Optical excitation of surface plasmons the ability of thefree electrons in the metal to respond If we now apply Maxwells equation VE=0 we find to the externally imposed fields. If the electrons are free (still of course constrained inside the metal) then they are able to respond with no scattering to the incident radiation giving an ideal metallic response. Such a E2=-E3 k that is E=0 everywhere inside the metal, must therefore have &=+oo. An ideal metal in which the electrons Then to find the relationship between H, and E respond perfectly to the applied external field, therefore use Maxwell's equation A E=-H:(Faraday's law H cancelling it, is the limit 8--oo. Such a material of course does not exist, for the free electrons inside a metal of electromagnetic induction) which with H= Ho gives the oscillating field. The electrons have a finite mass and they permittivities and the normal component of the suffer scattering with lattice vibrations (phonons), defects wavevectors in the two media and the surface This means that as we increase the frequency of the H (5a) incident radiation the free electrons progressively find =oF (5b) it harder to respond. Ultimately at high enough frequencies, low enough wavelengths, the metal becomes Finally we need to apply the boundary conditions at transparent and behaves more like a dielectric z=0. We know tangential H is continuous and so is tangential E, thus HyI=Hy2 and Ex1= Ex2 leading to the following simple relationship between the relative permittivities and the normal components of the wavevectors in both media 3. More detailed theory From this simplistic treatment of the free electrons in a metal it is easy to show that there is a limiting frequency, k,t k2,z the plasma frequency, (for many metals in the Also we have ultra-violet) above which the metal is no longer metallic In this article we shall concern ourselves only with kzi=i(k:,k2)2, requiring k2>E,k2(7a) frequencies below this limit, that is with long enough wavelengths so that a is largely real and negative. As mentioned for real metals there is resistive scattering and k22=i(k2-6,k2)/, requiring k2>E,k2,(7b) hence damping of the oscillations created by the incident where k= o/c. If the wave is truly a trapped surface E field. This damping causes an imaginary component wave with exponential decays into both media then we E to E. Before, however, concerning ourselves with the need iki >0 and ik22 <0. Thus both k, s are imaginary added complexity of E, let us examine the implications of with opposite signs and so E, and c, are of opposite sign having a dielectric with positive er adjacent to the metal with negative E tells us the surface mode wavevector k, is greater than Because of the requirement of the normal E fields to the maximum photon wavevector available in the create surface charges we need only consider p-polarized dielectric, VE,k. The second condition, for the metal, is electromagnetic waves. Further whatever form the automatically satisfied with E, negative surface wave takes it has to satisfy the electromagnetic We may substitute expressions(7) for k,i and k,, into wave equation in both media. If we take the x-y plane (6)to give to be the interface plane and the positive z half space as medium 2, then for wave propagation in the x direction only, we have And we then see for k to be real, the requirement for E,=(Ex, 0, E,)exp [i(k,x-an)] exp(ikz1z)(3a) a propagating mode, with e2 negative, is that le2l>E, H,=(0, H,i, O)exp [(k, x-or)] exp(ik, z), (3b) Thus we now have satisfied Maxwells equations ar boundary conditions to give a trapped surface wave, with E,=(E,2, 0, E, )exp [i(k, x-or)] exp(ik, z)(3c) real k, and appropriate kx, provided IE2l >&1 and e2 <0 Following the above analysis with purely real g values H2=(, Hy2, O)exp [i(k, x-an)] exp(ik,2 z).(3d) leads to a surface wave having purely real k, which is