© The Author(s), published by EDP Sciences, 2025

1 Introduction

Diffraction gratings are essential optical components for a wide range of applications in the extreme ultraviolet (EUV) domain, including interferometry and EUV metrology for X-ray Free Electron Lasers and table-top X-ray sources, as well as photolithography, imaging spectrometry, and adaptive optics for astronomical observations [1–6]. In the soft X-ray to EUV spectral range, gratings of various types, either transmission or reflective, have been employed in X-ray absorption spectroscopy, the development of EUV monochromators, and as beam splitters. Although transmission gratings have proven to be sufficiently efficient, their fragility and the complexity of their fabrication have limited their widespread adoption [7]. As a result, reflective gratings, due to their greater robustness, have become the dominant choice in recent years.

These components have been extensively studied, both theoretically and experimentally, in their two main configurations, classical and conical diffraction, under grazing incidence or with multilayer (ML) coatings deposited on their surfaces [8–10]. When operated at grazing incidence, they can achieve higher efficiencies, up to 35% at 25 nm in the first diffraction order, as reported by Braig et al. [9]. However, this configuration also introduces experimental drawbacks, such as distorted beam wavefronts and more complex experimental geometries for the overall setup. The deposition of multilayer coatings on the gratings has enabled their use at 45°, or even at normal incidence, and has made experimental setups more flexible, even though their efficiencies are lower [8, 11, 12]. As reported by Goray et al. [12], Mo/Si multilayers deposited on trapezoidal gratings were found to exhibit a diffraction efficiency of 14–16% for the +1st and −1st diffraction orders at 10° incidence around 15 nm, and an efficiency of 6–8% at 32° incidence. For the third order, however, efficiency drops to approximately 1.6%. Interestingly, in the same study, theoretical calculations based on rigorous coupled-wave models (RCWM) show that optimized multilayer (ML) gratings with ideal lamellar profiles can achieve up to 40% efficiency in the first diffraction orders around 20 nm at 10° incidence, and even higher efficiencies at 30°.

Moreover, a wavelength shift is predicted between the +1st and −1st orders, which becomes more pronounced as the angle of incidence increases. According to the authors, this wavelength separation is inversely proportional to the grating period. Additionally, asymmetry is also expected when varying the grating period, further highlighting the need to carefully optimize the grating geometry to maximize reflectivity. These results suggest that two key factors must be considered when evaluating grating performance, their profiles and overall geometry. Several studies have highlighted the significant impact of the structural profile on the efficiency of diffraction gratings. This influence is not limited to the initial substrate shape. As shown in our previous works [13–15], the grating profile undergoes notable changes during the multilayer deposition process for high-density gratings. This evolution of the grating geometry can alter the diffraction behavior, making theoretical predictions of efficiency more complex and requiring more sophisticated modeling approaches.

Conical diffraction has gained interest in recent years for monochromator setups, offering high spectral resolution, typically λ/Δλ > 300, and minimal temporal pulse broadening. However, aberration-free imaging of focal points remains challenging, particularly when using non-collimated beams or operating at grazing incidence [16]. In many experimental configurations, the presence of multiple optical elements reduces the photon flux at the instrument’s output. While this is generally not a critical issue for synchrotron facilities, it becomes a significant limitation for attosecond sources, such as high-harmonic generation sources. In this context, the development of gratings with the highest possible diffraction efficiencies is required. To the best of our knowledge there are only a few papers reporting on the experimental efficiency of gratings in conical diffraction in the EUV spectral domain either in grazing incidence [17] or with multilayer coatings [8].

In our previously published work, we reported on the design and optimization of high-groove-density Al/Mo/SiC periodic and aperiodic multilayer gratings for enhanced first-order diffraction efficiency at in the wavelength range 17–31 nm in the classical configuration [13, 18]. We investigated both the geometrical and multilayer parameters that influence peak efficiency and bandwidth and validated our experimental results using adapted theoretical models. Furthermore, we observed and characterized the evolution of the grating profile during multilayer deposition using atomic force microscopy (AFM) across various samples.

