## Abstract

Previous pulse-width measurement methods for ultrashort laser pulses have broadly employed nonlinear effects; thus any of these previous methods may experience problems relating to nonlinear effects. Here we present a new pulse-width measuring method based on the linear self-diffraction effect. Because the Talbot effect of a grating with ultrashort laser pulse illumination is different from that with continuous laser illumination, we are able to use this difference to obtain information about the pulse width. Three new techniques—the intensity integral technique, the intensity comparing ratio technique, and the two-dimensional structure technique— are introduced to make this method applicable. The method benefits from the simple structure of the Talbot effect and offers the possibility to extend the measurement of infrared and x-ray waves, for which currently used nonlinear methods do not work.

© 2002 Optical Society of America

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### Equations (11)

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(1)
$$r\left(t,\Delta \tau \right)=\mathrm{exp}\left[i{\omega}_{0}t-4\mathrm{ln}2{\left(\frac{t}{\Delta \tau}\right)}^{2}\right],$$
(2)
$$R(\omega ,\Delta \tau )=\frac{\Delta \tau}{4\sqrt{\pi \mathrm{ln}2}}\mathrm{exp}\left[\frac{-\Delta {\tau}^{2}{\left(\omega -{\omega}_{0}\right)}^{2}}{8\mathrm{ln}2}\right].$$
(3)
$$U\left(x,z,\omega \right)=\frac{\mathrm{exp}\left(i\frac{2\pi}{\lambda}z\right)}{\sqrt{i\lambda z}}{\int}_{-\infty}^{+\infty}{U}_{0}({x}_{0},\omega )\mathrm{exp}\left[-\frac{i\pi {\left(x-{x}_{0}\right)}^{2}}{\lambda z}\right]d{x}_{0},$$
(4)
$${U}_{0}\left(x\right)=\sum _{l}{A}_{l}\mathrm{exp}\left(i\frac{2\mathit{\pi lx}}{d}\right).$$
(5)
$$U\left(x,z,\omega \right)=\mathrm{exp}\left(i\frac{2\pi}{\lambda}z\right)\times \sum _{l}{A}_{l}\mathrm{exp}\left(i\frac{2\mathit{\pi lx}}{d}\right)\times \mathrm{exp}\left(\frac{i2\pi {l}^{2}z}{\frac{2{d}^{2}}{\lambda}}\right),$$
(6)
$$G\left(x,z,\omega ,\Delta \tau \right)=R(\omega ,\Delta \tau )U\left(x,z,\omega \right).$$
(7)
$$I\left(x,z,\mathrm{\Delta \tau}\right)=2\pi {\int}_{-\infty}^{+\infty}{\mid G\left(x,z,\omega ,\mathrm{\Delta \tau}\right)\mid}^{2}\mathrm{d\omega}.$$
(8)
$$I\left(x,z,\mathrm{\Delta \tau}\right)=\frac{\Delta {\tau}^{2}}{8\mathrm{ln}2}{\int}_{-\infty}^{+\infty}\mathrm{exp}\left[-\frac{\Delta {\tau}^{2}{\left(\omega -{\omega}_{0}\right)}^{2}}{8\mathrm{ln}2}\right]$$
(9)
$$\phantom{\rule{4.7em}{0ex}}\times \sum _{l,}^{+\infty}\sum _{m=-\infty}^{+\infty}{A}_{l}{A}_{m}\mathrm{exp}\left[i\frac{2\pi \left(l-m\right)x}{d}\right]\times \mathrm{exp}\left[\frac{i2\pi \left({l}^{2}-{m}^{2}\right)n{\omega}_{0}}{\omega}\right]\mathrm{d\omega}.$$
(10)
$$P({h}_{1},{h}_{2},\mathrm{\Delta \tau})={\int}_{{h}_{1}d}^{{h}_{2}d}I\left(x,z,\mathrm{\Delta \tau}\right)dx.$$
(11)
$$S\left(\Delta \tau \right)=\frac{P(\frac{1}{4},\frac{3}{4},\Delta \tau )}{P\left(-\frac{1}{4},\frac{1}{4},\Delta \tau \right)}.$$