4. Wave Optics
Spherical Wave, Image Formation, and Huygens’ Principle
Wavefront: a surface over which
the phase of a wave is constant
Huygens’ Principle
Linear Polarization
Circular/Elliptical Polarization
Unpolarized Light and Polarizer
Liquid Crystal Display (LCD)
3D Imaging by Polarizers
Reflection and Transmittance of Polarized Lights
Fresnel equations: rp 
t an 2  1 
t an 2  1 
2 sin  2 cos1
tp 
sin 1   2  cos1   2 
rs 
ts 
sin  2  1 
sin  2  1 
2 sin  2 cos1
sin 1   2 
Note:
p-polarization:
E-field  plane of incidence
s-polarization:
E-field  plane of incidence
Goos-Haenchen Shift
Optical Transfer Matrix to Analyze
Three-layer Film
Optical Transfer Matrix to Analyze
Three-layer Film (Cont’)
Antireflection Film
Antireflection Coatings on Solar Cells
High-reflectance Film
High-reflectance Film (Cont’)
Interference
Young’s Experiment
Interference —
superposition of two light
wave result in bright and
dark fringes
Conditions for Interference:
• same polarization
• same frequency
• constant phase
relationship (coherence)
Conditions for Interference


2r1
E1  A1cos 1t 
 1 
1




2r2
E2  A2 cos 2t 
  2 
2


If 1 = 2 = 
I  E1  E2
2
Bright fringes:  = 0, 2, 4,…(in phase)
 I1  I 2  2 I1 I 2 cos
I max  I1  I 2  2 I1I 2 (constructive interference)
where
A12
I1 
2
A22
I2 
2
  1   2 
Dark fringes:  = , 3, 5, …(out of phase)
I min  I1  I 2  2 I1I 2 (destructive interference)
2 r1  r2 

 1  2 
Fabry-Perot Interferometer
Fabry-Perot Interferometer (Cont’)
Fabry-Perot Interferometer (Cont’)
GaAs’s natural cleavage plane
is (1,1,0)-plane. Si’s and Ge’s
natural cleavage plane are
(1,1,1)-plane.
Mach-Zehnder Interferometer
Holography/Hologram
Recording
process
Reconstruction
process
3D Hologram Videos
Michelson Interferometer
Sagnac Effect and Ring Interferometer
N: Fringe number
Interferences of Coherent/Incoherent Waves
• Coherence: All component
electromagnetic waves are in
phase or in the same phase
difference.
• Interference of coherent
waves: Waves of different
frequencies interfere to form a
pulse if they are coherent.
• Interference of incoherent
waves: Spectrally incoherent
light interferes to form
continuous light with a
randomly varying phase and
amplitude.
Fresnel (Near-field) Diffraction
Fraunhofer (Far-field) Diffraction
Fraunhofer Diffraction Pattern of
a Rectangular Aperture
Fraunhofer Diffraction Pattern of
a Circular Aperture
Resolving Power of Imaging Systems
Rayleigh criterion
Resolution Limit
• Rayleigh criterion  two object point can be resolved by
the lens of an optical system
Minimum resolvable angular:
 min  1.22

D
D: diameter of open aperture
: wavelength of light source
Note: if < min, images cannot be resolved
Minimum resolvable separation:
For objective lens, h1,min  1.22
h1,min  1.22


 0.61
2 sin 
NA
 d1
D
where =h/d1
numerical aperture NA=sin1
Resolution of Human Eye
Resolving power of human eye  0.3 mrad
Resolution limit of human eye  0.075mm
Fourier Transform by a Convex Lens
Optical Fourier Transform
FTL
Input Plane
f
a(x,y)
Fourier Plane
f
A(u,v)
Optical Signal Processing
Examples of Optical Signal Processing
Examples of Optical Signal Processing
(Cont’)
Fourier Optics and Its applications
Optical Computing
Phase Contrast Microscopy
Appendix 4-1 Coherence
Coherence Function
Mutually coherent:
point sources u1(t1, x1, y1, z1) and u2(t1, x2, y2, z2 )
maintain a fixed phase relation
Mutual coherence function:
12 ()  u1 (t  )u2 (t )
T
1
 lim  u1 (t  )u2 (t )dt
T  T
0
temperalcross- correlation functionbetween u1 (t ), u2 (t )
Normalized mutual coherence function:
(complex degree of coherence or degree of correlation)
12 ()
12 () 
11 (0)22 (0)1/ 2
where 11() and 22() are the self-coherence functions
of u1(t) and u2(t)
Demonstration of Coherence
extended source
interference
pattern
Visibility of fringe:

I max  I min
I max  I min
I max : maximumintensityof fringe
I min : minimumintensityof fringe
If I1 = I2= I (best condition),  =  12() 
i.e., visibility of the fringe is a measure of the degree of coherence
Spatial Coherence
extended
source
Intensity distribution of the resultant fringe of two points on the extended source:

 d   
 d 


S  
I ' ()  S I1  I 2  2 I1 I 2 cos k   sinc

 r20 
 r10   


d
r10 

(fringesvanish)
S  s
where angular size of theextendedsource
S
s 
r10
extended source
sinc  
d  1.22
J1 Sd / r10 
(circularextendedsource)
Sd / r10
r10

 1.22
(fringesvanish)
S
s
Measurement of Spatial Coherence
Temporal Coherence
Visibility of the fringe is a measure of
the degree of temporal coherence 11()
at same point
Coherence length of the light source
c
2
lc 

f 
where
c : speed of light ( 3 108 m/s)
 : wavelengthof light (nm)
f : spectralwidth (Hz)
 : spectralwidth (nm)
Measurement of Temporal Coherence
Appendix 4-2 Fourier Transform
Fourier Transform Pairs
Basic Theorems of Fourier Transforms
Basic Theorems of Fourier Transforms (Cont’)
Application of Fourier TransformDistinguishing Similar Signals
Appendix 4-3 Phase Transform
Function of a Lens
Usage of a Thin Lens Phase Transformation
Phase transform function:
T(x,y)=exp[j(x,y)]
and
Phase variation: (x,y)=knt(x,y)
where
t(x,y): thickness function of lens
To find thickness function t(x,y)
Phase Transform Function of a Lens
2 1/ 2 
 
    

  
t 2 ( x, y)  t 2  R 2 1  1  
  R 2   



2 1/ 2 
 
    

t 1 ( x, y)  t 1  R 1 1  1     
  R 1   

 

where   x 2  y 2
Thickness function of lens: t ( x , y)  t 1 ( x , y)  t 2 ( x , y)
 1
1 



 R1 R 2 
(for paraxialapproximation : R 1 , R 2  )
2
 t 
2
Phase Transform Function of a Lens
Phase transform function:


 k 2

T( x, y)  exp jknt exp j
x  y2 
 2f

Note:
f > 0, convergent effect
f < 0, divergent effect