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Photonic Crystals: Periodic Surprises in Electromagnetism. Steven G. Johnson MIT. scattering. planewave. To Begin: A Cartoon in 2d. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •. •.

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to begin a cartoon in 2d3

planewave

...but for some l (~ 2a), no light can propagate: a photonic band gap

To Begin: A Cartoon in 2d

a

for most l, beam(s) propagate

through crystal without scattering

(scattering cancels coherently)

photonic crystals

1887

1987

(need a more complex topology)

Photonic Crystals

periodic electromagnetic media

with photonic band gaps: “optical insulators”

photonic crystals5

can trap light in cavities

and waveguides (“wires”)

magical oven mitts for

holding and controlling light

Photonic Crystals

periodic electromagnetic media

with photonic band gaps: “optical insulators”

photonic crystals6

But how can we understand such complex systems?

Add up the infinite sum of scattering? Ugh!

Photonic Crystals

periodic electromagnetic media

a mystery from the 19th century

+

+

+

+

+

current:

conductivity (measured)

mean free path (distance) of electrons

A mystery from the 19th century

conductive material

e–

e–

a mystery from the 19th century8

10’s

of

periods!

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

current:

conductivity (measured)

mean free path (distance) of electrons

A mystery from the 19th century

crystalline conductor (e.g. copper)

e–

e–

a mystery solved

electrons are waves (quantum mechanics)

1

waves in a periodic medium can propagate without scattering:

Bloch’s Theorem (1d: Floquet’s)

2

The foundations do not depend on the specificwave equation.

A mystery solved…
time to analyze the cartoon

planewave

Time to Analyze the Cartoon

a

for most l, beam(s) propagate

through crystal without scattering

(scattering cancels coherently)

...but for some l (~ 2a), no light can propagate: a photonic band gap

fun with math

+ constraint

eigen-state

eigen-operator

eigen-value

Fun with Math

First task:

get rid of this mess

0

dielectric function e(x) = n2(x)

hermitian eigenproblems

+ constraint

eigen-state

eigen-operator

eigen-value

Hermitian Eigenproblems

Hermitian for real (lossless) e

well-known properties from linear algebra:

w are real (lossless)

eigen-states are orthogonal

eigen-states are complete (give all solutions)

periodic hermitian eigenproblems
Periodic Hermitian Eigenproblems

[ G. Floquet, “Sur les équations différentielles linéaries à coefficients périodiques,” Ann. École Norm. Sup. 12, 47–88 (1883). ]

[ F. Bloch, “Über die quantenmechanik der electronen in kristallgittern,” Z. Physik 52, 555–600 (1928). ]

if eigen-operator is periodic, then Bloch-Floquet theorem applies:

can choose:

planewave

periodic “envelope”

Corollary 1: k is conserved, i.e.no scattering of Bloch wave

Corollary 2: given by finite unit cell,

so w are discrete wn(k)

periodic hermitian eigenproblems14

?

range of k?

Periodic Hermitian Eigenproblems

Corollary 2: given by finite unit cell,

so w are discrete wn(k)

band diagram (dispersion relation)

w3

map of

what states

exist &

can interact

w2

w

w1

k

periodic hermitian eigenproblems in 1d

Consider k+2π/a:

periodic!

satisfies same

equation as Hk

= Hk

k is periodic:

k + 2π/a equivalent to k

“quasi-phase-matching”

Periodic Hermitian Eigenproblems in1d

e1

e2

e1

e2

e1

e2

e1

e2

e1

e2

e1

e2

e(x) = e(x+a)

a

periodic hermitian eigenproblems in 1d16

band gap

Periodic Hermitian Eigenproblems in1d

k is periodic:

k + 2π/a equivalent to k

“quasi-phase-matching”

e1

e2

e1

e2

e1

e2

e1

e2

e1

e2

e1

e2

e(x) = e(x+a)

a

w

k

0

–π/a

π/a

irreducible Brillouin zone

any 1d periodic system has a gap
Any1d Periodic System has a Gap

[ Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philosophical Magazine 24, 145–159 (1887). ]

Start with

a uniform (1d) medium:

e1

w

k

0

any 1d periodic system has a gap18

bands are “folded”

by 2π/a equivalence

Any1d Periodic System has a Gap

[ Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philosophical Magazine 24, 145–159 (1887). ]

Treat it as

“artificially” periodic

e1

e(x) = e(x+a)

a

w

k

0

–π/a

π/a

any 1d periodic system has a gap19
Any1d Periodic System has a Gap

[ Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philosophical Magazine 24, 145–159 (1887). ]

Treat it as

“artificially” periodic

a

e(x) = e(x+a)

e1

w

0

π/a

x = 0

any 1d periodic system has a gap20

a

e(x) = e(x+a)

e1

e2

e1

e2

e1

e2

e1

e2

e1

e2

e1

e2

Any1d Periodic System has a Gap

[ Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philosophical Magazine 24, 145–159 (1887). ]

Add a small

“real” periodicity

e2 = e1 + De

w

0

π/a

x = 0

any 1d periodic system has a gap21

a

e(x) = e(x+a)

e1

e2

e1

e2

e1

e2

e1

e2

e1

e2

e1

e2

Any1d Periodic System has a Gap

[ Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philosophical Magazine 24, 145–159 (1887). ]

Splitting of degeneracy:

state concentrated in higher index (e2)

has lower frequency

Add a small

“real” periodicity

e2 = e1 + De

w

band gap

0

π/a

x = 0

some 2d and 3d systems have gaps
Some 2d and 3d systems have gaps

• In general, eigen-frequencies satisfy Variational Theorem:

“kinetic”

inverse

“potential”

bands “want” to be in high-e

…but are forced out by orthogonality

–> band gap (maybe)

algebraic interlude
algebraic interlude

algebraic interlude completed…

… I hope you were taking notes*

[ *if not, see e.g.: Joannopoulos, Meade, and Winn, Photonic Crystals: Molding the Flow of Light ]

2d periodicity e 12 1

a

2d periodicity, e=12:1

frequencyw(2πc/a) = a / l

irreducible Brillouin zone

G

G

X

M

M

E

gap for

n > ~1.75:1

TM

X

G

H

2d periodicity e 12 125

Ez

+

2d periodicity, e=12:1

Ez

(+ 90° rotated version)

G

G

X

M

E

gap for

n > ~1.75:1

TM

H

2d periodicity e 12 126

a

2d periodicity, e=12:1

frequencyw(2πc/a) = a / l

irreducible Brillouin zone

G

G

X

M

M

E

E

TM

TE

X

G

H

H

2d photonic crystal te gap e 12 1
2d photonic crystal: TE gap, e=12:1

TE bands

TM bands

E

TE

H

gap for n > ~1.4:1

3d photonic crystal complete gap e 12 1
3d photonic crystal: complete gap , e=12:1

I.

II.

gap for n > ~4:1

[ S. G. Johnson et al., Appl. Phys. Lett.77, 3490 (2000) ]

you too can compute photonic eigenmodes
You, too, can computephotonic eigenmodes!

MIT Photonic-Bands (MPB) package:

http://ab-initio.mit.edu/mpb

on Athena:

add mpb