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Abstract
X-ray mirrors with single-digit nanometer height errors are required to preserve the quality of ultra-intense photon beams produced at synchrotron or free electron laser sources. To fabricate suitable X-ray mirrors, accurate metrology data is needed for deterministic polishing machines. Fizeau phase-shifting interferometers are optimized to achieve accurate results under nulled conditions. However, for curved or aspheric mirrors, a limited choice of reference optic often necessitates measurement under non-nulled conditions, which can introduce retrace error. Using experimental measurements of a multi-tilted calibration mirror, we have developed an empirical model of Fizeau retrace error, based on Zernike polynomial fitting. We demonstrate that the model is in good agreement with measurements of ultra-high quality, weakly-curved X-ray mirrors with sags of only a few tens of microns. Removing the predicted retrace error improves the measurement accuracy for full aperture, single shot, Fizeau interferometry to < 2 nm RMS.
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
According to the Maréchal criterion [1], a wavefront is said to be diffraction limited after reflection from a mirror if the root-mean-squared (RMS) phase error is $< \lambda /(14\; \times 2\sin \theta )$, where λ is the wavelength of light, and $\theta$ is the angle of incidence of the mirror. To preserve the quality of ultra-low emittance synchrotron light and free electron laser (XFEL) sources, (λ ∼ 0.1 nm and θ ∼ 0.3 mrad), hard X-ray mirrors need height errors < 1 nm RMS [2]. Manufacturing such extreme optics requires high-quality metrology data to provide accurate feedback for deterministic polishing machines [3,4]. This makes it vitally important to minimize metrology feedback errors which will limit the final production quality and performance of the X-ray optic.
Many types of interferometer are based on the principle that light rays reflected from the reference and test surface follow the same optical path back to the detector. When this condition is satisfied, the interferometer is said to be nulled. Under such conditions, the path difference between the rays is simply the relative height difference between the two optical surfaces. In this case, many systematic errors cancel out. However, the reference and test surface often cannot be nulled, due to a limited choice of reference or compensator optic. Under a non-nulled geometry, rays from each surface take paths of different length and direction back through the optical components of the interferometer. Systematic measurement errors are introduced because aberrations in the optical system no longer cancel out. This phenomenon is known as retrace error [5,6], as illustrated in Fig. 1. To be specific, “true” retrace effects scale with the cavity length, whilst effects related to the imaging system do not (e.g., coma observed as a function of tilt angle). However, for brevity, we will refer to the combination of both types of errors as “retrace”.
Fig. 1. Abstracted visualization of the Fizeau experimental apparatus. When the interferometer’s reference optic (R) is not matched in curvature to the surface under test (SUT), the reflected light rays take different paths back through the optical system, creating so-called retrace error. Reducing systematic errors to < 5 nm is essential for accurately measuring X-ray mirrors used at synchrotron and XFEL facilities.
A popular method of minimising retrace error is stitching interferometry [7,8], whereby the optic is sequentially translated and / or pitched, and measured in numerous, overlapping, sub-aperture regions. After acquisition, the set of sub-apertures are stitched together to create a composite “panoramic” image of the full optical surface. Prior to each measurement, the current sub-aperture is nulled relative to the reference optic using motorized stages. This minimizes the amount of deviation from the null condition, and hence reduces the magnitude of retrace error. However, acquiring the larger number of sub-apertures can be time-consuming. Quantifying retrace error is also helpful to guide the number and overlap percentage of sub-apertures to suit each mirror under test.
