Abstract

We give a geometrical theory of resonances in Maxwell’s equations that generalizes the Mie formulae for spheres to all scattering channels of any dielectric or metallic particle without sharp edges. We show that the electromagnetic response of a particle is given by a set of modes of internal and scattered fields that are coupled pairwise on the surface of the particle and reveal that resonances in nanoparticles and excess noise in macroscopic cavities have the same origin. We give examples of two types of optical resonances: those in which a single pair of internal and scattered modes become strongly aligned in the sense defined in this paper, and those resulting from constructive interference of many pairs of weakly aligned modes, an effect relevant for sensing. This approach calculates resonances for every significant mode of particles, demonstrating that modes can be either bright or dark depending on the incident field. Using this extra mode information we then outline how excitation can be optimized. Finally, we apply this theory to gold particles with shapes often used in experiments, demonstrating effects including a Fano-like resonance.

© 2011 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Graham and R. Goodacre, “Chemical and bioanalytical applications of surface enhanced Raman scattering spectroscopy,” Chem. Soc. Rev. 37, 883–884 (2008).
    [CrossRef] [PubMed]
  2. T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
    [CrossRef] [PubMed]
  3. B. Lukyanchuk, N. Zheludev, S. Maier, N. Halas, P. Nordlander, H. Giessen, and C. Tow Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
    [CrossRef]
  4. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetic induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
    [CrossRef] [PubMed]
  5. J. Schuller, E. Barnard, W. Cai, Y. C. Jun, J. White, and M. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
    [CrossRef] [PubMed]
  6. Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12, 60–69 (2009).
    [CrossRef]
  7. J. Pendry, D. Schuring, and D. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [CrossRef] [PubMed]
  8. G. Roll and G. Schweiger, “Geometrical optics model of Mie resonances,” J. Opt. Soc. Am. A 17, 1301–1311 (2000).
    [CrossRef]
  9. G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. 330, 377–445 (1908).
    [CrossRef]
  10. Y. Han and Z. Wu, “Scattering of a spheroidal particle illuminated by a gaussian beam,” Appl. Opt. 40, 2501–2509 (2001).
    [CrossRef]
  11. M. I. Mishchenko, J. H. Hovernier, and L. D. Travis, eds., Light scattering by nonspherical particles: Theory, Measurements and Applications (Academic Press, 2000).
  12. C. Jordan, “Essai sur la géométrie à n dimension,” Bul. Soc. Math. France 3, 103–174 (1875).
  13. A. Knyazev, A. Jujusnashvili, and M. Argentati, “Angles between Infinite Dimensional Subspaces with Applications to the Rayleigh-Ritz and Alternating Projectors Methods,” J. Func. Anal. 259, 1323–1345 (2010).
    [CrossRef]
  14. K. Holms, B. Hourahine, and F. Papoff, “Calculation of internal and scattered fields of axisymmetric nanoparticles at any point in space,” J. Opt. A, Pure Appl. Opt. 11, 054009 (2009).
    [CrossRef]
  15. A. Aydin and A. Hizal, “On the completeness of the spherical vector wave functions,” J. Math. Anal. Appl. 117, 428–440 (1986).
    [CrossRef]
  16. A. Doicu, T. Wriedt, and Y. Eremin, Light Scattering by Systems of Particles (Springer, 2006).
    [CrossRef]
  17. Complete sets of functions exist on surfaces (Lyapunov surfaces) that are mathematically characterized by three conditions: the normal is well defined at every point; the angle between the normals at any two points on the surface is bounded from above by a function of the distance between these points; all the lines parallel to a normal at an arbitrary point on the surface intercept only once the patches of surface contained in balls centered at the point and smaller than a critical value [18].
  18. V. S. Vladimirov, Equations of mathematical physics (MIR, Moscow, 1984).
  19. A. Doicu and T. Wriedt, “Calculation of the T matrix in the null-field method with discrete sources,” J. Opt. Soc. Am. A 16, 2539–2544 (1999).
    [CrossRef]
  20. A. Doicu and T. Wriedt, “Extended boundary condition method with multipole sources located in the complex plane,” Opt. Commun. 139, 85–91 (1997).
    [CrossRef]
  21. T. Rother, M. Kahnert, A. Doicu, and J. Wauer, “Surface Green’s Function of the Helmholtz Equation in Spherical Coordinates,” Prog. Electromagn. Res. 38, 47–95 (2002).
    [CrossRef]
  22. A. Knyazev and M. Argentati, “Principal angles between subspaces in an A-based scalar product: algorithms and perturbation estimates,” SIAM J. Sci. Comput. 23, 2008–2040 (2002).
    [CrossRef]
  23. E. Hannan, “The general theory of canonical correlation and its relation to functional analysis,” J. Aust. Math. Soc. 2, 229–242 (1961/1962).
    [CrossRef]
  24. B. Hourahine, K. Holms, and F. Papoff, “Accurate near and far field determination for non spherical particles from Mie-type theory,” submitted (2011).
  25. The angles relevant to this work are the point angles 0 < ? < ?/2 of the infinite dimensional theory [13], together with the corresponding subspaces (principal modes) and their orthogonal complements (bi-orthogonal modes).
  26. G. New, “The origin of excess noise,” J. Mod. Opt. 42, 799–810 (1995).
    [CrossRef]
  27. W. J. Firth and A. Yao, “Giant excess noise and transient gain in misaligned laser cavities,” Phys. Rev. Lett. 95, 073903 (2005).
    [CrossRef] [PubMed]
  28. F. Papoff, G. D’Alessandro, and G.-L. Oppo, “State dependent pseudoresonances and excess noise,” Phys. Rev. Lett. 100, 123905 (2008).
    [CrossRef] [PubMed]
  29. M. I. Tribelsky and B. S. Lukyanchuk, “Anomalous light scattering by small particles,” Phys Rev. Lett. 97, 263902 (2006).
    [CrossRef]
  30. P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
    [CrossRef] [PubMed]
  31. P. G. Etchegoin, E. C. Le Ru, and M. Meyer, Erratum: “An analytic model for the optical properties of gold”. J. Chem. Phys. 127, 189901 (2007).
    [CrossRef]
  32. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
    [CrossRef]
  33. Evaluation of the data for the disc at 81 wavelengths required 418 seconds using the same machine as described in the caption of Table 1.
  34. J. Aizpurua, P. Hanarp, D. Sutherland, M. Kall, G. Bryant, and F. J. G. de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 057401 (2003).
    [CrossRef] [PubMed]
  35. H. Okamoto and K. Imura, “Near field optical imaging of enhanced electric fields and plasmon waves in metal nanostructures,” Prog. Surf. Sci. 84, 199–229 (2009).
    [CrossRef]

