Download Properties of Light

Chapter 2 - Properties of Light
Gabriel Popescu
University of Illinois at Urbana‐Champaign
y
p g
Beckman Institute
Quantitative Light Imaging Laboratory
Quantitative
Light Imaging Laboratory
http://light.ece.uiuc.edu
Principles of Optical Imaging
Electrical and Computer Engineering, UIUC
ECE 460 – Optical Imaging
2.1 Properties of EM Fields
2.1 Properties of EM Fields
 Amplitude A and phase φ
p
p
φ are random functions in both time
and space:

 
 
i (r , t )
E (r , t )  eA(r , t )).e
Chapter 2: Properties of Light
(1)
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ECE 460 – Optical Imaging
2.1 Properties of EM Fields
2.1 Properties of EM Fields
a)) Polarization:
 Gives the direction of field oscillation
 Generally, light is a transverse wave (unlike sound = longitudinal)

E
Propagation (k
 wave vector)
2
| k |

 Anisotropic materials: different optical properties along different axis  useful
Chapter 2: Properties of Light
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ECE 460 – Optical Imaging
2.1 Properties of EM Fields
2.1 Properties of EM Fields
a)) Polarization:
x , y
 There is always a basis for decomposing the field into 2 polarizations (eigen modes); equivalently (right, left) circular polarization is also a basis.
i l
l i ti i l
b i
 Dichroism: different absorption for different pol  one way to create polarizers:
y
p
 
ԕ
 Malus Law: Chapter 2: Properties of Light

E1
P
E2  E1 * cos 
P

E2
ԕ

E
(2)
demo available
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ECE 460 – Optical Imaging
2.1 Properties of EM Fields
2.1 Properties of EM Fields
a)) Polarization:
I  E ; I 2  I1 *cos 2 
2
I
Malus Law
Malus Law

 Birefringence – Different refr. index for different pol.
Chapter 2: Properties of Light
demo available
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ECE 460 – Optical Imaging
2.1 Properties of EM Fields
2.1 Properties of EM Fields
a)) Polarization:
 Natural Light  unpolarized  superposition Ex= Ey with no phase relationship between the two
 Circularly polarized  Ex= Ey, φx – φy = π/2 !
 Matrix formalism of polarization transformation (Jones 2x2, complex & Muller –
(Jones –
2x2 complex & Muller 4x4, real)
4x4 real)
We’llll do this later.
We
do this later
E  I  Stokes Vect.
Vect Dim 4
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Chapter 2: Properties of Light
 Ex ' 
 Ex 

J 
 Ey ' 
 Ey 
J ij  
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ECE 460 – Optical Imaging
2.1 Properties of EM Fields
2.1 Properties of EM Fields

V
b)) Amplitude:
p
 A(r , t )  

 m
A(t)
•Thermal source
k
t
A(x)
  k
A(t)
•Stabilized laser
t
A(x)
•Arbitrary field
Chapter 2: Properties of Light
x
•Plane Wave
x
demo available
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ECE 460 – Optical Imaging
2.1 Properties of EM Fields
2.1 Properties of EM Fields
k0
c)) Phase: [Φ]
[ ] = rad
Φ(t)
Φ(t)
o
•Thermal source
t
Φ(z)
•Laser at freq o
Φ =ωt
Φ(z)
t

•Random field
Chapter 2: Properties of Light
z
•Plane Wave
Φ =kz
z
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ECE 460 – Optical Imaging
2.1 Properties of EM Fields
2.1 Properties of EM Fields
c)) Phase: [Φ]
[ ] = rad
 For quasi-monochromatic fields, plane wave
 
  t  k  r

2 2 2
 k



 wave number
c
c
Tc 
1
  cT ; T 
;   2 N
N
Chapter 2: Properties of Light
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ECE 460 – Optical Imaging
2.2 The frequency domain representation
2.2 The frequency domain representation

Random variable E(t) has a frequency‐domain counterpart:
()
q
y
p
E ( )  A( )e

i (  )
(4)
Similarly E(x) has a frequency‐domain pair:
E ( )  A( )e
i (  )
(5)
 ( )  k  z  n( )  k0  z
k0
Chapter 2: Properties of Light
k0 
2

