Inverse source problem in nonhomogeneous background media

The scalar wave inverse source problem (ISP) is investigated for the case where the source is embedded in a non-homogeneous medium with known index of refraction proÞle n(r). It is shown that the solution to the ISP having minimum energy (so-called minimum energy source) can be obtained via a simple method of constrained optimization. This method is applied to the special case when the non-homogeneous background is spherically symmetric (n(r) = n(r)) and yields the minimum energy source in terms of a series of spherical harmonics and radial wave functions that are solutions to a Sturm-Liouville problem. The special case of a source embedded in a spherical region of constant index that differs from the background is treated in detail and results from computer simulations are presented for this case.