Inverse Source Problem in Nonhomogeneous Background Media

The scalar wave inverse source problem (ISP) of determining an unknown radiating source from knowledge of the field it generates outside its region of localization is investigated for the case in which the source is embedded in a nonhomogeneous medium with known index of refraction profile n(r). It is shown that the solution to the ISP having minimum energy (the 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 nonhomogeneous background is spherically symmetric (n(r)=n(r)), and it 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 is treated in detail, and results from computer simulations are presented for this case.