Propagation of electromagnetic waves in $PT$-symmetric hyperbolic structures

We investigate theoretically and numerically the propagation of electromagnetic waves in $PT$-symmetric periodic stacks composed of hyperbolic metamaterial layers separated by dielectric media with balanced loss and gain. We derive the characteristic frequencies governing the dispersion properties of the eigenwaves of $PT$-symmetric semiconductor-dielectric stacks. By tuning the loss/gain level and thicknesses of the layers, we study the evolution of the dispersion dependencies. We show that the effective-medium approach does not adequately describe the propagating waves in the $PT$-symmetric hypercrystals, even for wavelengths that are about 100 times larger than the period of the stack. We demonstrate the existence of anisotropic transmission resonances and above-unity reflection in $PT$-symmetric hyperbolic systems. The $PT$-symmetry-breaking transition of the scattering matrix is strongly influenced by the constitutive and geometrical parameters of the layers and the angles of wave incidence.

Figure 1

Geometry of the problem.

Figure 2

Effective permittivities ${\varepsilon }_{xx}^{\left(h\right)}$ (solid red line) and ${\varepsilon }_{zz}^{\left(h\right)}$ (dashed black line) for fine-stratified semiconductor medium with $a_1=0.1\mu \mathrm{m}$, (a) $a_1/a_2=2$, (b) $a_1/a_2=0.5$.

Figure 3

Dispersion dependence for the infinite periodic structure with $a_1=0.1\mu \mathrm{m},a_1/a_2=2,d_h=0.8\mathrm{mm},\alpha =2,{\varepsilon }^{\prime }=2.1$; (a),(b),(e) ${\varepsilon }^{\prime \prime }=0,$ (c),(f) ${\varepsilon }^{\prime \prime }=0.1$, (d),(g) ${\varepsilon }^{\prime \prime }=0.5$. (b)–(d) correspond to the area highlighted with a red rectangle in (a), (e)–(g) are for the area highlighted with a blue rectangle in (a). The transmission bands are indicated by blue hatching. The bands corresponding to different hyperbolic regimes for a uniaxial fine-stratified semiconductor medium are blue and pink shaded.

Figure 4

Effective permittivities ${\varepsilon }_{xx}^{\left(fs\right)}$ (solid red line) and ${\varepsilon }_{zz}^{(fs)}$ (dashed black line) for $PT$-symmetric fine-stratified medium with ${\varepsilon }^{\prime }=2.1,{\varepsilon }^{\prime \prime }=0.1,d_h=8\mu \mathrm{m},\alpha =2,a_1=0.1\mu \mathrm{m}$, (a) $a_1/a_2=2$, (b) $a_1/a_2=0.5$.

Figure 5

Isofrequency contours for (a) $\omega =1.005\times {10}^{11}{\mathrm{s}}^{-1}$; (b) $\omega =9.995\times {10}^{11}{\mathrm{s}}^{-1}$; $d_h=8\mu \mathrm{m},\alpha =2,a_1=0.1\mu \mathrm{m},a_1/a_2=2$. Dashed black line, ${\varepsilon }^{\prime }=2.1,{\varepsilon }^{\prime \prime }=0.1$; solid red line, ${\varepsilon }^{\prime }=0.01,{\varepsilon }^{\prime \prime }=0.1$; dash-dotted blue line, ${\varepsilon }^{\prime }=0.01,{\varepsilon }^{\prime \prime }=0.01$.

Figure 6

Reflectance of TM wave incident at ${\mathrm{\Theta }}_i={15}^{\circ }$ (solid red line) and ${\mathrm{\Theta }}_i={75}^{\circ }$ (dashed black line) on fine-stratified medium with ${\varepsilon }^{\prime }=2.1,{\varepsilon }^{\prime \prime }=0.1,a_1=0.1\mu \mathrm{m},a_1/a_2=2,d_h=8\mu \mathrm{m},\alpha =2$, and $L=1.2\mathrm{mm}$.

Figure 7

Reflectance of TM wave incident at ${\mathrm{\Theta }}_i={15}^{\circ }$ obtained by exact transfer-matrix approach $(|R^{\left(L\right)}{|}^2,$ solid blue line, $|R^{\left(R\right)}{|}^2,$ solid red line) and by effective-medium approach (dashed black line).

Figure 8

Reflectance of TM wave incident at ${\mathrm{\Theta }}_i={15}^{\circ }$ $(|R^{\left(L\right)}{|}^2,$ solid blue line, $|R^{\left(R\right)}{|}^2,$ dashed black line) on $PT$-symmetric periodic stack with $a_1=0.1\mu \mathrm{m},a_1/a_2=2,d_h=0.8\mathrm{mm},\alpha =2,{\varepsilon }^{\prime }=2.1,{\varepsilon }^{\prime \prime }=0.1$, and $N=5$. The bands corresponding to different hyperbolic regimes for a uniaxial fine-stratified semiconductor medium are blue and pink shaded.

Figure 9

Logarithm of the modulus of eigenvalues of the $S$ matrix for a $PT$-symmetric stack as a function of frequency at ${\varepsilon }^{\prime }=2.1,{\varepsilon }^{\prime \prime }=0.1,a_1=0.1\mu \mathrm{m},a_1/a_2=2,d_h=0.8\mathrm{mm},\alpha =2$; red lines, ${\mathrm{\Theta }}_i={15}^{\circ }$, black lines, ${\mathrm{\Theta }}_i={75}^{\circ }$; (a) $N=5$; (b) $N=50$. Solid and dashed curves of the same color correspond to the different eigenvalues of the same scattering matrix.