Researching | Far-zone behaviors of scattering-induced statistical properties of partially polarized spatially and spectrally partially coherent electromagnetic pulsed beam

Journals >Chinese Physics B >Volume 29 >Issue 10 >Page > Article

Chinese Physics B
Vol. 29, Issue 10, (2020)

Far-zone behaviors of scattering-induced statistical properties of partially polarized spatially and spectrally partially coherent electromagnetic pulsed beam

Yan Li, Ming Gao^†, Hong Lv, Li-Guo Wang, and Shen-He Ren

Author Affiliations

School of Optoelectronic Engineering, Xi’an Technological University, Xi’an 710021, China

show less

DOI: 10.1088/1674-1056/ab9de7 Cite this Article

Yan Li, Ming Gao, Hong Lv, Li-Guo Wang, Shen-He Ren. Far-zone behaviors of scattering-induced statistical properties of partially polarized spatially and spectrally partially coherent electromagnetic pulsed beam[J]. Chinese Physics B, 2020, 29(10): Copy Citation Text

show less

Abstract

In this study, we explore the far-zero behaviors of a scattered partially polarized spatially and spectrally partially coherent electromagnetic pulsed beam irradiating on a deterministic medium. The analytical formula for the cross-spectral density matrix elements of this beam in the spherical coordinate system is derived. Within the framework of the first-order Born approximation, the effects of the scattering angle θ, the source parameters (i.e., the pulse duration T0 and the temporal coherence length Tcxx), and the scatterer parameter (i.e., the effective width of the medium σR) on the spectral density, the spectral shift, the spectral degree of polarization, and the degree of spectral coherence of the scattered source in the far-zero field are studied numerically and comparatively. Our work improves the scattering theory of stochastic electromagnetic beams and it may be useful for the applications involving the interaction between incident light waves and scattering media.

Keywords

deterministic medium partially polarized spatially and spectrally partially coherent electromagnetic pulsed beam scattering statistical properties

$$ \begin{eqnarray}\begin{array}{lll} & & {{\boldsymbol{\varGamma }}}^{(0)}({{\boldsymbol{\rho }}}_{1},{{\boldsymbol{\rho }}}_{2},{t}_{1},{t}_{2})\\ & = & \left[\begin{array}{cc}{\varGamma }_{xx}^{(0)}({{\boldsymbol{\rho }}}_{1},{{\boldsymbol{\rho }}}_{2},{t}_{1},{t}_{2}) & {\varGamma }_{xy}^{(0)}({{\boldsymbol{\rho }}}_{1},{{\boldsymbol{\rho }}}_{2},{t}_{1},{t}_{2})\\ {\varGamma }_{yx}^{(0)}({{\boldsymbol{\rho }}}_{1},{{\boldsymbol{\rho }}}_{2},{t}_{1},{t}_{2}) & {\varGamma }_{yy}^{(0)}({{\boldsymbol{\rho }}}_{1},{{\boldsymbol{\rho }}}_{2},{t}_{1},{t}_{2})\end{array}\right],\end{array}\end{eqnarray}$$

(1)

View in Article

(1)

View in Article

$$ \begin{eqnarray}\begin{array}{lll} & & {\varGamma }_{\alpha \beta }^{(0)}({{\boldsymbol{\rho }}}_{1},{{\boldsymbol{\rho }}}_{2},{t}_{1},{t}_{2})\\ & = & {A}_{\alpha }{A}_{\beta }{B}_{\alpha \beta }\exp \left[-\displaystyle \frac{{{\boldsymbol{\rho }}}_{1}^{2}+{{\boldsymbol{\rho }}}_{2}^{2}}{4{w}_{0}^{2}}-\displaystyle \frac{{({{\boldsymbol{\rho }}}_{1}-{{\boldsymbol{\rho }}}_{2})}^{2}}{2{\delta }_{\alpha \beta }^{2}}\right]\\ & & \times \exp \left[-\displaystyle \frac{{t}_{1}^{2}+{t}_{2}^{2}}{2{T}_{0}^{2}}-\displaystyle \frac{{({t}_{1}-{t}_{2})}^{2}}{2{T}_{c\alpha \beta }^{2}}\right]\exp [{\rm{i}}{\omega }_{0}({t}_{1}-{t}_{2})],\\ & & (\alpha,\beta =x,y),\end{array}\end{eqnarray}$$

(2)

View in Article

$$ \begin{eqnarray}\begin{array}{lll} & & {W}_{\alpha \beta }^{(0)}({{\boldsymbol{\rho }}}_{1},{{\boldsymbol{\rho }}}_{2},{\omega }_{1},{\omega }_{2})\\ & = & \displaystyle \frac{1}{{(2\pi )}^{2}}\displaystyle \iint {\varGamma }_{\alpha \beta }^{(0)}({{\boldsymbol{\rho }}}_{1},{{\boldsymbol{\rho }}}_{2},{t}_{1},{t}_{2},)\\ & & \times \exp [-{\rm{i}}({\omega }_{1}{t}_{1}-{\omega }_{2}{t}_{2})]{\rm{d}}{t}_{1}{\rm{d}}{t}_{2},\end{array}\end{eqnarray}$$

(3)

View in Article

$$ \begin{eqnarray}\begin{array}{lll} & & {{\boldsymbol{W }}}^{(0)}({{\boldsymbol{\rho }}}_{1},{{\boldsymbol{\rho }}}_{2},{\omega }_{1},{\omega }_{2})\\ & = & \left[\begin{array}{cc}{W}_{xx}^{(0)}({{\boldsymbol{\rho }}}_{1},{{\boldsymbol{\rho }}}_{2},{\omega }_{1},{\omega }_{2}) & {W}_{xy}^{(0)}({{\boldsymbol{\rho }}}_{1},{{\boldsymbol{\rho }}}_{2},{\omega }_{1},{\omega }_{2})\\ {W}_{yx}^{(0)}({{\boldsymbol{\rho }}}_{1},{{\boldsymbol{\rho }}}_{2},{\omega }_{1},{\omega }_{2}) & {W}_{yy}^{(0)}({{\boldsymbol{\rho }}}_{1},{{\boldsymbol{\rho }}}_{2},{\omega }_{1},{\omega }_{2})\end{array}\right],\end{array}\end{eqnarray}$$

