1. Introduction

During the past decades, the problem of diffraction of short wavelength (hard X-ray to extreme ultraviolet (EUV)) radiation (wavelength $\lambda ~0.1-30$ nm) from rough surfaces and layered structures was analyzed in many journal papers and several monographs (e.g., see [1, 2 ] and references therein). On the one hand, quick progress in X-ray/EUV optics would be impossible without understanding the roughness effect on the reflectivity and coherent properties of an outgoing beam. On the other hand, the X-ray scattering technique is a unique instrument to study roughness with a lateral resolution comparable to that of atomic force microscopy (AFM) [3], and, in contrast to AFM, it allows for the investigation of latent interfaces and statistical interrelation (conformity) between the roughnesses of different interfaces [4].

Nevertheless, there are several questions that remain to be answered. One of them is the problem of X-ray/EUV radiation reflection from a rough surface at extremely small grazing angles. Currently, the problem is of essential practical importance for the development of optics for existing and future free electron lasers (FELs). The fact is that the mirrors placed in FEL beamlines have to withstand an extremely high radiation load and thus should absorb as small a portion of incident radiation power as possible. For this purpose, the grazing incidence mirrors should be installed with respect to the incident beam at a very small grazing angle θ ₀ of only a minor fraction of the critical angle ${\theta }_c$ of the total external reflection (TER). For example, the grazing incidence angle of distribution mirrors is planned to be 1 – 3.6 mrad only in the SASE1 beamline at the future European X-ray FEL (XFEL, Hamburg) [5], with the working photon energy lying in the E ~3 – 24 keV interval. Notice that the critical angle of TER at E = 3 keV is approximately 10 or 30 mrad for carbon or gold reflecting coating, respectively.

The simplest way to describe the roughness effect on the reflective and scattering properties of a single surface is to modify the Fresnel reflectivity $R_F({\theta }_0)$ by introducing the Debye-Waller (DW) [6] or Nevot-Croce (NC) [7] factors, which are valid in the limiting cases of extremely large or vanishingly small correlation length of the roughness ξ, respectively [8, 9 ]:

However, the following question arises: are the well-known DW and NC factors for the reflectivity and TIS still valid at extremely small grazing angles? Actually, the width of the angular distribution of the scattered intensity (scattering diagram) in the incidence plane is typically expressed via the correlation length of roughness as $\Delta \theta ~2\lambda /(\xi {\theta }_0)$. If the correlation length decreases down to ${\xi }_{crit}~2\lambda /{\theta }_0^2$, then the angular width of the scattered beam becomes comparable with the grazing incidence angle ${\theta }_0$. Further decrease in the correlation length results in the scattered beam “lying” on a surface; thus, we would expect the change in the scattering pattern and, in particular, increased scattering both in vacuum and into the matter depth. In the case of mirrors in the XFEL beamline ($E=3-24$ keV, ${\theta }_0=1$ mrad), the value of ${\xi }_{crit}$ is extremely large and equal to approximately 100 μm (at $E=24$ keV) or even 800 μm (at $E=3$ keV). The effect on the reflectivity and TIS value of long-scale roughness with a correlation length of tens of microns, not to mention hundreds of microns, is typically described by the Debye-Waller factor, but our brief consideration demonstrates that the situation is not so evident at extremely small grazing angles.

Inapplicability of the DW factor (1) for explanation of the total integrated scattering behavior at extremely small grazing angles was experimentally demonstrated in a number of papers (e.g., see [10, 11 ]). Qualitative theoretical explanation of this fact was first offered in [12, 13 ]. The goals of the present paper are the detailed analysis of X-ray diffraction from a rough surface at extremely small grazing angles, demonstration of the invalidity of both DW and NC factors in this angular region, and deduction of the simple expressions substituting Eqs. (1)-(2) at a vanishingly small grazing angle of an incident beam.

General formulas describing X-ray diffraction by a rough surface are briefly discussed in Section 2.

