Open Access
Add to BibSonomy
Abstract
Bragg mirrors with superlattices and depth grading of their period are studied numerically at sub-10-nm wavelengths, using La/B4C and Cr/Sc as bilayer materials at 6.64 and 3.12 nm, respectively. Superlattice-related reflection peaks (between normal incidence and total reflection) are very narrow, and sharpen with decreasing wavelength. Depth grading reduces and broadens the peaks. Together with weighting by source spectra, all-angle reflection is demonstrated. For the normal-incidence peaks, grading and weighting increases their overlap with source spectra and allows enhancement of the total collected power by up to 120% (at 6.64 nm) and 90% (3.12 nm) compared to ungraded mirrors. Interface roughness, determined from experimental data, approximately halves these enhancements.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
At wavelengths much shorter than the visible (below λ≈ 100 nm) the refractive index is dominated by contributions from atomic core level transitions. Its real part is very close to unity not allowing to design simple refractive elements like lenses. An alternative is to use the reflection at interfaces between different materials with a contrast of the real part of the refractive index as high as possible. Nevertheless, compared to the visible, this contrast is very low and also the reflectance of a single interface is low. In order to enhance the reflectance so-called Bragg mirrors consisting of many bilayers of different materials are used. With a bilayer thickness of about half the wavelength constructive interference causes the total reflectance being high [1]. In the extreme ultraviolet (EUV) at 13.5 nm the preferred bilayer materials are Mo and Si. Lithographic techniques using this wavelength in the illumination and projection systems are partly implemented by the microelectronics industry already [2,3]. The range of sub-10-nm wavelengths (beyond extreme ultraviolet – BEUV, soft X-rays) is recently gaining interest for further developments of the lithography [4] and applications like “at wavelength” metrology, surface patterning and microscopy [5,6]. Regarding the basic optical properties the index contrast of useable materials in the bilayers of BEUV mirrors is smaller than in the EUV with the wavelength range of high reflection being narrower. Due to these changes the basic properties of the Bragg mirrors are distinctly changed compared to the 13.5-nm case. The reduced wavelength is also a challenge for the fabrication. Since the bilayer thickness corresponds to about half the wavelength extremely high accuracy is necessary for thickness homogeneity and interface sharpness.
Standard Bragg mirrors consist of a multiple of bilayers whose width and periodicity are adjusted for the target wavelength. As mentioned above the index contrast of the materials of the bilayer should be large. Additionally the imaginary part (absorption) should be low. For applications the peak intensity of BEUV sources should coincide with ranges of high index contrast. Also weak absorption in air can be important for some applications. Examples are bilayers containing La and B (near 6.64 nm), the Tb plasma source (≈ 6.4 - 7 nm), and the water window (2.3–4.2 nm), respectively. Below 10 nm the index contrast is relatively large only near atomic absorption edges of the elements in the bilayer, as for boron near 6.64 nm or scandium near 3.12 nm. For high reflectance 200 bilayers or more are necessary compared to only about 50 for 13.5-nm EUV mirrors.
The purpose of the present work is to investigate numerically the basic and some application-related properties of BEUV Bragg mirrors modified regarding periodicity of the layers and variation of their thickness.
The properties of the simple bilayer structure of the standard mirror can be modified by various means [1,7]. In our previous work on EUV mirrors [7] we reported a numerical study regarding modifications like introduction of a periodic superstructure (“superlattice” [8–10]) and depth grading of the multilayer period. In the present work we apply these modifications to the BEUV. With the spectral and angular range of high reflection becoming increasingly narrower BEUV properties cannot be adopted straightforwardly from the EUV results. Beside the narrowness of the peaks, interface quality and low source power are of increased significance in the BEUV. Important new issues addressed are the possibility of broadening and overlapping superlattice related peaks between normal incidence and total reflection by depth grading, weighting of reflection spectra by BEUV source spectra and collecting more power from BEUV sources [8–15], non-ideal interfaces as an approach to realistic mirror structures. Thus, the results presented regard: (1) The dependence of superlattice related peaks and near-normal-incidence reflection peaks on the wavelength (near 3.12 and 6.64 nm). (2) The possibility of all-angle reflection of graded superlattice mirrors. (3) A quantitative analysis of the weighted and integrated near-normal-incidence reflectance of graded mirrors. (4) The influence of non-ideal interfaces for both types of reflection peaks. Accordingly, the paper is organized as follows. In section 2 details of the structure including the superstructure, the depth grading and non-ideal interfaces as well as the numerical procedure are described. Results are presented for two wavelength ranges near 6.64 nm and near 3.12 nm in the water window with La/B4C and Cr/Sc, respectively, as the bilayer materials, in subsection 3.1 for the superlattice mirrors, in subsection 3.2 for the mirrors without superlattice. The results on weighted and integrated reflectance regarding collecting power of BEUV sources with graded mirrors are presented in subsection 3.2.2. The effect of non-ideal interfaces is treated in subsections 3.1.1 and 3.2.3. Section 4 summarizes and draws conclusions.
2. Modifications of the multilayer period and the numerical procedure
The standard Bragg mirror has constant period of the bilayers and constant thickness of the individual layers. For La/B4C (this work), La/B and LaN/B highest reflectance is obtained at a wavelength near 6.64 nm above the boron-K absorption edge (6.6 nm) [16], for Cr/Sc near 3.12 nm above the scandium-L absorption edge (3.1 nm) [17]. For the thickness ratio of the upper (La; Cr) to the lower layer (B4C; Sc) 0.43/0.57 is chosen. For both type of mirrors this is very close to the ratio for highest reflectance. The reflectance deviates by less than 1% from the maximal values for ratios between 0.38/0.62 and 0.48/0.52. The bilayer thicknesses are given in the corresponding sections and figure captions. The number of bilayers for standard mirrors is 200 (La/B4C) and 400 (Cr/Sc) above which the peak reflectance values increase only very weakly.
In “superlattice mirrors” certain layers of one element are replaced periodically by layers of the other element (Fig. 1). E.g. in every fifth bilayer of a La/B4C mirror La (black squares) is replaced by B4C indicated by the grey squares (notation “superlattice-5”, SL-5). For superlattice-m the period is m times the period of the standard mirror. In section 3 results are shown for SL-5 mirrors. Compared to the standard mirrors the number of bilayers is increased to 250 (La/B4C) and 500 (Cr/Sc) which keeps the number of La/B4C and Cr/Sc interfaces constant (399 and 799, resp.).
