On obtaining high spectral resolution in extreme ultraviolet/soft X-ray monochromators operating off-plane diffraction in a divergent incident beam.

When the trajectory of an incident beam is oriented parallel to the grooves of a periodic grating structure the radiation beam is diffracted off-plane orthogonal to the plane of incidence. The diffraction efficiency in this condition is very high and in a grating with a sawtooth profile it can approach the reflection coefficient for a simple mirror, when the diffraction order of interest follows the direction for specular reflection at the flat part of the steps. When this concept is used in a plane grating in a monochromator for synchrotron radiation sources, the incident beam is almost always collimated in order to minimize any deterioration of the beam properties due to aberrations, which will be introduced in the diffraction process when an uncollimated beam is used. These aberrations are very severe when the groove density is constant. It will be shown that the effect of these aberrations can be corrected after the diffraction by the use of astigmatic focusing. The latter can be provided by a crossed mirror pair with different focal lengths in the corresponding orthogonal directions. Then a monochromator based on this concept can provide source size limited spectral resolution in an uncollimated incident beam. This is identical to the spectral resolution that can be provided by the same grating when operated at the same position in a collimated incident beam. The source size limited spectral resolution in this case corresponds to a high spectral resolving power of better than ΔE/E = 10 000 for photon energies around 300 eV in the soft X-ray range.

A high-efficiency and high-spectral-resolution EUV/soft X-ray monochromator based on off-plane diffraction.

The most efficient diffraction at a periodic grating structure is expected to take place when the incident radiation can be considered to have been specularly reflected off the inclined part of grooves that are positioned parallel to the trajectory of the incident beam. Very encouraging results for this configuration, in which the diffraction takes place off-plane, have been reported recently for a grating to be used in a spectrometer for space science investigations. This grating provided high efficiency for a relatively large groove density and a large blaze angle. High efficiency was observed even in higher diffraction orders up to the fourth order. Here the performance parameters, especially for the combination of diffraction efficiency and achievable spectral resolution, will be discussed for a grating used in a grazing-incidence plane-grating monochromator for monochromatization of synchrotron radiation in the extreme ultraviolet (EUV) and soft X-ray range with photon energies between 30 eV and 2000 eV. It is found that the instrument can provide competitive spectral resolution in comparison with the use of in-plane diffraction. In the case of comparable spectral resolution, the off-plane diffraction is found to provide superior efficiency.

On the advantages of operation in second-order diffraction of blazed gratings in soft X-ray monochromators.

The fact that a diffraction grating can provide twofold-smaller bandwidth when operated in second-order diffraction is long known and applied routinely in the laboratory for spectroscopy in the visible and ultraviolet spectral range. A similar routine operation in monochromators for the soft X-ray range is not reported yet. This study will thus address the feasibility of efficient diffraction of soft X-rays in the second order at reflection gratings when operated at grazing angles of incidence. The related systematic study could make profitable use of a recently introduced simple analytical equation for the prediction of the diffraction efficiency of blazed gratings with an ideal sawtooth profile. The predictions are then verified by use of rigorous calculations. The principle finding is that, by operation of gratings with lower groove densities, and thus with higher efficiencies, in higher order diffraction, one can extend the tuning in existing instruments with mechanical/optical limitations to larger photon energies. The performance in terms of transmission and spectral resolving power can be very similar to the performance of a grating with a larger groove density, which would otherwise have to be used for accessing the same energy range. This would allow operation of a single highly efficient grating over a larger photon energy interval at a modern synchrotron radiation source, e.g. from 0.3 to 2.2 keV. Without any requirement for a sophisticated grating exchange scheme, a related instrument promises to be sufficiently stable for the needs imposed by the improvements in source point stability at diffraction-limited storage rings.

On smart optimization of blazed soft X-ray gratings.

