Characterization of soft X-ray echo-enabled harmonic generation free-electron laser pulses in the presence of incoherent energy modulations

Echo-enabled harmonic generation free-electron lasers (EEHG FELs) are promising candidates to produce fully coherent soft x-ray pulses by virtue of efficient high harmonic frequency up-conversion from UV lasers. The ultimate spectral limit of EEHG, however, remains unclear, because of the broadening and distortions induced in the output spectrum by residual broadband energy modulations in the electron beam. We present a mathematical description of the impact of incoherent (broadband) energy modulations on the bunching spectrum produced by the microbunching instability through both the accelerator and the EEHG line. The model is in agreement with a systematic experimental characterization of the FERMI EEHG FEL in the photon energy range $130-210$ eV. We find that amplification of electron beam energy distortions primarily in the EEHG dispersive sections explains an observed reduction of the FEL spectral brightness that is proportional to the EEHG harmonic number. Local maxima of the FEL spectral brightness and of the spectral stability are found for a suitable balance of the dispersive sections' strength and the first seed laser pulse energy. Such characterization provides a benchmark for user experiments and future EEHG implementations designed to reach shorter wavelengths.

Echo-enabled harmonic generation free-electron lasers (EEHG FELs) are promising candidates to produce fully coherent soft x-ray pulses by virtue of efficient high harmonic frequency upconversion from UV lasers. The ultimate spectral limit of EEHG, however, remains unclear, because of the broadening and distortions induced in the output spectrum by residual broadband energy modulations in the electron beam. We present a mathematical description of the impact of incoherent (broadband) energy modulations on the bunching spectrum produced by the microbunching instability through both the accelerator and the EEHG line. The model is in agreement with a systematic experimental characterization of the FERMI EEHG FEL in the photon energy range 130 − 210 eV. We find that amplification of electron beam energy distortions primarily in the EEHG dispersive sections explains an observed reduction of the FEL spectral brightness that is proportional to the EEHG harmonic number. Local maxima of the FEL spectral brightness and of the spectral stability are found for a suitable balance of the dispersive sections' strength and the first seed laser pulse energy. Such characterization provides a benchmark for user experiments and future EEHG implementations designed to reach shorter wavelengths.

I. INTRODUCTION
Free electron lasers (FELs) have enabled a new way for researchers to explore electronic dynamics at molecular and atomic scales via femtosecond pulses, gigawatt peak powers, and tunable wavelengths in the range of extreme ultraviolet to hard x-rays [1]. Self-amplified spontaneous emission (SASE) FELs generate a spiky spectrum and therefore offer relatively poor longitudinal coherence [2,3]. Seeded FELs are, at present, the only devices producing stable pulses with good longitudinal coherence at wavelengths now approaching the water window [4,5].
Echo-enabled harmonic generation (EEHG) was conceived as a seeding method with excellent high harmonic conversion efficiency to generate transformlimited radiation pulses down to soft x-rays [6][7][8][9][10][11]. By utilizing two laser modulations and dispersive sections (DSs), a monochromatic (coherent) energy modulation is imprinted on to the relativistic electron beam and transformed to a high harmonic density modulation (see Fig.1). The beam then enters the undulator-radiator where the density-modulated (bunched) electrons radiate coherently at wavelengths up to ∼ 100 times shorter than that of the UV seeding lasers. With sufficient gain, the radiation can be amplified up to saturation.
Recently, the impact of phase variations in the seed lasers on EEHG performance was investigated experimentally [12], illustrating the capability to shape the EEHG FEL spectrum by tuning the second seed laser power and phase. The agreement between experimental data and the preceding theory [13][14][15] was obtained in a condition in which energy non-uniformities of the electron beam could be neglected. On the other hand, it is well know that energy distortions in the electron beam can impact the EEHG bunching spectrum (See, e.g., Refs. [16][17][18]). Particularly relevant to this work, the impact of incoherent energy modulations on EEHG performance was discussed theoretically in [19][20][21]. Here, we examine the details of these studies by means of a extensive theoretical formulation of the evolution of energy and density modulations in EEHG. In particular, we examine and compare the measured FEL performance with an analytical model that includes incoherent modulations in the electron beam longitudinal phase space that develop from the early beam acceleration process through the final EEHG transformations.
It is known that in seeded FEL systems, uncontrolled energy structures can lead to a broader FEL bandwidth and reduced peak spectral intensity. These energy structures, accumulated during beam manipulation in the accelerator, can hardly be removed completely. They can introduce extra frequencies into the FEL gain bandwidth that deteriorate the longitudinal coherence promised by external seeding [21][22][23]. Such structures are generally due to the build-up of beam (1). In particular, ∆p1,2 is the incoherent energy modulation (see eq.(2)) and bµ1,µ2 is the incoherent bunching factor (see eq. (13)). After DS2, the nano-bunched electron beam travels into the radiator (R) and emits coherent and powerful light pulse.
collective effects such as coherent synchrotron radiation (CSR) and longitudinal space charge (LSC) during acceleration and compression, both contributing to the so-called microbunching instability (MBI) [24][25][26] whose maximum gain, depending on the compression setting, is typically at final wavelengths λ 0 0.5µm. Here we report on results of a systematic investigation at the FERMI FEL operating in EEHG mode [10] where we find good agreement between theory and experimental data. These studies help to benchmark the analytic model which thus provides a practical tool for the design and optimization of EEHG sources at even shorter wavelengths.
The paper is organized as follows. Section II discusses the theory of MBI in EEHG. Section III presents the experiment results and compares them with theory. Conclusions are reached in Section IV.

