Electron Phase Slip in an Undulator with Dipole Field and BPM Errors

Transcription

1 CS-T--14 October 3, Electron Phase Slip in an Undlator with Dipole Field and BPM Errors Pal Emma SAC ABSTRACT A statistical analysis of a corrected electron trajectory throgh a planar ndlator is sed to predict the optimal beam position monitor (BPM) spacing. The ndlator is composed of mltiple modlar sections, each containing many dipoles with random field strength and roll angle errors. ocated between each section are inaccrate BPMs, steering correctors, and qadrpole magnets. An analytical formla for electron-to-photon phase errors is derived and is also sed to estimate the optimm BPM spacing. The rms trajectory amplitde is also predicted and the reslts are applied to the CS FE ndlator where the reqirements on electron trajectory straightness are very demanding.

2 1 Introdction The reqirements on the degree of straightness of an electron trajectory throgh a SASEbased FE ndlator can be qite demanding. For short wavelength ndlator radiation (1.5 Å in the CS [1]) a stringent reqirement exists on the relative phase relationship between the electron beam and the radiated photon beam. Undlator field and beam position monitor (BPM) accracy errors can force the electron beam to travel a longer path and ths phase-lag the photon beam. A significant phase error can move the electron beam away from resonance and negate the FE gain. Since the location and nmber of BPMs along the ndlator is an important factor in achieving a straight trajectory, it is sefl to have a simple way in which to estimate the expected electron-to-photon phase errors and the rms trajectory amplitde, both as a fnction of BPM separation distance, BPM resoltion, and dipole field errors. Previos work [] has examined the effects of dipole errors, bt the effects of inaccrate BPMs were not inclded. We derive an analytical formla which can be sed to estimate the optimal BPM separation along a planar ndlator given the expected BPM resoltion and dipole field errors. We also estimate the expected vale of the rms electron trajectory anywhere along the ndlator after steering sing misaligned BPMs. Misaligned qadrpole magnets can also affect the trajectory, bt steering corrections applied at, or very near, the qadrpoles can be sed to completely compensate this component of the trajectory. Since the limit of this compensation is solely dependent on the resoltion and transverse alignment of the BPMs, we can ignore qadrpole misalignments. They are implicitly inclded here in the steering corrections and the treatment of the BPM limitations. The effects of qadrpole magnets located in-between BPMs are, however, not covered here. Each qadrpole is assmed to have a nearby steering corrector and BPM in both planes. The existence of an optimal spacing can be imagined by considering each type of error (BPM and dipole) in isolation. If the BPMs are inaccrate (misaligned or resoltion limited), bt the dipole fields are perfect, then a shorter BPM spacing forces a higher freqency trajectory distortion after steering correction (see left plot of Figre 1) [3]. The higher freqency implies larger trajectory angles between BPMs and therefore larger phase lag with respect to the radiated photon beam [see Eq. (1)]. In contrast, steering correction sing perfect BPMs, bt imperfect dipole fields, will generate a larger cmlative trajectory deviation between BPMs with a larger BPM spacing, again casing an increased phase lag (see right plot of Figre 1). BPM errors demand a long BPM spacing, while dipole errors sggest a short BPM spacing (for a constant ndlator period). The goal here is to statistically estimate the optimm spacing, the expected trajectory amplitde, and the mean phase error.

