1. Introduction

To test the surface quality of fabricated lens, an interferometer is often used in optical testing. However, an interferometer does not measure the absolute phase of the tested optics but the optical path difference (OPD) between the reference optics and tested optics. Therefore, to precisely retrieve the phase of tested optics from the measurement result, having a high quality reference phase is essential. The reference surface figure error of most of the transmission spheres is typically around 1/10~1/30 waves. To achieve the measurement quality better than 1/100 waves, the reference calibration has become a necessary process. Jensen [1] proposed this concept in 1973 with the Twyman-Green interferometer. After that, many research groups proposed different configurations to reconstruct the reference phase by averaging the uncorrelated measurements. Creath and Wyant successfully measured a flat mirror with a surface roughness of 0.07 nm by incorporating the reference calibration [2]. Parks et al. used a chrome steel ball to calibrate the transmission sphere by placing at the center of curvature to form a confocal system [3]. Griesmann et al. developed a simple ball average by an auto mechanical controller that is easy to get sufficient measurements to reconstruct the reference phase of transmission sphere [4]. This ball averaging technique is also called the random ball test (RBT). However, there are several limitations of RBT. Although the RBT can average out the random noise, yet still cannot average out the errors from the interferometer, such as the retrace error and image distortion. The diffraction error also limits the effect of ball averaging [5].

The subaperture stitching interferometry (SSI) is now considered as a suitable metrology to measure either high numerical aperture optics or asphere [6–10]. In the SSI, the reference calibration is more critical than the full aperture interferometry since the reference error could be easily transferred through each subaperture in the stitching process. The aberrations induced by the misalignment of null optics may be coupled as surface error of subaperture at off-axis. Chen has offered a solution to the problem [11]. Smith and Burge derived an analytical expression for stitching error produced from noise in annular ring of subaperture [12]. The stitching process may accumulate the phase errors of the reference optics and consequently reduce the measurement precision. For adjacent subapertures, there are some discrepant error phases, such as the noise error phase, reference lens phase and retrace error phase. The continuity of two adjacent subaperture phases would be destroyed due to the contamination of error phases.

We had developed a vibration-modulated subaperture stitching interferometer [13,14]. This system included 4-axis motorized stages and a robust subaperture stitching algorithm. The interferogram of different subapertures are acquired on the fly when the tested lens is being rotated and vibrated by the stage bearing. A large amount of subaperture interferogram of difference phase modulation are recorded in a short time to cover the complete tested lens surface. The acquisition speed of subaperture interferogram is literally only limited by the camera bandwidth. Therefore, we come up with the idea to use this proposed interferometer to acquire numerous subaperture phases, much more than the prior methods have, to average out the measurement errors in the system [15,16]. It is also called the high overlapping density subaperture stitching interferometry (HOD-SSI). Because of the averaging effect, the HOD-SSI not only increases the stability and the accuracy of the stitched phase but also reduces the quality requirement of the reference optics. Therefore, a precise calibration of the transmission sphere is not critical anymore. The phase stitching process not only suppresses the measurement noise but also reconstruct the interferometer reference phase in the same time. Both the simulation and experimental results prove the feasibility of the proposed method.

2. Subaperture stitching interferometry

In the lens surface interferometry, aspherical surface testing has always been a challenging task due to the varied curvature of aspherical surfaces. The SSI is a flexible method to perform this challenging task. With proper design of the geometrical null path, the varying curvature of the aspherical surface can be measured subaperture by subaperture. Moreover, the SSI can have high measurement resolution and dynamic range when appropriately implemented. According to the best-fit curvature at each subaperture position, the 4-axis alignment stages, as in [14], null out the subaperture aberrations to the best focus with the fringe null. At the same time, piston, tilt and defocus will be introduced into the subaperture phase by the geometric calibration process.

