Properties of index transforms in modeling of nanostructures and plasmonic systems

In material structures with nanometer scale curvature or dimensions, electrons may be excited to oscillate in confined spaces. The consequence of such geometric confinement is of great importance in nano-optics and plasmonics. Furthermore, the geometric complexity of the probe-substrate/sample assemblies of many scanning probe microscopy experiments often poses a challenging modeling problem due to the high curvature of the probe apex or sample surface protrusions and indentations. Index transforms such as Mehler-Fock and Kontorovich-Lebedev, where integration occurs over the index of the function rather than over the argument, prove useful in solving the resulting differential equations when modeling optical or electronic response of such problems. By considering the scalar potential distribution of a charged probe in the presence of a dielectric substrate, we discuss certain implications and criteria of the index transform and prove the existence and the inversion theorems for the Mehler-Fock transform of the order m{small element of}N ⁰. The probe charged to a potential V ⁰, measured at the apex, is modeled, in the noncontact case, as a one-sheeted hyperboloid of revolution, and in the contact case or in the limit of a very sharp probe, as a cone. Using the Mehler-Fock integral transform in the first case, and the Fourier integral transform in the second, we discuss the necessary conditions imposed on the potential distribution on the probe surface.