Generalized Stokes parameters of rectangular hard-edge diffracted random electromagnetic beams

Zhangrong Mei

doi:10.1364/OE.18.027105

Optics Express
Vol. 18,
Issue 26,
pp. 27105-27111
(2010)
•https://doi.org/10.1364/OE.18.027105

Generalized Stokes parameters of rectangular hard-edge diffracted random electromagnetic beams

Zhangrong Mei

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Zhangrong Mei, "Generalized Stokes parameters of rectangular hard-edge diffracted random electromagnetic beams," Opt. Express 18, 27105-27111 (2010)

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About this Article
History
- Original Manuscript: November 11, 2010
- Manuscript Accepted: December 3, 2010
- Published: December 8, 2010

Abstract

Based on the complex Gaussian expansion method for two dimensional rectangular hard-edged aperture, the analytical formulas for the generalized Stokes parameters of random electromagnetic beams through a paraxial ABCD optical system with rectangular hard-edged aperture are derived. With the help of the analytical formulae, the changes in statistical properties of rectangular hard-edge diffracted random electromagnetic beams, such as in the spectral density, in the spectral degree of coherence, in the polarization properties and so on, can be determined. Numerical examples of such changes are presented and discussed.

1. Introduction

There is currently much interest in the investigation of statistical properties of random electromagnetic beams. Usual Stokes parameters have long been used successfully to characterize the state of polarization of an electromagnetic beam [1]. Recently proposed generalized Stokes parameters, treated as a two-point extension of the usual Stokes parameters, may be written in terms of the correlations between electric field components at a pair of points and can be used to characterize simultaneously the coherence and the polarization properties of the electromagnetic fields [2,3]. Much work has been carried out concerning the generalized Stokes parameters and employing them to characterized statistical properties of electromagnetic beams [4–8]. So far, all of the publications on the generalized Stokes parameters dealt with beams that were not diffracted by apertures, whereas the apertures are usually present in many practical applications. In this paper, we study the generalized Stocks parameters of rectangular hard-edged diffracted random electromagnetic beams. Analytical formulas for the generalized Stokes parameters of random electromagnetic beams through a paraxial ABCD optical system with rectangular hard-edged aperture are derived, and numerical examples are given to analyze the evolution behavior of their statistical properties.

2. Theory

Consider a stochastic, statistically stationary, electromagnetic beam truncated by a rectangular hard-edged aperture in the plane z = 0, which propagates close to the z direction into the half-space z>0. The generalized Stokes parameters of a pair of points r ₁ and r ₂ (the position vector r = (ρ, z), ρ = (x, y) is the transverse vector) are defined as the following formulas [2,3]:

S_{0} (r_{1}, r_{2}, ω) = 〈 E_{x}^{*} (r_{1}, ω) E_{x} (r_{2}, ω) 〉 + 〈 E_{y}^{*} (r_{1}, ω) E_{y} (r_{2}, ω) 〉,

S_{1} (r_{1}, r_{2}, ω) = 〈 E_{x}^{*} (r_{1}, ω) E_{x} (r_{2}, ω) 〉 - 〈 E_{y}^{*} (r_{1}, ω) E_{y} (r_{2}, ω) 〉,

S_{2} (r_{1}, r_{2}, ω) = 〈 E_{x}^{*} (r_{1}, ω) E_{y} (r_{2}, ω) 〉 + 〈 E_{y}^{*} (r_{1}, ω) E_{x} (r_{2}, ω) 〉,

S_{3} (r_{1}, r_{2}, ω) = i [〈 E_{y}^{*} (r_{1}, ω) E_{x} (r_{2}, ω) 〉 - 〈 E_{x}^{*} (r_{1}, ω) E_{y} (r_{2}, ω) 〉],

where the asterisk denotes the complex conjugate and the angular brackets stand for the average over the ensemble of realizations of the fluctuating electric field, x and y are Cartesian components in two mutually orthogonal directions perpendicular to z axis, E = (E _x, E _y) is a member of the statistical ensemble of the fluctuating component of the transverse electric field.

The propagation of light beams through a paraxial ABCD optical system with rectangular hard-edged aperture can be treated by the generalized Huygens-Fresnel diffraction integral [9]:

E_{i} (ρ, z, ω) = \frac{i}{λ B} \int_{- a}^{a} \int_{- b}^{b} E_{i} (ρ_{0}, 0, ω) \exp [- \frac{i k}{2 B} [A ρ_{0}^{2} - 2 ρ ρ_{0} + D ρ^{2}]] d^{2} ρ_{0},

where a and b are the half widths of the rectangular aperture in the x and y direction, k = 2π/λ, λ is the wavelength. A, B, C, D are elements of transfer matrix of the system after the aperture.

