Effects of Roughness
In order to understand the effect of roughness on surface plasmons, it is beneficial to first understand how a plasmon is coupled by a grating Figure2. When a photon is incident on a surface, the wave vector of the photon in the dielectric material is smaller than that of the SPP. In order for the photon to couple into a SPP, the wave vector must increase by . The grating harmonics of a periodic grating provide additional momentum parallel to the supporting interface to match the terms.
where is the wave vector of the grating, is the angle of incidence of the incoming photon, a is the grating period, and n is an integer.
Rough surfaces can be thought of as the superposition of many gratings of different periodicities. Kretschmann proposed that a statistical correlation function be defined for a rough surface
where is the height above the mean surface height at the position, and is the area of integration. Assuming that the statistical correlation function is Gaussian of the form
where is the root mean square height, is the distance from the point, and is the correlation length, then the Fourier transform of the correlation function is
where is a measure of the amount of each spatial frequency which help couple photons into a surface plasmon.
If the surface only has one Fourier component of roughness (i.e. the surface profile is sinusoidal), then the is discrete and exists only at, resulting in a single narrow set of angles for coupling. If the surface contains many Fourier components, then coupling becomes possible at multiple angles. For a random surface, becomes continuous and the range of coupling angles broadens.
As stated earlier, surface plasmons are non-radiative. When a surface plasmon travels along a rough surface, it usually becomes radiative due to scattering. The Surface Scattering Theory of light suggests that the scattered intensity per solid angle per incident intensity is
where is the radiation pattern from a single dipole at the metal/dielectric interface. If surface plasmons are excited in the Kretschmann geometry and the scattered light is observed in the plane of incidence (Fig. 4), then the dipole function becomes
with
where is the polarization angle and is the angle from the z-axis in the xz-plane. Two important consequences come out of these equations. The first is that if (s-polarization), then and the scattered light . Secondly, the scattered light has a measurable profile which is readily correlated to the roughness. This topic is treated in greater detail in reference.
Read more about this topic: Surface Plasmon
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