Reflection and Refraction — Question 7

Question 7

Repeated internal reflections restore part of a slabs transmitted power

A lossless plane slab lies between identical external media at normal incidence. Each face has power reflectance R from either side. Sum incoherent beam powers, so interference is averaged away. A wide detector collects every emerging forward beam.

1. Find the total forward transmitted fraction including all internal round trips.

2. Find the total reflected fraction and evaluate both for R=0.04.

Question 7 - Solution

Repeated internal reflections restore part of a slabs transmitted power

(a) Method and calculation

$$T_{\mathrm{total}}=(1-R)^2\sum_{m=0}^{\infty}R^{2m}=\frac{(1-R)^2}{1-R^2}.$$$$\boxed{T_{\mathrm{total}}=(1-R)/(1+R)}.$$

(b) Interpretation and further calculation

$$R_{\mathrm{total}}=R+\frac{(1-R)^2R}{1-R^2}=\boxed{2R/(1+R)}.$$$$\boxed{T_{\mathrm{total}}=12/13=0.923077,\quad R_{\mathrm{total}}=1/13=0.0769231}.$$

Physical check

The first forward beam alone contributes (1-R) squared; the remaining forward beams form a geometric series.