Nonparaxial propagation of an elegant Laguerre-Gaussian beam orthogonal to the optical axis of a uniaxial crystal

Yongzhou Ni; Guoquan Zhou

doi:10.1364/OE.20.017160

Optics Express
Vol. 20,
Issue 15,
pp. 17160-17173
(2012)
•https://doi.org/10.1364/OE.20.017160

Nonparaxial propagation of an elegant Laguerre-Gaussian beam orthogonal to the optical axis of a uniaxial crystal

Yongzhou Ni and Guoquan Zhou

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Yongzhou Ni and Guoquan Zhou, "Nonparaxial propagation of an elegant Laguerre-Gaussian beam orthogonal to the optical axis of a uniaxial crystal," Opt. Express 20, 17160-17173 (2012)

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Table of Contents Category
- Physical 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: May 23, 2012
- Revised Manuscript: June 28, 2012
- Manuscript Accepted: July 9, 2012
- Published: July 12, 2012

Abstract

Analytical expressions of the nonparaxial propagation of even and odd elegant Laguerre-Gaussian beams orthogonal to the optical axis of a uniaxial crystal are derived. The intensity distributions of even and odd elegant Laguerre-Gaussian beams and their three components propagating in uniaxial crystals orthogonal to the optical axis are illustrated by numerical examples. Even though one of the two transversal components of even and odd elegant Laguerre-Gaussian beams in the input plane is set to be zero, this transversal and the longitudinal components have nonzero intensities and cannot be neglected upon propagation inside the uniaxial crystal. The evolution laws of even and odd elegant Laguerre-Gaussian beams propagating in uniaxial crystals orthogonal to the optical axis are also demonstrated. The intensity distributions of even and odd elegant Laguerre-Gaussian beams can be modulated by the uniaxial crystal, which is beneficial to the some applications involving in the special beam profile.

1. Introduction

The cylindrically symmetric higher-order modes of laser cavities with spherical mirrors are the standard Laguerre-Gaussian beams. As an extension of the standard Laguerre-Gaussian beams, the elegant Laguerre-Gaussian beams, which are also the eigenmodes of the paraxial wave equation, have been introduced by T. Takenaka et al [1]. The elegant Laguerre-Gaussian beams differ from the standard Laguerre-Gaussian beams in that the former contains polynomials with a complex argument, while the argument in the latter is real. Within the paraxial and the non-paraxial frameworks, the properties of the elegant Laguerre-Gaussian beams propagating in free space, through a paraxial optical system, in turbulent atmosphere, and at a dielectric interface have been extensively investigated [2–17]. Also, the elegant Laguerre-Gaussian beam has been extended to the partially coherent case [18–20]. The relationship between the elegant Laguerre-Gaussian and the Bessel-Gaussian beams has been also examined [21]. Elegant Laguerre-Gaussian beams can be used as a new tool to describe the axisymmetric flattened Gaussian beam [22].

Research in the propagation of laser beams in a uniaxial crystal plays an important role in designing polarizing prism, amplitude- and phase-modulation devices. The propagation of laser beams in a uniaxial crystal can be treated by solving Maxwell’s equation in crystals. The propagation of various kinds of laser beams in uniaxial crystals has been widely examined [23–32]. In the remainder of this paper, therefore, the propagation of an elegant Laguerre-Gaussian beam in uniaxial crystals orthogonal to the optical axis is to be investigated. Moreover, here we consider the nonparaxial propagation of an elegant Laguerre-Gaussian beam orthogonal to the optical axis of a uniaxial crystal. Analytical propagation expressions of even and odd elegant Laguerre-Gaussian beams orthogonal to the optical axis of a uniaxial crystal are derived, and the corresponding figures are demonstrated.

2. Theoretical derivations

In the Cartesian coordinate system, the z-axis is taken to be the propagation axis. The optical axis of the uniaxial crystal coincides with the x-axis. The input plane is z = 0 and the observation plane is z. The dielectric tensor of the uniaxial crystal is described by

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

where n_o and n_e are the ordinary and the extraordinary refractive indices, respectively. The elegant Laguerre-Gaussian beam can be divided into two types: the even and the odd modes. First, we consider the even elegant Laguerre-Gaussian beam. The even elegant Laguerre-Gaussian beam considered here is linearly polarized in the x-direction and is incident on a uniaxial crystal in the plane z = 0. The even elegant Laguerre-Gaussian beam in the input plane z = 0 takes the form [1]

[\begin{matrix} E_{x} (ρ_{0}, 0) \\ E_{y} (ρ_{0}, 0) \end{matrix}] = [\begin{matrix} {(\frac{ρ_{0}}{w_{0}})}^{m} L_{n}^{m} (\frac{ρ_{0}^{2}}{w_{0}^{2}}) \exp (- \frac{ρ_{0}^{2}}{w_{0}^{2}}) \cos (m φ_{0}) \\ 0 \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix}],

where ρ₀ = x₀e_x + y₀e_y. Vectors e_x and e_y are the unit vectors in the x- and y-directions, respectively.

ρ_{0} = {(x_{0}^{2} + y_{0}^{2})}^{1 / 2}

and

φ_{0} = \tan^{- 1} (y_{0} / x_{0})

. Integers n and m are the radial and the angular mode numbers, respectively. The symbol w₀ is the waist radius of the Gaussian part.

L_{n}^{m}

is the associated Laguerre polynomial. We recall the following relation [33]:

ρ_{0}^{m} L_{n}^{m} (ρ_{0}^{2}) \cos (m φ_{0}) = \frac{{(- 1)}^{n}}{2^{2 n + m} n!} \sum_{l = 0}^{n} \sum_{s = 0}^{[m / 2]} {(- 1)}^{s} (\begin{matrix} n \\ l \end{matrix}) (\begin{matrix} m \\ 2 s \end{matrix}) H_{M_{1}} (x_{0}) H_{N_{1}} (y_{0}),

where M₁ = m + 2l-2s and N₁ = 2n-2l + 2s. The expression [m/2] denotes the integer part of m/2, and

H_{M_{1}}

is the Hermite polynomial of order M₁. The expressions

(\begin{matrix} n \\ l \end{matrix})

and

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

are binomial coefficients. Therefore, Eq. (2) can be rewritten as

[\begin{matrix} E_{x} (ρ_{0}, 0) \\ E_{y} (ρ_{0}, 0) \end{matrix}] = [\begin{matrix} \frac{{(- 1)}^{n}}{2^{2 n + m} n!} \sum_{l = 0}^{n} \sum_{s = 0}^{[m / 2]} {(- 1)}^{s} (\begin{matrix} n \\ l \end{matrix}) (\begin{matrix} m \\ 2 s \end{matrix}) H_{M_{1}} (\frac{x_{0}}{w_{0}}) H_{N_{1}} (\frac{y_{0}}{w_{0}}) \exp (- \frac{ρ_{0}^{2}}{w_{0}^{2}}) \\ 0 \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix}] .

