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

Introduction Design of the Linac Coherent Light Source (LCLS) requires short bunches (rms length 25-30 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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, 074802 (2002); also at BNL. LCLS DOE Review, April 24, 2002 G. Stupakov, SLAC

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

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

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