# Spatiospectral properties of redshifted free electron laser radiation

Marie Labat, Rui Prazeres, Eléonore Roussel, Yen-Yu Chang, Amin Ghaith, Maxwell Laberge, Susanne Schöbel, Ulrich Schramm, Patrick Ufer, Marie-Emmanuelle Couprie, et al.

► To cite this version:

Marie Labat, Rui Prazeres, Eléonore Roussel, Yen-Yu Chang, Amin Ghaith, et al. Spatiospectral properties of redshifted free electron laser radiation. Physical Review Research, 2025, 7 (2), pp.023061. 10.1103/PhysRevResearch.7.023061. [https://doi.org/10.1103/PhysRevResearch.7.023061]

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Distributed under a Creative Commons Attribution 4.0 International License
Spatiospectral properties of redshifted free electron laser radiation

Marie Labat<sup>1,*</sup> Rui Prazeres,<sup>2</sup> Eléonore Roussel,<sup>3</sup> Yen-Yu Chang,<sup>4</sup> Amin Ghaith,<sup>4</sup> Maxwell LaBerge,<sup>4</sup> Susanne Schöbel,<sup>4,5</sup> Ulrich Schramm,<sup>4,5</sup> Patrick Ufer,<sup>4,5</sup> Marie-Emmanuelle Couprie,<sup>1</sup> and Arie Irman<sup>4</sup>

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

<sup>2</sup>Institut de Chimie Physique, CNRS UMR 8000, Université Paris-Saclay, 91405 Orsay Cedex, France

<sup>3</sup>Univ. Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes et Molécules, F-59000 Lille, France

<sup>4</sup>Helmholtz-Zentrum Dresden - Rossendorf, Bautzner Landstrasse 400, 01328 Dresden, Germany

<sup>5</sup>Technische Universität Dresden, 01062 Dresden, Germany

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(Received 3 January 2025; accepted 25 March 2025; published 16 April 2025)

Free electron lasers (FELs) are advanced light sources delivering pulses with unprecedented intensity down to the hard x-ray range. Redshifted FEL represents a new scheme exploiting strongly chirped electron beams to emit tunable coherent radiation at wavelengths longer than the FEL's resonant condition. While theoretical studies and proposals for this scheme are rapidly advancing, experimental evidences are so far very limited. Here, we report on the nature of spatiospectral properties of a redshifted FEL, presenting analytical and numerical models benchmarked with experimental data from the COXINEL (coherence x-ray source inferred from electrons accelerated by laser) FEL beamline driven by a laser plasma accelerator at Helmholtz-Zentrum Dresden-Rossendorf (Germany). We reveal how those properties are governed by a peculiar, V-shape like, transverse spatial phase inherited from single-electron synchrotron radiation. Moving the observation point inside the undulator, i.e., backpropagating the radiation, it is shown how this synchrotron radiation inheritage imprints localized fringes on the spatiospectral intensity distribution, uniquely allowing to retrieve the FEL source point location. Building on these results, we anticipate a twofold impact towards development of the future high-average power cavity-based x-ray regenerative amplifier FEL and further advancement of a compact laser-plasma-based x-ray FEL.

DOI: 10.1103/PhysRevResearch.7.023061

I. INTRODUCTION

Free electron lasers (FELs) [1] represent unique research tools, providing intense pulses suitable for the investigation of matter at temporal and spatial resolution of atomic scales [2]. They constantly revolutionize user science.

FELs are instruments relying on a relativistic electron beam wiggling in the periodically alternating magnetic field of an undulator. While copropagating with electro-magnetic radiation, electrons and radiation interact, leading to an energy modulation of the electron beam at the so-called resonance wavelength [3],

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

where $\lambda_u$ is the undulator period, $K_u$ the undulator deflection parameter and $\gamma$ the electron beam Lorentz factor. This energy modulation is further converted into a density modulation under the natural undulator dispersion, leading to the emission of coherent radiation at $\lambda_R$. Provided sufficient electron

beam quality, driving the FEL gain, this coherent radiation further draws energy from the electrons and is exponentially amplified until saturation.

In the self-amplified spontaneous emission (SASE) configuration [4], the initial electromagnetic radiation is simply the spontaneous synchrotron radiation (SR) emitted in the undulator. Today, the majority of x-ray FEL facilities relies on this configuration, which provides partially longitudinally coherent pulses with a typical shot-to-shot intensity stability of 10 % limited by the inherent shot noise starting point, though without intrinsic limits in terms of radiation wavelength. Over the years, a variety of FEL schemes, such as single-spike emission [5] or self-seeding [6–9] have brought an improvement in coherence while the lack of shot-to-shot stability remains an issue. Generation of fully coherent and stable FEL light has been possible thanks to the use of coherent radiation, such as an external seed laser, as a seed to trigger and expedite the FEL amplification process [10,11]. Different seeding schemes, such as high-gain harmonic generation [11–13] and more recently echo-enabled-harmonic-generation [14–18], have enabled to address and improve the FEL pulse performance and quality, for instance by increasing their temporal coherence [19], shortening their duration [20] or offering multicolor [21,22] and variable polarization [23,24] features, but only down to the soft x-ray regime.

More recently, FEL developments have also turned to high repetition rate (MHz) operation, energy-consumption saving

<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>
<figure><img src="image_3.png" /><figcaption>FIG. 1. Basic features of a redshifted FEL. (a) Redshift mechanism. A strongly chirped electron beam is injected in the undulator, interacts with the copropagating radiation, leading to a density modulation at $\lambda_R$. The energy chirp then stretches this modulation period leading to the emission of coherent radiation at a wavelength $\lambda_{FEL} > \lambda_R$. (b) Spectral intensity of the FEL radiation in (dashed red) SASE and (solid red) seeded configuration at the undulator exit. (c) Transverse phase profile of the FEL radiation at the undulator exit at (blue) $\lambda_R$ the resonance wavelength and (red) $\lambda_{FEL}$.</figcaption></figure>

and compactness. One path towards high repetition rates and energy saving could be the use of recirculating machines instead of single-pass accelerators. Cavity-based FELs have been proposed, such as the x-ray regenerative amplifier FEL (XRAFEL) [25] and the XFEL oscillator (XFELO) [26]. Both are still confronted with the critical issue of radiation out-coupling. However, the use of a strongly chirped electron beam was recently suggested on an XRAFEL to purposefully redshift the FEL line out of the narrow reflection bandwidth of the Bragg mirror to ease light extraction [27]. Towards more compact FEL facilities, the use of laser plasma accelerators (LPAs) [28] is being considered [29]. Relying on high-intensity laser pulses focused in a plasma to create millimeter-scale accelerating structures supporting ultrahigh field gradients (>100 GV/m) [30,31], LPAs could help reduce the accelerator length especially for very high energies (beyond GeV electron beams). Since their first demonstration [32–34], LPAs have experienced tremendous progress greatly improving the delivered beam quality [35–38]. LPA-based FELs were recently demonstrated in the SASE configuration at 27 nm [39] and in the seeded configuration at 270 nm [40]. Together with other encouraging experiments [41–43], those results have renewed the interest for LPA based FELs. However, most LPAs still suffer from a large energy spread (typically ≈ 1%) dramatically reducing the FEL gain. In [40], to overcome this issue, the electron bunch of few femtosecond duration was strongly chirped to recover a small slice energy spread [44]. This strong chirp induced a stretching of the initial modulation wavelength leading to a redshift of the FEL line [45]. Redshifted FELs once more appear in the landscape of the present FEL developments.

