CN102175652A - Second-order perturbation calculation method of transmission characteristic of random rough surface - Google Patents

Abstract

The invention provides a second-order perturbation calculation method of the transmission characteristic of a random rough surface. The calculation method considering the influence of a second-order term solution on the transmission characteristics of the rough surface and providing a second-order transmission field, second-order transmissivity and second-order two-way transmission coefficients by utilizing a perturbation method enlarges the application scope, enhances the accuracy and meets the requirements for accuracy of microwave radiation remote-sensing quantitative information inversion.

Description

Second order perturbation method random rough face transmissison characteristic computing method

Technical field

The invention belongs to the electromagnetic radiation field, be specifically related to a kind of perturbation calculus method of random rough face transmissison characteristic.

Background technology

The application need of numerous subjects such as radio wave propagation, communication, Target Recognition classification, environmental system detection, remote sensing, biomedical diagnostic, construction material test, promoted the deep development of random rough surfaces electromagnetic radiation, scattering and transmission research. in analytical method is found the solution, most important two kinds of methods are Kirchhoff approximation method and perturbation method, and other method is all more or less relevant with these two kinds of basic skills.

The applicable elements of perturbation method is, the surface standard deviation is less than about 5% of electromagnetic wavelength, and surperficial average pitch and wave number and surface standard deviation are long-pending the same order of magnitude, and promptly is applicable to small scale roughness situation.Many application in microwave remote sensing all the small scale roughness can occur, as the small scale wave (capillary gravity wave) on sea, and more smooth exposed soil surface, and the regional area of moonscape etc.Studies show that when vertically observing the radiation brightness on atural object surface with microwave radiometer, this small scale roughness can produce material impact to radiation brightness.

Existing perturbation method can be considered the contribution of scatters of zeroth order, single order and second order, wherein to separate be that relevant component is separated when regarding uneven surface as plane to zeroth order, it is that the incoherent component of lowest-order is separated that single order is separated, it then is that the lowest-order of relevant component is corrected that second order is separated, for the energy conservation that keeps algorithm, the computational accuracy that improves reflectivity, emissivity has significance.Existing perturbation method research focuses mostly in the scattering research of uneven surface, research aspect transmission then is short of very much, in fact, the transmissison characteristic of uneven surface is the same with scattering properties important researching value, as in microwave radiation remote sensing to atural objects such as soil, lunar soil, these atural objects are usually visual makes shaggy non-isothermal layered medium, calculate the radiance temperature of these atural objects, must grasp the transmissison characteristic of its rough surface.Existing document only has the transmissison characteristic research of single order perturbation method, surpasses 20% error but only consider that zeroth order and single order are separated to cause, and can't satisfy the demand of practical application.And, have the foreign scholar to suppose recently based on Rayleigh to the second order perturbation method, only provided the two station of second order transmission coefficient, but do not derived the analytic formula of second order transmissivity at periodical media.The present invention is directed to the random rough face, based on Huygens' principle and extinction theorem, derive second order transmitted field, second order two station transmission coefficient and second order transmissivity with perturbation method, obtained complete second order transmissison characteristic and verified second order perturbation method energy conservation situation and the two station of second order transmission coefficient.

Summary of the invention

The present invention ignores second order term in the existing random rough face perturbation method transmission model and separates this weak point for solving, provide a kind of random rough face second order perturbation method transmissison characteristic computing method of considering second order transmission contribution, to satisfy the accuracy requirement of microwave radiation remote sensing quantitative information inverting.

Second order perturbation method random rough face transmissison characteristic computing method comprise the calculation procedure of transmitted field, transmissivity and two-way transmission coefficient, are specially:

(1) described uneven surface transmitted field is expressed as:

${\stackrel{\&OverBar;}{E}}_t^{\left(2\right)}\left(\stackrel{\&OverBar;}{r}\right)=\&Integral;{\stackrel{\&OverBar;}{\mathrm{dk}}}_{\&perp;}{e}^{{\stackrel{\&OverBar;}{\mathrm{ik}}}_{i\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}-{\mathrm{ik}}_{1z}z}\left\{{\hat{e}}_1(-k_{1z})\right[f_{\mathrm{ee}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{e}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_i)+f_{\mathrm{eh}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{h}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_i)]$

${+\hat{h}}_1(-k_{1z})[f_{\mathrm{he}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{e}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_i)+f_{\mathrm{hh}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{h}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_i)]$

Wherein,

$f_{\mathrm{ee}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})=\frac{k_{\mathrm{iz}}}{k_{\mathrm{iz}}+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{<figure-callout\; id="1"\; label="iz\; k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">iz</figure-callout>}}}\frac{k_{}^{}}{}\frac{{}_1^2-k^2}{k_{1z}+k_z}\{\cos ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)\&CenterDot;({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}-k_z)F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})$

$-2({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2)\&Integral;{\stackrel{\&OverBar;}{\mathrm{dk}}}_{\&perp;}^{\&prime;}F({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})$

$\&CenterDot;[-\sin ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})\sin ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i)\frac{k_z^{\&prime;}{k}_{1z}^{\&prime;}}{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2{k}_z^{\&prime;}+k^2{k}_{1z}^{\&prime;}}+\cos ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})\cos ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i)\frac{1}{k_{1z}^{\&prime;}+k_z^{\&prime;}}]\}$

$f_{\mathrm{eh}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})=\frac{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2}{k_{1z}+k_z}\frac{{\mathrm{<figure-callout\; id="1"\; label="kk\; iz\; k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">kk</figure-callout>}}_{\mathrm{iz}}}{k_{}^{}}\{({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}{k}_z)F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})\sin ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)$

$-2\&Integral;{\stackrel{\&OverBar;}{\mathrm{dk}}}_{\&perp;}^{\&prime;}F({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})[\sin ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_k^{\&prime;})\frac{k_1^2{k}_{\mathrm{\&rho;}}^{\&prime;}{k}_{\mathrm{\&rho;i}}}{k_{\mathrm{\&rho;}}^{\&prime;2}k+k_z^{\&prime;}{k}_{1z}^{\&prime;}}+\frac{({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2)k_{1\mathrm{zi}}}{k_1^2{k}_z^{\&prime;}+k^2{k}_{1z}^{\&prime;}}$

$\&CenterDot;(\sin ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})\cos ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i)k_z^{\&prime;}{k}_{1z}^{\&prime;}+\cos ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})\sin ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i)(k_{\mathrm{\&rho;}}^{\&prime;2}+k_z^{\&prime;}{k}_{1z}^{\&prime;})\left]\right\}$

$f_{\mathrm{he}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})=\frac{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2}{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}+k_{\mathrm{iz}}}\frac{k_1{\mathrm{<figure-callout\; id="1"\; label="k\; iz\; k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{\mathrm{iz}}}{k_{}^{}}\{(k^2-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}{k}_z)F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})\sin ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)$

$+2\&Integral;{\stackrel{\&OverBar;}{\mathrm{dk}}}_{\&perp;}^{\&prime;}F({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})[\sin ({\mathrm{\&phi;}}_k^{\&prime;}-{\mathrm{\&phi;}}_i)\frac{k_1^2{k}_{\mathrm{\&rho;}}^{\&prime;}{k}_{\mathrm{\&rho;}}}{k_{\mathrm{\&rho;}}^{\&prime;2}k+k_z^{\&prime;}{k}_{1z}^{\&prime;}}+\frac{({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2){\mathrm{<figure-callout\; id="1"\; label="k\; z\; k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_z}{k_{}^{}}$

$\&CenterDot;\left(k_z^{\&prime;}{k}_{1z}^{\&prime;}\sin ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})\cos ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i)+(k_{\mathrm{\&rho;}}^{\&prime;2}+k_z^{\&prime;}{k}_{1z}^{\&prime;})\cos ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})\sin ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i)\right)]$

$f_{\mathrm{hh}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})=\frac{({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2)}{({{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^{2}k}_z+{k^{2}k}_{1z})}\frac{k_1{\mathrm{<figure-callout\; id="1"\; label="k\; iz\; k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{\mathrm{iz}}}{k_{}^{}}\{k({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2{k}_z-{k^2\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})\sin ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)$

$+2\&Integral;{\stackrel{\&OverBar;}{\mathrm{dk}}}_{\&perp;}^{\&prime;}F({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})[{\mathrm{kk}}_z{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}\frac{({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2)}{k_z^{\&prime;}+k_{1z}^{\&prime;}}\sin ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})\sin ({\mathrm{\&phi;}}_k^{\&prime;}-{\mathrm{\&phi;}}_i)$

$+\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2{k}_z^{\&prime;}+k^2{k}_{1z}^{\&prime;}}[-k({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2)k_{\mathrm{\&rho;}}{k}_{\mathrm{\&rho;}}^{\&prime;2}{k}_{\mathrm{\&rho;i}}+k_{\mathrm{\&rho;}}{k}_{\mathrm{\&rho;}}^{\&prime;}{k}^3(k_z^{\&prime;}+k_{1z}^{\&prime;})k_{1\mathrm{zi}}\cos ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i)-k_{\mathrm{\&rho;}}^{\&prime;}{k}_{\mathrm{\&rho;i}}{\mathrm{<figure-callout\; id="1"\; label="kk"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">kk</figure-callout>}}_1^2$

$\&CenterDot;(k_z^{\&prime;}+k_{1z}^{\&prime;})k_z\cos ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})-{\mathrm{kk}}_z^{\&prime;}{k}_{1z}^{\&prime;}({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2)k_{\mathrm{<figure-callout\; id="1"\; label="z\; k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">z</figure-callout>}}{k}_{}{}_{1\mathrm{zi}}\cos ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})\cos ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i)\left]\right\}$

(2) described transmissivity is expressed as:

$t(\mathrm{\&pi;}-{\mathrm{\&theta;}}_i,{\mathrm{\&phi;}}_i)=\frac{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}}{k_{\mathrm{zi}}}|1+R_{\mathrm{ho}}{|}^2(\hat{e}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i)+\frac{k^2{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{<figure-callout\; id="1"\; label="zi\; k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">zi</figure-callout>}}}{{k_{}}^{}}|1+R_{\mathrm{vo}}{|}^2(\hat{h}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i)$

$+2\mathrm{Re}\left(\frac{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}}{k_{\mathrm{zi}}}(1+R_{\mathrm{ho}})f_{\mathrm{ee}}^{t\left(2\right)}\left(k_{i\&perp;}\right)\right)(\hat{e}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i)+2\mathrm{Re}\left(\frac{{\mathrm{<figure-callout\; id="1"\; label="kk"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">kk</figure-callout>}}_{1\mathrm{zi}}}{k_1{k}_{\mathrm{zi}}}(1+R_{\mathrm{vo}})f_{\mathrm{hh}}^{t\left(2\right)}\left(k_{i\&perp;}\right)\right)$

$\&CenterDot;(\hat{h}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i)+\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}\frac{k_{1\mathrm{zi}}}{k_{\mathrm{zi}}}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})\left[\right|f_{\mathrm{ee}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})\&CenterDot;(\hat{e}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i)+f_{\mathrm{eh}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})\&CenterDot;(\hat{h}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i){|}^2$

$+|f_{\mathrm{he}}^{t\left(1\right)}(\hat{e}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i)+f_{\mathrm{hh}}^{t\left(1\right)}(\hat{h}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i){|}^2]$

Wherein

Be the Fourier conversion of the related function of random fluctuation height,

$W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})=\frac{h^2{l}^2}{4\mathrm{\&pi;}}\exp (-\frac{(k_{\mathrm{\&rho;}}^2+k_{\mathrm{\&rho;i}}^2)l^2}{4}+\frac{k_{\mathrm{\&rho;}}{k}_{\mathrm{\&rho;i}}{l}^2}{2}\cos ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i));$

(3) described two-way transmission coefficient is expressed as:

In incident wave h polarization, the two-way transmission coefficient under the transmitted wave h polarization situation is:

$\frac{1}{4\mathrm{\&pi;}}{\mathrm{\&gamma;}}_{\mathrm{hh}}^t(\mathrm{\&pi;}-{\mathrm{\&theta;}}_t,{\mathrm{\&phi;}}_t;\mathrm{\&pi;}-{\mathrm{\&theta;}}_i,{\mathrm{\&phi;}}_i)=\left[\frac{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}}{k_{\mathrm{zi}}}\right|1+R_{\mathrm{ho}}{|}^2+1\mathrm{Re}\left(\frac{k_{1\mathrm{zi}}}{k_{\mathrm{zi}}}(1+R_{\mathrm{ho}})f_{\mathrm{ee}}^{t\left(2\right)}\left({\stackrel{\&OverBar;}{k}}_{i\&perp;}\right)\right)]\mathrm{\&delta;}({\cos \mathrm{\&theta;}}_t-\frac{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}}{k_1})\mathrm{\&delta;}({\mathrm{\&phi;}}_t-{\mathrm{\&phi;}}_i)$

$+\frac{k_1{k}_{1z}^2}{k_{\mathrm{iz}}}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})|f_{\mathrm{ee}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;}){|}^2$

In incident wave v polarization, under the transmitted wave h polarization situation,

$\frac{1}{4\mathrm{\&pi;}}{\mathrm{\&gamma;}}_{\mathrm{hv}}^t(\mathrm{\&pi;}-{\mathrm{\&theta;}}_t,{\mathrm{\&phi;}}_t;\mathrm{\&pi;}-{\mathrm{\&theta;}}_i,{\mathrm{\&phi;}}_i)=\frac{k_1{k}_{1z}^2}{k_{\mathrm{iz}}}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})|f_{\mathrm{eh}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;}){|}^2$

In incident wave h polarization, under the transmitted wave v polarization situation,

