Abstract

The calculation of the local density of states (LDOS) in lossy materials has long been disputed due to the divergence of the homogeneous Green function with equal space arguments. For arbitrary-shaped lossy structures, such as those of interest in nanoplasmonics, this problem is particularly challenging. A nondivergent LDOS obtained in numerical methods such as the finite-difference time-domain (FDTD) technique, at first sight appears to be wrong. Here we show that FDTD is not only an ideal choice for obtaining the regularized LDOS, but it can address the local-field problem for any lossy inhomogeneous material. We exemplify the case of a finite-size photon emitter (e.g., a single quantum dot) embedded within and outside a lossy metal nanoparticle and show excellent agreement with analytical results.

© 2012 Optical Society of America

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References

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  1. S. Scheel, L. Knöll, and D.-G. Welsch, Phys. Rev. A 60, 4094 (1999).
    [CrossRef]
  2. S. M. Barnett, B. Huttner, and R. Loudon, Phys. Rev. Lett. 68, 3698 (1992).
    [CrossRef]
  3. M. S. Tomaš, Phys Rev. A 63, 053811 (2001).
    [CrossRef]
  4. D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method (IEEE, 2000).
  5. O. Benson, Nature 480, 193 (2011).
    [CrossRef]
  6. E. M. Purcell, Phys. Rev. 69, 681 (1946).
  7. P. C. Chaumet, A. Sentenac, and A. Rahmani, Phys. Rev. E 70, 036606 (2004).
    [CrossRef]
  8. P. de Vries, D. V. van Coevorden, and A. Lagendijk, Rev. Mod. Phys. 70, 447 (1998).
    [CrossRef]
  9. L.-W. Li, P.-S. Kooi, M.-S. Leong, and T.-S. Yeo, IEEE Trans. Microw. Theory Technol. 42, 2302 (1994).
    [CrossRef]
  10. FDTD Solutions: www.lumerical.com.
  11. W. Yu and R. Mittra, IEEE Antennas Propag. Mag. 42, 28 (2000).
    [CrossRef]
  12. C. Van Vlack, P. T. Kristensen, and S. Hughes, Phys. Rev. B 85, 075303 (2012).
    [CrossRef]

2012

C. Van Vlack, P. T. Kristensen, and S. Hughes, Phys. Rev. B 85, 075303 (2012).
[CrossRef]

2011

O. Benson, Nature 480, 193 (2011).
[CrossRef]

2004

P. C. Chaumet, A. Sentenac, and A. Rahmani, Phys. Rev. E 70, 036606 (2004).
[CrossRef]

2001

M. S. Tomaš, Phys Rev. A 63, 053811 (2001).
[CrossRef]

2000

W. Yu and R. Mittra, IEEE Antennas Propag. Mag. 42, 28 (2000).
[CrossRef]

1999

S. Scheel, L. Knöll, and D.-G. Welsch, Phys. Rev. A 60, 4094 (1999).
[CrossRef]

1998

P. de Vries, D. V. van Coevorden, and A. Lagendijk, Rev. Mod. Phys. 70, 447 (1998).
[CrossRef]

1994

L.-W. Li, P.-S. Kooi, M.-S. Leong, and T.-S. Yeo, IEEE Trans. Microw. Theory Technol. 42, 2302 (1994).
[CrossRef]

1992

S. M. Barnett, B. Huttner, and R. Loudon, Phys. Rev. Lett. 68, 3698 (1992).
[CrossRef]

1946

E. M. Purcell, Phys. Rev. 69, 681 (1946).

Barnett, S. M.

S. M. Barnett, B. Huttner, and R. Loudon, Phys. Rev. Lett. 68, 3698 (1992).
[CrossRef]

Benson, O.

O. Benson, Nature 480, 193 (2011).
[CrossRef]

Chaumet, P. C.

P. C. Chaumet, A. Sentenac, and A. Rahmani, Phys. Rev. E 70, 036606 (2004).
[CrossRef]

de Vries, P.

P. de Vries, D. V. van Coevorden, and A. Lagendijk, Rev. Mod. Phys. 70, 447 (1998).
[CrossRef]

Hughes, S.

C. Van Vlack, P. T. Kristensen, and S. Hughes, Phys. Rev. B 85, 075303 (2012).
[CrossRef]

Huttner, B.

S. M. Barnett, B. Huttner, and R. Loudon, Phys. Rev. Lett. 68, 3698 (1992).
[CrossRef]

Knöll, L.

S. Scheel, L. Knöll, and D.-G. Welsch, Phys. Rev. A 60, 4094 (1999).
[CrossRef]

Kooi, P.-S.

L.-W. Li, P.-S. Kooi, M.-S. Leong, and T.-S. Yeo, IEEE Trans. Microw. Theory Technol. 42, 2302 (1994).
[CrossRef]

Kristensen, P. T.

C. Van Vlack, P. T. Kristensen, and S. Hughes, Phys. Rev. B 85, 075303 (2012).
[CrossRef]

Lagendijk, A.

P. de Vries, D. V. van Coevorden, and A. Lagendijk, Rev. Mod. Phys. 70, 447 (1998).
[CrossRef]

Leong, M.-S.

L.-W. Li, P.-S. Kooi, M.-S. Leong, and T.-S. Yeo, IEEE Trans. Microw. Theory Technol. 42, 2302 (1994).
[CrossRef]

Li, L.-W.

L.-W. Li, P.-S. Kooi, M.-S. Leong, and T.-S. Yeo, IEEE Trans. Microw. Theory Technol. 42, 2302 (1994).
[CrossRef]

Loudon, R.

