Generalized Stokes parameters of three-dimensional stochastic electromagnetic beams

Zhangrong Mei

doi:10.1364/OE.18.022826

1. Introduction

Most measure of the degree of coherence of light field was introduced within a scalar representation [1]. The definition of the degree of coherence was later refined and generalized in several ways [2–4]. The polarization properties of electromagnetic beams are important in a great variety of optical phenomena. Normally, the polarization statistics of electromagnetic beams at a point in space are described either by use of the coherent matrix or the two-dimensional (2D) Stokes parameters [5]. Recently, a general theory was presented by Wolf, known as the unified theory of coherence and polarization or the cross-spectral density matrix, made it possible to characterize the correlations between two mutually orthogonal spectral components of electric field at a pair of points in a plane perpendicular to the axis of the beam [6]. The Stokes parameters were also generalized from one-point quantities to two-point counterparts [7]. The 2D generalized Stokes parameters can be expressed in terms of the $2 \times 2$ cross-spectral density matrix of the field and can be also used to study both the coherent properties and the polarization properties of random electromagnetic beams. However, the 2D formulism is not adequate for the description of the statistics properties of arbitrary electromagnetic beams, such as the nonparaxial random electromagnetic beams, since three orthogonal components of vectorial field are present [8–10]. Consequently, the degree of coherence and the degree of polarization of the three-dimensional (3D) random electromagnetic beams must be extracted directly from the 3 × 3 cross-spectral density matrix of the field or from the 3D generalized Stokes parameters. In this Letter, we expand the 2D generalized Stokes parameters to the 3D case and investigate the statistics properties of 3D stochastic electromagnetic beams.

2. Theoretical analyses

The 2D generalized Stokes parameters can be expressed in terms of the 2 × 2 cross-spectral density matrix ${\overset{\leftrightarrow}{W}}_{2 D} (r_{1}, r_{2}, ω)$ of the field and the four Pauli matrices $σ_{i} (i = 0, 1, 2, 3)$ in a simple manner as follows

S_{i, 2 D} (r_{1}, r_{2}, ω) = tr [{\overset{\leftrightarrow}{W}}_{2 D} (r_{1}, r_{2}, ω) σ_{i}],

where

{\overset{\leftrightarrow}{W}}_{2 D} (r_{1}, r_{2}, ω) = [< E_{α}^{*} (r_{1}, ω) E_{β} (r_{2}, ω) >], (α, β = x, y)

, tr stands for trace. The 2D generalized Stokes parameters can be expanded in the linear combination of the four elements of the 2 × 2 cross-spectral density matrix [8].

In analogy with the 2D case, the 3D generalized Stokes parameters in terms of 3 × 3 cross-spectral density matrix ${\overset{\leftrightarrow}{W}}_{3 D} (r_{1}, r_{2}, ω)$ of the field and the nine Gell-Mann matrices λ_j(j = 0,1,…,8) [12] can be expressed to the following form

S_{j, 3 D} (r_{1}, r_{2}, ω) = tr [{\overset{\leftrightarrow}{W}}_{3 D} (r_{1}, r_{2}, ω) λ_{j}],

where

{\overset{\leftrightarrow}{W}}_{3 D} (r_{1}, r_{2}, ω) = [< E_{α}^{*} (r_{1}, ω) E_{β} (r_{2}, ω) >],

(α, β = x, y, z)

, which satisfy the relation

W_{α β}^{*} (r_{1}, r_{2}, ω) = W_{β α} (r_{1}, r_{2}, ω)

. The above relation (2) can be expanded in the form

S_{0, 3 D} (r_{1}, r_{2}, ω) = W_{x x, 3 D} (r_{1}, r_{2}, ω) + W_{y y, 3 D} (r_{1}, r_{2}, ω) + W_{z z, 3 D} (r_{1}, r_{2}, ω),

S_{1, 3 D} (r_{1}, r_{2}, ω) = W_{x y, 3 D} (r_{1}, r_{2}, ω) + W_{y x, 3 D} (r_{1}, r_{2}, ω),

