Repeatability analysis of one-dimensional angular-measurement-based stitching interferometry

Lei Huang; Josep Nicolas; Mourad Idir

doi:10.1364/OE.26.020192

1. Introduction

Mirrors are key optical components for synchrotron radiation facilities or free electron lasers in delivering the X-ray beam to the experimental station. The surface qualities of these mirrors directly affect the X-ray beam quality in terms of intensity and phase distributions.

Mirror metrology plays a significant role in either fabrication or inspection. Nowadays, the major techniques for synchrotron mirror metrology include the pointwise scanning techniques such as Long Trace Profiler (LTP)/Nanometer Optical component measuring Machine (NOM)/Nano-accuracy Surface Profiler (NSP) [1–4], the stitching interferometry [5–8], e.g. MSI [6] and RADSI [7, 8], and the stitching Shack-Hartmann [9, 10]. The LTP/NOM/NSP and stitching shack-Hartmann can give access to the mirror surface shape based on the numerical integration of the measured slope. Stitching interferometry measures the height values directly in each subset and provide the entire mirror height map via a stitching process.

One-dimensional (1D) Angular-measurement-based Stitching Interferometry (ASI) [11] was recently proposed for synchrotron mirror metrology. The ASI instrument mainly consists of an angular measuring device to tackle the stage motion error during the scan and an interferometer to capture the height distribution at each scanning position. In this work, a study on the measurement repeatability of the current 1D ASI system is conducted giving a better understanding of this system and further possible improvement. Both theoretical analysis and simulations with experimental signals are carried out to identify the dominating error source of the current ASI system and possible ways to make potential improvement.

2. Fundamental of 1D ASI

The 1D ASI mainly consists of an angular measuring device to measure the pitch error of the translation stage at each scanning step while the interferometer is acquiring the height data. The overlapping height data of the Surface Under Test (SUT) from the interferometer are used in conjunction with the tilts of each local profiles and then corrected before the piston adjustment. This kind of stitching process is called Angle Measuring Stitching (AMS).

The interferometer measures the local profile on the SUT as illustrated in the sketch of the 1D ASI system in Fig. 1. The angular measurement device records the pitch error of the translation stage during scanning. The measured pitch angle α is used to correct the tilts of the interferometer data φ (x). The tilt error introduced by the stage pitch t can be calculated from the measured pitch angle α as t = − tan (α). Now, the tilt-corrected profile of the nth subset h_n (x) can be calculated by correcting the local tilt from the interferometer data φ_n (x) as

h_{n} (x) = φ_{n} (x) - t_{n} \cdot x .

Fig. 1 The 1D ASI setup consists of an angular measurement device (e.g. an autocollimator with a flat mirror mounted on the stage) and an interferometer measuring local profiles of the SUT.

Download Full Size | PDF

The tilt-corrected height data h_n (x) are adjusted with their corresponding piston values by taking the average height difference of two adjacent subsets within their overlapping region $\bar{Δ h_{n}} = \bar{h_{n - 1} (x) - h_{n} (x)}$ . All subsets of the height data can be stitched one by one via the piston adjustment.

z_{n} (x) = h_{n} (x) + \bar{Δ h_{n}} .

Finally, the stitched subsets are merged into a single piece. A weight function is usually used within the overlaps for data merging to preserve continuity at the overlap boundaries.

3. Considerations on the repeatability of 1D ASI

In order to analyze the repeatability of 1D ASI, the AMS technique is compared with a similar data processing technique: Integration Of Slope (IOS) [12], which is widely used in wavefront sensing [13] and deflectometry [14, 15]. We will demonstrate these two techniques are almost equivalent in profiling the global shape. Therefore, the IOS theory can be used to study the ASI results and predict its performance in terms of repeatability.

