{"id":2057,"date":"2012-04-24T22:05:09","date_gmt":"2012-04-25T02:05:09","guid":{"rendered":"http:\/\/blogs.vassar.edu\/magnes\/?p=2057"},"modified":"2013-07-11T10:30:35","modified_gmt":"2013-07-11T14:30:35","slug":"derivation-of-transmitted-wave-equation","status":"publish","type":"post","link":"https:\/\/pages.vassar.edu\/magnes\/2012\/04\/24\/derivation-of-transmitted-wave-equation\/","title":{"rendered":"Derivation of Transmitted Wave Equation"},"content":{"rendered":"

(1) <\/span>   <\/span>\"\begin{equation*}<\/p>\n

$\\widetilde{P}^{NL}_{n} =$ Non-Linear part of Polarization Vector<\/p>\n

$\u00a0\\widetilde{E}_{n} =$ Electric Field vector<\/p>\n

$\\epsilon^{(1)}(\\omega_{n}) =$ Frequency dependent dielectric Tensor<\/p>\n

Equation (1) is derived from Maxwell\u2019s equation and is the equation for waves in medium. It is valid for each frequency component of the field.<\/p>\n

$\\widetilde{E}_{2}(z,t) =A_{2}e^{i(k_{2}z-wt)}, \u00a0\u00a0\\widetilde{P}_{j}(z,t) =P_{j}e^{-i\\omega_{j}t},$ \u00a0 \u00a0$P_{1} =4dA_{2}A^{*}_{1}e^{i(k_{2}-k_{1})z}, P_{2} =2dA^{2}_{1}e^{i2k_{1}z}$<\/p>\n

$\\widetilde{E}_{2}(z,t)$\u00a0will be my equation for the transmitted wave at frequency \u00a0propagating in the z direction,$\u00a0\\widetilde{P}_{2}(z,t)$ the nonlinear source term and $P_{2}, P_{1}$ \u00a0the amplitude of the nonlinear polarization and amplitude of incident beam respectively.<\/p>\n

(2) <\/span>   <\/span>\"\begin{equation*}<\/p>\n

$\\bigtriangledown^{2}$ can be replaced with $\u00a0\\frac{\\partial^2}{\\partial z^{2} }$<\/p>\n

(3) <\/span>   <\/span>\"\begin{equation*}<\/p>\n

(4) <\/span>   <\/span>\"\begin{equation*}<\/p>\n

So the wave equation becomes<\/p>\n

(5) <\/span>   <\/span>\"\begin{equation*}<\/p>\n

(6) <\/span>   <\/span>\"\begin{equation*}<\/p>\n

$k^{2}_{2}=\u00a0\\frac{\\omega^{2}_{2}}{c^{2}}\\epsilon^{(1)}(\\omega_{2})$ hence<\/p>\n

(7) <\/span>   <\/span>\"\begin{equation*}<\/p>\n

replace $\u00a0\\frac{\\partial}{\\partial}$ with $\\frac{d}{dz}$ because $A_{2}$ is only a function of $z$ and with the slowly-varying-amplitude approximation $|\\frac{d^{2}A_{2}}{dz^{2}}|<<|k_{2}\\frac{aA_{2}}{dz}|$ hence we get<\/div>\n

(8) <\/span>   <\/span>\"\begin{equation*}<\/p>\n","protected":false},"excerpt":{"rendered":"

(1)   $\\widetilde{P}^{NL}_{n} =$ Non-Linear part of Polarization Vector $\u00a0\\widetilde{E}_{n} =$ Electric Field vector $\\epsilon^{(1)}(\\omega_{n}) =$ Frequency dependent dielectric Tensor Equation (1) is derived from Maxwell\u2019s equation and is the equation for waves in medium. It is valid for each frequency component of the field. $\\widetilde{E}_{2}(z,t) =A_{2}e^{i(k_{2}z-wt)}, \u00a0\u00a0\\widetilde{P}_{j}(z,t) =P_{j}e^{-i\\omega_{j}t},$ \u00a0 \u00a0$P_{1} =4dA_{2}A^{*}_{1}e^{i(k_{2}-k_{1})z}, P_{2} =2dA^{2}_{1}e^{i2k_{1}z}$ $\\widetilde{E}_{2}(z,t)$\u00a0will […]<\/p>\n","protected":false},"author":1019,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4101,29905,29902],"tags":[],"class_list":["post-2057","post","type-post","status-publish","format-standard","hentry","category-advanced-em","category-spring-2012","category-tariq"],"_links":{"self":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts\/2057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/users\/1019"}],"replies":[{"embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/comments?post=2057"}],"version-history":[{"count":8,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts\/2057\/revisions"}],"predecessor-version":[{"id":2062,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts\/2057\/revisions\/2062"}],"wp:attachment":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/media?parent=2057"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/categories?post=2057"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/tags?post=2057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}