Plasmons and Nanofocusing
Surface plasmons have the unique ability to achieve very short wavelengths at modest optical frequencies. In addition to this, they can be stimulated with a purely optical field. They can thus mediate efficient optical coupling to the nanoscale.
Materials for Plasmonic Focusing Efficient Optical Coupling to the Nanoscale: PhD Dissertation

It is evident that deeply subwavelength focal spots cannot be formed through conventional focusing using a lens system or microscope objective. This is due, primarily, to the lack of highindex media at visible frequencies. What if, however, one was able to achieve a high effective index with conventional optical materials? That is the potential of surface plasmon optics. By employing geometries of conductors (such as metals or doped semiconductors) with dielectrics (such as air or glass), modes at optical frequencies can be created with effective indices of refraction that are orders of magnitude higher than those of the constituent materials. In fact, these indices can be so high as to create Xray wavelengths (less than 10nm) with visible frequencies. The reason surface plasmon modes can achieve anomalously high wavevectors at visible frequencies is because they are mediated by electrons rather than free space optical fields^{[i]}. Surface plasmons are electron oscillations^{[ii]}^{,[iii]} at optical frequencies which are localized to the interface of a material with a positive dielectric constant and that of a negative dielectric constant (as illustrated in Figure 1‑3). At wavevectors much smaller than the Fermi wavevector^{9} of the conductor, these modes can be well described by Maxwell’s Equations. Quantum mechanical considerations only become necessary at very short plasmon wavelengths, beyond the scope of this dissertation and are not required for determining the limits of efficient focusing. At low wavevectors, the behavior of these surface modes can be understood intuitively. In this regime, they are essentially transverse in character, although strictly speaking they have a small longitudinal component. These transverse fields generate a polarization in the dielectric which is aligned with the stimulating field. In the metal, however, the polarization will be in the opposite direction of the applied field owing to its negative dielectric constant. Now we have a situation where the stimulating field is creating equal and opposite electric displacements (D), in phase with each other across an interface. These opposing electric displacements serve to attract and confine the current to this interface, thus generating the collective electron oscillations of the surface plasmon.
Starting from Maxwell’s Equations, it is valuable to derive the characteristics of this simple plasmonic system. Again we take the free current to be zero and the relative permeability (μ_{r}) of all media to be unity. Following Economou^{[iv]}, we will assume translational symmetry and homogeneity in the direction (following the axes of Figure 1‑3) and propagation with wavevector k in the direction. The behavior in the direction is taken to be exponentially decaying away from the interface. Derived in the previous section, the wave equation tells us that the exponential decay constant in medium i must be
We now have enough information to determine the fields to within a scale factor. In the dielectric (medium 1), we define
The scalar quantity A represents a scale factor to be determined. From Gauss’s Law we may derive E_{z} _{} Likewise for the conductor, we may derive E_{x} and E_{z}. In this case, we set the scale factor to unity, as only the relative scale factor carries importance. Applying the electromagnetic boundary conditions at the interface allows one to then solve for A. The continuity of E_{z} and D_{z} yield
Some simple algebra may then be used to solve for k, finally generating the dispersion relation for these simple surface plasmon modes^{[v]}.
The wavevector is no longer a linear function of permittivity as in standard dielectrics. Because we have the sum of dielectrics of opposite sign in the denominator, very large wavevectors are possible.
These wavevectors, of course, are intimately tied to the dispersion of the constituent materials. One cannot discuss the plasmon dispersion relations without a model for the relative permittivities. Dielectric materials tend to have fairly constant permittivities over large bandwidths, while conductors tend to be very dispersive. To illustrate the dispersion relations of a typical material system, a Drude model^{[vi]} serves as a simple approximation for a metal. Here we will assume a lossless Drude metal and denote the plasma frequency as ω_{p.} Taking free space as the dielectric material allows us to generate a dispersion relation, plotted in Figure 1‑4. For generality, the frequency is plotted in units of the plasma frequency (ω_{p}) and the wavevector in units of ω_{p}/c. As a point of reference, the light line is plotted as a grey dashed line and represents the dispersion relation of an optical field propagating in the dielectric medium along the same direction as the surface plasmon.
While this is a very simplified material system, there are two important characteristics of surface plasmons which are evident in this model. The first is that the dispersion relation always lies at higher wavevectors than the light line. Due to the difference in wavevector, then, the plasmon field cannot efficiently couple to radiating modes. Conversely, freespace optical fields cannot directly stimulate surface plasmons unless a mechanism introduces additional momentum. Of course, this result is to be expected from our initial assumptions. By defining the mode to have an exponential decay normal to the surface, we assured an imaginary wavevector in this dimension. The absolute square of this positive quantity then adds to the lightline wavevector to determine k^{2} (as in Equation ), hence k must always be greater than that of the free space field.
Figure 1‑4: Dispersion relation for a Drude Metal plotted in units of the plasma frequency. The dashed line represents the light line. The second defining feature of this dispersion relation occurs as the frequency approaches 0.7 ω_{p}. Here the wavevector grows very large, orders of magnitude beyond the light line. It is precisely this feature which we will exploit to allow efficient focusing to the nanoscale. Although the permittivities of these materials may be modest, their geometry and interactions create an effective index much larger than that available from conventional transparent media. The reader may observe that these wavevectors seem to violate Heisenberg’s Uncertainty Principle^{[vii]} for optical fields in these materials. Because these modes are mediated by collective electron oscillations with subAngstrom wavelengths, they may achieve very large optical wavevectors. In the parlance of plasma physics, the energy is in the form of oscillating charge separations between the negative electrons and the positive ionic background of the metal, waves that extend to large wavevectors. Various groups have reported low loss plasmons which involve clever schemes to keep a vast majority of the optical energy in the dielectric materials rather than in the metals^{[viii]}. This, however, makes those modes incapable of achieving very large wavevectors. Next Section: Materials for Plasmonic Focusing [i] A. Graff, D. Wagner, H. Ditlbacher and U. Kreibig, European Physical Journal D, vol. 34, p.263 (2005).
[ii] C. Kittel, Introduction to SolidState Physics (John Wiley and Sons, New York, 1996).
[iii] H. Raether, Surface Plasmons, vol 111 of SpringerVerlag Tracts in Modern Physics (SpringerVerlag, New York, 1988).
[iv] E.N. Economou, Physical Review, vol 182, no. 2, p539 (1969).
[v] J.R. Sambles, G.W. Bradbery and F.Z. Yang, Contemporary Physics, vol 32, p.173 (1991).
[vi] A.H. Wilson, The Theory of Metals (Cambridge University Press, London, 1936).
[vii] P.A.M. Dirac, The Principles of Quantum Mechanics (Clarendon Press, Oxford, 1996).
[viii] K. Leosson, T. Nikolajsen, A. Boltasseva and S. I. Bozhevolnyi, Optics Express, vol. 14, p. 314 (2006).

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