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8/13/2019 Lecture 7 -- Theory of Periodic Structures
http://slidepdf.com/reader/full/lecture-7-theory-of-periodic-structures 1/23
2/7/20
ECE 5390 Special Topics:21st Century Electromagnetics
Instructor:
Office:Phone:
E‐Mail:
Dr. Raymond C. Rumpf
A‐337
(915) 747‐6958
[email protected] Spring 2012
Theory of Periodic
Structures
Lecture #7
Lecture 7 1
Lecture Outline• Periodic devices
• Math describing periodic structures
• Electromagnetic waves in periodic structures
• Electromagnetic bands
• Isofrequency contours
Lecture 7 Slide 2
8/13/2019 Lecture 7 -- Theory of Periodic Structures
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Periodic Devices
Examples of Periodic Electromagnetic
Devices
Lecture 7 Slide 4
Diffraction Gratings
Antennas
Waveguides Band Gap Materials
Frequency Selective
Surfaces
Metamaterials
Slow Wave Devices
8/13/2019 Lecture 7 -- Theory of Periodic Structures
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What is a Periodic Structure?
Lecture 7 Slide 5
Periodicity at
the
Atomic
Scale Larger
‐Scale
Periodicity
Materials are periodic at the atomic scale.
Metamaterials are periodic at a much larger scale, but smaller than a wavelength.
The math describing how things are periodic is the same for both atomic scale and larger scale.
Describing Periodic Structures• There is an infinite number of ways that
structures can be periodic.
• Despite this, we need a way to describe andclassify periodic lattices. We have to makegeneralizations to do this.
• We classify periodic structures into:
– 230 space groups
– 32 crystal classes
– 14 Bravais lattices
– 7 crystal systems
Lecture 7 Slide 6
Less specific.
More generalizations.
8/13/2019 Lecture 7 -- Theory of Periodic Structures
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Symmetry Operations
Lecture 7 Slide 7
Infinite crystals
are
invariant
under
certain
symmetry
operations
that
involve:
Pure Translation Pure Rotation
Pure Reflections Combinations
Definition of Symmetry
Categories• Space Groups
– Set of all possible combinations of symmetry operationsthat restore the crystal to itself.
– 230 space groups
• Bravais Lattices – Set of all possible ways a lattice can be periodic if
composed of identical spheres placed at the lattice points.
– 14 Bravais lattices
• Crystal Systems – Set of all Bravais lattices that have the same holohedry
(shape of the conventional unit cell)
– 7 crystal systems
Lecture 7 Slide 8
8/13/2019 Lecture 7 -- Theory of Periodic Structures
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8/13/2019 Lecture 7 -- Theory of Periodic Structures
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Math Describing
Periodic
Structures
Primitive Lattice Vectors
Lecture 7 Slide 12
Axis vectors most intuitively define the shape and orientation of the unit cell. They cannot
uniquely describe all 14 Bravais lattices.
Translation vectors connect adjacent points in the lattice and can uniquely describe all 14
Bravais lattices. They are less intuitive to interpret.
Primitive lattice vectors are the smallest possible vectors that still describe the unit cell.
8/13/2019 Lecture 7 -- Theory of Periodic Structures
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Non-Primitive Lattice Vectors
Lecture 7 Slide 13
Almost always,
the
label
“lattice
vector”
refers
to
the
translation
vectors, not the axis vectors.
A translation vector is any vector that connects two points in a lattice.
They must be an integer combination of the primitive translation
vectors.
1 2 3 pqr t pt qt rt
, 2, 1, 0,1, 2,
, 2, 1, 0,1, 2,
, 2, 1, 0,1, 2,
p
q
r
Primitive translation vector
Non‐primitive translation vector
Reciprocal Lattices (1 of 2)
Lecture 7 Slide 14
Each direct lattice has a unique reciprocal lattice so knowledge of one implies knowledge
of the other.
2t
8/13/2019 Lecture 7 -- Theory of Periodic Structures
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Reciprocal Lattices (2 of 2)
Lecture 7 Slide 15
Direct Lattice Reciprocal Lattice
Simple Cubic Simple Cubic
Body‐Centered Cubic Face‐Centered Cubic
Face‐Centered Cubic Body‐Centered Cubic
Hexagonal Hexagonal
Reciprocal Lattice Vectors
Lecture 7 Slide 16
The reciprocal lattice vectors can be calculated from the direct lattice vectors (and the
other way around) as follows:
2 3 3 1 1 21 2 3
1 2 3 1 2 3 1 2 3
2 3 3 1 1 21 2 3
1 2 3 1 2 3 1 2 3
2 2 2
2 2 2
t t t t t t T T T
t t t t t t t t t
T T T T T T t t t
T T T T T T T T T
There
also
exists
primitive
reciprocal
lattice
vectors.
