Abstract

We report a method of using a liquid-crystal spatial light modulator (LC-SLM) as reconfigurable multi-level interferogram-type computer generated holograms (ICGHs) to perform dynamic null tests for aspheric and free-form surfaces. With the proposed multi-level ICGHs encoding method, amplitude and accuracy of the applicable aberration of LC-SLMs are both suitable for interferometric test. No other equipment is required to monitor the dynamic phase of LC-SLM for guaranteeing test accuracy. Moreover, complicated phase response calibration of the LC-SLM is not required. Besides being used in collimated beams, the LC-SLM is demonstrated for the first time to be used in divergent beams; hence, concave surfaces with apertures larger than that of the LC-SLMs can be tested. For realizing practical tests, the calibration of inherit wavefront distortion of the LC-SLM, diffraction orders isolation, and alignment are analyzed in detail. Two free-form surfaces with about 20 μm departure from flat and spherical surfaces are successfully measured in collimated beam and divergent beam, respectively. Cross tests are provided to verify the test accuracy.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Aspheric and free-form surfaces have been successfully used in many important fields including illuminating and imaging. In order to guide the iterative fabrication process, surface figure of these surfaces must be measured ahead. Among various test techniques, interferometry is preferred for its higher accuracy and less measurement time compared with coordinate measurement [1] and shear interferometry [2], etc. The ability of interferometry to provide 3-D surface error information of the test surfaces has promoted great advances in optical fabrication and optical system. For testing flats and spherical surfaces, conventional phase-shifting interferometry utilizes a standard transmission flat or transmission sphere to generate high accuracy planar or spherical wavefront, respectively [1]. For testing aspheric and freeform surfaces, a null corrector either of refractive [3] or diffractive type [1] is additionally required. The null corrector can convert standard planar or spherical wavefront to aspheric or freeform wavefront which matches the ideal shape of the test surfaces. However, the null corrector is static and designed uniquely for the ideal test surface. Different tested surfaces require different null correctors, which can be cost prohibitive. Therefore, developing a flexible interferometric test method for aspheric and free-form surfaces is meaningful to enhance the test efficiency and reduce the cost.

For this purpose, it is desirable to have a means of producing accurate arbitrary wavefront shapes and to be able to reconfigure those wavefront shapes as flexibly as possible. One method is to introduce movement of traditional nulls or the test surface to generate variable aberrations [4–12]. For example, variable spherical aberration null by varying the axial spacing of two/three phase plates in a diverging beam was proposed by Hilbert and Rimmer [4]. However, only variable spherical aberration can be generated. Chen [6] proposed a variable off-axis aberration null. By counter-rotating a pair of Zernike plates, variable amount of Zernike terms Z4 and Z6 can be generated. It can correct most of the astigmatism and coma for off-axis subapertures of various aspheric shapes. However, all the above methods can only generate several particular kinds of aberrations. It is usually not enough for testing free-form surface. Another prevailing method is utilizing adaptive optics (AO) elements which can generate more flexible aberrations. AO elements mainly include deformation mirrors (DMs) and spatial light modulators (SLMs). However, these AO elements are manufactured for traditional AO applications such as telescopes, microscopes, and laser systems. The phase control accuracy is usually not the prime consideration of these systems. In fact, the phase control accuracy of AO elements is relatively low. However, relatively high accuracy (e.g. about λ/30 root mean square (RMS), λ = 632.8nm) is usually required by interferometric surface figure tests of optical surfaces. Therefore, troubleshooting the limited accuracy of AO elements is an urgent issue of AO elements based null methods. To improve test accuracy of methods using DMs as dynamic nulls, researchers have proposed utilizing Shack-Hartmann sensors [13,14], deflectometry systems [15] or interferometers [16,17] to monitor the shape of DMs. As alternatives to DMs, liquid-crystal spatial light modulators (LC-SLMs) have advantages of ease of controllability, a transmissive property, and the large number of actuators (typically several millions of pixels for SLMs, and typically several tens of actuators for DMs). However, utilizing SLMs to successfully test moderate (e.g. several tens microns departure) aspheric or free-form surfaces with relatively high test accuracy is challenging. The challenge mainly owes to phase control methods of SLMs.

The traditional control method is based on the electrically controlled birefringence effects [18]. Typically, the maximum phase retardance of SLMs is only about 2π. To generate a larger amount of aberrations, kinoform (i.e. phase-wrapping) technique is employed. This control method relies on the phase response. The phase response curve is nonlinear [19,20], spatially varying [19,20], and dependent on the incident angle [21] and polarization of the beam [22]. Its calibration is very complex [19–22]. Moreover, the phase jump from 0 to 2π (or from 2π to 0) will also introduce large error. Hence, the large aberration production accuracy can hardly meet the interferometric test requirements. Cao et al. [18], and Kacperski et al. [23] investigated using this method to control SLMs as adaptive nulls for realizing flexible tests. Although the reported accuracy can reach ~0.025λ RMS for testing a near flat surface with ~1.3λ departure [23], testing surfaces with moderate departure and cross-verified moderate accuracy is not reported to our best knowledge. Utilizing SLMs in divergent beam is also not reported due to the phase response curve is different under different incident angles. Another control method is to simply use the SLMs as diffractive optical elements. Researchers [24–26] have investigated utilizing two level ferroelectric liquid-crystal spatial light modulators (FLC-SLMs) as diffractive elements recording the binary pattern of the desired phase with tilt carrier. However, quantization errors are relatively large due to the only two levels of the binary pattern. The reported accuracy for generating ~30λ PV astigmatism is ~0.06λ RMS [26]. Moreover, this accuracy value is obtained after subtraction of unwanted Zernike terms. In summary, both above two encoding methods are not appropriate for testing moderate free-form surfaces with relatively high test accuracy to our best knowledge.

The motivation for conducting this research is to investigate using LC-SLMs in collimated and divergent beam for adaptive null tests of moderate free-form surfaces with moderate test accuracy. To troubleshoot the phase generation capacity and accuracy issues of using SLMs as adaptive nulls, we firstly showed how to use LC-SLMs as reconfigurable multi-level interferogram-type computer generated holograms (ICGHs) to produce arbitrary reconfigurable phase with moderate amplitude and accuracy in Section 2. To realize practical dynamic interferometry using the LC-SLMs, techniques for inherent wavefront distortion calibration, alignment, and overcoming the limited aperture of the LC-SLM were proposed. In Section 3, a LC-SLM encoded by the proposed multi-level ICGHs method was used to conduct dynamic null tests for two moderate free-form surfaces in collimated and divergent beams with verified high test accuracy. In Section 4, prospects of the proposed method are provided.

2. Principle

2.1. Encoding LC-SLMs as ICGHs and phase generation capacity and accuracy

The principle of ICGH is based on digital hologram [27]. Holography is a two-step process involving recording through the interference phenomenon and playback through the diffraction phenomenon. The principle applies to both flat and spherical reference wavefronts. Without loss of generality, the analysis is based on flat reference wavefront. Figure 1(a) shows the recording process. The object and reference wavefronts are flat wavefront with desired aberration and ideally flat wavefront, respectively. Their complex amplitudes at the recording plane are denoted as Uo(X, Y) and Ur(X, Y), respectively. The interferogram caused by these two wavefronts is generated by computers as

H=|Ur+Uo|2=|Ur|2+|Uo|2+Ur*Uo+UrUo*.
It includes three orders: the 0-order is composed of terms |Ur|2 + |Uo|2, the + 1 order is the term Ur*Uo, and the −1 order is the term UrUo*. The interferogram H is loaded to the LC-SLM. When the LC-SLM is illuminated by the reference wavefront, object wavefront is reconstructed as shown in Fig. 1(b). To realize separation of diffraction orders, the reference wavefront makes a small angle θ with the normal of recording plane.

 

Fig. 1 Principle of using LC-SLMs as ICGHs. Recording (a), and playback (b).

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The following Eqs. (2) and (3) represent the complex amplitudes of the reference and object wavefronts at the recording plane, respectively.

Ur=exp{i[2πλ(ysinθ+zcosθ)]},
Uo=exp{i[2πzλ+φo(x,y)]},
where φo(x, y) denotes the desired phase aberration.

Due to the finite number of phase levels (typically 8-bit gray levels) and pixels (generally about 106 pixels) of LC-SLMs, the interferogram H has to be quantified and discretized. For a LC-SLM with 8-bit gray levels and N × M pixels, substitute Eqs. (2) and (3) into Eq. (1). After some simplifications, it yields

H=[0.5{255+255cos2πλnsinθφo(m,n)}],
where φo(m, n) is the discretization of φo(x, y), (m; n)(–M/2, + M/2–1; –N/2, + N/2–1), (m; n) is the pixel coordinates. M and N denote pixel number in x and y direction, respectively. [ ∙ ] is the round (floor) operator. However, the quantization level of H is not 255 because the practical number of pixels to realize 1λ phase modulation (i.e., the number of pixels over which the phase changes by 2π) is much less than 255 when the gradient or amplitude of φo(m, n) is large. The applicable phase modulation φo would be larger if a smaller minimum quantization level Q is adopted. However, the quantization error will be larger if a smaller Q is set. Quantization error is caused by the anolog-to-digital conversion, i.e., the intensity value of the computer generated interferogram has to be rounded off and represented by finite gray levels. In [28], the phase quantization error in diffractive optics is analyzed theoretically. The relation of RMS value ε of the quantization error with Q can be obtained by simplifying the Eq. (68) in [28] as

ε=λ23Q.

