1. Introduction

X-ray Fabry–Perot resonators (XFPR) were first proposed in 1967 [1,2]. In contrast to the optical variety, an XFPR implements back diffraction from perfect crystals in which the Bragg angle approaches $\textrm{9}{\textrm{0}^\circ }$. The excited diffracted waves reflect back and forth within a gap between two crystals, resulting in interference and, consequently, X-ray resonance. By measuring the temporal response of Fabry-Perot interferometers, the storage of X-ray photons inside the X-ray resonators made of Si crystals and $\mathrm{\alpha }$-$\textrm{A}{\textrm{l}_\textrm{2}}{\textrm{O}_\textrm{3}}$ crystals, were observed by Liss et al. [3] and Shvyd’ko et al. [4], respectively. However, insufficient longitudinal coherence resulting from rather poor energy bandwidth of incident X-ray obstructed to directly observe resonant fringes from X-ray resonators.

Observable resonance fringes were only achieved in experiments after the longitudinal coherent length of the X-ray light source was taken into serious consideration. Chang et al. [5] first successfully observed resonance fringes from an XFPR by using Si (12 4 0) back diffraction at an incident X-ray energy of 14.4388 keV with a spectral resolution of 0.36 meV. The longitudinal coherent length of 1717µm allowed for sufficient coverage of the lithographically patterned XFPR.

Several X-ray optical devices were developed thereafter, including the multi-plate XFPR for pico-second ultra-fast clock [6], anomaly focusing of compound refractive lens (CRL) from X-ray back diffraction [7], single mode X-ray resonator and its application for a meV high-resolution monochromator. [8–10].

However, there remain practical challenges in developing advanced X-ray resonators. The intrinsic absorption and multiple diffractions of Si crystals degrade the reflectivity, resulting in poor optical finesse. Alternatively, Huang et al. [11] theoretically proposed to use diamond crystal due to its higher reflectivity than that of silicon. Tsai et al. [12] experimentally demonstrated the sapphire-based X-ray resonators in which only two-beam (0 0 0 30) back diffraction was excited. Nevertheless, deep lithography remains difficult and costly when manufacturing XFPR on these alternative materials. Inclined incident geometry was next proposed to reduce linear absorption. This geometry taken for the Bragg incidence (12 0 0) [13] and for the Laue incidence (0 -4 0) [14] were conducted with some degree of success.

Here, we suggest an alternative diffraction geometry. In contrast to normal-incidence geometry, X-ray back diffraction results from a grazing incident angle on the crystal surface and this is referred to as grazing-incidence X-ray back diffraction (GIXBD). The absorption coefficient is decreased while the incident X-ray is near the critical angle of the crystal. The multiple diffractions are consequently suppressed because of this surface-confined characteristic. An immediate advantage of surface confinement is to lower the fabrication barrier of deep lithography for complex devices.

The special properties of back diffraction at high asymmetry angles had been reported [15]. Si (008) reflection with an asymmetry angle of $\textrm{88}\textrm{.}{\textrm{5}^\circ }$ was used as a dispersive element for X-rays close to the back diffraction energy of 9.1315 keV [16]. It was later been integrated into an inelastic X-ray scattering setup with sub meV energy resolution [17].

Nevertheless, the novel diffraction phenomenon of GIXBD as the incident angle enters into the critical specular reflection region, and the miscut angle approaching $\textrm{9}{\textrm{0}^\circ }$ has not been fully treated by dynamical theory of X-ray diffraction in detail before. In this article, we perform a numerical simulation of Si (12 4 0) back diffraction based on the algorithm proposed by Stetsko and Chang [18] for solving dynamical X-ray diffraction equations. Only two-beam case is investigated. Comparisons between the normal and the grazing incidence are presented, including dispersion surfaces, resultant diffraction and angular dispersion rate (ADR).

