The optical coordinates used in the S800 are described in relation to a central trajectory passing through the center of the S800 magnets, and with the reference momentum given by p₀=qBρ₀, where ρ₀ is the central bend radius for the dipole, B is the dipole field and q is the ionic charge. These are the longitudinal distance z along the reference path, and the transversal distances x and y (perpendicular to z) with respect to the reference trajectory in the dispersive and non-dispersive directions, respectively. ¹⁾ The dispersive and non-dispersive angles a and b are referred with respect to the z axis in the xz-plane and yz-plane, respectively. The momentum coordinate δ is defined according to the equation δ + 1 = p / p₀, where p is the momentum of the particle. The energy coordinate d is defined as d= (E-E₀) / E₀, being E the energy of the particle and E₀ the energy equivalent to p₀.

With the exception of the sextupole in front of the S800 spectrograph and the trim coils, the aberrations—introduced mostly by the fringe fields of the magnets—are calculated and corrected analytically based on measured field maps. This method ²⁾ uses the ion optics code COSY Infinity ³⁾ to calculate the transfer matrix S from the target to the focal plane, including aberrations up to 5^th order. The precision attained with the 5^th order is comparable to the resolution of the position-sensitive detectors in the focal plane. The tracking algorithm inverts the matrix S before applying it to the coordinates measured at the focal plane on an event-by-event fashion, according to the equation (a_t y_t b_t d_t) = S^-1 (x_f a_f y_f b_f). The subindexes t and f refer to the positions behind the target and at the focal plane, respectively. The positions measured with the two CRDCs are used to determine the position x_f and y_f, and angles a_f and b_f at the focal plane. The main advantage of this method is the fast processing of the data as it avoids the tracking of each individual particle in the magnetic fields of the spectrograph. The standard S800 SpecTcl analysis software provides all necessary functions and interfaces to perform these calculations.

Since the energy coordinate d is one of the quantities deduced from the inverse map, the beam position x_t cannot be calculated and is assumed to be zero. This assumption implies that the final resolution is obtained by folding the finite size of the beam spot in that direction with the size obtained from the reconstruction, itself depending on the detector resolution and the order to which the calculation is performed. Note that this is still the case when the S800 is operated in dispersion-matched mode, where the incoming beam is momentum dispersed—and therefore rather large—on the target. In that case it is the size at the object location that matters since the entire S800 (analysis line + spectrograph) is achromatic. Because the shapes of the fringe fields vary significantly with the absolute strength of the magnets, inverse maps need to be calculated for each magnetic-rigidity setting of the spectrograph. Users can request these inverse maps for which only the magnet strengths, magnetic rigidity, and the particle (A,Z) are necessary. The calculation of the inverse is not automated (yet!) and therefore a latency of a few days is to be expected before receiving the map file.