Abstract

Traditional methods for distortion measurement of large-aperture optical systems are time-consuming and ineffective because they require each field of view to be individually measured using a high-precision rotating platform. In this study, a new method that uses a phase diffractive beam splitter (DBS) is proposed to measure the distortion of optical systems, which has great potential application for the large-aperture optical system. The proposed method has a very high degree of accuracy and is extremely economical. A high-precision calibration method is proposed to measure the angular distribution of the DBS. The uncertainty analysis of the factors involved in the measurement process has been performed to highlight the low level of errors in the measurement methodology. Results show that high-precision measurements of the focal length and distortion were successfully achieved with high efficiency. The proposed method can be used for large-aperture wide-angle optical systems such as those used for aerial mapping applications.

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

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References

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2019 (1)

X. Zhang, B. Chen, F. He, K. Song, L. He, S. Liu, Q. Guo, J. Li, X. Wang, H. Zhang, H. Wang, Z. Han, L. Sun, P. Zhang, S. Dai, G. Ding, L. Chen, Z. Wang, G. Shi, X. Zhang, C. Yu, Z. Yang, P. Zhang, and J. Wang, “Wide-field auroral imager onboard the Fengyun satellite,” Light: Sci. Appl. 8(1), 47 (2019).
[Crossref]

2018 (2)

C. Guo, Z. Zhang, D. Xue, L. Li, R. Wang, X. Zhou, F. Zhang, and X. Zhang, “High-performance etching of multilevel phase-type Fresnel zone plates with large apertures,” Opt. Commun. 407, 227–233 (2018).
[Crossref]

G. Yang, L. Miao, X. Zhang, C. Sun, and Y. Qiao, “High-accuracy measurement of the focal length and distortion of optical systems based on interferometry,” Appl. Opt. 57(18), 5217–5223 (2018).
[Crossref]

2017 (2)

2015 (2)

A. Harmat, M. Trentini, and I. S. Michael, “Multi-Camera tracking and mapping for unmanned aerial vehicles in unstructured environments,” J. Intell. Robot. Syst. 78(2), 291–317 (2015).
[Crossref]

A. Miks and P. Pokorny, “Use of diffraction grating for measuring the focal length and distortion of optical systems,” Appl. Opt. 54(34), 10200–10206 (2015).
[Crossref]

2013 (1)

T. Kouyama, A. Yamazaki, M. Yamada, and T. Imamura, “A method to estimate optical distortion using planetary images,” Planet. Space Sci. 86(15), 86–90 (2013).
[Crossref]

2012 (2)

A. Miks and J. Novak, “Dependence of camera lens induced radial distortion and circle of confusion on object position,” Opt. Laser Technol. 44(4), 1043–1049 (2012).
[Crossref]

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Stack. multiscale gigapixel photography,” Nature 486(7403), 386–389 (2012).
[Crossref]

2011 (1)

2009 (1)

2008 (1)

2007 (1)

2004 (1)

M. Sezgin and S. Bülent, “Survey over image thresholding techniques and quantitative performance evaluation,” J. Electron Imaging 13(1), 146–165 (2004).
[Crossref]

2003 (1)

K. Fuse, T. Hirai, T. Ushiro, T. Okada, K. Kurisu, and K. Ebata, “Design and performance of multilevel phase fan-out diffractive optical elements for laser materials processing,” J. Laser Appl. 15(4), 246–254 (2003).
[Crossref]

2002 (1)

S. H. Baik, S. K. Park, C. J. Kim, Y. S. Seo, and Y. J. Kang, “New centroid detection algorithm for the Shack-Hartmann wavefront sensor,” Proc. SPIE 4926, 251–260 (2002).
[Crossref]

1998 (1)

T. A. Clarke and J. F. Fryer, “The development of camera calibration methods and models,” Photogramm. Rec. 16(91), 51–66 (1998).
[Crossref]

Arfaoui, A.

Baik, S. H.

