Download Chapter 5: Jones Calculus and Its Application to Birefringent Optical

Chapter 5 Jones Calculus and Its Application
to Birefringent Optical Systems
Lecture 1 Wave plates
Wave plates (retardation plates) are optical elements used to transform the polarization
states of light. They are made from one or more pieces of birefringent crystals.
• Let us consider a plate made of a uniaxial crystal with a thickness of l. Usually the
plate is cut so that its optic axis lies in the plane of the plate surface.
• For a normally incident light, the polarization directions of the two eigenwaves both lie
in the surface of the plate, and are mutually orthogonal. One polarization direction
coincides with the optic axis, with a refractive index ne. The other is perpendicular to
the optic axis, with a refractive index no.
• The polarization direction with the larger refractive index is called the slow axis, and
the polarization direction with the smaller refractive index is called the fast axis,
regardless of whether it is an ordinary light or an extraordinary light. The refractive
indices are then designated as nf and ns, respectively.
1
In the frame of the slow and fast axes, or the s-f coordinate system, suppose the input light
is linearly polarized with the field
Ein  Es sˆ  E f fˆ eit , where Es and E f are both real.
Please note we are using the


exp[ i (t  kz)] convention .
At the output surface, the light field is changed into

 n l  
 n l 
Eout   Es exp   i s sˆ  E f exp   i f fˆ eit .
c 
c  



The phase difference (retardation or retardance) between the f and s polarization is then
Γ  ns  n f 
l
c
When Γ  (2m  1) , the plate is a half - wave plate .
When   (2m  1 / 2) , the plate is a quarter - wave plate.
f
f
s
s
f
l
s
2
Half-wave plates and quarter-wave plates
1) A half-wave plate converts a linearly polarized light into another linearly polarized
light, mirrored by the fast or slow axis.
2) A quarter-wave plate converts a linearly polarized light into a circularly polarized
light, when the input polarization is 45° to the fast and slow axes. At other azimuth
angles it converts a linearly polarized light into an elliptically polarized light oriented
along the fast or slow axes.
f
s
s
Half-wave plate
s
Quarter-wave plate
s
s
Quarter-wave plate
f
f
f
f
f
f
f
s
s
s
Quarter-wave plate
3
Zero-order and multiple-order wave plates
2
ns  n f l  (2m  1) with m  0
A half-wave plate with a retardation of  

is called a zero-order half-wave plate. The thickness of the plate is

500 nm
l

 25m (typically for crystal quartz).
2(ns  n f ) 2  0.01
This thickness is not easy to fabricate and not easy to handle.
We can use a thickness of l  (2m  1)

2(ns  n f )
, which is a multiple-order wave plate.
The wavelength sensitivity of a half-wave plate is

  2



l
n

n
s
f   l  2m  1.

  

Therefore a multiple-order wave plate has a limited bandwidth.
One technique to solve this problem is to make a compound zero-order wave plate
from two plates, with their optic axes intercrossed. The difference in thickness
between the two plates determines the overall retardation. Zero-order wave plates
thus have broad bandwidths.

2

n
s
 n f l1  l2 
4
Reading: Achromatic wave plates:
2
ns1 ( )  n f 1 ( )l1  n f 2 ( )  ns 2 ( )l2 
( , l1 , l2 ) 

Estimation of l1 and l2 (e.g., QWP):
In the wavelength range considered, the phase
retardation should be as close as possible to /2
(minimum rms, or similar criteria).
g (l1 , l2 )  
2 1000nm
1  700nm
y (e)
y (o)
MgF2
Crystal Quartz
x (o)
x (e)



(

,
l
,
l
)

d
1 2


2
2
g (l1 , l2 )

