Abstract

We study the spontaneous decay rate of a dipole emitter close to a metallic nanoparticle in the extreme near-field regime. The metal is modeled using a nonlocal dielectric function that accounts for the microscopic length scales of the free electron gas. We describe quantitatively the crossover between the macroscopic and microscopic regimes and the enhanced nonradiative decay due to microscopic interactions. Our theory is in agreement with results previously established in the asymptotic near- and far-field regimes.

© 2010 Optical Society of America

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  1. R. R. Chance, A. Prock, and R. Silbey, Adv. Chem. Phys. 37, 1 (1978).
    [CrossRef]
  2. P.Berman, ed., Cavity Quantum Electrodynamics (Academic, 1994).
  3. S. Kühn, U. Hakanson, L. Rogobete, and V. Sandoghdar, Phys. Rev. Lett. 97, 017402 (2006).
    [CrossRef] [PubMed]
  4. P. Anger, P. Bharadwaj, and L. Novotny, Phys. Rev. Lett. 96, 113002 (2006).
    [CrossRef] [PubMed]
  5. P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, Science 308, 1607 (2005).
    [CrossRef] [PubMed]
  6. T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, Nature Photon. 2, 234 (2008).
    [CrossRef]
  7. M. Thomas, J.-J. Greffet, R. Carminati, and J. R. Arias-Gonzales, Appl. Phys. Lett. 85, 3863 (2004).
    [CrossRef]
  8. S. Kühn, G. Mori, M. Agio, and V. Sandoghdar, Mol. Phys. 106, 893 (2008).
    [CrossRef]
  9. T. Pons, I. L. Medintz, K. E. Sapsford, S. Higashiya, A. F. Grimes, D. S. English, and H. Mattoussi, Nano Lett. 7, 3157 (2007).
    [CrossRef] [PubMed]
  10. J. Seelig, K. Leslie, A. Renn, S. Kühn, V. Jacobsen, M. van de Corput, C. Wyman, and V. Sandoghdar, Nano Lett. 7, 685 (2007).
    [CrossRef] [PubMed]
  11. G. W. Ford and W. H. Weber, Phys. Rep. 113, 195 (1984).
    [CrossRef]
  12. I. A. Larkin, M. I. Stockman, M. Achermann, and V. I. Klimov, Phys. Rev. B 69, 121403(R) (2004).
    [CrossRef]
  13. J. Vielma and P. T. Leung, J. Chem. Phys. 126, 194704 (2007).
    [CrossRef] [PubMed]
  14. T. L. Jennings, M. P. Singh, and G. F. Strouse, J. Am. Chem. Soc. 128, 5462 (2006).
    [CrossRef] [PubMed]
  15. O. Keller, Phys. Rep. 268, 85 (1996).
    [CrossRef]
  16. J. M. Wylie and J. E. Sipe, Phys. Rev. A 30, 1185 (1984).
    [CrossRef]
  17. H. Chew, J. Chem. Phys. 87, 1355 (1987).
    [CrossRef]
  18. R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, Opt. Commun. 261, 368 (2006).
    [CrossRef]
  19. J. E. Sipe, J. Opt. Soc. Am. B 4, 481 (1987).
    [CrossRef]
  20. C. Henkel and K. Joulain, Appl. Phys. B 84, 61 (2006).
    [CrossRef]
  21. P. O. Chapuis, S. Volz, C. Henkel, K. Joulain, and J.-J. Greffet, Phys. Rev. B 77, 035431 (2008).
    [CrossRef]
  22. The LM functions read fl(a,u)=1/2+[1−(a−u)2]/(8a)ln[(a−u+1)/(a−u−1)]+[1−(a+u)2]/(8a)ln[(a+u+1)/(a+u−1)] and ft(a,u)=3(a2+3u2+1)/8−3[1−(a−u)2]2/(32a)ln[(a−u+1)/(a−u−1)]−3[1−(a+u)2]2/(32a)ln[(a+u+1)/(a+u−1)]. The limit u-->0 has to be taken with a positive imaginary part so that ln[(a+/-u+1)/(a+/-u−1)]~ln|(a+1)/(a−1)|.

