# Tunability of a red-shifted free electron laser

Marie Labat, Eléonore Roussel, Philippe Berteaud, Frédéric Blache, Yen-Yu Chang, Jurjen Couperus Cabadağ, Amin Ghaith, Maxwell Laberge, Stéphane Lê, Susanne Schöbel, et al.

► To cite this version:

Marie Labat, Eléonore Roussel, Philippe Berteaud, Frédéric Blache, Yen-Yu Chang, et al. Tunability of a red-shifted free electron laser. Physical Review Accelerators and Beams, 2025, 28 (2), pp.020702. 10.1103/PhysRevAccelBeams.28.020702. [https://doi.org/10.1103/PhysRevAccelBeams.28.020702]

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<figure><img src="image_1.png" /></figure>
Tunability of a red-shifted free electron laser

Marie Labat<sup>1,*</sup> Eléonore Roussel<sup>2</sup> Philippe Berteaud,<sup>1</sup> Frédéric Blache,<sup>1</sup> Yen-Yu Chang<sup>3</sup>,

Jurjen Couperus Cabadağ<sup>3</sup>, Amin Ghaith,<sup>3</sup> Maxwell LaBerge<sup>3</sup>, Stéphane Lê,<sup>1</sup>

Susanne Schöbel<sup>3,4</sup> Ulrich Schramm<sup>3,4</sup> Patrick Ufer<sup>3,4</sup> Mathieu Valléau,<sup>1</sup>

Marie-Emmanuelle Couprie,<sup>1</sup> and Arie Irman<sup>3</sup>

<sup>1</sup>Synchrotron SOLEIL, L'Orme des Merisiers, Départementale 128, 91190 Saint-Aubin, France

<sup>2</sup>Univ. Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes et Molécules, F-59000 Lille, France

<sup>3</sup>Helmholtz-Zentrum Dresden - Rossendorf, Bautzner Landstrasse 400, 01328 Dresden, Germany

<sup>4</sup>Technische Universität Dresden, 01062 Dresden, Germany

<figure><img src="image_2.png" /></figure>

(Received 18 September 2024; accepted 22 January 2025; published 19 February 2025)

The development of red-shifted free electron lasers (FELs), based on strongly chirped electron beams, emerged a few years ago. Here we present a thorough study of the tunability of a seeded red-shifted FEL. An analytical model is introduced to describe the complex FEL resonance condition and its consequence for the FEL emission wavelength. This model is then benchmarked with experimental data from the COXINEL (COherence Xray source INferred from Electrons accelerated by Laser) FEL beamline driven by a laser plasma accelerator at Helmholtz-Zentrum Dresden-Rossendorf. We show that the FEL emission wavelength can be tuned not only with the undulator gap and taper geometry but also with the time delay between the seed laser pulse and the electron beam. Particularly a combination of individually chirped electron and seed beams promotes a wider tuning range compared with the usual undulator gap. This could serve as an additional yet simple tuning knob to tailor the FEL emission wavelength.

DOI: 10.1103/PhysRevAccelBeams.28.020702

# I. INTRODUCTION

Free electron lasers (FELs) [1–4] represent state-of-the-art coherent light sources uniquely expanding the range of available pulse parameters as compared to classical lasers. Operated down to the hard x-ray range, they provide ultrashort pulses with unprecedented peak power, allowing for the observation of matter at atomic temporal and spatial scales [5].

Similar to conventional lasers, FELs rely on a gain medium enabling the amplification process. The gain medium is provided through a relativistic electron beam wiggling in the periodically alternating magnetic field of an undulator. The amplification process manifests in an energy exchange between an initial copropagating radiation field and the wiggling electron beam. Withdrawing resonantly kinetic energy from the electrons, the radiation is exponentially amplified until saturation, while the FEL gain, or amplification power, is essentially driven by the electron beam quality. Such an interaction occurs at the so-called resonant wavelength [6]:

$$ \lambda_R = \frac{\lambda_u}{2\gamma^2} (1 + K_u^2/2), \quad (1) $$

where $\lambda_u$ is the undulator period, $K_u$ is the undulator deflection parameter, proportional to the undulator magnetic field, and $\gamma$ the Lorentz factor of the electrons. In the oscillator [7] configuration, the radiation being amplified is simply the spontaneous synchrotron radiation emitted by the electrons in the undulator and stored pass after pass of the beam in an optical cavity of critical spectral response. In the self-amplified spontaneous emission (SASE) [8,9] configuration, the radiation being amplified is also the spontaneous synchrotron radiation, however, within one single pass inside the undulator thanks to a higher gain. In the seeded configuration [10], it is an external coherent laser source tuned to $\lambda_R$. Seeding thus accelerates the first energy exchange stage but is also a powerful strategy to control the pulse shape and guarantee longitudinal coherence.

