1. Monochromatizing x-rays

X-rays are used to investigate the structure and dynamics of materials, providing high spatial resolution and chemical selectivity. In particular, third-generation synchrotron radiation sources can produce powerful soft and hard x-rays using insertion devices such as undulators. These undulators can provide tunable x-rays, and have become a mainstay in materials science and the life sciences. One of their characteristics is that they emit several harmonics of the fundamental radiation [1,2], each of which may be chosen for a given experiment. However, this requires rejection of the unwanted harmonics from the undulator spectrum. Common rejection techniques make use of crystal monochromators, multilayer mirrors or harmonic rejection mirrors. Crystal monochromators are commonly used when a very narrow bandwidth is required, typically on the order of ΔE/E = 10⁻⁴. Multilayer mirrors can be used to tailor the transmitted spectrum, resulting in two orders of magnitude higher photon flux, albeit at the expense of a significantly wider bandwidth [3,4]. Harmonic rejection mirrors, using the principle of total external reflection, can provide even higher throughput [5,6]. Although harmonic rejection ratios higher than 100:1 can be achieved in a single reflection [7], only higher harmonics can be rejected with this particular technique. All three of these selection methods suppress, rather than completely remove, the unwanted spectral components and the remaining single harmonic is not pure. This study was conducted to investigate and demonstrate the technical feasibility of using refractive optics as a method of dispersing and removing unwanted undulator harmonics, rather than merely suppressing them.

Already in 1926, Davis and Slack investigated the refraction of x-rays in aluminum [8]. In 1984 Deutsch and Hart carried out a similar experiment using a beryllium prism [9]. After an initial breakthrough by Snigirev et al. refractive x-ray lenses have become increasingly important for applications at synchrotrons [10] and their working range has been extended to softer x-rays by the use of beryllium [11]. Tur’yanskii demonstrated that x-ray prisms made of diamond could be used for spectrometric applications. He also proposed the use of beryllium and provided numerical results [12,13]. Zhong employed a compound acrylic prism and crystal monochromator assembly to reduce the content of unwanted harmonics from the transmission spectrum [14]. His setup, however, was limited to applications above 10keV due to high absorption losses as a result of the prism material. He therefore suggested a beryllium prism as possible improvement. In this study we have implemented a high throughput, high dispersion beryllium prism with a wide spectral operating range. We give experimental results, showing that a single beryllium prism is well suited for selection of an undulator harmonic in the x-ray regime between 3 and 12 keV. We were able to disperse incoming radiation while keeping the total losses below one order of magnitude and in particular around 50% for the intended application of harmonic selection. Disregarding scattering, this technique allows for an infinitely high rejection ratio by totally blocking the neighboring harmonics of a well-collimated beam.

The work presented here is part of an effort to develop a low-loss harmonic selection system for the FemtoMAX beamline at the MAX IV Short-Pulse Facility (SPF) [15]. The SPF provides 100 fs electron bunches at an energy of 3 GeV. The FemtoMAX beamline has been designed for laser pump/x-ray probe experiments, and will use two 5 m in-vacuum undulators with a 15 mm period, and gaps yielding a K-value ranging from 0.5 to 2.2. This will result in fundamental photon energies E_fund ranging from 1.8 keV to 5.5 keV. Higher harmonics provide a useful output up to 20 keV.

2. A first experimental trial

A proof-of-principle experiment was carried out at beamline D611 of the MAX II electron storage ring at the MAX IV laboratory in Lund, Sweden. The experimental setup is shown in Fig. 1. In order to achieve the required dispersion at around 50% throughput, the angle of incidence on the prism must be optimized for each photon energy. Typical grazing angles down to ϑ ≈1°, motivate the form of a cuboid (apex angle = 90°). The prism is made of I-70-H optical grade crystalline beryllium and has the dimensions 50 × 40 × 5 mm³. Both optical surfaces, i.e. the entrance surface measuring 50 × 40 mm² and the exit side wall measuring 40 × 5 mm², are polished to a scratch-dig of 20–10, with a surface roughness of 30 nm rms, measured by a Zygo interferometer.

 

Fig. 1 Side view of the experimental setup at the D611 beamline of the MAX II electron storage ring at the MAX IV laboratory. The x-ray beam (blue) arrives from the left and leaves the vacuum through a beryllium window. It is subsequently collimated by a slit and refracted by the prism. A phosphor screen whose emission is imaged onto a CCD serves as a detector. In addition, a representative beam profile and vertical intensity lineout are shown to confirm that a high beam quality can be maintained after passage through the prism.

