All-dielectric photonic quasicrystals may act as zero-refractive-index
homogeneous materials despite their lack of translational symmetry and
periodicity, stretching wavelengths to infinity and offering applications in
light wavefront sculpting and optical cloaking.Comment: 5 pages, 2 figure

D Shechtman

H Suchowski

J-K Yang

J-W Dong

L Dal Negro

ME Zoorob

P Moitra

R Lifshitz

R Maas

S Enoch

Svetlana V. Boriskina

SY Lee

X Huang

ZV Vardeny

Nature Photonics

English

arXiv

All-dielectric photonic quasicrystals may act as zero-refractive-index homogeneous materials despite their lack of translational symmetry and periodicity, stretching wavelengths to infinity and offering applications in light wavefront sculpting and optical cloaking

Boriskina, Svetlana V

DSpace@MIT

Quasicrystals:	  making	  invisible	  materials	  All-­‐dielectric	  photonic	  quasicrystals	  may	  act	  as	  zero-­‐refractive-­‐index	  homogeneous	  materials	  despite	  their	  lack	  of	  translational	  symmetry	  and	  periodicity.	  Zero-­‐index	  materials	  stretch	  wavelengths	  to	  infinity	  and	  offer	  applications	  in	  light	  wave	  front	  sculpting	  and	  optical	  cloaking.	  	  Svetlana	  V.	  Boriskina	  	  Optical	  materials	  that	  exhibit	  a	  vanishing	  refractive	  index1–4	  have	  recently	  moved	  from	  the	  realm	  of	  mathematical	  curiosity	  into	  the	  practical	  domain	  of	  functional	  photonic	  devices.	  As	  the	  refractive	  index	  is	  reduced	  to	  zero,	  both	  the	  phase	  velocity	  and	  the	  wavelength	  of	  propagating	  light	  waves	  stretch	  to	  infinity.	  Light	  can	  then	  propagate	  without	  any	  phase	  accumulation,	  leaping	  through	  the	  medium	  as	  if	  it	  were	  completely	  invisible	  for	  passing	  photons.	  Such	  materials	  can	  be	  created	  by	  arranging	  sub-­‐wavelength	  constituents	  into	  composite	  lattices,	  and	  offer	  application	  as	  directional	  light	  sources,	  innovative	  lenses	  and	  waveguides,	  insulators,	  optical	  cloaks,	  etc.	  To	  assign	  an	  effective	  index	  value	  to	  a	  composite	  metamaterial,	  it	  should	  be	  possible	  to	  treat	  the	  material	  as	  a	  homogeneous	  medium.	  While	  homogeneity	  is	  often	  associated	  with	  translational	  invariance,	  Jian-­‐Wen	  Dong	  and	  colleagues	  demonstrate5	  that	  artificial	  structures	  that	  lack	  translational	  periodicity	  yet	  exhibit	  long-­‐range	  order	  –	  known	  as	  quasicrystals	  -­‐	  can	  behave	  as	  zero-­‐index	  materials.	  This	  demonstration	  opens	  new	  opportunities	  for	  the	  development	  of	  all-­‐dielectric	  zero-­‐index	  materials	  by	  extending	  the	  concept	  to	  a	  wider	  class	  of	  photonic	  lattices	  other	  than	  periodic	  photonic	  crystals.	  Not	  long	  ago,	  quasicrystals	  themselves	  were	  considered	  a	  curiosity	  and	  deemed	  impossible,	  as	  long-­‐range	  order	  –	  which	  is	  a	  hallmark	  of	  crystallinity	  –	  was	  assumed	  to	  be	  inextricably	  linked	  to	  translational	  periodicity.	  