Final Examination
Introduction to Optics & Photonics
EECS 334
Instructor: Professor S.C. Rand
Date: April 20, 2007
Time: 1:30-3:30 p.m.
Duration: 2 hours
PLEASE read over the entire examination before you start. DO ALL QUESTIONS and
show all your work in submitted material to be eligible for partial credit. No textbook or
lecture materials or notes of any kind may be consulted. Additionally, no homework
solutions or electronic aids of any kind are permitted.
Honor Pledge: I have neither given nor received aid on this exam.
__
Name (Print)
____________________________________
Student ID
___________________
Signature
Total
Useful Formulae:
ni sin  i  nt sin  t
1 1 1
 
s s' f
1 1 n2  n1
 
s1 s 2'
n1
1
1
  
 R1 R2 
2 y 1 2 y

t 2 v 2 t 2
WE  1 2 E 2 ; WM  1 2 H 2 ; WTOT  2WE
EB
Power
S  Irradiance 
 EH ; S 
 EH
Area

v
v
  0 (1  ) ;    0 (1  )
c
c
 n
v g  v p (1 
)
n 
h

I  I min
s'
Magnification: m  i  i   ; Visibility: V  max
ho  o
s
I max  I min
I TOT  I1  I 2  2 I1 I 2 cos  , where δ is the phase difference between the waves
2
Phase difference δ and path difference Δ:   ( ) 

Interference in thin films: t   / 4n f and n f  n0 ns
T02
IT
, where T0  (tt ' t * t '*)1 / 2 ,

2
I 0 1  R0  2 R0 cos 
2
and   2n f t cos  

Multiple-beam interference: T ( ) 
R0  (rr ' r * r '*)1 / 2
m  2nd cos
 SFR
 R d  
c




 mF ; F 
; Q
;
2nd


 FW HM 1  R d
 FW HM
c

r
l  c 

; l

1 / 2
s
1.22
1 / 2 
D
 SFR 
2
 sin N   sin  
 where   12 ka sin  and   12 kb sin 
N slits: I P  I 0 
 
 sin     
Grating Eqn: m  a(sin  i  sin  m )
2
1. (10 marks total)
1A. Two identical concave mirrors of radii of curvature R are separated by a total
distance equal to R (diagram below). A light ray is emitted from a point P on the optical
axis that is centered between the mirrors at a small, arbitrary angle with respect to the
axis so that it strikes one of the mirrors to the right or the left.
(i) Carefully draw on the diagram the path subsequently followed by your chosen ray in
the region between the mirrors, including additional reflections if there are any.
(2 marks)
R
(ii) Based on your ray tracing above, comment on whether you think the mirrors form a
“stable” or “unstable” cavity for the light. That is, do the mirrors confine light effectively
between them or not? (2 marks)
1B. (i) For a Fabry-Perot etalon of index n and fixed thickness d, what change in
refractive index n would scan the transmission by one order ( m  1)? (2 marks)
(ii) A Fabry-Perot etalon is fabricated from a 1 mm thickness of glass by silvering its
two flat parallel, polished surfaces. The etalon is placed inside an oven to measure its
temperature drift, by monitoring transmission of a laser beam at =500 nm. The change
in refractive index of the glass with temperature T(K) is known to be n / T  10 5 K 1 .
If the F-P transmission is initially at a maximum and passes through 13 peaks over the
course of an hour, by how much did the temperature of the oven vary during this period?
(Ignore thermal expansion of the glass.) (4 marks)
2. (10 marks total)
Fresnel analyzed the geometry of diffraction of light from a source S passing through a
circular aperture by dividing up the aperture plane into circular zones of radius Rn, such
that the total pathlength rn  rn' differed by one half wavelength from the pathlength
( rn 1  rn' 1 ) associated with the next zone. The geometry is shown in the diagram below.
r'
S
r
R
h'
P
h
(a) Find an exact expression for rn  rn' in terms of the radius Rn to the boundary of the
nth Fresnel zone, and distances h and h'. (3 marks)
(b) Write down a similar expression for rn 1  rn' 1 in terms of Rn+1, h and h' using the
result in (a). (1 mark)
(c) Recall that the mean pathlength for successive Fresnel zones differs by /2. Find a
relationship between Rn+1 and Rn using (a) and (b), and simplify it for the usual case
when h,h'>>R. Note that this provides a relationship between radii of any two adjacent
zones. (The values of n are n=0,1,2,3...) (3 marks)
(d) Determine the radius of the boundary of the first Fresnel zone (n=1) for an
unobstructed circular aperture. (1 mark)
(e) State an application of the so-called Fresnel Zone Plate, constructed by making even
numbered Fresnel zones opaque. (1 mark)
(e) If a small circular aperture is illuminated uniformly by a plane wave, what is the
name of the mathematical transformation that describes the intensity pattern observed on
a distant screen? (1 mark)
3. (10 marks)
A solid state laser utilizes a Ti:sapphire rod in which the bandwidth of the spontaneous
fluorescence is 1000 GHz. Two flat cavity mirrors are glued directly to the rod to form
the cavity.
(a) To achieve optical gain in the laser medium, assuming there are N2 and N1 atoms (per
unit volume) in the upper and lower states of the emitting transition respectively, what
condition is necessary? (1 mark)
(b) If the laser rod 10 cm long, what is the frequency separation between modes of the
cavity? Assume the refractive index of the rod is n=1.7. (3 marks)
(c) If the diameter of the laser rod is 2 mm and its output wavelength is 3.0 microns, how
large will the output beam diameter be after propagating a distance of 4 km?
(4 marks)
(d) If the laser is allowed to run in a multi-mode condition by removing all frequency
selective elements in the cavity, what is its temporal coherence time? (2 marks)
4. (10 marks)
The Fraunhofer diffraction pattern from two microscopic circular holes separated by a
distance d is observed at a wavelength of 500 nm on a screen placed 50 cm from the
aperture plane. Inspection of the pattern shows that it is at the Rayleigh limit of
resolution (i.e. the principal maximum appears to consist of two peaks separated by a
small intensity dip).
(a) In one sentence, state the Rayleigh criterion for diffraction-limited resolution of two
sources such as these. (Answer must be different from information above.) (2 marks)
(b) If the separation of the two barely discernible peaks in the principal maximum of the
overall diffraction pattern is 1 mm, then what is the actual separation of the circular
holes? (8 marks)
5. Short problems (worth 1-3 marks each - total of 10 marks)
(a) State Babinet's Principle in a few words and give a simple (mathematical) derivation
of the principle, similar to that given in lecture. (2 marks)
(b) An aperture consists of 10 slits spaced by a distance of 100 microns. When the
aperture is uniformly illuminated by a plane wave at 500 nm, at what angle with respect
to the optical axis would the first minimum be observed? (3 marks)
(c) A researcher wants to produce a pulsed source of light with a central wavelength of
500 nm and a duration so short that it consists of only five periods of the electric field.
What bandwidth is necessary to accomplish this? (2 marks)
(d) An undergraduate research assistant inserts a sample of length 1 cm into one arm of a
Michelson interferometer. The interference pattern shifts by 2/3 of a fringe at a
wavelength of 500 nm. Assuming the sample has perfectly parallel end faces, what is the
refractive index of the sample? (3 marks)