First, let’s consider the following two reflection schemes from a variety of wave reflectivity models. The fist scheme is a conventional reflection scheme of the plane electromagnetic wave from planar mirror (see Figure 1). According to Figure 1, the equation holds, where the angles and are the angles of incidence and reflectance, respectively. The wave vectors of incident and reflected beams lie in this figure on the different sides of the normal line to the interface between two media. The vectors, and are unit vectors along positive directions of Ox-, Oy-, and Oz-axis, respectively.

The second scheme is a retro-reflection scheme of the plane electromagnetic wave (see Figure 2). According to Figure 2, the equation holds, where the angles and are the angles of incidence and retro-reflectance, respectively. The wave vectors of the incident and retro-reflected beams lie on the same side of the normal line to the interface between two media. These two reflection schemes presented in Figure 1 and Figure 2 are indistinguishable in the case of normal incidence when.

One may note that the coplanar geometry of the grazing-incidence hard X-ray backscattering diffraction (GIXB, see the scheme in Figure 3 and translations [30] [31] of our original papers) contains both coplanar reflection schemes presented in Figure 1 and Figure 2. The GIXB technique is a dynamical Bragg diffraction, which holds under

the conditions of total external reflection, and is extremely sensitive to variations of the spacing period of diffracting lattice planes, as well as to variations of the wavelength of incident X-radiation and the angle of incidence of the primary beam. Paper [32] contains the review of the theory and applications of GIXB technique in non-coplanar mode. Particularly, the specular beam suppression and enhancement phenomena are investigated in [32] for the case of non-coplanar GIXB by the crystal with a stacking fault.

The plane presented in Figure 3 is the X-ray entrance surface of the single-crystal wafer. Ox-axis is normal and Oy-axis is parallel to sample diffracting lattice planes. The GIXB setup performs when the magnitude of Bragg angle is close to 90˚, which leads to fulfillment of the following condition, where is a reciprocal lattice vector normal to real space diffracting lattice planes, as it is shown in Figure 3. The reciprocal space vector satisfies the orthogonality conditions:, where and are the unit vectors along the positive direction of Oy-axis and Oz-axis, respectively, is a spacing of the diffracting lattice planes of wafer, which are normal to the X-ray entrance surface. Let and are the angles of the wafer rotation around Oy and Oz coordinate axes, respectively:

where is the first-order kinematic Bragg angle, and is a unit vector along the positive directions of Ox-axis. If the exact condition of coplanar GIXB hold for region, and the angle of incidence of X-ray beam satisfies the first-order Bragg diffraction by lattice planes, then

Equation (4b) is the exact condition for the particular case of coplanar GIXB setup derived from general GIXB equations [29] [30] . In this case, the angle of incidence, at which proposed planar mirror reflects X-rays in reverse way, is close to right angle, unlike the homogeneous planar optical mirror, which does this only if the mirror is exactly perpendicular to the wave front, having a zero angle of incidence. The vector presented in Figure 3 is the wave vector of the incident X-ray beam, and the vectors and are the wave vectors of vacuum retro-reflected wave and specular (conventional reflected) wave, respectively. In this coplanar diffraction case the reciprocal space vector, as well as the vacuum wave vectors, , and, lie in the same plane which is parallel to x0z plane. The wave vector makes an angle with wave vector (see Figure 3). Consequently, there are no travelling waves along Oy-axis, so the mentioned wave vectors satisfy the following equations:

where

and are the wave number and frequency of the incident plane wave, respectively, and c is the speed of light in vacuum.

Now, let’s clarify the performance criteria of the planar grazing-angle incidence hard X-ray retro-reflector (GIRR):

1) A grazing-incidence planar retro-reflector is a mirror with a plane surface that totally reflects (or diffracts) the electromagnetic wave back to its source, and, consequently, the conventional mirror wave is completely suppressed (that is the reflectivity coefficient of conventional mirror wave is equal to zero, see Figure 2).

2) The wave vectors of incident and retro-reflected (or backward diffracted) waves are strictly collinear (see Figure 2).

Therefore, all the types of non-coplanar grazing-incidence hard X-ray diffraction geometries (for example, see [32] ) are not applicable for the GIRR since they do not satisfy the requirement 2). Fortunately, it is possible to transform the coplanar GIXB scheme given in Figure 3 into the required GIRR scheme as given in Figure 2.

・ Thus, the main goal of presented theoretical research is the elaboration and estimation of certain physical and technical solutions that can help to completely suppress a conventional mirror wave and, as a consequence, to increase the retro-ref- lectivity coefficient of the wave with wave vector (see Figure 3).

