resumo aula passada

1

Resumo aula passada

• Diferentes processos litográficos, por projeção, por contato, por mergulho, por escrita direta

• Evolução tamanho linha escrita• Litografia soft. Litografia nanoimpressa• SAW, dispositivos integrados• Sala limpa• Materiais fotônicos, MEMS – MOEMS

20110620

2

Cristais Fotônicos

Elétrons de um lado e fótons do outro lado,

junção de fóton + eletrônico (fotônico)

Temos elétrons em sólidos e fótons

em......materiais fotônicos

3

Referencias

• Fundamentals of Photonics (SALEH AND TEICH · Fundamentals of Photonics, Second Edition. ISBN: 978-0-471-35832-9 )– Ch. 9: Fiber Optics

• Photonic Crystals: Molding the Flow of Light(ISBN: 978-0-691-12456-8)– J. D. Joannopoulos

• Photonic crystal tutorials– Steven G. Johnson– http://ab-initio.mit.edu/photons/tutorial/

• MIT Photonic Bands Software– Free program for computing photonic crystal band structures– http://ab-initio.mit.edu/wiki/index.php/MIT_Photonic_Bands

http://ab-initio.mit.edu/photons/tutorial/

http://ab-initio.mit.edu/wiki/index.php/MIT_Photonic_Bands

4

Cristais fotônicos

Em Cristal Sólido

• elétrons • potencial periódico• banda de energia• defeitos: estados

dentro da banda proibida

Em Cristal Fotônico• fótons• modulação da constante

dielétrica• Banda de energia

fotônica = photonic band gap (PBG)

• defeitos: estados dentro da banda com direcionalidade bem definida

Yablonovitch, PRL 58 (1987) 2059; John, PRL 58 (1987) 2486

Analogia entre cristal sólido e cristal fotônico.

Analogias • portadores • estrutura• bandas• defeitos

5

Solid of N atomsTwo atoms Six atoms

Band Theory: “Bound” Electron Approach• For the total number N of atoms in a solid (1023 cm–3), N energy

levels split apart within a width E.– Leads to a band of energies for each initial atomic energy level (e.g.

1s energy band for 1s energy level).

Electrons must occupy different energies due to

Pauli Exclusion principle.

6

Filtro de Fabry-Perot C_MEMS

http://www.npphotonics.com/files/article/OEG20030324S0088.htm

http://www.npphotonics.com/files/article/OEG20030324S0088.htm

7

O seguinte é um seminário dado porSteven G. Johnson, MIT Applied

Mathematics

8

From electrons to photons: Quantum-inspired modeling in nanophotonics

Steven G. Johnson, MIT Applied Mathematics

9

Nano-photonic media (l-scale)

synthetic materials

strange waveguides

3d structures

hollow-core fibersoptical phenomena

& microcavities[B. Norris, UMN] [Assefa & Kolodziejski,

MIT]

[Mangan, Corning]

10

1887 1987

Photonic Crystals

periodic electromagnetic media

2-D

periodic intwo directions

3-D

periodic inthree directions

1-D

periodic inone direction

can have a band gap: optical “insulators”

11

Cristal fotônico 1D1887

1-D

periodic inone direction

e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 e1 e2

e(x) = e(x+a)a

12

Elétron numa rede periódica 1D

Elétron num ambiente livre

− ℏ2

2𝑚𝜕2 (𝑥 )𝜕 𝑥2 =𝐸 (𝑥 )

Solução: (𝑥 )=𝐴𝑒𝑖𝑘𝑥

𝐸=ℏ𝑘2

2𝑚

Ener

gia

E

k

− ℏ2

2𝑚𝜕2 (𝑥 )𝜕 𝑥2 +∑

𝐺𝑈𝐺𝑒− 𝑖𝐺𝑥 (𝑥 )=𝐸 (𝑥 )

𝐺=h 2𝜋𝑎

13

Elétron numa rede periódica – aproximação por poço retangular periódico

Mostra a existência de banda proibida, imposta por condições de contorno

14

Simulation of Band Gap Structures of 1D Photonic Crystal

• Journal of the Korean Physical Society, Vol. 52, February 2008, pp. S71S74

15

Electronic and Photonic Crystals

atoms in diamond structure

wavevector

elec

tron

ener

gy

Peri

odic

Med

ium

Blo

ch w

aves

:B

and

Dia

gram

dielectric spheres, diamond lattice

wavevector

phot

on fr

eque

ncy

interacting: hard problem non-interacting: easy problem

16

Electronic & Photonic Modelling

Electronic Photonic

• strongly interacting —tricky approximations

• non-interacting (or weakly), —simple approximations (finite resolution) —any desired accuracy

