research papers
Laserslicing at a lowemittance storage ring
^{a}Elettra – Sincrotrone Trieste SCpA, Basovizza, 34149 Trieste, Italy, and ^{b}Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA
^{*}Correspondence email: simone.dimitri@elettra.eu
Laserslicing at a diffractionlimited storage ring light source in the soft Xray region is investigated with theoretical and numerical modelling. It turns out that the slicing efficiency is favoured by the ultralow beam emittance, and that slicing can be implemented without interference to the standard multibunch operation. Spatial and spectral separation of the subpicosecond radiation pulse from a hundreds of picosecondlong background is achieved by virtue of 1:1 imaging of the radiation source. The spectral separation is enhanced when the radiator is a transverse gradient undulator. The proposed configuration applied to the Elettra 2.0 sixbend achromatic lattice envisages total slicing efficiency as high as 10^{−7}, one order of magnitude larger than the demonstrated stateoftheart, at the expense of pulse durations as long as 0.4 ps FWHM and average laser power as high as ∼40 W.
Keywords: storage ring; diffraction limit; laser slicing; undulator radiation; synchrotron light source; radiation background.
1. Introduction
The last decade (∼2010 to date) has seen a renaissance of accelerator physics studies and technological advances in synchrotron light sources, aiming to improve the average spectral ). Such a breakthrough in spectral and coherence comes, however, at cost of bunch durations typically longer than ∼15 ps r.m.s. (Martin, 2011). The natural bunch length is primarily set by the radiofrequency of the accelerating cavities, usually operating at 100 or 500 MHz. Additional bunch lengthening, on top of the aforementioned value and usually by a factor of two to four, is commonly induced by the electric field of a higherharmonic cavity with the aim of suppressing either transverse or longitudinal beam instabilities, and increasing the beam Touschek lifetime. Bunch lengthening is mandatory in DLSRs also for minimizing the transverse emittance growth otherwise induced by intrabeam scattering in high charge density bunches (Leemann, 2014). Increasing the spectral and the fraction of coherent photons in the Xray wavelength range at the expense of time resolution is therefore a general trend of lowemittance rings.
and the transversally coherent fraction of photons by several orders of magnitude, in both the soft and hard Xray wavelength ranges. A fourthgeneration of lowemittance or diffractionlimited storage rings (DLSRs) has been announced (Borland, 2014In the preceding decade (∼2000–2010), studies aimed in the opposite direction flourished: production of few tens of femtosecond long photon pulses in storage rings was proposed by Zholents & Zolotorev (1996), studied for implementation at several facilities (Zholents & Zolotorev, 1996; Ingold et al., 2001; Nadji et al., 2004; Yu et al., 2011; Lau et al., 2012; Prigent et al., 2013), and eventually implemented at a few (Schoenlein et al., 2000a,b; Khan et al., 2006; Ingold et al., 2001; Beaud et al., 2007; Holldack et al., 2014; Labat et al., 2018). This approach targeted timeresolved experiments requiring low photon pulse energy and moderate average For example, investigating reversible dynamics in molecular systems of different materials requires a nondestructive photon–matter interaction. Even in the case of samples surviving relatively high space charge effects in up to ablation may seriously compromise the collection of information about the sample properties. Storage rings therefore look like suitable light sources for experiments requiring a reduced peak but accompanied by pulse repetition rates up to hundreds of MHz. Wide wavelength tunability can be provided, e.g. by dipole magnets and variablegap insertion devices (IDs) in a series.
The next generation of Xray freeelectron lasers driven by superconducting linear accelerators will be able to offer similar performance at the expense of an attenuation of the radiation peak et al. (2019).
by many orders of magnitude, and up to 1 MHz repetition rate in continuouswave mode. The wavelength tunability in freeelectron lasers is typically limited to a few percent whereas, for example, extended Xray absorption finestructure and similar experimental techniques must access a far wider spectral range. For an exhaustive comparison of most advanced light sources, which is out of the aim of this article, we kindly send the reader to SchoenleinIn spite of some successful implementations of femtoslicing at presently running thirdgeneration synchrotron light sources, slicing has recently been given ever lower priority with respect to electron optics designs for diffractionlimited light sources. This trend is motived by the limited average
per unit relative bandwidth typically achieved with femtoslicing, and by the conflict of the relatively large dispersion function required for spatial and/or angular filtering of the short radiation pulse with the typically small dispersion of diffractionlimited optics designs.This work aims to demonstrate the possibility of improving the total slicing efficiency (and thereby the average spectral ; Karantzoulis, 2018), is considered as a case study. In Elettra 2.0, the dispersion function naturally excited in the arc cell is large enough to laterally separate by ∼0.5 mm the short radiation pulse from the hundreds of picosecondlong background. By imaging the radiation source with a 1:1 focusing system, the two light spots can be visualized downstream of the radiator; only one of the two is eventually selected for propagation to the sample, e.g. with a movable slit. The signaltonoise ratio (SNR) at the sample is additionally enhanced by the relative spectral separation of the radiation emitted in a transverse gradient undulator (TGU). A total slicing efficiency of ∼10^{−7} [present state of the art is 10^{−8} (Holldack et al., 2014) and typically approaching 10^{−9}] can be obtained with no impact on the lattice design; that is, the wigglermodulator and the TGUradiator can be installed without interference with the machine layout. In principle, the 12fold, sixbend lattice of Elettra 2.0 is suitable for the simultaneous operation of additional (up to 12) beamlines in short radiation pulse mode.
of the short radiation pulse) by one to two orders of magnitude with respect to existing solutions, at the expense of five to ten times longer (still subpicosecond) laser pulses and about four times higher average laser power (in the range 30–60 W). This work also shows that the implementation of laserslicing can be made transparent to the standard multibunch operation and optics design of a DLSR in the soft Xray region. The lowemittance optics design of Elettra 2.0, the current design of which is based on a sixbend achromatic cell (Karantzoulis & Barletta, 2019The theoretical framework for the electron beam dynamics in the presence of laserslicing in a storage ring was illustrated in the seminal work of Zholents & Zolotorev (1996). Here we elucidate the scaling laws of the laserinduced energy spread with laser sizes, electron beam sizes and wiggler length in a diffractionlimited storage ring (Subsection 2.1). We then discuss the different sources of radiation background from sliced and nonsliced electrons, either in the same bunch or in adjacent ones, and the main actions to be taken to minimize it. This discussion includes the proposal of a TGUradiator and specifications for detector temporal gating (Subsections 2.2 and 2.3). Section 3 presents laserslicing in Elettra 2.0. The electron beam dynamics and the effect of laserslicing are followed along the ring and on successive turns. Three possible scenarios are envisaged as a function of the laser and average laser power. Our findings are compared with results in the existing literature, both theoretical and experimental. We offer our conclusions in Section 4. We send the reader to Appendix A for explicit analytical expressions derived ab initio for the laserinduced energy modulation amplitude, the r.m.s. energy spread and the photon There, we put in evidence the approximations carried out by Zholents & Zolotorev (1996) and their range of validity, in comparison with expressions with exact numerical factors and including threedimensional particle beam sizes and laser diffraction. The simulation studies presented in this article were first set on the basis of the expressions in Appendix A, and found in good agreement with the theoretical predictions.
