research papers
Highefficiency Xray multilayercoated blazed gratings with shifted boundaries
^{a}St Petersburg Academic University, Khlopin St 8/3 Let A, St Petersburg 194021, Russian Federation, ^{b}ITMO University, Kronverkskiy prospekt 49, St Petersburg 197101, Russian Federation, and ^{c}Institute for Analytical Instrumentation, Rizhsky prospekt 26, St Petersburg 190103, Russian Federation
^{*}Correspondence email: lubov@spbau.ru, lig@pcgrate.com
A new design for a highefficiency multilayercoated blazed Xray grating with horizontalshifted (nonconformal) boundary profiles is proposed. The investigation of the grating design is carried out using an integrated approach based on rigorous numerical calculations of light diffraction by gratings with realistic boundary profiles obtained from simulations of multilayer grating growth. By varying the incidence angle of the deposition ^{−1} grating with a blaze angle of 1.76° and an antiblaze angle of 20°, working at a blaze wavelength of 1.3 nm and having shifted realistic boundary profiles, obtained using our integrated approach is 23.3%, while that of a grating with the ideal (triangular) boundary profile and the same shifts is 25.3%, and that of an ideal conformal profile is only 22.2%. The maximum absolute efficiency of 40.2% of a 2500 mm Cr/C grating with a blaze angle of 1.05° and a realistic antiblaze angle of 10°, working at a blaze wavelength of 0.83 nm and having ideal shifted boundaries, is higher than the maximum efficiency of the similar grating having ideal conformal boundaries with a nonrealistic antiblaze angle of 80°.
one can set the direction and magnitude of the boundary profile shifts over a wide range of values. It is shown that the diffraction efficiency of the blazed gratings with shifted boundary profiles may be substantially higher than the efficiency of gratings with conformal boundaries, which are, moreover, much more difficult to produce. Highefficiency gratings with shifted boundaries can be obtained when the deposition is mainly on the blaze facet with a high inclination of the deposition as opposed to widely used nearnormal deposition methods. The maximum absolute efficiency of a W/B4C 2500 mmKeywords: multilayercoated diffraction grating; soft Xray range; grating efficiency; growth modeling; multilayercoated blazed Xray grating; thinfilm growth simulations; diffraction efficiency; electromagnetic diffraction theory.
1. Introduction
Multilayercoated blazed gratings (MBGs) are considered the most promising optical structures for many hard and soft Xray applications (Rife et al., 1990; Spiller, 1994). One of the main requirements to be met by MBGs is the high diffraction efficiency η. This requirement can be achieved through improvements in the grating design and in the controlled fabrication of a multilayer structure, accounting for peculiarities of the growth process, such as smoothing of boundary profiles, interlayer mixing and surface roughness development. Recently, a number of studies have been conducted in which different ways to improve the efficiency of soft Xray MBGs have been considered (Voronov et al., 2015; Goray & Egorov, 2016; Senf et al., 2016). It was shown by Voronov et al., (2015, 2016) that gratings with high groove densities allow high grating efficiencies to be achieved at a very high resolution in the soft Xray and extreme ultraviolet range. It was demonstrated (Senf et al., 2016) that a sawtooth profile has to be ruled into a goldcoated surface and then reactive ion etching transfers the profile onto the substrate. Another way to increase grating efficiency in the soft Xray range is to implement another diffraction scheme, i.e. a conical one (Goray & Egorov, 2016). Compared with classical schemes (inplane diffraction), conical diffraction reduces negative effects, e.g. absorption and refraction, and eventually leads to a higher efficiency. However, deformation of MBG boundary profiles and their shifts during growth significantly decrease efficiency (Voronov et al., 2011a). The geometric shape of the profiles changes during growth, owing to smoothing, deposition source noise and various nonlocal effects such as shadowing and reemission (Pellicione & Lu, 2007). Deformation of the boundary profiles in the case of strong smoothing can transform the sawtooth profile into a sinusoidal one (Voronov et al., 2011b). Deposition noise not only affects the rootmeansquare surface roughness but also leads to the evolution of the initial surface topography into a complex structure characterized by a set of roughnesses and correlation lengths (Senthilkumar et al., 2005; Yashchuk et al., 2014; Goray & Lubov, 2015). Nonlocal effects can significantly transform the grating boundaries, since a local growth event occurring at some point on the profile surface influences the growth process at a great distance from the original event.
