Parity-time symmetry from stacking purely dielectric and magnetic slabs

We show that parity-time symmetry in matching electric permittivity to magnetic permeability can be established by considering an effective parity operator involving both mirror symmetry and coupling between electric and magnetic fields. This approach extends the discussion of parity-time symmetry to the situation with more than one material potential. We show that the band structure of a one-dimensional photonic crystal with alternating purely dielectric and purely magnetic slabs can undergo a phase transition between propagation modes and evanescent modes when the balanced gain or loss parameter is varied. The cross matching between different material potentials also allows exceptional points of the constitutive matrix to appear in the long-wavelength limit where they can be used to construct ultrathin metamaterials with unidirectional reflection.

Figure 1

(a) The permittivity $\left(\epsilon \right)$ and permeability $\left(\mu \right)$ profiles of a one-dimensional photonic crystal stacking alternating purely dielectric slab of $\epsilon =2.25+i\gamma$ and purely magnetic slab $\mu =2.25-i\gamma$ with equal filling ratio and varying $\gamma$. (b) The corresponding band structure: Bloch $k$ against normalized frequency and $\gamma$. The top surface of the plot labels the $\mathcal{PT}$-broken phase (generalized band gap).

Figure 2

(a) Forward and (b) backward reflectance (in natural log scale) for the photonic crystal in Fig. 1 with a finite thickness of four unit cells.

Figure 3

The degree of unidirectionality $\chi$ for $\mathcal{PT}$-symmetric photonic crystals by stacking alternating slabs with permittivity (permeability) ${\epsilon }_1\left({\mu }_1\right)$ and ${\epsilon }_2\left({\mu }_2\right)$. (a) Matching alternating slabs with ${\epsilon }_1=2.25+i\gamma ,{\epsilon }_2=2.25-i\gamma$, and ${\mu }_1={\mu }_2=1$. (b) Matching alternating slabs with ${\epsilon }_1=2.25+i\gamma ,{\mu }_2=2.25-i\gamma$, and ${\epsilon }_2={\mu }_1=1$. The black solid lines show the exceptional point (or curve) with unidirectional character in reflection.

Figure 4

Splitting of eigenvalues $(c_i^{-1},i=1,2)$ of the effective constitutive matrix $C^{-1}$. The scattered symbols (solid lines) are the numerical results from Eq. (6) [analytic effective medium results from Eq. (15)]. (a) and (c) show the real (black) and imaginary (red) parts of the eigenvalues vs $\gamma$ for the ideal $\left({\epsilon }_1={\mu }_2^{*}\right)$ and lossy cases (loss only in ${\mu }_2$), respectively. (b) and (d) are the corresponding plots of the eigenvalues in the complex plane. The arrows show the direction of increasing $\gamma$.