1. Introduction

Optical encoders are one of the most common displacement sensors for high precision applications [1, 2]. One of the key components in an optical encoder is a planar scale grating. The planar scale grating has periodic structures along the X- and Y-axes and is typically used as the measurement reference in an interferential scanning-type optical encoder [3, 4]. The phase shifts in the positive and negative first-order diffracted beams from the scale grating, which are induced by the displacement of the scale grating relative to the reading head of the encoder, are converted to the intensity changes of interference signals. The displacement of the scale grating is finally obtained by analyzing the interference signals. Since the period of the interference signal is determined by the pitch of the scale grating, the X- and Y-directional pitch deviations of the scale grating directly influence the measurement uncertainty of the encoder. In addition, the Z-directional out-of-flatness of the scale grating also influences the quality of the interference signal and further influences the measurement uncertainty of the encoder. It is thus necessary to evaluate the pitch deviations and out-of-flatness over the entire area of a planar scale grating.

The line scale comparator has long been used to evaluate the pitch deviations of one-axis linear scale gratings [5]. This method is very accurate with a direct linkage to the international length standard by employing a stabilized laser interferometer working in vacuum. It can provide not only the pitch deviations but also the absolute value of each scale pitch over a long range up to several meters. However, it is too expensive and complicated to expand this method from one-axis to two-axis for evaluating a planar scale grating. Also, the out-of-flatness of the scale grating is unmeasurable by this method. Scanning probe microscopy (SPM) is another technique that is able to evaluate the quality of a planar scale grating by the collected images [6]. However, it is impractical to make an evaluation over the entire grating area due to the limitation of SPM in both the scanning speed and range. We have proposed a noncontact method based on a Fizeau interferometer that can evaluate pitch deviations and out-of-flatness of planar scale gratings in Littrow setup over a large area in a short time [7]. However, it is noted that, the surface profile of the reference optical flat (a key component in the Fizeau interferometer) has not been taken into account in the proposed method, which therefore restricts its application in high accuracy measurements.

A Fizeau interferometer, which basically measures the optical path difference between a reference optical flat and the surface under test, is typically used for testing the flatness of a smooth surface [8]. Generally, the flatness of the reference flat is better than λ/20 (λ is the wavelength of the light source in the Fizeau interference). When the required measurement accuracy is higher than the flatness of the reference flat, it is necessary to calibrate the surface profile of the reference flat for compensation. The common methods used to calibrate the surface profile of the reference flat include the liquid-flat reference method [9–12] and the three-flat test method [13–19]. As for the former method, a liquid surface (normally mercury or oil) is used for the calibration or is directly employed as the reference flat surface in a Fizeau interferometer. A merit of this approach is that it relates the definition of flatness to the radius of the earth. Nevertheless, making accurate measurements with a liquid reference surface has to conquer a series of tough issues in practice, including meniscus effects, surface contamination and environmental fluctuations etc. As for the latter method, one needs to prepare three similar flats (including the target reference flat), then performs measurements by facing the flats two by two, and finally processes the interference fringes generated by each pair of flats. Although the latter approach provides a more practical way for flatness testing than operating with a liquid reference surface, multiple flips, rotations or translations of the flats need to be conducted to reconstruct their surface profiles, which makes this approach complicated in specific implementation. On the other hand, Littrow setup is often employed to inspect gratings [20, 21]. In this work, we propose a novel approach that can realize self-calibration of the Fizeau interferometer and the planar scale grating in Littrow setup. By using the proposed method, not only the pitch deviations and out-of-flatness of planar scale gratings can be evaluated, but also the surface profile of the reference optical flat in the Fizeau interferometer can be reconstructed in a much simpler and more efficient approach than the conventional calibration methods shown above.

