Midterm Examination 2
Introduction to Optics & Photonics
EECS 334
Instructor: Professor S.C. Rand
Date: November 24, 2009
Time: 10:30-12:30 p.m.
Duration: 2.0 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, with the exception of simple calculators, are
permitted. Having a ruler available for measurements and making clear diagrams is
recommended.
Honor Pledge: I have neither given nor received aid on this exam.
__
Name (Print)
____________________________________
Student ID
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Useful Formulae:
Imaging, refraction & propagation:
ni sin  i  nt sin  t ; ni sin  c  nt
1 1 1
 
s s' f
1 1 n2  n1  1
1
 ' 
  
s1 s 2
n1  R1 R2 
h

s'
Magnification: m  i  i  
ho  o
s
 2 ( z, t ) 1  2 ( z, t )
 2
z 2
v
t 2
WE  1 2 E 2 ; WM  1 2 H 2 ; WTOT  2WE
S  Irradiance  Power / Area  E  H ; S  EB /   EH
 n
v
v


v g  v p (1 
) ;    0 1   ;    0 1   ;
n 
c
c



; t  1

Coherence:
R
I max  I min
I max  I min
Coherent waves with random phase fluctuations emitted from two points S1, S2 of an
extended source and observed at a single point P:
I TOT  I 1  I 2   0 c Re E1 (t ) E 2* (t   )  I 1  I 2  2 I 1 I 2 Re( )
l  c ; Visibility: V 
 ( )  exp( i )( 0   ) /  0 and  0 is the source coherence time.
l
S1
s
S2
P1
Screen
r
P2
2
Incoherent waves emitted from two points S1, S2 of an extended source and observed
effectively at TWO points P1 and P2 by forming their fringe pattern:
I TOT  I 1  I 2   0 c Re E1,tot (t ) E 2*,tot (t   )  I 1  I 2  2 I 1 I 2 Re( ) , where
 ( )  sin( sl / r ) /( sl / r )
lt  r / s   / 1 / 2
Interferometry:
m  2nd cos
Interference in thin films: t   / 4n f and n f  n0 ns
T02
IT

, where T0  (tt ' t * t '*)1 / 2 ,
I 0 1  R02  2 R0 cos 
2
R0  (rr ' r * r '*)1 / 2 and   2n f t cos  

 SFR
 R d  

c

 SFR 


; Q
;
 mF ; F 
2nd


 FW HM 1  R d
 FW HM
Multiple-beam interference: T ( ) 
 sin N 
N slits: I P  I 0 

 sin  
Lasers:
I ( z )  I (0) exp( z )
  ( N 2  N1 )
2
2
 sin  

 where   12 ka sin  and   12 kb sin 



Bh n
2
g ( )  ( N 2  N1 )
c
8n 2
Diffraction:
Diffraction by a circular aperture of diameter D: 1 / 2  1.22 / D
Grating Eqn: m  a(sin  i  sin  m )
General:
Stefan-Boltzman Law: I  T 4
Wien displacement Law: max T  constant
Miscellaneous Mathematical Relations:
sin     31!  3  51!  5  ...
cos   1  21!  2  41!  4  ...
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1. (10 marks total)
Light from a star of unknown diameter located at a distance 5  1011 km is observed
through a filter at a wavelength of =400 nm. Starlight is collected by two small, stable
mirrors of variable spacing l in a plane perpendicular to the line of sight and redirected to
form a fringe pattern on a screen. When the mirrors are spaced by 50 cm, fringes with a
visibility of V=0.9 are observed. Determine the diameter of the star. (Note: If you don't
have a calculator, you may want to use the first two terms of the small angle expansions
for trigonometric functions to make the estimate. See formulas) (10 marks)
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2. (10 marks total)
(a) A 100 micron thick piece of glass (n=1.5) is inserted into an empty Fabry-Perot
cavity of length d=1 cm. At a wavelength of =500 nm, what is the change in mode
number m due to the insertion? Draw a picture and write an equation for m to get
started. (5 marks)
(b) In a second Fabry-Perot, the refractive index n=1.5 of material between the mirrors
slowly drifts by n  10 2 n . If the separation of the mirrors is 10 cm and the wavelength
of incident light is again =500 nm, how many transmission maxima are observed during
this change? (5 marks)
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3. (10 marks)
The oscillating mode spectrum of a gas laser is given in the figure below.
(1014s-1)
4.5
5
5.5
(a) Is this laser an ultraviolet, visible, near infrared or far infrared laser? Justify.
(1mark)
(b) Presuming it is a simple, linear 2-mirror laser, what is its length? (4 marks)
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(c) Assuming that the envelope of the emission spectrum results entirely from Dopplerbroadening of a narrowband emission, estimate the average velocity of atoms in the gas.
(3 marks)
(d) To achieve single-mode operation with an etalon inserted into the cavity, what should
the spectral free range and thickness of the etalon be? (2 marks)
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4. Short problems (worth 2-3 marks each - total of 20 marks)
4.1. (a) In an optical waveguide, such as a fiber, should the central core have a refractive
index higher, lower or equal to that of the cladding? (1 mark)
4.1. (b) In a desert mirage, state whether the index of the air nearest ground level is
higher, lower, or equal to that of the cooler air above and on this basis explain the
phenomenon. (An imaging diagram is adequate for the second part). (2 marks)
4.2. By using the uncertainty relation xk  1 (or by considering properties of the
Fourier transform pair x and k  2 /  ) justify whether precision optical metrology of
distance intervals should make use of a large or a small wavelength range ? (3 marks)
4.3. The light intensity inside a Fabry-Perot (FP) cavity is I c . If light of intensity I 0
impinges on a cavity with identical mirrors of 99% reflectivity and the FP is tuned to a
transmission peak (T=100%) so that the output is also I 0 , what is I c ? Justify. (2 marks)
4.4. Sketch the orientations of two prisms whose relative positioning can be used to
compensate normal dispersion inside an ultrashort pulse laser. (2 marks)
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4.5. In order to make the most efficient laser with continuous output, is it preferable to
choose a gain medium with 4 levels or to eliminate unnecessary levels by choosing a 2level system? Why? (3 marks)
4.6. Name three different laser gain media. (3 marks)
4.7. The angular spread of light due to diffraction is unavoidable and is completely
determined by the diameter D of the limiting aperture according to 1 / 2   / D . Use the
law of reversibility of linear optics to decide whether the diameter of a focal spot is
proportional to the diameter of the light beam impinging on the focusing lens. (2 marks)
4.8. The spectrum of a laser that initially is a delta function versus frequency becomes a
2
sin c function if it passes through an electro-optic shutter that opens and closes for a
brief time. True or false? Why? (2 marks)
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