Interferometric optical signature of electron microbunching in laser-driven plasma accelerators

We report observations of coherent optical transition radiation interferometry (COTRI) patterns generated by microbunched ~200-MeV electrons as they emerge from a laser-plasma accelerator. The divergence of the microbunched portion of electrons, deduced by comparison to an analytical COTRI model, is ~6x smaller than the ~3 mrad ensemble beam divergence, while the radius of the microbunched beam, obtained from COTR images on the same shot, is<3 microns. The combined results show that the microbunched distribution has estimated transverse normalized emittance ~0.5 mm mrad.

Periodic longitudinal density modulation of relativistic electron beams at optical wavelengths (microbunching) gives rise to coherent light emission in such forms as synchrotron radiation, including the free-electron laser (FEL) [1,2], and optical transition radiation (OTR) in its coherent form (COTR). The latter has been observed from FELs [3,4] and laser-driven plasma accelerators (LPAs) [5,6]. In the first case, the FEL mechanism fundamentally depends on growth of microbunching at the resonant wavelength and its harmonics [7]. In the second case, COTR in some configurations can characterize microbunched portions of the electrons [3][4][5][6].
Microbunching in an FEL oscillator was observed indirectly via the buildup of FEL output power to saturation [2]. The first direct time-resolved observation of microbunching in an FEL oscillator [8] used an off-phase final rf accelerator stage to streak a beam modulated at 60 µm wavelength, thereby mapping microbunch arrival time onto energy as displayed in an electron spectrometer. With the advent of self-amplified spontaneous emission (SASE) FELs with a single-pass through a long amplifier chain, FEL light and the electron beam became accessible after each undulator, enabling tracking of FEL power and microbunching. The first measurements of microbunching evolution at visible wavelengths [3,4] used COTR interferometry (COTRI) to track microbunched features uniquely through the exponential gain regime, through saturation, and into post saturation [4]. An analytical model of COTRI fringe patterns showed growth of a microbunched transverse core in the exponential gain regime, and its subsequent reduction after saturation [4,9]. Subsequently, such experiments were extended to vacuum ultraviolet wavelengths, further benchmarking this model [10,11]. These experiments foreshadowed the x-ray SASE FELs of today [2,[12][13][14][15][16] by benchmarking the GENESIS simulation code [17] used in their prediction and development [18].
In this Letter, we report new investigations of microbunching in laser-driven plasma accelerators (LPAs) [19], including the first evaluations of the beam emittance of this subset of electrons with COTR techniques. This microbunching is not accessible with betatron x-ray spectroscopy [20], pepper-pot measurements [21], scintillator-based methods [22], or other LPA beam diagnostics [23]. LPAs are currently a major initiative in advanced accelerator technology, with compact FELs [24,25] prominent amongst a growing list of potential applications. Detailed understanding of microbunching is critical to developing LPAs and LPA-based FELs. Past experiments have used COTR to deduce the presence of sub-bunches located in adjacent LPA buckets [26] or of an intrabunch slice of sub-per cent energy spread [27]. Here we use the interference of COTR from 2 tandem foils located downstream of the LPA to deduce the presence of visible-wavelength microbunching within the dominant quasi-monoenergetic component of an electron bunch that was ionization-injected [27,28] into, and accelerated to ~200 MeV within the leading bubble of a strongly nonlinear LPA. Indeed particle-in-cell simulations have predicted microbunching in LPAs [29,30]. Such a microbunched fraction might be used to seed an FEL. Our results show that visible-wavelength COTR gain relative to incoherent OTR rivals that obtained previously in a saturated SASE FEL. As a result, COTR is intense enough to distribute to multiple cameras with different frequency filters or imaging modalities on each shot, enabling thorough characterization. For example, one group of cameras detects COTR imaged from the surface of the first foil [hereafter "near-field" (NF) images], which when analyzed using a coherent point-spread function enables beam-size measurements at the LPA exit. Other cameras record the COTR in the focal plane of a collecting lens [hereafter "far-field" (FF) images], thereby measuring the angular distribution of radiation. We observed fringes in the latter data that are consistent with Wartski two-foil COTRI [31]. The first fringes shift outside of the single-foil 1/γ opening angle of the angular distribution pattern, where γ is the relativistic Lorentz factor of the electrons. Fringe visibility indicated microbunched electrons diverged less than the beam-ensemble observed in a downstream electron spectrometer.
