Open Access
Add to BibSonomy
Abstract
The most widespread verification method for optical elements is interferometry but, in the case of freeform surfaces, a strong deviation of the slope along the surface can create areas in which the fringe density is too high for the interferometer to resolve them. The most desirable solution is to create a null or near null interferogram introducing compensating elements like a spatial light modulator (SLM) that provides the flexibility to accommodate the measurement of a wide range of free-form surfaces. This paper shows the process for a convex freeform mirror metrology. The method consists of inserting the SLM in the optical path to compensate the freeform component of the surface to be verified and to generate a null of aberrations in the interferometer. The system is previously modelled in an optical design software to calculate the required phase to be introduced in the SLM to generate the null. The arrangement of the SLM makes possible to keep its position fixed and use the same setup to measure a wide range of freeform surfaces, limited by the dynamic range of the SLM. For each specific surface, it is necessary to introduce suitable elements to compensate the base surface, reserving the SLM for the freeform component compensation. The method is illustrated with the verification of a convex freeform mirror whose freeform component is described by the astigmatism Zernike polynomial Z5.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Freeform optics involves the breaking of the rotational symmetry in at least one of the surfaces that constitute an optical system. It generates a new aberrations field dependence in the image plane, and when the surface within the system is properly selected, it may produce the appearance of aberration nodes at field positions far from the optical axis [1,2]. This results in design solutions with a smaller number of elements and thus less mass and volume while increasing the field of view and the performance [3]. Some example applications include wide field of view telescopes with high optical quality [4,5], compact spectrometers or hyperspectral imagers with freeform surfaces [6–8] or head mounted displays for augmented reality [9]. The use of freeform optics represents great advantages during the design of optical systems. However, the introduction of freeform surfaces makes more complex the manufacturing process and most of the times requires customized techniques for the stand-alone testing of the freeform elements.
In both the manufacturing and the verification processes, the complexity increases as the local deviations of the slope along the surface increase and also if it does so in a non-continuous manner. Regarding verification methods, there are many options available, being non-contact measurements the most desirable. The most widespread non-contact methods are based on an optical probe, which provides a versatile method with nanometre accuracy, or on interferometry [10]. Interferometry provides non-contact and single shot measurements of the entire surface with an accuracy on the nanometre scale. The change in local deviations of the slope generates the increase of the interferogram fringe density making in many cases necessary to introduce compensating elements to reduce this fringe density so that the interferometer can resolve the wavefront error. In the case of freeform surfaces, this compensation was stated to be done by computer-generated holograms (CGHs) that provided high measurement accuracy. CGHs were introduced for interferometric testing of aspheric surfaces in the 70s [11,12] and they have been widely used for null interferometric testing of freeform surfaces [13]. However, CGHs are elements made specifically for the verification of a particular surface or element, i.e., they do not provide any flexibility in measurement. This flexibility is desirable for in-process measurements during manufacturing and for developing a universal methodology for freeform testing.
In search of such flexibility, adaptive optics elements such as deformable mirrors (DM) or spatial light modulators (SLM) have been explored and introduced in testing freeform optics. DMs have been used for generating a dynamic null in the measurement of freeform optical surfaces [14–16], both by themselves and in combination with other null lenses [17,18]. The main drawback of DMs is that their accuracy and dynamic range are limited by the number [19] and stroke [20] of the actuators, respectively. Regarding SLMs, they have higher accuracy than DMs due to the larger number of elements and the ability to generate phases larger than $2\pi$ radians by phase wrapping, as has been demonstrated in the correction of the human eye aberrations [21] or those induced by the atmospheric turbulences [22]. This makes SLMs most suitable for wavefront compensation when the freeform surface needs to be verified by null interferometry [23–26]. A method for freeform surface verification using a SLM was proposed [27] and implemeted [28] for the verification of a concave freeform mirror.
In this paper, it is shown a methodology for the verification of a wide number of freeform surfaces only limited by the dynamic range of the SLM. The proposed methodology is illustrated with the verification of a convex freeform mirror that has a hyperbolic base surface and a freeform profile defined by the sum of two Zernike polynomial components, those corresponding to $Z_{9}$ (spherical) and $Z_{5}$ (astigmatism), which forms part of a two-mirror telescope. The purpose of the freeform component is to generate a binodal astigmatism pattern to match the astigmatism nodes to the telescope working fields where the detectors need to be placed.
