Chapter 5
Computation and Properties of
Maps
Up to now, the equations of motion have only been solved perturbatively to
the first order. Yet the knowledge of the nonlinear part of the solution is
also needed to determine precisely the performance of a device. Traditionally,
this is done analytically using perturbation theory (see Section 4.5 for more
details). In the past, a tremendous amount of knowledge about aberrations
for various kinds of devices has been accumulated. Yet this approach is far
from being systematic, making the simulation prone to errors, and it is difficult to obtain an accurate solution for a realistic device where no analytical
solution exists. In this chapter, a modern method, the Differential Algebraic
(DA) technique, for computing the transfer map to arbitrary order, will be
described. But before we embark on this task, we will first classify aberrations
that can appear in transfer maps in terms of their symmetries.
5.1 Aberrations and Symmetries
Recall that the transfer map of an optical system relates final coordinates
to initial coordinates via
z f = M( z i ),
where z = (x, a, y, b, l, δ). In the previous chapters, we were concerned mostly
with the linearized part of the map, which describes the major part of the
motion and which can be described by transfer matrices. The matrix elements
were denoted as (x, a), etc.
In order to study the effects of the motion very precisely, it is necessary
to also consider higher order or nonlinear effects. For this purpose we Taylor
expand the map (in a rigorous sense the question whether the map can actually
be Taylor expanded is rather nontrivial, but we ignore this here), and use
names for the coefficients similar to what we had for the linear motion. We
115
DOI:10.1201/b12074-5
Computation and Properties of
Maps
Up to now, the equations of motion have only been solved perturbatively to
the first order. Yet the knowledge of the nonlinear part of the solution is
also needed to determine precisely the performance of a device. Traditionally,
this is done analytically using perturbation theory (see Section 4.5 for more
details). In the past, a tremendous amount of knowledge about aberrations
for various kinds of devices has been accumulated. Yet this approach is far
from being systematic, making the simulation prone to errors, and it is difficult to obtain an accurate solution for a realistic device where no analytical
solution exists. In this chapter, a modern method, the Differential Algebraic
(DA) technique, for computing the transfer map to arbitrary order, will be
described. But before we embark on this task, we will first classify aberrations
that can appear in transfer maps in terms of their symmetries.
5.1 Aberrations and Symmetries
Recall that the transfer map of an optical system relates final coordinates
to initial coordinates via
z f = M( z i ),
where z = (x, a, y, b, l, δ). In the previous chapters, we were concerned mostly
with the linearized part of the map, which describes the major part of the
motion and which can be described by transfer matrices. The matrix elements
were denoted as (x, a), etc.
In order to study the effects of the motion very precisely, it is necessary
to also consider higher order or nonlinear effects. For this purpose we Taylor
expand the map (in a rigorous sense the question whether the map can actually
be Taylor expanded is rather nontrivial, but we ignore this here), and use
names for the coefficients similar to what we had for the linear motion. We
115
DOI:10.1201/b12074-5