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Photonic Modeling and Design Lab.Graduate Institute of Photonics and Optoelectronics &Department of Electrical EngineeringNational Taiwan University
光子晶體光子晶體Photonic Crystals Photonic Crystals –– Chapter 8Chapter 8
Photonic Crystal SlabsPhotonic Crystal Slabs
台大光電所暨電機系
邱奕鵬Room 617, BL Building
(02) 3366-3603
Photonic Crystal Slabs
Why “SLAB”?
• Full 3D photonic crystals are too expensive
• Difficult to incorporate on a chip
Why “SLAB”?
• In-plane propagation• In-plane 2D photonic crystal
• Vertical index guiding
Why “SLAB”?• Easy to incorporate on a chip and can be
fabricated easier
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 / n o
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)
J
J
J
J
J
J
JJ
JJ J
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
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
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
J
J
J
J
J
J
JJ
JJ J
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
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
and
ga p�
a
range of frequenciesin which there are
no guided modes
slow-light band edge
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 obvious solution?
make2d 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
Projection!!
Photonic-Crystal Slabs
2d photonic bandgap + vertical index guiding
[ S. G. Johnson and J. D. Joannopoulos, Photonic Crystals: The Road from Theory to Practice“Guided modes in photonic crystal slabs,” Phy. Rev. B 60, 5751, (1999)]
Photonic-Crystal Slabs
2d photonic bandgap + vertical index guiding
[ S. G. Johnson and J. D. Joannopoulos, Photonic Crystals: The Road from Theory to Practice ]
r=0.2a
r=0.45a
Dielectric const. = 12 (both)
Two Dimensional LatticesTriangular Lattice
in z-direction of an length
Draw perpendicular bisectors of any lattice vector startingat origin
spacing
Symmetry in a Slab
2d: TM and TE modes
slab: odd (TM-like) and even (TE-like) modes
mirror planez = 0
E E
Like in 2d, there may only be a band gapin one symmetry/polarization
purely TM/TE inthe mirror plane
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
Shift of lightcone
n=1 n=1.5
Background n=1.0 Background n=1.0
SANDWICHESBackground n=1.5 Background n=1.5
2D band structure (kz=0, in-plane propagation)Slabs with periodic background
Slabs with symmetry-breaking background
Slabs with symmetry-breaking background
• The guided mode can no longer be classified as even or odd.
• There is no longer any band gap in the guided modes.
Slabs with symmetry-breaking background
• The guided modes are sufficiently localized within the slab.
• Buffer region.
Buffer region
Air-membrane Slabs
[ N. Carlsson et al., Opt. Quantum Elec. 34, 123 (2002) ]
who needs a substrate?
2µm
AlGaAs
Effects of slab thickness
• Too thick: Higher order modes can be created with little energy cost by adding horizontal nodal plane.
• Too thin: The slab will provide only a weak perturbation on the background dielectric constant.
r=0.2ar=0.45a
Dielectric const. = 12 (both)
Optimal slab thickness• λ/2: half the two-dimensional gap-bottom
wavelength.
• We only know the gap-bottom frequency.
• Need an effective dielectric constant to evaluate the wavelength.
Two dimensional wavefunction
Evaluate the dependence of its
frequency expectation value on the vertical
wavelength
Optimal slab thickness
Wavefunction
Eiganfunction Amplitude
Strongly localized within the slab
Optimal slab thickness
Only for divergenceless fields
Optimal slab thickness
ε
Effective dielectric constant
=5.06
ε=12
εrather than 3.92(uniform-weight mean)
=1.25εrather than 1.13
(uniform-weight inverse mean)
Optimal slab thickness
Photonic-Crystal Building Blocks
point defects(cavities)
line defects(waveguides)
Reduced-index waveguides Increased-index
waveguides
Dielectric-strip waveguides
Three criteria must be satisfied
• Waveguide must support true guided modes.
– Bloch
• The waveguide should be single-mode in the frequency range of interest.
• The guided mode should lie within the band gap of a photonic crystal in order to prohibit radiation losses.
• Conventional waveguides
To achieve optimal performance in many integrated-optical-circuit applications
Photonic-Crystal Building Blocks
point defects(cavities)
line defects(waveguides)
Reduced-index waveguides Increased-index
waveguides
Dielectric-strip waveguides
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 conservedwavevector—under thelight cone, no radiation
(r=0.10a)
We cannot completelyremove the rods—novertical confinement!
