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Snapshot phase-shifting lateral shearing interferometer
Optics and Lasers in Engineering 128 (2020) 106032
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Snapshot phase-shifting lateral shearing interferometer Daodang Wang a,b,∗, Chao Wang a,c, Xiaobo Tian b, Heng Wu b, Jian Liang b, Rongguang Liang b,∗ a
College of Metrology and Measurement Engineering, China Jiliang University, Hangzhou 310018, China James C. Wyant College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA c Guangxi Key Laboratory of Optoelectronic Information Processing, Guilin University of Electronic Technology, Guilin 541004, China b
a r t i c l e
i n f o
Keywords: Lateral shearing interferometer Adjustable shear ratio Snapshot phase shift Low-coherence light
a b s t r a c t We propose a novel, low-cost snapshot phase-shifting lateral shearing interferometer (SPLSI) with full-range continuously adjustable lateral shear ratio. A tunable shear plate, consisting of a polarizing beam splitter (PBS) plate and a reflective mirror, separates the orthogonally polarization states of the test wavefront into original and sheared wavefronts. The lateral shear ratio can be tuned by adjusting the spacing between PBS plate and the mirror. A pixelated polarization camera simultaneously captures four 𝜋/2 phase-shifted interferograms for phase-shifting interferometric measurement. In addition to tunable shear ratio and snapshot capture of phaseshifted interferograms, one more key feature is that SPLSI can measure the wavefront of low-coherence light. The feasibility of the proposed system is demonstrated experimentally. The proposed system provides a simple, compact, and fast way to achieve the instantaneous phase-shifting lateral shearing interferometric testing of wavefronts with minimal impact from environmental disturbance.
1. Introduction The interferometry is widely applied in accurate and noncontact testing of laser beams, optical elements and optical phases, etc. Various interferometric methods, such as the point-diffraction interferometers [1–3], radial and lateral shearing interferometers [4–8], have been proposed to achieve the wavefront measurement without the standard optics in traditional interferometers. As a self-reference testing technique, the lateral shearing interferometer (LSI) achieves the wavefront measurement by interfering the original wavefront with its lateral sheared copy, and the phase can be reconstructed from the gradients of test wavefront. It avoids the requirement of an ideal reference wavefront and is easy in alignment. As a key parameter in LSI, the shear ratio 𝛽 = u/T, which is defined as the ratio of shear amount u to the aperture diameter T of test wavefront on the image plane, determines the amount of wavefront information obtained. The increase in the shear ratio could lead to the decrease in actual interference region and the spatial resolution. Besides, a small shear ratio is typically required for the test wavefront with large distortions. Generally, the shear ratio should be determined according to the dynamic range, testing sensitivity, accuracy, and aperture of the wavefront under test [9,10]. A continuously variable shear ratio is required for the general wavefront measurement with LSI. The Mach-Zehnder configuration [11], gratings [7], parallel plates [12] and prisms [13] are commonly utilized to induce lateral shear, the shear ratio is typically pre-determined by the parameters of
∗
shearing devices, such as the grating period, plate thickness and tilt, etc. However, while Mach-Zehnder configuration can adjust the shear ratio, the system is complicated, bulky, and not common-path. Another limitation of the current LSI is that most systems are not suitable for non-coherence light, which has increasing applications in phase measurement. Though the quadriwave lateral shearing interferometer based on cross grating doesn’t depend on the bandwidth of light source, its image resolution is quite limited due to the low lateral sampling resolution (which is generally determined by the grating pitch) [7]. The phase-shifting interferometry (PSI) technique has also been applied in LSI to obtain the accurate wavefronts with various phaseshifting mechanisms, such as shifting of mirror, wedge plate [5,14], a double grating stepper [15], polarization phase shifter [16,17] and liquid-crystal retarder [12]. Most of these phase-shifting methods require precision mechanical motion with either a costly piezoelectric transducer or linear translator. While the liquid-crystal retarder doesn’t have the moving element, the phase shift is temperature dependent. What’s more, the multi-step phase shifting cannot be achieved instantaneously, and the environmental disturbance could introduce an additional error in the wavefront measurement, placing ultra-high requirement on the stability of the wavefront measuring system. In this paper, a low-cost, snapshot phase-shifting lateral shearing interferometer (SPLSI) is proposed to achieve the real-time measurement of wavefront based on self-reference testing technique. A tunable shear plate, which consists of a polarizing beam splitter (PBS) plate and a reflective mirror, is applied to separate the orthogonally polarization
Corresponding authors. E-mail addresses: [email protected] (D. Wang), [email protected] (R. Liang).
https://doi.org/10.1016/j.optlaseng.2020.106032 Received 14 November 2019; Received in revised form 15 January 2020; Accepted 17 January 2020 0143-8166/© 2020 Published by Elsevier Ltd.
D. Wang, C. Wang and X. Tian et al.
Fig. 1. Snapshot phase-shifting lateral shearing interferometer with continuous adjustable shear ratio: P, linear polarizer; PBS, polarizing beam splitter; M, Mirror; QWP, quarter-wave plate; L, imaging lens; PCam, polarization camera.
