Simulations of X-ray production for different undulator options 19 th of March 2015 Juergen Pfingstner

Content 1.Simulation setup and undulator layout 2.SASE simulations 3.Performance optimization 4.Conclusions and further work

1. Simulation setup and undulator design

The simulation code GENESIS Most popular code to simulate beam – light interaction in undulator. Very different scales of objects make standard EM codes impractical: -Undulator L g ≈ 1 m to 100 m. -Beam σ z ≈ 10 μm to 1 mm. -Light λ l ≈ 0.1nm to 10 μm. -12 orders of magnitude at LCLS. Problem is solved with several approximations: Beam: Period-averaged equations of motion (larger step size). Light: Par-axial approximation (no fast oscillating terms). Beam-light: Co-moving grid (no global grid). Several additional features: Beam creation or loading in different ways. Model for SASE process. MPI parallelized. Tools to implement advanced seeding schemes (to be implemented soon).

Simulation variants Steady-state simulation Beam is assumed to be infinitely. Therefore, only one micro- bunch has to be simulated with periodic boundary conditions. Very fast simulations (about 1 minute). Good approximation if beam is seeded with perfect laser, but not for SASE. Used for basic design especially with parameter scans. Time-dependent simulation: This means longitudinal changes of the beam. Many slices (order 10 4 ) along are simulated to resolve longitudinal behavior. Long calculation time (days on normal computer, hours on our super computer). Allows to simulate the SASE process. Used for final verification.

Used electron beams CLIC15: parameters from A. Aksoy -ε = 2.7μm -N = 250pC -σ z (full): 23μm (77fs) - = 3.3kA -E = 6.4GeV B1: full beam from A. Latina -ε = 3.2μm -N = 250pC -σ z (full) = 32μm (107fs) - = 2.4kA -E = 6Gev B5: full beam from A. Latina -Same as B1 but … -σ z (full) = 24μm (80fs) - = 3.1kA Comments: ε = ε x = ε y. CLIC15 was modeled Gaussian transversally and uniformly distributed longitudinally.

Undulator design and model Basic layout of one undulator section: Parameter choice: All length have to be chosen as multiples of λ u in simulations. λ u = 1.5 mm λ l = 1 Å (K u ≈ 1.4) L u = 4 m L g = 0.72 m L Q = 0.12 m β avg = 15 m B Q ≈ 16 T/m (dipole focusing not included yet). 13 Sections (length 57m) GENESIS model: 1.No gap: Basic functionality; focusing and undulator field interleaved (not the case in reality) and repeated. 2.Gap: External magnet file: possibility to model undulator and focusing field separately.

Steady-state simulations for different undulator setups (CLIC15) Two variants are used for the simulations: no gap and gap. With the gap the saturation length is longer (45m) than for no gap (35m). Difference more than gaps length. Probably reason: there is no beam - X-ray interaction in the gap. Hence, X-rays move faster than beam (phase error). This could be corrected with phase shifters (see later) m m Seed laser power: 3.2kW

Transversal X-ray intensity Field intensity in the centre (near field, dominant mode) is highly saturated to see also far field. Near field is about the same size as the electron beam size (20μm radius). Simulations resolve intensity variations, but assume the same frequency (Paraxial approximation).

Influence of space charge, energy loss and energy spread growth The increase of energy spread Δδ due to SR has negligible influence. The energy loss ΔE along the undulator due to SR has significant impact. Also space charge effects (SC) can be neglected. Therefore, the space charge effect has been turned off (much faster).

2. SASE simulations

SASE vs. seeded FEL (CLIC15) 10kW laser used for seeding. About equivalent to SASE. SASE and seeded (steady state) simulation give different results in saturation regime. Possible explanation: SASE contains more frequencies (due to phase jumps) and some parts are detuned (higher power, see later).

Power along electron bunch for SASE Coherence length Taken from “Free-Electron Lasers in the Ultraviolet and X- Ray Regime”, Schmuesser, Dohlus, Rossbach, Behrens. Charge profile is perfectly uniform along s (CLIC15). Typical spikes are visible due to SASE process with only locally coherent X- rays (shot noise). Model: Overlapping section of coherent light. New frequencies due to phase jumps.

X-ray spectrum and bandwidth Longitudinal spectrum of the X-rays at the end of the undulator. Number of spikes depend on coherence length. Width of spikes depends on beam length. Form the spectrum the relative X-ray bandwidth can be calculated. Saturation region: bandwidth has same value as FEL param. (as expected).

