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 > θc
are totally reflected
sinθc = no / ni
Snell’s Law:
ni sinθi = no sinθo
θo
< 1, so θc is real
θi
i.e. TIR can only guide
within higher index
unlike a band gap
Total Internal Reflection?
no
ni > 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?
no
ni > 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
no
ni > no
ray-optics picture is invalid on λ scale
(neglects coherence, near field…)
translational
symmetry
Snell’s Law is really
conservation of k|| and ω:
|ki| sinθi = |ko| sinθo
|k| = nω/c
(wavevector)
(frequency)
θi
θo
k||
conserved!
Waveguide Dispersion Relations
i.e. projected band diagrams
ω
higher-order modes
at larger ω, β
light cone
projection of all k⊥ in no
lig
no
ni > no
h
tl
ω
:
e
in
k
c
=
higher-index core
pulls down state
/ no
weakly guided (field mostly in no)
ω
/
k
c
=
ni
(
k||
(a.k.a. β)
∞)
Strange Total Internal Reflection
Index Guiding
0.5
0.45
light cone
frequency (c/a)
0.4
0.35
0.3
0.25
0.2
0.15
J
0J
0
J
J
J
a
J
0.1
0.05
J
J
J
J
J
0.05 0.1
0.15
0.2
0.25
0.3
0.35
wavenumber k (2π/a)
0.4
0.45
0.5
Conserved k and ω
+ higher index to pull down state
= localized/guided mode.
A Hybrid Photonic Crystal:
1d band gap + index guiding
0.5
0.45
light cone
frequency (c/a)
0.4
range of frequencies
in which there are
“band
gap”
0.35
no guided modes
0.3
0.25
0.2
0.15
J
0J
0
J
J
J
slow-light band edge
J
0.1
0.05
J
J
J
J
J
0.05 0.1
0.15
0.2
0.25
0.3
0.35
wavenumber k (2π/a)
a
0.4
0.45
0.5
A Resonant Cavity
– +
photonic band gap
index-confined
increased rod radius
pulls down “dipole” mode
(non-degenerate)
A Resonant Cavity
– +
photonic band gap
k not conserved
0.5
index-confined
0.45
light cone
The trick is to
keep the
radiation small…
(more on this later)
frequency (c/a)
0.4
“band
gap”
0.35
ω
0.3
0.25
0.2
0.15
J
0J
0
J
J
J
0.45
0.5
J
0.1
0.05
J
J
J
J
J
0.05 0.1
0.15
0.2
0.25
0.3
0.35
wavenumber k (2π/a)
0.4
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 λ
[ 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
Let’s start with the 2d
band diagram.
Square Lattice of
Dielectric Rods
(ε = 12, r=0.2a)
This is what we’d like
to have in 3d, too!
EEE E
J
J
E E
J
J
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E
EE
JEE
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E
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JJ E
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0.1
J
E J TM bands
J
E
TE bands
J
JE
E
JE
0J
frequency (c/a)
0.6
wavevector
A 2d band diagram in 3d
Let’s start with the 2d
band diagram.
Square Lattice of
Dielectric Rods
(ε = 12, r=0.2a)
No! When we include
out-of-plane propagation,
we get:
wavevector frequency
ω
frequency (c/a)
This is what we’d like
to have in 3d, too! 3D Structure:
ω+δω
projected band diagram fills gap!
EEE E
J
J
E E
J
J
J E E JJ JJJ J J J
E
EE
JEE
J
E
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J
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JJ E
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JJ EE
J
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J
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0.5
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JJ E
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0.4
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0.3
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J
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0.2
E
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J
J
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but
this
empty
0.1
J
E J TM bands
looks
useful…
J space
E
TE bands
J
JE
E
JE
0J
0.6
wavevector
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
Square Lattice of
Dielectric Rods
The Light Cone:
All possible states
propagating in the air
(ε = 12, r=0.2a, h=2a)
0.6
frequency (c/a)
0.5
0.4
0.3
0.2
0.1
E
0J
JEE JJJ
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E
J
JJ
E
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odd (TM-like) bands
JE
JE
even (TE-like) bands
JE
JE
JE
Γ
X
M
The Guided Modes:
Cannot couple to
the light cone…
—> confined to the slab
Thickness is critical.