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Radiation from a dipole magnet passes through an InSb crystal monochromator and leaves the vacuum system through a beryllium x-ray window. At this point, the radiation has a bandwidth of ΔE/E ≈2*10⁻⁴. A slit located 2 cm behind this window collimates the beam. As the beam is also collimated by a torodial mirror, this results in a super Gaussian vertical intensity distribution impinging on the prism, which is located 10 cm behind the collimation slit, which can be described by the expression I = I₀ exp(-(2r/ω₀)³), where ω₀ = 178 µm is the vertical spot size. The vertical position of the prism and the grazing angle ϑ can be adjusted during the experiment. A scintillator screen, located 0.98 m behind the prism, serves as an observation point for the beam deviation and transmission. Its emission is imaged 1:1 onto a Hamamatsu Orca CCD. A helium-filled tube is inserted between the prism and the detector to minimize absorption in the optical path.

Two measurements were carried out to evaluate the performance of such a prism harmonic separator. These measurements also enabled us to investigate the validity of our ray-tracing calculations, so that they can be used to extrapolate parameters for future applications. In the first experiment, beam deviation and throughput were measured at two grazing angles suitable for harmonic separation in the relevant spectral regime: ϑ₁ = (2.6 ± 0.4)° and ϑ₂ = (5.7 ± 0.4)°. Measurements were carried out at the two angles for 8 photon energies ranging from E = 3.7 keV to 11.5 keV. The prism was slowly inserted into the beam from below, for each combination of ϑ and E, in order to determine the optimum vertical position of the prism, i.e. the position at which the entire beam is refracted with the least absorption. The unrefracted signal, i.e., when the prism is still below the beam, serves as a reference for zero absorption and refraction.

Figure 2 shows the results of this experiment, together with the ray-tracing predictions. The errors on the measured data were derived from the 95% confidence interval of the nonlinear curve fits on the measured beam profiles. An error estimate on the grazing angle ϑ leads to an error margin in the theoretical results. When comparing experimental data and simulations, it should be noted that the incidence angle is experimentally determined using the shadow of the prism at ϑ = 0°, which yields an uncertainty of 0.4°. The vertical prism position used in the simulations is derived by analyzing the measured beam deviation as a function of vertical position, and is a free parameter in the calculations. As can be seen from the experimental data presented in Fig. 2, the throughput increases with photon energy, reaching a maximum of 86% for ϑ = 5.7°. The dispersion decreases with increasing energy, to 15 and 6 µrad/keV at 10.4 keV for ϑ = 2.6° and 5.7°, respectively. As larger grazing angles result in less dispersion at a higher throughput, an appropriate trade-off must be made. For the ray-tracing calculations implemented with MATLAB, the material composition given in Table 1 is used. The refractive index is then obtained together with tabulated data form Henke’s database [7]. The composition of the prism was determined previously from a piece of beryllium from the same block of material as that from which the prism was manufactured, and should closely resemble the composition used in the experiment. Agreement with the experimental results confirms this.

 

Fig. 2 Upper row: throughput as a function of energy. Lower row: prism dispersion. Shown in red are measured data with error bars. The solid and dashed lines show the results of the ray-tracing calculations and the corresponding error margin, respectively. These calculations are based on the measured material composition given in Table 1.

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Table 1. Prism Composition Given by the Manufacturer

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The discontinuity in the throughput curves in Fig. 2 at about 7 keV is the effect of minute iron impurities on the transmission. An increase in iron concentration from 0.017%_vol to 0.1%_vol would increase the absorption by at least 10% from above the absorption edge around 7 keV to at least 11 keV. This emphasizes the importance of a low iron concentration when using the prism in this regime.

In the second experiment, we measured the beam deflection 𝛷 while scanning the grazing angle ϑ at two energies (E₁ = 4.6 keV and E₂ = 6.0 keV) at different vertical prism positions. The results can be seen in Fig. 3. As expected, the beam refraction increased with decreasing grazing angle at the expense of throughput. Again, the experimental data are in agreement with the predictions assuming the material composition given in Table 1.

 

Fig. 3 Measured refraction $\Phi$ as a function of grazing angle ϑ. The black curves show the result of the calculations.

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3. Discussion of results

In both experiments, an increase in beam divergence was noted with increasing vertical prism position and thus increasing the beam propagation distance from the apex inside the material. The effect was most pronounced at lower energies. At a grazing angle of ϑ = 5.7° and an x-ray energy of 4.6 keV, the divergence increased by Δ𝛷_bulk = 140 µrad per mm from the apex, while the increase was only 11 µrad/mm at 11.6 keV. As this effect occurs equally in both dimensions it cannot be attributed to surface scattering. Most likely, this effect is caused by scattering from inclusions. According to the manufacturer, these may occur in a small number, occupying no more than 6*10⁻⁵ of the total volume, and result from the consolidation process of beryllium powder by hot isostatic pressing to produce the I-70-H optical grade beryllium.

