Absolute measurement of optical flats based on basic iterative methods

Haiyang Quan; Xi Hou; Fan Wu; Weihong Song

doi:10.1364/OE.23.016305

1. Introduction

Interferometric measurement of the surface form or figure of optical surfaces generally involves a comparison of the surface under test with a reference surface. For high-accuracy measurements, it is necessary to separate reference errors from test surface errors, especially when the reference errors have the same order magnitude as the test surface errors. Absolute test techniques involve multiple measurements, using a plurality of probes, artefacts, and/or set-ups [1], which actually form a linear system Ax = b, where x is the vector of unknown variables, b is the vector of observation values, and A is the system matrix. Over the last decade or so, a number of absolute test techniques have been presented in the literatures [2–14].

The key is how to solve such linear system of equations effectively. Typically the system matrices are large, sparse, and structured, resulting in more computational costs and memory space requirements. To solve such general, large sparse linear systems, direct methods are not economical and cannot get the exact solutions [15, 16]. While special direct methods such as QR decomposition [16] and LSQR algorithm [17] can save much working storage and computation cost, they cannot yet effectively solve the large-size problems [11]. In absolute measurement of optical surfaces, a fitting procedure based on Zernike polynomials is widely used to obtain the absolute Figs [4, 12–14]. However, usage of a fitting procedure to mathematically retrieve the absolute figures can only solve the modified (low spatial frequency filtered) components.

Generally, it is not possible to solve the problem directly. In this case, iterative algorithms can approach the correct solution using multiple iterations, which work by repeatedly improving the approximate solution until it is accurate enough. And the first application of iterative methods to the approximate solution of three-flat absolute test problem was proposed by Maurizio Vannoni and Giuseppe Molesini [18, 19]. The iterative algorithm has already been used for solving standard three-flat problem, both in horizontal and vertical configurations [20]; it was also used in some other application research field [21, 22]. And it was extended to solve the rotationally asymmetric components in three-flat test [23] and skip-flat test [24].

In the previous reports of iterative algorithm for absolute flatness measurement, most investigators paid much attention to the application of the exiting iterative algorithm proposed by Vannoni and Giuseppe Molesini. Few of them attached importance to the investigation of the iterative method itself, especially to the convergence of the iterative algorithm. For example, as we known, no one has pointed out what exactly the iterative method is and what principle the exact iterative method is.

In fact, the term “iterative method” refers to a wide range of techniques which use successive approximations to obtain more accurate solutions. Among them, two classes are most popular, including basic iterative methods (or stationary methods) and none-stationary methods (such as conjugate gradient method) [15, 25]. In this paper, we focus on the basic iterative methods: the Jacobi method, the Gauss-Seidel method, the Successive Over-relaxation (SOR) method, and the variants of them. Basic iterative methods are easy to understand and implement, and usually effective for absolute measurement of optical surfaces. Two main categories of basic iterative methods for absolute measurement of optical flats are introduced, namely block Jacobi SOR method and block SOR method. The convergence of both iterative methods is analyzed through both virtual experiment and real experiment. Factors that influence iterative performance have been discussed. The proposed basic iterative methods are generalized; are easy to understand and implement; and could be effectively used for almost all the absolute test models, including both absolute flatness measurement and absolute sphericity measurement.

2. Generalized iterative methods for absolute measurement of optical flats

2.1 The three-flat problem (modeling)

For classical three-flat problem, the general mathematical model of the absolute measurement of optical flats is:

{\begin{cases} K M = F_{y} K + M \\ L M = F_{y} L + M \\ L M_{R} = F_{y} L + R_{φ} M \\ L K = F_{y} L + K \end{cases}

Where K, L, and M are the surface errors of the three flats (measurand), and KM, LM, LM_R, and LK the measured combined surface errors. Here we use the symbolic linear operators proposed by Maurizio Vannoni in 2012 [26], say

F_{y} K

means a flip of K about y-axis, i.e.