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ECE 460 – Optical Imaging
2.2 The frequency domain representation
2.2 The frequency domain representation
a) Spectral amplitude:
2
 Optical Spectrum:
Optical Spectrum: S ( )  A( )
 Angular Spectrum: S ( )  A( ) 2
S(ω)
S(ξ)
ω0

k0
ω
ξ
0
ξ
1


m
 Spatial Frequency (connects to angular  
spectrum)
t
)
 Tipically: t  
Will follow similar equations
x‐ 
x ‐
 The information contained is the same (t, ) and (x, )


Chapter 2: Properties of Light
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ECE 460 – Optical Imaging
2.2 The frequency domain representation
2.2 The frequency domain representation
b)) Spectral phase:
p
p
 Phase delay of each spectral component
Optical Frequency
Φ(ω)
~ω
  chirp
Spatial Frequency
2
Φ(ξ)
~ξ
ω0
•Dispersive material
(linear chirp)
ω
ξ
•Defocused point source
(1st order aberration)

 Full similarity between (t, ) and (x, )

Chapter 2: Properties of Light
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A point is
mapped
to a blur
d 2
 
d
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ECE 460 – Optical Imaging
2.3 Measurable Quantities
2.3 Measurable Quantities
 The information about the system under investigation may y
g
y
be contained in polarization and:
 A(t), φ(t)
((t,, )


 A( ), φ( )
8 quantities
 A(x), φ(x)
A(x) φ(x)
(x, )
 A( ), φ( )
 
 Experimentally, we have access only to:
2
I  A(t )  time average Chapter 2: Properties of Light
demo available
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ECE 460 – Optical Imaging
2.3 Measurable Quantities
2.3 Measurable Quantities
 Experimentally, we have access only to:
p
y,
y
2
I  A(t )  time average
(6)
 i.e the phtodetectors ( photodiode, CCD, retina, etc) produce photoelectrons:
h  Ee  W

Photon incident
Electron
El
t
energy
kinetic
energy
Chapter 2: Properties of Light
(Einstein)
(7)
Work
demo available
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ECE 460 – Optical Imaging
2.3 Measurable Quantities
2.3 Measurable Quantities
 All detectors sensitive to power/energy
p
/
gy
 However, all 8 quantities can be accessed via various tricks
 Eg1: Want I( ) 
 measure I( ) and use a device with  ( )

 use interferometry  I( )
  E1 E2 cos(1  2 )
 Eg2: Want 
Chapter 2: Properties of Light
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ECE 460 – Optical Imaging
2.4 Uncertainty Principle
2.4 Uncertainty Principle
 Space ‐
p
momentum or energy‐time cannot be measured gy
simmultaneously with infinite accuracy
x  p  constant  h  Plank' s constant
E t  constant
 For photons: For photons:
E  
p  k ; p 
Chapter 2: Properties of Light
h

k
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ECE 460 – Optical Imaging
2.4 Uncertainty Principle
2.4 Uncertainty Principle
a)) t –
t  constant

t  2
 Implications: 1‐ short pulses require broad spectrum
2 high spectral resolution requires long time of
2‐high spectral resolution requires long time of measurement
Chapter 2: Properties of Light
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ECE 460 – Optical Imaging
2.4 Uncertainty Principle
2.4 Uncertainty Principle
b)) x –
ks
ki

ԕ
x q  
;
2sin( / 2)

x

xmin 
Chapter 2: Properties of Light
p  h(ks  ki )  hq
q


2

q  2k sin( )
2
1 ;


x
- meaning of resolution
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ECE 460 – Optical Imaging
2.4 Uncertainty Principle
2.4 Uncertainty Principle
k
diffraction
 Smaller aperture  Higher angles
k

 If aperture If aperture < , light doesn
, light doesn’tt go through (easily)
go through (easily)
2
 Eg: Microwave door Chapter 2: Properties of Light
demo available
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ECE 460 – Optical Imaging
2.4 Uncertainty Principle
2.4 Uncertainty Principle
 We will encounter these relationships many times later
 Fourier may have understood this uncertainity principle way before Heisenberg!
Chapter 2: Properties of Light
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