Week 13: Interference and Diffraction 453
colors, we are seeing thin film interference. Thin film interference gives color and life to
ornaments and has various other technological or social applications, even if those who
observe it don’t realize what it is.
We’d like to understand it and learn to recognize it and see one or two of its applications.
Fortunately, it is (at this point) quite simple. Here’s the idea.
In figures 192 and 193 a thin film of transparent material sits in between two other
transparent materials. Each material has its own index of refraction, and we will for the
moment use the convention thatn 1 is the index of refraction of the material the light is
comingfrom,n 2 is the index of the thin film itself, andn 3 is the index of the material
the light is goingto.
Incident light (often white light, a mixture of all the visible colors/wavelengths) is incident
approximately “normally” onto (coming in perpendicular to) the surface betweenn 1 and
n 2. Some fraction of this light reflects off of the interface; the rest istransmitted inton 2.
Of the light that makes it inton 2 and then is incident normally on the interface between
n 2 andn 3. Again, some fraction is reflected and some is transmitted. Finally, the light
that is reflected back up arrives at the interface betweenn 1 andn 2 a second time, this
time coming from below, and a fraction of it is transmitted back into mediumn 1 , where
the electromagnetic wavecombineswith the original reflected wave.
The interference we observe thus comes from adding two waves:Etot=E 12 sin(kr−ωt+δ 12 ) +E 23 sin(kr−ωt+δ 23 ) (1089)where (as we will see below) there is a chance of a phase shift occurring inbothreflected
waves compared to the phase of the incoming wave. Note also that itis almost certain
thatE 126 =E 23 , that is, the two reflected waves will very likely have somewhat different
amplitudes as they recombine.
Presuming that these two waves have at leastapproximatelyequal field amplitudes and
a consistent phase difference brought about at least partly by path difference (the wave
that traverses the film twice travels a distance 2dfarther than the wave that reflects of
of the first surface), this superposition will partially cancel or partially add the waves
for different wavelengths. Some wavelengths will be brightened, others diminished. The
reflected white light will therefore take on those characteristic mauves and greens and
poisonous shiny blues that are familiar to us all.
Of course, there are a fewdetailswe have to consider, and they are important; they are
why we needtwofigures (and two phase shifts) to demonstrate two of the four possible
patterns of sort order of the indices of refraction. In a nutshell,two things contribute to
the overall phase shift between the recombined waves – the phaseshift due to the path
difference in the mediumn 2 and a phase shift caused by reflecting off of a medium with a
higher index of refraction! Let’s begin by working out the former, as that is easiest, and
then we’ll talk extensively about the latter, as the phase shifts dueto reflection off of
the surfaces themselves will require us to go back to our intro physics 1 course and recall
e.g. thereflection of waves on stringsoff of interfaces between a light string (where the
speed of the wave is large) and a heavy string (where the speed of the wave is less).13.11.1: Phase Shift Due to Path Differencein the Thin Film!
This one, as promised, is easy. The wave that traverses the thin film(twice!) goes an
additional distance ∆r= 2dcompared to the wave that reflects off of the upper surface.
We are thus tempted to (after “reflection”^134 on what we have learned so far) to associate
with this path difference an additional phaseδpath=k(2d).(^134) Har, har...