Lattice Modules
239
As a result,
arcsin (x 0 + Δx)
= arcsin (x 0 )+arcsin
(x 0 + Δx)
1 − x 2
0 − x 0
1 − (x 0 + Δx)
2
= arcsin (x 0 )+arcsin
(x 0 + Δx)
1 − x 2
0 − x 0
1 − x 2
0
1 −
2x 0 Δx + Δx 2
1 − x 2
0
= 2 arcsin (x 0 )+arcsin
Δx
1−x 2
0 +x 0
1−x 2
0
2x 0 Δx+Δx
2
2 (1−x 2
0 )
+
4x
2
0 Δx
2
8 (1−x 2
0 )
2
= 2 arcsin (x 0 ) + Δx
1 − x 2
0 +
x 0
2
1 − x 2
0
2x 0 Δx + Δx
2
1 − x 2
0
+
x
2
0 Δx
2
(1 − x 2
0 )
2
= 2 arcsin (x 0 ) +
Δx
1 − x 2
0
+
x 0 Δx
2
2 (1 − x 2
0 )
3
2
.
To the second order, the path length difference is
l − l 0 = 2 4R 0
(1 + δ) arcsin
1 − δ + δ
2
sin φ 0
− φ 0
+ 2L 1
(1 + δ) cos φ 0
cos 2 φ 0 + 2δ + δ 2
− 1
= 2 4R 0
(1 + δ) arcsin
sin φ 0 −
δ − δ
2
sin φ 0
− φ 0
+ 2L 1
(1 + δ)
1 + (2δ + δ 2 ) / cos 2 φ 0
− 1
= 2 4R 0
(1 + δ)
φ 0 −
δ − δ
2
tan φ 0 +
1
2
δ
2 tan
3 φ 0
− φ 0
+ 2L 1
(1 + δ)
1 −
1
2
2δ + δ
2
cos 2 φ 0
+
3
8
4δ
2
cos 4 φ 0
− 1
= 2 4R 0
(φ 0 − tan φ 0 ) δ +
1
2
tan
3 φ 0
δ
2
− 2L 1 tan
2 φ 0
δ −
3
2
δ
2
cos 2 φ 0
.
For φ 0 1, only the terms of the lowest order in φ 0 are important and we
obtain
l − l 0 = 2 −2L 1 φ
2
0
δ −
3
2
δ
2
.
As an example, we discuss the parameters of the first bunch compressor
at the Linac Coherent Light Source (LCLS) at SLAC National Accelerator
Laboratory, California, USA, which operates at a beam energy of 250 MeV.
It is worth noting that the bend radius R 0 of 2.48 m (corresponding to a