## Abstract

The problem of X-ray diffraction from multilayer-coated blazed diffraction gratings is analyzed. Invalidity of the conventional condition of maximal diffraction efficiency observed in previous experiments is explained theoretically. This is attributed to two factors: contribution of anti-blaze facets to diffraction efficiency and effect of strongly asymmetric diffraction. We demonstrate that a proper choice of the multilayer d-spacing allows to design grating with the diffraction efficiency close to the maximal possible one throughout the tender X-ray range (*E*∼1-5 keV). An optimization procedure is suggested for the first time to choose the optimal grating parameters and the operation diffraction order to obtain a high fix-focus constant and high diffraction efficiency simultaneously in a wide spectral range.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Diffraction gratings are the key elements of soft X-ray (SXR) monochromators and spectrometers installed in synchrotron and free-electron laser beamlines. However, their efficiency decreases with increasing photon energy, and becomes lesser than 10%, even theoretically [1–3], in the 1−5 keV spectral range, called “the tender X-ray region”. The applicability of natural crystals as dispersion elements in this region is also very limited, especially for the lower spectral range. Notice that the diffraction efficiency can be increased through the use of the conical (out-of-plane) diffraction geometry, while at the cost of lower angular dispersion and decreasing spectral resolution [4,5].

At the same time, this energy region covers L and M absorption edges of most of the transition and rare-earth metals as well as K edges of lighter elements such as sulfur and phosphorus. Synchrotron sources have been widely used to investigate the atomic and electronic structure of these materials with X-ray spectroscopy, and to perform experiments on the study of X-ray magnetic circular dichroism. With the emerging new sources characterized by high brilliance and photon flux, the problem of low efficiency of classical single-coated diffraction gratings is becoming more and more urgent, because low grating efficiency limits the sensitivity and the resolution of the spectroscopy methods.

Deposition of a periodic multilayer structure (MS) onto saw-tooth or lamellar-like substrates results in the formation of a multilayer diffraction grating, allowing overcoming the problem of low efficiency even in the classical (in-plane) diffraction geometry case, if the grating grooves and multilayer structure parameters are chosen properly. As an example, lamellar grating demonstrated a peak efficiency of ∼27% at the 2.2 keV photon energy, and it is now used in the DEIMOS beamline of the SOLEIL synchrotron [6].

Blazed multilayer gratings (BMGs), which can achieve the highest theoretical efficiency [7], have been extensively studied. Voronov et al. used the anisotropic wet etching process to fabricate high-efficiency BMGs operating from SXR [8] to extreme ultraviolet region [9] with the maximum efficiency of 52% at 13.5 nm wavelength. The layer growth mechanism on the saw-tooth substrate was specially studied to avoid significant smoothing of the grating grooves [10]. Sokolov and Huang et al. developed BMGs operating in the tender X-ray region, using the well-known ruling technique [11,12]. A record diffraction efficiency of up to 60% was realized in the 3−4 keV spectral range [2], which is over one order of magnitude higher than that of conventional gold-coated blazed gratings. New possibilities can arise with further development of the vacancy epitaxy technology for fabrication of ultrahigh-line-density BMGs (>20,000 line/mm) [13].

The progress of the multilayer grating fabrication technology and the extension of the range of their applications call for further development of theoretical methods for optimizing the grating parameters, which are still far from completion. Conventional approaches for multilayer grating designing involve the use of complex computer programs based on rigorous diffraction theories, such as the coupled wave approach (CWA) [14] or the method of integral equations [15]. Optimization of the grating design requires lengthy computer calculations to compare the effects of different parameters. Moreover, the physical meaning of the obtained results is often not evident.

As an example, it is often argued that the maximal diffraction efficiency of BMG is achieved if the incidence and diffraction angles evaluated from the surface of a blaze facet are equal [16,17]. In other words, the diffraction wave is considered as a result of the specular reflection of the incident wave from the blaze facet. Assuming that the reflection from the multilayer-coated blaze facets occurs under the condition of the first order Bragg reflection, the simplest geometrical considerations result in the well-known condition of maximal diffraction $D\sin \alpha /d = |n|$, where *d* is the MS bilayer thickness, *D* and α are the grating period and blaze angle, respectively, and *n* is the diffraction order. Such a consideration first originated in the theory of traditional grazing incidence blaze gratings operating in the total external reflection (TER) region, where the penetration depth of an incident wave into the grating does not exceed several nanometers. If the grazing incidence angle is less than the anti-blaze angle, the anti-blaze facets are shadowed by the blaze ones, and thus, do not affect the incident beam.

However, the diffraction geometry is quite different for a multilayer-coated blazed grating, where the incoming beam falls onto the BMG outside the TER region and the wave penetrates through the whole MS stack. The beam thus propagates along the grating surface at a distance of several grating periods. Therefore, the interaction of the incident beam with the multilayer-coated anti-blaze facets cannot be neglected, and the condition of the maximal diffraction efficiency might differ from the traditional one. Indeed, there are experimental demonstrations of invalidity of these conventional conditions of maximal diffraction efficiency for BMGs operating in the SXR region. This effect was first observed and discussed in [8] and more recently in [2].

Besides the high efficiency, the most important issue in grating applications is to achieve a high spectral resolution of grating-based monochromators and spectrometers. As discussed in [18], the theoretical resolution of a monochromator operating in a combination of grating and other optical elements (plane and focusing constant mirrors, slits) is determined by the fix-focus constant *C _{ff}* = sin

*θ*

_{n}/sin

*θ*

_{0}(

*θ*

_{0}and

*θ*

_{n}are the grazing incidence and grazing diffraction angles, respectively) [19], which should be as high as possible, without causing essential degradation of the diffraction efficiency. However, the optimization of the BMG design to obtain a high

*C*-value in parallel to the high efficiency has not been well developed yet, especially for a grating operation in a wide spectral region.

_{ff}The present paper mainly aims to develop an analytic theory of X-ray diffraction from BMGs and physically clear methods of their parameter optimization. Particular attention is given to the optimization of BMGs operating in the whole tender X-ray range rather than optimization at a fixed wavelength.

We previously analyzed the optical performance of BMGs using a simplified grating model [7]. We assumed that the subsequent deposition of two materials onto a highly corrugated substrate results in their uniform intermixing above the anti-blaze facets, rather than forming a periodic structure. A sketch of the simplified model is presented in Fig. 1(a). The broken part of the MS gives no contribution to the wave reflection while absorbing the radiation, thus decreasing the BMG efficiency. Such a growth of the MS has been observed in several studies [20,21]. However, the opposite case, where a high-quality MS covers both blaze and anti-blaze facets (Fig. 1(b)), is most frequently observed in practice [2,9,10,21,22].

In section 2 of the present paper, we extend our theory to describe X-ray diffraction from such a BMG. Our analysis is based on the coupled waves approach (CWA), where the wave field inside the grating is represented as a superposition of waves with different diffraction orders. The CWA is a well-suited theory for calculations of multilayer grating performance in the X-ray region because the small polarizability of matter results in very narrow reflection and diffraction peaks and the negligible coupling of waves for the different diffraction orders outside the peaks. If the grating period is small enough, the diffraction peaks are shifted far from each other (in terms of the incidence angle), and multilayer grating operates in the single-order regime, which occurs when the incidence wave effectively excites only a single diffraction order. As a result, the infinite system of differential equations in the CWA reduces to a system of two equations, which can be solved in an explicitly analytical form [7,23,24]. The incident power is distributed over manifold diffraction orders in the multi-order regime, while the power is transformed into that of the single diffraction order resulting in a manifold increase in the diffraction efficiency [25].

In section 3, we discuss the problem of choosing an optimal bilayer thickness that provides the maximal diffraction efficiency at the given wavelength. We demonstrate that the optimal thickness may differ by almost 2 nm from that predicted by the conventional condition. The physical reasons of this fact are also discussed, including the effect of anti-blaze facets to X-ray diffraction and that of strongly asymmetrical diffraction, i.e., when the grazing diffraction angle *θ*_{n} differs essentially from the incidence angle *θ*_{0}, so that the fix-focus constant *C _{ff}* becomes large. Both these factors decrease the grating efficiency.

In section 4, we use the analytic theory to design the BMG for operation in the entire tender X-ray region spanning from 1 to 5 keV photon energy. We deduce very simple relations for immediately estimating the range of possible grating parameters for operation in a given spectral interval, without time-consuming numerical calculations. We demonstrate that a proper choice of the MS bilayer thickness allows designing a single BMG with the diffraction efficiency close to the maximal possible one throughout the tender X-ray range.