Over the past decades, several additional advanced FEL schemes were developed, such as self-seeding [11], high-gain harmonic generation [12], and echo-enabled harmonic generation [13], allowing for further performance increase and refined pulse tailoring. More recently, strongly chirped beam configurations were introduced, leading to a red-shifted FEL line [14]. This redshift can be used to facilitate FEL light extraction through a narrow-band mirror while using a recirculating accelerator [15]. Toward more compact

<footnote>*Contact author: marie.labat@synchrotron-soleil.fr</footnote>

<footnote>Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.</footnote>
FELs, a new category of FELs driven by laser plasma accelerators (LPAs) [16] instead of the traditional radio-frequency accelerators is emerging [17–22]. LPAs’ ultrahigh (>100 GV/m) accelerating gradients [23,24] could contribute to reducing the size of the accelerator facility, especially at high energies. Preliminary experiments with LPA beams only enabled the observation of spontaneous synchrotron radiation [25–29] due to the large energy spread and divergence of the LPA beams dramatically reducing the FEL gain. The use of a strong chirp, i.e., energy-time correlation, was then suggested to decompress the beam, i.e., to reduce the slice energy spread [30] at the cost of reducing the peak current. The first LPA based seeded FEL used this concept [19], leading, in addition to the seed centered at 269 nm, to the generation of a red-shifted coherent radiation around 275 nm.

FELs are highly tunable by adjusting the undulator deflection parameter for a fixed undulator period and/or the electron beam energy. Even in the seeded configurations, several techniques were developed such as accelerator tuning [31], frequency pulling [32], or frequency mixing [33,34] to recover tunability.

In this work, we explore the tunability of these emerging red-shifted FELs. An analytical model is introduced to understand the chirp dependency of the resonant condition and describe the resulting redshift of the FEL wavelength. The model predictions are then successfully confronted with experimental measurements on the LPA based seeded FEL of COXINEL (COherence Xray source INferred from Electrons accelerated by Laser) at HZDR (Helmholtz-Zentrum Dresden-Rossendorf). The present study demonstrates a superior tunability of red-shifted FELs with the injection of a chirped, i.e., time-dependent instantaneous frequency, seed.

## II. RED-SHIFTED FEL MODELING

### A. Red-shift mechanism

Red-shifted FELs rely on the same fundamental mechanism as standard FELs: an interaction between a bunch of relativistic electrons wiggling in an undulator and an electromagnetic radiation field at the resonant wavelength $\lambda_R$. This interaction first leads to an energy exchange, further converted by the undulator’s dispersive sinusoidal magnetic field into a density modulation at $\lambda_R$. This density modulation then allows for the emission of an additional coherent radiation at the same $\lambda_R$. Provided a sufficient FEL gain, mainly driven by the electron beam brightness, this coherent radiation is exponentially amplified until saturation. Red-shifted FELs differ from standard FELs in that their final wavelength is longer than the initial resonance wavelength. The additional mechanism responsible for this redshift is depicted in Fig. 1. The relativistic electron beam with uncorrelated phase space (a) first passes through a chicane. The chicane stretches the beam (b), inducing an

<figure><img src="image_3.png" /><figcaption>FIG. 1. Principle of a red-shifted FEL: the electron beam at accelerator exit (a) passes through a magnetic chicane inducing a strong energy chirp (b). Inside undulator, induced by radiation interaction, the electron beam gets density modulated (c) at $\lambda_R$, a modulation wavelength that is stretched (d) due to energy variation along $\lambda_R$ and undulator dispersion. The resulting emitted coherent radiation is redshifted (e). Insets: longitudinal energy distribution of the particles (i.e., phase space) at each stage of the red-shift process.</figcaption></figure>

energy and/or time correlation (chirp) while reducing the slice energy spread [30]. The resulting chirp can be described as

$$ \gamma(t) = \gamma_0(1 + ct/R_{56}) \quad (2) $$

with $\gamma_0$ the reference energy, $R_{56}$ the chicane strength, c the speed of light, and t the longitudinal position along the beam ($t < 0$ corresponds to the head of the beam). Conventionally, a 4-dipole symmetric magnetic chicane provides $R_{56} < 0$. Thus, at the exit of the chicane, the beam head features higher electron energies than the tail.

