216
An Introduction to Beam Physics
It turns out that this kind of system not only cancels dispersion, but also
cancels all second order geometrical aberrations. Recalling the equations of
motion (3.22), second order geometrical aberrations generated in the short
interval [s, s + ds] can be written as
x f = x i + hdsx i a i , a f = a i +
T a,k,l,m,n x
k
i a
l
i y
m
i b
n
i ,
y f = y i + hdsx i b i , b f = b i +
T b,k,l,m,n x
k
i a
l
i y
m
i b
n
i ,
where the summation
is taken over k, l, m, n from 0 to 2 such that k + l +
m + n = 2; so in the above,
reads as
2
k,l,m,n=0
k+l+m+n=2
.
This simplified description is used in the rest of this section unless otherwise
noted. Here the linear matrix from s to s+ds has been removed by the inverse
matrix. As a result, the second order map over the interval is lumped into a
point. Applying the transformation
⎛
⎜
⎜
⎜
⎝
x
a
y
b
⎞
⎟
⎟
⎟
⎠
= ˆ
A
⎛
⎜
⎜
⎜
⎝
x
a
y
b
⎞
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎝
1/
√
β x
0
0
0
α x /
√
β x
√
β x
0
0
0
0 1/
β y
0
0
0 α y /
β y
β y
⎞
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎝
x
a
y
b
⎞
⎟
⎟
⎟
⎠
,
we obtain
⎛
⎜
⎜
⎜
⎝
x f
a f
y f
b f
⎞
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎝
x/
√
β x
(α x x+β x a)/
√
β x
y/
β y
(α y y+β y b)/
β y
⎞
⎟
⎟
⎟
⎠
◦
⎛
⎜
⎜
⎜
⎝
x + hdsxa
a +
T a,k,l,m,n x
k a
l y
m b
n
y + hdsxb
b +
T b,k,l,m,n x
k a
l y
m b
n
⎞
⎟
⎟
⎟
⎠
◦
⎛
⎜
⎜
⎜
⎝
√
β x
x i
(−α x
x i + a i )/
√
β x
β y
y i
(−α y
y i +
b i )/
β y
⎞
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎝
x i +
hds/
√
β x
x i (−α x
x i + a i )
a i +
T a,k,l,m,n
x
k
i a
l
i
y
m
i
b
n
i
y i +
√
β x hds/β y
x i
−α y
y i + b i
b i +
T b,k,l,m,n
x
k
i a
l
i
y
m
i
b
n
i
⎞
⎟
⎟
⎟
⎟
⎟
⎠
,
where
T a,k,l,m,n and
T b,k,l,m,n are linear combinations T a,k,l,m,n and T b,k,l,m,n ,
each of which is multiplied by some powers of β x , β y , α x and/or α y . Now let
us consider a system that consists of n identical cells with the phase advances
μ x = μ y = 2π/n.
Defining
R
μ x
μ y
=
⎛
⎝
ˆ
R (μ x )
0
0
ˆ
R (μ x )
⎞
⎠
⎛
⎜
⎜
⎝
x
a
y
b
⎞
⎟
⎟
⎠ ,
An Introduction to Beam Physics
It turns out that this kind of system not only cancels dispersion, but also
cancels all second order geometrical aberrations. Recalling the equations of
motion (3.22), second order geometrical aberrations generated in the short
interval [s, s + ds] can be written as
x f = x i + hdsx i a i , a f = a i +
T a,k,l,m,n x
k
i a
l
i y
m
i b
n
i ,
y f = y i + hdsx i b i , b f = b i +
T b,k,l,m,n x
k
i a
l
i y
m
i b
n
i ,
where the summation
is taken over k, l, m, n from 0 to 2 such that k + l +
m + n = 2; so in the above,
reads as
2
k,l,m,n=0
k+l+m+n=2
.
This simplified description is used in the rest of this section unless otherwise
noted. Here the linear matrix from s to s+ds has been removed by the inverse
matrix. As a result, the second order map over the interval is lumped into a
point. Applying the transformation
⎛
⎜
⎜
⎜
⎝
x
a
y
b
⎞
⎟
⎟
⎟
⎠
= ˆ
A
⎛
⎜
⎜
⎜
⎝
x
a
y
b
⎞
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎝
1/
√
β x
0
0
0
α x /
√
β x
√
β x
0
0
0
0 1/
β y
0
0
0 α y /
β y
β y
⎞
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎝
x
a
y
b
⎞
⎟
⎟
⎟
⎠
,
we obtain
⎛
⎜
⎜
⎜
⎝
x f
a f
y f
b f
⎞
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎝
x/
√
β x
(α x x+β x a)/
√
β x
y/
β y
(α y y+β y b)/
β y
⎞
⎟
⎟
⎟
⎠
◦
⎛
⎜
⎜
⎜
⎝
x + hdsxa
a +
T a,k,l,m,n x
k a
l y
m b
n
y + hdsxb
b +
T b,k,l,m,n x
k a
l y
m b
n
⎞
⎟
⎟
⎟
⎠
◦
⎛
⎜
⎜
⎜
⎝
√
β x
x i
(−α x
x i + a i )/
√
β x
β y
y i
(−α y
y i +
b i )/
β y
⎞
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎝
x i +
hds/
√
β x
x i (−α x
x i + a i )
a i +
T a,k,l,m,n
x
k
i a
l
i
y
m
i
b
n
i
y i +
√
β x hds/β y
x i
−α y
y i + b i
b i +
T b,k,l,m,n
x
k
i a
l
i
y
m
i
b
n
i
⎞
⎟
⎟
⎟
⎟
⎟
⎠
,
where
T a,k,l,m,n and
T b,k,l,m,n are linear combinations T a,k,l,m,n and T b,k,l,m,n ,
each of which is multiplied by some powers of β x , β y , α x and/or α y . Now let
us consider a system that consists of n identical cells with the phase advances
μ x = μ y = 2π/n.
Defining
R
μ x
μ y
=
⎛
⎝
ˆ
R (μ x )
0
0
ˆ
R (μ x )
⎞
⎠
⎛
⎜
⎜
⎝
x
a
y
b
⎞
⎟
⎟
⎠ ,