A transverse spatially random monochromatic electric field vector, E(r,t), is composed of two mutually orthogonal polarization components \\(\\mathrm {E_i}\\)(r,t) and \\(\\mathrm {E_j}\\)(r,t) and represented as:<\/p>\n
$$\\begin{aligned} \\textrm{E}(\\textrm{r,t}) = \\mathrm {E_{i}} (\\textrm{r,t})\\hat{\\textrm{i}} + \\mathrm {E_{j}} (\\textrm{r, t})\\hat{\\textrm{j}}, \\end{aligned}$$<\/p>\n
\n (1)\n <\/p>\n
where \\({\\hat{\\textrm{i}}}\\) and \\({\\hat{\\textrm{j}}}\\) are the unit orthogonal polarization vector components, \\(\\textrm{r}\\) is the position vector on the observation plane, and \\(\\textrm{t}\\) is time. The coherence-polarization matrix of an inhomogeneously polarized random field is given by:<\/p>\n
$$\\begin{aligned} \\Gamma (r_1, r_2) = \\begin{bmatrix} \\langle \\mathrm {E_i^*}(\\mathrm {r_1}) \\mathrm {E_i}(\\mathrm {r_2}) \\rangle & \\langle \\mathrm {E_i^*}(\\mathrm {r_1}) \\mathrm {E_j}(\\mathrm {r_2}) \\rangle \\\\ \\langle \\mathrm {E_j^*}(\\mathrm {r_1}) \\mathrm {E_i}(\\mathrm {r_2}) \\rangle & \\langle \\mathrm {E_j^*}(\\mathrm {r_1}) \\mathrm {E_j}(\\mathrm {r_2}) \\rangle \\end{bmatrix}, \\end{aligned}$$<\/p>\n
where \\(“\\langle ~\\rangle ”\\) represents the ensemble average, and \\(\\mathrm {r_1}\\) and \\(\\mathrm {r_2}\\) are the two spatial position vectors on transverse plane. The mean intensity at position vector \\(\\textrm{r}\\) is given by:<\/p>\n
$$\\begin{aligned} \\langle \\textrm{I}(\\textrm{r}) \\rangle = \\langle |\\mathrm {E_i}(\\textrm{r})|^2 \\rangle + \\langle |\\mathrm {E_j}(\\textrm{r})|^2 \\rangle = \\text {tr} \\, \\Gamma (\\textrm{r,r}). \\end{aligned}$$<\/p>\n
The normalized two-point intensity correlation function \\(\\gamma _\\textrm{s}(\\mathrm {r_1}, \\mathrm {r_2})\\) of a non-uniform Gaussian random field is characterized as:<\/p>\n
$$\\begin{aligned} \\gamma _\\textrm{s}^{2}(\\mathrm {r_{1}}, \\textrm{r}_{2}) = \\frac{< \\Delta \\textrm{I}(\\textrm{r}_1) \\Delta \\textrm{I}(\\textrm{r}_2)>}{< \\textrm{I}(\\textrm{r}_1)><\\textrm{I}(\\textrm{r}_2)>}, \\end{aligned}$$<\/p>\n
\n (2)\n <\/p>\n
where \\(\\Delta \\textrm{I}(\\textrm{r}) = \\textrm{I}(\\textrm{r}) – <\\textrm{I}(\\textrm{r})>\\) is the spatial intensity fluctuation from its mean value and \\(<\\textrm{I}(\\textrm{r})>\\) is the average intensity at position \\(\\textrm{r}\\). The degree of polarization \\(\\textrm{P}(\\textrm{r})\\) at point \\(\\textrm{r}\\), is given by:<\/p>\n
$$\\begin{aligned} \\mathrm {P^{2}}(\\textrm{r})=2 \\gamma _\\textrm{s}^{2}(\\textrm{r,r}) -1. \\end{aligned}$$<\/p>\n
\n (3)\n <\/p>\n
If the spatial distribution of the polarization is completely random, i.e., \\(|\\mathrm {E_i}(\\textrm{r})|^2=|\\mathrm {E_j}(\\textrm{r})|^2\\) and \\(\\Gamma _\\mathrm {{ij}}(\\textrm{r,r}) = <\\mathrm {E_i^*}(\\textrm{r})\\mathrm {E_j}(\\textrm{r})>= 0\\), \\(\\gamma _\\textrm{s}^2(\\textrm{r,r}) = 0.5\\), and degree of polarization \\(\\textrm{P}(\\textrm{r}) = 0\\). For a fully coherent random field, \\(\\gamma _\\textrm{s}^2(\\textrm{r,r}) = 1\\) and the degree of polarization, \\(\\mathrm P(r)=1\\). The random monochromatic electric field \\(\\textrm{E}(\\textrm{r})\\) after passing through a polarizer with a pass-axis oriented at an angle \\({\\theta }\\) w.r.t. the x-axis, is:<\/p>\n
$$\\begin{aligned} \\mathrm {E_P}(\\textrm{r}) = [ \\textrm{cos}^{2}\\theta \\mathrm {E_i}(\\textrm{r}) + \\textrm{sin}\\theta \\textrm{cos}\\theta \\mathrm {E_j}((\\textrm{r}) ] \\hat{\\textrm{i}} + [ \\textrm{sin}\\theta \\textrm{cos}\\theta \\mathrm {E_i}(\\textrm{r}) + \\textrm{sin}^{2} \\theta \\mathrm {E_j}(\\textrm{r}) ]\\hat{\\textrm{j}} \\end{aligned}$$<\/p>\n
\n (4)\n <\/p>\n
The intensity, \\(\\textrm{I}\\), and the average intensity, \\(<\\textrm{I}(\\textrm{r})>\\), of field, \\(\\mathrm {E_P}(\\textrm{r})\\), can be written as<\/p>\n
