Abstract

Conventional optical imaging systems suffer from the presence of many imperfections, such as spherical aberrations, astigmatism, or coma. If the imaging system is corrected for spherical aberrations and fulfills the Abbe sine condition, perfect imaging is guaranteed between two parallel planes but only in a small neighborhood of the optical axis. It is therefore worth asking for optical systems that would allow for perfect imaging between arbitrary smooth surfaces without restrictions in shape or extension. In this Letter, we describe the application of transformation optics to design refractive index distributions that allow perfect, aberration-free imaging for various imaging configurations in Rn. A special case is the imaging between two extended parallel lines in R2, which leads to the well-known hyperbolic secant index distribution that is used for the fabrication of gradient index lenses.

© 2011 Optical Society of America

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References

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  1. D. Bachstein, K. Mantel, and U. Peschel, in Proceedings of the 5th EOS Topical Meeting on Advanced Imaging Techniques (2010).
  2. U. Leonhardt, New J. Phys. 11, 093040 (2009).
    [CrossRef]
  3. U. Leonhardt and T. G. Philbin, Phys. Rev. A 81, 011804(2010).
    [CrossRef]
  4. P. Benítez, J. C. Minano, and J. C. Gonzalez, Opt. Express 18, 7650 (2010).
    [CrossRef] [PubMed]
  5. U. Leonhardt and T. G. Philbin, in Progress in Optics, E.Wolf, ed. (Elsevier, 2009), Vol. 53, pp. 69–152.
    [CrossRef]
  6. U. Leonhardt, Science 312, 1777 (2006).
    [CrossRef] [PubMed]
  7. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
  8. A. Geiger, “Der Lichtstrahl in differentialgeometrischer Formulierung,” Bericht Nr. 140, Institut für Geodäsie und Photogrammetrie, ETH Zürich (1988), http://www.igp-data.ethz.ch/berichte/Graue_Berichte_PDF/140.pdf.
  9. A. Fletcher, T. Murphy, and A. Young, “Solutions of two optical problems,” Proc. R. Soc. London Ser. A 223, 216(1954).
    [CrossRef]
  10. A. J. Makowski, Ann. Phys. 324, 2465 (2009).
    [CrossRef]
  11. U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover, 2010).

2010 (2)

2009 (2)

U. Leonhardt, New J. Phys. 11, 093040 (2009).
[CrossRef]

A. J. Makowski, Ann. Phys. 324, 2465 (2009).
[CrossRef]

2006 (1)

U. Leonhardt, Science 312, 1777 (2006).
[CrossRef] [PubMed]

1954 (1)

A. Fletcher, T. Murphy, and A. Young, “Solutions of two optical problems,” Proc. R. Soc. London Ser. A 223, 216(1954).
[CrossRef]

Bachstein, D.

D. Bachstein, K. Mantel, and U. Peschel, in Proceedings of the 5th EOS Topical Meeting on Advanced Imaging Techniques (2010).

Benítez, P.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Fletcher, A.

A. Fletcher, T. Murphy, and A. Young, “Solutions of two optical problems,” Proc. R. Soc. London Ser. A 223, 216(1954).
[CrossRef]

Geiger, A.

A. Geiger, “Der Lichtstrahl in differentialgeometrischer Formulierung,” Bericht Nr. 140, Institut für Geodäsie und Photogrammetrie, ETH Zürich (1988), http://www.igp-data.ethz.ch/berichte/Graue_Berichte_PDF/140.pdf.

Gonzalez, J. C.

Leonhardt, U.

U. Leonhardt and T. G. Philbin, Phys. Rev. A 81, 011804(2010).
[CrossRef]

U. Leonhardt, New J. Phys. 11, 093040 (2009).
[CrossRef]

U. Leonhardt, Science 312, 1777 (2006).
[CrossRef] [PubMed]

U. Leonhardt and T. G. Philbin, in Progress in Optics, E.Wolf, ed. (Elsevier, 2009), Vol. 53, pp. 69–152.
[CrossRef]

U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover, 2010).

Makowski, A. J.

A. J. Makowski, Ann. Phys. 324, 2465 (2009).
[CrossRef]

Mantel, K.

D. Bachstein, K. Mantel, and U. Peschel, in Proceedings of the 5th EOS Topical Meeting on Advanced Imaging Techniques (2010).

Minano, J. C.

Murphy, T.

A. Fletcher, T. Murphy, and A. Young, “Solutions of two optical problems,” Proc. R. Soc. London Ser. A 223, 216(1954).
[CrossRef]

Peschel, U.

