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This paper is concerned with the development of an iterative algorithm for the reconstruction of the three-dimensional complex-valued refractive index of an object in electromagnetic scattering. The inverse problem is modelled by a second-order Maxwell equation for the electric field at fixed frequency with plane, time-harmonic microwave irradiation. The complex-valued scattered electric field is measured at some distance from the object for a finite number of directions of the incident waves. Each of these measurements is used to improve iteratively a starting guess for the true refractive index of the object. For each iteration one forward and one adjoint problem of a system of a second-order equation have to be solved. The complexity of each step is O(N3log(N)). The algorithm solves the full nonlinear inverse problem. It can be considered as a nonlinear generalization of the algebraic reconstruction technique in x-ray tomography. The practical usefulness of the algorithm is shown in three numerical simulations.

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