Fig. 1. Upper figure: Sequence of layer structures of a standard mirror, a graded mirror and a superlattice-5 mirror (schematic). Top layer (incidence of radiation) is at left side. Lower figure: Graded mirror: dependence of relative bilayer thickness on bilayer number for grading factor 1.05.
Depth grading of the multilayer period broadens the range of wavelength and angle of the reflectance peaks [18–20]. By applying an optimization procedure like minimization of a merit function a desired reflectance profile can be approached [21]. Optimization procedures can be numerically elaborate. In this work simple linear grading is used (Fig. 1). This is considered as sufficient because of the narrow width of the reflectance peaks in the BEUV. The thickness ratio of the top to the bottom layer is the grading factor which is varied between 1 (no grading) and 1.2. For the example of a graded La/B4C mirror with peak reflectance at 6.64 nm a grading factor 1.05 (Fig. 1, lower figure) means that the period changes from 3.305 (top) to 3.470 nm (bottom). In this case the step in thickness between two bilayers is less than 0.001 nm. This high accuracy is purely numerical and can only be considered as a nominal goal for the fabrication. Accordingly all the thickness values are given with corresponding high accuracy. The calculations are made for vacuum on both sides of the multilayer structures. From the optical point of view the substrate material is of minor significance since its contribution to the reflectance is very small due replacing a single interface only and due to reduced radiation intensity reaching it (because of the multiple reflections and the absorption in the multilayer). A frequently used substrate with very high surface quality is super-polished silicon.
Additionally to these modifications the mirror characteristics are related to those of BEUV sources by weighting the reflectance peaks with source spectra [22]. The weighted reflectance < R > is calculated as a function of incident angle via < R> = Σ[P(λi) R(λi)]/Σ[P(λi)], i.e. the normalized sum of the product of R and source power P. The wavelength λi runs from 6 to 8 nm (La/B4C) and 3 to 3.4 nm (Cr/Sc) which covers the full range of the reflectance of the mirrors. The emission spectra of the sources considered are far broader than the reflection spectra, so that Σ[P(λi)] is much larger than Σ[P(λi) R(λi)]. Therefore the values of < R > are small compared to the values of R. The purpose of the weighting is to show that with grading also parts of the source spectrum which are outside the reflectance peak of the standard mirror can be used. Qualitatively, one can expect that the effect of the weighting is small for a narrow spectrum of the source and increases with increasing width. BEUV light sources can have a small coherence length. This is implicitly taken into account by the width of the wavelength range in the weighting procedure.
The interface between the individual layers has been assumed above to be abrupt. A non-ideal interface can be due to, e.g., lateral interface roughness or inter-diffusion between two adjacent layers or both. This causes a density variation which - laterally averaged - is described in the following by the cumulative distribution function of the normal distribution [23]. The variation of the real part n of the refractive index follows this function (shown as the smooth curve in Fig. 2 for a La/B4C mirror). It is approximated by an intermediate layer between each of the layers in a discrete roughness model [24–26]. The intermediate layer (thickness dIL) consists of seven sublayers (Fig. 2). Seven sublayers are considered as sufficient since results of R(0°) for a La/B4C mirror with 21 sublayers are in accordance with the 7-sublayer calculation within less than 1%. Accordingly n varies in seven steps between the values of the individual layers (depending on wavelength). The roughness is described by the parameter σ (nm). Figure 2 assumes the general case of different thicknesses of the two layers in a bilayer; with increasing roughness the thinner layers can, at first, approach zero thickness or contain an admixture of the other material.
Fig. 2. Discrete roughness model with seven intermediate layers for a La/B4C mirror with values of the real part n of the refractive index at 6.64 nm. Without roughness the sharp La-B4C interfaces are at 2.03, 3.93 and 5.37 nm (La/B4C thickness ratio 0.43/0.57). The width dIL of the intermediate layers is 1.2 nm. The roughness parameter σ is 0.3 nm. The smooth curve is the cumulative distribution function.
In order to have the cumulative distribution function almost fully contained within the intermediate layer, 2σ has to be kept smaller than dIL. dIL is limited by the original thickness of the thinner La layer (for equal thicknesses of La and B4C by half the multilayer period). For dIL = 2σ the cumulative distribution function and the stepped curve would be cut off at the corresponding values of n. A relatively high jump to the pure layers would occur. With the value of dIL close to the original La thickness this jump appears close to the (remaining thin) La layer. With a smaller dIL the cut-off is closer to the center of the intermediate layer. If additionally dIL< 2σ the jump would be even higher. Avoiding a large unphysical jump with keeping 2σ smaller than dIL, experimental values of R(0°) much smaller than theoretical ones can only be simulated by assuming a reduction of the index contrast. As shown below (section 3.2.2) this reduction will be necessary for Cr/Sc mirrors and will be achieved by admixtures of Cr in Sc and vice versa leaving no pure Cr and Sc layers. For the La/B4C mirrors having the highest experimental R(0°) values such an admixture will not be necessary.
In the calculations the multiple scattering method (MSM) is used (MULTEM2 program [27,28]) which is applicable to layered structures like the bilayer structures considered here. It calculates the scattering transfer matrix for each individual layer and determines the total scattering matrix as the Redheffer star product of the individual matrices. The scattering transfer matrix is used to output transmittance, reflectance, and absorbance. In this publication results on the reflectance only are presented. The MULTEM2 program was modified [7] to take into account the wavelength dependence of the complex refractive index [29,30]. The intervals on the wavelength and angle scales are varied taking into account the sharpness of the reflectance peaks: 0.0005 nm - 0.005 nm and 0.05° – 1°, respectively. The reflectance results are obtained as a function of wavelength and angle of incidence. Due to numerical reasons the angle ranges are from -89.8° to 89.8° (La/B4C) and -89.95° to 89.95° (Cr/Sc), i.e. by one interval smaller than 90°. Some of the results are presented in colour plots [31]. In most of the figures cuts through the plots at constant wavelength or constant angle are shown.
3. Results
3.1 Superlattice mirrors
The range around 6.64 nm is interesting for sub-10-nm mirror design since it is near the boron-K absorption edge. There, the refractive index varies relatively strongly with wavelength [29]. The bilayer combination La/B4C gives sufficient index contrast for high reflectance in a narrow wavelength range above the edge [4,16,32]. In this section we present results regarding the effect of a superlattice including depth grading.
