Abstract

We propose a transformational design of an axi-symmetric gradient lens for electromagnetic waves. We show that a metamaterial consisting of toroidal air channels of diameters ranging from 23 nm to 190 nm in a matrix of Polymethylmethacrylate (PMMA) allows for a focussing effect of light over a large bandwidth i.e. [600 – 1000] nm. We finally propose a simplified design of lens allowing for a two-photon lithography implementation.

© 2011 OSA

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References

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  1. S. -C. S. Lin, T. J. Huang, J. H. Sun, and T. T. Wu, “Gradient-index phononic crystals,” Phys. Rev. B 79, 094302 (2009).
    [CrossRef]
  2. A. Climente, D. Torrent, and J. Sanchez-Dehesa, “Sound focusing by gradient index sonic lenses,” Appl. Phys. Lett. 97, 104103 (2010).
    [CrossRef]
  3. J. B. Pendry, D. Schurig, and D. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [CrossRef] [PubMed]
  4. U. Leonhardt, “Optical Conformal Mapping,” Science 312, 1777–1780 (2006).
    [CrossRef] [PubMed]
  5. C. A. Swainson (alias J. C. Maxwell), “Problems,” Cambridge Dublin Math. J. 8, 188–189 (1854).
  6. R. K. Luneburg, Mathematical theory of optics (University of California Press, Berkeley, 1964).
  7. U. Leonhardt, “Perfect imaging without negative refraction,” N. J. Phys. 11(9), 093040 (2009).
    [CrossRef]
  8. S. Guenneau, A. Diatta, and R. C. McPhedran, “Focussing: coming to the point in metamaterials,” J. Modern Opt. 57, 511–527 (2010).
    [CrossRef]
  9. P. Benitez, J. C. Minano, J. C. Gonzalez, and C. Juan, “Perfect focussing of scalar wave fields in three dimensions,” Opt. Express 18(8), 7650–7663 (2010).
    [CrossRef] [PubMed]
  10. R. MerlinComment on, “Perfect imaging with positive refraction in three dimensions,” Phys. Rev. A 82(5), 057801 (2010).
    [CrossRef]
  11. C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-index optics: Fundamentals and applications (Springer, New York, 2002).
  12. S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mat. 29, 1481–1490 (2007).
    [CrossRef]
  13. J. C. M. Garnett, “Colours in Metal Glasses and in Metallic Films,” Phil. Trans. Royal Soc. London, Ser. A 203, 385 (1904).
    [CrossRef]
  14. X. Hu, C. T. Chan, J. Zi, M. Li, and K. -M. Ho, “Diamagnetic Response of Metallic Photonic Crystals at Infrared and Visible Frequencies,” Phys. Rev. Lett. 96, 223901 (2006).
    [CrossRef] [PubMed]

2010

A. Climente, D. Torrent, and J. Sanchez-Dehesa, “Sound focusing by gradient index sonic lenses,” Appl. Phys. Lett. 97, 104103 (2010).
[CrossRef]

S. Guenneau, A. Diatta, and R. C. McPhedran, “Focussing: coming to the point in metamaterials,” J. Modern Opt. 57, 511–527 (2010).
[CrossRef]

R. MerlinComment on, “Perfect imaging with positive refraction in three dimensions,” Phys. Rev. A 82(5), 057801 (2010).
[CrossRef]

P. Benitez, J. C. Minano, J. C. Gonzalez, and C. Juan, “Perfect focussing of scalar wave fields in three dimensions,” Opt. Express 18(8), 7650–7663 (2010).
[CrossRef] [PubMed]

2009

U. Leonhardt, “Perfect imaging without negative refraction,” N. J. Phys. 11(9), 093040 (2009).
[CrossRef]

S. -C. S. Lin, T. J. Huang, J. H. Sun, and T. T. Wu, “Gradient-index phononic crystals,” Phys. Rev. B 79, 094302 (2009).
[CrossRef]

2007

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mat. 29, 1481–1490 (2007).
[CrossRef]

2006

X. Hu, C. T. Chan, J. Zi, M. Li, and K. -M. Ho, “Diamagnetic Response of Metallic Photonic Crystals at Infrared and Visible Frequencies,” Phys. Rev. Lett. 96, 223901 (2006).
[CrossRef] [PubMed]

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

U. Leonhardt, “Optical Conformal Mapping,” Science 312, 1777–1780 (2006).
[CrossRef] [PubMed]

2002

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-index optics: Fundamentals and applications (Springer, New York, 2002).

1964

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

1904

J. C. M. Garnett, “Colours in Metal Glasses and in Metallic Films,” Phil. Trans. Royal Soc. London, Ser. A 203, 385 (1904).
[CrossRef]

1854

C. A. Swainson (alias J. C. Maxwell), “Problems,” Cambridge Dublin Math. J. 8, 188–189 (1854).

Bao, C.