F_{y} K = K (- x, y)

;

R_{φ} M

means a flat M is rotated by a fixed angle

φ

about z–axis (54 degrees is widely used in practice).

After refining the above equations by several linear operators, we get the following refined model (named ‘Model $Ι$ ’):

Model I : {\begin{cases} K = \frac{A_{K} - (F_{y} L + F_{y} M)}{2} \\ L = \frac{A_{L} - [F_{y} K + F_{y} M + F_{y} R_{φ}^{} M)]}{3} \\ M = \frac{A_{M} - (F_{y} K + F_{y} L + R_{φ}^{- 1} F_{y} L)}{3} \end{cases}

Where A_K, A_L, and A_M are the components related to measured data, namely,

{\begin{cases} A_{K} = F_{y} (K M) + L K \\ A_{L} = F_{y} (L M) + F_{y} (L M_{R}) + F_{y} (L K) \\ A_{M} = K M + L M + R_{φ}^{- 1} (L M_{R}) \end{cases}

Also we get the following refined model (named ‘Model $Ι Ι$ ’):

Model I I : {\begin{cases} K = K + \frac{F_{y} [Δ (K M)] + Δ (L K)}{2} \\ L = L + \frac{F_{y} [Δ (L M)] + F_{y} [Δ (L M_{R})] + F_{y} [Δ (L K)]}{3} \\ M = M + \frac{Δ (K M) + Δ (L M) + R_{φ}^{- 1} [Δ (L M_{R})]}{3} \end{cases}

Where

Δ (\cdot)

is the residual error, which describes the difference between the measured data and the calculated one, namely,

{\begin{cases} Δ (K M) = K M - (F_{y} K + M) \\ Δ (L M) = L M - (F_{y} L + M) \\ Δ (L M_{R}) = L M_{R} - (F_{y} L + R_{φ} M) \\ Δ (L K) = L K - (F_{y} L + K) \end{cases}

In principle, Eqs. (2) (Model $Ι$ ) and Eqs. (4) (Model $Ι Ι$ ) are completely compatible. In practice, choosing Model $Ι Ι$ as the model seems better because Model $Ι$ would introduce more errors due to more operators like rotational interpolation errors. Moreover, Model $Ι$ needs more storage space for computer programming.

2.2 Basic iterative methods for the three-flat problem

With the refined model, it is easy to generalize the previous three iterative procedures mentioned earlier, namely, Jacobi, Gauss-Seidel, and SOR. For example, if the block Jacobi iteration is now applied, the Eqs. (2) or (4) are actually the iterative equations, namely the block Jacobi iteration. For Model $Ι$ , we rewrite it as follows:

Model I : {\begin{cases} K_{n e w} = \frac{A_{K} - (F_{y} L + F_{y} M)}{2} \\ L_{n e w} = \frac{A_{L} - [F_{y} K + F_{y} M + F_{y} (M_{R})]}{3} \\ M_{n e w} = \frac{A_{M} - (F_{y} K + F_{y} L + R_{φ}^{- 1} F_{y} L)}{3} \end{cases}

For Model $Ι Ι$ it is

Model Ι Ι : {\begin{cases} K_{n e w} = K + \frac{F_{y} [Δ (K M)] + Δ (L K)}{2} \\ L_{n e w} = L + \frac{F_{y} [Δ (L M)] + F_{y} [Δ (L M_{R})] + F_{y} [Δ (L K)]}{3} \\ M_{n e w} = M + \frac{Δ (K M) + Δ (L M) + R_{φ}^{- 1} [Δ (L M_{R})]}{3} \end{cases}

We can try to “speed up” the method by using the new value K_new instead of K in solving for L using the second equations, and the new value K_new and L_new instead of K and L in solving for M using the third equations. As for Model $Ι Ι$ , the updated value is included in the calculation of residual $Δ (\cdot)$ . Here the approximate solution is updated immediately after the new component is determined; it is called block Gauss-Seidel iteration. Obviously, one of the advantages of the method of Gauss-Seidel is that it requires only half of the memory used by Jacobi’s method. Typically the method of Gauss-Seidel converges faster than Jacobi’s method (when they both converge simultaneously), which can be observed in the following section.

