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.