From Homeopathy to Cloaking by Plasmonic Resonance

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From Homeopathy to Cloaking by Plasmonic Resonance
Ross McPhedran, Graeme Milton and Nicolae
Nicorovici
TexPoint fonts used in EMF AAAAAAAAAAA
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Outline

• Plasmons and resonances
• Arrays of Coated Cylinders vs Solid Cylinders
• What we liked (and the referees didnt)
  Homeopathy
• What we missed images where they shouldnt be
• Perfect Imaging and Cloaking a growth industry?

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Plasmons and resonances-1

• An interesting problem in technology given
  particles of dielectric constant ?a and volume
  fraction fa in a background material of
  dielectric constant ?b, what is the effective
  dielectric constant?
• Answer in 2D- cylinders-
• Called the Maxwell-Garnett formula

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Plasmons and resonances-2

• A resonance occurs when a response function
  becomes infinite output with no input
• For the Maxwell-Garnett formula, this occurs when
  the denominator is zero
• Notice this requires the relative dielectric
  constant to be real and negative
• This is called a surface plasmon resonance

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Dielectric Constant Real and Negative?

• Physical materials cannot have (unaided) ? real
  and negative
• If they could we would be able to construct
  electrostatic fields with total energy zero,
  positive energy in one region, negative in
  another
• Metals however can have ? with a large negative
  real part, and imaginary part not very large
• Particular case silver a good metal for
  plasmonics

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Dielectric Constant Real and Negative(2)?

• Abovereal part (red) and imaginary part (green)
  of ? for silver as a function of wavelength in
  micron.
• Below close-up
• where lt(?) is close to -1

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Dielectric Constant Real and Negative(3)?

• In summary ? cannot be real and negative unaided
• However, in some wavelength regions it can get
  close to being real and negative
• In such regions we get strong plasmon resonances
• We may also try to put gain into optical systems
  to effectively compensate for the imaginary part
  of ?

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Surface Plasmons

• Surface plasmons are in fact applied,
  particularly in sensitive biomolecule detection
• Their physics is interesting-particularly in
  clusters of particles

Dipole plasmon resonance for a cluster of three
cylinders
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The Array of Solid Cylinders (1)

• Consider the problem of calculating the
  efffective dielectric constant of an array of
  cylinders radius ra, dielectric constant ?a,
  spaced by d in a material with dielectric
  constant ?b
• Solve this using a multipole method due to
  Rayleigh
• Field around cylinders characterized by a vector
  of multipole coefficients B
• Field identity has the form

Boundary condition matrix whats there
Vector of applied field coefficients
Lattice sums for cylinder interaction whats
where
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The Array of Solid Cylinders (2)

• Resonant solutions are array plasmons
• Exist without an applied field

Here M is a diagonal matrix with lth element
ra is the cylinder radius
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The Array of Solid Cylinders (3)

• Resonances for the square array of solid
  cylinders concentrate around ?a-?b essential
  singularity

Vertical axis area fraction f horizontal axis-
resonant dielectric constant, measured from
?a-1. p is the order of the resonance
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Three Remarkable but Unremarked Papers (1)

• Nicorovici, N.A., McPhedran, R.C. and Milton,
  G.W. Transport Properties of a Three-Phase
  Composite Material The Square Array of Coated
  Cylinders, Proceedings of the Royal Society A
  442,599-620, 1993.
• Nicorovici, N.A., McPhedran, R.C. and Milton,
  G.W. Optical and Dielectric Properties of
  Partially Resonant Composites, Physical Review B,
  49, 8479-8482, 1994.
• Nicorovici, N.A., McKenzie, D.R., and McPhedran,
  R.C. Optical Resonances of Three-Phase
  Composites and Anomalies in Transmission, Optics
  Communications, 117, 151-169 (1995).
• These three papers treated arrays of coated
  cylinders, and studied their resonances
• The full significance of the results in them was
  not appreciated by the referees, the readers or
  even the authors

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Three Remarkable but Unremarked Papers (2)
Shell radius rs, dielectric constant ?s
Matrix dielectric constant ?m
Core radius rc, dielectric constant ?c
Line of core-shell resonances ?c?s0
Line of shell-matrix resonances ?s?m0
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Three Remarkable but Unremarked Papers (3)
The matrix equation for coated cylinders has the
same structure as for solid cylinders
Core-shell resonance
Shell-matrix resonance
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Shell-Core Resonance

• The shell hides itself (cloaks itself) by making
  the core behave as if it extended out to rs, not
  rc

Shell-core resonance
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Shell-Matrix Resonance

• The shell magnifies the core by making it behave
  as if it extended out to rs2/rc, not rc

Shell-matrix resonance
Homeopathic behaviour the smaller the core, the
bigger its effect after magnification by the
shell! Limited of course by the requirement that
rs2/rc not correspond to intersecting cylinders.
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Physics Behaving Badly!