2008 (3)

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, Nature Photon. 2, 234 (2008).
[CrossRef]

S. Kühn, G. Mori, M. Agio, and V. Sandoghdar, Mol. Phys. 106, 893 (2008).
[CrossRef]

P. O. Chapuis, S. Volz, C. Henkel, K. Joulain, and J.-J. Greffet, Phys. Rev. B 77, 035431 (2008).
[CrossRef]

2007 (3)

T. Pons, I. L. Medintz, K. E. Sapsford, S. Higashiya, A. F. Grimes, D. S. English, and H. Mattoussi, Nano Lett. 7, 3157 (2007).
[CrossRef] [PubMed]

J. Seelig, K. Leslie, A. Renn, S. Kühn, V. Jacobsen, M. van de Corput, C. Wyman, and V. Sandoghdar, Nano Lett. 7, 685 (2007).
[CrossRef] [PubMed]

J. Vielma and P. T. Leung, J. Chem. Phys. 126, 194704 (2007).
[CrossRef] [PubMed]

2006 (5)

T. L. Jennings, M. P. Singh, and G. F. Strouse, J. Am. Chem. Soc. 128, 5462 (2006).
[CrossRef] [PubMed]

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, Opt. Commun. 261, 368 (2006).
[CrossRef]

C. Henkel and K. Joulain, Appl. Phys. B 84, 61 (2006).
[CrossRef]

S. Kühn, U. Hakanson, L. Rogobete, and V. Sandoghdar, Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef] [PubMed]

P. Anger, P. Bharadwaj, and L. Novotny, Phys. Rev. Lett. 96, 113002 (2006).
[CrossRef] [PubMed]

2005 (1)

P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, Science 308, 1607 (2005).
[CrossRef] [PubMed]

2004 (2)

M. Thomas, J.-J. Greffet, R. Carminati, and J. R. Arias-Gonzales, Appl. Phys. Lett. 85, 3863 (2004).
[CrossRef]

I. A. Larkin, M. I. Stockman, M. Achermann, and V. I. Klimov, Phys. Rev. B 69, 121403(R) (2004).
[CrossRef]

1996 (1)

O. Keller, Phys. Rep. 268, 85 (1996).
[CrossRef]

1987 (2)

1984 (2)

J. M. Wylie and J. E. Sipe, Phys. Rev. A 30, 1185 (1984).
[CrossRef]

G. W. Ford and W. H. Weber, Phys. Rep. 113, 195 (1984).
[CrossRef]

1978 (1)

R. R. Chance, A. Prock, and R. Silbey, Adv. Chem. Phys. 37, 1 (1978).
[CrossRef]

Achermann, M.

I. A. Larkin, M. I. Stockman, M. Achermann, and V. I. Klimov, Phys. Rev. B 69, 121403(R) (2004).
[CrossRef]

Agio, M.

S. Kühn, G. Mori, M. Agio, and V. Sandoghdar, Mol. Phys. 106, 893 (2008).
[CrossRef]

Anger, P.

P. Anger, P. Bharadwaj, and L. Novotny, Phys. Rev. Lett. 96, 113002 (2006).
[CrossRef] [PubMed]

Arias-Gonzales, J. R.

M. Thomas, J.-J. Greffet, R. Carminati, and J. R. Arias-Gonzales, Appl. Phys. Lett. 85, 3863 (2004).
[CrossRef]

Bharadwaj, P.

P. Anger, P. Bharadwaj, and L. Novotny, Phys. Rev. Lett. 96, 113002 (2006).
[CrossRef] [PubMed]

Carminati, R.

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, Opt. Commun. 261, 368 (2006).
[CrossRef]

M. Thomas, J.-J. Greffet, R. Carminati, and J. R. Arias-Gonzales, Appl. Phys. Lett. 85, 3863 (2004).
[CrossRef]

Chance, R. R.

R. R. Chance, A. Prock, and R. Silbey, Adv. Chem. Phys. 37, 1 (1978).
[CrossRef]

Chapuis, P. O.