In section 5, we demonstrate that a proper choice of the grating parameters and the diffraction order of the grating operation yield a high *C _{ff}* value with high diffraction efficiency, comparable to the maximal possible one in a wide spectral range.

The results of the analytic theory are compared with those obtained with rigorous (in X-ray region) numerical calculations based on CWA throughout sections 3−5. We demonstrate a good agreement between the analytical and numerical calculations. As an additional verification of the analytic theory’s applicability, we demonstrate in section 6 that it quantitatively describes the experimental results obtained previously in [2]. The main results of this paper are summarized in section 7.

## 2. Extended analytic theory of X-ray diffraction from BMG

In this section, we develop an analytic theory of X-ray diffraction from BMG in the frame of an improved BMG model, as shown in Fig. 1(b), where the geometrical parameters of the grating are also indicated.

Some useful relations between the parameters are defined as follows:

*d*

_{α}and

*d*

_{β}indicate the MS periods counted perpendicular to the interfaces and placed above the blaze and anti-blaze facets, respectively;

*h*is the MS period along the vertical Z-axis (the same for multilayers deposited onto both facets);

*H*is the groove height;

*D*

_{α}and

*D*

_{β}indicate the lengths of the blaze and anti-blaze facets along the X-axis, and

*α*and

*β*are the blaze and anti-blaze angles of the groove profile respectively. We also introduce the grating ratio Γ to shorten the formulas below.

Similar to our previous papers [7,23–25], we assume here that an X-ray wave falls onto the grating at the grazing angle θ_{0} counting from the X-axis in clockwise direction and at diffraction angle θ_{n} evaluated in the anticlockwise direction, where *n* is the diffraction order.

The polarizability inside the single grating period ($x \in [ - {D_\alpha },{D_\beta }]$) can be represented as a Fourier series:

*u*

_{j}in the Fourier series has the following form in the simplest case of MS with abrupt interfaces:

_{A}and χ

_{S}indicate the polarizabilities of the absorbing and spacing materials, respectively; γ is the thickness ratio of MS, i.e., the ratio of the absorber thickness to the MS period; and $\bar{\chi }$ is the mean polarizability. The polarizability is expressed via the dielectric constant

*ɛ*as

*χ*= 1–

*ɛ*; thus, |χ|<<1 and Im

*χ*< 0. As in our previous paper, we neglect (cut) the saw-tooth relief on the substrate and grating top, which results in an essential simplification of the analytical formulas. Typically, the height

*H*of the saw-tooth relief does not exceed the thickness of 1−2 bilayers, while the total number

*N*of MS bilayers is several tens or even hundreds, so the small missing material on top of the grating scarcely affects the diffraction efficiency.

Then, we represent polarizability over the whole grating body $({x \in ( - \infty , + \infty ),\;\;z \in [0,L]} )$, where *L = Nh* is the total thickness of the multilayer structure counted along Z-axis, as

*h*is substituted with

*d*/cosα:

Equation (5) can be written in a compact form as

*η*as

*δθ*between the neighboring diffraction peaks, where both $\Delta {\theta _{BMG}}$ and

*δθ*are expressed in terms of the incidence angle. According to [23], “much less” means less by at least a factor of three, while the factor depends on the grazing incidence angle and the spectral range considered. Small polarizability of matter in X-ray region results in very narrow reflection and diffraction peaks, while the reflectivity and diffraction efficiency is very low outside the peaks. Therefore, if peaks are placed far apart, the incident beam may excite effectively the only diffraction order. The incident power thus is not distributed over manifold diffraction orders, but it is transformed into that of the single diffraction wave. Only in this regime, the diffraction efficiency is high and may achieve the reflectivity of the conventional multilayer mirror (MM), if the grating parameters are chosen properly [7,23,24]. Note that according to Eq. (9) the shorter the grating period

*D*is, the simpler it can be to fulfill the condition of the single-order regime. Then, we can directly apply the general formulas deduced in [7], but with the new parameter ${v_{j,\,n}}$ given in Eqs. (5)−(6). Notice that a possibility to increase the diffraction efficiency through the decrease in the grating period was also discussed in [26,27].

The final expression for the *n*^{th}-order diffraction efficiency of the single-order BMG is

*S*characterizes a decrease in the field amplitude with the grating depth, the Bragg parameter

*b*defines the deviation from the diffraction resonance occurring at $\sin {\theta _0} + \sin {\theta _n} \approx j\lambda /h$,

*L = Nh*is the total thickness of MS, and the parameter ${U_ \pm }$ describes the modulation (reflective and diffraction properties) of the structure:

*u*and

_{j}*ν*are determined in Eqs. (3) and (5),

_{j,n}*n*is the order of diffraction,

*j*is the order of Bragg reflection from the MS, and the diffraction angle

*θ*

_{n}is connected to the grazing incidence angle

*θ*

_{0}and the radiation wavelength via the grating equation In case of a semi-infinite MS ($|SL|\; \gg 1$), Eq. (10) reduces to the following simplest form: Then, we obtain the generalized Bragg condition of the maximal diffraction efficiency:

The generalized Bragg condition Eq. (14), together with the grating equation, determines uniquely the incidence and diffraction angles *θ*_{0} and *θ*_{n}, respectively, corresponding to the maximal *n*^{th}-order diffraction efficiency at a fixed wavelength. Both equations are invariant with respect to the replacement ${\theta _0} \to \pi - {\theta _n}$ and ${\theta _n} \to \pi - {\theta _0}$, and thus, establish conditions of resonant diffraction for both direct and inverse (expanding and compressing) diffraction geometries.

If condition Eq. (14) is fulfilled, we deduce a very simple expression for the peak value of the *n*^{th}-order diffraction efficiency:

*y*is represented as a product of three multipliers and the parameters

*f*and

*g*are completely determined by the polarizability of the MS materials:

*y*.

The first parameter *P*_{1} depends only on the MS composition and achieves the maximal value if the thickness ratio *γ* obeys the same equation $\tan (\pi j\gamma ) = \pi j(\gamma + g)$ as that appearing in the theory of conventional MMs [28].

The second parameter *P*_{2} is dictated by a diffraction asymmetry, and it depends on the MS and grating periods (*h* and *D*) via the grating equation and the Bragg condition of resonance diffraction Eq. (14), while it is independent of the grating type. *P*_{2} is close to unity if the incidence and diffraction angles are close to each other. *P*_{2} is related to the fix-focus constant ${C}_{ff} = \sin {\theta _n}/\sin {\theta _0}$ as

*C*-factor value, i.e., enhancement of the resolving power of the grating monochromator or spectrometer [18], results in reducing parameter

_{ff}*P*, i.e., decreasing diffraction efficiency. This is a general feature of multilayer grating of any type.

_{2}Only the third parameter ${P_{\,3}} = |{{\nu_{j, - n}}} |$ depends on the shape of grating grooves, and thus, dictates specific features of different multilayer gratings [7,23,24].

Assuming that the effect of absorption is small, the angular width of the diffraction peak in terms of the incidence angle can be expressed as

*h*[7]. By convention, we determine the total thickness of MS (along the Z-axis) necessary to achieve the saturated (maximal) value of the diffraction efficiency as

*L*= 1/

*S*determines the depth, where the incidence wave amplitude decreases by

*e*times under the condition of resonance diffraction Eq. (14) and the numerical coefficient 2.65 is introduced to provide the diffraction efficiency Eq. (10) of ∼0.97−0.98 of the ultimate value Eq. (15). If we put numerical coefficient in Eq. (19) equal to unity, the peak efficiency is only of order of 0.7 of the ultimate value Eq. (15).

All expressions deduced above are also valid for p-polarized radiation if we replace the MS modulation parameter ${U_ \pm }$ in Eqs. (10)−(11) by ${U_ \pm }\cos ({\theta _0} + {\theta _n})$, the parameter *y* in Eqs. (15) and (19) by $y\cos ({\theta _0} + {\theta _n})$, and the polarizability variation at interface ${\chi _A} - {\chi _S}$ in the generalized Bragg condition Eq. (14) by $({\chi _A} - {\chi _S})\cos ({\theta _0} + {\theta _n})$. This means that the diffraction efficiency of p-polarized radiation approaches zero if the diffracted beam propagates perpendicularly to the incident one. A more detailed consideration is given in [23].

