrix of a cylindrical vector partially coherent LG beam in the output plane

\begin{array}{l} W_{x x} (u_{1}, v_{1}, u_{2}, v_{2}, z) = \frac{1}{4} {(\frac{1}{λ | B |})}^{2} \frac{π^{2}}{M_{1 x x}} \frac{1}{2^{5 (n \pm 1 + 2 p) / 2} {(p!)}^{2}} {(1 - \frac{2}{M_{1 x x} w_{0}^{2}})}^{(n \pm 1 + 2 p) / 2} \\ \times \exp [\frac{i k D}{2 B} (u_{1}^{2} + v_{1}^{2})] \exp [- \frac{i k D^{*}}{2 B^{*}} (u_{2}^{2} + v_{2}^{2})] \exp [- \frac{k^{2}}{4 M_{1 x x} B^{2}} (u_{1}^{2} + v_{1}^{2})] \\ \times \exp {- \frac{k^{2}}{4 M_{2 x x}} {(\frac{u_{2}}{B^{*}} - \frac{u_{1}}{2 M_{1 x x} B σ_{x x}^{2}})}^{2}} \exp {- \frac{k^{2}}{4 M_{2 x x}} {(\frac{v_{2}}{B^{*}} - \frac{v_{1}}{2 M_{1 x x} B σ_{x x}^{2}})}^{2}} \\ \times \sum_{m = 0}^{p} \sum_{s = 0}^{n \pm 1} \sum_{l = 0}^{p} \sum_{h = 0}^{n \pm 1} \sum_{c_{1} = 0}^{2 m + n \pm 1 - s} \sum_{d_{1} = 0}^{c_{1} / 2} \sum_{e_{1} = 0}^{(2 l + n \pm 1 - h) / 2} \sum_{c_{2} = 0}^{2 p - 2 m + s} \sum_{d_{2} = 0}^{c_{2} / 2} \sum_{e_{2} = 0}^{(2 p - 2 l + h) / 2} (\begin{array}{l} 2 p - 2 m + s \\ c_{2} \end{array}) (\begin{array}{l} 2 m + n \pm 1 - s \\ c_{1} \end{array}) \\ \times (\begin{array}{l} p \\ m \end{array}) (\begin{array}{l} p \\ l \end{array}) (\begin{array}{l} n \pm 1 \\ s \end{array}) (\begin{array}{l} n \pm 1 \\ h \end{array}) [1 - {(- 1)}^{s}] [1 - {(- 1)}^{h}] (- 1)^{d_{1} + d_{2} + e_{1} + e_{2}} i^{s} (- i)^{h} (2 i)^{- (c_{1} + c_{2} - 2 d_{1} - 2 d_{2} + n \pm 1 + 2 p - 2 e_{1} - 2 e_{2})} \\ \times {(\frac{2 \sqrt{2}}{w_{0}})}^{2 p + n \pm 1 - 2 e_{1} - 2 e_{2}} \frac{c_{1}!}{d_{1}! (c_{1} - 2 d_{1})!} \frac{c_{2}!}{d_{2}! (c_{2} - 2 d_{2})!} \frac{(2 l + n \pm 1 - h)!}{e_{1}! (2 l + n \pm 1 - h - 2 e_{1})!} \frac{(2 p - 2 l + h)!}{e_{2}! (2 p - 2 l + h - 2 e_{2})!} \\ \times \frac{1}{{(\sqrt{M_{2 x x}})}^{c_{1} + c_{2} + n \pm 1 + 2 p - 2 d_{1} - 2 d_{2} - 2 e_{1} - 2 e_{2} + 2}} {(\frac{2}{{(w_{0}^{2} M_{1 x x}^{2} σ_{x x}^{4} - 2 M_{1 x x} σ_{x x}^{4})}^{1 / 2}})}^{c_{1} + c_{2} - 2 d_{1} - 2 d_{2}} \\ \times H_{c_{1} - 2 d_{1} + 2 l + n \pm 1 - h - 2 e_{1}} (\frac{k}{2 \sqrt{M_{2 x x}}} (\frac{u_{1}}{2 M_{1 x x} B σ_{x x}^{2}} - \frac{u_{2}}{B^{*}})) H_{2 m + n \pm 1 - s - c_{1}} (\frac{- i k u_{1}}{{(w_{0}^{2} M_{1 x x}^{2} B^{2} - 2 M_{1 x x} B^{2})}^{1 / 2}}) \\ \times H_{c_{2} - 2 d_{2} + 2 p - 2 l + h - 2 e_{2}} (\frac{k}{2 \sqrt{M_{2 x x}}} (\frac{v_{1}}{2 M_{1 x x} B σ_{x x}^{2}} - \frac{v_{2}}{B^{*}})) H_{2 p - 2 m + s - c_{2}} (\frac{- i k v_{1}}{{(w_{0}^{2} M_{1 x x}^{2} B^{2} - 2 M_{1 x x} B^{2})}^{1 / 2}}), \end{array}

with

M_{1 x x} = 1 / w_{0}^{2} + 1 / (2 σ_{x x}^{2}) - i k A / (2 B), M_{2 x x} = 1 / w_{0}^{2} + 1 / (2 σ_{x x}^{2}) + i k A^{*} / (2 B^{*}) - 1 / (4 M_{1 x x} σ_{x x}^{4}),

