Photonic Crystals: Principles and Applications

1
Photonic CrystalsPrinciples and Applications

• Steven G. Johnson
• MIT Applied Mathematics

2
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

3
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

4
To Begin A Cartoon in 2d
5
To Begin A Cartoon in 2d
a
for most l, beam(s) propagate through crystal
without scattering (scattering cancels coherently)
6
Photonic Crystals
periodic electromagnetic media
with photonic band gaps optical insulators
7
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)
8
Photonic Crystals
periodic electromagnetic media
9
Photonic Crystals
periodic electromagnetic media
10
A mystery from the 19th century
conductive material
e
e
11
A mystery from the 19th century
crystalline conductor (e.g. copper)
e
e
12
A mystery solved
13
Electronic and Photonic Crystals
atoms in diamond structure
Periodic Medium
Bloch waves Band Diagram
electron energy
wavevector
interacting hard problem
non-interacting easy problem
14
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
15
Fun with Math
First task get rid of this mess
0
dielectric function e(x) n2(x)
16
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)
17
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)
18
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
19
Periodic Hermitian Eigenproblems in 1d
e1
e2
e1
e2
e1
e2
e1
e2
e1
e2
e1
e2
e(x) e(xa)
a
20
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
21
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
22
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
23
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
24
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
25
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
26
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)
27
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

28
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
29
2d periodicity, e121
Ez
( 90 rotated version)
G
X
M
G
E
gap for n gt 1.751
TM
H
30
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
31
2d photonic crystal TE gap, e121
TE bands
TM bands
E
TE
gap for n gt 1.41
H
32
3d photonic crystal complete gap , e121
I.
II.
gap for n gt 41
S. G. Johnson et al., Appl. Phys. Lett. 77,
3490 (2000)
33
The Mother of (almost) All Bandgaps
The diamond lattice fcc (face-centered-cubic) wi
th two atoms per unit cell
a
Image http//cst-www.nrl.navy.mil/lattice/struk/a
4.html
34
The First 3d Bandgap Structure
K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys.
Rev. Lett. 65, 3152 (1990).
11 gap
overlapping Si spheres
MPB tutorial, http//ab-initio.mit.edu/mpb
35
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
36
A Schematic
M. Qi, H. Smith, MIT
37
7-layer E-Beam Fabrication
M. Qi, H. Smith, MIT
38
1.25 Periods of the Woodpile
S. Y. Lin et al., Nature 394, 251 (1998)
(4 log layers 1 period)
Si
http//www.sandia.gov/media/photonic.htm
39
Two-Photon Lithography
2-photon probability (light intensity)2
e
E0
Atom
40
Lithography is a Beast
S. Kawata et al., Nature 412, 697 (2001)
l 780nm resolution 150nm
7µm
(3 hours to make)
2µm
41
One-PhotonHolographic 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
42
One-PhotonHolographic Lithography
D. N. Sharp et al., Opt. Quant. Elec. 34, 3
(2002)
10µm
huge volumes, long-range periodic, fcc
latticebackfill for high contrast
43
Mass-production II Colloids
(evaporate)
silica (SiO2)
microspheres (diameter lt 1µm)
sediment by gravity into close-packed fcc lattice!
44
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
45
In Order To Forma More Perfect Crystal
figs courtesy D. Norris, UMN
meniscus
silica250nm
Convective Assembly
Nagayama, Velev, et al., Nature (1993) Colvin
et al., Chem. Mater. (1999)

• Capillary forces during drying cause assembly in
  the meniscus
• Extremely flat, large-area opals of controllable
  thickness

46
A Better Opal
fig courtesy D. Norris, UMN
47
Inverse-Opal Photonic Crystal
fig courtesy D. Norris, UMN
Y. A. Vlasov et al., Nature 414, 289 (2001).
48
You, too, can computephotonic eigenmodes!
MIT Photonic-Bands (MPB) package http//ab-initio
.mit.edu/mpb
49
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

