Dielectric antennas-a suitable platform for controlling magnetic dipolar emission : errata

We report and correct misprints in our manuscript. ©2012 Optical Society of America OCIS codes: (250.5403) Plasmonics; (160.3820) Magneto-optical materials; (260.2510) Fluorescence. References and links 1. M. K. Schmidt, R. Esteban, J. J. Sáenz, I. Suárez-Lacalle, S. Mackowski, and J. Aizpurua, “Dielectric antennas a suitable platform for controlling magnetic dipolar emission,” Opt. Express 20(13), 13636–13650 (2012). There is an error in the lower limit of the summation indices in Eqs. (1)-(8). Summations in all those expressions should be for n = 1, 2, ... instead of n = 0, 1, ... . Also, expressions in parentheses in Eqs. (3) and (7) were mistakenly squared. Corrected equations are listed:


Introduction
The idea of controlling the spontaneous decay of a molecule by placing it in an inhomogeneous medium has been extensively investigated since the pioneering work of Purcell [1].Recent works have been devoted to controlling both the shape [2,3], intensity [4][5][6] and directionality [7,8] of radiation from the dipolar electric emitter by means of coupling the emitter to a variety of metallic antennas.
In a classical description, an electric dipolar transition moment of the molecule is substituted by a dipolar emitter.A classical electrodynamics calculation of the electric fields originated by the dipole allows to calculate the modification of the decay rates, quantum yield and many other properties of the emitter induced by the environment [9][10][11][12][13].This classical description is known to have a clear correspondence with the quantum results [14][15][16].A particularly interesting system with an analytical solution, comprises an electric dipole in the vicinity of a spherical metallic nanoparticle [9], but more complex antenna architectures have also been studied [6][7][8].
An interesting aspect of dipolar emission is related to the magnetic versus electric nature of the dipolar emitter.Karaveli and Zia [17] have recently shown that, similar to the electric dipolar transition, spontaneous emission from certain molecules exhibiting a magnetic dipolar transition can be modified by carefully placing them near a flat gold mirror.Furthermore, as a growing interest has been given to the various magnetic phenomena occurring in nanoscale systems [18], such as the excitation of magnetic resonances in metallic [19] and dielectric particles [20][21][22], a possibility of designing simple platforms for controlling magnetic emission is thus at hand.
In this paper, we demonstrate the possibility of using spherical dielectric nanoparticles as antennas in the optical-infrared range, to modify the radiation from a localized magnetic dipolar emitter.We first present the analytical expressions describing the enhancement of decay rates of an electric dipolar emitter in the vicinity of a spherical nanoparticle and derive the equivalent expressions for the magnetic dipolar emitter.Submicron silicon particles of 230 nm radius are used here as a canonical example of dielectric antennas, since they exhibit spectrally wellseparated electric and magnetic dipolar resonances in the near infrared [20].Interestingly, for both electric and magnetic emitters, we observe a strong dependence of the excitations on the orientation of the emitter with respect to the surface of the antenna.This setup is then compared to a more familiar system where an electric dipolar emitter is placed near a small plasmonic nanoparticle acting as an optical electric nanoantenna.Finally, we show that a simple dipoledipole interaction model allows to describe in a simplified manner the enhancement of emission rates in several relevant situations.

