1. Introduction

In the X-ray region, the dielectric constant of any material is close to unity so that conventional diffraction gratings can only acquire high efficiency by operating in the total external reflection (TER) region. The TER region is characterized by small grazing incidence angles, leading to a small entrance aperture and throughput. Since the TER critical angle is inversely proportional to the photon energy, the diffraction efficiency starts to decrease when the photon energy reaches 1 keV and drops to only 20% when the energy exceeds 2 keV. In the hard X-ray region, crystals are typically used as diffractive optics. However, due to the limited lattice period [1,2], there is a gap between about 1 and 4 keV, also called the tender X-ray region, where the diffraction efficiency of both crystals and conventional diffraction gratings is low.

This predicament is broken with the appearance of X-ray interference multilayer mirrors that reflect efficiently at larger grazing incidence angles, up to normal incidence, in the soft X-ray region [3,4]. Combining the multilayer structure with diffraction gratings, we can form multilayer gratings that have a larger entrance aperture and a compact size, which is especially important for spectrometers utilized in astronomy. In addition, due to the interference effect, multilayer gratings are not restricted by the small TER critical angle and can attain diffraction efficiency one order of magnitude higher than that of conventional gratings in the tender X-ray region. This opens up new experimental possibilities when the multilayer gratings are used in monochromators or spectometers in synchrotron facilities [5].

Among all kinds of multilayer gratings, alternate multilayer gratings (AMG), whose schematic is shown in Fig. 1, are relatively easy to fabricate by depositing a periodic multilayer structure onto a lamellar-shaped substrate with the lamella depth being a fraction of the multilayer structure period [6]. Early in the development of AMGs, they were used in low energy range [7,8]. Recently, AMGs were designed and fabricated for application in the tender X-ray [6,9–11] and hard X-ray [12] region.

 

Fig. 1 Schematic of X-ray diffraction from an alternate multilayer grating.

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In parallel to AMGs, other types of multilayer gratings (such as blazed, lamella and sliced gratings) are under extensive development today to be used as monochromators and spectrometers in synchrotron radiation beamlines, space astronomy, plasma physics, and X-ray fluorescent analysis. These applications and their current state-of-art fabrication are described in a number of papers [13–22].

At present, theoretical analysis of multilayer gratings is usually based on sophisticated and time-consuming computer simulations [23,24]. In parallel, approaches analogous to those used in crystallography to describe diffraction of X-rays from crystals, are also developed [14].

One of the universal methods, which can be applied to multilayer grating of any type, is the coupled wave approach (CWA). CWA is a well-suited method to analyze diffraction efficiency of multilayer gratings operating in the hard and soft X-ray regions, where the dielectric constant of any material is close to unity and diffraction occurs over the whole depth of a multilayer structure exceeding the height of a grating surface relief by several tens or hundreds of times. In the recent studies [25–28], the CWA has been successfully applied to comparative analytical analysis of several types of multilayer gratings including lamellar, sliced, blazed, and blazed lamellar multilayer gratings (LMG, SMG, BMG, and BLMG, respectively). In the present paper, the same approach is used to obtain analytic expressions and analyze the diffraction efficiency and bandwidth of AMG operating in the single-order regime, when the only diffraction order is excited by an incidence wave. Notice that the single-order regime is the most interesting for practice because the incident radiation power is transformed to that of a single diffraction wave providing the highest diffraction efficiency close to the reflectivity of conventional multilayer mirror.

General formulae of X-ray diffraction from AMGs operating in the single-order regime are deduced in section 2. The peculiarities of AMG compared with other multilayer gratings are analyzed in section 3. The validity conditions of the analytical theory are discussed in section 4. The main results of our consideration are summarized in section 5.