(4)

View in Article

$$ \begin{eqnarray}\begin{array}{lll} & & {W}_{\alpha \beta }^{(0)}({{\boldsymbol{\rho }}}_{1},{{\boldsymbol{\rho }}}_{2},{\omega }_{1},{\omega }_{2})\\ & = & \displaystyle \frac{{A}_{\alpha }{A}_{\beta }{B}_{\alpha \beta }{T}_{0}}{2\pi {\Omega }_{0\alpha \beta }}\exp \left[-\displaystyle \frac{{{\boldsymbol{\rho }}}_{1}^{2}+{{\boldsymbol{\rho }}}_{2}^{2}}{4{w}_{0}^{2}}-\displaystyle \frac{{({{\boldsymbol{\rho }}}_{1}-{{\boldsymbol{\rho }}}_{2})}^{2}}{2{\delta }_{\alpha \beta }^{2}}\right]\\ & & \times \exp [-{T}_{\alpha \beta }({\omega }_{1},{\omega }_{2})],\end{array}\end{eqnarray}$$

(5)

View in Article

$$ \begin{eqnarray}{T}_{\alpha \beta }({\omega }_{1},{\omega }_{2})=\displaystyle \frac{{({\omega }_{1}-{\omega }_{0})}^{2}+{({\omega }_{2}-{\omega }_{0})}^{2}}{2{\varOmega }_{0\alpha \beta }^{2}}+\displaystyle \frac{{({\omega }_{1}-{\omega }_{2})}^{2}}{2{\varOmega }_{c\alpha \beta }^{2}},\end{eqnarray}$$

(6)

View in Article

(1)

View in Article

$$ \begin{eqnarray}\begin{array}{lll}{S}_{0}({\boldsymbol{\rho }},\omega ) & = & {\rm{Tr}}\left[{{\boldsymbol{W}}}^{(0)}({\boldsymbol{\rho }},{\boldsymbol{\rho }},\omega,\omega )\right]\\ & = & \displaystyle \frac{{A}_{x}^{2}{B}_{xx}{T}_{0}}{2\pi {\varOmega }_{0xx}}\exp \left[-\displaystyle \frac{{{\boldsymbol{\rho }}}^{2}}{2{w}_{0}^{2}}-\displaystyle \frac{{(\omega -{\omega }_{0})}^{2}}{{\varOmega }_{0xx}^{2}}\right]\\ & & +\displaystyle \frac{{A}_{y}^{2}{B}_{yy}{T}_{0}}{2\pi {\varOmega }_{0yy}}\exp \left[-\displaystyle \frac{{{\boldsymbol{\rho }}}^{2}}{2{w}_{0}^{2}}-\displaystyle \frac{{(\omega -{\omega }_{0})}^{2}}{{\varOmega }_{0yy}^{2}}\right],\end{array}\end{eqnarray}$$

(7)

View in Article

$$ \begin{eqnarray}{P}_{0}({\boldsymbol{\rho }},\omega )=\sqrt{1-\displaystyle \frac{4{\rm{Det}}{{\boldsymbol{W}}}^{(0)}({\boldsymbol{\rho }},{\boldsymbol{\rho }},\omega,\omega )}{{{\rm{Tr}}}^{2}{{\boldsymbol{W}}}^{(0)}({\boldsymbol{\rho }},{\boldsymbol{\rho }},\omega,\omega )}}.\end{eqnarray}$$

(8)

View in Article

$$ \begin{eqnarray}{P}_{0}({\boldsymbol{\rho }},\omega )=\left|\displaystyle \frac{\displaystyle \frac{{A}_{x}^{2}{B}_{xx}}{{\varOmega }_{0xx}}\exp \left[-\displaystyle \frac{{(\omega -{\omega }_{0})}^{2}}{{\varOmega }_{0xx}^{2}}\right]-\displaystyle \frac{{A}_{y}^{2}{B}_{yy}}{{\varOmega }_{0yy}}\exp \left[-\displaystyle \frac{{(\omega -{\omega }_{0})}^{2}}{{\varOmega }_{0yy}^{2}}\right]}{\displaystyle \frac{{A}_{x}^{2}{B}_{xx}}{{\varOmega }_{0xx}}\exp \left[-\displaystyle \frac{{(\omega -{\omega }_{0})}^{2}}{{\varOmega }_{0xx}^{2}}\right]+\displaystyle \frac{{A}_{y}^{2}{B}_{yy}}{{\varOmega }_{0yy}}\exp \left[-\displaystyle \frac{{(\omega -{\omega }_{0})}^{2}}{{\varOmega }_{0yy}^{2}}\right]}\right|;\end{eqnarray}$$

(9)

View in Article

$$ \begin{eqnarray}{P}_{0}({\boldsymbol{\rho }},\omega )=\displaystyle \frac{\sqrt{{p}_{2}^{2}(\omega )+4{p}_{3}^{2}(\omega )}}{{p}_{1}(\omega )},\end{eqnarray}$$

(10)