2. Theoretical background
2.1. Scaling laws in lowemittance rings
In laserslicing (Zholents & Zolotorev, 1996) an electron bunch interacts with an external, shorter laser pulse in a wiggler magnet installed in a dispersionfree section of a storage ring. Downstream, the laserinduced energy modulation translates into transversally spatial separation of the sliced electrons with respect to the beam core by virtue of a nonzero dispersion function. Here, electrons radiate in an undulator. Owing to the spatial and/or angular separation of sliced and nonsliced electrons, the short and the long radiation pulses can be separated and, depending on the photon transport layout, either the short or the long pulse reaches the detector.
In general, the higher the laserinduced energy modulation in the wiggler, the smaller the number of electrons pushed to large energy offsets, ΔE/E. However, a large energy offset is required to discriminate the short from background (this is the radiation from the beam core of the sliced bunch, as well as radiation from other bunches). This observation suggests a compromise between radiation and SNR. The experimental results collected so far suggest a value for the ratio of the laserinduced energy modulation over the equilibrium energy spread p = > 5, and a ratio of the induced energy spread over the equilibrium energy spread larger than 3. These values typically translate into energy modulation amplitude in the range 0.5–1.0%. Since the onaxis energy modulation depends upon the transverse sizes of neither the laser spot nor the electron beam, the laser–electron interaction should be designed in a way that maximizes the laserinduced energy spread for any given energy modulation amplitude. For a deeper look, we consider equation (10) in Appendix A in the approximation K ≫ 1. For any given laser pulse energy and laser wavelength, we find the following scaling law,
where we took advantage of the wiggler resonance condition. The relative energy spread shows, of course, the same dependence on K, electron beam energy and laser Its additional dependence on the electron beam and the laser beam transverse size can be discussed for two alternative scenarios, when either the laser spot is much larger than the electron beam or the laser is transversally matched to the electron beam envelope,
In the former case, equation (2a), a relatively long wiggler favours the accumulation of energy spread. Long wiggler periods (>10 cm) are allowed, which are compatible with relaxed Kvalues (<20), i.e. a weaker magnetic field (<2 T). The upper limit to the energy spread is given by the available longitudinal space for the wiggler, which can be several meters long (3–5 m). In the latter case, equation (2b), a squeezed laser size (<200 µm), corresponding to a very short Rayleigh length, forces the wiggler length to submeter length. This choice implies a short period (∼5 cm) and high Kvalues (>20), likely consistent with fields only available in superconducting or cryogenic devices (>2 T). Note that a higher electric field at the laser waist excited by a small laser transverse size does not necessarily imply a higher pvalue: in fact, it would determine a larger energy spread if the wiggler were short enough to keep its length comparable with the Rayleigh length. Otherwise, the expected gain in energy spread is diminished by the reduced coupling efficiency along the wiggler. Making the wiggler longer would not help much because the laser is diffracting rapidly, thus weakly coupling to the electrons far from the waist location.
Both expressions in equation (2) show that, by virtue of smaller electron beam sizes and for the same wiggler and laser parameters, a lowemittance storage ring tends to increase the laserinduced energy spread with respect to thirdgeneration light sources. In particular, the laser electric field can be locally enhanced by a smaller laser spot along the wiggler. A laser size as small as 500 µm r.m.s. (∼0.25 mm × 0.25 mm compared with ∼1 mm × 1 mm laser transverse section at present femtoslicing facilities) still generates an electric field that is sampled uniformly by the electrons. In this way a doublehorn energy distribution of the sliced electrons can be generated, as shown in the next section. Such a distribution determines, in turn, a larger fraction of sliced electrons at the maximum offenergy level with respect to that induced by a laser size perfectly matched to the electron beam size. Finally, a lower emittance favours the spatial/angular separation of the short pulse radiation from the whole bunch emission, i.e. a higher SNR.
2.2. Spatial, angular and spectral separation of emitted radiation
The spatial and angular separation of the radiation emitted by the sliced electrons entering a dipole magnet or an undulator is determined by the value of the energy dispersion function and its longitudinal derivative at the radiator,
where δ_{min} is intended to be the minimum relative energy offset within the energy distribution of lasersliced electrons.
The separation of radiation emitted by the lasersliced electrons from that emitted by all other electrons can be increased by the spectral difference of photons emitted, e.g. in a TGU, the field gradient being in the horizontal plane. A monochromator installed downstream of the TGU would then contribute to select the short radiation pulse, in addition to the spatial/angular separation described above. This separation increases the SNR with respect to the pure spatial/angular collimation of the radiation. A TGU is characterized by a magnetic field gradient = . We recall that, for K ≫ 1, the undulator resonance condition leads to (1/N_{u}) = . To have a net spectral separation, we require that the difference in photon wavelength with respect to the central undulator wavelength due to the lateral displacement of the particles be larger than the natural bandwidth of undulator radiation evaluated as if it were a standard undulator: , or, equivalently,
Inserting the definition of the pparameter, equation (4) becomes
The adoption of a TGU in a dispersive region introduces two perturbations to the beam dynamics. First, the electron optics and the closed orbit are distorted. Second, the damping partition numbers, i.e. the damping times, and the beam equilibrium emittances are modified. These considerations have been taken into consideration in the following study for Elettra 2.0.
2.3. Background radiation
Different types of radiation background and techniques to minimize it are described in Table 1. The background of type a and c (`core' background) is suppressed primarily by spatial and/or angular separation of the radiation. The same separation method applies to radiation background of type b from electrons at large betatron amplitudes (`tail' background). This method typically requires an angular separation of ∼1 mrad at the source, corresponding to several millimeters at the frontend of the beamline, where a slit can be applied to mask the unwanted radiation. Background of type b is also minimized by a small natural emittance (for any given betatron amplitude, fewer electrons reach large lateral and angular offsets from the core) and by spectral filtering of the emitted radiation with a monochromator (lasersliced, offenergy electrons emit at a slightly different central wavelength than onenergy electrons). Identical considerations apply to the background of type d.