If relaxation factor ω (also called acceleration parameter) is introduced, then we can obtain the block Jacobi SOR iteration and block SOR iteration respectively (i.e. block relaxation method) [15]. Thus introducing the relaxation factor ω leads to the following iterative scheme for Model $Ι$ :

Model Ι : {\begin{cases} K_{n e w} = (1 - ω) K + ω \frac{A_{K} - (F_{y} L + F_{y} M)}{2} \\ L_{n e w} = (1 - ω) L + ω \frac{A_{L} - [F_{y} K + F_{y} M + F_{y} (M_{R})]}{3} \\ M_{n e w} = (1 - ω) M + ω \frac{A_{M} - (F_{y} K + F_{y} L + R_{φ}^{- 1} F_{y} L)}{3} \end{cases}

And for Model $Ι Ι$ the block relaxation scheme is

Model I I : {\begin{cases} K_{n e w} = K + ω \frac{F_{y} [Δ (K M)] + Δ (L K)}{2} \\ L_{n e w} = L + ω \frac{F_{y} [Δ (L M)] + F_{y} [Δ (L M_{R})] + F_{y} [Δ (L K)]}{3} \\ M_{n e w} = M + ω \frac{Δ (K M) + Δ (L M) + R_{φ}^{- 1} [Δ (L M_{R})]}{3} \end{cases}

Where ω = 1, the block Jacobi SOR method simplifies to the block Jacobi method [Eqs. (6)], and the block SOR method simplifies to the block Gauss-Seidel method [Eqs. (7)]; generally, ω < 1 corresponds to under-relaxation with slower convergence while ω > 1 leads to over-relaxation and faster convergence [25]. In this paper, we classify the above basic iterative methods into two categories, namely block Jacobi SOR method (JSOR for short) and block SOR method (SOR for short), which can be effectively applied in the three-flat problem. The variants of them such as the Symmetric Gauss-Seidel and the Symmetric Successive Over-Relaxation (SSOR) iteration can be included in the classification [15]. The block Jacobi SOR method as relaxation factor ω = 0.1 is actually the iterative algorithm proposed by Vannoni and Giuseppe Molesini [18].

In general, it is not possible to compute in advance the value of relaxation factor ω that will maximize the rate of convergence (which is called optimal relaxation factor) [25]. In the following chapter, we will show how relaxation factor can be used to improve convergence of iterative methods.

To apply the above generalized basic iterative methods to absolute measurements, the following steps are implemented:

1. Model the measurement and apply some linear operations to the original model to get the refined model [see Eqs. (2) or (4)].
2. Define the measure and with an initial guess for the surface errors to be determined (zero matrices or random generated surfaces are usually used).
3. Compute the residuals which give the difference between the synthetic measurements and the experimental ones [see Eqs. (5)].
4. Update the trial surfaces using the above basic iterative methods according to the refined model [see Eqs. (8) or (9)].
5. Calculate merit function (MF) that uses RSS to combine independent residuals contributions. The iteration is repeated until the root-mean-square (rms) value of the MF reaches a minimum. Where $M F = \sqrt{Δ_{1}^{2} + Δ_{2}^{2} + ...}$ , △ is the residual.

3. Example results and discussions

In the following sections, we will show how to solve the three-flat problem by the generalized basic iterative methods mentioned earlier, namely, the block Jacobi SOR method and block SOR method. The data processing can be rapidly executed on a personal computer (with a 3.00 GHz CPU and 2.00 GB of memory). It is noted that later figures, and Tables 1-3 indicate the expected measurement results according to Eqs. (9) (Model $Ι Ι$ ).

