Transcript Document 7171595

Photonic Crystals:
Principles and Applications
Steven G. Johnson
MIT Applied Mathematics
Outline
•
•
•
•
•
•
Preliminaries: waves in periodic media
Photonic crystals in theory and practice
Bulk crystal properties
Intentional defects and devices
Index-guiding and incomplete gaps
Photonic-crystal and microstructured fibers
Outline
•
•
•
•
•
•
Preliminaries: waves in periodic media
Photonic crystals in theory and practice
Bulk crystal properties
Intentional defects and devices
Index-guiding and incomplete gaps
Photonic-crystal and microstructured fibers
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
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
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 in Nature
Peacock feather
Morpho butterfly
wing scale:
[ L. P. Biró et al.,
PRE 67, 021907
(2003) ]
6.21µm
[J. Zi et al, Proc. Nat. Acad. Sci. USA,
100, 12576 (2003) ]
[figs: Blau, Physics Today 57, 18 (2004)]
Photonic Crystals
periodic electromagnetic media
can
cavities
3D
Photrap
to niclight
C rystain
l with
De fe c ts
and waveguides (“wires”)
with photonic band gaps:
“optical insulators”
for holding and controlling light
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.
Electronic and Photonic Crystals
dielectric spheres, diamond lattice
wavevector
interacting: hard problem
photon frequency
electron energy
Bloch waves:
Band Diagram
Periodic
Medium
atoms in diamond structure
wavevector
non-interacting: “easy” problem
Time to Analyze the Cartoon
k
a
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
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)
 i 2  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)


i x
a
e

 cos
a
–π/a
0
π/a

i x
a
,e
 
x , sin
 a
k

x 

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

 
sin  x 
 a 
 
cos  x 
 a 
0
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
a
1 2 1 2 1 2 1 2  1 2 1 2

 
sin  x 
 a 
 
cos  x 
 a 
0
(x) = (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)
Outline
•
•
•
•
•
•
Preliminaries: waves in periodic media
Photonic crystals in theory and practice
Bulk crystal properties
Intentional defects and devices
Index-guiding and incomplete gaps
Photonic-crystal and microstructured fibers
2d periodicity,
picture.
=12:1
G
are
Qraphics
uickTim
needed
e™
decom
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
e™
decom
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
e™
decom
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
K'
U'' U W
W' K L
0.2
0.1
I: rod layer
II: hole layer
0
U’
L
G
X
W
K
gap for n > ~4:1
[ S. G. Johnson et al., Appl. Phys. Lett. 77, 3490 (2000) ]
The Mother of (almost) All Bandgaps
The diamond lattice:
fcc (face-centered-cubic)
with two “atoms” per unit cell
a
(primitive)
Recipe for a complete gap:
fcc = most-spherical Brillouin zone
+ diamond “bonds” = lowest (two) bands can concentrate in lines
Image: http://cst-www.nrl.navy.mil/lattice/struk/a4.html
The First 3d Bandgap Structure
K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).
are
G
Qraphics
uickTim
needed
e™
decom
toand
see
pressor
athis picture.
frequency (c/a)
(c/a)
frequency
0.6
J JJ
0.5 J J J JJ
0.4
J J J JJ
0.3
J
J J JJ JJ J
J J JJ J J
J
J
J
J
JJ J
J JJ J
JJ J JJ
11% gap
JJ J J
J J J
J JJ
J J
J J
J
J
J
0.2
X
U
L
J J
J JJ J JJ
J J J J J J J J JJ J J
J J JJ
J J
J J J J J J J J J JJ JJ
J
X
J
J
0
J
J
J
0.1
J
J
G
X
for gap at  = 1.55µm,
sphere diameter ~ 330nm
G
W
U L
K
W
K
overlapping Si spheres
MPB tutorial, http://ab-initio.mit.edu/mpb
Layer-by-Layer Lithography
• Fabrication of 2d patterns in Si or GaAs is very advanced
(think: Pentium IV, 50 million transistors)
…inter-layer alignment techniques are only slightly more exotic
So, make 3d structure one layer at a time
Need a 3d crystal with constant cross-section layers
A Schematic
[ M. Qi, H. Smith, MIT ]
7-layer E-Beam Fabrication
[ M. Qi, H. Smith, MIT ]
1.25 Periods of the Woodpile
(4 “log” layers = 1 period)
Si
http://www.sandia.gov/media/photonic.htm
[ S. Y. Lin et al., Nature 394, 251 (1998) ]
gap
Two-Photon Lithography
2-photon probability ~ (light intensity)2
2 hn = ∆E
hn
e
E0
hn
photon
photon
3d Lithography
Atom
lens
…dissolve unchanged stuff
(or vice versa)
some chemistry
(polymerization)
Lithography is a Beast
[ S. Kawata et al., Nature 412, 697 (2001) ]
 = 780nm
resolution = 150nm
7µm
(3 hours to make)
2µm
One-Photon
Holographic Lithography
[ D. N. Sharp et al., Opt. Quant. Elec. 34, 3 (2002) ]
Four beams make 3d-periodic interference pattern
k-vector differences give reciprocal lattice vectors (i.e. periodicity)
absorbing material
(1.4µm)
beam polarizations + amplitudes (8 parameters) give unit cell
One-Photon
Holographic Lithography
[ D. N. Sharp et al., Opt. Quant. Elec. 34, 3 (2002) ]
10µm
huge volumes, long-range periodic, fcc lattice…backfill for high contrast
Mass-production II: Colloids
(evaporate)
silica (SiO2)
microspheres (diameter < 1µm)
sediment by gravity into
close-packed fcc lattice!
Inverse Opals
[ figs courtesy
D. Norris, UMN ]
fcc solid spheres do not have a gap…
…but fcc spherical holes in Si do have a gap
sub-micron colloidal spheres
Template
(synthetic opal)
3D
Infiltration
complete band gap
Remove
Template
“Inverted Opal”
~ 10% gap between 8th & 9th bands
small gap, upper bands: sensitive to disorder
figs courtesy
In Order To Form D.[ Norris,
UMN ]
a More Perfect Crystal…
meniscus
65C
1 micron
silica
silica spheres
250nm
in ethanol
evaporate
solvent
80C
Convective Assembly
[ Nagayama, Velev, et al., Nature (1993)
Colvin et al., Chem. Mater. (1999) ]
Heat Source
• Capillary forces during drying cause assembly in the meniscus
• Extremely flat, large-area opals of controllable thickness
A Better Opal
[ fig courtesy
D. Norris, UMN ]
Inverse-Opal Photonic Crystal [ fig courtesy
D. Norris, UMN ]
[ Y. A. Vlasov et al., Nature 414, 289 (2001). ]
You, too, can compute
photonic eigenmodes!
MIT Photonic-Bands (MPB) package:
http://ab-initio.mit.edu/mpb
Outline
•
•
•
•
•
•
Preliminaries: waves in periodic media
Photonic crystals in theory and practice
Bulk crystal properties
Intentional defects and devices
Index-guiding and incomplete gaps
Photonic-crystal and microstructured fibers
The Story So Far…
a
Waves in periodic media can have:
• propagation with no scattering (conserved k)
• photonic band gaps (with proper  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