Should be about λ/2
(to have a gap
& be single-mode)
JE
Γ
M
Γ
X
Symmetry in a Slab
2d: TM and TE modes
r
E
r
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
Slab Gaps
Square Lattice of
Dielectric Rods
Triangular Lattice
of Air Holes
(ε = 12, r=0.2a, h=2a)
(ε = 12, r=0.3a, h=0.5a)
0.6
frequency (c/a)
0.5
0.4
0.3
0.2
0.1
E
0J
E EEE JEE JJE JE JE
JEE JJJ
E
J
E
E
JJ EJJ
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J
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cone
E
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J E JE
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J
E
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J
JE
JE J
E
odd (TM-like) bands
J
JE
even (TE-like) bands
JE
JE
JE
Γ
X
M
TM-like gap
0.5
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0.2 J E E
EJ EJ
E
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E E
J
E
E
JE
odd
(TM-like)
bands
J
0.1
JE
even (TE-like) bands E
JE
JE
JE
Γ
JE
E
0J
Γ
M
K
TE-like gap
Γ
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
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
~ λ/2, but λ/2 in what material?
effective medium theory: effective ε depends on polarization
35
E
E
30
gap size (%)
25
E
E
E
E
E
E
J
20
E
15
J
even (TE-like) gap
10
E
TE “sees” <ε>
~ high ε
5
J
J
J
J
J
J
odd (TM-like) gap
J
E
TM “sees” <ε-1>-1
~ low ε
J
0
0
0.5
1
1.5
slab thickness (a)
2
2.5
3
Photonic-Crystal Building Blocks
point defects
line defects
(cavities)
(waveguides)
A Reduced-Index Waveguide
0.44
frequency (c/a)
0.42
(r=0.10a)
We cannot completely
remove the rods—no
vertical confinement!
0.4
(r=0.12a)
Still have conserved
wavevector—under the
light cone, no radiation
0.38
(r=0.14a)
0.36
(r=0.16a)
0.34
0.32
0.3
(r=0.18a)
(r=0.2a)
0.35
0.4
0.45
wavevector k (2π/a)
0.5
Reduce the radius of a row of
rods to “trap” a waveguide mode
in the gap.
x→
Reduced-Index Waveguide Modes
x→
y→
z→
z→
y→
y→
y→
–1
Ez
+1
–1
Hz
+1
Experimental Waveguide & Bend
[ E. Chow et al., Opt. Lett. 26, 286 (2001) ]
1.2
transmission
bending
efficiency
1
0.8
0.6
0.4
0.2
0
1200
caution:
can easily be
multi-mode
E experiment
theory
EEE
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1µm
1µm
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SiO2
GaAs
1300
1400
1500
1600
wavelength (µm)
1700
1800
Inevitable Radiation Losses
whenever translational symmetry is broken
e.g. at cavities, waveguide bends, disorder…
coupling to light cone
= radiation losses
ω
(conserved)
k is no longer conserved!
All Is Not Lost
A simple model device (filters, bends, …):
Qr
Qw
1
Q
=1
+1
Qr Qw
Q = lifetime/period
= frequency/bandwidth
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:
r r
E( x ) =
Make defect weak
= delocalize mode
t r r r r
r
G
x
x
E
x
x
(
,
)
⋅
(
)
⋅
∆
ε
(
)
′
′
′
ω
∫
defect
far-field
(radiation)
Green’s function
(defect-free
system)
defect
near-field
(cavity mode)
Monopole Cavity in a Slab
Lower the ε of a single rod: push up
a monopole (singlet) state.
decreasing ε
(ε = 12)
Use small ∆ε: delocalized in-plane,
& high-Q (we hope)
Delocalized Monopole Q
1,000,000
ε=11 J
Qr
100,000
J ε=10
10,000
Jε=9
J ε=8
1,000
J ε=7
J
ε=6
100
0.0001
0.001
0.01
mid-gap
∆frequency above band edge (c/a)
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)
Super-Defect vs. Single-Defect Q
ε=11.5 G
1,000,000
ε=11 J
G
ε=11
100,000
Qr
G
ε=10
J ε=10
G
ε=9
10,000
G
Jε=9
ε=8
ε=7 G
J ε=8
1,000
J ε=7
J
ε=6
100
0.0001
0.001
0.01
mid-gap
∆frequency above band edge (c/a)
0.1
Super-Defect vs. Single-Defect Q
ε=11.5 G
1,000,000
ε=11 J
G
ε=11
100,000
Qr
G
ε=10
J ε=10
G
ε=9
10,000
G
Jε=9
ε=8
ε=7 G
J ε=8
1,000
J ε=7
J
ε=6
100
0.0001
0.001
0.01
mid-gap
∆frequency above band edge (c/a)
0.1
Super-Defect State
(cross-section)
∆ε = –3, Qrad = 13,000
Ez
(super defect)
still ~localized: In-plane Q|| is > 50,000 for only 4 bulk periods
(in hole slabs, too)
Hole Slab
ε=11.56
period a, radius 0.3a
thickness 0.5a
Q = 2500
near mid-gap (∆freq = 0.03)
Reduce radius of
7 holes to 0.2a
Very robust to roughness
(note pixellization, a = 10 pixels).
How do we compute Q?
(via 3d FDTD [finite-difference time-domain] simulation)
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 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?
