## Abstract

We theoretically propose a procedure based on a cascading genetic algorithm for the design of aperiodically quasi-phase-matched gratings for frequency conversion of optical ultrafast pulses during difference-frequency generation. By designing the sequence of a domain inversion grating, different wavelengths at the output idler pulse almost have the same phase response, so femtosecond laser pulses at wavelength
$800\text{\hspace{0.17em} nm}$ can be shifted to other wavelengths without group-velocity mismatch.

© 2007 Optical Society of America

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### Equations (16)

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(1)
$$\mathrm{d}\left(z\right)={\displaystyle \sum _{m=-\infty}^{+\infty}{\mathrm{d}}_{m}\left(z\right)}={\displaystyle \sum _{m=-\infty}^{+\infty}\left|{\mathrm{d}}_{m}\left(z\right)\right|}\text{exp}\left[i{K}_{0m}z+i{\phi}_{m}\left(z\right)\right].$$
(2)
$${\widehat{E}}_{m}\left(z,\omega \right)={\widehat{A}}_{m}\left(z,{\Omega}_{m}\right)\text{exp}\left[-ik\left({\omega}_{m}+{\Omega}_{m}\right)z\right],$$
(3)
$$\frac{\partial}{\partial z}\text{\hspace{0.17em}}{\widehat{A}}_{i}\left(z,{\Omega}_{i}\right)=-i\text{\hspace{0.17em}}\frac{{\mu}_{0}{{\omega}_{i}}^{2}}{2{k}_{i}}\text{\hspace{0.17em}}{\widehat{P}}_{\mathrm{N}\mathrm{L}}\left(z,{\Omega}_{i}\right)\text{exp}\left[ik\left({\omega}_{i}+{\Omega}_{i}\right)\right]z\text{,}$$
(4)
$$\frac{\partial}{\partial z}\text{\hspace{0.17em}}{\widehat{A}}_{s}\left(z,{\Omega}_{s}\right)=0\text{,}$$
(5)
$$\frac{\partial}{\partial z}\text{\hspace{0.17em}}{\widehat{A}}_{p}\left(z,{\Omega}_{p}\right)=0.$$
(6)
$${\widehat{P}}_{\mathrm{N}\mathrm{L}}\left(z,\Omega \right)=2{\epsilon}_{0}\mathrm{d}\left(z\right){\displaystyle {\int}_{-\infty}^{+\infty}{\widehat{A}}_{s}*}\left(z,-\Omega +\Omega \prime \right){\widehat{A}}_{p}\left(z,\Omega \prime \right)\times \text{exp}\left\{i\left[k\left({\omega}_{s}-\Omega +\Omega \prime \right)-k\left({\omega}_{p}+\Omega \prime \right)\right]z\right\}\mathrm{d}\Omega \prime \text{,}$$
(7)
$$\text{\hspace{0.17em}}{\widehat{A}}_{s}\left(z,\Omega \right)={\widehat{A}}_{s}\left(z=0,\Omega \right)={\widehat{A}}_{s}\left(\Omega \right)\text{,}$$
(8)
$${\widehat{A}}_{p}\left(z,\Omega \right)={\widehat{A}}_{p}\left(z=0,\Omega \right)={\widehat{A}}_{p}\left(\Omega \right)\text{,}$$
(9)
$${\widehat{A}}_{i}\left(L,\Omega \right)=-i\gamma {\displaystyle {\int}_{0}^{L}\mathrm{d}\left(z\right)\mathrm{d}z}{\displaystyle {\int}_{-\infty}^{+\infty}\mathrm{d}\Omega \prime}{\widehat{A}}_{s}*\left(\Omega \prime -\Omega \right){\widehat{A}}_{p}\left(\Omega \prime \right)\times \text{exp}\left[-i\Delta k\left(\Omega ,\Omega \prime \right)z\right]\text{,}$$
(10)
$$\Delta k\left(\Omega ,\Omega \prime \right)=k\left({\omega}_{p}+\Omega \prime \right)-k\left({\omega}_{i}+\Omega \right)-k\left({\omega}_{s}+\Omega \prime -\Omega \right).$$
(11)
$${\widehat{A}}_{p}\left(\Omega \right)={E}_{p}\delta \left(\Omega =0\right)\text{,}$$
(12)
$${\widehat{A}}_{i}\left(L,\Omega \right)=-i\gamma {\widehat{A}}_{s}*\left(-\Omega \right){E}_{p}{\displaystyle {\int}_{-\infty}^{+\infty}\mathrm{d}\left(z\right)}\text{exp}\left[-i\Delta k\left(\Omega \right)z\right]\mathrm{d}z\text{,}$$
(13)
$$\Delta k\left(\Omega \right)=k\left({\omega}_{p}\right)-k\left({\omega}_{i}+\Omega \right)-k\left({\omega}_{s}-\Omega \right).$$
(14)
$${B}_{s}\left(0,t\right)={E}_{s}\text{\hspace{0.17em}}\frac{{\tau}_{0}}{\sqrt{{{\tau}_{0}}^{2}+i{C}_{1}}}\text{\hspace{0.17em} exp}\left[-\frac{{t}^{2}}{2\left({{\tau}_{0}}^{2}+i{C}_{1}\right)}\right].$$
(15)
$${\widehat{A}}_{s}\left(\Omega \right)=\frac{1}{\sqrt{2\pi}}\text{\hspace{0.17em}}{E}_{s}{\tau}_{0}\text{\hspace{0.17em} exp}\left[-\text{1}/\text{2}\left({{\tau}_{0}}^{2}+i{C}_{1}\right){\Omega}^{2}\right].$$
(16)
$${\sigma}^{2}=\frac{1}{n}\left[{\left({\varphi}_{1}-\overline{\varphi}\right)}^{2}+{\left({\varphi}_{2}-\overline{\varphi}\right)}^{2}+\cdots +{\left({\varphi}_{n}-\overline{\varphi}\right)}^{2}\right].$$