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Complete Band Gaps:
You can leave home without them.
Photonic Crystals:Periodic Surprises in Electromagnetism
Steven G. Johnson
MIT
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How else can we confine light?
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Total Internal Reflection
ni > no
no
rays at shallow angles > care totally reflected
Snells Law:
io
ni sini = no sino
sinc = no / ni< 1, so c is real
i.e. TIR can only guidewithin higher indexunlike a band gap
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Total Internal Reflection?
ni > no
no
rays at shallow angles > care totally reflected
So, for example,a discontiguous structure cant possibly guide by TIR
the rays cant stay inside!
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Total Internal Reflection?
ni > no
no
rays at shallow angles > care totally reflected
So, for example,a discontiguous structure cant possibly guide by TIR
or can it?
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Total Internal Reflection Redux
ni > no
no
ray-optics picture is invalid on scale (neglects coherence, near field)
Snells Law is reallyconservation of k|| and :
io
|ki| sini = |ko| sino|k| = n/c
(wavevector) (frequency)
k||
translationalsymmetry
conserved!
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Waveguide Dispersion Relationsi.e. projected band diagrams
ni > no
no
k||
light lin
e: = c
k / no
light coneprojection of all k in no
(a.k.a. )
= ck / ni
higher-index corepulls down state
( )
higher-order modesat larger ,
weakly guided (field mostly in no)
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freq
uenc
y (c
/a)
wavenumber k (2/a)
light cone
Conserved k and + higher index to pull down state
= localized/guided mode.
Strange Total Internal ReflectionIndex Guiding
a
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A Hybrid Photonic Crystal:1d band gap + index guiding
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freq
uenc
y (c
/a)
wavenumber k (2/a)
light cone
band gap
a
range of frequenciesin which there are
no guided modes
slow-light band edge
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A Resonant Cavity
index-confined
photonic band gap
increased rod radiuspulls down dipole mode
(non-degenerate)
+
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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
freq
uenc
y (c
/a)
wavenumber k (2/a)
light cone
band gap
A Resonant Cavity
index-confined
photonic band gap
+
k not conservedso coupling to
light cone:
radiationThe trick is to
keep theradiation small
(more on this later)
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Meanwhile, back in reality
5 m
[ D. J. Ripin et al., J. Appl. Phys. 87, 1578 (2000) ]
d = 703nmd = 632nmd
Air-bridge Resonator: 1d gap + 2d index guiding
bigger cavity= longer
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Time for Two Dimensions
2d is all we really need for many interesting devicesdarn z direction!
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How do we make a 2d bandgap?
Most obvioussolution?
make2d patternreally 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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freq
uenc
y (c
/a)
TE bandsTM bands
Square Lattice ofDielectric Rods
( = 12, r=0.2a)
Lets start with the 2dband diagram.
This is what wed liketo have in 3d, too!
wavevector
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A 2d band diagram in 3d
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freq
uenc
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/a)
TE bandsTM bands
Square Lattice ofDielectric Rods
( = 12, r=0.2a)
Lets start with the 2dband diagram.
This is what wed liketo have in 3d, too!
wavevector
3D Structure:
No! When we includeout-of-plane propagation,
we get:
wavevectorfrequency
+
projected band diagram fills gap!
but this emptyspace looks useful
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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
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freq
uenc
y (c
/a)
even (TE-like) bandsodd (TM-like) bands
Square Lattice ofDielectric Rods( = 12, r=0.2a, h=2a)
light cone
The Light Cone:All possible statespropagating in the air
The Guided Modes:Cannot couple tothe light cone > confined to the slab
Thickness is critical.Should be about /2 (to have a gap & be single-mode)
X M X
M
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Symmetry in a Slab
2d: TM and TE modes
slab: odd (TM-like) and even (TE-like) modes
mirror planez = 0
rE
rE
Like in 2d, there may only be a band gapin one symmetry/polarization
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Slab Gaps
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0
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freq
uenc
y (c
/a)
even (TE-like) bandsodd (TM-like) bands
even (TE-like) bandsodd (TM-like) bands
X M M K
Square Lattice ofDielectric Rods( = 12, r=0.2a, h=2a)
Triangular Latticeof Air Holes
( = 12, r=0.3a, h=0.5a)
light cone light cone
TM-like gap TE-like gap
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Substrates, for the Gravity-Impaired
substrate
(rods or holes)
substrate breaks symmetry:some even/odd mixing kills gap
BUTwith strong confinement
(high index contrast)
mixing can be weak
superstrate restores symmetry
extruded substrate= stronger confinement
(less mixing evenwithout superstrate
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Extruded Rod Substrate
S. Assefa, L. A. Kolodziejski
high index
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Air-membrane Slabs
[ N. Carlsson et al., Opt. Quantum Elec. 34, 123 (2002) ]
who needs a substrate?
2m
AlGaAs
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Optimal Slab Thickness~ /2, but /2 in what material?
