Lecture 9: Birefringent Optical Elements Petr Kužel

Lecture 9: Birefringent optical elements Petr Kužel

• Polarizers • Phase retarders • Applications: Jones matrices

• Optical activity Basics

Birefringent optical elements: Plates or prisms which can change the polarization state of a beam in a well- defined way

Calcite (CaCO3): = = no 1.662, ne 1.488 − = − large birefringence (ne no 0.174): polarizers

Quartz (SiO2): = = no 1.546, ne 1.555 − = + smaller birefringence (ne no 0.009): phase retarders Polarizers couple of prisms (working on the total reflection principle) separated by • Canadian balm (n = 1.54) •Air layer

Dichroic (sheet) polarizers • Different absoption coefficient for both polarization Examples of polarizers

Name normal surface polarizer layout comment

! Large range of incidence angles Glan-Thompson (≈ 20°) O E

! Small range of incidence angles (≈ 5°) Glan-Taylor ! Brewster angle E incidence O ! Possible use: UV, high power ! Separation of the O and E beams Rochon O (typically 10°) E

! Symmetric separation O of the O et E beams Wollaston (typically 20°) E Optical elements: Jones matrices

J0 J1

M (2×2)

= ⋅ J1 M J0 Jones matrices of polarizers

1 0 0 0 P =   P =   x 0 0 y 0 1

cosψ − sin ψ 1 0  cosψ sin ψ   2 ψ ψ ψ =       =  cos sin cos  Pψ          sin ψ cosψ  0 0 − sin ψ cosψ sin ψ cosψ sin 2 ψ 

Parallel and perpendicular polarizations:

 2  c c c2 sc − s 0 c sc   =       =          2       sc s2   s  s  sc s   c  0

Crossed polarizers (angles ϕ et ϕ + π/2):  2   2 −    c sc  s sc 0 0     =    sc s2  − sc c2  0 0 Jones matrices of polarizers Sequence of 3 polarizers (ϕ-polarizer between two crossed polarizers):

ϕ 0 π/2

   2      1 0 c sc 0 0 0 cs       =   0 0  sc s2  0 1 0 0  If this sequence is applied on a linearly polarized beam: 0 cs 0 cs     =   0 0  1  0  Maximum transmission for ϕ = π/4, no transmission for ϕ = 0,π/2 Phase retardation plates Phase retardation (due to different optical path) for 2 orthogonal linear polarizations: π ()− ∆ϕ = 2 ne no d λ

Example: for a quarter-wave plate (∆ϕ = π/2) we need: d = 15.2 µm pour λ = 546.1 nm • higher order thick plates (∆ϕ = 5π/2, 9π/2…)

• 2 plates in optical contact with mutually perpendicular axis orientation and with a slightly different thickness axis

π()()− − ∆ϕ = ∆ϕ − ∆ϕ = 2 ne no d1 d2 1 2 1 2 λ ∆ϕ1 ∆ϕ2 Compensator

Babinet-Soleil: 2a axis axis

axis 1 2b displacement

Berek: axis Jones matrices of phase retarders

 iδ  e 0    0 1

 c s  iδ  c − s  2 iδ + 2 ()− iδ    e 0   = c e s cs 1 e       ()  δ δ  − s c  0 1  s c   cs 1− ei s2ei + c2  Phase retardation plate between 2 polarizers:  2 −   iδ   2   s sc e 0 c sc       − sc c2   0 1  sc s2 

The light does not pass for ϕ = 0, π/2 Maximum transmission for ϕ = π/4 (optical axis of the plate shows 45° with respect to the polarizing directions) Modulator: phase retarder between 2 polarizers

Optical axis of the plate shows 45° with respect to the polarizing directions

 2 −   iδ   2  c  iδ −   iδ −   s sc e 0 c sc   =  s(e 1) = 2 e 1         cs δ   δ  − sc c2   0 1  sc s2   s c(1− ei ) 4 1− ei 

Transmitted light intensity:

1 δ − δ δ − δ 1 I = ()()()()()ei −1 e i −1 + 1− ei 1− e i = ()1− cosδ 8 2 The dephasing δ can have 2 parts δ π • a large one which is constant (close to 0 = /2 — quarter-wave plate) δ = Γ ω Γ • a small one which is variable ( 1 sin mt, << 1): 1 1 1 I = ()1− cos(δ + δ ) = ()1+ sin(Γsin ω t) ≈ ()1+ Γsin ω t 2 0 1 2 m 2 m