Abstract

In this work we study the propagation characteristics of surface plasmon polaritons (surface eigenmodes) of Kretschmann attenuated total reflection structures with metamaterials. Contrary to the conventional case, in which surface polaritons with positive phase velocity appear at the boundary of a metallic guide, we consider a case where surface polaritons propagate along the boundary of a transparent metamaterial guide with negative refractive index. Depending on the choice of the metamaterial constitutive parameters, these polaritons can have either positive (forward) or negative (backward) phase velocity. For both situations we show numerical examples that illustrate the variation of the real and imaginary parts of the propagation constant with the guide width and the spatial distributions of energy.

© 2011 Optical Society of America

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  1. R. Ruppin, “Surface polaritons of a left-handed medium,” Phys. Lett. A 277, 61–64 (2000).
    [CrossRef]
  2. S. A. Darmanyan, M. Nevière, and A. A. Zakhidov, “Surface modes at the interface of conventional and left-handed media,” Opt. Commun. 225, 233–240 (2003).
    [CrossRef]
  3. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer Verlag, 2007).
  4. H. A. Atwater, “The promise of plasmonics,” Sci. Am. 296, 56–63(2007).
    [CrossRef] [PubMed]
  5. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).
  6. L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford Univ. Press, 2009).
  7. A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. 216, 398–410 (1968).
    [CrossRef]
  8. E. Kretschmann, “Die Bestimmung optischer Konstanten von Metallen durch Anregung von Oberflachenplasmaschwingugnen,” Z. Phys. 241, 313–324 (1971).
    [CrossRef]
  9. T. Tamir, “Beam and waveguide couplers,” in Integrated Optics, T.Tamir, ed. (Springer-Verlag, 1979), pp. 84–134.
  10. K. Park, B. J. Lee, C. J. Fu, and Z. M. Zhang, “Study of the surface and bulk polaritons with a negative index metamaterial,” J. Opt. Soc. Am. B 22, 1016–1023 (2005).
    [CrossRef]
  11. A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, “Generalized surface plasmon resonance sensors using metamaterials and negative index materials,” Prog. Electromagn. Res. 51, 139–152(2005).
    [CrossRef]
  12. F. Tao, H. F. Zhang, X. H. Yang, and D. Cao, “Surface plasmon polaritons of the metamaterial four-layered structures,” J. Opt. Soc. Am. B 26, 50–59 (2009).
    [CrossRef]
  13. H. F. Zhang, D. Cao, F. Tao, X. H. Yang, Y. Wang, X. N. Yan, and L. H. Bai, “Surface plasmon polaritons of symmetric and asymmetric metamaterial slabs,” Chin. Phys. B 19, 027301(2010).
    [CrossRef]
  14. M. Nevière, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R.Petit, ed. (Springer-Verlag, 1980), pp. 123–157.
    [CrossRef]
  15. D. Maystre, “General study of grating anomalies from electromagnetic surface modes,” in Electromagnetic Surface Modes, A.D.Boardman, ed. (Wiley, 1982), pp. 661–724.
  16. M. Cuevas and R. A. Depine, “The homogeneous problem for a corrugated metamaterial of arbitrary permittivity and permeability: choosing the proper Riemann surface,” Optik 122, 198–206 (2011).
    [CrossRef]
  17. L. C. Botten, M. S. Craig, and R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259(1983).
    [CrossRef]
  18. L. M. Delves and J. M. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comput. 21, 543–560(1967).
    [CrossRef]

2011 (1)

M. Cuevas and R. A. Depine, “The homogeneous problem for a corrugated metamaterial of arbitrary permittivity and permeability: choosing the proper Riemann surface,” Optik 122, 198–206 (2011).
[CrossRef]

2010 (1)

H. F. Zhang, D. Cao, F. Tao, X. H. Yang, Y. Wang, X. N. Yan, and L. H. Bai, “Surface plasmon polaritons of symmetric and asymmetric metamaterial slabs,” Chin. Phys. B 19, 027301(2010).
[CrossRef]

2009 (1)

2007 (1)

H. A. Atwater, “The promise of plasmonics,” Sci. Am. 296, 56–63(2007).
[CrossRef] [PubMed]

2005 (2)

A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, “Generalized surface plasmon resonance sensors using metamaterials and negative index materials,” Prog. Electromagn. Res. 51, 139–152(2005).
[CrossRef]

