In situ metrology for adaptive x-ray optics with an absolute distance measuring sensor array

Adaptive x-ray mirrors are emerging as one of the primary solutions for meeting the performance needs of the next generation of x-ray light sources. Currently, these mirrors operate open loop with intermittent feedback from invasive sensors that measure the beam quality. This paper outlines a novel design for real-time in situ metrology of the shape of these mirrors using an array of interferometric sensors that does not interrupt the x-ray beam. We describe a proof-of-principle demonstration which shows sub-nm agreement over a range of mirror deflection magnitudes and shapes as compared to simultaneous measurements by using a large-aperture Fizeau interferometer. © 2019 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5060954


I. INTRODUCTION
The advent of diffraction limited storage rings (DLSRs) and x-ray free-electron laser sources (XFELs) poses a series of new challenges for grazing incidence x-ray beamline optics to fully exploit the advantages of these sources. 1,2In addition, a growing set of demands on the optics originating from the science and usability needs of beamlines is leading to rapid progress in the design, manufacture, and deployment of these optics.
Evolving science needs are requiring greater flexibility from beamlines and unprecedented levels of performance.Some examples of functionalities engendered by these needs include nanometer level spot sizes and spot location stability for higher resolution imaging, 3,4 "zoom" capability to alter the spot sizes on demand to enable multi-scale imaging, 3 the ability to create tailored intensity profiles, 5,6 variable numerical aperture (NA) systems, 7 correction of aberrations in the beam wavefront, and dynamic spot size variation at frequencies up to 10 Hz. 8 Beamline usability is a growing consideration in the design and upgrade of user facilities to speed up beamline alignment, data collection, and analysis.This level of turnkey operation is becoming increasingly important as the user community becomes more diverse and multi-disciplinary with newer user communities often lacking the technical skills or inclination to operate a beamline. 8][11][12] Bimorph mirrors, in particular, appear to be a promising solution as they provide the most flexible control.These mirrors typically have a high actuator density (ranging from 8 to 32 depending on the mirror length) that allows for the correction of low and mid-spatial frequency errors, providing greater control over the mirror shape than mechanical benders.The most current generation of bimorphs 10,13 has addressed the limitations initially observed with the first generation and is capable of sub-nm figure control. 9ost adjustable mirror systems currently operate openloop, i.e., without any direct feedback.This leaves the operator blind to perturbations due to thermal changes in the ambient environment, localized heat loading from the x-ray beam

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scitation.org/journal/rsi(so called heat bumps), clamping and gravitational forces, and material and actuator creep. 9Devices which can supply direct feedback can only provide it intermittently and are typically invasive, 14 interrupting the experiment, and wasting valuable beam time.Other in situ feedback techniques rely on redirecting a portion of the photon flux, thereby making these photons unavailable to the experiment. 11For these reasons, a real-time in situ metrology technique for the measurement of mirror shape to maintain and dynamically control the beam shape, size, position, numerical aperture, and other parameters without interrupting the experiment is advantageous.
We propose and demonstrate the functioning of such an in situ metrology system that directly monitors the reflecting surface of a piezo bimorph mirror.The system we demonstrate has a high sensor density that complements the high actuator density of the bimorph.However, the methods described herein are perfectly general and apply to any type of fixed or deformable mirror.Such a system, when implemented, can enhance the long-term stability and dynamic functionality of these types of mirrors.

II. MEASUREMENT COMPARISON
The goal of this comparison is to demonstrate the capability of the sensor array by comparison to a standard metrology technique, i.e., Fizeau interferometry.6][17][18][19] While this sensor is an absolute sensor and in principle can measure the absolute figure of the mirror (described further in Subsection II A), this first proof-of-principle experiment focuses on the measurement of changes in mirror shape.

