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‘‘Twisted’ modes of split-Twisted’ modes of split-band-edge double-band-edge double-
heterostructure cavitiesheterostructure cavitiesSahand Mahmoodian
Andrey Sukhorukov, Sangwoo Ha, Andrei Lavrinenko, Christopher Poulton, Kokou Dossou, Lindsay Botten, Ross McPhedran, and C. Martijn de Sterke
Introduction
• Photonic Crystals (PC) are optical analogue of solid state crystals (cheesy definition)
• We can use effective mass theory to describe bound PC modes!
Photonic Crystal Slabs
• Periodic index creates optical bandgap.• Breaking the periodicity is used to construct
cavities and waveguides. • Out-of-plane confinement via TIR.
Double Heterostructure Cavities
• PCW with a region where structure is changed
• Like 1D finite potential it supports bound modes
• Modes have ultra-high quality factors (>106)- Very strong light-matter interaction
V
PC1 PC1PC2
Song et al Nat. Mat.
(2005)
Double Heterostructure Cavities
• Can also create DHCs in photosensitive chalcogenide glass
• Allows cavity profile to be tailored (minimize radiative losses)
Lee et al Opt. Lett.
(2009)
Split band-edge heterostructures
• Split band-edges - two degenerate band-edge modes.
Blue: nbg =3
Cyan: nbg = 3.005
What I’m going to show…
• Derive an effective mass theory for split-band-edge DHCs.
• Solve equations giving two modes• Nature of modes depends on how the
cavity is created (apodized or unapodized).
Degenerate effective mass theory
• Governing equations (2D)
• Bloch mode expansion
“Writing” the cavity
Degenerate effective mass theory
Weak coupling and shallow perturbation, we write:
• Two coupled equations (one for each minimum
1 2
Degenerate effective mass theory
• Going back to real space…
• Parabolic approximation:
Band-edge frequency Band-edge curvature (effective mass)
ω - cavity mode frequency
1 2
Degenerate effective mass theory
• Solution of equation gives frequency of modes and envelope functions
• We have created a theory that gives the fields and frequency of split band-edgeDHC modes.
1 2
Solutions and results
• Frequency of cavity modes as a function of cavity width:Blue – theory
Red - numerics
Unapodized cavity Gaussian apodized cavity
nbg=3
nhole=1
ncavity=3.005
Solutions and results
• The unapodized cavity:
• Nature of dispersion curve indicates a resonance-like effect.
Degeneracies correspond to zero off-diagonal terms.
= 0
= 0
Reciprocal space point of view
• We solve the problem with off-diagonal terms set to zero and look at cross coupling as a function of cavity width:
1 2Blue – width 10.5d
Green – width 8d
= 0
= 0
Reciprocal space point of view
• Now the same, but with a Gaussian apodized cavity.
• No nodes! No resonances!