Collision-mediated radiation due to Airy-soliton interaction in a nonlinear Kerr medium

We study the interaction of a copropagating finite-energy Airy pulse and a soliton in a Kerr medium under third-order dispersion. It is observed that a strong radiation appears when a self-accelerating Airy pulse collides with a delayed soliton in time domain. We confirm both analytically and numerically that this radiation can only be initiated in the environment of third-order dispersion (TOD). The radiation frequency is red or blue shifted depending on the numeric sign of TOD parameter. We adopt a simple Si-based waveguide to validate the normalized parameter that we use for the analysis. We develop a theory to explain the collision-mediated radiation and our theoretical results are found to be in good agreement with the direct numerical modeling of the generalized nonlinear Schrödinger equation.

Figure 1

(a) Temporal and (b) spectral distribution of the pulses at the launching point with $\mathrm{\Delta }\tau =15$. (c) Initial spectrum of a truncated Airy pulse (with $a=0.1$) and (d) soliton where $u_s=\text{sech}(\tau -\mathrm{\Delta }\tau )$.

Figure 2

(a) Temporal and (b) spectral dynamics of the total field with ${\delta }_3=0$. The temporal delay between the Airy pulse and soliton is $\mathrm{\Delta }\tau =15$. The value of $P$ is 1 and $a=0.1$. Note that at the collision point, ${\tau }_p\simeq \mathrm{\Delta }\tau$ and there is not any radiation in the spectral evolution.

Figure 3

XFROG spectrograms at different spatial positions (a) at initial point ($\xi =0$), (b) at the collision point ($\xi ={\xi }_c$), (c) after the collision point ($\xi =15$), and (d) at output $(\xi =25$).

Figure 4

(a) The block diagram and geometry of the proposed Si-based waveguide. The dimensions are $w_1=1500\mathrm{nm}$; $w_2=110\mathrm{nm}$; $w_3=230\mathrm{nm}$, and $h=410\mathrm{nm}$. The confinement of the fundamental TE mode at the operating wavelengths (${\lambda }_1=2425\mathrm{nm}$ and ${\lambda }_2=3200$ nm) are shown in the inset. (b) Dispersion profile of the waveguide. The operating wavelengths are indicated by the red solid dots. For ${\lambda }_1$ the TOD parameter is positive and for ${\lambda }_2$ TOD parameter is negative. Dynamics of a truncated Airy pulse for (c) ${\delta }_3>0(=0.08$), (d) without TOD (${\delta }_3=0$), and (e) negative ${\delta }_3<0(=-0.08$).

Figure 5

The (a) temporal and (b) spectral dynamics of the pulses with negative ${\delta }_3$ ($=-0.08$). The initial separation between the pulse $\mathrm{\Delta }\tau =15$. For airy pulse $a=0.25$.

Figure 6

The XFROG spectrograms at different distances. (a) At $\xi =0$ where two pulse are temporally separated ($\mathrm{\Delta }\tau =15$). (b) At $\xi =10$ ($\xi <{\xi }_c$) where only the soliton mediated weak radiation is visible. (c) At the collision point ($\xi ={\xi }_c$) where Airy pulse collide with soliton and leads to a strong radiation. (d) At output ($\xi ={\xi }_f$) where the coexistence of strong (indicated by the red dotted line) and weak radiation is evident. The value ${\delta }_3=-0.08$, and the truncation parameter $a=0.25$.

Figure 7

The spectral dynamics of the pulses for a fixed TOD parameter (${\delta }_3=-0.08$) with different temporal separations (a) $\mathrm{\Delta }\tau =10$, (b) $\mathrm{\Delta }\tau =15$, (c) $\mathrm{\Delta }\tau =20$, and (d) $\mathrm{\Delta }\tau =25$. The radiation is generated at different collision points based on the initial separation of the pulses. However, the spectral location of the radiation is hardly affected by delay $\mathrm{\Delta }\tau$.

Figure 8

(a) The spectral dynamics before the collision point ($\xi <{\xi }_c$) with ${\delta }_3=-0.08$. The spectral position of the weak radiation is verified by plotting the phase matching curve for the weak radiation. The final spectral distribution is inserted at the top of the figure. (b) The comparison of analytically and numerically found results for all the frequencies obtained due to the perturbations present in the system.

Figure 9

(a) Collision distance (${\xi }_c$) as a function of initial temporal separation ($\mathrm{\Delta }\tau$) between the pulses. The analytical formula (blue solid line) is verified by numerically data (red dots). (b) The variation of collision-mediated strong radiation frequency $\left({\omega }_s\right)$ with ${\xi }_c$ (for fixed ${\delta }_3=-0.08$). The radiation frequency ${\omega }_s$ hardly changes (around $1\%$) with ${\xi }_c$.

Figure 10

The (a) temporal and (b) spectral dynamics of the system for a fixed value of ${\delta }_3=0.04$. The final temporal and spectral distributions are attached with the figures. (c) The XFROG spectrogram of the system at the final spatial point. The patch of the strong radiation is evident on the blue side of the spectrum. (d) The comparison of analytical values of ${\omega }_s$ (blue solid line) and numerically found (red dots) values with respect to ${\delta }_3$.

Figure 11

The dynamics of the pulses for different TOD parameters in temporal [0.06 for (a) and 0.08 for (c)] and in spectral domain [0.06 for (b) and 0.08 for (d)]. The truncation parameter of the Airy pulse is $a=0.1$ and time separation of the pulses is 40 at the input. We can see that the position of collisions of two pulses is different for different TOD parameters.

Figure 12

The spectrograms of the pulses for different TOD parameters in spectral domain [0.06 for (a) and 0.08 for (b)]. The truncation parameter of the Airy pulse is $a=0.1$ and time separation of the pulses is 40 at the input. The patch of SR is indicated by the arrows in the figures.

Figure 13

The comparison between the analytical solution (solid line) and the numerical solutions (dots) of the inverted Airy pulse in presence of positive third-order dispersion with ${\delta }_3=0.06$ for (a) and ${\delta }_3=0.08$ for (b). The solutions are taken at a fixed distance ($\xi =40$).

Figure 14

(a) Variation of ${\omega }_s$ with TOD parameter ${\delta }_3$. The theoretical result (solid blue line) agrees well with numerical data (dots). (b) ${\omega }_s$ as a function of ${\xi }_c^{\prime }$. ${\omega }_s$ is hardly affected by point of collision.