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 1

2. 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/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_Bandshttp://ab-initio.mit.edu/wiki/index.php/MIT_Photonic_Bands 3

4. 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. 5 Solid of N atoms Two atomsSix atoms Band Theory: “Bound” Electron Approach For the total number N of atoms in a solid (10 23 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. 6 Filtro de Fabry-Perot C_MEMS http://www.npphotonics.com/files/article/OEG20030324S0088.htm

7. 7 O seguinte é um seminário dado por Steven G. Johnson, MIT Applied Mathematics

8. 8 From electrons to photons: Quantum-inspired modeling in nanophotonics Steven G. Johnson, MIT Applied Mathematics

9. 9 Nano-photonic media ( -scale) synthetic materials strange waveguides 3d structures hollow-core fibers optical phenomena & microcavities [B. Norris, UMN] [Assefa & Kolodziejski, MIT] [Mangan, Corning]

10. 10 18871987 Photonic Crystals periodic electromagnetic media can have a band gap: optical “insulators”

11. Cristal fotônico 1D 11 11 22 11 22 11 22 11 22 11 22 11 22  (x) =  (x+a) a

12. Elétron numa rede periódica 1D 12 Elétron num ambiente livre Solução:

13. Elétron numa rede periódica – aproximação por poço retangular periódico 13 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 14

15. 15 Electronic and Photonic Crystals atoms in diamond structure wavevector electron energy Periodic Medium Bloch waves: Band Diagram dielectric spheres, diamond lattice wavevector photon frequency interacting: hard problem non-interacting: easy problem

16. 16 Electronic & Photonic Modelling ElectronicPhotonic 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

17. 17 Fun with Math 0 dielectric function  (x) = n 2 (x) First task: get rid of this mess eigen-operatoreigen-value eigen-state + constraint

18. 18 Electronic & Photonic Eigenproblems ElectronicPhotonic 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. 19 A 2d Model System square lattice, period a dielectric “atom”  =12 (e.g. Si) a a E H TM

20. 20 Periodic Eigenproblems if eigen-operator is periodic, then Bloch-Floquet theorem applies: 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)

21. 21 Solving the Maxwell Eigenproblem H(x,y) e i(k  x –  t) 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

22. 22 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 kxkx kyky first Brillouin zone = minimum |k| “primitive cell”  M X irreducible Brillouin zone: reduced by symmetry

23. 23 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 solve: finite matrix problem:

24. 24 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 — must satisfy constraint: Planewave (FFT) basis constraint: uniform “grid,” periodic boundaries, simple code, O(N log N) Finite-element basis constraint, 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. 25 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 Faster way: — start with initial guess eigenvector h 0 — iteratively improve — O(Np) storage, ~ O(Np 2 ) time for p eigenvectors Slow way: compute A & B, ask LAPACK for eigenvalues — requires O(N 2 ) storage, O(N 3 ) time (p smallest eigenvalues)

26. 26 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 Many iterative methods: — Arnoldi, Lanczos, Davidson, Jacobi-Davidson, …, Rayleigh-quotient minimization

27. 27 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 Many iterative methods: — Arnoldi, Lanczos, Davidson, Jacobi-Davidson, …, Rayleigh-quotient minimization for Hermitian matrices, smallest eigenvalue  0 minimizes: minimize by preconditioned conjugate-gradient (or…) “variational theorem”

28. 28 Band Diagram of 2d Model System (radius 0.2a rods,  =12) E H TM a frequency  (2πc/a) = a /  X M  XM  irreducible Brillouin zone gap for n > ~1.75:1

29. 29 Origin of the Band Gap Hermitian eigenproblems: solutions are orthogonal and satisfy a variational theorem ElectronicPhotonic minimize kinetic + potential energy (e.g. “bonding” state) minimize: field oscillations field in high  higher bands orthogonal to lower — must oscillate (high kinetic) or be in low  (high potential) (e.g. “anti-bonding” state)

