An X-ray interference-monochromator combining a Fabry-Perot resonator (FPR) and a double-crystal monochromator (DCM) is proposed and realized for obtaining single-mode X-rays with 3.45 meV energy resolution. The monochromator is based on the generation of cavity interference fringes from a FPR and single-mode selection of the transmission spectrum by a DCM of a nearly backward symmetric reflection geometry. The energy of the monochromator can be tuned within 2500 meV(= ΔE) by temperature control of the FPR and the DCM crystals in the range of ΔT = 70 K at room temperature. The diffraction geometry and small size of the optical components used make the interference-monochromator very easy to be adapted in modern synchrotron beamlines and X-ray optics applications.
© 2016 Optical Society of America
Combination of several optical components often leads to improved optical systems for various applications in fundamental and applied research. In the case of hard X-rays, the use of diffraction devices that consist of several crystals is a common practice for having better optical properties. For example, the energy resolutions, ΔE/E, of the spectra from double crystal monochromators (DCM) and of channel-cut crystal monochromators, are around 10−4 to 10−6. When higher energy resolutions are desired, the more complex designs of the diffraction devices, such as multi-bounce monochromators  and back-reflection optics [2–4], are usually applied to reach the energy resolution of 2 × 10−8 or better. On the other hand, the use of gratings and resonance devices can also improve the energy resolutions a great deal. In decades, the theoretical studies of Fabry-Perot resonators (FPRs)  for hard X-rays have shown that resonance fringes with bandwidths of sub meV are possible [6–12]. The attempts to realize hard X-ray resonators were reported in the literature . In 2005, a direct measurement of the resonance fringes in energy spectrum from Si-based resonators was demonstrated [14,15]. The two-plate resonator using back reflection at energy near 14.4388 keV showed resonance fringes inside the total reflection range of Si (12 4 0). A multicavity X-ray Fabry-Perot resonator using diamond crystals has been theoretically proposed . Recently inclined-incidence resonators were proposed and realized and ultrahigh efficiency and resolution better than normal-incidence resonators have been reached .
The interference pattern of a resonator consists of several equally spaced fringes, which is different from the single diffraction peak from a monochromator. In order to achieve single-mode selection, two coupled Si Fabry-Perot resonators (FPRs) have recently been used as a demonstration . However, in practice, when a broad band incident beam is used, the output would contain unwanted diffraction intensity background around the selected central peak. In this paper, we propose an interference-monochromator so that only a single-resonance-peak can be singled out, which can be used for other X-ray optics investigations. The design ideas, the theoretical simulations, the experimental requirements and the measured results of a single-resonance-mode monochromator are reported below.
2. Design concept of interference-monochromators
The interference-monochromator is composed of a Fabry-Perot resonator and a double-crystal monochromator. The former is to generate resonance fringes with high resolution, the latter is to purify the resonance spectrum to form a single-mode distribution.
Figure 1(a) is a schematic of an interference-monochromator which is composed of a hard X-ray FPR and a DCM with two steering crystal plates (SPs). The FPR consists of two parallel crystal plates of silicon, with the crystallographic orientation indicated in Fig. 1(b), where the thicknesses of the crystal plates and the separated distance between the two plates in the FPR are noted as T and dg, respectively. The Si (12 4 0) as back reflection  for 14.4388 keV X-rays is employed. These two crystal plates diffract the incident X-rays via (12 4 0) in nearly normal incidence geometry, as the reflecting mirrors of an optical Fabry-Perot resonator for the visible light. When the X-rays are incident onto the FPR, the successively diffracted beams propagating forward and backward between the two crystal plates are interfered with each other, then the resonance fringes appear on the transmittance and reflectance spectra of the FPR. The steering crystal plates of the DCM use also Si (12 4 0) to reflect twice the transmitted beam from the FPR back to its incident direction, denoted as the bounced beam. Both the transmitted and bounced beams are monitored by an avalanche photo-diode (APD) via a horizontal β scan. The FPR and the steering crystal plate SP2 are located in stage 1 and the steering plate SP1 is put in stage 2. Both sample stages are arranged right after the high-resolution monochromator (HRM) in beam line TPS-09A at Taiwan Photon Source in the National Synchrotron Radiation Research Center (NSRRC), Taiwan [Fig. 1(c)].
