X-ray mirrors are widely used at synchrotron facilities for micro- and nano-focusing due to the achromatic nature of total external reflection. In response to demand and increased metrology capabilities, the optical quality of mirrors has gradually improved, with sub-microradian optical slope errors now commonplace. The recent introduction of “super-polishing” techniques has dramatically improved slope errors to only tens of nanoradians [1]. Nonetheless, even using such state-of-the-art mirrors, the focal spot size is often limited by wavefront aberrations from upstream optics or transmission windows caused by photon heat loads, vibrations, or mechanical clamping distortions.

At modern synchrotron facilities, increasing numbers of active optics, including piezo bimorph mirrors and mechanically bendable mirrors, are employed to correct wavefront errors or provide variable sizes of x-ray beams [2–4]. For example, the diameter of a hard x-ray focal spot has been reduced to $<10\mathrm{nm}$ using a deformable mirror to restore the wavefront shape with phase retrieval methods [2]. At-wavelength metrology techniques are leading candidates to achieve distortion-free and coherence-preserving x-ray optics, and push the resolution and sensitivity to the nanometer scale [3,5,6].

Over the last two decades, various at-wavelength metrology methods have been developed and extensively used to characterize and optimize x-ray mirrors. The pencil beam technique is frequently used for optimizing active optics, due to its simplicity [7], but its angular resolution suffers for mirrors with short focal lengths [6]. Hartmann x-ray wavefront sensors have been demonstrated for the characterization of active x-ray optics, but the ultimate resolution is inherently limited by the pitch of the microlens array [8]. Although Fresnel propagation iterative algorithms and ptychography have proven useful for the alignment and characterization of x-ray mirrors, several hundred images need to be collected for a successful reconstruction [9,10]. Recently, the near-field speckle-based technique has been used to perform the on-line characterization of hard x-ray reflective optics with high angular accuracy and a simple setup. However, the membrane has to be scanned to acquire dozens of images and accurately recover the mirror shapes [11,12]. In addition, an interferometer using a single grating was employed to measure the wavefront distortion from a deformable mirror [13]. However, it can be difficult using a single grating with smaller beam divergence from the focal plane of the mirror, due to the trade-off between sufficient illumination area on the grating and the detectable interferograms.

In contrast, a more sophisticated x-ray grating interferometer (GI) consists of a phase grating and an absorption grating, which can be freely placed downstream and out of the focal plane. Moiré fringes, generated from superimposition of the phase grating self-image and the absorption grating and set to the fractional Talbot distance, can be clearly resolved by x-ray detectors. This GI has been widely used for x-ray phase contrast imaging [14–18], as well as for online metrology [5,15]. In this Letter, we use the GI to optimize the shape of a bimorph mirror by minimizing slope errors using Moiré fringe analysis. Taking advantage of the rotating shearing interferometer technique, the optimal focal size was quickly achieved in only two iterations.

Experiments were performed with Diamond Light Source’s I13 beamline (Coherence branchline), using x rays from an undulator source on the 3 GeV storage ring. The vertical source size was 18.8 μm FWHM, and an x-ray energy of 8 keV was selected using a Si(111) pseudo-channel-cut crystal monochromator. As shown in Fig. 1, the bimorph mirror and the GI were each mounted on independent motorized towers.

 

Fig. 1. Optical layout of the in situ optimization of a bimorph mirror with 8 piezo actuators $A_1$, $A_2,\dots ,A_8$ using a GI. The phase grating is noted $G_1$ and the absorption grating $G_2$.

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The novel mirror under investigation was produced by combining bimorph technology and Elastic Emission Machining “super-polishing” [1,3,4]. The 8 piezo electrodes enable the substrate to be bent to a large range of cylinders or ellipses, with a residual figure error of $<1\mathrm{nm}$ (rms) [3]. The mirror’s figure was pre-polished as a tangential ellipse, whose foci are located at distances $p=41.5\mathrm{m}$ (source) and $q=0.4\mathrm{m}$ (image) from the bimorph mirror when no voltages are applied to its electrodes. The grazing angle of incidence at the center of the mirror is $\theta =3\mathrm{mrad}$. When the mirror was mounted on the coherence branchline of the I13 beamline, the distance $L_1$ between the source and the mirror was $\sim 217\mathrm{m}$, and the focal distance $L_2$ was set to 380 mm. The radiation source is, therefore, not located at the focus of the mirror’s pre-polished elliptical surface; however, the mirror’s shape can be corrected by the application of suitable voltages to its electrodes.

The GI consists of a first phase grating acting as a beam splitter and a second absorption grating acting as a transmission mask. Although transmission is reduced by adding the absorption grating, the combined effects of the two gratings on an x-ray beam produce Moiré fringes that can be resolved using a 2D detector with pixels much larger than the grating pitch. The phase grating $G_1$, with a pitch of $d_1=3.99\mathrm{\mu m}$, was mounted on a rotation stage. The absorption grating $G_2$, with a pitch of $d_2=2.00\mathrm{\mu m}$, was fabricated by depositing electroplated gold onto a silicon grating [19]. The silicon substrates for both gratings were etched down to a 50 μm thickness to minimize x ray absorption. The grating periodicity can be manufactured with nanometer accuracy; hence, distortion of the x-ray wavefront from grating defects was minimal and, thus, neglected. The grating $G_1$ was placed at a distance $L_3=400\mathrm{mm}$ downstream of the mirror focal plane, and the inter-grating distance $G_1$ to $G_2$ was set to $L_4=123\mathrm{mm}$, corresponding to the 7th fractional Talbot order.