states of the test wavefront into original and sheared wavefronts. The lateral shear ratio can be tuned by adjusting the spacing between PBS plate and the mirror. Besides, a pixelated polarization camera is adopted to realize snapshot and fast measurement, by which the influence of environmental disturbance can be minimized. There are three main contributions of this work: (1) the shear ratio of the proposed SPLSI is tunable from zero to any desired value (the maximum allowable value would be 1 for the coherent light source and Δ/T for the low-coherence light source with the coherence length Δ smaller than beam size T, respectively); (2) the system is applicable for measuring the wavefront of lowcoherence light; (3) only one capture is needed to obtain four 𝜋/2 phaseshifted interferograms simultaneously. All these features make it more practical and suitable for self-reference-based wavefront measurement. Sections 2 and 3 present the principle of the proposed SPLSI and error analysis. Section 4 shows the experimental results of shearing wavefront measurement from both high- and low-coherence light sources. Finally, some concluding remarks are drawn in Section 5. 2. Principle 2.1. System layout The schematic diagram of the proposed SPLSI is shown in Fig. 1. The test wavefront W(x, y) from the collimated laser beam at 633 nm goes through a linear polarizer P, and then is incident on the novel tunable shear plate which consists of a PBS plate (FPB639-21, Thorlabs Inc., the center wavelength 633 nm, bandwidth +17 nm/-4.5 nm, and the thickness 2 mm) and a reflective mirror. The linearly polarized beam is separated into s- and p-polarized beams, and they are reflected and transmitted by the PBS, respectively, serving as the original and sheared copies of test wavefront. The transmitted p-polarized beam is reflected by a plane mirror M parallel to the PBS plate and then passes through the PBS plate. A quarter-wave plate (QWP) with its fast axis oriented at 45° to x-axis is used to transform the orthogonally linearly polarized shearing beams to be oppositely circularly polarized. To achieve instantaneous interferometric measurement of wavefront, a pixelated polarization camera PCam from FLIR Systems Inc. is used to capture the four phase-shifted interferograms with a single shot [18,19]. Inside PCam, a Sony’s polarization image sensor (SONY IMX250MZR, resolution: 2448 × 2048 pixels, pixel size: 3.45 μm) is used, in which a micro-polarizer array with four linear polarizers at 0°, 45°, 90° and 135° adjacent to each other is fabricated onto the detector array, forming a 2 × 2 superpixel, as shown in Fig. 1. It enables the snapshot capture and motionless phase shifting, making the system insensitive to environmental disturbance. With the polarization beam splitting surface of the PBS plate facing the reflective mirror as shown in Fig. 1, the proposed wavefront shearing method has three unique features. The first is that the shear
Optics and Lasers in Engineering 128 (2020) 106032
ratio is tunable from zero to any desired value, simply by changing the spacing between the PBS plate and the reflective mirror M. The full range [0, 1] of shear ratio can be covered with the high-coherence light source; and for the low-coherence beams with their coherence length Δ smaller than the beam size, the corresponding tunable shear ratio range would be [0, Δ/T]. The second feature is that the dispersion is inherently compensated as both the original and sheared copies of test wavefront pass through the PBS plate twice, potentially suitable for measuring the wavefront of low-coherence light. One more feature of the proposed SPLSI is that only one capture is needed to obtain four 𝜋/2 phase-shifted interferograms simultaneously. The proposed system is insensitive to environmental disturbance, fast in measurement and large in the range of adjustable shear ratio. It should be noted that the coherence of light source strictly limits the maximum adjustable shear amount and shear ratio. According to the proposed system configuration in Fig. 1, the optical path difference√(OPD) (equals to the shear amount u) between two shearing beams is 2𝑑, where d is the distance between the PBS and mirror M. To obtain valid interference, the OPD should be smaller than the coherence length of light source.√Thus, the bandwidth Δ𝜆 of light source should be Δ𝜆 ≤ 𝜆2 ∕𝑢max = 𝜆2 ∕( 2𝑑), where umax is the maximum allowable shear amount and 𝜆 is the central wavelength of light source. 2.2. Operating principle The polarization and propagation of the beams can be described with Jones matrices. Without loss of generality, we take the shear in x direction as the case for analysis. In ideal case, the Jones matrices for the original test wave (Eot , x ) and its sheared copy (Est , x ) after the quarterwave plate QWP can be represented as { 𝐸𝑜𝑡,𝑥 =𝐴′ 𝑜𝑡,𝑥 ⋅ exp [𝑖𝑘𝑊 (𝑥, 𝑦)] ⋅ [1, −𝑖]𝑇 , (1) 𝐸𝑠𝑡,𝑥 =𝐴′ 𝑠𝑡,𝑥 ⋅ exp [𝑖𝑘𝑊 (𝑥 − 𝑢, 𝑦)] ⋅ [1, 𝑖]𝑇 where u is the shear amount, k is the wavenumber,𝐴′𝑜𝑡,𝑥 and 𝐴′𝑠𝑡,𝑥 are the corresponding Jones matrix coefficients. Thus, we have the Jones ′ , 𝐸 ′ ) after passing through matrices for the two interfering waves (𝐸𝑜𝑡,𝑥 𝑠𝑡,𝑥 the micro-polarizer array in PCam, { ′ 𝐸 𝑜𝑡,𝑥,𝑗 = 𝑄𝑗 ⋅ 𝐸𝑜𝑡,𝑥 , (2) 𝐸 ′ 𝑠𝑡,𝑥,𝑗 = 𝑄𝑗 ⋅ 𝐸𝑠𝑡,𝑥 where j = 1, 2, 3, 4, Qj is the Jones matrix for the micro-polarizer in PCam, with the transmission axis oriented at 0°, 45°, 90° and 135°, respectively. ′ of original and sheared test waves (𝐸 ′ The superposition 𝐸𝑥,𝑗 𝑜𝑡,𝑥,𝑗 , ′ 𝐸𝑠𝑡,𝑥,𝑗 ) in different micro-polarizer orientations, and the corresponding intensity Ix, j recorded on the camera can be defined as { ′ 𝐸 𝑥,𝑗 = 𝐸 ′ 𝑜𝑡,𝑥,𝑗 + 𝐸 ′ 𝑠𝑡,𝑥,𝑗 , (3) 𝐼𝑥,𝑗 = |𝐸 ′ 𝑥,𝑗 (1)|2 + |𝐸 ′ 𝑥,𝑗 (2)|2 where Ix, j is the jth (j = 1, 2, 3, 4) phase-shifted x shearing interferogram corresponding to various linear polarization (0°, 45°, 90°, 135°). Similarly, the four phase-shifted y shearing interferograms corresponding to various linear polarization can be described as Iy , j . Thus, the shearing wavefronts ΔWx , ΔWy (ΔWx = W(x, y)-W(x-u, y), ΔWy = W(x, y)-W(x, y-u)) in the x and y shearing interferograms can be calculated with four-step phase shifting algorithm, ( ) 𝐼𝑟,2 − 𝐼𝑟,4 1 Δ𝑊𝑟 = tan−1 , (4) 𝑘 𝐼𝑟,1 − 𝐼𝑟,3 where 𝑟 = 𝑥, 𝑦 is the shearing direction. Based on the measured shearing wavefronts ΔWx and ΔWy , the test wavefront W(x, y) can be reconstructed with differential Zernike polynomials fitting method [8]. 3. Error consideration Several factors may introduce measurement errors for the wavefront in the proposed configuration, mainly including those from the polarizing components (PBS, QWP), mirror M and polarization camera PCam.
