Optical Resonances of Cylinder and Sphere Clusters in the Quasistatic Limit

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Optical Resonances of Cylinder and Sphere
Clusters in the Quasistatic Limit
L. Field1, N. A. Nicorovici1,2, and R. C.
McPhedran1 1The University of Sydney, 2University
of Technology, Sydney
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Overview
Optical resonances solutions of the
electrostatic potential problem for clusters
consisting of dielectric spheres or cylinders,
with zero applied field. For metallic systems
the optical resonant behaviour is due to plasmon
resonances which depend on the metal and on the
size and shape of the components. These
resonances are well known in electrostatics, and
correspond to the poles in the Bergman-Milton
bounds or in multipole formulae of the Rayleigh
type for effective transport properties. The
method is of great interest in surface plasmon
resonance spectroscopy, and method of
images. Here, we present results of studies of
optical resonances of small clusters of cylinders
and spheres. We show analytic, numerical, and
asymptotic results.
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Some References
G. W. Milton, The Theory of Composites (Cambridge
University Press, Cambridge, New York, 2002). R.
C. McPhedran and D. R. McKenzie, Appl. Phys., 23,
223-235 (1980). R. C. McPhedran, Proc. R. Soc.
Lond. A 408, 31-43 (1986). R. C. McPhedran and
G. W. Milton, Proc. R. Soc. Lond. A 411, 313-326
(1987). R. C. McPhedran, L. Poladian, and G. W.
Milton, Proc. R. Soc. Lond. A 415, 185-196
(1988). L. Poladian, Q. J. Mech. Appl. Math., 41,
395-417 (1988). L. Poladian, Ph.D. thesis, The
University of Sydney, Australia (1990).
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Theory (1)
Electric potential multipole expansions
for cylinder q (q 1, N)
Here, Cq denotes the cross section of cylinder q.
We also have the Wijngaard type expansion, valid
in the region between the cylinders
In Dq\Cq, V(z) V(q)ext(z), i.e., V(z) is the
analytic continuation of V(q)ext(z).
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Theory (2)

• V(q)int(z) and V(q)ext(z) have to the satisfy
  the boundary conditions on ?Cq.
• If we consider relative dielectric constants e
  ecylinder/ematrix (assumed to be real) then the
  multipole coefficients of the real part of the
  electric potential will satisfy the equations
• According to Kellers theorem the multipole
  coefficients of the imaginary part of the
  electric potential will satisfy the equations
• Now, from V(z) V(q)ext(z), with
  (1e)/(1-e) and normalising the B coefficients by
  al we obtain the Rayleigh identity

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Theory (3)

• The optical resonances of a cluster of
  cylinders are given by the Rayleigh identity with
  E0.
• By separating the real and imaginary parts,
  we are led to a linear eigenvalue problem of the
  form
• Then, from the values of we obtain the
  relative dielectric constants of the cylinders
  (e) for a resonant system.

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Cluster of 3 Cylinders (1)

• First, we consider a cluster of three
  cylinders located at the corners of an
  equilateral triangle. The cylinders are of radius
  a 0.3, and the edge of the triangle is d 1.
• The two figures show the equipotential lines
  for the complementary structures
• cylinders of relative dielectric constant e,
  embedded in a matrix of relative dielectric
  constant 1,
• cylinders of relative dielectric constant 1,
  embedded in a matrix of relative dielectric
  constant e.
• The fundamental resonance of these systems
  occurs at

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Cluster of 3 Cylinders (2)
In this figure we show that, at resonance, the
equipotential lines for the two problems are
orthogonal. Hence, the contour lines of the
second problem may give an indication about the
orientation of the field lines for the first
problem and vice versa.
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Analytic Continuation
Using a truncated expression of the external
potential V(z) to plot the potential inside the
cylinders we can visualise the distribution of
image sources. In our case the accumulation
contour for images is the arc of the circle of
radius limited by the lines between the
centres of the cylinders.
(L. Poladian, PhD thesis, The University of
Sydney (1990))
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Cluster of 4 Cylinders
Next, we consider a cluster of four cylinders
located at the corners of a square. The cylinders
are of radius a 0.3, and the edge of the square
is d 1. We solved the problems for the
complementary structures and found the
fundamental resonance of both systems at
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Analytic Continuation
Now, the images accumulate up to an arc of the
circle of radius also limited by the lines
between the centres of the cylinders.
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Clusters of Spheres (1)
For spheres, the numerical implementation of the
multipole method is more complicated than for
cylinders because the coefficients of the
boundary conditions depend on the multipole order
and this leads to larger linear systems that
represent the Rayleigh identity.
Note that there is no Kellers theorem for
three-dimensional structures. The systems
considered are pairs of spheres of relative
dielectric constant e embedded in a matrix of
relative dielectric constant 1. For a pair of
identical spheres of radius a 1 at a distance d
3, we found the fundamental resonance at
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Clusters of Spheres (2)
For a pair of spheres of radii a1 1, a2 2 at
a distance d 4, having the same relative
dielectric constant we found the fundamental
resonance at
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Clusters of Spheres (3)
Two higher resonances of a pair of spheres
having different radii but the same relative
dielectric constant.
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Conclusions

• This analysis can be extended in many ways.
• Coated cylinders imaging is much more difficult
  physics is very rich.
• Cylinders tending to plane interfaces (large
  radius limit).
• Coated spheres spheres tending to half-spaces.
• Effective asymptotic methods for finite clusters
  linked with images.
• Correspondence for the Helmholtz equation.