2. Method

2.1 Basic principle

Let the Z-directional out-of-flatness and the X-, Y-directional pitch deviations of the planar scale grating under test be $e_Z(x,y)$, $e_X(x,y)$ and $e_Y(x,y)$, respectively, and the Z-directional out-of-flatness (i.e. the surface profile) of the reference optical flat in the Fizeau interferometer be $e_R(x,y)$. Figure 1 illustrates the evaluation procedure for the Z-directional out-of-flatness $e_Z(x,y)$. As depicted in Fig. 1, the wavefront of the zero-order diffracted beam from the grating superimposed by the light beam from the reference flat is measured by the Fizeau interferometer. The zero-order phase output $I_0(x,y)$ from the interferometer is

 

Fig. 1 Measurement of the wavefront of the zero-order diffracted beam.

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To evaluate the X-directional pitch deviation $e_X(x,y)$ of the planar scale grating, it is required to collect wavefronts of the X-directional positive and negative first-order diffracted beams, respectively. As an example, Fig. 2 depicts the measurement of the wavefront of the X-directional positive first-order diffracted beam. As shown in Fig. 2, the planar scale grating is inclined to superimpose the X-directional positive first-order diffracted beam with the light beam reflected from the reference flat of the Fizeau interferometer. The inclination angle is set to be half of the first-order diffraction angle θ, where θ = arcsin(λ/g) and g is the nominal period of the planar scale grating along the X-direction. The X-directional pitch deviation $e_X(x,y)$ of the planar scale grating will cause a phase shift in the wavefront of the X-directional positive first-order diffracted beam. Considering the change in the coordinate system after the inclination of the planar scale grating, as depicted in Fig. 2, the X-directional positive first-order phase output $I_{X+1}(x,y)$ from the Fizeau interferometer will be

 

Fig. 2 Measurement of the wavefront of the X-directional positive first-order diffracted beam.

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By the same token, the Y-directional pitch deviation $e_Y(x,y)$ can be obtained by measuring the wavefronts of the Y-directional positive and negative first-order diffracted beams, respectively, as follows:

2.2 One-dimensional (1D) case

Firstly, we investigate the results of $e_Z(x,y)$, $e_X(x,y)$ and $e_Y(x,y)$ at any slice of $x=x_i$ or $y=y_j$ (i and j are integers) of the phase outputs of the Fizeau interferometer. We take the slice of $y=y_j$ as an example. In this case, $e_Z(x,y)$, $e_X(x,y)$ and $e_Y(x,y)$ will only vary with x. For simplicity, we omit the argument y in Eqs. (5) and (9) and rewrite them as

Without loss of generality, we use n-degree polynomials to fit these discrete sampling points $(x_i,g_X(x_i))$ and $(x_i,f_X(x_i))$ by

According to Eqs. (11) and (13), we will have $e_X(x)={\displaystyle {\sum }_{k=0}^n{c}_k{\alpha }^k{x}^k}$. To obtain $e_Z(x)$, we assume that $e_Z(x)={\displaystyle {\sum }_{k=0}^n{a}_k{x}^k}$, where a_k are the undetermined coefficients. Note that here the degree of the fitting polynomial to $e_Z(x)$ is equal to that to $f_X(x)$. In fact, it can be readily demonstrated that the degree of the fitting polynomial to $e_Z(x)$ will be no larger than that to $f_X(x)$ by Eq. (12). The substitution of $e_Z(x)={\displaystyle {\sum }_{k=0}^n{a}_k{x}^k}$ into Eq. (12) leads to

2.3 Two-dimensional (2D) case

For simplicity, we denote the right sides of Eqs. (5), (8), (9) and (10) as $g_X(x,y)$, $g_Y(x,y)$, $f_X(x,y)$ and $f_Y(x,y)$, respectively. Similarly, a series of discrete values of $g_X(x_i,y_j)$, $g_Y(x_i,y_j)$, $f_X(x_i,y_j)$ and $f_Y(x_i,y_j)$ (i = 1, 2, …, M; j = 1, 2, …, N) can be obtained according to the phase outputs $I_0(x_i,y_j)$ and $I_{X\pm 1}(x_i,y_j)$ in practical implementation. Without loss of generality, we use n-degree bivariate polynomials to fit these discrete sampling points (x_i, y_j, $g_X(x_i,y_j)$), (x_i, y_j, $g_Y(x_i,y_j)$), (x_i, y_j, $f_X(x_i,y_j)$) and (x_i, y_j, $f_Y(x_i,y_j)$) by

