Femtosecond timing-jitter between photo-cathode laser and ultra-short electron bunches by means of hybrid compression

The SPARC photo-injector consists in a S-band 1.6 cell BNL/UCLA/SLAC type RF-gun providing 120 MV m⁻¹ peak electric field on the built-in metallic (Cu) PC. Electrons are extracted by means of UV laser pulses ($\lambda =266$ nm) whose shape and duration ($0.1\mbox{--}10$ ps FWHM) can be tailored to the needs of the aforementioned applications [23, 24]. They are accelerated up to 5.3 MeV in the gun [25] and then injected into three S-band sections (called S1, S2 and S3 in the following). S1 is also used as RF-compressor by means of VB [26, 27]. Solenoid coils embedding the first two sections can provide additional magnetic focusing during VB process and control of emittance and envelope oscillations [28]. A diagnostics transfer line, consisting in a spectrometer and a RF-deflector (RFD), allows a complete 6D beam characterization (LPS, projected and slice emittance [29, 30]).

When the photo-injector is not operated in the FEL mode, the beam can be bent by a dipole magnet either by $14^\circ$ towards the dogleg or by $25^\circ$ towards the plasma acceleration and Thomson scattering beamlines. The dogleg is sketched in figure 2. It consists of three dispersion-matching quadrupoles placed between the two dipoles and five more focusing quadrupoles in the final straight path, that allow to match the dispersion-free beam through the beamline. Downstream the dogleg both the bunch length and the ATJ can be measured with an electro-optic sampling (EOS) system [31]. The bunch length measurement can be cross-checked through auto-correlation of coherent transition radiation (CTR) spectrum in THz range [32].

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Being ${\bf{X}}=[x,x^{\prime} ,y,y^{\prime} ,z,\delta ]$ the six-element vector representing the coordinates of a test particle (with $x,y,z$ the positions, $x^{\prime} ,y^{\prime}$ the divergences and δ the fractional longitudinal momentum deviation), the transformation of ${\bf{X}}$ produced by a system of magnetic elements can be represented as a power series expansion of the trace space coordinates. Therefore we can assume

$$\begin{eqnarray}&&{({{\bf{X}}}_{f})}_{i}={R}_{{ij}}{({{\bf{X}}}_{{\bf{0}}})}_{j}+{T}_{{ijk}}{({{\bf{X}}}_{{\bf{0}}})}_{j}{({{\bf{X}}}_{{\bf{0}}})}_{k}+{U}_{{ijkl}}{({{\bf{X}}}_{{\bf{0}}})}_{j}{({{\bf{X}}}_{{\bf{0}}})}_{k}{({{\bf{X}}}_{{\bf{0}}})}_{l}\,+\,\cdots ,\end{eqnarray} \tag{ 1 }$$

where ${{\bf{X}}}_{{\bf{0}}}$ is the initial coordinate and R_ij, T_ijk, U_ijkl are the transport matrices of increasing order. The longitudinal coordinate z of the single particle evolves as

$$\begin{eqnarray}&&{z}_{f}\approx {z}_{0}+{R}_{56}{\delta }_{0}+{T}_{566}{\delta }_{0}^{2},\end{eqnarray} \tag{ 2 }$$

where ${\delta }_{0}={\sigma }_{{\rm{E}}}/{E}_{0}$ and ${\sigma }_{{\rm{E}}}=E-{E}_{0}$ is the energy deviation of the particle from the reference energy E₀. In equation (2) higher order effects (mostly due to the U₅₆₆₆ term⁶ ) have been neglected. The energy deviation of the single particle depends on its longitudinal position with respect to the RF wave during the acceleration process. Therefore ${\delta }_{0}$ is a function of z

$$\begin{eqnarray}&&{\delta }_{0}({z}_{0})={\delta }_{{\rm{u}}}+{h}_{1}{z}_{0}+{h}_{2}{z}_{0}^{2}\,+\,\cdots ,\end{eqnarray} \tag{ 3 }$$