Figure 3 shows the results for the reflectance of the superlattice-5 mirror in the wavelength range 6–7 nm and the dependence on incident angle for s and p polarization. La is replaced by B4C in every 5th bilayer. Due to the small index contrast between La and B4C a large number of bilayers has to be used in order to achieve high reflectance. For R(0°) > 0.7 at normal incidence at least 200 bilayers are necessary. For the results in Fig. 3 250 bilayers are used (see section 2). In the angle dependence four additional peaks (generally m-1 for superlattice-m) appear between normal incidence and total reflection. Their peak reflectance varies without grading between 0.28 and 0.45 (s polarization); the FWHM of the peak at 53° is 0.45° (Fig. 4(a)) and 0.03 nm (Fig. 4(c)), respectively, for the peak at 77.8° it is 1° (Fig. 4(b)) and 0.05 nm (Fig. 4(d)). Grading with factor 1.2 reduces the height of the peak and broadens the peak strongly at the lower angles (37° – 66°). At 77.8° the influence is almost negligible. This is discussed below after the Cr/Sc results.
Fig. 3. Reflectance R of La/B4C superlattice-5 mirrors around 6.64 nm. a) Colour plot of R versus wavelength and angle. No grading (grading factor 1). b) Angle dependence at 6.64 nm. Bilayer parameters see Fig. 4. Grading factor 1 (full curve), 1.2 (dotted). Positive (negative) angles: s (p) polarization.
Fig. 4. a, b) Angle dependence of R of the La/B4C SL-5 mirrors at 6.64 nm extended near 53° and 77.8° (s polarization). c, d) Wavelength dependence of R at 53° and 77.8°. Number of bilayers: 250. Bilayer thickness: 3.33 nm (grading factor 1, dashed curves), 3.305 to 3.470 nm (grading factor 1.05, dotted), 3.266 nm to 3.919 nm (grading factor 1.2, full).
On the wavelength scale the corresponding positions of the four superlattice peaks for normal incidence are at 8.17, 11.06, 15.96, and 29 nm. The full angular and wavelength dependence is similar to the one for Mo/Si mirrors in the EUV (Eq. (1) and Fig. 4 in [7]).
At normal incidence the peak reflectance R(0°) is 0.72 without grading (Fig. 3(b)). This value is lower than R(0°) of the standard mirror (0.77, Fig. 7(a)) with the same number of interfaces. The difference is attributed to the larger total thickness of the SL-5 mirror (higher absorption) reducing the contributions from the deeper layers. Also the (very small) change of the optical path in those layers, where La is replaced by B4C, slightly disturbs the constructive interference. The weak minimum in the angular dependence at 0° depends on the exact wavelength position. At the wavelength chosen the width of the normal-incidence peak is broadest.
For the water window (2.34–4.2 nm) Cr/V, Cr/Ti and Cr/Sc are material combinations of interest [4,26]. Relevant absorption edges are at about 2.4 nm (V), 2.7 nm (Ti) and 3.1 nm (Sc). Here we consider Cr/Sc. The highest reflectance of the standard mirror is achieved at 3.12 nm. The results, particularly the height of the reflectance peaks are very sensitive to the layer thickness. Therefore, for maximal reflectance the layer thicknesses of the superlattice (and also of the graded standard mirrors, section 3.2) are carefully adjusted so that the normal-incidence peaks are at the same wavelength as the one of the standard mirror without grading. Superlattice effects (Fig. 5) are qualitatively similar to those at 6.64 nm, i.e with almost the same angular peak positions but with lower peak values (Fig. 6) and narrower peak widths (see below). The exact positions of the SL-peaks depend on the real part of the refractive index at the wavelength chosen. The corresponding positions of the four SL-peaks on the wavelength scale at normal incidence are 3.88, 5.16, 7.68, and 14.70 nm.
Fig. 5. Reflectance R of Cr/Sc superlattice-5 mirrors. a) Colour plot of R versus wavelength and angle. b) Angle dependence at 6.64 nm. a) and b) no grading (grading factor 1). c, d) Angle dependence at 3.12 nm extended near 53° and 78°, respectively (s polarization). Grading factors: 1 (dashed curves), 1.05 (dotted), 1.2 (full). Number of bilayers: 500. Bilayer thickness: 1.56 nm (grading factor 1), 1.55 to 1.627 (grading factor 1.05), 1.54 to 1.848 nm (grading factor 1.2). Positive (negative) angles: s (p) polarization.
Fig. 6. Maximal values of the reflectance, Rp, and FWHM (°) of the superlattice peaks versus angle (s polarization). Grading factor 1 (+), 1.05 (x), 1.2 (o). a, b) La/B4C mirrors at 6.64 nm. c, d) Cr/Sc mirrors at 3.12 nm.
The angle dependence of the maximal values, Rp, and of the full-width-at-half-maximum (FWHM) of the superlattice-related peaks is summarized in Fig. 6 for both type of mirrors. Rp increases with the angle of incidence qualitatively in accordance with the Fresnel behaviour of the reflectance at a single interface for s polarization. The effect of grading on Rp is strong at the lowest angle and decreases with increasing angle nearly disappearing at the highest one. The reason for the decrease is related to the increasing optical path in the layers, to the Fresnel behaviour of the reflectance at the interfaces and to absorption. At low angle the deeper (and thicker) layers contribute distributing the reflectance over a wider range of angles and reducing Rp. At high angle less radiation penetrates to the deeper layers and is more strongly attenuated; therefore the main contributions to the reflectance come from the upper layers that are of similar thickness for different grading. The FWHM has a corresponding dependence on grading – the width increases with the reduction of Rp and depends strongly on grading at low angle becoming almost independent of it at the highest angle.
3.1.1 Weighting with source spectra and all-angle reflection
In the EUV at 13.5 nm depth grading of the superlattice structure leads to a strong broadening and weak overlap of the SL peaks [7]. All-angle reflection was achieved by combining SL-5 and SL-4 in a single structure (“SL-54”), grading it with factor 1.4 and weighting the reflectance with a Sn spectrum [33]. The purpose of combining two SL structures and grading is to increase the number and width of the SL peaks, respectively. At 6.64 nm and 3.12 nm grading also causes a strong broadening (except at the highest angles, Fig. 6). The overlap of the superlattice peaks is very weak and pronounced minima remain in R(0°) in between. Nevertheless, with the weighting all-angle reflection occurs as in the EUV (details of the weighting see section 2). The SL-54 structure alternates between the subunits of SL-5 and SL-4 starting with SL-5. The weighting is performed with a Bi spectrum [34] for a Cr/Sc mirror and a Tb spectrum [35] for a La/B4C mirror (details on the spectra in subsection 3.2.2). Figure 7 shows as an example the result of the weighted reflectance < R > of the La/B4C SL-54 mirror with grading factor 1.4. The SL peak structure is still present in < R>. For s polarization there is non-zero reflectance at all angles with the lowest minima being about 14–28% of the peak value at 0°. Also for the Cr/Sc mirror with grading factor 1.4 all-angle-reflection is observed with the minima values being 5–10%.
