## Abstract

A newly developed method for the characterization of the optical nonlinearities and dynamics is applied to Kerr liquids. The time-resolved pump-probe system based on the $4f$ nonlinear imaging technique with phase object is used to obtain the diffraction pattern of the nonlinear filter induced in the liquid $\mathrm{C}{\mathrm{S}}_{2}$ placed in the Fourier plane by a charge-coupled device at various delay times. A theory based on two-beam coupling in perpendicular linear polarizations is used to interpret the measurement results. Good agreement is obtained between theory and experiment, suggesting a new method for simultaneous measurements of both magnitude and sign of the intensity-dependent refractive index as well as the dynamics of the Kerr liquids.

© 2009 Optical Society of America

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### Equations (17)

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(1)
$$E(\stackrel{\u20d7}{r},r,t)=\mathrm{Re}\left[A(r,t)\mathrm{exp}\left(i\{\stackrel{\u20d7}{k}\cdot \stackrel{\u20d7}{r}-[\omega +\Delta \omega \left(t\right)]t\}\right)\right],$$
(2)
$$E(\stackrel{\u20d7}{r},r,t,\tau )={A}_{e}(r,t)\mathrm{exp}\left(i\{{\stackrel{\u20d7}{k}}_{e}\cdot \stackrel{\u20d7}{r}-[\omega +\Delta \omega \left(t\right)]t\}\right)+{A}_{p}(r,t)\mathrm{exp}\left(i\{{\stackrel{\u20d7}{k}}_{p}\cdot \stackrel{\u20d7}{r}-[\omega +\Delta \omega (t-\tau )](t-\tau )\}\right),$$
(3)
$${A}_{p0}(r,t)={A}_{0}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(-{t}^{2}\u22152{\tau}_{p}^{2})\mathrm{circ}(r\u2215{R}_{a})\mathrm{exp}\left[i\phi \mathrm{circ}(r\u2215{L}_{p})\right],$$
(4)
$${A}_{p0}(\rho ,t)=\frac{2\pi}{\lambda {f}_{1}}{\int}_{0}^{{R}_{a}}r{A}_{p0}(r,t){J}_{0}\left(2\pi r\rho \right)\mathrm{d}r,$$
(5)
$${A}_{pL}(\rho ,t)={A}_{p0}(\rho ,t)\mathrm{exp}\left(i\Delta {\phi}_{\mathrm{NL}}\right),$$
(6)
$${A}_{pL}(r,t)=2\pi \lambda {f}_{2}{\int}_{0}^{\infty}\rho {A}_{pL}(\rho ,t){J}_{0}\left(2\pi r\rho \right)\mathrm{d}\rho ,$$
(7)
$${F}_{L}\left(r\right)=\frac{{\int}_{-\infty}^{+\infty}{\mid {A}_{pL}(r,t)\mid}^{2}\mathrm{d}t}{{A}_{0}^{2}{\pi}^{1\u22152}{\tau}_{p}},$$
(8)
$$I={A}_{e}{A}_{e}^{*}+{A}_{p}{A}_{p}^{*}+{A}_{e}{A}_{p}^{*}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(i\stackrel{\u20d7}{q}\cdot \stackrel{\u20d7}{r})\mathrm{exp}(-i\omega \tau )\mathrm{exp}(-i\Omega t)+\mathrm{c.c.}$$
(9)
$${\tau}_{\mathrm{rot}}\frac{\mathrm{d}{n}_{\mathrm{NL}}}{\mathrm{d}t}+{n}_{\mathrm{NL}}={n}_{2}I,$$
(10)
$$\frac{\mathrm{d}{A}_{p}}{\mathrm{d}z}=i\alpha {n}_{2}k({I}_{e}+{I}_{p}+\frac{{I}_{e}}{1+i\Omega {\tau}_{\mathrm{rot}}}){A}_{p},$$
(11)
$$\frac{\mathrm{d}{I}_{p}}{\mathrm{d}z}=\alpha {n}_{2}k\frac{2\Omega {\tau}_{\mathrm{rot}}}{1+{\left(\Omega {\tau}_{\mathrm{rot}}\right)}^{2}}{I}_{e}{I}_{p},$$
(12)
$$\frac{\mathrm{d}{\phi}_{p}}{\mathrm{d}z}=\alpha {n}_{2}k({I}_{e}+{I}_{p}+\frac{{I}_{e}}{1+{\left(\Omega {\tau}_{\mathrm{rot}}\right)}^{2}}).$$
(13)
$${I}_{pL}={I}_{p0}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(\alpha gL\right),$$
(14)
$$g=\frac{2{n}_{2}k\Omega {\tau}_{\mathrm{rot}}{I}_{e}}{1+{\left(\Omega {\tau}_{\mathrm{rot}}\right)}^{2}},$$
(15)
$${\phi}_{pL}=\alpha {n}_{2}k[{I}_{e}(1+\frac{1}{1+{\left(\Omega {\tau}_{\mathrm{rot}}\right)}^{2}})L+{I}_{p0}\frac{\mathrm{exp}\left(\alpha gL\right)-1}{\alpha g}].$$
(16)
$$\mathrm{NT}={E}_{p}\u2215{E}_{p0}=\frac{2\pi {\int}_{-\infty}^{\infty}\mathrm{d}t{\int}_{0}^{\infty}r{I}_{pL}\mathrm{d}r}{2\pi {\int}_{-\infty}^{\infty}\mathrm{d}t{\int}_{0}^{\infty}r{I}_{p0}\mathrm{d}r},$$
(17)
$${I}_{e0}(z,r,t)={I}_{0}\frac{{\omega}_{0}^{2}}{{\omega}^{2}\left(z\right)}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(-\frac{2{r}^{2}}{{\omega}^{2}\left(z\right)})\mathrm{exp}(-\frac{{t}^{2}}{\tau _{p}{}^{2}}),$$