Silicon chip birefringence
Waveplates (optical retarders)
Asymmetric 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.
To analyze, we break light into
components along and perp to the
fast axis. Phase difference between
the fast and slow light after the WP
in terms of thickness:
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 q from the fast axis,
we will end up with ________
polarized light
Summary: QWP’s usual purpose is to change between
linear and circular pol, which means the lin-pol line has to
be at 45 deg to the fast/slow axes. Other orientations
give elliptical.
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 at
a general angle q 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.
Summary: HWP’s usual purpose is to rotate linear
polarization to a new line, by 2q.
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(q  q rot )  r cos(q )cos(q rot )  r sin(q )sin(q rot ) 
y '  r sin(q  q rot )  r sin(q )cos(q rot )  r cos(q )sin(q rot ) 
 cos q rot
R
  sinq rot
sinq rot 

cos q rot 
transforms a vector from the original basis to the vector in
the rotated basis. V '  RV
R 1 
 cos q rot

 sinq rot
 sinq rot 

cos q 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  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 q

 sin q
 sin q  1 0  cos q


cos q  0 0   sin q
 cos 2 q

sin q cos q
sin q 

cos q 
sin q cos q 

sin 2 q  JM for linear polarizer
JM for Waveplates
For waveplates, q 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 p/2.
1 0 


 0 i 
What does the l/2 plate Jones
matrix look like in the x’,y’
coordinate system? It delays the
slow (y’) component by p.
1 0 


 0 1
JM for Waveplates
 cos q

 sin q
 sin q   J x

cos q   J xy
J xy   cos q

J y    sin q
sin q 

cos q 
JM for quarter-wave plate
 cos 2 q  i sin 2 q

1  i  sin q cos q
1  i  sin q cos q 

sin q  i cos q 
JM for half-wave plate
cos 2q

 sin 2q
sin 2q 

 cos 2q 
2
2
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
If a circularly polarized beam in the horizontal plane
strikes a vertical mirror at say 45%, what will the final
state be?
JM for Reflection (vertical mirrors)
 t H

 0
0   t p

tV   0
0

ts 
Notes
Order of matrices matters!
Fraction of intensity transmitted: compare initial and final vector
squared magnitudes
Convention: choose x always so it stays on either your left or right hand
as you follow the beam around reflections in a plane.