Abstract

We present the theory of ray-optical transformation optics (RTO) with ideal thin lenses and show that ideal-thin-lens RTO devices are omnidirectional lenses. Key to designing such devices are two theorems, the loop-imaging theorem, and the edge-imaging theorem, which ensure that the interior physical space is distorted in the same way for all viewing directions. We discuss the possibility of realising such devices using lens holograms or Fresnel lenses, as both are in principle capable of changing the directions of rays incident from a specific point precisely like an ideal thin lens, thereby enabling macroscopic and broad-band RTO devices that work for at least one viewing position. Even when restricted in this way, our work opens up new possibilities in ray optics. Our devices have the potential to form the basis of new microscope objectives, virtual-reality headsets, and medical spectacles.

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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    [Crossref]
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2017 (2)

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]

J. C. Miñano, P. Benitez, D. Grabovičkić, P. Zamora, M. Buljan, and B. Narasimhan, “Time multiplexing for increased FOV and resolution in virtual reality,” Proc. SPIE 10335, 1033504 (2017).
[Crossref]

2016 (9)

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]

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

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 (2016).
[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] [PubMed]

F. Monticone and A. Alù, “Invisibility exposed: physical bounds on passive cloaking,” Optica 3, 718–724 (2016).
[Crossref]

J. S. Choi and J. C. Howell, “Digital integral cloaking,” Optica 3, 536–540 (2016).
[Crossref]

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, T. Sharpe, and J. Courtial, “Large-scale, white-light, transformation optics using integral imaging,” J. Opt. 18, 044009 (2016).
[Crossref]

T. Tyc, S. Oxburgh, E. N. Cowie, G. J. Chaplain, G. Macauley, C. D. White, and J. Courtial, “Omni-directional transformation-optics cloak made from lenses and glenses,” J. Opt. Soc. Am. A 33, 1032–1040 (2016).
[Crossref]

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]

2015 (3)

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] [PubMed]

G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat. Nanotech. 10, 308 (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] [PubMed]

2014 (4)

S. Oxburgh, T. Tyc, and J. Courtial, “Dr TIM: Ray-tracer TIM, with additional specialist capabilities,” Comp. Phys. Commun. 185, 1027–1037 (2014).
[Crossref]

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

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

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, and J. Courtial, “Transformation optics with windows,” Proc. SPIE 9193, 91931E (2014).
[Crossref]

2013 (4)

H. Chen, B. Zheng, L. Shen, H. Wang, X. Zhang, N. Zheludev, and B. Zhang, “Ray-optics cloaking devices for large objects in incoherent natural light,” Nature Commun. 4, 2652 (2013).
[Crossref]

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

S. Oxburgh and J. Courtial, “Perfect imaging with planar interfaces,” J. Opt. Soc. Am. A 30, 2334–2338 (2013).
[Crossref]

J. Lemos and E. Eggenberger, “Clinical utility and assessment of cyclodeviation,” Current Opinion in Ophthalmology 24, 558–565 (2013).
[Crossref] [PubMed]

2012 (1)

2011 (1)

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] [PubMed]

2006 (2)

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

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

1994 (1)

S. W. Hell, S. Lindek, C. Cremer, and E. H. K. Stelzer, “Measurement of the 4pi-confocal point spread function proves 75 nm axial resolution,” Appl. Phys. Lett. 64, 1335–1337 (1994).
[Crossref]

1908 (1)

G. Lippmann, “La photographie intégrale,” C. R. Hebd. Acad. Sci. 146, 446–451 (1908).

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]

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

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] [PubMed]

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] [PubMed]

Alù, A.

Ambrosio, A.

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

Antoniou, G.

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, T. Sharpe, and J. Courtial, “Large-scale, white-light, transformation optics using integral imaging,” J. Opt. 18, 044009 (2016).
[Crossref]

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, and J. Courtial, “Transformation optics with windows,” Proc. SPIE 9193, 91931E (2014).
[Crossref]

Arbabi, A.

Arbabi, E.

Belín, J.

Benitez, P.

J. C. Miñano, P. Benitez, D. Grabovičkić, P. Zamora, M. Buljan, and B. Narasimhan, “Time multiplexing for increased FOV and resolution in virtual reality,” Proc. SPIE 10335, 1033504 (2017).
[Crossref]

Buljan, M.

