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Validation of simultaneous reverse optimization reconstruction algorithm in a practical circular subaperture stitching interferometer
Optics Communications 403 (2017) 41–49
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Validation of simultaneous reverse optimization reconstruction algorithm in a practical circular subaperture stitching interferometer Lei Zhang a, *, Dong Li b , Yu Liu a , Jingxiao Liu a , Jingsong Li a , Benli Yu a a b
Key Laboratory of Opto-electronic information Acquisition and Manipulation Ministry of Education, Anhui University, Hefei, 230601, China Research Center of Laser Fusion, China Academy of Engineering Physics, Mianyang, 621900, China
a r t i c l e
i n f o
Keywords: Subaperture stitching Aspheric surface test Interferometer modeling Calibration
a b s t r a c t We demonstrate the validity of the simultaneous reverse optimization reconstruction (SROR) algorithm in circular subaperture stitching interferometry (CSSI), which is previously proposed for non-null aspheric annular subaperture stitching interferometry (ASSI). The merits of the modified SROR algorithm in CSSI, such as auto retrace error correction, no need of overlap and even permission of missed coverage, are analyzed in detail in simulations and experiments. Meanwhile, a practical CSSI system is proposed for this demonstration. An optical wedge is employed to deflect the incident beam for subaperture scanning by its rotation and shift instead of the six-axis motion-control system. Also the reference path can provide variable Zernike defocus for each subaperture test, which would decrease the fringe density. Experiments validating the SROR algorithm in this CSSI is implemented with cross validation by testing of paraboloidal mirror, flat mirror and astigmatism mirror. It is an indispensable supplement in SROR application in general subaperture stitching interferometry. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Accurate interferometric testing for large and deep aspheric surfaces has always been a challenge. Traditional interferometric testing approach is limited by the spatial-frequency resolution of the interferometric system. The subaperture stitching interferometry (SSI), as an alternative, was proposed in 1980s [1–4] to overcome the limitation. According to the differences of subaperture shape, circle subaperture and annular subaperture were developed for aspheric surfaces test. Annular subaperture stitching interferometry (ASSI) [5–8] employs a transmission sphere to generate spherical wavefronts of different radii of curvature to match the slopes in different annuli on the aspheric surface, and then combines the results of many subaperture measurements to obtain the full aperture surface map. But it applies only to rotational symmetric aspheric surfaces. Circular subaperture stitching interferometry (CSSI) [1–4,9–13] employs an optimal transmission sphere to generate spherical wavefronts of different radii of curvature to match the circular areas of different locations at surface aperture. It is now commercial available by QED Technologies, which has the test capacity of mild aspherical surfaces [11,12]. But its complex six-axis motioncontrol has always limited the test accuracy due to the mechanical error introduced during alignment. Moreover, each circular subaperture * Corresponding author.
E-mail address: [email protected] (L. Zhang). http://dx.doi.org/10.1016/j.optcom.2017.07.004 Received 23 April 2017; Received in revised form 14 June 2017; Accepted 2 July 2017 0030-4018/© 2017 Elsevier B.V. All rights reserved.
suffers retrace error due to the non-null test although the variable optic null (VON) [13] is employed for nullifying the test in some extent. It is obvious the stitching algorithm has played a vital role in the application of SSI. Many landmark algorithms have been proposed for subaperture stitching. From the Kwon Thunen method [14] and the simultaneous fit method [15], to the discrete phase method [16], the multi-aperture overlap-scanning technique [17] and the subaperture stitching and localization (SASL) algorithm [18], without exception, depend on either complex mathematic calculation or subaperture overlap for misalignment correction. Moreover, the retrace error is not treated suitably in most algorithms. In our previous work, we proposed a simultaneous reverse iterative optimizing reconstruction (SROR) algorithm [19] based on system modeling in a non-null ASSI [19,20]. All the subaperture measurement configurations were simulated with a multi-configuration model. The Zernike standard coefficients of full aperture figure error can be extracted directly from those of subaperture wavefronts by simultaneous ray tracing of the multi-configuration model, without any separate treatment for retrace error and subaperture misalignments (the misalignment was modeled). This method concurrently accomplished subaperture retrace error and misalignment correction, requiring neither complex mathematical stitching algorithms nor subaperture overlaps.
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Optics Communications 403 (2017) 41–49
In this paper, we continue to show the validity of the SROR algorithm in CSSI with a practical interferometer. The practical CSSI employs an optical wedge to deflect the incident beam and scan subaperture by its rotation and shift instead of the six-axis motion-control system. The wedge can provide the Zernike tilt and coma while the reference path provides Zernike defocus for each subaperture, which would decrease the fringe density obviously. The modified SROR algorithm then is employed for circular subaperture stitching. The retrace error and misalignments are corrected concomitantly with no consideration of overlap an even few missed coverage. With the previous work [19] and this one, the SROR algorithm is proved effective for general SSI. In Section 2, we outlines the practical CSSI system and the principle of the modified SROR method; detailed simulation analysis characterizing the SROR method is discussed in Section 3; the experiment verification is presented in Section 4; Section 5 presents a conclusion. 2. The practical CSSI with modified SROR method Fig. 1 illustrates the CSSI system for mild aspheric metrology. As is shown in Fig. 1(a), the collimated laser beam from the beam expander is split into two by a beam splitter. One is converged by an aplanatic lens and reflected by a reference mirror mounted on a piezoelectric actuator, serving as the reference beam; the other traveling though the transmission sphere (TS) and optical wedge is then reflected by the test aspheric, forming the test beam after traveling though the TS again. The reference and test beams interfere at the beam splitter and the resulted interferogram is imaged onto the detector (here CCD) by the imaging lens. The tunable laser is able to execute phase shift by wavelength changing. When testing, the deflected beam scan subapertures by the rotation and shift of the optical wedges instead of the six-axis motion-control system. As is shown in Fig. 1(b), the optical wedge deflect the incident beam to test a non-centric subaperture and thus scan all subapertures in an annular region. Meanwhile, with the wedge moving gradually away from the tested surface along the optical axis, the scanning region expands to cover the full aperture. Note that the fringe density of off-axis subaperture interferograms is mainly attributed to the low-order Zernike aberrations such as the wavefront tilt, defocus, coma and astigmatism. The optical wedge, not only deflect the beam for scanning, but just introduce wavefront tilt and coma aberrations at the direction of beam deflection when the incident wave is the spherical wavefront. The Zernike tilt and coma in the off-axis subaperture interferograms would be compensated in some extent with the wedge shift and selected wedge angle. While the reference mirror can be translated along the optical axis around the focus of the aplanat by an axial driving mechanism. It provides Zernike defocus to reduce fringe density of each circular subaperture as well. Therefore, some mild aspheric surface can be tested in the CSSI with subapertures of measurable fringe. The combination of movable reference mirror and the wedge, acting like the variable optical nulling (VON) in QED’s technology [13], decreases the fringe density of subaperture interferograms. Surface figure information is then obtained over subapertures with rational treatment for the misalignment and retrace error. Note that the optical wedge and movable reference mirror would introduce residual higher-order aberrations accompanying with the low-order aberrations compensation. These higher-order aberrations then would be treated together with the retrace error in the SROR algorithm proposed in nonnull ASSI previously. The SROR algorithm was proved effective in ASSI and now would be validated in CSSI with some modification. The SROR algorithm is based on system models. The non-null interferometer system modeling and calibration has been researched in our previous work [19,21–23]. In our method, a multi-configuration model is setup in a ray tracing program as is shown in Fig. 2, in which each configuration is a complete non-null interferometric system, characterizing a subaperture measurement. The only difference in all the interferometric configurations is the location of reference mirror and optical wedge.
Fig. 1. The sketch of the practical CSSI.