 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)
Superprisms
from divergent dispersion (band curvature)
[Kosaka, PRB 58, R10096 (1998).]
Negative Refraction
[ Veselago, 1968
negative , m ]
does not require
curved lens
opposite of ordinary lens:
only images close objects
can exceed classical
diffraction limit
Negative Refraction
with (all-dielectric) Photonic Crystals
[Luo et al, MIT]
Here, using positive effective index but negative “effective mass”
Negative Refraction
with (all-dielectric) Photonic Crystals
[Luo et al, MIT]
 contours
in (kx,ky) space
Here, using positive effective index but negative “effective mass”
periodicity:
unusual dispersion without scattering
Super-lensing
[Luo, PRB 68, 045115 (2003).]
image
Classical diffraction limit comes from
loss of evanescent waves
… can be recovered by
resonant coupling to surface states
2/3 diffraction limit
(needs band gap)
Outline
•
•
•
•
•
•
Preliminaries: waves in periodic media
Photonic crystals in theory and practice
Bulk crystal properties
Intentional defects and devices
Index-guiding and incomplete gaps
Photonic-crystal and microstructured fibers
cavity
waveguide
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
Cavity Modes
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
Help!
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
Cavity Modes
• • •
• • •
• • •
• • •
• • •
• • •
finite region –> discrete 
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
Cavity Modes: Smaller Change
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
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 )
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?
“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) ]
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
Slow Light Enhances Everything
Get a factor of 1/vg enhancement of:
Nonlinearity, gain (e.g. DBR lasers),
magneto-optic effects, loss…
Whoops!
…but device
length decreases by vg too
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.
Outline
•
•
•
•
•
•
Preliminaries: waves in periodic media
Photonic crystals in theory and practice
Bulk crystal properties
Intentional defects and devices
Index-guiding and incomplete gaps
Photonic-crystal and microstructured fibers
How else can we confine light?
Total Internal Reflection
no
ni > no
rays at shallow angles > qc
are totally reflected
sinqc = no / ni
< 1, so qc is real
Snell’s Law:
ni sinqi = no sinqo
qo
qi
i.e. TIR can only guide
within higher index
unlike a band gap
Total Internal Reflection?
no
ni > no
rays at shallow angles > qc
are totally reflected
So, for example,
a discontiguous structure can’t possibly guide by TIR…
the rays can’t stay inside!
Total Internal Reflection?
no
ni > no
rays at shallow angles > qc
are totally reflected
So, for example,
a discontiguous structure can’t possibly guide by TIR…
or can it?
Total Internal Reflection Redux
no
ni > no
ray-optics picture is invalid on  scale
(neglects coherence, near field…)
translational
symmetry
Snell’s Law is really
conservation of k|| and :
|ki| sinqi = |ko| sinqo
|k| = n/c
(wavevector)
(frequency)
qi
k||
qo
conserved!
Waveguide Dispersion Relations
i.e. projected band diagrams

light cone
projection of all k in no
higher-order modes
at larger ,b
higher-index core
pulls down state
no
ni > no
weakly guided (field mostly in no)
k||
(
(a.k.a. b)
)
Strange Total Internal Reflection
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
Index Guiding
0.5
0.45
lig
ht cone
0.4
0.35
frequency (c/a)
0.3
0.25
J
0.2
J
J
a
J
J
0.15
J
0.1
0.05
J
J
J
J
0J
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)
Conser v
ed
+ h
i gher i n
dext o
k
pull down
=
l oca l i ze d/ g
ui ded mode
an
d

s ta t e
.
A Hybrid Photonic Crystal:
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
1d band gap + index guiding
0.5
0.45
l i ght cone
0.4
0.35
range of frequencies
in which there are
“band
no guided modes
gap”
frequency (c/a)
0.3
0.25
J
0.2
J
J
J
J
0.15
J
0.1
0.05
J
J
slow-light band edge
J
J
0J
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)
a
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
A Resonant Cavity
– +
photonic bandgap
index-confined
increased rod radius
pulls down “dipole” mode
(non-degenerate)
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
A Resonant Cavity
– +
photonic bandgap
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
k not conserved
0.5
index-confined
0.45
l i ght cone
0.4
0.35

