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Complete Band Gaps:
You can leave home without them.
Photonic Crystals:Periodic Surprises in Electromagnetism
Steven G. Johnson
MIT
How else can we confine light?
Total Internal Reflection
ni > no
no
rays at shallow angles > θc
are totally reflected
Snell’s Law:
θi
θo
ni sinθi = no sinθo
sinθc = no / ni
< 1, so θc is real
i.e. TIR can only guidewithin higher indexunlike a band gap
Total Internal Reflection?
ni > no
no
rays at shallow angles > θc
are totally reflected
So, for example,a discontiguous structure can’t possibly guide by TIR…
the rays can’t stay inside!
Total Internal Reflection?
ni > no
no
rays at shallow angles > θc
are totally reflected
So, for example,a discontiguous structure can’t possibly guide by TIR…
or can it?
Total Internal Reflection Redux
ni > no
no
ray-optics picture is invalid on λ scale (neglects coherence, near field…)
Snell’s Law is reallyconservation of k|| and ω:
θi
θo|ki| sinθi = |ko| sinθo
|k| = nω/c
(wavevector) (frequency)
k||
translationalsymmetry
conserved!
Waveguide Dispersion Relationsi.e. projected band diagrams
ni > no
no
k||
ω
light line: ω
= ck / no
light coneprojection of all k⊥ in no
(a.k.a. β)
ω = ck / ni
higher-index corepulls down state
( ∞)
higher-order modesat larger ω, β
weakly guided (field mostly in no)
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freq
uenc
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/a)
wavenumber k (2π/a)
light cone
Conserved k and ω+ higher index to pull down state
= localized/guided mode.
Strange Total Internal ReflectionIndex Guiding
a
A Hybrid Photonic Crystal:1d band gap + index guiding
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freq
uenc
y (c
/a)
wavenumber k (2π/a)
light cone
“band gap”
a
range of frequenciesin which there are
no guided modes
slow-light band edge
A Resonant Cavity
index-confined
photonic band gap
increased rod radiuspulls down “dipole” mode
(non-degenerate)
– +
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freq
uenc
y (c
/a)
wavenumber k (2π/a)
light cone
“band gap” ω
A Resonant Cavity
index-confined
photonic band gap
– +
k not conservedso coupling to
light cone:
radiationThe trick is to
keep theradiation small…
Meanwhile, back in reality…
5 µm
[ 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 λ
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 obvioussolution?
make2d patternreally 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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TE bandsTM bands
Square Lattice ofDielectric Rods
(ε = 12, r=0.2a)
Let’s start with the 2dband diagram.
This is what we’d liketo have in 3d, too!
wavevector
A 2d band diagram in 3d
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uenc
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/a)
TE bandsTM bands
Square Lattice ofDielectric Rods
(ε = 12, r=0.2a)
Let’s start with the 2dband diagram.
This is what we’d liketo have in 3d, too!
wavevector
3D Structure:
No! When we includeout-of-plane propagation,
we get:
wavevectorfrequency
ω
ω+δω
projected band diagram fills gap!
but this emptyspace looks useful…
Photonic-Crystal Slabs
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
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freq
uenc
y (c
/a)
even (TE-like) bandsodd (TM-like) bands
Square Lattice ofDielectric Rods(ε = 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 λ/2 (to have a gap & be single-mode)
Γ X M ΓΓ X
M
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
Slab Gaps
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freq
uenc
y (c
/a)
even (TE-like) bandsodd (TM-like) bands
even (TE-like) bandsodd (TM-like) bands
Γ X M Γ Γ M K Γ
Square Lattice ofDielectric Rods(ε = 12, r=0.2a, h=2a)
Triangular Latticeof Air Holes
(ε = 12, r=0.3a, h=0.5a)
light cone light cone
TM-like gap TE-like gap
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
Extruded Rod Substrate
S. Assefa, L. A. Kolodziejski
high index
Air-membrane Slabs
[ N. Carlsson et al., Opt. Quantum Elec. 34, 123 (2002) ]
who needs a substrate?
2µm
AlGaAs
Optimal Slab Thickness~ λ/2, but λ/2 in what material?
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15
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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” <ε-1>-1
~ low εTE “sees” <ε>
~ high ε
effective medium theory: effective ε depends on polarization
Photonic-Crystal Building Blocks
point defects(cavities)
line defects(waveguides)
0.32
0.34
0.36
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0.4
0.42
0.44
0.3 0.35 0.4 0.45 0.5
freq
uenc
y (c
/a)
wavevector k (2π/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 conservedwavevector—under thelight cone, no radiation
(r=0.10a)
We cannot completelyremove the rods—novertical confinement!
