## Abstract

Analytical formulas for the cross-spectral density matrix of stochastic electromagnetic Gaussian Schell-model (EGSM) beams passing through an astigmatic optical system are derived. We show both analytically and by numerical examples the effects of astigmatism on spectra, coherence and polarization of stochastic electromagnetic EGSM beams propagating through an astigmatic lens. A comparison with the aberration-free case is made, and shows that the astigmatism has significant effect on the spectra, coherence and polarization.

©2009 Optical Society of America

## 1. Introduction

It has been known that spectra, coherence and polarization of stochastic electromagnetic beams can experience changes on propagation in free space [1-5], through turbulent atmosphere [6-9], through a gradient-index fiber, through axially symmetrical and nonsymmetrical optical systems [10-15], and through misaligned optical systems [16]. However, the study of stochastic electromagnetic beams has been restricted to the aberration-free beams and optical systems.

This paper is aimed at studying the effect of astigmatism in the astigmatic optical system on the spectra, coherence and polarization of stochastic electromagnetic beams. To illustrate the results we will consider the properties of a stochastic electromagnetic Gaussian Schell-model (EGSM) beam through an astigmatic lens. Some typical examples are illustrated and necessary explanations are also given for the physical phenomena. All the work will be done within the framework of the paraxial approximation.

## 2. Theoretical analysis

We consider an EGSM beam generated by a source located in the plane *z*=0, with its axis along the *z* direction, whose cross-spectral density matrix [17] of the light in the plane *z*=0 is given by

where [6]

$$\left(i=x,y,\phantom{\rule{.2em}{0ex}}j=x,y\right)$$

Here (*x*́_{1},*y*́_{1}), (*x*́_{2}, *y*́_{2}) are transverse position coordinates of the two points (perpendicular to the *z* axis), σ is the waist width of the beam, and *δ _{ij}* are related to the coherence length. The parameters

*A*,

_{i}*A*,

_{j}*B*and the variances

_{ij}*σ*,

*δ*are assumed to be independent of position but may depend on the frequency.

_{ij}The spectral density of the field at the point (*x*, *y*) can be calculated from the general formula [17]

where Tr is the trace of the cross-spectral density matrix.

The spectral degree of coherence of the field at a pair of field points (*x*
_{1}, *y*
_{1}) and (*x*
_{2}, *y*
_{2}) is defined by the formula [17]

The spectral degree of polarization of the field at the point (*x*, *y*) is given by the expression [17]

where Det is the determinant of the cross-spectral density matrix.

Within the framework of the paraxial approximation, according to the propagation law of the cross-spectral density matrix [17, 18], the element of the cross-spectral density matrix *W*↔ (*x*
_{1},*y*
_{1},*x*
_{2},*y*
_{2},*z*,*ω*) of EGSM beams through an astigmatic optical system at two points (*x*
_{1}, *y*
_{1}) and (*x*
_{2}, *y*
_{2}) in a transverse plane *z* can be written as

$$\phantom{\rule{1.2em}{0ex}}\times \mathrm{exp}\{-\frac{\mathit{ik}}{2B}[\left(A\left({x}_{1}^{\prime 2}+{y}_{1}^{\prime 2}\right)-2\left({x}_{1}{x}_{1}^{\prime}+{y}_{1}{y}_{1}^{\prime}\right)+D\left({x}_{1}^{2}+{y}_{1}^{2}\right)\right)$$

$$\phantom{\rule{2.2em}{0ex}}-\left(A\left({x}_{2}^{\prime 2}+{y}_{2}^{\prime 2}\right)-2\left({x}_{2}{x}_{2}^{\prime}+{y}_{2}{y}_{2}^{\prime}\right)+D\left({x}_{2}^{2}+{y}_{2}^{2}\right)\right)\left]\right\}d{x}_{1}^{\prime}d{y}_{1}^{\prime}d{x}_{2}^{\prime}d{y}_{2}^{\prime}$$

where *k*=*ω*/*c* denotes the wave number, *c* is the velocity of light in vacuum. *A*, *B* and *D* are elements of the transfer *ABCD* matrix. In Eq.(6) the astigmatism in the *ABCD* system is considered, and characterized by a term exp[-*ikC*
_{6}(*x*́^{2} - *y*́^{2})], *C*
_{6} being the astigmatic coefficient [19], but other phase aberrations can be neglected.