“band
gap”
The trick is to
keep the
radiation small…
(more on this later)
frequency (c/a)
0.3
0.25
J
0.2
J
J
J
J
0.15
J
0.1
0.05
J
J
J
J
0J
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)
so coupling to
light cone:
radiation
Meanwhile, back in reality…
Air-bridge Resonator: 1d gap + 2d index guiding
5 µm
d
d = 632nm
d = 703nm
bigger cavity
= longer 
[ D. J. Ripin et al., J. Appl. Phys. 87, 1578 (2000) ]
Time for Two Dimensions…
2d is all we really need for many interesting devices
…darn z direction!
How do we make a 2d bandgap?
Most obvious
solution?
make
2d pattern
really tall
How do we make a 2d bandgap?
If height is finite,
we must couple to
out-of-plane wavevectors…
kz not conserved
A 2d band diagram in 3d
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
Let’ s st ar t wi th t h
e 2d
Squ
a r e Lat ti ce of
ban
d dia gr am .
Di el ectr i c Rods
( = 12, r=0.2
This i s what we’ d li ke
a)
EE
J
J
EEE
J
E
JEE
E
JJ JJ E
EE
JE
J JEJEJ J J J J J J J
J
E
E
E
J
E
E
J
JJ E
EEJ J JJ J
JJ
EE
JEJ J
JJ JJ
JJ
J
0.5
J
J
E
J
E
E
E
E
JEJ J J J
E
JJ
E
E
JJ E
E
J
E
J J EEEE
JJ
E
J
E
J
E
JE
JE
J
0.4
E
E
E
E
E
0.3
JJJJJ JJJ E
E
J
J
J
JJJ
J
E JJ
J E
J
0.2
E J
J
J
EJ
JE
TM
bands
0.1 EJ
J
J
TEbands
E
E
J
J
J
J
E
0E
0.6
frequency (c/a)
t ohave i n3d, t oo!
wavevector
A 2d band diagram in 3d
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
Let’ s st ar t wi th t h
e 2d
Squ
a r e Lat ti ce of
ban
d dia gr am .
Di el ectr i c Rods
( = 12, r=0.2
This i s what we’ d li ke
a)
EE
J
J
EEE
J
E
JEE
E
JJ JJ E
EE
JE
J JEJEJ J J J J J J J
J
E
E
E
J
E
E
J
JJ E
EEJ J JJ J
JJ
EE
JEJ J
JJ JJ
JJ
J
0.5
J
J
E
J
E
E
E
E
JEJ J J J
E
JJ
E
E
JJ E
E
J
E
J J EEEE
JJ
E
J
E
J
E
JE
JE
J
0.4
E
E
E
E
E
0.3
JJJJJ JJJ E
E
J
J
J
JJJ
J
E JJ
J E
J
0.2
E J
J
J
EJ
JE
but
this
empty
TM
bands
0.1 EJ
J
J
TEbands looks useful… E
space
E
J
J
J
J
E
0E
0.6
frequency (c/a)
t ohave i n3d, t oo!
projected band diagram fills gap!
wavevector
Photonic-Crystal Slabs
G
are
Qraphics
uickTim
needed
e™
decom
toand
see
pressor
athis picture.
2d photonic bandgap + vertical index guiding
[ S. G. Johnson and J. D. Joannopoulos, Photonic Crystals: The Road from Theory to Practice ]
Rod-Slab Projected Band Diagram
G
X
M
G
M
G
X
Symmetry in a Slab
2d: TM and TE modes
E
E
mirror plane
z=0
slab: odd (TM-like) and even (TE-like) modes
Like in 2d, there may only be a band gap
in one symmetry/polarization
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
Slab Gaps
Squar e Lat ti ce of
Tr ia ngu
l ar Lat ti ce
Di el ectr i c Rods
of Ai r Hol es
( = 12, r=0.2
a, h=2 a)
JJ EE
J E
JE
JEE
JEE
EE
JJ
JEE
E
E
JJ
JJ JEE
E
E
J
J
E
J
EE
E
E
J
J
E
JE
E
J
J
E
JJE
J
J
J
J
JEE
EE
E
J
J
J
J
E
E
light
E
cone
J
E
E
J
J
J
JE
J JE
E
J
E
EE
JJ JJ JJ E
JJ E
EE
E
JJJE
E
J
J
J
J
J
EE
E
J
J J JJ EEE
J
J
J
J
E
E
E
J
E
J
J
J
J
E
JEE
E
E
JEE
E
J J EE
JJ JJJEJE
E
J
E
E
J
J
J
J
J
J
E
J
E
E
EEE
EEEE
J
E
E
E
E
J
E
J
J
E
J
E
E
E
J EE
E E
JJEE
JE
0.5
J JEE
JE
E
J JEJ
JE
EE
JEJ E
JEJE
J J J J J JEJEJE
E
E
J
J
J
J
J
J
E
E
E
E
J EJE
JJJ
JE
JE
JE
EE
J JE
JE
J
J
J JEJE
E
E
J
E
E
J
E
E
JJ JEJEJE EEEE EEEEE
JJE
JE E
J
EJ E
E
JE E
EEEEEE
J
0.4
J
E JE
E
JE E
J JE
E
E
J JE
E
JE EE
JE
E
J
J
J
J
J
J
J
E JE
E
J
JE
E
E
JJ
E
J
J
J
J
JE J
J
E
J
J
0.3
J
J
J
J
JE
E
J EJ J
EE
J
JJ
JJ EJ
E
J J
JJ J
0.2 J E
EE
J
E
E
J
JE
JE J
E
E
J
odd(TM-like) bands
J
0.1 E
even(TE-like) bands
J
E
J
E
J
J
J
E
0E
G
G
X
M
frequency (c/a)
0.6
TM-like gap
( = 12, r=0.3
a, h=0.5 a)
JJ J J J JEE
J J EE
0.5 J JEE
J J
JJ J J JEE
JEJ
EJEE
JEE
E
JJ JJJ E
J EJJ E
J EEE
JJE
E
J
E
E
J
E
E
J
J
J
E
J
J
E
E
E
J
EE
JE
J
EE
J
J
E
EJ E
J Jlight
E
J
J
E
J
J JEJcone
E
J
E
J
E
E
J
J
EE
EE
J
E
JJ E
E
E
JJ E
E
J
J
E
E
J
J
J J EJEE
J
JJ J JEE
E
JJE
JJ JEE J JE
E
JJJ JE
E
JEEJEEE
J
EEJEJ E
E
E
J
J
J
JEE
JEE
J JJ
J JEE
JE J E
E
J
E
E
J
E
E
E
J
J
J
0.4
J JE
EEE
J J JEE
E
JE
JJEJEJEJEJE
E
J
J
J
E
E
J EJ JEJ J
E
E
JEJ
E
E
J
J JEE
JE
EE
E
JJ JJ J
JE
EEEE
EE
J JJ E
J JJ EE
JJ JJ JJ EE
E
J
J
E
J E
J
J
J
J J
J J
0.3
J JE J
J JEJ
J
E
EE EE J J
JE J
EEEEE
EE
EJ
E
J
E
J E
E
JJ E
J
J
E E
J EEJ E
EJJ J
0.2 J E E
EJ E
E JE
E
E E
J
E
E
JE
odd
(TM
-like)
bands
0.1
JE
JE
even(TE-like) bands
JE
JE
J
J
E
0E
G
G
M
K
TE-like gap
Substrates, for the Gravity-Impaired
(rods or holes)
superstrate restores symmetry
substrate
substrate breaks symmetry:
some even/odd mixing “kills” gap
BUT
with strong confinement
(high index contrast)
mixing can be weak
“extruded” substrate
= stronger confinement
(less mixing even
without superstrate
Extruded Rod Substrate
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
QuickTime™ and a
Photo - JPEG decompressor
are needed to see this picture.
high index
S. Assefa, L. A. Kolodziejski
(GaAs on AlOx)
Air-membrane Slabs
who needs a substrate?
AlGaAs
2µm
[ N. Carlsson et al., Opt. Quantum Elec. 34, 123 (2002) ]
Optimal Slab Thickness
~ /2, but /2 in what material?
effective medium theory: effective  depends on polarization
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
35
E
E
E
30
E
25
gap size (%)
E
E
E
E
J
20
J
J
J
E
J
15
even(TE-like) gap
10
E
5
TE “sees” <>
~ high 
J
J
J
odd(TM-like) gap
J
E
TM “sees” <-1>-1
~ low 
J
0
0
0.5
1
1.5
slab thickness (a)
2
2.5
3
Photonic-Crystal Building Blocks
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
point defects
(cavities)
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
line defects
(waveguides)
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
are
G
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
A Reduced-Index Waveguide
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
0.44
0.42
(r=0.10a)
We cannot completely
remove the rods—no
vertical confinement!
frequency (c/a)
0.4
(r=0.12a)
0.38
Still have conserved
wavevector—under the
light cone, no radiation
(r=0.14a)
0.36
(r=0.16a)
0.34
0.32
0.3
(r=0.18a)
(r=0.2a)
0.35
0.4
wavevector k (2š/a)
0.45
0.5
Reduce the radius of a row of
rods to “trap” a waveguide mode
in the gap.
Reduced-Index Waveguide Modes
G
are
Qraphics
uickTim
needed
e™
decom
toand
see
pressor
athis picture.
G
are
Qraphics
uickTim
needed
e™
decom
toand
see
pressor
athis picture.
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
are
G
Qraphics
uickTim
needed
e™
decom
toand
see
pressor
athis picture.
x