2010 (3)

B. Lukyanchuk, N. Zheludev, S. Maier, N. Halas, P. Nordlander, H. Giessen, and C. Tow Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
[CrossRef]

J. Schuller, E. Barnard, W. Cai, Y. C. Jun, J. White, and M. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef] [PubMed]

A. Knyazev, A. Jujusnashvili, and M. Argentati, “Angles between Infinite Dimensional Subspaces with Applications to the Rayleigh-Ritz and Alternating Projectors Methods,” J. Func. Anal. 259, 1323–1345 (2010).
[CrossRef]

2009 (4)

K. Holms, B. Hourahine, and F. Papoff, “Calculation of internal and scattered fields of axisymmetric nanoparticles at any point in space,” J. Opt. A, Pure Appl. Opt. 11, 054009 (2009).
[CrossRef]

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12, 60–69 (2009).
[CrossRef]

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetic induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

H. Okamoto and K. Imura, “Near field optical imaging of enhanced electric fields and plasmon waves in metal nanostructures,” Prog. Surf. Sci. 84, 199–229 (2009).
[CrossRef]

2008 (2)

F. Papoff, G. D’Alessandro, and G.-L. Oppo, “State dependent pseudoresonances and excess noise,” Phys. Rev. Lett. 100, 123905 (2008).
[CrossRef] [PubMed]

D. Graham and R. Goodacre, “Chemical and bioanalytical applications of surface enhanced Raman scattering spectroscopy,” Chem. Soc. Rev. 37, 883–884 (2008).
[CrossRef] [PubMed]

2007 (1)

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, Erratum: “An analytic model for the optical properties of gold”. J. Chem. Phys. 127, 189901 (2007).
[CrossRef]

2006 (4)

M. I. Tribelsky and B. S. Lukyanchuk, “Anomalous light scattering by small particles,” Phys Rev. Lett. 97, 263902 (2006).
[CrossRef]

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
[CrossRef] [PubMed]