Fig. 7. Reflectance < R > of the La/B4C mirror with grading factor 1.4 weighted by a Tb emission spectrum [35]. Ideal and non-ideal interfaces (dashed and full curve, respectively).
The effect of non-ideal interfaces is calculated for the La/B4C mirror with the roughness parameter σ = 0.6 nm (cf. subsection 3.2.3.1). Near normal incidence < R > is approximately halved. However, at higher angles where the SL peaks occur the influence is very weak. Since σ = 0.6 nm is determined from a fit to a measured normal incidence spectrum (cf. subsection 3.2.3.1), this result shows that all-angle reflection can also be expected to exist with realistic mirror structures.
Also, the height and width of the normal-incidence peak is significantly influenced by the grading; this is studied in detail for mirrors without superlattice in the next section.
3.2 Depth-graded mirrors without superlattice
The purpose of the work described in this section is to investigate the conditions under which an enhancement of the power collected from a BEUV source is possible. The basis are reflectance results for near-normal incidence that are described first. Then, the reflectance spectra are weighted by BEUV source spectra. The weighted reflectance spectra < R > and their angular dependence are analysed to give quantitative results for the power which can be collected from the BEUV source.
3.2.1 Near-normal incidence reflectance
For La/B4C mirrors the effect of grading on the wavelength dependence at normal incidence and on the angle dependence at the peak wavelength (6.64 nm) is shown in Fig. 8. The thicknesses of the bilayers for the graded mirrors are adjusted so that the peaks appear at 6.64 nm as for the ungraded standard mirror. At normal incidence the reflectance peak broadens with grading mainly on the long-wavelength side with oscillations superimposed. Without grading (grading factor 1), the short-period oscillations are due to interference from the end faces of the total structure. The larger-period oscillations in the graded mirrors occur due to interference of waves reflected from different bilayers. With increasing grading (1.05 and 1.2) the FWHM increases from 0.06 nm (standard mirror) to 0.09 and 0.11 nm. As in Fig. 3(b) the appearance of a weak minimum in the angular dependence at 0° depends on the exact value of the wavelength; here it is chosen to give a broad normal-incidence peak. In the angle dependence (Fig. 8(b)) the width at half maximum (for each of the polarizations) increases from 6° (grading factor 1) to 10° (grading factor 1.05) and 15°(-)/20°(+) (grading factor 1.2) with tails up to ±15°, ±18° and ±32°, respectively. The normal-incidence reflectance R(0°) is reduced from 0.77 to 0.6 and 0.3, respectively.
Fig. 8. Effect of depth grading on the reflectance of La/B4C mirrors without a superlattice. Grading factor 1 (dashed), 1.05 (dotted), 1.2 (full). a): Wavelength dependence at normal incidence near 6.64 nm. b): Angle dependence at 6.64 nm. Number of bilayers: 200. Bilayer thickness: 3.344 nm (grading factor 1), 3.305–3.470 nm (grading factor 1.05), 3.266–3.919 nm (grading factor 1.2).
A broadening of the reflectance peak would also occur when simply reducing the number of bilayers. However, this is accompanied by a much larger reduction of the peak reflectance than caused by the grading. E.g., the normal-incidence peak of the graded mirror (grading factor 1.2) has FWHM = 0.11 nm and a peak value of 0.3. For the La/B4C standard mirror with 12 bilayers the peak value is only 0.06 for the same FWHM.
In the water window at 3.12 nm the wavelength dependence of the reflectance of the Cr/Sc mirror at normal incidence shows long tails on the long-wavelength side (Fig. 9) comparable to the 6.64-nm case. The FWHM increases from 0.015 (grading factor 1) to 0.03 and 0.05 nm for grading factor 1.05 and 1.2, respectively. In the angle dependence the width at half maximum (for each polarization) increases from 6° to about 8° and 12°, respectively. The effect of the grading on the reduction of the peak reflectance is stronger than at 6.64 nm. E.g., the effect on R(0°) at 3.12 nm and grading factor 1.05 is comparable to that one at 6.64 nm and grading factor 1.2.
Fig. 9. Reflectance of standard and graded Cr/Sc mirrors without a superlattice: a) R at normal incidence at wavelengths near 3.12 nm. b) R as a function of angle at 3.12 nm. Total number of bilayers: 400. Bilayer thicknesses: 1.56 nm (grading factor 1, dashed curves), 1.55–1.628 nm (grading factor 1.05, dotted curves), 1.54–1.848 nm (grading factor 1.2, full curves).
3.2.2 Relation of mirror and source characteristics—weighted reflectance
The widths of the spectra of BEUV sources depend on the excited species and the method of the excitation [5,6,34,35]. For a broad source spectrum a graded mirror can give access to parts of the spectrum which are outside the reflectance peak of a standard mirror. As shown above the grading not only broadens the reflectance peak but it also reduces its height. I.e., these two effects have opposite influence on the power which can be collected from the source spectrum. In order to find conditions where grading can lead to higher collected power the reflectance peaks are weighted with the source spectra as described in section 2.
For La/B4C the weighting of the reflectance is done by spectra of Tb and Gd (emission spectra in the range 6 - 8 nm [34] with different peak positions) and for Cr/Sc by a Bi spectrum (intensity in the whole water window [33]). The grading factor is 1.2 (La/B4C, Tb and Gd) and 1.05 (Cr/Sc, Bi). These grading factors have similar effect on R at normal and near-normal incidence for the mirrors at the two different wavelengths. Also, a stronger effect is investigated for Cr/Sc with grading factor 1.2.