View in Article

$$ \begin{eqnarray}\begin{array}{lll}{p}_{1}(\omega ) & = & \displaystyle \frac{{A}_{x}^{2}{B}_{xx}}{{\varOmega }_{0xx}}\exp \left[-\displaystyle \frac{{(\omega -{\omega }_{0})}^{2}}{{\varOmega }_{0xx}^{2}}\right]\\ & & +\displaystyle \frac{{A}_{y}^{2}{B}_{yy}}{{\varOmega }_{0yy}}\exp \left[-\displaystyle \frac{{(\omega -{\omega }_{0})}^{2}}{{\varOmega }_{0yy}^{2}}\right],\\ {p}_{2}(\omega ) & = & \displaystyle \frac{{A}_{x}^{2}{B}_{xx}}{{\varOmega }_{0xx}}\exp \left[-\displaystyle \frac{{(\omega -{\omega }_{0})}^{2}}{{\varOmega }_{0xx}^{2}}\right]\\ & & -\displaystyle \frac{{A}_{y}^{2}{B}_{yy}}{{\varOmega }_{0yy}}\exp \left[-\displaystyle \frac{{(\omega -{\omega }_{0})}^{2}}{{\varOmega }_{0yy}^{2}}\right],\\ {p}_{3}^{2}(\omega ) & = & \displaystyle \frac{{A}_{x}^{2}{A}_{y}^{2}{|{B}_{xy}|}^{2}}{{\varOmega }_{0xy}^{2}}\exp \left[-\displaystyle \frac{2{(\omega -{\omega }_{0})}^{2}}{{\varOmega }_{0xy}^{2}}\right],\end{array}\end{eqnarray}$$

(11)

View in Article

$$ \begin{eqnarray}\begin{array}{lll} & & \mu ({\boldsymbol{\rho }},{\boldsymbol{\rho }},{\omega }_{1},{\omega }_{2})\\ & = & \displaystyle \frac{\sqrt{{\rm{Tr}}[{{\boldsymbol{W}}}^{(0)}({\boldsymbol{\rho }},{\boldsymbol{\rho }},{\omega }_{1},{\omega }_{2}){{\boldsymbol{W}}}^{(0)\ast }({\boldsymbol{\rho }},{\boldsymbol{\rho }},{\omega }_{1},{\omega }_{2})]}}{\sqrt{{\boldsymbol{S}}({\boldsymbol{\rho }},{\omega }_{1}){\boldsymbol{S}}({\boldsymbol{\rho }},{\omega }_{2})}}.\end{array}\end{eqnarray}$$

(12)

View in Article

$$ \begin{eqnarray}{{\boldsymbol{W}}}^{(i)}({{\boldsymbol{r}}}_{1},{{\boldsymbol{r}}}_{2},{{\boldsymbol{s}}}_{01},{{\boldsymbol{s}}}_{02},{\omega }_{1},{\omega }_{2})=\left[\begin{array}{cc}{W}_{xx}({{\boldsymbol{r}}}_{1},{{\boldsymbol{r}}}_{2},{{\boldsymbol{s}}}_{01},{{\boldsymbol{s}}}_{02},{\omega }_{1},{\omega }_{2}) & \,\,\,{W}_{xy}({{\boldsymbol{r}}}_{1},{{\boldsymbol{r}}}_{2},{{\boldsymbol{s}}}_{01},{{\boldsymbol{s}}}_{02},{\omega }_{1},{\omega }_{2})\\ {W}_{yx}({{\boldsymbol{r}}}_{1},{{\boldsymbol{r}}}_{2},{{\boldsymbol{s}}}_{01},{{\boldsymbol{s}}}_{02},{\omega }_{1},{\omega }_{2}) & \,\,\,{W}_{yy}({{\boldsymbol{r}}}_{1},{{\boldsymbol{r}}}_{2},{{\boldsymbol{s}}}_{01},{{\boldsymbol{s}}}_{02},{\omega }_{1},{\omega }_{2})\end{array}\right],\end{eqnarray}$$

(13)

View in Article

$$ $\begin{array}{lll}{W}_{\alpha \beta }^{(i)}({{\boldsymbol{r}}}_{1},{{\boldsymbol{r}}}_{2},{{\boldsymbol{s}}}_{01},{{\boldsymbol{s}}}_{02},{\omega }_{1},{\omega }_{2}) & = & \displaystyle \mathop{\iint }\limits_{|{{\boldsymbol{s}}}_{01\perp }{|}^{2}\le 1}\,\,\displaystyle \mathop{\iint }\limits_{|{{\boldsymbol{s}}}_{02\perp }{|}^{2}\le 1}{{\rm{d}}}^{2}{{\boldsymbol{s}}}_{01\perp }{{\rm{d}}}^{2}{{\boldsymbol{s}}}_{02\perp }{A}_{\alpha \beta }({{\boldsymbol{s}}}_{01\perp },{{\boldsymbol{s}}}_{02\perp },{\omega }_{1},{\omega }_{2})\exp [-{\rm{i}}({k}_{1}{{\boldsymbol{s}}}_{01}\cdot {{\boldsymbol{r}}}_{1}-{k}_{2}{{\boldsymbol{s}}}_{02}\cdot {{\boldsymbol{r}}}_{2})],\,\,\,\end{array}$$$

(14)

View in Article

$$ \begin{eqnarray}\begin{array}{lll} & & {A}_{\alpha \beta }({{\boldsymbol{s}}}_{01\perp },{{\boldsymbol{s}}}_{02\perp },{\omega }_{1},{\omega }_{2})\\ & = & {\left(\displaystyle \frac{{k}_{1}{k}_{2}}{4{\pi }^{2}}\right)}^{2}{\displaystyle \iint }_{-\infty }^{\infty }{{\rm{d}}}^{2}{{\boldsymbol{\rho }}}_{1}{{\rm{d}}}^{2}{{\boldsymbol{\rho }}}_{2}{W}_{\alpha \beta }^{(0)}({{\boldsymbol{\rho }}}_{1},{{\boldsymbol{\rho }}}_{2},{\omega }_{1},{\omega }_{2})\\ & & \times \exp [-{\rm{i}}({k}_{2}{{\boldsymbol{s}}}_{02\perp }\cdot {{\boldsymbol{\rho }}}_{2}-{k}_{1}{{\boldsymbol{s}}}_{01\perp }\cdot {{\boldsymbol{\rho }}}_{1})].\end{array}\end{eqnarray}$$

(15)

