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The reconstruction of surface tangential components of the electromagnetic field from near-field measurements

We consider the problem of reconstructing the surface tangential components of the electromagnetic field from electric (or magnetic) measurements on a nearby surface. Mathematically, the electric (or magnetic) measurements satisfy the Maxwell system. We show that any solution to this equation admits a unique representation by a magnetic dipole, so that the problem is reduced to the solution of a linear integral equation of the first kind. This integral equation is discretized using boundary element methods and conjugate gradients are utilized as a regularization method for the numerical solution. We study uniqueness of the reconstruction and obtain stability estimates. In addition, we report numerical results applied to two cases: a sphere and a cylinder with flat end-caps.