Abstract

We describe the most general homogenous, planar, light-ray-direction-changing sheet that performs one-to-one imaging between object space and image space. This is a nontrivial special case (of the sheet being homogenous) of an earlier result [Opt. Commun. 282, 2480 (2009)]. Such a sheet can be realized, approximately, with generalized confocal lenslet arrays.

© 2013 Optical Society of America

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References

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  1. J. C. Maxwell, “Problems (3),” The Cambridge and Dublin Mathematical Journal 8, 188 (1853).
  2. J. C. Maxwell, “Solutions of problems (prob. 3, vol. viii. p. 188),” The Cambridge and Dublin Mathematical Journal 9, 9–11 (1854).
  3. R. K. Luneburg, Mathematical Theory of Optics (Brown University, 1944), pp. 189–213.
  4. J. E. Eaton, “On spherically symmetric lenses,” IRE Trans. Antennas Propag. 4, 66–71 (1952).
    [CrossRef]
  5. T. Tyc, L. Herzánová, M. Šarbort, and K. Bering, “Absolute instruments and perfect imaging in geometrical optics,” New J. Phys. 13, 115004 (2011).
    [CrossRef]
  6. M. Šarbort and T. Tyc, “Spherical media and geodesic lenses in geometrical optics,” J. Opt. 14, 075705 (2012).
    [CrossRef]
  7. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
    [CrossRef]
  8. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
    [CrossRef]
  9. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247–8256 (2006).
    [CrossRef]
  10. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
    [CrossRef]
  11. U. Leonhardt, “Perfect imaging without negative refraction,” New J. Phys. 11, 093040 (2009).
    [CrossRef]
  12. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
    [CrossRef]
  13. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [CrossRef]
  14. J. Courtial, “Ray-optical refraction with confocal lenslet arrays,” New J. Phys. 10, 083033 (2008).
    [CrossRef]
  15. A. C. Hamilton and J. Courtial, “Metamaterials for light rays: ray optics without wave-optical analog in the ray-optics limit,” New J. Phys. 11, 013042 (2009).
    [CrossRef]
  16. J. Courtial, “Geometric limits to geometric optical imaging with infinite, planar, non-absorbing sheets,” Opt. Commun. 282, 2480–2483 (2009).
    [CrossRef]
  17. 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]
  18. A. C. Hamilton and J. Courtial, “Generalized refraction using lenslet arrays,” J. Opt. A 11, 065502 (2009).
    [CrossRef]
  19. J. Courtial, “Standard and non-standard metarefraction with confocal lenslet arrays,” Opt. Commun. 282, 2634–2641 (2009).
    [CrossRef]
  20. T. Maceina, G. Juzeliūnas, and J. Courtial, “Quantifying metarefraction with confocal lenslet arrays,” Opt. Commun. 284, 5008–5019 (2011).
    [CrossRef]
  21. S. Oxburgh, C. D. White, G. Antoniou, and J. Courtial, “Generalised law of refraction for generalised confocal lenslet arrays,” Opt. Commun. (to be published).
  22. D. Lambert, A. C. Hamilton, G. Constable, H. Snehanshu, S. Talati, and J. Courtial, “TIM, a ray-tracing program for METATOY research and its dissemination,” Comput. Phys. Commun. 183, 711–732 (2012).
    [CrossRef]
  23. S. Oxburgh, T. Tyc, and J. Courtial, “Dr TIM: ray-tracer TIM, with additional specialist capabilities,” Comput. Phys. Commun. (submitted).
  24. 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]

2012 (3)

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

D. Lambert, A. C. Hamilton, G. Constable, H. Snehanshu, S. Talati, and J. Courtial, “TIM, a ray-tracing program for METATOY research and its dissemination,” Comput. Phys. Commun. 183, 711–732 (2012).
[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]

2011 (3)

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]

T. Tyc, L. Herzánová, M. Šarbort, and K. Bering, “Absolute instruments and perfect imaging in geometrical optics,” New J. Phys. 13, 115004 (2011).
[CrossRef]

