CSR Benchmark Test-Case Results Paul Emma SLAC January 4, 2 BERLIN CSR Workshop
Chicane CSR Test-Case Chicane parameters symbol value unit Bend magnet length (not curved length) L B.5 m Drift length (projected; B-B2 & B3-B4) L 5 m Drift length (B2-B3) L c. m Post-chicane drift length (after B4) L f 2. m Bend angle per dipole magnet θ 2.77 deg Bend radius of each dipole magnet R.35 m Momentum compaction factor R 56 25. mm 2 nd -order momentum compaction factor T 566 +37.5 mm Total projected-length of chicane L tot 3. m Vertical half-gap of bend magnets g 2.5 mm Electron beam parameters symbol value unit Nominal energy E 5. GeV bunch charge q.5 &. nc Incoherent rms relative energy spread ( E/E ) u-rms 2. 6 Linear energy-z correlation a +36. m Total initial rms relative energy spread ( E/E ) rms.72 % Initial rms bunch length σ zi 2 µm Final rms bunch length σ zf 2 µm Initial rms norm. emittances γε x,y.,. µm Initial beta-functions at st bend entrance β x,y 4, 3 m Initial α-functions at st bend entrance α x,y +2.6, +. L L L L B B B L L B B θ B2 L c B3 θ B4 Use line-charge CSR γ transient model described in LCLS-TN--2 (Stupakov/Emma, Dec. 2) [same now used in Elegant] based on TESLA-FEL-96-4 (Saldin et al., Nov. 996) (T 566 included, no ISR * added) * incoherent synchrotron radiation L f
Initial Gaussian Distribution Prior to Chicane E/E [%] 2 perfectly linear correlation E/E [%] 2 Energy distribution (σ E /E =.72 %) σ E /E =.72 % 2 bunch head 2 f(s) [/mm] Longitudinal distribution (σ s = 2 µm) 2.5.5 σ s = 2 µm 5 5 N E = 5 GeV
Second Order Compression Included: T 566 4 T 566 [m] /mm 3 2 T 566 566 3R 56 /2 2 4 6 8 2 4 6 S /m λ(s) (arb.) 2.5 2.5.5 after drift-3 before drift-3 leads to slight bunch shape distortion..5.5.
β x [m] 4 3 2 B Beta and Dispersion Functions Horizontal Beta Function (β x, RED/dash=CSR) B2 B3 CSR-altered β x B4 linear β x 2 4 6 8 2 4 6 S /m η x [m].2. Horizontal dispersion (η x ) η x-max 267 mm linear η x 2 4 6 8 2 4 6 S /m
σ z [mm].2.5. B Bunch Length and ز 56 σ s = 2 µm RMS bunch length B2 B3.5 σ s = 2 µm 2 4 6 8 2 4 6 S /m B4 R 56 [m]...2 B Momentum compaction factor (R 56 ) B2 B3.3 2 4 6 8 2 4 6 S /m B4 R 56 = 25 mm
Final input) δ - phase space (gaussian ε/ε =.52; ε c /ε =.499 Energy distribution (σ E /E =.76 %) 2 2 σ E /E =.76 % E/E [%] f(s) [/mm] 2 bunch head 4.. Longitudinal distribution (σ 25 z =2.3 µm) σ 2 s = 2.3 µm 5 5 ( E/E ) CSR [%] E/E [%] 2 4 2 N x 4 CSR induced Energy Gradient.2.2.4...6..
Energy Spread and Emittance (gaussian) E/E and ( E/E ) 2 /2 [%].4.2 CSR energy loss (DASH) and rms spread (SOLID) accumulated B B2 B3 2 4 6 8 2 4 6 S /m B4 Ε /Ε.43% σ δ.2% γε x [µm].6.4.2 B Bend plane normalized emittance B2 B3 B4 γε x.52 µm 2 4 6 8 2 4 6 S /m
Total RMS Relative Energy Spread (including chirp ) σ E /E /%.78.76.74 B Total RMS Relative Energy Spread B2 B3 B4 2 4 6 8 2 4 6 S /m
Chicane CSR-wake Movie (gaussian)
Chicane CSR-integrated integrated-wake (gaussian)
Final x-x Phase Space (gaussian input) γε CSR =.45 µm; β CSR =.368 m; α CSR =.99 (GRN=CSR,BLUE=nom,RED=tot) 5 5 γε.52 µm x /µrad 5 γε =. µm γε CSR.45 µm β opt α opt opt.37 m opt. 5 3 2 x /µm 2 3