K. Park, B. J. Lee, C. J. Fu, and Z. M. Zhang, “Study of the surface and bulk polaritons with a negative index metamaterial,” J. Opt. Soc. Am. B 22, 1016–1023 (2005).
[CrossRef]

2003 (1)

S. A. Darmanyan, M. Nevière, and A. A. Zakhidov, “Surface modes at the interface of conventional and left-handed media,” Opt. Commun. 225, 233–240 (2003).
[CrossRef]

2000 (1)

R. Ruppin, “Surface polaritons of a left-handed medium,” Phys. Lett. A 277, 61–64 (2000).
[CrossRef]

1983 (1)

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259(1983).
[CrossRef]

1971 (1)

E. Kretschmann, “Die Bestimmung optischer Konstanten von Metallen durch Anregung von Oberflachenplasmaschwingugnen,” Z. Phys. 241, 313–324 (1971).
[CrossRef]

1968 (1)

A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. 216, 398–410 (1968).
[CrossRef]

1967 (1)

L. M. Delves and J. M. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comput. 21, 543–560(1967).
[CrossRef]

Atwater, H. A.

H. A. Atwater, “The promise of plasmonics,” Sci. Am. 296, 56–63(2007).
[CrossRef] [PubMed]

Bai, L. H.

H. F. Zhang, D. Cao, F. Tao, X. H. Yang, Y. Wang, X. N. Yan, and L. H. Bai, “Surface plasmon polaritons of symmetric and asymmetric metamaterial slabs,” Chin. Phys. B 19, 027301(2010).
[CrossRef]

Botten, L. C.

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259(1983).
[CrossRef]

Cao, D.

H. F. Zhang, D. Cao, F. Tao, X. H. Yang, Y. Wang, X. N. Yan, and L. H. Bai, “Surface plasmon polaritons of symmetric and asymmetric metamaterial slabs,” Chin. Phys. B 19, 027301(2010).
[CrossRef]

F. Tao, H. F. Zhang, X. H. Yang, and D. Cao, “Surface plasmon polaritons of the metamaterial four-layered structures,” J. Opt. Soc. Am. B 26, 50–59 (2009).
[CrossRef]

Craig, M. S.

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259(1983).
[CrossRef]

Cuevas, M.

M. Cuevas and R. A. Depine, “The homogeneous problem for a corrugated metamaterial of arbitrary permittivity and permeability: choosing the proper Riemann surface,” Optik 122, 198–206 (2011).
[CrossRef]

Darmanyan, S. A.

S. A. Darmanyan, M. Nevière, and A. A. Zakhidov, “Surface modes at the interface of conventional and left-handed media,” Opt. Commun. 225, 233–240 (2003).
[CrossRef]

Delves, L. M.

L. M. Delves and J. M. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comput. 21, 543–560(1967).
[CrossRef]

Depine, R. A.

M. Cuevas and R. A. Depine, “The homogeneous problem for a corrugated metamaterial of arbitrary permittivity and permeability: choosing the proper Riemann surface,” Optik 122, 198–206 (2011).
[CrossRef]

Fu, C. J.

Ishimaru, A.

A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, “Generalized surface plasmon resonance sensors using metamaterials and negative index materials,” Prog. Electromagn. Res. 51, 139–152(2005).
[CrossRef]

Jaruwatanadilok, S.

A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, “Generalized surface plasmon resonance sensors using metamaterials and negative index materials,” Prog. Electromagn. Res. 51, 139–152(2005).
[CrossRef]

Kretschmann, E.

E. Kretschmann, “Die Bestimmung optischer Konstanten von Metallen durch Anregung von Oberflachenplasmaschwingugnen,” Z. Phys. 241, 313–324 (1971).
[CrossRef]

Kuga, Y.

A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, “Generalized surface plasmon resonance sensors using metamaterials and negative index materials,” Prog. Electromagn. Res. 51, 139–152(2005).
[CrossRef]

Lee, B. J.

Lyness, J. M.

L. M. Delves and J. M. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comput. 21, 543–560(1967).
[CrossRef]

Maier, S. A.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer Verlag, 2007).

Maystre, D.

D. Maystre, “General study of grating anomalies from electromagnetic surface modes,” in Electromagnetic Surface Modes, A.D.Boardman, ed. (Wiley, 1982), pp. 661–724.

McPhedran, R. C.

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259(1983).
[CrossRef]

Nevière, M.

