Dielectric particles supporting both magnetic and electric Mie resonances are shown to be able to either reflect or collect the light emitted by a single photon source. An analytical model accurately predicts the scattering behavior of a single dielectric particle electromagnetically coupled to the electric dipole transition moment of a quantum emitter. We derive near field extensions of the Kerker conditions in order to determine the conditions that strongly reduce scattering in either the forward or backward directions. This concept is then employed to design a lossless dielectric collector element whose directivity is boosted by the coherent scattering of both electric and magnetic dipoles.
© 2012 OSA
Emission direction control is of crucial importance when dealing with single photon sources. Different kinds of metallic nanoantennas including Yagi Uda [1–5] or patch  antennas have been proposed over the last 5 years to shape emission patterns. All of these geometries rely on the coupling between the electric transition dipole moment of the emitter with the electric dipolar or multipolar resonances supported by coupled metallic nanostructures. Compact designs exploiting the interferences between electric induced dipoles on neighboring metallic particles have been proposed [7, 8]. These antennas exhibit significant gains in directivity, but the losses inherent to the use of metals in optics spoil the quantum yield of the emitter, particularly problematic when the emitter is very close to the surface of the metallic elements [9–12].
Lossless dielectric materials on the other hand do not suffer from this handicap, and a quantum emitter can furthermore be brought as close as necessary to an antenna element when looking to optimize directivity [13,14]. Hybrid metallo-dielectric antennas are good candidates for designing compact optical antennas that are both radiative and highly directive since they combine the property of metallic particles to strongly increase the radiative properties of single photon sources with that of dielectric particles to redirect the emitted light into narrow lobes via morphological (Mie type) resonances . When illuminated from the far field region, the high quality factor of the dielectric resonator permits the collection of large amounts of incident light while a plasmonic resonator can concentrate this energy into nanometric volumes [15,16]. Unfortunately, decreasing the size of the dielectric collector weakens the strength of multipolar resonances, resulting in a decrease of the directivity gain.
We propose to circumvent this problem by using interference effects between the electric dipolar transition moment and the dipolar response of a neighboring collector particle element, the induced dipoles being both electric and magnetic in nature. The choice of the material is of crucial importance since this new kind of collector element must exhibit both magnetic and electric dipolar resonances without losses. The active field of metamaterials has investigated magnetodielectric and polaritonic materials to create left handed materials based on the resonant excitation of magnetic dipoles in dielectric resonators of high permittivity [17–19]. Unfortunately, such materials suffer from high losses in the infrared and optical frequency range. But recently, semiconductors with moderate (lossless) refractive indices in these spectral ranges have however been demonstrated to exhibit homogeneous negative magnetic permeability [20,21] and Evlyukhin et al. were recently able to fabricate single particles of silicon and to characterize their electric and magnetic resonances .
The interest of combining electric and magnetic dipoles to control the direction of light scattering was first proposed by Kerker in the case of a far field illumination . He demonstrated that by controlling the amplitude and the phase of the electric and magnetic dipoles of a single magnetodielectric particle illuminated by a plane wave, it is possible to suppress either the backward or the forward scattering. The so-called first Kerker condition is that there is a zero backward scattering when the condition ε = μ is satisfied (ε and μ are the relative permittivity and permeability of the sphere, respectively) and the second Kerker condition, derived in the small particle approximation, is that of vanishing forward scattered power when ε = (4 − μ )/(2μ + 1). Recently, an ensemble of magnetodielectric spheres were used to create a Fano-like transmission line shape . The coherent combination of electric and magnetic Mie dipolar resonances in purely dielectric germanium spheres leads to intriguing scattering properties, offering nearly perfectly vanishing scattering in either the backward or forward direction  and “generalised” Kerker conditions have been derived . Metallo-dielectric core-shell particles that offer tunable magnetic and electric resonances are also investigated to conceive negative index materials or to control the light scattering by a chain of self-similar particles illuminated by a propagating wave [27,28]. The electric and magnetic Mie resonances of a single Si nanoparticle were also recently applied to promote either the electric or magnetic dipolar transition of trivalent lanthanide ions in the near infrared regime .
In this study, we first show that a single dielectric particle can either reflect or collect the light emitted by a neighboring electric dipole. The difference of phase between the emitting and the induced dipoles is highly dependent on the distance  and we display the reflecting and collecting behavior of a single GaP sphere as a function of its radius and its distance from the emitting electric dipole with the use of the Generalized Mie Theory. We also provide an analytic dipolar model that provides insights into the role of the magnetic dipole in the collecting behavior. The derivation of this analytical model allows us to demonstrate that the coupling between the emitter and both electric and magnetic induced dipoles in the particle boosts the directivity with respect to that obtainable using one induced dipole moment only. We finally apply these findings to design a highly compact hybrid antenna about λ/2n in size that exhibits gains in directivity higher than 6 dBi over a wide range of wavelengths.
