Open Access
Add to BibSonomy
Abstract
Advances in x-radiation sources put greater demands on x-ray optics. Fabrication errors in optical elements lead to deformation of the radiation wavefront, which prevents diffraction-limited imaging of the source. A new adaptable x-ray phase compensator using refracting elements has been developed, fabricated, and tested. The compensator makes a sinusoidal correction to the x-ray wavefront with variable amplitude, period, and phase. The adaptable compensator was used to correct two planar compound refractive lenses and a Kirkpatrick–Baez mirror system on an x-ray beamline. Wavefront measurements showed a reduction in the rms wavefront error by a factor of seven for the lenses and three for the mirror system, reducing rms wavefront errors down to of order $ \lambda /100 $. This concept could be used with optics on existing x-ray beamlines to enable diffraction-limited focusing.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
Modern x-ray synchrotron radiation (SR) sources and x-ray free-electron laser (XFEL) sources are achieving new levels of x-ray coherent flux. X-ray optical elements, such as totally externally reflecting mirrors at grazing incidence angle or refractive lenses, are commonly used to condition the x-ray beam from the source for experiments, for example, to focus the x-rays into a small focal spot at a sample. Obtaining a focus at the diffraction limit is possible only with highly perfect optics; however, small residual fabrication errors can prevent the ultimate diffraction-limited focused beam from being obtained. A solution to this problem has been recently proposed [1,2] in which a refractive phase plate is used to compensate for the residual fabrication errors to generate a nearly perfect optical system. Exciting results have been obtained from such corrector-based x-ray optics. However, fabrication of these corrective optical elements requires precise knowledge of the imperfections in the x-ray optical element to be corrected—obtained by accurate metrology prior to fabrication of the corrective optics. Such corrector optics work only for the optical element for which they have been designed and hence have limited applicability.
Here, we describe a new form of adaptable corrective optics that allows dynamic wavefront corrections. The adaptable corrector consists of a pair of refractors with a calculated thickness profile such that changing their positioning allows the wavefront correction to be varied in a precise way. The concept is similar to variable focal length lenses that vary the lateral displacement between two cubic profile refractors to generate a variable parabolic component to the wavefront [3,4]. A benefit of adaptable correctors is that a priori knowledge of the optics imperfections is not required, and the same set of correctors can be used for a range of x-ray optical elements. In this work we demonstrate their applicability to x-ray compound refractive lenses (CRLs) and x-ray mirrors and show a reduction in wavefront errors by a factor of seven for the lenses and three for the mirrors. We hope that this new development will have an impact on the operation and usage of x-ray beamlines by allowing perfect focusing to be achieved at all times. Furthermore, the performance of the beamlines can be significantly enhanced “on the fly” to, for example, compensate for changes in thermal power load, drift in optical alignment, or wavefront distortion from upstream optics.
2. OPTICS FOR FOCUSING X-RAYS
Nano-focusing x-ray optics should have a high efficiency to collect the maximum coherent flux and have a high numerical aperture (NA) to give a small diffraction-limited focus size. The NA is defined for a focusing element with axial symmetry [5] as $ {\rm NA} = n \sin \theta $, where $ n $ is the refractive index of the surrounding medium, $ \theta $ is the half-angular aperture of the focused radiation, and the limit for the size of the focus spot is $ s \sim \lambda /(2\,{\rm NA}) $. This expression is also known as the Abbe limit [5] and was originally derived as the resolution limit for imaging under coherent illumination.
