Abstract

Recently researchers demonstrated, both theoretically and in the microwave range experimentally, subwavelength focusing of evanescent waves by patterned plates. The present Letter extends these ideas and the design procedure to scatterers of arbitrary shapes and to the optical range of wavelengths. The analytical study is supported by numerical results. The most intriguing feature of the proposed design is that, in the framework of classical electrodynamics of continuous media, focusing can in principle be arbitrarily sharp, subject to the constraints of fabrication.

© 2009 Optical Society of America

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  1. N. I. Zheludev, Nature Mater. 7, 420 (2008).
    [CrossRef]
  2. J. Pendry, Science 322, 71 (2008).
    [CrossRef] [PubMed]
  3. F. Huang and N. I. Zheludev, Nano Lett. 9, 1249 (2009).
    [CrossRef] [PubMed]
  4. N. I. Zheludev, University of Southampton, UK (personal communication, 2009).
  5. J. B. Pendry and D. R. Smith, Phys. Today 57, 37 (2004).
    [CrossRef]
  6. J. Dai, F. Čajko, I. Tsukerman, and M. I. Stockman, Phys. Rev. B 77, 115419 (2008).
    [CrossRef]
  7. R. Merlin, Science 317, 927 (2007).
    [CrossRef] [PubMed]
  8. A. Grbic, L. Jiang, and R. Merlin, Science 320, 511 (2008).
    [CrossRef] [PubMed]
  9. L. E. Helseth, Opt. Commun. 281, 1981 (2008).
    [CrossRef]
  10. S. I. Bozhevolnyi and B. Vohnsen, Phys. Rev. Lett. 77, 3351 (1996).
    [CrossRef] [PubMed]
  11. S. I. Bozhevolnyi, University of Southern Denmark (personal communication, 2009).
  12. G. Della Valle, T. Sondergaard, and S. I. Bozhevolnyi, Opt. Express 16, 6867 (2008).
    [CrossRef] [PubMed]
  13. The magnetic field can itself be viewed as a (scaled) stream function.
  14. I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, Phys. Rev. B 72, 155412 (2005).
    [CrossRef]
  15. G. Shvets and Y. A. Urzhumov, Phys. Rev. Lett. 93, 243902 (2004).
    [CrossRef]
  16. True for all natural materials at optical frequencies.
  17. As in , in the wave case small negative values of ϵshell″(ϕ) in Eq. are mathematically possible but could in practice be ignored.
  18. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002).
  19. J. T. Oden and S. Prudhomme, Comput. Math. Appl. 41, 735 (2001).
    [CrossRef]

2009 (1)

F. Huang and N. I. Zheludev, Nano Lett. 9, 1249 (2009).
[CrossRef] [PubMed]

2008 (6)

N. I. Zheludev, Nature Mater. 7, 420 (2008).
[CrossRef]

J. Pendry, Science 322, 71 (2008).
[CrossRef] [PubMed]

J. Dai, F. Čajko, I. Tsukerman, and M. I. Stockman, Phys. Rev. B 77, 115419 (2008).
[CrossRef]

A. Grbic, L. Jiang, and R. Merlin, Science 320, 511 (2008).
[CrossRef] [PubMed]

L. E. Helseth, Opt. Commun. 281, 1981 (2008).
[CrossRef]

G. Della Valle, T. Sondergaard, and S. I. Bozhevolnyi, Opt. Express 16, 6867 (2008).
[CrossRef] [PubMed]

2007 (1)

R. Merlin, Science 317, 927 (2007).
[CrossRef] [PubMed]

2005 (1)

I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, Phys. Rev. B 72, 155412 (2005).
[CrossRef]

2004 (2)

G. Shvets and Y. A. Urzhumov, Phys. Rev. Lett. 93, 243902 (2004).
[CrossRef]

J. B. Pendry and D. R. Smith, Phys. Today 57, 37 (2004).
[CrossRef]

2001 (1)

J. T. Oden and S. Prudhomme, Comput. Math. Appl. 41, 735 (2001).
[CrossRef]

1996 (1)

S. I. Bozhevolnyi and B. Vohnsen, Phys. Rev. Lett. 77, 3351 (1996).
[CrossRef] [PubMed]

Bozhevolnyi, S. I.

