Introduction

Extreme ultraviolet (XUV) light sources facilitate technological advances and scientific discoveries across multiple disciplines. XUV pulses are used to image “molecular movies”1,2,3, interrogate high-energy-density and warm dense matter4,5,6, perform nanometer-scale photolithography and imaging of microprocessors7,8,9, and resolve attosecond dynamics of bound electrons10. The importance of these applications has, in turn, fueled the development of XUV sources, including free-electron lasers11,12,13,14,15,16, high harmonic generation (HHG) in plasmas and gases17,18,19,20,21, in-band emission from laser-produced tin plasmas22, and collisional soft x-ray lasers23,24,25. Despite rapid progress in the development of these sources and advances in laser technology that could substantially increase their performance, a sizable gap exists between the intensities available at optical frequencies and those available in the XUV (Fig. 1). Bridging this intensity gap would improve the signal-to-noise ratio in critical XUV diagnostics and enable novel regimes of strong-field ionization26,27 and quantum electrodynamics28,29.

Fig. 1: Amplitude of sources ranging from x-ray to the long-wave infrared.
figure 1

Here, the normalized vector potential is \({a}_{0}=8.5\times 1{0}^{-10}{[I({{{\rm{W}}}}/{{{{\rm{cm}}}}}^{2})]}^{1/2}\lambda \,(\mu {{{\rm{m}}}})\), where I is intensity and a0 = 1 (solid black line) is the amplitude threshold for relativistic electron motion. A sizable gap in amplitude and intensity exists between available XUV and near-optical sources. This gap exists for both conventional pulses (open symbols) and those with spatio-polarization structure (solid symbols). Photon acceleration of structured optical pulses in an electron beam–driven plasma wave offers a tunable-wavelength source that can bridge this gap (span of light to dark blue solid X symbols). Experimentally demonstrated sources include conventional lasers (Lasers), perturbative harmonic generation (PHG)76, free-electron lasers (FELs)11,12,13,14,15,16, high harmonic generation (HHG)17,18,19,20,21, in-band emission from laser-produced tin plasmas (Tin)22, and collisional soft X-ray lasers (Soft)23,24,25; the purple dashed–solid and dashed–open triangle symbols represent simulations of HHG with incident lasers intensities of 5 × 1019 W/cm2 (ref. 77) and 2 × 1022 W/cm2 (ref. 78), respectively, and the red dashed diamond symbol represents a simulated plasma accelerator–driven x-ray source79.

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Alongside their ability to create high intensities, optical sources allow for the use of conventional, adaptive, or metasurface optics to manipulate the spatial, temporal, or polarization structure of the pulses they produce30,31,32,33. These optics have the advantage of linearity, which helps ensure that the manipulation is controlled and deterministic. This contrasts with the inherently nonlinear process of HHG that has been used to generate structured XUV pulses20,21, where the topological charge is usually related to the harmonic number. Within the field of structured light, vector vortex beams34,35—which generalize and couple the spatial, orbital-angular-momentum, and polarization degrees of freedom—have found utility in optical trapping36,37, spectroscopy38, quantum information39,40,41, particle acceleration42, and material processing43. Overcoming the “structure gap,” i.e., the large discrepancy in intensity and availability of structured XUV pulses compared to structured optical pulses, with new techniques for creating vector vortex XUV pulses would expand their flexibility as diagnostics and drivers of novel light–matter interactions.

Photon acceleration44,45,46,47,48,49,50,51 has the potential to convert high-intensity optical pulses to ultrafast, high-intensity XUV pulses while preserving the initial spatio-polarization structure. Photon acceleration refers to the temporal refraction of photons in a time-dependent refractive index. Just as a spatially varying refractive index changes the propagation direction of light rays, a time-varying refractive index changes the frequency of light. In a recently proposed photon acceleration scheme, a linearly polarized laser pulse co-propagates with and frequency upshifts in the density modulation of a plasma wave driven by a relativistic electron beam52,53,54. This scheme has two beneficial properties that offer a path to surmounting the intensity and structure gaps: first, a high-intensity laser pulse can extract energy from the plasma wave and maintain its amplitude as it upshifts, resulting in a high-intensity XUV pulse. Second, the plasma wave provides a guiding structure that can prevent diffraction and preserve the spatio-polarization structure of the high-intensity pulse.

Here, we demonstrate the production of near-diffraction-limited, high-intensity, attosecond XUV pulses with predetermined spatio-polarization structure. Quasi-3D particle-in-cell (PIC) simulations show that the plasma wave driven by a 50-GeV, 5.8-nC electron beam can guide and photon accelerate a relativistically intense, vector vortex pulse from a wavelength of 800 nm to 36 nm over 1.2 cm. The initial pulse corresponds to the output of a Ti:sapphire laser; the electron beam parameters are similar to those previously achieved at SLAC55 and were chosen to mitigate beam evolution over the length of the accelerator. The optical wavelength of the pulse allows for preliminary structuring with linear optical elements to form the vector vortex profile. This profile is preserved, the duration is compressed to attoseconds, and the intensity is enhanced by more than two orders of magnitude during the acceleration. To match the velocity of the accelerating pulse to that of the plasma wave, the background plasma density decreases over the length of the accelerator52. The PIC simulations were conducted with OSIRIS56,57 and were made possible by (i) the quasi-3D geometry, which uses cylindrical coordinates with an expansion in azimuthal modes, and (ii) a Lorentz-boosted frame, which significantly reduces the computational cost of resolving the wavelength of the XUV pulse. The simulation results indicate that photon acceleration can bridge the intensity and structure gaps between optical and XUV pulses (Fig. 1).