2. Dynamical treatment of GIXBD

The dynamical theory of X-ray diffraction describes the interaction of X-ray waves inside a large, perfect crystal. The translational periodic nature of the crystal implies that the solutions for multiple scatterings of X-ray waves inside the crystal satisfy the forms of the Bloch waves. Diffractions occur when Bragg’s law is satisfied, thus conserving the wave moment of the incident beam with the diffraction beam through the reciprocal lattice vectors of the crystal. As a result, there is a set of coupled equations of wavefields corresponding to the interactions of N-beam (m = 0, 1, …, N − 1) diffractions inside the crystal. The fundamental equations are formulated in the following conventional forms [19]:

The fundamental equations are 3N coupled equations. Traditionally, these 3N equations can be solved by taking an approximation for the dispersion of ${K_{{h_m}}}$. By matching the boundary conditions of electric and magnetic fields inside and outside the crystals, one can in theory obtain the measurable diffraction intensities outside the crystals. For examples, such as cases only considering one incident beam and one diffracted beam (a two-beam system), the procedure mentioned above has achieved great success and forms the basis of modern X-ray optical devices [20]. However, this equation is difficult to solve for more complicated systems, such as multi-beam diffractions and grazing incidence geometry, particularly when considering polarization of X-rays.

As with most dynamical systems, the dynamical behavior of X-rays inside the crystal gives rise to an eigenvalue problem that can be solved numerically. Stetsko and Chang [18] reformulated Eq. (1) by adopting reference Cartesian coordinates where the z axis is defined parallel to the crystal surface’s outward normal vector n. The wavevectors and electric fields inside the crystal are accordingly decomposed into x, y, and z components. By adopting $\hat{{\boldsymbol x}}$, $\hat{{\boldsymbol y}}$, and $\hat{{\boldsymbol z}}$ as unit vectors along the x, y, and z axes, respectively, the incident wavevector can be explicitly expressed as:

The wavevectors in the crystal are expressed as:

The ${z_j}$ in Eq. (3) give rise to the dispersion of ${{\boldsymbol K}_{{{\boldsymbol h}_{\boldsymbol m}}}}$, owing to the interactions of electric fields inside the crystal. It is convenient to assign the reciprocal lattice O as the origin of the reference coordinate and the x axis as the normal of the incident plane, and the y axis is accordingly defined as $\hat{{\boldsymbol z}} \times \hat{{\boldsymbol x}}$. The fundamental equations (1) can be rearranged to give the eigenvalue equations [18]:

Figure 1 illustrates the diffraction geometry on the (y, z) plane of a two-beam exact back diffraction of Si (12 4 0) at incident energy 14.4388 keV. Here, O and G are the reciprocal points of the origin, (0 0 0) and (12 4 0), respectively. The miscut angle ${\theta _m}$ is defined as the angle between the surface normal and the G vector.

 

Fig. 1. Schematic diagram of exact two-beam back diffraction in Cartesian coordinates expressed in real and reciprocal spaces simultaneously. Here, ${z_1}$ and ${z_2}$ represent the excited tie points derived from solutions of the fundamental equations.

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The two blue dotted circles centered at O and G with radius k (the amplitude of wavevectors in air) represent the possible incident and diffraction wavevectors, respectively, outside the crystal. The Laue points La are defined as the intersections of the two dotted circles. The two solid circles centered at O and G with radius K (the average amplitude of wavevectors inside the crystal), represent possible incident and diffraction wavevectors, respectively, inside the crystal. K and k are related by the non-dispersed refractive index ${n_0}$, K =${n_0}$k. We note that ${n_0} \cong 1 + \frac{1}{2}{\chi _{O - O}} = 1 - \frac{1}{2}{\theta _c}^2$, where ${\theta _c} = \textrm{0}\textrm{.121}{\textrm{9}^\circ }$ is the critical angle of Si at 14.4388 keV. The dynamical interactions inside the crystal disperse the distribution of ${K_O}$ and ${K_G}$ near the regions where the two solid circles touch, as shown in red, and this is referred to as the dispersion surface. The corresponding refraction index in the region is dispersed accordingly, as will be explained in a later section.

The reciprocal lattice points O and G are expressed as (0, 0, 0) and (X_G, Y_G, Z_G). The strongest diffraction intensity often occurs at the middle point between O and G, which is called the Lorentz point ${L_O}$.

