Use of eigenvectors in understanding and correcting storage ring orbits

Abstract

The response matrix A is defined by the equation X = A Θ , where Θ is the kick vector and X is the resulting orbit vector. Since A is not necessarily a symmetric or even a square matrix we symmetrize it by using A T A. Then we find the eigenvalues and eigenvectors of this A T A matrix. The physical interpretation of the eigenvectors for circular machines is discussed. The task of the orbit correction is to find the kick vector Θ for a given measured orbit vector X . We are presenting a method, in which the kick vector is expressed as linear combination of the eigenvectors. An additional advantage of this method is that it yields the smallest possible kick vector to correct the orbit. We will illustrate the application of the method to the NSLS X-ray and UV storage rings and the resulting measurements. It will be evident, that the accuracy of this method allows the combination of the global orbit correction and local optimization of the orbit for beam lines and insertion devices. The eigenvector decomposition can also be used for optimizing kick vectors, taking advantage of the fact that eigenvectors with corresponding small eigenvalues generate negligible orbit changes. Thus, one can reduce a kick vector calculated by any other correction method and still stay within the tolerance for orbit correction. The use of eigenvectors in accurately measuring the response matrixand the use of the eigenvalue decomposition orbit correction algorithm in digital feedback is discussed.