## Abstract

Based on Fresnel–Kirchhoff diffraction theory, a diffraction model of nonlinear optical media interacting with a Gaussian beam has been set up that can interpret the Z-scan phenomenon in a new way. This theory not only is consistent with the conventional Z-scan theory for a small nonlinear phase shift but also can be used for larger nonlinear phase shifts. Numerical computations indicate that the shape of the Z-scan curve is greatly affected by the value of the nonlinear phase shift. The symmetric dispersionlike Z-scan curve is valid only for small nonlinear phase shifts $(|\mathrm{\Delta}{\varphi}_{0}|\pi ),$ but, with increasingly larger nonlinear phase shifts, the valley of the transmittance is severely suppressed and the peak is greatly enhanced. The power output through the aperture will oscillate with increasing nonlinear phase shift caused by the input laser power. The aperture transmittance will attenuate and saturate with increasing Kerr constant.

© 2003 Optical Society of America

Full Article |

PDF Article
### Equations (12)

Equations on this page are rendered with MathJax. Learn more.

(1)
$$T(z,\mathrm{\Delta}{\varphi}_{0})\approx 1+\frac{4\mathrm{\Delta}{\varphi}_{0}x}{({x}^{2}+9)({x}^{2}+1)},$$
(2)
$$E(r,z)=E(0,0)\frac{{\omega}_{0}}{\omega (z)}exp[-\alpha (z-{Z}_{1})/2]exp\left[-\frac{{r}^{2}}{{\omega}^{2}(z)}\right]\times exp\left[-\frac{{\mathit{ikr}}^{2}}{2R(z)}-i\varphi (z)\right],$$
(3)
$$\mathrm{\Delta}n=\gamma I,$$
(4)
$$\mathrm{\Delta}n(r,z)=\frac{2\gamma P}{\pi {\omega}^{2}(z)}exp[-\alpha (z-{Z}_{1})]exp\left[-\frac{2{r}^{2}}{{\omega}^{2}(z)}\right].$$
(5)
$$\mathrm{\Delta}\varphi (r,{Z}_{1})=\frac{2\pi}{\mathrm{\lambda}}{\int}_{{Z}_{1}}^{{Z}_{1}+d}\mathrm{\Delta}n(r,z)\mathrm{d}z.$$
(6)
$$\mathrm{\Delta}\varphi (r,{Z}_{1})\approx \mathrm{\Delta}{\varphi}_{0}exp\left(-\frac{2{r}^{2}}{{{\omega}_{1}}^{2}}\right),$$
(7)
$$\mathrm{\Delta}{\varphi}_{0}=\frac{4\gamma P[1-exp(-\alpha d)]}{\mathrm{\lambda}\alpha {{\omega}_{1}}^{2}}.$$
(8)
$$\tilde{U}(r,{Z}_{1})=exp[-i\mathrm{\Delta}\varphi (r,{Z}_{1})].$$
(9)
$$\tilde{V}(r,{Z}_{1})=E(0,0)\frac{{\omega}_{0}}{{\omega}_{1}}exp(-\alpha d/2)\times exp\left(-\frac{{r}^{2}}{{{\omega}_{1}}^{2}}\right)exp\left(-\frac{{\mathit{ikr}}^{2}}{2{R}_{1}}\right).$$
(10)
$$\tilde{E}(\rho )=\frac{-i}{\mathrm{\lambda}}\int \int F({\theta}_{0},\theta )\tilde{V}(r)\tilde{U}(r)\frac{exp(\mathit{ikD})}{D}r\mathrm{d}r\mathrm{d}\phi .$$
(11)
$$I(\rho ,{Z}_{1},{Z}_{2})=\frac{P}{2\pi {\mathrm{\lambda}}^{2}{{\omega}_{1}}^{2}}exp(-\alpha d)\times {\left|{\int}_{0}^{2\pi}{\int}_{0}^{\sqrt{5}\omega ({Z}_{1})}\left(\frac{1}{D}+\frac{{Z}_{2}}{{D}^{2}}\right)\times exp\left(-\frac{{r}^{2}}{{{\omega}_{1}}^{2}}\right)exp\left\{i\left[\frac{2\pi}{\mathrm{\lambda}}D-\frac{{\mathit{kr}}^{2}}{2{R}_{1}}-\mathrm{\Delta}{\varphi}_{0}exp\left(-\frac{2{r}^{2}}{{{\omega}_{1}}^{2}}\right)\right]\right\}r\mathrm{d}r\mathrm{d}\phi \right|}^{2}.$$
(12)
$${P}_{A}({Z}_{1},{Z}_{2},a,\mathrm{\Delta}{\varphi}_{0})=2\pi {\int}_{0}^{a}I(\rho ,{Z}_{1},{Z}_{2})\rho \mathrm{d}\rho .$$