30. 30 Origin of Gap in 2d Model System E H TM  XM  EzEz –+ EzEz gap for n > ~1.75:1 lives in high  orthogonal: node in high 

31. 31 The Iteration Scheme is Important (minimizing function of 10 4 –10 8 + variables!) Steepest-descent: minimize (h +  f) over  … repeat Conjugate-gradient: minimize (h +  d) — d is  f + (stuff): conjugate to previous search dirs Preconditioned steepest descent: minimize (h +  d) — d = (approximate A -1 )  f ~ Newton’s method Preconditioned conjugate-gradient: minimize (h +  d) — d is (approximate A -1 ) [  f + (stuff)]

32. 32 The Iteration Scheme is Important (minimizing function of ~40,000 variables) # iterations % error preconditioned conjugate-gradient no conjugate-gradient no preconditioning

33. 33 The Boundary Conditions are Tricky  E || is continuous E  is discontinuous (D  =  E  is continuous) Any single scalar  fails: (mean D) ≠ (any  ) (mean E) Use a tensor  E || EE

34. 34 The  -averaging is Important resolution (pixels/period) % error backwards averaging tensor averaging no averaging correct averaging changes order of convergence from ∆x to ∆x 2 (similar effects in other E&M numerics & analyses)

35. 35 Gap, Schmap? a frequency   XM  But, what can we do with the gap?

36. 36 Intentional “defects” are good microcavities waveguides (“wires”)

37. 37 Intentional “defects” in 2d (Same computation, with supercell = many primitive cells)

38. 38 Microcavity Blues For cavities (point defects) frequency-domain has its drawbacks: Best methods compute lowest-  bands, but N d supercells have N d modes below the cavity mode — expensive Best methods are for Hermitian operators, but losses requires non-Hermitian

39. 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 many sharp peaks, one peak per mode complex  n [ Mandelshtam, J. Chem. Phys. 107, 6756 (1997) ] signal processing decay rate in time gives loss

40. 40 Signal Processing is Tricky complex  n ? signal processing Decaying signal (t) Lorentzian peak (  ) FFT a common approach: least-squares fit of spectrum fit to:

41. 41 Fits and Uncertainty Portion of decaying signal (t) Unresolved Lorentzian peak (  ) actual signal portion problem: have to run long enough to completely decay There is a better way, which gets complex  to > 10 digits

42. 42 Unreliability of Fitting Process  = 1+0.033i  = 1.03+0.025i sum of two peaks Resolving two overlapping peaks is near-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 peaks to > 10 digits

43. 43 Quantum-inspired signal processing (NMR spectroscopy): Filter-Diagonalization Method (FDM) [ Mandelshtam, J. Chem. Phys. 107, 6756 (1997) ] Given time series y n, write: …find complex amplitudes a k & frequencies  k by a simple linear-algebra problem! Idea: pretend y(t) is autocorrelation of a quantum system: say: time-∆t evolution-operator:

44. 44 Filter-Diagonalization Method (FDM) [ Mandelshtam, J. Chem. Phys. 107, 6756 (1997) ] We want to diagonalize U: eigenvalues of U are e i  ∆t …expand U in basis of |  (n∆t)>: U mn given by y n ’s — just diagonalize known matrix!

45. 45 Filter-Diagonalization Summary [ Mandelshtam, J. Chem. Phys. 107, 6756 (1997) ] U mn given by y n ’s — just diagonalize known matrix! A few omitted steps: —Generalized eigenvalue problem (basis not orthogonal) —Filter y n ’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. 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/

47. Photonic Crystals: Periodic Surprises in Electromagnetism Steven G. Johnson MIT A “Defective” Lecture

48. The Story So Far… a Waves in periodic media can have: propagation with no scattering (conserved k) photonic band gaps (with proper  function) Eigenproblem gives simple insight: Hermitian –> complete, orthogonal, variational theorem, etc.  k Bloch form: band diagram

49. Properties of Bulk Crystals by Bloch’s theorem (cartoon) conserved frequency  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 medium for propagation