Retrace error for interferometers is a well-studied topic, with numerous experimental and theoretical studies. Liu et al. [9] used ray-tracing to successfully model retrace error of a Twymann-Green interferometer for a sphere with a radius of curvature of R ∼ 2 m. Huang [10] derived general analytical forms of retrace error for Fizeau interferometry, which agreed with experimental data for a highly-curved sphere. Unfortunately, the accuracy of such predictions is not sufficient for our application, as contributions from aberrations within the imaging system were assumed negligible. It was recently shown by Wu et al. [11] that retrace error from imperfect internal optics can be significant when trying to achieve nanometer-level accuracy. Liu et al. [12], and Gappinger and Greivenkamp [13] used reverse optimization to calibrate non-nulled measurements to accurately predict retrace error. However, this requires knowledge of the internal optics of the interferometer, which are often unknown. It is therefore practical to use an empirical method, based entirely on measurement data. Shahinian et al. [14] proposed a method that reduced the RMS of the retrace error of a coherent scanning interferometer from 18 nm to 2 nm over a field of view (FoV) of a few millimeters. This was achieved by isolating the retrace error in the measurement data and finding a function that relates the Chebyshev polynomial decomposition of the retrace error to the angle of a flat, test mirror. In our study, we build upon these works to improve the accuracy to nanometer-levels for full aperture (∼150 mm diameter), single-shot, Fizeau interferometry.
2. Methodology
2.1 Experimental setup
A 150 mm diameter, super-flat calibration mirror (CM) (from ASML) with surface errors (without removal of power) of < 5 nm peak-to-valley (PV) was used to characterize the retrace error of a Zygo Verifire HDX Fizeau interferometer in the Optics Metrology Lab at Diamond Light Source [15]. The Fizeau’s transmission flat (from ADE Phaseshift) only has a surface quality of < 30 nm PV, but has been thoroughly characterized by averaging multiple measurements of several super-polished X-ray mirrors. After subtraction of the reference optic’s polishing errors, the residual height error improved to < 1 nm PV [8], as observed by sub-nm agreement in the absolute height profiles of several X-ray mirrors measured in two orientations (“A to B”, and rotated 180° in “B to A”). The CM was mounted as the “surface under test” on a motorized goniometer which translated and rotated it relative to the Fizeau beam. The experimental setup, as shown in Fig. 2, is mounted on a passively-damped, pneumatic optical table, surrounded by a thermal enclosure. To enable surface errors of the transmission optic and the CM to be removed from the subsequent scans, the CM was also measured using translation stitching, where each sub-aperture is measured under nulled conditions. This “flat-field” image was removed from all subsequent non-nulled scans to extract the retrace error.
Fig. 2. Experimental setup to characterize the retrace error of a 150 mm diameter Fizeau interferometer. The calibration mirror (CM) was measured at numerous combinations of pitch (rotation of mirror about Y axis) and roll (rotations of mirror about X axis) angles using a motorized goniometer. Retrace error was isolated from each single-shot measurement by comparison with a sub-aperture translation stitching measurement of the CM under nulled conditions.
Approximately 900 Fizeau measurements of the CM were acquired, each with the optic at a different pitch ${\theta _y}$ (rotation about the Y axis) or roll angle ${\theta _x}$ (rotation about the X axis). Angular ranges of ± 3000 µrad encompass the slope variation of most medium- and hard-energy X-ray mirrors used at Diamond. Figure 3 shows the point cloud of pitch and roll angles at which the CM was measured. Supplementary Materials show the evolution of the retrace error as the CM is pitched or yawed. These videos demonstrate that coma is the dominant contributor to the overall retrace error.
Fig. 3. Point-cloud distribution of ∼ 900 pitch (${\theta _y}$) and roll (${\theta _x}$) combinations used to measure a high-quality calibration mirror to characterize the retrace and imaging errors of the Fizeau interferometer. Visualization 1 and Visualization 2 show the sequence of measured retrace errors as the mirror is pitched or rolled.
To reduce the influence of slow, turbulent airflows within the optical cavity of the interferometer, each measurement was the average of ∼ 200 Fizeau images acquired over ∼ 3-minute periods. A low-pass filter was also applied to the height data to remove spatial wavelengths < 1 mm. To avoid edge effects, analysis was performed over the central 95% of the interferometer’s circular FoV. Retrace error was extracted from each measured interferogram by: removing a high-quality topography map of the CM; subtracting the surface errors of the transmission reference optic; and performing plane correction to remove tilt in the x and y directions.