In the present work, we present a comprehensive study of Al/Mo/SiC periodic and aperiodic multilayer gratings under oblique incidence (45 degrees). On one hand, oblique incidence enables the characterization in both classical and conical mountings, which provides additional information for the modeling of the multilayer grating. On the other hand, oblique incidence multilayer optics are commonly employed in experimental setups for EUV radiation, including monochromators [19], spectrometer [20, 21], interferometers [22], and polarizing beamsplitters [23]. At ~45° incidence, the efficient coupling between multilayer Bragg reflection and grating diffraction ensures broadband usability in compact configurations as well as an effective spatial separation of the diffracted beams for application such as beamsplitters. Because 45° is close to the Brewster angle, this geometry is well suited for application on linearly polarized sources such as high harmonic generation and free electron laser sources. Furthermore, the high groove density (3,600 line/mm) of the gratings used in this study provides efficient dispersion of the EUV radiation, which is critical for advanced instrumentation including high-resolution monochromators and spectrometers and beam splitters for plasma diagnostics.

In this study, we measured and modeled the diffraction efficiencies of the Al/Mo/SiC multilayer gratings in 3 orders for both classical and conical diffraction. Previous models optimized to fit the +1-diffraction order in Ref. [13], appeared to be insufficient to simulate accurately the asymmetry between +1 and −1 orders revealed by our new measurements. To further investigate the behavior of the gratings, we performed experimental measurements using monochromatic synchrotron radiation and grazing-incidence X-ray reflectometry (GIXR) with a Cu Kα source. Different theoretical models are employed and compared with experimental data, to better understand the limitations of each approach and to identify the origins of certain features observed in the GIXR measurements. In addition, the characterization of the grating profiles after deposition is presented using transmission electron microscopy (TEM), and the results are compared with previously published AFM data. The accuracy of the structural parameters, particularly the grating slopes, is also discussed. Finally, these parameters of grating substrates and multilayer gratings are used to refine the interpretation of experimental diffraction efficiency across various experimental configurations, where multiple physical phenomena may arise and different parameters may contribute differently to each of them.

2 Experimental setup and modeling tools

The grating substrates consisted of 3,600 lines/mm holographic gratings fabricated by Carl ZEISS Jena GmbH on fused silica. Details on the sample preparation can be found in Ref. [18]. The multilayer deposition on the grating substrates was done in an ISO6 cleanroom at Laboratoire Charles Fabry utilizing a Plassys MP800 magnetron sputtering machine. By employing a mask in front of the silica grating, two different multilayer coatings were deposited on the two halves of each silica grating sample. The materials used in the deposition were Si-doped (1.5 wt.%) Al target (99.99% purity), and pure SiC and Mo targets (respectively 99.5% and 99.95% purity).

A Discover D8 diffractometer from BRUKER company has been used for GIXR analyses. The machine uses a Cu Kα radiation source (λ = 0.154 nm), a collimating Göbel mirror, a rotary absorber, Soller and divergence slits, and a scintillator. A diaphragm of 0.5 mm has been added to limit the spot size on the sample. The reflectance curves have been acquired in specular configuration with grazing angles varying from 0 to 5 degrees by a step of 0.01 degree. We employed two models for the analysis of the reflectance curves: An effective medium approximation (EMA) model simulated using IMD software [24], and a Rigorous coupled-wave analysis (RCWA) by using a homemade MATLAB code based on Reticolo software [25].

The grating substrates have been previously characterized before and after ML deposition by AFM using an AFM NX20 from Park System located in an ISO7 cleanroom at SOLEIL Synchrotron. Analysis of the 2 × 2 μm² images in non-contact mode provides the slope of the trapeze (α), FWHM fill factor (ff), and groove depth (d) parameters [13].

Additionally, the multilayer gratings were characterized by using EUV radiation at the Metrology and Tests beamline of the SOLEIL synchrotron. The measurement parameters for the classical configuration were thoroughly discussed in a previous paper [26]. For the present study, the gratings were fixed at a 45-degree incidence angle. In the classical configuration, a detector scan was performed at each energy, and the diffraction orders were measured using a detector scan within the plane of incidence. In contrast, for the conical configuration, the detector was translated out of the plane of incidence to measure the diffracted orders at each energy. The polarization factor in the photon energy range of our measurements was estimated to be 96% TE [13].

Two grating samples, consisting of Al/Mo/SiC periodic (N = 16) and aperiodic multilayer structures, were characterized using TEM. Cross-sectional specimens were prepared with an FEI ThermoFisher Helios Nanolab 660, and a protective layer of platinum (Pt) was deposited on the surface of the multilayer to protect it during the ion beam etching process. High-angle annular dark field (HAADF) images were obtained using a detector that collects highly scattered electrons at large angles, where the differential scattering cross-section is proportional to the square of Z, thus providing Z‑contrast images with high‑Z materials appearing bright and low‑Z materials appearing dark. Energy dispersive X-ray spectrometry (EDX) analysis was performed in in scanning TEM (STEM) mode using an FEI ThermoFisher Titan3 G2 80–300 microscope equipped with a Cs probe corrector and a Super X EDX detector, operating at an acceleration voltage of 300 kV.

