Computation and Properties of Maps
139
where m = [n/2] and m > 1. It is clear that the inverse of a Taylor map
exists as long as the inverse of its linear part exists. In later chapters, we will
encounter a number of problems that have to be solved through finding the
inverse of a map.
5.4.3 Reversion of Maps
Throughout the development of beam optical systems, mirror symmetry has
been frequently used. Recently, mirror symmetric systems are being studied
in great detail in the search for high order achromats. Among the various
symmetry arrangements, reversion is the most commonly used. For a system
which is the reversion of another one, the transfer map is the map obtained by
going through the system in the reverse direction. The reversed motion can
be described by first reversing time, i.e., switching all the signs of p x and p y ,
then going through the inverse map, and finally re-reversing time. The time
reversal operation can be performed easily using Differential Algebra, and the
inversion of the transfer map is done as described in Section 5.4.2.
Specifically, reversion entails that, if a particle enters and exits the forward
system at an initial point (x i , a i , y i , b i , l i , δ i ) and a final point (x f , a f , y f , b f , l f ,
δ f ), respectively, it will exit the reversed system at (x i , −a i , y i , −b i , −l i , δ i )
after entering this system at (x f , −a f , y f , −b f , −l f , δ f ). This determines the
reversion transformation:
ˆ
R =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1
0 0 0
0
0
0 −1 0 0
0
0
0
0 1 0
0
0
0
0 0 −1
0
0
0
0 0 0 −1 0
0
0 0 0
0
1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
and hence the map of a reversed system:
M
R = ( ˆ
R) ◦ M
−1
◦ ( ˆ
R
−1 ).
In DA representation, M
R
n is computed through concatenation, where
M
R
n = ( ˆ
R) ◦ M
−1
n ◦ ( ˆ
R
−1 ).
In fact, the second composition ( ˆ
R) ◦ M
−1
n can be done by simply changing
the signs of the rows for a, b and l.
An interesting point worth noting is that ˆ
R is not symplectic. In fact, it
satisfies the following relation
ˆ
R
T ˆ
J ˆ
R = − ˆ
J,
which we call an anti-symplectic relation.
139
where m = [n/2] and m > 1. It is clear that the inverse of a Taylor map
exists as long as the inverse of its linear part exists. In later chapters, we will
encounter a number of problems that have to be solved through finding the
inverse of a map.
5.4.3 Reversion of Maps
Throughout the development of beam optical systems, mirror symmetry has
been frequently used. Recently, mirror symmetric systems are being studied
in great detail in the search for high order achromats. Among the various
symmetry arrangements, reversion is the most commonly used. For a system
which is the reversion of another one, the transfer map is the map obtained by
going through the system in the reverse direction. The reversed motion can
be described by first reversing time, i.e., switching all the signs of p x and p y ,
then going through the inverse map, and finally re-reversing time. The time
reversal operation can be performed easily using Differential Algebra, and the
inversion of the transfer map is done as described in Section 5.4.2.
Specifically, reversion entails that, if a particle enters and exits the forward
system at an initial point (x i , a i , y i , b i , l i , δ i ) and a final point (x f , a f , y f , b f , l f ,
δ f ), respectively, it will exit the reversed system at (x i , −a i , y i , −b i , −l i , δ i )
after entering this system at (x f , −a f , y f , −b f , −l f , δ f ). This determines the
reversion transformation:
ˆ
R =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1
0 0 0
0
0
0 −1 0 0
0
0
0
0 1 0
0
0
0
0 0 −1
0
0
0
0 0 0 −1 0
0
0 0 0
0
1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
and hence the map of a reversed system:
M
R = ( ˆ
R) ◦ M
−1
◦ ( ˆ
R
−1 ).
In DA representation, M
R
n is computed through concatenation, where
M
R
n = ( ˆ
R) ◦ M
−1
n ◦ ( ˆ
R
−1 ).
In fact, the second composition ( ˆ
R) ◦ M
−1
n can be done by simply changing
the signs of the rows for a, b and l.
An interesting point worth noting is that ˆ
R is not symplectic. In fact, it
satisfies the following relation
ˆ
R
T ˆ
J ˆ
R = − ˆ
J,
which we call an anti-symplectic relation.