According to Eqs. (5), (8), (16) and (17), we will have $e_X(x,y)={\tilde{x}}^{\text{T}}{C}_{X}y$ and $e_Y(x,y)=x^{\text{T}}{C}_Y\tilde{y}$, where $\tilde{x}={[1,\alpha x,{(\alpha x)}^2,\dots ,{(\alpha x)}^n]}^{\text{T}}$ and $\tilde{y}={[1,\alpha y,{(\alpha y)}^2,\dots ,{(\alpha y)}^n]}^{\text{T}}$, respectively. As for $e_Z(x,y)$, we assume that $e_Z(x,y)=x^{\text{T}}Ay$ (also with a degree of n), where A has a similar form to $C_X$ in Eq. (16) and contains the undetermined coefficients. The substitution of $e_Z(x,y)=x^{\text{T}}Ay$ into Eqs. (9) and (10) leads to

3. Results and discussions

3.1 Feasibility verification

Simulations were performed to verify the feasibility of the proposed method. In the simulations, the wavelength of the light source was λ = 632.8 nm, and the grating periods along the X- and Y-directions were both g = 1.0 μm. We will firstly give the expressions of the form errors $e_Z(x,y)$, $e_X(x,y)$, $e_Y(x,y)$ and $e_R(x,y)$, and then substitute them into Eqs. (1), (3), (4), (6) and (7) to simulate the phase outputs $I_0(x,y)$, $I_{X\pm 1}(x,y)$ and $I_{Y\pm 1}(x,y)$ of the Fizeau interferometer. Then, we try to reconstruct $e_Z(x,y)$, $e_X(x,y)$, $e_Y(x,y)$ and $e_R(x,y)$ from the simulated phase outputs by using the proposed method. Finally, we make a comparison between the reconstructed and the given form errors to judge the feasibility of the proposed method in noise-free and noisy cases. For simplicity, we assume that the ranges of x and y are both in [−1, 1]. For a general range of [a, b] (a, b ∈ ℝ), x and y can be normalized by a proper constant. We should note that the normalization process only adjusts the coefficient values but not the function values of the polynomials fitted to $e_Z(x,y)$, $e_X(x,y)$, $e_Y(x,y)$ and $e_R(x,y)$. It is also worth pointing out that the normalization process should not change the origin of the coordinate system. The reason will be discussed in Section 3.2. In addition, for the sake of clarity, low degrees of polynomials are given for the form errors $e_Z(x,y)$, $e_X(x,y)$, $e_Y(x,y)$ and $e_R(x,y)$ in the following examples. It should be noted that the proposed method is also applicable for higher-degree of polynomials due to the explicit relations shown in Eqs. (15), (23) and (24).

Example 1: $e_Z(x)$ = 52.86 – 152.50x² + 18.29x³ + 108.60x⁴, $e_X(x)$ = 62.42 – 224.88x – 142.13x² + 582.23x³ + 114.49x⁴ – 307.26x⁵, and $e_R(x)$ = 12.67 + 33.86x² + 3.33x³

As can be observed from the polynomials given in Example 1, except $e_X(x)$, both $e_Z(x)$ and $e_R(x)$ do not contain the linear terms. We firstly examine whether or not the proposed method could be used to accurately reconstruct the polynomial coefficients in the noise-free case. Figure 3 presents the comparison between the input and reconstructed results for $e_Z(x)$, $e_X(x)$ and $e_R(x)$. It is noted from Fig. 3 that the PV value for $e_R(x)$ is about 37.2 nm, which is comparative to the typical PV value (λ/20) of the reference optical flat in the Fizeau interferometer. As shown in Fig. 3, the reconstructed results are exactly the same to the input results, which is also verified by comparing the reconstructed polynomial coefficients with the corresponding polynomial coefficients given in Example 1.