This study focusses on three samples: Two Al/Mo/SiC periodic multilayer gratings (with 6 and 16 periods, respectively) and one aperiodic multilayer grating composed of 12 layers. The structure of these samples was previously investigated in Ref. [13]. A summary of the principal parameters of the multilayer gratings, as determined from RCWA modeling, is presented here: The multilayer stack was initially optimized on a flat substrate, yielding layer thicknesses of 8.00 nm for Al, 2.68 nm for Mo, and 3.69 nm for SiC. AFM measurements showed that the surface roughness of the top grating increased from 0.23 nm before deposition to 0.42 nm after deposition, while that of the bottom grating increased from 0.32 nm to 0.47 nm after deposition. The top oxide layer thickness was fixed at 1.7 nm, while the grating depth determined through fitting, was found to be 21.8 ± 0.7 nm, for the 3 samples. AFM measurements revealed that the top surface profile of the N = 16 grating changed from a trapezoidal to a sinusoidal shape, whereas the N = 6 and aperiodic gratings retained a trapezoidal profile. Among the studied configurations, the N = 6 multilayer grating demonstrated the best performance, achieving both high diffraction efficiency and the broadest bandwidth.

3 Modeling of the multilayer grating diffraction efficiencies

Figure 1 presents a schematic diagram of Al/Mo/SiC multilayer gratings in two distinct mountings: Classical diffraction and conical diffraction. Figure 1a illustrates the classical mounting, where the incident beam is oriented perpendicular to the grating grooves. In this configuration, the diffracted orders are distributed along within the 〈YZ〉 plane (plane of incidence). When the incident beam interacts with the grating grooves at a grazing angle (θ), it is diffracted into multiple orders (0, ±1, ±2, etc.), with each order (i) defined by its diffraction angle (ϕ_i), as illustrated in Figure 1a.

Fig. 1 Schematic diagram of a Al/Mo/SiC multilayer grating with N = 4 in (a) classical and (b) conical mountings. For classical mounting the diffracted orders are in the plane of incidence 〈YZ〉. For conical mounting, (c) and (d) represent projections of +1 order in 〈YZ〉 and 〈XZ〉 planes, respectively.

In contrast, Figure 1b depicts the conical mounting, in which the incident beam is parallel to the grating grooves, resulting in diffraction out of the incidence plane. Thus, each diffracted order (i) is defined by 2 angles: The angle (ϕ_i) in the plane of incidence and an azimuthal diffraction angle (ψ_i) (Figs. 1b–1d).

To differentiate between the classical and conical mountings in the subsequent analysis, the angle Ω is introduced. Ω represents the orientation of the plane of incidence relative to the groove axis. In classical configuration Ω = 90 degrees, whereas in conical configuration Ω = 0 degrees.

For both configurations (classical and conical), the position of the ith order of diffraction is determined by the expressions of ϕ_i and ψ_i provided in equations (1) and (2), respectively [27], where λ and p denote respectively the wavelength of the incident X-rays and the period of the grating grooves.$${\varphi }_i=\arcsin \left(\sqrt{{\sin }^2\left(\theta \right)-{\left(\frac{\mathrm{i\lambda }}{p}\right)}^2-\frac{2\mathrm{i\lambda }\sin \left(\mathrm{\Omega }\right)\cos \left(\theta \right)}{p}}\right),$$(1) $${\mathrm{\Psi }}_i=\arctan \left(\frac{\mathrm{i\lambda }\cos \left(\mathrm{\Omega }\right)}{\mathrm{i\lambda }\sin \left(\mathrm{\Omega }\right)+p\cos \left(\theta \right)}\right).$$(2)

In Figure 2a, we plotted the diffraction angles for the 0th and ±1 orders in a classical mounting computed with equation (1) for a wavelength of 0.154 nm. For the 0th order, the diffraction angle is equal to the incidence grazing angle, indicating specular reflection. The +1 order can exit from the multilayer only for incident angles above 1.9 degrees. When θ is inferior to this value, the +1 order does not exist anymore and the interior of the square root in equation (1) becomes negative. For the −1 order, the diffraction starts at ~2 degrees when the incident angle is 0 degrees.

Fig. 2 Variation of the diffracted angle ϕ (a) with grazing angle θi at λ = 0.154 nm and (b) with wavelength at θi = 45 degrees in case of classical mounting. The dots in (b) correspond to diffracted angles measured at SOLEIL synchrotron.

