Polarization evolution of radially polarized partially coherent vortex fields: role of Gouy phase of Laguerre–Gauss beams

R. Martínez-Herrero; F. Prado

doi:10.1364/OE.23.005043

Optics Express
Vol. 23,
Issue 4,
pp. 5043-5051
(2015)
•https://doi.org/10.1364/OE.23.005043

Polarization evolution of radially polarized partially coherent vortex fields: role of Gouy phase of Laguerre–Gauss beams

R. Martínez-Herrero and F. Prado

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R. Martínez-Herrero and F. Prado, "Polarization evolution of radially polarized partially coherent vortex fields: role of Gouy phase of Laguerre–Gauss beams," Opt. Express 23, 5043-5051 (2015)

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- Physical Optics
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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: January 2, 2015
- Manuscript Accepted: February 11, 2015
- Published: February 18, 2015

Abstract

In the framework of the paraxial approximation, we derive the analytical expressions for describing the effect of the Gouy phase of Laguerre–Gauss beams on the polarization evolution of partially coherent vortex fields whose electric field vector at some transverse plane exhibits a radially polarized behavior. At each transverse plane, the polarization distribution across the beam profile is characterized by means of the percentage of irradiance associated with the radial or azimuthal components. The propagation laws for these percentages are also presented. As an illustrative example, we analyze a radially polarized partially coherent vortex beam.

1. Introduction

Radially polarized vortex beams have attracted increasing attention in recent years as a consequence of their analytical and practical possibilities in applications such as lithography, optical trapping, optical data storage, and confocal microscopy [1–7]. In this context, it is noteworthy that while a beam can exhibit a radial (or azimuthal) polarization distribution throughout the initial cross-section, in general, it does not retain this characteristic (radial or azimuthal) polarization upon paraxial propagation [8].

In this study, we attempt to analyze the evolution of the polarization structure of free-propagating partially coherent paraxial vortex beams whose electric field vector at some transverse plane exhibits a purely radial polarization distribution. A number of parameters have been proposed to characterize the coherence and polarization features of a paraxial light field [9]. In this work, we focus on describing the polarization features by means of global parameters that have been recently introduced in the literature [9,10]. Interestingly, we observe that the Gouy phase of the Laguerre–Gauss modes (GPLG) plays an essential role in the polarization evolution of partially coherent multimode vortex beams.

This paper is organized as follows. In Section 2, the key definitions concerning the study are provided. The evolution of the polarization structure of partially coherent, radially polarized paraxial vortex beams is discussed in Section 3, and an illustrative example of such beams is analyzed. Finally, the main conclusions are summarized in Section 4.

2. Key definitions

Let us consider a light beam propagating essentially along the z axis. Since we utilize the paraxial approach, the longitudinal component is negligible, and the electric field vector is given as

\vec{E} (\vec{r}, z) = (E_{r a d} (\vec{r}, z), E_{a z} (\vec{r}, z))

Here,

E_{r a d}

and

E_{a z}

are the radial and azimuthal field components, respectively, and

\vec{r} = (r \cos θ, r \sin θ),

denotes the position vector at the transversal plane. In this study, we address stationary fields (at least in the wide sense of the term), and we consider the space–frequency domain. Consequently, the coherence-polarization properties of the beam can be obtained from the cross-spectral density matrix

\hat{W}

given by [9,11]

{\hat{W}}_{i j} ({\vec{r}}_{1}, {\vec{r}}_{2}, z) = 〈 E_{i} {({\vec{r}}_{1}, z)}^{*} E_{j} ({\vec{r}}_{2}, z) 〉

Here, the angle brackets symbolize the ensemble averages, the asterisk denotes conjugation, and

i, j = r a d, a z

In order to obtain global information on the evolution of the polarization state, we will handle the overall parameters defined in [9,10], $i, j = r a d, a z$

{\tilde{ρ}}_{j} (z) = \frac{\int_{0}^{\infty} \int_{0}^{2 π} ρ_{j} (r, θ, z) I (r, θ, z) r d r d θ}{\int_{0}^{\infty} \int_{0}^{2 π} I (r, θ, z) r d r d θ}

where

I (r, θ, z)

denotes the irradiance at each point of the beam profile. We have

I (r, θ, z) = < {| \vec{E} (r, θ, z) |}^{2} >

and

ρ_{j} (r, θ, z) = \frac{< {| E_{j} (r, θ, z) |}^{2} >}{< {| \vec{E} (r, θ, z) |}^{2} >} ⥄ ⥄

Thus, we note that

ρ_{j}

provides information about the percentage of the irradiance associated with the radial and azimuthal components. Accordingly, at each transverse plane z, the parameters

{\tilde{ρ}}_{r a d}

and

{\tilde{ρ}}_{a z}

allow the characterization of the overall radial and azimuthal polarization content of the vectorial field upon averaging

ρ_{r a d} (r, θ, z)

over the region of the beam cross-section wherein the irradiance is significant. Moreover, since

0 \leq {\tilde{ρ}}_{r a d}, {\tilde{ρ}}_{a z} \leq 1,

and

{\tilde{ρ}}_{r a d} + {\tilde{ρ}}_{a z} = 1

, it is sufficient to analyze the behavior of only one of these two parameters.