On substituting Eq.(2) into Eq.(6), after prolonged integral calculations, the element of cross-spectral density matrix *W*↔ (*x*
_{1},*y*
_{1},*x*
_{2},*y*
_{2},*z*,*ω*) of EGSM beams at two points (*x*
_{1}, *y*
_{1}) and (*x*
_{2}, *y*
_{2}) at the *z* plane is given by

$$\phantom{\rule{8.2em}{0ex}}\times \mathrm{exp}\left[-\frac{{\left({x}_{1}-{x}_{2}\right)}^{2}}{2{\delta}_{\mathit{ij}}^{2}{Q}_{x}}-\frac{{\left({y}_{1}-{y}_{2}\right)}^{2}}{2{\delta}_{\mathit{ij}}^{2}{Q}_{y}}\right]\mathrm{exp}\left[-\frac{\mathit{i\omega}\left({x}_{1}^{2}-{x}_{2}^{2}\right)}{2c{R}_{x}}-\frac{\mathit{i\omega}\left({y}_{1}^{2}-{y}_{2}^{2}\right)}{2c{R}_{y}}\right]$$

where

and

Assume the optical system consists of an astigmatic lens located at the plane *z*=0 and the separation between the lens and observation *z* plane, we have

where *f* is the focal length of the lens. *Q _{x}*,

*Q*and

_{y}*R*,

_{x}*R*in Eqs.(8) and (9) change to

_{y}and

Equation (7) is the analytical formula for an EGSM beam passing through an astigmatic lens. With the help of Eqs.(7), (11), (12) and (3)-(5), we can determine how various spectra, coherence and polarization of EGSM beams passing through an astigmatic lens.

## 3. Numerical calculations and discussions

To simplify the analysis, we assume that *A _{x}*=

*A*=

_{y}*A*,

*B*=1,

_{xx}*B*=B,

_{yy}*B*=

_{xy}*B*=0. On substituting from Eqs.(1) and (2) into Eq.(5), we readily obtain the expression for the polarization at the plane

_{yx}*z*=0 as

We will now study numerically the behavior of spectra, coherence and polarization of EGSM beams passing through an astigmatic lens using Eqs.(7), (11), (12) and (3)-(5). We choose *P*
^{(0)}=0.5 (i.e., the case of a partially polarized beam) and present some numerical examples in Figs.1-10 to show the influence of astigmatic coefficient *C*
_{6} on the spectral degree of polarization, spectra and coherence of EGSM beams, respectively.

In Fig.1 we present the spectral degree of polarization of the EGSM beam in the *x*-*z* plane. It can be seen from Fig.1 that the astigmatism affects the spectral degree of polarization, and the distribution of spectral degree of polarization is different for different values of *C*
_{6}. In Fig.2 we give the spectral degree of polarization of the EGSM beam in the *y*-*z* plane. The corresponding results for the aberration-free case of *C*
_{6}=0 the distribution of spectral degree of polarization in the *y*-*z* plane depicted in Fig. 2(a) is same behavior in the *x*-*z* plane in Fig. 1(a). For the cases of *C*
_{6} ≠ 0, the distribution of the spectral degree of polarization in the *x*-*z* plane is different from that in the *y*-*z* plane, and the distribution of spectral degree of polarization in the *y*-*z* plane is nearly *z* axis as *z*>400mm. The main reason for this phenomenon is the astigmatism of lens. In addition, from Fig.2 we see that there is a narrow band of higher degree of polarization close to *z*=0.32m for *C*
_{6}=0.3×10^{-3}mm^{-1} in Fig. 2(c) and *z*=0.3m for *C*
_{6}=0.5×10^{-3}mm^{-1} in Fig. 2(d), and numerical results show that the strongly elliptical plots appear in the *x*-*y* plane. The results can be interpreted as follows. In the *y*-*z* plane, for a fixed value *y* near *z*-axis, there is a maximum of the spectral degree of polarization close *z*=0.3m at larger the astigmatism of lens for *C*
_{6}=0.3×10^{-3}mm^{-1} and 0.5×10^{-3}mm^{-1} in Figs. 2(c) and 2(d).

Figure 3 and Fig.4 give the distribution of spectral degree of polarization in the *x*-*y* plane for the aberration-free case of *C*
_{6}=0 and aberration case of *C*
_{6}=0.3×10^{-3}mm^{-1}, respectively. In Fig.3, for the aberration-free case of *C*
_{6}=0 the distribution of spectral degree of polarization in the *x*-*y* plane is of circular symmetry. In Fig.4, for the aberration case of *C*
_{6}=0.3×10^{-3}mm^{-1} the distribution of spectral degree of polarization is of elliptical distributions at *z*=300mm, 500mm and 600mm(see Figs.4(a), 4(c) and 4(d)) because of the astigmatism of lens. In Fig.5, the EGSM source is changed with the correlation length *δ _{yy}* increased from 0.2 to 0.4mm.