50
The Story So Far
Waves in periodic media can have
propagation with no scattering (conserved k)
photonic band gaps (with proper e function)
Eigenproblem gives simple insight
51
Properties of Bulk Crystals
by Blochs theorem
(cartoon)
band diagram (dispersion relation)
photonic band gap
conserved frequency w
conserved wavevector k
52
Superprisms
from divergent dispersion (band curvature)
Kosaka, PRB 58, R10096 (1998).
53
Negative Refraction
Veselago, 1968 negative e, m
opposite of ordinary lens only images close
objects
does not require curved lens
can exceed classical diffraction limit
54
Negative Refractionwith (all-dielectric)
Photonic Crystals
Luo et al, MIT
Here, using positive effective index but negative
effective mass
55
Negative Refractionwith (all-dielectric)
Photonic Crystals
Luo et al, MIT
w contours in (kx,ky) space
Here, using positive effective index but negative
effective mass
56
periodicityunusual dispersion without scattering
57
Super-lensing
Luo, PRB 68, 045115 (2003).
image
Classical diffraction limit comes from loss of
evanescent waves
2/3 diffraction limit
58
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

59
Cavity Modes
Help!
60
Cavity Modes
finite region gt discrete w
61
Cavity Modes Smaller Change
62
Cavity Modes Smaller Change
Bulk Crystal Band Diagram
frequency (c/a)
L
G
G
X
M
63
Cavity Modes Smaller Change
Defect Crystal Band Diagram
frequency (c/a)
L
Defect bands are shifted up (less e)
G
G
X
M
64
Single-Mode Cavity
Bulk Crystal Band Diagram
frequency (c/a)
A point defect can push up a single mode from the
band edge
G
G
X
M
(k not conserved)
65
Single-Mode Cavity
Bulk Crystal Band Diagram
frequency (c/a)
A point defect can pull down a single mode
G
G
X
M
M
(k not conserved)
X
G
66
Tunable Cavity Modes
frequency (c/a)
Ez
monopole
dipole
67
Tunable Cavity Modes
band 1 at M
band 2 at Xs
multiply by exponential decay
Ez
monopole
dipole
68
Defect Flavors
69
Projected Band Diagrams
1d periodicity
M
X
G
So, plot w vs. kx onlyproject Brillouin zone
onto GX
70
Air-waveguide Band Diagram
any state in the gap cannot couple to bulk
crystal gt localized
71
(Waveguides dont really need a complete gap)
Fabry-Perot waveguide
Well exploit this later, with photonic-crystal
fiber
72
So What?


