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
k
scattering
planewave
E, H ~ e
i (k x t )
k  /c 
2

To Begin: A Cartoon in 2d
k
a
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planewave
E, H ~ e
i (k x t )
k  /c 
2

for most , beam(s) propagate
through crystal without scattering
(scattering cancels coherently)
...but for some  (~ 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
cavities
3D
Photrap
to niclight
C rystain
l with
De fe c ts
and waveguides (“wires”)
magical oven mitts for
holding and controlling light
with photonic band gaps: “optical insulators”
Photonic Crystals
periodic electromagnetic media
Hig h in d e x
o f re fra c tio n
Lo w ind e x
o f re fra c tio n
3D Pho to nic 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–
E
+
current:
+
J  E
conductivity (measured)
mean free path (distance) of electrons
A mystery from the 19th century
crystalline conductor (e.g. copper)
+ + + + + + + +
e–
e–
E
+
+
+
+
+
+
+
+
+
+
+
+
+
+
10’s
+
of
+ periods!
+
+
+
+
+
+
+
+
current:
J  E
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
k
a
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planewave
E, H ~ e
i (k x t )
k  /c 
2

for most , beam(s) propagate
through crystal without scattering
(scattering cancels coherently)
...but for some  (~ 2a), no light can propagate: a photonic band gap
Fun with Math
1

 E  
H i H
c t
c
0 
1
 H  
E  J  i E
c t
c
First task:
get rid of this mess
dielectric function (x) = n2(x)
 
    H    H
 c 

1
eigen-operator
2
eigen-value
+ constraint
 H  0
eigen-state
Hermitian Eigenproblems
 
    H    H
 c 

1
eigen-operator
2
eigen-value
+ constraint
 H  0
eigen-state
Hermitian for real (lossless) 
well-known properties from linear algebra:
 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:

i k x t
H(x ,t)  e
planewave

Hk (x )
periodic “envelope”
Corollary 1: k is conserved, i.e. no scattering of Bloch wave
Corollary 2: H k given by finite unit cell,
so  are discrete n(k)
Periodic Hermitian Eigenproblems
Corollary 2: H k given by finite unit cell,
so  are discrete n(k)
band diagram (dispersion relation)
3

map of
what states
exist &
can interact
2
1
k
?
range of k?
Periodic Hermitian Eigenproblems in 1d
H(x)  e Hk (x)
ikx
1 2  1 2 1 2 1 2 1 2 1 2
a
Consider k+2π/a: e
i(k 
2
)x
a
(x) = (x+a)
2


i
x
ikx
H 2  (x)  e e a H 2  (x)
k
k




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
1  2 1 2 1 2 1 2 1 2 1 2
k is periodic:
k + 2π/a equivalent to k
“quasi-phase-matching”
a
(x) = (x+a)

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:
1

0

k
1
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
1
a
(x) = (x+a)



a
e

 cos 
 a
–π/a
0
π/a
x


a
x
,e

 
x, sin  x 

 a 
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
1

0
 
sin  x 
 a 
 
cos  x 
 a 
π/a
x=0
(x) = (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
2 = 1 + D
(x) = (x+a)
1 2 1 2 1 2 1 2 1 2 1 2

0
a
 
sin  x 
 a 
 
cos  x 
 a 
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
2 = 1 + D
state concentrated in higher index (2)
has lower frequency
a
(x) = (x+a)
1 2 1 2 1 2 1 2 1 2 1 2

 
sin  x 
 a 
 
cos  x 
 a 
band gap
0
Splitting of degeneracy:
π/a
x=0
Some 2d and 3d systems have gaps
• In general, eigen-frequencies satisfy Variational Theorem:
1(k )  min
2
E1
E1  0
 2 (k )  min
2


  ik  E1
E
1
2
2
“kinetic”
c
2
inverse
“potential”
" " bands “want” to be in high-
E2
E 2  0
*
 E1  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,
picture.
=12:1
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis
a
frequency  (2πc/a) = a / 
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
Photonic BandGap
0.2
TMbands
0.1
0
irreducible Brillouin zone
M
k
G
X
G
TM
X
E
H
M
G
gap for
n > ~1.75:1
2d periodicity,
picture.
=12:1
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis
1
0.9
0.8
Ez
0.7
0.6
0.5
(+ 90° rotated version)
0.4
0.3
Photonic BandGap
0.2
TMbands
0.1
Ez
0
G
–
+
TM
X
E
H
M
G
gap for
n > ~1.75:1
2d periodicity,
picture.
=12:1
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis
a
frequency  (2πc/a) = a / 
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
Photonic BandGap
TEbands
0.2
TMbands
0.1
0
irreducible Brillouin zone
M
k
G
X
G
TM
X
E
H
G
M
E
TE
H
2d photonic crystal: TE gap, =12:1
TE bands
TM bands
E
TE
H
gap for n > ~1.4:1
3d photonic crystal: complete gap , =12:1
I.
II.
0.8
0.7
0.6
21% gap
0.5
0.4
z
L'
0.3
U'
G
X
U'' U W K'
W' K L
0.2
0.1
I: rod layer
II: hole layer
0
UХ
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