## Geometrical Mie theory for resonances in nanoparticles of any shape |

Optics Express, Vol. 19, Issue 22, pp. 21432-21444 (2011)

http://dx.doi.org/10.1364/OE.19.021432

Acrobat PDF (7099 KB)

### 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

## 1. Introduction

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]

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]

*a priori*choice of the property which being monitored to determine the resonance. For example a resonance associated with a strong surface field, that itself is not efficient at transporting energy to infinity, would not appear as an obvious feature in any far field efficiency spectra used to define resonance; nevertheless such resonances can be extremely important in near field applications or through interference with other channels which themselves are able to transport energy into the far field.

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]

## 2. Theory

*E,H*are continuous on passing through the particle boundaries, and the energy scattered by a particle flows towards infinity. The interaction of the particle with an incident field is determined by finding appropriate solutions of the Maxwell’s equations in the internal and the external media that satisfy these boundary conditions. We use [14

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]

*F*= [

*E*,

*H*]

*for electromagnetic fields. Their projections,*

^{T}*f*, onto the boundary of the particle are surface fields each with four components, two electric and two magnetic, that form a space ℋ where scalar products are defined in terms of overlap integrals on the surface of the particle,

*j*labels the components,

*f*

^{*}is the complex conjugate of

*f*and we sum over repeated indexes. In this formalism the boundary conditions become which has a simple geometrical meaning in ℋ : the projection,

*f*

^{0}, of the incident field,

*F*

^{0}(

*x*), onto the surface is equal to the difference between the projections of the internal and scattered fields,

*f*and

^{i}*f*. This suggests that an incident field with small tangent components can excite large internal and scattered surface fields provided that these two fields closely match. This happens when the “angle” between these two fields, and therefore their difference, is small. The angles in question can be rigorously defined as the angles between standing and outgoing waves that are solutions of the Maxwell’s equations for the internal and external media respectively, and that form two subspaces of ℋ. For each particle, these angles and the associated waves characterize completely the particle’s electromagnetic response, which can be determined with arbitrary precision from

^{s}*any*complete set of solutions of the Maxwell equations for the internal and external media.

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].

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]

*f*with four components, and completeness in the space ℋ of the surface fields

*f*is provided by the union of internal and scattered fields and not by either the scattered or the internal field separately. This point is illustrated by the spherical particles considered in Mie theory, where both internal and scattered modes are necessary to form a complete basis.

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]

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]

*N*, we achieve this through the matrix decomposition where

*Q*,

^{i}*Q*are invertible matrices that can be found through SVD or QR decomposition [22

^{s}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]

*U*,

^{i}*U*are unitary matrices whose columns are the orthogonal internal and scattering modes respectively. Scalar products between internal and scattering modes form a matrix with decomposition where

^{s}*C*a diagonal matrix with positive elements, and

*V*,

^{i}*V*are unitary matrices acting on the internal and scattered fields, respectively. These identities enable us to simplify the Gram matrix through the transformation which leads to the matrix equation On the left hand side of Eq. (7), 1 is the identity matrix and ϒ =

^{s}*U*and Σ =

^{i}V^{i}*U*are matrices whose columns are formed by the so called principal internal and scattering modes {

^{s}V^{s}*i*} and {

_{n}*s*}, which are one of the main tools in this theory.

_{n}**a**

*,*

^{i}**a**

*are the coefficients of the principal modes in the field’s expansion. The most important part of our theory is that, because matrix*

^{s}*C*is diagonal, principal modes are coupled pairwise, i.e., each mode is orthogonal to all but at most one function in the other space. This is the essential feature of the multipoles used in Mie’s theory for spheres. The positive diagonal elements of

*C*define the principal angles,

*ξ*, between

_{n}*s*and

_{n}*i*as follows The terms on the right-hand side of Eq. (8) are the principal cosines [12]: cos(

_{n}*ξ*) and sin (

_{n}*ξ*) are the statistical correlation [23

_{n}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]

*s*and

_{n}*i*.