The propagation of the even elegant Laguerre-Gaussian beam orthogonal to the optical axis of a uniaxial crystal obeys the following equation [34]:

\begin{array}{l} E (ρ, z) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d^{2} k \exp (i k \cdot ρ) \exp (i k_{e z} z) (\begin{matrix} {\tilde{E}}_{x} (k) \\ - \frac{k_{x} k_{y}}{k_{0}^{2} n_{o}^{2} - k_{x}^{2}} {\tilde{E}}_{x} (k) \\ - \frac{k_{e z} k_{x}}{k_{0}^{2} n_{o}^{2} - k_{x}^{2}} {\tilde{E}}_{x} (k) \end{matrix}) \\ \begin{matrix}  \end{matrix} + \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d^{2} k \exp (i k \cdot ρ) \exp (i k_{o z} z) (\begin{matrix} 0 \\ \frac{k_{x} k_{y}}{k_{0}^{2} n_{o}^{2} - k_{x}^{2}} {\tilde{E}}_{x} (k) + {\tilde{E}}_{y} (k) \\ - \frac{k_{y}}{k_{o z}} [\frac{k_{x} k_{y}}{k_{0}^{2} n_{o}^{2} - k_{x}^{2}} {\tilde{E}}_{x} (k) + {\tilde{E}}_{y} (k)] \end{matrix}), \end{array}

where k = k_xe_x + k_ye_y, ρ = xe_x + ye_y, and k₀ = 2π/λ is the wave number with λ being the optical wavelength.

{\tilde{E}}_{j} (k)

is the two-dimensional Fourier transform of the transverse components of the optical field in the input plane z = 0 and is given by

{\tilde{E}}_{j} (k) = \frac{1}{{(2 π)}^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E_{j} (ρ_{0}, 0) \exp [- i (k_{x} x_{0} + k_{y} y_{0})] d x_{0} d y_{0},

where j = x or y (hereafter). k_ez and k_oz are defined by

k_{e z} = {[k_{o}^{2} n_{e}^{2} - (n_{e}^{2} / n_{o}^{2}) k_{x}^{2} - k_{y}^{2}]}^{1 / 2}, k_{o z} = {(k_{o}^{2} n_{0}^{2} - k_{x}^{2} - k_{y}^{2})}^{1 / 2} .

When the beam waist radius of the even elegant Laguerre-Gaussian beam is comparable with the wavelength, the paraxial approach is not accurate enough. A nonparaxial laser beam can be produced from a paraxial laser beam by its focusing with a high numerical aperture, and is commonly encountered in holography microscopy and microlithography. When the field in a uniaxial crystal is nonparaxial, Eq. (5) can be expressed as a sum of the dominant paraxial result and a nonparaxial correction term [34]:

\begin{array}{l} E (ρ, z) = \exp (i k_{0} n_{e} z) \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d^{2} k \exp (i k \cdot ρ) \exp (- i \frac{n_{e}^{2} k_{x}^{2} + n_{o}^{2} k_{y}^{2}}{2 k_{0} n_{e} n_{o}^{2}} z) (\begin{matrix} {\tilde{E}}_{x} (k) \\ - \frac{k_{x} k_{y}}{k_{0}^{2} n_{o}^{2}} {\tilde{E}}_{x} (k) \\ - \frac{n_{e} k_{x}}{k_{0}^{} n_{o}^{2}} {\tilde{E}}_{x} (k) \end{matrix}) \\ \begin{matrix}  \end{matrix} + \exp (i k_{0} n_{o} z) \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d^{2} k \exp (i k \cdot ρ) \exp (- i \frac{k_{x}^{2} + k_{y}^{2}}{2 k_{0} n_{o}} z) (\begin{matrix} 0 \\ \frac{k_{x} k_{y}}{k_{0}^{2} n_{o}^{2}} {\tilde{E}}_{x} (k) + {\tilde{E}}_{y} (k) \\ - \frac{k_{y}}{k_{o} n_{o}} {\tilde{E}}_{y} (k) \end{matrix}) . \end{array}

E_x(ρ, z) is just the paraxial result. E_y(ρ, z) and E_z(ρ, z) are the first nonparaxial correction to the paraxial result. The nonparaxial correction contains all the nonparaxial features of the beam and can be easily evaluated from the knowledge of the paraxial result [34]. The nonparaxiality couples the Cartesian components of the field. The resultant longitudinal component is greater than the correction to the transverse component orthogonal to the optical axis. By using the property of the Fourier transforms [35], the three components of the nonparaxial field in the uniaxial crystal orthogonal to the optical axis can also be rewritten as

E_{x} (ρ, z) = \frac{k_{0} n_{o}}{2 π i z} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E_{x} (ρ_{0}, 0) Λ_{e} (ρ, ρ_{0}) d x_{0} d y_{0},

\begin{array}{l} E_{y} (ρ, z) = \frac{i k_{0} n_{o}}{2 π z^{3}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E_{x} (ρ_{0}, 0) (x - x_{0}) (y - y_{0}) [Λ_{e} (ρ, ρ_{0}) - Λ_{o} (ρ, ρ_{0})] d x_{0} d y_{0} \\ \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} & + \end{matrix} \frac{k_{0} n_{o}}{2 π i z} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E_{y} (ρ_{0}, 0) Λ_{o} (ρ, ρ_{0}) d x_{0} d y_{0}, \end{array}

E_{z} (ρ, z) = \frac{i k_{0} n_{o}}{2 π z^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} [E_{x} (ρ_{0}, 0) (x - x_{0}) Λ_{e} (ρ, ρ_{0}) + E_{y} (ρ_{0}, 0) (y - y_{0}) Λ_{o} (ρ, ρ_{0})] d x_{0} d y_{0},

with

Λ_{e} (ρ, ρ_{0})

and

Λ_{o} (ρ, ρ_{0})

being given by

Λ_{e} (ρ, ρ_{0}) = \exp (i k_{0} n_{e} z) \exp {- \frac{k_{0}}{2 i z n_{e}} [n_{o}^{2} {(x - x_{0})}^{2} + n_{e}^{2} {(y - y_{0})}^{2}]},

Λ_{o} (ρ, ρ_{0}) = \exp (i k_{0} n_{o} z) \exp {- \frac{k_{0} n_{o}}{2 i z} [{(x - x_{0})}^{2} + {(y - y_{0})}^{2}]} .