When the chirped beam enters and wiggles inside the undulator in the presence of a radiation set at $\lambda_R$, it experiences energy and density modulation at $\lambda_R$ (c), as on standard FELs. However, the strongly chirped electron bunch is stretched due to the undulator’s dispersion, leading to a stretching of the modulation wavelength too (d). As a result, the final coherent radiation is emitted at a wavelength slightly longer than the initial modulation wavelength $\lambda_R$, i.e., being redshifted (d). This red-shifted FEL wavelength can be analytically derived according to [14]

$$ \Delta\lambda_{\text{FEL}} = \frac{(1 + K_u^2/2)L_{\text{eff}}}{\gamma_0^2 R_{56}} \lambda_{\text{seed}}. \quad (3) $$

$L_{\text{eff}}$ is the effective undulator length inducing the beam modulation stretching. This length is essentially governed by the efficiency of the interaction inside the undulator, determined by the 3D (temporal, spectral, and spatial) overlap between seed and electron beam, while under present experimental conditions, no dependence on seed laser power was observed. At maximum efficiency, $L_{\text{eff}}$ nearly equals $L_u$ the undulator length. Modifying our notations $R_{56}$ and $L_{\text{eff}}$ into $1/h$ and $R_{56}$, respectively, the above formula is fully equivalent to the one derived
in [15]. Note that the same phenomenon can be obtained in both SASE and seeded configuration and was already exploited on standard FELs [35,36].

## B. Analytical modeling of the resonance wavelength

The tunable seeded red-shifted FELs are based on the combination of individually chirped electron and seed beams. The electron beam is chirped in a chicane, leading to a longitudinal variation of the energy $\gamma(t)$ of the particles. Thus, the electron beam resonant wavelength $\lambda_R$ varies along the beam. Inserting Eq. (2) into Eq. (1) gives

$$ \lambda_R(t) = \lambda_u(1 + K_u^2/2)/(2\gamma(t)^2). \quad (4) $$

The seed can also be chirped via a dispersive medium, leading to a similar time-dependent wavelength $\lambda_{\text{seed}}(t)$. Taking into account the delay $\tau$ between seed and electron beam, the seed wavelength can be described by

$$ \lambda_{\text{seed}}(t, \tau) = \lambda_{\text{seed-0}} + (t - \tau)/D_{\lambda}, \quad (5) $$

with $\lambda_{\text{seed-0}}$ the seed wavelength at $t = \tau = 0$ and $D_{\lambda}$ the seed group delay dispersion.

As a result, the resonance condition can only be fulfilled, for each delay $\tau$, at one single longitudinal position $t_0$ along the beam, as shown in Fig. 2, satisfying $\lambda_{\text{seed}}(t_0, \tau) = \lambda_R(t_0)$, i.e.,

$$ \lambda_{\text{seed-0}} + (t_0 - \tau)/D_{\lambda} = \lambda_u \times (1 + K_u^2/2)/(2\gamma(t_0)^2). \quad (6) $$

<figure><img src="image_4.png" /><figcaption>FIG. 2. FEL resonance condition and consequent FEL tunability. Electron beam longitudinal charge distribution (green area), resonance wavelength along the beam for two different $K_u$ values (green lines): $K_{u0}$ (continuous line) and $K_{u1} > K_{u0}$ (dashed line). Seed longitudinal intensity (blue area), seed longitudinal wavelength (blue lines) for two different $\tau$ values: $\tau_0$ (continuous line) and $\tau_1 > \tau_0$ (dashed line). Resonance location $t_0$ found at crossing of $\lambda_R(t)$ and $\lambda_{\text{seed}}(t)$ (black vertical lines), shifting with $K_u$ (dashed line) and $\tau$ (dotted line). Spectral shift of the resonance wavelength (red arrow).</figcaption></figure>

For one given $K_u$ value enabling to meet the resonance condition at $t = 0$ and $\tau = 0$, i.e., satisfying

$$ \lambda_{\text{seed-0}} = \lambda_u \times (1 + K_u^2/2)/(2\gamma_0^2), \quad (7) $$

a first expression of $t_0$ is obtained [19]:

$$ t_0 = \left[ \frac{1}{D_{\lambda}} + \frac{\lambda_u c (1 + K_u^2/2)}{\gamma_0^2 R_{56}} \right]^{-1} \times \frac{\tau}{D_{\lambda}}. \quad (8) $$

A more general expression of $t_0$, i.e., valid for any $K_u$ value, is derived in the following. Introducing the expression of $\gamma(t_0)$ into Eq. (6) leads to

$$ \lambda_{\text{seed-0}} + \frac{t_0 - \tau}{D_{\lambda}} = \frac{\lambda_u}{2\gamma(t_0)^2} \times (1 + K_u^2/2) \\ = \frac{\lambda_u}{2\gamma_0^2(1 + ct_0/R_{56})^2} \times (1 + K_u^2/2). \quad (9) $$