3.1. Similarity between AMS and IOS

Slope metrology such as LTP/NOM/NSP is another well-established technique for synchrotron mirror inspection. In order to get a height profile of the SUT, the measured slope data need to be converted into height values using a dedicated IOS technique. Here we will show that the AMS and IOS are almost equivalent in reconstructing global profiles. The slope metrology for synchrotron mirror is typically carried out by only scanning the tangential slope values along a single line. The measured slope s can be expressed as

s = z^{'} + e_{s} + e_{r},

where

z^{'} = \frac{\partial z}{\partial x}

stands for the first derivative of z along x-direction, e_s is the repeatable systematic error, and e_r is stands for the random error including the low-frequency drift error and the random slope noise. As a more general IOS solution, the study on 2D wavefront reconstruction has been studied since 1980s [13]. This 1D scanning of the one-directional slope simplifies the classical Southwell’s algorithm in [13] into a relaxed 1D format. Since there is no additional constraint from other directions in 1D IOS, the error terms in Eq. (3) can directly propagate into the height results. The relation between height and its first derivative is expressed as

z_{B} - z_{A} = \frac{{z^{'}}_{A} + {z^{'}}_{B}}{2} Δ x + O (Δ x^{3})

, where z_A and z_B are the height values at location A and B as shown in Fig. 2(a), z′_A and z′_B are the first derivatives at A and B, respectively. Δx is the space between two adjacent slope samples. The truncation error of the Southwell’s expression is O (Δx³). Hence, as shown in Fig. 2(a), the connections between the unknown height values and the measured slopes can be established as

z_{B} - z_{A} \overset{!}{=} \frac{s_{A} + s_{B}}{2} Δ x,

where s_A and s_B are the measured slope values at location A and B. The symbol

\overset{!}{=}

stands for “being equal in a least-squares sense”, mainly for the two-or three-dimensional IOS.

Fig. 2 The slope and height values are usually connected by a finite difference expression in IOS (a), while the relations between tilt-corrected profiles are established by the piston calculated inside overlapping region in AMS (b).

Download Full Size | PDF

On the other hand, the tilt-corrected height profiles of two overlapping subsets “a” and “b” in the AMS of the 1D ASI as illustrated in Fig. 2(b) can be expressed with polynomials as $z_{a} (x) = \sum_{n = 0}^{\infty} a_{n} x^{n}$ and $z_{b} (x) = \sum_{n = 0}^{\infty} b_{n} x^{n}$ , respectively. After a rigid transformation introduced by the stage motion, the height difference within the overlap should theoretically contain linear terms only, i.e. z_a (x) − z_b (x) = (a₁ − b₁)x + a₀ − b₀, however, due to the measurement errors, such as retrace errors of the interferometer [16, 17], the higher-order terms still exist in the height difference, which means $z_{a} (x) - z_{b} (x) = \sum_{n = 0}^{\infty} (a_{n} - b_{n}) x^{n}$ . The AMS results at locations A and B which are the centers of pieces “a” and “b” can be expressed as $z_{A} = \sum_{n = 0}^{\infty} a_{n} {(- \frac{Δ x}{2})}^{n}$ and $z_{B} = \sum_{n = 0}^{\infty} b_{n} {(- \frac{Δ x}{2})}^{n} + \bar{Δ z}$ , where $\bar{Δ z} = \bar{z_{a} (x) - z_{b} (x)}$ is the average height difference in the overlapping distance d and it will be the piston adjustment amount for the piece “b”. $\bar{Δ z}$ can be calculated via

\begin{matrix} \bar{Δ z} = \frac{\int_{- d / 2}^{d / 2} z_{a} (x) - z_{b} (x) d x}{d} \\ = \frac{\sum_{n = 0}^{\infty} \frac{a_{n} - b_{n}}{n + 1} x^{n + 1} |_{- d / 2}^{d / 2}}{d} \\ = \sum_{n = 0}^{\infty} \frac{a_{n} - b_{n}}{n + 1} {(\frac{d}{2})}^{n}, n = 0, 2, 4 \dots \end{matrix}

The height increment from location A to B across the overlapping region is

\begin{array}{l} z_{B} - z_{A} = \bar{Δ z} + \sum_{n = 0}^{\infty} b_{n} {(\frac{Δ x}{2})}^{n} - \sum_{n = 0}^{\infty} a_{n} {(- \frac{Δ x}{2})}^{n} \\ = (a_{1} + b_{1}) \frac{Δ x}{2} + \sum_{n = 2}^{\infty} \frac{a_{n} - b_{n}}{n + 1} {(\frac{d}{2})}^{n} + \sum_{n = 2}^{\infty} b_{n} {(\frac{Δ x}{2})}^{n} - \sum_{n = 2}^{\infty} a_{n} {(- \frac{Δ x}{2})}^{n} . \end{array}

In a practical measurement with d ≤ Δx, d² ≤ Δx² ≪ Δx and the coefficients a_n and b_n (n ≥ 2) are commonly much smaller than a₁ and b₁. Therefore the height increment z_B − z_A can be approximated as

z_{B} - z_{A} \approx \frac{a_{1} + b_{1}}{2} Δ x .

