Spatio-temporal coupling of random electromagnetic pulses interacting with reflecting gratings

Min Yao; Yangjian Cai; Olga Korotkova; Qiang Lin; Zhaoying Wang

doi:10.1364/OE.18.022503

1. 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].

Recently a class of electromagnetic pulses that exhibit non-stationary variations in time and are also spatially random was introduced by Paakkonen et al. [15] (see also [16], where only temporally random fields are discussed). Matrix treatment involving $6 \times 6$ tensors which are constructed from ABCD cells was suggested for such fields and their free-space evolution was treated [17]. 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 [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.

2. 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]

\begin{array}{l} Γ_{0 α β} (r_{01}, τ_{01}; r_{02}, τ_{02}) = A_{α} A_{β} B_{α β} \exp [- \frac{r_{01}^{2} + r_{02}^{2}}{4 σ_{I 0 α β}^{2}} - \frac{{(r_{01} - r_{02})}^{2}}{2 δ_{I 0 α β}^{2}}] \\ \times \exp [- \frac{τ_{01}^{2} + τ_{02}^{2}}{2 σ_{τ 0 α β}^{2}} - \frac{{(τ_{01} - τ_{02})}^{2}}{2 δ_{τ 0 α β}^{2}}] \exp [i ω_{0} (τ_{01} - τ_{02})], \end{array}

where

τ_{0} = c (t - t_{0})

and

r_{0}

represent longitudinal and transverse positions in the pulsed beams, respectively;

c = λ_{0} ν_{0}

is light velocity in vacuum, and

t_{0}

is an arrival time at a certain transverse plane;

σ_{I 0 α β}

and

δ_{I 0 α β}

are the r.m.s. widths of the intensity and of the degree of coherence in transverse plane, respectively;

σ_{τ 0 α β}

denotes the r.m.s. longitudinal width, being the averaged pulse duration;

δ_{τ 0 α β}^{}

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

Γ_{0 α β} ({\bar{r}}_{0}) = A_{α} A_{β} B_{α β} \exp [- \frac{i k_{0}}{2} {\bar{r}}_{0}^{T}^{} {\bar{Q}}_{0 α β}^{- 1} {\bar{r}}_{0}] \exp [i ω_{0} (τ_{01} - τ_{02})], (α = x, y; β = x, y)

with

{\bar{Q}}_{0 α β}^{- 1} = (\begin{matrix} - \frac{i}{k_{0}} (\frac{1}{2} σ_{I 0 α β}^{- 2} + δ_{I 0 α β}^{- 2}) I & 0_{21} & \frac{i}{k_{0}} δ_{I 0 α β}^{- 2} I & 0_{21} \\ 0_{12} & - \frac{i}{k_{0}} (σ_{τ 0 α β}^{- 2} + δ_{τ 0 α β}^{- 2}) & 0_{12} & \frac{i}{k_{0}} δ_{τ 0 α β}^{- 2} \\ \frac{i}{k_{0}} δ_{I 0 α β}^{- 2} I & 0_{21} & - \frac{i}{k_{0}} (\frac{1}{2} σ_{I 0 α β}^{- 2} + δ_{I 0 α β}^{- 2}) I & 0_{21} \\ 0_{12} & \frac{i}{k_{0}} δ_{τ 0 α β}^{- 2} & 0_{12} & - \frac{i}{k_{0}} (σ_{τ 0 α β}^{- 2} + δ_{τ 0 α β}^{- 2}) \end{matrix})

Here I is the

2 \times 2

identity matrix.

0_{21}

is

2 \times 1

zero matrix, and

0_{12}

is

1 \times 2

zero matrix.

{\bar{r}}_{0}^{T} = (\begin{matrix} {\tilde{r}}_{01}^{T} & {\tilde{r}}_{02}^{T} \end{matrix})

and

{\tilde{r}}_{0 i}^{T} = (\begin{matrix} x_{0 i} & y_{0 i} & τ_{0 i} \end{matrix})

(i = 1, 2)

;

k_{0} = ω_{0} / c

is the wave number and the corresponding

ω_{0}

is the central angular frequency;

A_{x}

and

A_{y}

are the amplitudes of x and y components of the electric field, respectively;

B_{α β} = | B_{α β} | \exp (i ϕ_{α β}) = B_{β α}^{*}

is the correlation coefficient between

E_{x}

and

E_{y}

field components.

