1 resumo anterior modulação eletromecânica – chopper eletro-óptica efeito pockel efeito kerr...

1

Resumo anterior

• Modulação Eletromecânica – chopper Eletro-óptica

Efeito Pockel Efeito Kerr

Magneto-óptica Efeito Faraday Magneto-Kerr

Acusto-óptica Reflexão de Bragg, célula de Bragg Xstal de quartzo, PZT

• Fotolitografia. Demo de materias p/ fotolitografia: placa cobreadas de fenolite, fibra de vidro, face simples , face duplas. Fotolito.

• Litografia20110613

2

Eletro-óptico Pockel e Kerr

Pockel

Kerr

3

Magnetooptical Kerr Effect = MOKE

4

O passo a passo da litografia

• Ver em: http://www.ee.byu.edu/cleanroom/lithography.phtml e procurar por Basic Lithography Tutorial é um java script com animação.

http://www.ee.byu.edu/cleanroom/lithography.phtml

5

SPM lithography

8

Litografia de imersão

Limite de resolução para litografia é usando a eq de Rayleigh:

W é a largura da linha impressa.

Onde k1 é o fator de resolução, l é o comprimento de onda da radiação de exposição e NA é a apertura numérica.

A colocação de água aumenta a NA (nsenq)

9

Litografia de imersão

10

Evolução da largura de linha mínima e l

• O fator de resolução k1 é um fator complexo que depende de várias variáveis no processo de fotolitografia: qld do fotoresist, técnicas de melhoramento da resolução, tipo de mascaras, tipo de iluminação, entre outros.

11

Evolução de NA e k1

Laser de ArF=> 193 nm

G-line => 436nm

I-line => 365nm

12

Imersão

13

Existem vários tipos de processos litográficos

• Feixe de elétrons• EUV• SPL• Raio-X• Mas.......

22

Independente da forma como é realizada....

• Observar o produto final objetivo dos efeitos que desejamos

23

Óptica integrada

• Desenvolvimento de dispositivos ópticos miniaturizados, em escala de micro – nano, de alta funcionalidade sobre substratos.

• Nesta área é possível distinguir: Circuitos ópticos integrados. A luz é confinada em guias de onda

de filmes finos, depositados ou cavados no substrato (vidro, xstal dielétrico, semicondutor).

Dispositivos ópticos planares

24

Canais de guia de onda fio de cobre

Condições: índice de refraçãomaior na guia do resto domaterial.

As guias de onda são feitas por deposição de material sobre o topo dosubstrato e posteriormente atacadoquimicamente para retirar o resto do material. Pode ser ao contrário tb, Fazendo os sulcos por ataque químicoe posteriormente preenchido com material da guia de onda

25

Outra guia de onda

26

Guia de onda em LiNbO3

• Tiras de Ti depositado no padrão de guia de onda desejado sobre substrato de LiNbO3 puro

• Aquecimento => difusão• Obtenção de guia de onda semicircular

27

Phase Shifter

A mudança de fase vem unicamente do efeito Pockels, campo elétrico provocado através dos dois fios de ouro, que fazem a mudança do índice de refração. Tem sido obtidos moduladores de até 40Gb/s.

28

SAW = Surface Acoustic Wave

Dispositivos acusto-ópticos são fabricados por processos fotolitográficos

Tipicamente consiste de dois conjuntos de transdutores interdigitais.

Um transdutor converte a energia do sinal elétrico em energia mecânica ondulatória.

O outro transdutor faz o processo reverso.

29

Diferentes arranjos dos

eletrodos

IDT =InterDigital Transducer

30

SAW

31

SAW+fibra = demux = dispositivo integrado

http://fb6www.uni-paderborn.de/ag/ag-sol/research/acousto/convert.htm

Conversor de polarização acusto-óptico

http://fb6www.uni-paderborn.de/ag/ag-sol/research/acousto/convert.htm

32

Filtros TE e TM

Y2O3 = 17nm

Al = 100nm

Comprimento = 1,5mm extinção 20dB TM e 0,5dB atenuação TE

Troca de Li por H

Região H aumenta ne

no diminui

TE se acopla na região H, extinção 25dB

TM atenua 1dB

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Materiais eletro-ópticos

Table 1. Electro-Optic Materials

Material AbbreviationChemical Formula

Transmission Range (mm)

Bandwidth (MHz)

Index ofRefraction

no,ne atwavelength

(mm)

