Absolute measurement of optical flat surface shape based on the conjugate differential method

Ya Huang; Jun Ma; Rihong Zhu; Caojin Yuan; Lei Chen; Huijuan Cai; Weiyuan Sun

doi:10.1364/OE.23.029687

1. Introduction

Optical interferometry is the most commonly used method to measure the plane mirror surface shape with high accuracy. The precision of the method is limited by the accuracy of the reference surface. As the quality accuracy requirement of optical flat surface shape improve continually even to nanometer or sub-nanometer, we need to obtain the absolute surface shape by excluding the effects of the reference surface and interferometric system.

The liquid surface can be considered as an ideal reference flat over hundreds of millimeters apertures since it have the same radius with earth [1–3 ]. However, it is still difficult to apply in practice for neither the stability nor the ability to eliminate the systematic error in the test results. From single line to the entire surface, traditional three-flat test [4] and the improved methods [5–12 ] gradually used in the absolute test of plane surface. All methods require three surfaces with similar high accuracy of surface shape and same diameter. The test procedures are very complex. It takes a long time for reconstructing constant temperature after adjustment. However, the result by three-flat method is considered as the criterion of the absolute surface shape of the surface under test.

The differential method (pseudo shear method) uses simple optical device and concise mathematical expression, which is firstly proposed by P. B. Keenan [13]. E. E. Bloemhof adopted the differential method to measure the absolute surface shape of optical flat using only two auxiliary measurements (small-amplitude lateral shifts of the test optic in each direction) without any extra element [14, 15 ]. Jun Ma et al improved the differential method to the conjugate differential method and used it to measure the absolute surface shape of the cone mirror [16]. Weihong Song and Dongqi Su et al used the shift-rotation method to measure the plane optical element [17, 18 ]. The approach may also be used to test other types of components such as the cylinder one.

In this paper the conjugate differential method is introduced to measure the absolute surface of flat mirror. It will be especially helpful to execute the absolute measurement for the large flat (or approximately flat) mirrors, including the ones used in the astronomical telescope (often with the diameter greater than 1m). The traditional three flats method cannot be used in these occasions. The differential data in two orthogonal directions can be measured, and then the absolute surface can be retrieved through several algorithms such as the Fourier transform, Zernike polynomial fitting and Hudgin model method. The potential errors caused by the measurement noise and adjustment are simulated to analyze the precision of this method, and the surface reconstruction algorithms are also compared to select the most suitable one.

2. Theoretical analysis

The conjugate differential method is an improvement of the differential method based on interferometric testing. To get the absolute surface shape, both of them need to obtain the differential data of the test surface but there is the significant difference. As shown in Fig. 1 , there are three interferometric measurements that represent the optical path difference (OPD) between the test surface in five different locations and the reference surface respectively in differential method. But there are four interferometric measurements. That is, with the central measurement result marked as $Φ (x, y)$ , holding the reference surface and move the test surface with a certain translation δ along the positive and negative directions respectively in two orthogonal directions. And then a series of measurement data $Φ (x - δ, y)$ , $Φ (x + δ, y)$ , $Φ (x, y - δ)$ and $Φ (x, y + δ)$ can be obtained in these locations. Different from the differential method which need three times measurements such as $Φ (x, y)$ , $Φ (x - δ, y)$ and $Φ (x, y - δ)$ , the conjugate differential method gives up the central measurement and adopt the data $Φ (x - δ, y)$ , $Φ (x + δ, y)$ , $Φ (x, y - δ)$ and $Φ (x, y + δ)$ .

Fig. 1 (a) Differential method and (b) conjugate differential method experimental process schematic diagram.

