1. Introduction

Absolute test aims to remove the systematic errors of the instrument without relying upon externally calibrated artifacts [1]. This approach becomes important when no calibration standards with sufficiently quantified accuracy are available because the required uncertainty in the measurement is of the same order as the instrument errors. Most of absolute tests methods [2–16] are based on the reversal principle [17] that uses 180° rotation of test part. Some averaging or integrating methods [7, 8, 10, 11, 16] need more rotations. The problem is that the previous absolute test methods assume no azimuthal position error during part rotation. If the test part is small enough, the azimuthal position errors can be greatly reduced by using an ultra-precision rotator. For large optics whose diameters are 0.6 m and over, however, exact rotations are physically difficult. Moreover, when higher order spatial frequency terms are required for rigorous inspection or alignment[18], rotation should be very precise because these higher order terms are particularly sensitive to azimuthal position errors. Motivated by this, we propose a new algorithm that permits part rotation to be made at arbitrary azimuthal positions. This generalized algorithm adopts least squares technique to determine the true azimuthal positions of part rotation and consequently eliminates testing errors caused by rotation inaccuracy. In addition it offers a great advantage of reducing the required number of part rotations.

2. Basic theory

The resulting wavefront W from any interferometric optical testing is composed of two partial wavefront components of

where T is the systematic error of the instrument including the reference surface, while P is the surface of the part to be measured. The above simple linear superposition of the two wavefronts is not strictly true but generally valid if they are of small orders as usual in most cases of optical shop testing. To separate the two wavefronts from each other, P is rotated about the optical axis by some physical means while T remains stationary. Then, let W_j be the wavefront of W sampled when P is stationed at an azimuthal position of α_j . The subscript j indicates the rotation index ranging from 0 to N-1, where α₀ =0 and N is the total number of part rotations. T undergoing no changes vanishes in the difference wavefront D_j , which is intermediately defined as the subtraction of

Now, with the intention of reconstruct P₀ from D_j , the expression of Zernike polynomials is adopted to decompose P₀ such as

where r and θ are the normalized radial and angular coordinates; $R_l^k$(r) the radial polynomials; c_lk and d_lk the coefficients of the angular terms. In line with P₀ expressed in Eq. (3), the rotated wavefront P_j is also described as

Equations (3) and (4) indicate that the Zernike coefficients c_lk and d_lk comply with the well-known transformation rule of vector rotation when they are regarded as the two orthogonal magnitude components of a two-dimensional vector. Then, substituting both the Zernike expressions of P₀ and P_j into Eq. (2) allows the coefficients of the difference wavefront Dj to be obtained such as Δc_lk≡c_lk-c’_lk and Δd_lk≡d_lk-d’ lk. Therefore, once D_j have actually been sampled and fitted to solve for Δc_lk and Δd_lk , the coefficients of the original part wavefront P₀ are readily determined as

Now we proceed to the averaging algorithm[8, 11, 16] that takes multiple part rotations to give the arithmetic mean of the measured wavefronts as the solution. To explain its underlying principles, by substituting Eqs. (3) and (4) into Eq. (2), the difference wavefront is rearranged in the form of

Then, summing up all the difference wavefronts leads to the expression of

The key idea of the averaging algorithm is to take the azimuthal angles to be equally spaced such as α_j=2πj/N, with the intention of making the most of the invariant properties of the sine and cosine harmonic functions of

Thus, if both the sums of sine and cosine terms are zero, Eq. (7) leads to the final form of the averaging algorithm of

This result was first proposed by Evans and Kestner [8], and named the multi-step averaging algorithm. In comparison with the algorithm using only 180° rotation, this arithmetic algorithm improves calibration accuracy and achieves a low uncertainty because any sampling error in a single wavefront is averaged out. However the error due to the rotation inaccuracy is still remaining.