This	  view	  has	  been	  shattered	  by	  Dan	  Shechtman’s	  discovery	  of	  an	  ‘impossible	  material’	  with	  10-­‐fold	  rotational	  symmetry6,	  which	  spurred	  a	  lot	  of	  debate,	  starting	  with	  ridicule	  and	  ending	  with	  the	  Noble	  Prize	  in	  Chemistry.	  Quasicrystals	  discovery	  was	  so	  controversial	  because	  local	  rotational	  symmetries	  other	  than	  2-­‐,	  3-­‐,	  4-­‐	  and	  6-­‐fold	  ones	  are	  incompatible	  with	  translational	  symmetry	  as	  illustrated	  in	  Fig.	  1a,b. Once	  accepted	  however,	  it	  led	  to	  a	  new	  definition	  of	  a	  crystal	  as	  ‘any	  solid	  having	  an	  essentially	  discrete	  diffraction	  pattern,’	  such	  as	  those	  shown	  in	  Fig.	  1d,e	  for	  a	  periodic	  and	  a	  5-­‐fold	  quasiperiodic	  lattice.	  	  	  Photonic	  analogs	  of	  crystals	  and	  quasicrystals7,8	  –	  artificial	  arrangements	  of	  two	  or	  more	  materials	  with	  different	  refractive	  indices	  –	  now	  find	  applications	  across	  a	  wide	  range	  of	  light-­‐based	  technologies.	  They	  are	  capable	  of	  molding	  the	  flow	  of	  light	  in	  many	  ways,	  from	  forbidding	  the	  light	  propagation	  to	  enhancing	  and	  shaping	  light	  emission.	  Interestingly,	  owing	  to	  their	  higher	  rotational	  symmetries,	  photonic	  quasicrystals	  are	  more	  ‘homogeneous’	  than	  their	  periodic	  counterparts	  for	  light	  propagating	  at	  different	  angles.	  As	  a	  result,	  they	  allow	  formation	  of	  isotropic	  and	  complete	  bandgaps	  (for	  both	  light	  polarizations)	  by	  using	  low-­‐index	  materials,	  which	  is	  not	  possible	  with	  periodic	  photonic	  crystals9.	  Furthermore,	  quasicrystals	  are	  not	  limited	  by	  strict	  constrains	  on	  the	  positions	  of	  Bragg	  peaks	  in	  their	  diffraction	  patterns,	  making	  them	  well-­‐suited	  for	  phase	  matching	  for	  nonlinear	  optical	  effects	  such	  as	  frequency	  conversion10.	  The	  rich	  spectrum	  of	  optical	  modes	  with	  various	  degrees	  of	  spatial	  localization	  also	  makes	  quasicrystals	  useful	  platforms	  for	  multi-­‐mode	  lasing11	  and	  multi-­‐color	  optical	  sensing12.	  	  Figure	   1|	   Quasicrystals	   provide	   long-­‐range	   order	   without	   structural	   periodicity.	   a,	   Periodic	  lattices	  with	   translational	  periodicities	   can	  be	   formed	  by	   tiling	  unit	   cells	  having	   local	   rotational	  symmetries.	  Only	  2-­‐,	  3-­‐,	  4-­‐	  and	  6-­‐fold	  symmetries	  are	  allowed,	  such	  as	  e.g.,	  in	  a	  two-­‐dimensional	  square-­‐lattice	  crystal.	  b,	  5-­‐,	  7-­‐fold	  and	  all	  higher	  local	  rotation	  symmetries	  are	  disallowed,	  as	  you	  cannot	   tile	   individual	   unit	   cells	  without	   leaving	   empty	   spaces.	   c,	   Quasicrystals	  with	   high-­‐order	  rotational	   symmetries	   –	   such	   as	   the	   Penrose	   lattice	   with	   5-­‐fold	   symmetry	   –	   can	   however	   be	  constructed	  by	  non-­‐periodic	  tiling	  of	  more	  than	  one	  type	  of	  tiles.	  