Electric and magnetic dipolar decay rates in the presence of a particle
Recent developments in the understanding and engineering of magnetic phenomena in nanooptics call for analytical methods to describe the properties of their building blocks.An example of that is the enhancement of the radiation rate from a magnetic dipole placed in an inhomogeneous environment.The analytical formulas describing the enhancement of radiative and nonradiative decay rates of an electric dipole positioned in the vicinity of a spherical nanoparticle have been derived by Ruppin [9].Following this work, we derive analogous expressions for the magnetic dipole.First, the dipolar electromagnetic field is expanded into a series of vector spherical harmonics, and the response from the sphere to each harmonic is obtained following the Mie formalism [23][24][25].Integrating the flow of Poynting vector through the full solid angle in the far-field gives the radiated power.The ohmic losses associated with induced currents within the particle give the rate of energy dissipation.The sum of the radiative decay rate Γ RAD and the non-radiative decay rate Γ NON−RAD gives the total decay rate Γ TOT .The detailed derivation is presented in the Appendix.
Below, we list the expressions for the radiative and total decay rates enhancements of the electric (superscript 'e') and magnetic ('m') dipoles.Two orientations of the emitter are considered and denoted with superscripts in the decay rates: perpendicular ('⊥') and parallel ('||') to the closest surface of the spherical antenna.The dipole is positioned at the distance z from the center of a homogeneous sphere of radius a and dielectric constant ε.The system is embedded in an environment characterized by the real-valued dielectric function ε e .The wavevector k = 2π/λ √ ε e is associated with the vacuum radiation wavelength of the emitter λ .The decay rates are normalized to the decay rate of the dipole in the homogeneous environment Γ 0 .Within the linear response theory, the obtained enhancement factors are independent of the strength of the emitter and are given by (3) where ℜ(z) denotes the real part of z.
For the magnetic emitter, the respective enhancement factors are given by (see Appendix for a detailed derivation) Bessel-Ricatti functions ψ and ζ are defined as ψ(x) = x j n (x) and ζ (x) = xh  n (x) of the first kind.The definitions of Mie coefficients a n and b n used here are adopted from Bohren and Huffman [23]: with where M is defined as the relative refractive index.In all the following considerations, we will assume a vacuum environment characterized by ε e = 1.All the derivatives in the above expressions are calculated with respect to the arguments in parentheses.The Mie coefficients can be attributed to the efficiency of excitation of the corresponding electric (a n ) and magnetic (b n ) Mie modes.The first two Mie resonances n = 1, 2 correspond to the dipolar and quadrupolar modes, respectively.Notably, the expressions for the decay of an electric emitter (Eqs.( 1)-( 4)) differ from those for the magnetic emitter (Eqs.( 5)-( 8)).It is possible to relate one with another by exchanging the Mie coefficients a n ↔ b n [22].Furthermore, in the absence of absorption (real √ ε), the enhancements of total and radiative decay rates can be shown to be equal, using the properties of the Mie coefficients [26].

