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Abstract
Deflectometric slope profiler is an essential technique for accessing the surface metrology of mirrors used in synchrotron radiation beamlines. To increase the upper spatial frequency bandwidth limits of deflectometric slope profiler, reducing the beam spot size on the mirrors is necessary. In this paper, we introduce a profiler system: the focusing long trace profiler (FLTP). It contains a newly developed optical head capable of raising upper spatial frequency bandwidth limits by using a focused beam instead of a collimated beam to scan the sample. This feature has been proven in a numerical simulation experiment, where a spatial resolution of up to around 0.05 mm was reached when the sample is set at focus plane. The system is implemented and characterized in several experiments; calibration of the focusing optical head shows that it can achieve a high angular accuracy of sub-50 nrad root-mean-square (rms) and defocusing of sample under test (SUT) has no effect on the measurement results; the measurement tests also demonstrate the system’s advantage in highly curved mirror profile metrology.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Both advanced synchrotron radiation (SR) [1] and X-ray free-electron laser [2] sources have facilitated the development of X-ray optics with sub-nm or 50-nrad root-mean-square (rms) accuracy to preserve the high quality of beamline light. Full characterization measurement of the X-ray mirror is extremely important for both high-quality fabrication and beamline performance evaluation. High spatial resolution and accuracy are critical for dedicated metrology techniques or equipment because of the surface-error frequency characteristic. In the power spectral density (PSD) plot of a certain surface, the surface errors are typically divided into three regimes: high (roughness), mid (ripples or waviness), or low spatial frequency (form or figure errors) [3,4]. These errors which are characterized by their wavelengths on the mirror surface, make specific contributions to the system performance. For example, physical optics propagation simulations [5,6] have shown that the specifications of X-ray mirrors are highly dependent on the mirror PSD and X-ray beamline applications. Thus, measurements to evaluate the surface quality of X-ray optics must be performed over a wide spatial frequency range.
Commercially available interferometers with high lateral resolution and accuracy are widely used in mirror surface metrology. For X-ray optics, microstitching interferometer and relative-angle-determinable stitching interferometer have been developed. The difficulty of the system development includes the stable stage, sophisticated absolute measurement considering gravitational effects [7], and systematic error in the stitching process. One-dimensional stitching interferometry based on angular measurement was developed to correct the tilt error introduced by the stage in the stitching process [8]. Based on the redundancy of the captured subset data, the BNL group [9] proposed self-calibration stitching algorithms to estimate and correct repeatable high-order additive systematic errors. The alternative is the deflectometric surface profile measurement system that provides long trace and absolute measurements. Such systems have been used to develop a precise standard for flatness that is traceable in absolute terms [10,11].
In the past decades, different slope- or angle-measuring instrument designs, long trace profiler (LTP), and nanometer optical component measuring machine (NOM), have been developed and constructed in most SRs worldwide. In a typical LTP [12], a laser beam from the optical head (an angular measuring device) is used to scan the sample. The pencil-beam interferometer principles were adopted, and the slope or angle was recovered by fitting the dark area of the interference fringe pattern using a second-order polynomial [13]. As a second-generation slope measuring profiler, the NOM [14] utilizes a commercial electronic autocollimator (AC) with a scanning penta-prism to inspect the surface under test (SUT). An aperture of 2 mm was ensured to limit the light beam size. The system can measure a flat or curved surface with a radius down to ∼ 5m with ≤50 nrad rms accuracy or in the sub-nm range. To reduce the difficulty of calibration for systematic error caused by changing path length and tilt angle, a slope-measuring profiler with two autocollimators (AC) serving as a reference and sample beams and the sample-beam AC maintaining a fixed distance from the mirror [15,16] were developed. However, these instrument designs cannot provide surface information errors for spatial frequencies from 1 mm-1 to 10 mm-1 owing to the convolution effect of the beam spot size.
