Photonic Crystals:
Periodic Surprises in Electromagnetism
Steven G. Johnson
MIT
A “Defective” Lecture
cavity
waveguide
The Story So Far…
a
Waves in periodic media can have:
• propagation with no scattering (conserved k)
• photonic band gaps (with proper e function)
G
are
Qraphics
uickTim
needed
e™
decom
toand
see
pressor
athis picture.
1
Eigenproblem gives simple insight:
band diagram
0.9
0.8
0.7
Bloch form:
H e
i(k x t)
Hk (x )
2



1

 n (k )
(
ik
)

(
ik
)

H


 Hk


k


e
 c 
ˆ

k

0.6
0.5
0.4
0.3
Photonic BandGap
0.2
TMbands
0.1
0
Hermitian –> complete, orthogonal, variational theorem, etc.
k
are
G
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
Properties of Bulk Crystals
by Bloch’s theorem
conserved frequency 
band diagram (dispersion relation)
photonic band gap
synthetic medium
for propagation
conserved wavevector k
(cartoon)
backwards slope:
negative refraction
d/dk  0: slow light
(e.g. DFB lasers)
strong curvature:
super-prisms, …
(+ negative refraction)
Applications of Bulk Crystals
using near-band-edge effects
Zero group-velocity d/dk: distributed feedback (DFB) lasers
divergent dispersion
(i.e. curvature):
Superprisms
[Kosaka, PRB 58, R10096 (1998).]
super-lens
negative group-velocity or
negative curvature (“eff. mass”):
V eselag o (1 9 6 8)
obje ct
im ag e
negat ive
ref ract ion
me diu m
Negative refraction,
Super-lensing
[ C. Luo et al.,
Appl. Phys. Lett. 81, 2352 (2002) ]
sourc e
im a g e
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Cavity Modes
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Help!
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Cavity Modes
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finite region –> discrete 
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Cavity Modes: Smaller Change
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Cavity Modes: Smaller Change
are
G
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
Bulk Crystal Band Diagram
0.6
L
frequency (c/a)
0.5
0.4
Photonic Band Gap
0.3
0.2
0.1
0
G
X
M
M
k
G
X
G
Cavity Modes: Smaller Change
are
G
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
Defect Crystal Band Diagram
0.6
J
J
J
J
frequency (c/a)
0.5
L
Defect bands are
shifted up (less e)
with discrete k

# ~ L
2
(k ~ 2 /  )
0.4
JJ
J
J
JJJJJ
∆k ~ π / L
Photonic Band Gap
0.3
escapes: J
0.2
J
J
J
0.1
J
JJJJJJJ
J
J
J
J
J
JJJ
JJ
JJJJJJJJ
J
J
JJJJ
J
JJ
J
k not conserved
at boundary, so
not confined outside gap
J
0J
G
confined
modes
J
J
J
J
J
J
X
M
M
k
G
X
G
Single-Mode Cavity
are
G
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
Bulk Crystal Band Diagram
0.6
A point defect
can push up
a single mode
from the band edge
field decay ~
  0
frequency (c/a)
0.5
0.4
Photonic Band Gap

0.3
0
0.2
0.1
0
G
X
M
curvature
G
M
k
G
X
(k not conserved)
“Single”-Mode Cavity
are
G
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
Bulk Crystal Band Diagram
0.6
A point defect
can pull down
a “single” mode
frequency (c/a)
0.5
…here, doubly-degenerate
(two states at same )
0.4
Photonic Band Gap
0.3
0.2
0.1
0
G
X
M
X
k
G
G
M
X
(k not conserved)
Tunable Cavity Modes
0.5
air b ands
frequency (c/a)
Air Defect
Dielect ric Defect
0.4
0.3
dielect ric b ands
0.2
0
0.1
0.2
0.3
0.4
Radius of Defect (r/ a)
Ez:
monopole
dipole
Tunable Cavity Modes
band #1 at M
band #2 at X’s
multiply by exponential decay
Ez:
monopole
dipole
are
G
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
Defect Flavors
are
G
Qraphics
uickTim
needed
decom
e™
to
and
see
pressor
athis
picture.
G
are
Qraphics
uickTim
needed
e™
decom
to
and
see
pressor
athis picture.
a
Projected Band Diagrams
M
k
conserved k!
X
G
conserved
So, plot  vs. kx only…project Brillouin zone onto G–X:
gives continuum of bulk states + discrete guided band(s)
not conserved
1d periodicity
Air-waveguide Band Diagram
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
0.5
0.45
J
0.4
J
ban
d ga
p
J
0.35
frequency (c/a)
0.3
0.25
0.2
0.15
J
J
J
J
J
J
st a
t es of the
bulk cry s t a
l
0.1
0.05
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
wavenumber k (2š/a)
any state in the gap cannot couple to bulk crystal –> localized
(Waveguides don’t really need a
complete gap)
Fabry-Perot waveguide:
We’ll exploit this later, with photonic-crystal fiber…
So What?
Review: Why no scattering?
forbidden by Bloch
(k conserved)
forbidden by gap
(except for finite-crystal tunneling)
Benefits of a complete gap…
broken symmetry –> reflections only
effectively one-dimensional
Lossless Bends
[ A. Mekis et al.,
Phys. Rev. Lett. 77, 3787 (1996) ]
symmetry + single-mode + “1d” = resonances of 100% transmission
Waveguides + Cavities = Devices
“tunneling”
Ugh, must we simulate this to get the basic behavior?
No! Use “coupling-of-modes-in-time” (coupled-mode theory)…
[H. Haus, Waves and Fields in Optoelectronics]
“Coupling-of-Modes-in-Time”
(a form of coupled-mode theory)
[H. Haus, Waves and Fields in Optoelectronics]
s1+
s1–
input
a
s2–
resonant cavity
frequency 0, lifetime t
da
2
2
 i 0 a  a 
s1
dt
t
t
s1  s1 
2
t
a,
s2 
2
t
a
output
|s|2 = flux
|a|2 = energy
assumes only:
• exponential decay
(strong confinement)
• conservation of energy
• time-reversal symmetry
“Coupling-of-Modes-in-Time”
(a form of coupled-mode theory)
[H. Haus, Waves and Fields in Optoelectronics]
input
s1+
s1–
a
s2–
resonant cavity
frequency 0, lifetime t
1
transmission T
= | s2– | / | s1+ |
2
2
output
|s|2 = flux
|a|2 = energy
T = Lorentzian filter
1
2

