Light knocks electrons out of a crystal, and they fly toward the detector like arrows toward a target. From their trajectories, physicists reconstruct how electrons behaved inside the material: with what energy and in which direction they moved. But switch on a magnetic field, and the trajectories bend. The arrows start to curve, the target turns into a warped mirror, and the picture blurs.

This problem confronted a group of researchers when they added a small solenoid to their ARPES setup. ARPES is a technique that measures the energy and momentum of electrons knocked out by light. The solenoid produced a magnetic field of up to 10 millitesla, roughly two hundred times weaker than the field of a refrigerator magnet. Yet even such a weak field noticeably distorted the electron paths, and the spectra lost sharpness.

The solution was not to remove the field but to calculate its effect and correct the image. The physicists simulated electron trajectories in the magnetic field using a computer program that divides space into small cells. Then they constructed an inverse transformation: knowing where an electron hit the detector, one can compute where it originated. This is similar to how a restorer reconstructs true facial features from a distorted reflection in an old mirror.

The test was performed on the topological insulator Bi2Se3, a material that conducts electricity only on its surface while remaining an insulator inside. At fields up to 8.6 millitesla, they recovered the Fermi surface and the dispersion of topological surface states with an accuracy of one degree in angle. This means the distortions were almost completely compensated, and the picture became as sharp as without the field.

In this method, the magnetic field acts like a lens: at some points it focuses electrons, at others it scatters them. At focal points, angular resolution is completely lost, but measurements are possible between them. The physicists mapped these points and showed that by choosing the geometry wisely, one can avoid the traps. Now any researcher with an ARPES setup and a solenoid can apply the same correction and peer into the electronic structure of materials under a magnetic field.