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1/23/2018
1
ECE 5322 21st Century Electromagnetics
Instructor:Office:Phone:E‐Mail:
Dr. Raymond C. RumpfA‐337(915) 747‐[email protected]
Electromagnetic Properties of Materials – Part 2
Nonlinear and Anisotropic Materials
Lecture #3
Lecture 3 1
Lecture Outline
• Nonlinear materials
• Anisotropic materials
• Tensor rotation and “unrotation”
• Dispersion relation and index ellipsoids
Lecture 3 Slide 2
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Nonlinear Materials
Nonlinear Materials
Lecture 3 Slide 4
All materials are nonlinear; some just have stronger nonlinear behavior than others.
For radio frequencies, materials tend to breakdown before they exhibit nonlinear properties. Nonlinear properties are commonly exploited in optics.
In general, the polarization of a material is a nonlinear function of the electric field and can be expressed as…
1 2 32 30 0 0e e eP E E E
Linear responseThis is what we have done so far.
Nonlinear response
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“Potential Well” Description
Lecture 3 Slide 5
r
V is the “push” required to displace the charge by a distance r.
Linear materials have a parabolic potential well.This leads to a resonator that oscillates as a perfect sinusoid.
2V r
“Potential Well” for Nonlinear Materials
Lecture 3 Slide 6
Linear “parabolic” region for small r.
Nonlinear for large r.
As bound charges are pushed very hard, the restoring force is no longer linear.
This results in anharmonic oscillation.
This type of resonance does not oscillate as a sinusoid. It can be seen as a series of sinusoids.
The oscillator is effectively resonating at multiple frequencies at the same time.
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Nonsymmetric Potentials
Lecture 3 Slide 7
This is a non‐symmetrical potential and therefore exhibits a preferred direction for polarization.
The result is a rectified signal along with potentially other frequency content.
DC component
Applications of Nonlinear Materials
Lecture 3 Slide 8
(1) Materials• Ordinary linear materials
(2) Materials• Rectification – can create a DC potential from an optical wave• Frequency doubling (second harmonic generation) – can make lasers at otherwise impossible wavelengths.
• Parametric mixing – can provide sum and difference frequencies• Pockel’s effect – can introduce birefringence from an applied electric field.
(3) Materials• Kerr effect – field dependent dielectric constant • Third harmonic generation – can generate very short wavelength waves.
• Raman scattering – the mechanical vibration of a molecule can shift the frequency of the wave
• Brillouin scattering – dielectric response changes with applied pressure
• Two photon absorption – two photons are absorbed simultaneously where as a single photon would not be absorbed.
Lacks inversion symmetry
Has inversion symmetry
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Anisotropic Materials
Anisotropic Materials
Lecture 3 Slide 10
In some materials, charges are more easily displaced along certain directions. For this reason, the material can become polarized in a direction slightly different than the applied field. In this case, the susceptibility is a tensor quantity.
Imagine pushing at an angle against a sliding glass door.The door has a preferred direction for displacement.Real materials are not this dramatic!
0 eP E
Isotropic materials are good for simple transmission of electromagnetic waves.
Anisotropic materials are good for controlling and manipulating the waves.
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Atomic Scale Picture
Lecture 3 Slide 11
Charges will oscillate differently in these two directions. Therefore, the susceptibility changes as a function of direction in the lattice.
large
small
Tensor Relations for Anisotropic Materials
Lecture 3 Slide 12
This is most often treated through a dielectric tensor.
r e I
0
x xx xy xz x
y yx yy yz y
z zx zy zz z
D E
D E
D E
The constitutive relation between E and D is then
The dielectric tensor has Hermitian symmetry for the general lossy case.*
ij ji This means the field components are no longer independent. But…it presents many new possibilities!!
0
x xx xy xz x
y yx yy yz y
z zx zy zz z
P E
P E
P E
The material polarization is now expressed as:
Materials can become polarized in directions that are slightly different than the electric field!