The first attempts to calculate the diffraction efficiencies of gratings in the soft X-ray range were made on a scalar model. The results were simple analytical equations, that always severely overestimated the performance of real objects. In this respect, computer programs were found to be more successful, which rigorously consider all diffracted and refracted waves. Consequently soft X-ray gratings are presently optimized using these tools, which requires rather extensive calculations for any instrument optimization as general trends are not immediately obvious. Here it will be shown that the results of the rigorous calculations for gratings with blaze or sawtooth profile can be approximated rather well with a simple analytical equation. This equation contains three multiplicative factors, which deal independently with the effect of the reflectivity, the blaze angle and the groove density. This opens the possibility to initially ignore the effects of the blaze angle and thus to start an optimization in a very general way. Such optimization can be based on isoreflectivity curves and it can then provide `blaze maximum efficiency maps', i.e. simple images. In these latter images, one can identify directly the optimum parameters for a grating, i.e. the groove density providing best efficiency for a requested spectral resolving power. Only successively will the blaze angle have to be fixed. Its choice is then not the result of an extensive optimization process but of a simple calculation applied for the photon energy at which maximum efficiency performance is requested. The maps presented here are used for the optimization of a medium-resolving-power soft X-ray monochromator, which can scan the photon energy range 300-2000 eV.

Zooming X-rays with a single rotation in X-ray prism zoom lenses (XPZL).

Prism arrays arranged to form a slightly open alligator mouth were found to focus incident X-rays, as with increasing distance from the object symmetry axis these rays hit an increasing number of refracting prism tips. Such an object is then formally a refractive lens. Due to the strong energy dependence of the refractive index of material for X-rays a refractive X-ray lens is chromatically focusing. The attractive feature of the alligator lens is the inherent zoomability possible as the mouth can easily be opened or closed. However, the required tolerances for the jaw rotations and the jaw positioning are so stringent, that the routine use of such systems has not been reported yet. This study will show that the related technical problems can be overcome by proper object fabrication. In fact the here presented objects can already be aligned in the production stage. Then the assembly can be made with simple tools. And the zooming is achieved by just a simple rotation. The transmission through the devices was found to be as expected, and thus performance-wise these objects can directly compete with other refractive X-ray focusing systems.

Rigorous calculations and synchrotron radiation measurements of diffraction efficiencies for tender X-ray lamellar gratings: conical versus classical diffraction.

When reflection gratings are operated at grazing incidence in the extreme off-plane configuration and the incident beam trajectory is parallel to the grooves, the diffraction into the first order can be more efficient than in the classical orientation. This situation is referred to as the conical diffraction case. In the classical configuration the grooves are perpendicular to the incident beam and thus an efficiency-reducing shadowing effect will be observed at very grazing angles. It was recently shown that a laminar grating could provide symmetric and relatively high efficiencies in conical diffraction for diffraction even of photons with large energies of the order of 4 and 6 keV. For photon energies in the tender X-ray range, accurate computing tools for the calculation of diffraction efficiencies from gratings with simple coatings have not been available. Promising results for this spectral range now require the development of tools for modelling the diffraction efficiency expected in optical instrumentation, in which the provision of high efficiency in the indicated spectral range is mandatory. This is the case when weak sources are to be investigated, like in space science. In this study it will be shown that scalar calculations are not appropriate for this purpose, while newly introduced rigorous calculations based on the boundary integral equation method, implemented in the PCGrate® code, can provide predictions that are in agreement with observed diffraction efficiencies. The agreement is achieved by modelling the exact surface profile. This applies for both the conical diffraction configuration and for the classical in-plane configuration, in which a significantly lower efficiency was obtained. Even though the profile of the presented grating was not perfect, but significantly distorted, the calculations show that efficiency-wise the structure provided already more than 75% of the ideally expected efficiency for conical diffraction. This is a very promising result for further optimization of diffraction gratings for use in the tender X-ray range.

On amplitude beam splitting of tender X-rays (2-8†keV photon energy) using conical diffraction from reflection gratings with laminar profile.

Conical diffraction is obtained when a radiation beam impinges onto a periodically ruled surface structure parallel or almost parallel to the ruling. In this condition the incident intensity is diffracted through an arc, away from the plane of incidence. The diffracted intensity thus lies on a cone, which leads to the name `conical diffraction'. In this configuration almost no part of the ruled structure will produce any shadowing effect for the incident or the diffracted beam. Then, compared with a grating in the classical orientation, relatively higher diffraction efficiencies will be observed for fewer diffraction orders. When the incident beam is perfectly parallel to the grooves of a rectangular grating profile, the symmetry of the setup causes diffraction of the intensity symmetrically around the plane of incidence. This situation was previously tested experimentally in the VUV spectral range for the amplitude beam splitting of a radiation beam with a photon energy of 25 eV. In this case the ideally expected beam splitting efficiency of about 80% for the diffraction into the two first orders was confirmed for the optimum combination of groove depth and angle of grazing incidence. The feasibility of the amplitude beam splitting for hard X-rays with 12 keV photon energy by use of the same concept was theoretically confirmed. However, no related experimental data are presented yet, not even for lower energy soft X-rays. The present study reports the first experimental data for the conical diffraction from a rectangular grating profile in the tender X-ray range for photon energies of 4 keV and 6 keV. The expected symmetries are observed. The maximum absolute efficiency for beam splitting was measured to be only about 30%. As the reflectivity of the grating coating at the corresponding angle of grazing incidence was found to be only of the order of 50%, the relative beam splitting efficiency was thus 60%. This is to be compared also here with an ideally expected relative efficiency of 80%. It is predicted that a beam splitting efficiency exceeding 50% should be possible by use of more appropriate materials.

