SPOTSIZE STABILIZATION STUDIES FOR THE TESLA BEAM DELIVERY SYSTEM A. Sery Branch of the Institute of Nuclear Physics, 142284 Protvino, Moscow region, Russia Abstract Studies of the ground motion induced spotsize growth in the interaction region of the TESLA linear collider and some tools to recover are presented here. Analytical re- sults are given and compared with simulations by particle tracking. Performance of different procedures, such as or- bit correction, adaptive alignment and knob scan, studied by tracking simulations, is reported. 1 INTRODUCTION The main parts of the TESLA Beam Delivery System[1] are the Collimation Section, Tuning and Diagnostic Section and the Final Focus Sytem (Fig. 1). It is the BDS where the most important tolerances on transverse misalignments of focusing elements are located. 0 500 1000 0 50 100 150 β 1/2 (m 1/2 ) β y 1/2 β x 1/2 0 500 1000 -0.1 0.0 0.1 η x s (m) η (m) Figure 1: Optics of the TESLA Beam Delivery System[1]. Ground motion is one of the main sources that will pro- duce, after certain time, an intolerable value of misalign- ments resulting in reduction of luminosity via the beam offset and distortion at the Interaction Point. The fast or- bit feedback, which is possible for TESLA due to uniquely long bunch separation, will cure the beam-beam offset. We will concentrate therefore only on the spotsize stabilization. When the beamline is affected by ground motion only (no correction applied yet), the free evolution of the beam spotsize can be evaluated analytically using the ground mo- tion spectrum P (ω,k) and the corresponding spectral re- sponse functions, determined in linear approximation[2]. This analytical treatment is incorporated into the “FFADA” program[3]. The analytical results provide quite helpful in- formation on the critical time scales, however the question “whether the linear approximation is sufficient?” should not be forgotten. Semi-analytical methods, appeared recently, are able in some cases to evaluate efficiency of a correction in lin- ear collider. The generalized spectral approach[4] can give clear analytical expression for several correction method applied to a regular linac. The method proposed in[5] is more general and may be of great help if fully developed. However, when a correction is applied to a BDS to stabi- lize the spotsize, simulations must be used to determine the procedure performance since such procedures may be quite complex and the focusing structure is very irregular. 2 FREE EVOLUTION OF THE BEAM Since the critical time scales are quite large for the beam size growth, it is sufficient to consider only the diffu- sive “ATL” ground motion[6]. The motion assumed to have the same coefficient A in both horizontal and verti- cal planes. Neighboring elements (such as quadrupole – sextupole pairs or the final doublet) were assumed to be placed on the same support. The beam parameters used in simulations were: 250 GeV/beam, ε x =2.8 · 10 -11 m, ε y =5· 10 -13 m, β * x = 25 mm, β * y =0.7 mm, σ E = 10 -3 . After the beam tracked through the misaligned beamline, the offset was removed and the average beam matrix ele- ments, for example σ xy = xy, were computed. Tracking was performed then with different seeds that gave the rms beam matrix hσ 2 xy i 1/2 . It was convenient to normalized it to the nominal values in the point of observation: hσ 2 xy i 1/2 n = hσ 2 xy i 1/2 / √ σ xx0 σ yy0 . Let us first consider only the the FFS part of the BDS. The analytically found dispersion, waist shift and coupling (which the spot size growth is determined by) versus the A · T coefficient are presented on Fig. 2 in comparison with the tracking results. One can see that the linear approxima- tion and tracking are in perfect agreement for the FFS part of BDS. If one assumes A = 10 -5 μm 2 s -1 m -1 then the critical time scale for 2% luminosity loss is about 200 s for the FFS part of the TESLA BDS. However, when the complete TESLA BDS was stud- ied by tracking, much faster spotsize degradation has been found (Fig. 3). The terms like yy 0 , yx 0 and yx of the nor- malized beam matrix, which were proportional to A · T that indicates on nonlinear effects, have shortened the resulting 10 -14 10 -13 10 -12 10 -11 10 -10 10 -9 10 -1 10 0 10 1 10 2 Normalized beam matrix yδ yδ yx’ yx’ yy’ yy’ y 2 y 2 AT (cm) Figure 2: Elements of the normalized rms beam matrix σ n for the FFS part of the TESLA BDS versus AT . Linear model prediction (lines) and tracking results (symbols). 557 0-7803-4376-X/98/$10.00  1998 IEEE