In this study, we investigated a newly developed surface slope metrology system to improve the lateral spatial resolution. In contrast to previous efforts, we propose a novel optical head that generates a focusing light beam rather than a collimated light beam for sampling the surface under test. Additionally, a Fourier-transform (FT)-based algorithm yielding a significantly high positioning accuracy is employed. We show the proof of the high spatial resolution concept via a numerical simulation experiment by measuring the chirped surface shape that covers a broad spatial frequency bandwidth. We also demonstrate the accuracy of the system through two experimental tests: (1) characterization of the optical head by cross-comparison with a commercial high-precision goniometer that indicates the main systematic error of the system; (2) surface profile metrology measurement that validates the performance of the integrated system. This work improves the spatial resolution of deflectometric slope profiler to the level of laser interferometer while maintaining high measurement accuracy. It can be crosschecked with other surface metrology instruments that have matched frequency band. We believe that this work will be of great significance for high accuracy absolute measurement [17].
2. Groundwork concepts and design
2.1 Principle of focusing optical head
Similar to other deflectometric surface profile measurement systems, the key instrument of the system is the optical head, a type of optical head. The basis of the focusing optical head proposed in this paper is essentially the F-theta principle. Figure 1 shows a simplified model of the designed optical head.
Fig. 1. General analysis model of deflectometric angle sensor. When ${L_1} = {L_3} = f$, it is corresponding to the autocollimator.
This model integrates both the illumination and detection functions along the same optical axis without considering physical realization. It is obvious that this model can also be applied to two other angle sensors: the AC ($\textrm{ }{L_1} = {L_3} = f$[see Fig. 1 for definitions]) and LTP optical head ($\textrm{ }{L_1} = {f_1};\textrm{ }{L_3} = {f_2})$, where the illuminating and detecting parts use different lenses. In the illuminating optical path, a ray with height ${x_0}$ and angle ${u_0}\; $ traverses the lens and illuminates the SUT. In the detection process, the deflected ray is collected by the lens and intersects the detection plane. Using the paraxial-ray tracing equation, the ray position on detector x can be expressed as
For tilt angle variation of $\varDelta \theta $ by the SUT, the ray position shifts by
that is identical to the optical head in LTP or electronic AC. It is obvious that the spatial resolution is improved if SUT is placed at the image plane of the source. This is the basic concept of the design in this study. In addition, two important conclusions can be drawn for profiler applications from these two equations. First, the spot position has no correlation with the SUT position relative to ${L_2}$; thus, the angle sensor is insensitive to the unavoidable defocus during curved surface scanning. The second is that the ray direction u0 in the front focal plane (the aperture location) is only determined by the source and detector plane. If the position of the source is stable, we can remove the issue of instability of the laser pointing error encountered in LTP by inserting an aperture to limit the beam in the detector plane. There are two main systematic errors. The first is the slope-profiler carriage pitch error that reduces the effectiveness of the calibration process. The second is the error of ${L_3}$. When ${L_3} = f$ is not perfectly fulfilled and ${L_3} = ({1 + \mathrm{\xi }} )f$ where $\mathrm{\xi }$ stands for a small quantity, the Eq. (3) will becomeIt is shown that the system’s angular response is related to the path length if the error of ${L_3}$ exists. When the SUT is set in the focusing plane (${L_2} = 2f$), the relative error is $3\xi \mathrm{\Delta }\theta $. If $f$= 300mm and the error of ${L_3}$ is kept below 0.1mm, the relative error will be less than 0.1%.
2.2 System setup (profiler)
Figure 2 shows the lab-built metrology system that is similar to a conventional LTP system for synchrotron mirrors. The system comprises a motion platform and two different deflectometric angle optical heads: an electronic autocollimator (AC) and a lab-built optical head. The motion platform (Q-sys, Netherlands) is a granite base plate with a fixed granite bridge and a machined aluminum carriage supported and guided by air bearings. The carriage carries the optical head for SUT scanning. The motion position and pitch errors are 2 µm and 6 µrad peak-to-valley (PV), respectively. An Elcomat HR external electronic AC (Moller-Wedel, Germany) on the riser combined with a mirror (D65, Moller-Wedel, Germany) on the carriage is used to monitor the optical head movement pitch error. The calibration accuracy is up to 50 nrad for a 10-µrad small angular range.
