Coherence Requirements for Various Seeding Schemes G. Penn 2.5 1 1 2. 1 1 sase 4 high current 4 low current SSSFEL12 Trieste 1 December 212 # photons / mev 1.5 1 1 1. 1 1 5. 1 9 1238.5 1239 1239.5 124 124.5 1241 1241.5 photon energy (ev)
Coherence is a major goal of seeding Laser seed phase errors multiplied by harmonic jump straightforward and unavoidable, not considered here small effect from laser seed power profile Shot noise required seed power grows with square of harmonic Electron beam slice variations (scale > L coop ) focus on slice energy peak current and emittance weaker G. Penn 2
Start with HGHG laser seed modulator radiator, pre-bunched beam chicane phase offsets ~ R 56 Δη Phase variations from energy profile dominated by chicane θ = 2π λ r R 56 η Δη = rel slice energy offset z = R 56 η avoids confusion when wavelength changes (arbitrary sign choices, normal chicane R 56 >) G. Penn 3
Phase Errors Still Grow after Chicane initial bunched beam à radiation phase a good fit is θ 2π λ u L u η, L u < 4.5L g 1.2 2π λ Lu u 3 4 L g η, Lu > 4.5L g. by analogy, define as z R FEL rad η shorthand: R value for radiator with pre-bunching G. Penn 4
R FEL for prebunched beams R FEL rad λr levels off at ~ R 56 for undulators is low gain, R FEL is half of R 56 from beamline L u is magnetic length λ u L u, L u < 4.5L g 1.2 λ r λ Lu u 3 4 L g, Lu > 4.5L g 1.2 L sat λ r /λ u after saturation L u (1 + a 2 u)/γ 2 =2L u λ r /λ u more accurate, use L u + L drift /(1+a u2 ) same result for modulator initial radiation field à modulation phase G. Penn 5
Prebunched beam: fit to simulations Amplitude of Phase Modulation.6.5.4.3.2.1 e-beam: 2.4 GeV,.6 micron, 6 A, 25 kev spread und: 2 mm pd, 3 m sections, tuned for 1 nm output 1 kev energy modulation is applied: track phases GENESIS low gain linear regime uniform with breaks.1 1 2 3 4 z (m) G. Penn 6 Amplitude of Phase Modulation.6.5.4.3.2.1 GENESIS low gain linear regime.1 1 2 3 4 z (m) after saturation, phase modulation levels off
Close to 1D FEL theory 3 modes, only 1 growing growth rates up to linear in Δη: initial mix for bunching with no modulation or field differences from numerical fit: Γ + = 1 2L 1D Γ = 1 2L 1D Γ = 1 2L 1D replace L g with L 1D, and 1.2 factor with 2 2/3 1+ i 4 3 3 ik u η 1+ i 4 3 3 ik u η 2i 3 4 3 ik u η 2/3 because bunch & modulation affected by dispersion, but not radiation; 4/3 à 1.2 due to diffraction? G. Penn 7
2-stage HGHG example 2.4 GeV, 2 nm seed laser, 15 kev σ E, fresh-bunch offsets before and after fresh bunch uncorrelated only look at errors up to FB delay mod1 ß rad1à mod2 rad2à h 1 =12 FB h 2 =8 delay G. Penn 8
Sensitivity to energy offsets Total effect: R FEL = Rmod FEL + R 56 + Rrad FEL 26.5 µm = 9 µm + 16 µm + 1.5 µm first chicane usually largest term R 56 λ seed 2πη M η M = rel energy modulation, typically ρ FEL energy modulation of 3.5 MeV yields R 56 =16 µm also some bunching in modulator G. Penn 9
Output Phase and Spectrum output phase (radian) 4 3 2 1-1 -2-3 1 kev offsets: 1.1 nm shifts in location of microbunch 1 st stage (16 nm) and 2 nd stage (2 nm) at 2 nm > +/- 3 radian; R FEL = 23 µm 16 nm 2 nm # photons / mev 7. 1 9 6. 1 9 5. 1 9 4. 1 9 3. 1 9 2. 1 9 1. 1 9 significant spreading of spectrum -4-14 -12-1 -8-6 -4-2 593 594 595 596 597 t (fs) photon energy (ev) using avg phase; on-axis phase looks a little worse G. Penn 1
