Lecture 2 : Propagation in
anisotropic media
• Linear anisotropic media
• Characteristic surfaces:
– Index ellipsoïd
– Index surface
– Radial velocity surface or Ray surface
NW wave optics (polarization)
For more details about the calculations in these
lectures:
Optical waves in crystals, Yariv & Yeh (more details about some of the
calculations done in this chapter, but does not mention the radial velocity
surface)
Optics, Born&Wolf (details of the calculation of the ray surface/ radial velocity
surface from page 792 of the 7th edition)
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2. Propagation in anisotropic media
NW wave optics (polarization)
3
Review about isotropic media
• Maxwell’s Equations
(in a medium without sources)
• Polarization : when it travels through a medium, the EM wave induces a
polarization that adds up in phase to the one in vacuum:
(the medium is non-magnetic:
)
• We try to find a solution to Maxwell’s equations
as a plane wave :
(other fields: same type of expression)
• We get :
• E and D are parallel and transverse
• B and H are perpend to them, also transverse
• The Poynting vector
is parallel to
NW optique physique II
2. Propagation in anisotropic media
Review about isotropic media
Which EM waves can propagate in the medium ?
• From Maxwell’s equations
because
• On the other hand,
• So :
- All transverse polarizations are possible
- Dispersion relationship:
• In general, we define the index of refraction:
• We get :
• Phase velocity :
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2. Propagation in anisotropic media
NW optique physique II
Linear anisotropic media
• In an anisotropic medium, the susceptibility, and thus the permittivity, are
no longer scalar, but tensors (3x3 matrices):
-> the medium is still linear, but no longer isotropic.
• If there is no absorption, it can be shown that the
permittivity tensor is symmetric
• It thus has three real eigenvalues,
corresponding to orthogonal eigenvectors.
In the basis of these eigenvectors, the matrix can
be written as :
• By analogy with the isotropic medium, we define indices:
The indices ni are called the principal indices of refraction.
In an anisotropic medium, the index « seen » by an EM
wave depends on its polarization state.
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2. Propagation in anisotropic media
NW optique physique II
Linear anisotropic media
Classification of materials depending on their
permittivity tensor
Isotropic medium with index n
Uniaxial anisotropic medium,
with optic axis Oz
Biaxial anisotropic medium
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NW optique physique II
2. Propagation in anisotropic media
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Linear anisotropic media
What is the structure of an EM wave propagating in an
anisotropic medium ?
-> D is no longer parallel to E
-> D and B are still orthogonal to k
-> D and H are orthogonal
-> E and B are orthogonal
D, E and k are
in the same
plane,
orthogonal to B

k2    
1   
=
D=−
2 E − u .E u
2 k × k × E
µ0ω
µ0ω
(
)
[ ( ) ]
• D is the projection of E in the plane
orthogonal to u (to within a mult. factor)
€
€ vector (direction of « light ray»)
• The Poynting
is no longer parallel to the wave vector
(direction of propagation of the phase)
NW optique physique II
2. Propagation in anisotropic media
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Linear anisotropic media
Which waves can propagate in a given direction?
• Definitions:
k0 = ω ε 0 µ 0 =
ω
c
n is the index «seen» by the
wave with direction u
and
 
 


2
2
u
u
• From Maxwell: D = ε 0 n E − P ( E ) = ε 0 n P⊥ E
[
€
]

u
⊥ is
P
the projector on the
plane perpendicular to u
• Also:
€
: Permeability tensor.
In the€
eigenvectors
basis,
• Condition on D:




D
u

2 = P⊥ [η] D = [ Au ] D
n
The solutions for D are
eigenvectors of matrix Au
2. Propagation in anisotropic media
NW optique physique II
Linear anisotropic media




D
u

2 = P⊥ [η] D = [ Au ] D
n
• Eigenvectors : polarization states (vector D) that can propagate in
the medium without changing their shape
€
Different from the isotropic case: not all polarization states can propagate
• Eigenvalues : corresponding indices of refraction 1/n2 ( and phase
velocities c/n)
The polarization states and the
corresponding indices of refraction depend
on the direction of propagation u
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NW optique physique II
2. Propagation in anisotropic media
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Linear anisotropic media
Can we show some specific properties of these eigenstates?