Results

Photon-accelerated vector vortex beams

In a photon accelerator, a moving refractive index gradient alters the local phase velocity of a light wave. If the light wave co-travels with a negative (positive) refractive index gradient in a normally (anomalously) dispersive medium, the phase fronts compress, causing a continual increase in the frequency and group velocity. The local frequency ω and wave vector k of the light wave are related through the refractive index η(ω, ζ, z) = ck/ω, where z is the propagation axis, ζ = t − z/vI is the moving-frame coordinate, and vI is the velocity of the refractive index gradient. In the ray description of light, the frequency acts as the Hamiltonian ω = ck/η(ω, ζ, z) and thus evolves according to

$$\frac{{{{\rm{d}}}}\omega }{{{{\rm{d}}}}t}=-\frac{\omega }{\eta }\frac{\partial }{\partial \zeta }\eta (\omega ,\zeta ,z),$$
(1)

where ∂t = ∂ζ has been used47. Equation (1) shows that photon acceleration to high frequencies over short distances requires large and ultrafast gradients in the refractive index (∂ζη).

Plasma is composed of highly mobile free electrons that can support large, ultrafast gradients in the refractive index. This same property allows plasma to maintain its dielectric properties in the presence of high-intensity light waves. An unmagnetized plasma exhibits normal dispersion characterized by the refractive index \(\eta ={(1-n/\gamma {n}_{{{{\rm{c}}}}})}^{1/2}\), where n is the electron density, nc = meε0ω2/e2 is the critical density, and γ is the electron energy normalized to mec2. Plasma-based photon accelerators exploit this density dependence in the refractive index. Relativistic charged-particle beams or intense laser pulses can drive large-amplitude, moving density modulations δn(ζ, z) = n(ζ, z) − n0(z), where n0(z) is the upstream, unperturbed plasma density. The moving refractive index gradient associated with these modulations (∂ζη ∝ ∂ζδn) travels at a velocity that is nearly equal to that of the driver vd and slightly faster than that of a trailing light wave vg, i.e., vI ≈ vd ≳ vg. Relativistic electron beams, in particular, are resilient to nonlinear modifications and can drive a steady, ultrafast accelerating gradient over the distances required for a trailing laser pulse to reach high frequencies.

A key advantage of using plasma-based photon acceleration to generate XUV pulses is that the acceleration process preserves the structure of the trailing pulse. Photon acceleration is an inherently linear process: the properties of the pulse are modified by the linear refractive index η, which is independent of the pulse structure and amplitude. While the pulse can nonlinearly modify the refractive index, these modifications are nonessential—and usually detrimental—to the process. To preserve the polarization and orbital angular momentum of a vector vortex pulse, the linear refractive index must be non-birefringent and independent of azimuth. An unmagnetized plasma, as considered here, is non-birefringent, and a trailing laser pulse aligned to the axis of a cylindrically symmetric electron beam ensures that the refractive index is independent of azimuth. Slight misalignment of the laser pulse and electron beam would only cause small changes to the vector vortex structure that are linearly proportional to the misalignment. The linearity of photon acceleration contrasts with HHG20,21, where a nonlinear current or polarization density that depends on the structure and amplitude of the initial pulse typically produces XUV pulses with higher-order vortex structure.

A simulation of an electron beam–driven photon accelerator is displayed in Fig. 2. An ultrashort electron beam and a trailing laser pulse with vector vortex structure (a–c) propagate through background plasma with a ramped density profile (d), resulting in a frequency upshift of the pulse (e). The space-charge force of the electron beam expels background electrons from its path. The expelled electrons are then attracted back to the propagation axis by the ions. This results in a nonlinear plasma wave, shown by the brown/blue surface plots of the electron density in (a–c). The density modulation of the plasma wave provides the moving refractive index gradient needed to guide and accelerate the trailing vector vortex pulse. The front half of the pulse straddles the rising edge of the density modulation and frequency upshifts due to the larger and smaller phase velocities behind and ahead of the rising edge, respectively. Similarly, the rear half of the pulse straddles the falling edge of the density modulation and experiences a frequency downshift. These frequency shifts are observed on either side of the density spike in (a).

Fig. 2: Electron beam–driven photon acceleration of a vector vortex laser pulse.
figure 2

a–c An electron beam (blue spheres) traveling through a plasma excites a moving electron density modulation (brown/blue surface) that photon accelerates a trailing vector vortex pulse (electric field in red, green, and purple). The three frames correspond to sequential locations along the accelerator and are demarcated by the red, green, and purple symbols in (d) and (e). d The tailored density decreases to synchronize the motion of the electron density gradient and the temporal centroid of the vector vortex pulse. e The maximum frequency upshifts by a factor of 27. Here, the vector vortex pulse is a superposition of orthogonally polarized Laguerre–Gaussian modes with azimuthal mode numbers ℓ = 2 and  −2 and radial mode number zero. The vector vortex structure is preserved throughout the process. The images were generated from the output of quasi-3D, boosted-frame PIC simulations (see “Methods“).

