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.

Motivation

Conventional Optics

Plasmons and Nanofocusing

Materials for Plasmonic Focusing

Efficient Optical Coupling to the Nanoscale: PhD Dissertation

Curriculum Vitae

UCLA Optoelectronics Group

SINAM efforts on Plasmons

Nanophotonics MURI

 

 

 

 

           It is evident that deeply sub-wavelength focal spots cannot be formed through conventional focusing using a lens system or microscope objective. This is due, primarily, to the lack of high-index 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 X-ray wavelengths (less than 10nm) with visible frequencies.

            The reason surface plasmon modes can achieve anomalously high wave-vectors 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 13). At wave-vectors much smaller than the Fermi wave-vector9 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 wave-vectors, 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 (μr) of all media to be unity. Following Economou[iv], we will assume translational symmetry and homogeneity in the direction (following the axes of Figure 13) and propagation with wave-vector 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 Ez

 

                                                         

Likewise for the conductor, we may derive Ex and Ez. 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 Ez and Dz 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 wave-vector 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 wave-vectors are possible.

 

These wave-vectors, 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 ωp.

                 Taking free space as the dielectric material allows us to generate a dispersion relation, plotted in Figure 14. For generality, the frequency is plotted in units of the plasma frequency (ωp) and the wave-vector in units of ωp/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 wave-vectors than the light line. Due to the difference in wave-vector, then, the plasmon field cannot efficiently couple to radiating modes. Conversely, free-space 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 wave-vector in this dimension. The absolute square of this positive quantity then adds to the light-line wave-vector to determine k2 (as in Equation ), hence k must always be greater than that of the free space field.

 

Figure 14: 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 ωp. Here the wave-vector 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 wave-vectors seem to violate Heisenberg’s Uncertainty Principle[vii] for optical fields in these materials. Because these modes are mediated by collective electron oscillations with sub-Angstrom wavelengths, they may achieve very large optical wave-vectors. 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 wave-vectors. 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 wave-vectors.

 
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 Springer-Verlag Tracts in Modern Physics (Springer-Verlag, 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).