Using the following mathematical formulae [35]:

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

x H_{M} (x) = \frac{1}{2} H_{M + 1} (x) + M H_{M - 1} (x),

the three components of the even elegant Laguerre-Gaussian beam in the uniaxial crystal orthogonal to the optical axis are found to be

E_{x} (ρ, z) = \frac{{(- 1)}^{n} k_{0} n_{o}}{2^{2 n + m + 1} n! i π z} \exp (i e k_{0} n_{o} z) \sum_{l = 0}^{n} \sum_{s = 0}^{[m / 2]} {(- 1)}^{s} (\begin{matrix} n \\ l \end{matrix}) (\begin{matrix} m \\ 2 s \end{matrix}) T_{M_{1}} (x) U_{N_{1}} (y),

\begin{array}{l} E_{y} (ρ, z) = \frac{{(- 1)}^{n} i k_{0} n_{o}}{2^{2 n + m + 1} n! π z^{3}} \sum_{l = 0}^{n} \sum_{s = 0}^{[m / 2]} {(- 1)}^{s} (\begin{matrix} n \\ l \end{matrix}) (\begin{matrix} m \\ 2 s \end{matrix}) {\exp (i e k_{0} n_{o} z) {x y T_{M_{1}} (x) U_{N_{1}} (y) - w_{0} x \\ \begin{matrix}  \end{matrix} \times T_{M_{1}} (x) [0.5 U_{N_{1} + 1} (y) + N_{1} U_{N_{1} - 1} (y)] - w_{0} y U_{N_{1}} (y) [0.5 T_{M_{1} + 1} (x) + M_{1} T_{M_{1} - 1} (x)] \\ \begin{matrix}  \end{matrix} + w_{0}^{2} [0.5 T_{M_{1} + 1} (x) + M_{1} T_{M_{1} - 1} (x)] [0.5 U_{N_{1} + 1} (y) + N_{1} U_{N_{1} - 1} (y)]} - \exp (i k_{0} n_{o} z) {x y \\ \begin{matrix}  \end{matrix} \times V_{M_{1}} (x) V_{N_{1}} (y) - w_{0} x V_{M_{1}} (x) [0.5 V_{N_{1} + 1} (y) + N_{1} V_{N_{1} - 1} (y)] - w_{0} y V_{N_{1}} (y) [0.5 V_{M_{1} + 1} (x) \\ \begin{matrix}  \end{matrix} + M_{1} V_{M_{1} - 1} (x)] + w_{0}^{2} [0.5 V_{M_{1} + 1} (x) + M_{1} V_{M_{1} - 1} (x)] [0.5 V_{N_{1} + 1} (y) + N_{1} V_{N_{1} - 1} (y)]}}, \end{array}

\begin{array}{l} E_{z} (ρ, z) = \frac{{(- 1)}^{n} i k_{0} n_{o}}{2^{2 n + m + 1} n! π z^{2}} \exp (i e k_{0} n_{o} z) \sum_{l = 0}^{n} \sum_{s = 0}^{[m / 2]} {(- 1)}^{s} (\begin{matrix} n \\ l \end{matrix}) (\begin{matrix} m \\ 2 s \end{matrix}) {x T_{M_{1}} (x) U_{N_{1}} (y) - w_{0} U_{N_{1}} (y) \\ \begin{matrix} \begin{matrix} \times \end{matrix} \end{matrix} [0.5 T_{M_{1} + 1} (x) + M_{1} T_{M_{1} - 1} (x)]}, \end{array}

with

T_{μ} (x)

U_{μ} (y)

V_{μ} (x)

, and

V_{μ} (y)

being given by

T_{μ} (x) = w_{0} \sqrt{π α} {(1 - α)}^{μ / 2} \exp (- a x^{2}) H_{μ} [\frac{α z_{r} n_{o}^{} x}{i e z w_{0} {(1 - α)}^{1 / 2}}],

U_{μ} (y) = w_{0} \sqrt{π β} {(1 - β)}^{μ / 2} \exp (- b y^{2}) H_{μ} [\frac{β z_{r} e n_{o} y}{i z w_{0} {(1 - β)}^{1 / 2}}],

V_{μ} (j) = w_{0} \sqrt{π γ} {(1 - γ)}^{μ / 2} \exp (- c j^{2}) H_{μ} [\frac{γ z_{r} n_{0} j}{i z w_{0} {(1 - γ)}^{1 / 2}}],

where µ is an arbitrary integer. The auxiliary parameters are defined as

e = \frac{n_{e}}{n_{o}}, z_{r} = \frac{k_{0} w_{0}^{2}}{2}, α = {(1 - \frac{i z_{r} n_{o}^{}}{e z})}^{- 1}, β = {(1 - \frac{i e z_{r} n_{o}}{z})}^{- 1}, γ = {(1 - \frac{i z_{r} n_{0}}{z})}^{- 1},

a = - α {(\frac{z_{r} n_{o}^{}}{i e z w_{0}})}^{2} + \frac{k_{0} n_{o}^{}}{2 i e z}, b = - β {(\frac{e z_{r} n_{o}}{i z w_{0}})}^{2} + \frac{e k_{0} n_{o}}{2 i z}, c = - γ {(\frac{z_{r} n_{o}}{i z w_{0}})}^{2} + \frac{k_{0} n_{o}}{2 i z} .

where z_r is the Rayleigh length of the Gaussian part. As shown in Eqs. (19)-(21), the polynomials in three components of the even elegant Laguerre-Gaussian beam have complex argument, which can be compared with those of the standard elegant Laguerre-Gaussian beam propagating in the uniaxial crystal orthogonal to the optical axis by using the method suggested in Ref [36].

The odd elegant Laguerre-Gaussian beam in the input plane z = 0 is described by

[\begin{matrix} E_{x} (ρ_{0}, 0) \\ E_{y} (ρ_{0}, 0) \end{matrix}] = [\begin{matrix} {(\frac{ρ_{0}}{w_{0}})}^{m} L_{n}^{m} (\frac{ρ_{0}^{2}}{w_{0}^{2}}) \exp (- \frac{ρ_{0}^{2}}{w_{0}^{2}}) \sin (m φ_{0}) \\ 0 \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix}] .