A. Theoretical Background
The evolution of the electron beam longitudinal phase space through the EEHG line in the presence of energy distortions is described by the following equations [20,21], : (1) P 2 =P 1 + A 2 (z 1 ) sin(k s2 z 1 ) + ∆p 2 (z 1 ), z 2 =z 1 + B 2 P 2 /k s1 , where A 1,2 (z) = ∆E 1,2 (z)/σ E is the normalized coherent energy modulation from seed lasers, and B 1,2 = k s1 R (1,2) 56 σ E /E is the normalized energy dispersion in the chicanes, E is the electron beam mean energy and σ E is the RMS slice energy spread, and ∆p 1,2 (z) the energy distortions of the electron beam distribution. The role of these terms is carefully investigated in the following.
In this description, ∆p 1 represents any energy structure accumulated in the electron beam up to the entrance of the first EEHG chicane B 1 . ∆p 2 is used to capture the integrated effect of CSR from B 1 and of LSC in second modulator. They can be expressed as the superposition of monochromatic modulations of different amplitudes [21,27]: where φ 1,2µ is a random phase. When the energy distortions ∆p 1,2 are ignored and in the assumption of uniform lasers A 1,2 (z) = A 1,2 (i.e., seed durations much longer than bunch duration), the Fourier transform of the electron beam density distribution -the so-called bunching factor -can be calculated at the exit of the second EEHG modulator according to [6]: where a E = n + mk s2 /k s1 is the harmonic number with integer numbers n and m. The EEHG wave number is k E = a E k s1 , and ζ E = nB 1 + a E B 2 . This factor is known to help characterize the EEHG performance and is optimized approximately at ζ E = j n,1 /A 1 where j n,1 is the first root of J n .
If the energy distortion ∆p 2 and second laser A 2 are sufficiently slowly-varying longitudinally that we can approximate their functional dependence as z 1 ≈ z, then the EEHG bunching factor close to the harmonic peak becomes [15,21]: where f (z) is electron beam density distribution function, and B = B 1 + B 2 . In the first two gain lengths in the radiator (R in Fig.1) the intensity of the FEL radiation is estimated to grow like ∝ z 2 |b n,m (k)| 2 . In the limit of negligible slippage the final radiation spectral pulse properties are given by |b n,m (k)| 2 . Thus, by virtue of the general expression for the energy modulations given in (2), this equation can be used to quantify the spectral effect of broadband energy modulations induced by MBI on the FEL output.

