Spatiotemporal quasisolitons and resonant radiation in arrays of silicon-on-insulator photonic wires

We have analyzed the conditions for low-power spatiotemporal soliton formation in arrays of evanescently coupled silicon-on-insulator photonic wires. We have verified that pronounced soliton effects can be observed in the presence of realistic loss, two-photon absorption, and higher-order dispersions. A soliton in an $N$-wire array can excite $N$ resonant frequencies, but some of these may be suppressed due to the soliton having zero projection onto the corresponding radiation supermodes. This results in pronounced differences between the radiation spectra observed from solitons excited at the edge and in the center of arrays.

Figure 1

(Color online) Symmetric (left) and antisymmetric (right) mode profiles, displayed over a $2.4\mu \mathrm{m}\times 1\mu \mathrm{m}$ cross section. Silica is beneath the horizontal line, with the rectangles denoting the silicon. The electric field vector is split into Cartesian components, with transverse components parallel to the silica-air interface shown top, transverse perpendicular components shown middle, and longitudinal components shown bottom. Color saturation gives the absolute value. The + and − signs (and blue and red regions where color is provided) denote the relative phase. (For clarity, the saturation of the middle figure has been doubled.)

Figure 2

(Color online) $1∕L_{\mathrm{D}}\left(\lambda \right)$ for the two waveguide widths. Anomalous (normal) dispersion is represented by solid (dashed) lines. The vertical bar highlights the $1.5\text{−}\mu \mathrm{m}$ pump wavelength. The leading (scaled) dispersion coefficients are $d_2=-0.5$, $d_3=-0.00326$, $d_4=0.00148$, and $d_5=-6.16\times {10}^{-5}$ for the $380\text{−}\mathrm{nm}$-wide wire and $d_2=-0.5$, $d_3=0.00615$, $d_4=7.00\times {10}^{-4}$, and $d_5=-4.06\times {10}^{-6}$ for the $420\text{−}\mathrm{nm}$-wide wire.

Figure 3

(Color online) $L_{\mathrm{C}}\left(\lambda \right)$ for the two waveguide widths. The vertical bar highlights the $1.5\text{−}\mu \mathrm{m}$ pump wavelength. The leading (scaled) coupling coefficients are $c_0=0.336$, $c_1=-0.106$, $c_2=0.0187$, and $c_3=-0.00213$ for the $380\text{−}\mathrm{nm}$-wide wire and $c_0=0.337$, $c_1=-0.0868$, $c_2=0.0124$, and $c_3=-0.00129$ for the $420\text{−}\mathrm{nm}$-wide wire.

Figure 4

(Color online) Normalized energy of edge and central solitons $f_N=U∕\sqrt{c_0}$, as a function of $V=q∕c_0$ shown with $N=\infty$ for both types of solitons and with $N=3$ and $N=5$ for edge and central solitons, respectively. The unstable $(\partial U∕\partial q<0)$ solitons are denoted by dashed lines. The energy unit $P_0{T}_0$ is typically in the vicinity of $100–1000\mathrm{fJ}$.

Figure 5

Effect of the array edges on the minimum soliton energy. To compare the cases of the central and edge solitons, the quantities are plotted against the number of wires separating the pump channel from each edge of the array for the central soliton case and from the far edge of the array for the edge soliton case. The energy unit $P_0{T}_0$ is typically in the vicinity of $100–1000\mathrm{fJ}$.

Figure 6

Parameter ${\alpha }_{\max }$ (defining maximum soliton duration) as a function of the number of wires between the maximum intensity wire and the array boundary as in Fig. 5.

Figure 7

Evolution of a sech-like pulse (with power $3.5P_0$) sent into a single wire. Center (a) and edge (b) wire excitation results in soliton formation. When nonlinearity is turned off (c) and (d), the pulses disperse and diffract. Power axis is in units of $P_0$. Propagation distance is $2.7\mathrm{mm}$. The model does not include linear loss or TPA.

Figure 8

$100\text{−}\mathrm{fs}$ pulse evolution in a $420\text{−}\mathrm{nm}$-wide wire over one dispersion length $\left(0.69\mathrm{mm}\right)$. At very low power (a) the pulse broadens. At $1.5P_0$ (b) a quasisoliton (with an approximate duration of $100\mathrm{fs}$) is formed. At $3.5P_0$ (c), the pulse is compressed to $34\mathrm{fs}$. Increasing the power to $7P_0$ (d) results in pulse splitting (soliton fission). Power axis is in units of $P_0$. The model includes linear loss and TPA.

Figure 9

Result of a $100\text{−}\mathrm{fs}$ pulse with peak power $3.5P_0$ being fired into the central wire $(n=8)$ of a 15-wire array (of $220\mathrm{nm}\times 420\mathrm{nm}$ wires placed $700\mathrm{nm}$ apart). Propagation distance is $2.7\mathrm{mm}$. With nonlinearity (a) the light is localized in the input wire, and broadening in time and space is suppressed, indicating quasisoliton formation. When nonlinerity is turned off (b) diffraction has transferred nearly all the light from the pump wire into nearest and next-nearest neighbors. Power axis is in units of $P_0$. The model includes linear losses and TPA.

Figure 10

The same as Fig. 9, but with the pulse fired into the edge wire $(n=1)$ of a 15-wire array. (a) shows the quasisoliton formation for $3.5P_0$ input power, and (b) shows linear diffraction. The model includes linear losses and TPA. Wires 8–15 (which contain almost no light) are omitted for clarity.

Figure 11

(Color online) Normalized eigenvectors for Eq. (10). Shown for (top to bottom) $N=3$, 4, 5, and 7. The eigenvectors are arranged by order of their corresponding eigenvalue, which increases from left to right. Modes with negative eigenvalues are shown in red (dark gray), while modes with positive eigenvalues are shown in blue (light gray).

Figure 12

(Color online) Resonance conditions for $220\mathrm{nm}\times 380\mathrm{nm}$ wires placed $700\mathrm{nm}$ apart for systems with (top, left) 3, (top, right) 5, (bottom, left) 7, and (bottom, right) $\infty$ wires. The soliton wave number $q=1.1$ is given by the horizontal dotted line. Antisymmetric modes are shown as dashed and symmetric modes as solid lines. Modes with negative eigenvalues are shown in red (dark gray) and those with positive eigenvalues in blue (light gray).

Figure 13

Power spectrum summed over all wires (after $2.4\text{−}\mathrm{mm}$ propagation, $\zeta =4$) for $N=3$ array of $220\mathrm{nm}\times 380\mathrm{nm}$ wires with $700\mathrm{nm}$ separation. Shown for center-wire input (top) and edge-wire input (bottom). As predicted, all three Čerenkov peaks are present for the edge soliton, but one of these is suppressed for the central soliton. The input pulse is taken as $\sqrt{3.5P_0}\mathrm{sech}\left(\tau \right)$.

Figure 14

The same as Fig. 13, but for $N=5$. All five Čerenkov peaks are present for the edge soliton, but two of them are suppressed for the central soliton.

Figure 15

The same as Figs. 13, 14, but for $N=7$. All seven Čerenkov peaks are present for the edge soliton, but three of them are suppressed for the central soliton. Peaks 1 and 2 cannot be resolved here, but by observing the modal profiles it can be shown that two resonances are indeed present.