Abstract

Phase elements can be used in optical systems to achieve similar design goals to traditional geometric optical elements. If we replace traditional geometrical optical elements in optical systems by phase elements (such as diffractive optical elements and metasurfaces) which have phase functions loaded on the geometric surface substrates, it is possible to generate imaging optical systems that offer better performance, increased compactness, lighter weight, and easier alignment and manufacturing than conventional imaging systems. Here we propose a design method for imaging systems consisting of freeform-surface-substrate phase elements. The design process begins from an initial system that uses simple geometric planes or other predefined geometric surfaces without phase functions. After point-by-point construction and iteration steps, the geometric substrate surfaces and closed-form phase functions can then be calculated quickly and efficiently. The resulting design can be used as a good starting point for further optimization. To illustrate the generality and feasibility of the proposed design method, we present two high-performance compact systems as design examples. Both systems meet the design requirements, with small distortions after optimization. Their modulation transfer function (MTF) curves are close to the diffraction limit. This design framework can be used to design next-generation imaging systems using phase elements for applications including near-eye-displays, high-performance cameras and remote sensing and detection. The proposed method also offers insight into design of imaging systems that are constrained to conformal substrate shapes or integrated substrates.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Design method of nonsymmetric imaging systems consisting of multiple flat phase elements

Tong Yang, Dewen Cheng, and Yongtian Wang
Opt. Express 26(19) 25347-25363 (2018)

Freeform imaging spectrometer design using a point-by-point design method

Tong Yang, Dewen Cheng, and Yongtian Wang
Appl. Opt. 57(16) 4718-4727 (2018)