Final x-x Phase Space (gaussian & optimal β, α ) γε CSR =.42 µm; β CSR =.368 m; α CSR =.3 (GRN=CSR,BLUE=nom,RED=tot) 5 5 emittance growth can be reduced by choosing optimal β-functions γε.5 µm x /µrad 5 γε =. µm γε CSR.45 µm β β opt α α opt 5 3 2 x /µm 2 3
Beta and emittance (gaussian & optimal β, α ) β x [m] 4 3 2 Horizontal Beta Function (β x, RED/dash=CSR) too big? β min.6 m β β opt α α opt 2 4 6 8 2 4 6 S /m.6 Bend plane normalized emittance γε x [µm].4.2 γε x.5 µm 2 4 6 8 2 4 6 S /m
..5 CSR wakefields (gaussian bend- to drift-2) bend- ( )( L =.4 m.8.6.4 drift- ( 2)( L = 5 m.2.2.5 N bin = 6, smoothed over 4. 6 4 2 2 4 6.4.6.8 6 4 2 2 4 6.3.2. bend-2 2 ( )( L =.4 m.2.5. drift-2 2 ( )( L = m..2.5.5..3.5.4 6 4 2 2 4 6.2 6 4 2 2 4 6
.5.5 CSR wakefields (gaussian bend-3 3 to drift-4) bend-3 3 ( 2)( L =.4 m.5 drift-3 3 ( 4)( L = 5 m.5.5.5 2 2.5 6 4 2 2 4 6 8.5 4 2 2 4 6 8 3 2 bend-4 4 ( 2)( L =.4 m.5 drift-4 4 ( 2)( L = 2 m.5 2.5 3 4 2 2 4 6 8 2 4 2 2 4 6 8
Compressing Uniform Distribution 25 Longitudinal distribution function 2 f(s) [/mm] 5 5.4.2.2.4
Final δ - phase space Uniform input dist. ε/ε =.8; ε c /ε =.2 Energy distribution (σ E /E =.72%) E/E [%] E/E [%] σ E /E =.72 % f(s) [/mm].4.2.2.4 Long. dist. (σ 25 z =2.2 µm) 2 5 5 σ s = 2.2 µm.4.2.2.4 ( E/E ) CSR [%] 5 N CSR induced Energy Gradient.5.5.4.2.2.4
Energy Spread and Emittance (uniform( uniform) E/E and ( E/E ) 2 /2 [%] γε x [µm].4.2.6.4.2 CSR energy loss (DASH) and rms spread (SOLID) accumulated 2 4 6 8 2 4 6 Bend plane normalized emittance emittance growth reduced compared to gaussian Ε /Ε.46% σ δ.7% γε x.2 µm 2 4 6 8 2 4 6 S /m
Chicane CSR-wake Movie Uniform Dist.
Chicane CSR-integrated integrated-wake Uniform Dist.
Final x-x Phase Space (uniform( input) γε CSR =.69 µm; β CSR = 3.872 m; α CSR =.58 (GRN=CSR,BLUE=nom,RED=tot) 5 5 γε.2 µm x /µrad 5 γε =. µm γε CSR.7 µm β opt α opt opt 3.9 m opt.5 5 3 2 x /µm 2 3
.5.4.3.2...2.3 N bin = 6, smoothed over 4 CSR wakefields (uniform( bend- to drift-2) bend- ( )( L =.4 m.8.6.4.2.2.4.6 drift- ( 2)( L = 5 m.4 2.5.5.5.5 2.8 2.5.5.5.5 2.6.4 bend-2 2 ( )( L =.4 m.3.2 drift-2 2 ( )( L = m.2..2.4..2.6.3.8 2.5.5.5.5 2.4 2.5.5.5.5 2
3 2 CSR wakefields (uniform( bend-3 3 to drift-4) bend-3 3 ( 2)( L =.4 m.5 drift-3 3 ( 4)( L = 5 m.5 2 3 2 2 3.5 2 2 3 6 4 2 2 bend-4 4 ( 2)( L =.4 m.5.5.5 drift-4 4 ( 2)( L = 2 m 4.5 6 2 2 3 2 2 2 3
Betatron Amplitude per Bunch Slice (x 2 + [xα + x β] 2 ) /2 /(βε) /2 (x 2 + [xα + x β] 2 ) /2 /(βε) /2 4 3 2.5.5. 4 3 2 Sliced Normalized Centroid Offsets (σ x units) λ(s) Sliced Normalized Centroid Offsets (σ x units) λ(s) gaussian uniform.4.2.2.4.6
Final δ - phase space - Single-Bend E/E [%].2.2 ε/ε =.2; ε c /ε =.37 E/E [%] Energy distribution (σ /E =. 4 ) E.2.2 σ E /E =. %.4.. Longitudinal distribution (σ z =2. µm) 2 f(s) [/mm] 5 5 σ s = 2. µm ( E/E ) CSR [%].4 5 N CSR induced Energy Gradient...2...3..
Energy Spread and Emittance Single Bend E/E and ( E/E ) 2 /2 [%] γε x [µm] CSR energy loss (DASH) and rms spread (SOLID) accumulated.2 (24σ s R 2 ) /3 E /E.5..5.6.4.2 steady-state state ( E/E E /E ) 2 /2 σ δ =.%.5 S /m.5 Bend plane normalized emittance.8 bend magnet.5.5 S /m
CSR-Wake Movie - Single-Bend
LCLS BC2 CSR-integrated integrated-wake (tracked dist.)