D. Wang, C. Wang and X. Tian et al.
Optics and Lasers in Engineering 128 (2020) 106032
Fig. 3. Acquired phase-shifted lateral shearing interferograms of laser beam with polarization camera: (a)–(d) interferograms in 0°, 45°, 90° and 135° channels, and (e) the corresponding measured shearing wavefront.
Fig. 2. Phase errors due to imperfection of PBS and QWP.
The polarizing components are key in the proposed SPLSI, because they could affect both the sheared test wavefront and phase-shifted interferograms, introducing additional phase error. To minimize the phase errors due to the imperfection of polarizing components, a high-extinctionratio PBS plate (with the transmissions for p- and s-polarization at the working wavelength 633 nm being 97.6% and 0.00004%, respectively) and a high-performance QWP (with the retardance 0.2495 waves) are employed in the proposed SPLSI. Fig 2 shows the phase errors introduced by the PBS and QWP. According to Fig. 2, the peak to valley (PV) value of the phase error introduced by PBS is 7.9 × 10−3 waves, and that corresponding to QWP is 2.0 × 10−4 waves, which are both negligible. To minimize the effect of the mirror M on test wavefront, a highperformance plane mirror (BBSQ1-E02, Thorlabs Inc., Ravg > 99% for s- and p- polarization for angles of incidence from 0 to 45°) with surface flatness better than 𝜆/10 at 633 nm is adopted in the system. For the pixelated polarization camera PCam, the imperfect performance of micro-polarizer array could result in the error factors such as the crosstalk among polarization channels, photon response nonuniformity (PRNU), micro-polarizer extinction ratio nonuniformity (ERNU) and micro-polarizer orientation misalignment (POMA). Various calibration methods can be applied to address this issue [19–21]. However, the development of polarization image sensor (Sony, IMX250MZR CMOS), in which the micro-polarizer array is formed on chip under on-chip microlens layer, has significantly improved the performance of polarization camera. The major error factor in the PCam that should be considered is the field of view (FOV) error (that is the phase distribution over a 2 × 2 superpixel region is not constant). The linear-spline interpolation of neighboring pixels with the same polarizing orientation over a 3 × 3 pixel grid [20] can be applied to retrieve the missing intensity information of all the pixels. According to Ref. [22], high calibration accuracy for FOV error can be achieved even for the high local wavefront tilt of 0.12 waves/pixel, and the PV value of residual phase error decreases from over 0.2800 waves to 0.0078 waves. Another possible error source is the inaccuracy in determination of shear ratio 𝛽. Generally, the shear ratio can be simply and accurately determined by examining the interference pattern, in which the aperture diameter T and shear amount u can be determined in pixels. The relative error in the shear ratio 𝛽 can be expressed as [ ]1∕2 Δ𝛽∕𝛽 = (Δ𝑇 ∕𝑇 )2 + (Δ𝑢∕𝑢)2 , (5) where Δ𝛽, ΔT and Δu are the errors in shear ratio, aperture diameter and shear amount. For the polarization camera, the fractional errors of aperture radius and shear amount increase to 1.0 relative to that of 0.5 in traditional camera. Thus, the relative error Δ𝛽/𝛽 in shear ratio is less than 1% for the aperture diameter of 790 pixels and shear amount of 105 pixels, corresponding to the absolute shear ratio error Δ𝛽 = 0.0013
at the shear ratio 𝛽 = 0.1329. According to Ref. [23], the relative error less than 1% in wavefront reconstruction can be expected. Thus, the shear ratio can be determined with sufficient precision in practical application. Besides, the loss of information, in either shearing wavefront measurement or test wavefront reconstruction, should be considered in the proposed SPLSI. An inherent problem with the lateral shearing interferometry is some loss of information about the u-period parts of test wavefront, where u is the shear amount. The combination of naturalextension-based analysis of lateral shearing interferograms with suitably chosen shear amounts [24,25] provides a feasible solution for the inherent loss of information in traditional LSI, by which the exact reconstruction of the whole test wavefront can be achieved. Due to the sampling with superpixels in polarization camera, the binning effect could also result in some loss of high-frequency information, which is similar with FOV error. According to Ref. [22], the spline-interpolation method provides a feasible way to compensate the missing information in polarization, with which high calibration accuracy of 0.0078 waves can be achieved even for the high local wavefront tilt of 0.12 waves/pixel. 4. Experimental results To analyze the performance of the proposed SPLSI, an experimental setup has been built to measure the x shearing wavefronts, in which the collimated wavefront from a reflective collimator based on a 90° off-axis parabolic mirror is taken as the test wavefront. Based on the collimated laser beam at 633 nm, the four 𝜋/2 phase-shifted interferograms extracted from the single-shot image are shown in Fig. 3(a)–(d), and the corresponding reconstructed wavefront is shown in Fig. 3(e). According to Fig. 3, precise polarization phase shift is achieved with the proposed method. Besides, obvious tooling marks in the reflective collimator due to diamond turning processing can be seen in the acquired figures, and the root mean square (RMS) value of the shearing wavefront is 0.0212 μm. To further analyze the effect of shear ratio on the shearing wavefront measurement result, the spacing between the PBS and mirror M is adjusted to obtain various shear ratios. Fig 4 shows the acquired shearing interferograms and the corresponding wavefronts under various shear ratios. Fig 5 is the RMS values of the shearing wavefront corresponding to the shear ratio range [0, 0.18]. According to Figs. 4 and 5, the RMS value of shearing wavefront grows with the increase in the shear ratio, and greater shearing wavefront aberration can be seen with a larger shear ratio, indicating higher testing sensitivity for the same input test wavefront; however, the increase in the shear ratio would result in the decrease in effective testing aperture. Besides, smaller shear ratio could make the shearing wavefront lower in magnitude (null wavefront aberration corresponding to the shear ratio 0), enabling larger testing dynamic range. By increasing the spacing between the PBS and mirror M, the shear ratio can be further increased. The proposed SPLSI enables
D. Wang, C. Wang and X. Tian et al.
Optics and Lasers in Engineering 128 (2020) 106032
Fig. 4. Acquired lateral shearing interferograms and wavefronts corresponding to various lateral shear ratios: (a)–(f) acquired interferograms, (g)–(l) the corresponding measured shearing wavefronts.