Experimental results are interpreted using a COTRI model [4,9] to extract electron divergence and pointing from FF data. Combined frequency-filtered beam-size and divergence data yield single-shot, transverse emittance estimates of these microbunched electrons generated within the plasma bubble.
Our experiments used pulses from the DRACO laser (central wavelength 800 nm, energy up to 4 J on target, pulse length 27 fs (FWHM), and peak power 150 TW [28,32] at Helmholtz-Zentrum Dresden-Rossendorf. These were focused to ~20 µm (FWHM) at the center of a 3-mmlong He gas jet (with 3% Nitrogen and ~0.5-mm-long entrance and exit ramps) to drive LPAs in a self-truncated ionization-injection regime [27] in plasma of density n e ~3 x 10 18 cm -3 [28]. A 75-µm-thick Al laser foil 700 µm from the exit of the jet, tilted ~3 o off normal (Fig. 1a), blocked the drive laser pulse. An aluminized Kapton foil located 1 mm downstream blocked j x B electrons of low energy and associated COTR from the back of the blocking foil [33] as shown in Fig. 1b. Indeed, direct laser excitation of the blocking foil with the gas jet turned off yielded no detectable OTR. With the gas jet turned on, LPA electrons generated forward OTR/COTR from the aluminized back surface of the Kapton foil. Several dozen of these foil pairs were mounted on a 15-cm diameter wheel, which rotated a fresh pair into position for each shot. A 200-µm-thick polished Si wafer oriented at 45 o degrees to the beam direction at distance L=18.5 mm downstream of the Kapton foil redirected the foil's OTR/COTR to a 4-cm focal length microscope objective with collection angle 0.14 rad, which relayed it to one group of chargecoupled device (CCD) cameras (12-bit, 3.75-µm square pixels) via beam splitters (Fig. 1a) to record NF images (Fig. 1c). Additionally, the electron bunch generated backward (reflected) COTR from the front surface of the Si wafer (Fig.1a). COTR from these two interfaces, which the microscope objective and an additional 15-cm focal length lens (see Fig. 1a) relayed to another CCD, formed interference fringes in FF images (Fig. 1d). For results reported here, one CCD recorded un-polarized FF images at observation wavelength λ= 633 ± 5 nm, while two others recorded NF images through orthogonal linear polarization analyzers [parallel (x) and perpendicular (y) to the drive laser polarization] at λ= 600 ± 5 nm, which were indistinguishable from 633 nm NF images. When the laser focus in the gas jet was adjusted to positions that yielded poly-energetic electron distributions similar to the background in the present LPA output, but with no quasi-monoenergetic peak, we observed OTR signals ~100x weaker than those reported here. Thus, the quasi-monoenergetic peak is the source of reported COTR signals.
These signals were intense enough to necessitate neutral density filters to prevent camera saturation. Nevertheless, when we operated the accelerator in different regimes, e.g. by removing the nitrogen dopant and relying on self-injection, we observed strong associated variations in COTR signal strength normalized to accelerated charge. Since foil and laser parameters were unchanged, these variations suggested that the LPA process --not interaction of electrons with foils or reflected laser fields ---created microbunching responsible for observed COTR. Beam scattering by a foil can reduce coherent emission from microbunching when the projected multiple scattering angle exceeds the OTR opening angle 1/γ [34]. The latter is 2.3 mrad, while the Bethe-Ashkin formula [35] yields a lower value (1.1 mrad) of the former for the Al foil, a higher value (2.6 mrad) for the 45 o Si mirror. To corroborate this conclusion, we measured space/angle-integrated forward COTR spectra in a downstream IR-to-UV spectrometer [36].
With the Si wafer temporarily removed, we observed strong IR and visible light down to λ ~300 nm. When we re-inserted the Si wafer and placed a new thin OTR foil downstream of it, however, only IR light remained strong. This showed that microbunching responsible for visible COTR survived transit through ≤ 75 µm Al, but not ≥200 µm/cos45 o Si foils. We will present complete COTR spectra and analysis in a planned forthcoming paper.
Currents induced when a charged particle beam enters and exits a foil generate, respectively, forward and backward optical transition radiation [37][38][39]. The backward radiation cone of half angle 1/γ is generated around the specular reflection direction while forward radiation is generated in a 1/γ cone around the beam direction. Thus, the configuration in Fig. 1 generates OTR at 90 o to the beam direction, enabling minimally invasive OTR detection and imaging.
