Geometrical Mie theory for resonances in nanoparticles of any shape F. Papoff∗ and B. Hourahine SUPA, Department of Physics, University of Strathclyde, Glasgow, G4 0NG, UK ∗[email protected]

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 Optical Society of America OCIS codes: (290.0290) Scattering; (290.5825) Scattering theory; (160.4236) Nanomaterials; (160.3900) Metals.

References and links 1. D. Graham and R. Goodacre, “Chemical and bioanalytical applications of surface enhanced Raman scattering spectroscopy,” Chem. Soc. Rev. 37, 883–884 (2008). 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). 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). 4. N. Liu, L. Langguth, T. Weiss, J. K¨astel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetic induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009). 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). 6. Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12, 60–69 (2009). 7. J. Pendry, D. Schuring, and D. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). 8. G. Roll and G. Schweiger, “Geometrical optics model of Mie resonances,” J. Opt. Soc. Am. A 17, 1301–1311 (2000). 9. G. Mie, “Beitr¨age zur optik tr¨uber medien, speziell kolloidaler metall¨osungen,” Ann. Phys. 330, 377–445 (1908). 10. Y. Han and Z. Wu, “Scattering of a spheroidal particle illuminated by a gaussian beam,” Appl. Opt. 40, 2501– 2509 (2001). 11. M. I. Mishchenko, J. H. Hovernier, and L. D. Travis, eds., Light scattering by nonspherical particles: Theory, Measurements and Applications (Academic Press, 2000).

#151293 - $15.00 USD

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Received 20 Jul 2011; revised 23 Sep 2011; accepted 28 Sep 2011; published 17 Oct 2011

24 October 2011 / Vol. 19, No. 22 / OPTICS EXPRESS 21432

12. C. Jordan, “Essai sur la g´eom´etrie a` 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). 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). 15. A. Aydin and A. Hizal, “On the completeness of the spherical vector wave functions,” J. Math. Anal. Appl. 117, 428–440 (1986). 16. A. Doicu, T. Wriedt, and Y. Eremin, Light Scattering by Systems of Particles (Springer, 2006). 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). 20. A. Doicu and T. Wriedt, “Extended boundary condition method with multipole sources located in the complex plane,” Opt. Commun. 139, 85–91 (1997). 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). 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). 23. E. Hannan, “The general theory of canonical correlation and its relation to functional analysis,” J. Aust. Math. Soc. 2, 229–242 (1961/1962). 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 < ξ < π/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). 27. W. J. Firth and A. Yao, “Giant excess noise and transient gain in misaligned laser cavities,” Phys. Rev. Lett. 95, 073903 (2005). 28. F. Papoff, G. D’Alessandro, and G.-L. Oppo, “State dependent pseudoresonances and excess noise,” Phys. Rev. Lett. 100, 123905 (2008). 29. M. I. Tribelsky and B. S. Lukyanchuk, “Anomalous light scattering by small particles,” Phys Rev. Lett. 97, 263902 (2006). 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). 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). 32. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998). 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). 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).

1.

Introduction

The interaction of light with wavelength sized particles has been intensely investigated for more than a century, continuing to provide interesting and surprising results. This coupling is essential in single molecule spectroscopy and single photon processes [1, 2], while interference between different scattering channels of particles produces classical analogues of quantum processes [3, 4], and carefully designed particles underpin the physical realization of metamaterials [5–7]. The basis all of these effects is that the particle-light interaction, which depends on the composition and shape of the particle and on the properties the incident light, can become very strong around resonances. For particles much larger than the wavelength of light, resonances are described by closed orbits of light rays [8] inside the particle. This geometric approach becomes less and less effective as the size of the particle decreases, eventually requir#151293 - $15.00 USD

(C) 2011 OSA

Received 20 Jul 2011; revised 23 Sep 2011; accepted 28 Sep 2011; published 17 Oct 2011

