Reflection and Refraction — Question 6

Question 6

An electric-field transmission coefficient is not a power fraction

At normal incidence between lossless nonmagnetic media, the electric amplitude coefficients are $r=(n_1-n_2)/(n_1+n_2)$ and $t=2n_1/(n_1+n_2)$. Intensity for peak electric amplitude E is $n\epsilon_0cE^2/2$. Take n1=1 and n2=1.5.

1. Derive the reflected and transmitted power fractions, retaining the intensity prefactor.

2. Evaluate both fractions and the field coefficients, and verify conservation of energy.

Question 6 - Solution

An electric-field transmission coefficient is not a power fraction

(a) Method and calculation

$$R=|r|^2,\quad T=\frac{n_2}{n_1}|t|^2=\boxed{\frac{4n_1n_2}{(n_1+n_2)^2}}.$$$$R+T=\frac{(n_1-n_2)^2+4n_1n_2}{(n_1+n_2)^2}=1.$$

(b) Interpretation and further calculation

$$\boxed{r=-0.20,\quad t=0.80}.$$$$\boxed{R=0.040,\quad T=0.960},\quad t^2=0.640\ne T.$$

Physical check

The reflected field changes sign for entry into the larger index; the transmitted energy depends on both field amplitude and medium impedance.