P. O. Chapuis, S. Volz, C. Henkel, K. Joulain, and J.-J. Greffet, Phys. Rev. B 77, 035431 (2008).
[CrossRef]

Chew, H.

H. Chew, J. Chem. Phys. 87, 1355 (1987).
[CrossRef]

Eisler, H. -J.

P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, Science 308, 1607 (2005).
[CrossRef] [PubMed]

English, D. S.

T. Pons, I. L. Medintz, K. E. Sapsford, S. Higashiya, A. F. Grimes, D. S. English, and H. Mattoussi, Nano Lett. 7, 3157 (2007).
[CrossRef] [PubMed]

Ford, G. W.

G. W. Ford and W. H. Weber, Phys. Rep. 113, 195 (1984).
[CrossRef]

Greffet, J. -J.

P. O. Chapuis, S. Volz, C. Henkel, K. Joulain, and J.-J. Greffet, Phys. Rev. B 77, 035431 (2008).
[CrossRef]

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, Opt. Commun. 261, 368 (2006).
[CrossRef]

M. Thomas, J.-J. Greffet, R. Carminati, and J. R. Arias-Gonzales, Appl. Phys. Lett. 85, 3863 (2004).
[CrossRef]

Grimes, A. F.

T. Pons, I. L. Medintz, K. E. Sapsford, S. Higashiya, A. F. Grimes, D. S. English, and H. Mattoussi, Nano Lett. 7, 3157 (2007).
[CrossRef] [PubMed]

Hakanson, U.

S. Kühn, U. Hakanson, L. Rogobete, and V. Sandoghdar, Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef] [PubMed]

Hecht, B.

P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, Science 308, 1607 (2005).
[CrossRef] [PubMed]

Henkel, C.

P. O. Chapuis, S. Volz, C. Henkel, K. Joulain, and J.-J. Greffet, Phys. Rev. B 77, 035431 (2008).
[CrossRef]

C. Henkel and K. Joulain, Appl. Phys. B 84, 61 (2006).
[CrossRef]

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, Opt. Commun. 261, 368 (2006).
[CrossRef]

Higashiya, S.

T. Pons, I. L. Medintz, K. E. Sapsford, S. Higashiya, A. F. Grimes, D. S. English, and H. Mattoussi, Nano Lett. 7, 3157 (2007).
[CrossRef] [PubMed]

Jacobsen, V.

J. Seelig, K. Leslie, A. Renn, S. Kühn, V. Jacobsen, M. van de Corput, C. Wyman, and V. Sandoghdar, Nano Lett. 7, 685 (2007).
[CrossRef] [PubMed]

Jennings, T. L.

T. L. Jennings, M. P. Singh, and G. F. Strouse, J. Am. Chem. Soc. 128, 5462 (2006).
[CrossRef] [PubMed]

Joulain, K.

P. O. Chapuis, S. Volz, C. Henkel, K. Joulain, and J.-J. Greffet, Phys. Rev. B 77, 035431 (2008).
[CrossRef]

C. Henkel and K. Joulain, Appl. Phys. B 84, 61 (2006).
[CrossRef]

Keller, O.

O. Keller, Phys. Rep. 268, 85 (1996).
[CrossRef]

Klimov, V. I.

I. A. Larkin, M. I. Stockman, M. Achermann, and V. I. Klimov, Phys. Rev. B 69, 121403(R) (2004).
[CrossRef]

Kühn, S.

S. Kühn, G. Mori, M. Agio, and V. Sandoghdar, Mol. Phys. 106, 893 (2008).
[CrossRef]

J. Seelig, K. Leslie, A. Renn, S. Kühn, V. Jacobsen, M. van de Corput, C. Wyman, and V. Sandoghdar, Nano Lett. 7, 685 (2007).
[CrossRef] [PubMed]

S. Kühn, U. Hakanson, L. Rogobete, and V. Sandoghdar, Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef] [PubMed]

Larkin, I. A.

