Homogeneous fields: Double expansion method, 3D printing/CNC realization, and verification by atomic magnetometry

In low-field magnetic resonance applications there is often an interest in creating homogeneous magnetic fields over unusual geometries, particularly when quantum magnetometers are involved. In this paper a design method is proposed, where both the surface current and magnetic field are expanded to find current coefficients that cancel out higher order field terms. Two coils are designed using this double expansion methodology: (1) a tuning field for a half-meter-long atomic magnetometer array and (2) a null field for a magnetometer to operate adjacent to an excitation solenoid. The field verification of the former shows the accuracy of CNC milling and the method proposed; a close analysis of the field signature in the latter revealed the limitations of 3D printing for precise scientific applications. Both coils are designed to be fifth-order error systems or better.