Since $ct_0/R_{56} \ll 1$, the derivation of $t_0$ can be continued at first order as follows:

$$ \lambda_{\text{seed-0}} + \frac{t_0 - \tau}{D_{\lambda}} \approx \frac{\lambda_u}{2\gamma_0^2} \left[ 1 - \frac{2ct_0}{R_{56}} \right] \times (1 + K_u^2/2) $$

$$ \frac{t_0}{D_{\lambda}} + \lambda_{\text{seed-0}} - \frac{\tau}{D_{\lambda}} \approx -\lambda_u ct_0 / (\gamma_0^2 R_{56}) \times (1 + K_u^2/2) \\ + \lambda_u / (2\gamma_0^2) \times (1 + K_u^2/2) $$

$$ t_0 \left[ \frac{1}{D_{\lambda}} + \lambda_u c / (\gamma_0^2 R_{56}) \times (1 + K_u^2/2) \right] \\ \approx \frac{\tau}{D_{\lambda}} - \lambda_{\text{seed-0}} + \lambda_u / (2\gamma_0^2) \times (1 + K_u^2/2). $$

Finally, $t_0$ can be expressed as

$$ t_0 = \left[ \frac{1}{D_{\lambda}} + \frac{\lambda_u c (1 + K_u^2/2)}{\gamma_0^2 R_{56}} \right]^{-1} \\ \times \left[ \frac{\tau}{D_{\lambda}} + \frac{\lambda_u}{2\gamma_0^2} \left( 1 + \frac{K_u^2}{2} \right) - \lambda_{\text{seed-0}} \right]. \quad (10) $$

Introducing $t_0$ into Eq. (3), one finally gets the refined formula for the FEL wavelength redshift:

$$ \Delta\lambda_{\text{FEL}}(\tau, K_u) = \frac{(1 + K_u^2/2)L_{\text{eff}}}{\gamma_0^2 R_{56}} \left( \lambda_{\text{seed-0}} + \frac{t_0 - \tau}{D_{\lambda}} \right), \quad (11) $$

while the final FEL wavelength is

$$ \lambda_{\text{FEL}}(\tau, K_u) = \lambda_R(t_0) + \Delta\lambda_{\text{FEL}}(\tau, K_u). \quad (12) $$

Note that this model only describes the resonance and FEL wavelengths, not the tunability range of the FEL since seed
and electron beam temporal envelops are not taken into account.

As shown in Fig. 2, the final FEL wavelength is expected to be tunable within the FEL gain bandwidth not only with the undulator’s deflection parameter $K_u$ but also with the seed delay $\tau$. From Eq. (11) and Fig. 2, the chirp $D_\lambda$ of the seed also appears as a key parameter: it could drive the FEL final wavelength and probably act on the temporal range of interaction, i.e., on the final FEL pulse duration [37–39]. These points are to be addressed in a future work.

The undulator’s deflection parameter can be modified for the undulator as a whole or it can be varied along the undulator. In this so-called tapered case $K_u$ is $z$ dependent. In the simple linear taper case, $K_u$ follows:

$$K_u(z) = K_u(z=0) + z \times \frac{\Delta K_u}{L_u} \quad (13)$$

with $\Delta K_u$ the variation of $K_u$ from undulator entrance to exit and $z$ the longitudinal position along the undulator, $z=0$ being the undulator entrance. In the first approximation, the final FEL wavelength is expected to follow Eq. (12) substituting $K_u$ with $K_u(L_u)$. However, the taper also modifies the FEL gain. Indeed, with a positive taper ($\Delta K_u > 0$), the undulator gap decreasing toward the exit), the resonant wavelength increases along the undulator, i.e., possibly helps maintain the resonance condition with the red-shifting FEL and eventually enhances the FEL growth. Conversely, with a negative taper ($\Delta K_u < 0$), the undulator gap increasing toward the exit), the resonant wavelength decreases, i.e., possibly aggravates the loss of the resonance condition and eventually inhibits the FEL. A more accurate modeling of the effect of the taper would require a change of the undulator-average stretching calculation method, which is beyond the scope of this paper.

### III. EXPERIMENTAL SETUP

The experiment was performed at the Helmholtz-Zentrum Dresden-Rossendorf (HZDR), combining the Dresden laser acceleration source (DRACO) of HZDR [40]

with the COXINEL (COherence Xray source INferred from Electrons accelerated by Laser) beam manipulation line [41,42]. The experimental setup is shown in Fig. 3(a) with parameters of interest listed in Table I.

**LPA beam.** The LPA is driven by the 100-TW-class arm of DRACO and operated in a tailored self-truncated ionization-induced injection scheme [43]. The LPA performance is optimized for high-charge and low-divergence beams by shaping the gas-density profile directly at the accelerator exit to induce a plasma lens effect [44,45]. The LPA is tuned in order to produce beams with a central energy $E_e$ of 188 MeV with 5% energy spread, resulting in a charge density of typically 4.5 pC/MeV, while the measured divergences and emittances are (1.5;1) mradrms and (1.5;1) mm mrad in the horizontal and vertical planes [19], respectively.