Fig. 6. Acquired phase-shifted lateral shearing interferograms of LED beam with polarization camera: (a)–(d) interferograms in 0°, 45°, 90° and 135° channels, and (e) the corresponding measured shearing wavefront.
Fig. 5. RMS values of shearing wavefront under various shear ratios.
the snapshot phase-shifting lateral shearing interferometric testing with continuously adjustable shear ratios in the range [0, 1] for the coherent light source, and it also provides a feasible way for experimental study on the optimization of shear ratio in the wavefront testing based on LSI. Besides, it enables the optimal testing sensitivity and dynamic range with arbitrarily optional shear ratios. It should be noted that additional measurement with the shear in y direction is required to reconstruct the wavefront aberration under test. It could be achieved either by separating the test beam into two arms, one for x shear measurement and the other for y shear measurement, or by rotating the test wavefront 90° with the same setup in Fig. 1. To validate the potential of the proposed SPLSI in the application of low-coherence light, a collimated wavefront from a light emitting diode (LED) source (with the center wavelength 530 nm) combining a band filter (LL01-532-12.5, Semrock, the center wavelength 532 nm and bandwidth 2 nm) is measured, corresponding to the maximum allowable shear amount umax = 0.141 mm. To match the operating wavelength, another PBS plate (FPB532-23, Thorlabs Inc., the center wavelength 532 nm, bandwidth +9 nm/-14 nm, and the thickness 2 mm) is utilized in experimental setup shown in Fig. 1, replacing the one working at 633 nm wavelength. Based on the LED source, the four 𝜋/2 phaseshifted interferograms extracted from the single-shot image are shown in Figs. 6(a)–(d), respectively, and the corresponding reconstructed shearing wavefront (with RMS value 0.0138 μm) is shown in Fig. 6(e). According to Fig. 6, a clear fringe pattern can be obtained with the lowcoherence LED light, even though the fringe contrast is relatively low with respect to that with a coherent laser beam. Thus, the proposed
SPLSI enables the LSI-based wavefront testing with a low-coherence light source, providing a feasible way to avoid the speckle noise and the potential application in biomedical imaging. By further narrowing the bandwidth of light source, a better fringe contrast can be expected. 5. Conclusion In summary, the proposed SPLSI based on polarization beam splitter plate and polarization camera enables the simultaneous phase-shifting lateral shearing interferometric testing with a single-shot interferogram, as well as the continuously adjustable shear ratio with the minimal available value being 0. The feasibility of the proposed method is experimentally validated. The proposed system is insensitive to environmental disturbance, fast in measurement and large in the range of adjustable shear ratio, providing a general way for lateral shearing interferometric wavefront testing with the light sources of various coherence lengths. Thus, it provides a feasible and effective way for the instantaneous testing of wavefronts. Declaration of Competing Interest None. CRediT authorship contribution statement Daodang Wang: Conceptualization, Methodology, Investigation, Writing - original draft. Chao Wang: Software, Validation. Xiaobo Tian: Visualization, Investigation. Heng Wu: Writing - review & editing. Jian Liang: Writing - review & editing. Rongguang Liang: Conceptualization, Writing - review & editing, Supervision, Funding acquisition.
D. Wang, C. Wang and X. Tian et al.
Funding National Natural Science Foundation of China (NSFC) (51775528, 61805048); Guangxi Key Laboratory of Optoelectronic Information Processing (GD18205); National Science Foundation (NSF) (1455630, 1918260); National Institutes of Health (NIH) (S10OD018061). References [1] Wang D, Yang Y, Chen C, Zhuo Y. Point diffraction interferometer with adjustable fringe contrast for testing spherical surfaces. Appl Opt 2011;50:2342–8. [2] Wang D, Xu Y, Liang R, Kong M, Zhao J, Zhang B, Li W. High-precision method for submicron-aperture fiber point-diffraction wavefront measurement. Opt Express 2016;24:7079–90. [3] Voznesenskiy N, Voznesenskaia M, Jha D. Testing high accuracy optics using the phase shifting point diffraction interferometer. Proc SPIE 2018;10829:1082902. [4] Schreiber H, Schwider J. Lateral shearing interferometer based on two Ronchi phase gratings in series. Appl Opt 1997;36:5321–4. [5] Lee HH, You JH, Park SH. Phase-shifting lateral shearing interferometer with two pairs of wedge plates. Opt Lett 2003;28:2243–5. [6] Toto-Arellano NI, Rodriguez-Zurita G, Meneses-Fabian C, Vázquez-Castillo JF. A single-shot phase-shifting radial-shearing interferometer. J Opt A: Pure Appl Opt 2009;11:045704. [7] Bon P, Maucort G, Wattellier B, Monneret S. Quadriwave lateral shearing interferometry for quantitative phase microscopy of living cells. Opt Express 2009;17:13080–94. [8] Ling T, Yang Y, Yue C, Liu D, Ma Y, Bai J, Wang K. Common-path and compact wavefront diagnosis system based on cross grating lateral shearing interferometer. Appl Opt 2014;53:7144–52. [9] Rimmer MP, Wyant JC. Evaluation of Large Aberrations Using a Lateral-Shear Interferometer Having Variable Shear. Appl Opt 1975;14:142–50.
