Silicon chip birefringence
Jones Matrix
 Jx

 J xy
J xy   A

J y   Bei

  New state

JM for linear polarizer
Horizontal transmission (trans. axis along x)
1 0


0
0


Vertical transmission (trans. axis along y)
0 0


0
1


Arbitrary angles for polarizers
Rotation of coordinates
x '  r cos(   rot )  r cos( )cos( rot )  r sin( )sin( rot ) 
y '  r sin(   rot )  r sin( )cos( rot )  r cos( )sin( rot ) 
 cos  rot
R
  sin rot
sin rot 

cos  rot 
transforms a vector from the original basis to the vector in
the rotated basis. V '  RV
R 1 
 cos  rot

 sin rot
 sin rot 

cos  rot 
transforms a vector from the rotated basis to the vector in
1
the original basis. V  R V '
Linear polarizer at arbitrary angles
1 0
Polarizer looks like  0 0  in “rotated” coordinates if x’ is


aligned with the transmission axis. Let’s get it in the x, y
system:
M  R 1 M ' R
transforms a matrix (operator) from the
original basis to the matrix in the rotated
basis.
 cos 

 sin 
 sin   1 0  cos 


cos   0 0   sin 
 cos 2 

sin  cos 
sin  

cos  
sin  cos  

sin 2   JM for linear polarizer
Waveplates (optical retarders)
Uniaxial crystals cut so optic axis is in the plane of the plate. Light
comes in perpendicular to the plate.
Light travels fastest if E is aligned with the fast axis (bold blue line).
The optic axis is the fast axis if ____
a) no > ne.
b) no < ne.
Phase difference between the fast
and slow light after the WP in terms
of thickness:
  (kd )
OPL  (nd )
Quarter-wave plates
Choose thickness so phase difference between fast and slow light is ____
If we start with linear polarization at
45o from the fast axis, we will end up
with ________ polarized light
a) linearly
b) circularly
c) elliptically
Hint, figure out the components
(Jones vector) in the x’, y’
coordinate system, and then do the
phase shift.
Quarter-wave plates
If we start with linear polarization at
90o from the fast axis, we will end
up with ________ polarized light
a) linearly
b) circularly
c) elliptically
Quarter-wave plates
If we start with linear polarization at
general angle  from the fast axis,
we will end up with ________
polarized light
a) linearly
b) circularly
c) elliptically
Half-wave plates
Choose thickness so phase difference between fast and slow light is ____
If we start with linear polarization at
45o from the fast axis, we will end up
with ________ polarized light
a) linearly
b) circularly
c) elliptically
Hint, figure out the components
(Jones vector) in the x’, y’
coordinate system, and then do the
phase shift.
Half-wave plates
If we start with linear polarization
along the x axis, and the fast axis is
rotated a general angle , we will
end up with ________ polarized
light
a) linearly
b) circularly
c) elliptically
JM for Waveplates
For waveplates,  is orientation of fast axis vs the x (H) axis.
What does the l/4 plate Jones matrix look like in the x’,y’
coordinate system? It delays the slow (y’) component by ______.
What does the l/2 plate Jones matrix look like in the x’,y’
coordinate system? It delays the slow (y’) component by ______.
JM for Waveplates
 cos 

 sin 
 sin    J x

cos    J xy
J xy   cos 

J y    sin 
sin  

cos  
JM for quarter-wave plate

cos 2   i sin 2 
sin  cos   i sin  cos  


2
2
sin   i cos 
sin  cos   i sin  cos 

Main use of a QWP:
JM for half-wave-plate
cos 2

 sin 2
sin 2  Main use of a HWP:

 cos 2 
Polarization demos again
Does our transparency sheet act more like QWP
or HWP?
http://optics.byu.edu/animation/polarwav.
mov
Notes
Order of matrices matters!
Fraction of intensity transmitted: compare initial and final vector
squared magnitudes
Choose x to stay on your right hand as you follow the beam around
reflections.
If a R-cir beam strikes a metal mirror at normal
incidence, what will the resulting beam be?
a. R-cir
b. L-cir
c. linearly polarized