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Principle Axes
Lecture 3 Slide 13
It is always possible to choose a different coordinate system such that the dielectric tensor becomes diagonal.
0
0 0
0 0
0 0
a a a
b b b
c c c
D E
D E
D E
ˆˆˆˆ
ˆˆ
ax
y b
z c
a, b, and c are called the “Principal Axes of the Crystal.”a, b, and c are not necessarily at 90° to each other.
Alternative Description: There are only three degrees of freedom for 3D tensors. Numbers can only occur in the off‐diagonal elements when the tensor is rotated.
Maxwell’s Equations with Anisotropic Materials
Lecture 3 Slide 14
Maxwell’s equations remain unchanged.
E j B
H j D
0
0
D
B
The constitutive relations now include tensors
D E
B H
x xx xy xz x
y yx yy yz y
z zx zy zz z
D E
D E
D E
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Symmetry and Anisotropy
Lecture 3 Slide 15
Anisotropy Crystal Symmetry Dielectric Properties
Isotropic Cubic
Uniaxial HexagonalTetragonal
Biaxial MonoclinicTriclinicOrthorhombic
0 0
0 0
0 0
0 0
0 0
0 0
positive birefringence
negative birefringence
O
O
E
E O
E O
0 0
0 0
0 0
a
b
c
a b c
Tensor Rotation and “Unrotation”
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Definition of a Rotation Matrix
ab
ab R
Rotation matrix [R] is defined as
The rotation matrix should notchange the amplitude. This implies that [R] is unitary.
1H
H H
R R
R R R R
b a
I
Lecture 3 17
Derivation of a 2D Rotation Matrix
a
Start with vector at angle .
ˆ ˆ
ˆ ˆcos sin
x yx y
a
a
y
a a
x
Apply trig identities
ˆ cos cos sin sin
ˆsin cos cos sin
ˆ cos sin
ˆcos sin
x y
y x
xa
a a
a
y
x
ya
b
cos sin
sin cosx
R
yy
xb
b
a
a
Lecture 3 18
a
b
Add angle to rotate the vector
ˆ ˆcos sina yb x
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3D Rotation Matrices
1 0 0
0 cos sin
0 sin cosxR
Rotation matrices for 3D coordinates can be written directly from the previous result.
cos 0 sin
0 1 0
sin 0 cosyR
cos sin 0
sin cos 0
0 0 1zR
a
b
b b
a
a
Lecture 3 19
Rotation Matrix that Rotates Onto
Lecture 3 20
a
b
ˆˆ a b
a ba b
Normalize Vectors
Algorithm
3 2
3 1
2 1
ˆˆ
i.e. sin
ˆˆ i.e. cos
0
0
0
ab
ab
v a b
s v
c a b
v v
v v v
v v
2
2
1 cR I v v
s
ˆˆR a b
[R] is such that
1/23/2018
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Tensor Rotation
Tensors are rotated in a slightly different manner than vectors.
1
zR
zR R
Lecture 3 21
Combinations of Rotations
For vectors…
z y xRb aR R
For tensors…
11 1
z y x x y zR R R R R R R
Lecture 3 22
Suppose we wish to first rotate about the x‐axis by some angle, second rotate about the y‐axis by some angle, and third rotate about the z‐axis by some angle.
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Composite Rotation Matrix
We can combine multiple rotation matrices into a single composite rotation matrix [R].
ab R
Tensor rotation using the composite rotation matrix is
1R R R Lecture 3 23
y z xR R R R Vector rotation using the composite rotation matrix is
This equation implies we will rotate first about the x‐axis, second about the z‐axis, and third about the y‐axis.
Lecture 1 24
Animation of Tensor Rotation
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Order of Rotations Matter
45 220 10z y xb aR R R
Lecture 3 25
The order that we multiply the rotation matrices controls the order that the rotations are performed.
1. First rotates about the x‐axis by 10°.2. Second rotates about the y‐axis by 220°.3. Third rotates about the z‐axis by ‐45°.