3 Figre 1. Steered ndlator trajectory with BPM resoltion of 1 m rms, bt perfect dipole fields (left), and steered trajectory with perfect BPM accracy, bt.1 % rms relative dipole field errors (right). A BPM, steering corrector, and qadrpole magnet are located approximately every meters. Between BPMs there are 63 ndlator periods of length 3 cm and field 1.3 T (K 3.7) at 14.3 GeV. Trajectory Analysis A simplified planar ndlator section is shown schematically below in Figre. The fll ndlator will be composed of mltiple sch sections. BPM & steering (& qad?) x BPM & steering (& qad?) e λ... s i = Figre. Simplified planar ndlator section sed to analyze the electron trajectory. The section has a length with / ndlator periods and a BPM and steering corrector placed between each section. Qadrpole magnets may be located at or near the BPMs. Their effects are implicitly addressed here. The section length is given by, the ndlator period is, and there are / = / periods over the section ( dipoles). A BPM and a dipole steering element (or moveable qadrpole) are placed at each section bondary. For simplicity, this model incldes no period breaks at the BPMs, and the details of wiggler termination at the start and end of a 3

4 section are ignored. Qadrpole magnets, if sed, are assmed to be placed at or near the BPMs, bt are not addressed here since their misalignments simply change the steering corrections reqired, and the focsing is not relevant for the single particle trajectory. The effect of an incoming betatron oscillation is also ignored since it can be removed at the ndlator inpt with pstream steering or added later to these reslts as an independent effect. It is assmed that no significant focsing exists between the qadrpoles (i.e. within the ndlator section), bt sch effects might be added in a ftre modified analysis. A perfect ndlator section (and a perfect initial trajectory) will prodce a nominal reference trajectory which is composed of the small transverse oscillations normally associated with an ndlator. This reference trajectory is sbtracted off and only the difference trajectory prodced by small dipole field errors and inaccrate steering is examined. Examples of sch difference trajectories are shown in Figre 1 [4]. For a sinsoidal varying field with peak B DQGUHODWYHIHOGHUURUV B j /B ), each dipole error prodces an additional e í transverse kick at the center of the j th dipole of B λ K θ j = j j π =, (1) ( ) ( B B ) ( ) B B Bρ γ where (% ) is the standard magnetic rigidity, is the orentz energy factor, K is the dimensionless ndlator parameter (K 3.7 for the CS), K eb λ π mc, () m is the electron rest mass, c is the speed of light, and e is the electron charge. Ignoring any weak focsing of the ndlator fields (as in ref. []), each pstream kick at location j ( j < ) displaces the electron beam at a downstream location i (j < i ) by x ij = j (s i í s j ) = j (i í j7khvxpryhudooxsvwuhdpgvsodfhphqwv x ij ; pls the displacement, c i/, prodced by an initial beam angle, c, at j = ; pls an initial BPMlimited position, b 1, prodces a trajectory position at location i of x i λ i = θ ci + θ j + j= 4 ( i j) b1. (3) The angle c (at j = ) is the sm of (1) an incoming angle from the previos section, pls () a correction angle sed to steer the trajectory to the next BPM (at section s end). The offset b 1 is the initial e í beam position at j = reslting from pstream steering of the trajectory to the first BPM offset, b 1. The BPM offset, b 1, can be interpreted as a transverse alignment error of the BPM, a noise component of the BPM reading, or both combined.

5 The angle, c, is now defined by steering the trajectory so that the next BPM reads zero. Since this next BPM has a different random offset, b, than the first BPM, the steering correction will prodce x = b. Using Eq. (3) with i = and x = b, and solving for the angle prodces 1 θ c = ( b b1 ) θ j ( j). (4) λ j= Eq. (4) is now sbstitted into (3) and the steered trajectory at any location, i, is given by x i λ i i i = + λ j= j= ( b b1 ) θ j ( j) + θ j ( i j) b1. (5) After some rearrangement Eq. (5) can be pt into a form where one sm extends from j = to i í 1, and a second sm extends from j = i to. x i 1 λ i i = + j= j= i ( i ) jθ j i θ j ( j) + 1 b1 b i. (6) This represents the beam position at the i th dipole and can now be sed to calclate the trajectory and the electron-to-photon phase slip over an ensemble of ndlator sections with random errors. We start the analysis with the phase slip considerations. 3 Trajectory Indced Phase Slip Errors The path length difference of an electron with speed v c and angle x ZWKUHVSHFWWRD photon with zero angle, over a length l, is given, for small angles, by x l s = l 1+ x l. (7) The beam angle, x i, throgh the i th pole-center to pole-center separation of length, /, is ( ) i i+ 1 i λ x = x x. (8) Here we approximate the small kicks over a dipole field error as eqivalent thin-kicks located at the dipole centers. This approximation will be jstified in nmerical comparisons to follow. The positions x i and x i+1 are taken from Eq. (6) and after some rearrangements, the beam angle x i can be written as 5