S_{2, 3 D} (r_{1}, r_{2}, ω) = i [W_{y x, 3 D} (r_{1}, r_{2}, ω) - W_{x y, 3 D} (r_{1}, r_{2}, ω)],

S_{3, 3 D} (r_{1}, r_{2}, ω) = W_{x x, 3 D} (r_{1}, r_{2}, ω) - W_{y y, 3 D} (r_{1}, r_{2}, ω),

S_{4, 3 D} (r_{1}, r_{2}, ω) = W_{x z, 3 D} (r_{1}, r_{2}, ω) + W_{z x, 3 D} (r_{1}, r_{2}, ω),

S_{5, 3 D} (r_{1}, r_{2}, ω) = i [W_{x z, 3 D} (r_{1}, r_{2}, ω) - W_{z x, 3 D} (r_{1}, r_{2}, ω)],

S_{6, 3 D} (r_{1}, r_{2}, ω) = W_{y z, 3 D} (r_{1}, r_{2}, ω) + W_{z y, 3 D} (r_{1}, r_{2}, ω),

S_{7, 3 D} (r_{1}, r_{2}, ω) = i [W_{y z, 3 D} (r_{1}, r_{2}, ω) - W_{z y, 3 D} (r_{1}, r_{2}, ω)],

S_{8, 3 D} (r_{1}, r_{2}, ω) = \sqrt{3} [W_{x x, 3 D} (r_{1}, r_{2}, ω) + W_{y y, 3 D} (r_{1}, r_{2}, ω) - 2 W_{z z, 3 D} (r_{1}, r_{2}, ω)] / 3.

The above definitions of 3D generalized Stokes parameters conveniently lead to the S _0,3D(r, ω)being equal to the spectral density of the field, the spectral degree of coherence of the field at a pair of point being given by the formula

μ_{3 D} (r_{1}, r_{2}, ω) = S_{0, 3 D} (r_{1}, r_{2}, ω) / (\sqrt{S_{0, 3 D} (r_{1}, r_{1}, ω)} \sqrt{S_{0, 3 D} (r_{2}, r_{2}, ω)}) .

The spectral degree of polarization at point r and frequency ω, $P_{3 D} (r, ω)$ , of the random electromagnetic beams that has three vector components can written in terms of the 3D generalized Stokes parameters as [11]

P_{3 D} (r, ω) = \frac{\sqrt{3}}{2} {(\sum_{j = 1}^{8} S_{j, 3 D}^{2} (r, ω))}^{1 / 2} / S_{0, 3 D} (r, ω) .

Let us now consider an electromagnetic Gaussian Shell-model source, the elements of its cross-spectral density matrix in the source plane are given by the expressions [12]

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

where the coefficients

A_{α}, A_{β}, B_{α β}

and the variances

σ_{α}^{2}, σ_{β}^{2}, δ_{α β}^{2}

are independent of position but may depend on frequency, the transverse vector

ρ \equiv (x, y)

.

The propagation of each of the elements of the cross-spectral density matrix of the electric field at a pair of points $r_{1}$ and $r_{2}$ (the position vector specifies by $r \equiv (ρ, z)$ ) in the half space $z > 0$ can be treated by the generalized Rayleigh-Sommerfeld formulas [13]:

W_{α β} (r_{1}, r_{2}, ω) = {(\frac{z}{λ})}^{2} \int \int_{} W_{α β}^{(0)} (ρ_{10}, ρ_{20}, ω) {\exp [i k (R_{2} - R_{1})]} / R_{2}^{2} R_{1}^{2} d^{2} ρ_{10} d^{2} ρ_{20}, (α, β = x, y),

\begin{array}{l} W_{α z} (r_{1}, r_{2}, ω) = - \frac{z}{λ^{2}} \int \int_{} [W_{α x}^{(0)} (ρ_{10}, ρ_{20}, ω) (x_{2} - x_{20}) + W_{α y}^{(0)} (ρ_{10}, ρ_{20}, ω) (y_{2} - y_{20})] \\ \times {\exp [i k (R_{2} - R_{1})]} / R_{2}^{2} R_{1}^{2} d^{2} ρ_{10} d^{2} ρ_{20} (α = x, y), \end{array}