T. Maceina, G. Juzeliūnas, and J. Courtial, “Quantifying metarefraction with confocal lenslet arrays,” Opt. Commun. 284, 5008–5019 (2011).
[CrossRef]

2009 (5)

U. Leonhardt, “Perfect imaging without negative refraction,” New J. Phys. 11, 093040 (2009).
[CrossRef]

A. C. Hamilton and J. Courtial, “Metamaterials for light rays: ray optics without wave-optical analog in the ray-optics limit,” New J. Phys. 11, 013042 (2009).
[CrossRef]

J. Courtial, “Geometric limits to geometric optical imaging with infinite, planar, non-absorbing sheets,” Opt. Commun. 282, 2480–2483 (2009).
[CrossRef]

A. C. Hamilton and J. Courtial, “Generalized refraction using lenslet arrays,” J. Opt. A 11, 065502 (2009).
[CrossRef]

J. Courtial, “Standard and non-standard metarefraction with confocal lenslet arrays,” Opt. Commun. 282, 2634–2641 (2009).
[CrossRef]

2008 (1)

J. Courtial, “Ray-optical refraction with confocal lenslet arrays,” New J. Phys. 10, 083033 (2008).
[CrossRef]

2007 (1)

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef]

2006 (3)

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

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

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247–8256 (2006).
[CrossRef]

2005 (1)

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[CrossRef]

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef]

1952 (1)

J. E. Eaton, “On spherically symmetric lenses,” IRE Trans. Antennas Propag. 4, 66–71 (1952).
[CrossRef]

1854 (1)

J. C. Maxwell, “Solutions of problems (prob. 3, vol. viii. p. 188),” The Cambridge and Dublin Mathematical Journal 9, 9–11 (1854).

1853 (1)

J. C. Maxwell, “Problems (3),” The Cambridge and Dublin Mathematical Journal 8, 188 (1853).

Aieta, F.

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]

Alekseyev, L. V.

Antoniou, G.

S. Oxburgh, C. D. White, G. Antoniou, and J. Courtial, “Generalised law of refraction for generalised confocal lenslet arrays,” Opt. Commun. (to be published).

Bering, K.

T. Tyc, L. Herzánová, M. Šarbort, and K. Bering, “Absolute instruments and perfect imaging in geometrical optics,” New J. Phys. 13, 115004 (2011).
[CrossRef]

Capasso, F.

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]

Constable, G.

D. Lambert, A. C. Hamilton, G. Constable, H. Snehanshu, S. Talati, and J. Courtial, “TIM, a ray-tracing program for METATOY research and its dissemination,” Comput. Phys. Commun. 183, 711–732 (2012).
[CrossRef]

Courtial, J.

D. Lambert, A. C. Hamilton, G. Constable, H. Snehanshu, S. Talati, and J. Courtial, “TIM, a ray-tracing program for METATOY research and its dissemination,” Comput. Phys. Commun. 183, 711–732 (2012).
[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. Maceina, G. Juzeliūnas, and J. Courtial, “Quantifying metarefraction with confocal lenslet arrays,” Opt. Commun. 284, 5008–5019 (2011).
[CrossRef]

J. Courtial, “Standard and non-standard metarefraction with confocal lenslet arrays,” Opt. Commun. 282, 2634–2641 (2009).
[CrossRef]

J. Courtial, “Geometric limits to geometric optical imaging with infinite, planar, non-absorbing sheets,” Opt. Commun. 282, 2480–2483 (2009).
[CrossRef]

A. C. Hamilton and J. Courtial, “Metamaterials for light rays: ray optics without wave-optical analog in the ray-optics limit,” New J. Phys. 11, 013042 (2009).
[CrossRef]

A. C. Hamilton and J. Courtial, “Generalized refraction using lenslet arrays,” J. Opt. A 11, 065502 (2009).
[CrossRef]

J. Courtial, “Ray-optical refraction with confocal lenslet arrays,” New J. Phys. 10, 083033 (2008).
[CrossRef]

S. Oxburgh, C. D. White, G. Antoniou, and J. Courtial, “Generalised law of refraction for generalised confocal lenslet arrays,” Opt. Commun. (to be published).

