## Abstract

We consider the self-similar amplification of two optical pulses of different wavelengths in order to investigate the effects of a collision between two similaritons. We theoretically demonstrate that similaritons are stable against collisions in a Raman amplifier: similaritons evolve separately in the amplifier without modification of the scaling of their temporal width and chirp and by conserving their velocities, only interact during their overlap and regain their parabolic form after collision. We show both theoretically and experimentally that the collision of two similaritons induces a sinusoidal modulation inside the overlap region, whose frequency decreases during the interaction. Theoretical and experimental studies of the pulse spectrum evidence that similaritons interact with each other through cross phase modulation.

© 2005 Optical Society of America

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### Equations (9)

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(1)
$$\Delta {T}_{s}\left(z\right)=\mid \Delta T\left(z\right)\mid -2{T}_{p}\left(z\right)$$
(2)
$$=\mid \Delta {T}_{0}-{\beta}_{2}\Omega \phantom{\rule{.2em}{0ex}}z\mid -2{T}_{p}\left(z\right)$$
(3)
$$i\frac{\partial \psi}{\partial z}=\frac{{\beta}_{2}}{2}\frac{{\partial}^{2}\psi}{\partial {T}^{2}}-\gamma {\mid \psi \mid}^{2}\psi +i\frac{g}{2}\psi .$$
(4)
$$\Delta {T}_{L}=-\Delta T\left(z=L\right)$$
(5)
$${\mid \psi \left(t\right)\mid}^{2}=2{\mid {A}_{p}\mid}^{2}\{1-\frac{1}{{T}_{p}^{2}}\left({t}^{2}+\frac{\Delta {T}^{2}}{4}\right)$$
(6)
$$+\mathrm{cos}\left(2\pi \phantom{\rule{.2em}{0ex}}{f}_{s}\phantom{\rule{.2em}{0ex}}\Delta T\right)\sqrt{1-{\left(\frac{t+\Delta T/2}{{T}_{p}}\right)}^{2}}\sqrt{1-{\left(\frac{t+\Delta T/2}{{T}_{p}}\right)}^{2}}\}$$
(7)
$${f}_{s}=\frac{1}{2\pi}\left(\Omega \phantom{\rule{.2em}{0ex}}+\phantom{\rule{.2em}{0ex}}{C}_{p}\phantom{\rule{.2em}{0ex}}\Delta T\right).$$
(8)
$$\{\begin{array}{c}i\frac{\partial {\psi}_{-}}{\partial z}=i\frac{g}{2}{\psi}_{-}+i\frac{\delta}{2}\frac{\partial {\psi}_{-}}{\partial t}+\frac{{\beta}_{2}}{2}\frac{{\partial}^{2}{\psi}_{-}}{\partial {t}^{2}}-\gamma \phantom{\rule{.2em}{0ex}}{\psi}_{-}\left({\mid {\psi}_{-}\mid}^{2}+2{\mid {\psi}_{+}\mid}^{2}\right)\\ i\frac{\partial {\psi}_{+}}{\partial z}=i\frac{g}{2}{\psi}_{+}-i\frac{\delta}{2}\frac{\partial {\psi}_{+}}{\partial t}+\frac{{\beta}_{2}}{2}\frac{{\partial}^{2}{\psi}_{+}}{\partial {t}^{2}}-\gamma \phantom{\rule{.2em}{0ex}}{\psi}_{+}\left({\mid {\psi}_{+}\mid}^{2}+2{\mid {\psi}_{-}\mid}^{2}\right)\end{array}$$
(9)
$$\psi \left(z,t\right)={\psi}_{+}\left(z,t\right)\mathrm{exp}\left(-\mathit{\Omega t}\mathit{/}\mathit{2}\right)+{\psi}_{-}\left(z,t\right)\mathrm{exp}\mathit{(}\mathit{\Omega t}\mathit{/}\mathit{2}\mathit{)}.$$