Optics and Lasers in Engineering 128 (2020) 106032 [10] Chanteloup JC. Multiple-wave lateral shearing interferometry for wave-front sensing. Appl Opt 2005;44:1559–71. [11] Paez G, Strojnik M, Torales Garcia G. Vectorial shearing interferometer. Appl Opt 2000;39:5172–8. [12] Griffin DW. Phase-shifting shearing interferometer. Opt. Lett 2001;26(3):140–1. [13] Cho W-J, Kim S-W. Stable lateral-shearing interferometer for production-line inspection of lenses. Opt Eng 1997;36:896–900. [14] Song JB, Lee YW, Lee IW, Lee YH. Simple phase-shifting method in a wedge-plate lateral-shearing interferometer. Appl Opt 2004;43:3989–92. [15] Schreiber H, Schwider J. Lateral shearing interferometer based on two Ronchi phase gratings in series. App Opt 1997;36:5321–4. [16] Kothiyal MP, Delisle C. Shearing interferometer for phase shifting interferometry with polarization phase shifter. Appl Opt 1985;24:4439–42. [17] Gu L, Liu L, Hu S, Zeng A, Huang H. Polarization phase-shifting lateral shearing interferometer with two polarization beam splitter plates. Opt Rev 2017;24:600–4. [18] Tian XB, Zhang Y, Sohn A, Spires OJ, Liang RG. Dual-mode snapshot interferometric system for on-machine metrology. Opt Eng 2019;58:044104. [19] Wang DD, Liang RG. Simultaneous polarization Mirau interferometer based on pixelated polarization camera. Opt Lett 2016;41:41–4. [20] Novak M, Millerd J, Brock N, North-Morris M, Hayes J, Wyant J. Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer. Appl Opt 2005;44:6861–8. [21] Tian XB, Tu XZ, Zhang JC, Spires O, Brock N, Pau S, Liang RG. Snapshot multi-wavelength interference microscope. Opt Express 2018;26:18279–91. [22] Wang D, Tian X, Xu P, Liang J, Wu H, Spires O, Liang R. Compact snapshot multiwavelength interferometer. Opt Lett 2019;44:4463–6. [23] Harbers G, Kunst PJ, Leibbrandt GW. Analysis of lateral shearing interferograms by use of Zernike polynomials. Appl Opt 1996;35:6162–72. [24] Elster C, Weingärtner I. Solution to the shearing problem. Appl Opt 1999;38:5024–31. [25] Elster C, Weingärtner I. Exact wave-front reconstruction from two lateral shearing interferograms. J Opt Soc Am A 1999;16:2281–5.
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Snapshot phase-shifting lateral shearing interferometer Daodang Wang a,b,∗, Chao Wang a,c, Xiaobo Tian b, Heng Wu b, Jian Liang b, Rongguang Liang b,∗ a
College of Metrology and Measurement Engineering, China Jiliang University, Hangzhou 310018, China James C. Wyant College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA c Guangxi Key Laboratory of Optoelectronic Information Processing, Guilin University of Electronic Technology, Guilin 541004, China b
a r t i c l e
i n f o
Keywords: Lateral shearing interferometer Adjustable shear ratio Snapshot phase shift Low-coherence light
a b s t r a c t We propose a novel, low-cost snapshot phase-shifting lateral shearing interferometer (SPLSI) with full-range continuously adjustable lateral shear ratio. A tunable shear plate, consisting of a polarizing beam splitter (PBS) plate and a reflective mirror, separates the orthogonally polarization states of the test wavefront into original and sheared wavefronts. The lateral shear ratio can be tuned by adjusting the spacing between PBS plate and the mirror. A pixelated polarization camera simultaneously captures four 𝜋/2 phase-shifted interferograms for phase-shifting interferometric measurement. In addition to tunable shear ratio and snapshot capture of phaseshifted interferograms, one more key feature is that SPLSI can measure the wavefront of low-coherence light. The feasibility of the proposed system is demonstrated experimentally. The proposed system provides a simple, compact, and fast way to achieve the instantaneous phase-shifting lateral shearing interferometric testing of wavefronts with minimal impact from environmental disturbance.
1. Introduction The interferometry is widely applied in accurate and noncontact testing of laser beams, optical elements and optical phases, etc. Various interferometric methods, such as the point-diffraction interferometers [1–3], radial and lateral shearing interferometers [4–8], have been proposed to achieve the wavefront measurement without the standard optics in traditional interferometers. As a self-reference testing technique, the lateral shearing interferometer (LSI) achieves the wavefront measurement by interfering the original wavefront with its lateral sheared copy, and the phase can be reconstructed from the gradients of test wavefront. It avoids the requirement of an ideal reference wavefront and is easy in alignment. As a key parameter in LSI, the shear ratio 𝛽 = u/T, which is defined as the ratio of shear amount u to the aperture diameter T of test wavefront on the image plane, determines the amount of wavefront information obtained. The increase in the shear ratio could lead to the decrease in actual interference region and the spatial resolution. Besides, a small shear ratio is typically required for the test wavefront with large distortions. Generally, the shear ratio should be determined according to the dynamic range, testing sensitivity, accuracy, and aperture of the wavefront under test [9,10]. A continuously variable shear ratio is required for the general wavefront measurement with LSI. The Mach-Zehnder configuration [11], gratings [7], parallel plates [12] and prisms [13] are commonly utilized to induce lateral shear, the shear ratio is typically pre-determined by the parameters of
∗
shearing devices, such as the grating period, plate thickness and tilt, etc. However, while Mach-Zehnder configuration can adjust the shear ratio, the system is complicated, bulky, and not common-path. Another limitation of the current LSI is that most systems are not suitable for non-coherence light, which has increasing applications in phase measurement. Though the quadriwave lateral shearing interferometer based on cross grating doesn’t depend on the bandwidth of light source, its image resolution is quite limited due to the low lateral sampling resolution (which is generally determined by the grating pitch) [7]. The phase-shifting interferometry (PSI) technique has also been applied in LSI to obtain the accurate wavefronts with various phaseshifting mechanisms, such as shifting of mirror, wedge plate [5,14], a double grating stepper [15], polarization phase shifter [16,17] and liquid-crystal retarder [12]. Most of these phase-shifting methods require precision mechanical motion with either a costly piezoelectric transducer or linear translator. While the liquid-crystal retarder doesn’t have the moving element, the phase shift is temperature dependent. What’s more, the multi-step phase shifting cannot be achieved instantaneously, and the environmental disturbance could introduce an additional error in the wavefront measurement, placing ultra-high requirement on the stability of the wavefront measuring system. In this paper, a low-cost, snapshot phase-shifting lateral shearing interferometer (SPLSI) is proposed to achieve the real-time measurement of wavefront based on self-reference testing technique. A tunable shear plate, which consists of a polarizing beam splitter (PBS) plate and a reflective mirror, is applied to separate the orthogonally polarization
Corresponding authors. E-mail addresses: [email protected] (D. Wang), [email protected] (R. Liang).
https://doi.org/10.1016/j.optlaseng.2020.106032 Received 14 November 2019; Received in revised form 15 January 2020; Accepted 17 January 2020 0143-8166/© 2020 Published by Elsevier Ltd.