Ammonium dihydrogen phosphate

ADP NH4H2PO4 0.3 - 1.2 to 5001.51, 1.47

at 1.06

Potassiumdihydrogen phosphate

KDP KH2PO4 0.25 - 1.7 > 1001.51, 1.47

at 0.55

Potassium dideuterium phosphate

KD*P KD2PO4 0.3 - 1.1 to 3501.49, 1.46

at 1.06

Lithium niobate LN LiNbO3 0.5 - 2 to 80002.23, 2.16

at 1.06

Lithium tantalate — LiTaO3 0.4 - 1.1 to 10002.14, 2.143

at 1.00

Cadmium telluride

— CdTe 2 - 16 to 1000no = 2.6

at 10

34

Table 2. Acousto-Optic Materials

MaterialChemical formula

Spectral range (mm)

Figure of merit M2

(10- 15 m2/W)

Bandwidth (MHz)

Typical drive

power (W)

IndexofRefraction

AcousticVelocity(m/sec)

Fused silica/quartz

SiO2 0.3 - 1.5 1.6 to 20 61.46

(634,3 nm)5900

Gallium arsenide

GaAs 1.0 - 11 104 to 350 13.37

(1.15 mm)5340

Gallium phosphide

GaP 0.59 - 1.0 45 to 1000 503.31

(1.15 mm)6320

Germanium Ge 2.5 - 15 840 to 5 504.0

(10.6 mm)5500

Lead molybdate

PbMoO4 0.4 - 1.2 50 to 50 1 - 22.26

(633 nm)3630

Telluriumdioxide

TeO2 0.4 - 5 35 to 300 1 - 22.26

(633 nm)4200

Lithiumniobate

L6Nb03 0.5-2 7 > 300 50-1002.20

(633nm)6570

35

Dispositivos

• Podem ser fabricados sobre substratos planares usando processos litográficos comuns e tecnologia de filmes finos.

• Litografia por feixe de elétrons ou por laser • Métodos epitaxiais para fabricação de fontes, detectores

e circuitos opto-eletrônicos.• AsGa, Si, InP

ramificação

36

Acoplamentos

Acopladores direcionais como ramificadorespor acoplamento de ondas evanescentes entre acopladores adjacentes

Acoplador por reflexão de Bragg

37

Chaveamento

Interferômetro Mach-Zehnder

Aquecedor (ms) ou onda acústica (piezo) (ms) ou sinal elétrico num dos braços

38

Outro tipo de chaveamento

39

MZI com filtro de Bragg

40

Algumas aplicações

• Filtros

• MUX/DEMUX

• Chaveadores

• Amplificadores ópticos

• Acopladores

41

O que transferir e como

Meta com óptica integrada

42

Materiais

43

Materiais

44

Componentes fotônicos integrados

45

Mais componentes fotônicos

46

Filtro sintonizável com SAW

47

EDFA integrado

48

Outros processos

49

Comparando duas formas de fazer litografia

50

Escrita direta com laser• Spot ~1 - 5um • Tightly Focuses,

modulated He-Cd or Argon-ion laser scanned across photresists surface

• Up to 256 phase levels

• Serial Process • Difficult to

accurately transfer structure into substrate

• Direct ablation of polyimide layer on substrate using an excimer laser is also possible

• Pattern can be transferred to a VHOE by processing in a 4f optical processor.

VHOE(volume holographic optical element)

51

Processo de photoresist para litografia

52

Recobrimento do photoresist por spinner

53

Método de replicação

54

Fontes de luz em litografia

G-line 436nm

I-line 365nm

KrF Excimer 248nm

ArF Excimer 193nm

EUV < 13nm

55

Materiais Fotônicos

56

Materiais Fotônicos

Dispositivos Fotônicos

Cristais Fotônicos

Algumas referencias

• http://mems.colorado.edu/c1.res.ppt/cat.g.shtml• Cid Araujo, Óptica Não-linear. VIII Escola J.A. Swieka 2002• http://www.photonic-lattice.com/en/Tech01.html

http://mems.colorado.edu/c1.res.ppt/cat.g.shtml

http://www.photonic-lattice.com/en/Tech01.html

57

Fotônica

• Ciência e tecnologia baseada em e relacionada com o controle e processamento de radiação eletromagnética, fóton de luz, incorporando óptica – eletrônica - ciência dos materiais – laser – memória – processamentos.

• É o equivalente óptico da eletrônica

65

Trabalhar em qq sala?