$\frac{1}{4\mathrm{\&pi;}}{\mathrm{\&gamma;}}_{\mathrm{vh}}^t(\mathrm{\&pi;}-{\mathrm{\&theta;}}_t,{\mathrm{\&phi;}}_t;\mathrm{\&pi;}-{\mathrm{\&theta;}}_i,{\mathrm{\&phi;}}_i)=\frac{k_1{k}_{1z}^2}{k_{\mathrm{iz}}}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})|f_{\mathrm{he}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;}){|}^2$

In incident wave v polarization, under the transmitted wave v polarization situation,

$\frac{1}{4\mathrm{\&pi;}}{\mathrm{\&gamma;}}_{\mathrm{vv}}^t(\mathrm{\&pi;}-{\mathrm{\&theta;}}_t,{\mathrm{\&phi;}}_t;\mathrm{\&pi;}-{\mathrm{\&theta;}}_i,{\mathrm{\&phi;}}_i)=\left[\frac{k^2{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{<figure-callout\; id="1"\; label="zi\; k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">zi</figure-callout>}}}{k_{}^{}}\right|1+R_{\mathrm{vo}}{|}^2+2\mathrm{Re}\left(\frac{{\mathrm{<figure-callout\; id="1"\; label="kk"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">kk</figure-callout>}}_{1\mathrm{zi}}}{k_1{k}_{\mathrm{iz}}}(1+R_{\mathrm{vo}})f_{\mathrm{hh}}^{t\left(2\right)}\left({\stackrel{\&OverBar;}{k}}_{i\&perp;}\right)\right)]$

$\&CenterDot;\mathrm{\&delta;}(\cos {\mathrm{\&theta;}}_t-\frac{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}}{k_1})\mathrm{\&delta;}({\mathrm{\&phi;}}_t-{\mathrm{\&phi;}}_i)+\frac{k_1{k}_{1z}^2}{k_{\mathrm{iz}}}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})|f_{\mathrm{hh}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;}){|}^2$

A wherein, b are respectively two kinds of situations of v polarization and h polarization, and the expression mirror should be with θ to transmission direction _t, φ _tForm:

Wherein

The time, θ _tJust point to transmission direction, Re () expression is got real part to the number in the bracket;

$f_{\mathrm{ab}}^{t\left(2\right)}\left({\stackrel{\&OverBar;}{k}}_{i\&perp;}\right)=<f_{\mathrm{ab}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})>a,b=v,h$

$f_{\mathrm{ee}}^{t\left(2\right)}\left({\stackrel{\&OverBar;}{k}}_{i\&perp;}\right)=\frac{k_{\mathrm{iz}}({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2)}{{(k_{\mathrm{iz}}+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})}^2}\{({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}-k_{\mathrm{iz}}){\&Integral;}_{-\&infin;}^{\&infin;}d{\stackrel{\&OverBar;}{k}}_{\&perp;}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})-2({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2){\&Integral;}_{-\&infin;}^{\&infin;}d{\stackrel{\&OverBar;}{k}}_{\&perp;}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})$

$\&CenterDot;[{\sin }^2({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)\frac{k_z{k}_{1\mathrm{<figure-callout\; id="1"\; label="z\; k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">z</figure-callout>}}}{k_{}^{}}+{\cos }^2({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)\frac{1}{k_{1z}+k_z}]\}$

$f_{\mathrm{hh}}^{t\left(2\right)}\left({\stackrel{\&OverBar;}{k}}_{i\&perp;}\right)=\frac{k_1({{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1}^2-k^2)k_{\mathrm{iz}}}{{({{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1}^2{k}_{\mathrm{iz}}+k^2{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})}^2}\{{\&Integral;}_{-\&infin;}^{\&infin;}d{\stackrel{\&OverBar;}{k}}_{\&perp;}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})k({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2{k}_{\mathrm{iz}}-k^2{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})+$

$2{\&Integral;}_{-\&infin;}^{\&infin;}d{\stackrel{\&OverBar;}{k}}_{\&perp;}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})[{-\mathrm{kk}}_{\mathrm{iz}}{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}\frac{({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2)}{k_2+k_{1z}}{\sin }^2({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)+$

$\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2{k}_z+k^2{k}_{1z}}(-k({{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1}^2-k^2)k_{\mathrm{\&rho;i}}^2{k}_{\mathrm{\&rho;}}^2+k_{\mathrm{\&rho;}}{k}_{\mathrm{\&rho;i}}k(k_z+k_{1z})(k^2{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2{k}_{\mathrm{iz}})$

$\&CenterDot;\cos ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)-{\mathrm{kk}}_z{k}_{1z}{\mathrm{<figure-callout\; id="1"\; label="k\; iz\; k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{\mathrm{iz}}{k}_{}{}_{1\mathrm{zi}}({{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1}^2-k^2){\cos }^2({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)\left)\right]\}$

The incident wave vector

The horizontal component of incident wave vector is

k _ix＝ksinθ _icosφ _i，k _iy＝ksinθ _isinφ _i，k _iz＝kcosθ _i，k _ρi＝ksinθ _i，

The mould of incident wave horizontal component

The scattering wave vector

The scattering wave horizontal component

The mould of scattering wave horizontal component

The mould of scattering wave vertical component

The transmitted wave vector

The horizontal component of transmitted wave vector is

The mould of transmitted wave horizontal component

The mould of transmitted wave vertical component

${\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}=\sqrt{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2+k_{i\&perp;}^2},$

The incident angle θ of incident wave _iAnd position angle The angle of transmission θ of transmitted wave _tAnd position angle

With

Represent an initial point and a position vector of putting respectively,

When the incident field is the TE polarization

When the incident field is the TM polarization

μ _oAnd μ ₁Represent the magnetic permeability of medium 0 and medium 1 respectively,

ε _oAnd ε ₁Represent the conductivity of medium 0 and medium 1 respectively,

W is an angular frequency,

η ₁Be the wave impedance of medium 1,

The root-mean-square height h of uneven surface and persistence length l,

$\hat{e}\left(k_z\right)=\hat{k}\&times;\hat{z}/|\hat{k}\&times;\hat{z}|=(1/k_{\mathrm{\&rho;}})(\hat{x}{k}_y-\hat{y}{k}_x),$

$\hat{e}\left({-k}_z\right)=(1/k_{\mathrm{\&rho;}})(\hat{x}{k}_y-\hat{y}{k}_x),$

$\hat{e}\left({-\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}\right)={\hat{k}}_1\&times;\hat{z}/|{\hat{k}}_1\&times;\hat{z}|=(1/{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{\&rho;i}})(\hat{x}{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{yi}}-\hat{y}{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{xi}}),$

${\hat{e}}_1\left(k_{1z}\right)={\hat{k}}_1\&times;\hat{z}/|{\hat{k}}_1\&times;\hat{z}|=(1/k_{\mathrm{\&rho;}})(\hat{x}{k}_y-\hat{y}{k}_x),$

$\hat{h}\left(k_z\right)=\frac{1}{k}\hat{e}\&times;\hat{k}=\frac{{-k}_z}{{\mathrm{kk}}_{\mathrm{\&rho;}}}(\hat{x}{k}_x+\hat{y}{k}_y)+\frac{k_{\mathrm{\&rho;}}}{k}\hat{z},$

$\hat{h}\left({-k}_z\right)=\frac{k_z}{{\mathrm{kk}}_{\mathrm{\&rho;}}}(\hat{x}{k}_x+\hat{y}{k}_y)+\frac{k_{\mathrm{\&rho;}}}{k}\hat{z},$

${\hat{h}}_1\left(k_{1z}\right)=\frac{1}{k_1}{\hat{e}}_1\&times;{\hat{k}}_1=\frac{{-k}_{1z}}{k_1{k}_{\mathrm{\&rho;}}}(\hat{x}{k}_x+\hat{y}{k}_y)+\frac{k_{\mathrm{\&rho;}}}{k_1}\hat{z},$

${\hat{h}}_1\left({-k}_{1z}\right)=\frac{k_{1z}}{k_1{k}_{\mathrm{\&rho;}}}(\hat{x}{k}_{1x}+\hat{y}{k}_{1y})+\frac{k_{\mathrm{\&rho;}}}{k_1}\hat{z},$

${\hat{h}}_1\left({-\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}\right)=\frac{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}}{k_1{k}_{\mathrm{\&rho;i}}}(\hat{x}{k}_{\mathrm{xi}}+\hat{y}{k}_{\mathrm{yi}})+\frac{k_{\mathrm{\&rho;i}}}{k_1}\hat{z},$

Fourier transform

$\frac{1}{{4\mathrm{\&pi;}}^2}\&Integral;d{\stackrel{\&OverBar;}{r}}_{\&perp;}{e}^{-i{\stackrel{\&OverBar;}{k}}_{\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}}{f}^2\left({\stackrel{\&OverBar;}{r}}_{\&perp;}\right)=F^2\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right),$

$F^2\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)={\&Integral;}_{-\&infin;}^{\&infin;}d{{\stackrel{\&OverBar;}{k}}^{\&prime;}}_{\&perp;}F\left({{\stackrel{\&OverBar;}{k}}^{\&prime;}}_{\&perp;}\right)F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{{\stackrel{\&OverBar;}{k}}^{\&prime;}}_{\&perp;}),$

Be the Fresnel transmission coefficient of TE ripple,

Be the Fresnel transmission coefficient of TM ripple,

φ ' _k, k ' _ρ, k ' _zAnd k ' _1zIt is the needed intermediate variable of integration.

The present invention has following advantage compared to existing technology:

At the random rough face, existing perturbation method has only provided the computing formula of single order transmitted field and single order transmissivity, and the scope of application of single order transmitting formula is:

Even if but in this scope, the transmissivity and the reflectivity that calculate do not meet energy conservation, with the error that causes near 20%, can't satisfy the demand of practical application.The present invention utilizes perturbation method, provided the computing formula of second order transmitted field, second order transmissivity and second order two-way transmission coefficient through strict theoretical, and verified the energy conservation of single order, second order perturbation method, the result shows that the scope of application of second order transmitting formula is wideer than the single order, and in its scope of application, the transmissivity and the reflectivity that calculate meet energy conservation, can satisfy the demand of practical application.

Description of drawings

Fig. 1 is a two-dimensional random uneven surface synoptic diagram;

Fig. 2 is at the energy conservation proof diagram of roughness in the single order scope of application, and wherein Fig. 2 (a) is the v polarization, and Fig. 2 (a) is the h polarization.

Fig. 3 is the energy conservation proof diagram that exceeds the single order scope of application in roughness, and wherein Fig. 3 (a) is the v polarization, and Fig. 3 (a) is the h polarization.

Embodiment

At first calculate the second order transmitted field, calculate the second order transmissivity then.

One, transmitted field is found the solution

Suppose to have a plane wave: Incided by medium 0 and to enter on the random rough face in the medium 1, the specific inductive capacity of medium 0 is ε ₀, the DIELECTRIC CONSTANT of medium 1 ₁Incident wave vector wherein

The horizontal component of incident wave vector is

The random rough face is set up three-dimensional cartesian coordinate system, by z=f (x, y) random function is described this uneven surface, to function get ensemble average get＜f (x, y) 〉=0.(as Fig. 1) f _MinAnd f _MaxReplace uneven surface f (x, minimum y) and maximal value respectively.

Utilize Huygens' principle and delustring principle to know the scattering electric field in the medium 0

And magnetic field

With transmission electric field in the medium 1

And magnetic field

Satisfy:

${\stackrel{\&OverBar;}{E}}_i\left(\stackrel{\&OverBar;}{r}\right)+{\&Integral;}_{S^{\&prime;}}d{S}^{\&prime;}\{\mathrm{iw}{\mathrm{\&mu;}}_o\stackrel{\stackrel{\&OverBar;}{\&OverBar;}}{G}(\stackrel{\&OverBar;}{r},{\stackrel{\&OverBar;}{r}}^{\&prime;})\&CenterDot;[\hat{n}\&times;\stackrel{\&OverBar;}{H}\left({\stackrel{\&OverBar;}{r}}^{\&prime;}\right)]+\&dtri;\&times;\stackrel{\stackrel{\&OverBar;}{\&OverBar;}}{G}(\stackrel{\&OverBar;}{r},{\stackrel{\&OverBar;}{r}}^{\&prime;})\&CenterDot;[\hat{n}\&times;\stackrel{\&OverBar;}{E}\left({\stackrel{\&OverBar;}{r}}^{\&prime;}\right)\left]\right\}=\left\{\begin{array}{ccc}\stackrel{\&OverBar;}{E}& z>f\left(\stackrel{\&OverBar;}{r}\right)& \left(a\right)\\ 0& z<f\left(\stackrel{\&OverBar;}{r}\right)& \left(b\right)\end{array}\right.$

${\&Integral;}_{S^{\&prime;}}d{S}^{\&prime;}\{\mathrm{iw}{\mathrm{\&mu;}}_1{\stackrel{\stackrel{\&OverBar;}{\&OverBar;}}{G}}_1(\stackrel{\&OverBar;}{r},{\stackrel{\&OverBar;}{r}}^{\&prime;})\&CenterDot;[{\hat{n}}_d\&times;{\stackrel{\&OverBar;}{H}}_1\left({\stackrel{\&OverBar;}{r}}^{\&prime;}\right)+\&dtri;\&times;{\stackrel{\stackrel{\&OverBar;}{\&OverBar;}}{G}}_1(\stackrel{\&OverBar;}{r},{\stackrel{\&OverBar;}{r}}^{\&prime;})\&CenterDot;[{\hat{n}}_d\&times;{\stackrel{\&OverBar;}{E}}_1\left({\stackrel{\&OverBar;}{r}}^{\&prime;}\right)]\}=\left\{\begin{array}{ccc}0& z>f\left(\stackrel{\&OverBar;}{r}\right)& \left(a\right)\\ {\stackrel{\&OverBar;}{E}}_1\left(\stackrel{\&OverBar;}{r}\right)& z<f\left(\stackrel{\&OverBar;}{r}\right)& \left(b\right)\end{array}\right.---\left(2\right)$

S ' expression is carried out integration to whole uneven surface,

Be the bin unit normal vector that points to medium 0, The unit normal vector of medium 1 is pointed in expression.