50. Applications of Bulk Crystals Zero group-velocity d  /dk: distributed feedback (DFB) lasers negative group-velocity or negative curvature (“eff. mass”): Negative refraction, Super-lensing 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) ]

51. Photonic Crystals: Periodic Surprises in Electromagnetism Steven G. Johnson MIT Fabrication of Three-Dimensional Crystals Those Clever Experimentalists

52. 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:

53. frequency (c/a) The First 3d Bandgap Structure K. 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 for gap at = 1.55µm, sphere diameter ~ 330nm

54. Lembram? 54

55. 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 bcc silica & latex spheres via micromanipulation …dissolve latex & sinter (heat and fuse) silica make Si inverse (12% gap)

56. Make that? Are you crazy? …maybe! 5µm dissolve latex 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). ]

57. A Layered Structure We’ve Seen Already (diamond-like: rods ~ “bonds”) A B C [ S. G. Johnson et al., Appl. Phys. Lett. 77, 3490 (2000) ] Up to ~ 27% gap for Si/air hole layer

58. Making Rods & Holes Simultaneously substrate top view side view Si

59. Making Rods & Holes Simultaneously substrate AAAA AAAA AAA AAAA AAA AAAA AAA expose/etch holes

60. Making Rods & Holes Simultaneously substrate AAAA AAAA AAA AAAA AAA AAAA AAA backfill with silica (SiO 2 ) & polish

61. Making Rods & Holes Simultaneously substrate AAAA AAAA AAA AAAA AAA AAAA AAA deposit another Si layer layer 1

62. Making Rods & Holes Simultaneously BBBB BBBB BBBB BBB BBB BBB substrate layer 1 AAAA BBBB AAAA AAA AAAA AAA AAAA AAA dig more holes offset & overlapping

63. Making Rods & Holes Simultaneously BBBB BBBB BBBB BBB BBB BBB substrate layer 1 AAAA BBBB AAAA AAA AAAA AAA AAAA AAA backfill

64. Making Rods & Holes Simultaneously BBBB BBBB BBBB BBB BBB BBB CCCC CCCC CCCC CCCC CCCC CCCC substrate layer 1 layer 2 layer 3 AAAA BBBB CCCC AAAA AAAA AAA AAAA AAA AAAA AAA etcetera (dissolve silica when done) one period

65. Making Rods & Holes Simultaneously BBBB BBBB BBBB BBB BBB BBB CCCC CCCC CCCC CCCC CCCC CCCC substrate layer 1 layer 2 layer 3 AAAA BBBB CCCC AAAA AAAA AAA AAAA AAA AAAA AAA etcetera one period hole layers

66. Making Rods & Holes Simultaneously BBBB BBBB BBBB BBB BBB BBB CCCC CCCC CCCC CCCC CCCC CCCC substrate layer 1 layer 2 layer 3 AAAA BBBB CCCC AAAA AAAA AAA AAAA AAA AAAA AAA etcetera one period rod layers

67. Tb existe forma alternativa 67

68. A More Realistic Schematic [ M. Qi, H. Smith, MIT ]

69. e-beam Fabrication: Top View [ M. Qi, H. Smith, MIT ]

70. e-beam Fabrication: Side Views (cleaving worst sample) [ M. Qi, H. Smith, MIT ]

71. Adding “Defect” Microcavities 450nm 740nm 580nm 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 72

73. 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

74. 2µm Lithography is a Beast [ S. Kawata et al., Nature 412, 697 (2001) ] = 780nm resolution = 150nm 7µm (3 hours to make)

75. 2µm For a physicist, this is cooler… [ S. Kawata et al., Nature 412, 697 (2001) ] (300nm diameter coils, suspended in ethanol, viscosity-damped)

76. A Two-Photon Woodpile Crystal [ B. H. Cumpston et al., Nature 398, 51 (1999) ] (much work on materials with 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. 77

78. Como Químicos e Físicos se entendem sobre estado sólido 78

79. 79 Próxima aula Apresentação de temas