B. Bunching Phase
The z-dependent additional bunching phase due to electron beam energy distortions in Eq.(4) is: From this, one can obtain the moments of the spectral bunching distribution and gain insight into the relative magnitude of the contributions from ∆p 1,2 [21]. MBI-induced energy modulations accumulated up to the exit of the first modulator, ∆p 1 , are multiplied by the small scaling parameter |ζ E | 1. Linear, quadratic, and sinusoidally-shaped initial modulations were investigated in [20], where it was shown that the smallness of ζ E accounts for the insensitivity of the EEHG bunching spectrum to small initial perturbations. However, as discussed in [21], energy modulations ∆p 2 that develop between the EEHG chicanes are multiplied by the much larger factor a E B 2 ≈ m/A 2 1, and therefore can have a noticeable impact on the final bunching spectrum at high harmonics. The is the bandwidth due to the nonlinear phase structure, where brackets denote integration over the z-dependent amplitudes in the integrand in (4). Assuming the bunching longitudinal envelope is determined by the second seed laser and that it is a TL Gaussian pulse, the relative bandwidth in the case of optimized bunching absent MBI can be approximated as [15]: whereσ ks2 is the relative second seed bandwidth. Inserting Eq.
(2) for broadband energy distortions into the phase in (5), the instantaneous spatial bunching frequency is k z = k E + ψ (z), where: is the z-derivative of the additional phase. The mean bunching frequency is then k z . Thus, ψ gives the spectral shift from k E , and σ ψ gives the excess bandwidth due to the distortions. Assuming that the characteristic MBI wavelengths are small compared the length of the bunching envelope (e.g, k µ σ ks ) and that the individual phases φ 1,2µ are uncorrelated over µ, bandwidth of |b n,m (k)| 2 is therefore:

C. Bunching Amplitude
Similarly to the phase, the bunching factor in Eq.(4) for generic energy distortions is here specialized for MBI-induced energy modulations described in Eq.(2). It becomes: With the definition of the bunching spectrum b n,m (k), we can now quantify the presence of sidebands and/or of a broader spectral pedestal in EEHG. The EEHG bunching amplitude evaluated for k = a E k 1s can be calculated when whereb n,m is the z-integration of Eq.(10) for k = a E k 1s and demonstrates the bunching factor when MBI is absent.
In above equation we use the Bessel function relationship for integer ν value J −ν (x) = (−1) ν J ν (x). In the case of long seed lasers (i.e A 1,2 (z) = A 1,2 ) and a uniform electron beam, it is easy to see thatb n,m =b n,m . Note that the bunching is suppressed at the roots of the two Bessel functions. Assuming that the arguments of J l1 in Eq.(11) are less than 1, the high order of Bessel functions can be ignored and the leading term can be expanded around 0. In doing so, the bunching factor can be simplified to:

D. Modeling the Microbunching Instability
For numerical modeling, the expressions of the perturbed bunching factor can now be made explicit for the MBI-induced broadband energy modulations accumulated up to the second EEHG DS. At typical frequencies k µ 1/σ z , we can write [21,28]: where p 1,2 , b µ1,µ2 (k µ ), and Z 1,2 (k) are the broadband energy modulation, broadband bunching factor, and LSC impedance per unit length in the first and second modulator of length L 1,2 . I and I A = 17045A are the bunch peak current and the Alfven current, and Z 0 = 377Ω is the free space impedance. The in-vacuum LSC impedance Z(k µ ) through the modulator is [28,29]: with the effective beam transverse size r b = 0.8735 x β x + y β y , where x,y and β x,y are emittance betatron functions in x and y directions, γ z = γ/ 1 + K 2 u /2 is the longitudinal Lorentz factor inside an undulator with the (peak) untapered undulator parameter K u , and C is compression factor.
The MBI-induced energy modulation ∆p 1,2 in Eq.(2) are calculated numerically by means of a comprehensive linear gain model of the instability from beam injection into the accelerator up to the undulator line, including longitudinal energy-dispersion function (R 56 ) and CSR in the magnetic compressor, LSC and intrabeam scattering, and beam heating at low energy [30,31]. The model starts from a shot-noise like initial bunching factor and provides the bunching factor at any point along the line as well.
In the simplified case of single-stage beam compression and assuming linear gain regime of the instability, the amplification of density modulation, or gain [24,26], in the presence of an arbitrary incoming energy distribution V (P 0 ) [29], goes like: where k µf = k µ0 /(1+hR 56 ), k µ0 and k µf are modulation wavelength before and after the beam compression, and h is the initial beam energy chirp. Equation (15) suggests that the gain can be reduced by energy Landau damping, i.e., by increasing the beam uncorrelated energy spread at relatively low beam energies. Indeed, this is now accomplished at several FEL facilities through the laser heater (LH) system [28,[32][33][34], whose accurate control has become a tool to optimize the FEL spectral brightness in the presence of MBI.

