Representation of an arbitrary beam transfer matrix and solution of the beam matching problem using equally spaced thin-lens quadrupoles

1
Representation of an arbitrary beam transfer
matrix and solution of the beam matching problem
using equally spaced thin-lens quadrupoles
Sergey Orlov
XFEL Beam Dynamics Group Meeting, 6 September 2010
2
Acknowledgements
Being a summer student I would like to thank all
members of the Machine Physics Group for their
hospitality, support, and enormous help with
solving many organizational questions. I
would like to thank the head of DESY Summer
Student Program Professor Meyer for giving me the
opportunity to come here and get such great
experience not only from professional point of
view. Last but not least, I would like to
thank my supervisor V.V. Balandin for his
wonderful thoughts and fantastical support in all
questions that I had, for many hours of patient
explanations, corrections and clarifications, as
well as for guidance throughout the whole summer
student program.
3
(No Transcript)
4
Mathematical formulation of the problem of
representation of an arbitrary uncoupled beam
transfer matrix by drifts and quadrupole thin
lenses
5
What is actually known about this problem
There are several papers, whose authors tried to
find solution of this problem in the form of
exact analytical formulas, with most advanced
probably being the following

• Y-Chiu Chao and John Irwin, Solution of a
  three-thin-lens system with arbitrary transfer
  properties, SLAC-PUB-5834, October 1992.
• O. Napoly, Thin lens telescopes for final focus
  systems,
• CERN/LEP-TH/89-69, CLIC Note 102, November
  1989.

6
What we have to solve
7
What is known about polynomial systems
8
What is known about polynomial systems
Surprisingly, for general polynomial system there
exists the similar algorithm, which allows to
construct the equivalent system, which is called
Gröbner basis, and it is an analogy of
triangular form of linear system. So, for
general polynomial system one can also answer in
algorithmic way about absence or existence and
number of its solutions. Nowadays, the
reduction of original system to its Gröbner form
can be done with the help of computers using such
formula manipulators as Maple or Mathematica.
Unfortunately, this equivalence of original
system and its Gröbner form takes place only over
the field of complex numbers and appears to be
not very useful for us, as far as we are
interested in physical (real) solutions.
Nevertheless, the use of computer assistance is
not completely useless in our problem. Because
the absence of complex solutions means also the
absence of real solutions, computer helped us to
construct examples of particular matrices, which
can not be represented with certain number of
thin lenses.
9
Example of matrix which can not be represented
by 3 thin lenses and 3 variable drift spaces
This matrix can not be represented with less than
5 thin lenses and 4 variable drift spaces (in
total, 9 parameters). This example was found with
the help of Maple and checked once more using
Mathematica program. This example clearly
shows that the simple degree-of-freedom count
not always leads to the correct answers.
10
Equally Spaced Quadrupoles
As far as the hope to give the main job to the
computer failed, the only way left to get some
insides into the problem is to make hand
calculations. For this purpose we consider the
system constructed from equally spaced thin
lenses. Such systems, as we will see, have very
pleasant symmetry, which significantly simplifies
hand calculations.
As one see, the matrix of thin lens
sandwiched between two equal drift spaces is
similar to the matrix , which depends on the
single parameter only.
11
Equally Spaced Quadrupoles Equations in new
variables
12
One Dimensional Case
Let us consider first the one dimensional
problem, when we are interested, for example,
only in horizontal motion. This consideration is
useful not only from methodological point of view
but we will use it later for the solution of
complete 2D problem. We start from
consideration of three thin lenses from that
physical point of view of degree-of-freedom
count.
Due to the simple form of the matrix ,
this system can be easily solved by hand with
respect to variables , and we obtain the
following possibilities when , we
have unique solution and when , the
solution either does not exist or is non-unique.
Geometrical interpretation 22 symplectic matrix
depends on 3 parameters. In the space of these 3
parameters there is a plane, and on this plane
there is a line. If matrix parameters are outside
the plane, then solution is unique. If parameters
are on the plane but outside line solution does
not exist. And if we are on the line, we have
infinite number of solutions.
In order to have solution for an arbitrary
matrix , we need to add one more parameter.
It could be the distance between lenses or one
more thin lens. In both cases solution will be
non-unique for each input matrix.
13
Though 1D problem can be easily solved by hand,
the complete 2D problem still remains too
complicated for the hand solution. It is mainly
connected with the fact that thin lens can not
act in two planes independently if it focuses
beam horizontally, then it defocuses beam
vertically and vice versa. Fortunately, we have
found four-lens combination which has transfer
matrix similar to the transfer matrix of a
single thin lens but its actions can be chosen
independently for both transverse planes.
Let us consider four-lens block, where two lenses
in the middle are set to the constant values
, (or
, ) By direct matrix
multiplication one can show that
where and are linear functions of
parameters , and and are some
constants. Because 22 matrix which connects
and with and is non-degenerated, the
values of parameters and can be chosen
independently.
14
Solution of 2D problem by its reduction to the
two 1D problems
For solution of 1D problem we need three thin
lenses plus one additional parameter for making
. So if we take our three four-lens
blocks, we will have three independent lenses for
each plane (the fact that in these equivalent
lenses we have constants c and d not equal to 1
is not essential). In order to satisfy conditions
and , we need one
more parameter, which again could be chosen
either as a distance between lenses or we may
add one more lens. So we have found thin lens
solution of the problem of representation of an
arbitrary transfer matrix, which requires 13
thin lenses or 12 lenses with possibility to use
distance between them as an additional
parameter. Note that in the case of 13 lenses
considered solution actually uses only 7
parameters in order to represent an arbitrary
transfer matrix because 6 lenses are set to the
constant strengths (in the case of 12 lenses 7
parameters depend on input matrix and other 6
just on distance between lenses). Although
the total number of lenses used in this solution
is probably still not the minimal needed, we
have simple explicit formulas for the lens
strengths as functions of coefficients of input
transfer matrix. And as concerning number of
variable parameters in our system (7 parameters
in the case of 13 lenses), it is the minimum
possible value because we have an example of
transfer matrix which can not be represented by
using six lenses with fixed distance between
them, namely
15
Beam Matching Problem
Let us assume that we have two sets of Twiss
parameters given in the two points of a beam
line. The beam matching problem is the problem of
finding transfer matrix , such that the
following identity holds
The first question is if this problem has
symplectic solution at all for arbitrary two sets
of Twiss parameters. The answer is Yes, it is
well known, and the general symplectic solution
of the matching problem can be expressed in the
following form
and depends on two arbitrary parameters (phase
advances). Because our previous solution for
an arbitrary transfer matrix certainly gives
also the solution of the matching problem, our
current goal is to solve matching problem with
less thin lenses using freedom in choosing the
phase advances.
16
Thin lens solution of the matching problem
We will use the same scheme as before and will
find first the number of thin lenses required for
solution of 1D problem, and then we will use our
four-lens blocks for solution of 2D problem.
Matching equations for 1D problem
where are the coefficients of matrix .
It is not difficult to show that this problem
can be solved with 2 lenses, if and only if
. So, as before, one more parameter for
solution of 1D matching problem is still needed.
Again, it could be one more lens or variable
distance. Therefore, using two four-lens
blocks plus one additional parameter we can get a
complete solution of 2D beam matching problem (so
in our solution we need 9 lenses with 4 again
staying fixed). And also we have an example of
Twiss parameters which can not be matched with 4
thin lenses
17
Summary
18
Thank you very much for your attention!