Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam

Gaofeng Wu; Yangjian Cai

doi:10.1364/OE.19.008700

1. Introduction

Recently, stochastic electromagnetic beam attracts more and more attention due to its importance in theories of coherence and polarization of light and in some applications, e.g., free-space optical communications, laser radar systems, optical imaging and remote sensing [1–20]. Electromagnetic Gaussian Schell-model (EGSM) beam was introduced as an extension of scalar GSM beam [2–4]. Numerous theoretical and experimental papers relating to EGSM beams have been published in the past several years [1–20]. It was shown that the EGSM beams have advantage over scalar GSM beams for overcoming or reducing the influence of turbulence [15–21], which makes them attractive for free-space optical communications, laser radar systems and remote sensing [21–23]. The radiation force of EGSM beams on a Rayleigh dielectric sphere is explored in [24], and it was found that we can increase trapping ranges by choosing suitable source polarization and coherence, which makes them useful in optical trapping. It was revealed in [25] that one may reduce/enhance the ghost image and its visibility by choosing suitable polarization of an EGSM source. Thus, it is of practical importance to modulate the properties of a stochastic electromagnetic beam. The statistics properties of EGSM beams in a Gaussian cavity were examined in [26–29], and it was found that we can modulate or control the spectral, coherence, polarization and propagation factor of an EGSM beam by a Gaussian cavity. More recently, spatio-temporal coupling of an EGSM pulse interacting with reflecting gratings was explored [30]. Most of previous papers were devoted to stochastic electromagnetic Gaussian Schell-model (EGSM) beam. Only few attentions were paid to stochastic electromagnetic beam with special beam profile [31,32]. Up to now, no results have been reported on generation of stochastic electromagnetic beam with special beam profile, such as stochastic electromagnetic flat-topped or dark hollow beams.

In many applications, such as free-space optical communication, atomic optics, binary optics, optical trapping, material thermal processing and inertial confinement fusion, light beams with special profiles such as flat-topped (FT) beam and dark hollow (DH) beam are required [33–36]. Recently, more and more attention is being paid to partially coherent beams with special beam profiles, such as partially coherent flat-topped beam and partially coherent dark hollow beam [37–52]. Theoretical models for partially coherent FT beam and partially coherent DH beam have been proposed [37–41]. Different methods, such as mode conversion, optical holography, transverse-mode selection, hollow-fiber method, computer-generated holography, optical resonators with binary optical elements, have been developed to generate coherent and partially coherent FT or DH beam experimentally [33,42,43]. Propagation properties of partially coherent FT beam and partially coherent DH beam have been studied in detail [44–50].

In previous literatures, both theoretical and experimental results have demonstrated that phase apertures can be used for selecting the lowest transverse mode in a cavity, and enhancing the fundamental mode volume by a factor as high as 12 [51,52]. A phase aperture can be formed by depositing a dielectric coating through a mask with a single aperture on a substrate of fused quartz [49]. To our knowledge, no results have been reported up until now on the propagation properties of an EGSM beam truncated by a circular phase aperture. In fact, little attention was paid even to the propagation of coherent beams through a phase aperture [53,54]. In this paper, we study the propagation of an EGSM beam truncated by a circular phase aperture in free space. Analytical propagation formula for an EGSM beam truncated by a phase aperture is derived, which provides a convenient and fast way for studying the propagation and transformation of an EGSM beam. We find that the phase aperture can be used to modulate the spectral intensity, polarization and coherence of an EGSM beam, and it can be used to generate electromagnetic partially coherent FT beam and DH beam, which will be useful in some applications, such as optical trapping, material thermal processing and free-space optical communication.

2. Statistics properties of an EGSM beam truncated by a circular phase aperture in free space

Figure 1 shows the schematic diagram of an EGSM beam truncated by a circular phase aperture whose phase delay is ϕ and radius is a propagating in free space. Within the validity of the paraxial approximation, the propagation of a partially coherent beam truncated by a circular phase aperture in free space can be studied with the help of the following generalized Huygens-Fresnel integral [55]

\begin{array}{l} W (ρ_{x 1}, ρ_{y 1}, ρ_{x 2}, ρ_{y 2}, z) = {(\frac{1}{λ z})}^{2} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} T (x_{1}, y_{1}) T^{*} (x_{2}, y_{2}) W (x_{1}, y_{1}, x_{2}, y_{2}, 0) \\ \times \exp [- \frac{i k}{2 z} (x_{1}^{2} - x_{2}^{2} - 2 x_{1} ρ_{x 1} + 2 x_{2} ρ_{x 2} + ρ_{x 1}^{2} - ρ_{x 2}^{2})] \\ \times \exp [- \frac{i k}{2 z} (y_{1}^{2} - y_{2}^{2} - 2 y_{1} ρ_{y 1} + 2 y_{2} ρ_{x 2} + ρ_{y 1}^{2} - ρ_{y 2}^{2})] d x_{1} d x_{2} d y_{1} d y_{2}, \end{array}

where

k = 2 π / λ

is the wave number with λ being the wavelength,

W (x_{1}, y_{1}, x_{2}, y_{2}, 0)

and

W (ρ_{x 1}, ρ_{y 1}, ρ_{x 2}, ρ_{y 2}, z)

are the cross-spectral densities in the input and output planes,

x_{i}, y_{i}

and

ρ_{x i}, ρ_{y i}

are the position coordinates in the input and output planes.