24 October 2011 / Vol. 19, No. 22 / OPTICS EXPRESS 21433

ing the solution of Maxwell’s equations. Mie-type solutions [9, 10], based on symmetry and coordinate separability, provide analytical description of resonances for a few specific shapes of particle. For spheres, internal and scattered fields are expanded by electric and magnetic multipoles and each multipole in one field is coupled exclusively to the corresponding multipole in the other field. Resonances are independent of the incident field and occur when the coefficients of one pair of multipoles reaches a maximum for the particular values of the particle radius, permittivity and susceptibility. For particles of arbitrary shape, resonances are always (to the best of our knowledge) defined implicitly by maxima in properties such as the far field extinction or scattering efficiency spectra. This is because, unlike the sphere, there is no decomposition of the internal and scattered fields into partial waves that are pairwise coupled. Hence monitoring of calculated specific properties of the scattered field are instead used empirically to define resonances. While there are several methods that can find spectra and their maxima [11], this approach to resonances is unsatisfying because it depends on the incident field; further it fails to recognize the consequences of resonances associated with fields which are strong only in one region of space, and finally requires an 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. Here we introduce a hermitian operator to define field expansions and resonances for any particle, where Mie’s treatment of the sphere is a special case of this more general theory. We use the mathematical framework of the angles between subspaces of functions [12, 13] to reveal the geometrical nature of resonances in Maxwell’s equations. This allows us to predict resonances of particles independently of the particular incident field, obtain the near and far features of particular modes and choose incident fields to optimize their excitation. 2.

Theory

The theory we develop applies to general metallic and dielectric particles without sharp edges in cases where the interaction between light and matter inside the particle is described by a local macroscopic permittivity and susceptibility. For metallic particles, this means that the free propagation length of carriers is smaller than the skin depth and all the characteristic lengths of the particle. The tangential components of electric and magnetic fields 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] six component vectors F = [E, H]T for electromagnetic fields. Their projections, f , onto the boundary of the particle are surface fields each with four components, two electric and two magnetic, that form a space H whereR scalar products are defined in terms of overlap integrals on the surface of the particle, f · g = S f j∗ g j ds, where the index j labels the components, f ∗ is the complex conjugate of f and we sum over repeated indexes. In this formalism the boundary conditions become f0 = fi− fs ,

(1)

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 i and f s . 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 #151293 - $15.00 USD

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Received 20 Jul 2011; revised 23 Sep 2011; accepted 28 Sep 2011; published 17 Oct 2011

24 October 2011 / Vol. 19, No. 22 / OPTICS EXPRESS 21434

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 any complete set of solutions of the Maxwell equations for the internal and external media. There exist several sets of exact solutions of the Maxwell equations that are linearly independent and complete [15, 16] on surfaces without sharp edges [17]. We choose two sets of ∞ electric and magnetic multipoles, {i˜n }∞ n=1 for internal fields and {s˜n }n=1 for scattered fields, centered at different positions within the particle [19, 20]. Any function in H can be approximated to arbitrary precision by a sufficiently large, but finite, number of multipoles [21]; that ∞ is, {i˜n }∞ n=1 ∪ {s˜n }n=1 is complete and no function in this set is the closure of the linear combinations of all the remaining functions. We remark that both the internal and scattered fields exist in real metallic and dielectric particles and fulfill the boundary conditions in Eq. (1) for the electric and magnetic tangential components. For this reason the interaction of light with these particles is determined by surface fields 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. One can show [21] that the coefficients of the internal and scattered fields, {a˜in , a˜sn }, that minimize the discrepancy between an incident field and the expansion of internal and scattered fields, | f 0 + ∑Nn=1 a˜sn s˜n − a˜in i˜n |, are the solutions of † i † 0 ϒ˜ ϒ˜ ϒ˜ † Σ˜ ϒ˜ f a˜ (2) = ˜† 0 , s ˜Σ† ϒ˜ Σ˜ † Σ˜ −˜a Σ f i/s i/s ˜ Σ˜ being matrices whose columns are the functions {i˜n }N and with a˜ i/s = [a˜1 , .., a˜N ]T and ϒ, n=1 N {s˜n }n=1 . The linear independence and completeness of the functions used guarantees that the Gram matrix in Eq. (2), i.e. the matrix of all possible inner products, can be inverted. However, this is numerical challenging [20] and for this reason this approach has received little attention. Here we instead take advantage of the block structure of the matrix in Eq. (2) to provide a complete characterization of the geometry of the internal and scattered fields and analytical expression for the coefficient of the expansions of the fields. The first step is to find orthogonal modes for the scattering and internal fields: for any number of multipoles, N, we achieve this through the matrix decomposition

