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Iterative correction method of a retrace error in interferometry

Chenhui Hu, Lei Chen, Donghui Zheng, Yuntao Wang, Zhiyao Ma, and Zhe Zhang

Open Access Open Access

Abstract

In interferometer measurements, the inconsistency of the optical range through which the reference and test lights pass introduces a retrace error in the phase measurement. In this study, we propose an iterative retrace error correction method in interferometry. A black-box model is established based on the linear and squared relationships between the retrace error and the tilt of the testing surface. The error correction phase is obtained using the least-squares method; thereafter, the global tilt is determined to iteratively correct the retrace error. The root mean square (RMS) of the residuals was > 3.2 × 10−5λ, >6.4 × 10−3λ, and >1.4 × 10−3λ in the simulation, experimentally computed retrace error correction in the planar measurement, and spherical measurement, respectively, proving that the retrace error can be effectively corrected.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

The interferometer works in the zero-fringe state; when the reference and test lights pass through the same optical path after being reflected through the transmission flat (TF), no retrace error [1] occurs in the measurement results. However, the working of interferometers at zero fringe is generally not guaranteed, especially in space carrier interferometers and non-null spherical tests, and the inconsistent optical path returned by the reference and test beams introduces a retrace error in the determination of the phase, which affects the measurement accuracy. Especially for high-precision measurements of ultra-smooth element surfaces, deviations in the quality of the collimating objective and imaging element in the interferometer optical path will amplify the retrace error of the light due to the steepness of the surface of the measured part, and ultimately affect the measurement results of the interferometer. Kinnstaetter [2] discussed the accuracy of the interferometer and noted that the aberration in the interferometer can cause fringe distortion and thus introduce a phase error. Liu [3] divided the retrace error into retrace element errors and retrace path errors; moreover, retrace path errors can be divided into retrace phase errors and retrace coordinate errors. The retrace error of non-null aspheric testing was analyzed using the ray-tracing method. Using empirical mapping of interferometer errors, Evans [4] proposed a correction method with aberration expansion and multiple measurements in advance for a tilted off-axis plane, reducing the error deviation from 150 nm to approximately 30 nm. The correction method is applicable to black-box systems; however, it is limited to low spatial frequencies and does not correct retrace element errors. Deck [5] Proposed a calibration method, calibration data that provide information related to the field- and orientation-dependent systematic errors of an optical profile are obtained using the optical profiler, in order to correct the errors. Shahinian [6] proposes a technique based on empirical data that can be used to correct the retrace error due to the non-null measurement of free-form surfaces. Using the third-order aberration theory, Murphy [7] proposed a calibration method. However, this method does not allow full error characterization of the system, and it becomes complicated when the imaging configuration has more than one single lens. Huang [8] applied the fourth-order aberration formula to study the retrace error and discussed various cases in which the aberration of the imaging system was combined with the surface of the test plate. Silin [9] proposed a Φ630 mm phase-shifted interferometer to compensate for the retrace error as a linear equation related to the inclination angle. In a perturbation method in the black-box model [10,11], the interferometer is described by its characteristic functions. He [12] used the point characteristic function method to simulate the wavefront of a Fieazu interferometer and investigate the retrace error with and without the element error. To correct the retrace error induced by the error value of the test part, Greivenkamp et al. first proposed the inverse optimization method involving optimally matching the data from the system model and experiment, and ray-tracing and optimizing the model several times [13,14]. However, the computation is complicated and the model changes with the interferometer structure. Notably, for the tracing model, system calibration is very important and has garnered considerable attention [3,15–17]. The aforementioned studies on the retrace error are based on the interferometer system model for ray tracing to compensate for the retrace error. The computation parameters need to correspond to the actual interferometer structural parameters and the computation is complicated. Similar to the other methods of determining the retrace error, the measured phase was considered as the ideal surface, the retrace element error caused by the measured surface error was not considered, and the retrace error correction was incomplete.

In this study, we propose an iterative correction method of the retrace error in the Fieazu interferometer plane and spherical measurements that can determine the error-corrected phase results from multiple phase results with retrace errors. The proposed method is based on the relationship between the retrace error and tilt angle of the measured object in the Fieazu interferometer. It establishes a system of equations to determine the retrace error correction results and iteratively updates the global tilt angle to determine the error correction phase and phase sensitivity factor by considering the retrace error caused by the surface tilt angle.

2. Principle

2.1 Retrace error of interferometric measurements

2.1.1 Retrace error in plane interferometry

In plane interferometry, owing to the tilt of the reflection flat (RF), the field of view of the reference and test beams after reflection through the different optical elements in the interferometer are not the same, resulting in varying wavefront aberrations of the two beams at the end. In this study, we consider the interferometer as a black box, and the wavefront aberration between the reference and test beams can be obtained according to the description of the Seidel aberration. When a slight deviation exists in the vertical axis distance Δx (the field of view angle is changed and the field of view angle = vertical axis distance/collimating objective focal length), Δx ≪ x0, the reference beam wavefront $\triangle {W_n}({{x_0}} )$ and the test beam wavefront $\triangle {W_n}({{x_0} + \triangle x} )$ interfere with the introduction of additional wavefront error. The wavefront error of the interferometer due to the tilt of the testing surface can be obtained according to the first three orders of the low-order wave aberration as [18]:

$$\begin{aligned} W({\rho ,\theta ,{x_0}} )&= \sum\limits_{n = 1}^5 {({\triangle {W_n}({{x_0} + \triangle x} )- \triangle {W_n}({{x_0}} )} )} \\ &\approx {W_{222}}({2\triangle x{x_0}} ){\rho ^2}{\cos ^2}\theta + {W_{131}}\triangle x{\rho ^3}\cos \theta \\ &+ {W_{220}}({2\triangle x{x_0}} ){\rho ^2} + {W_{311}}({3\triangle xx_0^2} )\rho \cos \theta \end{aligned}, $$
where ${W_{222}}$, ${W_{131}}$, ${W_{220}}$, and ${W_{311}}$ denote the wavefront aberration coefficients corresponding to astigmatism, coma, field curvature, aberration, and spherical aberration, respectively; x0 is the spatial position of the image plane, which corresponds to the spatial position of the parallel light focused after passing through the lens; and ρ and θ are the normalized pupil polar coordinate positions.