S. H. Baik, S. K. Park, C. J. Kim, Y. S. Seo, and Y. J. Kang, “New centroid detection algorithm for the Shack-Hartmann wavefront sensor,” Proc. SPIE 4926, 251–260 (2002).
[Crossref]

Bauer, M.

Brady, D. J.

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Stack. multiscale gigapixel photography,” Nature 486(7403), 386–389 (2012).
[Crossref]

Bülent, S.

M. Sezgin and S. Bülent, “Survey over image thresholding techniques and quantitative performance evaluation,” J. Electron Imaging 13(1), 146–165 (2004).
[Crossref]

Chen, B.

X. Zhang, B. Chen, F. He, K. Song, L. He, S. Liu, Q. Guo, J. Li, X. Wang, H. Zhang, H. Wang, Z. Han, L. Sun, P. Zhang, S. Dai, G. Ding, L. Chen, Z. Wang, G. Shi, X. Zhang, C. Yu, Z. Yang, P. Zhang, and J. Wang, “Wide-field auroral imager onboard the Fengyun satellite,” Light: Sci. Appl. 8(1), 47 (2019).
[Crossref]

Chen, L.

X. Zhang, B. Chen, F. He, K. Song, L. He, S. Liu, Q. Guo, J. Li, X. Wang, H. Zhang, H. Wang, Z. Han, L. Sun, P. Zhang, S. Dai, G. Ding, L. Chen, Z. Wang, G. Shi, X. Zhang, C. Yu, Z. Yang, P. Zhang, and J. Wang, “Wide-field auroral imager onboard the Fengyun satellite,” Light: Sci. Appl. 8(1), 47 (2019).
[Crossref]

Clarke, T. A.

T. A. Clarke and J. F. Fryer, “The development of camera calibration methods and models,” Photogramm. Rec. 16(91), 51–66 (1998).
[Crossref]

Dai, S.

X. Zhang, B. Chen, F. He, K. Song, L. He, S. Liu, Q. Guo, J. Li, X. Wang, H. Zhang, H. Wang, Z. Han, L. Sun, P. Zhang, S. Dai, G. Ding, L. Chen, Z. Wang, G. Shi, X. Zhang, C. Yu, Z. Yang, P. Zhang, and J. Wang, “Wide-field auroral imager onboard the Fengyun satellite,” Light: Sci. Appl. 8(1), 47 (2019).
[Crossref]

Desaulniers, P.

Ding, G.

X. Zhang, B. Chen, F. He, K. Song, L. He, S. Liu, Q. Guo, J. Li, X. Wang, H. Zhang, H. Wang, Z. Han, L. Sun, P. Zhang, S. Dai, G. Ding, L. Chen, Z. Wang, G. Shi, X. Zhang, C. Yu, Z. Yang, P. Zhang, and J. Wang, “Wide-field auroral imager onboard the Fengyun satellite,” Light: Sci. Appl. 8(1), 47 (2019).
[Crossref]

Dong, M. L.

Ebata, K.

K. Fuse, T. Hirai, T. Ushiro, T. Okada, K. Kurisu, and K. Ebata, “Design and performance of multilevel phase fan-out diffractive optical elements for laser materials processing,” J. Laser Appl. 15(4), 246–254 (2003).
[Crossref]

Fang, Z.

Feller, S. D.

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Stack. multiscale gigapixel photography,” Nature 486(7403), 386–389 (2012).
[Crossref]

Fryer, J. F.

T. A. Clarke and J. F. Fryer, “The development of camera calibration methods and models,” Photogramm. Rec. 16(91), 51–66 (1998).
[Crossref]

Fuse, K.

K. Fuse, T. Hirai, T. Ushiro, T. Okada, K. Kurisu, and K. Ebata, “Design and performance of multilevel phase fan-out diffractive optical elements for laser materials processing,” J. Laser Appl. 15(4), 246–254 (2003).
[Crossref]

Gehm, M. E.