2. Analytic theory of alternate multilayer gratings

The schematic of the AMG is shown in Fig. 1. The substrate represents a lamellar structure with the period D, the groove width ΓD, and the groove depth h. The substrate is coated by a periodic multilayer structure consisting of alternating layers of heavy (absorber) and light (spacer) materials with abrupt interfaces between them. The multilayer is characterized by the period d, the thickness of absorbing layer γd, the total number of bi-layers N, and the complex polarizability χ_A and χ_S of absorber and spacer. The depth-distribution of the dielectric constant inside the multilayer stack is described by the function ${\epsilon }_{ML}\left(z\right)\equiv 1-{\chi }_{ML}\left(z\right)$, where χ_ML(z) is the multilayer structure polarizability at 0 ≤ z ≤ L, while χ_ML(z < 0) = 0 and χ_ML(z > L) = 1 – ε_sub, where ε_sub is the dielectric constant of substrate material.

Taking into account the shift of neighboring multilayer stacks by h, the distribution of the dielectric constant in space is written as

The CWA is based on the Rayleigh representation of the total field as a superposition of different order diffraction waves [29] assuming a plane monochromatic wave to fall on the grating surface at the grazing incidence angle θ₀:

Substituting Eqs. (1)-(3) into the 2D wave equation (s-polarized radiation case) we obtain the following system of 1D equations for the field functions F_n

In the limiting cases, when either h = 0 or Г = 0 or 1, system of Eq. (4) reduces to the standard 1D wave equation describing the reflectivity from conventional mutilayer mirror:

Note that system (4)-(5) can be used for calculations of diffraction from AMG with arbitrary multilayer coating, both periodic in depth and depth-graded, the last being of extreme interest for designing broadband multilayer gratings operating in a wide spectral interval [27].

Below we compare results of analytical calculations of AMG diffraction efficiency with computer simulations via numerical solving the system of Eqs. (4)-(5), where 21 diffraction orders (−10th to + 10th) are taken into account.

To find the approximate analytic solution of system (4)-(5) we assume that the material polarizability inside multilayer stack is a periodic function of z and expand it into the Fourier series:

Then, following [26], we represent the nth order diffraction wave F_n(z) as a superposition of two waves propagating in opposite directions along Z-axis

Substituting Eqs. (6)-(8) into Eq. (4), we obtain the infinite system of the first-order differential equations:

Note that the system (9) is totally correct and the second derivatives of the wave functions F_n(z) were not neglected in Eqs. (9), because of condition (8) imposed on the amplitudes A_n and C_n (see [26] for more details).

For simplification, an approximation is made that the AMG structure is flattened at the top and bottom. The approximation is quite reasonable in the soft X-ray region, where the penetration depth of an incident wave covers several tens to hundreds of bilayers, overwhelmingly exceeding the missing fraction of bi-layer thickness. Under this circumstance, the boundary conditions are written as

Taking the parameters w_j and v_j contained in Eqs. (9)-(10) to be proportional to the polarizability of a material, which is much less than unity, we can conclude that the field amplitude derivatives $|d{A}_n/dz|,|d{C}_n/dz|~k\overline{\chi }/\sin {\theta }_n$ are very small and thus the amplitudes A_n(z) and C_n(z) change slowly with z compared to the quickly oscillating exponents on the right side of Eqs. (9), whose derivatives are equal to $|d\exp (i{\kappa }_{n}z)/dz|=k\sin {\theta }_n.$ This statement is valid until the grazing incidence and diffraction angles exceed essentially the critical angle of TER ${\theta }_{С}$:

Summands containing quickly oscillating exponents influence the amplitudes A_n(z) and C_n(z) only weakly and can therefore be neglected excluding the resonant case, when one of the exponential multipliers in Eqs. (9) is transformed into a slowly oscillating function (the single order regime of the diffraction). When the diffraction order is non-zero, a resonant interaction between incident and the nth-order diffraction wave occurs if the generalized Bragg condition of diffraction is fulfilled

Then Eqs. (9) are transformed into a system of only two differential equations

The final expression for the non-zero nth-order diffraction efficiency of AMG operating in the single-order regime is written as:

The parameter 1/S determines the penetration depth of an incident wave into the grating structure, the Bragg parameter b defines the deviation from the diffraction resonance, and the modulation parameter U _± characterizes the maximal possible diffraction efficiency of multilayer grating. In particular, if U _± = 0, the diffraction efficiency is equal to zero.

