Lecture 3: Crystal Optics
Homogeneous, Anisotropic Media
Outline
Introduction
material equations for homogeneous anisotropic media
1
2
3
4
5
~ = 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
1
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
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Lecture 3: Crystal Optics
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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
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6
~ 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
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Lecture 3: Crystal Optics
2 2
sx n x
Lecture 3: Crystal Optics
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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
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Lecture 3: Crystal Optics
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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
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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
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=
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
=
Lecture 3: Crystal Optics
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
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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
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Lecture 3: Crystal Optics
~ o = no E
~o k E
~o
displacement vector D
~ o k ~k
Poynting vector S
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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
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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
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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
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c
no
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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
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Lecture 3: Crystal Optics
n1 = no : index of refraction of ordinary wave
n2 : index of refraction of extraordinary wave
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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
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”
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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
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determine dispersion angle α and add to θ2 to obtain direction of
extraordinary ray
Lecture 3: Crystal Optics
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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
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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
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Lecture 3: Crystal Optics
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