It is common for x-ray focusing optics to be split into separate in-line elements for focusing the beam in two orthogonal directions, for example, the vertical ($ y $) and horizontal ($ x $) directions for an incident beam propagating along $ z $. Examples are Kirkpatrick–Baez (KB) mirror pairs [6], which use a vertically deflecting vertically focusing mirror (VKB) and a horizontally deflecting horizontally focusing mirror (HKB) and crossed planar CRLs, which use a vertically focusing lens stack and a horizontally focusing lens stack. The optical system can then be characterized by a different NA for each of the two orthogonal focusing directions. For both mirror or lens systems, the achievable NA depends on the x-ray refractive index, which is conventionally written for x-rays of energy $ E $ as
For mirrors operating in total external reflection, the real part deviation $ \delta (E) $ for the reflecting surface determines the maximum grazing angle of incidence at which total external reflection occurs through the Fresnel equations [5]. As $ \delta (E) $ for rhodium (a common x-ray mirror coating) is of order $ {10^{ - 5}} $ in the x-ray region, a typical grazing angle at the mirror surface is $ \theta = 3 \,\, {\rm mrad} $, and the NA is constrained to be $ {\rm NA} \lesssim 1.5 \times {10^{ - 3}} $. This results in the smallest achievable diffraction-limited focus spot size of $ s \sim 300\,\, \lambda $. For CRLs, x-ray absorption in the lens depends on the imaginary part of the refractive index decrement $ \beta (E) $, and the focusing strength depends on $ \delta (E) $. For a given focal length, x-ray absorption limits the aperture, and the smallest achievable diffraction-limited focus size is $ s \sim 1000\,\, \lambda $. The challenge for the design and manufacture of x-ray optics is to achieve a high NA in order to reduce the diffraction-limited focus size. This has led to the development of specialized x-ray optics such as graded multilayer mirrors [7], tapered CRLs [8], and multilayer Laue lenses [9].3. WAVEFRONT CORRECTING OPTICS
Obtaining a focus with size limited only by diffraction is possible only with highly perfect optics. Imperfections in the optics change the path length and perturb the x-ray wave field. For mirrors, the imperfection is the error in the surface figure and for lenses, a modification of lens thickness profile or a material density variation.
X-ray optical elements may be analyzed by their effect on the radiation wavefront. The wavefront of the radiation field is a surface composed of the locust of points of constant phase [10]. With perfect optics, the wavefront after the final optical element propagating to the focus is a converging spherical surface centered on the focus position where each part of the wavefront arrives in phase and interferes constructively. The wavefront error can be defined as the distance difference, along the propagation direction, between the actual wavefront and this ideal surface. In Cartesian $ (x,y,z) $ coordinates, for an optical axis along the $ z $ direction, the wavefront error can be expressed as a function $ w(x,y) $. A constant offset and a linear variation of the wavefront error correspond to a constant phase shift and a tilt of the radiation field, respectively, which do not affect the focus profile and can be ignored. A parabolic term in the wavefront error is equivalent to a change in the radius of the converging spherical wave, which corresponds therefore to a change in the longitudinal position of the focus and may therefore also be ignored. X-ray mirrors operating at grazing angle of incidence introduce a wavefront distortion if the incident angle deviates from the value used to specify the elliptical mirror surface. This allows the next term, the cubic term, to also be removed from the wavefront error by adjusting the mirror pitch angle. Planar x-ray lenses, due to fabrication methods, have highly symmetrical wavefront errors, so that in this case, the cubic term is small. The next term in the wavefront error is the quartic term, and the objective of x-ray wavefront correction is to reduce the quartic and higher terms in the wavefront error.
The wavefront error is characterized by the root mean squared (rms) value, which for a wavefront error $ w(x,y) $ with zero mean defined on an aperture $ {x_1} \le x \le {x_2} $, $ {y_1} \le y \le {y_2} $ is given by
For mirrors, the grazing angle of incidence results in an illuminated area on the mirror surface that is highly extended along the mirror length, and therefore the figure error along the mirror length is more significant than the figure error in the transverse direction. For a planar lens, when the lens profile is uniform in the non-focusing direction, only distortion of the wavefront in the focusing direction will be present. The wavefront error $ w(x,y) $ can therefore be decomposed into separate one-dimensional contributions from the vertically and horizontally focusing optics, $ {w_v}(y) $ and $ {w_h}(x) $, respectively:
Correction of the wavefront is most conveniently done by corrective optics located a short distance upstream of the focusing optics where the incident wavefront is more nearly plane and there is likely to be available space for insertion of additional optics. The aim is to modify the wavefront before the focusing optics, such that the wavefront error introduced by the optical element is cancelled out. Propagation of the wavefront across a one-dimensional focusing element retains the form of Eq. (4) when the corrective optics can be separated into two separate elements that each apply a correction in one of the orthogonal directions only.Various methods have been used for wavefront correction of x-ray optics using refractive optics or reflective optics to apply phase changes to the wavefront. The methods fall into two general categories. The first uses custom-designed and fabricated optics, specific to the optical element being corrected to provide a static correction. The second category uses a deformable optical element that can be dynamically adapted to provide a variable correction. Static correction methods require an initial measurement of the wavefront using at-wavelength methods. The measured wavefront is propagated numerically through the optical system to the intended location of the corrective optics and used to calculate the parameters of the corrector. Once fabricated, the corrective optics are installed, and at-wavelength wavefront measurements are used to align the corrective optics and confirm a reduction in the wavefront error. Examples of this method include the use of micro-fabricated refractive structures for correcting mirror optics and lens optics [1,11] and for correcting two-dimensional beryllium CRLs [2]. For dynamic correction methods, at-wavelength wavefront measurements are used to optimize the wavefront perturbation in order to minimize the measured wavefront error downstream of the focusing optics. Examples of this method are the use of a piezoelectric bimorph deformable mirror to correct the wavefront from multilayer mirrors [7,12,13].
