Physics of Degenerate Defect Modes in Photonic Crystal Bandgaps

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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
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Photonic Crystals

• Photonic Crystals (PCs) spatially periodic
  refractive index
• Man-made also exist in nature (butterflies,..)
• 1D, 2D and 3D PCs

2D
1D
3D
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Bandgaps

• Periodicity Bragg reflection
• Bragg reflection adds coherently at particular
  frequencies.
• Light cannot propagate in PC bandgap

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Photonic Crystal Devices

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

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Outline

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

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(Photonic) Crystal Overview

• Periodicity ? Blochs Thm ? form of modes

Reciprocal Lattice
PC Lattice
Lattice Vectors R
Reciprocal Lattice Vectors G
Fourier
Transform
Brillouin Zone
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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
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Origin of Bandgaps

• Variational Theorem ? Modes maximize electric
  energy (minimizing frequency) and are orthogonal
  to lower modes.

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Defect Modes in Bandgaps

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

n1.8
Bandgap
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Degenerate Defect Modes

• X and Y points in BZ have degenerate
• Bloch modes.
• Defect modes also
• degenerate
• Modes have different symmetry

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Where now

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

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Two Defects

• Coupling ? frequency split

Red points Two Defects four periods aparts
Blue line One Defect
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Tight-Binding Treatment of Defects

• Treat coupling using a tight-
• binding (supermodes) method
• Similar to coupled waveguides

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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
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Tight Binding

• Tight-binding ansatz in governing equation.
• Multiply by Ezi and integrate over space.
• K, J, I are overlap integrals, N is
  normalisation.

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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
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Tight Binding Treatment

• Solutions are now simple
• Fields are odd and even superpositions of
  isolated modes with same symmetry.

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Tight-Binding Treatment
Red points Two Defects four periods aparts
Cyan Two Defects Tight Binding calculation
Blue line One Defect
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Symmetry

• Applies to defects on any symmetry plane
• Treatment extends to hexagonal lattices

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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.