2. Reflecting or collecting light with an electro-magnetic resonant dielectric particle
2.1. Tailoring the directivity of the antenna by tuning the emitter-particle distance
We consider an electric dipole emitting at a vacuum wavelength of λ0 = 550 nm coupled with a single spherical particle made of GaP  whose refractive index is taken equal to 3.45 and the entire system is embedded in a homogeneous dielectric medium of refractive index 1.45. The coupling is transverse, meaning that the dipolar emitter moment is normal to the emitter-sphere separation axis. The emitting dipole is located at the origin, while the spherical particle is centered at a position +d along the z axis. The ratio of the collected power (power emitted in the z > 0 half-space) over the total emitted power is displayed in Fig. 1 as a function of both spherical resonator radius and distance between the emitter and the surface of the sphere. When this ratio is under (respectively over) 50%, the sphere is thus reflecting (respectively collecting) the light emitted by the dipolar source. Figure 1(b) is similar to Fig. 1(c), but it is obtained using an analytic dipolar model derived in the following to unveil the physics behind the ability of dielectric particles to control the direction of light emission. We remark a strong collector behavior of the sphere when the radius is around 85 nm and the emitter is located at or near the surface of the particle. Conversely, a strong reflector behavior of the sphere occurs for a radius of 90 nm and an emitter-sphere separation of 85 nm. Radiation diagrams are illustrated in Fig. 2 for a 85 nm GaP sphere placed respectively at z = 10 nm and z = 100 nm from an emitter confirm that a single sphere can efficiently collect or reflect the emitted light.
The Generalized Mie Theory permits to easily filter the electric and magnetic contributions. We take benefit of this method to calculate the gain in directivity (Fig. 3) of a single particle when both electric and magnetic Mie resonances of the particle are taken into account (Fig. 3(a)), and when the electric (Fig. 3(b)) and magnetic (Fig. 3(c)) contributions only are considered. The gain in directivity expressed in isotropic decibels (dBi), DdBi, is defined as DdBi = 10log10 (4π P/Γrad) where P is the power per steradian emitted in the direction of the +z axis and Γrad the total power emitted in the medium (for comparison, an isolated electric dipole radiates with a directivity of 1.76 dBi). We see that the gain in directivity can be over 6.5 dBi without filtering while it is limited to 5.2 dBi and 4.4 dBi without electric and magnetic contributions respectively. This result hilights that the coherent scattering of both magnetic and electric Mie resonances boosts the directivity of optical antennas. The next section aims to explain this result by analytically extending the Kerker conditions to near-field optics.
2.2. Analytical derivation of the radiation pattern of the antenna
Let us now derive analytical formulas for the Poynting vector in the far field region when an electric dipole emitter is transversely coupled to a sphere that is dominantly characterized by electric and magnetic dipolar responses. Fig. 4 defines the angular coordinates used in this section. The surrounding medium has relative permittivity εm and vacuum permeability μ0. The electric and magnetic fields produced by an electric dipole with dipolar moment denoted p, placed at origin are given by :33] while its amplitude whose correction is proportional to d/r is not modified in the far-field approximation: 32]:
The amplitude of the emitting electric dipole is taken equal to unity with a phase reference equal to zero: pem = x̂. The cartesian directions +x,+y will be the phase references respectively for the electric and magnetic dipoles. Using Eqs. (1) and (2), the field produced by the emitter at a distance d on the +ẑ direction can be cast:12] and provide the relation between the field of the emitter at this position and the electric and magnetic induced dipoles (we use dimensionless polarizabilities for notational simplicity): Eqs. (3) to (8), and replacing pin and min by their expressions in Eqs. (9) and (10), the far fields from the 3 dipoles together can be cast:
The time-averaged Poynting vector can be cast:
θ̂x × φ̂x and θ̂y × φ̂y both equal r̂ since (r̂, θ̂i,φ̂i) is an orthonormal base (i = x,y,z). For notational simplicity we choose to use the reduced cartesian coordinates, i.e. the x,y,z coordinates of the unit radial vector r̂. With (l,m,n)=(x,y,z), (y,z,x) or (z,x,y) we have:
In this equation, the first term in the brackets on the right-hand side results from the emission and interference in the far field region between the two electric dipoles, both being oriented along the x axis. The second term originates from the emission of the magnetic induced dipole (oriented on the y axis), while the last term corresponds to the interference between the induced magnetic dipole and the two electric dipoles. This latter term is null in the z = 0 plane since the electric and magnetic fields produced by the magnetic dipole on the one hand, and the two electric dipoles on the other hand, are orthogonal in this plane.
2.3. Derivation of the Kerker’s conditions in near-field optics
The collector behavior of the dielectric antenna will usually be optimized when the Poynting vector P(0,0,−r) directed towards the −z direction is minimized :Eq. (13)) is not fully verified, the condition on the phases (Eq. (12)) is well satisfied, and the radiation pattern in Fig. 2(a) shows a sharp minimum in the backward direction.