J. C. Miñano, P. Benitez, D. Grabovičkić, P. Zamora, M. Buljan, and B. Narasimhan, “Time multiplexing for increased FOV and resolution in virtual reality,” Proc. SPIE 10335, 1033504 (2017).
[Crossref]

Capasso, 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, A. Ambrosio, P. Kanhaiya, and F. Capasso, “Broadband and chiral binary dielectric meta-holograms,” Science Advances 2, e1501258 (2016).
[Crossref] [PubMed]

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] [PubMed]

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] [PubMed]

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

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

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] [PubMed]

Chaplain, G. J.

Chen, H.

H. Chen, B. Zheng, L. Shen, H. Wang, X. Zhang, N. Zheludev, and B. Zhang, “Ray-optics cloaking devices for large objects in incoherent natural light,” Nature Commun. 4, 2652 (2013).
[Crossref]

Chen, W. T.

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] [PubMed]

Choi, J. S.

Courtial, J.

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, T. Sharpe, and J. Courtial, “Large-scale, white-light, transformation optics using integral imaging,” J. Opt. 18, 044009 (2016).
[Crossref]

T. Tyc, S. Oxburgh, E. N. Cowie, G. J. Chaplain, G. Macauley, C. D. White, and J. Courtial, “Omni-directional transformation-optics cloak made from lenses and glenses,” J. Opt. Soc. Am. A 33, 1032–1040 (2016).
[Crossref]

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]

S. Oxburgh, T. Tyc, and J. Courtial, “Dr TIM: Ray-tracer TIM, with additional specialist capabilities,” Comp. Phys. Commun. 185, 1027–1037 (2014).
[Crossref]

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, and J. Courtial, “Transformation optics with windows,” Proc. SPIE 9193, 91931E (2014).
[Crossref]

S. Oxburgh and J. Courtial, “Perfect imaging with planar interfaces,” J. Opt. Soc. Am. A 30, 2334–2338 (2013).
[Crossref]

J. Courtial and T. Tyc, “Generalised laws of refraction that can lead to wave-optically forbidden light-ray fields,” J. Opt. Soc. Am. A 29, 1407–1411 (2012).
[Crossref]

T. Tyc, S. Oxburgh, C. D. White, E. N. Cowie, and J. Courtial, “On the loop-imaging theorem of transformation optics,” in preparation (2018).

J. Courtial, S. Oxburgh, E. N. Cowie, C. D. White, and T. Tyc, “Omni-directional cloaking using ideal lenses,” in preparation (2018).

Cowie, E. N.

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]

T. Tyc, S. Oxburgh, E. N. Cowie, G. J. Chaplain, G. Macauley, C. D. White, and J. Courtial, “Omni-directional transformation-optics cloak made from lenses and glenses,” J. Opt. Soc. Am. A 33, 1032–1040 (2016).
[Crossref]

J. Courtial, S. Oxburgh, E. N. Cowie, C. D. White, and T. Tyc, “Omni-directional cloaking using ideal lenses,” in preparation (2018).

T. Tyc, S. Oxburgh, C. D. White, E. N. Cowie, and J. Courtial, “On the loop-imaging theorem of transformation optics,” in preparation (2018).

Cremer, C.

S. W. Hell, S. Lindek, C. Cremer, and E. H. K. Stelzer, “Measurement of the 4pi-confocal point spread function proves 75 nm axial resolution,” Appl. Phys. Lett. 64, 1335–1337 (1994).
[Crossref]

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] [PubMed]

Eggenberger, E.

J. Lemos and E. Eggenberger, “Clinical utility and assessment of cyclodeviation,” Current Opinion in Ophthalmology 24, 558–565 (2013).
[Crossref] [PubMed]

Faraon, A.

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] [PubMed]

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] [PubMed]

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] [PubMed]

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] [PubMed]

Goodman, D. S.

D. S. Goodman, “General principles of geometric optics,” in “Handbook of Optics. Volume I: Fundamentals, Techniques, and Design,” 2nd ed., M. Bass, E. W. V. Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw-Hill, 1995), chap. 1.15, pp. 1.60–1.68.

Grabovickic, D.

J. C. Miñano, P. Benitez, D. Grabovičkić, P. Zamora, M. Buljan, and B. Narasimhan, “Time multiplexing for increased FOV and resolution in virtual reality,” Proc. SPIE 10335, 1033504 (2017).
[Crossref]

Hell, S. W.

