1. INTRODUCTION

High-numerical-aperture (NA) spherical lenses are required for the design of wide-angle or high-magnification optical products such as endoscopes, microscopes, and compact cameras. The repeatability of spherical surface measurement has reached ${\sim}0.1\;{\rm nm} $, as achieved using a phase-shifting Fizeau interferometer, which improves the quality control of optical products. There are several error sources in spherical testing, including the spatial nonuniformity of the phase shift, misalignment-induced aberrations, and instantaneous tilt and defocus during phase modulation. In high-NA optical lens surface tests, the spatial nonuniformity of the phase shift within the observation aperture is critical for mechanical modulation. The amount of phase shift in the marginal region of the aperture decreases in proportion to the direction cosine between the beam direction and the translation direction of the transmission sphere [1]. This nonuniformity can couple with the multiple-reflection light between the transmission sphere and test surface, resulting in phase measurement errors. These phase measurement errors become 1 nm root-mean-square (rms) or larger when the nonuniformity exceeds ${-}30\%$ (${\rm NA} = {0.71}$) [2].

A Fizeau interferometer with a wavelength tuning source is another option for spherical testing. There is no spatial nonuniformity of the phase shift in this configuration. However, the intensity variation of the source during wavelength scanning causes another type of error in the measured phase [3].

Iterative approaches [4–8] can be used to estimate the spatial nonuniformity of the phase shift if a sufficient number of interference images have been recorded. Because the decrease of the phase shift due to geometry can be easily predicted, the initial value of the iterative parameter can be easily determined. Iterative approaches were first formulated for a two-beam interferometer and then for a Fizeau cavity, for which the parameter equations become nonlinear.

Many phase-shifting algorithms that can compensate for phase-shift errors have been proposed. The coupling error between the phase-shift error and multiple-reflection light has been discussed [2,9] and compensated for to be first order by several recent algorithms such as the 15-frame algorithm by de Groot [2], the $4N - 3$-frame algorithm by Kim et al. [10], and the 13-frame algorithm by the present authors [11]. However, in a previous study, we showed that when the NA reaches 0.86 and the corresponding phase-shift error becomes ${-}50\%$, these linear approximations are no longer sufficient, with typical phase measurement errors of 5 nm rms or larger [12].

We previously proposed a synthetic approach [12] in which the observation aperture is divided into several annular regions and the object phase in each region is calculated by an individual algorithm specifically designed for the central value of the phase shift interval in the region. All the algorithms shared the same set of interference fringes recorded in a single phase-shift sequence. With this technique, the maximum nonuniformity of the phase shift decreased from ${-}50\%$ to ${-}8\%$, which significantly reduced the error in the measured phase. However, we observed an error in the discrete phase change of less than 1 nm at the regional boundary. Because the algorithms have different sensitivities to the phase-shift error, a discrete change of the error can occur at the boundary.

Figure 1 shows an example of the sensitivity difference. Figures 1(a) and 1(b) show the measured deviations of the same spherical surface with an NA of 0.86 obtained using a synthetic calculation. In the inner half of the apertures (${0}\; \lt \;r\; \lt \;{\rm half}\;{\rm radius}$), the phases were calculated using two different 90°-step 11-frame algorithms (see Table 1, 11-frames $A$ and $B$); in the outer half (half radius ${\lt}\;r$), the phases were calculated using a 72°-step algorithm (see Table 1, 11-frame $C$). To observe the boundary errors, the large background signal associated with the object shape should be subtracted from these results. The object deviation calculated using algorithm $C$ over the whole aperture [see Fig. 1(c)] was thus subtracted from the two results. Figures 1(d) and 1(e) show the differences. A comparison of these figures indicates that the gap error at the $A - C$ boundary is much smaller than that at the $B - C$ boundary. It will be shown that the dc bias in the calculated phase is proportional to the error coefficient ${K_0}$. Because algorithms $B$ and $C$ have different error coefficients, namely, ${K_0} = 81.8$ and 52.0 nm, respectively, we expect a discrete phase change of ${-}0.62\;{\rm nm} $ caused by the piezoelectric transducer (PZT) nonlinearity of ${-}2.34\%$. Figure 1(f) shows the X cross section of the $B - C$ boundary shown in Fig. 1(e), where we applied a ${\pm}20$-pixel spatial average along the $y$ direction to suppress random noise. We observed a discrete phase change of ${\sim}- 0.6\;{\rm nm} $ at the $B - C$ boundary. The magnitude of this error, which typically occurs in open-loop PZT modulation, was not negligible in recent high-precision spherical tests [15,16].