d,e,	  Diffraction	  patterns	  of	  both	  periodic	   crystals	   (d)	   and	   quasicrystals	   (e)	   are	   discrete	   and	   feature	   sharp	   peaks	   arranged	   in	  periodic	  lattices	  for	  crystals	  or	  incommensurate	  lattices	  for	  quasicrystals.	  	  Periodic	  photonic	  crystals	  can	  act	  as	  homogeneous	  materials,	  with	  effective	  refractive	  index	  ranging	  from	  positive	  –	  typical	  for	  conventional	  materials	  –	  to	  negative,	  which	  make	  materials	  capable	  of	  bending	  light	  the	  ‘wrong’	  way.	  By	  tuning	  their	  parameters	  to	  yield	  zero	  effective	  index,	  photonic	  crystals	  can	  even	  be	  made	  ‘invisible’	  for	  light	  waves13	  (Fig.	  2a).	  Importantly,	  photonic	  crystals	  can	  act	  as	  ‘double-­‐zero’	  materials,	  with	  effectively	  zero	  permittivity	  ε	  and	  permeability	  µ,	  if	  they	  are	  tuned	  to	  exhibit	  Dirac	  cone	  dispersion	  at	  the	  Brillouin	  zone	  center	  (k=0)	  at	  a	  non-­‐zero	  frequency13.	  Double-­‐zero	  materials	  provide	  impedance	  matching	  in	  addition	  to	  the	  phase	  invariance	  achievable	  in	  ‘single-­‐zero’	  materials	  with	  either	  ε=0	  or	  µ=0.	  Dong	  and	  co-­‐workers	  now	  demonstrate5	  that	  photonic	  quasicrystals	  may	  also	  behave	  as	  ‘double-­‐zero’	  materials,	  and	  this	  unique	  property	  stems	  from	  existence	  of	  Dirac	  points	  at	  k=0	  in	  their	  dispersion	  (Figs	  2b,c).	  	  Figure	  2|	  Quasicrystals	  with	  Dirac	  cone	  dispersion	  at	  k=0	  act	  as	  zero	  effective	  index	  materials.	  a,	  Light	  waves	  maintain	  spatially	  uniform	  phase	  distribution	  across	  zero-­‐index	  material,	   in	  striking	  contrast	  from	  both	  positive-­‐	  and	  negative-­‐index	  media	  as	  illustrated	  for	  plane	  wave	  refraction	  at	  the	  interface.	  b,c,	  Quasicrystals	  with	  high	  rotational	  symmetry	  like	  the	  8-­‐fold	  lattice	  shown	  in	  the	  inset	   provide	   enough	   symmetry	   to	   protect	   Dirac	   cones.	   Dirac	   point	   caused	   by	   accidental	  degeneracy	   is	   formed	  by	   intersecting	  bands	  with	   linear	   dispersion	   as	  well	   as	   an	   additional	   flat	  band.	   d,	   Plane	   wave	   propagating	   through	   a	   prism	   constructed	   from	   a	   quasicrystal	   material	  undergoes	   zero	   refraction	  at	   interfaces.	  e,	   ‘Invisible’	   zero-­‐index	  materials	   can	  make	  embedded	  objects	   invisible	   to	   light	   as	   demonstrated	   by	   cloaking	   of	   an	   obstacle	   inside	   the	   quasicrystal	  developed	  by	  Dong	  et	  al5.	  The	  images	  in	  panels	  b(inset)	  and	  c	  are	  modified	  from	  ref.	  5,	  APS.	  	  The	  relation	  between	  the	  Dirac	  cone	  and	  zero	  refractive	  index	  may	  not	  be	  immediately	  obvious.	  	  However,	  it	  is	  easy	  to	  see	  that	  if	  ε=µ=0	  at	  a	  non-­‐zero	  frequency	  ω0,	  the	  photon	  dispersion	  near	  this	  frequency	  is	  a	  linear	  function	  ω=ω0+νgk	  (where	  k	  	  is	  the	  wavevector	  and	  νg	  is	  the	  photon	  group	  velocity).	  