Sub-picosecond compression by velocity bunching in a photo-injector

We present an experimental evidence of a bunch compression scheme that uses a traveling wave accelerating structure as a compressor. The bunch length issued from a laser-driven radio-frequency electron source was compressed by a factor>3 using an S-band traveling wave structure located immediately downstream from the electron source. Experimental data are found to be in good agreement with particle tracking simulations.


I. INTRODUCTION
In the recent years there has been an increasing demand on ultrashort electron bunches to drive short-wavelength free-electron lasers and study novel accelerating techniques such as plasma-based accelerators [1,2]. Short bunches are commonly obtained by magnetic compression. In this latter scheme, the bunch is compressed using a series of dipoles arranged in a chicane configuration such to introduce an energy-dependent pathlength. Therefore an electron bunch having the proper time-energy correlation can be shortened in the chicane. However, problems inherent to magnetic compression such as momentum spread and transverse emittance dilution due to the bunch selfinteraction via coherent synchrotron radiation [4] has brought back the idea of bunching the beam with radio-frequency (rf) structures [5].
It was recently proposed to incorporate the latter method (henceforth named velocity bunching) into the next photo-injector designs [6]. The velocity bunching relies on the phase slippage between the electrons and the rf-wave that occurs during the acceleration of non ultra-relativistic electrons.
In this paper after presenting a brief analysis of the velocity bunching scheme, we report on its exploration at the deep ultraviolet free-electron laser (DUV-FEL) facility of Brookhaven National Laboratory (BNL). The measurements are compared with numerical simulations performed with the computer program ASTRA [7].

II. ANALYSIS OF THE VELOCITY BUNCHING TECHNIQUE
In this section we elaborate a simple model that describes how the velocity bunching works. A more detailed discussion is given in Reference [6].
An electron in an rf traveling wave accelerating structure experiences the longitudinal electric field: where E o is the peak field, k the rf wavenumber and ψ o the injection phase of the electron with respect to the rf wave. Let ψ(z, t) = ωt − kz + ψ o be the relative phase of the electron w.r.t the wave. The evolution of ψ(t, z) can be expressed as a function of z solely: Introducing the parameter α . = eEo kmc 2 , we write for the energy gradient [3]: The system of coupled differential equations (2) and (3) with the initial conditions γ(z = 0) = γ o and ψ(z = 0) = ψ o describe the longitudinal motion of an electron in the rf structure. Such a system is solved using the variable separation technique to yield: Or, expliciting ψ as a function of γ: Here the constant of integration is set by the initial conditions of the problem [16]: The latter equation gives insights on the underlying mechanism that provides compression. In order to get a simpler model, we consider the limit: ψ ∞ . = lim γ→∞ ψ(γ) = arccos cos(ψ o ) + 1 2αγo ; we have assumed γ o is larger than unit and did the approximation . After differentiation of Eq. 5, given an initial phase dψ o and energy dγ o extents we have for the final phase extent: Hence depending upon the incoming energy and phase extents, the phase of injection in the rf structure ψ o can be tuned to minimize the phase extent after extraction i.e. to ideally (under single-particle dynamics) make dψ ∞ → 0. We note that there are two contributions to dψ ∞ : the first term ∂ψ ∞ /∂ψ o comes from the phase slippage (the injection and extraction phases are generally different). The second term ∂ψ ∞ /∂γ o is the contribution coming from the initial energy spread.
To illustrate the compression mechanism we consider a two macro-particles model. In Figure 1 we present results obtained by numerically integrating the equation of motion for two non-interacting macro-particles injected into a 3 m long traveling wave structure. Given the incoming phase ∆ψ o and energy ∆γ o spreads between the two macro-particles, and the accelerating gradient of the structure (taken to be 20 MV/m), we can optimize the injection phase to minimize the bunch length at the structure exit.

