## Abstract

The dark spot effect downstream from the nonlinear hot image is accounted for in this paper. The conditions for the formation of dark spot are carefully discussed. The explanation is based on analytical analysis, and the results are verified by numerical simulations. The dependence of the location of the dark spot on the nonlinear phase delay seems to suggest a probable method for measuring the nonlinear refractive coefficient of materials.

© 2012 Optical Society of America

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### Equations (11)

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(1)
$$t({x}_{1},{y}_{1})=1+\alpha p({x}_{1},{y}_{1}),$$
(2)
$${U}_{o}\approx U({x}_{2},{y}_{2})\mathrm{exp}[\mathrm{i}\delta \mathrm{n}{k}_{0}d]=({U}_{b}+{U}_{s})\mathrm{exp}\left[\mathrm{i}{k}_{0}\frac{\gamma {|U({x}_{2},{y}_{2})|}^{2}}{2}d\right],$$
(3)
$${U}_{o}\approx \mathrm{exp}(iB)[{U}_{b}+(1+iB){U}_{s}+iB{U}_{b}^{2}{U}_{s}^{*}].$$
(4)
$${v}_{1}=\mathrm{exp}(iB)iB{U}_{b}^{2}{U}_{s}^{*}=-\frac{{\alpha}^{*}B}{\lambda {z}_{1}}\mathrm{exp}[i(k{z}_{1}+B)]\mathrm{exp}[-\frac{\mathrm{i}k}{2{z}_{1}}({x}_{2}^{2}+{y}_{2}^{2}\left)\right]{P}^{*}(\frac{{x}_{2}}{\lambda {z}_{1}},\frac{{y}_{2}}{\lambda {z}_{1}}),$$
(5)
$$-\frac{{\alpha}^{*}B}{i{\lambda}^{2}{z}_{1}{z}_{2}}\text{\hspace{0.17em}}\mathrm{exp}[i(k{z}_{1}+k{z}_{2}+B)]\times \iint \mathrm{exp}[-\frac{\mathrm{i}k}{2{z}_{1}}({x}_{2}^{2}+{y}_{2}^{2}\left)\right]{P}^{*}(\frac{{x}_{2}}{\lambda {z}_{1}},\frac{{y}_{2}}{\lambda {z}_{1}})\mathrm{exp}[\frac{\mathrm{i}k}{2{z}_{2}}({({x}_{3}-{x}_{2})}^{2}+{({y}_{3}-{y}_{2})}^{2})]\mathrm{d}{x}_{2}\mathrm{d}{y}_{2}.$$
(6)
$$i{\alpha}^{*}B\text{\hspace{0.17em}}\mathrm{exp}[i(2k{z}_{1}+B)]\mathrm{exp}\left[\frac{\mathrm{i}k}{2{z}_{1}}\right({x}_{3}^{2}+{y}_{3}^{2}\left)\right]p({x}_{3},{y}_{3}).$$
(7)
$$i{\alpha}^{*}B\text{\hspace{0.17em}}\mathrm{exp}[i(2k{z}_{1}+B)]p({x}_{3},{y}_{3}).$$
(8)
$$i{\alpha}^{*}AB\frac{\mathrm{exp}[i(2k{z}_{1}+B+k{z}_{c})]}{i\lambda {z}_{c}}=\frac{AB}{\lambda {z}_{c}}{\alpha}^{*}\text{\hspace{0.17em}}\mathrm{exp}[i(2k{z}_{1}+B+k{z}_{c})],$$
(9)
$$\frac{AB}{\lambda {z}_{c}}{\alpha}^{*}=-1.$$
(10)
$${z}_{c}=\frac{|\alpha |AB}{\lambda}.$$
(11)
$${z}_{c}=\frac{2\pi B{r}^{2}}{\lambda}.$$