Physics 471 Final Exam
Instructor: John Colton
Winter 2008
Name: ________________________
I promise I do not have any “illegal” constants/formulas stored in my calculator:
(signed) ________________________
Instructions: Closed book. 3 hour time limit, 1% penalty per minute over. Calculators permitted. Show your
work. Include units where appropriate. If additional space is needed, you may use the backs of pages.
Formulas:
p 

Nq 2
m 0
Lorentz model: ro 

/4:

cos 2   i sin 2 
sin  cos   i sin  cos  


 sin  cos   i sin  cos 

sin 2   i cos 2 


 cos 2 sin 2 

/2: 
 sin 2  cos 2 
qe
Eo
and
2
me o   2  i
p2
 0 2   2  i
2
2
2
uy
u
u
1
 2 x 2  2
 2 z 2
2
2
n
n  nx
n  ny
n  nz
p
  2  i
cos  2
n
Fresnel Eqns:  
,   2
n1
cos 1
 
2
(p-polarization) r 
,t
 
 
2
1  
(s-polarization) r 
,t
1  
1  
Jones vectors
 A 
general, standard form:  i 
 Be 
2
Metals:  
Uniaxial, optic axis  to surface:
no n e
n  no ,
2
2
no sin 2  2  ne cos 2  2
n
sin 1
(p-polar)
tan  2  e
no n 2  sin 2 
e
tan   
no
ne
1
sin 1
ne  sin 2 1
2
(p-polar)
Two interfaces:
t13 
 1 
RCP: 1  i ( 90) 
2 e

1  1 
LCP:
 i ( 90) 
2 e

T13 
e
ik2 d cos 2
t12t 23
 r21r23e ik2d cos 2
n3 cos  3
2
t13 
n1 cos 1
Tmax
1  F sin 2
Jones matrices
 cos 2 
linear pol: 
 sin  cos 
2
sin  cos  

sin 2  
Tmax
1

2
2
n cos  3 t12 t 23
T12T23
 3

2
n1 cos 1 1  r21 r23 
1  R21 R23


2
F
1
n  2n  0
2c
Gaussian wavepacket, through thickness z:

4 r21 r23
1  r
21
r23 
2
  2z
T   1 2
  2k 2 d cos  2   21   23
4
 FW MH 
F
2
 FW MH 
n2 d cos  2 F
 FSR 
E t , z  
2n2 d cos  2
2 1/ 4
 ( ) 


cos  j
Mj 
  in cos sin 
j
j
j

 n 0 cos  0

 n 0 cos  0

I
e
1  z
t
2T 2  vg
0
 d
vg  
Re( k )
 d
 0





 I ( )e
i
d

( )dt  2 0 1  Re  ( ) 

I det ( )  2 I 1  Re  ( )
FT NormSig 
 2 0    I    I   
2
Young:
 i sin  j 

n j cos j 
cos  j 

1  N
1
  M j 
 1 j 1
n
cos
 N 1
 N 1
det

1

 kyh

I det (h)  2 I 0 1  cos
    (pt source)
D



Extended source:

0

0 
 0   I ( y )dy 

Linear dispersion:
1
e

ikyh
D 

ikhy 
R
 0 
I det (h)  2 I oneslit 1  Re  (h)

 
Rays: R(r )  n(r ) sˆ(r )
ABCD Matrices:
 ( h) 


d
Re( k )
 r
d
 0
I e


s-polar:


2 kimag (0 )r
2

 i sin  j cos  j 

 cos  j



nj
Mj 

 in j sin  j


cos  j
 cos 

j


 cos  N 1 0 
 n0 cos  0  N
1

  M j 

A
0 
2n0 cos  0  n0  cos  0  j 1
 n N 1
t 
e




 0   I ( )d
t13 = 1/a11
r = a21/a11
j = kjljcosj
p-polar:
1
2n 0 cos  0
1   
i
i   z
tan 1  
t
2
2 T 2  vg
Michelson:
I det ( )  2I 0 (1  cos  ) (single )
Band of ’s:
2
Multilayers:
A
E0 e
i  kz 0t 
 2
E(t  t , r0 )
I ( y)e
dy
1 d 
Translation 

