## Abstract

It is shown that the dispersion-managed nonlinear pulse solutions can be viewed as nonlinear Bloch waves with a periodic scattering potential that is set up self-consistently by the wave itself. The pulses are shown to be chirp-free at the center of each dispersion segment. The essential physical mechanism is explained by the interaction of the $m=0$ and the $m=2$ Hermite–Gaussian components of the pulse.

© 1999 Optical Society of America

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### Equations (11)

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(1)
$$\frac{\partial u}{\partial z}=j\frac{{k}^{\u2033}(z)}{2}\frac{{\partial}^{2}}{\partial {t}^{2}}u-j\delta |u{|}^{2}u.$$
(2)
$${\psi}_{m}^{(\pm )}(t,z)={\left(\frac{\mathit{jb}}{\pm z+\mathit{jb}}\right)}^{1/2}{H}_{m}\left(\frac{t}{{\tau}_{\pm}}\right)\times exp\left(\frac{-{\mathit{jt}}^{2}}{\pm z+\mathit{jb}}\right)exp[\pm \mathit{jm}\varphi (z)],$$
(3)
$${\tau}_{\pm}^{2}={\tau}_{o\pm}^{2}\left(1+\frac{{z}^{2}}{{b}^{2}}\right);\hspace{1em}b=\frac{{\tau}_{o\pm}^{2}}{|{k}^{\u2033}\mp \mathrm{\Delta}{k}^{\u2033}|};\hspace{1em}\theta ={tan}^{-1}\left(\frac{z}{b}\right),$$
(4)
$$\sum _{n}{a}_{n}^{-}(L/2){\psi}_{n}^{-}(t,L/2)=\sum _{m}{a}_{m}^{+}(-L/2){\psi}_{m}^{+}(t,-L/2).$$
(5)
$${a}_{m}^{+}(-L/2)=\sum _{n}{a}_{n}^{-}(L/2)\int \mathrm{d}t\times {\psi}_{m}^{+*}(-L/2){\psi}_{n}^{-}(L/2)/\int \mathrm{d}t|{\psi}_{m}^{+}(-L/2){|}^{2}.$$
(6)
$$u(t,z)=\sum _{m-\mathrm{even}}{a}_{m}^{(\pm )}(z){\psi}_{m}^{(\pm )}(t,z)$$
(7)
$$\frac{\mathrm{d}}{\mathrm{d}z}{a}_{m}^{(\pm )}(z)=-j\delta \sum _{p,q,r-\mathrm{even}}{c}_{\mathit{mpqr}}^{(\pm )}(z){a}_{p}^{(\pm )*}(z){a}_{q}^{(\pm )}(z){a}_{r}^{(\pm )}(z),$$
(8)
$${c}_{\mathit{mpqr}}^{(\pm )}(z)=\int \mathrm{d}t{\psi}_{m}^{(\pm )*}(t,z){\psi}_{p}^{(\pm )*}(t,z){\psi}_{q}^{(\pm )}(t,z)$$
(9)
$$\times {\psi}_{r}^{(\pm )}(t,z)/\int \mathrm{d}t|{\psi}_{m}^{(\pm )}(t,z){|}^{2}$$
(10)
$$=\frac{exp[\pm j(q+r-m-p)\varphi (z)]}{[1+(z/b{)}^{2}{]}^{1/2}}\int \mathrm{d}{\mathit{xH}}_{m}(x)$$
(11)
$$\times {H}_{p}(x){H}_{q}(x){H}_{r}(x)/\int \mathrm{d}x|{H}_{m}(x){|}^{2}.$$