Fano-like interference of plasmon resonances at a single rod-shaped nanoantenna 2 F Lo´pez-Tejeira, R Paniagua-Dom´ınguez, 1 0 R Rodr´ıguez-Oliveros and J A S´anchez-Gil 2 Instituto de Estructura de la Materia (IEM-CSIC), Consejo Superior de n Investigaciones Cient´ıficas, Serrano 121, E-28006 Madrid, Spain a J E-mail: [email protected] 8 1 Abstract. Singlemetallicnanorodsactingashalf-waveantennasintheopticalrange ] exhibit an asymmetric, multi-resonant scattering spectrum that strongly depends on s c both their length and dielectric properties. Here we show that such spectral features i t can be easily understood in terms of Fano-like interference between adjacent plasmon p resonances. Onthebasisofanalyticalandnumericalresultsfordifferentgeometries,we o . demonstrate that Fano resonances may appear for such single-particle nanoantennas s c provided that interacting resonances overlap in both spatial and frequency domains. i s y h p PACS numbers: 73.20.Mf, 78.67.Qa, 84.40.Ba [ 2 v 1 5 5 3 . 1 1 1 1 : v i X r a 2 1. Introduction Different experimental [1]-[5] and purely theoretical [6]-[10] investigations have shown that metallic nanorods act as standing-wave resonators for localized plasmon resonances in the optical regime, thus exhibiting geometrical half-wavelength resonances with spectral positions depending mainly on the length of the rods. This particular type of so-called “optical nanoantennas” have raised the prospect of significant improvements in fields such as photodetection [11], field-enhanced spectroscopy [12] or control of emission direction in single-molecule light sources [13]. Generally speaking, most of device-oriented studies are focused on nanoantennas operating at the dipole-like resonance. However, structures with a high aspect ratio may support additional resonances that have usually been the subject of a more fundamental research work. Hence, several authors have already elucidated the scaling properties of high-order longitudinal modes, as well as their dependence on shape, size, orientation and dielectric environment by means of diverse approaches and techniques [2, 7], [14]-[17]. Nevertheless, a relevant issue has yet to be addressed for multi- resonant nanoantennas, that is the emergence of asymmetric line profiles in single- particle extinction or scattering spectra. Interestingly, such a feature seems to go almost unnoticed for the nanoplasmonics community, despite being apparent in some previous references [2, 7, 15, 17]. In fact, to the best of our knowledge, the only explicit report on the occurrence of Fano-like asymmetric line shapes in the scattering spectra of a single silver nanorod can be found in a recent paper by Reed et al. [18], though the emphasis is put on dimer structures therein. Inthiswork, weshowthattheseasymmetriclineprofilescanbeeasilyunderstoodin terms of the so-called Fano-like interference between localized plasmon resonances that has been recently reported for a variety of coupled metal nanoparticles [19]-[22]. Being more precise, we present a simplified analytical model that describes spectral features of a single rod-shaped nanoantenna in terms of Fano-like interference. Contrary to the common assumption that interference does not play any role in total scattering or extinction of a single metallic surface, we find a good agreement with numerical results, whichareattainedthroughtheseparationofvariables (SVM),finiteelement (FEM)and surface integral equation (SIEM) methods (see Appendix A for a succinct description of calculation techniques). Furthermore, we make use of explicit expressions for light scattering by spheroids to conclude that not only spectral but also spatial overlap (i.e. non-orthogonality) between interacting modes underlies the emergence of such single- rod resonances. This points out the need of being extremely cautious when applying the premises of standard Mie theory to particles that significantly depart from sphericity. 