3. Theoretical simulation
In order to design suitable X-ray FPR and DCM for single-mode operation, numerical simulations based on the dynamical diffraction theory for a single Cartesian coordinate system  and the recursion relations for X-ray diffracted waves in a layered crystal system  are carried out. For Si (12 4 0) back reflection, the photon energy 14.4388 keV is chosen for the incident beam. Figure 2(a) shows the simulated transmission spectrum (the solid curve) of (12 4 0) for a typical FPR with T1/dg/T2 = 70/45/70 μm. The calculated transmission spectrum of the same back reflection from a single silicon crystal of 140 μm (i.e., T1/dg/T2 = 70/0/70 μm with dg = 0) is shown in dashed curve for comparison. If the dashed curve is subtracted from the solid curve, then a single peak remains at the center of the spectrum. In practice, the subtraction can be achieved by optical means. That is, when the transmission spectrum of FPR is reflected twice by the (12 4 0) of a silicon crystal of 500 μm thick back to its original path [Fig. 2(b)], then a single-mode spectrum, shown in Fig. 2(c), can be realized in the transmission direction. Hence the use of the conjugated two steering Si plates, together with a FPR can select the resonance spectrum to become single-mode.
4. Energy tunability
The energy tunability of the interference-monochromator can be achieved using the lattice parameter control method of crystals. Referring to Bragg’s law, 2dhkl sin(θ) = hc/E, when the reflection, (hkl), and the incident angle, θ, of the X-ray are determined, the diffracting energy, E, depends on the interatomic distance, dhkl. The interatomic distance is a function of temperature 𝒯 due to thermal expansion effect. Considering the linear thermal expansion coefficient α, the lattice parameter dhkl (𝒯0 + Δ𝒯) can be described as dhkl (𝒯0 + Δ𝒯) = (1 + αΔ𝒯)dhkl (𝒯0), where 𝒯0 is the reference temperature and Δ𝒯 is the among of the temperature difference. Hence, the diffracting energy is also a function of temperature, i.e., E(𝒯0 + Δ𝒯) = hc/[2(1 + αΔ𝒯)dhkl (𝒯0) sin(θ)]. Accordingly, the estimated diffracting energy shift versus temperature change is shown as a solid line in Fig. 3, where the lattice constant of silicon is 5.43095 Å at 300 K and (hkl) is (12 4 0). The diffracting energy of Si (12 4 0) with the Bragg angle of 89.98° is 14437.36 eV (ΔE ≡ 0 at 300 K) and the linear expansion coefficient of silicon is 2.6 × 10−6 K−1. This calculation indicates that the energy tuning range could be above 2500 meV in a temperature interval of 70 K around 300 K.
The benefit of using the lattice parameter control method to tune the energy is that the optical geometry of the interference-monochromator is fixed during the energy tuning processes. In comparison with conventional monochromators, in order to tune the output energies, the crystals have to be rotated and/or translated to change the diffraction geometries. The geometrical perturbations always induce the beam path variations. In contrast, the energy tuning processes of the interference-monochromator is to increase and/or to decrease the sample temperature at a determined geometry as described. Free from the perturbations due to geometrical changes, the interference-monochromator can be considered as a band-pass filter rather than a conventional monochromator for hard X-rays with ultra-high energy resolution.
The 70/45/70 FPR and the DCM with two steering crystal plates SPs were manufactured monolithically on a silicon wafer by microelectronic lithography processes. The crystal orientations of the FPR are noted in Fig. 1(b). The dimensions of the substrate for FPRs and SPs are around 340 μm in thickness, 4 cm in width and 1 cm in length. Without counting the substrate, the widths of the two crystal plates of the FPR and the SPs are 1000 μm and 4000 μm, respectively, and the heights are both 160 μm [Fig. 1(b)]. The etched two silicon steering plates are put into two separate sample stages. As shown in Fig. 1(a), stage 1 includes the FPR and the SP2 and stage 2 contains the SP1. Both the FPR, SP1, and SP2 follow the optical path indicated for X-ray diffraction.
The experiments were carried out on the undulator beamline TPS-09A at Taiwan Photon Source (TPS) in Taiwan. Figure 1(c) illustrates the experimental setup where a Si (1 1 1) DCM, a horizontal focusing mirror (HFM), a four-bounce high resolution monochromator (HRM) and two diffractometers for stage 1 and 2, are arranged. The Si (1 1 1) DCM is employed to choose the wavelength of and the HFM is applied to converge the beamsize of the incident beam. However, the energy resolution of the Si (1 1 1) DCM is around 2 eV at 14.4373 keV that is insufficient to observe the resonance spectra of the FPR, hence the four-bounce HRM with energy resolution 0.36 meV was used. The HRM is composed of two pairs of asymmetrically cut crystals using Si (4 2 2) reflection at the first pair and Si (11 5 3) reflection at the second pair of crystals. The stage 1 and 2 were mounted on a 4-circle and a 9-circle diffractometer, respectively, manufactured by Huber, where the distance L between the centers of the two diffractometers is 2.48 m. At the 9-circle diffractometer, an avalanche photodiode (APD) manufactured by FMB Oxford was mounted on the 2θ and β arms that move along the vertical and horizontal directions.
