## Abstract

A novel subaperture stitching interferometry is developed to measure the surface deformation of the lens by utilizing the mechanical vibration induced from a motorized stage. The interferograms of different subapertures are acquired on the fly while the tested optics is rotating against its symmetrical axis. The measurement throughput and the subaperture positioning accuracy are improved simultaneously by adopting both the synchronous rotational scanning mechanism and the non-uniform phase shifting algorithm. The experimental measurement shows the stitched phase RMS error of 0.0037 waves proving the feasibility of the proposed phase acquisition method.

©2013 Optical Society of America

## 1. Introduction

With the development of optoelectronics technology, the demand of high precision asphere application grows at a fast pace. In the production of asphere optics, the difficulty of the testing determines the major cost of the asphere optics. The fabrication of high precision asphere can be very costly if the product volume is small and the asphere is difficult to be measured. Therefore, a cost effective and versatile asphere testing method is desired for the small volume asphere applications.

In the existing asphere testing methods, auxiliary optics such as the null correctors or the computer generated hologram (CGH) generate the aspherical reference test wavefront to null out the dense fringe in the interferometer testing [1–3]. But the dedicated auxiliary optics are, for most of the time, costly to make and require long leading time for design, fabrication, and even further proof verification. Thus, a flexible testing method is desirable to meet the demand of fast-turning industrial applications in a short time manner. The subaperture interferometric optical testing methods [4] have been widely used in the optical shop to meet such demand. This approach was first developed in testing large optical flat [5] and later applied to the testing of aspherical optics. In one of the recent studies, a high precision six-axis motion stage is utilized to provide the full six degrees of freedom of geometrical null requirement for mild asphere testing [6, 7]. Other studies use the symmetrical z-axis scanning method at the cost of two interferometer systems to test the axial symmetrical optics [8, 9].

Subaperture interferometry acquires the local interference phase maps from all portions of the tested surface and stitches into a complete tested surface map. To our best knowledge, current subaperture testing methods all require to position the tested lens and perform phase shifts after then to acquire the subaperture phase from the interferograms. Not only the phase shifting process takes time, but a comparable long waiting time is also required to cease the stage vibration motion and bring each subaperture positioning motion to a completely dead stop. As a result, the phase acquisition process significantly increases the measurement time especially when the number of the required subaperture is large or the aspherical departure is high. Furthermore, the subaperture position uncertainty is likely to increase in the routine start-and-stop mechanical positioning process. Therefore, we propose a novel subaperture phase acquisition method to increase the measurement throughput. By adopting the random phase shifting method and utilizing the stage mechanical vibrations as the random phase shifts, the interference phase is acquired on the fly while the lens is rotating. Consequently, such measurement improvement is not at the cost of any additional hardware.

## 2. Vibration modulated subaperture stitching interferometer

#### 2.1 Geometrical null and rotational measurement

In the subaperture interferometry, the interference beam retro-reflects back to the interferometer upon incidence on the tested surfaces. The wavefront of the returning beam is geometrically nulled by moving the tested optics or the interferometer reference optics. In our proposed subaperture interferometer setup, a device composed of multiple stages is designed to do the task by moving only the tested lens. This device includes one coplanar stage and two rotational stages. The corresponding stage functions are three axes of motion for the optical null and one axis for rotational measurement. The proposed interferometer is shown in Fig. 1(a).

Based on the optical prescription of the tested lens and the F-number of the reference lens, the stages are commanded to the predicted optical null position. As shown in Fig. 1(b), the tested lens is nulled on the rotational stage and rotates against its symmetrical axis while its local surface radius of curvature matches the testing laser beam. Utilizing the symmetry property of the asphere, the ring surface of tested optics at the same radial height can be measured if the testing beam is properly nulled and aligned to the tested surface. Therefore, the complete lens surface profile errors can be measured by the combination of the four-axis stages. The phase stitching method in section 2.3 is then applied to form a complete lens surface measurement result after all the subapertures of the complete lens surface are measured.

