folded bands in metamaterial photonic crystals
DESCRIPTION
Folded Bands in Metamaterial Photonic Crystals. Parry Chen 1 , Ross McPhedran 1 , Martijn de Sterke 1 , Ara Aasatryan 2 , Lindsay Botten 2 , Chris Poulton 2 , Michael Steel 3. 1 IPOS and CUDOS, School of Physics, University of Sydney, NSW 2006, Australia - PowerPoint PPT PresentationTRANSCRIPT
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
Metamaterial Photonic Crystals
• Metamaterials– Negative refractive index– Composed of artificial atoms
• Photonic Crystals– Periodic variation in refractive index– Coherent scattering influences
propagation of light
Contents of Presentation
1. Folded Bands and their Structures– Negative index metamaterial photonic
crystals
2. Give a mathematical condition and physical interpretation
– Give condition based on energy flux theorm
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
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
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
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
ω
Re(
ε)
ω
Im(ε
)
• A linear combination of Lorentz oscillators also satisfies Kramers-Kronig
Impact of Loss and Dispersion
• k is complex• Slow light significantly impacted by loss• Fast light relatively unaffected by loss
Lossless
Lossy
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 = ∞
Energy Velocity
USvv Eg
Rigorous argument for lossless case• Relation between group velocity, energy velocity, energy flux and density
0:0 Svg 0: Uvg
Energy Velocity
22HEU
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
Must have opposite group indexes for <U> = 0
In lossy media, a different expression for U is necessary
0: UvgCondition required:
Energy Velocity
22 )()(HEU
• Field localized in lossy positive ng: band shows lossy positive vg
• Field localized in lossy negative ng: band shows lossy negative vg
21 EnZU g
U influenced by dispersion
d
nd
cng
)(1
• Negative group index results in negative U
• vg and ng are changes in k and n as functions of frequency, respectively
Folded Bands• Folded bands must have infinite vg
• Both positive and negative ng present
Conclusions
• 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
Phenomena
1D Zero-average-n Photonic Band Gap (I)
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 vacuum (P) and metamaterial (N) layers
P PN NN
1D Zero-n Photonic Band Gap (II)
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
Alternating positive (P) and negative (N) group velocity
P PN N
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
Treat as Homogeneous Medium
Dispersion relation for positive index lossless homogeneous medium
cnk Infinite vg
requires
0ddk
0)(1
d
nd
cng
c
nk ave baave ffn )1(
f
f
n
n
gb
ga
1
Where two materials present, average index gives dispersion relation
Ratio of group indexes gives infinite vg
Single Constituent
Group velocities of opposite sign required
Dual Constituents
n
d
dn
0ddk
k
ω
ω
ε
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