Equation (1) can be obtained from the component tilt introduced by the wavefront error and the imaging point location of the field of view angle, that is, the vertical axis distance, which has an approximately linear relationship. The resulting aberrations are mainly power, astigmatism, and coma. Therefore, the retrace error in planimetry is the internal optics of the interferometer for the introduction of additional aberrations caused by the tilted beam [19]. The results of Eq. (1) are consistent with the empirical model obtained by Evens [4] where the different aberrations are linearly related to the field-of-view angle (RF tilt). According to the above theory, the retrace error in laser planar interferometry is related to the relative tilt angle of the testing surface and the standard TF, and the larger the tilt angle, the more fringes are introduced and the larger the wavefront error.

2.1.2 Retrace error in spherical interferometry

In addition to the retrace errors caused by the aberration of the optical element of the interferometer and the surface error of the testing surface, the retrace error caused by the inconsistency of the reflected beam of the reference and test parts must also be considered in the spherical measurement.

According to the geometric optical path of the spherical reference mirror and the testing surface shown in Fig. 1, the expression for the retrace error due to the test wavefront and cavity geometry can be obtained as follows [20]:

$$OPD = 2 \cdot \overline {AH} - \overline {ABD} = 2 \cdot ({{R_1}\textrm{ + }{R_2}} )- 2\left( {{R_1}\cos (\alpha )\textrm{ + }{R_2}\cos \left( {{{\sin }^{ - 1}}\left( {\frac{{{R_1}\sin (\alpha )}}{{{R_2}}}} \right)} \right)} \right), $$
where α denotes the test wavefront tilt angle, R1 is the reference spherical lens radius, and R2 is the radius of the test spherical mirror.

 figure: Fig. 1.

Fig. 1. Schematic of spherical measurement of wavefront tilt.

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In this section, only the retrace errors caused by the test wavefront and cavity geometry are discussed along with the graphical errors caused by the reference wavefront surface errors. Assuming that the test wavefront reference surface is perfect, interferometer imaging is set to focus precisely on the surface under test, and the imaging surface is conjugated to the test piece position. Equation (2) can be further reduced to an approximation to obtain the relationship between the wavefront tilt angle α and optical range as follows:

$$OPD \approx {\alpha ^2}{R_1}({{{({\rho + 1} )} / \rho }} ), $$
where $\rho = {{{R_2}} / {{R_1}}}$.

Based on the optical path distance function in Eq. (3), the retrace error in the spherical measurement can be obtained, which is not only related to the radius of curvature of the testing surface but also approximately linearly related to the square of the wavefront tilt. The wavefront tilt and tilt of the testing surface are linearly related, whereas the theoretical retrace error is squared with the tilt of the testing surface.

2.2 Principle of iterative correction algorithm for retrace error

According to the previous analysis, the linear and quadratic relationships between the retrace error and the tilt of the testing surface in the interferometer plane and spherical measurements, respectively, can be obtained. Therefore, we consider the interferometer to be a black-box system model and establish a system of equations between the retrace error and the tilt of the measured object as follows:

$$\left\{ {\begin{array}{c} {\sum\limits_{n = 0}^q {\sum\limits_{m = 0}^n {{C_{{x^n}{y^m}}}\theta_{x1}^n\theta_{y1}^m = {\varphi_1}} } }\\ \vdots \\ {\sum\limits_{n = 0}^q {\sum\limits_{m = 0}^n {{C_{{x^n}{y^m}}}\theta_{xp}^n\theta_{yp}^m = {\varphi_p}} } }\\ \vdots \\ {\sum\limits_{n = 0}^q {\sum\limits_{m = 0}^n {{C_{{x^n}{y^m}}}\theta_{x3q}^n\theta_{y3q}^m = {\varphi_{3q}}} } } \end{array}} \right., $$
where ${C_{{x^0}{y^0}}}$ denotes the phase after correction of the retrace error, when n and m are not all zero, ${C_{{x^n}{y^m}}}$ is the sensitivity factor matrix of the corresponding interferometer system; and ${\theta _{xp}}$ and ${\theta _{yp}}$ are the global tilts in the measurement results (including the degree of tilt caused by the phase in the local area). The p = 1,2,3…,3q, the term q represents the last power of the tilt angle in the black-box model. In planar measurement, the retrace error obtained from Eq. (1) is linearly related to the tilt angle; therefore, q = 1. The sensitivity factor matrices in the measurement are ${C_{x1}}$ and ${C_{y1}}$. In the spherical measurement, according to the results of Eq. (3), a quadratic relationship exists between the retrace error and amount of tilt; therefore, q = 2. The sensitivity factor matrix in spherical measurement has ${C_{x1}}$, ${C_{y1}}$, ${C_{{x^2}1}}$, ${C_{{y^2}1}}$, and ${C_{xy}}$, a total of 5 terms.
$$\begin{array}{l} {\theta _{xp}} = {\theta _{gxp\_p}} + {\theta _{\; gxt\_p}}\\ {\theta _{yp}} = {\theta _{gyp\_p}} + {\theta _{gyt\_p}} \end{array}, $$
where ${\theta _{gxp\_p}}$ and ${\theta _{gyp\_p}}$ denote the amount of local tilt caused by the magnitude of power change due to the tilt of the measured part in x and y directions, respectively; in the plane measurement, ${\theta _{gxp\_p}} = {\theta _{gyp\_p}} = 0$, ${\theta _{\; gxt\_p}}$, and ${\theta _{gyt\_p}}$ represent the tilts of the testing surface in the measurement result corresponding to the tilt term of the Zernike fit in the measurement phase.