 

Fig. 1. Measured deviations of spherical surface obtained using the synthetic method. (a) Algorithms $A$ and $C$, (b) algorithms $B$ and $C$, (c) algorithm $C$, (d) difference between (a) and (c), (e) difference between (b) and (c), and (f) X cross-sectional view of boundary in (e), where a ${\pm}20$-pixel spatial average was applied along the $y$ direction. The error coefficient ${K_0}$ for algorithms $A,B$, and $C$ is 52.8, 81.8, and 52.0 nm, respectively.

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Table 1. Algorithms: Sampling Windows and Their Error Coefficients to Phase-Shift Error ($\lambda = 633 \; {\rm nm}$, $R = 0.04$)

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In this paper, we propose a method for aligning the error coefficients of a set of algorithms to the phase-shift error. As shown in Fig. 1, the dc bias error in the measured phase caused by PZT nonlinearity is important in the synthetic calculation even though it is known as a “spurious piston” and is not relevant in conventional measurements. The dc bias phase is proportional to the momentum inertia of the sampling window of an algorithm. Although the dc bias can be eliminated through algorithm design [15–17], this increases sensitivity to random noise and other ripple noise.

We thus attempt to keep coefficient ${K_0}$ aligned to a certain value. We systematically adjust the momentum and other error coefficients based on the convolution of the sampling window with a three-frame processing window. Early works on algorithm design based on convolution attempted to increase the order of zeros at the harmonic frequencies of the frequency transfer function (FTF) [13,18]. It is interesting to note that the proposed convolution not only increases the order of zeros but also allows the position of zeros on the frequency axis to be manipulated. This manipulation, which can be used to directly adjust coefficient ${K_0}$, can be realized by fixing a single parameter for the processing window. Five algorithms that have the same error coefficients but different optimal phase shifts are derived and used as examples. These algorithms are applied to the synthetic measurement of a spherical surface with an NA of  0.86.

2. SYNTHETIC APERTURE PHASE-SHIFT MEASUREMENT

A. Optical Setup for Spherical Surface Fizeau Measurement

We first describe the signal of a Fizeau interferometer and the phase-shift errors caused by the direction cosine of the illuminating beams and the nonlinear response of a PZT. Figure 2 shows the optical setup for a spherical concave test with a mechanical phase shift. The source is a volume Bragg grating stabilized single-mode diode laser with wavelength $\lambda = 633\;{\rm nm} $ (Necsel IP, SLM-632.8-L-PMF). The output beam is transmitted through a rotating ground glass diffuser and a multi-mode fiber to reduce the lateral coherence of the beam. Lateral coherence is the correlation within the wavefront plane normal to the beam direction; it can cause speckle noise and internally scattered light noise [11]. The output from the fiber is transmitted through a polarization beam splitter, and the linearly polarized beam is transmitted through a quarter-wave plate to change its polarization from linear to circular. The beam is then collimated to illuminate the transmission sphere and the test surface. The reflections from both surfaces, which return along the original path, are transmitted through the quarter-wave plate again to acquire orthogonally linear polarization, and then reflected by the polarization beam splitter and combined to form interference fringes on a CCD camera (IMPERX, GEV-B1621W-TC0, $1632 \times 1232$ pixels). The test surface and transmission sphere are placed horizontally. The transmission sphere is translated along the optical axis by a PZT to introduce a phase shift. The interference images are recorded during the phase shift over equal time intervals. The relative phase shift between frames is adjusted to $\pi /2$ radians on the axis.

 

Fig. 2. Optical setup for phase-shift Fizeau interferometer. PBS, polarization beam splitter; QWP, quarter-wave plate; TS, transmission sphere; PZT, piezoelectric transducer.

