Abstract

In lens systems, the constituent lenses usually share a common optical axis, or at least a common optical-axis direction, and such combinations of lenses are well understood. However, in recent proposals for lens-based transformation-optics devices [Opt. Express 26, 17872 (2018) [CrossRef]  ], the lenses do not share an optical-axis direction. To facilitate the understanding of such lens systems, we describe here combinations of two ideal lenses in any arbitrary arrangement as a single ideal lens. This description has the potential to become an important tool in understanding novel optical instruments enabled by skew-lens combinations.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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References

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  1. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
    [Crossref]
  2. M. Šarbort and T. Tyc, “Spherical media and geodesic lenses in geometrical optics,” J. Opt. 14, 075705 (2012).
    [Crossref]
  3. A. H. Layard, Discoveries in the Ruins of Nineveh and Babylon (G. P. Putnam, 1853), pp. 197–198.
  4. M. Khorasaninejad, F. Aieta, P. Kanhaiya, M. A. Kats, P. Genevet, D. Rousso, and F. Capasso, “Achromatic metasurface lens at telecommunication wavelengths,” Nano Lett. 15, 5358–5362 (2015).
    [Crossref]
  5. M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
    [Crossref]
  6. E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, and A. Faraon, “Multiwavelength polarization-insensitive lenses based on dielectric metasurfaces with metamolecules,” Optica 3, 628–633 (2016).
    [Crossref]
  7. W. T. Chen, A. Y. Zhu, V. Sanjeev, M. Khorasaninejad, Z. Shi, E. Lee, and F. Capasso, “A broadband achromatic metalens for focusing and imaging in the visible,” Nat. Nanotechnol. 13, 220–226 (2018).
    [Crossref]
  8. J. S. Choi and J. C. Howell, “Paraxial ray optics cloaking,” Opt. Express 22, 29465–29478 (2014).
    [Crossref]
  9. T. Tyc, S. Oxburgh, E. N. Cowie, G. J. Chaplain, G. Macauley, C. D. White, and J. Courtial, “Omnidirectional transformation-optics cloak made from lenses and glenses,” J. Opt. Soc. Am. A 33, 1032–1040 (2016).
    [Crossref]
  10. J. Courtial, T. Tyc, J. Bělín, S. Oxburgh, G. Ferenczi, E. N. Cowie, and C. D. White, “Ray-optical transformation optics with ideal thin lenses makes omnidirectional lenses,” Opt. Express 26, 17872–17888 (2018).
    [Crossref]
  11. W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, 2000).
  12. R. A. Buchroeder, “Tilted component optical systems,” Ph.D. thesis (University of Arizona, 1976).
  13. J. Bělín and J. Courtial, “Mathematica notebooks related to imaging with two skew ideal lenses,” figshare (2018), https://doi.org/10.6084/m9.figshare.7195760.v1 .
  14. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), chap. 4.3.1.
  15. G. J. Chaplain, G. Macauley, J. Bělín, T. Tyc, E. N. Cowie, and J. Courtial, “Ray optics of generalized lenses,” J. Opt. Soc. Am. A 33, 962–969 (2016).
    [Crossref]
  16. C. Hahlweg, W. Zhao, and H. Rothe, “Fourier planes vs. Scheimpflug principle in microscopic and scatterometric devices,” Proc. SPIE 8127, 812708 (2011).
    [Crossref]
  17. P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. M. Arroyo, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489–1503 (2004).
    [Crossref]
  18. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
    [Crossref]
  19. X. Ni, A. V. Kildishev, and V. M. Shalaev, “Metasurface holograms for visible light,” Nat. Commun. 4, 2807 (2013).
    [Crossref]
  20. N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13, 139–150 (2014).
    [Crossref]
  21. G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat. Nanotechnol. 10, 308–312 (2015).
    [Crossref]
  22. L. Wang, S. Kruk, H. Tang, T. Li, I. Kravchenko, D. N. Neshev, and Y. S. Kivshar, “Grayscale transparent metasurface holograms,” Optica 3, 1504–1505 (2016).
    [Crossref]
  23. P. Genevet, F. Capasso, F. Aieta, M. Khorasaninejad, and R. Devlin, “Recent advances in planar optics: from plasmonic to dielectric metasurfaces,” Optica 4, 139–152 (2017).
    [Crossref]
  24. F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347, 1342–1345 (2015).
    [Crossref]
  25. M. Khorasaninejad, A. Ambrosio, P. Kanhaiya, and F. Capasso, “Broadband and chiral binary dielectric meta-holograms,” Sci. Adv. 2, e1501258 (2016).
    [Crossref]
  26. H.-T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: physics and applications,” Rep. Prog. Phys. 79, 076401 (2016).
    [Crossref]
  27. T. Scheimpflug, “Improved method and apparatus for the systematic alteration or distortion of plane pictures and images by means of lenses and mirrors for photography and for other purposes,” GB patent1196 (May12, 1904).

