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
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Metamaterial Photonic Crystals

• Metamaterials
• Negative refractive index
• Composed of artificial atoms

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

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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

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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, e
-1.8, µ -5
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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, e -3,
µ -3
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Kramers-Kronig

• Negative e and µ due to resonance, dispersion
  required
• Need to satisfy causality Kramers-Kronig
  relations with loss

• Lorentz oscillator satisfies Kramers-Kronig

Im(e)
Re(e)
?
?

• A linear combination of Lorentz oscillators also
  satisfies Kramers-Kronig

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Impact of Loss and Dispersion
Lossless Lossy

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

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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 8
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Energy Velocity

• Rigorous argument for lossless case
• Relation between group velocity, energy velocity,
  energy flux and density

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Energy Velocity
Condition required
Must have opposite group indexes for ltUgt 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 ? changes, leading to transitions
  in modal vg between positive and negative values

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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

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Folded Bands

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

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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 ? changes
• Rigorous mathematical condition derived for
  lossless dispersive materials

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1D Zero-average-n Photonic Band Gap (I)
Alternating vacuum (P) and metamaterial (N) layers
P
P
N
N
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

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1D Zero-n Photonic Band Gap (II)
Alternating positive (P) and negative (N) group
velocity
P
P
N
N

• Band diagram shows unusual topologies
• Bands fold
• Bands do not span k
• Positive and negative group velocity
• Bands cluster around k0
• Effect not related to zero-average-n

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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

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Treat as Homogeneous Medium
Single Constituent

• Dispersion relation for positive index lossless
  homogeneous medium

Infinite vg requires
?
e
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
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Non-Metamaterial Systems

• Simulated folded bands in positive n media
• Polymer rods in silicon background
• Embedded quantum dots dispersive e
• 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