Three-dimensional pure permanent magnet undulator design theory

Expressions for fields due to a point charge in 3D and due to a line charge in 2D are compared. Extensions to dipoles are made with emphasis on the relationship between dipole orientation and field component magnitudes. Differences between the effects on fields of dipole rotations in 2D and in 3D are highlighted and formulas for maximizing individual field components are given. A final macrogeometry extension is made and a closed-form expression is developed to calculate the field due to an arbitrary 3D configuration of permanent magnet (PM) blocks. The field optimization theory is applied to the design of the ALS elliptically polarizing undulator (EPU). Utilizing 3D field enhancement, peak on-axis field in practical designs can be increased typically by 5% to 40% or more over their 2D counterparts. The theory is generally applicable to any pure (i.e., no soft magnetic material) PM design.