6 1 xi = j j j + b b j= 1 j= i+ 1 λ θ θ ( 1). (9) We now write the electron-to-photon phase error at location idv i = s i / r = k r s i, where r is the FE radiation wavelength (1.5 Å for the CS) and k r is the radiation wavenmber. i x ϕi = krλ (1) 4 The total phase error over the ndlator section is jst the sm of individal phase errors. Combining Eqs. (9) and (1), the total phase error, s, over the ndlator section, after steering sing the inaccrate section-bonding BPMs, is krλ ϕs = θ j θ j + 4 i= 1 j= 1 j= i+ 1 λ j ( b b1). (11) We now move toward a statistical analysis of the trajectory and therefore consider b 1, b, and the set of angles j as random, ncorrelated variables (also with zero mean vale). θ j = θ k θ, b1 = b b, θ jθ k = b1b = θ jbl = (1) This is an important step. It implies that the rms BPM errors sed here, b 1/, are the ncorrelated, random component of the relative misalignments of two BPMs separated by one ndlator section. This incldes static, relative, ncorrelated misalignments as well as BPM readback noise. It does not inclde, nor shold it inclde, any relative misalignments on a scale longer than an ndlator section or any correlated component of the misalignments. Realistic simlations of beam-based alignment (BBA) [5], [6] over the CS ndlator indicate that the relative, ncorrelated, random misalignment of two adjacent BPMs can be redced to very nearly the fndamental BPM noise level (e.g. 1 m rms). We, therefore, do not need to evalate the BPM alignment with respect to a straight line over the length of the entire ndlator. We assme that BBA has been performed adeqately, which is anyway a probable reqirement for SASE satration with sch short wavelength FEs. The total phase error in Eq. (11) is now maniplated sing Eqs. (1) and the following relations for the sms of powers of integers. n n n j = n( n+ 1 ), j = n( n+ 1)( n+ 1 ), j = n ( n+ 1) (13) j= 1 j= 1 j= 1 6

7 The mean (averaged over many random ndlator sections) total phase error, ϕ s, of one ndlator section with / periods ( 1), random dipole errors, and resoltion limited BPMs is given by ϕ s ( 5) λ = kr + θ + b 4 λ. (14) This is the mean phase error per ndlator section. The total mean phase error,, over the entire ndlator length, (= s ), which is composed of s sections, is = s /. 1 ϕ k r θ + 6λ b Here we have scaled Eq. (14) by /, introdced the ndlator section length = 1 /, and assmed >> 5. So far, this reslt incldes trajectory errors in the wiggle-plane only (which we assme to be the horizontal plane). Eqation (15) shows the tradeoffs anticipated in the choice of an optimm section length,. The section length appears in the nmerator of the dipole error term (i.e. the term), bt in the denominator of the BPM resoltion term (i.e. the b term). Therefore, as described in the introdction, a shorter section length increases the phase lag indced by BPM errors, bt decreases the phase lag indced by dipole errors (for a constant period). The behavior of Eq. (15), per meter of ndlator, is shown in Figre 3 for a BPM resoltion of b 1/ = 1 P DQG IRU UHODWYH GSROH HUURUV RI B/B ) rms =.1 %,.5 %, and.5 %. Here the ndlator period is 3 cm, the radiation wavelength is 1.5 Å, the ndlator parameter is K = 3.7, and the energy is 14.3 GeV. The optimm BPM spacing,, moves from 1.8P DW B/B ) rms =.1 %, to 4.4 m at B/B ) rms =.5 %. The data points shown with error bars are a Monte Carlo compter calclation [3@ DW B/B ) rms =.1 % over 1 ndlator sections, which is sed for nmerical confirmation of Eq. (15). ote, the rms kick angles, 1/, are related to the rms relative dipole field errors as θ 1 1 K ( BB) (15) =. (16) γ The reslt of Eq. (15) can be differentiated with respect to and set eqal to zero in order to calclate the optimal BPM spacing, opt. In addition, we se Eq. (16) to make the reslt explicit in terms of the dipole field errors. opt 3λγ K b ( B B ) 13 (17) 7