x
y
x

z
z
y
y
y
–1
Ez
+1
–1
Hz
+1
Experimental Waveguide & Bend
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
[ E. Chow et al., Opt. Lett. 26, 286 (2001) ]
caution:
can easily be
multi-mode
1.2
t ra
nsmission
bending efficiency
1
0.8
0.6
0.4
0.2
0
1200
E
experiment
theory
EE
EE
E
EE
E
EEE E
E
E E EEwaveguide mode
E
AlO
E
EE
E
E
E
EE
1µm
E
1µm
E
E
EE
E
E
E
E
E
E
E
E
E
E EE E EEEEE
E
E E EE
E
E
EE EEEE
EEEE
E EE EEEE
E EEE
E
E EE E
E
EE
E
E E
E
E
E
EE
E
E
E
E
E
E
EE
E
E
E
E
E
EEE
E
EEEEEEEEEE
E EE
EE
E EEEE
E
E
E
E EEE E EE
bandgap
EE E
SiO2
GaAs
1300
1400
1500
wa
velen
gt h( µm)
1600
1700
1800
Inevitable Radiation Losses
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
whenever translational symmetry is broken
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
e.g. at cavities, waveguide bends, disorder…
coupling to light cone
= radiation losses

(conserved)
k is no longer conserved!
All Is Not Lost
A simple model device (filters, bends, …):
Qr
Qw
Input Waveguide
1
Q
=1
+1
Qr Qw
Q = lifetime/period
= frequency/bandwidth
Cavity
Output Waveguide
We want: Qr >> Qw
1 – transmission ~ 2Q / Qr
worst case: high-Q (narrow-band) cavities
Semi-analytical losses
Make field inside
defect small
= delocalize mode
A low-loss
strategy:
E(x ) 
Make defect weak
= delocalize mode
G
(x
,
x

)

E
(
x

)

D

(
x

)


defect
far-field
(radiation)
Green’s function
(defect-free
system)
defect
near-field
(cavity mode)
Monopole Cavity in a Slab
Lower the  of a single rod: push up
a monopole (singlet) state.
decreasing 
( = 12)
Use small D: delocalized in-plane,
& high-Q (we hope)
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
Delocalized Monopole Q
1,000,000
=11 J
100,000
10,000
Q
r
J =10
J =9
J =8
1,000
J =7
J
=6
100
0.0001
0.001
² f reque
ncyabov
e ba
nd e
dge( c/ a
)
0.01
mid-gap
0.1
Super-defects
Weaker defect with more unit cells.
More delocalized
at the same point in the gap
(i.e. at same bulk decay rate)
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
Super-Defect vs. Single-Defect Q
1,000,000
=11.5
G
=11 J
G
=11
100,000
G
=10
G
=9
10,000
Q
r
J =10
J =9
G
=8
=7 G
J =8
1,000
J =7
J
=6
100
0.0001
0.001
² f reque
ncyabov
e ba
nd e
dge( c/ a
)
0.01
mid-gap
0.1
Super-Defect State
(cross-section)
D = –3, Qrad = 13,000
Ez
(super defect)
still ~localized: In-plane Q|| is > 50,000 for only 4 bulk periods
G
are
Qraphics
uickTim
needed
e™
decom
toand
see
pressor
athis picture.
(in hole slabs, too)
Q= 2500
nea r mi d-gap( ² f re q =0. 0
3)
Ho
le S
l ab

pe
r iod
=1
1. 56
a
, r adiu
s 0.3
a
th
ickne
ss 0
.5
a
Re
ducer ad
ius o
f
7holest o0. 2
a
Ve
ry
(n
ot e
ro
bust
pix
elization
to
ro
ughn
ess
,
a
=10pixels
).
How do we compute Q?
(via 3d FDTD [finite-difference time-domain] simulation)
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
1
excite cavity with dipole source
(broad bandwidth, e.g. Gaussian pulse)
… monitor field at some point
…extract frequencies, decay rates via
signal processing (FFT is suboptimal)
[ V. A. Mandelshtam, J. Chem. Phys. 107, 6756 (1997) ]
Pro: no a priori knowledge, get all ’s and Q’s at once
Con: no separate Qw/Qr, Q > 500,000 hard,
mixed-up field pattern if multiple resonances
How do we compute Q?
(via 3d FDTD [finite-difference time-domain] simulation)
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
2
excite cavity with
narrow-band dipole source
(e.g. temporally broad Gaussian pulse)
— source is at 0 resonance,
which must already be known (via
…measure outgoing power P and energy U
Q = 0 U / P
Pro: separate Qw/Qr, arbitrary Q, also get field pattern
Con: requires separate run 1 to get 0,
long-time source for closely-spaced resonances
1
)
Can we increase Q
without delocalizing?
Semi-analytical losses
Another low-loss
strategy:
E(x ) 
exploit cancellations
from sign oscillations
G
(x
,
x

