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An off-axis total external reflection zone plate is applied to wavelength-dispersive X-ray spectrometry in the range from 7.8 keV to 9.0 keV. The resolving power E/ΔE of up to 1.1 × 102, demonstrated in a synchrotron proof-of-concept experiment, competes well with existing energy-dispersive instruments in this spectral range. In conjunction with the detection efficiency of (2.2 ± 0.6)%, providing a fairly constant count rate across the 1.2 keV band, the temporal pulse elongation to no more than 1.5 × 10−15 s opens the door to wide-range, ultra-fast hard X-ray spectroscopy at free-electron lasers (FELs).
© 2015 Optical Society of America
1. Introduction
Over the recent years, soft X-ray absorption spectroscopy has evolved into an important research method in many areas of analytical science, e.g. for the investigation of dynamical phenomena in magnetic materials [1].
The ultra-fast, time-resolved operational mode of the used spectrometer as required for experiments of that kind became accessible with the development of parallel detection. Traditional configurations are commonly based on, for example, the Rowland circle geometry or related concepts and require the wavelength-dependent adjustment of the grating or Bragg crystal. In contrast, simultaneous recording schemes contain no movable parts and enable the single-shot detection of complete spectra (“one pulse, one spectrum”) [2]. The prerequisite for this method is a two-dimensional imaging detector, e.g. a charge-coupled device (CCD), which is able to cover the whole X-ray photon energy range of interest. Recent results at the BESSY II femtoslicing facility proved the feasibility of the novel approach successfully, showing a significant performance gain in optical pump / soft X-ray probe experiments [3].
The extension of time-resolved spectroscopy with parallel spectral recording to energies beyond ∼ 2 keV is currently limited by the available instrumentation technology. Energy-dispersive detectors provide a typical resolving power E/ΔE ≲ 50 in this regime [4]. Hence, crystals are commonly used as the wavelength-dispersive optical element in a corresponding setup. To provide the required bandwidth, a crystal must be illuminated with a highly divergent beam [5] or bent to a small radius of curvature [6]. In each case however, a considerable temporal elongation of X-ray pulses (“pulse stretching”), typically in the order of (10–100) femtoseconds (fs), is introduced due to the relatively large depth of the crystal involved in the diffraction process.
In this paper, we use off-axis total external reflection zone plates (RZPs) [7] as an alternative solution for that purpose [8]. Already in use for soft X-ray spectroscopy due to the high efficiency [2] and signal-to-noise ratio of such all-in-one two-dimensional focusing optics [3, 9] and even yet implemented in their hard X-ray Bragg reflection progenitor version [10], an appropriate RZP also maintains the intrinsic pulse length on the femtosecond scale in combination with a sufficient resolving power. As a proof of principle for an exemplary device, we present design constraints, fabrication, measurement results and an outlook to further steps in Sect. 2–5, respectively.
2. Motivation and design
Regarding the typical length τ ≲ 10−14 s of FEL pulses [11] as used in modern hard X-ray time-resolved spectroscopy, the temporal elongation by the diffractive optical element (DOE) plays a critical role. This additional pulse stretching Δτ, usually of pronounced relevance toward the Fourier transform limit, grows proportional to the number N of illuminated grating lines in dispersion direction. In general, it can be related to observable beam properties also for non-ideal instrumental and optical conditions such as an extended, spatially incoherent source via the inequality
in the (±1)st order at an energy E, where h denotes Planck’s constant and the “=” case is valid at the diffraction limit. For an application near 10 keV for instance, N should not exceed ∼ 2 × 103, to ensure Δτ ≲ 1 fs. In our proof-of-principle design, we choose an operational, covered energy range ΔEc of approximately 1 keV around the design energy E0 = 8.3 keV, i.e. the Ni K-edge which serves as a paradigm for that of other transition metal elements such as Mn, Fe or Cu in the hard X-ray band. To confine the pulse elongation safely for the whole considered spectral range, to exploit the peculiar features of an RZP [2, 3, 8, 9] and to match the technical specifications of the beamline, the instrument is configured as an off-axis device with N = 7 × 102. As sketched in Fig. 1, the (−1)st diffraction order is used for reasons as follows. Under the given design constraints, the off-axis separation X0 represents one of few degrees of freedom. As it grows from the (−1)st over the 0th to the (+1)st order, a continuous decrease of the meridional full width at half maximum (FWHM) focus diameter (E0) is observed. Table 1 gives an overview, obtained by ray tracing for an incoherent source of 200 μm in diameter (FWHM). Such an entrance pupil width turns out to be necessary to get reasonable statistics for signal registration on the detector under the available technical conditions (Sect. 4). As a consequence of this fact and in conjunction with the conjugated optical configuration with a finite source distance R1, the μm-sized focal spots of the present experiment cannot exploit the great potential of grazing incidence RZPs to generate near diffraction-limited, tiny focal spots using micro-, rather than nano-structured DOEs. In fact, the ideal point spread function (PSF) of our design, calculated either by the simple Abbe formula or – more accurately – the Fresnel-Kirchhoff integral, would yield a sagittal FWHM of , whereas (FWHM) could be achieved for a perfect spherical point source at the design energy.