The initial models of aspheric surface, reference mirror and optical wedge in all configurations are ideal. However, the relative misalignments would be induced in experiment by the movements of the reference mirror and optical wedge for different subapertures test although an initial calibration is performed. A completely perfect calibration in experiment is unachievable usually. That is there are some slight deviations in the locations of the reference mirror and optical wedge between the model and experiment. Fortunately, these slight deviation can be made up by fine revisions of the model, that guarantee the high consistency of misalignments of the reference mirror, optical wedge and aspheric in the model with those in experiment. In our previously reported method, these misalignments are calculated beforehand and then revised in the model. However, this method is not applicable because the couple of the misalignments of the reference mirror, optical wedge and test surface impedes the accurate misalignment calculation. Therefore, we improved it by a more convenient revision approach in this paper. Unlike the calibration of subaperture misalignments modeling by specific misalignment calculation in ASSI [19], we improved the method by embedding the misalignment model calibration into the SROR algorithm as is shown in Fig. 2. Take the optical wedge as example, each subaperture wavefront resulting from a special configuration in the experiment is employed for evaluating the misalignment aberrations of the optical wedge in this configuration. The specific misalignment calculation is not necessary. The Zernike coefficients of these misalignment aberrations from experiment would be set as optimization objectives in the corresponding configuration in the model, while the specific misalignments can be extracted as the optimization variables. Then the model of optical wedge can be revised automatically by the reverse ray tracing optimization. The calibration method apply to the reference mirror and aspheric surface as well. The tested aspheric surface modeling should be revised firstly with the configuration without optical wedge, and this revision need only once because the aspheric under test does not need to move in other configurations. In view of the error couple, the subsequent revision 42
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Fig. 2. The multi-configuration model for ray tracing in SROR.
would aim simultaneously at the reference mirror and the optical wedge in different configurations. The simultaneous calibration in multi-configuration would decrease the error couple [19,24]. The final misalignments of all the reference mirror, optical wedge and aspheric in the model would achieve high consistency of those in experiment. Of course, an initial calibration for the reference mirror and the wedge should be implemented to reduce the error coupling. The overall description can be expressed as: { 𝐸m𝑖𝑠_𝑚 = 𝐸 m𝑖𝑠_𝑚 (1) 𝐸ret_𝑚 = 𝐸 ret_𝑚
implemented on all the subapertures simultaneously in the multiconfiguration model rather than each subaperture separately. We redefine a global optimization function ) ]2 [ ( ( )2 (𝐸asp ⊕ 𝐸asp ⊕ ⋯ ⊕ 𝐸asp ) 𝑈 = 𝐸ASP − 𝐸 ASP = − 𝐸 asp ⊕ 𝐸 asp ⊕ ⋯ ⊕ 𝐸 asp =
𝑚=1
𝐸ASP = 𝐸 ASP . (2)
(7)
To define more clearly, Zernike polynomials are employed to describe the wavefront and figure error. In this way, the full aperture figure error can be reconstructed in form of Zernike standard polynomials without special stitching process. Obviously, we need not special mathematic algorithm for stitching process. The SROR method is capable of sewing together all the subapertures with retrace errors and misalignment errors correction at the same time without overlap.
where, 𝐸asp_𝑚 is the subaperture figure error in experiment. We extract the 𝐸asp_𝑚 as { ( ) 𝐸asp_𝑚 ≅ 𝑓 −1 (𝑊𝑚 )− 𝐸m𝑖𝑠_𝑚 − 𝐸ret_𝑚 , (3) 𝐸 asp_𝑚 ≅ 𝑓 −1 𝑊 𝑚 − 𝐸 m𝑖𝑠_𝑚 − 𝐸 ret_𝑚 where 𝐸 asp_𝑚 and 𝑊 𝑚 are the simulated counterparts in the model corresponding to experiment, respectively. Of course, the initial values of 𝐸 asp_𝑚 and𝐸 m𝑖𝑠_𝑚 are zero in the model. A closed feedback system is set up to change the simulated subaperture figure error (𝐸 asp_𝑚 ) in the ray tracing program, making the simulated subaperture test wavefront (𝑊 𝑚 ) approaching to the actual one (𝑊𝑚 ) in the experiment constantly. We defined an optimized objective function with Eq. (3) to describe this process as ( )2 𝑈 = 𝐸asp_𝑚 − 𝐸 asp_𝑚 [ ( ) ) ]2 ( −1 (4) (𝑓 (𝑊𝑚 )− 𝐸m𝑖𝑠_𝑚 − 𝐸ret_𝑚 ) = − 𝑓 −1 𝑊 𝑚 − 𝐸 m𝑖𝑠_𝑚 − 𝐸 ret_𝑚
3. Simulation and discussions We carried out a numerical simulation for validation of SROR algorithm in CSSI. The tested surface and figure error added artificially (Fig. 3(a)) are all same with those in Ref. [16] for comparison. A matched TS with F/# 0.3 and a wedge with 14◦ wedge angle are selected in the simulation. The CSSI system was modeled in ray trace program. The upper limit of the fringe frequency was defined as 0.125𝜆∕𝑝𝑖𝑥𝑒𝑙 to promise a large redundancy in range of the resolving capacity of the detector. Four subapertures were obtained with corresponding interferograms in Fig. 3(b). Random slight misalignments (0 ∼ 0.2𝜆 tilts) and Gaussian distribution random noise (zero-mean and noise-to-signal ratio (NSR) 0.01) were added to each subaperture to simulate actual experiments condition. Then, a four-construction model is set up according to nominal parameters of the simulated experiment. Note that the additional misalignments would be removed by the calibration of model with an initial reverse optimization [23], aiming at make misalignments of each subaperture in the model accord with those in the simulated experiment. After the SROR algorithm performed, the full aperture figure error is extracted, showing in Fig. 3(c). Fig. 3(d) is the result cited from the Ref. [20], which is extracted by the ‘‘ROR-stitching’’ algorithm for full aperture in non-null ASSI. Figs. 3(e) and (f) present the reconstruction
= min . Then, Eq. (4) would be simplified by Eq. (1) as ( )2 𝑈 = 𝐸asp_𝑚 − 𝐸 asp_𝑚 [ ( )]2 ( ) = min . = 𝑓 −1 𝑊𝑚 − 𝑓 −1 𝑊 𝑚
(6)
𝑚=1
where, the 𝐸ASP and 𝐸 ASP are the full aperture figure error in experiment and simulation, respectively. The M is subaperture number and the sign ⊕ means the operation of addition in morphology, respectively. When Eq. (6) is satisfied well, i.e. all the simulated subaperture wavefront (𝑊 𝑚 ) are closed to those (𝑊𝑚 ) in the experiment simultaneously, we obtain
where, 𝐸mis_𝑚 and 𝐸ret_𝑚 are the subaperture misalignments and retrace error in the experiment, while 𝐸 m𝑖𝑠_𝑚 and 𝐸 ret_𝑚 are the simulated counterparts in the model corresponding to experiment, respectively. Based on the idea of ray tracing, each subaperture wavefront 𝑊𝑚 can be expressed by an implicit function as ( ) 𝑊𝑚 ≅ 𝑓 𝐸asp_𝑚 + 𝐸mis_𝑚 + 𝐸ret_𝑚 ,
𝑀 ( 𝑀 [ ( )]2 )2 ∑ ∑ ( ) 𝐸asp_𝑚 − 𝐸 asp_𝑚 = 𝑓 −1 𝑊𝑚 − 𝑓 −1 𝐸 𝑚 = 𝑚𝑖𝑛,
(5)
That is we consider 𝐸 asp_𝑚 as the actual 𝐸asp_𝑚 when 𝑊 𝑚 is close enough to 𝑊𝑚 . If all the simulated subaperture test wavefronts are close enough to experimental ones, the full aperture figure error in simulation would be able to characterize the actual one. That is, the optimization is 43
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Fig. 3. The reconstruction results, (a) the full-aperture map of the true figure error, (b) is the subaperture partition, (c) and (d) are reconstructed subaperture maps in CSSI and ASSI, (e) and (f) are corresponding reconstructed errors. The aperture is normalized.
Table 1 The specific parameters of the true figure error, result in ASSI and CSSI.
