58 Beam-based Correction and Optimization for Accelerators
where h = (h
+
x , h
−
x , h
+
y , h
−
y )
T are the original coordinates, ζ = (ζ
+
x , ζ
−
x , ζ
+
y ,
ζ
−
y )
T are the normal form coordinates, and
ζ
±
x,y =
2I x,y e
∓iψx,y ,
(2.112)
with new action-angle coordinates (I x , ψ x ) and (I y , ψ y ). The generating function, F , is chosen to make the motion in the new coordinate as simple as
possible. In the non-resonant case, the beam motion in the new resonance
basis will be a simple rotation. It can be shown that the coefficients, f jklm
and h jklm , are connected through [36]
f jklm =
h jklm
1 − e i2π[(j−k)νx+(l−m)νy] ,
(2.113)
where only the resonance driving terms (j = k or l = m or both) are kept.
The beam motion in the original coordinates can be determined from the
inverse coordinate transformation, i.e., e
:F : ζ
−
x , hence [10]
h
−
x ≈ ζ
−
x + [F, ζ
−
x ] = ζ
−
x − 2i
jklm
jf jklm (ζ
+
x )
j−1 ζ
−k
x (ζ
+
y )
l ζ
−m
y
,
(2.114)
and similarly for the vertical plane. The new resonance basis coordinates after
N turns will be
ζ
−
x,y (N ) =
2I x,y e
i(2πνx,yN +ψx0,y0) ,
(2.115)
where the tunes may include any nonlinear detuning. Therefore, the original
resonance basis coordinates after N turns are given by [10]
h
−
x (N ) =
2I x e
i(2πνx+ψx0) − 2i
jklm
jf jklm (2I x )
j+k−1
2
(2I y )
l+m
2
· e
i[(1−j+k)(2πνxN +ψx0)+(m−l)(2πνyN +ψy0)] ,
(2.116)
h
−
y (N ) =
2I y e
i(2πνy+ψy0) − 2i
jklm
lf jklm (2I x )
j+k
2 (2I y )
l+m−1
2
· e
i[(k−j)(2πνxN +ψx0)+(1−l+m)(2πνyN +ψy0)] .
(2.117)
With Eqs. (2.110), (2.113), and (2.116)-(2.117), the observed beam motion
is related to the RDTs in the generating function of the one-turn Lie map. Each
term in Eqs. (2.116)-(2.117) corresponds to a spectral line on the turn-by-turn
orbit data, while a spectral line typically has contributions from many terms.
For example, the third order RDT h 3000 , which is proportional to sextupole
strengths, drives the resonance 3ν x = p and its corresponding spectral line on
the horizontal turn-by-turn orbit data is 1 − 2ν x .
where h = (h
+
x , h
−
x , h
+
y , h
−
y )
T are the original coordinates, ζ = (ζ
+
x , ζ
−
x , ζ
+
y ,
ζ
−
y )
T are the normal form coordinates, and
ζ
±
x,y =
2I x,y e
∓iψx,y ,
(2.112)
with new action-angle coordinates (I x , ψ x ) and (I y , ψ y ). The generating function, F , is chosen to make the motion in the new coordinate as simple as
possible. In the non-resonant case, the beam motion in the new resonance
basis will be a simple rotation. It can be shown that the coefficients, f jklm
and h jklm , are connected through [36]
f jklm =
h jklm
1 − e i2π[(j−k)νx+(l−m)νy] ,
(2.113)
where only the resonance driving terms (j = k or l = m or both) are kept.
The beam motion in the original coordinates can be determined from the
inverse coordinate transformation, i.e., e
:F : ζ
−
x , hence [10]
h
−
x ≈ ζ
−
x + [F, ζ
−
x ] = ζ
−
x − 2i
jklm
jf jklm (ζ
+
x )
j−1 ζ
−k
x (ζ
+
y )
l ζ
−m
y
,
(2.114)
and similarly for the vertical plane. The new resonance basis coordinates after
N turns will be
ζ
−
x,y (N ) =
2I x,y e
i(2πνx,yN +ψx0,y0) ,
(2.115)
where the tunes may include any nonlinear detuning. Therefore, the original
resonance basis coordinates after N turns are given by [10]
h
−
x (N ) =
2I x e
i(2πνx+ψx0) − 2i
jklm
jf jklm (2I x )
j+k−1
2
(2I y )
l+m
2
· e
i[(1−j+k)(2πνxN +ψx0)+(m−l)(2πνyN +ψy0)] ,
(2.116)
h
−
y (N ) =
2I y e
i(2πνy+ψy0) − 2i
jklm
lf jklm (2I x )
j+k
2 (2I y )
l+m−1
2
· e
i[(k−j)(2πνxN +ψx0)+(1−l+m)(2πνyN +ψy0)] .
(2.117)
With Eqs. (2.110), (2.113), and (2.116)-(2.117), the observed beam motion
is related to the RDTs in the generating function of the one-turn Lie map. Each
term in Eqs. (2.116)-(2.117) corresponds to a spectral line on the turn-by-turn
orbit data, while a spectral line typically has contributions from many terms.
For example, the third order RDT h 3000 , which is proportional to sextupole
strengths, drives the resonance 3ν x = p and its corresponding spectral line on
the horizontal turn-by-turn orbit data is 1 − 2ν x .