III. EXPERIMENTAL RESULTS
The measurement was carried out at the DUV-FEL facility of Brookhaven national laboratory [11].
A block diagram of the linear accelerator is given in Fig. 2 To investigate the velocity bunching scheme, the linac section L1 was used as a buncher: its phase was varied and, for each phase setting, the section L2 was properly phased to maximize the beam energy with sections L3 and L4 turned off. The magnetic bunch compressor was turned off during the measurement. The nominal settings for the different rf and photo-cathode drive-laser parameters are gathered in Table I  The measurements of bunch length that follow are compared with simulations performed with the program ASTRA [7]. ASTRA is a macro-particle tracking code based on a rotational symmetric space charge algorithm. It incorporates a detailed model for the traveling wave accelerating structure [8,9]. To perform the simulations we used the parameters values of Table I. The laser transverse distribution was modeled by a radially uniform transverse distribution with 0.75 mm radius, and the time profile, measured using a single shot cross-correlation technique, was directly loaded into the simulations.
Both time-and frequency-domain techniques were used to characterize the bunching process as the phase of the linac L1 was varied.
The time-domain charge density was directly measured using the so-called zero-phasing method [12,13]. In the present case, we use the linac section L3 to cancel the incoming time-energy correlation, and operate the linac L4 at zero-crossing to introduce a linear time-dependent energy chirp along the bunch (we have investigated both zero-crossing points). The bunch is then directed to a beam viewer ("YaG monitor" in Fig. 2) downstream from a 72 • angle spectrometer. The viewer, located at a dispersion (horizontal) of η = 907 mm, allows the measurement of the bunch energy distribution.
Unlike in Reference [12], the longitudinal phase space of beams issued from an rf electron source is not perfectly linear: because of the longitudinal space charge forces, the phase space generally has a third order distortion [10]. To analyze the impact of such a distortion on our bunch length measurement method, it is interesting to consider the Gaussian normalized longitudinal phase space (s, δ) density: Here σ s and σ δ are the bunch rms length and rms uncorrelated fractional momentum spread and h 1 , h 3 are constants that quantify the linear and third order correlations of the longitudinal phase space. The zero-phasing measurement can then be analyzed in term of a sequence of numerical calculation based on Eq. 7: by computing and comparing the time and fractional momentum spread projections associated to P(s, δ + C o × s). The constant C o depends on the incoming beam energy E o , the accelerating voltage of the zero-phased linac section, the rf wavenumber k rf , and dispersion [12]: C o = ± Eo ηV rf k rf the ± sign reflects the two possible zero-crossing points.
An example of such a calculation is presented in Fig. 3. no fold over. Hence we expect the bunch time-profile reported hereafter to be longer than in reality.
As the phase of the linac section L1 was varied and L2 tuned to maximize the energy gain, the beam energy was measured. The so-obtained energy variation versus the phase of the linac L1 is compared with the simulations for the nominal operating point (see Table I) in Fig. 4 and the corresponding plot for the bunch length is shown in Fig. 5. As predicted, we observed that operating the linac at lower phases (thereby giving the bunch head less energy than the tail) provides some compression.
The parametric dependence of the rms bunch length on the phase of linac L1 is found to be in good agreement with the simulation predictions. Two cases of measured and simulated bunch timeprofile are presented in Fig. 6. Again, the agreement between simulation and experiment is fairly good taking into account the uncertainties associated to the zero-phasing method. Noteworthy is the achieved peak current of ∼150 A.
The frequency-domain technique is based on the measurement of the coherent radiation emitted by the electron bunch via some electromagnetic process. In the coherent regime (i.e. for frequencies as P ∼ Q 2 ωu ω l dω|S(ω)| 2 ∝ Q 2 /σ t (see annex for details). The typical signal observed as the charge is varied is presented in Fig. 7: the observed nonlinear behavior confirms that the emitted radiation is not incoherent. From simulation we expect the power to scale as P ∝ Q 1.37 (see annex for details) a number close to the one resulting from the fit of the data: P ∝ Q 1.57 .
In Figure 8, the measured bolometer output signal versus the phase of L1 is compared with the expectation (1) calculated from the simulated phase space density and (2) computed from the measured bunch time profile obtained by zero-phasing. As expected the increase of the coherent signal is an unambiguous signature of the bunch being compressed (the charge was monitored during the measurement and remained constant to 200±20 pC).
The data points computed from the measured time profile f meas were obtained by numerically computing the Fourier transform of the bunch time-profile (using a FFT algorithm) and by performing the integration: where R(ω) stands for the frequency response of the detection system.
To generate the data points from the simulated phase space distributions f simu we write the time-profile, S(t) as a Klimontovitch distribution: N being the number of macro-particle used (50000 in the simulations presented hereafter) and t i the time of arrival of the i-th macro-particle. Eq. (9) allows to write the integrated power as: Though Figure 8 shows the signal increases as the bunch is compressed, there are discrepancies between the measurement and the two calculations for the short bunch case, we believe this is due to the lack of a precise knowledge of the transmission line frequency response.

IV. CONCLUSION
We have measured the bunch length dependence on the phase of a traveling wave accelerating structure located just downstream from an rf electron source. We could compress the bunch by a factor ¿3, down to ∼0.5 ps, for a bunch charge of 200 pC. In our experimental setup, a stronger compression is currently difficult to achieve without significantly impinging the transverse phasespace quality. The linac section used for the compression also plays a crucial role in achieving low emittance since it quickly accelerates the beam at energies of approximately 60 MeV thereby freezing the transverse phase space. Hence operating the first linac far off-crest reduces the final energy and impact the emittance since transverse space charge forces scale as 1/γ 2 . An improvement of our experiment would be to surround the accelerating structure used as a bunch compressor with a solenoidal lens to enable a better control of the beam transverse envelope and emittance [14,15].

This work was sponsored by US-DOE grant number DE-AC02-76CH00016 and by the Deutsches
Elektronen-Synchrotron institute. We are indebted to Luca Serafini of Univ. Milano for carefully reading and commenting the manuscript.

Appendix: Dependence of radiated power on bunch charge
Let's consider the case of a Gaussian distribution: The corresponding bunch form factor takes the form: and the integrated bunch form factor in the [ω l , ω u ] frequency interval is: The integration of the latter equation can be written in term of "error" function: Taking into account the limit of the erf function, lim z→∞ erf(z) = 1, and lim z→0 erf(z) = 0, and assuming the frequency range is so that σ t ω u ≫ 1 and σ t ω l ≪ 1, we finally have for the radiated power: Figure 9 shows the dependence of the bunch length versus the charge expected from simulations .
We find σ t ∝ Q 0.43 and thus we would expect the radiated power to be P ∝ Q 1.57 which is close to the value deduced from the fit of the measurement presented in Fig. 7: P ∝ Q 1.37 .