The true figure error SROR in CSSI ROR-stitching in ASSI Reconstruction error in CSSI Reconstruction error in ASSI
PV (𝜆)
rms (𝜆)
0.5364 0.5397 0.5309 9.34e−3 9.58e−3
0.087 0.088 0.088 9.15e−4 9.67e−4
errors in ASSI and CSSI, respectively. From Fig. 3, we conclude that the test result by SROR in CSSI has a high consistency with true figure error and those by ROR-stitching algorithm ASSI. The further specific comparison is listed in Table 1, which illustrates the effectiveness of SROR method in CSSI. The SROR algorithm can retrieve Zernike coefficients of full aperture directly from those of subapertures based on system modeling. The algorithm accuracy may be susceptible to system modeling error, noise and subaperture layout (subaperture number, overlap and uncovering ratio). These influence are discussed below with simulation, respectively. 3.1. Modeling error and noise The resulted wavefront error due to the system modeling error is analyzed in Refs [22,23]. Unlike the traditional non-null interferometer, the modeling error of the optical wedge should be treated separately. The resulted wavefront errors due to modeling errors of the optical wedge are illustrated in Fig. 4, including the lateral translation error, rotation error, axial translation error and tilt error. It is obvious that the modeling errors of the optical wedge affect the result severely and the calibration is indispensable. Take the lateral translation error for example, the residual error after reverse optimization calibration is no more than 0.04 PV value in case of 2 mm modeling error. Of course, the final error would increase with the configuration number due to the error accumulation. In Ref. [20], we conclude that the ROR-stitching algorithm is relatively insensitive to random noise in NASSI. This is because the Gaussian noise we added to each subaperture is high frequency noise, which has little influence on low frequency figure error expressed by Zernike polynomials. In this paper, the performance of SROR was evaluated in different noise levels as well. Some zero-mean Gaussian distributions, acting as random noise, were added to each subaperture data. For convenience, the NSR (noise-to signal ratio) are defined as𝜎∕rms, where
Fig. 4. The modeling errors of the optical wedge and the corresponding wavefront aberrations introduced. (a)–(d) mean the lateral translation error, rotation error, axial translation error and tilt error, respectively. 44
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Fig. 6. The rms value of reconstruction error with different subaperture number in case of 0.01 x tilt, 0.1 NSR Gaussian distribution and vibration of 0.01 amplitude and 4 normalized frequency for each subaperture, respectively.
accumulation. In view of the excellent ability of retrace error correction of the SROR algorithm, fewer subaperture is an elegant choice in the range of system resolution tolerance. The simulation analysis in Fig. 6 proves this conclusion as well. 3.3. Overlap factor and uncovering ratio
Fig. 5. The performance of SROR in CSSI in (a) different NSR (Gaussian noise), (b) different amplitudes at the sample frequency of CCD and (c) different normalized frequencies (to sample frequency of CCD) at 0.01𝜆 amplitude.
The SROR algorithm is executed based on ray tracing of multiconfiguration model, which extract the Zernike coefficient of full aperture figure error directly from subapertures wavefront data. It is independent of subaperture overlap. Fig. 7(a) illustrates the reconstruction error with overlap factor increasing, which shows us the indistinctive change of errors. The variation of PV and rms value of errors are less than 3 × 10−4 𝜆 and 4 × 10−5 𝜆, respectively. The insensitivity of SROR algorithm to subaperture overlap is proved by simulation. In testing, subapertures may not cover the full aperture of the test surface. The residual areas may need several additional subapertures. The SROR possesses the ability of reconstruction for full aperture figure error from incompletely covered subapertures layout. Because the SROR algorithm extract the Zernike coefficient of full aperture figure error directly from subaperture wavefront data. Few uncovered areas may not affect the resulted Zernike coefficients, providing that the figure error of full aperture has a continuous slow change. Fig. 7(b) presents the reconstruction accuracy of SROR algorithm changing with uncovered ratio (ratio between uncovered areas and the full aperture). The accuracy decrease with the uncovered ratio increasing. Fortunately, the SROR algorithm keeps its accuracy decreasing under 0.01𝜆 rms with no more than 10% uncovering ratio, which is proved by the simulation and presented in Fig. 7(b). Note that it may become worse if there are some saltation in the figure error under test.
𝜎 is the standard deviation of Gaussian noise. The PV and rms value of reconstruction error in different NSR were illustrated in Fig. 5(a), where five results per noise level are plotted. As is shown in Fig. 5(a), the PV and rms value of reconstruction error increase with NSR gradually accompanying with some fluctuation. After calculation, we concluded that the relative PV and rms value would still be less than 3.0% and 2.7% even the NSR reached 0.5. That is, the SROR algorithm is relatively insensitive to random noise as well. In addition, the vibration and air fluctuations affect the results with middle frequency distribution. It happened in period of phase shifting and more sampling numbers induce less errors. The influence in single test has been discussed in Ref. [25]. We evaluate the performance of SROR method in simulation with these distribution of different amplitudes and frequencies with 16 sampling numbers for each subaperture test in four step phase shifting. Figs. 5(b) and (c) show the results. In fact, the experiment in Section 4 was carried out on a gas buoy platform in a super clean room. There is little influence of vibration and air fluctuations with constant frequency, except for a sudden vibration. The Figs. 5(b) and (c) show us the final error increases with the vibration amplitude rise and the most large error occurred at 2 times of sampling frequency of CCD, respectively [26]. 3.2. Subaperture number
3.4. Overall consideration As is known, the higher the limiting frequency is allowed, the less the subaperture would be necessary but more retrace error be suffered. Thus an overall consideration for equilibrium is necessary. We evaluated the reconstruction error of SROR algorithm in CSSI in different subaperture number. Fig. 6 presents the rms values of reconstruction error in different subaperture number with the same misalignments and noise level, respectively. Obviously, the misalignments and noise would contribute to inaccuracy for error accumulation with increasing subaperture number. That is, if the upper limit of fringe frequency was higher, the subaperture number would decrease with more retrace error suffered. However, more subapertures means more adjustment error and noise
From the above analysis, we conclude that: ∙ Less subapertures are available by the large limiting frequency of interferometer provide that each subaperture interferogram is distinguishable. It means less accumulation of modeling errors and noise. The resulted larger retrace error of each subaperture is the subordinate concern due to the excellent ability of retrace error correction of SROR algorithm. ∙ The overlap is insignificant in SROR algorithm. Thus less subapertures are available by a complementary subaperture layout with slight area uncovered. 45
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Optics Communications 403 (2017) 41–49
Fig. 8. The experiment setup of the CSSI.
Fig. 7. The reconstruction error in case of (a) different subaperture overlap factors and (b) uncovering ratios.
∙ The small uncovered ratio is preferred for accuracy. But it is a contradictory with small subaperture number. According to Fig. 7(b), the accuracy decreasing no more than 0.01𝜆PV with 10% uncovering ratio. Therefore, some basic subapertures with several complementary small aperture can achieve less subaperture with more than 90% covering ratio.
Fig. 9. The initial calibration of CSSI in experiment for (a) the reference mirror, (b) the test surface and (c) the optical wedge.