It is not difficult to realize the coefficients of the linear terms a₁ and b₁ in Eq. (7) can be considered as the slopes measured by ASI and they are essentially of the same meaning as s_A and s_B in Eq. (4). Comparing Eq. (7) and Eq. (4), the height reconstruction in AMS and IOS are very similar to each other. Figure 3 demonstrates an example of comparable reconstructions with the same ASI experimental data in either height (by AMS) or slope (by IOS), respectively.

Fig. 3 Similar global height profiles (c) with comparable repeatability (d) can be obtained by the AMS with subset profiles after tilt correction (a) or by the IOS with the corresponding slopes (b).

Download Full Size | PDF

Since the AMS and IOS are almost equivalent and comparable in reconstructing the global surface figure, we utilize the theory of IOS to study the repeatability of the ASI under different measurement conditions.

3.2. Theory of IOS repeatability

For an slope metrology instrument running in a stable environment, the repeatability of slope measurement is mainly determined by the measurement noise n_s of the slope measuring instrument. The RMS of the slope noise σ_s within a scan length L is defined as

σ_{s}^{2} = \frac{1}{L} \int_{- L / 2}^{L / 2} {| n_{s} (x) |}^{2} d x .

Here we have assumed the mean value of n_s (x) is zero. According to Parseval’s theorem, the RMS of the slope noise σ_s can be re-organized in terms of its Fourier transform as

σ_{s}^{2} = \frac{1}{L} \int_{- \infty}^{\infty} {| F_{s} (u) |}^{2} d u = \frac{1}{L} \int_{- \frac{N}{2} Δ u}^{\frac{N}{2} Δ u} {| F_{s} (u) |}^{2} d u .

where F_s(u) is the Fourier transform of the slope noise n_s (x), N is the sampling number, and

Δ u = \frac{1}{L}

is the spectrum resolution.

If we assume the slope measurement noise is zero-mean and uncorrelated, i.e. with a constant amplitude in the spatial frequency domain |F_s(u)| = A ≥ 0, we can simplify Eq. (9) as

σ_{s}^{2} = \frac{1}{L} \int_{- \frac{N}{2} Δ u}^{\frac{N}{2} Δ u} {| F_{s} (u) |}^{2} d u \approx \frac{1}{L} \int_{- \frac{N}{2} Δ u}^{\frac{N}{2} Δ u} A^{2} d u = \frac{N Δ u}{L} A^{2} = \frac{N}{L^{2}} A^{2} .

The constant amplitude A ≥ 0 is consequently calculated as $A = \frac{σ_{s} L}{\sqrt{N}} = σ_{s} Δ x \sqrt{N}$ . According to the property of Fourier transform for integration, the relation between the Fourier transform of the height noise F_z(u) and slope noise spectrum F_s(u) is $F_{z} (u) = \frac{F_{s} (u)}{i 2 π u}$ . Since F_s(u) is constant, we have the following expression for the spectral density of the height noise introduced by the slope measurement noise.

| F_{z} (u) | = \frac{| F_{s} (u) |}{2 π | u |} \approx \frac{A}{2 π | u |} = \frac{σ_{s} Δ x \sqrt{N}}{2 π | u |} .

The RMS value of the height noise σ_z can be expressed as

\begin{array}{l} σ_{z}^{2} = \frac{1}{L} \int_{- \frac{N}{2} Δ u}^{\frac{N}{2} Δ u} {| F_{s} (u) |}^{2} d u \\ = \frac{1}{L} {(\frac{σ_{s} Δ x \sqrt{N}}{2 π})}^{2} \int_{- \frac{N}{2} Δ u}^{\frac{N}{2} Δ u} {| u |}^{- 2} d u . \end{array}

The term within the integral sign is symmetric with respect to the origin, so the we can integrate from 0 to $\frac{N}{2} Δ u$ . In addition, the integral is improper, as it diverges at zero. This is because of the undetermined constant term associated to integration. In our case, we assume it to be zero, so we start integrating at the lowest non-zero spatial frequency within our sample, as follows

σ_{z}^{2} = \frac{2}{L} {(\frac{σ_{s} Δ x \sqrt{N}}{2 π})}^{2} \int_{Δ u}^{\frac{N}{2} Δ u} u^{- 2} d u .