A. Sensor technology
The sensor functions in a Fizeau optical configuration with interference between reflections from the sensor reference surface and the target creating the measurement signal, as shown in Fig. 1.Unlike most interferometric fiber sensors which can only measure displacement ∆d, this sensor can initially establish the absolute distance d of a target relative to the reference surface of the sensor, as shown in Fig. 1, and track incremental displacements ∆d therefrom.The absolute distance d is determined by an automated implementation of the method of exact fractions. 20The sensor light source cycles through a sequence of wavelengths multiple times to establish d for each of the measurement channels simultaneously as illustrated in Fig. 1(b).This capability enables unique functionalities including the absolute measurement of mirror figure as discussed later.
The ZPS sensor system is specifically designed for applications requiring tens of measurement channels (up to 64), making it well suited for measuring the shape, position, and orientation of adaptive x-ray optics.The compact size of the sensor, shown in Fig. 2(a), allows for densely packed arrays to enable sampling of small spatial periods.In addition to its compact size, the sensor has several features that make it uniquely suited for this application.The dependence of sensor noise on the operating bandwidth over the full 3.5 mm ± 0.6 mm measurement range is captured by the 0.02 nm Hz −1/ 2 (3σ) noise figure specification.For example, for an operating bandwidth of 1 kHz, the sensor noise is ∼0.63 nm (3σ).The specified noise performance applies over an angular acceptance range of ±1 mrad although the sensor can function beyond this range.The 3.5 mm standoff is fixed and set by the internal optics of the system.The maximum data rate of 208 kHz and the bandwidth of 104 kHz further enable rapid simultaneous synchronized data collection (regardless of the number of channels) providing the option of averaging to further drive down the noise.
The system is also capable of very high stability (<1.0 nm/day), and to exploit this fully requires mounting strategies with commensurate stabilities.The sensor design makes provision for this as shown in Fig. 2 the measurement is made) that is physically accessible such that it can be placed into direct physical contact against the metrology frame under the influence of an axial preload force.This eliminates any adhesives or other components prone to drift at the interface between the measurement reference and the metrology frame.This mounting strategy also removes the influence of the thermally induced dimensional changes of the sensor body from the measurement as these changes do not affect the location of the sensor reference which remains in contact with the metrology frame, while dimensional changes of the sensor body are accommodated by the preload mechanism.

B. Measurement principle
The measurement architecture is based on an array of sensors that measures the displacement of various parts of the reflecting surface of an x-ray optic.This could, for example, provide in situ feedback for the closed loop control of the mirror shape, as shown in Fig. 3(a).
This so-called "bed-of-nails" approach to mirror shape metrology is a technique that is used for measuring optics during fabrication 21 and is quite widespread in the astronomical optics community. 22A version of this measurement using only three sensors has been used to measure the curvature

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scitation.org/journal/rsichanges of synchrotron optics. 23The proposed method relies on a stable independent metrology frame that is not susceptible to thermal deformations or influenced by the mirror deformations to hold the sensors.Careful design of the metrology frame for thermal and temporal stability is key to the measurement.The metrology frame mount is designed to decouple it from deformations of the mirror frame while at the same time holding the frame in a stable fashion at a fixed distance parallel to the mirror.The sensors mounted in the frame directly measure the reflecting surface of the mirror.The grazing incidence x-ray beam passes beneath the metrology frame with the array of sensors ideally positioned directly above the beam.The 3.5 mm standoff provides sufficient clearance for most hard x-ray beams as they pass beneath the frame at grazing incidence.Configured as such, the array centerline passes through the x-ray beam in compliance with the Abbe principle 24 to provide feedback on the shape of the portion of the mirror reflecting the x-rays, as shown in Fig. 3(b).This aspect of the design is key to ensuring that mirror twist or tilt θ does not cause a measurement error ε due to the sensor offset a, as shown in Fig. 3(c).A metrology frame with a central channel for x-ray beam clearance with sensors on either side can accommodate soft x-ray applications with larger grazing angles and bigger beams.
Figure 4 shows an extension of this concept to the measurement of other deformations as well as rigid body motions in all six degrees-of-freedom (DOF).For example, a pair of sensors at each end can provide information about mirror twist along its length.The absolute nature of the sensors enables the measurement of position and orientation in addition to the absolute shape of the mirror.This sub-nm absolute position repeatability can aid in the reestablishment of a previously optimized mirror shape, position, and orientation, thereby shortening or eliminating the mirror optimization process after a stripe change or mirror swap.Furthermore, a calibrated metrology frame enables the determination of the absolute shape of the mirror in situ post-mounting in the operating environment, providing an accurate measurement of the mirror shape that is predictive of its performance.This is in contrast to the ex situ techniques currently used which measure the mirror under mounting conditions different from their in-use mounting conditions in a laboratory environment, rendering these techniques blind to the final low-order as-mounted mirror shape. 25

III. COMPARISON MEASUREMENT
The goal of this comparison is to demonstrate the performance of the sensor array by comparison to a large aperture Zygo 18" Fizeau interferometer.