In Figure 2b, the diffraction angles for the 0th and ±1st orders are shown for EUV wavelengths in the range 17–25 nm at an incident angle of 45 degrees in a classical configuration. As expected, the 0 order corresponds to specular reflection and its angle is constant, while ±1 orders demonstrate the wavelength-dependent diffraction behavior of equation (1).

Figures 3a and 3b show the diffraction angles ϕ_i and ψ_i computed respectively from equations (1) and (2) for the conical configuration. The calculation was performed for an incident angle of 45 degrees. The zero-order angles are consistent with specular reflection. Due to the symmetry relative to the plane of incidence (plane (YZ) in Fig. 1), the ϕ_i diffraction angles for ±1 orders are identical; their values remain close to the specular reflection (O order) and decrease slightly with increasing wavelength (Fig. 3a). In contrast, at Figure 3b the ϕ_i diffraction orders for ±1 order have opposite values with a significant wavelength-dependence.

Fig. 3 Variation of the diffracted angles (a) ϕ and (b) ψ with wavelength at θi = 45 degrees in case of conical mounting. The dots in (b) correspond to diffracted angles measured at SOLEIL synchrotron. See Figure 1 for the definition of ψ and ϕ angles.

The diffraction angles measured for the −1 and +1 orders of the periodic N = 16 ML grating at SOLEIL synchrotron are also plotted in Figures 2b and 3b They show a good agreement with theoretical values, with a difference of less than 0.05 degree, which remains within the accuracy of alignment of the sample (±0.1 degree).

In conclusion, in the classical mounting, the difference in peak wavelength observed between the +1 and −1 diffraction orders arise naturally from their different diffracted angles and the corresponding Bragg conditions. This will also result in a difference in the diffraction efficiency of the +1 and −1 orders. By contrast, the conical configuration is geometrically symmetric; therefore, we expect the same diffraction efficiency for the +1 and −1 orders if the grating profile is symmetrical.

For the modeling of the multilayer grating efficiencies, we used a homemade MATLAB code (based on RCWA Reticolo software [20]), which has been described in previous publications [13]. However, this code does not account for the effect of interfacial defects (roughness and/or interdiffusion) on the surface of the grating and between individual layers in the calculation of diffraction efficiency. To address this limitation, a Debye–Waller (DW) factor [28] has been applied to the output of the RCWA calculation, as indicated in equation (3).$$E_{\mathrm{DW}}=E_{\mathrm{RCWA}}\times \exp \left(-{\left[\frac{4\pi \cos \left(\theta \right){\sigma }_{\mathrm{DW}}}{\lambda }\right]}^2\right).$$(3)

Here, E_DW represents the diffraction efficiency with interfacial defects of width σ_DW, while E_RCWA is the efficiency calculated using the RCWA method for ideal smooth interfaces. The interfacial defects impact is modeled through an exponential attenuation term, where the factor 4πcos(θ)/λ determines the sensitivity to interfacial defects, leading to a stronger reduction in efficiency for higher σ_DW values, incidences closer to normal or shorter wavelengths.

4 Characterization of the grating profile evolution

The evolution of the grating profile with the multilayer deposition has been previously studied by AFM surface characterization performed on various samples [13]. In order to verify these results, we used TEM analyses as a direct and independent measurement of the ML grating profiles. Figure 4 shows the HAADF and EDX cross-section images of the 16-period (N = 16) and aperiodic multilayer gratings. For the periodic multilayer (N = 16), Al, Mo and SiC layers are well defined throughout the stack but we can clearly see a smoothening of the grating profile when the number of deposited period increases (Figs. 4a–4b). Actually, for N = 6 the surface profile remains similar to the initial substrate profile, while for N = 16 the surface profile is significantly smoothened. This is consistent with the AFM and EUV diffraction efficiency results reported in [13]: N = 6 corresponds to the optimal number of periods in terms of +1 order diffraction efficiency, while for N = 16 the +1 order diffraction efficiency decreases.

Fig. 4 TEM images of (a, b) the 16-period multilayer grating (N = 16) and (c, d) the aperiodic multilayer grating (12 layers): (a, c) HAADF, and (b, d) EDX combined images for Al (blue), Mo (green) and Si (red) atoms.

In the case of the aperiodic multilayer, the first Al layer on the substrate appears to be not continuous, probably because it is too thin. This creates a rough interface with the next Mo layer which propagates to the following SiC layer. Interestingly, this interfacial roughness is smoothened out by the subsequent deposited layer and the top surface appears to be as smooth as in the periodic multilayer case.