_{n}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]

*ξ*are invariant under unitary transformation of the multipoles and they completely characterize the geometry of the subspaces of the internal and scattered solutions in ℋ. This geometry is induced by the particular scattering particle through the surface integrals of the scalar products; its relevance to scattering and resonances has not been previously realized. The importance of the principal cosines is twofold: Theoretically they provide analytic equations for the coefficients of the internal and scattered principal modes, generalizing the Mie formulae and clarifying the nature of all scattering channels of a particle. Numerically, they allow us to reduce large matrices to their sub-blocks and eliminate the need for numerical inversion to determination of the mode coefficients. For spherical particles, each pair of modes corresponds to a pair of electric or magnetic multipoles of Mie theory. For non-spherical particles, principal modes are instead combinations of different multipoles (although in some cases there can be dominant contributions from a specific multipole).

*i*′

*=*

_{n}*i*– cos(

_{n}*ξ*)

_{n}*s*,

_{n}*s*′

*=*

_{n}*s*– cos(

_{n}*ξ*)

_{n}*i*are bi-orthogonal to

_{n}*i*,

_{n}*s*(

_{n}*i*′

*·*

_{n}*s*=

_{n}*s*′

*·*

_{n}*i*= 0) with

_{n}*i*′

*·*

_{n}*i*=

_{n}*s*′

*·*

_{n}*s*= sin

_{n}^{2}(

*ξ*). Both the principal or the bi-orthogonal modes fully specify the response of the particle at any point outside and inside the particle. This is shown by recasting the expansions of internal and scattered field as where

_{n}*G*(

_{S}*x*,

*s*) is the surface Green’s function [14

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]

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]

*(*

^{i}*x*) (𝒯

*(*

^{s}*x*)) is 1 inside (outside) the particle and null elsewhere. In practice, because the principal modes are combinations of known solutions of the Maxwell’s equations, propagation of the fields away from the surface (

*I*(

*x*) and

*S*(

*x*)) is performed for Eq. (11) by evaluation of Bessel or Hankel functions and vector spherical harmonics (all at a very low computational cost). Eq. (11) shows that the convergence of principal modes and principal angles as

*N*→ ∞ is a consequence of the convergence of the surface Green’s function [21

**38**, 47–95 (2002). [CrossRef]

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

*N*→ ∞, even if

*θ*,

_{n}*i*,

_{n}*s*change [25].

_{n}*f*

^{0}onto non-orthogonal vectors,

*i*and

_{n}*s*, while sin (

_{n}*ξ*) is defined as the Peterman factor [26

_{n}26. G. New, “The origin of excess noise,” J. Mod. Opt. **42**, 799–810 (1995). [CrossRef]

*ξ*) << 1 is at the origin of large surface fields in nanoparticles as well as large transient gain and excess noise [27

_{n}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]

*ξ*), the coefficient

_{n}*a*,

^{i}*a*are of the same order, but this is not the case for weakly aligned pairs, which can have qualitatively different absorption and differential scattering cross sections (DSCS). Moreover, Eqs. (9) and (11) show that modes can be “dark” for

^{s}*specific*incident fields but couple well to other incident fields.

*n̂*is the outward (inward) pointing normal to the surface for scattered (internal) modes and

^{s/i}*ξ*change with the frequency of incident light. Internal and scattered coefficients diverge when the denominators of Eqs. (9) and (10) vanish. This happens when a pair of normalized internal and scattering modes are parallel. For a sphere the angular dependence of internal and scattered modes can be factored out and the condition

_{n}*i*=

_{n}*s*can be recast in terms of the amplitude of the electric and magnetic components giving the usual Mie resonance condition, which can be interpreted geometrically in terms of alignment between internal and scattered modes. For spheres, the condition

_{n}*i*=

_{n}*s*occurs at complex wavelengths; for real wavelengths, resonances correspond to minima of the principal angles. This is also generally true for any smooth particle because the linear independence and completeness of the principal modes makes perfect alignment impossible. So, as with spherical particles [29

_{n}29. M. I. Tribelsky and B. S. Lukyanchuk, “Anomalous light scattering by small particles,” Phys Rev. Lett. **97**, 263902 (2006). [CrossRef]

*ξ*≠ 0) of pairs in ℋ, which are also minima of the eigenvalues of the hermitian operator in Eq. (7).