Radiating dipoles in the presence of a high-index dielectric particle (Si sphere)
It has been previously shown that a high-permittivity sub-micron size nanoparticle can exhibit strong and spectrally separated electric and magnetic dipolar resonances in the near-infrared [20].In Fig. 1(a) we show the normalized plane wave scattering cross section spectrum of a silicon sphere of radius a = 230 nm (relative refractive index M = 3.5), separated into dominant dipolar (n = 1) and quadrupolar (n = 2) contributions.The scattering cross section under plane wave illumination (Fig. 1(b)) can be decomposed into electric (σ s (a n )) and magnetic (σ s (b n )) contributions according to Spectra in Fig. 1(a) show the scattering efficiency of the nanoparticle defined as the scattering cross section normalized by the geometrical cross section C s = σ s /(πa 2 ).In the investigated range of wavelengths, three peaks at λ = 1160 nm, λ = 1350 nm and λ = 1680 nm can be identified and attributed to, respectively, the quadrupolar magnetic b 2 , the dipolar electric a 1 and the dipolar magnetic b 1 resonances.11).Spectra of the radiative rate enhancements shown in (c) were calculated using Eqs.( 2),( 4),( 6),( 8) for the electric (solid lines) and magnetic (dashed lines) emitter.The emitter is oriented either perpendicularly (blue lines) or parallelly (red lines) with respect to the closest surface of the sphere.The refractive index of the silicon nanosphere is 3.5, while the distance from the emitter to the surface of the sphere is set to 50 nm.
In Fig. 1(c), we present the spectra of the enhancement of the decay rates for both electric and magnetic emitters positioned at 50 nm distance from the surface of the silicon sphere (Fig. 1(d)).The enhancements for electric (solid lines) and magnetic (dashed lines) dipoles are shown for the two orientations of both types of emitters -parallel (red lines) and perpendicular (blue lines) to the closest surface of the nanoparticle (see schematics in Fig. 1(d)).For each configuration, modes excited by the emitter can be identified by the correspondence between the spectral positions of the peaks in the decay rate enhancement spectra in Fig. 1(c) and the maxima of Mie coefficients in Fig. 1(a).
We observe in Fig. 1(c) that both the orientation of the emitters as well as their electric or magnetic nature determine which modes of the sphere are excited.An electric emitter oriented parallelly (solid red line) excites the magnetic dipolar b 1 (at λ = 1680 nm), quadrupolar b 2 (at λ = 1160 nm) and -although very weakly -the electric dipolar a 1 (at λ = 1350 nm) resonance, while the same emitter aligned perpendicularly (solid blue line) couples only to the broad electric dipolar a 1 mode at λ = 1350 nm.The complementary behavior is present for the magnetic emitter: for the perpendicular orientation (blue dashed line), only the magnetic b 1 , b 2 modes are excited, while in the parallel orientation (red dashed line) the emitter couples to both the magnetic and electric modes.In principle, this effect can thus be used to discriminate between the two types of emitters.To further investigate the selectivity of the emission, we plot in Fig. 2 the enhancements of the decay rates of the electric (p emi ) and magnetic (m emi ) dipolar emitter as a function of the radiation wavelength λ and the radius a of the dielectric nanosphere.Since Eqs. ( 1)-( 10) depend on the parameter ka through the Mie coefficients, spectral positions of the Mie resonances form straight lines as marked in the maps in Fig. 2 by dashed lines defined by a ∝ λ .The dependence of Eqs. ( 1)-( 10) on the distance from the dipole to the center of the sphere z does not affect significantly the spectral features of the enhancements, but governs the relative strength of the resonances.Furthermore, for the largest sphere, many high-order resonances appear clearly in the high-energy region of spectra, as these modes can be efficiently activated for small values of (z − a)/a.
Similarly to Fig. 1(c), for perpendicular orientation, the electric emitter (Fig. 2(a)) couples only to the electric modes (marked as a n ), and excites mostly magnetic contribution when oriented parallelly (Fig. 2(b)).In the latter case however, an electric contribution is also present, which is mostly visible for the high order modes.Similarly, the perpendicular magnetic emitter excites magnetic modes exclusively (Fig. 2(c)), whereas in parallel orientation it couples to the electric and to the magnetic modes (Fig. 2(d)).Thus, the orientation and nature of the emitters allows to select which modes of the dielectric sphere are excited.
This modal selectivity can be understood by tracing the presence of Mie coefficients in Eqs. ( 1)-( 8) for each case.For the emitter perpendicular to the surface of the antenna (Eqs. (1),( 2),( 5),( 6)), only one type of Mie coefficients is present: electric a n terms for the electric emitter and magnetic b n terms for the magnetic emitter.On the other hand, the equations for the emitter oriented parallelly (Eqs.( 3),( 4),( 7),( 8)) include both electric and magnetic coefficients, therefore both types of resonances are excited.These properties reflect the symmetry of Eqs. ( 1)-( 4) and Eqs. ( 5)-( 8) with respect to the exchange of the electric and magnetic Mie coefficients in the sums.
So far, we have identified the peaks in Fig. 1 and Fig. 2 by studying different contributions to the sum in Eq. ( 11).This identification can be confirmed by plotting the distributions of the fields induced in each case.In Fig. 3 the induced electric (a,c,e,g) and magnetic field (b,d,f,h) distributions are shown for two resonances of the Si sphere -electric a 1 ((a-d), denoted in the schematics as p ind ) and magnetic b 1 ((e-h), m ind ) dipolar modes.The excitations are due to the emitter of the electric (p emi ) or magnetic (m emi ) nature, oriented as shown in the schematic of Fig. 3.The fields have been obtained from a vector harmonic decomposition (see Appendix for details).Dipolar electric resonances at λ = 1350 nm are excited by the perpendicularly oriented electric emitter (Fig. 3(a), 3(b)) or parallelly oriented magnetic emitter (Fig. 3(c), 3(d)).The dipolar magnetic resonance at λ = 1680 nm is induced both by the electric emitter oriented parallelly (Fig. 3(e), 3(f)) and by the magnetic emitter oriented perpendicularly (Fig. 3(g), 3(h)) to the surface of the sphere.The distributions of the fields shown in Fig. 3 clearly indicate the dipolar nature of the induced resonances and qualitatively agree well with the field distributions at the resonances obtained with excitation of the Si sphere by a plane wave [20].