We extracted the maximum values of the slope for each deposited period from the TEM HAADF images for the periodic and aperiodic cases. Lines were visually aligned with the maximum slope of the profile for each multilayer period, and the angle relative to the horizontal reference line was calculated using Fiji software [29]. Multiple slopes were analyzed for each cross-section and averaged. The results are plotted in Figures 5a and 5b respectively and are compared with previous measurements of the surface profiles by AFM [13]. Interestingly, the agreement between TEM and AFM slope values remains within the error bars for slopes lower than α ~45 degrees. However, for steeper slopes (α > 45 degrees), AFM seems to underestimate the slope values. This effect of “smoothening” of steep grating profiles by AFM has been explained in Ref. [30] and is probably due to the convolution of the actual surface profile with the shape of the AFM tip.

Fig. 5 Slope of the groove profile (α) as a function of the number of deposited periods N (resp. number of deposited layers) measured by TEM and by AFM for (a) the 16-period and (b) the aperiodic ML gratings. For TEM, the error bars on the slope (α) are calculated by averaging the relative standard deviation at each N. For AFM, the error bars correspond to the standard deviations on the measurements over 6 grating periods (i.e., 12 different α values) for each N.

5 Characterization of grating substrates and multilayer gratings with specular GIXR

We measured a grating substrate before multilayer deposition by GIXR in order to confirm the surface profile parameters previously determined by AFM [13]. Figure 6 shows the specular reflectance (0 order) versus grazing angle for the grating in classical mounting. Note that the angular acceptance of the slit in front of the detector is less than 0.03 degree so that the +1 and –1 diffracted order (see Fig. 1) are not collected when measuring in specular mode. The shape of the curve resembles the one of a single layer deposited on a plane substrate. Indeed, we can use effective medium approximation (EMA) to model the grating surface profile with a single EMA layer. The thickness of this EMA layer should be equal to the height of the groove and its optical constant a weighted average of the ones of SiO₂ and air. In practice, we use a layer of SiO₂ and tune its optical constant by modifying its density ρ_EMA. As shown in Figure 6, the simulation using an EMA layer with d_EMA = 22 nm and ρ_EMA = 0.835 g/cm³ agrees well with the experimental data for the position of the critical angle and of the Kiessig fringes. We have also plotted in Figure 6 the simulation of the reflectance with RCWA using the parameters reported in Table 1. The grating depth (22 nm) and the fill-factor (0.38) were deduced from the EMA model; for the grating slope, which is not accessible with EMA model, we used the value previously measured by AFM: 47 degrees. Note that the RCWA reflectance plotted in Figure 6 is not sensitive to the slope value. The curves simulated with EMA and RCWA are very similar, which confirms the validity of the EMA model. The main interest in using EMA model instead of RCWA is that it is much less time consuming since it is a 1D calculation.

Fig. 6 GIXR experimental results in classical configuration and RCWA and EMA models of a plane grating (3,600 l/mm) without ML.

The AFM values from previous work [13] are also reported in Table 1 for comparison. The grating depth measured by AFM is in very good agreement with the one determined either by EMA or RCWA models. A discrepancy exists however for the value of the fill factor: EMA and RCWA models give 0.38 while AFM measured 0.51. A possible explanation is that the top silica layer may have a lower density than the bulk silica due to the polishing and etching processes.

We plotted in Figure 7 the GIXR measurements after deposition of the multilayer on the grating substrate for N = 6, N = 16 and aperiodic cases. On Figure 7a (N = 6), we can see that the RCWA model reproduces well all the features of the experimental data except for the intensity of the Bragg peaks. For each case, we were able to improve significantly the fit to the experimental data by adding a DW factor to the RCWA model to account for the interfacial defects. The parameters used in the models are listed in Table 2. The ML and grating parameters have been determined in a previous study [13]. Therefore, the only free parameter in the fit of GIXR data is the DW factor. We found that for the 3 samples, a value of 0.8 nm provides a very good fit to the experimental data (see Fig. 7). This value is much higher than the surface roughness measured by AFM after deposition on the grating samples (between 0.4 and 0.5 nm). Thus, the higher value of DW factor used in the model may account for other defects of the multilayer gratings that are not included in the initial model.

Fig. 7 GIXR measurements for (a, b) N = 6, (c) N = 16 and (d) aperiodic multilayers on gratings in classical mounting. RCWA model of the multilayer grating and IMD simulation of the multilayer on a flat SiO2 substrate are also plotted in each case. (b) is a close view of the total reflection plateau of (a) and shows an IMD simulation of the multilayer on a flat SiO2 substrate with a top EMA layer to account for the surface profile.

