Soliton Pulse Propagation in the Presence of Disorder-Induced Multiple Scattering in Photonic Crystal Waveguides

We introduce a new coupled mode theory to model nonlinear Schrödinger equations for counterpropagating Bloch modes that include disorder-induced multiple scattering effects on nonlinear soliton propagation in photonic crystal waveguides. We also derive subunit-cell coupling coefficients and use these to introduce a generalized length scale associated with each coupling effect. In particular, we define a multiple-scattering length scale that quantifies the spatial extent of a disorder-induced cavity mode. Our numerical simulations of nonlinear pulse propagation are in excellent qualitative agreement with recent experiments and provide insight into how structural disorder inhibits soliton propagation and other nonlinear propagation effects in photonic crystal waveguides.

Figure 1

Simulated spatial profiles at various times of a soliton injected from the left side of a W1 PCW in the absence (left) and presence (right) of disorder-induced multiple scattering and localization. For comparing the two schematics, the two leftmost pulses (blue) are of the same magnitude.

Figure 2

Dispersion, group index (${\beta }_1$), and GVD (${\beta }_2$) characteristics of the W1 PCW with markers indicating the three values considered in this work.

Figure 3

Unchirped Gaussian pulse propagating in a W1 with ${\beta }_1=24.69$, ${\beta }_2=-9.744{\mathrm{ps}}^2/\mathrm{mm}$, $S=2.4$ and no disorder. Top: temporal snapshots in space taken at the beginning ($x=0$) and end of the PCW ($x=1$) showing temporal pulse compression. Bottom: spectral snapshots showing slight spectral broadening and fine structure characteristic of large anomalous GVD.

Figure 4

Using the same dispersion parameters and initial conditions as in Fig. 3, but now in the presence of only disorder-induced multiple scattering with rms roughness $\sigma =0.017a$. The rise of a backwards traveling pulse due to multiple scattering is also shown (pink,light). Top: temporal snapshots showing pulse degradation. Bottom: spectral snapshots showing strong backreflection and weak transmission.

Figure 5

Transmission (left) and reflection (right) spectra of an unchirped Gaussian pulse propagating in the presence of coupling to the counterpropagating mode via multiple scattering, SPM and XPM, for three different group indices: 8.093 (blue,bottom), 13.782 (green,middle) and 24.69 (red,top). The PCW is $251$ unit cells long with rms roughness fixed at $0.017a$. All curves are on the same scale.

Figure 6

For ${\beta }_1=24.69$ temporal and spectral profiles for the forward (blue, dark-solid) and the backward (pink, light-solid) mode envelopes at two different points inside the PCW. Strong multiple scattering dominates over SPM/XPM effects leading to sharp spectral peaks corresponding to the formation of localized modes.