Linear	  dispersion	  is	  of	  course	  a	  characteristic	  feature	  of	  the	  Dirac	  point,	  which	  yields	  many	  unique	  transport	  characteristics	  of	  graphene	  and	  its	  photonic	  analogs.	  However,	  the	  mere	  existence	  of	  the	  Dirac	  cone	  in	  the	  bandstructure	  is	  not	  sufficient	  for	  the	  photonic	  lattice	  to	  have	  zero	  effective	  index.	  Dong	  et	  al5	  explain	  that	  not	  only	  a	  Dirac	  cone	  at	  k=0	  is	  required	  but	  also	  it	  should	  be	  formed	  by	  crossings	  of	  the	  monopole	  and	  dipole	  branches	  in	  the	  dispersion	  diagram,	  and	  not	  higher-­‐order	  multipoles.	  	  The	  latter	  condition	  guarantees	  applicability	  of	  the	  effective	  medium	  approximation,	  while	  the	  former	  ensures	  that	  the	  effective	  index	  is	  local	  (i.e.,	  does	  not	  depend	  on	  the	  wavevector).	  	  The	  authors	  provide	  a	  recipe	  for	  the	  Dirac	  cone	  formation,	  which	  is	  based	  on	  tuning	  the	  quasicrystal	  parameters	  to	  achieve	  	  ‘accidental	  degeneracy’	  of	  its	  modes.	  This	  is	  different	  from	  the	  familiar	  Dirac	  cones	  in	  graphene	  etc.,	  which	  form	  at	  the	  Brillouin	  zone	  edge,	  while	  nondegenerate	  bands	  at	  the	  zone	  center	  are	  required	  by	  symmetry	  to	  be	  parabolic.	  The	  accidental	  modal	  degeneracy,	  which	  only	  occurs	  at	  certain	  parameters	  of	  the	  photonic	  lattice,	  results	  in	  the	  formation	  of	  a	  Dirac	  cone	  accompanied	  by	  a	  flat	  band	  that	  crosses	  the	  cone	  at	  the	  Dirac	  point	  (Figs.	  2b,c).	  The	  same	  group	  has	  previously	  used	  this	  recipe	  to	  design	  and	  experimentally	  demonstrate	  a	  double-­‐zero	  material	  with	  a	  periodic	  photonic	  crystal	  lattice13,	  and	  now	  have	  extended	  this	  concept	  to	  a	  much	  broader	  range	  of	  materials.	  The	  conical	  dispersions	  are	  protected	  by	  the	  lattice	  symmetry,	  and	  this	  work	  demonstrates	  that	  quasicrystals	  possess	  enough	  rotational	  symmetry	  to	  provide	  such	  protection.	  Furthermore,	  the	  authors	  show	  that	  quasicrystal	  approximants	  of	  different	  sizes	  feature	  Dirac	  cones,	  albeit	  at	  different	  optimal	  parameters	  of	  the	  lattice.	  This	  is	  an	  important	  finding	  as	  quasicrystals	  of	  increased	  size	  feature	  higher	  level	  of	  structural	  complexity	  and	  long-­‐range	  correlations,	  which	  sets	  them	  apart	  from	  their	  periodic	  counterparts.	  To	  probe	  if	  quasicrystals	  can	  act	  as	  double-­‐zero	  effective	  media,	  Dong	  and	  colleagues	  designed	  lattices	  with	  12-­‐fold	  and	  8-­‐fold	  symmetries	  for	  operation	  at	  frequencies	  about	  10.5GHz	  and	  fabricated	  them	  from	  alumina	  rods	  sandwiched	  between	  metal	  plates.	  The	  authors	  showed	  that	  the	  quasicrystal	  eigenmodes	  at	  the	  Dirac	  point	  have	  almost	  constant	  field	  intensity	  and	  phase	  throughout	  the	  whole	  lattice.	  