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Folded Bands in Metamaterial Photonic Crystals

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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

- Metamaterials
- Negative refractive index
- Composed of artificial atoms

- Photonic Crystals
- Periodic variation in refractive index
- Coherent scattering influences propagation of light

Folded Bands and their Structures

Negative index metamaterial photonic crystals

Give a mathematical condition and physical interpretation

Give condition based on energy flux theorm

- 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

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

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

- 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

Lossless

Lossy

- k is complex
- Slow light significantly impacted by loss
- Fast light relatively unaffected by loss

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 = ∞

Rigorous argument for lossless case

- Relation between group velocity, energy velocity, energy flux and density

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

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

- Folded bands must have infinite vg
- Both positive and negative ng present

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

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

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

- 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

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

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