3. Radially polarized partially coherent vortex beams

Let us now consider a well-behaved radially polarized beam at $z = 0$ . In this case, the cross-spectral density matrix is given as [9, 11]

\hat{W} ({\vec{r}}_{1}, {\vec{r}}_{2}, 0) = Γ ({\vec{r}}_{1}, {\vec{r}}_{2}, 0) r_{1} r_{2} (\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix})

Here,

Γ

represents a nonnegative definite function satisfying the relation

Γ^{*} ({\vec{r}}_{1}, {\vec{r}}_{2}, 0) = Γ ({\vec{r}}_{2}, {\vec{r}}_{1}, 0)

. For this class of fields, the electromagnetic degree of coherence

μ ({\vec{r}}_{1}, {\vec{r}}_{2}, 0)

[12] takes the form

μ^{2} ({\vec{r}}_{1}, {\vec{r}}_{2}, 0) = \frac{{| Γ ({\vec{r}}_{1}, {\vec{r}}_{2}, 0) |}^{2}}{Γ ({\vec{r}}_{1}, {\vec{r}}_{1}, 0) Γ ({\vec{r}}_{2}, {\vec{r}}_{2}, 0)}

By means of a procedure that is analogous to the one developed in reference 13, it can be proved that for this type of field, the cross-spectral density matrix in a given plane

z

can be written as

W_{r a d, r a d} ({\vec{r}}_{1}, {\vec{r}}_{2}, z) = [r_{1} r_{2} + \frac{i z}{k} (r_{2} \frac{\partial}{\partial r_{1}} - r_{1} \frac{\partial}{\partial r_{2}}) + \frac{z^{2}}{k^{2}} \frac{\partial^{2}}{\partial r_{1} \partial r_{2}}] Γ ({\vec{r}}_{1}, {\vec{r}}_{2}, z)

W_{a z, a z} ({\vec{r}}_{1}, {\vec{r}}_{2}, z) = \frac{z^{2}}{k^{2} r_{1} r_{2}} \frac{\partial^{2}}{\partial θ_{1} \partial θ_{2}} Γ ({\vec{r}}_{1}, {\vec{r}}_{2}, z)

W_{r a d, a z} ({\vec{r}}_{1}, {\vec{r}}_{2}, z) = W_{a z, r a d}^{*} ({\vec{r}}_{2}, {\vec{r}}_{1}, z) = (- \frac{i z}{k} r_{1} r_{2} \frac{\partial}{\partial θ_{2}} + \frac{z^{2}}{k^{2} r_{2}} \frac{\partial^{2}}{\partial r_{1} \partial θ_{2}}) Γ ({\vec{r}}_{1}, {\vec{r}}_{2}, z)

Equations (8)-(10) indicate in general that at each transverse plane defined by

z \neq 0

, the beam is no longer radially polarized. In order to obtain global information on the evolution of the polarization characteristics, we focus on the overall parameter

{\tilde{ρ}}_{a z}

. From Eqs. (3), (6), and (9), we obtain

{\tilde{ρ}}_{a z} (z) = \frac{z^{2} \int_{0}^{\infty} \int_{0}^{2 π} r^{- 2} {(\frac{\partial^{2} Γ ({\vec{r}}_{1}, {\vec{r}}_{2}, z)}{\partial θ_{1} \partial θ_{2}})}_{r_{1} = r_{2} = r} r d r d θ}{k^{2} \int_{0}^{\infty} \int_{0}^{2 π} r^{2} Γ (\vec{r}, \vec{r}, 0) r d r d θ}

Let us now consider fields with a spiral charge m, i.e., beams of the form

Γ ({\vec{r}}_{1}, {\vec{r}}_{2}, z) = Γ (r_{1}, r_{2}, z) e^{i m (θ_{2} - θ_{1})}

A suitable basis for describing these fields is the set of Laguerre–Gauss modes defined as [14]

u_{p}^{m} (r, θ, z) = f_{p}^{m} (r, z) e^{i m θ}

Where,

f_{p}^{m} (r, z) = \frac{a_{m p}}{ω (z)} {(\frac{\sqrt{2} r}{ω (z)})}^{| m |} L_{p}^{| m |} (\frac{2 r^{2}}{ω {(z)}^{2}}) e^{- \frac{i k r^{2}}{2 q (z)}} e^{i φ_{m p} (z)}

Here,

ω (z)

denotes the beam size,

q (z)

the complex beam parameter,

L_{p}^{| m |}

the associated Laguerre polynomial,

a_{m p}

the normalization factor, and

φ_{m p} (z)

the Gouy phase shift defined as

φ_{m p} (z) = \frac{(2 p + m + 1) α (z)}{ω (z)}

with

\tan (α (z)) = \frac{z}{z_{0}},

wherein

z_{0}

denotes the Rayleigh range.

