Abstract

We propose a three dimensional optical instrument with an isotropic gradient index in which all ray trajectories form Lissajous curves. The lens represents the first absolute optical instrument discovered to exist without spherical symmetry (other than trivial cases such as the plane mirror or conformal maps of spherically-symmetric lenses). An important property of this lens is that a three-dimensional region of space can be imaged stigmatically with no aberrations, with a point and its image not necessarily lying on a straight line with the lens center as in all other absolute optical instruments. In addition, rays in the Lissajous lens are not confined to planes. The lens can optionally be designed such that no rays except those along coordinate axes form closed trajectories, and conformal maps of the Lissajous lens form a rich new class of optical instruments.

© 2015 Optical Society of America

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References

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  1. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
    [Crossref] [PubMed]
  2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [Crossref] [PubMed]
  3. U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110–112 (2009).
    [Crossref]
  4. M. Born and E. Wolf, Principles of optics (Cambridge University, 2006).
  5. J. C. Miñano, “Perfect imaging in a homogeneous three-dimensional region,” Opt. Express 14, 9627–9635 (2006).
    [Crossref]
  6. T. Tyc, L. Herzánová, M. Šarbort, and K. Bering, “Absolute instruments and perfect imaging in geometrical optics,” New J. Phys. 13, 115004 (2011).
    [Crossref]
  7. R. K. Luneburg, Mathematical theory of optics (University of California, 1964).
  8. T. Tyc and A. J. Danner, “Frequency spectra of absolute optical instruments,” New J. Phys. 14, 085023 (2012).
    [Crossref]
  9. T. Tyc, “Spectra of absolute instruments from the WKB approximation,” New J. Phys. 15, 065005 (2013).
    [Crossref]
  10. H. Goldstein, C. Poole, and J. Safko, Classical mechanics (Addison-Wesley, 2001).
  11. U. Leonhardt and T. Philbin, Geometry and light: The science of invisibility (Dover, Mineola, 2010).
  12. M. Šarbort and T. Tyc, “Spherical media and geodesic lenses in geometrical optics,” J. Opt. 14, 075705 (2012).
    [Crossref]

2013 (1)

T. Tyc, “Spectra of absolute instruments from the WKB approximation,” New J. Phys. 15, 065005 (2013).
[Crossref]

2012 (2)

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

T. Tyc and A. J. Danner, “Frequency spectra of absolute optical instruments,” New J. Phys. 14, 085023 (2012).
[Crossref]

2011 (1)

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

2009 (1)

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110–112 (2009).
[Crossref]

2006 (3)

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]

J. C. Miñano, “Perfect imaging in a homogeneous three-dimensional region,” Opt. Express 14, 9627–9635 (2006).
[Crossref]

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]

Born, M.

M. Born and E. Wolf, Principles of optics (Cambridge University, 2006).

Danner, A. J.

T. Tyc and A. J. Danner, “Frequency spectra of absolute optical instruments,” New J. Phys. 14, 085023 (2012).
[Crossref]

Goldstein, H.

H. Goldstein, C. Poole, and J. Safko, Classical mechanics (Addison-Wesley, 2001).

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]

Leonhardt, U.

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110–112 (2009).
[Crossref]

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

U. Leonhardt and T. Philbin, Geometry and light: The science of invisibility (Dover, Mineola, 2010).

Luneburg, R. K.

R. K. Luneburg, Mathematical theory of optics (University of California, 1964).

Miñano, J. C.

Pendry, J. B.

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

Philbin, T.

U. Leonhardt and T. Philbin, Geometry and light: The science of invisibility (Dover, Mineola, 2010).

Poole, C.

H. Goldstein, C. Poole, and J. Safko, Classical mechanics (Addison-Wesley, 2001).

Safko, J.

H. Goldstein, C. Poole, and J. Safko, Classical mechanics (Addison-Wesley, 2001).

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] [PubMed]

Smith, D. R.

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

Tyc, T.

T. Tyc, “Spectra of absolute instruments from the WKB approximation,” New J. Phys. 15, 065005 (2013).
[Crossref]

T. Tyc and A. J. Danner, “Frequency spectra of absolute optical instruments,” New J. Phys. 14, 085023 (2012).
[Crossref]

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]

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110–112 (2009).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of optics (Cambridge University, 2006).

J. Opt. (1)

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

New J. Phys. (3)

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. Tyc and A. J. Danner, “Frequency spectra of absolute optical instruments,” New J. Phys. 14, 085023 (2012).
[Crossref]

T. Tyc, “Spectra of absolute instruments from the WKB approximation,” New J. Phys. 15, 065005 (2013).
[Crossref]

Opt. Express (1)

Science (3)

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]

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110–112 (2009).
[Crossref]

Other (4)

M. Born and E. Wolf, Principles of optics (Cambridge University, 2006).

H. Goldstein, C. Poole, and J. Safko, Classical mechanics (Addison-Wesley, 2001).

U. Leonhardt and T. Philbin, Geometry and light: The science of invisibility (Dover, Mineola, 2010).

R. K. Luneburg, Mathematical theory of optics (University of California, 1964).

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Figures (4)

Fig. 1
Fig. 1

Ray trajectories in a) Maxwell’s fisheye, b) the Luneburg lens, c) a Lissajous lens with a = 1, b = 2, and d) a Lissajous lens with a = 5/3, b = 2. The solid simple curves represent the line on which n = 1 in the optical case, and the outer dotted lines represent the n = 0 lens boundary. In (a–c), points A and B represent example source and image, respectively.

Fig. 2
Fig. 2

a) Lissajous lens with a = 1, b = 2, c = 1, b) Lissajous-like lens with a = 1, b → ∞, c = 1. The surface plots are the surfaces of unity index. The full red lines mark intersections of these surfaces with the xy plane while the dotted red lines mark the places in this plane where n = 0.

Fig. 3
Fig. 3

Equivalent geodesic surface to a Lissajous lens with a = 1, b = 2 for the region of the lens where n ≥ 1.

Fig. 4
Fig. 4

Time evolution of a Gaussian pulse propagating in a Lissajous lens with a = 1, b = 2, illustrating stigmatic imaging.

Equations (4)

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L = m 2 ( x ˙ 2 + y ˙ 2 + z ˙ 2 ) 2 ( x 2 a 2 + y 2 b 2 + z 2 c 2 ) ,
n ( x , y , z ) = 2 ( x a ) 2 ( y b ) 2 ( z c ) 2 ;
ψ x ( x ) = H n ( x k / a ) e k x 2 / ( 2 a ) ψ y ( y ) = H m ( y k / b ) e k y 2 / ( 2 b ) ψ z ( z ) = H p ( z k / c ) e k z 2 / ( 2 c )
k = ω C = n + 1 / 2 a + m + 1 / 2 b + p + 1 / 2 c

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