To separate the transmitted beam and bounced beam, the incident angle for the (12 4 0) back reflection at the FPR was set at 90 − Δθ, where Δθ is the deviation angle of the incident beam from the surface normal of the FPR. Here we chose Δθ = 0.02°, which is sufficient to decouple the bounced beam from the transmitted beam [Fig. 1(a)]. Experimentally, the separation about 1.8 mm between the transmitted and bounced beams was recorded on the β scan as shown in Fig. 4. Both stage 1 and stage 2 noted in Figs. 1(a) and 1(c) can be adjusted independently. Therefore, the spectra at different parts of the optical components at chosen detector positions and the status of the stages can be easily monitored.
In order to experimentally control the temperatures of the stages, two temperature control systems using PID method (proportional-integral-derivative tuning method) are employed. Each system consists of a multi-meter, Keithley 2700, with the temperature resolution of 0.001 K, a DC power supply, Keithley 2260B, and a thermoelectric cooler with the power of 27 W. The feedback rate of the systems is 3.3 Hz. The standard deviations of the measured temperatures at the stages are better than 0.006 K in 5 minutes. The two temperature control systems can work individually and corporately; they can be used to compensate for the surrounding temperature differences of the two stages to ensure the occurrence of correct energy of the (12 4 0) reflection.
6. Results and discussion
The transmission spectrum of (12 4 0) from the FPR alone measured at the transmitted beam position by the β-scan is shown as the blue solid curve in Fig. 5(a). The transmission spectrum of (12 4 0) reflection from the steering plate SP1 alone is the red dashed curve shown in the same figure [Fig. 5(a)], measured also at the transmitted beam position. These results are consistent with the simulations given in Figs. 2(a) and 2(b). The curve with open circles in Fig. 5(b) is the effective spectra of the interference-monochromator, i.e., the incident X-rays going through the (12 4 0) reflection of the FPR and reflected consecutively by the (12 4 0) of the two steering crystal plates, SP1 and SP2, which are measured at the bounced-beam position. The combination of a FPR and a Si (12 4 0) DCM results in a single resonance peak spectrum as the simulation, Fig. 2(c), indicated. The measured bandwidth of the single peak is around 3.45 meV. Thus single-mode operation with a relatively fine energy-resolution is achieved as the theoretical simulation predicted.
The energy tunability of the interference-monochromator has also been successfully demonstrated. The squares in Fig. 3 are the measured energies of (12 4 0) reflection at difference temperatures. The measured results are satisfied with the estimated trend of the solid line in Fig. 3. Figure 5(b) and the inserted figure in Fig. 3 are the spectra from the same interference-monochromator at temperatures 312.67 K and 301.95 K respectively, where the energy shift between them is 296.34 meV. The highest temperature point in our experiments was 368.67 K that induced the maximum energy shift of 2474.48 meV.
The interference-monochromator, consisting of a FPR and a DCM containing two steering silicon crystal plates, has been designed, fabricated, demonstrated, and studied. The bandwidth of the single-mode peak on the output spectrum is around 3.45 meV, which is a resonance peak selected from the resonance spectrum of the FPR by the X-ray optical means proposed. It is understood that, conventionally, the transmittance spectra of FPRs are multi-peak like spectra due to cavity resonance effects. And the energy resolution of the interference fringes can be relatively small according to design chosen. However, for the application as monochromator, a single peak spectrum is most desired. Here the proposed interference-monochromator, utilizing the conjugated combination of FPR and DCM leads to a monochromatic single-mode X-ray beam with high energy resolution. The energy tunability of the interference-monochromator using lattice parameter control method has also been demonstrated. The energy shift of (12 4 0) reflection was above 2500 meV in a 70 K temperature variation.
Since the Si (12 4 0) back reflection is used for both the FPR and the nearly backward symmetric DCM, the optical properties of the incident light source, such as beam size, divergence, transverse coherence etc., could be maintained. Moreover, the compact size of the optical components used, such as FPR and the steering crystal plates, and the fixed optical geometry during energy tuning is another benefit. Therefore, the interference-monochromator could be installed easily at beamlines in modern synchrotron facilities as an optical component to provide high energy resolution light sources.
Ministry of Science and Technology, R.O.C. (MOST).
The authors are indebted to the technical assistance from the NSRRC.
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