Hence, an adequate Q should consider both the phase production capacity and accuracy. Commonly, λ/30 RMS is a moderate accuracy value of interferometry. To limit phase production error within this value, Q = 8 was calculated from Eq. (5). Q = 8 means there should be at least 8 pixels to realize 1λ phase modulation. Briefly speaking, the above analysis concludes that the maximum phase gradient norm is 1/8 to achieve accuracy of λ/30 RMS.

The maximum amplitudes of Zernike (Wyant’s ordering) modes [29] for Q = 8 (i.e., the maximum phase gradient norm is 1/8) were calculated mathematically in Matlab to evaluate the nulling capability of LC-SLMs. The calculation was conducted for evaluating the ability of HoloeyeTM LC 2012 (a model of LC-SLM). HoloeyeTM LC 2012 is equipped with 1024 × 768 pixels. The central circular region with diameter of 722 pixels (avoiding the edge effect of the SLM) is chosen as the active region for our calculation. The calculation was conducted by Matlab as follows. Firstly, one certain Zernike term aberration (like Z6 term alone) with normalized coefficient (i.e., coefficient of 1) was generated in Matlab on a data grid of 722 × 722. Then its gradient norm distribution is obtained using gradient operator in Matlab. Finally, the allowable max coefficient value for this certain Zernike term is obtained by using 1/8 to divide the maximum value of the gradient norm. The calculation is only up to the Z9 (Primary spherical aberration), which is usually the highest nulling term. Table 1 presents the calculated results for Z2~Z9. For instance, Table 1 shows the LC-SLM can produce astigmatism terms (Z5/Z6) with PV value of about 36λ for Q = 8. Primary spherical aberration (i.e., Z9) is usually the dominate aberration for testing aspheric surfaces. To produce a larger amount of Z9 term, power term (i.e., Z4) was deliberately introduced in the calculation (see the last column of Table 1) to reduce gradient magnitude of the phase. Zernike aberrations with amplitudes larger than those shown in Table 1 can also be produced. However, it along with a reduction of Q value, which means the larger phase error. For production of Zernike aberrations with amplitudes smaller than those shown in Table 1, the phase generation accuracy will be better than λ/30 RMS theoretically, since more than 8 pixels are utilized to realize 1λ phase modulation.

Tables Icon

Table 1. The theoretically calculated maximum amplitudes of adaptable Zernike modes for LC 2012 when Q = 8

The above analysis was conducted theoretically. Experiments were further conducted to investigate the practical phase generation capacity and accuracy of LC 2012. The experiments were conducted in an indirect way based on the linear superposition theory. The simplest form of an ICGH is a linear grating of sinusoidal profile with a constant quantization level. According to the linear superposition theory, a general ICGH can be viewed as a collection of orthogonal sinusoidal gratings with variable spatial frequencies and quantization levels. Therefore, the performance of using the LC-SLM to produce different amounts of Z2 (tilt) and Z3 (tip) terms can be used to evaluate its performance of generating various Zernike modes with different amplitudes. The apparatus for this experiment is shown in Fig. 2(a). The schematic of the experimental set up is shown in Fig. 2(b). The 4” Zygo GPI is equipped with a transmissive flat (TF). The SLM is placed between the interferometer and a high accuracy retro-flat. The red solid line represents the beam of the interferometer. The beam of the interferometer transmits the SLM and reflects by the retro-flat. Then the beam transmits the SLM again and goes back to the interferometer. The aberration introduced by the SLM can be measured. The experiment procedures are as follows. Firstly, the distances from the LC-SLM to the TF and the retro-flat were measured by LenScan LS 600 [30]. LenScan LS600 can measure center thickness of optical elements and air gaps along the optical axis based on low coherence interferometry. Its measurement range is 600mm with absolute accuracy of ± 1μm. The collimator of the LenScan LS 600 was removed from the test system after the distances are measured. Note, the beam of the LenScan LS 600 is represented by green solid line in Fig. 2(b). Secondly, the transmissive inherit wavefront distortion (WFD) of the LC-SLM was measured without loading any dynamic phase to the SLM. Thirdly, gray maps encoding the WFD null phase (for nulling the WFD of the LC-SLM) and different amounts of Z2/Z3 (coefficients ranged from 2λ to 28λ in 2λ increment) were loaded to the LC-SLM sequentially. Phase measurements were also performed sequentially. A combination of + 1 order and 0-order was utilized in the double pass to reduce the fringes density. Finally, the theoretical Z2/Z3 terms measurement results from ray tracing model (simulated by Zemax) were subtracted from corresponding practical results to calculate the Z2/Z3 terms generation accuracy. In the ray tracing model, the distances from the LC-SLM to the TF and the retro-flat were set as the measurement results by LenScan LS 600.

 

Fig. 2 The experimental apparatus (a), and schematic of the experimental set up (b) for experimentally investigating the production of Z2 and Z3 terms using LC-SLM (LC 2012) in a collimated beam.

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The Z2/Z3 terms generation accuracy results are shown in Fig. 3. It verifies the conclusion that the larger Q is, the smaller phase error is. However, to limit the error value within λ/30 RMS, the maximum coefficient of Z2/Z3 terms is only about 28λ, which indicates that Q is about 12. According to the above linear superposition principle, the practically adaptable maximum amplitudes of Z2~Z9 for restricting the RMS error within λ/30 should be about two thirds of those presented in Table 1. The reduction may due to the black matrix structure and pixel patch which are not analyzed theoretically in this paper.

 

Fig. 3 The experimentally obtained accuracy for generating different amounts of Z2/Z3 using LC 2012.

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2.2. Test system design and techniques to perform dynamic tests using LC-SLMs

Using the above encoding method, LC-SLMs can be used to perform dynamic null tests of moderate free-form surfaces with moderate accuracy. However, to make practical measurement using LC-SLMs, issues about test system design, diffraction orders isolation, inherit WFD calibration, limited aperture and alignment of LC-SLMs should be firstly addressed. These techniques are analyzed as following.

2.2.1. The test system configuration design

The test systems of using LC-SLMs as ICGHs in collimated beam and divergent beam to perform dynamic null tests for aspheric and free-form surfaces are shown in Figs. 4(a) and 4(b), respectively. The test systems are both based on a commercially available Fizeau interferometer such as Zygo GPI. Test configuration shown in Fig. 4(a) is usually used for testing near flat free-form surfaces. Note, a beam expander or sub-aperture stitching technique [31–34] can be additionally utilized to overcome the limited aperture of LC-SLMs. Test configuration shown in Fig. 4(b) is commonly used for testing concave aspheric and free-form surfaces. The LC-SLM can be located near the convergence point, hence aspheric and free-form surfaces with aperture larger than that of the LC-SLM can be tested. For a particular free-form surface, the test configuration is firstly chosen from Fig. 4(a) and 4(b) according to the base shape of the free-form is flat or sphere.

 

Fig. 4 The test systems of using LC-SLMs as ICGHs in collimated beams (a), and divergent beams (b) to perform dynamic null tests for aspheric and free-form surfaces.

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2.2.2. The test null phase design

The desired phase φd(x, y) to null the inherent aberration (departure) of the test surface is designed by optical design software such as Zemax. The design method can follow the common CGH design procedures for aspheric and free-form surfaces [1]. During the null phase design process, tilt carrier is adopted. The test surface is tilted so that undesired diffraction orders can be blocked by the build-in spatial filter of commercial interferometers. When the LC-SLM is used in divergent beam, a spatial filter with variable size is additionally used. It locates near the convergence point as shown in Fig. 4(b). The size of the spatial filter can be adjusted to be smaller than that of the interferometers, so the required tilt carrier can be smaller.

2.2.3. Inherit WFD calibration

Due to thickness variations of glass substrate and LC layer of the LC-SLMs, the LC-SLMs will introduce inherent WFD to the incident wavefront. This is similar with the substrate errors of traditional CGHs.

By replacing the test surface shown in Fig. 4(a) with a high accuracy retro-flat, the transmissive wavefront error of the LC-SLM is obtained. It is exactly the WFD when the LC-SLM is used in collimated beams. However, this method cannot be directly applied to measure the WFD when the LC-SLM is used in divergent beams. Because if we replace the test surface with a high accuracy retro-sphere (also known as in situ measurement), the measurement result includes both the WFD and a certain amount of spherical aberration. The spherical aberration is introduced due to LC-SLM nominal thickness. To obtain the WFD caused by imperfect fabrication of the LC-SLM, we have to calculate the right amount of spherical aberration. The calculation can be conducted according to the nominal thickness of the LC-SLM. Then the theoretical spherical aberration is removed from the measurement result. However, this process will also introduce error due to incorrect mapping. Zhou has investigated this issue [35], and we will follow his conclusion to obtain the WFD when the LC-SLM is utilized in divergent beam. His conclusion is that ‘when the CGH (LC-SLM in our case) is used in a system faster than F/2.5, it is better to test its substrate (WFD in our case) in a collimated beam and then back out the substrate error (WFD in our case). When the CGH is used in a system slower than F/2.5, it can be tested in situ and then remove the spherical aberration.’.

2.2.4. Design the ICGH gray map for test surfaces

After WFD phase φw(x, y) is obtained, ICGH gray map encoding the desired null phase φd(x, y), WFD self-nulling phase φw(x, y) and carrier can be obtained according to Eq. (4). After the ICGH gray map is loaded to LC-SLM, the LC-SLM can be used as a traditional CGH for dynamic test of the free-form.

To obtain high-accuracy test result, alignment of LC-SLM is important. Alignment of the LC-SLM in collimated beam and divergent beam is realized by using a retro-flat and retro-sphere, respectively. The detailed alignment technique is shown in the following experiment part.

3. Experiments

To validate the feasibility of using LC-SLMs in collimated and divergent beam as reconfigurable ICGHs for adaptive null tests of moderate free-form surfaces with moderate test accuracy, experiments on two test surfaces were conducted.

