Excitation of terahertz modes localized on a layered superconductor: Anomalous dispersion and resonant transmission

We study theoretically the optic transmission through a slab of layered superconductor separated from two dielectric leads by spatial gaps. Based on the transfer matrix formalism along with the Josephson plasma electrodynamic approach, we derive analytic expressions for the transmittance and identify the conditions for the perfect transmission. The special interest of the study is focused on the resonant transmission, which occurs when the wave does not propagate in the spatial gaps. Far from the resonance, the transmittance is exponentially small due to the total internal reflection from the lead-gap interface. However, the excitation of electromagnetic modes localized on the layered superconductor gives rise to a remarkable resonant enhancement of the transmission. Moreover, this phenomenon is significantly modified for the layered superconductors in comparison with usual dielectrics or conductors. The dispersion curves for the modes localized on the layered superconductor are proved to be nonmonotonic, thus resulting in the specific dependence of the transmittance $T$ on the incidence angle $\theta$. In particular, we predict the onset of two resonant peaks in the $T\left(\theta \right)$ dependence and their subsequent merge into the broadened single peak with increasing of the wave frequency. Our analytical results are demonstrated by numerical data.

Figure 1

A sketch of the setup.

Figure 2

A schematic diagram for the wave reflection at the interface between semi-infinite dielectric media $a$ and $b$.

Figure 3

The normalized wave frequency $\mathrm{\Omega }$ vs the dimensionless wave number $k_z{\lambda }_c$ for the localized modes, Eq. (26b). The dashed red curves correspond to the surface modes at $\mathrm{\Omega }<1$ with indices $n=0,1$, and the solid green curves correspond to the waveguide modes with $n=1,2,3,...$ at $\mathrm{\Omega }=\omega /{\omega }_J>1$. The thin blue straight line is the light line, $\mathrm{\Omega }=c{k}_z/{\omega }_J\sqrt{{\varepsilon }_b}$. The inset shows the dispersion curves for surface waves and the waveguide modes with $n=0$ and $n=1$ in the narrow range near $\mathrm{\Omega }=1$. Parameters: ${\varepsilon }_b=1,{\varepsilon }_c=15,d_c/{\lambda }_c=1,\gamma ={\lambda }_c/{\lambda }_{ab}=15$.

Figure 4

The principal distribution of the magnetic field in the wave transferred through the system. The thick solid line in the $b_L-c-b_R$ subsystem represents the exited localized mode causing the resonant transmission.

Figure 5

Transmittance $T$ vs the incidence angle $\theta$ and the normalized frequency $\mathrm{\Omega }=\omega /{\omega }_J$. The solid green lines represent the dispersion curves for the localized modes in the system with infinite thicknesses of the spatial $b$ gaps, see Eqs. (26) and Fig. 3. Panel (b) shows the region around the maximum in the dispersion curve with $n=3$. Four horizontal straight lines in panel (b) correspond to the values of $\mathrm{\Omega }$ used in Fig. 6. The solid red circle marks the top point ${\theta }_{\max }=1.0057$ and ${\mathrm{\Omega }}_{\max }=2.5576$ on the dispersion curve for setup with finite $d_b$, Eq. (34). Parameters: ${\varepsilon }_a=20,d_b/{\lambda }_c=1/7$. The other parameters are the same as in Fig. 3.

Figure 6

Transmittance $T$ vs the incidence angle $\theta$ for four values of the normalized frequency: $\mathrm{\Omega }=2.55$ (dot-dashed brown curve), $\mathrm{\Omega }=2.556$ (solid blue curve), $\mathrm{\Omega }={\mathrm{\Omega }}_{\max }=2.5576$ (dashed red curve), and $\mathrm{\Omega }=2.559$ (dotted green curve). The solid red circle marks the top point ${\theta }_{\max }=1.0057$ and $T_{\max }=1$ of the broadened peak. The other parameters are the same as in Fig. 5.

Figure 7

Transmittance $T$ vs the incidence angle $\theta$ for two values of the normalized frequency, $\mathrm{\Omega }=2.556$ and $\mathrm{\Omega }={\mathrm{\Omega }}_{\max }=2.5576$: respectively, solid blue and dashed red curves related to ${\nu }_x={\nu }_z=0$, and dotted blue and dot-dashed red curves associated with ${\nu }_x={\nu }_z={10}^{-3}$. The other parameters are the same as in Fig. 6.