ABCD optical system

follows that
͗Ei*͑␳1, ␳2, z, ␻͒Ej͑␳1, ␳2, z, ␻͒͘
͵͵ 1
= ͑␭B͒2
͗Ei*͑␳1Ј , ␳2Ј , 0, ␻͒Ej͑␳1Ј , ␳2Ј , 0, ␻͒͘
ͭ ik
ϫexp − 2B ͓A͑␳1Ј2 − ␳2Ј2͒ − 2͑␳1Ј␳1 − ␳2Ј␳2͒
ͮ + D͑␳12 − ␳22͔͒ d2␳1Јd2␳2Ј.
The two-point Stokes parameters are defined as follows [1]:
S0͑r1,r2,␻͒ = ͗Ex*͑r1,␻͒Ex͑r2,␻͒͘ + ͗Ey*͑r1,␻͒Ey͑r2,␻͒͘, ͑1a͒
S1͑r1,r2,␻͒ = ͗Ex*͑r1,␻͒Ex͑r2,␻͒͘ − ͗Ey*͑r1,␻͒Ey͑r2,␻͒͘, ͑1b͒
S2͑r1,r2,␻͒ = ͗Ex*͑r1,␻͒Ey͑r2,␻͒͘ + ͗Ey*͑r1,␻͒Ex͑r2,␻͒͘, ͑1c͒
S3͑r1,r2,␻͒ = i͓͗Ey*͑r1,␻͒Ex͑r2,␻͒͘ − ͗Ex*͑r1,␻͒Ey͑r2,␻͔͒͘, ͑1d͒
where the asterisk denotes the complex conjugate and the angular brackets stand for the average over the ensemble of realizations of the fluctuating electric fients in two mutually orthogonal directions perpendicular to the z axis. E = ͑Ex , Ey͒ is a member of the statistical ensemble of the fluctuating component of the transverse electric field. Within the framework of the paraxial approximation, a diffraction integral formula written in terms of ABCD elements is given as follows [12–14]:
OCIS codes: 260.5430, 030.1640.
1. INTRODUCTION
Stokes parameters are usually used to determine the polarization properties of an electromagnetic beam. Recently, the four Stokes parameters were generalized from one-point quantities into two-point counterparts, and their propagation properties through free space were also investigated [1]. Conservation laws for Stokes parameters of stochastic electromagnetic free fields were derived with the help of the angular spectrum [2]. It is interesting that two planar sources whose Stokes parameters are the same can generate beams with different degrees of polarization [3]. Stokes parameters in Young’s interference experiment were also investigated [4,5]. It has been shown that polarization properties of a stochastic electromagnetic beam, isotropic or anisotropic, can experience drastic changes on propagation in free space and in some optical systems [6–9]. The changes in the spectrum, polarization, and coherence of a partially coherent beam propagating through a gradient-index fiber have also been investigated [10]. Recently, the changes in the state of polarization were considered, and the size, shape, and orientation of the polarization ellipse of a partially coherent beam propagating through free space were investigated [11].
On the basis of the generalized diffraction integral formula for an ABCD optical system in the spatial domain, a propagation law for the generalized Stokes parameters of a stochastic electromagnetic beam passing through an ABCD optical system is obtained. We describe the Stokes parameters of the source as linear combinations of the elements of the cross-spectral density matrix, and study the changes in the spectral degree of polarization and in the state of the polarization ellipse of a stochastic electromagnetic Gaussian Schell-model beam propagating through a gradient-index fiber with the help of generalized Stokes parameters and the crossspectral density matrix. The medium has significant effect on the change of the spectral degree of polarization. However, when the correlation coefficients of the source satisfy the relation ␦xx = ␦yy = ␦xy = ␦yx, the medium does not influence the spectral degree of polarization. © 2008 Optical Society of America
2. THEORY
We consider an electromagnetic beam generated by a planar, secondary, stochastic, statistically stationary source located in the plane z = 0 and propagating close to the z direction into the half-space z Ͼ 0 (see Fig. 1).
1084-7529/08/081944-5/$15.00 © 2008 Optical Society of America
Y. Zhu and D. Zhao
Vol. 25, No. 8 / August 2008 / J. Opt. Soc. Am. A 1945
ͱS12͑␳,z,␻͒ + S22͑␳,z,␻͒ + S32͑␳,z,␻͒
P͑␳,z,␻͒ =
, ͑5͒
S0͑␳,z,␻͒
ͩ ͪ 1
S2͑␳,z,␻͒
␪ = arctan
,
͑6͒
2
S1͑␳,z,␻͒
ͩ ͪ 1
S3͑␳,z,␻͒
ͱ ␧ = arcsin
. ͑7͒
2
S12͑␳,z,␻͒ + S22͑␳,z,␻͒ + S32͑␳,z,␻͒
Fig. 1. Notation illustrated.
Yingbin Zhu and Daomu Zhao*
Department of Physics, Zhejiang University, Hangzhou 310027, China *Corresponding author: zhaodaomu@
Received March 18, 2008; revised May 26, 2008; accepted June 3, 2008; posted June 6, 2008 (Doc. ID 94021); published July 10, 2008
͑3͒
So the Stokes parameters propagating through a paraxial ABCD optical system can be treated with the following generalized formula (similar to [1]):