By using this orthonormal basis, the function $Γ$ given in Eq. (12) can be written as follows

Γ ({\vec{r}}_{1}, {\vec{r}}_{2}, z) = \sum_{p, q}^{} B_{p q}^{m} u_{p}^{m} {(r_{1}, θ_{1}, z)}^{*} u_{q}^{m} (r_{2}, θ_{2}, z)

In the above expansion, the coefficients

B_{p q}^{m}

are defined by

B_{p q}^{m} = \int_{0}^{\infty} \int_{0}^{\infty} Γ (r_{1}, r_{2}, 0) f_{p}^{m} (r_{1}, 0) f_{q}^{m} {(r_{2}, 0)}^{*} r_{1} r_{2} d r_{1} d r_{2}

From Eq. (18) and upon taking into account the properties of the function

Γ

, it can be proven that

0 \leq {| B_{p, q}^{m} |}^{2} \leq B_{p, p}^{m} B_{q, q}^{m}

From the physical point of view,

B_{p p}^{m}

provides information about the weight of the p mode in the beam, and

\frac{{| B_{p q}^{m} |}^{2}}{B_{p p}^{m} B_{q q}^{m}}

can be understood as a measure of the coherence between the modes. In the remainder of the paper, we define the modes p and q as coherent (incoherent) when

{| B_{p, q}^{m} |}^{2} = B_{p, p}^{m} B_{q, q}^{m}

(B_{p, q}^{m} = δ_{p, q} B_{p, p}^{m})

; in all other cases, the modes are said to be partially coherent.

Taking Eqs. (11) and (17) into account, it can be proved (see Appendix) that for vortex beams with a topological charge m, the parameter ${\tilde{ρ}}_{a z} (z)$ can be written as

{\tilde{ρ}}_{a z} (z) = \frac{m^{2} z^{2}}{z^{2} + z^{2}_{0}} \frac{\sum_{p, q}^{} B_{p q}^{m} a_{m p} a_{m q} I_{m p q} e^{- 2 i α (z) (p - q)}}{\sum_{p, q}^{} B_{p q}^{m} a_{m p} a_{m q} γ_{m p q}}

with

I_{m p q} = \int_{0}^{\infty} x^{| m | - 1} L_{p}^{| m |} (x) L_{q}^{| m |} (x) e^{- x} d x = \sum_{b = 0}^{\min (p, q)} \frac{(| m | - 1 + b)!}{b!}

γ_{m p q} = \int_{0}^{\infty} x^{| m | + 1} L_{p}^{| m |} (x) L_{q}^{| m |} (x) e^{- x} d x = (| m | + 2 p + 1) \frac{(| m | + p)!}{p!} δ_{p q} - δ_{| p - q | = 1} \frac{(| m | + p)!}{(p - 1)!}

Equation (20) is one of the main results of this paper. It enables us to analyze the evolution of the azimuthal polarization content in terms of the intramodal coherence and the Gouy phase of each Laguerre–Gauss mode (GPLG).

In particular, certain interesting conclusions can be inferred about the role of the GPLG in the overall polarization features of the beam. For monomodal beams, i.e., for beams whose spatial structure is of the form

Γ ({\vec{r}}_{1}, {\vec{r}}_{2}, z) = B_{p p}^{m} u_{p}^{m} {(r_{1}, θ_{1}, z)}^{*} u_{p}^{m} (r_{2}, θ_{2}, z),

the parameter

{\tilde{ρ}}_{a z} (z)

takes on the simple form

{\tilde{ρ}}_{a z} (z) = \frac{m^{2} z^{2} I_{m p p}}{(z^{2} + z^{2}_{0}) γ_{m p p}}

From Eq. (24), it can be inferred that the overall azimuthal polarization content of the field does not depend on the GPLG, and that it monotonically increases with

z

_, reaching the asymptotic value

\frac{m^{2} I_{m p p}}{γ_{m p p}}

. Subsequently, for larges values of topological charge m and for large distances (in comparison with the Rayleigh range), the beam tends to be azimuthally polarized. Figure 1a illustrates this behavior. Moreover,

{\tilde{ρ}}_{a z}

is an increasing function of the topological charge (for fixed

p

), as shown in Fig. 1b.