Fig. 1 Experimental setup. The RZP section of length L and width W is illuminated under the grazing angle α0 and the spectra are recorded around the exit angle β0 for E0 along the dispersion direction ydet in the detector plane below the 0th order specular reflection (not shown). Dimensions and angles are not to scale. Table 2 summarizes the design parameters.
Table 1. Focal spot size for various X0 separations
Table 2. Spectrometer design parameters
For monochromatic and spatially fixed point focusing, specialized one-dimensional hard X-ray RZPs have been realized with an outermost zone width of 0.7 μm to be used under parallel beam illumination, i.e. in the telescopic mode [12]. The measured focus width of 14.4 nm may even be improved down to < 5 nm for a modified setup, following the authors of this study.
However, unlike advanced imaging devices like that or its mirror-based counterpart [13] for which the spot size is of relevance, spectroscopy as performed in our initial, simplified demonstration is based on an optimal or at least sufficient energy resolution, large free spectral range and maximal efficiency as well. How these values change with X0 for an arrangement with otherwise fixed parameters as sketched in Fig. 1, is shown in Fig. 2. On the left, ray tracing results for the absolute energy resolution ΔEres. (red curves, measured in [eV]) and the proximate free spectral range ΔEFSR (gray) within which the resolving power E/ΔEres. is at least half as good as for the design energy, are plotted for an assumed constant grating length of 10mm. The hatched region around X0 = 0 is hence excluded, to avoid the 0th order background [9]. As it will be analyzed in Sect. 4, ΔEres. is given as the product of the diffractive dispersion in units of [eV/μm] and the focus width from Tab. 1. Since the former one scales inversely with the on-axis, X0-dependent RZP line density [3] and thus strongly rises in the regime |X0| → 0, the effect of the monotone improvement of the latter one (spatial resolution) with off-axis separations X0 (Tab. 1) on ΔEres. is enhanced for X0 > 0, but partly compensated in X0 < 0. The resolving power . hence shows a monotonic growth with |X0|, albeit of different magnitude for the (−1)st and (+1)st order.
Fig. 2 Influence of the off-axis distance X0 on the spectral width ΔE(FSR) (left picture, semi-logarithmic scale) and on the effective area Aeff (right). See the text for further details.
Fig. 3 Ray tracing simulation for five selected energies between 7.8 keV and 9.0 keV. The geometrical aspect ratios of the focal spots are to scale; and the color encodes the relative photon runtime from the source to the detector with respect to the arithmetic mean for the rays associated with E0.
Figure 2 indicates a strong preference for X0 ≪ 0, rather than X0 ≲ 0, in which positive displacements X0 ≲ 15 mm are even prohibited due to inaccessible line densities dl > 2 × 104 mm−1 and exit angles β beyond the critical angle θc ≈ 0.5° for total external reflection (blue region in Fig. 2). Indeed, the ratio ΔEFSR/ΔEres., to be regarded as one measure for the spectroscopic capabilities of the instrument, improves to 11 or more for X0 < −50mm. On the right of Fig. 2, the effective area Aeff is depicted in a two-fold way: The product of the projected area Aproj. = L sin α0 and the geometrical mean ⟨ℜ⟩geom. ≡ [ℜ (α0) ℜ (β0)]1/2 of the mirror reflectivity remains nearly constant within X0 < 0, whereas the analogous quantity using the diffraction efficiency increases with the on-axis grating period ∝ 1/dl (X0) toward its maximum at X0 ≈ −20 mm with for binary phase gratings.
After all, the chosen value for X0 and the geometrical RZP width W, listed together with other key parameters in Tab. 2 in formal accordance with the notation from previous work [14], are taken as a compromise between performance and realizability by e-beam lithography with writable line densities up to 1.7 × 104 mm−1 at most.