Therefore, in the subaperture stitching process, the random piston, two directional tilts and defocus are used as the four compensators C₁ ~C₄ to minimize the phase discrepancy between the overlapping pixels. The acquired interference phase ${\varphi }_i$ at each overlapped pixel will be optimized in the least-squares sense to minimize the norm of phase discrepancy S_i between the overlapped phases. Therefore, the subaperture stitching algorithm for a single pixel i is represented by the function S_i,

After solving the values of four compensators, the stitched phase is simply mean of the compensated phase ${\widehat{\varphi }}_{i,k}$as

The standard deviation of the compensated phase ${\widehat{\varphi }}_{i,k}$ is one kind of quality measure for subaperture stitching and phase measurement. It is thus further defined as the subaperture stitching error ${\sigma }_i$ of each global stitched pixel i:

3. Reference optics wavefront reconstruction method

The high quality reference optics is critical to the measurement accuracy of interferometry. Poor quality or uncalibrated reference optics will lower the measurement accuracy. For the subaperture stitching interferometry, this problem has to be especially addressed, otherwise, the reference error would potentially propagate into the stitched phase map or the whole tested optical surface.

In the aspect of reference calibration, the random ball testing (RBT) method has been a popular method to calibrate the interferometer reference optics. By measuring the surface of high quality ceramics ball and randomly rotating the ball, these measured phase errors reflected from the ball surface $W_T$ will be averaged out to a zero mean after numerous measurements. The reference optics wavefront can therefore be obtained. The measured phase $W_M$ can be represented as

In a similar way, the HOD-SSI also shares the concept of RBT in the error averaging over numerous random measurements. The advantage of the averaging effect of HOD-SSI has been presented in our previous studies [15,16]. The results showed the additional effects of reference error suppression and the measurement accuracy improvement when more subapertures are used in the stitching algorithm. The HOD-SSI therefore effectively resists the error from propagation along the subaperture out of the reference error when compared to the regular normal overlapping density SSI method.

We now consider the measurement in the acquired interference phase of a SSI interferometer. For each measured pixel phase in the interferometer, the tested optics phase $T_i$and reference phase $R_{j,k}$ along with the four compensators determines the geometrical-null optimized interference phase ${\varphi }_{i,k}$. Where, the subscript i is the coordinate index of tested optics, subscript j is the coordinate index of reference optics, and the subscript k is the index for each overlapping subapertures.

After least square fitting the four compensators to minimize the norm of merit function S shown in the Eq. (1), the fitting residue ${\gamma }_{i,k}$ for each pixel measurement is calculated. It is defined as the difference between the compensated measured subaperture phase ${\widehat{\varphi }}_{i,k}$ and the stitched phase ${\Phi }_i$, that is, ${\gamma }_{i,k}={\Phi }_i-{\widehat{\varphi }}_{i,k}$. Therefore, Eq. (7) can be rewritten as the following equation:

Now the reference phase $R_{j,k}$ is determined by two components: The stitched phase error of corresponding tested optics phase${\delta }_i$ and the fitting residue ${\gamma }_{i,k}$ in each overlapped subaperture measurement. Therefore, $R_{j,k}$ varies in each subaperture measurement according to the induced measurement error in each subaperture stitching. If we average Eq. (8) over multiple subaperture measurements, the mean of reference phase is the sum of the mean of the tested optics stitched phase error and mean of the fitting residue over multiple measurements.

The mean of residue $\overline{{\gamma }_{i,k}}$can be derived by resorting the residue in the interferometer aperture coordinate after the corresponding residues in the tested optics coordinate are calculated. The residue is therefore the major component being used for reconstruction of the reference phase. On the other hand, the mean of the stitched phase error $\overline{{\delta }_i}$ depends on the overlapping locations i on the tested optics. The subaperture overlapping locations on the tested optics were determined by the scanning path optimized for the geometrical null during the SSI testing. Therefore, the mean of the stitched phase error $\overline{{\delta }_i}$ is dependent on the scanning path of the considered reference phase.