)

E
(
x

)

D

(
x

)


defect
far-field
(radiation)
Green’s function
(defect-free
system)
defect
near-field
(cavity mode)
Need a more
compact representation
G
are
Qraphics
uickTim
needed
e™
decom
toand
see
pressor
athis picture.
Cannot can
c el
Radi ati onpat te r nf r om
in
f i nit el ym any
lo
c al iz eds ourc e
— u
se
& cance l la r ges t moment
mul t ipol e expans ion
E
…
(
x
)
int eg
r al s
Multipole Expansion
G
are
Qraphics
uickTim
needed
e™
decom
toand
see
pressor
athis picture. [ Jackson, Classical Electrodynamics ]
radiated field =
+
dipole
+…
+
quadrupole
hexapole
Each term’s strength = single integral over near field
…one term is cancellable by tuning one defect parameter
Multipole Expansion
G
are
Qraphics
uickTim
needed
e™
decom
toand
see
pressor
athis picture. [ Jackson, Classical Electrodynamics ]
radiated field =
+
dipole
+…
+
quadrupole
hexapole
peak Q (cancellation) = transition to higher-order radiation
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
Multipoles in a 2d example
– +
photonic bandgap
index-confined
increased rod radius
pulls down “dipole” mode
(non-degenerate)
as we change the radius,  sweeps across the gap
Q = 1,773
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
r = 0.35 a
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
2d multipole
cancellation
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
30,000
J
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
r = 0.40
a
20,000
15,000
Q
10,000
a
Q = 6,624
r = 0.375
Q = 28,700
25,000
J
5,000
0
0.25
J
J
0.29
f re
quen
cy (c/ a)
J
J
0.33
J
J
0.37
cancel a dipole by opposite dipoles…
cancellation comes from
opposite-sign fields in adjacent rods
… changing radius changed balance of dipoles
3d multipole cancellation?
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
enlarge center & adjacent rods
vary side-rod  slightly
for continuous tuning
quadrupole mode
(balance central moment with opposite-sign side rods)
2000
JJ
Q
1500
1000
J
J
500
J
(Ez cross section)
J
0
0.319243
gap bottom
J
J
JJ
0.390536
frequency(c/a)
gap top
3d multipole cancellation
Q = 1925
far field |E|2
near field Ez
Q = 408
nodal planes
(source of high Q)
Q = 426
An Experimental (Laser) Cavity
elongate row of holes
[ M. Loncar et al., Appl. Phys. Lett. 81, 2680 (2002) ]
cavity
Elongation p is a tuning parameter for the cavity…
…in simulations, Q peaks sharply to ~10000 for p = 0.1a
(likely to be a multipole-cancellation effect)
* actually, there are two cavity modes; p breaks degeneracy
An Experimental (Laser) Cavity
elongate row of holes
[ M. Loncar et al., Appl. Phys. Lett. 81, 2680 (2002) ]
Hz (greyscale)
cavity
Elongation p is a tuning parameter for the cavity…
…in simulations, Q peaks sharply to ~10000 for p = 0.1a
(likely to be a multipole-cancellation effect)
* actually, there are two cavity modes; p breaks degeneracy
An Experimental (Laser) Cavity
[ M. Loncar et al., Appl. Phys. Lett. 81, 2680 (2002) ]
cavity
(InGaAsP)
Q ~ 2000 observed from luminescence
quantum-well lasing threshold of 214µW
(optically pumped @830nm, 1% duty cycle)
How can we get arbitrary Q
with finite modal volume?
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
Only one way:
a full 3d band gap
Now there are two ways.
[ M. R. Watts et al., Opt. Lett. 27, 1785 (2002) ]
The Basic Idea, in 2d
G
are
Qraphics
uickTim
needed
e™
decom
toand
see
pressor
athis picture.
G
are
Qraphics
uickTim
needed
e™
decom
toand
see
pressor
athis picture.
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
star t wit h:
jun
ct ionof t wowave
guide
s
G
are
Qraphics
uickTim
needed
e™
decom
toand
see
pressor
athis picture.


1

1

2

1
t ju
a
nct ion
pe
r f ec
t l ym atched

'
2
Nor ad
i at ion
i f t hem o
desar e
'

'
1
Perfect Mode Matching
G
are
Qraphics
uickTim
needed
e™
decom
toand
see
pressor
athis picture.
G
are
Qraphics
uickTim
needed
e™
decom
toand
see
pressor
athis picture.
re
quir e
s:
sa
me
dif er e
nt ialequa
t ions
nd
a
bo
unda
ry c
onditions
kTim
hics
eeded
decom
e™
toand
see
pressor
athis picture.


1
'
1



2
1

'
2

'
1
Mat chdi f er e
nt i a
l eq
uat ions…
…closely related to separability


2



1

'
2


'
1
[ S. Kawakami, J. Lightwave Tech. 20, 1644 (2002) ]
Perfect Mode Matching
G
are
Qraphics
uickTim
needed
e™
decom
toand
see
pressor
athis picture.
G
are
Qraphics
uickTim
needed
e™
decom
toand
see
pressor
athis picture.
re
quir e
s:
sa
me
dif er e
nt ialequa
t ions
nd
a
bo
unda
ry c
onditions
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.


1
'
1



2
1

'
2

'
1
Mat chbou
ndary co
ndi ti ons
:
f iel dm us
t beTE
(
E
ut o
o
f plane, i n2d)
(note switch in TE/TM
convention)
TE modes in 3d
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
ssor
is picture.
fo
r
cy
l i nd
r i ca
l
wave
gui d
es,
“a
zi mut ha
l l ypol a
r i ze
d”
TE
mode
s
0n
A Perfect Cavity in 3d
G
are
Qraphics
uickTim
needed
e™
decom
toand
see
pressor
athis picture.
(~ VCSEL + perfect lateral confinement)
Perfect index
confinement
(no scattering)
+
R
N layers
1d band gap
=
3d confinement
defect
N layers
A Perfectly Confined Mode
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.