References

  • View by:
  • |
  • |
  • |

  1. H. Huang and H. Hua, “High-performance integral-imaging-based light field augmented reality display using freeform optics,” Opt. Express 26(13), 17578–17590 (2018).
    [Crossref]
  2. D. Cheng, Y. Wang, C. Xu, W. Song, and G. Jin, “Design of an ultra-thin near-eye display with geometrical waveguide and freeform optics,” Opt. Express 22(17), 20705–20719 (2014).
    [Crossref]
  3. Z. Zheng, X. Liu, H. Li, and L. Xu, “Design and fabrication of an off-axis see-through head-mounted display with an x-y polynomial surface,” Appl. Opt. 49(19), 3661–3668 (2010).
    [Crossref]
  4. Y. Zhong and H. Gross, “Initial system design method for non-rotationally symmetric systems based on Gaussian brackets and Nodal aberration theory,” Opt. Express 25(9), 10016–10030 (2017).
    [Crossref]
  5. T. Yang, J. Zhu, W. Hou, and G. Jin, “Design method of freeform off-axis reflective imaging systems with a direct construction process,” Opt. Express 22(8), 9193–9205 (2014).
    [Crossref]
  6. Q. Meng, H. Wang, K. Wang, Y. Wang, Z. Ji, and D. Wang, “Off-axis three-mirror freeform telescope with a large linear field of view based on an integration mirror,” Appl. Opt. 55(32), 8962–8970 (2016).
    [Crossref]
  7. A. Bauer, E. M. Schiesser, K. P. Thompson, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018).
    [Crossref]
  8. J. Zhu, W. Hou, X. Zhang, and G. Jin, “Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view,” J. Opt. 17(1), 015605 (2015).
    [Crossref]
  9. D. Reshidko and J. Sasian, “Method for the design of nonaxially symmetric optical systems using free-form surfaces,” Opt. Eng. 57(10), 1 (2018).
    [Crossref]
  10. K. P. Thompson and J. P. Rolland, “Freeform Optical Surfaces: A Revolution in Imaging Optical Design,” Opt. Photonics News 23(6), 30–35 (2012).
    [Crossref]
  11. T. Yang, D. Cheng, and Y. Wang, “Design method of nonsymmetric imaging systems consisting of multiple flat phase elements,” Opt. Express 26(19), 25347–25363 (2018).
    [Crossref]
  12. T. Zhan, J. Xiong, Y. H. Lee, R. Chen, and S. T. Wu, “Fabrication of Pancharatnam-Berry phase optical elements with highly stable polarization holography,” Opt. Express 27(3), 2632–2642 (2019).
    [Crossref]
  13. R. Tian, J. Liu, X. Li, X. Wang, and Y. Wang, “Design and fabrication of complicated diffractive optical elements on multiple curved surfaces,” Opt. Express 23(26), 32917–32925 (2015).
    [Crossref]
  14. J. P. B. Mueller, N. A. Rubin, R. C. Devlin, B. Groever, and F. Capasso, “Metasurface polarization optics: independent phase control of arbitrary orthogonal states of polarization,” Phys. Rev. Lett. 118(11), 113901 (2017).
    [Crossref]
  15. A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013).
    [Crossref]
  16. N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
    [Crossref]
  17. G. D. Wassermann and E. Wolf, “On the Theory of Aplanatic Aspheric Systems,” Proc. Phys. Soc., London, Sect. B 62(1), 2–8 (1949).
    [Crossref]
  18. D. Knapp, “Conformal Optical Design,” Ph.D. Thesis, University of Arizona (2002).
  19. P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. M. Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1503 (2004).
    [Crossref]
  20. T. Yang, J. Zhu, X. Wu, and G. Jin, “Direct design of freeform surfaces and freeform imaging systems with a point-by-point three-dimensional construction-iteration method,” Opt. Express 23(8), 10233–10246 (2015).
    [Crossref]
  21. T. Yang, G. Jin, and J. Zhu, “Automated design of freeform imaging systems,” Light: Sci. Appl. 6(10), e17081 (2017).
    [Crossref]
  22. J. Mendes-Lopes, P. Benítez, J. C. Miñano, and A. Santamaría, “Simultaneous multiple surface design method for diffractive surfaces,” Opt. Express 24(5), 5584–5590 (2016).
    [Crossref]
  23. V. Perlick, Ray optics Fermat's principle, and applications to general relativity. (Springer, 2000).
  24. W. B. Wetherell and D. A. Womble, “All-reflective three element objective,” U.S. Patent 4,240, 707 (1980).
  25. Optical Research Associates, Code V Reference Manual (Synopsys Inc., 2012).
  26. T. Yang, J. Zhu, and G. Jin, “Compact freeform off-axis three-mirror imaging system based on the integration of primary and tertiary mirrors on one single surface,” Chin. Opt. Lett. 14(6), 060801 (2016).
    [Crossref]

2019 (1)

2018 (4)

T. Yang, D. Cheng, and Y. Wang, “Design method of nonsymmetric imaging systems consisting of multiple flat phase elements,” Opt. Express 26(19), 25347–25363 (2018).
[Crossref]

H. Huang and H. Hua, “High-performance integral-imaging-based light field augmented reality display using freeform optics,” Opt. Express 26(13), 17578–17590 (2018).
[Crossref]

A. Bauer, E. M. Schiesser, K. P. Thompson, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018).
[Crossref]

D. Reshidko and J. Sasian, “Method for the design of nonaxially symmetric optical systems using free-form surfaces,” Opt. Eng. 57(10), 1 (2018).
[Crossref]

2017 (3)

Y. Zhong and H. Gross, “Initial system design method for non-rotationally symmetric systems based on Gaussian brackets and Nodal aberration theory,” Opt. Express 25(9), 10016–10030 (2017).
[Crossref]

J. P. B. Mueller, N. A. Rubin, R. C. Devlin, B. Groever, and F. Capasso, “Metasurface polarization optics: independent phase control of arbitrary orthogonal states of polarization,” Phys. Rev. Lett. 118(11), 113901 (2017).
[Crossref]