Finally, if we put the blaze angle α = 0 and the grating ratio Γ = 1, Eqs. (10)−(19) are transformed into the well-known formulas describing X-ray reflection from the conventional MM [28]. Equations (10)−(19) allow us to design BMG and analyze its optical performance in a very simple manner.

Below, for definiteness, we will consider two BMGs operating in the tender X-ray range (photon energy *E* = 1–5 keV). The first one, denoted as G0, is characterized by 2400 line/mm grove density, i.e., the grating period *D* = 416.67 nm, blaze angle *α* = 0.98°, and anti-blaze angle *β* = 90°. The second grating G1 has the same values of *D* and *α* and a small anti-blaze angle *β* = 2.3°, so Г ≈ 0.7. Both gratings are covered by Cr/C MS (the thickness ratio $\gamma = {d_{Cr}}/d = 0.4$ is fixed) assuming that the number of bilayers *N* is large enough for obtaining a saturated value of the diffraction efficiency and applying Eqs. (14)−(18) for analysis of their optical performance. In the calculations, the optical constants of materials are taken from [29]. A grating similar to G1 (the same D, α, β, and Cr/C multilayer coating) was fabricated and studied experimentally in [2], while grating G0 is presented for comparison as it can be considered ideal for achieving the maximal diffraction efficiency due to the absence of anti-blaze facets.

First, we need to check the validity of the single-order regime condition for the studied gratings. We calculate the diffraction efficiency of different orders for grating G1 versus the grazing incidence angle for two photon energies *E* = 1 keV and 5 keV by substituting the bilayer thickness *h* = 6.16 nm and 5.33 nm, respectively. As we will discuss below, these values of *h* are optimal for achieving the maximal -1^{st} order peak efficiency of grating G1.

The results of the calculations performed with analytical Eqs. (13)−(14) for *E *= 1 keV (colored solid curves) are shown in Fig. 2(a). In addition, the diffraction efficiency is calculated numerically by using CWA [14], where 11 diffraction orders (from *n* = -5 to + 5) are taken into account, assuming Cr or C layer to be placed on the MS top (colored symbols or black dotted curves in Fig. 2(a), respectively). It is seen that the diffraction peaks are rather far from each other, so in the narrow angular range θ_{0} ∼ 5.35 - 5.45 degrees, where the -1^{st}-order diffraction peak is high (∼15%), the other orders’ diffraction efficiency is extremely low (<0.1%). Therefore, if the X-ray wave falls onto the grating within this angular interval, only the (-1^{st})-order diffraction wave is effectively excited, so the condition of the single-order regime is fulfilled. In addition, the analytical calculations agree well with the numerical ones, while the analytic equations are independent of the material placed on the MS top. Typically, the analytic curve lies between two curves calculated with CWA assuming Cr or C to be placed on the MS top.

The angular distance between the diffraction peaks, ∼*h*/*D*, is independent of the wavelength to the first approximation, while the angular width of the peaks decreases proportionally to λ^{2} (Eq. (18)). Therefore, the conditions of the single-order regime are satisfied within much higher accuracy at an increasing photon energy calculated at *E* = 5 keV, as shown in Fig. 2(b). The results of numerical calculations are not shown in Fig. 2(b) because the curves are indistinguishable in the graph.

For comparison, the reflectivity of the conventional Cr/C MM with bilayer thickness *h* is also shown in Fig. 2. As seen, the -1^{st} order diffraction peak is essentially lower than the MM reflectivity (by a factor of about two) at *E* = 1 keV, while they are quite comparable at *E* = 5 keV. This fact can be explained as follows.

The peak value of the diffraction efficiency Eqs. (15)−(16) is determined by the grating parameters via the product of the geometrical factors, ${P_2} \cdot {P_3}$. In addition, the efficiency depends on the optical constants of the materials via the parameters *f* and *g*. Following [7], we can find from Eqs. (15)−(16) the universal relation between the peak efficiency and decreasing product ${P_2} \cdot {P_3}$ caused by a variation in the grating parameters. The relation is demonstrated by Fig. 3 for two values of the energy *E* = 1 keV and 5 keV. The parameter $|f|$ is quite different and equal to ∼1.7 and 24, respectively, because of the sharp decrease in X-ray absorption with increasing photon energy. The efficiency is normalized to the reflectivity of the conventional MM at ${P_2} \cdot {P_3} = 1$.

Figure 3 clearly demonstrates that the decrease in the diffraction efficiency caused by the reduced value of ${P_2} \cdot {P_3}$, as compared to the conventional MM reflectivity, occurs considerably quicker at *E* = 1 keV when the parameter $|f|$ is small. In particular, decreasing the product ${P_2} \cdot {P_3}$ from 1 to 0.7 results in an efficiency drop by a factor of ∼2 at *E* = 1 keV, while the efficiency decreases by a factor of only 0.9 at *E* = 5 keV. The maximal value of parameter *P*_{3} is just ∼0.7 for grating G1.

The physical reason for this fact is explained by Eq. (19). Indeed, for a large parameter $|f |\cdot y\, \gg 1$, the denominator in Eq. (19) is proportional to ${\mathop{\rm Im}\nolimits} \bar{\chi } \cdot |f|\cdot y \,\sim \,|{\mathop{\textrm {Re}}\nolimits} ({\chi _A} - {\chi _S})|$, and as a first approximation, is independent of the radiation absorption determined by ${\mathop{\rm Im}\nolimits} \bar{\chi }$. The penetration depth of the incident wave is thus limited by the interference of waves reflected from different interfaces rather than the absorption effect. A decrease in parameter $|f| y$ results in reducing the interference effect and enhancing the absorption. Finally, when the parameter decreases down to $|f| y\sim 1$, the penetration depth of the incident wave is limited by absorption in a large part.

This consideration is fully applicable for conventional MMs. Indeed, Fig. 2 demonstrates that the Cr/C MM reflectivity is very high (∼91%) at *E* = 5 keV (parameter $|f |\cdot y\, \gg 1$, absorption effect is small) and much lower (∼28% only) at *E* = 1 keV ($|f| y\sim 1$, absorption effect is much larger). This situation is adversely affected for BMG, because the product ${P_2} \cdot {P_3}$ is always less than unity, which results in additional reduction in the interference effect and enhancement of absorption as compared to the MM case. An increase in absorption with decreasing ${P_2} \cdot {P_3}$ is far more pronounced at *E* = 1 keV, because parameter $|f| y\sim 1$ is small even for the conventional Cr/C MM (the case of ${P_2} \cdot {P_3} = 1$). The main adverse factor resulting in reducing the interference and increasing the absorption in grating G1 is the effect of anti-blaze facets. This effect disappears for grating G0, where the maximal value of parameter *P*_{3} is equal to unity, resulting in the G0 grating efficiency approaching MM reflectivity if the parameter *P*_{2} is close to unity.

Figure 3 indicates that the grating efficiency is always lower than the reflectivity of the conventional MM but can be close to it if the grating parameters are chosen properly. The problem of optimal choice of the grating parameters is discussed in the next sections.

## 3. Maximal diffraction efficiency: choice of optimal bilayer thickness

The first problem, which we will analyze further, is the choice of the optimal MS period providing the maximal value of diffraction efficiency. The problem proves to be not trivial as it may appear at the first glance.

Our goal involves the determination of the MS bilayer thickness *h* providing the maximal value of the parameter $y = {P_{\,1}} \cdot {P_{\,2}} \cdot {P_{\,3}}$ in Eq. (16), where *P*_{1} is independent of *h*. Parameter ${P_{\,3}}(h) = |{{\nu_{j, - n}}} |$ defined by Eq. (6) for G0 grating is shown in Fig. 4(a) at E = 1 keV versus the MS period *h*. It achieves the maximum value equaled to unity, if

*h*= 7.1 nm if |

*n*| =

*j*= 1. Condition Eq. (20) means that an integer number of bilayers equaled to the diffraction order |

*n*| (at

*j*= 1) are placed inside the saw-tooth facet, so that the shift Δ

*d*between the interfaces in the neighboring grooves shown in Fig. 1(a) is equal to zero. The well-known Eq. (20) is often considered in the literature [16,17] as the necessary condition to achieve the maximal diffraction efficiency for all BMGs.