The background of type e (`halo' background) is due to radiation emitted by those electrons that, after interacting with the laser, maintain their offenergy amplitude of transverse oscillation over a time scale shorter than the transverse damping time, τ_{x}. In one turn, those electrons shift longitudinally by ∼1 ps or so by virtue of the storage ring momentum compaction and of their energy offset. Hence, the radiation emitted at turns successive to the interaction with the laser should not be collected. It has been demonstrated (Streun, 2003; Schick et al., 2016) that alternating interactions with M bunches in the ring (slicing in `sequence mode') allows one to increase the repetition rate of sliced pulses up to the level of ∼100 Hz × M, provided that fast shutters or gating of the detector with rise and fall time shorter than ∼432/M ns (864 ns is the revolution period in Elettra 2.0) are available to block halo radiation from the other bunches. Stateoftheart synchronization of femtosecond laser amplifier with the storage ring allows the slicing process to alter between three bunches on the multibunch train with a mutual temporal delay of 12 ns (Holldack et al., 2014). We propose to push this scheme to four or ten bunches devoted to slicing, and therefore we will require gating rise and fall time of the order of 40–100 ns.
As explained by Streun (2003), chromatic filamentation, due to the unavoidable nonlinear optics elements in the ring lattice, contributes to generate a uniform halo of particles around the beam core at time scales shorter than τ_{x}. On the one hand, the formation of a halo is expected to generate background of constant intensity. On the other hand, filamentation randomizes the electrons' position and angle at the source point such that, in a realistic scenario of finite spatial and angular acceptance of the beamline, the background signal intensity collected at the detector tends to oscillate, while being exponentially suppressed overall on a time scale intermediate to the revolution period (∼1 µs) and the damping time (∼10 ms) (Holldack et al., 2014; Schick et al., 2016). We estimate the condition for full filamentation by the relative betatron phase advance and the particle relative energy offset, Δμ_{x}δ ≥ 1. If T_{riv} is the revolution period, T_{fil} is the time needed to achieve full filamentation, and Q_{x} = is the betatron tune, we can rewrite that condition
For parameters typical in a <3 GeV lowemittance ring such as Q_{x} ≃ 30, ≃ 0.4% and T_{riv} ≃ 1 µs, we estimate T_{fil} ≥ 1 µs. We expect an overall transverse damping of the background signal on a time scale intermediate to T_{fil} and τ_{x}, e.g. a fraction of a millisecond or so. This order of magnitude matches the observations reported by Holldack et al. (2014) and Schick et al. (2016). In conclusion, the repetition rate of short radiation pulses generated by the same bunch and detected at one beamline with much reduced background signal is expected to be in the frequency range 1–10 kHz. This single pulse repetition rate is then multiplied by the number of sliced bunches in the stored train. The effective total repetition rate will define the laser frequency, if not additionally constrained by the maximum practical average power of the laser.
3. Laserslicing at Elettra 2.0
3.1. Singlebunch dynamics
In this section we analyse the dynamics of a single electron bunch through the wiggler (modulator) and a downstream undulator (radiator) in the Elettra 2.0 lattice. We assume radiation emitted in the keVphoton energy range. We then follow the beam dynamics by particle tracking for three consecutive turns. The wiggler is assumed to be installed in one of the 4.62 mlong dispersionfree straight sections of Elettra 2.0. The radiator is installed in a 1.68 mlong dispersive region inside the cell. This arrangement does not require any modification to the lowemittance optics of Elettra 2.0. Moreover, the ∼3 mlong modulator can share the straight section with an additional ∼1 m short undulator, e.g. in canted configuration. The radiator is installed in a short dispersive straight section typically not used by IDbased photon beamlines. No additional dipole magnets or correctors are required for radiation separation, as we will see in the following.
A list of parameters of Elettra 2.0 and of the associated laserslicing setup is given in Table 2. At this stage of the study, the filling pattern is assumed to be uniform, i.e. all bunches have the same 0.8 nC charge and their separation is 2 ns (the RF frequency is 500 MHz). The bunch length at equilibrium, as reported in Table 2, is computed with the lengthening effect of a thirdharmonic cavity already in place. Radiation background is not considered in this section, which primarily aims to validate the analytical laserinduced energy modulation (see Appendix A) with particle tracking. Additional considerations about background and hybrid filling pattern will be discussed in Section 4.

The elegant code (Borland, 2000) was used to track a single electron bunch from the entrance of the wiggler to the same point after three turns. Fig. 1 shows the electron beamline from the dispersionfree straight section, where the wiggler is installed, to the end of the sixbend arc cell. The radiator is installed in the middle of the arc, where the dispersion function (blue line) is maximal. The transverse r.m.s. beam size at equilibrium in the wiggler and in the radiator is approximately 60 µm in the horizontal plane and 7 µm in the vertical (1% betatron coupling is assumed).
The effect of the interaction with a 0.8 ps FWHMlong, 3 mJenergy Gaussian and Fourier transform limited laser pulse is illustrated in Fig. 2. The peak energy deviation and the r.m.s. energy spread induced by the laser interaction are, respectively, nine and three times larger than the natural energy spread, in agreement with the analytical prediction. Since the laser spot size that maximizes the interaction with the electrons is approximately 550 µm (as predicted by tracking studies and in rough accordance with the theoretical prediction, see Appendix A), and hence much larger than the electron size, the electrons are collected into a doublehorn energy distribution (see inset in the top plot). This distribution is expected to facilitate suppression of background radiation from nonsliced electrons (see Section 3 later). On a singlepass basis, the interaction increases the projected value of the r.m.s. relative energy spread from 0.07% to 0.08%. After three turns, the sliced electrons are longitudinally shifted from the interaction region due to the momentum compaction of the storage ring. Fig. 3 shows how the longitudinal phase space evolves immediately after the laser interaction and for the successive two turns. Depletion of electrons from the region of laser interaction induces current ripples at its edges.
3.2. Coherent THz emission
The depletion of electrons one turn after the laser interaction generates a current ripple that emits coherent THz radiation. The depletion persists for a few thousand of turns, and its detection can be used to synchronize the laser pulse with the electron bunch, as predicted by Schoenlein et al. (2000a,b), experimentally observed by Holldack et al. (2006) and Byrd et al. (2006), and successfully applied at other facilities (Labat et al., 2018). Fig. 4 shows the bunch current profile (normalized to the peak value) simulated after a single laser interaction at successive points of maximum dispersion, in the middle of the arcs, and downstream of the wiggler. The plots show that the formation of a 1 pslong hole in the current profile is completed in one turn; i.e. the contrast of charge density becomes maximum and the current spikes become symmetric around the interaction region, one turn after the laser–electron interaction. Fig. 5 shows, at the top, the corresponding normalized current profile in the region of laser interaction, immediately after laserslicing, and for the successive two turns. The lower plot shows the Fourier transform of the longitudinal charge distribution (bunching factor) at each turn. Enhanced radiation in the frequency range 0.5–3 T Hz is expected.