I. A. Larkin, M. I. Stockman, M. Achermann, and V. I. Klimov, Phys. Rev. B 69, 121403(R) (2004).
[CrossRef]

Leslie, K.

J. Seelig, K. Leslie, A. Renn, S. Kühn, V. Jacobsen, M. van de Corput, C. Wyman, and V. Sandoghdar, Nano Lett. 7, 685 (2007).
[CrossRef] [PubMed]

Leung, P. T.

J. Vielma and P. T. Leung, J. Chem. Phys. 126, 194704 (2007).
[CrossRef] [PubMed]

Martin, O. J. F.

P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, Science 308, 1607 (2005).
[CrossRef] [PubMed]

Mattoussi, H.

T. Pons, I. L. Medintz, K. E. Sapsford, S. Higashiya, A. F. Grimes, D. S. English, and H. Mattoussi, Nano Lett. 7, 3157 (2007).
[CrossRef] [PubMed]

Medintz, I. L.

T. Pons, I. L. Medintz, K. E. Sapsford, S. Higashiya, A. F. Grimes, D. S. English, and H. Mattoussi, Nano Lett. 7, 3157 (2007).
[CrossRef] [PubMed]

Mori, G.

S. Kühn, G. Mori, M. Agio, and V. Sandoghdar, Mol. Phys. 106, 893 (2008).
[CrossRef]

Mühlschlegel, P.

P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, Science 308, 1607 (2005).
[CrossRef] [PubMed]

Novotny, L.

P. Anger, P. Bharadwaj, and L. Novotny, Phys. Rev. Lett. 96, 113002 (2006).
[CrossRef] [PubMed]

Pohl, D. W.

P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, Science 308, 1607 (2005).
[CrossRef] [PubMed]

Pons, T.

T. Pons, I. L. Medintz, K. E. Sapsford, S. Higashiya, A. F. Grimes, D. S. English, and H. Mattoussi, Nano Lett. 7, 3157 (2007).
[CrossRef] [PubMed]

Prock, A.

R. R. Chance, A. Prock, and R. Silbey, Adv. Chem. Phys. 37, 1 (1978).
[CrossRef]

Renn, A.

J. Seelig, K. Leslie, A. Renn, S. Kühn, V. Jacobsen, M. van de Corput, C. Wyman, and V. Sandoghdar, Nano Lett. 7, 685 (2007).
[CrossRef] [PubMed]

Rogobete, L.

S. Kühn, U. Hakanson, L. Rogobete, and V. Sandoghdar, Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef] [PubMed]

Sandoghdar, V.

S. Kühn, G. Mori, M. Agio, and V. Sandoghdar, Mol. Phys. 106, 893 (2008).
[CrossRef]

J. Seelig, K. Leslie, A. Renn, S. Kühn, V. Jacobsen, M. van de Corput, C. Wyman, and V. Sandoghdar, Nano Lett. 7, 685 (2007).
[CrossRef] [PubMed]

S. Kühn, U. Hakanson, L. Rogobete, and V. Sandoghdar, Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef] [PubMed]

Sapsford, K. E.

T. Pons, I. L. Medintz, K. E. Sapsford, S. Higashiya, A. F. Grimes, D. S. English, and H. Mattoussi, Nano Lett. 7, 3157 (2007).
[CrossRef] [PubMed]

Seelig, J.

J. Seelig, K. Leslie, A. Renn, S. Kühn, V. Jacobsen, M. van de Corput, C. Wyman, and V. Sandoghdar, Nano Lett. 7, 685 (2007).
[CrossRef] [PubMed]

Segerink, F. B.

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, Nature Photon. 2, 234 (2008).
[CrossRef]

Silbey, R.

R. R. Chance, A. Prock, and R. Silbey, Adv. Chem. Phys. 37, 1 (1978).
[CrossRef]

Singh, M. P.

T. L. Jennings, M. P. Singh, and G. F. Strouse, J. Am. Chem. Soc. 128, 5462 (2006).
[CrossRef] [PubMed]

Sipe, J. E.