Based on the following expansion [33]:

ρ_{0}^{m} L_{n}^{m} (ρ_{0}^{2}) \sin (m φ_{0}) = \frac{{(- 1)}^{n}}{2^{2 n + m} n!} \sum_{l = 0}^{n} \sum_{s = 0}^{[(m - 1) / 2]} {(- 1)}^{s} (\begin{matrix} n \\ l \end{matrix}) (\begin{matrix} m \\ 2 s + 1 \end{matrix}) H_{M_{1} - 1} (x_{0}) H_{N_{1} + 1} (y_{0}),

Equation (25) can be rewritten as

[\begin{matrix} E_{x} (ρ_{0}, 0) \\ E_{y} (ρ_{0}, 0) \end{matrix}] = [\begin{matrix} \frac{{(- 1)}^{n}}{2^{2 n + m} n!} \sum_{l = 0}^{n} \sum_{s = 0}^{[(m - 1) / 2]} {(- 1)}^{s} (\begin{matrix} n \\ l \end{matrix}) (\begin{matrix} m \\ 2 s + 1 \end{matrix}) H_{M_{1} - 1} (\frac{x_{0}}{w_{0}}) H_{N_{1} + 1} (\frac{y_{0}}{w_{0}}) \exp (- \frac{ρ_{0}^{2}}{w_{0}^{2}}) \\ 0 \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix}] .

The three components of the odd elegant Laguerre-Gaussian beam in the uniaxial crystal orthogonal to the optical axis turn out to be

E_{x} (ρ, z) = \frac{{(- 1)}^{n} k_{0} n_{o}}{2^{2 n + m + 1} n! i π z} \exp (i e k_{0} n_{o} z) \sum_{l = 0}^{n} \sum_{s = 0}^{[(m - 1) / 2]} {(- 1)}^{s} (\begin{matrix} n \\ l \end{matrix}) (\begin{matrix} m \\ 2 s + 1 \end{matrix}) T_{M_{1} - 1} (x) U_{N_{1} + 1} (y),

\begin{array}{l} E_{y} (ρ, z) = \frac{{(- 1)}^{n} i k_{0} n_{o}}{2^{2 n + m + 1} n! π z^{3}} \sum_{l = 0}^{n} \sum_{s = 0}^{[(m - 1) / 2]} {(- 1)}^{s} (\begin{matrix} n \\ l \end{matrix}) (\begin{matrix} m \\ 2 s + 1 \end{matrix}) {\exp (i e k_{0} n_{o} z) {x y T_{M_{1} - 1} (x) U_{N_{1 + 1}} (y) \\ \begin{matrix}  \end{matrix} - w_{0} x T_{M_{1} - 1} (x) [0.5 U_{N_{1} + 2} (y) + (N_{1} + 1) U_{N_{1}} (y)] - w_{0} y U_{N_{1} + 1} (y) [0.5 T_{M_{1}} (x) + (M_{1} - 1) \\ \begin{matrix}  \end{matrix} \times T_{M_{1} - 2} (x)] + w_{0}^{2} [0.5 T_{M_{1}} (x) + (M_{1} - 1) T_{M_{1} - 2} (x)] [0.5 U_{N_{1} + 2} (y) + (N_{1} + 1) U_{N_{1}} (y)]} \\ \begin{matrix}  \end{matrix} - \exp (i k_{0} n_{o} z) {x y V_{M_{1} - 1} (x) V_{N_{1} + 1} (y) - w_{0} x V_{M_{1} - 1} (x) [0.5 V_{N_{1} + 2} (y) + (N_{1} + 1) V_{N_{1}} (y)] \\ \begin{matrix}  \end{matrix} - w_{0} y V_{N_{1} + 1} (y) [0.5 V_{M_{1}} (x) + (M_{1} - 1) V_{M_{1} - 2} (x)] + w_{0}^{2} [0.5 V_{M_{1}} (x) + (M_{1} - 1) V_{M_{1} - 2} (x)] \\ \begin{matrix}  \end{matrix} \times [0.5 V_{N_{1} + 2} (y) + (N_{1} + 1) V_{N_{1}} (y)]}}, \end{array}

3. Numerical calculations and analyses

Based on the analytical formulae derived in the last section, now we investigate the propagation properties of an elegant Laguerre-Gaussian beam orthogonal to the optical axis of a uniaxial crystal. Here we mainly pay attention to the influence of the uniaxial crystal on the propagation of an elegant Laguerre-Gaussian beam. The calculation parameters are chosen as follow. m = 3, n = 2, w₀ = λ, and the ordinary refractive index is fixed to be 2.616. The extraordinary refractive index is variable. Figures 1 –3 represent the contour graphs of the intensity distribution of the three components of an even elegant Laguerre-Gaussian beam propagating in the uniaxial crystal orthogonal to the optical axis at several observation planes. e is set to be 1.5 in Figs. 1–3. The observation planes are z = 0.1z_r, z = z_r, z = 3z_r, and z = 5z_r, respectively. The intensity is proportional to the scale of 1/λ², which is not marked in the figures. The beam profile of the x-component of an even elegant Laguerre-Gaussian beam has six lobes. Upon propagation orthogonal to the optical axis of the uniaxial crystal, the spreading of the beam profile in the y-direction is far slower than that in the x-direction, which is caused by the anisotropic effect of the crystal. As a result, the beam profile in the uniaxial crystal is elongated in the x-direction. The magnitude of the intensity of the six lobes close to the input plane is equivalent. However, the equality of magnitude of the intensity of the six lobes is broken upon propagation in the uniaxial crystal. At first the magnitude of the intensity of the two middle lobes is smaller than that of the other four lobes. When the observation plane is far enough, the magnitude of the intensity of the four top and bottom lobes is smaller than that of the two middle lobes. Though the y-component of an even elegant Laguerre-Gaussian beam in the input plane is set to be zero, it is no longer equal to zero upon propagation in the uniaxial crystal, which can be interpreted as follow. The y-component is affected by the x-component because of the dependence on ${\tilde{E}}_{x} (k)$ , and this corresponds to a change of polarization state of the radiation that occurs during propagation in the uniaxial crystal [32]. The beam profile of the y-component of an even elegant Laguerre-Gaussian beam apparently varies upon propagation in the uniaxial crystal. Upon propagation in the uniaxial crystal, the two central lobes including the top and bottom side lobes will diminish until disappear. Instead, the four bilateral lobes gradually become the dominant lobes. Compared with the magnitude of the x-component, the magnitude of the y-component is small. When propagating from the observation plane z = 0.1z_r to the observation plane z = z_r, the beam profile of the longitudinal component undergoes a process of the split of the lobes. When propagating from the observation plane z = z_r to the observation plane z = 3z_r, the two lobes in the y-axis gradually diminish until disappear, and the two lobes in the x-axis evolve from the accessory lobes into the dominant lobes. When the observation plane is larger than 3z_r, the beam profile of the longitudinal component keeps stable. The magnitude of the longitudinal component is smaller than that of the x-component but larger than that of the y-component. The contour graph of the intensity distribution of an even elegant Laguerre-Gaussian beam propagating in the uniaxial crystal orthogonal to the optical axis at several observation planes is shown in Fig. 4 where e = 1.5. As ${| E_{x} |}^{2} > {| E_{z} |}^{2} > {| E_{y} |}^{2}$ , the beam profile of an even elegant Laguerre-Gaussian beam is close to that of the x-component. However, the details in the beam profile between the even elegant Laguerre-Gaussian beam and its x-component are different. Moreover, the magnitude of the intensity of the even elegant Laguerre-Gaussian beam is larger than that of its x-component, which sometimes cannot be distinguished through the comparison of the corresponding figures as the label of the intensity is marked in the rank.

