Characterization and tuning of ultrahigh gradient permanent magnet quadrupoles

The application of quadrupole devices with high field gradients and small apertures requires precise control over higher order multipole field components. We present a new scheme for performance control and tuning, which allows the illumination of most of the quadrupole device aperture because of the reduction of higher order field components. Consequently, the size of the aperture can be minimized to match the beam size achieving field gradients of up to $500\mathrm{T}{\mathrm{m}}^{-1}$ at good imaging quality. The characterization method based on a Hall probe measurement and a Fourier analysis was confirmed using the high quality electron beam at the Mainz Microtron MAMI.

Figure 1

Design of a miniature PMQ is shown with 12 wedges of permanent magnet. The inner radius of the aperture is $r_i=3\mathrm{mm}$ and the outer radius is $r_o=10\mathrm{mm}$. The arrows point in the magnetization direction.

Figure 2

Scheme of measuring the magnetic vector field in cylindrical coordinates using a Hall probe with the arrow being the surface normal. The radial component (a) as well as the azimuthal component (b) are obtained separately.

Figure 3

Part (a) shows a field measurement using the azimuthal component in cylindrical coordinates and part (b) its outermost ring at ${\rho }_0=1\mathrm{mm}$ is plotted against one rotation and used to expand the field coefficients $a_l$ and $b_l$, shown in (c) magnitude $\sqrt{a_l^2+b_l^2}$ and (d) phase $\arctan (b_l/a_l)$.

Figure 4

Part (a) shows the azimuthal HOMFC from Fig. 3 ($\overrightarrow{B}=\sum _{l=3}^6[B_{l\rho }(\rho ,\phi ){\overrightarrow{e}}_{\rho }+B_{l\phi }(\rho ,\phi ){\overrightarrow{e}}_{\phi }]$, note the absolute scale compared to Fig. 3). Part (b) shows the remainder after the $k$th field component.

Figure 5

Measurements at the Mainz Microtron MAMI are shown with two different beam configurations: The small electron beam configuration (panel I) is shown with the horizontal (Ia) and vertical (Ib) beam plane at the waist. The measured beam (black), the calculated beam using expanded fields (blue), and calculated beam for an ideal quadrupole field (red) are shown for comparison. The calculated beam envelope (Ic) and the emittance (Id) are plotted against the propagation direction of the beam. The vertical lines mark the PMQ positions. The large beam configuration is shown correspondingly (panel II). The green curve in panel IIa is computed from the expanded fields, but with a different focal spot slightly moved towards the PMQs by 1 mm or 0.5% of the focal length.

Figure 6

The introduction of HOMFC with the PMQ having the same orientation as in Fig. 1. Panel I: Magnitude and phases of the calculated magnetic field using the ideal arrangement of permanent magnet wedges. Panel II: Displacing a single pair of tuning wedges by $150\mu \mathrm{m}$ introduces a dominant sextupole (IIa). Independent from the specific pairs of tuning wedges displaced, panels IIb–IIe show the effect on the phases of the introduced field components by moving tuning wedges at $\alpha =0°$, 90°, 180°, 270°. Panel III: Two opposite pairs of tuning wedges are displaced by $150\mu \mathrm{m}$, this introduces a dominant octupole. Panel IIIa shows the effect on the magnitude of the field components. Panel IIIb–IIIc show the effect on the phases of the introduced field components moving tuning wedges at $\alpha =0°$ and 90° affecting $a_4$. Moving the positioning wedges as shown in panels IIId–IIIe affect $b_4$. Panel IV: All pairs of tuning wedges are displaced by $150\mu \mathrm{m}$; this introduces a dominant dodecapole. The effect on the magnitudes are shown in panel IVa. Panels IVb and IVc show distinct pairs of tuning wedges which are moved and the effect on magnitudes and phases of the field components. The application of distance holders might be required as in Fig. 7b.

Figure 7

Part (a) shows an arrangement of permanent magnet wedges and the center cylinder is shown schematically. The arrows point in the direction of the magnetic forces which act centrifugally and centripetally on the wedges. (b) Example for distance holders (e.g. $50–100\mu \mathrm{m}$ aluminum foil) of the positioning wedges from the cylinder center. (c) Positioning and tuning screws acting on the wedges.

Figure 8

HOMFC for the proof of principle are illustrated, which are obtained from azimuthal field measurements. Part (a) shows the HOMFC within a newly assembled quadrupole and (b) shows the PMQ tuned with significantly reduced HOMFC. The statistical errors obtained from independent measurements are between 0.1% and 0.7% for the values of the HOMFC, the phases are off between 0.3° and 1°, and are thus negligible.