A. Measurement setup
Figure 5 shows the physical realization of the concept shown in Fig. 3.The metrology frame is parallel to the reflecting surface of the mirror as it would be in service and constrained kinematically with respect to the mirror at the 3.5 mm working distance of the sensor.The sensor measurement beams exit through small apertures in the side of the metrology frame closest to the mirror (the bottom as depicted in Fig. 5) and in a direction nominally orthogonal to the mirror surface.The footprint of an individual sensor beam on the mirror is ∼0.18 mm in diameter.
Figure 6 shows a simplified representation of the full setup.The deformable mirror mounts to a common aluminum baseplate through a tip/tilt/z adjustment based on three adjustable balls in three-vee grooves (represented by a triangular block in the figure for simplicity) akin to a type II Kelvin clamp 26 which aligns the mirror in rotation about x and y and in translation in the z direction.This alignment sets the mirror surface at the sensor standoff and aligns it in angle to within a fraction of the ±1 mrad angular acceptance range of the sensors.The metrology frame mounts to the baseplate through three bipods.The metrology frame in this realization has two parallel rows of sensors that straddle an aperture in the metrology frame to enable a simultaneous Fizeau measurement.Section III A 2 below provides further details about this design choice.In actual use on a beamline, a single linear array of sensors would suffice to measure the mirror shape along the footprint of the beam as explained in Fig. 3

(b).
Sections III A 1 and III A 2 describe each of the key components of the setup in greater detail.

Bimorph mirror
The mirror used in this experiment is a vertical focusing mirror (VFM) of a Kirkpatrick-Baez mirror pair manufactured by Thales-SESO and driven by a piezo power supply from FMB Oxford.The mirror is ∼450 mm long and is a first-generation piezo bimorph with twelve actuators distributed along its length and sandwiched between two glass face sheets.The mirror has a broad off-center platinum coating.An aluminum frame surrounds the mirror and connects it to a mounting flange which has four clearance holes for mounting screws.Since details of the internal connections between the mirror and the frame are unknown, the mirror and mirror frame were modeled as a lumped mass in the finite element analysis (FEA) of the setup.
The undistorted radius of curvature (RoC) of the mirror is ∼3.2 km.The mirror can be driven over a range of radii ranging from RoC = ∞ to ∼6 km.