Fig. 1 Contour graph of the intensity of the x-component of an even elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1z_r. (b) z = z_r. (c) z = 3z_r. (d) z = 5z_r.

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Fig. 2 Contour graph of the intensity of the y-component of an even elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1z_r. (b) z = z_r. (c) z = 3z_r. (d) z = 5z_r.

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Fig. 3 Contour graph of the intensity of the longitudinal component of an even elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1z_r. (b) z = z_r. (c) z = 3z_r. (d) z = 5z_r.

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Fig. 4 Contour graph of the intensity of an even elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1z_r. (b) z = z_r. (c) z = 3z_r. (d) z = 5z_r.

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Figure 5 represents the contour graph of the intensity of an even elegant Laguerre-Gaussian beam in the observation plane z = 3z_r of different uniaxial crystals. The uniaxial crystals can be divided into two kinds. One is the positive uniaxial crystal, which corresponds to e>1. The other is the negative uniaxial crystal, which corresponds to e<1. As to a uniaxial crystal, the value of e is fixed. For simplicity, here a model which is the combination of two kinds of uniaxial crystals is used to illustrate the effect of the uniaxial crystals. e = 1 corresponds to the isotropic mediums. When e = 1, the beam profile of the even elegant Laguerre-Gaussian beam is symmetric and the magnitude of the intensity of the six lobes is equivalent. When e is not equal to unity, the beam profile becomes an astigmatic beam. When e<1, the beam profile is elongated in the y-direction, and the magnitude of the intensity of the top and bottom lobes is larger than that of the two middle lobes. When e>1, the beam profile is elongated in the x-direction, and the magnitude of the intensity of the top and bottom lobes is smaller than that of the two middle lobes. With increasing the deviation of e from unity, the elongation of the beam spot augments, and the difference of the magnitude of the intensity between the middle lobes and the other lobes also increases.

Fig. 5 Contour graph of the intensity of an even elegant Laguerre-Gaussian beam propagating in the observation plane z = 3z_r of different uniaxial crystals. (a) e = 0.6. (b) e = 0.8. (c) e = 1.0. (d) e = 1.2.

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The contour graphs of the intensity distribution of the three components of an odd elegant Laguerre-Gaussian beam propagating in the uniaxial crystal orthogonal to the optical axis at several observation planes are shown in Figs. 6 –8 where e = 1.5. When z = 0.1z_r, the beam profile of the x-component of the odd elegant Laguerre-Gaussian beam is that of the even elegant Laguerre-Gaussian beam rotating 90° from the y-axis. Upon propagation in the uniaxial crystal, the magnitude of the intensity of the lobes in the y-axis quickly decreases and is smaller than that of the four bilateral lobes. When z = 0.1z_r, the beam profile of the y-component of the odd elegant Laguerre-Gaussian beam is somewhat similar to that of the even elegant Laguerre-Gaussian beam rotating 90° from the y-axis. However, the magnitude of the intensity slightly decreases. When z = z_r or z = 3z_r, the beam profile of the y-component of the odd elegant Laguerre-Gaussian beam is completely different from that of the even elegant Laguerre-Gaussian beam. In these cases, the beam profile of the y-component of the odd elegant Laguerre-Gaussian beam takes on an annular shape. When z = 5z_r, the beam profile of the y-component of the odd elegant Laguerre-Gaussian beam is somewhat similar to that of the even elegant Laguerre-Gaussian beam. The beam profile of the longitudinal component of the odd elegant Laguerre-Gaussian beam is always completely different from that of the even elegant Laguerre-Gaussian beam. The beam profile of the longitudinal component of the odd elegant Laguerre-Gaussian beam has eight lobes. When z = 3z_r, each two lobes is mutually connected to form a group. Upon propagation in the uniaxial crystal, the magnitude of the intensity of the four central lobes diminishes and that of the four bilateral lobes increases. Figure 9 represents the contour graph of the intensity distribution of an odd elegant Laguerre-Gaussian beam propagating in the uniaxial crystal orthogonal to the optical axis at several observation planes. Comparing Fig. 9(d) with Fig. 6(d), the magnitude of the intensity denotes that the y- and z-components cannot be ignored. Figure 10 shows the contour graph of the intensity of an odd elegant Laguerre-Gaussian beam in the observation plane z = 3z_r of different uniaxial crystals. When e<1, the magnitude of the intensity of the lobes in the y-axis is larger than that of the four bilateral lobes. When e>1, the magnitude of the intensity of the lobes in the y-axis is smaller than that of the four bilateral lobes. When e<1, the beam profile of the odd elegant Laguerre-Gaussian beam and its components is elongated in the y-direction. When e>1, the beam profile of the odd elegant Laguerre-Gaussian beam and its components is elongated in the x-direction. Comparing Figs. 1–5 with Figs. 6–10, one can find that the even and odd elegant Laguerre-Gaussian beams have their respective evolution laws.

Fig. 6 Contour graph of the intensity of the x-component of an odd elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1z_r. (b) z = z_r. (c) z = 3z_r. (d) z = 5z_r.

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Fig. 7 Contour graph of the intensity of the y-component of an odd elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1z_r. (b) z = z_r. (c) z = 3z_r. (d) z = 5z_r.

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Fig. 8 Contour graph of the intensity of the longitudinal component of an odd elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1z_r. (b) z = z_r. (c) z = 3z_r. (d) z = 5z_r.

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Fig. 9 Contour graph of the intensity of an odd elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1z_r. (b) z = z_r. (c) z = 3z_r. (d) z = 5z_r.

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Fig. 10 Contour graph of the intensity of an odd elegant Laguerre-Gaussian beam propagating in the observation plane z = 3z_r of different uniaxial crystals. (a) e = 0.6. (b) e = 0.8. (c) e = 1.0. (d) e = 1.2.

