Abstract

The Luneburg lens is an aberration-free lens that focuses light from all directions equally well. We fabricated and tested a Luneburg lens in silicon photonics. Such fully-integrated lenses may become the building blocks of compact Fourier optics on chips. Furthermore, our fabrication technique is sufficiently versatile for making perfect imaging devices on silicon platforms.

© 2011 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
  2. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).
  3. U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover, 2010).
  4. U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110–112 (2009).
    [CrossRef]
  5. P. A. Huidobro, M. L. Nesterov, L. Martin-Moreno, and F. J. Garcia-Vidal, “Transformation optics for plasmonics,” Nano Lett. 10, 1985–1990 (2010).
    [CrossRef] [PubMed]
  6. Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10, 1991–1997 (2010).
    [CrossRef] [PubMed]
  7. V. N. Smolyaninova, I. I. Smolyaninov, A. V. Kildishev, and V. M. Shalaev, “Maxwell fish-eye and Eaton lenses emulated by microdroplets,” Opt. Lett. 35, 3396–3398 (2010).
    [CrossRef] [PubMed]
  8. S. Combleet, Microwave Optics: The Optics of Microwave Antenna Design (Academic Press, 1976).
  9. M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, 1981).
  10. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129–132 (2010).
    [CrossRef]
  11. H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun. 1, 124 (2010).
    [CrossRef] [PubMed]
  12. T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. (2011), doi:10.1038/nnano.2010.282.
    [CrossRef] [PubMed]
  13. M. Lipson, “Guiding, modulating and emitting light on silicon—challenges and opportunities,” J. Lightwave Technol. 23, 4222–4238 (2005).
    [CrossRef]
  14. J. C. Minano, “Perfect imaging in a homogeneous threedimensional region,” Opt. Express 14, 9627–9635 (2006).
    [CrossRef] [PubMed]
  15. U. Leonhardt, “Perfect imaging without negative refraction,” N. J. Phys. 11, 093040 (2009).
    [CrossRef]
  16. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
    [CrossRef] [PubMed]
  17. E. Colombini, “Design of thin-film Luneburg lenses for maximum focal length control,” Appl. Opt. 20, 3589–3593 (1981).
    [CrossRef] [PubMed]
  18. E. Colombini, “Index-profile computation for the generalized Luneburg lens,” J. Opt. Soc. Am. 71, 1403–1405 (1981).
  19. S. K. Yao and D. B. Anderson, “Shadow sputtered diffraction-limited waveguide Luneburg lenses,” Appl. Phys. Lett. 33, 307–309 (1978).
    [CrossRef]
  20. S. K. Yao, D. B. Anderson, R. R. August, B. R. Youmans, and C. M. Oania, “Guided-wave optical thin-film Luneburg lenses: fabrication technique and properties,” Appl. Opt. 18, 4067–4079 (1979).
    [CrossRef] [PubMed]
  21. F. Zernike, “Luneburg lens for optical waveguide use,” Opt. Commun. 12, 379–381 (1974).
    [CrossRef]
  22. S. Takahashi, C. Chang, S. Y. Yang, and G. Barbastathis, “Design and fabrication of dielectric nanostructured Luneburg lens in optical frequencies” in Optical MEMS and Nanophotonics, (IEEE Photonics Society, 2010), Paper Th1–1, pp. 177–178.
  23. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
    [CrossRef] [PubMed]
  24. L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
    [CrossRef]
  25. T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
    [CrossRef] [PubMed]
  26. L. H. Gabrielli, U. Leonhardt, and M. Lipson, “Perfect imaging in the optical domain using dielectric materials,” arXiv:1007.2564.
  27. D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test, (SPIE tutorial text, Washington, 2004).
  28. C. Reardon, A. Di Falco, K. Welna, and T. F. Krauss, “Integrated polymer microprisms for free space optical beam deflecting,” Opt. Express 17, 3423–3428 (2009).
    [CrossRef]

2011 (1)

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. (2011), doi:10.1038/nnano.2010.282.
[CrossRef] [PubMed]

2010 (6)

P. A. Huidobro, M. L. Nesterov, L. Martin-Moreno, and F. J. Garcia-Vidal, “Transformation optics for plasmonics,” Nano Lett. 10, 1985–1990 (2010).
[CrossRef] [PubMed]

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10, 1991–1997 (2010).
[CrossRef] [PubMed]

V. N. Smolyaninova, I. I. Smolyaninov, A. V. Kildishev, and V. M. Shalaev, “Maxwell fish-eye and Eaton lenses emulated by microdroplets,” Opt. Lett. 35, 3396–3398 (2010).
[CrossRef] [PubMed]

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129–132 (2010).
[CrossRef]

H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun. 1, 124 (2010).
[CrossRef] [PubMed]

