Download Polarization, waveplates

Chapter 9 - Polarization
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
Polarization
 The x‐y components of the field can be expressed via a 2D y
p
p
“Jones” vector:
i

Ax e
 Ex 
J  
i

E
 y   Ay e
x

y




(9 1)
(9.1)
Time variation:
Time variation:
Ex  Ax cos(t  x )
E y  Ay cos((t   y )
Chapter 9: Polarization
(9.2)
2
ECE 460 – Optical Imaging
Polarization

Eliminate time:
Ex  Ax cos t  cos x   sin t  sin x  
E y  Ay sin t  sin  y   cos t  cos  y  

(9.3)
Take as exercise to prove that the trajectory of the E‐field is:
2
Ey2
2 Ex E y
Ex
2
cos
sin






2
2
Ax Ay
Ax
Ay
(9.4)
where    y  x
Chapter 9: Polarization
3
ECE 460 – Optical Imaging
Polarization
y

This is an ellipse:
p
θ
x
 In general, it’s rotated by angle θ
 This is what we would “see” looking straight at the incoming beam
 There are few particular cases:
a) Linear polarization: Linear polarization:   0,
0
2
 Ex E y 


 0  line
 Ax Ay 


Chapter 9: Polarization
4
ECE 460 – Optical Imaging
Polarization
b)) Circular polarization:
p
   
Ax  Ay
2
  clockwise
2
   clockwise
2
 E x 2  E y 2  A2
   
2
c)) Straight ellipse: Ax  Ay
Chapter 9: Polarization
5
Jones Calculus for
Birefringent Optical Systems
Principles of Optical Imaging
Electrical and Computer Engineering, UIUC
ECE 460 – Optical Imaging
Introduction
 ABCD matrices told us about amplitude distribution on p
propagation
 Jones “calculus” deals with polarization changes for polarization sensitive optical elements
l i ti
iti
ti l l
t
input
output ?
polarizes axis Direfringent crystal
along lines cc-axis
axis along vector
(uniaxial)
Jones Calculus
7
ECE 460 – Optical Imaging
Introduction
 Jones “calculus” developed around 1940
p
 Based on 2x2 matrices (similat to ABCD)
 Need 2 to describe Z polarization
 Light travels 1 of 2 tranverse normal modes
 Birefringent crystals play role because two modes travel at different phase velocities
different phase velocities
 Uniaxials simple – c‐axis in plate surface
Jones Calculus
8
ECE 460 – Optical Imaging
Retardation Plates (Waveplates)
Retardation Plates (Waveplates)
 Change polarization state
g p
 Assumption: no reflections at surface of elements (can use anti‐reflection cratings)
 Use Jones matrices discussed before  1 
E  V E ( x, y, z )e j (t kz )  cc
2 Slow variation O.K.
describes
polarization
l i ti
Jones Calculus
 Vx 
2
2
; V

V
 
x
y 1
 Vy 
9
ECE 460 – Optical Imaging
Retardation Plates (Waveplates)
Retardation Plates (Waveplates)
 E.g. g
V
 x‐polarized  x   1 
   
 Vy   0 
1 1 
 j
2 
 y‐polarized  Vx 
0
  
 Vy  1 
1 1 
 
2  j
Jones Calculus
10
ECE 460 – Optical Imaging
Retardation Plates (Waveplates)
Retardation Plates (Waveplates)
x,y,z
laboratory
frame of reference
f – “fast” axis
s – “slow” axis
ns  n f
ff,s,z
sz
crystal
frame of reference
Jones Calculus
11
ECE 460 – Optical Imaging
Retardation Plates (Waveplates)
Retardation Plates (Waveplates)
 cos
 
 V f    sin
 From before  Vs 
just inside the crystal
sin   Vx 
 

cos   Vy 
R ( )
just outside the crystal
 Vs & Vf propagate independently in the crystal
j kL
 After a distance L: Vs '  Vs e  jn
s 0
Vf '  Vf e
Rewrite: Vs '  Vs
1
1
 j ( ns  n f ) k0 L  j ( ns  n f ) k0 L
e 2
e 2
Vf ' Vf
Jones Calculus
 jn f k0 L
1
1
 j ( ns  n f ) k0 L  j ( ns  n f ) k0 L
e 2
e 2
12
ECE 460 – Optical Imaging
Retardation Plates (Waveplates)
Retardation Plates (Waveplates)
1
  (ns  n f )k0 L
 Define   (ns  n f )k0 L
2
  j