73
Review Why no scattering?
74
Benefits of a complete gap
broken symmetry gt reflections only
75
Lossless Bends
A. Mekis et al., Phys. Rev. Lett. 77, 3787
(1996)
symmetry single-mode 1d resonances of
100 transmission
76
Waveguides Cavities Devices
tunneling
Ugh, must we simulate this to get the basic
behavior?
77
Coupling-of-Modes-in-Time(a form of
coupled-mode theory)
H. Haus, Waves and Fields in Optoelectronics
s1
a
input
output
s1
s2
resonant cavity frequency w0, lifetime t
s2 flux
a2 energy
assumes only exponential decay (strong
confinement) conservation of energy
time-reversal symmetry
78
Coupling-of-Modes-in-Time(a form of
coupled-mode theory)
H. Haus, Waves and Fields in Optoelectronics
s1
a
input
output
s1
s2
resonant cavity frequency w0, lifetime t
s2 flux
a2 energy
1
T Lorentzian filter
transmission T s2 2 / s1 2
w
w0
79
A Menagerie of Devices
l
Page 4
80
Wide-angle Splitters
S. Fan et al., J. Opt. Soc. Am. B 18, 162
(2001)
81
Waveguide Crossings
S. G. Johnson et al., Opt. Lett. 23, 1855
(1998)
82
Waveguide Crossings
83
Channel-Drop Filters
S. Fan et al., Phys. Rev. Lett. 80, 960 (1998)
84
Enough passive, linear devices
Photonic crystal cavities tight confinement (
l/2 diameter) long lifetime (high Q
independent of size) enhanced nonlinear
effects
85
A Linear Nonlinear Filter
in
out
86
A Linear Nonlinear Transistor
Logic gates, switching, rectifiers,
amplifiers, isolators,
feedback
Linear response Lorenzian Transmisson
shifted peak
Bistable (hysteresis) response
Power threshold is near optimal (mW for Si and
telecom bandwidth)
87
Enough passive, linear devices
Photonic crystal cavities tight confinement (
l/2 diameter) long lifetime (high Q
independent of size) enhanced nonlinear
effects
88
Cavities Cavities Waveguide
tunneling
coupled-cavity waveguide (CCW/CROW) slow light
zero dispersion
A. Yariv et al., Opt. Lett. 24, 711 (1999)
89
Enhancing tunability with slow light
M. Soljacic et al., J. Opt. Soc. Am. B 19, 2052
(2002)
90
periodicitylight is slowed, but not reflected
91
Slow Light Enhances Everything
Get a factor of 1/vg enhancement of
Nonlinearity, gain (e.g. DBR lasers), magneto-opt
ic effects, loss
but device length decreases by vg too
92
Uh oh, we live in 3d
93
2d-like defects in 3d
M. L. Povinelli et al., Phys. Rev. B 64, 075313
(2001)
94
3d projected band diagram
frequency (c/a)
frequency (c/a)
95
2d-like waveguide mode
96
2d-like cavity mode
97
The Upshot
To design an interesting device, you need only
single-mode (usually)
symmetry
resonance
(ideally) a band gap to forbid losses
Oh, and a full Maxwell simulator to get Q
parameters, etcetera.
98
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

99
How else can we confine light?
100
Total Internal Reflection
no
ni gt no
sinqc no / ni
lt 1, so qc is real
i.e. TIR can only guide within higher
index unlike a band gap
101
Total Internal Reflection?
no
ni gt no
So, for example, a discontiguous structure cant
possibly guide by TIR
the rays cant stay inside!
102
Total Internal Reflection?
no
ni gt no
So, for example, a discontiguous structure cant
possibly guide by TIR
or can it?
103
Total Internal Reflection Redux
no
ni gt no
ray-optics picture is invalid on l scale
(neglects coherence, near field)
104
Waveguide Dispersion Relationsi.e. projected
band diagrams
w
light cone projection of all k? in no
light line w ck / no
k
( ?)
(a.k.a. b)
105
Strange Total Internal Reflection
a
106
A Hybrid Photonic Crystal1d band gap index
guiding
range of frequencies in which there are no guided
modes
slow-light band edge
a
107
A Resonant Cavity
108
A Resonant Cavity