A Reduced-Index Waveguide
The thickness of the vein is the
critical parameter!
Two Dimensional LatticesTriangular Lattice
in z-direction of an length
Draw perpendicular bisectors of any lattice vector startingat origin
spacing
Reduced-Index Waveguide Modes
y →
x →
? +1Ez
y →
y →
x →
? +1Hz
z →
y →
Increased-index waveguide
• Modes that lie below both the light cone and the slab band continuum.
• A state that is pulled down into the gap from the upper slab bands.
Two sorts of guided modes
Index-guiding both horizontal and vertical
Index-guiding only vertically
higherDegenerate pairM
Increased-index waveguide
Non-degenerate
Single mode
Increased-index waveguide
Degenerate Multi-mode
Increased-index waveguide
Strip waveguides in photonic-crystal slabs
Strip waveguides in photonic-crystal slabs
Fundamental modeSecond
exciting mode
Need for absolute band gap
• Refractive index contrast is as high as possible
• Brillouin zone is close to a circle
• Shape of scattering objects matches the symmetry of Brillouin zone
• The PhC is comprised of isolated dielectric islands connected by narrow veins (this implies a high filling-factor for the low-index material)
Design of photonic crystals
• Thickness
• Size of “air rods”
• Angular orientation with respect to the lattice
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
Quality factor Q
• The number of cycles for its energy to decay by
• The decay rate 1/Q is the bandwidth at half maximum of any filter or other device based on the cavity
π2−e
A Resonant Cavity
index-confined
photonic band gap
increased rod radiuspulls down “dipole” mode
(non-degenerate)
– +
J
J
JJ
JJ
JJ J J J
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
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
and
ga p� ω
A Resonant Cavity
index-confined
photonic band gap
– +
k not conservedso coupling to
light cone:
radiationThe trick is to
keep theradiation small…
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
Input Waveguide Cavity Output Waveguide
Q
Increase Q⊥
The more spatially delocalized the mode is, the more localized it will be in wave vector space.
A. Mode delocalizationB. Multipole cancellations
Semi-analytical losses
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
Mode frequency
(ε = 12)
Monopole Cavity in a Slab
decreasing ε
Lower the ε of a single rod: push upa monopole (singlet) state.
Use small Δε: delocalized in-plane,& high-Q (we hope)
Increase Q
Enlarge the defect size
Weaker defect with more unit cells.
More delocalizedat the same point in the gap
(i.e. at same bulk decay rate)
Super-defects
Super defect v.s. single defect Super-Defect State(cross-section)
Δε = –3, Qrad = 13,000
(super defect)
still ~localized: In-plane Q|| is > 50,000 for only 4 bulk periods
Ez
(in hole slabs, too) 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
Another low-lossstrategy:
exploit cancellationsfrom sign oscillations
far-field(radiation) Green’s function
(defect-freesystem)
near-field(cavity mode)defect
Mode frequency
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
cancel a dipole by opposite dipoles…
cancellation comes fromopposite-sign fields in adjacent rods
… changing radius changed balance of dipoles
3d multipole cancellation
near
fie
ld E
zfa
r fi
eld
|E|2
Q = 408 Q = 426Q = 1925
nodal planes(source of high Q)
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) ]
A Perfect Cavity in 3d
R
N layers
N layers
defect
Perfect indexconfinement
(no scattering)
+1d band gap
=3d confinement
(~ VCSEL + perfect lateral confinement)
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 o
f air
ω= ck
||
ω
k��||
modesin crystal
TE
TM
Omnidirectional Reflection[ J. N. Winn et al, Opt. Lett. 23, 1573 (1998) ]
light
line o
f air
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
Smaller index, n
1
0%
10%
20%
30%
40%
50%
Δλ/λmid
6 9 1 2 1 50
5 0
1 0 0
0
5 0
1 0 00
5 0
1 0 00
5 0
1 0 00
5 0
1 0 0
normal
450 s
450 p
800 p
800 s
Refle
cta
nce
(%)
Wavelength (microns)Wavelength (microns)
Te / polystyrene
Ref
lect
ance
(%
)
contours of omnidirectional gap size