**Beam transport.** The LPA beam is transported and manipulated through the COXINEL beamline. A triplet of variable permanent magnet quadrupoles [46], located behind the LPA, first handles the electron beam divergence for chromatic emittance growth mitigation. The beam is then chirped (decompressed) in a four-dipole-magnet chicane, before passing through four quadrupoles to create a chromatic sliding focusing [47] inside the undulator. This undulator consists of 97 periods of 20 mm length providing a maximum deflection parameter $K_u$ of 2.47 at 4 mm gap. Thanks to these manipulations, the beam quality inside the undulator is noticeably improved [19]: the slice energy spread is reduced down to 0.1%, leading to a peak current of 130 A to a Pierce parameter [9] of 0.003 and a 3D gain length [48] of 0.6 m. Electrons are finally deflected with a dipole, allowing for photon-beam diagnostics.

**Seed.** The seed is generated by frequency conversion of a small fraction of the driver laser beam, providing a pulse energy $E_{seed}$ of 0.5 µJ within a bandwidth $\Delta\lambda_{seed}$ of 3.9 nm-FWHM at a central wavelength $\lambda_{seed-0}$ of 269 nm. The seed is stretched to an estimated duration ($\Delta T_{seed}$) of 1 ps-FWHM, inducing a strong linear chirp of the seed wavelength, as described in Eq. (5).

<figure><img src="image_5.png" /><figcaption>FIG. 3. (a) Experimental layout. The IR laser (DRACO) is focused into a gas jet to produce an LPA beam, further transported through COXINEL beamline consisting of quadrupoles (Quads), a four-bending magnet chicane, an undulator, and an ending dump dipole. Seed generated from a fraction of the main IR laser and injected into COXINEL beamline using a flat mirror inside chicane. FEL radiation is imaged at undulator exit by a pair of lenses with focal length $f$, (L1) with $f = 0.4$ m and (L2) with $f = -0.2$ m, onto a 2D spectrometer with a magnification of 2. (b) Example of spectro-spatial spectrometer record (using an 0.05 mm wide entrance slit) in FEL operation, with (L2) set for imaging at undulator exit, with result of image analysis for FEL wavelength extraction: maximum (open circle) and median (filled circle) FEL wavelength. Image displayed on a logarithmic scale, with a color scale on the image on the right side.</figcaption></figure>
<table><caption>TABLE I. Parameters of the red-shifted FEL components. Beam parameters are given at the LPA source point.</caption><thead><tr><th>Parameter</th><th>Notation</th><th>Value</th></tr></thead><tbody><tr><td>Energy</td><td><i>E</i><sub>e</sub></td><td>188.8 MeV</td></tr><tr><td>Relative energy spread</td><td><i>σ</i><sub>e</sub></td><td>5%-rms</td></tr><tr><td>Duration</td><td><i>σ</i><sub>z</sub></td><td>2 µm-rms</td></tr><tr><td>Charge</td><td><i>Q</i></td><td>100 pC</td></tr><tr><td>Charge density</td><td><i>dQ</i>/<i>dE</i></td><td>4.5 pC/MeV</td></tr><tr><td>Divergence</td><td><i>σ</i>'<sub>x0</sub>, <i>σ</i>'<sub>y0</sub></td><td>(1.5;1) mrad-rms</td></tr><tr><td>Normalized emittance</td><td><i>ε</i><sub>x0</sub>, <i>ε</i><sub>y0</sub></td><td>(1.5;1) mm mrad</td></tr><tr><td>Chicane strength</td><td><i>R</i><sub>56</sub></td><td>-1.8 mm</td></tr><tr><td>Undulator period</td><td><i>λ</i><sub>u</sub></td><td>20 mm</td></tr><tr><td>Number of periods</td><td><i>N</i><sub>u</sub></td><td>97</td></tr><tr><td>Nominal deflection parameter</td><td><i>K</i><sub>u</sub></td><td>2.31 (gap 4.4 mm)</td></tr><tr><td>Maximum deflection parameter</td><td><i>K</i><sub>u</sub></td><td>2.47 (gap 4.0 mm)</td></tr><tr><td>Central seed wavelength</td><td><i>λ</i><sub>seed-0</sub></td><td>269 nm</td></tr><tr><td>Seed bandwidth</td><td><i>Δλ</i><sub>seed</sub></td><td>3.9 nm</td></tr><tr><td>Seed group delay dispersion</td><td><i>D</i><sub>λ</sub></td><td>-0.296 ps/nm</td></tr><tr><td>Seed duration</td><td></td><td>1 ps-FWHM</td></tr><tr><td>Pulse energy</td><td><i>E</i><sub>seed</sub></td><td>0.5 µJ</td></tr><tr><td>Waist position</td><td></td><td>1 m</td></tr><tr><td>Rayleigh length</td><td></td><td>5 m</td></tr></tbody></table>

