Paul Steinhardt Princeton...

Paul SteinhardtPrinceton University

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

Until 1984, the common view was that solids are either crystalline or amorphous …

QUASICRYSTALS

• Translational Order

• Rotational Symmetry

D. Levine and PJS (1984)


• Structure can be reduced to repeating units

QUASICRYSTALS

• Translational Order . . . But QUASIPERIODIC instead of PERIODIC





QUASICRYSTALS


• Rotational Symmetry . . .


• Rotational Symmetry . . . But with FORBIDDEN symmetry


QUASICRYSTALS


• Rotational Symmetry . . .


• Rotational Symmetry . . . But with FORBIDDEN symmetry

• Structure can be reduced to a finite number of repeating units

|| - Space(Real Space)

⊥⊥⊥⊥ - Space⊥⊥⊥⊥ - Space(Perp Space)

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)

Quasi-unit Cell Picture

Gummelt Tile

Quasi-unit Cell Picture & Energetics

P.J. Steinhardt, H.-C. Jeong (1996)

Gummelt Tile

Quasi-unit Cell Picture & Energetics

Alternatively, possibly useful for self-assembled f abrication w/patchy sphere colloids

Y. Roichman, et al. (2005)

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)



••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


mean wider band gap (at mean wider band gap (at

least for some dielectric least for some dielectric


Challenges

•• Computationally costly to Computationally costly to

simulate in 2D . . . simulate in 2D . . .

impossible in 3D?impossible in 3D?










•• 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






Challenges


















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

Quasicrystal beats diamond

Diamond

Quasicrystal

Challenges




Motivations












at bestat best









Paul Steinhardt Princeton...

Documents

Transcript of Paul Steinhardt Princeton...