According to the general theory of the single-order multilayer gratings [26], diffraction efficiency (15) achieves the maximal value for semi-infinite multilayer coating, when the generalized Bragg condition of diffraction $\mathrm{Im}\left(U_{+}{U}_{-}-b^2\right)=0$ is fulfilled, and it can be written in the explicit form:

The maximal non-zero order diffraction efficiency R_n is written by the same manner as that for multilayer gratings of other types

Notice that Eqs. (15)-(20) are also valid for p-polarized radiation, if we change the dielectric modulation of the multilayer structure ${\chi }_A-{\chi }_S$ by $({\chi }_A-{\chi }_S)\cos ({\theta }_0+{\theta }_n)$ (see [25] for more details). Hence, the diffraction efficiency of p-polarized radiation goes to zero if the diffracted beam propagates perpendicular to the incident one, i.e. at ${\theta }_0+{\theta }_n=\pi /2.$

3. Peculiarities of AMG compared to other multilayer gratings

First of all, we note that for all types of multilayer gratings considered in [26], the expression for the zero-order diffraction efficiency (the specular reflectivity) can be obtained from Eqs. (15)-(16), if we tend the diffraction order n to zero. However, it is not the case of AMG. In fact, the specular reflectance achieves the maximal value if the Bragg condition of reflection is fulfilled:

Then, keeping only resonant summands in Eqs. (9), we obtain a system of two equations

According to Eqs. (10) parameters $w_{\pm j}=0$ if the following two conditions are fulfilled

To understand the phenomenon in detail we return to the CWA Eq. (4). The interaction between different diffraction orders in the body of AMG is described by the sum of m on the right side of Eq. (4). As we are interested in the specular reflection (n = 0) from the single-order AMG, we exclude this sum from consideration. Then Eq. (4) reduces to 1D wave equation describing the wave reflection from a periodic multilayer structure with the following depth-distribution of the polarizability:

For periodic X-ray multilayer mirrors, it is well known that if the jth harmonic u_j vanishes the jth Bragg peak disappears. Notice that the first Fourier-harmonic of polarizability can never become zero for an ordinary multilayer structure.

The same consideration is valid for a multilayer structure with the polarizability (25) except that there is an additional possibility to obtain zero harmonic w_j by proper choice of the lamella height: jh = d/2. The statement is valid for the first harmonic as well, resulting in the disappearance of the specularly reflected peak in Fig. 2(a). One can easily check that, if h = d/2, the period of the function ${\chi }_{eff}(z)$ in Eq. (25) is equal to d/2, while the period of the function ${\chi }_{ML}\left(z\right)$ is d. Therefore, even though a wave is reflected effectively from a conventional multilayer mirror at a grazing incidence angle lying inside the first Bragg peak, the wave falling at the same angle on the structure (25) with a halved period is not reflected from it.

 

Fig. 2 Diffraction efficiency of different orders (from −3rd to + 3rd) at 278 eV photon energy (s-polarized radiation case) versus grazing incidence angle for Cr/C AMG with different Г ratio: Г = 1/2 for (a) and Г = 0.4 for (b). Other geometrical parameters are: D = 300 nm, d = 5 nm, γ = 0.4, N = 150, h = d/2. Results shown in black dashed curves were obtained by numerical calculations basing on the CWA method. The colored solid lines represent analytical results obtained from Eq. (15).

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Next, if the reflectivity of AMG is equal to zero due to conditions (24) the incident wave should be transformed into a transmitted one in the single-order regime. In the case considered, the transmittance (23) is converted to a very simple expression

Moreover, if the first condition in (24) is fulfilled, the parameter $U_{\pm }$ in Eq. (16), and hence the diffraction efficiency, is equal to zero for all even diffraction orders. An example is presented in Fig. 2(a), where the diffraction efficiencies of different orders (from −3rd to + 3rd) are shown as a function of the grazing incidence angle at 278 eV photon energy (C-K_α characteristic emission line, s-polarized radiation case). Calculations were performed for Cr/C AMG with the following parameters: D = 300 nm, Г = 1/2, d = 5 nm, γ = 0.4, N = 150, h = d/2. As seen, only odd diffraction orders exist in this case.