Magnetic emitter in the vicinity of a plasmonic nanoparticle
Small metallic spherical nanoparticles do not exhibit significant magnetic resonances in the visible spectral range, but they present strong plasmonic resonances of electric nature that have been shown to strongly modify the decay rates of nearby electric dipolar emitters.In this section, we demonstrate that the plasmonic mode also affects the emission from a magnetic emitter.
In Fig. 4(a) we show the scattering efficiency spectra of a silver spherical 50 nm radius nanoparticle illuminated by a plane wave (Fig. 4(b)).The decomposition of the scattering efficiency into different contributions, as previously introduced for the silicon sphere in Fig. 1(a), indicates that the scattering properties of the nanoparticle are due to the excitation of the dipolar electric a 1 mode with a broad maximum at around λ ≈ 430 nm.
The spectral dependence of the radiative and total decay rate enhancements in this situation is shown in Fig. 4(c) and Fig. 4(d), respectively.The radiative behavior of the electric (solid line) and magnetic (dashed line) dipoles resembles the features observed in a dielectric sphere: both the perpendicular electric (blue solid line) and parallel magnetic (red dashed line) emitters couple well to an electric a 1 mode, while the parallel electric emitter (red solid line) couples also to the magnetic b 1 mode.The radiation enhancement in the latter case is much weaker because magnetic modes in plasmonic structures are strongly damped.The enhancement of the total decay rate (Fig. 4(d)) is dominated by coupling to higher order modes.These modes are very weakly radiative, but can couple to the dipolar emitter [12] and lead to efficient nonradiative decay.Nevertheless, the excitation of the dipolar electric and magnetic modes exhibits analogous dependence on the orientation and nature of the dipolar emitter as observed for the enhancement of the radiative rates.This suggests that, although the performance of the silicon antennas presented here is better than that of their metallic counterparts to control the emission from magnetic emitters, plasmonic antennas could be also considered as appropriate building blocks to this end.