• lengthscale dependent (from Planck’s h)

• scale-invariant —e.g. size 10 l 10

Option 1: Numerical “experiments” — discretize time & space … go

Option 2: Map possible states & interactions using symmetries and conservation laws: band diagram

17

Fun with Math

E 1

ct

H i

c

H

H e

1ct

E

J i

ceE

0

dielectric function e(x) = n2(x)

First task:get rid of this mess

1e

H

c

2 H

eigen-operator eigen-value eigen-state

H 0+ constraint

18

Electronic & Photonic Eigenproblems

1e

H

c

2 H

Electronic Photonic

2

2m2 V

E

simple linear eigenproblem(for linear materials)

nonlinear eigenproblem(V depends on e density ||2)

—many well-known computational techniques

Hermitian = real E & , … Periodicity = Bloch’s theorem…

19

A 2d Model System

square lattice,period a

dielectric “atom”e=12 (e.g. Si)

a

a

E

HTM

20

Periodic Eigenproblems

if eigen-operator is periodic, then Bloch-Floquet theorem applies:

H (x , t)e

ik x t H k (

x )can choose:

periodic “envelope”planewave

Corollary 1: k is conserved, i.e. no scattering of Bloch wave

Corollary 2: given by finite unit cell,so are discrete n(k)H k

21

Solving the Maxwell Eigenproblem

H(x,y) ei(kx – t)

ik 1e ik Hn

n2

c 2 Hn

ik H 0

where:

constraint:

1

Want to solve for n(k),& plot vs. “all” k for “all” n,

Finite cell discrete eigenvalues n

Limit range of k: irreducible Brillouin zone

2 Limit degrees of freedom: expand H in finite basis

3 Efficiently solve eigenproblem: iterative methods

QuickTime™ and aGraphics decompressorare needed to see this picture.00.10.20.30.40.50.60.70.80.91Photonic Band GapTM bands

22

Solving the Maxwell Eigenproblem: 1

1 Limit range of k: irreducible Brillouin zone



—Bloch’s theorem: solutions are periodic in k

kx

ky

first Brillouin zone= minimum |k| “primitive cell”

2aG

M

X

irreducible Brillouin zone: reduced by symmetry

23

Solving the Maxwell Eigenproblem: 2a


2 Limit degrees of freedom: expand H in finite basis (N)


H H(xt ) hmbm (x t )m1

N

solve: ˆ A H 2 H

Ah 2Bh

Am bmˆ A b Bm bm bf g f * g

finite matrix problem:

24

Solving the Maxwell Eigenproblem: 2b




( ik)H 0— must satisfy constraint:

Planewave (FFT) basis

H(x t ) HGeiGxt

G

HG G k 0constraint:

uniform “grid,” periodic boundaries,simple code, O(N log N)

Finite-element basisconstraint, boundary conditions:

Nédélec elements[ Nédélec, Numerische Math.

35, 315 (1980) ]

nonuniform mesh,more arbitrary boundaries,

complex code & mesh, O(N)[ figure: Peyrilloux et al.,

J. Lightwave Tech.21, 536 (2003) ]

25

Solving the Maxwell Eigenproblem: 3a




Ah 2Bh

Faster way:— start with initial guess eigenvector h0

— iteratively improve— O(Np) storage, ~ O(Np2) time for p eigenvectors

Slow way: compute A & B, ask LAPACK for eigenvalues— requires O(N2) storage, O(N3) time

(p smallest eigenvalues)

26

Solving the Maxwell Eigenproblem: 3b




Ah 2BhMany iterative methods:

— Arnoldi, Lanczos, Davidson, Jacobi-Davidson, …, Rayleigh-quotient minimization

27

Solving the Maxwell Eigenproblem: 3c




Ah 2BhMany iterative methods:

— Arnoldi, Lanczos, Davidson, Jacobi-Davidson, …, Rayleigh-quotient minimization

for Hermitian matrices, smallest eigenvalue 0 minimizes:

02 min

h

h' Ahh' Bh

minimize by preconditioned conjugate-gradient (or…)

“variationaltheorem”

28

Band Diagram of 2d Model System(radius 0.2a rods, e=12)

E

HTM

a

freq

uenc

y

(2π

c/a)

= a

/ l

G X

MG X M Girreducible Brillouin zone

k


gap forn > ~1.75:1

31

The Iteration Scheme is Important(minimizing function of 104–108+ variables!)

