PHYC 3540
Summary
I.
II.
Propagation of light
a. History, wave vs. particle picture
b. Fermats Principle – actual path is that for which the OPL is an extremum
c. Huygen’s principle – every point on wavefront acts as a source
i. Large aperture (>>) – rectilinear propagation (Geometric optics)
ii. Small aperture (   ) – spreading of wavefront (Diffraction)
d. Wavefronts, Rays – Laws of reflection and refraction (Snels law)
Geometric Optics
a. Imaging – Cartesian surfaces, Paraxial ray approximation
b. Refraction at a Spherical interface
n
n'
s
s'
R
n n' n'n n n'
 
 
s s'
R
f
f'
n n'

 P  Power
f
f'
c. Thin lens
n n' n L  n n L  n' n n'
 

 
P
s s'
R1
R2
f
f'
Magnificat ion
M 
h'
ns '

h
n' s
for
n
n'
F’
F
s
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all
s'
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imaging
d. Principal Planes (H,H’)
i. Used to represent any optical element or system and are located
such that the system images an object according to:
n n' n n'
  
P
s s' f
f'
ii. The principal planes used to represent a spherical interface (or thin
lens) must coincide at the interface (or centre of the thin lens).
H
H’
n'
n
f
f'
s'
s
e. Combination of two
systems
n
n2
H1
H1’
H
n'
H’
H2’
H2
s
s'
f
f'
i. Power of combined system P  P1  P2 
where, P1 
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dP1 P2
n2
n
n n2
n'
 ' and P2  2  '
f1
f2
f1
f2
ii. The location of H, H’ are determined with respect to H1 and H2’ as
f'
 P  n' 
h'  H 2 ' H '  d  1     d
f1 '
 P  n2 
follows:
f
 P  n 
h  H 1 H  d  2    d
f2
 P  n2 
f. Matrix methods in paraxial optics
i. Matrices
1 L 
Translation matrix 

0 1 
 1
Refraction matrix  n  n'
 Rn '
1
Reflection matrix  2
 R
0
n
n' 
0

1

0
 1
Thin lens matrix  1 1
 f



ii. Cardinal point locations
P  n'  1  A
h'  d 1   
P  n2 
C
hd
P2
P
 n

 n2
 
n
 D  n'
 
C

g. Stops in optical systems
i. Aperture stop
 Element that limits the bundle of rays (and hence light
gathering power or speed) collected by optical system
 Is that element whose entrance pupil subtends the smallest
angle at the object point in question
 Exit pupil is the image of the aperture stop in all elements
following it
 Entrance pupil is the image of the aperture stop in all
elements preceding it
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III.
ii. Field stop
 Element of the optical system that limits the field of view
 That element whose entrance window subtends the smallest
angle at the centre of the entrance pupil
 Exit window is the image of the field stop in all elements
following it.
 Entrance window is the image of the field stop in all
elements preceding it
iii. Chief ray, marginal rays – definition of edge of field of view
iv. Angular field of view
 Image space – angle formed by the entrance window as
viewed from the centre of the entrance pupil
 Object space – angle formed by the exit window as viewed
from the centre of the exit pupil
h. Applications
i. Cameras (aperture settings (f#), exposure, telephoto lens)
ii. Human eye – myopia, hyperopia
iii. Hand magnifiers, eyepieces, compound microscope
iv. Telescope, binoculars
Physical optics (wave properties are important)
a. Wave equation – plane waves, spherical waves, cylindrical waves
b. Amplitude reflection and transmission co-efficients
12 
v2  v1
v1  v2
;  12 
2v2
v1  v2
c. Electromagnetic waves (E  B; E, B in phase, speed c/n)
i. Irradiance
I
1
vEo 2
2
; E o  Amplitude
ii. Transmittance
T
I2
n1n2
n2 2
4




