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.     

Efficiency and coherence preservation studies of Be refractive lenses for XFELO application

Performance tests of parabolic beryllium refractive lenses, considered as X‐ray focusing elements in the future X‐ray free‐electron laser oscillator (XFELO), are reported. Single and double refractive lenses were subject to X‐ray tests, which included: surface profile, transmissivity measurements, imaging capabilities and wavefront distortion with grating interferometry. Optical metrology revealed that surface profiles were close to the design specification in terms of the figure and roughness. The transmissivity of the lenses is >94% at 8 keV and >98% at 14.4 and 18 keV. These values are close to the theoretical values of ideal lenses. Images of the bending‐magnet source obtained with the lenses were close to the expected ones and did not show any significant distortion. Grating interferometry revealed that the possible wavefront distortions produced by surface and bulk lens imperfections were on the level of ～λ/60 for 8 keV photons. Thus the Be lenses can be succesfully used as focusing and beam collimating elements in the XFELO.     

High‐energy X‐ray optics with silicon saw‐tooth refractive lenses

Silicon saw‐tooth refractive lenses have been in successful use for vertical focusing and collimation of high‐energy X‐rays (50–100 keV) at the 1‐ID undulator beamline of the Advanced Photon Source. In addition to presenting an effectively parabolic thickness profile, as required for aberration‐free refractive optics, these devices allow high transmission and continuous tunability in photon energy and focal length. Furthermore, the use of a single‐crystal material (i.e. Si) minimizes small‐angle scattering background. The focusing performance of such saw‐tooth lenses, used in conjunction with the 1‐ID beamline's bent double‐Laue monochromator, is presented for both short (～1:0.02) and long (～1:0.6) focal‐length geometries, giving line‐foci in the 2 µm–25 µm width range with 81 keV X‐rays. In addition, a compound focusing scheme was tested whereby the radiation intercepted by a distant short‐focal‐length lens is increased by having it receive a collimated beam from a nearer (upstream) lens. The collimation capabilities of Si saw‐tooth lenses are also exploited to deliver enhanced throughput of a subsequently placed small‐angular‐acceptance high‐energy‐resolution post‐monochromator in the 50–80 keV range. The successful use of such lenses in all these configurations establishes an important detail, that the pre‐monochromator, despite being comprised of vertically reflecting bent Laue geometry crystals, can be brilliance‐preserving to a very high degree.     

X‐ray focusing by the system of refractive lens(es) placed inside asymmetric channel‐cut crystals

An X‐ray one‐dimensionally focusing system, a refracting–diffracting lens (RDL), composed of Bragg double‐asymmetric‐reflecting two‐crystal plane parallel plates and a double‐concave cylindrical parabolic lens placed in the gap between the plates is described. It is shown that the focal length of the RDL is equal to the focal distance of the separate lens multiplied by the square of the asymmetry factor. One can obtain RDLs with different focal lengths for certain applications. Using the point‐source function of dynamic diffraction, as well as the Green function in a vacuum with parabolic approximation, an expression for the double‐diffracted beam amplitude for an arbitrary incident wave is presented. Focusing of the plane incident wave and imaging of a point source are studied. The cases of non‐absorptive and absorptive lenses are discussed. The intensity distribution in the focusing plane and on the focusing line, and its dependence on wavelength, deviation from the Bragg angle and magnification is studied. Geometrical optical considerations are also given. RDLs can be applied to focus radiation from both laboratory and synchrotron X‐ray sources, for X‐ray imaging of objects, and for obtaining high‐intensity beams. RDLs can also be applied in X‐ray astronomy.     

X‐ray harmonics rejection on third‐generation synchrotron sources using compound refractive lenses

A new method of harmonics rejection based on X‐ray refractive optics has been proposed. Taking into account the fact that the focal distance of the refractive lens is energy‐dependent, the use of an off‐axis illumination of the lens immediately leads to spatial separation of the energy spectrum by focusing the fundamental harmonic at the focal point and suppressing the unfocused high‐energy radiation with a screen absorber or slit. The experiment was performed at the ESRF ID06 beamline in the in‐line geometry using an X‐ray transfocator with compound refractive lenses. Using this technique the presence of the third harmonic has been reduced to 10^?3. In total, our method enabled suppression of all higher‐order harmonics to five orders of magnitude using monochromator detuning. The method is well suited to third‐generation synchrotron radiation sources and is very promising for the future ultimate storage rings.     

Kinoform optics applied to X‐ray photon correlation spectroscopy

Moderate‐demagnification higher‐order silicon kinoform focusing lenses have been fabricated to facilitate small‐angle X‐ray photon correlation spectroscopy (XPCS) experiments. The geometric properties of such lenses, their focusing performance and their applicability for XPCS measurements are described. It is concluded that one‐dimensional vertical X‐ray focusing via silicon kinoform lenses significantly increases the usable coherent flux from third‐generation storage‐ring light sources for small‐angle XPCS experiments.     