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The irradiance signal of the Fizeau interferometer observed at position $(x,y)$ on the detector is given by [19]

The intensity of Eq. (1) can be expanded into a series of $\sqrt {{R_1}{R_2}} \cos (\alpha + {\varphi _0})$ terms, expressed as

B. Linear Error Coefficients in Phase-Shift Measurement

Here, we analytically derive the first-order error coefficients caused by the direction cosine and nonlinear phase-shift errors. The object phase, defined in Eq. (6), is calculated using an $M$-frame phase-shifting algorithm:

Substituting Eqs. (2) and (6) into Eq. (7) and expanding the sinusoidal terms in the order of coefficients ${\varepsilon _i}$ using the approximations $\sin \varepsilon \alpha = \varepsilon \alpha$ and $\cos \varepsilon \alpha = 1$, we obtain the linear terms of the phase measurement error as

Equation (10) shows the lowest-order errors that are linear with respect to coefficients $\varepsilon _i^\prime $ or reflection index $R$. The residual errors not shown in this equation are higher-order terms that include ${\varepsilon _i}{\varepsilon _j}$ or ${\varepsilon _i}{R^2}$.

Table 1 shows, as examples, the specific values of these coefficients for conventional algorithms and the three algorithms shown in Fig. 1. Many error compensation algorithms have been designed to eliminate the first coefficient $J$ because the gain error $\varepsilon _1^\prime $ is usually dominant among the phase-shift errors. An experimental value for the second coefficient $\varepsilon _2^\prime $ is ${\sim}- 0.03$. Coefficient $\varepsilon _1^\prime $ exceeds ${-}0.4$ due to the direction cosine.

Each term in Eq. (10), except for the second term, depends on the object phase ${\varphi _0}$. The second term ${K_0}\varepsilon _2^\prime $ is a spurious piston whose typical value is ${\sim}{-} 1.5\;{\rm nm} $ (for ${K_0} = 50\;{\rm nm} $ and ${\varepsilon _2} = - 0.03$). As long as a single algorithm is used, this term contributes to a bias, most of which cannot be distinguished from piston or defocus components. This bias thus does not deform the object shape. It becomes important only when we measure the absolute phase, such as the absolute optical thickness [17].

In this paper, however, we use several algorithms depending on the position in the observation aperture. The bias is no longer uniform, as shown in Fig. 1. We therefore try to match the ${K_0}$ values for all algorithms to make the bias uniform, as discussed in Section 2.D. Coefficients $J$, ${K_1}$, ${L_1}$, and ${L_2}$ must be as small as possible (ideally zero). These coefficients are closely related to the frequency response and the characteristic polynomials of the algorithm, as briefly described in the next section.

C. Frequency Response and Characteristic Polynomials of the Algorithm

Here, we briefly describe the FTF [21] and the characteristic polynomials [22] for further discussion of the convolution technique. The algorithm can be rewritten in integral form as

The Fourier transforms of these functions [23] are then derived as

The Fourier transform gives the necessary condition for extracting the object phase as

The signal spectrum $K(\nu)$ is discrete. Its peaks are distributed at $\nu = 0, \pm 1, \pm 2, \ldots$. To extract the correct object phase, the FTF should become zero at these harmonic frequencies except for the fundamental frequency $\nu = 1$. The zero positions of the FTF can be classified by the characteristic polynomials that are derived from the function as

Apart from the irrelevant phase factor ${z^{m - 1}}$, the zero positions of the FTF can be identified on the unit circle in the complex plane as the roots of polynomial equations: $P(z) = 0$. The irrelevant factor ${z^{m - 1}}$ results from the Hermitian definition of the phase shift ${\alpha _{0r}}$, so that both ${F_1}$ and $i{F_2}$ become real numbers. Surrel [22] showed that when the polynomials have double and triple roots at frequency $\nu = - 1(z = \exp (- 2\pi i/N))$, zero coefficients $J$ and ${K_1}$ are obtained, respectively. Similarly, when the polynomials have double roots at both $\nu = \pm 2$, the algorithm has zero coefficients ${L_1}$ and ${L_2}$.

