The Magnetic Field of the Hybrid Undulator U27

In order to ensure sufficient overlap of the electrons with the optical beam, the center of the electron trajectory should be at the same position independently of electron energy and of the strength of the magnetic field. The principle of the magnetic structure of such a nearly "passive" undulator is shown in fig. 1. The structure consists of NdFeB permanent magnets and poles of decarborized iron. The field of these arrangement has been investigated by the code RADIA [1]. Corresponding results have been reported [2]. To ensure the correct slope and position of the electron beam at the undulator exit one has to add an additional magnet (m₂) and an iron pole (p₂) on either side producing appropriate fringe fields.

The undulator was adjusted for a gap g = 12 mm by varying the y-coordinate of p₂ and the distance between m₂ and p₂ (see fig. 1). The resulting displacement and the slope of the trajectory at the exit can be corrected by coils mounted at the entrance sides of both sections and allowing to inject the electrons at a small angle to the z-axis.

Fig. 1: Principle of the magnetic structure of U27. The blue rectangles represent NdFeB permanent magnets, the red ones show poles of decarborized iron. The symbols p₂ and m₂ denote poles and magnets of reduced thicknesses mounted at the same girder like the rest of the structure.

The undulator field was measured and tuned at HASYLAB (DESY) by means of the Hall probe as shown in fig. 2. After transporting the undulator to Rossendorf control mesurement of the undulator fields have been performed using the pulsed-wire method.

At DESY the probe was mounted on a goniometer with six degrees of freedom for right adjustment within the magnetic structures. The poles were tuned with the aim to get a minimum electron displacement at the exit sides of the modules as well as a peak field roughness lower than 0.4%. As a result of the measurements the fig. 3 shows the field component B_y(0,0,z) in the middle plane of both undulator units for a gap of g = 12 mm and a distance d = 250 mm between the sections (part a) and the first integral over the measured field (part b). The trajectory of a reference electron with an energy of E = 20 MeV in the wiggle plane is shown in part (c). Only weak magnetic fields of about 2 mT are necessary to keep the electrons within the optical mode. Fig.4 shows the magnetic field distribution B_y(x,0,z₅) perpendicular to the electron beam at different gaps , where z₅ is the z coordinate of the 5th pole.

Fig. 2: The Hall probe mounted on a goniometer for field measurements on the axis of one of the undulator sections.

 

 

In order to estimate the influence of the remaining field inhomogeneities on the lasing process, we determined for various gap widths the maxima (minima) B_i of the field, their average values B_av and the differences B_i - B_av. For g = 14 mm the results are shown in fig. 5. The standard deviations σ_B of the B_av contribute to the inhomogeneous line broadening of the emitted light corresponding to the following formula:

Fig. 5: Maxima and minima of the magnetic field B_y (z) in the middle plane of the whole undulator. For both sections the average values B_av and their standard deviations σ are denounced. The values σ_B/B_av characterizing the field roughness are given in parenthesis.

The fast Fourier transform obtained from the measured field distribution B_y(z) is shown in the fig. 6, which indicates for gap widths of g = 8, 12, 16 and 20 mm the absolute values |f(n)| of the contributions of the first, third and fifth harmonics to the magnetic field. The employed FFT - procedure delivers the wave numbers k_z as a multiple of the quantity k₀:

where L is the length (mm) of the interval used for the analysis. For each gap a window of 1690 points and a length of 885.0 mm was analysed containing 31 full magnetic periods. The quantity k₀ has therefore the value k₀ = 0.007099 mm^-1. Since the momenta A₃ in f(n) exhibit always a positive sign, the measured field B_y(z) is a bit more flat and more broad than a pure sine function. The reason for the appearence of higher harmonics is the width w_z = 5 mm of the iron poles in z-direction, influencing the form of the field mostly for lower gap values.

If the FEL works with maximum power the energy factor γ of the electrons decreases by dγ = γ/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. The electron energy changes continuously from the undulator entrance to the exit leading to a resonance wavelength λ(z) which depends on the coordinate z along the undulator

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

from which follows

For hybrid undulators, the Halbach equation allows to estimate the tapering of the gap g needed for the variation dB of the field corresponding to equation (4). Figure 7 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.

For variable gaps the radiation wavelength is changed and consequently the phasing between the two undulator systems has to be changed as well. This has to be done by properly choosing the electron flight path d between the two sections.
The optical phase in an undulator has been analyzed in detail by Walker [3]. It can be expressed by

Here z is the coordinate along the beam axis, λ is the radiation wavelength, γ is the electron kinetic energy in units of its rest mass and Θ is the electrons deflection angle. The terms in the bracket have the following descriptive meaning: the first gives the contribution to the phase, if the electron travels just a distance z in free space, the second term represents the additional contribution due to the magnetic field. The function Θ(z) can be derived from B_y(z) by

Using the eqs. (5) and (6) the phasing of the two undulator sub-systems can be determined experimentally by measuring B_y(z) along the axis of the entire undulator.
The phase on the poles in the second section varies with the distance d, which depends on the K-value and has to be chosen appropriately. Moreover, phase matching is obtained periodically after an increase of d by Δd = λ_u(1+K_rms ²). The deviation of the optical phase angle from its nominal value obtained for a perfect undulator is denoted by ΔΨ(z). For a gap g = 17 mm measurements for ΔΨ(z) are shown in fig. 8. The K-value for this case was 0.6688. The nominal phase deviation on the poles is nearly zero, only at the end poles before and after the interspace the phases deviate from zero. The proper choice of the distances d as functions of the K-value as determined experimentally is shown in fig. 9. Six different curves have been measured. They are shifted horizontally by Δd = λ_u(1+K_rms ²(g)). These curves have to be used to choose phasing distances d_ph in dependence on the gap g.

References

[1] P. Elleaume, O. Chubar and J. Chavanne, J. Synchr. Rad. 5 (1998) 481

[2] P. Gippner, W. Seidel and A. Schamlott, Annual Report 1998, FZD-271, p.16

[3] R. P. Walker, Nucl. Instr. Meth. A335 (1993) 328