Dipole-dipole interaction
The coupling in the previous examples was often dominated by dipolar modes induced in the sphere.It is convenient to study in more detail the distance dependence of the total decay rate.By limiting the sum in Eqs. ( 1)-( 8) to the n = 1 terms and taking the explicit form of the spherical Hankel function h (1) Fig. 4. Spectral features of the scattering efficiency of a 50 nm radius silver nanosphere in vacuum illuminated by a plane wave (a) and the radiative (c) and total (d) decay rate enhancements of an electric and magnetic emitter positioned in the vicinity of the particle.The geometries of the systems for calculating the plane wave extinction and the properties of dipolar emission are depicted in (b) and (e), respectively.In (a), the total scattering efficiency C s is split into different contributions: electric dipolar (a 1 ), magnetic dipolar (b 1 ), electric quadrupolar (b 2 ), according to Eq. (11).In plots (c) and (d), the enhancement factors of the, respectively, radiative and total decay rates of the dipoles are shown.In both plots, the spectra were calculated for the electric (solid lines) and magnetic (dashed lines) emitters.The orientation of the emitter with respect to the surface of the sphere is either perpendicular (blue lines) or parallel (red lines).The refractive index of the uniform silver nanosphere is taken from [27].decay rates within the dipole-dipole approximation: where the electric α E and the magnetic α M dipolar polarizabilities are associated with the first order Mie coefficients according to α E = ik 3 /(6π)a 1 and α M = ik 3 /(6π)b 1 , respectively.ℑ(z) denotes the imaginary part of z.
As in the exact formulas Eqs. ( 3),( 4),( 7),( 8), only Eqs. ( 13) and ( 15) depend on both magnetic (α M ) and electric (α E ) contributions.The perpendicular electric (magnetic) emitter couples only with the electric (magnetic) dipolar mode.In the cases when a single dipolar mode of the sphere, either electric or magnetic, dominates the response at a given wavelength, it is possible to describe the coupling at that wavelength as a simple dipole-dipole interaction.
To illustrate this situation, we show in Fig. 5 the enhancement of the decay rate as a function of the distance between the dipole and the center of the nanoparticle.Results were obtained with the use of the exact formulas given by Eqs. ( 1)-( 8) (solid black lines) as well as with the dipolar-interaction approximation (dashed red lines).We consider the 230 nm radius silicon and the 50 nm silver sphere as in the previous cases.To extract the dominant contributions, we choose the wavelengths corresponding to the dipolar electric a 1 and dipolar magnetic b 1 resonances, and set the polarizabilities of the complementary nature to 0 in Eqs. ( 12)- (15).Namely, in the case of the Si sphere, α E is neglected for λ = 1680 nm (corresponding to b 1 resonance) and α M is ignored for λ = 1350 nm (a 1 ).For the Ag sphere, we put the magnetic dipolar polarizability α M = 0 for λ = 430 nm (coinciding with the electric dipolar resonance a 1 ).
The dipole-dipole interaction model very accurately reproduces the exact results in all considered cases.In Fig. 5(a), we plot the decay rate enhancements of an electric emitter, perpendicular to the surface of the silicon sphere, radiating at λ = 1350 nm (dipolar electric antenna resonance).The disagreement between the curves can be attributed to the influence of the quadrupolar electric mode, not considered in the dipole-dipole approximation.In the analogous situation of an identically oriented magnetic emitter at the magnetic dipolar resonance (λ = 1680 nm, Fig. 5(d)), the agreement with the analytical solution is almost perfect, as no higher order magnetic modes contribute in this spectral range.Small differences due to the neglected dipolar polarizabilities can be seen for emitters oriented parallelly to the surface of Fig. 5. Distance dependence of the decay rate enhancements calculated using the exact formulas (solid lines) and the dipolar interaction approach (dashed lines).In plots (a-d), the emitter is positioned near the 230 nm radius silicon sphere and for (e-f) -near the 50 nm radius silver sphere.The wavelength of radiation matches the dipolar electric mode at 1350 nm for silicon (a-b) and 420 nm for silver sphere (e-f) or the dipolar magnetic mode in silicon antenna at 1680 nm (c-d).In each case, only the dominant induced dipole is considered for the dipolar approximation, while the mode of the complementary nature (magnetic or electric) is neglected.The insets show the orientation and the electric of magnetic nature of both the emitter (p emi or m emi , respectively) and the induced dipolar mode (p ind or m ind , respectively) in the antenna.The distance is measured between the dipole and the center of the antenna.For the silver antenna (Fig. 5(e)-5(f)), we show the distance dependence of the total decay rate enhancement for the radiation wavelength of 430 nm, which corresponds to the excitation of the dominant dipolar electric mode of the Ag particle.Almost perfect agreement between the exact solution and the results of the dipole-dipole interaction model is obtained both for the electric (Fig. 5(e)) and the magnetic emitter (Fig. 5(f)) oriented perpendicularly or parallelly to the surface of the antenna, respectively.The disagreements are mostly due to high order non-radiative contributions that become more significant for very short separation distances.However, we emphasize that the agreement between the simple model and the complete analytical solution is very good in all considered cases.