The trick is to keep the radiation small (more
on this later)
109
Meanwhile, back in reality
Air-bridge Resonator 1d gap 2d index guiding
5 µm
d 703nm
d 632nm
bigger cavity longer l
D. J. Ripin et al., J. Appl. Phys. 87, 1578
(2000)
110
Time for Two Dimensions
2d is all we really need for many interesting
devices darn z direction!
111
How do we make a 2d bandgap?
Most obvious solution? make 2d pattern really
tall
112
How do we make a 2d bandgap?
If height is finite, we must couple
to out-of-plane wavevectors
kz not conserved
113
A 2d band diagram in 3d
114
A 2d band diagram in 3d
115
Photonic-Crystal Slabs
2d photonic bandgap vertical index guiding
S. G. Johnson and J. D. Joannopoulos, Photonic
Crystals The Road from Theory to Practice
116
Rod-Slab Projected Band Diagram
M
X
G
117
Symmetry in a Slab
2d TM and TE modes
118
Slab Gaps
TM-like gap
TE-like gap
119
Substrates, for the Gravity-Impaired
(rods or holes)
superstrate restores symmetry
substrate breaks symmetry some even/odd mixing
kills gap
extruded substrate stronger confinement
BUT with strong confinement (high index
contrast) mixing can be weak
(less mixing even without superstrate
120
Extruded Rod Substrate
(GaAs on AlOx)
121
Air-membrane Slabs
who needs a substrate?
AlGaAs
2µm
N. Carlsson et al., Opt. Quantum Elec. 34, 123
(2002)
122
Optimal Slab Thickness
l/2, but l/2 in what material?
gap size ()
slab thickness (a)
123
Photonic-Crystal Building Blocks
point defects (cavities)
line defects (waveguides)
124
A Reduced-Index Waveguide
(r0.2a)
Reduce the radius of a row of rods to trap a
waveguide mode in the gap.
125
Reduced-Index Waveguide Modes
126
Experimental Waveguide Bend
E. Chow et al., Opt. Lett. 26, 286 (2001)
1µm
bending efficiency
127
Inevitable Radiation Losseswhenever
translational symmetry is broken
e.g. at cavities, waveguide bends, disorder
coupling to light cone radiation losses
w (conserved)
k is no longer conserved!
128
All Is Not Lost
A simple model device (filters, bends, )
worst case high-Q (narrow-band) cavities
129
Semi-analytical losses
far-field (radiation)
defect
Greens function (defect-free system)
near-field (cavity mode)
130
Monopole Cavity in a Slab
Lower the e of a single rod push up a monopole
(singlet) state.
decreasing e
Use small De delocalized in-plane, high-Q
(we hope)
131
Delocalized Monopole Q
e11
e10
e9
e8
e7
e6
mid-gap
132
Super-defects
Weaker defect with more unit cells. More
delocalized at the same point in the gap (i.e. at
same bulk decay rate)
133
Super-Defect vs. Single-Defect Q
e11.5
e11
e11
e10
e10
e9
e9
e8
e7
e8
e7
e6
mid-gap
134
Super-Defect State(cross-section)
De 3, Qrad 13,000
Ez
(super defect)
still localized In-plane Q is gt 50,000 for
only 4 bulk periods
135
(in hole slabs, too)
136
How do we compute Q?
(via 3d FDTD finite-difference time-domain
simulation)
137
How do we compute Q?
(via 3d FDTD finite-difference time-domain
simulation)
excite cavity with narrow-band dipole source
(e.g. temporally broad Gaussian pulse)
source is at w0 resonance, which must already
be known (via )
138
Can we increase Qwithout delocalizing?
139
Semi-analytical losses
Another low-loss strategy
exploit cancellations from sign oscillations
far-field (radiation)
defect
Greens function (defect-free system)
near-field (cavity mode)
140
Need a morecompact representation
141
Multipole Expansion
Jackson, Classical Electrodynamics
radiated field
dipole
quadrupole
hexapole
Each terms strength single integral over near
field
one term is cancellable by tuning one defect
parameter
142
Multipole Expansion
Jackson, Classical Electrodynamics
radiated field
dipole
quadrupole
hexapole
peak Q (cancellation) transition to
higher-order radiation
143
Multipoles in a 2d example
as we change the radius, w sweeps across the gap
144
2d multipolecancellation
145
cancel a dipole by opposite dipoles
cancellation comes from opposite-sign fields in
adjacent rods changing radius changed balance
of dipoles
146
3d multipole cancellation?
enlarge center adjacent rods
quadrupole mode
vary side-rod e slightly for continuous tuning
(balance central moment with opposite-sign side
rods)
(Ez cross section)
gap top
gap bottom
147
3d multipole cancellation
Q 408
Q 426
Q 1925
near field Ez
far field E2
nodal planes (source of high Q)
148
An Experimental (Laser) Cavity
M. Loncar et al., Appl. Phys. Lett. 81, 2680
(2002)
elongate row of holes
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
149
An Experimental (Laser) Cavity
M. Loncar et al., Appl. Phys. Lett. 81, 2680
(2002)
elongate row of holes
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
150
An Experimental (Laser) Cavity
M. Loncar et al., Appl. Phys. Lett. 81, 2680
(2002)
cavity
(InGaAsP)
quantum-well lasing threshold of 214µW (optically
pumped _at_830nm, 1 duty cycle)
151
How can we get arbitrary Qwith finite modal
volume?
152
The Basic Idea, in 2d
153
Perfect Mode Matching
closely related to separability S. Kawakami,
J. Lightwave Tech. 20, 1644 (2002)
154
Perfect Mode Matching
(note switch in TE/TM convention)
155
TE modes in 3d
156
A Perfect Cavity in 3d
( VCSEL perfect lateral confinement)
157
A Perfectly Confined Mode
158
Q limited only by finite size
159
Q-tips
160
Forget these devices
I just want a mirror.
ok
161
Projected Bands of a 1d Crystal(a.k.a. a Bragg
mirror)
incident light
light line of air w ck
k conserved
162
Omnidirectional Reflection
J. N. Winn et al, Opt. Lett. 23, 1573 (1998)
w
in these w ranges, there is no overlap between
modes of air crystal
light line of air w ck
TM
TE
modes in crystal
k
needs sufficient index contrast nhi gt nlo gt 1
163
Omnidirectional Mirrors in Practice
Y. Fink et al, Science 282, 1679 (1998)
Te / polystyrene
contours of omnidirectional gap size
Reflectance ()
Dl/lmid
164
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

