Matrix optics is applied to a class of random, in time and space, electromagnetic pulsed beam-like (REMPB) radiation interacting with linear optical elements. A order matrix describing transformation of a six-dimensional state vector including four spatial and two temporal positions within the field is used to derive conditions for spatio-temporal coupling. An example is included which deals with a spatio-temporal coupling in a typical REMPB on reflection from a reflecting grating. Electromagnetic nature of such interaction is explored via considering dependence of the degree of polarization of the reflected REMPB on its source and on the structure of the grating.
©2010 Optical Society of America
Matrix optics techniques serve as an extremely convenient tool in dealing with transformation of electromagnetic radiation by linear optical elements . Matrices of various dimensions and structures have been proposed for description of local transformation of beams and pulses of deterministic and random nature. Perhaps the most important classic theories belong to Jones  and Muller , which deal with modulation of electromagnetic fields and Stokes vectors, respectively, of stationary waves, providing with single-position transformations. The Jones and Mueller calculi were generalized to two-positions and unified by Korotkova and Wolf ,  taking into account transformation of field correlations. The very convenient tensor method was used by Lin and Cai  for description of a vide class of electromagnetic Gaussian Schell-model sources. This tensor method has proven to be a very convenient tool for solving problems relating to random light interaction with resonators  as well as LIDAR systems operating in turbulent atmosphere . All the aforementioned references, however, dealt with wide-sense stationary fields and did not include temporal variations. Matrix description of pulsed fields via 3x3 matrices was first introduced by Martinez ,  and extended to a more complete approach by Kostenbauder  which employed 4x4 matrices, and also to a more convenient ABCD (2x2) matrices by Dijaili et al. . Other extensions were proposed in Refs , .
Recently a class of electromagnetic pulses that exhibit non-stationary variations in time and are also spatially random was introduced by Paakkonen et al.  (see also , where only temporally random fields are discussed). Matrix treatment involving tensors which are constructed from ABCD cells was suggested for such fields and their free-space evolution was treated . Propagation, generation, coherent mode-expansion and ghost interference of such electromagnetic pulses were investigated widely [18–28].
Spatio-temporal coupling of electromagnetic pulsed waves on interaction with dispersive systems was known for long time (e.g. in  on focusing by a lens). The matrix approach to such coupling was introduced by Lin et al. , where the formalism of 6x6 matrices was used for reflection-induced coupling by a grating. That analysis was limited, however, to initially deterministic pulses, both in time and space. In a somewhat different perspective reflection of electromagnetic random (stationary) wave from a polarization grating was discussed by Piquero et al. , highlighting the aspect of polarization modulation.
We generalize the results of  to the class of REMPB and derive general conditions to be satisfied by the transformation matrix in order to describe interaction of REMPB with dispersive linear media, with and without spatio-temporal coupling. We also provide an example which provides coupling conditions for REMPB interacting with a reflecting grating and explore the effect of interaction and subsequent free-space propagation of the pulse from the grating on pulse’s degree of polarization.
We begin by reviewing matrix formalism of Ref . applied for REMPB, in which we employ the Gaussian Schell-model for temporal and spatial fluctuations of Refs . and . The notations are explained in Fig. 1 . We assume that the initial field does not carry spatiotemporal coupling. Hence, the elements of the mutual correlation matrix (MCM) of the pulsed beams are given by the expressions [27,28]6,17,30], Eq. (1) can be expressed in following tensor form
Just after interacting with a linear dispersive medium the mutual correlation matrix of the Gaussian Schell-model REMPB can be expressed by the formula (see Ref . for details of derivation):Eqs. (4) and (5) are also the matrices of the form30]. for matrices of typical dispersive elements. We will confine our analysis to the case when the reference plane is perpendicular to the direction of propagation of the field. Thus the optical elements must satisfy the relations, , and [17,30].
Matrix (6) may provide the information about the spatio-temporal coupling in pulsed beams on interacting with dispersive linear systems. If elements ,of this matrix remain non-zero [compare with corresponding elements of matrix (3)], then no coupling takes place, and it does otherwise.
On propagation from a dispersive element in free space the correlation matrix of the pulse undergoes the following change:19]1]. However, if the coupling has already occurred it may be further modified by free space propagation.
Using the Fourier-transform, the elements of the correlation matrices Γ0, Γ1 and Γ2 can be replaced by their counterparts in space-frequency domain:Eq. (10) can be used for calculation of all the second-order statistical properties of pulsed beams in what follows we will only consider the changes in the degree of polarization [32,33]:
3. Spatio-temporal coupling of REMPB reflected from a reflecting grating
To illustrate the general theory of Section 2 we will now provide an example dealing with spatio-temporal coupling and the evolution of the degree of polarization of a REMPB reflected from a reflecting grating.
