X-ray lens figure errors retrieved by deep learning from several beam intensity images.

The phase problem in the context of focusing synchrotron beams with X-ray lenses is addressed. The feasibility of retrieving the surface error of a lens system by using only the intensity of the propagated beam at several distances is demonstrated. A neural network, trained with a few thousand simulations using random errors, can predict accurately the lens error profile that accounts for all aberrations. It demonstrates the feasibility of routinely measuring the aberrations induced by an X-ray lens, or another optical system, using only a few intensity images.

An automated approach to the alignment of compound refractive lenses.

Compound refractive lenses (CRLs) are established X-ray focusing optics, and are used to focus the beam or image the sample in many beamlines at X-ray facilities. While CRLs are quite established, the stack of single lens elements affords a very small numerical aperture because of the thick lens profile, making them far more difficult to align than classical optical lenses that obey the thin-lens approximation. This means that the alignment must be very precise and is highly sensitive to changes to the incident beam, often requiring regular readjustments. Some groups circumvent the full realignment procedure by using engineering controls (e.g. mounting optics) that sacrifice some of the beam's focusing precision, i.e. spot size, or resolution. While these choices minimize setup time, there are clear disadvantages. This work presents a new automated approach to align CRLs using a simple alignment apparatus that is easy to adapt and install at different types of X-ray experiments or facilities. This approach builds on recent CRL modeling efforts, using an approach based on the Stochastic Nelder-Mead (SNM) simplex method. This method is outlined and its efficacy is demonstrated with numerical simulation that is tested in real experiments conducted at the Advanced Photon Source to confirm its performance with a synchrotron beam. This work provides an opportunity to automate key instrumentation at X-ray facilities.

On the theory of synchrotron radiation nanofocusing with planar compound refractive lenses.

Two new methods of computer simulation of synchrotron radiation nanofocusing with planar compound refractive lenses (PCRLs) are presented. The methods are based on the results of analytical theory. In contrast to previous works, the new methods take into account the PCRL aperture. It is especially important at high photon energies, when absorption is low and the calculations based on analytical theory, i.e. without taking into account the aperture, give incorrect results. A computer program was created and specific results were obtained for a silicon PCRL having an aperture of 50 µm, element length of 102 µm and minimum thickness of 2 µm. For an energy of 50 keV and number of elements 300, it focuses the beam to 31 nm size at a distance of one and a half times its length. Analysis of the calculation accuracy for the proposed methods is performed, as well as a demonstration of the capabilities of the computer program.

Feasibility of X-ray beam nanofocusing with compound refractive lenses.

A more general analytical theory of X-ray beam propagation through compound refractive lenses (CRLs) than the earlier study by Kohn [(2003). JETP, 97, 204-215] is presented. The problem of nanofocusing with CRLs is examined in detail. For a CRL with a relatively large aperture the focusing efficiency is limited by the radiation absorption in the lens material. The aperture does not affect the focusing process and it is replaced by the effective aperture. The X-ray transverse beam size at the focus is then by a factor of γ = ß/δ times smaller than the transverse beam size just behind the CRL. Here, δ and ß are the real and imaginary parts of the CRL material refractive index n = 1 - δ + iß. In this instance, to improve focusing efficiency, it is advantageous to decrease the CRL aperture and increase the photon energy E. However, with increasing photon energy, the material absorption decreases, which results in the CRL aperture impact on the transverse beam size. The latter leads to the fact that with a proper CRL length the beam size is independent of both the aperture and photon energy but depends only on the CRL material electron density and is approximately equal to wc = λ/(8δ)1/2, where λ denotes the radiation wavelength, as predicted by Bergemann et al. [(2003). Phys. Rev. Lett, 91, 204801].

Diamond nanofocusing refractive X-ray lenses made by planar etching technology.

The manufacturing steps and first tests of a refractive lens made of polycrystalline diamond are described. A fabrication process based on electron-beam lithography and deep reactive ion etching is introduced. Experimental tests on beamline ID13 at the ESRF have been performed. A spot size of 360 nm (FWHM) at an energy E = 24.3 keV is observed.