View in Article

$$ \begin{eqnarray}\begin{array}{lll} & & {A}_{\alpha \beta }({{\boldsymbol{s}}}_{01\perp },{{\boldsymbol{s}}}_{02\perp },{\omega }_{1},{\omega }_{2})\\ & = & \displaystyle \frac{{A}_{\alpha }{A}_{\beta }{B}_{\alpha \beta }{T}_{0}}{2\pi {\varOmega }_{0\alpha \beta }}{\left(\displaystyle \frac{{k}_{1}{k}_{2}}{2\pi }\right)}^{2}{w}_{0}^{2}{\sigma }_{\alpha \beta }^{2}\exp [-{T}_{\alpha \beta }({\omega }_{1},{\omega }_{2})]\\ & & \times \exp \left[-\displaystyle \frac{{w}_{0}^{2}{({k}_{1}{{\boldsymbol{s}}}_{01\perp }-{k}_{2}{{\boldsymbol{s}}}_{02\perp })}^{2}}{2}\right.\\ & & \left.-\displaystyle \frac{{\sigma }_{\alpha \beta }^{2}{({k}_{1}{{\boldsymbol{s}}}_{01\perp }+{k}_{2}{{\boldsymbol{s}}}_{02\perp })}^{2}}{8}\right],\end{array}\end{eqnarray}$$

(16)

View in Article

$$ \begin{eqnarray}\displaystyle \frac{1}{{\sigma }_{\alpha \beta }^{2}}=\displaystyle \frac{1}{4{w}_{0}^{2}}+\displaystyle \frac{1}{{\delta }_{\alpha \beta }^{2}}.\end{eqnarray}$$

(17)

View in Article

$$ \begin{eqnarray}\begin{array}{lll} & & {{\boldsymbol{E}}}^{(s)}(r{\boldsymbol{s}},\omega )\\ & = & -{\boldsymbol{s}}\times \left[{\boldsymbol{s}}\times \displaystyle \mathop{\int }\limits_{D}F({{\boldsymbol{r}}}^{\prime},\omega ){{\boldsymbol{E}}}^{(i)}({{\boldsymbol{r}}}^{\prime},{{\boldsymbol{s}}}_{0},\omega )G(r{\boldsymbol{s}},{{\boldsymbol{r}}}^{\prime},\omega ){{\rm{d}}}^{3}{{\boldsymbol{r}}}^{\prime}\right],\end{array}\end{eqnarray}$$

(18)

View in Article

$$ \begin{eqnarray}G(r{\boldsymbol{s}},{{\boldsymbol{r}}}^{\prime},\omega )=\displaystyle \frac{\exp ({\rm{i}}kr)}{r}\exp [-{\rm{i}}k{\boldsymbol{s}}\cdot {{\boldsymbol{r}}}^{\prime}].\end{eqnarray}$$

(19)

View in Article

$$ \begin{eqnarray}\begin{array}{lll} & & {E}_{x}^{(s)}(r{\boldsymbol{s}},\omega )=\displaystyle {\int }_{D}F({{\boldsymbol{r}}}^{\prime},\omega )G(r{\boldsymbol{s}},{{\boldsymbol{r}}}^{\prime},\omega )\left[(1-{s}_{x}^{2}){E}_{x}^{(i)}({{\boldsymbol{r}}}^{\prime},{{\boldsymbol{s}}}_{0},\omega )-{s}_{x}{s}_{y}{E}_{y}^{(i)}({{\boldsymbol{r}}}^{\prime},{{\boldsymbol{s}}}_{0},\omega )\right]{{\rm{d}}}^{3}{{\boldsymbol{r}}}^{\prime},\end{array}\end{eqnarray}$$

(20a)

View in Article

$$ \begin{eqnarray}\begin{array}{lll} & & {E}_{y}^{(s)}(r{\boldsymbol{s}},\omega )=\displaystyle {\int }_{D}F({{\boldsymbol{r}}}^{\prime},\omega )G(r{\boldsymbol{s}},{{\boldsymbol{r}}}^{\prime},\omega )\left[-{s}_{x}{s}_{y}{E}_{x}^{(i)}({{\boldsymbol{r}}}^{\prime},{{\boldsymbol{s}}}_{0},\omega )-(1-{s}_{y}^{2}){E}_{y}^{(i)}({{\boldsymbol{r}}}^{\prime},{{\boldsymbol{s}}}_{0},\omega )\right]{{\rm{d}}}^{3}{{\boldsymbol{r}}}^{\prime},\end{array}\end{eqnarray}$$

(20b)

View in Article

$$ \begin{eqnarray}\begin{array}{lll} & & {E}_{z}^{(s)}(r{\boldsymbol{s}},\omega )=\displaystyle {\int }_{D}F({{\boldsymbol{r}}}^{\prime},\omega )G(r{\boldsymbol{s}},{{\boldsymbol{r}}}^{\prime},\omega )\left[-{s}_{x}{s}_{z}{E}_{x}^{(i)}({{\boldsymbol{r}}}^{\prime},{{\boldsymbol{s}}}_{0},\omega )-{s}_{y}{s}_{z}{E}_{y}^{(i)}({{\boldsymbol{r}}}^{\prime},{{\boldsymbol{s}}}_{0},\omega )\right]{{\rm{d}}}^{3}{{\boldsymbol{r}}}^{\prime},\end{array}\end{eqnarray}$$

(20c)

View in Article

$$ \begin{eqnarray}\begin{array}{lll}{E}_{\theta }^{(s)}(r{\boldsymbol{s}},\omega ) & = & {\rm{\cos }}\theta \cos \varphi {E}_{x}^{(s)}(r{\boldsymbol{s}},\omega )+\cos \theta \sin \varphi {E}_{y}^{(s)}(r{\boldsymbol{s}},\omega )\\ & & -\sin \theta {E}_{z}^{(s)}(r{\boldsymbol{s}},\omega ),\end{array}\end{eqnarray}$$

(21a)