The parameter *P*_{3} in Eq. (5) is more complex for G1 grating and contains two summands, demonstrating the contribution of both grating facets to the diffraction efficiency due to the interference of the diffracted waves. The parameter is shown in Fig. 4(b) versus the MS period (at *E* = 1 keV). In addition, the modulus of each summand in Eq. (5) is shown separately (curves 1 and 2). Curve 1 demonstrates that the contribution of the blaze facets to *P*_{3} is similar to that of curve *P*_{3} in Fig. 4(a), while its maximum is decreased by a factor of $\Gamma \approx 0.7$. The effect of anti-blaze facets (curve 2) on the maximal value of parameter *P*_{3} is insignificant, which results in a certain shift of the optimal period *h* to about 6.2 nm. Parameter *P*_{3} is independent of the wavelength, and thus, remains the same in the case of the gratings operation at 5 keV.

Using Eq. (7), we find that the optimal bilayer thickness (h_{0}) is changed as compared to Eq. (20), due to the effect of anti-blaze facets, and is expressed as

_{n}tends to unity with increasing diffraction order |

*n*|, so that η

_{±1}= 0.871, η

_{±2}= 0.958, η

_{±3}= 0.971, etc., if the grating ratio Γ = 0.7. Thus, condition Eq. (20) for the optimal MS period is not fully correct, which negates the contribution of the antiblaze facets to X-ray diffraction. The value of parameter

*P*

_{3}, and hence, the diffraction efficiency, is changed insignificantly with varying the MS period from 7.1 to 6.2 nm at

*E*= 1 keV (Figs. 4(a) and 4(b)).

However, there is another factor, parameter *P*_{2}, which affects the optimal bilayer thickness and diffraction efficiency far more. It depends on the MS period *h* via the generalized Bragg condition of resonance diffraction Eq. (14). Hence, variation in the period results in changing *P*_{2}, and thus, varying the diffraction efficiency. We consider the case of a small ${\theta _0},\;{\theta _n} \ll 1$ (in radians), while assuming that both angles exceed the critical angle of TER, so that the refraction effect can be neglected to a first approximation. Then, the grating equation and the generalized Bragg condition of diffraction are written as

*P*

_{2}determined in Eq. (16) is transformed to the simplest form, which, nevertheless, is suitable for preliminary qualitative analysis The simplified function

*P*

_{2}(

*h*) is shown in Fig. 4. The product

*P*

_{2}

*P*

_{3}, which determines the diffraction efficiency, is also shown in Fig. 4. As seen, the optimal

*h*providing the maximal diffraction efficiency is essentially changed down to 5.3 nm for grating G1 at

*E*= 5 keV (Fig. 4(d)) and to 5.6 nm for grating G0 at the same photon energy (Fig. 4(c)). According to the traditional concept (Eq. (20)), the optimal bilayer thickness should be 7.1 nm, i.e., 1.5−1.8 nm larger. The effect of

*P*

_{2}on the optimal

*h*is insignificant at

*E*= 1 keV due to the larger wavelength in Eq. (24) (Figs. 4(a) and 4(b)). Thus, the conventional condition Eq. (20) can be incorrect even for an ideal BMG G0, due to the effect of

*P*

_{2}. An essential shift in the optimal bilayer thickness is only observed if the

*C*-value exceeds unity essentially, i.e., the effect is a consequence of strongly asymmetric diffraction and is not caused by specific geometrical properties of a multilayer-coated grating. Thus, the notion of the specular reflection of X-rays from the blaze facets cannot be applied to the analysis of the maximal diffraction efficiency of a multilayer-coated BMG, which thus is not totally the same as the blaze grating in the traditional understanding of this word.

_{ff}Next, Eq. (24) demonstrates a critical value of the MS period

when the parameter*P*

_{2}achieves zero value. Physically, this means that increasing the MS period

*h*above

*h*

_{max}results in an impossibility in obeying the grating equation and the Bragg condition of the resonance diffraction simultaneously, because either incidence angle

*θ*

_{0}or diffraction angle

*θ*

_{n}becomes negative. For gratings G0 and G1, the critical MS period ${h_{\max }} \approx 7.2$ or 16.1 nm at the photon energy

*E*= 5 or 1 keV and |

*n*| =

*j*= 1.

However, a sharp decrease in the diffraction efficiency occurs at an MS period smaller than *h*_{max}. According to Eq. (23), an increase in period *h* results in a decrease in the resonance angle *θ*_{0} (at negative diffraction order *n*). When *θ*_{0} approaches the critical angle of TER, ${\theta _c} = \sqrt {{\mathop{\textrm {Re}}\nolimits} \bar{\chi }}$, the specular reflected wave (0^{th} diffraction order) is effectively excited and takes away an essential part of the incident radiation power. Consequently, the diffraction efficiency decreases sharply, while our analytical approach does not consider an exiting specular reflected wave. Therefore, we determine the critical MS period as ${h_c} < {h_{\max }}$ assuming that the optimal grazing incidence angle *θ*_{0} (the case of negative *n*) is decreased down to *θ*_{c} at *h* = *h*_{c}. Its value proves to be equal to *h*_{c} = 6.4 nm at E = 5 keV and 12.6 nm at E = 1 keV. For clarity, all parameters introduced above, as well as the optimal bilayer thickness *h*_{opt} = 5.3 nm, are shown in Fig. 4(d) for grating G1 operating at E = 5 keV.

To illustrate the foregoing considerations of the impact on the efficiency of different grating parameters, as well as to demonstrate the accuracy of the analytic expressions, several examples are presented below.

Figure 5 shows the -1^{st} order diffraction efficiency of G0 and G1 gratings versus the grazing incidence angle at a fixed photon energy *E* = 1 keV. Calculations are performed with different MS bilayer thicknesses indicated (in nm) in the graphs by numbers above the diffraction peaks. We use both the simple analytic formulas deduced above (Eqs. (10)−(11), colored curves) and the numerical calculations based on CWA (black dashed curves), where 11 diffraction orders (from -5^{th} to + 5^{th}) are taken into account.

First, according to the previous analysis, the optimal bilayer thickness of grating G0 is equal to ∼7.1 nm and is close to the value predicted by Eq. (20), because of the minor effect of *P*_{2} at *E* = 1 keV. The optimal thickness is shifted down to 6.2 nm (Eq. (21)) for grating G1 due to the effect of the anti-blazed facets. The diffraction efficiency changes weakly with the bilayer thickness varied from 7.1 to 6.2 nm.

Second, the efficiency of grating G1 is reduced by a factor of almost two as compared to grating G0. This is explained by the fact that the maximal value of parameter *P*_{3} is equal to 0.7 for grating G1 instead of 1 for grating G0. According to Fig. 3, even a small decrease in the *P*_{3} value results in an essential drop in the efficiency at *E* = 1 keV. A quicker decrease in the peak efficiency for grating G0 at $h \le 5$ nm as compared to grating G1 is also explained by Fig. 4 owing to a sharper drop in parameter *P*_{3} with decreasing *h* for G0.

Third, if the bilayer thickness is not very large, so that the diffraction peak is placed far from the TER region (the critical angle ${\theta _c} = \sqrt {\bar{\chi }} = {2.30^ \circ }$ at *E* = 1 keV, where $\bar{\chi }$ is the average polarizability of multilayer structure), the agreement between the analytic and numerical calculations is excellent. However, the accuracy of the analytic formulas becomes worse when the diffraction peak approaches the TER region, as seen for the efficiency of the grating with *h *= 8 nm. The reasons causing inaccuracies of the analytic approach are discussed in more detail in [24].

The same conclusions can be drawn based on the analysis of diffraction peaks at *E* = 5 keV (Fig. 6). The agreement between the analytic and numerical calculations is excellent at *h* ≤ 5 nm, and becomes worse when the diffraction peak approaches the TER region. The critical angle of TER is equal to 0.455° at *E* = 5 keV. The difference between the curves calculated analytically and numerically at *h* = 6 nm is explained by the interrelation between the -1^{st} and 0^{th} order diffraction waves inside the multilayer grating. Note that the 0^{th} order diffraction efficiency (not shown in the graph) is 10% to 3% in the 0.5° to 0.65° interval of the grazing incidence angle, which results in a 5% decrease in the -1^{st} order diffraction efficiency.

The optimal bilayer thickness is only 5.6 nm for grating G0 and 5.3 nm for grating G1 at 5 keV. If, according to the traditional concept (Eq. (20)), the bilayer thickness is set to 7.1 nm, the -1^{st} diffraction peak will be shifted to the TER region and its efficiency will decrease down to a very low value of ∼16%. This is because the 0^{th} order diffraction wave is effectively excited with the efficiency exceeding 50%, i.e., the main part of the incident power is transformed to that of the specularly reflected wave. The -1^{st} and 0^{th} diffraction efficiencies at *h* = 7.1 nm are calculated numerically because the analytic approach is invalid inside the TER region. This is the main shortcoming of the analytic approach, which, nevertheless, is quite suitable for the preliminary optimization of the BMG design.

