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1
A Defective Lecture
Photonic CrystalsPeriodic Surprises in
Electromagnetism
Steven G. Johnson MIT
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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
3
Properties of Bulk Crystals
by Blochs theorem
(cartoon)
band diagram (dispersion relation)
photonic band gap
conserved frequency w
conserved wavevector k
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Applications of Bulk Crystals
using near-band-edge effects
Zero group-velocity dw/dk distributed feedback
(DFB) lasers
C. Luo et al., Appl. Phys. Lett. 81, 2352
(2002)
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Cavity Modes
Help!
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Cavity Modes
finite region gt discrete w
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Cavity Modes Smaller Change
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Cavity Modes Smaller Change
Bulk Crystal Band Diagram
frequency (c/a)
L
G
G
X
M
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Cavity Modes Smaller Change
Defect Crystal Band Diagram
frequency (c/a)
L
Defect bands are shifted up (less e)
G
G
X
M
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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)
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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
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Tunable Cavity Modes
frequency (c/a)
Ez
monopole
dipole
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Tunable Cavity Modes
band 1 at M
band 2 at Xs
multiply by exponential decay
Ez
monopole
dipole
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Defect Flavors
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Projected Band Diagrams
1d periodicity
M
X
G
So, plot w vs. kx onlyproject Brillouin zone
onto GX
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Air-waveguide Band Diagram
any state in the gap cannot couple to bulk
crystal gt localized
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(Waveguides dont really need a complete gap)
Fabry-Perot waveguide
Well exploit this later, with photonic-crystal
fiber
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So What?


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Review Why no scattering?
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Benefits of a complete gap
broken symmetry gt reflections only
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Lossless Bends
A. Mekis et al., Phys. Rev. Lett. 77, 3787
(1996)
symmetry single-mode 1d resonances of
100 transmission
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Waveguides Cavities Devices
tunneling
Ugh, must we simulate this to get the basic
behavior?
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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
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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
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A Menagerie of Devices
l
Page 4
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Wide-angle Splitters
S. Fan et al., J. Opt. Soc. Am. B 18, 162
(2001)
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Waveguide Crossings
S. G. Johnson et al., Opt. Lett. 23, 1855
(1998)
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Waveguide Crossings
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Channel-Drop Filters
S. Fan et al., Phys. Rev. Lett. 80, 960 (1998)
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Channel-Drop Filters
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Enough passive, linear devices
Photonic crystal cavities tight confinement (
l/2 diameter) long lifetime (high Q
independent of size) enhanced nonlinear
effects
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A Linear Nonlinear Filter
in
out
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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)
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Enough passive, linear devices
Photonic crystal cavities tight confinement (
l/2 diameter) long lifetime (high Q
independent of size) enhanced nonlinear
effects
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Cavities Cavities Waveguide
tunneling
coupled-cavity waveguide (CCW/CROW) slow light
zero dispersion
A. Yariv et al., Opt. Lett. 24, 711 (1999)
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Enhancing tunability with slow light
M. Soljacic et al., J. Opt. Soc. Am. B 19, 2052
(2002)
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periodicitylight is slowed, but not reflected
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Uh oh, we live in 3d
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2d-like defects in 3d
M. L. Povinelli et al., Phys. Rev. B 64, 075313
(2001)
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3d projected band diagram
frequency (c/a)
frequency (c/a)
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2d-like waveguide mode
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2d-like cavity mode
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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.