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

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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):

Γ_{1 α β} ({\bar{r}}_{1}) = A_{α} A_{β} B_{α β} {[\det ({\bar{A}}_{1} + {\bar{B}}_{1} {\bar{Q}}_{0 α β}^{- 1})]}^{- 1 / 2} \exp [- \frac{i k_{0}}{2} {\bar{r}}_{1}^{T}^{} {\bar{Q}}_{1 α β}^{- 1} {\bar{r}}_{1}],

where

{\bar{r}}_{1}^{T} = (\begin{matrix} {\tilde{r}}_{11}^{T} & {\tilde{r}}_{12}^{T} \end{matrix})

, det stands for the determinant of a matrix and

{\bar{Q}}_{1 α β}^{- 1} = ({\bar{C}}_{1} + {\bar{D}}_{1} {\bar{Q}}_{0 α β}^{- 1}) {({\bar{A}}_{1} + {\bar{B}}_{1} {\bar{Q}}_{0 α β}^{- 1})}^{- 1} .

{\bar{Q}}_{0 α β}^{- 1}

and

{\bar{Q}}_{1 α β}^{- 1}

denote spatiotemporal complex curvature tensor of input and output plane.

{\bar{Q}}_{1 α β}^{- 1}

is a

6 \times 6

matrix of the form

{\bar{Q}}_{1 α β}^{- 1} = (\begin{matrix} q_{11} & q_{12} & q_{13} & q_{14} & q_{15} & q_{16} \\ q_{21} & q_{22} & q_{23} & q_{24} & q_{25} & q_{26} \\ q_{31} & q_{32} & q_{33} & q_{34} & q_{35} & q_{36} \\ q_{41} & q_{42} & q_{43} & q_{44} & q_{45} & q_{46} \\ q_{51} & q_{52} & q_{53} & q_{54} & q_{55} & q_{56} \\ q_{61} & q_{62} & q_{63} & q_{64} & q_{65} & q_{66} \end{matrix}) .

In Eqs. (4) and (5)

{\bar{A}}_{1}, {\bar{B}}_{1}, {\bar{C}}_{1} and {\bar{D}}_{1}

are also the

6 \times 6

matrices of the form

{\bar{A}}_{1} = (\begin{matrix} {\tilde{A}}_{11} & 0_{33} \\ 0_{33} & {\tilde{A}}_{12} \end{matrix}), {\bar{B}}_{1} = (\begin{matrix} {\tilde{B}}_{11} & 0_{33} \\ 0_{33} & - {\tilde{B}}_{12}^{*} \end{matrix}), {\bar{C}}_{1} = (\begin{matrix} {\tilde{C}}_{11} & 0_{33} \\ 0_{33} & - {\tilde{C}}_{12}^{*} \end{matrix}), D_{1} = (\begin{matrix} {\tilde{D}}_{11} & 0_{33} \\ 0_{33} & {\tilde{D}}_{12} \end{matrix})

where “*” denotes complex conjugate.

{\tilde{A}}_{11}, {\tilde{B}}_{11}, {\tilde{C}}_{11}, {\tilde{D}}_{11}

and

{\tilde{A}}_{12}, {\tilde{B}}_{12}, {\tilde{C}}_{12}, {\tilde{D}}_{12}

are the

3 \times 3

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

{\tilde{A}}_{11} = {\tilde{A}}_{12} = {\tilde{A}}_{1}

,

{\tilde{B}}_{11} = {\tilde{B}}_{12} = {\tilde{B}}_{1}

,

{\tilde{C}}_{11} = {\tilde{C}}_{12} = {\tilde{C}}_{1}

and

{\tilde{D}}_{11} = {\tilde{D}}_{12} = {\tilde{D}}_{1}

[17,30].