\begin{array}{l} W_{x y} (u_{1}, v_{1}, u_{2}, v_{2}, z) = \frac{B_{x y}}{4 i} {(\frac{1}{λ | B |})}^{2} \frac{π^{2}}{M_{1 x y}} \frac{1}{2^{5 (n \pm 1 + 2 p) / 2} {(p!)}^{2}} {(1 - \frac{2}{M_{1 x y} w_{0}^{2}})}^{(n \pm 1 + 2 p) / 2} \\ \times \exp [\frac{i k D}{2 B} (u_{1}^{2} + v_{1}^{2})] \exp [- \frac{i k D^{*}}{2 B^{*}} (u_{2}^{2} + v_{2}^{2})] \exp [- \frac{k^{2}}{4 M_{1 x y} B^{2}} (u_{1}^{2} + v_{1}^{2})] \\ \times \exp {- \frac{k^{2}}{4 M_{2 x y}} {(\frac{u_{2}}{B^{*}} - \frac{u_{1}}{2 M_{1 x y} B σ_{x y}^{2}})}^{2}} \exp {- \frac{k^{2}}{4 M_{2 x y}} {(\frac{v_{2}}{B^{*}} - \frac{v_{1}}{2 M_{1 x y} B σ_{x y}^{2}})}^{2}} \\ \times \sum_{m = 0}^{p} \sum_{s = 0}^{n \pm 1} \sum_{l = 0}^{p} \sum_{h = 0}^{n \pm 1} \sum_{c_{1} = 0}^{2 m + n \pm 1 - s} \sum_{d_{1} = 0}^{c_{1} / 2} \sum_{e_{1} = 0}^{(2 l + n \pm 1 - h) / 2} \sum_{c_{2} = 0}^{2 p - 2 m + s} \sum_{d_{2} = 0}^{c_{2} / 2} \sum_{e_{2} = 0}^{(2 p - 2 l + h) / 2} (\begin{array}{l} 2 p - 2 m + s \\ c_{2} \end{array}) (\begin{array}{l} 2 m + n \pm 1 - s \\ c_{1} \end{array}) \\ \times (\begin{array}{l} p \\ m \end{array}) (\begin{array}{l} p \\ l \end{array}) (\begin{array}{l} n \pm 1 \\ s \end{array}) (\begin{array}{l} n \pm 1 \\ h \end{array}) [1 - {(- 1)}^{s}] [1 + {(- 1)}^{h}] (- 1)^{d_{1} + d_{2} + e_{1} + e_{2}} i^{s} (- i)^{h} (2 i)^{- (c_{1} + c_{2} - 2 d_{1} - 2 d_{2} + n \pm 1 + 2 p - 2 e_{1} - 2 e_{2})} \\ \times {(\frac{2 \sqrt{2}}{w_{0}})}^{2 p + n \pm 1 - 2 e_{1} - 2 e_{2}} \frac{c_{1}!}{d_{1}! (c_{1} - 2 d_{1})!} \frac{c_{2}!}{d_{2}! (c_{2} - 2 d_{2})!} \frac{(2 l + n \pm 1 - h)!}{e_{1}! (2 l + n \pm 1 - h - 2 e_{1})!} \frac{(2 p - 2 l + h)!}{e_{2}! (2 p - 2 l + h - 2 e_{2})!} \\ \times \frac{1}{{(\sqrt{M_{2 x y}})}^{c_{1} + c_{2} - 2 d_{1} - 2 d_{2} + n \pm 1 + 2 p - 2 e_{1} - 2 e_{2} + 2}} {(\frac{2}{{(w_{0}^{2} M_{1 x y}^{2} σ_{x y}^{4} - 2 M_{1 x y} σ_{x y}^{4})}^{1 / 2}})}^{c_{1} + c_{2} - 2 d_{1} - 2 d_{2}} \\ \times H_{c_{1} - 2 d_{1} + 2 l + n \pm 1 - h - 2 e_{1}} (\frac{k}{2 \sqrt{M_{2 x y}}} (\frac{u_{1}}{2 M_{1 x y} B σ_{x y}^{2}} - \frac{u_{2}}{B^{*}})) H_{2 m + n \pm 1 - s - c_{1}} (\frac{- i k u_{1}}{{(w_{0}^{2} M_{1 x y}^{2} B^{2} - 2 M_{1 x y} B^{2})}^{1 / 2}}) \\ \times H_{c_{2} - 2 d_{2} + 2 p - 2 l + h - 2 e_{2}} (\frac{k}{2 \sqrt{M_{2 x y}}} (\frac{v_{1}}{2 M_{1 x y} B σ_{x y}^{2}} - \frac{v_{2}}{B^{*}})) H_{2 p - 2 m + s - c_{2}} (\frac{- i k v_{1}}{{(w_{0}^{2} M_{1 x y}^{2} B^{2} - 2 M_{1 x y} B^{2})}^{1 / 2}}), \end{array}

with

M_{1 x y} = 1 / w_{0}^{2} + 1 / (2 σ_{x y}^{2}) - i k A / (2 B), M_{2 x y} = 1 / w_{0}^{2} + 1 / (2 σ_{x y}^{2}) + i k A^{*} / (2 B^{*}) - 1 / (4 M_{1 x y} σ_{x y}^{4}),

W_{y x} (u_{1}, v_{1}, u_{2}, v_{2}, z) = {[W_{x y} (u_{1}, v_{1}, u_{2}, v_{2}, z)]}^{*},

\begin{array}{l} W_{y y} (u_{1}, v_{1}, u_{2}, v_{2}, z) = \frac{1}{4} {(\frac{1}{λ | B |})}^{2} \frac{π^{2}}{M_{1 y y}} \frac{1}{2^{5 (n \pm 1 + 2 p) / 2} {(p!)}^{2}} {(1 - \frac{2}{M_{1 y y} w_{0}^{2}})}^{(n \pm 1 + 2 p) / 2} \\ \times \exp [\frac{i k D}{2 B} (u_{1}^{2} + v_{1}^{2})] \exp [- \frac{i k D^{*}}{2 B^{*}} (u_{2}^{2} + v_{2}^{2})] \exp [- \frac{k^{2}}{4 M_{1 y y} B^{2}} (u_{1}^{2} + v_{1}^{2})] \\ \times \exp {- \frac{k^{2}}{4 M_{2 y y}} {(\frac{u_{2}}{B^{*}} - \frac{u_{1}}{2 M_{1 y y} B σ_{y y}^{2}})}^{2}} \exp {- \frac{k^{2}}{4 M_{2 y y}} {(\frac{v_{2}}{B^{*}} - \frac{v_{1}}{2 M_{1 y y} B σ_{y y}^{2}})}^{2}} \\ \times \sum_{m = 0}^{p} \sum_{s = 0}^{n \pm 1} \sum_{l = 0}^{p} \sum_{h = 0}^{n \pm 1} \sum_{c_{1} = 0}^{2 m + n \pm 1 - s} \sum_{d_{1} = 0}^{c_{1} / 2} \sum_{e_{1} = 0}^{(2 l + n \pm 1 - h) / 2} \sum_{c_{2} = 0}^{2 p - 2 m + s} \sum_{d_{2} = 0}^{c_{2} / 2} \sum_{e_{2} = 0}^{(2 p - 2 l + h) / 2} (\begin{array}{l} 2 p - 2 m + s \\ c_{2} \end{array}) (\begin{array}{l} 2 m + n \pm 1 - s \\ c_{1} \end{array}) \\ \times (\begin{array}{l} p \\ m \end{array}) (\begin{array}{l} p \\ l \end{array}) (\begin{array}{l} n \pm 1 \\ s \end{array}) (\begin{array}{l} n \pm 1 \\ h \end{array}) [1 + {(- 1)}^{s}] [1 + {(- 1)}^{h}] (- 1)^{d_{1} + d_{2} + e_{1} + e_{2}} i^{s} (- i)^{h} (2 i)^{- (c_{1} + c_{2} - 2 d_{1} - 2 d_{2} + n \pm 1 + 2 p - 2 e_{1} - 2 e_{2})} \\ \times {(\frac{2 \sqrt{2}}{w_{0}})}^{2 p + n \pm 1 - 2 e_{1} - 2 e_{2}} \frac{c_{1}!}{d_{1}! (c_{1} - 2 d_{1})!} \frac{c_{2}!}{d_{2}! (c_{2} - 2 d_{2})!} \frac{(2 l + n \pm 1 - h)!}{e_{1}! (2 l + n \pm 1 - h - 2 e_{1})!} \frac{(2 p - 2 l + h)!}{e_{2}! (2 p - 2 l + h - 2 e_{2})!} \\ \times \frac{1}{{(\sqrt{M_{2 y y}})}^{c_{1} + c_{2} - 2 d_{1} - 2 d_{2} + n \pm 1 + 2 p - 2 e_{1} - 2 e_{2} + 2}} {(\frac{2}{{(w_{0}^{2} M_{1 y y}^{2} σ_{y y}^{4} - 2 M_{1 y y} σ_{y y}^{4})}^{1 / 2}})}^{c_{1} + c_{2} - 2 d_{1} - 2 d_{2}} \\ \times H_{c_{1} - 2 d_{1} + 2 l + n \pm 1 - h - 2 e_{1}} (\frac{k}{2 \sqrt{M_{2 y y}}} (\frac{u_{1}}{2 M_{1 y y} B σ_{y y}^{2}} - \frac{u_{2}}{B^{*}})) H_{2 m + n \pm 1 - s - c_{1}} (\frac{- i k u_{1}}{{(w_{0}^{2} M_{1 y y}^{2} B^{2} - 2 M_{1 y y} B^{2})}^{1 / 2}}) \\ \times H_{c_{2} - 2 d_{2} + 2 p - 2 l + h - 2 e_{2}} (\frac{k}{2 \sqrt{M_{2 y y}}} (\frac{v_{1}}{2 M_{1 y y} B σ_{y y}^{2}} - \frac{v_{2}}{B^{*}})) H_{2 p - 2 m + s - c_{2}} (\frac{- i k v_{1}}{{(w_{0}^{2} M_{1 y y}^{2} B^{2} - 2 M_{1 y y} B^{2})}^{1 / 2}}), \end{array}

with

M_{1 y y} = 1 / w_{0}^{2} + 1 / (2 σ_{y y}^{2}) - i k A / (2 B), M_{2 y y} = 1 / w_{0}^{2} + 1 / (2 σ_{y y}^{2}) + i k A^{*} / (2 B^{*}) - 1 / (4 M_{1 y y} σ_{y y}^{4}) .

