Abstract

We have investigated the transmission and reflection properties of structures incorporating left-handed materials with graded index of refraction. We present an exact analytical solution to Helmholtz’ equation for a graded index profile changing according to a hyperbolic tangent function along the propagation direction. We derive expressions for the field intensity along the graded index structure, and we show excellent agreement between the analytical solution and the corresponding results obtained by accurate numerical simulations. Our model straightforwardly allows for arbitrary spectral dispersion.

© 2009 Optical Society of America

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References

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  1. C. M. Soukoulis, M. Kafesaki, and E. N. Economou, "Negative-Index Materials: New Frontiers in Optics," Adv. Mater. 18, 1941-1952 (2006).
    [CrossRef]
  2. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ? and," Sov. Phys. Usp 10, 509-514 (1968).
    [CrossRef]
  3. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low Frequency Plasmons in ThinWire Structures," J. Phys.: Cond. Matter 10, 4785-4788 (1998).
    [CrossRef]
  4. 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- 2084 (1999).
    [CrossRef]
  5. R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292, 77-79 (2001).
    [CrossRef] [PubMed]
  6. P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, "Low Loss Metamaterials Based on Classical Electromagnetically Induced Transparency," Phys. Rev. Lett. 102, 053901 (2009).
    [CrossRef] [PubMed]
  7. J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  8. N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
    [CrossRef] [PubMed]
  9. Z. Jacob, L. V. Alekseyev, and E. Narimanov, "Optical Hyperlens: Far-field imaging beyond the diffraction limit," Opt. Express 14, 8247-8256 (2006).
    [CrossRef] [PubMed]
  10. K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "‘Trapped rainbow’ storage of light in metamaterials," Nature 450, 397-401 (2007).
    [CrossRef] [PubMed]
  11. N. Engheta, "Circuits with Light at Nanoscales: Optical Nanocircuits Inspired by Metamaterials," Science 317, 1698-1702 (2007).
    [CrossRef] [PubMed]
  12. P. Tassin, X. Sahyoun, and I. Veretennicoff, "Miniaturization of photonic waveguides by the use of left-handed materials," Appl. Phys. Lett. 92, 203111 (2008).
    [CrossRef]
  13. U. Leonhardt, "Optical Conformal Mapping," Science 312, 1777-1780 (2006).
    [CrossRef] [PubMed]
  14. J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 1780-1782 (2006).
    [CrossRef] [PubMed]
  15. S. A. Ramakrishna and J. B. Pendry, "Spherical perfect lens: Solutions of Maxwell’s equations for spherical geometry," Phys. Rev. B 69, 115115 (2004).
    [CrossRef]
  16. A. O. Pinchuk and G. C. Schatz, "Metamaterials with gradient negative index of refraction," J. Opt. Soc. Am. A 24, A39-A44 (2007).
    [CrossRef]
  17. C. G. Parazzoli, B. E. C. Koltenbah, R. B. Greegor, T. A. Lam, and M. H. Tanielian, "Eikonal equation for a general anisotropic or chiral medium: application to a negative-graded index-of-refraction lens with an anisotropic material," J. Opt. Soc. Am. B 23, 439-450 (2006).
    [CrossRef]
  18. N. Dalarsson, M. Maksimovic, and Z. Jaksic, "A Simplified Analytical Approach to Calculation of the Electromagnetic Behavior of Left-Handed Metamaterials with a Graded Refractive Index Profile," Science of Sintering 39, 185-191 (2007).
    [CrossRef]
  19. D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, "A gradient index metamaterial," Phys. Rev. E 71, 036609 (2005).
    [CrossRef]
  20. N. M. Litchinitser, A. I. Maimistov, I. R. Gabitov, R. Z. Sagdeev, and V. M. Shalaev, "Metamaterials: electromagnetic enhancement at zero-index transition," Opt. Lett. 33, 2350-2352 (2008).
    [CrossRef] [PubMed]
  21. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

2009 (1)

P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, "Low Loss Metamaterials Based on Classical Electromagnetically Induced Transparency," Phys. Rev. Lett. 102, 053901 (2009).
[CrossRef] [PubMed]

2008 (2)

P. Tassin, X. Sahyoun, and I. Veretennicoff, "Miniaturization of photonic waveguides by the use of left-handed materials," Appl. Phys. Lett. 92, 203111 (2008).
[CrossRef]