J. Pendry, D. Schuring, and D. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

2005 (1)

W. J. Firth and A. Yao, “Giant excess noise and transient gain in misaligned laser cavities,” Phys. Rev. Lett. 95, 073903 (2005).
[CrossRef] [PubMed]

2003 (1)

J. Aizpurua, P. Hanarp, D. Sutherland, M. Kall, G. Bryant, and F. J. G. de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 057401 (2003).
[CrossRef] [PubMed]

2002 (2)

T. Rother, M. Kahnert, A. Doicu, and J. Wauer, “Surface Green’s Function of the Helmholtz Equation in Spherical Coordinates,” Prog. Electromagn. Res. 38, 47–95 (2002).
[CrossRef]

A. Knyazev and M. Argentati, “Principal angles between subspaces in an A-based scalar product: algorithms and perturbation estimates,” SIAM J. Sci. Comput. 23, 2008–2040 (2002).
[CrossRef]

2001 (1)

2000 (1)

1999 (1)

1997 (1)

A. Doicu and T. Wriedt, “Extended boundary condition method with multipole sources located in the complex plane,” Opt. Commun. 139, 85–91 (1997).
[CrossRef]

1995 (1)

G. New, “The origin of excess noise,” J. Mod. Opt. 42, 799–810 (1995).
[CrossRef]

1986 (1)

A. Aydin and A. Hizal, “On the completeness of the spherical vector wave functions,” J. Math. Anal. Appl. 117, 428–440 (1986).
[CrossRef]

1908 (1)

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

1875 (1)

C. Jordan, “Essai sur la géométrie à n dimension,” Bul. Soc. Math. France 3, 103–174 (1875).

Aizpurua, J.

J. Aizpurua, P. Hanarp, D. Sutherland, M. Kall, G. Bryant, and F. J. G. de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 057401 (2003).
[CrossRef] [PubMed]

Aoki, T.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
[CrossRef] [PubMed]

Argentati, M.

A. Knyazev, A. Jujusnashvili, and M. Argentati, “Angles between Infinite Dimensional Subspaces with Applications to the Rayleigh-Ritz and Alternating Projectors Methods,” J. Func. Anal. 259, 1323–1345 (2010).
[CrossRef]

A. Knyazev and M. Argentati, “Principal angles between subspaces in an A-based scalar product: algorithms and perturbation estimates,” SIAM J. Sci. Comput. 23, 2008–2040 (2002).
[CrossRef]

Aydin, A.

A. Aydin and A. Hizal, “On the completeness of the spherical vector wave functions,” J. Math. Anal. Appl. 117, 428–440 (1986).
[CrossRef]

Barnard, E.

J. Schuller, E. Barnard, W. Cai, Y. C. Jun, J. White, and M. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef] [PubMed]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
[CrossRef]

Bowen, W. P.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
[CrossRef] [PubMed]

Brongersma, M.

J. Schuller, E. Barnard, W. Cai, Y. C. Jun, J. White, and M. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef] [PubMed]

Bryant, G.

J. Aizpurua, P. Hanarp, D. Sutherland, M. Kall, G. Bryant, and F. J. G. de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 057401 (2003).
[CrossRef] [PubMed]

Cai, W.

J. Schuller, E. Barnard, W. Cai, Y. C. Jun, J. White, and M. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef] [PubMed]

D’Alessandro, G.

F. Papoff, G. D’Alessandro, and G.-L. Oppo, “State dependent pseudoresonances and excess noise,” Phys. Rev. Lett. 100, 123905 (2008).
[CrossRef] [PubMed]

Dayan, B.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
[CrossRef] [PubMed]

de Abajo, F. J. G.

J. Aizpurua, P. Hanarp, D. Sutherland, M. Kall, G. Bryant, and F. J. G. de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 057401 (2003).
[CrossRef] [PubMed]

Doicu, A.

T. Rother, M. Kahnert, A. Doicu, and J. Wauer, “Surface Green’s Function of the Helmholtz Equation in Spherical Coordinates,” Prog. Electromagn. Res. 38, 47–95 (2002).
[CrossRef]

A. Doicu and T. Wriedt, “Calculation of the T matrix in the null-field method with discrete sources,” J. Opt. Soc. Am. A 16, 2539–2544 (1999).
[CrossRef]

A. Doicu and T. Wriedt, “Extended boundary condition method with multipole sources located in the complex plane,” Opt. Commun. 139, 85–91 (1997).
[CrossRef]

A. Doicu, T. Wriedt, and Y. Eremin, Light Scattering by Systems of Particles (Springer, 2006).
[CrossRef]

Eremin, Y.