8 For an expected level of random, ncorrelated dipole errors, and a given BPM resoltion, the optimm BPM spacing can be predicted sing Eq. (17). 8 Mean phase slip per meter vs. BPM spacing 7 6 φ x / [deg/m] /m Figre 3. Mean phase error per meter of ndlator [Eq. (15)] vs. ndlator section length,, for a BPM resoltion of b 1/ = 1 P DQG GSROH IHOG HUURUV RI B/B ) rms =.1 % (solid/red),.5 % (dash/ble), and.5 % (dash-dot/green). The optimm BPM spacing,, moves from 1.8 m at.1 % to 4.4 m at.5 %. Points with error bars are a Monte Carlo compter calclation at.1 % sed as a cross-check ( = 3 cm, r = 1.5 Å, K = 3.7, and PF = 14.3 GeV). The argment can be extended into both horizontal and vertical planes by adding p the mean phase lag vales per plane (i.e. xy = x + y ). The additional mean phase lag de to vertical trajectory errors can be described by interpreting in Eq. (15) as a vertical kick, y, de to a small random dipole roll-angle error,, and the BPM term as the vertical resoltion (eqal to the horizontal resoltion). The total mean phase lag over the length of the ndlator, inclding horizontal and vertical trajectory errors, is then K ϕxy k r BB + + b 3 γ λ where the rms dipole roll angles, 1/ (<< 1), are given by 1 ( ) ψ θ 1 1 y ψ, (18) K =. (19) γ Eq. (18) is simply twice that of Eq. (15) when B/B ) =. In this case, the optimal spacing of Eq. (17) is still valid. If the rms field strength errors are significantly different from the rms roll angle errors, then Eq. (17) becomes 8

9 opt 6λγ b K ( B B ) + ψ 13. () From this point on we treat the case where = B/B ), for simplicity. ow sbstitting opt from Eq. (17) into of Eq. (18) gives the minimm mean phase lag achieved at = opt. ϕ 4 3K b ( B B ) = k 4 λγ xy min r The mean phase error shold be kept below some tolerance level throgh qality control of both B/B ) and b. Here we assme that a total phase error over the length of the ndlator of xy is a reasonable tolerance goal. This has been briefly stdied for the CS parameters where it was fond that a net phase error increased the gain length by abot 3 % [7]. This point needs frther stdy, bt we contine here with this simplifying assmption. In any case, the minimm phase lag is still achieved at the optimal BPM spacing described here. If Eq. (1) is set to, the pper limit on the tolerable dipole field errors (and the roll angle errors) can be calclated, given the BPM resoltion and radiation wavelength, r, sing ( BB) 1 γ K 3λ λr 8 b 3 1 tol (1). () This reslt can be re-sbstitted into Eq. (17) and the optimal BPM spacing can be calclated from knowledge only of the BPM resoltion. opt 6 1 = b (3) λ r In this case, the dipole field (and roll angle) tolerances are dependent variables prescribed by Eq. (). The ndlator design can therefore be initiated sing jst the knowledge of the expected BPM resoltion. As an example, Figre 4 shows the optimal BPM spacing [Eq. (3)] and the tolerable dipole field errors [Eq. ()] as a fnction of BPM resoltion. The optimal BPM spacing is linearly dependent on the BPM resoltion, b 1/. For this plot: = 1 m, r = 1.5 Å, = 3 cm, K = 3.7, and PF = 14.3 GeV. For example, a 4- m BPM resoltion reqires 9

