1 Resumo anterior Modulação Eletromecânica – chopper Eletro-óptica Efeito Pockel Efeito Kerr...
Transcript of 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.
5
SPM lithography
6
7
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.......
14
15
16
17
18
19
20
21
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
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
33
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
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
58
59
60
61
62
63
64
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
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
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
71
Mais uma geral
72
Com que roupa vou?
73
74
75
76
77
78
79
80
MEMS = Micro Electro Mechanical System
81
Engrenagens - acaro
82
Situação atual
Design Fabricação
Vendas de substratos Mascaras
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
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
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
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
2 Limit degrees of freedom: expand H in finite basis
3 Efficiently solve eigenproblem: iterative methods
—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
1 Limit range of k: irreducible Brillouin zone
2 Limit degrees of freedom: expand H in finite basis (N)
3 Efficiently solve eigenproblem: iterative methods
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
1 Limit range of k: irreducible Brillouin zone
2 Limit degrees of freedom: expand H in finite basis
3 Efficiently solve eigenproblem: iterative methods
( 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
1 Limit range of k: irreducible Brillouin zone
2 Limit degrees of freedom: expand H in finite basis
3 Efficiently solve eigenproblem: iterative methods
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
1 Limit range of k: irreducible Brillouin zone
2 Limit degrees of freedom: expand H in finite basis
3 Efficiently solve eigenproblem: iterative methods
Ah 2BhMany iterative methods:
— Arnoldi, Lanczos, Davidson, Jacobi-Davidson, …, Rayleigh-quotient minimization
110
Solving the Maxwell Eigenproblem: 3c
1 Limit range of k: irreducible Brillouin zone
2 Limit degrees of freedom: expand H in finite basis
3 Efficiently solve eigenproblem: iterative methods
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
QuickTime™ and aGraphics decompressorare needed to see this picture.00.10.20.30.40.50.60.70.80.91Photonic Band GapTM bands
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
(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
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
QuickTime™ and aGraphics decompressorare needed to see this picture.00.10.20.30.40.50.60.70.80.91Photonic Band GapTM bands
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 /
127
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!
128
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
129
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/
130
Apresentação de temas