T. Yang, G. Jin, and J. Zhu, “Automated design of freeform imaging systems,” Light: Sci. Appl. 6(10), e17081 (2017).
[Crossref]

2016 (3)

2015 (3)

2014 (3)

2013 (1)

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref]

2012 (1)

K. P. Thompson and J. P. Rolland, “Freeform Optical Surfaces: A Revolution in Imaging Optical Design,” Opt. Photonics News 23(6), 30–35 (2012).
[Crossref]

2010 (1)

2004 (1)

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. M. Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1503 (2004).
[Crossref]

1949 (1)

G. D. Wassermann and E. Wolf, “On the Theory of Aplanatic Aspheric Systems,” Proc. Phys. Soc., London, Sect. B 62(1), 2–8 (1949).
[Crossref]

Arroyo, R. M.

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. M. Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1503 (2004).
[Crossref]

Associates, Optical Research

Optical Research Associates, Code V Reference Manual (Synopsys Inc., 2012).

Bauer, A.

A. Bauer, E. M. Schiesser, K. P. Thompson, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018).
[Crossref]

Benítez, P.

Blen, J.

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. M. Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1503 (2004).
[Crossref]

Boltasseva, A.

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref]

Capasso, F.

J. P. B. Mueller, N. A. Rubin, R. C. Devlin, B. Groever, and F. Capasso, “Metasurface polarization optics: independent phase control of arbitrary orthogonal states of polarization,” Phys. Rev. Lett. 118(11), 113901 (2017).
[Crossref]

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref]

Chaves, J.

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. M. Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1503 (2004).
[Crossref]

Chen, R.

Cheng, D.

Devlin, R. C.

J. P. B. Mueller, N. A. Rubin, R. C. Devlin, B. Groever, and F. Capasso, “Metasurface polarization optics: independent phase control of arbitrary orthogonal states of polarization,” Phys. Rev. Lett. 118(11), 113901 (2017).
[Crossref]

Dross, O.

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. M. Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1503 (2004).
[Crossref]

Falicoff, W.

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. M. Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1503 (2004).
[Crossref]

Gimenez-Benitez, P.

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. M. Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1503 (2004).
[Crossref]

Groever, B.

J. P. B. Mueller, N. A. Rubin, R. C. Devlin, B. Groever, and F. Capasso, “Metasurface polarization optics: independent phase control of arbitrary orthogonal states of polarization,” Phys. Rev. Lett. 118(11), 113901 (2017).
[Crossref]

Gross, H.

Hernandez, M.

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. M. Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1503 (2004).
[Crossref]

Hou, W.

J. Zhu, W. Hou, X. Zhang, and G. Jin, “Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view,” J. Opt. 17(1), 015605 (2015).
[Crossref]

T. Yang, J. Zhu, W. Hou, and G. Jin, “Design method of freeform off-axis reflective imaging systems with a direct construction process,” Opt. Express 22(8), 9193–9205 (2014).
[Crossref]

Hua, H.

Huang, H.

Ji, Z.

Jin, G.

Kildishev, A. V.

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref]

Knapp, D.

D. Knapp, “Conformal Optical Design,” Ph.D. Thesis, University of Arizona (2002).

Lee, Y. H.

Li, H.

Li, X.

Liu, J.

Liu, X.

Mendes-Lopes, J.

Meng, Q.

Miñano, J. C.

J. Mendes-Lopes, P. Benítez, J. C. Miñano, and A. Santamaría, “Simultaneous multiple surface design method for diffractive surfaces,” Opt. Express 24(5), 5584–5590 (2016).
[Crossref]

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. M. Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1503 (2004).
[Crossref]

Mueller, J. P. B.

J. P. B. Mueller, N. A. Rubin, R. C. Devlin, B. Groever, and F. Capasso, “Metasurface polarization optics: independent phase control of arbitrary orthogonal states of polarization,” Phys. Rev. Lett. 118(11), 113901 (2017).
[Crossref]

Perlick, V.