 

Fig. 3 Comparison between the input and reconstructed results for $e_Z(x)$, $e_X(x)$ and $e_R(x)$ of Example 1 in the noise-free case.

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Then, we try to examine the feasibility of the proposed method in the noisy case. To this end, noise was added to the phase outputs $I_0(x,y)$, $I_{X\pm 1}(x,y)$ and $I_{Y\pm 1}(x,y)$. The simulated phase outputs with noise were supposed to be subject to a normal distribution with the mean of $I_i(x,y)$ and the standard deviation of $\gamma \left|I_i(x,y)\right|$, where i = 0, X ± 1, Y ± 1 and $\gamma$ represents the noise level. The mean of $I_i(x,y)$ was directly obtained by substituting the expressions of $e_Z(x,y)$, $e_X(x,y)$, $e_Y(x,y)$ and $e_R(x,y)$ into Eqs. (1), (3), (4), (6) and (7). As an example, Figs. 4(a) and 4(b) present the simulated phase outputs with the noise level of γ = 0.02 and γ = 0.05, respectively. Figures 4(c) and 4(d) present the comparison between the input and reconstructed results for $e_Z(x)$, $e_X(x)$ and $e_R(x)$ under 2% and 5% noise levels, respectively. As can be observed, noise affects the accuracy of the reconstructed polynomials. Under the 2% noise level, the reconstructed results still show good agreement with the input results. In comparison, the accuracy of the reconstructed results, especially for $e_Z(x)$ and $e_R(x)$, is reduced but is still acceptable under the 5% noise level. We can also observe from Figs. 4(c) and 4(d) that the reconstructed results for $e_R(x)$ seem to be more noisy than those for $e_Z(x)$. It is because that here the zero-order phase output $I_0(x,y)$ contained noise and $e_R(x)$ was directly obtained according to Eq. (2). Polynomial fitting can be performed to the reconstructed results of $e_R(x)$ to make them smoother. As for $e_X(x)$, the reconstructed results always show good agreement with the input results even under the 5% noise level. The different performances for $e_Z(x)$, $e_X(x)$ and $e_R(x)$ can be interpreted according to Eqs. (2) and (5). From Eq. (5), we know that $e_X(x)$ is uncorrelated with $e_Z(x)$ and $e_R(x)$, while $e_Z(x)$ and $e_R(x)$ are linearly correlated according to Eq. (2). A small decrease (or increase) in $e_Z(x)$ will lead to the same amount of decrease (or increase) in $e_R(x)$. In the following examples, we would like to further examine the feasibility of the proposed method in the more noisy case, namely under the 5% noise level. It can be expected that the actual noise level of a Fizeau interferometer would be better than the more noisy case.

 

Fig. 4 (a) and (b) simulated phase outputs, where the black solid lines were generated by adding noise to the corresponding phase outputs with (a) a 2% noise level (γ = 0.02) and (b) a 5% noise level (γ = 0.02); Comparison between the input and reconstructed results for $e_Z(x)$, $e_X(x)$ and $e_R(x)$ of Example 1 in the noisy cases: (c) 2% noise level and (d) 5% noise level.

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Example 2: $e_Z(x)$ = 34.32 – 1.57x – 175.68x² − 10.92x³ + 119.27x⁴, $e_R(x)$ = 12.67 + 1.86x + 33.86x² + 3.33x³, and $e_X(x)$ is the same to that in Example 1