In contrary to Fig. 5, the diffraction efficiency of grating G1 is only slightly lower than that of grating G0. This is explained by Fig. 3 demonstrating a very slow decrease in efficiency with decreasing product *P*_{2}*P*_{3} at *E *= 5 keV.

## 4. Optimization of highly efficient BMG operating at different orders in the entire tender X-ray range

We demonstrated above that, first, the maximal value of parameter *P*_{3} is achieved if the bilayer thickness *h*_{0} obeys Eq. (21), and second, the parameter *P*_{2} achieves the zero value at the critical bilayer thickness *h*_{max} determined by Eq. (25), so that the diffraction efficiency drops sharply at larger *h*. Therefore, a high efficiency can only be achieved if *h*_{0} < *h*_{max}, i.e., the maximum of *P*_{3} is placed from the left side of the critical thickness *h*_{max}. Meanwhile, the minimal bilayer thickness *h*_{min} is limited by technological possibilities, which means *h*_{0} > *h*_{min}. Then, using Eqs. (21) and (25), we find that the following conditions should be fulfilled:

*λ*

_{min}= 0.043 nm if

*h*

_{min}= 3 nm and |

*n*| =

*j*= 1. The value of

*λ*

_{min}is proportional to the diffraction order |

*n*| and inversely proportional to the grating period

*D*, i.e., the shorter the radiation wavelength is, the longer should be the grating period. This statement, seemingly paradoxical at the fist glance, is the direct consequence of the grating Eq. (12) considered together with the Bragg resonance condition Eq. (14) to achieve a reasonable efficiency.

Using Eq. (26) and putting λ = 0.248 nm (*E* = 5 keV), *h*_{min} = 3 nm, *D* = 416.67 nm, and *j* = 1, we find the following operating ranges of the blaze angle α:

^{1/2}with increasing wavelength. Therefore, grating G1 (α = 0.98°) can operate in the whole X-ray tender range (

*E*= 1−5 keV, λ = 1.24−0.248 nm) using the first and second diffraction orders only. The left inequality breaks down for grating G1 at |

*n*| = 3 as the optimal bilayer thickness becomes less than

*h*

_{min}= 3 nm.

Figure 7 represents a graphical illustration of these considerations. The product *P*_{2}*P*_{3} is shown to be dependent on the MS bilayer thickness. The parameter *P*_{2} is calculated via Eq. (16) and cut at the value of *h* = *h*_{c} corresponding to the wave incidence at the critical angle of TER.

The maximal values of product *P*_{2}*P*_{3} are close to each other for all three diffraction orders at *E* = 1 keV. Therefore, the peak values of the diffraction efficiency are practically the same (Fig. 8(a)), while the number of bilayers necessary for achieving the saturated value of the diffraction efficiency (Eq. (19)) is increased with the diffraction order |*n*|, because of the smaller optimal bilayer thickness, thus increasing the grazing incidence angle. A decrease in the angular bandwidth of BMG with increasing diffraction order |*n*| is explained by a decrease in the bandwidth of the conventional MS with increasing Bragg angle *θ*_{B} (the second expression in Eq. (18)). The peak value of product *P*_{2}*P*_{3} increases with the diffraction order |*n*| increasing at *E* = 5 keV, because of the increasing value of *P*_{2} at *h* = *h*_{opt}. However, the efficiency dependence on *P*_{2}*P*_{3} is weak according to Fig. 3, so the peak efficiency is almost the same for different orders |*n*| (Fig. 8(b)). However, the high efficiency of the -2^{nd} and -3^{rd} diffraction orders requires a larger total thickness of MS *L*_{sat} than that for the -1^{st} order, which can bring more difficulty to the fabrication. Therefore, the use of the (-1^{st}) diffraction order is preferable if the grating efficiency is a crucial parameter.

Table 1 shows the geometrical and optical grating parameters for two photon energies *E* = 1 and 5 keV and for different diffraction orders. It also demonstrates an invalidity of the traditional assumption of the specular reflection of an incident beam from the blaze facets under conditions of maximal diffraction efficiency. As an example, in the case of the grating optimized to the maximal (-1st) order efficiency at E = 5 keV, the grazing incidence angle on the blaze facet ${\theta _0} + \alpha \approx {1.71^ \circ }$ is 1.5 times more than the grazing reflection angle ${\theta _{ - 1}} - \alpha \approx {1.13^ \circ }$.

Despite the similar theoretical efficiency of the different orders, the impact of interfacial roughness on the diffraction efficiency increases with decreasing bilayer thickness, i.e., with increasing diffraction order (Table 1). The roughness effect can be easily analyzed in the limiting case of the delta-correlated roughness with negligible correlation length in both interface plane and vertical direction (along Z-axis in Fig. 1). In this case, the roughness effect is equivalent to that of a smooth transition layer formed at the interfaces and aroused as a result of statistical averaging of roughness. In the particular case of Gaussian law of the roughness height distribution, the coefficient *u*_{j} in the Fourier series Eq. (3), as well as in the final Eqs. (10)−(13), is replaced by [30]

*h*is negligible. The dashed curves in Fig. 8 illustrate the roughness effect on the diffraction efficiency of different orders. They are calculated using Eq. (10) taking into account Eq. (29) and setting σ = 0.3 nm. The dashed curves are shifted to right for clarity. As seen, the roughness effect is essentially weaker at higher photon energy. The fact is that the delta-correlated roughness does not scatter an incidence radiation, but increases its transmittance into the MS depth, because of decreasing modulation of the MS (Eq. (29)) [31]. Consequently, first, the angular bandwidth of BMG is decreased, and second, the roughness effect can be partially compensated by increasing the number of MS bilayers in the case of low radiation absorption (high photon energy). Figure 8 demonstrates onсe more that, if the grating efficiency is the crucial parameter, the use of the -1

^{st}diffraction order is preferable, especially in the low-energy part of the tender X-ray range. The values of the diffraction efficiency shown in Table 1 are the maximum possible ones because the optimal bilayer thickness is individually chosen for a particular photon energy. The maximal values of (-1

^{st}) diffraction efficiency are shown in Fig. 9(a) (green stars). Number of bilayers

*N*= 100 is enough to obtain the saturated value of the efficiency.

To design a BMG operating in the entire tender X-ray range, we choose a particular value of the bilayer thickness *h* = 6 nm keeping the same *N* = 100. This value of *h* is close to the optimal one at *E* = 1 keV, while slightly differing from that at *E* = 5 keV. As the dependence of the efficiency on parameter *P*_{2}*P*_{3} is weak in the high-energy region (Fig. 3), the -1^{st} order diffraction efficiency will not differ strongly from the maximal achievable one. Indeed, curve 2 in Fig. 9(a), calculated at a fixed *h* = 6 nm, coincides practically with curve 1. The maximal difference between the curves is achieved at *E* = 5 keV and is less than 1%.

The chosen number of bilayers *N* = 100 is too large in practice, because of the saw-tooth profile smoothening with MS growth. If we decrease *N* down to 35, the diffraction efficiency is only slightly decreased at a high photon energy (curve 4 in Fig. 9(a)), because the saturated number of bilayers (Eq. (19)) is changed from *N*_{sat} = 35 at *E* = 1 kev to 48 at *E* = 5 keV. Note that grating G1 with the deposited 35 Cr/C bilayers was successfully fabricated and studied in [2].

So far, we have analyzed the problem of choosing grating parameters to obtain the maximal diffraction efficiency. However, as mentioned above, the *C _{ff}*-factor should be as high as possible to achieve a high spectral resolution of grating-based monochromators and spectrometers, without essential degradation of the diffraction efficiency.

Table 1 demonstrates that the *C _{ff}*-value is decreased with increasing diffraction order, which is inconsistent with the theory of conventional incidence gratings. However, this fact can be easily explained based on our simplified analysis. Indeed, Eq. (23) shows that

*C*-factor is determined by the product |

_{ff}*n*|

*h*

^{2}. If the bilayer thickness

*h*is fixed, the

*C*-value is indeed increased with |

_{ff}*n*|. However, the optimal bilayer thickness providing the maximum efficiency decreases sharply with the diffraction orders (Table 1). As a result, the product |

*n*|

*h*

^{2}decreases with increasing |

*n*|, leading to a decrease in the

*C*-value indicated in Table 1. In other words, it is not possible to obtain the maximal efficiency and high

_{ff}*C*factor simultaneously in this case.