4. Experiment until the Zernike coefficient of the tilt wavefront aberration is less than 0.1 𝜆 after the iterative alignment. Subsequently, the test surface are located and calibrated by the corresponding aberration coefficients of the central interferogram, which is illustrated in Fig. 9(b). The system model would provide a distinguishable interferogram (central area of normalized radius r). The same area of the interferogram in experiment is extracted for comparison, which refers to surface decentration and tilt. The wavefront aberration coefficients of the two sub-area would act as the criteria for comparison. In fact, the alignment accuracy is limited due to the mechanical accuracy although the misalignments of the test surface can be calculated from the aberration coefficients. But the alignment in the model is easy and accurate. Therefore, the surface posture is iterative revised in the model until the resulted misalignment aberration coefficients are close to those in experiment in the same subarea. The detail calibration can be found in Ref. [23]. The next step is calibration for the optical wedge which is presented in Fig. 9(c). The back surface of the wedge is positioned at the focus of TS. In this experiment, the focal length of TS is 240 mm and thus the position accuracy achieves 0.02 mm by the wavefront aberration evaluation. Then, the standard spherical mirror is employed for the calibration of the wedge decentration. Rotate the wedge until the differences of PV value of the resulted aberrations of wavefronts in
An experiment is set up to validate the SROR algorithm in the practical CSSI, which is presented in Fig. 8. A tunable diode laser from Germanic TEM corporation with central wavelength 633 nm is employed for the phase shift. The moveable reference mirror is driven by a piezoelectric actuator equipped with a micrometer provides 12 mm coarse travel with 1 μm resolution and 15 μm fine piezo travel with 10 nm resolution. The rotating optical wedges with 25.4 mm aperture and 10.8◦ wedge angle is employed for ray deflection, which is located in a precise rotation stage, which provide continuous 360◦ motorized rotation with 20 arcsec achievable incremental motion. A linear guide is used for the translation of the optical wedge. In the experiment, the paraboloidal surface, flat mirror and astigmatism mirror are tested respectively. 4.1. Initial alignment and calibration For element misalignments in the interferometer, an initial alignment is executed. First is the alignment for reference mirror with a standard spherical mirror, which is presented in Fig. 9(a). The reference mirror is driven to shift along the optical axis. After an iterative alignment for the tilt of the mirror, the wavefront tilt would maintain in a low level during the mirror’s axial translation. The calibration is completed 46
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Optics Communications 403 (2017) 41–49
Fig. 10. CSSI model and interferograms, (a) is the CSSI model, (b) presents the initial interferograms of four subapertures in the four-configuration model.
different rotation angles (0◦ , 90◦ , 180◦ and 270◦ ) are less than 0.05𝜆. The optical wedge is located in a precise rotating sleeve, whose tilt errors and eccentricity errors are reduced to 30’’ and micron dimension, respectively. The rms value of the corresponding wavefront error is less than 10−3 𝜆. From the above calibration, a relative accurate system modeling is achievable with the aid of modern fabrication and computer modeling techniques. More accurate modeling calibration would be described hereinafter.
Fig. 11. Corresponding Zernike coefficients of four subapertures.
4.2. The paraboloidal surface test A paraboloidal surface with 280 mm vertex radius and 50.8 mm aperture is tested. The system model was set up in the ray tracing program according to the experiment parameters. Fig. 10(a) illustrates the layout of the four subapertures partition (covering ratio 90%) with resulted interferograms presented in Fig. 10(b). Then, the experiment was implemented according to the subaperture partition in the model. The corresponding Zernike coefficients of four subaperture wavefronts in the model and experiment are presented in Fig. 11 in which the yellow line means initial Zernike coefficients of subaperture wavefronts in the model while the blue dash line means the Zernike coefficients of subaperture wavefronts in experiments. In fact, the misalignment coefficients of each subaperture in experiments have a little deviation with those in the model due to the initial calibration accuracy and the adjustment error of the test surface. Therefore, a more accurate calibration is executed by reverse ray tracing [23], to calibrate the misalignments of the reference mirror, optical wedge and test surface in the model to approach the pose of those in the experiment. This calibration provides consistency of misalignments of subapertures between the model and experiment. After the calibration, the corresponding Zernike coefficients characterizing subaperture misalignments in the model (the red line in Fig. 11) are all close to those in experiments, indicating now the pose of the three elements in four-configuration model is close to those in the experiment. Fig. 12 provides an exemplificative calibration result, in which Figs. 12(a) and (b) are the initial subaperture interferogram in the model and the corresponding one in the experiment while Fig. 12(c) is the one after calibration in the model. It is obvious in Fig. 13 that there
Fig. 12. An exemplificative calibration (a) the initial subaperture interferogram, (b) the subaperture interferogram in the experiment, (c) the subaperture interferogram after calibration in the model.
is difference between Figs. 12(a) and (b), which means misalignments between the model and experiment. After calibration, Fig. 12(c) has a higher similarity with Fig. 12(b), which means a preferable alignment. This calibration in the model is obviously easier than the one in the experiment. Then, the SROR method was implemented for full aperture figure error reconstruction. The optimization function was set up. With the simultaneous iterative ray tracing, the simulated figure error approached the actual one, accompanied with all subaperture Zernike coefficients in the model approaching those in the experiment. The resulted Zernike coefficients of the full aperture figure error was then extracted and shown in Fig. 13(a), with the figure error in Fig. 13(b). The paraboloid surface was also tested with a ZYGO interferometer by the aberration free method for cross validation. The resulted surface figure is presented in Fig. 13(c), which has a basic consistency with Fig. 13(b). The specific parameters of the final result in the experiment is listed in Table 2 for further validation. The effectiveness of the SROR algorithm in CSSI is proved obviously. 47
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Fig. 13. Test results of the paraboloidal mirror, (a) and (b) is the final result of Zernike standard coefficients and figure error of full aperture by SROR, (c) is the surface figure error tested by ZYGO interferometer.
Fig. 14. Test results of the paraboloidal mirror with different covering ratio. Table 3 The PV and rms value of the test result by SROR in CSSI and ZYGO interferometer.
Table 2 The PV and rms value of the test result by SROR in CSSI and ZYGO interferometer.
Paraboloidal mirror
SROR in CSSI ZYGO interferometer
PV (𝜆)
rms (𝜆)
0.5021 0.4560
0.0458 0.0500
Fig. 14 illustrates experiment results of different subaperture partition with different covering ratio. The PV and rms value of error achieve steady minimum values at covering ratio about 92.5%. The error even has slight rise with the increasing covering ratio (when larger than 92.5%) due to the error accumulation.
PV (𝜆)
rms (𝜆)
Flat mirror
SROR in CSSI ZYGO interferometer
0.3451 0.3380
0.0352 0.0350
Astigmatism mirror
SROR in CSSI ROR
0.7180 0.7480
0.0940 0.1050
powerless for an astigmatism mirror [28]. The specific parameters of final results in the two experiments are listed in Table 3 for further validation. The effectiveness of the SROR algorithm in CSSI is further proved.
4.3. Flat mirror and an astigmatism mirror test 5. Conclusion Another two validations were implemented with testing of a flat mirror and an astigmatism mirror. The flat mirror aperture is 50 mm × 50 mm. Of course the test beam employed is a plane wavefront with 20 mm circular aperture and nine subapertures can cover more than 93% of the full aperture (Fig. 15(a)). Note that the full aperture Zernike polynomial employed in SROR is not applicable due to its non-orthogonality in the square aperture. A nonrecursive matrix orthonormal method [27] is employed for polynomial orthogonalization based on orthonormal Zernike basis. Resulted orthonormal polynomials are obtained and embed in SROR algorithm. Fig. 15(b) presents the final full aperture map of figure error, which has a basic consistency with that by ZYGO interferometer (Fig. 15(c)). The astigmatism mirror has a 30 mm aperture with 201 mm × radius and 200 mm y radius. Eight subapertures can cover more than 94% of the full aperture (Fig. 15(d)). The final result is shown in Fig. 15(e), with a cross validation result by the ROR algorithm for its full aperture, which is shown in Fig. 15(f). Note that we do not obtain the result from ZYGO interferometer as a contrast because the ZYGO interferometer is
We demonstrate the validity of the SROR algorithm in CSSI. The modified SROR algorithm is improved in misalignment correction. A practical CSSI system is proposed for this demonstration, in which an optical wedge is employed to deflect the incident beam for subaperture scanning by its rotation and shift instead of the six-axis motion-control system. The SROR algorithm’s merits such as auto retrace error correction, no need of overlap and even permission of missed coverage are analyzed in detail with simulation. Experiments validating the SROR algorithm in CSSI is implemented with cross validation. We now conclude that the SROR algorithm applied for ASSI and CSSI, with no overlap, small subaperture number and even few missed coverage. Acknowledgments This work was partly supported by the National Natural Science Foundation of China (61675005, 61440010) and The Doctoral Start-up Foundation of Anhui University (No. J01003208). 48
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Fig. 15. Test results of the flat mirror and astigmatism mirror, in which (a), (b) and (c) are results of the flat mirror and (d), (e) and (f) are results of the astigmatism mirror. (a) is the subaperture partition with more than 93% covering ratio, (b) is figure error of full aperture by SROR, (c) is the surface figure error tested by ZYGO interferometer, (d) is the subaperture partition with more than 94% covering ratio, (e) is figure error of full aperture by SROR algorithm, (f) is the surface figure error by ROR method.