Sala limpa• Qualidade, temperatura e unidade do ar altamente

controlada para evitar contaminação, e.g.: Centros cirúrgicos (aquela estória de infecção hospitalar) Laboratórios de processamentos litográficos

• Remoção de partículas e impurezas, inclusive bactérias, através de processos de filtragem

• Classificação de Sala Limpa: Americano: Federal Standard 209 Europeu: ISO 14644-1

66

Classes de sala limpa, padrão Americano

Número máximo de partículas no ar (partículas por pé cúbico de ar)

ClasseTamanho da Partícula

0.1 μm 0.2 μm 0.3 μm 0.5 μm 5.0 μm

1 35 7.5 3 1

10 350 75 30 10

100 750 300 100

1,000 1,000 7

10,000 10,000 70

100,000 100,000 700

http://www.engineeringtoolbox.com/clean-rooms-d_932.html

http://www.engineeringtoolbox.com/clean-rooms-d_932.html

67

Classes de sala limpa, padrão ISO

CLASS Número de partículas por metro cúbico por tamanho micrométrico

0.1 um 0.2 um 0.3 um 0.5 um 1 um 5 um

ISO 1 10 2

ISO 2 100 24 10 4

ISO 3 1,000 237 102 35 8

ISO 4 10,000 2,370 1,020 352 83

ISO 5 100,000 23,700 10,200 3,520 832 29

ISO 6 1,000,000 237,000 102,000 35,200 8,320 293

ISO 7 352,000 83,200 2,930

ISO 8 3,520,000 832,000 29,300

ISO 9 35,200,000 8,320,000 293,000

http://www.particle.com/whitepapers_met/Cleanroom%20Standards.htm

http://www.particle.com/whitepapers_met/Cleanroom%20Standards.htm

68

Outras normas ISO para sala limpa

ISO Document Title

ISO-14644-1 Classification of Air Cleanliness

ISO-14644-2 Cleanroom Testing for Compliance

ISO-14644-3 Methods for Evaluating & Measuring Cleanrooms & Associated Controlled Environments

ISO-14644-4 Cleanroom Design & Construction

ISO-14644-5 Cleanroom Operations

ISO-14644-6 Terms, Definitions & Units

ISO-14644-7 Enhanced Clean Devices

ISO-14644-8 Molecular Contamination

ISO-14698-1 Biocontamination: Control General Principles

ISO-14698-2 Biocontamination: Evaluation & Interpretation of Data

ISO-14698-3 Biocontamination: Methodology for Measuring Efficiency of Cleaning Inert Surfaces

69

Tamanhos de partículas (mm)

70

Outra representação dos tamanhos

HEPA = high efficiency particulate air

http://en.wikipedia.org/wiki/Particulate

71

Mais uma geral

72

Com que roupa vou?

80

MEMS = Micro Electro Mechanical System

81

Engrenagens - acaro

82

Situação atual

Design Fabricação

Vendas de substratos Mascaras

http://www.mems-exchange.org/services/design

http://www.mems-exchange.org/services/fabrication

http://www.mems-exchange.org/services/wafer_sales

http://www.mems-exchange.org/services/mask_making

83

MEMS @ NIST

http://mems.nist.gov/

http://mems.nist.gov/

84

Detecção de fluorescência

85

MEMS compatible micro-GRIN lenses for fiberto chip coupling of light

• Michael Zickar, Wilfried Noell, Cornel Marxer, Nico de Rooij. Institute of Microtechnology (IMT), University of Neuchatel, Switzerland

86

Acoplamento lente grin –fibra óptica

87

Novos materiais óptica integrada

http://www.solgel.com/articles/june01/owghyb.asp

http://www.solgel.com/articles/june01/owghyb.asp

88

Outro sistema

89

Microfone óptico

http://www.imt.tu-bs.de/imt/en/institut/mitarb/feldmann/projekte/mikrofon

Fig. 1: optical principle

Fig. 2: basic build-up

Structuring the optical fibers to produce a prism by polishing the chip

optical work bench with discrete optical components

Integrated optics like lenses, prisms and waveguides

http://www.imt.tu-bs.de/imt/en/institut/mitarb/feldmann/projekte/mikrofon

90

Cristais Fotônicos

Elétrons de um lado e fótons do outro

lado, junção de fóton + eletrônico

Temos elétrons em sólidos e fótons

em......materiais fotônicos

91

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

92

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.