**FEL main diagnostic.** The FEL radiation at the undulator exit is collected by a pair of spherical lenses, allowing for near-field imaging of the radiation at the undulator exit onto a two-dimensional (2D) UV spectrometer. The spectral resolution of the spectrometer is driven by the aperture of its entrance slit. The recorded images therefore correspond to the 2D spatio-spectral intensity distribution of the FEL radiation, that is, vertical position y versus spectral distribution in wavelength λ, as shown in Fig. 3(b). These intensity patterns typically consist of a main spot centered at λ = 296 nm, the seed, and of a secondary spot found redshifted to ≈274 nm, the FEL. The FEL spot is displaced vertically by ≈ −0.4 mm due to undulator imperfections causing a kick on the electron beam orbit. Finally, for spectrometer entrance slit apertures below 0.1 mm, interference fringes in between seed and FEL can be observed, corresponding to the temporal coherence signature of our seeded FEL [14,19].

**FEL data analysis.** The analysis of the FEL raw data for the following studies is nontrivial and therefore detailed below in four main steps: (i) FEL raw experimental data consist of 2D images recorded on the 2D spectrometer. These raw images are first treated and corrected to take into account the spectrometer spectral response (including spectrometer inner grating, focusing mirror, and CCD respective spectral responses). Examples of final images are shown in Fig. 4 with in (a) the seed only and in (b) the seed and the FEL. (ii) The next step is to extract an FEL spectral profile. Since for the present experimental configuration, the spectro-spatial FEL signal is always weaker than the seed one, the seed centroid is first identified

<figure><img src="image_6.png" /><figcaption>FIG. 4. Example of FEL data analysis. Spectro-spatial intensity distributions recorded in near-field on the 2D ending spectrometer with (a) seed only (b) seed and FEL. Images are normalized after treatment and displayed on a logarithmic scale with a colorbar given in inset. Image treatment: removal of hot pixels and background, masking of the left part of the image suffering from readout defects, smoothing of the intensity distribution using a median filter with a kernel of 3 pixels, and correction to take into account the spectrometer spectral response. FEL analysis in (b): seed centroid (blue dot) detected by scanning the treated image with a 10 × 10 pixels square to locate the maximum intensity region while identifying the seed centroid as the center of mass of this region. Seed area (blue ellipse) and vertical position (blue dashed line) $y_{FEL*}$ for seed profile extraction. FEL area (red contour) defined restricting the out-of-ellipse area to $\lambda > 271$ nm and to $y_{FEL-G} \pm 10$ pixels, with $y_{FEL-G}$ a guess of the FEL vertical location. FEL centroid (red dot) identified using the same previously mentioned square scan technique and (red dashed line) FEL vertical position $y_{FEL}$. (c) Spectral profile of (blue) the seed extracted at $y_{FEL*}$ and (red) the FEL extracted at $y_{FEL}$ integrating over $\pm 5$ pixels besides respective ys, with FEL (open circle) maximum and (filled circle) median wavelengths. (d) FEL isolated trace with the same (open circle) and (filled circle) FEL wavelengths.</figcaption></figure>

