1. Introduction

An optical Fabry-Perot (FP) resonator [1] is an indispensable optical component widely used in telecommunications, lasers, and spectroscopy to control and measure the wavelengths of electromagnetic waves and related physical properties [2]. The simplest form of the Fabry-Perot resonator involves a pair of reflecting mirrors reflecting light back and forth within a cavity to form resonance under specific conditions. The Fabry-Perot resonator is applicable for wavelengths ranging from visible to soft X-rays [3]. For hard X-rays, the use of Bragg back diffraction (Bragg angle ~90°) from a series of atomic planes acting as reflecting mirrors has been proposed and attempted since 1967 [4–12] because the large extinction length allows the X-rays to penetrate even relatively thick crystals. The attempts were not successful until 2000, when Shvyd’ko et al. [13] observed the storage of X-ray photons in a few tens of back-and-forth reflection cycles from two-plate crystals (thickness: 50 ~150 mm) in time-resolved transmission measurements, but the experimental results do not show resonance fringes, primarily because the energy resolution is insufficient for the free spectral range [14].

In 2005, Chang et al. directly observed cavity resonance fringes in a Si crystal of smaller size (thickness: 40 ~150 μm) [15, 16]. When an ultrahigh-resolution monochromator was inserted upstream, Si resonators using (12 4 0) back diffraction showed resonance fringes inside the total reflection range near 14.4388 keV with the energy resolution ΔE of 0.36 meV and finesse of 2.3. The barely satisfactory finesse could be attributed to the following factors of crystal-based resonators: (1) The FP finesse and efficiency (peak transmission) have a trade-off relationship. Since the FP finesse is governed by crystal reflectivity, increasing the reflectivity implies the increase in the crystal thickness and absorption at the cost of efficiency. (2) Multiple-beam diffraction seems inevitable for crystal-based resonators. For the high-energy X-rays used and the high symmetry of the diamond structure of Si, (0 0 0) transmission and Si (12 4 0) back diffraction were excited with additional 22 beam diffractions simultaneously at 14.4388 keV [17], which lowered the intensities, thereby affecting the performance of resonance. The multicavity resonators recently proposed by Huang et al. [14] also theoretically face the problem of four-beam diffraction at (0 0 0), (2 2 4), (0 0 4), and (2 2 0) of diamond (224) at 8.5146 keV.

To solve the aforementioned problems, in this study, instead of a conventional cavity with normal incidence, we use an inclined incident beam along one of the multiple diffraction directions to generate back diffraction in a crystal cavity for resonance and demonstrate FP resonance with ultrahigh efficiency and highly purified resolving power in the sub-meV range. That is, we change the path of incidence to remove the absorption in the first crystal that would be found in normal-incidence crystal-based resonators. The experiment described below, using an inclined-incidence beam that fulfills the Bragg condition for (12 0 0) and for backscattering (12 4 0) simultaneously, shows intense interference fringes.

2. Model of inclined-incidence X-ray resonators

The idea of using an inclined-incidence X-ray resonator was derived from the inevitable multiple-beam diffractions accompanied with back diffraction. The multiple-beam diffractions occur when two or more atomic planes of a crystal satisfy the Bragg’s law, namely more than one reciprocal lattice points lie on the Ewald sphere in reciprocal space simultaneously. For a normal-incidence resonator of Si (12 4 0) at 14.4388 keV, in addition to the (0 0 0) direct incident beam and (12 4 0) primary diffraction, there are 22 diffractions called secondary diffractions occurring with the Si (12 4 0) back diffraction, leading to resonance [15] (The 22 beams that accompany Si (12 4 0) back diffraction can be classified into 9 coplanar diffractions C1–C9. C1: (12 0 0), (0 4 0), (4 −4 0), (8 8 0), (8 −4 0) and (4 8 0); C2: (12 2 −2) and (0 2 2); C3: (6 4 6) and (6 0 −6); C4: (8 2 −6) and (4 2 6); C5: (8 2 6) and (4 2 −6); C6: (6 4 −6) and (6 0 6); C7: (12 2 2) and (0 2 −2); C8: (6 8 −2) and (6 −4 2); C9: (6 8 2) and (6 −4 −2) [17]). Based on the principle of reciprocity of light, each reflected beam of this 24-beam case can be treated as the incident beam for diffraction. For example, in Fig. 1(a), ${\overrightarrow{K}}_L$ is one of the multiply diffracted beams generated by Bragg diffraction from the atomic plane L (dashed gray line). That is, ${\overrightarrow{K}}_L$ is the wave vector of reflection L, directed from the center of the Ewald sphere to the reciprocal lattice point L in reciprocal space. For inclined incidence, the incident beam [Fig. 1(b)] along the reversed direction, −${\overrightarrow{K}}_L$, is reflected from the second plate of the resonator towards the first plate normal to the crystal face via reflection L. Then, the (12 4 0) back diffraction is excited from the first plate and reflected back and forth in the resonator. In Fig. 1(b), the inclined incident beam skips the entrance to the first plate (i.e., suffers no absorption from the first plate) and retains most of its intensity inside the resonator to enhance the efficiency of interference. In other words, the thickness of the first plate can be considerably increased to enhance the reflectivity of the resonator. Furthermore, since the incident beam is not along the direction of the back diffraction, the background of the beam transmitted through the resonator can be effectively suppressed.