D
u

2 = P⊥ [η] D = [ Au ] D
n

u
⊥ is
P
the projector on the plane perpendicular
to u, i.e. on the phase plane.
• In a basis consisting of 2 vectors in the phase plane plus the vector u, the
matrix can be written as:
€
€
€
[ ]=
A u

u
P⊥
⎡1 0 0⎤ ⎡η11 η12 η13 ⎤
⎡η11 η12 η13 ⎤
⎢
⎥ ⎢
⎥
⎢
⎥
[η] = ⎢ 0 1 0⎥ ⎢η12 η22 η23 ⎥ = ⎢η12 η22 η23 ⎥
⎢⎣ 0 0 0⎥⎦ ⎢⎣η13 η23 η33 ⎥⎦
⎢⎣ 0
0
0 ⎥⎦
[η] is not diagonal in this
basis but it remains
symmetric (like [ε])
Only the restriction of [Au] in the phase plane plays a role. We thus
have to find the two eigenvectors and eigenvalues of a 2x2 symmetric
matrix
⇒ 2 real eigenvalues 1/n’² and 1/n’’²(we will show later
that they must be positive)
⇒ orthogonal eigenvectors D’⊥D’’
Since the matrix is also real, D’ and D’’ are real (linear polarizations)
2. Propagation in anisotropic media
NW optique physique II
Linear anisotropic media
Can we calculate more precisely the eigenvalues?
• The two eigenvalues 1/n2 depend on the
direction of u (direction of propagation) :
• To calculate them, we need to solve the
equation:
• We have to calculate the determinant of a 3x3 matrix. We do this
calculation in the (x,y,z) frame where [ε] and [η] are diagonal. After some
tedious calculations, we get Fresnel’s equation for the indices of
refraction:
This equation can turn into an equation of the type n4-Sn²+P=0:
⇒ we get two solutions n’² and n’’² (we showed before that they must be real)
⇒ we can show now that they are positive (S>0 and P>0):
⇒ n’ and n’’ are real numbers (propagation without absorption)
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2. Propagation in anisotropic media
NW optique physique II
Linear anisotropic media
We will now show an easier, graphical way to
represent the solutions to Fresnel’s equation and the
corresponding eigen polarizations
Several characteristic surfaces
• Index ellipsoïd
• Index surface
• Radial velocity surface or Ray surface
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2. Propagation in anisotropic media
NW optique physique II
Characteristic surface 1: Index ellipsoïd
Definition
Let us recall the wave equation in terms of D and E:
Scalar-multiplying it by D we obtain :
For a given D vector, there is only one index n of propagation, given
by the above equation.
The surface we obtain when we report the value of the index n in the
corresponding direction of D is called the index ellipsoïd and its
equation is:
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NW optique physique II
2. Propagation in anisotropic media
Index Ellipsoïd – directions of E and B
For a given direction of D corresponding to a point M on the
ellipsoïd:
1) E is normal to the ellipsoïd at point M
Proof:
⎡ X 0 n x2 ⎤
⎢
2 ⎥
Y
n
Vector normal to the index ellipsoïd at point (X0,Y0,Z0): ⎢ 0 y ⎥
⎢⎣ Z 0 n z2 ⎥⎦
⎡ Dx n x2 ⎤
⎡ X 0 n x2 ⎤
1 ⎢
D ⎢
2 ⎥
2 ⎥
Y
n
€ ⎢Dy n y ⎥ =
ε0
ε 0 n ⎢ 0 y2 ⎥
2
⎢⎣ Dz n z ⎥⎦
⎢⎣ Z 0 n z ⎥⎦
Vector E corresponding to this direction
of D (X0,Y0,Z0):
€
2) B, orthogonal to E, is thus tangent to the ellipsoïd at point M
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NW optique physique II
2. Propagation in anisotropic media
Index Ellipsoïd – graphical determination of eigenpolarizations
For a given direction of the wave vector k:
• D is in the phase plane (orthogonal to k) thus on the intersection
of the index ellipsoïd with the plane orthogonal to k which is an
ellipse
• D is orthogonal to B, which is tangent to the ellipsoïd, thus the
only possibilities for D’ and D’’ are the two principal axes of the
ellipse
Intersection of the ellipsoïd
by the phase plane Π
E
B
k
D’
B
D’’
Π
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NW optique physique II
2. Propagation in anisotropic media
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Index Ellipsoïd in the case of a uniaxial medium
Uniaxial medium: nx= ny = no
Positive uniaxial ne > no
nz=ne
optic axis z
Ellipsoïd invariant under rotation about the z axis:
Do ⊥ k & optic axis
De ⊥ k & Do
or De projection of the
optic axis on the
phase plane Π
Eo // Do
Ee normal to the
ellipsoïd
2. Propagation in anisotropic media
NW optique physique II
Characteristic surface 2: Index surface
Definition
We have shown that for a fixed direction of propagation k, there
are two possible values for index of refraction n’ and n’’
The index surface represents the two values of the index in the
direction of k (locus of points N satisfying ON=n(u)u). It thus has
two shells.
Writing ON=(X=nα,Y=nβ,Z=nγ) where X²+Y²+Z²=n², we can
write Fresnel’s equation relative to the wave normals as a function
of X, Y and Z. We get a complicated equation from which we can
