Bloch oscillations and resonant radiation of light propagating in arrays of nonlinear fibers with high-order dispersion

Bloch oscillations of spatiotemporal light wave packets in arrays of nonlinear fibers with high-order dispersion are studied. The light wave experiences discrete spatial diffraction along the waveguide array coordinate together with continuous temporal dispersion including higher-order terms. When a gradient in the waveguide light confinement strength is considered, the wave packet features robust long-lived Bloch oscillations with temporal and spatial spreading in the presence of a Kerr nonlinearity. The effect of spatial Bloch oscillations on the emission of the dispersive radiation by the solitary wave is analyzed. It is shown that Bloch oscillations result in the generation of new frequencies. The condition of resonant emission of dispersive waves is derived and it is demonstrated that it matches very well with the results of direct numerical simulations.

Figure 1

Dynamics of the field for quadratic dispersion (${\beta }_3=0$). The initial time shift of the pulse is $t_0=0$. The evolutions of $|{\psi }_n(t=0)|$ and $|{\psi }_{n=0}\left(t\right)|$ as function of the propagation distance $z$ are shown in (a) and (b) correspondingly. (c) Zoomed quasistationary oscillations of $|{\psi }_n(t=0)|$. (d) Field distribution $|{\psi }_n\left(t\right)|$ at $z=88$.

Figure 2

(a) Dynamics of the spectrum of ${\psi }_{n_c}\left(t\right)$ as function of the propagation distance $z$. At each propagation distance, $n_c\left(z\right)$ corresponds to the central point of the pulse and is defined as $n_c\left(z\right)=⌈\int {\sum }_n\left(\right|{\psi }_n{\left(t\right)|}^{2}n)dt/\int {\sum }_n|{\psi }_n\left(t\right){|}^{2}dt)-\frac{1}{2}⌉$. (b) Spatial-temporal spectrum of the field ${\psi }_n\left(t\right)$ at $z=88$. $q$ is the spatial wave vector corresponding to the $n$ axis. (c) Graphical solution of the resonance condition, Eq. (9), with $K_s=0.25$ where the value of $K_s$ is an estimate based on the numerical simulations. The horizontal green lines correspond to the resonances between different Bloch harmonics of the solitary pulse and the radiated wave. The spectra are calculated for the fields illustrated in Fig. 1.

Figure 3

Same as Fig. 1 but for ${\beta }_3=0.45$ and an initial time shift of $t_0=180$.

Figure 4

Same as Fig. 2 but for ${\beta }_3=0.45$ and an initial time shift of $t_0=180$.

Figure 5

Spatial-temporal spectra of the field at the propagation distance $z=88$ for different gradient strengths $\gamma$.

Figure 6

Dependencies of the spectral distances between the neighboring lines as functions of the parameter $\gamma$ defining the spatial frequency of the BOs. (a) Corresponds to the spectral line marked ${\omega }_1$ in Fig. 5. The dark solid line corresponds to the first-order approximation for the distance between neighboring lines. The red dashed line shows the second-order approximation for the distance to the neighboring line having lower frequency (the line is the Cherenkov radiation line and marked ${\omega }_{ch-1}$ in Fig. 5). The spectral distances extracted from direct numerical simulations are shown by red circles. The blue dashed line marked ${\mathrm{\Delta }}_{1-2}$ shows the second-order approximation for the distance to the neighboring spectral line having higher frequency (the line is marked ${\omega }_2$ in Fig. 5). (b) Same as (a), but for the line marked ${\omega }_2$ in Fig. 5. The curve ${\mathrm{\Delta }}_{1-2}$ and the circles are for the spectral distance to the neighboring line having lower frequency and the blue curve ${\mathrm{\Delta }}_{2-3}$ and the blue circles are for the neighboring spectral line having higher frequency (the line is marked ${\omega }_3$ in Fig. 5).

Figure 7

The evolution of ${\psi }_{n=n_c}\left(t\right)$ along time is shown in panel (a). Panel (b) shows the evolution of the temporal spectrum of ${\psi }_{n=n_c}\left(t\right)$. The field intensity and the spatial-temporal spectrum are shown in panels (c) and (d) for a propagation distance of $z=88$. The third-order dispersion is ${\beta }_3=0.45$, the wave vector of the initial envelope is $k_0=0$, and the amplitude of the initial pulse is $a_0=0.7$.

Figure 8

The same as Fig. 7 but for an initial field distribution with $k_0=\pi /2$.