In spite of the low momentum compaction of Elettra 2.0 (see Table 2) – momentum compaction in DLSRs is at least one order of magnitude lower than in thirdgeneration light sources – the reduction of the singlebunch threshold for the burst emission of coherent synchrotron radiation (microwave instability) (Venturini & Warnock, 2002; Bane et al., 2010) is alleviated by the relatively long bunch duration ensured by the high harmonic cavity. The 1D CSR (coherent synchrotron radiation) instability threshold in Elettra 2.0 at 2.0 GeV turns out to be ∼7 mA for the single bunch average current, where no 2D suppression of CSR emission by the vacuum chamber is taken into account. This estimate provides a safe margin for stable slicing operation with increased charge in sliced bunches.
3.3. Electron beam: spatial and energy distribution
The upper plot of Fig. 6 shows that for the electron beam and laser parameters of Table 2 the sliced electrons suitable for spatial and spectral separation of the emitted radiation are sitting at positive lateral coordinates, between 200 and 500 µm, with respect to the longitudinal axis of the beam, at the location of the radiator. The lower plot of Fig. 6 quantifies the percentage of offenergy particles with respect to the amount of particles onenergy. The total percentage of particles with a positive pfactor larger than five, i.e. energy modulation amplitude exceeding 0.4%, is approximately 0.5%. Fig. 7 illustrates the transverse phase space of the beam at the radiator for the first three consecutive turns after interaction with the laser. According to these results, we anticipate that a slit selecting lateral coordinates x > 250 µm in the proximity of the source point or, in a practical scenario, of its 1:1 image downstream of the radiator, would suffice for selecting sliced electrons with p > 5. A singleturn gated detector could then be used to eliminate radiation pulses longer than 1 ps, which are emitted by sliced electrons on successive turns.
Most of the shortpulse radiation is emitted by electrons with a relative energy deviation ΔE/E ≥ 0.5%. For an undulator field gradient of 100 T m^{−1} and a dispersion function of 0.06 m at the radiator, the relative variation of undulator parameter from the onaxis value sampled by the offenergy electrons would be 3%, i.e. the relative photon energy difference would be approximately 6%. For comparison, the same 25period undulator with no field gradient would provide a fullwidth spectral bandwidth of the order of 1/N_{u} = 4%. In that case, the separation in photon energy of the short pulse from background would only be due to the energy difference of the modulated electrons, i.e. 1%, and therefore within the natural bandwidth of the undulator.
3.4. Undulator radiation: spatial and spectral distribution
Radiation emission from a linearly polarized planar undulator was simulated with the SPECTRA code (Tanaka & Kitamura, 2001). This preliminary study aims to illustrate the scheme of suppression of the long radiation pulse (background) by virtue of the pure spatial separation of the short and long pulse, and removal of the long pulse by a slit. This provides a first estimate of the ratio of the two pulse intensities at the 1:1 focusing plane.
The undulator parameters are listed in Table 2; the electron beam and the laser parameters are those of Option 3 of Table 3 (see below). This takes into account a hybrid filling pattern of the stored bunched, i.e. higher bunch charge for bunches involved in the slicing. By virtue of a higher laser peak power – we now consider a 0.4 ps FWHMlong laser pulse instead of the 0.8 pslong pulse in Table 2 – the relative energy modulation amplitude becomes 0.9% and the horizontal distribution of sliced electrons at the radiator location extends up to 660 µm off the beam axis. Approximately 0.5% of the total bunch charge is the fraction of particles with energy deviation ΔE/E > 0.8%, and can be found at horizontal coordinates larger than 400 µm at the radiator.

The thirdharmonic spectral , the blue peak is the radiation pattern of the entire bunch train, averaged over one orbit; it is centred at x = −480 µm. A horizontal slit absorbs the radiation at horizontal coordinates x < −180 µm (or 300 µm from the undulator central axis). In red is the emission from a single sliced bunch, whose centre was chosen as the origin of the xaxis (see inset). The spatial distribution of the sliced electron bunch is assumed to be Gaussian with a standard deviation σ_{x} = 70 µm and dominated by the chromatic particles motion (the actual sliced electron distribution would be an asymmetric Gaussian still peaked where the short radiation pulse is in Fig. 8; such an asymmetry, however, is not that pronounced, as shown in Fig. 7). The ratio of the spatial spectral emitted by the sliced (single bunch) and nonsliced electrons (train of bunches) over one orbit and calculated on the emission axis of the short pulse (x = y = 0) is at least as large as 1000.
spatial distribution (949 eV photon energy) was calculated at the virtual source point. Such a spatial distribution can be replicated downstream with a 1:1 optical focusing system, with a horizontal slit placed at the focal plane. In Fig. 8In order to identify major distortions possibly induced by slope errors of the first mirror of the 1:1 focusing system, the radiation spatial distribution at the focal plane (where the slit is supposed to be installed) was simulated with the Shadow code (Cerrina & Rio, 2009). Radiation sources are still modelled as Gaussian distributions, which allows us to point out distortions or asymmetries of the radiation spot due to the mirror slope error. A cylindrical mirror working at 1° grazing incidence angle (α) and providing a 1:1 horizontal sagittal focusing (with 0.1 µrad tangential and 1 µrad sagittal r.m.s. slope error) is placed 7.5 m downstream of the radiator. Fig. 9 shows the simulated distributions on the focal plane, 15 m from the radiator. The choice of sagittal focusing has been dictated by the ratio of tangential and sagittal scattering, which is lower in the second case by a factor of about 1/tanα (Raimondi & Spiga, 2015). Heatload calculations reveal that a watercooling system with standard parameters at Elettra is sufficient for the mirror, and therefore no image quality degradation due to thermal effects is expected. This simulation does not highlight any serious concern for the slope error of 1 µrad r.m.s., and confirms the feasibility of the focusing scheme.
In addition, the scattering contribution originating from the microroughness of the mirror surface was evaluated with WISEr (Raimondi et al., 2013; Raimondi & Spiga, 2015). This allows us to quantify the contamination of the long pulse average tail onto the short pulse peak. The computation was performed in the worstcase scenario of longitudinal scattering, assuming a microroughness r.m.s. amplitude of 0.5 Å. Considering that sagittal scattering is lower by about a factor of 57 as compared with longitudinal scattering, we estimate that the ratio between the long pulse peak intensity and its sagittal scattering contribution ∼500 µm away from the optical axis is about 1.6 × 10^{10} (see Fig. 10). With a slicing efficiency of 10^{−7} (see Table 3 below), the predicted signaltonoise ratio (short pulse intensity over background) at such a position is therefore ∼1000 in a realistic situation and with a highquality mirror surface.
