face and exponentially decays away from the interface. The mathematical form of surface plasmons can be obtained by solving Maxwell's equations at a metal-dielectric interface:
Assuming the metal-dielectric interface lies in the xy plane at z = 0, it can be shown that x-propagating surface waves of the following form are solutions to Maxwell's equations:
with Ex,m = Ex,d (E|| continuous), &epsilonmEz,m = &epsilondEz,d (D&perp continuous), and Hy,m = Hy,d (B|| continuous); all other field components are zero. The subscript j is used to denote fields in the metal or dielectric. The dispersion relation for surface plasmons can be obtained by inserting the equations for E and H into Maxwell's equations and enforcing the boundary conditions:
The dispersion relation, assuming &epsilonm = 1 - (&omegap/&omega)2, is shown in Fig. 1, along with the free-space and bulk plasmon dispersion relations. The surface plasmon dispersion relation always lies to the right of the free-space dispersion relations; as a result, free-space radiation does not have enough momentum to launch surface plasmons.
Additionally, the magnitude of kx for the surface plasmon asymptotically approaches &infin as &omega approaches &omegasp, where &epsilonm = -&epsilond. For &epsilonm = 1 - (&omegap/&omega)2, &omegasp = &omegap / &radic(1 + &epsilond). As kx increases, the surface plasmon wavelength decreases and &kappaz increases, resulting in a rapidly varying surface wave that is more tightly confined to the surface. For smaller values of &omega, the surface plasmon wave vector approaches the light line, the surface plasmon and free-space wavelengths become more similar, and the surface wave becomes less and less strongly confined to the surface.
For the simple free-electron model considered thus far, there is no loss in the system and the surface wave will have an infinite propagation length. In real materials, however, there is loss and the wave will only propagate a finite length. The propagation length will decrease as the surface wave becomes more tightly bound to the interface (as &omega approaches &omegasp), as the energy of the field becomes more concentrated in the metal. Consequently, there is a trade-off between confinement and propagation distance.
A final property of interest that can be obtained from the dispersion diagram shown in figure A-2 is the group velocity of the surface wave. Any information carried by the wave will propagate at the wave group velocity, vg = (&part &omega / &part k). As &omega approaches &omegasp, vg asymptotically approaches zero.
The dispersion diagram plots shown above were generated using matplotlib, an open source plotting library written in Python; the source code of the python script for the plots can be downloaded here. The images of the above equations were generated using latex and latex2html; the .tex file can be downloaded here. The figure of the Kretschmann-Raether coupler was drawn in Inkscape, an open source vector graphics editor; the svg file can be downloaded here.
© 2007 Justin White. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
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