Abstract

Broadband optical pulses with attosecond to femtosecond durations provide unique opportunities for studies of time-resolved electron dynamics. However, focusing these pulses—typically ranging from the vacuum ultraviolet to the soft-X-ray region—remains challenging. Conventional refractive lenses are not suitable owing to large dispersion and strong absorption, whereas reflective optics do not suffer from these issues but have high losses. Here we demonstrate a tunable hydrogen plasma lens to focus broadband extreme-ultraviolet attosecond pulses with energies of around 20 eV and 80 eV. Simulation results suggest that the stretching of attosecond pulses is negligible, and temporal compression is possible when atto-chirp is included. A key advantage of the plasma lens is its compatibility with nonlinear frequency conversion processes like high-harmonic generation. The different focusing properties of the fundamental and harmonic frequencies allow for an efficient separation of these components. Consequently, the transmission of high-harmonic generation beamlines can be increased to more than 80% and this approach can be suitable for applications requiring high photon flux.

Main

Refractive lenses are a straightforward method to focus light without altering its propagation direction. Their use, however, is problematic for ultrashort laser pulses with femtosecond durations. Due to dispersion, different frequency components propagate at different group velocities inside the lens, leading to pulse stretching at the focus and reducing the temporal resolution in pump–probe experiments1. On a microscopic level, refraction is typically governed by the interaction of light with bound electrons within the material. In response to the incident oscillating electromagnetic field, these electrons start to oscillate and emit radiation with the same frequency but with a phase delay.

When dealing with even shorter light pulses in the attosecond regime2,3,4,5,6, an additional challenge for using refractive lenses arises: the spectra of these pulses typically fall within the extreme-ultraviolet (XUV) and soft-X-ray regions of the electromagnetic spectrum, where matter is highly absorbing. In the lower part of the XUV region, up to about 25 eV, this problem has been addressed through the development of gas-phase lenses7 and metalenses8. Yet, like their counterparts in other spectral regions, these lenses are designed for specific wavelength ranges and are not suitable for focusing extremely short light pulses in the attosecond regime. Instead, mirrors are commonly used to focus attosecond pulses, although they come with several disadvantages: mirrors suffer from low reflectivities, rapid degradation effects9 and may require sophisticated alignment strategies10,11.

Plasma lens

Here we propose and demonstrate a refractive plasma lens for focusing broadband XUV pulses with an initial duration in the attosecond range. The refractive index n originating from the interaction of an electromagnetic field with free electrons is given by12

$$n=\sqrt{1-{\omega }_{{\rm{p}}}^{{2}/{\omega }}{2}},$$

(1)

$${\omega }_{{\rm{p}}}^{{2}={n}_{{\rm{e}}}{e}}{2}/{\varepsilon }_{0}{m}_{{\rm{e}}}.$$

(2)

Here ωp is the plasma frequency and ω is the angular frequency of the external electromagnetic field. The former depends on the free electron density ne, the electron charge e, the permittivity in free space ε0 and the electron mass me. For frequencies exceeding the plasma frequency, the refractive index has real values below unity, leading to a phase advance of the incident electromagnetic field. To exploit these properties of the propagation of light in free electron density for focusing attosecond light pulses, a concave radial electron density profile with a minimum electron density on the optical axis is required. Optimal performance is achieved for a parabolic electron density profile13,14. Unlike conventional optics, the plasma lens is not affected by XUV-induced damage, as the plasma can be replenished with every laser shot. In this context, plasma lenses have also been proposed for focusing high-power laser pulses15,16,17.

The plasma lens is ideally suited for focusing attosecond pulses comprising a broad spectral bandwidth. Although the refractive index varies across the spectrum, pulse stretching within the plasma lens under our experimental conditions (ωp = 0.56 PHz and photon energy of 80 eV) is small. Moreover, the negative dispersion of plasma offers the possibility to temporally compress positively chirped attosecond pulses, as discussed later.

To create conditions suitable for focusing broadband XUV pulses, we generated a plasma from hydrogen molecules using capillary discharge18,19,20,21 (Fig. 1). Due to the low electron binding energies, hydrogen can be nearly fully ionized13,18. The absence of bound electrons results in an exceptionally high transmission of the XUV pulses, which is an attractive feature of plasma that sets it apart from all other forms of matter.

Fig. 1: Scheme of the plasma lens experiment.

Capillary discharge source consisting of a sapphire block that has a 5-cm-long, 300-µm-diameter channel. Hydrogen is delivered via four symmetric inlets. Copper electrodes are attached to both ends, applying a current pulse to ignite a plasma. An XUV pulse produced by HHG is transmitted through the channel, and a knife edge is placed at the focal plane to measure the XUV focus size. Following diffraction from a grating, the XUV spectrum is recorded using an MCP/phosphor screen assembly, and the data are acquired using a charge-coupled device camera. a, Illustrative parabolic plasma density profile, ne (purple curve), across the capillary profile following plasma cooling at the channel walls. The corresponding refractive index, n, for 80-eV XUV light is shown in blue. b, NIR beam profiles in the XUV focal plane without (red curve) and with (yellow curve) plasma, showing that the NIR beam is effectively defocused after the plasma lens.