Fig. 1 Monomodal beams: (a) asymptotic behavior of the parameter ${\tilde{ρ}}_{a z}$ as function of spiral charge m, (b) parameter ${\tilde{ρ}}_{a z}$ in terms of propagation distance $z$ in units of Rayleigh range $z_{0}$ . The p index is fixed as $p = 1$ , and the topological charge m takes on the values of 1, 3, and 4.

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When the Laguerre–Gauss functions are the eigenfunctions of $Γ (r_{1}, r_{2}, z)$ [11, 15,16], i.e., the beam can be expressed as the incoherent superposition of Laguerre–Gauss modes, then $B_{p, q}^{m} = δ_{p, q} B_{p, p}^{m}$ , and the parameter ${\tilde{ρ}}_{a z} (z)$ behaves in a manner similar to that observed in the monomode case. In other multimodal situations, the structure of ${\tilde{ρ}}_{a z} (z)$ depends on the intramodal coherence and the spatial structure of the involved modes. In this case, the GPLG plays an essential role in the evolution of the polarization of the beam. To illustrate this behavior, we analyze the superposition of two modes.

Let us consider a radially polarized beam at the initial plane defined by z = 0, characterized by a function $Γ$ given as

\begin{array}{l} Γ ({\vec{r}}_{1}, {\vec{r}}_{2}, 0) = \\ u_{q}^{m} {(r_{1}, θ_{1}, 0)}^{*} u_{q}^{m} (r_{2}, θ_{2}, 0) + u_{p}^{m} {(r_{1}, θ_{1}, 0)}^{*} u_{p}^{m} (r_{2}, θ_{2}, 0) \\ + \cos τ (u_{q}^{m} (r_{1}, θ_{1}, 0) * u_{p}^{m} (r_{2}, θ_{2}, 0) + u_{p}^{m} (r_{1}, θ_{1}, 0) * u_{q}^{m} (r_{2}, θ_{2}, 0)) \end{array}

Then we are dealing with a beam with topological charge m that is a superposition of two spatial modes

p

and

q

(p > q)

. By selecting the value of

τ

(0 \leq τ \leq \frac{π}{2})

_, we can cover the range from coherent

(τ = 0)

to incoherent (

τ = \frac{π}{2}

) mode superposition. In this case, by using Eqs. (20) and (25), we obtain the following expression for the parameter

{\tilde{ρ}}_{a z} (z)

{\tilde{ρ}}_{a z} (z) = \frac{m^{2} z^{2}}{z^{2} + z^{2}_{0}} (\frac{ρ_{0} + 2 ρ_{1} \cos τ \cos (2 (p - q) α (z))}{σ_{0} - 2 σ_{1} \cos τ})

Where

ρ_{0} = a_{m p p}^{2} I_{m p p} + a_{m q q}^{2} I_{m q q}

ρ_{1} = 2 \sqrt{a_{m p p} a_{m q q}} I_{m p q}

σ_{0} = a_{m p p}^{2} γ_{m p p} + a_{m q q}^{2} γ_{m q q}

σ_{1} = 2 \sqrt{a_{m p p} a_{m q q}} \frac{(m + p)!}{(p - 1)!} δ_{| p - q | = 1}

Consequently, from Eqs. (26) and (30), we have that for

p = q + 1

, the overall content of azimuthal polarization is independent of the coherence between the modes at a distance

z_{1}

, as given by

\cos (2 α (z_{1})) = - \frac{σ_{1} ρ_{0}}{σ_{0} ρ_{1}}

However, since for

p > q + 1

we have

σ_{1} = 0

, there exist a number of distances, namely,

z_{k}

k = 1... (p - q)

_{such that}

z_{k} = z_{0} \tan (\frac{(2 k - 1) π}{4 (p - q)})

for which

\cos (2 (p - q) α (z)) = 0

, and therefore, the overall content of azimuthal polarization is independent of the coherence between the modes.

In both cases, in the range $0 < z < z_{k}$ , the behavior of ${\tilde{ρ}}_{a z} (z)$ depends on the intermodal coherence and the GPLG. For distances $z > > z_{k}$ , the azimuthal polarization content reaches an asymptotic value that depends on the intermodal coherence and spiral charge. In particular, for odd $p - q$ , ${\tilde{ρ}}_{a z}$ increases with $τ$ _, and therefore, low coherence between modes implies greater “destruction” of the initial radial polarization. In contrast, for even $p - q$ _{, ${\tilde{ρ}}_{a z}$} decreases with $τ$ , and consequently, low intermodal coherence implies a small degree of destruction of the initial radial polarization. These results are illustrated in Figs. 2a and 2b.

