Design of high-order superconducting planar gradiometers with shaped asymmetric near-source response

We describe the solution of the general problem of designing an N-th order planar gradiometer withm \geq Ncrossovers, where the position of (m-N) of these crossovers is selected on near-source response criteria. The ability to pre-select certain crossover positions is valuable in many applications (e.g. biomedical, NDT) because it allows us to control specific properties of the near-source response such as (i) response peaking at specified locations (ii) zero-response at certain locations (iii) minimal coupling to near-sources, at each end of a gradiometer of arbitrary order. Examples are given for the important cases m = N+1 and m = N+2.