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0
5
10
15
20
25
30
35
0 0.5 1 1.5 2 2.5 3
even (TE-like) gap odd (TM-like) gap
slab thickness (a)
gap
size
(%
)
TM sees -1~ low
TE sees ~ high
effective medium theory: effective depends on polarization
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Photonic-Crystal Building Blocks
point defects(cavities)
line defects(waveguides)
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0.32
0.34
0.36
0.38
0.4
0.42
0.44
0.3 0.35 0.4 0.45 0.5
freq
uenc
y (c
/a)
wavevector k (2/a)
A Reduced-Index Waveguide
Reduce the radius of a row ofrods to trap a waveguide modein the gap.
(r=0.2a)
(r=0.18a)
(r=0.16a)
(r=0.12a)
(r=0.14a)
Still have conservedwavevectorunder thelight cone, no radiation
(r=0.10a)
We cannot completelyremove the rodsnovertical confinement!
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Reduced-Index Waveguide Modes
x
z
y
x
1 +1Ez
z
y x
z
y
x
1 +1Hz
z
y
-
Experimental Waveguide & Bend
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0
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1
1.2
1200 1300 1400 1500 1600 1700 1800
tran
smis
sion
wavelength (m)
E experiment
theory
band gap
waveguide mode
[ E. Chow et al., Opt. Lett. 26, 286 (2001) ]
1m 1m
GaAs
AlO
SiO2
be
nd
ing
eff
icie
ncy
caution:can easily bemulti-mode
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Inevitable Radiation Losseswhenever translational symmetry is broken
e.g. at cavities, waveguide bends, disorder
k is no longer conserved!
(conserved)
coupling to light cone= radiation losses
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All Is Not Lost
Qw
A simple model device (filters, bends, ):
Qr
Q1
Qr1
Qw1= +
Q = lifetime/period = frequency/bandwidth
We want: Qr >> Qw
1 transmission ~ 2Q / Qr
worst case: high-Q (narrow-band) cavities
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Semi-analytical losses
r r t r r r r rE x G x x E x x
defect
( ) ( , ) ( ) ( )=
far-field(radiation) Greens function
(defect-freesystem)
near-field(cavity mode)
defect
A low-lossstrategy:
Make field insidedefect small
= delocalize mode
Make defect weak= delocalize mode
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( = 12)
Monopole Cavity in a Slab
decreasing
Lower the of a single rod: push upa monopole (singlet) state.
Use small : delocalized in-plane, & high-Q (we hope)
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1,000
10,000
100,000
1,000,000
0.0001 0.001 0.01 0.1
Qr
frequency above band edge (c/a)
Delocalized Monopole Q
=6
=7
=8
=9
=10
=11
mid-gap
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Super-defects
Weaker defect with more unit cells.
More delocalizedat the same point in the gap(i.e. at same bulk decay rate)
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Super-Defect vs. Single-Defect Q
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1,000
10,000
100,000
1,000,000
0.0001 0.001 0.01 0.1
Qr
frequency above band edge (c/a)
=6
=7
=8
=9
=10
=11
=7
=8
=9
=10
=11
=11.5
mid-gap
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Super-Defect vs. Single-Defect Q
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1,000
10,000
100,000
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0.0001 0.001 0.01 0.1
Qr
frequency above band edge (c/a)
=6
=7
=8
=9
=10
=11
=7
=8
=9
=10
=11
=11.5
mid-gap
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Super-Defect State(cross-section)
= 3, Qrad = 13,000
(super defect)
still ~localized: In-plane Q|| is > 50,000 for only 4 bulk periods
Ez
-
(in hole slabs, too)
Hole Slab=11.56
period a, radius 0.3athickness 0.5a
Reduce radius of7 holes to 0.2a
Q = 2500near mid-gap (freq = 0.03)
Very robust to roughness(note pixellization, a = 10 pixels).
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How do we compute Q?
1excite cavity with dipole source
(broad bandwidth, e.g. Gaussian pulse)
monitor field at some point
(via 3d FDTD [finite-difference time-domain] simulation)
extract frequencies, decay rates viasignal processing (FFT is suboptimal)
[ V. A. Mandelshtam, J. Chem. Phys. 107, 6756 (1997) ]
Pro: no a priori knowledge, get all s and Qs at once
Con: no separate Qw/Qr, Q > 500,000 hard, mixed-up field pattern if multiple resonances
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How do we compute Q?
2excite cavity with
narrow-band dipole source(e.g. temporally broad Gaussian pulse)
(via 3d FDTD [finite-difference time-domain] simulation)
source is at 0 resonance,which must already be known (via )1
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 to get 0, long-time source for closely-spaced resonances
1
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Can we increase Qwithout delocalizing?
