Abstract

Within an effective medium theory, we numerically study by means of a finite element method the transmission properties of prisms and slabs of media with negative refractive index. The constitutive parameters employed are similar to those of recent experiments that confirmed the existence of negative refraction as well as the focusing property of flat slabs. In this way, we further analyze in detail the influence of diffraction and scattering due to the large wavelength of the radiation in use, and its suppression by employing waveguide configurations with absorbing walls. Also, we address the effects of different amounts of absorption on both the angle of refraction, (for which we derive a new refraction law in prisms), and on the position, resolution and isoplantism of the focus produced by flat slabs.

© 2004 Optical Society of America

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References

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  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]
  2. R.A. Shelby, D.R. Smith, S.C. Nemat-Nasser and S. Schultz, �??Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,�?? Appl. Phys. Lett. 78, 489-91 (2001).
    [CrossRef]
  3. J.B. Pendry, A.J. Holden, D.J. Robbins and W.J. Stewart �??Magnetism from conductors and enhanced nonlinear phenomena,�?? IEEE Trans. Microwave Theory Tech. 47, 2075-2074 (1999).
    [CrossRef]
  4. P. Markos and C.M. Soukoulis, �??Transmission studies of left-handed materials,�?? Phys. Rev. B 65, 033401 (2001).
    [CrossRef]
  5. R.W. Ziolkowski and E. Heyman, �??Wave propagation in media having negative permittivity and permeability,�?? Phys. Rev. E 64, 056625 (2001).
    [CrossRef]
  6. P. Markos and C.M. Soukoulis, �??Numerical studies of left-handed materials and arrays of split ring resonators,�?? Phys. Rev. E 65, 036622 (2002).
    [CrossRef]
  7. N. Garcia and M. Nieto-Vesperinas, �??Is there an experimental verification of a negative index of refraction yet?�?? Opt. Lett. 27, 885-7 (2002).
    [CrossRef]
  8. P.M. Valanju, R.M.Walser and A.P. Valanju, �??Wave refraction in negative-index media: always positive and very inhomogenous,�?? Phys. Rev. Lett. 91, 187401 (2002).
    [CrossRef]
  9. M.W. McCall, A. Lakhtakia and W.S. Weiglhofer, �??The negative index of refraction demystified,�?? European Journal of Physics, 23, 353 - 359 (2002).
    [CrossRef]
  10. V.G. Veselago, �??The electrodynamics of substances with simultanenous negative values of ε and μ,�?? Sov. Phys. Usp. 10, 509-514 (1968).
    [CrossRef]
  11. C.G. Parazzoli, R.B. Greegor, K. Li, B.E.C. Koltenbah and M. Tanielian, �??Experimental verification and simulation of negative index of refraction using Snell�??s law,�?? Phys. Rev. Lett. 90, 107401 (2003).
    [CrossRef] [PubMed]
  12. R.B. Greegor, C.G. Parazzoli, K. Li, and M.H. Tanielian, �??Origin of dissipative losses in negative index of refraction materials,�?? Appl. Phys. Lett. 82, 2356-2358 (2003).
    [CrossRef]
  13. A.A. Houck, J.B. Brock and I.L. Chuang, �??Experimental observations of a left-handed material that obeys Snell�??s law�?? Phys. Rev. Lett. 90, 137401 (2003).
    [CrossRef] [PubMed]
  14. J.B. Pendry �??Negative refraction makes a perfect lens�?? Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  15. N. Garcia and M. Nieto-Vesperinas, �??Left-handed materials do not make a perfect lens,�?? Phys. Rev. Lett. 88, 207403 (2002).
    [CrossRef] [PubMed]
  16. J.B. Pendry �??Comment on �??Left-handed materials do not make a perfect lens,�?? Phys. Rev. Lett. 91, 099701 (2003).
    [CrossRef] [PubMed]
  17. M. Nieto-Vesperinas and N. Garcia �??Nieto-Vesperinas and Garcia reply,�?? Phys. Rev. Lett. 91, 099702 (2003).
    [CrossRef]
  18. A.L. Pokrovsky and A.L. Efros �??Electrodynamics of metallic photonics crystals and the problem of left-handed materials,�?? Phys. Rev. Lett. 89, 093901 (2002).
    [CrossRef] [PubMed]
  19. P. Kolinko and D.R. Smith, �??Numerical study of electromagnetic waves interacting with negative index materials,�?? Opt. Express 11, 640-648 (2003), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-640">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-640</a.
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  20. D.R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S.A. Ramakrishna, and J.B. Pendry, �??Limitations on subdiffraction imaging with a negative refractive index slab,�?? Appl. Phys. Lett. 82, 1506-1508 (2003).
    [CrossRef]
  21. 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), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-696">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-696</a>
    [CrossRef] [PubMed]
  22. A.A. Houck, Harvard University, Department of Physics, Cambridge, Massachusetts 02138 (personal communication, 2003).
  23. C. Caloz, C.-C. Chang and T. Itoh, �??Full-wave verification of the fundamental properties of left-handed materials in waveguide configurations,�?? J. Appl. Phys. 90, 5483-5486 (2001).
    [CrossRef]
  24. M. Born and E. Wolf, Principles of Optics, Chap. 13, pp.613 and 615-617 (Pergamon Press, Oxford,1993).

