A Simple Introduction to Interference
First, consider two wave pulses on a string, approaching each other.
Assume that
each moves with speed meter per second. The figure shows the string at time
. The
effect of each wave pulse on the string (which is the medium for these wave pulses) is to
displace it up or down. The pulses have the same shape, except for their orientation.
Assume that each pulse displaces the string a maximum of
on the x axis is in meters.
meters, and that the scale
Part A
At time
, what will be the displacement
at point
Express your answer in meters, to two significant figures.
=0
?
Correct
Part B
Choose the picture that most closely represents what the rope will actually look like at
time
.
DCorrect
The same process of superposition is at work when we talk about continuous waves
instead of wave pulses. Consider a sinusoidal wave as in the figure.
Part C
How far
to the left would the original sinusoidal wave have to be shifted to give a
wave that would completely cancel the original? The variable in the picture denotes the
wavelength of the wave.
Express your answer in terms of .
=0.5*lambdaCorrect
Part D
In talking about interference, particularly with light, you will most likely speak in terms
of phase differences, as well as wavelength differences. In the mathematical description
of a sine wave, the phase corresponds to the argument of the sine function. For example,
in the function
, the value of
at a particular point is the phase of the wave
at that point. Recall that in radians a full cycle (or a full circle) corresponds to
How many radians would the shift of half a wavelength from the previous part
correspond to?
Express your answer in terms of .
phase difference =pi radiansCorrect
radians.
Part E
The phase difference of radians that you found in the previous part provides a criterion
for destructive interference. What phase difference corresponds to completely
constructive interference (i.e., the original wave and the shifted wave coincide at all
points)?
Express your answer as a number in the interval
phase difference =0 radiansCorrect
.
Part F
Since sinusoidal waves are cyclical, a particular phase difference between two waves is
identical to that phase difference plus a cycle. For example, if two waves have a phase
difference of
, the interference effects would be the same as if the two waves had a
phase difference of
. The complete criterion for constructive interference between
two waves is therefore written as follows:
Write the full criterion for destructive interference between two waves.
Express your answer in terms of and .
phase difference =pi+pi*2*n
Correct
The phase for a plane wave is a somewhat complicated expression that depends on both
position and time. For most interference problems, you will work at a specific time and
with coherent light sources, so that only geometric considerations are relevant. Consider
two light rays propagating from point A to point B in the figure, which are
ray follows a straight path, and the other travels at a
apart. One
angle to that path and then
reflects off a plane surface to point B. Both rays have wavelength .
Part G
Find the phase difference between these two rays at point B.
Express your answer in terms of .
phase difference =3*pi/2 radiansCorrect
Part H
Suppose that the reflected ray receives an extra half-cycle phase shift when it reflects.
What is the new phase shift at point B?
Express your answer in terms of .
phase difference =5*pi/2 radiansCorrect
Whenever light reflects from a transparent interface, moving from lower index of
refraction to higher index of refraction, it gets an extra half cycle phase difference. Being
able to accurately find the phase differences between waves at various points will be
useful in both interference and diffraction problems.
Why Butterfly Wings Shimmer
Part A
Assume that light is incident normal to the surface of a film of thickness . How much
farther does the light reflected from the back surface travel than the light reflected from
the front surface?
Express your answer in terms of .
2*dCorrect
Part B
For constructive interference to occur, the difference between the two paths must be an
integer multiple of the wavelength of the light (as is true in any interference problem), i.e.
the general criterion for constructive interference is
, where is a
positive integer. This is usually stated in the slightly more explicit form
.
Given the thickness of the film , what is the longest wavelength that can exhibit
constructive interference?
Express your answer in terms of .
=2*dCorrect
Part C
If you have a thin film of thickness
, what is the third-longest wavelength
of light that exhibits constructive interference with the reflected light?
Note that this corresponds to
.
Express your answer in nanometers to three significant figures.
=200
Correct
Part D
The criterion for destructive interference is very similar to the criterion for constructive
interference. For destructive interference to occur, the difference between the two paths
must be some integer number of wavelengths plus half a wavelength:
,
or
,
where is a nonnegative integer. What is the second-longest wavelength
that
will not be visible (i.e., will have strong destructive interference for the reflected waves)
when reflected from a film of thickness
?
Note that the longest wavelength corresponds to
for destructive interference. This is
why the notation used for the second-longest wavelength is
instead of
.
Express your answer in nanometers to three significant figures.
=400.0Correct
The blue morpho butterfly lives in tropical rainforests and can have a wingspan greater
than 15 centimeters.
The brilliant blue color of its wings is a result of thin-film
interference. A pigment would not produce such vibrant, pure colors. What cannot be
conveyed by a picture is that the colors vary with the viewing angle, which causes the
shimmering iridescence of the actual butterfly.
The scales of the butterfly's wings consist of two thin layers of keratin (a transparent
substance with index of refraction greater than one), separated by a 200-nanometer gap
filled with air.
Part E
What wavelength of light would be strongly reflected at normal incidence? The keratin
layers are thin enough that you can think of them simply as marking the surfaces of a
200-nanometer "film" of air.
Express your answer in nanometers to two significant figures.
=400
Correct
This wavelength is near the cutoff between visible (violet) and ultraviolet light, so the
shorter wavelengths that are strongly reflected will not be relevant to what humans see
when they look at the butterfly.
To understand why the color changes with viewing angle, try drawing a diagram of light
incident on a thin film at a large angle. The distance within the film will be increased, but
the light reflected from the front surface will have to travel further to the observer outside
of the film than the light reflected from the back surface. The increased distance outside
of the material for the front surface reflection actually makes the net path-length
difference smaller than it would be for normal incidence. As viewing angle increases, the
largest wavelength that experiences constructive interference gets shorter. Thus, while the
butterfly is blue at normal incidence, at large angle of incidence no visible light is short
enough wavelength to be strongly reflected.
Part F
The wavelength that we have been discussing is technically the wavelength of light
within the medium of the film. It is important to remember this, since most thin-film
problems will involve films with index of refraction different from that of air. Suppose
that the butterfly gets wet, thus filling the gaps between the keratin sheets with water
(
). What wavelength
in air will be strongly reflected now?
Note that the wavelength within the film was already determined in the previous problem.
Express your answer in nanometers to two significant figures.
=530
Correct
Part G
Several thin films are stacked together in each butterfly wing scale. How would these
multiple layers of thin films affect the light reflected by the butterfly's wings?
More 400-nanometer light would be reflected, because some of the light transmitted
through the first layer could be reflected by the second layer, light transmitted by the
second layer could be reflected by the third, etc.
Correct
Layering of thin films is used to make wavelength-specific mirrors, used widely in laser
applications, that are far more reflective than metal mirrors. Layered films are also used
to make antireflective coatings for camera lenses.