-
Semi-analytical losses
r r t r r r r rE x G x x E x x
defect
( ) ( , ) ( ) ( )=
far-field(radiation) Greens function
(defect-freesystem)
near-field(cavity mode)
defect
Another low-lossstrategy:
exploit cancellationsfrom sign oscillations
-
Need a morecompact representation
Cannot cancel infinitely many E(x) integrals
Radiation pattern from localized source
use multipole expansion & cancel largest moment
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Multipole Expansion[ Jackson, Classical Electrodynamics ]
+ + +
radiated field =
dipole quadrupole hexapole
Each terms strength = single integral over near fieldone 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
index-confined
photonic band gap
increased rod radiuspulls down dipole mode
(non-degenerate)
+
as we change the radius, sweeps across the gap
-
2d multipolecancellation
JJJ
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0
5,000
10,000
15,000
20,000
25,000
30,000
0.25 0.29 0.33 0.37
Q
frequency (c/a)
r = 0.35a
Q =
1,7
73
r = 0.40a
Q =
6,6
24
r = 0.375a
Q =
28,
700
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cancel a dipole by opposite dipoles
cancellation comes fromopposite-sign fields in adjacent rods
changing radius changed balance of dipoles
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3d multipole cancellation?
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0
500
1000
1500
2000
0.319243 0.390536
Q
frequency (c/a) gap topgap bottom
enlarge center & adjacent rods
vary side-rod slightlyfor continuous tuning
quadrupole mode
(Ez cross section)
(balance central moment with opposite-sign side rods)
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3d multipole cancellationne
ar f
ield
E zfa
r fie
ld |E
|2
Q = 408 Q = 426Q = 1925
nodal planes(source of high Q)
-
An Experimental (Laser) Cavity[ M. Loncar et al., Appl. Phys. Lett. 81, 2680 (2002) ]
elon
gate
row
of
hole
s
Elongation p is a tuning parameter for the cavity
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) ]
elon
gate
row
of
hole
s
Elongation p is a tuning parameter for the cavity
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
Hz (greyscale)
-
An Experimental (Laser) Cavity[ M. Loncar et al., Appl. Phys. Lett. 81, 2680 (2002) ]
cavity
quantum-well lasing threshold of 214W(optically pumped @830nm, 1% duty cycle)
(InGaAsP)
Q ~ 2000 observed from luminescence
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How can we get arbitrary Qwith finite modal volume?
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
12 < 1
1'
2' < 1'
start with:junction of two waveguides
No radiation at junctionif the modes are perfectly matched
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Perfect Mode Matching
12 < 1
1'
2' < 1'
requires:same differential equations and boundary conditions
Match differential equations2 1 = 2' 1' closely related to separability[ S. Kawakami, J. Lightwave Tech. 20, 1644 (2002) ]
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Perfect Mode Matching
12 < 1
1'
2' < 1'
requires:same differential equations and boundary conditions
Match boundary conditions:field must be TE
(E out of plane, in 2d)
(note switch in TE/TMconvention)
-
TE modes in 3d
for
cylindrical waveguides,
azimuthally polarized
TE0n modes
-
A Perfect Cavity in 3d
R
N layers
N layers
defect
Perfect indexconfinement
(no scattering)
+1d band gap
=3d confinement
(~ VCSEL + perfect lateral confinement)
-
A Perfectly Confined Mode
1', 2' = 4, 1
1, 2 = 9, 6/2 core
E energy density, vertical slice
-
Q limited only by finite size
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GG G
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J
J
J
J
J
J
10
100
1000
10000
100000
1000000
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Q
cladding R / core radius
N = 5
N = 10
~ fixed mode vol.V = (0.4)3
-
Q-tipsThree independent mechanisms for high Q:
Delocalization : trade off modal size for Q
Multipole Cancellation : force higher-order far-field pattern
Qr grows monotonically towards band edge
Qr peaks inside gap
New nodal planes appear in far-field pattern at peak
Mode Matching : allows arbitrary Q, finite V
Requires special symmetry & materials
-
Forget these devices
I just want a mirror.
ok
-
k||
modesin crystal
TE
TM
Projected Bands of a 1d Crystal(a.k.a. a Bragg mirror)
incident light
k|| conserved
(normalincidence)
1d b
and
gap
light
line
of a
ir =
ck||
-
k||
modesin crystal
TE
TM
Omnidirectional Reflection[ J. N. Winn et al, Opt. Lett. 23, 1573 (1998) ]
light
line
of a
ir =
ck||
in these ranges,there is
no overlapbetween modesof air & crystal
all incident light(any angle, polarization)
is reflectedfrom flat surface
needs: sufficient index contrast & nhi > nlo > 1
-
[ Y. Fink et al, Science 282, 1679 (1998) ]
Omnidirectional Mirrors in Practice
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
Inde
x ra
tio,n
2 /n
1
Smaller index, n1
0%
10%
20%
30%
40%
50%
/mid
6 9 12 150
5 0
10 0
0
5 0
10 0
0
5 0
10 0
0
5 0
10 0
0
5 0
10 0
normal
450 s
450 p
800 p
800 s
Refle
cta
nc
e (
%)
Wavelength (microns)
Te / polystyrene
Ref
lect
ance
(%
)
contours of omnidirectional gap size
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