\begin{array}{l} W_{z z} (r_{1}, r_{2}, ω) = \frac{1}{λ^{2}} \int \int_{} [W_{x x}^{(0)} (ρ_{10}, ρ_{20}, 0) (x_{1} - x_{10}) (x_{2} - x_{20}) + 2 W_{x y}^{(0)} (ρ_{10}, ρ_{20}, 0) (x_{1} - x_{10}) \\ \times (y_{2} - y_{20}) + W_{y y}^{(0)} (ρ_{10}, ρ_{20}, 0) (y_{1} - y_{10}) (y_{2} - y_{20})] {\exp [i k (R_{2} - R_{1})]} / R_{2}^{2} R_{1}^{2} d^{2} ρ_{10} d^{2} ρ_{20} . \end{array}

On substituting from Eq. (6) into Eq. (7), we obtain the elements of the cross-spectral density matrix of a stochastic electromagnetic beam in the output plane as follows:

W_{α β} (r_{1}, r {}_{2}, ω) = A_{α} A_{β} B_{α β} z^{2} {\exp [i k (r_{2} - r_{1})] / r_{1}^{2} r_{2}^{2}} \exp (- F_{α β}) / P_{α β}, (α, β = x, y),

\begin{array}{l} W_{α z} (r_{1}, r {}_{2}, ω) = - \frac{A_{α} z \exp [i k (r_{2} - r_{1})]}{r_{1}^{2} r_{2}^{2}} {\frac{A_{x} B_{α x} J_{α x}}{P_{α x}} \exp (- F_{α x}) - \frac{A_{y} B_{α y} J_{α y}}{P_{α y}} \exp (- F_{α y})} \\ (α = x, y), \end{array}

\begin{array}{l} W_{z z} (r_{1}, r {}_{2}, ω) = \exp [i k (r_{2} - r_{1})] / (r_{1}^{2} r_{2}^{2}) {A_{x}^{2} Q_{x x} \exp (- F_{x x}) / P_{x x} + A_{y}^{2} Q_{y y} \exp (- F_{y y}) / P_{y y} \\ + A_{x} A_{y} (B_{x y} + B_{x y}) Q_{x y} \exp (- F_{x y}) / P_{x y}}, \end{array}

where

F_{α β} = k [C_{1 α β} ρ_{2}^{2} / r_{2}^{2} + C_{2 α β} ρ_{1}^{2} / r_{1}^{2} - k f_{α β}^{2} (x_{1} x_{2} + y_{1} y_{2}) / (r_{1} r_{2})] / P_{α β},

Q_{x y} = (1 - i / (2 C_{1 x y} r)) x_{1} y_{2} + i k f_{x y}^{2} y_{2} G_{x y}^{(x)} / P_{x y} + H_{x y}^{(x)} G_{x y}^{(y)} - 2 k f_{x y}^{2} C_{1 x y} G_{x y}^{(x)} G_{x y}^{(y)} / P_{x y}^{2},

Q_{α α} = (1 - \frac{i}{2 C_{1 α α} r_{1}}) α_{1} α_{2} - (\frac{i f_{α α}^{2} k α_{2}}{P_{α α}} + H_{α α}^{(α)}) G_{α α}^{(α)} - \frac{2 C_{1 α α} k f_{α α}^{2}}{P_{α α}^{2}} {(G_{α α}^{(α)})}^{2} + \frac{f_{α α}^{2}}{P_{α α}},

and

f_{j} = 1 / (k σ_{α})

,

f_{α β} = 1 / (k δ_{α β})

,

C_{j α β} = k f_{α β}^{2} / 2 + k f_{j}^{2} / 4 - i {(- 1)}^{j} / (2 r_{j}), (j = 1, 2)

,

P_{α β} = 4 C_{1 α β} C_{2 α β} - k^{2} f_{α β}^{4}

,

H_{α β}^{(γ)} = 2 i C_{1 α β} γ_{1} / P_{α β} + γ_{1} / (P_{α β} r_{1})

,

J_{α γ} = 2 i C_{1 α γ} G_{α γ}^{(γ)} / P_{α x} + γ_{2}

,

G_{α β}^{(γ)} = γ_{2} / r_{2} - k f_{α β}^{2} γ_{1} / (2 C_{1 α β} r_{1}), (γ = x, y)

.

