The generalized Stokes parameters of 2D stochastic electromagnetic beams are developed to the 3D case, which can be addressed as certain linear combinations of the 3 × 3 cross-spectral density matrix in terms of the nine Gell-Mann matrices. Using the electromagnetic Gaussian Shell-model source as an example, we investigate their precise propagation laws of coherence properties and polarization properties with the help of the 3D generalized Stokes parameters. Some numerical examples and detailed comparisons of the obtained results with the 2D case are made. It is shown that 3D generalized Stokes parameters are required for the exact description of stochastic electromagnetic beams.
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Most measure of the degree of coherence of light field was introduced within a scalar representation . 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 . 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 . The Stokes parameters were also generalized from one-point quantities to two-point counterparts . The 2D generalized Stokes parameters can be expressed in terms of the 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 of the field and the four Pauli matrices in a simple manner as follows8].
In analogy with the 2D case, the 3D generalized Stokes parameters in terms of 3 × 3 cross-spectral density matrix of the field and the nine Gell-Mann matrices λj(j = 0,1,…,8)  can be expressed to the following form
The spectral degree of polarization at point r and frequency ω, , of the random electromagnetic beams that has three vector components can written in terms of the 3D generalized Stokes parameters as 
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 
The propagation of each of the elements of the cross-spectral density matrix of the electric field at a pair of points and (the position vector specifies by ) in the half space can be treated by the generalized Rayleigh-Sommerfeld formulas :
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 zR = 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 = 15zR for different parameters fj and fαβ 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 fj and fαβ 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 fj and fαβ, 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 = 15zR when the value of parameters fj and fαβ 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.
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 fj and fαβ, so that for this case the 2D results hold true. Whereas for the large values of parameters fj and fαβ, the initial spectral degrees of coherence exist obvious difference, but both of the 2D and 3D results tend to 1 with increasing propagation distance.
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 = 15zR, the dashed lines are the 2D degree of polarization derived from the formula . For the smaller value of fj and fαβ, it is straightforward to show that in this case the two degree of polarization, 3D and 2D, are connected as . 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 fj and fαβ 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 fxx = fyy = fxy = fyx, ie, δxx = δyy = δxy = δyx, 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 fj and fαβ 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 fj and fαβ, 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.
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 = 15zR 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.
In summary of the results in Figs. 1–6, we conclude that the fj and fαβ are key parameters for determine the propagation behavior of 3D random electromagnetic beams. fj is proportional to the ratio of the wavelength to the beam width, and fαβ is proportional to the ratio of the wavelength to the correlation length. The large values of parameters fj mean the beam width that is comparable with or less than λ results in the beam nonparaxiality. The large values of parameters fαβ mean weak coherence that leads to the large divergence angle. Here, the longitudinal component must be taken into consideration.
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 fj and fαβ 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.
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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