In the case of a hypothetical point source and perfect optics, Eq. (1) predicts with Δτ0 = N · h/E the lower limit on the time elongation to Δτ0 = 0.35 fs, if the configuration is operated at the design energy E0. Under the actual experimental conditions with the wide pinhole source and considered over the whole scanned spectral range ΔEc however, the relative optical path length distribution from the source to the detector plane in Fig. 1 needs to be evaluated in general by ray tracing. Figure 3 shows the result for the real Gaussian source with its size of 200 μm (FWHM) again, transferred into the vacuum photon runtime. Near E0, the pulse will be stretched to only several 10−16 s within the focal intensity distribution and by no more than 1.5 × 10−15 s over the whole energy range. Relation (1) is fulfilled in the sense that the extended source, which significantly prohibits an exploitation of the theoretically possible, maximal resolving power (see Sect. 4), is in the same way responsible for the time elongation beyond the ideal sub-femtosecond limit from above.
3. Fabrication and metrology
After defining the parameters of the individual lenses according to Tab. 2, their structure is calculated using an efficient software package that has been specifically developed for the structural design and simulation of the imaging properties of diffractive optics at the Institut für Nanometeroptik und Technologie (INT) in close collaboration with the Institute of Microelectronics Technology and High Purity Materials (RAS) in Chernogolovka, Russia [3, 15]. This software operates with a point source and consists of several parts:
- the ZON software, to calculate the structure of an optical element in transmission and reflection;
- the KRGF software, to simulate the diffraction properties of an ideal optical element with a point source;
- the DS software, to calculate the diffraction properties by taking into consideration material and shadowing effects.
In the frame of the present investigation, only the ZON software is used. It calculates the distribution of the phase shift in the plane of the zone plate, which is necessary for the transformation of the outgoing wave from a point source into a wave converging to the focal point.
The polygonal grating line structure obtained in this way is subsequently converted by the software “Nanomaker” (Interface Ltd.) for the high voltage e-beam writer (Vistec EBPG5000plusES). To suppress aberrations caused by low spatial frequency surface distortions, a plane super-polished Si substrate of 6 mm thickness and 1 inch in diameter is used. Its slope error is specified to 0.6arcsec rms ("root-mean-square"). Instead of a fully variable laminar profile as it may be obtained by a rigorous coupled wave analysis (RCWA), we constrain to a near-optimal binary structure where both the etch depth of 10 nm and the 50% duty cycle are set as constant over the whole RZP. These parameters provide a sufficient efficiency, as the simulation results in Fig. 4 demonstrate for an ideal binary profile. After manufacturing [8], the Au coating of 60 nm corresponds to the 20-fold penetration depth of 8.3 keV X-rays under 0.3° grazing incidence and hence hides the Si substrate.
Fig. 4 Calculated on-axis efficiency of the laminar Au profile in (−1)st order for the design energy E0 at the geometrical RZP center X0 (Tab. 2). No surface roughness is presumed.
Atomic force microscopy (AFM) scans of inner and outer grating areas are shown in Fig. 5, from which the right one (b) reveals the characteristically steep inclination of the outermost lines up to 89.7° with respect to the on-axis zero slope of the elliptic line shapes. In fact, the nearly in-line orientation of these lines with the incident beam gives rise to efficient conical diffraction in the absence of shadowing. However, the ultimate line density of ≲1.1 × 104 mm−1 comes at the expense of a degraded profile shape in the proximate outer third of the RZP area, nearly sinusoidal with a nominal, 17% relative efficiency loss off the laminar phase grating performance in scalar theory. Figure 6 illustrates the height profile for each case from Fig. 5, gathered from single-pixel rows across the grating lines. The AFM recordings well confirm the aimed groove depth – we find ⟨Δt⟩L = (10.1 ± 1.2) nm and ⟨Δt⟩H = (10.6 ± 0.7) nm, respectively. For an RZP as characterized so far, RCWA simulations predict a (−1)st order diffraction efficiency of 17.8% at 8.3 keV, averaged over the whole grating area.
Fig. 5 AFM images of the grating profile. The pictures are taken at low (a) and high (b) line density regions of the RZP, as indicated by yellow rectangles in the schematic central sketch, not drawn to scale here for better illustration.
Fig. 6 Grating profile cross section, evaluated from the AFM images in regions of low (L) and high (H) line density, as indicated in Fig. 5(a) and 5(b), respectively. The mean etch depth is accordingly marked by ⟨Δt⟩L/H for the two domains.
