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

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References

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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)

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]

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]

2015 (2)

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]

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]

2014 (2)

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]

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]

2011 (2)

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]

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]

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)

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]

2007 (1)

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]

2006 (1)

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

2005 (1)

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]

1998 (1)

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]

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)

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

1970 (1)

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

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.

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]

Booth, M. J.

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]

Boruah, B. R.

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]

Cao, Z.

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]

Cashmore, M. T.

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]

Chen, D.

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]

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.

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]

Chen, Q.

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]

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.

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]

Cheng, H.

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]

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.

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]

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.

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]

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.

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]

Fan, Z.

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]

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.

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]

Fuerschbach, K.

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]

Gao, B.

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]

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.

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]

Guo, H.

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]

Hall, S. R.

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]

Hangying Zhang, H. Z.

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]

Hao, Q.

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]

He, T.

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]

He, Y.

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]

Hehui Geng, H. G.

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]

Hilbert, R. S.

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

Hong Zhao, H. Z.

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]

Hou, X.

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]

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]

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.

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]

Hu, Y.

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]

Huang, L.

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]

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]

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]

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.

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]

Jingsong Li, J. L.

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]

Kacperski, J.

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

Kaznatcheev, K.

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]

Kim, D. W.

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]

Kujawinska, M.

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

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.

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]

Li, D.

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]

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]

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.

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]

Liang, R.

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]

Liu, C.

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]

Liu, D.

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]

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]

Liu, Y.

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]

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]

Love, G. D.

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]

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]

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.

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]

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.

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]

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.

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]

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]

Noll, R.

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

Peng, J.

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]

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]

Peng, X.

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]

Qin, S.

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]

Rimmer, M. P.

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

Rolland, J. P.

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]

Sheng Zhou, S. Z.

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]

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.

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]

Tian, C.

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]

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]

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.

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]

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.

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]

Wang, Q.

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]

Wang, S.

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]

Wei, T.

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]

Wilson, T.

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]

Wu, F.

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]

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.

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]

Xuan, L.

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]

Xue, J.

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]

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.

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]

Yang, Y.

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]

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]

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.

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]

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.

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]

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]

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.

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]

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]

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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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