Wigner time for electromagnetic radiation in plasma

Wave tunneling is an intriguing phenomenon spanning different branches of physics, from quantum mechanics to classical electrodynamics and optics. The Wigner (or phase) time is proved to be an adequate measure to describe wave transit through a potential barrier or material layer in the tunneling regime. Here we analytically and numerically calculate the Wigner time for electromagnetic-radiation propagation through the layer of both lossless and lossy plasmas. It is shown that the plasma frequency is the key parameter governing the value of Wigner time, allowing us to interpret tunneling as due to the reaction of plasma as a whole. We analyze the Wigner time for obliquely incident waves of transverse electric (TE) and transverse magnetic (TM) polarizations and discuss the meaning of negative Wigner times appearing in the lossy case in the low-frequency range and close to the plasma frequency. The results show that plasma deserves attention as a perspective object for tunneling studies.

Figure 1

Frequency dependence of the Wigner time for normally incident radiation.

Figure 2

Frequency dependence of the Wigner time for $\theta =\pi /3$ and $\gamma /{\omega }_p=0.01$ calculated with the approximate and exact expressions. Solid lines are for the TE waves and dashed lines are for the TM waves.

Figure 3

Frequency dependence of the Wigner time for (a) different losses at $\theta =\pi /3$ and (b) different angles at $\gamma /{\omega }_p=0.05$. Solid lines are for the TE waves and dashed lines are for the TM waves.