References
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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Validation of simultaneous reverse optimization reconstruction algorithm in a practical circular subaperture stitching interferometer Lei Zhang a, *, Dong Li b , Yu Liu a , Jingxiao Liu a , Jingsong Li a , Benli Yu a a b
Key Laboratory of Opto-electronic information Acquisition and Manipulation Ministry of Education, Anhui University, Hefei, 230601, China Research Center of Laser Fusion, China Academy of Engineering Physics, Mianyang, 621900, China
a r t i c l e
i n f o
Keywords: Subaperture stitching Aspheric surface test Interferometer modeling Calibration
a b s t r a c t We demonstrate the validity of the simultaneous reverse optimization reconstruction (SROR) algorithm in circular subaperture stitching interferometry (CSSI), which is previously proposed for non-null aspheric annular subaperture stitching interferometry (ASSI). The merits of the modified SROR algorithm in CSSI, such as auto retrace error correction, no need of overlap and even permission of missed coverage, are analyzed in detail in simulations and experiments. Meanwhile, a practical CSSI system is proposed for this demonstration. An optical wedge is employed to deflect the incident beam for subaperture scanning by its rotation and shift instead of the six-axis motion-control system. Also the reference path can provide variable Zernike defocus for each subaperture test, which would decrease the fringe density. Experiments validating the SROR algorithm in this CSSI is implemented with cross validation by testing of paraboloidal mirror, flat mirror and astigmatism mirror. It is an indispensable supplement in SROR application in general subaperture stitching interferometry. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Accurate interferometric testing for large and deep aspheric surfaces has always been a challenge. Traditional interferometric testing approach is limited by the spatial-frequency resolution of the interferometric system. The subaperture stitching interferometry (SSI), as an alternative, was proposed in 1980s [1–4] to overcome the limitation. According to the differences of subaperture shape, circle subaperture and annular subaperture were developed for aspheric surfaces test. Annular subaperture stitching interferometry (ASSI) [5–8] employs a transmission sphere to generate spherical wavefronts of different radii of curvature to match the slopes in different annuli on the aspheric surface, and then combines the results of many subaperture measurements to obtain the full aperture surface map. But it applies only to rotational symmetric aspheric surfaces. Circular subaperture stitching interferometry (CSSI) [1–4,9–13] employs an optimal transmission sphere to generate spherical wavefronts of different radii of curvature to match the circular areas of different locations at surface aperture. It is now commercial available by QED Technologies, which has the test capacity of mild aspherical surfaces [11,12]. But its complex six-axis motioncontrol has always limited the test accuracy due to the mechanical error introduced during alignment. Moreover, each circular subaperture * Corresponding author.
E-mail address: [email protected] (L. Zhang). http://dx.doi.org/10.1016/j.optcom.2017.07.004 Received 23 April 2017; Received in revised form 14 June 2017; Accepted 2 July 2017 0030-4018/© 2017 Elsevier B.V. All rights reserved.
suffers retrace error due to the non-null test although the variable optic null (VON) [13] is employed for nullifying the test in some extent. It is obvious the stitching algorithm has played a vital role in the application of SSI. Many landmark algorithms have been proposed for subaperture stitching. From the Kwon Thunen method [14] and the simultaneous fit method [15], to the discrete phase method [16], the multi-aperture overlap-scanning technique [17] and the subaperture stitching and localization (SASL) algorithm [18], without exception, depend on either complex mathematic calculation or subaperture overlap for misalignment correction. Moreover, the retrace error is not treated suitably in most algorithms. In our previous work, we proposed a simultaneous reverse iterative optimizing reconstruction (SROR) algorithm [19] based on system modeling in a non-null ASSI [19,20]. All the subaperture measurement configurations were simulated with a multi-configuration model. The Zernike standard coefficients of full aperture figure error can be extracted directly from those of subaperture wavefronts by simultaneous ray tracing of the multi-configuration model, without any separate treatment for retrace error and subaperture misalignments (the misalignment was modeled). This method concurrently accomplished subaperture retrace error and misalignment correction, requiring neither complex mathematical stitching algorithms nor subaperture overlaps.
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In this paper, we continue to show the validity of the SROR algorithm in CSSI with a practical interferometer. The practical CSSI employs an optical wedge to deflect the incident beam and scan subaperture by its rotation and shift instead of the six-axis motion-control system. The wedge can provide the Zernike tilt and coma while the reference path provides Zernike defocus for each subaperture, which would decrease the fringe density obviously. The modified SROR algorithm then is employed for circular subaperture stitching. The retrace error and misalignments are corrected concomitantly with no consideration of overlap an even few missed coverage. With the previous work [19] and this one, the SROR algorithm is proved effective for general SSI. In Section 2, we outlines the practical CSSI system and the principle of the modified SROR method; detailed simulation analysis characterizing the SROR method is discussed in Section 3; the experiment verification is presented in Section 4; Section 5 presents a conclusion. 2. The practical CSSI with modified SROR method Fig. 1 illustrates the CSSI system for mild aspheric metrology. As is shown in Fig. 1(a), the collimated laser beam from the beam expander is split into two by a beam splitter. One is converged by an aplanatic lens and reflected by a reference mirror mounted on a piezoelectric actuator, serving as the reference beam; the other traveling though the transmission sphere (TS) and optical wedge is then reflected by the test aspheric, forming the test beam after traveling though the TS again. The reference and test beams interfere at the beam splitter and the resulted interferogram is imaged onto the detector (here CCD) by the imaging lens. The tunable laser is able to execute phase shift by wavelength changing. When testing, the deflected beam scan subapertures by the rotation and shift of the optical wedges instead of the six-axis motion-control system. As is shown in Fig. 1(b), the optical wedge deflect the incident beam to test a non-centric subaperture and thus scan all subapertures in an annular region. Meanwhile, with the wedge moving gradually away from the tested surface along the optical axis, the scanning region expands to cover the full aperture. Note that the fringe density of off-axis subaperture interferograms is mainly attributed to the low-order Zernike aberrations such as the wavefront tilt, defocus, coma and astigmatism. The optical wedge, not only deflect the beam for scanning, but just introduce wavefront tilt and coma aberrations at the direction of beam deflection when the incident wave is the spherical wavefront. The Zernike tilt and coma in the off-axis subaperture interferograms would be compensated in some extent with the wedge shift and selected wedge angle. While the reference mirror can be translated along the optical axis around the focus of the aplanat by an axial driving mechanism. It provides Zernike defocus to reduce fringe density of each circular subaperture as well. Therefore, some mild aspheric surface can be tested in the CSSI with subapertures of measurable fringe. The combination of movable reference mirror and the wedge, acting like the variable optical nulling (VON) in QED’s technology [13], decreases the fringe density of subaperture interferograms. Surface figure information is then obtained over subapertures with rational treatment for the misalignment and retrace error. Note that the optical wedge and movable reference mirror would introduce residual higher-order aberrations accompanying with the low-order aberrations compensation. These higher-order aberrations then would be treated together with the retrace error in the SROR algorithm proposed in nonnull ASSI previously. The SROR algorithm was proved effective in ASSI and now would be validated in CSSI with some modification. The SROR algorithm is based on system models. The non-null interferometer system modeling and calibration has been researched in our previous work [19,21–23]. In our method, a multi-configuration model is setup in a ray tracing program as is shown in Fig. 2, in which each configuration is a complete non-null interferometric system, characterizing a subaperture measurement. The only difference in all the interferometric configurations is the location of reference mirror and optical wedge.
Fig. 1. The sketch of the practical CSSI.