2. Methodology
In this paper, an adaptive method for freeform surface verification is presented. The method is based on converting the wavefront generated by the freeform surface in a plane wavefront. We have started modelling the testing setup with an optical design tool (Code V, Synopsis) and then created the laboratory setup and carried out the measurements. Results will be then analysed by comparing the measurement with the design prediction.
The method is based on the concept shown in Fig. 1 [27]. It uses the collimated beam from an interferometer that goes through a polarizer to select the working polarization and reaches a SLM slightly tilted (9.5 degrees, below 10 degrees as suggested in [29]) and reflects the beam in direction to the freeform surface. The freeform element closes the double-pass cavity so the beam goes back to the interferometer.
The intention is to create a fixed setup that allows to easily measure a wide range of freeform surface. The main issue for that purpose is not the freeform component of the surface, the main issue is that the setup cannot deal with any kind of base surface (we can always consider the freeform surface as the sum of a base surface and the freeform profile itself). The element could be refractive or reflective, the base surface could be a plane, spheric, conic or even aspherical and, additionally, it could be convex or concave. Thus, the first part of the setup (from the interferometer to the SLM) is fixed while the rest of the setup will be adapted to each surface. Once the polarizer and the SLM are in place, different additional elements will be required to allow the measurement. The set interferometer-polarizer-SLM works just as an interferometer for measuring non-freeform elements, additional elements are used to compensate for the base surface or for closing the double-pass cavity. The only difference is that the SLM provides the option of compensating for the freeform component too.
Once the polarizer and the SLM are in place, before introducing the freeform surface, the double-pass cavity is closed using a flat mirror so the reference wavefront error of the cavity is measured. After measuring this reference, the setup is ready and the freeform surface can be inserted.
The next step is to insert the freeform surface into the assembly with the necessary elements to compensate for the base surface and isolating the freeform component. If the base surface is flat, additional elements are not needed and the freeform surface is placed in the setup by itself, unless the magnification between the surface and the SLM is not 1:1 so a beam expander would be required. The SLM will compensate for the freeform component and a null will be measured with the interferometer. The introduction of the proper phase into the SLM starts with the design of the setup in Code V assuming the SLM to be a flat mirror. Then, the SLM is described as a Fringe Zernike polynomial surface in Code V and the corresponding Zernike coefficients set as variables and optimized up to generate a flat wavefront at the exit pupil of the system. The optimized phase generated by the optimized Zernike coefficients obtained in Code V is then introduced into the real SLM, in the laboratory, and a measurement is made without modifying any SLM parameter: a flat wavefront is expected.
In practice, the surface used to demonstrate the validity of the above method is the secondary mirror of a Cassegrain telescope, a convex freeform mirror whose base surface is a conic described by a radius of curvature $R = 20.8955$ $mm$ and a conic constant $K = -2.1118$. Its freeform component is described by the sum of the Zernike polynomials $Z_{5}$ and $Z_{9}$ in its Fringe base with their corresponding coefficients $C_{5}^{FF}= 3.3422$ $\mu m$ and $C_{9}^{FF}= 0.2445$ $\mu m$ (FF refers to freeform). Strictly speaking, the only freeform contribution is described by $Z_{5}$, since $Z_{9}$ is not. The semi-aperture of this mirror, which is also the normalisation radius $r_{norm}$ for the Zernike polynomials, is $9.1816$ $mm$. The sagitta of the mirror surface can be obtained as follows:
Fig. 2. Freeform mirror sagitta. (a) Full sagitta. (b) Zernike polynomial $Z_{9}$ sagitta contribution. (c) Zernike polynomial $Z_{5}$ sagitta contribution.
For this mirror, in order to compensate the wavefront error induced by the base surface we built a two-element optical subsystem integrated by a commercial aspheric lens ($0.33$ numerical aperture, $25$ $mm$ aperture, 49104 Edmund Scientific) and a spherical mirror specifically designed for this application ($106$ $mm$ diameter and a radius of curvature of $65$ $mm$). This mirror let us closing the double-pass interferometric cavity. Its large diameter is due to the convex shape of mirror under test. With these elements, the resulting design of the setup is shown in Fig. 3. The collimated beam coming from the SLM reaches the aspheric lens, is reflected on the freeform mirror and the double-pass cavity is closed with the spherical mirror. As can be seen, the spherical mirror has a central hole that generates a central obscuration without data.
3. Set-up modelling
So far, all the elements, components and procedure to generate a null interferogram have been identified. The next step is to create the optical model and estimate the tolerances that ensure compliance with specifications to later compare with the results obtained in the laboratory.