Table 1. Simulation results for the two basic iterative methods with different relaxation factors

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Table 2. rms reconstruction error as a function of the number of iterations for different iterative methods

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Table 3. Experimental results for the two basic iterative methods with different relaxation factors

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3.1 Simulation results

To validate the basic iterative methods described in the last section, we use Zernike polynomials with zero piston and tilt to generate random surfaces of the three flats K, L, and M, whose circular pupils are all 169 pixels in diameter. The first 36 Zernike polynomial terms are used to construct the virtual flats. Then we get the virtual experimental results of the classical three-flat problem, i.e. KM, LM, LM_R, and LK, which have been shown in Fig. 1.

Fig. 1 Example of randomly generated surfaces K, L, M and virtual experimental results KM, LM, LM_R, and LK.

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The results of a numerical comparison of the two basic iterative methods discussed above are tabulated in Table 1. The reconstruction error is the difference between the reconstructed surface using the basic iterative methods and the simulated surface. From Table 1, we can clearly see that convergence depends on the iterative methods and relaxation factors when the same reconstruction accuracy is expected. The block Jacobi SOR methods (ω = 0.1, ω = 0.5, and ω = 0.92) converge to the termination metric threshold of 0.01 nm in 829 iterations, 165 iterations, and 89 iterations respectively, whereas the block SOR methods (ω = 1.0 and ω = 1.5) converge in 43 iterations and 19 iterations respectively. Table 1 also indicates that higher expected accuracy increases the iteration number and elapsed time (when a smaller rms MF threshold is used as an iteration termination condition, such as 0.001 nm). Similarly, the block SOR method with the optimal relaxation factor provides the greatest speed of convergence. When it is compared with the block Jacobi SOR method (ω = 0.1), the block SOR method with the optimal relaxation factor (ω = 1.5) accelerates the iteration by factors approximately 50 without losing the accuracy (see Fig. 2).

Fig. 2 Plots of the rms MF versus the number of iterations [Figs. (a) and (b)] and the measurement error maps of M [Figs. (c) and (d)] reconstructed by the block Jacobi SOR method (ω = 0.1) and the block SOR method (ω = 1.5), respectively. (An rms value of MF threshold of 0.01 nm is used as an iteration termination condition).

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As to both block Jacobi SOR method and block SOR method, the speed of convergence depends critically on the relaxation factor ω; typically the block SOR method converges faster than the block Jacobi SOR method. After several trials, we found ω = 0.92 (JSOR) and ω = 1.5 (SOR) are nearly the optimal for our case. Notice that when ω = 1 (JSOR) or ω > 1(JSOR), the iteration does not converge.

Table 2 gives the comparison of the two iterative methods when the number of iterations holds the same. The block Jacobi SOR method (ω = 0.1) converges but leaves a much bigger residual error than the other iterative methods. According to the obtained values, the block SOR method with the optimal relaxation factor (ω = 1.5) provides the highest accuracy with the least iterations, which also can be clearly indicated from Fig. 3. The reconstruction accuracy is almost two order magnitudes higher than the block Jacobi SOR method (ω = 0.1).

Fig. 3 Comparison of convergence rate for different iterative methods (Simulation results).

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In theory, these iterative methods are equivalent, giving the same correct solution, if the number of iterations is large enough. But the block SOR method (with optimal relaxation factor) converges much faster than the block Jacobi SOR method. This is much advantageous for the large-size problems.

3.2 Real experimental results

We have also tested the generalized iterative methods on real experimental results in order to further validate the basic iterative methods for solving the classical three-flat problem. Three flats were measured in pairs with a 6-inch phase-shifting Fizeau interferometer (with the optical axis horizontal). A 5-Axis mount was used to hold the return flat. The experimental operations and laboratory temperature were though not well controlled, the test results were still proper for the validation of methods.

