## Abstract

It has previously been realized that angle-resolved spectra (or far-field spectra) reflect in detail the complex dynamics of femtosecond pulses propagating in nonlinear dispersive media [J. Opt. Soc. Am. B 22, 862 (2005)], and thus represent a potentially useful experimental diagnostic tool. In this paper we extend an effective three-wave mixing approach previously applied to nonlinear X-waves [Phys. Rev. Lett. **92**, 253901 (2004)] to the analysis of far-field spectra. Theoretical justification for the approach is presented, and a practical method is proposed that makes it possible to interpret all features of far-field spectra, and to extract quantitative information about the underlying intra-pulse dynamics.

© 2005 Optical Society of America

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### Equations (15)

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(1)
$$E\left(x,y,z,t\right)=\sum _{u,v,\omega}{A}_{u,v,\omega}\left(z\right){e}^{i{k}_{z}(\omega ,u,v)z-i\left(\mathit{\omega t}-\mathit{ux}-\mathit{vy}\right)},$$
(2)
$${k}_{z}(\omega ,{k}_{x},{k}_{y})\equiv \sqrt{\frac{{\omega}^{2}\epsilon \left(\omega \right)}{{c}^{2}-{k}_{x}^{2}-{k}_{y}^{2},}}$$
(3)
$${\partial}_{z}{A}_{u,v,\omega}\left(z\right)=\frac{i{\omega}^{2}}{2{c}^{2}{k}_{z}(\omega ,u,v)}{e}^{-i{k}_{z}(\omega ,u,v)}\times \int \frac{\text{d}x\text{d}y\text{d}t}{V}{e}^{i\left(\mathit{\omega t}-\mathit{ux}-\mathit{vy}\right)}\Delta \chi \left(x,y,z,t\right)E\left(x,y,z,t\right).$$
(4)
$$\Delta \chi \left(x,y,z,t\right)\approx \sum _{r=\mathrm{1,2}}\Delta {\chi}_{r}\left(\frac{x,y,z,t-z}{{\nu}_{r}}\right).$$
(5)
$$\Delta {\chi}_{r}\left(\frac{x,y,z,t-z}{{\nu}_{r}}\right)=\sum _{m,n,\tilde{\omega}}\Delta {\chi}_{m,n,\tilde{\omega}}^{\left(r\right)}\left(z\right){e}^{-i\tilde{\omega}\left(\frac{t-z}{{\nu}_{r}}\right)+i\left(\mathit{mx}+\mathit{ny}\right)}.$$
(6)
$${A}_{{k}_{x},{k}_{y},\omega}\left(z\right)=\frac{i{\omega}^{2}}{2{c}^{2}{k}_{z}(\omega ,{k}_{x},{k}_{y})}\times \int \text{d}z\sum _{r,u,v,\Omega}\Delta {\chi}_{{k}_{x}-u,{k}_{y}-v,\omega -\Omega}^{\left(r\right)}\left(z\right){A}_{u,v,\Omega}\left(z\right)\times $$
(7)
$$\phantom{\rule{5em}{0ex}}\mathrm{exp}\left[\mathit{iz}\left(-{k}_{z}(\omega ,{k}_{x},{k}_{y})z+\frac{\left(\omega -\Omega \right)z}{{v}_{r}}+{k}_{z}(\Omega ,u,v)\right)\right]$$
(8)
$$\mid -{k}_{z}(\omega ,{k}_{x},{k}_{y})+{k}_{z}\left(\Omega ,u,v\right)+\frac{\omega -\Omega}{{v}_{r}}\mid \le \frac{2\pi}{{l}_{r}},r=\mathrm{1,2},$$
(9)
$$\mid -{k}_{z}(\omega ,{k}_{x},{k}_{y})+{k}_{z}\left({\omega}_{0},0,0\right)+\frac{\omega -{\omega}_{0}}{{v}_{r}}\mid \le \frac{2\pi}{{l}_{r}},r=\mathrm{1,2},$$
(10)
$$\tau \left(z\right)={\tau}_{0}+\left(\frac{1}{{v}_{g}}-\frac{1}{{v}_{r}}\right)\left(z-{z}_{0}\right)+{\alpha \left(z-{z}_{0}\right)}^{2},$$
(11)
$$\frac{1}{{l}_{r}^{2}}\to \frac{1}{{l}_{r}^{2}}+{\alpha}^{2}{l}_{r}^{2}{\left(\omega -{\omega}_{0}\right)}^{2}$$
(12)
$$E\left(x,y,z,t\right)\approx {e}^{-i{\omega}_{o}t+i{k}_{z}({\omega}_{o},\mathrm{0,0})z}$$
(13)
$$\Delta \chi \left(x,y,z,t\right)\approx \mathrm{exp}\left[-\frac{{\left(z-{z}_{0}\right)}^{2}}{{l}^{2}}-\frac{{\left(t-\tau \left(z\right)\right)}^{2}}{{w}_{t}^{2}}\right]$$
(14)
$$\mathrm{exp}\left[\frac{-4{(-{k}_{z}\left(\omega ,{k}_{x},{k}_{y}\right)+{k}_{z}\left({\omega}_{0},\mathrm{0,0}\right)+\frac{\omega -{\omega}_{0}}{{v}_{r}})}^{2}}{\frac{1}{{l}^{2}}+{\alpha}^{2}{l}^{2}{\left(\omega -{\omega}_{0}\right)}^{2}}\right]$$
(15)
$${(-{k}_{z}\left(\omega ,{k}_{x},{k}_{y}\right)+{k}_{z}({\omega}_{0},\mathrm{0,0})+\frac{\omega -{\omega}_{0}}{{v}_{r}})}^{2}<\frac{P}{4}\left(\frac{1}{{l}^{2}}+{\alpha}^{2}{l}^{2}{\left(\omega -{\omega}_{0}\right)}^{2}\right)$$