Lecture 3: Crystal Optics

Homogeneous, Anisotropic Media

Outline

Introduction material equations for homogeneous anisotropic media

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~ = E ~ D ~ = µH ~ B

Homogeneous, Anisotropic Media Crystals Plane Waves in Anisotropic Media Wave Propagation in Uniaxial Media Reflection and Transmission at Interfaces

Christoph U. Keller, Utrecht University, [email protected]

Lecture 3: Crystal Optics

tensors of rank 2, written as 3 by 3 matrices : dielectric tensor µ: magnetic permeability tensor

examples: crystals, liquid crystals external electric, magnetic fields acting on isotropic materials (glass, fluids, gas) anisotropic mechanical forces acting on isotropic materials

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Properties of Dielectric Tensor

Lecture 3: Crystal Optics

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Lecture 3: Crystal Optics

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Uniaxial Materials

Maxwell equations imply symmetric dielectric tensor   11 12 13  = T =  12 22 23  13 23 33

isotropic materials: nx = ny = nz for any coordinate system anisotropic materials: nx 6= ny 6= nz uniaxial materials: nx = ny 6= nz

symmetric tensor of rank 2 ⇒ coordinate system exists where tensor is diagonal

ordinary index of refraction: no = nx = ny

orthogonal axes of this coordinate system: principal axes

extraordinary index of refraction: ne = nz

elements of diagonal tensor: principal dielectric constants 3 principal indices of refraction in coordinate system spanned by principal axes  2  nx 0 0 ~ =  0 n2 0  E ~ D y 2 0 0 nz

rotation of coordinate system around z does not change anything most materials used in polarimetry are (almost) uniaxial

x, y , z because principal axes form Cartesian coordinate system Christoph U. Keller, Utrecht University, [email protected]

Christoph U. Keller, Utrecht University, [email protected]

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Christoph U. Keller, Utrecht University, [email protected]

Crystals

Plane Waves in Anisotropic Media

Crystal Axes Terminology

Displacement and Electric Field Vectors

optic axis is the axis that has a different index of refraction

~ E, ~ H ~ plane-wave ansatz for D,

also called c or crystallographic axis

~ =E ~ 0 ei (~k ·~x −ωt ) E ~ =D ~ 0 ei (~k ·~x −ωt ) D ~ =H ~ 0 ei (~k ·~x −ωt ) H

fast axis: axis with smallest index of refraction ray of light going through uniaxial crystal is (generally) split into two rays ordinary ray (o-ray) passes the crystal without any deviation

~ = 0) no net charges in medium (∇ · D

extraordinary ray (e-ray) is deviated at air-crystal interface two emerging rays have orthogonal polarization states

~ · ~k = 0 D

common to use indices of refraction for ordinary ray (no ) and extraordinary ray (ne ) instead of indices of refraction in crystal coordinate system

~ perpendicular to ~k D ~ and E ~ not parallel ⇒ E ~ not perpendicular to ~k D

ne < no : negative uniaxial crystal

wave normal, energy flow not in same direction, same speed

ne > no : positive uniaxial crystal Christoph U. Keller, Utrecht University, [email protected]

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Christoph U. Keller, Utrecht University, [email protected]

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~ and E ~ Relation between D combine Maxwell, material equations in principal coordinate system    ~ i = n2 E ~ i − si E ~ · ~s i = 1···3 µDi = µi E

Magnetic Field constant, scalar µ, vanishing current ~ kB ~ density ⇒ H ~ =0⇒H ~ ⊥ ~k ∇·H ~ = 1 ∂ D~ ⇒ H ~ ⊥D ~ ∇×H

~s = ~k /|~k |: unit vector in direction of wave vector ~k n: refractive index associated with direction ~s, i.e. n = n(~s) ~i 3 equations for 3 unknowns E

c ∂t

~ = − µ ∂ H~ ⇒ H ~ ⊥E ~ ∇×E c ∂t ~ E, ~ and ~k all in one plane D, ~ B ~ perpendicular to that plane H,

~ assuming E ~ 6= ~0 ⇒ Fresnel equation eliminate E

~ = cE ~ ×H ~ Poynting vector S 4π ~ (in ~ and H ~ ⇒S perpendicular to E general) not parallel to ~k

sy2 sz2 1 sx2 + + = 2 , 2 2 2 n − µx n − µy n − µz n with ni2 = µi

energy (in general) not transported in direction of wave vector ~k Christoph U. Keller, Utrecht University, [email protected]