We propose a solution to this problem through the coplanar GIXB that holds in a single-crystal wafer covered by an ultra-thin, non-diffracting layer of low-absorbing material (see Figure 4). The total retro-reflection can be achieved by ensuring that the

intensity of conventional mirror wave is reduced by destructive interference between the wave reflected from wafer and the wave reflected from cover layer. A scheme of proposed GIRR is presented in Figure 4, where the plane is the interface between single-crystal wafer and thin non-diffracting cover layer of thickness, so the plane is the X-ray entrance surface. The other parameters, angles, and vectors are the same as in Figure 3 (see text in subsection 2.1 for the details).

In a particular case of the specular beam suppression mode [32] [33] , the scheme of coplanar grazing-incidence X-ray backscattering diffraction (GIXB) presented in Figure 4 corresponds to total retro-reflection.

The mentioned above grazing-incidence retro-reflection from the proposed planar X-ray optical device is an X-ray analogue of the reflection phenomenon when a light beam is propagating in a photonic crystal, and is incident upon the plane interfaces between the crystal and a uniform dielectric [34] . The authors of paper [34] showed that neither the phase velocity nor the group velocity directions of reflected beam satisfies Snell’s law. The system exhibits remarkable and unusual reflection effects. In particular, total internal reflection is absent except at discrete angular values. The direction of the reflected beam can also be pinned along special crystal directions, independent of the orientation of the interface. At grazing incidences, strong backward reflection occurs (see figure 4 in [34] ). These effects may be important for creating integrated photonic circuits, and for on-chip image transfer [34] .

The study of wave propagation in one-dimensional periodic media was pioneered by G. Floquet in 1883 [35] . This theory was extended for three-dimensional periodic media by F. Bloch in 1928 [36] . Bloch proved that waves in such a medium can propagate without scattering, their behaviour governed by a periodic envelope function multiplied by a plane wave.

The same technique can be applied to electromagnetism by considering Maxwell’s equations as an eigenvalues and eigenfuctions problem by analogue with Schrödinger’s equation (see [30] - [33] ). Using such approach, we treat the dynamical diffraction of X-rays by a set of diffracting lattice planes of perfect crystal as the superposition of X-ray “Bloch waves” in a medium with harmonically varying dielectric susceptibility (polarizability). Therefore, the problem of X-ray wave field propagation through an arbitrary set of diffracting lattice planes of periodic structure with the symmetry centre can be brought mathematically to an analogous problem of the solution of stationary Schrödinger equation with cosine-like coefficient [30] . Our method is based on eigenvalues and eigenfunctions problem solution technique. This technique is applicable for both media―the cover layer and crystalline wafer. The common hard X-ray dynamical diffraction theories [10] - [13] did not use this advanced technique. We transform the Maxwell’s equations to the set of differential equations [30] involving the Mathieu or Hill equations [37] - [39] .

We seek a non-trivial y-component of the stationary X-ray wave field strength inside thin non-diffracting cover layer in the following Fourier integral form:

where

is equal to either or, where corresponds to the electric field strength vector of the s-polarized X-ray wave field inside a thin non-diffracting cover layer, corresponds to the magnetic field strength vector of the p-polarized X-ray wave field inside a thin non-diffracting cover layer. are the unknown weight functions. is the value of polarizability averaged throughout cover layer.

If absorption is not taken into account, and crystalline wafer has a center of symmetry, then the following relations describe components of the stationary X-ray wave fields inside the crystalline wafer [29] [30] :

where

is the real part of the crystal polarizability averaged throughout the crystal lattice cell, is the real part of the Fourier coefficient of the crystal polarizability corresponding to a set of diffracting lattice planes. Functions and are the ordinary Mathieu functions of first kind to which eigenvalues correspond [35] - [37] . and are the unknown weight functions which have to be determined from boundary conditions.

The choice of germanium single-crystal wafer for use in GIRR is stipulated by the following reason:

・ Germanium has relatively low melting point and was the first material that could be grown dislocation free. With the improvement of crystal growth, dislocation-free wafers became available and are nowadays the standard in the case of 200 and 400 mm diameter germanium and silicon substrates [40] - [43] .

The selection of beryllium as a base material for non-diffracting cover layer is stipulated by the following:

・ Beryllium is a metal that has low density and low atomic mass, and hence very low absorption level of X-rays, making beryllium the preferred choice for X-ray tube windows where the energy low absorption is desired.

・ The surface of beryllium metal, like the aluminum, is covered by a thin layer of oxide that helps protect the metal from attack by acids. Due to this coating, corrosion and oxidation in air is minimal up to temperatures of about 760˚C. Beryllium metal does not react with water or steam, even if the metal is heated to red heat.

The deposition of the beryllium thin layer can, for example, be realized via Atomic Layer Deposition (ALD) technology:

・ ALD technology allows semiconductor manufacturers to choose from a wide field of deposition precursors for the application of any thin film in use today on the surface of a large wafer with atomic layer precision.