As shown in Fig. 1, the simulation conventionally starts from the entrance point E defined as (X_c, Y_c, Z_c) by drawing a straight line parallel to the surface normal (z axis) and passing through the Laue point ${L_a}$.

The entrance point E can be explicitly expressed as:

In order to satisfy the tangential continuity of wavevectors at the interface, the tie points lie along the plumb line at the intersections with the dispersion surfaces and connect to reciprocal lattice points. These constitute the set of wavevectors inside the crystal ${K_{O1,\; 2}}$ (${K_{G1,\; 2}}$), as depicted by the purple dashed lines in Fig. 1.

As the incident angle ${\theta _i}$ increases, the black line moves horizontally along the y axis, thus creating new tie points. The green line in Fig. 1 shows the new ${\theta _i}$, which forms a new set of tie points, and ${\theta _f}$.

Despite that we treat the grazing-incidence diffraction such that ${\theta _m}$ approaches $\textrm{9}{\textrm{0}^\circ }$, the calculation can be adopted for any miscut angle without modification, including ${\theta _m} = $0° for normal-incidence cases.

The wavefields outside the crystal can be derived by matching the boundary conditions of the electric and magnetic fields at the interface [18]. The polarization of incident X-rays, the specular reflection near the critical angle, and the beam divergence must be taken into consideration. In particular, at a grazing incidence, the contribution of specular reflection strongly affects the manner of diffraction, as will be seen later. To conserve the total energy, the intensity of the $m$-th diffraction beam is expressed as:

3. Two-beam grazing-incidence X-ray back diffraction

To begin, we perform a numerical simulation with an exact two-beam Si (12 4 0) X-ray back diffraction, which has been extensively investigated in details in normal-incidence geometry by experiments [22] and theoretical simulation [23]. The incident X-ray energy is fixed at 14.4388 keV and the Bragg angle ${\theta _B}$ is $\textrm{89}\textrm{.878}{\textrm{1}^\circ }$. The critical angle ${\theta _c}$ of Si at this energy is $\textrm{0}\textrm{.121}{\textrm{9}^\circ }$, as mentioned earlier. The crystal thickness is $100\,\mathrm{\mu}\mathrm{m}$ directed downward along the z axis and the crystal is considered to be infinite in extent along the x and y axes. Moreover, $\mathrm{\sigma }$-polarization of the incident X-rays is assumed, in which the electric field is perpendicular to the scattering plane. $\phi $ is set as ${\textrm{0}^\circ }$.

The simulations were implemented under MATLAB environment. The simulations were started by assigning the geometry parameters, electric susceptibility, etc. in the matrix Q of Eq. (6). The eigenvalues ${z_j}$ of Eq. (5) and eigenvectors $E(j )$ were solved accordingly by using linear algebra, as described in section 2. By matching boundary conditions, one can obtain the diffraction intensities outside the crystals. The simulation step width for ${\theta _i}$ was $\textrm{0}\textrm{.000}{\textrm{8}^\mathrm{^\circ }}$, well enough to compare with future experimental verification. The diffraction intensities were calculated in accuracy to $\textrm{1}{\textrm{0}^{\textrm{ - 5}}}$, finer than practical experimental measurements $\textrm{1}{\textrm{0}^{\textrm{ - 3}}}$. It is noticed that the inclusion of the imaginary part of atomic scattering factors assures convergence during calculation.

Figure 2 illustrates the simulated dispersion surfaces (a), (c), and (e), via the incident angle ${\theta _i}$, and their corresponding specular reflectivities (red line) and back diffraction intensities (black line) outside the crystal (b), (d), and (f), with miscut angles ${\theta _m} = {\textrm{0}^\circ }$, $\textrm{89}\textrm{.878}{\textrm{1}^\circ }$, and $\textrm{9}{\textrm{0}^\circ }$, respectively. The ${\theta _m} = \textrm{}{\textrm{0}^\circ }$ angle corresponding to the NIXBD can be used as a comparison [23].

 

Fig. 2. Simulated dispersion surfaces and reflectivity of exact two-beam back diffractions at ${\theta _m}$ angles of (a, b) ${\textrm{0}^\circ }$, (c, d) $\textrm{89}\textrm{.878}{\textrm{1}^\circ }$, and (e, f) $\textrm{9}{\textrm{0}^\circ }$.

