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
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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 2
Chapter 2: Properties of Light
Ex ' Ex J Ey ' Ey J ij 6
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 12
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
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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