With

Green function in expression medium 0 and the medium 1.

According to electric field and the continuous boundary condition of magnetic field tangential component, can make

$d{{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\stackrel{\&OverBar;}{a}\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right)=d{S}^{\&prime;}\mathrm{\&eta;}\hat{n}\&times;\stackrel{\&OverBar;}{H}\left({\stackrel{\&OverBar;}{r}}^{\&prime;}\right)=d{S}^{\&prime;}\mathrm{\&eta;}\hat{n}\&times;{\stackrel{\&OverBar;}{H}}_1\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right),d{{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\stackrel{\&OverBar;}{b}\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right)=d{S}^{\&prime;}\hat{n}\&times;\stackrel{\&OverBar;}{E}\left({\stackrel{\&OverBar;}{r}}^{\&prime;}\right)=d{S}^{\&prime;}\hat{n}\&times;{\stackrel{\&OverBar;}{E}}_1\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right)---\left(3\right)$

$\hat{n}\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right)\&CenterDot;\stackrel{\&OverBar;}{a}\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right)=0,\hat{n}\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right)\&CenterDot;\stackrel{\&OverBar;}{b}\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right)=0$

Because

Can release:

$a_z\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right)=(\hat{x}\frac{\&PartialD;f\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right)}{{\&PartialD;x}^{\&prime;}}+\hat{y}\frac{\&PartialD;f\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right)}{{\&PartialD;y}^{\&prime;}})\&CenterDot;\stackrel{\&OverBar;}{a}\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right),b_z\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right)=(\hat{x}\frac{\&PartialD;f\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right)}{{\&PartialD;x}^{\&prime;}}+\hat{y}\frac{\&PartialD;f\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right)}{{\&PartialD;y}^{\&prime;}})\&CenterDot;\stackrel{\&OverBar;}{b}\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right)---\left(5\right)$

Then by (1b) and (2a) can push away:

${\stackrel{\&OverBar;}{E}}_i\left(\stackrel{\&OverBar;}{r}\right)=\frac{1}{8{\mathrm{\&pi;}}^2}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}{e}^{i{\stackrel{\&OverBar;}{k}}_{i\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}}{e}^{-{\mathrm{ik}}_{z}z}\frac{k}{k_z}\&Integral;d{{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}{e}^{-i{\stackrel{\&OverBar;}{k}}_{i\&perp;}\&CenterDot;{{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}}{e}^{i{k}_{z}f\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right)}\&times;\left\{\right[\hat{e}(-k_z)\hat{e}(-k_z)+\hat{h}(-k_z)\&CenterDot;\hat{h}(-k_z)]\&CenterDot;\stackrel{\&OverBar;}{a}\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right)---\left(6\right)$

$+[-\hat{h}\left({-k}_z\right)\hat{e}\left({-k}_z\right)+\hat{e}(-k_z)\&CenterDot;\hat{h}(-k_z)]\&CenterDot;\stackrel{\&OverBar;}{b}\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right)\}$

$0=\frac{1}{8{\mathrm{\&pi;}}^2}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}{e}^{i{\stackrel{\&OverBar;}{k}}_{i\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}}{e}^{{\mathrm{ik}}_{1z}z}\frac{k_1}{k_{1z}}\&Integral;d{{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}{e}^{-i{\stackrel{\&OverBar;}{k}}_{i\&perp;}\&CenterDot;{{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}}{e}^{-{\mathrm{ik}}_{1z}f\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right)}\&times;\left\{\frac{k}{k_1}\right[{\hat{e}}_1\left(k_{1z}\right){\hat{e}}_1\left(k_{1z}\right)+{\hat{h}}_1\left(k_{1z}\right)\&CenterDot;{\hat{h}}_1\left(k_{1z}\right)]\&CenterDot;\stackrel{\&OverBar;}{a}\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right)---\left(7\right)$

$+[-{\hat{h}}_1\left({\mathrm{kk}}_{1z}\right){\hat{e}}_1\left(k_{1z}\right)+{\hat{e}}_1\left(k_{1z}\right)\&CenterDot;{\hat{h}}_1\left(k_{1z}\right)]\&CenterDot;\stackrel{\&OverBar;}{b}\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right)\}$

Scattering wave vector wherein

$\stackrel{\&OverBar;}{k}={\stackrel{\&OverBar;}{k}}_{\&perp;}+\hat{z}{k}_z=\hat{x}{k}_x+\hat{y}{k}_y+\hat{z}{k}_z,k=\mathrm{\&omega;}\sqrt{{\mathrm{\&mu;}}_0{\mathrm{\&epsiv;}}_0}---\left(8\right)$

The horizontal component of scattering wave vector is

${\stackrel{\&OverBar;}{k}}_{\&perp;}=\hat{x}{k}_x+\hat{y}{k}_y---\left(9\right)$

Transmitted wave vector wherein

${\stackrel{\&OverBar;}{k}}_1={\stackrel{\&OverBar;}{k}}_{\&perp;}+\hat{z}{k}_{1z}=\hat{x}{k}_x+\hat{y}{k}_y+\hat{z}{k}_{1z},{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1=\mathrm{\&omega;}\sqrt{{\mathrm{\&mu;}}_0{\mathrm{\&epsiv;}}_0}---\left(10\right)$

The horizontal component of transmitted wave vector is

${\stackrel{\&OverBar;}{k}}_{1\&perp;}=\hat{x}{k}_x+\hat{y}{k}_y=-{\stackrel{\&OverBar;}{k}}_{\&perp;}--\left(11\right)$

The mould of horizontal component

$k_{\mathrm{\&rho;}}=\sqrt{k_x^2+k_y^2},k_z=\sqrt{k^2-k_{\mathrm{\&rho;}}^2},k_{1z}=\sqrt{{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1}^2-k_{\mathrm{\&rho;}}^2}---\left(12\right)$

$\hat{e}\left(k_z\right)=\hat{k}\&times;\hat{z}/|\hat{k}\&times;\hat{z}|=(1/k_{\mathrm{\&rho;}})(\hat{x}{k}_y-\hat{y}{k}_x)---\left(13\right)$

$\hat{h}\left(k_z\right)=\frac{1}{k}\hat{e}\&times;\hat{k}=\frac{{-k}_z}{{\mathrm{kk}}_{\mathrm{\&rho;}}}(\hat{x}{k}_x+\hat{y}{k}_y)+\frac{k_{\mathrm{\&rho;}}}{k}\hat{z}---\left(14\right)$

${\hat{e}}_1\left(k_{1z}\right)={\hat{k}}_1\&times;\hat{z}/|{\hat{k}}_1\&times;\hat{z}|=(1/k_{\mathrm{\&rho;}})(\hat{x}{k}_y-\hat{y}{k}_x)---\left(15\right)$

${\hat{h}}_1\left(k_{1z}\right)=\frac{1}{k_1}{\hat{e}}_1\&times;{\hat{k}}_1=\frac{{-k}_{1z}}{{k_{1}k}_{\mathrm{\&rho;}}}(\hat{x}{k}_x+\hat{y}{k}_y)+\frac{k_{\mathrm{\&rho;}}}{k_1}\hat{z}---\left(16\right)$

We can determine the surface field of uneven surface with (6) and (7), promptly solve two unknown quantitys

With

In case we determine the surface field of this uneven surface, the scattered field of medium 0 and the transmitted field of medium 1 just can by (1a) and (2b) formula determine.

Can know by inference by (1a) with (2b)

${\stackrel{\&OverBar;}{E}}_s\left({\stackrel{\&OverBar;}{r}}_{\&perp;}\right)=-\frac{1}{8{\mathrm{\&pi;}}^2}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}{e}^{i{\stackrel{\&OverBar;}{k}}_{\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}}{e}^{{\mathrm{ik}}_{z}z}\frac{k}{k_z}\&Integral;d{{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;\&perp;}{e}^{-i{\stackrel{\&OverBar;}{k}}_{\&perp;}\&CenterDot;{{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}}{e}^{-{\mathrm{ik}}_{z}f\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right)}$

$\&times;\left\{\right[\hat{e}\left(k_z\right)\hat{e}\left(k_z\right)+\hat{h}\left(k_z\right)\hat{h}\left(k_z\right)]\&CenterDot;\stackrel{\&OverBar;}{a}\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right)---\left(17\right)$

$+[-\hat{h}\left(k_z\right)\hat{e}\left(k_z\right)+\hat{e}\left(k_z\right)\hat{h}\left(k_z\right)]\&CenterDot;\stackrel{\&OverBar;}{b}\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right)\}$

${\stackrel{\&OverBar;}{E}}_t\left({\stackrel{\&OverBar;}{r}}_{\&perp;}\right)=\frac{1}{8{\mathrm{\&pi;}}^2}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}{e}^{i{\stackrel{\&OverBar;}{k}}_{\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}}{e}^{{\mathrm{ik}}_{1z}z}\frac{k_1}{k_{1z}}\&Integral;d{{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;\&perp;}{e}^{-i{\stackrel{\&OverBar;}{k}}_{i\&perp;}\&CenterDot;{{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}}{e}^{-{\mathrm{ik}}_{1z}f\left({{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}\right)}$

$\&times;\left\{\frac{k}{k_1}\right[{\hat{e}}_1(-k_{1z}){\hat{e}}_1\left({-k}_{1z}\right)+{\hat{h}}_1\left({-k}_{1z}\right){\hat{h}}_1(-k_{1z})]\&CenterDot;\stackrel{\&OverBar;}{a}\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right)---\left(18\right)$

$+[-{\hat{h}}_1(-k_{1z}){\hat{e}}_1(-k_{1z})+{\hat{e}}_1\left({-k}_{1z}\right){\hat{h}}_1(-k_{1z})]\&CenterDot;\stackrel{\&OverBar;}{b}\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right)\}$

Utilize the perturbation method calculating of further deriving below, decompose

With

As follows:

$\stackrel{\&OverBar;}{a}\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right)=\underset{m=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{\stackrel{\&OverBar;}{a}}^{\left(m\right)}\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right),\stackrel{\&OverBar;}{b}\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right)=\underset{m=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{\stackrel{\&OverBar;}{b}}^{\left(m\right)}\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right)---\left(19\right)$

Here

Representative

The component on m rank.

Can obtain by Taylor series expansion:

$e^{\&PlusMinus;{\mathrm{ik}}_{z}f\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right)}=\underset{m=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{{[\&PlusMinus;{\mathrm{ik}}_{z}f\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right)]}^m}{m!},e^{\&PlusMinus;{\mathrm{ik}}_{1z}f\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right)}=\underset{m=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{{[\&PlusMinus;{\mathrm{ik}}_{1z}f\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right)]}^m}{m!}---\left(20\right)$

In perturbation method, And Slope Parameters all to require be the small scale parameter, promptly false

Utilize (5) and (19) can obtain following relation:

${\stackrel{\&OverBar;}{a}}_z^{\left(0\right)}\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right)={\stackrel{\&OverBar;}{b}}_z^{\left(0\right)}\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right)=0---\left(21\right)$

$a_z^{\left(m\right)}\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right)=(\hat{x}\frac{\&PartialD;f\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right)}{\&PartialD;x^{\&prime;}}+\hat{y}\frac{\&PartialD;f\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right)}{\&PartialD;y^{\&prime;}})\&CenterDot;{\stackrel{\&OverBar;}{a}}_{\&perp;}^{(m-1)}\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right),b_z^{\left(m\right)}\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right)=(\hat{x}\frac{\&PartialD;f\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right)}{\&PartialD;x^{\&prime;}}+\hat{y}\frac{\&PartialD;f\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right)}{\&PartialD;y^{\&prime;}})\&CenterDot;{\stackrel{\&OverBar;}{b}}_{\&perp;}^{(m-1)}\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right)---\left(22\right)$

(19) and (20) formula is brought into (5), and (6) are with (7) and keep identical exponent number, can calculate each rank surface field, and with surface field substitution (17), (18) formula then can obtain the scattering and the transmitted field on each rank.Owing to utilize perturbation method to study the characteristic of second order transmitted field, so be that example illustrates with the second order.

