Derivation of Transmitted Wave Equation

(1)   \begin{equation*} \bigtriangledown \times \bigtriangledown \times \widetilde{E}_{n} + \frac{\epsilon^{(1)}(\omega_{n})}{c^{2}} \bullet \frac{\partial^2 \widetilde{E}_{n}}{\partial t^2} = \frac{-4\pi}{c^2} \frac{\partial^2 \widetilde{P}^{NL}_{n}}{\partial t^2} \end{equation*}

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

$ \widetilde{E}_{n} =$ Electric Field vector

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

Equation (1) is derived from Maxwell’s equation and is the equation for waves in medium. It is valid for each frequency component of the field.

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

$\widetilde{E}_{2}(z,t)$ will be my equation for the transmitted wave at frequency  propagating in the z direction,$ \widetilde{P}_{2}(z,t)$ the nonlinear source term and $P_{2}, P_{1}$  the amplitude of the nonlinear polarization and amplitude of incident beam respectively.

(2)   \begin{equation*} -\bigtriangledown^{2} \widetilde{E}_{n}-\frac{\omega^{2}_{n}}{c^{2}}\epsilon^{(1)}(\omega_{n}) \bullet \widetilde{E}_{n}= \frac{4\pi \omega^{2}_{n}}{c^2} \widetilde{P}^{NL}_{n}} \end{equation*}

$\bigtriangledown^{2}$ can be replaced with $ \frac{\partial^2}{\partial z^{2} }$

(3)   \begin{equation*} \frac{\partial^2}{\partial z^{2}}\widetilde{E}_{2}= \frac{\partial}{\partial z} [ \frac{\partial A_{2}}{\partial z}e^{i(k_{2}z-wt)} +ik_{2}A_{2}e^{i(k_{2}z-wt)}] \end{equation*}

(4)   \begin{equation*} \frac{\partial^2}{\partial z^{2}}\widetilde{E}_{2}= [ \frac{\partial^{2} A_{2}}{\partial z^{2}} +ik_{2}\frac{\partial A_{2}}{\partial z}+ ik_{2} \frac{\partial A_{2}}{\partial z} -k^{2}_{2}A_{2}]e^{i(k_{2}z-wt)} \end{equation*}

So the wave equation becomes

(5)   \begin{equation*} -[ \frac{\partial^{2} A_{2}}{\partial z^{2}} + 2ik_{2} \frac{\partial A_{2}}{\partial z}-k^{2}_{2}A_{2}]e^{i(k_{2}z-wt)} -\frac{\omega^{2}_{2}}{c^{2}}\epsilon^{(1)}(\omega_{2})A_{2}e^{i(k_{2}z-wt)} =\frac{4\pi \omega^{2}_{n}}{c^2}\bullet2dA^{2}_{1}e^{i(2k_{1}z-\omega_{2}t)} \end{equation*}

(6)   \begin{equation*} -[ \frac{\partial^{2} A_{2}}{\partial z^{2}} + 2ik_{2} \frac{\partial A_{2}}{\partial z}-k^{2}_{2}A_{2} -\frac{\omega^{2}_{2}}{c^{2}}\epsilon^{(1)}(\omega_{2})A_{2}]e^{i(k_{2}z-wt)} =\frac{8\pi \omega^{2}_{2}}{c^2}dA^{2}_{1}e^{i(2k_{1}z-\omega_{2}t)} \end{equation*}

$k^{2}_{2}= \frac{\omega^{2}_{2}}{c^{2}}\epsilon^{(1)}(\omega_{2})$ hence

(7)   \begin{equation*} [ \frac{\partial^{2} A_{2}}{\partial z^{2}} + 2ik_{2} \frac{\partial A_{2}}{\partial z}]e^{i(k_{2}z-wt)} =-\frac{8\pi \omega^{2}_{2}}{c^2}dA^{2}_{1}e^{i(2k_{1}z-\omega_{2}t)} \end{equation*}

replace $ \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

(8)   \begin{equation*} \frac{dA_{2}}{dz}= \frac{4 \pi id \omega^{2}_{2}}{k_{2}c^{2}}A^{2}_{1}e^{i\bigtriangleup kz} \end{equation*}

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1 thought on “Derivation of Transmitted Wave Equation”

  1. joandrade

    When you say equation 1 is “derived from Maxwell’s equation,” are you referring to the four Maxwell equations we learned about in Griffiths, or to just the Ampere-Maxwell equation (the fourth of Maxwell’s equations), or to some other equation? Aside from a few in-line symbols (when you were trying to use the Latex form) missing from your process, your derivation looks very complicated and you should be proud that you were able to work through it! Perhaps you could expand upon your work by further explaining the procedure you followed to get from equation to equation, without assuming your audience knows the logic of each step..

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