As mentioned, the SLM is introduced in the design as a flat mirror and described as a Code V Fringe Zernike surface with the coefficients set as variables. Figure 4(a) shows the SLM phase obtained after the optimization process. It is basically defined by the astigmatism coefficient $C_{5}^{SLM} = 2.9203$ $\mu m$ and the spherical aberration coefficient $C_{9}^{SLM} = 0.3783$ $\mu m$. The remaining coefficients are practically negligible. The wavefront error at the exit pupil of the system after the SLM optimisation is shown in Fig. 4(b), it has $RMS = 0.0372$ $\lambda$. Further optimisation could have been carried out until a completely flat wavefront was achieved (much lower RMS), but it was decided to stop here because the RMS value is already below the error that can be measured in the lab and this avoids introducing higher order Zernike polynomials into the SLM.
Fig. 4. (a) Phase on the SLM obtained through optimization process. (b) Wavefront at the exit pupil of the set up once the modulator phase has been optimized.
Thus, the design has provided the phase to be introduce in the SLM in the laboratory (Fig. 4(a)) and the wavefront (Fig. 4(b)) that is expected to be obtained at the laboratory. However, the expected wavefront needs to take into account the tolerances on the freeform mirror and on the different elements composing the system so, a tolerance analysis of this setup is needed. The tolerances considered are those defined by the limits and accuracy of the mechanical adjustments which are, in relation to misalignments, decentres of the lens and the spherical mirror ($20$ $\mu m$), their movements along the optical axis ($20$ $\mu m$) and their tilts ($0.002$ $radians$), and in relation to fabrication errors, the sagitta error of the lens ($7.5$ $\mu m$), its central thickness ($0.1$ $mm$) and the radius of curvature of the spherical mirror ($0.1$ $mm$) and the values were provided by the manufacturer. Tilts and decentres of the freeform mirror are used as compensators. Additionally, tolerances in decentres ($20$ $\mu m$) and in the tilt angle ($0.01$ $radians$) of the SLM are considered. With this analysis, we obtain that the RMS of the wavefront error at the exit of the system goes from $0.0372$ $\lambda$ in the nominal design to $0.2892$ $\lambda$ with a probability of $97.7\%$ (according to a TOR analysis in Code V). Here, a significant increase of the RMS is observed. In any case, it is worth noting that the RMS that would be obtained with this set up in the case of having an inactive SLM is $7.9552$ $\lambda$, so the SLM provides a large correction capability, even after the tolerance analysis.
4. Laboratory setup
Once the optical design and the tolerance analysis have been performed, the laboratory setup can be implemented. In our case, we used a Fizeau interferometer (ZYGO Phase Shifting GPI-XP). This equipment provides a high-quality 4-inch aperture collimated beam.
The SLM is a phase modulator based on liquid crystals working in reflection (Hamamatsu model X8267 with 1024x768 pixels, $19x26$ $\mu m$ pixel pitch and an effective area of $20x20$ $mm$). To introduce the required phase in the SLM, we used a custom software coded in MATLAB (Mathworks) to generate the phase from the Zernike coefficients, translate it to grey levels and send it to the SLM. In addition, a calibration of the SLM was carried out for a working wavelength of $632.8$ $nm$. At this wavelength, a modulation greater than $2\pi$ is generated, so the grey level at which the modulation reaches $2\pi$ was measured and the dynamic range reduced to adjust it to that level. The calibration was done as proposed in [30]. The maximum phase deviation that the SLM can provide is directly related to the number of pixels. According to Nyquist criterion, the maximum wavefront (peak-to-valley) that the SLM can generate is $W_{max}=N\frac {\lambda }{2}$, where $N$ is the number of pixels and $\lambda$ is the working wavelength ($\lambda =632.8$ $nm$). The maximum slope that the SLM can provide is $\theta _{max}=\arctan {\frac {\lambda }{2p}}$, where $p$ is the pixel pitch. With the SLM used here, the values are $W_{max}=243$ $\mu m$ and $\theta _{max}=0.696^{\circ }$. These values could be increased just by using a SLM with a higher number of pixels.
First of all, the linear polariser is inserted in the optical path to select the working polarisation. Then, the SLM is introduced. Before starting, the SLM shall be aligned to the interferometer beam to perform the wavelength calibration. Then, the SLM is tilted to the working angle and the interferometric cavity is closed with a high-quality flat mirror to measure the reference of the cavity. Once this reference is measured, the setup is ready to introduce the freeform surface with the aspheric lens and the spherical mirror.
