Abstract

Employing artificially structured metamaterials provides a means of circumventing the limits of conventional optical materials. Here, we use transformation optics (TO) combined with nanolithography to produce a planar Luneburg lens with a flat focal surface that operates at telecommunication wavelengths. Whereas previous infrared TO devices have been transformations of free-space, here we implement a transformation of an existing optical element to create a new device with the same optical characteristics but a user-defined geometry.

© 2012 OSA

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References

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  1. E. W. Marchland, Gradient Index Optics (Academic Press, 1978).
  2. R. Luneburg, Mathematical Theory of Optics (Brown Univ. Press, 1944).
  3. S. P. Morgan, “General solution of the Luneberg lens problem,” J. Appl. Phys. 29(9), 1358–1368 (1958).
    [CrossRef]
  4. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
    [CrossRef] [PubMed]
  5. L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461–463 (2009).
    [CrossRef]
  6. M. Gharghi, C. Gladden, T. Zentgraf, Y. Liu, X. Yin, J. Valentine, and X. Zhang, “A carpet cloak for visible light,” Nano Lett. 11(7), 2825–2828 (2011).
    [CrossRef] [PubMed]
  7. L. Gabrielli and M. Lipson, “Transformation optics on a silicon platform,” J. Opt. 13(2), 024010 (2011).
    [CrossRef]
  8. A. Di Falco, S. C. Kehr, and U. Leonhardt, “Luneburg lens in silicon photonics,” Opt. Express 19(6), 5156–5162 (2011).
    [CrossRef] [PubMed]
  9. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
    [CrossRef] [PubMed]
  10. D. A. Roberts, N. Kundtz, and D. R. Smith, “Optical lens compression via transformation optics,” Opt. Express 17(19), 16535–16542 (2009).
    [CrossRef] [PubMed]
  11. D. Schurig, “An aberration-free lens with zero f-number,” New J. Phys. 10(11), 115034 (2008).
    [CrossRef]
  12. J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
    [CrossRef] [PubMed]
  13. N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express 17(17), 14872–14879 (2009).
    [CrossRef] [PubMed]
  14. N. I. Landy, N. Kundtz, and D. R. Smith, “Designing three-dimensional transformation optical media using quasiconformal coordinate transformations,” Phys. Rev. Lett. 105(19), 193902 (2010).
    [CrossRef] [PubMed]
  15. D. R. Smith, Y. Urzhumov, N. B. Kundtz, and N. I. Landy, “Enhancing imaging systems using transformation optics,” Opt. Express 18(20), 21238–21251 (2010).
    [CrossRef] [PubMed]
  16. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010).
    [CrossRef] [PubMed]
  17. J. Hunt, N. Kundtz, N. Landy, V. Nguyen, T. Perram, A. Starr, and D. R. Smith, “Broadband wide angle lens implemented with dielectric metamaterials,” Sensors (Basel, Switzerland) 11(8), 7982–7991 (2011).
    [CrossRef] [PubMed]
  18. A. Subashiev and S. Luryi, “Modal control in semiconductor optical waveguides with uniaxially patterned layers,” J. Lightwave Technol. 24(3), 1513–1522 (2006).
    [CrossRef]
  19. N. Grigoropoulos and P. Young, “Low cost non radiative perforated dielectric waveguides,” in Proceedings of 33rd European Microwave Conference (2003), 439–442.
  20. J. Hunt, N. Kundtz, N. Landy, and D. R. Smith, “Relaxation approach for the generation of inhomogeneous distributions of uniformly sized particles,” Appl. Phys. Lett. 97(2), 024104 (2010).
    [CrossRef]

2011 (4)

M. Gharghi, C. Gladden, T. Zentgraf, Y. Liu, X. Yin, J. Valentine, and X. Zhang, “A carpet cloak for visible light,” Nano Lett. 11(7), 2825–2828 (2011).
[CrossRef] [PubMed]

L. Gabrielli and M. Lipson, “Transformation optics on a silicon platform,” J. Opt. 13(2), 024010 (2011).
[CrossRef]

J. Hunt, N. Kundtz, N. Landy, V. Nguyen, T. Perram, A. Starr, and D. R. Smith, “Broadband wide angle lens implemented with dielectric metamaterials,” Sensors (Basel, Switzerland) 11(8), 7982–7991 (2011).
[CrossRef] [PubMed]

A. Di Falco, S. C. Kehr, and U. Leonhardt, “Luneburg lens in silicon photonics,” Opt. Express 19(6), 5156–5162 (2011).
[CrossRef] [PubMed]

