Ross McPhedran, Graeme Milton and
Nicolae Nicorovici

Outline
• Plasmons and resonances
• Arrays of Coated Cylinders vs Solid Cylinders
• What we liked (and the referees didn’t):
Homeopathy
• What we missed: images where they shouldn’t be
• Perfect Imaging and Cloaking: a growth industry?

Plasmons and resonances-1
• An interesting problem in technology: given particles of dielectric constant
 a and volume fraction f a in a background material of dielectric constant
 b
, what is the effective dielectric constant?
• Answer: in 2D- cylinders-
² ef f
² b
= 1 +
2f a
( ² a
+ ² b
) =( ² a
¡ ² b
) ¡ f a
• Called the Maxwell-Garnett formula

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:
³
² a = ¡
1+ f a
´
² b
1¡ f a
• Notice: this requires the relative dielectric constant to be real and negative
• This is called a surface plasmon resonance

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

Dielectric Constant Real and
Negative(2)?
• Above:real part (red) and imaginary part
(green) of
 for silver as a function of wavelength in micron.
• Below: close-up where < (

) is close to
-1

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


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

The Array of Solid Cylinders (1)
• Consider the problem of calculating the efffective dielectric constant of an array of cylinders: radius r a
, dielectric constant
 spaced by d in a material with dielectric constant
 a
, 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:
(M + S)B = E
Boundary condition matrix: what’s there
Lattice sums for cylinder interaction: what’s where
Vector of applied field coefficients

The Array of Solid Cylinders (2)
• Resonant solutions are array plasmons
• Exist without an applied field:
M B = ¡ SB
Here M is a diagonal matrix with lth element
M l
=
² a
+ ² b
² b
¡ ² a r
1
4 l ¡ 2 a r a is the cylinder radius

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

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

Shell: radius r s
, dielectric constant
 s
Three Remarkable but Unremarked
Papers (2)
Core: radius r c
, dielectric constant
 c
Matrix: dielectric constant
 m
Line of shell-matrix resonances:
 s
+
 m
=0
Line of core-shell resonances:
 c
+
 s
=0

Three Remarkable but Unremarked
Papers (3)
The matrix equation for coated cylinders has the same structure as for solid cylinders:
G
2l ¡ 1
=
(G + S)B = E r r
4 l ¡ 2 c
4 l ¡ 2 c
( ² s
¡ ² c
) ( ² m
¡ ² s
) + r
( ² s
¡ ² c
) ( ² m
+ ² s
) + r
4 l ¡ 2 s
4 l ¡ 2 s
( ² s
+ ² c
) ( ² m
+ ² s
( ² s
+ ² c
) ( ² m
¡ ² s
)
)
Core-shell resonance:
² s
+ ² c
= 0; G
2l ¡ 1
=
( ² m
+ ² c
)
( ² m
¡ ² c
)
Shell-matrix resonance:
² s
+ ² m
= 0; G
2l ¡ 1
= r r
4 l ¡ 2 c
4 l ¡ 2 s
( ² m
+ ² c
)
( ² m
¡ ² c
)

Shell-Core Resonance
• The shell hides itself (cloaks itself) by making the core behave as if it extended out to r s
, not r c
Shell-core resonance

Shell-Matrix Resonance
• The shell magnifies the core by making it behave as if it extended out to r s
2 /r c
, not r c
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 r s
2 /r c intersecting cylinders.
not correspond to

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 didn’t like homeopathy, so
PRB was the result

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!

Ghost Sources
• Interaction of a charge at r
0 and enlarged cylinders with the coated a 2 /r
0 r
0 q g q g q
The ghost source is in the matrix medium if r
0
> a; a
2 r
0
> r s or a < r
0
< a
2 r s

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.

Ghost Sources (3)
• The procedure of letting
 s
=-1+i
 s
’’ with  s
” small and positive yielded the following picture:
Potential converges as loss decreases
Potential diverges as loss decreases
GS1
GS2 a r
0 r crit
Note that there are two ghost sources, bounding the region in which the field is non-convergent r cr i t
= a
2 r s

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
Pendry’s 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)

Perfect Imaging (2)
The Pendry picture of perfect imaging: a slab of lefthanded material is used. A source is placed at d
0
.Two images are formed at x=-d
0 and x=d
0
-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).

Metamaterials
• Pendry’s 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 today’s 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)

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

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

Cloaking via Resonant Interactions
(2)
•
• For the cylindrical lens, cloaking occurs for distances r
*
=r s
2
0
/r c less than r
# distances less than r
Here r , and
* r
# if
 c p
= if
 c
=
 m
, and for
  m r 3 s
=r c
• If r c
!
1 , r s
= r c
+ d, t hen r
#
!
r s
+ (d=2)
• Note that r crit is the image of r s the image of r c in r
# in a, and also

The Animation
The following animation was provided by
Nicolae Nicorovici. It shows a coated cylinder with
 c
=1,
 s
=-1+i*10 -7 , r s
=4,r c
=2 placed in a uniform electric field. A polarizable molecule moves from the right. The dashed line marks the circle r=r
#.
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.

Animation

Response of Polarizable Molecule

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