where ${\delta }_{{\rm{u}}}$ denotes the uncorrelated energy offset and h₁ and h₂ are the first and second order chirp terms, respectively. Dropping the single particle assumption, equations (2) and (3) can be combined to quantify the expected bunch length ${\sigma }_{z,f}$ at the end of a dispersive beamline [35]

$$\begin{eqnarray}&&{\sigma }_{z,f}^{2}={R}_{56}^{2}{\sigma }_{{\delta }_{{\rm{u}}}}^{2}+{(1+{h}_{1}{R}_{56})}^{2}{\sigma }_{z,i}^{2}+3{({h}_{2}{R}_{56}+{T}_{566}{h}_{1}^{2})}^{2}{\sigma }_{z,i}^{4},\end{eqnarray} \tag{ 4 }$$

where ${\sigma }_{{\delta }_{{\rm{u}}}}$ is the uncorrelated (normalized) energy spread and ${\sigma }_{z,i}$ is the initial bunch length. According to equation (4), the minimum of ${\sigma }_{z,f}$ can be found by canceling the last two terms. This leads to

$$\begin{eqnarray}&&{h}_{1}=-\displaystyle \frac{1}{{R}_{56}},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{h}_{2}=-\displaystyle \frac{{T}_{566}}{{R}_{56}}{h}_{1}^{2}=-\displaystyle \frac{{T}_{566}}{{R}_{56}^{3}}\end{eqnarray} \tag{ 6 }$$

and the compression results more efficient at high energies (${\sigma }_{{\delta }_{u}}\to 0$).

The previous formulation has to be modified if the beam energy $\tilde{E}$ has an offset ${\rm{\Delta }}=(\tilde{E}-{E}_{0})/{E}_{0}$ with respect to the reference energy E₀ [33]. In this case, a particle with arbitrary energy E has an energy error $\widetilde{\delta }=(E-\tilde{E})/\tilde{E}$ relative to the central energy of the beam, and an energy error δ relative to the design energy of the beamline. By applying the coordinate transformation $\delta \to (\tilde{E}/{E}_{0})\widetilde{\delta }+{\rm{\Delta }}$, equation (2) modifies as

$$\begin{eqnarray}&&{z}_{f}\approx {z}_{0}+\widetilde{{Q}_{5}}+{\tilde{R}}_{56}\widetilde{\delta }+{\tilde{T}}_{566}{\widetilde{\delta }}^{2},\end{eqnarray} \tag{ 7 }$$

where $\widetilde{{Q}_{5}}={R}_{56}{\rm{\Delta }}+{T}_{566}{{\rm{\Delta }}}^{2}$ and the linear and nonlinear transport terms are rewritten as follows:

$$\begin{eqnarray}&&{\tilde{R}}_{56}=\displaystyle \frac{\tilde{E}}{{E}_{0}}({R}_{56}+2{T}_{566}{\rm{\Delta }}),\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{\tilde{T}}_{566}={\left(\displaystyle \frac{\tilde{E}}{{E}_{0}}\right)}^{2}{T}_{566}.\end{eqnarray} \tag{ 9 }$$

Equation (7) represents the effective longitudinal transformation. Therefore, equation (4) has to be modified by replacing the ${R}_{56}$ and ${T}_{566}$ terms with the ones in equations (8) and (9).

Following equation (5), the bunch compression in a dogleg with ${R}_{56}\lt 0$ is obtained if the chirp is positive (particles with higher energies on the head) and equal to ${h}_{1}\approx \tfrac{1}{{\sigma }_{z}}\tfrac{{\sigma }_{{\rm{E}}}}{{E}_{0}}=-\tfrac{1}{{R}_{56}}$. This can be fulfilled with a proper choice of energy (E₀), energy spread (${\sigma }_{{\rm{E}}}$) and length (${\sigma }_{z}$). Short beams with positive chirp can be obtained with the VB scheme, using S1 as a RF compressor.