,
 ,  
1
2
,
'

'
 ,  1
2

1
Ee
ner g
y de
nsit y
, ve
r t iclaslice
/ 2cor e
Q limited only by finite size
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
100
0000
~ fi xedm o
de v
ol .
J
3

V =( 0.4
)
N
=10
J
100
000
J
J
100
00
Q
J
G
J
G
G
4
4.5
G
G
5.5
6
G
N
=5
JG
100
0
JG
G
J
100
JG
10
1
1.5
2
2.5
cladingR / cor er ad
ius
3
3.5
5
Q-tips
G
are
Qraphics
uickTim
needed
e™
decom
toand
see
pressor
athis picture.
Th
r ee
in
depe
nden
t
mech
ani s
m sf or hi ghQ :
De
l oca
l i za
t i on
:
Q
t radeof f m od
al s
i ze
r ows
g
fo
r Q
mo
not o
ni ca
lly
towards b
andedge
r
Mul t ipol eCan
cel lat i o
n
: f or c
e
Q
e aks
p
in
s i de
higher- or d
er far - fi el d
ap
g
r
New no
dal pl an
es
ModeM atchi n
g
appear i nf ar- f i el d p
a t t er n at pea k
: al l o
ws
Requi re s specia l sym met ry & m at er ia l s
arbi t rar yQ , f i nit e V
at t e
p
rn
Forget these devices…
I just want a mirror.
ok
Projected Bands of a 1d Crystal
(a.k.a. a Bragg mirror)

k
||
conserved
1d band gap
incident light
TM
TE
modes
in crystal
(normal
incidence)
k
||
Omnidirectional Reflection
[ J. N. Winn et al, Opt. Lett. 23, 1573 (1998) ]

in these  ranges,
there is
no overlap
between modes
of air & crystal
all incident light
(any angle, polarization)
is reflected
from flat surface
TM
TE
modes
in crystal
needs: sufficient index contrast & nhi > nlo > 1
k
||
Omnidirectional Mirrors in Practice
[ Y. Fink et al, Science 282, 1679 (1998) ]
Te / polystyrene
contours of omnidirectional gap size
10 0
normal
50
0
D/mid
Reflectance (%)
(%)
Reflectance
10 0
450 s
50
0
10 0
450 p
50
0
10 0
800 s
50
0
10 0
800 p
50
0
6
9
12
Wav
elength (microns)
(microns)
Wavelength
15
Outline
•
•
•
•
•
•
Preliminaries: waves in periodic media
Photonic crystals in theory and practice
Bulk crystal properties
Intentional defects and devices
Index-guiding and incomplete gaps
Photonic-crystal and microstructured fibers
Optical Fibers Today
(not to scale)
losses ~ 0.2 dB/km
more complex profiles
to tune dispersion
“high” index
doped-silica core
n ~ 1.46
silica cladding
n ~ 1.45
at =1.55µm
(amplifiers every
50–100km)
“LP01”
confined mode
field diameter ~ 8µm
protective
polymer
sheath
[ R. Ramaswami & K. N. Sivarajan, Optical Networks: A Practical Perspective ]
but this is
~ as good as
it gets…
The Glass Ceiling: Limits of Silica
Loss: amplifiers every 50–100km
…limited by Rayleigh scattering (molecular entropy)
…cannot use “exotic” wavelengths like 10.6µm
Nonlinearities: after ~100km, cause dispersion, crosstalk, power limits
(limited by mode area ~ single-mode, bending loss)
also cannot be made (very) large for compact nonlinear devices
Radical modifications to dispersion, polarization effects?
…tunability is limited by low index contrast
Long Distances
High Bit-Rates
Compact Devices
Dense Wavelength Multiplexing (DWDM)
Breaking the Glass Ceiling:
Hollow-core Bandgap Fibers
Bragg fiber
1000x better
loss/nonlinear limits
[ Yeh et al., 1978 ]
(from density)
1d
crystal
+ omnidirectional
= OmniGuides
2d
crystal
Photonic Crystal
(You can also
put stuff in here …)
PCF
[ Knight et al., 1998 ]
Breaking the Glass Ceiling II:
Solid-core Holey Fibers
solid core
holey cladding forms
effective
low-index material
Can have much higher contrast
than doped silica…
strong confinement = enhanced
nonlinearities, birefringence, …
[ J. C. Knight et al., Opt. Lett. 21, 1547 (1996) ]
Universal Truths: Bloch’s Theorem
an arbitrary-shaped fiber
(1) Linear, time-invariant system:
(nonlinearities are small correction)
z
frequency  is conserved
cladding
(2)
z-invariant system:
(bends etc. are small correction)
wavenumber b is conserved
(previously called k)
core
electric (E) and magnetic (H) fields can be chosen:
E(x,y) ei(bz – t),
H(x,y) ei(bz – t)
Sequence of Computation
1
Plot all solutions of infinite cladding as  vs. b

“light cone”
b
empty spaces (gaps): guiding possibilities
2
Core introduces new states in empty spaces
— plot (b) dispersion relation
3
Compute other stuff…
Conventional Fiber: Uniform Cladding
c
2
2

b  kt
n
cb

n
uniform cladding, index n
b
kt
(transverse wavevector)

light cone
light line:
=cb/n
b
Conventional Fiber: Uniform Cladding
c
2
2

b  kt
n
cb

n
uniform cladding, index n
b

light cone
higher-order
core with higher index n’
pulls down
index-guided mode(s)
fundamental
 = c b / n'
b
PCF: Periodic Cladding
periodic cladding (x,y)
b
a
Bloch’s Theorem for periodic systems:
fields can be written:
E(x,y) ei(bz+kt xt – t), H(x,y) ei(bz+kt xt – t)
periodic functions
on primitive cell
transverse (xy)
Bloch wavevector kt
primitive cell
1
2
satisfies
 k t ,b   k t ,b  H  2 H