3.1. Experiment of using LC-SLM in divergent beam for free-form with spherical surface base

The 1# test surface was tested in divergent beam. It is a bi-conic Zernike surface, which is described as

z=cxx2+cyy21+1(1+kx)cx2x2(1+ky)cy2y2+j=1nAjZj,
where z is the sag of the surface, cx and cy are the vertex curvatures in x direction, and y direction, respectively. kx and ky are the conic constants in x direction, and y direction, respectively. Zj is the jth Standard Zernike polynomial (Noll’s ordering) [36] and Aj is the coefficient of Zj.

The shape parameter values for 1# surface are cx = −1/199.103 mm−1, cy = −1/200.906mm−1, kx = ky = 0, A5 = 1.317e−3mm, A6 = 0.015mm, A8 = 1.111e−4, A12 = 1.121e−5mm. The aperture diameter is 100mm. The maximum sag is about 6.358mm. The departure from its fit sphere (R = 200mm) is shown in Fig. 5(a) with ~26λ PV. Astigmatism is the dominant aberration. The departure after astigmatism removed is mainly coma aberration as shown in Fig. 5(b).

 

Fig. 5 Departure (a), and the departure after astigmatism removed (b) of 1# surface.

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Design of the test scheme for 1# surface is similar with the conventional methods of designing CGHs as stated in Section 2. The designed optical layout is shown in Fig. 6(a). To realize separation of diffraction orders, tilt carrier (Z2 with coefficient of 100λ) was adopted. The designed distances from the LC-SLM to the convergence point and 1# surface are 50.68mm, and 148.085mm, respectively. The designed null phase after tilt removed is shown in Fig. 6(b). Test system residual wavefront error is shown in Fig. 6(c) with 0.006λ RMS. Then the null phase, the WFD self-null phase and the tilt carrier were encoded into an ICGH gray map according to Eq. (4).

 

Fig. 6 The designed test system optical layout (a), null phase after tilt removed (b), and test system theoretical residual wavefront error (c) for testing 1# surface. The designed LC-SLM alignment system optical layout (d), null phase after tilt removed (e), and theoretical residual wavefront error (f) for aligning LC-SLM.

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Alignment of the LC-SLM is performed by replacing the test surface with a high accuracy retro-sphere [a Zygo 6” F/1.1 transmissive sphere (TS)]. The basic idea is that the position and posture of the LC-SLM are designed to be same for the test system and the alignment system. In the alignment system, the alignment of the retro-sphere relative to the interferometer with a standard TS is easy to be performed. Moreover, the surface figure accuracy of both the retro-sphere and the standard TS are both very high. Hence, if the tested phase error is smaller than a threshold, the alignment of the LC-SLM is finished. Since the position and posture of the LC-SLM are designed to be same for the test system and the alignment system, the alignment of the LC-SLM for the test system is thought to be finished too. Radius of curvature of the aplanatic surface of the retro-sphere is 123.3mm. The designed optical layout for alignment is shown in Fig. 6(d). To realize separation of diffraction orders, tilt carrier (Z2 with coefficient of 100λ) was also adopted. Note, the distance from the LC-SLM to the convergence point is same with that of the test system shown in Fig. 6(a). It means the alignment was conducted in situ, i.e., the nominal position and posture of the LC-SLM are same for both the alignment and measurement systems. The designed null phase (after tilt removed) for alignment is shown in Fig. 6(e) with coma as dominant aberration for nulling tilted spherical surface. Alignment system residual wavefront error is shown in Fig. 6(f) with 0.0018λ RMS.

To test 1# surface, alignment of the LC-SLM was firstly conducted. Experimental alignment system shown in Fig. 7(a) was set up according to the optical layout shown in Fig. 6(d). The utilized interferometer is a 4” Zygo GPI equipped with an F/1.5 TS. The gray map encoding the null phase for tilted sphere, the WFD phase and tilt carrier was loaded to the LC-SLM. Then the LC-SLM was adjusted according to misalignment analysis to null the fringes until the measured phase was less than a threshold. The measured phase and fringes after LC-SLM is aligned are shown in Figs. 7(b) and 7(c), respectively.

 

Fig. 7 Experimental apparatus (a), the measured phase (b), and fringes (c) of aligning the LC-SLM for 1# surface.

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After alignment of the LC-SLM, the retro-sphere was replaced by 1# surface. Experimental test system shown in Fig. 8 was set up according to the optical layout shown in Fig. 6(a). The LC-SLM (LC-SLM) is placed between the interferometer (4” Zygo GPI equipped with a CCD with resolution of 640 × 480 and an F/1.5 TS, same with the alignment system) and 1# surface. The converging beam [represented by red beam in Fig. 8] of the interferometer transmits the spatial filter which locates at the convergence point. Then the beam becomes a divergent beam and transmits through the LC-SLM. After diffraction of LC-SLM, the divergent beam is transformed, and the transformed beam reflects from 1# surface. Finally, the test beam transmits the SLM again and goes back to the interferometer and interferences with the reference beam. When the gray map for the test surface is loaded to the LC-SLM, null test condition is met. The surface figure error of 1# surface can be measured after fine adjustment of 1# surface. The surface figure with 0.182λ RMS and fringes were obtained as shown in Figs. 9(a) and 9(b), respectively. To realize cross-test, 1# surface was deliberately designed to be testable by a Zygo Verfire AsphereTM interferometer equipped with a CCD with higher resolution of 1000 × 1000. Utilizing an F/1.1 TS, non-null test was conducted. Figures 9(c) and 9(d) show the fringes and phase map of the non-null test results, respectively. After retrace error calibration [37], surface figure is obtained as shown in Fig. 9(e) with 0.195λ RMS. By comparing results shown in Figs. 9(a) and 9(e), it can be concluded that surface figure distribution and PV/RMS value of the two methods results are very similar. Point-to-point difference (PPD) between the two results is further obtained to quantificationally evaluate accuracy of the proposed method. It is shown in Fig. 9(f) with 0.039λ RMS.

 

Fig. 8 Experimental apparatus of utilizing LC-SLM for testing 1# surface.

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Fig. 9 Surface figure (a), and fringes (b) of 1# surface tested by utilizing the LC-SLM. Fringes (c), and test result (d) of the non-null test by Verfire AsphereTM. Reconstructed surface figure (e) of the non-null test. (f) is PPD map between (a) and (e).

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3.2. Experiment of using LC-SLM in collimated beam for free-form with flat surface base

2# test surface was tested in collimated beam. It is a concave φ-polynomial (Zernike) surface, which is described by

z=cρ21+1(1+k)c2ρ2+j=1nCjZj,
where z is the sag of surface, c is the vertex curvature, k is the conic constant, ρ is the radial component in the aperture, Zj is the jth Zernike polynomial (Wyant’s ordering) [29] and Cj is the coefficient of Zj.

Shape parameter values for 2# surface are c = 0 (flat surface base shape), C4 = 14.939λ (power term), C9 = −4.979λ (primary spherical aberration term). The aperture diameter is 26mm. The surface sag is shown in Fig. 10(a). The maximum departure is about 30λ. Power and primary spherical aberration are the dominant aberration terms. The departure after power removed is mainly spherical aberration as shown in Fig. 10(b). Figure 10(c) shows the fringes of utilizing Zygo 4” GPI interferometer (CCD resolution: 640 × 480) with a standard TF to test the surface. The departure is beyond the dynamic range.

 

Fig. 10 Surface sag (a), surface sag after power removed (b), and fringes of testing 2# surface by an interferometer with a TF (c).

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The process of designing null phase and the gray map encoding the null phase is similar with that for 1# surface except that a collimated beam is utilized. Tilt carrier (Z2) with coefficient of 215λ was adopted. The residual wavefront error is about 0.001λ RMS, which is negligible. Then the null phase, the WFD self-null phase and the carrier were encoded into an ICGH gray map according to Eq. (4).

The experimental apparatus for testing 2# surface is shown in Fig. 11. The LC-SLM (LC-SLM) is placed between the interferometer (4” Zygo GPI equipped with a CCD with resolution of 640 × 480 and a TF) and 2# surface. The collimated beam [represented by red beam in Fig. 11] of the interferometer transmits the SLM. After diffraction of SLM, the collimated beam is transformed, and the transformed beam reflects from 2# surface. Finally, the test beam transmits the SLM again and goes back to the interferometer and interferences with the reference beam. When the gray map for the test surface is loaded to the LC-SLM, null test condition is met. The surface figure error of 2# surface can be measured after fine adjustment.

 

Fig. 11 The experimental apparatus for testing 2# surface by LC-SLM.

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Before measurement, a high accuracy retro-flat (a standard TF) was placed after the LC-SLM for measuring WFD of the LC-SLM and aligning the LC-SLM. The LC-SLM rotation around optical axis was aligned by applying 0° coma after self-nulling the WFD. Then rotate the LC-SLM until the detected 90° coma was less than a threshold. Tilt of the LC-SLM was aligned by tilting the LC-SLM until the fringes formed by the reflected wavefront from the front surface of the LC-SLM and the reference wavefront were sparse. Note, the LC-SLM was then slightly tilted for eliminating ghost reflections. Because the beam aperture is larger than the LC-SLM aperture, the decentration alignment is avoided.