D. Bachstein, K. Mantel, and U. Peschel, in Proceedings of the 5th EOS Topical Meeting on Advanced Imaging Techniques (2010).

Philbin, T. G.

U. Leonhardt and T. G. Philbin, Phys. Rev. A 81, 011804(2010).
[CrossRef]

U. Leonhardt and T. G. Philbin, in Progress in Optics, E.Wolf, ed. (Elsevier, 2009), Vol. 53, pp. 69–152.
[CrossRef]

U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover, 2010).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Young, A.

A. Fletcher, T. Murphy, and A. Young, “Solutions of two optical problems,” Proc. R. Soc. London Ser. A 223, 216(1954).
[CrossRef]

Ann. Phys. (1)

A. J. Makowski, Ann. Phys. 324, 2465 (2009).
[CrossRef]

New J. Phys. (1)

U. Leonhardt, New J. Phys. 11, 093040 (2009).
[CrossRef]

Opt. Express (1)

Phys. Rev. A (1)

U. Leonhardt and T. G. Philbin, Phys. Rev. A 81, 011804(2010).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

A. Fletcher, T. Murphy, and A. Young, “Solutions of two optical problems,” Proc. R. Soc. London Ser. A 223, 216(1954).
[CrossRef]

Science (1)

U. Leonhardt, Science 312, 1777 (2006).
[CrossRef] [PubMed]

Other (5)

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

A. Geiger, “Der Lichtstrahl in differentialgeometrischer Formulierung,” Bericht Nr. 140, Institut für Geodäsie und Photogrammetrie, ETH Zürich (1988), http://www.igp-data.ethz.ch/berichte/Graue_Berichte_PDF/140.pdf.

D. Bachstein, K. Mantel, and U. Peschel, in Proceedings of the 5th EOS Topical Meeting on Advanced Imaging Techniques (2010).

U. Leonhardt and T. G. Philbin, in Progress in Optics, E.Wolf, ed. (Elsevier, 2009), Vol. 53, pp. 69–152.
[CrossRef]

U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover, 2010).

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Figures (4)

Fig. 1
Fig. 1

Statement of the problem: two hypersurfaces H 1 and H 2 of arbitrary shape shall be perfectly imaged onto each other, i.e., aberration free even for wide bundles of rays.

Fig. 2
Fig. 2

From the fish eye to the hyperbolic secant distribution. (a) Object and image points for the Maxwell fish eye on the x 2 axis. The light rays between corresponding points form circles, possibly of infinite radius. (b) Space is “stretched out” by a transformation to polar coordinates. (c) An additional logarithmic stretching yields the point-to-point correspondence of two parallel lines with an inversion symmetry with respect to the line joining the gray points. The composite transformation is conformal.

Fig. 3
Fig. 3

Imaging geometry for the hyperbolic secant distribution. (a) Two parallel lines are imaged perfectly onto each other with a magnification factor of 1 . The coordinates r , φ have been renamed x 1 , x 2 to indicate that they are interpreted as Cartesian. The additional periodically repeating image lines are not shown. (b) With the same distribution, lines that are inclined can also be imaged perfectly onto each other.

Fig. 4
Fig. 4

2D ray-tracing simulation of a GRIN element with hyperbolic secant distribution. The system is illuminated by a point source on the axis. Multiple focal points are created in the GRIN element. In two dimensions, refraction at the boundaries is responsible for the aberrations of the system.

Equations (11)

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n ( x ) = n 0 1 + x 2 x 0 2 ,
g i j = n 2 ( x ) δ i j .
g ˜ i j ( x ) = λ ( x ) g i j ( x ( x ) ) ,
g ˜ i j ( x ) = λ ( x ) n 2 ( x ( x ) ) δ i j .
n ˜ ( x ) = λ ( x ) n ( x ) .
d s 2 = n 2 ( x ) ( d x 1 2 + d x 2 2 ) ,
d s 2 = n 2 ( r , φ ) ( d r 2 + r 2 d φ 2 ) ,
d s 2 = n 2 ( r , φ ) ( e r ) 2 ( d r 2 + d φ 2 ) ,
n ˜ = n ( r , φ ) e r = e r 1 + e 2 r = 1 2 cosh ( r ) ,
n ˜ ( x , y ) = n 0 r k + 1 + r k + 1 , r = x 2 + y 2 .
z s = exp ( 2 s + 1 k i π ) 1 ( z ) * , s = 1 , k .

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