S. Oxburgh, T. Tyc, and J. Courtial, “Dr TIM: ray-tracer TIM, with additional specialist capabilities,” Comput. Phys. Commun. (submitted).

Eaton, J. E.

J. E. Eaton, “On spherically symmetric lenses,” IRE Trans. Antennas Propag. 4, 66–71 (1952).
[CrossRef]

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[CrossRef]

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.

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]

Hamilton, A. C.

D. Lambert, A. C. Hamilton, G. Constable, H. Snehanshu, S. Talati, and J. Courtial, “TIM, a ray-tracing program for METATOY research and its dissemination,” Comput. Phys. Commun. 183, 711–732 (2012).
[CrossRef]

A. C. Hamilton and J. Courtial, “Metamaterials for light rays: ray optics without wave-optical analog in the ray-optics limit,” New J. Phys. 11, 013042 (2009).
[CrossRef]

A. C. Hamilton and J. Courtial, “Generalized refraction using lenslet arrays,” J. Opt. A 11, 065502 (2009).
[CrossRef]

Herzánová, L.

T. Tyc, L. Herzánová, M. Šarbort, and K. Bering, “Absolute instruments and perfect imaging in geometrical optics,” New J. Phys. 13, 115004 (2011).
[CrossRef]

Jacob, Z.

Juzeliunas, G.

T. Maceina, G. Juzeliūnas, and J. Courtial, “Quantifying metarefraction with confocal lenslet arrays,” Opt. Commun. 284, 5008–5019 (2011).
[CrossRef]

Kats, M. A.

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]

Lambert, D.

D. Lambert, A. C. Hamilton, G. Constable, H. Snehanshu, S. Talati, and J. Courtial, “TIM, a ray-tracing program for METATOY research and its dissemination,” Comput. Phys. Commun. 183, 711–732 (2012).
[CrossRef]

Lee, H.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[CrossRef]

Leonhardt, U.

U. Leonhardt, “Perfect imaging without negative refraction,” New J. Phys. 11, 093040 (2009).
[CrossRef]

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

Liu, Z.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (Brown University, 1944), pp. 189–213.

Maceina, T.

T. Maceina, G. Juzeliūnas, and J. Courtial, “Quantifying metarefraction with confocal lenslet arrays,” Opt. Commun. 284, 5008–5019 (2011).
[CrossRef]

Maxwell, J. C.

J. C. Maxwell, “Solutions of problems (prob. 3, vol. viii. p. 188),” The Cambridge and Dublin Mathematical Journal 9, 9–11 (1854).

J. C. Maxwell, “Problems (3),” The Cambridge and Dublin Mathematical Journal 8, 188 (1853).

Narimanov, E.

Oxburgh, S.

S. Oxburgh, T. Tyc, and J. Courtial, “Dr TIM: ray-tracer TIM, with additional specialist capabilities,” Comput. Phys. Commun. (submitted).

S. Oxburgh, C. D. White, G. Antoniou, and J. Courtial, “Generalised law of refraction for generalised confocal lenslet arrays,” Opt. Commun. (to be published).

Pendry, J. B.

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

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef]

Šarbort, M.

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

T. Tyc, L. Herzánová, M. Šarbort, and K. Bering, “Absolute instruments and perfect imaging in geometrical optics,” New J. Phys. 13, 115004 (2011).
[CrossRef]

Schurig, D.

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

Smith, D. R.

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

Snehanshu, H.

D. Lambert, A. C. Hamilton, G. Constable, H. Snehanshu, S. Talati, and J. Courtial, “TIM, a ray-tracing program for METATOY research and its dissemination,” Comput. Phys. Commun. 183, 711–732 (2012).
[CrossRef]

Sun, C.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[CrossRef]

Talati, S.

D. Lambert, A. C. Hamilton, G. Constable, H. Snehanshu, S. Talati, and J. Courtial, “TIM, a ray-tracing program for METATOY research and its dissemination,” Comput. Phys. Commun. 183, 711–732 (2012).
[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.

