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Abstract
We demonstrate how the individual mirrors of a high-quality Kirkpatrick–Baez (KB) mirror system can be aligned to each other to create an optimally focused beam, through minimizing aberrations in the phase of the ptychographically reconstructed pupil function. Different sources of misalignment and the distinctive phase artifacts they create are presented via experimental results from the alignment of the KB mirrors at the NanoMAX diffraction endstation. The catalog of aberration artifacts can be used to easily identify which parameter requires further tuning in the alignment of any KB mirror system.
Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
1. Introduction
In the synchrotron and XFEL community Kirkpatrick–Baez (KB) mirror systems [1] are widely used to focus X-ray beams [2–10]. With a growing number of diffraction limited storage rings around the world, and the increased coherent fraction in the x-ray light they produce, many more beamlines will be able to access micro and nano-focused beams. To achieve this, many instruments have opted for or will opt for utilizing KB mirror systems, due to their high efficiency and achromatic focusing capabilities. To obtain the best focusing performance the highest quality mirror surfaces paired with high precision alignment of the two mirrors is demanded [11,12]. With the increasing quality of the mirror surfaces due to the use of deterministic polishing techniques, the requirements for high precision alignment rise as well. In the presented work, we will focus on the latter.
At the diffraction endstation [13] of the NanoMAX beamline [2], a KB mirror system has been in operation for about five years. Ptychographic reconstruction of the coherent wavefront at the sample position has routinely been used to monitor the alignment quality of the KB mirror system [2,14]. The whole KB mirror system is housed in one vacuum chamber which has been described in detail by Johansson [2]. After a recent intervention, a thorough repetition of the alignment of the KB mirror system system in all critical degrees of freedom was required.
In the past, knife-edge scans [15,16], speckle tracking methods [17] or the reconstructed beam intensity profiles from ptychographic reconstructions of the probing beam and subsequent numerical propagation [18,19], have been used for aligning the pitch of KB mirrors and to quantify the mirror’s surface errors. In the following, we describe how the high-precision alignment of all critical degrees of freedom was performed at the NanoMAX diffraction endstation using the Fourier representation of the ptychographically reconstructed complex wavefront at the focus position (further referred to as the pupil function), as it allows to efficiently identify and quantify the aberrations corresponding to specific mirror misalignments. To our knowledge it is the first time this method and the step by step alignment procedure of a KB mirror system in all degrees of freedom is reported. The shown concepts and procedures are generally applicable to any KB mirror systems using any wavefront sensing technique. We start by introducing the KB mirror system at the NanoMAX beamline, the critical parameters to be adjusted and define the coordinate system in which the results and arguments are presented.
2. KB mirror system at NanoMAX
The KB mirror system at the diffraction endstation of the NanoMAX beamline consists of two meridionally focusing elliptical mirrors M1 and M2. M1, positioned upstream of M2, focuses the incoming beam in the vertical ($y$) direction and reflects it downwards. M2, positioned downstream of M1, focuses the beam in the horizontal ($x$) direction and reflects the beam to the right, meaning towards the MAX IV 3 GeV storage ring. Ideally the two mirrors M1 and M2 are spaced apart by a distance $d_{12} = f_1 - f_2$ ($f_1$ and $f_2$ being the focal lengths of M1 and M2 respectively) along the incoming beam ($z$) direction, so that the two one-dimensional foci intersect to create one two-dimensional focus. Figure 1 illustrates the described arrangement. A aperture defining slit system consisting of 4 independent slit blades is placed just upstream of M1, while a pinhole is placed between M2 and the focal point. The slits and the mirrors are placed inside the same vacuum chamber, while the pinhole is positioned outside of the chamber.
Fig. 1. Scheme of the KB mirror system and coordinate system at NanoMAX in the top view (a), the side view (b) and in a 3D rendering (c). The vertically focusing mirror M1 (with focal length $f_1$) is positioned upstream and deflects the beam towards the negative $y$–direction (downwards). The horizontally focusing mirror (with focal length $f_2$) is positioned downstream and deflects the beam in towards the negative $x$–direction (towards the MAX IV 3 GeV storage ring). Both mirrors are separated along the $z$–axis by the distance $d_{12} = f_1 - f_2$. The deflection angles of M1 and M2 are strongly exaggerated.
