17. Jones Matrices & Mueller Matrices
Jones Matrices
Rotation of coordinates - the rotation matrix
Stokes Parameters and unpolarized light
Mueller Matrices
Sir George G. Stokes
(1819 - 1903)
R. Clark Jones
(1916 - 2004)
Hans Mueller
(1900 - 1965)
Jones vectors describe the polarization
state of a wave
Define the polarization state of a field as a 2D vector—
“Jones vector” —containing the two complex amplitudes:
 Ex 
 1 
E     Ex 


 Ey 
 E y Ex 
1
Ex  E y
2
2
 Ex 
E 
 y
(normalized to length of unity)
1 
0
 
A few examples:
0° linear (x) polarization: Ey /Ex = 0
linear (arbitrary angle) polarization: Ey /Ex = tan 
right or left circular polarization: Ey /Ex = ±j
 1 
 tan  


1
 j 
 
To model the effect of a medium on light's
polarization state, we use Jones matrices.
Since we can write a polarization state as a (Jones) vector, we use
matrices, A, to transform them from the input polarization, E0, to the
output polarization, E1.
 a11
A
 a21
E1  AE0
This yields:
This should be
thought of as a
transfer function.
E1x  a11 E0 x  a12 E0 y
E1 y  a21 E0 x  a22 E0 y
For example, an x-polarizer can be written:
So:
a12 
a22 
1 0 
Ax  

0
0


1 0   E0 x   E0 x 
E1  A x E0  
E    

0 0   0 y   0 
Other Jones matrices
A y-polarizer:
A half-wave plate:
0 0 
Ay  

0
1


A HWP
1 0 


0
1



A half-wave plate rotates 45-degreepolarization to -45-degree, and vice versa.
A quarter-wave plate:
A QWP
1 0 


0

j


R. Clark Jones
(1916 - 2004)
1 0  1  1 
0 1 1   1

   
1 0   1  1
0 1  1  1

   
1 0  1  1 
0  j  1    j 

   
The orientation of a wave plate
matters.
0° or 90° Polarizer
Remember that a quarter-wave plate
only converts linear to circular if the
input polarization is ±45°.
If it sees, say, x polarization,
the input is unchanged.
Wave plate
w/ axes at
0° or 90°
1 0  1  1 
0  j  0   0 

   
AQWP
Jones matrices are an extremely useful way to keep track of all this.
A wave plate example
What does a quarter-wave plate do if the input polarization is linear
but at an arbitrary angle?
1

1 0   1  
0 j   tan      j tan   


 

AQWP
Ein
Eout
For arbitrary , this is an elliptical polarization.
 = 30°
 = 45°
 = 60°
Jones Matrices for standard components
Rotated Jones matrices
What about when the polarizer or wave plate responsible for
the transfer function A is rotated by some angle, ?
Rotation of a vector by an angle  means multiply by the rotation matrix:
rotated Jones vector
of the input
where:
E0 '  R   E0
and
E1 '  R   E1
rotated Jones vector
of the output
cos( )  sin( ) 
R    

 sin( ) cos( ) 
Rotating E1 by  and inserting the identity matrix R()-1 R(), we have:
1
E1 '  R   E1  R   AE0  R   A  R   R    E0


1
1
  R   AR     R   E0    R   AR    E0 '  A ' E0 '




Thus:
A '  R   AR  
1
Rotated Jones matrix for a polarizer
Example: apply this to an x polarizer. A '  R   A R  
1
cos( )  sin( )  1 0   cos( ) sin( ) 
Ax    
 0 0    sin( ) cos( ) 


sin(
)
cos(
)




cos( )  sin( )  cos( ) sin( ) 

 0

sin(

)
cos(

)
0



 cos 2 ( )
cos( ) sin( ) 


2
sin ( ) 
cos( ) sin( )
So, for example:
1/ 2 1/ 2 
Ax  45   

1/
2
1/
2



1  
Ax    


0


for a small
angle 
To model the effect of many media on light's
polarization state, we use many Jones matrices.
The aggregate effect of multiple components or objects can be
described by the product of the Jones matrix for each one.
input

E0
transfer function

A1
A2
A3
output

E1
E1  A 3 A 2 A1 E0
The order may look counter-intuitive, but order matters!
x
Multiplying Jones Matrices
y
Crossed polarizers:
E0
E1  A y A x E0
x-pol
E1
y-pol
0 0  1 0  0 0 
Ay Ax  





0
1
0
0
0
0


 

rotated
x-pol
Uncrossed polarizers
(by a slight angle ):
E0
0 0   1    0
A y A x   




0 1   0  
 Ex   0
A y A x     
 E y  
so no light leaks through.
E1
0
0 
0   Ex   0 
 