Metrology frame and bipods
The metrology frame is a separate structure that holds the sensor array.An independent metrology frame is chosen over one that couples directly to the mirror frame.While the latter has the potential for a shorter and stiffer structural loop, it leaves open the possibility of the transfer of deformations of the mirror frame during mirror operation distorting the metrology frame and contaminating the measurement.Three symmetric bipods which are the flexure equivalent of a kinematic type II Kelvin clamp serve to exactly constrain the metrology frame relative to the mirror.The kinematic nature of the bipod coupling minimizes deformation of the frame due to differential expansion between the frame and the baseplate and any other deformations of the baseplate.
Aluminum (6061-T6) is the material of choice for the metrology frame for several reasons including cost and ease of manufacture.While aluminum with its high coefficient of thermal expansion (CTE) may seem a counterintuitive choice for a component with nm level stability requirements, this choice is justifiable based on several design considerations.The frame must simultaneously satisfy three requirements: (1) a high resonance frequency for the first non-rigid body mode to reject the effects of seismic vibrations ⇒ a high Young's modulus E to maximize the stiffness and a low density ρ to minimize the mass (maximum E/ρ); (2) dimensional stability in the face of temperature gradients ⇒ a high thermal conductivity λ to prevent the formation of thermal gradients within the structure (which promote bending), in conjunction with a low coefficient of thermal expansion α (maximum λ/α); and (3) low mass to maximize the resonance of the first mode of the rigid body motions on the bipod supports to also reject the effects of seismic vibration (maximum 1/ρ).Table I lists these ratios along with cost and machinability criteria for aluminum and contrasts them with Zerodur ® (a natural but expensive choice for this application), Super Invar, and stainless steel.It becomes readily evident that while Zerodur is a clear choice based on material property ratios, aluminum represents an acceptable tradeoff for a first proof-of-principle experiment.The thermal properties of aluminum are not as bad as they would seem if judged based on the CTE alone which is ∼730 times greater than Zerodur.When thermal conductivity is also taken into account, the performance difference is only approximately a factor of seven worse than Zerodur.
The design of the bipods and metrology frame satisfies the conflicting requirement for a compliant bipod design in the non-constrained directions to minimize the effects of differential thermal expansion between the baseplate and the metrology frame and for a stiff design in the constraint directions to achieve an adequately high first resonance of the rigid body vibrations of the metrology frame.Another  The metrology frame in this realization has two parallel rows of sensors that straddle an aperture in the metrology frame to enable a simultaneous Fizeau measurement in compliance with the Abbe principle, as illustrated in Fig. 7.The frame has two rows of 19 sensors each with a spacing of approximately 24 mm to span the length of the ∼450 mm long mirror.Only the central 17 in each row participates in the measurement due to data drop-outs in the Fizeau data from high local slopes from sharp edge roll-off at the mirror extremities.A pairwise average of corresponding sensors in the upper and lower arrays provides an effective measurement axis midway between the two arrays and coincident with the centerline of the Fizeau measurement, thereby satisfying the Abbe principle.This strategy compensates for any tilt in the mirror surface in the sagittal direction at the location of any sensor pair as the mirror deforms.Any higher order changes in shape in the sagittal direction during deformation will contribute to an error in the measurement.The two arrays are positioned such that the sensor beams impinge a few mm inside the edge of the coated region, while the central portion remains unobscured for the Fizeau measurement.
The sensor mount exploits a chamfer around the edge of the accessible sensor reference surface and registers it against the metrology frame by preloading it into a spherical cup (inset of Fig. 7).The sensor beam passes through a 1 mm diameter aperture at the bottom of the cup.The cup constrains this end of the sensor radially and axially while also allowing sensor tip and tilt for alignment purposes.A spring mechanism (not shown) constrains the other end of the sensor in the radial direction while providing an axial preload that registers the sensor into the spherical cup.This constrains five of the six DOF of the sensor, while friction between the preload spring mechanism and the sensor prevents motion along the sixth DOF (rotation about the sensor axis).Manipulation in the radial direction of the spring mechanism that constrains the rear-end facilitates adjustment of sensor pointing.The location of the spherical cups fixes the spacing between the sensors and their axial position within the metrology frame, both of which are fixed for a given metrology frame.The adjustable mirror mounting mechanism provides the adjustment to set the axial position and the tip-tilt orientation of the mirror as a whole relative to the sensor array.Locking of the preload mechanism upon sensor alignment fixes the radial constraint at the rear end thereby setting the sensor pointing.A common target in the form of a quarter-wave optical flat assures co-alignment of the measurement axes of all the sensors to <0.3 mrad during initial alignment at the time of sensor installation in the frame.Maximizing the return signal which peaks when the optical axis of the sensor is perpendicular to the target provides the feedback required for adjusting the sensor pointing.The entire setup mounts to a tip/tilt mount as shown on the left in Fig. 6(b).The setup is aligned to the Fizeau interferometer by minimizing the number of fringes observed on the mirror surface through the aperture in the metrology frame.As seen in the detailed uncertainty analysis (Sec.V), Fizeau interferometer focus is critical and consequently the Fizeau interferometer is carefully focused by using a fiducial that is placed within 0.1 mm of the mirror surface.A 24 h stabilization period precedes the start of measurements.
The basis for comparison is simultaneous measurements by using the Fizeau interferometer and ZPS sensor array at each mirror state.The Fizeau measurement is the result of 32 phase averages to minimize the effect of air turbulence and vibration.Each sensor data point in each measurement is an average of 50 k data points sampled at 2 kHz.The sampling rate is chosen to match the time taken for a Fizeau measurement of approximately 25 s.Five measurements are made at each mirror configuration.A low-pass filter with a 24 Hz cutoff on each of the array measurement channels minimizes contributions at the line frequency of 60 Hz and its harmonics from acoustic vibrations originating in the lighting and other machinery in the facility while still providing adequate bandwidth for the essentially static changes in mirror shape.A compact refractometer located on the metrology frame corrects the sensor array data for index changes due to pressure and temperature.
A typical measurement sequence starts by allowing the mirror to stabilize at its initial shape followed by simultaneous Fizeau and sensor array measurements to provide a measure of the initial mirror shape.In the measurement data shown below, the initial shape of the mirror was its free shape, i.e., the shape with all actuators set to 0 V.A measurement sequence is comprised of progressively larger voltage inputs to a particular set of actuators to generate deformations of varying magnitudes for a given shape; this process is repeated with four different configurations to achieve four distinct mirror shapes.No attempt was made to optimize the mirror to any particular shape as the goal was to create a shape change that could be measured by both instruments.A stabilization period of a few minutes follows each shape change to allow the transients resulting from the change to settle.This stabilization period is long enough to allow the relatively fast transients to settle and short enough to minimize the effects of thermal changes in the environment that cause the rest of the setup to drift.A second set of simultaneous measurements records the deformed shape of the mirror.The mirror is then set to the next voltage amplitude, and the measurements are repeated.Each measurement sequence concludes by returning the mirror to its undeformed shape, measurements of which become the reference for the next shape's measurement sequence.

