Photonic Crystals: Periodic Surprises in Electromagnetism Steven G. Johnson MIT

Photonic Crystals:
Periodic Surprises in Electromagnetism
Steven G. Johnson
MIT
To Begin: A Cartoon in 2d
r
k
scattering
planewave
r r
r r
i ( k ⋅ x -wt )
E, H ~ e
r
2p
k =w /c =
l
To Begin: A Cartoon in 2d
r
k
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
a
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
planewave
r r
r r
i ( k ⋅ x -wt )
E, H ~ e
r
2p
k =w /c =
l
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
Photonic Crystals
periodic electromagnetic media
1887
1987
1-D
2-D
periodic in
one direction
periodic in
two directions
3-D
periodic in
three directions
with photonic band gaps: “optical insulators”
(need a
more
complex
topology)
Photonic Crystals
periodic electromagnetic media
can trap light in cavities
3D Photonic C rysta l with Defects
and waveguides (“wires”)
magical oven mitts for
holding and controlling light
with photonic band gaps: “optical insulators”
Photonic Crystals
periodic electromagnetic media
High index
of refra ction
Low index
of refra ction
3D Photonic C rysta l
But how can we understand such complex systems?
Add up the infinite sum of scattering? Ugh!
A mystery from the 19th century
conductive material
+
+
e–
+
e–
r
E
+
current:
+
r
r
J = sE
conductivity (measured)
mean free path (distance) of electrons
A mystery from the 19th century
crystalline conductor (e.g. copper)
+ + + + + + + +
e–
e–
r
E
+
+
+
+
+
+
+
+
+
+
+
+
+
+
10’s
+
of
+ periods!
+
+
+
+
+
+
+
+
current:
r
r
J = sE
conductivity (measured)
mean free path (distance) of electrons
A mystery solved…
1
electrons are waves (quantum mechanics)
2
waves in a periodic medium can
propagate without scattering:
Bloch’s Theorem (1d: Floquet’s)
The foundations do not depend on the specific wave equation.
Time to Analyze the Cartoon
r
k
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
a
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
planewave
r r
r r
i ( k ⋅ x -wt )
E, H ~ e
r
2p
k =w /c =
l
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
r r
1∂ r
w r
—¥E=H =i H
c ∂t
c
r r
1 ∂ r r0 w r
—¥H =e
E + J = i eE
c ∂t
c
First task:
get rid of this mess
dielectric function e(x) = n2(x)
2 r
r
Êw ˆ
1
—¥ —¥H =Á ˜ H
Ë c¯
e
eigen-operator
eigen-value
+ constraint
r
—⋅ H = 0
eigen-state
Hermitian Eigenproblems
2 r
r
Êw ˆ
1
—¥ —¥H =Á ˜ H
Ë c¯
e
eigen-operator
eigen-value
+ constraint
r
—⋅ H = 0
eigen-state
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
[ 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:
r r
r r
i ( k ⋅ x -wt ) r
r
H( x ,t) = e
Hkr ( x )
planewave
periodic “envelope”
Corollary 1: k is conserved, i.e. no scattering of Bloch wave
r
Corollary 2: Hrk given by finite unit cell,
so w are discrete wn(k)
Periodic Hermitian Eigenproblems
r
Corollary 2: Hrk given by finite unit cell,
so w are discrete wn(k)
band diagram (dispersion relation)
w3
w
map of
what states
exist &
can interact
w2
w1
k
?
range of k?