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The measurement result of initial position can be represented as

Φ (x, y) = ϕ_{t e s t} (x, y) + ϕ_{r e f e r e n c e} (x, y) + ϕ_{noise} (x, y) + ϕ_{adjustment} (x, y)

where

ϕ_{t e s t} (x, y)

is the test surface shape,

ϕ_{r e f e r e n c e} (x, y)

is the reference surface shape,

ϕ_{n o i s e} (x, y)

is the measurement noise, and

ϕ_{a d j u s t m e n t} (x, y)

is the adjustment error. Assuming the reference surface is fixed and the measurement time is short,

ϕ_{r e f e r e n c e} (x, y)

can be considered as the constants. Considering the noise and adjustment errors are usually small, these two terms are ignored here to simplify the expression [14,16 ].

So the measurement result of initial position can be written as

Φ (x, y) = ϕ_{t e s t} (x, y) + ϕ_{r e f e r e n c e} (x, y)

After translating the test surface along x direction and y direction respectively, we can obtain four measurements results.

\begin{array}{l} Φ (x - δ, y) = ϕ_{t e s t} (x - δ, y) + ϕ_{r e f e r e n c e} (x, y) \\ Φ (x + δ, y) = ϕ_{t e s t} (x + δ, y) + ϕ_{r e f e r e n c e} (x, y) \\ Φ (x, y + δ) = ϕ_{t e s t} (x, y + δ) + ϕ_{r e f e r e n c e} (x, y) \\ Φ (x, y - δ) = ϕ_{t e s t} (x, y - δ) + ϕ_{r e f e r e n c e} (x, y) \end{array}

The two orthogonal surface shape differences can be obtained by subtracting two measurement surfaces of the conjugate positions.

\begin{array}{l} Φ (x + δ, y) - Φ (x - δ, y) = ϕ_{t e s t} (x + δ, y) - ϕ_{t e s t} (x - δ, y) \\ Φ (x, y + δ) - Φ (x, y - δ) = ϕ_{t e s t} (x, y + δ) - ϕ_{t e s t} (x, y - δ) \end{array}

Absolute surface shape gradients of the surface under test can be obtained as follows.

\begin{array}{l} d w_{x} (x, y) = \frac{ϕ_{t e s t} (x + δ, y) - ϕ_{t e s t} (x - δ, y)}{2 δ} = \frac{Φ (x + δ, y) - Φ (x - δ, y)}{2 δ} \\ d w_{y} (x, y) = \frac{ϕ_{t e s t} (x, y + δ) - ϕ_{t e s t} (x, y - δ)}{2 δ} = \frac{Φ (x, y + δ) - Φ (x, y - δ)}{2 δ} \end{array}

The absolute surface shape $w$ of the surface under test can be obtained by differential wavefront reconstruction algorithms from $d w_{x} (x, y)$ and $d w_{y} (x, y)$ . In comparison, the two differential data in the traditional differential method are $d w_{x} (x - δ / 2, y)$ and $d w_{y} (x, y - δ / 2)$ respectively since the central surface data in Fig. 1 is used. That means there is a small offset between two differential data while not exist in the conjugate method. Therefore, it is the advantage of the conjugate differential method, although one more measurement is necessary.

The translation amount $δ$ is an important parameter for the conjugate differential method, so we need to synthetically evaluate the translation error of the motion devices. Although the influence of the same translation error is minor since $δ$ is large amount, the differential amount is also bigger so that the high frequency information will be blurred. However, since $δ$ is small amount, the noise and the other tiny errors will make significant effects. Considering the nominal limiting spatial resolution limited to the size of CCD camera pixels, the shifts should be recommended choosing the value corresponding to single pixel variation of interferometer camera [14,16 ].

Compared with the differential method, the conjugate differential method slips the initial measurement and choses two pairs of conjugate positions which can be considered to be independent with each other. It greatly reduces the sensitive to the mechanical drift during the measurement, which improves the accuracy compared with the differential method.

To get the absolute test surface shape from its gradients, the differential wavefront reconstruction algorithms play important roles and affect the final results directly. Currently, there are three primary differential wavefront reconstruction methods which are Fourier transform method [19,20 ], Zernike polynomial fitting method [19,21 ] and Hudgin model method [14,15,22 ]. To analyze the effect of the differential wavefront reconstruction algorithms, we will reconstruct the absolute surface shape from the same data in three methods above and give the comparison in the simulations and experiments.

