‘Twisted’ modes of split-band-edge double-heterostructure cavities Sahand Mahmoodian

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 (>106) • Very strong light-matter interaction PC2 PC1 PC1 V 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: nbg =3 Cyan: nbg = 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 ω - cavity mode frequency • Going back to real space… • Parabolic approximation: 1 2 Band-edge frequency Band-edge curvature (effective mass)

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 nbg=3 nhole=1 ncavity=3.005 Gaussian apodized cavity Unapodized cavity

14. Cavity modes |Ey| nbg=3 nhole=1 ncavity=3.005 Cavity length = 9d

15. Solutions and results • The unapodized cavity: • Nature of dispersion curve indicates a resonance-like effect. Degeneracies correspond to zero off-diagonal terms. = 0 = 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: = 0 = 0 1 2 Blue – width 10.5d Green – width 8d

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.