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In order for the laser pulse to continually upshift, it must remain overlapped with the rising edge of the density modulation. As the pulse upshifts in frequency, its subluminal group velocity vg ≈ cη, increases. If the unperturbed plasma density n0 were uniform, the location of the maximum density gradient would travel at the speed of the electron beam vd = vI ≈ c, which is slightly faster than the group velocity of the pulse vg. Thus, despite accelerating, the pulse would recede with respect to the maximum gradient and eventually slip into the falling edge of the modulation and begin downshifting. To ensure continual upshifting, the unperturbed density profile n0(z) can be tailored52,54 so that the maximum density gradient (i.e., where δn  = 0) tracks the temporal centroid of the pulse (Fig. 2d). The location behind the electron beam where δn = 0 depends on the wavelength of the plasma wave, which is proportional to \(1/\sqrt{{n}_{0}}\). In a density downramp, the plasma wave elongates as the density decreases, causing the location of the maximum gradient to recede. Thus, the profile of n0(z) can be tailored with a density downramp so that the pulse and maximum gradient recede at the same rate with vI(z) = vg(z).

Determination of the tailored density profile requires solving a system of two coupled equations. The first equation describes the spatial evolution of the frequency and is obtained from Eq. (1):

$$\frac{{{{\rm{d}}}}\omega }{{{{\rm{d}}}}z}=\frac{{e}^{2}}{2{\varepsilon }_{0}{m}_{{{{\rm{e}}}}}{c}^{2}{k}_{z}}\frac{\partial (\delta n/\gamma )}{\partial \zeta },$$
(2)

where kz is the longitudinal wave vector and dz/dt = vg = c2kz/ω has been used. The second equation is derived by setting the location of the temporal centroid ζc(z) equal to the location where the density modulation of the plasma wave is zero: ζδ(z) satisfying δn[ζδ(z), z] = 0. In differential form, this condition is dζc/dz = dζδ/dz, or

$$\frac{1}{{v}_{{{{\rm{g}}}}}}-\frac{1}{c}=\frac{{{{\rm{d}}}}{\zeta }_{\delta }}{{{{\rm{d}}}}{n}_{0}}\frac{{{{\rm{d}}}}{n}_{0}}{{{{\rm{d}}}}z}\quad \Rightarrow \quad \frac{{{{\rm{d}}}}{n}_{0}}{{{{\rm{d}}}}z}\approx \frac{{n}_{0}}{2{n}_{{{{\rm{c}}}}}(\omega )}{\left[\frac{{{{\rm{d}}}}{\zeta }_{\delta }}{{{{\rm{d}}}}{n}_{0}}\right]}^{-1},$$
(3)

where the approximations vd = c and γ(ζδ) = 1 have been used. Closing the system of Eqs. (2) and (3) requires either analytic expressions or numerical solutions for the density modulation δn and location of its zero ζδ52. These can be obtained from a quasistatic model of the electron beam–driven plasma wave (see “Methods”).

The exact density profile n0(z) and maximum frequency shift depend on the parameters of the electron beam and trailing laser pulse. The charge of the electron beam should be large enough to create a nonlinear density depression and spike that are wide radially (brown and blue in Fig. 2a–c), but not so large that the density spike is too narrow longitudinally. A wide density depression acts like a waveguide that confines the trailing laser pulse transversely and mitigates diffraction as it accelerates. This is particularly important for higher-order vector vortex modes, which have a larger radial extent. A slightly diffuse density spike provides a longer accelerating region in ζ than a narrow spike, which reduces sensitivity to the exact location of the trailing pulse. The energy of the beam must be sufficiently high that it is resilient to the electrostatic fields of the plasma wave over the length of the accelerator. A 50-GeV, 5.8-nC drive beam that roughly corresponds to previous SLAC parameters55 satisfies these conditions and provides a steady accelerating structure over 1.2 cm in a background plasma density on the order of 1019 cm−3.

Because the photon accelerator preserves the structure of the trailing pulse, linear optical techniques based on conventional, adaptive, or metasurface optics can be used to prepare the initial structure of an optical pulse that is injected into the accelerator. The optical pulses used here correspond to the output of a commercially available Ti:sapphire laser system and have a vacuum wavelength λ = 800 nm, spot size w = 5.5 μm, full width at half maximum (FWHM) duration of τ = 31 fs, and energy E = 32 mJ. The pulses are initialized with different vector vortex profiles, which feature a polarization vector that varies in the plane perpendicular to the propagation axis. The profiles are constructed by superposing Laguerre–Gaussian modes in distinct polarization states35. Here, the orbital angular momentum value ℓ is changed while only the zeroth-order radial mode is used. The vector vortex structure can then be characterized using the bra–ket notation \(\left\vert \ell ,P\right\rangle\), where P can take the values V, H, R, or L for vertical, horizontal, right circular, or left circular polarization, respectively.