T (x_{1}, y_{1})

is the transmission function of the phase aperture with

T (x_{1}, y_{1}) = \exp (- i ϕ)

for

x_{1}^{2} + y_{1}^{2} \leq a^{2}

and

T (x_{1}, y_{1}) = 1

for

x_{1}^{2} + y_{1}^{2} > a^{2}

.

T (x_{1}, y_{1})

can be expressed in the following alternative form

T (x_{1}, y_{1}) = 1 + [\exp (- i ϕ) - 1] H (x_{1}, y_{1}),

where

H (x_{1}, y_{1})

is the transmission function of a circular hard aperture with

H (x_{1}, y_{1}) = 1

for

x_{1}^{2} + y_{1}^{2} \leq a^{2}

and

H (x_{1}, y_{1}) = 0

for

x_{1}^{2} + y_{1}^{2} > a^{2}

. Then the term

T (x_{1}, y_{1}) T^{*} (x_{2}, y_{2})

in Eq. (1) equals to

\begin{array}{l} T (x_{1}, y_{1}) T^{*} (x_{2}, y_{2}) = 1 + [\exp (- i ϕ) - 1] H (x_{1}, y_{1}) + [\exp (i ϕ) - 1] H^{*} (x_{2}, y_{2}) \\ + [\exp (- i ϕ) - 1] [\exp (i ϕ) - 1] H (x_{1}, y_{1}) H^{*} (x_{2}, y_{2}), \end{array}

H (x_{1}, y_{1})

can be expanded as the following finite sum of complex Gaussian functions [56,57]

H (x_{1}, y_{1}) = \sum_{m = 1}^{M} A_{m} \exp [- \frac{B_{m}}{a_{}^{2}} (x_{1}^{2} + y_{1}^{2})],

where

A_{m}

and

B_{m}

are the expansion and Gaussian coefficients, which can be obtained by optimization computation, a table of

A_{m}

and

B_{m}

can be found in Ref [56]. For a hard aperture, M = 10 assures a very good description of the diffracted beam [56,57]. Although the method of complex Gaussian expansion of a circular hard aperture is an approximately method, both numerical results [56,57] and experimental results [58] have shown that this method is a reliable and fast method for treating the propagation of a truncated beam.

Fig. 1 Schematic diagram of an EGSM beam truncated by a phase aperture propagating in free space.

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After some arrangement, Eq. (1) can be expressed in the following alternative tensor form

W (\tilde{ρ}, z) = \frac{1}{λ^{2} {(\det \tilde{B})}^{1 / 2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} T (r_{1}) T^{*} (r_{2}) W (\tilde{r}, 0) \exp [- \frac{i k}{2} ({\tilde{r}}^{T} {\tilde{B}}^{- 1} \tilde{r} - 2 {\tilde{r}}^{T} {\tilde{B}}^{- 1} \tilde{ρ} + {\tilde{ρ}}^{T} {\tilde{B}}^{- 1} \tilde{ρ})] d \tilde{r},

where

{\tilde{r}}^{T} = (\begin{matrix} r_{1}^{T} & r_{2}^{T} \end{matrix}) = (\begin{matrix} x_{1} & y_{1} & x_{2} & y_{2} \end{matrix})

,

{\tilde{ρ}}^{T} = (\begin{matrix} ρ_{1}^{T} & ρ_{2}^{T} \end{matrix}) = (\begin{matrix} ρ_{x 1} & ρ_{y 1} & ρ_{x 2} & ρ_{y 2} \end{matrix})

,

d \tilde{r} = d x_{1} d x_{2} d y_{1} d y_{2}

, and

\tilde{B} = (\begin{matrix} z I & 0 I \\ 0 I & - z I \end{matrix}) .

Equation (3) can be expressed in the following alternative form

\begin{array}{l} T (r_{1}) T^{*} (r_{2}) = 1 + [\exp (- i ϕ) - 1] \sum_{m = 1}^{M} A_{m} \exp (- \frac{i k}{2} {\tilde{r}}^{T} {\tilde{B}}_{m} \tilde{r}) + [\exp (i ϕ) - 1] \sum_{p = 1}^{M} A_{p}^{*} \exp [- \frac{i k}{2} {\tilde{r}}^{T} {\tilde{B}}_{p} \tilde{r}] \\ + [\exp (- i ϕ) - 1] [\exp (i ϕ) - 1] \sum_{m = 1}^{M} \sum_{p = 1}^{M} A_{m} A_{p}^{*} \exp [- \frac{i k}{2} {\tilde{r}}^{T} {\tilde{B}}_{m p} \tilde{r}], \end{array}

where

{\tilde{B}}_{m} = (\begin{matrix} - \frac{2 i B_{m}}{k a_{}^{2}} I & 0 I \\ 0 I & 0 I \end{matrix}), {\tilde{B}}_{p} = (\begin{matrix} 0 I & 0 I \\ 0 I & - \frac{2 i B_{p}^{*}}{k a_{}^{2}} I \end{matrix}), {\tilde{B}}_{m p} = (\begin{matrix} - \frac{2 i B_{m}}{k a_{}^{2}} I & 0 I \\ 0 I & - \frac{2 i B_{p}^{*}}{k a_{}^{2}} I \end{matrix}) .

