Silicon chip birefringence
Asymmetric crystals cut so optic axis is in the plane of the plate. Light comes in perpendicular to the plate.
Waveplates (optical retarders)
The optic axis is the fast axis if ____ a) no > ne. b) no < ne.
Light travels fastest if E is aligned with the fast axis (bold blue line)
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:
Choose thickness so phase difference between fast and slow light is ____
Quarter-wave plates
If we start with linear polarization at 45o from the fast axis, we will end up with ________ polarized lighta) linearly b) circularlyc) 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 lighta) linearly b) circularlyc) 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
If we start with linear polarization at 45o from the fast axis, we will end up with ________ polarized lighta) linearly b) circularlyc) elliptically
Hint, figure out the components (Jones vector) in the x’, y’ coordinate system, and then do the phase shift.
Choose thickness so phase difference between fast and slow light is ____
Half-wave platesIf we start with linear polarization at a general angle q from the fast axis, we will end up with ________ polarized lighta) linearly b) circularlyc) 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 2 .q
Jones Matrix
New statex xy
ixy y
J J A
J J Be
JM for linear polarizer
Horizontal transmission (trans. axis along x)1 0
0 0
Vertical transmission (trans. axis along y) 0 0
0 1
Rotation of coordinates
Arbitrary angles for polarizers
rot rot rot
rot rot rot
x r r r
y r r r
' cos( ) cos( )cos( ) sin( )sin( )
' sin( ) sin( )cos( ) cos( )sin( )
rot rot
rot rot
Rcos sin
sin cos
transforms a vector from the original basis to the vector in the rotated basis.
1 R
transforms a vector from the rotated basis to the vector in the original basis.
rot rot
rot rot
cos sin
sin cos
V VR'
1V VR '
Linear polarizer at arbitrary angles
transforms a matrix (operator) from the original basis to the matrix in the rotated basis.
cos sin 1 0 cos sin
sin cos 0 0 sin cos
1 0
0 0
Polarizer looks like if x’ is aligned with the
transmission axis. Let’s get it in the x, y system:
1 M R M R'
2
2
cos sin cos
sin cos sin
JM for linear polarizer
JM for Waveplates
What does the /4 l plate Jones matrix look like in the x’,y’ coordinate system? It delays the slow (y’) component by /2.p
For waveplates, q is orientation of fast axis vs the x (H) axis.
What does the /2 l plate Jones matrix look like in the x’,y’ coordinate system? It delays the slow (y’) component by p.
1 0
0 i
1 0
0 1
JM for Waveplates
JM for quarter-wave plate
cos sin cos sin
sin cos sin cosx xy
xy y
J J
J J
2 2
2 2
cos sin 1 sin cos
1 sin cos sin cos
i i
i i
2 2
2 2
cos sin
sin cos
JM for half-wave plate
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)
0 0
0 0H p
V s
t t
t t
NotesOrder 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.