[see blue dot in (b)]. A predefined ellipse, corresponding to the seed spectro-spatial area, is then used to crop the image around this seed centroid [see blue ellipse in (b)]. The inside of the ellipse is assumed to correspond to the seed contribution, while the FEL contribution, thanks to its redshift, is assumed to be found outside of this ellipse. Since the FEL is known to be redshifted, the out-of-ellipse area is first restricted spectrally to $\lambda > 271$ nm. Since the FEL is also known to be vertically shifted with respect to
the seed, the out-of-ellipse area is then restricted vertically to a narrow band centered on $y_{FEL-G}$, a guess of the vertical FEL location driven by the first shots of each series (the FEL vertical position is indeed very stable once the electron beam orbit is set). A predefined FEL area is obtained [see red contour in (b)], in which the FEL centroid is detected [see red dot in (b)]. The vertical position of this centroid is defined as the final FEL vertical location $y_{FEL}$ displayed with red dashed line in (b). The FEL spectral profile is finally extracted from the 2D spectro-spatial images along $y_{FEL}$ as shown in (c). (iii) The FEL intensity is then defined as the number of counts obtained from the integration of the image within the previously defined FEL area though only taking into account pixel counts above 20% of the maximum number of counts in the area. (iv) The determination of an FEL wavelength is the most difficult step of the analysis, essentially due to the weakness and the spectral width of the FEL line. We chose to keep two definitions for the FEL wavelength. First, the median FEL wavelength $\lambda_{median}$ is defined as the median $\lambda$ value of the FEL area defined previously [see (filled circle) in (c) and (d)]. The second FEL definition requires the calculation of the FEL isolated spectral profile, i.e., the FEL spectral profile subtracted from the main seed spectral profile. Since the seed intensity distribution fluctuates shot-to-shot, the seed spectral profile has to be extracted from the same image as the FEL one. Assuming a y symmetric seed distribution, one can consider that the seed-only spectral profile at $y_{FEL}$ is the same as the seed spectral profile at $y_{FEL*}$, where $y_{FEL*}$ is the symmetric of $y_{FEL}$ with respect to the seed centroid [see blue dashed line in (b)]. The seed does not exhibit indeed any transverse chirp (spatial-wavelength dependence) as shown in (a) and the FEL spectral modulation has vanished at this distance as shown in (b). The seed spectral profile is therefore extracted from the 2D spectro-spatial images integrating the images along $y_{FEL*}$ as shown in (c). Subtracting the seed spectral profile to the FEL spectral profile leads to the isolated trace shown in (d). The second FEL definition, the maximum FEL wavelength $\lambda_{max}$, is finally defined as the wavelength at the isolated trace maximum intensity location above 273 nm [see (open circle) in (c) and (d)]. The two wavelength definitions typically differ by 1 nm.

#### IV. TUNABILITY WITH UNDULATOR DEFLECTION PARAMETER $K_u$

The experimental tunability of the red-shifted FEL with the undulator deflection parameter $K_u$ is shown in Fig. 5. $K_u$ is varied from 2.25 up to 2.425, the minimum and maximum $K_u$ values before FEL extinction, while a set of ten spectrometer images was recorded for each $K_u$ value. Figures 5(a) and 5(b) first present a sample of the results at three different $K_u$ values (i–iii), with the spectro-spatial intensity distribution of the best (highest FEL intensity) shot (a) and all spectral profiles with possibly FEL

<figure><img src="image_7.png" /><figcaption>FIG. 5. FEL tunability versus $K_u$. (a) Typical single-shot experimental observations of spectro-spatial intensity distributions while varying the undulator $K_u$ value from (i) 2.428 (gap = 4.1 mm), to (ii) 2.347 (gap = 4.3 mm) and (iii) 2.27 (gap = 4.5 mm). Images displayed on a logarithmic scale with a color scale in inset. Interference fringes are no longer visible because the spectrometer entrance slit was opened to 0.4 mm, preventing the resolution of the sub-nm fringing pattern. (b) Series of FEL spectra recorded at same $K_u$ values as in (a,i–iii), with corresponding (open circle) maximum and (filled circle) median extracted wavelengths. (c) FEL wavelength versus $K_u$ from experiment with (open circle) maximum and (filled circle) median wavelengths, and from (dashed lines) analytical model using Eq. (12). FEL intensity at maximum wavelength versus $K_u$ (c-inset). Parameters are given in Table I.</figcaption></figure>

associated (filled circle) median and (open circle) maximum wavelengths (b) extracted following the data analysis method described in Fig. 4. When $K_u$ decreases from (i) to (iii), the FEL redshift decreases, in very good agreement with the prediction of Eq. (12) shown with a white in (a) and black in (b) dashed line. In (c), the complete tuning curve is presented. The experimental measurements, illustrated with (filled circle) median and (open circle) maximum wavelengths, follow the model (black line) of Eq. (12). The shot-to-shot fluctuations of the FEL wavelength are mainly due to the shot-to-shot fluctuations of the electron beam quality, essentially charge density at the reference energy, affecting the efficiency of the interaction inside the undulator, i.e., the red-shift mechanism driving the FEL wavelength. The inset in (c) also presents the
evolution of the FEL intensity versus $K_u$. The shot-to-shot intensity fluctuations are also due to the electron beam variations. The observed tuning range for $K_u$ spans from 2.2 up to 2.425. This range corresponds to an FEL variation of ≈3 nm. Whereas in standard seeded FELs, the output wavelength is determined by the seed wavelength, in the case of red-shifted FELs, a slight tunability versus $K_u$ is obtained, which is, by itself, a valuable result.