Starting from the cathode, the bunch is accelerated in the RF-gun where space-charge forces gradually lead to its lengthening, depending on the initial transverse and longitudinal profiles [36]. Figure 3 reports the simulated LPS of a 50 pC beam for several injection phases in S1. Early (late) particles correspond greater (smaller) time values on the horizontal axis. The simulation is performed with general particle tracer (GPT) [37], a PIC code that accounts for space-charge effects. On the cathode we used a laser pulse with longitudinal (${\sigma }_{{\rm{t}}}=450$ fs) and transverse (${\sigma }_{x,y}=150\,\mu {\rm{m}}$) gaussian profiles. The beam is then accelerated, resulting in a duration ${\sigma }_{{\rm{t}}}\approx 830$ fs (${\sigma }_{z}\approx 250\,\mu {\rm{m}}$) at the gun exit. In VB the beam is injected in S1 forward-off-crest, i.e. towards negative RF phases. The process is based on a correlated time-velocity chirp in the electron bunch, in such a way that electrons on the tail of the bunch are faster than electrons in the bunch head. This leads to a rotation of the LPS if the injected beam is slower than the phase velocity of the RF wave. So when it is injected at the zero-crossing field phase, it slips back to phases where the field is accelerating, but it is simultaneously chirped and compressed. As showed in figure 3, by moving towards negative phases the chirp is negative up to the maximum compression (${\phi }_{S1}=-88^\circ$ with respect to the maximum energy phase), corresponding to a bunch duration ${\sigma }_{{\rm{t}}}\approx 50$ fs (${\sigma }_{z}\approx 15\,\mu {\rm{m}}$). Then, at even more negative phases (over-compression), it slightly elongates while the chirp becomes positive.

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Longitudinal beam dynamics is sensitive to RF field fluctuations in the gun and accelerating sections. Fluctuations in the magnetic field of dispersive elements located along the machine can contribute too [38]. The energy, energy spread and duration of the beam are consequently affected, depending on the photo-injector configuration. The beam time of arrival is also influenced, resulting in an ATJ at the end of the line. We can define the ATJ as the shot-to-shot time of arrival fluctuation of the beam center of mass with respect to a fixed position. The ATJ is produced by several sources, e.g. the changing of the laser arrival time on the PC (${\rm{\Delta }}{t}_{\mathrm{laser}}$), or instabilities in the timing (${\rm{\Delta }}{t}_{\mathrm{RF}}$) of the RF system. In our discussion we do not consider fluctuations in the amplitude of RF and magnetic fields. If compared to the other jitter sources described in the following, their contribution to the overall ATJ is negligible⁷ .

At SPARC_LAB, the synchronization system operates by distributing a RF signal generated in a μ-wave reference master oscillator (RMO) through a coaxial cable star network. The client lock-in is then performed with electronic PLLs, resulting in less than 50 fs (rms) timing-jitter between the RMO and the PC laser system [39]. Downstream the photo-injector, the ATJ arises from three main sources: the PC laser and the two S-band klystrons; the first (K1) feeds the gun, S3 and the RFD while the other (K2) powers S1 and S2. For a given configuration, the beam arrival time variation ${\rm{\Delta }}{t}_{\mathrm{linac}}$ can be expressed as a linear combination

$$\begin{eqnarray}&&{\rm{\Delta }}{t}_{\mathrm{linac}}\approx \displaystyle \sum _{i=1}^{3}{c}_{i}{\rm{\Delta }}{t}_{i},\end{eqnarray} \tag{ 10 }$$

where $i=1,2,3$ refer, respectively, to the PC laser, K1 and K2 terms If the dogleg is included, equation (10) becomes

$$\begin{eqnarray}&&{\rm{\Delta }}{t}_{\mathrm{dogleg}}\approx \displaystyle \sum _{i=1}^{3}({c}_{i}+{h}_{1,i}{R}_{56}){\rm{\Delta }}{t}_{i}.\end{eqnarray} \tag{ 11 }$$