The Interrelation of four diffraction channels (coherent reflectance, coherent transmittance, diffuse scattering in vacuum, and scattering into the matter depth) is analyzed in Section 3 for different limiting cases of large and small correlation lengths of roughness and large and extremely small grazing angles of incident radiation.

Transformation of one limiting diffraction regime into another one with variation in the correlation length of roughness is analyzed in Section 4. Conditions of applicability of the DW and NC factors are also discussed.

The main results of the paper are summarized in Section 5.

2. General formulae

Suppose that the boundary between vacuum and matter is a rough surface described by the stochastic function $z=\zeta (\overrightarrow{\rho })$, where the vector $\overrightarrow{\rho }\equiv (x,y)$ lies in the plane of a perfectly smooth surface, the Z-axis is directed into the depth of matter, and the dielectric permittivity is changed abruptly at the surface from unity in vacuum to ε in matter. Because we are interested in very smooth surfaces, which are only used in X-ray optics, and small grazing angles, we neglect the polarization effects and use very simple and vivid perturbation theory on the roughness height for analysis of X-ray scattering. The theory is well-known and has been widely used in visible optics [6] for almost 90 years [14]. Analysis of its applicability in the X-ray region was performed in [15].

In frames of the first-order perturbation theory, the scattering diagrams in vacuum ${\Phi }_{-}(\theta ,\phi )$ and in the depth of matter ${\Phi }_{+}(\theta ,\phi )$ are written as follows:

 

Fig. 1 Scheme illustrating reflection of X-ray wave from a rough surface.

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The statistical parameters of the roughness are described by the 2D Power Spectral Density function (PSD-function)

After algebraic manipulations, the expressions for the Total Integrated Scattering (TIS) in vacuum ${\text{TIS}}_{-}$ and in the depth of matter ${\text{TIS}}_{+}$ can be written in the following form:

In the frame of the perturbation theory, the coherent reflectance R and transmittance T are written as

Notice that transmittance $T_F$ characterizes the flux through the perfectly smooth surface z = 0, while the flux in the depth of an absorbing medium is described by $T_F\cdot \exp \left[-\mathrm{Im}{\kappa }_{+}(q_0)z\right]$, where $\mathrm{Im}{\kappa }_{+}(q_0)>0$ and tends to zero in the matter depth. Similarly, the angular distribution of the flux scattered into the depth of absorbing matter ${\Phi }_{+}(\theta ,\phi )$ should be replaced by ${\Phi }_{+}(\theta ,\phi )\cdot \exp \left[-\mathrm{Im}{\kappa }_{+}(q)z\right]\underset{z\to \infty }{\to 0}$. Nevertheless, we will analyze scattering into the depth of absorbing medium assuming radiation absorption to be low and roughness to be small. Then, we can consider ${\Phi }_{+}(\theta ,\phi )$ as a scattered flux through the plane z = const placed below the rough relief ($z\ge \max \left[\xi (x,y)\right]$ in Fig. 1), but it is so close to the surface that the absorption effect is negligible.

One can verify that Eqs. (7)-(10) provide the fulfilment of the energy conservation law, namely, the validity of the equality $R+T+{\text{TIS}}_{-}+{\text{TIS}}_{+}\equiv 1$ for non-absorbing matter. To prove the last equality, it is necessary to consider two cases separately: the wave incidence onto the surface inside the total external reflection (TER) region, when $\sin {\theta }_0<\sqrt{1-\epsilon }$ and $\mathrm{Re}(q_0)\equiv 0$, and outside it, where $\mathrm{Im}(q_0)\equiv 0$. Similarly, the integration area in the plane of the scattering vector $\overrightarrow{q}$ should be divided into two areas, where $q>k\sqrt{\epsilon }$ or $q<k\sqrt{\epsilon }$. More detailed discussion of the energy conservation law in the theory of X-ray diffraction from arbitrary rough-layered medium is given in [16].