0 1 
Quadratic dispersion:
1
2
k  k0    0      0 
vg
1
Flat surface refraction 
0

1 1
 n  n   
0
vg c
0
n1 

n2 
1
0

Curved surface refraction  1  n1  n1 
   1

R n

  2  n2 
2




2
 2 J1 k  a  


 k a 
R = positive for convex, negative for concave
 1

 1 f

Spherical mirror/thin lens
Top hat: a 2 

0

1

1
f lens
 n
 1
1 
  2  1   , R = positive for curving
 n1  R1 R2 
Spectrometer:  
away; negative for curving towards
R
f mirror  , R = positive for concave
2
p1 = (1–D)/C, p2 = (1–A)/C
 min 
Gaussian Beams:
Diffraction formulas
Fresnel-Kirchoff:
E ( x, y , z  d )  
i


E ( x , y , z  0)
e ikR
dx dy 
R
aperture
Fresnel:
E ( x, y, d )  

k 2 2
i
x y
ikd 2 d
ie e
d

i
k
 E( x, y,0)e 2d
x 
2
 y2

e
i
Plank:  f 
k
 xx  yy  
d
E ( x, y, d )  
i
aperture

k 2 2
x y
2d

d
 E( x, y ,0)e
i
k
 xx yy 
d

e
dx dy 
aperture
z 

8hf
3

i ( t  t 0 )
d  2 (t  t 0 )

Fourier Transforms: (without factors of sqrt(2) )
Comb function (N total deltas):
1 
c 3 e hf k B T  1
 = 5.669610-8 W/m2K4
Integrals:
dxdy
Fraunhofer:
ie ikd e
ik 2
 2
ikz 
i tan  
w
2
2R
 z0 
E  x , y , z   E0 0 e w e
w
2
2
2
kw0
z0
with z 0 
, R z
, w  w0 1  z 2
2
z
z0
Aq  B
; q = z + iz0
q2  1
Cq1  D
A D
1
2
Cavity stability:  1 
1.22
l
xh