Steepest-descent: minimize (h + af) over a … repeat

02 min

h

h' Ahh'Bh

f (h)

Conjugate-gradient: minimize (h + ad)— d is f + (stuff): conjugate to previous search dirs

Preconditioned steepest descent: minimize (h + ad) — d = (approximate A-1) f ~ Newton’s method

Preconditioned conjugate-gradient: minimize (h + ad)— d is (approximate A-1) [f + (stuff)]

32

The Iteration Scheme is Important(minimizing function of ~40,000 variables)QuickTime™ and aGraphics decompressorare needed to see this picture.EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑJJJJJJJJJJJJJJJJJJJJJJJJJJJJ0.0000010.000010.00010.0010.010.111010010001000010000010000001101001000

# iterations

% e

rror

preconditionedconjugate-gradient no conjugate-gradient

no preconditioning

33

The Boundary Conditions are Tricky

e?

E|| is continuous

E is discontinuous

(D = eE is continuous)

Any single scalar e fails: (mean D) ≠ (any e) (mean E)

Use a tensor e:

ee

e 1 1

E||

E

34

The e-averaging is ImportantQuickTime™ and aGraphics decompressorare needed to see this picture.BBBBBBBBBBBBBJJJJJJJJJJJJJHHHHHHHHHHHHH0.010.111010010100

resolution (pixels/period)

% e

rror

backwards averaging

tensor averaging

no averaging

correct averagingchanges order of convergencefrom ∆x to ∆x2

(similar effectsin other E&M

numerics & analyses)

35

Gap, Schmap?

a

freq

uenc

y

G X M G


But, what can we do with the gap?

36

Intentional “defects” are good

3D Photonic Crystal with Defects

microcavities waveguides (“wires”)

37

Intentional “defects” in 2dQuickTime™ and aGraphics decompressorare needed to see this picture.a

QuickTime™ and aGraphics decompressorare needed to see this picture.QuickTime™ and aGraphics decompressorare needed to see this picture.QuickTime™ and aGraphics decompressorare needed to see this picture.(Same computation, with supercell = many primitive cells)

38

Microcavity Blues

For cavities (point defects)frequency-domain has its drawbacks:

• Best methods compute lowest- bands, but Nd supercells have Nd modes below the cavity mode — expensive

• Best methods are for Hermitian operators, but losses requires non-Hermitian

39

Time-Domain Eigensolvers(finite-difference time-domain = FDTD)

Simulate Maxwell’s equations on a discrete grid,+ absorbing boundaries (leakage loss)

• Excite with broad-spectrum dipole ( ) source

Response is manysharp peaks,

one peak per modecomplex n [ Mandelshtam,

J. Chem. Phys. 107, 6756 (1997) ]

signal processing

decay rate in time gives loss

40

QuickTime™ and aGraphics decompressorare needed to see this picture.EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE05010015020025030035040045000.511.522.533.54Signal Processing is Tricky

complex n

?signal processingQuickTime™ and aGraphics decompressorare needed to see this picture.EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE-1-0.8-0.6-0.4-0.200.20.40.60.8012345678910

Decaying signal (t) Lorentzian peak ()

FFT

a common approach: least-squares fit of spectrum

fit to:

A( 0)2 G2

41

QuickTime™ and aGraphics decompressorare needed to see this picture.EEEEEEEEEEE05000100001500020000250003000035000400000.50.60.70.80.911.11.21.31.41.5Fits and UncertaintyQuickTime™ and aGraphics decompressorare needed to see this picture.EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE-1-0.8-0.6-0.4-0.200.20.40.60.81012345678910

Portion of decaying signal (t) Unresolved Lorentzian peak ()

actual

signalportion

problem: have to run long enough to completely decay

There is a better way, which gets complex to > 10 digits

42

Unreliability of Fitting ProcessQuickTime™ and aGraphics decompressorare needed to see this picture.EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE0200400600800100012000.50.60.70.80.911.11.21.31.41.5 = 1+0.033i

= 1.03+0.025i

sum of two peaks

Resolving two overlapping peaks isnear-impossible 6-parameter nonlinear fit

(too many local minima to converge reliably)

Sum of two Lorentzian peaks ()

There is a better way, which gets

complex for both peaksto > 10 digits

43

Quantum-inspired signal processing (NMR spectroscopy):

Filter-Diagonalization Method (FDM)[ Mandelshtam, J. Chem. Phys. 107, 6756 (1997) ]

Given time series yn, write:

yn y(nt) ake i k nt

k

…find complex amplitudes ak & frequencies k

by a simple linear-algebra problem!

