Coupled Mode Analysis
Chapter 4
Physics 208, Electro-optics
Peter Beyersdorf
Document info
Ch 4, 1
Class Outline
The dielectric tensor
Plane wave propagation in anisotropic media
The index ellipsoid
Phase velocity, group velocity and energy
Crystal types
Propagation in uniaxial crystals
Propagation in biaxial crystals
Optical activity and Faraday rotation
Coupled mode analysis of wave propagation
Ch 4, 2
The Dielectric Tensor
ε relates the electric field to the electric
displacement by
! = "E
! = "0 E
! + P!
D
In anisotropic materials the polarization may
not be in the same direction as the driving
electric field. Why?
Ch 4, 3
Anisotropic Media
Charges in a material are
the source of polarization.
They are bound to neighboring
nuclei like masses on springs.
In an anisotropic material
the stiffness of the springs
is different depending on the
orientation.
The wells of the electrostatic potential that the
charges sit in are not symmetric and therefore the
material response (material polarization) is not
necessarily in the direction of the driving field.
Ch 4, 4
Analogies
What every-day phenomena can have a response
in a direction different than the driving force?
Ch 4, 5
The Dielectric Tensor
The susceptibility tensor χ relates the
polarization of the material to the driving
electric field by
!
P! = "0 χ̄E
or
 

Px
χ11
Py  = !0 χ21
Pz
χ31
χ12
χ22
χ32
 
χ13
Ex
χ23  Ey 
χ33
Ez
! = "E
! = "0 E
! + P!
Thus the material permittivity ε in D
is also a tensor !¯ = !0
!
"
¯
I + χ̄
Ch 4, 6
Tensor Notation
For brevity, we often express tensor equations
such as
in the form
!
"
!
¯
!
D = "0 I + χ̄ E
Di = !0 (δij + χij )Ej
where summation over repeated indices is
assumed, so that this is equivalent to
Di =
!
!0 (δij + χij )Ej
j
Ch 4, 7
Dielectric Tensor Properties
The dielectric tensor is Hermetian such that
!ij = !∗ji
In a lossless material all elements of ε are real
so the tensor is symmetric!! !ij
! =
! !ji! and can
be described by 6 (rather than 9) elements

!11
!12
!13
!12
!22
!23

!13
!23 
!33
Ch 4, 8
Plane Wave Propagation in
Anisotropic Media
For a given propagation direction in a crystal (or
other anisotropic material) the potential wells
for the charges will be ellipsoidal, and so there
will be two directions for the driving field for
which the material response is in the same
direction.
These two polarization states each have a
(different) phase velocity associated with them.
Light polarized at either of these angles will
propagate through the material with the
polarization unchanged.
Ch 4, 9
Wave Equation in
Anisotropic Materials
From Maxwell’s equations in a dielectric (ρ=J=0),
→
using d/dt→iω and ∇→ik we have
!
∂
B
! ×E
!+
∇
∂t
! + iω B
!
i!k × E
!k × E
! = −ωµH
!
= 0
!
∂
D
! ×H
! −
∇
∂t
= J!
= 0
! − iω D
!
i!k × H
= J!
!k × H
! = ω#E
!
!k × !k × E
! + ω 2 µ#E
!
=
0
Which is the wave equation for a plane wave, most easily
analyzed in the principle coordinate system where ε is
Ch 4,10
diagonal (i.e. a coordinate system aligned to the crystal axes)
Wave Equation in
Anisotropic Materials
or
 2
ω µ"x − ky2 − kz2

ky kx
kz kx
!k × !k × E
! + ω 2 µ#E
!
kx ky
ω 2 µ"y − kx2 − kz2
kz ky
=
0
 
kx kz
Ex
  Ey  = 0
ky kz
Ez
ω 2 µ"z − kx2 − ky2
for which the determinant of the matrix must be zero for
solutions (other than k=ω=0) to exist
! 2
!ω µ"x − ky2 − kz2
!
!
ky kx
!
!
kz kx
kx ky
ω 2 µ"y − kx2 − kz2
kz ky
!
!
kx kz
!
!=0
ky kz
!
2!
2
2
ω µ"z − kx − ky
This defines two surfaces in k-space called the normal shells.
Ch 4, 11
Normal Shells
In a given direction, going out from the
origin, a line intersect this surface at
two points, corresponding to the
magnitude of the two k-vectors (and
hence the two values of the phase
velocity) wave in this direction can
have.
kz
ky
kx
The two directions where the surfaces
meet are called the optical axes.
Waves propagating in these directions
will have only one possible phase
velocity.
Ch 4, 12
Wave Equation in
Anisotropic Materials
or
 2
ω µ"x − ky2 − kz2