4. ADAPTABLE REFRACTIVE WAVEFRONT COMPENSATORS
We have developed a new form of refractive optics consisting of two planar structures that allow dynamic wavefront corrections. For correction of the wavefront in the vertical direction $ y $, the structures have a thickness profile given by $ {t_1}(y) $ and $ {t_2}(y) $, where
For $ {y_0} $ at the center of the beam aperture and for longer-period-fabricated structures, the period $ \Lambda $ is significantly larger than $ |y - {y_0}| $, and the contribution from the power-six and higher terms is small compared to the power-four term. The correction then reduces to a fourth -order polynomial. After removing the terms up to cubic (see Section 3), the correction is a power function $ a{y^4} $, where the amplitude $ a $ is the single variable parameter. The corrector is then the analogue for quartic wavefront corrections of Lohmann–Alvarez refractive lenses, which produces a variable parabolic correction [3,4].
Wavefront errors originate in the fabrication of the optical elements. In the case of curved x-ray mirrors, to reduce figure errors to the nanometer scale, manufacturers often use surface metrology to control deterministic polishing by, for example, ion beam figuring (IBF) or elastic emission machining (EEM), while for planar lenses, thickness errors could, for example, be caused by relaxation of the polymer during fabrication. In either case, it is not possible to be certain of the error source, and it is difficult to arrive at a general form for the resulting wavefront error.
The motivation for using a sequence of sinusoidal correctors is that the wavefront error defined over the finite range of the aperture of the optical element can be expressed as a Fourier series—a summation of sine and cosine functions with harmonic periods. It is common for x-ray mirror surfaces to be characterized by the power spectral distribution (PSD) of the surface (see, e.g., [14]). The PSD when plotted on a log–log scale typically exhibits a linear relation with negative slope indicating a negative power law dependence of the Fourier components, so that the amplitude of the Fourier components reduces as the period reduces. Correction would therefore be done with a finite number of in-line correctors corresponding to the lowest-order terms of the Fourier decomposition of the wavefront error. For correction by just a single corrector, we have, however, taken a pragmatic approach, where the corrector period as well as amplitude and phase are optimized by an algorithm that minimizes the rms value of the corrected wavefront using least square fitting to the measured wavefront. Allowing the corrector period to be optimized invariably gives a significant reduction in the rms wavefront error.
The corrector structures were fabricated at the x-ray lithography beamline at Indus-2 [15] using the Lithographie, Galvanoformung and Abformung (LIGA) process [16]. The refracting material is the polymer SU-8, which is composed of light elements (carbon, oxygen, hydrogen) and therefore has a favorable ratio of refraction to absorption. SU-8 is also relatively resistant to damage from exposure to x-rays. The LIGA process creates high-aspect-ratio planar structures laid down on a flat substrate, usually a single crystal silicon wafer. The structures have parallel, highly smooth side walls perpendicular to the substrate with structure depths of over 1 mm and areas of over $ 100 \,\, {\rm mm} \times 100\,\, {\rm mm} $ being possible. The structures have sub-micrometer precision, which, given the weak refraction effect for x-rays, is more than sufficient for applying picometer-level wavefront corrections. The surface roughness is below 100 nm [17].
Fig. 1. Knife-edge scanning method with a pixel detector to determine the wavefront error in vertical direction. Intensity is measured by detector pixel at position $ {P_2} $. $ {P_2} $ is projected back along the wavefront normal to point $ {P_1} $, intersecting the focal plane at a point offset by distance $y$ from the optical axis.
