Directive and enhanced spontaneous emission using shifted cubes nanoantenna

Localized surface plasmon resonances in metallic nanostructures generate electromagnetic fields that enhance the linear and nonlinear optical effects near the surface of metals.¹ These resonances are capable of locally intensifying the electromagnetic field by many orders of magnitude and can provide light localization below the diffraction limit.^2–5 This characteristic has been applied for novel light sources and optoelectronic devices,⁶ ultrasensitive chemical and biomolecular detectors,^7–9 enhanced scattering,^10–12 metamaterials,¹³ and quantum information.¹⁴ 

An interesting phenomenon resulting from the near-field interaction of adjacent metallic nanoparticles is the optical field enhancement that takes place in the dielectric gap region between them.¹⁵ The gap is simply a capacitance that, when the separation between nanoparticles is small enough, leads to light concentration into deep-subwavelength volumes.^16–20 The strongly confined electromagnetic field can be used to improve the spontaneous emission rate of quantum emitters (QE) placed in the gap region.^21,22 This rate is characterized in terms of the so-called Purcell factor (⁠$Fp$⁠),^23–25 given by

where $Q$ is the quality factor of the resonator, $λ$ the resonance wavelength, $n$ the refractive index of the bulk material without the resonator, and $Veff$ is the mode volume—the spatial confinement of the electromagnetic field (⁠$E(r)$⁠) in the resonator—which is defined as

where $ε(r)$ is the material permittivity and the integration is performed over the volume containing the resonator.

Thus, increasing the quality factor and decreasing the mode volume lead to higher Purcell factors. To achieve this goal, many research groups have studied different nanoantenna configurations, such as bowtie,²⁶ conically shaped,²⁷ nano-patch antennas,^{17,21,22,28–31} and others.^32,33 The small mode volumes reported in these works are due to the concentration of the electromagnetic field at the sharp corners and edges of the metallic nanoparticles.

Of particular interest are the nanoantennas based on cubic-shaped nanoparticles³⁴ (Fig. 1(a)). These nanoparticles have recently been applied as nano-patch antennas (Fig. 1(b)) to increase the spontaneous emission rate of QE^21,22,28,35 characterized in terms of the $Fp$ or the localized density of states (LDOS).³⁶ However, these quantities do not provide information about the radiation pattern of the nanoantenna, an important parameter in the design of nanophotonic devices.³⁷ For example, on-chip applications require directive in-plane rather than out-of-plane emission in order to efficiently couple light to integrated waveguides.

In this work, we theoretically demonstrate that dimers formed by two metallic nanocubes present higher directivity and Purcell factor compared to the nano-patch antenna. After investigating the variation of $Fp$ for the nano-patch antenna as a function of the separation between the nanocube and the metallic film, we compare this configuration with a nanocube dimer and study the dependence of $Fp$ on the lateral shift, $d$⁠, between the cubes (Figs. 1(c)–1(e)). This system is reminiscent of the shift-bar particle introduced in Refs. 38 and 39. To the best of our knowledge, QE have not been coupled to the shift cube system to analyze the emission rate, the coupling efficiency, and the directivity. We also analyze a more realistic situation by rounding the edges and corners of the particles. Finally, using analytical expressions,⁴⁰ we compare the directivity of the radiation pattern and the antenna coupling efficiency in the two nanoantenna configurations.

We first consider the modes and Purcell factor of a single silver (Ag) nanocube of length $a=80$ nm surrounded by a dielectric homogeneous medium. A Drude model⁴¹ was used for the dielectric function of silver with $ε(ω)=ε∞−(ωp2)/((ω2−iΓω))$⁠, where $ε∞=3.92$⁠, $ωp2=1.33×1016$ (rad/s), and $Γ=2.73×1013$ (rad/s). The refractive index of the surrounding dielectric medium in all the simulations was fixed to 1.5, close to the values of standard polymers used for colloidal solutions. However, this choice does not affect the conclusions of this paper.

The first three modes of the single nanocube are illustrated in Figs. 2(a)–2(c). Simulations were performed using COMSOL Multiphysics (a finite element method (FEM)-based software). From the charge distribution of these modes, we identify the vertical and horizontal electric dipoles (Figs. 2(a) and 2(b)), and an electric quadrupole (Fig. 2(c)). We must note that each mode is degenerated due to the symmetry of the system,³ but since they are equivalent in terms of the Purcell factor, we only represent one of them.

The quadrupolar (Q) mode presents smaller mode volume and larger quality factor compared to the dipolar (D) modes. This reduction in the mode volume stems from the confinement of the electromagnetic field at the corners of the nanocube, thereby leading to a large Purcell factor, $Fp=1750$⁠, at the wavelength of $λ=583$ nm.