Fig. 2 (a) Parameter ${\tilde{ρ}}_{a z}$ versus propagation distance $z$ in units of Rayleigh range for a beam (superposition of two spatial modes p = 1, q = 0) with topological charge m = 2. The modes are assumed to be coherent ( $τ = 0$ ), partially coherent ( $τ = \frac{π}{3}$ ), and incoherent ( $τ = \frac{π}{2}$ ). (b) Parameter ${\tilde{ρ}}_{a z}$ versus propagation distance $z$ in units of Rayleigh range for a beam (superposition of two spatial modes p = 2, q = 0) with topological charge m = 2. The modes are assumed to be coherent ( $τ = 0$ ), partially coherent ( $τ = \frac{π}{3}$ ), and incoherent ( $τ = \frac{π}{2}$ ).

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4. Concluding remarks

Within the paraxial approximation, we derived the analytical expressions for describing the polarization evolution of partially coherent vortex beams whose electric field vector at some transverse plane exhibits a radially polarized behavior. We showed that the Gouy phase of the Laguerre–Gauss modes plays an essential role in the polarization evolution of partially coherent multimode vortex beams. In fact, when there are only two modes involved, the GPLG affects the asymptotic behavior of the polarization, and causes that the polarization becomes independent of the intermodal coherence for certain specific distances.

Appendix

From Eq. (11), in order to evaluate the parameter ${\tilde{ρ}}_{a z} (z)$ , it is required to calculate the following integrals

A (z) = \int_{0}^{\infty} \int_{0}^{2 π} r^{- 2} {(\frac{\partial^{2} Γ ({\vec{r}}_{1}, {\vec{r}}_{2}, z)}{\partial θ_{1} \partial θ_{2}})}_{r_{1} = r_{2} = r} r d r d θ

B = \int_{0}^{\infty} \int_{0}^{2 π} r^{2} Γ (\vec{r}, \vec{r}, 0) r d r d θ

Taking into account Eq. (17), we have

\begin{array}{l} A (z) = \frac{2 π m^{2}}{ω^{2} (z)} \\ \sum_{p, q} B_{p q}^{m} a_{m p} a_{m q} e^{- i (φ_{m p} (z) - φ_{m q} (z))} \int_{0}^{\infty} r^{- 2} {(\frac{\sqrt{2} r}{ω (z)})}^{2 | m |} L_{p}^{| m |} (\frac{2 r^{2}}{ω^{2} (z)}) L_{q}^{| m |} (\frac{2 r^{2}}{ω^{2} (z)}) \exp (- \frac{2 r^{2}}{ω^{2} (z)}) r d r \end{array}

and

\begin{array}{l} B = \frac{2 π}{ω^{2} (0)} \\ \sum_{p, q} B_{p q}^{m} a_{m p} a_{m q} \int_{0}^{\infty} r^{2} {(\frac{\sqrt{2} r}{ω (0)})}^{2 | m |} L_{p}^{| m |} (\frac{2 r^{2}}{ω^{2} (0)}) L_{q}^{| m |} (\frac{2 r^{2}}{ω^{2} (0)}) \exp (- \frac{2 r^{2}}{ω^{2} (0)}) r d r \end{array}

Upon using the adimensional variables

x = \frac{2 r^{2}}{ω^{2} (z)}

and

x_{0} = \frac{2 r^{2}}{ω^{2} (0)}

, Eqs. (35) and (36) become

\begin{array}{l} A (z) = \frac{π m^{2}}{ω^{2} (z)} \\ \sum_{p, q} B_{p q}^{m} a_{m p} a_{m q} e^{- i (φ_{m p} (z) - φ_{m q} (z))} \int_{0}^{\infty} x^{| m | - 1} L_{p}^{| m |} (x) L_{q}^{| m |} (x) \exp (- x) d x \end{array}

B = \frac{π ω^{2} (0)}{4} \sum_{p, q} B_{p q}^{m} a_{m p} a_{m q} \int_{0}^{\infty} x_{0}^{| m | + 1} L_{p}^{| m |} (x_{0}) L_{q}^{| m |} (x_{0}) \exp (- x_{0}) d x_{0}

Considering the properties of the associated Laguerre polynomial [17], it can be proven that

I_{m p q} = \int_{0}^{\infty} x^{| m | - 1} L_{p}^{| m |} (x) L_{q}^{| m |} (x) e^{- x} d x = \sum_{a = 0}^{\min (p, q)} \frac{(| m | - 1 + a)!}{a!},

and

γ_{m p q} = \int_{0}^{\infty} x^{| m | + 1} L_{p}^{| m |} (x) L_{q}^{| m |} (x) e^{- x} d x = (| m | + 2 p + 1) \frac{(| m | + p)!}{p!} δ_{p q} - δ_{| p - q | = 1} \frac{(| m | + p)!}{(p - 1)!}