In this paper, we further investigate the physics of redshifted FEL. We demonstrate that redshifted FEL exhibits a V-shaped transverse spatial phase owing to single-electron SR inheritance, and show how this phase shape drives its spatiospectral distribution. As the central finding, we reveal that, in backpropagation, this inherited peculiar phase imprints localized fringes onto the FEL spatiospectral distribution, allowing for the retrieval of the FEL source point location inside the undulator. This paper relies on analytical and numerical models as well as experimental measurements performed on the LPA based seeded FEL of COXINEL (coherent xray source inferred from electrons accelerated by laser) implemented at the Helmholtz-Zentrum Dresden-Rossendorf (HZDR).

## II. REDSHIFTED FEL FUNDAMENTALS

The mechanism responsible for the FEL redshift is recalled in Fig. 1(a). The initial LPA beam, exhibiting a large energy spread, although without longitudinal energy correlation (chirp), is first decompressed in a magnetic chicane. The resulting strongly energy-chirped electron beam is injected into the undulator and interacts with the spontaneous SR or an external coherent source, leading to its density modulation at the resonance wavelength $\lambda_R$. This strong chirp then induces a stretching of this modulation. As a result, the coherent radiation at the undulator exit is emitted at a wavelength $\lambda_{FEL}$ longer than $\lambda_R$, i.e., redshifted with respect to $\lambda_R$. The magnitude of this redshift $\Delta\lambda_{FEL}$ can be analytically derived according to [45]

$$ \Delta\lambda_{FEL} = \frac{(1 + K_u^2/2)L_{eff}}{\gamma^2 R_{56}} \lambda_R, \quad (2) $$

with $R_{56}$ as the chicane strength parametrizing pulse stretching. $L_{eff}$ is the effective undulator length inducing the beam modulation stretching. This length is essentially driven by the efficiency of the interaction inside the undulator, determined by the 3D (temporal, spectral, and spatial) overlap between seed and electron beam. At maximum efficiency, $L_{eff}$ nearly equals the undulator length $L_u$. Thus, for a given efficiency inside the undulator, the magnitude of the redshift is simply driven by the undulator and the electron beam energy chirp.

Typical redshifted FEL spectra are depicted in Fig. 1(b) for both SASE and seeded configurations. While in the SASE case (dashed red line) the spectrum exhibits random spikes, a signature of limited temporal coherence inherited from the shot-noise starting point, in the seeded case (continuous red line) the FEL line exhibits a well-defined fully coherent radiation profile. For low-gain FELs, as depicted in Fig. 1(b), the residual seed is still visible besides the FEL line, while for high-gain FELs, the seed contribution falls into noise level.

Figure 1(c) finally introduces the transverse phase of the redshifted FEL. At $\lambda_R$, this phase is rather smoothly curved (blue line) but at $\lambda_{FEL}$, i.e., on the red side of the resonance, it becomes sharply “V” shaped (red line). These peculiar shapes of transverse phase are the result of single-electron SR inheritance. FELs are well known indeed to inherit most of their spatial properties from SR [24,46,47].
<figure><img src="image_4.png" /><figcaption>FIG. 2. Single-electron SR transverse properties at (a) the exit of the undulator and (b) the middle of the undulator. Intensity distribution at (i) $\lambda_R = 268$ nm and (ii) $\lambda = 276.6$ nm. (iii) Transverse phase along y at x = 0 and at (continuous line) $\lambda_R = 268$ nm and (dashed line) $\lambda = 276.6$ nm. Calculation using SR model (see Appendix A) with parameters of Table I.</figcaption></figure>

In [48], A. Hofmann introduces an analytical expression to describe the electric field of the SR emitted in the far-field by a single-electron wiggling inside a planar undulator. Starting from this expression, we model the SR electric field in the near-field, i.e., *virtually* emitted at any point inside the undulator (see Appendix A). As shown in Fig. 2(a), the single-electron SR transverse intensity distribution at the undulator exit exhibits a simple bell shape without intensity modulation at $\lambda_R$ and slightly modulated at $\lambda > \lambda_R$. The transverse phases extracted along x = 0 at $\lambda_R$ and $\lambda > \lambda_R$ [see white lines in (a-i) and (a-ii)] are then shown in (a-iii). From a smoothly curved

<table><caption>TABLE I. Parameters of the COXINEL FEL components at HZDR. Beam parameters are given at source point. Deflection parameter given at the minimum undulator gap of 4 mm.</caption><thead><tr><th>Parameter</th><th>Notation</th><th>Value</th></tr></thead><tbody><tr><td>Energy</td><td>E<sub>e</sub></td><td>188.8 MeV</td></tr><tr><td>Relative energy spread</td><td>σ<sub>e</sub></td><td>5%</td></tr><tr><td>Duration</td><td>σ<sub>z</sub></td><td>2 µm-rms</td></tr><tr><td>Charge</td><td>Q</td><td>100 pC</td></tr><tr><td>Charge density</td><td>dQ/dE</td><td>4.5 pC/MeV</td></tr><tr><td>Divergence</td><td>σ'<sub>x0</sub>, σ'<sub>y0</sub></td><td>(1.5;1) mrad-rms</td></tr><tr><td>Emittance</td><td>ε<sub>x0</sub>, ε<sub>y0</sub></td><td>(1.5;1) mm mrad</td></tr><tr><td>Chicane strength</td><td>R<sub>56</sub></td><td>1.8 mm</td></tr><tr><td>Undulator period</td><td>λ<sub>u</sub></td><td>20 mm</td></tr><tr><td>Number of periods</td><td>N<sub>u</sub></td><td>99</td></tr><tr><td>Undulator deflection parameter</td><td>K<sub>u</sub></td><td>2.31</td></tr><tr><td>Central seed wavelength</td><td>λ<sub>seed-0</sub></td><td>269 nm</td></tr><tr><td>Seed bandwidth</td><td>Δλ<sub>seed</sub></td><td>3.9 nm</td></tr><tr><td>Seed group delay dispersion</td><td>D<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>E<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>

