Complete Band Gaps: You can leave home without them.

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Complete Band GapsYou can leave home without
them.
Photonic CrystalsPeriodic Surprises in
Electromagnetism
Steven G. Johnson MIT
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How else can we confine light?
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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
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Total Internal Reflection?
no
ni gt no
So, for example, a discontiguous structure cant
possibly guide by TIR
the rays cant stay inside!
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Total Internal Reflection?
no
ni gt no
So, for example, a discontiguous structure cant
possibly guide by TIR
or can it?
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Total Internal Reflection Redux
no
ni gt no
ray-optics picture is invalid on l scale
(neglects coherence, near field)
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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)
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Strange Total Internal Reflection
a
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A Hybrid Photonic Crystal1d band gap index
guiding
range of frequencies in which there are no guided
modes
slow-light band edge
a
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A Resonant Cavity
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A Resonant Cavity

The trick is to keep the radiation small (more
on this later)
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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)
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Time for Two Dimensions
2d is all we really need for many interesting
devices darn z direction!
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How do we make a 2d bandgap?
Most obvious solution? make 2d pattern really
tall
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How do we make a 2d bandgap?
If height is finite, we must couple
to out-of-plane wavevectors
kz not conserved
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A 2d band diagram in 3d
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A 2d band diagram in 3d
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Photonic-Crystal Slabs
2d photonic bandgap vertical index guiding
S. G. Johnson and J. D. Joannopoulos, Photonic
Crystals The Road from Theory to Practice
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Rod-Slab Projected Band Diagram
M
X
G
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Symmetry in a Slab
2d TM and TE modes
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Slab Gaps
TM-like gap
TE-like gap
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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
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Extruded Rod Substrate
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Air-membrane Slabs
who needs a substrate?
AlGaAs
2µm
N. Carlsson et al., Opt. Quantum Elec. 34, 123
(2002)
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Optimal Slab Thickness
l/2, but l/2 in what material?
gap size ()
slab thickness (a)
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Photonic-Crystal Building Blocks
point defects (cavities)
line defects (waveguides)
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A Reduced-Index Waveguide
(r0.2a)
Reduce the radius of a row of rods to trap a
waveguide mode in the gap.
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Reduced-Index Waveguide Modes
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Experimental Waveguide Bend
E. Chow et al., Opt. Lett. 26, 286 (2001)
1µm
bending efficiency
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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!
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All Is Not Lost
A simple model device (filters, bends, )
worst case high-Q (narrow-band) cavities
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Semi-analytical losses
far-field (radiation)
defect
Greens function (defect-free system)
near-field (cavity mode)
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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)
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Delocalized Monopole Q
e11
e10
e9
e8
e7
e6
mid-gap
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Super-defects
Weaker defect with more unit cells. More
delocalized at the same point in the gap (i.e. at
same bulk decay rate)
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Super-Defect vs. Single-Defect Q
e11.5
e11
e11
e10
e10
e9
e9
e8
e7
e8
e7
e6
mid-gap
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Super-Defect vs. Single-Defect Q
e11.5
e11
e11
e10
e10
e9
e9
e8
e7
e8
e7
e6
mid-gap
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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
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(in hole slabs, too)
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How do we compute Q?
(via 3d FDTD finite-difference time-domain
simulation)
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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 )
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Can we increase Qwithout delocalizing?
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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)
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Need a morecompact representation
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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
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Multipole Expansion
Jackson, Classical Electrodynamics
radiated field
dipole
quadrupole
hexapole
peak Q (cancellation) transition to
higher-order radiation
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Multipoles in a 2d example
as we change the radius, w sweeps across the gap
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2d multipolecancellation
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cancel a dipole by opposite dipoles
cancellation comes from opposite-sign fields in
adjacent rods changing radius changed balance
of dipoles
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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
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3d multipole cancellation
Q 408
Q 426
Q 1925
near field Ez
far field E2
nodal planes (source of high Q)
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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
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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
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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)
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How can we get arbitrary Qwith finite modal
volume?
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The Basic Idea, in 2d
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Perfect Mode Matching
closely related to separability S. Kawakami,
J. Lightwave Tech. 20, 1644 (2002)
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Perfect Mode Matching
(note switch in TE/TM convention)
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TE modes in 3d
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A Perfect Cavity in 3d
( VCSEL perfect lateral confinement)
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A Perfectly Confined Mode
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Q limited only by finite size
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Q-tips
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Forget these devices
I just want a mirror.
ok
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Projected Bands of a 1d Crystal(a.k.a. a Bragg
mirror)
incident light
light line of air w ck
k conserved
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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
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Omnidirectional Mirrors in Practice
Y. Fink et al, Science 282, 1679 (1998)
Te / polystyrene
contours of omnidirectional gap size
Reflectance ()
Dl/lmid