Reflection by Conducting Surfaces

Suppose that the region $z<0$ is a vacuum, and the region $z>0$ is occupied by a good conductor of conductivity $\sigma$ . Consider a linearly polarized plane wave normally incident on the interface. Let the wave electric and magnetic fields in the vacuum region take the form

$$E_x(z,t) = E_i\,\cos[k_0\,(c\,t-z)] + E_r\,\cos[k_0\,(c\,t+z)+\phi_r],$$ (1042)

$$H_y(z,t) = E_i\,Z_0^{\,-1}\,\cos[k_0\,(c\,t-z)] - E_r\,Z_0^{\,-1}\,\cos[k_0\,(c\,t+z)+\phi_r],$$ (1043)

where $k_0=\omega/c$ is the vacuum wavenumber. Here, $E_i$ and $E_r$ are the amplitudes of the incident and reflected waves, respectively, whereas $Z_0=\sqrt{\mu_0/\epsilon_0}$ . The wave electric and magnetic fields in the conductor are written

$$E_x(z,t) = E_t\,{\rm e}^{-z/d}\,\cos(\omega\,t-z/d+\phi_t),$$ (1044)

$$H_y(z,t) = E_t\,Z_0^{\,-1}\,\alpha^{-1}\,{\rm e}^{-z/d}\,\cos(\omega\,t-z/d-\pi/4+\phi_t),$$ (1045)

where $E_t$ is the amplitude of the evanescent wave that penetrates into the conductor, $\phi_t$ is the phase of this wave with respect to the incident wave, and

$$\alpha =\left(\frac{\epsilon_0\,\omega}{\sigma}\right)^{1/2}\ll 1.$$ (1046)

The appropriate matching conditions are the continuity of $E_x$ and $H_y$ at the vacuum/conductor interface ($z=0$ ). In other words,

$$E_i\,\cos(\omega\,t) + E_r\,\cos(\omega\,t+\phi_r) = E_t\,\cos(\omega\,t+\phi_t),$$ (1047)

$$\alpha\left[E_i\,\cos(\omega\,t) - E_r\,\cos(\omega\,t+\phi_r)\right] =E_t\,\cos(\omega\,t-\pi/4+\phi_t).$$ (1048)

Equations (1049) and (1050), which must be satisfied at all times, can be solved, in the limit $\alpha\ll 1$ , to give

$$E_r \simeq-\left(1-\sqrt{2}\,\alpha\right)\,E_i,$$ (1049)

$$\phi_r \simeq - \sqrt{2}\,\alpha,$$ (1050)

$$E_t \simeq 2\,\alpha\,E_i,$$ (1051)

$$\phi_t \simeq \frac{\pi}{4}-\frac{\alpha}{\sqrt{2}}.$$ (1052)

Hence, the coefficient of reflection becomes

$$R \simeq\left(\frac{E_r}{E_i}\right)^2\simeq 1-2\sqrt{2}\,\alpha =1- \left(\frac{8\,\epsilon_0\,\omega}{\sigma}\right)^{1/2}.$$ (1053)

According to the previous analysis, a good conductor reflects a normally incident electromagnetic wave with a phase shift of almost $\pi$ radians (i.e., $E_r\simeq -E_i$ ). The coefficient of reflection is just less than unity, indicating that, while most of the incident energy is reflected by the conductor, a small fraction of it is absorbed.

High quality metallic mirrors are generally coated in silver, whose conductivity is $6.3\times 10^7\,(\Omega\,{\rm m})^{-1}$ . It follows, from Equation (1055), that at optical frequencies ( $\omega = 4\times 10^{15}\,{\rm rad.\,s}^{-1}$ ) the coefficient of reflection of a silvered mirror is $R\simeq 93.3$ percent. This implies that about $7$ percent of the light incident on the mirror is absorbed, rather than being reflected. This rather severe light loss can be problematic in instruments, such as astronomical telescopes, that are used to view faint objects.