3. Simulation

In order to verify the feasibility of the conjugate differential method, the simulated test surface A and reference surface B are generated by the random Zernike coefficients, which are shown in Fig. 2(a) and Fig. 2(b). The PV and RMS values of the test surface are 118.5 nm and 15.2 nm. The test results can be represented by the difference between A and B. Assuming B is immobile in the initial position, move A with single pixel in the four directions and obtain the two sets of conjugate measurement results of $Φ (x - δ, y)$ , $Φ (x + δ, y)$ , $Φ (x, y - δ)$ and $Φ (x, y + δ)$ . The surface gradients $d w_{x} (x, y)$ and $d w_{y} (x, y)$ of the test surface can be obtained by Eq. (4) and Eq. (5). The absolute surface can be obtained according to the differential wavefront reconstruction algorithms. Then the reconstruction results are merited with the deviations between the calculated results and the ideal ones. The measurement noise and adjustment error are also introduced in the simulations respectively to observe that whether they have significant affections on the result.

Fig. 2 The ideal (a) test surface and (b) reference surface (units: nm)

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3.1 Reconstructed results with no error

In the ideal case without any noise or adjustment error, reconstructed surface by conjugate differential method are shown in Fig. 3(a)-(c) . They are reconstructed from the conjugate differential data with Fourier transform method, Zernike Fitting method and Hudgin model method in sequence. The residual errors of these algorithms are merited by the deviation with the ideal test surface, which are shown in Fig. 3(d)-(f). In this situation, all three methods can reconstruct the test surface correctly, with the RMS value of retrieving errors all less than 0.2 nm, especially Zernike fitting method. So it can be seen that in the ideal case the major error is caused by the inherent error of the algorithms itself and the truncation error in computational procedure. The detailed calculated results are listed in Table 1 .

Fig. 3 Reconstructed surface by conjugate differential method with (a) Fourier transform method, (b) Zernike Fitting method and (c) Hudgin model method. (d), (e) and (f) show the residual errors corresponding to these three wavefront reconstruction methods (units: nm).

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Table 1. Wavefront reconstruction errors of the test surface (Ideal PV = 118.5nm and RMS = 15.2nm, the values showed with types of PV/RMS unit: nm)

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Reconstructed surface by differential method are also provided as shown in Fig. 4(a)-(c) . They are also reconstructed from the differential data by the three algorithms above, and the residual errors of these algorithms are shown in Fig. 4(d)-(f). With the same translation value (i.e. the shearing amount), it shows lower accuracy than the conjugate differential method by comparing the residual errors of same reconstructed algorithms. It is because that the offset between two differential data in the differential method is evaded in the conjugate one, as explained before. The detailed parameters and also listed in Table 1.

Fig. 4 Reconstructed surface by differential method with (a) Fourier transform method, (b) Zernike Fitting method and (c) Hudgin model method. (d), (e) and (f) show the residual errors corresponding to these three wavefront reconstruction methods (units: nm).

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3.2 Reconstructed results with noise

The CCD noise can be considered as a sort of random error. The random noise amplitude is set to 2 nm PV at first, and the residual errors between the reconstructed surface and the ideal test surface are shown in Fig. 5 with the RMS value less than 0.6 nm. Besides, their error trends with different noise amplitude are given in Fig. 6 , and it can be seen that their accuracies are still acceptable with the normal noise level. The Zernike fitting algorithm filters the noise error so the deviation is better suppressed. But on the other hand, it is potential to blur the high frequency information in the test surface. The operator can make the choice based on the measuring requirement. For example, if the major component of the test surface is the low frequency one and the noise level is high, the Zernike fitting algorithm will be more suitable than the other two algorithms.

Fig. 5 Residual errors of (a) Fourier transform method, (b) Zernike Fitting method and (c) Hudgin model method reconstructed by conjugate differential method with 2 nm noise (units: nm)

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Fig. 6 Residual errors trend curves with different random noise.

