## Abstract

Z-scan measurement with a slit instead of a circular aperture is proposed and analyzed for an astigmatic Gaussian beam. This new technique can accurately determine not only the nonlinear refractive index but also the ellipticity and positions of $x\u2013y$ foci of the astigmatic beam.

© 2000 Optical Society of America

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### Equations (10)

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(2)
$$=E(0,0,z,t)exp(-\alpha L/2)$$
(3)
$$\times \sum _{m=0}^{\infty}{\left[\frac{{w}_{m0x}(z){w}_{m0y}(z)}{{W}_{\mathit{mx}}(z){w}_{\mathit{my}}(z)}\right]}^{1/2}exp[i{\eta}_{m}(z)]\times exp\left[-\frac{{\mathit{ikx}}^{2}}{2{q}_{\mathit{mx}}(z)}-\frac{{\mathit{iky}}^{2}}{2{q}_{\mathit{my}}(z)}\right]\frac{[i\mathrm{\Delta}{\varphi}_{0}(z,t){]}^{m}}{m!},$$
(4)
$${T}_{y}(z)=\frac{{\int}_{-{y}_{0}/2}^{{y}_{0}/2}{\int}_{-\infty}^{\infty}|{E}_{a}(x,y,z,\mathrm{\Delta}{\mathrm{\Phi}}_{0}){|}^{2}\mathrm{d}x\mathrm{d}y}{{\int}_{-{y}_{0}/2}^{{y}_{0}/2}{\int}_{-\infty}^{\infty}|{E}_{a}(x,y,z,0){|}^{2}\mathrm{d}x\mathrm{d}y},$$
(5)
$$\xi =(0.398+0.662x-0.0375{x}^{2}){S}^{-k},$$
(6)
$$x=\frac{\mathrm{\Delta}{T}_{y}}{\mathrm{\Delta}{T}_{x}},\hspace{1em}k=-0.0167+\frac{0.319}{4(S-0.0972{)}^{2}+0.0162}.$$
(7)
$$\frac{\mathrm{\Delta}{T}_{y}}{\mathrm{\Delta}{\mathrm{\Phi}}_{0}}=\left\{0.309-0.0804exp\left[-\frac{(\xi -1)}{0.472}\right]\right\}{S}^{-k},$$
(8)
$$k=-0.119+\frac{0.259}{4(S-0.957{)}^{2}+0.0234}.$$
(9)
$${\mathrm{\Delta}}_{\mathit{xy}}=w_{0y}{}^{2}f(\xi ),$$
(10)
$$f(\xi )=1.231-0.594\xi +0.494{\xi}^{2},$$