Spectral filtering of the background radiation can be added to the spatial filtering. In Fig. 11, the (normalized to peak value) observed along a direction at x = 480 µm from the radiator axis is calculated with SPECTRA for, respectively, onenergy nonsliced electrons (black), offenergy sliced electrons (red), and for the sliced electrons passing through a TGU with relative field gradient α = 100 m^{−1} (green). A pure horizontal field gradient is simulated in SPECTRA: though this model does not simulate in a consistent manner the focusing effect of the TGU on the electrons motion, it is able to correctly predict the spectral properties of the emitted radiation. The impact of the TGU focusing properties on the beam dynamics was treated with the elegant code, as reported below.
The central photon energy difference of radiation emitted by sliced and nonsliced electrons is increased from 1.5% for a standard undulator with uniform field to 9% for a TGU. Owing to the field gradient, the bandwidth of the signal emitted by the sliced electrons is expected to be larger than the bandwidth from a planar undulator. Nevertheless, since the horizontal size of the sliced electrons participating to most of the short pulse emission is of the order of 50 µm only, the actual bandwidth enlargement due to the field gradient is small (<10%) – and barely visible in Fig. 11 – compared with the natural bandwidth of a pure planar undulator emission.
The impact of a TGU with α = 100 m^{−1} installed in a dispersive region (η_{x} = 0.06 m) on the beam closed orbit, linear optics and partition numbers of Elettra 2.0 were evaluated through particle tracking. The orbit deflection is easily recovered by steering magnets installed all across the TGU section. The linear optics asymmetry is partially compensated by tweaking quadrupole magnets' strengths; the residual linear optics asymmetry (betabeating) induced by the TGU is ∼1%. The horizontal emittance grows by ∼5%. This effect, however, is comparable with that of a realistic full set of beamline undulators and machine errors. Still, these preliminary considerations indicate that the installation of several TGUs in analogous dispersive regions for, for example, supplying multiple short pulse beamlines, would start affecting the at a nonnegligible level. In this case, and depending upon users requirements, dedicated ultralowemittance runs could be conceived as an alternative operation to slicing; this would still be operated in a multibunch filling pattern, in a `moderate' regime.
3.5. Hybrid filling pattern for improved slicing efficiency
The short pulse radiation performance reported in Table 2 is calculated for a uniformly filled train of bunches, i.e. same bunch charge for all bunches. However, equation (29) suggests that, to increase the total slicing efficiency, dedicated highcharge bunches should be used (Holldack et al., 2014; Schick et al., 2016). Multibunch coupled instabilities are enhanced by higher charge density, and typically are minimized by hybrid filling pattern, i.e. by temporal separation of ≥10 ns between bunches devoted to slicing and to all the others in a train. In the case of multiple bunches designed for slicing, such temporal separation must match the temporal gating of the detector, which is adopted to minimize the radiation background from sliced electrons on successive turns (see Section 1). Moreover, owing to the higher charge density, the sliced bunches will show a higher transverse emittance at equilibrium, primarily as a consequence of intrabeam scattering (Leemann, 2014). However, if the number of high charge bunches is small compared with the total charge, photon beamlines sensitive to diffractionlimited optics will not be affected considerably on a multiturn multibunch basis.
As an example of hybrid filling pattern for Elettra 2.0, we consider four high chargebunches. The filling pattern is reduced from the nominal 93% to 91% (392 RF buckets filled), while the total average current remains at the nominal 400 mA level. The time separation of the special bunches from the others is ±10 ns. Alternatively, a single high chargebunch separated by ±40 ns from the rest of the train could be considered. Assuming a sixtimes higher bunch charge, the transverse emittances are estimated to double, while the energy spread and the bunch length at equilibrium are almost unchanged. As already discussed in Section 2, no microwave instability build up is foreseen at this current level. A 10 kHz laser repetition rate and a few mJ laser pulse energy over a ∼0.5 ps laser produces an energy modulation amplitude of the order of 1%. The total slicing efficiency is at the level of 10^{−7}.
A more accurate evaluation of practical options for Elettra 2.0 and of expected laserslicing performance is summarized in Table 3. For the sake of brevity, parameters not listed in Table 3 are assumed to be the same as in Table 2. On the basis of the observations in Section 2 on background mitigation, Option 2 results to be the most promising scenario. A 10 kHz repetition rate with slicing in sequence mode is similarly announced by Holldack et al. (2014). In detail, the following options are considered:
(i) All bunches have the nominal charge of 0.8 nC; ten bunches devoted to slicing are separated by ±6 ns from leading and trailing bunches; the filling pattern is reduced to 86% (344 mA). Laser parameters are: 0.8 ps FWHM .
3 mJ pulse energy, 10 kHz repetition rate (30 W average power). This case is simulated in Section 3(ii) Four bunches, each with a charge of 4.8 nC, are selected for slicing; each bunch is separated by ±10 ns from leading and trailing bunches. The filling pattern is 92.5% (376 mA). Laser parameters are: 0.4 ps FWHM
4 mJ pulse energy and 10 kHz repetition rate (40 W average power).(iii) Same as in (ii), but now with increased laser average power: 3 mJ pulse energy, 20 kHz repetition rate (60 W average power).
4. Final remarks
The analytical expressions in Appendix A have been benchmarked with theoretical and experimental studies reported in the literature (Zholents & Zolotorev, 1996; Ingold et al., 2001, 2007; Nadji et al., 2004; Yu et al., 2011; Lau et al., 2012; Schoenlein et al., 2000a; Khan et al., 2006; Holldack et al., 2014; Labat et al., 2018). For each case, the laserinduced energy modulation and the total slicing efficiency were calculated with equation (10) and equation (29). We typically found a discrepancy <5% and for few cases in the range 15–20% for the energy modulation amplitude, and by a factor <2 for the total slicing efficiency. This result translates into a discrepancy in the predicted average spectral within ∼50%. Part of the discrepancy, sometimes referring to reported measured quantities, can be attributed to the uncertainty about the full list of parameters actually adopted by the authors; often different laser options are investigated and not all the details of the machine set up are given at the same time. Moreover, it is unclear if different authors calculated the energy modulation amplitude as given here in equation (11) or with the exact expression in equation (10). Our conclusion is that our modelling captures the salient features of the laser–electron interaction and predicts the radiation and the related spectral properties from the radiator with an uncertainty smaller than a factor two.