The incidence angles of the two mirrors, meaning the rotation of M1 around its $x$–axis (M1$_{rx}$) and the rotation of M2 around its $y$–axis (M2$_{ry}$) are the most important and most sensitive alignment parameters. At NanoMAX these two rotations are referred to as M1-pitch and M2-pitch respectively. Any misalignment of these angles will result in the source point and the intended focal point not being in the two foci of the elliptical shape of the mirror surface, effectively focusing the beam to another position along the $z$–axis and distorting the focus shape. Typically the tolerance for these angles is a few tens of nanoradians [20–22]. For that reason, the mirror mountings at NanoMAX have a coarse adjustment screw pushing on a lever arm, as well as a piezo actuator on the same lever arm for motorized fine adjustments.
The adjustment of the relative alignment of the two mirrors in their rotations around the $z$–axis, is the second most important alignment parameter. As the two mirrors only need to be aligned relative to each other, only M1 has a rotational freedom around its $z$–axis (M1$_{rz}$), which is referred to as M1-roll. The tolerance for this angle is less critical, being typically in the order of tens of microradians [20–22]. A misalignment of M1-roll would distort the focal spot shape and size. Again, a manual adjustment screw for coarse alignment and a piezo actuator for fine alignment are in use at NanoMAX to facilitate the rotation of M1 around its $z$–axis.
The angular alignment of M1 around its $y$–axis and M2 around its $x$–axis (the mirror yaws) are the least critical angles, with tolerances of a few milliradians [20,21]. For that reason, these degrees of freedom, only have manual adjustment screws in the NanoMAX KB mirror system. The distance $d_{12}$ between M1 and M2 along the $z$–axis can also be adjusted manually using a screw, but is also not motorized.
In the following the complete alignment of the KB-mirror system is presented, starting from the coarse alignment steps and then progressively adjusting degrees of freedom with smaller and smaller effects on the resulting X-ray focus, until a diffraction limited focal-spot size has been achieved.
3. Coarse alignment of the vacuum chamber
The first step of the realignment of KB mirror system, was to coarse align the whole vacuum chamber. To this end, each fine alignment piezo actuator (M1$_{rx}$, M2$_{ry}$ and M1$_{rz}$) on the mirrors themselves was driven to the center of its travel range. The entry slits, just upstream of M1, where opened maximally and the pinhole downstream of the vacuum chamber was moved out, to allow beam to pass on the side of both mirrors. To protect the detector from damage caused by a too intense beam, attenuators were used to reduce the photon flux.
An Eiger2 X 4M detector (DECTRIS, Switzerland), placed 3.62 m downstream of the designed focal position, recorded the direct beam, the two beams only reflected by either M1 or M2 as well as the beam reflected by both mirrors (see Fig. 2). The spatial separations ($D_1$ and $D_2$) of the beams reflected on just one of the mirrors to the beam reflected on both mirrors can be used to calculate the beam deflection angles in horizontal and vertical direction. The difference to the design deflection angle was then applied as an angular correction of the mirror chamber, by calculating the necessary amount of rotation for the spindle of each mounting leg. Subsequently the height $d_1$ and width $d_2$ of the beam reflected on both mirrors were used to check if the whole mirror surfaces were illuminated. If not, an additional translation of the chamber would have been required. Finally, the aperture defining slits and cleanup pinhole were moved in again and the attenuators were removed.
Fig. 2. (a) Schematic view of the coarse alignment of the whole KB mirror system, showing how a direct (straight) beam is created by increasing the beam acceptance aperture upstream of the mirrors; (b) recorded detector image showing the direct beam (top left), the two beams reflected by only on one of the mirrors (top right and bottom left) and the beam reflected on both mirrors (bottom right).
4. Fine alignment of the mirrors
With the vacuum chamber coarsly aligned, the fine alignment was left to the motorized stages inside the vacuum chamber. The first thing to do, was to adjust the two mirror pitches M1$_{rx}$ and M2$_{ry}$) so that the foci of the two mirrors coincide along the $z$–axis. This adjustment is the most critical. It was necessary for all following alignment steps. Due to the very low tolerances, this is also an alignment step, which needs to be repeated regularly at the beamline [2] to correct for very slow drifts.