D. Wang, C. Wang and X. Tian et al.
Fig. 1. Snapshot phase-shifting lateral shearing interferometer with continuous adjustable shear ratio: P, linear polarizer; PBS, polarizing beam splitter; M, Mirror; QWP, quarter-wave plate; L, imaging lens; PCam, polarization camera.
states of the test wavefront into original and sheared wavefronts. The lateral shear ratio can be tuned by adjusting the spacing between PBS plate and the mirror. Besides, a pixelated polarization camera is adopted to realize snapshot and fast measurement, by which the influence of environmental disturbance can be minimized. There are three main contributions of this work: (1) the shear ratio of the proposed SPLSI is tunable from zero to any desired value (the maximum allowable value would be 1 for the coherent light source and Δ/T for the low-coherence light source with the coherence length Δ smaller than beam size T, respectively); (2) the system is applicable for measuring the wavefront of lowcoherence light; (3) only one capture is needed to obtain four 𝜋/2 phaseshifted interferograms simultaneously. All these features make it more practical and suitable for self-reference-based wavefront measurement. Sections 2 and 3 present the principle of the proposed SPLSI and error analysis. Section 4 shows the experimental results of shearing wavefront measurement from both high- and low-coherence light sources. Finally, some concluding remarks are drawn in Section 5. 2. Principle 2.1. System layout The schematic diagram of the proposed SPLSI is shown in Fig. 1. The test wavefront W(x, y) from the collimated laser beam at 633 nm goes through a linear polarizer P, and then is incident on the novel tunable shear plate which consists of a PBS plate (FPB639-21, Thorlabs Inc., the center wavelength 633 nm, bandwidth +17 nm/-4.5 nm, and the thickness 2 mm) and a reflective mirror. The linearly polarized beam is separated into s- and p-polarized beams, and they are reflected and transmitted by the PBS, respectively, serving as the original and sheared copies of test wavefront. The transmitted p-polarized beam is reflected by a plane mirror M parallel to the PBS plate and then passes through the PBS plate. A quarter-wave plate (QWP) with its fast axis oriented at 45° to x-axis is used to transform the orthogonally linearly polarized shearing beams to be oppositely circularly polarized. To achieve instantaneous interferometric measurement of wavefront, a pixelated polarization camera PCam from FLIR Systems Inc. is used to capture the four phase-shifted interferograms with a single shot [18,19]. Inside PCam, a Sony’s polarization image sensor (SONY IMX250MZR, resolution: 2448 × 2048 pixels, pixel size: 3.45 μm) is used, in which a micro-polarizer array with four linear polarizers at 0°, 45°, 90° and 135° adjacent to each other is fabricated onto the detector array, forming a 2 × 2 superpixel, as shown in Fig. 1. It enables the snapshot capture and motionless phase shifting, making the system insensitive to environmental disturbance. With the polarization beam splitting surface of the PBS plate facing the reflective mirror as shown in Fig. 1, the proposed wavefront shearing method has three unique features. The first is that the shear
Optics and Lasers in Engineering 128 (2020) 106032
ratio is tunable from zero to any desired value, simply by changing the spacing between the PBS plate and the reflective mirror M. The full range [0, 1] of shear ratio can be covered with the high-coherence light source; and for the low-coherence beams with their coherence length Δ smaller than the beam size, the corresponding tunable shear ratio range would be [0, Δ/T]. The second feature is that the dispersion is inherently compensated as both the original and sheared copies of test wavefront pass through the PBS plate twice, potentially suitable for measuring the wavefront of low-coherence light. One more feature of the proposed SPLSI is that only one capture is needed to obtain four 𝜋/2 phase-shifted interferograms simultaneously. The proposed system is insensitive to environmental disturbance, fast in measurement and large in the range of adjustable shear ratio. It should be noted that the coherence of light source strictly limits the maximum adjustable shear amount and shear ratio. According to the proposed system configuration in Fig. 1, the optical path difference√(OPD) (equals to the shear amount u) between two shearing beams is 2𝑑, where d is the distance between the PBS and mirror M. To obtain valid interference, the OPD should be smaller than the coherence length of light source.√Thus, the bandwidth Δ𝜆 of light source should be Δ𝜆 ≤ 𝜆2 ∕𝑢max = 𝜆2 ∕( 2𝑑), where umax is the maximum allowable shear amount and 𝜆 is the central wavelength of light source. 2.2. Operating principle The polarization and propagation of the beams can be described with Jones matrices. Without loss of generality, we take the shear in x direction as the case for analysis. In ideal case, the Jones matrices for the original test wave (Eot , x ) and its sheared copy (Est , x ) after the quarterwave plate QWP can be represented as { 𝐸𝑜𝑡,𝑥 =𝐴′ 𝑜𝑡,𝑥 ⋅ exp [𝑖𝑘𝑊 (𝑥, 𝑦)] ⋅ [1, −𝑖]𝑇 , (1) 𝐸𝑠𝑡,𝑥 =𝐴′ 𝑠𝑡,𝑥 ⋅ exp [𝑖𝑘𝑊 (𝑥 − 𝑢, 𝑦)] ⋅ [1, 𝑖]𝑇 where u is the shear amount, k is the wavenumber,𝐴′𝑜𝑡,𝑥 and 𝐴′𝑠𝑡,𝑥 are the corresponding Jones matrix coefficients. Thus, we have the Jones ′ , 𝐸 ′ ) after passing through matrices for the two interfering waves (𝐸𝑜𝑡,𝑥 𝑠𝑡,𝑥 the micro-polarizer array in PCam, { ′ 𝐸 𝑜𝑡,𝑥,𝑗 = 𝑄𝑗 ⋅ 𝐸𝑜𝑡,𝑥 , (2) 𝐸 ′ 𝑠𝑡,𝑥,𝑗 = 𝑄𝑗 ⋅ 𝐸𝑠𝑡,𝑥 where j = 1, 2, 3, 4, Qj is the Jones matrix for the micro-polarizer in PCam, with the transmission axis oriented at 0°, 45°, 90° and 135°, respectively. ′ of original and sheared test waves (𝐸 ′ The superposition 𝐸𝑥,𝑗 𝑜𝑡,𝑥,𝑗 , ′ 𝐸𝑠𝑡,𝑥,𝑗 ) in different micro-polarizer orientations, and the corresponding intensity Ix, j recorded on the camera can be defined as { ′ 𝐸 𝑥,𝑗 = 𝐸 ′ 𝑜𝑡,𝑥,𝑗 + 𝐸 ′ 𝑠𝑡,𝑥,𝑗 , (3) 𝐼𝑥,𝑗 = |𝐸 ′ 𝑥,𝑗 (1)|2 + |𝐸 ′ 𝑥,𝑗 (2)|2 where Ix, j is the jth (j = 1, 2, 3, 4) phase-shifted x shearing interferogram corresponding to various linear polarization (0°, 45°, 90°, 135°). Similarly, the four phase-shifted y shearing interferograms corresponding to various linear polarization can be described as Iy , j . Thus, the shearing wavefronts ΔWx , ΔWy (ΔWx = W(x, y)-W(x-u, y), ΔWy = W(x, y)-W(x, y-u)) in the x and y shearing interferograms can be calculated with four-step phase shifting algorithm, ( ) 𝐼𝑟,2 − 𝐼𝑟,4 1 Δ𝑊𝑟 = tan−1 , (4) 𝑘 𝐼𝑟,1 − 𝐼𝑟,3 where 𝑟 = 𝑥, 𝑦 is the shearing direction. Based on the measured shearing wavefronts ΔWx and ΔWy , the test wavefront W(x, y) can be reconstructed with differential Zernike polynomials fitting method [8]. 3. Error consideration Several factors may introduce measurement errors for the wavefront in the proposed configuration, mainly including those from the polarizing components (PBS, QWP), mirror M and polarization camera PCam.