Rocking curve and spatial coherence properties of a long X-ray compound refractive lens.

Semi-analytical theory of a long set of X-ray compound refractive lenses (CRLs) based on recurrence relations is developed further. The geometrical aperture, angular divergence of incident radiation and source size were accurately taken into account. Using this theory it is possible to calculate the width of the rocking curve of a long (40.7 cm) Be CRL which coincides with experimental data obtained earlier. By this approach the transverse coherence length for the X-ray beam after passing a set of CRLs of arbitrary complexity has been estimated. It is shown that at the focus this coherence length is equal to a diffraction-limited beam size (beam size in the case of a point source) and has minimal difference with the real beam size.

Computer simulations of X-ray spherical wave dynamical diffraction in one and two crystals in the Laue case.

This article reports computer simulations of X-ray spherical wave dynamical diffraction in one and two single crystals in the Laue case. An X-ray compound refractive lens (CRL) as a secondary radiation source of spherical waves was considered for the first time and in contrast to previous simulations with the assumption of the use of a slit. The main properties of the CRL as a secondary source are discussed and two focusing phenomena are analysed. The first one is the diffraction focusing effect for one single crystal in the reflected beam and in the case of a large source-to-detector distance. The second one is the same but for two single crystals and for the twice-reflected beam in the case of a short distance between the source and detector. The first effect is well pronounced in the case of strong absorption. However, it may also be used as an element of an energy spectrometer in the medium and even weak absorption case. The second effect will appear in the case of weak absorption. It is shown that it is not effective to use it in an energy spectrometer. In the case of weak absorption the transverse size of the diffraction focused beam will oscillate together with the reflected beam integral intensity. The oscillation period is close to the extinction length.

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%.

Refraction and ultra-small-angle scattering of X-rays in a single-crystal diamond compound refractive lens.

In this work a double-crystal setup is employed to study compound refractive lenses made of single-crystal diamond. The point spread function of the lens is calculated taking into account the lens transmission, the wavefront aberrations, and the ultra-small-angle broadening of the X-ray beam. It is shown that, similarly to the wavefront aberrations, the ultra-small-angle scattering effects can significantly reduce the intensity gain and increase the focal spot size. The suggested approach can be particularly useful for the characterization of refractive X-ray lenses composed of many tens of unit lenses.

Effective aperture of X-ray compound refractive lenses.

A new definition of the effective aperture of the X-ray compound refractive lens (CRL) is proposed. Both linear (one-dimensional) and circular (two-dimensional) CRLs are considered. It is shown that for a strongly absorbing CRL the real aperture does not influence the focusing properties and the effective aperture is determined by absorption. However, there are three ways to determine the effective aperture in terms of transparent CRLs. In the papers by Kohn [(2002). JETP Lett. 76, 600-603; (2003). J. Exp. Theor. Phys. 97, 204-215; (2009). J. Surface Investig. 3, 358-364; (2012). J. Synchrotron Rad. 19, 84-92; Kohn et al. (2003). Opt. Commun. 216, 247-260; (2003). J. Phys. IV Fr, 104, 217-220], the FWHM of the X-ray beam intensity just behind the CRL was used. In the papers by Lengeler et al. [(1999). J. Synchrotron Rad. 6, 1153-1167; (1998). J. Appl. Phys. 84, 5855-5861], the maximum intensity value at the focus was used. Numerically, these two definitions differ by 50%. The new definition is based on the integral intensity of the beam behind the CRL over the real aperture. The integral intensity is the most physical value and is independent of distance. The new definition gives a value that is greater than that of the Kohn definition by 6% and less than that of the Lengeler definition by 41%. A new approximation for the aperture function of a two-dimensional CRL is proposed which allows one to calculate the two-dimensional CRL through the one-dimensional CRL and to obtain an analytical solution for a complex system of many CRLs.

Simulating and optimizing compound refractive lens-based X-ray microscopes.