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Stack. multiscale gigapixel photography,” Nature 486(7403), 386–389 (2012).
[Crossref]

Golish, D. R.

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Stack. multiscale gigapixel photography,” Nature 486(7403), 386–389 (2012).
[Crossref]

Grießbach, D.

Guo, C.

C. Guo, Z. Zhang, D. Xue, L. Li, R. Wang, X. Zhou, F. Zhang, and X. Zhang, “High-performance etching of multilevel phase-type Fresnel zone plates with large apertures,” Opt. Commun. 407, 227–233 (2018).
[Crossref]

Z. Zhang, C. Guo, R. Wang, H. Hu, X. Zhou, T. Liu, D. Xue, X. Zhang, F. Zhang, and X. Zhang, “Hybrid-level Fresnel zone plate for diffraction efficiency enhancement,” Opt. Express 25(26), 33676–33687 (2017).
[Crossref]

Guo, Q.

X. Zhang, B. Chen, F. He, K. Song, L. He, S. Liu, Q. Guo, J. Li, X. Wang, H. Zhang, H. Wang, Z. Han, L. Sun, P. Zhang, S. Dai, G. Ding, L. Chen, Z. Wang, G. Shi, X. Zhang, C. Yu, Z. Yang, P. Zhang, and J. Wang, “Wide-field auroral imager onboard the Fengyun satellite,” Light: Sci. Appl. 8(1), 47 (2019).
[Crossref]

Han, Z.

X. Zhang, B. Chen, F. He, K. Song, L. He, S. Liu, Q. Guo, J. Li, X. Wang, H. Zhang, H. Wang, Z. Han, L. Sun, P. Zhang, S. Dai, G. Ding, L. Chen, Z. Wang, G. Shi, X. Zhang, C. Yu, Z. Yang, P. Zhang, and J. Wang, “Wide-field auroral imager onboard the Fengyun satellite,” Light: Sci. Appl. 8(1), 47 (2019).
[Crossref]

Harmat, A.

A. Harmat, M. Trentini, and I. S. Michael, “Multi-Camera tracking and mapping for unmanned aerial vehicles in unstructured environments,” J. Intell. Robot. Syst. 78(2), 291–317 (2015).
[Crossref]

He, F.

X. Zhang, B. Chen, F. He, K. Song, L. He, S. Liu, Q. Guo, J. Li, X. Wang, H. Zhang, H. Wang, Z. Han, L. Sun, P. Zhang, S. Dai, G. Ding, L. Chen, Z. Wang, G. Shi, X. Zhang, C. Yu, Z. Yang, P. Zhang, and J. Wang, “Wide-field auroral imager onboard the Fengyun satellite,” Light: Sci. Appl. 8(1), 47 (2019).
[Crossref]

He, L.

X. Zhang, B. Chen, F. He, K. Song, L. He, S. Liu, Q. Guo, J. Li, X. Wang, H. Zhang, H. Wang, Z. Han, L. Sun, P. Zhang, S. Dai, G. Ding, L. Chen, Z. Wang, G. Shi, X. Zhang, C. Yu, Z. Yang, P. Zhang, and J. Wang, “Wide-field auroral imager onboard the Fengyun satellite,” Light: Sci. Appl. 8(1), 47 (2019).
[Crossref]

Hermerschmidt, A.

Hirai, T.

K. Fuse, T. Hirai, T. Ushiro, T. Okada, K. Kurisu, and K. Ebata, “Design and performance of multilevel phase fan-out diffractive optical elements for laser materials processing,” J. Laser Appl. 15(4), 246–254 (2003).
[Crossref]

Hu, H.

Imamura, T.

T. Kouyama, A. Yamazaki, M. Yamada, and T. Imamura, “A method to estimate optical distortion using planetary images,” Planet. Space Sci. 86(15), 86–90 (2013).
[Crossref]

Kang, Y. J.

