Abstract

We discuss by a Poynting vector analysis how the losses of a negative-index material (NIM) affect the resolution performances of a Veselago–Pendry lens, and we analyze those performances in the framework of the Abbe criterion. The limits of both high losses and low losses are explored. We find that the impedance-matched NIM is able to resolve 30% better than the limit imposed by the Abbe criterion even when the imaginary part of the refractive index (the material losses) exceeds the absolute value of the real part of the refractive index. The NIM is described by a lossy Drude model with equal permittivity and permeability. By increasing the damping parameter of the Drude model, we also explore the regime where both permittivity and permeability are positive and point out the conditions under which the metamaterial is still able to superresolve.

© 2008 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. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000) and references therein.
    [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. C. G. Parazzoli, R. B. Greegor, K. Li, K. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell's law," Phys. Rev. Lett. 90, 107401-1-4 (2003).
    [CrossRef] [PubMed]
  5. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic response of metamaterials at 100 terahertz," Science 306, 1351-1353 (2004).
    [CrossRef] [PubMed]
  6. G. D'Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, "Bright and dark gap solitons in a negative index Fabry-Perot etalon," Phys. Rev. Lett. 93, 213902-1-4 (2004).
    [CrossRef] [PubMed]
  7. G. D'Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, "TE and TM guided modes in an air waveguide with a negative-index-material cladding," Phys. Rev. E 71, 046603-1-7 (2005).
    [CrossRef]
  8. G. D'Aguanno, N. Akozbek, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, "Dispersion-free pulse propagation in a negative-index material," Opt. Lett. 30, 1998-2000 (2005).
    [CrossRef] [PubMed]
  9. M. Bloemer, G. D'Aguanno, M. Scalora, and N. Mattiucci, "Broadband omnidirectional reflection from negative index materials," Appl. Phys. Lett. 87, 261921-1-3 (2005).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  12. S. Zhang, W. Fan, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, "Near-infrared double negative metamaterials," Opt. Express 13, 4922-4930 (2005).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  14. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, "Negative-index metamaterial at 780 nm wavelength," Opt. Lett. 32, 53-55 (2007).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  17. K. Aydin, I. Bulu, and E. Ozbay, "Subwavelength resolution with a negative index metamaterial superlens," Appl. Phys. Lett. 90, 254102-1-3 (2007).
    [CrossRef]
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  19. X. Wang, Z. F. Ren, and K. Kempa, "Unrestricted superlensing in a triangular two-dimensional photonic crystal," Opt. Express 12, 2919-2924 (2004).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  23. R. W. Ziolkowski, "Propagation in and scattering from a matched metamaterial having a zero index of refraction," Phys. Rev. E 70, 046608-1-12 (2004).
    [CrossRef]
  24. Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985).
  25. The term "canalization" has been first used in P. A. Belov, C. R. Simovski, and P. Ikonen, "Canalization of subwavelength images by electromagnetic crystals," Phys. Rev. B 71, 193105-1-4 (2005).
    [CrossRef]
  26. H. Raether, Surface Plasmons (Springer-Verlag, 1988).
  27. M. Tsang and D. Psaltis, "Reflectionless evanescent-wave amplification by two dielectric planar waveguides," Opt. Lett. 31, 2741-2743 (2006).
    [CrossRef] [PubMed]
  28. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  29. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).
  30. Y. Zhao, P. Belov, and Y. Hao, "Accurate modeling of the optical properties of left-handed media using a finite-difference time-domain method," Phys. Rev. E 75, 037602 (2007).
    [CrossRef]
  31. A. A. Sukhorukov, I. V. Shadrivov, and Yu. S. Kivshar, "Wave scattering by metamaterial wedges and interfaces," Int. J. Numer. Model. 19, 105 (2006).
    [CrossRef]

2007 (5)

M. J. Bloemer, G. D'Aguanno, N. Mattiucci, M. Scalora, and N. Akozbek, "Broadband super-resolving lens with high transparency for propagating and evanescent waves in the visible range," Appl. Phys. Lett. 90, 174113-1-3 (2007).
[CrossRef]

K. Aydin, I. Bulu, and E. Ozbay, "Subwavelength resolution with a negative index metamaterial superlens," Appl. Phys. Lett. 90, 254102-1-3 (2007).
[CrossRef]

Y. Zhao, P. Belov, and Y. Hao, "Accurate modeling of the optical properties of left-handed media using a finite-difference time-domain method," Phys. Rev. E 75, 037602 (2007).
[CrossRef]

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, "Negative-index metamaterial at 780 nm wavelength," Opt. Lett. 32, 53-55 (2007).
[CrossRef]