3. Least square algorithm

The key idea of the newly proposed algorithm is that the rotated angles α_j are treated as additional unknowns together with the coefficients c_lk and d_lk . Then their actual values are determined from the measured wavefronts D_j using least squares technique. Our new algorithm beings with decomposing the part wavefront in terms of the angular order in k. Letting L(k) be the maximum radial order to be considered for each k, the partial sum is made up in detail such as

In Eq. (10), ${\xi }_l^k$ denotes another form of Zernike coefficients. Similarly, the sampled wavefront difference D_j is also expressed as D_j (r, θ)=${\sum }_{k=1}^K$ $D_j^k$ (r, θ) in which

For convenience, the subscript i is newly introduced to replace the notation (r,θ) such as $D_{\mathit{\text{ij}}}^k$ =${\sum }_l^{L\mathit{\left(}k\mathit{\right)}}$ $X_{\mathit{\text{lj}}}^k$$Z_{\mathit{\text{li}}}^k$. Now, let ${\widehat{D}}_{\mathit{ij}}^{\mathit{\text{k}}}$ be the actually measured value of ${\widehat{D}}_{\mathit{ij}}^{\text{k}}$ . Then computation for Zernike fitting of ${\widehat{D}}_{\mathit{ij}}^{\text{k}}$ and subsequent partial summing of the coefficients with same order of k allows $X_{\mathit{\text{lj}}}^k$ of Eq. (11) to be computed as ${\widehat{X}}_{\mathit{\text{lj}}}^k$ . In doing that, if the values of α_j are not correctly estimated, the computed values of ${\widehat{X}}_{\mathit{\text{lj}}}^k$ never equal the true values of $X_{\mathit{\text{lj}}}^k$ . The computational errors in ${\widehat{X}}_{\mathit{\text{lj}}}^k$ are arranged in the form of two cost functions, which are defined for each k such as

and

The former $E_l^k$ represents the partial sum of errors induced in the radial coefficients of Zernike fitting by inaccurate estimation of rotation angles α_j . On the other hand, the latter $E_j^k$ is the partial sum of errors resulting in the j-th wavefront. The cost functions should be minimized to determine the true values of the unknowns of ${\mathrm{\xi }}_{0\mathrm{l}}^{\mathrm{k}}$, $\tilde{\xi }$^k_0l and α_j . The necessary conditions are derived as

The above conditions are arranged in the form of matrix equations such as

No analytical solutions are found for the above simultaneous equations, thus iterative numerical technique is adapted. For each k, an initial guess is made for the azimuthal positions α_j so that ${\xi }_{\mathit{0}l}^k$ and $\tilde{\xi }$^k_0l are computed from Eq. (15). Then by using computed values of ${\xi }_{\mathit{0}l}^k$ and $\tilde{\xi }$^k_0l, the azimuthal positions α_j are upgraded from Eq. (16). Next step is go back to Eq. (15) with the new values of α_j and repeat the computation of ${\xi }_{\mathit{0}l}^k$ and $\tilde{\xi }$^k_0l, and α_j is adjusted again. The iterative computation between Eqs. (15) and (16) continues until the change of α_j converges into a predefined small value. Total computation time is influenced by the number of terms of Zernike polynomials in consideration, the number of rotation N, and the initially guess of α_j . Finally, with the converged values of ${\xi }_{\mathit{0}l}^k$ , the part wavefront P ₀ is reconstructed.

4. Simulation and experiment

The proposed algorithm, hereafter referred to as the least squares algorithm, has been tested to verify its advantages and usefulness through computer simulation. Fig. 1 describes a case study in which the performances of the previous averaging and least squares algorithms are compared when there are significant amounts of azimuthal position errors in part rotations. Fig. 1(a) illustrates the instrument error to be eliminated while the part wavefront is assumed perfectly flat. The number of steps was taken as 6 equally for both the algorithms and the maximum Zernike order of interest was k=5. Comparison reveals that the least squares algorithm effectively removes almost all the instrument error although part rotations are not accurately induced as intended. On the other hand, the averaging algorithm is limited in restoring the part wavefront especially around the circumference of the measured area.