#### 2.2 Acquiring the interferograms on the fly during the lens rotation

During the testing, the tested lens is held by a lens holder installed on the rotational stage. The rotating asphere is therefore unavoidably immersed in the mechanical vibrations induced from the stage actuating mechanics during the interferometer measurement. The potential vibration impact on the interference phase is decomposed and illustrated in Fig. 2. Consider the relative movement between the laser focus spot and the tested lens due to the vibration: During the testing, the laser beam retro-reflects back to the interferometer and has its focal spot near the center of curvature of the local tested surface as shown in Fig. 2(b). Therefore, in phase shifting measurement, the longitudinal vibration motion introduces the random piston phase errors and the transverse vibration motion introduces the random two directional tilting phase errors. The other three angular vibration motions with respect to the center of curvature cause the subaperture angular positioning errors. These positioning errors are simply the subaperture coordinate’s errors in the phase stitching process. However, the angular vibrations are completely random and the amplitudes are much smaller when compared with the numerical aperture of the subaperture testing beam. As a result, the angular vibration motions are ignorable and the most significant vibration impacts on the phase shifting measurement are the random piston and random tilt phase errors.

One can use the dynamic interferometry to take the measurement of the interference phase in the presence of vibrations to overcome the phase errors [10]. However, instead of doing so, we apply the non-uniform phase-shifting algorithm to solve such problem. This algorithm was initially developed in our lab for the calibration of the piezo actuator of the phase shifting module for the reference optics [11]. It solves the following multiple nonlinear phase shifting equations simultaneously for each measured pixel located at the interferogram intensity pixel coordinate (*x*, *y*):

*i*denotes the

*i*-th randomly phase-shifted interferogram, ${I}^{dc}(x,y)$ and ${I}^{ac}(x,y)$ are the background and modulation intensity respectively, $\varphi (x,y)$ is the phase-difference distribution between two beams, ${p}_{i}$ is the random piston errors in the

*i*-th frame, ${T}_{i}^{x}$ and ${T}_{i}^{y}$ are the two directional random tilt errors, and

*M*is the total number of acquired interferogram intensity frames and its minimum requirement is three frames.

Therefore, the algorithm is insensitive to the random piston and random tilt phase errors which are both commonly seen phase errors induced from the piezo actuator response difference during the phase shifting process. Such phase shifting algorithm is very suitable to deal with the phase errors induced from vibrations As a result, instead of adopting high quality premium air-bearing rotational stages in the measurement, we exploit the mechanical vibration of the regular ball-bearing rotation stage to induce the random phase modulation for the interference phase measurement. Since the phase shifting process is not required, a laser Fizeau interferometer without the phase shifting module is utilized to provide the reference beam in the proposed subaperture interferometer.

In addition to the non-uniform phase shifting algorithm, a novel synchronous rotational subaperture phase acquisition method is also developed to increase the measurement throughput. This phase acquisition method acquires the interference phase and records the subaperture rotational position. By rotating the tested lens against its symmetrical axis, the CCD camera acquires the interferogram at the same subaperture location for as many times as the rotational cycles. With a synchronized CCD camera, a complete set of subapertures at the same azimuthal height can be continuously acquired on the fly. Also, since the rotational stage is at a constant rotational speed, the output rotational angle of each subaperture is precisely determined by the encoder to just a few arc seconds precision, eliminating the possible backlashing errors during the repeatedly stop-start stage re-positioning motion. As a result, the precisely determined subaperture coordinate significantly reduces the position uncertainty in the later phase stitching step. The multiple interferograms at the same rotational angle is re-arranged and sorted as a set of randomly phase-shifted interferogram. The interference phase is then calculated by the non-uniform phase shifting algorithm [11].

Figure 3 shows the acquired interferograms of a spherical lens described in section 3. There are 60 interferograms located at 12 subapertures that are evenly spaced every 30 rotational degrees. Therefore, each subaperture has five randomly phase shifted interferograms. Due to the misalignment between the optical axis and the rotational stage, we can see the varying tilt fringes dominating the measured interference phase.