Because the above calculation is based on the premise that the testing surface is ideal, the inconsistency of the test beam return path is mainly caused by the global attitude of the testing surface and the component aberration in the interferometer. The retrace-path error generated by the global tilt can be determined and corrected, as described above. However, in practical applications, low- and medium-frequency information exists on the testing surface, and the local area tilt changes the return direction of the test beam, thus changing the optical path in the local area. This is not feasible for high-precision plane interferometry; therefore, the tilt of the local area of the measured surface needs to be determined and updated to the new tilt coefficient in real time, the retrace error caused by the local area corrected through several iterations, and the retrace error correction phase finally determined.

According to the retrace error correction equation, further correction is made to the inclination angle determination formula (5) in the retrace error computation equation, and the inclination angle is corrected according to the value of the inclination angle deviation determined from the obtained phase. Thus, the inclination angle correction equation is expressed as follows:

$$\begin{array}{l} \theta _{xp}^{(k )} = \frac{{({\varphi_0^{({k - 1} )}({x + 1} )- \varphi_0^{({k - 1} )}(x )} )\lambda }}{c} + {\theta _{gxp\_p}} + {\theta _{\; gxt\_p}}\\ \theta _{yp}^{(k )} = \frac{{({\varphi_0^{({k - 1} )}({y + 1} )- \varphi_0^{({k - 1} )}(y )} )\lambda }}{c} + {\theta _{gyp\_p}} + {\theta _{gyt\_p}} \end{array}, $$
where $\theta _{xp}^{(k )}$ and $\theta _{yp}^{(k )}$ denote the global tilt after k iterations in in x and y directions, respectively, c is the calibration factor, which indicates the actual measurement size corresponding to a single pixel, λ is the operating wavelength of the interferometer, $\varphi _0^{({k - 1} )}$ is the error-free phase computed in the k-1th iteration (eliminating the overall tilt behind the shape), and $\varphi _0^{({k - 1} )} = 0$ in the first iteration.

Because the use of differencing in the determination of the local tilt results in the loss of data on the leftmost side of the surface data, it is necessary to save the leftmost data of the surface results before each iteration to cover the error due to data loss in subsequent computations. The iterative process can be established using the angle correction formula expressed in Eq. (6). With each iteration, a new retrace error correction phase result can be obtained, and the local inclination parameter introduced by the surface error is determined using the phase gradient. Then, the inclination angle is updated and iterated again to determine the phase that eliminates the retrace error until the iterative convergence determination condition is satisfied.

The convergence conditions of the iteration are expressed as follows:

$$\frac{{\sum {||{\varphi_0^{(k )} - \varphi_0^{({k - 1} )}} ||_2^2} }}{{\sum {{{({\varphi_0^{({k - 1} )}} )}^2}} }} < \varepsilon, $$
where $\varphi _0^{(k )}$ denotes the retrace error correction phase determined in the kth iteration, $\varphi _0^{({k - 1} )}$ is the result of the k-1th retrace error correction phase, and ε is the threshold of convergence, which is generally set at 10−3.

3. Simulation

To verify the performance of the iterative correction method of the retrace error, we determined the theoretical simulation phase for plane measurements. The results for the established retrace error sensitivity factor matrix are shown in Fig. 2. The calibration factor was c = 4 × 10−4 m/pixel, whereas the interferometer operating wavelength was λ = 6.32 × 10−7 m. The simulated phase includes power, astigmatism, coma, and spherical aberrations, and can be expressed as follows:

$$\phi ({x,y} )= {a_1}({{x^2} + {y^2}} )+ {a_2}({{x^2} - {y^2}} )+ {a_3}x({x^2} + {y^2}) + {a_4}{({x^2} + {y^2})^2},$$
where a1, a2, a3, and a4 denote the power, astigmatism, coma, and spherical aberration coefficients, respectively. We set a1 = 0.125, a2 = 0.125, a3 = 0.25, and a4 = 0.11. The simulated phase is shown in Fig. 2(c). Retrace errors were introduced in the measurement results for varying carrier frequencies, and the specific values are listed in Table 1. The results of determining the phase with the retrace error based on the sensitivity factor matrix are shown in Figs. 4(a)–(c). To analyze the similarity of the wavefront before and after the correction of the retrace error, the wavefront correlation coefficient was introduced to evaluate the similarity between the corrected phase and the zero-fringe wavefront. The wavefront correlation coefficient [21] is defined as follows:
$$r = \frac{{\sum\limits_m {\sum\limits_n {({{A_{mn}} - \bar{A}} )({{B_{mn}} - \bar{B}} )} } }}{{\sqrt {\left( {\sum\limits_m {\sum\limits_n {{{({{A_{mn}} - \bar{A}} )}^2}} } } \right)\left( {\sum\limits_m {\sum\limits_n {{{({{B_{mn}} - \bar{B}} )}^2}} } } \right)} }}, $$
where Amn and Bmn represent two wavefront datasets; $\bar{A}$ and $\bar{B}$ are the average of wavefront data; and m and n are the row and column values for the wavefront, respectively. The correlation coefficients of wavefronts are obtained by the point-to-point operation of two wavefronts datasets, which can be used to evaluate the similarity of the two wavefronts. The correlation coefficients of the simulated error and error-free surfaces are summarized in Table 1.

 figure: Fig. 2.