In general, we do not get the same result when the order of rotation is changed.
y x x yR R R R
Numerical Examples (1 of 2)
Lecture 3 26
1 0 0
Given: 0 2 0
0 0 3r
Rotate about x‐axis by 20°
1 0 0
20 0 0.9397 0.3420
0 0.3420 0.9397xR
1 0 0
0 2.1170 0.3214
0 0.3214 2.8830r
Rotate about y‐axis by 45°
0.7071 0 0.7071
45 0 1 0
0.7071 0 0.7071yR
2 0 1
0 2 0
1 0 2r
Rotate about z‐axis by 60°
0.5000 0.8660 0
60 0.8660 0.5000 0
0 0 1zR
1.7500 0.4330 0
0.4330 1.2500 0
0 0 3r
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Numerical Examples (2 of 2)
Lecture 3 27
1 0 0
Given: 0 2 0
0 0 3r
Rotate first about x‐axis by 20° and second about the y‐axis by 45°
0.7071 0.2418 0.6645
45 20 0 0.9397 0.3420
0.7071 0.2418 0.6645y xR R
1.9415 0.2273 0.9415
0.2273 2.1170 0.2273
0.9415 0.2273 1.9415r
Rotate first about z‐axis by 60° and second about the y‐axis by 45°
0.3536 0.6124 0.7071
45 60 0.8660 0.5000 0
0.3536 0.6124 0.7071y zR R
2.3750 0.3062 0.6250
0.3062 1.2500 0.3062
0.6250 0.3062 2.3750r
Rotate first about x‐axis by 20°, second about the y‐axis by 45°, and third about the z‐axis by 60°
0.3536 0.6929 0.6284
60 45 20 0.6124 0.6793 0.4044
0.7071 0.2418 0.6645z y xR R R
2.2699 0.0377 0.6676
0.0377 1.7886 0.7017
0.6676 0.7017 1.9415r
Tensor Unrotation (1 of 2)
Lecture 3 28
A tensor can always be diagonalized along its principle axes.
0 0
0 0
0 0 c
a
b
ˆ is along
ˆ is along
ˆ is along
Principle A
xes:
b
c
a a
c
b
But suppose we are given a general nine‐element tensor.
rot
xx xy xz
yx yy yz
zx zy zz
How do we determine a, b, c and the principle axes a, b, and c?
Convention:
ca b
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Tensor Unrotation (2 of 2)
Lecture 3 29
We can relate the original tensor and the rotated tensor through the composite rotation matrix [R].
1
rot R R
We calculate the eigen‐vectors and eigen‐values of [rot].
rotrot
rot
is the eigen-vector matrix of
is the eigen-value matrix of
R
Determining Principle Axes (1 of 2)
Lecture 3 30
What are the principle axes of [rot]?
Let’s put the original principal axes a, b, and c into a matrix.
ˆ ˆ
ˆ
x
y
x
y
z
x
y
zz
a
P a
a
b
b
b
b
c
c
c
ca
Now let’s rotate them according to [R] so they correspond to that of [rot].
rot PP R
We are not rotating a tensor here so we don’t use [R][P][R]-1.
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Determining Principle Axes (2 of 2)
Lecture 3 31
For the special case of orthorhombic symmetry, the principle axes start off as
0
0
1
1
0
0
ˆ
ˆ
0
1
0
ˆ
P
a cb
[P] is just the identity matrix here. The principle axes of the rotated tensor are then
rot RP I R We now have a second interpretation of the eigen‐
vector matrix [R]. These are the principle axes of the rotated tensor (for orthorhombic symmetry).
Directly Setting the Principle Axes
Lecture 3 32
We may know the orientation that we would like to place our tensor directly in terms of the desired crystal axes a, b, and c.
We place these into a matrix and interpret it as the composite rotation matrix.
ˆ ˆ
ˆ
x
y
z
y
z
x x
y
z
c
c
c
c
a
a
a
a
R
b
b
b
b
The rotated tensor can be directly calculated from [R].
