Question

A plane electromagnetic wave propagates from one dielectric to another at normal incidence. Calculate the ratio of the indices of refraction of the two dielectrics for which the reflection and transmission coefficients are both equal to 0.5.

Answer 1

The reflection coefficient in this case is given by

$$R = \left( \frac{n_1 - n_2}{n_1 + n_2} \right)^2 \, ,$$

where $n_1$ and $n_2$ are the respective indices of refraction of two dielectrics. Denoting the ratio $n_1 / n_2 = r$, we obtain the equation

$$R = \left( \frac{r - 1}{r + 1} \right)^2 = \frac12 \, ,$$

which results in a quadratic equation $r^2 - 6 r + 1 = 0$. Solving it, we obtain $r = 3 \pm 2 \sqrt{2}$. Both solutions are admissible and indicate the symmetry of the problem: if $n_1 / n_2 = r = 3 + 2 \sqrt{2}$, then $n_2 / n_1 = 1 / r = 3 - 2 \sqrt{2}$.

Answer: $3 \pm 2 \sqrt{2}$.