Frequency Domain Berry Curvature Effect on Time Refraction

Abstract

We demonstrate that there exist frequency domain Berry curvature in the wave function of photons in dispersive optical systems. This property arises from the frequency dispersion of its dielectric function, which makes Maxwell equations a non-standard eigenvalue equation, with the eigenvalue (frequency) appearing inside the operator itself. We study this new Berry curvature effect on time refraction of magnetoplasmon-polariton as an example. It can induce deflection in the trajectory of a photon and make the ray swing.

Figures (4)

• Figure 1: (a) The temporal modulation $\xi$ is applied at $t=0$ and the pulse's velocity acquires an anomalous component that is perpendicular to the group velocity. (b) $\xi$ is removed at $t=t_{f}$ and the pulse loses the anomalous velocity immediately.
• Figure 2: (a) Group velocity $\mathbf{v}_{g}$. (b) Berry curvature $\Omega_{\mathbf{k} \omega}=(\Omega_{k_{y} \omega},\Omega_{k_{x} \omega})$ in $k_{x}-k_{y}$ plane at $t=0$ with parameters $\omega_{B}=\omega_{p}(0)$, $\xi=0.01\omega_{p}^{2}(0)$ and $\omega=1.75\omega_{p}(0)$. The red dashed circles represent the on-shell condition $k(\omega=1.75\omega_{p}(0))=1.059\omega_{p}(0)/c$.
• Figure 3: Dispersion curves (a) without external magnetic field and (b) with finite external magnetic field $\omega_{B}=\omega_{p}$. (c) The group velocity $\mathbf{v}_{g}$ and chirp rate $P$. (d) Magnitudes of Berry curvatures $\Omega_{\mathbf{k} \omega}$ and $\Omega_{\mathbf{k} t}$ of the upper branch TE mode calculated under the on-shell condition \ref{['onshell_condition']} as a single-valued function of $\omega$ at $t=0$ with $\xi=0.01\omega^{2}_{p}(0)$.
• Figure 4: (a) The trajectories of $\hat{\theta}$ center (solid line) and $\hat{\theta}^{\prime}$ center (dashed line) for a pulse with initial frequency $\omega(0)=2\omega_{p}(0)$ emitted at $t = 0$. (b) Ray swings (solid lines) for continuously emitted pulses with the same initial frequency of $\omega(0) = 3.6\omega_{p}(0)$ The dashed line represents the trajectory of the first emitted pulse. In both (a) and (b), the color maps the frequency of the pulse at the moment it reaches that point, and the black rays denote the pulse's trajectory in the absence of the time modulation. The variations of (c) the initial deflection angle and (d) the total displacement with the initial frequency, where $\gamma$ and $\delta r_{y}$ corresponds to the center of $\hat{\theta}$, and $\gamma^{\prime}$ and $\delta r_{y}^{\prime}$ represent the center of $\hat{\theta}^{\prime}$.