Reverse optimal design of longitudinal gradient coils for miniature atomic sensors

Atomic sensors exhibit extensive application potential across various fields owing to their high precision and small volume. However, ensuring the uniformity of the internal magnetic field within the vapor during the miniaturization process remains a critical challenge. To address this challenge, this paper proposes a novel design method that uses biplanar coils arranged in a predefined circular pattern to generate a longitudinal gradient magnetic field while ensuring the preservation of a central light-passing hole as per the design requirements. Compared to the conventional gradient coil design method, this method can minimize the aspect ratio to unity while realize magnetic field corrections. Meanwhile, the regularization method was introduced to tackle the ill-conditioned problems where a unique solution may not be achievable. Simulation results show that the proposed method demonstrates an excellent linear gradient. A scaled-up prototype (5:1) was fabricated using printed circuit board (PCB) technology to avoid manual winding errors. Experimental measurements confirm that the achieved magnetic field uniformity aligns closely with the theoretical predictions.