4.1 Mirror pitches M1$_{rx}$ and M2$_{ry}$
The misalignment of the vertical and the horizontal focus positions along the $z$–axis was measured using a ptychographic reconstruction of the complex wavefront near the focus position. A Siemens star test sample was placed near the suspected focus position in the beam. A ptychography scan was performed and the complex wavefront at the sample position was reconstructed (for details see section 5.2). The wavefront was numerically propagated upstream and downstream. The intensity profiles integrated along the $x$–axis and $y$–axis revealed the positions of the vertical focus and the horizontal focus along the $z$–axis (see Fig. 3(a)). Instead of shifting just one of the mirrors, to move its focus position on top of the other mirror’s focus, we opted for shifting both mirrors, so that both foci ended up ${500}\;\mathrm{\mu}\textrm{m}$ upstream of the sample - a plane marked with an optical on-axis microscope present at the beamline. Knowing the source to mirror distance, the focal distance, the design incident angle at the center of the mirror and the length of the lever arm manipulating the pitch of the mirror, the necessary piezo motor movement to achieve a certain focus shift along the $z$-axis can be calculated. In the first approximation, these calculation are a multiplication with a factor depending on the the given geometry of the mirrors (see section 5.3 for details). As these adjustments are performed so frequently at the beamline, these factors have been implemented as convenient macros in the beamlines control system [23]. In the presented case, the required commands were: ’%m1shift 206’ and ’%m2shift −118’ to shift the focus of M1 ${206}\;\mathrm{\mu}\textrm{m}$ downstream and shift the focus of M2 ${118}\;\mathrm{\mu}\textrm{m}$ upstream respectively. Another ptychographic scan was performed afterwards. The extracted profiles are shown in Fig. 3 and show the $z$–positions of the vertical and horizontal focus before (a) and after (b) pitch adjustment.
Fig. 3. Ptychographically reconstructed beam intensity profiles before (a) and after (b) the adjustment of M1$_{rx}$ and M2$_{ry}$. The dotted orange lines mark the positions of the vertical focus and horizontal focus defined by the pitch of M1 and M2 respectively. The dotted green lines mark the position of the sample during the ptychographic measurement and thus the plane in which the complex wavefront has been reconstructed. From there the profiles where extracted by numerically propagating the reconstructed wavefront.
At this point it was unclear if the chosen focal position matched the correct distances $f_1$ from M1 and $f_2$ from M2, and if the distance $d_{12}$ between the mirrors was correct. For this, it was necessary to check the shape and structure of the wavefront at the focus position. This was achieved by numerically propagating the reconstructed complex probe at the sample position ${500}\;\mathrm{\mu}\textrm{m}$ upstream to the focus position (see Fig. 4(a)) and applying the two-dimensional Fourier transform to retrieve the pupil function (see Fig. 4(b)).
Fig. 4. (a) Real-space representation of the complex wavefront at the focus position (of the not yet perfectly aligned KB mirror system) and (b) its Fourier representation - the pupil function. The shown hue encodes the phase of the complex images, while the saturation represents the amplitude.
The reconstructed focus was still distorted and too large, when compared to a diffraction limited focal spot previously achieved at NanoMAX [14]. While the phase and amplitude of the real-space representation of the focus profile is difficult to evaluate, the pupil function is far simpler to interpret. For a KB-mirror system with matched numerical-apertures, like this one, it would ideally be a perfect square with a flat phase. The saddle-function, aligned across the two diagonals (matching the 2D Legendre polynomial $L_5$ and similar to the spherical Zernike polynomial $Z^{-2}_{2}$ for an oblique astigmatism), in the phase of the pupil function is an indicator for the remaining misalignment - the misaligned relative rotation around the $z$–axis (roll) between the two mirrors M1 and M2.
4.2 Mirror roll M1$_{rz}$
As described above, only M1 has a manual and a motorized rotation adjustment around the $z$–axis for this relative alignment of the two mirrors. The piezo-motor used for this has a travel range of ${30}\;\mathrm{\mu}\textrm{m}$ and acts on a ${84}\;\textrm{mm}$ long lever arm, which allows for a motorized angular fine adjustment of M1$_{rz}$ of $\pm {178.6}\;\mathrm{\mu}\textrm{rad}$ around its center position.
To align the M1$_{rz}$, we recorded ptychographic scans at seven positions across the motorized angular range. Due to mechanical cross-talk of the different degrees of freedom and the low tolerances for the incidence angles, an alignment of the two mirror pitches M1$_{rx}$ and M2$_{ry}$ was required every time M1$_{rz}$ was changed. The reconstructed complex probes of each ptychographic scan were again numerically propagated to the focus position. The amplitude of the beam at the focus positions and the complex pupil function for these seven angular positions are shown in Fig. 5.