2018 (2)

W. T. Chen, A. Y. Zhu, V. Sanjeev, M. Khorasaninejad, Z. Shi, E. Lee, and F. Capasso, “A broadband achromatic metalens for focusing and imaging in the visible,” Nat. Nanotechnol. 13, 220–226 (2018).
[Crossref]

J. Courtial, T. Tyc, J. Bělín, S. Oxburgh, G. Ferenczi, E. N. Cowie, and C. D. White, “Ray-optical transformation optics with ideal thin lenses makes omnidirectional lenses,” Opt. Express 26, 17872–17888 (2018).
[Crossref]

2017 (1)

2016 (7)

2015 (3)

G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat. Nanotechnol. 10, 308–312 (2015).
[Crossref]

F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347, 1342–1345 (2015).
[Crossref]

M. Khorasaninejad, F. Aieta, P. Kanhaiya, M. A. Kats, P. Genevet, D. Rousso, and F. Capasso, “Achromatic metasurface lens at telecommunication wavelengths,” Nano Lett. 15, 5358–5362 (2015).
[Crossref]

2014 (2)

J. S. Choi and J. C. Howell, “Paraxial ray optics cloaking,” Opt. Express 22, 29465–29478 (2014).
[Crossref]

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

2013 (1)

X. Ni, A. V. Kildishev, and V. M. Shalaev, “Metasurface holograms for visible light,” Nat. Commun. 4, 2807 (2013).
[Crossref]

2012 (1)

M. Šarbort and T. Tyc, “Spherical media and geodesic lenses in geometrical optics,” J. Opt. 14, 075705 (2012).
[Crossref]

2011 (2)

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref]

C. Hahlweg, W. Zhao, and H. Rothe, “Fourier planes vs. Scheimpflug principle in microscopic and scatterometric devices,” Proc. SPIE 8127, 812708 (2011).
[Crossref]

2006 (1)

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref]

2004 (1)

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

Aieta, F.

P. Genevet, F. Capasso, F. Aieta, M. Khorasaninejad, and R. Devlin, “Recent advances in planar optics: from plasmonic to dielectric metasurfaces,” Optica 4, 139–152 (2017).
[Crossref]

M. Khorasaninejad, F. Aieta, P. Kanhaiya, M. A. Kats, P. Genevet, D. Rousso, and F. Capasso, “Achromatic metasurface lens at telecommunication wavelengths,” Nano Lett. 15, 5358–5362 (2015).
[Crossref]

F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347, 1342–1345 (2015).
[Crossref]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref]

Ambrosio, A.

M. Khorasaninejad, A. Ambrosio, P. Kanhaiya, and F. Capasso, “Broadband and chiral binary dielectric meta-holograms,” Sci. Adv. 2, e1501258 (2016).
[Crossref]

Arbabi, A.

Arbabi, E.

Arroyo, R. M.

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

Belín, J.

Blen, J.

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

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), chap. 4.3.1.

Buchroeder, R. A.

R. A. Buchroeder, “Tilted component optical systems,” Ph.D. thesis (University of Arizona, 1976).

Capasso, F.

W. T. Chen, A. Y. Zhu, V. Sanjeev, M. Khorasaninejad, Z. Shi, E. Lee, and F. Capasso, “A broadband achromatic metalens for focusing and imaging in the visible,” Nat. Nanotechnol. 13, 220–226 (2018).
[Crossref]

P. Genevet, F. Capasso, F. Aieta, M. Khorasaninejad, and R. Devlin, “Recent advances in planar optics: from plasmonic to dielectric metasurfaces,” Optica 4, 139–152 (2017).
[Crossref]

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
[Crossref]

M. Khorasaninejad, A. Ambrosio, P. Kanhaiya, and F. Capasso, “Broadband and chiral binary dielectric meta-holograms,” Sci. Adv. 2, e1501258 (2016).
[Crossref]

F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347, 1342–1345 (2015).
[Crossref]

M. Khorasaninejad, F. Aieta, P. Kanhaiya, M. A. Kats, P. Genevet, D. Rousso, and F. Capasso, “Achromatic metasurface lens at telecommunication wavelengths,” Nano Lett. 15, 5358–5362 (2015).
[Crossref]

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

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref]

Chaplain, G. J.

