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Schrodinger-Maxwell analogy
Photonic Band Structure
)()()]([ 22
2
rErrVmh rrr ψψ =+∇−
quantum
“scalar” (spin-1/2)
complicated by e-e interaction
fundamental scale
massive quanta, parabolic disp.
analogous to electronic band structure
atomic/molecular structures
)()(]([2
)(1 rH
crHr
rvrvrrr
=×∇×∇ ωε
classical
vector (TM + TE polarizations)
linear (no γγγγ−−−−γγγγ interaction))()()(
2
rEc
rrErrrrrrr
=×∇×∇ ωεsolution scales*
massless quanta, linear disp.
can design continuous structures
Photonic Band Gap Formation
ε2ε1
Bragg scattering mechanism
aa rr
“large scale” resonance of the array
Mie scattering mechanism
Largest gap: Bragg and Mie scattering reinforce each other
aa
rr
“small scale” resonance of scatterers
Band gaps and Bragg scattering1D Photonic Crystals:
a
k
Bragg scattering condition:
2a=nλ
k’k0 π/a
ω
In higher dimensions, In higher dimensions, kk ..GG = |G|= |G|22/2 or /2 or 2a=nλ
k’ k
In reciprocal space:
Gxnakkr)rr
==− )/2()'( π
2ππππ/a
Irreducible Brillouin Zone :Brillouin Zone
G
In higher dimensions, In higher dimensions, kk ..GG = |G|= |G| /2 or /2 or |k|~1/cosθ ωωωω ~ 1/cosθ
Schrodinger-Maxwell analogy
3D band gap3D band gap
)()()]([ 22
2
rErrVmh rrr ψψ =+∇−
( ) )()(]([ 2)(
1 rHrH cr
rvrvrrr
ωε =×∇×∇
analogous to electronic band structurePhotonic Band Structure
CloseClose--packed FCC lattice of packed FCC lattice of air spheres in siliconair spheres in silicon((fccfcc or inverted opal)or inverted opal)
( ) )()(]([ )( rHrH crε =×∇×∇
Two Dimensional Photonic Crystals
Transverse-magnetic (TM) modes
Transverse-electric (TE) modes
⊥E
||E
⊥E ||E
Two Dimensional Photonic Crystals
TM, TE and Full photonic band gaps
Isolated scatterer and connected network architectures
Boundary conditions: Eand D|| are continuous
TE GapTM Gap
Full Gap
QUASICRYSTALS
• Translational Order
• Rotational Symmetry
D. Levine and PJS (1984)
• Rotational Symmetry
• Structure can be reduced to repeating units
QUASICRYSTALS
• Translational Order . . . But QUASIPERIODIC instead of PERIODIC
• Rotational Symmetry
D. Levine and PJS (1984)
• Rotational Symmetry
• Structure can be reduced to repeating units
QUASICRYSTALS
• Translational Order . . . But QUASIPERIODIC instead of PERIODIC
• Rotational Symmetry . . .
D. Levine and PJS (1984)
• Rotational Symmetry . . . But with FORBIDDEN symmetry
• Structure can be reduced to repeating units
QUASICRYSTALS
• Translational Order . . . But QUASIPERIODIC instead of PERIODIC
• Rotational Symmetry . . .
D. Levine and PJS (1984)
• Rotational Symmetry . . . But with FORBIDDEN symmetry
• Structure can be reduced to a finite number of repeating units
5DHypercubic
Lattice
2D ||-Space
(The Penrose Tiling)
3DPerp-Space
(The Penrose Tiling)
Proves:long-range translational orderquasiperiodicityBragg peak diffractionany rotational symmetry possible
AlAl 6060LiLi 3030CuCu1010 AlMnPd
Al63Fe24Cu13
L. Bindi, PJS, N. Yao & P. Lu (2009)
Al72Ni20Co8 First Natural QC
AlAl 7272NiNi 2020CoCo88
High Angle Annular Dark Field
Quasi-unit Cell Picture
P.J. Steinhardt, H.-C. Jeong, K. Saitoh, M. Tanaka, E. Abe, A.P. TsaiNature 396, 55 -57 (1998)
Gummelt Tile
Quasi-unit Cell Picture & Energetics
Alternatively, possibly useful for self-assembled f abrication w/patchy sphere colloids
Why consider photonic quasicrystals?
kkkkyyyyyyyy
kkkkxxxxxxxx
Square lattice (4-fold)
yyyy
xxxx
For crystals, higher rotational symmetry means wider bandgap!
But any symmetry beyond 6-fold is forbidden.
(no restriction for quasicrystals, though!)
Hexagonal lattice (6-fold)
xxxx
Motivations
•• Higher symmetry may Higher symmetry may
produce wider band gap produce wider band gap
(at least for some dielectric (at least for some dielectric
constant ratios)constant ratios)
Why consider photonic quasicrystals?
constant ratios)constant ratios)
••More circular symmetry More circular symmetry
may mean more highly may mean more highly
isotropic gap.isotropic gap.