 0
l1

  l1=603 m, l2=477 m.
g (l1 , l2 )
 0

l2
Question: Why do we need two materials?
dn f 1 
 dn
 dn
dn  
( , l1 , l2 )
2 
l1    f 2  s 2 l2 
  2 (ns1  n f 1 )l1  (n f 2  ns 2 )l2    s1 

 
d 
d  
 d
 d
For a normal compound zero-order wave plate (one material), l1-l2 is fixed, thus
 /  is fixed. For achromatic wave plates (two materials), l1 and l2 can be chosen
to minimize  /  , which greatly expands the applicable bandwidth.
5
Lecture 2 Jones matrix
5.1 Jones matrix formulation
While it is not difficult to track the polarization of light passing through an individual
wave plate or polarizer using junior algebra, when there is a combination of several such
optical elements, and a certain goal is aimed, the algebra involved can be complicated.
Jones calculus is created to study the transformation of polarization using linear algebra,
where the polarization of light is represented by a Jones vector, and the function of an
optical element is represented by a 2×2 matrix.
A fixed lab coordinate system (instead of the
principle axes of the crystal) is normally used. The
azimuth angle of the retardation plate is defined as
the angle from the lab x axis to the slow axis of the
crystal. The light is propagating in the z direction.
6
Wave plates:
Let us first derive the 2×2 matrix for a wave plate. Suppose the input light has an arbitrary
polarization state (Vx ,Vy )T. At the surface of the plate this light need to be decomposed into
the two eigenwaves, which is a transformation to the s-f coordinate system.
 Vs   cos
   
V f    sin 
sin  Vx 
Vx 
 cos sin  




 R( ) , with R( )  

.
V
cos Vy 

sin

cos



 y
The polarization state after the plate, in the s-f coordinates, is then
0
 V ' s   exp  insl / c 
 Vs  i  ei / 2
 Vs 
0  Vs 

  
   e 
   W0  
i / 2 


V
'
0
exp

in

l
/
c
V
e V f 
f
 0
 f 
 f 
V f 
Here Γ  ns  n f 
l
c
is the phase retardation, and  
1
ns  n f l is the average
2
c
 i / 2

e
0 
 is the Jones matrix for the wave plate expressed in
phase change. W0  ei 
i / 2 
e 
 0
its own principle s-f coordinate system.
In the x-y coordinate system the polarization state after the plate is
Vx '   cos
   
V '  sin 
 y 
 sin   Vs ' 
 Vs ' 
   R( ) 
cos V f ' 
V f ' 
7
The overall effect of the retardation plate is then
Vx ' 
V 
 cos
   R(  )W0 R( ) x , with R( )  
  sin 
Vy ' 
Vy 
 i / 2
sin  
0 
i  e
.
, W0  e 
i / 2 
cos 
e 
 0
The e-i factor can be dropped if we are not dealing with interference. Therefore a
retardation plate is characterized by its phase retardation  and azimuth angle  , and is
represented in the lab frame by
W  R( )W0 R( )
Note that the transformation is unitary: W+W=1. It does not change the inner product
between two Jones vectors. This is because a wave plate does not absorb light.
Linear polarizers (analyzers):
An ideal linear polarizer with its transmission axis on the x axis is
1 0
, with  as the absolute phase change.
P0  e i 
 0 0
For a linear polarizer oriented at an azimuth angle , the Jones matrix is
P  R( ) P0 R( )
1 0
 0 0
Particularly Px  
, Py  
.
 0 0
0 1
8
Combination of wave plates and polarizers:
For a series of wave plates and polarizers, we need to just multiply the Jones matrix
of individual element in sequence.
Wave plates examples:
Half-wave plate:
A half-wave plate with its slow axis oriented at  =45°. The input light is linearly
polarized in the vertical direction:
0  1  1 1  0  i 
1 1  1 e  i / 2




  