To estimate the influence of scattering from surface imperfections, quantified by the frequency-dependent rms roughness σν, on the focal spot size, the contrast and the efficiency, the power spectral density (PSD) is computed from the AFM data and plotted in Fig. 7. Besides the line peaks near 103 mm−1 and 104 mm−1, the PSD reveals at ν ≈ 2.4 × 104 mm−1 the footprint of the Au grains with a typical diameter of ≈ 40 nm. The overall rms roughness of the RZP is given as [16]. After subtraction of the line peak contributions, we find σtot = ±1.0nm. Via the common Strehl ratio [17], the total scattered intensity (TIS) amounts to 17%. Since the smooth-surface condition [17] is already fulfilled for σtot, it is so for each frequency component σν too, and we conclude that the spatial resolution in the image plane is not degraded by low-spatial frequency components of the PSD. On the other hand, the steep decline ∝ ν−5 beyond 2 × 104 mm−1 indicates small roughness amplitudes on length scales below the minimal groove width – visualized also in the comparatively smooth shape of the high line density profile scan from Fig. 6. The TIS is thus expected to mainly broaden the PSF by ≈ 4%, rather than being diffusively distributed via wide-angle scattering. Accordingly, the roughness diminishes the diffraction efficiency by 1.5%, due to the loss in reflectivity.
Fig. 7 Normalized power spectral density (PSD) function across the grating lines, based on the AFM data from Fig. 5 close to the optical axis (red) and near the outer edge (blue).
4. Measurements and results
An efficiency of 17.5% is measured at E0 for the grating itself, 0.1% less than predicted by the RCWA code together with the surface roughness. Over the full 1.2 keV band on the other hand, the linear rise of the simulated diffraction efficiency with the energy from 16.1% to 19.4% is in part amplified by the increasing CCD quantum efficiency (analog-to-digital units, ADU) of the CCD camera in use (Rigaku Xsight Micron) between 10% and 14%. This allows for an estimation of the slightly energy-dependent total detection efficiency
to (2.2 ± 0.6)%. A sufficient sensitivity is hence assured over the accessible bandwidth of more than 1 keV.
Experiments are carried out at the KMC-2 beamline, one of the BESSY II storage ring facilities [18, 19], stabilized and monochromatized to δ E ≲ 1.5 eV. Its maximal photon flux of 7 × 109 s−1/100mA within ΔEc will support the measurements. The imaging CCD camera is equipped with a scintillator screen to provide a field of view (FOV) of 0.67 mm (H) × 0.90 mm (V) and an effective spatial resolution of 1.6 μm at 8 keV. A series of 17 recordings with an increment of 75 eV and the same integration time of ~ 5 min covers the range from 7.8 keV to 9.0 keV, as shown in Fig. 8. Including the strong fluctuations of the diffracted power beyond 8.9 keV, likely caused by a minor mechanical malfunction of the beamline optics as discussed below, the otherwise fairly constant count rate varies statistically by only ±32% around its mean. Unlike the focal spot size in dispersion direction, which broadens as expected from the design energy toward the low and high energy end of the scanned range due to the intrinsic chromatic aberration of the RZP, the horizontal width, i.e. perpendicular to ydet in Fig. 8, increases from 6.5 μm at 7.8 keV over 18 μm at E0 up to ≳20 μm at 9.0 keV. These observations contradict the predictions from ray tracing simulations for perfect beamline optics: A near-linear and symmetric broadening ∝ |E − E0| with a minimum of 2 μm at E0 is indeed missed and may be mainly attributed to two reasons:
- With 2.5mrad, the sagittal beam divergence as provided by the KMC-2 bending magnet dipole source, corresponding to the horizontal detector coordinate, is 5 times larger than the meridional divergence and only limited by the entrance pupil, i.e. the pinhole width.
- The double-crystal monochromator [18, 19] suffers from an almost inevitable mechanical tolerance, both in rotation and translation of the crystals during an energy scan. Accordingly, the beam varies not only in its angular orientation, but also moves around in its lateral position within the sagittal plane.
Fig. 8 CCD focal plane recordings for photon energies between 7.8 keV and 9.0 keV. Each image is composed from 131 (H) × 151 (V) pixel of 0.65 μm in size. The intensity is color-encoded on the same scale, where brighter tones indicate higher intensity.
We emphasize the only marginal effect of such shortcomings on the aimed scientific task, i.e. a proof-of-concept feasibility study of wavelength-dispersive ultra-fast hard X-ray spectroscopy. It will be shown in the following by comparing theory and measurements, that they hardly deteriorate the one-dimensional intensity distribution along the relevant dispersion axis ydet, as it results from an integration over the CCD pixel rows in horizontal direction.