10 rms dipole field errors % (and rms roll errors mrad), and has an optimal BPM spacing of 8 m, while a 1- m BPM resoltion reqires rms dipole field errors % (and rms roll errors mrad), and has an optimal BPM spacing of m..15 Dipole Errors and BPM Resoltion for CS Parameters 15 opt.1 1 B/B /% opt /m.5 5 B/B BPM resoltion /µm Figre 4. Tolerable dipole field errors (left scale; solid/ble) and optimal section length (right scale; dash/green) vs. BPM resoltion for a net mean phase lag over the ndlator length ( = 1 m, r = 1.5 Å, = 3 cm, K = 3.7, and PF = 14.3 GeV). ikewise, if the dipole errors are known in advance, and the reqired BPM resoltion is soght, then Eq. (1) can be rearranged as b 1 tol 3γ λ λr 8K 3 ( B B ) 3, (4) where the tolerance of xy is still assmed. From this, the optimal BPM spacing can be taken directly from the dipole errors, rather than from the BPM resoltion as in Eq. (3). opt λλγ r = K ( B B ) (5) These reslts can be compared to the nmerical stdy done by H.-D. hn sing FRED- 3D which is pblished in the CS Design Stdy Report [1]. Here the optimal BPM spacing was fond by applying.1 % random dipole errors and varios BPM resoltion limitations. At a BPM resoltion of 1 m, the optimal spacing was fond arond m, which is in reasonable agreement with 1.7 m from Eq. (17). 1

11 The dipole errors, however, can typically be controlled to a mch better level by magnetic measrements and shimming techniqes [8], [9], [1]. Using sch techniqes can greatly redce the ncorrelated random component of the dipole field errors, which are sed here to calclate the optimal spacing, and as sch will tend to arge for mch larger BPM spacing. The focsing lattice mst, however, also be taken into accont when increasing the BPM/qadrpole spacing, since, for a FODO-lattice, this will also increase the mean beta fnction, or increase the ratio of max / min. A focsing lattice composed of qadrpole triplets can take advantage of the longer section lengths [11], bt the internal alignment tolerances within the triplet assembly can be extremely tight [1]. 4 Trajectory Amplitde We can also analyze the amplitde of the electron trajectory, which may have some significance when considering the spatial overlap of the electron and photon beams. The beam position at the i th dipole in Eq. (6) is now sed to calclate the rms trajectory amplitde over an ensemble of ndlator sections with random errors. The form in Eq. (6) is convenient since it is a linear combination of ncorrelated, random variables and its variance, x, is simply the sm in qadratre of the components. The variance, or the expectation vale of the sqare of the electron trajectory over an ensemble of sections, is then written as x [ i ] i 1 λ 1 i = θ ( i ) j i ( j) b ( i) j= j= i where we have sed Eq. (1) for the ncorrelated random variables., (6) Eq. (6) is redced to a polynomial in i by applying the smmation relations of Eq. (13). The rms kick angles are also replaced by the rms dipole errors given by Eq. (16). λ K B 3 1 i = 4 + ( 1) γ B x i i i i b i i (7) The index, i, is replaced by the longitdinal axis, s (= /), along the ndlator section, and the nmber of dipoles, (= / ), is replaced by the section length,. With zeromean errors, = b =, the rms trajectory is: x rms = x 1/, and Eq. (7) becomes 3 B 3 λ λ ( ) ( ) ( ) ( ) B K xrms () s = s 8s 16s + 8 s+ + 3λγ ( ) ( ) b s s 1 +. (8) 11