In above derivations, we have used the following integral and expansion formulae [48,49]

\int_{- \infty}^{\infty} \exp [- {(x - y)}^{2}] H_{n} (a x) d x = \sqrt{π} {(1 - a^{2})}^{n / 2} H_{n} (\frac{a y}{{(1 - a^{2})}^{1 / 2}}),

\int_{- \infty}^{\infty} x^{n} \exp [- {(x - β)}^{2}] d x = {(2 i)}^{- n} \sqrt{π} H_{n} (i β),

H_{n} (x + y) = \frac{1}{2^{n / 2}} \sum_{k = 0}^{n} (\begin{array}{l} n \\ k \end{array}) H_{k} (\sqrt{2} x) H_{n - k} (\sqrt{2} y),

H_{n} (x) = \sum_{k = 0}^{n / 2} {(- 1)}^{k} \frac{n!}{k! (n - 2 k)!} {(2 x)}^{n - 2 k} .

The intensity distribution of a cylindrical vector partially coherent LG beam in the output plane is expressed as

I (u, v, z) = W_{x x} (u, v, u, v, z) + W_{y y} (u, v, u, v, z) .

The effective beam size of a cylindrical vector partially coherent LG beam in the output plane is defined as

W_{s z} (z) = \sqrt{\frac{2 \int s^{2} 〈 I (u, v, z) 〉 d u d v}{\int 〈 I (u, v, z) 〉 d u d v}}, (s = u, v) .

The degree of polarization a cylindrical vector partially coherent LG beam in the output plane is expressed as [24,25]

P (u, v, z) = \sqrt{1 - \frac{4 \det [\hat{W} (u, v, u, v, z)]}{{T r [\hat{W} (u, v, u, v, z)]}^{2}}} .

Applying Eqs. (15)–(18) and Eqs. (23)–(25), we can study the statistics properties of a cylindrical vector partially coherent LG beam propagating through paraxial optical system in free space conveniently.

3. Statistics properties of a cylindrical vector partially coherent LG beam in free space

In this section, as a numerical example, we study the statistics properties of a cylindrical vector partially coherent LG beam on propagation in free space by using the formulae derived in Section 2.The ray transfer matrix relating to free-space propagation between the source plane ( $z = 0$ ) and the output plane ( $z \geq 0$ ) takes the form

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

Substituting Eq. (26) into Eqs. (15), (18) and (23), we calculate in Fig. 1 the normalized intensity distribution (cross line

v = 0

) of a cylindrical vector partially coherent LG beam for different values of the correlation coefficients

σ_{x x}, σ_{y y}

at several propagation distances in free space with

p = 1, n \pm 1 = 1, w_{0} = 2 m m, λ = 632.8 n m .

As is seen from Fig. 1, the intensity distribution properties of a cylindrical vector partially coherent LG beam are very different from those of a cylindrical vector coherent LG beam, and is closely determined by the correlation coefficients

σ_{x x}, σ_{y y}

. For a cylindrical vector coherent LG beam, its initial source beam profile remains invariant on propagation although its beam spot increases, which agree well with previous results of [1–21]. For a cylindrical vector partially coherent LG beam, its initial source beam profile doesn’t remain invariant on propagation, but gradually disappears on propagation and eventually takes a Gaussian shape. At suitable propagation distance, a flat-topped beam profile can be formed (see Figs. 1(b) and 1(c)). As the initial correlation coefficients

σ_{x x}, σ_{y y}

decrease (i.e., the initial degree of coherence decreases), the transition from a cylindrical vector LG beam into a Gaussian beam occurs more quickly and the beam spreads more rapidly. Comparing Fig. 1 in this paper and Fig. 1 of [50], we find that the evolution properties of the intensity distribution of a cylindrical vector partially coherent LG beam are similar to that of a partially coherent standard LG beams. It is known that we can trap the Rayleigh dielectric particle with the refractive index smaller than the ambient by a beam with zero central intensity [51], and trap the particle with the refractive index larger than the ambient by a Gaussian beam [43]. One finds from Fig. 1 that it is possible to perform beam shaping by degrading the coherence of a cylindrical vector beam in the far field. We know that the beam profile of a focused laser beam at the focal plane is similar to its beam profile in the far field, thus we can modulate the beam profile of a focused cylindrical vector beam at the focal plane by controlling its initial coherence, a Gaussian beam can be formed through focusing a cylindrical vector LG beam with low coherence, and a beam with zero central intensity can be formed through focusing a cylindrical vector LG beam with high coherence. Thus we can expect to trap the particle with the refractive index smaller than the ambient by a focused cylindrical vector LG beam with high coherence, and trap the particle with the refractive index larger than the ambient by a focused cylindrical vector LG beam with low coherence.

Fig. 1 Normalized intensity distribution (cross line $v = 0$ ) of a cylindrical vector partially coherent LG beam for different values of the correlation coefficients $σ_{x x}, σ_{y y}$ at several propagation distances in free space (a) $z = 0$ , (b) $z = 3.2 m$ , (c) $z = 6.5 m$ , (d) $z = 20 m$ .

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To learn about the spreading properties of the cylindrical vector partially coherent LG beam in free space, substituting Eqs. (15), (18), (23) and (26) into Eq. (24), we calculate the effective beam size ( $W_{u z} = W_{v z} = W_{z}$ ) of a cylindrical vector partially coherent LG beam versus the propagation distance z for different values of the correlation coefficients $σ_{x x}, σ_{y y}$ and the mode orders $p, n \pm 1$ in free space with $λ = 632.8 n m$ . For the convenience of comparison, we have chosen $w_{0} = 10 m m$ for $p = 1, n \pm 1 = 1$ , $w_{0} = 8.615 m m$ for $p = 2, n \pm 1 = 1$ , $w_{0} = 7.07 m m$ for $p = 3, n \pm 1 = 1$ , $w_{0} = 8.944 m m$ for $p = 1, n \pm 1 = 2$ , and $w_{0} = 8.166 m m$ for $p = 1, n \pm 1 = 3$ , so the cylindrical vector partially coherent LG beams with different mode orders have the same effective beam sizes in the source plane ( $z = 0$ ). One sees that the spreading properties of the cylindrical vector LG beam in free space are closely determined by the correlation coefficients and the mode orders. A cylindrical vector LG beam with low coherence spreads more rapidly than that with high coherence as expected. When the initial correlation coefficients are large (i.e., initial degree of coherence is high), the cylindrical vector partially coherent beam spreads more rapidly as the mode orders increase (see Figs. 2 (a) and 2(b)). When the initial degree of coherence is low, the cylindrical vector partially coherent beams with different mode orders exhibit almost the same spreading features (see Fig. 2(c)). We can explain this phenomenon as follows: it is known that the spreading properties of a cylindrical vector coherent LG beam due to free-space diffraction are closely determined by its mode orders, and the beam with higher mode orders spreads more rapidly. For a cylindrical vector partially coherent LG beam, its spreading properties are determined by its mode orders and its initial coherence together. When the initial coherence is high, the effect of the mode orders plays a dominant role, and the effect of partially coherence can be neglected, then its spreading properties are similar to that of a cylindrical vector coherent LG beam. When the initial coherence is low, the effect of the partially coherence plays a dominant role, and the effect of the mode orders can be neglected.

