Computation of magnetic fields from field components on a plane grid

An algorithm is presented to calculate the field components of the magnetic field [Bx(x,y,z),By(x,y,z),Bz(x,y,z)] at a point (x,y,z) in space, from the knowledge of the components [Bx(x,y=0,z),By(x,y=0,z),Bz(x,y=0,z)] on a “reference plane”, which is normal to the y-axis at y=0. The algorithm, which is a general one and is not restricted to fields with mid-plane symmetry is based on the Maclaurin series expansion of the magnetic field components at any point in space in terms of the distance (y) of the point from the reference plane. The coefficients of the Maclaurin series expansion are expressed in terms of the on-plane field components and their partial derivatives with respect to spatial coordinates (x,z). The field components are usually generated from magnetic field measurements on a rectangular grid on the plane. This algorithm was employed in 1986 in the RAYTRACE computer code to help calculate the optical properties of magnets and of the Alternating Gradient Synchrotron (AGS) at the Brookhaven National Laboratory (BNL). A general mathematical formulation of this algorithm based on the differential algebraic method was presented by Makino in 2011. This paper presents the step by step derivation of the algorithm and provides the necessary formulas to be introduced by the reader in any computer code which requires the field components generated by magnetic devices. In addition provides an example of the use of the algorithm and its limitations as applied to a Halbach type magnet with or without median plane symmetry.