Fig. 2. Simulation results: sensitivity factor matrix in the (a) x-direction and (b) y-direction; (c) peak-to-valley value (PV) of simulated phase = 0.1316λ, root mean square (RMS) = 0.0254λ.

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According to the surface correlation coefficient, as the carrier frequency increases, the surface correlation coefficient decreases, which means that the effect of the retrace error is greater. The simulated results with retrace errors in Figs. 4(a), (b), and (c) are substituted into the flowchart shown in Fig. 3. In the flowchart, W1∼n denotes the raw results of multiple measurements of the simulation (phase results with a tilt term). The retrace-error-corrected phase was determined after four iterations, as shown in Fig. 4(e), and the point-to-point computation was performed with the theoretical error-free phase to obtain the algorithm to determine the residual results, as shown in Fig. 4(f). The final simulated phases, error-corrected phases, and corresponding residual phases are illustrated in Fig. 4.

 figure: Fig. 3.

Fig. 3. Flowchart of the iterative correction method of retrace error.

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 figure: Fig. 4.

Fig. 4. Simulation result: (a)–(c) results with retrace error phase φ1∼3 at varying carrier frequencies; (d) theoretical error-free phase φ0; (e) retrace error-corrected result φ'0; (f) remaining residuals after algorithm correction Res0; (g)–(i) retrace errors Res1∼3 under varying carrier frequency conditions.

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Table 1. Simulated surface parameters and surface correlation coefficients between simulated and error-free surfaces.

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From the three residual results illustrated in Figs. 4(g)–(i), the retrace error increases with increasing carrier frequency, which means that the retrace error is related to the tilt of the test surface. After correcting the retrace error, the obtained error-corrected phase was consistent with the theoretical error-free phase. The PV and RMS of the obtained phase are listed in Table 2. The error-corrected phase differs from the theoretical error-free phase by approximately 0.0012λ in the PV value and 0.0001λ in the RMS value. The residual distribution obtained from the point-to-point computation of the theoretical error-free phase is shown in Fig. 4(f), and the value of the residual RMS was 3.18 × 10−5λ. The simulation proved the effectiveness of the method for correcting retrace errors.

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Table 2. Simulation results of PV and RMS for different phases (λ = 632.8 nm).

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To further analyze the difference between the retrace error-corrected phase and the theoretical error-free phase, we compared the original three input phases, corrected phases, and error-free phases with Zernike coefficients characterized by computing the first nine low-order Zernike coefficients for each of the five terms, as shown in Fig. 5. Significant differences between the three input original phases and the error-free phases in Zernike terms 3, 4, 5, and 6 were observed and some of the term coefficients exhibited sign flips owing to the retrace error. The Zernike coefficients of the corrected retrace errors were consistent with the theoretical values, and the maximum deviation did not exceed 4 × 10−4. A comparison of the results of the Zernike coefficients of the two phases can further prove the effectiveness of the retrace error correction method.

 figure: Fig. 5.

Fig. 5. Distribution of the first nine orders of Zernike coefficients of the obtained phase.

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4. Experiment

To verify the practical performance of the method, we performed retrace error correction experiments in a Φ100 mm horizontal interferometer from Zygo for plane and spherical measurements.

4.1 Plane measurement

4.1.1 Retrace error correction in Zygo interferometer plane measurement

We performed a plane retrace error correction experiment on a Φ100 mm horizontal interferometer from Zygo. Φ100 mm TF and RF were used in the experiment, and different interferogram were obtained by changing the tilt of the RF three times to introduce different frequency-carrying amounts, as shown in Fig. 6. The calculated phase is shown in Figs. 7(a)–(c). The wavefront measured under the zero-fringe condition after self-collimation of the interferometer is generally taken as the wavefront without retrace error [22]. Figure 7(d) illustrates the zero-fringe measurement results, which theoretically do not contain the retrace error in the obtained results. The point-to-point subtracted residuals of the three measurements with varying tilt amounts and the zero fringe measurements are shown in Figs. 7(g)–(i). The results of the three measurements are substituted into the flowchart of the above algorithm to obtain the results of the corrected retrace error, as shown in Fig. 7(e), and the residuals of the surface measured with zero fringe, as shown in Fig. 7(f).

 figure: Fig. 6.

Fig. 6. The interferogram of (a)φ0, (b)φ1, (c)φ2, (d)φ3,

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 figure: Fig. 7.

Fig. 7. Experimental results: (a)–(c) results of the phase with retrace error φ1∼3 at varying carrier frequencies; (d) theoretical error-free phase φ0; (e) results of φ'0 after correction of the retrace error; (f) remaining residuals after algorithm correction Res0; and (g)–(i) results of retrace error Res1∼3 at varying carrier frequencies.