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3.3 Reconstructed results with translation error

Adjustment error is usually caused by the small undesirable offset of the adjusting device, such as the translation rail. Assuming that there is the adjustment error caused by 0.02mm translation deviation, which is close to the precision level of the normal translation stage, the reconstructed surface errors are shown in Fig. 7 . It can be seen that the accuracies of the three algorithms are still close in this condition. The results with different translation errors are included in Fig. 8 , where the performance of Zernike Fitting algorithm is a little better. It can be concluded that the accuracies decrease slowly with the translation error smaller that 0.1mm. This precision is easy to be satisfied with the translation stage, and it is notable that the tolerance will be even looser when the test optics is larger.

Fig. 7 Residual errors of (a) Fourier transform method, (b) Zernike Fitting method and (c) Hudgin model method reconstructed by conjugate differential method with 0.02 mm translation along x direction (units: nm).

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Fig. 8 Residual errors trend curves with different translation error along x direction.

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3.4 Reconstructed results with rotation error

The rotation error also can be introduced in the measurement by imperfect translation fixture, especially for the large mirror. In Fig. 9 , the surface errors of the three algorithms with the rotation error of 0.4° are presented, and the trend curves with different rotation error are also give in Fig. 10 . It can be seen that the results are still accurate if the rotation error is less than 1°, which is easy to ensure with the suitable mechanic construction.

Fig. 9 Residual errors of (a) Fourier transform method, (b) Zernike Fitting method and (c) Hudgin model method reconstructed by conjugate differential method with 0.4° rotation about z axes (units: nm).

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Fig. 10 Residual errors trend curves with different rotation deviation about z axes.

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3.5 Analysis and comparison

In this section, the conjugate differential method and the differential method are used to reconstruct the surface shape for comparison at first. Both of them can retrieve the surface correctly, but the conjugate method acquires better results for all the three reconstruction algorithms because it can evade the offset between the two orthogonal differential data.

Then the conjugate differential method is tested in the simulations with the errors of different kinds and amplitudes, and the three algorithms are used to reconstruct the surface from the differential data. Generally speaking, the conjugate method still works well under these error conditions with different reconstruction algorithms, therefore its stability is validated. Zernike fitting algorithm often acquires relatively better accuracy due to its robustness, which means it can be used as the first choice if the loss of high frequency information is acceptable. In comparison, if the high frequency components are important, Hudgin model algorithm can be selected since its precision is a little better and there is not the edge effects suffered in the Fourier transform algorithm.

4. Experiments

In the experiments a flat mirror is tested on a commercial Fizeau type interferometer. The test mirror is translated along four lateral directions to obtain the phase data with one pixel shift distance. As descript in Fig. 1, five phase data in different locations can be obtained marked as $Φ (x, y)$ , $Φ (x - δ, y)$ , $Φ (x + δ, y)$ , $Φ (x, y - δ)$ and $Φ (x, y + δ)$ , which are shown in Fig. 11 and $Φ (x, y)$ is not used in the conjugate differential method and the measurement result is hyalinized for caution. The absolute surface shape reconstructed by three differential wavefront reconstruction algorithms method are shown in Fig. 12 .

Fig. 11 Measurement results at five positions (the values showed in the textbox with types of PV/RMS and units of nm).

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Fig. 12 The absolute surface shape reconstructed by conjugate differential method with (a) Fourier transform method, (b) Zernike Fitting method and (c) Hudgin model method (units: nm).

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To verify the reliability of the conjugate differential method, we extract the 1D surface profiles at x = 0 from reconstructed surfaces and compare them with the surface shape obtained by the traditional three-flat method [4]. The contrary graph is shown in Fig. 13 , and the detailed PV/RMS values of all results are given in Table 2 . Through the comparison, we can clearly see that the results of the conjugate method accord well with the three-flat one.

Fig. 13 Surface shape result at x = 0 by conjugate differential method, differential method and traditional three-flat method.