S. W. Hell, S. Lindek, C. Cremer, and E. H. K. Stelzer, “Measurement of the 4pi-confocal point spread function proves 75 nm axial resolution,” Appl. Phys. Lett. 64, 1335–1337 (1994).
[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,” Science Advances 2, e1501258 (2016).
[Crossref] [PubMed]

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] [PubMed]

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] [PubMed]

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

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] [PubMed]

Kenney, M.

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

Khorasaninejad, M.

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, A. Ambrosio, P. Kanhaiya, and F. Capasso, “Broadband and chiral binary dielectric meta-holograms,” Science Advances 2, e1501258 (2016).
[Crossref] [PubMed]

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] [PubMed]

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] [PubMed]

Kildishev, A. V.

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

Kivshar, Y. S.

Kravchenko, I.

Kruk, S.

Lemos, J.

J. Lemos and E. Eggenberger, “Clinical utility and assessment of cyclodeviation,” Current Opinion in Ophthalmology 24, 558–565 (2013).
[Crossref] [PubMed]

Leonhardt, U.

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

Li, G.

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

Li, T.

Lindek, S.

S. W. Hell, S. Lindek, C. Cremer, and E. H. K. Stelzer, “Measurement of the 4pi-confocal point spread function proves 75 nm axial resolution,” Appl. Phys. Lett. 64, 1335–1337 (1994).
[Crossref]

Lippmann, G.

G. Lippmann, “La photographie intégrale,” C. R. Hebd. Acad. Sci. 146, 446–451 (1908).

Macauley, G.

Miñano, J. C.

J. C. Miñano, P. Benitez, D. Grabovičkić, P. Zamora, M. Buljan, and B. Narasimhan, “Time multiplexing for increased FOV and resolution in virtual reality,” Proc. SPIE 10335, 1033504 (2017).
[Crossref]

Monticone, F.

Mühlenbernd, H.

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

Narasimhan, B.

J. C. Miñano, P. Benitez, D. Grabovičkić, P. Zamora, M. Buljan, and B. Narasimhan, “Time multiplexing for increased FOV and resolution in virtual reality,” Proc. SPIE 10335, 1033504 (2017).
[Crossref]

Neshev, D. N.

Ni, X.

X. Ni, A. V. Kildishev, and V. M. Shalaev, “Metasurface holograms for visible light,” Nature Communications 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] [PubMed]

Orife, E.

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, T. Sharpe, and J. Courtial, “Large-scale, white-light, transformation optics using integral imaging,” J. Opt. 18, 044009 (2016).
[Crossref]

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, and J. Courtial, “Transformation optics with windows,” Proc. SPIE 9193, 91931E (2014).
[Crossref]

Oxburgh, S.

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, T. Sharpe, and J. Courtial, “Large-scale, white-light, transformation optics using integral imaging,” J. Opt. 18, 044009 (2016).
[Crossref]

T. Tyc, S. Oxburgh, E. N. Cowie, G. J. Chaplain, G. Macauley, C. D. White, and J. Courtial, “Omni-directional transformation-optics cloak made from lenses and glenses,” J. Opt. Soc. Am. A 33, 1032–1040 (2016).
[Crossref]

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, and J. Courtial, “Transformation optics with windows,” Proc. SPIE 9193, 91931E (2014).
[Crossref]

S. Oxburgh, T. Tyc, and J. Courtial, “Dr TIM: Ray-tracer TIM, with additional specialist capabilities,” Comp. Phys. Commun. 185, 1027–1037 (2014).
[Crossref]

S. Oxburgh and J. Courtial, “Perfect imaging with planar interfaces,” J. Opt. Soc. Am. A 30, 2334–2338 (2013).
[Crossref]

J. Courtial, S. Oxburgh, E. N. Cowie, C. D. White, and T. Tyc, “Omni-directional cloaking using ideal lenses,” in preparation (2018).

T. Tyc, S. Oxburgh, C. D. White, E. N. Cowie, and J. Courtial, “On the loop-imaging theorem of transformation optics,” in preparation (2018).

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

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] [PubMed]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

Shalaev, V. M.

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

Sharpe, T.

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, T. Sharpe, and J. Courtial, “Large-scale, white-light, transformation optics using integral imaging,” J. Opt. 18, 044009 (2016).
[Crossref]

Shen, L.

H. Chen, B. Zheng, L. Shen, H. Wang, X. Zhang, N. Zheludev, and B. Zhang, “Ray-optics cloaking devices for large objects in incoherent natural light,” Nature Commun. 4, 2652 (2013).
[Crossref]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

Stelzer, E. H. K.