 12
12 21
I1
n1
(n1  n2 ) 2
iii. Reflectance (12)
2
 n  n1 
I
2
  12 2   21
R  R   2
I 1  n2  n1 
d. Interference
i. Two source interference
 Irradiance
I  I 1  I 2  2 I 1 I 2 cos 2  1 
where,
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 2  1    k (r2  r1 )  ( 2   1 )
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
If 1 = 2 and I1=I2=Io there are maxima located at,

sin    m
a
 Examples
 Double slit, Lloyds mirror, radio towers
 Dielectric layers, e.g. Haidinger bands, fringes of
equal thickness observed in oil films, air wedges
 Antireflection coatings
 Newton’s rings
ii. Multiple beam interference (coated dielectric layers)
 Transmitted irradiance




2
T Io 
1

IT 
2 

4
R

(1  R)
sin 2 
1
2
2
(1  R)

where,

2
 k o nd cos  ' and n, ’ pertain to the
dielectric layer.
 Irradiance maxima occur when
2nd cos '  mo

Fabry-Perot Interferometer
 Finesse =

separation between fringes  R

fringe width ( FWHM )
1 R
 Free spectral range
 FSR 
2
2nd
; v~FSR 
1
2nd
 Fringe width
free spectral range
2
 

finesse
2nd
 Spectral Resolution


2nd

 min

v~
  ~  2ndv~
v
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e. Diffraction
i. Fresnel–Kirchoff diffraction formula (Mathematical statement of
Huygen’s principle)
 ikEo
E
2
e ik ( r  r ')
aperture rr ' F  dA
where,
F   
1
cos(nˆ, r )  cos(nˆ, r ' )  obliquity
2
factor
P
r
r'
n
ii. Fraunhofer limit (plane waves)
diffraction integral reduces to

I 0
when
2
where,  
sin    m
kb sin 
2

b
Circular aperture (diameter do = 2ao)
 2 J   
I  I o  1 
  
2
where,  = kaosin
1.22
ao
Airy disc; Spatial resolution of optical devices;
focussing limit
N slits (width b, separation a)
I 0

aperture
Single slit (width b)
 sin  

I  I o 
  

E  C '  e ikr dA
when sin  
2
 sin    sin N 
 
I  I o 
 where,
     
2
Principal maxima when sin    p
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

a
ka sin 
2

Diffraction gratings
Grating dispersion
Spectral resolution   mN
f. Polarization
i. Jones vector for polarization
 E ox 
General 
i 
 E oy e 
0 
Vertical  
1 
Horizontal
1 
0 
 
cos  
Linearly polarized at an angle α, (1) m 

 sin  
1 1
Left Circularly polarized
  (i.e. counterclockwise)
2 i 
1 1
  (i.e. clockwise)
2  i 
 A 
1
Left Elliptically polarized
2
2
2  B  iC 

A  B C 
 A 
1
Right Elliptically polarized


A 2  B 2  C 2  B  iC 
Right Circularly polarized
C 

B
and the angle of inclination, α, of the ellipse is given by
2 Eox Eoy
tan 2  
2
Eox  Eoy2
ii. Jones matrices for optical elements
 Linear polarizer (Transmission axis at an angle  wrt xaxis)
 cos 2 
sin  cos 


sin 2  
sin  cos
 Phase retarders
e i x
0 
In general, 
where, x and y represent
i y 
 0 e 
the advance in phase of the components
 Quarter wave plate (|y - x| = /2)
0
i 1
e 4
 Slow axis vertical
0 i 
where, E ox  A
e
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E oy  B 2  C 2
 i
4
  tan 1 
1 0 
0  i  Slow axis horizontal


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 Half wave plate (|y - x| = )
0
i 1
e 2
 Slow axis vertical
0  1
1 0 
0  1 Slow axis horizontal


Rotator (light polarized at an angle  is rotated by an angle
)
cos   sin  
 sin  cos  


e

 i
2
g. Holography
i. Hologram
R( x, y, z )  r ( x, y, z )e i ( x , y , z )  re ikz
O( x, y, z )  o( x, y, z )e i ( x , y , z )  oe ikr
where
r  x2  y2  z2
I ( x, y )  O  R  OO *  RR *  OR *  O * R
2
where, R – reference wave, O – object wave, I - intensity
ii. Reconstructed wave
2
R( x, y )t p  (tb  BOO * ) R  BR 2O*  B R O
where, t – transmission, first term – direct wave, second
term – conjugate wave, third term – object wave
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