7-2E. Photonic crystals

Transcription

1 7-2E. Photonic crystals Purdue Univ, Prof. Shalaev, Univ Central Florida, CREOL, Prof Kik,

2 3-D 2-D Λ 1-D

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9 Consider a two-dimensional photonic crystal

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13 Bloch theorem

14 Bloch theorem

15 Bloch theorem

16 Bloch theorem

17 Bloch theorem

18 Bloch theorem

19 Photonic Bandstructure Dispersion curve = Photonic band structure Bandgap #2 Bandgap (no transmission) Standing wave v group =0 Long wavelength limit: effective index

20 Dispersion curve = Photonic band structure

21 Remind the Dispersion Curve of Slab Waveguide Dispersion curve = Photonic band structure Because guiding modes redistribute themselves with frequency, for small ω, the dispersion curve of guiding modes approaches the cladding line; Band structure For large ω, it approaches the For large ω, it approaches the core line.

22 Dispersion curve = Photonic band structure

23 Origin of Photonic Band Gap (PBG) Light in 1-D photonic crystal H L H L H L Photonic band gap

24 Photonic band gap

25 Photonic band gap Bragg Reflection λ B = 2 nd Sin( θ ) λ B B ~2d k B 2π = = λ B π d

26 Photonic band gap Bragg Diffraction Wavelength corresponds to the period. Reflected waves are in phase. Wave does not propagate inside. Wavelength does not correspond to the period Reflected waves are not in phase. Wave propagates through.

27 Photonic band gap Electron Energy gap E 2 h 2 = h k 2m Gap in energy spectra of electrons arises in periodic structure

28 PBG formation Photonic band gap 1. Dispersion curve for free space 3. At the band edges, standing waves form, with the energy being either in the high or the low index regions 2. In a periodic system, y when half the wavelength corresponds to the periodicity λ 2 = a k = π a the Bragg effect prohibits photon propagation. 4. Standing waves transport no energy with zero group velocity

29 Dispersion relation Dispersion curve = Photonic band structure ω n 1 : high h index material n 2 : low index material 4. Standing waves transport no energy with zero group velocity Air standing band wave in n 2 n 1 n 2 n 1 n 2 n 1 n 2 n 1 Stop band bandgap standing wave in n 1 Dielectric band 0 π/a k

30 Dispersion Relation Dispersion curve = Photonic band structure Plot the dispersion curves for both the positive and the negative sides, and then shift the curve segments with k >π/a upward or downward one reciprocal lattice vectors. This reduced range of wave vectors is called the Brillouin zone

31 2-D Photonic Crystals 1. In 2-D PBG, different layer spacing, a, can be met along different direction. Strong interaction occurs when λ/2 = a. 2. PBG (Photonic band gap) = stop bands overlap in all directions

32 2D Photonic band structure Band Diagram Air band Stop band Dielectric band

33 2D Photonic band structure

34 2D Photonic band structure

35 2D Photonic band structure

36 2D Photonic band structure

37 Four Possible Functionalities of PBG 1. Use of Stop Band 1. Stop band 1. Stop Band: Use PBG as high reflectivity omni-directional mirror (PBG waveguides) ) Stop band

38 2. Dielectric band 2. Use of Dielectric Band 2. Dielectric Band: Uses the strong dispersion available in a photonic crystal (dispersion engineering with form birefringence) Dielectric band

39 2. Dielectric band

40 Remind the dispersion relation in bulk media 2. Dielectric band 1. In a homogeneous material in absence of material dispersion n(ω)=constant =n, the dispersion i diagram is simply a straight line: ω=kc/n. 2. In 2D systems, one can think of this line as a cone. For a given frequency ω, this cone becomes a constant frequency circle.

41 2. Dielectric band ky kx

42 Wave propagation in k-space 2. Dielectric band Real space The wave vector diagram tells us the direction and magnitude of the refracted and reflected beams. Their direction is normal to the iso-frequency curve and corresponds to Snell s law.

43 2. Dielectric band

44 2. Dielectric band

45 2. Dielectric band

46 2. Dielectric band

47 3. Air band 3. Use of Air Band 3. Air Band : Couples to radiative modes for light extraction from high-efficiency LEDs and fiber coupling. Air band

48 3. Air band

49 4. Defect band 4. Use of Defect Band 4. Defect Band : Couples to waveguide/cavity modes for spectral control such as PBG point defect laser or PBG line defect filter, etc. Defect band

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51 Line Defect PBG Waveguide 4. Defect band Defect modes in stop band Dispersion diagram of W1 line-defect photonic crystal waveguide: Waveguide modes exist within the bandgap. Photons are prohibited in the 2D PBG, which lead to lossless confinement of photons in the line defect area.

52 Defects in PBG 4. Defect band

53 4. Defect band

54 4. Defect band

55 4. Defect band

56 4. Defect band

57 3D Photonic band structure 3D Photonic materials S.Noda, Nature (1999) K. Robbie, Nature (1996) E. Yablonovitch, PRL(1989)

58 Artificial Phonic Structure E.Yablonovitch et al., PRL (1987, 1991) 3D Photonic band structure Fabrication of artificial fcc material and band gap structure for such material.

59 Bragg diffraction through all electromagnetic region

60 Natural Opals

61 3D Photonic band structure Artificial Opal Artificial opal sample (SEM Image) Several cleaved planes of fcc structure are shown

62 Fabrication of artificial opals 3D Photonic band structure Silica spheres settle in close packed hexagonal layers There are 3 in-layer position A red; B blue; C green; Layers could pack in fcc lattice: ABCABC or ACBACB hcp lattice: ABABAB

63 3D Photonic band structure Inverted Opals Inversed opals obtain greater dielectric contrast than opals.

64 3D Photonic band structure Band structure of diamond lattice Ph i b d f di d l i ( f i i d 3 45) Photonic band structure of diamond lattice (refractive index ~3.45) John et. al. PRE (1998)

65 Photonic Crystal Fibers PCF

66 PCF

67 The fiber supports a single mode over the range of at least nm! PCF

68 PCF

69 PCF

70 PCF

71 PCF

72 PCF