Regarding the 6.x-nm range, the Tb and Gd spectra of vacuum spark excited plasmas are about five times broader (at half the peak intensity) than the reflectance peaks of La/B4C mirrors with grading factor 1.2 (Fig. 10(a) and (b), compare the reflectance of the ungraded mirrors in Figs. 8(a) and 12(a)), but the long-wavelength tails of the reflectance peaks cover a large part of the emission spectra. The short-wavelength edge of the Tb spectrum is below 6.64 nm, which is the peak wavelength of the optimal La/B4C mirror. The edge of the Gd spectrum is above 6.64 nm and for the weighting procedure the reflectance peak is shifted to 7 nm (Fig. 10(b)), i.e. the thicknesses of the bilayers in the La/B4C mirror are slightly increased which also causes a reduction of the peak reflectance. Figures 10(c) and (d) show the results for the weighted reflectance < R > in the range of -40° to +40° incident angles. The weak asymmetry of the angle dependence in Figs. 10(c) and 10(d) is due to the reduced reflectance for p polarization near the Brewster angle (≈ 45°). The detailed shape of the < R > curve depends on the peak position of the reflectance R relative to the source spectrum. With the graded mirrors < R > is increased and the range is extended to higher angles compared to weighting the standard-mirror reflectance.
Fig. 10. Weighting of the reflectance by source spectra [35]. a): Tb spectrum and La/B4C reflectance (curve with peak at 6.64 nm, same as in Fig. 5, grading factor 1.2). Bilayer parameters as in Fig. 5. b): Gd spectrum and La/B4C reflectance (curve with peak at 7 nm, grading factor 1.2), bilayer thickness: 3.451 nm to 4.142 nm. c) and d): Weighted reflectance < R > for Tb and Gd spectrum, respectively. Grading factors for dashed curves: 1, full curves: 1.2.
A measure for the total source power which can be collected by a mirror on the wavelength and angular scale is < R>int, the weighted reflectance integrated over the angles with non-zero reflectance. For the mirrors studied the range is from -40° to +40° (wavelength range 6 - 8 nm and 3 - 3.4 nm, resp.). For the La/B4C mirrors with grading factor 1.2 int increases approximately by a factor of 2.2 and 1.5 for the Tb and Gd source, respectively, compared to the ungraded mirrors.
For the 3.12-nm range the Bi spectrum [33] used for the weighting is a typical broad spectrum in the water window range, like those of W and Pb [6]. It is essentially constant over the full width of the reflectance peak. The weighted < R > at 0° is significantly larger for grading factor 1.05 than for the ungraded case but the increase is reduced for grading factor 1.2 (Fig. 11). int is larger by 56% for grading factor 1.05 than without grading. For grading factor 1.2, despite the strong reduction of R(0°) and the smaller increase of < R>(0°) the increase of < R>int is 90% originating from the strong reflectance contributions up to 32°.
Fig. 11. Weighted reflectance < R > of standard and graded Cr/Sc mirrors for the Bi spectrum. R(0°) reflectance spectra, bilayer parameters and Bi spectrum see Fig. 8. Grading factors for dashed curve: 1, dotted curve: 1.05; full curve: 1.2.
3.2.3 Influence of a non-ideal interface
For taking into account the effect of a non-ideal interface particularly on the weighted reflectance < R > and the integrated reflectance < R>int (subsection 3.2.2) parameter values of such an interface have to be chosen. In subsection 3.2.3.1 they are determined from a simulation of experimental data of the normal-incidence reflectance R(0°) by the discrete roughness model (section 2). The results described in the following two subsections are compiled in Table 1 at the end of this section.
Table 1. Properties of graded and ungraded mirrors with ideal and non-ideal interfaces. Tb, Gd, and Bi refer to the spectra used for the weighting. Definitions of < R > and < R>int see text (section 2 and 3.2.2).
3.2.3.1 Normal-incidence reflectance
A non-ideal interface is simulated by intermediate layers in the discrete roughness model described in section 2. In Fig. 12(a) results of R(0°) for La/B4C mirrors with the reflectance maxima at 6.64 nm are shown. The highest theoretical value of the maximum, Rp(0°), is 0.77, the highest experimental one 0.64 [32]. The reduction from 0.77 to 0.64 can be reproduced by assuming an intermediate layer with width dIL = 1.4 nm and σ = 0.6 nm (Fig. 12(a), curve 3). With this dIL the thickness of the narrower pure layer (La) is almost zero (0.038 nm). This means that in real structures with a peak reflectance of about 0.64 there might be no pure La layer but instead one with a small admixture of B4C. Since the discrete roughness model does not distinguish between lateral interface roughness, inter-diffusion or both, “admixture” means a laterally averaged density (see section 2). The result applies to a thickness ratio 0.43/0.57 of La/B4C. With equal thicknesses thin pure layers of both materials (≈ 0.27 nm) would remain.
Fig. 12. Influence of roughness on (a) normal-incidence reflectance R(0°) of La/B4C and (b) weighted reflectance < R > for Tb source (cf. Figure 10(a)). No roughness (σ = 0): 1 … grading factor 1, 2 … grading factor 1.2. With roughness (σ = 0.6 nm): 3 … grading factor 1, 4 … grading factor 1.2.
For mirrors in the water window the theoretical peak reflectances Rp(0°) are higher than 0.5. However, experimental values of Cr/Sc mirrors are only 0.32 (optimized with B4C diffusion barriers [36]) or 0.2 and lower without a barrier [4,17,26,36,37,39]. Reductions from the theoretical value of 0.63 for Cr/Sc to 0.32 or 0.2 can be achieved within the discrete roughness model without a barrier layer as follows. 2σ is kept smaller than dIL (see text above Fig. 2) and dILhas to be smaller than the thickness of the (thinner) La layer (0.67 nm). With dIL = 0.6 nm Rp(0°) = 0.32 can be simulated by using σ = 0.2 nm and the “pure” Cr and Sc layers having an admixture of 25% Sc and 18% Cr, respectively (Fig.14a, dashed curve). For a peak reflectance of 0.2 the values are 35% and 26%, respectively [38]. The difference in the admixtures is assumed according to the different thicknesses of the two layers. For equal thicknesses of the Cr and Sc layers the admixtures would be equal (≈ 30% for Rp(0°) = 0.2).
3.2.3.2 Weighted reflectance
For calculating the effect of the roughness on the weighted reflectance < R > the values of dIL and σ from the R(0°) analysis are used (subsection 3.2.3.1), i.e. σ = 0.6 nm for La/B4C and σ = 0.2 nm plus admixtures for Cr/Sc. The questions addressed are: a) How is the weighted reflectance < R > altered due to roughness? b) Does the grading-induced increase of the integrated reflectance < R>int (subsection 3.2.2, no roughness) remain in the presence of roughness?
Figures 12(b) and 13(b) (for the Tb and Gd source, resp.) compare < R > of the La/B4C mirrors for grading factors 1 and 1.2, curves 1 and 2 (without roughness), curves 3 and 4 (with roughness). Figure 14 shows the roughness results for the Cr/Sc mirror; the results without roughness (Fig. 11) are not included in this figure because of the large difference of the < R > scales.
