1. ‘Twisted’ modes of split-band- edge double-heterostructure cavities Sahand Mahmoodian Andrey Sukhorukov, Sangwoo Ha, Andrei Lavrinenko, Christopher Poulton, Kokou Dossou, Lindsay Botten, Ross McPhedran, and C. Martijn de Sterke

2. Our University and Group

3. Introduction Photonic Crystals (PC) are optical analogue of solid state crystals (cheesy definition) We can use effective mass theory to describe bound PC modes!

4. Photonic Crystal Slabs Periodic index creates optical bandgap. Breaking the periodicity is used to construct cavities and waveguides. Out-of-plane confinement via TIR.

5. 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 (>10 6 ) -Very strong light-matter interaction V PC1 PC2 Song et al Nat. Mat. (2005)

6. 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)

7. Split band-edge heterostructures Split band-edges - two degenerate band- edge modes. Blue: n bg =3 Cyan: n bg = 3.005

8. 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).

9. Degenerate effective mass theory Governing equations (2D) Bloch mode expansion “Writing” the cavity

10. Degenerate effective mass theory Weak coupling and shallow perturbation, we write: Two coupled equations (one for each minimum 1 2

11. 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

12. 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-edge DHC modes. 1 2

13. Solutions and results Frequency of cavity modes as a function of cavity width: Blue – theory Red - numerics Unapodized cavity Gaussian apodized cavity n bg =3 n hole =1 n cavity = 3.005

14. Cavity modes n bg =3 n hole =1 n cavity =3.005 Cavity length = 9d |E y |

15. Solutions and results The unapodized cavity: Nature of dispersion curve indicates a resonance-like effect. Degeneracies correspond to zero off-diagonal terms. = 0

16. 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 2 Blue – width 10.5d Green – width 8d = 0

17. Reciprocal space point of view Now the same, but with a Gaussian apodized cavity. No nodes! No resonances!

18. 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.