The measurement of the red-shifted FEL tunability with the undulator taper $\Delta K_u$ is then shown in Fig. 6. As shown in Fig. 5, panel (a) presents the best spectro-spatial intensity distributions, while panel (b) displays all recorded spectral profiles exhibiting an FEL signal, for different $\Delta K_u$ values. The FEL spectral centroid redshifts, as predicted, when the taper is increased along lines from (i) $\Delta K_u = -0.038$ to (v) $\Delta K_u = 0.08$. Because of the FEL spectral profile distortion with taper, the FEL wavelength extraction was abandoned. It still clearly appears that, as expected, the FEL spectral centroid slightly deviates in absolute from the predictions of Eq. (12), depicted by vertical lines. However, all other relative tendencies are found in the predicted

<figure><img src="image_8.png" /><figcaption>FIG. 6. FEL tunability versus $\Delta K_u$. (a) Typical single-shot experimental observations of spectro-spatial intensity distributions while varying the undulator taper value $\Delta K_u$ from (i) -0.038, (ii) 0., (iii) 0.008, (iv) 0.04, and (v) 0.08. Images displayed on a logarithmic scale with a color scale in inset. Interference fringes are no longer visible because the spectrometer entrance slit was opened to 0.4 mm, preventing to resolve the sub-nm fringing pattern. (b) Single-shot (thin black line) and average (thick black line) FEL spectra recorded at the same $\Delta K_u$ values as in (a,i-v). Model prediction from Eq. (12) is shown in (a) and (b) using $K_u$ at undulator entrance with (dashed line) and $K_u$ at undulator exit, i.e., including taper, with (blue line). Parameters are given in Table I.</figcaption></figure>

direction, including the inhibition of the FEL for a negative tapering, and the transitory enhancement of the FEL for a positive tapering. Whereas tapering is mainly used in standard FELs to compensate for an energy loss in the electron beam [49], it is used here to compensate for a shift in the FEL wavelength.

## V. TUNABILITY WITH SEED DELAY $\tau$

The red-shifted FEL tunability with $\tau$, the delay between chirped seed and electron beam, is now explored. The measurements are shown in Fig. 7 with examples of spectro-spatial intensity distributions (a), spectra and extracted wavelengths (b), and the full tuning curve (c) with corresponding intensity (inset). The agreement with the model of Eq. (12) is very good. As expected, the FEL wavelength shifts with $\tau$. The tuning range (c) in $\tau$ is ≈1.5 ps, leading to an FEL wavelength variation of more than 5 nm. The tunability capability offered by seed delay (5 nm) is significantly higher than the one by undulator deflection parameter (3 nm).

<figure><img src="image_9.png" /><figcaption>FIG. 7. FEL tunability versus $\tau$. (a) Examples of spectro-spatial intensity distributions for $\tau$ value from (i) +500 fs, to (ii) 0 fs and (iii) -500 fs. Images displayed on a logarithmic scale with a color scale in inset. (b) Series of FEL spectra recorded at same $\tau$ values as in (a,i-iii), with corresponding (open circle) maximum and (filled circle) median extracted wavelengths. (c) FEL wavelength versus $\tau$ from experiment with (open circle) maximum and (filled circle) median wavelengths, and from (dashed line) analytical model using Eq. (12). FEL intensity at maximum wavelength versus $\tau$ (c-inset). Parameters are given in Table I.</figcaption></figure>
## VI. CONCLUSION

In summary, we have introduced an analytical model to describe the complex resonant wavelength of a seeded red-shifted FEL and its consequence for the FEL emission. It reveals that by using, in addition to a chirped electron beam, a chirped seed pulse, the FEL emission wavelength can be tuned with the time delay between the seed pulse and the electron beam. The model is compared to systematic measurements at the LPA-based seeded FEL of COXINEL by varying undulator parameters, such as the undulator gap and taper configuration, as well as the time delay. The agreement between model and experiment demonstrates that the time delay of the chirped seed, essentially equivalent to the spectral tuning of a narrow-band seed of equal spectral intensity, represents a new tuning knob. In our specific electron beam and seed duration/chirp configuration, this new knob even provided a wider wavelength range, i.e., superior tunability, than the undulator gap. In the case of low-energy electron beam-based FELs, whereas tuning with the undulator gap or even taper can affect the beam dynamics such as focusing, tuning with the seed delay offers the advantage of leaving the electron beam dynamics nonimpacted.

## ACKNOWLEDGMENTS

The COXINEL project was supported by the European Research Council for the Advanced Grants COXINEL [340015; Principal Investigator (PI), M.-E. Couprie]. The laser-electron acceleration project at Helmholtz-Zentrum Dresden-Rossendorf is fully supported by the Helmholtz Association under the program “Matter and Technology,” the topic “Accelerator Research and Development.” E. R. is supported by the LABEX CEMPI project (ANR-11-LABX-0007), the ANR-DFG ULTRASYNC project (ANR-19-CE30-0031), the Ministry of Higher Education and Research, Hauts de France Council, and the European Regional Development Fund (ERDF) through the Contrat de Projets Etat-Région (CPER) WaveTech. M.-E. C. and M. Labat are supported by Laserlab-Europe V (Contract No. 871124).

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