12 For an ndlator section with many dipole periods (i.e. << ), the rms trajectory anywhere along a section ( s ) then simplifies to 3 8 K xrms () s BB s + b 1 s + b s 3 λγ ( ) ( ) { } ( ). (9) The peak, mid-section vale of the rms trajectory, xˆ rms, is taken at s/ = ½. 3 1 K xˆ rms ( B B ) + b (3) 3λγ The rms trajectory in Eq. (9) is a fnction of s since the rms is taken over an ensemble of ndlator sections rather than over s. We can also integrate ot the s-dependence by calclating the rms of Eq. (9) over the entire section length. This prodces a single global rms vale for the entire ndlator trajectory integrated over both s and over many random seeds. The global rms vale is given by integrating the sqare of Eq. (9) over the section length rms + s K rms ( ). (31) 3 5λγ x x ds B B b Eq. (9) is compared with compter generated trajectories in Figre 5 sing 1 random seeds and for varios vales of b. The simlations propagate the beam trajectory continosly throgh 1 consective ndlator sections, ignoring any weak field gradients, bt otherwise sing an accrate model. These are difference trajectories with respect to the reference particle and do not inclde the nominal trajectory wiggles. Eqation (9) is shown as a dashed crve (red) in each plot, and the rms over 1 compter generated steered sections is shown as a solid crve (ble). Since / (<< 1), the reslts of Eq. (8) and (9) are virtally identical and agree well with the time consming compter calclations. The compter calclated vales of the global rms (rms over the entire simlated trajectory) are also listed at the top of each plot. These vales each agree with Eq. (31) to within <%. Eq. (31) demonstrates the somewhat obvios fact that the trajectory can be dominated by large BPM misalignments after applying one-to-one steering. That is, for BPMs of poor resoltion, b K ( B B ) > γ 15λ, (3) the trajectory amplitde is BPM-dominated and the relative dipole errors play a less important role. In this case, the rms of the trajectory simplifies to 1

13 1 1 x rms b. (33) s 3 For a -m BPM spacing, Eq. (3) evalates to ~.8 m at = 3 cm, K = 3.7, and PF = 14.3 GeV, and.1 % rms relative dipole errors, which means that a 1- m resoltion BPM, even if it is perfectly aligned, will still dominate the trajectory errors. 1 b 1/ =. µm, x rms 1/ =.6 µm 1 b 1/ =.5 µm, x rms 1/ =.63 µm.8.8 x rms /µm.6.4 x rms /µm s/ s/ b 1/ = 1. µm, x rms 1/ = 1. µm 1 b 1/ = 1. µm, x rms 1/ = 7.97 µm x rms /µm 1 x rms /µm s/ s/ Figre 5. Trajectory rms over an ndlator section showing the analytical reslt of Eq. (9) (dashed/red) and a compter calclation over 1 random seeds (solid/ble). The for figres are for for different vales of rms BPM errors, b 1/ (=,.5, 1 and 1 m), as shown on the plots, and = 3 cm, = 1.9 m, K = 3.7, ( %% ) 1/ =.1%, and PF = 14.3 GeV. It may also be interesting to ask which of the plots in Figre 5 best represent the minimal phase error conditions. Sbstitting opt from Eq. (3) into in Eq. (3) and sing Eq. () to describe the dipole errors, we find the peak rms-trajectory amplitde, at s/ = ½, is simply the BPM resoltion xˆ rms opt 1/ b, (34) and the global rms over the entire ndlator, Eq. (31), is very nearly the BPM resoltion. 13