To localize the field via capacitive effects and increase $Fp$⁠, the nanocube was placed in front of a Ag film at a distance, $g$⁠, forming a nano-patch antenna.^21,22,28 This configuration will be subsequently denoted as Face-Surface (FS). Figs. 2(d)–2(f) show the first three modes of FS, which are similar to those of the single nanocube in terms of charge distribution (see inset). However, the energy distribution over the nanocube is now non-uniform and concentrated in the gap between the metallic film and the nanocube. The Purcell factor, quality factor, and mode volume for all modes are listed in Table I. The Q mode has the largest $Fp$ not only because of the largest quality factor but also due to the smaller mode volume arising from the high energy concentration at the corners in the gap region.^21,22,28 In contrast, for the D modes, the energy is mainly spread at the surface of the nanocube in the gap region.

In the remaining of the paper, we will focus on the Q mode. This mode has also been used in Ref. 42 to produce Fano lineshapes.

Fig. 3 shows the variation of $Fp$ for the Q mode in FS configuration as a function of the gap separation, $g$⁠. When the gap is small, the absorption of the electromagnetic field is larger than the stored energy in the cavity.⁴³ By increasing $g$⁠, the losses due to absorption are reduced and the Purcell factor reaches a maximum value around $g=10$ nm (⁠$Fp=2925$⁠, $λ=660$ nm). Beyond this gap distance, light confinement in the cavity is reduced, until $Fp$ tends toward the value of the isolated nanocube.

To further increase $Fp$⁠, we propose an alternative configuration by replacing the metallic film by a second Ag nanocube, obtaining a dimer nanoantenna (Fig. 1(c)). The choice of this design is realistic and achievable by new fabrication techniques based on bottom-up approaches and is amenable to large scale fabrication.^44–47 When the faces of the two nanocubes are aligned (Fig. 1(c)), the configuration is denoted as Face-Face (FF).

We then analyze the variation of the $Fp$ as a function of the lateral shift, $d$⁠, between the nanocubes (Fig. 1(d)). When their edges are aligned (Fig. 1(e)), the configuration is denoted as Edge-Edge (EE). In order to compare them with the nano-patch antenna, we set the gap distance between the nanocubes to $g=10$ nm, the value that produced the largest Purcell factor in the FS configuration.

When the nanocubes are shifted, the mode volume starts to decrease as the capacitance formed by the two nanocubes is now composed of less surfaces facing each other. For $d=80$ nm (EE), we obtained $Fp=5350$ at $λ=662$ nm. The results are summarized in Fig. 4. As shown in Fig. 4, in the FF configuration, the energy of the mode at the gap region is concentrated into the eight internal corners of the dimer, while in the EE configuration, the energy is mainly confined at the four internal corners due to the reduction in the mode volume.

A comparison of the results shows that the Purcell factor for the EE configuration is larger by a ratio of more than three- and almost two-times compared to the FF and the FS, respectively. While these results were obtained for identical nanocubes, realistic experimental samples can present fabrication imperfections such as rounded corners, size mismatch, misalignment, or roughness. The $Fp$⁠, however, mainly arises from the small mode volume at the sharp edges of the nanocubes, making the rounded corners the main fabrication challenge to overcome.⁴⁸ 

We illustrate these limitations by calculating the Purcell factor for the FF and EE configurations when the nanocubes are size mismatched. In the FF configuration, by increasing the size of one of the nanocubes, the larger nanocube tends towards a metallic ground plane for the other nanocube; therefore, FF asymptotically becomes FS. In Table II, we present the variation of the Purcell factor when one of the nanocubes has a size mismatch of 20 nm, 10 nm, and 0 nm compared to the other one. The field distributions of the Q mode for these four different nanoantennas are plotted in Fig. 5.

Similarly, by increasing the size mismatch between the nanocubes in EE configuration, the Purcell factor decreases. This stems from the fact that an increment of the distance between the edges increases the mode volume and decreases the Purcell factor (Equations (1) and (2)). The computed Purcell factors for a size mismatch of 10 nm and 20 nm between the nanocubes are given in Table III, and the corresponding field distributions of the Q mode are depicted in Fig. 6.

Finally, we performed a quantitative comparison of the Purcell factor between FS and EE, as a function of the radius of curvature, $r$⁠, of the edges and corners (Fig. 7) of the nanocubes.