The capillary discharge source consists of a sapphire block that contains a 5-cm-long, 300-µm-diameter channel. Hydrogen is supplied via four inlets and ionized by applying a discharge current pulse using electrodes attached at both ends of the capillary (Methods). Following the ignition of plasma, its temperature distribution becomes inhomogeneous due to energy exchange with the relatively cooler capillary walls, resulting in the formation of a parabolic plasma density profile13,22. An attractive feature of the plasma lens is the ability to control its focal length by varying the hydrogen pressure inside the capillary.

Experimental demonstration of focusing broadband XUV pulses

The experiments were performed utilizing an 18-m-long beamline that was optimized for the generation of intense attosecond pulses via high-harmonic generation (HHG)23,24 (Methods). Following spectral filtering using an ultrathin Sn foil, XUV pulses centred at 20 eV were focused by the plasma lens, which was mounted at a distance of 13 m from the HHG target. To characterize the focus, a knife edge, placed 20 cm after the lens, was scanned across the XUV beam profile. The XUV beam was diffracted from a grating onto a microchannel plate (MCP)/phosphor screen assembly (Fig. 1). The measurements were conducted during the peak plasma transmission window, approximately 160 ns after the discharge (Supplementary Fig. 4). Figure 2a shows the spatial profiles of the XUV beam for an evacuated capillary (yellow triangles) and following plasma generation using a hydrogen pressure of 63 mbar (purple circles). Focusing of the XUV beam is clearly observed, characterized by a decrease in the 1/e2 radius from 97 ± 2 µm to 40 ± 2 µm. The focused radius as a function of photon energy and hydrogen pressure is depicted in Fig. 2b, showing good agreement with the numerical results obtained using a wave propagation method (dashed line; Supplementary Section 8 provides further details).

Fig. 2: Experimental realization of the plasma lens.

a, XUV beam profiles measured 20 cm after the lens, for an evacuated capillary (yellow triangles) and following plasma ignition at a hydrogen pressure of 63 mbar (purple circles). The plasma lens reduces the beam radius from 97 ± 2 µm to 40 ± 2 µm. The values were obtained by performing Gaussian fits (solid curves) to the knife-edge scan data. Observed XUV transmission during plasma operation was approximately 62%, indicating that hydrogen was not fully ionized. b, Spectrally resolved fitted XUV beam radii as a function of hydrogen pressure inside the capillary. Each value is obtained from a single knife-edge scan, with the central point given by the fitted radius and the error bars representing the standard deviation from the fit. The dashed curve shows the simulated XUV beam waist, calculated for a photon energy of 21 eV.

Although the focusing of XUV beams in the 20–25-eV region using lenses has been demonstrated previously7,8, no lenses have so far been developed for higher XUV photon energies. XUV light at around 70–100 eV (corresponding to wavelengths of 12.4–17.7 nm) is of particular interest, as it enables the interaction with core electrons in atoms and molecules25,26. Furthermore, XUV lithography, a critical technology for fabricating integrated circuits with structure sizes in the few-nanometre range, relies on light at a wavelength of 13.5 nm (refs. 27,28).

We have generated broadband XUV pulses centred at 80 eV (Fig. 3a) using HHG in Ne and a Zr foil for spectral filtering. Since the refractivity decreases with increasing photon energies for otherwise identical conditions (equation (1)), this configuration results in a longer XUV focal length. The XUV focus size was measured directly on an MCP/phosphor screen assembly that was placed 120 cm behind the lens (Fig. 3c). The pressure-dependent XUV beam radius is presented in Fig. 3b, showing good agreement between the experiment and simulation. The relatively large XUV focus waist of 80 ± 1 µm is a result of the small numerical aperture used in this experiment. Figure 3d presents a spectrally resolved simulation of the XUV beam profile under the same conditions. The beam waist radius integrated over the entire bandwidth is only slightly larger than the beam waist radius at 80 eV (80 µm compared with 79 µm), showing that detrimental effects due to chromatic aberrations are small.

Fig. 3: Focusing XUV pulses at 80 eV.

a, XUV spectrum obtained from HHG in Ne measured after a Zr filter. b, Comparison of the experimentally obtained XUV beam radii (violet circles) and the simulated XUV beam radii (dashed line) as a function of hydrogen pressure. Each experimental point is given by the mean of Gaussian-fitted radii from 15 measured XUV beam profiles, with error bars representing the standard deviation of the fitted values. c, XUV beam profile measured in the presence of plasma at a hydrogen pressure of 70 mbar. d, Simulated spectrally resolved XUV beam profile for the same experimental conditions as in c, showing a waist of 80 µm. e, Spectrally resolved XUV beam profile, calculated for a capillary with a length of 10 cm and a pressure of 200 mbar, showing a waist of 27 µm.