2.2 Empirical model
Retrace error in each of the ∼ 900 measurements was represented by the sum of the first 25 Zernike polynomials in the Wyant index [16]:
To create an empirical model of the retrace error from Eq. (1), the amplitude coefficients ${c_i}$(${\theta _x}$, ${\theta _y}$) for each of the 25 Zernike polynomials were sequentially plotted against the discrete angle value of ${\theta _x}$ and ${\theta _y}$ at which the calibration mirror was measured. As ${Z_0}$ is the piston term, and ${c_0}$, defines the height offset of the image, relationships were only determined for ${Z_1}$ to ${Z_{24}}$. Fitting low-order polynomials to each relationship for ${c_i}$ (${\theta _x}$, ${\theta _y}$), as a function of ${\theta _x}$ and ${\theta _y}$, enabled interpolative calculation of the retrace error at any given angle within the FoV. Coma terms, ${Z_6}$ and ${Z_7}$ in the Wyant index, were the largest contributors (∼ 20% each) to the overall retrace error. Figure 4 shows linear (1st order polynomial) relationships between the 6th and 7th Zernike polynomial coefficients and the pitch and roll angles of the CM respectively, i.e., ${c_6} \propto {\theta _y}$, and ${c_7} \propto {\theta _x}$. These findings are in good agreement with the literature [5,14,17]. Higher-order polynomial fits (up to 4th order) were required to adequately map the measurements for non-coma Zernike amplitude coefficients.
Fig. 4. Fitting the measured retrace error of the Fizeau data with selected Zernike polynomials revealed linear relationships: between ${c_6}$, the Zernike coefficient for coma in the horizontal (coma H) and the pitch of the mirror (left image); and between ${c_7}$, Zernike coefficient for coma in the vertical (coma V), and the roll of the mirror (right image). Inset images in each plot show the shape of ${Z_6}$ and ${Z_7}$ respectively.
Aside from the coma terms, there were also significant contributions to the total retrace error from ${Z_1}$ (10%), ${Z_2}$ (10%), and the circularly-symmetric Zernike ${Z_3}$ (15%). Terms ${Z_8}$, ${Z_9}$, ${Z_{13}}$ and ${Z_{14}}$ each contributed ∼ 3% to the overall retrace error. As seen in Fig. 5, 6 and 7, many of the Zernike amplitudes exhibit non-linear relationships with pitch and roll. For example, as shown in Fig. 5, the relationship between Zernike amplitude ${c_3}$ is a tilted spheric with respect to the pitch and roll angles of the surface under test. We speculate that the tilt of the spheric is caused by small misalignment of the transmission flat relative to the direction of the Fizeau beam. Such relationships are largely oversimplified by low-order polynomial fitting. Despite this, the model still provides nanometer-level agreement with the measured retrace error. Further research is required to understand whether higher-order, small-scale structures in the Zernike coefficient plots are observed for optics with different reflectivity or curvature.
Fig. 5. 3D scatterplot showing the tilted spheric relationship between ${c_3}$, the amplitude coefficient of the 3rd Zernike, and the pitch and roll angles of the test mirror. The inclination and curvature of the relationship are not well understood. The mesh represents the low-order polynomial fit used for interpolation in our model.
Fig. 6. 3D scatterplot showing the inclined planar relationship between ${c_{13}}$, the amplitude coefficient of the 13th Zernike, and the pitch and roll angles of the test mirror. The inclination angle of the plane, and the observed bump at the centre of the map, are not well understood.
Fig. 7. 3D scatterplot showing the complicated relationship between ${c_8}$, the amplitude coefficient of the 8th Zernike, and the pitch and roll angles of the test mirror. The general relationship is a twisted plane, but the additional oscillations are not well understood.