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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tran
smis
sion
wavelength (µm)
E experiment
theory
band gap
waveguide mode
[ E. Chow et al., Opt. Lett. 26, 286 (2001) ]
1µm 1µm
GaAs
AlO
SiO2
be
nd
ing
eff
icie
ncy
caution:can easily bemulti-mode
Inevitable Radiation Losseswhenever translational symmetry is broken
e.g. at cavities, waveguide bends, disorder…
k is no longer conserved!
ω(conserved)
coupling to light cone= radiation losses
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
Semi-analytical losses
r r t r r r r rE x G x x E x x
defect
( ) ( , ) ( ) ( )= ′ ⋅ ′ ⋅ ′∫ ω ε∆
far-field(radiation) Green’s function
(defect-freesystem)
near-field(cavity mode)
defect
A low-lossstrategy:
Make field insidedefect small
= delocalize mode
Make defect weak= delocalize mode
(ε = 12)
Monopole Cavity in a Slab
decreasing ε
Lower the ε of a single rod: push upa monopole (singlet) state.
Use small ∆ε: delocalized in-plane, & high-Q (we hope)
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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
ε=6
ε=7
ε=8
ε=9
ε=10
ε=11
mid-gap
Super-defects
Weaker defect with more unit cells.
More delocalizedat the same point in the gap(i.e. at same bulk decay rate)
Super-Defect vs. Single-Defect Q
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0.0001 0.001 0.01 0.1
Qr
∆frequency above band edge (c/a)
ε=6
ε=7
ε=8
ε=9
ε=10
ε=11
ε=7
ε=8
ε=9
ε=10
ε=11
ε=11.5
mid-gap
Super-Defect vs. Single-Defect Q
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Qr
∆frequency above band edge (c/a)
ε=6
ε=7
ε=8
ε=9
ε=10
ε=11
ε=7
ε=8
ε=9
ε=10
ε=11
ε=11.5
mid-gap
Super-Defect State(cross-section)
∆ε = –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 Slabε=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).
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 ω’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?
2excite cavity with
narrow-band dipole source(e.g. temporally broad Gaussian pulse)
(via 3d FDTD [finite-difference time-domain] simulation)
— source is at ω0 resonance,which must already be known (via )1
…measure outgoing power P and energy U
Q = ω0 U / P
Pro: separate Qw/Qr, arbitrary Q, also get field pattern
Con: requires separate run to get ω0, long-time source for closely-spaced resonances
1
Can we increase Qwithout delocalizing?
Semi-analytical losses
r r t r r r r rE x G x x E x x
defect
( ) ( , ) ( ) ( )= ′ ⋅ ′ ⋅ ′∫ ω ε∆
far-field(radiation) Green’s 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
Multipole Expansion[ 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[ Jackson, Classical Electrodynamics ]
+ + + …
radiated field =
dipole quadrupole hexapole
peak Q (cancellation) = transition to higher-order radiation
Multipoles in a 2d example
index-confined
photonic band gap
increased rod radiuspulls down “dipole” mode
(non-degenerate)
– +
as we change the radius, ω sweeps across the gap
2d multipolecancellation
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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
cancel a dipole by opposite dipoles…
cancellation comes fromopposite-sign fields in adjacent rods
… changing radius changed balance of dipoles
3d multipole cancellation?
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1500
2000
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Q
frequency (c/a) gap topgap bottom
enlarge center & adjacent rods
vary side-rod ε slightlyfor continuous tuning
quadrupole mode
(Ez cross section)
(balance central moment with opposite-sign side rods)
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 214µW(optically pumped @830nm, 1% duty cycle)
(InGaAsP)
Q ~ 2000 observed from luminescence
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
ε1
ε2 < ε1
ε1'
ε2' < ε1'
start with:junction of two waveguides
No radiation at junctionif the modes are perfectly matched
Perfect Mode Matching
ε1
ε2 < ε1
ε1'
ε2' < ε1'
requires:same differential equations and boundary conditions
Match differential equations…ε2 − ε1 = ε2' − ε1' …closely related to separability
[ S. Kawakami, J. Lightwave Tech. 20, 1644 (2002) ]
Perfect Mode Matching
ε1
ε2 < ε1
ε1'
ε2' < ε1'
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
ε1', ε2' = 4, 1
ε1, ε2 = 9, 6λ/2 core
E energy density, vertical slice
Q limited only by finite size
G
G
G
G
G
GG
GG G
J
J
J
J
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 = (0.4λλλλ)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
ω
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 ω =
ck||
ω
k||
modesin crystal
TE
TM
Omnidirectional Reflection[ J. N. Winn et al, Opt. Lett. 23, 1573 (1998) ]
light
line
of a
ir ω =
ck||
in these ω 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%
∆λ/λmid
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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