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Abstract
A pure adaptive interferometer is proposed for optical free form surfaces metrology without auxiliary devices such as the wavefont sensors and deflectometry systems for DM monitoring. In this method, the DM surface monitoring and free form surface measurement are achieved simultaneously in only one interferometer. The polarizing optics divide the interferometer into two partial common path interferometric system, which provide the null test for tested free form surface and non-null test for the DM surface. The final figure error of the free form surface is extracted by ray tracing. Experiments proving the feasibility of this interferometer is shown.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The freeform optical element has been widely used in illumination [1], displays [2] and imaging systems [3] due to its high performance in beam shaping and aberration correction. It has been an important member in modern optics, following after the rotational symmetric aspheric optics. Advanced design and fabrication technologies for high-quality optical freeform surfaces have been developed. Nevertheless, the metrology approach has not yet kept pace in past few years, and thus limits the application of freeform optics [4].
The best choice for precision optical surfaces test is the non-contact metrology, led by various interferometry, which have elegant performance in modern optical testing such as spherical and aspherical surfaces test [5]. Especially, with designed null [6,7] and non-null compensators [8], the interferometry for aspherical surfaces now achieves considerable accuracy. However, it is powerless for free from surfaces test because of the difficult of non-rotational symmetric wavefront aberrations compensation in a traditional interferometer, even in a subaperture stitching interferometer [9–11]. For these aberrations, the special compensator (such as CGH [12, 13]) has to be produced, which would loses the flexibility greatly. The tilted wave interferometer (TWI) [14–16] would increase the flexibility in some extent. But the lateral resolution of the microlens array limits the accuracy. The deflectometry [17–19], with excellent flexibility, has been used to test free form surfaces in recent years. But the delicate calibration process has always been its bottleneck in testing accuracy.
Therefore, the balance of the flexibility and accuracy has been researchers’ pursuit, which refers to the flexible null interferometry. Thus the deformable mirror (DM) started coming into researcher’s sight due to its high performance in flexible aberration correcting. In recent years, the commercial available DMs have arisen in aspheric and free form optics interferometry. In 2004, Pruss [20] made effort to employ a membrane DM to realize the alterable Zernike defocus compensation for flexible aspheric surfaces test. This attempt inspired the flexible test for free form surfaces. However, this idea does not have prominent developments in next few years because the DM surface in work is also a free form surface and the surface accuracy is unpredictable.
Thus, the DM surface figure monitoring had been the urgent issue in the flexible free form interferometer. In 2014, Fuerschbach [21] carried out the null test for an φ polynomial mirror with the assistant of a DM. In this test, the DM surface was measured by the Zygo inteferometer in advance and thus unable to achieve the real time test. The final accuracy would be reduced by this non-real time test due to the instability of the DM surface. Moreover, this null test still relied on the specially designed off-axis configuration, which hindered flexibility in some extent. In 2016, Huang. proposed an adaptive null interferometric method based on the DM for free form surfaces test with unknown shapes [22], in which the DM and null optics were employed for null test for the free form surface in an Zygo interferometer while a deflectometry system (DS) was employed for DM deformation monitoring in real time. Then, the final surface figure was acquired by the ray tracing. However, this method would suffer the sophisticated calibration and unsatisfactory precision of the auxiliary DS.
In this paper, we proposed the pure adaptive free form surface interferometry (AFI), in which a polarized system design achieves simultaneous measurement of the free form surface and the DM surface in real time, without any other auxiliary devices such as the wavefront sensor and the DS for DM monitoring. The free form surface is tested in a null configuration while the DM surface is tested in a non-null configuration. The final figure error of the tested surface would be extracted by the ray tracing with the two results. The polarized optics are employed to eliminate the crosstalk of the common-path part of the two configurations. Because the commercial available DMs now have limited stroke, on the order, of 40 microns maximum (about 63λ with λ = 632.8nm in this paper) depending on the aberration type (except for tip and tilt), it would be within the dynamic range of a common interferometer. Thus the DM monitoring would not suffer the low-spatial resolution of the Shark-Hartmann wavefront sensor and the sophisticated calibration and unsatisfactory precision of the DS. It is in favor of the interferometer instrumentation in optical shop test.