165
Optical Fibers Today(not to scale)
silica cladding n 1.45
R. Ramaswami K. N. Sivarajan, Optical
Networks A Practical Perspective
166
The Glass Ceiling Limits of Silica
Radical modifications to dispersion, polarization
effects? tunability is limited by low index
contrast
High Bit-Rates
Compact Devices
Long Distances
Dense Wavelength Multiplexing (DWDM)
167
Breaking the Glass CeilingHollow-core Bandgap
Fibers
Photonic Crystal
168
Breaking the Glass Ceiling IISolid-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)

169
Universal Truths Blochs Theorem
an arbitrary-shaped fiber
(1) Linear, time-invariant system (nonlinearities
are small correction)
z
frequency w 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 wt),
H(x,y) ei(bz wt)
170
Sequence of Computation
Plot all solutions of infinite cladding as w vs. b
1
w
light cone
b
empty spaces (gaps) guiding possibilities
Core introduces new states in empty spaces
plot w(b) dispersion relation
2
Compute other stuff
3
171
Conventional Fiber Uniform Cladding
uniform cladding, index n
kt
b
(transverse wavevector)
w
light cone
light line w c b / n
b
172
Conventional Fiber Uniform Cladding
uniform cladding, index n
b
w
light cone
higher-order
core with higher index n pulls down index-guided
mode(s)
fundamental
w c b / n'
b
173
PCF Periodic Cladding
periodic cladding e(x,y)
Blochs Theorem for periodic systems
fields can be written
b
a
E(x,y) ei(bzkt xt wt),
H(x,y) ei(bzkt xt wt)
transverse (xy) Bloch wavevector kt
periodic functions on primitive cell
primitive cell
where
satisfies eigenproblem (Hermitian if lossless)
constraint
174
PCF Holey Silica Cladding
n1.46
2r
a
r 0.1a
light cone
w (2pc/a)
w bc
b (2p/a)
175
PCF Holey Silica Cladding
n1.46
2r
a
r 0.17717a
light cone
w (2pc/a)
w bc
b (2p/a)
176
PCF Holey Silica Cladding
n1.46
2r
a
r 0.22973a
light cone
w (2pc/a)
w bc
b (2p/a)
177
PCF Holey Silica Cladding
n1.46
2r
a
r 0.30912a
light cone
w (2pc/a)
w bc
b (2p/a)
178
PCF Holey Silica Cladding
n1.46
2r
a
r 0.34197a
light cone
w (2pc/a)
w bc
b (2p/a)
179
PCF Holey Silica Cladding
n1.46
2r
a
r 0.37193a
light cone
w (2pc/a)
w bc
b (2p/a)
180
PCF Holey Silica Cladding
n1.46
2r
a
r 0.4a
light cone
w (2pc/a)
w bc
b (2p/a)
181
PCF Holey Silica Cladding
n1.46
2r
a
r 0.42557a
light cone
w (2pc/a)
w bc
b (2p/a)
182
PCF Holey Silica Cladding
n1.46
2r
a
r 0.45a
light cone
w (2pc/a)
w bc
b (2p/a)
183
PCF Holey Silica Cladding
n1.46
2r
a
r 0.45a
light cone
w (2pc/a)
air light line w bc
b (2p/a)
figs West et al, Opt. Express 12 (8), 1485
(2004)
184
PCF Guided Mode(s)
J. Broeng et al., Opt. Lett. 25, 96 (2000)
2.4
fundamental 2nd order guided modes
2.0
air light line
w (2pc/a)
fundamental air-guided mode
1.6
bulk crystal continuum
1.2
0.8
1.11 1.27 1.43 1.59 1.75 1.91
2.07 2.23 2.39
b (2p/a)
185
Guided Mode in a Solid Core
small computation only lowest-w band!
( one minute, planewave)
holey PCF light cone
flux density