M. Šarbort and T. Tyc, “Spherical media and geodesic lenses in geometrical optics,” J. Opt. 14, 075705 (2012).
[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, L. Herzánová, M. Šarbort, and K. Bering, “Absolute instruments and perfect imaging in geometrical optics,” New J. Phys. 13, 115004 (2011).
[CrossRef]

S. Oxburgh, T. Tyc, and J. Courtial, “Dr TIM: ray-tracer TIM, with additional specialist capabilities,” Comput. Phys. Commun. (submitted).

White, C. D.

S. Oxburgh, C. D. White, G. Antoniou, and J. Courtial, “Generalised law of refraction for generalised confocal lenslet arrays,” Opt. Commun. (to be published).

Xiong, Y.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef]

Yu, N.

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]

Zhang, X.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[CrossRef]

Comput. Phys. Commun. (1)

D. Lambert, A. C. Hamilton, G. Constable, H. Snehanshu, S. Talati, and J. Courtial, “TIM, a ray-tracing program for METATOY research and its dissemination,” Comput. Phys. Commun. 183, 711–732 (2012).
[CrossRef]

IRE Trans. Antennas Propag. (1)

J. E. Eaton, “On spherically symmetric lenses,” IRE Trans. Antennas Propag. 4, 66–71 (1952).
[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. A (1)

A. C. Hamilton and J. Courtial, “Generalized refraction using lenslet arrays,” J. Opt. A 11, 065502 (2009).
[CrossRef]

J. Opt. Soc. Am. A (1)

New J. Phys. (4)

U. Leonhardt, “Perfect imaging without negative refraction,” New J. Phys. 11, 093040 (2009).
[CrossRef]

J. Courtial, “Ray-optical refraction with confocal lenslet arrays,” New J. Phys. 10, 083033 (2008).
[CrossRef]

A. C. Hamilton and J. Courtial, “Metamaterials for light rays: ray optics without wave-optical analog in the ray-optics limit,” New J. Phys. 11, 013042 (2009).
[CrossRef]

T. Tyc, L. Herzánová, M. Šarbort, and K. Bering, “Absolute instruments and perfect imaging in geometrical optics,” New J. Phys. 13, 115004 (2011).
[CrossRef]

Opt. Commun. (3)

J. Courtial, “Geometric limits to geometric optical imaging with infinite, planar, non-absorbing sheets,” Opt. Commun. 282, 2480–2483 (2009).
[CrossRef]

J. Courtial, “Standard and non-standard metarefraction with confocal lenslet arrays,” Opt. Commun. 282, 2634–2641 (2009).
[CrossRef]

T. Maceina, G. Juzeliūnas, and J. Courtial, “Quantifying metarefraction with confocal lenslet arrays,” Opt. Commun. 284, 5008–5019 (2011).
[CrossRef]

Opt. Express (1)

Phys. Rev. Lett. (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef]

Science (5)

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[CrossRef]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[CrossRef]

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

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

The Cambridge and Dublin Mathematical Journal (2)

J. C. Maxwell, “Problems (3),” The Cambridge and Dublin Mathematical Journal 8, 188 (1853).

J. C. Maxwell, “Solutions of problems (prob. 3, vol. viii. p. 188),” The Cambridge and Dublin Mathematical Journal 9, 9–11 (1854).

Other (3)

R. K. Luneburg, Mathematical Theory of Optics (Brown University, 1944), pp. 189–213.

S. Oxburgh, C. D. White, G. Antoniou, and J. Courtial, “Generalised law of refraction for generalised confocal lenslet arrays,” Opt. Commun. (to be published).

S. Oxburgh, T. Tyc, and J. Courtial, “Dr TIM: ray-tracer TIM, with additional specialist capabilities,” Comput. Phys. Commun. (submitted).

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Figures (5)

Fig. 1.
Fig. 1.