Fig. 2 Effective beam size of a cylindrical vector partially coherent LG beam versus the propagation distance z for different values of the correlation coefficients $σ_{x x}, σ_{y y}$ and the mode orders $p, n \pm 1$ in free space.

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Now we study the polarization properties of a cylindrical vector partially coherent LG beam on propagation in free space. Substituting Eqs. (15)–(18) and (26) into Eq. (25), we calculate in Fig. 3 the degree of polarization (cross line, $v = 0$ ) of a cylindrical vector partially coherent LG beam for different values of the initial correlation coefficients $σ_{x x}, σ_{x y}, σ_{y y}$ at several propagation distances in free space with $λ = 632.8 n m$ , $w_{0} = 2 m m$ , $B_{x y} = 1$ , $p = 1$ and $n \pm 1 = 1$ . Our numerical results (not present here to save space) show that the degree of polarization of a cylindrical vector coherent LG beam in the source plane equals 1 for all the points across the entire transverse plane, and it’s value remains invariant during propagation, which means the polarization structure of a cylindrical vector coherent LG beam will not be destroyed during propagation in free space as expected [1–21]. For a cylindrical vector partially coherent LG beam, although the degree of polarization equals 1 for all the points across the entire transverse plane and is independent of the initial correlation coefficients $σ_{x x}, σ_{x y}, σ_{y y}$ in the source plane (see Fig. 3(a) and Fig. 3(e)), it doesn’t remain invariant during propagation, on the contrary, the degree of polarization varies on propagation and is closely determined by the initial correlation coefficients (i.e., the initial degree of coherence). From Figs. 3(b)–3(d) and 3(f)–3(h), one sees that a dip appears in the distribution of the degree of polarization of a cylindrical vector partially coherent LG beam on propagation, in other words, the degree of polarization of the on-axis point becomes zero after propagation and the degree of polarization of the off-axis point rises gradually towards the edges of the off-axis regions. The width of the dip increases during propagation, and its value increases as the initial correlation coefficients $σ_{x x}, σ_{y y}$ decrease. Thus, one comes to the conclusion that the polarization structure of a cylindrical vector partially coherent LG beam is destroyed during propagation in free space (i.e., a cylindrical vector partially coherent LG beam is depolarized during propagation), and the cylindrical vector LG beam becomes a partially polarized beam. One finds from Fig. 4 that the degree of polarization a cylindrical vector partially coherent LG beam on propagation in free space is also closely related with the mode order $n \pm 1$ and the width of the dip increases as the mode order $n \pm 1$ increases, but the degree of polarization is almost independent of the mode order p. Thus, we may control the polarization properties of a cylindrical vector LG beam by choosing suitable initial degree of coherence and mode order $n \pm 1$ .

Fig. 3 Degree of polarization (cross line, $v = 0$ ) of a cylindrical vector partially coherent LG beam for different values of the initial correlation coefficients $σ_{x x} = σ_{x y} = σ_{y y}$ at several propagation distances in free space. In Fig. 3(a)–3(d), $σ_{x x} = σ_{x y} = σ_{y y} = 1 m m$ , in Fig. 3(e)–3(h), $σ_{x x} = σ_{x y} = σ_{y y} = 0.5 m m$ .

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Fig. 4 Degree of polarization (cross line, v = 0) of a partially coherent cylindrically polarized LG beams at $z = 6.5 m$ in free space for different mode orders of p and $n \pm 1$ with $λ = 632.8 n m$ , $w_{0} = 2 m m$ , $B_{x y} = 1$ , $σ_{x x} = σ_{x y} = σ_{y y} = 1 m m$ .

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4. Summary

We have proposed theoretical model to describe cylindrical vector partially coherent beam based on the unified theory of coherence and polarization, and have derived the analytical formulae for the elements of the cross-spectral density matrix of such beam propagating through paraxial ABCD optical system. As numerical examples, we have studied the properties of the intensity, spreading and degree of polarization of a cylindrical vector partially coherent LG beam on propagation in free space, and have carried out the corresponding comparison with those of a cylindrical vector coherent LG beam. We have found that the properties of a cylindrical vector partially coherent LG beam on propagation in free space are much different those of a cylindrical vector coherent LG beam. By degrading the coherence of a cylindrical vector LG beam, we can shape the beam profile of such beam, and alter its polarization structure. Our results may find applications in connection with laser beam shaping and optical trapping.

Acknowledgments

Yangjian Cai acknowledges the support by the National Natural Science Foundation of China (NSFC) under Grant No. 10904102, the Foundation for the Author of National Excellent Doctoral Dissertation of PR China under Grant No. 200928, the Natural Science of Jiangsu Province under Grant No. BK2009114, the Huo Ying Dong Education Foundation of China under Grant No. 121009, the Key Project of Chinese Ministry of Education under Grant No. 210081 and the Project for Constructing Superiority Branch of Learning of Universities of Jiangsu Province. Chengliang Zhao acknowledges the support by the NSFC under Grant No. 61008009 and the Universities Natural Science Research Project of Jiangsu Province under Grant No. 10KJB140011. Min Yao acknowledges the support by Scientific Research Fund of Zhejiang Provincial Education Department, China (Grant No. Y200908631).

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Fig. 1 Normalized intensity distribution (cross line

v = 0

) of a cylindrical vector partially coherent LG beam for different values of the correlation coefficients

σ_{x x}, σ_{y y}

at several propagation distances in free space (a)

z = 0

, (b)

z = 3.2 m

, (c)

z = 6.5 m

, (d)

z = 20 m

View in Article | Download Full Size | PDF

Fig. 2 Effective beam size of a cylindrical vector partially coherent LG beam versus the propagation distance z for different values of the correlation coefficients

σ_{x x}, σ_{y y}

and the mode orders

p, n \pm 1

in free space.

