Abstract

A study of the corrections for some low-order aberrations of multilayer flat lenses based on the introduction of negative-index materials is presented. With the aberration coefficients of a multilayer flat lens that we provided, numerical solutions for the parameters of the double-layer three-layer, and four-layer flat lenses are investigated, respectively, to correct the aberrations, under some conditions, up to orders as high as possible. We find that with the increment of the layer number, the spherical and oblique aberrations can be corrected up to higher orders though the corresponding ranges of the refractive indices and working distances become narrower.

© 2006 Optical Society of America

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  1. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsi and μ," Sov. Phys. Usp. 10, 509-514 (1968).
    [CrossRef]
  2. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
    [CrossRef] [PubMed]
  3. R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
    [CrossRef] [PubMed]
  4. A. K. Iyer, P. C. Kremer, and G. V. Eleftheriades, "Experimental and theoretical verification of focusing in a large, periodically loaded transmission line negative refractive index metamaterial," Opt. Express 11, 696-708 (2003).
    [CrossRef] [PubMed]
  5. B. Gralak, S. Enoch, and G. Tayeb, "Anomalous refractive properties of photonic crystals," J. Opt. Soc. Am. A 17, 1012-1020 (2000).
    [CrossRef]
  6. S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, "Refraction in media with a negative refractive index," Phys. Rev. Lett. 90, 107402-107405 (2003).
    [CrossRef] [PubMed]
  7. P. Vodo, P. V. Parimi, W. T. Lu, and S. Sridhar, "Focusing by planoconcave lens using negative refraction," Appl. Phys. Lett. 86, 201108-201110 (2005).
    [CrossRef]
  8. V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, "Plasmon modes and negative refraction in metal nanowire composites," Opt. Express 11, 735-745 (2003).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  11. N. Fang and X. Zhang, "Imaging properties of a metamaterial superlens," Appl. Phys. Lett. 82, 161-163 (2003).
    [CrossRef]
  12. A. L. Pokrovsky and A. L. Efros, "Lens based on the use of left-handed materials," Appl. Opt. 42, 5701-5705 (2003).
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    [CrossRef]
  15. C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, and M. H. Tanielian, "Performance of a negative index of refraction lens," Appl. Phys. Lett. 84, 3232-3234 (2004).
    [CrossRef]
  16. J. Braat, "Analytical expressions for the wave-front aberration coefficients of a tiled plane-parallel plate," Appl. Opt. 36, 8459-8466 (1997).
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  17. M. Bom and E. Wolf, Principles of Optics (Pergamon, 1980), pp. 135-137.

2005 (1)

P. Vodo, P. V. Parimi, W. T. Lu, and S. Sridhar, "Focusing by planoconcave lens using negative refraction," Appl. Phys. Lett. 86, 201108-201110 (2005).
[CrossRef]

2004 (3)

L. Liu and S. He, "Near-field optical storage system using a solid immersion lens with a left-handed material slab," Opt. Express 12, 4835-4840 (2004).
[CrossRef] [PubMed]

D. Schurig and D. R. Smith, "Negative index lens aberrations," Phys. Rev. E 70, 065601-065604 (2004).
[CrossRef]

C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, and M. H. Tanielian, "Performance of a negative index of refraction lens," Appl. Phys. Lett. 84, 3232-3234 (2004).
[CrossRef]

2003 (6)

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

2000 (3)

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

B. Gralak, S. Enoch, and G. Tayeb, "Anomalous refractive properties of photonic crystals," J. Opt. Soc. Am. A 17, 1012-1020 (2000).
[CrossRef]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

1997 (1)

1968 (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsi and μ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Bom, M.

M. Bom and E. Wolf, Principles of Optics (Pergamon, 1980), pp. 135-137.

Braat, J.

Economou, E. N.

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, "Refraction in media with a negative refractive index," Phys. Rev. Lett. 90, 107402-107405 (2003).
[CrossRef] [PubMed]

Efros, A. L.

Eleftheriades, G. V.

Enoch, S.

Fang, N.

N. Fang and X. Zhang, "Imaging properties of a metamaterial superlens," Appl. Phys. Lett. 82, 161-163 (2003).
[CrossRef]

Foteinopoulou, S.

