Abstract
A planar surface on which a scalar wave satisfies the Neumann boundary conditions does not support a surface wave. However, a structured Neumann surface can support such a wave. By means of a Rayleigh equation for a scalar field in the region above the two-dimensional rough surface of a semi-infinite medium on which the Neumann boundary condition is satisfied, we derive the dispersion relation for surface waves on both doubly periodic and randomly rough surfaces. Dispersion curves for these waves on doubly periodic surfaces with three forms of the surface profile function are presented together with dispersion curves for surface waves on a two-dimensional randomly rough surface.
Highlights
In recent years interest has arisen in surface waves that propagate on impenetrable surfaces, in particular on perfectly conducting surfaces [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]
These surface waves cannot exist if the perfectly conducting surface on which they propagate is planar: the surface must be periodically [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18] or randomly [1,15] rough. The interest in such waves on periodically rough perfectly conducting surfaces is due to the fact that their dispersion curves mimic those of surface plasmon polaritons on planar metallic surfaces, even though a perfect conductor has no electronic plasma that can display collective oscillations
By varying the periodic structuring of a perfectly conducting surface, for example, by changing its period, or the heights and depths of its protuberances or indentations, or by changing the forms of these features, one can design these surface waves to possess specified properties, such as the frequency range in which they exist. Such ‘‘tuning’’ of the properties of these surface waves is not possible for surface plasmon polaritons on planar metallic surfaces. The existence of these waves on structured perfectly conducting surfaces has created the possibility of surface plasmon polaritons on periodically structured metallic surfaces in the gigahertz and terahertz frequency ranges, in which a metal is well represented by a perfect conductor
Summary
In recent years interest has arisen in surface waves that propagate on impenetrable surfaces, in particular on perfectly conducting surfaces [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]. By varying the periodic structuring of a perfectly conducting surface, for example, by changing its period, or the heights and depths of its protuberances or indentations, or by changing the forms of these features, one can design these surface waves to possess specified properties, such as the frequency range in which they exist Such ‘‘tuning’’ of the properties of these surface waves is not possible for surface plasmon polaritons on planar metallic surfaces. We obtain the dispersion relation for surface waves on a two-dimensional randomly rough Neumann surface on the basis of the small roughness approximation to the Rayleigh equation. For both types of two-dimensional surface roughness we find that a Neumann surface supports surface waves
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