Fig. 13. Influence of roughness on (a) normal-incidence reflectance R(0°) of La/B4C (peak of R(0°) shifted to 7 nm) and (b) weighted reflectance < R > for Gd source (cf. Figure 10(b)). Bilayer thickness: 3.537 nm (grading factor 1), 3.451 nm to 4.142 nm (grading factor 1.2). Notation 1-4 as in Fig. 12.
Fig. 14. Influence of roughness on (a) normal-incidence reflectance R(0°) of Cr/Sc mirrors and (b) weighted reflectance < R > for Bi source. With roughness (σ = 0.2 nm + admixtures, details see text), grading factors for dashed curve: 1, dotted curve: 1.05; full curve: 1.2. No roughness see Figs. 9 and 11.
In Table 1 the results of both types of mirrors are compiled. As in the case of R(0°) the angle dependence of < R > is qualitatively similar for La/B4C with grading factor 1.2 and Cr/Sc with grading factor 1.05 (in comparison to grading factor 1). Without roughness, despite the strong reduction of the peak values of R(0°) due to grading (Table 1, 3rd column) the weighted reflectance < R > is larger with grading than without at all angles (Figs. 11, 12(b), 13(b)). As an example for strong grading Cr/Sc with grading factor 1.2 is considered; <R > at 0° is reduced below the value for grading factor 1.05, but in the tails at angles between 8° and 32° <R > is higher (Fig. 11).
The non-ideal interface with roughness (σ = 0.6 nm) reduces < R>(0°) of the La/B4C mirrors with grading factor 1 from 0.058 to 0.036, i.e. by a factor 0.62, and with grading factor 1.2 by a factor 0.45 (Tb weighting). For Gd weighting the reduction factor is 0.55 and 0.48, respectively. The tails towards large angles remain but are weaker than without roughness. For the Cr/Sc mirrors with grading factor 1, 1.05 and 1.2 roughness (σ=0.2 nm) plus Cr/Sc mixing reduces < R>(0°) by factors 0.28, 0.21, and 0.19.
For < R>int the non-ideal interface (roughness) has pronounced effects. At the same grading (last column in Table 1) it reduces < R>int, e.g. by a factor 0.64 for La/B4C (Tb weighting) with grading factor 1 and factor 0.2 for Cr/Sc (Bi weighting) with grading factor 1.2. However, for the same interfaces (no roughness or roughness, resp.) the grading always has the effect to increase < R>int (5th column in Table 1). Without roughness the increase is up to a factor 2.21 (for La/B4C with Tb weighting) and 1.90 (Cr/Sc, Bi) for grading factor 1.2. With roughness this increase is smaller than without roughness but it is still remarkable, viz. up to factors 1.64 and 1.54, respectively.
4. Summary and conclusions
Modifications of the multilayer structure – superlattice, depth grading and non-ideal interfaces – have been applied to La/B4C and Cr/Sc Bragg mirrors in the BEUV range for peak-reflectance wavelengths near 6.64 nm and at 3.12 nm, respectively.
4.1 Superlattice mirrors
Due to the properties of the refractive index of the bilayer materials in Bragg mirrors in the BEUV the reflectance peaks have a very narrow width. The results reported here for SL-5 mirrors show that this is also the case for the reflectance peaks specific for SL-mirrors, located at angles between normal incidence and total reflection. The results on depth grading of SL-mirrors show that the broadened peaks remain narrower (and weaker) than in the EUV. They do not overlap even when increasing the number of SL peaks by combining two superlattice structures (SL-4 and SL-5) and grading with a strong factor 1.4. Nevertheless, weighting the reflectance spectra with source spectra leads to all-angle reflection with minima between 5 and 10% (Cr/Sc) and 14–28% (La/B4C) of the value at normal incidence. Non-ideal interfaces do not destroy all-angle reflection; the reflectance is significantly reduced at small incident angles but only very weakly above about 40°. Weighting and all-angle reflection are inevitably connected with some spectral width. In the cases considered here it is 3 to 3.4 nm (Cr/Sc) and 6 to 8 nm (La/B4C).
Without grading the m-1 peaks of superlattice-m mirrors offer the possibility to work with multiple sharp angular bands with a single optical component. They are an alternative to a stack of multilayers with different periods [1] or to multiperiod multilayers [10]. The superlattice reflectance peaks can be positioned at certain angles by a proper choice of the superlattice period. By additionally adjusting the bilayer thickness d the angle of a peak can be further tuned. E.g., for a SL-3 La/B4C mirror the 2nd order SL-peak is tuned to the Brewster angle (≈45°) with d = 3.14 nm and a linear polarizer can be designed.
4.2 Mirrors without superlattice
On the standard mirror without a superlattice the influence of depth grading on the normal-incidence peaks as well as the relation of the spectral characteristics of the reflection peaks and BEUV sources was studied. The depth grading causes a broadening on the wavelength as well as on the angular scale at the expense of a lower peak reflectance. Weak grading, e.g. a La/B4C mirror with factor 1.05, reduces the maximal reflectance Rp(0°) from 0.77 to 0.6 and increases the FWHM from 0.06 to 0.09 nm so that different mirrors in an optical system can be more easily matched without large loss in radiation power [40]. Stronger grading (1.2 for La/B4C, 1.05 and 1.2 for Cr/Sc) reduces Rp(0°) to 0.34 and lower values and strongly increases the FWHM.
4.2.1 Weighted and integrated reflectance
Despite the reduction of Rp(0°) but due to the increased peak width the grading has an advantageous effect on the weighted reflectance < R > which relates the spectral characteristics of mirror and source. This can be useful when a broader spectrum and/or more radiation power reflected by a mirror is needed for an application. Source spectra of different width are considered: a) Tb and Gd spectra (with different peak positions relative to the highest reflectance peak at 6.64 nm) which are roughly 5 times broader at half maximum than the reflectance peaks of the La/B4C mirrors, b) Bi spectrum which has little intensity variation over the full width of the reflectance peak of the Cr/Sc mirror. Only for source spectra broader than the reflectance peaks an increase of < R > with grading can be expected.