For the four real experimental results KM, LM, LM_R (54 degrees), and LK, we reconstruct the absolute surfaces based on the two basic iterative methods. The reconstruction results are tabulated in Table 3. Here we get the same conclusion as shown in Table 1 that convergence depends on the iterative methods and relaxation factors when the same reconstruction accuracy is expected. As for the case of ω = 1.34 (SOR), the rms value of MF converges rapidly to the threshold of 0.5 nm (just at the noise level) in only 5 iterations (less than 7 seconds). The block SOR method with the optimal relaxation factor (ω = 1.34) converges nearly 50 times fewer iterations than the block Jacobi SOR method (ω = 0.1). Take ω = 0.1 (JSOR) and ω = 1.34 (SOR) as the example, the difference between the two reconstruction maps has an rms of 0.15 nm (Fig. 4). After several trials, we found ω = 1.34 (SOR) is nearly the optimal for this case.

Fig. 4 Reconstruction map of M using (a) the block Jacobi SOR method (ω = 0.1) (PVr = 17.58 nm, rms = 4.29 nm) and (b) the block SOR method (ω = 1.34) (PVr = 17.45 nm, rms = 4.27 nm); (c) difference map of the two methods (PVr = 0.85 nm, rms = 0.15 nm). Map resolution 951 × 951 points.

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Just like Fig. 3, we have the comparison of convergence rate for different iterative methods using real experimental results (see Fig. 5). Similarly, it is concluded that the block SOR method using the optimal relaxation factor provides the greatest speed of convergence; fewer than 10 iterations provide sub-nanometer surface accuracy. And it saves much more computational costs and memory space.

Fig. 5 Comparison of convergence rate for different iterative methods (Experimental results).

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Provided that iterations converge to the same termination metric threshold of 0.5 nm, these iterative methods approach the similar correct solution. In reality, if the computing time is long enough, iterations for all the iterative methods will first converge to a minimum and then diverge a little. However, this is not necessary and not suggested for practical implementation.

3.3 Choosing the value of relaxation factor ω

The SOR Method is devised by applying extrapolation to the Gauss-Seidel method, which takes the form of a weighted average between the previous iteration and the computed Gauss-Seidel iteration [Eqs. (8) or Eqs. (9)]. Thus one main advantage of SOR is that it accelerates the rate of convergence by introducing a relaxation factor. For the optimal choice of ω, SOR may converge faster than Gauss-Seidel by an order of magnitude [25]. According to the reference [25], however, “In general, it is not possible to compute in advance the value of ω that is optimal with respect to the rate of convergence of SOR. Even when it is possible to compute the optimal value for ω, the expense of such computation is usually prohibitive”. Although we estimated an optimal value by a heuristic method in the previous sections [25], the expense was also prohibitive. In practice, Monte Carlo simulation can be conducted to evaluate the performance of the generalized iterative algorithms, as well as the appropriate choice of ω for SOR.

During each Monte Carlo trial, we use a random selection of 36-terms Zernike polynomials to generate surfaces of the three original flats K, L, and M (169 pixels in diameter). For all the simulations, the same iteration termination condition (an rms MF threshold 0.01 nm) is used, which can ensure the same reconstruction accuracy of the iterations.

Firstly, we use three different basic iterative methods to compute the number of iterations for each Monte Carlo trial: Jacobi iteration with ω = 0.1 [JSOR (ω = 0.1)], Gauss-Seidel iteration [SOR (ω = 1.0)], and SOR iteration with relaxation factor ω = 1.5 [SOR (ω = 1.5)]. After 1,000 trials, we get the statistic result shown in Fig. 6. The Monte Carlo simulation indicates that the SOR method with an inexpensive rough estimate of ω (such as 1.5) accelerates the iteration with the least iterations of the three methods.

Fig. 6 Monte Carlo simulation results for three different basic iterative methods [JSOR (ω = 0.1), SOR (ω = 1), and SOR (ω = 1.5)] (To make it more visible in the figure, only 100 trials are show).

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Then we repeat the Monte Carlo simulation by using different values of ω (1.0 to 1.8, the interval is 0.05). After 1,000 trials, we get the statistic results shown in Fig. 7 and Fig. 8. Figure 7 shows that the SOR with ω = 1.5 is generally the best one (mean value = 15.6 and standard deviation = 4.4); this can also be indicated by the observed number of SOR iterations versus ω for the statistical mean values of Monte Carlo trials (Fig. 8, dotted line).

