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Supercontinuum generation in nonlinear optical fibers

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

Cooperation with: M. Kroh, H.G. Weber (Fraunhofer-Institut für Nachrichtentechnik, Heinrich-Hertz-Institut (HHI) Berlin)

Supported by: Terabit Optics Berlin (project B4)

Description:

The propagation of short and intense optical pulses through a nonlinear optical fiber is described by the nonlinear Schrödinger equation (NLSE). Our model uses the following form of the NLSE for the complex envelope A(z,$\tau$) of an optical pulse which includes group velocity dispersion up to fourth order as well as self-steepening and Raman scattering:

$${\frac{{\partial A}}{{\partial z}}} {\frac{{\rm i}}{{2}}} \beta_{2}^{} {\frac{{\partial^2 A}}{{\partial \tau^2}}} {\frac{{1}}{{6}}} \beta_{3}^{} {\frac{{\partial^3 A}}{{\partial \tau^3}}} {\frac{{\rm i}}{{24}}} \beta_{4}^{} {\frac{{\partial^4 A}}{{\partial \tau^4}}} {\frac{{1}}{{2}}} \alpha$$ =

$$\gamma {\frac{{\gamma}}{{\omega_0}}} {\frac{{\partial}}{{\partial \tau}}} \left(\vphantom{ \vert A\vert^2 A}\right. \left.\vphantom{ \vert A\vert^2 A}\right) \gamma {\frac{{\partial}}{{\partial \tau}}} \left(\vphantom{ \vert A\vert^2}\right. \left.\vphantom{ \vert A\vert^2}\right)$$ (1)

The details of the derivation of this equation are given in [1]. The numerical investigation of Eq. 1 in view of the supercontinuum (SC) generation [3] requires a high accuracy as well as a high resolution. We use a de-aliased pseudospectral method with integration performed by an eighth-order Runge-Kutta integration scheme using adaptive stepsize control [1,2] which is more precise than the widely-used split-step method. Experimental and numerical investigations [3,4,5] show the importance of higher-order dispersion terms. It turns out to be highly questionable whether the use of even higher-order dispersion terms resulting from a Taylor expansion in the Fourier domain is reasonable. This led us to a review of the modeling process to come up with more general equations. Concerning nonlinear Schrödinger equations, one may start with the following scheme:

$$\mathcal {N} \mathcal {L}$$ (2)

where u : $\IR$ x $\IR^{+}_{}$$\to$$\IC$, ($\tau$, z) $\mapsto$ u($\tau$, z). The linear operator $\mathcal {L}$ describing the dispersion is determined by the dispersion relation $\omega$($\xi$) in the Fourier domain as follows: $\widehat{{\mathcal{L}(u)}}$($\xi$) = $\omega$($\xi$)$\widehat{{u}}$($\xi$). In the case of $\omega$ being a polynomial (e.g., from a Taylor expansion), $\mathcal {L}$ is a differential operator and the equation is therefore local. Choosing non-polynomial functions for $\omega$ leads to pseudo-differential operators and nonlocal equations on which we focused our analysis. Similar to that it is worth to ask for generalizations of the nonlinearity $\mathcal {N}$(u). We have investigated the case $\mathcal {N}$(u)($\tau$, z) = u($\tau$, z)$\left(\vphantom{R(\tau)*\vert u(\tau,z)\vert^2}\right.$R($\tau$)*| u($\tau$, z)|²$\left.\vphantom{R(\tau)*\vert u(\tau,z)\vert^2}\right)$, which is suggested by the physical model. The convolution with R describes the response of the material and we have R($\tau$) = $\delta_{\tau}^{}$ in the standard case where only instantaneous response is considered ( $\mathcal {N}$(u) = u| u|²).

On the other hand, we validated our algorithm by the substantial example of supercontinuum generation [4,5], which apparently comprises all physical effects modeled by Eq. 1. We demonstrate in [3] that the modulation instability (MI), which is a special four-wave-mixing process, can be responsible for ultrabroadband octave-spanning continua for pico- and subpicosecond pulses in the anomalous as well as in the normal dispersion region. We illustrate the effect of MI in Figure 1.

Fig. 1: Spectra of an initial sech pulse with $\tau_{0}^{}$ = 1ps with a peak input power P₀ = 400 W and T_R = 3 fs at different propagation distances. The initial spectrum has a sech profile (black). After 0.3 m (green), the MI generates two sidebands at $\mp$50 THz (Stokes and anti-Stokes components), which are multiplied after 0.5 m (blue). Further phase-matched four-wave mixing then explosively excites new frequencies and hence broadens and completes the spectrum afterwards (red).