Note that in the RCWA simulation used here, the grating profile is the same at each interface: The model does not consider the evolution of profile revealed by TEM analyses and described in the previous section. Therefore, the slope values reported in Table 2 correspond to an “average” of the slopes in the ML grating structure. These average values are consistent with the measured values reported in Figure 5.

We also plotted in Figure 7 the simulated GIXR curves of the corresponding multilayer on a flat SiO₂ substrate for comparison. It is interesting to notice that for N = 6, the 1st and subsequent odd Bragg orders of the multilayer are extinguished when the multilayer is deposited on the grating: This means that all the incident photon flux is distributed in the other diffraction orders (≠ 0) of the grating. For N = 16 however, the 1st and subsequent odd Bragg orders are not totally extinguished, and this can be explained by the evolution of the grating profile as we increase the number of deposited periods (see Fig. 4).

One can notice that the measurements and the RCWA simulations display a sag in the total reflection plateau region (at grazing angles around 0.25 degree), which is not present in the case of the multilayer on a flat substrate simulation. This feature appears on all multilayer gratings. In order to understand its origin, we simulated with IMD a multilayer on a flat SiO₂ substrate with a top EMA layer to account for the surface profile. The idea is similar to the EMA model used in the case of the grating substrate (see Fig. 6) but here the EMA layer is a mixture of Al, Mo, SiC top layers and air (see schematic in Fig. 7b). The thickness of the EMA layer has been fixed to the value of the groove depth, and its refractive index is the effective refractive index of one multilayer period divided by 2 (we consider here a fill factor of 0.5). As shown in Figure 7b (which represents a close view of the total reflection plateau for N = 6), this model allows to reproduce the sag feature in the plateau. This feature actually corresponds to the critical angle of the effective surface layer of the multilayer grating. RCWA simulations (not shown here) indicate that less than 2% of the flux is diffracted in the −1 order at the angles corresponding to the sag in the 0 order. This means that most of this decrease of specular intensity in the total reflection region is mainly due to absorption in the top effective surface layer of the multilayer grating.

6 EUV diffraction efficiencies in classical and conical mountings

We measured the diffraction efficiencies of the 6-period ML grating and the aperiodic ML gratings in the −1, 0 and +1 orders and in classical mounting from 17 eV to 25 eV at SOLEIL synchrotron at 45 degrees incident angle. The results are plotted in Figure 8. We also plotted the RCWA simulation using a model optimized to fit the +1 order in our previous work [13]. The position in wavelength of the peak efficiency of each diffracted order is in very good agreement for both samples. One can notice that the +1 and −1 orders don’t have their peak efficiency at the same wavelength. For the periodic case for example, at λ = 18 nm, the +1 order presents an experimental peak efficiency of ~18% while the −1 order is about 10 times lower. This effect is due to the fact the efficiency of ML gratings is based on volume diffraction (as opposed to surface diffraction) and that the +1 and −1 diffracted orders don’t have the same exit angle inside the ML. Therefore, the Bragg conditions for the ML cannot be satisfied at the same wavelength for both orders [31]. For the periodic ML grating (Fig. 8a), we can see that the model fits very well with the 0-order efficiency spectrum. The fit of the 0 order is not as good for the aperiodic ML grating (Fig. 8b) and this may be due to the defects revealed by TEM analyses on the 1st deposited period of the aperiodic ML. For both samples however, there is a large discrepancy between the RCWA model and the experimental data for the −1 order, the measured peak efficiency being ~30% lower (relative) than the modeled ones.

Fig. 8 SXR measurements and RCWA model for (a) the periodic multilayer grating N = 6 and (b) the aperiodic multilayer grating in classical mounting.

The DW factor was not included in the RCWA simulations in Figure 8. According to the average roughness previously determined on these ML gratings (about 0.45 nm), we can expect peak efficiencies to be lower by about 5% (relative) [15]. This may explain the difference between experimental and simulated data for the +1-diffraction order, but it will not improve significantly the discrepancy for the −1-diffraction order.

In order to better understand this discrepancy between experimental and simulated efficiencies in the −1-diffraction order, we measured the efficiency of the 6-period ML grating in conical mounting. The results are plotted on Figure 9 for the −1, 0 and +1 diffraction orders. As for the classical mounting, we observe a much lower peak efficiency in the −1 order than in the +1 order, while theoretically we expect the same efficiency for both orders due to the symmetry of conical diffraction.