Fig. 7 Monte Carlo simulation results for SOR with different relaxation factors (To make it more visible in the figure, only 100 trials and 6 different relaxation factors are show).

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Fig. 8 Observed number of SOR iterations vs. relaxation factor ω for the Monte Carlo means (dotted line) with 1σ error bars (σ is the standard deviation of the 1,000 trials).

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A simple choice of ω = 1.5 as the relaxation factor is reasonable and economical for practical implementation, even though ω = 1.5 is not the optimal value for some samples. The number of iterations shows no big difference between ω = 1.5 and the optimal one – as for the sample 59, SOR (ω = 1.4) converges in 21 iterations while SOR (ω = 1.5) converges in 22 iterations (Fig. 7). An appropriate choice of ω between 1.3 and 1.7 is suggested. Still, SOR with simple choice of ω performs better than Jacobi iteration and Gauss-Seidel iteration.

4. Error considerations

It is obvious from the example results that both basic iterative methods can produce accurate output and the block SOR method with an inexpensive choosing of relaxation factor is the most advantageous for practical implementation. However, the real example results imply that it is necessary to investigate errors that can influence performance if higher accuracy is expected. In the real iteration for the absolute measurement of optics, there are mainly four sources of errors that complicate the iteration process: errors due to modeling, random noise, surface qualities of the flats under test and errors due to experimental operations.

4.1 Model-induced errors

As described previously, Model $Ι$ [Eqs. (8)] would introduce more errors due to more operators like rotational interpolation errors. This is shown in Fig. 6. The results indicated that the iteration cannot converge to the termination metric threshold 0.01 nm. As shown in Fig. 9(a), the rms MF reaches minimum in 9 iterations, but after 9 iterations the rms MF increases.

Fig. 9 (a) Plot of the rms MF versus the number of iterations and (b) the measurement error map of M in the case of Model $Ι$ . The iteration number is 100. (The block SOR method and ω = 1.5).

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However, interpolation error due to the rotation operation of flat M can be compensated as follows [27]:

M_{n e w} = (1 - ω) M + ω \frac{A_{M} - (F_{y} K + F_{y} L + R_{φ}^{- 1} F_{y} L)}{3} - ω \frac{R_{φ}^{- 1} M_{R} - M}{3}

After compensating the rotational interpolation error, we get the same result as Figs. 2(b) and 2(d), in which Model $Ι Ι$ was applied. This is true for both basic iterative methods with different relaxation factors and different iteration termination conditions.

Note that if the model itself cannot estimate some error terms, then these error terms will also be present in the estimated surface error using the basic iterative methods [8, 11]. As discussed in reference [29], angular terms $k \frac{2 π}{θ_{0}} θ (k = 1, 2, 3, ...)$ of rotation asymmetric part cannot be reduced by measurement at one angle of rotation $θ_{0}$ (θ is azimuthal frequency in Zernike polynomials). Then the error terms $20 n θ (n = 1, 2, 3, ...)$ will be missed if one angle 54 degree is chosen. In this case, additional measurements (such as multiple rotations) can be included in the model if higher accuracy is required [27, 29]. Iterative method itself can still solve the problem by only minor modification of the iteration, if additional measurements are added. Thus to modify the model is an alternative way to improve accuracy.

4.2 Noise

Interferometric measurements suffer from noise due to environmental influences, such as air turbulence, vibrations, detector noise and so on. The noise of a single measurement can be determined by taking two measurements M₁ and M₂ in immediate succession, subtract the difference of the two resulting maps, and calculate the standard deviation of the difference map, i.e.,

σ_{n o i s e} = \frac{r m s (M_{1} - M_{2})}{\sqrt{2}}

Then we can repeat the virtual measurements KM, LM, LM_R, and LK with different noise levels. And by repeating the block SOR iteration procedures (using Model $Ι Ι$ ), we get the results of the effect of random noise on iterative surface reconstruction (see Table 4). The table shows the rms of the error map is just at the noise level, while the iterating number is getting better (fewer iterations) when the noise is higher. As for the real example in the case of ω = 1.34 (SOR), the rms value of MF converges rapidly to the noise level (0.5 nm or so) in only 5 iterations (Fig. 5).