View in Article

$$ \begin{eqnarray}\begin{array}{lll}{E}_{\varphi }^{(s)}(r{\boldsymbol{s}},\omega ) & = & -\sin \varphi {E}_{x}^{(s)}(r{\boldsymbol{s}},\omega )+\cos \varphi {E}_{y}^{(s)}(r{\boldsymbol{s}},\omega ),\end{array}\end{eqnarray}$$

(21b)

View in Article

$$ \begin{eqnarray}\begin{array}{lll}{E}_{\theta }^{(s)}(r{\boldsymbol{s}},\omega ) & = & \displaystyle {\int }_{D}{{\rm{d}}}^{3}{{\boldsymbol{r}}}^{\prime}F({{\boldsymbol{r}}}^{\prime},\omega )G(r{\boldsymbol{s}},{{\boldsymbol{r}}}^{\prime},\omega )\\ & & \times [{\rm{\cos }}\theta \cos \varphi {E}_{x}^{(i)}({{\boldsymbol{r}}}^{\prime},{{\boldsymbol{s}}}_{0},\omega )\\ & & +{\rm{\cos }}\theta \sin \varphi {E}_{y}^{(i)}({{\boldsymbol{r}}}^{\prime},{{\boldsymbol{s}}}_{0},\omega )],\end{array}\end{eqnarray}$$

(22a)

View in Article

$$ \begin{eqnarray}\begin{array}{lll}{E}_{\varphi }^{(s)}(r{\boldsymbol{s}},\omega ) & = & \displaystyle {\int }_{D}F({{\boldsymbol{r}}}^{\prime},\omega )G(r{\boldsymbol{s}},{{\boldsymbol{r}}}^{\prime},\omega )\\ & & \times [-\sin \varphi {E}_{x}^{(i)}({{\boldsymbol{r}}}^{\prime},{{\boldsymbol{s}}}_{0},\omega )\\ & & +{\rm{\cos }}\varphi {E}_{y}^{(i)}({{\boldsymbol{r}}}^{\prime},{{\boldsymbol{s}}}_{0},\omega )]{{\rm{d}}}^{3}{{\boldsymbol{r}}}^{\prime},\end{array}\end{eqnarray}$$

(22b)

View in Article

$$ \begin{eqnarray}\begin{array}{lll} & & {{\boldsymbol{W}}}^{(s)}(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})\\ & = & [{W}_{\alpha \beta }^{(s)}(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})]\\ & = & [\langle {E}_{\alpha }^{(s)\ast }(r{{\boldsymbol{s}}}_{1},{\omega }_{1}\rangle )\langle {E}_{\beta }^{(s)}(r{{\boldsymbol{s}}}_{2},{\omega }_{2})\rangle ],\\ & & \,\,(\alpha,\beta =\theta,\varphi ).\,\,\,\,\end{array}\end{eqnarray}$$

(23)

View in Article

$$ \begin{eqnarray}\begin{array}{lll} \quad {W}_{\theta \theta }^{(s)}(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})\\ = \displaystyle \frac{{{\rm{e}}}^{[-{\rm{i}}({k}_{1}-{k}_{2})r]}}{{r}^{2}}[\cos {\theta }_{1}\cos {\varphi }_{1}\cos {\theta }_{2}\cos {\varphi }_{2}{F}_{xx}(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})\\ \quad +\cos {\theta }_{1}\cos {\varphi }_{1}\cos {\theta }_{2}\sin {\varphi }_{2}{F}_{xy}(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})\\ \quad +\cos {\theta }_{1}\sin {\varphi }_{1}\cos {\theta }_{2}\cos {\varphi }_{2}{F}_{yx}(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})\\ \quad +\cos {\theta }_{1}\sin {\varphi }_{1}\cos {\theta }_{2}\sin {\varphi }_{2}{F}_{yy}(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})],\\ \end{array}\end{eqnarray}$$

(24a)

View in Article

$$ \begin{eqnarray}\begin{array}{lll} & & {W}_{\theta \varphi }^{(s)}(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})\\ & = & \displaystyle \frac{{{\rm{e}}}^{[-{\rm{i}}({k}_{1}-{k}_{2})r]}}{{r}^{2}}[-\cos {\theta }_{1}\cos {\varphi }_{1}\sin {\varphi }_{2}{F}_{xx}(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})\\ & & +\cos {\theta }_{1}\cos {\varphi }_{1}\cos {\varphi }_{2}{F}_{xy}(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})\\ & & -\cos {\theta }_{1}\sin {\varphi }_{1}\sin {\varphi }_{2}{F}_{yx}(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})\\ & & +\cos {\theta }_{1}\sin {\varphi }_{1}\cos {\varphi }_{2}{F}_{yy}(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})],\end{array}\end{eqnarray}$$

(24b)

View in Article

$$ \begin{eqnarray}\begin{array}{lll} & & {W}_{\varphi \theta }^{(s)}(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})\\ & = & \displaystyle \frac{{{\rm{e}}}^{[-{\rm{i}}({k}_{1}-{k}_{2})r]}}{{r}^{2}}[-\sin {\varphi }_{1}\cos {\theta }_{2}\cos {\varphi }_{2}{F}_{xx}(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})\\ & & -\sin {\varphi }_{1}\cos {\theta }_{2}\sin {\varphi }_{2}{F}_{xy}(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})\\ & & +\cos {\varphi }_{1}\cos {\theta }_{2}\cos {\varphi }_{2}{F}_{yx}(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})\\ & & +\cos {\varphi }_{1}\cos {\theta }_{2}\sin {\varphi }_{2}{F}_{yy}(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})],\end{array}\end{eqnarray}$$

(24c)

View in Article

$$ \begin{eqnarray}\begin{array}{lll}\quad {W}_{\varphi \varphi }^{(s)}(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})\\ =\displaystyle \frac{{{\rm{e}}}^{[-{\rm{i}}({k}_{1}-{k}_{2})r]}}{{r}^{2}}[\sin {\varphi }_{1}\sin {\varphi }_{2}{F}_{xx}(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})\\ \quad -\sin {\varphi }_{1}\cos {\varphi }_{2}{F}_{xy}(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})\\ \quad -\cos {\varphi }_{1}\sin {\varphi }_{2}{F}_{yx}(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})\\ \quad +\cos {\varphi }_{1}\cos {\varphi }_{2}{F}_{yy}(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})].\end{array}\end{eqnarray}$$