here I is a

2 \times 2

identity matrix,

The second-order statistical properties of an EGSM beam is generally characterized by the 2 × 2 cross-spectral density (CSD) matrix $\overset{\leftrightarrow}{W} (r_{1}, r_{2}, 0)$ specified at any two points with position vectors $r_{1}$ and $r_{2}$ in the source plane with elements [1–4,26]

W_{α β}^{} (\tilde{r}, 0) = A_{α} A_{β} B_{α β} \exp [- \frac{i k}{2} {\tilde{r}}^{T} M_{0 α β}^{- 1} \tilde{r}], (α = x, y; β = x, y),

where

A_{α}

is the square root of the spectral density of electric field component

E_{α}

,

B_{α β} = | B_{α β} | \exp (i φ)

is the correlation coefficient between the

E_{x}

and

E_{y}

field components, T stands for vector transposition and

M_{0 α β}^{- 1}

is the

4 \times 4

matrix of the form

M_{0 α β}^{- 1} = (\begin{matrix} \frac{1}{i k} (\frac{1}{2 σ_{a}^{2}} + \frac{1}{δ_{α β}^{2}}) I & \frac{i}{k δ_{α β}^{2}} I \\ \frac{i}{k δ_{α β}^{2}} I & \frac{1}{i k} (\frac{1}{2 σ_{β}^{2}} + \frac{1}{δ_{α β}^{2}}) I \end{matrix}),

where

σ_{α}

is the r.m.s width of the spectral density along α direction,

δ_{x x}

,

δ_{y y}

and

δ_{x y}

are the r.m.s widths of auto-correlation functions of the x component of the field, of the y component of the field and of the mutual correlation function of x and y field components, respectively. In Eqs. (9) and (10) as well as in all the formulas below the explicit dependence of the parameters

σ_{x}

,

σ_{y}

,

δ_{x x}

,

δ_{y y}

and

δ_{x y}

on the frequency was omitted for simplicity.

Substituting Eqs. (7) and (9) into Eq. (5), we obtain (after tedious vector integration and tensor operation) the following expression for the elements of the cross-spectral density matrix of the truncated EGSM beam in the output plane

\begin{array}{l} W_{α β} (\tilde{ρ}, z) = A_{α} A_{β} B_{α β} {\det {[\tilde{I} + \tilde{B} M_{0 α β}^{- 1}]}^{- 1 / 2} \exp [- \frac{i k}{2} {\tilde{ρ}}^{T} M_{1 α β}^{- 1} \tilde{ρ}] \\ + [\exp (- i ϕ) - 1] \sum_{m = 1}^{M} A_{m} \det {[\tilde{I} + \tilde{B} M_{0 α β}^{- 1} + \tilde{B} {\tilde{B}}_{m}]}^{- 1 / 2} \times \exp [- \frac{i k}{2} {\tilde{ρ}}^{T} M_{2 α β}^{- 1} \tilde{ρ}] \\ + [\exp (i ϕ) - 1] \sum_{p = 1}^{M} A_{p}^{*} \det {[\tilde{I} + \tilde{B} M_{0 α β}^{- 1} + \tilde{B} {\tilde{B}}_{p}]}^{- 1 / 2} \exp [- \frac{i k}{2} {\tilde{ρ}}^{T} M_{3 α β}^{- 1} \tilde{ρ}] + \\ [\exp (- i ϕ) - 1] [\exp (i ϕ) - 1] \sum_{m = 1}^{M} \sum_{p = 1}^{M} A_{m} A_{p}^{*} \det {[\tilde{I} + \tilde{B} M_{0 α β}^{- 1} + \tilde{B} {\tilde{B}}_{m p}]}^{- 1 / 2} \exp [- \frac{i k}{2} {\tilde{ρ}}^{T} M_{4 α β}^{- 1} \tilde{ρ}]}, \end{array}

where

\tilde{I}

is a

4 \times 4

identity matrix

\begin{array}{l} M_{1 α β}^{- 1} = {(M_{0 α β} + \tilde{B})}^{- 1}, M_{2 α β}^{- 1} = {[{(M_{0 α β}^{- 1} + {\tilde{B}}_{m})}^{- 1} + \tilde{B}]}^{- 1} \\ M_{3 α β}^{- 1} = {[{(M_{0 ε β}^{- 1} + {\tilde{B}}_{p})}^{- 1} + \tilde{B}]}^{- 1}, M_{4 α β}^{- 1} = {[{(M_{0 α β}^{- 1} + {\tilde{B}}_{m p})}^{- 1} + \tilde{B}]}^{- 1} . \end{array}

