Abstract

We introduce a family of spherically symmetric gradient-index lenses forming a perfect (sharp) image of any point of the space in three dimensions, including the points of a homogeneous (constant refractive index) region. The only previously known example of an optical instrument with such properties is the plane mirror (or combinations thereof).

© 2006 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, (Pergamon, Oxford, 1989).
  2. G. W. Forbes and J. K. Wallace, "Can the bounds to system performance in geometrical optics be attained?," J. Opt. Soc. Am. A 12, 2064-2071 (1995) http://www.opticsinfobase.org/abstract.cfm?URI=josaa-12-9-2064
    [CrossRef]
  3. M. Herzberger, Modern Geometrical Optics, (Interscience, New York, 1958).
  4. S. Cornbleet, Microwave and Geometrical Optics (Academic, 1994).
  5. R.K. Luneburg, Mathematical Theory of Optics, (University of California Press, Los Angeles 1964).
  6. E. Kreyszig, Differential Geometry, (Dover, New York, 1991).
  7. D.J. Struik, Lectures on Classical Differential Geometry, (Dover, New York, 1988).
  8. J. Eaton, "On spherically symmetric lenses," IRE Transactions on Antennas and Propagation, 4, 66-71 (1952), http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=25697&arnumber=1144350&count=27&index=14
  9. A.F. Kay, "Spherically Symmetric Lenses," IRE Transactions on Antennas and Propagation, 7, 32-38, (1959), http://ieeexplore.ieee.org/search/wrapper.jsp?arnumber=1144648
    [CrossRef]
  10. S.P. Morgan, "Generalizations of Spherically Symmetric Lenses," IRE Transactions on Antennas and Propagation, 7, 342-345 (1959), http://ieeexplore.ieee.org/search/wrapper.jsp?arnumber=1144697
    [CrossRef]
  11. J.C. Miñano, P. Benítez, A. Santamaría, "Hamilton-Jacobi equation in momentum space," Opt. Express 14, 9083-9092 (2006).
    [CrossRef] [PubMed]
  12. P. Uslenghi, "Electromagnetic and Optical Behavior of Two Classes of Dielectric Lenses," IEEE Transactions on Antennas and Propagation, 17, 235-236 (1969), http://ieeexplore.ieee.org/search/wrapper.jsp?arnumber=1139390
    [CrossRef]
  13. U. Leonhardt, "Optical Conformal Mapping", Science,  312, 1777-1780 (2006), http://www.sciencemag.org/cgi/rapidpdf/1126493?ijkey=afgHAtDOTcMj.&keytype=ref&siteid=sci
    [CrossRef] [PubMed]
  14. U. Leonhardt, "Notes on Conformal Invisibility Devices", New Journal of Physics 8, 118 (2006), http://stacks.iop.org/1367-2630/8/118
    [CrossRef]
  15. M. Kerker, "Invisible bodies," J. Opt. Soc. Am. 65, 376- (1975) http://www.opticsinfobase.org/abstract.cfm?URI=josa-65-4-376
    [CrossRef]
  16. G. Gbur, "Nonradiating sources and other 'invisible' objects", in E. Wolf (Ed.), Prog. in Optics, vol. 45 (Elsevier, Amsterdam, 2003), p. 273.
    [CrossRef]
  17. E. Wolf, T. Habashy, "Invisible bodies and uniqueness of the inverse scattering problem," J. of Modern Optics,  40, 785-792 (1993) http://taylorandfrancis.metapress.com/openurl.asp?genre=article&issn=0950-0340&volume=40&issue=5&spage=785.
    [CrossRef]

2006

U. Leonhardt, "Optical Conformal Mapping", Science,  312, 1777-1780 (2006), http://www.sciencemag.org/cgi/rapidpdf/1126493?ijkey=afgHAtDOTcMj.&keytype=ref&siteid=sci
[CrossRef] [PubMed]

U. Leonhardt, "Notes on Conformal Invisibility Devices", New Journal of Physics 8, 118 (2006), http://stacks.iop.org/1367-2630/8/118
[CrossRef]

J.C. Miñano, P. Benítez, A. Santamaría, "Hamilton-Jacobi equation in momentum space," Opt. Express 14, 9083-9092 (2006).
[CrossRef] [PubMed]

1995

1993

E. Wolf, T. Habashy, "Invisible bodies and uniqueness of the inverse scattering problem," J. of Modern Optics,  40, 785-792 (1993) http://taylorandfrancis.metapress.com/openurl.asp?genre=article&issn=0950-0340&volume=40&issue=5&spage=785.
[CrossRef]

Benítez, P.

Forbes, G. W.

Habashy, T.

E. Wolf, T. Habashy, "Invisible bodies and uniqueness of the inverse scattering problem," J. of Modern Optics,  40, 785-792 (1993) http://taylorandfrancis.metapress.com/openurl.asp?genre=article&issn=0950-0340&volume=40&issue=5&spage=785.
[CrossRef]

Leonhardt, U.

U. Leonhardt, "Notes on Conformal Invisibility Devices", New Journal of Physics 8, 118 (2006), http://stacks.iop.org/1367-2630/8/118
[CrossRef]

U. Leonhardt, "Optical Conformal Mapping", Science,  312, 1777-1780 (2006), http://www.sciencemag.org/cgi/rapidpdf/1126493?ijkey=afgHAtDOTcMj.&keytype=ref&siteid=sci
[CrossRef] [PubMed]

Miñano, J.C.

Santamaría, A.

Wallace, J. K.

Wolf, E.

