1. Surface Roughness Impedance in the LCLS Undulator Beam Pipe Gennady Stupakov, SLAC April 24, 2002
Introduction
Small-angle approximation
Surface roughness measurements and estimate of the impedance
Case σz > bump size
Synchronous mode in a pipe with rough surfaces
Conclusions
LCLS DOE Review, April 24, 2002
G. Stupakov, SLAC

2. Introduction
Design of the Linac Coherent Light Source (LCLS) requires short bunches (rms length μm) and small energy spread in the beam (< 0.1%).
It was initially estimated that the longitudinal impedance due to wall roughness in the undulator may be the dominant source of impedance in the LCLS. A small bunch length and close proximity of the wall in the undulator increase the effect of roughness impedance which is usually negligible in accelerators.
During last 5 years several theories of the roughness impedance have been developed which treat different aspects of the beam – rough-wall interaction. The theories are often complicated and sometimes seemingly contradict to each other. However, in my opinion, they are mostly consistent, but may differ in assumptions of roughness properties.
LCLS DOE Review, April 24, 2002
G. Stupakov, SLAC

3. Resistive wall wakefields, for a Gaussian axial distribution
Resistive wall wakefields, for a Gaussian axial distribution. Given are the average energy loss,<d >, the rms energy spread, sd , and the relative correlated emittance growth, De /e0, of a 100 μm betatron oscillation (from LCLS Design Study Report)
<d>/ %
sd / %
De /e0 / %
Resistive Wall, SS
0.34
0.35
260
Resistive Wall, Cu
0.06 8
LCLS DOE Review, April 24, 2002
G. Stupakov, SLAC

4. LCLS DOE Review, April 24, 2002
G. Stupakov, SLAC

5. LCLS DOE Review, April 24, 2002
G. Stupakov, SLAC

6. Small-Angle Approximation, sz Bump Size
In this limit the impedance is inductive. Inductance per unit length of the pipe
LCLS DOE Review, April 24, 2002
G. Stupakov, SLAC

7. Undulator Pipe Surface Roughness Measurements
Surface profile was measured for an undulator pipe using Atomic Force Microscope at NIST, Boulder, Colorado (Stupakov, Thomson, Carr & Walz, 1999)
A high quality Type 316-L stainless steel tubing from the VALEX Corporation with an outer diameter of 6.35 mm and a wall thickness of 0.89 mm was used. The pipe had the best commercial finish corresponding to Ra = 125 nm.
LCLS DOE Review, April 24, 2002
G. Stupakov, SLAC

8. LCLS DOE Review, April 24, 2002
G. Stupakov, SLAC

9. LCLS DOE Review, April 24, 2002
G. Stupakov, SLAC

10. LCLS DOE Review, April 24, 2002
G. Stupakov, SLAC

11. Product Lb for Large Samples.
Sample Size
hrms (nm)
(Lb)1 (pH)
(Lb)2 (pH)
108 × 108 mm 98
4.1 × 10-4
2.8 × 10-4
85 × 85 mm 84
4.0 × 10-4
3.9 × 10-4
65 × 65 mm
109
3.5 × 10-4
1.9 × 10-3
104 × 104 mm
186
7.0 × 10-4
2.7 × 10-4
LCLS DOE Review, April 24, 2002
G. Stupakov, SLAC

12. Case sz < g
The theory has been developed only for a wall with a sinusoidal corrugation
There are synchronous modes in a corrugated waveguide with the wavelength λ > 2/κ. Excitation of those modes results in a resistive part of the impedance.
The wake
where
LCLS DOE Review, April 24, 2002
G. Stupakov, SLAC

13. Gaussian bunch with RMS length σz
Roughness wake for a Gaussian bunch
LCLS DOE Review, April 24, 2002
G. Stupakov, SLAC

14. Gaussian bunch with RMS length σz
LCLS DOE Review, April 24, 2002
G. Stupakov, SLAC

15. Rectangular Bunch Profile
A rectangular axial profile gives a better approximation for the LCLS bunch. LCLS nominal parameters: N = 6 ·109, h = 0.28 μm (corresponding to the RMS roughness of 0.2 μm),
g = 2π/κ = 100 μm, L = 100 m, E = 15 GeV, b = 2.5 mm.
The averages energy loss is 2 · 10-3 %. The RMS energy spread is 10-4 %.
LCLS DOE Review, April 24, 2002
G. Stupakov, SLAC

16. Synchronous Mode
A low-frequency synchronous mode, λ >> g, with vph = c can propagate in a pipe with rough surfaces, (A. Novokhatski & A. Mosnier, 1997). This theory cannot be derived in the perturbation approach. Interacting with the beam, this mode produces an impedance that is not purely inductive any more.
Novokhatsky et al., 1998
LCLS DOE Review, April 24, 2002
G. Stupakov, SLAC

17. Theory of Synchronous Mode
K. Bane & A. Novokhatski (1999) modeled roughness as axisymmetric steps on the surface, assuming that d, g, p << b
The mode wavelength is
hence ω0 ∼ 1/√δ. Wakefunction
and the loss factor K (per unit length)
Surprisingly, K does not depend on the roughness properties.
LCLS DOE Review, April 24, 2002
G. Stupakov, SLAC

18. Theory of Synchronous Mode
Effect of 50 nm roughness (from SLAC-AP-117)
BN estimate the tolerance δ < 10 – 20 nm.
This model does not take into account the large “aspect ratio” of roughness.
Experimental observation of synchronous mode wake.elds: M. Huning et al., PRL, 88, (2002); also at BNL.
LCLS DOE Review, April 24, 2002
G. Stupakov, SLAC

19. Synchronous Mode in a Pipe with Corrugated Walls
The mode frequency ω0 depends on κ and h. Longitudinal wake (Stupakov, 1999)
LCLS DOE Review, April 24, 2002
G. Stupakov, SLAC

20. Synchronous Mode in a Pipe with Corrugated Walls
Typical parameters for roughness: h = 0.5 μm, b = 2 mm, κ = 2π / 50 μm-1.
Corresponding loss-factor parameter f  2·10-3
LCLS DOE Review, April 24, 2002
G. Stupakov, SLAC

21. Conclusions
After more than 5 years of theoretical research, we have a good understanding and a theoretical picture of the wake generated by a rough surface in the undulator.
Theoretical predictions strongly depend on the ”aspect ratio” g/h of roughness. Typically this is a large number. Measured by AFM for a SS pipe with a best finish g/h ∼ 102 – 103.
The predictions for the current design of the LCLS undulator based on the measured roughness profile (RMS height h∼ 100 nm, g ∼ 20 – 100 μm) show that the roughness impedance is smaller than the resistive impedance of the copper wall. Further surface measurements and/or wakefield experiments are desirable to validate this conclusion.
LCLS DOE Review, April 24, 2002
G. Stupakov, SLAC