The aspheric lens is placed and the cavity is closed with the 10 mm spherical reflector. This configuration, as it is shown in Fig. 5, will be our reference because it takes into account the wavefront error introduced by the polarizer and the SLM when inactive at the working angle, but additionally it also considers the error induced by the aspheric lens.
Once the reference has been measured, the freeform mirror is inserted into the assembly and the cavity is closed with the spherical mirror, as shown in the schematic in Fig. 3. First, the spherical mirror is placed in its nominal position, on a 5-axis stage, and the cavity is closed with a high-quality plane mirror. As expected, the light beam does not cover the full aperture of the spherical mirror, but previous CodeV analyses have shown that this configuration provides high quality when the system is well aligned and allows us to achieve correction in tip&tilt and transverse displacement of the spherical mirror (Fig. 6).
Second, we introduce the freeform mirror and adjust its tilt, offset and axial position with respect to the spherical mirror minimizing the coma and sperical aberration observed in the interferometer. Finally, the unit including the spherical mirror and the freeform mirror are moved axially together to align their relative position with the aspheric lens minimizing defocus aberration in the interferometer. As can be seen, the alignment is an interactive process that is performed with the help of CodeV. The laboratory setup is shown in Fig. 7.
Fig. 7. Laboratory set up with the SLM in measurement configuration corresponding to the layout shown in Fig. 3.
5. Results
Starting from the configuration shown in Fig. 5, the wavefront map corresponding to the measurement of reference is shown in Fig. 8(a). Once measured, the freeform and the spherical mirrors were introduced in the set up as shown in Fig. 7. In this configuration, the phase that was obtained by the optimization process is introduced in the modulator (Fig. 4(a)). Theoretically, by doing this, the aberrations of the assembly should be corrected and a null wavefront should be obtained. The resulting wavefront is shown in Fig. 8(b), corresponding to an $RMS = 0.27$ $\lambda$. From this measurement, the reference (Fig. 8(a)) is subtracted resulting in $RMS = 0.13$ $\lambda$. This value is within the tolerances range of the setup design in which the RMS could reach up to $0.2892$ $\lambda$. The measurement uncertainty is mainly defined by three componets: the pixelation of the SLM, the quantization of the phase on the SLM to 256 grey levels and the turbulences due to the long optical path. The contribution of the pixelation and quantization are analysed and quantified during the SLM calibration while the turbulence effects are quantified by the standard deviation of a large set of measurements. In our case, the resulting test accuracy is dominated by turbulences being the others almost negligible, so the final uncertainty is equivalent to $0.09$ $\lambda$ RMS.
It is worth noting that if, besides analysing the RMS, the Zernike coefficients describing the wavefront are evaluated, it is seen that astigmatism, spherical aberration and second order astigmatism are very well corrected, values obtained for these aberrations are very close to zero, $C_{5}^{wfe}= 0.02 \pm 0.01$ $\lambda$, $C_{9}^{wfe} = 0.03 \pm 0.02$ $\lambda$ and $C_{12}^{wfe} = 0.03 \pm 0.02$ $\lambda$. The procedure demonstrates that it has been possible to verify the freeform component of the mirror within the measurement error.
6. Discussion
So far, the work shows the methodology followed for the freeform optical surface testing by interferometric techniques and its adaptation to convex surface testing. This methodology can be adapted to the verification of a wide range of freeform components due to the flexibility that the SLM provides. The verification of the freeform surface with this method must always be done by comparing the measurements obtained in the laboratory with the analysis of the system conducted in an optical design tool. It is essential to use an optical model since it is necessary to optimize the SLM phase to compensate for the freeform contribution.
The main advantage of this method is that it constitutes a generic method that allows the verification of a much wider range of freeform surfaces. However, it leads to long optical paths involving a large contribution of turbulences and resulting in a test accuracy, in our case, of $0.09$ $\lambda$.