 

Fig. 1 Comparison of the incident path of a conventional normal-incidence cavity (left) and an inclined incidence cavity (right). At normal incidence, the light is injected into the cavity at 90°. At inclined incidence, the incident beam is injected along the reversed direction of one of the multiply diffracted beams, −${\overrightarrow{K}}_L$.

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3. Design of the Si-based resonator and diffraction experiment

The experiment on the inclined-incidence resonator was performed at the Taiwan undulator beamline BL12XU at the Spring-8 synchrotron facility in Japan. The experimental setup [Fig. 2] and the design of inclined-incidence hard-X-ray resonators [Fig. 3] are combined to fulfill the following experimental criteria: (a) the energy resolution of the incident photon beam ΔE < free spectral range E_d, (b) spectral width Γ < E_d for observing the FP resonance in energy scan [15]. That is, the temporal coherent length of the X-rays is larger than the round-trip distance of X-rays travelling inside the resonator. At 14.4388 keV, an energy resolution of 0.36 meV was achieved using a Si (1 1 1) double-crystal monochromator (DCM) and a four-crystal ultrahigh-resolution monochromator (HRM) consisting of two pairs of (4 2 2) and (11 5 3) asymmetric reflections in a ( + − − + ) geometry with the asymmetry factor b₁ = 0.19, b₂ = 0.0469 and b₃ = b₄ = 15 accordingly [15]. It corresponds to a normal distribution of 1717 μm in the longitudinal coherent length with a beam size of 100 μm in the vertical direction and 600 μm in the horizontal direction [18]. The resonator was placed at the center of an eight-circle diffractometer for the alignment of Si (12 4 0) back diffraction by Δθ_h which rotates about the vertical axis and coincides with the crystal’s (0 0 1) reciprocal lattice vector. An avalanche photodiode (APD) adjusted by beta and a pin diode were used to collect the reflected and transmitted signals, respectively. At beta = 0, the APD is in the path of the direct beam.

 

Fig. 2 Experimental setup for the inclined resonator.

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Fig. 3 Design and orientation of a normal-incidence (a) and an inclined-incidence resonator (b) in real space. (c) and (d): Top view of the diffracted wave vector and lattice point in reciprocal space for the cavity shown in (a) and (b). The upper figures depict the normal-incidence geometry and the lower ones depict the inclined incidence.

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The sample was prepared from a silicon (0 0 1) crystal wafer through microelectronic lithography to a size of a few hundred microns [Fig. 3(a) and 3(b)]. The sidewall of plates was etched along [3 1 0] for back reflection Si (12 4 0) at 14.4388 keV. The incident beam ${\overrightarrow{K}}_N$ [Fig. 3(a)] denotes the wave vector at normal incidence and ${\overrightarrow{K}}_I$ [Fig. 3(b)] denotes an inclined incidence at an angle 36.87° with respect to [-3 −1 0]. The height h of the etched crystal plates was 180 μm for compatibility with the incident beam size. The widths w₁ and w₂ of the crystal plates were 1000 μm and 2000 μm, respectively, to prevent the incident beam from being blocked by the first plate. The gap between the two plates was 240 μm. The thickness of the second plate was 100 μm, whereas that of the first plate was 200 μm to enhance reflectivity. The designed free spectral range E_d of this resonator was 2.03 meV, as calculated using E_d = hc / (2L), where h and c are the Planck’s constant and the speed of light, respectively. L ≈d_g + 2d_e(0) [19] is the effective gap of the cavity, where d_g is the gap between the 2 plates and d_e(0) is the extinction length of the Bragg diffraction considered. The value of d_e(0) is 34.4 μm for Si (12 4 0).