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The PV and RMS values of the experimentally measured phases are listed in Table 3. According to the point-to-point subtracted residuals of the direct measurement results and zero fringe measurement results illustrated in Figs. 7(g)–(i) and the corresponding residuals Res1, Res2, and Res3 listed in Table 3, the PV and RMS parameters of the three, can be obtained that with an increase of the introduced carrier frequency, and the retrace error contained in the actual measured surface subsequently increased. The variance between the PV of the phase before and after correction was approximately 3.1 × 10−3λ, whereas the variance between the RMS was 7×10−4λ. In comparison with the original three error phases, the corrected phase was closer to the error-free phase in terms of PV and RMS. The calculated wavefront correlation coefficients are listed in Table 4. The corrected wavefront correlation coefficient improved to 0.992, which was higher than the wavefront correlation coefficient determined from the original error phase, and the phase was closer to the error-free phase.

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Table 3. Experimental results of PV and RMS for different phases (λ = 632.8 nm)

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Table 4. Wavefront correlation coefficients before and after error correction.

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To further analyze the retrace error in the measurement results at varying carrier frequencies and the correction effect, the first nine Zernike coefficients were determined for the results shown in Fig. 7, as illustrated in Fig. 8. After comparing the Zernike coefficients of the zero-fringe phase and the three original phases, we observed the following: in the plane, the retrace error was mainly in the power and astigmatism, followed by a slight inconsistency in the coma, and the experimental results the plane retrace error in the principle of analysis were consistent. Second, from the power term, which accounted for the largest proportion of the retrace error, the out-of-focus coefficient in the zero-fringe phase result was 0.04565. Comparing the power coefficients before and after correction, the difference between the corrected corresponding coefficient and the coefficient of the zero-fringe phase was the smallest, being 0.00615. Comparing the Zernike coefficients calculated for the zero-fringe phase and the other phases, the corrected coefficients were closer to the zero-fringe phase results than the data with retrace errors, which further demonstrates the effectiveness of the iterative retrace error correction method in planimetry.

 figure: Fig. 8.

Fig. 8. Distribution of the first nine orders of Zernike coefficients for different phase results in planimetric measurements.

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4.1.2 Retrace error correction in Φ300 mm vertical interferometer plane measurement

To further verify the general applicability of the method, we conduct the retrace error experiments in the Φ300mm vertical Fizeau interferometer [23,24]. The experimental procedure is the same as that in the plane measurement of Zygo interferometer. The tilt attitude of the test flat is changed to introduce different carrier frequency amounts to collect the interferogram, as shown in Fig. 9. The interferogram is calculated [25] to obtain three measurements, as shown in Figs. 10(a)∼(c). Figure 10(d) shows the zero-fringe measurement results, and the retrace error is not included in the measurement results theoretically. The point-to-point subtracted residuals of the three measurements with different tilt amounts and the zero-fringe measurement results are shown in Figs. 10(g)∼(i), and the results of the three measurements are substituted into the flow chart of the above algorithm to obtain the results corrected for the retrace error as shown in Fig. 10(e). The residuals corresponding to the error-corrected phase and zero-fringe measurement phase are shown in Fig. 10(f).

 figure: Fig. 9.

Fig. 9. The interferogram of (a)φ0, (b)φ1, (c)φ2, (d)φ3

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 figure: Fig. 10.

Fig. 10. Experimental results: (a)–(c) results of the phase with retrace error φ1∼3 at varying carrier frequencies; (d) theoretical error-free phase φ0; (e) results of φ'0 after correction of the retrace error; (f) remaining residuals after algorithm correction Res0; and (g)–(i) results of retrace error Res1∼3 at varying carrier frequencies.

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According to the residual results in Figs. 10(f)-(i), it can be obtained that the residuals of the corrected phase results are smaller than the original phase results, and from the PV and RMS of the calculated residuals in Table 5, the PV of the corrected phase residuals is better than 0.01λ, and the error is reduced by at least 30% compared to the phase results of three measurements. This further demonstrates the general applicability of this iterative correction method for the retrace error in interferometry.

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Table 5. Experimental results of PV and RMS for different phases (λ = 632.8 nm)

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4.2 Spherical measurement

To investigate the practical effect of this iterative correction method on the retrace error in spherical measurements, we performed spherical measurement experiments using a Fieazu horizontal interferometer from Zygo. The reflective lens used in the experiment was a standard F-number 0.75 lens from Zygo, and the testing lens was a F-number 1.5 lens. Different interferogram were obtained by changing the lateral and longitudinal defocus of the testing lens six times, as shown in Fig. 11. The calculated phase is shown in Figs. 12(a)–(f). Figure 12 (d) shows the results of the null measurement, and the retrace error was theoretically not included in the measurement results.

 figure: Fig. 11.

Fig. 11. The interferogram of (a)φ1, (b)φ2, (c)φ3, (d)φ4, (e)φ5, (c)φ6,

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 figure: Fig. 12.

Fig. 12. Experimental results: (a)–(f) results with retrace error phases φ1−6 at varying carrier frequencies; (h) theoretical error-free phase φ0; (i) retrace error-corrected result φ'0; and (j) remaining residuals after algorithm correction Res0.

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The results of the six spherical measurements were substituted into the flowchart of the above algorithm to obtain the phase results corrected for the retrace error, as shown in Fig. 12 (i), whereas the phase residuals with the null measurement are shown in Fig. 12 (j). The phase parameters are listed in Table 6. By comparing the phase distribution tendencies of the six measured phases and the null measurement, the phase distribution gradually deviated from the null measurement results as the measured part attitude changed and the introduced carrier frequency increased. From the phase parameters, the PV and RMS of the measured phase increased with an increase in the carrier frequency.