Figure 3 displays the results of quasi-3D PIC simulations performed with OSIRIS that demonstrate photon acceleration of vector vortex pulses from the optical to the XUV (see “Methods” for simulation details). Three equal-energy vector vortex pulses were considered to cover different combinations of orbital angular momentum and polarization: (a, e) a linearly polarized Gaussian pulse \(\left\vert 0,H\right\rangle\) with an intensity I = 2.1 × 1018 W/cm2; (b, f) a radially polarized pulse \(\left\vert 1,R\right\rangle +\left\vert -1,L\right\rangle\) with I = 7.9 × 1017 W/cm2; and (c, g) a pulse that has “spiderweb” polarization \(\left\vert 2,H\right\rangle +\left\vert -2,V\right\rangle\) with I = 5.8 × 1017 W/cm2. The top row shows their transverse polarization and intensity profiles near the end of the accelerator at the time slice where the intensity is the highest. The insets show the phase of the (a, c) horizontal and (b, d) radial electric field components, demonstrating preservation of the structure. In (a), (b), and (d), the variation across the regions of near-constant phase is less than 2°, 3°, and 12°, respectively. The bottom row shows the evolution of the spectra and energies along the length of the accelerator. In all cases, the pulses maintain their initial vector vortex structure, increase in average frequency (red dashed–dotted line) by more than a factor of 20, reach up to 370 times their initial intensity, and maintain their initial amplitude a0 to within a factor of 2, where \({a}_{0}=8.5\times 1{0}^{-10}{[I({{{\rm{W}}}}/{{{{\rm{cm}}}}}^{2})]}^{1/2}\lambda (\mu {{{\rm{m}}}})\) is the normalized vector potential.

Fig. 3: Spatio-polarization profiles and spectral evolution of vector vortex pulses accelerated in an electron beam–driven plasma wave.
figure 3

a–d The transverse intensity and polarization profile near the end of the accelerator for the temporal slice with the maximum intensity. The insets show the phase of the (a, c) horizontal and (b, d) radial components of the electric field. e–h The spectral energy density along the length of the accelerator. An electron beam with uniform emittance drives a density modulation that photon accelerates (a, e) a Gaussian pulse with linear polarization, (b, f) a radially polarized pulse, and (c, g) a pulse with “spiderweb” polarization. In each case, the initial vector vortex structure is preserved, and the average frequency upshifts by more than a factor of 20. d, h When the slice emittance of the electron beam is structured to compensate the transverse forces of the plasma wave, the radially polarized pulse upshifts by a factor of 26. In the bottom row, the red dashed–dotted line is the average frequency, and the purple line is the pulse energy within the first period of the plasma wave behind the electron beam. The intensity in (a) is scaled down by a factor of 3 for visualization.

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Quality of the photon-accelerated pulses

Photon acceleration converts the optical, radially polarized pulse to an ultrashort, high-intensity, structured XUV pulse with a central wavelength of 36 nm, intensity I = 2.5 × 1020 W/cm2, near-transform-limited FWHM duration of 710 as, and flat phase fronts. Figure 4 examines the evolution and properties of this pulse (Fig. 3b, f) in greater detail. The magnitude of the Wigner distributions W for the pulse is displayed in Fig. 4a, revealing that the pulse is initially divided by the density spike. The portion in front of the spike accelerates and compresses temporally. The compression can be observed in the phase-space rotation of the Wigner distribution from a temporally (horizontally) extended structure to a spectrally (vertically) extended structure.

Fig. 4: Quality of a radially polarized XUV pulse produced by photon acceleration in a beam-driven plasma wave.
figure 4

The pulse is the same as that presented in Fig. 3b, f, where the electron beam had uniform emittance. a Wigner distributions W at four z positions along the accelerator, showing the frequency upshift, increase in bandwidth, and temporal compression of the pulse. The dashed lines mark the location of the density spike. b The electric field of the pulse at the beginning (top) and end (bottom) of the accelerator. The phase fronts are nearly flat, and the final intensity profile (black solid line) is nearly transform-limited (purple dashed line). c The transverse intensity profile of the pulse (top right) when focused by an ideal f/3 lens compared to a diffraction-limited profile (bottom left). The spot size is only 75% larger than the diffraction limit, indicating the quality of the phase fronts. d The intensity (top) and electric field (bottom) of the pulse when focused by an f/3 plasma lens, which provides a potential experimental path to producing higher-intensity XUV light.

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Figure 4b shows that the frequency-upshifted pulse has flat phase fronts and a near-transform-limited duration. The top and bottom panels compare the electric fields of the pulse initially and near the end of the accelerator, respectively. The radial profile of the density modulation is locally parabolic near the center of the accelerating pulse (see Fig. 2a–c), which acts to guide the pulse and maintain flat phase fronts. The temporal compression of both the phase fronts and overall duration is apparent in Fig. 4b. The temporal profile of the XUV pulse at its peak intensity (black solid line) is close to the transform-limited profile (purple dashed line). The transform-limited profile was obtained by Fourier transforming the field to the frequency domain, setting the spectral phase to zero, and transforming back to the time domain.

To demonstrate the quality of the phase fronts, Fig. 4c compares the transverse profile of the XUV pulse in the focal plane of an f/3 lens (upper right) to the diffraction-limited profile of an ideal radially polarized beam (bottom left). The accelerated XUV pulse has a spot size of 120 nm, which is only 75% larger than the diffraction-limited spot. The focused profile of the XUV pulse was obtained by applying the phase of an ideal f/3 lens and then simulating the propagation to the focal plane using the unidirectional pulse propagation equation (UPPE)58.

