E ect Of Quadrant Bow On Delta Undulator Phase Errors

Transcription

1 LCLS-TN-15-1 E ect Of Qadrant Bow On Delta Undlator Phase Errors Zachary Wolf SLAC Febrary 18, 015 Abstract The Delta ndlator qadrants are tned individally and are then assembled to make the tned ndlator. Bow of the qadrants on the tning stand can case large phase errors in the assembled ndlator. The e ect of qadrant bow is stdied in this note and limits on bow are derived in order to limit phase errors. 1 Introdction 1 The Delta ndlator is tned by individally tning each of the for qadrants. The permanent magnet material of the qadrants has relative permeability approximately eqal to 1, so the magnetic eld of the assembled ndlator is approximately the same as the sperposition of the elds from the for qadrants. If each qadrant is tned to give straight trajectories and small phase errors, the assembled ndlator shold give straight trajectories and small phase errors to good approximation. The method of tning each qadrant individally is very sensitive to the position of the Hall probe relative to the qadrant. The magnetic eld from a qadrant decays exponentially with distance away from the qadrant. If the qadrants are systematically bent dring tning, phase errors are introdced in the assembled ndlator as explained below. Similarly, if the ndlator is assembled with the qadrants bent, phase errors are introdced. When tning the Delta ndlator, it was noticed that the weight of the qadrants introdced a systematic bow in their shape while they were being tned. In this note, we detail the e ect of this bow, and we set limits on the bow in order to limit the reslting phase errors. Qadrant Bow Dring Tning Dring the tning of the Delta ndlator, the qadrants were held in a kinematic mont and their weight cased a vertical bow in their shape. Figre 1 shows CMM measrements of the bow of qadrant. The size of the bow is approximately 30 microns. The bow dring tning leads to eld errors in the assembled ndlator as otlined in gre. When the qadrant is placed on the bench, it assmes the bowed shape shown in part A) of the gre. The measrement probe follows a straight line over the magnets and we assme it is at the beam axis location in the center of the qadrant. The beam axis is indicated by the z-axis in the gre. The qadrant is tned by "virtal shimming", that is, by moving the permanent magnets. E ectively, the magnets are moved so that they follow a straight line despite the crvatre in the 1 Work spported in part by the DOE Contract DE-AC0-76SF This work was performed in spport of the LCLS project at SLAC. A. Temnykh, Physical Review Special Topics-Accelerators and Beams 11, 1070 (008). 1

2 Figre 1: When the qadrants were placed on the bench, they bowed by approximately 30 microns. magnet keeper. This is shown in part B) of the gre. When the keeper is straightened in the assembled ndlator, the magnets move relative to the beam axis and e ectively reverse their original crvatre as shown in part C) of the gre. The magnets are frther from the beam axis at the ends of the ndlator and the elds are weaker there. The change in eld strength along the ndlator cases phase errors, as discssed below. 3 Phase Errors In The Assembled Undlator Sppose the magnet arrays in the assembled ndlator assme the inverse of the bow of the keeper when the qadrant was placed on the measrement bench, as discssed in the previos section. We derive below the phase errors reslting from sch a bow in all for qadrants. We consider the Delta ndlator in planar, vertical eld mode, bt the reslts are easily extended to other polarization modes. The slippage between two points a and b in a planar ndlator with vertical magnetic eld is given by 3 Z b 1 Sab = + 1 x0 dz (1) a where is the Lorentz factor and x 0 is the slope of the horizontal trajectory in the vertical ndlator eld. The horizontal trajectory slope is given by x 0 = e K cos(k z) () where K e is the local ndlator parameter at the z-position of interest, and k = =, where is the ndlator period. If we consider two points a and b spaced apart by one period, the slippage 3 Z. Wolf, "Introdction To LCLS Undlator Tning", LCLS-TN-0-7, Jne, 00.

3 Figre : A) The qadrants assme a bowed shape when placed on the bench. B) Tning the qadrant e ectively moves the magnets to a straight line, even thogh the magnet keeper is bowed. C) When the qadrant strctres are placed in the ndlator, the keepers assme a straight shape, bt the magnet positions have a bow. in the period is given by S = K e (3) The vertical magnetic eld can be considered as an ideal eld pls small errors that change with position. We assme the magnetic eld keeps its sinsoidal z-dependence to good approximation and the small errors make eld changes over long distances compared to a period. The local ndlator parameter K e can be written as an ideal vale K, pls a small position dependent change K. ek = K + K () The error in the eld cases a change in the slippage per period compared to the ideal case of S Since the K vale is proportional to the eld, we have = KK (5) K K = (6) So S = K The change in the phase corresponding to the slippage change is given by P = r S (7) where, to good approximation, we can se the radiation wavelength from the ideal ndlator, which is given by r = K (8) 3

