If the FEL works with maximum power the energy factor g of the electrons decreases by Dg = g/2 N_u (for the first harmonic) along the electron path due to the interaction of the electron beam with the electromagnetic field, where N_u = 68 is the total number of magnetic periods in the two undulator units [1]. The electron energy changes continuously from the undulator entrance to the exit leading to a resonance wavelength l(z) which depends on the coordinate z along the undulator

l(z) = lu 2g2(z) (1+ 1 2 K2).

(1)

Within a certain interval of l this effect can be compensated by differentially increasing the gap, and hence decreasing the magnetic field along the undulator. To compensate the energy loss dg by a reduction dB of the magnetic field one has to ensure

dl = ¶l ¶g dg + ¶l ¶K dK = 0

(2)

from which follows

dB B = 2+K2 K2 dg g .

(3)

For hybrid undulators, the Halbach equation allows one to estimate the widening of the gap g needed for the variation dB of the field corresponding to equation (3). Figure 1 shows the effect of field tapering in both sections of the undulator U27, which would be used in a situation typical for high intensity lasing.

Fig. 1  (a) Measured field distribution B_y(z) in the middle plane for a gap width g = 12 mm and a tapering of 0.3 mm on the exits of both undulator sections.
(b) Second field integral I₂ of the undulator U27 as a function of the coordinate z. Within the scale used an influence of the tapering on the form of I₂(z) is not noticeable. For conversion of I₂ in mm the formula x[mm] = 300·I₂[T mm²]/E[MeV] must be applied.

References

[1] P. Gippner, E. Grosse, W. Seidel, J. Pflüger, this Report  IKH 05/21/01 © P. Gippner