Introducing the rectangular hard-edged aperture function

F (ρ) = {\begin{matrix} 1, | x | \leq a, | y | \leq b \\ 0, | x | > a, | y | > b \end{matrix},

then the finite integral in Eq. (2) can be expressed as the following infinite integral

E_{i} (ρ, z, ω) = \frac{i}{λ B} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} F (ρ_{0}) E_{i} (ρ_{0}, 0, ω) \exp [- \frac{i k}{2 B} [A ρ_{0}^{2} - 2 ρ ρ_{0} + D ρ^{2}]] d^{2} ρ_{0} .

According to the method proposed by Wen and Breazeale, the rectangular hard-edged aperture function can be expanded into two finite sums of complex Gaussian functions [9]

F (ρ) = \sum_{h = 1}^{N} \sum_{g = 1}^{N} A_{h} A_{g} \exp [- B_{h} x^{2} / a^{2} - B_{g} y^{2} / b^{2}],

where A_h _, _g and B_h _, _g denote the expansion and Gaussian coefficients, respectively, which could be obtained by optimization-computation directly [10]. From Eq. (4) it follows that

\begin{array}{l} 〈 E_{i}^{*} (ρ_{1}, ρ_{2}, z, ω) E_{j} (ρ_{1}, ρ_{2}, z, ω) 〉 \\ = \frac{1}{{(λ B)}^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} F (ρ_{10}, ρ_{20}) 〈 E_{i}^{*} (ρ_{10}, ρ_{20}, 0, ω) E_{j} (ρ_{10}, ρ_{20}, 0, ω) 〉 \\ \times \exp {- i k [A (ρ_{10}^{2} - ρ_{10}^{2}) - 2 (ρ_{10} ρ_{1} - ρ_{20} ρ_{2}) + D (ρ_{10}^{2} - ρ_{10}^{2})] / 2 B} d^{2} ρ_{10} d^{2} ρ_{20}, \end{array}

where

F (ρ_{10}, ρ_{20}) = F^{*} (ρ_{10}) F (ρ_{20})

is the four-dimensional aperture function and express as the following forms:

F (ρ_{10}, ρ_{20}) = \sum_{h_{1} = 1}^{N} \sum_{g_{1} = 1}^{N} \sum_{h_{2} = 1}^{N} \sum_{g_{2} = 1}^{N} A_{h_{1}}^{*} A_{g_{1}}^{*} A_{h_{2}} A_{g_{2}} \exp [- \frac{B_{h_{1}}^{*}}{a^{2}} x_{10}^{2} - \frac{B_{g_{1}}^{*}}{b^{2}} y_{10}^{2} - \frac{B_{h_{2}}}{a^{2}} x_{20}^{2} - \frac{B_{g_{2}}}{b^{2}} y_{20}^{2}],

According to Eqs. (1)a)-(1d), the generalized Stokes parameters are just certain linear combinations of correlation functions $〈 E_{i}^{*} (r_{1}, ω) E_{j} (r_{2}, ω) 〉$ (i = x,y; j = x,y), it follows that each of them satisfies the same propagation law, i.e., that

\begin{array}{l} S_{α} (ρ_{1}, ρ_{2}, z, ω) = \frac{1}{{(λ B)}^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} F (ρ_{10}, ρ_{20}) S_{α}^{(0)} (ρ_{10}, ρ_{20}, ω) \exp {- \frac{i k}{2 B} [A (ρ_{10}^{2} - ρ_{10}^{2}) \\ - 2 (ρ_{10} ρ_{1} - ρ_{20} ρ_{2}) + D (ρ_{10}^{2} - ρ_{10}^{2})] \begin{matrix}  \end{matrix}} d^{2} ρ_{10} d^{2} ρ_{20} (α = 0, 1, 2, 3) . \end{array}

3. An example

Let us now assume that a beam is generated by an electromagnetic Schell-model source, the four Stokes parameters in the source plane are given as follows [3]:

S_{0}^{(0)} (ρ_{10}, ρ_{20}, ω) = W_{x x}^{(0)} (ρ_{10}, ρ_{20}, ω) + W_{y y}^{(0)} (ρ_{10}, ρ_{20}, ω),

