Abstract

The optical power flow around a plasmonic particle has been a topic of research interest over the years [see e.g., Am. J. Phys. 51, 323 (1983) ; Opt. Express 13, 8372 (2005) ]. Here we revisit this problem with an emphasis on higher-order resonances, and we present the theoretical results of our analysis for such power-flow distribution for plasmonic nanoparticles at their multipolar resonances. Results for the second and third orders of resonance show optical power-flow patterns that are significantly different from that of the first-order resonance inside and around plasmonic superdirective nanoparticles, with multicenter vortices, saddle points, and saddle lines and with an anomalous circulation of power resembling higher-order modes in a resonant cavity. A potential application of these optical flow patterns to trap or move a neighboring nanoparticle is also briefly suggested.

© 2007 Optical Society of America

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References

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  1. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  2. M. Kerker, "Founding fathers of light scattering and surface-enhanced Raman scattering," Appl. Opt. 30, 4699-4705 (1991).
    [CrossRef] [PubMed]
  3. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  4. M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, "Electromagnetic energy transport via linear chains of silver nanoparticles," Opt. Lett. 23, 1331-1333 (1998).
    [CrossRef]
  5. N. Engheta, A. Salandrino, and A. Alù, "Circuit elements at optical frequencies: nano-inductors, nano-capacitors and nano-resistors," Phys. Rev. Lett. 95, 095504 (2005).
    [CrossRef] [PubMed]
  6. A. Alù and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016623 (2005).
    [CrossRef]
  7. M. G. Silveirinha, A. Alù, and N. Engheta, "Parallel plate metamaterials for cloaking structures," Phys. Rev. E 75, 036603 (2007).
    [CrossRef]
  8. A. Alù and N. Engheta, "Plasmonic materials in transparency and cloaking problems: mechanism, robustness, and physical insights," Opt. Express 15, 3318-3332 (2007).
    [CrossRef] [PubMed]
  9. A. Alù and N. Engheta, "Enhanced directivity from sub-wavelength infrared/optical nano-antennas loaded with plasmonic materials or metamaterials," IEEE Trans. Antennas Propag. (to be published).
  10. G. Mie, "Considerations on the optics of turbid media, especially colloidal metal sols," Ann. Phys. 25, 377-442 (1908).
    [CrossRef]
  11. C. F. Bohren, "How can a particle absorb more than the light incident on it?," Am. J. Phys. 51, 323-327 (1983).
    [CrossRef]
  12. Z. B. Wang, B. S. Lukyanchuk, M. H. Hong, Y. Lin, and T. C. Chong, "Energy flow around a small particle investigated by classical Mie theory," Phys. Rev. B 70, 035418 (2004).
    [CrossRef]
  13. M. V. Bashevoy, V. A. Federov, and N. I. Zheludev, "Optical whirlpool on an absorbing metallic nanoparticle," Opt. Express 13, 8372-8379 (2005).
    [CrossRef] [PubMed]
  14. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
  15. A. Alù and N. Engheta, "Polarizabilities and effective parameters for collections of spherical nano-particles formed by pairs of concentric double-negative (DNG), single-negative (SNG) and/or double-positive (DPS) metamaterial layers," J. Appl. Phys. 97, 094310 (2005).
    [CrossRef]
  16. M. I. Tribelskii, "Resonant scattering of light by small particles," Sov. Phys. JETP 59, 534-536 (1984).
  17. By "incident on the particle" the problem is considered in terms of geometrical optics.
  18. P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
    [CrossRef]
  19. D. Bohm and E. P. Gross, "Theory of plasma oscillations. A. Origin of medium-like behavior," Phys. Rev. 75, 1851-1864 (1949).
    [CrossRef]
  20. M. Abramowitz and I. A. Stegun, eds., "Spherical Bessel functions," in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972), Subsection 10.1, pp. 437-442.
  21. We note that the analysis in expands the whole scattered field in the Taylor approximation, mixing together the different Mie spherical harmonics. Although this procedure eventually ensures numerical convergence, it requires a much higher number of terms in order to predict the correct results near the resonance of the nanoparticle, similar to the cases considered here. This explains the disagreement between our full-wave results and the approximate calculation in . Also contrary to the claim mentioned in that a correct evaluation of the power-flow distribution would require multiple Mie orders, we believe that this is not the case, in the sense that, even if in their Taylor expansion higher-order terms are needed, with a full-wave Mie harmonic expansion one would require one single resonant harmonic to predict the overall power distribution, as has been shown here.
  22. S. J. Oldenburg, G. D. Hale, J. B. Jackson, and N. J. Halas, "Light scattering from dipole and quadrupole nanoshell antennas," Appl. Phys. Lett. 75, 1063-1065 (1999).
    [CrossRef]