However, if $\Gamma \ne 1/2$, the diffraction waves of zero and all even orders are excited by an incident wave. An example is given in Fig. 2(b) for the same grating as in Fig. 2(a), but with Г = 0.4. Note that the disappearing of zero and even orders diffraction waves is observed, if only representation (2)-(3) for distribution of polarizability inside the grating body is valid and all even summands in series (3) turns to zero at Г = 1/2. In particular, it means that interfaces between neighboring multilayer stacks are abrupt as it is shown in Fig. 1.

At the same time, the deposition of materials onto lamellar walls and surface diffusion of adatoms during multilayer structure growth results in a gradual transformation of the rectangular substrate profile into a trapezoidal and then sinusoidal-like one [11]. This means that interfaces between absorber and spacer do not change abruptly at the boundaries between neighboring multilayer stacks and multilayer structure is not completely periodic along Z-axis. Hence, representation (2)-(3) is not fully correct and thus zero and even diffraction orders should be always observed for real AMGs, while we would expect their effective suppression if the geometrical parameters of AMG are chosen properly. This would be useful for grating monochromators placed in synchrotron beamlines, where suppression of higher orders undulator harmonics is of extreme practical importance. In addition, analysis of zero and even orders diffraction waves may provide information about imperfections of AMG internal structure.

Next, as for any multilayer grating, the peak diffraction efficiency is determined by the parameter $y=P_1\cdot P_2\cdot P_3\le 1,$ with the larger the value of y, the higher the diffraction efficiency. The multiplier P₁ is completely determined by the multilayer structure parameters and achieves the maximal value if the thickness ratio γ obeys the equation $\tan \left(\pi j\gamma \right)=\pi j\left(\gamma +g\right)$providing the maximal reflectivity of conventional multilayer mirror (MM) [30]. The multiplier P₂ is also the same for any multilayer grating and is determined by the grating period. Only multiplier P₃ is different for different types of multilayer gratings. Note that for all gratings analyzed in [26] (BMG, SMG, LMG, and BLMG) the multiplier P₃ approaches unity if the grating parameters are properly chosen. Therefore, the diffraction efficiency of all these gratings may be very close to the reflectivity of conventional MM if the incidence and diffraction angles are close to each other (i.e. the grating period is not too small) and thus parameter P₂ is close to unity. However, it is not the case for AMG. In fact, even under the optimal diffraction conditions (|n| = 1, Г = 1/2, jh = d/2) the maximal value of P₃ = 2/π and, hence, the diffraction efficiency of AMG is always less than the multilayer mirror reflectivity. Examples are presented in Figs. 3-4.

 

Fig. 3 −1st order diffraction efficiency of Cr/C AMG and BMG and reflectivity of corresponding MM versus grazing incidence angle at 3keV photon energy (s-polarized radiation case). The geometrical grating parameters are: D = 300 nm, d = 5 nm, γ = 0.4 N = 100. For AMG, Г = 1/2, h = d/2, and for BMG, the blaze angle is 0.95°. Results shown in black dashed curves were obtained by numerical calculations basing on the CWA method. The colored solid lines represent analytical results obtained with Eq. (15).

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Fig. 4 −1st order diffraction efficiency of W/C AMG and BMG and reflectivity of corresponding MM versus grazing incidence angle at 278 eV photon energy (s-polarized radiation case). The γ ratio is equal to 0.15 (a) or 0.5 (b). Other grating geometrical parameters are: D = 300 nm, N = 200, d = 5 nm. For AMG, Г = 1/2, h = d/2, and for BMG, the blaze angle is 0.95°. Results shown in black dashed curves were obtained by numerical calculations basing on the CWA method. The colored solid lines represent analytical results obtained with Eq. (15).