Fig. 2. (a) Surface slope metrology system in optical metrology laboratory (Beijing Synchrotron Radiation Facility; BSRF). (b) Detailed schematic of developed optical head with focused beam sampling of SUT. Dashed yellow and solid blue lines indicate the illumination light path and deflected light path for detection, respectively.
A fiber-coupled diode laser outputs divergent light at a wavelength λ = 637 nm. The fiber mode field diameter and numerical aperture are 4.5 µm and 0.12, respectively. Following the reflection by the mirror and a cubic beam splitter prism, the light is transmitted to the center of a chromatic lens with a focal length f = 300 mm. This significantly decreases the spherical aberration, for a source in the paraxial domain. Most importantly, an aperture with a small hole is placed at the lens front focal plane to limit the spot size on the lens and project a clear spot on the detector. The light converges after the lens and illuminates or samples the SUT with a certain spot size. The deflected light is collected by an achromatic lens and recorded by a SCA1400-17gm (Basler, Germany) charge-coupled device (CCD) camera in the front focal plane after transmission via the beam splitter. The frame acquisition speed is ∼50 Hz for the selected ∼500$\textrm{ } \times \textrm{ }$500 region of interest (ROI). Similar to the electronic AC, a perfect image of the aperture is obtained and recorded. The image position shift is related to the angular variation of the SUT. According to the following analysis, there is no limit on the SUT position, provided that the deflected light can be collected by the lens. The minimum sampling spot size is obtained once the SUT is placed on the optical fiber pigtail port conjugate plane, corresponding to the highest spatial resolution. Considering the diffraction limit caused by the aperture and Fourier lens, the spot size or resolution of the system is as small as tens of microns. In addition, a Fourier-transform-based positioning algorithm is adopted to ensure beam-spot positioning accuracy. According to the Fourier transform shift theorem [18], a shift in the spatial domain corresponds to a linear phase term in the frequency domain. We consider ${I_1}(x )$ and ${I_2}(x )= {I_1}({x + \varDelta x} )$ as the detected light spot intensity functions before and after the light spot shift, respectively. The phase shift is $\varDelta \mathrm{\varphi } = 2\mathrm{\pi }\varDelta x{f_x}$ where the ${f_x}$ represent a certain spatial frequency of the light spot intensity functions in x direction; therefore, we can recover the position shift by fitting the phase shift with a first-order polynomial equation in the frequency domain, according to
According to the focusing feature of the optical head, our new profiler system is called the Focusing Long Trace Profiler (FLTP). In the following part, we will abbreviate FLTP.
3. Spatial frequency response characteristics
First, we conducted a simulation experiment to investigate the spatial resolution of the new profiler by measuring its modulation transfer function (MTF). The simulation model is based on physical optics. The system parameter in Section 2 is adopted here. Referring to the work of Advanced Light Source [19–21], a chirped sample whose slope profile has a constant amplitude and increasing frequency and covers a specific frequency band in a limited length is selected as the sample surface. Therefore, the frequency bandwidth characteristics of the measurement system can be obtained from a single-scan measurement. The one-dimensional slope equation of the chirped sample used is
where A0 is the amplitude, ω1 and φ make up the argument of the sine function and determine the frequency increasing rate and initial phase. To fully cover the working frequency band of the new profiler in a limited domain, the argument is designed as a third-order polynomial. In this sample, the values of the parameters in Eq. (6) are A0 = 100 µrad, ω1 = 0.004 mm-3, and φ = 1 mm. A scanning step size of 0.004 mm is used. The intrinsic slope of SUT and profiler measurement result are shown in Fig. 3.Fig. 3. (a) Slope profile of chirp sample formed by Eq. (6) and its measurement result in scanning simulation. (b) PSD curves of (a), with horizontal axis transformed into spatial resolution. (c) Square root of the ratio of PSD in (b), regarded as the MTF curve of measurement system.