V. Perlick, Ray optics Fermat's principle, and applications to general relativity. (Springer, 2000).

Reshidko, D.

D. Reshidko and J. Sasian, “Method for the design of nonaxially symmetric optical systems using free-form surfaces,” Opt. Eng. 57(10), 1 (2018).
[Crossref]

Rolland, J. P.

A. Bauer, E. M. Schiesser, K. P. Thompson, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018).
[Crossref]

K. P. Thompson and J. P. Rolland, “Freeform Optical Surfaces: A Revolution in Imaging Optical Design,” Opt. Photonics News 23(6), 30–35 (2012).
[Crossref]

Rubin, N. A.

J. P. B. Mueller, N. A. Rubin, R. C. Devlin, B. Groever, and F. Capasso, “Metasurface polarization optics: independent phase control of arbitrary orthogonal states of polarization,” Phys. Rev. Lett. 118(11), 113901 (2017).
[Crossref]

Santamaría, A.

Sasian, J.

D. Reshidko and J. Sasian, “Method for the design of nonaxially symmetric optical systems using free-form surfaces,” Opt. Eng. 57(10), 1 (2018).
[Crossref]

Schiesser, E. M.

A. Bauer, E. M. Schiesser, K. P. Thompson, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018).
[Crossref]

Shalaev, V. M.

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref]

Song, W.

Thompson, K. P.

A. Bauer, E. M. Schiesser, K. P. Thompson, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018).
[Crossref]

K. P. Thompson and J. P. Rolland, “Freeform Optical Surfaces: A Revolution in Imaging Optical Design,” Opt. Photonics News 23(6), 30–35 (2012).
[Crossref]

Tian, R.

Wang, D.

Wang, H.

Wang, K.

Wang, X.

Wang, Y.

Wassermann, G. D.

G. D. Wassermann and E. Wolf, “On the Theory of Aplanatic Aspheric Systems,” Proc. Phys. Soc., London, Sect. B 62(1), 2–8 (1949).
[Crossref]

Wetherell, W. B.

W. B. Wetherell and D. A. Womble, “All-reflective three element objective,” U.S. Patent 4,240, 707 (1980).

Wolf, E.

G. D. Wassermann and E. Wolf, “On the Theory of Aplanatic Aspheric Systems,” Proc. Phys. Soc., London, Sect. B 62(1), 2–8 (1949).
[Crossref]

Womble, D. A.

W. B. Wetherell and D. A. Womble, “All-reflective three element objective,” U.S. Patent 4,240, 707 (1980).

Wu, S. T.

Wu, X.

Xiong, J.

Xu, C.

Xu, L.

Yang, T.

Yu, N.

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref]

Zhan, T.

Zhang, X.

J. Zhu, W. Hou, X. Zhang, and G. Jin, “Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view,” J. Opt. 17(1), 015605 (2015).
[Crossref]

Zheng, Z.

Zhong, Y.

Zhu, J.

Appl. Opt. (2)

Chin. Opt. Lett. (1)

J. Opt. (1)

J. Zhu, W. Hou, X. Zhang, and G. Jin, “Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view,” J. Opt. 17(1), 015605 (2015).
[Crossref]

Light: Sci. Appl. (1)

T. Yang, G. Jin, and J. Zhu, “Automated design of freeform imaging systems,” Light: Sci. Appl. 6(10), e17081 (2017).
[Crossref]

Nat. Commun. (1)

A. Bauer, E. M. Schiesser, K. P. Thompson, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018).
[Crossref]

Nat. Mater. (1)

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref]

Opt. Eng. (2)

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. M. Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1503 (2004).
[Crossref]

D. Reshidko and J. Sasian, “Method for the design of nonaxially symmetric optical systems using free-form surfaces,” Opt. Eng. 57(10), 1 (2018).
[Crossref]

Opt. Express (9)