Interference of forward OTR from the aluminized Kapton with the backward OTR from the silicon reflector produces fringes with peak maxima at p = 1/2, 3/2, 5/2, etc., related to foil separation L by L = pγ 2 λ. Choosing L = 18.5 mm enabled focusing near-field optics on the first foil, while still providing good fringe contrast.
The number W 1 of OTR photons that a single electron generates per unit frequency ω per unit where ħ is Planck's constant/2π, e is the electron charge, c is the speed of light, and θ x and θ y are radiation angles [9]. The black curve in Fig. 2a shows the single-foil OTR angular distribution where ∥,! , is the reflection coefficient of the second foil for parallel and perpendicular polarization components, respectively. I(k) is given by [31] = 4 sin ! [ where = = 2 , and we used a small-angle approximation. The dashed red curve in Fig.   2a shows the corresponding two-foil OTR angular distribution for L= 18.5 mm and λ= 633 ± 5 nm. At γ values of interest, intensity asymmetry of the first peaks of the parallel polarization component in Fig. 2a are negligible (see Eq. 1 ref. [38]). Strong fringe modulation is seen in this example where a Gaussian beam divergence of 0.2 mrad was convolved with (2). The coherence function can be defined as is a product (where we assume separability of ( )) of Fourier transforms g(  into account the single-electron, single-foil OTR source energy [40] and the measured charge in the quasi-monoenergetic peak based on a calibrated LANEX screen [41] in the spectrometer on the same shot (see Supplementary Material for details). Thus coherent enhancements dominate over incoherent OTR for these conditions.
The FF image in Fig. 1d has 8 to 9 visible fringes. The azimuthal asymmetry in fringe intensity probably results from transverse distortions of the COTR source from a Gaussian charge distribution. Here, to simplify analysis, we averaged the data in Fig. 1d azimuthally. We then assess the resulting fringe pattern (Fig. 3a, black curve) by comparison to analytical results in Fig. 2b. Since the outermost fringes (Fig. 3a, inset) are more sensitive to σ θ than the first two fringes, whereas amplitudes of the latter are more sensitive to neglected non-Gaussian features of the beam shape, we analyzed σ θ quantitatively by fitting azimuthally symmetric COTRI model calculations for various σ θ to the 4 th thru 9 th azimuthally averaged fringes. The dashed red curve in Fig. 3a shows the complete best-fit COTRI curve. Full analysis yields σ θ = mrad (see Supplementary Material for details). This is significantly smaller than the ensemble divergence of 3 mrad measured at the electron spectrometer without foils present [28]. Calculated and measured fringe peak positions agree well, and the number of fringes detectable (8 to 9) in the azimuthal average can only be explained with a sub-mrad divergence value. This number of fringes requires σ x,y < 6 µm based on modeling results in Fig. 2b. Analysis of linear polarized NF images (e.g. Fig. 1c) confirms this conclusion. Fig. 3b compares a COTR model calculation for σ x = 2.75 µm (red dotted) to a y-averaged version of x-polarized NF data in Fig. 1c (Fig. 3b, black curve). Here, calculated curves take into account the finite optical collection angle via Eq. 26 of Ref. [42]. The left-right asymmetry of the NF data, again, probably results from distortions of the beam charge profile from a Gaussian shape, which we Interpreting the latter as the accelerator exit, and extrapolating σ x =2.75 !!.!" !!.!" back to z = 0 using = 0.48 !.!" !.!" mrad yields estimated rms beam radius σ x (z = 0) ≈ 1.5 µm at the accelerator exit for this shot. This value is consistent with the range of rms beam radii inside, and near the end of, this accelerator determined independently by analyzing betatron x-ray spectra [43], results which will be published separately.
To illustrate wider COTRI diagnostic possibilities for LPA beams with more complex structure, Fig. 4a shows FF data without linear polarizer for a different shot. Here, a dark node runs vertically through the interference pattern, a feature not seen in Fig. 1d, nor in most shots. The corresponding NF image (Fig. 4b) shows two pairs of point-spread-function lobes separated by ~6 µm along x, instead of one pair as in Fig. 1c, suggesting that two beamlets emerged sideby-side from the LPA. Such bi-modal beam distributions along the laser polarization have been observed in PIC simulations of bubble-regime LPAs (see Fig. 6 of Ref. [44]). Fig. 4c shows a reconstruction of the main qualitative features of Fig. 4a by modeling two beamlets having a phase difference of 0.75 π with angular trajectories differing by 2 mrad, each with σ θ = 0.6 mrad.