We define the Fourier transform of surface field:

$\stackrel{\&OverBar;}{A}\left(\stackrel{\&OverBar;}{k}\right)=\frac{1}{{\left(2\mathrm{\&pi;}\right)}^2}\&Integral;d{\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\stackrel{\&OverBar;}{a}\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right)e^{-i{\stackrel{\&OverBar;}{k}}_{\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}},\stackrel{\&OverBar;}{B}\left(\stackrel{\&OverBar;}{k}\right)=\frac{1}{{\left(2\mathrm{\&pi;}\right)}^2}\&Integral;d{\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\stackrel{\&OverBar;}{b}\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right)e^{-i{\stackrel{\&OverBar;}{k}}_{\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}}---\left(23\right)$

Below

With Be respectively right

Fourier transform, at spectral domain, (23) can obtain in conjunction with (22) formula:

$A_z\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)=i{\&Integral;}_{-\&infin;}^{\&infin;}d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}){\stackrel{\&OverBar;}{A}}_{\&perp;}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}),B_z\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)=i{\&Integral;}_{-\&infin;}^{\&infin;}d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}){\stackrel{\&OverBar;}{B}}_{\&perp;}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})---\left(24\right)$

Because what derive is second order, so Taylor expansion is also approximate to second order promptly:

$e^{\&PlusMinus;{\mathrm{ik}}_{1z}f\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right)}=1\&PlusMinus;{\mathrm{ik}}_{1z}f\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right)-\frac{k_{1z}^2}{2}{f}^2\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right)---\left(25\right)$

Again to f, f ²Carry out Fourier transform,

$\frac{1}{4{\mathrm{\&pi;}}^2}\&Integral;d{\stackrel{\&OverBar;}{r}}_{\&perp;}{e}^{-i{\stackrel{\&OverBar;}{k}}_{\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}}f\left({\stackrel{\&OverBar;}{r}}_{\&perp;}\right)=F\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right),\frac{1}{4{\mathrm{\&pi;}}^2}\&Integral;d{\stackrel{\&OverBar;}{r}}_{\&perp;}{e}^{-i{\stackrel{\&OverBar;}{k}}_{\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}}{f}^2\left({\stackrel{\&OverBar;}{r}}_{\&perp;}\right)=F^2\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)---\left(26\right)$

Can prove

$F^2\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)={\&Integral;}_{-\&infin;}^{\&infin;}d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}F\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})---\left(27\right)$

(23) and (24) and above-mentioned Taylors approximation substitution (6) (7) formula can be obtained:

${\stackrel{\&OverBar;}{E}}_i\left(\stackrel{\&OverBar;}{r}\right)=\frac{1}{2}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}{e}^{i{\stackrel{\&OverBar;}{k}}_{i\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}-{\mathrm{ik}}_{z}z}\frac{k}{k_z}\left\{\right[\hat{e}(-k_z)\hat{e}(-k_z)+\hat{h}(-k_z)\&CenterDot;\hat{h}(-k_z)]\&CenterDot;[\stackrel{\&OverBar;}{A}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)+{\mathrm{ik}}_z\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{A}}_{\&perp;}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)$

$\&CenterDot;F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})-\frac{k_z^2}{2}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{A}}_{\&perp;}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})]+[-\hat{h}(-k_z)\hat{e}(-k_z)+\hat{e}(-k_z)\&CenterDot;\hat{h}(-k_z)]---\left(28\right)$

$\&CenterDot;[\stackrel{\&OverBar;}{B}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)+{\mathrm{ik}}_z\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{B}}_{\&perp;}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})-\frac{k_z^2}{2}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{A}}_{\&perp;}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})]\}$

$0=\frac{1}{2}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}{e}^{i{\stackrel{\&OverBar;}{k}}_{\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}+{\mathrm{ik}}_{1z}z}\frac{k_1}{k_{1z}}\left\{\frac{k}{k_1}\right[\hat{e}\left(k_{1z}\right){\hat{e}}_1\left(k_{1z}\right)+{\hat{h}}_1\left(k_{1z}\right)\&CenterDot;{\hat{h}}_1\left(k_{1z}\right)]\&CenterDot;[\stackrel{\&OverBar;}{A}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)-{\mathrm{ik}}_{1z}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{A}}_{\&perp;}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)$

$\&CenterDot;F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})-\frac{k_{1z}^2}{2}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{A}}_{\&perp;}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})]+[-{\hat{h}}_1\left(k_{1z}\right){\hat{e}}_1\left(k_{1z}\right)+{\hat{e}}_1\left(k_{1z}\right)\&CenterDot;{\hat{h}}_1\left(k_{1z}\right)]---\left(29\right)$

$[\stackrel{\&OverBar;}{B}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)-{\mathrm{ik}}_{1z}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{B}}_{\&perp;}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})-\frac{k_{1z}^2}{2}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{B}}_{\&perp;}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})]\}$

In like manner can obtain approximate transmitted field to second order is:

${\stackrel{\&OverBar;}{E}}_t\left(\stackrel{\&OverBar;}{r}\right)=\frac{1}{2}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}{e}^{i{\stackrel{\&OverBar;}{k}}_{\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}}{e}^{-i{k}_{1z}z}\frac{k_1}{k_{1z}}\left\{\frac{k}{k_1}\right[{\hat{e}}_1(-k_{1z}){\hat{e}}_1(-k_{1z})+{\hat{h}}_1(-k_{1z}){\hat{h}}_1\left(k_{1z}\right)]$

$[\stackrel{\&OverBar;}{A}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)+{\mathrm{ik}}_{1z}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\stackrel{\&OverBar;}{A}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})-\frac{k_{1z}^2}{2}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\stackrel{\&OverBar;}{A}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})]---\left(30\right)$

$+[-{\hat{h}}_1(-k_{1z}){\hat{e}}_1(-k_{1z})+{\hat{e}}_1(-k_{1z}){\hat{h}}_1(-k_{1z})][\stackrel{\&OverBar;}{B}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)+{\mathrm{ik}}_{1z}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\stackrel{\&OverBar;}{B}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})$

$-\frac{k_{1z}^2}{2}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\stackrel{\&OverBar;}{B}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})\left]\right\}$

(5) formula is carried out Fourier transform can release the horizontal component of uneven surface and the relation between the z axle component:

$A_z^{\left(m\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)=i{\&Integral;}_{-\&infin;}^{\&infin;}d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}){\stackrel{\&OverBar;}{A}}_{\&perp;}^{(m-1)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)---\left(31\right)$

$B_z^{\left(m\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)=i{\&Integral;}_{-\&infin;}^{\&infin;}d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}){\stackrel{\&OverBar;}{B}}_{\&perp;}^{(m-1)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)---\left(32\right)$

The z axle component that is to say m rank uneven surface can be calculated by the horizontal component of m-1 rank uneven surface.Then (6) formula can be write as:

${\&Integral;}_{-\&infin;}^{\&infin;}d{\stackrel{\&OverBar;}{k}}_{\&perp;}{e}^{i{\stackrel{\&OverBar;}{k}}_{i\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}-{\mathrm{ik}}_{z}z}{\hat{e}}_i\mathrm{\&delta;}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})=\frac{\mathrm{<figure-callout\; id="8"\; label="i"\; filenames="DEST\_PATH\_110322152532.png,DEST\_PATH\_110322152538.png"\; state="\{\{state\}\}">i</figure-callout>}}{8{\mathrm{\&pi;}}^2}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}{e}^{i{\stackrel{\&OverBar;}{k}}_{i\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}-{\mathrm{ik}}_{z}z}\frac{k}{k_z}\&Integral;d{{\stackrel{\&OverBar;}{r}}^{\&prime;}}_{\&perp;}{e}^{-i{\stackrel{\&OverBar;}{k}}_{i\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}}(1+{\mathrm{ik}}_{z}f\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right)-\frac{k_z^2}{2}{f}^2\left({\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}\right))$

$\&times;\left\{\right[\hat{e}(-k_z)\hat{e}(-k_z)+\hat{h}(-k_z)\&CenterDot;\hat{h}(-k_z)]\&CenterDot;\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\stackrel{\&OverBar;}{A}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)e^{+i{\stackrel{\&OverBar;}{k}}_{\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}}---\left(33\right)$

$+[-\hat{h}(-k_z)\hat{e}(-k_z)+\hat{e}(-k_z)\&CenterDot;\hat{h}(-k_z)]\&CenterDot;\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\stackrel{\&OverBar;}{B}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)e^{+i{\stackrel{\&OverBar;}{k}}_{\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}^{\&prime;}}\}$

Abbreviation (33) can obtain:

${\hat{e}}_i\mathrm{\&delta;}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})=\frac{1}{2}\frac{k}{k_z}[\hat{e}(-k_z)\hat{e}(-k_z)+\hat{h}(-k_z)\&CenterDot;\hat{h}(-k_z)]$

$\&CenterDot;[\stackrel{\&OverBar;}{A}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)+{\mathrm{ik}}_z\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\stackrel{\&OverBar;}{A}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})-\frac{k_z^2}{2}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\stackrel{\&OverBar;}{A}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})]---\left(34\right)$

$+\frac{1}{2}\frac{k}{k_z}[-\hat{h}(-k_z)\hat{e}(-k_z)+\hat{e}(-k_z)\hat{h}(-k_z)]+\frac{1}{2}\frac{k}{k_z}[-\hat{h}(-k_z)\hat{e}(-k_z)+\hat{e}(-k_z)\hat{h}(-k_z)]$

$\&CenterDot;[\stackrel{\&OverBar;}{B}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)+{\mathrm{ik}}_z\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\stackrel{\&OverBar;}{B}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})-\frac{k_z^2}{2}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\stackrel{\&OverBar;}{B}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})]$

Can obtain with identical abbreviation mode (7) formula

$0=\frac{1}{2}\frac{k}{k_{1z}}[{\hat{e}}_1\left(k_{1z}\right)\hat{e}\left(k_{1z}\right)+{\hat{h}}_1\left(k_{1z}\right)\&CenterDot;{\hat{h}}_1\left(k_{1z}\right)]$

$\&CenterDot;[\stackrel{\&OverBar;}{A}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)-{\mathrm{ik}}_{1z}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\stackrel{\&OverBar;}{A}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})-\frac{k_{1z}^2}{2}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\stackrel{\&OverBar;}{A}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})]---\left(35\right)$

$+\frac{1}{2}\frac{k_1}{k_{1z}}[-{\hat{h}}_1\left(k_{1z}\right){\hat{e}}_1(-k_{1z})+{\hat{e}}_1\left(k_{1z}\right){\hat{h}}_1\left(k_{1z}\right)]$

$\&CenterDot;[\stackrel{\&OverBar;}{B}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)-{\mathrm{ik}}_{1z}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\stackrel{\&OverBar;}{B}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})-\frac{k_{1z}^2}{2}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\stackrel{\&OverBar;}{B}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})]$

Following sublevel is found the solution (34) and (35) equation, and provides the derivation of single order and second order transmitted field.

1, zeroth order and single order transmitted field are found the solution

(1) zeroth order field

Can derive zero Jie's transmitted field by (30) formula is:

${\stackrel{\&OverBar;}{E}}_t^{\left(0\right)}\left(\stackrel{\&OverBar;}{r}\right)=\frac{1}{2}d{\stackrel{\&OverBar;}{k}}_{\&perp;}{e}^{i{\stackrel{\&OverBar;}{k}}_{\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}-{\mathrm{ik}}_{1z}z}\frac{k_1}{k_{1z}}\left\{\frac{k}{k_1}\right[{\hat{e}}_1(-k_{1z}){\hat{e}}_1(-k_{1z})+{\hat{h}}_1(-k_{1z}){\hat{h}}_1\left(k_{1z}\right)]\&CenterDot;{\stackrel{\&OverBar;}{A}}^{\left(0\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)---\left(36\right)$

$+[-{\hat{h}}_1(-k_{1z}){\hat{e}}_1(-k_{1z})+{\hat{e}}_1(-k_{1z}){\hat{h}}_1(-k_{1z})]\&CenterDot;{\stackrel{\&OverBar;}{B}}^{\left(0\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)\}$

0 rank transmission is separated the corresponding flat ripple and is incided a transmission on the plane, (34) and the expression formula of (35) these two extinction theorems is approximate can get to zeroth order:

${\hat{e}}_i\mathrm{\&delta;}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})=\frac{k}{2k_z}[\hat{e}(-k_z)\hat{e}(-k_z)+\hat{h}(-k_z)\hat{h}(-k_z)]\&CenterDot;{\stackrel{\&OverBar;}{A}}_{\&perp;}^{\left(0\right)}---\left(37\right)$

$+\frac{k}{k_z}[-\hat{h}(-k_z)\hat{e}(-k_z)+\hat{e}(-k_z)\hat{h}(-k_z)]\&CenterDot;{\stackrel{\&OverBar;}{B}}_{\&perp;}^{\left(0\right)}$

$0=\frac{k}{2k_{1z}}[{\hat{e}}_1\left(k_{1z}\right){\hat{e}}_1\left(k_{1z}\right)+{\hat{h}}_1\left(k_{1z}\right){\hat{h}}_1\left(k_{1z}\right)]\&CenterDot;{\stackrel{\&OverBar;}{A}}_{\&perp;}^{\left(0\right)}---\left(38\right)$

$+\frac{k}{2k_{1z}}[-{\hat{h}}_1\left(k_{1z}\right){\hat{e}}_1(-k_{1z})+{\hat{e}}_1\left(k_{1z}\right){\hat{h}}_1\left(k_{1z}\right)]\&CenterDot;{\stackrel{\&OverBar;}{B}}_{\&perp;}^{\left(0\right)}$

Can obtain zero Jie's surface field by (36) and (37) With Substitution (36) formula can be released the expression formula of zeroth order transmitted field:

${\stackrel{\&OverBar;}{E}}_t^{\left(0\right)}=\left\{(1+R_{h0})\right[\hat{e}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i]{\hat{e}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})+\frac{k}{k_1}(1+R_{v0})[\hat{h}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i\left]{\hat{h}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\right\}{e}^{i{\stackrel{\&OverBar;}{k}}_{i\&perp;}{\stackrel{\&OverBar;}{r}}_{\&perp;}-{\mathrm{<figure-callout\; id="1"\; label="ik"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">ik</figure-callout>}}_{1\mathrm{zi}}z}---\left(39\right)$

R wherein _H0Be the Fresnel reflection coefficient of TE ripple, R _V0Be the Fresnel reflection coefficient of TM ripple,

$R_{h0}=\frac{k_{\mathrm{iz}}-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{iz}}}{k_{\mathrm{iz}}+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{iz}}};$ $R_{v0}=\frac{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2{k}_{\mathrm{iz}}-k^2{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{<figure-callout\; id="1"\; label="iz\; k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">iz</figure-callout>}}}{k_{}^{}}$

(2) derivation of single order transmitted field

Can derive the single order transmitted field by (30) formula is

${\stackrel{\&OverBar;}{E}}_t^{\left(1\right)}\left(\stackrel{\&OverBar;}{r}\right)=\frac{1}{2}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}{e}^{i{\stackrel{\&OverBar;}{k}}_{\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}-i{k}_{1z}z}\frac{k_1}{k_{1z}}\left\{\frac{k}{k_1}\right[{\hat{e}}_1(-k_{1z}){\hat{e}}_1(-k_{1z})+{\hat{h}}_1(-k_{1z}){\hat{h}}_1\left(k_{1z}\right)]$

