Numerical Calculation of Near Fields of Tapered Fiber Probe in Presence of a Sample by Borgnis Potentials

A method is presented for the calculation of vector near fields of a tapered fiber probe in presence of a rotationally asymmetric sample by numerically solving boundary integral equations of two scalar Borgnis potentials. For the convenience of the description of all the boundaries, cylindrical coordinates and Cartesian coordinates are, respectively, used in the domains, where the rotationally symmetric tapered fiber probe and the rotationally asymmetric sample are located. The Borgnis potentials on the surface of the fiber probe are harmonically expanded in the azimuthal direction of the cylindrical coordinates so that the fiber probe can be treated as a 2-D structure. Only the Borgnis potentials on the surface of the sample, rather than those on all surfaces, are first calculated. The Borgnis potentials on other surfaces could be finally calculated by their relationship with the solved Borgnis potentials on the surface of the sample. With this method, the memory requirement can be considerably reduced in calculation and the method can easily extend to the solution of electromagnetic fields in structures much more complex.