Folded Bands in Metamaterial
Photonic Crystals
Parry Chen1, Ross McPhedran1, Martijn de Sterke1, Ara Aasatryan2, Lindsay Botten2, Chris
Poulton2, Michael Steel3
1IPOS
2CUDOS,
3MQ
and CUDOS, School of Physics, University of Sydney, NSW 2006, Australia
School of Mathematical Sciences, University of Technology, Sydney, NSW 2007, Australia
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
–
2.
Negative index metamaterial photonic
crystals
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 Square array
Cylinder radius: a = 0.3d
Bands do not span Brillouin zone
Metamaterial rods in air:
Bands cluster at high symmetry points
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 Square array
Cylinder radius: a = 0.3d
Bands do not span Brillouin zone
Metamaterial rods in air:
Bands cluster at high symmetry points
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 KramersKronig
Im(ε)
Re(ε)
•
ω
ω
•
A linear combination of Lorentz
oscillators also satisfies KramersKronig
Impact of Loss and Dispersion
Lossless
Lossy
•
•
•
k is complex
Slow light significantly impacted by loss
Fast light relatively unaffected by loss
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
Rigorous argument for lossless case
• Relation between group velocity, energy velocity, energy flux and density
vg  vE  S
vg  0 :
S 0
U
vg   :
U 0
Energy Velocity
Condition required: vg   :
2
U   E  H
U 0
2
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
Energy Velocity
U influenced by dispersion
U
 ( ) 2  (  ) 2
E 
H


1
U  Z ng E
2
ng 
1 d ( n )
c d
• 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
• Folded bands must have infinite vg
• Both positive and negative ng present
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
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
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
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
Single Constituent
Dispersion relation for positive index lossless homogeneous medium
k  n c
Infinite vg requires
dk
1 d ( n )
0
 0 ng 
c
d

d
dn
n

d

ε
ω
ω
Dual Constituents
Where two materials present, average index gives dispersion relation
n 
nave  f  a  (1  f )  b
k  ave
c
Ratio of group indexes gives infinite vg
nga
f
dk


0
ngb
1 f
d
Group velocities of opposite sign required
k
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