Finally, since

{\tilde{ρ}}_{a z} (z)

can be written as

{\tilde{ρ}}_{a z} (z) = \frac{z^{2} A (z)}{k^{2} B}

we obtain the following expression for the overall degree of azimuthal polarization:

{\tilde{ρ}}_{a z} (z) = \frac{4 z^{2} m^{2}}{k^{2} ω^{2} (0) ω^{2} (z)} \frac{\sum_{p, q}^{} B_{p, q}^{m} a_{m p} a_{m q} I_{m, p, q} e^{- 2 i α (z) (p - q)}}{\sum_{p, q}^{} B_{p, q}^{m} a_{m p} a_{m q} γ_{m, p, q}}

Subsequently, we obtain Eq. (20) by using the parabolic evolution of

ω^{2} (z)

and the definition of the Rayleigh range.

Acknowledgments

The present work has been supported by the Spanish Ministerio de Ciencia e Innovación projects FIS2010-17543 and FIS2013-46475.

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Fig. 1 Monomodal beams: (a) asymptotic behavior of the parameter

{\tilde{ρ}}_{a z}

as function of spiral charge m, (b) parameter

{\tilde{ρ}}_{a z}

in terms of propagation distance

z

in units of Rayleigh range

z_{0}

. The p index is fixed as

p = 1

, and the topological charge m takes on the values of 1, 3, and 4.

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Fig. 2 (a) Parameter

{\tilde{ρ}}_{a z}

versus propagation distance

z

in units of Rayleigh range for a beam (superposition of two spatial modes p = 1, q = 0) with topological charge m = 2. The modes are assumed to be coherent (

τ = 0

), partially coherent (

τ = \frac{π}{3}

), and incoherent (

τ = \frac{π}{2}

). (b) Parameter

{\tilde{ρ}}_{a z}

versus propagation distance

z

in units of Rayleigh range for a beam (superposition of two spatial modes p = 2, q = 0) with topological charge m = 2. The modes are assumed to be coherent (

τ = 0

), partially coherent (

τ = \frac{π}{3}

), and incoherent (

τ = \frac{π}{2}

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Equations (42)

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\vec{E} (\vec{r}, z) = (E_{r a d} (\vec{r}, z), E_{a z} (\vec{r}, z))

{\hat{W}}_{i j} ({\vec{r}}_{1}, {\vec{r}}_{2}, z) = 〈 E_{i} {({\vec{r}}_{1}, z)}^{*} E_{j} ({\vec{r}}_{2}, z) 〉

{\tilde{ρ}}_{j} (z) = \frac{\int_{0}^{\infty} \int_{0}^{2 π} ρ_{j} (r, θ, z) I (r, θ, z) r d r d θ}{\int_{0}^{\infty} \int_{0}^{2 π} I (r, θ, z) r d r d θ}

I (r, θ, z) = < {| \vec{E} (r, θ, z) |}^{2} >

ρ_{j} (r, θ, z) = \frac{< {| E_{j} (r, θ, z) |}^{2} >}{< {| \vec{E} (r, θ, z) |}^{2} >} ⥄ ⥄

\hat{W} ({\vec{r}}_{1}, {\vec{r}}_{2}, 0) = Γ ({\vec{r}}_{1}, {\vec{r}}_{2}, 0) r_{1} r_{2} (\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix})

μ^{2} ({\vec{r}}_{1}, {\vec{r}}_{2}, 0) = \frac{{| Γ ({\vec{r}}_{1}, {\vec{r}}_{2}, 0) |}^{2}}{Γ ({\vec{r}}_{1}, {\vec{r}}_{1}, 0) Γ ({\vec{r}}_{2}, {\vec{r}}_{2}, 0)}

W_{r a d, r a d} ({\vec{r}}_{1}, {\vec{r}}_{2}, z) = [r_{1} r_{2} + \frac{i z}{k} (r_{2} \frac{\partial}{\partial r_{1}} - r_{1} \frac{\partial}{\partial r_{2}}) + \frac{z^{2}}{k^{2}} \frac{\partial^{2}}{\partial r_{1} \partial r_{2}}] Γ ({\vec{r}}_{1}, {\vec{r}}_{2}, z)

W_{a z, a z} ({\vec{r}}_{1}, {\vec{r}}_{2}, z) = \frac{z^{2}}{k^{2} r_{1} r_{2}} \frac{\partial^{2}}{\partial θ_{1} \partial θ_{2}} Γ ({\vec{r}}_{1}, {\vec{r}}_{2}, z)

