Abstract

The propagation of a surface plasmon polariton on a planar metal surface perturbed by N equally spaced rectangular grooves, each with the same width but with varying depths, is investigated by the finite-difference time-domain method. For a linear dependence of the depth of the nth groove on n, the transmissivity of the surface plasmon polariton and of the power radiated into the vacuum above the surface, as functions of its frequency, consist of N equally spaced dips and peaks, respectively. These are the signatures of the surface plasmon polariton analog of a Wannier–Stark ladder.

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  1. G. H. Wannier, Elements of Solid State Theory (Cambridge University, 1959), p. 190.
  2. F. Bloch, Z. Phys. 22, 555 (1928).
  3. E. E. Méndez, F. Agulló-Rueda, and J. M. Hong, Phys. Rev. Lett. 60, 2426 (1988).
    [CrossRef]
  4. S. Sapienza, P. Constantino, D. Wiersma, M. Chulinyan, C. J. Oton, and L. Pavesi, Phys. Rev. Lett. 91, 263902 (2003).
    [CrossRef]
  5. J. Feldman, K. Leo, J. Shah, D. A. B. Miller, J. E. Cunningham, T. Meier, G. von Plessen, A. Shulze, P. Thomas, and S. Schmitt-Rink, Phys. Rev. B 46, 7252 (1992).
    [CrossRef]
  6. K. Leo, K. Haring Bolivar, F. Brügemann, R. Schwedler, and K. Köhler, Solid State Commun. 84, 943 (1992).
    [CrossRef]
  7. S. R. Wilkinson, C. F. Bharucha, K. W. Madison, Q. Niu, and M. G. Raizen, Phys. Rev. Lett. 76, 4512 (1996).
    [CrossRef]
  8. G. Monsivais, M. del Castillo-Mussot, and F. Claro, Phys. Rev. Lett. 64, 1433 (1990).
    [CrossRef]
  9. C. M. de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, Phys. Rev. E 57, 2365 (1998).
    [CrossRef]
  10. J. L. Mateos and G. Monsivais, Physica A 207, 445 (1994).
    [CrossRef]
  11. L. Gutierrez, A. Daz-de-Anda, J. Flores, R. A. Méndez-Sánchez, G. Monsivais, and A. Morales, Phys. Rev. Lett. 97, 114301 (2006).
    [CrossRef]
  12. P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
    [CrossRef]
  13. OptiFDTD 2012 Optiwave, Version 11.2.
  14. A. Taflove and S. C. Hagness, Computational Electromagnetics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
  15. F. López-Tejeira, F. J. Garca-Vidal, and L. Martín-Moreno, Phys. Rev. B 72, 161405 (2005).
    [CrossRef]
  16. A. A. Maradudin, T. A. Leskova, E. E. García-Guerrero, and E. R. Méndez, Low Temp. Phys. 36, 815 (2010).
    [CrossRef]

2010 (1)

A. A. Maradudin, T. A. Leskova, E. E. García-Guerrero, and E. R. Méndez, Low Temp. Phys. 36, 815 (2010).
[CrossRef]

2006 (1)

L. Gutierrez, A. Daz-de-Anda, J. Flores, R. A. Méndez-Sánchez, G. Monsivais, and A. Morales, Phys. Rev. Lett. 97, 114301 (2006).
[CrossRef]

2005 (1)

F. López-Tejeira, F. J. Garca-Vidal, and L. Martín-Moreno, Phys. Rev. B 72, 161405 (2005).
[CrossRef]

2003 (1)

S. Sapienza, P. Constantino, D. Wiersma, M. Chulinyan, C. J. Oton, and L. Pavesi, Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

1998 (1)

C. M. de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, Phys. Rev. E 57, 2365 (1998).
[CrossRef]

1996 (1)

S. R. Wilkinson, C. F. Bharucha, K. W. Madison, Q. Niu, and M. G. Raizen, Phys. Rev. Lett. 76, 4512 (1996).
[CrossRef]

1994 (1)

J. L. Mateos and G. Monsivais, Physica A 207, 445 (1994).
[CrossRef]

1992 (2)

J. Feldman, K. Leo, J. Shah, D. A. B. Miller, J. E. Cunningham, T. Meier, G. von Plessen, A. Shulze, P. Thomas, and S. Schmitt-Rink, Phys. Rev. B 46, 7252 (1992).
[CrossRef]

K. Leo, K. Haring Bolivar, F. Brügemann, R. Schwedler, and K. Köhler, Solid State Commun. 84, 943 (1992).
[CrossRef]

1990 (1)

G. Monsivais, M. del Castillo-Mussot, and F. Claro, Phys. Rev. Lett. 64, 1433 (1990).
[CrossRef]

1988 (1)

E. E. Méndez, F. Agulló-Rueda, and J. M. Hong, Phys. Rev. Lett. 60, 2426 (1988).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

1928 (1)

F. Bloch, Z. Phys. 22, 555 (1928).

Agulló-Rueda, F.