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Table 6. Experimental results of PV and RMS for different phases (λ = 632.8 nm)

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The wavefront correlation coefficients of the phase and null tests of the phase computations before and after the error correction are listed in Table 7. The wavefront correlation coefficient of the original phase deviates from one because of the effect of the retrace error. The wavefront correlation coefficient of the phase determination after retrace error correction was 0.98, which is most similar to the phase result of the null test.

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Table 7. Wavefront correlation coefficients before and after error correction.

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To analyze the retrace error in the spherical measurement, the residuals were obtained by point-to-point subtraction of the phase junction and the null measurement results of the six times measured in different attitudes of the measured parts, as shown in Figs. 13(a)–(f). The results of the six residuals show that the main expression of the retrace error in the nonnull measurement in the spherical measurement is dominated by coma [26]. Comparing the six original residuals with the corrected phase residuals, the distribution of the coma in the corrected phase residuals was not clear, the residual PV was 0.0085λ, and the RMS was 0.0014λ. The coma tendency observed in residuals in the original phase was clear, and the determined residual PV and RMS values are listed in Table 8. The PV of the calculated original residual phase was larger than 0.02λ, whereas the RMS was larger than 0.0031λ, which was approximately two to five times larger than the phase result after retrace error correction. According to the experimental results, the iterative correction method of the retrace error has a better correction effect for the spherical retrace error.

 figure: Fig. 13.

Fig. 13. Experimental results: (a)–(f) Retrace error Res1−6 in the results of varying carrier frequency phases.

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Table 8. Experimental results of PV and RMS of the residual phase of the six original phases.

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To further analyze the correction effect of the retrace error, the low-order Zernike coefficients were determined for the above phase results, as shown in Fig. 14. According to the coefficients corresponding to the original measurement data φ1–6 and the Zernike coefficients of the phase of the null test, we can observe that in the spherical measurement, the retrace error is mainly in the form of coma, followed by astigmatism. According to the determined Zernike coefficients, the deviation between the original data and the null test is the largest in the Zernike coefficients corresponding to the coma term, reaching 0.0209, which is approximately seven times that of the null test. When comparing the Zernike coefficients determined from the corrected results and the phase results of the null test, the variance in the coma coefficient in the corrected phase was reduced to 0.0015, and the determined deviation was reduced from seven times the original zero measurement coefficient to a value of 0.5. Additionally, the Zernike coefficients determined from the corrected phase converged to the null test in both directions of the astigmatism coefficients. The effectiveness of this iterative correction method for spherical retrace error correction was further demonstrated by the experimental results.

 figure: Fig. 14.

Fig. 14. Experimental results of the first nine Zernike coefficients for different phases.

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5. Discussion

The retrace error iterative correction method can iteratively determine the retrace error and correct the phase using multiple interferometry measurements. To further analyze the performance of the proposed method, the carrier frequency direction, effect of the algorithm iteration, and effect of the sensitivity factor matrix on the retrace error correction must be analyzed separately. The iterative nature of the algorithm used in this study makes it difficult to derive a specific formula; therefore, we used simulation and experimental analysis methods.

5.1 Effect of the number of iterations on the correction of retrace error

The iterative convergence method for the retrace error is based on an iterative method. To determine the iterative convergence condition of the method, the number of iterations and the iterative error were simulated, and the relationship between the number of iterations and computed PV and RMS of the residuals, and the evaluation function in the plane and spherical measurements were obtained, as shown in Fig. 15. According to the results shown in Fig. 15, the merit function and RMS of the residual error exhibit nonlinear convergence for both measurement types. According to the simulation results, the convergence speed of the method is fast, the method in plane measurement is guaranteed to be less than 10−3 after nine iterations and finally converges to 4 × 10−4, the RMS of the iterative error is maintained at 3.1 × 10−4 λ and PV at 2 × 10−3 λ. In addition, the iteration tends to converge, and the iterative error meets the demand of actual measurement. In the spherical measurement, the method converges to 1.7 × 10−5 after three iterations with a merit function less than 10−4. The RMS of the residual error is maintained at 3.5 × 10−4 λ and PV at 2 × 10−3 λ, and iterative convergence is observed.

 figure: Fig. 15.

Fig. 15. Residual PV, RMS, and convergence determination function values versus number of iterations in (a) plane and (b) spherical measurements.

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5.2 Effect of the tilt error on the error correction

The iterative correction method of the retrace error is mainly based on the determination of the retrace error with the tilt angle of the testing surface; hence, the impact of the surface tilt angle measurement error on the retrace error correction needs to be further considered. Therefore, the error source was introduced in the simulation results of the inclination data, and the relationship between the residual error and inclination angle after the correction of the retrace error was determined, as shown in Fig. 16.

 figure: Fig. 16.

Fig. 16. Relationship between tilt error and retrace error correction residuals.

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The tilt angle of the testing surface can be determined from the measured phase inclination, and the tilt angle error obtained based on this principle was within ± 3 µrad (± 1 fringe). According to the results of the simulation, the maximum PV of the phase residual of the retrace error correction was 1.4 × 10−3 λ (0.88 nm) and the RMS was 2 × 10−4 λ when the actual tilt error was within ± 3 µrad. Therefore, the phase-correction error introduced by the tilt calculation error for the measured part was within the acceptable range, and the tilt error had a negligible effect on the proposed method.

