Photonic Crystals:
Periodic Surprises in Electromagnetism
Steven G. Johnson
MIT
Complete Band Gaps:
You can leave home without them.
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 l scale
(neglects coherence, near field…)
translational
symmetry
Snell’s Law is really
conservation of k|| and w:
|ki| sinqi = |ko| sinqo
|k| = nw/c
(wavevector)
(frequency)
qi
k||
qo
conserved!
Waveguide Dispersion Relations
i.e. projected band diagrams
w
light cone
projection of all k in no
higher-order modes
at larger w, 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
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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
w
s ta t e
.
A Hybrid Photonic Crystal:
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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
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A Resonant Cavity
– +
photonic bandgap
index-confined
increased rod radius
pulls down “dipole” mode
(non-degenerate)
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A Resonant Cavity
– +
photonic bandgap
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k not conserved
0.5
index-confined
0.45
l i ght cone
0.4
0.35
w
“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 l
[ 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
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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
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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
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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
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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
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E
JEE
E
E
JEE
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JJ JJJEJE
E
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E
E
J
J
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J
J
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J
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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
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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
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QuickTime™ and a
Photo - JPEG decompressor
are needed to see this picture.
high index
S. Assefa, L. A. Kolodziejski
Air-membrane Slabs
who needs a substrate?
AlGaAs
2µm
[ N. Carlsson et al., Opt. Quantum Elec. 34, 123 (2002) ]
Optimal Slab Thickness
~ l/2, but l/2 in what material?
effective medium theory: effective  depends on polarization
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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
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pressor
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point defects
(cavities)
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pressor
athis picture.
line defects
(waveguides)
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pressor
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are
G
Qraphics
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pressor
athis picture.
A Reduced-Index Waveguide
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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
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pressor
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pressor
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x
are
G
Qraphics
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pressor
athis picture.
x
x
y
z
z
y
y
y
–1
Ez
+1
–1
Hz
+1
Experimental Waveguide & Bend
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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
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E EE E
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EE
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E E
E
E
E
EE
E
E
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E
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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
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pressor
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whenever translational symmetry is broken
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pressor
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e.g. at cavities, waveguide bends, disorder…
coupling to light cone
= radiation losses
w
(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

)



(
x

)
w

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 : delocalized in-plane,
& high-Q (we hope)
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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)
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pressor
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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
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pressor
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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)
 = –3, Qrad = 13,000
Ez
(super defect)
still ~localized: In-plane Q|| is > 50,000 for only 4 bulk periods
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pressor
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(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)
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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 w’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)
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athis picture.
2
excite cavity with
narrow-band dipole source
(e.g. temporally broad Gaussian pulse)
— source is at w0 resonance,
which must already be known (via
…measure outgoing power P and energy U
Q = w0 U / P
Pro: separate Qw/Qr, arbitrary Q, also get field pattern
Con: requires separate run 1 to get w0,
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

)



(
x

)
w

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, w 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.









'

Nor ad
i at ion
i f t hem o
desar e
'
t ju
a
nct ion
pe
r f ec
t l ym atched

'

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.



'






'


'

Mat chdi f er e
nt i a
l eq
uat ions…
…closely related to separability






'



'

[ 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.



'






'


'

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.
l


,
, 


,
'

'
, 



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
l
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)
w
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) ]
w
in these w 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
l/lmid
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