Plasmonic nanoparticles are commonly used to tune and direct the radiation from electric dipolar emitters. Less progress has been made towards understanding complementary systems of magnetic nature. However, it has been recently shown that high-index dielectric spheres can act as effective magnetic antennas. Here we explore the concept of coupling dielectric magnetic antennas with either an electric or magnetic dipolar emitter in a similar fashion to the purely electric systems reported previously. We investigate the enhancement of radiation from systems comprising admixtures of these electric and magnetic elements and perform a full study of its dependence on the distance and polarization of the emitter with respect to the antenna. A comparison to the plasmon antennas reveals remarkable symmetries between electric and magnetic systems, which might lead to novel paradigms in the design of nanophotonic devices that involve magnetic activity.
© 2012 Optical Society of America
CorrectionsM. 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: errata," Opt. Express 20, 18609-18610 (2012)
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 . Recent works have been devoted to controlling both the shape [2, 3], intensity [4–6] 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 [9–13]. This classical description is known to have a clear correspondence with the quantum results [14–16]. A particularly interesting system with an analytical solution, comprises an electric dipole in the vicinity of a spherical metallic nanoparticle , but more complex antenna architectures have also been studied [6–8].
An interesting aspect of dipolar emission is related to the magnetic versus electric nature of the dipolar emitter. Karaveli and Zia  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 , such as the excitation of magnetic resonances in metallic  and dielectric particles [20–22], 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 well-separated electric and magnetic dipolar resonances in the near infrared . 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 dipole-dipole interaction model allows to describe in a simplified manner the enhancement of emission rates in several relevant situations.
2. Electric and magnetic dipolar decay rates in the presence of a particle
Recent developments in the understanding and engineering of magnetic phenomena in nano-optics 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 non-radiative decay rates of an electric dipole positioned in the vicinity of a spherical nanoparticle have been derived by Ruppin . 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 [23–25]. 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 ΓRAD and the non-radiative decay rate ΓNON–RAD gives the total decay rate ΓTOT. 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 εe. The wavevector 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 Γ0. Within the linear response theory, the obtained enhancement factors are independent of the strength of the emitter and are given by
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) = xjn(x) and with the use of spherical Bessel jn(x) and Hankel functions of the first kind. The definitions of Mie coefficients an and bn used here are adopted from Bohren and Huffman :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 an ↔ bn . Furthermore, in the absence of absorption (real ), the enhancements of total and radiative decay rates can be shown to be equal, using the properties of the Mie coefficients .
3. 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 . 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 (σs(an)) and magnetic (σs(bn)) 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 Cs = σs/(πa2). 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 b2, the dipolar electric a1 and the dipolar magnetic b1 resonances.
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 b1 (at λ = 1680 nm), quadrupolar b2 (at λ = 1160 nm) and - although very weakly - the electric dipolar a1 (at λ = 1350 nm) resonance, while the same emitter aligned perpendicularly (solid blue line) couples only to the broad electric dipolar a1 mode at λ = 1350 nm. The complementary behavior is present for the magnetic emitter: for the perpendicular orientation (blue dashed line), only the magnetic b1, b2 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 (pemi) and magnetic (memi) 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 an), 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 an terms for the electric emitter and magnetic bn 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 a1 ((a–d), denoted in the schematics as pind) and magnetic b1 ((e–h), mind) dipolar modes. The excitations are due to the emitter of the electric (pemi) or magnetic (memi) 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 .
4. 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 a1 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 a1 mode, while the parallel electric emitter (red solid line) couples also to the magnetic b1 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  and lead to efficient non-radiative 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.
5. 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 , we obtain the following expressions for the enhancement of the decay rates within the dipole-dipole approximation:
As in the exact formulas Eqs. (3),(4),(7),(8), only Eqs. (13) and (15) depend on both magnetic (αM) and electric (αE) 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 a1 and dipolar magnetic b1 resonances, and set the polarizabilities of the complementary nature to 0 in Eqs. (12)–(15). Namely, in the case of the Si sphere, αE is neglected for λ = 1680 nm (corresponding to b1 resonance) and αM is ignored for λ = 1350 nm (a1). For the Ag sphere, we put the magnetic dipolar polarizability αM = 0 for λ = 430 nm (coinciding with the electric dipolar resonance a1).
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 the sphere: the magnetic emitter coupled to the silicon antenna at its electric dipolar resonance (λ = 1350 nm, Fig. 5(b)) and the electric emitter at the magnetic dipolar resonance (λ = 1680 nm, Fig. 5(c)).
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.
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.
Enhancements of the total and radiative decay rates of a magnetic dipolar emitter positioned in the vicinity of a sphere and fields distributions in the system
A formal derivation of a radiative and non-radiative decay rate enhancement is presented for a magnetic dipolar emitter m positioned at a location given by vector r′ near a homogeneous sphere of radius a and dielectric function ε, embedded in a medium of dielectric function εe. The sphere is centered at the origin of the coordinate system. As previously, the radiation of the dipolar emitter will be characterized by a wavevector , where λ denotes its vacuum radiation wavelength. The derivation is based on the work by Ruppin  in which the author considers an electric dipolar emitter positioned in the vicinity of a spherical particle.
The magnetic field of the magnetic dipolar emitter m can be expanded into a series of vector spherical harmonics Mν and Nν 
The above expansion and its coefficients can be derived by a transformation9], with the substitution p → m/c . Z is the impedance of the medium and c - the speed of light in the medium.
The scattered field and the field inside the sphere are expanded in a similar manner
The electric fields Edip, Esc and Etr of the magnetic dipole can be calculated using a relationEqs. (16), (25), (26) and the corresponding electric field expansions given by Eq. (30), we obtain from the boundary conditions in Eqs. (28) and (29) the following relations: Eqs. (9) and (10). The coefficients for the transmitted fields αn and βn are defined as 23], which differ from those used by Ruppin  and - more recently - Mertens et al.  by interchanging an ↔ −bn and αn ↔ −βn.
Evaluating the Poynting vector of the incident and scattered fields and integrating its flux over a very large sphere we obtain the power radiated from the systemEqs. (20)–(23), and normalizing the results to the power radiated by a unitary quantum efficiency dipole in the homogeneous medium, we obtain the radiative and total decay rate enhancements of the magnetic dipole (Eqs. (5)–(8)). It should be noted that the evaluation of these quantities requires notably fewer spherical harmonics to be calculated in comparison to an exact description of the strongly inhomogeneous near fields induced in the proximity of the sphere.
We note that our expressions are consistent with those introduced by quantum mechanical considerations of the magnetic dipolar transition between two states of the molecule in the vicinity of a lossless spherical nanoparticle .
Financial support from the ETORTEK 2011 “nanoiker” of the Department of the Industry of the Basque Country Government, the WELCOME program “Hybrid nanostructures as a stepping-stone towards efficient artificial photosynthesis” awarded by the Foundation for Polish Science, the Spanish MEC FIS2010-19609-C02-02, Consolider NanoLight (CSD2007-00046) and FIS2009-13430-C02 research grants and the Comunidad de Madrid Microseres-CM Program (S2009/TIC- 1476) are gratefully acknowledged. RE and JA acknowledge support from the Spanish National project Euroinvestigación EUI200803816 CUBiHOLE.
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