#### 2.3 Phase Stitching Process

After the interference phase map of each subaperture is calculated by the non-uniform phase shifting algorithm, the subaperture phase maps are stitched into a complete lens surface phase map. In the phase stitching process, the random piston, two directional tilts, and defocus wavefronts in the subapertures are used as the linear compensators to minimize the overall compensated phase difference of the overlapped pixels from different subapertures [12, 13]. Given the number of subapertures *N*, during the optimization process, the merit function *S* is defined as the sum of squared errors of the compensated phase at all the overlapped pixels as

*i*-th subaperture and ${\phi}_{j}(k)$ is the phase map of the

*j*-th subaperture at the same pixel

*k*. The four parameters $\delta ,\alpha ,\beta ,\kappa $ are the coefficients of the compensators: piston, two directional tilts, and defocus respectively. In addition, the sub-aperture radial position is scanned to minimize the positioning uncertainty [14].

After the optimization process, the standard deviation of the difference of the compensated phase from different subapertures is used as the figure of merit to determine the phase stitching quality of each overlapped pixel *k*,

*k*, ${\widehat{\phi}}_{i}\left(k\right)$ is the compensated phase of the

*i*-th subaperture after the optimization process, and ${\phi}_{avg}\left(k\right)$ is the average compensated phase from the $N\left(k\right)$ subapertures. The standard deviation $\sigma \left(k\right)$ of the stitched phase can be used as the indicator for the quality of the stitched phase pixel wise.

## 3. The measurement of a spherical surface

A high quality spherical lens with 89 mm radius of curvature is used as the tested object for the verification proof of the proposed method. The lens is first tested with an F/1.5 reference sphere by the four-step phase shifting method in full aperture. Later, the reference optics is changed to a slower F/3.3 one. The proposed vibration modulated subaperture interferometry is then applied. The measurement result by the full aperture method is shown in Fig. 4(a) while the stitched phase measurement result is shown in Fig. 4(b) yielding similar surface profile. Therefore, the proposed method is convincing given that both reference optics having about typical 0.005 waves RMS wavefront errors.

In this measurement, there are 24 subapertures used to cover the complete lens surface. The number of overlapped subapertures of each tested surface pixel is shown in Fig. 4(d). Counting that each subaperture has five phase shifting interferograms, there are total 120 interferograms acquired. The F/3.3 reference optics emits the beam cone size of 17.23 degree. The CCD camera has a square resolution of 480 × 480 in the interested subaperture region with the frame exposure time set at 89 μs. The rotational stage takes 30seconds for one rotation scan, i.e 12 degrees/sec. Therefore, the corresponding tested surface image moving speed at the CCD sensor plane is 12 × 480/17.23 = 334 pixels/sec. Inversely, the image of the locally tested surface travels over one pixel distance in 3 ms which is significantly longer than the exposure time 89 μs. Therefore, in this experimental setup, the shutter time is sufficiently fast not only to freeze the vibration induced fringe motion but also enough to keep the native lateral resolution of the captured frame.

The stitching error map, as suggested by Eq. (3), is shown in Fig. 4(c). The RMS phase stitching error is lower than 4/1000 waves, proving the attainable accuracy of the proposed phase acquisition method and the stitching algorithm.

## 4. Conclusion

In this research, the vibration induced from the rotational stage is utilized as the random phase modulator in the interference phase acquisition process. The interferogram is acquired on the fly while the tested lens is rotating. Therefore, the measurement throughput and subaperture positioning is significantly improved without any additional optics or hardwares. The phase stitching error of a spherical surface proves the attainable accuracy of the proposed method.

Not only applicable to the spherical surface shown, the proposed method can also be utilized in the subaperture interferometry of asphere since the random phase shifting algorithm works with the dense fringe distribution as well. However, there are some factors that cannot be overlooked when the asphericity is high. For example, the retrace errors have to be considered due to the departure of the common optical path inside the interferometer. This retrace errors induced phase has to be calibrated before the phase stitching process. Also, the subaperture coordinate has to be precisely determined to eliminate the phase stitching errors induced from the high asphericity.

## Acknowledgments

The authors would like to thank the National Science Council of Taiwan for financial support under Contract No. NSC 100-2627-E-008-001.

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