The spectral density and the degree of polarization of an EGSM beam at point ρ are defined by the expressions [1]

I (ρ, z) = Tr \overset{\leftrightarrow}{W} (ρ, ρ, z),

and

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

The degree of coherence of the EGSM beam at a pair of transverse points

ρ_{1} and ρ_{2}

is defined by the formula [1]

μ (ρ_{1}, ρ_{2}, z) = \frac{Tr \overset{\leftrightarrow}{W} (ρ_{1}, ρ_{2}, z)}{\sqrt{Tr \overset{\leftrightarrow}{W} (ρ_{1}, ρ_{1}, z) Tr \overset{\leftrightarrow}{W} (ρ_{2}, ρ_{2}, z)}} .

The state of polarization of an EGSM beam can be described with the polarization ellipse associated with the completely polarized portion of the EGSM beam. It is known that the cross-spectral density matrix of an EGSM beam can be locally represented as a sum of a completely polarized beam and a completely unpolarized beam [14,17,27]

\overset{\leftrightarrow}{W} (ρ, ρ, z) = {\overset{\leftrightarrow}{W}}^{(u)} (ρ, ρ, z) + {\overset{\leftrightarrow}{W}}^{(p)} (ρ, ρ, z),

where

{\overset{\leftrightarrow}{W}}^{(u)} (ρ, ρ, z) = (\begin{matrix} A (ρ, ρ, z) & 0 \\ 0 & A (ρ, ρ, z) \end{matrix}), {\overset{\leftrightarrow}{W}}^{(p)} (ρ, ρ, z) = (\begin{matrix} B (ρ, ρ, z) & D (ρ, ρ, z) \\ D^{*} (ρ, ρ, z) & C (ρ, ρ, z) \end{matrix}),

with

\begin{array}{l} A (ρ, ρ, z) = \frac{1}{2} [W_{x x} (ρ, ρ, z) + W_{y y} (ρ, ρ, z) + \sqrt{{[W_{x x} (ρ, ρ, z) - W_{y y} (ρ, ρ, z)]}^{2} + 4 {| W_{x y} (ρ, ρ, z) |}^{2}}], \\ B (ρ, ρ, z) = \frac{1}{2} [W_{x x} (ρ, ρ, z) - W_{y y} (ρ, ρ, z) + \sqrt{{[W_{x x} (ρ, ρ, z) - W_{y y} (ρ, ρ, z)]}^{2} + 4 {| W_{x y} (ρ, ρ, z) |}^{2}}], \\ C (ρ, ρ, z) = \frac{1}{2} [W_{y y} (ρ, ρ, z) - W_{x x} (ρ, ρ, z) + \sqrt{{[W_{x x} (ρ, ρ, z) - W_{y y} (ρ, ρ, z)]}^{2} + 4 {| W_{x y} (ρ, ρ, z) |}^{2}}], \\ D (ρ, ρ, z) = W_{x y} . \end{array}

The elements of the matrix

{\overset{\leftrightarrow}{W}}^{(p)}

may be written as products of “equivalent monochromatic field” components, say

E_{x}

and

E_{y}

[14,17,27]. The quadratic form associated with such a matrix can be shown to represent the ellipse, known as a spectral polarization ellipse, of the form

C (ρ, ρ, z) ε_{x}^{(r) 2} (ρ, ρ, z) - 2 Re D (ρ, ρ, z) ε_{x}^{(r)} (ρ, ρ, z) ε_{y}^{(r)} (ρ, ρ, z) + B (ρ, ρ, z) ε_{y}^{(r) 2} (ρ, ρ, z) = {[Im D (ρ, ρ, z)]}^{2}

where Re and Im stand for real and imaginary parts of complex numbers and

ε_{x}^{(r)} (ρ, ρ, z) = \sqrt{B (ρ, ρ, z)} \cos (ω t + δ_{x})

,

ε_{y}^{(r)} (ρ, ρ, z) = \sqrt{C (ρ, ρ, z)} \cos (ω t + δ_{y})

and

δ_{y} - δ_{x} = \arg [D (ρ, ρ, z)]

. The major and minor semi-axes of the ellipse, A₁ and A₂, as well as its degree of ellipticity, ε, and its orientation angle, θ, can be related directly to the elements of the cross-spectral density matrix

{\overset{\leftrightarrow}{W}}^{(p)}

with the help of the expressions

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

ε (ρ, ρ, z) = \frac{A_{2} (ρ, ρ, z)}{A_{1} (ρ, ρ, z)},

θ (ρ, ρ, z) = \frac{1}{2} \arctan [\frac{2 Re W_{x y} (ρ, ρ, z)}{W_{x x} (ρ, ρ, z) - W_{y y} (ρ, ρ, z)}] .

