Question

If a thin film of soap hanging in the air (n = 1.35) reflects dominantly red light (667 nm) What is the minimum thickness of the film?

Now this film is on a sheet of glass, (n = 1.49), which is the longest wavelength that will now be predominantly reflected?

Answer 1

Making a couple of assumptions:
you are in are in a position to see the light reflected from both the front and back surfaces of the film and that the two rays constructively interfere for this wavelength

for the hanging film
the light reflected from the front surface is phase inverted because it is going from low n to higher n
and
the ray from the back is not inverted (nigh n to lower n)
so
the path difference needs to be 1/2 wavelength to put them back in phase (constructive interference)
that means the minimum thickness must be 1/4 wavelength
1/4 * 667 = 166.75 nm

when backed by glass the ray from the rear surface phase shifted (lower n to higher n)
so the total path length must be 1 wavelength to put the two rays back in phase
so the thickness must be 1/2 wavelength
1/2 * 667 = 333.5 nm