C. Data processing
The difference between the shape changes measured by using the two instruments serves as the criteria for comparison.In each case, the Fizeau and sensor array measure an initial and deformed shape of the mirror; the difference between the initial and deformed mirror measurements (∆S A and ∆S F for the array and the Fizeau interferometer, respectively) represents the shape change.The difference ∆S between the two measurements of shape change ∆S A and ∆S F quantifies the agreement between the two methods with zero difference representing perfect agreement.Figure 8 shows a schematic representation of the overall data collection and processing workflow.
For the difference ∆S to be meaningful, it must be evaluated at the same point on the mirror surface by both techniques.This requires transferring the locations of the

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measurements points of the sensors into the coordinate system of the Fizeau interferometer (or vice-versa).A combination of two measurements establishes the locations of the sensors in the array in the coordinate system of the Fizeau interferometer.First, two fiducials on the metrology frame which are within the field of view of the Fizeau interferometer and visible simultaneously with the mirror establish the location of the entire frame in the Fizeau coordinate system.These fiducials consist of two circular apertures in the body of the metrology frame behind which mount two retroreflectors which guarantee a reflection even in the presence of misalignment between the metrology frame and the transmission flat of the Fizeau interferometer.Figure 9(a) shows the intensity image of the Fizeau field-of-view (FOV) with the fiducials and the fringe pattern.Fringes are visible both on the surface of the mirror and on the fiducials.The latter are problematic when using the intensity image to establish fiducial locations by centroiding as they bias the centroid evaluation.Averaging all the frames in the Fizeau measurement eliminates the fringes and produces well-defined images of the fiducials as seen in Fig. 9(b).Thresholding and then calculating the centroid of the group of pixels that defines each fiducial in the fringe-free image produces the locations of the fiducials in pixel space.Second, measurements on a coordinate measuring machine (CMM) establish the locations of each of the sensor cups which locate the sensors in the metrology frame relative to the fiducials in length units.The final conversion of the location of the sensor from length units to pixels requires scaling the measured distances by the lateral calibration constant (LCC).The known distance between the two fiducials and the corresponding distance in pixels from the centroid calculation leads to the LCC.Calculation of each of the sensor locations in pixel space also identifies the row of pixels in the Fizeau data that correspond to the effective line of measurement (ELM) between the two arrays.Distortion in the Fizeau imaging system affects the apparent fiducial locations and consequently the LCC value.Distortion correction is required to determine an accurate value for the LCC, and information about the distortion is derived from the coma that results in the tilt direction when measuring a tilted flat. 27A series of measurements was taken immediately prior to the main experiment in order to characterize the distortion.The intensity maps are corrected based on this information prior to the determination of the LCC and the fiducial locations.
Figure 10 shows a pictorial representation of the processing to obtain the two sets of data for comparison.Figure 10(a) shows the band of Fizeau data centered about the row of pixels closest to the ELM.This band consists of five rows of pixels on either side of the row of pixels closest to the ELM for a total of 11 rows.Based on a pixel pitch of 0.5 mm, averaging these 11 rows pointwise along the x coordinate collapses this band of pixels into a single row of data that represents an average over a ∼5.5 mm wide strip on the mirror surface [Fig.10(b)].This averaging accounts for the slight misalignment between the ELM and the nearest row of pixels and also helps to minimize the effects of local deviations due to scratches and dust.Figure 10(c) shows the pairwise averaging of the sensor data from the two linear arrays that produce the corresponding sensor array data denoted by the crosses in Fig. 10(d).This approach ensures that the comparison complies with the Abbe principle.Five sets of data are acquired for each test condition, and the five difference results are averaged on a point-bypoint basis.The effects of rigid body motions of the mirror and the frame, i.e., tilt and piston of the mirror and metrology frame, are subtracted from the difference data.These terms result from the rigid body motions of the mirror as it deforms and those of the metrology frame due to thermally induced changes in the lengths of bipods.Their removal is justified as neither impacts the measurement of shape change.