In practice, achromatic focusing of such intense and broadband XUV pulses to near-diffraction-limited spots presents a technological challenge. The existing approaches to focusing short-wavelength light, such as toroidal mirrors or Fresnel zone plates59,60,61, cannot withstand high intensities or achieve diffraction-limited spots at small f numbers. As an alternative, the XUV pulses could be focused by a plasma lens62,63. Plasma lenses are advantageous because they are resistant to high intensities, and can therefore be placed in the far field, and are replaceable on a shot-to-shot basis. The downside is that they are inherently chromatic. UPPE simulations show that an f/3 plasma lens could focus a radially polarized XUV pulse produced by the photon accelerator to a 200-nm spot size, 800-as FWHM duration, and intensity I = 2.6 × 1022 W/cm2 (Fig. 4d). If an achromatic focusing scheme could be devised, the focused pulse would have a 120-nm spot size, 680-as duration, and intensity I = 1.3 × 1023 W/cm2.

Both the initial optical pulse and final XUV pulse have a relativistic amplitude a0 ≈ 0.6. Despite this high amplitude and regardless of the frequency, the pulse does not appreciably modify the density gradient responsible for the acceleration, i.e., the photon acceleration process is robust to high-intensity radiation. When the initial amplitude of the optical pulse is increased beyond a0 ≈ 0.6, however, the ponderomotive force of the witness pulse modifies the electron density in the spike, which disrupts the acceleration process. Figure 5 shows how the final average wavelength and intensity of the radially polarized XUV pulse depend on the initial a0. Ideal intensity scaling (dashed line) persists up to a0 ≈ 0.6, after which the final wavelength begins to increase and the intensity is limited. Even at a0 = 10, the frequency upshift is greater than a factor of 15. A similar trend is observed for other initial pulse durations and for linearly polarized Gaussian pulses, as verified by additional simulations (see Supplementary Note 1).

Fig. 5: Robustness of the photon accelerator to relativistically intense pulses.
figure 5

The final maximum intensity and average wavelength of the radially polarized pulse near the end of the accelerator as a function of the initial a0. The dashed line shows the ideal intensity scaling \(I\propto {a}_{0}^{2}\). The beam-driven plasma wave is resilient to the relativistic amplitude of the accelerated pulse up to a0 ≈ 0.6. Beyond this threshold, the ponderomotive force of the pulse deforms the electron density gradient enough to disrupt the acceleration.

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Evolution of the drive beam

The maximum frequency upshift of the laser pulse is limited by the evolution of the electron beam. The background electrons expelled by the beam and the surplus of ions that remain generate transverse and longitudinal fields that increase from the head of the beam to the tail. As a result, the back of the beam experiences the largest focusing and decelerating forces. The focusing, in particular, causes the back of the beam to pinch. This changes the location where the density modulation is zero ζδ, so that it no longer tracks the temporal centroid ζc of the trailing pulse.

Over the width of the beam, the focusing force responsible for the pinching Fr can be approximated as \({F}_{r}\approx -\frac{1}{2}\alpha (\zeta ,z){m}_{{{{\rm{e}}}}}{\omega }_{{{{\rm{p0}}}}}^{2}r\), where r is the radial coordinate, \({\omega }_{{{{\rm{p0}}}}}={[{e}^{2}{n}_{0}(0)/{m}_{{{{\rm{e}}}}}{\varepsilon }_{0}]}^{1/2}\) is the plasma frequency at z = 0, and the coefficient α(ζ, z) depends on the background plasma and beam parameters. The spot size of the electron beam σ(ζ, z) evolves in response to this force as follows64:

$$\frac{{{{{\rm{d}}}}}^{2}\sigma }{{{{\rm{d}}}}{z}^{2}}+\frac{1}{\gamma }\frac{{{{\rm{d}}}}\gamma }{{{{\rm{d}}}}z}\frac{{{{\rm{d}}}}\sigma }{{{{\rm{d}}}}z}=\sigma \left[\frac{{\epsilon }_{{{{\rm{n}}}}}^{2}(\zeta )}{{\gamma }^{2}(\zeta ,z){\sigma }^{4}}-{k}_{\beta }^{2}(\zeta ,z)\right],$$
(4)

where \({k}_{\beta }(\zeta ,z)=\sqrt{\alpha (\zeta ,z)/2\gamma (\zeta ,z)}{\omega }_{{{{\rm{p0}}}}}/c\) and ϵn(ζ) is the normalized emittance. If the plasma density and beam energy are constant, the beam will propagate with a constant (matched) spot size when the focusing force compensates the natural expansion of the beam, i.e., kβ = ϵn/γσ2. For the conditions considered here, however, Fr does not vary linearly with r across the entire beam, the background density n0 depends on z, and the electron energy γ depends on both ζ and z. This makes it difficult to structure the emittance so that exact matching is satisfied everywhere.

Approximate matching can be achieved by choosing the emittance to match the average focusing force, i.e., ϵn = γkβσ2. The uniform-emittance electron beam used in the simulations had a normalized emittance determined by the average force: \({\epsilon }_{{{{\rm{n}}}}}={({\gamma }_{0}\bar{\alpha }/2)}^{1/2}{\sigma }_{0}^{2}{\omega }_{{{{\rm{p0}}}}}/c=\) 1.2 mm-rad, where σ0 is the initial beam radius, γ0 is the initial electron energy, and \(\bar{\alpha }\) is the average of α over density and ζ (see “Methods” for details). Simulations of the beam propagating through the full density ramp shown in Fig. 2d confirmed that this value of emittance slowed the evolution of the beam and optimized the length of the photon accelerator.