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4. Conclusions

The analytical nonparaxial formulae of even and odd elegant Laguerre-Gaussian beams propagating in a uniaxial crystal orthogonal to the optical axis have been derived, respectively. The intensity distributions of even and odd elegant Laguerre-Gaussian beams and their three components propagating in a uniaxials crystal orthogonal to the optical axis have been demonstrated by numerical examples. Though the y-component of even and odd elegant Laguerre-Gaussian beams in the input plane is set to be zero, the y-component emerges upon propagation in the uniaxial crystal. Moreover, the y- and z-components cannot be ignored. The even and odd elegant Laguerre-Gaussian beams propagating in uniaxial crystal become astigmatic beams. The beam profile of even and odd elegant Laguerre-Gaussian beams in the uniaxial crystal is elongated in the x- or y-direction, which is determined by the ratio of the extraordinary refractive index to the ordinary refractive index. With the increase of the deviation of the ratio of the extraordinary refractive index to the ordinary refractive index from unity, the elongation of the beam profile also increases. The even and odd elegant Laguerre-Gaussian beams propagating in the uniaxial crystal have their respective evolution laws. As the beam profile of even and odd elegant Laguerre-Gaussian beams can be modulated by the uniaxial crystal, this research is beneficial to the some applications involving in the special beam profile.

Acknowledgments

This research was supported by National Natural Science Foundation of China under Grant Nos. 10974179 and 61178016. The authors are indebted to the reviewer for valuable comments and suggestions.

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Fig. 1 Contour graph of the intensity of the x-component of an even elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1z_r. (b) z = z_r. (c) z = 3z_r. (d) z = 5z_r.

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Fig. 2 Contour graph of the intensity of the y-component of an even elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1z_r. (b) z = z_r. (c) z = 3z_r. (d) z = 5z_r.

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Fig. 3 Contour graph of the intensity of the longitudinal component of an even elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1z_r. (b) z = z_r. (c) z = 3z_r. (d) z = 5z_r.

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Fig. 4 Contour graph of the intensity of an even elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1z_r. (b) z = z_r. (c) z = 3z_r. (d) z = 5z_r.

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Fig. 5 Contour graph of the intensity of an even elegant Laguerre-Gaussian beam propagating in the observation plane z = 3z_r of different uniaxial crystals. (a) e = 0.6. (b) e = 0.8. (c) e = 1.0. (d) e = 1.2.

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Fig. 6 Contour graph of the intensity of the x-component of an odd elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1z_r. (b) z = z_r. (c) z = 3z_r. (d) z = 5z_r.

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Fig. 7 Contour graph of the intensity of the y-component of an odd elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1z_r. (b) z = z_r. (c) z = 3z_r. (d) z = 5z_r.

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Fig. 8 Contour graph of the intensity of the longitudinal component of an odd elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1z_r. (b) z = z_r. (c) z = 3z_r. (d) z = 5z_r.

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Fig. 9 Contour graph of the intensity of an odd elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1z_r. (b) z = z_r. (c) z = 3z_r. (d) z = 5z_r.

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Fig. 10 Contour graph of the intensity of an odd elegant Laguerre-Gaussian beam propagating in the observation plane z = 3z_r of different uniaxial crystals. (a) e = 0.6. (b) e = 0.8. (c) e = 1.0. (d) e = 1.2.

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Equations (29)

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

[\begin{matrix} E_{x} (ρ_{0}, 0) \\ E_{y} (ρ_{0}, 0) \end{matrix}] = [\begin{matrix} {(\frac{ρ_{0}}{w_{0}})}^{m} L_{n}^{m} (\frac{ρ_{0}^{2}}{w_{0}^{2}}) \exp (- \frac{ρ_{0}^{2}}{w_{0}^{2}}) \cos (m φ_{0}) \\ 0 \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix}],

ρ_{0}^{m} L_{n}^{m} (ρ_{0}^{2}) \cos (m φ_{0}) = \frac{{(- 1)}^{n}}{2^{2 n + m} n!} \sum_{l = 0}^{n} \sum_{s = 0}^{[m / 2]} {(- 1)}^{s} (\begin{matrix} n \\ l \end{matrix}) (\begin{matrix} m \\ 2 s \end{matrix}) H_{M_{1}} (x_{0}) H_{N_{1}} (y_{0}),

[\begin{matrix} E_{x} (ρ_{0}, 0) \\ E_{y} (ρ_{0}, 0) \end{matrix}] = [\begin{matrix} \frac{{(- 1)}^{n}}{2^{2 n + m} n!} \sum_{l = 0}^{n} \sum_{s = 0}^{[m / 2]} {(- 1)}^{s} (\begin{matrix} n \\ l \end{matrix}) (\begin{matrix} m \\ 2 s \end{matrix}) H_{M_{1}} (\frac{x_{0}}{w_{0}}) H_{N_{1}} (\frac{y_{0}}{w_{0}}) \exp (- \frac{ρ_{0}^{2}}{w_{0}^{2}}) \\ 0 \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix}] .

\begin{array}{l} E (ρ, z) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d^{2} k \exp (i k \cdot ρ) \exp (i k_{e z} z) (\begin{matrix} {\tilde{E}}_{x} (k) \\ - \frac{k_{x} k_{y}}{k_{0}^{2} n_{o}^{2} - k_{x}^{2}} {\tilde{E}}_{x} (k) \\ - \frac{k_{e z} k_{x}}{k_{0}^{2} n_{o}^{2} - k_{x}^{2}} {\tilde{E}}_{x} (k) \end{matrix}) \\ \begin{matrix}  \end{matrix} + \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d^{2} k \exp (i k \cdot ρ) \exp (i k_{o z} z) (\begin{matrix} 0 \\ \frac{k_{x} k_{y}}{k_{0}^{2} n_{o}^{2} - k_{x}^{2}} {\tilde{E}}_{x} (k) + {\tilde{E}}_{y} (k) \\ - \frac{k_{y}}{k_{o z}} [\frac{k_{x} k_{y}}{k_{0}^{2} n_{o}^{2} - k_{x}^{2}} {\tilde{E}}_{x} (k) + {\tilde{E}}_{y} (k)] \end{matrix}), \end{array}

{\tilde{E}}_{j} (k) = \frac{1}{{(2 π)}^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E_{j} (ρ_{0}, 0) \exp [- i (k_{x} x_{0} + k_{y} y_{0})] d x_{0} d y_{0},

k_{e z} = {[k_{o}^{2} n_{e}^{2} - (n_{e}^{2} / n_{o}^{2}) k_{x}^{2} - k_{y}^{2}]}^{1 / 2}, k_{o z} = {(k_{o}^{2} n_{0}^{2} - k_{x}^{2} - k_{y}^{2})}^{1 / 2} .