From the one-dimensional focal spot size (FWHM) along the direction ỹ – we abbreviate the ydet-axis in Fig. 8 by ỹ in the following – the spectral resolution is evaluated to
and , describing the theoretical dispersion for the mean line density dl (Xm) ≈ 70 mm−1 [3] at the optical median Xm of the RZP [14]. In analogy to the coordinate convention, we omit the subscript(s) for the spectral width in part for the sake of simplicity, such that ΔE denotes the spectral resolution ΔEres. from Sect. 2, if not stated differently.For our setup, the dispersion from Eq. (2) increases nearly linear, but relatively weak within the considered energy range and is written as
where the coefficients are found as a0 = 20.8 μm−1 eV, a1 = 9 × 10−3 μm−1 and a2 = 4 × 10−7 μm−1 eV−1. The relative difference between Eq. (3) and the corresponding empirical results dE/dỹ|ex. is plotted as a function of the energy in Fig. 9. Whereas the experimental results match the analytical prediction of 20.8 eV/μm at E0 to a high accuracy, the deviations of opposite sign in the order of ≲ 4% toward lower and higher energies are most likely caused by the additional focal spot shift perpendicular to the dispersion direction in Fig. 8, due to the mentioned mechanical imperfection of the optical elements at the KMC-2 beamline.Fig. 9 Relative deviation of the measured energy dispersion dE/dỹ|ex., as evaluated from the ỹ-positions of the peaks in Fig. 8 (black dots), from the value expected by Eq. (2).
Fig. 10 Resolving power between 7.8 keV and 9.0 keV, evaluated from the focal spot FWHM of the CCD images (Fig. 8) in dispersion direction (black dots). The data (black dots) are fitted (red line) by the model defined in Eq. (6) and compared to ray tracing results (gray dots and dashed line).
A simplified model for the spectrometer performance relies on its functional behaviour, as depicted in Fig. 1. In agreement with experimental observations – an FWHM between 3.4 μm at 8.4 keV and (1.8−2.6) × larger values toward 9.0 keV and 7.8 keV are derived from the CCD images, respectively – a Gaussian beam analogy lets the spot size to be estimated as
where the constant (Sect. 2) is given by the instrumental specifications. For an enhanced flexibility in the adaption of the model to the data, the reference energy E⋆ which yields the minimal spot size , is moreover considered as a free parameter. Far away from E, the spot size – denoted by in this regime – is expected to increase like . The slightly varying, asymmetric weight factor accounts for the linear increase of the spot size with the energy in the geometrical limit of sufficiently large energy separations δE ≡ E – E⋆ ≳ σ⋆. From the theorem of intersecting lines and the linearity between the focal distance and E, we obtain , where L sin β0 represents the projected meridional aperture. With the definition in Eq. (5), this leads to . The dispersion term , as introduced in Eq. (2), is balanced, i.e. almost equalized over the whole energy range of 1.2 keV within an rms error amplitude of ±2.5% by . After some straightforward manipulations, we hence cancel them by each other in the final expression for the resolving power , that turns out to follow a near-Lorentzian shape, whose maximum at E⋆ ≈ E0, considered as the third free fit parameter in Eq. (6), is analytically expressed asEq. (6) may be adapted to the measured data using E⋆, and σ⋆. Figure 10 illustrates the result of the least-squares fit. With an rms error of the fit residuals for the empirical data set and ±0.7 in case of the analogous ray tracing results, the model as proposed in Eq. (6) may be accepted with a sufficient reliability for this spectrometer. In particular, the experimental data fit reveals a free spectral range of 1.24 keV, i.e. the full energy band as it was aimed for in the design studies.
Whereas near the outer edges of this band the simulation coincides with the experimental data fit function to the resolving power, a relative difference of 14% near its maximum around 8.4 keV might be explained in a natural way with an effectively smaller source size than the nominal pinhole width would suggest. Partial spatial coherence can be ruled out to a far extent, since the 200 μm wide source illuminates only about 5% of the grating in a distance of 10m coherently [20]. In fact, a substitutional incoherent source of (175 ± 8) μm in size would be consistent with the observed discrepancy. Following basic geometrical relations [3] and as confirmed by ray tracing within a 3.6% tolerance, the spatial resolution
grows in proportion to the source size at least far above the diffraction limit of 0.51 μm for a fictive point source.For more reliable statements, a detailed knowledge of the intensity and wavefront distribution of the beam, as it propagates from the source, is required however and would strongly support the physical interpretation of measured results in upcoming experiments.