S_{1}^{(0)} (ρ_{10}, ρ_{20}, ω) = W_{x x}^{(0)} (ρ_{10}, ρ_{20}, ω) - W_{y y}^{(0)} (ρ_{10}, ρ_{20}, ω),

S_{2}^{(0)} (ρ_{10}, ρ_{20}, ω) = W_{x y}^{(0)} (ρ_{10}, ρ_{20}, ω) + W_{y x}^{(0)} (ρ_{10}, ρ_{20}, ω),

S_{3}^{(0)} (ρ_{10}, ρ_{20}, ω) = i [W_{y x}^{(0)} (ρ_{10}, ρ_{20}, ω) - W_{x y}^{(0)} (ρ_{10}, ρ_{20}, ω)],

where

W_{i j}^{(0)} (ρ_{10}, ρ_{20}, ω)

are the elements of the cross-spectral density matrix and have the following forms [11,12]:

W_{i j}^{(0)} (ρ_{10}, ρ_{20}, ω) = \sqrt{S_{i}^{(0)} (ρ_{10}, ω) S_{j}^{(0)} (ρ_{20}, ω)} η_{i j}^{(0)} (ρ_{20} - ρ_{10}, ω),

S_i ⁽⁰⁾and S_j ⁽⁰⁾ is the spectral density of the component E_i and E_j of the electric field in the source plane, and η_ij ⁽⁰⁾ is the spectral degree of correlation between the components E_i and E_j. If the spectral density and the spectral degree of correlation are both represented by Gaussian distributions, then the source is called the electromagnetic Gaussian Schell-model source and the elements of its cross-spectral density matrix are given by the following expressions [13]:

W_{i j}^{(0)} (ρ_{10}, ρ_{20}, ω) = A_{i} A_{j} B_{i j} \exp (- \frac{ρ_{10}^{2}}{4 σ_{i}^{2}}) \exp (- \frac{ρ_{20}^{2}}{4 σ_{j}^{2}}) \exp (- \frac{{| ρ_{20} - ρ_{10} |}^{2}}{2 δ_{i j}^{2}}),

where the coefficients A_i, A_j, B_ij and the variances σ_i ², σ_j ², δ_ij ² are independent of position but may depend on frequency.

On substituting from Eqs. (7), (9) and (11) into Eq. (8), and after performing tedious integration, we obtain for the four generalized Stokes parameters of rectangular hard-edged aperture diffracted random electromagnetic beams the formulas

\begin{array}{l} S_{0} (ρ_{1}, ρ_{2}, z, ω) = \exp [- i k D (ρ_{1}^{2} - ρ_{2}^{2}) / (2 B)] \sum_{h_{1}}^{N} \sum_{h_{2}}^{N} \sum_{g_{1}}^{N} \sum_{g_{2}}^{N} A_{h_{1}} A_{h_{2}} A_{g_{1}} A_{g_{2}} \\ \times [A_{x}^{2} \exp (- F_{x x}) / \sqrt{P_{x x}^{(h)} P_{x x}^{(g)}} + A_{y}^{2} \exp (- F_{y y}) / \sqrt{P_{y y}^{(h)} P_{y y}^{(g)}}], \end{array}

\begin{array}{l} S_{1} (ρ_{1}, ρ_{2}, z, ω) = \exp [- i k D (ρ_{1}^{2} - ρ_{2}^{2}) / (2 B)] \sum_{h_{1}}^{N} \sum_{h_{2}}^{N} \sum_{g_{1}}^{N} \sum_{g_{2}}^{N} A_{h_{1}} A_{h_{2}} A_{g_{1}} A_{g_{2}} \\ \times [A_{x}^{2} \exp (- F_{x x}) / \sqrt{P_{x x}^{(h)} P_{x x}^{(g)}} - A_{y}^{2} \exp (- F_{y y}) / \sqrt{P_{y y}^{(h)} P_{y y}^{(g)}}], \end{array}

\begin{array}{l} S_{2} (ρ_{1}, ρ_{2}, z, ω) = \exp [- i k D (ρ_{1}^{2} - ρ_{2}^{2}) / (2 B)] \sum_{h_{1}}^{N} \sum_{h_{2}}^{N} \sum_{g_{1}}^{N} \sum_{g_{2}}^{N} A_{h_{1}} A_{h_{2}} A_{g_{1}} A_{g_{2}} [A_{x} A_{y} B_{x y} \\ \times \exp (- F_{x y}) / \sqrt{P_{x x}^{(h)} P_{x x}^{(g)}} + A_{y} A_{x} B_{y x} \exp (- F_{y x}) / \sqrt{P_{y y}^{(h)} P_{y y}^{(g)}}], \end{array}