They	  also	  demonstrated	  zero	  refraction	  of	  plane	  waves	  propagating	  through	  a	  prism	  constructed	  from	  a	  quasicrystal	  material	  (Fig.	  2d).	  Experimental	  demonstration	  of	  the	  cloaking	  of	  an	  obstacle	  (a	  metal	  rod)	  inside	  the	  quasicrystal	  (Fig.	  2e)	  was	  also	  consistent	  with	  the	  zero	  index	  behavior	  of	  the	  material.	  Finally,	  another	  proof	  that	  the	  quasicrystals	  had	  zero	  effective	  index	  was	  delivered	  by	  measuring	  asymmetric	  wave	  transport	  through	  an	  inversion-­‐symmetry-­‐breaking	  prism	  with	  a	  quasicrystal	  lattice.	  Dong	  et	  al	  explain	  that	  conditions	  to	  achieving	  zero-­‐index	  behavior	  in	  quasicrystals	  are	  more	  stringent	  than	  for	  opening	  up	  photonic	  bandgaps	  or	  even	  than	  for	  realizing	  negative	  refractive	  index.	  High	  level	  of	  symmetry	  is	  required	  to	  protect	  conical	  dispersion,	  and	  while	  8-­‐fold	  and	  12-­‐fold	  quasicrystals	  were	  shown	  to	  form	  Dirac	  cones,	  the	  same	  feature	  was	  not	  observed	  in	  5-­‐fold	  Penrose	  lattices	  shown	  in	  Fig.	  1c.	  Other	  limitations	  of	  this	  approach	  include	  failure	  of	  the	  effective	  index	  approximation	  for	  low-­‐index	  structures	  as	  well	  as	  for	  vein	  rather	  than	  discrete-­‐rod	  lattices.	  Despite	  these	  challenges,	  this	  research	  expands	  the	  range	  of	  geometries	  for	  double-­‐zero	  materials	  design.	  It	  is	  intriguing	  if	  aperiodic	  structures	  with	  long-­‐range	  order	  other	  than	  quasicrystals8	  can	  provide	  symmetry	  protection	  required	  to	  achieve	  zero	  refractive	  index.	  Interesting	  future	  directions	  include	  extension	  to	  the	  visible	  band	  and	  to	  the	  third	  dimension,	  which	  may	  unveil	  new	  opportunities	  for	  such	  materials.	  Svetlana	  V.	  Boriskina	  is	  at	  the	  Department	  of	  Mechanical	  Engineering,	  Massachusetts	  Institute	  of	  Technology,	  Cambridge,	  Massachusetts,	  02139,	  USA.	  e-­‐mail:	  sborisk@mit.edu	  	  References	  1.	   Enoch,	  S.,	  Tayeb,	  G.,	  Sabouroux,	  P.,	  Guérin,	  N.	  &	  Vincent,	  P.	  A	  Metamaterial	  for	  directive	  emission.	  Phys.	  Rev.	  Lett.	  89,	  213902	  (2002).	  2.	   Suchowski,	  H.,	  O’Brien,	  K.,	  Wong,	  Z.	  J.,	  Salandrino,	  A.,	  Yin,	  X.	  &	  Zhang,	  X.	  Phase	  mismatch-­‐free	  nonlinear	  propagation	  in	  optical	  zero-­‐index	  materials.	  Science.	  342,	  1223–6	  (2013).	  3.	   Moitra,	  P.,	  Yang,	  Y.,	  Anderson,	  Z.,	  Kravchenko,	  I.	  I.,	  Briggs,	  D.	  P.	  &	  Valentine,	  J.	  Realization	  of	  an	  all-­‐dielectric	  zero-­‐index	  optical	  metamaterial.	  Nat.	  Photonics	  7,	  791–795	  (2013).	  4.	   Maas,	  R.,	 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Quasicrystals: Making invisible materials

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Making invisible materials

Quasicrystals: Making invisible materials

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