Question #43173

A uniform plane electromagnetic wave traveling in vacuum (characteristic impedance $Z_0 = 120\pi \ \Omega$) along the $+z$-direction is normally incident on a semiconductor interface at $z=0$. The incident electric field is given by $\vec{E}_i = E_0 \cos(\omega t - kz) \hat{x}$. Due to charge carrier dynamics, the semiconductor surface cannot be treated as a perfect conductor. Instead, it is modeled with a complex surface impedance $Z_s = R_d + jX_s$, where the dynamic resistance is $R_d = 60\pi \ \Omega$ and the surface reactance is $X_s = 60\pi\sqrt{3} \ \Omega$ ($j = \sqrt{-1}$). The reflected electric field at the interface is expressed as $\vec{E}_r(z=0) = E_r \cos(\omega t + \phi) \hat{x}$, with $E_r > 0$ and $\phi \in (-\pi, \pi]$. What is the value of the phase shift $\phi$?