Spatio-temporal Coupling of Random Electromagnetic Pulses Interacting With Reflecting Gratings

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 6 6 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 OCIS codes: (030.1670) Coherent optical effects; (350.5500) Propagation References and links 1. S. M. Wang and D. M. Zhao, Matrix Optics, (Springer, 2000). 2. For the collection of original papers by R. C. Jones see W. 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Introduction
Matrix optics techniques serve as an extremely convenient tool in dealing with transformation of electromagnetic radiation by linear optical elements [1].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 [2] and Muller [3], 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 [4], [5] taking into account transformation of field correlations.The very convenient tensor method was used by Lin and Cai [6] 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 [7] as well as LIDAR systems operating in turbulent atmosphere [8].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 [9], [10] and extended to a more complete approach by Kostenbauder [11] which employed 4x4 matrices, and also to a more convenient ABCD (2x2) matrices by Dijaili et al. [12].Other extensions were proposed in Refs [13], [14].
Spatio-temporal coupling of electromagnetic pulsed waves on interaction with dispersive systems was known for long time (e.g. in [29] on focusing by a lens).The matrix approach to such coupling was introduced by Lin et al. [30], 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. [31], highlighting the aspect of polarization modulation.
We generalize the results of [30] 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.

Theory
We begin by reviewing matrix formalism of Ref [30].applied for REMPB, in which we employ the Gaussian Schell-model for temporal and spatial fluctuations of Refs [27].and [28].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]

G
stands for the r.m.s.width of longitudinal degree of coherence, describing the state of temporal coherence.According to the tensor method [6,17,30], Eq. ( 1) can be expressed in following tensor form Here I is the 2 2 u identity matrix.21 0 is 2 1  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 [17].for details of derivation): , det stands for the determinant of a matrix and Q and 1 1DE Q denote spatiotemporal complex curvature tensor of input and output plane. .q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q In Eqs. ( 4) and ( 5)  , , , where "*" denotes complex conjugate.matrices of the dispersive optical elements (see Appendix A of Ref [30].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 11  [17,30].Matrix ( 6) may provide the information about the spatio-temporal coupling in pulsed beams on interacting with dispersive linear systems.If elements 13 16 23 26 , , , , q q q q 31 32 34 35 , , , , q q q q 43 46 53 56 61 62 64 , , , , , , q q q q q q q , 65 q of this matrix remain non-zero [compare with corresponding elements of matrix (3)], then no coupling takes place, and it does otherwise.
In the most cases, there is no spatiotemporal coupling in pulsed beams on free-space propagation [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 Γ 0 , Γ 1 and Γ 2 can be replaced by their counterparts in space-frequency domain: Even though matrices in 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]: where 0, 1, 2. j We believe the evolution of the degree of polarization on interaction with dispersive systems was not previously studied.

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 ydirection.where \ and I are the incident and the reflective angles, respectively (see Fig. 2), which satisfy the grating equation with G d being the groove spacing and m being the diffraction order.The matrix describing reflecting grating (Eq.( 12)) was first derived in [12] 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.Using Eqs. ( 4)-( 7) and ( 12) we can easily obtain the elements of the correlation matrix just after reflection: where 0 0 0 0 0 0 0 0 .0 0 0 0 0 0 0 0 q q q q q q q q q q q q q q q q q q q q Here 1 sin / sin g \ I and 2 cos cos / sin g \ I I . It can be seen from Eq. ( 16) that since 13 q and 16 q 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 2 0 r , 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 form and the degree of polarization becomes It is seen from Eqs. ( 17 On substituting Eq. ( 20) into Eq.( 18), we find that the result for the degree of polarization of the pulsed beam agrees with that of reference [27] by Ding et al.
If \ I z 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 expression where DE : is a parameter independent of the propagation distance z, and given by the formula Thus the degree of polarization tends to a constant value after the pulsed beam travels in free space at a sufficiently large distance.It is well known that the degree of polarization of a coherent electromagnetic beam remains invariant on propagation in free space, while the degree of polarization of a stochastic electromagnetic beam varies on propagation in the near field and in the intermediate propagation distances [32].The degree of coherence

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: 3/ 2,

Conclusion
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.
.m.s.widths of the intensity and of the degree of coherence in transverse plane, respectively; 0 W DE V denotes the r.m.s.longitudinal width, being the averaged pulse duration; 0 W DE

Z
is the central angular frequency; x A and y A are the amplitudes of x and y components of the electric field, respectively;

Fig. 1 .
Fig.1.Propagation scheme for a pulse beam through dispersive optical system and free space.

D
are also the 6 6 u matrices of the form

Fig. 2 .
Fig. 2. Illustrating notation for reflection of a pulse from a reflecting grating.The submatrices of 6x6 matrix describing reflecting grating have the form[12,30] 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.

Fig. 3 .Fig. 4 .Fig. 5 .
Fig. 3. Dependence of the degree of polarization of a REMPB on the propagation distance z and the pulse duration 0 t V for different diffraction orders m with 0 1 I

Fig. 11 .
Fig. 11.Dependence of the degree of polarization P on the r.m.s.width of spatial degree of coherence 0 I xx G (19)9)that if the incident angle and the reflected angle coincide, i.e. if