Thus, an investigation of the design of MBGs should include both an accurate calculation of diffraction characteristics, taking into account realistic boundary profiles, and a simulation of the evolution of the initial profile during growth. The most effective approach to calculating diffraction grating efficiencies would be to use rigorous numerical methods, since they, unlike analytical and approximate calculations, allow one to take complex boundary profiles, random roughness and changes in the composition of layers exactly into account (Goray & Schmidt, 2014). Boundary profiles can either be obtained from a numerical calculation of the growth process or derived from (TEM) measurements. However, TEM determination of boundary profiles is destructive and ineffective, since such measurements are necessary for each growth experiment and for many periods and grating boundaries, the number of which can reach hundreds. It is also necessary not only to determine boundary profiles and their random roughnesses but also the composition of layers, since this affects complexvalued refractive indices (Gullikson, 2019). Therefore, a theoretical approach based on the modeling of layer growth, with subsequent computations of the scattering intensity of a simulated structure, is more effective in the design of multilayer gratings working in the shortest wavelength range.
For the first time, to the best of our knowledge, this integrated approach was proposed by Goray & Lubov (2013) to calculate the absolute efficiencies of multilayer Mo/Si and Al/Zr gratings, working in the extreme ultraviolet range. This approach was also successfully applied to the consideration of the growth of rough multilayer mirrors and the calculation of scattering intensities in the soft Xray and extreme ultraviolet ranges (Goray & Lubov, 2015). The use of this approach allowed the shift and deformation of the profile during the growth of Al/Zr gratings to be explained (Voronov et al., 2011a,b). It was shown that deposition of the material on the asymmetrical profile leads to a horizontal shift in the boundary profiles. However, the dependence of the profile shift on the incidence angle of the deposition remains unknown. Calculation of this dependence allows one to choose the optimal incidence angle and estimate how the initial profile changes during growth. Moreover, the question arises of how the diffraction efficiency depends on the shifts of the boundary profiles. According to the prediction of scalar diffraction theory, for blazed gratings with angle ζ at the top of a profile equal to 90°, 100% efficiency is achieved with a conformal [see Fig. 1(a)] arrangement of the layers at one polarization (Voronov et al., 2015; Nevière & Montiel, 1996). However, for stateoftheart shortperiod soft Xray gratings with ζ > 90° and β (the antiblaze angle) equal to several degrees or several tens of degrees, the configuration of the layers corresponding to maximum efficiency is unknown. Therefore, to determine the configuration of the grating layers that corresponds to the maximum efficiency, it is necessary to calculate and compare diffraction efficiencies of gratings with conformal and nonconformal [see Fig. 1(b)] boundary profiles.
In this manuscript, on the basis of the simulations of the growth of W/B_{4}C and Cr/C MBGs and rigorous calculations of their diffraction efficiencies, we demonstrate that MBGs with shifted boundary profiles may have higher diffraction efficiencies than similar MBGs with conformal layers.
2. Growth simulation
The growth process is considered within the continuous approach (Pellicione & Lu, 2007; Villain, 1991). This approach allows us to model MBGs growth over large scales, both spatially, with the grating period d ≃ 10^{1}–10^{4} nm, and temporally, with a growth time of the order of 10^{2}–10^{3} s. This approach has proven effective in the study of growth on patternedsurface substrates (Goray & Lubov, 2013; De Virgilis et al., 2003; Ballestad et al., 2005; Castez, 2010; Harrison & Bradley, 2017).