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2 2

sx n x

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“ ”“ ” “ ”“ ” “ ”“ ” 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 n − ny n − nz + sy ny n − nx n − nz + sz nz n − nx n − ny = 0

Christoph U. Keller, Utrecht University, [email protected]

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Electric Field in Anisotropic Material

Non-Absorbing, Non-Active, Anisotropic Materials

with arbitrary constant a, electric field vector given by  ~ = a E 

sx n2 −nx2 sy n2 −ny2 sz n2 −nz2

~k not parallel to a principal axis ⇒ ratio of 2 electric field components k and l
sk n2 − µl Ek
= El sl n2 − µk

  

quadratic equation in n ⇒ generally two solutions for given direction ~s

ratio is independent of electric field components

electric field can also be written as

in non-absorbing, non-active, anisotropic material, 2 waves propagate that have different linear polarization states and different directions of energy flows ~ corresponding to 2 solutions are direction of vibration of D



~k = E

~ · ~s n2~sk E

n2 and i real ⇒ ratios are real ⇒ electric field is linearly polarized



n2 − µk

orthogonal to each other (without proof)

system of 3 equations can be solved for Ek denominator vanishes if ~k parallel to a principal axis ⇒ treat

~1 · D ~2 = 0 D

separately Christoph U. Keller, Utrecht University, [email protected]

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Wave Propagation in Uniaxial Media

z = ne2

θ: angle between wave vector and optic axis φ: azimuth angle in plane perpendicular to optic axis

second form of Fresnel equation reduces to   i   h  n2 − no2 no2 sx2 + sy2 n2 − ne2 + sz2 ne2 n2 − no2 = 0

1 n22

two solutions n1 , n2 given by

Christoph U. Keller, Utrecht University, [email protected]

=

n2 (θ) =

n12 = no2 sx2 + sy2 ne2

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(unit) wave vector direction in spherical coordinates     sx sin θ cos φ ~s =  sy  =  sin θ sin φ  sz cos θ

x = y = no2

=

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Propagation in General Direction

Introduction uniaxial media ⇒ dielectric constants:

1 n22

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+

sz2 no2

Lecture 3: Crystal Optics

cos2 θ sin2 θ + no2 ne2 no ne q no2 sin2 θ + ne2 cos2 θ

take positive root, negative value corresponds to waves propagating in opposite direction 11

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Ordinary Beam ordinary beam speed independent of wave vector direction    ~ i − si E ~ · ~s , i = 1 · · · 3 to hold for any ~ i = n2 E for µDi = µi E

Ordinary and Extraordinary Rays from before 1 n22

=

n2 (θ) =

~ o · ~s = 0 and Eo,z = 0 direction ~s, E

cos2 θ sin2 θ + no2 ne2 no ne q no2 sin2 θ + ne2 cos2 θ

electric field vector of ordinary beam (with real constant ao 6= 0)   sin φ ~ o = ao  − cos φ  E 0

n2 varies between no for θ = 0 and ne for θ = 90◦ first solution propagates according to ordinary index of refraction, independent of direction ⇒ ordinary beam or ray

ordinary beam is linearly polarized ~ o perpendicular to plane formed by E wave vector ~k and c-axis

second solution corresponds to extraordinary beam or ray index of refraction of extraordinary beam is (in general) not the extraordinary index of refraction

Christoph U. Keller, Utrecht University, [email protected]

Lecture 3: Crystal Optics

~ o = no E ~o k E ~o displacement vector D ~ o k ~k Poynting vector S 13

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Dispersion Angle

Extraordinary Ray

~ = angle between E ~ and D ~ angle between ~k and Poynting vector S = dispersion angle ~
~ e Ee × D ne2 − no2 (ne2 − no2 ) tan θ sin 2θ tan α = = 2 = ~e · D ~e 2 no2 sin2 θ + ne2 cos2 θ ne + no2 tan2 θ E

~ e · ~k = 0 and D ~e · D ~ o = 0 ⇒ unique solution (up to real since D constant ae )   cos θ cos φ ~ e = ae  cos θ sin φ  D − sin θ

equivalent expression

~e since Ee · Do = 0, De = E  2  ne cos θ cos φ ~ e = a  n2 cos θ sin φ  E e −no2 sin θ