_{ff}According to Eq. (17), an increase in the *C _{ff}*-value can only be provided by decreasing parameter

*P*

_{2}, which results in decreased efficiency. Indeed, if we choose the bilayer thickness

*h*= 6 nm, the

*C*-value for the -1

_{ff}^{st}order is somewhat increased in the high-energy part of the tender range (curve 2 in Fig. 9(b)), because of increasing

*h*above the optimal value.

A further increase in the *C _{ff}*-value is possible if we use the -2

^{nd}order diffraction and a larger

*h*= 4.45 nm, compared to the optimal values indicated in Table 1. Then, the

*C*-value is enhanced up to 5.1 at

_{ff}*E*= 5 keV (curve 3 in Fig. 9(b)), while the diffraction efficiency shown in Fig. 9(a) is modestly decreased. This benefits from the slow variation in efficiency with d-spacing at 5 keV. The number of bilayers is set to

*N*= 200 (curve 3), which is sufficient for obtaining the saturated value of efficiency, and to

*N*= 47 (curve 5), resulting in the same total MS thickness of ∼210 nm as that for the (-1

^{st}) order curve 4.

In the current design, the use of the (-3^{rd}) diffraction order does not generate a higher *C _{ff}* than the (-1

^{st}) order at

*h*= 6 nm, because an increase in the bilayer thickness will cause a dramatic decrease in the -3

^{rd}order diffraction efficiency.

Therefore, we analyze the BMG optical performance by changing the bilayer thickness *h* and keeping the blaze angle α constant. We have discussed how the optimal bilayer thickness should be chosen to provide the maximal diffraction efficiency. However, the problem of the increasing *C _{ff}* factor has not been appropriately solved, especially in the low energy part of the tender X-ray range.

## 5. Maximal C_{ff} and high diffraction efficiency: choice of optimal grating parameters and operation diffraction order

In this section, we demonstrate that a proper choice of the grating parameters (blaze angle, bilayer thickness, and grating period), as well as the operation diffraction order, gives a possibility to essentially increase the *C _{ff}* factor in the operating spectral range and maintain high enough diffraction efficiency.

The diffraction efficiency is determined by the product *P*_{2}*P*_{3}, which thus should be as large as possible. However, a high fix-focus constant *C*_{ff} requires a decrease in parameter *P*_{2}. Therefore, we should take the grating parameters that provide the maximal value of *P*_{3}, i.e., choose the blaze angle according to Eq. (21):

*C*

_{ff}-value. As Eqs. (21) and (31) are independent of the wavelength, the optimal blaze angle is the same in the entire spectral range of the grating operation.

Next, the simplified Eq. (24) shows that *P*_{2} is decreased with increasing bilayer thickness *h* at a fixed grating period. Therefore, we should choose the maximal possible bilayer thickness *h* to obtain a high *C*_{ff}-value. As discussed above, the maximal value of *h = h _{c}* is determined under the condition that the grazing incidence angle

*θ*

_{0}achieves the critical angle of TER

*θ*

_{c}; a further increase in

*h*results in a sharp drop in the diffraction efficiency. Using the simplified Eq. (23) and putting the Bragg reflection order

*j*= 1, we find that

*θ*

_{c}(λ) is proportional to λ, the bilayer thickness ${h_c}(\lambda )$ is monotonically decreased with decreasing λ. Therefore, if we need to optimize BMG parameters for operation in the whole X-ray tender range and obtain the maximal

*C*

_{ff}-value in parallel to the high diffraction efficiency, we should choose the bilayer thickness as the optimal one for the shortest wavelength ${h_{opt}} = {h_c}({\lambda _{\min }})$, where

*λ*

_{min}= 0.248 nm (

*E*= 5 keV) in our case. Note again that an increase in the bilayer thickness above the indicated value excites the specular reflected wave in the high-energy part of the tender X-ray range, and thus, produces a sharp decrease in the efficiency of the diffracted wave, which is not interesting for the grating applications.

Based on the simplified qualitative analysis performed above, we suggest the following general procedure for the grating parameter optimization. For definiteness, we fix the grating ratio Γ = 0.7 and the Bragg reflection order *j* = 1.

First, we put the grazing incidence angle ${\theta _0} = {\theta _c}(\lambda = 0.248\textrm{ nm}) = {0.455^ \circ }$ and the parameter ${P_3} = |{\nu _{j, - n}}|$ into the maximal value coinciding practically with Г. Using the grating equation, we determine the value of the diffraction angle *θ*_{n} depending on the ratio *D*/|*n*|. Then, the right side of the generalized Bragg condition Eq. (14) depends only on the grating parameter *D*/|*n*| via the angle *θ*_{n}, and thus, we find the universal dependence of the optimal bilayer thickness *h*_{opt} on parameter *D*/|*n*|. The dependence is shown in Fig. 10(a), curve 1.

Second, using Eq. (31), we determine the product *η*_{n}tan(*α*_{opt}) that provides the maximal value of ${P_3} \approx \Gamma $, shown in Fig. 10(b), curve 1, versus the same parameter *D*/|*n*|. Note that the optimal blaze angle α_{opt} depends not only on the value of *D*/|*n*| but also on the diffraction order *n* via the parameter *η*_{n}. The obtained values of *h*_{opt} and *α*_{opt} are the optimal ones because it is impossible to shift up the whole curve *C*_{ff}(*E*) without an essential decrease in the efficiency.

As an example, we first consider the grating with a groove density of 2400 line/mm operating in the whole X-ray tender range (*E* = 1–5 keV) in the -1^{st} diffraction order, i.e., *D*/|*n*| = 416.67 nm. Figure 10 shows the optimal parameters *h*_{opt} = 6.40 nm and α_{opt} = 1.01°, which totally correspond to the experimental parameters of the best grating studied in [2], i.e., *h* = 6.3–6.4 nm and α = 0.96–0.99°, while optimization of the grating is performed in [2] using time-consuming numerical calculations. The optimal bilayer thickness *h*_{opt} differs slightly from that (∼6.16 nm) obtained in section 3, where optimization is performed to achieve the maximal possible diffraction efficiency and the *C*_{ff} factor is not considered.

As the optimal grating parameters *h*_{opt} and *α*_{opt} are determined by Fig. 10, we calculate both the diffraction efficiency and the *C*_{ff}-value for several gratings with different periods *D* operating in different diffraction orders *n*. Their parameters are presented in Table 2. The calculated efficiencies and *C*_{ff} factors are shown in Figs. 11(a) and 11(b). In addition, the saturated thickness of MS (Eq. (19)), another important parameter for grating fabrication, is presented in Fig. 11(c). The curve number in Fig. 11 corresponds to the line number in Table 2.

First, we note that the right part of the generalized Bragg condition Eq. (14), as well as *h*_{opt}, depends on the only geometrical parameter *D*/|*n*|. Hence, the resonance angles *θ*_{0} and *θ*_{n} are determined uniquely by the same parameter at any wavelength. Consequently, both the efficiency R_{n}(*E*) and fix-focus constant *C*_{ff}(*E*), as well as the wave penetration depth into MS, are the same for gratings with the same value of *D*/|*n*|. In particular, gratings with the parameters shown in lines 2−3 or 4−6 are characterized by the same efficiency and *C*_{ff}-factor in the whole X-ray tender range.

To check this statement, we calculate numerically with CWA the efficiency and *C*_{ff}-factor for two gratings with different geometrical parameters, but with the same ratio *D*/|*n*| = 200 nm (lines 2 and 3 in Table 2). The results are shown as circles in Fig. 11. As seen, the analytic calculations agree well with the numerical ones, except for the high-energy part of the tender X-ray range, where excitation of the 0^{th} order diffraction wave results in a sharp decrease in efficiency.

The *C*_{ff}-factor at the high-energy edge of the operating interval is equal to

*E*=

*E*

_{max }= 5 keV from 4.5 to 8.9 with decreasing ratio

*D*/|

*n*| from 400 to 100 nm (Fig. 11(b), gratings 1 to 6).