After alignment of the LC-SLM, the retro-flat was replaced by 2# surface. The ICGH gray map for 2# surface was loaded to the LC-SLM. Then 2# surface was adjusted according to misalignment aberration analysis results. Finally, surface figure test result was obtained as shown in Fig. 12(a) with 0.281λ RMS. Figure 12(b) shows the corresponding sparse fringes which imply that null test (complete compensation) is realized. For cross-verification, the surface was tested by LuphoScan 260. LuphoScan 260 platforms are scanning interferometers based on an optical point probe that utilizes multi-wavelength interferometry (MWLI) technology. As shown in Fig. 13. Four motion stages (two linear stages: Z, R; and two rotary stages: C, T) are incorporated to performing scanning test of surfaces with diameters up to 260 mm and with slope up to 90°. LuphoScan 260 can provide an absolute measurement accuracy of better than ± 50 nm (3σ). To obtain the surface figure error of 2# surface, the profile of 2# surface was firstly measured by LuphoScan 260. Note, 2# surface was treated as a flat surface during the measurement. The experiment apparatus is shown in Fig. 13. During the measurement, the rotation stage C rotates 2# surface. The linear stages T translates the probe following the profile of an ideal flat surface. Therefore, a spiral scan is performed over the whole surface. After the profile of 2# surface was obtained, theoretical shape of 2# surface was subtracted from the measurement result to obtain surface figure error of 2# surface. The surface figure error result is shown in Fig. 14(a) with 0.317λ RMS. It is similar with Fig. 12(a) in surface figure distribution. The PPD map between Figs. 12(a) and 14(a) is presented in Fig. 14(b) with 0.039λ RMS.

 

Fig. 12 Surface figure test result (a), and sparse fringes (b) of 2# surface by using the LC-SLM.

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Fig. 13 Experimental apparatus of utilizing LuphoScan 260 for testing 2# surface.

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Fig. 14 Surface figure test result obtained by LuphoScan (a), and the PPD map between the two tests results (b).

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4. Discussion

LC-SLMs are AO elements able to generate very flexible aberrations. By addressing the issues about the accuracy, null capacity, WFD calibration, limited aperture and alignment, the LC-SLMs can be used in interferometric tests conveniently. The prospects are fascinating. Besides being utilized alone as variable nulls for free-form surfaces, the LC-SLMs can realize null test of aspheric and free-form surfaces with larger departure by nulling the residual wavefront error of non-null tests with a partial null [8,10,11]. It would be preferred to DMs because no phase monitoring devices are required, and its transmissive property will simplify the test system. Moreover, in-process free-form surfaces with unknown severe local surface figure error can be measured if we combine the ‘LC-SLM & partial null’ method with the adaptive wavefront interferometry technique that we reported recently [38]. These techniques are under research by us.

As for using LC-SLMs alone as variable null reported in this paper, the critical limit is set by the available LC-SLMs. The resolution and pixel pitch of SLMs determine the null ability. Using the specific HoloeyeTM LC 2012 model, which has a resolution of 1024 × 768 pixel and 36μm pixel pitch, freeform surfaces with ∼20 μm departure can be tested with moderate test accuracy. Recently, SLMs with 4096 × 2160 (4K) resolution and 3.74 μm pixel pitch [39] are available. We could expect freeform optics with larger departure (e.g., 60 μm) can be measured.

Compared with the dynamic null methods using DMs. The prime advantage of our method is the simplicity of the test system. Our method can test surfaces using SLMs without shape monitoring systems. The moderate phase control accuracy of encoding SLMs as ICGHs can fulfill the accuracy requirements of most tests. However, the control accuracy of using DMs alone (i.e., using DMs by open-loop control) is very low. Hence a shape monitoring system such as Shack-Hartmann sensors, deflectometry systems or wavefront interferometers are required to monitor the shape of DMs. This will complicate the test system. In terms of phase control capacity, the SLMs can generate null phase with amplitude of about 40 waves. It is similar with that of most DMs.

5. Conclusion

We have described an adaptive null interferometry method only using a LC-SLM in collimated or divergent beams as reconfigurable multi-level interferogram-type CGHs for aspheric and free-form surfaces metrology. The magnitude and accuracy of the phase production ability of the LC-SLM encoded by the proposed method are analyzed theoretically and experimentally. Test schemes with detailed techniques about test system optical design, inherent wavefront distortion calibration, overcoming limited aperture, and alignment are established. Two free-form surfaces with about 20 μm departure from flat or spherical surfaces were successfully measured with moderate accuracy. The test flexibility and efficiency would be enhanced with reduced cost for moderate free-form surfaces using the presented method.

Funding

Natural Science Foundation of Hunan Province (2016JJ1003); Science Challenge Program of China (TZ2018006).

Acknowledgments

We want to thank Mr. Jinfeng Lu who has graduated from NUDT for his earlier investigations about using the SLMs. Ms. Wanxia Deng at Oulu University is appreciated for exchanging ideas about how to write papers well. Mr. Kang Kuang and Mr. Linchao Zhang are appreciated for their efforts of fabricating and testing the free-form surfaces. We also wish to thank Dr. Cashmore from NPL who have provided us with advice about using SLMs.

References

1. D. Malacara, Optical Shop Testing (Wiley, 2007).

2. S. Xue, S. Chen, D. Zhai, and F. Shi, “Quasi-absolute surface figure test with two orthogonal transverse spatial shifts,” Opt. Commun. 389, 133–143 (2017). [CrossRef]  

3. S. Xue, S. Chen, Y. Tian, J. Lu, and H. Hu, “Verification and in situ calibration of large-aperture null correctors for convex aspheric mirrors,” Meas. 106, 79–87 (2017). [CrossRef]  

4. R. S. Hilbert and M. P. Rimmer, “A variable refractive null lens,” Appl. Opt. 9(4), 849–852 (1970). [CrossRef]   [PubMed]  

5. M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010). [CrossRef]  

6. S. Chen, S. Xue, Y. Dai, and S. Li, “Subaperture stitching test of convex aspheres by using the reconfigurable optical null,” Opt. Laser Technol. 91, 175–184 (2017). [CrossRef]  

7. L. Zhang, D. Li, Y. Liu, Y. Bai, J. Li, and B. Yu, “Flexible interferometry for optical aspheric and free form surfaces,” Opt. Rev. 24(6), 677–685 (2017). [CrossRef]  

8. J. Peng, D. Chen, H. Guo, J. Zhong, and Y. Yu, “Variable optical null based on a yawing CGH for measuring steep acylindrical surface,” Opt. Express 26(16), 20306–20318 (2018). [CrossRef]   [PubMed]  

9. Y. Dou, Q. Yuan, Z. Gao, H. Yin, L. Chen, Y. Yao, and J. Cheng, “Partial null astigmatism-compensated interferometry for a concave freeform Zernike mirror,” J. Opt. 20(6), 065702 (2018). [CrossRef]  

10. S. Xue, S. Chen, and G. Tie, “Near-null interferometry using an aspheric null lens generating a broad range of variable spherical aberration for flexible test of aspheres,” Opt. Express 26(24), 31172–31189 (2018). [CrossRef]   [PubMed]  

11. C. Tian, Y. Yang, T. Wei, and Y. Zhuo, “Nonnull interferometer simulation for aspheric testing based on ray tracing,” Appl. Opt. 50(20), 3559–3569 (2011). [CrossRef]   [PubMed]  

12. Q. Hao, S. Wang, Y. Hu, H. Cheng, M. Chen, and T. Li, “Virtual interferometer calibration method of a non-null interferometer for freeform surface measurements,” Appl. Opt. 55(35), 9992–10001 (2016). [CrossRef]   [PubMed]  

13. L. Z. Lei Zhang, S. Z. Sheng Zhou, D. L. Dong Li, J. L. Jingsong Li, and B. Y. Benli Yu, “Model-based adaptive non-null interferometry for freeform surface metrology,” Chin. Opt. Lett. 16(8), 081203 (2018). [CrossRef]  

14. Y. He, L. Huang, X. Hou, W. Fan, and R. Liang, “Modeling near-null testing method of a freeform surface with a deformable mirror compensator,” Appl. Opt. 56(33), 9132–9138 (2017). [CrossRef]   [PubMed]  

15. L. Huang, H. Choi, W. Zhao, L. R. Graves, and D. W. Kim, “Adaptive interferometric null testing for unknown freeform optics metrology,” Opt. Lett. 41(23), 5539–5542 (2016). [CrossRef]   [PubMed]  

16. K. Fuerschbach, K. P. Thompson, and J. P. Rolland, “Interferometric measurement of a concave, φ-polynomial, Zernike mirror,” Opt. Lett. 39(1), 18–21 (2014). [CrossRef]   [PubMed]  

17. L. Zhang, S. Zhou, D. Li, Y. Liu, T. He, B. Yu, and J. Li, “Pure adaptive interferometer for free form surfaces metrology,” Opt. Express 26(7), 7888–7898 (2018). [CrossRef]   [PubMed]  

18. Z. Cao, L. Xuan, L. Hu, Y. Liu, Q. Mu, and D. Li, “Investigation of optical testing with a phase-only liquid crystal spatial light modulator,” Opt. Express 13(4), 1059–1065 (2005). [CrossRef]   [PubMed]  

19. Z. Zhao, Z. Xiao, Y. Zhuang, H. Zhang, and H. Zhao, “An interferometric method for local phase modulation calibration of LC-SLM using self-generated phase grating,” Rev. Sci. Instrum. 89(8), 083116 (2018). [CrossRef]   [PubMed]  

20. Z. Z. Zixin Zhao, Y. Z. Yiying Zhuang, Z. X. Zhaoxian Xiao, H. Z. Hangying Zhang, C. F. Chen Fan, H. G. Hehui Geng, and H. Z. Hong Zhao, “Characterizing a liquid crystal spatial light modulator at oblique incidence angles using the self-interference method,” Chin. Opt. Lett. 16(9), 090701 (2018). [CrossRef]  

21. H. Chen, J. Yang, H. Yen, Z. Hsu, Y. Huang, and S. Wu, “Pursuing high quality phase-only liquid crystal on silicon (LCoS) devices,” Appl. Sci. (Basel) 8(11), 2323 (2018). [CrossRef]  