2. Principle
As is described above, the real time and accurate measurement for DM surface figure is the crucial issues in the adaptive interferometer for freeform surfaces. The previous attempt either lost real time performance [21], or relied on the extrinsic assist [22]. As a result, sophisticated calibration and unsatisfactory precision would be suffered. To address this issue, we measure the free form surface and the DM surface in work simultaneously in only one interferometer.
Figure 1(a) illustrates the AFI system layout, which is based on a special designed polarized interferometer. The system physically consists of the interferometric system (left of the green dotted line) and the polarized compensation system (right of the green dotted line), as is shown in the Fig. 1(a). In the interferometric system, the laser is reflected to the beam expander by a small reflector. The collimated laser beam from the beam expander is split into two by a beam splitter (BS 1). One is reflected by a reference mirror mounted on a piezoelectric ceramic transducer (PZT), serving as the reference beam; the other travels into the polarized compensation system. The polarized compensation system sends back two beams, which are the null wavefront from the tested free form surface after compensation and the non-null wavefront from the DM surface, respectively. The two beams interfere with the reference beam and the resulted interferograms are imaged at the CCD1 and CCD 2, respectively. In fact, the two CCDs are belong to two separate interferometric configurations, which is called the free form surface interferometric configuration and the DM monitoring interferometric configuration. Figures 1(b) and 1(c) present the two partial-common path interferometric configurations, which are separated by polarized optics.
Fig. 1 The system layout of the AFI. (a) The AFI sketch. (b) The free form surface interferometric configuration (c) the DM surface monitoring interferometric configuration. For the balanced fringe contrast, the transmittance of the BS 1 and BS 2 are all 90%.
Figure 1(b) illustrated the free form surface interferometric configuration. The transmitted beam from BS1 (90% transmittance) arrives at the polarized beam splitter 1 (PBS1). The p-polarized part travels through the PBS1 and a λ/4 wave plate (WP) and then arrives at the DM surface. The reflected p-polarized beam travels through the λ/4 WP again and becomes the s-polarized beam. Thus this s-polarized beam would not pass through the PBS 1 and all reflected by the PBS 1. The reflected s-polarized beam travels through the BS 2 (90% transmittance) and the null optics and then arrives at the tested free form surface. The reflected s-polarized beam from the free form surface would backtrack and reflected by the DM again. The polarized direction of the resulted beam arrived at the PBS 1 would be rotated 90°once again due to another round-trip traveling through the λ/4 WP. That is the beam at the PBS 1 becomes the p-polarized one once again and all travels through the PBS 1. Half part of the p-polarized beam reflected by the BS 1 and then all travels though the PBS 2, as the p-polarized test beam. Note that PBS2 is the same as PBS 1 and the reference beam traveling through PBS 2 are p-polarized one too. The CCD 2 receives the interferogram of the p-polarized test beam and the reference beam by the imaging lens. The CCD 2 provides the feedback and drives the DM deformation to pursue the null fringe. In this process, the system aberrations are compensated by DM and the null optics. Note that the null optics does not mean the strict “null” compensator but refers to some relative tolerant near null optics such as the partial null lens [8, 23, 24] or partial compensating CGH. The undercompensation is permitted and would be supplemented by the DM. Thus the design of the null optics would has a large toleration with more simple structure. The DM deformation should be monitored in real time due to its instability. That is, interferograms in CCD 1 and CCD 2 should be obtained simultaneously. The DM monitoring alleviated in previous literature is either incapable of in situ action, or depend on other non-interferometric devices. We complete the in situ DM monitoring accompany with the freeform surface test in only one interferometer as is shown in Fig. 1(c).