 Vs '   j  e 2
0   Vs   j  Vs 

e 
   e W0  


Vf 
Vf 
j
V f ' 


 0
2 
e


W0 ‐ describes birefringent medium
 To get “output”, return to lab frame of reference
Jones Calculus
13
ECE 460 – Optical Imaging
Retardation Plates (Waveplates)
Retardation Plates (Waveplates)
 Vx '   cos  sin   Vs ' 
 Vs ' 
 

  R( ) 


 Vy '   sin cos   V f ' 
V f ' 
 Vx ' 
 Vx 
    R( )  W0  R( )  
 Vy ' 
 Vy 
order of doing products of matrices
 Note: each matrix is unitary R ( )  1
 Polarization remain orthogonal
Polarization remain orthogonal
Jones Calculus
14
ECE 460 – Optical Imaging
Polarizers
x‐axis Px  e
 j p
y‐axis Py  e  j
p
1
0

0
0

0
0 
0
1 
 p ‐ phase change phase change
through polarizer
?
 Vx ' 
 Vx 
   R( )  P0  R( )  
 Vy ' 
 Vy 
Jones Calculus
15
ECE 460 – Optical Imaging
Half (1/2) Wave Plate
Half (1/2) Wave Plate
   L 
e
s

j

2
f
Jones Calculus
2
j
y
x

(ns  n f )
W  R ( )  W0  R ( )
 cos  sin   j 0  cos sin 

 0 j   sin cos 
sin

cos





 cos  sin   cos  sin 
 j
  sin cos 
s
in

cos




 cos 2   sin 2 
2sin cos 
  j

2
2
 2sin cos
sin   cos  

16
ECE 460 – Optical Imaging
½ Wave Plate
½ Wave Plate
 cos 2
W   j
 sin 2
sin 2 
 cos 2 
rotates a plane of rotates
a plane of
Polarization by 2 x Ψ
angle between lab g
and crystal frames
 Another important factor is the transmission through the element
T
E'
E
Jones Calculus
2
2
2

Vx '  Vy '
2
Vx  Vy
2
2
17
ECE 460 – Optical Imaging
½ Wave Plate
½ Wave Plate
 E.g.
g 1  input
p
0
 
 Vx ' 
 cos 2
    j
 sin 2
 Vy ' 
sin 2  1 
 cos 2 
0
  j
 




o
 cos 2   0 
 sin 2  if  45 1 
 Just get net polarization rotation
cos 2 2  sin 2 2  1  T  1
Jones Calculus
18
ECE 460 – Optical Imaging
½ Wave Plate
½ Wave Plate
Jones Calculus
19
ECE 460 – Optical Imaging
½ Wave Plate
½ Wave Plate
 E.g.
g 1 1  input
p
 j
2 
 Vx '   cos 2
 
 Vy '   sin 2
sin 2 1  1
j  cos 2  j sin 2 





2  sin 2  j cos 2 
 cos 2  j  2
1
T  ((cos 2  j sin
i 2 )(cos
)( 2  j sin
i 2 )  (sin
( i 2  j cos 2 )(sin
)( i 2  j cos 2 )
2
1
 (cos 2 2  sin 2 2  cos 2 2  sin 2 2 )  1
2
 Is the output still circulary polarized?
 Depends on Ψ
Depends on Ψ
Jones Calculus
20
ECE 460 – Optical Imaging
½ Wave Plate
½ Wave Plate
 Choose  

4
 Vx ' 
j  j
1 1 

 


 j
V
'
1
2
2
y
 
 
 
(1) still circulary polarized
( ) reverses sense of rotation
(2)

(3) for   , get elliptical polarization!
4
Jones Calculus
21
ECE 460 – Optical Imaging
¼ Wave Plate
¼ Wave Plate

e

2
j
L

4
 cos

4

4
(ns  n f )
 j sin
Assume Ψ= 450:

