454 Week 13: Interference and Diffraction
As it turns out, this heuristic guess isalmostcorrect! But as the saying goes, “almost”
only counts in horseshoes and hand grenades^135. The problem is that the path difference
accumulateswhile the wave is in the thin film! To get the phase difference right, then,
we have to use the wavelength (and hence wave number)in the thin film mediumn 2 , not
the one we used in the originating mediumn 1 , or worse, the one that the light would
have in a vacuum!
You should recall that:
λ 2 =λ
n 2(1090)
whereλis the wavelength of the light in a vacuum. This leads to a wavenumber of:k 2 =^2 πn^2
λ(1091)
and a phase shift of:
δpath=k 2 (2d) (1092)Basically, the wave that traverses the thin film accumulates phase at the spatial rate of
k 2 , notk,k 1 , ork3.
Using k instead of k 2 is a very common mistake made by students of physics!
Don’t let it be you!
Next, let’s examine the phase shifts due to the actual reflections themselves.13.11.2: Phase Shifts Due to Reflections at the Surfaces
As you should remember from the treatment of waves in the first half of this course (see
my^136 book online if all of this eludes you.), a wave pulse on a string that partially
reflects off of the junction with aheavierstring (slower speed)flips over, where a wave
pulse on a heavier string that partially reflects of off the junction with a lighter one does
not. The transmitted wave pulse in both cases does not flip.
Exactly the same thing happens for harmonic wave trains or wave pulses in the case of
light. If a harmonic light wave reflects off of a denser medium (which usually has a higher
index of refraction and a slower velocity of light) the reflected waveinverts. Inversion is
basically multiplication by a minus sign, or equivalently (for harmonic waves) shifting the
phaseof the reflected wave byπor the heuristic equivalent half-wavelength. If a harmonic
light wave reflects off of a lighter medium (lower index of refraction) the reflected wave
does not flip, it retains it’s original phase.
There are thus four permutations of sort order for the indices ofrefractionn 1 ,n 2 ,n 3.
They are:
Istrongly recommendthat when you solve a problem involving thin film interference, you
circle the reflectionsthat have a phase shiftδij=πand write a little “π” next to each
one, as I did in figures??and 193 above. Then you are less likely to forget to include it
in your overall computation and understanding of the total relative phase shift.Leaving
out one or more of these phase shifts(and getting the max’s and min’s backwards
as a result)is another common error. Don’t do it!
Now we are ready to put all of this together and and determine the heuristic conditions
for maxima and minima. We’ll do this twice, once for each of the two “opposite” rules
one gets for max’s and min’s.(^135) ...and possibly even other things that begin with ‘h’, such as hydrogen bombs. Being “almost” hit by a hydrogen
bomb can ruin your whole day...
(^136) http://www.phy.duke.edu/ rgb/Class/introphysics1.php Introductory Physics 1