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Physics of Degenerate Defect Modes in Photonic Crystal Bandgaps

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Photonic Crystals (PCs): spatially periodic refractive index. Man-made; also exist in nature (butterflies,..) 1D, 2D and 3D PCs. 2D. 3D. 1D. Bandgaps ... – PowerPoint PPT presentation

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Title: Physics of Degenerate Defect Modes in Photonic Crystal Bandgaps


1
Physics of Degenerate Defect Modes in Photonic
Crystal Bandgaps
Sahand Mahmoodian1
Ross McPhedran1, Martijn de Sterke1, Kokou
Dossou2, Chris Poulton2, Lindsay Botten2 1 -
University of Sydney, 2 - University of
Technology Sydney
2
Photonic Crystals
  • Photonic Crystals (PCs) spatially periodic
    refractive index
  • Man-made also exist in nature (butterflies,..)
  • 1D, 2D and 3D PCs

2D
1D
3D
3
Bandgaps
  • Periodicity Bragg reflection
  • Bragg reflection adds coherently at particular
    frequencies.
  • Light cannot propagate in PC bandgap

4
Photonic Crystal Devices
  • Perturbations or defects in
  • periodic lattice used for PC devices.
  • Require good understanding of defects and their
    modes

5
Outline
  • Overview of PC theory
  • Defect Modes
  • Degenerate Modes
  • Coupling of Degenerate Modes
  • Outlook and Conclusion

6
(Photonic) Crystal Overview
  • Periodicity ? Blochs Thm ? form of modes

Reciprocal Lattice
PC Lattice
Lattice Vectors R
Reciprocal Lattice Vectors G
Fourier
Transform
Brillouin Zone
7
Bands and Brillouin Zone
  • Non-redundant set of wave-vectors in First
    Brillouin zone.
  • Bands and
  • Bandgaps
  • Irreducible Brillouin
  • Zone (BZ) edge

Square Lattice Photonic Crystal
8
Origin of Bandgaps
  • Variational Theorem ? Modes maximize electric
    energy (minimizing frequency) and are orthogonal
    to lower modes.

9
Defect Modes in Bandgaps
  • Defect by changing index of single cylinder
  • ve index change ? lower freq.
  • -ve index change ? higher freq.

n1.8
Bandgap
10
Degenerate Defect Modes
  • X and Y points in BZ have degenerate
  • Bloch modes.
  • Defect modes also
  • degenerate
  • Modes have different symmetry

11
Where now
  • Looked at single defect
  • Need understanding of multiple defects to
    construct devices (eg. waveguide)
  • Use single defect info for multiple defects?

12
Two Defects
  • Coupling ? frequency split

Red points Two Defects four periods aparts
Blue line One Defect
13
Tight-Binding Treatment of Defects
  • Treat coupling using a tight-
  • binding (supermodes) method
  • Similar to coupled waveguides

14
Tight Binding Photonic Crystals
  • Two identical defects infinite PC
  • Ez polarisation
  • Supermode via Tight-Binding

Periodic Permittivity
Governs Single Isolated Defect
Change in permitivitty
Overall Frequency
Governs Both Defects
15
Tight Binding
  • Tight-binding ansatz in governing equation.
  • Multiply by Ezi and integrate over space.
  • K, J, I are overlap integrals, N is
    normalisation.

16
Tight-Binding Treatment
  • Solutions of 4x4 matrices
  • tend to be messy.
  • But many elements vanish
  • because
  • Defects are arranged in symmetric fashion.
  • No cross coupling.

Modes have different symmetry
17
Tight Binding Treatment
  • Solutions are now simple
  • Fields are odd and even superpositions of
    isolated modes with same symmetry.

18
Tight-Binding Treatment
Red points Two Defects four periods aparts
Cyan Two Defects Tight Binding calculation
Blue line One Defect
19
Symmetry
  • Applies to defects on any symmetry plane
  • Treatment extends to hexagonal lattices

20
Conclusion
  • Analysed coupling of degenerate defects.
  • Degeneracy doesnt cause cross-coupling when
    defects are positioned symmetrically
  • Treatment OK for both polarisations
  • Symmetry arguments applies, provided defects
    preserve symmetry of lattice.
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