The grating profile can be considered isotropic and twodimensional; therefore, we represent the profile height h as a function of coordinate x and time t. The evolution of the boundary profile is calculated with the help of the Mullins equation (Mullins, 1957), in which the local surface curvature K(x, t) is strictly taken into account:
Here g(x, t) is the density of the deposition and ν_{4} is the smoothing coefficient, determined by the surface and surface free energy (Mullins, 1957). The explicit expression for K(x, t) can be found in Polyanin & Manzhirov (2006). The of atoms on the surface of a grating with a triangular profile is defined by the blaze angle φ, β, and the deposition incidence angle α measured from the horizon counterclockwise, where α = 0 corresponds to the horizontal striking only the antiblaze facet. In these calculations, we do not consider shadowing effects; therefore, φ, β and α should be related by φ < α and β < 180 − α. The deposition to the blaze facet can be written (taking into account the condition of nonshadowing) as (Goray & Lubov, 2013)
Here, I_{0} is the value of the deposition emitted from the source and Similarly, for the deposited on the antiblaze facet, we have
Here, Other mechanisms of layer growth are not considered in this work, for example, when growth occurs along the local surface normal [Kardar–Parisi–Zhang equation (Kardar et al., 1986)] or when diffusion between the layers with a sharp change in composition is significant [Cahn–Hilliard equation (Cahn & Hilliard, 1958)]. This is because the purpose of our research is to investigate how a change in the incidence angle of the deposition affects the shift of boundaries. The shift, as shown by Goray & Lubov (2013), is caused by a nonuniform distribution of the material on the profile surface, which is, in turn, determined by α. In the case of other growth mechanisms, the boundary profile shift still occurs and only the shift values change.
The smoothing coefficients (ν_{4}) in the growth simulation were chosen on the basis that in order to obtain high diffraction efficiencies it is necessary to ensure that smoothing is sufficiently small in such a growth mode Significant smoothing of the grating boundary profile means that a nonoptimal growth mode is realized (Goray & Lubov, 2013).
3. Results and discussion
3.1. W/B_{4}C grating
The grating growth was calculated for a W/B_{4}C MBG with the following parameters (Voronov et al., 2016): φ = 1.76°, β = 20°, d = 402 nm (d is the grating period), Γ = 0.5 (Γ is the ratio of the thickness of a layer to the period of the multilayer coating), L = 5.78 nm (L is the period of the multilayer coating) and n = 18 (n is the number of bilayers). The shift of the boundary profiles Δd_{i}(α) for the ith layer boundary (i = 1, 2,…, n) can be calculated using . Here, h_{s} and h_{i} are the x coordinates of the maximum profile height of the substrate and ith layer boundary, respectively.
Fig. 2 shows the dependences of Δd on α and i. Calculations were carried out with I_{0} = 1 nm s^{−1}, with ν_{4} = 25 nm^{4} s^{−1} in the first layer and ν_{4} = 20 nm^{4} s^{−1} in the second layer. As can be seen from Fig. 2(a), the dependence of Δd_{1} on α is almost linear in the range α = 60–140° and increases rapidly for α > 140°, whereas the dependence of Δd on i (with α = 150°) is linear for all i, as shown in Fig. 2(b). The shift in the profile is caused by the nonuniform deposition of the material (Goray & Lubov, 2013), and therefore the increase in profile height is also nonuniform. The nonuniformity leads to redistribution of the material and smoothing of the profile. The magnitude of the shift depends on the ratio of the fluxes to different facets of the profile. Since the fluxes to the facets are determined by φ, β and α, equality of the fluxes is achieved when f(x) = sin[α – φ(x,t)]/sin[α + β(x,t)] = 1. The behavior of f(x) is linear when α = 60–120° for the considered profile angles and increases rapidly for larger α. However, the transition of f(x) from linear to nonlinear behavior depends on β. The smaller the value of β, the greater the value of α at which the transition to nonlinear behavior occurs. During smoothing of the profile, β becomes smaller and therefore the transition from linear behavior to nonlinear occurs at larger β.
The linear dependence of Δd on i is mostly caused by the establishment of a steadystate smoothing mode, at which deposition fluxes on the facets near the maximum profile height become approximately equal some time after the beginning of deposition.