An efficient plane-grating monochromator based on conical diffraction for continuous tuning in the entire soft X-ray range including tender X-rays (2-8†keV).

Recently it was verified that the diffraction efficiency of reflection gratings with rectangular profile, when illuminated at grazing angles of incidence with the beam trajectory along the grooves and not perpendicular to them, remains very high for tender X-rays of several keV photon energy. This very efficient operation of a reflection grating in the extreme off-plane orientation, i.e. in conical diffraction, offers the possibility of designing a conical diffraction monochromator scheme that provides efficient continuous photon energy tuning over rather large tuning ranges. For example, the tuning could cover photon energies from below 1000 eV up to 8 keV. The expected transmission of the entire instrument is high as all components are always operated below the critical angle for total reflection. In the simplest version of the instrument a plane grating is preceded by a plane mirror rotating simultaneously with it. The photon energy selection will then be made using the combination of a focusing mirror and exit slit. As is common for grating monochromators for soft X-ray radiation, the minimum spectral bandwidth is source-size-limited, while the bandwidth can be adjusted freely to any larger value. As far as tender X-rays (2-8 keV) are concerned, the minimum bandwidth is at least one and up to two orders of magnitude larger than the bandwidth provided by Si(111) double-crystal monochromators in a collimated beam. Therefore the instrument will provide more flux, which can even be increased at the expense of a bandwidth increase. On the other hand, for softer X-rays with photon energies below 1 keV, competitive relative spectral resolving powers of the order of 10000 are possible.

On symmetric X-ray beam splitting with high efficiency by use of reflection gratings with rectangular profile in the extreme off-plane configuration.

In order to be reflected or diffracted off a surface structure soft X-rays and hard X-rays need to impinge at grazing angles of incidence onto the surface. In case of a reflection grating of highly symmetric structure with rectangular groove profile these grooves can be oriented parallel to the beam trajectory. In such a symmetric situation the distribution of the diffracted intensity with respect to the plane of incidence is then expected to be symmetric. This is indeed observed with symmetrically oriented diffraction peaks. It can be predicted that for appropriate structure parameters the intensity can be contained mostly in two symmetrically oriented diffraction peaks. This will also be the case for hard X-rays. The diffraction efficiency will be particularly high, when the angle of grazing incidence is chosen in the total reflection regime below the critical angle of the grating coating. These predictions were experimentally verified in this work for hard X-rays with photon energies between 4 keV and 12.4 keV. In the experiment of the order of 30% of the incident intensity was diffracted into the two first orders. This is to be compared to reflectivities of the order of 50% measured at the same coating in an unruled area of the substrate. Consequently the relative structural diffraction efficiency for each first order was about 30%, while ideally it could have been 40%. The presented grating structure will thus be a rather efficient amplitude beam splitter for hard X-rays, e.g. in the coherent beam from a free electron laser. In addition such object could then be used as the first component in Michelson interferometers for the beam characterisation or for introducing a time delay between two coherent beams.

Bidimensional focusing of x rays by refraction in an edge.

When an x-ray beam passes through the tip of a triangular prism, i.e., an edge, it undergoes two consecutive refraction processes. This will also happen when the incident beam is not perpendicular to the tip but when the beam progresses at a very small inclination to it. It will be shown that in such a condition, when both interfaces adjacent to the tip have concave surfaces, decoupled focusing in two orthogonal directions can be introduced in the transmitted x-ray beam. The limitations for this application are discussed, and focusing of x rays to spots with diffraction limited sizes of the order of 100 nanometers is found to be feasible. The feasibility of bidimensional focusing by use of such a device was experimentally verified.

