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Longitudinal Matching, or How to move things around by a hairs-width at the speed of light….

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DC Gun SRF Linac Dump IR Wiggler Bunching Chicane Longitudinal Matching Scenario Requirements on phase space: high peak current (short bunch) at FEL – bunch length compression at wiggler using quads and sextupoles to adjust compactions “small” energy spread at dump – energy compress while energy recovering – “short” RF wavelength/long bunch, large exhaust p/p (~10%) get slope, curvature, and torsion right (quads, sextupoles, octupoles) E E E E E E

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Why a long bunch at injection? Space Charge… Longitudinal Space Charge…

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Space Charge – Esp. LSC – Down Linac Had a number of issues in linac during commissioning: – Why was the bunch “too long” at the wiggler? bunch length at wiggler “too long” even when fully “optimized” (with good longitudinal emittance out of injector) – could only get fsec rms, needed 200 fsec – Why did the “properly tuned lattice” not fully compress the bunch? M 55 measurement showed proper injector-to-wiggler transfer function, but beam didn’t “cooperate”… minimum bunch length at “wrong” compaction – Why was the beam momentum spread asymmetric around crest? dp/p ahead of crest ~1.5 x smaller than after crest; average ~ PARMELA We blamed wakes, mis-phased cavities, fundamental design flaws, but in reality it was LSC… PARMELA simulation (C. Hernandez-Garcia) showed LSC-driven growth in correlated & uncorrelated dp/p; magnitudes consistent with observation Simulation showed uncorrelated momentum spread (which dictates compressed bunch length) tracks correlated (observable) momentum spread

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Space-Charge Induced Degradation of Longitudinal Emittance Mechanism: self-fields cause bunch to “spread out” – Head of bunch accelerated, tail of bunch decelerated, causing correlated energy slew Ahead of crest (head at low energy, tail at high) observed momentum spread reduced After crest (head at high energy, tail at low) observed energy spread increased – “Intrinsic” momentum spread similarly aggravated (driving longer bunch) Simple estimates => imposed correlated momentum spread ~1/L b 2 and 1/r b 2 – The latter observed – bunch length clearly match-dependent – The former quickly checked…

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Solution Additional PARMELA sims (C. Hernandez-Garcia) showed injected bunch length could be controlled by varying phase of the final injector cavity. – bunch length increased, uncorrelated momentum spread fell (but emittance increased) – reduced space charge driven effects – both correlated asymmetry across crest and uncorrelated induced momentum spread When implemented in accelerator: – final momentum spread increased from ~1% (full, ahead of crest) to ~2%; – bunch length of ~800–900 fsec FWHM reduced to ~500 fsec FWHM (now typically 350 fsec) – bunch compressed when “decorrelated” injector-to-wiggler transfer function used (“beam matched to lattice”)

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Happek Scan

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Key Points “Lengthen thy bunch at injection, lest space charge rise up to smite thee” (Pv. 32:1, or Hernandez-Garcia et al., Proc. FEL ’04) “best” injected emittance DOES NOT NECESSARILY produce best DELIVERED emittance! LSC effects visible with streak camera E t E t

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Streak Camera Data from IR Upgrade -5 o -6 o 0o0o -1 o -2 o -3 o -4 o (t,E) vs. linac phase after crest (data by S. Zhang, v.g. from C. Tennant)

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+5 o +6 o 0o0o +1 o +2 o +3 o +4 o Streak Camera Data from IR Upgrade (t,E) vs. linac phase, before crest asymmetry between + and - show effect of longitudinal space charge after 10 MeV (data by S. Zhang, v.g. from C. Tennant)

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±4 and ±6 degrees off crest “+” on rising, “-” on falling part of waveform L bunch consistent with dp/p and M 56 from linac to observation point dp/p(-)>dp/p(+) on “-” side there are electrons at energy higher than max out of linac distribution evolves “hot spot” on “-” side (kinematic debunching, beam slides up toward crest…) => LSC a concern… +4 o -4 o -6 o +6 o

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Longitudinal Matching: The Concept Basic idea: turn the long, low momentum spread bunch that comes out of the injector (maybe with chirp and curvature) into short bunch at the FEL How? Parallel-to-point image in longitudinal phase space – Low momentum spread/long => “parallel” beam – Put beam through lens (linac, off crest longitudinal focusing) and then a longitudinal drift (beamline with compaction, M 56 ≠0 injected beam linac transport with nonzero compaction wiggler

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Longitudinal Matching: The Math Injected beam – with chirp and curvature { Linac acts on injected beam { Recirculator acts on accelerated beam { E z zz E=Cz+Kz 2 EE eoeo t E

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Can expand E out -E o in powers of z/ RF (small parameter) E out -E o = E z[C- E(2 / RF ) sin o ] + z 2 [K-½ E(2 / RF ) 2 cos o ] + … and shovel into z WIG, group by powers of z and E/E o (another small parameter) to get z WIG = M 56 ( E/E o ) + T 566 ( E/E o ) 2 + z {1 + M 56 [(C/E o )-( E/E o )(2 / RF ) sin o ] +2 T 566 ( E/E o ) [(C/E o )-( E/E o )(2 / RF ) sin o ] } + ( z) 2 { [(K/E o )-½ ( E/E o ) (2 / RF ) 2 cos o ][M T 566 ( E/E o )] + T 566 [(C/E o )-( E/E o )(2 / RF ) sin o ] 2 } + …