The ${h}_{1,i}$ terms are related to energy fluctuations

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}{E}_{0}}{{E}_{0}}\approx c\displaystyle \sum _{i=1}^{3}{h}_{1,i}{\rm{\Delta }}{t}_{i},\end{eqnarray} \tag{ 12 }$$

where c is the speed of light and E₀ is the bunch energy. If the laser and RF fields are delayed all together by a given value, the beam arrival time is delayed by the same amount while the final energy remains unchanged. Therefore the following conditions apply:

$$\begin{eqnarray}&&\displaystyle \sum _{i=1}^{3}{c}_{i}=1\qquad \mathrm{and}\qquad \displaystyle \sum _{i=1}^{3}{h}_{1,i}=0.\end{eqnarray} \tag{ 13 }$$

All ${\rm{\Delta }}{t}_{i}$ values are measured with respect to the RMO. Since they are mostly uncorrelated, we can write the standard deviations values of equations (10) and (11) as

$$\begin{eqnarray}&&{\sigma }_{{t}_{\mathrm{linac}}}^{2}\approx \displaystyle \sum _{i=1}^{3}{c}_{i}^{2}{\sigma }_{{t}_{i}}^{2},\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&{\sigma }_{{t}_{\mathrm{dogleg}}}^{2}\approx \displaystyle \sum _{i=1}^{3}{({c}_{i}+{h}_{1,i}{R}_{56})}^{2}{\sigma }_{{t}_{i}}^{2}.\end{eqnarray} \tag{ 15 }$$

They represent the expected absolute ATJ (with respect to the RMO) at the linac and dogleg exit respectively. Their evaluation is provided in section 4.

The PC laser is configured as reported in table 1. Setting the linac for maximum energy gain (on crest configuration), the bunch has 164 MeV energy (90 keV energy spread), 1 ps (rms) duration and $1.1\,\mu {\rm{m}}$ (rms) normalized emittance. The beam is then longitudinally compressed with VB and its LPS is manipulated in order to match the dogleg (see 3.1). The final energy (81.2 MeV) and energy spread (0.4 MeV) are obtained by tuning the S2 and S3 phases. With 81.2 MeV energy, the dogleg transport matrix terms result ${R}_{56}=-4.5\,{\rm{mm}}$ and ${T}_{566}=-83.6\,{\rm{cm}}$. According to equation (5), such R₅₆ value requires an ${h}_{1}=-{R}_{56}^{-1}\approx 222\,{{\rm{m}}}^{-1}$ chirp at the dogleg entrance. In order to avoid an excessive emittance growth, the S1 embedded solenoids are turned on, in order to adopt the emittance compensation scheme [28]. The achieved emittance is $1.7\,\mu {\rm{m}}$ (rms). The resulting bunch (68 fs, rms) with positive chirp is obtained by moving the S1 injection phase by $-88^\circ$ with respect to its on crest value, i.e. $1^\circ$ beyond the maximum compression (50 fs, rms). All relevant parameters are reported in table 1.

Table 1.  Beam parameters in the VB configuration. The values reported between brackets refer to simulated data.

Laser parameters	Value (sim.)	Unit
X(Y) spot (rms)	230 ± 5 (230)	$\mu {\rm{m}}$
Pulse duration (rms)	450 ± 50 (450)	fs
Beam parameters	Value (sim.)	Unit
Charge	50 ± 2 (50)	pC
Energy	81.2 ± 0.1 (81)	MeV
Energy spread (rms)	400 ± 10 (410)	keV
Duration	68 ± 18 (65)	fs
Norm. emittance (rms)	1.7 ± 0.2 (1.8)	$\mu {\rm{m}}$

Figure 4(a) shows the simulated time-energy distribution. Experimentally the LPS is retrieved by measuring the bunch time and energy profiles with the RFD and a magnetic spectrometer, respectively. By means of three quadrupoles, the resulting beam is then imaged on a screen located 3 m downstream the spectrometer. The effects of the quadrupoles and the RFD on the simulated beam LPS have been considered in figure 4(b). Its overall shape and inner structure are in good agreement with the measured LPS in figure 4(c). By performing a 2nd order polynomial fit to the simulated data in figure 4(a), we found ${h}_{1}=204.1\,{{\rm{m}}}^{-1}$, ${h}_{2}=1.5\times {10}^{7}\,{{\rm{m}}}^{-2}$ and 60 keV uncorrelated energy spread (see equation (3)). According to equation (4), with such values the expected bunch duration downstream the dogleg is ${\sigma }_{{\rm{t}}}\approx 77$ fs, indicating that the nonlinear T₅₆₆ and h₂ terms introduce non-negligible effects.