IV. MEASUREMENT RESULTS
The measurand in this experiment is the pointwise difference between the sensor array and the Fizeau interferometer at the effective measurement points of the sensor array.Since the measurand is the pointwise difference, a vector of repeatability values corresponding to each of the sensors best characterizes the repeatability of the difference.The repeatability of the measurement is a key measure of the quality of the measurement, and Fig. 11 shows one example of the short-term repeatability over the time taken to perform one comparison (approximately 5 min).The standard deviation of the measured values from five measurements characterizes the pointwise repeatability which is below 0.4 nm (1σ) in the worst case.
The graphs in Figs.12-15 show the results of the comparison between the Fizeau and the sensor array for a range of actuator configurations and deformation magnitudes.In each case, the difference shown is relative to the natural shape of the mirror, i.e., the shape with all actuators set to 0 V.The graphic above the graphs in each figure provides a visual representation of the energized actuators, where red denotes an active actuator and gray denotes an inactive one set to 0 V.The graph on the left shows the Fizeau measurement of the deformed mirror shape, while the graph on the right shows the average difference between the sensor array and the Fizeau      II summarizes the differences between the sensor array and the Fizeau interferometer.

V. MEASUREMENT UNCERTAINTY
Agreement between two measurement techniques, especially ones with commensurate measurement uncertainties, should be judged in context of these uncertainties.The first and most critical step in any analysis of measurement  uncertainty is the identification of the measurand. 28As stated in Sec.IV, the measurand is the pointwise difference between the change measured by the sensor array at a point midway between each sensor pair and the corresponding Fizeau measurement.
While there are numerous sources of uncertainty, there are only a handful that dominate.Furthermore, since the measurement uncertainty varies as a function of position along the mirrors, a single number, while convenient, does not capture this dependence on position.This is because some of the contributors depend on the local slope change when the mirror deforms, e.g., the uncertainty due to focusing the Fizeau, while others are position dependent like the contributions due to the bending of the metrology frame or the Fizeau transmission flat.For this reason, the uncertainty is described by a value at each sensor location i at which the difference d i is calculated, as shown in Fig. 16, wherein the uncertainty associated with the deformation which produces the largest slope change, i.e., the +100 V case with all actuators energized, is shown.The ISO Guide to the Expression of Uncertainty in Measurement (GUM) 29 informs the evaluation of the measurement uncertainty, and we report a combined standard uncertainty in the pointwise difference U C (d i ) with a coverage factor of k = 2 or at a confidence level of 95%.Table III lists the main sources of uncertainty in descending order of magnitude with corresponding values for the average uncertainty over all the sensor locations as a means of providing a single number to give a sense for the magnitude of each contribution.Since these terms are uncorrelated for the most part, the overall uncertainty in Table III is the average calculated after quadrature combination of the individual terms except as noted below.A detailed discussion of each contributor follows.
The contributions from the uncertainty in focusing the Fizeau interferometer dominate the measurement uncertainty.Errors in focus lead to a contribution that is dependent on the local slope change and hence varies from zero at the FIG.16.Overall measurement uncertainty as a function of sensor location.Sensor 10 is located at the centre of the mirror as shown in Fig. 10(c).
center to a maximum at the ends of the mirror in a parabolic fashion.This uncertainty contribution increases dramatically for larger deflections, and therefore, the testing is restricted to small mirror deflections to limit this contribution from the Fizeau measurement to the nanometer level.The resulting 2σ uncertainty term from this effect is approximately 1.12 nm.
The second largest contributions come from deformations in the metrology frame and the Fizeau transmission flat due to temperature gradients across the depth of these components.While the high thermal conductivity of the aluminum metrology frame minimizes this effect (see Sec. III A 2), the frame shows a parabolic deformation with a thermal sensitivity of 40 nm of sag per degree Kelvin of temperature gradient.The fused silica transmission flat exhibits a similar contribution of 26.5 nm of sag per degree Kelvin of temperature gradient.Based on the observed gradients of 20 mK during the measurement, these lead to sag contributions of ∼0.8 nm and ∼0.5 nm of sag, respectively.An assumption of correlation for these contributions is justified since one side of the flat and the frame are likely to see the same changes in temperature changes and hence the same changes in gradient.Even though the correlation is likely to be somewhat less than perfect due to the differing thermal inertias of the two components, these terms are added arithmetically in the uncertainty evaluation to A