A high‐accuracy complex‐phase method of simulating X‐ray propagation through a multi‐lens system

The propagation of X‐ray waves through an optical system consisting of many X‐ray refractive lenses is considered. For solving the problem for an electromagnetic wave, a finite‐difference method is applied. The error of simulation is analytically estimated and investigated. It was found that a very detailed difference grid is required for reliable and accurate calculations of the propagation of X‐ray waves through a multi‐lens system. The reasons for using a very detailed difference grid are investigated. It was shown that the wave phase becomes a function, very quickly increasing with increasing distance from the optical axis, after the wave has passed through the multi‐lens system. If the phase is a quickly increasing function of the coordinates perpendicular to the optical axis, then the electric field of the wave is a quickly oscillating function of these coordinates, and thus a very detailed difference grid becomes necessary to describe such a wavefield. To avoid this difficulty, an equation for the phase function is proposed as an alternative to the equation of the electric field. This allows reliable and accurate simulations to be carried out when using the multi‐lens system. An equation for the phase function is derived and used for accurate simulations. The numerical error of the suggested method is estimated. It is shown that the equation for the phase function allows efficient simulations to be fulfilled for the multi‐lens system.     

Optical performance of materials for X‐ray refractive optics in the energy range 8–100 keV

A quantitative analysis of the crucial characteristics of currently used and promising materials for X‐ray refractive optics is performed in the extended energy range 8–100 keV. According to the examined parameters, beryllium is the material of choice for X‐ray compound refractive lenses (CRLs) in the energy range 8–25 keV. At higher energies the use of CRLs made of diamond and the cubic phase of boron nitride (c‐BN) is beneficial. It was demonstrated that the presence of the elements of the fourth (or higher) period has a fatal effect on the functional X‐ray properties even if low‐Z elements dominate in the compound, like in YB₆₆. Macroscopic properties are discussed: much higher melting points and thermal conductivities of C and c‐BN enable them to be used at the new generation of synchrotron radiation sources and X‐ray free‐electron lasers. The role of crystal and internal structure is discussed: materials with high density are preferable for refractive applications while less dense phases are suitable for X‐ray windows. Single‐crystal or amorphous glass‐like materials based on Li, Be, B or C that are free of diffuse scattering from grain boundaries, voids and inclusions are the best candidates for applications of highly coherent X‐ray beams.     

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.     

The fractional Fourier transform as a simulation tool for lens‐based X‐ray microscopy

The fractional Fourier transform (FrFT) is introduced as a tool for numerical simulations of X‐ray wavefront propagation. By removing the strict sampling requirements encountered in typical Fourier optics, simulations using the FrFT can be carried out with much decreased detail, allowing, for example, on‐line simulation during experiments. Moreover, the additive index property of the FrFT allows the propagation through multiple optical components to be simulated in a single step, which is particularly useful for compound refractive lenses (CRLs). It is shown that it is possible to model the attenuation from the entire CRL using one or two effective apertures without loss of accuracy, greatly accelerating simulations involving CRLs. To demonstrate the applicability and accuracy of the FrFT, the imaging resolution of a CRL‐based imaging system is estimated, and the FrFT approach is shown to be significantly more precise than comparable approaches using geometrical optics. Secondly, it is shown that extensive FrFT simulations of complex systems involving coherence and/or non‐monochromatic sources can be carried out in minutes. Specifically, the chromatic aberrations as a function of source bandwidth are estimated, and it is found that the geometric optics greatly overestimates the aberration for energy bandwidths of around 1%.     

Diamond kinoform hard X‐ray refractive lenses: design,nanofabrication and testing

Motivated by the anticipated advantageous performance of diamond kinoform refractive lenses for synchrotron X‐ray radiation studies, this report focuses on progress in designing, nanofabricating and testing of their focusing performance. The method involves using lift‐off and plasma etching to reproduce a planar definition of numerically determined kinoform refractive optics. Tests of the focusing action of a diamond kinoform refractive lens at the APS 8‐ID‐I beamline demonstrate angular control of the focal spot.     

Large‐aperture prism‐array lens for high‐energy X‐ray focusing

A new prism‐array lens for high‐energy X‐ray focusing has been constructed using an array of different prisms obtained from different parabolic structures by removal of passive parts of material leading to a multiple of 2π phase variation. Under the thin‐lens approximation the phase changes caused by this lens for a plane wave are exactly the same as those caused by a parabolic lens without any additional corrections when they have the same focal length, which will provide good focusing; at the same time, the total transmission and effective aperture of this lens are both larger than those of a compound kinoform lens with the same focal length, geometrical aperture and feature size. This geometry can have a large aperture that is not limited by the feature size of the lens. Prototype nickel lenses with an aperture of 1.77 mm and focal length of 3 m were fabricated by LIGA technology, and were tested using CCD camera and knife‐edge scan method at the X‐ray Imaging and Biomedical Application Beamline BL13W1 at Shanghai Synchrotron Radiation Facility, and provided a focal width of 7.7 µm and a photon flux gain of 14 at an X‐ray energy of 50 keV.     

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 µm. 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.     

