Lecture 7 -- Theory of Periodic Structures

8/13/2019 Lecture 7 -- Theory of Periodic Structures

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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



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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



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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.



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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





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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.



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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



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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



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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.



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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.



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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.



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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.



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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



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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



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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.



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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



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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.



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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



2/7/20

Standard View of IsofrequencyContours

Lecture 7 Slide 45

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Lecture 7 -- Theory of Periodic Structures

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