imagine the shape of the normal surface by choosing nx<ny<nz
and studying its intersection with each of the coordinate planes x,
y or z=0. We obtain for example in the xz plane:
a circle with radius ny
an ellipse of equation x²/nz²+z²/nx²=1
Using circular permutations we can obtain the three intersections.
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2. Propagation in anisotropic media
NW optique physique II
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Shape of the index surface when nx<ny<nz
There are two directions ON and ON’ for k that satisfy n’=n’’: this is
why the medium is called BIAXIAL.
NW optique physique II
2. Propagation in anisotropic media
Shape of the index surface when nx= ny = no
nz=ne
z
z
Sphere with
radius no
no
no
ne
y
ne
no
x
ne
ellipse
circle
x
Ellipsoïd with axial
symmetry around
the z axis
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Uniaxial medium with
optic axis z
NW optique physique II
2. Propagation in anisotropic media
Positive and negative uniaxial crystals
Negative uniaxial ne < no
Positive uniaxial ne > no
calcite
no= 1,658 ne=1,486
Quartz
no= 1,544
(λ=589nm)
ne=1,553
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NW optique physique II
2. Propagation in anisotropic media
Characteristic surface 3:
Radial velocity surface
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or Ray surface
For each EM wave defined by (k,D) propagating with index n, there is
a similar description where:
(k,D) is replaced by (R, E)
Index n is replaced by the « apparent » radial velocity defined by:
D
θ E
k,n
θ
R,vr
Σ0
Σ1
2. Propagation in anisotropic media
NW optique physique II
Radial velocity surface
Definition
In a very similar way as we defined the indices surface, we
define a radial velocity surface by reporting in the direction
of the ray (Poynting vector R) the value of the radial
velocity vr=v/cosθ (where θ is the angle between k and S).
The shape of this surface is determined from Fresnel’s
equation relative to the radial velocities which has a
similar expression as the one relative to the wave
normals:
We get a surface with two shells very similar to the normal
surface, but with its characteristic length along the
directions of the axes x,y,z equal to c/ni or rather 1/ni.
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2. Propagation in anisotropic media
NW optique physique II
Radial velocity surface for a uniaxial medium
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NW optique physique II
2. Propagation in anisotropic media
Property of the radial velocity surface
The radial velocity surface gives access, for a given
direction of the Poynting vector R (light ray), to the
corresponding directions of Do , Eo , De , Ee :
For the ordinary wave: Eo ⊥ z & R
Do // Eo
For the extraordinary wave:
Ee ⊥ Eo & R,
De tangent to the extraordinary shell
of the radial velocity surface (see
demonstration in the next slide)
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2. Propagation in anisotropic media
NW optique physique II
Finding the direction of De from the velocity surface
Relationship between De and Ee :
Coordinates of the Poynting vector
in the (x,y,z) reference frame:
Extraordinary shell of the
velocity surface:
Vector normal to that shell in the direction of R:
Finally:
De is tangent to the extraordinary shell of
the radial velocity surface
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2. Propagation in anisotropic media
NW optique physique II
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Summary about the characteristic surfaces
• Index ellipsoïd:
We plot in the direction of D (linear polarisation state that can propagate in the
medium) the corresponding value of the index.
It is the simplest surface (only one shell) to represent the dielectric
properties of the medium
• Radial velocity surface:
We plot in the direction of the ray R (direction of propagation of energy,
direction of the Poynting vector) the two possible values of the radial velocities
corresponding to the two eigen polarization states that can propagate with this
direction of ray.
It is a more complex surface, with two shells, useful to construct the
refracted or reflected rays in an anisotropic medium
• Index surface:
We plot in the direction of the wave vector k (perpendicular to the wave surface)
the two possible values of the index of refraction corresponding to the two eigen
polarization states that can propagate in that direction of k.
It also has two shells, it will be useful later to calculate more easily the
path or phase differences when propagating through an anisotropic
medium