An achromatic X-ray lens

Diffractive and refractive optical elements have become an integral part of most high-resolution X-ray microscopes. However, they suffer from inherent chromatic aberration. This has to date restricted their use to narrow-bandwidth radiation, essentially limiting such high-resolution X-ray microscopes to high-brightness synchrotron sources. Similar to visible light optics, one way to tackle chromatic aberration is by combining a focusing and a defocusing optic with different dispersive powers. Here, we present the first successful experimental realisation of an X-ray achromat, consisting of a focusing diffractive Fresnel zone plate (FZP) and a defocusing refractive lens (RL). Using scanning transmission X-ray microscopy (STXM) and ptychography, we demonstrate sub-micrometre achromatic focusing over a wide energy range without any focal adjustment. This type of X-ray achromat will overcome previous limitations set by the chromatic aberration of diffractive and refractive optics and paves the way for new applications in spectroscopy and microscopy at broadband X-ray tube sources.

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X-ray techniques for the non-invasive investigation of the inner structure and elemental composition of matter at the micro- and nanoscale require high-performance X-ray optics. For this purpose, various types of reflective, refractive, and diffractive optical elements have been developed in the last decades1,2,3. Reflective X-ray optics rely on grazing incidence configuration and require complicated geometries to produce a magnified image of an extended field of view4,5, resulting in long focal lengths that are incompatible with a compact setup. While these limitations do not apply to refractive and diffractive optics, making them more suitable for the formation of magnified X-ray images, they suffer from inherent chromatic aberration, meaning that X-rays of different energies are not focused to the same focal plane. As a consequence, the performance of these optics for multi-energy or spectroscopic measurements at synchrotrons and for high-resolution microscopy at polychromatic X-ray tube sources has to date been severely limited.

In the visible-light regime, chromatic aberration of refractive lenses was observed already centuries ago, impairing the performance of telescopes6. In the mid-18th century, Chester Moor Hall found a solution for tackling the chromaticity by stacking a focusing lens made of crown glass and a defocusing lens made of flint glass to form an achromatic doublet7. Owing to the dispersion being stronger for flint glass than for crown glass, a proper combination of the two lenses provided identical focal lengths for two distinct wavelengths and low chromatic aberration for the wavelength range between them, despite the strong chromaticity of each individual lens. The span between these two wavelengths over which the remaining change in focal length lies within the depth of focus (DOF) is commonly defined as the achromatic range.

The concept of an achromatic doublet can be transferred to the X-ray regime. For X-rays, the refraction and absorption in matter are described by the complex refractive index n, which is related to the atomic scattering factor fa = f1 + if2 of the atoms in the given material:

n=1−δ+iβ=1−renaλ22π(f1+if2)   (1)

where (1 − δ) and β are the real and imaginary parts of n, λ is the X-ray wavelength, re the classical radius of the electron, and na the number of atoms per volume. For the majority of the X-ray regime, the real part f1 of the atomic scattering factor changes only little with λ, meaning that the material dispersion D of f1 is close to zero, D = (Δf1/f1)/(Δλ/λ) ≈ 0, and δ can be approximated as δ ∝ λ2 for all materials (see Eq. (1)). Only near the absorption edges, D reaches large positive or even negative values, leading to anomalous dispersion. Combining two refractive X-ray lenses from different materials therefore cannot provide achromatic behaviour over an extended range of X-ray wavelengths.

Refractive and diffractive X-ray lenses, on the other hand, scale differently regarding their chromaticity, which opens up the fundamental possibility of combining them to compensate for chromatic aberration and form an X-ray achromat8,9, see Fig. 1a. In the theoretical work by Wang et al.8, a general expression providing achromatic behaviour is derived for the ratio of the focal lengths fRL of the RL and fFZP of the diffractive FZP:

fRLfFZP=−(2+D)  (2)

A dispersion of D ≪ − 2 can be found for wavelengths near the absorption edges in the extreme ultra-violet and the soft X-ray regimes. Wang et al. conclude that one can thus form an achromat by combining a focusing FZP with a weakly focusing RL. This elegant solution, however, is limited to narrow wavelength ranges where a material suitable for the fabrication of the refractive corrector shows strongly negative dispersion. In the broad wavelength ranges far from the absorption edges, where D ≈ 0, the focal length of a FZP is proportional to the inverse wavelength, fFZP ∝ λ−1, whereas the focal length of a RL scales as fRL ∝ λ−2. Therefore, according to Eq. (2), an X-ray achromat needs to be composed of a focusing diffractive part (FZP), see Fig. 1b, and a defocusing refractive part (RL), see Fig. 1c, fulfilling the relation fRL = −2fFZP. The focal length fA of such an achromat is then given by: fA = 2fFZP. These relations are valid for the case of direct contact of the two elements. When taking into account their separation along the beam direction, we arrive at slightly modified expressions, see Methods and Supplementary Material. This type of X-ray achromat has previously been proposed in the context of telescopes in X-ray astronomy10,11,12. More recently, the promising potential of X-ray achromats has been discussed for microscopy and spectroscopy and moreover for focusing of short pulses without distortion in time9,13,14. However, an experimental realisation has not been reported to date.