2007

M. G. Silveirinha, A. Alù, and N. Engheta, "Parallel plate metamaterials for cloaking structures," Phys. Rev. E 75, 036603 (2007).
[CrossRef]

A. Alù and N. Engheta, "Plasmonic materials in transparency and cloaking problems: mechanism, robustness, and physical insights," Opt. Express 15, 3318-3332 (2007).
[CrossRef] [PubMed]

2005

M. V. Bashevoy, V. A. Federov, and N. I. Zheludev, "Optical whirlpool on an absorbing metallic nanoparticle," Opt. Express 13, 8372-8379 (2005).
[CrossRef] [PubMed]

N. Engheta, A. Salandrino, and A. Alù, "Circuit elements at optical frequencies: nano-inductors, nano-capacitors and nano-resistors," Phys. Rev. Lett. 95, 095504 (2005).
[CrossRef] [PubMed]

A. Alù and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016623 (2005).
[CrossRef]

A. Alù and N. Engheta, "Polarizabilities and effective parameters for collections of spherical nano-particles formed by pairs of concentric double-negative (DNG), single-negative (SNG) and/or double-positive (DPS) metamaterial layers," J. Appl. Phys. 97, 094310 (2005).
[CrossRef]

2004

Z. B. Wang, B. S. Lukyanchuk, M. H. Hong, Y. Lin, and T. C. Chong, "Energy flow around a small particle investigated by classical Mie theory," Phys. Rev. B 70, 035418 (2004).
[CrossRef]

2000

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

1999

S. J. Oldenburg, G. D. Hale, J. B. Jackson, and N. J. Halas, "Light scattering from dipole and quadrupole nanoshell antennas," Appl. Phys. Lett. 75, 1063-1065 (1999).
[CrossRef]

1998

1991

1984

M. I. Tribelskii, "Resonant scattering of light by small particles," Sov. Phys. JETP 59, 534-536 (1984).

1983

C. F. Bohren, "How can a particle absorb more than the light incident on it?," Am. J. Phys. 51, 323-327 (1983).
[CrossRef]

1972

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

1949

D. Bohm and E. P. Gross, "Theory of plasma oscillations. A. Origin of medium-like behavior," Phys. Rev. 75, 1851-1864 (1949).
[CrossRef]

1908

G. Mie, "Considerations on the optics of turbid media, especially colloidal metal sols," Ann. Phys. 25, 377-442 (1908).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, eds., "Spherical Bessel functions," in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972), Subsection 10.1, pp. 437-442.

Alù, A.

A. Alù and N. Engheta, "Plasmonic materials in transparency and cloaking problems: mechanism, robustness, and physical insights," Opt. Express 15, 3318-3332 (2007).
[CrossRef] [PubMed]

M. G. Silveirinha, A. Alù, and N. Engheta, "Parallel plate metamaterials for cloaking structures," Phys. Rev. E 75, 036603 (2007).
[CrossRef]

A. Alù and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016623 (2005).
[CrossRef]

A. Alù and N. Engheta, "Polarizabilities and effective parameters for collections of spherical nano-particles formed by pairs of concentric double-negative (DNG), single-negative (SNG) and/or double-positive (DPS) metamaterial layers," J. Appl. Phys. 97, 094310 (2005).
[CrossRef]

N. Engheta, A. Salandrino, and A. Alù, "Circuit elements at optical frequencies: nano-inductors, nano-capacitors and nano-resistors," Phys. Rev. Lett. 95, 095504 (2005).
[CrossRef] [PubMed]

A. Alù and N. Engheta, "Enhanced directivity from sub-wavelength infrared/optical nano-antennas loaded with plasmonic materials or metamaterials," IEEE Trans. Antennas Propag. (to be published).

Aussenegg, F. R.

Bashevoy, M. V.

Bohm, D.