•• New types of structures New types of structures
with different modes & with different modes &
defects may enable new defects may enable new
applications applications
Motivations
•• Higher symmetry may Higher symmetry may
produce wider band gap produce wider band gap
(at least for some dielectric (at least for some dielectric
constant ratios)constant ratios)
Why consider photonic quasicrystals?
constant ratios)constant ratios)
••More circular symmetry More circular symmetry
may mean more highly may mean more highly
isotropic gap.isotropic gap.
•• New types of structures New types of structures
with different modes & with different modes &
defects may enable new defects may enable new
applications applications
Do quasicrystals have higher symmetry?obvious in 2d, but in 3d ?? What is BZ??
2255
33
22 5533 22////
55
5252ΓΓΓΓ
LUU
LK K
WWUUXX
Do quasicrystals have higher symmetry?obvious in 2d, but…
25
32
53 2//
5
ΓΓΓΓW W X W WK
UU
UUXX
Brillouin zone of Diamond, FCC lattice
Triacontahedron
from L to W29% difference
from 2|| to 5 direction17.5% difference
Motivations
•• Higher symmetry may Higher symmetry may
mean wider band gap (at mean wider band gap (at
least for some dielectric least for some dielectric
constant ratios)constant ratios)
Challenges
•• Computationally costly to Computationally costly to
simulate in 2D . . . simulate in 2D . . .
impossible in 3D?impossible in 3D?
Why consider photonic quasicrystals?
constant ratios)constant ratios)
••More circular symmetry More circular symmetry
may mean more highly may mean more highly
isotropic gap.isotropic gap.
•• New types of structures New types of structures
with different modes & with different modes &
defects may enable new defects may enable new
applications applications
•• Dense set of Dense set of Bragg peaksBragg peaks, ,
so so is there is there really a really a bandgapbandgap
after allafter all? What is the ? What is the
analogue of a BZ???analogue of a BZ???
•• Not clear how to find Not clear how to find
optimum structure optimum structure –– costly costly
at bestat best
Motivations
•• Higher symmetry may Higher symmetry may
mean wider band gap (at mean wider band gap (at
least for some dielectric least for some dielectric
constant ratios)constant ratios)
Why consider photonic quasicrystals?
Challenges
•• Computationally costly to Computationally costly to
simulate in 2D . . . simulate in 2D . . .
impossible in 3D?impossible in 3D?
constant ratios)constant ratios)
••More circular symmetry More circular symmetry
may mean more highly may mean more highly
isotropic gap.isotropic gap.
•• New types of structures New types of structures
with different modes & with different modes &
defects may enable new defects may enable new
applications applications
•• Dense set of Dense set of Bragg peaksBragg peaks, ,
so so is there is there really a really a bandgapbandgap
after allafter all? What is the ? What is the
analogue of a BZ???analogue of a BZ???
•• Not clear how to find Not clear how to find
optimum structure optimum structure –– costly costly
at bestat best
••StereolithographyStereolithography
d=1cm
The brutest of brute force “calculational methods”:. . .use the experiment as the theoretical simulati on !!
Layer thickness 100 mm 1
2.72
/ 0.00035( )d df GHz
εε −
== −
1
1
0.149( )
/ 0.0042( )
m
d df m GHz
σσ
−
−
= Ω ⋅= Ω ⋅ ⋅
n=1.65-0.025i at 33 GHz
Microwave transmission measurements
Rotate the sample around one rotation symmetry axis and record transmission spectrum every 2 degrees.
λ ~ d
Kα band 15 - 26 GHz,1.15 - 2 cm K band 26 to 42 GHz.0.71 - 1.15 cm
Microwave transmission measurements
Recall that, for Recall that, for each BZ each BZ plane,plane,|k|=|kmin|/cosθ f ~ fmin/cosθ
Measured transmission (color) as a function of f and θ
Visualizing the Brillouin zone of Icosahedral QC
2255
33
22 5533 22////
555-fold rotation
2255
3322
5533 22////
552-fold rotation
Plot in polar coordinates:Measured transmission as a function of f=r and θ=θ
Challenges
•• Computationally costly to Computationally costly to
simulate in 2D . . . simulate in 2D . . .
impossible in 3D?impossible in 3D?
Motivations
•• Higher symmetry may Higher symmetry may
mean wider band gap (at mean wider band gap (at
least for some dielectric least for some dielectric
constant ratios)constant ratios)
Why consider photonic quasicrystals?
•• Dense set of Dense set of Bragg peaksBragg peaks, ,
so so is there is there really a really a bandgapbandgap
after allafter all? What is the ? What is the
analogue of a BZ???analogue of a BZ???
•• Not clear how to find Not clear how to find
optimum structure optimum structure –– costly costly
at bestat best
constant ratios)constant ratios)
••More circular symmetry More circular symmetry
may mean more highly may mean more highly
isotropic gap.isotropic gap.
•• New types of structures New types of structures
with different modes & with different modes &
defects may enable new defects may enable new
applications applications