An emittance profile that is tailored in ζ can further slow the beam evolution, enabling higher-intensity and shorter-wavelength XUV pulses. The tailored-emittance electron beam used for Fig. 3d, h had a slice-dependent normalized emittance given by \({\epsilon }_{{{{\rm{n}}}}}(\zeta )={[{\gamma }_{0}\tilde{\alpha }(\zeta )/2]}^{1/2}{\sigma }_{0}^{2}{\omega }_{{{{\rm{p0}}}}}/c\), where \(\tilde{\alpha }(\zeta )\) is the average of α over density (see “Methods” for details). Comparing Fig. 3d, h and b, f shows that the tailored-emittance profile increases the energy and frequency of the XUV pulse by 12% (final average wavelength λ = 32 nm).

Discussion

A relativistic electron beam can drive a plasma wave that accelerates high-intensity vector vortex laser pulses from the optical to the XUV over a few centimeters. The efficiency of the frequency conversion can be defined in several ways. In the case of the radially polarized pulse, the energy of the electron beam decreased from 290 to 280 J, while the pulse energy in the first period of the plasma wave increased from 14 to 87 mJ. This transfer of energy from the beam to the pulse corresponds to an efficiency of 7.3 × 10−3, which compares favorably to other schemes for generating XUV light17,18,22.

The frequency conversion in a plasma wave–based photon accelerator conserves the number of photons and phase periods46,47,65. The conservation of photon number can be used to determine (approximately) the fraction of initial photons that are trapped and accelerated by the plasma wave, or “trapping efficiency.” The number of photons \({{{\mathcal{N}}}}\propto \omega \tau {w}^{2}{a}_{0}^{2}\) and energy \(E\propto {\omega }^{2}\tau {w}^{2}{a}_{0}^{2}\) are related by \({{{\mathcal{N}}}}=E/\omega\). If the entirety of the initial pulse contained in the first plasma period were accelerated, the final (f) energy would exceed the initial (i) energy by the ratio of the final and initial frequencies: Ef = (ωf/ωi)Ei. As observed in Fig. 3, however, the final energies of the XUV pulses are somewhat lower: Ef/Ei ≈ 6, while ωf/ωi ≈ 22. This is because a portion of the photons eligible for acceleration in the first period of the plasma wave escape laterally due to diffraction. This effect is most pronounced for the “spiderweb” pulse, whose radial extent is comparable to the transverse width of the density spike (Fig. 3c, g). Evaluating the right-hand side of the relation \({{{{\mathcal{N}}}}}_{{{{\rm{f}}}}}/{{{{\mathcal{N}}}}}_{{{{\rm{i}}}}}=({\omega }_{{{{\rm{i}}}}}/{\omega }_{{{{\rm{f}}}}})({E}_{{{{\rm{f}}}}}/{E}_{{{{\rm{i}}}}})\) with the results for the radially polarized pulse indicates that approximately 30% of the initially eligible photons were trapped and accelerated. Conservation of the number of phase periods ∝ ωτ can be used with the photon number to show that wa0 should also be conserved. Despite the initial escape of photons, a0 is roughly constant because the electron density modulation from the plasma wave slightly focuses the pulse.

The background plasma density n0(z) was tailored to optimize the frequency-upshift of a pulse trapped in the first period of the plasma wave [Eqs. (2) and (3)]. Because the initial pulse spanned multiple periods (see Fig. 2a), a substantial portion of the pulse was also trapped in the second period. Despite the suboptimal density profile for acceleration in the second period, this portion upshifted to the XUV. For the radially polarized pulse pictured in Fig. 3b, f, the second pulse reached an average wavelength λ = 51 nm, energy E = 120 mJ, and intensity I = 2.7 × 1020 W/cm2. In principle, an initial pulse with a longer duration that overlapped many periods could produce a train of XUV pulses separated by ≈20 fs, but with gradually increasing wavelengths. Such a train may have utility as a sequence of probes for ultrafast, time-resolved measurements.

The design of the photon accelerator was motivated by electron-beam parameters previously achieved at SLAC and laser pulses produced by commercially available Ti:sapphire amplifiers66. The 50-GeV electron beam evolved minimally over 1.2 cm: the electrons in the tail of the beam lost at most 4 GeV, or 8% of their initial energy. This suggests that lower-energy drive beams with  >4-GeV electrons could be used to obtain comparable frequency shifts. However, mitigating beam evolution at lower energies may require a slice-by-slice transverse profile and emittance as determined by a more complete theoretical model, e.g., a generalization of Eq. (4) to higher radial moments. Optimal structuring of drive beams is an outstanding area of research in beam dynamics that also has applications to collider physics67. Such an investigation is outside the scope of this work, but it provides a starting point for future studies.

