Localization and time-reversal of light through dynamic modulation

We study the dynamics of a waveguide made of coupled resonators with a sinusoidal modulation of the resonance frequencies. We present a modulation scheme that achieves complete dynamic localization and is experimentally suitable for optical cavities. Furthermore, we highlight the importance of the way the modulation is turned on and off. One striking consequence is that, although a state returns to its starting amplitude at the end of every cycle, it can also be fully time-reversed if the modulation is turned off midcycle. Finally, we show that localization is always achieved when the modulation envelope is adiabatic with respect to the oscillation frequency. The results are experimentally feasible using existing integrated photonic technologies and are relevant to applications, such as tunable delay lines, dispersion compensation, and imaging.

Figure 1

(a) Schematic of the system of coupled resonators with a coupling constant $J$ and a time-dependent resonance frequency of ${\omega }_0+{\omega }_n\left(t\right)$. We set the distance between resonators to 1 and furthermore consider a spatially periodic modulation with period $a=2$. (b) Band diagram of the Floquet quasienergies for ${\omega }_n\left(t\right)=A_{0}cos(\mathrm{\Omega }t+n\pi )$, for $J=0.1\mathrm{\Omega }$, and for several different values of $A_0$. (c) The same as (b) for $J=0.4\mathrm{\Omega }$. The dashed lines illustrate the $A_0=0$ dispersion folded into the temporal Brillouin zone.

Figure 2

Wave-packet propagation ${\left|\psi (x,t)\right|}^2$ inside the CCW with $J=0.1\mathrm{\Omega }$ with a modulation ${\omega }_n=A_{0}sin(\mathrm{\Omega }t+n\pi ),A_0=1.2\mathrm{\Omega }$, turned on and off at times $10T$ and $20T$, respectively. (a) Real-space propagation. (b) Resonance frequency vs time for the cavities at positions $x=0$ and $x=1$. (c) Field intensity in the cavity at $x=0$ for a pulse propagating in an unmodulated (red) and a modulated (blue) chain.

Figure 3

(a) Fourier-space-time evolution ${\left|\psi (k,t)\right|}^2$ for the wave packet and modulation scheme of Fig. 2. (b) Time evolution of a starting state $\psi (k,0)=1$ for $k=[0,\pi ],\psi (k,0)=0$ otherwise, under the modulation ${\omega }_n\left(t\right)=A_{0}sin(\mathrm{\Omega }t+n\pi )$ for $2A_0/\mathrm{\Omega }=2.4$. (c) The same as in (b) for $2A_0/\mathrm{\Omega }=4$.

Figure 4

(a) Propagation of the same starting wave packet as in Fig. 2 but with the modulation applied in the interval $20T<t<25.3T$ as shown in (b). (c) Light from a Gaussian source at $x=0$ under a localizing modulation that is adiabatically turned on and off as shown in (d) where the black line is the envelope function $A\left(t\right)$. (e) and (f) The same as in (c) and (d), but the width $t_w$ of Eq. (10) is ten times shorter.