shape at resonance, the phase evolves towards a strongly “V” shape on the red side of resonance, i.e., where the redshifted FEL line develops, as predicted in Fig. 1(c). In the middle of the undulator [see Fig. 2(b)], the single-electron SR transverse intensity distribution exhibits high-contrast rings. At $\lambda_R$ (b-i), these rings develop more than half a millimeter away from the optical axis, but on the red side of the resonance, i.e., at $\lambda > \lambda_R$, the spatial frequency of the rings increases, leading to a dense fringing pattern very close to the optical axis (b-ii). We derived an analytical formula to approximate this intensity profile in one dimension, along y at x = 0 (see Appendix A),

$$B(y) \propto J_0 \left( \frac{2\pi \theta_0 y}{\lambda_R} \right) \quad (3)$$

with $\theta_0$ as the opening angle at the observation wavelength $\lambda$,

$$\theta_0 = \sqrt{2 \frac{\lambda - \lambda_R}{\lambda_u}} \quad (4)$$

Equation (3) clearly links the spatial frequency of the rings with the observation wavelength $\lambda$: the more $\lambda$ is redshifted with respect to resonance, the wider $\theta_0$ and as a result the higher the frequency of the Bessel rings. The transverse phases extracted as previously [along white lines in (b-i) and (b-ii)] are shown in (b-iii). Both phases exhibit strong discontinuities leading to zeros in intensity, i.e., finally drawing rings in the xy plane.

### III. EXPERIMENTAL SETUP

The experiment was performed at HZDR, combining the Dresden laser acceleration source (DRACO) [49] with the COXINEL beam manipulation line [50–52]. The experimental setup is shown in Fig. 3(a) with parameters of interest listed in Table I.

The LPA is driven by the 100-TW-class arm of DRACO and operated in a tailored self-truncated ionization-induced injection scheme [53]. 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 [36,54]. The LPA is also 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 are (1.5;1) mrad-rms in the horizontal and vertical planes [40], respectively.

The LPA beam is transported and manipulated through the COXINEL beamline. A triplet of quadrupoles, located behind the LPA, first handles the electron beam divergence for chromatic emittance growth mitigation. The beam is then decompressed in a four-dipole-magnet chicane, introducing the strong energy chirp described as

$$\gamma(t) = \gamma_0(1 + ct/R_{56}) \quad (5)$$

with $t$ as the longitudinal position along the beam ($t < 0$ corresponds to the head of the beam), $\gamma_0$ as the reference electron beam energy at $t = 0$ and $c$ as the speed of light. After passing through four quadrupoles to create a chromatic sliding focusing [55], the beam enters an undulator consisting of 97 periods of 20-mm length and providing a maximum
<figure><img src="image_5.png" /><figcaption>FIG. 3. (a) Experimental layout. The high-power 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. Undulator output radiation is imaged onto a 2D spectrometer using a pair of lenses (L1) and (L2), with (L2) mounted on a translation stage to be able to scan the virtual object plane. (b) Example of spatiospectral intensity distribution of the seeded redshifted FEL recorded with the 2D spectrometer. Image displayed in logarithmic scale according to color bar in insert.</figcaption></figure>

deflection parameter $K_u$ of 2.47 at 4-mm gap. Thanks to these manipulations, the beam quality inside the undulator is dramatically improved [40], essentially the slice energy spread dropping down to 0.1%, leading to a Pierce parameter [56] of 0.003 and a gain length [57] of 0.6 m. Electrons are finally deflected with a dipole, allowing for photon-beam diagnostics.

The seed is generated by frequency conversion of a small fraction of the driver laser, 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 according to

$$ \lambda_{seed}(t) = \lambda_{seed-0} + (t - \tau)/D_{\lambda} \quad (6) $$

where $\lambda_{seed-0}$ is the seed wavelength at $t = \tau = 0$, $\tau$ is the delay with respect to the electron beam and $D_{\lambda}$ is the group-delay dispersion.

The radiation at the undulator exit is collected and imaged by a pair of spherical lenses onto the entrance slit of a two-dimensional (2D) UV spectrometer (see details in Appendix C). While the first convex lens position is fixed, the second concave lens can be moved longitudinally, allowing for near-field imaging of the radiation at any virtual object plane back inside the undulator. This system does not allow looking back in time at the radiation building process along the undulator, but virtually backpropagating in space the radiation collected at the undulator exit. An example of spectrometer record is shown in Fig. 3(b). The image corresponds to the 2D spatiospectral intensity distribution, that is, vertical position y versus spectral distribution in wavelength $\lambda$, of the radiation at the undulator exit. The 2D pattern consists of two spots: the seed centered at 269 nm and the redshifted FEL at 274 nm. The FEL spot is slightly vertically misaligned with respect to the seed owing to a kick on the electron orbit caused by the undulator imperfections. In between seed and FEL spots, vertical fringes can be distinguished. Seed and FEL are indeed two coherent spectrally spaced pulses, therefore producing interference fringes [45]. Note that these fringes can be observed only in the small-gain regime. For higher gains, the seed falling within noise, the fringes would disappear.

All experimental results presented below are spatiospectral intensity distributions recorded with this system in single shot. Because of the shot-to-shot fluctuations, an averaging of the distributions would blur the fine structures of interest.

Nevertheless, these shot-to-shot fluctuations, essentially owing to LPA beam energy and charge jitter, mainly affect the intensity of the FEL, not its spatiospectral shaping. The experimental results presented in the following are therefore a selection of representative single shots.

