An inverse problem for the magnetic force microscopy of a superconducting sphere

A fundamental inversion problem for the magnetic force microscopy (MFM) of a spherical superconductor in the Meissner state is formulated. Based upon a dipole model of the MFM tip, it is shown how a penetration depth (r) depending upon the radial coordinate can be recovered from one-dimensional force measurements. The ill-posedness of the problem for this particular geometry is discussed. A differential equations approach is followed and the connection with the radial Schrödinger equation is described. Special forward problems illustrate part of the inversion procedure. The inversion results give a means to recover for a superconductor of finite radius, in an axisymmetric geometry.