Download - 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
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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
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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
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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
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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.
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Filtro de Fabry-Perot C_MEMS
http://www.npphotonics.com/files/article/OEG20030324S0088.htm
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O seguinte é um seminário dado porSteven G. Johnson, MIT Applied
Mathematics
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From electrons to photons: Quantum-inspired modeling in nanophotonics
Steven G. Johnson, MIT Applied Mathematics
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Nano-photonic media (l-scale)
synthetic materials
strange waveguides
3d structures
hollow-core fibersoptical phenomena
& microcavities[B. Norris, UMN] [Assefa & Kolodziejski,
MIT]
[Mangan, Corning]
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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”
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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
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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
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Elétron numa rede periódica – aproximação por poço retangular periódico
Mostra a existência de banda proibida, imposta por condições de contorno
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Simulation of Band Gap Structures of 1D Photonic Crystal
• Journal of the Korean Physical Society, Vol. 52, February 2008, pp. S71S74
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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
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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
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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
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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…
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A 2d Model System
square lattice,period a
dielectric “atom”e=12 (e.g. Si)
a
a
E
HTM
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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
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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
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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
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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:
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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) ]
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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)
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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
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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 0 minimizes:
02 min
h
h' Ahh' Bh
minimize by preconditioned conjugate-gradient (or…)
“variationaltheorem”
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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
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
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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)]
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# iterations
% e
rror
preconditionedconjugate-gradient no conjugate-gradient
no preconditioning
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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
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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)
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Gap, Schmap?
a
freq
uenc
y
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?
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Intentional “defects” are good
3D Photonic Crystal with Defects
microcavities waveguides (“wires”)
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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)
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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
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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
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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
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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
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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
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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 /
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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 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!
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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/
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.
QuickTime™ and aGraphics decompressorare needed to see this picture.00.10.20.30.40.50.60.70.80.91Photonic Band GapTM bands
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
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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
Making Rods & Holes Simultaneously
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
Making Rods & Holes Simultaneously
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
Making Rods & Holes Simultaneously
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
Making Rods & Holes Simultaneously
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
Making Rods & Holes Simultaneously
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
Making Rods & Holes Simultaneously
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
Making Rods & Holes Simultaneously
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
Making Rods & Holes Simultaneously
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
[ M. Qi, H. Smith, MIT ]
e-beam Fabrication: Side Views(cleaving worst sample)
[ M. Qi, H. Smith, MIT ]
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
[ M. Qi, H. Smith, MIT ]
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