J. E. Sipe, J. Opt. Soc. Am. B 4, 481 (1987).
[CrossRef]

J. M. Wylie and J. E. Sipe, Phys. Rev. A 30, 1185 (1984).
[CrossRef]

Stefani, F. D.

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, Nature Photon. 2, 234 (2008).
[CrossRef]

Stockman, M. I.

I. A. Larkin, M. I. Stockman, M. Achermann, and V. I. Klimov, Phys. Rev. B 69, 121403(R) (2004).
[CrossRef]

Strouse, G. F.

T. L. Jennings, M. P. Singh, and G. F. Strouse, J. Am. Chem. Soc. 128, 5462 (2006).
[CrossRef] [PubMed]

Taminiau, T. H.

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, Nature Photon. 2, 234 (2008).
[CrossRef]

Thomas, M.

M. Thomas, J.-J. Greffet, R. Carminati, and J. R. Arias-Gonzales, Appl. Phys. Lett. 85, 3863 (2004).
[CrossRef]

van de Corput, M.

J. Seelig, K. Leslie, A. Renn, S. Kühn, V. Jacobsen, M. van de Corput, C. Wyman, and V. Sandoghdar, Nano Lett. 7, 685 (2007).
[CrossRef] [PubMed]

van Hulst, N. F.

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, Nature Photon. 2, 234 (2008).
[CrossRef]

Vielma, J.

J. Vielma and P. T. Leung, J. Chem. Phys. 126, 194704 (2007).
[CrossRef] [PubMed]

Vigoureux, J. M.

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, Opt. Commun. 261, 368 (2006).
[CrossRef]

Volz, S.

P. O. Chapuis, S. Volz, C. Henkel, K. Joulain, and J.-J. Greffet, Phys. Rev. B 77, 035431 (2008).
[CrossRef]

Weber, W. H.

G. W. Ford and W. H. Weber, Phys. Rep. 113, 195 (1984).
[CrossRef]

Wylie, J. M.

J. M. Wylie and J. E. Sipe, Phys. Rev. A 30, 1185 (1984).
[CrossRef]

Wyman, C.

J. Seelig, K. Leslie, A. Renn, S. Kühn, V. Jacobsen, M. van de Corput, C. Wyman, and V. Sandoghdar, Nano Lett. 7, 685 (2007).
[CrossRef] [PubMed]

Adv. Chem. Phys. (1)

R. R. Chance, A. Prock, and R. Silbey, Adv. Chem. Phys. 37, 1 (1978).
[CrossRef]

Appl. Phys. B (1)

C. Henkel and K. Joulain, Appl. Phys. B 84, 61 (2006).
[CrossRef]

Appl. Phys. Lett. (1)

M. Thomas, J.-J. Greffet, R. Carminati, and J. R. Arias-Gonzales, Appl. Phys. Lett. 85, 3863 (2004).
[CrossRef]

J. Am. Chem. Soc. (1)

T. L. Jennings, M. P. Singh, and G. F. Strouse, J. Am. Chem. Soc. 128, 5462 (2006).
[CrossRef] [PubMed]

J. Chem. Phys. (2)

J. Vielma and P. T. Leung, J. Chem. Phys. 126, 194704 (2007).
[CrossRef] [PubMed]

H. Chew, J. Chem. Phys. 87, 1355 (1987).
[CrossRef]

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

Mol. Phys. (1)

S. Kühn, G. Mori, M. Agio, and V. Sandoghdar, Mol. Phys. 106, 893 (2008).
[CrossRef]

Nano Lett. (2)

T. Pons, I. L. Medintz, K. E. Sapsford, S. Higashiya, A. F. Grimes, D. S. English, and H. Mattoussi, Nano Lett. 7, 3157 (2007).
[CrossRef] [PubMed]

J. Seelig, K. Leslie, A. Renn, S. Kühn, V. Jacobsen, M. van de Corput, C. Wyman, and V. Sandoghdar, Nano Lett. 7, 685 (2007).
[CrossRef] [PubMed]

Nature Photon. (1)