S. H. Baik, S. K. Park, C. J. Kim, Y. S. Seo, and Y. J. Kang, “New centroid detection algorithm for the Shack-Hartmann wavefront sensor,” Proc. SPIE 4926, 251–260 (2002).
[Crossref]

Kim, C. J.

S. H. Baik, S. K. Park, C. J. Kim, Y. S. Seo, and Y. J. Kang, “New centroid detection algorithm for the Shack-Hartmann wavefront sensor,” Proc. SPIE 4926, 251–260 (2002).
[Crossref]

Kittle, D. S.

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Stack. multiscale gigapixel photography,” Nature 486(7403), 386–389 (2012).
[Crossref]

Kouyama, T.

T. Kouyama, A. Yamazaki, M. Yamada, and T. Imamura, “A method to estimate optical distortion using planetary images,” Planet. Space Sci. 86(15), 86–90 (2013).
[Crossref]

Krüger, S.

Kurisu, K.

K. Fuse, T. Hirai, T. Ushiro, T. Okada, K. Kurisu, and K. Ebata, “Design and performance of multilevel phase fan-out diffractive optical elements for laser materials processing,” J. Laser Appl. 15(4), 246–254 (2003).
[Crossref]

Li, J.

X. Zhang, B. Chen, F. He, K. Song, L. He, S. Liu, Q. Guo, J. Li, X. Wang, H. Zhang, H. Wang, Z. Han, L. Sun, P. Zhang, S. Dai, G. Ding, L. Chen, Z. Wang, G. Shi, X. Zhang, C. Yu, Z. Yang, P. Zhang, and J. Wang, “Wide-field auroral imager onboard the Fengyun satellite,” Light: Sci. Appl. 8(1), 47 (2019).
[Crossref]

Li, L.

C. Guo, Z. Zhang, D. Xue, L. Li, R. Wang, X. Zhou, F. Zhang, and X. Zhang, “High-performance etching of multilevel phase-type Fresnel zone plates with large apertures,” Opt. Commun. 407, 227–233 (2018).
[Crossref]

Li, X.

Liu, S.

X. Zhang, B. Chen, F. He, K. Song, L. He, S. Liu, Q. Guo, J. Li, X. Wang, H. Zhang, H. Wang, Z. Han, L. Sun, P. Zhang, S. Dai, G. Ding, L. Chen, Z. Wang, G. Shi, X. Zhang, C. Yu, Z. Yang, P. Zhang, and J. Wang, “Wide-field auroral imager onboard the Fengyun satellite,” Light: Sci. Appl. 8(1), 47 (2019).
[Crossref]

Liu, T.

Lu, N. G.

Marks, D. L.

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Stack. multiscale gigapixel photography,” Nature 486(7403), 386–389 (2012).
[Crossref]

Miao, L.

Michael, I. S.

A. Harmat, M. Trentini, and I. S. Michael, “Multi-Camera tracking and mapping for unmanned aerial vehicles in unstructured environments,” J. Intell. Robot. Syst. 78(2), 291–317 (2015).
[Crossref]

Miks, A.

A. Miks and P. Pokorny, “Use of diffraction grating for measuring the focal length and distortion of optical systems,” Appl. Opt. 54(34), 10200–10206 (2015).
[Crossref]

A. Miks and J. Novak, “Dependence of camera lens induced radial distortion and circle of confusion on object position,” Opt. Laser Technol. 44(4), 1043–1049 (2012).
[Crossref]

Novak, J.

A. Miks and J. Novak, “Dependence of camera lens induced radial distortion and circle of confusion on object position,” Opt. Laser Technol. 44(4), 1043–1049 (2012).
[Crossref]

Okada, T.

K. Fuse, T. Hirai, T. Ushiro, T. Okada, K. Kurisu, and K. Ebata, “Design and performance of multilevel phase fan-out diffractive optical elements for laser materials processing,” J. Laser Appl. 15(4), 246–254 (2003).
[Crossref]

Park, S. K.