A. Bandwidth Enlargement and Central Frequency Fluctuation
The EEHG experiment was conducted with an electron beam accelerated through the FERMI linac to the final energy of E=1.32 GeV. The beam normalized emittance measured in front of the undulator amounts to approximately 1 mm mrad in both transverse planes. The electron bunch is compressed by a factor C ∼10 to reach a final peak current in the core of I=700 A. The spectral content of the energy distortion amplitudes predicted by the MBI model for two different LH energy spread settings is shown in the right Fig. 2 subplots.
The integrated impact of these distortions matches well the measured FEL bandwidth, which is substantially reduced for a LH-induced energy spread ≥ 30 keV. The model allows us to explain the observations on the basis of MBI-induced energy modulations augmented by the first EEHG dispersive section, where p 2 (λ µ ) in absent of the first seed results always larger than p 1 (λ µ ).
The different MBI sensitivity of the EEHG bandwidth for the cases n = −1 and n = −2 is explained by means of Eq. (8). On the one side, p 1 (λ µ ) is multiplied by the EEHG scaling factor, which therefore can be modified to change the sensitivity of the final bunching to the electron beam energy perturbations coming from the accelerator. On the other side, p 2 (λ µ ) is multiplied by a E B 2 , with |B 2 | ≈ nB 1 /a E , such that a higher value of |n| forces larger values R (2) 56 of the second dispersion section. Equation (7) suggests that, by virtue of larger values of B 2 in the presence of MBI, frequency fluctuations in the configuration n = −2 for fixed ζ E , should be larger than in n = −1.
tends to 1. Likewise, when the MBI is more pronounced at low LH pulse energies, the product function approaches zero. At the same time, owing to the large LH-induced energy spread (larger than 40 keV), the FEL gain is diminished and therefore the FEL intensity is reduced. We note that in the Γ function, p 2 (λ µ ) is multiplied by a E B 2 , which explains the different behaviour of the function for n = −1 and n = −2, in agreement with the experimental observation.

C. Impact of first Seed Laser
It is well-known that in the processes of harmonic emission driven by an external laser, the seed laser-induced energy modulation has to exceed the uncorrelated energy spread of the beam at the undulator entrance.
Moreover, the EEHG bunching becomes less sensitive to MBI with increased laser modulations. This leads to the question if and to which extent the seeding laser pulse energy could be increased in order to counteract the effect of MBI, before preventing any further lasing by exceeding the FEL normalized energy bandwidth.
To answer this question, Fig.5-left plot illustrates the FEL intensity recorded as function of the first seed laser pulse energy, for two values of the LH pulse energy. Since the bunching is more sensitive to the coherent energy modulation in the first modulator at higher harmonics, the experiment was done for harmonic 18. The beam energy was 1.1 GeV and the compression factor about 7, for approximately 550 A in the bunch core. EEHG was set in configuration n = −1 (R (1) 56 = 1.9 mm and R (2) 56 = 98 µm). The figure shows that, once the FEL emission is built up for the seed laser pulse energy of ∼ 15-20 µJ, the intensity is weakly affected by even larger seed energies. In particular, a stronger seed laser is not able to recover the intensity reduction due to a weaker heating effect. The right plots provide the theoretical explanation of the experimental data. They show the spectrum of broadband energy modulation at the exit of the second modulator for different coherent energy modulations from the first seed laser, at two LH pulse energies. The contour plots are generated by inserting the energy distribution modified by the first seed laser into Eq. (15), which allows us to calculate the MBI gain at the exit of the first DS. Using such spectral gain function as an input to Eq.(13), the spectrum of the MBI-induced broadband energy modulation at the exit of the second modulator is eventually derived as function of the first seed pulse energy. An increase of the first coherent modulation is effective in removing incoherent energy modulations at initial wavelengths shorter than 10 µm, or ∼ 1 µm at the entrance of the undulator. However, the effect becomes negligible at immediately longer wavelengths.
We plugged these two sets of energy modulation and different first seed energies into Eq. function (Γ), see bottom plots. It is shown that, while a strong beam heating is able to shift the product function to 1 or so, an increase of A 1 is not able to recover a unitary product function (compare red starts and blue stars for σ LH = 16 keV and σ LH = 24 keV, respectively). The second plot of the second row compares the bunching factor with (dashed lines with stars, Eq.(13)) and without MBI (doted lines with circle, Eq. (3)). The plot illustrates the degradation of the bunching factor by MBI, at different coherent energy modulation amplitudes from the first seed. Finally, we find that the measured FEL intensity (left plot) at the two LH pulse energies is in agreement with the theoretical behaviour of the bunching factor: by increasing the first seed energy, the FEL intensity grows up, to eventually fall down for exceeding seeding energies.