1.46 bc/w 1.46 neff
fundamental mode (two polarizations)
n1.46
2r
endlessly single mode Dneff decreases with l
a
r 0.3a
l / a
186
Bragg Fiber Cladding
Bragg fiber gaps (1d eigenproblem)
at large radius, becomes planar
w
nhi 4.6
nlo 1.6
b
radial kr (Bloch wavevector)
0 by conservation of angular momentum
kf
wavenumber b
b 0 normal incidence
187
Omnidirectional Cladding
Bragg fiber gaps (1d eigenproblem)
w
omnidirectional (planar) reflection
e.g. light from fluorescent sources is trapped
for nhi / nlo big enough and nlo gt 1
J. N. Winn et al, Opt. Lett. 23, 1573 (1998)
wavenumber b
b
b 0 normal incidence
188
An Easier Problem Bragg-fiber Modes
In each concentric region, solutions are Bessel
functions c Jm (kr) d Ym(kr) ? eimf
angular momentum
At circular interfaces match boundary
conditions with 4 ? 4 transfer matrix
search for complex b that satisfies finite at
r0, outgoing at r?
Johnson, Opt. Express 9, 748 (2001)
189
Hollow Metal Waveguides, Reborn
OmniGuide fiber modes
wavenumber b
190
An Old Friend the TE01 mode
lowest-loss mode, just as in metal
non-degenerate mode cannot be split no
birefringence or PMD
191
All Imperfections are Small
(or the fiber wouldnt work)
Material absorption small imaginary De
Nonlinearity small De E2
Acircularity (birefringence) small e boundary
shift
Bends small De Dx / Rbend
Roughness small De or boundary shift
Weak effects, long distances hard to compute
directly use perturbation theory
192
Perturbation Theory
Given solution for ideal system compute
approximate effect of small changes
solves hard problems starting with easy
problems provides (semi) analytical insight
193
Perturbation Theoryfor Hermitian eigenproblems
given eigenvectors/values
find change for small
194
Perturbation Theoryfor electromagnetism
e.g. absorption gives imaginary Dw decay!
195
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
196
Suppressing Cladding Losses
Mode Losses Bulk Cladding Losses
EH11
Large differential loss
TE01 strongly suppresses cladding
absorption (like ohmic loss, for metal)
TE01
l (mm)
197
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)
198
Suppressing Cladding Nonlinearity
Mode Nonlinearity Cladding Nonlinearity
TE01
Will be dominated by nonlinearity of air 10,000
times weaker than in silica fiber (including
factor of 10 in area)
l (mm)
nonlinearity Db(1) / P g
199
Acircularity Perturbation Theory
(or any shifting-boundary problem)
e2
e1
just plug Des 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)
200
Acircularity Perturbation Theory
(or any shifting-boundary problem)
e2
e1
(continuous field components)
Dh
S. G. Johnson et al., PRE 65, 066611 (2002)
201
Yes, but how do you make it?
figs courtesy Y. Fink et al., MIT
202
Fiber Draw Tower _at_ MITbuilding 13, constructed
20002001
6 meter (20 feet) research tower
figs courtesy Y. Fink et al., MIT
203
A Drawn Bandgap Fiber
figs courtesy Y. Fink et al., MIT