c
eigenproblem
(Hermitian
constraint: 
k t ,b  H  0
if lossless)
where:
k t ,b    ikt  ibzˆ
PCF: Holey Silica Cladding
r = 0.1a
 (2πc/a)
light cone
b (2π/a)
2r
n=1.46
a
PCF: Holey Silica Cladding
r = 0.17717a
 (2πc/a)
light cone
b (2π/a)
2r
n=1.46
a
PCF: Holey Silica Cladding
r = 0.22973a
 (2πc/a)
light cone
b (2π/a)
2r
n=1.46
a
PCF: Holey Silica Cladding
r = 0.30912a
 (2πc/a)
light cone
b (2π/a)
2r
n=1.46
a
PCF: Holey Silica Cladding
r = 0.34197a
 (2πc/a)
light cone
b (2π/a)
2r
n=1.46
a
PCF: Holey Silica Cladding
r = 0.37193a
 (2πc/a)
light cone
b (2π/a)
2r
n=1.46
a
PCF: Holey Silica Cladding
r = 0.4a
 (2πc/a)
light cone
b (2π/a)
2r
n=1.46
a
PCF: Holey Silica Cladding
r = 0.42557a
 (2πc/a)
light cone
b (2π/a)
2r
n=1.46
a
PCF: Holey Silica Cladding
n=1.46
2r
a
r = 0.45a
light cone
 (2πc/a)
gap-guided modes
go here
index-guided modes
go here
b (2π/a)
PCF: Holey Silica Cladding
r = 0.45a
light cone
2r
n=1.46
a
above air line:
 (2πc/a)
guiding in air core
is possible
[ figs: West et al,
Opt. Express 12 (8), 1485 (2004) ]
b (2π/a)
below air line: surface states of air core
PCF Guided Mode(s)
[ J. Broeng et al., Opt. Lett. 25, 96 (2000) ]
2.4
fundamental & 2nd order
 (2πc/a)
guided modes
2.0
fundamental
air-guided
mode
1.6
1.2
0.8
1.11 1.27
bulk
crystal
continuum
1.43 1.59
1.75
1.91
b (2π/a)
2.07 2.23 2.39
Guided Mode in a Solid Core
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
small computation: only lowest- band!
(~ one minute, planewave)
0.12
1.46 – bc/ = 1.46 – neff
holey PCF light cone
flux density
0.1
0.08
0.06
fundamental
mode
0.04
(two polarizations)
2r
0.02
0
0.3
endlessly single mode: Dneff decreases with 
0.4
0.5
0.6
0.7
0.8
/ a
0.9
1
1.1
1.2
n=1.46
a
r = 0.3a
Bragg Fiber Cladding
are
G
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
at large radius,
becomes ~ planar
nhi = 4.6
Bragg fiber gaps (1d eigenproblem)

nlo = 1.6
b
radial kr
(Bloch wavevector)
kf
0 by conservation
of angular momentum
wavenumber b
b = 0: normal incidence
Omnidirectional Cladding
are
G
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
omnidirectional
(planar) reflection
e.g. light from
fluorescent sources
is trapped
Bragg fiber gaps (1d eigenproblem)

for nhi / nlo
big enough
and nlo > 1
[ J. N. Winn et al,
Opt. Lett. 23, 1573 (1998) ]
b
wavenumber b
b = 0: normal incidence
An Easier Problem: Bragg-fiber Modes
In each concentric region,
solutions are Bessel functions:
c Jm (kr) + d Ym(kr)
 eimf
 2
k      b 2
c 
“angular momentum”
At circular interfaces
match boundary conditions
with 4  4 transfer matrix
…search for complex b that satisfies: finite at r=0, outgoing at r=
[ Johnson, Opt. Express 9, 748 (2001) ]
Hollow Metal Waveguides, Reborn
G
are
Qraphics
uickTim
needed
e™
decom
toand
see
pressor
athis picture.
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
metal waveguide modes
OmniGuide fiber modes
frequency 
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
1970’s microwave tubes
@ Bell Labs
wavenumber b
wavenumber b
modes are directly analogous to those in hollow metal waveguide
An Old Friend: the TE01 mode
lowest-loss mode,
just as in metal
(near) node at interface
= strong confinement
= low losses
non-degenerate mode
— cannot be split
= no birefringence or PMD
E
All Imperfections are Small
(or the fiber wouldn’t work)
• Material absorption: small imaginary D
• Nonlinearity: small D ~ |E|2
• Acircularity (birefringence): small  boundary shift
• Bends: small D ~ Dx / Rbend
• Roughness: small D or boundary shift
Weak effects, long distances: hard to compute directly
— use perturbation theory
Perturbation Theory
Given solution for ideal system
compute approximate effect
of small changes
…solves hard problems starting with easy problems
& provides (semi) analytical insight
Perturbation Theory
for Hermitian eigenproblems
Oˆ u  u u
Du & D u for small DOˆ
given eigenvectors/values:
…find change
Solution:
DOˆ
(1)
(2 )
Du  0  Du  Du  
expand as power series in
Du
(1)
u DOˆ u

uu
(1)
& D u  0  D u 
(first order is usually enough)
Perturbation Theory
for electromagnetism
D
c H DAˆ H

2 H H
2
(1)
  D E

2  E2
Db
(1)
 D / vg
(1)
2
…e.g. absorption
gives
imaginary D
= decay!
d
vg 
db
A Quantitative Example
…but what about
the cladding?
Gas can have
low loss
& nonlinearity
…some field
penetrates!
& may need to use
very “bad” material
to get high index contrast
Suppressing Cladding Losses
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
-2
1x1
0
Mode Losses
÷
Bulk Cladding Losses
EH11
-3
1x1
0
Large differential loss
TE01 strongly suppresses
cladding absorption
-4
1x1
0
TE01
(like ohmic loss, for metal)
1x1
0
-5
1.2
1.6
2
 (mm)
2.4
2.8
Quantifying Nonlinearity
Db~ power P ~ 1 / lengthscale for nonlinear effects
g = Db / P
= nonlinear-strength parameter determining
self-phase modulation (SPM), four-wave mixing (FWM), …
(unlike “effective area,”
tells where the field is,
not just how big)
Suppressing Cladding Nonlinearity
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
-6
1x1
0
Mode Nonlinearity*
÷
Cladding Nonlinearity
-7
1x1
0
TE01
Will be dominated by
nonlinearity of air
-8
1x1
0
~10,000 times weaker
than in silica fiber
(including factor of 10 in area)
1x1
0
* “nonlinearity” = Db(1) / P = g
-9
1.2
1.6
2
 (mm)
2.4
2.8
Acircularity & Perturbation Theory
(or any shifting-boundary problem)
D= 1 – 2
2
1
D= 2 – 1
… just plug D’s into
perturbation formulas?
FAILS for high index contrast!
beware field discontinuity…
fortunately, a simple correction exists
[ S. G. Johnson et al.,
PRE 65, 066611 (2002) ]
Acircularity & Perturbation Theory
(or any shifting-boundary problem)
D= 1 – 2
2
D= 2 – 1
1
(continuous field components)
Dh
D
(1)