The initial models of aspheric surface, reference mirror and optical wedge in all configurations are ideal. However, the relative misalignments would be induced in experiment by the movements of the reference mirror and optical wedge for different subapertures test although an initial calibration is performed. A completely perfect calibration in experiment is unachievable usually. That is there are some slight deviations in the locations of the reference mirror and optical wedge between the model and experiment. Fortunately, these slight deviation can be made up by fine revisions of the model, that guarantee the high consistency of misalignments of the reference mirror, optical wedge and aspheric in the model with those in experiment. In our previously reported method, these misalignments are calculated beforehand and then revised in the model. However, this method is not applicable because the couple of the misalignments of the reference mirror, optical wedge and test surface impedes the accurate misalignment calculation. Therefore, we improved it by a more convenient revision approach in this paper. Unlike the calibration of subaperture misalignments modeling by specific misalignment calculation in ASSI [19], we improved the method by embedding the misalignment model calibration into the SROR algorithm as is shown in Fig. 2. Take the optical wedge as example, each subaperture wavefront resulting from a special configuration in the experiment is employed for evaluating the misalignment aberrations of the optical wedge in this configuration. The specific misalignment calculation is not necessary. The Zernike coefficients of these misalignment aberrations from experiment would be set as optimization objectives in the corresponding configuration in the model, while the specific misalignments can be extracted as the optimization variables. Then the model of optical wedge can be revised automatically by the reverse ray tracing optimization. The calibration method apply to the reference mirror and aspheric surface as well. The tested aspheric surface modeling should be revised firstly with the configuration without optical wedge, and this revision need only once because the aspheric under test does not need to move in other configurations. In view of the error couple, the subsequent revision 42
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Fig. 2. The multi-configuration model for ray tracing in SROR.
would aim simultaneously at the reference mirror and the optical wedge in different configurations. The simultaneous calibration in multi-configuration would decrease the error couple [19,24]. The final misalignments of all the reference mirror, optical wedge and aspheric in the model would achieve high consistency of those in experiment. Of course, an initial calibration for the reference mirror and the wedge should be implemented to reduce the error coupling. The overall description can be expressed as: { 𝐸m𝑖𝑠_𝑚 = 𝐸 m𝑖𝑠_𝑚 (1) 𝐸ret_𝑚 = 𝐸 ret_𝑚
implemented on all the subapertures simultaneously in the multiconfiguration model rather than each subaperture separately. We redefine a global optimization function ) ]2 [ ( ( )2 (𝐸asp ⊕ 𝐸asp ⊕ ⋯ ⊕ 𝐸asp ) 𝑈 = 𝐸ASP − 𝐸 ASP = − 𝐸 asp ⊕ 𝐸 asp ⊕ ⋯ ⊕ 𝐸 asp =
𝑚=1
𝐸ASP = 𝐸 ASP . (2)
(7)
To define more clearly, Zernike polynomials are employed to describe the wavefront and figure error. In this way, the full aperture figure error can be reconstructed in form of Zernike standard polynomials without special stitching process. Obviously, we need not special mathematic algorithm for stitching process. The SROR method is capable of sewing together all the subapertures with retrace errors and misalignment errors correction at the same time without overlap.
where, 𝐸asp_𝑚 is the subaperture figure error in experiment. We extract the 𝐸asp_𝑚 as { ( ) 𝐸asp_𝑚 ≅ 𝑓 −1 (𝑊𝑚 )− 𝐸m𝑖𝑠_𝑚 − 𝐸ret_𝑚 , (3) 𝐸 asp_𝑚 ≅ 𝑓 −1 𝑊 𝑚 − 𝐸 m𝑖𝑠_𝑚 − 𝐸 ret_𝑚 where 𝐸 asp_𝑚 and 𝑊 𝑚 are the simulated counterparts in the model corresponding to experiment, respectively. Of course, the initial values of 𝐸 asp_𝑚 and𝐸 m𝑖𝑠_𝑚 are zero in the model. A closed feedback system is set up to change the simulated subaperture figure error (𝐸 asp_𝑚 ) in the ray tracing program, making the simulated subaperture test wavefront (𝑊 𝑚 ) approaching to the actual one (𝑊𝑚 ) in the experiment constantly. We defined an optimized objective function with Eq. (3) to describe this process as ( )2 𝑈 = 𝐸asp_𝑚 − 𝐸 asp_𝑚 [ ( ) ) ]2 ( −1 (4) (𝑓 (𝑊𝑚 )− 𝐸m𝑖𝑠_𝑚 − 𝐸ret_𝑚 ) = − 𝑓 −1 𝑊 𝑚 − 𝐸 m𝑖𝑠_𝑚 − 𝐸 ret_𝑚
3. Simulation and discussions We carried out a numerical simulation for validation of SROR algorithm in CSSI. The tested surface and figure error added artificially (Fig. 3(a)) are all same with those in Ref. [16] for comparison. A matched TS with F/# 0.3 and a wedge with 14◦ wedge angle are selected in the simulation. The CSSI system was modeled in ray trace program. The upper limit of the fringe frequency was defined as 0.125𝜆∕𝑝𝑖𝑥𝑒𝑙 to promise a large redundancy in range of the resolving capacity of the detector. Four subapertures were obtained with corresponding interferograms in Fig. 3(b). Random slight misalignments (0 ∼ 0.2𝜆 tilts) and Gaussian distribution random noise (zero-mean and noise-to-signal ratio (NSR) 0.01) were added to each subaperture to simulate actual experiments condition. Then, a four-construction model is set up according to nominal parameters of the simulated experiment. Note that the additional misalignments would be removed by the calibration of model with an initial reverse optimization [23], aiming at make misalignments of each subaperture in the model accord with those in the simulated experiment. After the SROR algorithm performed, the full aperture figure error is extracted, showing in Fig. 3(c). Fig. 3(d) is the result cited from the Ref. [20], which is extracted by the ‘‘ROR-stitching’’ algorithm for full aperture in non-null ASSI. Figs. 3(e) and (f) present the reconstruction
= min . Then, Eq. (4) would be simplified by Eq. (1) as ( )2 𝑈 = 𝐸asp_𝑚 − 𝐸 asp_𝑚 [ ( )]2 ( ) = min . = 𝑓 −1 𝑊𝑚 − 𝑓 −1 𝑊 𝑚
(6)
𝑚=1
where, the 𝐸ASP and 𝐸 ASP are the full aperture figure error in experiment and simulation, respectively. The M is subaperture number and the sign ⊕ means the operation of addition in morphology, respectively. When Eq. (6) is satisfied well, i.e. all the simulated subaperture wavefront (𝑊 𝑚 ) are closed to those (𝑊𝑚 ) in the experiment simultaneously, we obtain
where, 𝐸mis_𝑚 and 𝐸ret_𝑚 are the subaperture misalignments and retrace error in the experiment, while 𝐸 m𝑖𝑠_𝑚 and 𝐸 ret_𝑚 are the simulated counterparts in the model corresponding to experiment, respectively. Based on the idea of ray tracing, each subaperture wavefront 𝑊𝑚 can be expressed by an implicit function as ( ) 𝑊𝑚 ≅ 𝑓 𝐸asp_𝑚 + 𝐸mis_𝑚 + 𝐸ret_𝑚 ,
𝑀 ( 𝑀 [ ( )]2 )2 ∑ ∑ ( ) 𝐸asp_𝑚 − 𝐸 asp_𝑚 = 𝑓 −1 𝑊𝑚 − 𝑓 −1 𝐸 𝑚 = 𝑚𝑖𝑛,
(5)
That is we consider 𝐸 asp_𝑚 as the actual 𝐸asp_𝑚 when 𝑊 𝑚 is close enough to 𝑊𝑚 . If all the simulated subaperture test wavefronts are close enough to experimental ones, the full aperture figure error in simulation would be able to characterize the actual one. That is, the optimization is 43
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Fig. 3. The reconstruction results, (a) the full-aperture map of the true figure error, (b) is the subaperture partition, (c) and (d) are reconstructed subaperture maps in CSSI and ASSI, (e) and (f) are corresponding reconstructed errors. The aperture is normalized.
Table 1 The specific parameters of the true figure error, result in ASSI and CSSI.