Compared with Example 1, Example 2 presents polynomials for $e_Z(x)$ and $e_R(x)$ with small linear terms. We want to examine feasibility of the proposed method when the grating under test and the reference flat have a small inclination. Figures 5(a) and 5(b) present the comparison between the input and reconstructed results for $e_Z(x)$ and $e_R(x)$ in the noise-free and noisy (with a 5% noise level) cases, respectively. As discussed above, since $e_X(x)$ is uncorrelated with $e_Z(x)$ and $e_R(x)$, different polynomials for $e_Z(x)$ and $e_R(x)$ will have no impact on the reconstructed result for $e_X(x)$. We therefore did not present the corresponding result for $e_X(x)$. As can be observed from Fig. 5, the reconstructed results still show good agreement with the input results in both noise-free and noisy cases. In the noise-free case, as shown in Fig. 5(a), the reconstructed polynomials for $e_Z(x)$ and $e_R(x)$ are $e_Z(x)$ = 34.32 – 175.68x² − 10.92x³ + 119.27x⁴ and $e_R(x)$ = 12.67 + 3.43x + 33.86x² + 3.33x³, respectively. By comparing with the input expressions, we know that, except for the linear terms, other polynomial coefficients are exactly the same to the corresponding input polynomial coefficients. Moreover, the linear term in $e_Z(x)$ is finally incorporated into the counterpart in $e_R(x)$ due to the linear correlation between $e_Z(x)$ and $e_R(x)$. Anyway, Fig. 5 suggests that the proposed method is still able to reconstruct $e_Z(x)$ and $e_R(x)$ when the grating and the reference flat have a small inclination. However, it should be noted that, when the inclination of the grating is larger (corresponding to larger linear terms), the reconstructed results for $e_R(x)$ will become worse due to the incorporation of the linear term of $e_Z(x)$.

 

Fig. 5 Comparison between the input and reconstructed results for $e_Z(x)$ and $e_R(x)$ of Example 2 in (a) the noise-free case and (b) the noisy case with a 5% noise level.

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Example 3: $e_Z(x,y)$ = 0.27 – 22.70x + 2.86y + 128.61x² + 15.75y² + 18.00x³ – 0.46x²y + 2.40xy² – 2.53xy² – 134.53x⁴ + 10.53x³y + 14.04x²y² + 10.53xy³ – 23.87y⁴, $e_X(x,y)$ = 0.56 – 125.80x + 2.82y + 62.26x² + 1.19xy + 11.12y² + 122.53x³ – 0.51x²y + 1.54xy² – 3.07xy² – 40.67x⁴ + 0.80x³y – 4.98x²y² – 1.87xy³ – 7.33y⁴, $e_Y(x,y)$ = 0.35 – 130.23x – 1.01y + 61.98x² + 5.26xy + 13.68y² + 127.33x³ + 1.09x²y + 1.20xy² – 0.53xy² – 39.87x⁴ – 4.27x³y – 5.63x²y² – 0.93xy³ – 9.87y⁴, and $e_R(x,y)$ = 7.37 + 1.03x + 3.89y + 0.06x² + 20.46y² – 0.13x³ + 0.29x²y – 0.40xy² – 3.33xy² + 40.93x⁴ + 0.40x³y – 17.39x²y² – 0.93xy³ + 0.93y⁴

Example 3 presents the expressions for $e_Z(x,y)$, $e_X(x,y)$, $e_Y(x,y)$ and $e_R(x,y)$ in the 2D case. We firstly try to reconstruct the polynomial coefficients in the noise-free case. As expected, the reconstructed polynomial coefficients are exactly the same to the corresponding input polynomial coefficients. Therefore, we did not present the comparison results between the input and reconstructed expressions in the noise-free case. We then try to reconstruct the polynomial coefficients in the noisy case. Figure 6 presents the comparison between the input and reconstructed results in the noisy case with a 5% noise level, where the 3rd column corresponds to the difference between the input (shown in the 1st column) and reconstructed (shown in the 2nd column) results, namely $\Delta e_i(x,y)$ = $e_i(x,y)$ –${\tilde{e}}_i(x,y)$ (i = Z, X, Y, R). As can be observed, the reconstructed results show good agreement with the corresponding input results. Note that, in Example 3, the polynomials for both $e_Z(x,y)$ and $e_R(x,y)$ do not contain the saddling term (namely the xy term). Similar to Example 2, it can be demonstrated that the proposed method is still applicable for $e_Z(x,y)$ and $e_R(x,y)$ with small saddling terms.

