PHY 142!
Assignment 10!
Summer 2013
Reading: Wave Optics 1, 2
Key concepts: Superposition; phase difference; amplitude and intensity; thin film interference;
Fraunhofer diffraction; gratings; resolving power.
1.!
2.!
3.!
Questions about interference of waves.
a.!
Why do we not see interference patterns in the light from the two
headlights of a distant car?
b.!
You are standing outdoors near a corner of a building. Two people around
the corner are talking. You can hear them but not see them. How do you
hear them? [Reflection from other objects is negligible.]
c.!
You know the two people are talking, but you are unable to understand
their conversation. Why? [They are speaking your language.]
Questions about thin films.
a.!
When a film of water on glass evaporates the reflected light becomes very
bright just before the film is gone. Why?!
b.!
When a soap film in air gets thin just before breaking the reflected light is
very dim. Why?
c.!
A circular film of oil on water is thinnest near the edge. In the reflected
light, would one see blue or red colors in that region of the film?
In Assignment 8 you examined the intensities of the reflected
and transmitted rays shown. To analyze the interference of the
waves you must know the amplitudes. Since I  A 2 The
relations between amplitude magnitudes are:
a
b
c
A0
Arefl = R ⋅ Ainc , Atrans = T ⋅ Ainc .
a.!
Show that if R << 1 then we have Ab ≈ Aa >> Ac . [This
a′ b ′ c ′
is why we considered only waves a and b in analyzing
interference in the reflected light from a thin film.]
b.!
Show that in this case Aa′ ≈ A0 >> Ab′ >> Ac ′ . [This is why interference
effects in the transmitted light are negligible for ordinary films.]
c.!
Show that if R ≈ 1 we have Aa ≈ A0 >> Ab >> Ac , so usually nearly all the
light is reflected and interference is (usually) negligible.
PHY 142!
4.!
Assignment 10!
Summer 2013
The exception to the conclusion in 3(c) occurs when all the transmitted waves
interfere constructively.
a.!
Write these amplitudes in terms of A0 and R.
b.!
Assuming all these waves are in phase, find the total transmitted
1
amplitude. Use the fact that if x < 1 then 1 + x + x 2 + x 3 + … =
. What
1− x
does this say about the transmitted light? What about the reflected light?!
5.!
We examine the case R ≈ 1 in more detail. The transmitted waves have (complex)
amplitudes
Ea′ = A0T, Eb′ = A0T ⋅ Re iδ , Ec ′ = A0T ⋅ R 2 e 2iδ ,
!
and so on. Here δ is the phase difference caused by the wave traveling back and
forth between the surfaces; thus δ = 2nkt , where n is the film’s refractive index
and t is its thickness.
6.!
Itrans
(1 − R)2
=
. [Remember e iδ + e −iδ = 2cos δ .]
2
I0
1 + R − 2R cos δ
a.!
Show that
b.!
Let R = 0.99 . Use a graphing calculator to plot this formula as a function
of δ for π ≤ δ ≤ 3π . Note the extremely narrow peak at δ = 2π .
c.!
In (b) take n = 1.5 and t = 200 nm. For what visible wavelength will there
be nearly total transmission? If you look at an incident beam of white light
transmitted through this system, what will you see?
In an attempt to make a very narrow beam of light, an engineer passes a plane
wave through a circular hole of small diameter. To his dismay, as he makes the
hole smaller the beam that emerges spreads out more from the original direction
of the wave.
a.!
Why is this happening?
b.!
Is it peculiar to light, or a general property of waves?
c.!
Show that the product of the diameter of the opening and the angular
width δθ of the beam is a constant (for given wavelength).
PHY 142!
7.!
Assignment 10!
Summer 2013
If light is interpreted as a stream of particle-like photons, the spread δθ in the
directions they move is interpreted as a spread δ p⊥ in the component p⊥ of their
momentum (p) perpendicular to the original direction of the beam. That is, before
they went through the circular opening they were all moving in the same
direction, but afterwards their directions vary within the range δθ .
Approximately we have δ p⊥ = p ⋅ δθ . The momentum of a photon is related to its
wavelength by the quantum formula p = h / λ , where h is Planck’s constant.
Show that D ⋅ δ p⊥ ≈ 1.22h . [This is an example of the Uncertainty Principle.]