Fig. 1: Concept of the X-ray achromat and experimental setup.

a Principle of achromatic focusing: The chromaticity of the defocusing refractive lens (RL) acts as a corrector for the chromatic behaviour of the focusing Fresnel zone plate (FZP). b Scanning electron microscopy (SEM) image of a nickel FZP fabricated by electron-beam lithography and nickel electroplating, as used for the comparison measurements. c SEM image of the RL consisting of four stacked paraboloids 3D-printed using two-photon polymerisation lithography. d Sketch of the experimental setup for scanning transmission X-ray microscopy (STXM) and ptychography using the achromat as a focusing optic.

Here, we present a compact optical system that can achieve achromatic focusing, delivering images of consistently high quality, over an X-ray photon energy range from 5.8 keV to 7.3 keV.

 

Results

The presented X-ray achromat consisted of a FZP fabricated using electron-beam lithography and nickel electroplating and a 3D-printed RL made by two-photon polymerisation. The achromat was installed as a focusing optic at the cSAXS beamline of the Swiss Light Source (Villigen, Switzerland) with the experimental setup shown in Fig. 1d. The fabrication, properties, and arrangement of the optical components are described in more detail in the Methods section. In order to demonstrate the achromatic behaviour of the optic, STXM as well as ptychography measurements were performed at multiple X-ray energies between 5.2 keV and 8.0 keV, with the sample placed in the focal plane of the achromat at its optimum energy of 6.4 keV. For comparison, reference measurements were taken with a conventional FZP (see Fig. 1b) instead of the achromat. Both optics had the same numerical aperture, limiting the achievable spot size to about 500 nm.

 

Figure 2a presents the STXM images of the Siemens star test sample shown in Fig. 2b acquired with the achromat as a focusing optic at different X-ray energies. No significant change of the image quality is visible between 6.0 keV and 7.2 keV. The significant advantage of using the achromat is clear when directly comparing its performance with STXM data obtained with the FZP as an optical element, see Fig. 2c. While the achromat delivers images of consistently high contrast and spatial resolution over a wide energy range, the images taken with the FZP exhibit severe blurring already at X-ray energies deviating 200 eV from its design energy of 6.2 keV. The achromat is capable of resolving line widths below 400 nm over its achromatic range, however, it does not reach the image quality of the FZP at its design energy. This is due to shape imperfections in the refractive element resulting in aberrations and will be improved in future designs of the achromat.

Fig. 2: Demonstration of STXM imaging at different energies using the achromat.

a STXM images of the Siemens star sample shown in panel b obtained with the achromat, indicating an achromatic range of  > 1 keV around the optimum energy of ~ 6.4 keV. b SEM image of the Siemens star test sample. The radial lines and spaces (L/S) at the outer and inner concentric rings have widths of 400 nm and 200 nm, respectively, see red arrows. c Comparison of the STXM results in the energy range of 6.0 keV to 6.4 keV obtained with the achromat (top) and the conventional FZP (bottom). While the contrast of the FZP images changes rapidly with the energy, the image quality achieved with the achromat varies only little.

 

The focusing properties of the achromat can also be analysed by looking at the evolution of the X-ray wavefield with the energy. For this purpose, ptychography measurements (see Supplementary Fig. 1 in the Supplementary Material) were conducted at the same energies as the STXM data, which allows for retrieving the illuminating X-ray wavefield. The intensity distribution of the wavefield at the sample position was then propagated computationally along the optical axis z to create an X-ray beam caustic at each energy. Cuts along the xz plane, see coordinate system in Fig. 1d, through the caustics of the achromat are shown in Fig. 3a, where the focus position is indicated with a red dashed line. A direct comparison of the caustics obtained with the achromat and the FZP, see Fig. 3b, confirms the significant gain in achromatic range achieved by the achromat, as already observed in the STXM images in Fig. 2.

Fig. 3: Evolution of the X-ray beam profile with the energy measured with ptychography.

a Caustics at energies from 5.2 keV to 8.0 keV obtained with the achromat. The red dashed line indicates the location of the focal plane at the different energies. b Comparison of the caustics obtained with the achromat and the FZP. While the position of the focal plane remains almost constant with the energy for the achromat (red dashed line), it changes rapidly for the FZP (blue dashed line). c Calculated curves (solid and dashed lines) and experimental data (dots) for the focal length versus energy for the FZP (blue) and the achromat (red; solid: based on Eq. (6), dashed: based on tabulated refractive index values for the calculation of fRL at each energy).