U. K. Chettiar, A. V. Kildishev, H.-K. Yuan, W. Cai, S. Xiao, V. P. Drachev, and V. M. Shalaev, "Dual-band negative index metamaterial: double negative at 813 nm and single negative at 772 nm," Opt. Lett. 32, 1671-1673 (2007).
[CrossRef] [PubMed]

2006 (3)

2005 (8)

The term "canalization" has been first used in P. A. Belov, C. R. Simovski, and P. Ikonen, "Canalization of subwavelength images by electromagnetic crystals," Phys. Rev. B 71, 193105-1-4 (2005).
[CrossRef]

S. Zhang, W. Fan, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, "Near-infrared double negative metamaterials," Opt. Express 13, 4922-4930 (2005).
[CrossRef] [PubMed]

G. D'Aguanno, N. Akozbek, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, "Dispersion-free pulse propagation in a negative-index material," Opt. Lett. 30, 1998-2000 (2005).
[CrossRef] [PubMed]

M.-C. Yang and K. J. Webb, "Poynting vector analysis of a superlens," Opt. Lett. 30, 2382-2384 (2005).
[CrossRef] [PubMed]

V. M. Shalaev, W. Cai, U. K. Chettiar, H. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, "Negative index of refraction in optical metamaterials," Opt. Lett. 30, 3356-3358 (2005).
[CrossRef]

G. D'Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, "TE and TM guided modes in an air waveguide with a negative-index-material cladding," Phys. Rev. E 71, 046603-1-7 (2005).
[CrossRef]

M. Bloemer, G. D'Aguanno, M. Scalora, and N. Mattiucci, "Broadband omnidirectional reflection from negative index materials," Appl. Phys. Lett. 87, 261921-1-3 (2005).
[CrossRef]

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett. 95, 137404-1-4 (2005).
[CrossRef] [PubMed]

2004 (5)

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic response of metamaterials at 100 terahertz," Science 306, 1351-1353 (2004).
[CrossRef] [PubMed]

G. D'Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, "Bright and dark gap solitons in a negative index Fabry-Perot etalon," Phys. Rev. Lett. 93, 213902-1-4 (2004).
[CrossRef] [PubMed]

R. Merlin, "Analytical solution of the almost-perfect-lens problem," Appl. Phys. Lett. 84, 1290-1293 (2004) and references therein.
[CrossRef]

R. W. Ziolkowski, "Propagation in and scattering from a matched metamaterial having a zero index of refraction," Phys. Rev. E 70, 046608-1-12 (2004).
[CrossRef]

X. Wang, Z. F. Ren, and K. Kempa, "Unrestricted superlensing in a triangular two-dimensional photonic crystal," Opt. Express 12, 2919-2924 (2004).
[CrossRef] [PubMed]

2003 (1)

C. G. Parazzoli, R. B. Greegor, K. Li, K. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell's law," Phys. Rev. Lett. 90, 107401-1-4 (2003).
[CrossRef] [PubMed]

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 (1)

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

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]

Appl. Phys. Lett. (4)

M. J. Bloemer, G. D'Aguanno, N. Mattiucci, M. Scalora, and N. Akozbek, "Broadband super-resolving lens with high transparency for propagating and evanescent waves in the visible range," Appl. Phys. Lett. 90, 174113-1-3 (2007).
[CrossRef]

K. Aydin, I. Bulu, and E. Ozbay, "Subwavelength resolution with a negative index metamaterial superlens," Appl. Phys. Lett. 90, 254102-1-3 (2007).
[CrossRef]

M. Bloemer, G. D'Aguanno, M. Scalora, and N. Mattiucci, "Broadband omnidirectional reflection from negative index materials," Appl. Phys. Lett. 87, 261921-1-3 (2005).
[CrossRef]

R. Merlin, "Analytical solution of the almost-perfect-lens problem," Appl. Phys. Lett. 84, 1290-1293 (2004) and references therein.
[CrossRef]

Int. J. Numer. Model. (1)

A. A. Sukhorukov, I. V. Shadrivov, and Yu. S. Kivshar, "Wave scattering by metamaterial wedges and interfaces," Int. J. Numer. Model. 19, 105 (2006).
[CrossRef]

Opt. Express (2)

Opt. Lett. (7)

Phys. Rev. B (1)

The term "canalization" has been first used in P. A. Belov, C. R. Simovski, and P. Ikonen, "Canalization of subwavelength images by electromagnetic crystals," Phys. Rev. B 71, 193105-1-4 (2005).
[CrossRef]

Phys. Rev. E (3)

R. W. Ziolkowski, "Propagation in and scattering from a matched metamaterial having a zero index of refraction," Phys. Rev. E 70, 046608-1-12 (2004).
[CrossRef]