E. E. Méndez, F. Agulló-Rueda, and J. M. Hong, Phys. Rev. Lett. 60, 2426 (1988).
[CrossRef]

Bharucha, C. F.

S. R. Wilkinson, C. F. Bharucha, K. W. Madison, Q. Niu, and M. G. Raizen, Phys. Rev. Lett. 76, 4512 (1996).
[CrossRef]

Bloch, F.

F. Bloch, Z. Phys. 22, 555 (1928).

Bright, J. N.

C. M. de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, Phys. Rev. E 57, 2365 (1998).
[CrossRef]

Brügemann, F.

K. Leo, K. Haring Bolivar, F. Brügemann, R. Schwedler, and K. Köhler, Solid State Commun. 84, 943 (1992).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Chulinyan, M.

S. Sapienza, P. Constantino, D. Wiersma, M. Chulinyan, C. J. Oton, and L. Pavesi, Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Claro, F.

G. Monsivais, M. del Castillo-Mussot, and F. Claro, Phys. Rev. Lett. 64, 1433 (1990).
[CrossRef]

Constantino, P.

S. Sapienza, P. Constantino, D. Wiersma, M. Chulinyan, C. J. Oton, and L. Pavesi, Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Cunningham, J. E.

J. Feldman, K. Leo, J. Shah, D. A. B. Miller, J. E. Cunningham, T. Meier, G. von Plessen, A. Shulze, P. Thomas, and S. Schmitt-Rink, Phys. Rev. B 46, 7252 (1992).
[CrossRef]

Daz-de-Anda, A.

L. Gutierrez, A. Daz-de-Anda, J. Flores, R. A. Méndez-Sánchez, G. Monsivais, and A. Morales, Phys. Rev. Lett. 97, 114301 (2006).
[CrossRef]

de Sterke, C. M.

C. M. de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, Phys. Rev. E 57, 2365 (1998).
[CrossRef]

del Castillo-Mussot, M.

G. Monsivais, M. del Castillo-Mussot, and F. Claro, Phys. Rev. Lett. 64, 1433 (1990).
[CrossRef]

Feldman, J.

J. Feldman, K. Leo, J. Shah, D. A. B. Miller, J. E. Cunningham, T. Meier, G. von Plessen, A. Shulze, P. Thomas, and S. Schmitt-Rink, Phys. Rev. B 46, 7252 (1992).
[CrossRef]

Flores, J.

L. Gutierrez, A. Daz-de-Anda, J. Flores, R. A. Méndez-Sánchez, G. Monsivais, and A. Morales, Phys. Rev. Lett. 97, 114301 (2006).
[CrossRef]

Garca-Vidal, F. J.

F. López-Tejeira, F. J. Garca-Vidal, and L. Martín-Moreno, Phys. Rev. B 72, 161405 (2005).
[CrossRef]

García-Guerrero, E. E.

A. A. Maradudin, T. A. Leskova, E. E. García-Guerrero, and E. R. Méndez, Low Temp. Phys. 36, 815 (2010).
[CrossRef]

Gutierrez, L.

L. Gutierrez, A. Daz-de-Anda, J. Flores, R. A. Méndez-Sánchez, G. Monsivais, and A. Morales, Phys. Rev. Lett. 97, 114301 (2006).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electromagnetics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Hammon, T. E.

C. M. de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, Phys. Rev. E 57, 2365 (1998).
[CrossRef]

Haring Bolivar, K.

K. Leo, K. Haring Bolivar, F. Brügemann, R. Schwedler, and K. Köhler, Solid State Commun. 84, 943 (1992).
[CrossRef]

Hong, J. M.

E. E. Méndez, F. Agulló-Rueda, and J. M. Hong, Phys. Rev. Lett. 60, 2426 (1988).
[CrossRef]

Johnson, P. B.

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Köhler, K.

K. Leo, K. Haring Bolivar, F. Brügemann, R. Schwedler, and K. Köhler, Solid State Commun. 84, 943 (1992).
[CrossRef]

Krug, P. A.

C. M. de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, Phys. Rev. E 57, 2365 (1998).
[CrossRef]

Leo, K.

K. Leo, K. Haring Bolivar, F. Brügemann, R. Schwedler, and K. Köhler, Solid State Commun. 84, 943 (1992).
[CrossRef]

J. Feldman, K. Leo, J. Shah, D. A. B. Miller, J. E. Cunningham, T. Meier, G. von Plessen, A. Shulze, P. Thomas, and S. Schmitt-Rink, Phys. Rev. B 46, 7252 (1992).
[CrossRef]

Leskova, T. A.