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Optics and Lasers in Engineering 128 (2020) 106032
Fig. 3. Acquired phase-shifted lateral shearing interferograms of laser beam with polarization camera: (a)–(d) interferograms in 0°, 45°, 90° and 135° channels, and (e) the corresponding measured shearing wavefront.
Fig. 2. Phase errors due to imperfection of PBS and QWP.
The polarizing components are key in the proposed SPLSI, because they could affect both the sheared test wavefront and phase-shifted interferograms, introducing additional phase error. To minimize the phase errors due to the imperfection of polarizing components, a high-extinctionratio PBS plate (with the transmissions for p- and s-polarization at the working wavelength 633 nm being 97.6% and 0.00004%, respectively) and a high-performance QWP (with the retardance 0.2495 waves) are employed in the proposed SPLSI. Fig 2 shows the phase errors introduced by the PBS and QWP. According to Fig. 2, the peak to valley (PV) value of the phase error introduced by PBS is 7.9 × 10−3 waves, and that corresponding to QWP is 2.0 × 10−4 waves, which are both negligible. To minimize the effect of the mirror M on test wavefront, a highperformance plane mirror (BBSQ1-E02, Thorlabs Inc., Ravg > 99% for s- and p- polarization for angles of incidence from 0 to 45°) with surface flatness better than 𝜆/10 at 633 nm is adopted in the system. For the pixelated polarization camera PCam, the imperfect performance of micro-polarizer array could result in the error factors such as the crosstalk among polarization channels, photon response nonuniformity (PRNU), micro-polarizer extinction ratio nonuniformity (ERNU) and micro-polarizer orientation misalignment (POMA). Various calibration methods can be applied to address this issue [19–21]. However, the development of polarization image sensor (Sony, IMX250MZR CMOS), in which the micro-polarizer array is formed on chip under on-chip microlens layer, has significantly improved the performance of polarization camera. The major error factor in the PCam that should be considered is the field of view (FOV) error (that is the phase distribution over a 2 × 2 superpixel region is not constant). The linear-spline interpolation of neighboring pixels with the same polarizing orientation over a 3 × 3 pixel grid [20] can be applied to retrieve the missing intensity information of all the pixels. According to Ref. [22], high calibration accuracy for FOV error can be achieved even for the high local wavefront tilt of 0.12 waves/pixel, and the PV value of residual phase error decreases from over 0.2800 waves to 0.0078 waves. Another possible error source is the inaccuracy in determination of shear ratio 𝛽. Generally, the shear ratio can be simply and accurately determined by examining the interference pattern, in which the aperture diameter T and shear amount u can be determined in pixels. The relative error in the shear ratio 𝛽 can be expressed as [ ]1∕2 Δ𝛽∕𝛽 = (Δ𝑇 ∕𝑇 )2 + (Δ𝑢∕𝑢)2 , (5) where Δ𝛽, ΔT and Δu are the errors in shear ratio, aperture diameter and shear amount. For the polarization camera, the fractional errors of aperture radius and shear amount increase to 1.0 relative to that of 0.5 in traditional camera. Thus, the relative error Δ𝛽/𝛽 in shear ratio is less than 1% for the aperture diameter of 790 pixels and shear amount of 105 pixels, corresponding to the absolute shear ratio error Δ𝛽 = 0.0013
at the shear ratio 𝛽 = 0.1329. According to Ref. [23], the relative error less than 1% in wavefront reconstruction can be expected. Thus, the shear ratio can be determined with sufficient precision in practical application. Besides, the loss of information, in either shearing wavefront measurement or test wavefront reconstruction, should be considered in the proposed SPLSI. An inherent problem with the lateral shearing interferometry is some loss of information about the u-period parts of test wavefront, where u is the shear amount. The combination of naturalextension-based analysis of lateral shearing interferograms with suitably chosen shear amounts [24,25] provides a feasible solution for the inherent loss of information in traditional LSI, by which the exact reconstruction of the whole test wavefront can be achieved. Due to the sampling with superpixels in polarization camera, the binning effect could also result in some loss of high-frequency information, which is similar with FOV error. According to Ref. [22], the spline-interpolation method provides a feasible way to compensate the missing information in polarization, with which high calibration accuracy of 0.0078 waves can be achieved even for the high local wavefront tilt of 0.12 waves/pixel. 4. Experimental results To analyze the performance of the proposed SPLSI, an experimental setup has been built to measure the x shearing wavefronts, in which the collimated wavefront from a reflective collimator based on a 90° off-axis parabolic mirror is taken as the test wavefront. Based on the collimated laser beam at 633 nm, the four 𝜋/2 phase-shifted interferograms extracted from the single-shot image are shown in Fig. 3(a)–(d), and the corresponding reconstructed wavefront is shown in Fig. 3(e). According to Fig. 3, precise polarization phase shift is achieved with the proposed method. Besides, obvious tooling marks in the reflective collimator due to diamond turning processing can be seen in the acquired figures, and the root mean square (RMS) value of the shearing wavefront is 0.0212 μm. To further analyze the effect of shear ratio on the shearing wavefront measurement result, the spacing between the PBS and mirror M is adjusted to obtain various shear ratios. Fig 4 shows the acquired shearing interferograms and the corresponding wavefronts under various shear ratios. Fig 5 is the RMS values of the shearing wavefront corresponding to the shear ratio range [0, 0.18]. According to Figs. 4 and 5, the RMS value of shearing wavefront grows with the increase in the shear ratio, and greater shearing wavefront aberration can be seen with a larger shear ratio, indicating higher testing sensitivity for the same input test wavefront; however, the increase in the shear ratio would result in the decrease in effective testing aperture. Besides, smaller shear ratio could make the shearing wavefront lower in magnitude (null wavefront aberration corresponding to the shear ratio 0), enabling larger testing dynamic range. By increasing the spacing between the PBS and mirror M, the shear ratio can be further increased. The proposed SPLSI enables
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Optics and Lasers in Engineering 128 (2020) 106032
Fig. 4. Acquired lateral shearing interferograms and wavefronts corresponding to various lateral shear ratios: (a)–(f) acquired interferograms, (g)–(l) the corresponding measured shearing wavefronts.