In the limiting case of the zero correlation length of roughness $\xi \to 0$, the function ${\text{PSD}}_{2D}(\overrightarrow{\nu })\to 0$ and the integral summands in Eqs. (9)-(10) disappear. The remaining summands coincide with the absolute values, which can be verified using the explicit expressions (11) for the Fresnel reflectance and transmittance. They are nothing but the first terms of the NC factor (2) expansion in powers of the rms roughness σ:

In the opposite limiting case of extremely large correlation length $\xi \to \infty$, the PSD-function is very narrow. Then, putting $q=q_0$ in the integrals in Eqs. (7)-(10) and widening the integration interval over q up to infinity, we obtain the well-known expressions

The validity of the DW and NC factors in the limiting cases of extremely large and vanishingly small correlation length of the single surface roughness was proved in several papers (e.g., see [8, 9 ]) based on the approach developed in [17]. Equations (12)-(13) demonstrate that the same conclusion is a simple and evident result of the perturbation theory. Moreover, Eqs. (12)-(13) are valid for arbitrary roughness height probability distribution, while Eqs. (1)-(2) – for normal distribution only. However, an accurate quantitative condition of subdividing the correlation length into large and small ones was not analyzed in details in the literature. In the next two sections, we will determine the conditions of validity of Eqs. (12)-(13) more accurately and will demonstrate that they depend both on the radiation wavelength and the grazing angle of the incoming beam and not only on the correlation length of roughness.

3. Interrelation of diffraction channels

Once an X-ray wave has reflected from a rough surface, the power of the incident beam is distributed among four diffraction channels, namely, coherent reflection, coherent transmittance, diffuse scattering in vacuum, and scattering into the matter depth. The radiation flux distribution between different channels depends essentially on the roughness parameters (correlation length) and the value of the grazing incidence angle of the incoming beam.

Evidently, there is no essential problem to calculate numerically reflectance, transmittance, and integral scattering using Eqs. (7)-(10) , if one or another model of the PSD-function is chosen. Below, for illustrative calculations, we will apply the ABC-model, widely used in visible and X-ray optics, of the 2D PSD-function and the corresponding 1D one [18]:

Illustrative calculations based on the model (14) of the PSD-function will be performed in the next section. Here, we deduce simple and vivid analytic expressions that allow us to analyze the radiation flux distribution between diffraction channels for several limiting cases

First, we perform more accurate estimations of integrals in Eqs. (7)-(10) , limiting consideration of isotropic surfaces so that the last equality in Eq. (5) is valid. Taking into account that only the argument of the PSD-function depends on the azimuth scattering angle φ, we perform an integration over φ in Eqs. (7)-(10) in the following form:

It is reasonable to assume that the correlation length of roughness is incomparably larger than the wavelength of X-rays and that the grazing incidence and scattering angles are small: ${\theta }_0,\theta <<1$. Therefore, the contribution of small values of $\rho ~q^{-1}~\lambda$ to the integral on the right side of Eq. (15) is negligible. Then, we can use asymptotic expansion of the Bessel functions to perform integration on Eq. (15) in an explicit, approximate form:

Notice again that going from the 2D PSD-function integrated over the azimuth scattering angle to the 1D PSD-function is valid for isotropic surfaces, assuming that $q_0\xi >>1$ and $q\xi >>1$. These inequalities are obeyed for X-rays, while they may be invalid for visible light and, especially, at normal incidence of incoming radiation.

We introduce the following two dimensionless parameters:

These parameters are proven to determine totally all features of X-ray scattering by a rough surface. The physical sense of the parameter ${\mu }_0$ is evident. The angular width of the scattering diagram in the incidence plane is typically equal to $\Delta \theta \approx 2\lambda /(\xi \sin {\theta }_0)$, hence, the parameter ${\mu }_0\approx {\theta }_0/\Delta \theta$. Therefore, the condition ${\mu }_0>>1$ means that the scattering pattern width is small in comparison with the grazing angle of the incident beam, i.e., the scattered radiation propagates at relatively large angles with respect to the surface. However, if ${\mu }_0\le 1$, then the scattered beam “lies” on the surface. Evidently, we can expect an appearance of specific features in the scattering distribution in this case.