Effectively, we are treating the interferometer as a black-box and measuring its output for a range of given inputs. Empirically-derived relationships extracted from the Fizeau data (i.e., the curved, twisted, or tilted planes shown in Fig. 5 to 7) are interpolated to predict the retrace error for an optic of any shape, given its slope profile (1st spatial derivative of height) and location within the FoV. Our model does not consider the influence of the local “curvature” (2nd spatial derivative of height) of the mirror as a function of x and y, or distance to the mirror. These topics could be considered in a future project.
3. Results: performance of Zernike model
The effectiveness of the empirically-derived model of the Fizeau’s retrace error from measurement of the CM was quantified by measuring two high-grade, weakly-curved, X-ray optics. The first optic was an uncoated silica, tangential cylinder (i.e., curved along the mirror’s length, but flat in the orthogonal, sagittal direction), with a length of 200 mm, a radius of curvature R of ∼ 95 m, and a sag of 52 µm. The second mirror was an uncoated silicon (single crystal), tangential ellipse (source to mirror distance p = 46 m, mirror to focus position q = 0.4 m, angle of incidence $\theta $ = 3 mrad), with a length of 160 mm, R ∼ 264.4 m at the centre of the mirror, and a sag of ∼ 12 µm. For each optic, the measured retrace error was approximated as the difference between a single shot, full-aperture image (which includes retrace error) and a stitched measurement of the same mirror using multiple, overlapped, small, nulled sub-apertures (minimal retrace but includes the shape and figure error of the optic). Based on the stitched data, the local pitch and roll angles of each mirror were calculated for every pixel in the Fizeau’s FoV. These angles (as a function of position x and y) were inserted into Eq. (1) to predict the retrace error for each optic. Predictions were then compared with the corresponding single-shot, full-aperture measurement.
3.1 Cylindrically curved mirror: purposefully pitched
To challenge the retrace model, the cylindrical mirror was purposefully pitched by 2500 µrad to induce a large retrace error. As seen in Fig. 8, the 2D empirical model predicts a retrace error of ∼ 83 nm peak-to-valley (PV) and 14.9 nm RMS, which is in good agreement with the measurement of ∼ 97 nm PV and 15.1 nm RMS. After removing the model’s predicted retrace error from the single-shot measurement, the 2D height residual was improved by a factor of five to ∼ 2.5 nm RMS. Three, 1D line profiles of the retrace error were extracted from the 2D plot. The 1D retrace errors, shown in Fig. 9, were 10.423, 10.277, and 12.012 nm RMS respectively. After removal of the model’s predicted retrace error, the 1D height residuals improve by a factor greater than five to 1.376, 2.245 and 1.893 nm RMS.
Fig. 8. 2D Fizeau retrace measurement (upper plot) and predicted retrace error (lower plot) of a cylindrical optic pitched by 2500 µrad. Both plots employ the same color scale. Three, extracted 1D line slices are shown in Fig. 9.
Fig. 9. Three 1D line profiles extracted from Fig. 8 show good agreement between the measured retrace error (blue curves) of a tilted cylindrical mirror and the prediction of the empirical model (orange curves).
3.2 Cylindrically curved mirror: at normal incidence
The cylindrical mirror (R ∼ 95 m) was next measured with its centre at normal incidence to the optical axis of the Fizeau. Figure 10 shows the interferogram to highlight the high density of fringes (∼ 70) within the FoV. For the pitched case in Section 3.1, there were several hundred fringes within the interferogram.
Fig. 10. 2D interferogram of the cylindrical mirror, as measured under normal incidence, showing a high density of fringes towards the edges of the field of view.