The integral is analytically solved, and we get

σ_{z}^{2} = \frac{2}{L} {(\frac{σ_{s} Δ x \sqrt{N}}{2 π})}^{2} (\frac{1}{Δ u} - \frac{2}{N Δ u}) .

After some simplifications, and applying that the frequency resolution is the inverse of the sample length, i.e. $Δ u = \frac{1}{L}$ , we have

σ_{z}^{2} = \frac{σ_{s}^{2} Δ x^{2} (N - 2)}{2 π^{2}} .

Finally, the height noise RMS σ_z can be rewritten as the following form, if we primarily use the sampling step Δx and and the scanning length L in analysis.

\begin{matrix} σ_{z} = \frac{\sqrt{2} σ_{s}}{2 π} \sqrt{Δ x (L - 2 Δ x)} \\ \approx \frac{\sqrt{2} σ_{s}}{2 π} \sqrt{Δ x L}, when L ≫ 2 Δ x . \end{matrix}

Equation (16) indicates the height repeatability σ_z is proportional to the slope repeatability σ_s. In addition, when the spatial sampling step Δx is fixed, σ_z is almost proportional to $\sqrt{L}$ . When the scanning length L is determined, sigma_z is almost proportional to $\sqrt{Δ x}$ . It is noteworthy that these analytical results are based on the assumption that the slope noise is zero-mean and uncorrelated noise with a constant amplitude, which can be considered as ideal measurement situations within a stable measurement environment.

3.3. Study on the ASI repeatability with IOS theory

In order to study the measurement repeatability of the current ASI system, we utilize 24-hour experimental stationary autocollimator (AC) readings and white light interferometer (WLI) tilt data shown in Fig. 4 to simulate numbers of repeating ASI scans on a perfectly flat mirror. The objective of the WLI is 2.5× and the field of view is about 5.6 mm × 4.2 mm. Similar to real ASI measurements, the AC readings are averaged during each WLI acquisition. The WLI takes about 7 seconds to get one tilt result, so the AC readings are averaged over 7 seconds in order to match the real ASI measurement. The SUT slope is calculated by subtracting the AC data from the tilt data from WLI. System drifts in the temperature readings, AC readings and WLI tilt values are observed in Fig. 4.

Fig. 4 Real temperature, AC angle and WLI tilt data in a 24-hour stationary test in our current ASI measurement environment. AC readings are averaged over 7 seconds.

Download Full Size | PDF

Assuming 15 mins per scan, these 24-hour real data can be divided into 96 ASI scans. The mirror slopes are calculated from the AC readings and WLI tilt data, and the slope discrepancies from their mean values are shown in Fig. 5(a). With the current ASI configuration, more than 210 mm can be scanned in 15 mins. The average slope repeatability of these 96 scans is 30 nrad RMS [see Fig. 5(b)].

Fig. 5 By using the real 24-hour stationary data, 96 sets of 15 mins virtual scans are simulated: (a) slope discrepancies of these 96 repeating measurements, (b) slope repeatability of each scan, (c) spectrum of the slope profiles, (d) amplitudes of the lowest frequency component, (e) height discrepancies, and (f) height repeatability of each scan.

Download Full Size | PDF

Figure 5(c) shows the Fourier spectrum of the slope profiles. The low-frequency components are generally stronger than the high-frequency ones. Figure 5(d) displays the amplitudes of the lowest frequency term in Fig. 5(c) of these 96 scans. By using the IOS technique, the resultant height discrepancies from their mean value of these 96 scans are shown in Fig. 5(e) while the height repeatability of each scan is displayed in Fig. 5(f) with an average RMS value of about 0.31 nm. The height repeatability in Fig. 5(f) shows strong correlation with the amplitudes of the lowest frequency term in Fig. 5(d). The low frequency drift error is the major factor dominating the height repeatability values. The simulation matches our real experimental scans. Figure 6 shows a set of 70 real ASI scans of a 200-mm-long flat mirror. This flat mirror has been measured by other instruments, such as Fizeau interferometer and NSP, and the shape error is about 2 nm PV. The average height repeatability of these 70 ASI scans is about 0.31 nm RMS.

