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Ray Optics
•
•
n1 sin 1  n2 sin 2
Snell’s Law:
Sign Convention
– Mirrors
• R Negative for Concave (Positive for Convex)
• z Negative on Right (Positive on Left)
– Lenses (positive and negative)
•
•
•
•
•
•
•
z 1, f 1
z 2, f 2
R
y1, y2
+ left of Vertex
+ right of Vertex
+ if C is right of Vertex
+ above optical axis
Imaging Equation
1 1 1
 
z1 z2 f
where f mirror  
or fthinlens
Magnification
Ray Transfer Matrix
R
2
`
n
 1 1 
  l  1  
 nm   R1 R2 
– Stability
– Periodic
1
Simple Optical Components
Free-Space Propagation
Refraction at a Planar Boundary
Refraction as a Spherical Boundary
convex, R>0; concave, R<0
Transmission Through a Thin Lens
convex, f>0; concave, f<0
1 d 
M

0 1 
1
M
0
0
n1 
n2 
 1
M    n1 n2 
  n2 R
1
M 1
 f
0
1
Refraction from a Planar Mirror
1 0 
M

0 1 
Refraction from a Spherical Mirror
 1 0
M  2

 R 1
convex, R>0; concave, R<0
0
n1 
n2 

2
Wave Optics
• Wavefunctions & Wave Equations
• Elementary Waves
Spherical Wave
–
–
–
–
Spherical
Paraboloidal
Plane
Paraxial
U (r ) 
A  jkr
e
r
(Fresnel Approx to a spherical wave)
A  j k z  jk x 2zy
U (r )  e e
z
Plane Waves
U (r )  Ae j k r
Paraboloidal Wave
2
2
• Complex Amplitude Transmittance, t(x,y)
• Example: Gaussian Beam
– q(z) = z + j z0
– R(z), W(z), ABCD Law
– Hermite-Gaussian Beams
 W0   2 x 
 2 y


2
U l ,m ( x, y, z )  Al ,m 
 j (l  m  1)   z  
 Gl 
 Gm 
 exp  jkz  jk
2 R( z )


W ( z ) 
W  z   W ( z ) 
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Derivation of the Gaussian Beam
U (r )  A(r)e jkz
Plane Waves
Paraboloidal Wave
A(r ) 
(Fresnel Approx to a spherical wave)
Gaussian Wave
z  q ( z )  z  jz 0
A
e
z
x2  y 2
 jk
2z
A
A(r ) 
e
q( z )
 jk
x2  y 2
2q( z )



W0
2 
2
U (r)  A0
exp  2  exp  jkz  jk
 j ( z ) 
W ( z)
2 R( z )
 W ( z) 


1
1


j
q( z ) R( z )
W 2 ( z )
  z 
W ( z )  W0 1   
  z0 
2



1
2
  z0  2 
R( z )  z 1    
  z  
 z
 ( z )  tan  
 z0 
1
z 
W0   4 0 
  
1
2
Transmission Through Optical Components
Gaussian Beam
W1
R1
q1
A B 
 C D


W2
R2
q2
Aq1  B
q2 
Cq1  D
1
1


j
q( z ) R( z )
W 2 ( z )
Applies to thin optical components and to propagation in homogeneous medium of paraxial waves
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Resonators
• RTPS = 2 q
• Intensity
–
–
–
–
I=
Finesse
Q
Free Spectral Range
FWHM
I max
1   2F /   sin 2  /  F 
2
I max 
I0
(1  r ) 2
r  r1r2
F
 r1/2
1 r
• Resonator Stability (gi)
• Matching Gaussian Wavefronts
to Mirror Surfaces
6
Resonator Stability Diagram
g2
Symmetrical concentric
(R1 = R2 = -d/2)
e
1
-1
d
a
b
0
c
Symmetrical confocal
(R1 = R2 = -d)
g1
1
0  g1 g 2  1
where g1,2  1 
d
R1,2
Concave / convex
(R1 < 0, R2 > 0
Planar
(R1 = R2 = ∞)
Confocal / planar
(R1 = -d, R2 = ∞)
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Photon Optics
• Photon
– Energy
– Position
– Momentum
• Photon Streams
–
–
–
–
–
Optical Energy
Optical Power
Photon Flux Density
Photon Flux
Photon Number
• Coherent
• Thermal
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Photon-Number Statistics
Coherent Light
Poisson Distribution
n n exp  n 
p ( n) 
n!
n  0,1, 2,...
Mean & Variance