The weighted reflectance < R > for grading factor 1.2 (La/B4C, Tb and Gd spectrum) and 1.05 (Cr/Sc, Bi spectrum) is significantly larger at all angles than without grading. The strong grading factor 1.2 for Cr/Sc reduces < R > at 0° but in the tails at angles from 8° to 32° the values are above those for grading factor 1 and 1.05. <R > contains contributions from the reflectance spectra between 6.2 and 7.5 nm (La/B4C, Tb), 6.5 and 7.8 nm (La/B4C, Gd), and 3 and 3.4 nm (Cr/Sc, Bi) with the main contributions from about half these widths. The angular ranges where < R > is larger for the graded mirrors are reflected in the values of the integrated reflectance < R>int being larger by factors between about 1.5 and 2.2 than without grading for the cases considered here. The integration is from -40° to +40° which covers the full width of the near-normal-incidence reflectance peaks. For a range narrower than -10° to +10°, which might be relevant for weakly diverging or converging beams, the advantage of grading remains.
4.2.2 Non-ideal interfaces
For the calculation of < R > and < R>int of realistic samples it is taken into account that the interfaces are not ideal (intermixing between layers, interface roughness). This is observed to become increasingly important with reduced bilayer thickness [4,17,26,36,37,39]. The necessary intermixing and roughness parameters are determined from simulations of experimental results of the normal-incidence reflectance. For La/B4C (peak at 6.64 nm, R≈ 0.6) the width of the intermediate interface layer is almost as large as the La thicknesses. This leads to significantly reduced thicknesses of the pure La and B4C layers. For Cr/Sc (3.12 nm) no such pure layers are left in case of reflectance values of 0.3 and below. Strong admixtures of Sc in Cr and Cr in Sc have to be assumed.
Regarding < R>int, despite non-ideal interfaces the advantage of the graded mirrors over the ungraded ones remains but is reduced. The enhancement factors are between about 1.3 and 1.6 (1.5 and 2.2 without roughness). Thus, a benefit of exploiting more total intensity from a source spectrum and eventually increasing the throughput of an optical system should be possible for fabricated graded La/B4C mirrors. Also for the narrower integration range (-10° to +10°) the enhancement is maintained for the weaker gradings (1.2, La/B4C and 1.05, Cr/Sc) but lost for the stronger grading (1.2, Cr/Sc). For Cr/Sc and other mirrors for the shorter wavelengths in the BEUV the power advantage could only become effective if the quality of the deposited layers can be improved.
Concerning application-related properties of the depth modified mirrors one has to keep in mind that applications with several reflections on a series of mirrors would reduce the final power appreciably if each reflectance is below about 0.6. However, for single or few-mirror reflections the graded mirrors can have distinct power advantages depending on the range of wavelengths and angles used.
Disclosures
The authors declare no conflicts of interest.
Data availability
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.
References
1. Q. Huang, V. Medvedev, R. van de Kruijs, A. Yakshin, E. Louis, and F. Bijkerk, “Spectral tailoring of nanoscale EUV and soft x-ray multilayer optics,” Appl. Phys. Rev. 4(1), 011104 (2017). [CrossRef] and references therein.
2. S. K. Moore, “EUV lithography finally ready for fabs,” IEEE Spectrum 55(1), 46–48 (2018). [CrossRef]
3. M. LaPedus, “EUV Challenges And Unknowns At 3 nm and Below,” (2020) https://semiengineering.com/euv-challenges-and-unknowns-at-3nm-and-below/
4. T. Feigl, H. Pauer, T. Fiedler, M. Perske, and P. Naujok, “Multilayer Optics for 1 nm to 13.5 nm: Can we reduce the Lithography Wavelength further?” https://www.euvlitho.com/2018/S18.pdf
5. P. Sheridan, A. Manzoni, A. O’Connor, C. Rogers, D. Rogers, D. Skoko, G. Murray, I. Tobin, J. Costello, J. Howard, K. Fahy, M. Donnellan, S. Brady, T. Donnelly, T. McEnroe, W. Fyans, G. Joseph, M. Olszewski, and F. O’Reilly, “A Water Window Source for Soft-X-Ray Microscopy and other Applications,” https://www.euvlitho.com/2018/S16.pdf
6. G. O’Sullivan, P. Dunne, D. Kilbane, L. Liu, R. Lokasani, E. Long, B. W. Li, T. McCormack, F. O’Reilly, J. Shiel, E. Sokell, C. Suzuki, T. Wu, and T. Higashiguchi, “Source Development for Extreme Ultraviolet Lithography and Water Window Imaging,” AIP Conf. Proc. 1811(1), 170001 (2017). [CrossRef]
7. R. Meisels and F. Kuchar, “EUV Bragg reflectors with photonic superlattices,” Optics Express 25(26), 32215–32226 (2017). [CrossRef] The second term on the right side of Eq. (1) should be squared.
8. The term superlattice has been also used with a different meaning, viz. for the simple bilayer structure superimposed on the natural crystal structure [9] and for multiperiodic multilayers [10].
9. A. F. Jankowski, P. Sandoval, and J. P. Hayes, “Superlattice effects on solid-state amorphization,” Nanostruct. Mater. 5(5), 497–503 (1995). [CrossRef]
10. S. Yulin, T. Feigl, T. Kuhlmann, N. Kaiser, A. I. Fedorenko, V. V. Kondratenko, O. V. Poltseva, V. A. Sevryukova, A. Yu. Zolotaryov, and E. N. Zubarev, “Interlayer transition zones in Mo/Si superlattices,” J. Appl. Phys. 92(3), 1216–1220 (2002). [CrossRef]
11. Increasing the integrated reflectance and collecting more power with graded mirrors has been frequently discussed in the literature (reviewed in [1]). Quantitative results of the integrated reflectance were reported, e.g., in [12] (adaption of a broad-band mirror to a Xe plasma source) and in [13] (estimate of the throughput assuming a flat source spectrum) for 13.5 nm and normal incidence, in [14] for 3.6 nm and grazing incidence. Wang and Michette [15] made detailed calculations of the effect of grading on the normal-incidence reflectance throughput in the 13–19 nm range, i.e. performing weighting using a Rhenium source spectrum. In our work the weighted reflectance spectra < R > and their angular dependence are analysed to give quantitative results for the power which can be collected from the BEUV source.