 

Fig. 6 Comparison between the input and reconstructed results for $e_Z(x,y)$, $e_X(x,y)$, $e_Y(x,y)$ and $e_R(x,y)$ of Example 3 in the noisy case with a 5% noise level. The 1st and 2nd columns correspond to the input and reconstructed results, respectively, and the 3rd column corresponds to the difference between the input and reconstructed results.

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3.2 Impact of the rotation center

It is required to make the origin of the coordinate system in coincidence with the rotation center of the tilt stage when applying the proposed method in the whole implementation process. This can be interpreted as follows. For simplicity, we take the 1D case as an example. A similar discussion can also be made for the 2D case. According to Eq. (12), it can be readily derived that, when the rotation center of the tilt stage coincides with the origin of the coordinate system, the first-order derivative of $f_X(x)$ at x = 0 will be equal to 0, namely ${f^{\prime }}_X(0)=0$. Since $f_X(x)={\displaystyle {\sum }_{k=0}^n{b}_k{x}^k}$, then ${f^{\prime }}_X(0)=b_1=0$, which indicates that there is no linear term in $f_X(x)$. Moreover, since ${f^{\prime }}_X(0)=0$ is always valid to any types of expressions for $f_X(x)$, the orthogonal polynomials, such as the Chebyshev polynomials and the Hermite polynomials [25], will also lead to a zero linear term in $f_X(x)$ ultimately. However, when there is an offset between the rotation center of the tilt stage and the origin of the coordinate system (let’s denote the offset as β), we will have the positive and negative first-order phase outputs at the slice of y = y_j as follows

According to the above discussion, attention should be paid to the relative position of the rotation center of the tilt stage and the origin of the coordinate system when using the proposed method. On the other hand, provided that the rotation center of the tilt stage coincides with the origin of the coordinate system, the property of the first-order derivative of $f_X(x)$ at x = 0, namely ${f^{\prime }}_X(0)=0$, can be used to guide the selection of the degree of the polynomial fitted to $f_X(x)$. Specifically, besides the fitting performance (typically estimated by the RMS error between the polynomial and the experimental data) of the polynomial fitted to $f_X(x)$, we should choose the degree of a polynomial that has a linear term as small as possible. Once the degree of the polynomial fitted to $f_X(x)$ is determined, the degree of the polynomial fitted to $e_Z(x)$ will also be determined. In addition, the property of ${f^{\prime }}_X(0)=0$ also indicates that $f_X(x)$ shall have an extremum at x = 0, provided that the second-order derivative of $f_X(x)$ at x = 0 is not equal to zero. Therefore, if $f_X(x)$ has a unique extremum within the range of interest, the extremum must appear at the origin of the coordinate system.

According to Eq. (27), we know that there will be a nonzero linear term in ${\tilde{f}}_X(x)$ when there is an offset β ($\beta \ne 0$) between the rotation center of the tilt stage and the origin of the coordinate system. The property of ${{\tilde{f}}^{\prime }}_X(0)\ne 0$ suggests that we might determine the linear term of $e_Z(x)$ if we have known the offset β. Specifically, substituting ${\tilde{f}}_X(x)={\displaystyle {\sum }_{k=0}^n{\tilde{b}}_k{x}^k}$ and $e_Z(x)={\displaystyle {\sum }_{k=0}^n{a}_k{x}^k}$ into Eq. (27), we have

Step 1: Measure the zero-order phase output $I_0(x,y)$;

Step 2: Coincide the rotation center of the tilt stage with the origin of the coordinate system, and measure the positive and negative first-order phase outputs $I_{X\pm 1}(x,y)$;

Step 3: Calculate the coefficients a_k (k = 0, 2, 3, …, n) of the polynomial fitted to $e_Z(x)$ by Eqs. (12) and (15);

Step 4: Move the rotation center of the tilt stage a distance of β with respect to the origin of the coordinate system, and measure the new positive and negative first-order phase outputs ${\tilde{I}}_{X\pm 1}(x,y)$;

Step 5: Calculate the coefficient a₁ of the linear term of the polynomial fitted to $e_Z(x)$ by Eqs. (27) and (29).