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For a general two-beam diffraction, there are eight eigenvalues, corresponding to the eight excited modes (tie points) inside the crystal under which the $\mathrm{\sigma }$- and $\pi $-polarization incidences (four for each polarization) are both considered. We conventionally assign the odd-number modes to the $\mathrm{\sigma }$-polarization incidence and the even-number modes to the $\pi $-polarization. They are degenerate in pairs. When only the $\mathrm{\sigma }$-polarization incidence is considered, only the four odd modes can be excited.

For the NIXBD example (Fig. 2(a)), there are only two modes, 3 and 5, strongly intersecting with each other near the ${L_O}$ point while modes 1 and 7 are far away from the ${L_O}$ point and have a negligible contribution to the diffraction.

The crystal thickness ($100\;\mathrm{\mu}\mathrm{m}$) gives rise to the oscillating pattern (Fig. 2(b)) of NIXBD, as reported earlier [22,23]. The maximum reflectivity (∼0.6) is thus lower than that for the very thick crystal (∼0.87). As ${\theta _m}$ approaches $\textrm{89}\textrm{.878}{\textrm{1}^\circ }$ (Fig. 2(c)), the dispersion surface inclines and the diffraction intensity width shrink accordingly. The diffraction intensity is almost the same as that of NIXBD for a very thick crystal [23]. However, the specular reflectivity starts to have a contribution as shown by the red lines in Fig. 2(d). This leads to increase specular reflection but in turn reduces the intensity of back diffraction. It is obvious that, when ${\theta _m}$ $= \textrm{9}{\textrm{0}^\circ }$, the specular reflectivity dominates, yet there is still around 10% of the intensity that can be attributed to X-ray back diffraction.

Figure 3(a) illustrates the dependence of the miscut angle ${\theta _m}$ on the (12 4 0) intensity distribution. We note that the maximum intensity (the black line in Fig. 3(a)) follows a simple linear ${\theta _i}\; $–${\theta _m}$ dependence until the ${\theta _i}\; $ approaches the critical angle ${\theta _c}$. This kinematic trend is the trajectory of the Lorentz point Lo, which can be derived using Snell’s law as follows.

 

Fig. 3. (a) The distribution map of intensity over ${\theta _i}$ and ${\theta _m}$. The line of best fit is plotted as a black line and can be well derived using the continuity of tangential components at the crystal boundary. The yellow line indicates the critical angle at which specular reflection occurs. (b) The 3D dispersion surface where two regions are most distorted, corresponding to the stronger diffraction intensity. The 2D intensity maps of (c) specular reflection and (d) Si (12 4 0) back diffraction are also presented. The miscut angle ${\theta _m}$ for (b), (c) and (d) is $\textrm{9}{\textrm{0}^\circ }$.

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The Lorentz point ${L_O}$ at the center of $\overline {OG} $ occurs at the exact position where the ${K_O}$-sphere and ${K_G}$-sphere intersect. This is expected to be the point at which the maximum diffraction intensity occurs. Snell’s law is applied to satisfy the continuity of the tangential components of the wavevectors inside and outside the crystal (referring to Fig. 1), which are related by the non-dispersive refractive index ${n_0}$:

Here, ${n_0}$ can be approximated to $\cos {\theta _c}$ since ${\theta _c}$ is an extremely small value. At the Lorentz point ${L_O}$, ${\theta _r}$=$\textrm{9}{\textrm{0}^\circ }$ − ${\theta _m}$. Thus, the relationship between ${\theta _i}$ and ${\theta _m}$ is:

However, the kinematic relationship shown in Eq. (15) cannot explain the intensity of back diffraction leaking into the region ${\theta _i} < \; {\theta _c}$, as indicated by the yellow line in Fig. 3(a). This is the dynamical result of the interaction of the incident beam and diffraction beam inside the crystal that is obtained by solving for the eigenvalues of Eq. (5) and (6) associated with the dispersion surface and the corresponding eigenfunctions Eq. (7) and (8) inside the crystal. The standard boundary conditions for Maxwell’s equations are then applied to obtain the electric fields outside the crystal and the corresponding diffraction intensities [18]. As the incident angle approaches the critical angle, the specular reflection becomes crucial when matching the boundary conditions.