## IV. TRANSVERSE SHAPING

In a first step, the effect of the transverse phase profile on the transverse spatiospectral distribution is investigated. Using a semi-analytical seeded FEL model [45], the FEL is assumed to be the simple sum of two electromagnetic fields, the seed and an additional coherent emission, while each field is described by a 3D Gaussian including a chirp on the seed longitudinal phase [45]. Both seed and coherent pulses are assigned parabolic transverse phases. The spatiospectral intensity distribution predicted by this initial model is recalled in Fig. 4(a). Using a parabolic phase [see inset in (a)], smoothly curved parabolic fringes are obtained. The model was then refined to enable a specific phase shape definition for the coherent emission contribution. The sharp V phase profile previously modeled at 276 nm and recalled in inset

<figure><img src="image_6.png" /><figcaption>FIG. 4. Spatiospectral intensity distributions at undulator exit obtained with the double-pulse analytical model using a transverse (a) quadratic and (b) V-shaped phase for the FEL pulse. Input phases shown in the inset. Intensity distributions displayed normalized to their maximum and in logarithmic scale according to color bar in insert. Other parameters in Table I.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>FIG. 5. Spatiospectral intensity distributions at undulator exit obtained in (a) numerical simulation with perfect orbit, (b) numerical simulation taking into account experimental orbit, and (c) experiment. Intensity distributions displayed normalized to their maximum and in logarithmic scale according to color bar in inset. Red-dashed lines depict the X shape of the fringe pattern. [(a),(b)] (Inset) Transverse phases at 269 and 276 nm from simulation (black lines) along lines in (a) and (b) and from SR model (blue lines). Other parameters in Table I.</figcaption></figure>

in (b) was used as input to define the coherent emission field transverse phase, finally leading to the spatiospectral intensity distribution shown in (b). The fringes are no longer parabolic but triangularly shaped. The transverse phase profile drives the curvature of the interference fringes, so that redshifted FELs should exhibit triangularly shaped fringes.

We now move on to numerical simulations (see Appendix B). The spatiospectral intensity distribution simulated at the undulator exit assuming a perfect electron beam orbit is shown in Fig. 5(a). The global pattern consists as expected of the residual seed centered at 269 nm, the redshifted coherent emission centered around 275 nm, together with interference fringes in between. More importantly, those fringes are clearly triangularly shaped, as predicted. The vertical phase (at $x = 0$) of the electromagnetic field simulated at the undulator exit was extracted at different wavelengths and compared to the SR one [see inset in (a)]. At the resonance wavelength (dashed lines), the phases cannot be compared since it is mainly the seed phase, which is observed in the numerical case. However, at the FEL wavelength, i.e., 276 nm, the two phases match, confirming FEL inheritance of the single-electron SR transverse phase. Figure 5(b) shows the spatiospectral intensity distribution simulated including the electron beam orbit kick. Even though slightly distorted, the interference fringes still exhibit a triangular curvature while the transverse phase

still corresponds to the SR V-shaped phase [see inset in (b)]. Finally Fig. 5(c) presents an experimental data of the spatiospectral intensity distribution while imaging the undulator exit. The measured pattern is found in excellent agreement with the simulated one (b). Figures 4 and 5 confirm that redshifted FELs inherit the single-electron SR phase, a V-shaped phase since being selected on the red side of the resonance. This V phase drives in the case of a seeded configuration peculiar triangularly shaped vertical fringes in the FEL spatiospectral distribution at undulator exit.

## V. SPECTRAL SHAPING

We now explore the spectral shaping of the spatiospectral distribution. In Fig. 5, the fringes in between seed and coherent emission appear with a variable spacing, decreasing towards longer wavelengths, and with a flip of curvature towards shorter wavelengths forming a cross shape [see red-dashed cross in Fig. 5(a)] in between seed and coherent emission spectral centroids. Such a shape is typical in the frame of spectral interferometry with a chirped reference pulse [58]. Considering the seed as the reference pulse, it is the strong chirp of the seed, which causes a spectral dependance of the fringes spacing, while the center of the cross should sit where the two pulses, i.e., seed and coherent emission, spectrally overlap.

The spectral overlap occurs at resonance, i.e., where the seed wavelength equals the resonance wavelength of the electron beam. However, since the seed is strongly chirped, the seed wavelength $\lambda_{\text{seed}}$ is longitudinally, or time, dependent [see Eq. (6)] and since the electron beam is also strongly chirped [see Eq. (5)] the resonance wavelength is also time dependent,

$$ \lambda_R(t) = \frac{\lambda_u}{2\gamma(t)^2} (1 + K_u^2/2). \quad (7) $$

This longitudinal dependence of the seed and electron beam resonance wavelengths is illustrated in Fig. 6(a). As a result, the resonance (or spectral overlap) only occurs at the longitudinal position $t_0$ satisfying $\lambda_{\text{seed}}(t_0, \tau) = \lambda_R(t_0)$. The analytic expression of $t_0$ is derived in [59],

$$ t_0 = \left[ \frac{1}{D_{\lambda}} + \frac{\lambda_u c (1 + K_u^2/2)}{\gamma_0^2 R_{56}} \right]^{-1} \times \left[ \frac{\tau}{D_{\lambda}} + \frac{\lambda_u}{2\gamma_0^2} \left( 1 + \frac{K_u^2}{2} \right) - \lambda_{\text{seed}-0} \right]. \quad (8) $$

Inserting Eq. (8) into Eq. (7) enables to predict the spectral location $\lambda_R(t_0)$ of the overlap (see red-dashed lines in Fig. 6), while varying $\tau$ the delay between seed and electron beam as depicted in Figs. 6(a-i)-6(a-iii). If the cross shape sits indeed at the seed and coherent emission spectral overlap, it should be observed at $\lambda_R(t_0)$ and shift with the seed delay.

Figure 6(b) presents the evolution of the simulated spatiospectral intensity distribution of the ideal FEL (without orbit kick) while varying the seed delay. When $\tau$ increases, the cross shape shifts spectrally towards longer wavelength, sitting exactly at the predicted resonance location $\lambda_R(t_0)$. The addition of the orbit kick in the simulation [see Fig. 6(c)]
<figure><img src="image_8.png" /></figure>

FIG. 6. (a) Schematical scenario of the resonance wavelength versus delay. [(b)–(d)] Spectrospatial intensity distributions at the undulator exit as a function of seed delay: (i) $\tau = -500$ fs, (ii) $\tau = 0$ fs, and (iii) $\tau = +250$ fs. Data from (b) numerical simulation using a perfect beam orbit, (c) numerical simulations using the experimental beam orbit (with a kick), and (d) experimental measurements. Intensity distributions normalized and displayed in logarithmic scale, with color scale in the insets. (Vertical-red dashed lines) indicating the spectral location of the resonance predicted by the model in (a), (dashed rectangles) surrounding the resonance location. Other parameters in Table I.

shifts the cross shape only vertically. The corresponding measurements shown in Fig. 6(d) are found in very satisfactory agreement given the refinement of the desired structures.

Figure 6 confirms our full understanding of the complex shaping of the redshifted spatiospectral intensity distribution at the undulator exit. Following previous studies on reference pulse 1D spectrotemporal interferometry [60,61], we also introduce chirped reference pulse 2D spatiospectral interferometry, a tool that could be used to monitor for instance the time jitter between the seed and the electron beam.

## VI. BACKPROPAGATION FEATURES

Up to now, we focused on radiation spatiospectral intensity distributions at the undulator exit. We will now study their behavior in backpropagation.