22. J. Xu, S. Qin, C. Liu, S. Fu, and D. Liu, “Precise calibration of spatial phase response nonuniformity arising in liquid crystal on silicon,” Opt. Lett. 43(12), 2993–2996 (2018). [CrossRef]   [PubMed]  

23. J. Kacperski and M. Kujawinska, “Active, LCoS based laser interferometer for microelements studies,” Opt. Express 14(21), 9664–9678 (2006). [CrossRef]   [PubMed]  

24. M. A. Neil, M. J. Booth, and T. Wilson, “Dynamic wave-front generation for the characterization and testing of optical systems,” Opt. Lett. 23(23), 1849–1851 (1998). [CrossRef]   [PubMed]  

25. B. R. Boruah, G. D. Love, and M. A. Neil, “Interferometry using binary holograms without high order diffraction effects,” Opt. Lett. 36(12), 2357–2359 (2011). [CrossRef]   [PubMed]  

26. M. T. Cashmore, S. R. Hall, and G. D. Love, “Traceable interferometry using binary reconfigurable holograms,” Appl. Opt. 53(24), 5353–5358 (2014). [CrossRef]   [PubMed]  

27. P. Picart, New Techniques in Digital Holography (ISTE & Wiley, 2015).

28. M. A. Golub, I. N. Sisakian, and V. A. Soifer, “Phase quantization and discretization in diffractive optics,” Proc. SPIE 1334, 188–199 (1990). [CrossRef]  

29. R. Shannon and J. Wyant, Applied Optics and Optical Engineering (Academic, 1992).

30. FOGALE nanotech, “Sensors & systems documentations,”http://www.fogale.fr/brochures.html.

31. X. Hou, F. Wu, L. Yang, and Q. Chen, “Experimental study on measurement of aspheric surface shape with complementary annular subaperture interferometric method,” Opt. Express 15(20), 12890–12899 (2007). [CrossRef]   [PubMed]  

32. J. Xue, L. Huang, B. Gao, K. Kaznatcheev, and M. Idir, “One-dimensional stitching interferometry assisted by a triple-beam interferometer,” Opt. Express 25(8), 9393–9405 (2017). [CrossRef]   [PubMed]  

33. S. Chen, S. Xue, Y. Dai, and S. Li, “Subaperture stitching test of large steep convex spheres,” Opt. Express 23(22), 29047–29058 (2015). [CrossRef]   [PubMed]  

34. J. Peng, Q. Wang, X. Peng, and Y. Yu, “Stitching interferometry of high numerical aperture cylindrical optics without using a fringe-nulling routine,” J. Opt. Soc. Am. A 32(11), 1964–1972 (2015). [CrossRef]   [PubMed]  

35. P. Zhou, “Error analysis and data reduction for interferometric surface measurements,” Doctoral dissertation, The university of Arizona, US (2009).

36. R. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207–211 (1976). [CrossRef]  

37. D. Liu, Y. Yang, C. Tian, Y. Luo, and L. Wang, “Practical methods for retrace error correction in nonnull aspheric testing,” Opt. Express 17(9), 7025–7035 (2009). [CrossRef]   [PubMed]  

38. S. Xue, S. Chen, Z. Fan, and D. Zhai, “Adaptive wavefront interferometry for unknown free-form surfaces,” Opt. Express 26(17), 21910–21928 (2018). [CrossRef]   [PubMed]  

39. Holoeye Corporation, “GAEA 10 Megapixel Phase Only Spatial Light Modulator,” https://holoeye.com/spatial-light-modulators/gaea-4k-phase-only-spatial-light-modulator/.

References

  • View by:
  • |
  • |
  • |

  1. D. Malacara, Optical Shop Testing (Wiley, 2007).
  2. S. Xue, S. Chen, D. Zhai, and F. Shi, “Quasi-absolute surface figure test with two orthogonal transverse spatial shifts,” Opt. Commun. 389, 133–143 (2017).
    [Crossref]
  3. S. Xue, S. Chen, Y. Tian, J. Lu, and H. Hu, “Verification and in situ calibration of large-aperture null correctors for convex aspheric mirrors,” Meas. 106, 79–87 (2017).
    [Crossref]
  4. R. S. Hilbert and M. P. Rimmer, “A variable refractive null lens,” Appl. Opt. 9(4), 849–852 (1970).
    [Crossref] [PubMed]
  5. M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
    [Crossref]
  6. S. Chen, S. Xue, Y. Dai, and S. Li, “Subaperture stitching test of convex aspheres by using the reconfigurable optical null,” Opt. Laser Technol. 91, 175–184 (2017).
    [Crossref]
  7. L. Zhang, D. Li, Y. Liu, Y. Bai, J. Li, and B. Yu, “Flexible interferometry for optical aspheric and free form surfaces,” Opt. Rev. 24(6), 677–685 (2017).
    [Crossref]
  8. J. Peng, D. Chen, H. Guo, J. Zhong, and Y. Yu, “Variable optical null based on a yawing CGH for measuring steep acylindrical surface,” Opt. Express 26(16), 20306–20318 (2018).
    [Crossref] [PubMed]
  9. Y. Dou, Q. Yuan, Z. Gao, H. Yin, L. Chen, Y. Yao, and J. Cheng, “Partial null astigmatism-compensated interferometry for a concave freeform Zernike mirror,” J. Opt. 20(6), 065702 (2018).
    [Crossref]
  10. S. Xue, S. Chen, and G. Tie, “Near-null interferometry using an aspheric null lens generating a broad range of variable spherical aberration for flexible test of aspheres,” Opt. Express 26(24), 31172–31189 (2018).
    [Crossref] [PubMed]
  11. C. Tian, Y. Yang, T. Wei, and Y. Zhuo, “Nonnull interferometer simulation for aspheric testing based on ray tracing,” Appl. Opt. 50(20), 3559–3569 (2011).
    [Crossref] [PubMed]
  12. Q. Hao, S. Wang, Y. Hu, H. Cheng, M. Chen, and T. Li, “Virtual interferometer calibration method of a non-null interferometer for freeform surface measurements,” Appl. Opt. 55(35), 9992–10001 (2016).
    [Crossref] [PubMed]
  13. L. Z. Lei Zhang, S. Z. Sheng Zhou, D. L. Dong Li, J. L. Jingsong Li, and B. Y. Benli Yu, “Model-based adaptive non-null interferometry for freeform surface metrology,” Chin. Opt. Lett. 16(8), 081203 (2018).
    [Crossref]
  14. Y. He, L. Huang, X. Hou, W. Fan, and R. Liang, “Modeling near-null testing method of a freeform surface with a deformable mirror compensator,” Appl. Opt. 56(33), 9132–9138 (2017).
    [Crossref] [PubMed]
  15. L. Huang, H. Choi, W. Zhao, L. R. Graves, and D. W. Kim, “Adaptive interferometric null testing for unknown freeform optics metrology,” Opt. Lett. 41(23), 5539–5542 (2016).
    [Crossref] [PubMed]
  16. K. Fuerschbach, K. P. Thompson, and J. P. Rolland, “Interferometric measurement of a concave, φ-polynomial, Zernike mirror,” Opt. Lett. 39(1), 18–21 (2014).
    [Crossref] [PubMed]
  17. L. Zhang, S. Zhou, D. Li, Y. Liu, T. He, B. Yu, and J. Li, “Pure adaptive interferometer for free form surfaces metrology,” Opt. Express 26(7), 7888–7898 (2018).
    [Crossref] [PubMed]
  18. Z. Cao, L. Xuan, L. Hu, Y. Liu, Q. Mu, and D. Li, “Investigation of optical testing with a phase-only liquid crystal spatial light modulator,” Opt. Express 13(4), 1059–1065 (2005).
    [Crossref] [PubMed]
  19. Z. Zhao, Z. Xiao, Y. Zhuang, H. Zhang, and H. Zhao, “An interferometric method for local phase modulation calibration of LC-SLM using self-generated phase grating,” Rev. Sci. Instrum. 89(8), 083116 (2018).
    [Crossref] [PubMed]
  20. Z. Z. Zixin Zhao, Y. Z. Yiying Zhuang, Z. X. Zhaoxian Xiao, H. Z. Hangying Zhang, C. F. Chen Fan, H. G. Hehui Geng, and H. Z. Hong Zhao, “Characterizing a liquid crystal spatial light modulator at oblique incidence angles using the self-interference method,” Chin. Opt. Lett. 16(9), 090701 (2018).
    [Crossref]
  21. H. Chen, J. Yang, H. Yen, Z. Hsu, Y. Huang, and S. Wu, “Pursuing high quality phase-only liquid crystal on silicon (LCoS) devices,” Appl. Sci. (Basel) 8(11), 2323 (2018).
    [Crossref]
  22. J. Xu, S. Qin, C. Liu, S. Fu, and D. Liu, “Precise calibration of spatial phase response nonuniformity arising in liquid crystal on silicon,” Opt. Lett. 43(12), 2993–2996 (2018).
    [Crossref] [PubMed]
  23. J. Kacperski and M. Kujawinska, “Active, LCoS based laser interferometer for microelements studies,” Opt. Express 14(21), 9664–9678 (2006).
    [Crossref] [PubMed]
  24. M. A. Neil, M. J. Booth, and T. Wilson, “Dynamic wave-front generation for the characterization and testing of optical systems,” Opt. Lett. 23(23), 1849–1851 (1998).
    [Crossref] [PubMed]
  25. B. R. Boruah, G. D. Love, and M. A. Neil, “Interferometry using binary holograms without high order diffraction effects,” Opt. Lett. 36(12), 2357–2359 (2011).
    [Crossref] [PubMed]
  26. M. T. Cashmore, S. R. Hall, and G. D. Love, “Traceable interferometry using binary reconfigurable holograms,” Appl. Opt. 53(24), 5353–5358 (2014).
    [Crossref] [PubMed]
  27. P. Picart, New Techniques in Digital Holography (ISTE & Wiley, 2015).
  28. M. A. Golub, I. N. Sisakian, and V. A. Soifer, “Phase quantization and discretization in diffractive optics,” Proc. SPIE 1334, 188–199 (1990).
    [Crossref]
  29. R. Shannon and J. Wyant, Applied Optics and Optical Engineering (Academic, 1992).
  30. FOGALE nanotech, “Sensors & systems documentations,” http://www.fogale.fr/brochures.html .
  31. X. Hou, F. Wu, L. Yang, and Q. Chen, “Experimental study on measurement of aspheric surface shape with complementary annular subaperture interferometric method,” Opt. Express 15(20), 12890–12899 (2007).
    [Crossref] [PubMed]
  32. J. Xue, L. Huang, B. Gao, K. Kaznatcheev, and M. Idir, “One-dimensional stitching interferometry assisted by a triple-beam interferometer,” Opt. Express 25(8), 9393–9405 (2017).
    [Crossref] [PubMed]
  33. S. Chen, S. Xue, Y. Dai, and S. Li, “Subaperture stitching test of large steep convex spheres,” Opt. Express 23(22), 29047–29058 (2015).
    [Crossref] [PubMed]
  34. J. Peng, Q. Wang, X. Peng, and Y. Yu, “Stitching interferometry of high numerical aperture cylindrical optics without using a fringe-nulling routine,” J. Opt. Soc. Am. A 32(11), 1964–1972 (2015).
    [Crossref] [PubMed]
  35. P. Zhou, “Error analysis and data reduction for interferometric surface measurements,” Doctoral dissertation, The university of Arizona, US (2009).
  36. R. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207–211 (1976).
    [Crossref]
  37. D. Liu, Y. Yang, C. Tian, Y. Luo, and L. Wang, “Practical methods for retrace error correction in nonnull aspheric testing,” Opt. Express 17(9), 7025–7035 (2009).
    [Crossref] [PubMed]
  38. S. Xue, S. Chen, Z. Fan, and D. Zhai, “Adaptive wavefront interferometry for unknown free-form surfaces,” Opt. Express 26(17), 21910–21928 (2018).
    [Crossref] [PubMed]
  39. Holoeye Corporation, “GAEA 10 Megapixel Phase Only Spatial Light Modulator,” https://holoeye.com/spatial-light-modulators/gaea-4k-phase-only-spatial-light-modulator/ .