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, "Refraction in media with a negative refractive index," Phys. Rev. Lett. 90, 107402-107405 (2003).
[CrossRef] [PubMed]

Gralak, B.

Greegor, R. B.

C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, and M. H. Tanielian, "Performance of a negative index of refraction lens," Appl. Phys. Lett. 84, 3232-3234 (2004).
[CrossRef]

He, S.

Iyer, A. K.

Kremer, P. C.

Li, K.

C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, and M. H. Tanielian, "Performance of a negative index of refraction lens," Appl. Phys. Lett. 84, 3232-3234 (2004).
[CrossRef]

Liu, L.

Lu, W. T.

P. Vodo, P. V. Parimi, W. T. Lu, and S. Sridhar, "Focusing by planoconcave lens using negative refraction," Appl. Phys. Lett. 86, 201108-201110 (2005).
[CrossRef]

P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, "Imaging by flat lens using negative refraction," Nature 426, 404-404 (2003).
[CrossRef] [PubMed]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Nielsen, J. A.

C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, and M. H. Tanielian, "Performance of a negative index of refraction lens," Appl. Phys. Lett. 84, 3232-3234 (2004).
[CrossRef]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Parazzoli, C. G.

C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, and M. H. Tanielian, "Performance of a negative index of refraction lens," Appl. Phys. Lett. 84, 3232-3234 (2004).
[CrossRef]

Parimi, P. V.

P. Vodo, P. V. Parimi, W. T. Lu, and S. Sridhar, "Focusing by planoconcave lens using negative refraction," Appl. Phys. Lett. 86, 201108-201110 (2005).
[CrossRef]

P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, "Imaging by flat lens using negative refraction," Nature 426, 404-404 (2003).
[CrossRef] [PubMed]

Pendry, J. B.

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

Podolskiy, V. A.

Pokrovsky, A. L.

Sarychev, A. K.

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Schurig, D.

D. Schurig and D. R. Smith, "Negative index lens aberrations," Phys. Rev. E 70, 065601-065604 (2004).
[CrossRef]

Shalaev, V. M.

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Smith, D. R.

D. Schurig and D. R. Smith, "Negative index lens aberrations," Phys. Rev. E 70, 065601-065604 (2004).
[CrossRef]

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Soukoulis, C. M.

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, "Refraction in media with a negative refractive index," Phys. Rev. Lett. 90, 107402-107405 (2003).
[CrossRef] [PubMed]

Sridhar, S.

P. Vodo, P. V. Parimi, W. T. Lu, and S. Sridhar, "Focusing by planoconcave lens using negative refraction," Appl. Phys. Lett. 86, 201108-201110 (2005).
[CrossRef]

P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, "Imaging by flat lens using negative refraction," Nature 426, 404-404 (2003).
[CrossRef] [PubMed]

Tanielian, M. H.

C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, and M. H. Tanielian, "Performance of a negative index of refraction lens," Appl. Phys. Lett. 84, 3232-3234 (2004).
[CrossRef]

Tayeb, G.

Thompson, M. A.

C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, and M. H. Tanielian, "Performance of a negative index of refraction lens," Appl. Phys. Lett. 84, 3232-3234 (2004).
[CrossRef]

Veselago, V. G.

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsi and μ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Vetter, A. M.

C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, and M. H. Tanielian, "Performance of a negative index of refraction lens," Appl. Phys. Lett. 84, 3232-3234 (2004).
[CrossRef]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Vodo, P.

P. Vodo, P. V. Parimi, W. T. Lu, and S. Sridhar, "Focusing by planoconcave lens using negative refraction," Appl. Phys. Lett. 86, 201108-201110 (2005).
[CrossRef]

P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, "Imaging by flat lens using negative refraction," Nature 426, 404-404 (2003).
[CrossRef] [PubMed]

Wolf, E.

M. Bom and E. Wolf, Principles of Optics (Pergamon, 1980), pp. 135-137.

Zhang, X.