On substituting from Eq. (8) into Eq. (3), the nine generalized Stokes parameters of a stochastic electromagnetic beam can be obtained as linear combinations of the nine elements of the 3 × 3 cross-spectral density matrix, then can be used to study both the coherence properties and the polarization properties of the 3D stochastic electromagnetic beams.

3. Numerical calculation results and comparative analyses

We apply the preceding analysis to illustrate how the statistical properties of the 3D stochastic electromagnetic beams change upon propagation in free space and compare the results with that of the 2D case. In the following figures, we use the wavelength λ and Rayleigh distance specified by z_R = πσ ²/λ to normalize the corresponding transverse and longitudinal distances. Figure 1 is the spectral density of a 3D stochastic electromagnetic beam in the plane z = 15z_R for different parameters f_j and f_αβ value. For the convenience of comparison the corresponding 2D results are also given in the figures, which are represented by the dashed lines. One can see from Fig. 1 that when the value of parameters f_j and f_αβ is very small, the spectral density of 3D beams coincide with the 2D results quite well, so that for this case the 2D results hold true and the z component is very small and can be negligible. However, for the large values of parameters f_j and f_αβ, the difference between the 3D results and 2D results become obvious, the longitudinal component also become large and cannot be neglected. Figure 2 is the contour graphs of the spectral density S _0,2D and S _0,3D of a stochastic electromagnetic beam in the plane z = 15z_R when the value of parameters f_j and f_αβ is larger. From Fig. 2(a), we can see when the results were considered 2D case, the spectral density distribution S _0,2D is axially rotational symmetry. However, when we consider 3D field distribution, shown as Fig. 2(b), the spectral density distribution S _0,3D loses their circular symmetry and become inclined elliptical symmetry due to the existence of the longitudinal component.

Fig. 1 The spectral density of a stochastic electromagnetic beam in the plane z = 15z_R, The source is assumed to be a Gaussian Shell-model source with A_x = 1.5, A_y = 1, B_xy = 0.3exp(iπ/6), B_yx = 0.3exp(-iπ/6), (a) f ₁ = f ₂ = 0.005, f_xx = f_yy = f_xy = f_yx = 0.015, (b) f ₁ = f ₂ = 0.2, f_xx = f_yy = f_xy = f_yx = 0.45.

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Fig. 2 The contour graphs of the spectral densityof a stochastic electromagnetic beam in the plane z = 15z_R. The source parameters are the same as in Fig. 1(b). (a) S _0, _2D, (b) S _0, _3D.

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Figure 3 represents the changes in the spectral degree of coherence along z-axis direction of a stochastic electromagnetic Gaussian Schell-model beam when the pair of field points are located symmetrically with respect to the z-axis, i. e. ρ₂ = -ρ₁. The 3D results of spectral degree of coherence coincide with the 2D results quite well for the small values of parameters f_j and f_αβ, so that for this case the 2D results hold true. Whereas for the large values of parameters f_j and f_αβ, the initial spectral degrees of coherence exist obvious difference, but both of the 2D and 3D results tend to 1 with increasing propagation distance.

Fig. 3 The changes in the spectral degree of coherence along z-axis direction of a stochastic electromagnetic Gaussian Schell-model beam. The source parameters are the same as in Fig. 1 except for f_xy = 0.6f_xx. Pairs of field point $ρ_{2} = - ρ_{1} = (0.2 m m, 0.2 m m)$ .