In Fig.6, we further investigate the behavior of the spectral degree of polarization for the on-axis point in the focused field. In Fig.6(a), we plot the variation of the on-axis spectral degree of polarization with different values of *C*
_{6}=0, 0.1×10^{-3}mm^{-1}, 0.3×10^{-3}mm^{-1} and 0.5×10^{-3}mm^{-1}. We find that maxima of the spectral degree of polarization along the *z* axis become smaller with larger values of *C*
_{6}. In particular, the distributions of spectral degree of polarization are split into two lines. In Fig.6(b), the EGSM source is changed with the correlation length *δ _{yy}* increased from 0.2 to 0.4mm. Comparing with

*δ*=0.2mm, we find that maxima of the spectral degree of polarization along the

_{yy}*z*axis become smaller with larger value of

*δ*=0.4mm.

_{yy}In Fig.7 we investigate the behavior of the spectral density for the on-axis point. In Fig.7(a), we plot the variation of the on-axis spectral density with aberration-free case of *C*
_{6}=0 and three aberration case of *C*
_{6}=0.1×10^{-3}mm^{-1}, 0.3×10^{-3}mm^{-1} and 0.5×10^{-3}mm^{-1}. The variation of the on-axis spectral density with the three different coherence lengths *δ _{xx}* and

*δ*is also illustrated in Fig.7(b). As can be seen, the maxima of axial spectral density change and the position of maximum of axial spectral density moves toward the lens, depending on

_{yy}*C*

_{6},

*δ*and

_{xx}*δ*. From Fig.7(a), it is shown that as

_{yy}*C*

_{6}increases, the maxima of the spectral density along the

*z*axis become smaller and the position of the maximum of axial spectral density moves toward the lens. From Fig.7(b), we find that the maxima of the spectral density along the

*z*axis become larger with larger values of

*δ*and

_{xx}*δ*.

_{yy}In Fig.8 and 9, we give the spectral degree of coherence |*μ*(0,0,*x*,0,*z*,*ω*)| at a pair of points (0,0) and (*x*,0) and at a pair of points (0,0) and (0,*y*) in the plane *z*=300mm, 400mm, 500mm and 600mm for different values of *C*
_{6}=0, 0.1×10^{-3}mm^{-1}, 0.3×10^{-3}mm^{-1} and 0.5×10^{-3}mm^{-1}, respectively. It is shown that for aberration-free case of *C*
_{6}=0 and smaller aberration case of *C*
_{6}=0.1×10^{-3}mm^{-1}, the coherence width (the width of the spectral degree of coherence curve) decreases with propagation distance increasing from *z*=300mm to 400mm, and the coherence width increases with propagation distance increasing from *z*=400mm to 600mm. For larger aberration case of *C*
_{6}=0.3×10^{-3}mm^{-1} and 0.5×10^{-3}mm^{-1}, coherence width increases with increment of propagation distance *z* on the *x* axis, but coherence width decreases with increment of propagation distance *z* on the *y* axis. We further investigate the behavior of spectral degree of coherence with different values of *δ _{xx}* and

*δ*in the geometrical focal plane

_{yy}*z*=400mm (see Fig. 10). It is seen that the coherence width along the

*x*axis and along the

*y*axis change not obviously for larger values of

*δ*and

_{xx}*δ*, and the coherence width along the

_{yy}*x*axis is noticeably larger than that along the

*y*axis for smaller values of

*δ*=0.6mm and

_{xx}*δ*=0.2mm.

_{yy}## 4. Conclusions

In conclusion, our formulas provide a convenient and effective way to study the effects of astigmatism on spectra, coherence and polarization of electromagnetic Gaussian Schell-model beams through an astigmatic optical system. We conclude by saying that we have demonstrated that the astigmatism affects the spectra, coherence and polarization of the beam, and we illustrated these results by computed curves. The phenomena are explained as optical system aberration-induced spectra, coherence and polarization changes. Finally, it is to be noted that these results may be important for many applications, such as tracking, remote sensing, and optical communication. In addition, the astigmatism discussed in this paper is primary astigmatism. However, other types of astigmatism (e.g. triangular astigmatism) may affect the spectra, coherence and polarization of stochastic electromagnetic beams which will be considered in future work.

## Acknowledgments

This research was supported by the National Natural Science Foundation of China under grant 60678055, by the Program for New Century Excellent Talents in University of Henan Province No. 2006HANCET-09, and by the Foundation of State Key Laboratory of Laser Technology.

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