The parameter ${\mu }_c$ is also understandable. Suppose that an X-ray beam falls onto a surface at the grazing angle ${\theta }_0$ close to the critical angle of TER ${\theta }_c=|1-\epsilon {|}^{1/2}$, which is a typical situation for X-ray experiments. If the parameter ${\mu }_c$ is large ($\left|{\mu }_c\right|>>1$), the angular width of the PSD-function is small compared with the angle ${\theta }_c$, characterizing the typical variation scale of the electrodynamics factor placed in front of the PSD-function in Eq. (3). Hence, the shape of the scattering distribution is determined mainly by the PSD-function in this case. In contrast, if the parameter ${\mu }_c$ is small ($\left|{\mu }_c\right|\le 1$), the shape hardly depends on the electrodynamics factor, which results in the appearance of peculiar effects such as the Yoneda [20] and “anti-Yoneda” [10] peaks.

The parameter ${\mu }_c$ characterizes the roughness correlation length $\xi$, which we will call “small” if $\left|{\mu }_c\right|\le 1$ and “large” if $\left|{\mu }_c\right|>>1$. Because the parameter ${\mu }_c$ is proportional to the radiation wavelength, the roughness correlation length of a specific surface may be “large” for soft X-rays, while it is “small” for hard X-rays. As an illustration, a transition from a “large” correlation length to a “small” one ($\left|{\mu }_c\right|=1$) occurs at $\xi ~6-60$ μm at λ = 0.1 nm and at $\xi ~0.15-0.6$ μm only at λ = 10 nm, the critical value of ξ being larger for lighter materials.

We now represent the 1D PSD-function as ${\text{PSD}}_{1D}(p)={\sigma }^2\xi F(p\xi )$, where the dimensionless function F is assumed to decrease with increasing argument and to obey the normalization condition ${\displaystyle {\int }_0^{\infty }F(\tau )d\tau =1}$. In frame of the model (14), the function $F(\tau )$ has the following form:

Instead of integration over $q\in [0,k]$ in Eqs. (7)-(10) , we perform integration over the parameter $\tau =p\xi =|q-q_0|\xi /(2\pi )$ and re-write the equations in the form most suitable for subsequent analysis:

When deducing (19)-(22), we assumed that small scattering angles only give a contribution to the integrals such that the following inequality is valid: $k-q<<k$. Then, ${\kappa }_{-}(q)\xi \approx \sqrt{4\pi k\xi ({\mu }_0\pm \tau )}$ and ${\kappa }_{+}(q)\xi \approx \sqrt{4\pi k\xi ({\mu }_0-{\mu }_c\pm \tau )}$, where plus or minus is chosen according to the positive or negative value of the difference $q_0-q$. The upper integration limit $\xi q_0/(2\pi )~\xi /\lambda$ is typically very large for X-ray radiation and can be set to infinity.

Using Eqs. (19)-(22) , we consider several limiting cases depending on the values of the parameters ${\mu }_0$ and ${\mu }_c$.

Case 1: Large grazing incidence angle: ${\mu }_0>>1$

This case is typical for visible optics and corresponds to the situation discussed above, when the scattered radiation propagates at large angles to the surface. Maximal scattered radiation intensity is achieved in the direction of the specular reflectance, i.e., at $\theta ={\theta }_0$. The angular width of the scattering diagram is determined by the width of the PSD-function and is equal to $\Delta \theta =2\lambda /(\xi \sin {\theta }_0)$. Eliminating grazing incidence angles close to the critical angle of TER from consideration, i.e., assuming $|{\mu }_0-{\mu }_c|>>1$, neglecting the variable $\tau$ in comparison with the large parameters ${\mu }_0$ and $|{\mu }_0-{\mu }_c|$, and tending the upper limits in all integrals in Eqs. (19)-(22) to infinity, we immediately obtain the same well-known expressions (13). Therefore, the validity of the DW factor is determined only by the condition: the width of the scattering diagram should be small in comparison with the grazing incidence angle of the incoming beam. This condition is obeyed either for a large enough correlation length of roughness $\xi$ and the fixed grazing incidence angle ${\theta }_0$ or at a fixed correlation length and large enough angle ${\theta }_0$.