The true figure error SROR in CSSI ROR-stitching in ASSI Reconstruction error in CSSI Reconstruction error in ASSI
PV (𝜆)
rms (𝜆)
0.5364 0.5397 0.5309 9.34e−3 9.58e−3
0.087 0.088 0.088 9.15e−4 9.67e−4
errors in ASSI and CSSI, respectively. From Fig. 3, we conclude that the test result by SROR in CSSI has a high consistency with true figure error and those by ROR-stitching algorithm ASSI. The further specific comparison is listed in Table 1, which illustrates the effectiveness of SROR method in CSSI. The SROR algorithm can retrieve Zernike coefficients of full aperture directly from those of subapertures based on system modeling. The algorithm accuracy may be susceptible to system modeling error, noise and subaperture layout (subaperture number, overlap and uncovering ratio). These influence are discussed below with simulation, respectively. 3.1. Modeling error and noise The resulted wavefront error due to the system modeling error is analyzed in Refs [22,23]. Unlike the traditional non-null interferometer, the modeling error of the optical wedge should be treated separately. The resulted wavefront errors due to modeling errors of the optical wedge are illustrated in Fig. 4, including the lateral translation error, rotation error, axial translation error and tilt error. It is obvious that the modeling errors of the optical wedge affect the result severely and the calibration is indispensable. Take the lateral translation error for example, the residual error after reverse optimization calibration is no more than 0.04 PV value in case of 2 mm modeling error. Of course, the final error would increase with the configuration number due to the error accumulation. In Ref. [20], we conclude that the ROR-stitching algorithm is relatively insensitive to random noise in NASSI. This is because the Gaussian noise we added to each subaperture is high frequency noise, which has little influence on low frequency figure error expressed by Zernike polynomials. In this paper, the performance of SROR was evaluated in different noise levels as well. Some zero-mean Gaussian distributions, acting as random noise, were added to each subaperture data. For convenience, the NSR (noise-to signal ratio) are defined as𝜎∕rms, where
Fig. 4. The modeling errors of the optical wedge and the corresponding wavefront aberrations introduced. (a)–(d) mean the lateral translation error, rotation error, axial translation error and tilt error, respectively. 44
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Fig. 6. The rms value of reconstruction error with different subaperture number in case of 0.01 x tilt, 0.1 NSR Gaussian distribution and vibration of 0.01 amplitude and 4 normalized frequency for each subaperture, respectively.
accumulation. In view of the excellent ability of retrace error correction of the SROR algorithm, fewer subaperture is an elegant choice in the range of system resolution tolerance. The simulation analysis in Fig. 6 proves this conclusion as well. 3.3. Overlap factor and uncovering ratio
Fig. 5. The performance of SROR in CSSI in (a) different NSR (Gaussian noise), (b) different amplitudes at the sample frequency of CCD and (c) different normalized frequencies (to sample frequency of CCD) at 0.01𝜆 amplitude.
The SROR algorithm is executed based on ray tracing of multiconfiguration model, which extract the Zernike coefficient of full aperture figure error directly from subapertures wavefront data. It is independent of subaperture overlap. Fig. 7(a) illustrates the reconstruction error with overlap factor increasing, which shows us the indistinctive change of errors. The variation of PV and rms value of errors are less than 3 × 10−4 𝜆 and 4 × 10−5 𝜆, respectively. The insensitivity of SROR algorithm to subaperture overlap is proved by simulation. In testing, subapertures may not cover the full aperture of the test surface. The residual areas may need several additional subapertures. The SROR possesses the ability of reconstruction for full aperture figure error from incompletely covered subapertures layout. Because the SROR algorithm extract the Zernike coefficient of full aperture figure error directly from subaperture wavefront data. Few uncovered areas may not affect the resulted Zernike coefficients, providing that the figure error of full aperture has a continuous slow change. Fig. 7(b) presents the reconstruction accuracy of SROR algorithm changing with uncovered ratio (ratio between uncovered areas and the full aperture). The accuracy decrease with the uncovered ratio increasing. Fortunately, the SROR algorithm keeps its accuracy decreasing under 0.01𝜆 rms with no more than 10% uncovering ratio, which is proved by the simulation and presented in Fig. 7(b). Note that it may become worse if there are some saltation in the figure error under test.
𝜎 is the standard deviation of Gaussian noise. The PV and rms value of reconstruction error in different NSR were illustrated in Fig. 5(a), where five results per noise level are plotted. As is shown in Fig. 5(a), the PV and rms value of reconstruction error increase with NSR gradually accompanying with some fluctuation. After calculation, we concluded that the relative PV and rms value would still be less than 3.0% and 2.7% even the NSR reached 0.5. That is, the SROR algorithm is relatively insensitive to random noise as well. In addition, the vibration and air fluctuations affect the results with middle frequency distribution. It happened in period of phase shifting and more sampling numbers induce less errors. The influence in single test has been discussed in Ref. [25]. We evaluate the performance of SROR method in simulation with these distribution of different amplitudes and frequencies with 16 sampling numbers for each subaperture test in four step phase shifting. Figs. 5(b) and (c) show the results. In fact, the experiment in Section 4 was carried out on a gas buoy platform in a super clean room. There is little influence of vibration and air fluctuations with constant frequency, except for a sudden vibration. The Figs. 5(b) and (c) show us the final error increases with the vibration amplitude rise and the most large error occurred at 2 times of sampling frequency of CCD, respectively [26]. 3.2. Subaperture number
3.4. Overall consideration As is known, the higher the limiting frequency is allowed, the less the subaperture would be necessary but more retrace error be suffered. Thus an overall consideration for equilibrium is necessary. We evaluated the reconstruction error of SROR algorithm in CSSI in different subaperture number. Fig. 6 presents the rms values of reconstruction error in different subaperture number with the same misalignments and noise level, respectively. Obviously, the misalignments and noise would contribute to inaccuracy for error accumulation with increasing subaperture number. That is, if the upper limit of fringe frequency was higher, the subaperture number would decrease with more retrace error suffered. However, more subapertures means more adjustment error and noise
From the above analysis, we conclude that: ∙ Less subapertures are available by the large limiting frequency of interferometer provide that each subaperture interferogram is distinguishable. It means less accumulation of modeling errors and noise. The resulted larger retrace error of each subaperture is the subordinate concern due to the excellent ability of retrace error correction of SROR algorithm. ∙ The overlap is insignificant in SROR algorithm. Thus less subapertures are available by a complementary subaperture layout with slight area uncovered. 45
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Optics Communications 403 (2017) 41–49
Fig. 8. The experiment setup of the CSSI.
Fig. 7. The reconstruction error in case of (a) different subaperture overlap factors and (b) uncovering ratios.
∙ The small uncovered ratio is preferred for accuracy. But it is a contradictory with small subaperture number. According to Fig. 7(b), the accuracy decreasing no more than 0.01𝜆PV with 10% uncovering ratio. Therefore, some basic subapertures with several complementary small aperture can achieve less subaperture with more than 90% covering ratio.
Fig. 9. The initial calibration of CSSI in experiment for (a) the reference mirror, (b) the test surface and (c) the optical wedge.
4. Experiment until the Zernike coefficient of the tilt wavefront aberration is less than 0.1 𝜆 after the iterative alignment. Subsequently, the test surface are located and calibrated by the corresponding aberration coefficients of the central interferogram, which is illustrated in Fig. 9(b). The system model would provide a distinguishable interferogram (central area of normalized radius r). The same area of the interferogram in experiment is extracted for comparison, which refers to surface decentration and tilt. The wavefront aberration coefficients of the two sub-area would act as the criteria for comparison. In fact, the alignment accuracy is limited due to the mechanical accuracy although the misalignments of the test surface can be calculated from the aberration coefficients. But the alignment in the model is easy and accurate. Therefore, the surface posture is iterative revised in the model until the resulted misalignment aberration coefficients are close to those in experiment in the same subarea. The detail calibration can be found in Ref. [23]. The next step is calibration for the optical wedge which is presented in Fig. 9(c). The back surface of the wedge is positioned at the focus of TS. In this experiment, the focal length of TS is 240 mm and thus the position accuracy achieves 0.02 mm by the wavefront aberration evaluation. Then, the standard spherical mirror is employed for the calibration of the wedge decentration. Rotate the wedge until the differences of PV value of the resulted aberrations of wavefronts in
An experiment is set up to validate the SROR algorithm in the practical CSSI, which is presented in Fig. 8. A tunable diode laser from Germanic TEM corporation with central wavelength 633 nm is employed for the phase shift. The moveable reference mirror is driven by a piezoelectric actuator equipped with a micrometer provides 12 mm coarse travel with 1 μm resolution and 15 μm fine piezo travel with 10 nm resolution. The rotating optical wedges with 25.4 mm aperture and 10.8◦ wedge angle is employed for ray deflection, which is located in a precise rotation stage, which provide continuous 360◦ motorized rotation with 20 arcsec achievable incremental motion. A linear guide is used for the translation of the optical wedge. In the experiment, the paraboloidal surface, flat mirror and astigmatism mirror are tested respectively. 4.1. Initial alignment and calibration For element misalignments in the interferometer, an initial alignment is executed. First is the alignment for reference mirror with a standard spherical mirror, which is presented in Fig. 9(a). The reference mirror is driven to shift along the optical axis. After an iterative alignment for the tilt of the mirror, the wavefront tilt would maintain in a low level during the mirror’s axial translation. The calibration is completed 46
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Fig. 10. CSSI model and interferograms, (a) is the CSSI model, (b) presents the initial interferograms of four subapertures in the four-configuration model.