A. Doicu, T. Wriedt, and Y. Eremin, Light Scattering by Systems of Particles (Springer, 2006).
[CrossRef]

Etchegoin, P. G.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, Erratum: “An analytic model for the optical properties of gold”. J. Chem. Phys. 127, 189901 (2007).
[CrossRef]

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

Firth, W. J.

W. J. Firth and A. Yao, “Giant excess noise and transient gain in misaligned laser cavities,” Phys. Rev. Lett. 95, 073903 (2005).
[CrossRef] [PubMed]

Fleischhauer, M.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetic induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

Giessen, H.

B. Lukyanchuk, N. Zheludev, S. Maier, N. Halas, P. Nordlander, H. Giessen, and C. Tow Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
[CrossRef]

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetic induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

Goodacre, R.

D. Graham and R. Goodacre, “Chemical and bioanalytical applications of surface enhanced Raman scattering spectroscopy,” Chem. Soc. Rev. 37, 883–884 (2008).
[CrossRef] [PubMed]

Graham, D.

D. Graham and R. Goodacre, “Chemical and bioanalytical applications of surface enhanced Raman scattering spectroscopy,” Chem. Soc. Rev. 37, 883–884 (2008).
[CrossRef] [PubMed]

Halas, N.

B. Lukyanchuk, N. Zheludev, S. Maier, N. Halas, P. Nordlander, H. Giessen, and C. Tow Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
[CrossRef]

Han, Y.

Hanarp, P.

J. Aizpurua, P. Hanarp, D. Sutherland, M. Kall, G. Bryant, and F. J. G. de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 057401 (2003).
[CrossRef] [PubMed]

Hannan, E.

E. Hannan, “The general theory of canonical correlation and its relation to functional analysis,” J. Aust. Math. Soc. 2, 229–242 (1961/1962).
[CrossRef]

Hizal, A.

A. Aydin and A. Hizal, “On the completeness of the spherical vector wave functions,” J. Math. Anal. Appl. 117, 428–440 (1986).
[CrossRef]

Holms, K.

K. Holms, B. Hourahine, and F. Papoff, “Calculation of internal and scattered fields of axisymmetric nanoparticles at any point in space,” J. Opt. A, Pure Appl. Opt. 11, 054009 (2009).
[CrossRef]

B. Hourahine, K. Holms, and F. Papoff, “Accurate near and far field determination for non spherical particles from Mie-type theory,” submitted (2011).

Hourahine, B.

K. Holms, B. Hourahine, and F. Papoff, “Calculation of internal and scattered fields of axisymmetric nanoparticles at any point in space,” J. Opt. A, Pure Appl. Opt. 11, 054009 (2009).
[CrossRef]

B. Hourahine, K. Holms, and F. Papoff, “Accurate near and far field determination for non spherical particles from Mie-type theory,” submitted (2011).

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
[CrossRef]

Imura, K.

H. Okamoto and K. Imura, “Near field optical imaging of enhanced electric fields and plasmon waves in metal nanostructures,” Prog. Surf. Sci. 84, 199–229 (2009).
[CrossRef]

Jordan, C.

C. Jordan, “Essai sur la géométrie à n dimension,” Bul. Soc. Math. France 3, 103–174 (1875).

Jujusnashvili, A.

A. Knyazev, A. Jujusnashvili, and M. Argentati, “Angles between Infinite Dimensional Subspaces with Applications to the Rayleigh-Ritz and Alternating Projectors Methods,” J. Func. Anal. 259, 1323–1345 (2010).
[CrossRef]

Jun, Y. C.

J. Schuller, E. Barnard, W. Cai, Y. C. Jun, J. White, and M. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef] [PubMed]

Kahnert, M.

T. Rother, M. Kahnert, A. Doicu, and J. Wauer, “Surface Green’s Function of the Helmholtz Equation in Spherical Coordinates,” Prog. Electromagn. Res. 38, 47–95 (2002).
[CrossRef]

Kall, M.

