Measurement of Beta-Function and Phase Using the Response Matrix

A new method for extracting the beta-function and phase for the beam position monitors (BPMs) and the corrector magnets from the measured response matrix is presented. The response matrix relates beam motion at the BPM locations to changes in corrector magnet strengths. Using the model beta and phase as me initial values, new values are obtained by iteration. The accuracy of beta and phase thus calculated is limited by the accuracy of response matrix measurement and calibration of BPMs and correctors. The scaling ambiguity in the beta-function is resolved by matching the beta product and phase advance across a drift region. A by-product of this technique is an accurate determination of the betatron tune, and in principle, quadrupole strengths can be calculated from the betas and phases. This method is applied to data obtained from the X-ray ring at the National Synchrotron Light Source at Brookhaven. The possibility of applying the results to lattice-debugging will also be discussed. I. INTRODUCTION In the past, several methods for me measurement of accelerator beta functions and betatron phase have been suggested and applied. Accelerators having independently controllable quadrupole magnets can make use of the variation of tune with quadrupole magnet strength to extract the value of beta at the location of the quad, as is done at the Cornell Electron Storage Ring (l). A technique was suggested by Harrison and Peggs (23 whereby betas and phases could be determined from closed orbit measurements resulting from varying two steering correctors in turn, provided that the values of beta and phase at those two correctors could be found from some other technique. The beta function and phase at the locations of all beam position monitors (BPMs) could in principle be found in this way. In this paper, a technique is presented for determining the best fit betas and phases at all BPMs and all steering correctors from measured response matrix data. The response matrix relates beam motion at the BPM locations to changes in steering corrector strengths. This technique is similar to that of Ref. (2), but makes use of all steering correctors rather than just two. Because the problem is severely over- constrained, the method of singular value decomposition (SVD) of a matrix is used to minimize the difference between the resulting ring model and the measured response matrix data. One by-product of this technique is an accurate determination of the betatron tune. The technique is applied to data obtained from the X-ray ring at the National Synchrotron Light Source at Brookhaven. Results are obtained which agree qualitatively with measurements made there two and a half years ago using the technique of Ref. (2), together with the least squares fitting, by Decker and Swenson, (3) II. THEORY Suppose that there are M BPMs and N correctors in the storage ring. Both M and N are larger than 1. Let pi and v, be the amplitude and phase functions at tbe location of the i-tb BPM. p, and r+rcj are similarly defined for the j-tb corrector. The response matrix Rd corresponding to the beam motion at the i-m BPM per unit angle of kick by the j-th corrector is then given by (4)