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Table 2. PV and RMS of the surface shape at x = 0

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It also can be seen that the profiles reconstructed by the Fourier transform algorithm and Hudgin model algorithm are very close, where both the low order information and the high frequency one are retrieved well. The RMS values of the surface shape at x = 0 reconstructed by using Fourier transform method and Hudgin model method are very close. However, the Zernike fitting algorithm gives more smooth surface than the other two methods.

The surface profile at x = 0 and its PV/RMS value obtained by the differential method are also given in Fig. 13 and Table 2, where only the result reconstructed by the Hudgin model algorithm is used for the simplicity. Its shearing amount is 2 pixels here, which is the same as the conjugate method so they can be compared with the same standard. It can be deduced that the profile obtained by the conjugate differential method is more close to the differential one.

5. Conclusion

In this paper the conjugate differential method is introduced into the absolute flat mirror testing, which is accurate and easy to be implemented. The conjugate differential method is developed from the differential method and it enhances the precision of differential data and reduces the coupling between the test steps, so the reliability of the tests is improved. The absolute surface shape results of the conjugate differential method accord well with the result of the three-flat test in the experiment, where the deviation of RMS value is about 0.3 nm. In the simulations, several algorithms are used to reconstruct the surface from the differential data obtained by the conjugate method under different error conditions such as the random noise and adjustment error, where the results are still accurate so the stability is validated. Besides, the Zernike Fitting algorithm is generally more robust to reconstruct the surface from the differential data in the conjugate method deduced by the simulation and experiment results, while the Hudgin model algorithm can retrieve the high frequency surface information well. The conjugate method also can be used to measure the absolute surface shape of cylindrical mirror, cone mirror.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (NSFC) (61205162, U1231112), the National Key Scientific Instrument and Equipment Development Project (2013YQ150829) and the Province and University Education Cooperation Project of Yunnan province (2014IB017).

References and links

1. L. Rayleigh, “Interference bands and their applications,” Nature 48(1235), 212–214 (1893). [CrossRef]

2. R. Bünnagel, H. A. Oehring, and K. Steiner, “Fizeau interferometer for measuring the flatness of optical surfaces,” Appl. Opt. 7(2), 331–335 (1968). [CrossRef] [PubMed]

3. M. Vannoni and G. Molesini, “Validation of absolute planarity reference plates with a liquid mirror,” Metrologia 42(5), 389–393 (2005). [CrossRef]

4. G. Schulz and J. Schwider, “Precise measurement of planeness,” Appl. Opt. 6(6), 1077–1084 (1967). [CrossRef] [PubMed]

5. B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23(4), 379–383 (1984). [CrossRef]

6. C. Ai and J. C. Wyant, “Absolute testing of flatness decomposed to even and odd functions,” Proc. SPIE 1776, 73–83 (1992). [CrossRef]

7. C. J. Evans and R. N. Kestner, “Test optics error removal,” Appl. Opt. 35(7), 1015–1021 (1996). [CrossRef] [PubMed]

8. K. R. Freischlad, “Absolute interferometric testing based on reconstruction of rotational shear,” Appl. Opt. 40(10), 1637–1648 (2001). [CrossRef] [PubMed]

9. M. F. Küchel, “A new approach to solve the three flat problem,” Optik (Stuttg.) 112(9), 381–391 (2001). [CrossRef]

10. U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. 45(23), 5856–5865 (2006). [CrossRef] [PubMed]

11. C. Xu, L. Chen, and J. Yin, “Method for absolute flatness measurement of optical surfaces,” Appl. Opt. 48(13), 2536–2541 (2009). [CrossRef] [PubMed]

12. Z. Han, L. Chen, T. Wulan, and R. Zhu, “The absolute flatness measurements of two aluminum coated mirrors based on the skip flat test,” Optik (Stuttg.) 124(19), 3781–3785 (2013). [CrossRef]