D. Bohm and E. P. Gross, "Theory of plasma oscillations. A. Origin of medium-like behavior," Phys. Rev. 75, 1851-1864 (1949).
[CrossRef]

Bohren, C. F.

C. F. Bohren, "How can a particle absorb more than the light incident on it?," Am. J. Phys. 51, 323-327 (1983).
[CrossRef]

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Chong, T. C.

Z. B. Wang, B. S. Lukyanchuk, M. H. Hong, Y. Lin, and T. C. Chong, "Energy flow around a small particle investigated by classical Mie theory," Phys. Rev. B 70, 035418 (2004).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Engheta, N.

A. Alù and N. Engheta, "Plasmonic materials in transparency and cloaking problems: mechanism, robustness, and physical insights," Opt. Express 15, 3318-3332 (2007).
[CrossRef] [PubMed]

M. G. Silveirinha, A. Alù, and N. Engheta, "Parallel plate metamaterials for cloaking structures," Phys. Rev. E 75, 036603 (2007).
[CrossRef]

A. Alù and N. Engheta, "Polarizabilities and effective parameters for collections of spherical nano-particles formed by pairs of concentric double-negative (DNG), single-negative (SNG) and/or double-positive (DPS) metamaterial layers," J. Appl. Phys. 97, 094310 (2005).
[CrossRef]

A. Alù and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016623 (2005).
[CrossRef]

N. Engheta, A. Salandrino, and A. Alù, "Circuit elements at optical frequencies: nano-inductors, nano-capacitors and nano-resistors," Phys. Rev. Lett. 95, 095504 (2005).
[CrossRef] [PubMed]

A. Alù and N. Engheta, "Enhanced directivity from sub-wavelength infrared/optical nano-antennas loaded with plasmonic materials or metamaterials," IEEE Trans. Antennas Propag. (to be published).

Federov, V. A.

Gross, E. P.

D. Bohm and E. P. Gross, "Theory of plasma oscillations. A. Origin of medium-like behavior," Phys. Rev. 75, 1851-1864 (1949).
[CrossRef]

Halas, N. J.

S. J. Oldenburg, G. D. Hale, J. B. Jackson, and N. J. Halas, "Light scattering from dipole and quadrupole nanoshell antennas," Appl. Phys. Lett. 75, 1063-1065 (1999).
[CrossRef]

Hale, G. D.

S. J. Oldenburg, G. D. Hale, J. B. Jackson, and N. J. Halas, "Light scattering from dipole and quadrupole nanoshell antennas," Appl. Phys. Lett. 75, 1063-1065 (1999).
[CrossRef]

Hong, M. H.

Z. B. Wang, B. S. Lukyanchuk, M. H. Hong, Y. Lin, and T. C. Chong, "Energy flow around a small particle investigated by classical Mie theory," Phys. Rev. B 70, 035418 (2004).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Jackson, J. B.

S. J. Oldenburg, G. D. Hale, J. B. Jackson, and N. J. Halas, "Light scattering from dipole and quadrupole nanoshell antennas," Appl. Phys. Lett. 75, 1063-1065 (1999).
[CrossRef]

Johnson, P. B.

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Kerker, M.

Krenn, J. R.

Leitner, A.

Lin, Y.

Z. B. Wang, B. S. Lukyanchuk, M. H. Hong, Y. Lin, and T. C. Chong, "Energy flow around a small particle investigated by classical Mie theory," Phys. Rev. B 70, 035418 (2004).
[CrossRef]

Lukyanchuk, B. S.

Z. B. Wang, B. S. Lukyanchuk, M. H. Hong, Y. Lin, and T. C. Chong, "Energy flow around a small particle investigated by classical Mie theory," Phys. Rev. B 70, 035418 (2004).
[CrossRef]

Mie, G.

G. Mie, "Considerations on the optics of turbid media, especially colloidal metal sols," Ann. Phys. 25, 377-442 (1908).
[CrossRef]

Oldenburg, S. J.

S. J. Oldenburg, G. D. Hale, J. B. Jackson, and N. J. Halas, "Light scattering from dipole and quadrupole nanoshell antennas," Appl. Phys. Lett. 75, 1063-1065 (1999).
[CrossRef]

Pendry, J. B.

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Quinten, M.

Salandrino, A.

