Planar Coil Optimization in a Magnetically Shielded Cylinder

Hybrid magnetic shields with both active field generating components and high-permeability shielding are increasingly needed for a variety of technologies experiments that require precision-controlled environments. However, the fields generated by interact passive shield, distorting desired profiles. Consequently, optimization to generate user-specified target must include coupling components. Here, we consider planar systems, on which an arbitrary static current flows, coupled closed cylindrical shield. We modify Green's function vector potential match boundary conditions shield's interior surface, enabling us construct inverse problem design coils inside shields. validate our methodology designing two bi-planar hybrid active--passive constant transverse field, $\mathbf{B}=\mathbf{\hat{x}}$, linear gradient, $\mathbf{B}=(-x~\mathbf{\hat{x}}-y~\mathbf{\hat{y}}+2z~\mathbf{\hat{z}})$, respectively. For inverse-optimized profiles agree well forward numerical simulations. Our is accurate flexible, facilitating miniaturization high-performance strict constraints spatial limitations.