Remarkably, even when the miscut angle is as large as ${\theta _m} = \textrm{89}\textrm{.878}{\textrm{1}^\circ }$, the reflectivity from back diffraction can be as high as 0.8. The reflectivity of back diffraction becomes weaker as the miscut angle approaches $\textrm{9}{\textrm{0}^\circ }$, subject to the increasing role of specular reflectivity. At ${\theta _m} = \textrm{9}{\textrm{0}^\circ }$, the reflectivity of back diffraction reduces to 0.1, also seen in Fig. 2(f).

Figure 3(b) depicts the dispersion surface for the case when ${\theta _m} = \textrm{9}{\textrm{0}^\circ }$ along the incident angle ${\theta _i}$ and the azimuthal angle $\phi $. The cross-section at $\phi = {\textrm{0}^\circ }$ is already shown in Fig. 2(e). The sheets with gray, red, blue, and cyan colors correspond to the dispersion sheets for modes (7, 8), (3, 4), (5,6), and (1,2), respectively. As mentioned earlier, the modes (5,6) and (1,2) are with negative absorption coefficients of $\mu ={-} 4\mathrm{\pi Im}({{z_j}} )$ which are unphysical and should not be considered [21]. At $\phi = {\textrm{0}^\circ }$, modes 3 and 4 are dominant. As $\phi $ increases, the modes 7 and 8 start interacting. The most interaction occurs at $\phi ={\pm} \textrm{0}\textrm{.02}{\textrm{8}^\circ }$ where the four modes merge into one, as seen in Fig. 3(b). The corresponding reflectivities for specular reflection and back diffraction are shown in Figs. 3(c) and (d), respectively.

The bending distribution (indicated by the black line) can again be explained using Snell’s law by modifying Eq. (15) with the contribution from $\phi $:

The grazing-incidence X-ray diffraction (GIXD) [19] can be used to qualitatively interpret the intensity distribution in Fig. 3(d). After setting the incident X-ray energy much higher than 14.4388 keV, two GIXD cases are found at the two sites around $\phi = {\textrm{0}^\circ }$, similar to the case in conventional GIXD. The two GIXD cases approach each other as the incident energy approaches 14.4388 keV, giving rise to the overlapping intensity at $\phi = {\textrm{0}^\circ }$°. As compared to the dispersion surface demonstrated in Fig. 3(b), the merge of the dispersion sheets of mode (3,4) and (7,8) gives rise to the higher intensity at $\phi ={\pm} \textrm{0}\textrm{.02}{\textrm{8}^\circ }$.

As highlighted by Shvyd’ko et al. [15–18], the back diffraction from highly asymmetry-cut crystal exhibits large angular dispersion rate (ADR). Si (008) reflection with a miscut angle of $\textrm{88}\textrm{.}{\textrm{5}^\circ }$ gives an 8.5 µrad/meV at 9.1315 keV [15]. Instead of using traditional DuMond diagram analysis, we directly examined the ADR of GIXBD Si (12 4 0) from our dynamical simulation. By varying the incident energy $\mathrm{\Delta E}$, the intensity map of back diffraction over $\mathrm{\Delta E}$ and $\delta \theta \equiv {\theta _f} - {\theta _i}$ at particular ${\theta _m}$ is plotted. It is noticed that ${\theta _i}$ is related to ${\theta _m}$ through Eq. (16). Figure 4(a) depicts the example case of ${\theta _m} = \textrm{9}{\textrm{0}^\circ }$. The criteria for finding ADR are to first take the $\delta \textrm{E}$ at the half reflectivity of back diffraction and then project it to find $\delta \theta $ as shown in Fig. 4(a). ADR is defined as ADR = $\delta \textrm{E}$ / $\delta \theta $. Figure 4(b) demonstrates the ADR and the maximum reflectivity at various ${\theta _m}$. It is noticed that the ADR exhibits a nonlinearly manner that it approaches 64 µrad/meV at ${\theta _m} = \textrm{9}{\textrm{0}^\circ }$, more than 10 times larger than that of 5.5 µrad/meV at ${\theta _m} = \textrm{88}\textrm{.}{\textrm{5}^\circ }$. The same algorithm and procedure are used to calculate the ADR for the Si (008) reflection at miscut angle ${\theta _m} = \textrm{88}\textrm{.}{\textrm{5}^\circ }$ (critical angle $\textrm{0}\textrm{.19}{\textrm{3}^\circ }$) at energy 9.1315 keV, obtaining 6 µrad/meV, close to the reported 8.5 µrad/meV [15].