N. Fang and X. Zhang, "Imaging properties of a metamaterial superlens," Appl. Phys. Lett. 82, 161-163 (2003).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (3)

C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, and M. H. Tanielian, "Performance of a negative index of refraction lens," Appl. Phys. Lett. 84, 3232-3234 (2004).
[CrossRef]

N. Fang and X. Zhang, "Imaging properties of a metamaterial superlens," Appl. Phys. Lett. 82, 161-163 (2003).
[CrossRef]

P. Vodo, P. V. Parimi, W. T. Lu, and S. Sridhar, "Focusing by planoconcave lens using negative refraction," Appl. Phys. Lett. 86, 201108-201110 (2005).
[CrossRef]

J. Opt. Soc. Am. A (1)

Nature (1)

P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, "Imaging by flat lens using negative refraction," Nature 426, 404-404 (2003).
[CrossRef] [PubMed]

Opt. Express (3)

Phys. Rev. E (1)

D. Schurig and D. R. Smith, "Negative index lens aberrations," Phys. Rev. E 70, 065601-065604 (2004).
[CrossRef]

Phys. Rev. Lett. (3)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

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

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, "Refraction in media with a negative refractive index," Phys. Rev. Lett. 90, 107402-107405 (2003).
[CrossRef] [PubMed]

Science (1)

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsi and μ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Other (1)

M. Bom and E. Wolf, Principles of Optics (Pergamon, 1980), pp. 135-137.

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Figures (6)

Fig. 1
Fig. 1

Schematic showing the geometry involved in the calculation of the aberration coefficients introduced by a k-layer flat lens. The principal ray S P 1 P 2 P k + 1 F α with an incident angle α is taken as a reference ray for analyzing the wavefront aberration of a general ray S G 1 G 2 G k + 1 F m at an angle u, where F α is the center point of the reference sphere and F m is the point of intersection when a perpendicular line is dropped from F α onto the general ray. The position of a random point within the Exp is described as a polar coordinate ( ρ , θ ) with a normalized radius ρ and an azimuthal angle θ.

Fig. 2
Fig. 2

Several low-order aberration coefficients contributed by the ith layer with thickness d i of a MFL are shown as functions of its refractive index, n i , for NA = 0.5 and α = 0.1   rad . (a) The spherical aberration of a marginal ray W sm - i and some low-order spherical aberration coefficients, W 40 - i , W 60 - i , and W 80 - i ; (b) several oblique aberration coefficients, W 31 - i , W 51 - i , W 71 - i , W 42 - i , and W 62 - i in the oblique incidence case. Note that the unit of the y axis is 10 3 d i and the coefficients, W 71 - i , W 42 - i , and W 62 - i are multiplied 20 times for they are too small to display clearly in the same graph.

Fig. 3
Fig. 3

Quantities n 2 , d w , and W sm are shown with the variation of n 1 for the DLFLs that are free from the fourth-order spherical aberration and the third-order oblique aberrations in the case that d 1 = d 2 = d and NA = 0.5 . The thick solid, thin solid, and dashed curves indicate the three kinds of DLFLs with d w = 0 , d w > 0 , and d w < 0 , respectively.

Fig. 4
Fig. 4

Quantities n 2 , d w , and W sm are shown as functions of n 1 for the DLFLs free from the fourth-order spherical aberration and the third-order oblique aberrations in the case that d 1 = 0.4 d , d 2 = 0.6 d , and NA = 0.5 . There always exist three solutions when | n 1 | > 0.8233 , but only one solution when | n 1 | < 0.8233 . The solid and dashed curves indicate the parameters of such DLFLs with d w > 0 and d w < 0 , respectively.

Fig. 5
Fig. 5

Quantities n 2 , n 3 , d w , and W sm are shown with the variation of n 1 for the fourth- and sixth-order spherical aberration-free TLFLs in the cases that d 1 = d 2 = d 3 = ( 1 / 3 ) d and NA = 0.5 . There are two solutions of n 2 and n 3 when | n 1 | > 0 . 9 3 3 2 and no solutions for | n 1 | < 0 . 9 3 3 2 (excluding those n 2 , 3 = ± 1 , ± n 1 ). The solid and dashed curves indicate the parameters of such TLFLs with d w > 0 and d w < 0 , respectively.