The condition in Eq. (13) is that of a total destructive interference between the two electric dipoles and the magnetic dipole in the −z direction. The analogy with the generalized first Kerker condition  is clear. The first Kerker condition was defined for the case of a magneto-dielectric sphere illuminated by a plane wave and results in zero backscattering from the sphere . However, when the sphere is illuminated from the near field region, the emitting dipole must also be taken into account. In order to cancel the light emitted by the dipole in the backscattering direction, the induced dipole must scatter in this direction and thus applying the first generalised Kerker condition will not yield a zero backscattering radiation diagram. Equation (13) is thus an extension of the first Kerker condition when dealing with optical antennas.
In the top row of Fig. 5, we show that neither the electric nor the magnetic induced dipoles can explain the collecting behavior of the spheres. If the sphere only behaves as an induced electric dipole (Fig. 5(a)), the small distance between the emitter and the sphere together with a phase of the electric polarizability of ≅ 0.35π > π/4, results in a reflector behavior . On the other hand, if the sphere only behaves as a magnetic induced dipole (Fig. 5(b)), the emitting electric dipole (oriented on the x̂ axis) and the magnetic dipole (oriented on the ŷ axis) would not interfere in the z = 0 plane, resulting in the majority of the far-field radiation being emitted in this plane. When both electric and magnetic induced dipoles are considered however (Fig. 5(c)), interferences result in a lobe in the +z direction, with little power emitted in the −z direction.
Canceling the forward scattering would require the fulfillment of the condition:26], which leads to minimal forward-scattered power by the sphere when it is illuminated with a plane wave. This condition (Eq. (14)) is not satisfied here (indeed, we see in Fig. 2(b) that the forward scattering is not eliminated). Once again however, neither the electric nor the magnetic induced dipoles can fully explain by themselves the radiation diagram observed for a a ≈ 85 nm radius GaP sphere placed at 100 nm from the emitter (see Fig. 5(d–f)), while their combined influence produces a pronounced reflector behavior.
3. Hybrid metallo-dielectric antenna exhibiting both electric and magnetic resonances
We now illustrate the gain in directivity offered by the magneto-electric collector to design a compact, radiative, and directive nanoantenna. The term “compact” is justified by the fact that the maximal antenna size is about λ /2n for its designed emission frequencies. The antenna is composed of a GaP collector sphere 150 nm in diameter coupled with a quantum emitter located in the 8 nm nanogap of a silver particle dimer with radii 30 nm [34, 35]. The role of the silver dimer coupled to the emitter is to highly enhance the decay rate of the dipolar transition. Its emission pattern remains similar to that offered by a single electric dipole in a homogeneous medium. The emitter is separated by 30 nm from the surface of the dielectric particle and the embedding medium has a refractive index of 1.45 (Fig. 6(a)). Figure 6(b) displays the emission pattern and confirms the high directivity offered by this hybrid metallo-dielectric antenna.
The radiative decay rate enhancement is the ratio of the radiative power of the antenna to the radiative power of the emitter placed in the homogeneous background, Γrad/Γ0. The quantum efficiency, η, is the ratio of the radiative power by the total dissipated power, η = Γrad/Γtot . Figure 6(c) shows that the lossless magneto-electric collector permits to further enhance the radiative decay rates over a wide range of wavelengths (as compared to the isolated metallic dimer antenna, see Fig. 6(c)). This compact hybrid antenna exhibits a gain in directivity higher than 6 dBi, a radiative decay rate enhancement factor larger than 103, and a quantum efficiency above 55% over a wide range of wavelengths with all three properties being satisfied for a 30 nm range, centered around 530 nm. Similarly to the case of a metallic collector composed of a self-similar chain of particles [1–5], the gain in directivity of this antenna could be further enhanced by considering a self similar chain of dielectric particles [37, 38].
In this study, a single particle element is shown to either reflect or collect the light emitted by a single photon source. The directivity can be tuned with a large amplitude by controlling the distance between the emitter and the particle element at a λ/6n scale. This property is fully predicted by the accurate dipolar model derived in this study. This analytical model permitted us to extend the so-called Kerker conditions to nano-optics where optical antennas are illuminated by electric dipolar transition moments. We took benefit of the GMT to show that combining electric and magnetic Mie resonances in optical antennas permits to boost the gain in directivity. We applied this formalism to design a hybrid nanoantenna whose size around λ/2n does not forbid gains in directivity higher than 6 dBi over a wide spectral range. The detailed analytical derivation presented in this study highlights the key role played by the coherent scattering between magnetic and electric dipoles in the control of the light scattered by single quantum emitters and we believe the latter will soon play a major role in the field of optical antennas.
The authors acknowledge Stefan Varault for his careful reading of the analytical derivations and Sebastien Bidault for stimulating discussions. This research was funded by the French Agence Nationale de la Recherche under Contract No. ANR-11-BS10-002-02 TWINS.
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