Fig. 5. Rotation series of M1$_{rz}$ over the whole travel range of the piezo motor. Top row: amplitudes at the focus position. Bottom row: complex pupil functions. The hue represents the phase while the saturation represents the amplitude.
It can be seen that the saddle-shaped aberrations could not be completely corrected for using the motorized adjustment of M1$_{rz}$ due to its limited travel range. By extrapolating from the measured data points the required angular adjustment for eliminating the astigmatism was estimated (see section 5.4 for details). Using this estimate, the M1$_{rz}$ was changed using the manual adjusted screw. The procedure of performing ptychographic scans at multiple positions of M1$_{rz}$ across the travel range of the piezo motor was subsequently repeated. The results are presented in Fig. 6(a). The sign of the phase gradient of saddle function changes between the measurements at the two ends of the piezo motor travel range. Thus the optimal point could now be reached using the motorized fine alignment. Now interpolating the strength of the found aberration as a function of M1$_{rz}$ allowed to estimate the optimal motor position (see method section for details). After moving the piezo motor to the estimated optimal M1$_{rz}$ mirror roll, the wavefront at the focus position was measured again. The result is shown in Fig. 6(b).
Fig. 6. (a) Second rotation series of M1$_{rz}$ over the whole travel range of the piezo motor. Top row: amplitudes at the focus position. Bottom row: complex pupil functions. (b) Complex wavefront measured at the interpolated focus position with with optimal M1$_{rz}$ alignment. The hue represents the phase while the saturation represents the amplitude.
4.3 Optimal $f_1$ and $f_2$
The focus amplitudes now show the expected cross, created by the convolution of two one-dimensional sinc-functions (Fourier transform of a square). The small side peaks outside the main horizontal and vertical axis are clearly visible. Their symmetry is adjusted in the last step: finding the optimal $z$-position(s) for the two foci. In the final scan series, we varied the $z$-position of both the vertical and the horizontal focus together by adjusting the two pitches M1$_{rx}$ and M2$_{ry}$ (see above). The amplitudes at the focus position (see Fig. 7 top row) show the expected asymmetric deformation of the side peaks at the focus position [24]. When focusing too close to the mirrors, the side peak intensities are increased on the sides towards the mirrors (plus $x$ and plus $y$ in the presented case) and reduced on the sides facing away from the mirrors. The pupil functions show an increasing deformation of the phase profile with increased misplacement along the $z$-axis (see Fig. 7 bottom row). These two-dimensional phase deformations can be separated into two independent one-dimensional components for the horizontal coma and the vertical coma. In the used base of 2D Legendre polynomials [25–27], these correspond to the terms $L_7$ and L$_{10}$ respectively (see section 5.5 for details). Their equivalent for spherical Zernike polynomials are $Z^{-1}_{3}$ and $Z^{1}_{3}$.
Fig. 7. Top row: Measured amplitude profile at the focus position for different $z$-positions of two foci; Bottom row: the respective complex pupil functions. The hue represents the phase while the saturation represents the amplitude.
In case $d_{12}$ were optimally set, the optimal $z$-position for the horizontal focus and the vertical focus would be the same. Thus it is crucial to identify the optimal $z$–positions for the two focusing directions independently, as their difference equals the error of the inter-merror spacing $d_{12}$. This was done by interpolating to the $z$-position in which the respective one-dimensional coma were zero (see section 5.5 for details). The measurement showed a missmatch of the two $z$-positions, and thus an error of $d_{12}$, of ${290}\;\mathrm{\mu}\textrm{m}$, which is approximately the depth of field of each of the foci. Changing this distance is possible via another manual adjustment screw. As the adjustment would have required an additional venting of the chamber, and most likely a repetition of the fine adjustment of all the previously mentioned adjustment parameters because of a slight cross-talk among the different degrees of freedom, it was decided to not adjust $d_{12}$. Instead, the mirror pitches M1$_{rx}$ and M2$_{ry}$ where used to place $z$-position of the focus of both mirrors right between the optimal positions of the individual mirrors. Thus the focus of each mirror would be very slightly aberrated. The result was then confirmed via a ptychographic measurement. The reconstructed wavefront at the focus position is shown in Fig. 8(a) and slight vertical and horizontal coma can be seen in the phase of the pupil function (see Fig. 8(b))
Fig. 8. (a) The reconstructed wavefront at the focus position of the aligned KB mirror system at the diffraction endstation of the NanoMAX beamline and (b) the respective pupil function. The shown hue encodes the phase of the complex images, while the saturation represents the amplitude.