 α = θ − arctan

no2 tan θ ne2



for given ~k in principal axis system, α fully determines direction of energy propagation in uniaxial medium

~o · E ~e = 0 uniaxial medium ⇒ E ~ e · ~k 6= 0 however, E

for θ approaching π/2, α = 0 for θ = 0, α = 0

Christoph U. Keller, Utrecht University, [email protected]

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Christoph U. Keller, Utrecht University, [email protected]

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Propagation Direction of Extraordinary Beam ~ and optic axis angle θ0 between Poynting vector S tan θ0 =

Propagation Along c Axis

no2 tan θ ne2

plane wave propagating along c-axis ⇒ θ = 0 ordinary and extraordinary beams propagate at same speed

c no

electric field vectors are perpendicular to c-axis and only depend on azimuth φ

ordinary and extraordinary wave do (in general) not travel at the same speed

ordinary and extraordinary rays are indistinguishable

phase difference in radians between the two waves given by

uniaxial medium behaves like an isotropic medium

ω (n2 (θ)de − no do ) c

example: “c-cut” sapphire windows

do,e : geometrical distances traveled by ordinary and extraordinary rays

Christoph U. Keller, Utrecht University, [email protected]

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Christoph U. Keller, Utrecht University, [email protected]

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Propagation Perpendicular to c Axis plane wave propagating perpendicular to c-axis ⇒ θ = π/2   sin φ ~ o =  − cos φ  E 0

Phase Delay between Ordinary and Extraordinary Rays ordinary and extraordinary wave propagate in same direction ordinary ray propagates with speed

extraordinary beam propagates at different speed

c ne

~ o, E ~ e perpendicular to each other ⇒ plane wave with arbitrary E polarization can be (coherently) decomposed into components ~ o and E ~e parallel to E

~ o perpendicular to plane formed by ~k and c-axis E electric field vector of extraordinary wave   0 ~e =  0  E 1

2 components will travel at different speeds (coherently) superposing 2 components after distance d ⇒ phase difference between 2 components ωc (ne − no )d radians phase difference ⇒ change in polarization state

~ e parallel to c-axis E

basis for constructing linear retarders

direction of energy propagation of extraordinary wave parallel to ~k ~e k D ~e since E Christoph U. Keller, Utrecht University, [email protected]

c no

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Christoph U. Keller, Utrecht University, [email protected]

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Reflection and Transmission at Uniaxial Interfaces Summary: Wave Propagation in Uniaxial Media ordinary ray propagates like in an isotropic medium with index no

General case from isotropic medium (nI ) into uniaxial medium (no , ne ) θI : angle between surface normal and ~kI for incoming beam

extraordinary ray sees direction-dependent index of refraction no ne n2 (θ) = q no2 sin2 θ + ne2 cos2 θ n2 no ne θ

θ1,2 : angles between surface normal and wave vectors of (refracted) ordinary wave ~k1 and extraordinary wave ~k2 phase matching at interface requires

direction-dependent index of refraction of the extraordinary ray ordinary index of refraction extraordinary index of refraction angle between extraordinary wave vector and optic axis

~kI · ~x = ~k1 · ~x = ~k2 · ~x ~x : position vector of a point on interface surface

extraordinary ray is not parallel to its wave vector angle between the two is dispersion angle tan α =

nI sin θI = n1 sin θ1 = n2 sin θ2

(ne2 − no2 ) tan θ ne2 + no2 tan2 θ

Christoph U. Keller, Utrecht University, [email protected]

Lecture 3: Crystal Optics

n1 = no : index of refraction of ordinary wave n2 : index of refraction of extraordinary wave 21

Christoph U. Keller, Utrecht University, [email protected]

Lecture 3: Crystal Optics

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Extraordinary Ray Refraction for General Case Ordinary and Extraordinary Rays

s cx cy

ordinary wave ⇒ Snell’s law sin θ1 =

“

no2 − ne2

± no

cot θ2 =

nI sin θI n1

sin2 θI

“ ”“ ” − no2 − ne2 − no2 cx2 + cy2 ”

propagation vector of extraordinary ray Sx

=

Sy

=

Sz

=

nI sin θI = n2 (θ(θ2 )) sin θ2 (in general) θ2 and therefore ~k2 will not determine direction of extraordinary beam since Poynting vector (in general) not parallel to wave vector