The measurement result presents different performances in different spatial frequency domains, as shown in Fig. 3(a). In the low-frequency domain, the measured data matches the real slope profile. As the spatial frequency increases, the shape is maintained, but the amplitude is scaled. Near the position where u = 42 mm, the measurement result scales to an extremely small value and gradually becomes distorted until it is entirely reverse-phase (see the enlarged view). This attenuation process is clearly evident, particularly in the frequency domain, as depicted in Fig. 3(b), where the PSD curves of the measurement result and real slope profile are plotted. Figure 3(c) shows the ratio of the measured value to the intrinsic value. The horizontal axis is transformed into a spatial period, corresponding to the changing period of the chirped slope profile. Due to the reverse-phase phenomenon, surface areas with spatial frequency greater than 20 mm-1 cannot be measured, implying that the spatial resolution is limited to around 0.05 mm. There is no random error introduced in the simulation. As a result, the actual spatial resolution of FLTP will be greater considering the signal-to-noise ratio in real measurements.
The curve showed in Fig. 3(c) can function as a correction factor curve. In actual measurements with the profile unknown, slope information whose spatial frequency is lower than the cut-off frequency, can be reconstructed by data processing with corresponding correction factors. The correction factors are real numbers without phase information. In Fig. 4(a), we correct the slope in Fig. 3(a) with the correction factor curve. The results show that the loss of phase information hardly affects the correction performance. The same correction factor curve (after interpolation) is also used to correct another chirped sample with different parameters in Fig. 4(b) and obtain a satisfactory result. This proves that the correction factor curve represents an inherent feature of the profiler, and it can function on the reconstruction of any profile within the spatial frequency bandwidth.
Fig. 4. (a) Same sample slope in Fig. 3(a) reconstructed (red) by correction factor in Fig. 3(c). Anti-phase phenomenon still exists. (b) A new sample whose slope expression is α(u) 0.0001·sin[0.05·2π·(u+7)2] reconstructed by interpolated correction factor in Fig. 3(c).
4. Experiment verification
4.1 Optical head characterization
The lab-built optical head performance was first checked by considering the ability of precise angle measurement and influence of defocusing. The optical head was supported by a lab jack, and the sample mirror was a λ/20 mirror (Edmund, USA) with a Polaris mirror mount (Thorlabs, USA). The entire system was placed inside a hutch on a vibration-isolated table. The entire optical path was shielded using paper tubes to reduce the air-convection-induced disturbance of the optical path.
The optical head accuracy was measured by computing the difference between the angles generated by a high-precision goniometer KTG-15d (Kohzu, Japan) and those measured using the optical head that is also related to the calibration process. The step angle variation had ≤25-nrad resolution. Figure 5(a) and 5(b) show the results for the angle ranges of 30 and 400 µrad, with 1.12-µrad and 24.2-µrad step sizes, respectively. To ensure a stable environment, these measurements were performed within a short time period of ∼20 min. The five-round calibration repeatability for both cases was ∼20 nrad, close to the stability measurement result shown in Fig. 5. The flat calibration curve in Fig. 5(a) exhibits the deviations for different rotation angles of ∼20 nrad rms and 60-nrad PV, indicating that the optical head functions efficiently to measure a flat mirror with a slope error < 50nrad rms. Similar to other surface profilers, such as the interferometers, LTP, or NOM, the ≤2 µrad systematic error shown in Fig. 5(b) is significant, owing to the imperfect optics in the optical head.
Fig. 5. Calibration data for optical head X-axis obtained for sampling steps of: (a) 1.12 and (b) 24.2 µrad. For each case, the standard deviation from the multiple round measurement is also plotted in the lower graph, indicating the calibration repeatability.
