Light Guiding by Effective Gauge Field for Photons

We propose a waveguiding mechanism based on the effective gauge potential for photons. The waveguide geometry consists of core and cladding regions with the same underlying dispersion relation, but subject to different gauge potentials. This geometry can be realized in a dynamically modulated resonator lattice and provides a conceptually straightforward and dynamically reconfigurable mechanism for generating a one-way waveguide.

Popular Summary

Optical waveguides, which route light between optical devices much like metal wires routing electrons on a silicon chip, are the fundamental building blocks of integrated photonics. The most widely used waveguide is a dielectric waveguide in which a high-index core is surrounded by a low-index cladding; light is confined in the core because of total internal reflection. We propose a new waveguiding mechanism based on an effective gauge field for photons. Unlike dielectric waveguides, our proposed waveguide allows light to propagate in only one direction, eliminating undesirable back reflections.

We provide a theoretical framework to understand a gauge-field waveguide, where a gauge field can displace, in wave vector space, otherwise identical states in the core and cladding, thus producing light confinement in the core. Even though photons are neutral and therefore have no naturally analogous magnetic field, effective gauge fields for photons can be realized by controlling the phase of dynamic modulations that drive photonic transitions. We use standard waveguide theory to describe an inhomogeneous effective gauge-field waveguide constructed from a square lattice of dynamically coupled resonators, and we illustrate its novel features such as nonreciprocity and single-mode, one-way guiding over certain frequency ranges. In such a system, the effective gauge field for photons is mapped to the modulation phases, which one can readily control in real time, allowing for the reconfiguration of the routing scheme dynamically, without changing the underlying device structure.

Our work highlights new possibilities in controlling the flow of light and exemplifies a rich set of gauge-field physics that can be studied in photonic systems. Experimental realization of our proposal could be feasible with the rapid development of on-chip silicon photonics.

Figure 1

Structures and isofrequency contours corresponding to two different waveguiding mechanisms: (a) index guiding, where the core region has a high refractive index and the cladding has a low refractive index; (b) gauge-field guiding, where the core region has a nonzero gauge potential $\mathbf{A}$ and the cladding has a zero gauge potential. The dashed line on the isofrequency contours demonstrates the absence of $k_y$-matching states in the cladding region.

Figure 2

Dispersion relation of the guided mode (red solid line) for (a) a conventional waveguide with cladding index $n=1$ and core index $n=1.4$, (b) a gauge-field waveguide with $\varphi =0.3\pi$, (c) a gauge-field waveguide with $\varphi =0.53\pi$, with the inflection point [22] marked by a black circle, (d) a gauge-field waveguide with $\varphi =\pi$. The pink and blue regions show the light cones for the core and cladding regions, respectively. Isofrequency contours for (e) small and (f) large $\varphi$ at the same frequency $\omega$. Panel (e) has only guided mode with $v_g>0$ while panel (f) has guided modes with $v_g>0$ and $v_g<0$.

Figure 3

Projected band diagrams for a modulated square lattice (a) with no gauge field and (b) with an effective gauge field of $\varphi =0.5\pi$. The insers show the lattice configuration with the corresponding modulation phases.

Figure 4

(a) A square lattice of resonators with resonant frequencies ${\omega }_A$ (red) and ${\omega }_B$ (blue) described by the time-dependent Hamiltonian of Eq. (4). The label on each vertical bond denotes the phase of the dynamic modulation between nearest neighbors along the $y$ direction. All the horizontal couplings have a zero modulation phase. (b) A square lattice of identical resonators with nearest-neighbor couplings described by Eq. (5), which is obtained after performing the rotating wave approximation on Eq. (4). In the core region, there is a direction-dependent coupling phase ($+\varphi$ hopping upward and $-\varphi$ hopping downward), which corresponds to a uniform gauge potential. Thus, under rotating wave approximation, the modulation phase in (a) is mapped into a time-independent gauge potential in (b).

Figure 5

Guided mode dispersions obtained by solving the Hamiltonian in Eq. (6) with a core region width $d=2a$ ($N=2$) and a cladding region width of $10a$ on each side for $\varphi =A_{y}a=[1/3,2/3,1,4/3,5/3]\pi$ (a)–(e).

Figure 6

Guided mode wave function (amplitudes as a function of $x_n$) for $\varphi =\pi$ at (a) the highest band at $k_y=\pi /a$ and (b) the lowest band at $k_y=0$. The calculation is based on $N=1$ in the core region and $10a$ on each side in the cladding region.

Figure 7

Guided mode dispersion for $\varphi =0.5\pi$ and $N=1$ obtained by (a) numerically solving the tight-binding model Hamiltonian in Eq. (6) with cladding region width of $10a$ on each side and (b) waveguide theory calculation.

Figure 8

Cutoff ${\varphi }_c$ for higher-order modes when $N\ge 2$. The mode order varies from $s=2$ (in blue) to $s=5$ (in red). Circles represent numerical results from tight-binding model. Curves represent analytical results from waveguide theory.