The reconstructed wavefront was used to measure the focus size: ${86}\;\textrm{nm} \times {86}\;\textrm{nm}$ (full width at half maximum), slightly improving upon previously reported results [14] and matching the theoretical limit for the achievable focus size [11] (see section 5.6 for details). A permanently installed optical on-axis microscope was used to mark the X-ray beams focus position in space and subsequently align the rotation center of the goniometer [13] to the X-ray beam focus, making the NanoMAX diffraction endstation ready for user experiments.
4.4 Simulations
The found wave front aberrations in the pupil function connected to specific misalignments of a KB mirror system can be explained geometrically by path length differences, and thus relative phase differences, of rays passing through the KB mirror system at different positions. To demonstrate this, we modeled perfect elliptical mirrors and measured the path lengths of different rays from the source to a screen downstream of M2 by ray-tracing using xrt [28]. As argued above, the wavefront downstream of M2 and the pupil function used above only differ in the parabolic (defocus) components of the phase profile. By removing any absolute, linear and parabolic contributions in the ray-traced relative path length profile, the left over features in the pupil function for the ideal mirror setup, as well as the cases for a misaligned roll, pitch, yaw and shift along the optical axis of M2 respectively (see Fig. 9) were extracted. The last, an error in $d_{12}$, was combined with an realignment of the focus position using the M2 pitch, as it was done throughout the whole experiment.
Fig. 9. Simulated path length differences (top) and respective phase shifts at 10 keV for the ideal mirror setup and M2 misaligned in pitch, roll, yaw and shift along the optical axis (plus refocusing with the pitch). The magnitude for the pitch and roll misalignments were chosen to be similar to the experimental results. The magnitude of the misalignment of the yaw, was chosen to create aberrations of a similar strength. The shown hue encodes the phase of the complex images, while the saturation represents the amplitude.
It can be seen that the expected relative sensitivities to angular misalignments of pitch, roll and yaw were confirmed. The relative strength of the first 15 Legendre polynomials used to fit the phase aberrations is plotted next to the phase images and confirms the correlation of L$_{10}$ and the M1 pitch, of L$_{5}$ and the M1 roll. Furthermore it shows that a strong misalignment of the M1 yaw creates aberrations connected to both L$_{5}$ and L$_{10}$.
This can be understood geometrically: Adjusting the roll of M1, changes the length of semi-minor axis of the cylindrical ellipse and results in the rays hitting the mirror earlier on one side than on the other side. The former does not change the focal length nor focal points of the ellipse, while the latter creates the features related to L$_{5}$. Adjusting the yaw of M1 on the other hand, changes the length of the semi-major axis of the ellipse and also makes the rays hit one side of the mirror earlier than the other side. The former does slightly change the focal length of the mirror and places the optimal source and focus point at different $z$-positions. This is equivalent to a wrong pitch and mirror spacing d$_{12}$, which produces the known aberrations matching L$_{10}$. That is why a misalignment in yaw, results in a combination of both L$_{5}$ and L$_{10}$ aberrations.
Finally, we also confirmed that a wrong inter-mirror spacing d$_{12}$ can be identified via the existence of the horizontal and vertical coma terms L$_{7}$ and L$_{10}$. The mirrors can be made to focus to the same $z$–position by adjusting their pitches. But at least one of them will show the respective coma-aberration if d$_{12}$ is wrong.
5. Methods
5.1 Mirror design parameters
The design parameters of M1 and M2, required for the quantification of the presented results are collected in Table 1. The values are an excerpt from a previous publication [2]. Radii are given as cylinders fitted to the elliptical mirror profiles at the center of the mirrors.
Table 1. Design parameters of the two mirrors M1 and M2.