Matrix (6) may provide the information about the spatio-temporal coupling in pulsed beams on interacting with dispersive linear systems. If elements $q_{13}, q_{16}, q_{23}, q_{26},$ $q_{31}, q_{32}, q_{34}, q_{35},$ $q_{43}, q_{46}, q_{53}, q_{56}, q_{61}, q_{62}, q_{64}$ , $q_{65}$ 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:

Γ_{2 α β} ({\bar{r}}_{2}) = A_{α} A_{β} B_{α β} {[\det ({\bar{A}}_{1} + {\bar{B}}_{1} {\bar{Q}}_{0 α β}^{- 1})]}^{- 1 / 2} {[\det ({\bar{A}}_{2} + {\bar{B}}_{2} {\bar{Q}}_{1 α β}^{- 1})]}^{- 1 / 2} \exp [- \frac{i k_{0}}{2} {\bar{r}}_{2}^{T} {\bar{Q}}_{2 α β}^{- 1} {\bar{r}}_{2}],

where

{\bar{r}}_{2}^{T} = (\begin{matrix} {\tilde{r}}_{21}^{T} & {\tilde{r}}_{22}^{T} \end{matrix})

and

{\bar{Q}}_{2 α β}^{- 1} = ({\bar{C}}_{2} + {\bar{D}}_{2} {\bar{Q}}_{1 α β}^{- 1}) {({\bar{A}}_{2} + {\bar{B}}_{2} {\bar{Q}}_{1 α β}^{- 1})}^{- 1} .

The submatrices of a matrix describing free space propagation have the form [19]

{\tilde{A}}_{21} = {\tilde{D}}_{21} = (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}), {\tilde{B}}_{21} = (\begin{matrix} z & 0 & 0 \\ 0 & z & 0 \\ 0 & 0 & 1 \end{matrix}), {\tilde{C}}_{21} = (\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) .

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 Γ₀, Γ₁ and Γ₂ can be replaced by their counterparts in space-frequency domain:

W_{j α β} (r_{j 1}, ω_{1}; r_{j 2}, ω_{2}) = \frac{1}{2 π} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} Γ_{j α β} (r_{j 1}, τ_{j 1}; r_{j 2}, τ_{j 2}) \exp [- (ω_{1} τ_{j 1} - ω_{2} τ_{j 2})] d τ_{j 1} d τ_{j 2},

where

j = 0, 1, 2.

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]:

P_{j}^{ω} (r_{j}, ω) = \sqrt{1 - \frac{4 D e t [{\overset{\leftrightarrow}{W}}_{j} (r_{j}, ω, r_{j}, ω)]}{{T r [{\overset{\leftrightarrow}{W}}_{j} (r_{j}, ω, r_{j}, ω)]}^{2}}},

where

j = 0, 1, 2.

We believe the evolution of the degree of polarization on interaction with dispersive systems was not previously studied.

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 Illustrating notation for reflection of a pulse from a reflecting grating.

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The submatrices of 6x6 matrix describing reflecting grating have the form [12,30]

{\tilde{A}}_{11} = (\begin{matrix} - \frac{\sin ϕ}{\sin ψ} & 0 & 0 \\ 0 & 1 & 0 \\ \frac{\cos ψ - \cos ϕ}{\sin ψ} & 0 & 1 \end{matrix}), {\tilde{B}}_{11} = {\tilde{C}}_{11} = 0, {\tilde{D}}_{11} = (\begin{matrix} - \frac{\sin ψ}{\sin ϕ} & 0 & \frac{\cos ψ - \cos ϕ}{\sin ϕ} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}),

where ψ and ϕ are the incident and the reflective angles, respectively (see Fig. 2), which satisfy the grating equation

d_{G} (\cos ϕ - \cos ψ) = m λ_{0},

with

d_{G}

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:

Γ_{1 α β} ({\bar{r}}_{1}) = A_{α} A_{β} B_{α β} \frac{\sin ψ}{\sin ϕ} \exp [- \frac{i k_{0}}{2} {\bar{r}}_{1}^{T} {\bar{Q}}_{1 α β}^{- 1} {\bar{r}}_{1}],

where

{\bar{Q}}_{1 α β}^{- 1}

has the form

{\bar{Q}}_{1 α β}^{- 1} = (\begin{matrix} q_{11} & 0 & q_{13} & q_{14} & 0 & q_{16} \\ 0 & q_{22} & 0 & 0 & q_{25} & 0 \\ q_{31} & 0 & q_{33} & q_{34} & 0 & q_{36} \\ q_{41} & 0 & q_{43} & q_{44} & 0 & q_{46} \\ 0 & q_{52} & 0 & 0 & q_{55} & 0 \\ q_{61} & 0 & q_{63} & q_{64} & 0 & q_{66} \end{matrix}) .

with

\begin{array}{l} q_{11} = q_{44} = - i \frac{g_{1}^{2}}{2 k_{0}} (σ_{I 0 α β}^{- 2} + 2 δ_{I 0 α β}^{- 2}) - i \frac{g_{2}^{2}}{k_{0}} (σ_{τ 0 α β}^{- 2} + δ_{τ 0 α β}^{- 2}), \\ q_{13} = q_{31} = q_{46} = q_{64} = - i \frac{g_{2}}{k_{0}} (σ_{τ 0 α β}^{- 2} + δ_{τ 0 α β}^{- 2}), \\ q_{14} = q_{41} = \frac{i}{k_{0}} (g_{1}^{2} δ_{I 0 α β}^{- 2} + g_{2}^{2} δ_{τ 0 α β}^{- 2}), q_{16} = q_{34} = q_{43} = q_{61} = i \frac{g_{2}}{k_{0}} δ_{τ 0 α β}^{- 2}, \\ q_{22} = q_{55} = - \frac{i}{2 k_{0}} (σ_{I 0 α β}^{- 2} + 2 δ_{I 0 α β}^{- 2}), q_{25} = q_{52} = \frac{i}{k_{0}} δ_{I 0 α β}^{- 2}, \\ q_{33} = q_{66} = - \frac{i}{k_{0}} (σ_{τ 0 α β}^{- 2} + δ_{τ 0 α β}^{- 2}), q_{36} = q_{63} = \frac{i}{k_{0}} δ_{τ 0 α β}^{- 2} . \end{array}

Here

g_{1} = \sin ψ / \sin ϕ

and

g_{2} = (\cos ψ - \cos ϕ) / \sin ϕ

. It can be seen from Eq. (16) that since

q_{13}

and

q_{16}

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

{\bar{r}}_{2} = 0

, 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

\begin{array}{l} Γ_{2 α β} = A_{α} A_{β} B_{α β} g_{1} {[1 + \frac{z^{2}}{4 k_{0}^{2}} σ_{I 0 α β}^{- 2} (4 δ_{I 0 α β}^{- 2} + σ_{I 0 α β}^{- 2})]}^{- 1 / 2} \\ \times {1 + \frac{z^{2}}{4 k_{0}^{2}} (g_{1}^{2} σ_{I 0 α β}^{- 2} + 2 g_{2}^{2} σ_{τ 0 α β}^{- 2}) [g_{1}^{2} (4 δ_{I 0 α β}^{- 2} + σ_{I 0 α β}^{- 2}) + 2 g_{2}^{2} (2 δ_{τ 0 α β}^{- 2} + σ_{τ 0 α β}^{- 2})]}^{- 1 / 2}, \end{array}

and the degree of polarization becomes

P^{τ} (z) = \sqrt{1 - \frac{4 D e t ({\overset{\leftrightarrow}{Γ}}_{2})}{T r ({\overset{\leftrightarrow}{Γ}}_{2})}} = | \frac{Γ_{2 x x} - Γ_{2 y y}}{Γ_{2 x x} + Γ_{2 y y}} | = | \frac{A_{x}^{2} Δ_{x x}^{} - A_{y}^{2} Δ_{y y}^{}}{A_{x}^{2} Δ_{x x}^{} + A_{y}^{2} Δ_{y y}^{}} |