S. A. Darmanyan, M. Nevière, and A. A. Zakhidov, “Surface modes at the interface of conventional and left-handed media,” Opt. Commun. 225, 233–240 (2003).
[CrossRef]

M. Nevière, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R.Petit, ed. (Springer-Verlag, 1980), pp. 123–157.
[CrossRef]

Otto, A.

A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. 216, 398–410 (1968).
[CrossRef]

Park, K.

Raether, H.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

Ruppin, R.

R. Ruppin, “Surface polaritons of a left-handed medium,” Phys. Lett. A 277, 61–64 (2000).
[CrossRef]

Shamonina, E.

L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford Univ. Press, 2009).

Solymar, L.

L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford Univ. Press, 2009).

Tamir, T.

T. Tamir, “Beam and waveguide couplers,” in Integrated Optics, T.Tamir, ed. (Springer-Verlag, 1979), pp. 84–134.

Tao, F.

H. F. Zhang, D. Cao, F. Tao, X. H. Yang, Y. Wang, X. N. Yan, and L. H. Bai, “Surface plasmon polaritons of symmetric and asymmetric metamaterial slabs,” Chin. Phys. B 19, 027301(2010).
[CrossRef]

F. Tao, H. F. Zhang, X. H. Yang, and D. Cao, “Surface plasmon polaritons of the metamaterial four-layered structures,” J. Opt. Soc. Am. B 26, 50–59 (2009).
[CrossRef]

Wang, Y.

H. F. Zhang, D. Cao, F. Tao, X. H. Yang, Y. Wang, X. N. Yan, and L. H. Bai, “Surface plasmon polaritons of symmetric and asymmetric metamaterial slabs,” Chin. Phys. B 19, 027301(2010).
[CrossRef]

Yan, X. N.

H. F. Zhang, D. Cao, F. Tao, X. H. Yang, Y. Wang, X. N. Yan, and L. H. Bai, “Surface plasmon polaritons of symmetric and asymmetric metamaterial slabs,” Chin. Phys. B 19, 027301(2010).
[CrossRef]

Yang, X. H.

H. F. Zhang, D. Cao, F. Tao, X. H. Yang, Y. Wang, X. N. Yan, and L. H. Bai, “Surface plasmon polaritons of symmetric and asymmetric metamaterial slabs,” Chin. Phys. B 19, 027301(2010).
[CrossRef]

F. Tao, H. F. Zhang, X. H. Yang, and D. Cao, “Surface plasmon polaritons of the metamaterial four-layered structures,” J. Opt. Soc. Am. B 26, 50–59 (2009).
[CrossRef]

Zakhidov, A. A.

S. A. Darmanyan, M. Nevière, and A. A. Zakhidov, “Surface modes at the interface of conventional and left-handed media,” Opt. Commun. 225, 233–240 (2003).
[CrossRef]

Zhang, H. F.

H. F. Zhang, D. Cao, F. Tao, X. H. Yang, Y. Wang, X. N. Yan, and L. H. Bai, “Surface plasmon polaritons of symmetric and asymmetric metamaterial slabs,” Chin. Phys. B 19, 027301(2010).
[CrossRef]

F. Tao, H. F. Zhang, X. H. Yang, and D. Cao, “Surface plasmon polaritons of the metamaterial four-layered structures,” J. Opt. Soc. Am. B 26, 50–59 (2009).
[CrossRef]

Zhang, Z. M.

Chin. Phys. B (1)

H. F. Zhang, D. Cao, F. Tao, X. H. Yang, Y. Wang, X. N. Yan, and L. H. Bai, “Surface plasmon polaritons of symmetric and asymmetric metamaterial slabs,” Chin. Phys. B 19, 027301(2010).
[CrossRef]

Comput. Phys. Commun. (1)

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259(1983).
[CrossRef]

J. Opt. Soc. Am. B (2)

Math. Comput. (1)

L. M. Delves and J. M. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comput. 21, 543–560(1967).
[CrossRef]

Opt. Commun. (1)

S. A. Darmanyan, M. Nevière, and A. A. Zakhidov, “Surface modes at the interface of conventional and left-handed media,” Opt. Commun. 225, 233–240 (2003).
[CrossRef]

Optik (1)

M. Cuevas and R. A. Depine, “The homogeneous problem for a corrugated metamaterial of arbitrary permittivity and permeability: choosing the proper Riemann surface,” Optik 122, 198–206 (2011).
[CrossRef]