S. W. Hell, S. Lindek, C. Cremer, and E. H. K. Stelzer, “Measurement of the 4pi-confocal point spread function proves 75 nm axial resolution,” Appl. Phys. Lett. 64, 1335–1337 (1994).
[Crossref]

Tang, H.

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] [PubMed]

Tyc, T.

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]

T. Tyc, S. Oxburgh, E. N. Cowie, G. J. Chaplain, G. Macauley, C. D. White, and J. Courtial, “Omni-directional transformation-optics cloak made from lenses and glenses,” J. Opt. Soc. Am. A 33, 1032–1040 (2016).
[Crossref]

S. Oxburgh, T. Tyc, and J. Courtial, “Dr TIM: Ray-tracer TIM, with additional specialist capabilities,” Comp. Phys. Commun. 185, 1027–1037 (2014).
[Crossref]

J. Courtial and T. Tyc, “Generalised laws of refraction that can lead to wave-optically forbidden light-ray fields,” J. Opt. Soc. Am. A 29, 1407–1411 (2012).
[Crossref]

J. Courtial, S. Oxburgh, E. N. Cowie, C. D. White, and T. Tyc, “Omni-directional cloaking using ideal lenses,” in preparation (2018).

T. Tyc, S. Oxburgh, C. D. White, E. N. Cowie, and J. Courtial, “On the loop-imaging theorem of transformation optics,” in preparation (2018).

Wang, H.

H. Chen, B. Zheng, L. Shen, H. Wang, X. Zhang, N. Zheludev, and B. Zhang, “Ray-optics cloaking devices for large objects in incoherent natural light,” Nature Commun. 4, 2652 (2013).
[Crossref]

Wang, L.

White, C. D.

T. Tyc, S. Oxburgh, E. N. Cowie, G. J. Chaplain, G. Macauley, C. D. White, and J. Courtial, “Omni-directional transformation-optics cloak made from lenses and glenses,” J. Opt. Soc. Am. A 33, 1032–1040 (2016).
[Crossref]

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, T. Sharpe, and J. Courtial, “Large-scale, white-light, transformation optics using integral imaging,” J. Opt. 18, 044009 (2016).
[Crossref]

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, and J. Courtial, “Transformation optics with windows,” Proc. SPIE 9193, 91931E (2014).
[Crossref]

J. Courtial, S. Oxburgh, E. N. Cowie, C. D. White, and T. Tyc, “Omni-directional cloaking using ideal lenses,” in preparation (2018).

T. Tyc, S. Oxburgh, C. D. White, E. N. Cowie, and J. Courtial, “On the loop-imaging theorem of transformation optics,” in preparation (2018).

Yu, N.

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

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] [PubMed]

Zamora, P.

J. C. Miñano, P. Benitez, D. Grabovičkić, P. Zamora, M. Buljan, and B. Narasimhan, “Time multiplexing for increased FOV and resolution in virtual reality,” Proc. SPIE 10335, 1033504 (2017).
[Crossref]

Zentgraf, T.

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

Zhang, B.

H. Chen, B. Zheng, L. Shen, H. Wang, X. Zhang, N. Zheludev, and B. Zhang, “Ray-optics cloaking devices for large objects in incoherent natural light,” Nature Commun. 4, 2652 (2013).
[Crossref]

Zhang, S.

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

Zhang, X.

H. Chen, B. Zheng, L. Shen, H. Wang, X. Zhang, N. Zheludev, and B. Zhang, “Ray-optics cloaking devices for large objects in incoherent natural light,” Nature Commun. 4, 2652 (2013).
[Crossref]

Zheludev, N.

H. Chen, B. Zheng, L. Shen, H. Wang, X. Zhang, N. Zheludev, and B. Zhang, “Ray-optics cloaking devices for large objects in incoherent natural light,” Nature Commun. 4, 2652 (2013).
[Crossref]

Zheng, B.

H. Chen, B. Zheng, L. Shen, H. Wang, X. Zhang, N. Zheludev, and B. Zhang, “Ray-optics cloaking devices for large objects in incoherent natural light,” Nature Commun. 4, 2652 (2013).
[Crossref]

Zheng, G.

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

Zhu, A. Y.