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[CrossRef] [PubMed]

2009 (5)

C. Reardon, A. Di Falco, K. Welna, and T. F. Krauss, “Integrated polymer microprisms for free space optical beam deflecting,” Opt. Express 17, 3423–3428 (2009).
[CrossRef]

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

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

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[CrossRef] [PubMed]

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
[CrossRef]

2006 (1)

2005 (1)

2000 (1)

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

1981 (2)

1979 (1)

1978 (1)

S. K. Yao and D. B. Anderson, “Shadow sputtered diffraction-limited waveguide Luneburg lenses,” Appl. Phys. Lett. 33, 307–309 (1978).
[CrossRef]

1974 (1)

F. Zernike, “Luneburg lens for optical waveguide use,” Opt. Commun. 12, 379–381 (1974).
[CrossRef]

Anderson, D. B.

August, R. R.

Bartal, G.

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10, 1991–1997 (2010).
[CrossRef] [PubMed]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[CrossRef] [PubMed]

Brenner, P.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[CrossRef] [PubMed]

Cardenas, J.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
[CrossRef]

Colombini, E.

Cui, T. J.

H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun. 1, 124 (2010).
[CrossRef] [PubMed]

Di Falco, A.

C. Reardon, A. Di Falco, K. Welna, and T. F. Krauss, “Integrated polymer microprisms for free space optical beam deflecting,” Opt. Express 17, 3423–3428 (2009).
[CrossRef]

Ergin, T.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[CrossRef] [PubMed]

Gabrielli, L. H.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
[CrossRef]

Garcia-Vidal, F. J.

P. A. Huidobro, M. L. Nesterov, L. Martin-Moreno, and F. J. Garcia-Vidal, “Transformation optics for plasmonics,” Nano Lett. 10, 1985–1990 (2010).
[CrossRef] [PubMed]

Huidobro, P. A.

P. A. Huidobro, M. L. Nesterov, L. Martin-Moreno, and F. J. Garcia-Vidal, “Transformation optics for plasmonics,” Nano Lett. 10, 1985–1990 (2010).
[CrossRef] [PubMed]

Kildishev, A. V.

Krauss, T. F.

C. Reardon, A. Di Falco, K. Welna, and T. F. Krauss, “Integrated polymer microprisms for free space optical beam deflecting,” Opt. Express 17, 3423–3428 (2009).
[CrossRef]

Kundtz, N.

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129–132 (2010).
[CrossRef]

Leonhardt, U.

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

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

Li, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[CrossRef] [PubMed]

Lipson, M.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
[CrossRef]

M. Lipson, “Guiding, modulating and emitting light on silicon—challenges and opportunities,” J. Lightwave Technol. 23, 4222–4238 (2005).
[CrossRef]

Liu, Y.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. (2011), doi:10.1038/nnano.2010.282.
[CrossRef] [PubMed]

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10, 1991–1997 (2010).
[CrossRef] [PubMed]

Ma, H. F.

H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun. 1, 124 (2010).
[CrossRef] [PubMed]

Martin-Moreno, L.

P. A. Huidobro, M. L. Nesterov, L. Martin-Moreno, and F. J. Garcia-Vidal, “Transformation optics for plasmonics,” Nano Lett. 10, 1985–1990 (2010).
[CrossRef] [PubMed]

Mikkelsen, M. H.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. (2011), doi:10.1038/nnano.2010.282.
[CrossRef] [PubMed]

Minano, J. C.

Nesterov, M. L.

P. A. Huidobro, M. L. Nesterov, L. Martin-Moreno, and F. J. Garcia-Vidal, “Transformation optics for plasmonics,” Nano Lett. 10, 1985–1990 (2010).
[CrossRef] [PubMed]

Oania, C. M.

Pendry, J. B.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[CrossRef] [PubMed]

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

Poitras, C. B.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
[CrossRef]

Reardon, C.

C. Reardon, A. Di Falco, K. Welna, and T. F. Krauss, “Integrated polymer microprisms for free space optical beam deflecting,” Opt. Express 17, 3423–3428 (2009).
[CrossRef]

Shalaev, V. M.

Smith, D. R.

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129–132 (2010).
[CrossRef]

Smolyaninov, I. I.

Smolyaninova, V. N.

Stenger, N.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[CrossRef] [PubMed]

Tyc, T.

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

Valentine, J.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. (2011), doi:10.1038/nnano.2010.282.
[CrossRef] [PubMed]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[CrossRef] [PubMed]

Wegener, M.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[CrossRef] [PubMed]

Welna, K.

C. Reardon, A. Di Falco, K. Welna, and T. F. Krauss, “Integrated polymer microprisms for free space optical beam deflecting,” Opt. Express 17, 3423–3428 (2009).
[CrossRef]

Yao, S. K.