ϒ˜ = U i Qi , Σ˜ = U s Qs ,

(3) (4)

where Qi , Qs are invertible matrices that can be found through SVD or QR decomposition [22] and U i ,U s 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 † U i U s = V iCV s† , (5) where C a diagonal matrix with positive elements, and V i ,V s are unitary matrices acting on the internal and scattered fields, respectively. These identities enable us to simplify the Gram

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Received 20 Jul 2011; revised 23 Sep 2011; accepted 28 Sep 2011; published 17 Oct 2011

24 October 2011 / Vol. 19, No. 22 / OPTICS EXPRESS 21435

matrix through the transformation " # " −1 † † −1 ϒ˜ † ϒ˜ ϒ˜ † Σ˜ V i Qi 0 Qi V i † † −1 ˜ ˜ ˜ ˜ Σ ϒ Σ Σ 0 0 V s† Qs† which leads to the matrix equation 1 C†

C 1

ai −as

=

0

#

Qs−1V s

ϒ† f 0 Σ† f 0

=

1 C†

C 1

,

(6)

,

(7)

On the left hand side of Eq. (7), 1 is the identity matrix and ϒ = U iV i and Σ = U sV s are matrices whose columns are formed by the so called principal internal and scattering modes {in } and {sn }, which are one of the main tools in this theory. ai , as are the coefficients of the principal modes in the field’s expansion. The most important part of our theory is that, because matrix 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, ξn , between sn and in as follows in · sn = cos (ξn ). (8) The terms on the right-hand side of Eq. (8) are the principal cosines [12]: cos (ξn ) and sin (ξn ) are the statistical correlation [23] and the orthogonal distance between sn and in . From the general theory of angles between subspaces [13] and the definition above, the angles 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). The interaction of particles with light can √ now be interpreted in terms of eigenvalues and 2, of the hermitian operator in Eq. (7); providing orthogonal eigenvectors, w± = (i ± s )/ n n n interesting analogies between the electromagnetic response of a classical particles with the response of atoms or molecules. However, away from the surface one measures either internal or scattered fields, so we transform the eigenfunctions, {w± n }, to find the coefficients of the principal modes: ain asn

in − cos (ξn )sn 0 i0 · f = 0 n · f 0, 2 in · in sin (ξn ) sn − cos (ξn )in 0 s0 = − · f = − 0 n · f0 . 2 sn · sn sin (ξn ) =

(9) (10)

Here i0n = in − cos (ξn )sn , s0n = sn − cos (ξn )in are bi-orthogonal to in , sn (i0n · sn = s0n · in = 0) with i0n · in = s0n · sn = sin2 (ξn ). 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

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Received 20 Jul 2011; revised 23 Sep 2011; accepted 28 Sep 2011; published 17 Oct 2011

24 October 2011 / Vol. 19, No. 22 / OPTICS EXPRESS 21436

the expansions of internal and scattered field as i T (x)In (x)i0n (s) T s (x)Sn (x)s0n (s) F s/i (x) = GS (x, s) · f0 (s) = − · f0 (s), i0n · in s0n · sn

(11)

where GS (x, s) is the surface Green’s function [14,21] of the particle and T 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] for any complete set of solutions of the Maxwell equations. This convergence can be monitored by the surface residual | f 0 + ∑Nn=1 asn sn − ain in |, which provides an upper bound for the maximum error of scattered and internal fields that decreases with the distance from the surface [16, 24]. Furthermore, the form of Eq. (9) remains unchanged as N → ∞, even if θn , in , sn change [25]. The modal decomposition in Eq. (11) has several unique advantages. The left-hand terms in the scalar products depend exclusively on the particle, so this allows us to strongly compress the description of scattering by identifying the modes that are coupled to a given field and discarding the others, and also to optimize an incident field in order to excite a specific principal mode. We remark that ain , asn in Eqs. (9) and (10) are found by projecting the incident field f 0 onto non-orthogonal vectors, in and sn , while sin (ξn ) is defined as the Peterman factor [26] that gives the order of magnitude of transient gain and excess noise in unstable cavity modes. Therefore the presence of strongly aligned vectors with sin (ξn )

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 Optical Society of America OCIS codes: (290.0290) Scattering; (290.5825) Scattering theory; (160.4236) Nanomaterials; (160.3900) Metals.