with

\begin{array}{l} Δ_{α β}^{} = {[1 + \frac{z^{2}}{4 k_{0}^{2}} σ_{I 0 α β}^{- 2} (4 δ_{I 0 α β}^{- 2} + σ_{I 0 α β}^{- 2})]}^{- 1 / 2} \\ \times {1 + \frac{z^{2}}{4 k_{0}^{2}} (g_{1}^{2} σ_{I 0 α β}^{- 2} + 2 g_{2}^{2} σ_{τ 0 α β}^{- 2}) [g_{1}^{2} (4 δ_{I 0 α β}^{- 2} + σ_{I 0 α β}^{- 2}) + 2 g_{2}^{2} (2 δ_{τ 0 α β}^{- 2} + σ_{τ 0 α β}^{- 2})]}^{- 1 / 2} . \end{array}

It is seen from Eqs. (17)-(19) that if the incident angle and the reflected angle coincide, i.e. if

ψ = ϕ

, then

g_{1} = 1

and

g_{2} = 0

and, hence is not spatial-temporal coupling. Equation (19) then reduces to

Δ_{α β} = {[1 + \frac{z^{2}}{4 k_{0}^{2}} σ_{I 0 α β}^{- 2} (4 δ_{I 0 α β}^{- 2} + σ_{I 0 α β}^{- 2})]}^{- 1} .

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 $ψ \neq ϕ$ 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

\lim_{z \to \infty} P^{τ} (z) = | \frac{A_{x}^{2} Ω_{x x} - A_{y}^{2} Ω_{y y}}{A_{x}^{2} Ω_{x x} + A_{y}^{2} Ω_{y y}} |,

where

Ω_{α β}

is a parameter independent of the propagation distance z, and given by the formula

\begin{array}{l} Ω_{α β} = {σ_{I 0 α β}^{- 2} (4 δ_{I 0 α β}^{- 2} + σ_{I 0 α β}^{- 2}) (g_{1}^{2} σ_{I 0 α β}^{- 2} + 2 g_{2}^{2} σ_{τ 0 α β}^{- 2})}^{- 1 / 2} \\ \times {[g_{1}^{2} (4 δ_{I 0 α β}^{- 2} + σ_{I 0 α β}^{- 2}) + 2 g_{2}^{2} (2 δ_{τ 0 α β}^{- 2} + σ_{τ 0 α β}^{- 2})]}^{- 1 / 2} \end{array}

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

γ ({\bar{r}}_{1}, {\bar{r}}_{2}) = T r [\overset{\leftrightarrow}{Γ} ({\bar{r}}_{1}, {\bar{r}}_{2})] / \sqrt{T r [\overset{\leftrightarrow}{Γ} ({\bar{r}}_{1}, {\bar{r}}_{1})] T r [\overset{\leftrightarrow}{Γ} ({\bar{r}}_{2}, {\bar{r}}_{2})]}

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: $A_{x} = \sqrt{3 / 2},$ $A_{y} = 1$ , $B_{x x} = B_{y y} = 1,$ $B_{x y} = B_{y x} = 0$ , $σ_{I 0 x x}^{} = σ_{I 0 y y}^{} = σ_{I 0}^{}$ , $δ_{I 0 y y}^{} = 2 δ_{I 0 x x}^{}$ , $σ_{τ 0 x x}^{} = σ_{τ 0 y y}^{} = σ_{τ 0}^{}$ , $δ_{τ 0 x x}^{} = δ_{τ 0 y y}^{} = δ_{τ 0}^{}$ , $ω_{0} = 2.36 r a d f s^{- 1}$ , $d_{G} = 1 / 600 m m$ , $ψ = 30^{0}$ . Also, we set $σ_{t 0}^{} = σ_{τ 0}^{} / c$ and $δ_{t 0}^{} = δ_{τ 0}^{} / c$ , where $σ_{t 0}^{}$ and $δ_{t 0}^{}$ 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 $σ_{t 0}$ , Fig. 4 that of r.m.s. width $σ_{I 0}$ of the intensity, Fig. 5 that of r.m.s. width of temporal degree of coherence $δ_{t 0}$ and Fig. 6 that of r.m.s. width of spatial degree of coherence $δ_{I 0 x x}$ (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.