A. A. Maradudin, T. A. Leskova, E. E. García-Guerrero, and E. R. Méndez, Low Temp. Phys. 36, 815 (2010).
[CrossRef]

López-Tejeira, F.

F. López-Tejeira, F. J. Garca-Vidal, and L. Martín-Moreno, Phys. Rev. B 72, 161405 (2005).
[CrossRef]

Madison, K. W.

S. R. Wilkinson, C. F. Bharucha, K. W. Madison, Q. Niu, and M. G. Raizen, Phys. Rev. Lett. 76, 4512 (1996).
[CrossRef]

Maradudin, A. A.

A. A. Maradudin, T. A. Leskova, E. E. García-Guerrero, and E. R. Méndez, Low Temp. Phys. 36, 815 (2010).
[CrossRef]

Martín-Moreno, L.

F. López-Tejeira, F. J. Garca-Vidal, and L. Martín-Moreno, Phys. Rev. B 72, 161405 (2005).
[CrossRef]

Mateos, J. L.

J. L. Mateos and G. Monsivais, Physica A 207, 445 (1994).
[CrossRef]

Meier, T.

J. Feldman, K. Leo, J. Shah, D. A. B. Miller, J. E. Cunningham, T. Meier, G. von Plessen, A. Shulze, P. Thomas, and S. Schmitt-Rink, Phys. Rev. B 46, 7252 (1992).
[CrossRef]

Méndez, E. E.

E. E. Méndez, F. Agulló-Rueda, and J. M. Hong, Phys. Rev. Lett. 60, 2426 (1988).
[CrossRef]

Méndez, E. R.

A. A. Maradudin, T. A. Leskova, E. E. García-Guerrero, and E. R. Méndez, Low Temp. Phys. 36, 815 (2010).
[CrossRef]

Méndez-Sánchez, R. A.

L. Gutierrez, A. Daz-de-Anda, J. Flores, R. A. Méndez-Sánchez, G. Monsivais, and A. Morales, Phys. Rev. Lett. 97, 114301 (2006).
[CrossRef]

Miller, D. A. B.

J. Feldman, K. Leo, J. Shah, D. A. B. Miller, J. E. Cunningham, T. Meier, G. von Plessen, A. Shulze, P. Thomas, and S. Schmitt-Rink, Phys. Rev. B 46, 7252 (1992).
[CrossRef]

Monsivais, G.

L. Gutierrez, A. Daz-de-Anda, J. Flores, R. A. Méndez-Sánchez, G. Monsivais, and A. Morales, Phys. Rev. Lett. 97, 114301 (2006).
[CrossRef]

J. L. Mateos and G. Monsivais, Physica A 207, 445 (1994).
[CrossRef]

G. Monsivais, M. del Castillo-Mussot, and F. Claro, Phys. Rev. Lett. 64, 1433 (1990).
[CrossRef]

Morales, A.

L. Gutierrez, A. Daz-de-Anda, J. Flores, R. A. Méndez-Sánchez, G. Monsivais, and A. Morales, Phys. Rev. Lett. 97, 114301 (2006).
[CrossRef]

Niu, Q.

S. R. Wilkinson, C. F. Bharucha, K. W. Madison, Q. Niu, and M. G. Raizen, Phys. Rev. Lett. 76, 4512 (1996).
[CrossRef]

Oton, C. J.

S. Sapienza, P. Constantino, D. Wiersma, M. Chulinyan, C. J. Oton, and L. Pavesi, Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Pavesi, L.

S. Sapienza, P. Constantino, D. Wiersma, M. Chulinyan, C. J. Oton, and L. Pavesi, Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Raizen, M. G.

S. R. Wilkinson, C. F. Bharucha, K. W. Madison, Q. Niu, and M. G. Raizen, Phys. Rev. Lett. 76, 4512 (1996).
[CrossRef]

Sapienza, S.

S. Sapienza, P. Constantino, D. Wiersma, M. Chulinyan, C. J. Oton, and L. Pavesi, Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Schmitt-Rink, S.

J. Feldman, K. Leo, J. Shah, D. A. B. Miller, J. E. Cunningham, T. Meier, G. von Plessen, A. Shulze, P. Thomas, and S. Schmitt-Rink, Phys. Rev. B 46, 7252 (1992).
[CrossRef]

Schwedler, R.

K. Leo, K. Haring Bolivar, F. Brügemann, R. Schwedler, and K. Köhler, Solid State Commun. 84, 943 (1992).
[CrossRef]

Shah, J.

J. Feldman, K. Leo, J. Shah, D. A. B. Miller, J. E. Cunningham, T. Meier, G. von Plessen, A. Shulze, P. Thomas, and S. Schmitt-Rink, Phys. Rev. B 46, 7252 (1992).
[CrossRef]

Shulze, A.