References and links 1. D. Graham and R. Goodacre, “Chemical and bioanalytical applications of surface enhanced Raman scattering spectroscopy,” Chem. Soc. Rev. 37, 883–884 (2008). 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). 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). 4. N. Liu, L. Langguth, T. Weiss, J. K¨astel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetic induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009). 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). 6. Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12, 60–69 (2009). 7. J. Pendry, D. Schuring, and D. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). 8. G. Roll and G. Schweiger, “Geometrical optics model of Mie resonances,” J. Opt. Soc. Am. A 17, 1301–1311 (2000). 9. G. Mie, “Beitr¨age zur optik tr¨uber medien, speziell kolloidaler metall¨osungen,” Ann. Phys. 330, 377–445 (1908). 10. Y. Han and Z. Wu, “Scattering of a spheroidal particle illuminated by a gaussian beam,” Appl. Opt. 40, 2501– 2509 (2001). 11. M. I. Mishchenko, J. H. Hovernier, and L. D. Travis, eds., Light scattering by nonspherical particles: Theory, Measurements and Applications (Academic Press, 2000).

#151293 - $15.00 USD

(C) 2011 OSA

Received 20 Jul 2011; revised 23 Sep 2011; accepted 28 Sep 2011; published 17 Oct 2011

24 October 2011 / Vol. 19, No. 22 / OPTICS EXPRESS 21432

12. C. Jordan, “Essai sur la g´eom´etrie a` 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). 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). 15. A. Aydin and A. Hizal, “On the completeness of the spherical vector wave functions,” J. Math. Anal. Appl. 117, 428–440 (1986). 16. A. Doicu, T. Wriedt, and Y. Eremin, Light Scattering by Systems of Particles (Springer, 2006). 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). 20. A. Doicu and T. Wriedt, “Extended boundary condition method with multipole sources located in the complex plane,” Opt. Commun. 139, 85–91 (1997). 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). 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). 23. E. Hannan, “The general theory of canonical correlation and its relation to functional analysis,” J. Aust. Math. Soc. 2, 229–242 (1961/1962). 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 < ξ < π/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). 27. W. J. Firth and A. Yao, “Giant excess noise and transient gain in misaligned laser cavities,” Phys. Rev. Lett. 95, 073903 (2005). 28. F. Papoff, G. D’Alessandro, and G.-L. Oppo, “State dependent pseudoresonances and excess noise,” Phys. Rev. Lett. 100, 123905 (2008). 29. M. I. Tribelsky and B. S. Lukyanchuk, “Anomalous light scattering by small particles,” Phys Rev. Lett. 97, 263902 (2006). 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). 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). 32. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998). 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). 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).

1.

Introduction

The interaction of light with wavelength sized particles has been intensely investigated for more than a century, continuing to provide interesting and surprising results. This coupling is essential in single molecule spectroscopy and single photon processes [1, 2], while interference between different scattering channels of particles produces classical analogues of quantum processes [3, 4], and carefully designed particles underpin the physical realization of metamaterials [5–7]. The basis all of these effects is that the particle-light interaction, which depends on the composition and shape of the particle and on the properties the incident light, can become very strong around resonances. For particles much larger than the wavelength of light, resonances are described by closed orbits of light rays [8] inside the particle. This geometric approach becomes less and less effective as the size of the particle decreases, eventually requir#151293 - $15.00 USD

(C) 2011 OSA

Received 20 Jul 2011; revised 23 Sep 2011; accepted 28 Sep 2011; published 17 Oct 2011