Periodically structured X-ray waveguides.

The properties of X-ray vacuum-gap waveguides (WGs) with additional periodic structure on one of the reflecting walls are studied. Theoretical considerations, numerical simulations and experimental results confirm that the periodic structure imposes additional conditions on efficient propagation of the electromagnetic field along the WGs. The transmission is maximum for guided modes that possess sufficient phase synchronism with the periodic structure (here called `super-resonances'). The field inside the WGs is essentially given at low incidence angle by the fundamental mode strongly coupled with the corresponding phased-matched mode. Both the simulated and the experimental diffraction patterns show in the far field that propagation takes place essentially only for low incidence angles, confirming the mode filtering properties of the structured X-ray waveguides.

Concepts for flexible and efficient monochromatization of X-rays by refraction to a relative bandwidth of the order of 0.5%.

Recently it was shown experimentally that regular arrays of tiny prisms can be used as X-ray monochromators providing a spectral bandwidth of below 2%. Successively the measured spectral transmission functions of monochromators operated under different conditions were found to be in agreement with expectations within an analytical model. This type of monochromator focuses chromatically and thus necessitates the use of an exit slit for the monochromatization. This contribution will show that the lower limit for the achievable bandpass can be predicted under practically feasible experimental conditions. Refractive monochromators based on prism arrays are found to be feasible solutions for monochromatization with high transmission to a spectral bandwidth of the order of 0.5%. The bandwidth can easily be increased by adjusting the exit slit setting accordingly. Consequently, the presented refractive devices would make for tunable monochromators with tunable bandwidth, which provides more flexibility for an intermediate bandwidth of <1%, which multilayer monochromators have difficulty providing routinely.

On easily tunable wide-bandpass X-ray monochromators based on refraction in arrays of prisms.

Refractive lenses focus X-rays chromatically owing to a significant variation of the refractive index of the lens material with photon energy. Then, in combination with an exit slit in the focal plane, such lenses can be used as monochromators. The spectral resolution obtainable with refractive lenses based on prism arrays was recently systematically investigated experimentally. This contribution will show that a wide-bandpass performance can be predicted with a rather simple analytical approach. Based on the good agreement with the experimental data, one can then more rapidly and systematically optimize the lens structure for a given application. This contribution will then discuss more flexible solutions for the monochromator operation. It will be shown that a new monochromator scheme could easily provide tuning in a fixed-exit slit.

On aberrations in saw-tooth refractive X-ray lenses and on their removal.

The X-ray lens, which is composed of opposing canted saw-tooth structures, originally assembled from cut-out pieces from long-playing records, is understood by recognizing that an incident plane X-ray wave will traverse a varying number of triangular prisms in them. The refraction will deflect any beam towards the prism tips and the variation of the deflection angle, which grows linearly with the number of traversed prisms, can result in X-ray focusing. The structure offers focusing flexibility by simply changing the taper angle. This report will discuss the aberrations arising in the saw-tooth structure in its simplest form with identical prisms. It is found that the saw-tooth structures in low-Z materials with focal length below 1 m provide less flux density in the focal spot than stacks of one-dimensionally focusing refractive lenses with identical transmission function. This is due to excessive aberrations in the regular structure, which are absent in stacks of concave lenses, and which limit the focusing to spot sizes of just submicrometre dimensions, as measured experimentally for some lenses. It will be shown that this limitation can be overcome by appropriately modifying the prism shape. Then the image size could be reduced by about an order of magnitude to the diffraction limit with competitive numbers even below 0.1 µm. Microfabrication techniques are identified as the appropriate means for producing the structures.

Diffraction of partially coherent X-rays in clessidra prism arrays.

When small triangular prisms are arranged in arrays which have an overall appearance like an hourglass (in Italian: clessidra) they can focus X-rays owing to a combined action of diffraction and refraction. From the optical point of view these objects can be regarded as a Fresnel variant of concave transmission lenses. Consequently they can provide larger apertures than purely refractive lenses. However, one has to recognize that clessidra lenses will strongly diffract as the lens structure is periodic in the direction perpendicular to the incident beam. In experiments the diffraction is reduced because it is difficult to illuminate the large apertures with a full spatially coherent wavefront. So the illumination is at best partially coherent. In order to interpret available experimental data for this condition, diffraction theory has been applied appropriately to the clessidra structure, taking into account the limited spatial coherence. The agreement between the theoretical simulations and experimental data is very good, keeping the lens properties at their projected values and allowing for only two free model parameters. The first is the lateral spatial coherence; the second is a lens defect, a rounding of all edges and tips in the structure. Both values obtained from the simulations have been found to be in agreement with expectations.