・ ALD technology can also be used to construct complex, compound film structures with a level of control and conformity that was previously unavailable or impractical.

・ ALD offers the opportunity to create precisely controlled ultra-thin films coating up to 300 mm diameter single-crystal wafers.

・ General ALD process benefits are the excellent process control with wafer-to-wafer repeatability with accuracy less than ±1%, up to 300 mm diameter wafer with typical uniformity accuracy less than ±2%, excellent step coverage even inside high aspect ratio structures, low film impurities, etc. (for example, see [44] ).

The improvement of tensile strain, crystal quality, and surface morphology of germanium wafer after its covering with thin beryllium layer can be achieved by the thermal annealing. For example, the tensile strain and crystal surface quality, of 500 nm thick Ge films were improved after rapid thermal annealing at 900˚C for a short period less than 20 s [45] . The films were grown on Si(001) substrates by ultra-high vacuum chemical vapour deposition. These improvements are attributed to relaxation and defect annihilation in the Ge films. However, after prolonged more than 20 s rapid thermal annealing, tensile strain and crystal quality degenerated. This phenomenon results from intensive Si-Ge mixing at high temperature.

The solid solubility of beryllium in germanium is extremely small up to temperatures of about 900˚C, and, consequently, the mutual diffusion process of atoms between germanium wafer and beryllium layer has very low rate (see Fig. 15.11 in [46] ).

・ This type of properties opens the opportunity of removing the interface tensile strain and improving the crystal quality and morphology of subsurface region of germanium wafer covered by thin beryllium layer through the application of a longer thermal annealing method at, for example, 700˚C practically without any appreciable inter-diffusion between these two materials. The essential reduction of the tensile strain along the Be-Ge large interface will increase the survivability of the coverage layer.

The main problem of experimental research involving the grazing-incidence hard X-ray optics is the small collection efficiency due to the small critical angle, since the collection area of grazing incidence mirrors is the projection of the mirror surface onto the aperture plane.

A Japanese optics company, JTEC Corporation, fabricates the mirrors for synchrotrons and other X-ray laser research facilities [47] such as Japan’s Spring-8 Angstrom Compact Free-Electron Laser (SACLA) and the European X-ray Free-Electron Laser (EXFEL), located in Hamburg, Germany and due to come online in 2017. Recently, scientists installed new mirrors to improve the quality of the X-ray laser beam at the Department of Energy’s SLAC National Accelerator Laboratory. The meter-long mirrors are the ultimate in flatness, smooth to within the height of one atom or one-fifthof a nanometer [48] . Each mirror is made from an individual silicon crystal, artificially grown in a lab. After the mirror is polished with conventional techniques, the company uses a process called elastic emission machining, where a jet of ultra-pure water containing fine particles removes any remaining imperfections atom by atom. The same advanced technology may be used for the GIRR manufacturing.

An alternative method to long plane retro-reflectors is a nesting multiple GIRRs that also increases the collection area, and a thin metallic coating will further increase the critical angle and thus collection efficiency (see Figure 5).

Let’s discuss a diffraction in case where the exact condition of coplanar GIXB holds inside bulk germanium wafer without cover layer, and the angle of incidence of X-ray primary beam satisfies the first-order Bragg diffraction by the lattice planes:

The reflectivity coefficients from a single-crystal wafer and visibility of images obtained from subsurface non-diffracting regions of wafer are described by the following formulas:

where n = 0, 1, 2, the function is the X-ray reflectivity coefficient of a single-crystal wafer without a cover layer in the case when the Bragg condition is not satisfied, the functions and are the retro-reflectivity and reflectivity coefficients, respectively [30] [31] . The graphs of reflectivity coefficients and visibility of images depending on Bragg angle are presented in Figure 6.

These graphs are computed using the values of the incidence angle taken from the angular region presented below:

Equation (28a) can be rewritten in the following form using Equation (3):

where, is the germanium lattice constant at the room temperature, and [49] [50] .

The polarizability Fourier components, , , and are calculated based on the method of computer simulation of experimental results for the complex kinematic scattering parameters of X-rays [51] . The maximum value of the retro-reflectivity coefficient corresponds to , and the visibility function (see Figure 6). Also, note that, so the mirror wave field reflected from crystal wafer does not get completely suppressed.

One may write the following relation between and polarization modes of electric wave field propagating inside the crystalline wafer [10] - [13] :

Since GIXB setup performs only if the magnitude of Bragg angle is close to 90˚, then the square of polarization factor, and, according to (29), the intensities of X-ray wave fields inside the crystalline wafer are practically the same for both and polarization modes.

The diffraction angles can be determined through the system of analyzer crystals with an accuracy of at least, and can be obtained experimentally under the conditions of high diffraction intensity in grazing-angle incidence geometry, notwithstanding the fact that the cap layer thickness is less than 5 nm [52] . Therefore, the presence of a thin cap layer may essentially change the reflectivity properties of single- crystal substrate when the GIXB holds inside.