We first compute the electric field of the spontaneous SR at the middle of the undulator, extract its component along $y$ (at $x = 0$) at the resonance wavelength, and propagate it using Fourier optics (see Appendix C) from $z = 1$ m to $z = -3$ m ($z = 0$ corresponding to the undulator exit). The intensity of this ($z, y$) distribution is shown in Fig. 7(a-i). As in the case

<figure><img src="image_9.png" /></figure>

FIG. 7. [(a)–(d)] Intensity distributions in (z, y) obtained by propagating the radiation field extracted along y at $x = 0$ from (a-i) SR model at $\lambda = 269$ nm, i.e., at the resonance wavelength, (b-i) SR model at $\lambda = 275$ nm, (c-i) numerical simulation without orbit kick at $\lambda = 275$ nm, and (d-i) numerical simulation including the experimental orbit kick at $\lambda = 275$ nm. [(a-ii)–(d-ii)] Phase profile along y extracted along the propagation at locations indicated in (a-i) to (d-i) with white lines. Other parameters in Table I.

of standard Gaussian beams, the obtained pattern is symmetric with respect to the center of the undulator. However, it also exhibits two waists, at the undulator entrance and exit, resulting from the undulator SR specific extended source point feature. The pattern also presents fringes with a high contrast in the middle of the undulator corresponding to the rings presented in Fig. 2(b-i). The phases extracted at different z locations along this propagation [see white lines in (a)] are given in Fig. 7(a-ii), echoing Fig. 2(b-iii). It appears that because of its slightly V shape, the SR transverse phase aliases besides undulator entrance and exit points, causing zeros in intensity, i.e., fringes in the (z, y) distribution and rings in the (x, y) distribution.

We then compute and propagate the electric field of the spontaneous SR at the redshifted FEL wavelength. The (z, y) intensity distribution and associated transverse phases of Figs. 7(b-i) and 7(b-ii) are obtained. The now strongly V-shaped transverse phase of SR aliases with a higher spatial frequency, giving a dense fringe pattern inside the undulator corresponding to the high-frequency rings of Fig. 2(b-ii). As at the resonance wavelength, the (z, y) distribution remains symmetric with respect to the middle of the undulator, but exhibits a *butterfly* shape. Beyond the undulator, the
<figure><img src="image_10.png" /><figcaption>FIG. 8. Spatiospectral intensity distributions versus *z*: (a) simulated and (b) measured at (i) $z = -1.5$ m, (ii) $z = -1.0$ m, (iii) $z = -0.4$ m, and (iv) $z = -0.1$ m. (a/b-i/ii/iii/iv-z) Zoom inside the white rectangles of (a/b-i/ii/iii/iv). All distributions normalized to their maximum and displayed in logarithmic scale, color scales being given in insert. Calculation parameters in Table I.</figcaption></figure>

intensity becomes null close to axis over an increasing vertical range while the distance to the undulator increases, owing to the off-axis nature of SR emitted on the red side of the resonance.

Propagation is now applied to the seeded FEL simulated electric field leading to Figs. 7(c-i) and 7(c-ii). The (z, y) intensity distribution presents similar butterfly shape and high-frequency fringes as the one of SR at the same wavelength, fringes again resulting from the transverse phase aliases. However, the spatial extension in *z* and *y* of the fringe pattern is significantly reduced and shifted towards the undulator exit. Coherent SR is indeed not produced all along *z* in the undulator as uncoherent SR, but only once beam modulation is achieved, i.e., in the last part of the undulator. Coherent SR is nor emitted by the whole electron beam distribution along *y*, but only where the charge density is high enough to guarantee the required FEL gain [62,63]. As a result, the redshifted FEL exhibits a source point with an original signature given by high-contrast horizontal fringes. These fringes are intrinsically here even on standard FELs, but beyond the FEL gain bandwidth in *y*. Only very high-frequency fringes, i.e., phase aliases, enabled by the strongly V-shaped phase of redshifted FELs, can fall in the *y* observation range. One can use Eq. (3) to easily predict fringes location and avoid other misleading cause (diffraction, etc.).

For the sake of completeness, the FEL case with orbit kick is shown in Figs. 7(d-i) and 7(d-ii). The kick only distorts the butterfly structure and introduces a *z* disymmetry, but does not affect the observation of the fringes at the FEL source point.

## VII. OBSERVATION OF FEL SOURCE POINT

Having demonstrated numerically that the redshifted FEL source point exhibits an original signature, high-contrast fringes owing to its transverse phase peculiar shaping, we now move on to experimental evidence.

Figure 8(a) first presents the simulated spatiospectral distributions backpropagated to (iv) $z = -0.1$ m (i.e., close to undulator exit), (iii) $z = -0.4$ m, (ii) $z = -1.0$ m, and (i) $z = -1.5$ m (i.e., close to undulator entrance). To each full-scale image (i to iv) is associated a zoom-image (i-z to iv-z),

highlighting a peculiar substructure. Immediately before the undulator exit (iv), weak fringes are observed as expected from Fig. 7(d). At $z = -0.4$ m (iii), the fringes reach their maximum contrast, signing the source point location. Beyond this source point (ii, i), the fringes disappear indeed, while the distribution starts exhibiting an opening shape corresponding to the beginning of the butterfly structure of Fig. 7(d). The remaining rings observed nearby the undulator entrance (i) are not yet fully understood.

Corresponding measurements shown in Fig. 8(b) are in excellent agreement with simulations, up to the most refined substructures and all along the backpropagation. Fringes are observed as expected over a very limited *z* range, allowing the identification of our FEL source location, at $z = -0.4$ m.

To further confirm this source point retrieval method, the vertical profiles extracted at maximum fringes contrast location from (black line) experiment, (blue line) numerical simulation with orbit kick and (red line) SR model, all at 275 nm, are compared in Fig. 9. The agreement on the first zeros of the $y > 0$ fringes is more than satisfying. The residual disagreement on the first zeros of the $y < 0$ fringes can be attributed to the orbit kick. The analytical prediction of Eq. (3) (red-dashed line) elegantly matches all

<figure><img src="image_11.png" /><figcaption>FIG. 9. Intensity profiles along *y* at 275 nm extracted from (red-dashed line) SR model, (red line) Bessel approximation of SR model according to Eq. (3), (blue line) numerical simulation including the experimental orbit kick, (black line) experiment. Calculation parameters in Table I.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>FIG. 10. Numerical simulation of high-gain FELs in (a) seeded and (b) SASE configuration. (i) Peak power along undulator. (ii) Intensity distribution in (z, y) obtained by propagating the radiation field extracted along y at x = 0 and λ = 273 nm. (iii) Spectrospatial intensity distribution at x = 0 and (i) z = -0.35 m, (ii) z = -0.19 m, i.e., at source point. Calculation parameters in Table I except for the charge: Q = 1 nC in the seeded and Q = 5 nC in the SASE cases.</figcaption></figure>

first zeros predicted by the SR model, as well as experi-
mental and numerical FEL results. This simple analytical
formula can significantly ease the search of a redshifted FEL
source point.