 surf.
2

1
2
2 
DhD E||  D D 



E
2
[ S. G. Johnson et al.,
PRE 65, 066611 (2002) ]
Yes, but how do you make it?
[ figs courtesy Y. Fink et al., MIT ]
1
3
find compatible materials
(many new possibilities)
2
Make pre-form
(“scale model”)
chalcogenide glass, n ~ 2.8
+ polymer (or oxide), n ~ 1.5
fiber drawing
Fiber Draw Tower @ MIT
building 13, constructed 2000–2001
~6 meter
(20 feet)
research
tower
[ figs courtesy Y. Fink et al., MIT ]
A Drawn Bandgap Fiber
[ figs courtesy Y. Fink et al., MIT ]
• Photonic crystal structural
uniformity, adhesion,
physical durability through
large temperature excursions
white/grey
= chalco/polymer
Band Gap Guidance
Wavevector
Transmission
window can be
shifted by scaling
(different draw speed)
original (blue)
& shifted (red)
transmission:
Transmission (arb. u.)
1.2
0.8
0.4
0.0
10000
[ figs courtesy Y. Fink et al., MIT ]
8000
6000
4000
-1
Wavenumber (cm )
2000
High-Power Transmission
at 10.6µm (no previous dielectric waveguide)
Polymer losses @10.6µm ~ 50,000dB/m…
…waveguide losses ~ 1dB/m
Log. of Trans. (arb. u.)
-3.0
Transmission (arb. u.)
8
6
4
-3.5
[ B. Temelkuran et al.,
Nature 420, 650 (2002) ]
-4.0
slope = -0.95 dB/m
R2 = 0.99
-4.5
2.5
3.0
3.5
Length (meters)
4.0
2
0
5
5
6
6
7
7
8
8
9
9 10 10
10 11 11
11 12 12
12
Wavelength(mm)
(mm)
Wavelength
[ figs courtesy Y. Fink et al., MIT ]
Experimental Air-guiding PCF
Fabrication (e.g.)
silica glass tube (cm’s)
(outer
cladding)
~50 µm
fiber
draw
~1 mm
fuse &
draw
Experimental Air-guiding PCF
[ R. F. Cregan et al., Science 285, 1537 (1999) ]
10µm
5µm
Experimental Air-guiding PCF
[ R. F. Cregan et al., Science 285, 1537 (1999) ]
transmitted intensity
after ~ 3cm
 (c/a) (not 2πc/a)
State-of-the-art air-guiding losses
[Mangan, et al., OFC 2004 PDP24 ]
hollow (air) core (covers 19 holes)
guided field profile:
(flux density)
3.9µm
1.7dB/km
BlazePhotonics
over ~ 800m @1.57µm
State-of-the-art air-guiding losses
larger core =
less field penetrates
cladding
ergo,
roughness etc.
produce lower loss
13dB/km
1.7dB/km
Corning
BlazePhotonics
over ~ 100m @1.5µm
over ~ 800m @1.57µm
[ Smith, et al., Nature 424, 657 (2003) ]
[ Mangan, et al., OFC 2004 PDP24 ]
State-of-the-art air-guiding losses
larger core = more surface states crossing guided mode
… but surface states can be removed by proper crystal termination
[ West, Opt. Express 12 (8), 1485 (2004) ]
100nm
20nm
13dB/km
1.7dB/km
Corning
BlazePhotonics
over ~ 100m @1.5µm
over ~ 800m @1.57µm
[ Smith, et al., Nature 424, 657 (2003) ]
[ Mangan, et al., OFC 2004 PDP24 ]
Index-Guiding PCF & microstructured fiber:
Holey Fibers
solid core
holey cladding forms
effective
low-index material
Can have much higher contrast
than doped silica…
strong confinement = enhanced
nonlinearities, birefringence, …
[ J. C. Knight et al., Opt. Lett. 21, 1547 (1996) ]
Holey Projected Bands, Batman!
 (c/a) (not 2πc/a)
(Schematic)
band gaps
are
unused
bulk
crystal
continuum
guided band
lies below
“crystal light line”
b (a–1)
Holey Fiber PMF
(Polarization-Maintaining Fiber)
birefringence B = Dbc/
= 0.0014
(10 times B of silica PMF)
Loss = 1.3 dB/km @ 1.55µm
over 1.5km
no longer degenerate with
Can operate in a single polarization, PMD = 0
(also, known polarization at output)
[ K. Suzuki, Opt. Express 9, 676 (2001) ]
Nonlinear Holey Fibers:
Supercontinuum Generation
(enhanced by strong confinement + unusual dispersion)
e.g. 400–1600nm “white” light:
from 850nm ~200 fs pulses (4 nJ)
[ W. J. Wadsworth et al., J. Opt. Soc. Am. B 19, 2148 (2002) ]
Endlessly Single-Mode
[ T. A. Birks et al., Opt. Lett. 22, 961 (1997) ]
at higher 
(smaller ),
the light is more
concentrated in silica
…so the effective
index contrast is less
…and the fiber can stay
single mode for all !
http://www.bath.ac.uk/physics/groups/opto
Low Contrast Holey Fibers
[ J. C. Knight et al., Elec. Lett. 34, 1347 (1998) ]
The holes can also form an
effective low-contrast medium
i.e. light is only affected slightly
by small, widely-spaced holes
This yields
large-area, single-mode
fibers (low nonlinearities)
…but bending loss is worse
~ 10 times standard fiber mode diameter
Holey Fiber Losses
Best reported results:
0.28 dB/km @1.55µm
[ Tajima, ECOC 2003 ]
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
TheThe
Upshot
Upshot:
Potential new regimes for fiber operation,
even using very poor materials.
The Story of Photonic Crystals
Finding Materials –> Finding Structures
Further Reading
Reviews:
• J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals:
Molding the Flow of Light (Princeton Univ. Press, 1995).
• S. G. Johnson and J. D. Joannopoulos, Photonic Crystals: The Road from
Theory to Practice (Kluwer, 2002).
• K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).
• P. Russell, “Photonic-crystal fibers,” Science 299, 358 (2003).
This Presentation, Free Software, …
http://ab-initio.mit.edu/photons/tutorial