The structures were laid down on a 100 mm silicon wafer substrate and consisted of a sequence of 46 structures of the form given by Eq. (5) (the “A” sequence) and Eq. (6) (the “B” sequence). The structure amplitude A was 40 µm, and the period increased from $ \Lambda = 40 \,\, \unicode{x00B5}{\rm m} $ to $ \Lambda = 360 \,\, \unicode{x00B5}{\rm m} $, increasing by 5% between adjacent structures in the sequence. Each structure was 750 µm long, sufficient to cover the aperture of the optical elements, and was separated from adjacent structures by a rectangular block. This block strengthened the structure sequence and gave a reference position for alignment. After fabrication, the “A” sequence and “B” sequence of structures were separated by cleaving the silicon substrate, and each was then mounted on separate translation and tilt stages, allowing fine angular and positional adjustment relative to the x-ray beam. For these measurements, a single pair of structure sequences was fabricated to make one adaptable compensator for correction of a single sinusoidal spatial period from the wavefront error in one direction only (i.e., vertical or horizontal). The correction direction was selected by mounting the compensator in either the vertical or horizontal orientation. Future measurements could use many in-line compensators, which would enable simultaneous correction in both planes and compensation of a number of different spatial periods.
5. WAVEFRONT MEASUREMENT
In order to correct the wavefront, it is essential to be able to measure the wavefront error. The method conventionally used to measure the intensity distribution at the focal plane is to step an absorbing knife edge into the beam while measuring the transmitted intensity on a spatially integrating x-ray detector positioned downstream. Each step of the knife edge cuts a section of the x-ray beam. The intensity profile is given by the intensity change versus knife-edge position. We have developed a method for measuring the wavefront error in which the spatially integrating detector is replaced by a pixel array detector.
At the pixel detector, at a sufficiently large distance from the focus, the wavefront is locally plane with rectilinear propagation along the wavefront normal direction. This allows the wavefront at each pixel of the detector to be projected back along the propagation direction to the focal plane and then back further to a plane perpendicular to the optical axis at the optical element. This is illustrated for an optical element focusing in the vertical ($ y $) direction in Fig. 1. The pixel detector is positioned at a distance $ {D_2} $ downstream from the focal plane, and the focal plane is at a distance $ {D_1} $ downstream from the center of the optical element. Position $ {P_2} $ on the pixel detector is projected backwards to a point displaced by vertical distance $ y $ from the optical axis at the focal plane and then to point $ {P_1} $ on the plane perpendicular to the optical axis at the center of the focusing optical element. As $ y $ is small (being within the focal spot), the wavefront position at the detector ($ {y_2} $) and the projected position at the optical element ($ {y_1} $) are related by $ - {y_1}/{D_1} = {y_2}/{D_2} $. For each pixel in the detector, the position $ y $ of the wavefront projected to the focal plane is determined by stepping the knife edge through the beam at the focal plane and determining the position at which intensity is obscured from the pixel. This position is obtained by fitting an appropriate “step” function (such as based on a cumulative normal distribution) to the pixel intensity versus knife-edge position and then locating the edge as the position where the intensity is reduced to 50% of maximum. This procedure is repeated for every pixel within the beam aperture of the optical element. The distance $ y $ gives the angular error of the wavefront. The angular displacement of the wavefront at position $ {y_1} $ on the propagation plane at the optical element is given by $ {d}w({y_1})/{d}{y_1} = y/{D_1} $. The wavefront error at the optical element $ w({y_1}) $ is then obtained by integration.
Fig. 2. Schematic of the experimental layout showing a section of the two structure sequences with the KB mirror pair focusing the beam. Inset shows the vertical motion $ {y_0} $ that changes the phase and $ \epsilon $ that changes the amplitude of the sinusoidal wavefront correction.
For a wavefront with zero error, the knife edge in the focal plane would simultaneously obscure the x-ray beam from every detector pixel at the same position $ y = 0 $, and therefore the wavefront error obtained by integration would, as expected, be constant. The technique is highly sensitive, as the measurement is relative to the ideal wavefront for perfect focusing and there is no need to subtract a term to account for uniform curvature of the wavefront. If the knife edge is displaced in $ z $ from the focal plane, then a parabolic component is introduced to the measured wavefront error, which increases linearly with the displacement. In addition, for a focusing elliptical mirror, a cubic component in the wavefront error increases linearly with deviation of the mirror pitch angle from its optimum value. This allows accurate alignment of the optical element. Of particular relevance to implementing wavefront corrective optics, the knife-edge longitudinal position can be adjusted to remove the parabolic contribution, and for a focusing mirror, the pitch angle can be adjusted to remove the cubic component from the wavefront error.