Tighter focusing of the XUV beam can be achieved in the future by increasing the length of the plasma lens. Our simulations suggest that for a capillary length of 10 cm, an XUV beam waist radius of 27 µm is achievable when using a hydrogen pressure of 200 mbar (Fig. 4e). In this case, the focal length is reduced to 41 cm. Detrimental effects due to chromatic aberrations remain small in this case as well: the waist at 80 eV is 23 µm, compared with 27 µm when integrated over the full spectrum. Importantly, discharge capillaries initially filled with hydrogen at pressures of up to 330 mbar have already been demonstrated29. Additionally, even stronger focusing may be achievable in the future by using laser-heated capillaries, which further increase the on-axis plasma temperature and steepen the refractive index gradient across the capillary profile30,31,32.

Fig. 4: Simulated temporal profiles of attosecond pulses focused by a plasma lens.

a, Temporal profiles of a transform-limited 90-as pulse centred at 80 eV before (blue) and after (violet) passing through a plasma lens filled with 70 mbar of hydrogen, resulting in negligible stretching to 96 as. b, Simulation of an initially chirped 190-as pulse propagating through the same plasma lens, demonstrating compression to 165 as. This suggests that a plasma lens can be used for atto-chirp compensation. Norm., normalized.

Temporal properties of attosecond pulses focused by a plasma lens

To study the temporal properties of attosecond pulses focused by the plasma lens, we have performed pulse propagation simulations (Supplementary Section 11). First, we considered a transform-limited attosecond pulse with a central photon energy of 80 eV and a full-width at half-maximum duration of 90 as passing through a plasma lens filled with 70 mbar of hydrogen. As shown in Fig. 4a, the pulse experiences negligible stretching to 96 as in the XUV focal plane. Our analysis indicates that the group delay dispersion is the main contribution to pulse stretching (Supplementary Section 12).

However, when accounting for the positive chirp (known as the atto-chirp)33 acquired by attosecond pulses during HHG, the hydrogen plasma may be used to temporally compress the pulses34. Under typical experimental conditions, our simulations show that pulses with an initial duration of 190 as may be compressed to 165 as (Fig. 4b). At a higher pressure of 200 mbar, further compression down to 127 as is anticipated (Supplementary Fig. 10), suggesting that the attosecond plasma lens could enhance the temporal resolution in pump–probe experiments.

Discussion

The properties of the plasma lens may be exploited in the future to substantially increase the transmission of HHG beamlines. Following XUV pulse generation via HHG, the attosecond pulses co-propagate with the more powerful near-infrared (NIR) driving pulses. To avoid strong NIR laser fields being present in the focal plane, thin metal filters are typically used to attenuate the NIR power. However, this approach comes with substantial XUV transmission losses. A key advantage of the plasma lens is that the focusing properties of NIR and XUV beams are entirely different. The focal plane of the NIR beam is located inside the capillary, and the NIR beam is divergent after exiting the plasma lens (Fig. 1b). As a result, an estimated NIR intensity of only 108 W cm−2 is obtained at the XUV focal plane (Supplementary Section 9). This value is negligible in typical pump–probe experiments. Consequently, the use of thin metal filters may become unnecessary in the future. In addition, we benefit from the low XUV absorption within the lens. The plasma lens enables a beamline transmission exceeding 80%, compared with typical values on the order of 10%–20% (refs. 35,36). This provides opportunities for photon-demanding attosecond applications, such as applications in attochemistry using molecules that can only be supplied at low densities37,38,39.

Another drawback of thin metal filters is their low damage threshold, limiting their use to sufficiently low NIR intensities. To attenuate the NIR power, additional mirrors may be used40, but this further reduces the XUV beam transmission. Alternatively, the filter must be positioned sufficiently far from the HHG source. By removing the need for a filter, a plasma lens can be placed much closer to the HHG source, enabling a more compact setup.

For the proof-of-principle experiments described here, the capillary discharge source was operated at 20 Hz. Capillary discharge sources operating at kilohertz repetition rates have already been demonstrated41, which could enhance the XUV photon flux available in experiments and may, therefore, enable the temporal characterization of focused XUV pulses using attosecond streaking or RABBITT3,4,42. Alternatively, lasers may be used for plasma generation at kilohertz repetition rates. Laser-produced plasmas with suitable characteristics have already been demonstrated for waveguiding applications43,44,45,46, where axicon lenses were used to create extended plasma channels with minimal electron density along the optical axis.