Fig. 6
Fig. 6

Quantities n 2 , n 3 , n 4 , d w , and W sm are shown as functions of n 1 for the FLFLs, which are free from the aberrations up to eigth order in the case that d 1 = d 2 = d 3 = d 4 = ( 1 / 4 ) d and NA = 0.5 . There are two solutions of n 2 , n 3 , and n 4 when | n 1 | > 0 . 9 5 9 9 and no solutions for | n 1 | < 0 . 9 5 9 9 (excluding those of n 2 , 3 , 4 = ± 1 , ± n 1 ). The solid and dashed curves indicate the parameters of such FLFLs with d w > 0 and d w < 0 , respectively.

Tables (1)

Tables Icon

Table 1 Expressions for Some Low-Order Aberration Coefficients for a k -Layer Flat Lens with a Small α

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

W = [ S G 1 G 2 G k + 1 F m ] [ S P 1 P 2 P k + 1 F α ] .
W ( u , α ) = i = 1 k d i n i ( 1 sin 2 u / n i 2 ) 1 / 2 + d i ( 1 cos α cos u n i 2 ) n i ( 1 sin 2 α / n i 2 ) 1 / 2 .
ρ = r / r max = sin u / sin u max = sin u / NA ,
W ( ρ , θ ) = W l m ρ 1 cos m θ = W 00 + W 20 ρ 2 + W 11 ρ cos θ + W 40 ρ 4 + W 31 ρ 3 cos θ + W 22 ρ 2 cos 2 θ + W 60 ρ 6 + W 51 ρ 5 cos θ + W 42 ρ 4 cos 2 θ + W 33 ρ 3 cos 3 θ + W 80 ρ 8 + W 71 ρ 7 cos θ + .
W ( ρ , θ ; α ) = i = 1 k d cos α ( n i 2 sin 2 a ) 1 / 2 [ ρ NA   sin   α   cos   θ + cos   α ( 1 ρ 2 NA 2 ) 1 / 2 ] + d ( 1 n i     2 ) ( n i     2 sin 2 α ) 1 / 2 + d { n i     2 sin 2 α [ ρ 2 NA 2 ( cos 2 α sin 2 α cos 2 θ ) ρ NA   sin   2 α   cos   θ ( 1 ρ 2 NA 2 ) 1 / 2 ] } 1 / 2 .
W sm = l = 0 + W l 0 = W 40 + W 60 + W 80 +
= i = 1 k d i ( 1 1 NA 2 ) / n i - n i d i × [ 1 1 - ( NA / n i ) 2 ] ,
d w = i = 1 k ( d i n i ) > s 1 > 0 ,
W 40 = ( NA ) 4 i = 1 k ( n i 2 1 ) d i / ( 8 n i 3 ) = 0.
W 40 = ( NA ) 4 i = 1 3 ( n i 2 1 ) / ( 8 n i 3 ) d i = 0 ,
W 60 = ( NA ) 6 i = 1 3 ( n i 4 1 ) / ( 16 n i 5 ) d i = 0 ,
W 51 = 2 ( α NA ) W 40 6 ( α / NA ) W 60 , W 42 = 3 α 2 W 40 + 6 ( α / NA ) 2 W 60 , W 33 = ( α 3 / NA ) W 40 2 ( α / NA ) 3 W 60 .
W 40 = ( NA ) 4 i = 1 4 ( n i 2 1 ) / ( 8 n i 3 ) d i = 0 ,
W 60 = ( NA ) 6 i = 1 4 ( n i 4 1 ) / ( 16 n i 5 ) d i = 0 ,
W 80 = ( NA ) 8 i = 1 4 5 ( n i 6 1 ) / ( 128 n i 7 ) d i = 0
W 71 = 0.5 ( α NA 3 ) W 40 + 3 ( α NA ) W 60 8 ( α / NA ) W 80 , W 62 = 7.5 α 2 W 60 + 12 ( α / NA ) 2 W 80 , W 53 = 1.5 α 3 NA W 40 18 ( α 3 / NA ) W 60 + 24 ( α / NA ) 2 W 80 , W 44 = W 53 / 4 .
i = 1 k ( n i 2 1 ) / ( n i 3 ) = 0 , i = 1 k ( n i 4 1 ) / ( n i 5 ) i = 0 ,
i = 1 k ( n i 6 1 ) / ( n i 7 ) i = 0 ,   ,
i = 1 k ( n i     2 k 2 1 ) / ( n i     2 k 1 ) = 0 ,

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