The results obtained by this method allow to confirm that the freeform contribution is within the tolerance range because the tolerances on the surface have been transferred to the wavefront by the tolerance analysis. In addition, the reverse process can be done to find the deviation of the freeform contribution from the nominal value. For this purpose, the wavefront error measured in the laboratory has been introduced with the inverted phase in the exit pupil of the optical model to obtain the residual due to the difference between the Code V simulation and the measured data (residual RMS of $0.1709$ $\lambda$). This residual appears with two contributions. One is due to misalignments or surface errors and the other is due to differences in the coefficients describing the freeform component with respect to the nominal values. To identify the former, an optimization is performed setting the possible misalignments as variable parameters to eliminate its contribution to the residual (resulting residual RMS of $0.1203$ $\lambda$). For the latter, a second optimisation is performed setting variable now the freeform coefficients to know how much of the residual can be absorbed by the coefficients tolerance (resulting residual RMS of $0.1107$ $\lambda$). At this point, the final residuals include mainly the higher order contribution and the measurement noise. The values obtained for the coefficients are $C_{5}^{FF}= 3.3142$ $\mu m$ and $C_{9}^{FF}= 0.2544$ $\mu m$. The nominal values for the coefficients and their tolerances are $C_{5}^{FF}= 3.34 \pm 0.04$ $\mu m$ and $C_{9}^{FF}= 0.24 \pm 0.02$ $\mu m$. These results are summarized in Table 1. As expected, values are within the specified range of tolerances demonstrating that the method is suitable for the verification of the freeform mirror.
Table 1. Summary of the process to derive the freeform coefficients from the residual RMS ($\lambda =632.8$$nm$).
Additionally, to validate the method, a cross test was performed. For this telescope, during the design process, the deviation of the local slopes along the whole surface was limited by using Q-polynomials for describing the freeform component during the optimization of the optical system. For that reason, the freeform component of this mirror could be measured directly by interferometry (the interferometer resolves the fringe pattern) and this measurement was used to validate the SLM method. For the measurement, the aspherical lens and the spherical mirror remain in place to compensate the base surface and the system is directly placed in front of the interferometer, as shown in Figs. 9(a),(b). The measured wavefront error (Fig. 9(c)) was compared to the one predicted by Code V (Fig. 9(c)) and, again, a tolerance analysis was performed to show that the freeform mirror surface was within tolerance range. In this case, the uncertainty in the measurement is $0.02$ $\lambda$, due to the much shorter interferometric cavity and the absence of the SLM.
Fig. 9. (a) Cross test Code V layout. (b) Laboratory setup. (c) Code V wavefront error. (d) Measured wavefront error. The astigmatism Zernike coefficient for both wavefront is shown along with the RMS.
7. Conclusion
It has been presented how the verification capability of freeform convex surfaces has been developed. Introducing the SLM allows to measure any freeform surface within the limits of the dynamic range of the SLM. Having the capacity of verifying a wide range of freeform surfaces by interferometry is very useful as it represents a high-precision, non-contact, single-shot verification method. Therefore, the capabilities presented here are important to work regularly with systems containing such surfaces.
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Theory of aberration fields for general optical systems with freeform surfaces,” Opt. Express 22(22), 26585–26606 (2014). [CrossRef]
2. E. M. Schiesser, A. Bauer, and J. P. Rolland, “Estimating field-dependent nodal aberration theory coefficients from zernike full-field displays by utilizing eighth-order astigmatism,” J. Opt. Soc. Am. A 36(12), 2115–2128 (2019). [CrossRef]
3. Q. Meng, H. Wang, W. Liang, Z. Yan, and B. Wang, “Design of off-axis three-mirror systems with ultrawide field of view based on an expansion process of surface freeform and field of view,” Appl. Opt. 58(3), 609–615 (2019). [CrossRef]