S. H. Baik, S. K. Park, C. J. Kim, Y. S. Seo, and Y. J. Kang, “New centroid detection algorithm for the Shack-Hartmann wavefront sensor,” Proc. SPIE 4926, 251–260 (2002).
[Crossref]

Pokorny, P.

Qiao, Y.

Scheele, M.

Schischmanow, A.

Seo, Y. S.

S. H. Baik, S. K. Park, C. J. Kim, Y. S. Seo, and Y. J. Kang, “New centroid detection algorithm for the Shack-Hartmann wavefront sensor,” Proc. SPIE 4926, 251–260 (2002).
[Crossref]

Sezgin, M.

M. Sezgin and S. Bülent, “Survey over image thresholding techniques and quantitative performance evaluation,” J. Electron Imaging 13(1), 146–165 (2004).
[Crossref]

Shi, G.

X. Zhang, B. Chen, F. He, K. Song, L. He, S. Liu, Q. Guo, J. Li, X. Wang, H. Zhang, H. Wang, Z. Han, L. Sun, P. Zhang, S. Dai, G. Ding, L. Chen, Z. Wang, G. Shi, X. Zhang, C. Yu, Z. Yang, P. Zhang, and J. Wang, “Wide-field auroral imager onboard the Fengyun satellite,” Light: Sci. Appl. 8(1), 47 (2019).
[Crossref]

Smith, W. J.

W. J. Smith, Modern optical engineering, (McGraw-Hill), Chap. 3 (2000).

Song, K.

X. Zhang, B. Chen, F. He, K. Song, L. He, S. Liu, Q. Guo, J. Li, X. Wang, H. Zhang, H. Wang, Z. Han, L. Sun, P. Zhang, S. Dai, G. Ding, L. Chen, Z. Wang, G. Shi, X. Zhang, C. Yu, Z. Yang, P. Zhang, and J. Wang, “Wide-field auroral imager onboard the Fengyun satellite,” Light: Sci. Appl. 8(1), 47 (2019).
[Crossref]

Stack, R. A.

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Stack. multiscale gigapixel photography,” Nature 486(7403), 386–389 (2012).
[Crossref]

Sun, C.

Sun, L.

X. Zhang, B. Chen, F. He, K. Song, L. He, S. Liu, Q. Guo, J. Li, X. Wang, H. Zhang, H. Wang, Z. Han, L. Sun, P. Zhang, S. Dai, G. Ding, L. Chen, Z. Wang, G. Shi, X. Zhang, C. Yu, Z. Yang, P. Zhang, and J. Wang, “Wide-field auroral imager onboard the Fengyun satellite,” Light: Sci. Appl. 8(1), 47 (2019).
[Crossref]

Sun, P.

Thibault, S.

Trentini, M.

A. Harmat, M. Trentini, and I. S. Michael, “Multi-Camera tracking and mapping for unmanned aerial vehicles in unstructured environments,” J. Intell. Robot. Syst. 78(2), 291–317 (2015).
[Crossref]

Ushiro, T.

K. Fuse, T. Hirai, T. Ushiro, T. Okada, K. Kurisu, and K. Ebata, “Design and performance of multilevel phase fan-out diffractive optical elements for laser materials processing,” J. Laser Appl. 15(4), 246–254 (2003).
[Crossref]

Vera, E. M.

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, “Stack. multiscale gigapixel photography,” Nature 486(7403), 386–389 (2012).
[Crossref]

Wang, H.

X. Zhang, B. Chen, F. He, K. Song, L. He, S. Liu, Q. Guo, J. Li, X. Wang, H. Zhang, H. Wang, Z. Han, L. Sun, P. Zhang, S. Dai, G. Ding, L. Chen, Z. Wang, G. Shi, X. Zhang, C. Yu, Z. Yang, P. Zhang, and J. Wang, “Wide-field auroral imager onboard the Fengyun satellite,” Light: Sci. Appl. 8(1), 47 (2019).
[Crossref]

Wang, J.