D. Alignment of Error Coefficients by Convolution

Here, we derive five algorithms that have zero coefficient $J$ and matched coefficient ${K_0}$. Suppose that a $p + q - 1$-frame sampling window $\{{w_n}\}$ is generated by the convolution of a $p$-frame window $\{{u_{{n^\prime}}}\}$ with a $q$-frame window $\{{\nu _{{n^{{\prime \prime}}}}}\}$ as

We are now going to derive 11-frame algorithms with divisors $N = 4,5,6,7,8$. We start with an $N$-frame rectangular window $(1,1, \ldots ,1)$ or an $N + 1$-frame trapezoidal window $(1,2,2, \ldots ,2,1)$ depending on whether $N$ is odd or even.

Table 2 shows the starting windows and additional three-frame processing windows for each divisor. We convolve the starting window with the processing window several times to acquire the necessary characteristics, namely, $J,{K_1}$ elimination and ${K_0}$ adjustment. The final 11-frame windows need to have double or triple roots at frequency $\nu = - 1(z = \exp (- 2\pi i/N))$ in their polynomials and have matched coefficients ${K_0}$. We prepare a three-frame processing window as

Table 2. Processing Windows for Algorithm Derivation

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Fig. 3. Positions of two roots ${\rho ^{1 + k}}$ and ${\rho ^{1 - k}}$ of characteristic polynomials in complex plane where processing window is ${W^{(k)}} = (1, - 2\cos (2\pi k/N),1)$ for $N = 8$ and $k = 1.8$.

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Let us discuss the case $N = 4$ as an example. We start with a five-frame trapezoidal window $(1,2,2,2,1)$ that is represented by the convolution of ${W^{(1)}} = (1,0,1)$ and ${W^{(2)}} = (1,2,1)$. This window is known as the Schwider–Hariharan algorithm [13]. The characteristic polynomials are obtained from Eq. (25) as $P(z) = - ({z^2} - 1)(z + i{)^2}$, which has double roots at $\nu = - 1(z = \exp (- 2\pi i/4))$. The starting window thus gives a zero-coefficient $J.$ We then proceed to the triple roots, which can eliminate coefficient ${K_1}$. Taking the convolution with the processing window for $k = 2$ as

For one of the two remaining convolutions, we could choose the first convolution with ${W^{(1)}}$ to simultaneously eliminate coefficients ${L_1}$ and ${L_2}$. However, the resultant momentum ${K_0}$ becomes so large that we cannot adjust it using the final convolution. Therefore, we discuss a ${K_0}$ adjustment. We will show that the last case $N = 8$ has no degree of freedom for ${K_0}$ adjustment and gives a ${K_0}$ coefficient of 51.4 nm. Therefore, we attempt to converge all other ${K_0}$ values to this value. We broaden the definition of parameter $k$ from an integer to a real number. Coefficient ${K_0}$ of the resultant window after convolution is a function of parameter $k$ and decreases monotonically as $k$ varies from zero to $N/2$. It can easily be found that we obtain the matched momentum by convoluting twice with the processing window for $k = 1.41$ as

Equation (33) is the final window for $N = 4$, which gives $J = {K_1} = 0$ and ${K_0} = 51.4\;{\rm nm} $. Figure 4 shows the positions of the roots of the characteristic polynomials for this window. The additional four roots are located outside the harmonic signal peaks on the unit circle.

 

Fig. 4. Root positions and FTFs for five algorithms, where sampling weights are shown in Table 3: (${{\rm a}_{1,2}}$) $N = 4$, (${{\rm b}_{1,2}}$) $N = 5$, (${{\rm c}_{1,2}}$) $N = 6$, (${{\rm d}_{1,2}}$) $N = 7$, and (${{\rm e}_{1,2}}$) $N = 8$. The ordinate of the FTF is compressed as $(1/2)\log 10[|F(\nu)| + 1]$, where the sampling weights are normalized with ${\rm sumwr} = 99$ for clarity.

