## Abstract

We consider a programmable, phase, and amplitude femtosecond pulse shaper based on a two-dimensional (2D) reflective liquid-crystal (LC) spatial light modulator (SLM). A new zero-order pulse shaping scheme is introduced and compared to the first-order scheme, both theoretically and experimentally, using liquid crystal on silicon 2D SLM. While the spectral components of the pulse are spread across the horizontal dimension, we use the vertical direction for modulation of both spectral phases and amplitudes. It was found that while zero-order approach provided better light efficiency (67% versus 43%), the first-order scheme has superior dynamic range of amplitude modulation.

© 2007 Optical Society of America

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### Equations (12)

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(1)
$$T(z,\omega )=[\mathrm{rect}\left(\frac{z}{2z0}\right)\bullet {e}^{i(A\left(\omega \right)\mathrm{rect}\left(\frac{z}{{z}_{0}}\right)+B\left(\omega \right))}]*\sum _{n=-\infty}^{+\infty}\delta (z-2{z}_{0}n).$$
(2)
$$T(z,\omega )=\sum _{n=-\infty}^{+\infty}{C}_{n}{e}^{i\frac{2\pi n}{P}z}.$$
(3)
$${C}_{0}\left(\omega \right)=\frac{1}{2{z}_{0}}{\int}_{-{z}_{0}}^{{z}_{0}}T(z,\omega )\mathrm{d}z=\frac{1}{2}(1+{e}^{iA\left(\omega \right)}){e}^{iB\left(\omega \right)}.$$
(4)
$${I}_{0}\sim {\mid {C}_{0}\left(\omega \right)\mid}^{2}={\mathrm{cos}}^{2}\left(\frac{A\left(\omega \right)}{2}\right).$$
(5)
$$A\left(\omega \right)=2\phantom{\rule{0.2em}{0ex}}\mathrm{arccos}\left[\tau \left(\omega \right)\right],$$
(6)
$$B\left(\omega \right)=\phi \left(\omega \right)-\mathrm{arg}(1+{e}^{iA\left(\omega \right)})=\phi \left(\omega \right)-\frac{A\left(\omega \right)}{2}.$$
(7)
$$T(z,\omega )=[{e}^{i(\frac{2\pi \alpha \left(\omega \right)}{P}z+\pi )}\bullet \mathrm{rect}\left(\frac{z}{P}\right)]*\sum _{n=-\infty}^{+\infty}\delta (z-nP)*\delta (z-s\left(\omega \right)),$$
(8)
$${E}_{\mathrm{form}}({k}_{z},\omega )\propto \mathrm{sinc}\left(\frac{{k}_{z}}{1\u2215P}\right)*\delta ({k}_{z}-\alpha \left(\omega \right)\u2215P).$$
(9)
$$E({k}_{z},\omega )\propto \{\mathrm{sinc}\left(\frac{{k}_{z}-\alpha \left(\omega \right)\u2215P}{1\u2215P}\right)\bullet {e}^{i2\pi s\left(\omega \right){k}_{z}}\bullet \sum _{n=-\infty}^{+\infty}\delta ({k}_{z}-n\u2215P)\}*\stackrel{\u0303}{A}\left(\frac{{k}_{z}}{1\u2215D}\right).$$
(10)
$${E}^{+1}({k}_{z},\omega )\propto \mathrm{sinc}[1-\alpha \left(\omega \right)]\bullet {e}^{i\frac{2\pi s\left(\omega \right)}{P}}\bullet \stackrel{\u0303}{A}\left(\frac{{k}_{z}-1\u2215P}{1\u2215D}\right),$$
(11)
$${\Phi}_{\mathrm{max}}\left(\omega \right)-2\pi \ge \frac{2\pi}{N}\phantom{\rule{0.3em}{0ex}}\forall \omega .$$
(12)
$${\eta}_{q}={\mathrm{sinc}}^{2}\left(\frac{q}{{2}^{N}}\right)\frac{{\mathrm{sinc}}^{2}(q-\frac{{\Phi}_{0}}{2\pi})}{{\mathrm{sinc}}^{2}\left(\frac{q-\frac{{\Phi}_{0}}{2\pi}}{{2}^{N}}\right)},$$