Periodic Hermitian Eigenproblems in 1d
e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 e1 e2
ikx
H(x) = e Hk (x)
a
Consider k+2π/a: e
i( k +
2p
)x
a
e(x) = e(x+a)
2p
È
˘
i
x
ikx
H 2p (x) = e Í e a H 2p (x)˙
k+
k+
Í
˙˚
Î
a
a
k is periodic:
k + 2π/a equivalent to k
“quasi-phase-matching”
periodic!
satisfies same
equation as Hk
= Hk
Periodic Hermitian Eigenproblems in 1d
e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 e1 e2
k is periodic:
k + 2π/a equivalent to k
“quasi-phase-matching”
a
e(x) = e(x+a)
w
band gap
–π/a
0
π/a
irreducible Brillouin zone
k
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, 145–159 (1887). ]
Start with
a uniform (1d) medium:
e1
w
0
k
w=
e1
k
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, 145–159 (1887). ]
Treat it as
“artificially” periodic
bands are “folded”
by 2π/a equivalence
e1
a
e(x) = e(x+a)
w
+
p
x
a
e
Êp
Æ cosÁ
Ëa
–π/a
0
π/a
-
p
x
a
,e
ˆ
Êp ˆ
x˜ , sinÁ x ˜
¯
Ëa ¯
k
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, 145–159 (1887). ]
Treat it as
“artificially” periodic
a
e1
w
0
Êp ˆ
sinÁ x ˜
Ëa ¯
Êp ˆ
cosÁ x ˜
Ëa ¯
π/a
x=0
e(x) = e(x+a)
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, 145–159 (1887). ]
Add a small
“real” periodicity
e2 = e1 + De
e(x) = e(x+a)
e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 e1 e2
w
0
a
Êp ˆ
sinÁ x ˜
Ëa ¯
Êp ˆ
cosÁ x ˜
Ëa ¯
π/a
x=0
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, 145–159 (1887). ]
Add a small
“real” periodicity
e2 = e1 + De
state concentrated in higher index (e2)
has lower frequency
a
e(x) = e(x+a)
e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 e1 e2
w
Êp ˆ
sinÁ x ˜
Ëa ¯
Êp ˆ
cosÁ x ˜
Ëa ¯
band gap
0
Splitting of degeneracy:
π/a
x=0
Some 2d and 3d systems have gaps
• In general, eigen-frequencies satisfy Variational Theorem:
r 2
w1 ( k ) = min
r
E1
r
—⋅eE1 = 0
r r 2 “kinetic”
— + ik ¥ E1
2
c
r 2
Ú e E1 inverse
Ú(
)
“potential”
r 2
w 2 ( k ) = min
"L" bands “want” to be in high-e
r
E2
r
—⋅eE 2 = 0
*
Ú eE1 ⋅ E 2 = 0 …but are forced out by orthogonality
–> band gap (maybe)
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
frequency w (2πc/a) = a
/l
1
0.9
0.8
0.7
0.6
0.5
0.4
Photonic Band Gap
0.3
0.2
TM bands
0.1
0
irreducible Brillouin zone
M
r
k
G
X
G
TM
X
E
H
M
G
gap for
n > ~1.75:1
2d periodicity, e=12:1
1
0.9
0.8
Ez
0.7
0.6
0.5
(+ 90° rotated version)
0.4
Photonic Band Gap
0.3
0.2
TM bands
0.1
Ez
0
G
–
+
TM
X
E
H
M
G
gap for
n > ~1.75:1
2d periodicity, e=12:1
a
frequency w (2πc/a) = a
/l
1
0.9
0.8
0.7
0.6
0.5
0.4
Photonic Band Gap
0.3
TE bands
0.2
TM bands
0.1
0
irreducible Brillouin zone
M
r
k
G
X
G
TM
X
E
H
G
M
E
TE
H
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
I.
II.
0.8
0.7
0.6
21% gap
0.5
0.4
z
0.3
U'
G
X
K'
U'' U W
W' K L
0.2
0.1
I: rod layer
II: hole layer
L'
0
U’
L
G
X
W
K
gap for n > ~4:1
[ S. G. Johnson et al., Appl. Phys. Lett. 77, 3490 (2000) ]
You, too, can compute
photonic eigenmodes!
MIT Photonic-Bands (MPB) package:
http://ab-initio.mit.edu/mpb
on Athena:
add mpb