$$\\begin{aligned} & \\textrm{I} = \\textrm{cos}^2\\theta \\textrm{E}_\\textrm{i}\\textrm{E}_\\textrm{i}^* + \\textrm{sin}\\theta \\textrm{cos}\\theta \\textrm{E}_\\textrm{i}^*\\textrm{E}_\\textrm{j} + \\textrm{sin}\\theta \\textrm{cos}\\theta \\textrm{E}_\\textrm{i}\\textrm{E}_\\textrm{j}^* + \\textrm{sin}^2\\theta \\textrm{E}_\\textrm{j}\\textrm{E}_\\textrm{j}^*, \\end{aligned}$$<\/p>\n
\n (5)\n <\/p>\n
$$\\begin{aligned} & <\\textrm{I}>= \\textrm{cos}^2\\theta \\Gamma _{\\textrm{ii}}+\\textrm{sin}\\theta \\textrm{cos}\\theta \\Gamma _{\\textrm{ij}}+\\textrm{sin}\\theta \\textrm{cos}\\theta \\Gamma _{\\textrm{ji}} + \\textrm{sin}^2\\theta \\Gamma _{\\textrm{jj}}, \\end{aligned}$$<\/p>\n
\n (6)\n <\/p>\n
where \\(\\Gamma _{\\textrm{ij}}\\) are the element of the coherence-polarization matrix of the random field, E(r,\u00a0t), and are defined as<\/p>\n
$$\\begin{aligned} \\Gamma _{\\textrm{ij}}(\\textrm{r}_1,\\textrm{r}_2) = <\\textrm{E}_\\textrm{i}^*(\\textrm{r}_1)\\textrm{E}_\\textrm{j}(\\textrm{r}_2)>. \\end{aligned}$$<\/p>\n
Assuming that the speckle field follows the Gaussian statistics, the cross-correlation between two speckle patterns, recorded for the pass-axis of the polarizer at \\({\\theta _1}\\) and \\({\\theta _2}\\), is calculated by:<\/p>\n
$$\\begin{aligned} < \\Delta \\textrm{I}(\\textrm{r}_1) \\Delta \\textrm{I}(\\textrm{r}_2)> = (\\textrm{cos}\\theta _1 \\textrm{cos}\\theta _2\\Gamma _{\\textrm{ii}}+\\textrm{cos}\\theta _1 \\textrm{sin}\\theta _2\\Gamma _{\\textrm{ij}}+\\textrm{sin}\\theta _1 \\textrm{cos}\\theta _2\\Gamma _{\\textrm{ji}}+\\textrm{sin}\\theta _1 \\textrm{sin}\\theta _2 \\Gamma _{\\textrm{jj}})^2 \\end{aligned}$$<\/p>\n
\n (7)\n <\/p>\n
where \\(\\Delta I(r_i)= I(r_i)- \\) represents the intensity fluctuation from its means value. The normalized cross-correlation is expressed by:<\/p>\n
$$\\begin{aligned} \\gamma _\\textrm{c}^{2}(\\textrm{r}_{1}, \\textrm{r}_{2}) = \\frac{< \\Delta \\textrm{I}(\\textrm{r}_1) \\Delta \\textrm{I}(\\textrm{r}_2)>}{\\sigma (\\textrm{r}_1)\\sigma (\\textrm{r}_2)}, \\end{aligned}$$<\/p>\n
\n (8)\n <\/p>\n
where \\(\\sigma (\\textrm{r}_\\textrm{i}) = \\textrm{cos}^2\\theta _\\textrm{i} \\Gamma _{\\textrm{ii}}+\\textrm{sin}\\theta _\\textrm{i} \\textrm{cos}\\theta _\\textrm{i}\\Gamma _{\\textrm{ij}}+\\textrm{sin}\\theta _\\textrm{i} \\textrm{cos}\\theta _\\textrm{i} \\Gamma _{\\textrm{ji}} + \\textrm{sin}^2\\theta _\\textrm{i} \\Gamma _{\\textrm{jj}}\\) is the ensemble average.<\/p>\n","protected":false},"excerpt":{"rendered":"A transverse spatially random monochromatic electric field vector, E(r,t), is composed of two mutually orthogonal polarization components \\(\\mathrm…\n","protected":false},"author":2,"featured_media":203494,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[24],"tags":[17365,4230,4231,21570,11659,2302,90,56,54,55],"class_list":{"0":"post-203493","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-physics","8":"tag-applied-optics","9":"tag-humanities-and-social-sciences","10":"tag-multidisciplinary","11":"tag-optical-materials-and-structures","12":"tag-optics-and-photonics","13":"tag-physics","14":"tag-science","15":"tag-uk","16":"tag-united-kingdom","17":"tag-unitedkingdom"},"_links":{"self":[{"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/posts\/203493","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/comments?post=203493"}],"version-history":[{"count":0,"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/posts\/203493\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/media\/203494"}],"wp:attachment":[{"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/media?parent=203493"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/categories?post=203493"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/tags?post=203493"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}