Light-ray trajectories (red lines) through an imaging sheet (thick, vertical, turquoise line). The imaging sheet refracts like a combination of an idealized thin lens of focal length f and CLAs [14] (which strain image space by a factor η in the axial direction). F and G are the object- and image-sided focal points, located, respectively, a distance f in front of the sheet and a distance g=ηf behind it. The solid rays are involved in imaging P to P; rays 1 to 3 are the principal rays through the object- and image-sided focal points and through the intersection between the sheet and the optical axis, respectively. Note that ray 3 does not pass straight through the sheet, but other rays do (for example, ray 4). Apart from passing through P and P, ray 5 is not special in any way; we call the point where it intersects the sheet R. Ray 6 (dashed) is the principal ray through R that passes through the object-sided focal point and is parallel to the optical axis in image space.

Fig. 2.
Fig. 2.

Light-ray trajectories (red lines) through a homogeneous imaging sheet (thick, vertical, turquoise line). In this example, the law of refraction from a point, R, on the (inhomogeneous) imaging sheet shown in Fig. 1 applies across the entire sheet. Two rays, marked 5 and 6, with the same incident and outgoing directions as the corresponding rays through R shown in Fig. 1 are used to construct the position of the image of a point Q, Q. Ray 7 is an example of an additional light ray that passes through Q. As the sheet images all points, after transmission through the sheet, the ray (or its straight-line continuation) passes through Q.

Fig. 3.
Fig. 3.

Fine structure of the gCLAs relevant in the context of this paper. The gCLAs consist of arrays of telescopelets, each comprising two lenslets sharing a common focal plane (dotted vertical line), forming a sheet. One telescopelet is surrounded by a dashed gray line. The focal length of the lenslets is f1 and f2=ηf1, respectively. Both optical axes of the two lenslets are perpendicular to the plane on which the sheet of gCLAs is centered (here the z=0 plane). The optical axis of the second lenslet is offset with respect to that of the first by δxf1 in the x direction and by δyf1 in the y direction.

Fig. 4.
Fig. 4.

Law of refraction for a homogeneous imaging sheet. The relationship between the vectors d and d, which point in the direction of the incident and outgoing light-ray directions, respectively, can be derived from the light-ray trajectory (dotted line) through a point P and its image, P, via an arbitrary point S on the imaging sheet. The diagram is drawn for η>0, and so the image is on the same side of the sheet as the object. The object is drawn to be real, and the image is virtual.

Fig. 5.
Fig. 5.

Generalized gCLAs as homogeneous imaging sheets. In these ray-tracing simulations, an extended object is seen through an example of gCLAs. The depth of focus is relatively short, blurring any part of the image that is not in focus. Focusing on different planes brings different parts of the object into sharp focus, demonstrating that the gCLAs image all those different parts (as they should—as imaging sheets, they must image all object space). The extended object is a three-dimensional lattice of colored cylinders. The parameters of the gCLAs are η=2, δx=0.2, and δx=0.3; they are positioned in a plane a distance of 10 floor-tile lengths in front of the camera. The focusing distance is z (again in floor-tile lengths). The simulations were performed using the open-source ray-tracing program TIM [22,23].

Equations (16)

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

xx=ffo,yy=ffo,fo+gi=1,
xx=ff+z,yy=ff+z,fz+gz=1.
x˜=xRx,y˜=yRy,z˜=z,x˜=xRx,y˜=yRy,z˜=z,
Rx+x˜Rx+x˜=ff+z˜,Ry+y˜Ry+y˜=ff+z˜.
tx=Rxf,ty=Ryf.
x˜=ff+z˜(ftx+x˜)ftx=ff+z˜(x˜z˜tx).
x˜=x˜z˜tx.
y˜=y˜z˜ty.
z˜=ηz˜,
x=xztx,y=yzty,z=ηz.
P=Pz(txty1η),
dx=dxdzδxη,dy=dydzδyη,dz=dz,
dx=Sxx,dy=Syy,dz=z.
dx=Sxx,dy=Syy,dz=z.
dxSxx+ztxη=dxdztxη,dySyy+ztyη=dydztyη,dzz=dz.
δx=tx,δy=ty,

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