T. Yang, J. Zhu, W. Hou, and G. Jin, “Design method of freeform off-axis reflective imaging systems with a direct construction process,” Opt. Express 22(8), 9193–9205 (2014).
[Crossref]

D. Cheng, Y. Wang, C. Xu, W. Song, and G. Jin, “Design of an ultra-thin near-eye display with geometrical waveguide and freeform optics,” Opt. Express 22(17), 20705–20719 (2014).
[Crossref]

T. Yang, J. Zhu, X. Wu, and G. Jin, “Direct design of freeform surfaces and freeform imaging systems with a point-by-point three-dimensional construction-iteration method,” Opt. Express 23(8), 10233–10246 (2015).
[Crossref]

R. Tian, J. Liu, X. Li, X. Wang, and Y. Wang, “Design and fabrication of complicated diffractive optical elements on multiple curved surfaces,” Opt. Express 23(26), 32917–32925 (2015).
[Crossref]

J. Mendes-Lopes, P. Benítez, J. C. Miñano, and A. Santamaría, “Simultaneous multiple surface design method for diffractive surfaces,” Opt. Express 24(5), 5584–5590 (2016).
[Crossref]

Y. Zhong and H. Gross, “Initial system design method for non-rotationally symmetric systems based on Gaussian brackets and Nodal aberration theory,” Opt. Express 25(9), 10016–10030 (2017).
[Crossref]

H. Huang and H. Hua, “High-performance integral-imaging-based light field augmented reality display using freeform optics,” Opt. Express 26(13), 17578–17590 (2018).
[Crossref]

T. Yang, D. Cheng, and Y. Wang, “Design method of nonsymmetric imaging systems consisting of multiple flat phase elements,” Opt. Express 26(19), 25347–25363 (2018).
[Crossref]

T. Zhan, J. Xiong, Y. H. Lee, R. Chen, and S. T. Wu, “Fabrication of Pancharatnam-Berry phase optical elements with highly stable polarization holography,” Opt. Express 27(3), 2632–2642 (2019).
[Crossref]

Opt. Photonics News (1)

K. P. Thompson and J. P. Rolland, “Freeform Optical Surfaces: A Revolution in Imaging Optical Design,” Opt. Photonics News 23(6), 30–35 (2012).
[Crossref]

Phys. Rev. Lett. (1)

J. P. B. Mueller, N. A. Rubin, R. C. Devlin, B. Groever, and F. Capasso, “Metasurface polarization optics: independent phase control of arbitrary orthogonal states of polarization,” Phys. Rev. Lett. 118(11), 113901 (2017).
[Crossref]

Proc. Phys. Soc., London, Sect. B (1)

G. D. Wassermann and E. Wolf, “On the Theory of Aplanatic Aspheric Systems,” Proc. Phys. Soc., London, Sect. B 62(1), 2–8 (1949).
[Crossref]

Science (1)

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref]

Other (4)

D. Knapp, “Conformal Optical Design,” Ph.D. Thesis, University of Arizona (2002).

V. Perlick, Ray optics Fermat's principle, and applications to general relativity. (Springer, 2000).

W. B. Wetherell and D. A. Womble, “All-reflective three element objective,” U.S. Patent 4,240, 707 (1980).

Optical Research Associates, Code V Reference Manual (Synopsys Inc., 2012).

Supplementary Material (1)