Focusing femtosecond X‐ray free‐electron laser pulses by refractive lenses

The possibility of using a parabolic refractive lens with initial X‐ray free‐electron laser (XFEL) pulses, i.e. without a monochromator, is analysed. It is assumed that the measurement time is longer than 0.3 fs, which is the time duration of a coherent pulse (spike). In this case one has to calculate the propagation of a monochromatic wave and then perform an integration of the intensity over the radiation spectrum. Here a general algorithm for calculating the propagation of time‐dependent radiation in free space and through various objects is presented. Analytical formulae are derived describing the properties of the monochromatic beam focused by a system of one and two lenses. Computer simulations show that the European XFEL pulses can be focused with maximal efficiency, i.e. as for a monochromatic wave. This occurs even for nanofocusing lenses.     

Large‐acceptance diamond planar refractive lenses manufactured by laser cutting

For the first time, single‐crystal diamond planar refractive lenses have been fabricated by laser micromachining in 300 µm‐thick diamond plates which were grown by chemical vapour deposition. Linear lenses with apertures up to 1 mm and parabola apex radii up to 500 µm were manufactured and tested at the ESRF ID06 beamline. The large acceptance of these lenses allows them to be used as beam‐conditioning elements. Owing to the unsurpassed thermal properties of single‐crystal diamond, these lenses should be suitable to withstand the extreme flux densities expected at the planned fourth‐generation X‐ray sources.     

Spectrometer for hard X‐ray free‐electron laser based on diffraction focusing

X‐ray free‐electron lasers (XFELs) generate sequences of ultra‐short spatially coherent pulses of X‐ray radiation. A diffraction focusing spectrometer (DFS), which is able to measure the whole energy spectrum of the radiation of a single XFEL pulse with an energy resolution of ΔE/E? 2 × 10^?6, is proposed. This is much better than for most modern X‐ray spectrometers. Such resolution allows one to resolve the fine spectral structure of the XFEL pulse. The effect of diffraction focusing occurs in a single‐crystal plate due to dynamical scattering, and is similar to focusing in a Pendry lens made from a metamaterial with a negative refraction index. Such a spectrometer is easier to operate than those based on bent crystals. It is shown that the DFS can be used in a wide energy range from 5 keV to 20 keV.     

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.     

A simple x‐ray monochromator based on an alligator lens

Alligator lenses, i.e. two inclined arrays of sawteeth or prisms, which face each other, can focus x‐rays with photon energies >4 keV. The inclination angle can be changed easily, and thus either the focal length for fixed photon energy or the photon energy in a fixed slit position can be varied. The material distribution in this condition is approximately parabolic along the teeth axis, hence it is of the shape required for aberration‐free focusing at a given photon energy. As the refractive index varies significantly with photon energy in the x‐ray range, these lenses suffer from chromatic aberrations, if illuminated with white x‐rays. In combination with a slit such a lens can therefore be used as an easily insertable inline monochromator. In this work, a simple universal function for the dependence of the transmission on the photon energy was derived for this application. The required tolerances for the shape of the sawteeth are found to be compatible with standard workshop machining procedures. A laboratory‐made lens of Plexiglas is shown to increase the flux density in a laboratory setup by a factor of 3, i.e. 50% of the expected result for a perfect lens. The discrepancy can be consistently ascribed to macroscopic defects of the sawteeth tips. Expectations for the performance of these lenses as monochromators at synchrotron radiation sources are presented. A single Be alligator lens is expected to provide tuning between at least 8 keV and 20 keV photon energy with a bandpass of 6%, sufficient for XRF and SAXS experiments. Consequently, such a lens pair is all that is needed for building simple synchrotron radiation beamlines for special x‐ray experiments. Copyright © 2004 John Wiley & Sons, Ltd.     

30‐Lens interferometer for high‐energy X‐rays

A novel high‐energy multi‐lens interferometer consisting of 30 arrays of planar compound refractive lenses is reported. Under coherent illumination each lens array creates a diffraction‐limited secondary source. Overlapping such coherent beams produces an interference pattern demonstrating strong longitudinal functional dependence. The proposed multi‐lens interferometer was tested experimentally at the 100 m‐long ID11 ESRF beamline in the X‐ray energy range from 30 to 65 keV. The interference pattern generated by the interferometer was recorded at fundamental and fractional Talbot distances. An effective source size (FWHM) of the order of 15 µm was determined from the first Talbot image, proving the concept that the multi‐lens interferometer can be used as a high‐resolution tool for beam diagnostics.     

Application of kinoform lens for X‐ray reflectivity analysis

In this paper the first practical application of kinoform lenses for the X‐ray reflectivity characterization of thin layered materials is demonstrated. The focused X‐ray beam generated from a kinoform lens, a line of nominal size ～50 µm × 2 µm, provides a unique possibility to measure the X‐ray reflectivities of thin layered materials in sample scanning mode. Moreover, the small footprint of the X‐ray beam, generated on the sample surface at grazing incidence angles, enables one to measure the absolute X‐ray reflectivities. This approach has been tested by analyzing a few thin multilayer structures. The advantages achieved over the conventional X‐ray reflectivity technique are discussed and demonstrated by measurements.