Appl. Phys. Lett. (3)

R.B. Greegor, C.G. Parazzoli, K. Li, and M.H. Tanielian, �??Origin of dissipative losses in negative index of refraction materials,�?? Appl. Phys. Lett. 82, 2356-2358 (2003).
[CrossRef]

R.A. Shelby, D.R. Smith, S.C. Nemat-Nasser and S. Schultz, �??Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,�?? Appl. Phys. Lett. 78, 489-91 (2001).
[CrossRef]

D.R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S.A. Ramakrishna, and J.B. Pendry, �??Limitations on subdiffraction imaging with a negative refractive index slab,�?? Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

European Journal of Physics (1)

M.W. McCall, A. Lakhtakia and W.S. Weiglhofer, �??The negative index of refraction demystified,�?? European Journal of Physics, 23, 353 - 359 (2002).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

J.B. Pendry, A.J. Holden, D.J. Robbins and W.J. Stewart �??Magnetism from conductors and enhanced nonlinear phenomena,�?? IEEE Trans. Microwave Theory Tech. 47, 2075-2074 (1999).
[CrossRef]

J. Appl. Phys. (1)

C. Caloz, C.-C. Chang and T. Itoh, �??Full-wave verification of the fundamental properties of left-handed materials in waveguide configurations,�?? J. Appl. Phys. 90, 5483-5486 (2001).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. B (1)

P. Markos and C.M. Soukoulis, �??Transmission studies of left-handed materials,�?? Phys. Rev. B 65, 033401 (2001).
[CrossRef]

Phys. Rev. E (2)

R.W. Ziolkowski and E. Heyman, �??Wave propagation in media having negative permittivity and permeability,�?? Phys. Rev. E 64, 056625 (2001).
[CrossRef]

P. Markos and C.M. Soukoulis, �??Numerical studies of left-handed materials and arrays of split ring resonators,�?? Phys. Rev. E 65, 036622 (2002).
[CrossRef]

Phys. Rev. Lett. (8)

P.M. Valanju, R.M.Walser and A.P. Valanju, �??Wave refraction in negative-index media: always positive and very inhomogenous,�?? Phys. Rev. Lett. 91, 187401 (2002).
[CrossRef]

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

A.A. Houck, J.B. Brock and I.L. Chuang, �??Experimental observations of a left-handed material that obeys Snell�??s law�?? Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

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

N. Garcia and M. Nieto-Vesperinas, �??Left-handed materials do not make a perfect lens,�?? Phys. Rev. Lett. 88, 207403 (2002).
[CrossRef] [PubMed]

J.B. Pendry �??Comment on �??Left-handed materials do not make a perfect lens,�?? Phys. Rev. Lett. 91, 099701 (2003).
[CrossRef] [PubMed]

M. Nieto-Vesperinas and N. Garcia �??Nieto-Vesperinas and Garcia reply,�?? Phys. Rev. Lett. 91, 099702 (2003).
[CrossRef]

A.L. Pokrovsky and A.L. Efros �??Electrodynamics of metallic photonics crystals and the problem of left-handed materials,�?? Phys. Rev. Lett. 89, 093901 (2002).
[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 simultanenous negative values of ε and μ,�?? Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Other (2)

M. Born and E. Wolf, Principles of Optics, Chap. 13, pp.613 and 615-617 (Pergamon Press, Oxford,1993).

A.A. Houck, Harvard University, Department of Physics, Cambridge, Massachusetts 02138 (personal communication, 2003).

Supplementary Material (6)

» Media 1: MPG (834 KB)     
» Media 2: MPG (693 KB)     
» Media 3: MPG (940 KB)     
» Media 4: MPG (566 KB)     
» Media 5: MPG (376 KB)     
» Media 6: MPG (351 KB)     

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Figures (19)

Fig. 1.
Fig. 1.

Maps of the modulus of the total electric field for (a) a lossless prism (n 2 = -0.35) in free vacuum environment and (b) the same prism in a waveguide of absorbing walls, illuminated by a plane wave from below of λ = 3cm; (c) and (d) the same as (a),(b), respectively, but with an absorbing prism (n 2 κ 2 = 0.25). The dimensions of the prism are seen in the horizontal and vertical axes.

Fig. 2.
Fig. 2.

(Movie 834 KB, 693 KB, 939 KB, 565 KB) Same as Fig.1 but for the electric field of the time-harmonic wave E = ∣E∣cos(ϕ+ωt).

Fig. 3.
Fig. 3.

Refraction angle θ t versus n 2 κ 2 for prisms with φ = 18 and 26 degrees (n 2 = -0.35).

Fig. 4.
Fig. 4.