8.!
A TV signal is received by an old fashioned antenna connected to the set. The
reception from a station whose antenna is 15 miles away is rather weak. When an
airplane files over the line from the transmitting antenna to the receiver, the
received signal alternately rises and falls in strength. Explain what is happening.
9.!
Questions about resolving power.
a.!
The resolution of an optical microscope can be increased by immersing the
objective lens and the sample in a transparent oil. Why does this work?
b.!
Radio telescopes make images of distant objects using the waves emitted
in the radio part of the spectrum. If a radio telescope receiving waves with
λ = 3 cm is to have the same resolution the 5 m (diameter) Hale telescope
has with 500 nm light, how large must the diameter of the antenna be?
c.!
The electrons in an electron microscope are accelerated through a potential
difference sufficient to give them a momentum p = mc . [This does not
mean they are traveling at speed c; the formula p = mv is valid only for
speeds much smaller than c.] Find their deBroglie wavelength from the
formula λ = h/ p , where p is the momentum and h is Planck’s constant.
Ans: 2.43 × 10−3 nm.
10.!
Questions about gratings.
a.!
A grating’s ability to resolve two spectral lines with nearly the same
wavelength is improved if the intensity peaks are narrower. Which factor
in the formula for resolving power R = mN expresses this fact?
b.!
This ability is also improved if the peaks are farther apart in angle. Which
factor express this fact?
c.!
Show that no matter how many lines a grating has, the visible spectrum in
3rd order always overlaps that in 2nd order.
PHY 142!
11*.!
12*.!
13*.!
Assignment 10!
Summer 2013
A wedge-shaped air film is produced by
light
slipping a sheet of paper of thickness t between
two flat glass plates as shown. Light of
t
wavelength 500 nm comes from above at
normal incidence, and the pattern in the
reflected light is observed. There are 400 bright bands across the length of the
plate, and there is a dark band at the end where the paper is inserted. The plates
are 10 cm long and the glass has refractive index 1.4.
a.!
What is t? Ans: 0.1 mm.
b.!
Suppose the region between the plates were filled with oil of refractive
index 1.25. Then how many bright bands would be seen? Ans: 500.
c.!
Suppose the plates are 20 cm long, but the same sheet of paper is slipped
between them as shown. How would this affect the pattern observed
(without the oil)?
A lens is coated so as to reduce reflection in the infrared and ultraviolet regions.
The coating causes destructive interference in the reflected light for wavelengths
1080 nm and 360 nm but no wavelength in between.
a.!
If the coating has refractive index 1.4 and the lens has refractive index 1.6,
what is the minimum thickness of the coating? Ans: 193 nm.
b.!
For what visible wavelength is there constructive interference in the
reflected light? Ans: 540 nm.
A grating 2 cm wide is used to determine wavelengths of visible light.
a.!
If two wavelengths with λ av = 500 nm and Δλ = 0.025 nm are to be
resolved in second order, What is the largest that the spacing d between
the rulings can be? Ans: 2 × 103 nm.
b.!
Will the entire visible spectrum [400 to 700 nm] be seen in third order for
this grating? Explain how you know.
c.!
How small can Δλ be for the situation in (a) if the third order spectrum is
used?
PHY 142!
Assignment 10!
Summer 2013
14.!
Real light sources never produce only one frequency, so the interference pattern
of a film is never as sharp as the equations suggest. This “fuzziness” increases as
the thickness of the film increases. You are to show why.
!
Consider a thin air film between glass plates. Let the incident light consist of all
frequencies between f and f + Δf . For frequency f the phase difference between
the waves reflected back into the top glass plate from the top and bottom of the
air film is δ1 = 2mπ (where m is an integer) so there is constructive interference in
the reflected light. For frequency f + Δf , we have δ 2 = (2m + 1)π , giving
destructive interference. For this light source the total reflected intensity will be
the same as if there were no interference at all.
a.!
Write the general formula for the phase difference between the waves
reflected at the top and bottom of the film in terms of frequency f and
thickness t of the film.
b.!
Find the value of Δf in terms of t for which the situation described occurs.
c.!
Take the two wavelengths involved to be 500 nm and 501 nm. For what
value of t will the situation described occur? Ans: 62.5 µm.
!
[! A film this thick or thicker will not give a discernible interference pattern
for this source.]