Exceptional Contours and Band Structure Design in Parity-Time Symmetric Photonic Crystals

We investigate the properties of two-dimensional parity-time symmetric periodic systems whose non-Hermitian periodicity is an integer multiple of the underlying Hermitian system’s periodicity. This creates a natural set of degeneracies that can undergo thresholdless $\mathcal{P}\mathcal{T}$ transitions. We derive a $\mathbf{k}·\mathbf{p}$ perturbation theory suited to the continuous eigenvalues of such systems in terms of the modes of the underlying Hermitian system. In photonic crystals, such thresholdless $\mathcal{P}\mathcal{T}$ transitions are shown to yield significant control over the band structure of the system, and can result in all-angle supercollimation, a $\mathcal{P}\mathcal{T}$-superprism effect, and unidirectional behavior.

Figure 1

(a) Schematic of the 2D PhC comprised of square rods with side length $0.6a$ of a dielectric, ${ϵ}_{\text{die}}=12$, embedded in air, ${ϵ}_{\text{air}}=1$, with a square primitive cell side length of $a$. The primitive cell is indicated in gray, while the supercell contains two primitive cells and is marked with a dashed border. When $\tau \ne 0$, the red rods contain gain, while the cyan rods contain loss. (b) Schematic of the underlying 2D Hermitian PhC. (c),(e) Real part of the frequencies for the first (blue) and second (red) supercell TM bands when $\tau =0$ and $\tau =1.5$. Locations where the bands have merged are shown in magenta. (d),(f) Real part of the frequencies for the fifth (blue) and sixth (red) supercell TM bands when $\tau =0$ and $\tau =1.5$. (g) Imaginary part of the frequencies for the first (blue) and second (red) supercell TM bands when $\tau =1.5$. Black denotes no imaginary component. (h) Imaginary part of the frequencies for the fifth (blue) and sixth (red) supercell TM bands when $\tau =1.5$.

Figure 2

(a) Nonunitary behavior as a function of incident angle $\varphi$ and $\mathcal{P}\mathcal{T}$ symmetry breaking parameter $\tau$ for an $s$-polarized plane wave source with $\omega =0.185(2\pi c/a)$ incident upon a PhC slab infinite in the $x$ direction and with 50 layers in the $y$ direction for the same system shown in Fig. 1. $\varphi =0$ corresponds to normal incidence in the $y$ direction. The PhC slab is surrounded by a passive dielectric with $ϵ=3$. Values of 0 correspond to unitary behavior, while $[-1,0)$ signifies absorption, and $(0,\infty ]$ signifies amplification. The reflection and transmission coefficients were calculated using the Fourier modal method as implemented in ${\mathrm{S}}^4$ [43]. (b),(c) Plot of the real part of the electric field for the same structure with $\tau =0.65$, and $\varphi =40.9°$ (b), or $\varphi =63.8°$ (c). Field plots were generated using the freely available maxwellfdfd software package [44].

Figure 3

(a) Schematic of the 2D PhC comprised of square rods with side length $0.6a$ of a dielectric, ${ϵ}_{\text{die}}=12$, embedded in air, ${ϵ}_{\text{air}}=1$, with a square primitive cell side length of $a$. The primitive cell is indicated in gray, while the supercell contains two primitive cells and is marked with a dashed border. When $\tau \ne 0$, the red rods contain gain, while the cyan rods contain loss. (b),(c) Real part of the frequencies for the first (blue) and second (red) supercell TM bands when $\tau =0$ and $\tau =1.5$. Locations where the bands have merged are shown in magenta. (d) Imaginary part of the frequencies for the first (blue) and second (red) supercell TM bands when $\tau =1.5$. Black denotes no imaginary component.