ky kx
kz kx
!k × !k × E
! + ω 2 µ#E
!
kx ky
ω 2 µ"y − kx2 − kz2
kz ky
=
0
 
kx kz
Ex
  Ey  = 0
ky kz
Ez
ω 2 µ"z − kx2 − ky2


 
kx
Ex
k2 −ω 2 µ"x

ky
 Ey  = E0 
 k2 −ω2 µ"y 
kz
Ez
2
2
k −ω µ"z
where E02 ≡ Ex2 + Ey2 + Ez2
Ch 4,13
Example
Find the possible phase velocities (vp=ω/k) for a
wave propagating along the x-axis in a crystal,
and the associated polarization directions.
1
vp = √
µ!y
1
or vp = õ!
z
polarized in the y or z direction
respectively
Ch 4,14
Wave Propagation in Anisotropic
Materials
! ·D
! =ρ
Gauss’ law ∇
in a dielectric (ρ=0) can be written as
! =0
i!k · D
implying the propagation direction k̂ is orthogonal
!.
to the displacement vector D
!=E
! ×H
!
But the Poynting vector S
which gives the direction of energy flow is
! , not the
orthogonal to the electric field E
displacement vector. Thus energy does not flow
in the direction of the wave’s propagation if the
polarization of the wave (Ê ) is not an eigenvector of the material’s dielectric tensor!
Ch 4,15
Spatial Walkoff
Image 8.26 from Hecht
Ch 4, 16
Eigenstates of a material
For any direction of propagation there exists two
polarization directions in which the displacement
vector is parallel to the transverse component of
the electric field. These are the eigenstates of
the material.
→
→
Requiring D and E be parallel in the matrix
equation
! = "¯E
!
D
_
is equivalent to finding the eigenvalues and
eigenvectors of ε satisfying
λi u"i = #̄"ui
Ch 4,17
Index Ellipsoid
Index of refraction as a function of polarization angle
n1 •
!k
n2 •
The polarization directions that have a max and min
index of refraction form the major and minor axes of
an ellipse defining n(θ) the index fora wave with the
electric displacement vector at an angle of θ in the
transverse plane.
Ch 4,18
Index Ellipsoid
The sum of all index ellipses plotted in three
dimensions is the index ellipsoid, defined by
x2
y2
z2
+ 2 + 2 =1
2
nx
ny
nz
Ch 4,19
Example
Find the angle of the direction of the optical
axis for Lithium Niobate
Ch 4,20
Example
Find the angle of the direction of the optical
axis for Topaz
Ch 4,21
Crystal Types
The index ellipsoid is determined by the three
principle indices of refraction nx, ny and nz.
A crystal with nx≠ny≠nz will have ____
optical axes
A crystal with nx=ny≠nz will have ____
optical axes
A crystal with nx=ny=nz will have _____
optical axes
Ch 4,22
Crystal Properties
The geometry of a crystal determines the form
of the dielectric tensor
Ch 4,23
Crystal Properties
The geometry of a crystal determines the form
of the dielectric tensor
Isotropic

n2
! = !0  0
0
0
n2
0

0
0
n2
Ch 4,24
Crystal Properties
The geometry of a crystal determines the form
of the dielectric tensor
Unaxial

n20
! = !0  0
0
0
n20
0

0
0
n2e
Ch 4,25
Crystal Properties
The geometry of a crystal determines the form
of the dielectric tensor
Biaxial

n2x
! = !0  0
0
0
n2y
0

0
0
n2z
Ch 4,26
Biaxial Crystals
By convention nx<ny<nz so the optical axis is
always in the xz plane
Ch 4,27
Uniaxial Crystals
By convention no nx=ny and ne nz
The optical axis in a uniaxial crystal is always in
the z-direction
A negative uniaxial crystal is one in which ne<no
while a positive uniaxial crystal has ne>no.
Ch 4,28
Example
A wave is propagating in the [1,1,1] direction in
Mica, what are the two principle indices of
refraction and in which direction are the
eigenpolarizations?
Ch 4,29
Example
A wave is propagating in the [1,1,1] direction in
Mica, what are the two principle indices of
refraction and in which polarization directions do
these correspond to?
plane of polarization is given by !k · !rr.k=0
= 0 so x+y+z=0.
The intersection of this plane and the index ellipsoid, given by
x2
y2
z2
+ 2 + 2 =1
n2x
ny
nz
is,
x2
y2
(x + y)2
+ 2 +
=1
n2x
ny
n2z
Find extremes of r2=x2+y2+z2 subject to the preceding two constraints to find
directions of transverse eigenpolarizations. Plug directions back into r=(x2+y2
+z2)1/2 to find index for each polarization
Ch 4,30
Example Solution in Mathematica
Ch 4,31
Index of extraordinary ray
The index of refraction of the ordinary ray is
always equal to the index seen when the ray
propagates along the optical axis
The extra-ordinary ray sees an index of
refraction that is a function of the ray
propagation angle from the optical axis. In a
uniaxial crystal
1
cos2 θ sin2 θ
=
+
2
2
ne (θ)
no
n2e
Ch 4,32
Double Refraction
Snell’s law requires the tangential component of
the k-vector be continuos across a boundary
θi
kisinθi
n
knsinθn
θn
θi
kisinθi
ne
no
knsinθn
θe
θo
Ch 4,33
Conical Refraction
The normal surface is a surface
of constant ω in k-space. Group
velocity vg=∇kω is perpendicular
to the normal surfaces, but in a
biaxial crystal the normal
surfaces are singular along the
optical axis.
The surrounding region is
conical, thus a wave along the
optical axis will spread out
conically
Ch 4,34
Optical Activity
Quartz and other materials with
a helical molecular structure
exhibit “Optical Activity”, a
rotation of the plane of
polarization when passing
through the crystal.
Ch 4,35
Optical Activity
Optically active materials can be thought of as
birefringent, having a different index of
refraction for right-circular polarization and for
left-circular polarization.
The specific rotary power describes how much
rotation there is per unit length
π
ρ = (nl − nr )
λ
The source of this optical activity is the induced
dipole moment of the molecule formed by
changing magnetic flux through the helical
molecule
Ch 4,36
Optical Activity
The displacement vector thus has an additional contribution in a
direction perpendicular to the electric field
! = "E
! + i"0 G
! ×E
!
D
2 ! is called the gyration vector and
! != (g
where ! G
!
!
!
!
k
k
/k
)k̂  
ij
i
j
0
 