N. M. Litchinitser, A. I. Maimistov, I. R. Gabitov, R. Z. Sagdeev, and V. M. Shalaev, "Metamaterials: electromagnetic enhancement at zero-index transition," Opt. Lett. 33, 2350-2352 (2008).
[CrossRef] [PubMed]

2007 (4)

N. Dalarsson, M. Maksimovic, and Z. Jaksic, "A Simplified Analytical Approach to Calculation of the Electromagnetic Behavior of Left-Handed Metamaterials with a Graded Refractive Index Profile," Science of Sintering 39, 185-191 (2007).
[CrossRef]

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "‘Trapped rainbow’ storage of light in metamaterials," Nature 450, 397-401 (2007).
[CrossRef] [PubMed]

N. Engheta, "Circuits with Light at Nanoscales: Optical Nanocircuits Inspired by Metamaterials," Science 317, 1698-1702 (2007).
[CrossRef] [PubMed]

A. O. Pinchuk and G. C. Schatz, "Metamaterials with gradient negative index of refraction," J. Opt. Soc. Am. A 24, A39-A44 (2007).
[CrossRef]

2006 (5)

2005 (2)

D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, "A gradient index metamaterial," Phys. Rev. E 71, 036609 (2005).
[CrossRef]

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

2004 (1)

S. A. Ramakrishna and J. B. Pendry, "Spherical perfect lens: Solutions of Maxwell’s equations for spherical geometry," Phys. Rev. B 69, 115115 (2004).
[CrossRef]

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).
[CrossRef] [PubMed]

1999 (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- 2084 (1999).
[CrossRef]

1998 (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low Frequency Plasmons in ThinWire Structures," J. Phys.: Cond. Matter 10, 4785-4788 (1998).
[CrossRef]

1968 (1)

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

Alekseyev, L. V.

Boardman, A. D.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "‘Trapped rainbow’ storage of light in metamaterials," Nature 450, 397-401 (2007).
[CrossRef] [PubMed]

Dalarsson, N.

N. Dalarsson, M. Maksimovic, and Z. Jaksic, "A Simplified Analytical Approach to Calculation of the Electromagnetic Behavior of Left-Handed Metamaterials with a Graded Refractive Index Profile," Science of Sintering 39, 185-191 (2007).
[CrossRef]

Economou, E. N.

P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, "Low Loss Metamaterials Based on Classical Electromagnetically Induced Transparency," Phys. Rev. Lett. 102, 053901 (2009).
[CrossRef] [PubMed]

C. M. Soukoulis, M. Kafesaki, and E. N. Economou, "Negative-Index Materials: New Frontiers in Optics," Adv. Mater. 18, 1941-1952 (2006).
[CrossRef]

Engheta, N.

N. Engheta, "Circuits with Light at Nanoscales: Optical Nanocircuits Inspired by Metamaterials," Science 317, 1698-1702 (2007).
[CrossRef] [PubMed]

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Gabitov, I. R.

Greegor, R. B.

Hess, O.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "‘Trapped rainbow’ storage of light in metamaterials," Nature 450, 397-401 (2007).
[CrossRef] [PubMed]

Holden, A. J.

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- 2084 (1999).
[CrossRef]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low Frequency Plasmons in ThinWire Structures," J. Phys.: Cond. Matter 10, 4785-4788 (1998).
[CrossRef]

Jacob, Z.

Jaksic, Z.

N. Dalarsson, M. Maksimovic, and Z. Jaksic, "A Simplified Analytical Approach to Calculation of the Electromagnetic Behavior of Left-Handed Metamaterials with a Graded Refractive Index Profile," Science of Sintering 39, 185-191 (2007).
[CrossRef]

Kafesaki, M.

C. M. Soukoulis, M. Kafesaki, and E. N. Economou, "Negative-Index Materials: New Frontiers in Optics," Adv. Mater. 18, 1941-1952 (2006).
[CrossRef]

Koltenbah, B. E. C.

Koschny, T.

P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, "Low Loss Metamaterials Based on Classical Electromagnetically Induced Transparency," Phys. Rev. Lett. 102, 053901 (2009).
[CrossRef] [PubMed]

Lam, T. A.