_{n}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]

## 3. Numerical validation

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]

**38**, 47–95 (2002). [CrossRef]

*N*→ ∞. We calculate

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

^{−2},4] times the wavelength of light. The timings for these calculations are shown in Table 1.

## 4. Resonances in gold nanorods and nanodiscs

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]

^{−1}(

*ξ*), i.e. the largest value of Eqs. (9) and (10) for |

*f*

^{0}| = 1; while the shading of the traces overlaid on top show, for each wavelength, the values of the intrinsic mode fluxes

*and Φ*

^{i}*, which has a resonance at 613 nm. Most of the other mode pairs are strongly absorbing and not able to effectively scatter energy away from the particle. However, one poorly aligned mode pair is capable of strongly radiating if excited, but very weakly absorbs. We also clearly identify a subset of absorbing pairs that become resonant at short wavelengths around 525 nm.*

^{s}*specific*axial incident field overlaid on top. This illustrates the

*excitation paths*of the modes as the wavelength changes. We can see that the short-wavelength absorbing resonance around 525 nm is not effectively excited by the specific incident field used; similarly the weakly aligned mode pair which does not pass through a resonance in this range and is capable of strong scattering but weak absorption shows excitation, but only of its internal mode. In contrast the the internal and scattered amplitudes of the pair which reaches resonance at 613 nm is strongly excited by this field.

*ξ*) at the resonance. The internal field also contains a second more weakly aligned excited mode, where its counterpart in the scattering field is not excited. This mode does not absorb much energy on its own, as shown in Fig. 3(f), but affects the surface current, which can be found using these fields and the Ohm equation. The internal and scattered near field of the resonant pair, an electric dipole, is shown in Fig. 2(c). Other strongly aligned mode pairs are not excited, i.e. are dark: this is because they rapidly vary at the surface as in Fig. 2(d), so do not couple to the smoothly varying incident field. The appreciable asymmetry of the absorption and scattering efficiencies are explained by the asymmetry in the principal cosine of the resonant mode as a function of wavelength. For non axial incident light, the main peak in Fig. 2(a) becomes smaller, due to a weaker coupling with this resonant mode (with a corresponding decrease in the amplitudes of Figs. 3(b) and 3(e).

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]

## 5. Conclusion

## References and links

1. | D. Graham and R. Goodacre, “Chemical and bioanalytical applications of surface enhanced Raman scattering spectroscopy,” Chem. Soc. Rev. |

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 |

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. |

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. |

5. | J. Schuller, E. Barnard, W. Cai, Y. C. Jun, J. White, and M. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. |

6. | Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today |

7. | J. Pendry, D. Schuring, and D. Smith, “Controlling electromagnetic fields,” Science |

8. | G. Roll and G. Schweiger, “Geometrical optics model of Mie resonances,” J. Opt. Soc. Am. A |

9. | G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. |

10. | Y. Han and Z. Wu, “Scattering of a spheroidal particle illuminated by a gaussian beam,” Appl. Opt. |

11. | M. I. Mishchenko, J. H. Hovernier, and L. D. Travis, eds., |

12. | C. Jordan, “Essai sur la géométrie à n dimension,” Bul. Soc. Math. France |

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. |

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. |

15. | A. Aydin and A. Hizal, “On the completeness of the spherical vector wave functions,” J. Math. Anal. Appl. |

16. | A. Doicu, T. Wriedt, and Y. Eremin, |

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, |

19. | A. Doicu and T. Wriedt, “Calculation of the T matrix in the null-field method with discrete sources,” J. Opt. Soc. Am. A |

20. | A. Doicu and T. Wriedt, “Extended boundary condition method with multipole sources located in the complex plane,” Opt. Commun. |