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Figure 4 shows changes in the spectral degree of polarization along z-axis of a stochastic electromagnetic Gaussian Schell-model beam and the corresponding transverse distribution in the plane z = 15z_R, the dashed lines are the 2D degree of polarization derived from the formula $P_{2 D}^{2} (r, ω) = \sum_{j = 1}^{3} S_{j, 2 D}^{2} (r, ω) / S_{0, 2 D}^{2} (r, ω)$ . For the smaller value of f_j and f_αβ, it is straightforward to show that in this case the two degree of polarization, 3D and 2D, are connected as $P_{3 D}^{2} = 3 P_{2 D}^{2} / 4 + 1 / 4$ [8]. Therefore, for this case, although the two definitions for the spectral degree of polarization take on different values, but they have the same evolution behavior, expressed as Figs. 4(a) and 4(c). However, when the parameters f_j and f_αβ are large, P _3D and P _2D are not related as the above equation and have the different evolution behavior, expressed as Figs. 4(b) and 4(d). Furthermore, when the correlation parameters of the source satisfy the relation f_xx = f_yy = f_xy = f_yx, ie, δ_xx = δ_yy = δ_xy = δ_yx, the 2D spectral degree of polarization P _2D does not change upon propagation and is uniform for all the points at the any propagation plane. Under this sufficiency conditions, the 3D spectral degree of polarization P _3D for the small values of parameters f_j and f_αβ is also invariant since the z component is very small and can be negligible. But an interesting result is that for the large values of parameters f_j and f_αβ, the longitudinal component become large and cannot be neglected, the P _3D changes upon propagation and is non-uniform at the transversal section of the beam under this case.

Fig. 4 (a) and (b) are the changes in the spectral degree P of polarization along z-axis direction of a stochastic electromagnetic Gaussian Schell-model beam, (c) and (d) are the transverse distribution of P in the plane z = 15z_R. The source parameters are the same as in Fig. 1 except for f_αβ, f_xy = f_yx showed in figures, (a) and (c) f_xx = f_yy = 0.015, (b) and (d) f_xx = f_yy = 0.45.

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For vividly comparing the difference of P _2D and P _3D, three-dimensional distributions of spectral degree of polarization and corresponding contour graphs of 2D and 3D stochastic electromagnetic beams in the plane z = 15z_R are plotted in Figs. 5 and 6 . It can be seen from Figs. 5 and 6 that the difference between P _2D and P _3D is very obvious. The 2D spectral degree of polarization P _2D is Gaussian distribution and axially rotational symmetry, but the 3D spectral degree of polarization P _3D is not Gaussian distribution and loses their circular symmetry. The P _3D first decreases to a minimum and then rises with increasing transverse coordinate. This is due to the existence of the longitudinal component of the 3D light field.

Fig. 5 There-dimensional distributions of spectral degree of polarization P _2D and corresponding contour graphs of a stochastic electromagnetic beam in the plane z = 15z_R. f_xy = 0.6f_xx and the other source parameters are the same as in Fig. 4(d).

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Fig. 6 As Fig. 5 but for P _3D, (a) A_x = 1.5, A_y = 1; (b) A_x = 1, A_y = 1; (c) A_x = 1, A_y = 1.5.

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In summary of the results in Figs. 1–6, we conclude that the f_j and f_αβ are key parameters for determine the propagation behavior of 3D random electromagnetic beams. f_j is proportional to the ratio of the wavelength to the beam width, and f_αβ is proportional to the ratio of the wavelength to the correlation length. The large values of parameters f_j mean the beam width that is comparable with or less than λ results in the beam nonparaxiality. The large values of parameters f_αβ mean weak coherence that leads to the large divergence angle. Here, the longitudinal component must be taken into consideration.

4. Conclusions

In conclusion, we have expanded the 2D generalized Stokes parameters to the 3D case and analyzed comparatively the peculiar statistical behavior of 3D random electromagnetic beams with the help of numerical examples by applying the generalized Rayleigh-Sommerfeld integral formulas and extending the concept of the degree of polarization and coherence in 2D to 3D fields. The results show that the statistical behavior of 3D random electromagnetic beams clearly different from 2D situation when the parameters f_j and f_αβ are larger. It means that the 2D→3D transformation is necessary and the 3D propagation behavior of stochastic electromagnetic beams should be taken into consideration under this case. We hope our analysis might serve to understand the physical statistical properties of 3D stochastic electromagnetic beams and inspire related studies of 3D stochastic electromagnetic beams.

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).

References and links

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Generalized Stokes parameters of three-dimensional stochastic electromagnetic beams

Abstract

1. Introduction

2. Theoretical analyses

3. Numerical calculation results and comparative analyses

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (6)

Equations (23)

Optics Express