 

Fig. 4. (a) The distribution map of intensity at miscut angle ${\theta _m}$ = $\textrm{9}{\textrm{0}^\circ }$, over the varying incident energy $\mathrm{\Delta E}$ and the difference angle $\delta \theta \equiv {\theta _f} - {\theta _i}\; $ between the exit angle ${\theta _f}$ and the incident angle ${\theta _i}$. The red line depicts the intensity of Si (12 4 0) versus $\mathrm{\Delta E}$. The vertical line marks the half intensity of Si (12 4 0) at $\delta \textrm{E}$ = 3.1 meV. The intersection to intensity map gives the $\delta \theta = \textrm{0}\textrm{.0114}{\textrm{4}^\circ }$. The angular dispersion rate (ADR) is defined as ADR = $\delta \textrm{E}$ / $\delta \theta $, in this case equals 64 µrad/meV. (b) The ADR and the maximum Si (12 4 0) reflectivity at various ${\theta _m}$.

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The GIXBD can be directly measured on a bared-surface silicon wafer, by adopting the ultra-high energy resolution ($\mathrm{\Delta E}$ /E = $\mathrm{2\ \times 1}{\textrm{0}^{\textrm{ - 8}}}$) experimental set up reported before [5]. For the sake of testing the concept GIXBD for XFPR, we propose to fabricate a device by modern lithography technology. The design is demonstrated in Supplement 1 (Fig. S1). The resonance is expected to occur within a narrow trench of width 6 µm. The miscut angle is ${\theta _m} = \textrm{89}\textrm{.878}{\textrm{1}^\circ }$, the optimal ${\theta _m}$ as demonstrated in Fig. 4(b). The ADR at this angle provides an angular tolerance of 15.8 µrad, corresponding to the energy bandwidth 0.36 meV, ensuring the parallelism caused by etching error. The resonance fringes are to measure by varying the incident energy or rotating the sample. Other than XFPR, one can design high energy resolution optics such like monochromators, spectral analyzer, by employing the novel optical characters of GIXBD.

4. Conclusion

Based on the framework of the dynamical theory of X-ray diffraction, the grazing-incidence X-ray back diffraction (GIXBD) for Si (12 4 0) was investigated. By solving the eigenEqs. (5)–(8) in Cartesian coordinate form [18], the dispersion surfaces and the excited electric fields inside the crystal were obtained. The refractive characteristics of back diffraction were dependent on the miscut angle ${\theta _m}$ versus the incident angle ${\theta _i}$ and can be interpreted by the kinematic Snell’s law, as well as detailed by the dynamical theory of diffraction. The reflectivity of back diffraction can be as high as 0.8 with a miscut angle ${\theta _m} = \textrm{89}\textrm{.878}{\textrm{1}^\circ }$ and even 0.1 at ${\theta _m} = \textrm{9}{\textrm{0}^\circ }$.

The maximum reflectivity occurs at azimuthal angles where the four eigenmodes merge together. The large angular area for GIXBD allows the incident beam to have large divergent angles, which is beneficial to the fabrication of complex optical devices. [17]

The thin extinction depth in GIXBD is essential for applying X-ray back diffraction in the development of high-efficiency X-ray resonators and other X-ray optical devices. The angular dispersion rate (ADR) exhibits nonlinearly dependence over large miscut angle $\textrm{}{\theta _m}$. The large ADR as well as the high diffraction reflectivity at miscut angle ${\theta _m} = \textrm{89}\textrm{.878}{\textrm{1}^\circ }$ is anticipated to design high efficiency optical devices.