Two undulators (U27 and U50) are intended to be installed at the beam of ELBE to produce intense coherent radiation in the mid and far infrared range. To estimate the quality of the undulator we have measured the field strength B along the undulator axis z. In particular we consider the standard deviations s_B and s_Y of the magnetic field amplitude B and of the optical phase Y from their values in case of an ideal magnet. In an undulator with period length l_U and rms parameter K_rms, the optical phase is given by [1]

where z(t) is the electron coordinate along the undulator axis and x¢ º dx/dz = e/(gmc)ò^zdz¢ B_y(z¢) is the angle the electron makes with the z-axis in the transverse magnetic field B_y (m, -e and g denote mass, charge and energy of the electron; l is the resonance wavelength l = (1+K²_rms) l_U/2g²). In an ideal planar undulator, the phase (1) oscillates uniformly with half the undulator period around a constant value if the resonance condition is fulfilled. For a realistic undulator, strength and period of the field scatter around the values in case of ideal field.

Fig. 1 Deviations dY,dB of phase (1) and field amplitudes from their values in case of perfectly sinusoidal magnetic field as measured for the U50 undulator at the gap g=20 mm. s_Y,B denotes the variance of the corresponding quantity.

Fig. 2 The same as in Fig. 1 for the two units of the U27 undulator at the gap g=14 mm. Similar results have been obtained for other gaps.

Apart from the drift length between the two undulator units, where the electron does not interact with the field, the phase deviation for U27 (Fig. 2) is significantly smaller than for U50 (Fig. 1). That is the result of the meticulous tuning procedure applied to the U27 undulator.
The effect of the field errors on the lasing process has to be related to the electron energy spread s_E. Both the deviations from their average values cause a line broadening s_l in the spontaneous emission spectrum which is also relevant for the degradation of the laser gain

sl l = 2 sE E + 2Krms2 1 +Krms2 sB B .

(2)

For the ELBE beam (s_E » 90 keV) the first term in eq. (2) amounts to roughly 0.5% at E=40 MeV so that s_B/B » 0.5% is only just small enough not further to reduce the laser gain considerably. On the other hand, the investigations of Bobbs et al. [2] have shown that a phase deviation smaller than 10^o does not significantly affect the laser gain.

References
[1] B. Faatz, J. Pflüger and P. Pierini, Nucl. Instr. Meth. A 375 (1996) 441-444
[2] B. L. Bobbs et al., Nucl. Instr. Meth. A 296 (1990) 574-578  IKH 05/22/01 © R. Wünsch