Chaves, J.

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

Chen, H.-T.

H.-T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: physics and applications,” Rep. Prog. Phys. 79, 076401 (2016).
[Crossref]

Chen, W. T.

W. T. Chen, A. Y. Zhu, V. Sanjeev, M. Khorasaninejad, Z. Shi, E. Lee, and F. Capasso, “A broadband achromatic metalens for focusing and imaging in the visible,” Nat. Nanotechnol. 13, 220–226 (2018).
[Crossref]

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
[Crossref]

Choi, J. S.

Courtial, J.

Cowie, E. N.

Devlin, R.

Devlin, R. C.

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
[Crossref]

Dross, O.

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

Falicoff, W.

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

Faraon, A.

Ferenczi, G.

Gaburro, Z.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref]

Genevet, P.

P. Genevet, F. Capasso, F. Aieta, M. Khorasaninejad, and R. Devlin, “Recent advances in planar optics: from plasmonic to dielectric metasurfaces,” Optica 4, 139–152 (2017).
[Crossref]

F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347, 1342–1345 (2015).
[Crossref]

M. Khorasaninejad, F. Aieta, P. Kanhaiya, M. A. Kats, P. Genevet, D. Rousso, and F. Capasso, “Achromatic metasurface lens at telecommunication wavelengths,” Nano Lett. 15, 5358–5362 (2015).
[Crossref]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref]

Gimenez-Benitez, P.

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

Hahlweg, C.

C. Hahlweg, W. Zhao, and H. Rothe, “Fourier planes vs. Scheimpflug principle in microscopic and scatterometric devices,” Proc. SPIE 8127, 812708 (2011).
[Crossref]

Hernández, M.

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

Horie, Y.

Howell, J. C.

Kamali, S. M.

Kanhaiya, P.

M. Khorasaninejad, A. Ambrosio, P. Kanhaiya, and F. Capasso, “Broadband and chiral binary dielectric meta-holograms,” Sci. Adv. 2, e1501258 (2016).
[Crossref]

M. Khorasaninejad, F. Aieta, P. Kanhaiya, M. A. Kats, P. Genevet, D. Rousso, and F. Capasso, “Achromatic metasurface lens at telecommunication wavelengths,” Nano Lett. 15, 5358–5362 (2015).
[Crossref]

Kats, M. A.

M. Khorasaninejad, F. Aieta, P. Kanhaiya, M. A. Kats, P. Genevet, D. Rousso, and F. Capasso, “Achromatic metasurface lens at telecommunication wavelengths,” Nano Lett. 15, 5358–5362 (2015).
[Crossref]

F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347, 1342–1345 (2015).
[Crossref]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref]

Kenney, M.

G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat. Nanotechnol. 10, 308–312 (2015).
[Crossref]

Khorasaninejad, M.

W. T. Chen, A. Y. Zhu, V. Sanjeev, M. Khorasaninejad, Z. Shi, E. Lee, and F. Capasso, “A broadband achromatic metalens for focusing and imaging in the visible,” Nat. Nanotechnol. 13, 220–226 (2018).
[Crossref]

P. Genevet, F. Capasso, F. Aieta, M. Khorasaninejad, and R. Devlin, “Recent advances in planar optics: from plasmonic to dielectric metasurfaces,” Optica 4, 139–152 (2017).
[Crossref]

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
[Crossref]

M. Khorasaninejad, A. Ambrosio, P. Kanhaiya, and F. Capasso, “Broadband and chiral binary dielectric meta-holograms,” Sci. Adv. 2, e1501258 (2016).
[Crossref]

M. Khorasaninejad, F. Aieta, P. Kanhaiya, M. A. Kats, P. Genevet, D. Rousso, and F. Capasso, “Achromatic metasurface lens at telecommunication wavelengths,” Nano Lett. 15, 5358–5362 (2015).
[Crossref]

Kildishev, A. V.

X. Ni, A. V. Kildishev, and V. M. Shalaev, “Metasurface holograms for visible light,” Nat. Commun. 4, 2807 (2013).
[Crossref]

Kivshar, Y. S.

Kravchenko, I.

Kruk, S.

Layard, A. H.

A. H. Layard, Discoveries in the Ruins of Nineveh and Babylon (G. P. Putnam, 1853), pp. 197–198.

Lee, E.