î
! ×E
! = Gx
G
Ex
ĵ
Gy
Ey
0
k̂
Gz  =  Gz
−Gy
Ez
−Gz
0
Gx
Gy
Ex
!
−Gx  Ey  = [G]E
0
Ez
so we can define an effective dielectric tensor
!! = ! + i!0 [G] so that
! = "! E
!
D
allowing the wave equation to be solved for eigenpolarizations of
propagation and two corresponding indices of refraction that
depend on [G]
Ch 4,37
Faraday Rotation
A form of optical activity induced by an external
magnetic field.
ρ=VB
where V is called the Verdet constant. The sense
of rotation depends on the direction of
propagation relative to the direction of the
magnetic field
Ch 4,38
Faraday Rotation
The motion of the charges in a material driven by the electric
! ! resulting in an induced
field feel a Lorentz force! q!
! v !× !B
!! ×! E
!! such that
dipole with a term proportional to!! B
where γ=-Vn0λ0/π.
! = "E
! + i"0 γ B
! ×E
!
D
For Faraday rotation the imaginary term is proportional to B,
while for optical activity it is proportional to k.
Thus a wave double passing (in opposite directions) a material
will net zero polarization rotation from optical activity, but twice
the one-way rotation due to Faraday rotation.
Ch 4,39
Exercise
Given an isotropic material that is perturbed such
that
! = "E
! + i∆"E
!
D
where Δε is

0
−Gz
0
∆! = !o  Gz
−Gy Gx

Gy
−Gx 
0
find the eigenpolarizations for the electric field and
associated indices of refraction. If a linearly
polarized plane wave were to propagate along the
direction of G through a thickness d of such
material, what would the output wave look like?
Ch 4,40
Exercise
Ch 4,41
Coupled Mode Analysis
When the eigenstates of an unperturbed system
are known, a small perturbation can be treated
as a coupling between these states.
Rather than rediagonalizing the matrix for ε one
could solve the wave equation with the
perturbed matrix ε+Δε
with ε
E = A1 ei(k1 z−ωt) ê1 + A2 ei(k2 z−ωt) ê2
E = A3 ei(k3 z−ωt) ê3 + A4 ei(k4 z−ωt) ê4
or
with ε+Δε
E = A1 (z)ei(k1 z−ωt) ê1 + A2 (z)ei(k2 z−ωt) ê2
Ch 4,42
Exercise
Given an isotropic material that is perturbed such
that
! = "E
! + i∆"E
!
D
where Δε is

0
−Gz
0
∆! =  Gz
−Gy Gx

Gy
−Gx 
0
find the coupled mode expression for the electric
field for a linearly polarized plane wave
propagating along the direction of G.
Ch 4,43
Coupled Mode Analysis Example
The 1-dimensional wave equation
in isotropic media
Ch 4,44
Coupled Mode Analysis Example
This is from wave equation solutions
to the unperturbed material
Ch 4,45
Coupled Mode Analysis Example
Envelope varies slowly
compared to optical frequency
These coupled differential
equations reduce to coupled
equations that can be solved
using linear algebra when we
use phasors to represent the
fields
Ch 4,46
Coupled Mode Analysis Example
Ch 4,47
Diagonalization
Consider the same problem where

2
n
! = !0 i G
0
−iG
n2
0

0
0
n2
Find the normal modes of propagation by finding
the eigenvectors of the dielectric tensor
Ch 4,48
Diagonalization
Consider the same problem where

2
n
! = i!0 G
0
−i!0 G
n2
0

0
0
n2
Find the normal modes of propagation by finding
the eigenvectors of the dielectric tensor
λ=n2, n2-εG,n2+εG
→
u=[0,0,1], [i,1,0], [-i,1,0]
Ch 4,49
Summary
Material polarization is not necessarily parallel
to electric field in anisotropic materials
permittivity of material (ε) is a tensor
relating E to D
Direction of wave vector and energy flow
can be different
Normal shells and index ellipsoid are
geometrical constructions to visualize the
propagation parameters of a wave
Ch 4,50
References
Yariv & Yeh “Optical Waves in Crystals” chapter 4
Hecht “Optics” section 8.4
Fowles “Modern Optics” section 6.7
Ch 4,51