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Green's function for a sharpened and metal-coated dielectric probe.

In apertureless scanning-probe optical microscopy and in the case of more traditional scanned optical probes coated with a metal that is thin near the probe tip (in lieu of an aperture), samples are probed via interaction between the probe and surface. In the nanometer-scale region between the tip and the sample, the field can be approximated by quasi-electrostatic analytics. Hence, the coated probe can be modeled as in the present case as a hyperboloid of revolution without the need for hyperboloidal wave functions in the near zone. The solutions to Laplace's equation and in general Green's function with the application of the boundary conditions, therefore, yield an appropriate approximation and allow a completely analytical solution for the resonance effects upon the probe tip to be obtained. The large field enhancements due to the sharpness of the tip and to surface plasmon fields may thus be analytically examined.