N. Engheta, A. Salandrino, and A. Alù, "Circuit elements at optical frequencies: nano-inductors, nano-capacitors and nano-resistors," Phys. Rev. Lett. 95, 095504 (2005).
[CrossRef] [PubMed]

Silveirinha, M. G.

M. G. Silveirinha, A. Alù, and N. Engheta, "Parallel plate metamaterials for cloaking structures," Phys. Rev. E 75, 036603 (2007).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, eds., "Spherical Bessel functions," in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972), Subsection 10.1, pp. 437-442.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Tribelskii, M. I.

M. I. Tribelskii, "Resonant scattering of light by small particles," Sov. Phys. JETP 59, 534-536 (1984).

Wang, Z. B.

Z. B. Wang, B. S. Lukyanchuk, M. H. Hong, Y. Lin, and T. C. Chong, "Energy flow around a small particle investigated by classical Mie theory," Phys. Rev. B 70, 035418 (2004).
[CrossRef]

Zheludev, N. I.

Am. J. Phys.

C. F. Bohren, "How can a particle absorb more than the light incident on it?," Am. J. Phys. 51, 323-327 (1983).
[CrossRef]

Ann. Phys.

G. Mie, "Considerations on the optics of turbid media, especially colloidal metal sols," Ann. Phys. 25, 377-442 (1908).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

S. J. Oldenburg, G. D. Hale, J. B. Jackson, and N. J. Halas, "Light scattering from dipole and quadrupole nanoshell antennas," Appl. Phys. Lett. 75, 1063-1065 (1999).
[CrossRef]

J. Appl. Phys.

A. Alù and N. Engheta, "Polarizabilities and effective parameters for collections of spherical nano-particles formed by pairs of concentric double-negative (DNG), single-negative (SNG) and/or double-positive (DPS) metamaterial layers," J. Appl. Phys. 97, 094310 (2005).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev.

D. Bohm and E. P. Gross, "Theory of plasma oscillations. A. Origin of medium-like behavior," Phys. Rev. 75, 1851-1864 (1949).
[CrossRef]

Phys. Rev. B

Z. B. Wang, B. S. Lukyanchuk, M. H. Hong, Y. Lin, and T. C. Chong, "Energy flow around a small particle investigated by classical Mie theory," Phys. Rev. B 70, 035418 (2004).
[CrossRef]

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Phys. Rev. E

A. Alù and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016623 (2005).
[CrossRef]

M. G. Silveirinha, A. Alù, and N. Engheta, "Parallel plate metamaterials for cloaking structures," Phys. Rev. E 75, 036603 (2007).
[CrossRef]

Phys. Rev. Lett.

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

N. Engheta, A. Salandrino, and A. Alù, "Circuit elements at optical frequencies: nano-inductors, nano-capacitors and nano-resistors," Phys. Rev. Lett. 95, 095504 (2005).
[CrossRef] [PubMed]

Sov. Phys. JETP

M. I. Tribelskii, "Resonant scattering of light by small particles," Sov. Phys. JETP 59, 534-536 (1984).

Other

By "incident on the particle" the problem is considered in terms of geometrical optics.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

M. Abramowitz and I. A. Stegun, eds., "Spherical Bessel functions," in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972), Subsection 10.1, pp. 437-442.

We note that the analysis in expands the whole scattered field in the Taylor approximation, mixing together the different Mie spherical harmonics. Although this procedure eventually ensures numerical convergence, it requires a much higher number of terms in order to predict the correct results near the resonance of the nanoparticle, similar to the cases considered here. This explains the disagreement between our full-wave results and the approximate calculation in . Also contrary to the claim mentioned in that a correct evaluation of the power-flow distribution would require multiple Mie orders, we believe that this is not the case, in the sense that, even if in their Taylor expansion higher-order terms are needed, with a full-wave Mie harmonic expansion one would require one single resonant harmonic to predict the overall power distribution, as has been shown here.

A. Alù and N. Engheta, "Enhanced directivity from sub-wavelength infrared/optical nano-antennas loaded with plasmonic materials or metamaterials," IEEE Trans. Antennas Propag. (to be published).

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Figures (8)

Fig. 1
Fig. 1

Power flow around a spherical nanoparticle with ϵ = 2.223 ϵ 0 , k 0 a = 0.3 , under plane-wave incidence at its first resonance in (a) the E plane and (b) the H plane. Brighter (darker in the grayscale version) arrows correspond to higher values of power density flow.