In Eq. (20) signs “+” and “-” between the two square roots correspond to

A_{1}

(major semi-axis) and

A_{2}

(minor semi-axis), respectively. With the help of Eqs. (11)–(22), we can study the statistics properties of an EGSM beam truncated by a circular phase aperture in free space numerically. In the following numerical examples, we set

σ_{x} = σ_{y} = σ = 1 m m

,

λ = 632.8 n m

,

A_{x} = 0.5, A_{y} = 1,

A_{i}

and

B_{i}

are chosen to be the same as those in Ref [56]. with M = 10.

Figure 2 shows the normalized spectral intensity (contour graph) and the corresponding cross line (y = 0) of an EGSM beam truncated by a circular phase aperture at several propagation distances in free space with $a = 1 m m, δ_{x x} = 3 m m, δ_{y y} = 3.5 m m, ϕ = 6.15 π / 4.$ One finds from Fig. 2 that the evolution properties of the intensity distribution of the truncated EGSM beam on propagation in free space are quite interesting. The Gaussian beam profile of the EGSM beam disappears gradually on propagation in the near field, and the central intensity decreases gradually on propagation. At certain propagation, electromagnetic partially coherent DH beam can be formed. With the further increase of the propagation distance, the hollow beam profile disappears gradually, and in the far field the truncated EGSM beam retrieves its Gaussian beam profile again. To understand this propagation phenomenon, we start from the fact that the phase of the cross region of the truncated EGSM beam within the circular phase aperture was different from that outside the circular phase aperture, and the total field of the truncated EGSM beam on propagation can be regarded as the superposition of the light from the region within the phase aperture and that from the region outside the phase aperture. There is interference between the light from the region within the phase aperture and that from the region outside the phase aperture due to the phase difference, and we may call this self-interference. Self-interference of a truncated EGSM beam leads to its interesting propagation properties as shown in Fig. 2. Figure 3 shows the normalized spectral intensity (cross line y = 0) of a truncated EGSM beam at z = 6m for different values of the phase delay ϕ and correlation coefficients $δ_{x x}$ and $δ_{y y}$ with $a = 1 m m$ . As shown in Fig. 3, at fixed propagation distance, the beam profile of a truncated EGSM beam varies with the change of the phase delay of the phase aperture or the change of the correlation coefficients $δ_{x x}$ and $δ_{y y}$ of the input beam. For fixed values of the correlation coefficients $δ_{x x}$ and $δ_{y y}$ , we can generate electromagnetic partially coherent DH or FT beam by varying the phase delay of the phase aperture. For fixed value of the phase delay, we can generate electromagnetic partially coherent DH or FT beam by varying the correlation coefficients $δ_{x x}$ and $δ_{y y}$ of the input beam. From Eq. (13), it is known that the spectral intensity of the truncated EGSM beam is expressed as the sum of two components $W_{x x} (\tilde{ρ}, z)$ and $W_{y y} (\tilde{ρ}, z)$ . $W_{x x} (\tilde{ρ}, z)$ and $W_{y y} (\tilde{ρ}, z)$ are closely determined with $δ_{x x}$ and $δ_{y y}$ , respectively. Different values of $δ_{x x}$ and $δ_{y y}$ lead to different evolution properties of $W_{x x} (\tilde{ρ}, z)$ and $W_{y y} (\tilde{ρ}, z)$ . Thus, by varying $δ_{x x}$ and $δ_{y y}$ , the superposition of $W_{x x} (\tilde{ρ}, z)$ and $W_{y y} (\tilde{ρ}, z)$ varies, and we can modulate the spectral intensity of an truncated EGSM beam through controlling the values of $δ_{x x}$ and $δ_{y y}$ . What’s more, we know that we can trap the Rayleigh dielectric particle with the refractive index smaller than the ambient by a DH beam [33], and trap the particle with the refractive index larger than the ambient by a Gaussian or FT beam [59]. Thus, our results in this paper may be applied for trapping two kinds of Rayleigh dielectric particles.

Fig. 2 Normalized spectral intensity (contour graph) and the corresponding cross line (y = 0) of an EGSM beam truncated by a circular phase aperture at several propagation distances in free space.

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Fig. 3 Normalized spectral intensity (cross line y = 0) of a truncated EGSM beam at z = 6m for different values of the phase delay ϕ and correlation coefficients $δ_{x x}$ and $δ_{y y}$ .

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To learn about the coherence properties of the truncated EGSM beam on propagation in free space. We calculate in Fig. 4 the modulus of the degree of coherence between two transverse positions $ρ_{1} = (1, 0) m m, ρ_{2} = (- 1, 0) m m$ of a truncated EGSM beam versus the propagation distance for different values of the phase delay ϕ with $a = 1 m m,$ $δ_{x x} = 3 m m, δ_{y y} = 3.5 m m .$ It is clear from Fig. 4 that the evolution properties of the degree of coherence of a truncated EGSM beam are much different from that of an EGSM beam without truncation, and is closely determined by the phase delay. For ϕ = 0 (without phase aperture), the spectral degree of coherence increases monotonically, and approaches to 1 in the far field. For $ϕ \neq 0$ , the degree of coherence exhibits growth with oscillations, particularly in the near field, and it saturates in the far field. Figure 5 shows the modulus of the degree of coherence between two transverse positions $ρ_{1} = (1, 0) m m, ρ_{2} = (- 1, 0) m m$ of a truncated EGSM beam versus the propagation distance for different values of the correlation coefficients $δ_{x x}, δ_{y y}$ with $a = 1 m m, ϕ = 6.15 π / 4.$ We see that the evolution properties of the degree of coherence are also related with the initial values of the correlation coefficients. From above discussions, we see that we can modulate or manipulate the coherence properties of a truncated EGSM beam by a circular phase aperture.

