## Abstract

We describe a spectrogram-based simulated annealing algorithm for designing quasi-phase-matched crystals capable of producing second harmonic generation pulses of any chosen amplitude and phase profile. The approach applies a new and rapid analytic method for calculating the amplitude and phase of the second harmonic generation pulses generated by a quasi-phase-matched crystal containing an arbitrary grating design. The performance of the algorithm is illustrated by examples of femtosecond second harmonic pulses designed according to various target shapes including single, double and triple Gaussian pulses, positive and negative linear chirp and square, triangular and stepped profiles.

© 2005 Optical Society of America

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### Equations (8)

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(1)
$${E}_{\mathit{crys}}\left(\Omega \right)=-\frac{\kappa {d}_{\mathit{ijk}}}{\Delta k\left(\Omega \right)}\left[1-{\left(-1\right)}^{n}\mathrm{exp}\left(i\Delta k\left(\Omega \right){Q}_{n}\right)+\sum _{m=1}^{n-1}2{\left(-1\right)}^{m}\mathrm{exp}\left(i\Delta k\left(\Omega \right){Q}_{m}\right)\right]$$
(2)
$${E}_{2}\left(\Omega \right)=F\left\{{E}_{1}{\left(t\right)}^{2}\right\}{E}_{\mathit{crys}}\left(\Omega \right)$$
(3)
$${E}_{2}\left(t\right)={F}^{-1}\left\{{E}_{2}\left(\Omega \right)\right\}$$
(4)
$$i\frac{\partial {A}_{1}(z,t)}{\partial z}+i{k}_{1}^{\prime}\frac{\partial {A}_{1}(z,t)}{\partial t}-\frac{1}{2}{k}_{1}^{\u2033}\frac{{\partial}^{2}{A}_{1}(z,t)}{\partial {t}^{2}}+\sigma \left(z\right){\Gamma}_{1}{A}_{1}^{*}(z,t){A}_{2}(z,t)\mathrm{exp}\left(i\Delta {k}_{o}z\right)=0$$
(5)
$$i\frac{\partial {A}_{2}(z,t)}{\partial z}+i{k}_{2}^{\prime}\frac{\partial {A}_{2}(z,t)}{\partial t}-\frac{1}{2}{k}_{2}^{\u2033}\frac{{\partial}^{2}{A}_{2}(z,t)}{\partial {t}^{2}}+\sigma \left(z\right){\Gamma}_{2}{A}_{1}^{2}(z,t)\mathrm{exp}(-i\Delta {k}_{o}z)=0$$
(6)
$${I}_{\mathit{FROG}}(\Omega ,\tau )={\mid \underset{-\infty}{\overset{\infty}{\int}}{E}_{\mathit{sig}}(t,\tau )\mathrm{exp}(-i\Omega t)\mathit{dt}\mid}^{2}$$
(7)
$${E}_{\mathit{sig}}(t,\tau )=E\left(t\right){\mid E\left(t-\tau \right)\mid}^{2}$$
(8)
$${e}_{k}=\sqrt{\sum _{\tau}\sum _{\Omega}{\mid {\hat{I}}_{\mathit{FROG}}^{\mathrm{SHG}}-{\hat{I}}_{\mathit{FROG}}^{\mathit{target}}\mid}^{2}}$$