E. Wolf, T. Habashy, "Invisible bodies and uniqueness of the inverse scattering problem," J. of Modern Optics,  40, 785-792 (1993) http://taylorandfrancis.metapress.com/openurl.asp?genre=article&issn=0950-0340&volume=40&issue=5&spage=785.
[CrossRef]

J. of Modern Optics

E. Wolf, T. Habashy, "Invisible bodies and uniqueness of the inverse scattering problem," J. of Modern Optics,  40, 785-792 (1993) http://taylorandfrancis.metapress.com/openurl.asp?genre=article&issn=0950-0340&volume=40&issue=5&spage=785.
[CrossRef]

J. Opt. Soc. Am. A

New Journal of Physics

U. Leonhardt, "Notes on Conformal Invisibility Devices", New Journal of Physics 8, 118 (2006), http://stacks.iop.org/1367-2630/8/118
[CrossRef]

Opt. Express

Science

U. Leonhardt, "Optical Conformal Mapping", Science,  312, 1777-1780 (2006), http://www.sciencemag.org/cgi/rapidpdf/1126493?ijkey=afgHAtDOTcMj.&keytype=ref&siteid=sci
[CrossRef] [PubMed]

Other

M. Kerker, "Invisible bodies," J. Opt. Soc. Am. 65, 376- (1975) http://www.opticsinfobase.org/abstract.cfm?URI=josa-65-4-376
[CrossRef]

G. Gbur, "Nonradiating sources and other 'invisible' objects", in E. Wolf (Ed.), Prog. in Optics, vol. 45 (Elsevier, Amsterdam, 2003), p. 273.
[CrossRef]

M. Born, E. Wolf, Principles of Optics, (Pergamon, Oxford, 1989).

M. Herzberger, Modern Geometrical Optics, (Interscience, New York, 1958).

S. Cornbleet, Microwave and Geometrical Optics (Academic, 1994).

R.K. Luneburg, Mathematical Theory of Optics, (University of California Press, Los Angeles 1964).

E. Kreyszig, Differential Geometry, (Dover, New York, 1991).

D.J. Struik, Lectures on Classical Differential Geometry, (Dover, New York, 1988).

J. Eaton, "On spherically symmetric lenses," IRE Transactions on Antennas and Propagation, 4, 66-71 (1952), http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=25697&arnumber=1144350&count=27&index=14

A.F. Kay, "Spherically Symmetric Lenses," IRE Transactions on Antennas and Propagation, 7, 32-38, (1959), http://ieeexplore.ieee.org/search/wrapper.jsp?arnumber=1144648
[CrossRef]

S.P. Morgan, "Generalizations of Spherically Symmetric Lenses," IRE Transactions on Antennas and Propagation, 7, 342-345 (1959), http://ieeexplore.ieee.org/search/wrapper.jsp?arnumber=1144697
[CrossRef]

P. Uslenghi, "Electromagnetic and Optical Behavior of Two Classes of Dielectric Lenses," IEEE Transactions on Antennas and Propagation, 17, 235-236 (1969), http://ieeexplore.ieee.org/search/wrapper.jsp?arnumber=1139390
[CrossRef]

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Figures (9)

Fig. 1.
Fig. 1.

The Eaton lens is an AIHR of type –RV, i.e., magnification m=-1 and virtual image. This lens forms a virtual image of any point x o on the point x i=-x o.

Fig. 2.
Fig. 2.

Partially mirror-coated Luneburg lens is another AIHR of type -RV, i.e., magnification m=-1 and a virtual image. This lens forms a virtual image of any point x o on the point x i=-x o.

Fig. 3.
Fig. 3.

Trajectories of rays in the meridian plane (x3=0 and so p3=0) for the Maxwell fish-eye lens (γ=1) (left); and the trajectories of the same rays (same color indicates the same ray) in the momentum space (right). The bold arrow tip is vector p and the hollow arrow tip is vector x.

Fig. 4.
Fig. 4.

Trajectories of rays in the meridian plane (x 3=p 3=0) for the generalized Maxwell fish-eye lens (γ=0.5) (left); and the trajectories of the same rays in the momentum space (right).The bold arrow tip is vector p and the hollow arrow tip is vector x.

Fig. 5.
Fig. 5.

A refractive-index distribution η(| x |)=(2/| x |-1)½ for 1≤| x |≤2 and η(| x |)=1 for | x |≤1 gives sharp imaging of any point x o (at the point x i=-x o) including the points in the n=1 region.

Fig. 6.
Fig. 6.

A refractive-index distribution η(| x |)=[1/(3u)-u]2 being u=[-1/|x|+(1/|x|2+1/27)½] for | x |>1 and η(| x |)=1 for | x |≤1, gives perfect self-imaging of any point, including the n=1 points.

Fig. 7.
Fig. 7.

The trajectories of rays in the invisible lens form a virtual image of any point on itself x i=x o.

Fig. 8.
Fig. 8.

Trajectories of rays in the meridian plane (x 3=0, p 3=0) for the Maxwell fish-eye lens distribution in | x |≤1 surrounded by a mirror at | x |=1 (left); and the trajectories of the same rays in the momentum space (right) for which corresponds a refractive index distribution η=(2/|p|-1)½ when | p |>1 and η=1 otherwise.

Fig. 9.
Fig. 9.

Continuous transition of the vectors x and p in a reflection caused by a sharp but continuous decay of the refractive index n. The vector N is parallel to the gradient of n.

Equations (3)

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n ( x ) = 2 x γ 1 1 + x 2 γ
x 1 = ( 1 sin θ cos θ ) 1 γ cos φ γ x 2 = ( 1 sin θ θ ) 1 γ sin φ γ
n 2 ( x 1 , x 2 , 0 ) [ ( d x 1 ) 2 + ( d x 2 ) 2 ] = ( d θ ) 2 + ( cos θ d φ ) 2 γ 2

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