24 October 2011 / Vol. 19, No. 22 / OPTICS EXPRESS 21433

ing the solution of Maxwell’s equations. Mie-type solutions [9, 10], based on symmetry and coordinate separability, provide analytical description of resonances for a few specific shapes of particle. For spheres, internal and scattered fields are expanded by electric and magnetic multipoles and each multipole in one field is coupled exclusively to the corresponding multipole in the other field. Resonances are independent of the incident field and occur when the coefficients of one pair of multipoles reaches a maximum for the particular values of the particle radius, permittivity and susceptibility. For particles of arbitrary shape, resonances are always (to the best of our knowledge) defined implicitly by maxima in properties such as the far field extinction or scattering efficiency spectra. This is because, unlike the sphere, there is no decomposition of the internal and scattered fields into partial waves that are pairwise coupled. Hence monitoring of calculated specific properties of the scattered field are instead used empirically to define resonances. While there are several methods that can find spectra and their maxima [11], this approach to resonances is unsatisfying because it depends on the incident field; further it fails to recognize the consequences of resonances associated with fields which are strong only in one region of space, and finally requires an 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. Here we introduce a hermitian operator to define field expansions and resonances for any particle, where Mie’s treatment of the sphere is a special case of this more general theory. We use the mathematical framework of the angles between subspaces of functions [12, 13] to reveal the geometrical nature of resonances in Maxwell’s equations. This allows us to predict resonances of particles independently of the particular incident field, obtain the near and far features of particular modes and choose incident fields to optimize their excitation. 2.

Theory

The theory we develop applies to general metallic and dielectric particles without sharp edges in cases where the interaction between light and matter inside the particle is described by a local macroscopic permittivity and susceptibility. For metallic particles, this means that the free propagation length of carriers is smaller than the skin depth and all the characteristic lengths of the particle. The tangential components of electric and magnetic fields 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] six component vectors F = [E, H]T for electromagnetic fields. Their projections, f , onto the boundary of the particle are surface fields each with four components, two electric and two magnetic, that form a space H whereR scalar products are defined in terms of overlap integrals on the surface of the particle, f · g = S f j∗ g j ds, where the index j labels the components, f ∗ is the complex conjugate of f and we sum over repeated indexes. In this formalism the boundary conditions become f0 = fi− fs ,

(1)

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 i and f s . 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 #151293 - $15.00 USD

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Received 20 Jul 2011; revised 23 Sep 2011; accepted 28 Sep 2011; published 17 Oct 2011

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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 any complete set of solutions of the Maxwell equations for the internal and external media. There exist several sets of exact solutions of the Maxwell equations that are linearly independent and complete [15, 16] on surfaces without sharp edges [17]. We choose two sets of ∞ electric and magnetic multipoles, {i˜n }∞ n=1 for internal fields and {s˜n }n=1 for scattered fields, centered at different positions within the particle [19, 20]. Any function in H can be approximated to arbitrary precision by a sufficiently large, but finite, number of multipoles [21]; that ∞ is, {i˜n }∞ n=1 ∪ {s˜n }n=1 is complete and no function in this set is the closure of the linear combinations of all the remaining functions. We remark that both the internal and scattered fields exist in real metallic and dielectric particles and fulfill the boundary conditions in Eq. (1) for the electric and magnetic tangential components. For this reason the interaction of light with these particles is determined by surface fields 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. One can show [21] that the coefficients of the internal and scattered fields, {a˜in , a˜sn }, that minimize the discrepancy between an incident field and the expansion of internal and scattered fields, | f 0 + ∑Nn=1 a˜sn s˜n − a˜in i˜n |, are the solutions of † i † 0 ϒ˜ ϒ˜ ϒ˜ † Σ˜ ϒ˜ f a˜ (2) = ˜† 0 , s ˜Σ† ϒ˜ Σ˜ † Σ˜ −˜a Σ f i/s i/s ˜ Σ˜ being matrices whose columns are the functions {i˜n }N and with a˜ i/s = [a˜1 , .., a˜N ]T and ϒ, n=1 N {s˜n }n=1 . The linear independence and completeness of the functions used guarantees that the Gram matrix in Eq. (2), i.e. the matrix of all possible inner products, can be inverted. However, this is numerical challenging [20] and for this reason this approach has received little attention. Here we instead take advantage of the block structure of the matrix in Eq. (2) to provide a complete characterization of the geometry of the internal and scattered fields and analytical expression for the coefficient of the expansions of the fields. The first step is to find orthogonal modes for the scattering and internal fields: for any number of multipoles, N, we achieve this through the matrix decomposition