The iterative retrace error correction method is a least squares solution method based on inclination, so multiple equations need to be solved when solving the phase of error correction. To ensure the stability of the calculation, there are certain requirements for the measurement attitude of the measured part. In plane measurement, there are three equations in the set of equations of the least squares method, and in order to achieve the condition that the set of equations has a solution of full rank, according to Eq. (4) needs to satisfy:

$$rank\left( {\left[ {\begin{array}{ccc} 1&{{\theta_{x1}}}&{{\theta_{y1}}}\\ 1&{{\theta_{x2}}}&{{\theta_{y2}}}\\ 1&{{\theta_{x3}}}&{{\theta_{y3}}} \end{array}} \right]} \right) = 3,$$
Where rank() is the function that calculates the rank of the matrix. The inclination angle of the measured part needs to be independent of each other when measuring it three times. That is, the inclination angles of the measured parts need to be guaranteed to be independent of each other for three measurements. For spherical measurements, it is necessary to ensure that the six measurements of inclination are independent of each other.
$$rank\left( {\left[ {\begin{array}{cccccc} 1&{{\theta_{x1}}}&{{\theta_{y1}}}&{{\theta_{x1}}{\theta_{y1}}}&{\theta_{x1}^2}&{\theta_{y1}^2}\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 1&{{\theta_{x6}}}&{{\theta_{y6}}}&{{\theta_{x6}}{\theta_{y6}}}&{\theta_{x6}^2}&{\theta_{y6}^2} \end{array}} \right]} \right) = 6$$

Satisfying the above conditions, the stability of the calculation can be guaranteed when solving the system of equations by least squares during the iterative process.

5.3 Effect of sensitivity factor matrix on the correction of retrace error

The iterative correction method can also obtain the sensitivity factor matrix for the retrace error in the interferometer system while effectively correcting the retrace error. To study the effect of the retrace error sensitivity factor matrix on the correction of the retrace error, we determined the wavefront correlation coefficients for the experimental data. Therefore, the effect of the retrace error in different interferometer measurement types can be obtained, as shown in Fig. 17.

 figure: Fig. 17.

Fig. 17. Effect of the sensitivity factor matrix on the actual correction of the retrace error under different tilts.

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The wavefront correlation coefficients were determined for 75 sets of phases (the number of fringes was distributed between 0 and 95) and zero-fringe phase in the plane measurement as shown in Fig. 17. In the plane measurement, the determined wavefront correlation coefficients before correction were kept above 0.95 before 65 fringes owing to the small retrace error, and the variance between the wavefront correlation coefficients determined before and after the correction was small. When the test surface was tilted until the fringe was greater than 65, the correlation coefficient of the wavefront before correction was less than 0.95, and the retrace error became larger; thereafter the wavefront correlation coefficient after correction of the retrace error was greater than the result obtained before correction, and the effect of correction of the retrace error became clearer. The wavefront correlation coefficients of the 26 sets of data in the spherical measurement (the number of fringes was distributed from 0 to 53) and the computed wavefronts at zero fringe are shown in Fig. 17. The wavefront correlation coefficients were always greater than the wavefront correlation coefficients before correction, and all of them were greater than 0.9. Therefore, the sensitivity factor matrix computed using the proposed method has a certain correction effect on the retrace error.

6. Conclusions

In conclusion, we proposed an iterative correction method of the retrace error, which can correct the retrace error from the phase measurement results of varying carrier frequencies and obtain the sensitivity factor matrix for the correction of the retrace error in subsequent measurement results. This method is fast in terms of iterations, and the RMS of the residual error is guaranteed to be greater than 3 × 10−5 λ after a maximum of 10 iterations in the simulation results. We performed experiments in a Φ100 mm horizontal interferometer from Zygo, and the RMS of the residual error was greater than 6.4 × 10−3λ for plane measurement and 1.4 × 10−3λ for spherical measurement. Experimental results on a Φ300 mm vertical Fizeau interferometer further verify the universal applicability of this method. According to the experimental results, the determined retrace error sensitivity factor also had a certain correction effect on the retrace error, and the wavefront correlation coefficient between the corrected and zero-fringe phases was greater than 0.9. The proposed method corrects the tilt generated by the phase of the measured element to the global tilt angle by iteration, which can correct the retrace element error generated by the local surface. The theoretical and experimental results demonstrate that the proposed method can effectively correct the retrace error in the plane and spherical measurements in an interferometer without changing the hardware facilities. This method can not only relax the zero-fringe restrictions in plane measurement and spherical measurement, but also ensure the measurement accuracy, so it is expected to become a method to improve the measurement accuracy in phase-shifting interferometers.

Funding

National Natural Science Foundation of China (62005122, U1731115); Natural Science Foundation of Jiangsu Province (BK20200458); Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX20_0263).

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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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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Figures (17)
Fig. 1.
Fig. 1. Schematic of spherical measurement of wavefront tilt.

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Fig. 2.
Fig. 2. Simulation results: sensitivity factor matrix in the (a) x-direction and (b) y-direction; (c) peak-to-valley value (PV) of simulated phase = 0.1316λ, root mean square (RMS) = 0.0254λ.

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Fig. 3.
Fig. 3. Flowchart of the iterative correction method of retrace error.

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Fig. 4.
Fig. 4. Simulation result: (a)–(c) results with retrace error phase φ1∼3 at varying carrier frequencies; (d) theoretical error-free phase φ0; (e) retrace error-corrected result φ'0; (f) remaining residuals after algorithm correction Res0; (g)–(i) retrace errors Res1∼3 under varying carrier frequency conditions.

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Fig. 5.
Fig. 5. Distribution of the first nine orders of Zernike coefficients of the obtained phase.

