Designing Static Fields for Unilateral Magnetic Resonance by a Scalar Potential Approach

We present a method for designing single-sided magnets suitable for unilateral magnetic resonance (UMR) measurements. The method uses metal pole pieces to shape the field from permanent magnets in a target region. The pole pieces are shaped according to solutions to Laplace´s equation, and can be designed using a combination of analytical methods and numerical optimization. The design leads to analytical expressions for the pole piece shapes and magnetic field. Here, we develop the method in Cartesian, polar, and spherical coordinates, and discuss the merits of each system. Finite magnet size has a substantial effect on the field quality in many cases, according to our simulations. We found that in order to achieve a compact magnet in which the static field closely matches that specified, a full 3-D design approach is necessary. A magnet designed by our method produces a static field with a constant gradient over a region 2 cm in diameter and 2 mm thick. This leads to a compact cylindrical magnet just over 11 cm in diameter, topped with a single metal pole piece. The design is validated through simulation. The simulated field is found to agree closely with that specified analytically through the design procedure