• This sort of homeopathic behaviour is not what we
  expect of physical systems
• It has occurred because we have chosen material
  properties lying right on a resonant line
• Remember however we can approach arbitrarily
  closely to the resonant line (particularly with
  system gain)
• Potential for ultra-sensitive detectors, etc???
• Our PRL referees didnt like homeopathy, so PRB
  was the result

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Some Subtleties We Overlooked

• We showed that shell matrix resonance could make
  a coated cylinder behave in electrostatics like a
  larger solid cylinder
• Suppose then we bring a charge or dipole close to
  the coated cylinder. It will interact with it in
  the same way as it would with the solid cylinder.
• This interaction is taken into account by having
  an image charge inside the solid cylinder.
• The sublety now the image charge can be inside
  the equivalent solid cylinder, but outside the
  coated cylinder, and so accessible to observation!

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Ghost Sources

• Interaction of a charge at r0 with the coated and
  enlarged cylinders

a2/r0
r0
qg
qg
q
The ghost source is in the matrix medium if
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Ghost Sources (2)

• How can we study such singular situations? With
  care.
• Two reasonable procedures exist make the shell
  region slightly lossy, study fields in this
  physical situation, then take the limit as loss
  goes to zero.
• The second procedure switch on a source at some
  finite time, and observe how fields evolve.
• Either procedure will generate fields which
  contain finite energy.
• Clearly, if there is a ghost source in the
  matrix, it corresponds to a response field with
  infinite electrostatic energy such fields will
  not be generated in lossy systems or in finite
  time simulations.

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Ghost Sources (3)

• The procedure of letting ?s-1i?s with ?s
  small and positive yielded the following picture

Potential converges as loss decreases
Note that there are two ghost sources, bounding
the region in which the field is non-convergent
rcrit
GS1
a
GS2
r0
Potential diverges as loss decreases
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Perfect Imaging (1)

• Note that as we tend to GS1 from larger radii,
  this ghost source tends to a true line source as
  the loss tends to zero
• This could have been the first example of perfect
  imaging of a point or line source
• Perfect imaging however really caught on with
  Pendrys remarkable paper in 2000 on perfect
  imaging by a slab of left-handed material
• To go from what we have said about coated
  cylinders to slabs, let the cylinder radius tend
  to infinity as the cylinder centre tends to
  (infinity)

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Perfect Imaging (2)
The Pendry picture of perfect imaging a slab of
left-handed material is used. A source is placed
at d0.Two images are formed at x-d0 and xd0-2
d. Note that these images are just ghost sources
of the type we have described. The correspondence
is discussed in detail in Milton et al, Proc.
Roy. Soc. A 461, 3999-4034 (2005).
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Metamaterials

• Pendrys paper on perfect imaging and
  left-handed material spawned the creation of a
  new field metamaterials.
• This field does have antecedents e.g. artificial
  dielectrics for radar, composite materials for
  solar energy absorption, etc
• What is new is its combination with ideas of
  negative refraction due to Veselago and with
  todays powerful techniques for creation of
  microstructural optical elements
• The field is a playground for exploring radical
  new ideas- e.g cloaking
• Cloaking is the design of optical systems which
  can hide an object from observation (often only
  for probes in a narrow frequency range, or a
  narrow range of directions)

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Cloaking via Control of Refractive Index
From Pendry, Schurig and Smith, Science, 23 June
2006. A metamaterial shield is used in which the
refractive index goes to zero in an annulus,
cloaking the region inside the annulus
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Cloaking via Resonant Interactions

• Milton and Nicorovici, Proc Roy Soc, published
  on-line in May 2006
• Proved that cloaking can occur both for coated
  cylinders and the plane Pendry-Veselago lens,
  provided the polarizable line source is located
  close enough to the cloaking system.
• For the plane lens, cloaking occurs for sources
  half the lens thickness away from its face

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Cloaking via Resonant Interactions (2)

• For the cylindrical lens, cloaking occurs for
  distances r0 less than r if ?c?m, and for
  distances less than r if ?c? ?m
• Here rrs2/rc , and
• If
• Note that rcrit is the image of rs in a, and also
  the image of rc in r

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The Animation

• The following animation was provided by Nicolae
  Nicorovici. It shows a coated cylinder with ?c1,
  ?s-1i10-7, rs4,rc2 placed in a uniform
  electric field. A polarizable molecule moves from
  the right. The dashed line marks the circle rr.
  The polarizable molecule has a strong induced
  dipole moment and perturbs the field around the
  coated cylinder strongly. It then enters the
  cloaking region, and it and the coated cylinder
  do not perturb the external field.

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Animation
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Response of Polarizable Molecule
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Conclusions

• Metamaterials are a current hot topic in
  electromagnetics.
• They offer many exciting possibilities for exotic
  and perhaps useful physics.
• The big challenge with them is compensating for
  the loss inevitably present
• Their properties are very sensitive to very small
  amounts of loss, so this is indeed a formidable
  challenge
• If it can be done, we can look forward to many
  surprising achievements.
• This work was supported by the Australian
  Research Council