The change in α affects not only the position of the boundaries of the layer but also the profile height. The change in profile height Δh_{i}(α) for the ith layer boundary can be calculated using Fig. 3 shows the dependences of h_{i} and Δh_{i} on α and i. The inset figure shows parts of the initial profile and topmost profiles at α = 60° and 150°, shifted for convenience of comparison. As one can see from Fig. 3, the greater the value of α, the smaller the value of h_{1}(α) during growth.
It is worth noting that calculations made for different values of ν_{4} have shown that an increase in ν_{4} leads to: (i) an additional shift of the boundaries to the left (along the blaze facet) and (ii) stronger profile smoothing and a greater decrease in profile height.
The diffraction efficiencies of gratings with realistic and ideal boundary profiles were calculated using the rigorous boundary integral equation method (Goray & Schmidt, 2014). This method has been successfully used in the modeling of efficiencies and scattering intensities of different multilayer structures used in the soft Xray and extreme ultraviolet regions, e.g. gratings, rough mirrors or zone plates (Goray & Schmidt, 2014; Seely et al., 1999; Goray et al., 2006; Goray, 2010). The diffraction efficiencies of the MBGs were calculated using the commercial program PCGrateSX (version 6.7; https://pcgrate.com).
The results of the calculations demonstrate that the shift of the grating boundaries has a significant effect on η. A shift of the profile to the right, Δd < 0, leads to a reduction in efficiency, while a shift to the left, Δd > 0, leads to an increase in efficiency; see Table 1. For gratings with ideal shifted boundaries, values of Δd_{i}(α) were taken from simulation results corresponding to realistic shifted boundaries. The results presented in Table 1 show that the grating efficiencies with both realistic η_{r} and ideal η_{id} boundaries change in the same way. Moreover, the efficiency of the gratings with ideal conformal boundaries is smaller than the efficiency of gratings with shifted realistic boundaries for α = 150°. Such efficiency behavior is unexpected and suggests opportunities for the fabrication of new, more efficient and easytoproduce MBGs.

For a more detailed understanding of the efficiency behavior, highaccuracy computations for MBGs with realistically grown profiles were made at α = 135–150°. Fig. 4 presents the efficiencies of the gratings with conformal and nonconformal boundaries. As can be seen from Fig. 4, the efficiency generally increases with increasing deposition angle; however, η(α) may have local minima and maxima. The ratio between η_{id} and η_{r} is greater for small α. Hence, it can be concluded that the rapid decrease in efficiency is mostly caused by the decrease in profile height. The difference in realistic profile heights h(x, t) of the topmost boundaries for α = 60° and 150° is only 0.26 nm or 2% of the initial height, and the value of the efficiency is more than halved (56%), whereas for the gratings with ideal shifted boundaries, Δh = 0, the efficiency decreases by only 14%. We suppose that such dependences are caused by the complex interplay between boundary shifts and changes in profile height during growth. Indeed, the efficiency of an MBG depends on various processes occurring during diffraction, i.e. absorption, multiple reflection and refraction, shadowing, etc. Meanwhile a contribution of all these processes to the efficiency is affected by the structure of the layers, i.e. the boundary shifts and depths of the layers on the blaze and antiblaze facets. Therefore, although the growth of MBGs with the layer structure leads to an increase in efficiency with increasing α, a specific set of structural parameters might not be optimal.
In Fig. 5, results of calculations of the diffraction efficiency of the W/B_{4}C grating with ideal shifted boundaries and the constant shift Δd_{i} for all boundaries versus shift of the boundary profile are shown. As one can see, the behavior is nonlinear and there is an optimal shift value which corresponds to the maximal efficiency and minimal absorption.
3.2. Cr/C grating
In connection with the results obtained for W/B_{4}C MBGs, the question arises whether a similar change in the efficiency is observed for other sawtooth gratings with nonconformal geometries. In order to study the influence of the shift of the boundary profiles, calculations of growth and diffraction efficiencies were performed for the Cr/C MBG. Parameters of the grating included φ = 1.05°, d = 400 nm and β = 10°; multilayer Cr/C parameters were taken from the literature (Senf et al., 2016): Γ = 0.6, L = 7.3 nm and n = 20. This grating has a flatter profile and can work at an optimal wavelength of 0.83 nm (1.5 keV) in the −1st order. Calculations of the grating growth were carried out with I_{0} = 1 nm s^{−1}, with ν_{4} = 25 nm^{4} s^{−1} in the first layer and ν_{4} = 20 nm^{4} s^{−1} in the second layer, which is the same for W/B_{4}C gratings.