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Fig. 6.
Fig. 6. The interferogram of (a)φ0, (b)φ1, (c)φ2, (d)φ3,

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Fig. 7.
Fig. 7. Experimental results: (a)–(c) results of the phase with retrace error φ1∼3 at varying carrier frequencies; (d) theoretical error-free phase φ0; (e) results of φ'0 after correction of the retrace error; (f) remaining residuals after algorithm correction Res0; and (g)–(i) results of retrace error Res1∼3 at varying carrier frequencies.

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Fig. 8.
Fig. 8. Distribution of the first nine orders of Zernike coefficients for different phase results in planimetric measurements.

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Fig. 9.
Fig. 9. The interferogram of (a)φ0, (b)φ1, (c)φ2, (d)φ3

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Fig. 10.
Fig. 10. Experimental results: (a)–(c) results of the phase with retrace error φ1∼3 at varying carrier frequencies; (d) theoretical error-free phase φ0; (e) results of φ'0 after correction of the retrace error; (f) remaining residuals after algorithm correction Res0; and (g)–(i) results of retrace error Res1∼3 at varying carrier frequencies.

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Fig. 11.
Fig. 11. The interferogram of (a)φ1, (b)φ2, (c)φ3, (d)φ4, (e)φ5, (c)φ6,

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Fig. 12.
Fig. 12. Experimental results: (a)–(f) results with retrace error phases φ1−6 at varying carrier frequencies; (h) theoretical error-free phase φ0; (i) retrace error-corrected result φ'0; and (j) remaining residuals after algorithm correction Res0.

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Fig. 13.
Fig. 13. Experimental results: (a)–(f) Retrace error Res1−6 in the results of varying carrier frequency phases.

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Fig. 14.
Fig. 14. Experimental results of the first nine Zernike coefficients for different phases.

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Fig. 15.
Fig. 15. Residual PV, RMS, and convergence determination function values versus number of iterations in (a) plane and (b) spherical measurements.

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Fig. 16.
Fig. 16. Relationship between tilt error and retrace error correction residuals.

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Fig. 17.
Fig. 17. Effect of the sensitivity factor matrix on the actual correction of the retrace error under different tilts.

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Tables (8)

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Table 1. Simulated surface parameters and surface correlation coefficients between simulated and error-free surfaces.

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Table 2. Simulation results of PV and RMS for different phases (λ = 632.8 nm).

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Table 3. Experimental results of PV and RMS for different phases (λ = 632.8 nm)

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Table 4. Wavefront correlation coefficients before and after error correction.

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Table 5. Experimental results of PV and RMS for different phases (λ = 632.8 nm)

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Table 6. Experimental results of PV and RMS for different phases (λ = 632.8 nm)

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Table 7. Wavefront correlation coefficients before and after error correction.

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Table 8. Experimental results of PV and RMS of the residual phase of the six original phases.

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Equations (11)

Equations on this page are rendered with MathJax. Learn more.

W ( ρ , θ , x 0 ) = ∑ n = 1 5 ( △ W n ( x 0 + △ x ) − △ W n ( x 0 ) ) ≈ W 222 ( 2 △ x x 0 ) ρ 2 cos 2 θ + W 131 △ x ρ 3 cos ⁡ θ + W 220 ( 2 △ x x 0 ) ρ 2 + W 311 ( 3 △ x x 0 2 ) ρ cos ⁡ θ ,
O P D = 2 ⋅ A H ¯ − A B D ¯ = 2 ⋅ ( R 1  +  R 2 ) − 2 ( R 1 cos ⁡ ( α )  +  R 2 cos ⁡ ( sin − 1 ( R 1 sin ⁡ ( α ) R 2 ) ) ) ,
O P D ≈ α 2 R 1 ( ( ρ + 1 ) / ρ ) ,
{ ∑ n = 0 q ∑ m = 0 n C x n y m θ x 1 n θ y 1 m = φ 1 ⋮ ∑ n = 0 q ∑ m = 0 n C x n y m θ x p n θ y p m = φ p ⋮ ∑ n = 0 q ∑ m = 0 n C x n y m θ x 3 q n θ y 3 q m = φ 3 q ,
θ x p = θ g x p _ p + θ g x t _ p θ y p = θ g y p _ p + θ g y t _ p ,
θ x p ( k ) = ( φ 0 ( k − 1 ) ( x + 1 ) − φ 0 ( k − 1 ) ( x ) ) λ c + θ g x p _ p + θ g x t _ p θ y p ( k ) = ( φ 0 ( k − 1 ) ( y + 1 ) − φ 0 ( k − 1 ) ( y ) ) λ c + θ g y p _ p + θ g y t _ p ,
∑ | | φ 0 ( k ) − φ 0 ( k − 1 ) | | 2 2 ∑ ( φ 0 ( k − 1 ) ) 2 < ε ,
ϕ ( x , y ) = a 1 ( x 2 + y 2 ) + a 2 ( x 2 − y 2 ) + a 3 x ( x 2 + y 2 ) + a 4 ( x 2 + y 2 ) 2 ,
r = ∑ m ∑ n ( A m n − A ¯ ) ( B m n − B ¯ ) ( ∑ m ∑ n ( A m n − A ¯ ) 2 ) ( ∑ m ∑ n ( B m n − B ¯ ) 2 ) ,
r a n k ( [ 1 θ x 1 θ y 1 1 θ x 2 θ y 2 1 θ x 3 θ y 3 ] ) = 3 ,
r a n k ( [ 1 θ x 1 θ y 1 θ x 1 θ y 1 θ x 1 2 θ y 1 2 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 1 θ x 6 θ y 6 θ x 6 θ y 6 θ x 6 2 θ y 6 2 ] ) = 6
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