Study the step-by-step buildup of the field inside an etalon and the etalon’s transmitted field, when the input field is suddenly turned on. More specifically, carry out the following steps.

(a) When the wave first turns on, the transmitted field inside the etalon, at point A of Fig. 9.7, is ψ_a = tψ_i, which is very small if the reflectivity is high so that |t|« 1. Show (with the aid of Fig. 9.7) that, after one round-trip-travel time in the etalon, the transmitted field at A is ψ_a = tψ_i + (r')²e^iφtψ_i. Show that for high reflectivity and on resonance, the tiny transmitted field has doubled in amplitude and its energy flux has quadrupled.

(b) Compute the transmitted field ψ_a at A after more and more round trips, and watch it build up. Sum the series to obtain the steady-state field ψ_a. Explain the final, steady-state amplitude: why is it not infinite, and why, physically, does it have the value you have derived?

(c) Show that, at any time during this buildup, the field transmitted out the far side of the etalon is ψt = t' ψ_ae^iks1 [Eq. (9.30b)]. What is the final, steady-state transmitted field? Your answer should be Eqs. (9.33a).

FIGURE 9.7.

V be-iks = t'eik (51-52), = teiks, (9.30b)