A flatter initial profile in the case of Cr/C MBG leads to less smoothing of the profile during growth, and hence to a lesser effect of the deformation of the profile on the efficiency. Indeed, for α = 150° the reduction in height Δh for the Cr/C MBG is about 0.3 nm for L = 7.3 nm and n = 20, whereas, for the W/B_{4}C MBG with L = 5.78 nm and n = 18, the value of Δh is equal to 0.39 nm (see Table 1).
It should be noted that the profile deformation during growth, i.e. the boundary profile shift and the change in its height are determined by the geometry of the profile and the relaxation mechanism, which were the same in both MBG cases. In this regard, the dependencies of changes in the profile height and shifts on α and i are similar for W/B_{4}C and Cr/C MBGs and therefore the results of growth calculations are presented here only for the W/B_{4}C MBGs.
For Cr/C gratings with ideal shifted boundaries, Δd_{i} values were held constant for all layers of the considered MBG. The results of the Cr/C efficiency calculations, as in the case of W/B_{4}C gratings, demonstrate that the shift of the grating boundaries to the left, Δd > 0, leads to an increase in efficiency (Fig. 6). In opposition to the shifts to the left, even a small shift to the right leads to a significant decrease in efficiency.
As shown in Fig. 6, the maximal absolute efficiency (40.2%) of the MBG with a realistic antiblaze angle of 10° and ideal shifted boundaries is higher than the maximal efficiency (39.7%) of similar grating having ideal conformal boundaries with a nonrealistic antiblaze angle of 80°. Moreover, these efficiencies are only slightly higher than the maximal efficiency of 39.2% of the MBG with realistic shifted boundaries [Fig. 6(b)]. The maximal efficiency of the Cr/C MBG with realistic boundaries is reached at α = 155°, while at α = 160°, the diffraction efficiency decreases significantly and the absorption increases. We also considered the case of shifting upper boundary profiles to the left by more than the period of grating d = 400 nm. For example, at the angle α = 160° for the fourth layer from the top, the value of the shift from the initial (substrate) profile position is equal to 408 nm, and the topmost layer is shifted by 435 nm. In the case of such a strong shift, upper layers of the grating, which impact efficiency the most, could be considered as shifted to the right relative to the initial (substrate) profile position.
4. Conclusions
Thus, we found that MBGs with conformal boundaries do not have the highest efficiency in the soft Xray range; on the contrary, MBGs with appropriately horizontalshifted boundary profiles are more efficient, taking into account realistic antiblaze angle values and the decreasing absorption. Moreover, the efficiency of gratings with realistic boundary profiles has a complex dependence on profile shifts and height reductions. Therefore, any search for optimal MBGs should be based on exact calculations or measurements of the grating boundary
and efficiency.We have demonstrated that the diffraction efficiency of the MBGs with shifted boundary profiles may be substantially higher than the efficiency of gratings with conformal boundaries, which are, moreover, much more difficult to produce. Based on our results, a new design of the soft Xray MBGs could be proposed. The principal new idea of the proposed design is to fabricate MBGs with the layers highly shifted toward the direction of the incidence radiation, i.e. along the antiblaze facet of the profile. The strong shift of the boundary profiles in the desired direction is achieved when deposition occurs predominantly on the blaze facet. In the manufacture of such gratings, it is necessary to set the incidence angle of the deposition in the range at which most of particles are deposited on the blaze facet, while the antiblaze facet is not shaded from the deposition flux.
Funding information
This work was partially supported by the Ministry of Education and Science of the Russian Federation, the Russian Foundation for Basic Research (RFBR) (grant No. 162911697) and Russian Science Foundation (RSF) (grant No. 191200270).
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