The DM monitoring interferometric configuration is illustrated in Fig. 1(c). The beam propagation from the laser source to the BS 2 is the common path with the freeform surface interferometric configuration as the Fig. 1(b). As the s-polarized beam arrived at the BS 2, the reflected part travels through a λ/2 WP and becomes the p-polarized beam. Then the p-polarized beam would all travel through the PBS 2, which is called monitoring beam. The s-polarized part of the reference beam would reflected by the PBS 2. The p-polarized monitoring beam and the s-polarized reference beam travel through a polaroid and thus interfere with each other at the direction of the polarization axis. The CCD 1 receives the non-null fringes which characterize the first reflected wavefront of the DM deformation. Note that the wavefront in the interferometer is reflected by the DM for two times while the CCD 1 only measures the first reflection, which reduces the load of CCD 1. The commercial available DMs now have limited stroke, on the order, of 40 microns maximum depending on the aberration type (except for tip and tilt). It induces about maximum 63 fringes (λ = 632.8 nm), within the coverage of the dynamic range in a traditional interferometer (CCD 1). It is noteworthy that such a DM would achieve aberration compensation with 80 microns PV value by two reflections in this polarized system. With the assistant of null optics for the aspheric base compensation, the free form surface with 80 um departure from its aspheric base is measurable in null condition.
To sum up, the CCD 1 and CCD 2 measure the DM and the free form surface simultaneously in only interferometry. The CCD 2 measures the null fringes after compensation by the DM and null optics while the CCD 1 measures the non-null fringes due to the DM surface deformation. The final figure of the tested free form surface would be obtained by the ray tracing for the two results as is shown in Fig. 2. In this way, the measurement difficulty of the tested free form surface is shifted to the DM monitoring. Note that a big difficulty in the non-null test for freeform surface is the tested surface misalignment calibration [25]. In this null test interferometer, the misalignment calibration is shifted from the tested free form surface to the DM as well. Although the in-work DM, as a freeform surface too, is hard to be calibrated. But it can be easily calibrated in case of no deformation or other rotational symmetric deformation in advance.
3. The DM deformation monitoring in interferometry
Due to the non-null fringes at the CCD 1, the DM deformation surface in work have to be extracted by the ray tracing method based on the DM monitoring system model as the Fig. 1(c). However, the DM, as a free form surface in work, is difficult to be modeled due to its six-axis freedom degree. Therefore, the posture and position should be determined before deformation. That is, it would be calibrated in advance.
3.1 DM Calibration
Before the DM calibration, a standard flat mirror, instead of the DM surface, would be employed for system error storage as is shown in Fig. 3(a). The measured system error is set as the surface sag of a “dummy surface” in the monitoring system model for subsequent ray tracing. Then the DM calibration is carried out. The standard flat mirror in Fig. 3(a) is replaced by the DM with flat surface as is shown in Fig. 3(b). In this case, the tilt of the flat surface of DM is easily to be aligned by wavefront aberration estimation. Note that the DM surface with no voltage driven is not an accurate flat. The flat of the DM would be obtained in a close-loop condition (10 nm rms error) with the system error deduction in the monitoring system. Subsequently, a defocus deformation of the DM is provided and the DM surface decentration can be also calibrated by the tested wavefront evaluation, as is shown in Fig. 3(c). The DM in experiment would be aligned until the coma coefficients of the tested wavefront less than a threshold value (such as 0.02 λ). The DM-providing defocus with 3λ PV value would merely induce 6 nm rms error, which would bring about minuscule calibration error. Then the DM in model is aligned until the coma coefficients of the tested wavefront equal to the one in experiment. Now we consider the decentration of the DM surface in experiment is little and has the same value with the one in the model. The next step is the DM axial position determination. In general, the defocus aberration can be used to solve the axial position of DM by ray tracing. But the DM surface sag is unknown accurately together with its axial position. That is the DM surface figure would be an aspheric surface when providing the defocus aberration. The surface figure should be solved together with the axial position. Therefore, single measurement for ray tracing, which means one mathematic function, would not be able to address this problem. A series of axial shifts with known displacement distance would work, as is shown in Fig. 3(d), which was proposed to measure the paraxial radius and axial position of aspheric surface in our previous work [26]. The relative accuracy about 0.02% is proved. The principle is to set up multiple measurements for ray tracing with known constraints. These constraints are the axial displacement distance measured by the precise shifter or the displacement measurement interferometer (DMI, such as the Renishaw DMI). The multi-configuration model is set up, in which the multiple displacements of DM act as the constraint while the DM initial axial position () as the variable. When all the simulated test wavefronts are close enough to experimental ones, the DM axial position () in simulation would be consistent with the actual one (). Now, the DM calibration is complete. Note that the previous researches on the adaptive interferometry [21,22] must calibrate the DM posture and position as well.
