Excitation of surface Josephson plasma waves in layered superconductors

This is a theoretical study of the resonant suppression of the specular reflection of terahertz waves in layered superconductors due to the excitation of surface Josephson plasma waves (SJPWs). Here, we consider in detail the specific case of SJPW excitations by evanescent electromagnetic waves via the attenuated total reflection of incident waves in a dielectric prism. We also derive the dispersion relation for surface waves propagating along the vacuum-superconductor interface parallel to the $\mathbf{a}\mathbf{b}$ plane. We show that, due to the SJPW excitation, the reflectivity of the incident wave depends resonantly on both its frequency and incident angle. We find the optimal conditions for the best matching of the incident wave and SJPWs, as well as for the total suppression of the specular reflection.

Figure 1

(Color online) Geometry for studying surface waves.

Figure 2

(Color online) Geometry considered here (Otto configuration): a dielectric prism is separated from a layered superconductor by a vacuum gap of thickness $h$. An electromagnetic wave with incident angle $\theta >{\theta }_t$ can excite SJPWs that satisfy the following resonant condition: $\omega \sin \theta ∕c=q$. Here ${\mathbf{k}}^i$ and ${\mathbf{k}}^r$ are the wave vectors of the incident and reflected waves associated with the magnetic field amplitudes ${\mathbf{H}}^i$ and ${\mathbf{H}}^r$. The resonant excitation of SJPWs by the incident wave produces a strong suppression of the reflected wave. This method for producing surface waves is known as the “attenuated-total-reflection” method.

Figure 3

(Color online) The dependence of the reflectivity coefficient ${\mid R\mid }^2$ on the incident angle $\theta$, calculated for the parameters $b=0.7$, $\Gamma =0.005$, $r={10}^{-6}$, $1-{\Omega }^2=1.2\times {10}^{-5}$, ${\epsilon }_s=20$, and $\epsilon =4$. The thickness of the vacuum gap is one wavelength, $hk=2\pi$. The solid red curve presents the results of numerical calculations using Eqs. (22, 23, 24). The dashed black curve (that almost overlaps the red curve) describes the analytically obtained asymptotic dependence [Eqs. (31, 32, 33, 34)]. The vertical line at $\vartheta =30°$ corresponds to the limiting angle of the total internal reflection. The blue thin solid curve presents the Fresnel reflectivity coefficient.

Figure 4

The reflectivity coefficient in the plane $(\theta ,(1-\Omega ))$ shown in gray levels for the same values of the parameters as in Fig. 3. The dispersion relation of the dielectric-vacuum-layered superconductor is presented by the solid curve.

Figure 5

The frequency dependence of the reflectivity coefficient ${\mid R\mid }^2$ obtained for $\theta =31.867°$. Other parameters are the same as in Fig. 3.

Figure 6

(Color online) The magnetic field distribution (a) for the nonresonant case, $\theta \ne {\theta }_{\mathrm{res}}$, and (b) for the resonant condition, $\theta ={\theta }_{\mathrm{res}}=31.867°$. Other parameters are the same as in Fig. 3.