Singular spectrum analysis filtering and Fourier inversion: an efficient and fast way to improve resolution and quality of current density maps with low-cost Hall scanning systems

We provide a Biot–Savart inversion scheme that, for any two-dimensional, or bulk with planar crystallization, high-temperature superconducting (HTS) sample, determines current density maps with a higher resolution and accuracy than previous procedures and at a fraction of its computational cost.

The starting point of our scheme is a Hall scanning microscopy map of the out-of-plane component of the magnetic field generated by the current. Such maps are noisy in scans of real samples with commercial-grade equipment, and their error is the limiting factor in any Biot–Savart inversion scheme. The main innovation of our proposed scheme is a singular spectrum analysis (SSA) filtering of the Hall probe maps, which cancels measurement errors such as noise or drifts without introducing any artifacts in the field map.

The SSA filtering of the Hall probe data is so successful in this task that the resulting magnetic field map does not require an overdetermined QR inversion, allowing Fourier inversion of the Biot–Savart problem.

Our implementation of SSA filtering of the Hall scan measurements, followed by Biot–Savart inversion using the fast Fourier transform (FFT), is applied to both simulations and real samples of HTS tape stacks. The algorithm works in cases where ill conditioning ruled out the application of Fourier inversion, and achieves a finer resolution for a fraction of the cost of the QR inversion used to date. The computation passes physical and statistical validity tests in all cases, and in three-dimensional samples it is shown to yield the average, with a depth-dependent weight, of the current density circulating in the different layers of the sample.