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As previously mentioned, the bunch timing-jitter can be measured either with the RFD or the EOS [9]. In the first case the jitter is relative to the RF reference while in the second one the is relative to the EOS laser system. As pointed out in section 3.3, the total ATJ can be expressed as a linear combination of several jitter sources. For the evaluation of the c_i and ${h}_{1,i}$ coefficients, a GPT simulation has been performed. Here the PC laser time of arrival is, in turn, delayed or anticipated with respect to the linac RF accelerating phase. Finally, the beam time of arrival and its central energy are recorded on a fixed screen located at the exit of the linac Results are reported in table 2.

Table 2.  ATJ coefficients (equation (15)) from measurements and simulations (reported in brackets). The S1 injection phase in VB is set at $-88^\circ$ with respect to the on crest one.

Parameter	On crest	VB
c1	0.7 ± 0.2 (0.66)	−0.1 ± 0.4 (−0.14)
c2	0.3 ± 0.1 (0.34)	−0.2 ± 0.5 (−0.07)
c3	0.1 ± 0.9 (0.01)	1.2 ± 0.5 (1.18)
${h}_{\mathrm{1,1}}$ (${{\rm{m}}}^{-1}$)	−1.8 ± 1.2 (−0.8)	−93 ± 22 (−90)
${h}_{\mathrm{1,2}}$ (${{\rm{m}}}^{-1}$)	−0.6 ± 1.2 (−0.4)	41 ± 23 (38)
${h}_{\mathrm{1,3}}$ (${{\rm{m}}}^{-1}$)	2.8 ± 2.5 (1.2)	42 ± 35 (52)

The simulated coefficients have been cross-checked with experimental measurements obtained by acquiring ten consecutive images on the first Ce:YAG screen located after the main dipole, with RFD turned on. In this case each image contains both the temporal and energy information. The c₁ (h₁) coefficient is obtained by measuring the beam time of arrival (energy) while varying the PC laser timing on the cathode. Similarly, the ${c}_{\mathrm{2,3}}$ (${h}_{\mathrm{2,3}}$) coefficients are evaluated, respectively, by changing the RF field phase (i.e. its timing) on the K1 (gun) and K2 (S1) lines. Such coefficients allow to estimate all the jitter sources contained in equation (14). Table 2 highlights two main aspects. First, in the on crest case, the PC laser timing-jitter is compressed by a factor c₁ = 0.66 in the gun [42], meaning that beam dynamics is mostly determined by the PC laser (${c}_{1}\gt {c}_{2}\gt {c}_{3}$). Second, in VB the longitudinal compression strongly links the beam to the S1 fields (${c}_{3}\gg {c}_{\mathrm{1,2}}$) but a correlation between the PC laser and the beam energy still holds (${h}_{\mathrm{1,1}}=-90$). As explained in section 3.3, the SPARC_LAB power system employs two klystrons. If we measure the beam ATJ (downstream the linac) relative to the the RF system (K1 line) with the RFD, equation (14) becomes

$$\begin{eqnarray}&&{\sigma }_{{t}_{\mathrm{linac}}}^{2}\approx {c}_{1}^{2}{\sigma }_{{t}_{{\rm{L}}}}^{2}+{({c}_{2}-1)}^{2}{\sigma }_{{t}_{{\rm{K}}1}}^{2}+{c}_{3}^{2}{\sigma }_{{t}_{{\rm{K}}2}}^{2}.\end{eqnarray} \tag{ 16 }$$

From equation (16), we can discriminate the contributions of the various jitter sources. For instance, if the linac is operated in the on crest configuration, the PC laser is the leading jitter source (${\sigma }_{{t}_{{\rm{L}}}}$). Therefore the beam time of arrival mainly follows the PC laser timing. On the contrary, in VB regime the beam timing is strongly linked to the RF fields in S1, and its ATJ measured with RFD is actually due to ${\sigma }_{{t}_{{\rm{K}}2}}$. In this case the PC laser timing has a negligible effect.