Example 4: $e_Z(x)$ = 21.03 – 87.15x – 182.43x² + 110.07x³ + 139.20x⁴, $e_R(x)$ = 12.67 + 15.86x + 33.86x² – 8.46x³, and $e_X(x)$ is the same to that in Example 1

We present Example 4 to investigate the feasibility of the above steps to obtain the linear term of $e_Z(x)$. Compared with Example 2, the expressions for $e_Z(x)$ and $e_R(x)$ have larger linear terms. Figures 7(a) and 7(b) present the comparison between the input and reconstructed results for $e_Z(x)$ and $e_R(x)$ in the noise-free and noisy (with a 5% noise level) cases, respectively. To obtain the linear term of $e_Z(x)$, the offset β was set to be β = 0.05 in the simulation. As can be observed from Fig. 7(a), in the noise-free case, the reconstructed results are exactly the same to the input results, which is also verified by comparing the reconstructed polynomial coefficients with the corresponding input polynomial coefficients. In the noisy case, as shown in Fig. 7(b), the reconstructed results also exhibit good agreement with the corresponding input results. The comparison result presented in Fig. 7 therefore verifies the feasibility of the above steps in obtaining the linear term of $e_Z(x)$. Note that here the present steps to obtain the linear term of $e_Z(x)$ are for the 1D case. It should be pointed out that similar steps can also be derived for the 2D case to obtain the saddling term of $e_Z(x,y)$. In addition, since all the coefficients of the polynomial fitted to $e_Z(x)$ at any slice of y = y_j are now obtained by the above five steps, we can also perform 2D polynomial fitting to the values of $e_Z(x)$ at different slices to obtain $e_Z(x,y)$.

 

Fig. 7 Comparison between the input and reconstructed results for $e_Z(x)$ and $e_R(x)$ of Example 4 in (a) the noise-free case and (b) the noisy case with a 5% noise level.

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4. Conclusions

A self-calibration method of the Fizeau interferometer and the planar scale grating has been proposed in this work. Detailed formulations for both 1D and 2D cases have been derived to evaluate the Z-directional out-of-flatness and the X-, Y-directional pitch deviations of the planar scale grating by measuring wavefronts of the zero-order as well as the positive and negative first-order diffracted beams of the grating. The simulation results in both noise-free and noisy cases have verified the feasibility of the proposed method. As for the linear term in the 1D case and the saddling term in the 2D case of the polynomial fitted to the Z-directional out-of-flatness of the planar scale grating, another measurement of the positive and negative first-order phase outputs needs to be carried out by moving the rotation center of the tilt stage a known distance with respect to the origin of the coordinate system.

Besides the evaluation of planar scale gratings, the surface profile of the reference optical flat in the Fizeau interferometer can also be calibrated in the meanwhile by using the proposed method. Compared with the existing calibration methods, such as the liquid-flat reference method and the three-flat test method, the proposed method is much simpler and more efficient for implementation, and is thus expected to gain wider applications in flatness calibration.

It should be noted that this paper is a report of the first step of the research where the proposal and preliminary verification of the proposed method in realizing self-calibration of the Fizeau interferometer and the planar scale grating were primarily focused. Further experimental demonstration and quantitative analysis of error factors influencing measurement uncertainty of the proposed method need to be carried out as future works in the next step of the research where the orthogonal polynomials, such as the Chebyshev polynomials and the Hermite polynomials [25], will also be used to further improve the robustness of convergence of the functions fitting to the real form errors.