This example thus illustrates that FF COTRI patterns contain signatures of the phase and trajectory of multiple microbunched beamlets, when present. Combined FF and NF data indicate single-beamlet normalized emittance similar to that obtained from the data in Fig. 3.
In summary, we have identified visible wavelength microbunching in electron beamlets accelerated in LPAs with concomitant COTR gain over incoherent OTR exceeding 10 5 . We

Effect of detection bandwidth on COTRI fringe visibility.
Generally in a FF angular distribution pattern, the optical bandwidth of the filter through which the radiation is detected influences fringe visibility. Fig. S1 illustrates this effect for the case of COTR generated in two tandem films separated by L = 18.5 mm by a Gaussian electron bunch of energy E e = 200 MeV with σ r = 2µm, σ θ = 0.5 mrad. Calculated COTRI patterns are shown for λ = 633 nm and detection bandwidths ∆λ FWHM = 0, 10 and 40 nm. The patterns are calculated by convolving the field from a single wavelength [Eq. 8 of Wartski, J. Appl. Phys. (1975)] with a Gaussian wavelength distribution of the given bandwidth. The results show that for ∆λ FWHM = 10 nm, the bandwidth chosen for experimental results presented in the main text, fringe visibility is close to that of ideal monochromatic detection. Thus observed fringe visibility can be interpreted purely in terms of e-beam parameters. For ∆λ FWHM = 40 nm, on the other hand, significant fringe visibility is lost because of detection bandwidth alone.

Calibration of COTR gain, microbunching fraction.
COTR gain N b 2 /N and microbunching fraction N b /N were determined by calibrating the integrated optical energy deposited in the FF detector, the throughput of the optical path from OTR foils to detector, and the total charge Q within the quasi-monoenergetic 200 MeV peak of the LPA output that was detected in the electron spectrometer. The integrated FF signal in Fig. 1d corresponds to deposited optical energy 7.4 pJ at λ = 633 nm. This is based on an in-house calibration of the CCD camera's detection efficiency using a HeNe laser of independently measured power, and spot size similar to that of the detected signal. Measured and/or published transmission and reflectivity coefficients of individual elements ---microscope objective, beam splitters, silicon wafer, bandpass filter and neutral density filters ---of the aluminized-Kapton-foil-todetector relay line yielded overall throughput 0.013% within the bandwidth 628 < λ < 638 nm, implying 58 nJ generated within this bandwidth at the foil. A single 200 MeV electron produces 8.1 × 10 -14 nJ of radiation within this bandwidth and the solid angle of our collection system at one foil [Eq. 1 of Schroeder, Phys. Rev. 2004] or twice this at two foils, smaller by a factor 3.6 × 10 14 than the generated energy. For predominantly coherent OTR, we conclude N b = [3.6 × 10 14 ] 1/2 ≈ 1.8 × 10 7 electrons, or charge eN b ≈ 3 pC. The total charge Q = Ne = 235 pC, corresponding to N = 1.47 × 10 9 electrons, in the quasi-monoenergetic peak was determined from the intensity of light emission that it stimulated in a calibrated luminescent screen (Lanex) located at the detection plane of an electron spectrometer. Absolute calibration of the Lanex screen is described in detail in Kurz et al., Rev. Sci. Instrum. 89, 093303 (2018). Thus N b /N = 0.013 and N b 2 /N = 2.4 × 10 5 .

Analysis of COTRI data.