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Next, we briefly discuss the case of $N = 5$. We start with a five-frame rectangular window $(1,1,1,1,1)$ that is represented by the convolution of ${W^{(1)}} = (1, - 0.618,1)$ and ${W^{(2)}} = (1,1.618,1)$. This window originally has a single root at $\nu = - 1(z = \exp (- 2\pi i/5))$ in the polynomials. Performing two convolutions with the processing window ${W^{(2)}}$ generates a nine-frame window that has triple roots at $z = \exp (- 2\pi i/5)$ and thus gives zero-coefficients $J$ and ${K_1}$. We convolve this window with the window ${W^{(1.29)}} = (1,0.093,1)(k = 1.29)$ to adjust the momentum. The resultant 11-frame window for $N = 5$ is obtained as

For divisors $N = 6$ or larger, we similarly convolve each starting window with the processing window ${W^{(2)}}$ to increase the order of roots at $\nu = - 1$ to eliminate coefficient $J$, and then choose an adequate window ${W^{(k)}}$ to adjust the momentum coefficient ${K_0}$. For the divisor $N = 8$, because the starting window is a nine-frame window and has no freedom to adjust ${K_0}$, we convolve it only once to eliminate coefficient $J$.

Table 3 shows the resultant five windows and their error coefficients, where we assume that $\lambda = 633\;{\rm nm} $ and $R = 0.04$. Coefficients $J$ are eliminated for all windows, and coefficients ${K_1}$ are eliminated for $N = 4$ and 5. Coefficients ${K_0}$ are adjusted to 51.4 nm. The spurious piston error due to ${K_0}$ mismatch is therefore expected to be eliminated to first order. The signal-to-noise ratio (SNR) in the table is the sensitivity to the random noise defined in [15]. The maximum value is less than the number of frames $(M = 11)$.

Table 3. 11-Frame Algorithms for Five Regions and Their Error Coefficients ($\lambda = 633\;{\rm nm}$, $R = 0.04$)

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Table 4. 11-Frame Algorithms for Five Regions and Their Error Coefficients Generated Using Our Previous Method ($\lambda = 633\;{\rm nm}$, $R = 0.04$)

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Coefficients ${L_{1,2}}R$ are not discussed here because they are smaller than the other coefficients and because the number of frames is limited to 11. To eliminate these coefficients, we need to increase the number of image frames to 13, 15, or more and introduce processing windows ${W^{(1)}} = (1, - 2\cos (2\pi /N),1)$ and ${W^{(3)}} = (1, - 2\cos (6\pi /N),1)$. For the divisor $N \le 6$, one additional convolution with ${W^{(1)}}$ is sufficient to eliminate ${L_{1,2}}$. For the divisor $N \gt 6$, two additional convolutions with ${W^{(1)}}$ and ${W^{(3)}}$ are necessary to eliminate both coefficients.

Finally, we calculate the FTF for the five algorithms. Figure 4 shows the FTFs and the positions of the polynomial roots. The FTFs are zero at the negative fundamental frequency $\nu /{\nu _0} = - 1$ and in its vicinity, which shows robustness against the phase-shift error.

E. Numerical Calculation of Phase Measurement Errors

We numerically evaluate the phase measurement errors during the test of a spherical surface with NA = 0.86 and ${R_1} = 3.5\%$ compared to a transmission sphere with ${R_2} = 7.9\%$. For simplicity, we assume that the phase-shift error ${\varepsilon _1}$ is caused by only the direction cosine and that the PZT nonlinearity ${\varepsilon _2}$ is ${-}3\%$. Coefficient $\varepsilon _1^\prime $ for each region is then given by $(N/{N_{\rm{min}}})\cos \theta - 1$, where ${N_{\rm{min}}} = 4$.

Table 3 shows the five aligned algorithms. For comparison, Table 4 shows five algorithms (and their error coefficients) derived using our previous method [12], in which a couple of rectangular windows are convoluted with the processing window ${W^{(2)}}$ as

Figure 5 shows the distribution of ${K_0}$ values for both sets of algorithms. The previous algorithms have uneven ${K_0}$ values, especially at the boundary between $N = 5$ and $N = 6$. Because the bias phase in each region is ${K_0}{\varepsilon _2}\cos \theta$, a gap error of 0.2 nm is expected at this boundary for a nonlinearity of ${\varepsilon _2} = - 3\%$.