Idea: pretend y(t) is autocorrelation of a quantum system:

ˆ H i t

say:

yn (0) (nt) (0) ˆ U n (0)

time-∆t evolution-operator:

ˆ U e i ˆ H t /

44

Filter-Diagonalization Method (FDM)[ Mandelshtam, J. Chem. Phys. 107, 6756 (1997) ]

yn (0) (nt) (0) ˆ U n (0)

ˆ U e i ˆ H t /

We want to diagonalize U: eigenvalues of U are ei∆t

…expand U in basis of |(n∆t)>:

Um,n (mt) ˆ U (nt) (0) ˆ U m ˆ U ˆ U n (0) ymn1

Umn given by yn’s — just diagonalize known matrix!

45

Filter-Diagonalization Summary[ Mandelshtam, J. Chem. Phys. 107, 6756 (1997) ]

Umn given by yn’s — just diagonalize known matrix!

A few omitted steps: —Generalized eigenvalue problem (basis not orthogonal) —Filter yn’s (Fourier transform):

small bandwidth = smaller matrix (less singular)

• resolves many peaks at once • # peaks not known a priori • resolve overlapping peaks • resolution >> Fourier uncertainty

46

Do try this at home

Bloch-mode eigensolver: http://ab-initio.mit.edu/mpb/

Filter-diagonalization: http://ab-initio.mit.edu/harminv/

Photonic-crystal tutorials (+ THIS TALK): http://ab-initio.mit.edu/

/photons/tutorial/

Photonic Crystals:Periodic Surprises in Electromagnetism

Steven G. JohnsonMIT

A “Defective” Lecture

cavity waveguide

The Story So Far…a

Waves in periodic media can have:

• propagation with no scattering (conserved k)• photonic band gaps (with proper e function)

Eigenproblem gives simple insight:

( i

k )1

e( i

k )

H k

n (k )

c

2 H k

ˆ k Hermitian –> complete, orthogonal, variational theorem, etc.


k

H ei(

k x t ) H k (

x )Bloch form:

band diagram

Properties of Bulk Crystalsby Bloch’s theorem

QuickTime™ and aGraphics decompressorare needed to see this picture. (cartoon)

cons

erve

d fr

eque

ncy

conserved wavevector k

photonic band gap

band diagram (dispersion relation)

d/dk 0: slow light(e.g. DFB lasers)

backwards slope:negative refraction

strong curvature:super-prisms, …

(+ negative refraction)

synthetic mediumfor propagation

Applications of Bulk Crystals

Zero group-velocity d/dk: distributed feedback (DFB) lasers

negative group-velocity ornegative curvature (“eff. mass”):Negative refraction,Super-lensing

source imageobject image

negat ive refraction medium

super-lens Veselago (1968)

divergent dispersion (i.e. curvature): Superprisms

[Kosaka, PRB 58, R10096 (1998).]

using near-band-edge effects

[ C. Luo et al.,Appl. Phys. Lett. 81, 2352 (2002) ]

Photonic Crystals:Periodic Surprises in Electromagnetism

Steven G. JohnsonMIT

Fabrication of Three-Dimensional Crystals

Those Clever Experimentalists

Image: http://cst-www.nrl.navy.mil/lattice/struk/a4.html

The Mother of (almost) All Bandgaps

The diamond lattice:

fcc (face-centered-cubic)with two “atoms” per unit cell

(primitive)a

fcc = most-spherical Brillouin zone

+ diamond “bonds” = lowest (two) bands can concentrate in lines

Recipe for a complete gap:

QuickTime™ and aGraphics decompressorare needed to see this picture.JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ00.10.20.30.40.50.6freq

uenc

y (c

/a)

GXXULWKfreq

uenc

y (c

/a)

The First 3d Bandgap StructureK. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).

11% gap

overlapping Si spheres

MPB tutorial, http://ab-initio.mit.edu/mpb

L

GW

X

UK

for gap at l = 1.55µm,sphere diameter ~ 330nm

54

Lembram?

Make that? Are you crazy? …maybe!