Whereas in the case of a standard Gaussian radiation field
where the source point would simply correspond to a mini-
mum size along y, in the case of redshifted FELs an additional
signature is obtained: fringes owing to the peculiar V phase
aliasing in backpropagation. The tracking of the location of
these fringes appears as a new method to retrieve redshifted
FEL source point. Such a method is especially useful in the
case of low-gain seeded FELs, for which the transverse size
might be affected by a residual seed.

VIII. EXTENSION TO SASE AND HIGH-GAIN REGIME

We demonstrated that the source point location of a red-
shifted FEL can be retrieved tracking localized high-contrast
fringes thanks to an SR inheritance; however, all in the low-
gain regime. Since this signature results from standard FEL
basics, it should be preserved in the high-gain regime and
moving on to the SASE configuration.

Figure 10(a) first presents the simulated case of a high-gain
seeded FEL. Starting from the low-gain FEL parameters of
Table I, the charge is increased up to 1 nC. As shown in
(a-i), the initial seed power is exponentially amplified and the
FEL reaches saturation nearby the undulator exit. The electric
field at the undulator exit is extracted at the FEL wavelength,
273 nm, and propagated from z = +0.5 m down to z = -1 m
to get the (z, y) intensity pattern presented in (a-ii). A very

similar shape is obtained as in low gain, and localized fringes
are observed at z ≈ 0.35 m, signing the FEL source point
location. The (λ, y) intensity distribution at this z location, see
(a-iii), also exhibits the expected fringes.

In Fig. 10(b) is then presented the simulated case of a
high-gain SASE FEL. Once again starting from the low-gain
FEL parameters of Table I, the charge is increased up to 5 nC.
To reach saturation within the 2-m-long undulator, the charge
required in SASE is higher than the one required previously
in seeded configuration since SASE does not benefit from
the speeding effect of a seed in the bunching process. With
a 5-nC charge, saturation is reached nearby the undulator exit,
as shown in (b-i). Again, extracting the FEL electric field at
the undulator exit and backpropagating it, localized fringes
are obtained with a maximum contrast reached at z = -0.2 m
as shown in (b-ii). The (λ, y) intensity distribution at this z
location (b-iii) exhibits two spots corresponding to two spikes
in the SASE spectrum, while each spot is surrounded by the
expected fringes.

Figure 10 demonstrates that our source point retrieval
method, identifying highest fringes contrast location in back-
propagation, works up to high gain and for both SASE and
seeded FEL configurations.

IX. CONCLUSIONS

In conclusion, we reported on the nature of spatiospec-
tral structures of a redshifted FEL, relying on analytical and
numerical models as well as measurements. We revealed
how those structures are governed by a peculiar, V-shaped,
transverse phase inherited from synchrotron radiation. The
understanding of these structures at the undulator exit enabled
us to propose an original method to retrieve the FEL source
point location: by moving the observation point inside the
undulator, i.e., backpropagating the radiation. We believe that
these results are highly valuable for the further understanding
of these recent redshifted FELs, and will pave the way towards
development of the future high average power cavity-based
x-ray regenerative amplifier FEL and further advancement of
a compact laser-plasma-based x-ray FEL.

ACKNOWLEDGMENTS

The COXINEL project was supported by European Re-
search Council for the Advanced Grants COXINEL [340015;
Principal Investigator (PI), M.-E.C.]. The laser-electron ac-
celeration project at Helmholtz-Zentrum Dresden-Rossendorf
is fully supported by the Helmholtz Association under the
program “Matter and Technology”, topic “Accelerator Re-
search and Development”. E.R. is supported by the LABEX
CEMPI project (ANR-11-LABX-0007), the ANR-DFG UL-
TRASYNC 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-R'egion (CPER) WaveTech.
M.-E.C. and M.L. are supported by Laserlab-Europe V (Con-
tract No. 871124). Authors also thank O. Chubar for fruitful
discussions about SRW capabilities and coherence definitions
in the frame of synchrotron radiation.
DATA AVAILABILITY

The data that support the findings of this article are not publicly available. The data are available from the authors upon reasonable request.

APPENDIX A: SR MODELING

1. SR in far-field

In [48], A. Hofmann introduces an analytical derivation of the synchrotron radiation (SR) emitted in the far-field by a single electron in a planar undulator,

$$E(\omega, \theta_x, \theta_y) = \frac{-ek_u \gamma^{*3}}{\pi \sqrt{2\pi} \epsilon_0 r_p} \frac{2\gamma^* \theta \cos(\phi) \Sigma_{11} - K_u^* \Sigma_{12}}{(1 + \gamma^{*2}\theta^2)^2} \\ \times \frac{\pi N_u}{\omega_R} \times \frac{\sin\left(\frac{\Delta\omega}{\omega_R} \pi N_u\right)}{\frac{\Delta\omega}{\omega_R} \pi N_u} \quad (A1)$$

with $\omega$ the SR angular frequency and ($\theta_x, \theta_y$) the angular coordinates. $k_u = 2\pi/\lambda_u$ is the undulator wave number, $\gamma^* = \frac{\gamma}{\sqrt{1+K_u^2/2}}$ the Lorentz factor $\gamma$ normalized, $r_p$ the norm of the vector pointing from the middle of the undulator to the observer, associated to $\theta$ and $\phi$, respectively, the polar and azimuthal angles, with $\theta = \sqrt{\theta_x^2 + \theta_y^2}$, $\epsilon_0$ the vacuum permit- tivity and $e$ the electron charge. $\omega_R$ corresponds to the angular frequency at resonance and $\Delta\omega$ to the frequency offset with respect to $\omega_R$,

$$\Delta\omega = \omega - \omega_R. \quad (A2)$$

Finally, $\Sigma_{11,12}$ are a combination of Bessel functions $J_n$ of order $n$ with $K_u$, $\theta$, and $\gamma$ as arguments,

$$\Sigma_{11} = \sum_{n=-\infty}^{+\infty} J_n(a_u) J_{1+2n}(b_u), \\ \Sigma_{12} = \sum_{n=-\infty}^{+\infty} \frac{2(1+2n)}{b_u} J_n(a_u) J_{1+2n}(b_u),$$