 

Fig. 1. Comparison of simulation results when true values of α_j are not equally spaced mistakenly as 0°, 61°, 121°, 179°, 241°, 299°. (a) Original wavefront generated for simulation with all the Zernike coefficients being 0.1λ for k=1-5. (b) Fringe map of the original wavefront (P-V: 1.118µm, rms: 0.110λ). (c) The wavefront error extracted by the 6-step averaging (P-V: 0.017µm, rms: 0.001λ). (d) The wavefront error computed by the least-squares algorithm (P-V: 0.001µm, rms: 3×10^-7λ).

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Fig. 2. Suppression capabilities of high frequency components when αj are intentionally taken as 0°, 61°, 121°, 179°, 241°, 299°. (a) Original wavefront generated for simulation with all the Zernike coefficients being 0.1λ for k=1-8 (P-V: 1.763µm, rms: 0.11λ). (b) The wavefront error extracted by the 6-step averaging (P-V: 0.409µm, rms: 0.001λ). (c) The wavefront error computed by the least squares algorithm (P-V: 0.012µm, rms: 2×10^-5λ).

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Figure 2 describes another case in which higher order instrument errors are dominant in the range of k=6 to 8 as shown in (a). If the number of part rotation is taken as 6, the averaging algorithm fails to remove the 6th angular harmonic error components as illustrated in (b). On the other hand, the least squares algorithm suppresses the higher order instrument errors even with the same number of part rotations, demonstrating that higher order surface irregularities of the part are examined accurately. Detailed numerical data for comparison are listed in the Fig. 2.

 

Fig. 3. Simulated result after random noise inserting.

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Figure 3 shows the simulation result after random noise inserting in the original wavefront. The simulation procedure is here. 1) The original wavefront was generated with all the Zernike coefficients being 0.1λ for k=1-8 It has 1.763 µm in P-V value and 0.11 λ in RMS as show in Fig. 2(a). The diameter was 480 pixels. 2) Using the original wavefront, six rotated wavefronts were generated. The rotated angles were 0°, 60°, 120°, 180°, 240°, and 300°. 3) Using C++ code, six different random noise sets were automatically generated within the wavefront area. The P-V value of the noise was 0.01736 µm (1 % magnitude of the original wavefront) 4) Each random noise set (Step 3) was added on each rotated wavefront (Step 2) pixel by pixel. 5) We applied both algorithms, and calculated the residual errors due to the random noise. 6) We repeated the simulation procedure with different noise magnitude (2%, 3%, 4%, 5%, and 10% of the original wavefront.) The result is shown in Fig. 3. Our algorithm can reduce the random noise effect by 94.4 %.

To check our algorithm by experiment we tested 0.6 m spherical concave mirror with a Fizeau interferometer. Actual testing diameter was about 0.5 m, and the rotated angles αj were taken as 0°, 90°, 180°, 270° (4-step) instead of 6-step for experimental convenience. The least square algorithm gave us the estimated values of α_j as 0°, 92.17°, 180.98°, 271.20°. Table 1 shows typical aberration coefficients.

Table 1. Typical aberration coefficients before and after absolute tests. The unit is λ (632.8 nm).

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5. Conclusions

Our intention in this paper is to improve the previous absolute test. The least squares algorithm proposed in this paper eliminates testing errors caused by rotation inaccuracy and also offers a great advantage of reducing the required number of part rotations drastically when higher order spatial frequency terms are of particular importance. The latter benefit is obtained by imposing a predetermined small amount of intentional offset in the azimuthal positions during part rotations. To check the performance of the suggested algorithm, we applied it to 0.6 m spherical concave mirror testing. Furthermore, we have a plan to apply the new algorithm to a 300 mm wafer flatness test and a 0.9 m aspheric concave mirror [19] test.