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] [PubMed]

Appl. Phys. Lett. (1)

S. W. Hell, S. Lindek, C. Cremer, and E. H. K. Stelzer, “Measurement of the 4pi-confocal point spread function proves 75 nm axial resolution,” Appl. Phys. Lett. 64, 1335–1337 (1994).
[Crossref]

C. R. Hebd. Acad. Sci. (1)

G. Lippmann, “La photographie intégrale,” C. R. Hebd. Acad. Sci. 146, 446–451 (1908).

Comp. Phys. Commun. (1)

S. Oxburgh, T. Tyc, and J. Courtial, “Dr TIM: Ray-tracer TIM, with additional specialist capabilities,” Comp. Phys. Commun. 185, 1027–1037 (2014).
[Crossref]

Current Opinion in Ophthalmology (1)

J. Lemos and E. Eggenberger, “Clinical utility and assessment of cyclodeviation,” Current Opinion in Ophthalmology 24, 558–565 (2013).
[Crossref] [PubMed]

J. Opt. (1)

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, T. Sharpe, and J. Courtial, “Large-scale, white-light, transformation optics using integral imaging,” J. Opt. 18, 044009 (2016).
[Crossref]

J. Opt. Soc. Am. A (4)

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] [PubMed]

Nat. Mater. (1)

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

Nat. Nanotech. (1)

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

Nature Commun. (1)

H. Chen, B. Zheng, L. Shen, H. Wang, X. Zhang, N. Zheludev, and B. Zhang, “Ray-optics cloaking devices for large objects in incoherent natural light,” Nature Commun. 4, 2652 (2013).
[Crossref]

Nature Communications (1)

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

Opt. Express (1)

Optica (5)

Proc. SPIE (2)

J. C. Miñano, P. Benitez, D. Grabovičkić, P. Zamora, M. Buljan, and B. Narasimhan, “Time multiplexing for increased FOV and resolution in virtual reality,” Proc. SPIE 10335, 1033504 (2017).
[Crossref]

S. Oxburgh, C. D. White, G. Antoniou, E. Orife, and J. Courtial, “Transformation optics with windows,” Proc. SPIE 9193, 91931E (2014).
[Crossref]

Science (5)

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] [PubMed]

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

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

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

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] [PubMed]

Science Advances (1)

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

Other (4)

J. Courtial, S. Oxburgh, E. N. Cowie, C. D. White, and T. Tyc, “Omni-directional cloaking using ideal lenses,” in preparation (2018).

D. S. Goodman, “General principles of geometric optics,” in “Handbook of Optics. Volume I: Fundamentals, Techniques, and Design,” 2nd ed., M. Bass, E. W. V. Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw-Hill, 1995), chap. 1.15, pp. 1.60–1.68.

T. Tyc, S. Oxburgh, C. D. White, E. N. Cowie, and J. Courtial, “On the loop-imaging theorem of transformation optics,” in preparation (2018).

J. Courtial, T. Tyc, S. Oxburgh, J. Bělín, E. N. Cowie, and C. D. White, “Mathematica notebooks with detailed loop-imaging-theorem calculations,” figshare, https://dx.doi.org/10.6084/m9.figshare.4269701.v1 (2016).

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Figures (7)

