Question #43742

A p-polarized plane electromagnetic wave is incident from vacuum ($n_1 = 1$) onto a non-magnetic dielectric medium with refractive index $n_2 = \sqrt{3}$. The interface at $z=0$ is coated with an ultra-thin metamaterial that acts as a localized displacement current source, providing an effective surface current density $\vec{K} = \sigma_s \vec{E}_t$, where $\vec{E}_t$ is the tangential electric field and $\sigma_s$ is a real constant. The incident magnetic field is given by $\vec{H}_i = H_0 \cos(k_x x + k_z z - \omega t) \hat{y}$. It is observed that the reflected wave is completely extinguished when the angle of incidence is $\theta_i = \arcsin(12/13)$. Given the impedance of free space is $Z_0 = \sqrt{\mu_0/\epsilon_0}$, what is the value of the dimensionless parameter $\sigma_s Z_0$?