W_{r a d, a z} ({\vec{r}}_{1}, {\vec{r}}_{2}, z) = W_{a z, r a d}^{*} ({\vec{r}}_{2}, {\vec{r}}_{1}, z) = (- \frac{i z}{k} r_{1} r_{2} \frac{\partial}{\partial θ_{2}} + \frac{z^{2}}{k^{2} r_{2}} \frac{\partial^{2}}{\partial r_{1} \partial θ_{2}}) Γ ({\vec{r}}_{1}, {\vec{r}}_{2}, z)

{\tilde{ρ}}_{a z} (z) = \frac{z^{2} \int_{0}^{\infty} \int_{0}^{2 π} r^{- 2} {(\frac{\partial^{2} Γ ({\vec{r}}_{1}, {\vec{r}}_{2}, z)}{\partial θ_{1} \partial θ_{2}})}_{r_{1} = r_{2} = r} r d r d θ}{k^{2} \int_{0}^{\infty} \int_{0}^{2 π} r^{2} Γ (\vec{r}, \vec{r}, 0) r d r d θ}

Γ ({\vec{r}}_{1}, {\vec{r}}_{2}, z) = Γ (r_{1}, r_{2}, z) e^{i m (θ_{2} - θ_{1})}

u_{p}^{m} (r, θ, z) = f_{p}^{m} (r, z) e^{i m θ}

f_{p}^{m} (r, z) = \frac{a_{m p}}{ω (z)} {(\frac{\sqrt{2} r}{ω (z)})}^{| m |} L_{p}^{| m |} (\frac{2 r^{2}}{ω {(z)}^{2}}) e^{- \frac{i k r^{2}}{2 q (z)}} e^{i φ_{m p} (z)}

φ_{m p} (z) = \frac{(2 p + m + 1) α (z)}{ω (z)}

\tan (α (z)) = \frac{z}{z_{0}},

Γ ({\vec{r}}_{1}, {\vec{r}}_{2}, z) = \sum_{p, q}^{} B_{p q}^{m} u_{p}^{m} {(r_{1}, θ_{1}, z)}^{*} u_{q}^{m} (r_{2}, θ_{2}, z)

B_{p q}^{m} = \int_{0}^{\infty} \int_{0}^{\infty} Γ (r_{1}, r_{2}, 0) f_{p}^{m} (r_{1}, 0) f_{q}^{m} {(r_{2}, 0)}^{*} r_{1} r_{2} d r_{1} d r_{2}

0 \leq {| B_{p, q}^{m} |}^{2} \leq B_{p, p}^{m} B_{q, q}^{m}

{\tilde{ρ}}_{a z} (z) = \frac{m^{2} z^{2}}{z^{2} + z^{2}_{0}} \frac{\sum_{p, q}^{} B_{p q}^{m} a_{m p} a_{m q} I_{m p q} e^{- 2 i α (z) (p - q)}}{\sum_{p, q}^{} B_{p q}^{m} a_{m p} a_{m q} γ_{m p q}}

I_{m p q} = \int_{0}^{\infty} x^{| m | - 1} L_{p}^{| m |} (x) L_{q}^{| m |} (x) e^{- x} d x = \sum_{b = 0}^{\min (p, q)} \frac{(| m | - 1 + b)!}{b!}

γ_{m p q} = \int_{0}^{\infty} x^{| m | + 1} L_{p}^{| m |} (x) L_{q}^{| m |} (x) e^{- x} d x = (| m | + 2 p + 1) \frac{(| m | + p)!}{p!} δ_{p q} - δ_{| p - q | = 1} \frac{(| m | + p)!}{(p - 1)!}

Γ ({\vec{r}}_{1}, {\vec{r}}_{2}, z) = B_{p p}^{m} u_{p}^{m} {(r_{1}, θ_{1}, z)}^{*} u_{p}^{m} (r_{2}, θ_{2}, z),

{\tilde{ρ}}_{a z} (z) = \frac{m^{2} z^{2} I_{m p p}}{(z^{2} + z^{2}_{0}) γ_{m p p}}

\begin{array}{l} Γ ({\vec{r}}_{1}, {\vec{r}}_{2}, 0) = \\ u_{q}^{m} {(r_{1}, θ_{1}, 0)}^{*} u_{q}^{m} (r_{2}, θ_{2}, 0) + u_{p}^{m} {(r_{1}, θ_{1}, 0)}^{*} u_{p}^{m} (r_{2}, θ_{2}, 0) \\ + \cos τ (u_{q}^{m} (r_{1}, θ_{1}, 0) * u_{p}^{m} (r_{2}, θ_{2}, 0) + u_{p}^{m} (r_{1}, θ_{1}, 0) * u_{q}^{m} (r_{2}, θ_{2}, 0)) \end{array}