2010 (4)

D. R. Smith, Y. Urzhumov, N. B. Kundtz, and N. I. Landy, “Enhancing imaging systems using transformation optics,” Opt. Express 18(20), 21238–21251 (2010).
[CrossRef] [PubMed]

J. Hunt, N. Kundtz, N. Landy, and D. R. Smith, “Relaxation approach for the generation of inhomogeneous distributions of uniformly sized particles,” Appl. Phys. Lett. 97(2), 024104 (2010).
[CrossRef]

N. I. Landy, N. Kundtz, and D. R. Smith, “Designing three-dimensional transformation optical media using quasiconformal coordinate transformations,” Phys. Rev. Lett. 105(19), 193902 (2010).
[CrossRef] [PubMed]

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010).
[CrossRef] [PubMed]

2009 (4)

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 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(8), 461–463 (2009).
[CrossRef]

N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express 17(17), 14872–14879 (2009).
[CrossRef] [PubMed]

D. A. Roberts, N. Kundtz, and D. R. Smith, “Optical lens compression via transformation optics,” Opt. Express 17(19), 16535–16542 (2009).
[CrossRef] [PubMed]

2008 (2)

D. Schurig, “An aberration-free lens with zero f-number,” New J. Phys. 10(11), 115034 (2008).
[CrossRef]

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

2006 (2)

1958 (1)

S. P. Morgan, “General solution of the Luneberg lens problem,” J. Appl. Phys. 29(9), 1358–1368 (1958).
[CrossRef]

Bartal, G.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[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(8), 461–463 (2009).
[CrossRef]

Di Falco, A.

Gabrielli, L.

L. Gabrielli and M. Lipson, “Transformation optics on a silicon platform,” J. Opt. 13(2), 024010 (2011).
[CrossRef]

Gabrielli, L. H.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461–463 (2009).
[CrossRef]

Gharghi, M.

M. Gharghi, C. Gladden, T. Zentgraf, Y. Liu, X. Yin, J. Valentine, and X. Zhang, “A carpet cloak for visible light,” Nano Lett. 11(7), 2825–2828 (2011).
[CrossRef] [PubMed]

Gladden, C.

M. Gharghi, C. Gladden, T. Zentgraf, Y. Liu, X. Yin, J. Valentine, and X. Zhang, “A carpet cloak for visible light,” Nano Lett. 11(7), 2825–2828 (2011).
[CrossRef] [PubMed]

Hunt, J.

J. Hunt, N. Kundtz, N. Landy, V. Nguyen, T. Perram, A. Starr, and D. R. Smith, “Broadband wide angle lens implemented with dielectric metamaterials,” Sensors (Basel, Switzerland) 11(8), 7982–7991 (2011).
[CrossRef] [PubMed]

J. Hunt, N. Kundtz, N. Landy, and D. R. Smith, “Relaxation approach for the generation of inhomogeneous distributions of uniformly sized particles,” Appl. Phys. Lett. 97(2), 024104 (2010).
[CrossRef]

Kehr, S. C.

Kundtz, N.

J. Hunt, N. Kundtz, N. Landy, V. Nguyen, T. Perram, A. Starr, and D. R. Smith, “Broadband wide angle lens implemented with dielectric metamaterials,” Sensors (Basel, Switzerland) 11(8), 7982–7991 (2011).
[CrossRef] [PubMed]

J. Hunt, N. Kundtz, N. Landy, and D. R. Smith, “Relaxation approach for the generation of inhomogeneous distributions of uniformly sized particles,” Appl. Phys. Lett. 97(2), 024104 (2010).
[CrossRef]

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010).
[CrossRef] [PubMed]

N. I. Landy, N. Kundtz, and D. R. Smith, “Designing three-dimensional transformation optical media using quasiconformal coordinate transformations,” Phys. Rev. Lett. 105(19), 193902 (2010).
[CrossRef] [PubMed]

D. A. Roberts, N. Kundtz, and D. R. Smith, “Optical lens compression via transformation optics,” Opt. Express 17(19), 16535–16542 (2009).
[CrossRef] [PubMed]

Kundtz, N. B.

Landy, N.

J. Hunt, N. Kundtz, N. Landy, V. Nguyen, T. Perram, A. Starr, and D. R. Smith, “Broadband wide angle lens implemented with dielectric metamaterials,” Sensors (Basel, Switzerland) 11(8), 7982–7991 (2011).
[CrossRef] [PubMed]

J. Hunt, N. Kundtz, N. Landy, and D. R. Smith, “Relaxation approach for the generation of inhomogeneous distributions of uniformly sized particles,” Appl. Phys. Lett. 97(2), 024104 (2010).
[CrossRef]

Landy, N. I.