(via 3d FDTD [finite-difference time-domain] simulation)
2
excite cavity with
narrow-band dipole source
(e.g. temporally broad Gaussian pulse)
— source is at ω0 resonance,
which must already be known (via
…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 1 to get ω0,
long-time source for closely-spaced resonances
1
)
Can we increase Q
without delocalizing?
Semi-analytical losses
Another low-loss
strategy:
r r
E( x ) =
exploit cancellations
from sign oscillations
t r r r r
r
G
x
x
E
x
x
(
,
)
⋅
(
)
⋅
∆
ε
(
)
′
′
′
ω
∫
defect
far-field
(radiation)
Green’s function
(defect-free
system)
defect
near-field
(cavity mode)
Need a more
compact 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
– +
photonic band gap
index-confined
increased rod radius
pulls down “dipole” mode
(non-degenerate)
as we change the radius, ω sweeps across the gap
Q = 1,773
r = 0.35a
2d multipole
cancellation
30,000
J
25,000
Q = 28,700
r = 0.375a
20,000
Q 15,000
10,000
Q = 6,624
r = 0.40a
J
5,000
0
0.25
JJ
0.29
J
J
0.33
J
0.37
frequency (c/a)
J
cancel a dipole by opposite dipoles…
cancellation comes from
opposite-sign fields in adjacent rods
… changing radius changed balance of dipoles
3d multipole cancellation?
enlarge center & adjacent rods
vary side-rod ε slightly
for continuous tuning
quadrupole mode
(balance central moment with opposite-sign side rods)
2000
JJ
Q
1500
1000
J
J
500
(Ez cross section)
0
J J
J
J
JJ
0.319243
gap bottom
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?
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
start with:
junction of two waveguides
ε1
ε 1'
ε2 < ε1
ε 2' < ε 1'
No radiation at junction
if the modes are perfectly matched
Perfect Mode Matching
requires:
same differential equations and boundary conditions
ε1
ε 1'
ε2 < ε1
ε 2' < ε 1'
Match differential equations…
ε2 − ε1 = ε2' − ε1'
…closely related to separability
[ S. Kawakami, J. Lightwave Tech. 20, 1644 (2002) ]
Perfect Mode Matching
requires:
same differential equations and boundary conditions
ε1
ε 1'
ε2 < ε1
ε 2' < ε 1'
Match boundary conditions:
(note switch in TE/TM
field must be TE
convention)
(E out of plane, in 2d)
TE modes in 3d
for
cylindrical waveguides,
“azimuthally polarized”
TE0n modes
A Perfect Cavity in 3d
(~ VCSEL + perfect lateral confinement)
Perfect index
confinement
(no scattering)
+
R
N layers
1d band gap
=
3d confinement
defect
N layers
A Perfectly Confined Mode
ε1, ε2 = 9, 6
λ/2 core
ε1', ε2' = 4, 1
E energy density, vertical slice
Q limited only by finite size
~ fixed mode vol.
λ)3
V = (0.4λ
1000000
100000
J
Q
10000
1000
GJ
N = 10
J
J
G
G
G
4
4.5
5
J
G
G
5.5
6
N=5
G
J
100
10
GJ
J
G
J
GJ
1
1.5
2
2.5
3
3.5
cladding R / core radius
Q-tips
Three independent mechanisms for high Q:
Delocalization: trade off modal size for Q
Qr grows monotonically towards band edge
Multipole Cancellation: force higher-order far-field pattern
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
Projected Bands of a 1d Crystal
(a.k.a. a Bragg mirror)
ω
incident light
|
1d band gap
k | conserved
ck |
f
o
e
air
in
l
t
gh
li
TM
TE
modes
in crystal
(normal
incidence)
ω
=
k|
Omnidirectional Reflection
[ J. N. Winn et al, Opt. Lett. 23, 1573 (1998) ]
ω
in these ω ranges,
there is
no overlap
between modes
of air & crystal
all incident light
(any angle, polarization)
is reflected
from flat surface
ck |
|
f
o
e
air
in
l
t
gh
li
TM
TE
modes
in crystal
needs: sufficient index contrast & nhi > nlo > 1
ω
=
k|
Omnidirectional Mirrors in Practice
[ Y. Fink et al, Science 282, 1679 (1998) ]
Te / polystyrene
contours of omnidirectional gap size
10 0
3
50%
0
40%
2.6
2.4
10 0
(%)
Reflec tanc e (%)
Reflectance
Index ratio, n2 / n1
2.8
normal
50
30%
2.2
20%
2
10%
1.8
0%
1.6
∆λ/λmid
1.4
450 s
50
0
10 0
450 p
50
0
10 0
800 s
50
0
1.2
10 0
1
1
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Smaller index, n1
2
800 p
50
0
6
9
12
Wavelength (microns)
15