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z WIG thus depends only on powers of ( E/E o ) – i.e. very small numbers - and a polynomial in z. Proper choice of parameters (M 56, T 566, o, C, K,…) can make those terms very small – or vanish – making z WIG independent of z – i.e., all lengths at injection go to zero offset at the wiggler: the bunch is compressed, giving high peak current Simple Case Take C=0, K=0 and E very small (i.e., make the injector designer’s life hard…) z WIG = z {1 - M 56 [( E/E o )(2 / RF ) sin o ]} + ( z) 2 { [-½ ( E/E o ) (2 / RF ) 2 cos o ]M 56 + T 566 [( E/E o )(2 / RF ) sin o ] 2 } + …

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Take And get z WIG = 0 – a very compressed (and curvature corrected) bunch… Numerical Example (JLab UV FEL Driver) E o =135 MeV, o =-10 o, E=(135-9)/cos 10 o = 128 MeV, RF = 0.2 m M 56 = (0.2/2 )(135/128)(1/sin (-10 o )) = m T 566 = ½ (2 /0.2) 2 (128/135) cos(-10 o )( m) 3 = m

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Summary: Longitudinal Match to FEL Inject long, low-energy-spread bunch to avoid LSC problems – need o rms with 1497 MHz 135 pC in our machine Chirp on the rising part of the RF waveform – counteracts LSC – phase set-point then determined by required momentum spread at wiggler Compress (to required order, including curvature/torsion compensation) using recirculator compactions M 56, T 566, W 5666,… Entire process generates a parallel-to-point longitudinal image from injector to wiggler

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Energy Recovery Can do the same thing to compress energy spread from FEL during energy recovery. Bunch length compression: long bunch + small dp/p => chirped beam => compression by momentum compaction Energy compression: short bunch + large dp/p => decompress using momentum compaction => differential energy recovery to compress energy spread

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Longitudinal Matching Schematic Longitudinal Matching for ERL-Driven FEL E E E “oscillator” “amplifier” E E injector dump wiggler linac E

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Energy Recovery: Details Longitudinal Match to Wiggler Inject long, low-energy-spread bunch to avoid LSC problems – need o rms with 1497 MHz 135 pC in our machine Chirp on the rising part of the RF waveform – counteracts LSC – phase set-point then determined by required momentum spread at wiggler Compress (to required order, including curvature/torsion compensation) using recirculator compactions M 56, T 566, W 5666,… Entire process generates a parallel-to-point longitudinal image from injector to wiggler

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Longitudinal Match to Dump FEL exhaust bunch is short & has very large energy spread (10-15%) => Must energy compress during energy recovery to avoid beam loss linac during energy recovery; this defines the longitudinal match to dump – Highest energy must be phase-synchronous with (or precede) trough of RF wave-form – Transport momentum compactions must match the slope (M 56 ), curvature (T 566 ), torsion (W 5666 ),… of the RF waveform Recovered bunch centroid usually not 180 o out of phase with accelerated centroid – Not all RF power recovered, but get as close as possible (recover ahead of trough), because… – Additional forward RF power required for field control, acceleration, FEL operation; more power needed for larger phase misalignments For specific longitudinal match, energy & energy spread at dump does not depend on lasing efficiency, exhaust energy, or exhaust energy spread – Only temporal centroid and bunch length change as lasing conditions change The match constitutes a point-to-parallel image from wiggler to dump

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Energy Compression Beam central energy drops, beam energy spread grows Recirculator energy must be matched to beam central energy to maximize acceptance Beam rotated, curved, torqued to match shape of RF waveform Maximum energy can’t exceed peak deceleration available from linac – Corollary: entire bunch must preced trough of RF waveform E t E t All e - after trough go into high- energy tail at dump E t

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Higher Order Corrections Without nonlinear corrections, phase space becomes distorted during deceleration Curvature, torsion,… can be compensated by nonlinear adjustments – differentially move phase space regions to match gradient required for energy compression E t Required phase bite is cos -1 (1- E FEL /E); this is >25 o at the RF fundamental for 10% exhaust energy spread, >30 o for 15% –typically need 3 rd order corrections (octupoles) –also need a few extra degrees for tails, phase errors & drifts, irreproducible & varying path lengths, etc, so that system operates reliably In this context, harmonic RF very hard to use…

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JLab IR Demo Dump core of beam off center, even though BLMs showed edges were centered (high energy tail)

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Nonlinearity Control Validated By Measurement Figure 1: Inner sextupoles to g-cm and trim quads to -215 g Figure 2: trim quads at -185 g with same sextupoles Figure 3: trim quads at -245 g Figure 4: quads at -215, but sextupoles 3000 g below design, at g-cm Figure 5: where we left it: trim quads -215 g sextupoles at g-cm arrival launch

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Injector to Wiggler Transport

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If you do it right linac produces stable ultrashort pulses Can regularly achieve 300 fs FWHM electron pulses ~150 fsec rms

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Injector to Reinjection Transport

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