Leonhardt, U.

Li, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

Lipson, M.

L. Gabrielli and M. Lipson, “Transformation optics on a silicon platform,” J. Opt. 13(2), 024010 (2011).
[CrossRef]

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461–463 (2009).
[CrossRef]

Liu, Y.

M. Gharghi, C. Gladden, T. Zentgraf, Y. Liu, X. Yin, J. Valentine, and X. Zhang, “A carpet cloak for visible light,” Nano Lett. 11(7), 2825–2828 (2011).
[CrossRef] [PubMed]

Luryi, S.

Morgan, S. P.

S. P. Morgan, “General solution of the Luneberg lens problem,” J. Appl. Phys. 29(9), 1358–1368 (1958).
[CrossRef]

Nguyen, V.

J. Hunt, N. Kundtz, N. Landy, V. Nguyen, T. Perram, A. Starr, and D. R. Smith, “Broadband wide angle lens implemented with dielectric metamaterials,” Sensors (Basel, Switzerland) 11(8), 7982–7991 (2011).
[CrossRef] [PubMed]

Padilla, W. J.

Pendry, J. B.

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Perram, T.

J. Hunt, N. Kundtz, N. Landy, V. Nguyen, T. Perram, A. Starr, and D. R. Smith, “Broadband wide angle lens implemented with dielectric metamaterials,” Sensors (Basel, Switzerland) 11(8), 7982–7991 (2011).
[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(8), 461–463 (2009).
[CrossRef]

Roberts, D. A.

Schurig, D.

D. Schurig, “An aberration-free lens with zero f-number,” New J. Phys. 10(11), 115034 (2008).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Smith, D. R.

J. Hunt, N. Kundtz, N. Landy, V. Nguyen, T. Perram, A. Starr, and D. R. Smith, “Broadband wide angle lens implemented with dielectric metamaterials,” Sensors (Basel, Switzerland) 11(8), 7982–7991 (2011).
[CrossRef] [PubMed]

D. R. Smith, Y. Urzhumov, N. B. Kundtz, and N. I. Landy, “Enhancing imaging systems using transformation optics,” Opt. Express 18(20), 21238–21251 (2010).
[CrossRef] [PubMed]

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010).
[CrossRef] [PubMed]

J. Hunt, N. Kundtz, N. Landy, and D. R. Smith, “Relaxation approach for the generation of inhomogeneous distributions of uniformly sized particles,” Appl. Phys. Lett. 97(2), 024104 (2010).
[CrossRef]

N. I. Landy, N. Kundtz, and D. R. Smith, “Designing three-dimensional transformation optical media using quasiconformal coordinate transformations,” Phys. Rev. Lett. 105(19), 193902 (2010).
[CrossRef] [PubMed]

D. A. Roberts, N. Kundtz, and D. R. Smith, “Optical lens compression via transformation optics,” Opt. Express 17(19), 16535–16542 (2009).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Starr, A.

J. Hunt, N. Kundtz, N. Landy, V. Nguyen, T. Perram, A. Starr, and D. R. Smith, “Broadband wide angle lens implemented with dielectric metamaterials,” Sensors (Basel, Switzerland) 11(8), 7982–7991 (2011).
[CrossRef] [PubMed]

Subashiev, A.

Urzhumov, Y.

Valentine, J.

M. Gharghi, C. Gladden, T. Zentgraf, Y. Liu, X. Yin, J. Valentine, and X. Zhang, “A carpet cloak for visible light,” Nano Lett. 11(7), 2825–2828 (2011).
[CrossRef] [PubMed]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

Yin, X.

M. Gharghi, C. Gladden, T. Zentgraf, Y. Liu, X. Yin, J. Valentine, and X. Zhang, “A carpet cloak for visible light,” Nano Lett. 11(7), 2825–2828 (2011).
[CrossRef] [PubMed]

Zentgraf, T.

M. Gharghi, C. Gladden, T. Zentgraf, Y. Liu, X. Yin, J. Valentine, and X. Zhang, “A carpet cloak for visible light,” Nano Lett. 11(7), 2825–2828 (2011).
[CrossRef] [PubMed]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

Zhang, X.