\begin{array}{l} S_{3} (ρ_{1}, ρ_{2}, z, ω) = i \exp [- i k D (ρ_{1}^{2} - ρ_{2}^{2}) / (2 B)] \sum_{h_{1}}^{N} \sum_{h_{2}}^{N} \sum_{g_{1}}^{N} \sum_{g_{2}}^{N} A_{h_{1}} A_{h_{2}} A_{g_{1}} A_{g_{2}} [A_{y} A_{x} B_{y x} \\ \times \exp (- F_{y x}) / \sqrt{P_{x x}^{(h)} P_{x x}^{(g)}} - A_{x} A_{y} B_{x y} \exp (- F_{x y}) / \sqrt{P_{y y}^{(h)} P_{y y}^{(g)}}], \end{array}

where

F_{i j} = \exp [\frac{k}{2 B} (\frac{x_{1}^{2}}{C_{h_{1}}} + \frac{y_{1}^{2}}{C_{g_{1}}}) + \frac{1}{P_{i j}^{(h)}} {(\frac{k C_{h_{1}} x_{2}}{2 B} - \frac{x_{1}}{2 δ_{i j}^{2}})}^{2} + \frac{1}{P_{i j}^{(g)}} {(\frac{k C_{g_{1}} y_{2}}{2 B} - \frac{y_{1}}{2 δ_{i j}^{2}})}^{2}],

and

P_{i j}^{(γ)} = C_{γ_{1}} C_{γ_{2}} - B^{2} / (δ_{i j}^{4} k^{2}),

Equation (12) can be used to determine how the generalized Stokes parameters of rectangular hard-edged aperture diffracted random electromagnetic beam change upon propagation. The result is shown in Fig. (1) . The matrix elements are A = 1, B = z, C = 0, D = 1. In the near-field, the generalized Stokes parameters of beam truncated by different size apertures are coincident. However, the difference between them appears gradually with increasing of propagation distance, the values of Stokes parameters decrease with decreasing size of aperture in the same transmission plane. Finally, they are also converging. Figure 2 is the three-dimensional spectral density distributions and corresponding contour graphs of a rectangular hard-edged aperture (a = b = 5mm) diffracted random electromagnetic beam in the different transverse plane. When the transmission distance is not very large, the intensity distribution is the rectangular symmetry. With the increasing of propagation distance, the light spot shape gradually changes to the circular symmetry and becomes Gaussian distribution.

Fig. 1 Changes in the Stokes parameters along z-axis direction of a rectangular hard-edge diffracted random electromagnetic beam on propagation in free space. The source is assumed to be a Gaussian Schell-model source with λ = 632.8nm, A_x = 1.5, A_y = 1.0, B_xy = 0.3exp(iπ/3), B_yx = 0.3exp(-iπ/3), σ_x = σ_y = 10mm, δ_xx = 0.15mm, δ_xy = δ_yx = 0.25mm, δ_yy = 0.2mm.

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Fig. 2 Three-dimensional spectral density distributions and corresponding contour graphs of a hard-edged aperture (a = b = 5mm) diffracted random electromagnetic beam in the transverse plane z = 10m (a1-a2) and z = 100m (b1-b2). The source parameters are the same as in Fig. 1.

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The spectral degree of polarization of the random electromagnetic beams can be written in terms of the generalized Stokes parameters as [12]

P^{2} (ρ, z, ω) = \sum_{i = 1}^{3} S_{i}^{2} (ρ, z, ω) / S_{0}^{2} (ρ, z, ω) .

Figure 3 shows the changes in the spectral degree of polarization along z-axis direction of different aperture size diffracted random electromagnetic beams. The figure shows that the degree of polarization of truncated beams is influenced by the aperture size. But the spectral degree of polarization of random electromagnetic beams diffracted by apertures of different sizes have similar evolution behavior, i.e., they are invariable in the near field, with the increasing of propagation distance z, they first decrease and arrive at a minimum value, and then increase with increasing of z, finally reach a maximum constant. Figure 4 is the three-dimensional spectral degree of polarization distributions and corresponding contour graphs of random electromagnetic beam diffracted by a rectangular hard-edged aperture with a = b = 5mm in the different transverse plane. Similarly, the transverse spectral degree of polarization distributions of rectangular hard-edged aperture diffracted random electromagnetic beams is the rectangular symmetry in the near field and the circular symmetry in the far field.