To compare the COTRI data in Fig. 1d with the model, we averaged over azimuthal variations in the data. To implement this average, we first fit a circle to the outermost visible fringe in Fig. 1d. From this we determined the center of the interference pattern. The data in Fig. 1d was then averaged over radial integration paths to produce the azimuthally averaged data shown by the black curve in Fig. 3a, where it is plotted against the emission angle θ. The common center of the circle was fine-tuned by varying its location within a 7x7-pixel area around the central minimum of the interference pattern and identifying the pixel that maximized visibility of the outermost fringes. The resulting plot provided the basis for further analysis, which proceeded in the following logical sequence: a. Insensitivity of COTRI fringes to γ. Eq. (3) shows that there is a γ-dependent phase shift in the COTRI fringe intensity. In our experiments, COTR originates mainly from electrons in the quasi-monoenergetic peak, for which 400 < γ < 500. Fig. S2a plots COTRI intensity generated in two tandem films separated by L = 18.5 mm by Gaussian electron bunches of γ = 300, 400 and 500 with σ r = 2µm, σ θ = 0 mrad. A small phase shift is visible for γ = 300 (dashed blue curve), but intensity and fringe visibility become nearly γ-independent for γ = 400, 500 and higher. Consequently fringe visibility in the range of our experiments is unaffected by small electron energy broadening. Fig. S2b shows this explicitly by plotting COTRI intensity at γ = 400 for bandwidths ∆ γ = 0, 50 and 250. The first two plots are indistinguishable from each other. Thus the ~10% energy bandwidth of the quasimonoenergetic electrons in our experiment does not affect fringe visibility, which therefore is determined by alone. b. Determination of divergence and its uncertainty from fringe visibility. Since COTRI fringe visibility is insensitive to in the range < 4 µm (see main text, Fig. 2b, positive angles), we fixed the beam radius at ≈ 2 µm for purposes of fitting azimuthally averaged COTRI data in Fig. 3a with as the sole variable. As a preliminary step, we subtracted a θ-dependent background from the data (see Fig. S3a) and obtained the result in Fig. S3b. The less than unity modulation depth in the raw data results from finite pixel size, optics misalignment, aberrations, and other imperfections in the imaging system. The backgroundsubtracted data was then fit to a family of model COTRI curves with variable . Fig. S4 shows results of this fit. The outer fringe visibility is more sensitive to beam divergence than the inner fringes. Also, due to coherence effects, the inner fringes are more sensitive to high spatial frequency features within the transverse momentum distribution than the outer fringes [Eq. 8 of Wartski, J. Appl. Phys. (1975)]. We chose to fit our model to the signal beyond the third peak, thereby selecting for the region that greatest sensitivity to total bunch divergence. The χ 2 value of the fit is minimized when = 0.48 mrad (see Fig. S4b). We bounded the uncertainty range at values (0.39 and 0.58) that yielded twice the minimum value of χ 2 , yielding = 0.48 !.!" !.!" mrad. If we fit fringes at a higher or lower cutoff angle, this χ 2 best fit could vary by a few tenths of mrad, but remains below 0.7 mrad regardless of this choice.

Analysis of near-field COTR data.
We determined bunch size σ r and its uncertainty by comparing polarized NF data (e.g. Fig. 1c of main text) to predictions of a COTR model, using σ r as a variable fitting parameter. We first integrated polarized NF signals along a direction (y) perpendicular to the polarization analyzer's transmission axis, to obtain a double-lobed intensity pattern along x (Fig. 3b, main text). We model this pattern by σ θ σ θ σ θ σ θ superposing TR fields from individual electrons in the bunch. Electron energy and the numerical aperture of the imaging optic determine each electron's imaged TR field pattern, according to Eq. 26 of Castellano Phys. Rev. ST Accel. Beams 1998. Because the observed intensity pattern had lobes of unequal height (see Fig. 3b), we modeled the beam as a skewed normal distribution where is the width parameter, the shape parameter and (x) is the Gauss error function. The standard deviation of this distribution is ! = 1 − ! ! ! ! !!! ! , which for < 0.4 becomes approximately equal to ω: i.e. ! ≈ . The ratio of lobe heights proved sensitive to the skew parameter , but insensitive to the distribution width σ x . Thus we first fit the observed lobe height ratio (0.64 for the data in Fig. 3b), yielding = 0.15. As Fig.  S5ba shows, this determination was nearly independent of σ x . Using this value for , we then generated COTR patterns for varying σ x . Fig. S5c shows examples for σ x = 2.45, 2.75 and 3.2 µm. We found that χ 2 was minimized at σ x =2.75 µm, and that bounding σ x values 2.45 and 3.2 µm yielded twice the minimum χ 2 (see Fig. S5b). Thus we cite rms beam xradius σ x = 2.75 !!.!" !!.!" at position z ≈ 2.5 mm from the exit of the accelerator, where the TR foil was located.