Lee, H.

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Leonhardt, U.

U. Leonhardt, "Optical Conformal Mapping," Science 312, 1777-1780 (2006).
[CrossRef] [PubMed]

Litchinitser, N. M.

Maimistov, A. I.

Maksimovic, M.

N. Dalarsson, M. Maksimovic, and Z. Jaksic, "A Simplified Analytical Approach to Calculation of the Electromagnetic Behavior of Left-Handed Metamaterials with a Graded Refractive Index Profile," Science of Sintering 39, 185-191 (2007).
[CrossRef]

Mock, J. J.

D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, "A gradient index metamaterial," Phys. Rev. E 71, 036609 (2005).
[CrossRef]

Narimanov, E.

Parazzoli, C. G.

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

S. A. Ramakrishna and J. B. Pendry, "Spherical perfect lens: Solutions of Maxwell’s equations for spherical geometry," Phys. Rev. B 69, 115115 (2004).
[CrossRef]

J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

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- 2084 (1999).
[CrossRef]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low Frequency Plasmons in ThinWire Structures," J. Phys.: Cond. Matter 10, 4785-4788 (1998).
[CrossRef]

Pinchuk, A. O.

Ramakrishna, S. A.

S. A. Ramakrishna and J. B. Pendry, "Spherical perfect lens: Solutions of Maxwell’s equations for spherical geometry," Phys. Rev. B 69, 115115 (2004).
[CrossRef]

Robbins, D. J.

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- 2084 (1999).
[CrossRef]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low Frequency Plasmons in ThinWire Structures," J. Phys.: Cond. Matter 10, 4785-4788 (1998).
[CrossRef]

Sagdeev, R. Z.

Sahyoun, X.

P. Tassin, X. Sahyoun, and I. Veretennicoff, "Miniaturization of photonic waveguides by the use of left-handed materials," Appl. Phys. Lett. 92, 203111 (2008).
[CrossRef]

Schatz, G. C.

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]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, "A gradient index metamaterial," Phys. Rev. E 71, 036609 (2005).
[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.

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, "A gradient index metamaterial," Phys. Rev. E 71, 036609 (2005).
[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]

Soukoulis, C. M.

P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, "Low Loss Metamaterials Based on Classical Electromagnetically Induced Transparency," Phys. Rev. Lett. 102, 053901 (2009).
[CrossRef] [PubMed]

C. M. Soukoulis, M. Kafesaki, and E. N. Economou, "Negative-Index Materials: New Frontiers in Optics," Adv. Mater. 18, 1941-1952 (2006).
[CrossRef]

Starr, A. F.

D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, "A gradient index metamaterial," Phys. Rev. E 71, 036609 (2005).
[CrossRef]

Stewart, W. J.

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- 2084 (1999).
[CrossRef]

Sun, C.

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Tanielian, M. H.

Tassin, P.

P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, "Low Loss Metamaterials Based on Classical Electromagnetically Induced Transparency," Phys. Rev. Lett. 102, 053901 (2009).
[CrossRef] [PubMed]

P. Tassin, X. Sahyoun, and I. Veretennicoff, "Miniaturization of photonic waveguides by the use of left-handed materials," Appl. Phys. Lett. 92, 203111 (2008).
[CrossRef]

Tsakmakidis, K. L.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "‘Trapped rainbow’ storage of light in metamaterials," Nature 450, 397-401 (2007).
[CrossRef] [PubMed]

Veretennicoff, I.

P. Tassin, X. Sahyoun, and I. Veretennicoff, "Miniaturization of photonic waveguides by the use of left-handed materials," Appl. Phys. Lett. 92, 203111 (2008).
[CrossRef]

Veselago, V. G.

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

Zhang, L.

P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, "Low Loss Metamaterials Based on Classical Electromagnetically Induced Transparency," Phys. Rev. Lett. 102, 053901 (2009).
[CrossRef] [PubMed]

Zhang, X.