Q  0t
FWHM
0

4
2
t

2
4
   0   2
t
…quality factor Q
A Menagerie of Devices
l
1.55 microns
Wide-angle Splitters
[ S. Fan et al., J. Opt. Soc. Am. B 18, 162 (2001) ]
Waveguide Crossings
[ S. G. Johnson et al., Opt. Lett. 23, 1855 (1998) ]
Waveguide Crossings
G
are
Qraphics
uickTim
needed
e™
decom
toand
see
pressor
athis picture.
1
1x1
0.8
0.6
3x3
t hroughput
0.4
empty
0.2
5x5
5x5
empty
1x10 -2
1x1
1x10 -4
3x3
1x10 -6
5x5
3x3
cr o
s s t al k
empty
0
1x10 0
1x1
1x10 -8
1x10 -10
0.32
0.33
0.34
f re quency( c/ a)
0.35
0.36
0.37
0.38
Channel-Drop Filters
[ S. Fan et al., Phys. Rev. Lett. 80, 960 (1998) ]
Channel-Drop Filters
QuickTime™ and a
Video decompressor
are needed to see this picture.
QuickTime™ and a
Video decompressor
are needed to see this picture.
Enough passive, linear devices…
Photonic crystal cavities:
tight confinement (~ /2 diameter)
+ long lifetime (high Q independent of size)
= enhanced nonlinear effects
e.g. Kerr nonlinearity, ∆n ~ intensity
A Linear Nonlinear Filter
in
out
Linear response:
Lorenzian Transmisson
shifted peak
+ nonlinear
index shift
A Linear Nonlinear “Transistor”
Logic gates, switching,
rectifiers, amplifiers,
isolators, …
numerical
+ feedback
Linear response:
Lorenzian Transmisson
Bistable (hysteresis) response
Power threshold is near optimal
(~mW for Si and telecom bandwidth)
shifted peak
Enough passive, linear devices…
Photonic crystal cavities:
tight confinement (~ /2 diameter)
+ long lifetime (high Q independent of size)
= enhanced nonlinear effects
Photonic crystal waveguides:
tight confinement (~ /2 diameter)
+ slow light (e.g. near band edge)
= enhanced nonlinear effects
Cavities + Cavities = Waveguide
“tunneling”
coupled-cavity waveguide (CCW/CROW): slow light + zero dispersion
[ A. Yariv et al., Opt. Lett. 24, 711 (1999) ]
Enhancing tunability with slow light
[ M. Soljacic et al., J. Opt. Soc. Am. B 19, 2052 (2002) ]
periodicity:
light is slowed, but not reflected
Uh oh, we live in 3d…
C
rod layer
B
A
hole layer
(fcc crystal)
2d-like defects in 3d
[ M. L. Povinelli et al., Phys. Rev. B 64, 075313 (2001) ]
modify single layer
of holes or rods
3d projected band diagram
0.5
3D Photonic Crystal
J
J
0.5
J
J
J
J
0.3
0.2
0.1
J
TM gap
J
frequency (c/a)
frequency (c/a)
0.4
2D Photonic Crystal
0.4
J
J
J
J
J
J
J
J
J
0.3
0.2
0.1
0
0
0
0.1
0.2
wavevector
0.3
0.4
kx ( 2/a)
0.5
0
0.1
0.2
wavevector
0.3
0.4
k x ( 2 /a)
0.5
2d-like waveguide mode
3D Photonic Crystal
2D Photonic Crystal
z
x
x
y
-1
Ez
y
1
-1
y
Ez
1
2d-like cavity mode
The Upshot
To design an interesting device, you need only:
symmetry
+ single-mode (usually)
+ resonance
+ (ideally) a band gap to forbid losses
Oh, and a full Maxwell simulator to get Q parameters, etcetera.