J. Aizpurua, P. Hanarp, D. Sutherland, M. Kall, G. Bryant, and F. J. G. de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 057401 (2003).
[CrossRef] [PubMed]

Kästel, J.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetic induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

Kimble, H. J.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
[CrossRef] [PubMed]

Kippenberg, T. J.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
[CrossRef] [PubMed]

Knyazev, A.

A. Knyazev, A. Jujusnashvili, and M. Argentati, “Angles between Infinite Dimensional Subspaces with Applications to the Rayleigh-Ritz and Alternating Projectors Methods,” J. Func. Anal. 259, 1323–1345 (2010).
[CrossRef]

A. Knyazev and M. Argentati, “Principal angles between subspaces in an A-based scalar product: algorithms and perturbation estimates,” SIAM J. Sci. Comput. 23, 2008–2040 (2002).
[CrossRef]

Langguth, L.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetic induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

Le Ru, E. C.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, Erratum: “An analytic model for the optical properties of gold”. J. Chem. Phys. 127, 189901 (2007).
[CrossRef]

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

Lippens, D.

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12, 60–69 (2009).
[CrossRef]

Liu, N.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetic induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

Lukyanchuk, B.

B. Lukyanchuk, N. Zheludev, S. Maier, N. Halas, P. Nordlander, H. Giessen, and C. Tow Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
[CrossRef]

Lukyanchuk, B. S.

M. I. Tribelsky and B. S. Lukyanchuk, “Anomalous light scattering by small particles,” Phys Rev. Lett. 97, 263902 (2006).
[CrossRef]

Maier, S.

B. Lukyanchuk, N. Zheludev, S. Maier, N. Halas, P. Nordlander, H. Giessen, and C. Tow Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
[CrossRef]

Meyer, M.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, Erratum: “An analytic model for the optical properties of gold”. J. Chem. Phys. 127, 189901 (2007).
[CrossRef]

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

Mie, G.

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

New, G.

G. New, “The origin of excess noise,” J. Mod. Opt. 42, 799–810 (1995).
[CrossRef]

Nordlander, P.

B. Lukyanchuk, N. Zheludev, S. Maier, N. Halas, P. Nordlander, H. Giessen, and C. Tow Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
[CrossRef]

Okamoto, H.

H. Okamoto and K. Imura, “Near field optical imaging of enhanced electric fields and plasmon waves in metal nanostructures,” Prog. Surf. Sci. 84, 199–229 (2009).
[CrossRef]

Oppo, G.-L.

F. Papoff, G. D’Alessandro, and G.-L. Oppo, “State dependent pseudoresonances and excess noise,” Phys. Rev. Lett. 100, 123905 (2008).
[CrossRef] [PubMed]

Papoff, F.

K. Holms, B. Hourahine, and F. Papoff, “Calculation of internal and scattered fields of axisymmetric nanoparticles at any point in space,” J. Opt. A, Pure Appl. Opt. 11, 054009 (2009).
[CrossRef]

F. Papoff, G. D’Alessandro, and G.-L. Oppo, “State dependent pseudoresonances and excess noise,” Phys. Rev. Lett. 100, 123905 (2008).
[CrossRef] [PubMed]

B. Hourahine, K. Holms, and F. Papoff, “Accurate near and far field determination for non spherical particles from Mie-type theory,” submitted (2011).

Parkins, A. S.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
[CrossRef] [PubMed]

Pendry, J.

J. Pendry, D. Schuring, and D. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

Pfau, T.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetic induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

Roll, G.

Rother, T.

T. Rother, M. Kahnert, A. Doicu, and J. Wauer, “Surface Green’s Function of the Helmholtz Equation in Spherical Coordinates,” Prog. Electromagn. Res. 38, 47–95 (2002).
[CrossRef]

Schuller, J.

J. Schuller, E. Barnard, W. Cai, Y. C. Jun, J. White, and M. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef] [PubMed]

Schuring, D.

J. Pendry, D. Schuring, and D. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

Schweiger, G.

Smith, D.

J. Pendry, D. Schuring, and D. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

Sutherland, D.

J. Aizpurua, P. Hanarp, D. Sutherland, M. Kall, G. Bryant, and F. J. G. de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 057401 (2003).
[CrossRef] [PubMed]

Tow Chong, C.

B. Lukyanchuk, N. Zheludev, S. Maier, N. Halas, P. Nordlander, H. Giessen, and C. Tow Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
[CrossRef]

Tribelsky, M. I.