Case 2: Large correlation length and extremely small grazing incidence angle: ${\mu }_c>>1$ and ${\mu }_0<<1$

In this case, typical for FEL/XFEL mirrors, the scattering diagram “lies” on the surface and the scattering pattern is changed: maximum of the scattering diagram is shifted to the range of larger scattering angles exceeding ${\theta }_0$, and the diagram width $\Delta \theta \approx \sqrt{2\lambda /\xi }$ is independent of the grazing incidence angle ${\theta }_0$. Setting ${\mu }_0=0$ and neglecting the variable $\tau$ in comparison with the large parameter ${\mu }_c$ in integrals in Eqs. (19)-(22) , we obtain

The main feature of Eqs. (23)-(24) is that the total integrated scattering in vacuum is proportional to $\sin {\theta }_0$ in the first power and not to ${\sin }^2{\theta }_0$ as in case 1, described by Eq. (13), i.e., the scattered intensity is increased. This effect was experimentally observed in several works [10, 11 ]. Moreover, the scattered intensity (23) depends on the roughness correlation length. If the correlation length increases indefinitely while the grazing incidence angle ${\theta }_0$ is fixed, the considered case ${\mu }_0<<1$ will be eventually converted to the previous case ${\mu }_0>>1$ and the TIS value will be determined by Eq. (13) and will not tend to zero.

Notice that in the case of non-absorbing matter and $\epsilon <1$, the scattering into the matter depth disappears. The correction to the transmittance even in the case of absorbing matter is of a higher order of smallness compared with the correction to the reflectivity and the TIS values.

Case 3: Small correlation length and small grazing incidence angle: ${\mu }_c<<1$ and ${\mu }_0<<1$

In this case, the PSD-function varies slowly with the scattering angle. Therefore, the shape of the scattering diagram hardly depends on the electrodynamics factor placed in front of the PSD-function in Eqs. (3)-(4) . As a result, a number of specific features arise in the scattering pattern, such as the peak and “anti-peak” of Yoneda [20, 10 ].

Expending Eqs. (7)-(10) to a series on small parameters ${\mu }_c$ and ${\mu }_0$, we obtain

Equations (25) show that the radiation power scattered to both vacuum and matter depth is small and diminishes with decreasing correlation length of roughness. Corrections (26)-(27) to the coherent reflection and transmission are the sum of two terms, the first of which is independent of the correlation length and is nothing but the NC factor (12). Notice that for the reflective coating with small absorption ($\gamma /\delta <<1$), only the second summand in Eq. (27) is of first importance. The NC formula (12) is then valid only for vanishingly small correlation lengths $\xi <\lambda /2\pi \cdot {\left(\gamma /f_2/{\delta }^{3/2}\right)}^2$, where we assume that ${\theta }_0<<{\theta }_c$. In particular, in the case of a carbon-coated mirror, the photon energy E = 8 keV, the fractal parameter $\alpha =0.5$, and according to the last inequality, the correlation length should be smaller than 0.12 nm, which is physically meaningless.

Next, we briefly consider two cases, when the X-ray beam falls onto a mirror inside and outside the TER region, and we assume small absorption of X-rays in matter, namely, $\gamma =\mathrm{Im}\epsilon <<\delta =\mathrm{Re}(1-\epsilon )$.

Case 3a. Extremely small grazing angle: ${\mu }_0<<{\mu }_c<<1$

In this case, Eqs. (25)-(27) are simplified:

Both scattering components, as well as an increase in the transmittance, are conditioned by a reflectivity decrease. As in case 2, the TIS value is proportional to $\sin {\theta }_0$ in the first power, while the dependence on the correlation length is inverted compared with Eq. (23). Moreover, the scattering intensity is proportional to the dielectric constant variation at a surface $\delta =\mathrm{Re}(1-\epsilon )$. As a result, the total integrated scattering of hard X-rays from a surface of a heavy material (Au, W) is 7 – 10 times greater than that from a light one (C, B₄C, fused quartz) at the same roughness parameters. Evidently, the transmittance is equal to zero for non-absorbing material ($\gamma =0$) because X-rays fall onto a surface in the TER angular region in the case considered.