In order to analyze the spatiotemporal coupling and its effects on transmission characteristics, we consider a REMPB reflected by the reflecting grating and propagating in the free space, as shown in Fig. 2 . The grooves of the grating are assumed to coincide with y-direction.Fig. 2), which satisfy the grating equationEq. (12)) was first derived in  by only considering the effect of interference and diffraction of reflecting grating on light, and the Fresnel reflection coefficient and the absorption of reflecting grating were not taken into consideration. This assumption has been adopted in many previous literatures. For the most general case, the Fresnel reflection coefficient and absorption of reflecting grating should be taken into consideration, while up to now, no such general matrix was developed, and we leave this for future study.Eq. (16) that since and are not zero, there exist spatiotemporal coupling. It means that only the pulsed beam of zero diffraction order m ([given values of ψ and φ]) has no spatiotemporal coupling. Also it is seen from Eqs. (15) and (16) that the spatial-temporal coupling of the pulsed beam at output plane depends on the diffraction order of grating, i.e. reflective angle of the pulsed beam. For the sake of convenience we confine our analysis to values of degree of polarization when , i.e. on the axis of the beam in space domain, and at the pulse center in time domain. For this position, on using Eqs. (8) and (9), we find that the correlation matrix takes the formEqs. (17)-(19) that if the incident angle and the reflected angle coincide, i.e. if , then and and, hence is not spatial-temporal coupling. Equation (19) then reduces toEq. (20) into Eq. (18), we find that the result for the degree of polarization of the pulsed beam agrees with that of reference  by Ding et al.
If then the spatial-temporal coupling occurs and, as it can be seen from the Eq. (18), it affects the evolution of degree of polarization. In particular in the limiting case of the far-field the degree of polarization is given by the expression32]. The degree of coherence of a REMPB increases on propagation, and the REMPB approaches to a coherent electromagnetic pulse beam in the far field, thus its degree of polarization tends to a constant value in the far field.
4. Numerical results
We will now discuss the results of the previous section with the help of numerical calculations of the degree of polarization of the REMPB after its interaction with a reflection grating. Unless it is specified otherwise in figure captions the following parameters of the source of the pulse and the grating were selected: , , , , , , , , . Also, we set and , where and represent the pulsed duration and the temporal degree of coherence respectively.
In Figs. 3 -6 we demonstrate with the help of three-dimensional plots of the degree of polarization varying with propagation distance z from the reflecting grating and several parameters of the pulse, for several chosen diffraction orders of the grating (m = 0, −1, −7). In particular, Fig. 3 shows the effect of pulse duration, Fig. 4 that of r.m.s. width of the intensity, Fig. 5 that of r.m.s. width of temporal degree of coherence and Fig. 6 that of r.m.s. width of spatial degree of coherence (while the other correlations are kept fixed). One sees from these plots that only in case when m = 0 no spatio-temporal coupling occurs in the reflected pulsed beam. In order to provide quantitatively better results we show in Figs. 7 -11 two dimensional plots the degree of polarization depending on various source and system parameters, for several fixed values of diffraction order m. In particular, in Fig. 7 the dependence of the degree of polarization on propagation distance from the source is given, in the rest of the figures the distance was kept fixed while the parameters of the source were varied. One notices that in some cases, such as in Fig. 7 and 9 the degree of polarization does not depend on propagation distance and r.m.s intensity in a monotonic way.
With the help of matrix optics, we have analyzed the interaction of a REMPB with linear optical systems. As an application example, the degree of polarization of a typical REMPB reflected from a reflecting grating was explored. It is found that spatio-temporal coupling occurs in the reflected REMPB of nonzero diffraction order, and the statistics properties of the reflected REMPB depend closely on the parameters of the source beam and the reflecting grating.
Min Yao acknowledges the support by Scientific Research Fund of Zhejiang Provincial Education Department, China (Grant No. Y200908631). Yangjian Cai acknowledges the support by the National Natural Science Foundation of China under Grant No. 10904102, the Foundation for the Author of National Excellent Doctoral Dissertation of PR China under Grant No. 200928, the Natural Science of Jiangsu Province under Grant No. BK2009114, the Huo Ying Dong Education Foundation of China under Grant No. 121009 and the Key Project of Chinese Ministry of Education under Grant No. 210081. O. Korotkova's research is funded by the US AFOSR (grant FA 95500810102) and US ONR (Grant N0018909P1903). Qiang Lin acknowledges the support by the National Natural Science Foundation of China under Grant No. 60925022. Zhaoying Wang acknowledges the support by the Zhejiang Provincial Qian-Jiang-Ren-Cai Project of China (2009R10034) and the Fundamental Research Funds for the Central Universities (2010QNA3024).
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