Plots of (a) lines of constant amplitude and (b) contour lines of electric field, for a LHM prism with 2 = -0.35 + 0.10i and ε = μ̂.

Fig. 5.
Fig. 5.

Geometry of the numerical simulation for the LHM slab.

Fig. 6.
Fig. 6.

Refraction angle θ t versus n 2 κ 2 for a θi = 15, n 2 = -0.35 and ε̂ = μ̂.

Fig. 7.
Fig. 7.

Maps of the modulus of the total electric field after passing through a LHM slab of n 2 = -0.35 and (a)n 2 κ 2 = 0.05 (b)n 2 κ 2 = 0.25 [compare with Fig. 3(b) of Ref.[13]], ε̂ = μ̂ in both cases (c) and(d) ray theory for the cases of (a) and (b), respectively.

Fig. 8.
Fig. 8.

Variation of the focus position with the LHM slab absorption n 2 κ 2 (n 2 = -0.35 and ε = μ̂ in all cases).

Fig. 9.
Fig. 9.

Peak intensity of the focus normalized to the source intensity versus n 2 κ 2 for n 2 = -0.35 and ε̂ = μ̂ in all cases.

Fig. 10.
Fig. 10.

Distance of focus to slab for different values of μ as n 2 κ 2 varies (n 2 = -0.35 in all cases).

Fig. 11.
Fig. 11.

Geometry of the numerical simulation with two slits.

Fig. 12.
Fig. 12.

Modulus of electric field after passing through the LHM slab according to the geometry of Fig. 11 for (a)n 2 κ 2 = 0 and (b)n 2 κ 2 = 0.25 (n 2 = -0.35 and ε̂ = μ̂). the distance between the centers of the slits is 10cm. (d) Object cross section along the slit plane and (c) cross section of (a) along the image plane situated at 17cm from the slab and (e) cross section of (b) along the image plane situated to 5cm from slab.

Fig. 13.
Fig. 13.

Same as Fig.12 but for two slits of different intensity.

Fig. 14.
Fig. 14.

Same as Fig.12 but for slits separated 5cm.

Fig. 15.
Fig. 15.

Same as Fig.13 but for slits separated 5cm.

Fig. 16.
Fig. 16.

(a) Map of the electric field modulus for two slits separated 10cm in front of a LHM slab (2 = -1.0454 + 0.0041i). (b) Distribution of electric field modulus along the slits (top) and along the image plane after the slab at 1.4cm from the last interface (bottom).

Fig. 17.
Fig. 17.

Same as Fig. 16 but for different intensity in the slits.

Fig. 18.
Fig. 18.

Same as Fig.16 but for slits separated 3cm. [A movie (376 KB) of (a) for the evolution with time of the time-harmonic electric field E = ∣E∣cos(ϕ+ωt) is available].

Fig. 19.
Fig. 19.

Same as Fig.17 but for slits separated 3cm. (A movie (350 KB) of (a) for the evolution with time of the time-harmonic electric field E = ∣E∣cos(ϕ+ωt) is available). In (b) black solid curve is for a slab 40cm width, blue dashed-point curve correspond to a slab 25cm width and red dashed curve is for a slab 15cm width.

Equations (15)

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s x t = sin θ t = n 2 ( 1 + 2 ) sin θ i ,
s z t = cos θ t = ( 1 sin 2 θ t ) 1 2 = ( 1 n 2 2 ( 1 κ 2 2 ) sin 2 θ i i 2 n 2 2 κ 2 sin 2 θ i ) 1 2 .
cos θ t = q exp ( ) .
q 2 cos 2 γ = 1 n 2 2 ( 1 κ 2 2 ) sin 2 θ i ,
q 2 sin 2 γ = 2 n 2 2 κ 2 sin 2 θ i .
k ( r · s t ) = ω c ( x s x t + z s z t ) = ω c [ x n 2 sin θ i + zq cos γ + i ( x n 2 κ 2 sin θ i + zq sin γ ) ] .
x n 2 sin θ i + zq cos γ = c 1 ,
x n 2 κ 2 sin θ i + zq sin γ = c 2 .
sin θ t = n 2 sin θ i n 2 2 sin 2 θ i + q 2 cos 2 γ .
E i = E 0 e i k i · r = E 0 e i ω c ( x n 1 sin θ i + z n 1 cos θ i ) ,
E i = E 0 t e i ( k t + i a t ) · r ,
k i x = k t x sin θ t = k i k t sin θ i ,
a t x = 0 a t z = a t ,
k t = { 1 2 [ ( n 2 2 n 2 2 κ 2 2 sin 2 θ i ) 2 + ( 2 n 2 2 κ 2 2 ) 2 + n 2 2 n 2 2 κ 2 2 + sin 2 θ i ] } 1 2 .
| t T E | = | 2 cos θ t cos θ t = z ^ 2 cos θ i | = | 2 1 + ( μ cos θ i ) / ( n ^ 2 cos θ t ) | ,

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