2018 (10)

J. Peng, D. Chen, H. Guo, J. Zhong, and Y. Yu, “Variable optical null based on a yawing CGH for measuring steep acylindrical surface,” Opt. Express 26(16), 20306–20318 (2018).
[Crossref] [PubMed]

Y. Dou, Q. Yuan, Z. Gao, H. Yin, L. Chen, Y. Yao, and J. Cheng, “Partial null astigmatism-compensated interferometry for a concave freeform Zernike mirror,” J. Opt. 20(6), 065702 (2018).
[Crossref]

S. Xue, S. Chen, and G. Tie, “Near-null interferometry using an aspheric null lens generating a broad range of variable spherical aberration for flexible test of aspheres,” Opt. Express 26(24), 31172–31189 (2018).
[Crossref] [PubMed]

L. Z. Lei Zhang, S. Z. Sheng Zhou, D. L. Dong Li, J. L. Jingsong Li, and B. Y. Benli Yu, “Model-based adaptive non-null interferometry for freeform surface metrology,” Chin. Opt. Lett. 16(8), 081203 (2018).
[Crossref]

L. Zhang, S. Zhou, D. Li, Y. Liu, T. He, B. Yu, and J. Li, “Pure adaptive interferometer for free form surfaces metrology,” Opt. Express 26(7), 7888–7898 (2018).
[Crossref] [PubMed]

Z. Zhao, Z. Xiao, Y. Zhuang, H. Zhang, and H. Zhao, “An interferometric method for local phase modulation calibration of LC-SLM using self-generated phase grating,” Rev. Sci. Instrum. 89(8), 083116 (2018).
[Crossref] [PubMed]

Z. Z. Zixin Zhao, Y. Z. Yiying Zhuang, Z. X. Zhaoxian Xiao, H. Z. Hangying Zhang, C. F. Chen Fan, H. G. Hehui Geng, and H. Z. Hong Zhao, “Characterizing a liquid crystal spatial light modulator at oblique incidence angles using the self-interference method,” Chin. Opt. Lett. 16(9), 090701 (2018).
[Crossref]

H. Chen, J. Yang, H. Yen, Z. Hsu, Y. Huang, and S. Wu, “Pursuing high quality phase-only liquid crystal on silicon (LCoS) devices,” Appl. Sci. (Basel) 8(11), 2323 (2018).
[Crossref]

J. Xu, S. Qin, C. Liu, S. Fu, and D. Liu, “Precise calibration of spatial phase response nonuniformity arising in liquid crystal on silicon,” Opt. Lett. 43(12), 2993–2996 (2018).
[Crossref] [PubMed]

S. Xue, S. Chen, Z. Fan, and D. Zhai, “Adaptive wavefront interferometry for unknown free-form surfaces,” Opt. Express 26(17), 21910–21928 (2018).
[Crossref] [PubMed]

2017 (6)

J. Xue, L. Huang, B. Gao, K. Kaznatcheev, and M. Idir, “One-dimensional stitching interferometry assisted by a triple-beam interferometer,” Opt. Express 25(8), 9393–9405 (2017).
[Crossref] [PubMed]

Y. He, L. Huang, X. Hou, W. Fan, and R. Liang, “Modeling near-null testing method of a freeform surface with a deformable mirror compensator,” Appl. Opt. 56(33), 9132–9138 (2017).
[Crossref] [PubMed]

S. Xue, S. Chen, D. Zhai, and F. Shi, “Quasi-absolute surface figure test with two orthogonal transverse spatial shifts,” Opt. Commun. 389, 133–143 (2017).
[Crossref]

S. Xue, S. Chen, Y. Tian, J. Lu, and H. Hu, “Verification and in situ calibration of large-aperture null correctors for convex aspheric mirrors,” Meas. 106, 79–87 (2017).
[Crossref]

S. Chen, S. Xue, Y. Dai, and S. Li, “Subaperture stitching test of convex aspheres by using the reconfigurable optical null,” Opt. Laser Technol. 91, 175–184 (2017).
[Crossref]

L. Zhang, D. Li, Y. Liu, Y. Bai, J. Li, and B. Yu, “Flexible interferometry for optical aspheric and free form surfaces,” Opt. Rev. 24(6), 677–685 (2017).
[Crossref]

2016 (2)

2015 (2)

2014 (2)

2011 (2)

2010 (1)

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

2009 (1)

2007 (1)

2006 (1)

2005 (1)

1998 (1)

1990 (1)

M. A. Golub, I. N. Sisakian, and V. A. Soifer, “Phase quantization and discretization in diffractive optics,” Proc. SPIE 1334, 188–199 (1990).
[Crossref]

1976 (1)

1970 (1)

Bai, Y.

L. Zhang, D. Li, Y. Liu, Y. Bai, J. Li, and B. Yu, “Flexible interferometry for optical aspheric and free form surfaces,” Opt. Rev. 24(6), 677–685 (2017).
[Crossref]

Bauer, M.

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Benli Yu, B. Y.

Booth, M. J.

Boruah, B. R.

Cao, Z.

Cashmore, M. T.

Chen, D.

Chen, H.

H. Chen, J. Yang, H. Yen, Z. Hsu, Y. Huang, and S. Wu, “Pursuing high quality phase-only liquid crystal on silicon (LCoS) devices,” Appl. Sci. (Basel) 8(11), 2323 (2018).
[Crossref]

Chen, L.

Y. Dou, Q. Yuan, Z. Gao, H. Yin, L. Chen, Y. Yao, and J. Cheng, “Partial null astigmatism-compensated interferometry for a concave freeform Zernike mirror,” J. Opt. 20(6), 065702 (2018).
[Crossref]

Chen, M.

Chen, Q.

Chen, S.

S. Xue, S. Chen, and G. Tie, “Near-null interferometry using an aspheric null lens generating a broad range of variable spherical aberration for flexible test of aspheres,” Opt. Express 26(24), 31172–31189 (2018).
[Crossref] [PubMed]

S. Xue, S. Chen, Z. Fan, and D. Zhai, “Adaptive wavefront interferometry for unknown free-form surfaces,” Opt. Express 26(17), 21910–21928 (2018).
[Crossref] [PubMed]

S. Chen, S. Xue, Y. Dai, and S. Li, “Subaperture stitching test of convex aspheres by using the reconfigurable optical null,” Opt. Laser Technol. 91, 175–184 (2017).
[Crossref]

S. Xue, S. Chen, D. Zhai, and F. Shi, “Quasi-absolute surface figure test with two orthogonal transverse spatial shifts,” Opt. Commun. 389, 133–143 (2017).
[Crossref]

S. Xue, S. Chen, Y. Tian, J. Lu, and H. Hu, “Verification and in situ calibration of large-aperture null correctors for convex aspheric mirrors,” Meas. 106, 79–87 (2017).
[Crossref]

S. Chen, S. Xue, Y. Dai, and S. Li, “Subaperture stitching test of large steep convex spheres,” Opt. Express 23(22), 29047–29058 (2015).
[Crossref] [PubMed]

Chen Fan, C. F.