Fig. 12 collects our calculations of the laserslicing performance in theoretical and experimental studies, including the three options illustrated in Table 3 for Elettra 2.0. We conclude that energy modulation amplitude of ∼1% can be reached in Elettra 2.0 with a 400 fs FWHMlong laser pulse and 3 mJ pulse energy in the radiator. In spite of an about three times lower laser peak power with respect to stateoftheart experiments (Holldack et al., 2014; Labat et al., 2018), the laser–electron interaction is predicted to be roughly twice as efficient, mainly by virtue of the about five times smaller laser spot size at the waist (higher electric field), as allowed by the diffractionlimited optics of the storage ring. Compared with stateoftheart experiments carried out at BESSYII (Holldack et al., 2014) and for a comparable energy modulation amplitude of 0.7%, laser slicing at Elettra 2.0 is expected to provide a ten times higher total slicing efficiency at the expense of a four times higher average laser power. In terms of per unit length of the Xray pulse, the proposed scheme is at the same level as the demonstrated stateoftheart. We stress that the hybrid filling pattern with high bunch charges, as well as slicing in sequence mode, is essential to the predicted performance.
One distinct feature of the proposed scheme is the transparency of the laserslicing scheme to the diffractionlimited optics design of Elettra 2.0. Namely, modifications to neither the magnetic lattice nor to the spaces planned for installation of IDs is required. The relatively low dispersion function at the radiator translates into a pure 0.5 mm spatial separation of the long and short radiation pulses (contrary to the configurations implemented, e.g. in BESSYII and Soleil). This impasse can be solved by 1:1 imaging of the radiation source. At the second focus of the imaging system, a slit can be used to select either the very intense 100 pslong radiation pulses or less intense 0.4 pslong pulses. The additional spectral separation allowed by emission in a TGU enhances the SNR at the detector by at least one additional order of magnitude in total density.
APPENDIX A
Theoretical model of laserslicing
A1. Energy modulation amplitude
We consider a linearly polarized wiggler (undulator parameter K ≫ 1). The number of photons emitted per wiggler period by a single electron by incoherent (spontaneous) emission at the fundamental wavelength is (Jackson, 1999)
where α ≃ 1/137 is the finestructure constant, K = 0.934B_{0}[T]λ_{u}[cm] with B_{0} the peak wiggler field, and [JJ] is the electronfield coupling parameter (Jackson, 1999). The wavelength of emission at the nth harmonic (n ≥ 1) is
where θ is the observation angle with respect to the wiggler axis. The total energy radiated along N_{u} wiggler periods at the fundamental wavelength (n = 1) is
We now consider a Gaussian and Fourier transform limited laser beam copropagating with the electron beam and superimposed on it through the wiggler. The laser central wavelength is the same as the fundamental wavelength of the onaxis ω_{R} = ω_{L}. The electrons wiggling in the insertion device interact with the transverse electric field of the laser, and the energy exchange is proportional to the laser energy within the laser relative spectral bandwidth, . The energy modulation imposed by the laser on the electron beam is (Zholents & Zolotorev, 1996) = and, therefore, by virtue of equation (9),
Here we used the definition of wiggler relative spectral bandwidth N_{u} ≅ ω_{R}/Δω_{R} and [JJ]^{2} → 1/2 for K > 3. Since the ratio of the laser and wiggler bandwidths is equivalent to the number of wiggler periods N_{u} over the number of laser cycles N_{L}, and since the energy modulation becomes rather insensitive to K for K ≫ 1, we can approximate equation (10) by
A2. Energy spread
In the approximation of a sinusoidal variation of the laser electric field with time, the r.m.s. energy spread associated with the onaxis energy modulation amplitude is = . A more realistic picture of the laser–electron interaction takes into account both the laser and the electron beam transverse size, σ_{E} and σ_{L}, as well as their variation along the wiggler. To do this, we revisit the expression given by Huang et al. (2004), initially derived, for example, by Tran & Wurtele (1989), which expresses the r.m.s. laserinduced energy spread as a function of the laser peak power P_{L}; this quantity is intended to be the laser energy per within the relative spectral bandwidth of the laser,
with
In equation (13) we introduced the following quantities: P_{0} = (m_{e}c^{2})^{2}/(αℏ) = 8.7 GW, L_{u} = N_{u}λ_{u} is the wiggler length, and the laser beam's intensity envelope is assumed to be round, which is not necessarily the case for the electron beam. The waist of the laser and of the electron beam is assumed to be in the middle of the wiggler, so that = is the laser transverse size at the waist calculated for the Rayleigh length that minimizes the laser spot at the wiggler edges, i.e. = L_{u}/2. It is worth recalling that, neglecting any focusing by the wiggler field of the electron beam envelope, the laser and the electron beam envelope evolves from the waist as follows,
where ∊ is the geometric or natural electron beam transverse emittance, β_{0} is the betatron function at the waist, and σ_{L,w} is the laser size at the waist. We point out that for a zeroemittance electron beam, and in the approximation = all along the wiggler; when the electron beam size matches the aforementioned `optimum' laser size. For a vanishing electron beam size and an optimum laser size, equation (12) shows that, as expected, the smaller the laser size in the wiggler, the larger the energy modulation will be by virtue of the increased electric field amplitude of the laser pulse.
Below we demonstrate that equation (12) is identical to the r.m.s. energy spread derived from equation (11) for negligible electron beam sizes and minimum average laser spot size { = } along the wiggler (K > 3),
Equation (15) shows that the energy modulation amplitude given by Zholents & Zolotorev (1996) is in fact the maximum amplitude expected in ideal conditions. In this article we adopted the more realistic expression in equation (10) for the onaxis energy modulation amplitude induced by the laser–electron interaction, and in equation (12) for the corresponding energy spread.
A3. Pulse length
The photon σ_{ph} is estimated by Zholents & Zolotorev (1996) as the quadratic sum of the laser σ_{L}, the slippage of radiation over the electrons along the wiggler (typically negligible for laser pulses of femtosecond duration or longer), and the elongation of the portion of lasersliced electrons due to the electron beam transport from the wiggler to the radiation source σ_{l},
σ_{L} depends on the transport matrix terms of the magnetic lattice from the wiggler to the radiation source that are coupled to the beam size, angular divergence (horizontal only in the absence of vertical dispersion) and relative energy spread evaluated at the wiggler location. In the linear approximation,
It can be shown (Di Mitri & Cornacchia, 2014) that, to first order in the particle coordinates, the first two terms on the righthand side of equation (17) are approximately equal to ∊_{x}〈H_{x}〉_{s}, where the H_{x} function is averaged over the length of the beamline from the wiggler to the radiation source,
and are the energy dispersion function and its longitudinal derivative; γ_{x}, α_{x}, β_{x} are the socalled Courant–Snyder or Twiss parameters. The last term on the righthand side of equation (17) can be written in terms of the storage ring momentum compaction and the length of the beamline from the wiggler to the radiation source,
The result is that a lowemittance storage ring favours the preservation of short radiation pulses through the ring by virtue of both small emittance and small average H_{x} function.