with

$$a_u = \frac{K_u^{*2}}{4(1 + \gamma^{*2}\theta^2)}, \\ b_u = \frac{2K_u^* \gamma^* \theta \cos(\theta)}{1 + \gamma^{*2}\theta^2},$$

and

$$K_u^* = \frac{K_u}{\sqrt{1 + K_u^2/2}}.$$

Introducing the off-axis resonance wavelength $\Lambda_R$,

$$\Lambda_R = \frac{\lambda_u}{2\gamma^2} \left(1 + \frac{K_u^2}{2}\right) + \frac{\lambda_u \theta^2}{2},$$

we define the wavelength offset with respect to resonance as

$$\Delta\lambda = \Lambda_R - \lambda.$$

<figure><img src="image_13.png" /><figcaption>FIG. 11. SR intensity distributions in the far-field. [(a),(b)] Angular distributions at (a) on-axis resonance $\lambda_R = 268$ nm and (b) off-axis resonance $\Lambda_R = 276.6$ nm. (c) Angulospectral distribution at $\theta_x = 0$. Calculation using Eq. (A3) with $\lambda_u = 20$ mm, $N_u = 99$, $K_u = 2.31$, $\gamma = 370.86$ corresponding to $E_e = 189$ MeV.</figcaption></figure>

Now substituting $\omega$ by $2\pi c/\lambda$ and $\Delta\omega$ by $\omega_R\Delta\lambda/\lambda$ in Eq. (A1), we obtain

$$E(\lambda, \theta_x, \theta_y) = \frac{-ek_u \gamma^{*3}}{\pi \sqrt{2\pi} \epsilon_0 r_p} \frac{2\gamma^* \theta \cos(\phi) \Sigma_{11} - K_u^* \Sigma_{12}}{(1 + \gamma^{*2}\theta^2)^2} \\ \times \frac{N_u \lambda_R}{2c} \times \frac{\sin\left(\frac{\Delta\lambda}{\lambda} \pi N_u\right)}{\frac{\Delta\lambda}{\lambda} \pi N_u}. \quad (A3)$$

The monochromatic angular intensity distributions of $E(\lambda, \theta_x, \theta_y)$ are shown in Fig. 11 at (a) on-axis ($\theta = 0$) and (b) off-axis ($\theta \neq 0$) resonance wavelengths. The off-axis resonance distribution (b) exhibits a main ring at $\theta_0$, corresponding to the angle satisfying the resonance condition

$$\theta_0 = \sqrt{\frac{2}{\lambda_u}} \Delta\lambda = \sqrt{\frac{2}{\lambda_u}} (\lambda - \lambda_R). \quad (A4)$$

In (c) is then shown the angulospectral ($\theta_x, \lambda$) intensity distribution. The typical SR moon shape is obtained, clearly exhibiting the angular dependency of the resonance wavelength.

All these far-field angulospectral distributions can be observed experimentally collecting the radiation with a lens while setting a 2D detector at the lens focal plane, as reported previously on COXINEL beamline [52].

2. SR in near-field at the middle of undulator

The inverse Fourier transform of $E(\lambda, \theta_x, \theta_y)$ corresponds to the spatospectral, near-field, distribution of the radiation $E(\lambda, x, y)$ at the middle of the undulator. Figure 12 shows the monochromatic spatial intensity distributions of $E(\lambda, x, y)$ at the same (a) on-axis and (b) off-axis resonance wavelengths as previously. The intensity distribution at off-axis resonance (b) exhibits a clear ringing pattern. Figure 12(c) also shows the spatospectral intensity distribution of $E(\lambda, x, y)$ at $x = 0$, exhibiting a horizontal, spectral dependant, fringing pattern.
<figure><img src="image_14.png" /><figcaption>FIG. 12. SR near-field intensity distributions at the middle of the undulator. [(a),(b)] Spatial distributions at (a) on-axis resonance $\lambda_R = 268$ nm and (b) off-axis resonance $\lambda_R = 276.6$ nm. (c) Spectrospatial distribution at $x = 0$. Calculation using the inverse Fourier transform of Eq. (A3) with parameters of Fig. 11.</figcaption></figure>

This peculiar pattern is responsible for the FEL horizontal fringes described in the main text.

This fringing pattern can actually be described by a simple Bessel function. Indeed, the SR far-field distribution at a wavelength $\lambda$ above resonance, as shown in Fig. 11(b), can be approximated by a ring distribution $\Theta(\theta_x, \theta_y)$ of radius $\theta_0$ according to

$$ \Theta(\theta_x, \theta_y) = 1, \text{ for } \theta_0 - \frac{\delta\theta}{2} < \sqrt{\theta_x^2 + \theta_y^2} < \theta_0 + \frac{\delta\theta}{2} \\ = 0, \text{ elsewhere,} \tag{A5} $$

where $\delta\theta$ is the angular thickness of the ring. Using the variables $(u, v)$ in Fourier space, with $u = \theta_x/\lambda$ and $v = \theta_y/\lambda$, the inverse Fourier transform of $\Theta$ can be defined as

$$ \tilde{\Theta}(x, y) = \iint \Theta(u, v) \times e^{2i\pi(ux+vy)} du dv. \tag{A6} $$

Using then the following variable change: $u = s \times \cos(\theta)$, $v = s \times \sin(\theta)$, $x = r \times \cos(\psi)$, $y = r \times \sin(\psi)$ and $\alpha = \psi - \theta$, Eq. (A6) can be reduced to a one-dimensional integral,

$$ \tilde{\Theta}(x, y) = s_0 \delta s \int e^{2i\pi s_0 r \times \cos(\alpha)} d\alpha, \tag{A7} $$

with $s_0 = \theta_0/\lambda$ the ring diameter and $\delta s = \delta\theta/\lambda$ the thickness of the ring in space domain. The real part of this one-dimensional integral corresponds to the first order of the Bessel function $J_0$, so that

$$ \mathrm{Re}[\tilde{\Theta}(x, y)] = 2\pi s_0 \delta s J_0 \left( \frac{2\pi \theta_0 r}{\lambda_R} \right) \tag{A8} $$

Restricting Eq. (A8) to $x = 0$, the following approximation of the vertical fringing pattern at $\lambda$ is obtained,

$$ B(y) = 2\pi s_0 \delta s J_0 \left( \frac{2\pi \theta_0 y}{\lambda_R} \right). \tag{A9} $$

<figure><img src="image_15.png" /><figcaption>FIG. 13. Vertical intensity profiles of the near-field SR in the middle of the undulator extracted at $x = 0$ and $\lambda = 276.6$ nm. Profiles displayed in (a) linear and (b) logarithmic scale, and calculated using (black line) the inverse Fourier transform of Eq. (A3) and (blue line) Eq. (A8). Calculation parameters of Fig. 11.</figcaption></figure>

In Fig. 13, the result of Eq. (A9) (blue line) and of the inverse Fourier transform of Eq. (A3) (black line) are confronted. The agreement is excellent up to the second zero intensity.