Fig. 3 Changes in the spectral degree of polarization along z-axis direction of a rectangular hard-edged aperture diffracted random electromagnetic beams on propagation in free space. All parameters are the same as in Fig. 1.

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Fig. 4 There-dimensional spectral degree of polarization distributions and corresponding contour graphs of a rectangular hard-edged aperture (a = b = 5mm) diffracted random electromagnetic beam in the transverse plane z = 10m (a1-a2) and z = 100m (b1-b2). The source parameters are the same as in Fig. 1.

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The spectral degree of coherence of the field at a pair of point r₁ = (ρ₁, z) and r₂ = (ρ₂, z) can be obtained in terms of the generalized Stokes parameters S ₀ by use of the formula [12]

μ (r_{1}, r_{2}, ω) = S_{0} (r_{1}, r_{2}, ω) / (\sqrt{S_{0} (r_{1}, r_{1}, ω)} \sqrt{S_{0} (r_{2}, r_{2}, ω)}) .

Figure 5 illustrates the evolution behavior in the spectral degree of coherence along z-axis direction of a rectangular hard-edged aperture diffracted random electromagnetic beams on propagation in free space when the pair of field points are located symmetrically with respect to the z-axis, i. e. ρ₂ = -ρ₁. The figure shows that the spectral degree of coherence of different size aperture diffracted random electromagnetic beams has similar evolution behavior in the near-axis points, nothing more than has a different value for different aperture size in the same propagation plane, that is, has a larger value to the smaller aperture size. However, the spectral degree of coherence of different size aperture diffracted random electromagnetic beams has a completely different transverse distribution, shown as Fig. 6 . The transverse spectral degree of coherence distributions of rectangular hard-edged aperture diffracted random electromagnetic beams is the rectangular symmetry for the smaller aperture size and the circular symmetry for the large size aperture.

Fig. 5 Changes in the spectral degree of coherence along z-axis direction of a rectangular hard-edge diffracted random electromagnetic beam on propagation in free space. Pairs of field point ρ ₂ = ρ ₁ = (0.05mm, 0.05mm, 0.05mm,0.05mm), the source parameters are the same as in Fig. 1.

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Fig. 6 There-dimensional spectral degree of coherence distributions and corresponding contour graphs of different rectangular hard-edged diffracted random electromagnetic Gaussian Schell-model beam in the transverse plane z = 10m when the pair of field points are located symmetrically with respect to the z-axis, i. e. ρ ₂ = -ρ ₁. (a1-a2) a = b = 5mm; (b1-b2) a = b = 50mm.

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4. Concluding remarks

The analytical propagation expressions and direct numerical calculations of the hard-edged aperture diffracted random electromagnetic beams are difficult to derive due to the fourfold finite integral limit. In the present paper we applied the complex Gaussian expansion method of two-dimensional rectangular aperture to study the generalized Stocks parameters of rectangular hard-edged aperture diffracted random electromagnetic beams and their statistical properties. The analytical expressions for the generalized Stocks parameters of rectangular hard-edged aperture diffracted random electromagnetic beams have been derived, which could easily reduce to the cases of unaperture. The obtained results provide more convenience for treating their propagation or transformation. With the help of numerical calculations we have studied their lognitudinal and transverse evolution behavior of coherence and polarization and other statistical properties. The results show the changes in the statistical properties can be influenced by the aperture size and the aperture can be used be as a kind of coherence and polarization modulator.

Acknowledgments

This work was supported by the Zhejiang Provincial Natural Science Foundation of China (Y6100605) and Huzhou Civic Natural Science Fund of Zhejiang Province of China (2009YZ01).

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Fig. 1 Changes in the Stokes parameters along z-axis direction of a rectangular hard-edge diffracted random electromagnetic beam on propagation in free space. The source is assumed to be a Gaussian Schell-model source with λ = 632.8nm, A_x = 1.5, A_y = 1.0, B_xy = 0.3exp(iπ/3), B_yx = 0.3exp(-iπ/3), σ_x = σ_y = 10mm, δ_xx = 0.15mm, δ_xy = δ_yx = 0.25mm, δ_yy = 0.2mm.

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