Fig. 3 The DM calibration process. (a) is the system error storage in the monitoring system, (b) is DM tilt calibration, (c) is the DM decentration calibration, (d) is the DM axial position determination.
3.2 DM Monitoring
When the DM is in work, DM surface figure have to be extracted by the ray tracing method based on the DM monitoring system model. The model based ray tracing method has been proposed for retrace error correction in non-null aspheric interferometry in our previous work [27–29], where the basic idea would been applied to the DM monitoring interferometry. The test wavefront in experiment at the CCD 1 can be expressed by an implicit function as
where , and are the DM surface figure, DM surface misalignment and other system error, respectively. The Eq. (1) in the multi-model would bewhere , , and are the simulated counterparts in the model corresponding to Eq. (1), respectively. The and are calibrated in case of no deformation (as a flat) as mentioned above, which is described asA closed feedback system is set up to change the DM surface figure error () in the model, making the simulated wavefront () approaching to the actual one () in the experiment. We defined an optimized objective function with Eqs. (1) and (2) to describe this process as
The would be able to characterize the actual one () if the Eq. (4) is satisfied. That is
4. Experiment validation
The experiment system layout is presented in Fig. 4. The employed laser is the HNL008LB from Thorlabs corp. with wavelength 632.8 nm. The beam traveling from the beam expander is about 30 mm. The tested surface is a bi-conic mirror with 52mm aperture, 244mm x radius and 250mm y radius. The nominal sag of the bi-conic surface can be mainly decomposed into the Zernike defocus and astigmatism. A null optics was designed to cover the defocus component, whose parameters are listed in Table 1. The AlpaoTM DM88 is employed to cover the main astigmatism component, with 88 actuators at 25mm aperture. The WPs are from Thorlabs corp. with 20-10 scratch-dig and λ/4 PV surface quality. The CCDs employed are MVC930DAM/C-GE30 from Microview.
Table 1. The Parameters of the Null Optics.
4.1 Experimental calibration
Firstly, the calibration like the Fig. 3(a) was carried out and the system errorwas measured and fitted with Zernike standard polynomials, which is illustrated in Fig. 5(a). This error was adhere to a “dummy surface” in the model. Then the DM misalignment was calibrated as is presented in Fig. 3(b). The tilt of the flat surface of DM was aligned and the resulted wavefront tilt aberration was less than 0.08 PV. Because the beam aperture is larger than the DM aperture, the decentration calibration for the DM surface was avoided. The next step was the DM axial position determination. The DM with defocus deformation shifted along the optical axis for seven times and each was monitored by a Renishaw DMI, as is shown in Fig. 5(b). Thus 8-configuration model was set up accordingly, in which the initial DM axial position was set as the variable. The in the model was optimized and the result is presented in Fig. 5(c), in which the fluctuation of is less than 0.03 mm within 4-8 configurations participating in optimization. The average value was obtained. This fluctuation would introduce wavefront error of about 0.009 rms value. Now we complete the DM calibration.
Fig. 5 Experimental results of initial calibration. (a) The Zernike coefficients of the system error of the DM monitoring system, (b) The experiment layout of the DM axial position () determination, (c) The results of the DM axial position determination in different configurations by ray tracing.
