220
B. J. Holzer et al.
The functions C(s) and S(s) are the cosine and sine like matrix elements of the
lattice element in the sense that e.g. C(s) = M[1,1], and the integral is executed over
one complete cell.
In the dispersion suppressor section, the dispersion D(s) starts with the value D 0
the end of the arc cell and is reduced to zero. Or turning it around and thinking from
right to left: the dispersion has to be created inside the suppressor part by proper
arrangement of the dipole magnets, starting from D = D’ = 0 in the straight section
to reach the values that correspond to the periodic dispersion of the arc cells. Solving
the equation above by integrating over a certain number of cells will determine the
bending strength 1/ρ and the number n of cells in the suppressor part that is needed
to fulfill the boundary condition and get the values of the dispersion in the following
periodic arc cell.
For a given phase advance ϕ c per cell two conditions for the dispersion matching
are obtained that combine the number of suppressor cells, n, and the strength of the
suppressor dipoles, δ supr :
2δ supr sin
2
nφ c
2
= δ arc
sin (nφ c ) = 0
δ supr =
1
2
δ arc .
(6.18)
If the phase advance per cell in the arc fulfills the condition sin(nφ c ) = 0, the
strength of the dipoles in the suppressor region is just half the strength of the arc
dipoles. In other words the phase advance has to fulfill the condition
nφ c = kπ, k = 1, 3, . . . .
(6.19)
There are a number of possible phase advances that fulfill that relation, but clearly
not every arbitrary phase is allowed. Possible constellations would be for example,
φ c = 90
◦ , n = 2 cells, or, φ c = 60
◦ , n = 3 cells in the suppressor.
Figure 6.11 shows such a half bend dispersion suppressor, starting from a FODO
structure with 60 ◦ phase advance per cell. The focusing strength of the FODO cells
before and after the suppressor are identical, with the exception that—clearly—the
FODO cells on the right are “empty”, i.e. they have no bending magnets.
It is evident that unlike to the suppressor scheme with quadrupole lenses now the
beta function is unchanged in the suppressor region.
Again this scheme has advantages:
• no additional quadrupole lenses are needed and no individual power supplies;
• in first order the β functions are unchanged; aperture needs and beam sensitivity
are not increased;
and disadvantages:
• it works only for certain values of the phase advance in the structure and therefore
restricts the free choice of the optics in the arc;
• special dipole magnets are needed (having half the strength of the arc types);
• the geometry of the ring is changed.
B. J. Holzer et al.
The functions C(s) and S(s) are the cosine and sine like matrix elements of the
lattice element in the sense that e.g. C(s) = M[1,1], and the integral is executed over
one complete cell.
In the dispersion suppressor section, the dispersion D(s) starts with the value D 0
the end of the arc cell and is reduced to zero. Or turning it around and thinking from
right to left: the dispersion has to be created inside the suppressor part by proper
arrangement of the dipole magnets, starting from D = D’ = 0 in the straight section
to reach the values that correspond to the periodic dispersion of the arc cells. Solving
the equation above by integrating over a certain number of cells will determine the
bending strength 1/ρ and the number n of cells in the suppressor part that is needed
to fulfill the boundary condition and get the values of the dispersion in the following
periodic arc cell.
For a given phase advance ϕ c per cell two conditions for the dispersion matching
are obtained that combine the number of suppressor cells, n, and the strength of the
suppressor dipoles, δ supr :
2δ supr sin
2
nφ c
2
= δ arc
sin (nφ c ) = 0
δ supr =
1
2
δ arc .
(6.18)
If the phase advance per cell in the arc fulfills the condition sin(nφ c ) = 0, the
strength of the dipoles in the suppressor region is just half the strength of the arc
dipoles. In other words the phase advance has to fulfill the condition
nφ c = kπ, k = 1, 3, . . . .
(6.19)
There are a number of possible phase advances that fulfill that relation, but clearly
not every arbitrary phase is allowed. Possible constellations would be for example,
φ c = 90
◦ , n = 2 cells, or, φ c = 60
◦ , n = 3 cells in the suppressor.
Figure 6.11 shows such a half bend dispersion suppressor, starting from a FODO
structure with 60 ◦ phase advance per cell. The focusing strength of the FODO cells
before and after the suppressor are identical, with the exception that—clearly—the
FODO cells on the right are “empty”, i.e. they have no bending magnets.
It is evident that unlike to the suppressor scheme with quadrupole lenses now the
beta function is unchanged in the suppressor region.
Again this scheme has advantages:
• no additional quadrupole lenses are needed and no individual power supplies;
• in first order the β functions are unchanged; aperture needs and beam sensitivity
are not increased;
and disadvantages:
• it works only for certain values of the phase advance in the structure and therefore
restricts the free choice of the optics in the arc;
• special dipole magnets are needed (having half the strength of the arc types);
• the geometry of the ring is changed.