• Photonic crystal structural uniformity, adhesion,
  physical durability through large temperature
  excursions

204
Band Gap Guidance
Transmission window can be shifted by
scaling (different draw speed)
original (blue) shifted (red) transmission
figs courtesy Y. Fink et al., MIT
205
High-Power Transmissionat 10.6µm (no previous
dielectric waveguide)
Polymer losses _at_10.6µm 50,000dB/m
waveguide losses 1dB/m
B. Temelkuran et al., Nature 420, 650 (2002)
figs courtesy Y. Fink et al., MIT
206
High-Power Transmissionat 10.6µm (no previous
dielectric waveguide)
207
Experimental Air-guiding PCF
Fabrication (e.g.)
208
Experimental Air-guiding PCF
R. F. Cregan et al., Science 285, 1537 (1999)
10µm
5µm
209
Experimental Air-guiding PCF
R. F. Cregan et al., Science 285, 1537 (1999)
transmitted intensity after 3cm
w (c/a) (not 2pc/a)
210
State-of-the-art air-guiding losses
Mangan, et al., OFC 2004 PDP24
hollow (air) core (covers 19 holes)
guided field profile (flux density)
1.7dB/km BlazePhotonics over 800m _at_1.57µm
3.9µm
211
State-of-the-art air-guiding losses
larger core less field penetrates cladding
ergo, roughness etc. produce lower loss
13dB/km Corning over 100m _at_1.5µm Smith, et
al., Nature 424, 657 (2003)
1.7dB/km BlazePhotonics over 800m _at_1.57µm
Mangan, et al., OFC 2004 PDP24
212
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 Corning over 100m _at_1.5µm Smith, et
al., Nature 424, 657 (2003)
1.7dB/km BlazePhotonics over 800m _at_1.57µm
Mangan, et al., OFC 2004 PDP24
213
Index-Guiding PCF microstructured fiberHoley
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)

214
Holey Projected Bands, Batman!
(Schematic)
band gaps are unused
bulk crystal continuum
w (c/a) (not 2pc/a)
guided band lies below crystal light line
b (a1)
215
Holey Fiber PMF (Polarization-Maintaining Fiber)
birefringence B Dbc/w 0.0014 (10 times B of
silica PMF)
Loss 1.3 dB/km _at_ 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)
216
Nonlinear Holey Fibers
Supercontinuum Generation
(enhanced by strong confinement unusual
dispersion)
e.g. 4001600nm white light
from 850nm 200 fs pulses (4 nJ)
W. J. Wadsworth et al., J. Opt. Soc. Am. B 19,
2148 (2002)
217
Endlessly Single-Mode
T. A. Birks et al., Opt. Lett. 22, 961 (1997)
at higher w (smaller l), the light is
more concentrated in silica
so the effective index contrast is less
and the fiber can stay single mode for all l!
http//www.bath.ac.uk/physics/groups/opto
218
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
219
Holey Fiber Losses
Best reported results 0.28 dB/km _at_1.55µm
Tajima, ECOC 2003
220
The Upshot
221
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