12. S. A. Yulin, T. Kuhlmann, T. Feigl, and N. Kaiser, “Spectral reflectance tuning of EUV mirrors for metrology applications,” Proc. SPIE 5037, 286–293 (2003). [CrossRef]
13. S. P. Vernon, D. G. Stearns, and R. S. Rosen, “Chirped multilayer coatings for increased x-ray throughput,” Opt. Lett. 18(9), 672–674 (1993). [CrossRef]
14. J. F. Seely, M. P. Kowalski, W. R. Hunter, and G. Gutman, “Reflectance of a wideband multilayer x-ray mirror at normal and grazing incidences,” Appl. Opt. 35(22), 4408–4412 (1996). [CrossRef]
15. Z. Wang and A. G. Michette, “Broadband multilayer mirrors for optimum use of soft x-ray source output,” J. Opt. A: Pure Appl. Opt. 2(5), 452–457 (2000). [CrossRef]
16. I. A. Makhotkin, E. Zoethout, E. Louis, A. M. Yakunin, S. Müllender, and F. Bijkerk, “Spectral properties of La/B–based multilayer mirrors near the boron K absorption edge,” Opt. Express 20(11), 11778–11786 (2012). [CrossRef]
17. S. Yulin, T. Feigl, and N. Kaiser, “High-performance Cr/Sc multilayers for the soft x-ray range,” Proc. SPIE 5963, 59631T (2005). [CrossRef]
18. Ping Lee, “Uniform and graded multilayers as x-ray optical elements,” Appl. Opt. 22(8), 1241–1246 (1983). [CrossRef]
19. A. E. Yakshin, I. V. Kozhevnikov, E. Zoethout, E. Louis, and F. Bijkerk, “Properties of broadband depth-graded multilayer mirrors for EUV optical systems,” Opt. Express 18(7), 6957–6971 (2010). [CrossRef]
20. Ref. 1, chapter II.
21. Ref. 1, citations 24-31.
22. M. Mitchell, B. Muftakhidinov, and T. Winchen, “Engauge Digitizer Software,” http://markummitchell.github.io/engauge-digitizer, Last Accessed: 16 Jan 2019. This software is used for requiring the data of the spectra.
23. D. G. Stearns, “The scattering of x rays from nonideal multilayer structures,” J. App. Phys. 65(2), 491–506 (1989). The error function used in this reference is essentially the cumulative distribution function of the normal distribution. Also other distributions in the intermediate layer were studied (sinusoidal, linear, exponential). It was found that the results are relatively insensitive to the precise shape of the profiles. [CrossRef]
24. D. L. Windt, “IMD—Software for modeling the optical properties of multilayer films,” Comput. Phys. 12(4), 360–370 (1998). [CrossRef]
25. https://www.classe.cornell.edu/∼dms79/refl/XR-Roughness.html. In this reference, microscopic roughness is defined as the roughness period being smaller than the coherence length of the radiation. Based on this definition all the roughness calculations cited here and our own calculations are performed.
26. A. Haase, S. Bajt, P. Hönicke, V. Soltwisch, and F. Scholze, “Multiparameter characterization of subnanometre Cr/Sc multilayers based on complementary measurements,” J. Appl. Crystallogr. 49(6), 2161–2171 (2016). [CrossRef]
27. N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113(1), 49–77 (1998). [CrossRef]
28. N. Stefanou, V. Yannopapas, and A. Modinos, “MULTEM 2: A new version of the program for transmission and band-structure calculations of photonic crystals,” Comput. Phys. Commun. 132(1-2), 189–196 (2000). [CrossRef]
29. http://henke.lbl.gov/optical_constants/. In [30] improved values for Cr were reported that differ from the Henke values particularly at absorption edges. For the wavelength range near 3.12 nm (Cr/Sc mirror), far away from Cr edges, the difference is negligible. The values of the refractive indices are given for a density corresponding to crystalline material. For amorphous material they would only deviate if the densities were different.
30. F. Delmotte, J. Meyer-Ilse, F. Salmassi, R. Soufli, C. Burcklen, J. Rebellato, I. Vickridge, E. Briand, and E. Gullikson, “Soft X-ray optical constants of sputtered chromium thin films with improved accuracy in the L and M absorption edge regions,” J. Appl. Phys. 124(3), 035107 (2018). [CrossRef]
31. J. W. Eaton, D. Bateman, S. Hauberg, and R. Wehbring, “GNU Octave version 6.1.0 manual: a high-level interactive language for numerical computations,” https://www.gnu.org/software/octave/doc/v6.1.0/ (2020).
32. D. S. Kuznetsov, A. E. Yakshin, J. M. Sturm, R. W. E. van de Kruijs, E. Louis, and F. Bijkerk, “High reflectance La/B based multilayer mirrors for 6.x nm wavelength,” Opt. Lett. 40(16), 3778–3781 (2015). [CrossRef]
33. F. Kuchar and R. Meisels, “Photonic superlattice multilayers for EUV lithography infrastructure,” Proc. SPIE 10775, 1077503 (2018). [CrossRef]
34. T. Higashiguchi, T. Otsuka, N. Yugami, W. Jiang, A. Endo, B. Li, P. Dunne, and G. O’Sullivan, “Feasibility study of broadband efficient “water window” source,” Appl. Phys. Lett. 100(1), 014103 (2012). [CrossRef] The Bi emission spectrum with the highest intensity [Fig. 2(a)] is used in the present work.
35. S. S. Churilov, R. R. Kildiyarova, A. N. Ryabtsev, and S. V. Sadovsky, “EUV spectra of Gd and Tb ions excited by laser-produced and vacuum spark plasmas,” Phys. Scr. 80(4), 045303 (2009). [CrossRef]
36. E. M. Gullikson, F. Salmassi, A. L. Aquila, and F. Dollar, “Progress in short period multilayer coatings for water window applications,” Lawrence Berkeley National Laboratory, http://escholarship.org/uc/item/8hv7q0hj (2008).
37. Q. Huang, Q. Yi, Z. Cao, R. Qi, R. A. Loch, P. Jonnard, M. Wu, A. Giglia, W. Li, E. Louis, F. Bijkerk, Z. Zhang, and Z. Wang, “High Reflectance Nanoscale V/Sc Multilayer for Soft X-ray Water Window Region,” Sci. Rep. 7(1), 12929 (2017). [CrossRef]
38. In [26] the (low) reflectance of 0.15 (at 3.14 nm) could be fitted by assuming a reduced index contrast in a model similar the one in the present work. The reduced index contrast was attributed to intermixing and interdiffusion. This supports the assumptions made in the present work regarding the admixtures of Sc in Cr and Cr in Sc.
39. F. Delmotte, C. Burcklen, E. Meltchakov, J. Rebellato, S. de Rossi, D. Dennetiere, B. Capitanio, and F. Polack, “Advanced multilayer development for the water window spectral region,” EUVL Workshop, Berkeley, 14 June 2019, http://www.euvlitho.com/2019/P52.pdf
40. The issue of low bandwidth (matching of mirrors, integrated power) is addressed in [4].