M. Gharghi, C. Gladden, T. Zentgraf, Y. Liu, X. Yin, J. Valentine, and X. Zhang, “A carpet cloak for visible light,” Nano Lett. 11(7), 2825–2828 (2011).
[CrossRef] [PubMed]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

J. Hunt, N. Kundtz, N. Landy, and D. R. Smith, “Relaxation approach for the generation of inhomogeneous distributions of uniformly sized particles,” Appl. Phys. Lett. 97(2), 024104 (2010).
[CrossRef]

J. Appl. Phys. (1)

S. P. Morgan, “General solution of the Luneberg lens problem,” J. Appl. Phys. 29(9), 1358–1368 (1958).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. (1)

L. Gabrielli and M. Lipson, “Transformation optics on a silicon platform,” J. Opt. 13(2), 024010 (2011).
[CrossRef]

Nano Lett. (1)

M. Gharghi, C. Gladden, T. Zentgraf, Y. Liu, X. Yin, J. Valentine, and X. Zhang, “A carpet cloak for visible light,” Nano Lett. 11(7), 2825–2828 (2011).
[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(7), 568–571 (2009).
[CrossRef] [PubMed]

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010).
[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(8), 461–463 (2009).
[CrossRef]

New J. Phys. (1)

D. Schurig, “An aberration-free lens with zero f-number,” New J. Phys. 10(11), 115034 (2008).
[CrossRef]

Opt. Express (4)

Phys. Rev. Lett. (2)

N. I. Landy, N. Kundtz, and D. R. Smith, “Designing three-dimensional transformation optical media using quasiconformal coordinate transformations,” Phys. Rev. Lett. 105(19), 193902 (2010).
[CrossRef] [PubMed]

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

Science (1)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Sensors (Basel, Switzerland) (1)

J. Hunt, N. Kundtz, N. Landy, V. Nguyen, T. Perram, A. Starr, and D. R. Smith, “Broadband wide angle lens implemented with dielectric metamaterials,” Sensors (Basel, Switzerland) 11(8), 7982–7991 (2011).
[CrossRef] [PubMed]

Other (3)

N. Grigoropoulos and P. Young, “Low cost non radiative perforated dielectric waveguides,” in Proceedings of 33rd European Microwave Conference (2003), 439–442.

E. W. Marchland, Gradient Index Optics (Academic Press, 1978).

R. Luneburg, Mathematical Theory of Optics (Brown Univ. Press, 1944).

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Figures (4)

Fig. 1
Fig. 1

Raytraces through the Luneburg index distributions. (a) The original 2D Luneburg lens, (b) the flattened Luneburg lens, and (c) the fabricated flattened Luneburg lens. The bold black line in (a) and (b) shows the transformed boundary. The range of indexes of the flattened lens, (b), have increased compared to the original lens, (a). In the fabricated lens, (c), indices less than 1.5 have been approximated as 1.5 to maintain waveguiding, and the transformation has been truncated with an index matching region that matches the transformation to the waveguide index.

Fig. 2
Fig. 2

Effective index dispersion. The dashed lines show the index of bulk Si and SiO2 while the solid lines show the effective mode index in the Si slab waveguide vs. frequency. The solid red region is the range of indexes achievable with the hole-array metamaterial used here. This region covers nearly the entire range of accessible indices and exhibits small dispersion.

Fig. 3
Fig. 3

Fabricated Luneburg sample. (a) Cut-away view of the lens fabricated in silicon-on-insulator (SOI). Inset shows an SEM image of the same region of the lens where the local-crystallinity of the hole distribution can be seen. (b) Schematic of the fabricated lens. The pattern shown was etched through the Si slab using EBL followed by DRIE, stopping abruptly at SiO2 layer - except for the input/output gratings which only partially penetrate the Si slab. Input waveguides were defined by etching air trenches in the silicon slab-waveguide. (c) SEM image of the fabricated lens.

Fig. 4
Fig. 4

Experimental characterization of the sample. (a) The optical circuit of our characterization setup. An amplified spontaneous emission source was used to illuminate the entire lens, while a 1.55 μm laser was focused to one of the four input gratings at a time. A CCD camera was used to image the lens and observe the location of the output beam. The half-wave plate and polarizer were oriented to partially filter the input illumination to reduce saturation of the CCD detector. (b)-(e) Images of the lens with the IR laser coupled to each of the four input gratings.

Tables (1)

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Table 1 Experimental Results for Transformed Luneburg Lens Beam Angle

Equations (3)

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n(r)= 2 ( r r lens ) 2 r r lens
' = μ ' = Λ Λ T |Λ| ,
n ' (x,y)= μ zz ' n(x,y)

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