Case 3b. The incident beam falls out of the TER region: ${\mu }_c<<{\mu }_0<<1$

Expressions describing X-ray scattering can be recast in the following forms:

In this case, only the first term in Eqs. (26)-(27) , describing the re-distribution of the intensity between coherent reflectance and transmittance, plays the main role. The first summand in Eqs. (30)-(31) is nothing but the first terms of NC factor expansion in powers of the rms roughness σ. Therefore, the small-scale roughness increases the transmittance of X-rays out of the TER region (at ${\theta }_0>{\theta }_c$) due to the formation of an “effective” transition layer, aroused as a result of the roughness averaging. Scattering both in vacuum and into the depth of material is small and is mainly conditioned by a slight decrease in the transmittance. Nevertheless, we keep the small second summand proportional to $\delta =\mathrm{Re}(1-\epsilon )$ in Eq. (30) for the specular reflectance to demonstrate the transition to the NC regime of diffraction. To provide the fulfillment of energy conservation, we keep the small summands proportional to δ in Eqs. (29) and (31) as well.

In the considered case of the grazing angle exceeding essentially the critical angle of TER, the NC factor for the reflectivity practically coincides with the DW factor. However, the diffraction occurs by a different manner compared with case 1: a decrease in reflectance is explained by an increase in transmittance and not by radiation scattering in vacuum.

Expressions for coherent transmittance and reflectance, as well as for the total integrated scattering in vacuum and into the depth of matter, are summarized in Table 1 for a number of limiting values of the parameters ${\mu }_0$ and ${\mu }_c$. As above, we assume the small absorption of X-rays in matter such that $\gamma =\mathrm{Im}\epsilon <<\delta =\mathrm{Re}\epsilon$ and, moreover, simplify Eqs. (29)-(31) for clarity, taking into account that ${\theta }_0>>{\theta }_c$. One can easily verify that the expressions obtained provide the fulfillment of the energy conservation law for non-absorbing matter, namely, ${\text{TIS}}_{-}+{\text{TIS}}_{+}=\delta R+\delta T$. The table demonstrates clearly the interrelation between diffraction channels consisting of the following:

Table 1. The total integrated scattering in vacuum (${\text{TIS}}_{-}$) and into the matter depth (${\text{TIS}}_{+}$), as well as corrections to the coherent reflectance ($\delta R$) and transmittance ($\delta T$), calculated for a number of limiting cases.

View Table

Therefore, our consideration demonstrates that the interrelation of diffraction channels is not trivial even in the simplest case of a single rough surface. Notice that several sets of roughness with different correlation lengths can typically be recognized on a sample surface, especially if the sample represents a layered film deposited onto a substrate. Low-frequency roughness is conformal with the substrate roughness and is usually characterized by a large correlation length, while deposition results in the development of intrinsic film roughness with small correlation length of a fraction of a micron, as a rule [21, 22 ]. Therefore, in practice, we will observe a combination of the cases considered above.

4. Discussion

The plane of the parameters $\xi -{\theta }_0$ (correlation length – grazing angle of incident radiation) in Fig. 2 is divided into four areas corresponding to the limiting cases considered above and presented in Table 1. The effect of surface roughness on X-ray reflection is described by the DW factor in the depth of area 1 and by the NC factor in areas 3a – 3b near the abscissas axis. However, none of these factors is valid in areas 2 and 3a near the ordinate axis, where the grazing incidence angle is small compared with the critical angle of TER. Notice again that only extremely small grazing angles are of the most practical interest for the development of optics for modern FEL/XFEL beamlines.

 

Fig. 2 Scheme illustrating different limiting cases of X-ray diffraction from a rough surface.