As seen in Fig. 11, the measured PV of retrace error was reduced by a factor greater than three compared to the tilted case (Section 3.1). However, the measured retrace error was still ∼ 25 nm PV and ∼ 3.2 nm RMS, which is significantly larger than the typical procurement tolerances for X-ray mirrors. After removing the model’s predicted retrace error from the measured value, the 2D height residual reduced to ∼ 2 nm RMS.
Fig. 11. 2D Fizeau retrace measurement (upper plot) and predicted retrace error (lower plot) of a cylindrical optic at normal incidence to the Fizeau beam. Both plots are on the same color scale. Three extracted 1D line slices are shown in Fig. 12.
Three, 1D line profiles of the retrace error were extracted from the 2D plot. As shown in Fig. 12, the 1D retrace errors were 2.537 nm, 1.532 nm, and 3.425 nm RMS. After removal of the model’s predicted retrace error, the 1D height residuals improved slightly to 1.302, 1.510, and 1.646 nm RMS.
Fig. 12. Three 1D line profiles extracted from Fig. 11 show agreement on the nm-scale between the measured retrace error (blue curves) of a cylindrical mirror and the prediction of the empirical model (orange curves).
To demonstrate the need to remove retrace error to achieve accurate Fizeau interferometry of high-quality X-ray mirrors, Fig. 13 shows the figure error of the cylinder (after subtraction of the same cylinder from all datasets) before and after removal of retrace error. Note that the measurement of the tilted cylinder (top left image) has a color bar scale of ±40 nm, whereas all other images have a smaller range of ±7.5 nm. The top right image demonstrates that the retrace correction is effective in revealing a more accurate representation of the optic’s figure errors.
Fig. 13. Fizeau measurement of the figure error of the cylindrical mirror (after removal of the curvature), before and after retrace correction. The upper row confirms that the retrace correction model is effective in revealing the “true” figure error of the mirror, even under extreme pitching. The lower row highlights that correction of retrace error is essential to accurately measure curved, X-ray mirrors with figure errors < 2 nm RMS.
3.3 Elliptically curved mirror
Finally, retrace error was investigated for an elliptically curved mirror. The weaker curvature of this mirror (R ∼ 264.5 m) creates a retrace error of only ∼ 7 nm PV, but this is still significant in accurately measuring state-of-the-art X-ray mirrors. As seen in Fig. 14 and 15, the predicted and measured retrace errors show reasonable agreement. By subtracting the predicted retrace from the measurement, the 2D residual errors were improved by ∼ 33% from 1.61 nm RMS to 1.07 nm RMS. Before subtraction of the retrace, the measured 1D lines profiles were 1.423, 1.543, and 1.608 nm RMS. After subtraction of the model’s prediction, the residual errors improved to 0.977, 0.614, and 0.786 nm RMS. The remaining residuals are likely due to measurement noise and high frequency errors that are difficult to represent with low-order Zernike polynomials. Despite this, the model can still reduce the retrace error of the ellipse line profiles to < 1 nm RMS.
Fig. 14. 2D Fizeau retrace measurement (upper plot) and predicted retrace error (lower plot) of a weakly curved, elliptical mirror at normal incidence to the Fizeau beam. Both plots are on the same color scale. Three, extracted 1D line slices are shown in Fig. 13.
Fig. 15. Three 1D line profiles extracted from Fig. 12 show agreement on the nm-scale between the measured retrace error (blue curves) of an elliptical mirror and the prediction of the empirical model (orange curves).
Ultimately, the limiting factors for our retrace model are the density of measured points (Fig. 3), noise in the Fizeau data, and low-order polynomial fitting of the Zernike coefficients. Our current model, based on a low-order polynomial fit over a coarse sampling grid of ± 3000 µrad will likely oversimplify any complex relationships, and exclude fine details such as the higher-frequency oscillations observed in Fig. 5 to 7. Similarly, in the case of the elliptical mirror, higher frequency terms observed in Fig. 14 are not recreated by the model. These features are likely caused by a small misalignment between the full aperture measurement and the stitched image. Although multiple Fizeau images were acquired to average away random noise and temporal drifts, any such errors will be incorporated into the Zernike model, thereby reducing its accuracy. To enhance the accuracy of retrace fitting for weakly-curved optics, a denser sampling of measurement points around the nulled point is recommended for future work to capture finer detail in the relationships between the Zernike coefficients and pitch and roll of the SUT. As shown in Table 1, the retrace error correction successfully reduce the residual errors for the 2D maps and the extracted 1D line profiles.