Fig. 3 Dependence of the degree of polarization of a REMPB on the propagation distance z and the pulse duration $σ_{t 0}$ for different diffraction orders m with $σ_{I 0}^{} = 1 m m$ , $δ_{I 0 x x}^{} = 1 m m$ , $δ_{t 0}^{} = 2 p s .$

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Fig. 6 Dependence of the degree of polarization P on the propagation distance z and the r.m.s. width of spatial degree of coherence $δ_{I 0 x x}$ for different diffraction orders m with $σ_{I 0}^{} = 1 m m$ , $σ_{t 0}^{} = 2 p s$ , $δ_{t 0}^{} = 2 p s .$

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Fig. 4 Dependence of the degree of polarization P of a REMPB on the propagation distance z and the r.m.s. width $σ_{I 0}$ of the intensity for different diffraction orders m with $δ_{I 0 x x}^{} = 1 m m$ , $σ_{t 0}^{} = 2 p s$ , $δ_{t 0}^{} = 2 p s .$

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Fig. 5 Dependence of the degree of polarization P on the propagation distance z and the r.m.s. width of temporal degree of coherence $δ_{t 0}$ for different diffraction orders m with $σ_{I 0}^{} = 1 m m$ , $δ_{I 0 x x}^{} = 1 m m$ , $σ_{t 0}^{} = 2 p s .$

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Fig. 7 Dependence of the degree of polarization P on the propagation distance z for different diffraction orders m with $σ_{I 0}^{} = 1 m m$ , $δ_{I 0 x x}^{} = 1 m m$ , $σ_{t 0}^{} = 2 p s$ , $δ_{t 0}^{} = 2 p s .$

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Fig. 11 Dependence of the degree of polarization P on the r.m.s. width of spatial degree of coherence $δ_{I 0 x x}$ for different diffraction orders m at $z = 200 m$ with $σ_{I 0}^{} = 1 m m$ , $σ_{t 0}^{} = 2 p s$ , $δ_{t 0}^{} = 2 p s .$

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Fig. 9 Dependence of the degree of polarization P on the r.m.s. width $σ_{I 0}$ of the intensity for different diffraction orders m at z = 200m with $δ_{I 0 x x}^{} = 1 m m$ , $σ_{t 0}^{} = 2 p s$ , $δ_{t 0}^{} = 2 p s .$

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Fig. 8 Dependence of the degree of polarization P on the pulse duration $σ_{t 0}$ for different diffraction orders m at z = 200m with $σ_{I 0}^{} = 1 m m$ , $δ_{I 0 x x}^{} = 1 m m$ , $δ_{t 0}^{} = 2 p s .$

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Fig. 10 Dependence of the degree of polarization P on the r.m.s. width of temporal degree of coherence $δ_{t 0}$ for different diffraction orders m at z = 200m with $σ_{I 0}^{} = 1 m m$ , $δ_{I 0 x x}^{} = 1 m m$ , $δ_{t 0}^{} = 2 p s .$

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5. 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.

Acknowledgments

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).

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. Swindel, Polarized light, (Dowden, Hutchinson & Ross, (Stroudsburg, Pennsylvania, 1975).

3. H. Mueller, “The foundations of optics,” J. Opt. Soc. Am. 38, 661–661 (1948); for account of the Mueller’s theory see also E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, Inc., New York, 1993). Chap. 5.