Simulated beam from RF gun not an accurate estimate of microbunching 241 ripples 12 kev (some is numerical) spectrum with ±1.5 radian phase modulation 12 fs pd kinetic energy (MeV) 24.5 24 2399.5 2399 slice avg -4-35-3-25-2-15-1-5 equiv bw of 25 fs FWHM t (fs) G. Penn 11
How to get more tolerance to chirp? More modulation, weaker chicane? modulator already contributes a lot self-bunching (no chicane) degrades performance Negative R 56 chicane? have to unwind bunching from modulator -18 µm = 9 µm - 29 µm + 2 µm more complex, usually not tunable either way, get less than 5% improvement needs further study G. Penn 12
Optical Klystron: new constraints similar configuration, just no harmonic jump saturate sooner V.N. Litvinenko, NIMA 34 (1991) 463 low-power HHG seeds M. Gullans et al., Opt. Commun. 274 (27) 167 oscillators, G. Dattoli et al., J. Quan. Elec. 31 (1995) 1584 optimal bunching when kr 56 σ η ~ 1 but phase errors are kr 56 Δ η, order unity when Δ η σ η if only care about power, use slice energy spread if care about coherence, use harmonic Δ η (typically, Δ G. Penn η ~ projected energy spread) 13
Self-seeding and R FEL only post-monochromator stage has an impact initial radiation à final radiation phase same analysis, different dependence R FEL ss, also close to 1D theory: low gain, no change in phase by definition of low gain linear regime would have 4/3 instead of 1.2 Lu < 1.5L g 1.2 λ r λ u Lu 3 2 L g, Lu > 1.5L g. G. Penn 14
LCLS HXRSS self-seeding electron beam: 13 GeV, 3 ka,.4 µm emit, parabolic profile radiate at.15 nm current (A) 3 25 2 15 1 5 end stage 1 -.5-1 -1.5-2 -7-6 -5-4 -3-2 -1-2.5 t (fs) output energy offset (MeV) output phase (radian) 4 3 2 1-1 -2-3 phase end of stage 2-4 -7-6 -5-4 -3-2 -1 G. Penn 15 t (fs) current (A) 3 25 2 15 1 5 end stage 2-2 -4-6 -8-1 -12-14 -16-18 -2-7 -6-5 -4-3 -2-1 -22 t (fs) output energy offset (MeV)
EEHG has intriguing result In echo scheme, mixing two waves: output k x =pk 2 -mk 1 p, m integers; k=2π/λ smaller phase errors in bunching than HGHG* so θ =(k x R 2 mk 1 R 1 ) η R FEL echo = R 2 mr 1 λ x /λ 1 two terms can partly cancel, but bunching b J m [(k x R 2 mk 1 R 1 ) η M1 ] if arg=, no bunching η M1 is 1st relative energy modulation G. Penn 16 * D. Xiang and G. Stupakov, PRST:AB 12 (29) 372
EEHG less sensitive to energy offsets usually m=1, Bessel function yields soft optimum k x R 2 k 1 R 1 1.8/η M1 very good scaling, almost cancels ratio of terms is 1.8(η M2 /η M1 )(λ 2 /λ x ) R FEL goes like output wavelength, not seed wavelength R FEL echo = R 2 mr 1 λ x /λ 1 G. Penn 17
Easy control over sign of R FEL useful feature, EEHG has two optimal settings for R 1 for given Δη, can choose sign of Δθ can look like negative R 56 get this for free by mixing two frequencies single electrons at higher energy shift to front, but regions of high current are shifted back in phase tunable: just cannot have R FEL =, else b= bunching scales linear or better with R FEL G. Penn 18
Use EEHG to tolerate big energy offsets seed 1 modulator seed 2 modulator radiator at 8 nm, pre-bunched beam 8 nm output 3 undulators to saturation beam: 2.4 GeV,.6 µm emit, 15 kev σ E 2 nm seeds, 3 kev modulation R 1 =7.5 mm, R 2 =28 micron optimum R 2 -R 1 λ x /λ 1 = 14 micron (with energy spread) close to HGHG case ~ 1/1 energy modulation G. Penn 19