M. I. Tribelsky and B. S. Lukyanchuk, “Anomalous light scattering by small particles,” Phys Rev. Lett. 97, 263902 (2006).
[CrossRef]

Vahala, K. J.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
[CrossRef] [PubMed]

Vladimirov, V. S.

V. S. Vladimirov, Equations of mathematical physics (MIR, Moscow, 1984).

Wauer, J.

T. Rother, M. Kahnert, A. Doicu, and J. Wauer, “Surface Green’s Function of the Helmholtz Equation in Spherical Coordinates,” Prog. Electromagn. Res. 38, 47–95 (2002).
[CrossRef]

Weiss, T.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetic induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

White, J.

J. Schuller, E. Barnard, W. Cai, Y. C. Jun, J. White, and M. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef] [PubMed]

Wilcut, E.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
[CrossRef] [PubMed]

Wriedt, T.

A. Doicu and T. Wriedt, “Calculation of the T matrix in the null-field method with discrete sources,” J. Opt. Soc. Am. A 16, 2539–2544 (1999).
[CrossRef]

A. Doicu and T. Wriedt, “Extended boundary condition method with multipole sources located in the complex plane,” Opt. Commun. 139, 85–91 (1997).
[CrossRef]

A. Doicu, T. Wriedt, and Y. Eremin, Light Scattering by Systems of Particles (Springer, 2006).
[CrossRef]

Wu, Z.

Yao, A.

W. J. Firth and A. Yao, “Giant excess noise and transient gain in misaligned laser cavities,” Phys. Rev. Lett. 95, 073903 (2005).
[CrossRef] [PubMed]

Zhang, F.

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12, 60–69 (2009).
[CrossRef]

Zhao, Q.

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12, 60–69 (2009).
[CrossRef]

Zheludev, N.

B. Lukyanchuk, N. Zheludev, S. Maier, N. Halas, P. Nordlander, H. Giessen, and C. Tow Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
[CrossRef]

Zhou, J.

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12, 60–69 (2009).
[CrossRef]

Ann. Phys. (1)

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

Appl. Opt. (1)

Bul. Soc. Math. France (1)

C. Jordan, “Essai sur la géométrie à n dimension,” Bul. Soc. Math. France 3, 103–174 (1875).

Chem. Soc. Rev. (1)

D. Graham and R. Goodacre, “Chemical and bioanalytical applications of surface enhanced Raman scattering spectroscopy,” Chem. Soc. Rev. 37, 883–884 (2008).
[CrossRef] [PubMed]

J. Aust. Math. Soc. (1)

E. Hannan, “The general theory of canonical correlation and its relation to functional analysis,” J. Aust. Math. Soc. 2, 229–242 (1961/1962).
[CrossRef]

J. Chem. Phys. (2)

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, Erratum: “An analytic model for the optical properties of gold”. J. Chem. Phys. 127, 189901 (2007).
[CrossRef]

J. Func. Anal. (1)

A. Knyazev, A. Jujusnashvili, and M. Argentati, “Angles between Infinite Dimensional Subspaces with Applications to the Rayleigh-Ritz and Alternating Projectors Methods,” J. Func. Anal. 259, 1323–1345 (2010).
[CrossRef]

J. Math. Anal. Appl. (1)

A. Aydin and A. Hizal, “On the completeness of the spherical vector wave functions,” J. Math. Anal. Appl. 117, 428–440 (1986).
[CrossRef]

J. Mod. Opt. (1)

G. New, “The origin of excess noise,” J. Mod. Opt. 42, 799–810 (1995).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

K. Holms, B. Hourahine, and F. Papoff, “Calculation of internal and scattered fields of axisymmetric nanoparticles at any point in space,” J. Opt. A, Pure Appl. Opt. 11, 054009 (2009).
[CrossRef]

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

Mater. Today (1)

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12, 60–69 (2009).
[CrossRef]

Nat. Mater. (3)

B. Lukyanchuk, N. Zheludev, S. Maier, N. Halas, P. Nordlander, H. Giessen, and C. Tow Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
[CrossRef]

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetic induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009).
[CrossRef] [PubMed]

J. Schuller, E. Barnard, W. Cai, Y. C. Jun, J. White, and M. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef] [PubMed]

Nature (1)

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
[CrossRef] [PubMed]

Opt. Commun. (1)

A. Doicu and T. Wriedt, “Extended boundary condition method with multipole sources located in the complex plane,” Opt. Commun. 139, 85–91 (1997).
[CrossRef]