$\&CenterDot;[{\stackrel{\&OverBar;}{A}}^{\left(1\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)+{\mathrm{ik}}_{1z}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{A}}^{\left(0\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})]+[-{\hat{h}}_1(-k_{1z}){\hat{e}}_1(-k_{1z})+{\hat{e}}_1(-k_{1z}){\hat{h}}_1(-k_{1z})]---\left(40\right)$

$\&CenterDot;[{\stackrel{\&OverBar;}{B}}^{\left(1\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)+{\mathrm{ik}}_{1z}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{B}}^{\left(0\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})]$

Can get to single order by the expression formula of (34) and (35) these two extinction theorems is approximate:

${\hat{e}}_i\mathrm{\&delta;}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})=\frac{k}{2k_z}[\hat{e}(-k_z)\hat{e}(-k_z)+\hat{h}(-k_z)\hat{h}(-k_z)]$

$\&CenterDot;[{\stackrel{\&OverBar;}{A}}^{\left(1\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)+{\mathrm{ik}}_{1z}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{A}}^{\left(0\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})]---\left(41\right)$

$+\frac{k}{k_z}[-\hat{h}(-k_z)\hat{e}(-k_z)+\hat{e}(-k_z)\hat{h}(-k_z)]$

$\&CenterDot;[{\stackrel{\&OverBar;}{B}}^{\left(1\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)+{\mathrm{ik}}_{1z}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{B}}^{\left(0\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})]$

$0=\frac{k}{2k_{1z}}[{\hat{e}}_1\left(k_{1z}\right){\hat{e}}_1\left(k_{1z}\right)+{\hat{h}}_1\left(k_{1z}\right){\hat{h}}_1\left(k_{1z}\right)]$

$\&CenterDot;[{\stackrel{\&OverBar;}{A}}^{\left(1\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)+{\mathrm{ik}}_{1z}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{A}}^{\left(0\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})]---\left(42\right)$

$+\frac{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1}{2k_{1z}}[-{\hat{h}}_1\left(k_{1z}\right){\hat{e}}_1(-k_{1z})+{\hat{e}}_1\left(k_{1z}\right){\hat{h}}_1\left(k_{1z}\right)]$

$\&CenterDot;[{\stackrel{\&OverBar;}{B}}^{\left(1\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)+{\mathrm{ik}}_{1z}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{B}}^{\left(0\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})]$

Can solve the surface field of single order by (41) and (42)

With

Again will With Substitution (40) can draw single order transmitted field expression formula:

${\stackrel{\&OverBar;}{E}}_t^{\left(1\right)}\left(\stackrel{\&OverBar;}{r}\right)=\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}{e}^{i{\stackrel{\&OverBar;}{k}}_{i\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}-{\mathrm{ik}}_{1z}z}\mathrm{iF}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})\left\{{\hat{e}}_1(-k_{1z})\right[f_{\mathrm{ee}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{e}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_1)$

$+f_{\mathrm{he}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{h}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_1)]+{\hat{h}}_1\left({-k}_{1z}\right)[f_{\mathrm{he}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{e}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_1)---\left(43\right)$

$+f_{\mathrm{hh}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{h}}_1\left({-\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}\right)\&CenterDot;{\hat{e}}_1)]$

When the incident field is the TE polarization

When the incident field is the TM polarization

We define and are released by the coordinate relation:

$f_{\mathrm{ee}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})=\frac{({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2)}{k_z+k_{1z}}\left(\frac{2k_{\mathrm{iz}}}{k_{\mathrm{iz}}+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}}\right)\cos ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)---\left(44\right)$

In like manner:

$f_{\mathrm{he}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})=\frac{k_z{k}_1({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2)}{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2{k}_z+{k^{2}k}_{1z}}\left(\frac{2k_{\mathrm{iz}}}{k_{\mathrm{iz}}+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}}\right)\sin ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)---\left(45\right)$

$f_{\mathrm{eh}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})=\frac{k_{\mathrm{iz}}({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2)}{k_z+k_{1z}}\left(\frac{2{\mathrm{<figure-callout\; id="1"\; label="kk"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">kk</figure-callout>}}_{1\mathrm{<figure-callout\; id="1"\; label="zi\; k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">zi</figure-callout>}}}{{k_{}^{}}_{}}\right)\sin ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)---\left(46\right)$

$f_{\mathrm{hh}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})=\frac{({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2)}{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2{k}_z+{k^{2}k}_{1z}}\left(\frac{2k{k}_1{k}_{\mathrm{iz}}}{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2{k}_{\mathrm{iz}}+{k^2\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}}\right)\{k_{\mathrm{\&rho;}}{k}_{\mathrm{\&rho;i}}+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}{k}_z\cos ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)\}---\left(47\right)$

φ _k, φ _iRefer to transmission position angle and incident orientation angle respectively.

2, the second order transmitted field is found the solution

Can derive the second order transmitted field by (30) formula is

${\stackrel{\&OverBar;}{E}}_t^{\left(2\right)}\left(\stackrel{\&OverBar;}{r}\right)=\frac{1}{2}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}{e}^{i{\stackrel{\&OverBar;}{k}}_{i\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}-{\mathrm{ik}}_{1z}z}\frac{k_1}{k_{1z}}\left\{\frac{k}{k_1}\right[{\hat{e}}_1(-k_{1z}){\hat{e}}_1(-k_{1z})+{\hat{h}}_1(-k_{1z}){\hat{h}}_1\left(k_{1z}\right)]\&CenterDot;[{\stackrel{\&OverBar;}{A}}^{\left(2\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)$

$+{\mathrm{ik}}_{1z}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{A}}^{\left(1\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})-\frac{k_{1z}^2}{2}d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{A}}^{\left(0\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})]---\left(48\right)$

$+[-{\hat{h}}_1(-k_{1z}){\hat{e}}_1\left({-k}_{1z}\right)+{\hat{e}}_1(-k_{1z}){\hat{h}}_1(-k_{1z})]\&CenterDot;[{\stackrel{\&OverBar;}{B}}^{\left(2\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)+{\mathrm{ik}}_{1z}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{B}}^{\left(1\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)$

$\&CenterDot;F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})-\frac{k_{1z}^2}{2}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{B}}^{\left(0\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})]$

Can write out the expression formula of the extinction theorem of second order by (34) and (35) formula:

${\hat{e}}_i\mathrm{\&delta;}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})=\frac{k}{2k_z}[\hat{e}(-k_z)\hat{e}(-k_z)+\hat{h}(-k_z)\hat{h}(-k_z)]\&CenterDot;[{\stackrel{\&OverBar;}{A}}^{\left(2\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)+{\mathrm{ik}}_{\mathrm{iz}}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{A}}^{\left(1\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})$

$-\frac{k_{1z}^2}{2}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{A}}^{\left(0\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right){F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})}^{}]+\frac{k}{k_z}[-\hat{h}\left({-k}_z\right)\hat{e}\left({-k}_z\right)+\hat{e}\left({-k}_z\right)\hat{h}(-k_z)]---\left(49\right)$

$\&CenterDot;[{\stackrel{\&OverBar;}{B}}^{\left(2\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)+{\mathrm{ik}}_{1z}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{B}}^{\left(1\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})-\frac{k_{1z}^2}{2}d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{B}}^{\left(0\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})]$

$0=\frac{k}{2k_{1z}}[{\hat{e}}_1\left(k_{1z}\right){\hat{e}}_1\left(k_{1z}\right)+{\hat{h}}_1\left(k_{1z}\right){\hat{h}}_1\left(k_{1z}\right)]\&CenterDot;[{\stackrel{\&OverBar;}{A}}^{\left(2\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)+{\mathrm{ik}}_{1z}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{A}}^{\left(1\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})$

$-\frac{k_{1z}^2}{2}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{A}}^{\left(0\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})]+\frac{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1}{2k_{1z}}[-{\hat{h}}_1\left(k_{1z}\right){\hat{e}}_1(-k_{1z})+{\hat{e}}_1\left(k_{1z}\right){\hat{h}}_1\left(k_{1z}\right)]---\left(50\right)$

$[{\stackrel{\&OverBar;}{B}}^{\left(2\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)+i{k}_{1z}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{B}}^{\left(1\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})-\frac{k_{1z}^2}{2}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}{\stackrel{\&OverBar;}{B}}^{\left(0\right)}\left({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}\right)F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})]$

By (49), (50) two formulas can solve the surface field of second order

With

Again will

With Substitution (48) can be released out the second order transmitted field

${\stackrel{\&OverBar;}{E}}_t^{\left(2\right)}\left(\stackrel{\&OverBar;}{r}\right)=\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}{e}^{i{\stackrel{\&OverBar;}{k}}_{i\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}-{\mathrm{ik}}_{1z}z}\left\{{\hat{e}}_1(-k_{1z})\right[f_{\mathrm{ee}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{e}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_1)+f_{\mathrm{eh}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{h}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_i)]---\left(51\right)$

$+{\hat{h}}_1(-k_{1z})[f_{\mathrm{he}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{e}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_i)+f_{\mathrm{hh}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{h}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_i)]$

$f_{\mathrm{ee}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})=\frac{k_{\mathrm{iz}}}{k_{\mathrm{iz}}+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{<figure-callout\; id="1"\; label="iz\; k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">iz</figure-callout>}}}\frac{k_{}^{}}{}\frac{{}_1^2-k^2}{k_{1z}+k_z}\{\cos ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)\&CenterDot;({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}-k_z)F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})$

$-2({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2)\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}F({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})---\left(52\right)$

$\&CenterDot;[-\sin ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})\sin ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i)\frac{k_z^{\&prime;}{k}_{1z}^{\&prime;}}{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2{k}_z^{\&prime;}+k^2{k}_{1z}^{\&prime;}}+\cos ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})\cos ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i)\frac{1}{k_{1z}^{\&prime;}+k_z^{\&prime;}}]\}$

$f_{\mathrm{eh}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})=\frac{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2}{k_{1z}+k_z}\frac{k{\mathrm{<figure-callout\; id="1"\; label="k\; iz\; k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{\mathrm{iz}}}{k_{}^{}}\{\left({{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}{k}_z\right)F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})\sin ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)$

$-2\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}F({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})[\sin ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_k^{\&prime;})\frac{k_1^2{k}_{\mathrm{\&rho;}}^{\&prime;}{k}_{\mathrm{\&rho;i}}}{k_{\mathrm{\&rho;}}^{\&prime;2}k+k_z^{\&prime;}{k}_{1z}^{\&prime;}}+\frac{({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2)k_{1\mathrm{zi}}}{k_1^2{k}_z^{\&prime;}+k^2{k}_{1z}^{\&prime;}}---\left(53\right)$

$\&CenterDot;(\sin ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})\cos ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i)k_z^{\&prime;}{k}_{1z}^{\&prime;}+\cos ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})\sin ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i)(k_{\mathrm{\&rho;}}^{\&prime;2}+k_z^{\&prime;}{k}_{1z}^{\&prime;})\left]\right\}$

$f_{\mathrm{he}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})=\frac{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2}{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}+k_{\mathrm{iz}}}\frac{k_1{\mathrm{<figure-callout\; id="1"\; label="k\; iz\; k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{\mathrm{iz}}}{k_{}^{}}\{\left({k^2-\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}{k}_z\right)F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})\sin ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)$

$+2\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}F({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})[\sin ({\mathrm{\&phi;}}_k^{\&prime;}-{\mathrm{\&phi;}}_i)\frac{k_1^2{k}_{\mathrm{\&rho;}}^{\&prime;}{k}_{\mathrm{\&rho;}}}{k_{\mathrm{\&rho;}}^{\&prime;2}+k_z^{\&prime;}{k}_{1z}^{\&prime;}}+\frac{({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2){\mathrm{<figure-callout\; id="1"\; label="k\; z\; k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_z}{k_{}^{}}---\left(54\right)$

$\&CenterDot;(k_z^{\&prime;}{k}_{1z}^{\&prime;}\sin ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})\cos ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i)+(k_{\mathrm{\&rho;}}^{\&prime;2}+k_z^{\&prime;}{k}_{1z}^{\&prime;})\cos ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})\sin ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i))]$

$f_{\mathrm{hh}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})=\frac{({{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1}^2-k^2)}{({{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1}^2{k}_z+k^2{k}_{1z})}\frac{k_1{\mathrm{<figure-callout\; id="1"\; label="k\; iz\; k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{\mathrm{iz}}}{k_{}^{}}\{k({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2{k}_z-k^2{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})\cos ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)$

$+2\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}F({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})[{\mathrm{kk}}_z{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}\frac{({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2)}{k_z^{\&prime;}+k_{1z}^{\&prime;}}\sin ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})\sin ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i)---\left(55\right)$

$+\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2{k}_z^{\&prime;}+k^2{k}_{1z}^{\&prime;}}[-k({{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1}^2-k^2)k_{\mathrm{\&rho;}}{k^{\&prime;}}_{\mathrm{\&rho;}}^2{k}_{\mathrm{\&rho;i}}+k_{\mathrm{\&rho;}}{k}_{\mathrm{\&rho;}}^{\&prime;}{k}^3(k_z^{\&prime;}+k_{1z}^{\&prime;})k_{1\mathrm{zi}}\cos ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i)-k_{\mathrm{\&rho;}}^{\&prime;}{k}_{\mathrm{\&rho;i}}{\mathrm{<figure-callout\; id="1"\; label="kk"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">kk</figure-callout>}}_1^2$

$\&CenterDot;(k_z^{\&prime;}+k_{1z}^{\&prime;})k_z\cos ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})-{\mathrm{kk}}_z^{\&prime;}{k}_{1z}^{\&prime;}({{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1}^2-k^2)k_{\mathrm{<figure-callout\; id="1"\; label="z\; k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">z</figure-callout>}}{k}_{}{}_{1\mathrm{zi}}\cos \left({\mathrm{\&phi;}}_k{{-\mathrm{\&phi;}}_k}^{\&prime;}\right)\cos ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i)\left]\right\}$