4 So the change in the phase per period compared to the ideal case is P K = (1 + 1 K ) The elds from a Delta ndlator qadrant are given by (9) = B 0q exp( k r) cos(k (z z 0 )) (10) where r is measred away from the qadrant and z is measred along the qadrant. If the qadrant bows pward by +y q dring tning when it is placed on the bench, the distance from the magnets to the beam axis changes by r = y q. The eld on the beam axis changes by = +k y q (11) Positive position changes of the qadrant make the magnets move closer to the beam axis, increasing the eld from the qadrant. The ndlator will be tned to take ot this eld change. When the qadrant is assembled straight, the negative of this eld change will appear in the assembled ndlator. Br = k y q (1) ndlator When the assembled ndlator is in planar, vertical eld mode, the vertical ndlator eld is given by so By ndlator = p (13) Br = ndlator = k y q (1) assming the bend in the qadrants is the same for all qadrants. The change in the slippage per period in the assembled ndlator compared to the case when the qadrants are tned straight is then and the change in phase per period is S = K k y q (15) K P = (1 + 1 K ) k y q (16) For the Delta ndlator, the period is 0:03 m. Inserting a vale of K = 3 and a position change of y q = 10 microns gives a phase change of 1:16 degree per period. This di erence will accmlate as one moves down the ndlator. Sppose the ndlator has length L and we take z = 0 at the ndlator center. The accmlated slippage change along the ndlator from initial position z = L= where the slippage di erence is zero to an arbitrary z is given by S(z) = Z z L= K k y q (z 0 ) dz0 (17) The qadrants are spported at two points and bend on the test bench in an approximately qadratic manner. We take the position to be correct at the center, which means that the Hall probe is at Z. Wolf, "A Calclation Of The Fields In The Delta Undlator", LCLS-TN-1-1, Janary, 01.

5 the correct beam axis location at the center of the qadrant. Sppose the qadrant bends p by y q max at the ends. With the approximate qadratic bending, the bend in the qadrant is given by z y q (z 0 0 ) = y q max (18) L= From the ndlator entrance to an arbitrary location z, the bending of the qadrants dring tning cases a slippage change in the ndlator compared to the nbent case of S(z) = K k y q max 3L z 3 + (19) and the corresponding change in phase is P (z) = K k (1 + 1 y q max K ) 3L z 3 + (0) Withot errors, the ndlator has constant K vale along its length. The slippage as a fnction of z is given by S 0 (z) = K z (1) With the errors, the slippage as a fnction of z becomes S = S 0 + S: S(z) = K z K k y q max 3L z 3 + () and the phase as a fnction of z is P (z) = z k (1 + 1 y q max K ) 3L z 3 + K (3) In order to determine an e ective K vale for the ndlator, and also the rms phase error in the ndlator, a linear t is made to the slippage as a fnction of z. The linear t has slope M and the slope gives the e ective K vale, K eff, as follows: M = K eff () and The residals of the t give the phase errors. K eff = p ( M 1) (5) = r residals (6) where in this case we se the measred e ective K vale to determine the radiation wavelength r = K eff (7) The Delta ndlator has L = 96 and = :03 m. Take K = 3, and = 10 althogh the vale of drops ot of the phase calclations. We take y q max = 30 microns, which is the measred vale when the Delta qadrants were initially tned. The phase in the ndlator both 5

6 Figre 3: Phase in the ndlator both withot errors and with errors scaled by a factor of 10. withot errors and with scaled errors mltiplied by a factor of 10 for illstration is shown in gre 3. As yo can see, the phase change from the qadrant bow is small on the scale of the total phase change, bt it is visible at the ndlator exit. The change in phase is shown by itself in gre. These are the actal changes and the previos factor of 10 scaling is not inclded here or from this point on. When a linear t is made to the slippage as a fnction of z, the K vale shifts from its ideal vale of 3 to a vale of K eff = :9965, and the residals of the t give the phase errors. The phase errors are shown in gre 5. The rms phase error is 8: degrees. For small bow, the rms phase error scales linearly with the size of the qadrant bow dring tning, with slope :8 degrees per 10 m of bow. If we wish to limit the rms phase errors from qadrant bow to 5 degrees, the qadrants mst be tned with a bow of less than 18 m. Conclsion If the qadrants are tned with a bow de to their weight, it prodces a systematic o set of the magnet locations in the assembled ndlator at the ends of the ndlator. This prodces a phase error with a cbic z-dependence. The rms phase error is 8: degrees for the 30 m of bow observed in the qadrants in their kinematic monts. If we wish to limit the rms phase error to 5 degrees, the bow mst be kept below 18 m. Similarly, systematic bow of the qadrants in the assembled ndlator mst be kept below 18 m. Acknowledgements I am gratefl to Heinz-Dieter Nhn and Yrii Levashov for many discssions abot this work. 6

7 Figre : Change in phase from the qadrant bow. Figre 5: Phase error and K shift when the qadrants are tned with 30 m of bow. 7