X. Zhang, B. Chen, F. He, K. Song, L. He, S. Liu, Q. Guo, J. Li, X. Wang, H. Zhang, H. Wang, Z. Han, L. Sun, P. Zhang, S. Dai, G. Ding, L. Chen, Z. Wang, G. Shi, X. Zhang, C. Yu, Z. Yang, P. Zhang, and J. Wang, “Wide-field auroral imager onboard the Fengyun satellite,” Light: Sci. Appl. 8(1), 47 (2019).
[Crossref]

Wang, R.

C. Guo, Z. Zhang, D. Xue, L. Li, R. Wang, X. Zhou, F. Zhang, and X. Zhang, “High-performance etching of multilevel phase-type Fresnel zone plates with large apertures,” Opt. Commun. 407, 227–233 (2018).
[Crossref]

Z. Zhang, C. Guo, R. Wang, H. Hu, X. Zhou, T. Liu, D. Xue, X. Zhang, F. Zhang, and X. Zhang, “Hybrid-level Fresnel zone plate for diffraction efficiency enhancement,” Opt. Express 25(26), 33676–33687 (2017).
[Crossref]

Wang, X.

X. Zhang, B. Chen, F. He, K. Song, L. He, S. Liu, Q. Guo, J. Li, X. Wang, H. Zhang, H. Wang, Z. Han, L. Sun, P. Zhang, S. Dai, G. Ding, L. Chen, Z. Wang, G. Shi, X. Zhang, C. Yu, Z. Yang, P. Zhang, and J. Wang, “Wide-field auroral imager onboard the Fengyun satellite,” Light: Sci. Appl. 8(1), 47 (2019).
[Crossref]

Wang, Z.

X. Zhang, B. Chen, F. He, K. Song, L. He, S. Liu, Q. Guo, J. Li, X. Wang, H. Zhang, H. Wang, Z. Han, L. Sun, P. Zhang, S. Dai, G. Ding, L. Chen, Z. Wang, G. Shi, X. Zhang, C. Yu, Z. Yang, P. Zhang, and J. Wang, “Wide-field auroral imager onboard the Fengyun satellite,” Light: Sci. Appl. 8(1), 47 (2019).
[Crossref]

Wernicke, G.

Xue, D.

C. Guo, Z. Zhang, D. Xue, L. Li, R. Wang, X. Zhou, F. Zhang, and X. Zhang, “High-performance etching of multilevel phase-type Fresnel zone plates with large apertures,” Opt. Commun. 407, 227–233 (2018).
[Crossref]

Z. Zhang, C. Guo, R. Wang, H. Hu, X. Zhou, T. Liu, D. Xue, X. Zhang, F. Zhang, and X. Zhang, “Hybrid-level Fresnel zone plate for diffraction efficiency enhancement,” Opt. Express 25(26), 33676–33687 (2017).
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T. Kouyama, A. Yamazaki, M. Yamada, and T. Imamura, “A method to estimate optical distortion using planetary images,” Planet. Space Sci. 86(15), 86–90 (2013).
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X. Zhang, B. Chen, F. He, K. Song, L. He, S. Liu, Q. Guo, J. Li, X. Wang, H. Zhang, H. Wang, Z. Han, L. Sun, P. Zhang, S. Dai, G. Ding, L. Chen, Z. Wang, G. Shi, X. Zhang, C. Yu, Z. Yang, P. Zhang, and J. Wang, “Wide-field auroral imager onboard the Fengyun satellite,” Light: Sci. Appl. 8(1), 47 (2019).
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[Crossref]

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[Crossref]

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Figures (15)