Fig. 1
Fig. 1 Simple ideal-thin-lens structure, S, that forms an ideal-thin-lens RTO device. (a) A number of light-ray trajectories (red solid lines) through S, drawn for clarity in two dimensions (2D), are shown. The light-ray trajectories intersect at a physical-space position A inside S; the straight-line continuations (dotted red lines) of the outside segments of those same light-ray trajectories intersect in the virtual-space position A′. The ideal thin lenses (cyan lines) divide physical space inside S into cells numbered 1, 2, 3, etc.; cell 0 is the outside of the device. (b) In the full 3D structure, the respective roles of polygon vertices and edges in the 2D structure are played by polyhedron edges and faces. Each grey cylinder marks the common edge of two or more ideal thin lenses with a triangular clear aperture; wherever three of the cylinders form a triangle, this triangle forms the clear aperture of an ideal thin lens, of which there are 16 in total. For example, the shaded triangle with vertices V1, V2, and V3 forms the clear aperture of a lens labelled L, which corresponds to lens L01 in (a). V1 to V3 form an equilateral triangle; V4 to V6 lie on a straight line perpendicular to this triangle and intersecting it at its center. The principal points of all lenses in S lie on this straight line. Note that the principal point coincides with the center of the clear aperture only in the case of lens L.
Fig. 2
Fig. 2 Simulated visual effect of the ideal-lens structure S (Fig. 1(b)). In the upper images (a, c, e), S is absent, in the lower images (b, d, f) it is present. The left and center columns show the scene from two different viewing positions, from straight above in the case of the center column. The right column shows part of the images in the left column, magnified. The black and white sphere is positioned inside S, the rest of the scene (with the exception of S itself) is outside of S. Any object seen through S appears slightly darker as the ideal lenses that form S have been simulated to be slightly absorptive. The simulations were performed using an extended version of our custom raytracer Dr TIM [16].
Fig. 3
Fig. 3 Raytracing simulations of a phase-hologram realisation of the ideal-lens structure S shown in Fig. 2. When seen from the intended viewing position (a), the device does not distort the scene seen through it, exactly like the corresponding ideal-thin-lens device (b). This is not the case for any other viewing position; an example is shown in (c). The light-ray-direction change due to each hologram was calculated by treating each ray as a plane wave, adding the transverse phase gradient introduced by the hologram to the transverse wave numbers, and calculating the longitudinal wave number such that each plane wave retains its wavelength (i.e. such that k x 2 + k y 2 + k z 2 = ( 2 π / λ 2 )) — see [17] below Eq. (21). All simulations assumed perfect dispersion compensation for all visible light and were performed with an extended version of our custom raytracer Dr TIM [16].
Fig. 4
Fig. 4 Structure and parameters of the ideal-lens structure S. The red tubes indicate the edges of ideal thin lenses; each triangle formed by red tubes is the aperture of a lens, of which there are 16 in total. The lenses form a structure that can be understood as three nested tetrahedra that share a base (an equilateral triangle with vertices V1, V2 and V3, centered at P0, and with circumradius R), but that have different heights, respectively h1 (the tetrahedron with fourth vertex P1), h2 (fourth vertex P2), and h (the outermost tetrahedron with fourth vertex P3). Black spheres are centered on the positions P0 to P3, which are the positions of the principal points of the lenses. (Note that the principal point coincides with the aperture center only in the base lens.) The vertical line through the principal points is an axis of three-fold symmetry.
Fig. 5
Fig. 5 Application of the loop-imaging theorem to the edges in the ideal-lens structure. Each row corresponds to one edge geometry. The drawings sketch the edge geometries, indicating a suitable choice of coordinate system and the parameters. The raytracing images above the drawings show a cylinder-frame model of the ideal-lens structure highlighting (in red) the edges that possess the geometry shown in the corresponding drawing. (a) Three lenses, L1 to L3, intersect such that the angle from L2 to L1 is equal in magnitude but opposite in sign to the angle from L2 to L3. All three lenses share a common principal point, P, which lies on the intersection. (b) Four lenses, L4 to L7, intersect such that L5 and L7 lie in the same plane but on opposite sides of the intersection line, and L4 and L6 are arranged symmetrically with respect to this plane. The principal points lie on a straight line in the plane of L5 and L7. This straight line makes an angle δ with the intersection line. (c) Four lenses, L8 to L11, intersect. The principal points lie on a straight line that is perpendicular to the plane of L11. In all drawings, the coordinate system is chosen such that the lenses intersect along the z axis. In (a), the x axis was chosen such that the (x, z) plane is the plane of mirror symmetry.
Fig. 6
Fig. 6 Placement of the point A in the space between lenses L9 and L10 and construction of two images of A, namely A9,8, the image due to lenses L9 and L8, and A10,11, that due to L10 and L11. A lies on the intersection between the focal planes of lenses L9 and L10. For the four lenses to satisfy the loop-imaging-theorem condition, the locations of the images A9,8 and A10,11 have to coincide.
Fig. 7
Fig. 7 The different types of ideal lenses that make up the ideal-lens structure. Each frame shows a solid triangle in the plane of one of the corresponding lenses. As the ideal-lens structure is 3-fold rotationally symmetric around the vertical axis of the structure, there are three lenses of all types other than type D, of which there is only one lens.