4. L. Schifano, M. Vervaeke, D. Rosseel, J. Verbaenen, H. Thienpont, S. Dewitte, F. Berghmans, and L. Smeesters, “Freeform wide field-of-view spaceborne imaging telescope: From design to demonstrator,” Sensors 22(21), 8233 (2022). [CrossRef]
5. J. Zhang, Y. Zheng, C. Lin, Y. Han, and Y. Shi, “Large numerical aperture off-axis reflective telescope design with a freeform mirror based on aperture expansion strategy,” Appl. Opt. 62(6), 1510–1520 (2023). [CrossRef]
6. J. Zhang, C. Lin, Z. Ji, H. Wu, C. Li, B. Du, and Y. Zheng, “Design of a compact hyperspectral imaging spectrometer with a freeform surface based on anastigmatism,” Appl. Opt. 59(6), 1715–1725 (2020). [CrossRef]
7. Y. Liu, A. Bauer, T. Viard, and J. P. Rolland, “Freeform hyperspectral imager design in a cubesat format,” Opt. Express 29(22), 35915–35928 (2021). [CrossRef]
8. J. Zhang, Y. Zheng, C. Lin, Z. Ji, Y. Han, and Y. Shi, “Analysis and design of a wide-field and large-numerical-aperture compact imaging spectrometer with a freeform surface,” Appl. Opt. 61(33), 10021–10031 (2022). [CrossRef]
9. A. Bauer and J. P. Rolland, “Design of a freeform electronic viewfinder coupled to aberration fields of freeform optics,” Opt. Express 23(22), 28141–28153 (2015). [CrossRef]
10. S. Chen, S. Xue, D. Zhai, and G. Tie, “Measurement of freeform optical surfaces: trade-off between accuracy and dynamic range,” Laser Photonics Rev. 14(5), 1900365 (2020). [CrossRef]
11. A. J. MacGovern and J. C. Wyant, “Computer generated holograms for testing optical elements,” Appl. Opt. 10(3), 619–624 (1971). [CrossRef]
12. J. Wyant and V. Bennett, “Using computer generated holograms to test aspheric wavefronts,” Appl. Opt. 11(12), 2833–2839 (1972). [CrossRef]
13. P. Su, G. Kang, Q. Tan, and G. Jin, “Estimation and optimization of computer-generated hologram in null test of freeform surface,” Chin. Opt. Lett. 7(12), 1097–1100 (2009). [CrossRef]
14. 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]
15. L. Zhang, C. Li, X. Huang, Y. Zhang, S. Zhou, J. Li, and B. Yu, “Compact adaptive interferometer for unknown freeform surfaces with large departure,” Opt. Express 28(2), 1897–1913 (2020). [CrossRef]
16. L. Zhang and Y. Zhang, “Freeform surface interferometry with an adaptive ring-cavity compensator,” Surf. Topogr.: Metrol. Prop. 8, 025036 (2020). [CrossRef]
17. 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]
18. 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]
19. 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]
20. E. Dalimier and C. Dainty, “Comparative analysis of deformable mirrors for ocular adaptive optics,” Opt. Express 13(11), 4275–4285 (2005). [CrossRef]
21. P. M. Prieto, E. J. Fernández, S. Manzanera, and P. Artal, “Adaptive optics with a programmable phase modulator: applications in the human eye,” Opt. Express 12(17), 4059–4071 (2004). [CrossRef]
22. C. Rickenstorff, J. A. Rodrigo, and T. Alieva, “Programmable simulator for beam propagation in turbulent atmosphere,” Opt. Express 24(9), 10000–10012 (2016). [CrossRef]
23. 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]
24. S. Xue, S. Chen, G. Tie, and Y. Tian, “Adaptive null interferometric test using spatial light modulator for free-form surfaces,” Opt. Express 27(6), 8414–8428 (2019). [CrossRef]
25. S. Xue, S. Chen, G. Tie, Y. Tian, H. Hu, F. Shi, X. Peng, and X. Xiao, “Flexible interferometric null testing for concave free-form surfaces using a hybrid refractive and diffractive variable null,” Opt. Lett. 44(9), 2294–2297 (2019). [CrossRef]
26. S. Xue, W. Deng, and S. Chen, “Intelligence enhancement of the adaptive wavefront interferometer,” Opt. Express 27(8), 11084–11102 (2019). [CrossRef]
27. R. Chaudhuri, J. Papa, and J. P. Rolland, “System design of a single-shot reconfigurable null test using a spatial light modulator for freeform metrology,” Opt. Lett. 44(8), 2000–2003 (2019). [CrossRef]
28. R. Chaudhuri, A. Wansha, R. Porras-Aguilar, and J. P. Rolland, “Implementation of a null test for freeform optics using a high-definition spatial light modulator,” Opt. Express 30(24), 43938–43960 (2022). [CrossRef]
29. A. Lizana, N. Martín, M. Estapé, E. Fernández, I. Moreno, A. Márquez, C. Iemmi, J. Campos, and M. Yzuel, “Influence of the incident angle in the performance of liquid crystal on silicon displays,” Opt. Express 17(10), 8491–8505 (2009). [CrossRef]
30. M. Silva-López, N. Uribe-Patarroyo, and A. Álvarez-Herrero, “Advanced iterative algorithm for phase calibration of spatial light modulators integrated in optical instrumentation in a vibration environment,” Appl. Opt. 59(22), 6760–6764 (2020). [CrossRef]