(24d)

View in Article

$$ \begin{eqnarray}\begin{array}{lll} & & {F}_{\alpha \beta }(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})\\ & = & \displaystyle \mathop{\iint }\limits_{|{{\boldsymbol{s}}}_{01\perp }{|}^{2}\le 1}\,\,\displaystyle \mathop{\iint }\limits_{|{{\boldsymbol{s}}}_{01\perp }{|}^{2}\le 1}{\mathop{F}\limits^{\sim }}^{\ast }({{\boldsymbol{K}}}_{1},{\omega }_{1})\mathop{F}\limits^{\sim }({{\boldsymbol{K}}}_{2},{\omega }_{2})\\ & & \times {A}_{\alpha \beta }({{\boldsymbol{s}}}_{01\perp },{{\boldsymbol{s}}}_{02\perp },{\omega }_{1},{\omega }_{2}){{\rm{d}}}^{3}{{\boldsymbol{s}}}_{01\perp }{{\rm{d}}}^{3}{{\boldsymbol{s}}}_{02\perp },\\ & & (\alpha,\beta =x,y),\,\,\,\,\,\,\,\end{array}\end{eqnarray}$$

(25)

View in Article

$$ \begin{eqnarray}\mathop{F}\limits^{\sim }({\boldsymbol{K}},\omega )=\displaystyle {\int }_{D}F({{\boldsymbol{r}}}^{\prime},\omega )\exp [-{\rm{i}}{\boldsymbol{K}}\cdot {{\boldsymbol{r}}}^{\prime}]{{\rm{d}}}^{3}{{\boldsymbol{r}}}^{\prime}\end{eqnarray}$$

(26)

View in Article

$$ \begin{eqnarray}F({{\boldsymbol{r}}}^{\prime},\omega )=\left\{\begin{array}{ll}\displaystyle \frac{{k}^{2}}{4\pi }[{n}^{2}({{\boldsymbol{r}}}^{\prime},\omega )-1], & \,\,{{\boldsymbol{r}}}^{\prime}\in D,\\ 0, & \,\,{\rm{otherwise}}.\end{array}\right.\end{eqnarray}$$

(27)

View in Article

$$ \begin{eqnarray}F({{\boldsymbol{r}}}^{\prime},\omega )={F}_{0}{\rm{\exp }}\left[-\displaystyle \frac{{{\boldsymbol{r}}}^{\prime}{}^{2}}{2{\sigma }_{R}^{2}}\right],\end{eqnarray}$$

(28)

View in Article

$$ \begin{eqnarray}\begin{array}{lll} & & {\mathop{F}\limits^{\sim }}^{\ast }({{\boldsymbol{K}}}_{1},{\omega }_{1})\mathop{F}\limits^{\sim }({{\boldsymbol{K}}}_{2},{\omega }_{2})\\ & = & \displaystyle {\int }_{D}\displaystyle {\int }_{D}{F}^{\ast }({{\boldsymbol{r}}}^{\prime}{}_{1},{\omega }_{1})F({{\boldsymbol{r}}}^{\prime}{}_{2},{\omega }_{2})\\ & & \times \exp [{\rm{i}}({{\boldsymbol{K}}}_{1}\cdot {{\boldsymbol{r}}}^{\prime}{}_{1}-{{\boldsymbol{K}}}_{2}\cdot {{\boldsymbol{r}}}^{\prime}{}_{2})]{{\rm{d}}}^{3}{{\boldsymbol{r}}}^{\prime}{}_{1}{{\rm{d}}}^{3}{{\boldsymbol{r}}}^{\prime}{}_{2}\\ & = & {F}_{0}^{2}{(2\pi )}^{3}{\sigma }_{R}^{6}\exp \left[-\displaystyle \frac{{\sigma }_{R}^{2}}{2}({{\boldsymbol{K}}}_{1}^{2}+{{\boldsymbol{K}}}_{2}^{2})\right].\end{array}\end{eqnarray}$$

(29)

View in Article

$$ \begin{eqnarray}\begin{array}{lll} & & {F}_{\alpha \beta }(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})\\ & = & \displaystyle \frac{{k}_{1}^{2}{k}_{2}^{2}{A}_{\alpha }{A}_{\beta }{B}_{\alpha \beta }{T}_{0}{F}_{0}^{2}{\sigma }_{R}^{6}{w}_{0}^{2}{\sigma }_{\alpha \beta }^{2}}{{\varOmega }_{0\alpha \beta }}\exp [-{T}_{\alpha \beta }({\omega }_{1},{\omega }_{2})]\\ & & \times \displaystyle \mathop{\iint }\limits_{{p}_{1}^{2}+{q}_{1}^{2}\le 1}\,\,\displaystyle \mathop{\iint }\limits_{{p}_{2}^{2}+{q}_{2}^{2}\le 1}\exp \left\{-\displaystyle \frac{{\sigma }_{R}^{2}{k}_{1}^{2}}{2}{({s}_{1x}-{p}_{1})}^{2}+{({s}_{1y}-{q}_{1})}^{2}+{\left({s}_{1z}-\sqrt{1-{p}_{1}^{2}-{q}_{1}^{2}}\right)}^{2}\right\}\\ & & \times \exp \left\{-\displaystyle \frac{{\sigma }_{R}^{2}{k}_{2}^{2}}{2}{({s}_{2x}-{p}_{2})}^{2}+{({s}_{2y}-{q}_{2})}^{2}+{\left({s}_{2z}-\sqrt{1-{p}_{2}^{2}-{q}_{2}^{2}}\right)}^{2}\right\}\\ & & \times \exp \left\{-\displaystyle \frac{{w}_{0}^{2}}{2}[{k}_{1}^{2}({p}_{1}^{2}+{q}_{1}^{2})-2{k}_{1}{k}_{2}({p}_{1}{p}_{2}+{q}_{1}{q}_{2})+{k}_{2}^{2}({p}_{2}^{2}+{q}_{2}^{2})]\right\}\\ & & \times \exp \left\{-\displaystyle \frac{{\sigma }_{\alpha \beta }^{2}}{8}[{k}_{1}^{2}({p}_{1}^{2}+{q}_{1}^{2})+2{k}_{1}{k}_{2}({p}_{1}{p}_{2}+{q}_{1}{q}_{2})+{k}_{2}^{2}({p}_{2}^{2}+{q}_{2}^{2})]\right\}{\rm{d}}{p}_{1}{\rm{d}}{q}_{1}{\rm{d}}{p}_{2}{\rm{d}}{q}_{2},\,\,\,(\alpha,\beta =x,y),\end{array}\end{eqnarray}$$