S. M. Barnett, B. Huttner, and R. Loudon, Phys. Rev. Lett. 68, 3698 (1992).
[CrossRef]

Mittra, R.

W. Yu and R. Mittra, IEEE Antennas Propag. Mag. 42, 28 (2000).
[CrossRef]

Purcell, E. M.

E. M. Purcell, Phys. Rev. 69, 681 (1946).

Rahmani, A.

P. C. Chaumet, A. Sentenac, and A. Rahmani, Phys. Rev. E 70, 036606 (2004).
[CrossRef]

Scheel, S.

S. Scheel, L. Knöll, and D.-G. Welsch, Phys. Rev. A 60, 4094 (1999).
[CrossRef]

Sentenac, A.

P. C. Chaumet, A. Sentenac, and A. Rahmani, Phys. Rev. E 70, 036606 (2004).
[CrossRef]

Sullivan, D. M.

D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method (IEEE, 2000).

Tomaš, M. S.

M. S. Tomaš, Phys Rev. A 63, 053811 (2001).
[CrossRef]

van Coevorden, D. V.

P. de Vries, D. V. van Coevorden, and A. Lagendijk, Rev. Mod. Phys. 70, 447 (1998).
[CrossRef]

Van Vlack, C.

C. Van Vlack, P. T. Kristensen, and S. Hughes, Phys. Rev. B 85, 075303 (2012).
[CrossRef]

Welsch, D.-G.

S. Scheel, L. Knöll, and D.-G. Welsch, Phys. Rev. A 60, 4094 (1999).
[CrossRef]

Yeo, T.-S.

L.-W. Li, P.-S. Kooi, M.-S. Leong, and T.-S. Yeo, IEEE Trans. Microw. Theory Technol. 42, 2302 (1994).
[CrossRef]

Yu, W.

W. Yu and R. Mittra, IEEE Antennas Propag. Mag. 42, 28 (2000).
[CrossRef]

IEEE Antennas Propag. Mag.

W. Yu and R. Mittra, IEEE Antennas Propag. Mag. 42, 28 (2000).
[CrossRef]

IEEE Trans. Microw. Theory Technol.

L.-W. Li, P.-S. Kooi, M.-S. Leong, and T.-S. Yeo, IEEE Trans. Microw. Theory Technol. 42, 2302 (1994).
[CrossRef]

Nature

O. Benson, Nature 480, 193 (2011).
[CrossRef]

Phys Rev. A

M. S. Tomaš, Phys Rev. A 63, 053811 (2001).
[CrossRef]

Phys. Rev.

E. M. Purcell, Phys. Rev. 69, 681 (1946).

Phys. Rev. A

S. Scheel, L. Knöll, and D.-G. Welsch, Phys. Rev. A 60, 4094 (1999).
[CrossRef]

Phys. Rev. B

C. Van Vlack, P. T. Kristensen, and S. Hughes, Phys. Rev. B 85, 075303 (2012).
[CrossRef]

Phys. Rev. E

P. C. Chaumet, A. Sentenac, and A. Rahmani, Phys. Rev. E 70, 036606 (2004).
[CrossRef]

Phys. Rev. Lett.

S. M. Barnett, B. Huttner, and R. Loudon, Phys. Rev. Lett. 68, 3698 (1992).
[CrossRef]

Rev. Mod. Phys.

P. de Vries, D. V. van Coevorden, and A. Lagendijk, Rev. Mod. Phys. 70, 447 (1998).
[CrossRef]

Other

D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method (IEEE, 2000).

FDTD Solutions: www.lumerical.com.

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Figures (3)

Fig. 1.
Fig. 1.

Schematics of the regularized GF integrated over a finite volume, for (a) a homogeneous material and (b) a MNP. Schematics of the real cavity GF are shown inside (c) a homogeneous material and (d) a MNP. (e) ρy(r) as a function of height above and inside a MNP. FDTD calculations (log scale) are shown for 1 nm (blue circles) and 2 nm (orange crosses) grids. Outside (above) the MNP, these are compared with the analytic GF, which formally diverges over lossy materials (solid green line). The dashed vertical black lines show the cases considered in Fig. 2.

Fig. 2.
Fig. 2.

ρy(ω) as a function of frequency inside and outside the MNP. Blue circles (orange crosses) show a gridding of a/20 (a/10). The integration of the homogeneous GF over a cube of size a/20 (a/10) are given by blue–dark (orange–light) lines in (a), (b). (a) z/a=0 in log scale. (b) z/a=0.9 in log scale. (c) z/a=1.2 in linear scale; here, the green solid line shows the analytic results.

Fig. 3.
Fig. 3.

Real cavity model versus regularized GF calculations. In blue (right) we show the regularized GF for the homogeneous case [circles, Fig. 1(a)] and for the total GF inside a MNP [crosses, Fig. 1(b)] using FDTD with a gridding of a/20. In orange (left) we show the real cavity GF for the homogeneous case [circles, Fig. 1(c)] and for the total GF inside a MNP [crosses, Fig. 1(d)] using FDTD with a gridding of a/20, where we have inserted a spherical cavity filled with vacuum where the cavity volume gives the same effective volume as an FDTD grid cell, (a/20)3. The orange solid line corresponds to exact calculations of the homogeneous real cavity GF using analytical techniques [Fig. 1(c)].

Equations (2)

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Im[Giihom(r,r;ω)]=k03nμ/6π,
ρi(r;ω)=Im[Gii(r,r;ω)]Im[Giivac(r,r;ω)],

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