Topology and the optical Dirac equation

Through understanding Maxwell's equations as an effective Dirac equation (the optical Dirac equation), we reexamine the relationship between electromagnetic interface states and topology. We illustrate a simple case where electromagnetic material parameters play the roles of mass and energy in an equivalent Dirac equation. The modes trapped between a gyrotropic medium and a mirror are then the counterpart of those at a domain wall, where the mass of the Dirac particle changes sign. Considering the general case of arbitrary electromagnetic media, we provide an analytical proof relating the integral of the Berry curvature (the Chern number) to the number of interface states. We show that this reduces to the usual result for periodic media and also that the Chern number can be computed without knowledge of how the material parameters depend on frequency.

Figure 1

TM polarized electromagnetic waves in planar gyrotropic media obey a two-dimensional Dirac equation, where the energy and mass are given by the material parameters $\mathcal{E}=\sqrt{{\epsilon }_1}$ and $m=\alpha /\sqrt{{\epsilon }_1}$ [see (5)]. (a) Value of the wave vector (along say the $y$ axis) plotted as a function of $\sqrt{{\epsilon }_1}$ for fixed $\alpha =1$ with regions of bulk propagation in blue and the dispersion of the interface state in red (identical to the corresponding plot for the two-dimensional Dirac equation in the presence of a domain wall). (b) Magnitude and (c) phase maps (generated using comsol multiphysics) of the out-of-plane $H$ field for the two regions of parameter space indicated in the (a) (color indicates phase and brightness magnitude). The white dot shows the position of a point source used to generate the field, and a mirror is present along the line $x=-17.5\lambda$ ($\lambda =2\pi /k_0$).

Figure 2

(a) An electromagnetic interface state propagating in a planar gyrotropic medium (permittivity tensor $\mathbf{\epsilon }$) and bound to the interface of a mirror at $x=0$ is governed by a 2D Dirac equation, where the mass changes sign at $x=0$. The out-of-plane magnetic field combined with either of the two circulating electric-field components $E_{\pm }$ determines the respective spinor $\psi$ in the two half spaces $x>0$ and $x<0$. (b) Given the mapping onto the Dirac equation, the integral of the Berry curvature $\mathcal{F}$ in the space of parameters ${\epsilon }_1$, $k_x$, and $k_y$ determines the number of interface states. The two plots show the dispersion (see Fig. 1) with the coloring indicating the value of the Berry curvature [the integrand of (18)].

Figure 3

(a) Logarithm of reflection coefficient $ln\left[\right|r(\mathcal{E},k_y)\left|\right]$ for the medium with properties given by (24) and polarization $E_z=-tan\left(\theta \right){\eta }_0{H}_z$, verifying the prediction of (25) with two modes moving from $k_y,\mathcal{E}<0$ to $k_y,\mathcal{E}>0$ (another two such modes are evident in the orthogonal polarization). Between the two horizontal dashed white lines no propagation is allowed. (b) Functional form of $\alpha \left(x\right)$ which has been chosen as an arbitrary function that changes sign around $x=0$ (the function ${\alpha }^{\prime }$ plays no role for this polarization, but is also assumed to also change sign). Curves (i) and (ii) show the functional form of the two interface modes, with the corresponding parameters shown in (a).

Figure 4

(a) We consider the electromagnetic states bound to the interface between a mirror and a gyrotropic medium as equivalent to the modes of the 2D Dirac equation bound to the interface between regions where the mass changes sign. We approximate this interface as with a region where $m$ smoothly changes sign. In order to make the spectrum of our Dirac operator discrete, we also assume that the mass becomes infinite as $x\to \infty$ (without changing sign along the way). (b) The system shown in (a) supports propagating waves for ${\mathcal{E}}^2>m^2$ (i.e., when ${\epsilon }_1^2>{\alpha }^2$), shown as the blue shaded regions, with an interface state connecting these two regions. To work out the number of interface states that connect these regions in parameter space, we count the number of modes that cross the line $\mathcal{E}=\sqrt{{\epsilon }_1}=0$, where no propagation is allowed. This counting is done in terms of the difference in spectral asymmetry $\nu \left(k_y\right)$ [defined in (A2)] at $k\to +\infty$ and $k\to -\infty$, shown schematically by the vertical dashed lines. (c) Number of edge states $N$ given as the difference between two integrals over hypersurfaces of constant $k_y$ (labeled $A$). Using Stokes's theorem, this can be transformed into integrals over constant $x$ (labeled $B$).