6. EXPERIMENTAL
Measurements were performed on the Test Beamline B16 at Diamond Light Source [18], a bending magnet beamline that provided an unfocused monochromatic x-ray beam for these experiments. The adaptable wavefront compensator was fabricated at the x-ray lithography beamline at Indus-2. The compensator was used to correct a 3 mrad incidence angle KB mirror system belonging to the Test Beamline, and two SU-8 planar CRLs [19] containing 19 lenses (lens1) and 44 lenses (lens2). The experimental layout is shown in Fig. 2.
Each optic was mounted on a goniometer in the experimental hutch with the single adaptable compensator mounted a short distance upstream. For the KB mirrors, the compensator was first mounted in the vertical orientation to correct the VKB mirror and then remounted in the horizontal orientation to correct the HKB mirror. Measurements of the two LIGA lenses were made with each of the lenses focusing in the vertical direction for wavefront correction in the vertical direction.
The knife-edge technique using a vertically scanned, horizontally oriented knife edge measured the wavefront error for the vertically focusing optics and using a horizontally scanned, vertically oriented knife edge, measured the horizontally focusing optics. The knife edge was a precision-fabricated gold knife edge that was mounted on a piezo translation stage at the focal plane. The pixel detector was a direct-coupled fiber-optic CCD detector with 6.45 µm pixel resolution and $ 1360 \times 1040 $ pixels, which was mounted 545 mm downstream from the knife edge for the mirrors and 940 mm for the lenses. The wavefront rms error was calculated with Eq. (2), and this was used as a figure of merit for optimizing the wavefront correction.
Fig. 3. Wavefront error: the difference along the propagation direction between actual wavefront and the ideal spherical surface as measured by the knife-edge method as a function of transverse position. Blue: no wavefront correction; red: wavefront correction by the single adaptable refractive compensator.
7. RESULTS
Initial alignment of the focusing optics was performed using the wavefront measurements to locate the focal plane and in the case of the mirrors, to optimize the pitch angle. The measured wavefront error was then used to determine the structure period ($ \Lambda $) that would give the best wavefront compensation. Using this value, the sequence of structures was then translated to select the period from both the A and B structure sequences. The two structure sequences were mounted on separate piezo translation and rotation stages, allowing accurate alignment to the focusing optic. Using combined translation of both structures along their length to change $ {y_0} $ and differential motion to change $ \epsilon $ in Eq. (8), the wavefront error was optimized iteratively until a measured wavefront error with the smallest measured rms wavefront error was obtained.
The measured wavefronts before (blue line) and after (red line) correction of all the optical elements studied are shown in Fig. 3. The results are summarized in Table 1. As can be seen, there is a significant reduction in the wavefront error in all cases. Furthermore, the two measurements on the HKB mirror at an x-ray energy of 15 keV and 18 keV show a similar wavefront error before correction. To achieve the optimum correction, a different compensator amplitude ($ \epsilon $) was, however, required in order to account for the energy dependence of the refraction. This then produced a similar wavefront error at the two energies after correction. The measured wavefront errors (before and after correction) were also used to generate a fully coherent field amplitude at a plane centered on the focusing optic. This amplitude was propagated numerically through the focal plane at $ z = 0 $ using the Fresnel–Kirchhoff equation and is shown in Fig. 4. The reduction in wavefront error makes the beam caustic symmetric about the focal plane and leads to an increased intensity in the diffraction-limited focus. For these measurements, the focused beam profile size is dominated by the relatively large size of the Diamond dipole source ($ \sim 100 \,\, \unicode{x00B5}{\rm m} \times 40 \,\, \unicode{x00B5}{\rm m} $ FWHM).