Harmonics, bunching, and R FEL 2 choices to optimize bunching: 25th harmonic, bunching and R FEL vs ΔR 1 radiator adds 4 µm includes energy spread neglects scattering bunching bunching G. Penn 2.3.25.2.15.1.5 5 1 15 2 25 3 35 harmonic will operate here, -5 micron.1.5.2.1.1.2 relative shift in R56 smaller R 1 positive R FEL h=23, 25 larger R 1 negative R FEL h=25, 27 1 Recho R echo at optimum is -14 micron
EEHG simulations at 8 nm 1 consider initial energy modulation +/- 1 MeV optional +/- 1 kev/m wakefield includes ISR and rough IBS models 9 1 8 energy offset (MeV).5 -.5 5-5 wakefield (kev/m) output power (MW) 7 6 5 4-1 E wake -14-12 -1-8 -6-4 -2 t (fs) -1 red: just energy modulation G. Penn 21 3 2-14 -12-1 -8-6 -4-2 t (fs) green: added wakefield
Residual Phase Errors due to slippage, phase errors misaligned with energy output phase (radian) 1.5 -.5-1 -1.5-2 but if slippage > modulation scale length, acts like σ E wakefields generate uncorrected phase errors thanks to Paul Emma for pointing this out different radiated power generates extra energy offsets corr to R FEL = -1 µm -2.5-14 -12-1 -8-6 -4-2 G. Penn 22 t (fs) # photons / mev 1. 1 11 9. 1 1 8. 1 1 7. 1 1 6. 1 1 5. 1 1 4. 1 1 3. 1 1 2. 1 1 1. 1 1 154.7 154.8 154.9 155 155.1 155.2 155.3 photon energy (ev)
Push to 2 nm tweak design to reach 2 nm, using EEHG+HGHG try not to use fresh bunch otherwise slice energy before/after uncorrelated could use fresh bunch and just accept phase offsets from after delay earlier design seed 1 seed 2 radiator at 8 nm, pre-bunched beam modulator modulator 8 nm output 3 undulators to saturation G. Penn 23
Push to 2 nm tweak design to reach 2 nm, using EEHG+HGHG try not to use fresh bunch otherwise slice energy before/after uncorrelated could use fresh bunch and just accept phase offsets from after delay extra harmonic seed 1 seed 2 self-modulate, 8 nm radiator at 2 nm modulator modulator 7 undulators to saturation G. Penn 24
45 Results at 2 nm no wakes equivalent to R FEL ~ -1.5 µm average power (MW) 4 35 3 25 2 15 1 5 from start of final radiator 3 compare to HGHG, 23 µm 5 1 15 2 25 3 z (m) 6. 1 9 2 5. 1 9 output phase (radian) 1-1 -2-3 # photons / mev 4. 1 9 3. 1 9 2. 1 9 1. 1 9-4 -14-12 -1-8 -6-4 -2 t (fs) 618 618.5 619 619.5 62 62.5 621 621.5 photon energy (ev) G. Penn 25
output phase (radian) 4 3 2 1-1 -2-3 Results at 2 nm compare to perfect bunching at start of radiator R FEL for radiator alone is ~ 3 µm improvement ~ full sim radiator only sin(πl shift /L E ) 3 µm 2 µm # photons / mev 6. 1 9 5. 1 9 4. 1 9 3. 1 9 2. 1 9 1. 1 9 full sim radiator only -4-14 -12-1 -8-6 -4-2 t (fs) 618 618.5 619 619.5 62 62.5 621 621.5 photon energy (ev) G. Penn 26
Quick Summary rough scalings for different configurations: HGHG or OK, Δθ ~ (λ seed /λ out ) Δη/η M > (λ seed /λ out ) Δη/ρ want Δη < ρ / (λ seed /λ out ) selfseed or radiator, Δθ ~ (# gain lengths) Δη/3ρ want Δη < ρ 3 / (# gain lengths) EEHG, Δθ ~ (λ x /λ out ) Δη/η M1 à Δη < η M1 (λ out /λ x ) other slice variations, Δθ ~ (# gain lengths) Δρ/3ρ one region reaches saturation first: ~ 1 radian energy loss ~ ρ over a few gain lengths slice energy spread barely has any effect G. Penn 27 killer app for echo scheme?