\begin{array}{l} E (ρ, z) = \exp (i k_{0} n_{e} z) \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d^{2} k \exp (i k \cdot ρ) \exp (- i \frac{n_{e}^{2} k_{x}^{2} + n_{o}^{2} k_{y}^{2}}{2 k_{0} n_{e} n_{o}^{2}} z) (\begin{matrix} {\tilde{E}}_{x} (k) \\ - \frac{k_{x} k_{y}}{k_{0}^{2} n_{o}^{2}} {\tilde{E}}_{x} (k) \\ - \frac{n_{e} k_{x}}{k_{0}^{} n_{o}^{2}} {\tilde{E}}_{x} (k) \end{matrix}) \\ \begin{matrix}  \end{matrix} + \exp (i k_{0} n_{o} z) \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d^{2} k \exp (i k \cdot ρ) \exp (- i \frac{k_{x}^{2} + k_{y}^{2}}{2 k_{0} n_{o}} z) (\begin{matrix} 0 \\ \frac{k_{x} k_{y}}{k_{0}^{2} n_{o}^{2}} {\tilde{E}}_{x} (k) + {\tilde{E}}_{y} (k) \\ - \frac{k_{y}}{k_{o} n_{o}} {\tilde{E}}_{y} (k) \end{matrix}) . \end{array}

E_{x} (ρ, z) = \frac{k_{0} n_{o}}{2 π i z} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E_{x} (ρ_{0}, 0) Λ_{e} (ρ, ρ_{0}) d x_{0} d y_{0},

\begin{array}{l} E_{y} (ρ, z) = \frac{i k_{0} n_{o}}{2 π z^{3}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E_{x} (ρ_{0}, 0) (x - x_{0}) (y - y_{0}) [Λ_{e} (ρ, ρ_{0}) - Λ_{o} (ρ, ρ_{0})] d x_{0} d y_{0} \\ \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} & + \end{matrix} \frac{k_{0} n_{o}}{2 π i z} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E_{y} (ρ_{0}, 0) Λ_{o} (ρ, ρ_{0}) d x_{0} d y_{0}, \end{array}

E_{z} (ρ, z) = \frac{i k_{0} n_{o}}{2 π z^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} [E_{x} (ρ_{0}, 0) (x - x_{0}) Λ_{e} (ρ, ρ_{0}) + E_{y} (ρ_{0}, 0) (y - y_{0}) Λ_{o} (ρ, ρ_{0})] d x_{0} d y_{0},

Λ_{e} (ρ, ρ_{0}) = \exp (i k_{0} n_{e} z) \exp {- \frac{k_{0}}{2 i z n_{e}} [n_{o}^{2} {(x - x_{0})}^{2} + n_{e}^{2} {(y - y_{0})}^{2}]},

Λ_{o} (ρ, ρ_{0}) = \exp (i k_{0} n_{o} z) \exp {- \frac{k_{0} n_{o}}{2 i z} [{(x - x_{0})}^{2} + {(y - y_{0})}^{2}]} .

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

x H_{M} (x) = \frac{1}{2} H_{M + 1} (x) + M H_{M - 1} (x),

E_{x} (ρ, z) = \frac{{(- 1)}^{n} k_{0} n_{o}}{2^{2 n + m + 1} n! i π z} \exp (i e k_{0} n_{o} z) \sum_{l = 0}^{n} \sum_{s = 0}^{[m / 2]} {(- 1)}^{s} (\begin{matrix} n \\ l \end{matrix}) (\begin{matrix} m \\ 2 s \end{matrix}) T_{M_{1}} (x) U_{N_{1}} (y),

\begin{array}{l} E_{y} (ρ, z) = \frac{{(- 1)}^{n} i k_{0} n_{o}}{2^{2 n + m + 1} n! π z^{3}} \sum_{l = 0}^{n} \sum_{s = 0}^{[m / 2]} {(- 1)}^{s} (\begin{matrix} n \\ l \end{matrix}) (\begin{matrix} m \\ 2 s \end{matrix}) {\exp (i e k_{0} n_{o} z) {x y T_{M_{1}} (x) U_{N_{1}} (y) - w_{0} x \\ \begin{matrix}  \end{matrix} \times T_{M_{1}} (x) [0.5 U_{N_{1} + 1} (y) + N_{1} U_{N_{1} - 1} (y)] - w_{0} y U_{N_{1}} (y) [0.5 T_{M_{1} + 1} (x) + M_{1} T_{M_{1} - 1} (x)] \\ \begin{matrix}  \end{matrix} + w_{0}^{2} [0.5 T_{M_{1} + 1} (x) + M_{1} T_{M_{1} - 1} (x)] [0.5 U_{N_{1} + 1} (y) + N_{1} U_{N_{1} - 1} (y)]} - \exp (i k_{0} n_{o} z) {x y \\ \begin{matrix}  \end{matrix} \times V_{M_{1}} (x) V_{N_{1}} (y) - w_{0} x V_{M_{1}} (x) [0.5 V_{N_{1} + 1} (y) + N_{1} V_{N_{1} - 1} (y)] - w_{0} y V_{N_{1}} (y) [0.5 V_{M_{1} + 1} (x) \\ \begin{matrix}  \end{matrix} + M_{1} V_{M_{1} - 1} (x)] + w_{0}^{2} [0.5 V_{M_{1} + 1} (x) + M_{1} V_{M_{1} - 1} (x)] [0.5 V_{N_{1} + 1} (y) + N_{1} V_{N_{1} - 1} (y)]}}, \end{array}

\begin{array}{l} E_{z} (ρ, z) = \frac{{(- 1)}^{n} i k_{0} n_{o}}{2^{2 n + m + 1} n! π z^{2}} \exp (i e k_{0} n_{o} z) \sum_{l = 0}^{n} \sum_{s = 0}^{[m / 2]} {(- 1)}^{s} (\begin{matrix} n \\ l \end{matrix}) (\begin{matrix} m \\ 2 s \end{matrix}) {x T_{M_{1}} (x) U_{N_{1}} (y) - w_{0} U_{N_{1}} (y) \\ \begin{matrix} \begin{matrix} \times \end{matrix} \end{matrix} [0.5 T_{M_{1} + 1} (x) + M_{1} T_{M_{1} - 1} (x)]}, \end{array}

T_{μ} (x) = w_{0} \sqrt{π α} {(1 - α)}^{μ / 2} \exp (- a x^{2}) H_{μ} [\frac{α z_{r} n_{o}^{} x}{i e z w_{0} {(1 - α)}^{1 / 2}}],