A comprehensive optical description of compound refractive lenses (CRLs) in condensing and full-field X-ray microscopy applications is presented. The formalism extends ray-transfer matrix analysis by accounting for X-ray attenuation by the lens material. Closed analytical expressions for critical imaging parameters such as numerical aperture, spatial acceptance (vignetting), chromatic aberration and focal length are provided for both thin- and thick-lens imaging geometries. These expressions show that the numerical aperture will be maximized and chromatic aberration will be minimized at the thick-lens limit. This limit may be satisfied by a range of CRL geometries, suggesting alternative approaches to improving the resolution and efficiency of CRLs and X-ray microscopes.

Angular vibrations of cryogenically cooled double-crystal monochromators.

The effect of angular vibrations of the crystals in cryogenically cooled monochromators on the beam performance has been studied theoretically and experimentally. A simple relation between amplitude of the vibrations and size of the focused beam is developed. It is shown that the double-crystal monochromator vibrations affect not only the image size but also the image position along the optical axis. Several methods to measure vibrations with the X-ray beam are explained and analyzed. The methods have been applied to systematically study angular crystal vibrations at monochromators installed at the PETRA III light source. Characteristic values of the amplitudes of angular vibrations for different monochromators are presented.

Single-crystal diamond refractive lens for focusing X-rays in two dimensions. Erratum.

A correction is made to a citation in the article by Antipov et al. (2016) [J. Synchrotron Rad. 23, 163-168].

Compound refractive lenses as prefocusing optics for X-ray FEL radiation.

The performance of X-ray free-electron laser beamlines may be limited by the angular aperture. Compound refractive lenses (CRLs) can be employed to prefocus the X-ray beam, thereby increasing the beamline transmission. A prefocusing CRL was implemented in the X-ray transport of the Matter under Extreme Conditions Instrument at the Linac Coherent Light Source. A significant improvement in the beamline transmission was calculated over the 3-10 keV photon energy range. At 5 keV, the relative X-ray intensity was measured and a factor of four increase was seen in the beamline transmission. The X-ray focus was also determined by the ablation imprint method.

Single-crystal diamond refractive lens for focusing X-rays in two dimensions.

The fabrication and performance evaluation of single-crystal diamond refractive X-ray lenses of which the surfaces are paraboloids of revolution for focusing X-rays in two dimensions simultaneously are reported. The lenses were manufactured using a femtosecond laser micromachining process and tested using X-ray synchrotron radiation. Such lenses were stacked together to form a standard compound refractive lens (CRL). Owing to the superior physical properties of the material, diamond CRLs could become indispensable wavefront-preserving primary focusing optics for X-ray free-electron lasers and the next-generation synchrotron storage rings. They can be used for highly efficient refocusing of the extremely bright X-ray sources for secondary optical schemes with limited aperture such as nanofocusing Fresnel zone plates and multilayer Laue lenses.

Ronchi test for characterization of X-ray nanofocusing optics and beamlines.

A Ronchi interferometer for hard X-rays is reported in order to characterize the performance of the nanofocusing optics as well as the beamline stability. Characteristic interference fringes yield qualitative data on present aberrations in the optics. Moreover, the visibility of the fringes on the detector gives information on the degree of spatial coherence in the beamline. This enables the possibility to detect sources of instabilities in the beamline like vibrations of components or temperature drift. Examples are shown for two different nanofocusing hard X-ray optics: a compound refractive lens and a zone plate.

A planar parabolic refractive nickel lens for high-energy X-rays.

A compound refractive lens made of nickel and designed for focusing high-energy synchrotron X-rays is presented. The lens consists of 600 parabolic grooves and focuses X-rays in one plane only (planar lens). The lenses made and investigated by us earlier exhibited low transmission and irregularities in the focused beam profile. Since then, improvements in lens manufacturing technology have been made. The present lens gives an almost Gaussian profile and produces four times higher intensity at its maximum compared with the intensity of primary X-ray beams of 174 keV.