3 2. Single metallic nanorods acting as half-wave nanoantennas 2.1. Fano-like interference of longitudinal plasmon resonances Let us begin by briefly reviewing the basics of light scattering by a single metallic nanorod with dielectric function ε that is surrounded by a medium with constant m permittivity ε : On the assumption that rod diameter D is much smaller than its total d length L, the electromagnetic response to p-polarized light impinging perpendicular to the long side is fully governed by longitudinal modes. The fundamental resonance λ(1) res can thus be described as a dipolar excitation of charges at the rod surface, with its wavelength exhibiting a linear dependence on L , λ(1) ∝ 2L. For a perfectly conducting res material of negligible thickness, λ(1) is precisely equal to 2L, whereas λ(1) (cid:29) 2L at res res optical frequencies [7, 8, 9]. As rod length increases, additional resonances may appear. Following Khlebtsov and Khlebtsov [7], we assume the position of any longitudinal resonance to be described by the following approximate scaling law L λ(n) ≈ B +B (1) res 0 1nD where n is an odd integer and B ,B are coefficients that depend on the actual 0 1 geometry and dielectric environment of the system and have to be determined from linear regression. With respect to B ,B , we also have to notice that, although not used 0 1 in this paper, explicit expressions can be obtained within the framework of Novotny’s model for effective wavelength scaling at optical antennas [8]. As mentioned in the previous section, the interaction of adjacent resonances has been found to be compatible with a Fano-like interference model [23],[20]-[25], where the lower resonance plays the role of continuum in canonical Fano line shape [26]. Given that light scattering by a rod of finite length cannot be treated in a closed form within the framework of standard Mie theory [27], we assume a heuristic line shape for the first approach. Hence, Q ≈ |f(ω)|2 (2) sca with (cid:34) (cid:35) b qb 1 3 f(ω) ≡ A(ω)+F + (3) (ω −ω )+ib (ω −ω )+ib 1 1 3 3 where A(ω) is a slowly varying amplitude and F stands for the complex amplitude of the fundamental resonance, which is described in terms of its real central frequency ω 1 and spectral width b . Dimensionless real parameter q modulates the interaction with 1 the adjacent plasmon resonance (analogously defined by ω and b ), thus governing the 3 3 line shape asymmetry. 2.2. Prolate spheroidal nanorods From the computational point of view, the shape of half-wave nanoantennas is usually modeled by a right circular cylinder with either flat or hemispherical ends 4 45 L = 140 nm (a) 40 L = 180 nm 35 L = 220 nm L = 260 nm 30 L = 300 nm 25 sca 20 L = 340 nm Q 15 10 5 L = 380 nm 0 500 1000 1500 2000 2500 3000 Wavelength (nm) 2400 (b) 2200 2000 1800 1600 λ(n) = 198.92 +160.90 L/nD res m) 1400 n (s 1200 e λr 1000 800 n = 1 n = 3 600 Best fit to Eq. (1) 400 0 2 4 6 8 10 12 14 L/nD Figure 1. (a) Calculated scattering efficiency as a function of wavelength for a single Agspheroidsurroundedbyglass(ε =2.25). Incidentfieldisp-polarizedandimpinges d perpendicular to the rotation axis of the spheroid. Different curves correspond to increasing values of L, whereas D is set to 30 nm for all calculations. (b) Linear scaling of resonant wavelengths in panel (a) as a function of normalized aspect ratio L/nD. Dashed line marks the best fit to Eq. (1). 5 SVM 20 ω = 0.61695 eV 1 Best fit to Eq. (2) 15 ω = 1.48945 eV q = 0.82324 3 0.9 a 0.8 sc 10 Q 0.7 0.6 q 0.5 5 0.4 0.3 100 150 200 250 300 350 400 L (nm) 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Photon Energy (eV) Figure 2. Calculated scattering efficiency (solid line) as a function of photon energy for a single Ag spheroid (L = 340 nm; D = 30 nm) in ε = 2.25. Incident field is d p-polarized and impinges perpendicular to the rotation axis of the spheroid. Dashed line renders the best-fitting curve to Eq. (2). Obtained values of ω ,ω and q are also 1 3 shown. Lowerrightinsetpaneldepictstheobtainedvaluesof q forevery Linfigure1. (spherocylinder). However,wehavefoundtheprolatespheroidtobethemostconvenient geometry to start with, because of the following reasons: (i) Previous works [28, 29] on the basis of SVM have provided us with a very efficient approach to calculate extinction, scattering and absorption cross-sections even for very elongated nanoantennas; (ii) In addition to its low numerical cost, such a theoretical framework also makes apparent the origin of unexpected asymmetry in line profiles, as detailed hereafter; (iii) Prolate spheroids are not only of academic interest, as they accurately describe the so-called nanorice structures [17, 30]. In figure 1 we present the calculated scattering efficiency Q for a single silver sca spheroid surrounded by glass (ε = 2.25) under the assumption that incident field d is p-polarized