5.2 Ptychographic measurements and reconstructions
A Siemens star (see [14]) was used as a sample for all ptychography scans. The structures are designed to be ${1}\;\mathrm{\mu}\textrm{m}$ thick, are made from tungsten, repeat every ${2.5}\;\mathrm{\mu}\textrm{m}$ in both the vertical and horizontal direction and are placed on a diamond substrate. The sample had been placed ${500}\;\mathrm{\mu}\textrm{m}$ downstream of the focus position, increasing the beam size on the sample to about ${500}\;\textrm{nm}$ (FWHM). All ptychographic scans were recorded at an incident photon energy of ${10}\;\textrm{keV}$, in a step - settle - expose scheme with scan positions in the shape of a Fermat spiral [29], an average step size of ${300}\;\textrm{nm}$ and an exposure time of ${300}\;\textrm{ms}$ at each position. The scanned area was always confined to a square. For measurement of the focus profile at the end of the alignment (see Fig. 8), the scanned area was ${3}\;\mathrm{\mu}\textrm{m} \times {3}\;\mathrm{\mu}\textrm{m}$, while all other presented scans were recorded on an area of ${2}\;\mathrm{\mu}\textrm{m} \times {2}\;\mathrm{\mu}\textrm{m}$, resulting in $168$ and $55$ recorded diffraction patterns per scan respectively. The patterns were recorded by an EIGER2 X 4M pixel detector (DECTRIS, Switzerland) positioned ${3.62}\;\textrm{m}$ downstream of the sample.
The recorded diffraction patterns were cropped to a size of $1024 \times 1024$ pixel around the center of the direct beam, resulting in a pixel size of ${5.85}\;\textrm{nm}$ in the ptychographic reconstructions. The ptychographic reconstructions have been performed using the ptypy framework [30] using $24$ parallel threads on compute nodes with two Intel Xeon E5-2650 CPUs version4 and 256 GB RAM. The initial estimate for the object was completely non-absorbing and non-phase-shifting, while the inital estimate for the probe was a square of ${100}\;\textrm{nm}$ edge length, homogeneous amplitude and flat phase numerically propagated ${500}\;\mathrm{\mu}\textrm{m}$ in the downstream ($+z$) direction. The reconstructions were performed by running $200$ iterations using the difference-map reconstruction algorithm [31] followed by $800$ iterations using the maximum likelihood reconstruction algorithm [32]. Each reconstruction took about ${30}\;\textrm{min}$ to finish the $1000$ iterations. For each scan, the reconstructed complex-valued probe at the sample position was numerically propagated ${500}\;\mathrm{\mu}\textrm{m}$ upstream to the focus position using the angular spectrum method [33].
For Fig. 3, the numerical propagation was performed for $600$ distances equally spaced from ${900}\;\mathrm{\mu}\textrm{m}$ upstream of the sample to ${100}\;\mathrm{\mu}\textrm{m}$ downstream of the sample. The amplitude of the recovered three-dimensional representation of the beam caustic, was squared to recover intensities and then integrated along the $x$– and $y$–axes to obtain the vertical and horizontal intensity beam profiles shown in Fig. 3.
5.3 Mirror pitches
Both mirrors are meridionally focusing and can thus be described using the following equation:
To approximate the new $z$-position of focus for a change in pitch on the elliptic KB’s, the following equation links the change in the exit arm length, $r'$, to the grazing angle $\alpha$ on the mirror surface:
where $r$ is the distance between the source and the center of the mirror surface, $R$ is the radius of mirror surface curvature at the center of the mirror and $\alpha$ is the incidence angle of the incoming X-rays at the center of the mirror surface.Using Eq. (1), any found displacement $\Delta r'$ of the focus position along the $z$-axis can be converted to an angular error $\Delta \alpha$:
The motion $\Delta l$ of the piezo motors, to correct $\Delta \alpha$ can then be calculated by: where $L$ is the length of the lever arm the piezo motor is acting on. For the KB-mirror system at the diffraction endstation of the NanoMAX beamline, the approximated adjustments $\Delta l$ for a focus position shift of ${1}\;\mathrm{\mu}\textrm{m}$ along the $z$–axis are thus $-1381^{-1} \;\mathrm{\mu}\textrm{m}$ for M1$_{rx}$ and $-857^{-1} \;\mathrm{\mu}\textrm{m}$ for M2$_{ry}$.5.4 Mirror rolls
To align the mirror rolls to each other, ptychographic scans have been recorded at different M1$_{rz}$ positions across the motorized travel range. The ptychographically reconstructed probes were propagated to the focus position, because the Fourier transform of the wavefront at the focus position is a scaled image of the exit wavefront at the downstream end of the optics. As the described alignment procedure only requires a relative magnitude of the aberrations present in the wavefront, this method is much quicker than propagating the wavefront all the way to the downstream end of the optics, which would require an extensive zero-padding of the wavefront, a Fourier transform of a much larger array and the subsequent removal of the spherical term of the propagated wavefront [34]. To quantify the found aberrations in the reconstructed pupil function, the representation in the base of 2D Legendre polynomials [25–27] was used, as it is the most suited for the KB-mirror system [26] (square aperture and non-rotational symmetric optical elements). It was found that using the first $15$ 2D Legendre polynomials was sufficient to identify the present aberrations. The relative change of these $15$ 2D Legendre components with a change in M1$_{rz}$ is shown in Fig. 10(a) and (c). The optimal motor position for M1$_rz$ was then estimated by extra-/interpolating a line through the measured L$_5$ terms (see Fig. 10(b) and (d)) Venting the vacuum chamber (for the manual rotational adjustment) with dry nitrogen, opening the access port for the manual adjustment screw, rotating it ${15}^{\circ}$ counter clockwise and closing the port took about ${30}\;\textrm{min}$. The subsequent pumping to acceptable vacuum levels, required an additional ${2}\;\textrm{h}$.