1
rot R R
1/23/2018
17
Dispersion Relation &
Index Ellipsoids
The Wave Vector
Lecture 3 Slide 34
The wave vector (wave momentum) is a vector quantity that conveys two pieces of information:
1. Wavelength and Refractive Index – The magnitude of the wave vector tells us the spatial period (wavelength) of the wave inside the material. When the free space wavelength is known, |k| conveys the material’s refractive index n(more to be said later).
2. Direction – The direction of the wave is perpendicular to the wave fronts (more to be said later).
00
2 2 free space wavelengthkn
ˆˆ ˆa b ck k a k b k c
1/23/2018
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Dispersion Relations
Lecture 3 Slide 35
The dispersion relation for a material relates the wave vector to frequency.Essentially, it tells us the refractive index as a function of direction through a material.
It is derived by substituting a plane wave solution into the wave equation.
For an ordinary linear homogeneous and isotropic (LHI) material, the dispersion relation is:
2 2 2 2 20a b ck k k k n
2 2 2202
0a b ck k kk
n
This can also be written as:
Index Ellipsoids
Lecture 3 Slide 36
From the previous slide, the dispersion relation for a LHI material was:
This defines a sphere called an “index ellipsoid.”
The vector connecting the origin to a point on the surface of the sphere is the k‐vector for that direction. Refractive index is calculated from this.
For LHI materials, the refractive index is thesame in all directions.
Think of this as a map of the refractiveindex as a function of the wave’s directionthrough the medium.
2 2 2 2 20a b ck k k k n
ab
c
index ellipsoid
0k k n
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How to Derive the Dispersion Relation (1 of 2)
Lecture 3 Slide 37
The wave equation in a linear homogeneous anisotropic material is:
20 0 0rE k E
The solution to this equation is still a plane wave, but our allowed values for k (modes) are more complicated.
0 0ˆˆ ˆ jk r
a b cE E e E E a E b E c
We are ignoring the magnetic response here.
Substituting this solution into the wave equation leads to the following relation:
20 0 0 0 0rk k E k E k E
This equation has the form: ˆˆ ˆ 0a b c
Each (•••) term has the form: 0a b cE E E
Each vector component must be set to zero independently.
ˆ component: 0
ˆ component: 0
ˆ component: 0
a b c
a b c
a b c
a E E E
b E E E
c E E E
0a
b
c
E
E
E
Matrix form…
How to Derive the Dispersion Relation (2 of 2)
Lecture 3 Slide 38
This leads to the following general equation:
det 0
Solutions for k are the eigen‐values of the big matrix and derived by setting the determinant to zero.
2 2 2
2 2 22 2 2 2 2 20 0 0
1a b c
a b c
k k k
k k n k k n k k n
It can also be shown that given the wave vector, the polarization of the electric field is:
0 2 2 22 2 2 2 2 20 0 0
ˆˆ ˆa b c
a b c
k k kE a b c
k k n k k n k k n
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Generalized Dispersion Relation
Lecture 3 Slide 39
Given the dielectric tensor…,
The general form of the dispersion relation is:
This can be written in a more useful form as:
2 2 2
2 2 22 2 2 2 2 20 0 0
1a b c
a b c
k k k
k k n k k n k k n
2
2
2
0 0 0 0
0 0 0 0
0 0 0 0
a a
r b B
c C
n
n
n
2 2 2 2 2 2 2 2 222 40 02 2 2 2 2 2 2 2 2
1a b c b c a c a b
b c a c a b a b c
k k k k k k k k kk k k
n n n n n n n n n
Dispersion Relation for UniaxialCrystals
Lecture 3 Slide 40
Uniaxial crystals have
The general dispersion relation reduces to:
2 2 2 2 2 22 20 02 2 2
Sphere EllipseOrdinary Wave Extraordinary Wave
0a b c a b c
O E O
k k k k k kk k
n n n
ordinary refractive index
extraordinary refractive indexa b O O
c E E
n n n n
n n n
This has two solutions corresponding to the two polarizations (TE and TM).