Further optimization of the photon accelerator could be achieved by tuning the plasma, electron beam, or laser pulse parameters. The photon accelerator inherently provides a tunable-wavelength source: the length of the plasma can be adjusted to obtain any frequency between the initial and final frequencies. A wider electron beam with the same charge density would drive a plasma wave with a broader density spike, allowing for the acceleration of laser pulses with larger spot sizes and more energy. In principle, the initial temporal profile or phase of the laser pulse could be structured to control the final bandwidth or duration, either through linear or nonlinear interactions with the plasma wave. As a final example, simulations of vector vortex pulses initialized with λ = 400 nm (i.e., frequency-doubled Ti:sapphire pulses) yielded an XUV pulse with 4× higher peak intensity (1.5 × 1021 W/cm2), comparable final wavelength (47 nm), and comparable quality as pulses initialized with λ = 800 nm accelerated over the same distance. The higher intensity resulted from better trapping efficiency; the f/N was the same between the two cases, but longer-wavelength pulses are more susceptible to plasma refraction and lateral losses due to their larger spot size.

In summary, a nonlinear plasma wave driven by a relativistic electron beam can frequency upshift optical vector vortex pulses to the XUV while preserving the initial vector vortex structure and maintaining collimation. Boosted-frame, quasi-3D PIC simulations demonstrated the formation of relativistically intense (1020 W/cm2), attosecond (710 as), vector vortex XUV pulses with λ = 36-nm average wavelengths—frequency upshifts by more than a factor of 20. The pulses are nearly transform-limited in duration and have flat phase fronts that can be focused to within 75% of the diffraction-limited spot size. The photon acceleration process is largely independent of laser polarization and is mainly affected by the radial extent of the pulse, indicating that, in principle, any vector vortex profile could be accelerated to the XUV. The tunable XUV source proposed here bridges the gap in wavelength, structure, and intensity between traditional lasers and existing XUV light sources, providing new tools for scientific discovery.

Methods

Tailored density profile

Tailoring the longitudinal profile of the plasma density ensures that the accelerating pulse remains colocated with the density gradient of the plasma wave52. The density profile n0(z) for which the locations of the temporal centroid of the pulse ζc and zero density modulation ζδ are equal can be found by solving the coupled system of Eqs. (2) and (3). In order to close this system, analytic or numerical solutions are needed for ζδ(n0) and the density gradient \({\partial }_{\zeta }\delta n(\zeta ,{n}_{0}){| }_{\zeta = {\zeta }_{\delta }}\) as functions of the background density. Here, these solutions are obtained numerically using a 1D, quasistatic model54,68. In this model, an electron beam with density nb(ζ) drives a plasma wave whose normalized electrostatic potential ϕ(ζ, n0) = eΦ/mec2 evolves according to

$$\frac{{\partial }^{2}\phi }{\partial {\zeta }^{2}}=\frac{1}{2}{\omega }_{{{{\rm{p}}}}}^{2}\left[\frac{1}{{(1+\phi )}^{2}}-1+\frac{2{n}_{{{{\rm{b}}}}}(\zeta )}{{n}_{0}}\right],$$
(5)

where \({\omega }_{{{{\rm{p}}}}}={({e}^{2}{n}_{0}/{m}_{{{{\rm{e}}}}}{\varepsilon }_{0})}^{1/2}\). Upon solving this equation for ϕ(ζ, n0), the electron density modulation can be calculated as

$$\delta n(\zeta ,{n}_{0})=\frac{{n}_{0}}{2}\left\{\frac{1}{{\left[1+\phi (\zeta ,{n}_{0})\right]}^{2}}-1\right\}.$$
(6)

This expression can then be used to find ζδ(n0) and \({\partial }_{\zeta }\delta n(\zeta ,{n}_{0}){| }_{\zeta = {\zeta }_{\delta }}\).

For implementation in the PIC simulations, the numerically calculated density profile n0(z) is fit with the four-parameter equation

$$\frac{{n}_{0}(z)}{{n}_{0}(0)}=A+\frac{B}{{(C+z)}^{D}}\approx 0.2517+\frac{1.255}{{(20.53+z)}^{0.1710}},$$
(7)

where the right-most expression shows the values of the fitting parameters for the electron beam and initial laser pulse considered here. The fit agrees almost exactly with the numerically calculated profile and is shown in Fig. 6. Based on the results of the PIC simulations, the 1D quasistatic model and selected fit do an excellent job of approximating the ideal background density profile. In an experiment, the production of this continuous density profile may present a challenge. However, as demonstrated by Sandberg and Thomas53, comparable results can be obtained by using segments of constant-density plasma. This could be achieved experimentally with a sequence of gas cells or jets at different densities.

Fig. 6: Background plasma density profile.
figure 6

Detailed reproduction of Fig. 2d, showing the density profile used in the simulations [Eq. (7)].

Full size image

While 3D effects could modify the ideal density profile, these modifications are expected to be small. This is because the photon accelerator is designed to operate in a quasi-nonlinear regime of plasma wave excitation, where the longitudinal forces dominate the transverse forces. Moreover, the robust performance of the photon accelerator and its insensitivity to the exact profile (such as when using segments of constant density53) indicate that small corrections to account for 3D effects would not significantly impact the maximum frequency shift.