G. D'Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, "TE and TM guided modes in an air waveguide with a negative-index-material cladding," Phys. Rev. E 71, 046603-1-7 (2005).
[CrossRef]

Y. Zhao, P. Belov, and Y. Hao, "Accurate modeling of the optical properties of left-handed media using a finite-difference time-domain method," Phys. Rev. E 75, 037602 (2007).
[CrossRef]

Phys. Rev. Lett. (4)

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett. 95, 137404-1-4 (2005).
[CrossRef] [PubMed]

G. D'Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, "Bright and dark gap solitons in a negative index Fabry-Perot etalon," Phys. Rev. Lett. 93, 213902-1-4 (2004).
[CrossRef] [PubMed]

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

C. G. Parazzoli, R. B. Greegor, K. Li, K. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell's law," Phys. Rev. Lett. 90, 107401-1-4 (2003).
[CrossRef] [PubMed]

Science (2)

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic response of metamaterials at 100 terahertz," Science 306, 1351-1353 (2004).
[CrossRef] [PubMed]

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 (6)

M. Born and E. Wolf, Principles of Optics, 7th (expanded) edition (Cambridge U. Press, 1999).

M. Born and E. Wolf, Principles of Optics, 7th (expanded) edition (Cambridge U. Press, 1999), page 10.

H. Raether, Surface Plasmons (Springer-Verlag, 1988).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985).

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Figures (14)

Fig. 1
Fig. 1

A plane, monochromatic wave at the operative wavelength λ is incident on a screen of negligible thickness with two very small apertures or slits (P1 and P2) that act as pointlike sources (dimension of 1 500 th of the incident wavelength) whose mutual distance is D. A slab of NIM L = λ 9 in length, placed at a distance d 1 = λ 18 from the object plane, captures the light diffracted from the two slits and focuses it on the image plane placed at a distance d 2 = λ 18 from the end of the NIM slab. The image plane is chosen following the geometrical rule for the image formation: d 2 = L d 1 .

Fig. 2
Fig. 2

(a) Real (solid curve) and imaginary part (dashed curve) of the refractive index versus the damping term ( γ ̃ ) for a wavelength λ = 2 λ ep . The horizontal line with double arrows indicates the region where the metamaterial superresolves, i.e., when it resolves two point sources whose mutual distance is less than 0.6 λ with a visibility V 50 % . The superresolving region extends from γ ̃ = 0 + to γ ̃ 3.5 , which correspond respectively to a refractive index of n ̂ = 1 + i 0 + and n ̂ 0.92 + i 0.38 . (b) Figure of merit (FOM) versus the damping term ( γ ̃ ) . In both figures the vertical line indicates the position of γ ̃ = 1 2 , which corresponds to the transition of the metamaterial from a true NIM to a positive-index material (PIM).

Fig. 3
Fig. 3

Diffraction figures from the two pointlike sources on the image plane, respectively, for a (a) FOM 3500 and D Min = λ 16.5 , (b) FOM 350 and D Min = λ 12 , (c) FOM 35 and D Min = λ 8 , and (d) FOM 3.5 and D Min = λ 4.5 . The image contrast ( V ) is approximately 50% in all cases. The square modulus of the field is referred to the electric field for TE polarization or to the magnetic field for TM polarization. D Min is the minimum resolved distance that corresponds to a visibility V 50 % . In all the figures the maximum value of the field at the image plane has been normalized to 1.

Fig. 4
Fig. 4

1 D Min (solid circles) versus FOM according to a field analysis. For FOM 1 the minimum resolved distance ( D Min ) scales as λ ( 2 log ( FOM ) ) , while for FOM 1 the minimum resolved distance scales as 0.4 λ . The dashed horizontal line represents the limit imposed by the Abbe criterion.

Fig. 5
Fig. 5

Diffraction figures calculated through the Poynting vector for (a) a FOM 3500 and D Min = λ 15 , (b) FOM 350 and D Min = λ 11 , (c) FOM 35 and D Min = λ 7.5 , and (d) FOM 3.5 and D Min = λ 4.5 . In all the figures the maximum of the z component of the Poynting vector ( S z ) has been normalized to 1.

Fig. 6
Fig. 6

1 D Min (solid circles) versus FOM when the Poynting vector is considered. For FOM 1 the minimum resolved distance ( D Min ) scales as λ [ 1.8 log ( FOM ) ] , while for FOM 1 the minimum resolved distance scales as 0.4 λ . The dashed horizontal line represents the limit imposed by the Abbe criterion.

Fig. 7
Fig. 7

(a) Transmittance t 2 of the NIM versus k x k 0 for γ ̃ = 1 2 . (b) 3-D topographic plot of the transmittance versus k x k 0 and γ ̃ . The vertical solid line indicates the position of γ ̃ = 1 2 , where Re ( n ̂ ) = 0 .