Fig. 4 Modulus of the degree of coherence between two transverse positions $ρ_{1} = (1, 0) m m, ρ_{2} = (- 1, 0) m m$ of a truncated EGSM beam versus the propagation distance for different values of the phase delay ϕ.

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Fig. 5 Modulus of the degree of coherence between two transverse positions $ρ_{1} = (1, 0) m m, ρ_{2} = (- 1, 0) m m$ of a truncated EGSM beam versus the propagation distance for different values of the correlation coefficients $δ_{x x}, δ_{y y}$ .

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Figure 6 shows the degree of polarization (cross line y = 0) of a truncated EGSM beam at several propagation distances for different values of the phase delay ϕ with $a = 1 m m,$ $| B_{x y} | = 0.25, φ = π / 6$ , $δ_{x x} = 3 m m, δ_{y y} = 3.5 m m, δ_{x y} = 4 m m .$ The polarization properties the truncated EGSM beam in the source plane are uniform across the source plane. As shown in Fig. 6, after propagation in free space, the initial uniformly polarized EGSM beam becomes non-uniformly polarized. The distribution of the degree of polarization of a truncated EGSM beam on propagation is much different from that of an EGSM beam without truncation, and is closely determined by the phase delay ϕ of the circular phase aperture. To learn about the behavior of the on-axis polarization ellipse of the truncated EGSM beam on propagation. We calculate in Fig. 7 the polarization ellipse of the truncated EGSM beam at several propagation distances in free space with $a = 1 m m,$ $φ = π / 6, | B_{x y} | = 0.25,$ $δ_{x x} = 3 m m,$ $δ_{y y} = 3.5 m m,$ $δ_{x y} = 4 m m$ , $ϕ = 6.15 π / 4$ . Figure 8 shows the on-axis polarization ellipse of the truncated EGSM beam at z = 6m for different values of the phase delay ϕ in free space with $a = 1 m m,$ $φ = π / 6, | B_{x y} | = 0.25,$ $δ_{x x} = 3 m m, δ_{y y} = 3.5 m m, δ_{x y} = 4 m m$ . As shown in Figs. 7 and 8, the evolution properties of the on-axis polarization ellipse of the truncated EGSM beam are quite interesting and different from the evolution properties of an EGSM beam without truncation [14]. Both the orientation angle and degree of ellipticity vary on propagation, and are closely determined by the phase delay of the circular phase aperture. From above discussions, we can come to conclusion that we can modulate the degree of polarization and the polarization ellipse of a truncated EGSM beam by a circular phase aperture.

Fig. 6 Degree of polarization (cross line y = 0) of a truncated EGSM beam at several propagation distances for different values of the phase delay ϕ.

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Fig. 7 On-axis polarization ellipse of the truncated EGSM beam at several propagation distances in free space.

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Fig. 8 Polarization ellipse of the truncated EGSM beam at z = 6m for different values of the phase delay ϕ in free space.

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3. Propagation factor of an EGSM beam truncated by a circular phase aperture

The propagation factor (also known as the $M^{2}$ -factor) proposed by Siegman [60] is a particularly useful property of an optical laser beam, and plays an important role in the characterization of beam behavior on propagation. The definition of $M^{2}$ -factor was extended to the partially coherent beams in [61]. The propagation factor of an EGSM beam without truncation has been studied in [20], and the evolution properties of the propagation factor of an EGSM beam in a Gaussian cavity were analyzed in [29]. In this section, we study the propagation factor of an EGSM beam truncated by a circular phase aperture.

The $M^{2}$ -factor of a partially coherent beam is defined as follows [29]

M_{x}^{2} = 4 π Δ x Δ p_{x}, M_{y}^{2} = 4 π Δ y Δ p_{y},

where

Δ x = \sqrt{\frac{1}{J} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {(x - \bar{x})}^{2} W (x, y, x, y) d x d y},

Δ y = \sqrt{\frac{1}{J} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {(y - \bar{y})}^{2} W (x, y, x, y) d x d y},

Δ p_{x} = \sqrt{\frac{1}{J} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {(p_{x} - {\bar{p}}_{x})}^{2} \tilde{W} (p_{x}, p_{y}, - p_{x}, - p_{y}) d p_{x} d p_{y}},

Δ p_{y} = \sqrt{\frac{1}{J} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {(p_{y} - {\bar{p}}_{y})}^{2} \tilde{W} (p_{x}, p_{y}, - p_{x}, - p_{y}) d p_{x} d p_{y}},

and the normalization factor J is given by:

J = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} W (x, y, x, y) d x d y = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \tilde{W} (p_{x}, p_{y}, - p_{x}, - p_{y}) d p_{x} d p_{y},

\tilde{W}

is the Fourier transform of W. The mean values

\bar{x}, \bar{p}

are defined as follows

\bar{x} = \frac{1}{J} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} x W (x, y, x, y) d x d y,

{\bar{p}}_{x} = \frac{1}{J} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} p_{x} \tilde{W} (p_{x}, p_{y}, - p_{x}, - p_{y}) d p_{x} d p_{y} .