Phys Rev. Lett. (1)

M. I. Tribelsky and B. S. Lukyanchuk, “Anomalous light scattering by small particles,” Phys Rev. Lett. 97, 263902 (2006).
[CrossRef]

Phys. Rev. Lett. (3)

W. J. Firth and A. Yao, “Giant excess noise and transient gain in misaligned laser cavities,” Phys. Rev. Lett. 95, 073903 (2005).
[CrossRef] [PubMed]

F. Papoff, G. D’Alessandro, and G.-L. Oppo, “State dependent pseudoresonances and excess noise,” Phys. Rev. Lett. 100, 123905 (2008).
[CrossRef] [PubMed]

J. Aizpurua, P. Hanarp, D. Sutherland, M. Kall, G. Bryant, and F. J. G. de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 057401 (2003).
[CrossRef] [PubMed]

Prog. Electromagn. Res. (1)

T. Rother, M. Kahnert, A. Doicu, and J. Wauer, “Surface Green’s Function of the Helmholtz Equation in Spherical Coordinates,” Prog. Electromagn. Res. 38, 47–95 (2002).
[CrossRef]

Prog. Surf. Sci. (1)

H. Okamoto and K. Imura, “Near field optical imaging of enhanced electric fields and plasmon waves in metal nanostructures,” Prog. Surf. Sci. 84, 199–229 (2009).
[CrossRef]

Science (1)

J. Pendry, D. Schuring, and D. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

SIAM J. Sci. Comput. (1)

A. Knyazev and M. Argentati, “Principal angles between subspaces in an A-based scalar product: algorithms and perturbation estimates,” SIAM J. Sci. Comput. 23, 2008–2040 (2002).
[CrossRef]

Other (8)

A. Doicu, T. Wriedt, and Y. Eremin, Light Scattering by Systems of Particles (Springer, 2006).
[CrossRef]

Complete sets of functions exist on surfaces (Lyapunov surfaces) that are mathematically characterized by three conditions: the normal is well defined at every point; the angle between the normals at any two points on the surface is bounded from above by a function of the distance between these points; all the lines parallel to a normal at an arbitrary point on the surface intercept only once the patches of surface contained in balls centered at the point and smaller than a critical value [18].

V. S. Vladimirov, Equations of mathematical physics (MIR, Moscow, 1984).

M. I. Mishchenko, J. H. Hovernier, and L. D. Travis, eds., Light scattering by nonspherical particles: Theory, Measurements and Applications (Academic Press, 2000).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
[CrossRef]

Evaluation of the data for the disc at 81 wavelengths required 418 seconds using the same machine as described in the caption of Table 1.

B. Hourahine, K. Holms, and F. Papoff, “Accurate near and far field determination for non spherical particles from Mie-type theory,” submitted (2011).

The angles relevant to this work are the point angles 0 < ? < ?/2 of the infinite dimensional theory [13], together with the corresponding subspaces (principal modes) and their orthogonal complements (bi-orthogonal modes).

Supplementary Material (1)

» Media 1: AVI (284 KB)     

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

(Color online) Comparison between this theory (points) and exact Mie results (lines) for gold spheres of radius 10, 25, 50 and 400 nm. Here the refractive index of the external medium is n = 1.3

Fig. 2
Fig. 2

(Color online) Calculated modes of the gold nanodisc. a) Scattering, absorption and extinction efficiencies, showing the strong resonance at 613 nm. b) Differential scattering cross section (DSCS) as a function of wavelength, illustrating the dipole nature of this particular resonance. c) and d) The induced fields of the bright resonance and the most strongly aligned dark mode, both at 613 nm.

Fig. 3
Fig. 3

(Color online) Principal angle landscapes: the order of the mode pairs according to their principal cosines is on the x-axis, the wavelength of the incident field is on the y-axis and sin−1(ξ), the largest possible values of | a n i / s | for |f0| = 1, is on the z-axis. Landscapes are overlaid with traces shaded according to: a) the intrinsic mode fluxes Φ n s, normalized for each wavelength to the range [0,1], b) the amplitudes | a n s | (dependent on incident field), c) the mode fluxes | a n s | 2 Φ n s again normalized to [0, 1]. d)–f) as a)–c) but for internal modes. Discontinuities in the traces are due to crossing between different modes which occur when the values of their principal cosines become the same, this is visible since the ordering of the mode indices then change. a) and d) show that most pairs either do not transport energy or only show strong absorbance, except the one pair that is resonant at 613 nm. c) and f) show that this one resonant pair accounts for most of the energy absorbed and transported into the far region for this incident field. A resonance in the absorbing pairs is visible at short wavelengths (at ∼ 525 nm) in d), but not in e) and f) showing that those modes are not able to coupled to the particular illuminating axial incident field, but would be strongly absorbing if excited by other fields.