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The diffraction efficiency of Cr/C AMG and BMG and the reflectivity of conventional MM are shown in Fig. 3 as a function of the grazing incidence angle at 3 keV photon energy. Note that, as previously mentioned, in the 1-4 keV tender X-ray energy range, multilayer gratings are of great importance because neither conventional gratings with single-layer coating nor crystals function satisfactorily [5]. The geometrical parameters of the gratings used for calculations are the following: D = 300 nm, d = 5 nm, γ = 0.4. For AMG, h = d/2, Г = 1/2, and for BMG, blaze angle is chosen as tan⁻¹(d/D) = 0.95°. As expected, the −1st order peak diffraction efficiency of BMG is practically the same as the peak reflectivity of MM, while the peak efficiency of AMG is somewhat less (64% for AMG instead of 78% for BMG).

As discussed in [26], diminishing diffraction efficiency with decreasing parameter y depends essentially on the parameter f determined in (20), so the efficiency decrease is relatively weak if the parameter is large | f | >> 1; the parameter | f | = 10.7 for Cr/C multilayer gratings discussed above. However, if | f | ≤ 1, a sharp decrease in the diffraction efficiency may occur with decreasing parameter y. An example is presented in Fig. 4(a), where comparison of the diffraction efficiencies of W/C AMG and BMG operating at photon energy E = 278 eV is presented. Notice that the parameter | f | is as small as 0.8 in this case. As shown, the diffraction efficiency of AMG (14%) is 3.3 times less than that of BMG (46%); the last value coincides practically with the MM reflectivity. This fact should be considered when choosing a pair of materials for the reflecting coating of AMGs.

The optimal thickness ratio γ = 0.15, providing the maximal reflectivity of a conventional W/C multilayer mirror, was used when calculating diffraction efficiency in Fig. 4(a). Note that in a major part of previous papers considering AMGs [6,9–11] the thickness ratio γ was chosen to be 1/2 to provide a symmetric shape of the “unit cell”, the notion introduced in [6]. Our consideration demonstrates that this choice may be far from the optimal one. In fact, the only difference between gratings in Fig. 4(a) and Fig. 4(b) is the multilayer structure thickness ratio γ. The value of γ = 0.15 is optimal in Fig. 4(a), while γ = 1/2 in Fig. 4(b). As shown, both the diffraction efficiency of gratings and the MM reflectivity in Fig. 4(b) drop by a factor exceeding two compared to the case of optimal γ = 0.15.

4. The validity conditions of the single-order regime of AMG operation

The single-order regime analyzed in the present paper means that the incident beam excites only one diffraction wave. Only then simple analytic expressions (14)-(26) are valid. As the polarizabilty of any material in the soft X-ray region is much less than unity, the spectral and angular band of reflection from multilayer structure is very narrow. As a result, the diffraction pattern represents a sequence of narrow peaks [see Fig. 2(b)] with the angular distance between them (in terms of the incidence angle of the incoming beam) being equal to d/(jD) (see [26] for more details). Diffraction efficiency outside the peaks is negligibly low. Hence, the grating operates in the single-order regime if the angular width of the diffraction peak Δθ (full width at half-maximum) is several times smaller than the distance between neighboring peaks: $\Delta \theta <<d/\left(jD\right).$

An expression for the non-zero order diffraction peak width of AMG can be obtained on the basis of general formulae deduced in [26] on the assumption that absorption effects are small:

If, for simplicity, we neglect the difference between grazing incidence θ₀ and diffraction θ_n angles and consider the case of optimal diffraction conditions |n| = 1, Г = 1/2, and jh = d/2, the width of the AMG diffraction peak is equal to $\Delta \theta \approx \left(2/\pi \right)\cdot {\left(\Delta \theta \right)}_{MM},$ which corresponds to Fig. 3. Therefore, the condition of the single-order regime is written as $jD\cdot {\left(\Delta \theta \right)}_{MM}<<3d,$ because odd diffraction peaks only can be excited and so the angular distance between neighboring diffraction peaks is doubled in Fig. 2(a) compared to Fig. 2(b). A more rigid condition, which is valid for arbitrary AMG while assuming h to be of the order of d/2, has the following form

Operation of AMG in the single-order regime is illustrated by Figs. 2-4, where diffraction peaks were calculated numerically taking 21 diffraction orders (from −10th to + 10th) into account and analytically with the use of simple expressions (15), (23). As shown, the validity of the single-order diffraction condition (28) results in an excellent agreement between numerical and analytic calculations.