IV. CONCLUSIONS
Electron beam imperfections play a significant role in determining the spectral bandwidth and the pulse intensity of EEHG FEL emission. While EEHG is predicted to be more robust than other external seeding schemes to energy distortions that occur upstream, it is also anticipated that distortions that occur between the EEHG chicanes can significantly impact the FEL spectrum. This due in part to the combination of a strong magnetic chicane in the EEHG set up (compared to HGHG, for example) and longitudinal space charge forces acting through the second modulator that together can amplify incoherent microbunching generated upstream in the accelerator. Here, the role played by the instability in different EEHG configurations was illustrated with experimental data, and good agreement was found with start-to-end semi-analytical results for the perturbed bunching factor. As such, the model turns out to be a practical tool for the design and optimization of short wavelength EEHG sources. Notably, the higher the EEHG harmonic jump, the more sensitive the FEL sprectral brightness is to incoherent energy modulations through the second modulator. Moreover, different balances of the strength of the two dispersive sections change the impact of MBI onto the FEL spectrum and intensity, with smaller value of the |n|-factor less sensitive to incoherent energy modulations, as predicted. Finally, attempts to maximize the FEL intensity with a stronger coherent energy modulation from the first seed laser pulse were successful only for moderate or strong beam heating in the first stages of acceleration. This identifies a limitation in recovering optimal EEHG performance through the seed laser pulse energy. Further, it suggests a careful control of the instability in the accelerator, and a consequent optimization of the EEHG set up in the presence of relatively large heating levels.

Appendix A
The derivation of equations (13) follows the strategy proposed in [20]. Starting from the general expression (4) in presence of energy distortions, we retain only the lowest order contribution near the harmonic: b(k) ≈ +∞ −∞ dzχ(z, k)e i(k−k E +nψ1+mψ2)z × e i(−ζ E ∆p1(z)−a E B2∆p2(z1))z . (A1) From here on, we will assume that ∆p 1 is small enough not to alter significantly the phase space distribution after the first dispersive region. Also, we will further simplify our calculation taking ∆p 2 (z 1 ) = ∆p 2 (z). This assumption is true as long as ∆p 2 (z 1 ) is a sufficiently slowly-varying function.
Using the definition of ∆p 1,2 given in eq.(1), and the identity it is possible to expand the phase associated to broadband energy modulations, The maximum EEHG bunching amplitude evaluated for k = a E k 1s is: b n,m (a E k 1s ) =b n,m ∞ µ=0 J 0 (−ζ E p 1 (k µ )) J 0 (−a E B 2 p 2 (k µ )) .
(A3) Assuming that the energy distortions amplitudes, namely p 1,2 , are small compared to the energy modulations induced by the seed laser, we can expand the Bessel functions around 0. For a generic α, we have We truncate the series at second order in distortion amplitudes, obtaining b n,m (a E k 1s ) ≈b n,m ∞ µ=0 1 − 1 4 (ζ E p 1 (k µ )) 2 + (a E B 2 p 2 (k µ ))