A4. Equilibrium, repetition rate and modulation efficiency
We next write the equation for the variation of the relative energy spread with time in the presence of radiation damping and laser slicing. In the following, σ_{δ,0} is intended to be the natural (equilibrium) relative energy spread in the absence of laser excitation. Radiation damping tends to reduce the energy spread exponentially with time, on the typical time scale of τ_{D} ≃ 10 ms. Laserslicing enlarges the energy spread over of the bunch portion interacting with the laser, at the laser repetition rate 1/τ_{sl} = f_{L}/N_{b}, with N_{b} the number of bunches in the stored bunch train interacting with the laser, and f_{L} the laser frequency,
By imposing equation (20) equal to zero, we find the relative energy spread at equilibrium,
where, following Zholents & Zolotorev (1996), we defined the ratio of the laserinduced energy modulation over the new equilibrium energy spread, and therefore the energy spread growth rate,
From equation (21) we also have
Equation (23) can be used to specify the minimum value of τ_{g} for any maximum tolerable ratio σ_{δ,eq}/σ_{δ,0} (e.g. smaller than 1.5). Such a tolerance translates through equation (22) into a specification for the maximum laser pulse repetition rate.
The physical meaning of the pparameter is related to the number of electrons brought to the energy amplitude required for spatial/angular/spectral separation of the short pulse radiation from background radiation. As the energy modulation is sinusoidal with the electronlaser relative phase, the larger energy modulation amplitude – i.e. better separation from the nonsliced electrons – corresponds to fewer electrons moved to the offenergy level. In addition, radiation from only half of those electrons (either at the positive or negative offenergy level) will be collected. For these reasons, a larger energy separation leads to better signaltonoise ratio of the emitted radiation at the expense of lower Zholents & Zolotorev (1996) defined the electron capturing efficiency η_{1} in a way that, for a betatron size of the electron beam negligible with respect to the chromatic beam size at the radiator, i.e. σ_{x,tot} ≃ σ_{x,η}, it satisfies
Thus, η_{1} is maximum and equal to 1 for the amplitude of energy modulation equal to a quarter of the energy spread at equilibrium, in the presence of slicing. Experimental evidence (Schoenlein et al., 2000a; Ingold et al., 2007; Holldack et al., 2014; Labat et al., 2018) suggest a value p ≃ 4–10 for an acceptable SNR, and therefore the efficiency turns out to be bounded to the range η_{1} ≃ 0.1–0.2.
A5. Radiation from dipole magnet and undulator
The spectral energy distribution of radiation emitted by a single electron in a dipole magnet is (Jackson, 1999)
where = is the critical frequency of dipole magnet synchrotron radiation in terms of the beam energy Lorentz factor γ, the light speed in vacuum c, and the dipole bending radius R. K_{5/3} is the secondorder modified Bessel function. Thus, the number of photons emitted per unit spectral bandwidth is
or
Once divided by the beam revolution period in the storage ring and taking into account both the total number of N_{e} electrons in the bunch and the total slicing efficiency ξ_{sl} (see below), equation (27) turns into the expression given by Zholents & Zolotorev (1996) for the total number of photons emitted per unit time (intensity), per unit azimuthal angle (defined over the entire range 0–2π) and unit relative spectral bandwidth,
The slicing efficiency is defined as (Zholents & Zolotorev, 1996; Holldack et al., 2014)
where f_{riv} is the single bunch revolution frequency and I_{sl}/I_{b} is the ratio of average currents of the total stored beam and of a single sliced bunch. In other words, f_{riv}(I_{b}/I_{sl}) is the effective repetition rate of short photon pulses. Note that equation (22) holds for arbitrarily large f_{L}, which is assumed to be consistent with the maximum tolerable equilibrium energy spread [see equation (23)]. If, instead, f_{L} is limited by the maximum laser average power, the equilibrium energy spread will be determined by equations (22) and (23) (Zholents & Zolotorev, 1996).
ξ_{sl} applies identically to any parameter associated with the radiation emitted by a bunch in a storage ring (spectral angular intensity, etc.). The existing literature gives its value in the range 10^{−8}–10^{−9} for optimized practical situations. So, for example, the number of photons emitted per second at the nth harmonic by a lasersliced electron bunch in a short undulator and unit relative bandwidth, integrated over the central angular cone of amplitude
is
where I stands for the electron beam (average or peak) current, and [JJ]_{n} is the undulator's fieldelectron coupling factor calculated for the nth harmonic. The factor n^{2} [JJ]_{n}^{2}K^{2}/(1+K^{2}/2) in the first equality of equation (30) has a value around 0.4 for n = 3–9 (Kim, 1995).
Acknowledgements
Two of the authors (SDM and WAB) acknowledge Professor F. Parmigiani for stimulating briefings on the generation of short pulses, and E. Karantzoulis for providing the lattice of Elettra 2.0.
References
Bane, K., Cai, Y. & Stupakov, G. (2010). Phys. Rev. ST Accel. Beams, 13, 104402. Web of Science CrossRef Google Scholar
Beaud, P., Johnson, S. L., Streun, A., Abela, R., Abramsohn, D., Grolimund, D., Krasniqi, F., Schmidt, T., Schlott, V. & Ingold, G. (2007). Phys. Rev. Lett. 99, 174801. Web of Science CrossRef PubMed Google Scholar
Borland, M. (2000). elegant: A Flexible SDDSCompliant Code for Accelerator Simulation. Presented at the 6th International Computational Accelerator Physics Conference (ICAP2000), 11–14 September 2000, Darmstadt, Germany. Report No. LS287. Google Scholar
Borland, M. (2014). Synchrotron Radiat. News, 27(6), 2–3. Google Scholar