14 x 14 (35) rms b sopt This sitation is depicted qalitatively (not precisely optimm) in the lower-left plot of Figre 5, which shows the almost intitive reslt that minimal phase errors are achieved for the flattest trajectory rms. This sitation is very similar to the criterion for beta fnction variations in the ndlator where the best conditions reqire the smoothest beta fnctions. It is worth noting that for typical CS parameters and tolerances, the predicted rms trajectory amplitde is 1- m, yet the mean rms beam size is ~3 m. The electron/photon beam spatial overlap then appears to be qite good, yet the potential phase errors are beginning to affect the gain. For very short wavelength radiation the spatial overlap may be of less concern than the phase slip. Of corse, knowledge of the rms trajectory amplitde and phase errors is not enogh to flly evalate the FE performance. The beta fnction is also an important parameter which is related to the BPM/qadrpole spacing. For well controlled pole errors and conservative vales of BPM resoltion, these conclsions wold tend to psh the BPM spacing toward the longer limits set by mean beta fnction and max/ min considerations. 5 Conclsions For an FE ndlator composed of separate sections which are bonded by BPMs, steering correctors, and qadrpole magnets, the ndlator section length can be optimized analytically sing Eq. (3) [or (5)], with knowledge of the ndlator length, the radiation wavelength, and the expected BPM resoltion [or dipole field and roll errors]. The dipole field and roll angle tolerances [or BPM resoltion reqirements] are then given by Eq. () [or (4)]. This assmes that beam-based alignment has been performed to align the BPMs, with respect to their nearest neighbors, to the level of their resoltion. Well controlled pole errors, as might be prodced sing shimming methods, will increase the optimal section length by redcing the dipole field and roll errors. The mean beta fnction, and its variation, mst, however, also be considered. A FODOlattice for the CS ( 1 m, = 3 cm, r = 1.5 Å, K = 3.7, PF = 14.3 GeV), with (for example).5-% random ncorrelated pole field errors (and.5-mrad roll errors) has an optimal section length of 4. m (from the standpoint of trajectory errors alone), and a BPM resoltion reqirement of m. Sch a system will, on average, prodce a tolerable net phase lag of the electron beam over the length of the ndlator, with respect to the radiated x-ray beam, de to electron trajectory errors in both planes. A slightly shorter section length (e.g. 3-4 m) is, however, better sited to the reqirements on the mean beta fnction. 14

15 6 Acknowledgements I wold like to thank Ilan Ben-Zvi and Max Cornacchia for encoraging this work, and Heinz-Dieter hn for many helpfl comments. 7 References [1] CS Design Stdy Report, SAC-R-51, April [].-H. Y, et. al., Effect of Wiggler Errors on Free-Electron-aser Gain, Physical Review A, Vol. 45, o., Janary 15, 199. [3] These particle tracking calclations were performed sing the compter code Elegant written by Michael Borland at Argonne ational aboratory. [4] By examining the difference trajectory we have ignored the fact that it is also possible to find an electron trajectory which is advanced in phase, rather than retarded, with respect to the reference trajectory. Sch an nlikely trajectory will follow a straighter path than the wiggling reference trajectory. The mean phase, however, will always be retarded. This has been confirmed in compter tracking. [5] P. Emma, R. Carr, H.-D. hn, Beam Based Alignment For The CS FE Undlator, Proceedings of the 1998 Free Electron aser Conference, ewport ews, Virginia, Agst [6] P. Castro, TTF FE Beam-based Alignment by Dispersion Correction Using Micado Algorithm, TESA-FE 97-4, Agst, [7] H.-D. hn, private commnication, March. [8] I. Vasserman, A Shimming Techniqe for Improvement of the Spectral Performance of APS Undlator A, S-53, Janary 9, [9] C.M. Fortgang, R.W. Warren, Measrement and Correction of Magnetic Fields in Plsed Slotted-Tbe Microwigglers, IM A 341, pp , [1] R.W. Warren, imitations on the Use of the Plsed-Wire Field Measring Techniqe, IM A 7, pp , [11]. Vinokrov, private commnication, April. [1] P. Emma, H.-D. hn, Qadrpole Magnet Error Sensitivities for FODO-Cell and Triplet attices in the CS Undlator, CS-T--5, Febrary. 15