In our experiment, the atomic plane (12 0 0) was chosen to reflect the inclined incident beam ${\overrightarrow{K}}_I$ into the resonator. Figure 3(c) and 3(d) show the top view of the normal –incidence and inclined-incidence geometry in reciprocal space of the cavity shown in Fig. 3(a) and 3(b). In the normal incidence of Fig. 3(c), the Ewald sphere is constructed by the incident ${\overrightarrow{K}}_N$ and diffracted vectors of (12 4 0) back diffraction, and (12 0 0) is one of its multiple diffractions that intersect the sphere. When changing to the inclined incidence of Fig. 3(d), the (12 0 0) diffraction generated via the incident ${\overrightarrow{K}}_I$ is parallel to the reciprocal lattice vector of (12 4 0) (blue arrow). That is, the diffracted beam of (12 0 0) is incident on the atomic plane of (12 4 0) at nearly 90°, thereby generating resonance inside the resonator. Besides the (12 0 0) diffraction, there are 22 other diffractions that are simultaneously excited. In Fig. 3(d), only the incident (0 0 0), Bragg diffraction (12 0 0), and Laue (transmission) diffraction (0 −4 0) beams are indicated.

Figure 4 shows the experimental results of transmission from the 200 μm (first plate)/240 μm (gap)/100 μm (second plate) resonator in energy scans of the high-resolution monochromator for both inclined incidence [Fig. 4(a)] and normal [Fig. 4(b)], where the photon energy δE = E - 14.4388 keV. The transmission intensity of energy scan was obtained at Δθ_h = 0 and beta = 0 in the normal incidence case when the resonator was aligned first for (12 4 0) back diffraction to exhibit a broad dip at ~0.1° in the Δθ_h scan along [0 0 1]. For inclined incidence, both the crystal (Δθ_h axis) and detector (beta axis) were rotated by approximately 36.87° to align for (0 −4 0) and (12 0 0) in the transmitted and reflected directions, respectively. The energy scan was performed by tuning together the Bragg angles of the third and fourth crystals of the ultrahigh-resolution monochromator with a minimum step corresponding to 58.548 μeV in energy. The measured free spectral range E_d of the 200 μm/240 μm/100 μm resonator is 1.84 meV on average. As can be seen in Fig. 4(a), the interference fringes observed were intense and distinct at inclined incidence because most of the intensity remained inside the resonator to form constructive interference because of the two highly reflective Si plates. In contrast, normal incidence caused blurred fringes inside the range of (12 4 0) back diffraction because of the absorption in the plates of total thickness 300 μm [Fig. 4(b)]. The visibility V = (I_max − I_min)/(I_max + I_min) of the interference fringes was evaluated as 0.205 for inclined incidence and 0.007 for normal incidence. The efficiency of the resonator at inclined incidence [Fig. 4(a)] was substantially higher than that at normal incidence [Fig. 4(b)].

 

Fig. 4 The experimental results of the energy scan in the transmission of the 200 μm/240 μm/100 μm resonator: (a) inclined incidence; (b) normal incidence. δE is E - 14.4388 keV. The inclined-incidence resonator showed intense resonance fringes.

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4. Simulation of FP resonance from dynamical diffraction theory

The simulation of hard-X-ray resonance at inclined incidence in a two-crystal-plate resonator is based on the dynamical theory of X-ray diffraction [20, 21]. Because of (12 0 0) diffraction used at inclined incidence, the coplanar diffractions C1 including (0 0 0), (12 4 0), (12 0 0), (0 4 0), (4 −4 0), (8 8 0), (8 −4 0) and (4 8 0) are all participated in cavity resonance. It is usually a formidable task to calculate the FP resonance using 8-beam diffraction for both the first and second crystals in sequence for back-and-forth reflections. For simplicity, the characteristic of the FP resonance can be roughly approximate as the following steps: (1) The (0 0 0) inclined incidence on the second plate generates the (12 0 0) Bragg reflection and (0 −4 0) forward transmission which is relatively strong compared with other secondary diffractions. (2) The (12 0 0) reflection participates in the cavity resonance, i.e., it is reflected back and forth between the two crystal plates. The intensity of the (12 0 0) diffraction is approximately determined by solving a three-beam eigenvalue–eigenvector matrix, which includes the three reflections (12 0 0), (0 −4 0), and (0 0 0). The transmissivity of FP resonance can then be expressed as