NameDescription
» Data File 1       Detailed coefficients of phase functions

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1.
Fig. 1. Propagation of a general feature light ray Ri when passing through a surface with a phase function. ri and ri are the unit vectors for the incident direction and the outgoing direction of ray Ri when travelling at a surface. (a) Refractive surface. (b) Reflective surface.
Fig. 2.
Fig. 2. Structure of an FSP element. The solid brown curve is used to represent a general or regular phase profile (or phase function) loaded on the geometric surface substrate (plotted in blue).
Fig. 3.
Fig. 3. Design step for construction of the current geometric substrate shape. (a) Choose the first feature ray R1 and its corresponding feature point P1 in RS. (b) When the ith feature point Pi has been calculated, ∇ϕ(Pi) can be calculated and then we can obtain the normal vector Ni of Pi in RS. The next feature point Pi+1 can be determined by the “nearest-ray algorithm”. (c) When all the data points as well as their surface normals in RS have been calculated, fit the data points into a closed-form geometrical substrate shape considering both the coordinates and surface normals.
Fig. 4.
Fig. 4. Design step for construction of the current phase function. (a) Choose the first feature ray R1 and its corresponding feature point P1,ϕ in PS. (b) When Pi and Pi,ϕ have been obtained, calculate the surface normal Ni at point Pi. Then the “surface normal” of the phase profile Ni,ϕ at Pi,ϕ in PS can be obtained. The next feature ray Ri+1 as well as the next feature point Pi+1,ϕ in PS can be determined by the “nearest-ray algorithm” used in phase function calculation. (c) When all the data points as well as their surface normals in PS have been calculated, fit the data points into a closed-form phase function considering both the coordinates and surface normals.
Fig. 5.
Fig. 5. Flowchart for the construction and iterative processes applied to calculate geometric substrate shapes and phase functions.
Fig. 6.
Fig. 6. (a) Initial system of design example A using geometric planes. (b) System layout of design example A after point-by-point design process.
Fig. 7.
Fig. 7. (a) Change of σRMS with iterations for design example A. (b) Distortion grid of design example A after iterations.
Fig. 8.
Fig. 8. Final design result of design example A. (a) System layout. (b) MTF plot. (c) Distortion grid. (d) Ray aberration graph of the meridian direction. (e) Ray aberration graph of the sagittal direction.
Fig. 9.
Fig. 9. (a) Initial system of design example B. (b) System layout of design example B after the point-by-point design process.
Fig. 10.
Fig. 10. Distortion grid of design example B after iterations.
Fig. 11.
Fig. 11. Final design result of design example B. (a) Optical layout of the final system after optimization. (b) MTF plot. (c) Final distortion grid. (d) Ray aberration graph of the meridian direction. (e) Ray aberration graph of the sagittal direction.

Equations (11)

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

O P L ( S i , E i ) = n | P i S i | + n | E i P i | + m λ 2 π ϕ ( P i ) ,
O P L x = 0 , O P L y = 0.
n ( N i × r i ) = n ( N i × r i ) + m λ 2 π [ N i × T ϕ ( P i ) ] ,
T = [ 1 0 0 0 cos α sin α 0 sin α cos α ] .
n r i n r i m λ 2 π T ϕ ( P i ) = ε N i ,
[ m λ 2 π 0 N i x 0 m λ 2 π cos α N i y 0 m λ 2 π sin α N i z ] [ ϕ x | P i ϕ y | P i ε ] = [ n r i , x n r i , x n r i , y n r i , y n r i , z n r i , z ] .
σ RMS = μ = 1 Q T σ μ 2 Q T
I i , ideal  = [ I i , ideal, x I i , ideal, y I i , ideal,z ]  =  [ 1 0 0 0 cos ( α o ) sin ( α o ) 0 sin ( α o ) cos ( α o ) ] [ E F L tan ( ω x ω x , central ) E F L tan ( ω y ω y , central ) 0 ] + [ x o y o z o ]
ϕ ( x , y ) = A 2 y + A 3 x 2 + A 5 y 2 + A 7 x 2 y + A 9 y 3 + A 10 x 4 + A 12 x 2 y 2 + A 14 y 4 + A 16 x 4 y + A 18 x 2 y 3 + A 20 y 5 .
l 3 Φ = 1 2 d 1 c 1 2 d 2 c 2 + 2 d 2 c 1 + 4 d 1 d 2 c 1 c 2
Φ = 2 c 3 l 3 Φ + 2 c 1 2 c 2 + 4 d 1 c 1 c 2

Metrics