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, Nature Photon. 2, 234 (2008).
[CrossRef]

Opt. Commun. (1)

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, Opt. Commun. 261, 368 (2006).
[CrossRef]

Phys. Rep. (2)

O. Keller, Phys. Rep. 268, 85 (1996).
[CrossRef]

G. W. Ford and W. H. Weber, Phys. Rep. 113, 195 (1984).
[CrossRef]

Phys. Rev. A (1)

J. M. Wylie and J. E. Sipe, Phys. Rev. A 30, 1185 (1984).
[CrossRef]

Phys. Rev. B (2)

P. O. Chapuis, S. Volz, C. Henkel, K. Joulain, and J.-J. Greffet, Phys. Rev. B 77, 035431 (2008).
[CrossRef]

I. A. Larkin, M. I. Stockman, M. Achermann, and V. I. Klimov, Phys. Rev. B 69, 121403(R) (2004).
[CrossRef]

Phys. Rev. Lett. (2)

S. Kühn, U. Hakanson, L. Rogobete, and V. Sandoghdar, Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef] [PubMed]

P. Anger, P. Bharadwaj, and L. Novotny, Phys. Rev. Lett. 96, 113002 (2006).
[CrossRef] [PubMed]

Science (1)

P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, Science 308, 1607 (2005).
[CrossRef] [PubMed]

Other (2)

P.Berman, ed., Cavity Quantum Electrodynamics (Academic, 1994).

The LM functions read fl(a,u)=1/2+[1−(a−u)2]/(8a)ln[(a−u+1)/(a−u−1)]+[1−(a+u)2]/(8a)ln[(a+u+1)/(a+u−1)] and ft(a,u)=3(a2+3u2+1)/8−3[1−(a−u)2]2/(32a)ln[(a−u+1)/(a−u−1)]−3[1−(a+u)2]2/(32a)ln[(a+u+1)/(a+u−1)]. The limit u-->0 has to be taken with a positive imaginary part so that ln[(a+/-u+1)/(a+/-u−1)]~ln|(a+1)/(a−1)|.

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Figures (2)

Fig. 1
Fig. 1

Normalized decay rate Γ / Γ 0 close to a silver sphere ( R = 25   nm ) versus the distance z between the emitter and the sphere surface. Emission wavelength λ = 700   nm . The nanoparticle is described by a local (bulk) dielectric function with three different models: electric dipole model (black dashed line), Mie theory (blue solid line), and plane model (red dotted line).

Fig. 2
Fig. 2

Normalized decay rate Γ / Γ 0 in the extreme near-field regime for a silver plane. Emission wavelength λ = 700   nm . Red dotted line, bulk dielectric function. Blue solid line, nonlocal model with parameters ϵ b = 3.6 , ω p = 1.42 × 10 16 s 1 , and ν = 8.79 × 10 13 s 1 [11]. The relevant length scales are indicated on the horizontal axis. Inset, comparison between the quasi-static approximation and the full calculation using the nonlocal model.

Equations (5)

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

Γ Γ 0 = 1 + 3 2 ϵ 1 k 0 3 Re 0 K 3 q 1 ( K ) r p ( K ) exp [ 2 i q 1 ( K ) z ] d K .
r p ( K ) = q 1 ( K ) / ( ω ϵ 1 ) Z ( K ) q 1 ( K ) / ( ω ϵ 1 ) + Z ( K ) .
Z ( K ) = 2 i π ω 0 [ q 2 ϵ t ( k , ω ) ( k / k 0 ) 2 + K 2 ϵ l ( k , ω ) ] d q k 2 ,
ϵ l ( k , ω ) = ϵ b + 3 ω p 2 ω + i ν u 2 f l ( a , u ) ω + i ν f l ( a , u ) / f l ( a , 0 ) ,
ϵ t ( k , ω ) = ϵ b ω p 2 ω 2 ( ω + i ν ) { ω [ f t ( a , u ) 3 a 2 f l ( a , u ) ] + i ν [ f t ( a , 0 ) 3 a 2 f l ( a , 0 ) ] } ,

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