Conclusion
We have presented a detailed analysis of the radiative and non-radiative decay rates of a dipolar emitter positioned in the vicinity of a spherical particle.Based on Mie theory, we derived the general analytical expressions describing decay rates for both electric and magnetic dipole emitters.As we have shown, silicon nanospheres, with strong magnetic dipolar resonances in the near infrared, provide a canonical example of dielectric antennas that can be used to selectively enhance the magnetic dipolar emission.Interestingly, the resonant coupling between a dipolar (electric or magnetic) emitter and the dipolar Mie resonances is well described by a simple dipole-dipole interaction approach at certain wavelengths even when the emitter is in close proximity of the sphere surface.
Near the magnetic resonance, we found a strong enhancement of the decay rate of magnetic emitters which resembles the enhancement of electric dipole emission near a resonant plasmonic particle.However, while the total decay rate in plasmonic nano-antennas is often dominated by non-radiative channels, the total emission rate in loss-less silicon antennas is purely radiative.Our results show that nanosphere dielectric antennas are excellent platforms to enhance and manipulate magnetic dipolar emission with important possible applications as elements of infrared and telecommunication devices.

( 1 )
n (x) with the use of spherical Bessel j n (x) and Hankel functions h

Fig. 1 .
Fig.1.Spectral features of the scattering cross section of a 230 nm radius Si nanosphere in vacuum illuminated by a plane wave (a) and the decay rate enhancements of an electric or magnetic emitter positioned in the vicinity of the particle (c).The schematics of the systems are shown in (b) and (d), respectively.In (a), the total scattering efficiency C s is separated into different contributions: electric (a 1 ) and magnetic dipolar (b 1 ), electric (a 2 ) and magnetic quadrupolar (b 2 ), according to Eq.(11).Spectra of the radiative rate enhancements shown in (c) were calculated using Eqs.(2),(4),(6),(8) for the electric (solid lines) and magnetic (dashed lines) emitter.The emitter is oriented either perpendicularly (blue lines) or parallelly (red lines) with respect to the closest surface of the sphere.The refractive index of the silicon nanosphere is 3.5, while the distance from the emitter to the surface of the sphere is set to 50 nm.

Fig. 2 .
Fig. 2. Spectra of the decay rate enhancements of electric (p emi , (a,b)) or magnetic (m emi , (c,d)) nature near the silicon sphere of varying radius a in vacuum.Dipoles are oriented either perpendicularly (a,c) or parallelly (b,d) to the surface of the antenna, positioned at the fixed distance of 50 nm from its surface.Dashed lines correspond to the Mie resonances a n and b n , as denoted in each of the plots.Geometries of the setup are shown in the schematics.

Fig. 3 .
Fig. 3. Distribution of the electric (upper row) and magnetic (lower row) induced field amplitudes by an electric (p emi ) and magnetic (m emi ) dipole at the 50 nm distance from the surface of a 230 nm radius silicon sphere.Shown cross-sections contain the dipole and the center of the sphere.Plots (a-d) correspond to the dipolar electric mode at λ = 1350 nm, with the induced dipole denoted as p ind , excited by the perpendicular electric (a,b) or parallel magnetic (c,d) emitter.(e-h) illustrate the field distributions at the dipolar magnetic mode λ = 1680 nm (m ind ), induced by the parallel electric (e,f) or perpendicular magnetic (g,h) emitter.Schematics of the exciting and induced dipoles are shown at the top of the figure.Intensities of the induced fields E ind and H ind are normalized to the values of the fields E 0 and H 0 produced by the dipolar emitter in the absence of the particle and evaluated at the distance of 280 nm from the emitter in the direction perpendicular to its axis.The position of the normalization point is marked by a white cross in each case.