As shown in section 2, the elements of the cross-spectral density matrix of an EGSM beam truncated by a phase aperture at z = 0 can be expressed as follows

W_{T α β}^{} (r_{1}, r_{2}, 0) = T (r_{1}) T^{*} (r_{2}) W_{α β}^{} (r_{1}, r_{2}, 0), (α = x, y; β = x, y)

where

T (r_{1}) T^{*} (r_{2})

are given by Eq. (7) and

W_{α β}^{} (\tilde{r}, 0)

are given by Eq. (9). The trace of the cross-spectral density matrix of the truncated EGSM beam at z = 0 is expressed as

W_{T r}^{} (r_{1}, r_{2}, 0) = T (r_{1}) T^{*} (r_{2}) W_{x x}^{} (r_{1}, r_{2}, 0) + T (r_{1}) T^{*} (r_{2}) W_{y y}^{} (r_{1}, r_{2}, 0) .

To calculate the

M^{2}

-factor of a truncated EGSM beam, we should replace W in Eqs. (24)–(30) with

W_{T r}^{} (r, r, 0)

. After tedious operation, we obtain the following expression for the

M^{2}

-factor of a truncated EGSM beam

M^{2} = M_{x}^{2} = M_{y}^{2} = \frac{2}{J} \sqrt{(G_{x x} + G_{y y}) (Q_{x x} + Q_{y y})},

where

\begin{array}{l} G_{x x} = A_{x}^{2} {2 π σ^{4} + [\exp (- i ϕ) - 1] \sum_{m = 1}^{M} A_{m} \frac{π}{2 α_{x x 1}^{2}} + [\exp (i ϕ) - 1] \sum_{p = 1}^{M} A_{p}^{*} \frac{π}{2 α_{x x 2}^{2}} \\ + [\exp (- i ϕ) - 1] [\exp (i ϕ) - 1] \sum_{m = 1}^{M} \sum_{p = 1}^{M} A_{m} A_{p}^{*} \frac{π}{2 α_{x x 3}^{2}}}, \end{array}

\begin{array}{l} G_{y y} = A_{y}^{2} {2 π σ^{4} + [\exp (- i ϕ) - 1] \sum_{m = 1}^{M} A_{m} \frac{π}{2 α_{y y 1}^{2}} + [\exp (i ϕ) - 1] \sum_{p = 1}^{M} A_{p}^{*} \frac{π}{2 α_{y y 2}^{2}} \\ + [\exp (- i ϕ) - 1] [\exp (i ϕ) - 1] \sum_{m = 1}^{M} \sum_{p = 1}^{M} A_{m} A_{p}^{*} \frac{π}{2 α_{y y 3}^{2}}}, \end{array}

\begin{array}{l} Q_{x x} = A_{x}^{2} {\frac{π (2 β_{x x 0} δ_{x x}^{2} + 1)}{2 (2 β_{x x 0} δ_{x x}^{2} - 1)} + (\exp (- i ϕ) - 1) \sum_{m = 1}^{M} A_{m} \frac{π (4 β_{x x 0} β_{x x 1} δ_{x x}^{4} - 1)}{2 {(β_{x x 0} δ_{x x}^{2} + β_{x x 1} δ_{x x}^{2} - 1)}^{2}} + (\exp (i ϕ) - 1) \\ \times \sum_{p = 1}^{M} A_{p}^{*} \frac{π (4 β_{x x 0} β_{x x 2} δ_{x x}^{4} - 1)}{2 {(β_{x x 0} δ_{x x}^{2} + β_{x x 2} δ_{x x}^{2} - 1)}^{2}} + [\exp (- i ϕ) - 1] [\exp (i ϕ) - 1] \sum_{m = 1}^{M} \sum_{p = 1}^{M} A_{m} A_{p}^{*} \frac{π (4 β_{x x 1} β_{x x 2} δ_{x x}^{4} - 1)}{2 {(β_{x x 1} δ_{x x}^{2} + β_{x x 2} δ_{x x}^{2} - 1)}^{2}}, \end{array}