Interferograms were recorded with a high-resolution imaging x-ray microscope, consisting of a microscope objective, a Ce-doped YAG scintillator producing a visible light image of the x-ray beam intensity, a mirror reflecting at 90°, and a PCO 4000 CCD camera, onto which the $10\times$ objective focuses light. The pixel size of the CCD camera is $6.5\mathrm{\mu m}\times 6.5\mathrm{\mu m}$.

Before measuring the absolute wavefront slope of the reflected beam, a calibration scan was performed by rotating the phase grating angle $\beta$ along the $z$ axis. Such a scan permits recovery of the angle of the absorption grating $\alpha$ relative to the detector’s horizontal $x$ axis. In Fig 2, the inset interferogram image corresponds to one of the acquisitions taken at various values of $\beta$. The Moiré fringe periods $d_x$ and $d_y$ along $x$ and $y$ for each image, were extracted using Fourier analysis [5]. The values ${\gamma }_x$ and ${\gamma }_y$ correspond to the ratio between the absorption grating pitch $d_2$ and the components $d_x$ and $d_y$, respectively:

 

Fig. 2. Experimental data and fitted model for ${\gamma }_y$ as a function of ${\gamma }_x$. Moiré fringe periods $d_x$ and $d_y$ along the $x$ and $y$ directions are also shown in the inset image.

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To determine the optimum voltages to minimize slope errors, an initial interferogram was collected, with all piezo voltages set to 0 V. For each subsequent image, the voltage on electrode $j$ was incremented by $\nu$ (200 V), until, by the 9th image, all piezo voltages were at 200 V. The wavefront slopes $S_{ij}$ were calculated using the method described in reference [5] from the measurements at $n=691$ points, corresponding to an illuminated mirror length of $\sim 107\mathrm{mm}$, and stored in an $n\times j$ matrix. As shown in Fig. 3, the response of the $j$th electrode is calculated by subtracting the slope extracted from the $j$th and $(j+1)$th image. Mirror surface height changes are calculated by numerically integrating the slope differences and scaling these to the surface length. It can be seen in Fig. 3 that the height change induced by each piezo actuator is below $1\mathrm{nm}/\mathrm{V}$. An $n\times 8$ interaction matrix $M$, which defines the response of each electrode per unit voltage change, is constructed from:

 

Fig. 3. Piezo response function of the bimorph mirror: (Top) the slope and (bottom) height changes induced by applying a fixed voltage to each piezo actuator from $A_1$ to $A_8$, in sequence. Here, line 1–0 means the slope response of the 1st electrode by subtracting the 0th image (no voltages applied) from the 1st image (only apply voltage for 1st electrode). The height change caused by each piezo actuator was calculated by numerical integration of the slope change.

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Once the wavefront slope $S_{i0}$ is derived, the slope error $\mathrm{\Delta }{S}_{i0}$ can be calculated by subtracting the best fit linear term. As plotted in Fig. 4 (black, dashed line), the initial wavefront slope error $\mathrm{\Delta }{S}_{i0}$ was $3.11\mathrm{\mu rad}$ (rms).

 

Fig. 4. Wavefront slope error at zero volts (without correction), and the 1st and 2nd iteration corrections.

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The relationship between the mirror’s deformation and the applied piezo actuator’s voltage is sufficiently linear to permit the interaction matrix method to be applied without modification. Hence, the required voltage $V_j$ can be calculated by multiplying the Moore–Penrose pseudo-inverse matrix of $M^{+}$ with the wavefront slope error $\mathrm{\Delta }{S}_{i0}$ [4,20]:

To validate the optimized slope error, the focused beam size was measured by performing “knife edge” scans with a 200 μm diameter Au wire, as shown in Fig. 5. By setting the piezos of the bimorph mirror to the optimal voltages, determined using the GI, the measured FWHM size of the focused beam was reduced to 0.34 μm. The theoretical beam size at the focal position is 0.32 μm, accounting for the source size contribution (0.03 μm) and the slope error contribution (0.31 μm). This demonstrates that the measured beam size is consistent with the theoretical calculation, assuming a mirror slope error of $0.17\mathrm{\mu rad}$ (rms).

 

Fig. 5. Transmission signal from a gold wire scan (blue squares) in the focal plane of the bimorph mirror. A Gaussian fit (red line) of the derivative of the raw data (black dots) indicates a FWHM of 0.34 μm.

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We have shown that a GI can be used for fast optimization of an active optic, bimorph mirror. The GI can be freely located out of the focal plane, and the requirement of divergence matching of the two gratings pitches is relaxed by using Moiré fringe analysis [5]. For this experimental setup, the angular sensitivity of the GI is less than 50 nrad, and can be further improved by choosing a larger inter-grating distance $L_4$ [15]. Moreover, the technique can easily be extended to focus the beam in two dimensions (2D) by rotating the 1D grating 90°, or using a 2D grating [16–18]. This technique has the potential to create a user-defined level of defocus or beam-shaping for active optics. In addition, it can also be used to speed up mirror alignment by combining with the conventional knife edge scan technique. As only a single shot is required, the technique will be highly advantageous for x-ray free electron laser mirror commissioning and dynamical wavefront sensing.