Lecture 22: More on Interference• Last lecture we used phasors to predict the interference

pattern that would appears if light of a given wavelengthwere passed through three equally-spaced slits

• The result is:

• The phasors for each of the above features are:1. Primary maximum:2. Secondary maximum:3. Minimum:

Primarymaximum

Secondarymaximum

Minimum

! = 0,2" , 4"…

! = ±" ,±3" ,±5"…

! = ±2

3" ,±

4

3" ,±

8

3"…

Even More Slits• Here’s the same thing, with five equally-spaced slits:

• The trend is that the primary maxima become even biggerrelative to the secondary maxima– also, there are now more secondary maxima– primary maxima are narrower

Phase Changes and Reflection• When light reflects from a surface, not only does the

direction of propagation change, but the phase may alsochange

• You’ll need to take an advanced electromagnetism courseto see why this happens

• For now, we’ll just state the rules:1. When light reflects from a medium with larger n than the

one in which the light was originally traveling, the phaseof the reflected wave shift by 180o

• i.e., wave is turned upside-down2. When reflecting from a medium with smaller n, there is

no phase change

Interference in Thin Films• When we see a layer of oil on top of water, or a soap

bubble in air, we often see a “rainbow” effect• This can be understood using the principles of reflection

and interference of light• Consider a transparent film with index of refraction n and

thickness t in air:

tn

The reflected light is a combination of reflections from thetop and bottomsurfaces of the film

180o phasechange

No phasechange

• To keep the math simple, we’ll only consider rays that arenearly normal to the surface of the film

• If the difference in path length between the two rays is amultiple of the wavelength, the interference will bedestructive– this is due to the 180o phase change for the ray reflecting

from the top surface– note also that the wavelength that’s important here is the

wavelength inside the film• We found earlier that this wavelength is related to the

wavelength in air by:

!n=!air

n

• So the rays will cancel, giving minimal total reflection,when:

and the rays will interfere constructively, given maximalreflection, when:

• If the thickness of a film varies slightly, some areas mightreflect red wavelengths most strongly, and other areasblue, etc.– this is where the “rainbow” of colors comes from

2t = m!n=m!

air

n; m = 0,1,2…

2t =

m +1

2

!"#

$%&'

air

n; m = 0,1,2…

What if the film isn’t in air?• The film of oil we mentioned earlier was on top of water,

not suspended in air– how does this change the behavior?

• It depends on how the index of refraction of the oilcompares with that of water– if noil > nwater then nothing changes!– but if noil < nwater then we have:

noil

180o phasechange

nwater

nair = 1

180o phasechange

Phase change forboth reflected raysin this case

• With two phase changes, the conditions for maximum andminimum reflected are reversed:Maximum when:

Minimum when:

• Note that rays passing through the film also interfere:

2t = m!n=m!

air

n; m = 0,1,2…

2t =

m +1

2

!"#

$%&'

air

n; m = 0,1,2…

noil

nwater

nair = 1

Interferencehere too!

The Michelson Interferometer• In the 1800’s, A.A. Michelson figured out a way to use

interference of light to measure very small distances• Idea is as follows:

• The point at the center of the screen will be either dark orlight, depending on the difference between L1 and L2

Screen

Incoming beam

Mirrors

Half-mirror“beam splitter”

L1

L2

• If the path length difference changes by only one-fourth ofa wavelength, the center point will switch from light todark (or vice-versa)

• Since visible light has a wavelength of a few hundred nm,this device is sensitive to very small changes in distance– or in the wavelength (or speed) of light– in Michelson’s day, physicists thought that light waves must

propagate through some medium, the aether– since this aether must fill the whole universe, the Earth

would have to be moving through it– Michelson set out to use his interferometer to measure the

speed as which the Earth moves through the aether• He got an completely unexpected answer:

– The speed was zero!– In fact, there was no indication that the aether existed at all

• Due to this null result, physicists were forced to rethinktheir notions of how light worked

• Eventually led Einstein to develop the theory of relativity,one of the cornerstones of modern physics

The lesson is: Sometimes in science not seeingsomething is an important result!