‘‘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

Our University and Group

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

Cavity modes

nbg=3

nhole=1

ncavity=3.005

Cavity length = 9d

|Ey|

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!

Conclusion

• We have developed an effective mass formalism for split-band-edge DHCs.

• We showed that unapodized and apodized cavities have modes with different qualitative behaviour.

• Split-band-edge DHCs may prove useful when non-linearities are introduced.