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Simulation of pulse propagation in nonlinear optical fibers

Collaborator: U. Bandelow, A. Demircan, M. Kesting

Cooperation with: Fraunhofer-Institut für Nachrichtentechnik, Heinrich-Hertz-Institut (HHI), Berlin

Supported by: Terabit Optics Berlin (project B4)

Description: Wave propagation in dispersive nonlinear media has become a topic of intense research activities, in part stimulated by its potential application to optical fiber communication systems. Propagation of optical pulses in monomode optical fibers is mainly influenced by the group velocity dispersion and the refractive index nonlinearity. The propagation of sub-picosecond optical pulses is governed by a generalized nonlinear Schrödinger equation (NLSE) (1), which can be derived from the underlying Maxwell equations within the slowly varying envelope approximation (see [1]). This means that the pulse envelope A(z, t) modulating the underlying carrier wave exp[i(k0z - $ \omega_{0}^{}$t)] is assumed to be slowly varying in time and space. The pulse width has to be much longer than the carrier oscillation period and the spectral content of the field has to be narrower than the carrier frequency $ \omega_{0}^{}$ itself. This is satisfied for optical pulses with widths down to 10 fs.

The general form of the NLSE for the complex envelope A(z,$ \tau$) of a pulse is given by

$\displaystyle {\frac{{\partial A}}{{\partial z}}}$ = - $\displaystyle {\frac{{i}}{{2}}}$$\displaystyle \beta_{2}^{}$$\displaystyle {\frac{{\partial^2 A}}{{\partial \tau^2}}}$ + $\displaystyle {\frac{{1}}{{6}}}$$\displaystyle \beta_{3}^{}$$\displaystyle {\frac{{\partial^3 A}}{{\partial \tau^3}}}$ + $\displaystyle {\frac{{i}}{{24}}}$$\displaystyle \beta_{4}^{}$$\displaystyle {\frac{{\partial^4 A}}{{\partial \tau^4}}}$ - $\displaystyle {\frac{{1}}{{2}}}$$\displaystyle \alpha$A  
    + i$\displaystyle \gamma$| A|2A - a1$\displaystyle {\frac{{\partial}}{{\partial \tau}}}$$\displaystyle \left(\vphantom{ \vert A\vert^2 A}\right.$| A|2A$\displaystyle \left.\vphantom{ \vert A\vert^2 A}\right)$ - ia2A$\displaystyle {\frac{{\partial}}{{\partial \tau}}}$$\displaystyle \left(\vphantom{ \vert A\vert^2}\right.$| A|2$\displaystyle \left.\vphantom{ \vert A\vert^2}\right)$ , (1)

where the initial value problem A(0,$ \tau$) = f ($ \tau$) along z within a retarded time frame $ \tau$ = t - z/vg has to be solved. The linear terms on the right-hand side of Eq. (1) are the group velocity dispersion (GVD), namely second-order (SOD), third-order (TOD) and fourth-order dispersion (FOD) and the attenuation term corresponding to the fiber loss $ \alpha$. The main contribution to the group velocity dispersion is represented by the parameter $ \beta_{2}^{}$, which leads in general to a broadening of the pulse shape. TOD and FOD are higher-order effects originating from the wavelength dependence of the group velocity dispersion. These dispersive effects can distort ultrashort optical pulses in the linear as well as in nonlinear regimes. Another important fiber parameter is the measure of power loss during the transmission of optical signals inside the fiber, given by the attenuation constant $ \alpha$.

The first nonlinear term represents the self-phase modulation (SPM), which results from the intensity dependence of the refractive index. It is responsible for a large variety of phenomena, such as spectral broadening or optical solitons. The term proportional to a1 results from the intensity dependence of the group velocity and causes self-steepening and shock formation at the pulse edge. The last term considers the intrapulse Raman scattering and originates from the delayed response, which causes a self-frequency shift. a2 = $ \gamma$TR, where TR is related to the slope of the Raman gain. The intrapulse Raman scattering becomes a dominant perturbation for ultrashort pulses and is one of the most important limitations for ultrashort pulse propagation in optical fibers.

In [1] we have derived the NLSE (1) from the Maxwell equations in a non-standard way, showing that the neglect of the 2nd derivative with respect to z is no approximation as often claimed in the literature. In general a numerical approach is needed for an investigation of the generalized NLSE. For the numerical solution we have developed a code based on a standard dealiased pseudospectral method with a Runge-Kutta integration scheme ([1]). This scheme differs from the most commonly used split-step Fourier method and guarantees a higher accuracy, because no further approximations to (1) are used. Using our code we have investigated the impact of the various terms in (1) separately as well as their interplay, where we could reproduce previous analytical as well as experimental data ([1]). As the most prominent application we have studied the propagation of optical solitons. As an example the temporal evolution of a third-order soliton over one soliton period is drawn in Figure 1 (left). Perturbations caused by self-steepening and intrapulse Raman scattering break the degeneracy of solitons. The higher-order solitons decompose then into their constituents, which propagate at different speed. In the case of the intrapulse Raman scattering the low-intensity pulse is advanced, Figure 1 (right), whereas in the case of self-steepening both pulses are delayed.

Moreover, we have investigated the phenomenon of supercontinuum (SC) generation, where ultrabroad optical spectra are generated during the propagation of femto- or high-power picosecond pulses through dispersive nonlinear media ([2]). We have demonstrated that the primary mechanism for SC generation is the modulation instability (MI), accompanied by four-wave mixing. Higher-order effects, such as the self-steepening effect and the Raman effect were shown to be of minor influence on the generation of broad spectra. Raman scattering affects mainly the shape of the spectra. Because higher-order effects are not a prerequisite for the generation of SC, it is not restricted only to ultrashort sub-picosecond pulses. The MI enhances when higher-order dispersive terms are present, such that it can appear also in the normal dispersion regime ($ \beta_{2}^{}$ > 0), if the fourth-order dispersion coefficient $ \beta_{4}^{}$ is negative.


Fig. 1: Spatio-temporal evolution of a 3rd-order soliton in an optical fiber (left) and of a soliton decay induced by Raman scattering (right)
\makeatletter \@ZweiProjektbilderNocap[h]{.4\textwidth}{3d-soliton_ho_n_3_l.eps}{3d-soliton_tr-time_l.eps} \makeatother

References:

  1. U. BANDELOW, A. DEMIRCAN, M. KESTING, Simulation of pulse propagation in nonlinear optical fibers, WIAS Report no. 23, 2003 .
  2. A. DEMIRCAN, U. BANDELOW, Supercontinuum generation by the modulation instability, WIAS Preprint no. 881, 2003 .


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2004-08-13