Table 4. The effect of random noise on iterative surface reconstruction

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4.3 Surface qualities of the flats under test

As shown in Table 5, we use different surface qualities of the flats for simulation. For example, the ‘1-2’ represents the surface whose PV is about λ/50, while the ‘11-12’ represents the surface whose PV is about λ/10. The rest can be done in the same manner. Table 5 shows the effect of figure qualities on iterative surface reconstruction is almost negligible. Even for the 11-12 nm rms original surface with random noise whose standard deviation is 0.2 nm, the block SOR iteration can also retrieve the absolute figure at the noise level (see Fig. 10). The results indicate that the block SOR iteration can produce very robust results and very suitable for surfaces of different qualities.

Table 5. The effect of figure qualities on iterative surface reconstruction

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Fig. 10 (a) Plot of the rms MF versus the number of iterations and (b) the measurement error map of M. The iteration number is 29. (The block SOR method and ω = 1.5).

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4.4 Errors due to experimental operations

Errors due to experimental operations cannot be neglected when we perform the real measurements, but can be controlled. Among them, there are angle errors for the rotated flat M, misalignment of the axis of rotation for the rotated flat M and misalignment in the orientation in rotation of flat K when used in two different positions. After repeating the simulation using maps with different experimental operation errors, we find that the error of the iterative surface reconstruction is not sensitive to angle errors (Fig. 11), but sensitive to misalignments of the flats (Fig. 12). After lots of trials, it is recommended that the angle errors of the flat M should be less than 0.5 degrees, and the alignment errors of the flats M and K should be less than 1.0 pixel if the expected accuracy 0.1 nm is required. In practice, this can be guaranteed with the matching method [14]. Figures 11 and 12 also suggest that large CCD resolution can reduce the experimental errors a lot, which is not a problem for nowadays commercial interferometer.

Fig. 11 Plots of the rms of residual error map versus error of rotation angle. Map resolution 169 × 169 and 601 × 601 were simulated, respectively.

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Fig. 12 Plots of the rms of residual error map versus misalignment of (a) the rotated flat M and (b) the flat K. Map resolution 169 × 169 and 601 × 601 were simulated, respectively.

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Furthermore, retrace errors, temperature gradients in the flats, variation in flat figure due to remounting can also influence the performance of basic iterative methods. Thus a high repeatability interferometer, a temperature controlled laboratory, and high stability mounts are necessary for high accuracy surface measurements. Then the main problem is how to reduce errors due to environmental effects and experimental operations, both of which are also obstacles of other absolute test methods. As to the basic iterative methods, further development of much more robust iterative algorithms is an important topic for practical applications.

5. Conclusions

In this paper generalized basic iterative methods for absolute measurement of optical flats have been discussed. They are based on iterative scheme and can be classified into block Jacobi SOR method and block SOR method. Both simulation results and experimental results demonstrated that both of the methods can correctly reconstruct absolute figures with pixel-level spatial resolution; are easy to understand and implement; and computationally efficient. Compared to the block Jacobi SOR method, the block SOR method (with an inexpensive choosing of relaxation factor such as ω = 1.5) saves more computational costs and memory space requirements and converges much faster. One reason is that the approximate solution is updated immediately after the new component is determined. The other is that a good choice of relaxation factor accelerates the rate of convergence of the iterations to the solution. Furthermore, the generalized basic iterative methods can be effectively used for almost all the absolute test models, including both absolute flatness measurement and absolute sphericity measurement.