4.2 Results
Then the tested surface was placed into the system. The interferogram at the CCD 2 provided feedback to the DM to pursue the null fringe. The adaptive process was performed by the ALPAO Core Engine (ACE), a powerful software for adaptive optics. Of course, the tested surface was aligned accordingly. The changing process is illustrated in Fig. 6. Figure 6(a) is the interferogram with only a matched transmission sphere as compensator while Fig. 6(b) is the one with the only null optics. The ellipticalness of the two interferograms arise from the large astigmatism. Figure 6(c) refers to the interferogram with the null optics and part compensation of the DM. Figure 6(d) is the final null fringe with the complete compensation of the null optics and DM. The according interferograms at the CCD 1 for DM deformation monitoring was also obtained simultaneously and listed in the Figs. 6(e)-6(g).
Fig. 6 The interferogram at the CCD1 and CCD 2, respectively. (a) is the interferogram with only a matched transmission sphere as compensator, (b) is the interferogram with the only null optics, (c) refers to the interferogram with the null optics and part compensation of the DM, 6(d) is the final null fringes with the complete compensation of null optics and DM. 6(e)-6(g) are the interferograms at the CCD 1 for DM deformation monitoring according to the process from 6(b)-6(d).
The final DM surface figure was extracted by the ray tracing for monitoring interferometer according to the interferogram in Fig. 6(g) and the result is presented in Fig. 7(a), which refers to the deformation in form of Zernike defocus, astigmatism and spherical aberrations. The figure error of the tested free form surface was also extracted by ray tracing for the test system according to the null fringes in Fig. 6(d). Of course, the system error was calibrated and stored in the test system model in advance. After the ray tracing, the final null fringe in the model and the figure error result is presented in the Figs. 7(b) and 7(c).
Fig. 7 Experiment results. (a) DM surface figure, (b) The final null fringes in the model, (c) The map of figure error of the tested freeform surface.
For the more powerful cross validation, we also measured a paraboloidal mirror in the AFI and Zygo interferometer. The tested surface has a 50 mm aperture and about 17 μm departure from its vertical spherical surface. The DM was employed for Zernike defocus compensation. Figure 8(a) refers to the initial interferogram in the experiment with the same null optics above. The interferogram after DM compensation is shown in Fig. 8(b) while the one characterized DM surface is presented in Fig. 8(c). The test surface figure error was extracted and presented in Fig. 8(d), with a cross validation result shown in Fig. 8(e), which is the result in Zygo interferometer by the aberration-free method. Figure 8(f) is the direct difference map charactering the testing error. Specific parameters of the three maps are listed in Table 2, which show us the validity of the AFI.
Fig. 8 Test results of the paraboloidal surface by spherical aberrations compensation. (a) is the initial interferogram in the experiment, (b) is the interferogram after DM compensation in the experiment, (c) is the interferogram characterizing the DM surface, (d) is the surface figure error map in AFI, (e) is the figure error map by Zygo interferometer, (f) is the error map between (d) and (e).
Table 2. Specific Parameters of the Test Results by AFI and Zygo Interferometer
5. Conclusion
We proposed a pure interferometry method for optical free form surface adaptive measurement. As the adaptive null compensator, the DM surface is measured in interferometry with the free form surface measurement simultaneously, without other assistant devices such as the wavefront sensor and deflectometry system. The polarized design provides the interferometer two common-path interferometric configurations: the free form surface interferometric configuration and the DM monitoring interferometric configuration. The former measures the null aberrations after the compensation of null optics and DM surface. The latter affords the measurement of only one reflection of DM surface in real time, which expands the flexibility. The final figure error of the freeform surface is extracted by the ray tracing based on the system model, for which the DM is calibrated and measured accurately. Experiments proving the feasibility of the method are carried out to test a bi-conic surface and a paraboloidal surface. It is in favor of the interferometer instrumentation and would have enormous potential in optical shop test. For steeper free form surfaces metrology, the DM series would work with respective interferometric monitoring.
Funding
National Natural Science Foundation of China (61705002, 61675005); Opening project of Key Laboratory of Astronomical Optics & Technology in Nanjing Institute of Astronomical Optics & Technology of Chinese Academy of Sciences (CAS-KLAOT-KF201704); Doctoral Start-up Foundation of Anhui University (J01003208); National Key R&D Program of China (2016YFC0301900).
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