93

Filtro de Fabry-Perot C_MEMS

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

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

94

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

95

From electrons to photons: Quantum-inspired modeling in nanophotonics

Steven G. Johnson, MIT Applied Mathematics

96

Nano-photonic media (l-scale)

synthetic materials

strange waveguides

3d structures

hollow-core fibersoptical phenomena

& microcavities

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

[Mangan, Corning]

97

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”

98

Electronic and Photonic Crystals

atoms in diamond structure

wavevector

elec

tron

ene

rgy

Per

iod

ic M

ediu

mB

loch

wav

es:

Ban

d D

iagr

amdielectric spheres, diamond lattice

wavevector

phot

on f

requ

ency

interacting: hard problem non-interacting: easy problem

99

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 10

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

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

100

Fun with Math

E 1

c

t

H i

c

H

H

1

c

t

E

J i

c

E

0

dielectric function e(x) = n2(x)

First task:get rid of this mess

1

H

c

2 H

eigen-operator eigen-value eigen-state

H 0+ constraint

101

Electronic & Photonic Eigenproblems

1

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 & w, … Periodicity = Bloch’s theorem…

102

A 2d Model System

square lattice,period a

dielectric “atom”e=12 (e.g. Si)

a

a

E

HTM

103

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 w are discrete wn(k)H

k

104

Solving the Maxwell Eigenproblem

H(x,y) ei(kx – wt)

ik 1

ik Hn

n2

c 2 Hn

ik H 0

where:

constraint:

1

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

Finite cell discrete eigenvalues wn

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

105

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

106

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:

107

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) ]

108

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)

109

Solving the Maxwell Eigenproblem: 3b




Ah 2BhMany iterative methods:

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

110

Solving the Maxwell Eigenproblem: 3c




Ah 2BhMany iterative methods:

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

for Hermitian matrices, smallest eigenvalue w0 minimizes:

02 min

h

h' Ahh' Bh

minimize by preconditioned conjugate-gradient (or…)

“variationaltheorem”

111

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

E

HTM

a

freq

uenc

y w

(2π

c/a)

= a

/ l

G X

M

G X M Girreducible Brillouin zone

k


gap forn > ~1.75:1

114

The Iteration Scheme is Important

(minimizing function of 104–108+ variables!)

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

02 min

h

h' Ah

h'Bh f (h)

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

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

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

115

The Iteration Scheme is Important

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# iterations

% e

rror

preconditionedconjugate-gradient no conjugate-gradient

no preconditioning

116

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

1 1

E||

E

117

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)

118

Gap, Schmap?

a

freq

uenc

y w

G X M G


But, what can we do with the gap?

119

Intentional “defects” are good

3D Photonic Crystal with Defects

microcavities waveguides (“wires”)

120

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)

121

Microcavity Blues

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

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

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

122

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

Dw

Response is manysharp peaks,

one peak per modecomplex wn [ Mandelshtam,

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

signal processing

decay rate in time gives loss

123

QuickTime™ and aGraphics decompressorare needed to see this picture.EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE05010015020025030035040045000.511.522.533.54

Signal Processing is Tricky

complex wn

?

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 (w)

FFT

a common approach: least-squares fit of spectrum

fit to:

A

( 0)2 2

124

QuickTime™ and aGraphics decompressorare needed to see this picture.EEEEEEEEEEE05000100001500020000250003000035000400000.50.60.70.80.911.11.21.31.41.5

Fits and UncertaintyQuickTime™ and aGraphics decompressorare needed to see this picture.EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE-1-0.8-0.6-0.4-0.200.20.40.60.81012345678910Portion of decaying signal (t) Unresolved Lorentzian peak (w)

actual

signalportion

problem: have to run long enough to completely decay

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

125

Unreliability of Fitting ProcessQuickTime™ and aGraphics decompressorare needed to see this picture.EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE020040060080010001200

0.50.60.70.80.911.11.21.31.41.5w = 1+0.033i

w = 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 (w)

There is a better way, which gets

complex wfor both peaksto > 10 digits

126

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 wk

by a simple linear-algebra problem!

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

ˆ H it

say:

yn (0) (nt) (0) ˆ U n (0)

time-∆t evolution-operator:

ˆ U e i ˆ H t /

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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 eiw∆t

…expand U in basis of |(n∆t)>:

Um,n (mt) ˆ U (nt) (0) ˆ U m ˆ U ˆ U n (0) ym n 1

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

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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

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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/

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