Controlling the Flow of Light Using the Inhomogeneous Effective Gauge Field that Emerges from Dynamic Modulation

We show that the effective gauge field for photons provides a versatile platform for controlling the flow of light. As an example we consider a photonic resonator lattice where the coupling strength between nearest neighbor resonators are harmonically modulated. By choosing different spatial distributions of the modulation phases, and hence imposing different inhomogeneous effective magnetic field configurations, we numerically demonstrate a wide variety of propagation effects including negative refraction, one-way mirror, and on- and off-axis focusing. Since the effective gauge field is imposed dynamically after a structure is constructed, our work points to the importance of the temporal degree of freedom for controlling the spatial flow of light.

Figure 1

A resonator lattice consisting of two kinds of resonators (red and blue dots) with dynamically modulated nearest neighbor coupling. The lattice is divided into two regions: the left region has an effective gauge potential $A_y=0$ and the right region has an effective gauge potential $A_y=\varphi (y)/a$. Inset: the phase of the modulated coupling between nearest neighbor resonators corresponds to the rf wave that generates the modulation, and can be externally set by rf generators or rf phase shifters.

Figure 2

(a) Analysis of beam refraction for the structure in Fig. 1, for $\varphi (y)\equiv \varphi$. Circles are the constant quasienergy contours in the left and right regions, respectively. $k_{1,2}$ and $v_{1,2}$ are the incident and refracted momentum and group velocity of the beam. $k_1$ and $k_2$ have equal parallel components. $k_2^{\prime }$ and $v_2^{\prime }$ are the time reversal of the refracted beam that corresponds to $k_2$. (b) Simulation for the case of $\varphi =0.05$ showing positive refraction. (c) Simulation for the case of $\varphi =2$ demonstrating negative refraction.

Figure 3

Demonstration of the interface of the two regions in Fig. 1 functioning as a circulator. The right region has $\varphi =2$. Arrows indicate the incident direction of the beams.

Figure 4

A collimated beam is focused on axis (a) and off axis (b) by designing $\varphi (y)$ according to Eq. (5) in the right region of Fig. 1.