Two, two-way transmission coefficient and transmissivity

Can know the total transmission electric field that is accurate to second order by inference by the formula of above 0 rank, single order and second order is:

${\stackrel{\&OverBar;}{E}}_t\left(\stackrel{\&OverBar;}{r}\right)={\stackrel{\&OverBar;}{E}}_t^{\left(0\right)}\left(\stackrel{\&OverBar;}{r}\right)+{\stackrel{\&OverBar;}{E}}_t^{\left(1\right)}\left(\stackrel{\&OverBar;}{r}\right)+{\stackrel{\&OverBar;}{E}}_t^{\left(2\right)}\left(\stackrel{\&OverBar;}{r}\right)$

$=\left\{(1+R_{h0})\right[\hat{e}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i]{\hat{e}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})+\frac{k}{k_1}(1+R_{v0})[\hat{h}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i\left]{\hat{h}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\right\}{e}^{i{\stackrel{\&OverBar;}{k}}_{i\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}-i{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}z}$

$+\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}{e}^{i{\stackrel{\&OverBar;}{k}}_{i\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}-i{k}_{1z}z}\mathrm{iF}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})\left\{{\hat{e}}_1(-k_{1z})\right[f_{\mathrm{ee}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{e}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_1)+f_{\mathrm{ee}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})---\left(56\right)$

$\&CenterDot;({\hat{h}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_1)]+{\hat{h}}_1(-k_{1z})[f_{\mathrm{he}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{e}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_1)+f_{\mathrm{hh}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{h}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_1)\left]\right\}$

$+\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}{e}^{i{\stackrel{\&OverBar;}{k}}_{i\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}-{\mathrm{ik}}_{1z}z}\left\{{\hat{e}}_1(-k_{1z})\right[f_{\mathrm{ee}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{e}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_i)+f_{\mathrm{eh}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{h}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_i)]$

$+{\hat{h}}_1(-k_{1z})[f_{\mathrm{he}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{e}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_i)+f_{\mathrm{hh}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{h}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_i)]$

Again by formula

η ₁Be wave impedance, can know by inference:

${\stackrel{\&OverBar;}{H}}_t\left(\stackrel{\&OverBar;}{r}\right)={\stackrel{\&OverBar;}{H}}_t^{\left(0\right)}\left(\stackrel{\&OverBar;}{r}\right)+{\stackrel{\&OverBar;}{H}}_t^{\left(1\right)}\left(\stackrel{\&OverBar;}{r}\right)+{\stackrel{\&OverBar;}{H}}_t^{\left(2\right)}\left(\stackrel{\&OverBar;}{r}\right)$

$=e^{i{\stackrel{\&OverBar;}{k}}_{i\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}-i{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}z}\{-{\hat{h}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})(1+R_{h0})[\hat{e}(-k_{\mathrm{iz}})\&CenterDot;\hat{e}]+{\hat{e}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{iz}})\frac{k}{k_1}(1+R_{v0})[\hat{h}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i\left]\right\}$

$+\frac{1}{{\mathrm{\&eta;}}_1}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}{e}^{i{\stackrel{\&OverBar;}{k}}_{i\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}-{\mathrm{ik}}_{1z}z}\mathrm{iF}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})\{-{\hat{h}}_1(-k_{1z})[f_{\mathrm{ee}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{e}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_1)+f_{\mathrm{he}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})---\left(57\right)$

$\&CenterDot;({\hat{h}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_1)]+{\hat{e}}_1(-k_{1z})[f_{\mathrm{he}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{e}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_1)+f_{\mathrm{hh}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{h}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_1)\left]\right\}$

$+\frac{1}{{\mathrm{\&eta;}}_1}\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}{e}^{i{\stackrel{\&OverBar;}{k}}_{i\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}-{\mathrm{ik}}_{1z}z}\{-{\hat{h}}_1(-k_{1z})[f_{\mathrm{ee}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{e}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_1)+f_{\mathrm{eh}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{h}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_i)]$

$+{\hat{e}}_1(-k_{1z})[f_{\mathrm{he}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{e}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_i)+f_{\mathrm{hh}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{h}}_1(-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_i)]$

Incident wave on the unit area

The power of axle component is respectively:

${\stackrel{\&OverBar;}{S}}_i\&CenterDot;\hat{z}=-\frac{\cos {\mathrm{\&theta;}}_i}{2\mathrm{\&eta;}}---\left(58\right)$

$<{\stackrel{\&OverBar;}{S}}_t\&CenterDot;\hat{z}>=\frac{1}{2}\mathrm{Re}<{\stackrel{\&OverBar;}{E}}_t\&times;{\stackrel{\&OverBar;}{H}}_t^{*}>\&CenterDot;\hat{z}$

$=\frac{1}{2}\mathrm{Re}{\stackrel{\&OverBar;}{E}}_t^{\left(0\right)}\&times;{\stackrel{\&OverBar;}{H}}_t^{\left(0\right)*}\&CenterDot;\hat{z}+\frac{1}{2}\mathrm{Re}{\stackrel{\&OverBar;}{E}}_t^{\left(0\right)}\&times;<{\stackrel{\&OverBar;}{H}}_t^{\left(2\right)*}>\&CenterDot;\hat{z}+\frac{1}{2}\mathrm{Re}<{\stackrel{\&OverBar;}{E}}_t^{\left(2\right)}>\&times;{\stackrel{\&OverBar;}{H}}_t^{\left(0\right)*}\&CenterDot;\hat{z}---\left(59\right)$

$+\frac{1}{2}\mathrm{Re}<{\stackrel{\&OverBar;}{E}}_t^{\left(1\right)}\&times;{\stackrel{\&OverBar;}{H}}_t^{\left(1\right)*}>\&CenterDot;\hat{z}$

Wherein

For prolonging the booth vector in the average slope of incident wave,

The booth vector is prolonged on the average slope that is the transmitted wave in the medium 1.

$t(\mathrm{\&pi;}-{\mathrm{\&theta;}}_i,{\mathrm{\&phi;}}_i)=\frac{<{\stackrel{\&OverBar;}{S}}_t\&CenterDot;\hat{z}>}{{\stackrel{\&OverBar;}{S}}_i\&CenterDot;\hat{z}}$

$=\frac{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}}{k_{\mathrm{zi}}}{|1+R_{\mathrm{ho}}|}^2(\hat{e}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i)+\frac{k^2{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{<figure-callout\; id="1"\; label="zi\; k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">zi</figure-callout>}}}{{k_{}}^{}}{|1+R_{\mathrm{vo}}|}^2(\hat{h}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i)$

$+2\mathrm{Re}\left(\frac{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}}{k_{\mathrm{zi}}}(1+R_{\mathrm{ho}})f_{\mathrm{ee}}^{*t\left(2\right)}\left(k_{i\&perp;}\right)\right)(\hat{e}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i)+2\mathrm{Re}\left(\frac{{\mathrm{<figure-callout\; id="1"\; label="kk"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">kk</figure-callout>}}_{1\mathrm{zi}}}{k_1{k}_{\mathrm{zi}}}(1+R_{\mathrm{vo}})f_{\mathrm{hh}}^{*t\left(2\right)}\left(k_{i\&perp;}\right)\right)$

$\&CenterDot;(\hat{h}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i)+\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}\frac{k_{1\mathrm{zi}}}{k_{\mathrm{zi}}}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})[{|f_{\mathrm{ee}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})\&CenterDot;(\hat{e}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i)+f_{\mathrm{eh}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})\&CenterDot;(\hat{h}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i)|}^2$

$+{|f_{\mathrm{he}}^{t\left(1\right)}(\hat{e}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i)+f_{\mathrm{hh}}^{t\left(1\right)}(\hat{h}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i)|}^2---\left(60\right)$

Wherein

It is the Fourier conversion of the related function of random fluctuation height.Then

$W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})=\frac{h^2{l}^2}{4\mathrm{\&pi;}}\exp (-\frac{(k_{\mathrm{\&rho;}}^2+k_{\mathrm{\&rho;i}}^2)l^2}{4}+\frac{k_{\mathrm{\&rho;}}{k}_{\mathrm{\&rho;i}}{l}^2}{2}\cos ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)).$

So our more concern two-way transmission coefficient in application is with transmissivity t (π-θ _i, φ _i) write as two-way transmission coefficient

Form:

$t_b(\mathrm{\&pi;}-{\mathrm{\&theta;}}_i,{\mathrm{\&phi;}}_i)=\frac{1}{4\mathrm{\&pi;}}{\&Integral;}_0^{\mathrm{\&pi;}/2}d{\mathrm{\&theta;}}_t\sin {\mathrm{\&theta;}}_t{\&Integral;}_0^{2\mathrm{\&pi;}}d{\mathrm{\&phi;}}_t({\mathrm{\&gamma;}}_{\mathrm{ab}}^t(\mathrm{\&pi;}-{\mathrm{\&theta;}}_t,{\mathrm{\&phi;}}_t;\mathrm{\&pi;}-{\mathrm{\&theta;}}_i,{\mathrm{\&phi;}}_i)+{\mathrm{\&gamma;}}_{\mathrm{bb}}^t(\mathrm{\&pi;}-{\mathrm{\&theta;}}_t,{\mathrm{\&phi;}}_t;\mathrm{\&pi;}-{\mathrm{\&theta;}}_i,{\mathrm{\&phi;}}_i))---\left(61\right)$

A wherein, b are respectively two kinds of situations of v polarization and h polarization, and the expression mirror should be with θ to transmission direction _t, φ _tForm:

Wherein

θ _tJust point to transmission direction.

Polarize at incident wave h as can be known through deriving, the two-way transmission coefficient under the transmitted wave h polarization situation is:

$\frac{1}{4\mathrm{\&pi;}}{\mathrm{\&gamma;}}_{\mathrm{hh}}^t(\mathrm{\&pi;}-{\mathrm{\&theta;}}_t,{\mathrm{\&phi;}}_t;\mathrm{\&pi;}-{\mathrm{\&theta;}}_i,{\mathrm{\&phi;}}_i)=[\frac{k_{1\mathrm{zi}}}{k_{\mathrm{zi}}}{|1+R_{\mathrm{ho}}|}^2+2\mathrm{Re}\left(\frac{k_{1\mathrm{zi}}}{k_{\mathrm{zi}}}(1+R_{\mathrm{ho}})f_{\mathrm{ee}}^{t^{*}\left(2\right)}\left({\stackrel{\&OverBar;}{k}}_{i\&perp;}\right)\right)]\mathrm{\&delta;}(\cos {\mathrm{\&theta;}}_t-\frac{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}}{k_1})\mathrm{\&delta;}({\mathrm{\&phi;}}_t-{\mathrm{\&phi;}}_i)$

$+\frac{k_1{k}_{1z}^2}{k_{\mathrm{iz}}}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;}){\left|f_{\mathrm{ee}}^{t^{*}\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})\right|}^2---\left(62\right)$

In like manner, in incident wave v polarization, under the transmitted wave h polarization situation,

$\frac{1}{4\mathrm{\&pi;}}{\mathrm{\&gamma;}}_{\mathrm{hv}}^t(\mathrm{\&pi;}-{\mathrm{\&theta;}}_t,{\mathrm{\&phi;}}_t;\mathrm{\&pi;}-{\mathrm{\&theta;}}_i,{\mathrm{\&phi;}}_i)=\frac{k_1{k}_{1z}^2}{k_{\mathrm{iz}}}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;}){\left|f_{\mathrm{eh}}^{t^{*}\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})\right|}^2---\left(63\right)$

In incident wave h polarization, under the transmitted wave v polarization situation,

$\frac{1}{4\mathrm{\&pi;}}{\mathrm{\&gamma;}}_{\mathrm{vh}}^t(\mathrm{\&pi;}-{\mathrm{\&theta;}}_t,{\mathrm{\&phi;}}_t;\mathrm{\&pi;}-{\mathrm{\&theta;}}_i,{\mathrm{\&phi;}}_i)=\frac{k_1{k}_{1z}^2}{k_{\mathrm{iz}}}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})|f_{\mathrm{he}}^{t^{*}\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;}){|}^2---\left(64\right)$

In incident wave v polarization, under the transmitted wave v polarization situation,

$\frac{1}{4\mathrm{\&pi;}}{\mathrm{\&gamma;}}_{\mathrm{vv}}^t(\mathrm{\&pi;}-{\mathrm{\&theta;}}_t,{\mathrm{\&phi;}}_t;\mathrm{\&pi;}-{\mathrm{\&theta;}}_i,{\mathrm{\&phi;}}_i)=\left[\frac{k^2{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{<figure-callout\; id="1"\; label="zi\; k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">zi</figure-callout>}}}{k_{}^{}}\right|1+R_{\mathrm{vo}}{|}^2+2\mathrm{Re}\left(\frac{{\mathrm{<figure-callout\; id="1"\; label="kk"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">kk</figure-callout>}}_{1\mathrm{zi}}}{k_1{k}_{\mathrm{iz}}}(1+R_{\mathrm{vo}})f_{\mathrm{hh}}^{t^{*}\left(2\right)}\left({\stackrel{\&OverBar;}{k}}_{i\&perp;}\right)\right)]---\left(65\right)$

$\&CenterDot;\mathrm{\&delta;}(\cos {\mathrm{\&theta;}}_t-\frac{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}}{k_1})\mathrm{\&delta;}({\mathrm{\&phi;}}_t-{\mathrm{\&phi;}}_i)+\frac{k_1{k}_{1z}^2}{k_{\mathrm{iz}}}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})|f_{\mathrm{hh}}^{t^{*}\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;}){|}^2$

Wherein $f_{\mathrm{ab}}^{t\left(2\right)}\left({\stackrel{\&OverBar;}{k}}_{i\&perp;}\right)=<f_{\mathrm{ab}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})>a,b=v,h$