Fig. 1.
Fig. 1. Schematic of optical distortion.
Fig. 2.
Fig. 2. Schematic of optical distortion measurement with DBS.
Fig. 3.
Fig. 3. Schematic of angular calibration of the DBS.
Fig. 4.
Fig. 4. Flow chart showing the angular measurement process of the DBS.
Fig. 5.
Fig. 5. (a) the photograph of the DBS, and (b) the micrograph of the pattern.
Fig. 6.
Fig. 6. Experimental setup for angular measurement of the DBS.
Fig. 7.
Fig. 7. Alignments: (a) Optical axis is not aligned (b) Optical axis is aligned (c) Array center of the DBS is aligned.
Fig. 8.
Fig. 8. (a) Angular measurement initial position and scanning path (b) Angle detection path.
Fig. 9.
Fig. 9. Experimental setup of optical distortion measurement using the DBS.
Fig. 10.
Fig. 10. Image point distribution in optical distortion measurement.
Fig. 11.
Fig. 11. The influence of beam parallelism on distortion measurement.
Fig. 12.
Fig. 12. The defocus caused by non-parallel light on axis.
Fig. 13.
Fig. 13. (a) Fitted theoretical image height, actual image height, and the relative distortion in X direction. (b) Fitted theoretical image height, actual image height, and the relative distortion in Y direction.
Fig. 14.
Fig. 14. Utilized image points on X and Y axis in the fitting.
Fig. 15.
Fig. 15. (a) Distortion distribution of the tested lens.

Tables (4)

Tables Icon

Table 1. Measurement results of DBS angular distribution.

Tables Icon

Table 2. Centroid coordinates of image points.

Tables Icon

Table 3. X-direction image heights for fitting the distortion coefficient (Unit: pixel).

Tables Icon

Table 4. Y-direction image heights for fitting the distortion coefficient (Unit: pixel).

Equations (23)

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δ y z = y z f tan w .
t ( x , y ) = exp [ i ϕ ( x , y ) ] .
I ( x 0 , y 0 ) = | F T { t ( x , y ) } | 2 .
u w 1 = 0.25 3 0.14 .
u w 2 = σ n = 0.2 5 = 0.09 .
u w = u 2 w 1 + u 2 w 2 0.17 ,
I ( x , y ) = { 0 , P ( x , y ) T ( x , y ) , P ( x , y ) < T ( x , y ) P ( x , y ) T ( x , y ) .
{ x c = j = y 0 W y / 2 y 0 + W y / 2 i = x 0 W x / 2 x 0 + W x / 2 ( I i j w x i ) j = y 0 W y / 2 y 0 + W y / 2 i = x 0 W x / 2 x 0 + W x / 2 ( I i j w ) y c = j = y 0 W y / 2 y 0 + W y / 2 i = x 0 W x / 2 x 0 + W x / 2 ( I i j w y i ) j = y 0 W y / 2 y 0 + W y / 2 i = x 0 W x / 2 x 0 + W x / 2 ( I i j w ) .
u y z 1 = 4.4 μ m 50 × 3 0.05 μ m .
Δ = A B A B = d tan w .
d = f 2 n r 2 ρ Δ W ρ .
d = θ f 2 r ρ 2 .
Δ = θ f 2 tan w r .
u y z 2 = | y z w | u w = θ f 2 r cos 2 w u w .
u y z = u 2 y z 1 + u 2 y z 2 u y z 1 = 0.05 μ m .
f = ( y z i tan w i ) tan 2 w i .
{ u f = ( f y z i u y z ) 2 + ( f w i u w ) 2 f y z i = tan w i tan 2 w i f w i = y z i tan 2 w i 2 tan w i ( y z i tan w i ) ( cos w i tan 2 w i ) 2 .
u f = 6.82 μ m .
U f = u f f × 100 % = 0.019 % .
{ A P = X ¯ X A = ( x ( x 2 + y 2 ) 0 0 y ( x 2 + y 2 ) ) P = ( k 1 k 2 ) T .
P = ( 4.96 e 8 1.02 e 8 ) T .
u δ y z = u 2 y z + ( δ y z f ) 2 u 2 f + ( δ y z w ) 2 u 2 w .
u δ y z 0.28 μ m .

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