Photonic Crystals: Periodic Surprises in Electromagnetism

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Photonic CrystalsPeriodic Surprises in
Electromagnetism

• Steven G. Johnson
• MIT

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To Begin A Cartoon in 2d
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To Begin A Cartoon in 2d
a
for most l, beam(s) propagate through crystal
without scattering (scattering cancels coherently)
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Photonic Crystals
periodic electromagnetic media
with photonic band gaps optical insulators
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Photonic Crystals
periodic electromagnetic media
with photonic band gaps optical insulators
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Photonic Crystals
periodic electromagnetic media
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A mystery from the 19th century
conductive material
e
e
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A mystery from the 19th century
crystalline conductor (e.g. copper)
e
e
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A mystery solved
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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
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Fun with Math
First task get rid of this mess
0
dielectric function e(x) n2(x)
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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)
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Periodic Hermitian Eigenproblems
G. Floquet, Sur les équations différentielles
linéaries à coefficients périodiques, Ann. École
Norm. Sup. 12, 4788 (1883). F. Bloch, Über
die quantenmechanik der electronen in
kristallgittern, Z. Physik 52, 555600 (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)
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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
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Periodic Hermitian Eigenproblems in 1d
e1
e2
e1
e2
e1
e2
e1
e2
e1
e2
e1
e2
e(x) e(xa)
a
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Periodic Hermitian Eigenproblems in 1d
e1
e2
e1
e2
e1
e2
e1
e2
e1
e2
e1
e2
k is periodic k 2p/a equivalent to
k quasi-phase-matching
e(x) e(xa)
a
w
k
0
p/a
p/a
irreducible Brillouin zone
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Any 1d 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, 145159 (1887).
Start with a uniform (1d) medium
e1
w
k
0
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Any 1d 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, 145159 (1887).
Treat it as artificially periodic
e1
e(x) e(xa)
a
w
k
0
p/a
p/a
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Any 1d 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, 145159 (1887).
Treat it as artificially periodic
a
e(x) e(xa)
e1
w
0
p/a
x 0
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Any 1d 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, 145159 (1887).
Add a small real periodicity e2 e1 De
w
0
p/a
x 0
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Any 1d 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, 145159 (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
p/a
x 0
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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 gt band gap
(maybe)
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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
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2d periodicity, e121
a
frequency w (2pc/a) a / l
G
X
M
G
irreducible Brillouin zone
M
E
gap for n gt 1.751
TM
X
G
H
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2d periodicity, e121
Ez
( 90 rotated version)
G
X
M
G
E
gap for n gt 1.751
TM
H
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2d periodicity, e121
a
frequency w (2pc/a) a / l
G
X
M
G
irreducible Brillouin zone
M
E
E
TM
TE
X
G
H
H
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2d photonic crystal TE gap, e121
TE bands
TM bands
E
TE
gap for n gt 1.41
H
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3d photonic crystal complete gap , e121
I.
II.
gap for n gt 41
S. G. Johnson et al., Appl. Phys. Lett. 77,
3490 (2000)
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You, too, can computephotonic eigenmodes!
MIT Photonic-Bands (MPB) package http//ab-initio
.mit.edu/mpb
on Athena add mpb