Figure 5(a) and 5(b) demonstrate the simulation of the X-ray 200 μm/240 μm/100 μm resonator at inclined and normal incidence. Figure 5(a) shows the calculated transmissivity of the resonator at inclined incidence. The injecting source of FP resonance in Eq. (2) is r₁₂₀₀(100 μm), which is obtained from the Si (0 0 0) (12 0 0)(0 −4 0) three-beam diffraction calculation. To get clear resonance, the simulation of energy scan is proceeded in accordance with the condition: Δθ_h is 2 × 10⁻⁵ deg. off the (0 −4 0) center. The FP resonance has ultrahigh efficiency exceeding 200% on average with extremely low background due to the high reflectivity of Si (12 0 0). The low broad-band background mainly originates from (0 −4 0) Laue diffraction transmitting through the second plate with approximately 15% of the diffraction intensity and extends 175 meV in the Darwin curve. The simulation exhibits needle-like peaks with ultra-narrow bandwidth (0.16 meV in energy) because of the increased reflectivity. However, normal incidence [Fig. 5(b)] yields a low efficiency of approximately 4% and low transmissivity inside the range of Si (12 4 0). The peak efficiency drops due to the absorption of the thickened crystal plates.

 

Fig. 5 Simulation of transmissivity of the 200 μm/240 μm/100 μm resonator in (a) inclined and (b) normal incidence. Normal incidence yields a low efficiency of 4% because of the absorption of 300 µm in Si, whereas the resonance at inclined incidence exhibits an efficiency (peak transmission) exceeding 200%.

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5. Qualitative analysis of finesse and efficiency of resonators

The substantially enhanced efficiency of the inclined-incidence resonator can be illustrated by comparing the results of the qualitative analysis of a 200 μm/240 μm/100 μm resonator at normal incidence (A) to that at inclined incidence (B). Figure 6 shows the reflection/transmission curves of the source entering a resonator according to Eq. (1) and (2), which can be used to evaluate the percentage of the incident synchrotron radiation utilized for X-ray resonators. Resonator (A) (red line) transmits only 0.8% (dip of the curve) because the injecting source t₁₂₄₀(200 μm) is at normal incidence. On the other hand, resonator (B) (black line) at inclined incidence to Si (12 0 0), with 100-μm thickness, reflects 57% in the entire reflection range. That is, the injected beam is drastically enhanced by 71.25(57%/0.8%) times only by changing the incidence angle, thereby enhancing the resonator efficiency.

 

Fig. 6 Comparison of resonators (A) 200 μm/240 μm/100 μm at normal incidence and (B)200 μm/240 μm/100 μm at inclined incidence: the injecting source of the resonators are 0.8% of t₁₂₄₀(200 μm) in (A), 57% of r₁₂₀₀(100 μm) in (B).

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In addition to the efficiency, the energy resolution ΔE is also enhanced by the calculation using F = E_d / ΔE. The finesse F is π × $\sqrt{R}$/ (1 − R), where R is the reflectivity of the resonator [19]. The reflectivity of a single plate Si is 70.7% for 100 μm and 85.6% for 200 μm. The calculated finesse of a 200 μm/240 μm/100 μm resonator is 12.48, which corresponds to a ΔE of 0.16 meV for E_d = 2.03 meV, and extends the longitudinal coherent length <λ/(ΔE/E)> to 3800 μm in normal distribution. However, estimating ΔE from the numerical calculations for the case given in Fig. 5(a) is difficult because of the high absorption in crystal plates at normal incidence; this lowers the practicability of the resonators. Our results illustrate the superiority of the inclined-incidence resonators in terms of ultrahigh energy resolution and ultrahigh efficiency in peak transmission with low background.

6. Conclusion

We have proved the practicability of hard-X-ray resonators at inclined incidence, which show ultrahigh efficiency, highly purified resolving power in energy, and low background. The compact-sized resonator with these promising features obtained via the inclined-incidence geometry can be widely implemented to different energies and materials such as sapphire and diamond as high-resolution sub-meV monochromators, and as analyzers that preserve the energy flux in practical applications and prevent the participation of direct incident beam in the transmission spectrum of the FP resonance. Furthermore, the narrowband FP resonance with good temporal coherence facilitates coherence diffractive imaging for relatively thick samples of condensed matter and bio-medical materials. Longitudinal correlation could be applied to high-resolution scattering/diffraction, spectroscopy, phase-contrast imaging, and nuclear resonant scattering in fields such as biology, medical science, and crystallography.