ϒ˜ = U i Qi , Σ˜ = U s Qs ,

(3) (4)

where Qi , Qs are invertible matrices that can be found through SVD or QR decomposition [22] and U i ,U s 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 † U i U s = V iCV s† , (5) where C a diagonal matrix with positive elements, and V i ,V s are unitary matrices acting on the internal and scattered fields, respectively. These identities enable us to simplify the Gram

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matrix through the transformation " # " −1 † † −1 ϒ˜ † ϒ˜ ϒ˜ † Σ˜ V i Qi 0 Qi V i † † −1 ˜ ˜ ˜ ˜ Σ ϒ Σ Σ 0 0 V s† Qs† which leads to the matrix equation 1 C†

C 1

ai −as

=

0

#

Qs−1V s

ϒ† f 0 Σ† f 0

=

1 C†

C 1

,

(6)

,

(7)

On the left hand side of Eq. (7), 1 is the identity matrix and ϒ = U iV i and Σ = U sV s are matrices whose columns are formed by the so called principal internal and scattering modes {in } and {sn }, which are one of the main tools in this theory. ai , as are the coefficients of the principal modes in the field’s expansion. The most important part of our theory is that, because matrix 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, ξn , between sn and in as follows in · sn = cos (ξn ). (8) The terms on the right-hand side of Eq. (8) are the principal cosines [12]: cos (ξn ) and sin (ξn ) are the statistical correlation [23] and the orthogonal distance between sn and in . From the general theory of angles between subspaces [13] and the definition above, the angles 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). The interaction of particles with light can √ now be interpreted in terms of eigenvalues and 2, of the hermitian operator in Eq. (7); providing orthogonal eigenvectors, w± = (i ± s )/ n n n interesting analogies between the electromagnetic response of a classical particles with the response of atoms or molecules. However, away from the surface one measures either internal or scattered fields, so we transform the eigenfunctions, {w± n }, to find the coefficients of the principal modes: ain asn

in − cos (ξn )sn 0 i0 · f = 0 n · f 0, 2 in · in sin (ξn ) sn − cos (ξn )in 0 s0 = − · f = − 0 n · f0 . 2 sn · sn sin (ξn ) =

(9) (10)

Here i0n = in − cos (ξn )sn , s0n = sn − cos (ξn )in are bi-orthogonal to in , sn (i0n · sn = s0n · in = 0) with i0n · in = s0n · sn = sin2 (ξn ). 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

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the expansions of internal and scattered field as i T (x)In (x)i0n (s) T s (x)Sn (x)s0n (s) F s/i (x) = GS (x, s) · f0 (s) = − · f0 (s), i0n · in s0n · sn

(11)

where GS (x, s) is the surface Green’s function [14,21] of the particle and T 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] for any complete set of solutions of the Maxwell equations. This convergence can be monitored by the surface residual | f 0 + ∑Nn=1 asn sn − ain in |, which provides an upper bound for the maximum error of scattered and internal fields that decreases with the distance from the surface [16, 24]. Furthermore, the form of Eq. (9) remains unchanged as N → ∞, even if θn , in , sn change [25]. The modal decomposition in Eq. (11) has several unique advantages. The left-hand terms in the scalar products depend exclusively on the particle, so this allows us to strongly compress the description of scattering by identifying the modes that are coupled to a given field and discarding the others, and also to optimize an incident field in order to excite a specific principal mode. We remark that ain , asn in Eqs. (9) and (10) are found by projecting the incident field f 0 onto non-orthogonal vectors, in and sn , while sin (ξn ) is defined as the Peterman factor [26] that gives the order of magnitude of transient gain and excess noise in unstable cavity modes. Therefore the presence of strongly aligned vectors with sin (ξn )