In the RBT calibration, the ball is rotated and placed randomly for multiple times to ensure the measured ball surface phase error is averaged out. Similarly, if all the averaged stitched phase error is zero over the scanning path, the reference error R_j will simply equal the mean of residue $\overline{{\gamma }_{i,k}}$ . To meet this assumption, we further analyze the conditions for a zero mean of stitched phase error. The stitched phase error is the overall system level measurement error of a SSI system. It is determined by all the possible noise and errors within the SSI system such as the speckle noise, the subaperture alignment error, the retrace error, the geometrical null design and even the reference error we are looking for. It can be presented in the mid-high frequency errors and low frequency error. The mid-high frequency error is possibly induced from the random fringe speckle, the laser jittering, or even the detector noise during the long time subaperture scanning. Such errors can be either temporarily or spatially randomly distributed during each subaperture interferogram acquisition with a zero mean. Therefore, large number subaperture sampling by the proposed HOD-SSI will be helpful to suppress such high frequency errors.

On the other hand, the low frequency error of the stitched phase which presents in the form of aberration polynomials is related with the reference form error or the retracing error as we also indicated in previous study [16]. As shown in the following Fig. 1, there are two points on the reference optics coordinate labeled as the blue triangle and the red cross point. Both of the points have different averaging path along the surface of the tested optics. By designing the averaging path to be diversely spread over the tested optics surface, the stitched phase error will possibly have the mean close to zero in the presence of the reference form error. Most of the lower order reference form errors does not couple into the stitched phase after a diverse scanning path design with the exception of the astigmatism aberration. We will show that one of the reference form errors may still have influence over the tested surface even with a diverse scanning path design. The feasibility and limitation of the proposed method will be examined thoroughly by both the simulation and experiments in the next two sessions.

 

Fig. 1 Two reference locations, the blue triangle dot and red cross dot, are spread on the tested optics forming the averaging paths for constructing the reference phase, as presented by the dashed blue circles and dashed red circles on the tested optics coordinate, respectively.

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4. Numerical simulation of third order Zernike aberrations

In this session, we use five major third-order Zernike aberrations to simulate the low frequency reference phase to test the feasibility of the reference construction method of HOD-SSI. The simulated wavefront phase being stitched is a third order spherical aberration expressed as 3.5 × R⁴, where, R is the normalized radius. Therefore, the peak-valley of the stitched wavefront is 3.5 waves. The geometrical null process is designed to have 10 concentric scanning rings to cover the complete tested surface. In order to closely match the experiments shown in the next experimental session, the numbers of subapertures along the 10 rings are chosen as 5, 10, 15, 20, 25, 30, 35, 50, 60, and 75. Therefore, the total numbers of subapertures used is 325. We then compared the construction errors of the reference phase among the five basic Zernike aberrations. Each simulated Zernike aberration RMS error is set to be around 1/20 waves. We have the assumption that there is neither subaperture position uncertainty nor any retrace error contributed from the interferometer. It is worthwhile to find out the impact on reference phase reconstruction with the five basic Zernike aberrations individually. The simulation will also verify if the five third order Zernike aberrations can be suitably reconstructed by the proposed method.

Figure 2 shows the results of the reference phase reconstruction and the difference between the estimated and the ideal reference phase. From top to bottom of Fig. 2, the Zernike aberrations simulated are the astigmatism at zero degree, the astigmatism at 45 degrees, the primary Y coma, the primary X coma and the spherical aberration, respectively. Except the zero-degree astigmatism, all the Zernike aberrations can be reasonably reconstructed with small errors and prevented from being stitched into the tested phase. By comparing the RMS values of the ideal and difference phases, the reconstruction accuracy is about 83 percent for the primary Y coma and more than 95 percent for the other three cases. This simulation results indicates that in presence of zero-degree astigmatism, the reference has to be carefully calibrated. Otherwise, rotating the axis of reference astigmatism to 45 degrees will also be effective to prevent the reference coupling problem.

 

Fig. 2 The simulation results of the reference phase reconstruction presented with five third-order Zernike aberrations.