Fig. 9 SXR measurements for the periodic multilayer grating N = 6 in conical mounting (Ω ≈ 0°). The symbols with dashed line correspond to measurement after a 180° rotation of the sample (Ω = Ω0 + 180°).

The only possible explanation for this difference in peak efficiency between +1 and −1 order in conical mounting is a loss of symmetry, either in the experimental set-up or in the sample.

During the experiment reported in Figure 9, the sides of the rectangular grating substrate were aligned with the beam, but an offset angle may exist between the direction of the grooves and the side of the substrate. Let’s consider that the initial measurements were done with an azimuthal angle Ω₀ between the incident beam and the grooves. From the measured out-of-plane positions of the diffracted orders, we determined that the groove direction was aligned with the incident beam within 0.2°. Simulation confirms that Ω₀ values in this range cannot explain the difference in peak efficiency observed between +1 and −1 orders. Figure 10 shows the +1 and −1 order efficiency spectra simulated as a function of Ω. We can notice that close to conical diffraction (Ω = 0°) the position of both diffracted orders is sensitive to a small change in Ω and that they vary in opposite direction. However, the peak efficiency for both orders vary only slightly with Ω. Consequently, the misalignment of the grating groove with the direction of the incident beam is not responsible for the difference in peak efficiency observed in Figure 9.

Fig. 10 (a) +1 order and (b) −1 order efficiency spectra simulated as a function of Ω for the 6-period ML grating at 45° angle of incidence. The multilayer grating parameters used in this simulation correspond to the model of Figure 11.

Furthermore, we re-measured the grating efficiency after a 180° rotation of the sample (Ω = Ω₀ + 180°). If the efficiency difference observed between the +1 and −1 orders was due to an asymmetry in experimental setup, we would expect to obtain the same result for the 2 configurations (Ω₀ and Ω₀ +180°) because the tilt of the groove will always be in the same direction. However, we measured exactly the opposite effect: After the rotation of 180° the −1 order provides higher efficiency than the +1 order (Fig. 9). This means that the loss of symmetry is inherent to the sample itself.

In order to account for the asymmetry of the ML grating, we implemented a model with different slopes, respectively α₁ and α₂, on the left and right edges of the grooves (see inset in Fig. 11a). The grating parameters for these models are given in Table 3. In the same way as the previous model (see Fig. 8), the DW factor was not included in this modeling, as shown in Figure 11, the asymmetric groove model agrees well with experimental data for the 3 diffraction orders in the case of the periodic ML grating both in classical and conical mountings (Figs. 11a and 11b respectively). Furthermore, we used in this model the newly available optical constant for Al [32], which improves slightly the agreement in the spectral shape and near the Al L_2,3 absorption edge. The asymmetric groove model also provides better agreement with experimental data in the case of the aperiodic ML grating in classical and conical mountings (Figs. 11c and 11d respectively). Nevertheless, a discrepancy remains for 0 order and −1 order efficiencies. As explained earlier, this can be due to the defects observed by TEM for the first deposited layers of this aperiodic ML which are not taken into account in the model.

Fig. 11 SXR measurements and RCWA asymmetrical models for the periodic multilayer grating N = 6 at (a) classical and (b) conical mounting, and for the aperiodic multilayer grating at (c) classical and (d) conical mounting.

7 Conclusion

EUV gratings with periodic and aperiodic multilayer coatings were studied in both classical and conical mountings. The structural parameters and evolution of the grating profile within the multilayer coating were characterized by TEM and GIXR. We showed that a simple EMA model allows the GIXR data to fit the characterization of the grating parameters before coating and may also be useful to understand specific GIXR features in the case of multilayer gratings. The slopes of the trapezoidal grating profile within the multilayer have been determined with TEM imaging and compared with previous AFM measurements. The differences between both methods stay within the experimental uncertainties, except for the slopes higher than ~50 degrees for which AFM seems to produce underestimated values.

The measurements of the diffraction efficiency in −1, 0 and +1 orders in classical mountings were compared with a RCWA model and reveal a discrepancy between the +1 and −1 order efficiencies which was confirmed by measurements in conical mounting. Finally, a model based on asymmetrical slopes for the trapezoidal profile is proposed and shows a good agreement for the 3 diffraction orders in both classical and conical mounting. It would be interesting to further improve the model by including the evolution of the grating profile within the multilayer in order to study its effect on the diffraction efficiency.