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-index optics: Fundamentals and applications (Springer, New York, 2002).

Benitez, P.

Chan, C. T.

X. Hu, C. T. Chan, J. Zi, M. Li, and K. -M. Ho, “Diamagnetic Response of Metallic Photonic Crystals at Infrared and Visible Frequencies,” Phys. Rev. Lett. 96, 223901 (2006).
[CrossRef] [PubMed]

Climente, A.

A. Climente, D. Torrent, and J. Sanchez-Dehesa, “Sound focusing by gradient index sonic lenses,” Appl. Phys. Lett. 97, 104103 (2010).
[CrossRef]

Diatta, A.

S. Guenneau, A. Diatta, and R. C. McPhedran, “Focussing: coming to the point in metamaterials,” J. Modern Opt. 57, 511–527 (2010).
[CrossRef]

Garnett, J. C. M.

J. C. M. Garnett, “Colours in Metal Glasses and in Metallic Films,” Phil. Trans. Royal Soc. London, Ser. A 203, 385 (1904).
[CrossRef]

Gomez-Reino, C.

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-index optics: Fundamentals and applications (Springer, New York, 2002).

Gonzalez, J. C.

Guenneau, S.

S. Guenneau, A. Diatta, and R. C. McPhedran, “Focussing: coming to the point in metamaterials,” J. Modern Opt. 57, 511–527 (2010).
[CrossRef]

Ho, K. -M.

X. Hu, C. T. Chan, J. Zi, M. Li, and K. -M. Ho, “Diamagnetic Response of Metallic Photonic Crystals at Infrared and Visible Frequencies,” Phys. Rev. Lett. 96, 223901 (2006).
[CrossRef] [PubMed]

Hu, X.

X. Hu, C. T. Chan, J. Zi, M. Li, and K. -M. Ho, “Diamagnetic Response of Metallic Photonic Crystals at Infrared and Visible Frequencies,” Phys. Rev. Lett. 96, 223901 (2006).
[CrossRef] [PubMed]

Huang, T. J.

S. -C. S. Lin, T. J. Huang, J. H. Sun, and T. T. Wu, “Gradient-index phononic crystals,” Phys. Rev. B 79, 094302 (2009).
[CrossRef]

Ivanov, C. D.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mat. 29, 1481–1490 (2007).
[CrossRef]

Juan, C.

Kasarova, S. N.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mat. 29, 1481–1490 (2007).
[CrossRef]

Leonhardt, U.

U. Leonhardt, “Perfect imaging without negative refraction,” N. J. Phys. 11(9), 093040 (2009).
[CrossRef]

U. Leonhardt, “Optical Conformal Mapping,” Science 312, 1777–1780 (2006).
[CrossRef] [PubMed]

Li, M.

X. Hu, C. T. Chan, J. Zi, M. Li, and K. -M. Ho, “Diamagnetic Response of Metallic Photonic Crystals at Infrared and Visible Frequencies,” Phys. Rev. Lett. 96, 223901 (2006).
[CrossRef] [PubMed]

Lin, S. -C. S.

S. -C. S. Lin, T. J. Huang, J. H. Sun, and T. T. Wu, “Gradient-index phononic crystals,” Phys. Rev. B 79, 094302 (2009).
[CrossRef]

Luneburg, R. K.

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

McPhedran, R. C.

S. Guenneau, A. Diatta, and R. C. McPhedran, “Focussing: coming to the point in metamaterials,” J. Modern Opt. 57, 511–527 (2010).
[CrossRef]

Merlin, R.

R. MerlinComment on, “Perfect imaging with positive refraction in three dimensions,” Phys. Rev. A 82(5), 057801 (2010).
[CrossRef]

Minano, J. C.

Nikolov, I. D.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mat. 29, 1481–1490 (2007).
[CrossRef]

Pendry, J. B.

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

Perez, M. V.

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-index optics: Fundamentals and applications (Springer, New York, 2002).

Sanchez-Dehesa, J.

A. Climente, D. Torrent, and J. Sanchez-Dehesa, “Sound focusing by gradient index sonic lenses,” Appl. Phys. Lett. 97, 104103 (2010).
[CrossRef]

Schurig, D.

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

Smith, D.

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

Sultanova, N. G.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mat. 29, 1481–1490 (2007).
[CrossRef]

Sun, J. H.

S. -C. S. Lin, T. J. Huang, J. H. Sun, and T. T. Wu, “Gradient-index phononic crystals,” Phys. Rev. B 79, 094302 (2009).
[CrossRef]

Swainson, C. A.

C. A. Swainson (alias J. C. Maxwell), “Problems,” Cambridge Dublin Math. J. 8, 188–189 (1854).

Torrent, D.

A. Climente, D. Torrent, and J. Sanchez-Dehesa, “Sound focusing by gradient index sonic lenses,” Appl. Phys. Lett. 97, 104103 (2010).
[CrossRef]

Wu, T. T.