Youmans, B. R.

Zentgraf, T.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. (2011), doi:10.1038/nnano.2010.282.
[CrossRef] [PubMed]

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10, 1991–1997 (2010).
[CrossRef] [PubMed]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[CrossRef] [PubMed]

Zernike, F.

F. Zernike, “Luneburg lens for optical waveguide use,” Opt. Commun. 12, 379–381 (1974).
[CrossRef]

Zhang, X.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. (2011), doi:10.1038/nnano.2010.282.
[CrossRef] [PubMed]

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10, 1991–1997 (2010).
[CrossRef] [PubMed]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[CrossRef] [PubMed]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

S. K. Yao and D. B. Anderson, “Shadow sputtered diffraction-limited waveguide Luneburg lenses,” Appl. Phys. Lett. 33, 307–309 (1978).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

N. J. Phys. (1)

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

Nano Lett. (2)

P. A. Huidobro, M. L. Nesterov, L. Martin-Moreno, and F. J. Garcia-Vidal, “Transformation optics for plasmonics,” Nano Lett. 10, 1985–1990 (2010).
[CrossRef] [PubMed]

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10, 1991–1997 (2010).
[CrossRef] [PubMed]

Nat. Commun. (1)

H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun. 1, 124 (2010).
[CrossRef] [PubMed]

Nat. Mater. (2)

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[CrossRef] [PubMed]

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129–132 (2010).
[CrossRef]

Nat. Nanotechnol. (1)

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. (2011), doi:10.1038/nnano.2010.282.
[CrossRef] [PubMed]

Nat. Photonics (1)

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3, 461–463 (2009).
[CrossRef]

Opt. Commun. (1)

F. Zernike, “Luneburg lens for optical waveguide use,” Opt. Commun. 12, 379–381 (1974).
[CrossRef]

Opt. Express (2)

C. Reardon, A. Di Falco, K. Welna, and T. F. Krauss, “Integrated polymer microprisms for free space optical beam deflecting,” Opt. Express 17, 3423–3428 (2009).
[CrossRef]

J. C. Minano, “Perfect imaging in a homogeneous threedimensional region,” Opt. Express 14, 9627–9635 (2006).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

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

Science (2)

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[CrossRef] [PubMed]

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

Other (8)

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

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

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

S. Combleet, Microwave Optics: The Optics of Microwave Antenna Design (Academic Press, 1976).

M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, 1981).

L. H. Gabrielli, U. Leonhardt, and M. Lipson, “Perfect imaging in the optical domain using dielectric materials,” arXiv:1007.2564.

D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test, (SPIE tutorial text, Washington, 2004).

S. Takahashi, C. Chang, S. Y. Yang, and G. Barbastathis, “Design and fabrication of dielectric nanostructured Luneburg lens in optical frequencies” in Optical MEMS and Nanophotonics, (IEEE Photonics Society, 2010), Paper Th1–1, pp. 177–178.

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Figures (4)

Fig. 1
Fig. 1

Luneburg lens. The lens (blue disk) focuses all light rays (red) propagating in one direction at the point on its rim that lies in that direction. The underlying wave pattern shows the real part of the Fourier component of a plane wave with wavelength 0.5a incident from the right (calculated by partial-wave expansion). One sees that the focal spot is about half a wavelength wide. As the lens is rotationally symmetric it focuses light from all directions equally well.

Fig. 2
Fig. 2

The device. a: the waveguide confining the light also creates the refractive index profile (1) of the Luneburg lens for horizontal light propagation (Fig. 1) on a chip. The red curves show the vertical intensity distributions calculated with a modal solver. b: effective refractive index depending on the thickness of the silicon below the polymer in (a). c: measured silicon profile (green) versus the theoretical curve (red) required for implementing the Luneburg lens.

Fig. 3
Fig. 3

Light focusing. a: False-colour image of the observed intensity profile in the Luneburg lens (Fig. 2). Infrared light incident from the right is guided on the chip and focused in the lens (red spot). The lens itself is made visible by illuminating it with white light from the top. b: measured intensity profiles of the focused infrared light along the two lines indicated in (a). In our experiment the full width at half maximum (FWHM) of the focus is dominated by the width of the incident Gaussian beam.

Fig. 4
Fig. 4

Performance tests. The figure shows the observed infrared light focused in the Luneburg lens (yellow-red spots similar to Fig. 3, except that the lens is not made visible) depending on the propagation direction (red arrow with angle θ) and the offset Δx of the incident Gaussian beam (profile shown). ac: the focal spot lies in the propagation direction. d,e: the focal point is independent of the offset. Our device thus behaves as expected from a Luneburg lens.

Equations (1)

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n = 2 ( r / a ) 2 for r a ,

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