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Adv. Mater. (1)

C. M. Soukoulis, M. Kafesaki, and E. N. Economou, "Negative-Index Materials: New Frontiers in Optics," Adv. Mater. 18, 1941-1952 (2006).
[CrossRef]

Appl. Phys. Lett. (1)

P. Tassin, X. Sahyoun, and I. Veretennicoff, "Miniaturization of photonic waveguides by the use of left-handed materials," Appl. Phys. Lett. 92, 203111 (2008).
[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- 2084 (1999).
[CrossRef]

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

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

J. Phys.: Cond. Matter (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low Frequency Plasmons in ThinWire Structures," J. Phys.: Cond. Matter 10, 4785-4788 (1998).
[CrossRef]

Nature (1)

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "‘Trapped rainbow’ storage of light in metamaterials," Nature 450, 397-401 (2007).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (1)

S. A. Ramakrishna and J. B. Pendry, "Spherical perfect lens: Solutions of Maxwell’s equations for spherical geometry," Phys. Rev. B 69, 115115 (2004).
[CrossRef]

Phys. Rev. E (1)

D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, "A gradient index metamaterial," Phys. Rev. E 71, 036609 (2005).
[CrossRef]

Phys. Rev. Lett. (2)

P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, "Low Loss Metamaterials Based on Classical Electromagnetically Induced Transparency," Phys. Rev. Lett. 102, 053901 (2009).
[CrossRef] [PubMed]

J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Science (5)

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
[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]

N. Engheta, "Circuits with Light at Nanoscales: Optical Nanocircuits Inspired by Metamaterials," Science 317, 1698-1702 (2007).
[CrossRef] [PubMed]

U. Leonhardt, "Optical Conformal Mapping," Science 312, 1777-1780 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

Science of Sintering (1)

N. Dalarsson, M. Maksimovic, and Z. Jaksic, "A Simplified Analytical Approach to Calculation of the Electromagnetic Behavior of Left-Handed Metamaterials with a Graded Refractive Index Profile," Science of Sintering 39, 185-191 (2007).
[CrossRef]

Sov. Phys. Usp (1)

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

Other (1)

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

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Figures (3)

Fig. 1.
Fig. 1.

Propagation of an electromagnetic wave through a graded index structure with a hyperbolic tangent profile. This is the index profile assumed in this paper.

Fig. 2.
Fig. 2.

We assume in this paper that the effective permittivity εeff and permeability μeff vary along the propagation direction according to a hyperbolic tangent function.

Fig. 3.
Fig. 3.

Comparison of the analytical and numerical results for the electric field. We plot E(x)at t = 0. (a) Analytical solution for ρ = 10 μm. (b)Numerical solution for ρ = 10 μm. (c) Analytical solution for ρ = 1 μm. (b) Numerical solution for ρ = 1 μm.

Equations (15)

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

dE dx = iωμH ( x ) ,
dH dx = iωεE ( x ) ,
d 2 E d x 2 1 μ dx dE dx + ω 2 μεE ( x ) = 0 .
d 2 H d x 2 1 μ dx dH dx + ω 2 μεEH ( x ) = 0 ,
E ( x ) = μ ( x ) F ( x ) , H ( x ) = ε ( x ) G ( x ) .
d 2 F d x 2 + [ ω 2 με + 1 2 μ d 2 μ d x 2 3 4 μ 2 ( dx ) 2 ] F ( x ) = 0 ,
d 2 G d x 2 + [ ω 2 με + 1 2 ε d 2 ε d x 2 3 4 ε 2 ( dx ) 2 ] G ( x ) = 0 .
d 2 F d x 2 + k μ 2 ( x ) F ( x ) = 0 , d 2 G d x 2 + k ε 2 ( x ) G ( x ) = 0 ,
k μ 2 ( x ) = ω 2 με + 1 2 μ d 2 μ d x 2 3 4 μ 2 ( dx ) 2
k ε 2 ( x ) = ω 2 με + 1 2 ε d 2 μ d x 2 3 4 ε 2 ( dx ) 2
μ = μ 0 μ eff ( ω ) tanh ( ρx ) , ε = ε 0 ε eff ( ω ) tanh ( ρx ) ,
E ( x ) = E 0 [ cosh ( ρx ) ] ± i κ ρ , H ( x ) = H 0 [ cosh ( ρx ) ] ± i κ ρ ,
κ 2 = ω 2 ε eff ( ε ) μ eff ( ω ) .
E ( x ) = E 0 [ cosh ( ρx ) ] i κ ρ .
E ( ± ) = E 0 e i κ ρ ln 2 e ± κx ,

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