(30)

View in Article

$$ \begin{eqnarray}\begin{array}{lll} & & {F}_{\alpha \beta }(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})\\ & = & \displaystyle \frac{{k}_{1}^{2}{k}_{2}^{2}{A}_{\alpha }{A}_{\beta }{B}_{\alpha \beta }{T}_{0}{F}_{0}^{2}{\sigma }_{R}^{6}{w}_{0}^{2}{\sigma }_{\alpha \beta }^{2}}{{\varOmega }_{0\alpha \beta }}\exp \left[-{T}_{\alpha \beta }({\omega }_{1},{\omega }_{2})\right]\\ & & \times \displaystyle \frac{{\pi }^{2}}{{m}_{2\alpha \beta }{n}_{\alpha \beta }}\exp \left[\displaystyle \frac{{k}_{2}^{2}{\sigma }_{R}^{4}({s}_{2x}^{2}+{s}_{2y}^{2})}{4{m}_{2\alpha \beta }}+\displaystyle \frac{{\xi }_{x\alpha \beta }^{2}+{\xi }_{y\alpha \beta }^{2}}{4{n}_{\alpha \beta }}\right]\\ & & \times \exp \{-{\sigma }_{R}^{2}[{k}_{1}^{2}(1-{s}_{1z})+{k}_{2}^{2}(1-{s}_{2z})]\},\\ & & (\alpha,\beta =x,y),\end{array}\end{eqnarray}$$

(31)

View in Article

$$ \begin{eqnarray}\begin{array}{lll} & & {a}_{\alpha \beta }=\displaystyle \frac{{w}_{0}^{2}}{2}+\displaystyle \frac{{\sigma }_{\alpha \beta }^{2}}{8},\,\,\,{b}_{\alpha \beta }=\displaystyle \frac{{w}_{0}^{2}}{2}-\displaystyle \frac{{\sigma }_{\alpha \beta }^{2}}{8},\\ & & {m}_{j\alpha \beta }={k}_{j}^{2}{a}_{\alpha \beta }+{k}_{j}^{2}\displaystyle \frac{{\sigma }_{R}^{2}}{2}{s}_{jz}\,\,\,(j=1,2),\\ & & {n}_{\alpha \beta }={m}_{1\alpha \beta }-\displaystyle \frac{{k}_{1}^{2}{k}_{2}^{2}{b}_{\alpha \beta }^{2}}{{m}_{2\alpha \beta }},\\ & & {\xi }_{x\alpha \beta }={k}_{1}^{2}{\sigma }_{R}^{2}{s}_{1x}+\displaystyle \frac{{k}_{1}{k}_{2}^{3}{\sigma }_{R}^{2}{b}_{\alpha \beta }}{{m}_{2\alpha \beta }}{s}_{2x},\\ & & {\xi }_{y\alpha \beta }={k}_{1}^{2}{\sigma }_{R}^{2}{s}_{1y}+\displaystyle \frac{{k}_{1}{k}_{2}^{3}{\sigma }_{R}^{2}{b}_{\alpha \beta }}{{m}_{2\alpha \beta }}{s}_{2y}.\end{array}\end{eqnarray}$$

(32)

View in Article

$$ \begin{eqnarray}S(r{\boldsymbol{s}},\omega )={W}_{\theta \theta }^{(s)}(r{\boldsymbol{s}},r{\boldsymbol{s}},\omega,\omega )+{W}_{\varphi \varphi }^{(s)}(r{\boldsymbol{s}},r{\boldsymbol{s}},\omega,\omega ),\end{eqnarray}$$

(33)

View in Article

$$ \begin{eqnarray}\begin{array}{lll}P(r{\boldsymbol{s}},\omega ) & = & \sqrt{1-\displaystyle \frac{4{\rm{Det}}{{\boldsymbol{W}}}^{(s)}(r{\boldsymbol{s}},r{\boldsymbol{s}},\omega,\omega )}{{{\rm{Tr}}}^{2}{{\boldsymbol{W}}}^{(s)}(r{\boldsymbol{s}},r{\boldsymbol{s}},\omega,\omega )}}\\ & = & \displaystyle \frac{\sqrt{{\left({W}_{\theta \theta }^{(s)}-{W}_{\varphi \varphi }^{(s)}\right)}^{2}+4{W}_{\theta \varphi }^{(s)}{W}_{\varphi \theta }^{(s)}}}{{W}_{\theta \theta }^{(s)}+{W}_{\varphi \varphi }^{(s)}},\end{array}\end{eqnarray}$$

(34)

View in Article

$$ \begin{eqnarray}\begin{array}{lll} & & \mu (r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})\\ & = & \displaystyle \frac{\sqrt{{\rm{Tr}}[{{\boldsymbol{W}}}^{(s)}(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2}){{\boldsymbol{W}}}^{(s)\ast }(r{{\boldsymbol{s}}}_{1},r{{\boldsymbol{s}}}_{2},{\omega }_{1},{\omega }_{2})]}}{\sqrt{{\boldsymbol{S}}(r{{\boldsymbol{s}}}_{1},{\omega }_{1}){\boldsymbol{S}}(r{{\boldsymbol{s}}}_{2},{\omega }_{2})}}.\,\,\,\,\,\,\,\end{array}\end{eqnarray}$$

