Errors in Beam Based Alignment Measurements

1. Errors in Beam Based Alignment Measurements Beam Based Alignment Formulae Error by thin lens treatment Error in the Analysis by Optical Error F. Wileke, HERA Seminar Zeuthen, 7-9 January 2002

2. Beam Based Alignment Formulae Closed Orbit (L) =0 Revolution matrix closed orbit vector x,x‘ d(0)=d´(0)=0 Closed orbit after a change of a quadrupole strengths Dk F. Wileke, HERA Seminar Zeuthen, 7-9 January 2002

3. Comments for Slide 2: The change of the closed orbit by a change in a quadrupole strength is expanded up to first order in the quadrupole change. The center of changed quadrupole is taken at the beginning of the lattice. As Georg Hoffstaetter has found out, it is very advantageous for the treatment of beam based alignment to consider the magnet offset in the middle (longitudinal middle of the quadrupole. The vector d is the solution of the orbit for starting values x=x’=0. H are the inhomogeneities. The term (1-M)^-1 makes the solution ring-periodic. F. Wileke, HERA Seminar Zeuthen, 7-9 January 2002

4. lattice ½quad Half quadrupole matrix ½quad M0 Gh-1 Start: Middle oft the quad Dk*dG/dk Gh-1 x0 F. Wileke, HERA Seminar Zeuthen, 7-9 January 2002

5. Comments slide 4: In order to express the effect of a change quadrupole up to first order as a function of quadrupole offset and angle in the middle, one starts with the offset in the middle of the quadrupole, transforms it back to the beginning by half an inverse quadrupole matrix. Then one transforms the offset vector at the beginning through the whole quadrupole for the original and changed value. The difference is expressed by the difference matrix d/dkM. The resulting orbit change is transformed back to the middle of the quadrupole. Since we are limiting ourselves to first order in dk, the transformation back kann be performed with the unchange quadruple half matrix. The matrix G(l/2)-1(d/dkM)G(L/2)-1 has a antidiagonal structure which will ease the analysis and which will clearly separate angle and offset effects. From the final expression one can see that offset produce cosine like orbits wrt the quadrupole middle and the angle offset produces sine-like orbits wrt to the middle of the quad F. Wileke, HERA Seminar Zeuthen, 7-9 January 2002

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7. Comments slide 6: If the quadrupole strength is changed, the inhomogeneity vector d is also changing via the misalignment of this quadrupole. We find that if the offset and angle are taken in the middle, the change of d can be expressed by a similar antidiagonal matrix. Then we can absorb the d/dk d term and express the closed orbit change as a function of the sum of original orbit and misalignment in the test quadrupole F. Wileke, HERA Seminar Zeuthen, 7-9 January 2002

8. Error by Thin Lens Treatment Thick lens matrix Thick lens differential matrix Thin lens matrix Thin lens differential matrix Relative Pos Error F. Wileke, HERA Seminar Zeuthen, 7-9 January 2002

9. Comments slide 8: The first error considered is the error made by simplifying the analysis to a thinn lens treatment. The expressions relate the thick lens analysis of the orbit change caused by the quadrupole change with a thin lens analysis. Again, taking the offset and angle in the middle, the analysis simplifies due to the antidiagonal structure of the difference matrix. We see that the coefficient between the “true”offset obtained by thick lens analysis and the “false”offset obtained by thin lens offset has a simple analytical form which can be easily evaluated F. Wileke, HERA Seminar Zeuthen, 7-9 January 2002

10. Errors by thin lens treatment F. Wileke, HERA Seminar Zeuthen, 7-9 January 2002

11. Comments 10th Slide: This slide shows the errors made by thin lens treatment for the quadrupoles in the interaction regions F. Wileke, HERA Seminar Zeuthen, 7-9 January 2002

12. Error due to optics error in a quad (closed bumpmethod) xa+x´b dk2L2b2x0 This point implies large angle error dkLbx0 x dkLbx0 First corrector dkL b =0.5/m*200m*1%=1 F. Wileke, HERA Seminar Zeuthen, 7-9 January 2002

13. Comments 12th Slide: Here we consider the effect of a gradient error of a quadrupole which is between the test quadrupole and the bpms or the compensating kicks. In the IR, the The quadrupole are long, strong and the optics has a large beta. K*L*b can be easily 100 so that dklb=1 is possible. If the analysis point is at a distance of p from the test quad, and the error quad at a distance p/2, the diagram shows that the errors can be quite large. Instead of the yellow arrow which is calculated as an effect of the quadrupole offset without knowledge of the quadrupole error one has the Red arrow as the “true”effect. From this one would calculate a very large angle offset of the quadrupole which is not there. F. Wileke, HERA Seminar Zeuthen, 7-9 January 2002

14. Undisturbed transformation from komensating kicks to middle of test quad F. Wileke, HERA Seminar Zeuthen, 7-9 January 2002

15. Comments 14th slide: The analysis considered is to compensate the closed orbit distortion by two orbit correctors. The two kicks, the in-between-optics and the test quadrupole difference matrix allow to calculate the test quadrupole offset. The two matrices describe the transformation from the kicks to the beam offset in the test quad middle for the undistorted and the distorted transformation Comments 16th Slide: The distorted transformation is expressed by the sum of two transformations, one containing the gradient error. The result using the distorted and the undistorted transfromation is expressed by the matrix product in the red box. F. Wileke, HERA Seminar Zeuthen, 7-9 January 2002

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18. Comments 17th Slide: Finally, we do the same for the quadrupole offset calculation by multiplying the distorted and undistorted transformation with with the differential matrix respectively. The red box contains now the bba result with and without taking into account the quadrupole error between test quad and analysing kicks F. Wileke, HERA Seminar Zeuthen, 7-9 January 2002

19. This potentially small denominator enhances the angle error (infinite for thin lens) F. Wileke, HERA Seminar Zeuthen, 7-9 January 2002

20. Comments Slide 19: The matrix which relate the correct and the false result has a simple analytic form. The second row describes the errors of the angle. It can be very large for a short test quadrupole Comments rest of the slides: Erros of analysis for the test quadrupole GO and assuming any other quadrupole as a source of error is performed. An error of 1% is assumed. The errors of the offset can become as large as 20% And the errors of the slope can be as large as 100% F. Wileke, HERA Seminar Zeuthen, 7-9 January 2002

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30. Conclusion: Errors due to thin lens treatment are in the order of 20% Errors of the same order of magnitude are produced by a Gradient error of 1% in the analysis. Angle errors can become quite large. Facit: I see no reason why not to trust BBA measurement To a level of 20% or better! F. Wileke, HERA Seminar Zeuthen, 7-9 January 2002