pointwise estimate of the short-term measurement repeatability as shown in the graph in Fig. 11 is a component of the measurement uncertainty, albeit a relatively minor one.This captures the contributions of vibration (of acoustic and seismic origin), instrument noise (post-averaging), short-term thermal effects, air turbulence, and localized index variations in the space between the array and the mirror over the period taken to make a typical measurement of about 5 min.The most significant contributor is most likely the air turbulence in the ∼300 mm long Fizeau cavity.While the sensor array is also susceptible to air turbulence, this effect, which scales with the length of the air path, is orders of magnitude smaller for the 3.5 mm air path for the sensor array.The average 2σ contribution from these sources is ∼0.75 nm, while the 2σ standard deviation of the average of five measurements is ∼0.34 nm.
Another slope dependent contributor is the tilt-induced measurement error of the sensors in the array.A similarly minor contributor is the change in the sagittal mirror shape between any pair of sensors in addition to tilt in the sagittal direction.This contributes a small uncertainty to the comparison since such changes will only be measured by the Fizeau and not by the sensor array.The magnitude of the errors due to sensor tilt-dependence and sagittal figure change are approximately 0.28 nm and 0.24 nm 2σ, respectively.This uncertainty source would not be a contributor in a beamline application when using a single row of sensors looking directly through the x-ray beam at the region of the mirror reflecting the beam.
Numerous additional uncertainty sources were considered but found to be insubstantial and as such do not affect the overall uncertainty estimate.Uncertainties in the knowledge of the lateral calibration constant of the Fizeau interferometer and the spacing of the sensors in the array contribute in a similar parabolic fashion to the Fizeau interferometer focusing term although with a negligible contribution.The uncertainty that often dominates a Fizeau measurement, i.e., contributions due to the deviations of the transmission flat from a perfect plane do not contribute significantly as the measurements here measure changes in shape and these common mode errors drop out in the differences.The other contributions due to uncertainties in the measurement of displacement of the mirror surface by the sensor array that are typical of any displacement interferometer 30 or the measurement of figure with a Fizeau interferometer scale with the displacement and are vanishingly small for these small displacements.
Effects of bulk index changes on the sensor array while compensated by the dedicated refractometer are also not a major contributor as they enter the measurement as displacements of the entire array relative to the mirror, which amounts to a rigid body translation which does not affect the measured shape.Similarly, a linear index gradient along the length results in tilt which is also a rigid body motion that does not affect the measured shape.The repeatability measurements account for the higher order index variations which do affect the shape.Longer-term higher order index variations have an effect similar to metrology frame deformation and primarily result from temperature variations, pressure being virtually unchanged over the time period of a typical measurement sequence.The ∼20 mK 2σ temperature fluctuations observed result in a ∼0.07 nm 2σ contribution making it one of the minor contributors.

VI. DISCUSSION OF RESULTS
The results readily show agreement between measurements of the shape changes by the sensor array and the Fizeau interferometer to sub-nm levels.This agreement is well within the uncertainty of the measurement comparison for all the test cases.
In some of the test cases, a low-order parabolic term is evident in the difference.This is most likely to be due to the flexion of the metrology frame from thermal gradients across the frame and the Fizeau transmission flat (the reference for the Fizeau measurement).A reduction of this sub-nm effect is achievable by using a low-CTE material for the metrology frame, with improvements of an order of magnitude (or more) being easily achievable.For beamline applications with relatively large temperature excursions, a Zerodur frame can achieve the requisite sub-nm stability.
To get a sense for the capability of the sensor array approach, it is important to understand the uncertainty contributors that apply in the typical use case.As the analysis of the measurement uncertainty shows, the dominant contribution to the difference is from the uncertainty associated with focusing the Fizeau on the surface of the mirror.However, this source of uncertainty does not apply to the sensor array in a typical beamline application for measuring mirrors where the dominant contributors are deformation of the metrology frame and the tilt-dependent error of the sensor.Contributions due to air turbulence for the array alone are much smaller due to the short air path for the sensor array and completely vanish in the case of operation in a vacuum environment.The use of low CTE materials for the metrology frame reduces the former to sub-nm levels as discussed above, while the contribution from the sensor tilt-dependent error is expected to be at the sub-nm level for the slope changes typical of synchrotron mirror deformations.

