1. Chromatic Corrections in LCLS-II P. Emma, Y. Nosochkov, M. Woodley Mar
Nominal 4-GeV relative energy spread in LCLS-II ( % rms) is small and generates no significant chromatic aberrations
However, we would also like to run with an energy chirp (linear energy correlation along the bunch) to provide large BW FEL operations (up to 1% FWHM e- BW = 2% photon BW)
Mark Woodley did routine check of chromatic aberrations in dog-leg quads with 1% energy spread and found some issues…

2. Chromatic Aberrations
x = xb + hd (where d  DE/E)
f0 = 6.9 m
bx0 = 32 m
h = 0.44 m
sd = 0.3% (rms)
E = 4 GeV
gex0 = 0.35 mm
x  (1.3 mm)/(0.04 mm)  35
De/e0  25%
First Dog-Leg Parameters
>> 1
“2” here is derived for a Gaussian
energy spread where  4 = 3s 4

3. Nominal Energy Spread in First Dog-Leg at 4 GeV
(standard chirp at dog-leg – FEL-chirp not shown)
Nominal 0.14% rms relative energy spread at first dog-leg (100 pC)
High-charge (300 pC) has 0.22% rms energy spread
1% FW e- chirp  0.30% rms (max)

4. Optics in LCLS-II First Dog-Leg (common to SXR & HXR paths)
Bend-1 hy
Bend-2 hx by bx
M. Borland, “Elegant”

5. sE/E = 0.14% rms  Small Chromatic Emittance Growth
 gex0 = gey0 = 0.35 mm
Dispersion dominates projected emittance until end of final bend
Dex/ex0 = 1%
Dey/ey0 = 4%
Before dog-leg
After dog-leg
No problem with the nominal (0.14%) energy spread

6. sE/E = 0.3% rms  Large Chromatic Emittance Growth
0.3% rms energy spread is based on 1% FW e- chirp
Dex/ex0 = 21%
Dey/ey0 = 57%
 gex0 = gey0 = 0.35 mm
Almost 60% emittance growth at 0.3% rms energy spread
Before dog-leg
After dog-leg
Problem with large (0.3%) energy spread

7. 2nd Order Dispersion (sE/E = 0.3% rms)
Dy' ~ d 2
Similar effect in x
>> 1

8. Add 2 Sextupole Magnets to Rolled Optics
Minimize (exey)1/2 with 2 sext. strengths and 2 tilt angles (Elegant) SY SX

9. Emittance with 2 Sextupoles (0.30% rms Energy Spread)SY SX
SX: K2 = m-3
SY: K2 = m-3
SX: TILT = +19
SY: TILT = -13
Dex/ex0 = 0.1%
Dey/ey0 = 0.1%
after sextupoles optimized
We can imagine adding 2 more sextupoles for better local correction, but it’s not necessary.

10. Possible Sextupole Magnets (use existing design)9”
Built in 1990 for FFTB
(2.13” bore shown)
1.38” 3”
“1.38S3.00”
Use FFTB sextupole design “1.38S3.00” (L  0.1 m, r = 17.5 mm).
Stay-clear radius limit in dog-leg area is: Rbsc > 16 mm (so this is just OK).
Max. available strength: |G' |L = 2|B|L/r2  870 kG/m (at 8 A, |B|  1.33 kG).
Max. |K2| needed over all (8) new sextupoles is |K2|  24.7 m-3.
Max. available |K2|  2|B|/r2/(Br) = 65 m-3 at 4 GeV (or 26 m-3 at 10 GeV).