These results have improved our understanding of the optical behavior of ML coated gratings in different diffraction geometries and have provided further insights into the physical origin of order asymmetries and profile-induced effects. The accurate modelling of diffraction efficiencies through experimentally validated parameters is a step toward more reliable and predictive design tools for EUV optical components, for several applications such as monochromators and spectrometers, EUV telescopes and EUV interferometers. In conclusion, this work contributes not only to the precise metrology and modeling of EUV multilayer gratings but also supports their practical implementation in advanced light sources where both optical performance and structural control are important.

Acknowledgments

This work was performed under the auspices of the Institut d’Optique Graduate School, Université Paris-Saclay. The authors would like to thank Pascal Mercere (SOLEIL synchrotron, France) for his assistance during SOLEIL measurements and Maxime Vallet (CentraleSupélec, Université Paris-Saclay, France) for the TEM measurements.

Funding

This work has received support from CNES, convention N°241768/00.

Conflicts of interest

The authors declare no conflicts of interest.

Data availability statement

The data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Author contribution statement

Conceptualization, S.R. and F.D.; Methodology, A.M., S.R. and F.D.; Software, A.M. and S.R.; Validation, A.M., I.P. and F.D.; Formal Analysis, A.M., C.T. and F.D.; Investigation, A.M., I.P., B.C. and E.M.; Resources, E.M.; Data Curation, A.M.; Writing – Original Draft Preparation, A.M. and I.P.; Writing – Review & Editing, A.M., I.P. and F.D.; Supervision, S.R. and F.D.; Project Administration, F.D.; Funding Acquisition, F.D.

References

All Tables

All Figures

	Fig. 1 Schematic diagram of a Al/Mo/SiC multilayer grating with N = 4 in (a) classical and (b) conical mountings. For classical mounting the diffracted orders are in the plane of incidence 〈YZ〉. For conical mounting, (c) and (d) represent projections of +1 order in 〈YZ〉 and 〈XZ〉 planes, respectively.
In the text

	Fig. 2 Variation of the diffracted angle ϕ (a) with grazing angle θi at λ = 0.154 nm and (b) with wavelength at θi = 45 degrees in case of classical mounting. The dots in (b) correspond to diffracted angles measured at SOLEIL synchrotron.
In the text

	Fig. 3 Variation of the diffracted angles (a) ϕ and (b) ψ with wavelength at θi = 45 degrees in case of conical mounting. The dots in (b) correspond to diffracted angles measured at SOLEIL synchrotron. See Figure 1 for the definition of ψ and ϕ angles.
In the text

	Fig. 4 TEM images of (a, b) the 16-period multilayer grating (N = 16) and (c, d) the aperiodic multilayer grating (12 layers): (a, c) HAADF, and (b, d) EDX combined images for Al (blue), Mo (green) and Si (red) atoms.
In the text

Fig. 5 Slope of the groove profile (α) as a function of the number of deposited periods N (resp. number of deposited layers) measured by TEM and by AFM for (a) the 16-period and (b) the aperiodic ML gratings. For TEM, the error bars on the slope (α) are calculated by averaging the relative standard deviation at each N. For AFM, the error bars correspond to the standard deviations on the measurements over 6 grating periods (i.e., 12 different α values) for each N.

In the text

	Fig. 6 GIXR experimental results in classical configuration and RCWA and EMA models of a plane grating (3,600 l/mm) without ML.
In the text

Fig. 7 GIXR measurements for (a, b) N = 6, (c) N = 16 and (d) aperiodic multilayers on gratings in classical mounting. RCWA model of the multilayer grating and IMD simulation of the multilayer on a flat SiO2 substrate are also plotted in each case. (b) is a close view of the total reflection plateau of (a) and shows an IMD simulation of the multilayer on a flat SiO2 substrate with a top EMA layer to account for the surface profile.

In the text

	Fig. 8 SXR measurements and RCWA model for (a) the periodic multilayer grating N = 6 and (b) the aperiodic multilayer grating in classical mounting.
In the text

	Fig. 9 SXR measurements for the periodic multilayer grating N = 6 in conical mounting (Ω ≈ 0°). The symbols with dashed line correspond to measurement after a 180° rotation of the sample (Ω = Ω0 + 180°).
In the text

	Fig. 10 (a) +1 order and (b) −1 order efficiency spectra simulated as a function of Ω for the 6-period ML grating at 45° angle of incidence. The multilayer grating parameters used in this simulation correspond to the model of Figure 11.
In the text

	Fig. 11 SXR measurements and RCWA asymmetrical models for the periodic multilayer grating N = 6 at (a) classical and (b) conical mounting, and for the aperiodic multilayer grating at (c) classical and (d) conical mounting.
In the text