{\tilde{ρ}}_{a z} (z) = \frac{m^{2} z^{2}}{z^{2} + z^{2}_{0}} (\frac{ρ_{0} + 2 ρ_{1} \cos τ \cos (2 (p - q) α (z))}{σ_{0} - 2 σ_{1} \cos τ})

ρ_{0} = a_{m p p}^{2} I_{m p p} + a_{m q q}^{2} I_{m q q}

ρ_{1} = 2 \sqrt{a_{m p p} a_{m q q}} I_{m p q}

σ_{0} = a_{m p p}^{2} γ_{m p p} + a_{m q q}^{2} γ_{m q q}

σ_{1} = 2 \sqrt{a_{m p p} a_{m q q}} \frac{(m + p)!}{(p - 1)!} δ_{| p - q | = 1}

\cos (2 α (z_{1})) = - \frac{σ_{1} ρ_{0}}{σ_{0} ρ_{1}}

z_{k} = z_{0} \tan (\frac{(2 k - 1) π}{4 (p - q)})

A (z) = \int_{0}^{\infty} \int_{0}^{2 π} r^{- 2} {(\frac{\partial^{2} Γ ({\vec{r}}_{1}, {\vec{r}}_{2}, z)}{\partial θ_{1} \partial θ_{2}})}_{r_{1} = r_{2} = r} r d r d θ

B = \int_{0}^{\infty} \int_{0}^{2 π} r^{2} Γ (\vec{r}, \vec{r}, 0) r d r d θ

\begin{array}{l} A (z) = \frac{2 π m^{2}}{ω^{2} (z)} \\ \sum_{p, q} B_{p q}^{m} a_{m p} a_{m q} e^{- i (φ_{m p} (z) - φ_{m q} (z))} \int_{0}^{\infty} r^{- 2} {(\frac{\sqrt{2} r}{ω (z)})}^{2 | m |} L_{p}^{| m |} (\frac{2 r^{2}}{ω^{2} (z)}) L_{q}^{| m |} (\frac{2 r^{2}}{ω^{2} (z)}) \exp (- \frac{2 r^{2}}{ω^{2} (z)}) r d r \end{array}

\begin{array}{l} B = \frac{2 π}{ω^{2} (0)} \\ \sum_{p, q} B_{p q}^{m} a_{m p} a_{m q} \int_{0}^{\infty} r^{2} {(\frac{\sqrt{2} r}{ω (0)})}^{2 | m |} L_{p}^{| m |} (\frac{2 r^{2}}{ω^{2} (0)}) L_{q}^{| m |} (\frac{2 r^{2}}{ω^{2} (0)}) \exp (- \frac{2 r^{2}}{ω^{2} (0)}) r d r \end{array}

\begin{array}{l} A (z) = \frac{π m^{2}}{ω^{2} (z)} \\ \sum_{p, q} B_{p q}^{m} a_{m p} a_{m q} e^{- i (φ_{m p} (z) - φ_{m q} (z))} \int_{0}^{\infty} x^{| m | - 1} L_{p}^{| m |} (x) L_{q}^{| m |} (x) \exp (- x) d x \end{array}

B = \frac{π ω^{2} (0)}{4} \sum_{p, q} B_{p q}^{m} a_{m p} a_{m q} \int_{0}^{\infty} x_{0}^{| m | + 1} L_{p}^{| m |} (x_{0}) L_{q}^{| m |} (x_{0}) \exp (- x_{0}) d x_{0}

I_{m p q} = \int_{0}^{\infty} x^{| m | - 1} L_{p}^{| m |} (x) L_{q}^{| m |} (x) e^{- x} d x = \sum_{a = 0}^{\min (p, q)} \frac{(| m | - 1 + a)!}{a!},

γ_{m p q} = \int_{0}^{\infty} x^{| m | + 1} L_{p}^{| m |} (x) L_{q}^{| m |} (x) e^{- x} d x = (| m | + 2 p + 1) \frac{(| m | + p)!}{p!} δ_{p q} - δ_{| p - q | = 1} \frac{(| m | + p)!}{(p - 1)!}

{\tilde{ρ}}_{a z} (z) = \frac{z^{2} A (z)}{k^{2} B}

{\tilde{ρ}}_{a z} (z) = \frac{4 z^{2} m^{2}}{k^{2} ω^{2} (0) ω^{2} (z)} \frac{\sum_{p, q}^{} B_{p, q}^{m} a_{m p} a_{m q} I_{m, p, q} e^{- 2 i α (z) (p - q)}}{\sum_{p, q}^{} B_{p, q}^{m} a_{m p} a_{m q} γ_{m, p, q}}

Abstract

1. Introduction

2. Key definitions

3. Radially polarized partially coherent vortex beams

4. Concluding remarks

Appendix

Acknowledgments

References and links

Cited By

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Equations (42)

Optics Express