4. O. Korotkova and E. Wolf, “Effects of linear non-image forming devices on coherence and polarization properties of random electromagnetic beams. Part I. General theory,” J. Mod. Opt. 52, 2659–2671 (2005). [CrossRef]

5. O. Korotkova and E. Wolf, “Effects of linear non-image forming devices on spectra and on coherence and polarization properties ofstochastic electromagnetic beams. Part II. Examples,” J. Mod. Opt. 52, 2673–2685 (2005). [CrossRef]

6. Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002). [CrossRef]

7. M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008). [CrossRef] [PubMed]

8. O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009). [CrossRef]

9. O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24(12), 2530–2536 (1988). [CrossRef]

10. O. E. Martinez, “Matrix formalism for dispersive laser cavities,” IEEE J. Quantum Electron. 25(3), 296–300 (1989). [CrossRef]

11. A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26(6), 1148–1157 (1990). [CrossRef]

12. S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26(6), 1158–1164 (1990). [CrossRef]

13. Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Transformation of pulsed nonideal beams in a four-dimension domain,” Opt. Lett. 18(9), 669–671 (1993). [CrossRef] [PubMed]

14. P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16(4), 196–198 (1991). [CrossRef] [PubMed]

15. P. Paakkonen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002). [CrossRef]

16. Q. Lin, L. Wang, and S. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003). [CrossRef]

17. L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003). [CrossRef] [PubMed]

18. H. Lajunen, J. Tervo, and P. Vahimaa, “Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses,” J. Opt. Soc. Am. A 21(11), 2117–2123 (2004). [CrossRef]

19. Y. Cai and Q. Lin, “The fractional Fourier transform for a partially coherent pulse,” J. Opt. A, Pure Appl. Opt. 6(4), 307–311 (2004). [CrossRef]

20. J. Lancis, V. Torres-Company, E. Silvestre, and P. Andrés, “Space-time analogy for partially coherent plane-wave-type pulses,” Opt. Lett. 30(22), 2973–2975 (2005). [CrossRef] [PubMed]

21. H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 255(1-3), 12–22 (2005). [CrossRef]

22. H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22(8), 1536–1545 (2005). [CrossRef]

23. V. Torres-Company, G. Mínguez-Vega, J. Lancis, and A. T. Friberg, “Controllable generation of partially coherent light pulses with direct space-to-time pulse shaper,” Opt. Lett. 32(12), 1608–1610 (2007). [CrossRef] [PubMed]

24. A. T. Friberg, H. Lajunen, and V. Torres-Company, “Spectral elementary-coherence-function representation for partially coherent light pulses,” Opt. Express 15(8), 5160–5165 (2007). [CrossRef] [PubMed]

25. V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, “Ghost interference with classical partially coherent light pulses,” Phys. Rev. A 77(4), 043811 (2008). [CrossRef]

26. H. Lajunen, V. Torres-Company, J. Lancis, E. Silvestre, and P. Andrès, “Pulse-by-pulse method to characterize partially coherent pulse propagation in instantaneous nonlinear media,” Opt. Express 18(14), 14979–14991 (2010). [CrossRef] [PubMed]

27. C. Ding, L. Pan, and B. Lu, “Characterization of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams,” N. J. Phys. 11(8), 083001 (2009). [CrossRef]

28. C. Ding and B. Lu, “Spectral shifts and spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in the far field,” J. Opt. A, Pure Appl. Opt. 10(9), 095006 (2008). [CrossRef]

29. M. Kempe, U. Stamm, B. Wilhelmi, and W. Rudolph, “Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems,” J. Opt. Soc. Am. B 9(7), 1158–1165 (1992). [CrossRef]

30. Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Spatial-temporal coupling in grating-pair pulse compression system analysed by matrix optics,” Opt. Quantum Electron. 27(7), 679–692 (1995). [CrossRef]

31. G. Piquero, R. Borghi, and M. Santarsiero, “Gaussian Schell-model beams propagating through polarization gratings,” J. Opt. Soc. Am. A 18(6), 1399–1405 (2001). [CrossRef]

32. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic fields,” Phys. Lett. A 312(5-6), 263–267 (2003). [CrossRef]

33. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998). [CrossRef]

Spatio-temporal coupling of random electromagnetic pulses interacting with reflecting gratings

Abstract

1. Introduction

2. Theory

3. Spatio-temporal coupling of REMPB reflected from a reflecting grating

4. Numerical results

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (11)

Equations (22)

Optics Express