G. Della Valle, T. Sondergaard, and S. I. Bozhevolnyi, Opt. Express 16, 6867 (2008).
[CrossRef] [PubMed]

S. I. Bozhevolnyi and B. Vohnsen, Phys. Rev. Lett. 77, 3351 (1996).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, University of Southern Denmark (personal communication, 2009).

Cajko, F.

J. Dai, F. Čajko, I. Tsukerman, and M. I. Stockman, Phys. Rev. B 77, 115419 (2008).
[CrossRef]

Dai, J.

J. Dai, F. Čajko, I. Tsukerman, and M. I. Stockman, Phys. Rev. B 77, 115419 (2008).
[CrossRef]

Della Valle, G.

Fredkin, D. R.

I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, Phys. Rev. B 72, 155412 (2005).
[CrossRef]

Grbic, A.

A. Grbic, L. Jiang, and R. Merlin, Science 320, 511 (2008).
[CrossRef] [PubMed]

Helseth, L. E.

L. E. Helseth, Opt. Commun. 281, 1981 (2008).
[CrossRef]

Huang, F.

F. Huang and N. I. Zheludev, Nano Lett. 9, 1249 (2009).
[CrossRef] [PubMed]

Jiang, L.

A. Grbic, L. Jiang, and R. Merlin, Science 320, 511 (2008).
[CrossRef] [PubMed]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Mayergoyz, I. D.

I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, Phys. Rev. B 72, 155412 (2005).
[CrossRef]

Merlin, R.

A. Grbic, L. Jiang, and R. Merlin, Science 320, 511 (2008).
[CrossRef] [PubMed]

R. Merlin, Science 317, 927 (2007).
[CrossRef] [PubMed]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Oden, J. T.

J. T. Oden and S. Prudhomme, Comput. Math. Appl. 41, 735 (2001).
[CrossRef]

Pendry, J.

J. Pendry, Science 322, 71 (2008).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry and D. R. Smith, Phys. Today 57, 37 (2004).
[CrossRef]

Prudhomme, S.

J. T. Oden and S. Prudhomme, Comput. Math. Appl. 41, 735 (2001).
[CrossRef]

Shvets, G.

G. Shvets and Y. A. Urzhumov, Phys. Rev. Lett. 93, 243902 (2004).
[CrossRef]

Smith, D. R.

J. B. Pendry and D. R. Smith, Phys. Today 57, 37 (2004).
[CrossRef]

Sondergaard, T.

Stockman, M. I.

J. Dai, F. Čajko, I. Tsukerman, and M. I. Stockman, Phys. Rev. B 77, 115419 (2008).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Tsukerman, I.

J. Dai, F. Čajko, I. Tsukerman, and M. I. Stockman, Phys. Rev. B 77, 115419 (2008).
[CrossRef]

Urzhumov, Y. A.

G. Shvets and Y. A. Urzhumov, Phys. Rev. Lett. 93, 243902 (2004).
[CrossRef]

Vohnsen, B.

S. I. Bozhevolnyi and B. Vohnsen, Phys. Rev. Lett. 77, 3351 (1996).
[CrossRef] [PubMed]

Zhang, Z.

I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, Phys. Rev. B 72, 155412 (2005).
[CrossRef]

Zheludev, N. I.