4

1
(1  j )
2
  j
4
1

1
e




1


W  R (45o )  W0    R (45o )  
2 1 1  
2
 0

1  j 4 1  j 1  j 
 e 

1
1

j

j
2


Jones Calculus

0   1 1
   1 1 
j



4
e 
22
ECE 460 – Optical Imaging
¼ Wave Plate
¼ Wave Plate
j

 Vx  1   Vx '  e 4 1  j 1 
 E.g.      ;   
1  j 1 
'
V
V
0
2

 y    y 

1  j  j 4 1 

e  
2
 j
j

j 1  e 4



j  0 
2
1 
1 

j
j 
RCP

 Vx  1   Vx '  e 4 1  j 1 
 Eg. Eg.      ;   
1  j 1 
V
V
'

j
2

 y    y 

1 j  j 4 0

e  
2
1 
Jones Calculus
j
j

j  1 1  e 4  0





j  2   j  2 2  2  2 j 
y‐polarized
23
ECE 460 – Optical Imaging
¼ Wave Plate
¼ Wave Plate
Jones Calculus
24
ECE 460 – Optical Imaging
Waveplates and Polarizers
Waveplates and Polarizers
Jones Calculus
25
ECE 460 – Optical Imaging
Waveplates and Polarizers
Waveplates and Polarizers
  j
1 1 1  e 2
o
o
W  R(45 )  W0     R (45 )  
2 1 1  
 0


cos
2
2
 

2
j
sin



2
1
Px  W  Px  
0
1

0
Jones Calculus
 

 j sin   cos
2 
2
 


cos    j sin
2  
2

 j sin 
2

 
cos 
2 



cos

j
sin
i
0 
2
2  1




  0
0 

j
sin
cos



2
2 




cos
0


0 
cos
2


2


0 

j
sin
0   0




2

0  1 1
  1
1 
j

e 2 
0
0 

0

0
26
ECE 460 – Optical Imaging
Waveplates and Polarizers
Waveplates and Polarizers
 Case I: unpolarized light
p
g  Vx 
1  1
 
 1
V
2
 
 y




 Vx '  1  cos
1
0     cos 


2
2
 


linearly polarized: obvious !!




'
V
1
2
y


 
0
 0
0

2
T
Jones Calculus
Vx '  Vy '
2
Vx  Vy
2
2
1

 cos 2
2
2
½ lost at first polarizer!
27
ECE 460 – Optical Imaging
Waveplates and Polarizers
Waveplates and Polarizers
 Case II:  Vx  1 
  
 Vy   0 




 Vx '   cos
1
0     cos 


2
2
  





0

 Vy '   0
0
0


T  cos
2
2
Jones Calculus
28
ECE 460 – Optical Imaging
Waveplates and Polarizers
Waveplates and Polarizers
 Case III:  Vx 
1 1 
 
 j
V
2
 
 y




 Vx '  1  cos
1
0    1  cos 


2
2
 






 j
2
2

 Vy ' 
0
 0
0

1
2
T  cos
2
2
 For ½ wave plates, For ½ wave plates     T  0
Jones Calculus
29
ECE 460 – Optical Imaging
Birefringent Plate between Crossed Polarizers
d l i
Ψ= 45
Ψ=
450:



cos
 j sin 

2
2
W  R(45o )  W0     R (45o )  


  j sin 
cos 

2
2 



cos
 j sin 
 0
1
0
 0 0 


2
2
Py  W  Px  









0
1
0
0

j
sin

   j sin

cos  


2

2
2 
Jones Calculus
0

0

30
ECE 460 – Optical Imaging
Birefringent Plate between Crossed Polarizers
d l i
 Case I: unpolarized light
p
g  Vx 
1  1
 
 1
V
2
 
 y
0
0
0

 Vx '  1 
1


1




 




0  1
2   j sin
2   j sin 
 Vy ' 



2
2
1 2
T  sin
2
2
Jones Calculus
31
ECE 460 – Optical Imaging
Birefringent Plate between Crossed Polarizers
d l i
 Case II:  Vx  1 
  
 Vy   0 
0
 Vx '  
 

V
'

j
sin
 y  
2
0
0

1








0   0    j sin 


2

T  sin
2
2
Jones Calculus
32