Note that within the 1D quasistatic approximation, the electron density n(ζ, z) = n0(z) + δn(ζ, z) can be expressed in terms of the electrostatic potential and electron energy69:

$$n(\zeta ,z)=\frac{\gamma (\zeta ,z)}{1+\phi (\zeta ,z)}{n}_{0}(z).$$
(8)

As a result, the rate of frequency upshifting depends on the gradient of the potential:

$$\frac{{{{\rm{d}}}}\omega }{{{{\rm{d}}}}z}=\frac{{e}^{2}}{2{\varepsilon }_{0}{m}_{{{{\rm{e}}}}}{c}^{2}{k}_{z}}\frac{\partial }{\partial \zeta }\left(\frac{1}{1+\phi }\right).$$
(9)

The gradient of the potential ∂ζϕ has an extremum where δn = 0 [Eq. (6)]. Thus, in 1D, the location where δn = 0 corresponds to the location where the rate of frequency upshifting is the largest.

Particle-in-cell simulations

The bulk of the simulations presented in this work were performed using the PIC code OSIRIS56, which solves the fully relativistic equations of motion for particles and employs a finite-difference time-domain solver for the electromagnetic fields. The simulations were discretized in the quasi-3D geometry, which represents quantities as a truncated expansion in azimuthal modes57. This geometry is appropriate for systems with a high degree of cylindrical symmetry. The small cell sizes required to properly resolve the upshifted light make lab-frame simulations of the 2-cm-long accelerators prohibitively expensive, so simulations were performed in a Lorentz-boosted frame70,71,72 with γ = 15. Simulations in the boosted frame are susceptible to the numerical Cerenkov instability (NCI), which can cause spurious radiation to grow from noise. This was mitigated by using a customized solver73, which was found to suppress NCI while simultaneously correcting for numerical dispersion and time-staggering errors in the Lorentz force. Results were interpolated from the boosted frame to the lab frame when plotting the results.

The simulations used a moving window in the boosted frame of size 458 × 76.4 μm (14,896 × 1200 cells) in z and r, respectively. The time step was 5.73 as, and the total simulation duration was 2.91 ps. Electrons (ions) were simulated with 16–48 (8–16) particles per cell, depending on the number of azimuthal modes used in the quasi-3D geometry. The number of modes was one larger than the highest ℓ mode of each vector vortex pulse, i.e., modes 0, 1, and 2 were used for the radially polarized pulse. Truncating the number of azimuthal modes can result in inaccurate modeling of 3D effects, such as hosing instabilities. To verify that the number of modes was sufficient, convergence tests were performed by doubling the number of modes for the Gaussian pulse. Compared to the nominal number of modes, the central frequency and intensity of the final XUV pulse only changed by a few percent.

Each laser pulse had an initial wavelength λ = 800 nm, spot size w = 5.5 μm, energy E = 32 mJ, and FWHM duration of τ = 31 fs. The electron beam was initialized with a 0.64-μm length, σ0 = 13-μm e−1/2 radius, 5.8 nC of total charge, and an average electron energy of γ0 = 50 GeV. For the uniform-emittance cases, a normalized emittance ϵn = 1.2 mm-rad was used; for the tailored-emittance case, the normalized emittance varied linearly from 0.11 mm-rad at the head of the beam to 1.9 mm-rad at the tail. The background plasma density shown in Fig. 2d with n0(0) = 7 × 1019 cm−3 was used in all cases.

To determine the optimal emittance of the electron beam, simulations were performed using QPAD74—a quasistatic PIC code that employs azimuthal decomposition. The focused beam was propagated a short distance into plasmas with constant densities ranging from ne = 3 to 7 × 1019 cm−3. At each density, the transverse force of the plasma wave on the beam was fit to \({F}_{r}=-\frac{1}{2}\alpha (\zeta ,{n}_{0}){m}_{{{{\rm{e}}}}}{\omega }_{{{{\rm{p0}}}}}^{2}r\), where ωp0 is the plasma frequency for the maximum density n0(0) = 7 × 1019 cm−3, and α(ζ, n0) is a fitting parameter that depends on ζ and density. For linear plasma waves driven by a Gaussian beam, this parameter can be approximated as \(\alpha \approx 2{c}^{2}\delta n(\zeta ,{n}_{0})/{n}_{0}{\omega }_{{{{\rm{p0}}}}}^{2}{\sigma }^{2}\) 75. For the uniform-emittance cases, the emittance was calculated using the density- and ζ-averaged value \(\bar{\alpha }=4.8\times 1{0}^{-4}\) [\({\epsilon }_{{{{\rm{n}}}}}={({\gamma }_{0}\bar{\alpha }/2)}^{1/2}{\sigma }_{0}^{2}{\omega }_{{{{\rm{p}}}}0}/c\)]. For the tailored-emittance case, the slice emittance was calculated using the density-averaged values \(\tilde{\alpha }(\zeta )\), which increased from 0 at the head of the beam to 7 × 10−4 at the tail.

All PIC simulations were performed on the CPU nodes of Perlmutter at NERSC, each of which contains two AMD EPYC 7763 processors. The OSIRIS simulations took between 47 and 310 total node hours on either 8 or 16 compute nodes, depending on the number of azimuthal modes used. The QPAD simulations had a wall time of about 5 min on 4 compute nodes. Quasi-3D OSIRIS simulations performed without using a Lorentz-boosted frame would have taken on the order of 105–106 node hours, making them prohibitively expensive.