Finally, the optical head's angular response and spot images was also evaluated at a variety of positions. Spot images from experimental measurements and numerical simulations are shown in Fig. 6. The left two columns are images of spot on the SUT. Their 1-D intensity distributions are plotted in middle column. The right two columns are images of spot on the detector. It is apparent that the spot on the detector remains unchanged while the sample’s position changes along the optical axis. The simulation program based on physical optics is the same as the one used to measure the MTF of FLTP in Section-2. The 1-D intensity distribution comparison in Fig. 6 shows the consistency between simulation and actual results. Angular response data from both experimental measurements and numerical simulations are presented by the box plot in Fig. 7. The original data is the ratio of given tilt angle of SUT (within 10 arc sec range with step size 1 arc sec) and measurement result when the SUT is put in different positions ranging from the focal plane to the 200 mm defocus plane. It is shown that the angular response of the focusing optical head was linear. For experimental measurements and numerical simulations, the nominal ratio difference is approximately 3% and 0.4%, respectively. The results demonstrate that the measurement is not sensitive to the sample position as predicted in Section-2's theoretical analysis. The disparity is due to the optical head's alignment error. This results in a departure of the recorded curvature radius, but a slight change in residual error due to the tested mirror's small sag height. The following test established this effect.
Fig. 6. Comparison of spot images on the sample and detector when the sample is at a variety of focusing or defocusing positions in numerical simulation and experimental measurement.
Fig. 7. The angular response of the focusing optical head to the sample at different positions. (a) experimental measurements and (b) numerical simulations are used to generate the data.
4.2 Curved mirror surface shape metrology
In this section, we mounted the optical head on the motion platform (Fig. 2(a)) and performed surface profile metrology. To suppress the drift caused by the instable environment, eight-round measurements with a forward–backward–forward scanning direction order were performed. For each sampling point, 20 frames images are averaged to reduce the noise of the system, and the averaged time cost is about 3s.
To show the high spatial resolution and curved shape measurement capabilities, the first measurement was performed on a commercial spherical Mirror-A with curvature radius of 0.5m as shown in Table 1. Given the 10 mrad range of the optical head, the scan length is 5mm with step size of 0.05 mm, which corresponds to the 6.494µm sag height. Each scan took around 10 minutes. The residual profile of the mirror is shown in Fig. 8, where the height profile in Fig. 8(b) was obtained by integrating the measured slope profile in Fig. 8(a). In each graph, the difference from the mean values was also plotted. The repeatability of the measurement was about 150 nrad RMS which was significantly less than profile variation of 5.07 µrad RMS. Because to the small step, the height repeatability was as high as 0.02 nm RMS as shown in Fig. 8(d). To determine the error budget of the measurement, a cross check measurement using a Fizeau interferometer with a sampling pitch of 25 µm was also carried out. To reduce the effect of the different sampling resolution, interferometer data were smoothed using a three-point averaged method. The height difference of 0.62 nm RMS was obtained. The curvature radius of the mirror was about the 491mm which is also in the range of the error specification of the mirror.
Fig. 8. Measured mirror profile obtained based on the new surface slope profiler. (a) Measurement-trace slope profile and (b) height profile reconstructed via integration. differences between the individual slope/topography scans and the mean slope/topography are shown below. The residual error plots for slope(c) and height(d) were also presented.
Table 1. Parameters of the mirrors under test
In the second experiment, we chose a long tangentially cylindric mirror with curvature radius of 40 m as the specimen. The mirror was fabricated by ion beam figuring (IBF) process by the Zeiss company. To minimize the gravity deformation, the mirror was mounted on three-Bessel points. The trace of the measurement is 120 mm with step size of 0.2 mm, which corresponds to a 1.5mrad angular range. We shifted the mirror from focal plane to the 50 mm defocus plane. The residual slope and height profile are depicted in Fig. 9. The shape measured by a flag-type surface profiler (FSP) with traditional optical head using collimated beam is also given. The reproducibility of the residual slope/height error between the two setups is 0.05 µrad rms / 0.47 nm rms, which is less than the criterion for curved mirrors used in synchrotron radiation applications. Due to the different optics and optical path in the optical head, the discrepancy may reflect the systematic error of the metrology system. This measurement shows that the FSP has the flexibility to work at varying spatial resolutions, which is extremely important for balancing speed and spatial resolution in the scanning process.
