Folded Bands in Metamaterial Photonic Crystals

1. Folded Bands in Metamaterial Photonic Crystals Parry Chen1, Ross McPhedran1, Martijn de Sterke1, Ara Aasatryan2, Lindsay Botten2, Chris Poulton2, Michael Steel3 1IPOS and CUDOS, School of Physics, University of Sydney, NSW 2006, Australia 2CUDOS, School of Mathematical Sciences, University of Technology, Sydney, NSW 2007, Australia 3MQ Photonics Research Centre, CUDOS, and Department of Physics and Engineering, Macquarie University, Sydney, NSW 2109, Australia

2. Metamaterial Photonic Crystals • Metamaterials • Negative refractive index • Composed of artificial atoms • Photonic Crystals • Periodic variation in refractive index • Coherent scattering influences propagation of light

3. Contents of Presentation Folded Bands and their Structures Negative index metamaterial photonic crystals Give a mathematical condition and physical interpretation Give condition based on energy flux theorm

4. Numerical Methodology • Ready-to-use plane wave expansion band solvers do not handle negative index materials, dispersion or loss • Modal method: expand incoming and outgoing waves as Bessel functions • Handles dispersion and produces complex band diagrams

5. Lossless Non-Dispersive Band Diagrams Negative n photonic crystal • Infinite group velocity • Zero group velocity at high symmetry points • Positive and negative vg bands in same band • Bands do not span Brillouin zone • Bands cluster at high symmetry points Square array Cylinder radius: a = 0.3d Metamaterial rods in air: n = -3, ε = -1.8, μ = -5

6. Lossless Non-Dispersive Band Diagrams Negative n photonic crystal • Infinite group velocity • Zero group velocity at high symmetry points • Positive and negative vg bands in same band • Bands do not span Brillouin zone • Bands cluster at high symmetry points Square array Cylinder radius: a = 0.3d Metamaterial rods in air: n = -3, ε = -3, μ = -3

7. Kramers-Kronig • Negative ε and μ due to resonance, dispersion required • Need to satisfy causality Kramers-Kronig relations with loss • Lorentz oscillator satisfies Kramers-Kronig Im(ε) Re(ε) ω ω • A linear combination of Lorentz oscillators also satisfies Kramers-Kronig

8. Impact of Loss and Dispersion Lossless Lossy • k is complex • Slow light significantly impacted by loss • Fast light relatively unaffected by loss

9. Summary of Band Topologies Key topological features • Zero vg at high symmetry pts • Infinite vg points present When loss is added • Zero vg highly impacted • Infinite vg unaffected Vg = ∞

10. Energy Velocity Rigorous argument for lossless case • Relation between group velocity, energy velocity, energy flux and density

11. Energy Velocity Condition required: Must have opposite group indexes for <U> = 0 In lossy media, a different expression for U is necessary To obtain infinite vg • Group indexes of two materials must be opposite sign • Field density transitions between positive and negative ng as ω changes, leading to transitions in modal vg between positive and negative values

12. Energy Velocity U influenced by dispersion • Negative group index results in negative U • vg and ng are changes in k and n as functions of frequency, respectively • Field localized in lossy positive ng: band shows lossy positive vg • Field localized in lossy negative ng: band shows lossy negative vg

13. Folded Bands • Folded bands must have infinite vg • Both positive and negative ng present

14. Conclusions Phenomena • Metamaterial photonic crystals display folded bands that do not span the Brillouin zone • Contain infinite vg points • Infinite vg stable against dispersion and loss Phenomena • Structures contain both positive and negative ng materials • Field distribution transitions positive to negative ng as ω changes • Rigorous mathematical condition derived for lossless dispersive materials

15. 1D Zero-average-n Photonic Band Gap (I) Alternating vacuum (P) and metamaterial (N) layers N P N P N New zero-average-n band gap • Scale invariant, polarization independent • Robust against perturbations • Structure need not be periodic • Origin due to zero phase accumulation

16. 1D Zero-n Photonic Band Gap (II) Alternating positive (P) and negative (N) group velocity P N P N Band diagram shows unusual topologies • Bands fold • Bands do not span k • Positive and negative group velocity • Bands cluster around k=0 • Effect not related to zero-average-n

17. Numerical Methodology • Modal method: expand incoming and outgoing waves as Bessel functions • Lattice sums express incoming fields as sum of all other outgoing fields • Transfer Function method translates between rows of cylinders • Handles dispersion and produces complex band diagrams

18. Treat as Homogeneous Medium Dispersion relation for positive index lossless homogeneous medium Single Constituent Infinite vgrequires ε ω ω k Dual Constituents Where two materials present, average index gives dispersion relation Ratio of group indexes gives infinite vg Group velocities of opposite sign required

19. Non-Metamaterial Systems Simulated folded bands in positive n media • Polymer rods in silicon background • Embedded quantum dots: dispersive ε • Positive index medium, non-dispersive μ • Homogeneous medium: Maxwell-Garnett • Bands have characteristic zero and infinite vg • Loss affects zero vg but not infinite vg