W. T. Chen, A. Y. Zhu, V. Sanjeev, M. Khorasaninejad, Z. Shi, E. Lee, and F. Capasso, “A broadband achromatic metalens for focusing and imaging in the visible,” Nat. Nanotechnol. 13, 220–226 (2018).
[Crossref]

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref]

Li, G.

G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat. Nanotechnol. 10, 308–312 (2015).
[Crossref]

Li, T.

Macauley, G.

Miñano, J. C.

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

Mühlenbernd, H.

G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat. Nanotechnol. 10, 308–312 (2015).
[Crossref]

Neshev, D. N.

Ni, X.

X. Ni, A. V. Kildishev, and V. M. Shalaev, “Metasurface holograms for visible light,” Nat. Commun. 4, 2807 (2013).
[Crossref]

Oh, J.

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
[Crossref]

Oxburgh, S.

Rothe, H.

C. Hahlweg, W. Zhao, and H. Rothe, “Fourier planes vs. Scheimpflug principle in microscopic and scatterometric devices,” Proc. SPIE 8127, 812708 (2011).
[Crossref]

Rousso, D.

M. Khorasaninejad, F. Aieta, P. Kanhaiya, M. A. Kats, P. Genevet, D. Rousso, and F. Capasso, “Achromatic metasurface lens at telecommunication wavelengths,” Nano Lett. 15, 5358–5362 (2015).
[Crossref]

Sanjeev, V.

W. T. Chen, A. Y. Zhu, V. Sanjeev, M. Khorasaninejad, Z. Shi, E. Lee, and F. Capasso, “A broadband achromatic metalens for focusing and imaging in the visible,” Nat. Nanotechnol. 13, 220–226 (2018).
[Crossref]

Šarbort, M.

M. Šarbort and T. Tyc, “Spherical media and geodesic lenses in geometrical optics,” J. Opt. 14, 075705 (2012).
[Crossref]

Scheimpflug, T.

T. Scheimpflug, “Improved method and apparatus for the systematic alteration or distortion of plane pictures and images by means of lenses and mirrors for photography and for other purposes,” GB patent1196 (May12, 1904).

Shalaev, V. M.

X. Ni, A. V. Kildishev, and V. M. Shalaev, “Metasurface holograms for visible light,” Nat. Commun. 4, 2807 (2013).
[Crossref]

Shi, Z.

W. T. Chen, A. Y. Zhu, V. Sanjeev, M. Khorasaninejad, Z. Shi, E. Lee, and F. Capasso, “A broadband achromatic metalens for focusing and imaging in the visible,” Nat. Nanotechnol. 13, 220–226 (2018).
[Crossref]

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, 2000).

Tang, H.

Taylor, A. J.

H.-T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: physics and applications,” Rep. Prog. Phys. 79, 076401 (2016).
[Crossref]

Tetienne, J.-P.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref]

Tyc, T.

Wang, L.

White, C. D.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), chap. 4.3.1.

Yu, N.

H.-T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: physics and applications,” Rep. Prog. Phys. 79, 076401 (2016).
[Crossref]

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

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref]

Zentgraf, T.

G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat. Nanotechnol. 10, 308–312 (2015).
[Crossref]

Zhang, S.

G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat. Nanotechnol. 10, 308–312 (2015).
[Crossref]

Zhao, W.

C. Hahlweg, W. Zhao, and H. Rothe, “Fourier planes vs. Scheimpflug principle in microscopic and scatterometric devices,” Proc. SPIE 8127, 812708 (2011).
[Crossref]

Zheng, G.

G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat. Nanotechnol. 10, 308–312 (2015).
[Crossref]

Zhu, A. Y.

W. T. Chen, A. Y. Zhu, V. Sanjeev, M. Khorasaninejad, Z. Shi, E. Lee, and F. Capasso, “A broadband achromatic metalens for focusing and imaging in the visible,” Nat. Nanotechnol. 13, 220–226 (2018).
[Crossref]

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
[Crossref]

J. Opt. (1)

M. Šarbort and T. Tyc, “Spherical media and geodesic lenses in geometrical optics,” J. Opt. 14, 075705 (2012).
[Crossref]

J. Opt. Soc. Am. A (2)

Nano Lett. (1)

M. Khorasaninejad, F. Aieta, P. Kanhaiya, M. A. Kats, P. Genevet, D. Rousso, and F. Capasso, “Achromatic metasurface lens at telecommunication wavelengths,” Nano Lett. 15, 5358–5362 (2015).
[Crossref]

Nat. Commun. (1)