\begin{array}{l} Q_{y y} = A_{y}^{2} {\frac{π (2 β_{y y 0} δ_{y y}^{2} + 1)}{2 (2 β_{y y 0} δ_{y y}^{2} - 1)} + (\exp (- i ϕ) - 1) \sum_{m = 1}^{M} A_{m} \frac{π (4 β_{y y 0} β_{y y 1} δ_{y y}^{4} - 1)}{2 {(β_{y y 0} δ_{y y}^{2} + β_{y y 1} δ_{y y}^{2} - 1)}^{2}} + (\exp (i ϕ) - 1) \\ \times \sum_{p = 1}^{M} A_{p}^{*} \frac{π (4 β_{y y 0} β_{y y 2} δ_{y y}^{4} - 1)}{2 {(β_{y y 0} δ_{y y}^{2} + β_{y y 2} δ_{y y}^{2} - 1)}^{2}} + [\exp (- i ϕ) - 1] [\exp (i ϕ) - 1] \sum_{m = 1}^{M} \sum_{p = 1}^{M} A_{m} A_{p}^{*} \frac{π (4 β_{y y 1} β_{y y 2} δ_{y y}^{4} - 1)}{2 {(β_{y y 1} δ_{y y}^{2} + β_{y y 2} δ_{y y}^{2} - 1)}^{2}}, \end{array}

\begin{array}{l} J = A_{x}^{2} {2 π σ^{2} + [\exp (- i ϕ) - 1] \sum_{m = 1}^{M} A_{m} \frac{π}{α_{x x 1}} + [\exp (i ϕ) - 1] \sum_{p = 1}^{M} A_{p}^{*} \frac{π}{α_{x x 2}} \\ + [\exp (- i ϕ) - 1] [\exp (i ϕ) - 1] \sum_{m = 1}^{M} \sum_{p = 1}^{M} A_{m} A_{p}^{*} \frac{π}{α_{x x 3}}} + A_{y}^{2} {2 π σ^{2} + [\exp (- i ϕ) - 1] \sum_{m = 1}^{M} A_{m} \frac{π}{α_{y y 1}} \\ + [\exp (i ϕ) - 1] \sum_{p = 1}^{M} A_{p}^{*} \frac{π}{α_{y y 2}} + [\exp (- i ϕ) - 1] [\exp (i ϕ) - 1] \sum_{m = 1}^{M} \sum_{p = 1}^{M} A_{m} A_{p}^{*} \frac{π}{α_{y y 3}}}, \end{array}

with

\begin{array}{l} α_{α β 1} = \frac{1}{2 σ^{2}} + \frac{B_{m}}{a^{2}}, α_{α β 2} = \frac{1}{2 σ^{2}} + \frac{B_{p}^{*}}{a^{2}}, α_{α β 3} = \frac{1}{2 σ^{2}} + \frac{B_{m}}{a^{2}} + \frac{B_{p}^{*}}{a^{2}}, \\ β_{α β 0} = \frac{1}{4 σ^{2}} + \frac{1}{2 δ_{α β}^{2}}, β_{α β 1} = \frac{1}{4 σ^{2}} + \frac{1}{2 δ_{α β}^{2}} + \frac{B_{m}}{a^{2}}, β_{α β 2} = \frac{1}{4 σ^{2}} + \frac{1}{2 δ_{α β}^{2}} + \frac{B_{p}^{*}}{a^{2}} . \end{array}

With the help of Eqs. (32)–(38), we calculate in Fig. 9 the dependence of the

M^{2}

-factor of a truncated EGSM beam on the phase delay ϕ with

a = 1 m m,

δ_{x x} = 1.2 m m, δ_{y y} = 1 m m .

. Figure 10 shows the dependence of the

M^{2}

-factor of a truncated EGSM beam on the radius a with

ϕ = 3

,

δ_{x x} = 1.2 m m, δ_{y y} = 1 m m .

One finds from Figs. 9 and 10 that the self-interference of a truncated EGSM beam also induces the change of its propagation factor, and we can modulate the

M^{2}

-factor of a truncated EGSM beam by a circular phase aperture with suitable parameters of the phase delay and the radius of the aperture.

Fig. 9 Dependence of the $M^{2}$ -factor of a truncated EGSM beam on the phase delay ϕ.

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Fig. 10 Dependence of the $M^{2}$ -factor of a truncated EGSM beam on the radius a.

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4. Conclusion

We have obtained the analytical formula for an EGSM beam truncated by a circular phase aperture propagating in free, and studied the statistics properties the truncated EGSM beam on propagation in free space. We have found that the circular phase aperture can be used to modulate the spectral intensity, the degree of coherence, the degree of polarization and the polarization ellipse of an EGSM beam. Electromagnetic partially coherent DH or FT beam can be formed with the help of the circular phase aperture. The propagation factor of a truncated EGSM beam is also investigated, and it is found that the phase aperture can be used to modulate the propagation factor of an EGSM beam. Our results may be useful in some applications, such as optical trapping, material processing and free-space optical communications.

Acknowledgments

Yangjian Cai acknowledges the support by the National Natural Science Foundation of China 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 Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam

Abstract

1. Introduction

2. Statistics properties of an EGSM beam truncated by a circular phase aperture in free space

3. Propagation factor of an EGSM beam truncated by a circular phase aperture

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (10)

Equations (39)

Optics Express