Fig. 6 The repeatability of 70 actual ASI scans. (a) Height discrepancies from the mean value, and (b) the height repeatability in RMS.

Download Full Size | PDF

In addition to the 24-hour real stationary experimental data and WLI tilt data of the current ASI system [see Fig. 7(a), same as Fig. 4], some virtual signals with little drift [Fig. 7(b)] are simulated by removing the best fitted polynomials from the real AC and WLI data. Furthermore, virtual data with no drift [Fig. 7(c)] are calculated by simulating AC and WLI data with a white Gaussian noises with the same RMS values as the signals in Fig. 7(b). These 3 sets of data in different conditions are used to study the repeatability of the current ASI system and predict its potential improvement.

Fig. 7 Virtual scans by using (a) real experimental data, (b) virtual data with little drift, and (c) virtual data with no drift. Their corresponding height repeatability for a 15 mins scans (d)–(f) and 30 mins scans (g)–(i), respectively.

Download Full Size | PDF

As expected, Fig. 7 clearly demonstrates that the height repeatability σ_z becomes better once the drift error reduces from the real data case [Fig. 7(a)] to the virtual data with little drift [Fig. 7(b)] and achieving its limit with the virtual data with no drift [Fig. 7(c)].

As predicted by Eq. (16), with a fixed step size Δx, the height repeatability σ_z will become worse if the scanning range increases. Comparing Figs. 7(d)–7(f) with Figs. 7(g)–7(i), if the scanning time increases from 15 mins to 30 mins, i.e. the scanning range increases from 210 mm to 420 mm with the current scanning speed, the height repeatability σ_z will be larger in all simulated conditions. In fact, to scan a longer distance L with a certain speed, a longer scanning time is expected, and therefore, both measurement accuracy and repeatability will be affected more by the drift errors, though we focus on the measurement repeatability in this work.

Figure 8 shows the trends of the height repeatability increasing with the scanning time or distance. The repeatability results in simulation with the real data are much worse comparing to those of the virtual data with much less drift. It obviously reveals that the drift error is a major factor which dominates the current ASI measurement repeatability. For a short time (<15 mins) scan, our experimental record is very close to the simulation result. As illustrated in Fig. 8, once the system drift error is reduced to a negligible level, we may start to see the trend described by the analytic expression in Eq. (16). This requires the high-precision metrology instrument working in an extremely stable measurement environment.

Fig. 8 The average height repeatability is getting larger along with the scanning time increasing under different drift conditions. The scanning speed is based on the current ASI speed.

Download Full Size | PDF

4. Discussion

From the study on the theory of IOS repeatability and virtual scans composed with the real stationary data, it reveals that low-frequency drift errors in the current ASI system are dominating the repeatability of the stitching results. In our current setup, for an ASI scan which takes >15 mins per scan (about 210 mm), the influence by the low-frequency slope drift is much more severe. The best condition for our existing system is to limit the scanning range to less than 210 mm to achieve a 0.5 nm RMS level in repeatability.

A more stable environmental condition will make the ASI measurement more precise mainly for mirrors longer than 250 mm. The other possibility is to reduce the acquisition time and speed up each motion and measurement during the scanning. For the same scanning length, a higher scanning speed (including motion and acquisition) enables a shorter scanning time, which will reduce the influence from the system drift errors and end up with a more repeatable stitching results.

After the drift error is controlled down to a certain level, we may start to consider the possible improvement by changing the step size Δx according to Eq. (16). Of course, there is a trade-off between speed and repeatability.

5. Conclusion

This study is focused on the repeatability of ASI measurement. The similarity of AMS and IOS are analyzed in theory and demonstrated with real measurement data. We derive the repeatability of IOS under zero-mean uncorrelated noise assumption. The ASI measurement repeatability is simulated by using the real stationary signals. The simulation results and analytic expression indicate that the drift error is the main factor dominating the repeatability of the current system. For short scans (<15 mins per scan) the drift influence on the repeatability is not so severe, but if the scans with the current ASI system become longer than 15 mins per scan, the drift has more significant impact on the stitching results. After reducing the signal drift or speeding up the scan, other adjustment, such as changing the step size, may become more meaningful for more repeatable height results via the stitching process.

Funding

Department of Energy (DOE) Office of Science (DE-SC0012704).

Acknowledgments

This research used resources of the National Synchrotron Light Source II, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under Contract No. DE-SC0012704.