View in Article | Download Full Size | PDF

Fig. 3 Degree of polarization (cross line,

v = 0

) of a cylindrical vector partially coherent LG beam for different values of the initial correlation coefficients

σ_{x x} = σ_{x y} = σ_{y y}

at several propagation distances in free space. In Fig. 3(a)–3(d),

σ_{x x} = σ_{x y} = σ_{y y} = 1 m m

, in Fig. 3(e)–3(h),

σ_{x x} = σ_{x y} = σ_{y y} = 0.5 m m

View in Article | Download Full Size | PDF

Fig. 4 Degree of polarization (cross line, v = 0) of a partially coherent cylindrically polarized LG beams at

z = 6.5 m

in free space for different mode orders of p and

n \pm 1

with

λ = 632.8 n m

w_{0} = 2 m m

B_{x y} = 1

σ_{x x} = σ_{x y} = σ_{y y} = 1 m m

View in Article | Download Full Size | PDF

Equations (29)

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\vec{E} (r, ϕ, z) = \exp (- \frac{r^{2}}{w_{0}^{2}}) {(\frac{2 r^{2}}{w_{0}^{2}})}^{(n \pm 1) / 2} L_{p}^{n \pm 1} (\frac{2 r^{2}}{w_{0}^{2}}) {\begin{cases} \cos (n ϕ) {\vec{e}}_{ϕ} \mp \sin (n ϕ) {\vec{e}}_{r} \\ \pm \sin (n ϕ) {\vec{e}}_{ϕ} + \cos (n ϕ) {\vec{e}}_{r} \end{cases}},

\begin{array}{l} \vec{E} (r, ϕ, 0) = \exp (- \frac{r^{2}}{w_{0}^{2}}) {[\frac{2 r^{2}}{w_{0}^{2}}]}^{(n \pm 1) / 2} L_{p}^{n \pm 1} [\frac{2 r^{2}}{w_{0}^{2}}] {\begin{cases} \sin [(n \pm 1) ϕ] {\vec{e}}_{x} + \cos [(n \pm 1) ϕ] {\vec{e}}_{y} \\ \cos [(n \pm 1) ϕ] {\vec{e}}_{x} + \sin [(n \pm 1) ϕ] {\vec{e}}_{y} \end{cases}} = \exp (- \frac{r^{2}}{w_{0}^{2}}) \\ \times {[\frac{2 r^{2}}{w_{0}^{2}}]}^{(n \pm 1) / 2} L_{p}^{n \pm 1} [\frac{2 r^{2}}{w_{0}^{2}}] {\begin{cases} \frac{\exp [i (n \pm 1) ϕ] - \exp [- i (n \pm 1) ϕ]}{2 i} {\vec{e}}_{x} + \frac{\exp [i (n \pm 1) ϕ] + \exp [- i (n \pm 1) ϕ]}{2} {\vec{e}}_{y} \\ \frac{\exp [i (n \pm 1) ϕ] + \exp [- i (n \pm 1) ϕ]}{2} {\vec{e}}_{x} + \frac{\exp [i (n \pm 1) ϕ] - \exp [- i (n \pm 1) ϕ]}{2 i} {\vec{e}}_{y} \end{cases}}, \end{array}

{[\frac{2 r^{2}}{w_{0}^{2}}]}^{(n \pm 1) / 2} L_{p}^{n \pm 1} [\frac{2 r^{2}}{w_{0}^{2}}] \exp [i (n \pm 1) ϕ] = \frac{{(- 1)}^{p}}{2^{2 p + n \pm 1} p!} \sum_{m = 0}^{p} \sum_{s = 0}^{n \pm 1} i^{s} (\begin{array}{l} p \\ m \end{array}) (\begin{array}{l} n \pm 1 \\ s \end{array}) H_{2 m + n \pm 1 - s} (\frac{\sqrt{2} x}{w_{0}}) H_{2 p - 2 m + s} (\frac{\sqrt{2} y}{w_{0}}),

\begin{array}{l} \vec{E} (x, y, 0) = {\begin{matrix} E_{x} (x, y, 0) {\vec{e}}_{x} + E_{y} (x, y, 0) {\vec{e}}_{y} \\ E_{y} (x, y, 0) {\vec{e}}_{x} + E_{x} (x, y, 0) {\vec{e}}_{y} \end{matrix}} \\ = \exp (- \frac{x^{2} + y^{2}}{w_{0}^{2}}) \frac{{(- 1)}^{p}}{2^{2 p + n \pm 1} p!} {\begin{matrix} \frac{1}{2 i} \sum_{m = 0}^{p} \sum_{s = 0}^{n \pm 1} i^{s} [1 - {(- 1)}^{s}] (\begin{array}{l} p \\ m \end{array}) (\begin{array}{l} n \pm 1 \\ s \end{array}) H_{2 m + n \pm 1 - s} (\frac{\sqrt{2} x}{w_{0}}) H_{2 p - 2 m + s} (\frac{\sqrt{2} y}{w_{0}}) {\vec{e}}_{x} \\ \frac{1}{2} \sum_{m = 0}^{p} \sum_{s = 0}^{n \pm 1} i^{s} [1 + {(- 1)}^{s}] (\begin{array}{l} p \\ m \end{array}) (\begin{array}{l} n \pm 1 \\ s \end{array}) H_{2 m + n \pm 1 - s} (\frac{\sqrt{2} x}{w_{0}}) H_{2 p - 2 m + s} (\frac{\sqrt{2} y}{w_{0}}) {\vec{e}}_{x} \end{matrix} \\ \begin{matrix} + \frac{1}{2} \sum_{m = 0}^{p} \sum_{s = 0}^{n \pm 1} i^{s} [1 + {(- 1)}^{s}] (\begin{array}{l} p \\ m \end{array}) (\begin{array}{l} n \pm 1 \\ s \end{array}) H_{2 m + n \pm 1 - s} (\frac{\sqrt{2} x}{w_{0}}) H_{2 p - 2 m + s} (\frac{\sqrt{2} y}{w_{0}}) {\vec{e}}_{y} \\ + \frac{1}{2 i} \sum_{m = 0}^{p} \sum_{s = 0}^{n \pm 1} i^{s} [1 - {(- 1)}^{s}] (\begin{array}{l} p \\ m \end{array}) (\begin{array}{l} n \pm 1 \\ s \end{array}) H_{2 m + n \pm 1 - s} (\frac{\sqrt{2} x}{w_{0}}) H_{2 p - 2 m + s} (\frac{\sqrt{2} y}{w_{0}}) {\vec{e}}_{y} \end{matrix}} . \end{array}

\hat{W} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = (\begin{array}{l} W_{x x} (x_{1}, y_{1}, x_{2}, y_{2}, 0) W_{x y} (x_{1}, y_{1}, x_{2}, y_{2}, 0) \\ W_{y x} (x_{1}, y_{1}, x_{2}, y_{2}, 0) W_{y y} (x_{1}, y_{1}, x_{2}, y_{2}, 0) \end{array}),

W_{α β} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = 〈 E_{α} (x_{1}, y_{1}, 0) E_{β}^{*} (x_{2}, y_{2}, 0) 〉, (α = x, y; β = x, y),

W_{α β} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = \sqrt{E_{α} (x_{1}, y_{1}, 0)} \sqrt{E_{β} (x_{2}, y_{2}, 0)} μ_{α β} (x_{1} - x_{2}, y_{1} - y_{2}, 0), (α = x, y; β = x, y),