Fig. 8
Fig. 8

Schematic picture of the rays inside a slab with (a) ε ̂ = μ ̂ = n ̂ = 1 and (b) ε ̂ = μ ̂ = n ̂ = i .

Fig. 9
Fig. 9

1 D Min (solid circles) versus ∣FOM∣ for FOM < 0 . The dashed horizontal line represents the limit imposed by the Abbe criterion. Also indicated in the figure is the refractive index of the metamaterial at some particular points.

Fig. 10
Fig. 10

Effective index Re ( n ̂ eff ) and extinction coefficient Im ( n ̂ eff ) versus wavelength for SPPs at the air–silver interface. The shaded area indicates the region where SPPs exsist. In the figure are also indicated the wavelength of 337 nm , where Re ( ε Ag ) 1 , and the wavelength of 295 nm , where Re ( ε Ag ) 1.13 .

Fig. 11
Fig. 11

(a) Geometry used for superresolution from a single layer of silver at λ = 295 nm . (b) Image of the two pointlike sources for TM polarization. The visibility is approximately 52%. The vertical dashed lines indicate the position of the pointlike sources in the object plane. S z is normalized to 1.

Fig. 12
Fig. 12

Refraction angle of the Poynting vector versus the incident angle at the air–silver interface for a wavelength of 295 nm . The dashed curve is for TE polarization, and the solid curve is for TM polarization.

Fig. 13
Fig. 13

Transmittance t 2 of the silver layer at 295 nm vs k x k 0 .

Fig. 14
Fig. 14

Comparison between S z and H 2 for the geometry described in Fig. 11. We have used nondimensional units; i.e., we take ε 0 = μ 0 = c = 1 and unitary amplitude for the electric and magnetic field of the plane, monochromatic, TM-polarized wave incident on the screen. In those nondimensional units the intensity carried by the plane wave is S z , pw = 0.5 .

Equations (14)

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Re ( n ̂ ) = 1 + 2 γ ̃ 2 1 + 2 γ ̃ 2 , Im ( n ̂ ) = 2 2 γ ̃ 1 + 2 γ ̃ 2 , FOM = 1 2 γ ̃ 2 2 2 γ ̃ .
D Min { λ 2 log ( FOM ) λ 2 log ( Im ( n ̂ ) ) for FOM 1 , Im ( n ̂ ) 1 0.4 λ for 0 FOM 1 , Im ( n ̂ ) 1 } .
D Min { λ 1.8 log ( FOM ) λ 1.8 log ( Im ( n ̂ ) ) for FOM 1 , Im ( n ̂ ) 1 0.4 λ for 0 FOM 1 , Im ( n ̂ ) 1 } .
tan ( ϑ S ) = Re ( n ̂ ) sin ϑ 0 ( [ Re ( n ̂ ) ] 2 + [ Im ( n ̂ ) ] 2 ) Re ( 1 sin 2 ϑ 0 n ̂ 2 ) .
k ̂ SPP = k 0 ( ε ̂ 1 + ε ̂ ) ,
tan ( ϑ S ) = Re ( ε ̂ ) sin ϑ 0 ε ̂ 2 Re ( n ̂ ε ̂ 1 sin 2 ϑ 0 n ̂ 2 ) ( TM polarization ) ,
tan ( ϑ S ) = Re ( μ ̂ ) sin ϑ 0 μ ̂ 2 Re ( n ̂ μ ̂ 1 sin 2 ϑ 0 n ̂ 2 ) ( TE polarization ) .
H ̃ ( x , z , t ) = ( 1 2 ) [ H ( x , z ) exp ( i ω t ) + c.c. ] ,
H ( x , z L ) = + A ( k x ) t TM ( k x ) exp [ i ( k x x + k 0 2 k x 2 ( d 1 + z L ) ) ] d k x .
t TM ( k x ) = 2 2 cos ( n ̂ k 0 2 k x 2 n ̂ 2 L ) i ( n ̂ k 0 2 k x 2 n ̂ 2 ε ̂ k 0 2 k x 2 + ε ̂ k 0 2 k x 2 n ̂ k 0 2 k x 2 n ̂ 2 ) sin ( n ̂ k 0 2 k x 2 n ̂ 2 L ) .
A ( k x ) = FT ( t screen ( z = d 1 , x ) ) ,
t screen ( z = d 1 , x ) = { 0 < x < D 2 a 1 1 D 2 a 1 x D 2 0 D 2 < x < D 2 1 D 2 x D 2 + a 2 0 D 2 + a 2 < x < } .
× H = i ω E .
S = ( 1 2 ) Re [ E × H * ] .

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