X. Ni, A. V. Kildishev, and V. M. Shalaev, “Metasurface holograms for visible light,” Nat. Commun. 4, 2807 (2013).
[Crossref]

Nat. Mater. (1)

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

Nat. Nanotechnol. (2)

G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat. Nanotechnol. 10, 308–312 (2015).
[Crossref]

W. T. Chen, A. Y. Zhu, V. Sanjeev, M. Khorasaninejad, Z. Shi, E. Lee, and F. Capasso, “A broadband achromatic metalens for focusing and imaging in the visible,” Nat. Nanotechnol. 13, 220–226 (2018).
[Crossref]

Opt. Eng. (1)

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

Opt. Express (2)

Optica (3)

Proc. SPIE (1)

C. Hahlweg, W. Zhao, and H. Rothe, “Fourier planes vs. Scheimpflug principle in microscopic and scatterometric devices,” Proc. SPIE 8127, 812708 (2011).
[Crossref]

Rep. Prog. Phys. (1)

H.-T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: physics and applications,” Rep. Prog. Phys. 79, 076401 (2016).
[Crossref]

Sci. Adv. (1)

M. Khorasaninejad, A. Ambrosio, P. Kanhaiya, and F. Capasso, “Broadband and chiral binary dielectric meta-holograms,” Sci. Adv. 2, e1501258 (2016).
[Crossref]

Science (4)

F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347, 1342–1345 (2015).
[Crossref]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref]

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
[Crossref]

Other (6)

A. H. Layard, Discoveries in the Ruins of Nineveh and Babylon (G. P. Putnam, 1853), pp. 197–198.

W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, 2000).

R. A. Buchroeder, “Tilted component optical systems,” Ph.D. thesis (University of Arizona, 1976).

J. Bělín and J. Courtial, “Mathematica notebooks related to imaging with two skew ideal lenses,” figshare (2018), https://doi.org/10.6084/m9.figshare.7195760.v1 .

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), chap. 4.3.1.

T. Scheimpflug, “Improved method and apparatus for the systematic alteration or distortion of plane pictures and images by means of lenses and mirrors for photography and for other purposes,” GB patent1196 (May12, 1904).

Supplementary Material (1)

NameDescription
» Code 1       Mathematica notebooks containing detailed calculations of image positions for a general system of two skew ideal lenses.

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Figures (9)

Fig. 1.
Fig. 1. Telescope-like behavior of a combination of two skew lenses, L1 and L2. A bundle of parallel light rays (thin solid lines) is incident on lens L1 with a direction such that it is focused to a point on the intersection line I between the image-sided focal plane of lens 1, F1, and the object-sided focal plane of lens 2, F2 (dashed lines). Because I lies in the object-sided focal plane of L2, the bundle is parallel again after transmission through both lenses. The lens planes are shown as thick solid lines; P1 and P2 are the principal points of the two lenses.
Fig. 2.
Fig. 2. Geometry of a system of two coaxial lenses, L1 and L2, separated by a distance d. The optical axes of the individual lenses coincide with the z axis. The focal planes, F and F, and the principal planes, P and P, of the combined system are shown as dashed lines.
Fig. 3.
Fig. 3. Coordinates for single-lens imaging. A pair of conjugate positions, O and I, is shown. In a Cartesian (x,y,z) coordinate system, placed such that the lens is in the z=0 plane and the position of the principal point, P, coincides with the origin, the relationships between the coordinates of O and I have the standard form, given by Eqs. (6) and (7). The object and image positions can alternatively be expressed in skew coordinates with basis vectors u^, v^, and w^, with the imaging equations retaining their standard form. The figure shows a cross section in the y=v=0 plane.
Fig. 4.
Fig. 4. General system of two lenses L1 and L2 (thick cyan lines). Our choice of optical axis for this two-lens system, the two-lens optical axis (dashed line), passes through both principal points P1 and P2. ϕ1 and ϕ2 are the angles between the lens normals and the two-lens optical axis.
Fig. 5.
Fig. 5. System of two parallel but non-coaxial lenses. The red dashed lines correspond to the focal lines of this system and the green dashed lines correspond to its principal lines with principal points P and P. It is clear that point P is imaged to a point P by this optical device.
Fig. 6.
Fig. 6. Construction of the (a) image-sided and (b) object-sided two-lens focal planes, F and F. The image-sided two-lens focal plane, F, is the image due to L2 of the image-sided focal plane F1 of lens L1. Similarly, F is imaged by L1 to the object-sided focal plane F2 of L2. The image- and object-sided focal points, F and F, are the intersections of the corresponding focal planes with the optical axis.
Fig. 7.
Fig. 7. Construction of the two-lens object- and image-sided principal planes, P and P. P is the object-sided transverse plane through V, the line where the lens planes intersect; P is the image-sided transverse plane through V. The corresponding principal points P and P are the intersections of P and P with the optical axis. The intermediate image of the object-sided principal plane P due to lens L1 is the plane through both lines V and I. It intersects the optical axis at point wim.
Fig. 8.
Fig. 8. Geometry of a system of two 2D skew lenses, and definition of the two-lens lens-imaging coordinates. The lens-imaging basis is given by vectors u and w^ in object space and u and w^ in image space. The origins of the lens-imaging coordinates coincide with two-lens principal points P and P, respectively. For the purposes of calculations, a Cartesian coordinate system (x,z) has been chosen such that z-axis coincides with the generalized optical axis of the two-lens system (i.e., a line passing through both principal points P1 and P2 of the individual lenses) and the origin coincides with principal point P1 of lens L1.
Fig. 9.
Fig. 9. Definition of the object- and image-space lens-imaging coordinates in three dimensions. Object and image space are described in terms of skew coordinate systems with basis vectors u, v^, w^ (object space) and u, v^, w^ (image space). Note that the basis vectors in the u and u directions are not normalized. u lies on the intersection of the object-sided principal plane, P, and the median plane, M; u lies on the intersection of the image-sided principal plane, P, and the median plane, M. P and P are the object- and image-sided principal points.

