Magnetic Field Analysis of Helical Undulators

The analytical expression of the on-axis field for an infinitely long helical undulator, derived in terms of undulator period length, uniform current density, inner radius and dimensions of the coils, was compared with numerical analyses of model undulators based on the Biot-Savart law. The two calculations agreed within 110. The expression also showed that the on-axis field has the first harmonic only, and no higher harmonics exist, regardless of the coil dimensions. Calculations for 61-period models showed that the third and fifth harmonics were less than 210. For a finite-length model the end fields were adjusted by reducing the currents in a discrete number of steps for the last two or four periods of both ends. It was also shown as a practical method that increasing the coil winding radius can be a substitute for reducing the coil current.