## Abstract

We develop an intuitive approach for studying propagation of optical pulses through nonlinear dispersive media. Our new approach is based on the impulse response of linear systems, but we extend the impulse response function using a self-consistent time-transformation approach so that it can be applied to nonlinear media as well. Numerical calculations based on our new approach show excellent agreement with the generalized nonlinear Schrödinger equation in the specific case of the Kerr nonlinearity in both the normal and anomalous dispersion regimes. An important feature of our approach is that it works directly with the electric field associated with an optical pulse and can be applied to pulses of arbitrary width. Numerical calculations performed using single-cycle optical pulses show that our results agree with those obtained with the finite-difference time-domain technique using considerably more computing resources.

© 2012 Optical Society of America

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### Equations (15)

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(1)
$$y(t)={\int}_{-\infty}^{\infty}h(t,{t}^{\prime})x({t}^{\prime})\mathrm{d}{t}^{\prime},$$
(2)
$$\tilde{y}(\omega )=H(\omega )\tilde{x}(\omega ),$$
(3)
$${E}_{\text{out}}(t)={\int}_{-\infty}^{\infty}h(t-{t}^{\prime}){E}_{\text{in}}({t}^{\prime})\mathrm{d}{t}^{\prime}.$$
(4)
$${\tilde{E}}_{\text{out}}(\omega )=\mathrm{exp}[i\beta (\omega )L]{\tilde{E}}_{\text{in}}(\omega ),$$
(5)
$$h(t)=\frac{1}{2\pi}{\int}_{-\infty}^{\infty}\mathrm{exp}[i\beta (\omega )L-i\omega t]\mathrm{d}\omega .$$
(6)
$$h(t)=\delta (t-{T}_{r}),$$
(7)
$$\beta (\omega )\approx {\beta}_{0}+{\beta}_{1}(\omega -{\omega}_{0})+\frac{1}{2}{\beta}_{2}{(\omega -{\omega}_{0})}^{2},$$
(8)
$$h(t)=\sqrt{\frac{i}{2\pi {\beta}_{2}L}}\text{\hspace{0.17em}}\mathrm{exp}[\frac{{(t-{T}_{r})}^{2}}{2i{\beta}_{2}L}-i{\omega}_{0}(t-{T}_{r})],$$
(9)
$$h(t,{t}^{\prime})=\sqrt{\frac{i}{2\pi {\beta}_{2}L}}\phantom{\rule{0ex}{0ex}}\times \mathrm{exp}[\frac{{(t-{t}^{\prime}-{T}_{r}-{T}_{\text{nl}})}^{2}}{2i{\beta}_{2}L}-i{\omega}_{0}(t-{t}^{\prime}-{T}_{r}-{T}_{\text{nl}})].$$
(10)
$${t}_{1}=F({t}^{\prime})={t}^{\prime}+{T}_{r}+{T}_{\text{nl}}({t}^{\prime}).$$
(11)
$${E}_{\text{out}}(t)={\int}_{-\infty}^{\infty}h(t-{t}_{1}){E}^{\prime}({t}_{1})J({t}_{1})\mathrm{d}{t}_{1},$$
(12)
$$J({t}_{1})=d{t}^{\prime}/d{t}_{1}={(1+d{T}_{\text{nl}}/d{t}^{\prime})}^{-1}$$
(13)
$$E(z,t)=\mathrm{Re}[A(z,t-z/{v}_{g})\mathrm{exp}(i{\beta}_{0}z-{\omega}_{0}t)].$$
(14)
$$i\frac{\partial A}{\partial \xi}+\frac{1}{2}\frac{{\partial}^{2}A}{\partial {\tau}^{2}}+{\beta}_{0}{n}_{2}{L}_{D}{|A|}^{2}A=0,$$
(15)
$$i\frac{\partial A}{\partial \xi}+\frac{1}{2}\frac{{\partial}^{2}A}{\partial {\tau}^{2}}+i{\delta}_{3}\frac{{\partial}^{3}A}{\partial {\tau}^{3}}+{\beta}_{0}{n}_{2}{L}_{D}[{|A|}^{2}A+is\frac{\partial}{\partial \tau}({|A|}^{2}A)]=0,$$