All
reciprocal
lattice
vectors
must
be an integer combination of the primitive reciprocal lattice vectors.
1 2 3 PQRT PT QT RT , 2, 1, 0,1, 2,
, 2, 1, 0,1, 2,
, 2, 1, 0,1, 2,
P
Q
R
8/13/2019 Lecture 7 -- Theory of Periodic Structures
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K
Grating Vector
Lecture 7 Slide 17
2 K
ˆ ˆ ˆ x y z K K x K y K z
A grating
vector
is
very
much
like
a wave
vector
in
that
its
direction
is
normal to planes and its magnitude is 2 divided by the spacing
between the planes.
avg cosr r K r
ˆ ˆ ˆr xx yy zz
position vector
Reciprocal Lattice Vectors are
Grating Vectors
Lecture 7 Slide 18
Reciprocal lattice vectors are grating vectors!!
2a a
a
T K
There is a close and elegant relationship between
the reciprocal lattice vectors and wave vectors.
For this
reason,
periodic
structures
are
often
analyzed in reciprocal (grating vector) space.
8/13/2019 Lecture 7 -- Theory of Periodic Structures
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2/7/20
Primitive Unit Cells
Lecture 7 Slide 19
Primitive unit
cells
are
the
smallest
volume
of
space
that
can
be
stacked
onto
itself
(with no voids and no overlaps) to correctly reproduce the entire lattice.
The Wigner‐Seitz cell is one method of constructing a primitive unit cell. It is defined
as the volume of space around a single point in the lattice that is closer to that point
than any other point in the lattice.
These illustrations show the relationship between the conventional unit cell (shown as
frames) and the Wigner‐Seitz cell (shown as volumes) for the three cubic lattices.
Brillouin Zones
Lecture 7 Slide 20
The Brillouin zone is constructed in the same manner as the “Wigner‐Seitz,” but from
the reciprocal lattice.
The Brillouin zone is closely related to wave vectors and diffraction so analysis of
periodic structures is often performed in “reciprocal space.”
The Brillouin zone for a face‐
centered‐cubic lattice is a
“truncated” octahedron with 14
sides.
This is the most “spherical” of all the
Brillouin zones so
the
FCC
lattice
is
said to have the highest symmetry of
the Bravais lattices.
Of the FCC lattices, diamond has the
highest symmetry.
8/13/2019 Lecture 7 -- Theory of Periodic Structures
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Exploiting Additional Symmetry
Lecture 7 Slide 21
If the
field
is
known
at
every
point
inside
a single
unit
cell,
than
it
is
also
known
at
any
point in an infinite lattice because the field takes on the same symmetry as the lattice
so it just repeats itself.
Since the reciprocal lattice uniquely defines a direct lattice, knowing the solutions to
the wave equation at each point inside the reciprocal lattice unit cell also defines the
field everywhere in the infinite reciprocal lattice.
Many times, there is still additional symmetry to exploit so the smallest volume of
space that completely describes the electromagnetic wave can be smaller than the
unit cell itself.
The field in each of these squares
is a mirror image of each other.
Due to the symmetry in this example, the field at
any point in the entire lattice can be mapped to an
equivalent point in this triangle.
The Irreducible Brillouin Zone
Lecture 7 Slide 22
The smallest volume of space within the Brillouin zone that
completely characterizes the field inside a periodic structure is called
the irreducible Brillouin zone. It is smaller than the Brilluoin zone
when there is additional symmetry to exploit.
8/13/2019 Lecture 7 -- Theory of Periodic Structures
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ElectromagneticWaves in
Periodic
Structures
Fields are Perturbed by Objects
Lecture 7 Slide 24
k
A portion of the wave front is
delayed after travelling through
the dielectric object.
8/13/2019 Lecture 7 -- Theory of Periodic Structures
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Fields in Periodic Structures
Lecture 7 Slide 25
Waves in
periodic
structures
take
on
the same periodicity as their host.
k
The Bloch Theorem
Lecture 7 Slide 26
The field inside a periodic structure takes on the same symmetry and
periodicity of that structure according to the Bloch theorem.
j r A r e E r
Overall field Amplitude envelope
with same periodicity
and symmetry as the
device.
Plane‐wave like phase “tilt” term
8/13/2019 Lecture 7 -- Theory of Periodic Structures
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Example Waves in a PeriodicLattice
Lecture 7 27
Wave normally
incident
on
a periodic structure.
Wave incident
at
45° on
the
same periodic structure.