Byrd, J. M., Hao, Z., Martin, M. C., Robin, D. S., Sannibale, F., Schoenlein, R. W., Zholents, A. A. & Zolotorev, M. S. (2006). Phys. Rev. Lett. 96, 164801. Web of Science CrossRef PubMed Google Scholar
Cerrina, F. & Sanchez del Rio, M. (2009). In Handbook of Optics, Vol. V, 3rd ed., ch. 35, edited by M. Bass. New York: McGrawHill. Google Scholar
Di Mitri, S. & Cornacchia, M. (2014). Nucl. Instrum. Methods Phys. Res. A, 735, 60–65. CrossRef CAS Google Scholar
Holldack, K., Bahrdt, J., Balzer, A., Bovensiepen, U., Brzhezinskaya, M., Erko, A., Eschenlohr, A., Follath, R., Firsov, A., Frentrup, W., Le Guyader, L., Kachel, T., Kuske, P., Mitzner, R., Müller, R., Pontius, N., Quast, T., Radu, I., Schmidt, J.S., SchüßlerLangeheine, C., Sperling, M., Stamm, C., Trabant, C. & Föhlisch, A. (2014). J. Synchrotron Rad. 21, 1090–1104. Web of Science CrossRef CAS IUCr Journals Google Scholar
Holldack, K., Khan, S., Mitzner, R. & Quast, T. (2006). Phys. Rev. Lett. 96, 054801. Web of Science CrossRef PubMed Google Scholar
Huang, Z., Borland, M., Emma, P., Wu, J., Limborg, C., Stupakov, G. & Welch, J. (2004). Phys. Rev. ST Accel. Beams, 7, 074401. CrossRef Google Scholar
Ingold, G., Beaud, P., Johnson, S. L., Gromlind, D., Schlott, V., Schmidt,Th.&Streun,A.(2007).SynchrotronRad.News,20(5), 39. Google Scholar
Ingold, G., Streun, A., Singh, B., Abela, R., Beaud, P., Knopp, G., Rivkin, L., Schlott, V., Schmidt, Th., Sigg, H., van der Veen, J. F., Wrulich, A. & Khan, S. (2001). Proceedings of the 2001 Particle Accelerator Conference (PAC2001), 18–22 June 2001, Chicago, IL, USA, pp. 2656–2658. WPPH085. Google Scholar
Jackson, J. D. (1999). Classical Electrodynamics, 3rd ed. New York: John Wiley and Sons. Google Scholar
Karantzoulis, E. (2018). Nucl. Instrum. Methods Phys. Res. A, 880, 158–165. CrossRef CAS Google Scholar
Karantzoulis, E. & Barletta, W. (2019). Nucl. Instrum. Methods Phys. Res. A, 927, 70–80. CrossRef CAS Google Scholar
Khan, S., Holldack, K., Kachel, T., Mitzner, R. & Quast, T. (2006). Phys. Rev. Lett. 97, 074801. Web of Science CrossRef PubMed Google Scholar
Kim, K.J. (1995). Opt. Eng. 342, doi:10.1117/12.194193. Google Scholar
Labat, M., Brubach, J.B., Ciavardini, A., Couprie, M.E., Elkaim, E., Fertey, P., Ferte, T., Hollander, P., Hubert, N., Jal, E., Laulhé, C., Luning, J., Marcouillé, O., Moreno, T., Morin, P., Polack, F., Prigent, P., Ravy, S., Ricaud, J.P., Roy, P., Silly, M., Sirotti, F., Taleb, A., Tordeux, M.A. & Nadji, A. (2018). J. Synchrotron Rad. 25, 385–398. CrossRef CAS IUCr Journals Google Scholar
Lau, W. K., Chou, M. C., Hwang, C. S., Lee, A. P., Liu, Y. C., Luo, G. H. & Huang, N. Y. (2012). Proceedings of the Third International Particle Accelerator Conference (IPAC2012), 20–25 May, New Orleans, LA, USA, pp. 1671–1673. TUPPP027. Google Scholar
Leemann, S. (2014). Phys. Rev. ST Accel. Beams, 17, 050705. CrossRef Google Scholar
Martin, I. P. S. (2011). PhD Thesis, Wolfson College, University of Oxford, UK (https://ora.ox.ac.uk/objects/uuid:9ac0bcc2bedb46d0b95c22f4741f45a0). Google Scholar
Nadji, A., Chubar, O., Idir, M., Level, M.P., Loulerge, A., Moreno, T., Nadolski, L. & Polack, F. (2004). Proceedings of the 9th Particle Accelerator Conference (EPAC2004), 5–7 July 2004, Lucerne, Switzerland, pp. 2332–2334. THPKF029. Google Scholar
Prigent, P., Hollander, P., Labat, M., Couprie, M. E., Marlats, J. L., Laulhé, C., Luning, J., Moreno, T., Morin, P., Nadji, A., Polack, F., Ravy, S., Silly, M. & Sirotti, F. (2013). J. Phys. Conf. Ser. 425, 072022. CrossRef Google Scholar
Raimondi, L. & Spiga, D. (2015). A A, 573, A22. CrossRef Google Scholar
Raimondi, L., Svetina, C., Mahne, N., Cocco, D., Abrami, A., De Marco, M., Fava, C., Gerusina, S., Gobessi, R., Capotondi, F., Pedersoli, E., Kiskinova, M., De Ninno, G., Zeitoun, P., Dovillaire, G., Lambert, G., Boutu, W., Merdji, H., Gonzalez, A. I., Gauthier, D. & Zangrando, M. (2013). Nucl. Instrum. Methods Phys. Res. A, 710, 131–138. Web of Science CrossRef CAS Google Scholar
Schick, D., Le Guyader, L., Pontius, N., Radu, I., Kachel, T., Mitzner, R., Zeschke, T., SchüßlerLangeheine, C., Föhlisch, A. & Holldack, K. (2016). J. Synchrotron Rad. 23, 700–711. CrossRef CAS IUCr Journals Google Scholar
Schoenlein, R. W., Chattopadhyay, S., Chong, H. H., Glover, T. E., Heimann, P. A., Leemans, W. P., Shank, C. V., Zholents, A. A. & Zolotorev, M. S. (2000b). Appl. Phys. B, 71, 1–10. CrossRef CAS Google Scholar
Schoenlein, R. W., Chattopadhyay, S., Chong, H. H., Glover, T. E., Heimann, P. A., Shank, C. V., Zholents, A. A. & Zolotorev, M. S. (2000a). Science, 287, 2237–2240. CrossRef PubMed CAS Google Scholar
Schoenlein, R. W., Elsaesser, T., Holldack, K., Huang, Z., Kapteyn, H., Murnane, M. & Woerner, M. (2019). Philos. Trans. R. Soc. A, 377, 20180384. CrossRef Google Scholar
Streun, A. (2003). SLSFEMTO: beam halo formation and maximum repetition rate for laser slicing. Report SLSTMETA20030222. Paul Scherrer Institut, CH5232 Villigen PSI, Switzerland. Google Scholar
Tanaka, T. & Kitamura, H. (2001). J. Synchrotron Rad. 8, 1221–1228. Web of Science CrossRef CAS IUCr Journals Google Scholar
Tran, T. M. & Wurtele, J. S. (1989). Comput. Phys. Commun. 54, 263–272. CrossRef CAS Web of Science Google Scholar
Venturini, M. & Warnock, R. (2002). Phys. Rev. Lett. 89, 224802. Web of Science CrossRef PubMed Google Scholar
Yu, L. H., Blednykh, A., Guo, W., Krinsky, S., Li, Y., Shaftan, T., Tchoubar, O., Wang, G. M., Willeke, F. & Yang, L. (2011). Proceedings of the 2011 Particle Accelerator Conference (IPAC2011), 28 March–3 April 2011, New York, USA, pp. 2381–2383. THP136. Google Scholar
Zholents, A. A. & Zolotorev, M. S. (1996). Phys. Rev. Lett. 76, 912–915. CrossRef PubMed CAS Web of Science Google Scholar
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