On the use of clessidra prism arrays in long-focal-length X-ray focusing.

Clessidra (hour-glass) X-ray lenses have an overall shape of an old hour glass, in which two opposing larger triangular prisms are formed of smaller identical prisms or prism-like objects. In these lenses, absorbing and otherwise optically inactive material was removed with a material-removal strategy similar to that used by Fresnel in the lighthouse lens construction. It is verified that when the single prism rows are incoherently illuminated they can be operated as independent micro-lenses with coinciding image positions for efficient X-ray beam concentration. Experimental data for the line width and the refraction efficiency in one-dimensional focusing are consistent with the expectations. Imperfections in the structures produced by state-of-the-art deep X-ray lithography directed only 35% of the incident intensity away from the image and widened it by just 10% to 125 microm. An array of micro-lenses with easily feasible prism sizes is proposed as an efficient retrofit for the refocusing optics in an existing beamline, where it would provide seven-fold flux enhancement.

Diffraction theory applied to X-ray imaging with clessidra prism array lenses.

Clessidra (hourglass) lenses, i.e. two large prisms each composed of smaller identical prisms or prism-like objects, can focus X-rays. As these lenses have a periodic structure perpendicular to the incident radiation, they will diffract the beam like a diffraction grating. Refraction in the prisms is responsible for blazing, i.e. for the concentration of the diffracted intensity into only a few diffraction peaks. It is found that the diffraction of coherent radiation in clessidra lenses needs to be treated in the Fresnel, or near-field, regime. Here, diffraction theory is applied appropriately to the clessidra structure in order to show that blazing in a perfect structure with partly curved prisms can indeed concentrate the diffracted intensity into only one peak. When the lens is entirely composed of identical perfect prisms, small secondary peaks are found. Nevertheless, the loss in intensity in the central peak will not lead to any significant widening of this peak. Clessidras with perfect prisms illuminated by full coherent X-ray radiation can then provide spatial resolutions, which are consistent with the increased aperture, and which are far below the height of the single small prisms.

On the feasibility of large-aperture Fresnel lenses for the microfocusing of hard X-rays.

Like visible light, X-rays can also be focused by refraction in transmission lenses. For visible light this requires convex lenses while for X-rays one needs to use concave lenses instead. Both lens types can be lightened by the material removal strategy introduced by Fresnel, which results in a lens subdivided into zones. Until now, for the focusing of X-rays, stacks of standard lenses and of Fresnel lenses have mostly been produced. The first are dubbed compound refractive lenses, abbreviated as CRL. State-of-the-art systems of this kind now achieve almost theoretical performance for the focus size and the transmission. On the other hand, the latter Fresnel systems, which promise to provide larger apertures, are still in their infancy. This report discusses systematically the properties of two possible schemes for their realisation. It then compares the optimized apertures of these two schemes with those for CRLs. The best Fresnel lenses in this study are found to provide experimentally more than 50% of the expected refraction efficiency at 8.5 keV photon energy. The photon flux in their focus is then almost identical to that of perfect Be CRLs with the same focal length. This report will also interpret experimental data reported previously for other Fresnel lenses.

Prediction of the transmission through thin-film waveguides for X-ray microscopy.

Thin-film slab waveguides can confine incident X-ray beams in one direction in guiding layers as thin as 10 nm. Consequently they can provide attractive beam dimensions for microscopy purposes. This report presents a simple model and analytical equations for the transmission calculation, which provide results consistent with the rigorous calculations based on recursion techniques. By using these results the waveguide transmission can be compared directly with other microscopy objectives. Ideally X-ray waveguides can filter the spatially coherent content out of an incident radiation beam with an efficiency of 1. The transmissions measured for state-of-the-art one- and two-dimensional waveguides are found to correspond to experimental efficiencies of 0.5 for each confinement direction. Waveguides with thinner guiding layers cannot be used efficiently in highly collimated beams; instead the beam divergence in unfocused beamlines at state-of-the-art synchrotron radiation sources may eventually have to be increased to the larger angular acceptance of these waveguides by use of other focusing optics.