W  R( )W0 R( ) 
i / 2 
e  2   1 1   i 0 
2 1 1  0
 0  i  0    i 
1
      i  is a horizontal ly polarized light.
V '  

i
0

 1   0 
 0
For a general azimuth angle , a half-wave plate will rotate a horizontally or vertically
polarized light by 2.
A half-wave plate will change a left-handed circularly polarized light into a right-handed
circularly polarized light, and vice versa, regardless of the azimuth angle of the plate.
9
Quarter-wave plate:
A quarter wave plate with its slow axis oriented at  =45°. The input light is linearly
polarized in the vertical direction:
W  R(  )W0 R( )
1 1  1 e i / 4



2  1 1  0
1  1  i



2  i 1 
0  1  1 1



i / 4 
e  2   1 1
Our textbook
is wrong here.
1  1  i  0  1   i   i 1
V'

  
 
 
2   i 1  1 
2  1 
2  i 
is a left - handed circularly polarized light.
If the input light is horizontally polarized, it
will be changed into a right-handed
circularly polarized light.
10
5.2 Intensity transmission


For a Jones vector J  E x , E y  , the corresponding electric field is E  Ex xˆ  E y yˆ. eit .
The absolute intensity of the light is then
2
2
1
1
2
I  e E x xˆ  E y yˆ v  ev( E x  E y ).
2
2
For convenience, if we only care the relative intensity of light in one medium (e.g., air),
the factor (1/2)ev is a constant and can be dropped. We therefore define the intensity of
light as
2
I  Ex  E y  E  E  J   J.
2
 Ex ' 
 Ex 
For a Jones matrix transformation    M   , the intensity transmittance is then
Ey '
Ey 


2
2
Ex '  E y '
T
.
2
2
Ex  E y
11
Intensity transmission examples:
A birefringent plate sandwiched between two parallel polarizers:
Suppose the transmission axes of the polarizers are both vertical. The slow axis of the
birefringent plate is oriented at 45° from the x axis. The plate introduces a phase retardation
d
d
Γ  ns  n f
 2 ns  n f
c





The Jones matrix is
0  1  1 1  cos / 2  i sin  / 2
1 1  1 e i / 2





  

W  R( )W0 R( ) 
i / 2 




1
1

1
1

i
sin

/
2
cos

/
2
e  2
2
 0
 

Let the incident light be unpolarized, with unit intensity. After the first polarizer, the
1 0
Jones vector is
 . The electric field of the final transmitted light is
2 1
0
 0 0  cos / 2  i sin  / 2 1  0  1 









E'  







2  cos / 2
 0 1   i sin  / 2 cos / 2  2  1 
The intensity transmittance is
1
 1
d

T  E ' E '  cos2  cos2  ns  n f  .
2
2 2


12
A birefringent plate sandwiched between two crossed polarizers:
As above, but let the transmission axis of the final polarizer be horizontal. The electric
field of the final transmitted light is
 1 0  cos / 2  i sin  / 2 1  0   i  sin  / 2


  


E '  
0
2
 0 0   i sin  / 2 cos / 2  2  1 

The intensity transmittance is
1
 1
d

T  E ' E '  sin 2  sin 2  ns  n f  .
2
2 2


This is complementary to the case of parallel polarizers.
In both cases the transmission is a sinusoidal function of wave number 1/.
13
Lecture 3 Twisted nematic liquid crystals
Twisted nematic liquid crystal displays (TNLCDs) are currently the most commonly used
LCDs. The display panel consists of many
liquid crystal cells sandwiched between crossed
polarizers. Inside each cell the nematic liquid
crystals are aligned so that the director of the
first molecule is parallel to the transmission axis
of the polarizer. Each next molecule is gradually
twisted so that the director of the last molecule
in the cell is parallel to the transmission axis of
the analyzer.
How does an LCD work?
In the OFF-state the linearly polarized incident light is changed in the liquid crystal cell
so that its polarization follows the twist of the liquid crystal molecules. The polarization
is finally rotated to be parallel to the analyzer and light is therefore transmitted.
In the ON-state an electric field is added across the liquid crystal cell. The directors of the
liquid crystal molecules will tilt toward the electric field direction. In this way the actual
birefringence experienced by the incident light will decrease. The cell appears gray or
dark depending on the strength of the external electric field.
14
Light propagation in twisted anisotropic media
Jones matrix of a twisted nematic liquid crystal cell
Assume the twist is linear. Let  be the total twist angle
(clockwise) of the crystal. Let  be the total phase
retardation when the molecules are not twisted. We divide
the crystal into N small thin plates with equal thickness. In
the principal coordinate system of the first molecule, i.e.,
let its slow axis be the x-axis, and look toward the light
source, the overall matrix of the small plates is
 i / 2 N