Fig. 4
Fig. 4

Rounded nanorod when illuminated axially. a) (Multimedia online) Scattering, absorption and extinction efficiencies showing the 205 nm mode and weaker absorption peak at 486 nm. Excitation of other resonances for different angles of incidence is shown online. b) (Color online) DSCS showing the strong forward scattering of light by the particle.

Fig. 5
Fig. 5

(Color online) Principal angle landscapes as in Fig. 3 for the the rounded nanorod illuminated axially shown in Fig. 4. b) and c) show that the peak at 205 nm is a multimode resonance due to the excitation amplitudes of several principal modes that contribute similarly to the radiative flow, although two are also absorbing modes. The weak absorption peak at 515 nm is also due to a group of internal modes, which become more strongly aligned at around this resonance. In these pairs the coupling of internal and scattering modes is different and the far field is dominated by the radiative mode with the weakest alignment, as shown in c).

Fig. 6
Fig. 6

(Color online) The particle from Fig. 4, but illuminated from the side with an incident light polarization of 45° with respect to its long axis. a) Far field scattering efficiencies showing the presence of both a broad feature similar to Fig. 4 at around 200–450 nm and also a sharp resonance at 676 nm, the effect of changing the incident angle on these efficiencies is shown in Media 1. Note that the scattering efficiency shows a strongly asymmetric Fano-like resonance that is sharper than the resonance in the extinction spectrum. b) The DSCS for equatorial illumination. c) and d) Near field for the 480 nm rod, shown at the broad feature (207 nm) and at the “waveguide” Mie-like mode at 676 nm.

Fig. 7
Fig. 7

(Color online) Principal angle landscapes as in Fig. 3 for the the rounded nanorod illuminated from the side as in Fig. 6. The sharp mode is again a single well aligned principal pair, the best aligned of three (and the only one excited by this particular field). c) Shows that the energy is transported into the far field by the resonant mode and by a weakly aligned, non-resonant, scattering mode. These modes also interfere to give the the total scattering cross section and produce the characteristic asymmetric and sharpened Fano-like resonance shown in the scattering efficiencies of Fig. 6(a).

Tables (1)

Tables Icon

Table 1 Total Calculation Time Using this Theory for the Spectra Shown in Fig. 1, Evaluated at 81 Wavelengths Using an Intel Core 2 Duo 2.13 GHz System

Equations (12)

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

f 0 = f i f s ,
[ ϒ ˜ ϒ ˜ ϒ ˜ Σ ˜ Σ ˜ ϒ ˜ Σ ˜ Σ ˜ ] [ a ˜ i a ˜ s ] = [ ϒ ˜ f 0 Σ ˜ f 0 ] ,
ϒ ˜ = U i Q i ,
Σ ˜ = U s Q s ,
U i U s = V i C V s ,
[ V i Q i 1 0 0 V s Q s 1 ] [ ϒ ˜ ϒ ˜ ϒ ˜ Σ ˜ Σ ˜ ϒ ˜ Σ ˜ Σ ˜ ] [ Q i 1 V i 0 0 Q s 1 V s ] = [ 1 C C 1 ] ,
[ 1 C C 1 ] [ a i a s ] = [ ϒ f 0 Σ f 0 ] ,
i n s n = cos ( ξ n ) .
a n i = i n cos ( ξ n ) s n sin 2 ( ξ n ) f 0 = i n i n i n f 0 ,
a n s = s n cos ( ξ n ) i n sin 2 ( ξ n ) f 0 = s n s n s n f 0 .
F s / i ( x ) = G S ( x , s ) f 0 ( s ) = ( 𝒯 i ( x ) I n ( x ) i n ( s ) i n i n 𝒯 s ( x ) S n ( x ) s n ( s ) s n s n ) f 0 ( s ) ,
Φ n s / i = s Re ( n ^ s / i E n s / i × H n s / i * ) d s ,

Metrics