Fig. 6. Acquired phase-shifted lateral shearing interferograms of LED beam with polarization camera: (a)–(d) interferograms in 0°, 45°, 90° and 135° channels, and (e) the corresponding measured shearing wavefront.
Fig. 5. RMS values of shearing wavefront under various shear ratios.
the snapshot phase-shifting lateral shearing interferometric testing with continuously adjustable shear ratios in the range [0, 1] for the coherent light source, and it also provides a feasible way for experimental study on the optimization of shear ratio in the wavefront testing based on LSI. Besides, it enables the optimal testing sensitivity and dynamic range with arbitrarily optional shear ratios. It should be noted that additional measurement with the shear in y direction is required to reconstruct the wavefront aberration under test. It could be achieved either by separating the test beam into two arms, one for x shear measurement and the other for y shear measurement, or by rotating the test wavefront 90° with the same setup in Fig. 1. To validate the potential of the proposed SPLSI in the application of low-coherence light, a collimated wavefront from a light emitting diode (LED) source (with the center wavelength 530 nm) combining a band filter (LL01-532-12.5, Semrock, the center wavelength 532 nm and bandwidth 2 nm) is measured, corresponding to the maximum allowable shear amount umax = 0.141 mm. To match the operating wavelength, another PBS plate (FPB532-23, Thorlabs Inc., the center wavelength 532 nm, bandwidth +9 nm/-14 nm, and the thickness 2 mm) is utilized in experimental setup shown in Fig. 1, replacing the one working at 633 nm wavelength. Based on the LED source, the four 𝜋/2 phaseshifted interferograms extracted from the single-shot image are shown in Figs. 6(a)–(d), respectively, and the corresponding reconstructed shearing wavefront (with RMS value 0.0138 μm) is shown in Fig. 6(e). According to Fig. 6, a clear fringe pattern can be obtained with the lowcoherence LED light, even though the fringe contrast is relatively low with respect to that with a coherent laser beam. Thus, the proposed
SPLSI enables the LSI-based wavefront testing with a low-coherence light source, providing a feasible way to avoid the speckle noise and the potential application in biomedical imaging. By further narrowing the bandwidth of light source, a better fringe contrast can be expected. 5. Conclusion In summary, the proposed SPLSI based on polarization beam splitter plate and polarization camera enables the simultaneous phase-shifting lateral shearing interferometric testing with a single-shot interferogram, as well as the continuously adjustable shear ratio with the minimal available value being 0. The feasibility of the proposed method is experimentally validated. The proposed system is insensitive to environmental disturbance, fast in measurement and large in the range of adjustable shear ratio, providing a general way for lateral shearing interferometric wavefront testing with the light sources of various coherence lengths. Thus, it provides a feasible and effective way for the instantaneous testing of wavefronts. Declaration of Competing Interest None. CRediT authorship contribution statement Daodang Wang: Conceptualization, Methodology, Investigation, Writing - original draft. Chao Wang: Software, Validation. Xiaobo Tian: Visualization, Investigation. Heng Wu: Writing - review & editing. Jian Liang: Writing - review & editing. Rongguang Liang: Conceptualization, Writing - review & editing, Supervision, Funding acquisition.
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Funding National Natural Science Foundation of China (NSFC) (51775528, 61805048); Guangxi Key Laboratory of Optoelectronic Information Processing (GD18205); National Science Foundation (NSF) (1455630, 1918260); National Institutes of Health (NIH) (S10OD018061). References [1] Wang D, Yang Y, Chen C, Zhuo Y. Point diffraction interferometer with adjustable fringe contrast for testing spherical surfaces. Appl Opt 2011;50:2342–8. [2] Wang D, Xu Y, Liang R, Kong M, Zhao J, Zhang B, Li W. High-precision method for submicron-aperture fiber point-diffraction wavefront measurement. Opt Express 2016;24:7079–90. [3] Voznesenskiy N, Voznesenskaia M, Jha D. Testing high accuracy optics using the phase shifting point diffraction interferometer. Proc SPIE 2018;10829:1082902. [4] Schreiber H, Schwider J. Lateral shearing interferometer based on two Ronchi phase gratings in series. Appl Opt 1997;36:5321–4. [5] Lee HH, You JH, Park SH. Phase-shifting lateral shearing interferometer with two pairs of wedge plates. Opt Lett 2003;28:2243–5. [6] Toto-Arellano NI, Rodriguez-Zurita G, Meneses-Fabian C, Vázquez-Castillo JF. A single-shot phase-shifting radial-shearing interferometer. J Opt A: Pure Appl Opt 2009;11:045704. [7] Bon P, Maucort G, Wattellier B, Monneret S. Quadriwave lateral shearing interferometry for quantitative phase microscopy of living cells. Opt Express 2009;17:13080–94. [8] Ling T, Yang Y, Yue C, Liu D, Ma Y, Bai J, Wang K. Common-path and compact wavefront diagnosis system based on cross grating lateral shearing interferometer. Appl Opt 2014;53:7144–52. [9] Rimmer MP, Wyant JC. Evaluation of Large Aberrations Using a Lateral-Shear Interferometer Having Variable Shear. Appl Opt 1975;14:142–50.
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