The opposite case, when condition (28) is invalid, is presented in Fig. 5. Diffraction peaks are shown in the figure for the AMG similar with that in Fig. 2(b), but with the grating period increased up to D = 1200 nm. As shown, several diffraction orders are excited simultaneously at the given grazing angle of incoming beam with the incident power being distributed over all of them. As a result, the ± 1st order diffraction efficiency is decreased compared to the single-order regime. One more example of transition from multi-order to single-order regime with decreasing grating period is discussed in [28] as applied to operation of LMGs.

 

Fig. 5 Diffraction efficiency of different orders (from −4th to + 4th) at 278 eV photon energy versus grazing incidence angle for Cr/C AMG (s-polarized radiation case). Geometrical parameters are: D = 1200 nm, Г = 0.4, d = 5 nm, γ = 0.4, N = 150, h = d/2. Results shown in solid curves were obtained by numerical calculations basing on the CWA method. The black dashed curves represent the analytical calculations of ± 1st diffraction order efficiency obtained with Eq. (15).

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Our consideration demonstrates that fulfillment of condition (28) allow us to describe all features of AMG operation with very simple analytical formulae and without using rigorous numerical approaches based on sophisticated and time-consuming computer simulations, especially in the case of multilayer coatings consisting of several hundreds of bi-layers. Moreover, the single-order regime of grating operation is very promising from a practical point of view, because in this regime the incident radiation power is transformed into that of the single diffraction wave thus providing the maximal diffraction efficiency.

Nevertheless, the fulfillment of the necessary condition (28) does not always guarantee a high accuracy of analytical expressions. The fact is that condition (12) should be obeyed in addition to (28), i.e. both incident and diffraction waves must be propagated outside the TER region. In fact, Fig. 2(a) demonstrates an excellent agreement between analytical and numerical calculations. Both incident and diffraction waves propagate far from the TER region (critical angle of TER ${\theta }_c=\sqrt{\mathrm{Re}\overline{\chi }}$ is equal to 4.53°) and both conditions (28) and (12) are valid in the case considered. However, if we increase the multilayer structure period up to 15 nm, the Bragg condition of diffraction (13) can be fulfilled at small grazing angle only, which is close to the critical angle of TER. Figure 6(a) demonstrates that agreement between analytical and numerical calculations becomes worse. Moreover, in contrast to Figs. 2-4, the shape and height of the −1st order diffraction peak calculated numerically [see Fig. 6(a), curves 2 and 3] depends on material placed on the multilayer structure top (Cr or C), while the analytic expression (15) is independent of that. Notice that the same effect is observed for conventional MM [see Fig. 6(b)].

 

Fig. 6 Diffraction efficiency (0th and −1st order) of Cr/C AMG (a) and reflectivity of corresponding MM (b) at 200 eV photon energy versus grazing incidence angle (s-polarized radiation case). Geometrical parameters are: D = 300 nm, Г = 1/2, h = d/2, d = 15 nm, γ = 0.4, N = 40. Curves 1 were calculated with analytical formulae. Curves 2, 3 were calculated by numerical integration of Eqs. (4)-(5) taking into account 21 diffraction orders (−10th to + 10th) (graph a) or with the IMD software [32] (graph b). The uppermost layer of multilayer structure was Cr (curves 2) or C (curves 3).

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The main reason of this discrepancy is described in the following. We expand the structure polarizability distribution into a Fourier series (6), but to obtain an analytical solution for the wave diffraction or reflection we keep only one resonant Fourier harmonics in series (6) obeying the generalized Bragg condition (13) and thus reduce Eqs. (9) to the simplest system of two Eqs. (14). This assumption is quite reasonable if a wave falls onto MM far from the TER region [condition (12)]. However, if condition (12) is invalid, we cannot guarantee that there is only one summand oscillating slowly in the right part of system (9) or, in other words, that incident wave excites more than one diffraction wave.