S. -C. S. Lin, T. J. Huang, J. H. Sun, and T. T. Wu, “Gradient-index phononic crystals,” Phys. Rev. B 79, 094302 (2009).
[CrossRef]

Zi, J.

X. Hu, C. T. Chan, J. Zi, M. Li, and K. -M. Ho, “Diamagnetic Response of Metallic Photonic Crystals at Infrared and Visible Frequencies,” Phys. Rev. Lett. 96, 223901 (2006).
[CrossRef] [PubMed]

Appl. Phys. Lett.

A. Climente, D. Torrent, and J. Sanchez-Dehesa, “Sound focusing by gradient index sonic lenses,” Appl. Phys. Lett. 97, 104103 (2010).
[CrossRef]

Cambridge Dublin Math. J.

C. A. Swainson (alias J. C. Maxwell), “Problems,” Cambridge Dublin Math. J. 8, 188–189 (1854).

J. Modern Opt.

S. Guenneau, A. Diatta, and R. C. McPhedran, “Focussing: coming to the point in metamaterials,” J. Modern Opt. 57, 511–527 (2010).
[CrossRef]

N. J. Phys.

U. Leonhardt, “Perfect imaging without negative refraction,” N. J. Phys. 11(9), 093040 (2009).
[CrossRef]

Opt. Express

Opt. Mat.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mat. 29, 1481–1490 (2007).
[CrossRef]

Phil. Trans. Royal Soc. London, Ser. A

J. C. M. Garnett, “Colours in Metal Glasses and in Metallic Films,” Phil. Trans. Royal Soc. London, Ser. A 203, 385 (1904).
[CrossRef]

Phys. Rev. A

R. MerlinComment on, “Perfect imaging with positive refraction in three dimensions,” Phys. Rev. A 82(5), 057801 (2010).
[CrossRef]

Phys. Rev. B

S. -C. S. Lin, T. J. Huang, J. H. Sun, and T. T. Wu, “Gradient-index phononic crystals,” Phys. Rev. B 79, 094302 (2009).
[CrossRef]

Phys. Rev. Lett.

X. Hu, C. T. Chan, J. Zi, M. Li, and K. -M. Ho, “Diamagnetic Response of Metallic Photonic Crystals at Infrared and Visible Frequencies,” Phys. Rev. Lett. 96, 223901 (2006).
[CrossRef] [PubMed]

Science

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

U. Leonhardt, “Optical Conformal Mapping,” Science 312, 1777–1780 (2006).
[CrossRef] [PubMed]

Other

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

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-index optics: Fundamentals and applications (Springer, New York, 2002).

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Figures (6)

Fig. 1
Fig. 1

Numerical validation at wavelength λ = 700 nm for a GRIN lens of thickness and radius R 0 of 1000 nm. (a) Profile of the refractive index versus radius; (b) Three-dimensional plot of the norm of electric field; (c) Side view of the real part of y-component of electric field; (d) Side view of norm of electric field.

Fig. 2
Fig. 2

(a) Three-dimensional diagrammatic view of the structured GRIN lens; (b) Side view where R denotes the radius of the lens and r the radius of toroidal air channels; (c) Top view; (d) Table of filling fraction and effective index of refraction.

Fig. 3
Fig. 3

Numerical validation at wavelength λ = 700 nm for a GRIN lens of thickness of 600 nm and radius R 0 of 1000 nm. (a) The total energy density varies with optical axis (αβ). Dashed box indicates the location of GRIN lens; (b) Three-dimensional plot of the real part of the the y-component of electric field; (c) Side view of the y-component of electric field; (d) Side view of norm of electric field.

Fig. 4
Fig. 4

(a) The total energy density varies with the distance from lens along optical axis; (b) The distributions of total energy density on the focal plane β in panel (c); (c) Side view of total energy density; (d) The distributions of total energy density from panel (c).

Fig. 5
Fig. 5

Total energy density for wavelengths of (a) 600 nm; (b) 850 nm; (c) 1000 nm; (d) Variation of total energy density versus z axis. Dashed box indicates the location of lens.

Fig. 6
Fig. 6

Calculated magnitude of energy density (J/m 3) for a wavelength of 700 nm (a) Only PMMA; (b) Effective structure with inner air channels removed; (c) Complete structure; (d) The effective refractive index for each toroidal unit cell (dotted green line and solid red lines are the effective index in panels (b) and (c), respectively). The dashed blue curve is the ideal index from equation (1) for comparison. (e) Three-dimensional diagrammatic view of the split ring GRIN lens; (f) Three-dimensional plot of the energy density.

Equations (2)

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n ( r ) = n 0 sech ( α r ) where α = 1 R 0 cosh 1 ( n 0 n R ) ,
ɛ e ɛ ɛ e + ɛ = ɛ 0 ɛ ɛ 0 + ɛ f

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