Table 1. Measurement Summary of Wavefront Correction Using the Adaptable Refractive Compensator
Fig. 4. Calculated beam intensity of a fully coherent 15 keV x-ray beam propagating through the focal plane of the HKB mirror. The calculation uses the measured wavefront error, with residual second- and third-order terms removed by polynomial fitting, to modify the coherent field amplitude of the focused beam. The field amplitude is then propagated numerically, using the Fresnel–Kirchhoff equation, to positions $ z $ before and after the focal plane at $ z = 0 $. Intensity is plotted as a function of $ z $ and transverse distance in the focusing direction ($ y $). Left plot shows the intensity distribution with no wavefront correction; right plot shows intensity distribution with the wavefront compensator inserted.
8. DISCUSSION AND CONCLUSION
These novel optics combine the advantages of custom refractive structure compensators with the flexibility of adaptable optics. Using the single refractive compensator, the correction was most effective for the two planar CRLs, as they have a wavefront error that is closest to a sinusoidal profile. A significant reduction in the wavefront error (a factor of about three) was also obtained for the two mirrors. The HKB mirror gained most from the wavefront correction as this optic has a large figure error and also the highest NA of the optics tested. For these initial measurements, the single adaptable compensator allowed only a single spatial period of the wavefront to be corrected; however, multiple in-line adaptable compensators would allow additional spatial periods to be corrected to further reduce the wavefront error. Similarly, provision of orthogonally mounted compensators would allow simultaneous correction of the wavefront in both focusing directions. For an existing beamline, the wavefront compensators could be installed upstream of the focusing optics, and as the optical axis is not changed, the beamline can be operated without wavefront correction simply by translating the compensators out of the beam path. The alignment procedure was reliable and reproducible using the rms wavefront as a measure of wavefront compensation, and this method is well suited to an automated alignment procedure.
The LIGA technique is well suited to fabricating the corrective optics, as the fabricated structures have highly perpendicular smooth refracting surfaces. The structures are made with high spatial resolution and cover the large substrate, so the cost per individual structure is low. SU-8 has proved to be resistant to damage from the x-radiation on the dipole beamline. For sources with higher x-ray intensity such as XFELs, it would be possible to use electro deposition with LIGA or to use other fabrication techniques to produce structures made from other materials. The weakness of refraction for x-rays by SU-8 is an advantage, as it relaxes the tolerance with which the structures must be fabricated, while due to the small magnitude of the correction, the structures are sufficiently thin so that x-ray transmission through the prototype structures is about 98% at 15 keV. This allows the possibility of using a number of compensators in-line with the x-ray beam without a significant loss of intensity.
In conclusion, we have demonstrated that the same adaptable compensator can reduce the wavefront error from two different planar x-ray lenses and both elliptical mirrors in a KB pair and can also be reoptimized for a mirror at different x-ray energies by simply adjusting the compensator amplitude. A factor reduction in the rms wavefront error of between three and seven was obtained using a single adaptable compensator, and there is the prospect of greater improvements by adding additional compensators. A standard design could be developed for rapid deployment on various beamlines with diverse x-ray optics. The adaptable compensators can also be reoptimized to correct wavefront errors caused by drift in the optics and also allows upstream wavefront errors (e.g., from x-ray windows) to be corrected at the same time. Simulations show that wavefront correction can be achieved using the adaptable correctors for an x-ray source emitting fully coherent radiation over the aperture of the optics to allow x-ray focusing to approach the diffraction limit.
Funding
Diamond Light Source; European Union's Horizon 2020 Framework Programme, Marie Sklodowska-Curie actions (665593).
Acknowledgment
This work was carried out with the support of Diamond Light Source. The authors thank Oliver Fox and Andrew Malandain for the help and support during the measurements on the Test Beamline and Simon Alcock for proofreading. P. Mondal, P. Tiwari, and N. Khantwal are acknowledged for their technical assistance during the x-ray lithography fabrication at the Indus-2 synchrotron facility, and we are grateful to Arndt Last from IMT/KIT for providing the LIGA fabricated gold knife edge used in the at-wavelength measurements.
