Linear canted permanent magnet field derivation for low dispersion undulator design

In this study, we present analytical formulas for the 3D magnetic field of canted (right-angled trapezoid) bar magnets. These magnets are a variant of pure permanent magnets (PPMs) that introduce a quadrupole component in addition to the dipole field, allowing quad-like beam focusing without side magnets. Through conformal mapping and linear approximations, we derive a set of equations that accurately describe the spatial magnetic field in a canted magnet. These equations are validated through numerical simulations and electron beam trajectory analysis, confirming their reliability.Our results demonstrate the advantages of using canted magnets in canted magnet planar undulator structures. The quadrupole focusing effect decreases the horizontal beam divergence caused by strong longitudinal compression, eliminating the need for additional focusing magnets. This approach simplifies the undulator’s mechanical design while maintaining beam stability and improving radiation performance. The proposed analytical model provides a powerful tool for designing optimized undulators for compact FELs and synchrotron radiation sources.