J. Feldman, K. Leo, J. Shah, D. A. B. Miller, J. E. Cunningham, T. Meier, G. von Plessen, A. Shulze, P. Thomas, and S. Schmitt-Rink, Phys. Rev. B 46, 7252 (1992).
[CrossRef]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electromagnetics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Thomas, P.

J. Feldman, K. Leo, J. Shah, D. A. B. Miller, J. E. Cunningham, T. Meier, G. von Plessen, A. Shulze, P. Thomas, and S. Schmitt-Rink, Phys. Rev. B 46, 7252 (1992).
[CrossRef]

von Plessen, G.

J. Feldman, K. Leo, J. Shah, D. A. B. Miller, J. E. Cunningham, T. Meier, G. von Plessen, A. Shulze, P. Thomas, and S. Schmitt-Rink, Phys. Rev. B 46, 7252 (1992).
[CrossRef]

Wannier, G. H.

G. H. Wannier, Elements of Solid State Theory (Cambridge University, 1959), p. 190.

Wiersma, D.

S. Sapienza, P. Constantino, D. Wiersma, M. Chulinyan, C. J. Oton, and L. Pavesi, Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Wilkinson, S. R.

S. R. Wilkinson, C. F. Bharucha, K. W. Madison, Q. Niu, and M. G. Raizen, Phys. Rev. Lett. 76, 4512 (1996).
[CrossRef]

Low Temp. Phys. (1)

A. A. Maradudin, T. A. Leskova, E. E. García-Guerrero, and E. R. Méndez, Low Temp. Phys. 36, 815 (2010).
[CrossRef]

Phys. Rev. B (3)

F. López-Tejeira, F. J. Garca-Vidal, and L. Martín-Moreno, Phys. Rev. B 72, 161405 (2005).
[CrossRef]

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

J. Feldman, K. Leo, J. Shah, D. A. B. Miller, J. E. Cunningham, T. Meier, G. von Plessen, A. Shulze, P. Thomas, and S. Schmitt-Rink, Phys. Rev. B 46, 7252 (1992).
[CrossRef]

Phys. Rev. E (1)

C. M. de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, Phys. Rev. E 57, 2365 (1998).
[CrossRef]

Phys. Rev. Lett. (5)

L. Gutierrez, A. Daz-de-Anda, J. Flores, R. A. Méndez-Sánchez, G. Monsivais, and A. Morales, Phys. Rev. Lett. 97, 114301 (2006).
[CrossRef]

S. R. Wilkinson, C. F. Bharucha, K. W. Madison, Q. Niu, and M. G. Raizen, Phys. Rev. Lett. 76, 4512 (1996).
[CrossRef]

G. Monsivais, M. del Castillo-Mussot, and F. Claro, Phys. Rev. Lett. 64, 1433 (1990).
[CrossRef]

E. E. Méndez, F. Agulló-Rueda, and J. M. Hong, Phys. Rev. Lett. 60, 2426 (1988).
[CrossRef]

S. Sapienza, P. Constantino, D. Wiersma, M. Chulinyan, C. J. Oton, and L. Pavesi, Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Physica A (1)

J. L. Mateos and G. Monsivais, Physica A 207, 445 (1994).
[CrossRef]

Solid State Commun. (1)

K. Leo, K. Haring Bolivar, F. Brügemann, R. Schwedler, and K. Köhler, Solid State Commun. 84, 943 (1992).
[CrossRef]

Z. Phys. (1)

F. Bloch, Z. Phys. 22, 555 (1928).

Other (3)

G. H. Wannier, Elements of Solid State Theory (Cambridge University, 1959), p. 190.

OptiFDTD 2012 Optiwave, Version 11.2.

A. Taflove and S. C. Hagness, Computational Electromagnetics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

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Figures (5)

Fig. 1.
Fig. 1.

Surface structure employed to obtain a surface plasmon polariton Wannier–Stark ladder.

Fig. 2.
Fig. 2.

Transmittance, reflectance, and power radiated into the vacuum as functions of the normalized frequency of a surface plasmon polariton incident normally on a groove of width l=25nm and depth h=1000nm cut into a planar silver surface.

Fig. 3.
Fig. 3.

Dependence of the j=2 and j=3 resonance frequencies on the depth h of the groove of width l=25nm cut into a planar silver surface.

Fig. 4.
Fig. 4.

(a) Transmittance and the power radiated into the vacuum as functions of the normalized frequency. (b) Transmissivity of the surface plasmon polariton as a function of its frequency for several values of the parameter γ.

Fig. 5.
Fig. 5.

Difference between the frequencies of consecutive Wannier–Stark resonances as a function of γ.

Equations (3)

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

ω(j)(h)=ω0(j)[1+a(j)(h/h0)]1+a(j),
ωn(j)=ω0(j)[1+a(j)1+a(j)γ(j)n].
Δωn(j)=ωn+1(j)ωn(j)=ω0(j)a(j)1+a(j)γ(j)

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