0   E y   Ex 
y-pol
So Iout
≈ 2 Iin,x
z
x
Multiplying Jones Matrices
y
Now, it is easy to compute how
inserting a third polarizer
between two crossed polarizers
leads to larger transmission.
E0
x-pol
E1
45º-pol
y-pol
E1  A y A 45 A x E0
1
0 0   2
A y A 45 A x  
1
0
1


 2
Thus:
z
1 
0 0
2  1 0   
1




0
1 0 0   2 


2
 0 0   Ex ,in   0 

E1   1
 1


 2 0   E y ,in   Ex ,in 
2

The third polarizer, between the other two, makes the
transmitted wave non-zero.
Natural light (e.g., sunlight, light bulbs, etc.)
is unpolarized
The direction of the E vector is
randomly changing. But, it is
always perpendicular to the
propagation direction.
polarized light
natural light
Light with very complex polarization
vs. position is "unpolarized."
Light that has scattered multiple times, or that has scattered
randomly, often becomes unpolarized as a result.
Here, light from the blue sky is
polarized, so when viewed
through a polarizer it looks
much darker. Light from clouds
is unpolarized, so its intensity is
reduced by only 50%.
If the polarization vs. position is unresolvable, we call this
“unpolarized.” Otherwise, we refer to this light as “locally
polarized” or “partially polarized.”
When the phases of the x- and y-polarizations
fluctuate, we say the light is "unpolarized."

E ( z , t )  Re  E

 t  
Ex ( z , t )  Re E0 x exp  j  kz  t   x  t   
y
0y
exp  j  kz  t   y
where x(t) and y(t) are functions that vary on a time scale slower than
the period of the wave, but faster than you can measure.
The polarization state (Jones vector) is:
1


E

 0 y exp  j  t   j  t   
x
 y

 E0 x

In practice, the
amplitudes are also
functions of time!
As long as the time-varying relative phase, x(t)–y(t), fluctuates, the light
will not remain in a single polarization state and hence is unpolarized.
Stokes Parameters
We cannot use Jones vectors to describe something that is rapidly
fluctuating like this. So, to treat fully, partially, or unpolarized light, we
use a different scheme. We define "Stokes parameters."
Suppose we have four detectors, three with polarizers in front of them:
#0 detects total irradiance............................................I0
#1 detects horizontally polarized irradiance..........…...I1
#2 detects +45° polarized irradiance............................I2
#3 detects right circularly polarized irradiance.....…….I3
Note that these
quantities are timeaveraged, so even
randomly polarized
light will give a welldefined answer.
The Stokes parameters:
S0  I0
S1  2I1 – I0
S2  2I2 – I0
S3  2I3 – I0
Interpretation of the Stokes Parameters
The Stokes parameters:
S0  I0
S1  2I1 – I0
S2  2I2 – I0
S3  2I3 – I0
S0 = the total irradiance
S1 = the excess in intensity of light transmitted by a horizontal polarizer
over light transmitted by a vertical polarizer
S2 = the excess in intensity of light transmitted by a 45° polarizer over
light transmitted by a 135° polarizer
S3 = the excess in intensity of light transmitted by a RCP filter over light
transmitted by a LCP filter
What we mean when we say ‘unpolarized light’:
All three of these excess quantities are zero
Degree of polarization
If any of the excess quantities (S1, S2, or S3) are nonzero, then the wave has some degree of polarization.
We can quantify this by defining the “degree of
polarization”:
Degree of polarization =  S + S + S
2
1
2
2

2 1/2
3
/ S0
= 1 for polarized light
= 0 for unpolarized light
Note that this quantity can never be greater than unity,
since S0 is the total intensity.
This is not the same as the ‘degree of polarization’
defined in the homework problem, which was only
defined for fully polarized light.
The Stokes vector
We can write the four Stokes parameters in vector form:
 S0 
S 
S   1
 S2 
 
 S3 
The Stokes vector S contain information about both the
polarized part and the unpolarized part of the wave.
S = S(1) + S(2)
unpolarized part:
S
1
S  S 2  S 2  S 2 
1
2
3
 0

0




0




0
polarized part:
S
2
 S 2 S 2 S 2
2
3
 1

S1

 

S
2




S3
Stokes vectors
(and Jones
vectors for
comparison)
Sir George G. Stokes
(1819 - 1903)
Mueller Matrices multiply Stokes vectors
We can define matrices that multiply Stokes vectors,
just as Jones matrices multiply Jones vectors. These
are called Mueller matrices.
Sin
M1
M2
M3
Sout
To model the effects of more than one medium on the polarization
state, just multiply the input polarization Stokes vector by all of the
Mueller matrices:
Sout = M3 M2 M1 Sin
(just like Jones matrices multiplying Jones vectors, except that the
vectors have four elements instead of two)
Mueller Matrices
(and Jones
Matrices for
comparison)
With Stokes vectors and
Mueller matrices, we can
describe light with arbitrarily
complicated combination of
polarized and unpolarized light.
Hans Mueller
(1900 - 1965)