The plasma lens is suitable for focusing light at other wavelengths, including few-femtosecond deep-ultraviolet and vacuum-ultraviolet pulses, which can be efficiently generated in gas cells47 or through soliton dynamics in hollow-core fibres48,49,50,51,52. An important challenge is the attenuation of fundamental light, which can be addressed using mirrors at Brewster’s angle50. However, this approach suffers from notable reflection losses and mirror degradation. These obstacles can be overcome by using a plasma lens, which promises high transmission and effective defocusing of the driving laser. We have performed simulations confirming that the plasma lens can efficiently separate the fundamental NIR beam from the co-propagating vacuum-ultraviolet pulse, allowing for the effective isolation of vacuum-ultraviolet light (Supplementary Section 10).

In conclusion, we have proposed and demonstrated a lens that focuses broadband XUV pulses by interaction with free electrons in a nearly fully ionized plasma. The plasma lens is characterized by high transmission, immunity to XUV laser damage and inherent wavelength tunability.

Methods

Attosecond pulse generation

NIR driving laser pulses were obtained from a Ti:sapphire laser system operating at 1 kHz (Spitfire, Spectra Physics). In the current experiment, 37-fs-long pulses centred at 800 nm with a pulse energy of 9 mJ were used. To enable the generation of near-isolated attosecond pulses, the NIR pulses were temporally compressed down to 3.7 fs using a three-stage post-compression scheme, resulting in a pulse energy up to 4.9 mJ (ref. 23). Attosecond pulses were generated using an 18-m-long HHG beamline23,24. To this end, the NIR pulses were focused using a telescope consisting of a convex (f = −1 m) and a concave (f = 0.75 m) spherical mirror. HHG was performed in a 4-cm-long cell that was statically filled with gas. Argon was used for the generation of harmonics at around 20 eV and 60 eV, whereas neon was used for the generation of harmonics around 80 eV. The experimental vacuum chamber, in which the attosecond plasma lens was mounted, was placed approximately 13 m downstream from the HHG cell. The experimental setup for attosecond pulse generation is presented in Supplementary Fig. 1. To select specific spectral regions of the XUV pulses, either a 100-nm Sn foil or a 150-nm Zr foil was inserted. The measured XUV spectra are shown in Supplementary Fig. 2. For the knife-edge measurements conducted 20 cm after the capillary (Fig. 2), an attosecond pulse filtered by a Sn filter was used (Supplementary Fig. 2a), resulting in a full-width at half-maximum of 6.8 eV. The XUV spectrum centred at 80 eV, as used in Fig. 3, is shown in Supplementary Fig. 2b. Additionally, the dashed line in Supplementary Fig. 2b represents an XUV spectrum with a full-width at half-maximum of 22 eV, which was used for the calculations shown in Figs. 3 and 4. Transmission data for both filters were obtained from another work53.

Plasma lens pressure calibration

A constant hydrogen gas flow into the capillary was achieved using a thermal mass flow controller (Bronkhorst). A series of XUV light transmission measurements were performed to accurately determine the relationship between the hydrogen mass flow and the resulting gas density inside the capillary. To this end, the spectrally resolved transmission of the XUV light through the capillary was measured as a function of the hydrogen flow (Supplementary Fig. 3a). By applying Beer–Lambert’s law and using known molecular hydrogen cross-sections (σ)54, the neutral atomic hydrogen density na inside the capillary was determined according to

$${n}_{{\rm{a}}}=-\frac{2\mathrm{ln}[T]}{\sigma L},$$

where T is the measured transmission of a specific harmonic of XUV light and L is the length of the capillary. To calculate the pressure from the gas density, the ideal gas law was used, assuming a temperature of 295 K. Calculated hydrogen pressures are presented in Supplementary Fig. 3b. The measured data points were fitted with a power function of the form ax**b, where a and b are the fitting coefficients and x is the gas flow. The resulting calibration fit is displayed in Supplementary Fig. 3b as a red curve.

Hydrogen ignition

The hydrogen gas inside the capillary was ignited using a high-voltage current pulser, which generates submicrosecond-long electron pulses with a 500-A peak current at a repetition rate of up to 1 kHz. The current pulse was delivered to the lens through copper electrodes attached to the ends of the capillary, and the discharge current was monitored using an oscilloscope (Supplementary Fig. 4, shaded area). For the experiments presented in the main text, the pulser repetition rate was set to 20 Hz.

Data availability

All data used in the article or the Supplementary Information are accessible via Zenodo at https://doi.org/10.5281/zenodo.15180960 (ref. [55](https://www.nature.com/articles/s41566-025-01794-y#ref-CR55 “Svirplys, E. et al. Plasma lens for focusing attosecond pulses. Zenodo https://doi.org/10.5281/zenodo.15180960

(2025).“)).

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