    e
M   R   m    
 N   0
m 1 
0     
 R  m   
i / 2 N 
e
   N 
N
 e i / 2 N
 R( ) 
 0
0    
 R  
i / 2 N  
e
  N 
 i / 2 N

 cos e
N
 R( )
 sin  ei / 2 N
N

 sin
cos

N

N
e

m=1 is on the most right.
N
i / 2 N
ei / 2 N





N
15
Chebyshev’s identity: For a unimodular (AD-BC=1) 2×2 matrix,
 A B
1  A sin m  sin( m  1)

 

C
D
C sin m
sin




m
B sin m

A D
.
, with   cos1
D sin m  sin( m  1) 
2
It can be easily proved by mathematical induction.
The Jones matrix of the small plates is then
 i / 2 N

cos
e

N
M  R ( )
 sin  ei / 2 N
N



 sin e i / 2 N 
N

 i / 2 N 
cos e

N

N
 i / 2 N

cos
e
sin N  sin( N  1)

1
N

 R ( )
 i / 2 N
sin  
sin
e
sin N

N


 

with   cos 1  cos cos
.
N
2N 

 i / 2 N

e
sin N

N

 i / 2 N
cos e
sin N  sin( N  1) 
N

 sin
16
The Jones matrix of the twisted nematic liquid crystal cell is given by letting N approach
infinity in the above equation.
2
 1   2   2   1

 


1 
2
  cos  cos cos
   .
  cos 1     
  
N
2N 

2
 2  N   2 N    N
1
 i / 2 N
 i / 2 N


cos
e
sin
N


sin(
N

1
)


sin
e
sin
N



1
N
N


M  lim R( )
N 


sin  
sin e i / 2 N sin N
cos e i / 2 N sin N  sin( N  1) 

N
N


 sin X
sin X


2

 cos X  i


 
2 X
X
, with X   2   
 R( )
sin X
 sin X 
2


cos X  i

X
2 X 

 ei / 2
0 
.
Adiabatic following:
When    , M  R( )
i / 2 
e 
 0
Now if the incident light is linearly polarized in the slow (or fast) axis of the first molecule,
the polarization vector of the light inside the twisted crystal will follow the rotation of the
slow (or fast) principal axis of the local molecules.
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Twisted nematic liquid crystal with =/2:
 sin X
 sin X

cos
X

i

2 X
2 X
M 
 sin X
  cos X  i  sin X
2 X
2 X


  A
  
*
   B

2
2
B


   
, with X       .
A
2 2
Transmittance of twisted nematic liquid crystal with =/2:
1. Between crossed polarizers.

2 
sin 2 
1   /   
 A B  1   A 
2
2
.
 *
    * , T  B  1 
2
1   /  
  B A  0    B 
2. Between parallel polarizers.
 A
 *
 B

2 
sin 2 
1   /   
B  1   A 
2
2
.
    * , T//  A 
2
A  0    B 
1   /  
States of twisted nematic liquid crystal with =/2:
1. OFF-state: >>, T=1 for crossed polarizers, T=0 for parallel polarizers.
2. Fully ON-state:  0, T=0 for crossed polarizers, T=1 for parallel polarizers.
3. Gray-level state:  is controlled by external field so that the transmission is
continuously varied. This is how the twisted nematic liquid crystal display works.
18