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The above simulation results show that not all aberrations perform equally in the reference reconstruction. This issue can be analyzed from the view of the very beginning of phase stitching process. As shown in the Eq (1), the merit function S is determined by the difference of measured phase and the free compensator. With the Eq. (6), we can rewrite the Eq (1) as

When the reference error presents, the free compensators will minimize the phase mismatch induced from the reference error within the overlapped subaperture region. From the Eq (10), the difference in the overlapped subaperture region is simply the sum of the difference of the two shifted and rotated reference errors and the four compensators due to the subaperture alignment. Therefore, if the astigmatism presents in the reference error, the overlapped region of the subapertures shows tilt aberration. This phenomenon is quite similar as the shearing interferometer whereas the sheared phase is the reference error in this case. For the zero-degree astigmatism, if the overlap is sheared in the tangential direction, the tangential tilt aberration presents in the overlapped region. On the other hand, if the overlap is sheared in the sagittal direction, the sagittal tilt aberration presents in the overlapped region.

To visualize the effect, we modulate the phase into fringes as shown in the Fig. 3(a) (b).

 

Fig. 3 For the zero-degree astigmatism, (a) If the overlap is sheared in the tangential direction, the tangential tilt aberration presents in the overlapped region. (b)If the overlap is sheared in the sagittal direction, the sagittal tilt aberration presents in the overlapped region.

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Both the tangential tilt in tangential overlap direction and sagittal tilt in sagittal overlap direction are compensators being used in the geometrical null for aspherical optics. Therefore, there is no residue left after the stitching process for accurate estimate of the reference error. Also, depends on the asphere prescription, the mismatch from reference astigmatism will be compensated and contribute to the compensated phase. In addition, the stitched phase is the average of the compensated phase of all subapertures. At the end, there is minimum residue left for reference reconstruction and the reference error couples into the stitched phase of the tested optics. The zero-degree astigmatism therefore turns out to be one of the most critical reference aberration to be calibrated.

On the other hand, the 45-degree astigmatism generates the sagittal tilt in the tangential overlap direction and tangential tilt in the sagittal overlap direction. Though the two tilts are among the four compensators being used in phase stitching, they are nearly orthogonal to the compensators used in the geometrical null. As a result, there is more residue left after stitching and the mismatch of the reference error will less likely propagate into the stitched. The residue is therefore closer to the reference error for accurate reference reconstruction. Other aberrations such like coma aberration and spherical aberration have higher order aberrations other than the four compensator, is even more unlikely to introduce errors under high overlapping density stitching.

5. Experimental results

A laser Fizeau interferometer with a 4-inch diameter reference optics of F/3.3 was set up for the experiment. The interferometer cooperates with a rotational scanning stage to measure a large number of subaperture phases in a short time. In this experiment, we use a spherical surface with 13 mm radius of curvature and 25 mm diameter as the tested optics and to verify the feasibility of our proposed method. For the subaperture location, the tested surface is divided into 10 rings. The number of subapertures from the inner ring to the outer ring as shown in Fig. 4(a) are 6, 13, 19, 25, 31, 38, 44, 50, 57 and 63, respectively. Over 90% of the tested area is overlapped with up to 15 subapertures as shown in Fig. 4(b). Comparing with the conventional technology of subaperture stitching interferometer that has only 2 or 3 overlapped data in one pixel, the rotational scanning configuration of our vibration modulated interferometer has a great gain in the number of subapertures without increasing the acquisition time. When measuring the spherical surface, the convergence beam from the Fizeau interferometer will focus on the center of curvature at each subaperture location. Because of the characteristic of the spherical surface measurement, the retrace errors are negligible in the subaperture phase. That is to say, the measured phases only include the difference between the reference phase and tested phase. So it is a suitable configuration to reconstruct the reference phase. The stitched phase and the stitching error map are shown in Fig. 5(a) and 5(b). The tested optics has good surface quality with a RMS value around 1/20 waves and the peak to valley is just 0.327 waves. In additional, except the center of the tested surface, the stitching quality map is quite low and uniform to justify a well-stitching testing case.

 

Fig. 4 The experimental configuration: (a) the subaperture lattice; (b) the number of overlapped subapertures at each pixel.

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Fig. 5 Experimental results: (a) the stitched phase; (b) the stitching quality map.