Fig. 10. Estimation of the required roll adjustments of M1 (M1$_{rz}$): pre manual adjustment: (a) separation into the base of 2D Legendre polynomials with the term for the astigmatism (L$_5$) showing a strong correlation with the M1$_{rz}$-positions and (b) their extrapolation reveling that the optimal position could not be reached with the motorized M1$_{rz}$ adjustment (travel range ${0}\;\mathrm{\mu}\textrm{m}$ to ${30}\;\mathrm{\mu}\textrm{m}$). Post manual adjustment: (c) separation into the base of 2D Legendre polynomials showing that the sign of astigmatism (L$_5$) term switches inside the motorized range and (d) the interpolation of the astigmatism terms to estimate the optimal M1$_{rz}$ position.
5.5 Optimal $z$ positions
Similar to the adjustment of the mirror rolls, a series of ptychographic scans has been recorded in which the $z$-position of two foci had been varied. The reconstructed wavefronts have again been numerically propagated to the focus position and its Fourier transform represented in the base of 2D Legendre polynomials (see Fig. 11(a)). The found coefficients for L$_7$ and L$_{10}$, representing the horizontal and the vertical coma respectively, have then been linearly interpolated to find the optimal $z$-positions for each of the two foci (see Fig. 11(b) and (c)). We estimated that the optimal position for the vertical focus was ${357}\;\mathrm{\mu}\textrm{m}$ upstream of the marked $z$-position, while the optimal position for the horizontal position was only ${67}\;\mathrm{\mu}\textrm{m}$ upstream of the marked position.
Fig. 11. Estimation of the optimal $z$-positions for the horizontal focus and the vertical focus. (a) Separation into the base of 2D Legendre polynomials with the terms for the horizontal coma (L$_7$) and vertical coma (L$_{10}$) showing a strong correlation with the $z$-positions. (b) Linear interpolation of the extracted horizontal coma (L$_7$) term, giving the optimal $z$-positions for the horizontal focus. (c) Linear interpolation of the extracted vertical coma (L$_{10}$) term, giving the optimal $z$-positions for the vertical focus.