Tables (1)

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Table 1. Comparison between Scaled Buchroeder Coordinates and the Lens-Imaging Coordinates Introduced Herea

Equations (56)

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x=a1x+b1y+c1z+d1a0x+b0y+c0z+d0,
y=a2x+b2y+c2z+d2a0x+b0y+c0z+d0,
z=a3x+b3y+c3z+d3a0x+b0y+c0z+d0,
x=axz+d,y=byz+d,z=czz+d,
x=fxz+f,y=fyz+f,z=fzz+f,
1z+1z=1f
xx=yy=zz.
f=f1f2f1+f2d,
Pz=dff2,Pz=ddff1.
1zPz+1zPz=1f,
xx=yy=zPzzPz.
g=fcosα,
1w+1w=1g
uu=vv=ww
g1=f1cosφ1,g2=f2cosφ2,
f=g1g2g1+g2d,
Pw=dfg2,
Pw=ddfg1.
1wPw+1wPw=1f.
u=xztanα,v=y,w=zcosα.
1wPw+1wPw=1F,uu=vv=wPwwPw,
1zPz+1zPz=1f,xPztanαxPztanα=yy=zPzzPz.
Fw=Pwf,
Fw=Pw+f.
1Fw+1dg2=1g1,
Fw=g1dg1g2g1+g2d.
Fw=d+g1g2g2dg1+g2d.
wim=dg1g1+g2.
1Pw+1wim=1g1
Pw=dfg2,
1wimd+1Pwd=1g2,
(uw)=fw+f(uw),
x=ko(zPzw)objectspace,
x=ki(zPzw)imagespace,
koxIzI=cotφ1dg1g2dg2g1(cotφ1/cotφ2),
kixIzId=cotφ2dg1g2dg1g2(cotφ2/cotφ1),
x=u,x=u.
w=zPzxko,u=x.
(uw)=(101ko1)(xzPz).
(xzPz)=(101ki1)(uw).
x=fxf+zxkoPz,
z=Pz+f(zxkikokikoPz)f+zxkoPz.
IP=ff+(OP)·n^(OP),
Ti1=(100tanαki1sinα1ki0cosα),
To=(10001tanα01kocosα1cosα).
IP=Ti1(uvw)=fw+fTi1To(OP),
(xyPzsinαzPzcosα)=fw+f(100kokikikotanα10kokikiko01)(xyPzsinαzPzcosα).
x=axz+d,z=czz+d.
x=uuPcosθwsinθ,z=wcosθ,x=uuPcosθwsinθ,z=wcosθ,
uP=uP=f(1ki+1/ki1ko+1/ko)
tanθ=1ko,tanθ=1ki,
a=f1+1ki21+1ko2,
c=f1+1ki2,
d=f1+1ko2.
cotθ=z(zo+x/q,x)z(zo,0)x(zo+x/q,x)x(zo,0)=1ki+(1q1ko)ff+zoPz.
1q1ki=(1q1ko)ff+zoPz,

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