1. Dispersion correction in the bypass dogleg
Y. Nosochkov
February 16, 2017

2. Bypass dogleg layout
X&Y bending

3. Dogleg and bypass match section – nominal optics
Dogleg: four 90° FODO cells with 8 quads (same strength (±), one power supply), periodic beta, local dispersion in X & Y planes
Bypass beta match: 5 independent quads, no dispersion
dogleg
b match

4. Initial conditions (1)
Assume non-zero incoming X&Y dispersion and slope at entrance of the first dogleg bend: h0, h0, where Twiss functions are b0, a0
Without correction, the incoming dispersion will propagate downstream as
where m is phase advance, and the so-called dispersion invariant A determines the maximum dispersion amplitude
The same A value can be produced by different combinations of h0, and h0 which can be parameterized through two orthogonal functions
For a given A, a 360° range of angle q provides a complete range of h0, h0

5. Initial conditions (2)
Perform the study for the incoming dispersion invariant fixed at A = 25 mm
Generate 12 x 12 = 144 combinations of (hx0, hx0) and (hy0, hy0) settings with a 30° step of q angles (X&Y), covering the full 360° range in X&Y
After every 12 steps in X-angle (360°), the Y-angle moves by one step (30°) h0
h0a0+h0b0 A q

6. Method-1
Use 4 dogleg quads to cancel X&Y h and h due to the incoming dispersion
A quad with DK generates X&Y kicks according to
which produce dispersion
Quads also change b which will be corrected after the dogleg
Choose quads QDOG2 & QDOG4, and QDOG5 & QDOG7 based on their dispersion efficiency (hq√bq) and 90° separation for orthogonality (h0, h0)
This requires 4 new power supplies; the other 4 quads are on another supply 2 5 4 7

7. Dispersion correction
Perfect match in all 144 cases
However, the nominal quad strength at 10 GeV is already close to the magnet 50 kG limit, thus the required tuning strength range limits the energy to 8 GeV

8. Beta match using 4 bypass matching quads
Use 4 matching quads QL1P — QL4P (out of 5 available) downstream of the dogleg
MAD could not match about 30% of the cases
Quad strengths are ok up to 10 GeV

9. Beta match using 5 bypass matching quads
Use all 5 matching quads QL1P — QL5P downstream of the dogleg
Perfect match in all cases; also lower beta functions than with the 4 quad match
Quad strengths are ok up to 10 GeV

10. Method-2
To reduce the variation of quad strength for dispersion correction, use all 8 dogleg quadrupoles, but arrange them in 4 families: [QDOG1, QDOG5], [QDOG2, QDOG6], [QDOG3, QDOG7] and [QDOG4, QDOG8]
Same family quads in each pair are separated by 180°, therefore they are in the same phase amplifying each other’s effect (both dispersion and beta), potentially reducing the tuning strength range
This method requires 4 quad power supplies versus 5 supplies in method-1 2 5 4 7 1 3 6 8

11. Dispersion correction
Perfect match in all 144 cases
The quad tuning range is reduced a factor of 2 compared to method-1, increasing the maximum energy from 8 to 9 GeV (for 50 kG quad)

12. Beta match using 5 bypass matching quads
Use all 5 matching quads QL1P — QL5P downstream of the dogleg
Perfect match in all cases; also lower X-beta functions than in method-1
Quad strengths are ok up to 10 GeV

13. Conclusions
Dispersion in the bypass dogleg can be corrected using independently powered dogleg quadrupoles:
Using 4 independent quadrupoles (out of 8) requires total of 5 power supplies in the dogleg; for A = 25 mm the required tuning range limits the maximum energy to 8 GeV
Using all 8 dogleg quadrupoles arranged in 4 independent pairs requires 4 power supplies; for A = 25 mm the tuning range limits the maximum energy to 9 GeV
Distortion of dogleg beta function can be corrected in the downstream bypass matching section, where 5 quadrupoles have to be used for the match at all initial conditions; the tuning range is ok up to 10 GeV