5.6 Focus size
The amplitudes in the retrieved pupil functions do not form a square, but rather a rhombus. This is a result of the entry slit blades just upstream of M1 being parallel to their respectively opposing blade, but not perpendicular to the slit blades in the other direction. The reason for this is that in the implementation at NanoMAX, the slit blades are mounted as pairs of opposing blades. These opposing blades being parallel can be checked and adjusted prior to and kept during the mounting of the slits inside the chamber, while the perpendicularity of the two pairs of slits to each other can only be adjusted during the mounting in the small vacuum chamber. Thus the later is more prone to slight errors. As a result, the two streaks of the cross-shaped focus in the real-space representation intersect under the same angle close to, but not quite ${90}^{\circ}$. The very slight tilt and remaining coma did however not affect the size of the focus. The diffraction limited focal spot size (FWHM) of a KB mirror system with an rectangular aperture can be estimated by [11]:
where $f$ is the focal length of one of the mirrors, $\lambda$ is the wavelength of the focused photon beam and $d$ is the acceptance aperture of that mirror. Using the values in Table 1 we calculated the diffraction limited focal spot size $S_d$ to be ${89.5}\;\textrm{nm}$ for M1 and ${87.3}\;\textrm{nm}$ for M2 at the used photon energy of ${10}\;\textrm{keV}$. The amplitude of reconstructed wavefront at the focus position was squared to recover the two-dimensional intensity distribution at the focus position. Integrating along the vertical and horizontal direction resulted in the two one-dimensional focus profiles. These were normalized and subsequently interpolated by splines. The focus size was then measured as the distance between the two crossing of the $0.5$ marks of each spline interpolated focus profile. A symmetrical focus size of ${86}\;\textrm{nm}$ in both direction was measured this way. The measured focus sizes are very slightly smaller than the theoretical limits. However, the differences to the theoretical limits are smaller than the pixel size of the reconstructions (${5.85}\;\textrm{nm}$) and can thus be understood as a result of the spline interpolation.5.7 Ray-tracing simulations
For the ray-tracing simulations perfect elliptical mirror profiles with the parameters in Table 1 were modelled in xrt [28]. A point source was used as a starting point for every ray. The detector screen just downstream of M2 was divided into $512 \times 512$ pixel and recorded the path length of the rays bouncing on both mirrors and then intersecting with the pixels with a $64$ bit floating point precision. Constant offsets, linear and parabolic contributions were fitted and removed from the two-dimensional path length profiles to retrieve the contributions that make the aberrations in the focus. These relative path length differences were multiplied with the wave number $k$ to obtain relative phase shifts, as measured in the experiment.
6. Discussion
A detailed step by step procedure from the coarse alignment to the fine alignment of a KB mirror system has been presented. The influence of all degrees of freedom have been presented one by one. While it was presented in previous work how the misalignment of a specific degree of freedom changes the structure of the focus in real space [22], we linked it to the aberrations found in the reconstructed pupil functions and connected them to specific 2D Legendre polynomials [26].
The same wavefront aberrations can also be identified by interferometric wavefront sensing [35–37] or two-dimensional speckle-tracking methods [38–41]. However, ptychographic reconstructions of the wavefront offer a higher spatial resolution and a higher phase sensitivity, while not imposing constrains on beam divergence and maximum magnitude of the phase gradient [19,42]. When using less stable sources like free-electron lasers, which differ from shot to shot, the ptychographic method is not as easily used [43,44] as at synchrotron radiation sources where the provided beam is more stable. In that case speckle-tracking methods are the better choice [22].
Instead of identifying and removing one aberration term after the other, as presented in the manuscript, it should also be possible to retrieve the full information about the misalignment of all degrees of freedom from one single measurement. By separating the reconstructed wavefront at the focus position into the base of 2D Legendre polynomials, all misaligned degrees of freedom can be identified at once. As the measured wavefronts are quantitative, it should be possible to calculate the required adjustments to angles and distances and apply all correction at once. As however, the mechanics used to align the mirrors experience a slight cross-talk, a step by step approach is advisable.
The cross-talk issue could be circumvented by measuring the influence of each adjustment motor in terms of change in the 2D Legendre polynomial coefficients to describe the measured pupil function [35]. Fitting 2D Legendre polynomials to a single ptychographicly reconstructed pupil function and representing the fitted coefficients in the base of the measured adjustment-vectors, should suffice to align a previously characterized KB-system in one step, without the need for any ab initio calculations using the mirrors design parameters.
While the automatizing of the one-step fine-alignment remains work for the future, visually inspecting the pupil function has proven so useful at the NanoMAX beamline, that an automated generation of it was added to the ptychoViewer software [45] used at NanoMAX to inspect the results of ptychographic reconstructions. Even though the KB-mirror system at the NanoMAX diffraction endstation was used as an example to present the detailed step by step alignment procedure, the same can used at any other instruments as well that allows for the measurement of the pupil function downstream the KB-optics.
Funding
Svenska Forskningsrådet Formas (2019-02496); Vetenskapsrådet (2018-07152); VINNOVA (2018- 04969).
Acknowledgments
We acknowledge MAX IV Laboratory for time on Beamline NanoMAX under Proposal 20220196. Research conducted at MAX IV, a Swedish national user facility, is supported by the Swedish Research council under contract 2018-07152, the Swedish Governmental Agency for Innovation Systems under contract 2018-04969, and Formas under contract 2019-02496. The authors thank Frank Seiboth for the discussion on measuring aberrations in reconstructed wavefronts.
Disclosures
The authors declare no conflicts of interest.
Data availability
The raw data and the scripts used for the simulations underlying the results presented in this paper are available from [46].
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