U_{μ} (y) = w_{0} \sqrt{π β} {(1 - β)}^{μ / 2} \exp (- b y^{2}) H_{μ} [\frac{β z_{r} e n_{o} y}{i z w_{0} {(1 - β)}^{1 / 2}}],

V_{μ} (j) = w_{0} \sqrt{π γ} {(1 - γ)}^{μ / 2} \exp (- c j^{2}) H_{μ} [\frac{γ z_{r} n_{0} j}{i z w_{0} {(1 - γ)}^{1 / 2}}],

e = \frac{n_{e}}{n_{o}}, z_{r} = \frac{k_{0} w_{0}^{2}}{2}, α = {(1 - \frac{i z_{r} n_{o}^{}}{e z})}^{- 1}, β = {(1 - \frac{i e z_{r} n_{o}}{z})}^{- 1}, γ = {(1 - \frac{i z_{r} n_{0}}{z})}^{- 1},

a = - α {(\frac{z_{r} n_{o}^{}}{i e z w_{0}})}^{2} + \frac{k_{0} n_{o}^{}}{2 i e z}, b = - β {(\frac{e z_{r} n_{o}}{i z w_{0}})}^{2} + \frac{e k_{0} n_{o}}{2 i z}, c = - γ {(\frac{z_{r} n_{o}}{i z w_{0}})}^{2} + \frac{k_{0} n_{o}}{2 i z} .

[\begin{matrix} E_{x} (ρ_{0}, 0) \\ E_{y} (ρ_{0}, 0) \end{matrix}] = [\begin{matrix} {(\frac{ρ_{0}}{w_{0}})}^{m} L_{n}^{m} (\frac{ρ_{0}^{2}}{w_{0}^{2}}) \exp (- \frac{ρ_{0}^{2}}{w_{0}^{2}}) \sin (m φ_{0}) \\ 0 \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix}] .

ρ_{0}^{m} L_{n}^{m} (ρ_{0}^{2}) \sin (m φ_{0}) = \frac{{(- 1)}^{n}}{2^{2 n + m} n!} \sum_{l = 0}^{n} \sum_{s = 0}^{[(m - 1) / 2]} {(- 1)}^{s} (\begin{matrix} n \\ l \end{matrix}) (\begin{matrix} m \\ 2 s + 1 \end{matrix}) H_{M_{1} - 1} (x_{0}) H_{N_{1} + 1} (y_{0}),

[\begin{matrix} E_{x} (ρ_{0}, 0) \\ E_{y} (ρ_{0}, 0) \end{matrix}] = [\begin{matrix} \frac{{(- 1)}^{n}}{2^{2 n + m} n!} \sum_{l = 0}^{n} \sum_{s = 0}^{[(m - 1) / 2]} {(- 1)}^{s} (\begin{matrix} n \\ l \end{matrix}) (\begin{matrix} m \\ 2 s + 1 \end{matrix}) H_{M_{1} - 1} (\frac{x_{0}}{w_{0}}) H_{N_{1} + 1} (\frac{y_{0}}{w_{0}}) \exp (- \frac{ρ_{0}^{2}}{w_{0}^{2}}) \\ 0 \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix}] .

E_{x} (ρ, z) = \frac{{(- 1)}^{n} k_{0} n_{o}}{2^{2 n + m + 1} n! i π z} \exp (i e k_{0} n_{o} z) \sum_{l = 0}^{n} \sum_{s = 0}^{[(m - 1) / 2]} {(- 1)}^{s} (\begin{matrix} n \\ l \end{matrix}) (\begin{matrix} m \\ 2 s + 1 \end{matrix}) T_{M_{1} - 1} (x) U_{N_{1} + 1} (y),

\begin{array}{l} E_{y} (ρ, z) = \frac{{(- 1)}^{n} i k_{0} n_{o}}{2^{2 n + m + 1} n! π z^{3}} \sum_{l = 0}^{n} \sum_{s = 0}^{[(m - 1) / 2]} {(- 1)}^{s} (\begin{matrix} n \\ l \end{matrix}) (\begin{matrix} m \\ 2 s + 1 \end{matrix}) {\exp (i e k_{0} n_{o} z) {x y T_{M_{1} - 1} (x) U_{N_{1 + 1}} (y) \\ \begin{matrix}  \end{matrix} - w_{0} x T_{M_{1} - 1} (x) [0.5 U_{N_{1} + 2} (y) + (N_{1} + 1) U_{N_{1}} (y)] - w_{0} y U_{N_{1} + 1} (y) [0.5 T_{M_{1}} (x) + (M_{1} - 1) \\ \begin{matrix}  \end{matrix} \times T_{M_{1} - 2} (x)] + w_{0}^{2} [0.5 T_{M_{1}} (x) + (M_{1} - 1) T_{M_{1} - 2} (x)] [0.5 U_{N_{1} + 2} (y) + (N_{1} + 1) U_{N_{1}} (y)]} \\ \begin{matrix}  \end{matrix} - \exp (i k_{0} n_{o} z) {x y V_{M_{1} - 1} (x) V_{N_{1} + 1} (y) - w_{0} x V_{M_{1} - 1} (x) [0.5 V_{N_{1} + 2} (y) + (N_{1} + 1) V_{N_{1}} (y)] \\ \begin{matrix}  \end{matrix} - w_{0} y V_{N_{1} + 1} (y) [0.5 V_{M_{1}} (x) + (M_{1} - 1) V_{M_{1} - 2} (x)] + w_{0}^{2} [0.5 V_{M_{1}} (x) + (M_{1} - 1) V_{M_{1} - 2} (x)] \\ \begin{matrix}  \end{matrix} \times [0.5 V_{N_{1} + 2} (y) + (N_{1} + 1) V_{N_{1}} (y)]}}, \end{array}

\begin{array}{l} E_{z} (ρ, z) = \frac{{(- 1)}^{n} i k_{0} n_{o}}{2^{2 n + m + 1} n! π z^{2}} \exp (i e k_{0} n_{o} z) \sum_{l = 0}^{n} \sum_{s = 0}^{[(m - 1) / 2]} {(- 1)}^{s} (\begin{matrix} n \\ l \end{matrix}) (\begin{matrix} m \\ 2 s + 1 \end{matrix}) {x T_{M_{1} - 1} (x) U_{N_{1} + 1} (y) \\ \begin{matrix}  \end{matrix} - w_{0} U_{N_{1} + 1} (y) [0.5 T_{M_{1}} (x) + (M_{1} - 1) T_{M_{1} - 2} (x)]} . \end{array}

Abstract

1. Introduction

2. Theoretical derivations

3. Numerical calculations and analyses

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (10)

Equations (29)

Optics Express