different rotation angles (0◦ , 90◦ , 180◦ and 270◦ ) are less than 0.05𝜆. The optical wedge is located in a precise rotating sleeve, whose tilt errors and eccentricity errors are reduced to 30’’ and micron dimension, respectively. The rms value of the corresponding wavefront error is less than 10−3 𝜆. From the above calibration, a relative accurate system modeling is achievable with the aid of modern fabrication and computer modeling techniques. More accurate modeling calibration would be described hereinafter.
Fig. 11. Corresponding Zernike coefficients of four subapertures.
4.2. The paraboloidal surface test A paraboloidal surface with 280 mm vertex radius and 50.8 mm aperture is tested. The system model was set up in the ray tracing program according to the experiment parameters. Fig. 10(a) illustrates the layout of the four subapertures partition (covering ratio 90%) with resulted interferograms presented in Fig. 10(b). Then, the experiment was implemented according to the subaperture partition in the model. The corresponding Zernike coefficients of four subaperture wavefronts in the model and experiment are presented in Fig. 11 in which the yellow line means initial Zernike coefficients of subaperture wavefronts in the model while the blue dash line means the Zernike coefficients of subaperture wavefronts in experiments. In fact, the misalignment coefficients of each subaperture in experiments have a little deviation with those in the model due to the initial calibration accuracy and the adjustment error of the test surface. Therefore, a more accurate calibration is executed by reverse ray tracing [23], to calibrate the misalignments of the reference mirror, optical wedge and test surface in the model to approach the pose of those in the experiment. This calibration provides consistency of misalignments of subapertures between the model and experiment. After the calibration, the corresponding Zernike coefficients characterizing subaperture misalignments in the model (the red line in Fig. 11) are all close to those in experiments, indicating now the pose of the three elements in four-configuration model is close to those in the experiment. Fig. 12 provides an exemplificative calibration result, in which Figs. 12(a) and (b) are the initial subaperture interferogram in the model and the corresponding one in the experiment while Fig. 12(c) is the one after calibration in the model. It is obvious in Fig. 13 that there
Fig. 12. An exemplificative calibration (a) the initial subaperture interferogram, (b) the subaperture interferogram in the experiment, (c) the subaperture interferogram after calibration in the model.
is difference between Figs. 12(a) and (b), which means misalignments between the model and experiment. After calibration, Fig. 12(c) has a higher similarity with Fig. 12(b), which means a preferable alignment. This calibration in the model is obviously easier than the one in the experiment. Then, the SROR method was implemented for full aperture figure error reconstruction. The optimization function was set up. With the simultaneous iterative ray tracing, the simulated figure error approached the actual one, accompanied with all subaperture Zernike coefficients in the model approaching those in the experiment. The resulted Zernike coefficients of the full aperture figure error was then extracted and shown in Fig. 13(a), with the figure error in Fig. 13(b). The paraboloid surface was also tested with a ZYGO interferometer by the aberration free method for cross validation. The resulted surface figure is presented in Fig. 13(c), which has a basic consistency with Fig. 13(b). The specific parameters of the final result in the experiment is listed in Table 2 for further validation. The effectiveness of the SROR algorithm in CSSI is proved obviously. 47
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Fig. 13. Test results of the paraboloidal mirror, (a) and (b) is the final result of Zernike standard coefficients and figure error of full aperture by SROR, (c) is the surface figure error tested by ZYGO interferometer.
Fig. 14. Test results of the paraboloidal mirror with different covering ratio. Table 3 The PV and rms value of the test result by SROR in CSSI and ZYGO interferometer.
Table 2 The PV and rms value of the test result by SROR in CSSI and ZYGO interferometer.
Paraboloidal mirror
SROR in CSSI ZYGO interferometer
PV (𝜆)
rms (𝜆)
0.5021 0.4560
0.0458 0.0500
Fig. 14 illustrates experiment results of different subaperture partition with different covering ratio. The PV and rms value of error achieve steady minimum values at covering ratio about 92.5%. The error even has slight rise with the increasing covering ratio (when larger than 92.5%) due to the error accumulation.
PV (𝜆)
rms (𝜆)
Flat mirror
SROR in CSSI ZYGO interferometer
0.3451 0.3380
0.0352 0.0350
Astigmatism mirror
SROR in CSSI ROR
0.7180 0.7480
0.0940 0.1050
powerless for an astigmatism mirror [28]. The specific parameters of final results in the two experiments are listed in Table 3 for further validation. The effectiveness of the SROR algorithm in CSSI is further proved.
4.3. Flat mirror and an astigmatism mirror test 5. Conclusion Another two validations were implemented with testing of a flat mirror and an astigmatism mirror. The flat mirror aperture is 50 mm × 50 mm. Of course the test beam employed is a plane wavefront with 20 mm circular aperture and nine subapertures can cover more than 93% of the full aperture (Fig. 15(a)). Note that the full aperture Zernike polynomial employed in SROR is not applicable due to its non-orthogonality in the square aperture. A nonrecursive matrix orthonormal method [27] is employed for polynomial orthogonalization based on orthonormal Zernike basis. Resulted orthonormal polynomials are obtained and embed in SROR algorithm. Fig. 15(b) presents the final full aperture map of figure error, which has a basic consistency with that by ZYGO interferometer (Fig. 15(c)). The astigmatism mirror has a 30 mm aperture with 201 mm × radius and 200 mm y radius. Eight subapertures can cover more than 94% of the full aperture (Fig. 15(d)). The final result is shown in Fig. 15(e), with a cross validation result by the ROR algorithm for its full aperture, which is shown in Fig. 15(f). Note that we do not obtain the result from ZYGO interferometer as a contrast because the ZYGO interferometer is
We demonstrate the validity of the SROR algorithm in CSSI. The modified SROR algorithm is improved in misalignment correction. A practical CSSI system is proposed for this demonstration, in which an optical wedge is employed to deflect the incident beam for subaperture scanning by its rotation and shift instead of the six-axis motion-control system. The SROR algorithm’s merits such as auto retrace error correction, no need of overlap and even permission of missed coverage are analyzed in detail with simulation. Experiments validating the SROR algorithm in CSSI is implemented with cross validation. We now conclude that the SROR algorithm applied for ASSI and CSSI, with no overlap, small subaperture number and even few missed coverage. Acknowledgments This work was partly supported by the National Natural Science Foundation of China (61675005, 61440010) and The Doctoral Start-up Foundation of Anhui University (No. J01003208). 48
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Fig. 15. Test results of the flat mirror and astigmatism mirror, in which (a), (b) and (c) are results of the flat mirror and (d), (e) and (f) are results of the astigmatism mirror. (a) is the subaperture partition with more than 93% covering ratio, (b) is figure error of full aperture by SROR, (c) is the surface figure error tested by ZYGO interferometer, (d) is the subaperture partition with more than 94% covering ratio, (e) is figure error of full aperture by SROR algorithm, (f) is the surface figure error by ROR method.
References
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