and impinges perpendicular to the long side of the rod. Different curves correspond to increasing values of total length L within the [100,400] nm range, whereas the polar diameter D is set to 30 nm for all calculations. As can be seen, the position of resonances increases linearly with L and such displacement is fairly well described by (1) (see figure 1(b)). For L/D (cid:38) 5, the peaks arising from resonances with n = 1 and n = 3 are clearly apparent, as it is the asymmetry of the line shape between them. Inordertotestoutourapproach,letustakeacloserlooktothecurvecorresponding to 340 nm-long spheroid. Figure 2 renders the calculated scattering efficiency as a 6 function of energy and its best-fitting curve to (2). It may be seen that our heuristic Fano-like line shape agrees very well with the full electromagnetic calculation. Besides, the obtained value q = 0.82324 is consistent with asymmetric profiles being described by |q| ≈ 1 [19]. Such an agreement extends to the whole range of rod lengths in figure 1, as summarized in the inset panel (See Appendix B). Forthecaseofobliqueincidence(i.e. iftheangleαbetweentheincidentlightvector k and the rotation axis of the spheroid is not equal to 90o), “even” modes become inc accessible [2, 7, 17] and should therefore be incorporated to our analysis. Different curves in 3(a) correspond to the calculated Q for decreasing values of α and the same sca geometry, polarization and dielectric environment as in figure 2. Logarithmic scale is used for optimal visualization of less intense features. As can be seen, the oblique line shapes differ quite little from that for normal incidence at the vicinity of resonances with labels n = 1,3, except from that the intensities of “odd” peaks exhibit a linear dependencewithsin2α(seelowerleftinsetpanel). However,anextraresonancedevelops just on top of base line at ω ≈ 1.045 eV. Given that it is located within the [ω ,ω ] 1 3 range, we label it as n = 2. This new peak reaches its maximum for α = 45o and seems to be symmetrical with respect to its central frequency. Hence, we can conjecture that symmetry precludes interference between longitudinal modes with different parity, so that only an additive term accounts for the contribution of this resonance to Q , sca |F |2b2 Q (ω,α (cid:54)= 90o) ≈ |f(ω)|2 + 2 2 (4) sca b2 +(ω −ω )2 2 2 whereF ,ω ,b arethecomplexamplitude, thecentralfrequency, andthespectralwidth 2 2 2 of the resonance with label n = 2, respectively. Comparison between calculated Q (ω,α = 45o) and its best-fitting curve to (4) in sca figure 3(b) confirms our hypothesis of the lack of interaction between “odd” and “even” resonances. As summarized in the upper right inset panel, the only effect of oblique incidence (aside from the emergence of a new peak!) is the already mentioned global quenching of Q (ω(1),(3)) (i.e.,the intensities of “odd” peaks) and therefore that of q. sca res Being the understanding of line shape asymmetry the main goal of our work, we do not discuss on oblique incidence any further, except for the detailed description of the dependence of different parameters on α that is presented in Appendix B. However, we have to point out that inter-parity coupling may be allowed for a symmetry-broken configuration,suchasdepositingthenanospheroidontoadielectricsubstrate,asrecently proposed for the nanocube geometry [31]. In fact, we can easily envisage that the two different intra- and inter-parity mechanisms’ operating simultaneously opens a very interesting scenario that certainly warrants further investigation. Going beyond heuristic description brings us up against the actual meaning of (3). From a formal point of view, it is clear that f(ω) plays the role of effective polarizability inasomehowgeneralizedRayleigh-Ganstheoryfornonsphericalparticles. Nevertheless, there is still the concern of how to explain the emergence of asymmetry in line profiles. As previously mentioned, Fano resonances require an observable that is sensitive to interference, but standard Mie theory predicts the total scattering by a single metallic 7 (a) n = 1 n = 3 10 n = 2 1 α = 45o α = 30o Qsca 0.1 11124680 Qsca( ω(r1e)s, α ) α = 60o 1102 α = 75o 8 0.01 6 Q ( ω(3), α ) α = 90o sca res 4 2 0.2 0.4 0.6 0.8 1.0 sin2α 1E-3 0.5 1.0 1.5 2.0 Photon Energy (eV) L=340nm   H E D=30nm inc inc α  k 22 inc (b) 0.9 20 SVM (α = 45o) q = 0.69904 18 0.8 Best fit to Eq. (4) 16 q 0.7 14 ω = 0.61751 eV 1 0.6 12 a 0.0 0.2 0.4 0.6 0.8 1.0 Qsc 10 sin2α 8 ω = 1.48863 eV 3 6 4 ω = 1.04843 eV 2 2 0 0.5 1.0 1.5 Photon Energy (eV) Figure 3. (a)CalculatedQ asafunctionofphotonenergyforasingleAgspheroid sca (L = 340 nm; D = 30 nm) surrounded by ε = 2.25. Incident field is p-polarized and d impingeswithangleαwithrespecttotherotationaxis. Differentcurvescorrespondto decreasing values of α. The intensities of “odd” peaks as a function of α are presented at lower left inset panel. (b) Comparison between calculated Q and its best-fitting sca curve to Eq. (4) for α = 45o. Obtained values of ω ,ω ,ω and q are also shown. 