References and links

1. P. Z. Takacs, S.-n. Qian, and J. Colbert, “Design of a long trace surface profiler,” Proc. SPIE 0749, 59–64 (1987). [CrossRef]

2. F. Siewert, T. Noll, T. Schlegel, T. Zeschke, and H. Lammert, “The nanometer optical component measuring machine: a new sub-nm topography measuring device for x-ray optics at BESSY,” AIP Conf. Proc. 705(1), 847–850 (2004). [CrossRef]

3. S. Qian and M. Idir, “Innovative nano-accuracy surface profiler for sub-50 nrad rms mirror test,” Proc. SPIE 9687, 96870D (2016). [CrossRef]

4. S. Qian and B. Gao, “Nano-accuracy measurement technology of optical-surface profiles,” Proc. SPIE 9687, 96870E (2016). [CrossRef]

5. A. Vivo, B. Lantelme, R. Baker, and R. Barrett, “Stitching methods at the European Synchrotron Radiation Facility (ESRF),” Rev. Sci. Instruments 87(5), 051908 (2016). [CrossRef]

6. K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Ueno, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, “Microstitching interferometry for x-ray reflective optics,” Rev. Sci. Instruments 74(5), 2894–2898 (2003). [CrossRef]

7. H. Mimura, H. Yumoto, S. Matsuyama, K. Yamamura, Y. Sano, K. Ueno, K. Endo, Y. Mori, M. Yabashi, K. Tamasaku, Y. Nishino, T. Ishikawa, and K. Yamauchi, “Relative angle determinable stitching interferometry for hard x-rayreflective optics,” Rev. Sci. Instruments 76(4), 045102 (2005). [CrossRef]

8. H. Yumoto, H. Mimura, T. Kimura, S. Handa, S. Matsuyama, Y. Sano, and K. Yamauchi, “Stitching interferometric metrology for steeply curved x-ray mirrors,” Surf. Interface Analysis 40(6–7), 1023–1027 (2008). [CrossRef]

9. M. Idir, K. Kaznatcheev, G. Dovillaire, J. Legrand, and R. Rungsawang, “A 2D high accuracy slope measuring system based on a stitching Shack Hartmann Optical head,” Opt. Express 22(3), 2770–2781 (2014). [CrossRef] [PubMed]

10. L. Huang, J. Xue, and M. Idir, “Controlling X-ray deformable mirrors during inspection,” J. Synchrotron Radiat. 23(6), 1348–1356 (2016). [CrossRef] [PubMed]

11. L. Huang, J. Xue, B. Gao, and M. Idir, “One-dimensional angular-measurement-based stitching interferometry,” Opt. Express 26(8), 9882–9892 (2018). [CrossRef] [PubMed]

12. L. Huang, M. Idir, C. Zuo, K. Kaznatcheev, L. Zhou, and A. Asundi, “Comparison of two-dimensional integration methods for shape reconstruction from gradient data,” Opt. Lasers Eng. 64, 1–11 (2015). [CrossRef]

13. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70(8), 998–1006 (1980). [CrossRef]

14. L. Huang, J. Xue, B. Gao, C. Zuo, and M. Idir, “Zonal wavefront reconstruction in quadrilateral geometry for phase measuring deflectometry,” Appl. Opt. 56(18), 5139–5144 (2017). [CrossRef] [PubMed]

15. W. Li, T. Bothe, C. v. Kopylow, and W. P. O. Juptner, “Evaluation methods for gradient measurement techniques,” Proc. SPIE 5457, 300–311 (2004). [CrossRef]

16. C. B. Kreischer, “Retrace error: interferometry’s dark little secret,” Proc. SPIE 8884, 88840X (2013). [CrossRef]

17. H. Yiwei, X. Hou, Q. Haiyang, and W. Song, “Retrace error reconstruction based on point characteristic function,” Opt. Express 23(22), 28216–28223 (2015). [CrossRef] [PubMed]

Repeatability analysis of one-dimensional angular-measurement-based stitching interferometry

Abstract

1. Introduction

2. Fundamental of 1D ASI

3. Considerations on the repeatability of 1D ASI

3.1. Similarity between AMS and IOS

3.2. Theory of IOS repeatability

3.3. Study on the ASI repeatability with IOS theory

4. Discussion

5. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (8)

Equations (16)

Optics Express