Fig. 9. Measured mirror profile for Mirror-B by FSP and FLTP. (a) Measurement-trace slope profile and (b) height profile reconstructed via integration.
5. Discussions
Both simulation and experiments have proved that the new profiler based on focusing deflectometry is sufficiently practical and advanced for flat mirrors. It is worth noting that the system can provide an extremely high height resolution dh that is related to the measured slope α by dh = αdx, where dx is the spatial resolution. For the system reported in this paper, slope measurement accuracy α of sub-50 nrad and spatial resolution dx of 0.05 mm affords the height resolution dh of 0.0025 nm that is better than that of the interferometer. Therefore, it provides an easy method to check the interferometer performance.
With enhanced lateral resolution, increasing the sampling rate will facilitate curved mirror measurement. As curved mirrors usually have a small clear aperture, our system can acquire both measurement data with higher spatial resolution and higher accuracy in finite length. However, similar to the stitching interferometer, the stitching process is necessary, considering that the measurement range is limited by the detector.
Moreover, in angle measurement test, the FLTP system is demonstrated that it can project beams of various sizes by shifting the sample surface on the optical axis without affecting the measurement performance. This feature is extremely useful for long mirror characterization because it is time-consuming when a smaller spatial resolution (short step size) is required. Considering deformable mirrors such as bend mirrors and bimorph mirrors as examples, low spatial frequency profile features are the most interesting. A large scanning spot can facilitate improving the efficiency of the commissioning mirror system.
Finally, we want to discuss the measurement error that can be divided into random errors and systematic errors. Temperature instability is treated as the main random error source and has been proven to be under control. For systematic errors that are caused by the imperfection of optical components (e.g. lens, prism) and lateral beam shift, there are large differences between the flat and curved mirrors. For the flat mirror, the deflected beam on the lens and prism shifts by an extremely small distance that is a 12-µm shift for 10-µrad slope variation. Compared with the 8-mm beam on the lens, the 12-µm lateral shift is extremely small and causes an extremely small variation in the systematic error due to the convolution effect. For a curved mirror, a systematic error is unavoidable for a large lateral beam shift. In addition to the calibration of the system, because the system is simple and the systematic error is repeatable, the sources of systematic error can be found and corrected by numerical simulation using a commercial raytrace program [22].
6. Conclusion
The FLTP, a newly lab-built surface slope profiler that can achieve a high spatial resolution has been reported, in which the primary innovative factor is its optical head that can provide a focusing beam. In comparison to the collimated beam scheme wherein the autocollimator or optical head of LTP is used, the focusing beam can improve the spatial resolution while maintaining the signal-to-noise ratio. The MTF of the FLTP calibrated by the numerical simulation of scanning chirped sample demonstrates an ultimate spatial resolution of roughly 0.05 mm. In the laboratory, the profiler system was implemented and proved that the optical head has an accuracy of sub-50 nrad in the calibration experiments. We also established through experimental measurements and numerical simulations that the response of the optical head is not sensitive to the position of the mirror under test. The FTLP’s performance is verified by performing the metrology on two curved mirrors with curvature radius of 500 mm and 40 m, respectively and cross-checked with the interferometer and FSP. With improved spatial resolution and elimination of the necessity for the reference mirror, this new profiler obviously holds promise for freeform surface profile metrology. Additionally, in the same system configuration, the system is capable of low spatial resolution measurement if we adjust the sample position on the optical axis to obtain a larger beam. This feature enables the profiler system in improving the commissioning efficiency of X-ray mirror systems, such as bending or active mirrors.
Funding
National Natural Science Foundation of China (11505212, 11875059); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2019012).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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