,  
mN
md

e
 N t 0 
sin 

 2 
 t 
sin  0 
 2 

Single slit: asinc(kxa/2)
3
 Ax 2  Bx  C
dx 

A
e
B2
C
4A
, Re( A)  0
Final Exam – 110 total points possible
True/False. Please circle the correct answer. (2 pts each)
1. T or F: T or F: When light is incident upon a material interface at Brewster’s angle, only one
polarization can transmit.
2. T or F: Aside from a constant factor, the Fourier transform of a convolution is the
convolution of the Fourier transforms of the individual functions.
3. T or F: Spherical waves of the form
A
coskr  t  are exact solutions to Maxwell’s
r
equations.
4. T or F: The resolving power of a spectrometer used in a particular diffraction order depends
on the number of lines illuminated, but not on the wavelength or grating period.
5. T or F: The central peak of the Fraunhofer diffraction from two narrow slits separated by a
(a >> slit width) has the same width as the central diffraction peak from a single slit of width
a (width of the central peak being measured from first zero on left to first zero on right).
6. T or F: The function J1(x) crosses zero at 0, , , 3, etc.
Multiple Choice. Please circle the letter of the correct answer for full credit (2 pts each).
7. If sunlight is unpolarized coming from the sun, after it reflects off of a smooth surface when
the sun is in front of (and above) you, it will probably be:
a. more horizontally-polarized
b. more vertically -polarized
c. still unpolarized
8. In the analysis of a three-layer system, the electric field at the right side of the middle layer
was connected to the electric field at the left side of the middle layer via:
a. changing polarization
b. the Fresnel coefficients
c. a phase factor
d. a rubber band
9. The approximation made to derive the eikonal equation and hence Fermat’s principle of least
time was:
a. short wavelengths
b. long wavelengths
10. “Optical path length” depends on something besides length. That other thing is:
a. index of refraction
b. angle of incidence
c. frequency
4
11. If you want to focus a laser beam to a tight spot, spherical aberration can be partially
corrected by placing a plano-convex lens:
a. so the light strikes the flat side
b. so the light strikes the curved side
12. Whose law says that the amount of emitted blackbody radiation is proportional to T4?
a. Curie & Weiss’s
b. Planck’s
c. Rayleigh & Jeans’
d. Sommerfeldt’s
e. Stefan & Boltzmann’s
13. Which of these processes was not considered in the “Einstein A and B coefficients” analysis?
a. spontaneous absorption
b. stimulated absorption
c. spontaneous emission
d. stimulated emission
Problems. Please answer the following questions/solve the following problems.
14. (5 pts) At a certain frequency a material has n = 3 and = 4. (a) Find the complex
susceptibility . (b) Find the phase of the polarization relative to the phase of the electric
field.
5
15. (8 pts) Suppose you have a laser that is vertically polarized. What optical element could you
use to turn it into a right circularly polarized laser beam and how should it be arranged? Be as
specific as possible.
6
16. (5 pts) The first lens of a telescope has a diameter of 30 cm, which is the only place where
light is clipped. You wish to use the telescope to examine two stars in a binary system. The
stars are approximately 25 light-years away from Earth. (One light-year is 9.46051015 m.)
How far apart need the stars be from each other (in the perpendicular sense) for you to
distinguish them in the visible range of  = 500 nm?
17. (6 pts) A laser cavity is formed with four flat mirrors
and a lens of focal length f (see figure). Let the full
path around the cavity be L.
a. What is the round-trip ABCD matrix for the
cavity? Please start by having the light first
go through the lens.
b. What are the possible values for L if the
cavity is to be stable? Let the focal length of
the lens be 1 meter.
7
3a
18. (6 pts) Derive the Fraunhofer intensity pattern for the two
identical rectangular apertures as shown at the right, whose
centers are separated by 4a. Put your answer in terms of the
wavevector k, the distance to the screen d, the x- and ycoordinates on the screen, and the intensity at the center of the
pattern I0. Hint: you can start with the FT of a single slit
aperture function.
a
4a
3a
a
Extra Credit (3 pts) Use your Fourier transform intuition and/or an analysis of your above
answer to sketch what the Fraunhofer pattern would look like. Advice: It’s probably wisest to
save this for after you’ve finished the rest of the exam.
8
19. (14 pts) P-polarized light travelling inside a diamond (n = 2.4) strikes a surface
going to air (n = 1) at an angle of 50 with respect to the perpendicular. (a)
Draw the direction of the light’s oscillating electric field on the picture on the
paper. (b) Find the reflection and transmission coefficients, r and t, as well as
the reflectance and transmittance, R and T.
9
20. (14 pts) A thin glass film is suspended in air, with index n = 1.8 and thickness w. It has
s-polarized 633 nm light incident on it at an angle of 30 from the perpendicular.
a. What is the smallest non-zero thickness w that will give a maximum in the transmission?
b. Evaluate T13 for w = 0.5 m.
10
21. (14 pts) A telescope is formed with two thin lenses separated by the sum of their focal
lengths f1 and f2. Rays from a given far-away point all strike the first lens with essentially the
same angle 1. “Angular magnification,” M, is defined as 2/1 and quantifies the
telescope’s purpose of enlarging the apparent angle between points in the field of view.
Use ABCD-matrix formulation to derive M of this system in terms of f1 and f2.
11
22. (12 pts) Suppose you send a nearly non-diverging laser beam of width w = a into a lens of
focal length f. Use the ABCD law to determine the q-parameter and hence the nature of the
beam coming out of the lens. Specifically: (a) What will the beamwaist of the new beam be?
Write your answer in terms of f, a, and the wavelength of the laser. (b) How far after the lens
will the beam hit this waist? Show that if f << the Rayleigh range of the incident beam, this
answer becomes exactly what a Physics 123 student would expect for the distance to the
focus. Hint: for the incident beam to be “non-diverging”, it must be near its own waist as it
strikes the lens.
12