Cheng, H.

Cheng, J.

Y. Dou, Q. Yuan, Z. Gao, H. Yin, L. Chen, Y. Yao, and J. Cheng, “Partial null astigmatism-compensated interferometry for a concave freeform Zernike mirror,” J. Opt. 20(6), 065702 (2018).
[Crossref]

Choi, H.

Dai, Y.

S. Chen, S. Xue, Y. Dai, and S. Li, “Subaperture stitching test of convex aspheres by using the reconfigurable optical null,” Opt. Laser Technol. 91, 175–184 (2017).
[Crossref]

S. Chen, S. Xue, Y. Dai, and S. Li, “Subaperture stitching test of large steep convex spheres,” Opt. Express 23(22), 29047–29058 (2015).
[Crossref] [PubMed]

DeVries, G.

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Dong Li, D. L.

Dou, Y.

Y. Dou, Q. Yuan, Z. Gao, H. Yin, L. Chen, Y. Yao, and J. Cheng, “Partial null astigmatism-compensated interferometry for a concave freeform Zernike mirror,” J. Opt. 20(6), 065702 (2018).
[Crossref]

Fan, W.

Fan, Z.

Fleig, J.

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Forbes, G.

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Fu, S.

Fuerschbach, K.

Gao, B.

Gao, Z.

Y. Dou, Q. Yuan, Z. Gao, H. Yin, L. Chen, Y. Yao, and J. Cheng, “Partial null astigmatism-compensated interferometry for a concave freeform Zernike mirror,” J. Opt. 20(6), 065702 (2018).
[Crossref]

Golub, M. A.

M. A. Golub, I. N. Sisakian, and V. A. Soifer, “Phase quantization and discretization in diffractive optics,” Proc. SPIE 1334, 188–199 (1990).
[Crossref]

Graves, L. R.

Guo, H.

Hall, S. R.

Hangying Zhang, H. Z.

Hao, Q.

He, T.

He, Y.

Hehui Geng, H. G.

Hilbert, R. S.

Hong Zhao, H. Z.

Hou, X.

Hsu, Z.

H. Chen, J. Yang, H. Yen, Z. Hsu, Y. Huang, and S. Wu, “Pursuing high quality phase-only liquid crystal on silicon (LCoS) devices,” Appl. Sci. (Basel) 8(11), 2323 (2018).
[Crossref]

Hu, H.

S. Xue, S. Chen, Y. Tian, J. Lu, and H. Hu, “Verification and in situ calibration of large-aperture null correctors for convex aspheric mirrors,” Meas. 106, 79–87 (2017).
[Crossref]

Hu, L.

Hu, Y.

Huang, L.

Huang, Y.

H. Chen, J. Yang, H. Yen, Z. Hsu, Y. Huang, and S. Wu, “Pursuing high quality phase-only liquid crystal on silicon (LCoS) devices,” Appl. Sci. (Basel) 8(11), 2323 (2018).
[Crossref]

Idir, M.

Jingsong Li, J. L.

Kacperski, J.

Kaznatcheev, K.

Kim, D. W.

Kujawinska, M.

Kulawiec, A.

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Lei Zhang, L. Z.

Li, D.

Li, J.

L. Zhang, S. Zhou, D. Li, Y. Liu, T. He, B. Yu, and J. Li, “Pure adaptive interferometer for free form surfaces metrology,” Opt. Express 26(7), 7888–7898 (2018).
[Crossref] [PubMed]

L. Zhang, D. Li, Y. Liu, Y. Bai, J. Li, and B. Yu, “Flexible interferometry for optical aspheric and free form surfaces,” Opt. Rev. 24(6), 677–685 (2017).
[Crossref]

Li, S.

S. Chen, S. Xue, Y. Dai, and S. Li, “Subaperture stitching test of convex aspheres by using the reconfigurable optical null,” Opt. Laser Technol. 91, 175–184 (2017).
[Crossref]

S. Chen, S. Xue, Y. Dai, and S. Li, “Subaperture stitching test of large steep convex spheres,” Opt. Express 23(22), 29047–29058 (2015).
[Crossref] [PubMed]

Li, T.

Liang, R.

Liu, C.

Liu, D.

Liu, Y.

Love, G. D.

Lu, J.

S. Xue, S. Chen, Y. Tian, J. Lu, and H. Hu, “Verification and in situ calibration of large-aperture null correctors for convex aspheric mirrors,” Meas. 106, 79–87 (2017).
[Crossref]

Luo, Y.

Miladinovich, D.

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Mu, Q.

Murphy, P.

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Neil, M. A.

Noll, R.

Peng, J.

Peng, X.

Qin, S.

Rimmer, M. P.

Rolland, J. P.

Sheng Zhou, S. Z.

Shi, F.

S. Xue, S. Chen, D. Zhai, and F. Shi, “Quasi-absolute surface figure test with two orthogonal transverse spatial shifts,” Opt. Commun. 389, 133–143 (2017).
[Crossref]

Sisakian, I. N.

M. A. Golub, I. N. Sisakian, and V. A. Soifer, “Phase quantization and discretization in diffractive optics,” Proc. SPIE 1334, 188–199 (1990).
[Crossref]

Soifer, V. A.

M. A. Golub, I. N. Sisakian, and V. A. Soifer, “Phase quantization and discretization in diffractive optics,” Proc. SPIE 1334, 188–199 (1990).
[Crossref]

Thompson, K. P.

Tian, C.

Tian, Y.

S. Xue, S. Chen, Y. Tian, J. Lu, and H. Hu, “Verification and in situ calibration of large-aperture null correctors for convex aspheric mirrors,” Meas. 106, 79–87 (2017).
[Crossref]

Tie, G.

Tricard, M.

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Wang, L.

Wang, Q.

Wang, S.

Wei, T.

Wilson, T.

Wu, F.

Wu, S.

H. Chen, J. Yang, H. Yen, Z. Hsu, Y. Huang, and S. Wu, “Pursuing high quality phase-only liquid crystal on silicon (LCoS) devices,” Appl. Sci. (Basel) 8(11), 2323 (2018).
[Crossref]

Xiao, Z.

Z. Zhao, Z. Xiao, Y. Zhuang, H. Zhang, and H. Zhao, “An interferometric method for local phase modulation calibration of LC-SLM using self-generated phase grating,” Rev. Sci. Instrum. 89(8), 083116 (2018).
[Crossref] [PubMed]

Xu, J.

Xuan, L.

Xue, J.

Xue, S.

S. Xue, S. Chen, Z. Fan, and D. Zhai, “Adaptive wavefront interferometry for unknown free-form surfaces,” Opt. Express 26(17), 21910–21928 (2018).
[Crossref] [PubMed]

S. Xue, S. Chen, and G. Tie, “Near-null interferometry using an aspheric null lens generating a broad range of variable spherical aberration for flexible test of aspheres,” Opt. Express 26(24), 31172–31189 (2018).
[Crossref] [PubMed]

S. Xue, S. Chen, D. Zhai, and F. Shi, “Quasi-absolute surface figure test with two orthogonal transverse spatial shifts,” Opt. Commun. 389, 133–143 (2017).
[Crossref]

S. Xue, S. Chen, Y. Tian, J. Lu, and H. Hu, “Verification and in situ calibration of large-aperture null correctors for convex aspheric mirrors,” Meas. 106, 79–87 (2017).
[Crossref]

S. Chen, S. Xue, Y. Dai, and S. Li, “Subaperture stitching test of convex aspheres by using the reconfigurable optical null,” Opt. Laser Technol. 91, 175–184 (2017).
[Crossref]

S. Chen, S. Xue, Y. Dai, and S. Li, “Subaperture stitching test of large steep convex spheres,” Opt. Express 23(22), 29047–29058 (2015).
[Crossref] [PubMed]

Yang, J.

H. Chen, J. Yang, H. Yen, Z. Hsu, Y. Huang, and S. Wu, “Pursuing high quality phase-only liquid crystal on silicon (LCoS) devices,” Appl. Sci. (Basel) 8(11), 2323 (2018).
[Crossref]

Yang, L.

Yang, Y.

Yao, Y.

Y. Dou, Q. Yuan, Z. Gao, H. Yin, L. Chen, Y. Yao, and J. Cheng, “Partial null astigmatism-compensated interferometry for a concave freeform Zernike mirror,” J. Opt. 20(6), 065702 (2018).
[Crossref]

Yen, H.

H. Chen, J. Yang, H. Yen, Z. Hsu, Y. Huang, and S. Wu, “Pursuing high quality phase-only liquid crystal on silicon (LCoS) devices,” Appl. Sci. (Basel) 8(11), 2323 (2018).
[Crossref]

Yin, H.

Y. Dou, Q. Yuan, Z. Gao, H. Yin, L. Chen, Y. Yao, and J. Cheng, “Partial null astigmatism-compensated interferometry for a concave freeform Zernike mirror,” J. Opt. 20(6), 065702 (2018).
[Crossref]

Yiying Zhuang, Y. Z.

Yu, B.

L. Zhang, S. Zhou, D. Li, Y. Liu, T. He, B. Yu, and J. Li, “Pure adaptive interferometer for free form surfaces metrology,” Opt. Express 26(7), 7888–7898 (2018).
[Crossref] [PubMed]

L. Zhang, D. Li, Y. Liu, Y. Bai, J. Li, and B. Yu, “Flexible interferometry for optical aspheric and free form surfaces,” Opt. Rev. 24(6), 677–685 (2017).
[Crossref]

Yu, Y.

Yuan, Q.

Y. Dou, Q. Yuan, Z. Gao, H. Yin, L. Chen, Y. Yao, and J. Cheng, “Partial null astigmatism-compensated interferometry for a concave freeform Zernike mirror,” J. Opt. 20(6), 065702 (2018).
[Crossref]

Zhai, D.