21. | T. Rother, M. Kahnert, A. Doicu, and J. Wauer, “Surface Green’s Function of the Helmholtz Equation in Spherical Coordinates,” Prog. Electromagn. Res. |

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. | E. Hannan, “The general theory of canonical correlation and its relation to functional analysis,” J. Aust. Math. Soc. |

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 < |

26. | G. New, “The origin of excess noise,” J. Mod. Opt. |

27. | W. J. Firth and A. Yao, “Giant excess noise and transient gain in misaligned laser cavities,” Phys. Rev. Lett. |

28. | F. Papoff, G. D’Alessandro, and G.-L. Oppo, “State dependent pseudoresonances and excess noise,” Phys. Rev. Lett. |

29. | M. I. Tribelsky and B. S. Lukyanchuk, “Anomalous light scattering by small particles,” Phys Rev. Lett. |

30. | P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. |

31. | P. G. Etchegoin, E. C. Le Ru, and M. Meyer, Erratum: “An analytic model for the optical properties of gold”. J. Chem. Phys. |

32. | C. F. Bohren and D. R. Huffman, |

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. |

35. | H. Okamoto and K. Imura, “Near field optical imaging of enhanced electric fields and plasmon waves in metal nanostructures,” Prog. Surf. Sci. |

**OCIS Codes**

(160.3900) Materials : Metals

(290.0290) Scattering : Scattering

(160.4236) Materials : Nanomaterials

(290.5825) Scattering : Scattering theory

**ToC Category:**

Scattering

**History**

Original Manuscript: July 20, 2011

Revised Manuscript: September 23, 2011

Manuscript Accepted: September 28, 2011

Published: October 17, 2011

**Citation**

F. Papoff and B. Hourahine, "Geometrical Mie theory for resonances in nanoparticles of any shape," Opt. Express **19**, 21432-21444 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-21432

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### References

- D. Graham and R. Goodacre, “Chemical and bioanalytical applications of surface enhanced Raman scattering spectroscopy,” Chem. Soc. Rev.37, 883–884 (2008). [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,” Nature443, 671–674 (2006). [CrossRef] [PubMed]
- 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]
- Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today12, 60–69 (2009). [CrossRef]
- J. Pendry, D. Schuring, and D. Smith, “Controlling electromagnetic fields,” Science312, 1780–1782 (2006). [CrossRef] [PubMed]
- G. Roll and G. Schweiger, “Geometrical optics model of Mie resonances,” J. Opt. Soc. Am. A17, 1301–1311 (2000). [CrossRef]
- G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys.330, 377–445 (1908). [CrossRef]
- Y. Han and Z. Wu, “Scattering of a spheroidal particle illuminated by a gaussian beam,” Appl. Opt.40, 2501–2509 (2001). [CrossRef]
- M. I. Mishchenko, J. H. Hovernier, and L. D. Travis, eds., Light scattering by nonspherical particles: Theory, Measurements and Applications (Academic Press, 2000).
- C. Jordan, “Essai sur la géométrie à n dimension,” Bul. Soc. Math. France3, 103–174 (1875).
- 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]
- 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]
- A. Aydin and A. Hizal, “On the completeness of the spherical vector wave functions,” J. Math. Anal. Appl.117, 428–440 (1986). [CrossRef]
- 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).
- A. Doicu and T. Wriedt, “Calculation of the T matrix in the null-field method with discrete sources,” J. Opt. Soc. Am. A16, 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]
- 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]
- E. Hannan, “The general theory of canonical correlation and its relation to functional analysis,” J. Aust. Math. Soc.2, 229–242 (1961/1962). [CrossRef]
- 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 < ξ < π/2 of the infinite dimensional theory [13], together with the corresponding subspaces (principal modes) and their orthogonal complements (bi-orthogonal modes).
- G. New, “The origin of excess noise,” J. Mod. Opt.42, 799–810 (1995). [CrossRef]
- 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]
- 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]
- 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]
- 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.
- 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]
- 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]

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