Phys. Lett. A (1)

R. Ruppin, “Surface polaritons of a left-handed medium,” Phys. Lett. A 277, 61–64 (2000).
[CrossRef]

Prog. Electromagn. Res. (1)

A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, “Generalized surface plasmon resonance sensors using metamaterials and negative index materials,” Prog. Electromagn. Res. 51, 139–152(2005).
[CrossRef]

Sci. Am. (1)

H. A. Atwater, “The promise of plasmonics,” Sci. Am. 296, 56–63(2007).
[CrossRef] [PubMed]

Z. Phys. (2)

A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. 216, 398–410 (1968).
[CrossRef]

E. Kretschmann, “Die Bestimmung optischer Konstanten von Metallen durch Anregung von Oberflachenplasmaschwingugnen,” Z. Phys. 241, 313–324 (1971).
[CrossRef]

Other (6)

T. Tamir, “Beam and waveguide couplers,” in Integrated Optics, T.Tamir, ed. (Springer-Verlag, 1979), pp. 84–134.

M. Nevière, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R.Petit, ed. (Springer-Verlag, 1980), pp. 123–157.
[CrossRef]

D. Maystre, “General study of grating anomalies from electromagnetic surface modes,” in Electromagnetic Surface Modes, A.D.Boardman, ed. (Wiley, 1982), pp. 661–724.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford Univ. Press, 2009).

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer Verlag, 2007).

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Figures (8)

Fig. 1
Fig. 1

Geometry of the ATR configuration.

Fig. 2
Fig. 2

Existence regions of SPPs on the relative constitutive parameter space ε μ . The values used in the examples are indicated by stars.

Fig. 3
Fig. 3

(a) Real and (b) imaginary parts of the nondimensional SPP propagation constant κ for different widths of the metamaterial slab d and for material parameters in region A corresponding to p-polarized, forward SPPs.

Fig. 4
Fig. 4

Lines of current and absolute value of the normalized Poynting vector for a p-polarized, forward SPP. The values of the material parameters are the same as in Fig. 3 and d / λ = 0.56 . The dotted lines indicate the separation surfaces.

Fig. 5
Fig. 5

Absolute value of the normalized Poynting vector for a p-polarized, forward SPP near the separation surface 2–3. All the parameters correspond to those in Fig. 4.

Fig. 6
Fig. 6

(a) Real and (b) imaginary parts of the nondimensional SPP propagation constant κ for different widths of the metamaterial slab d and for material parameters in region B corresponding to s-polarized, backward SPPs.

Fig. 7
Fig. 7

Lines of current and absolute value of the normalized Poynting vector for a p-polarized, forward SPP. The values of the material parameters are the same as in Fig. 6 and d / λ = 0.81 . The dotted lines indicate the separation surfaces.

Fig. 8
Fig. 8

Absolute value of the normalized Poynting vector for an s-polarized, backward SPP near the separation surface 2–3. All the parameters correspond to those in Fig. 7.

Equations (10)

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φ = { r 1 e i ( α x + β 1 y ) y > 0 t 1 e i ( α x β 2 y ) + r 2 e i ( α x + β 2 y ) d < y < 0 t 2 e i ( α x β 3 y ) y < d ,
r 1 = t 1 + r 2 , t 1 e i β 2 d + r 2 e i β 2 d = t 2 e i β 3 d , Z 1 r 1 = Z 2 [ t 1 + r 2 ] , Z 2 [ t 1 e i β 2 d + r 2 e i β 2 d ] = Z 3 t 2 e i β 3 d ,
( 1 1 1 0 0 e i β 2 d e i β 2 d e i β 3 d Z 1 Z 2 Z 2 0 0 Z 2 e i β 2 d Z 2 e i β 2 d Z 3 e i β 3 d ) ( r 1 t 1 r 2 t 2 ) = ( 0 0 0 0 ) .
( Z 1 + Z 2 ) ( Z 2 + Z 3 ) + ( Z 1 Z 2 ) ( Z 2 Z 3 ) e 2 i β 2 d = 0 .
Re β j + Im β j 0 ,
Re β j + Im β j 0 .
( Z 1 + Z 2 ) ( Z 2 + Z 3 ) = 0 .
Z 1 + Z 2 = 0 ,
Z 2 + Z 3 = 0 .
S = 1 2 Re ( E × H * ) ,

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