## Abstract

We theoretically propose a new procedure of designing quasi-phase-matching (QPM) gratings for compressing optical ultra-short pulses during second-harmonic generation. The grating consists of blocks of crystal with same block length and the direction of spontaneous polarization of each block is determined by optimal algorithm, by which the sign of the nonlinear coefficient of each block is optimized to make the phase response of the grating same for different wavelength at second harmonic waves, so that the generated second harmonic pulse from the end of the crystal will be compressed. during nonlinear optical frequency conversion process.

© 2005 Optical Society of America

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### Equations (14)

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(1)
$$d\left(z\right)=\sum _{m=-\infty}^{+\infty}{d}_{m}(z)=\sum _{m=-\infty}^{+\infty}\mid {d}_{m}\left(z\right)\mid \mathrm{exp}\left[i{K}_{0m}z+i{\phi}_{m}\left(z\right)\right]$$
(2)
$$\hat{E}\left(z,\omega \right)=\hat{A}\left(z,\Omega \right)\mathrm{exp}\left[-\mathit{ik}\left({\omega}_{0}+\Omega \right)z\right],$$
(3)
$$E\left(z,t\right)=B\left(z,t\right)\mathrm{exp}\left(i{\omega}_{0}t-i{k}_{0}z\right),$$
(4)
$$\hat{E}\left(z,\omega \right)=\frac{1}{2\pi}\underset{-\infty}{\overset{+\infty}{\int}}E\left(z,t\right)\mathrm{exp}\left(-\mathit{i\omega t}\right)\mathit{dt}$$
(5)
$$\frac{\partial}{\partial z}{\hat{A}}_{1}\left(z,\Omega \right)=0$$
(6)
$$\frac{\partial}{\partial z}{\hat{A}}_{2}\left(z,\Omega \right)=-i\frac{{\mu}_{0}{\omega}_{2}^{2}}{2{k}_{2}}{\hat{P}}_{\mathit{NL}}\left(z,\Omega \right)\mathrm{exp}\left[\mathit{ik}\left({\omega}_{2}+\Omega \right)z\right],$$
(7)
$${\hat{P}}_{\mathit{NL}}\left(z,\Omega \right)={\epsilon}_{0}d\left(z\right)\underset{-\infty}{\overset{+\infty}{\int}}{\hat{A}}_{1}(z,\Omega \prime ){\hat{A}}_{1}\left(z,\Omega -\Omega \prime \right)\mathrm{exp}\left\{-i\left[k\left({\omega}_{1}+\Omega \prime \right)+k\left({\omega}_{1}+\Omega -\Omega \prime \right)\right]z\right\}d\Omega \prime $$
(8)
$${\hat{A}}_{1}\left(z,\Omega \right)={\hat{A}}_{1}\left(z=0,\Omega \right)$$
(9)
$${\hat{A}}_{2}\left(L,\Omega \right)=\underset{-\infty}{\overset{+\infty}{\int}}{\hat{A}}_{1}(\Omega \prime ){\hat{A}}_{1}\left(\Omega -\Omega \prime \right)\hat{d}\left[\Delta k\right(\Omega ,\Omega \prime \left)\right]d\Omega \prime ,$$
(10)
$$\Delta k(\Omega ,\Omega \prime )=k\left({\omega}_{1}+\Omega \prime \right)+k\left({\omega}_{1}+\Omega -\Omega \prime \right)-k\left({\omega}_{2}+\Omega \right).$$
(11)
$${\hat{A}}_{2}\left(L,\Omega \right)=-\mathit{i\gamma}{\hat{A}}_{1}^{2}\left(\frac{\Omega}{2}\right)\underset{-\infty}{\overset{+\infty}{\int}}d\left(z\right)\mathrm{exp}\left[i\left(2k\left({\omega}_{1}+\frac{\Omega}{2}\right)-k\left(2{\omega}_{1}+\Omega \right)\right)z\right]\mathit{dz}$$
(12)
$${B}_{1}\left(0,t\right)={E}_{0}\frac{{\tau}_{0}}{\sqrt{{\tau}_{0}^{2}+i{C}_{1}}}\mathrm{exp}\left(-\frac{{t}^{2}}{2\left({\tau}_{0}^{2}+i{C}_{1}\right)}\right)$$
(13)
$${\tau}_{1}=\sqrt{{\tau}_{0}^{2}+{\left(\frac{{C}_{1}}{{\tau}_{0}}\right)}^{2}}$$
(14)
$${\hat{A}}_{1}\left(\Omega \right)=\frac{1}{\sqrt{2\pi}}{E}_{0}{\tau}_{0}\mathrm{exp}\left[-\frac{1}{2}\left({\tau}_{0}^{2}+i{C}_{1}\right){\Omega}^{2}\right]$$