μ_{α β} (x_{1} - x_{2}, y_{1} - y_{2}, 0) = B_{α β} \exp [- \frac{{(x_{1} - x_{2})}^{2} + {(y_{1} - y_{2})}^{2}}{2 σ_{α β}^{2}}], (α = x, y; β = x, y),

\begin{array}{l} W_{x x} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = \frac{1}{4} \exp [- \frac{x_{1}^{2} + x_{2}^{2} + y_{1}^{2} + y_{2}^{2}}{w_{0}^{2}} - \frac{{(x_{1} - x_{2})}^{2} + {(y_{1} - y_{2})}^{2}}{2 σ_{x x}^{2}}] \\ \times \frac{1}{2^{4 p + 2 n \pm 2} {(p!)}^{2}} \sum_{m = 0}^{p} \sum_{s = 0}^{n \pm 1} \sum_{l = 0}^{p} \sum_{h = 0}^{n \pm 1} i^{s} {(- i)}^{h} [1 - {(- 1)}^{s}] [1 - {(- 1)}^{h}] (\begin{array}{l} p \\ m \end{array}) (\begin{array}{l} p \\ l \end{array}) (\begin{array}{l} n \pm 1 \\ s \end{array}) (\begin{array}{l} n \pm 1 \\ h \end{array}) H_{2 m + n \pm 1 - s} (\frac{\sqrt{2} x_{1}}{w_{0}}) \\ \times H_{2 l + n \pm 1 - h} (\frac{\sqrt{2} x_{2}}{w_{0}}) H_{2 p - 2 m + s} (\frac{\sqrt{2} y_{1}}{w_{0}}) H_{2 p - 2 l + h} (\frac{\sqrt{2} y_{2}}{w_{0}}), \end{array}

\begin{array}{l} W_{x y} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = \frac{B_{x y}}{4 i} \exp [- \frac{x_{1}^{2} + x_{2}^{2} + y_{1}^{2} + y_{2}^{2}}{w_{0}^{2}} - \frac{{(x_{1} - x_{2})}^{2} + {(y_{1} - y_{2})}^{2}}{2 σ_{x y}^{2}}] \\ \times \frac{1}{2^{4 p + 2 n \pm 2} {(p!)}^{2}} \sum_{m = 0}^{p} \sum_{s = 0}^{n \pm 1} \sum_{l = 0}^{p} \sum_{h = 0}^{n \pm 1} i^{s} {(- i)}^{h} [1 - {(- 1)}^{s}] [1 + {(- 1)}^{h}] (\begin{array}{l} p \\ m \end{array}) (\begin{array}{l} p \\ l \end{array}) (\begin{array}{l} n \pm 1 \\ s \end{array}) (\begin{array}{l} n \pm 1 \\ h \end{array}) H_{2 m + n \pm 1 - s} (\frac{\sqrt{2} x_{1}}{w_{0}}) \\ \times H_{2 l + n \pm 1 - h} (\frac{\sqrt{2} x_{2}}{w_{0}}) H_{2 p - 2 m + s} (\frac{\sqrt{2} y_{1}}{w_{0}}) H_{2 p - 2 l + h} (\frac{\sqrt{2} y_{2}}{w_{0}}), \end{array}

W_{y x} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = {[W_{x y} (x_{1}, y_{1}, x_{2}, y_{2}, 0)]}^{*},

\begin{array}{l} W_{y y} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = \frac{1}{4} \exp [- \frac{x_{1}^{2} + x_{2}^{2} + y_{1}^{2} + y_{2}^{2}}{w_{0}^{2}} - \frac{{(x_{1} - x_{2})}^{2} + {(y_{1} - y_{2})}^{2}}{2 σ_{y y}^{2}}] \\ \times \frac{1}{2^{4 p + 2 n \pm 2} {(p!)}^{2}} \sum_{m = 0}^{p} \sum_{s = 0}^{n \pm 1} \sum_{l = 0}^{p} \sum_{h = 0}^{n \pm 1} i^{s} {(- i)}^{h} [1 + {(- 1)}^{s}] [1 + {(- 1)}^{h}] (\begin{array}{l} p \\ m \end{array}) (\begin{array}{l} p \\ l \end{array}) (\begin{array}{l} n \pm 1 \\ s \end{array}) (\begin{array}{l} n \pm 1 \\ h \end{array}) H_{2 m + n \pm 1 - s} (\frac{\sqrt{2} x_{1}}{w_{0}}) \\ \times H_{2 l + n \pm 1 - h} (\frac{\sqrt{2} x_{2}}{w_{0}}) H_{2 p - 2 m + s} (\frac{\sqrt{2} y_{1}}{w_{0}}) H_{2 p - 2 l + h} (\frac{\sqrt{2} y_{2}}{w_{0}}) . \end{array}

\begin{array}{l} W_{1 x x} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = W_{y y} (x_{1}, y_{1}, x_{2}, y_{2}, 0), W_{1 y y} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = W_{x x} (x_{1}, y_{1}, x_{2}, y_{2}, 0), \\ W_{1 x y} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = W_{y x} (x_{1}, y_{1}, x_{2}, y_{2}, 0), W_{1 x y} (x_{1}, y_{1}, x_{2}, y_{2}, 0) = W_{x y} (x_{1}, y_{1}, x_{2}, y_{2}, 0) . \end{array}

\begin{array}{l} W_{α β} (u_{1}, v_{1}, u_{2}, v_{2}, z) = {(\frac{1}{λ | B |})}^{2} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} W_{α β} (x_{1}, y_{1}, x_{2}, y_{2}, 0) \\ \times \exp [\frac{i k}{2 B} (A x_{1}^{2} + A y_{1}^{2} - 2 x_{1} u_{1} - 2 y_{1} v_{1} + D u_{1}^{2} + D v_{1}^{2})] \\ \times \exp [- \frac{i k}{2 B^{*}} (A^{*} x_{2}^{2} + A^{*} y_{2}^{2} - 2 x_{2} u_{2} - 2 y_{2} v_{2} + D^{*} u_{2}^{2} + D^{*} v_{2}^{2})] d x_{1} d x_{2} d y_{1} d y_{2}, \end{array}

\begin{array}{l} W_{x x} (u_{1}, v_{1}, u_{2}, v_{2}, z) = \frac{1}{4} {(\frac{1}{λ | B |})}^{2} \frac{π^{2}}{M_{1 x x}} \frac{1}{2^{5 (n \pm 1 + 2 p) / 2} {(p!)}^{2}} {(1 - \frac{2}{M_{1 x x} w_{0}^{2}})}^{(n \pm 1 + 2 p) / 2} \\ \times \exp [\frac{i k D}{2 B} (u_{1}^{2} + v_{1}^{2})] \exp [- \frac{i k D^{*}}{2 B^{*}} (u_{2}^{2} + v_{2}^{2})] \exp [- \frac{k^{2}}{4 M_{1 x x} B^{2}} (u_{1}^{2} + v_{1}^{2})] \\ \times \exp {- \frac{k^{2}}{4 M_{2 x x}} {(\frac{u_{2}}{B^{*}} - \frac{u_{1}}{2 M_{1 x x} B σ_{x x}^{2}})}^{2}} \exp {- \frac{k^{2}}{4 M_{2 x x}} {(\frac{v_{2}}{B^{*}} - \frac{v_{1}}{2 M_{1 x x} B σ_{x x}^{2}})}^{2}} \\ \times \sum_{m = 0}^{p} \sum_{s = 0}^{n \pm 1} \sum_{l = 0}^{p} \sum_{h = 0}^{n \pm 1} \sum_{c_{1} = 0}^{2 m + n \pm 1 - s} \sum_{d_{1} = 0}^{c_{1} / 2} \sum_{e_{1} = 0}^{(2 l + n \pm 1 - h) / 2} \sum_{c_{2} = 0}^{2 p - 2 m + s} \sum_{d_{2} = 0}^{c_{2} / 2} \sum_{e_{2} = 0}^{(2 p - 2 l + h) / 2} (\begin{array}{l} 2 p - 2 m + s \\ c_{2} \end{array}) (\begin{array}{l} 2 m + n \pm 1 - s \\ c_{1} \end{array}) \\ \times (\begin{array}{l} p \\ m \end{array}) (\begin{array}{l} p \\ l \end{array}) (\begin{array}{l} n \pm 1 \\ s \end{array}) (\begin{array}{l} n \pm 1 \\ h \end{array}) [1 - {(- 1)}^{s}] [1 - {(- 1)}^{h}] (- 1)^{d_{1} + d_{2} + e_{1} + e_{2}} i^{s} (- i)^{h} (2 i)^{- (c_{1} + c_{2} - 2 d_{1} - 2 d_{2} + n \pm 1 + 2 p - 2 e_{1} - 2 e_{2})} \\ \times {(\frac{2 \sqrt{2}}{w_{0}})}^{2 p + n \pm 1 - 2 e_{1} - 2 e_{2}} \frac{c_{1}!}{d_{1}! (c_{1} - 2 d_{1})!} \frac{c_{2}!}{d_{2}! (c_{2} - 2 d_{2})!} \frac{(2 l + n \pm 1 - h)!}{e_{1}! (2 l + n \pm 1 - h - 2 e_{1})!} \frac{(2 p - 2 l + h)!}{e_{2}! (2 p - 2 l + h - 2 e_{2})!} \\ \times \frac{1}{{(\sqrt{M_{2 x x}})}^{c_{1} + c_{2} + n \pm 1 + 2 p - 2 d_{1} - 2 d_{2} - 2 e_{1} - 2 e_{2} + 2}} {(\frac{2}{{(w_{0}^{2} M_{1 x x}^{2} σ_{x x}^{4} - 2 M_{1 x x} σ_{x x}^{4})}^{1 / 2}})}^{c_{1} + c_{2} - 2 d_{1} - 2 d_{2}} \\ \times H_{c_{1} - 2 d_{1} + 2 l + n \pm 1 - h - 2 e_{1}} (\frac{k}{2 \sqrt{M_{2 x x}}} (\frac{u_{1}}{2 M_{1 x x} B σ_{x x}^{2}} - \frac{u_{2}}{B^{*}})) H_{2 m + n \pm 1 - s - c_{1}} (\frac{- i k u_{1}}{{(w_{0}^{2} M_{1 x x}^{2} B^{2} - 2 M_{1 x x} B^{2})}^{1 / 2}}) \\ \times H_{c_{2} - 2 d_{2} + 2 p - 2 l + h - 2 e_{2}} (\frac{k}{2 \sqrt{M_{2 x x}}} (\frac{v_{1}}{2 M_{1 x x} B σ_{x x}^{2}} - \frac{v_{2}}{B^{*}})) H_{2 p - 2 m + s - c_{2}} (\frac{- i k v_{1}}{{(w_{0}^{2} M_{1 x x}^{2} B^{2} - 2 M_{1 x x} B^{2})}^{1 / 2}}), \end{array}