VII. CONCLUSION
The results of the experiments described in this paper show that an array of ZPS position sensors has the capacity to measure changes to mirror shape to sub-nm levels for small mirror deformations (including the low-order shape).The contribution to the measurement uncertainty from the array alone is expected to be at the sub-nm level, the exact values depending on the operating environment.
An array monitoring system opens the door to more automation by feeding the mirror shape data into a servo controller for constant correction of mirror perturbations during beamline operation without user intervention.Use of the Rev. Sci.Instrum.90, 021703 (2019); doi: 10.1063/1.506095490, 021703-13

Review of Scientific Instruments
ARTICLE scitation.org/journal/rsiZPS sensor system with its sub-nm absolute position capability would also allow beamline operators to store and return to known good mirror configurations after mirror or stripe swaps.Calibration of the array against a known shape could also enable in situ metrology of the absolute mirror shape in its in-use mounted condition.In contrast to the current practice of characterizing the mirror shape (sometimes in its unclamped state) by ex situ techniques under laboratory conditions, this technique could provide mirror shape in the asclamped state in the service environment, allowing more rapid convergence of the shape to its operational state.Furthermore, this same metrology capability would allow characterization of the transfer function of a deformable mirror system for implementation of a closed loop control system.
The real time in situ metrology capability enabled by the sensor array system has the potential to lead to better mirror shape control, stability, and easy reconfigurability providing a better user experience to user communities that are not experts in beamline operation and increased beamline uptime making more time available for scientific investigation.

VIII. FUTURE WORK
Work is currently underway to compare shape changes over larger mirror deflections to measurements made using a slope measuring profilometer.Preliminary x-ray exposure testing of the sensors has been completed and plans are underway to implement the sensor array in an x-ray beamline.Testing in the actual beamline will evaluate x-ray hardness, confirm the <10 −9 Torr vacuum compatibility of the sensors, and evaluate the long-term stability of the entire system.

FIG. 3 .
FIG. 3. (a) Side view showing overall measurement architecture for feedback control of mirror shape.(b) End view showing measurement through the beam to comply with the Abbe principle.(c) Error ε resulting from twist or tilt θ and sensor offset a.

FIG. 5 .
FIG. 5. Disposition of metrology frame and mirror.The metrology frame sectioned to show only one row of sensors.
FIG. 6.(a) Overview of measurement setup in the orientation required to present it to a horizontal axis Fizeau interferometer.(b) Photograph with setup shown mounted on a tip/tilt mount for alignment to the Fizeau interferometer.Setup shown rotated away from the Fizeau interferometer to show metrology frame details.

FIG. 7 .
FIG. 7. Details of sensor mounting and Fizeau beam access.Detailed view shows interface between the sensor and the metrology frame.

FIG. 11 .
FIG. 11.(a) Repeatability of difference with mean subtracted.(b) Repeatability as quantified by the standard deviation as a function of sensor location.Sensor 10 is located at the centre of the mirror.Open circles represent data points, and the lines in between are only provided as an aid to the eye.
at the sensor locations (indicated by the open circles) over five measurements.Lines connecting the markers in the difference data are for visualization only and do not represent actual information about the mirror between adjacent points.Table

TABLE I .
Comparison of properties of candidate materials for the metrology frame.

TABLE II .
Summary of differences between Fizeau and sensor array.

TABLE III .
Uncertainty sources and average contributions.a conservative estimate.Both sources contribute nonuniformly along the mirror length with maxima at the ends and the center of the mirror.The resulting 2σ uncertainty contributions due to the deformation of the metrology frame and the Fizeau transmission flat are 0.48 nm and 0.34 nm, respectively. provide