5. Discussion and conclusion
Based on a diffractive-reflective zone plate in an off-axis grazing incidence configuration, the experimental realization of a wavelength-dispersive spectrometer with an energy resolving power of ~ 102 is demonstrated in the hard X-ray range from 7.8 keV to 9.0 keV. In particular, the ultimate Fano limit [21] of a Silicon drift detector as it is in common use for energy-dispersive spectroscopy with a hypothetical zero read-out noise is exceeded by a factor of 1.7 at least for E0. Realistic, commercially available state-of-the-art devices of this type [4] still provide a somewhat worse resolution of ≈ 160 eV around 8 keV. Hence, even in the current version of the instrument whose resolution is constricted by the source size, the wavelength-dispersive spectrometer performs better (near the design energy) than or at least comparable (below and above E0) to its energy-dispersive analogue.
A remarkable large free spectral range of ≳1 keV with a practically constant sensitivity is accessible due to an optimized design and fabrication process. From the scientific point of view, this feature allows for the continuous recording of spectra around the K-edge of transition metals, such as Ni in our demonstration design.
For a sufficiently low number of 7 × 102 grating lines and supported by the almost negligible penetration depth of the X-rays into the Au coating of 3 nm, the short pulse elongation time in the order of ~ 1.5 fs will maintain the time resolution in upcoming ultra-fast experiments at free-electron lasers to a far extent. An “extreme case” example may be considered for illustrative purposes: If convolved with a fairly short intrinsic pulse duration of 4.5 fs [11] under the assumption of Gaussian shapes, the original pulse would be elongated by likely marginal 5% during its detection.
Recent advancements in hard X-ray transmission zone plate (TZP) nano-fabrication technology [22] are, already a major breakthrough in the field of imaging, in principle applicable to the purpose of spectroscopy as well. With an outermost line density of 5 × 103 mm−1, the Ir-filled diamond TZP of 5 × 10−4 m in diameter is supposed to be illuminated in an off-centered segment region of the same numerical aperture: We estimate dE/dỹ|TZP ≳ 32.3 eV/μm and a focal distance of 0.35m, which would nonetheless imply an 60% loss in E/ΔE at least, compared to the RZP as used within this work. However, the rapid progress in the development of transmission zone plates still keeps the possibility that such a lens competes in future also in the field of hard X-ray spectrometry.
With the “photon sieve” [23], an attractive approach was introduced several years ago to relax the technological limitations on the aspect ratio of hard X-ray TZPs [24]. The flexibility, which arises from the customized, approximative replacement of the open Fresnel zones, enables furthermore an optimized design and improved optical performance, e.g. an enhanced signal-to-noise ratio (SNR) for a properly apodized aperture window function. If combined with the concept of reflection zone plates [25] under grazing incidence, the major practical advantage of an increased heat load, relevant for FEL applications, relies on the equivalence of nanomirrors and -absorbers in the photon sieve array [25], according to Babinet’s principle. On the other hand, the spatial and hence spectral resolution gain of a photon sieve, as compared to its conventional zone plate analogue, is induced by the enlarged effective aperture for the same minimum structure size. Typically, the focal spot width may shrink by up to 50% in the (±1)st diffraction order, the required precision in the nano patterning process via electron beam lithography presumed [23]. In view of the potential of reflective photon sieves with respect to the focus quality, this relatively new device deserves to be considered for future applications in time-resolved hard X-ray spectroscopy, as long as the moderate efficiency can be compensated by a brilliant source or the improved SNR of such an apodized DOE.
The most exciting prospect from this basic experimental and theoretical study is admittedly still hidden in the source diameter. As mentioned right afore in Sect. 4 as the result of straightforward Fresnel-Kirchhoff calculations [15], the focal spot may shrink to several 100 nm in its width along the dispersion direction for an appropriately modified experimental setup. The spectral resolving power is expected to increase by a multiple, while the pulse elongation could be kept as small as presently or even further shortened in this near diffraction-limited operational regime of the RZP. Such an advancement would clearly benefit the ambitious pathway to truly Fourier-limited experiments on the sub-femtosecond time scale.
Acknowledgments
The authors acknowledge support from the BMBF project “Next generation instrumentation for ultrafast X-ray science at accelerator-driven photon sources” (no. 05K12CB4) and a Marie Curie FP7-Reintegration-Grants within the 7th European Community Framework Program (project no. PCIG10-GA-2011-297905).
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