n   n p(n) n  0,1, 2,...
n 0

    n  n  p (n)  n
2
n
2
n 0
Signal-to-Noise Ratio
(mean)2 n 2
SNR 

n
variance  n2 9
Photon-Number Statistics
Thermal Light
– Boltzman Prob. Dist.
 E 
P( En )  exp   n  kB  1.38 1023 J/K
 kBT 
 nh
p(n)  exp  
 kBT
 
 h  

exp
 


 
 k BT  
n
n  0,1, 2,...
Bose-Einstein Distribution
1  n 
p ( n) 


n 1 n 1 
n
Mean & Variance
n
1
exp  h / k BT   1
  nn
2
n
2
Signal-to-Noise Ratio
SNR 
n
n 1
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Photons & Atoms
• Thermal
– Boltzmann Distribution
• Population Ratios
– Average Energy
– Spectral Energy Densities
• Atom – Photon Interactions
– Spontaneous Emission
– Absorption
– Stimulated Emission
• Probabilities of a Transition
– Einstein’s Coefficients
– Cross Section
– Lineshape
• Lifetime Broadening
• Collision Broadening
• Doppler Broadening
• Rate Equation
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Atom – Photon Interactions
Spontaneous Emission
• Probability Density of Spontaneous Emission into a
Single Prescribed Mode
2
• Probability Density of Spontaneous Emission into
any Prescribed Mode
h
1
Absorption
• Probability Density of Absorption of one photon from
a single mode containing n photons
2
h
• Probability Density of Absorption of one photon from
a stream of “single-mode” light by one atom
1
• Probability Density of Absorption of one photon in a
cavity of volume V containing multi-mode light
Stimulated Emission
2
h
1
h
h
• Probability Density of Stimulated Emission of one photon
into a single mode containing n photons
• Probability Density of Stimulated Emission of one photon
into a stream of “single-mode” light by one atom
• Probability Density of Stimulated Emission of one photon
into a cavity of volume V containing multi-mode light
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Interactions of Photons with Atoms
Spontaneous
Emission
Stimulated
Emission
(Absorption)
Prob Density into
a prescribed
mode
c
psp   ( )
V
Prob Density into
a prescribed
mode
c
pst   ( )
V
Where the transition cross section is
Prob Density of
One Photon into
Any Mode
1
Psp 
tsp
Prob Density into a
prescribed mode
with n photons
present
Monochromatic
Light
(prescribed mode)
c
Pst  n  ( )
V
Broadband
(any mode)
Wi 
Wi    ( )
n
tsp
2
 ( )  S g   
g ( )
8 tsp
with lineshape g() given by:
•Homogeneous broadening (Lorentzian): g ( ) 
 / 2
(  0 )2  ( / 2)2
 
1
2
1 1
  
 1  2 
•Inhomogeneous broadening (Collision): g ( ) 
 / 2
(  0 )2  ( / 2)2
 
1
2
1 1



2
f

col 


 1
2

•Inhomogeneous broadening (Doppler): g ( ) 
1
2 D

e
  0 2
2 D2
1/ 2
 D 
1k T 
8 ln 2  D   13B 
 M 
Rate Equation
Thermal Equilibrium
Blackbody
Avg Energy of a Mode
E  n h 
Thermal Light
h
eh / kBT  1
Rate Equation
d N2
N n N1 n N 2
 2 

dt
tsp
tsp
tsp
Radiation Spectral Energy Density
8 h 3
1
p( ) 
c3 eh / kBT  1
Equilibrium Condition
N2
n

N1 1  n
where n 
1
e h / kB T  1
where n is the average number
of photons in a mode of frequency 
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