This first solution is the same as for an isotropic material. It acts like it is propagating through a isotropic material with index nO so it is called the “ordinary wave.”
The second solution is an ellipsoid. Depending on its direction, the effective refractive index will be somewhere between nO and nE.
This has two solutions corresponding to the two polarizations (TE and TM).
1/23/2018
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Index Ellipsoids for UniaxialCrystals (1 of 2)
Lecture 3 Slide 41
2 2 2 2 2 22 20 02 2 2
0a b c a b c
O E O
k k k k k kk k
n n n
2 2 2202
0a b c
O
k k kk
n
2 2 2202 2
0a b c
E O
k k kk
n n
O En n O En n
Ordinary Wave Extraordinary Wave
Index Ellipsoids for UniaxialCrystals (2 of 2)
Lecture 3 Slide 42
ab
cObservations
• Both solutions share a common axis.• This “common” axis looks isotropic with refractive index n0 regardless of polarization.
• Since both solutions share a single axis, these crystals are called “uniaxial.”
• The “common” axis is called:oOptic axisoOrdinary axisoC axisoUniaxial axis
• Deviation from the optic axis will result in two separate possible modes.
On
On
On
En En
1/23/2018
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Dispersion Relation for Biaxial Crystals (1 of 2)
Lecture 3 Slide 43
Biaxial crystals have all unique refractive indices. Most texts adopt the convention where
The general dispersion relation cannot be reduced.
a b cn n n
ab
c
optic axes
Notes and Observations
• The convention na<nb<nc causes the optic axes to lie in the a‐c plane.
• The two solutions can be envisioned as one balloon inside another, pinched together so they touch at only four points.
• Propagation along either of the optic axes looks isotropic, thus the name “biaxial.”
Dispersion Relation for Biaxial Crystals (2 of 2)
Lecture 3 Slide 44
There are three special cases when it can be simplified.We can cause these conditions by launching electromagnetic waves at the proper orientation.
2 22 2 2 2 2
0 02 2
2 22 2 2 2 2
0 02 2
2 22 2 2 2 2
0 02 2
0 : 0
0 : 0
0 : 0
b ca b c a
c b
a cb a c b
c a
a bc a b c
b a
k kk k k k n k
n n
k kk k k k n k
n n
k kk k k k n k
n n
We see that each case has two separate solutions corresponding to the two polarizations (TE and TM).
1/23/2018
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Dispersion Surfaces of Magnetoelectric Materials
Lecture 3 Slide 45
Magnetoelectric materials can exhibit up to 16 singularities.
and D E H B H E
Alberto Favaro and Friedrich W. Hehl, “Light propagation in local and linear media: Fresnel‐Kummerwave surfaces with 16 singular points,” arXiv preprint arXiv:1510.05566 (2015).
Direction of Power Flow
Lecture 3 Slide 46
Isotropic Materials
k
P
x
y
k
P
Anisotropic Materials
x
y
Phase propagates in the direction of k. Therefore, the refractive index derived from |k| is best described as the phase refractive index. Velocity here is the phase velocity.
Energy propagates in the direction of P which is always normal to the surface of the index ellipsoid. From this, we can define a group velocity and a group refractive index.
1/23/2018
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Illustration of k versus
Lecture 3 Slide 47
Negative refraction into a photonic crystal.
Raymond C. Rumpf "Engineering the Dispersion and Anisotropy of Periodic Electromagnetic Structures," Solid State Physics, Vol. 66, pp. 213‐300, 2015.
Double Refraction in Anisotropic Materials
Lecture 3 Slide 48
Isotropic Materials
y
1k
x
2k
Anisotropic materials have two index ellipsoids – one for each polarization. Wave energy can split between the two produce an ordinary and an extraordinary wave.
y
1k
x
0k
Anisotropic Materials
ek