 

Fig. 5. Coefficient ${K_0}$ for sets of five algorithms $(N = 4,5,...8)$ shown in Tables 3 and 4.

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Figures 6(a) and 6(b) show the calculated phase measurement errors as a function of NA for the mismatched (Table 4) and matched algorithms (Table 3), respectively. In Fig. 6(a), each region has a different dc bias, which directly leads to profile error. In Fig. 6(b), in contrast, the bias for all regions converges to ${-}1.5\;{\rm nm} $, which is not relevant for the profile measurement. After the momentum alignment, the residual errors at the boundary between $N = 7$ and $N = 8$ in Fig. 6(b) are caused by ${K_1}{\varepsilon _2}\cos 2{\varphi _0}$, whose elimination was not discussed in Section 2.D.

 

Fig. 6. Phase measurement errors for the synthetic method for spherical surface with ${R_1} = 3.5\%$ compared with the transmission sphere with ${R_2} = 7.9\%$ with ${\varepsilon _2} = - 3\%$ and measured phase $\psi = 45$ degrees: (a) conventional sampling weights shown in Table 4 and (b) inertia-aligned sampling weights shown in Table 3.

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3. EXPERIMENTS

A glass spherical concave surface with a diameter of 17 mm, ${R_1} = 3.5\%$, and ${\rm NA} = {0.86}$, shown in Fig. 7, was compared to a transmission sphere with a diameter of 38 mm, ${R_2} = 7.9\%$ and the same NA. The optical setup of the measurement is shown in Fig. 2. The transmission sphere was translated toward the object surface by a PZT, and 11 interference images were recorded with an equal phase-shift interval of $\pi /2$. The nonlinearity of the open-loop PZT actuator exhibited temporal drift. For convenience, we recorded two sets of images with different nonlinearities, namely, ${\varepsilon _2} = - 0.22\%$ and ${-}3.58\%$, as shown in Fig. 8.

 

Fig. 7. Photograph of the spherical concave test object with a diameter of 17 mm and a radius of curvature of 10 mm (${\rm NA} = {0.86}$).

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Fig. 8. Measured fringe image sequence, where acquisition order is shown in the upper-left corner: (a) ${\varepsilon _2} = - 0.22\%$ and (b) ${\varepsilon _2} = - 3.58\%$.

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The PZT nonlinearities for these images were estimated by minimizing the sum-squared error defined as

Figure 9 shows the observed fringe luminance (dots), calculated luminance (curves), and estimated nonlinear phase shifts (broken curves). The last one, as the estimated nonlinear phase shifts, was normalized by the half width of the total phase shift $T$ by subtracting unperturbed phase shift ${\alpha _{0r}}$ from phase shift ${\alpha _r}$ to visualize the nonlinear components.

 

Fig. 9. Estimation result for nonlinear coefficient ${\varepsilon _2}$ and other coefficients from measured luminance value ${I_r}$: (a) ${\varepsilon _1 = 4.66\% },{\varepsilon _2 = - 0.22\% },{\varepsilon _3} = - 2.54\%$ and (b) ${\varepsilon _1} = 0.94\% ,{\varepsilon _2 = - 3.58\% },{\varepsilon _3 = - 0.95\%}$.

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The boundary position between the two regions of divisors $N$ and $N + 1$ is defined as [12]

 

Fig. 10. Annular sub-aperture division of observation aperture on the test object and corresponding phase-shift devisors for each region.

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The object phases were calculated using the five aligned algorithms in Table 3 and the five conventional algorithms in Table 4. The synthetic object phases were then unwrapped.