However, the *C*_{ff}-value is very low at the low-energy edge of the operating interval *E* = *E*_{min }= 1 keV, and is changed from 1.3 to 1.4 only when the ratio *D*/|*n*| is decreased from 400 to 100 nm (Fig. 11(b), gratings 1 to 6). This fact is easily explained by the simplified Eq. (30). Indeed, the parameter $|n|h_{opt}^2/D$ in Eq. (30) is small and is changed in a very narrow interval from 0.099 to 0.111 nm with *D*/|*n*| decreasing from 400 to 100. Considering that λ_{min} = 1.24 nm is much larger, we conclude that *C*_{ff} is close to unity (∼1.4) and almost unchanged. The only way to increase the *C*_{ff}-factor is to essentially increase the bilayer thickness as compared to the values shown in Fig. 10(a). However, the diffraction efficiency will be sharply decreased in the high-energy part of the operating interval in this case. Therefore, it is impossible to obtain high efficiency compared to the MM reflectivity and high *C*_{ff}-value in the whole X-ray tender range spanning from 1 to 5 keV.

Suppose now that the grating operating interval is narrower, so that *E*_{max} = 2.5 keV. Performing the procedure described above, we calculate universal dependencies of the grating parameters on *D*/|*n*| shown in Fig. 10, curves 2. As seen, the value of $h_{opt}^2$ is essentially increased. Consequently, the *C*_{ff}-factor is enhanced up to a practicably suitable value of 2 in the low-energy edge of the working interval (Fig. 11(b), gratings 7 to 12), while the diffraction efficiency is somewhat decreased (Fig. 11(a)) because of decreasing parameter *P*_{2}.

Further narrowing of the grating operating interval down to *E*_{max} = 1.5 keV results in increasing the *C*_{ff}-factor up to 3−3.5 at the low-energy edge (Fig. 11(b), gratings 13−16) with a simultaneous decrease in the diffraction efficiency (Fig. 11(a)).

Note that the increase in the *C*_{ff}-factor requires decreasing the bilayer thickness (Table 2) and increasing the total thickness of MS (Fig. 11(c)). Both these factors impose increasingly rigid requirements on the fabrication technology. The first one results in the enhancement of the roughness effect, while the second results in smoothening of the saw-tooth relief during MS deposition.

The radiation penetration depth into BMG shown in Fig. 11(c) is a nonmonotonic function of photon energy. This is explained by a concurrence of two factors. On the one hand, the matter polarizability and thereby a variation in the dielectric constants on interfaces between neighboring layers is decreased proportionally to 1/*E*^{2} with increasing photon energy, resulting in a decrease in interference effect, and thus, an increase in the penetration depth, as in the MM case. Second, according to Eq. (23), the resonant incidence angle decreases with increasing energy approaching the critical angle of TER, so the reflectivity from each interface is quickly enhanced, resulting in decreasing penetration depth. The values of *L*_{sat} indicated in Table 2 correspond to the maximum in the curves shown in Fig. 11(c).

For comparison, the peak reflectivity of the conventional Cr/C MM is also shown in Fig. 11(a) (curve MM). As seen, the diffraction efficiency is quite comparable with MM reflectivity in the entire tender X-ray region. The diffraction efficiency can be further enhanced if the antiblaze angle β, i.e., the grating ratio Γ, is increased compared to the values shown in Table 2.

We do not consider gratings with the period exceeding 400 nm because the conditions of the single-order regime may be invalid in this case, resulting in a sharp drop in efficiency.

Finally, basing on the results presented in Table 2 we find that the parameter

is changed within a rather narrow interval for all gratings indicated in the table. Thus, Eq. (34) determines the range of the optimal grating parameters for operation in the given spectral range, where the blaze angle is found via Eq. (31). Among the various parameters obeying Eq. (34), it is necessary to choose the minimal practicable value of*D*/|

*n*| provided the maximal

*C*

_{ff}-factor according to Eq. (33). Equation (34) shows that decreasing

*D*/|

*n*| results in decreasing the bilayer thickness

*h*, which limits the achievable value of the fix-focus constant.

## 6. Comparison with the experimental results

As a demonstration of our theory, we compare the simulated results based on the analytic approach with the experimental diffraction efficiency of a recently fabricated BMG, and study it in the tender X-ray range [2]. The BMG sample has the same structure parameters as G1 grating. Curve 1 in Fig. 12 is the measured reflectivity of the conventional MM (the bi-layer thickness *d* = 6.3 nm, the thickness of Cr layers *d*_{Cr} = 2.7 nm, and the number of bi-layers *N* = 35) versus the photon energy, while curve 2 is the experimental -1^{st} order diffraction efficiency of grating G1 covered by the same MS. For comparison, curves 3 and 4 show the calculated MM reflectivity and diffraction efficiency of grating G1 neglecting the effect of interlayers and interfacial roughness.

The experimental mirror reflectivity and grating efficiency are lower than the theoretical ones, which can be explained by the interfacial roughness effect. The sharp decrease in the experimental values at E = ∼3.2 keV is caused by the absorption edge of argon, which can be introduced into the MS during deposition. By assuming a 5.5 at.% argon concentration in the carbon layers of the MS and setting the delta-correlated rms roughness to σ = 0.48 nm (for MM) or σ = 0.60 nm (for BMG), the experimental curves are fitted adequately using Eqs. (10)−(11) and (29). This is a clear proof of our theory and the conclusions generated in this study. A decrease in the experimental grating efficiency at E > 4.5 keV is explained by the 0^{th} order diffraction wave exciting, which is not considered in the analytic simulation.

## 7. Conclusions

An analytic theory was developed for analyzing the X-ray diffraction from single-order BMGs taking into account the presence of the MS above both blaze and antiblaze facets.

The problem of the choice of the optimal bilayer thickness providing the maximal diffraction efficiency at the given wavelength was discussed. It was demonstrated that the optimal thickness may differ by almost 2 nm from that predicted by the conventional condition. The physical reasons of this fact were explained by a contribution of anti-blaze facets to the diffraction efficiency and the effect of strongly asymmetrical diffraction.

Analytic relations were deduced, allowing the estimation of the range of possible grating parameters for operation in a given spectral interval, without time-consuming numerical calculations. We demonstrated that a proper choice of the MS bilayer thickness allows to design a BMG with the diffraction efficiency close to the reflectivity of the conventional MM throughout the tender X-ray range.

Based on these, a very simple, logically justified, and physically appreciated optimization procedure was developed for designing BMGs with simultaneously high diffraction efficiency compared to the reflectivity of conventional MM and high fix-focus constant *C*_{ff} in a wide operating spectral interval. The obtained design is the optimal one, because it is impossible to shift the obtained curve *C*_{ff}(*E*) upward without an essential drop in efficiency.

The results of the analytic theory were compared with those obtained with rigorous numerical calculations based on CWA. A good agreement between the analytical and numerical calculations was demonstrated. In addition, the analytic theory was verified by the quantitative agreement between the theoretical calculation and the experimental data if interfacial roughness was considered.

Note that we by no means state that the analytic methods developed in the present study replace rigorous numerical calculations. Both approaches should be used in parallel: first, optimization of BMG is performed with a physically obvious analytic approach, and then, the results of analytic optimization should be checked, and if necessary, to refine by numerical calculations especially in the case, when the grazing incidence angle of the incoming beam approaches the TER region.

Although in the present paper, we applied the developed approach to the analysis of BMGs operating in the tender X-ray range, it can be applied in any spectral interval (from hard X-rays to extreme ultraviolet radiation) and for multilayer-coated gratings of any type, including alternate multilayer gratings [6,24] widely used currently in monochromators and spectrometers in parallel with BMGs.

## Funding

National Natural Science Foundation of China (61621001, U1732268); Shanghai Rising-Star Program (19QA1409200).

## Acknowledgments

One of the authors (IVK) acknowledges the Russian Ministry of Science and Higher Education for support of his work within the State assignment FSRC “Crystallography and Photonics” RAS.

## References

**1. **V. N. Strocov, T. Schmitt, U. Flechsig, T. Schmidt, A. Imhof, Q. Chen, J. Raabe, R. Betemps, D. Zimoch, J. Krempasky, X. Wang, M. Grioni, A. Piazzalunga, and L. Patthey, “High-resolution soft X-ray beamline ADRESS at the Swiss Light Source for resonant inelastic X-ray scattering and angle-resolved photoelectron spectroscopies,” J. Synchrotron Radiat. **17**(5), 631–643 (2010). [CrossRef]

**2. **A. Sokolov, Q. Huang, F. Senf, J. Feng, S. Lemke, S. Alimov, J. Knedel, T. Zeschke, O. Kutz, T. Seliger, G. Gwalt, F. Schafers, F. Siewert, I. V. Kozhevnikov, R. Qi, Z. Zhang, W. Li, and Z. Wang, “Optimized highly-efficient multilayer-coated blazed gratings for the tender X-ray region,” Opt. Express **27**(12), 16833–16846 (2019). [CrossRef]

**3. **X. Yang, H. Wang, M. Hand, K. Sawhney, B. Kaulich, I. V. Kozhevnikov, Q. Huang, and Z. Wang, “Design of the multilayer based collimated plane grating monochromator for tender X-ray range,” J. Synchrotron Radiat. **24**(1), 168–174 (2017). [CrossRef]

**4. **D. Maystre, M. Neviere, and R. Petit, Experimental Verifications and Applications of the Theory, in *Electromagnetic Theory of Gratings*, edited by R. Petit, Springer: Berlin, 1980.