1 2 3 Upper right inset panel depicts the obtained values of q for every α in (a). 8 surface to be proportional to the mere sum of intensities. Such a discrepancy can be easily explained for the case of prolate spheroidal particles by means of the SVM formalism. According to Reference [29], the scattering efficiency of a prolate spheroid for p-polarized light impinging at normal incidence is given by (cid:40) ∞ ∞ ∞ ∞ (cid:20) 4 (cid:88) (cid:88)(cid:88)(cid:88) (cid:16) (cid:17)∗ Q = 2 |b(d)|2N2(c )+Re i n−l k2a(d) a(d) ω(m)(c ,c )+ sca LDk2 l 1l d d ml mn ln d d d l=1 l=1 m=ln=m (cid:18) (cid:19) (cid:16) (cid:17)∗ (cid:16) (cid:17)∗ + ik b(d) a(d) κ(m)(c ,c )−a(d) b(d) κ(m)(c ,c ) + (5) d ml mn ln d d ml mn nl d d (cid:41) (cid:21) (cid:16) (cid:17)∗ + b(d) b(d) τ(m)(c ,c ) N (c )N (c ) ml mn ln d d ml d mn d Withoutenteringintofurtherdetails,letussaythatκ(m),τ(m) andω(m) aredifferent ln ln ln integralsofthenormalizedprolateangularspheroidalwavefunctionswithnormalization √ √ coefficients N (c ), where c = k /2 L2 −D2, k = ε 2π/λ and λ is the wavelength ij d d d d d of incoming (or outgoing) radiation. More importantly, a(d) and b(d), b(d) represent the ij l ij expansion coefficients of the azimuthal components of scattered electric and magnetic fields respectively. Single or double subscript marks whether the expansion accounts for the axisymmetric (i.e. the one that does not depend on the azimuthal angle) or non-axisymmetric (i.e. azimuth-dependent) part of the corresponding field. For the axisymmetric part, the introduction of Debye’s potential representation of vector fields (i.e. the potentials that solve the light scattering problem for spheres) leads to the term following the curly bracket in (5), which closely resembles Mie’s result. However, a properrepresentationofthenon-axisymmetricpartoftheelectromagneticfieldsrequires a combination of Debye’s potential with that used to deal with light scattering by an infinitely long cylinder, namely Hertz’s [32]. Hence double-subscript interference terms arise in Q , thus providing a mathematical description for the interaction between sca adjacent resonances in prolate spheroids. Note that such interference terms are not negligible only if the interacting fields are non-orthogonal, that is, spatial overlap between interfering plasmon modes is required for Fano-like interference. Although Eq. (5) is only valid for spheroids, we conjecture the underlying sphere- like vs. cylinder-like interference mechanism to operate for any single nanoantenna with a similar geometry. In order to determine if there is any real substance in our guess, we present in figure 4 the calculated scattering efficiency for a single 550x30 nm Ag nanoantenna embedded in ε = 2.25 assuming three different geometries: a circular d cylinder with flat ends, a spherocylinder and a prolate spheroid. Numerical values are obtained from either FEM or SVM. Logarithmic scale is used for the sake of a better comparison. As can be seen, the three curves can be fitted to Eq. (2) even for the case of a flat-ended cylinder, although only a modest q ≈ 0.15 is obtained for such a high aspect ratio. Hemispherical ends result in more than a 30 percent increase of asymmetry parameter, whichrisestoitsmaximumvalueofq ≈ 0.92forprolatespheroidalgeometry, where spherical and cylindrical features interact in the most efficient way. These results 9 100 q = 0.92291 10 q = 0.15169 a 1 c s Q 0.1 q = 0.20294 0.01 0.5 1.0 Photon Energy (eV) Figure 4. Calculated scattering efficiency as a function of photon energy for a single Ag nanorod (L = 550 nm; D = 30 nm) surrounded by glass (ε = 2.25). Incident d field is p-polarized and impinges perpendicular to the rotation axis. Black, red, and blue curves correspond to flat-ended cylinder, spherocylinder and prolate spheroid, respectively. Open symbols render the best-fitting curves. Obtained values of q are also shown. are consistent with our previous formal discussion on potential theory and point out the subtle balance of different contributions to light scattering. 