Mathematical Description of
Periodicity
Lecture 7 Slide 28
A structure is periodic if its material properties repeat. Given the
lattice vectors, the periodicity is expressed as
1 2 3 pqr pqr r t r t pt qt rt
Recall that it is the amplitude of the Bloch wave that has the same
periodicity as the structure the wave is in. Therefore,
1 2 3 pqr pqr r t A r t pt qt rt
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Example – 1D Periodicity
Lecture 7 Slide 29
For a device that is periodic only along one direction, these relations
reduce to
, , , ,
, , , ,
x
x
p y z x y z
A x p y z A x y z
x
x
x p x
A x p A x
more compact
notation
Many devices are periodic along just one dimension.
x
y
z
x
Electromagnetic
Bands
8/13/2019 Lecture 7 -- Theory of Periodic Structures
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Band Diagrams (1 of 2)
Lecture 7 Slide 31
Band diagrams
are
a compact,
but
incomplete,
means
of
characterizing the electromagnetic properties of a periodic structure.
It is essentially a map of the frequencies of the eigen‐modes as a
function of the Bloch wave vector .
Band Diagrams (2 of 2)
Lecture 7 Slide 32
To construct a band diagram, we make small steps around the perimeter of the
irreducible Brilluoin zone (IBZ) and compute the eigen‐values at each step. When
we plot all these eigen‐values as a function of , the points line up to form
continuous “bands.”
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Animation of the Construction of aBand Diagram
Lecture 7 Slide 33
Reading Band Diagrams
Lecture 7 Slide 34
• Band gaps – Absence of any bands within a range of frequencies indicates a band gap.
– A COMPLETE BAND GAP is one that exists over all possible Bloch wave vectors.
• Transmission/reflection spectra – Band gaps lead to suppressed transmission and enhanced reflection
• Phase velocity – The slope of the line connecting to the point on the band corresponds to phase
velocity.
– From this, we get the effective phase refractive index.
• Group velocity – The slope of the band at the point of interest corresponds to the group velocity.
– From this, we get the effective group refractive index.
• Dispersion – Any time the band deviates from the “light line” there is dispersion.
– The phase and group velocity are the same except when there is dispersion.
At least five electromagnetic properties can be estimated from a band
diagram.
8/13/2019 Lecture 7 -- Theory of Periodic Structures
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Reading Band Diagrams
Lecture 7 Slide 35
The Band Diagram is Missing
Information
Lecture 7 Slide 36
x
y
yk
xk
M
X
Direct lattice: We have an array
of air holes in a dielectric with
n=3.0.
Reciprocal lattice: We
construct the band diagram
by marching around the
perimeter of the irreducible
Brillouin zone.
The band extremes “almost” always occur at the key points of symmetry.
But we are missing information from inside the Brilluoin zone.
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The Complete Band Diagram
Lecture 7 Slide 37
02
a
c
a a
a
a 0
0
yk xk
The Full
Brillouin Zone…
yk
a
a
0
a 0 a
xk There is an infinite set of eigen‐frequencies
associated with each point in the Brillouin zone.
These form “sheets” as shown at right.
Isofrequency
Contours
8/13/2019 Lecture 7 -- Theory of Periodic Structures
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Index Ellipsoids for IsotropicMaterials
Lecture 7 Slide 39
The dispersion
relation
for
a LHI
material
is:
This defines a sphere called an “Index Ellipsoid.”
The vector connecting a point on the sphere
to the origin is the k ‐vector for that direction
from which refractive index can be calculated.
For LHI materials, the refractive index is the
same in all directions.
2 2 2 2 2
0a b ck k k k n
a
b
c
index ellipsoid
Index Ellipsoids for Uniaxial
Materials
Lecture 7 Slide 40
ab
c Observations
• 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 axis
oOrdinary axis
oC axis
oUniaxial axis
• Deviation from the optic axis will
result in two separate possible
modes.
On
O
n
On
E n E n
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Index Ellipsoids for BiaxialMaterials
Lecture 7 Slide 41
Biaxial materials
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.”
Direction of Power Flow
Lecture 7 Slide 42
Isotropic Materials
k
P
x
y
Anisotropic Materials
k
P
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.
8/13/2019 Lecture 7 -- Theory of Periodic Structures
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Isofrequency Contours FromFirst-Order Band
Lecture 7 Slide 43
02
a
c
a a
a
a 0
0
yk xk
02
a
c
a a
a
a 0
0
yk xk
Isofrequency Contours From
Second-Order Band
Lecture 7 Slide 44
02
a
c
a a
a
a 0
0
yk xk
02
a
c
a a
a
a 0
0
yk xk
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Standard View of IsofrequencyContours
Lecture 7 Slide 45