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Up to this point, we have used the simplified Eqs. (23)-(31) to analyze qualitatively the specific characteristics of X-ray scattering in a number of limiting cases. We now consider the behavior of the specular reflectance and integral scattering in vacuum with varying correlation lengths more accurately on the basis of the exact numerical integration of Eqs. (7) and (9) .

First, we consider Fig. 3 , where the total integrated scattering in vacuum TIS and the correction $\delta R$ to the specular reflectance of X-rays (3 keV photon energy) from an Au-coated mirror are shown to be dependent on the correlation length of roughness. The grazing angle of an incident beam ${\theta }_0=50$ mrad is twice that of the critical angle of TER (${\theta }_{с}=|1-\epsilon {|}^{1/2}=24.8$ mrad). The model (14) of the PSD-function is used for calculation, where the fractal parameter is set to $\alpha =0.8$. The correlation length ($\xi \approx 0.33$ μm), when the parameter ${\mu }_0$ takes the value of unity, is indicated by the vertical dashed-dotted line in Fig. 3. Horizontal dashed lines show the values of the DW and the NC factors calculated via Eqs. (12)-(13) . We choose gold coating as an illustration because this material is characterized by a large polarizability and thus, the conditions for case 2 can be more easily satisfied.

 

Fig. 3 The total integral scattering (TIS) and specular reflectance ($\delta R$) of an Au-coated mirror at the photon energy E = 3 keV and with the grazing angle of an incident beam of ${\theta }_0=50$ mrad versus the correlation length of roughness. The TIS and $\delta R$ values were normalized to the DW factor (Eq. (13)). The model (14) of the PSD-function was used, assuming the fractal parameter to be equal to $\alpha =0.8$. Curves 1 and 2 were calculated using simplified analytic expressions (25) and (27). The horizontal dashed lines show the value of $\delta R$ calculated for the limiting cases of very large and extremely small correlation lengths (Eqs. (12) and (13) ). The vertical dashed-dotted line shows the value of the correlation length when the parameter ${\mu }_0=1$.

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If the correlation length is large, i.e., the parameter ${\mu }_0>>1$, the simplest case 1 takes place, when neither the TIS value nor $\delta R$ depends on the correlation length and both of them are described by the DW factor. Figure 3 demonstrates that “much more” in the last inequality means “more by a factor of two, at least”. Decreasing correlation length results in going from region 1 to region 3b in Fig. 2. The characteristics of the scattering are then changed: the correction term to the specular reflectivity is decreased and is described by the NC factor at very small ($\xi <20-30$ nm) correlation lengths. The total integrated scattering tends to zero proportionally to $\sqrt{\xi }$. The solid curves in Fig. 3 were calculated numerically by using exact Eqs. (7) and (9) , and dotted curves 1 and 2 – using simplified Eqs. (25) and (27) . As seen, the simplified expressions are in reasonably good agreement with numerical calculation: the correction to the specular reflectivity $\delta R$ is ideally described by curve 2 everywhere in region 3b, while curve 1 is rather close to the TIS calculated numerically and coincides with it at the correlation length $\xi <30$ nm. Notice that the TIS value is equal to zero in the Nevot-Croce approximation.

We call attention to the maximum on the curve of the total integrated scattering at the correlation length $\xi ~1$ μm, where the TIS value exceeds $\delta R$. This fact means that the integral reflectivity $R_{\Sigma }=R_{spec}+{\text{TIS}}_{-}=R_F-\delta R+{\text{TIS}}_{-}$ is greater than the Fresnel reflectivity, i.e., a rough surface directs a larger part of an incident beam back to vacuum compared with an ideally smooth surface, while the effect is observed only in the limited interval of the correlation lengths. Essential exceeding of the experimental reflectivity measured out of the TER region over the Fresnel one was observed, e.g., in [23].