REFERENCES
1. K. Sawhney, D. Laundy, V. Dhamgaye, and I. Pape, “Compensation of x-ray mirror shape-errors using refractive optics,” Appl. Phys. Lett. 109, 051904 (2016). [CrossRef]
2. F. Seiboth, A. Schropp, M. Scholz, F. Wittwer, C. Rödel, M. Wünsche, T. Ullsperger, S. Nolte, J. Rahomäki, K. Parfeniukas, S. Giakoumidis, U. Vogt, U. Wagner, C. Rau, U. Boesenberg, J. Garrevoet, G. Falkenberg, E. C. Galtier, H. Ja Lee, B. Nagler, and C. G. Schroer, “Perfect x-ray focusing via fitting corrective glasses to aberrated optics,” Nat. Commun. 8, 14623 (2017). [CrossRef]
3. A. W. Lohmann, “A new class of varifocal lenses,” Appl. Opt. 9, 1669–1671 (1970). [CrossRef]
4. S. Barbero, “The Alvarez and Lohmann refractive lenses revisited,” Opt. Express 17, 9376 (2009). [CrossRef]
5. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999), vol. 10.
6. P. Kirkpatrick and A. V. Baez, “Formation of optical images by x-rays,” J. Opt. Soc. Am. 38, 766–774 (1948). [CrossRef]
7. H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2010). [CrossRef]
8. C. G. Schroer and B. Lengeler, “Focusing hard x rays to nanometer dimensions by adiabatically focusing lenses,” Phys. Rev. Lett. 94, 054802 (2005). [CrossRef]
9. H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, “Nanometer linear focusing of hard x rays by a multilayer Laue lens,” Phys. Rev. Lett. 96, 127401 (2006). [CrossRef]
10. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Butterworth-Heinemann, 1980), Vol. 2 of Course of theoretical physics.
11. D. Laundy, V. Dhamgaye, I. Pape, and K. J. Sawhney, “Refractive optics to compensate x-ray mirror shape-errors,” Proc. SPIE 10386, 103860B (2017). [CrossRef]
12. S. Matsuyama, T. Inoue, J. Yamada, J. Kim, H. Yumoto, Y. Inubushi, T. Osaka, I. Inoue, T. Koyama, K. Tono, H. Ohashi, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Nanofocusing of x-ray free-electron laser using wavefront-corrected multilayer focusing mirrors,” Sci. Rep. 8, 1–10 (2018). [CrossRef]
13. H. Jiang, N. Tian, D. Liang, G. Du, and S. Yan, “A piezoelectric deformable x-ray mirror for phase compensation based on global optimization,” J. Synchrotron Radiat. 26, 729–736 (2019). [CrossRef]
14. M. Sanchez del Rio, D. Bianchi, D. Cocco, M. Glass, M. Idir, J. Metz, L. Raimondi, L. Rebuffi, R. Reininger, X. Shi, F. Siewert, S. Spielmann-Jaeggi, P. Takacs, M. Tomasset, T. Tonnessen, A. Vivo, and V. Yashchuk, “DABAM: an open-source database of x-ray mirrors metrology,” J. Synchrotron Radiat. 23, 665–678 (2016). [CrossRef]
15. V. P. Dhamgaye, G. S. Lodha, B. Gowri Sankar, and C. Kant, “Beamline BL-07 at Indus-2: a facility for microfabrication research,” J. Synchrotron Radiat. 21, 259–263 (2014). [CrossRef]
16. E. W. Becker, W. Ehrfeld, P. Hagmann, A. Maner, and D. Münchmeyer, “Fabrication of microstructures with high aspect ratios and great structural heights by synchrotron radiation lithography, galvanoforming, and plastic moulding (LIGA process),” Microelectron. Eng. 4, 35–56 (1986). [CrossRef]
17. L. Jian, Y. M. Desta, J. Goettert, M. Bednarzik, B. Loechel, Y. Jin, G. Aigeldinger, V. Singh, G. Ahrens, G. Gruetzner, R. Ruhmann, and R. Degen, “SU-8 based deep x-ray lithography/LIGA,” Proc. SPIE 4979, 394 (2003). [CrossRef]
18. K. J. Sawhney, I. P. Dolbnya, M. K. Tiwari, L. Alianelli, S. M. Scott, G. M. Preece, U. K. Pedersen, and R. D. Walton, “A test beamline on diamond light source,” AIP Conf. Proc. 1234, 387–390 (2010). [CrossRef]
19. V. P. Dhamgaye, M. K. Tiwari, C. K. Garg, P. Tiwari, K. J. S. Sawhney, and G. S. Lodha, “Development of high aspect ratio X-ray parabolic compound refractive lens at Indus-2 using X-ray lithography,” Microsyst. Technol. 10, 2055 (2014). [CrossRef]