Here we analyze a particular case of coplanar GIXB by germanium single-crystal wafer covered with thin non-diffracting beryllium layer. Our goal is to find an opportunity to reduce the common reflected wave and increase the retro-reflected one.

The graphs of X-ray reflectivity coefficients, , as well as the retro-reflectivity coefficient and visibility function, are presented in Figure 7. These graphs depend on the thickness T of beryllium cap layer while the coplanar GIXB by lattice planes holds inside bulk germanium single-crystal wafer for the particular case of kinematic Bragg angle.

Curves in Figure 7 correspond to the incident X-ray wavelength and to absolute values of the electric susceptibility coefficients for beryllium layer and germanium wafer presented below:

, ,

, ,

,.

The coefficients, , , and visibility are oscillating functions with slowly damping amplitude (see Figure 7). Coefficients and have the same period and are mutually shifted for a half of a period. The zeros of reflectivity coefficient correspond to certain values of cover layer thicknesses, where, ,. In this particular case when, the identity holds because a complete suppression of the reflectivity coefficient in the direction of conventional mirror wave happens due to destructive interference between the X-ray wave reflected from crystal wafer and X-ray wave reflected from the cover layer. Also, in this particular case, the retro-reflectivity coefficient has local maximums, and visibility function is . It can be noted that the following approximate relations hold for the neighbour extremal values since these coefficients have the slowly damping amplitudes:

The relation (30) can be rewritten in the following form:

Hence, a small changes of the cover layer thickness equal to leads to essential redistribution in magnitudes of reflectivity and retro-reflectivity coefficients. Fortunately, the advanced nano-technologies of the present day allow the layer deposition and the thickness control with less than 1 nm accuracy. The magnitudes of local maximums of the retro-reflectivity coefficient are greater than the magnitudes of the neighbour local maximums of reflectivity coefficients .

The comparison of the corresponding graphs presented in Figure 6 and Figure 7 brings to the following relations:

The relation (32c) can be rewritten in the following form:

where j = 0, 1 (see Figure 7). The enhancement of retro-reflectivity coefficient of a single crystal wafer happens if it has a thin non-diffracting cover layer (see Equation (33)). This phenomenon is caused by the constructive interference between the X-ray wave field retro-reflected from crystal wafer and the X-ray waves multiply reflected from the top and bottom surfaces of the cover layer.

Now, let’s consider the X-ray reflectivity and retro-reflectivity coefficients and visibility function depending on the Bragg angle. In this particular case, the coplanar GIXB by lattice planes holds inside bulk germanium wafer covered with a thin non-diffracting beryllium layer of thickness. The corresponding graphs are presented in Figure 8. It can be noted from Figure 8 that the extreme values

of the reflectivity graphs, , as well as of the visibility function graph correspond to the magnitude of Bragg angle.

We present below three-dimensional graphs of the retro-reflectivity and reflectivity coefficients, as well as the visibility function depending on both variables and T (see Figures 9-12). According to the graphs presented in Figures 9-12, the following limit equations hold:

where T is an arbitrary value of the cover layer thickness. Note that the maximums on oscillating graphs of reflectivity and retro-reflectivity coefficients and of visibility function correspond to kinematic Bragg angle (see Figures 9-12).

The modern laboratory X-ray source has a high intensity X-ray beam close to 10⁹ photons s⁻¹ focused on the sample surface by using a Cu rotating anode X-ray generator operated at 40 kV - 30 mA (1.2 kW), according to paper [52] . The design, implementation, and performance of an X-ray monochromator with ultra-high energy resolution of at 14.41 keV is given in [53] .

The images of non-diffracting nano-sized subsurface inclusions, obtained by the hard X-rays scattering process, show a high-intensity background, so, besides a sufficient intensity and energy resolution, the advanced hard X-ray image registration system needs an all-optical filter that removes unwanted background or optical noise from the conventional specular wave and essentially increases the visibility of the informative signal.

In the retro-reflection performance mode, a planar GIRR removes the primary radiation from the common reflected beam propagating in the direction of the wave vector (see Figure 4). Consequently, using the proposed experimental scheme, researchers will be able to register weak signals scattered by non-diffracting inclusions located at wafer’s subsurface region (if there are any).

Finally, we can conclude that single crystal wafer covered by a thin cover layer with a certain thickness performs like an X-ray optical noise (or background) filter. Moreover, simultaneously with the increase in the magnitude of the visibility function, a diminishing of certain effects occurs, for example, in the probability of data read-out errors and data read-out time during the digital data read-out procedure from X-ray optical data carrier preliminary covered by a thin non-diffracting protective layer [33] .