The dependence on the correlation length of the specular reflectance and the integrated scattering becomes far more complex at extremely small grazing angles of an incident beam. This case is illustrated by Fig. 4 for the same Au-coated mirror and photon energy E = 3 keV as above, but at a grazing angle only as small as 1 mrad. Notice that the parameter ${\mu }_0$ is very small (approximately 0.1) at the maximal correlation length $\xi =100$ μm, as indicated in the figure; thus, the simplest case, case 1 can be realized only at extremely large correlation lengths of several cm or even more. Therefore, case 2 takes place in the example considered, and decreasing the correlation length results in increasing scattering, in accordance with Eq. (23). As seen, the total integrated scattering may be 4.5 times larger than the value predicted by the DW formula (13), and the correction to the specular reflectance may exceed it by a factor of 9.5. Notice that the maximal difference is observed at a value of the parameter ${\mu }_{с}~1$, i.e., in going from one limiting diffraction regime to another one. An increase in the integrated scattering compared with the DW value with decreasing grazing angle was observed in a number of papers (e.g., see [10, 11 ]).

 

Fig. 4 The total integral scattering (TIS) and specular reflectance ($\delta R$) of an Au-coated mirror at the photon energy E = 3 keV and the extremely small grazing angle of an incident beam ${\theta }_0=1$ mrad versus the correlation length of roughness. The TIS and $\delta R$ values were normalized to the DW factor (Eq. (13)). The model (14) of the PSD-function was used, assuming the fractal parameter to be equal to $\alpha =0.8$. Curves 1–4 were calculated using simplified analytic expressions (23), (25), (27). The horizontal dashed lines show the value of $\delta R$ calculated for the limiting cases of very large and extremely small correlation lengths (Eqs. (12) and (13) ). The vertical dashed-dotted line shows the value of the correlation length when the parameter ${\mu }_{с}=1$.

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A decrease in the correlation length down to $\xi \approx 1.34$ μm (then the parameter ${\mu }_{с}=1$) corresponds to going from region 2 to region 3a in Fig. 2. As a result, the dependence on the correlation length is inverted: both TIS and $\delta R$ are decreasing proportionally $\sqrt{\xi }$ with decreasing correlation length, in accordance with Eqs. (28). The specular reflectance tends to the value predicted by the NC factor and coincides with it at vanishingly small correlation lengths of less than 1 nm, a value that looks unrealistic.

Curves 1-4 were calculated with the use of simplified Eqs. (23) and (28) . They are in much better agreement with the numerical calculations compared with the DW or NC factors and can be used for the qualitative analysis of X-ray scattering by roughness at extremely small grazing angles.

5. Conclusions

Peculiarities of X-ray diffraction from a rough surface at extremely small grazing angles of an incident beam were theoretically studied. Until now, this case had not been discussed thoroughly in the literature.

Analysis of the interrelation of four diffraction channels (coherent reflectance, coherent transmittance, diffuse scattering in vacuum, and scattering into the matter depth) was performed for different limiting cases of large and small correlation lengths of roughness and large and extremely small grazing angles of incident radiation.

The validity of the DW factor was shown to be determined only by the following condition: the width of the scattering diagram should be small in comparison with the grazing incidence angle of an incoming beam. This condition is obeyed either for large enough correlation lengths of roughness $\xi$ and the fixed grazing incidence angle ${\theta }_0$ or at fixed correlation length and a large enough angle ${\theta }_0$.

The NC factor for the specular reflectivity proved to be valid for small-scale roughness only if the X-ray beam falls onto a surface outside the TER region. Under an X-ray incidence inside the TER region, the NC factor was shown to hold at a physically meaningless correlation length of roughness smaller than a fraction of a nm.

Both the DW and NC factors were demonstrated to describe improperly the features of X-ray diffraction at extremely small grazing angles. More appropriate simple analytic expressions for the specular reflectivity and total integrated scattering in vacuum were obtained instead of the DW and NC formulas.

Transformation of one limiting diffraction regime into another one with variation in the correlation length of roughness was analyzed. It was shown than the total integrated scattering and correction to the specular reflectance may be 5-10 times larger than the value predicted by the DW formula.

The results obtained in the paper will be used as a theoretical basis for subsequent experimental study of different X-ray diffraction regimes and investigations of interaction of highly intensive and totally coherent FEL and XFEL radiation with rough reflecting surfaces at extremely small grazing angles.