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Before reconstructing the reference phase by the HOD-SSI, the RBT was used to calibrate the reference optics as an alternative method to compare the result from the proposed method. Figure 6(a) shows the calibrated reference after the RBT calibration for 100 times of random measurements. Figure 6(b) shows the result of reference phase reconstruction by the proposed method. The difference between the RBT and HOD-SSI results is shown in the Fig. 6(c) by subtraction one with the other. The RMS of the difference between the two reconstructed phases is estimated to be around 0.0072 waves, which proves the feasibility of the proposed method in the application of a spherical surface testing. It’s interesting to see how closely the diffraction phase ripples reconstructed from both methods match each other. Also, the phase of the reference optics is mostly the lower order spherical aberration and the coma aberration as both aberrations are more commonly seen than the astigmatism aberration. While the spherical aberration is mostly from the figure error of the reference surface, the coma aberration is probably induced from the misaligned elements inside the interferometer.

 

Fig. 6 The reference phase measured by (a) the random ball test by 100 measurements; (b) the HOD-SSI with 346 subapertures stitching; (c) the difference between both methods.

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Based on Eq. (9), we know that the RMS error of the reconstructed reference phase is decreased with the increasing number of subaperture measurements N. To find the exact relation between the two, we tried to down sample the number of subapertures being used in stitching and observe the resulting difference of the reference phase between the two methods. Figure 7(a) shows the resulting curve of the number of subapertures N versus the RMS error of the phase difference between the RBT and the HOD-SSI methods.

 

Fig. 7 The RMS $\sigma$of the the proposed method have the relation with the number of measurements N as $\sigma =a+b/\sqrt{N}$. measurement in (a) linear scale; (b) log-log scale with the constant a omitted.

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In this experiment, there are maximum of 346 subapertures being used to cover the complete tested optics surface. The difference between the reference phases calibrated by RBT and HOD-SSI seems to be inversely proportional to the root square of measurement number N, as shown in Fig. 7(a).Therefore, we assume the RMS error σ have the relation with number of measurement N as $\sigma =a+b/\sqrt{N}$ . Where a and b are constants. The convergence curve shares the same similarity as in the error suppression as the RBT testing method. After least square fitting and removing the constant a, the curve is thus re-plotted in the log-log scale plot as shown in the in Fig. 7(b). The best fit green line is estimated to have the slope around −0.5 which therefore prove the inverse square root relation. As a result, the reference calibration is proven to have the similar error suppression characteristic as the RBT calibration method.

6. Conclusion

The proposed high overlapping density subaperture stitching interferometry (HOD-SSI) is found to have the self-calibration capability to reconstruct the reference optics phase without any additional optical instruments or calibration procedures to be implemented. By subtracting the stitched phase from the compensated phase, the reference phase can be estimated from the averaging of the difference between the two. Therefore, the calibration of the reference optics becomes less critical than before in the SSI measurement.

The third order Zernike aberrations are used as the reference phase to verify the feasibility of the proposed method in noiseless simulations. The results show that only the zero-degree astigmatism cannot be reconstructed properly. Otherwise, the other four types of aberrations, including the astigmatism at 45 degrees, the primary Y coma, the primary X coma and the spherical aberration, all have low errors in the reference phase reconstruction.

We conducted an experiment to compare the reference phase reconstruction results from the RBT and the HOD-SSI methods. The RMS difference between the reconstructed reference phase and the RBT phase is only 0.0072 waves. It yields a similar result considering that the reference optics is not calibrated at all and the existence of the multiple diffraction rings inside the reference sphere. We prove the practicability of the proposed method by comparing the reference phase acquired from RBT calibration against the one from the HOD-SSI.

Although the proposed method seems to have some limitation because the profile of the reference phase and the designated averaging path affect the reconstruction result. Yet it shows another capability of the HOD-SSI not only limited to the reference suppression but the reference reconstruction. This study demonstrates that the HOD-SSI not only can suppress the reference error but also can self-reconstruct the reference phase while stitching the phase profile.