Introductory Physics Volume Two

154 Wave Optics 7.10
The reason that we care the wavelength has changed, is that when
we use the result that ∆r = m λ 2
we need to use the wavelength where
the path difference ∆r occurs. In the case considered above, the extra
bit of path occurs within the oil, so we need to use the wavelength in
the oil.
In order to understand the root cause of the second complication,
consider a soap bubble again. If you make a soap bubble and watch it
carefully, you will notice that just before it pops, it appears that the top
of the bubble has a hole in it. There is not actually a hole, but the top
of the bubble looks like it has a hole because when the bubble gets very
thin, no light is reflected from the surface of the bubble. Let us consider
what our prediction would be for a very thin bubble. From our work
with the oil film we expect that we will get a maximum reflection when
2t = mλ/2, and m is even. Recall also that m = 0 gives a maximum,
so t ≈ 0 should give a maximum for all wavelengths. Thus we expect
the soap bubble to reflect very well when the film of bubble juice gets
very thin. The truth is just the opposite.
The key to this puzzle is to understand that when a wave is reflected
it can be inverted, that is the direction of the displacement can
be reversed. But a reversal is the same as a π phase shift.
Fact: Reflection Phase Shift
When a wave is reflected at an interface between two media there
will be a π phase shift in the reflected wave if the index of refraction
of the incident medium is lower than the index of refraction of the
reflecting medium. If the incident medium has a higher index,
there is no phase shift.
⊲ Problem 7.11
Explain using the reflection phase shift why a thin soap bubble reflects
no light.
§ 7.10 Single Slit Diffraction
If a wave is passed through a rectangular hole that is much narrower
in one direction than the other we find that we get an interference
pattern like we do with a two source. Such a hole is usually referred to
as a slit since it is so narrow in one direction.