Analytical Green´s function for tip-sample interaction in microwave impedance microscopy

Analysis and evaluation of tip-sample interaction in microwave microscopy are crucial to accurately measure the permittivity and conductivity of nanomaterial. However, even till now, an exact analytical solution for the tip sample interaction has yet to be found due to the complexity of various tips. In this paper, the exact three dimensional solution of Green´s function due to an arbitrary dipole is given under bispherical coordinate system, and this solution is verified numerically. Moreover, even though the solution is derived under the spherical model, it turns out that it can be applied to various tip models when sample is close to tip. After obtaining the analytical solution, we can use it to improve the resolution of microwave microscopy and save time and memory in three dimensional simulations.