$f_{\mathrm{ee}}^{t\left(2\right)}\left({\stackrel{\&OverBar;}{k}}_{i\&perp;}\right)=\frac{k_{\mathrm{iz}}({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2)}{{(k_{\mathrm{iz}}+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})}^2}\{({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}-k_{\mathrm{iz}}){\&Integral;}_{-\&infin;}^{\&infin;}d{\stackrel{\&OverBar;}{k}}_{\&perp;}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})-2({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2){\&Integral;}_{-\&infin;}^{\&infin;}d{\stackrel{\&OverBar;}{k}}_{\&perp;}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})---\left(66\right)$

$\&CenterDot;\left[{\sin }^2({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)\frac{k_z{k}_{1\mathrm{<figure-callout\; id="1"\; label="z\; k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">z</figure-callout>}}}{k_{}^{}}{\cos }^2({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)\frac{1}{k_{1z}+k_z}\right]\}$

$f_{\mathrm{hh}}^{t\left(2\right)}\left({\stackrel{\&OverBar;}{k}}_{i\&perp;}\right)=\frac{k_1({{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1}^2-k^2)k_{\mathrm{iz}}}{{({{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1}^2{k}_{\mathrm{iz}}+k^2{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})}^2}\{{\&Integral;}_{-\&infin;}^{\&infin;}d{\stackrel{\&OverBar;}{k}}_{\&perp;}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})k({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2{k}_{\mathrm{iz}}-k^2{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}})+$

$2{\&Integral;}_{-\&infin;}^{\&infin;}d{\stackrel{\&OverBar;}{k}}_{\&perp;}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})[-{\mathrm{kk}}_{\mathrm{iz}}{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}\frac{({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2-k^2)}{k_z+k_{1z}}{\sin }^2({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)+---\left(67\right)$

$\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2{k}_z+k^2{k}_{1z}}(-k({{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1}^2-k^2)k_{\mathrm{\&rho;i}}^2{k}_{\mathrm{\&rho;}}^2+k_{\mathrm{\&rho;}}{k}_{\mathrm{\&rho;i}}k(k_z+k_{1z})(k^2{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1^2{k}_{\mathrm{iz}}))$

$\&CenterDot;\cos ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)-{\mathrm{kk}}_z{k}_{1z}{k}_{\mathrm{iz}}{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_{1\mathrm{zi}}({{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_110322152532.png"\; state="\{\{state\}\}">k</figure-callout>}}_1}^2-k^2){\cos }^2({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)\left]\right\}$

Below the present invention program is verified:

For two-dimentional uneven surface as shown in Figure 1, incident wave incides on this uneven surface from free space (medium 0), wherein known media 0 μ ₀=4 π * 10 ^-7, needing setup parameter simultaneously is the incident angle θ of incident wave _iAnd position angle

The DIELECTRIC CONSTANT of the latter half medium ₁, the standard deviation σ of uneven surface and persistence length l.

At first utilize incident angle θ _iAnd position angle

Calculate: k _Ix=ksin θ _iCos φ _i, k _Iy=ksin θ _iSin φ _i, k _Iz=kcos θ _i, k _{ρ i}=ksin θ _i,

Utilize formula (56) can calculate transmissivity again.The implication and the value of each variable see correlation formula for details in the formula (56).

Second order dispersion field that perturbation method is calculated and second-order reflection rate have been derived by forefathers and have been obtained, be checking second order transmitted field and second order transmissivity that this paper derived, be provided with down in different parameters, press this paper method and calculate second order transmissivity (emissivity), calculate the second-order reflection rate by the formula that provides, with reflectivity and transmissivity addition, think and meet energy conservation that promptly the derivation result of this paper second order transmitted field and second order transmissivity is correct if equal 1.Be used for contrast, also emulation single order perturbation method transmissivity and reflectivity, and energy conservation situation.

1, the scope of application that meets the single order perturbation method when surfaceness.

Make DIELECTRIC CONSTANT=6, root-mean-square height h=0.0398 λ, persistence length l=λ, frequency f=37Ghz, wherein λ is the wavelength of free space under this frequency.This parameter setting is satisfied the scope of application of single order perturbation method.Fig. 2 has shown transmissivity (second order, single order), reflectivity (second order, single order), and transmissivity+reflectivity (second order, single order), changes situation about changing with incident angle.

The corresponding h polarization of Fig. 2 (a) situation meets the scope of application of single order perturbation method though find out surfaceness, and the single order perturbation method does not also meet energy conservation, and when 0 degree incident angle, error is 18%; And the second order perturbation method to all incident angles, comprises the glancing incidence situation when h polarizes, and all keeps energy conservation;

The corresponding v of Fig. 2 (b) polarization situation, the single order perturbation method polarizes at v, to all incident angle energy nonconservation all.The second order perturbation method polarizes at v, nonconservation of energy during glancing incidence, and when other most of incident angles, are conservations.

For the second order perturbation method, when glancing incidence, have only the h polarization to meet energy conservation, and the v polarization is not meet energy conservation.This point and existing document are consistent about the scope of application research conclusion of second order perturbation method.

Find out that from this example the transmissivity computational accuracy of second order perturbation method is greatly improved than the transmissivity computational accuracy of single order perturbation method.

At Fig. 3 (a), among 3 (b), specific inductive capacity is ε=6, root-mean-square height σ=0.07 λ, and persistence length l=0.15 λ, frequency f=37Ghz, λ are the wavelength of free space under this frequency.This surfaceness and correlation parameter have exceeded the scope of application of single order perturbation method.Fig. 2 (a), 2 (b), 3 (a), among 3 (b), r1, r2 represent single order and second order emissivity respectively; T1, t2 represent the transmissivity of single order and second order respectively.

2, when the scope of application of surfaceness single order perturbation method.

The corresponding h polarization of Fig. 3 (a) situation finds out that when surfaceness exceeds the scope of application of single order perturbation method the situation of the single order perturbation method nonconservation of energy is all more remarkable, and especially when 0 spent, error was near 40%; And the second order perturbation method to all incident angles, comprises the glancing incidence situation when h polarizes, and all keeps energy conservation;

The corresponding v of Fig. 3 (b) polarization situation, the single order perturbation method polarizes at v, and to all incident angle energy nonconservation all, error is many about 40%.The second order perturbation method polarizes at v, and nonconservation of energy during glancing incidence, error are less than 20%, and energy is a conservation when other most of incident angles.

Find out that from this example second order perturbation method transmissivity computing formula is than single order, the scope of application has enlarged, and precision has improved.

Claims (1)

1. second order perturbation method random rough face transmissison characteristic computing method comprise the calculation procedure of transmitted field, transmissivity and two-way transmission coefficient, are specially:

(1) described transmitted field is expressed as:

${\stackrel{\&OverBar;}{E}}_t^{\left(2\right)}\left(\stackrel{\&OverBar;}{r}\right)=\&Integral;{\stackrel{\&OverBar;}{\mathrm{dk}}}_{\&perp;}{e}^{{\stackrel{\&OverBar;}{\mathrm{ik}}}_{i\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}-{\mathrm{ik}}_{1z}z}\left\{{\hat{e}}_1(-k_{1z})\right[f_{\mathrm{ee}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{e}}_1(-k_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_i)+f_{\mathrm{eh}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{h}}_1(-k_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_i)]$

${+\hat{h}}_1(-k_{1z})[f_{\mathrm{he}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{e}}_1(-k_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_i)+f_{\mathrm{hh}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})({\hat{h}}_1(-k_{1\mathrm{zi}})\&CenterDot;{\hat{e}}_i)]$

Wherein,

$f_{\mathrm{ee}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})=\frac{k_{\mathrm{iz}}}{k_{\mathrm{iz}}+k_{1\mathrm{iz}}}\frac{k_1^2-k^2}{k_{1z}+k_z}\{\cos ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)\&CenterDot;(k_{1\mathrm{zi}}-k_z)F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})$

$-2(k_1^2-k^2)\&Integral;{\stackrel{\&OverBar;}{\mathrm{dk}}}_{\&perp;}^{\&prime;}F({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})$

$\&CenterDot;[-\sin ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})\sin ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i)\frac{k_z^{\&prime;}{k}_{1z}^{\&prime;}}{k_1^2{k}_z^{\&prime;}+k^2{k}_{1z}^{\&prime;}}+\cos ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})\cos ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i)\frac{1}{k_{1z}^{\&prime;}+k_z^{\&prime;}}]\}$

$f_{\mathrm{eh}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})=\frac{k_1^2-k^2}{k_{1z}+k_z}\frac{{\mathrm{kk}}_{\mathrm{iz}}}{k_1^2{k}_{\mathrm{iz}}+k^2{k}_{1\mathrm{zi}}}\{(k_1^2-k_{1\mathrm{zi}}{k}_z)F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})\sin ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)$

$-2\&Integral;{\stackrel{\&OverBar;}{\mathrm{dk}}}_{\&perp;}^{\&prime;}F({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})[\sin ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_k^{\&prime;})\frac{k_1^2{k}_{\mathrm{\&rho;}}^{\&prime;}{k}_{\mathrm{\&rho;i}}}{k_{\mathrm{\&rho;}}^{\&prime;2}k+k_z^{\&prime;}{k}_{1z}^{\&prime;}}+\frac{(k_1^2-k^2)k_{1\mathrm{zi}}}{k_1^2{k}_z^{\&prime;}+k^2{k}_{1z}^{\&prime;}}$

$\&CenterDot;(\sin ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})\cos ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i)k_z^{\&prime;}{k}_{1z}^{\&prime;}+\cos ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})\sin ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i)(k_{\mathrm{\&rho;}}^{\&prime;2}+k_z^{\&prime;}{k}_{1z}^{\&prime;})\left]\right\}$

$f_{\mathrm{he}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})=\frac{k_1^2-k^2}{k_{1\mathrm{zi}}+k_{\mathrm{iz}}}\frac{k_1{k}_{\mathrm{iz}}}{k_1^2{k}_z+k^2{k}_{1z}}\{(k^2-k_{1\mathrm{zi}}{k}_z)F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})\sin ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)$

$+2\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}F({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})[\sin ({\mathrm{\&phi;}}_k^{\&prime;}-{\mathrm{\&phi;}}_i)\frac{k_1^2{k}_{\mathrm{\&rho;}}^{\&prime;}{k}_{\mathrm{\&rho;}}}{k_{\mathrm{\&rho;}}^{\&prime;2}+k_z^{\&prime;}{k}_{1z}^{\&prime;}}+\frac{(k_1^2-k^2)k_z}{k_1^2{k}_z^{\&prime;}+k^2{k}_{1z}^{\&prime;}}$

$\&CenterDot;(k_z^{\&prime;}{k}_{1z}^{\&prime;}\sin ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})\cos ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i)+(k_{\mathrm{\&rho;}}^{\&prime;2}+k_z^{\&prime;}{k}_{1z}^{\&prime;})\cos ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})\sin ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i))]$

$f_{\mathrm{hh}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})=\frac{(k_1^2-k^2)}{({k_1^{2}k}_z+{k^{2}k}_{1z})}\frac{k_1{k}_{\mathrm{iz}}}{k_1^2{k}_{\mathrm{iz}}+k^2{k}_{1\mathrm{zi}}}\{k(k_1^2{k}_z-{k^{2}k}_{1\mathrm{zi}})F^{\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})\sin ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)$

$+2\&Integral;{\stackrel{\&OverBar;}{\mathrm{dk}}}_{\&perp;}^{\&prime;}F({\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{\&perp;}^{\&prime;})[{\mathrm{kk}}_z{k}_{1\mathrm{zi}}\frac{(k_1^2-k^2)}{k_z^{\&prime;}+k_{1z}^{\&prime;}}\sin ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})\sin ({\mathrm{\&phi;}}_k^{\&prime;}-{\mathrm{\&phi;}}_i)$

$+\frac{1}{k_1^2{k}_z^{\&prime;}+k^2{k}_{1z}^{\&prime;}}[-k(k_1^2-k^2)k_{\mathrm{\&rho;}}{k}_{\mathrm{\&rho;}}^{\&prime;2}{k}_{\mathrm{\&rho;i}}+k_{\mathrm{\&rho;}}{k}_{\mathrm{\&rho;}}^{\&prime;}{k}^3(k_z^{\&prime;}+k_{1z}^{\&prime;})k_{1\mathrm{zi}}\cos ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i)-k_{\mathrm{\&rho;}}^{\&prime;}{k}_{\mathrm{\&rho;i}}{\mathrm{kk}}_1^2$

$\&CenterDot;(k_z^{\&prime;}+k_{1z}^{\&prime;})k_z\cos ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})-{\mathrm{kk}}_z^{\&prime;}{k}_{1z}^{\&prime;}(k_1^2-k^2)k_z{k}_{1\mathrm{zi}}\cos ({\mathrm{\&phi;}}_k-{{\mathrm{\&phi;}}_k}^{\&prime;})\cos ({{\mathrm{\&phi;}}_k}^{\&prime;}-{\mathrm{\&phi;}}_i)\left]\right\}$

Described transmissivity is expressed as:

$t(\mathrm{\&pi;}-{\mathrm{\&theta;}}_i,{\mathrm{\&phi;}}_i)=\frac{k_{1\mathrm{zi}}}{k_{\mathrm{zi}}}|1+R_{\mathrm{ho}}{|}^2(\hat{e}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i)+\frac{k^2{k}_{1\mathrm{zi}}}{{k_1}^2{k}_{\mathrm{zi}}}|1+R_{\mathrm{vo}}{|}^2(\hat{h}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i)$