Table 1. Measured retrace error, before and after correction using the empirical model.a
4. Importance of removing retrace errors to achieve nanometer-level accuracy
Having demonstrated that predictions of the data-driven empirical model for retrace error are in good agreement with experimental measurements of weakly-curved X-ray mirrors, the model was then used to calculate retrace errors for a range of hypothetical mirrors with cylindrical radii varying between 9 m and 100 km. Predicted 1D retrace error profiles, as a function of the radius of the hypothetical mirror, are plotted in Fig. 16 (note, R < 100 m is not plotted to aid visibility of the retrace error for weaker curvatures). The corresponding RMS value for each curve is plotted in Fig. 17. Note that the retrace error for a mirror with a radius of 100 m is ∼ 5 nm RMS. This value drops to 3 nm RMS for R = 250 m. This highlights the importance of minimising retrace error to enable accurate Fizeau interferometry of weakly curved X-ray optics which typically require figure errors << 5 nm RMS. Retrace error minimization is achieved either by modelling, or by sub-aperture stitching of small, nulled regions.
Fig. 16. Model’s prediction of the retrace error as a function of radius of curvature (R > 100 m) for a range of hypothetical mirrors.
Fig. 17. Log-log plot of the retrace error RMS (of the profiles in Fig. 16) as a function of radius of curvature of the mirror. This demonstrates that retrace error needs to be considered even for weakly curved mirrors with figure errors of << 5 nm RMS.
According to our empirical model, the retrace error in Fig. 17 is predicted to asymptotically approach a value of ∼ 2.5 nm RMS, rather than the correct value of zero. This discrepancy is likely caused by insufficient calibration data recorded at small angles of pitch and roll, and slow-moving pockets of air turbulence which add noise into the measurement and model. However, this does not mean that accuracy limit of our model is 2.5 nm RMS, since we demonstrated 2D retrace errors < 1.5 nm RMS (see Table 1). Instead, the accuracy of the model is worsening at very small angles (between ± 5 µrad) for the reasons stated above. Future work could explore this narrow angular region to further improve the model.
5. Conclusions
We have developed an empirically-derived, Zernike-based, polynomial model of the systematic errors of a commercial Fizeau interferometer. Based on Fizeau measurements of a super-flat calibration mirror, oriented at multiple pitch and roll angles, we demonstrate that the model’s predictions of retrace error are in nanometer-level agreement with experimental measurements of two weakly curved X-ray mirrors. The model revealed a retrace error of ∼ 3 nm RMS for optics with a radius of curvature of 200 m, and an exponential increase for stronger curvatures. Calibration reduced retrace error to < 2 nm RMS for full aperture, single shot, Fizeau interferometry. This calibration procedure is essential to achieve the nanometer-level accuracy required for measuring state-of-the-art X-ray mirrors with polishing errors << 5 nm RMS. For sub-aperture stitching, the model also helps guide the size, number, and overlap percentage to ensure systematic errors remain below a desired level.
Funding
Science and Technology Facilities Council; Diamond Light Source.
Acknowledgments
This work was conducted during Katherine Morrow’s 3-month secondment to the Optics & Metrology group at Diamond Light Source. The Authors would like to thank Kawal Sawhney and Mark Roper for organizing the placement. We are grateful to Vivek Badami and colleagues at Zygo, and Ken Middleton at Lambda Photometrics for instructive discussions. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Disclosures
The authors declare no conflicts of interest.
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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