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Basic iterative methods	rms MF	0.01 nm			0.001 nm
Basic iterative methods	Relaxation factor	Iteration number	Elapsed time/seconds	M error ^a (PV/rms)/nm	Iteration number	Elapsed time/seconds	M error (PV/rms) /nm
JSOR	ω = 0.1	829	25.430367	0.6310/0.0314	4796	175.917465	0.1046/0.0051
	ω = 0.5	165	5.209105	0.6316/0.0314	959	29.285358	0.1045/0.0051
	ω = 0.92	89	2.792732	0.6335/0.0314	521	15.982607	0.1044/0.0051
SOR	ω = 1.0	43	1.607193	0.6670/0.0350	244	9.228667	0.1267/0.0052
SOR	ω = 1.5	19	0.833572	0.7272/0.0374	94	3.442501	0.1399/0.0062

Number of iterations	rms error/nm
Number of iterations	JSOR (ω = 0.1)	JSOR (ω = 0.5)	JSOR (ω = 0.92)	SOR (ω = 1.0)	SOR (ω = 1.5)
8	2.4854	0.9374	0.7674	0.6741	0.2395
16	1.6583	0.6825	0.4577	0.2829	0.0428
32	1.0759	0.4262	0.2047	0.0617	0.0245
64	0.7696	0.1826	0.0534	0.0245	0.0105
128	0.5152	0.0462	0.0235	0.0133	0.0049
256	0.2582	0.0223	0.0134	0.0049	0.0044

Basic iterative methods	rms MF	0.5 nm (noise level)
Basic iterative methods	Relaxation factor	Iteration number	Elapsed time /seconds	M Map /nm (PVr/rms) ^a
JSOR	ω = 0.1	232	250.046001	17.5786/4.2941
	ω = 0.5	46	48.795915	17.5791/4.2941
	ω = 0.92	19	21.18460	17.9853/4.4151
SOR	ω = 1.0	11	14.304560	17.5340/4.2913
SOR	ω = 1.34	5	6.844655	17.4466/4.2745

Figure qualities(rms)/nm	Number of iterations	M error (PV/rms)/nm
1-2	12	0.6243/0.0311
2-3	12	0.6206/0.0315
3-4	17	0.6801/0.0355
5-6	25	0.8327/0.0414
7-8	33	0.8808/0.0443
9-10	39	0.9373/0.0474
11-12	46	0.9800/0.0472

Basic iterative methods	rms MF	0.01 nm			0.001 nm
Basic iterative methods	Relaxation factor	Iteration number	Elapsed time/seconds	M error ^a (PV/rms)/nm	Iteration number	Elapsed time/seconds	M error (PV/rms) /nm
JSOR	ω = 0.1	829	25.430367	0.6310/0.0314	4796	175.917465	0.1046/0.0051
	ω = 0.5	165	5.209105	0.6316/0.0314	959	29.285358	0.1045/0.0051
	ω = 0.92	89	2.792732	0.6335/0.0314	521	15.982607	0.1044/0.0051
SOR	ω = 1.0	43	1.607193	0.6670/0.0350	244	9.228667	0.1267/0.0052
SOR	ω = 1.5	19	0.833572	0.7272/0.0374	94	3.442501	0.1399/0.0062

Absolute measurement of optical flats based on basic iterative methods

Abstract

1. Introduction

2. Generalized iterative methods for absolute measurement of optical flats

2.1 The three-flat problem (modeling)

2.2 Basic iterative methods for the three-flat problem

3. Example results and discussions

3.1 Simulation results

3.2 Real experimental results

3.3 Choosing the value of relaxation factor ω

4. Error considerations

4.1 Model-induced errors

4.2 Noise

4.3 Surface qualities of the flats under test

4.4 Errors due to experimental operations

5. Conclusions

References and links

Cited By

Figures (12)

Tables (5)

Equations (11)

Optics Express

Noise	Number of iterations	M error (PV/rms)/nm
0	19	0.7272/0.0374
0.01	19	0.7656/0.0396
0.02	28	0.6482/0.0406
0.03	51	0.5112/0.0537
0.05	33	0.7638/0.0819
0.08	25	1.0837/0.1218
0.1	22	1.2885/0.1472
0.2	12	2.1941/0.2578