A thin transparent coating of unknown thickness t (index of refraction nfilm=1.50) is deposited uniformly on a flat glass substrate of thickness 1.00cm and index nglass=1.60. A short laser pulse of frequency 6.0×1014Hz is directed perpendicularly onto the coated surface from air. The pulse passes through the coating, the substrate, and exits through the opposite face back into air, where its arrival time is measured relative to an identical reference pulse that travels the same geometric distance entirely through air.
(a) Derive an expression for the total transit time of the pulse from entry into the coating to exit from the far side of the glass substrate, in terms of t, nfilm, nglass, and any necessary physical constants.
(b) If the measured transit-time difference between the test pulse and the reference pulse is 4.15×10−11s, determine the thickness t of the coating.
(c) The indices of refraction are each known to within ±0.01. Estimate the resulting uncertainty in the calculated thickness t. State any assumptions you make.
(d) Explain qualitatively how the photon frequency and wavelength change (or remain unchanged) as the pulse crosses each interface: air→coating, coating→glass, and glass→air.