Figure 5 reports the beam time of arrival in the on crest and VB configurations for 40 consecutive shots measured with the RFD by streaking the beam on a Ce:YAG screen. The emitted light is then imaged on a CCD, where each pixel corresponds to 18 fs. The resulting rms is ${\sigma }_{{t}_{\mathrm{linac}}}\approx 34$ fs in both cases. According to equation (16) and table 2, this corresponds to a PC laser jitter of ${\sigma }_{{t}_{{\rm{L}}}}\approx 48$ fs and an RF system jitter of ${\sigma }_{{t}_{{\rm{K}}1}}\approx {\sigma }_{{t}_{{\rm{K}}2}}\approx 22$ fs, compatible with previous measurements [39]. Assuming the PC laser as the reference, equation (16) becomes

$$\begin{eqnarray}&&{\sigma }_{{t}_{\mathrm{linac}}}^{2}\approx {({c}_{1}-1)}^{2}{\sigma }_{{t}_{{\rm{L}}}}^{2}+{c}_{2}^{2}{\sigma }_{{t}_{{\rm{K}}1}}^{2}+{c}_{3}^{2}{\sigma }_{{t}_{{\rm{K}}2}}^{2},\end{eqnarray} \tag{ 17 }$$

corresponding to a relative ATJ of ${\sigma }_{{t}_{\mathrm{linac}}}\approx 60$ fs between the electron beam and the PC laser at linac exit.

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When a bunch is compressed in a non-isochronous magnetic line, its timing-jitter relative to the RF system (that has the initial energy chirp imprinted on it) is compressed too. On the contrary, the timing-jitter relative to the PC laser increases, since the beam results more linked to the RF system and uncorrelated from the laser time of arrival on the cathode. According to equation (15) we demonstrate that, with a proper correlation between shot-to-shot time of arrival and beam central energy, the hybrid scheme is able to manage simultaneously the bunch-to-laser jitter reduction and its longitudinal compression. Downstream the dogleg line, measurements have been performed by the EOS system. Since its probe laser is directly split from the PC laser, the ATJ relative to the PC laser is given by

$$\begin{eqnarray}&&{\sigma }_{{t}_{\mathrm{dogleg}}}^{2}\approx {({c}_{1}+{h}_{\mathrm{1,1}}{R}_{56}-1)}^{2}{\sigma }_{{t}_{{\rm{L}}}}^{2}+{({c}_{2}+{h}_{\mathrm{1,2}}{R}_{56})}^{2}{\sigma }_{{t}_{{\rm{K}}1}}^{2}+{({c}_{3}+{h}_{\mathrm{1,3}}{R}_{56})}^{2}{\sigma }_{{t}_{{\rm{K}}2}}^{2}.\end{eqnarray} \tag{ 18 }$$

For the relative ATJ reduction, we exploited the off-energy beam transport through the dogleg (see section 3.1) in order to manipulate the R₅₆. By setting the design energy 0.18 MeV below the 81.2 MeV nominal beam energy (${\rm{\Delta }}=2.2\times {10}^{-3}$), it results ${\tilde{R}}_{56}=-8.2\,{\rm{mm}}$ and ${\tilde{T}}_{566}=-83.9\,{\rm{cm}}$ with a ${\sigma }_{{\rm{t}}}\approx 90$ fs (rms) expected bunch duration at the dogleg exit (see equation (4)). According to equation (18) we foresee a relative ATJ of ${\sigma }_{{t}_{\mathrm{dogleg}}}\approx 26$ fs, about two times lower than ${\sigma }_{{t}_{\mathrm{linac}}}\approx 60$ fs obtained in section 4.2.