### 3. SR propagation and source point identification

It is worth recalling that Hofmann's derivations correspond to a description of the single-electron SR emitted in an undulator at its virtual source point (middle of undulator). Single-electron SR emitted in an undulator is intrinsically complex because the radiation results from an extended source point. A virtual source point located at the middle of the undulator, however, provides an excellent approximation.

Using Fourier optics [64], the SR near-field obtained at the middle of the undulator can be propagated to any up- or downstream $z$ location. Such manipulation enables to explore the behavior of the SR being propagated, and eventually identify some key features helping localizing the source point experimentally.

Figure 7 shows the SR near-field $E(\lambda, x = 0, y)$ intensity distribution propagated back and forth in $z$, using as reference $z = 0$ for the undulator exit, at (a) on-axis resonance wavelength and (b) off-axis resonance wavelength. In both cases, the patterns exhibit a clear symmetry besides $z = -1$ m, i.e., as expected besides the middle of the undulator. In (a), at on-axis resonance wavelength, the source point can be identified as the $z$ location of widest vertical intensity profile width. In (b), at off-axis resonance wavelength, the source point can be identified as the $z$ location of highest fringes contrast along $y$. Such a key tool for source point localization can be used in the frame of FELs, as demonstrated in the main text.

### 4. SR in near-field at exit of undulator

The SR near-field propagated at undulator exit is now described in more detail. The monochromatic spatial intensity distributions are shown in Fig. 14, once again at (a) on-axis and (b) off-axis resonance wavelengths. The on-axis pattern (a) is free of structures, strongly peaked at $x = y = 0$, while the off-axis pattern (b) exhibits a weak ringing pattern. The spatiospectral intensity distribution at $x = 0$ is shown in
<figure><img src="image_16.png" /></figure>

FIG. 14. SR near-field intensity distributions at the exit of the undulator. [(a),(b)] Spatial distributions at (a) on-axis resonance $\lambda_R = 268$ nm and (b) off-axis resonance $\Lambda_R = 276.6$ nm. (c) Spectrospatial distribution at $x = 0$. Calculation using the inverse Fourier transform of Eq. (A3) and Fourier optics for fourth-propagation with parameters of Fig. 11.

Fig. 14(c). The pattern exhibits an opening shape along $\lambda$, as the moon shape in far-field, though filled inside as expected from the (a) and (b) patterns.

More unexpected SR features of SR in near-field at undulator exit are shown in Fig. 15—in (a) the amplitudes and in (b) the phases of the radiation extracted at different wavelengths: from (black) at on-axis resonance to (blue) slightly at off-axis resonance, and finally to (red) at the off-axis resonance of our redshifted FEL. Both amplitudes and phases exhibit a clear spectral dependence. The amplitudes are decreasingly peaked while the wavelength increases, with a weak dip appearing at $y = 0$ on the red side of resonance. The phases evolve from a smoothly concave shape at on-axis resonance to a more and more V shape while the wavelength increases. This peculiar phase shape beyond the on-axis resonance wavelength was found precious to understand key features of redshifted FELs, as presented in the main text.

<figure><img src="image_17.png" /></figure>

FIG. 15. SR near-field (a) amplitude and (b) phase at undulator exit and $x = 0$ extracted at (black) $\lambda = 269$ nm, i.e., resonance wavelength, (blue) $\lambda = 273$ nm, i.e., slightly on red side of resonance, (red) $\lambda = 276$ nm, i.e., close to expected redshifted FEL wavelength. Calculation using the inverse Fourier transform of Eq. (A3) and Fourier optics.

## APPENDIX B: FEL MODELING

The spatiospectral distribution of the FEL at the undulator exit is numerically modelled in four steps.

In the first step, a 6D Gaussian electron-beam distribution with $10^6$ particles is generated using a set of rms values for beam positions, divergences, energy spread, and duration, with a given total bunch charge.

In the second step, this 6D distribution is transported down to the undulator entrance using BETA code [65], implementing a chromatic matching lattice for a beam energy of 188.8 MeV. The 6D distribution obtained at the undulator entrance is longitudinally sampled into 2000 slices, while each slice is described by the six rms values of its envelope. The duration of one slice is $4 \times 269$ nm, i.e., 3.6 fs. In parallel, the seed (amplitude and phase) is simply modelled in the frequency domain according to its central wavelength $\lambda_{\text{seed}-0}$, bandwidth $\Delta\lambda_{\text{seed}}$ of 3.9 nm-fwhm and group-delay dispersion $D_{\lambda}$ of 0.296 ps/nm. After Fourier transform, the seed pulse in the time domain is sampled into 2,000 slices, while each slice is described by its peak power and phase.

In the third step, both the seed and the electron beam longitudinal distributions are loaded into GENESIS [66] code to simulate the radiation generation process along the undulator. The result of this simulation is the 3D $(x, y, t)$ electric field of the radiation at the undulator exit, with $x$ (resp. $y$) the coordinate along the horizontal (resp. vertical) plane and $t$ the longitudinal (time) coordinate. With a simple Fourier transform, the 2D $(x = 0, y, t)$ field distribution is converted into a 2D $(y, \lambda)$ spatiospectral distribution.

To be comparable to the spatiospectral distributions recorded on the 2D ending spectrometer, in the fourth (last) step these spatiospectral distributions at undulator exit are backpropagated towards any $z$ location inside the undulator using standard Fourier optics [64].

## APPENDIX C: RADIATION IMAGING SYSTEM

At the undulator exit, the radiation is collected and imaged onto a 2D UV spectrometer. The imaging system consists of two lenses from EKSMA made of fused silica. The first lens (L1) is convex with a radius of curvature of 190.4 mm (ESKMA reference 110-1523E). Given the index of refraction of fused silica at 266 nm, $n = 1.4997$, this radius of curvature corresponds to a focal length of 381 mm. The second lens (L2) is concave with a radius of curvature of -95.2 mm (EKSMA reference 112-1419E), corresponding at 266 nm to a focal length of -190.5 mm.

While (L1) is placed at a fixed distance with respect to the undulator exit, (L2) is mounted after (L1) on a translation stage allowing to vary its distance from (L1) and therefore to vary the object plane imaged on the UV spectrometer set at a fixed location with respect to the undulator exit.

The magnification of this system was first computed as a function of the distance between (L1) and (L2) and then confronted to measurements using the seed laser. The comparison between calculation and measurements allowed to finely adjust the calibration of the imaging system.
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