Figures 11(a₁) and 11(b₁) show the object phases calculated using our previous algorithms for the images with ${\varepsilon _2} = - 0.22\%$ and ${-}3.58\%$, respectively. To visualize the phase measurement error of the sub-nanometer magnitude, we have to subtract the large background due to the object shape from the results. Figure 11(c₁) shows the difference between Figs. 11(a₁) and 11(b₁); it cancels the object shape component and thus represents the difference of PZT nonlinearities. We can clearly observe a gap error of ${0 {-} 0.2}\;{\rm nm}$ along the annular boundary between the regions of $N = 5$ and $N = 6$. In these regions, the previous algorithms have ${K_0}$ values of 57.6 and 51.0 nm, respectively. From the different magnitudes of the ${K_0}{\varepsilon _2}\cos \theta$ term, the gap is expected to be 0.2 nm, which is consistent with the above observation.

 

Fig. 11. Measured phase distributions of spherical surface with ${\rm NA} = {0.86}$ obtained using five 11-frame algorithms (after removal of tilt and defocus): (${{\rm a}_1}$, ${{\rm b}_1}$) sampling weights are shown in Table 4; (${{\rm a}_2}$, ${{\rm b}_2}$) sampling weights are shown in Table 3, where (${{\rm a}_{1,2}}$) is ${\varepsilon _2} = - 0.22\%$ and (${{\rm b}_{1,2}}$) is ${\varepsilon _2} = - 3.58\%$; (${{\rm c}_1}$) ${B_1} - {A_1}$ and (${{\rm c}_2}$) ${B_2} - {A_2}$; (d) cross-sections of (${{\rm c}_1}$) and (${{\rm c}_2}$) along the $x$ axis, where each pixel value is averaged over ${\pm}20$ pixels in the $y$-axis direction, and the defocus and piston are compensated for to match the edge shape for the two methods.

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Figures 11(a2), 11(b2), and 11(c2) show the calculated object phases for ${\varepsilon _2} = - 0.22\%$ and ${-}3.58\%$ and their difference obtained using the present algorithms, respectively. In Fig. 11(c₂), the boundary gap becomes obscure and is difficult to observe.

Figure 11(d) shows cross-sectional views of Figs. 11(c1) and 11(c2) along the $x$ axis, where we applied a spatial average of ${\pm}20$ pixels in the $y$ axis direction to suppress random noise. The defocus ${\rm Z3} = {0.15}\;{\rm nm}$ and the piston ${\rm Z0} = - {0.11}\;{\rm nm}$ from the Zernike polynomials [24] were added to the profile of the conventional method to match the edge shape at the aperture periphery for the two methods. We clearly observe that the surface profile obtained using the previous algorithms undergoes a systematic shift of 0.15–0.2 nm in the negative direction in the two regions of $N = 4$ and 5. The bias coefficient ${K_0}$ for the previous algorithms is ${\sim}58\;{\rm nm} $ for these regions. This value is 6–7 nm larger than the aligned value of 51.4 nm.

These observations show only the difference of the bias component in the object profile that depends on the nonlinearity ${\varepsilon _2}$ of the PZT response because the spatial nonuniformity in the bias value generally has an error of less than 1 nm and is much smaller than the object shape deviations. Nevertheless, we verified that our linear estimation of the error sensitivities to phase-shift nonlinearities is valid and well describes the bias variation. We can conclude that our systematic adjustment of the error coefficients decreases the spatial nonuniformity of the bias error, and thus improves the robustness of the synthetic method to a nonlinear phase shift.

4. CONCLUSION

In spherical surface tests by Fizeau interferometers with a mechanical phase shift, spatial nonuniformity in the phase shift within the observation aperture occurs in proportion to the direction cosine between the illumination beam direction and the translation direction of the transmission sphere. We divided the observation aperture into several annular regions and calculated the object phase using different algorithms designed for different optimal phase shifts. This synthetic calculation significantly decreases the error in the measured object phase; however, the dc bias phase (spurious piston) that differs between the algorithms causes a new type of spatially nonuniform error. The dc bias is proportional to the second-order nonlinearity of the PZT response and the momentum inertia of the sampling window of the algorithm. We aligned the momenta of the five algorithms by convolution of the sampling windows with our three-frame processing window. The processing window can also eliminate the linear error coefficients. The resultant five algorithms have the same bias and reduce the error coefficients to the first- and second-order nonlinearities of the phase shift. The numerical simulation and experimental demonstration showed that the algorithms can minimize the dc bias error in spherical testing for an NA of 0.86.