**5. **L. Goray, W. Jark, and D. Eichert, “Rigorous calculations and synchrotron radiation measurements of diffraction efficiencies for tender X-ray lamellar gratings: conical versus classical diffraction,” J. Synchrotron Radiat. **25**(6), 1683–1693 (2018). [CrossRef]

**6. **F. Choueikani, B. Lagarde, F. Delmotte, M. Krumrey, F. Bridou, M. Thomasset, E. Meltchakov, and F. Polack, “High-efficiency B_{4}C/Mo_{2}C alternate multilayer grating for monochromators in the photon energy range from 0.7 to 3.4 keV,” Opt. Lett. **39**(7), 2141–2144 (2014). [CrossRef]

**7. **X. Yang, I. V. Kozhevnikov, Q. Huang, and Z. Wang, “Unified analytic theory of single-order soft X-ray multilayer gratings,” J. Opt. Soc. Am. B **32**(4), 506–522 (2015). [CrossRef]

**8. **D. L. Voronov, F. Salmassi, J. Meyer-Ilse, E. M. Gullikson, T. Warwick, and H. A. Padmore, “Refraction effects in soft x-ray multilayer blazed gratings,” Opt. Express **24**(11), 11334–11344 (2016). [CrossRef]

**9. **D. L. Voronov, E. M. Gullikson, F. Salmassi, T. Warwick, and H. A. Padmore, “Enhancement of diffraction efficiency via higher-order operation of a multilayer blazed grating,” Opt. Lett. **39**(11), 3157–3160 (2014). [CrossRef]

**10. **D. L. Voronov, P. Gawlitza, R. Cambie, S. Dhuey, E. M. Gullikson, T. Warwick, S. Braun, V. V. Yashchuk, and H. A. Padmore, “Conformal growth of Mo/Si multilayers on grating substrates using collimated ion beam sputtering,” J. Appl. Phys. **111**(9), 093521 (2012). [CrossRef]

**11. **F. Siewert, B. Löchel, J. Buchheim, and F. Eggenstein, A. Firsov, G. Gwalt, O. Kutz, St. Lemke, B. Nelles, I. Rudolph, F. Schäfers, T. Seliger, F. Senf, A. Sokolov, Ch. Waberski, J. Wolf, T. Zeschke, I. Zizak, R. Follath, T. Arnold, F. Frost, F. Pietag, and A. Erko, “Gratings for synchrotron and FEL beamlines: a project for the manufacture of ultra-precise gratings at Helmholtz Zentrum Berlin,” J. Synchrotron Radiat. **25**(1), 91–99 (2018). [CrossRef]

**12. **F. Senf, F. Bijkerk, F. Eggenstein, G. Gwalt, Q. Huang, R. Kruijs, O. Kutz, S. Lemke, E. Louis, M. Mertin, I. Packe, I. Rudolph, F. Schäfers, F. Siewert, A. Sokolov, J. M. Sturm, C. Waberski, Z. Wang, J. Wolf, T. Zeschke, and A. Erko, “Highly efficient blazed grating with multilayer coating for tender X-ray energies,” Opt. Express **24**(12), 13220–13230 (2016). [CrossRef]

**13. **Q. Huang, Q. Jia, J. Feng, H. Huang, X. Yang, J. Grenzer, K. Huang, S. Zhang, J. Lin, H. Zhou, T. You, W. Yu, S. Facsko, P. Jonnard, M. Wu, A. Giglia, Z. Zhang, Z. Liu, Z. Wang, X. Wang, and X. Ou, “Realization of wafer-scale nanogratings with sub-50 nm period through vacancy epitaxy,” Nat. Commun. **10**(1), 2437 (2019). [CrossRef]

**14. **Commercial software “DiffractMOD/RSoft” distributed by LIGHT TEC, France.

**15. **L. I. Goray, “Numerical analysis of the efficiency of multilayer-coated gratings using integral method,” Nucl. Instrum. Methods Phys. Res., Sect. A **536**(1-2), 211–221 (2005). [CrossRef]

**16. **J. C. Rife, W. R. Hunter, T. W. Barbee Jr., and R. G. Cruddace, “Multilayer-coated blazed grating performance in the soft x-ray region,” Appl. Opt. **28**(15), 2984–2986 (1989). [CrossRef]

**17. **J. H. Underwood, C. K. Malek, E. M. Gullikson, and M. Krumrey, “Multilayer-coated echelle gratings for soft x-rays and extreme ultraviolet,” Rev. Sci. Instrum. **66**(2), 2147–2150 (1995). [CrossRef]

**18. **R. Follath and F. Senf, “New plane-grating monochromators for third generation synchrotron radiation light sources,” Nucl. Instrum. Methods Phys. Res., Sect. A **390**(3), 388–394 (1997). [CrossRef]

**19. **H. Petersen, C. Jung, C. Hellwig, W. B. Peatman, and W. Gudat, “Review of plane grating focusing for soft x-ray monochromators,” Rev. Sci. Instrum. **66**(1), 1–14 (1995). [CrossRef]

**20. **M. Prasciolu, A. Haase, F. Scholze, H. N. Chapman, and S. Bajt, “Extended asymmetric-cut multilayer X-ray gratings,” Opt. Express **23**(12), 15195–15204 (2015). [CrossRef]

**21. **D. L. Voronov, E. H. Anderson, R. Cambie, P. Gawlitza, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Development of near atomically perfect diffraction gratings for EUV and soft x-rays with very high efficiency and resolving power,” J. Phys.: Conf. Ser. **425**(15), 152006 (2013). [CrossRef]

**22. **D. L. Voronov, E. H. Anderson, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Control of surface mobility for conformal deposition of Mo–Si multilayers on saw-tooth substrates,” Appl. Surf. Sci. **284**, 575–580 (2013). [CrossRef]

**23. **I. V. Kozhevnikov, R. van der Meer, H. J. M. Bastiaens, K.-J. Boller, and F. Bijkerk, “Analytic theory of soft X-rays diffraction by lamellar multilayer gratings,” Opt. Express **19**(10), 9172–9184 (2011). [CrossRef]

**24. **X. Yang, I. V. Kozhevnikov, Q. Huang, H. Wang, M. Hand, K. Sawney, and Z. Wang, “Analytic theory of alternate multilayer gratings operating in single-order regime,” Opt. Express **25**(14), 15987–16001 (2017). [CrossRef]

**25. **I. V. Kozhevnikov, R. van der Meer, H. J. M. Bastiaens, K.-J. Boller, and F. Bijkerk, “High-resolution, high-reflectivity operation of lamellar multilayer amplitude gratings: identification of the single-order regime,” Opt. Express **18**(15), 16234–16242 (2010). [CrossRef]

**26. **L. I. Gorai, “Scalar and electromagnetic properties of X-ray diffraction gratings,” Bull. Russ. Acad. Sci.: Phys. **69**(2), 231–236 (2005).

**27. **D. L. Voronov, L. I. Goray, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “High-order multilayer coated blazed gratings for high resolution soft x-ray spectroscopy,” Opt. Express **23**(4), 4771–4790 (2015). [CrossRef]

**28. **I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of XUV multilayer optics,” Phys. Scr. **T17**, 137–145 (1987). [CrossRef]

**29. **Index of refraction in the X-ray region, http://henke.lbl.gov/opticalconstants/getdb2.html

**30. **S. V. Gaponov, V. M. Genkin, N. N. Salashchenko, and A. A. Fraerman, “Scattering of neutrons and x-radiation in the range 10-300 Å by periodic structures with rough boundaries,” JETP Lett. **41**(2), 63–65 (1985).

**31. **M. Wen, I. V. Kozhevnikov, and Z. Wang, “Reflection of X-rays from a rough surface at extremely small grazing angles,” Opt. Express **23**(19), 24220–24235 (2015). [CrossRef]