## Abstract

Parametric amplification in fibers with dispersion fluctuations is analyzed. The fluctuations are modelled as a stochastic process, with their size at a given position modelled as a Gaussian, and the autocorrelation decreasing exponentially. Two models are studied: in one the dispersion is piecewise constant, while in the other it is continuous. We find that the amplification does not depend on the models’ details and that only fluctuations with long correlation lengths affect the amplification significantly.

©2004 Optical Society of America

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### Equations (10)

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(1)
$$\Delta \beta =2{\beta}_{p}-{\beta}_{s}-{\beta}_{i},$$
(2)
$${A}_{p}=\sqrt{{P}_{0}}\mathrm{exp}\left(i\gamma {P}_{0}z\right),$$
(3)
$$\frac{d\mathbf{A}}{\mathit{dz}}=\left(\begin{array}{cc}i\Delta k& i\gamma {P}_{0}\\ -i\gamma {P}_{0}& -i\Delta k\end{array}\right)\phantom{\rule{.2em}{0ex}}\mathbf{A}.$$
(4)
$$\mathbf{A}\left(z\right)=\mathbf{MA}\left(0\right)\equiv \left[\begin{array}{cc}\mathrm{cosh}\alpha z+\frac{i\Delta k}{\alpha}\mathrm{sinh}\alpha z& \frac{i\gamma {P}_{0}}{\alpha}\mathrm{sinh}\alpha z\\ -\frac{i\gamma {P}_{0}}{\alpha}\mathrm{sinh}\alpha z& \mathrm{cosh}\alpha z-\frac{i\Delta k}{\alpha}\mathrm{sinh}\alpha z\end{array}\right]\phantom{\rule{.2em}{0ex}}\mathbf{A}\left(0\right),$$
(5)
$$G={\mathrm{cosh}}^{2}\alpha L+\frac{{\left(\Delta k\right)}^{2}}{{\alpha}^{2}}{\mathrm{sinh}}^{2}\alpha L,$$
(6)
$${f}_{G}\left(\delta k\right)=\frac{1}{\sigma \sqrt{2\pi}}\mathrm{exp}\left[-\frac{1}{2}{\left(\frac{\delta k}{{\sigma}^{2}}\right)}^{2}\right],$$
(7)
$$C\left(z\right)={\sigma}^{2}\mathrm{exp}\left(-\mid z\mid \u2044{L}_{c}\right),$$
(8)
$$P\left({L}_{s}\right)\propto {e}^{-{L}_{s}\u2044{L}_{c}}.$$
(9)
$$f\left(\delta k\left({z}_{1}\right)\mid \delta k\left({z}_{0}\right)\right)=\frac{1}{\sigma \sqrt{2\pi \left(1-{r}^{2}\right)}}{e}^{-\frac{1}{2{\sigma}^{2}\left(1-{r}^{2}\right)}{\left(\delta k\left({z}_{1}\right)-r\delta k\left({z}_{0}\right)\right)}^{2}},$$
(10)
$$G\left({L}_{c}\gg L\right)=\frac{1}{\sqrt{2\pi}\sigma}{\int}_{-\infty}^{+\infty}d(\Delta k){e}^{-{\left(\Delta k-\Delta \right)}^{2}\u20442{\sigma}^{2}}\mathrm{log}\left[{\mathrm{cosh}}^{2}\alpha L+\frac{{\left(\Delta k\right)}^{2}}{{\alpha}^{2}}{\mathrm{sinh}}^{2}\alpha L\right].$$