Abstract
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 pose a challenging modeling prob-
lem 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 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
∈ N
0 . The probe charged to a potential V0 , measured
at the apex, is modeled, in the non-contact 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.