M_{1 x x} = 1 / w_{0}^{2} + 1 / (2 σ_{x x}^{2}) - i k A / (2 B), M_{2 x x} = 1 / w_{0}^{2} + 1 / (2 σ_{x x}^{2}) + i k A^{*} / (2 B^{*}) - 1 / (4 M_{1 x x} σ_{x x}^{4}),

\begin{array}{l} W_{x y} (u_{1}, v_{1}, u_{2}, v_{2}, z) = \frac{B_{x y}}{4 i} {(\frac{1}{λ | B |})}^{2} \frac{π^{2}}{M_{1 x y}} \frac{1}{2^{5 (n \pm 1 + 2 p) / 2} {(p!)}^{2}} {(1 - \frac{2}{M_{1 x y} w_{0}^{2}})}^{(n \pm 1 + 2 p) / 2} \\ \times \exp [\frac{i k D}{2 B} (u_{1}^{2} + v_{1}^{2})] \exp [- \frac{i k D^{*}}{2 B^{*}} (u_{2}^{2} + v_{2}^{2})] \exp [- \frac{k^{2}}{4 M_{1 x y} B^{2}} (u_{1}^{2} + v_{1}^{2})] \\ \times \exp {- \frac{k^{2}}{4 M_{2 x y}} {(\frac{u_{2}}{B^{*}} - \frac{u_{1}}{2 M_{1 x y} B σ_{x y}^{2}})}^{2}} \exp {- \frac{k^{2}}{4 M_{2 x y}} {(\frac{v_{2}}{B^{*}} - \frac{v_{1}}{2 M_{1 x y} B σ_{x y}^{2}})}^{2}} \\ \times \sum_{m = 0}^{p} \sum_{s = 0}^{n \pm 1} \sum_{l = 0}^{p} \sum_{h = 0}^{n \pm 1} \sum_{c_{1} = 0}^{2 m + n \pm 1 - s} \sum_{d_{1} = 0}^{c_{1} / 2} \sum_{e_{1} = 0}^{(2 l + n \pm 1 - h) / 2} \sum_{c_{2} = 0}^{2 p - 2 m + s} \sum_{d_{2} = 0}^{c_{2} / 2} \sum_{e_{2} = 0}^{(2 p - 2 l + h) / 2} (\begin{array}{l} 2 p - 2 m + s \\ c_{2} \end{array}) (\begin{array}{l} 2 m + n \pm 1 - s \\ c_{1} \end{array}) \\ \times (\begin{array}{l} p \\ m \end{array}) (\begin{array}{l} p \\ l \end{array}) (\begin{array}{l} n \pm 1 \\ s \end{array}) (\begin{array}{l} n \pm 1 \\ h \end{array}) [1 - {(- 1)}^{s}] [1 + {(- 1)}^{h}] (- 1)^{d_{1} + d_{2} + e_{1} + e_{2}} i^{s} (- i)^{h} (2 i)^{- (c_{1} + c_{2} - 2 d_{1} - 2 d_{2} + n \pm 1 + 2 p - 2 e_{1} - 2 e_{2})} \\ \times {(\frac{2 \sqrt{2}}{w_{0}})}^{2 p + n \pm 1 - 2 e_{1} - 2 e_{2}} \frac{c_{1}!}{d_{1}! (c_{1} - 2 d_{1})!} \frac{c_{2}!}{d_{2}! (c_{2} - 2 d_{2})!} \frac{(2 l + n \pm 1 - h)!}{e_{1}! (2 l + n \pm 1 - h - 2 e_{1})!} \frac{(2 p - 2 l + h)!}{e_{2}! (2 p - 2 l + h - 2 e_{2})!} \\ \times \frac{1}{{(\sqrt{M_{2 x y}})}^{c_{1} + c_{2} - 2 d_{1} - 2 d_{2} + n \pm 1 + 2 p - 2 e_{1} - 2 e_{2} + 2}} {(\frac{2}{{(w_{0}^{2} M_{1 x y}^{2} σ_{x y}^{4} - 2 M_{1 x y} σ_{x y}^{4})}^{1 / 2}})}^{c_{1} + c_{2} - 2 d_{1} - 2 d_{2}} \\ \times H_{c_{1} - 2 d_{1} + 2 l + n \pm 1 - h - 2 e_{1}} (\frac{k}{2 \sqrt{M_{2 x y}}} (\frac{u_{1}}{2 M_{1 x y} B σ_{x y}^{2}} - \frac{u_{2}}{B^{*}})) H_{2 m + n \pm 1 - s - c_{1}} (\frac{- i k u_{1}}{{(w_{0}^{2} M_{1 x y}^{2} B^{2} - 2 M_{1 x y} B^{2})}^{1 / 2}}) \\ \times H_{c_{2} - 2 d_{2} + 2 p - 2 l + h - 2 e_{2}} (\frac{k}{2 \sqrt{M_{2 x y}}} (\frac{v_{1}}{2 M_{1 x y} B σ_{x y}^{2}} - \frac{v_{2}}{B^{*}})) H_{2 p - 2 m + s - c_{2}} (\frac{- i k v_{1}}{{(w_{0}^{2} M_{1 x y}^{2} B^{2} - 2 M_{1 x y} B^{2})}^{1 / 2}}), \end{array}