S. Xue, S. Chen, Z. Fan, and D. Zhai, “Adaptive wavefront interferometry for unknown free-form surfaces,” Opt. Express 26(17), 21910–21928 (2018).
[Crossref] [PubMed]

S. Xue, S. Chen, D. Zhai, and F. Shi, “Quasi-absolute surface figure test with two orthogonal transverse spatial shifts,” Opt. Commun. 389, 133–143 (2017).
[Crossref]

Zhang, H.

Z. Zhao, Z. Xiao, Y. Zhuang, H. Zhang, and H. Zhao, “An interferometric method for local phase modulation calibration of LC-SLM using self-generated phase grating,” Rev. Sci. Instrum. 89(8), 083116 (2018).
[Crossref] [PubMed]

Zhang, L.

L. Zhang, S. Zhou, D. Li, Y. Liu, T. He, B. Yu, and J. Li, “Pure adaptive interferometer for free form surfaces metrology,” Opt. Express 26(7), 7888–7898 (2018).
[Crossref] [PubMed]

L. Zhang, D. Li, Y. Liu, Y. Bai, J. Li, and B. Yu, “Flexible interferometry for optical aspheric and free form surfaces,” Opt. Rev. 24(6), 677–685 (2017).
[Crossref]

Zhao, H.

Z. Zhao, Z. Xiao, Y. Zhuang, H. Zhang, and H. Zhao, “An interferometric method for local phase modulation calibration of LC-SLM using self-generated phase grating,” Rev. Sci. Instrum. 89(8), 083116 (2018).
[Crossref] [PubMed]

Zhao, W.

Zhao, Z.

Z. Zhao, Z. Xiao, Y. Zhuang, H. Zhang, and H. Zhao, “An interferometric method for local phase modulation calibration of LC-SLM using self-generated phase grating,” Rev. Sci. Instrum. 89(8), 083116 (2018).
[Crossref] [PubMed]

Zhaoxian Xiao, Z. X.

Zhong, J.

Zhou, S.

Zhuang, Y.

Z. Zhao, Z. Xiao, Y. Zhuang, H. Zhang, and H. Zhao, “An interferometric method for local phase modulation calibration of LC-SLM using self-generated phase grating,” Rev. Sci. Instrum. 89(8), 083116 (2018).
[Crossref] [PubMed]

Zhuo, Y.

Zixin Zhao, Z. Z.

Appl. Opt. (5)

Appl. Sci. (Basel) (1)

H. Chen, J. Yang, H. Yen, Z. Hsu, Y. Huang, and S. Wu, “Pursuing high quality phase-only liquid crystal on silicon (LCoS) devices,” Appl. Sci. (Basel) 8(11), 2323 (2018).
[Crossref]

Chin. Opt. Lett. (2)

CIRP Ann. (1)

M. Tricard, A. Kulawiec, M. Bauer, G. DeVries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

J. Opt. (1)

Y. Dou, Q. Yuan, Z. Gao, H. Yin, L. Chen, Y. Yao, and J. Cheng, “Partial null astigmatism-compensated interferometry for a concave freeform Zernike mirror,” J. Opt. 20(6), 065702 (2018).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Meas. (1)

S. Xue, S. Chen, Y. Tian, J. Lu, and H. Hu, “Verification and in situ calibration of large-aperture null correctors for convex aspheric mirrors,” Meas. 106, 79–87 (2017).
[Crossref]

Opt. Commun. (1)

S. Xue, S. Chen, D. Zhai, and F. Shi, “Quasi-absolute surface figure test with two orthogonal transverse spatial shifts,” Opt. Commun. 389, 133–143 (2017).
[Crossref]

Opt. Express (10)

J. Peng, D. Chen, H. Guo, J. Zhong, and Y. Yu, “Variable optical null based on a yawing CGH for measuring steep acylindrical surface,” Opt. Express 26(16), 20306–20318 (2018).
[Crossref] [PubMed]

S. Xue, S. Chen, and G. Tie, “Near-null interferometry using an aspheric null lens generating a broad range of variable spherical aberration for flexible test of aspheres,” Opt. Express 26(24), 31172–31189 (2018).
[Crossref] [PubMed]

L. Zhang, S. Zhou, D. Li, Y. Liu, T. He, B. Yu, and J. Li, “Pure adaptive interferometer for free form surfaces metrology,” Opt. Express 26(7), 7888–7898 (2018).
[Crossref] [PubMed]

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J. Xue, L. Huang, B. Gao, K. Kaznatcheev, and M. Idir, “One-dimensional stitching interferometry assisted by a triple-beam interferometer,” Opt. Express 25(8), 9393–9405 (2017).
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S. Chen, S. Xue, Y. Dai, and S. Li, “Subaperture stitching test of large steep convex spheres,” Opt. Express 23(22), 29047–29058 (2015).
[Crossref] [PubMed]

D. Liu, Y. Yang, C. Tian, Y. Luo, and L. Wang, “Practical methods for retrace error correction in nonnull aspheric testing,” Opt. Express 17(9), 7025–7035 (2009).
[Crossref] [PubMed]

S. Xue, S. Chen, Z. Fan, and D. Zhai, “Adaptive wavefront interferometry for unknown free-form surfaces,” Opt. Express 26(17), 21910–21928 (2018).
[Crossref] [PubMed]

J. Kacperski and M. Kujawinska, “Active, LCoS based laser interferometer for microelements studies,” Opt. Express 14(21), 9664–9678 (2006).
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Opt. Laser Technol. (1)

S. Chen, S. Xue, Y. Dai, and S. Li, “Subaperture stitching test of convex aspheres by using the reconfigurable optical null,” Opt. Laser Technol. 91, 175–184 (2017).
[Crossref]

Opt. Lett. (5)

Opt. Rev. (1)

L. Zhang, D. Li, Y. Liu, Y. Bai, J. Li, and B. Yu, “Flexible interferometry for optical aspheric and free form surfaces,” Opt. Rev. 24(6), 677–685 (2017).
[Crossref]

Proc. SPIE (1)

M. A. Golub, I. N. Sisakian, and V. A. Soifer, “Phase quantization and discretization in diffractive optics,” Proc. SPIE 1334, 188–199 (1990).
[Crossref]

Rev. Sci. Instrum. (1)

Z. Zhao, Z. Xiao, Y. Zhuang, H. Zhang, and H. Zhao, “An interferometric method for local phase modulation calibration of LC-SLM using self-generated phase grating,” Rev. Sci. Instrum. 89(8), 083116 (2018).
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Figures (14)

Fig. 1
Fig. 1 Principle of using LC-SLMs as ICGHs. Recording (a), and playback (b).
Fig. 2
Fig. 2 The experimental apparatus (a), and schematic of the experimental set up (b) for experimentally investigating the production of Z2 and Z3 terms using LC-SLM (LC 2012) in a collimated beam.
Fig. 3
Fig. 3 The experimentally obtained accuracy for generating different amounts of Z2/Z3 using LC 2012.
Fig. 4
Fig. 4 The test systems of using LC-SLMs as ICGHs in collimated beams (a), and divergent beams (b) to perform dynamic null tests for aspheric and free-form surfaces.
Fig. 5
Fig. 5 Departure (a), and the departure after astigmatism removed (b) of 1# surface.
Fig. 6
Fig. 6 The designed test system optical layout (a), null phase after tilt removed (b), and test system theoretical residual wavefront error (c) for testing 1# surface. The designed LC-SLM alignment system optical layout (d), null phase after tilt removed (e), and theoretical residual wavefront error (f) for aligning LC-SLM.
Fig. 7
Fig. 7 Experimental apparatus (a), the measured phase (b), and fringes (c) of aligning the LC-SLM for 1# surface.
Fig. 8
Fig. 8 Experimental apparatus of utilizing LC-SLM for testing 1# surface.
Fig. 9
Fig. 9 Surface figure (a), and fringes (b) of 1# surface tested by utilizing the LC-SLM. Fringes (c), and test result (d) of the non-null test by Verfire AsphereTM. Reconstructed surface figure (e) of the non-null test. (f) is PPD map between (a) and (e).
Fig. 10
Fig. 10 Surface sag (a), surface sag after power removed (b), and fringes of testing 2# surface by an interferometer with a TF (c).
Fig. 11
Fig. 11 The experimental apparatus for testing 2# surface by LC-SLM.
Fig. 12
Fig. 12 Surface figure test result (a), and sparse fringes (b) of 2# surface by using the LC-SLM.
Fig. 13
Fig. 13 Experimental apparatus of utilizing LuphoScan 260 for testing 2# surface.
Fig. 14
Fig. 14 Surface figure test result obtained by LuphoScan (a), and the PPD map between the two tests results (b).

Tables (1)

Tables Icon

Table 1 The theoretically calculated maximum amplitudes of adaptable Zernike modes for LC 2012 when Q = 8

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

H = | U r + U o | 2 = | U r | 2 + | U o | 2 + U r * U o + U r U o * .
U r = exp { i [ 2 π λ ( y sin θ + z cos θ ) ] } ,
U o = exp { i [ 2 π z λ + φ o ( x , y ) ] } ,
H = [ 0.5 { 255 + 255 cos 2 π λ n sin θ φ o ( m , n ) } ] ,
ε = λ 2 3 Q .
z = c x x 2 + c y y 2 1 + 1 ( 1 + k x ) c x 2 x 2 ( 1 + k y ) c y 2 y 2 + j = 1 n A j Z j ,
z = c ρ 2 1 + 1 ( 1 + k ) c 2 ρ 2 + j = 1 n C j Z j ,

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