[ F. Garcia-Santamaria et al., APL 79, 2309 (2001) ]http://www.icmm.csic.es/cefe/Fab/Robot/robot_strategy.htm

fabrication schematic

carefully stack bccsilica & latex spheresvia micromanipulation

…dissolve latex

& sinter (heat and fuse) silica

make Si inverse(12% gap)

Make that? Are you crazy? …maybe!

5µm

dissolvelatex spheres

6-layer [001] silica diamond lattice

4-layer [111] silica diamond lattice

5µm

[ F. Garcia-Santamaria et al., Adv. Mater. 14 (16), 1144 (2002). ]

A Layered StructureWe’ve Seen Already

rod layer

hole layer

(diamond-like: rods ~ “bonds”)

A

B

C

[ S. G. Johnson et al.,Appl. Phys. Lett. 77, 3490 (2000) ]

Up to ~ 27% gapfor Si/air

hole layer

Making Rods & Holes Simultaneously

substrate

top view

side view

Si


substrate

A A A A

A A A A

A A A

A A A A

A A A

A A A A

A A A

expose/etchholes


substrate

A A A A

A A A A

A A A

A A A A

A A A

A A A A

A A A

backfill withsilica (SiO2)& polish


substrate

A A A A

A A A A

A A A

A A A A

A A A

A A A A

A A A

deposit anotherSi layer

layer 1


B B B B

B B B B

B B B B

B B B

B B B

B B B

substrate

layer 1

A A A A

B B B B

A A A A

A A A

A A A A

A A A

A A A A

A A A

dig more holesoffset& overlapping


B B B B

B B B B

B B B B

B B B

B B B

B B B

substrate

layer 1

A A A A

B B B B

A A A A

A A A

A A A A

A A A

A A A A

A A A

backfill


B B B B

B B B B

B B B B

B B B

B B B

B B B

C C C C

C C C C

C C C C

C C C C

C C C C

C C C C

substrate

layer 1

layer 2

layer 3

A A A A

B B B B

C C C C

A A A A

A A A A

A A A

A A A A

A A A

A A A A

A A A

etcetera

(dissolvesilicawhendone)

oneperiod


B B B B

B B B B

B B B B

B B B

B B B

B B B

C C C C

C C C C

C C C C

C C C C

C C C C

C C C C

substrate

layer 1

layer 2

layer 3

A A A A

B B B B

C C C C

A A A A

A A A A

A A A

A A A A

A A A

A A A A

A A A

etcetera oneperiod

hole layers


B B B B

B B B B

B B B B

B B B

B B B

B B B

C C C C

C C C C

C C C C

C C C C

C C C C

C C C C

substrate

layer 1

layer 2

layer 3

A A A A

B B B B

C C C C

A A A A

A A A A

A A A

A A A A

A A A

A A A A

A A A

etcetera oneperiod

rod layers

67

Tb existe forma alternativa

A More Realistic Schematic

[ M. Qi, H. Smith, MIT ]

e-beam Fabrication: Top View

5 m


e-beam Fabrication: Side Views(cleaving worst sample)


substrate

layer 1A A A A

A A A A

B B B B

C C C C

2layer 3

layer 5

layer 7

4

6

Adding “Defect” Microcavities450nm

740nm

580nm

B'

Easiest defect: don’t etch some B holes— non-periodically distributed: suppresses sub-band structure— low Q = easier to detect from planewave


72

Outros resultados

Yes, it works: Gap at ~4µm[ K. Aoki et al., Nature Materials 2 (2), 117 (2003) ]

1µm

50nm accuracy:

(gap effects are limited by finite lateral size)

20 layers

2µm

Lithography is a Beast[ S. Kawata et al., Nature 412, 697 (2001) ]

l = 780nm

resolution = 150nm

7µm

(3 hours to make)

2µm

For a physicist, this is cooler…[ S. Kawata et al., Nature 412, 697 (2001) ]

(300nm diameter coils, suspended in ethanol, viscosity-damped)

A Two-Photon Woodpile Crystal[ B. H. Cumpston et al., Nature 398, 51 (1999) ]

(much work on materialswith lower power 2-photon process)

Difficult topologies

• Arbitrary lattice• No “mask”• Fast/cheap prototyping

[ fig. courtesy J. W. Perry, U. Arizona ]

77

Atualmente existem vários métodos e processos para a construção de cristais

fotônicos, fora aqueles citados anteriormente.

É um novo campo com grandes expectativas.

78

Como Químicos e Físicos se entendem sobre estado sólido

79

Próxima aulaApresentação de temas

resumo aula passada

Documents