(35)

View in Article

$$ \begin{eqnarray}\begin{array}{lll} & & {F}_{\alpha \beta }(r{\boldsymbol{s}},r{\boldsymbol{s}},{\omega }_{1},{\omega }_{2})\\ & = & \displaystyle \frac{{A}_{\alpha }{A}_{\beta }{B}_{\alpha \beta }{T}_{0}{F}_{0}^{2}{\sigma }_{R}^{6}{w}_{0}^{2}{\sigma }_{\alpha \beta }^{2}}{{\varOmega }_{0\alpha \beta }}\exp [-{T}_{\alpha \beta }({\omega }_{1},{\omega }_{2})]\\ & & \times \displaystyle \frac{{\pi }^{2}}{{m}_{\alpha \beta }{n}_{\alpha \beta }}\exp \left[\left(\displaystyle \frac{{\sigma }_{R}^{4}}{4{m}_{\alpha \beta }}+\displaystyle \frac{{\mu }_{\alpha \beta }^{2}}{4{n}_{\alpha \beta }}\right)({s}_{x}^{2}+{s}_{y}^{2})\right]\\ & & \times \exp [-{\sigma }_{R}^{2}({k}_{1}^{2}+{k}_{2}^{2})(1-{s}_{z})],\\ & & (\alpha,\beta =x,y),\end{array}\end{eqnarray}$$

(36)

View in Article

$$ \begin{eqnarray}\begin{array}{lll} & & {a}_{\alpha \beta }=\displaystyle \frac{{w}_{0}^{2}}{2}+\displaystyle \frac{{\sigma }_{\alpha \beta }^{2}}{8},\,\,{b}_{\alpha \beta }=\displaystyle \frac{{w}_{0}^{2}}{2}-\displaystyle \frac{{\sigma }_{\alpha \beta }^{2}}{8},\\ & & {\mu }_{\alpha \beta }={k}_{1}^{2}{\sigma }_{R}^{2}+\displaystyle \frac{{k}_{1}{k}_{2}^{3}{\sigma }_{R}^{2}{b}_{\alpha \beta }}{{m}_{\alpha \beta }},\\ & & {m}_{\alpha \beta }={a}_{\alpha \beta }+\displaystyle \frac{{\sigma }_{R}^{2}}{2}{s}_{z},\,\,{n}_{\alpha \beta }={m}_{\alpha \beta }-\displaystyle \frac{{k}_{2}^{2}{b}_{\alpha \beta }^{2}}{{m}_{\alpha \beta }}.\end{array}\end{eqnarray}$$

(37)

View in Article

$$ \begin{eqnarray}\left\{\begin{array}{l}\displaystyle \frac{1}{4{w}_{0}^{2}}+\displaystyle \frac{1}{{\delta }_{\alpha \alpha }^{2}}\le \displaystyle \frac{2{\pi }^{2}}{{\lambda }^{2}},\,(\alpha =x,y),{\sigma }_{R}\gg \displaystyle \frac{{\lambda }_{0}}{\pi \sqrt{2}},\\ \max \left\{{\delta }_{xx},{\delta }_{yy}\right\}\le {\delta }_{xy}\le \min \left\{\displaystyle \frac{{\delta }_{xx}}{|{B}_{xy}|},\displaystyle \frac{{\delta }_{yy}}{|{B}_{xy}|}\right\},\\ \max \{{T}_{cxx},{T}_{cyy}\}\le {T}_{cxy}\le \min \{\displaystyle \frac{{T}_{cxx}}{|{B}_{xy}|},\displaystyle \frac{{T}_{cyy}}{|{B}_{xy}|}\}.\end{array}\right.\end{eqnarray}$$

(38)

View in Article

$$ \begin{eqnarray}{{\boldsymbol{J}}}^{(s)}(r{\boldsymbol{s}},\omega )={{\boldsymbol{W}}}^{(s)}(r{\boldsymbol{s}},r{\boldsymbol{s}},\omega,\omega ).\end{eqnarray}$$

(39)

View in Article

(1)

View in Article

$$ \begin{eqnarray}\begin{array}{lll}{P}^{2}(r{\boldsymbol{s}},\omega ) & = & 1-\displaystyle \frac{\det {{\boldsymbol{J}}}^{(s)}(r{\boldsymbol{s}},\omega )}{{{\rm{Tr}}}^{2}{{\boldsymbol{J}}}^{(s)}(r{\boldsymbol{s}},\omega )}\\ & = & 2\displaystyle \frac{{\rm{Tr}}{{\boldsymbol{J}}}^{(s)2}(r{\boldsymbol{s}},\omega )}{{{\rm{Tr}}}^{2}{{\boldsymbol{J}}}^{(s)}(r{\boldsymbol{s}},\omega )}-1.\,\,\,\end{array}\end{eqnarray}$$

(40)

View in Article

$$ \begin{eqnarray}{P}^{2}(r{\boldsymbol{s}},\omega )=2{\mu }^{2}(r{\boldsymbol{s}},r{\boldsymbol{s}},\omega,\omega )-1.\end{eqnarray}$$

(41)

View in Article

Figures&Tables (10)

References (28)

Copy Citation Text

Yan Li, Ming Gao, Hong Lv, Li-Guo Wang, Shen-He Ren. Far-zone behaviors of scattering-induced statistical properties of partially polarized spatially and spectrally partially coherent electromagnetic pulsed beam[J]. Chinese Physics B, 2020, 29(10):

Download Citation

Save the article for my favorites

Paper Information

Recommended Topics

Nuclear physics

Particles and fields

Particle and nuclear astrophysics and cosmology

Set citation alerts for the article

Please enter your email address