In fact, Fig. 6(a) demonstrates that the specularly reflected (zero order diffraction) wave is excited in parallel with the −1st order diffraction one, when the grazing angle of incident radiation approaches the TER region. Note that the higher order diffraction efficiencies are lower and not shown in Fig. 6(a). Hence, we should keep at least three waves in the system (9), namely, the incident wave A₀, the −1st order diffracted wave C_-1, and the specularly reflected wave C₀. The interplay between them via Eqs. (9) results in a change in the diffraction efficiency R_-1 at small grazing incidence angle (θ < 7°) compared to the single-order regime.

Next, consider the wave reflection from conventional MM. In this case, we put h = 0, w_j = u_j and keep only two waves in the CWA system of Eqs. (9), namely, the incident wave A₀ and the specularly reflected one C₀. However, an infinite series in j (Fourier harmonics of the MM polarizability) persists in the system (9). Neglecting all summands excluding the one which obeys the Bragg condition (21) proves to be an acceptable simplification if a wave falls onto MM far from the TER region. In this case, the MM reflectivity is actually determined by the single harmonics in the Fourier series (6) as well as the mean polarizability of MM and it is practically independent of the materials placed on the top of MM, because reflection is a cumulative effect of many tens or even hundreds of interfaces. However, near or inside the TER region the penetration depth of an incident wave is small and we observe difference in the MM reflectivities depending on the uppermost material, which is of the order of the reflectivity from a single interface. The same conclusion applies for diffraction efficiency of any multilayer grating if the incident or diffraction wave propagates near the TER region. Notice that simple analytic expression does not describe reflectivity at all in the TER region [see Fig. 6(b)].

5. Conclusions

Basing on the coupled wave approach, we have developed an analytic theory of soft X-ray diffraction from alternate multilayer gratings operating in the single-order regime, when an incident wave excites only one diffraction wave.

The optimal diffraction conditions providing the maximal diffraction efficiency were obtained: the grating should operate in the $\pm$1st diffraction orders, the lamella height should be half of the multilayer structure period (h = d/2), and the groove width should be half of the grating period (Г = 1/2). The generalized Bragg condition of diffraction (17) should be obeyed as well.

As for other multilayer gratings, the diffraction efficiency of AMG achieves the maximal possible value if the parameters of multilayer coating, primarily the thickness ratio γ, are chosen to provide the maximal reflectivity of conventional MM. It is noted that many papers assume explicitly or implicitly that the highest diffraction efficiency is achieved for AMG with a symmetric shape of the “unit cell”, i.e. at γ = 1/2. We demonstrated that this statement is incorrect and may result in twofold, at least, decrease in the diffraction efficiency compared to the optimal γ value.

Among advantages of AMGs we can indicate the following two: (a) relatively simple fabrication of the lamella relief on a grating substrate as compared with the saw-tooth relief on BMG substrate and (b) a possibility to decrease significantly the diffraction efficiency of the zero and all even orders. The latter one would be useful for grating monochromators placed in synchrotron beamlines, where suppression of higher order undulator harmonics is of extreme practical importance.

The main disadvantage of AMGs compared to multilayer gratings of other types is that the diffraction efficiency of AMG is always less than the reflectivity of conventional MM, even if the grating parameters and coating materials are chosen optimally. If the absorber layer of multilayer structure is chosen improperly (too large absorptivity), the maximal efficiency of AMG may be several times lower than the reflectivity of corresponding MM and the diffraction efficiency of multilayer gratings of other types.

The validity conditions of the analytic theory have also been discussed. First, the AMG geometrical parameters should obey the single-order condition (28) assuring the angular or spectral width of the diffraction peaks to be several times less compared to the distance between them. Additionally, the condition (12) should be fulfilled, i.e. both incident and diffracted waves should be propagated far enough from the TER region. Otherwise, the single-order regime is no longer valid, because the specularly reflected wave is always excited in addition to the diffracted wave.

In conclusion, we would like to emphasize that our consideration of multilayer gratings performed in the present and our previous works were based on, as a rule, a model of ideal gratings having no structural imperfections. The final choice of multilayer grating type as applied to a given practical application can be done only based on comparative analysis of possibilities from existing fabrication technologies for multilayer gratings of different types.