PHYSICAL REVIEW C VOLUME 47, NUMBER 2 FEBRUARY 1993 Reconstructive correction of aberrations in nuclear particle spectrographs M. Berz, K. Joh,* J. A. Nolen,* B. M. Sherrill, and A. F. Zeller Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824 (Received 24 August 1992) A method is presented that allows the reconstruction of trajectories in particle spectrographs and the reconstructive correction of residual aberrations that otherwise limit the resolution. Using a computed or fitted high order transfer map that describes the uncorrected aberrations of the spectrograph, it is possible to calculate a map via an analytic recursion relation that allows the computation of the corrected data of interest such as reaction energy and scattering angle as well as the reconstructed trajectories in terms of position measurements in two planes near the focal plane. The technique is only limited by the accuracy of the position measurements, the incoherent spot sizes, and the accuracy of the transfer map. In practice the method can be expressed as an inversion of a nonlinear map and implemented in the differential algebraic framework. The method is applied to correct residual aberrations in the S800 spectrograph which is under construction at the National Superconducting Cyclotron Laboratory at Michigan State University and to two other high resolution spectrographs. PACS number(s): 07.75.+h729.30.Aj I. INTRODUCTION Efficient modern high-resolution spectrographs for nu- clear physics usually offer large phase space acceptances, with solid angles of more than 10 msr and energy accep- tances of greater than 10%. One such spectrograph is the 5800 currently under construction at Michigan State University's National Superconducting Cyclotron Labo- ratory [1,2]. Such large acceptance, high-resolution spec- trographs require careful consideration and correction of aberrations. But because of the large phase space accep- tance, aberrations up to seventh order may contribute significantly and affect the resolution. The large phase space makes the correction process considerably more difficult and complex, and often pre- vents a complete correction of aberrations in the conven- tional sense. The conventional correction of aberrations with higher order hardware (sextupoles, octupoles, dipole edge curvatures, etc) will cure second or third order aber- rations, but at the expense of inducing large higher order terms. These terms will eventually limit the maximum solid angle and energy acceptance. It is often possible to circumvent or at least alleviate these problems by using additional information about the particles. In particular, one often measures not only their final positions but also their final slopes by means of a second detector. With this additional information one attempts to retroactively construct the whole trajectory of the particle. This information can be used both for the numerical correction of the quantities of interest, as well as to reveal additional properties such as the scat- tering angle, which is important in the study of many nuclear processes. Some proposed spectrographs, such as those under construction at CEBAF, will use hard- 'Present address: Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439. ware corrections to reduce aberrations up to fifth order and additionally trajectory reconstruction to meet their design goals [3]. A powerful numerical procedure to calculate a poly- nomial relationship between measured final coordinates and the quantities of interest is contained in the program MOTER [4,5]. Using magnetic-field specifications and ge- ometry as input, this program relates initial coordinates, selected randomly, to the corresponding final coordinates by numerically tracing each ray through the system. The coefficients of the polynomial are determined by a nu- merical fit to the data from the ray tracing results. The relevant coefficients to be included in this fitting proce- dure which are connected to the significant aberrations of the system are chosen by experience and empirical tri- als, and for reasons of computational expense there are limits to the number of coefficients that can be consid- ered [6]. Such trajectory reconstruction techniques are usually quite involved computationally. In the following, we present a new direct and efficient method based on differential algebraic (DA) techniques, which is also very useful in rapidly evaluating or optimizing spectrograph designs [7]. Recently we have shown that maps of particle optical systems can be computed to higher orders than previ- ously possible using DA methods [8-11]. Furthermore, the techniques also allow the computation of maps for complicated measured fields that can be treated only ap- proximately otherwise. In our particular case, these in- clude the fringe fields of the large aperture magnets re- quired for such particle spectrographs. As soon as the fields are known, it is now possible to compute all the aberrations that occur in a modern high-resolution spec- trograph without having to rely on tedious ray tracing. On the practical side this requires high order codes for the computation of highly accurate maps for realis- tic fields. The code COSY INFINITY [7,12-14] allows such computations in a powerful language environment. 47 537 1993 The American Physical Society