Fig. 2
Fig. 2

Same as in Fig. 1, but with material loss included. Here, the spherical metallic nanoparticle has permittivity ϵ = ( 2.223 + i 0.2 ) ϵ 0 , k 0 a = 0.3 . Brighter (darker in the grayscale version) arrows correspond to higher values of power density flow.

Fig. 3
Fig. 3

Power flow around a spherical nanoparticle with ϵ = 1.533 ϵ 0 , k 0 a = 0.3 , under plane-wave incidence at its second (i.e., quadrupolar) resonance in (a) the E plane and (b) the H plane. Brighter (darker in the grayscale version) arrows correspond to higher values of power density flow.

Fig. 4
Fig. 4

Amplitude of the perpendicular magnetic field distribution in (a) the E plane and of the electric field distribution on (b) the H plane for the nanoparticle at its quadrupolar resonance, corresponding to the geometry of Fig. 3. Panels (c) and (d) correspond to the octopolar resonance, for the geometry of Figs. 5, 6. Brighter (darker in the grayscale version) colors correspond to higher values of the field amplitude.

Fig. 5
Fig. 5

Power flow around a spherical nanoparticle with ϵ = 1.346 ϵ 0 , k 0 a = 0.3 , under plane-wave incidence at its third (octopolar) resonance in (a) the E plane and (b) the H plane. Brighter (darker in the grayscale version) arrows correspond to higher values of power density flow.

Fig. 6
Fig. 6

Same as in Fig. 5 but for a region of space closer to the nanosphere (here the total computational plane is 6 λ 0 × 6 λ 0 ). (a), the E plane; (b), the H plane. Brighter (darker in the grayscale version) arrows correspond to higher values of power density flow.

Fig. 7
Fig. 7

Same as in Fig. 3 but with small material loss included. Here, the spherical metallic nanoparticle has permittivity ϵ = ( 1.533 + i 5 10 4 ) ϵ 0 , k 0 a = 0.3 . Brighter (darker in the grayscale version) arrows correspond to higher values of power density flow.

Fig. 8
Fig. 8

Same as in Fig. 5 but with material loss included. Here, the spherical metallic nanoparticle has permittivity ϵ = ( 1.346 + i 10 6 ) ϵ 0 , k 0 a = 0.3 . Brighter (darker in the grayscale version) arrows correspond to higher values of power density flow.

Equations (10)

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

E i = E 0 ( n , m a n m T M N n m ( 1 ) + n , m a n m T E M n m ( 1 ) ) ,
H i = i E 0 η 0 ( n , m a n m T E N n m ( 1 ) + n , m a n m T M M n m ( 1 ) ) ,
E s = E 0 ( n , m c n T M a n m T M N n m ( 3 ) + n , m c n T E a n m T E M n m ( 3 ) ) ,
H s = i E 0 η 0 ( n , m c n T E a n m T E N n m ( 3 ) + n , m c n T M a n m T M M n m ( 3 ) ) .
c n T M = ϵ j n ( k a ) [ ( k 0 a ) j n ( k 0 a ) ] ϵ 0 [ ( k a ) j n ( k a ) ] j n ( k 0 a ) ϵ j n ( k a ) [ ( k 0 a ) h n ( k 0 a ) ] ϵ 0 [ ( k a ) j n ( k a ) ] h n ( k 0 a ) ,
c n T E = j n ( k a ) [ ( k 0 a ) j n ( k 0 a ) ] [ ( k a ) j n ( k a ) ] j n ( k 0 a ) j n ( k a ) [ ( k 0 a ) h n ( k 0 a ) ] [ ( k a ) j n ( k a ) ] h n ( k 0 a ) ,
ϵ j n ( k a ) [ ( k 0 a ) y n ( k 0 a ) ] = ϵ 0 [ ( k a ) j n ( k a ) ] y n ( k 0 a ) ( TM ) ,
j n ( k a ) [ ( k 0 a ) y n ( k 0 a ) ] = [ ( k a ) j n ( k a ) ] y n ( k 0 a ) ( TE ) ,
ϵ = n + 1 n ϵ 0 ,
Im [ ϵ ϵ 0 ] ( k 0 a ) 2 n + 1 n + 1 n 2 [ ( 2 n 1 ) ! ! ] 2 .

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