F. Huang and N. I. Zheludev, Nano Lett. 9, 1249 (2009).
[CrossRef] [PubMed]

N. I. Zheludev, Nature Mater. 7, 420 (2008).
[CrossRef]

N. I. Zheludev, University of Southampton, UK (personal communication, 2009).

Comput. Math. Appl. (1)

J. T. Oden and S. Prudhomme, Comput. Math. Appl. 41, 735 (2001).
[CrossRef]

Nano Lett. (1)

F. Huang and N. I. Zheludev, Nano Lett. 9, 1249 (2009).
[CrossRef] [PubMed]

Nature Mater. (1)

N. I. Zheludev, Nature Mater. 7, 420 (2008).
[CrossRef]

Opt. Commun. (1)

L. E. Helseth, Opt. Commun. 281, 1981 (2008).
[CrossRef]

Opt. Express (1)

Phys. Rev. B (2)

J. Dai, F. Čajko, I. Tsukerman, and M. I. Stockman, Phys. Rev. B 77, 115419 (2008).
[CrossRef]

I. D. Mayergoyz, D. R. Fredkin, and Z. Zhang, Phys. Rev. B 72, 155412 (2005).
[CrossRef]

Phys. Rev. Lett. (2)

G. Shvets and Y. A. Urzhumov, Phys. Rev. Lett. 93, 243902 (2004).
[CrossRef]

S. I. Bozhevolnyi and B. Vohnsen, Phys. Rev. Lett. 77, 3351 (1996).
[CrossRef] [PubMed]

Phys. Today (1)

J. B. Pendry and D. R. Smith, Phys. Today 57, 37 (2004).
[CrossRef]

Science (3)

J. Pendry, Science 322, 71 (2008).
[CrossRef] [PubMed]

R. Merlin, Science 317, 927 (2007).
[CrossRef] [PubMed]

A. Grbic, L. Jiang, and R. Merlin, Science 320, 511 (2008).
[CrossRef] [PubMed]

Other (6)

True for all natural materials at optical frequencies.

As in , in the wave case small negative values of ϵshell″(ϕ) in Eq. are mathematically possible but could in practice be ignored.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002).

N. I. Zheludev, University of Southampton, UK (personal communication, 2009).

S. I. Bozhevolnyi, University of Southern Denmark (personal communication, 2009).

The magnetic field can itself be viewed as a (scaled) stream function.

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Figures (4)

Fig. 1
Fig. 1

A thin shell (coating) can be capable of focusing light to an arbitrarily narrow spot if the angular variation of its dielectric function and/or thickness are judiciously chosen. See text for details.

Fig. 2
Fig. 2

ϵ shell Δ r r cyl versus angle for H f = 2 , r f = 1.2 r cyl and Δ ϕ = 2 π 16 , N = 12 harmonics.

Fig. 3
Fig. 3

ϵ shell Δ r r cyl vs. angle for H f = 10 , r f = 1.2 r cyl and Δ ϕ = 2 π 18 , N = 20 harmonics.

Fig. 4
Fig. 4

E r versus angle at the radius of the focus r f = 1.2 r cyl . H f = 10 , N = 20 , Δ ϕ = 2 π 18 . Solid curve, semianalytical solution. Coinciding dashed and dotted curves, no losses ( ϵ = 0 ) , FD grids n r × n ϕ = 150 × 480 and 300 × 720 . Empty squares, ϵ = 0.1 , grid 150 × 480 .

Equations (10)

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H out ( r f , ϕ ) = H f g ( ϕ ϕ f ) ,
H in ( r , ϕ ) = n = a n r n exp ( i n ϕ ) , r r cyl ,
H s ( r , ϕ ) = n = c n r n exp ( i n ϕ ) ,
H in r = ϵ in ϵ out H out r ,
a n = c n r cyl 2 n ϵ in ϵ out 1 , n ± 1 ,
a ± 1 = ϵ in ϵ out 1 ( c ± 1 r cyl 2 + 1 2 h 0 exp ( i ϕ 0 ) ) .
H out H in = i ω ϵ shell ( ϕ ) E ϕ Δ r ( ϕ ) .
H out H in = ϵ shell ( ϕ ) ϵ in out Δ r ( ϕ ) H in out r ,
c n = H f r f n g ̃ n , n ± 1 , c ± 1 = H f r f 1 2 h 0 r f exp ( i ϕ 0 ) ,
ϵ shell ( ϕ ) Δ r ( ϕ ) = ϵ in [ H out ( r cyl ) H in ( r cyl ) ] H in ( r cyl ) r .

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