University of California, Berkeley
Physics 110B Spring 2004 (Strovink)
EXAMINATION 4
Directions: Do all three problems, which have unequal weight. This is a closed-book closed-note
exam except for Griffiths, Pedrotti, a copy of anything posted on the course web site, and anything
in your own original handwriting (not Xeroxed). Calculators are not needed, but you may use one if
you wish. Laptops and palmtops should be turned off. Use a bluebook. Do not use scratch paper –
otherwise you risk losing part credit. Show all your work. Cross out rather than erase any work that
you wish the grader to ignore. Justify what you do. Express your answer in terms of the quantities
specified in the problem. Box or circle your answer.
Problem 1. (20 points)
A plane wave of initial irradiance I0propagat-
ing along ˆzis incident upon a screen lying in
the z= 0 plane. The screen is totally absorbing,
except that one quadrant of it (the piece with
0<x,y<∞) is missing. An observer stationed
at (0,0,z), where kz 1, detects an irradiance
I.WhatisI/I0, and why?
Problem 2. (40 points)
(a.) (20 points)
Consider four equally spaced long (∆y=∞)
thin slits, located in the z= 0 plane at x=±d
2
and x=±3d
2. As usual, tan ψx=dx
dz of the out-
going wavefront. Consider the aperture function
for these four slits to be the convolution of a
pair of δ-functions separated by dand another
pair of δ-functions separated by 2d(both pairs
are symmetric about x= 0). In the Fraunhofer
limit, write down the irradiance pattern
R(ψx)≡I(ψx)
I(ψx=0)
as the product of two two-slit R’s.
(b.) (20 points)
Consider an opaque baffle in the z= 0 plane in
which a 4×4 array of 16 tiny holes is drilled. The
holes are arranged on a square grid, with verti-
cal and horizontal hole-to-hole separation equal
to d. The entire array is centered at the origin
(where there is no hole). In the Fraunhofer limit,
defining tan ψx=dx
dz and tan ψy=dy
dz of the out-
going wavefront, deduce the resulting irradiance
pattern
R(ψx,ψ
y)≡I(ψx,ψ
y)
I(ψx=0,ψ
y=0) .
Please justify your answer.
Problem 3. (40 points)
A monochromatic beam traveling in medium
“0” is normally incident upon a substrate “T”.
Three films (“1”, “2”, and “3”) are interposed
between the two media, such that film 1 adjoins
medium0andfilm3adjoinsmediumT. The
refractive indices are frequency-independent and
equal, respectively, to n0,n1,n2,n3,andnT,
with n0=nT. You may assume that all mate-
rials are insulating and nonabsorbing, and that
they all have the same magnetic permeability.
Films “1” and “3” have thickness λi/4 (where
λiis the wavelength of the beam in the partic-
ular material of which that film is made), while
film “2” has thickness λi/2.
(a.) (20 points)
Work out a condition on n0,n1,n2,n3,andnT
that allows no light to be reflected.
(b.) (20 points)
Now the frequency of the monochromatic beam
is doubled (red light becomes blue light), while
all three films retain their same physical thick-
nesses (in meters). As a function of the (unre-
stricted) five refractive indices, what is the ratio
Rof the reflected to incident irradiance?