11. SXR-Chicane Optics (sE/E < 0.1% rms nominal)

12. sE/E = 0.3% rms  Chromatic Emittance Growth
SXR-Chicane
Dex/ex0 = 60% 
No sextupoles yet

13. Add 2 Sextupoles to SXR-Chicane
Minimize (exey)1/2 with 2 sext. strengths
SX1
SX2
Dex/ex0 = 0 
sextupoles optimized
SX1: K2 = 12.6 m-3, SX2: K2 = 11.7 m-3, L = 0.1 m, |Dx| < 0.3 mm, |Dy| < 0.3 mm, T566 = T566-0/2.4

14. SXR-Dog-Leg Optics (sE/E < 0.1% rms nominal)

15. sE/E = 0.3% rms  Chromatic Emittance Growth
Dex/ex0 = 51% 
No sextupoles yet

16. Minimize (exey)1/2 with 2 sext. strengths
Add 2 Sextupoles
Minimize (exey)1/2 with 2 sext. strengths
SX1
SX2
Dex/ex0 = 0.4% 
sextupoles optimized
SX1: K2 = m-3, SX2: K2 = m-3, L = 0.1 m, |Dx| < 0.3 mm, |Dy| < 0.5 mm, T566 = T566-0/3

17. HXR-Cross-Over Optics (sE/E < 0.1% rms nominal)
SX3
SX4

18. sE/E = 0.3% rms  Chromatic Emittance Growth
Dex/ex0 = 210% 
Dey/ey0 = 3% 
No sextupoles yet

19. Minimize (exey)1/2 with 2 sext. Strengths & 2 tilt angles
Add 2 Sextupoles
Minimize (exey)1/2 with 2 sext. Strengths & 2 tilt angles
SX1
SX2
Dey/ey0 = 3% 
Dex/ex0 = 0 
sextupoles optimized
SX1: K2 = m-3 (Tilt = ), SX2: K2 = m-3 (Tilt = ), L = 0.1 m, |Dx| < 0.25 mm, |Dy| < 0.25 mm, T566 = T566-0/3

20. No chromatic effects at sE/E = 0.3%
Existing HXR-DL2 (sE/E = 0.1% rms nominal)
No chromatic effects at sE/E = 0.3%

21. e- Chromatic Band-Pass of SXR Branch
(sextupoles ON & OFF)
gex
gey
gex - sextupoles OFF
gey - sextupoles OFF
gex - sextupoles ON
gey - sextupoles ON
sE/E0 
0.5% rms
HXR branch assumed to be a smaller issue

22. Dx & Dy alignment: < 0.3 mm at sE/E = 0.3% (De/e < 5%).
Sextupole Alignment Tolerances
Beam size in sextupoles is dominated by dispersion, so misalignments, Dx & Dy, dominantly cause dispersion errors in both planes.
∆𝜀 𝑥 𝜀 𝑥0  𝐾 2 𝐿 𝛽 𝑥 𝜂 𝑥 2 𝜎 𝛿 2 𝛽 𝑥 𝜀 𝑥0 ∆𝑥 2 ,
∆𝜀 𝑦 𝜀 𝑦0  𝐾 2 𝐿 2 𝛽 𝑥 𝛽 𝑦 𝜀 𝑥0 𝜀 𝑦 𝜂 𝑥 2 𝜎 𝛿 2 𝛽 𝑥 𝜀 𝑥0 ∆𝑦 2
Dx & Dy alignment: < 0.3 mm at sE/E = 0.3% (De/e < 5%).

23. Sextupole Magnet Summary
(L = 0.1 m, r = 17.5 mm)
Area
Name
Adjacent Quad K2
(m-3)
Tilt (deg)
Rbsc (mm)
First Dog-Leg SX
QDOG3
-20.6 19 16 SY
QDOG2
-9.8
-13 12
SXR-Chicane
SX1
QSP1S
12.6 14
SX2
QSP9S
11.7
SXR-Dog-Leg
SX5
QDL13
-8.24
SX6
QDL15
-24.7 9
HXR Cross-Over
SX3
QSP1H
-18.5
SX4
QSP3H
20.3 17 13

24. Summary 8 new sextupole magnets will allow up to 0.5% e- rms BW
All sextupoles now in MAD files, but deferred
Sextupole magnets likely based on existing “1.38S3.00” design
Need to be aligned to < 0.3 mm (when chirp is used)
Sextupoles do almost nothing with nominal < 0.1% energy spread
Magnet movers would allow linear dispersion tuning, but probably not worth the cost and time