Results are reported in figure 6 as a function of the RF field timing-jitter. There is a correlation between the shot-to-shot time of arrival (blue line) and energy (black-dashed line) of the over-compressed beam downstream the linac This means that bunches arriving earlier (later) have higher (lower) energies and they are delayed (anticipated) by the dogleg. The time reference is assumed to be the PC laser, thus a lower slope of the solid lines corresponds to a lower relative jitter. The ATJ reduction from ${\sigma }_{{t}_{\mathrm{linac}}}\approx 60$ fs down to ${\sigma }_{{t}_{\mathrm{dogleg}}}\approx 25$ fs is confirmed by the model of equation (11) (red line) and the GPT simulations⁸ (green line).

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The underlying principle for the simultaneous bunch compression and relative ATJ reduction relies on the differences in dynamics between particles in the same bunch and bunches in different shots. By means of the hybrid scheme we used VB in order to shorten the duration of the low energy (5.3 MeV) beam exiting from the gun. In these conditions space-charge forces strongly affect the beam LPS. On the contrary, the time-of-flight and mean energy are not perturbed. To clarify this principle we assumed each bunch centroid as a single particle so that the centroid distribution over consecutive shots can be considered as a unique space-charge-free beam, as showed by red dots in figure 7. The simulated single-shot bunch LPS evolution, including space-charge effects, is highlighted by blue dots. Figure 7(a) shows the beam LPS at the gun exit. As discussed in section 4.2, in this case the time of arrival is mainly linked to the release time from the cathode as reported in figure 7(d). The resulting correlation term is c₁ = 1.58 (c₁ = 0.66, see table 2). Downstream the gun, the beam is longitudinally compressed by VB and strongly linked to the RF field phase-jitter (figure 7(b)). Consequently, at the linac exit its time of arrival showed in figure 7(e) is not linked to the PC laser (${c}_{1}\approx -0.14$, see table 2). At this point, the correlation between the PC laser and the time-of-flight is restored by the dogleg. The different dynamics between the bunch inner structure and its longitudinal centroid allows us to reduce the ATJ relative to the PC laser in the dogleg while preserving its duration (figure 7(c)). Figure 7(f) shows that bunch length is compressed (c₁ = 0.12) while the correlation between the centroids (red dots) and the PC laser is partly recovered (c₁ = 0.56). This result represents a compromise between relative ATJ reduction and maximum achievable bunch compression.

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As explained in section 4.3, the off-energy setup of the dogleg allows to handle the R₅₆ term in order to achieve the relative timing-jitter reduction. The dogleg matching is obtained with the MAD-X code [43], by constraining to zero the dispersion and its first derivative downstream the second dogleg dipole. The matching parameters are then imported in GPT in order to perform a full simulation including space-charge. From the resulting LPS, shown in figure 8, we evaluated a final bunch duration of 92 fs (rms), in agreement with the one expected according to equation (4) (90 fs, rms). If we consider that in the on-crest configuration the bunch duration was 1 ps (rms), the corresponding compression factor using the hybrid scheme is $C\approx 11$.

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The Michelson interferometer, schematically shown in figure 9, consists of two highly polished mirrors and a $12\,\mu {\rm{m}}$ mylar layer acting as a beam-splitter for the incoming CTR. In our setup CTR is produced by electron bunches crossing an aluminum-coated silicon screen oriented at $45^\circ$ with respect to the beam line. Only the backward transition radiation is collected. It is extracted through a diamond window and collimated by a $90^\circ$ off-axis parabolic mirror towards a flat mirror reflecting the radiation to the interferometer. The light is then split in two beams. One is transmitted towards a fixed mirror while the other is reflected in the direction of a movable one. The beams are then recombined on the beam-splitter and are measured by a pyro-electric detector provided by Gentec^TM (0.5–30 THz spectral range, 140 kV W⁻¹ sensitivity). The radiation is collected by the detector at several positions of the movable mirror, producing an interference pattern used to reconstruct the beam temporal profile with a resolution ${\sigma }_{{\rm{t}}}\approx 20$ fs (rms) [44].