The MI can dominate higher-order effects such as third- and fourth-order dispersion, self-steepening, and Raman scattering, because it acts directly from the beginning on the high-order solitons. Therefore higher-order effects are not a prerequisite for the generation of SC, and it is hence not restricted only to ultrashort subpicosecond pulses, but appears for much longer pulses, too. The MI can also appear in the normal dispersion regime $\beta_{2}^{}$ > 0, if the fourth-order dispersion coefficient $\beta_{4}^{}$ is negative. In particular, we have verified quantitatively the experimental results in [4], where the MI sidebands are observed in the normal dispersion regime with negative $\beta_{4}^{}$. For these calculations a resolution of 2¹⁹ frequencies is required. The calculated wavelenghts of the Stokes ($\lambda_{s}^{}$) and anti-Stokes ($\lambda_{a}^{}$) components are represented in the following table:

P₀ [W] 100 150 200 250 300 350 400

$$\lambda_{a}^{}$$ 503.84 502.9 502.26 501.37 501.45 501.51 501.48

$$\lambda_{s}^{}$$ 903.79 906.84 907.97 911.84 911.75 911.40 911.49

in dependence on the optical input power P₀.

In applications, the effect of spectral broadening of a pulse propagating through a fiber is used for pulse compression. A schematic of the experimental setup is shown in Figure 2. A highly nonlinear fiber (HNLF) is used to induce a linear chirp in the spectral broadening process and the chirped pulses are compressed by dispersion compensation in a subsequent standard single mode fiber (SMF). Applications require flat and wideband spectra without phase instabilities. This is achieved by the self-phase modulation (i$\gamma$| A|²A) in Eq. 1 as the main broadening mechanism. The corresponding spectral broadening is confirmed by the measurements. One example is shown in Figure 3.

The experimental investigations at the HHI as well as our numerical simulations indicate an optimal pulse compressor in the vicinity of the zero-dispersion wavelength in the normal dispersion regime leading to a linear up-chirp across the entire pulse width.

Fig. 2: Upper: Experimental setup used in the HHI for pulse compression using a HNLF.
Lower left: measured pulse shape (black: before compression, blue: after compression).
Lower right: measured pulse spectrum (black: before compression, blue: after compression).
We acknowlegde the permission by Prof. Weber and M. Kroh (HHI) to show these figures.

We have simulated the spectral broadening for realistic fibers and have compared our results with measurements, results are plotted in Figure 3. A typical problem for such investigations is that the fiber coefficients of a realistic fiber are not exactly known. Therefore, we first used measurements at low optical power for a proper adjustment of the dispersion coefficients. The resulting spectrum is drawn in Figure 3, left. Then we used the simulation parameters obtained in this way for further simulations with higher optical input power. For medium input power (20.0 dBm sech input) we again obtained good agreement with the experiments. Furthermore, we could reconstruct the measured pulse compression down to 0.4 ps. We obtained good agreement with the measurements for even higher input power. Surprisingly, no further pulse compression could be achieved by increasing the input power, which is due to the occurrence of other disturbing nonlinearities. The latter can now be studied with our tools in order to design a proper HNLF.

Fig. 3: Measured optical spectra (black, HHI, for a train of identical pulses) and calculated optical spectra for a single pulse (red, based on Eq. 1) after propagation over 510 meters along a HNLF; left: low input power, right: high input power.

References:

1. U. BANDELOW, A. DEMIRCAN, M. KESTING, Simulation of pulse propagation in nonlinear optical fibers, WIAS Report no. 23, 2003 .

2. N. SEEHAFER, A. DEMIRCAN, Dynamo action in cellular pattern, Magnetohydrodynamics, 39(3) (2003), pp. 335-344.

3. A. DEMIRCAN, U. BANDELOW, Supercontinuum generation by the modulation instability, Optics Communications, 244(6) (2005), pp. 181-185.

4. J.D. HARVEY, R. LEONHARDT, S. COEN, G.K.L. WONG, J.C. KNIGHT, W.J. WADSWORTH, P.S.J. RUSSEL, Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber, Optics Letters, 28(22) (2003), pp. 2225-2227.

5. W.J. WADSWORTH, N. JOLY, J.C. KNIGHT, T.A. BIRKS, F. BIANCALANA, P.S.J. RUSSEL, Supercontinuum and four-wave mixing with q-switched pulses in endlessly single-mode photonic crystal fibres, Optics Express, 12(2) (2004), pp. 299-309.

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