2.3. Spherocylinder-shaped nanorods In figure 5 we present a separate plot of scattering efficiency as a function of photon energy for the spherocylinder-shaped nanorod in figure 4. As already mentioned, the asymmetric line profile can be fitted to (2) (dashed line). The obtained q = 0.20294 accounts for the moderate interaction between adjacent resonances located at ω(1) = res 0.45563 eV and ω(3) = 1.19918 eV. Please notice that spectral features at ω (cid:38) 1.7 res eV suggest the need to include subsequent resonances in (3), although that refinement is beyond the scope of our present work. Inset panels at the right hand side show the calculated Q curves (upper) and their corresponding q values (lower) for different rods sca with D = 30 nm and L within the [340,550] nm range. As expected, the interaction between adjacent resonances is significantly lower than that of prolate spheroids with the same L,D parameters. (Details on fitting are presented in Appendix B.) The normalized electric near-field distribution in the xz plane at most significant values of photon energy is presented in the three panels at the left hand side of figure 5. 10 Leaving aside the typical n-node quasi-standing-wave patterns at ω(1) and ω(3), let us res res concentrate on the plot for ω = 1.12139 eV. Given that Q reaches its local minimum sca within at this precise value, we expect some destructive interference to appear in spatial domain. Such interference pattern can be noticed for the field distribution inside the volume of the rod, which is limited by dashed lines. For the sake of clarity, we present in the bottom panel the line profiles at x = 0 (dotted lines on contour plots). In short, for ω = ω , interference cancels a significant part of the field intensity that is present at min the zones marked by descending arrows for ω = ω(3). On the other hand, field intensity res at the central part of the rod (ascending arrow) for ω = ω is slightly enhanced with min respect to that for ω = ω(3), thus resembling the field pattern corresponding to ω = ω(1). res res 2.4. The simplest case: a rectangular nanowire Forabetterunderstandingoftheelusivespatialoverlapbetweenmodes,wenowconsider light scattering by an infinitely long rectangular nanowire with large aspect ratio, which seems to be the simplest geometry that clearly shows such behavior. In figure 6 we plot the scattering efficiency (blue curve) and the near-field amplitude (black curve) at normal incidence calculated using SIEM for a silver rectangular nanowire with L = 600 nm and D = 10 nm surrounded by ε = 1. Near-field amplitude is now evaluated d 3 nm outside the end of the nanowire and normalized to the incident field at this point. However, the two curves are then re-normalized to unity to more clearly show their spectral shift. Such a change in the position of maxima, which has recently been explained in terms of driven and damped harmonic oscillators [33], will be relevant for our discussion. In this simple geometry, the very broad dipole-like resonance in Q at λ ≈ 1500 sca nm is more similar to the “canonical” Fano continuum than those in the previous configurations and therefore provides a strong spectral overlap with the narrower 3λ/2-mode. With respect to spatial overlap between different longitudinal plasmon resonances, we find that it can be monitored by means of the real part of the normal component of electric field (i.e. Enear in figure 6), which is directly proportional z to surface charge distribution. Given that, in the electrostatic limit, surface charge distribution can be approximated by a sum of sinusoidal functions with argument an integer multiple of πx/L, we finally arrive to the following expression N (cid:20) (cid:21) (cid:88) nπx Re[Enear/Einc](x) ≈ A sin (6) z z n L n=2k+1 for which the set of coefficients {A } can be straightforwardly determined from Fourier n analysis. According to this approximation, we expect surface charge distributions at the maxima of near-field amplitude (and not of scattering efficiency!) to have a net dipolar moment and therefore to exhibit a pattern with an odd number of nodes and