M_{1 x y} = 1 / w_{0}^{2} + 1 / (2 σ_{x y}^{2}) - i k A / (2 B), M_{2 x y} = 1 / w_{0}^{2} + 1 / (2 σ_{x y}^{2}) + i k A^{*} / (2 B^{*}) - 1 / (4 M_{1 x y} σ_{x y}^{4}),

W_{y x} (u_{1}, v_{1}, u_{2}, v_{2}, z) = {[W_{x y} (u_{1}, v_{1}, u_{2}, v_{2}, z)]}^{*},

\begin{array}{l} W_{y y} (u_{1}, v_{1}, u_{2}, v_{2}, z) = \frac{1}{4} {(\frac{1}{λ | B |})}^{2} \frac{π^{2}}{M_{1 y y}} \frac{1}{2^{5 (n \pm 1 + 2 p) / 2} {(p!)}^{2}} {(1 - \frac{2}{M_{1 y y} w_{0}^{2}})}^{(n \pm 1 + 2 p) / 2} \\ \times \exp [\frac{i k D}{2 B} (u_{1}^{2} + v_{1}^{2})] \exp [- \frac{i k D^{*}}{2 B^{*}} (u_{2}^{2} + v_{2}^{2})] \exp [- \frac{k^{2}}{4 M_{1 y y} B^{2}} (u_{1}^{2} + v_{1}^{2})] \\ \times \exp {- \frac{k^{2}}{4 M_{2 y y}} {(\frac{u_{2}}{B^{*}} - \frac{u_{1}}{2 M_{1 y y} B σ_{y y}^{2}})}^{2}} \exp {- \frac{k^{2}}{4 M_{2 y y}} {(\frac{v_{2}}{B^{*}} - \frac{v_{1}}{2 M_{1 y y} B σ_{y y}^{2}})}^{2}} \\ \times \sum_{m = 0}^{p} \sum_{s = 0}^{n \pm 1} \sum_{l = 0}^{p} \sum_{h = 0}^{n \pm 1} \sum_{c_{1} = 0}^{2 m + n \pm 1 - s} \sum_{d_{1} = 0}^{c_{1} / 2} \sum_{e_{1} = 0}^{(2 l + n \pm 1 - h) / 2} \sum_{c_{2} = 0}^{2 p - 2 m + s} \sum_{d_{2} = 0}^{c_{2} / 2} \sum_{e_{2} = 0}^{(2 p - 2 l + h) / 2} (\begin{array}{l} 2 p - 2 m + s \\ c_{2} \end{array}) (\begin{array}{l} 2 m + n \pm 1 - s \\ c_{1} \end{array}) \\ \times (\begin{array}{l} p \\ m \end{array}) (\begin{array}{l} p \\ l \end{array}) (\begin{array}{l} n \pm 1 \\ s \end{array}) (\begin{array}{l} n \pm 1 \\ h \end{array}) [1 + {(- 1)}^{s}] [1 + {(- 1)}^{h}] (- 1)^{d_{1} + d_{2} + e_{1} + e_{2}} i^{s} (- i)^{h} (2 i)^{- (c_{1} + c_{2} - 2 d_{1} - 2 d_{2} + n \pm 1 + 2 p - 2 e_{1} - 2 e_{2})} \\ \times {(\frac{2 \sqrt{2}}{w_{0}})}^{2 p + n \pm 1 - 2 e_{1} - 2 e_{2}} \frac{c_{1}!}{d_{1}! (c_{1} - 2 d_{1})!} \frac{c_{2}!}{d_{2}! (c_{2} - 2 d_{2})!} \frac{(2 l + n \pm 1 - h)!}{e_{1}! (2 l + n \pm 1 - h - 2 e_{1})!} \frac{(2 p - 2 l + h)!}{e_{2}! (2 p - 2 l + h - 2 e_{2})!} \\ \times \frac{1}{{(\sqrt{M_{2 y y}})}^{c_{1} + c_{2} - 2 d_{1} - 2 d_{2} + n \pm 1 + 2 p - 2 e_{1} - 2 e_{2} + 2}} {(\frac{2}{{(w_{0}^{2} M_{1 y y}^{2} σ_{y y}^{4} - 2 M_{1 y y} σ_{y y}^{4})}^{1 / 2}})}^{c_{1} + c_{2} - 2 d_{1} - 2 d_{2}} \\ \times H_{c_{1} - 2 d_{1} + 2 l + n \pm 1 - h - 2 e_{1}} (\frac{k}{2 \sqrt{M_{2 y y}}} (\frac{u_{1}}{2 M_{1 y y} B σ_{y y}^{2}} - \frac{u_{2}}{B^{*}})) H_{2 m + n \pm 1 - s - c_{1}} (\frac{- i k u_{1}}{{(w_{0}^{2} M_{1 y y}^{2} B^{2} - 2 M_{1 y y} B^{2})}^{1 / 2}}) \\ \times H_{c_{2} - 2 d_{2} + 2 p - 2 l + h - 2 e_{2}} (\frac{k}{2 \sqrt{M_{2 y y}}} (\frac{v_{1}}{2 M_{1 y y} B σ_{y y}^{2}} - \frac{v_{2}}{B^{*}})) H_{2 p - 2 m + s - c_{2}} (\frac{- i k v_{1}}{{(w_{0}^{2} M_{1 y y}^{2} B^{2} - 2 M_{1 y y} B^{2})}^{1 / 2}}), \end{array}

M_{1 y y} = 1 / w_{0}^{2} + 1 / (2 σ_{y y}^{2}) - i k A / (2 B), M_{2 y y} = 1 / w_{0}^{2} + 1 / (2 σ_{y y}^{2}) + i k A^{*} / (2 B^{*}) - 1 / (4 M_{1 y y} σ_{y y}^{4}) .

\int_{- \infty}^{\infty} \exp [- {(x - y)}^{2}] H_{n} (a x) d x = \sqrt{π} {(1 - a^{2})}^{n / 2} H_{n} (\frac{a y}{{(1 - a^{2})}^{1 / 2}}),

\int_{- \infty}^{\infty} x^{n} \exp [- {(x - β)}^{2}] d x = {(2 i)}^{- n} \sqrt{π} H_{n} (i β),

H_{n} (x + y) = \frac{1}{2^{n / 2}} \sum_{k = 0}^{n} (\begin{array}{l} n \\ k \end{array}) H_{k} (\sqrt{2} x) H_{n - k} (\sqrt{2} y),

H_{n} (x) = \sum_{k = 0}^{n / 2} {(- 1)}^{k} \frac{n!}{k! (n - 2 k)!} {(2 x)}^{n - 2 k} .

I (u, v, z) = W_{x x} (u, v, u, v, z) + W_{y y} (u, v, u, v, z) .

W_{s z} (z) = \sqrt{\frac{2 \int s^{2} 〈 I (u, v, z) 〉 d u d v}{\int 〈 I (u, v, z) 〉 d u d v}}, (s = u, v) .

P (u, v, z) = \sqrt{1 - \frac{4 \det [\hat{W} (u, v, u, v, z)]}{{T r [\hat{W} (u, v, u, v, z)]}^{2}}} .

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

Abstract

1. Introduction

2. Theory

3. Statistics properties of a cylindrical vector partially coherent LG beam in free space

4. Summary

Acknowledgments

References and links

Cited By

Figures (4)

Equations (29)

Optics Express