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5.1.1. Bunch duration measurements

Data have been acquired by changing the position of the interferometer movable mirror with $10\,\mu {\rm{m}}$ steps. At each position, five references and auto-correlation signals are acquired by two pyro-electric detectors and then averaged in order to take into account fluctuations in the beam charge (of the order of 5%). Each point is then used to make the interferogram of figure 10, from which the bunch frequency spectrum is retrieved. In order to reconstruct the bunch temporal profile from the interferogram, the effect of the finite-size of the CTR screen has been considered [45]. It introduces a suppression of the radiated intensity at low-frequencies (i.e. long wavelengths λ) when the extent of the particle field, which is of the order of $\gamma \lambda$, exceeds the dimension of the screen (30 × 30 mm). This is always the case for coherent radiation at THz frequency, and in our case it leads to a suppression of frequency components below 0.5 THz. To compensate these losses in the CTR spectrum, we introduced a low frequency Gaussian reconstruction. The electron bunch profile S(z) is then retrieved from the form factor, applying Kramers–Kronig relations [46]. The reconstructed bunch profile is shown in the inset of figure 10. The retrieved duration is ${\sigma }_{{\rm{t}}}=86\pm 7$ fs (rms), in good agreement with expectations of section 4.4. The result confirms that the bunch slightly elongated due to the larger R₅₆ term resulting from the dogleg off-energy setup.

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The EOS is a non-intercepting and single-shot device that allows to monitor the bunch longitudinal profile and time of arrival [47]. At SPARC_LAB we employ a $100\,\mu {\rm{m}}$-thick gallium phosphide (GaP) crystal and a Ti:Sa laser ($\lambda =800\,{\rm{nm}}$, 80 fs rms) as probe, fixing the EOS resolution on the bunch duration to ${\sigma }_{{\rm{t}}}\approx 90$ fs (rms). The EOS input layout is shown in figure 11(a). The probe laser is split directly from the PC laser oscillator and then amplified, resulting in a natural synchronization with the electron beam. This solution allows us to directly measure the relative timing-jitter between the PC laser and the beam, as described in section 4.3. The encoding of the beam longitudinal profile is then obtained with the spatial decoding setup [31], in which the laser crosses the nonlinear crystal at an angle of $30^\circ$. Finally, the laser is imaged on a CCD camera as shown in figure 11(b). Each pixel corresponds to 10 fs.

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5.2.1. Bunch duration measurements

The EOS station is located downstream the dogleg, 2 m far from the THz station. The EOS signals are obtained by using the $100\,\mu {\rm{m}}$-thick GaP crystal, located $400\,\mu {\rm{m}}$ far from the traveling electron beam. Figure 12(a) shows a single-shot EOS signal acquired with the CCD camera after background subtraction. The temporal profile is calculated by projecting the acquired images along the vertical axis. The resulting signal is then fitted with a gaussian function, as shown in figure 12(b). The signal shows two negative valleys separated by a central positive peak. This behavior is due to a quarter-wave plate inserted between the EOS crystal and the exit polarizer, whose optical axis is slightly tilted with respect to the input laser polarization, ensuring a better sharpness of the output signal. It results that the laser cumulated a mean phase delay of $4.8^\circ \pm 0.4^\circ$, corresponding to a bunch peak electric field of about 3.3 MV m⁻¹. By averaging all the acquired shots, the resulting mean signal width is ${\sigma }_{{\rm{t}}}=95\pm 5$ fs, in agreement with both GPT simulations (section 4.4) and CTR measurements (section 5.1.1).

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5.2.2. Relative ATJ reduction downstream the dogleg

Being a single-shot device, the EOS can also be used as a time of arrival monitor measuring the relative ATJ between the electron beam and the probe laser. In figure 13 the time of arrival of 330 consecutive shots is reported. The rms of the resulting distribution is ${\sigma }_{{t}_{\mathrm{dogleg}}}=19\pm 5$ fs. Since the PC laser is used as the EOS reference, this value exactly represents the relative ATJ. This result confirms the validity of the model discussed in section 4.3, where we foresaw the reduction of the relative ATJ from 60 fs (downstream the linac) to 25 fs (downstream the dogleg). Although this value is slightly larger than the the experimental one, its discrepancy is limited to about 20%, confirming the validity of our assumptions.

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