$+2\mathrm{Re}\left(\frac{k_{1\mathrm{zi}}}{k_{\mathrm{zi}}}(1+R_{\mathrm{ho}})f_{\mathrm{ee}}^{t\left(2\right)}\left(k_{i\&perp;}\right)\right)(\hat{e}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i)+2\mathrm{Re}\left(\frac{{\mathrm{kk}}_{1\mathrm{zi}}}{k_1{k}_{\mathrm{zi}}}(1+R_{\mathrm{vo}})f_{\mathrm{hh}}^{t\left(2\right)}\left(k_{i\&perp;}\right)\right)$

$\&CenterDot;(\hat{h}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i)+\&Integral;d{\stackrel{\&OverBar;}{k}}_{\&perp;}\frac{k_{1\mathrm{zi}}}{k_{\mathrm{zi}}}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})\left[\right|f_{\mathrm{ee}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})\&CenterDot;(\hat{e}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i)+f_{\mathrm{eh}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})\&CenterDot;(\hat{h}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i){|}^2$

$+|f_{\mathrm{he}}^{t\left(1\right)}(\hat{e}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i)+f_{\mathrm{hh}}^{t\left(1\right)}(\hat{h}(-k_{\mathrm{iz}})\&CenterDot;{\hat{e}}_i){|}^2]$

Wherein

Be the Fourier conversion of the related function of random fluctuation height,

$W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})=\frac{h^2{l}^2}{4\mathrm{\&pi;}}\exp (-\frac{(k_{\mathrm{\&rho;}}^2+k_{\mathrm{\&rho;i}}^2)l^2}{4}+\frac{k_{\mathrm{\&rho;}}{k}_{\mathrm{\&rho;i}}{l}^2}{2}\cos ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i));$

(3) described two-way transmission coefficient is expressed as:

In incident wave h polarization, the two-way transmission coefficient under the transmitted wave h polarization situation is:

$\frac{1}{4\mathrm{\&pi;}}{\mathrm{\&gamma;}}_{\mathrm{hh}}^t(\mathrm{\&pi;}-{\mathrm{\&theta;}}_t,{\mathrm{\&phi;}}_t;\mathrm{\&pi;}-{\mathrm{\&theta;}}_i,{\mathrm{\&phi;}}_i)=\left[\frac{k_{1\mathrm{zi}}}{k_{\mathrm{zi}}}\right|1+R_{\mathrm{ho}}{|}^2+1\mathrm{Re}\left(\frac{k_{1\mathrm{zi}}}{k_{\mathrm{zi}}}(1+R_{\mathrm{ho}})f_{\mathrm{ee}}^{t\left(2\right)}\left({\stackrel{\&OverBar;}{k}}_{i\&perp;}\right)\right)]\mathrm{\&delta;}({\cos \mathrm{\&theta;}}_t-\frac{k_{1\mathrm{zi}}}{k_1})\mathrm{\&delta;}({\mathrm{\&phi;}}_t-{\mathrm{\&phi;}}_i)$

$+\frac{k_1{k}_{1z}^2}{k_{\mathrm{iz}}}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})|f_{\mathrm{ee}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;}){|}^2$

In incident wave v polarization, under the transmitted wave h polarization situation,

$\frac{1}{4\mathrm{\&pi;}}{\mathrm{\&gamma;}}_{\mathrm{hv}}^t(\mathrm{\&pi;}-{\mathrm{\&theta;}}_t,{\mathrm{\&phi;}}_t;\mathrm{\&pi;}-{\mathrm{\&theta;}}_i,{\mathrm{\&phi;}}_i)=\frac{k_1{k}_{1z}^2}{k_{\mathrm{iz}}}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})|f_{\mathrm{eh}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;}){|}^2$

In incident wave h polarization, under the transmitted wave v polarization situation,

$\frac{1}{4\mathrm{\&pi;}}{\mathrm{\&gamma;}}_{\mathrm{vh}}^t(\mathrm{\&pi;}-{\mathrm{\&theta;}}_t,{\mathrm{\&phi;}}_t;\mathrm{\&pi;}-{\mathrm{\&theta;}}_i,{\mathrm{\&phi;}}_i)=\frac{k_1{k}_{1z}^2}{k_{\mathrm{iz}}}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})|f_{\mathrm{he}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;}){|}^2$

In incident wave v polarization, under the transmitted wave v polarization situation,

$\frac{1}{4\mathrm{\&pi;}}{\mathrm{\&gamma;}}_{\mathrm{vv}}^t(\mathrm{\&pi;}-{\mathrm{\&theta;}}_t,{\mathrm{\&phi;}}_t;\mathrm{\&pi;}-{\mathrm{\&theta;}}_i,{\mathrm{\&phi;}}_i)=\left[\frac{k^2{k}_{1\mathrm{zi}}}{k_1^2{k}_{\mathrm{iz}}}\right|1+R_{\mathrm{vo}}{|}^2+2\mathrm{Re}\left(\frac{{\mathrm{kk}}_{1\mathrm{zi}}}{k_1{k}_{\mathrm{iz}}}(1+R_{\mathrm{vo}})f_{\mathrm{hh}}^{t\left(2\right)}\left({\stackrel{\&OverBar;}{k}}_{i\&perp;}\right)\right)]$

$\&CenterDot;\mathrm{\&delta;}(\cos {\mathrm{\&theta;}}_t-\frac{k_{1\mathrm{zi}}}{k_1})\mathrm{\&delta;}({\mathrm{\&phi;}}_t-{\mathrm{\&phi;}}_i)+\frac{k_1{k}_{1z}^2}{k_{\mathrm{iz}}}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})|f_{\mathrm{hh}}^{t\left(1\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;}){|}^2$

A wherein, b are respectively two kinds of situations of v polarization and h polarization, and the expression mirror should be with θ to transmission direction _t, φ _tForm:

Wherein

The time, θ _tJust point to transmission direction, Re () expression is got real part to the number in the bracket;

$f_{\mathrm{ab}}^{t\left(2\right)}\left({\stackrel{\&OverBar;}{k}}_{i\&perp;}\right)=<f_{\mathrm{ab}}^{t\left(2\right)}({\stackrel{\&OverBar;}{k}}_{\&perp;},{\stackrel{\&OverBar;}{k}}_{i\&perp;})>a,b=v,h$

$f_{\mathrm{ee}}^{t\left(2\right)}\left({\stackrel{\&OverBar;}{k}}_{i\&perp;}\right)=\frac{k_{\mathrm{iz}}(k_1^2-k^2)}{{(k_{\mathrm{iz}}+k_{1\mathrm{zi}})}^2}\{(k_{1\mathrm{zi}}-k_{\mathrm{iz}}){\&Integral;}_{-\&infin;}^{\&infin;}d{\stackrel{\&OverBar;}{k}}_{\&perp;}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})-2(k_1^2-k^2){\&Integral;}_{-\&infin;}^{\&infin;}d{\stackrel{\&OverBar;}{k}}_{\&perp;}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})$

$\&CenterDot;[{\sin }^2({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)\frac{k_z{k}_{1z}}{k_1^2{k}_z+k^2{k}_{1z}}+{\cos }^2({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)\frac{1}{k_{1z}+k_z}]\}$

$f_{\mathrm{hh}}^{t\left(2\right)}\left({\stackrel{\&OverBar;}{k}}_{i\&perp;}\right)=\frac{k_1({k_1}^2-k^2)k_{\mathrm{iz}}}{{({k_1}^2{k}_{\mathrm{iz}}+k^2{k}_{1\mathrm{zi}})}^2}\{{\&Integral;}_{-\&infin;}^{\&infin;}d{\stackrel{\&OverBar;}{k}}_{\&perp;}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})k(k_1^2{k}_{\mathrm{iz}}-k^2{k}_{1\mathrm{zi}})+$

$2{\&Integral;}_{-\&infin;}^{\&infin;}d{\stackrel{\&OverBar;}{k}}_{\&perp;}W({\stackrel{\&OverBar;}{k}}_{\&perp;}-{\stackrel{\&OverBar;}{k}}_{i\&perp;})[{-\mathrm{kk}}_{\mathrm{iz}}{k}_{1\mathrm{zi}}\frac{(k_1^2-k^2)}{k_2+k_{1z}}{\sin }^2({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)+$

$\frac{1}{k_1^2{k}_z+k^2{k}_{1z}}(-k({k_1}^2-k^2)k_{\mathrm{\&rho;i}}^2{k}_{\mathrm{\&rho;}}^2+k_{\mathrm{\&rho;}}{k}_{\mathrm{\&rho;i}}k(k_z+k_{1z})(k^2{k}_{1\mathrm{zi}}-k_1^2{k}_{\mathrm{iz}})$

$\&CenterDot;\cos ({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)-{\mathrm{kk}}_z{k}_{1z}{k}_{\mathrm{iz}}{k}_{1\mathrm{zi}}({k_1}^2-k^2){\cos }^2({\mathrm{\&phi;}}_k-{\mathrm{\&phi;}}_i)\left)\right]\}$

The incident wave vector

The horizontal component of incident wave vector is

k _ix＝ksinθ _icosφ _i，k _iy＝ksinθ _isinφ _i，k _iz＝kcosθ _i，k _ρi＝ksinθ _i，

The mould of incident wave horizontal component

The scattering wave vector

The scattering wave horizontal component

The mould of scattering wave horizontal component

The mould of scattering wave vertical component

The transmitted wave vector

The horizontal component of transmitted wave vector is

The mould of transmitted wave horizontal component

The mould of transmitted wave vertical component

$k_{1\mathrm{zi}}=\sqrt{k_1^2+k_{i\&perp;}^2},$

The incident angle θ of incident wave _iAnd position angle

The angle of transmission θ of transmitted wave _tAnd position angle

With

Represent an initial point and a position vector of putting respectively,

When the incident field is the TE polarization

When the incident field is the TM polarization

μ _oAnd μ ₁Represent the magnetic permeability of medium 0 and medium 1 respectively,

ε _oAnd ε ₁Represent the conductivity of medium 0 and medium 1 respectively,

W is an angular frequency,

η ₁Be the wave impedance of medium 1,

The root-mean-square height h of uneven surface and persistence length l,

$\hat{e}\left(k_z\right)=\hat{k}\&times;\hat{z}/|\hat{k}\&times;\hat{z}|=(1/k_{\mathrm{\&rho;}})(\hat{x}{k}_y-\hat{y}{k}_x),$

$\hat{e}\left({-k}_z\right)=(1/k_{\mathrm{\&rho;}})(\hat{x}{k}_y-\hat{y}{k}_x),$

$\hat{e}\left({-k}_{1\mathrm{zi}}\right)={\hat{k}}_1\&times;\hat{z}/|{\hat{k}}_1\&times;\hat{z}|=(1/k_{1\mathrm{\&rho;i}})(\hat{x}{k}_{1\mathrm{yi}}-\hat{y}{k}_{1\mathrm{xi}}),$

${\hat{e}}_1\left(k_{1z}\right)={\hat{k}}_1\&times;\hat{z}/|{\hat{k}}_1\&times;\hat{z}|=(1/k_{\mathrm{\&rho;}})(\hat{x}{k}_y-\hat{y}{k}_x),$

$\hat{h}\left(k_z\right)=\frac{1}{k}\hat{e}\&times;\hat{k}=\frac{{-k}_z}{{\mathrm{kk}}_{\mathrm{\&rho;}}}(\hat{x}{k}_x+\hat{y}{k}_y)+\frac{k_{\mathrm{\&rho;}}}{k}\hat{z},$

$\hat{h}\left({-k}_z\right)=\frac{k_z}{{\mathrm{kk}}_{\mathrm{\&rho;}}}(\hat{x}{k}_x+\hat{y}{k}_y)+\frac{k_{\mathrm{\&rho;}}}{k}\hat{z},$

${\hat{h}}_1\left(k_{1z}\right)=\frac{1}{k_1}{\hat{e}}_1\&times;{\hat{k}}_1=\frac{{-k}_{1z}}{k_1{k}_{\mathrm{\&rho;}}}(\hat{x}{k}_x+\hat{y}{k}_y)+\frac{k_{\mathrm{\&rho;}}}{k_1}\hat{z},$

${\hat{h}}_1\left({-k}_{1z}\right)=\frac{k_{1z}}{k_1{k}_{\mathrm{\&rho;}}}(\hat{x}{k}_{1x}+\hat{y}{k}_{1y})+\frac{k_{\mathrm{\&rho;}}}{k_1}\hat{z},$

${\hat{h}}_1\left({-k}_{1\mathrm{zi}}\right)=\frac{k_{1\mathrm{zi}}}{k_1{k}_{\mathrm{\&rho;i}}}(\hat{x}{k}_{\mathrm{xi}}+\hat{y}{k}_{\mathrm{yi}})+\frac{k_{\mathrm{\&rho;i}}}{k_1}\hat{z},$

Fourier transform

$\frac{1}{{4\mathrm{\&pi;}}^2}\&Integral;d{\stackrel{\&OverBar;}{r}}_{\&perp;}{e}^{-i{\stackrel{\&OverBar;}{k}}_{\&perp;}\&CenterDot;{\stackrel{\&OverBar;}{r}}_{\&perp;}}{f}^2\left({\stackrel{\&OverBar;}{r}}_{\&perp;}\right)=F^2\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right),$

$F^2\left({\stackrel{\&OverBar;}{k}}_{\&perp;}\right)={\&Integral;}_{-\&infin;}^{\&infin;}d{{\stackrel{\&OverBar;}{k}}^{\&prime;}}_{\&perp;}F\left({{\stackrel{\&OverBar;}{k}}^{\&prime;}}_{\&perp;}\right)F({\stackrel{\&OverBar;}{k}}_{\&perp;}-{{\stackrel{\&OverBar;}{k}}^{\&prime;}}_{\&perp;}),$

Be the Fresnel transmission coefficient of TE ripple,

Be the Fresnel transmission coefficient of TM ripple,

φ ' _k, k ' _ρ, k ' _zAnd k ' _1zIt is the needed intermediate variable of integration.

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