` ´ sin α sin θ2 cx sin θ2 − cy cos θ2 q ` ´ cz2 + cx sin θ2 − cy cos θ2 2 ` ´ sin α cos θ2 cx sin θ2 − cy cos θ2 cos α sin θ2 − q ` ´ cz2 + cx sin θ2 − cy cos θ2 2 cos α cos θ2 +

cz ∗ q

sin α ` ´ cz2 + cx sin θ2 − cy cos θ2 2

~c optic axis vector ~c = (cx , cy , cz )T ~ propagation direction of extraordinary ray S ~ = (Sx , Sy , Sz )T S θI angle between ~kI and interface normal θ2 angle between ~ke and interface normal α dispersion angle

solve for θ2 ⇒ determine direction of Poynting vector special cases reduce complexity of equations

Lecture 3: Crystal Optics

“ ” 2 n2 +n2 c 2 n2 −n2 no e e x e o

“ no2 + cx2 ne2 − no2

law for extraordinary ray not trivial

Christoph U. Keller, Utrecht University, [email protected]

”

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Christoph U. Keller, Utrecht University, [email protected]

Lecture 3: Crystal Optics

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Normal Incidence

Optic Axis in Plane of Incidence and Plane of Interface θ + θ2 = π/2 ⇒ cot θ2 =

ne no

cot θ1

θ1 : angle between surface normal and ordinary ray or wave vector (sin θI = no sin θ1 ) extraordinary wave sees equivalent refractive index s   ne2 2 2 ny = ne + sin θI 1 − 2 no direction of Poynting vector

normal incidence ⇒ θI = 0, θ1 = θ2 = 0 choose plane formed by surface normal and crystal axis

Sx

= cos(θ2 + α)

both wave vectors and ordinary ray not refracted

Sy

= sin(θ2 + α)

extraordinary ray refracted by dispersion angle α  2  no α = θ − arctan tan θ ne2

Sz

= 0

Christoph U. Keller, Utrecht University, [email protected]

determine dispersion angle α and add to θ2 to obtain direction of extraordinary ray

Lecture 3: Crystal Optics

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Christoph U. Keller, Utrecht University, [email protected]

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Interface from Uniaxial Medium to Isotropic Medium ordinary ray follows Snell’s law transmitted extraordinary wave vector and ray coincide

Optic Axis Perpendicular to Plane of Incidence c-axis perpendicular to plane of incidence ⇒ θ = π2 , n2

π 2




exit of extraordinary wave on interface defined by extraordinary ray = ne

extraordinary wave vector follows Snell’s law with index n2 (θ)

nI sin θI = ne sin θ2

nI sin θE = n2 sin θU

extraordinary wave vector obeys Snell’s law with index ne θ=

π 2

⇒ dispersion angle α = 0

nI index of isotropic medium θE angle of wave/ray vector with surface normal in isotropic medium n2 , θU corresponding values for extraordinary wave vector in uniaxial medium

Poynting vector k wave vector, extraordinary beam itself obeys Snell’s law with ne double refraction only for non-normal incidence

n2 is function of θ normally already known from beam propagation in uniaxial medium θU is function of geometry of interface, plane-parallel slab of uniaxial medium, θE = θI , (in general) extraordinary beam displaced on exit

Christoph U. Keller, Utrecht University, [email protected]

Lecture 3: Crystal Optics

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Total Internal Reflection (TIR) TIR also in anisotropic media no 6= ne ⇒ one beam may be totally reflected while other is transmitted principal of most crystal polarizers example: calcite prism, normal incidence, optic axis parallel to first interface, exit face inclined by 40◦

40°

no, ne

⇒ extraordinary ray not refracted, two rays propagate according to indices no ,ne at second interface rays (and wave vectors) at 40◦ to surface

40° c

632.8 nm: no = 1.6558, ne = 1.4852 requirement for total reflection

nU nI

e

o

sin θU > 1

with nI = 1 ⇒ extraordinary ray transmitted, ordinary ray undergoes TIR Christoph U. Keller, Utrecht University, [email protected]

Lecture 3: Crystal Optics

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