Equations (28)

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

A = c 0 m c k j c j i A .
c 0 m c k j c j i = c 0 m c k j c j i
c 0 m c k j c j i c i j c j k c m 0 = I ,
Q = Q + ( Q P ) a ^ f ( Q P ) a ^ ( Q P ) ,
Q f 1 f 2 d Q ,
d = f 1 ( f 2 y ) 2 f 2 y cos α .
Z = d ( Q Q ) ,
f 1 + 2 f 2 cos α = 0 .
f 5 f 7 = h 5 h 7 , f 4 = f 7 ( 2 f 5 cos β + h 5 sin β sin δ ) f 5 f 7 .
f 5 = f 7 = h 7 2 sin δ tan β .
( x , y , z ) = ( x 0 t sin φ i , y 0 + t cos φ i , 0 ) ,
( x , y , z ) = ( x 0 f i sin φ i + t cos φ i , y i ± f i cos φ i + t sin φ i , s ) ,
( y y i ) = ( x x 0 ) tan φ i ± F i ,
x = F 9 + F 10 tan φ 9 tan φ 10 x 9 , 10 , y = F 9 tan φ 10 + F 10 tan φ 9 tan φ 9 tan φ 10 y 9 , 10 , z = t .
x = x 0 + t ( x 9 , 10 x 0 ) , y = y 10 + t ( y 9 , 10 y 10 ) , z = t + t 0 .
x = x 0 + t ( x 9 , 10 x 0 ) , y = y 11 + t ( y 9 , 10 y 10 ) , z = t + t 0 .
x = x 0 + s ( x 9 , 10 x 0 ) , y = y 8 + s ( y 9 , 10 y 9 ) , z = s + t 0 .
x 8 , 11 = x 0 tan φ 8 tan φ 9 + tan φ 10 tan φ 11 tan φ 10 tan φ 9 + tan φ 8 tan φ 11 ( tan φ 10 tan φ 9 ) 2 ( F 9 + F 10 ) ,
y 8 , 11 = x 0 tan φ 8 tan φ 10 tan φ 9 tan φ 11 tan φ 10 tan φ 9 + tan φ 8 tan φ 11 ( tan φ 10 tan φ 9 ) 2 ( F 9 tan φ 10 + F 10 tan φ 9 ) .
x 8 , 11 = F 8 + F 11 tan φ 11 tan φ 8 , y 8 , 11 = F 8 tan φ 11 + F 11 tan φ 8 tan φ 11 tan φ 8 .
F 8 + F 11 y 11 y 8 = y 8 + y 9 y 10 + y 11 y 9 y 10 + y 8 y 11 ( y 9 y 10 ) 2 ( F 9 + F 10 ) ,
F 8 y 11 + F 11 y 8 y 11 y 8 = y 11 y 9 y 8 y 10 y 9 y 10 + y 8 y 11 ( y 9 y 10 ) 2 ( F 9 y 10 + F 10 y 9 ) .
F 9 + F 8 y 8 y 9 = y 8 y 9 + y 10 + y 11 y 10 y 11 + y 9 y 8 ( y 10 y 11 ) 2 ( F 10 + F 11 ) ,
F 9 y 8 + F 8 y 9 y 8 y 9 = y 8 y 10 y 9 y 11 y 10 y 11 + y 9 y 8 ( y 10 y 11 ) 2 ( F 10 y 11 + F 11 y 11 ) .
f 8 = f 11 s 11 Δ y 8 , 9 Δ y 8 , 10 s 8 Δ y 9 , 11 Δ y 11 , 10 + x 0 Δ y 8 , 9 Δ y 8 , 11 s 8 Δ y 9 , 11 , f 9 = f 11 s 11 Δ y 8 , 9 Δ y 9 , 10 s 9 Δ y 8 , 11 Δ y 11 , 10 , f 10 = f 11 s 11 Δ y 8 , 10 Δ y 9 , 10 s 8 Δ y 8 , 11 Δ y 11 , 9 x 0 Δ y 10 , 9 Δ y 10 , 11 s 10 Δ y 9 , 11 ,
f D = h 1 h 1 h 1 h 1 .
f A = ( h 2 h ) ( f D ( h 1 h ) + h 1 h ) R h 1 h 2 4 h 2 + R 2 , f B = ( f D ( h 1 h ) + h 1 h ) R h 1 h 4 h 2 2 + R 2 , f C = ( h 1 h 2 ) ( f D ( h 1 h ) + h 1 h ) R h 2 h 4 h 1 2 + R 2 .
f E = ( h 1 h 2 ) ( f D ( h 1 h ) + h 1 h ) R 2 3 h 1 h 2 h , f F = ( h 2 h ) ( f D ( h 1 h ) + h 1 h ) R 2 3 h 1 h 2 h .