As mentioned earlier, there is a trade-off between throughput and dispersion. For comparison, the design requirement at the new FemtoMAX beamline for the angular separation of two neighboring odd undulator harmonics at a selection slit is Δ𝛷 ≈50 µrad, disregarding scattering. An undulator beam may have a divergence down to Δ𝛷_undulator ≈10 µrad [15]. However, a large beam diameter ω₀ will require a deep passage through the material and the above observed increase in divergence due to bulk scattering increases the necessary dispersion. In addition surface scattering might also be present to some degree. The latter has been investigated numerically with the commercially available ray-tracing software FRED from Photon Engineering LLC. It is found that a surface polishing to a scratch/dig of 20-10 renders this effect negligible. As most of the refraction takes place at the first surface with the shallow grazing angle, the requirements on surface flatness of the output side of the prism can be less stringent. Putting all of these considerations into one equation leads to the following approximation for the dispersion requirement for odd harmonics:

4. Application at a beamline

Absorption in the prism is the major source of loss. This can be reduced by adjusting the vertical position of the prism such that only part of the beam is refracted, while the path length of the more intense central part of the beam inside the prism is reduced. This results in decreased absorption losses, outweighing geometric losses. Alternatively, throughput can be maintained at a larger dispersion. The benefit in throughput is more pronounced for a Gaussian beam than for a super Gaussian, due to the fact that a Gaussian with the same spot size ω₀ carries more energy in its wings, and for high energies, a larger distance to the apex becomes necessary to keep geometric losses low.

Using the beam parameters for beamline D611, Fig. 4 shows the calculated throughput as a function of energy and distance from the prism apex for a fixed grazing angle ϑ = 5.7°, which provides high dispersion and low absorption losses for photon energies around 6 keV. It can be seen that higher energies benefit less, and that throughput is optimized by refracting as much of the incoming beam as possible. Lower energies, in particular those below 6 keV, benefit from lowering the prism, enhancing throughput by as much as 10%.

 

Fig. 4 Modeled throughput as a function of distance to the prism apex and photon energy. a) The incident beam profile with respect to the prism apex. The prism was continuously lowered from black to red. b) The throughput for the same series of prism apex distances taking absorption and geometric losses into account. The throughput can be enhanced when the reduction in propagation length outweighs the reduction in integrated refracted energy.

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As discussed previously, throughput is mainly determined by the minimum dispersion requirement (dϕ/dE)_prism given by Eq. (1). By opening the magnetic gap, the undulator emission lines are tuned to higher energies. As E_fund increases, the dispersion can be decreased. At a certain point the gap is closed again, and the next higher harmonic is used for the experiment, resetting E_fund albeit requiring a higher dispersion, leading to less throughput. Typically only odd harmonics are used for experiments as they provide a significantly higher photon flux than even ones [2,15].

The expected throughput at the currently built FemtoMAX beamline for grazing angles between ϑ = 0.5° and 45° was calculated using our MATLAB ray-tracing code. To cater for scattering effects, a separation of odd neighboring harmonics by twice the spot size ω_slit = ω₀ + l Δϕ_undulator is required at the slit l = 8.9 m behind the prism, as indicated by Eq. (2). The required minimum dispersion is generally less than 23 µrad/keV.

In these calculations, each energy requires the grazing angle and vertical prism position to be optimized for maximum throughput, while achieving the minimum dispersion requirement. With a spot size ω₀ = 188 µm at the prism and a full divergence of 12.6 µrad and depending on the harmonic in use, a working range from 3 keV to beyond 20 keV becomes possible. The results of these calculations are summarized in Fig. 5. For typical beam parameters it is found that undulator source size mainly affects the throughput, while the beam divergence is the crucial parameter to achieve a wide working range.

 

Fig. 5 The expected maximum achievable throughput at the FemtoMAX beamline when exceeding the minimum dispersion requirements for odd harmonics, calculated for prism grazing angles ϑ = 0.5°–45°. The bold regime of each plotted harmonic indicates possible undulator emission, while the dotted line is a guide for the eye, extrapolating the theoretical throughput. The harmonics are separated by two full spot sizes at a slit 8.9 m behind the prism. As shown, the required dispersion can be achieved with a throughput well above 50% over an energy range up to at least 20 keV.

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Heat transfer calculations show that a water-cooled prism setup could tolerate ≈ 1 kW beam power before reaching temperatures in excess of 800 °C, at which recrystallization starts. The prism harmonic separator is thus well-suited for use with sources that have low average power such as free-electron lasers (FEL) [16], including the LINAC Coherent Light Source (LCLS) at SLAC [17], the SACLA FEL at SPRING 8 [18], and the European XFEL Facility [19], or at short-pulse sources such as the currently built FemtoMAX beamline at the MAX IV SPF [15]. Undulators at storage rings can produce a radiative power well in excess of 10 kW, but it may also be possible to use this prism setup at these facilities by removing the low-photon-energy content from the beam with heat absorbers.

5. Conclusions

We have shown that a prism made of beryllium is well suited for high throughput, high dispersion applications for x-rays in the range from 3 to beyond 20 keV. We have demonstrated experimentally that the technique fits the requirements for a low-loss harmonic separator for undulator radiation as it yields more than 50% throughput over its working range while providing sufficient dispersion for spatial separation of the desired wavelength. The beryllium prism harmonic separator is thus an interesting alternative to well-established selection methods at modern high brightness x-ray sources.