13. P. B. Keenan, “Pseudo-shear interferometry,” Proc. SPIE 123(4), 2–9 (1984).

14. E. E. Bloemhof, “Absolute surface metrology by differencing spatially shifted maps from a phase-shifting interferometer,” Opt. Lett. 35(14), 2346–2348 (2010). [CrossRef] [PubMed]

15. E. E. Bloemhof, “Absolute surface metrology with a phase-shifting interferometer for incommensurate transverse spatial shifts,” Appl. Opt. 53(5), 792–797 (2014). [CrossRef] [PubMed]

16. J. Ma, C. Pruss, R. Zhu, Z. Gao, C. Yuan, and W. Osten, “An absolute test for axicon surfaces,” Opt. Lett. 36(11), 2005–2007 (2011). [CrossRef] [PubMed]

17. W. Song, F. Wu, X. Hou, W. Zhao, and Y. Wan, “Absolute measurement of flats with the method of shift-rotation,” Opt. Rev. 20(5), 374–377 (2013). [CrossRef]

18. D. Su, E. Miao, Y. Sui, and H. Yang, “Absolute surface figure testing by shift-rotation method using Zernike polynomials,” Opt. Lett. 37(15), 3198–3200 (2012). [CrossRef] [PubMed]

19. J. Nam and J. Rubinstein, “Numerical reconstruction of optical surfaces,” J. Opt. Soc. Am. A 25(7), 1697–1709 (2008). [CrossRef] [PubMed]

20. P. Bon, S. Monneret, and B. Wattellier, “Noniterative boundary-artifact-free wavefront reconstruction from its derivatives,” Appl. Opt. 51(23), 5698–5704 (2012). [CrossRef] [PubMed]

21. G. M. Dai, “Comparison of wavefront reconstructions with Zernike polynomials and Fourier transforms,” J. Refract. Surg. 22(9), 943–948 (2006). [PubMed]

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

Reconstruction method		Reconstructed surface	Residual error
Fourier transform method	Conjugate differential method	120.6 / 15.2	0.80 / 0.03
Fourier transform method	Differential method	119.8 / 15.0	1.24 / 0.15
Zernike fitting method	Conjugate differential method	119.8 / 15.1	0 / 0
Zernike fitting method	Differential method	120.0 / 15.0	0.83 / 0.14
Hudgin model method	Conjugate differential method	118.1 / 14.9	1.67 / 0.32
Hudgin model method	Differential method	117.3 / 14.8	2.51 / 0.46

Method	PV(nm)	RMS(nm)
Three-flat test	6.4	1.4
Conjugate differential Fourier transform method	4.9	1.3
Conjugate differential Zernike fitting method	4.2	1.1
Conjugate differential Hudgin model method	4.9	1.3
Differential Hudgin model method	5.4	1.2

Reconstruction method		Reconstructed surface	Residual error
Fourier transform method	Conjugate differential method	120.6 / 15.2	0.80 / 0.03
Fourier transform method	Differential method	119.8 / 15.0	1.24 / 0.15
Zernike fitting method	Conjugate differential method	119.8 / 15.1	0 / 0
Zernike fitting method	Differential method	120.0 / 15.0	0.83 / 0.14
Hudgin model method	Conjugate differential method	118.1 / 14.9	1.67 / 0.32
Hudgin model method	Differential method	117.3 / 14.8	2.51 / 0.46

Method	PV(nm)	RMS(nm)
Three-flat test	6.4	1.4
Conjugate differential Fourier transform method	4.9	1.3
Conjugate differential Zernike fitting method	4.2	1.1
Conjugate differential Hudgin model method	4.9	1.3
Differential Hudgin model method	5.4	1.2

Absolute measurement of optical flat surface shape based on the conjugate differential method

Abstract

1. Introduction

2. Theoretical analysis

3. Simulation

3.1 Reconstructed results with no error

3.2 Reconstructed results with noise

3.3 Reconstructed results with translation error

3.4 Reconstructed results with rotation error

3.5 Analysis and comparison

4. Experiments

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (13)

Tables (2)

Equations (5)

Optics Express