CLIC15 and B1 performance (steady- state/seeded) CLIC15 performs much better than B1. Main difference is the higher beam current. Also the smaller emittance helps. Andrea created right away a B5, which is shorter to increase the current. However, it is not clear if CLIC15 parameters are realistic or assumed.

Simulation of B5 (shorter bunch) Shape of SASE curves are different than steady-state. B5 is only very slightly better than B1. B5 is shorter than B1, but current is mainly the same apart from head and tail. At head and tail also energy spread has been increased (not efficient).

Necessary undulator length Minimum length: For proposed beam parameter, saturation length is about 35m. Small emittance increase prolongs also the saturation length. Therefore, the active undulator length should be probably not shorter than 40m (50m full). More relaxed length: To have a bit more margin and to be able to play with tapering and phase shifts (see next part) a bit longer undulator would be preferable, e.g. 50m (62m full). This could increase power (about factor 5 seems possible). SwissFEL uses 46m active undulator length (57 full). This seems to be a very good compromise.

3. Performance optimization

Detuning simulations (seeded) Seeding laser wave length determines wave length of X-rays (λ l = λ s ), since FEL acts as light amplifier. However, K u, λ u and γ determine the resonance wave length λ R of the FEL. Usually choice: λ R = λ s. But slight detuning (via beam energy γ) of the undulator to λ R < λ s improves FEL performance.

Explanation of detuning Taken from “Free-Electron Lasers in the Ultraviolet and X- Ray Regime”, Schmuesser, Dohlus, Rossbach, Behrens. From FEL theory the left curves can be derived. In the low gain regime (very small micro-bunching), no light amplification for λ R = λ s. In high gain regime (strong micro-bunching) the maximum gain is close to λ R = λ s, but at slightly detuned. This also explains why SASE performs better than a seeded FEL in terms of power. Some frequency components due to shot noise are detuned.

Tapering If the undulator strength K(Z) is weakened along the FEL, the output power can be strongly increased. This is usually done by changing the gap size of the undulator. With a linear taper (starting at m) the power can be increase from 1GW to 30-40GW. With a quadratic taper 50GW can be reached. Tapering is not the standard in nowadays FELs, but first experiments have been made and it is foreseen in LCLS2. Implications for the operation of the FEL and tolerances have to be investigated.

Explanation of tapering Taken from “Free- Electron Lasers in the Ultraviolet and X-Ray Regime”, Schmuesser, Dohlus, Rossbach, Behrens. Many papers claim tapering compensates for the energy loss of the beam. But this is only a small effect. More important is that micro-bunches are kept in the right have of the FEL bucket, where they transfer energy to the light wave. Tapering moves bucket to the left.

Phase shifters Phase shift between beam and X-rays in gaps is unavoidable: 360 o = 1 Å. Phase shifters (magnetic chicane) are used to shifty beam. Sometimes there are individual magnet designs, sometimes they are based on the magnets of the undulators. First and second field integral determine properties of phase shifter. Taken from: “A PERMANENT MAGNET PHASE SHIFTER FOR THE EUROPEAN X-RAY FREE ELECTRON LASER” by H. H. Lu et al.

Necessity of phase shifters Different phase shifts are tested in simulations. Significant changes in power are observable. It is possible to do better than perfect alignment 0 o. 100 o seems to be optimal. Explanation: Phase shift moves micro-bunch to optimal FEL bucket position. Similar effect as tapering. But systematic studies necessary. Phase shifts of several wavelengths are used in novel FEL ideas as iSASE and mode-locking.

4. Conclusions Got started with theory and simulations (GENESIS) of XFELs. First results: – Preliminary undulator design – Simulations of seeded and SASE FELs with beams from linac team. Collaborative efforts with the linac team: – Suggestions for the optimization of the e- beam. – Suggestions were right a way realized, but no improvement yet. – Linearity of bunch compressors will be crucial. Derived first hardware specifications (phase shifters) and looked into tapering and detuning.

Further work Review of X-ray experiments to define (or deliver input) the best wave length range. Soft X-ray design and integration of energy scans into the design. Detailed design on the hardware level: – Type of undulator, quadrupole, BPM, phase shifter. – Derive hardware specifications and tolerances. – Include wake fields in simulations. Derive cost model. Take a look at advanced concepts: seeding and shorter X- ray pulses. Evaluate interest and cost.

Many thanks for your attention!