As observed in Fig. 7, $Fp$ decreases with $r$⁠. This is because the energy density is distributed in a larger volume of the nanocubes when the corners are rounded.⁴⁹ The Purcell factor decreases faster in the EE configuration (red circles) than in the FS configuration (blue circles). When $r$ is about $10$ nm, $Fp$ is almost the same for both nanoantennas (⁠$Fp=170$⁠). We would like to note that this value of the Purcell factor (⁠$Fp=170$⁠) is larger than the one of the D mode in the FS configuration with sharp corners (⁠$Fp=60$⁠). For the D mode with a radius of curvature of $r=10$ nm, the Purcell factor is $Fp=34$⁠. If the fabricated nanocubes are such that their radius of curvature is smaller than $4$ nm, the difference between the analyzed nanoantenna configurations (FS, FF, and EE) becomes relevant.⁵⁰ These results will give guidelines in the fabrication of complex antenna systems.

Until now, we have calculated the Purcell factor for different nanoantenna structures. As defined in Equation (1), this number simultaneously relates the material properties, mode volume, and quality of the resonating structure. However, for applications requiring radiation of QE towards specific directions, as in integrated photonics, we need to consider both nanoantenna design and coupling efficiency from the QE to the resonator. Neither $Fp$ nor LDOS provide an intuitive insight about the directivity. Instead, antenna design parameters such as directivity, gain, and radiation efficiency need to be considered.

The directivity of an antenna is defined as the ratio of the radiation intensity in a given direction to the radiation intensity averaged over all directions. The directivity can be calculated by⁴⁰ 

where $U(θ,ϕ)$ is the radiation intensity, and $Prad$ the total radiated power. The gain of an antenna is defined as the ratio of the intensity in a given direction to the radiation intensity that would be obtained if the power accepted by the antenna was radiated isotropically. This parameter is given by⁴⁰ 

where $Pin$ is the input power. The relationship between gain and directivity is⁴⁰ 

where $e0=ec ed er$ is the antenna radiation efficiency, $ed$ the dielectric efficiency, $ec$ the conduction efficiency, and $er$ is the reflection (coupling) efficiency that corresponds to the coupling mismatch between the source and the antenna. The coupling efficiency, $er$⁠, is one of the key parameters that shows the amount of energy emitted from a QE coupled to the nanoantenna.

The radiation patterns of the studied nanoantennas (FS and EE) were numerically computed with an eigenmode solver (Figs. 8(a) and 8(b)), and from these results, we calculated the directivity and gain through Equations (3) and (4) (see the supplementary material).

Because the eigenmode solver is source-free (no QE is involved), the coupling mismatch efficiency, $er$⁠, is one, then $e0=eced$⁠. We term this factor, $ecd=eced$⁠, “material efficiency” since it contains the dissipation of energy into the materials that have been used. The results of the directivity and material efficiency for both EE and FS configurations are shown in Table IV.

As shown in Figs. 8(a) and 8(b), the dimer nanoantenna presents a more directive radiation pattern compared to the nano-patch antenna, because in the former one, the radiation is parallel to the principal axis of the EE dimer, while for FS, the radiation pattern is doughnut-shaped in the plane parallel to the metallic film. In addition, an intrinsic characteristic of a Q mode is that its radiation pattern is null in the plane or axis of symmetry (see, for example, Chapter 5 in Ref. 40). Hence, for FS, no radiation is observed along the $z$-axis, while for EE no radiation is observed in the $xz$-plane.

The mode mismatch efficiency, $er$⁠, is obtained from the radiation patterns when the structure is excited with a QE (Figs. 8(d) and 8(e)). The QE was simulated as a dipolar source of 7 nm length and placed near the position of the electric field hot-spots to efficiently excite the nanoantennas (Figs. 8(c) and 8(f)). We must note that the higher F_p and antenna radiation efficiency of the Q mode comes at the expense of difficulties to position emitters in the electric field hot-spot. The results of the directivity, material, and coupling mismatch efficiencies for both EE and FS configurations are shown in Table IV. The results show that $5%$ of the power radiated by the dipole is coupled to the EE nanoantenna, which is three times larger than the FS (⁠$1.6%$⁠).

Flowing information in specific directions could be advantageous for certain applications. For example, the dimer nanoantenna is more suited for integrated optics as it can be more efficiently excited through a waveguide compared to the nano-patch antenna, which requires a metallic film substrate.

In summary, we proposed the shifted-cube nanoantenna as a platform for directive and enhanced light-matter interaction. The system exhibits higher Purcell factor than nano-patch antennas, thanks to the reduction in the mode volume, in the Edge-Edge configuration.

By increasing the radius of curvature of the nanocubes, the mode volume increases, until the patch-nanoantenna and the shifted-cube nanoantenna present the similar $Fp$⁠. Such dimer structures will thus only be competitive if the quality of realized nanostructures is high.

We also showed that the Edge-Edge dimer presents a more directive radiation pattern as well as higher coupling efficiency to nanoscale emitters even though this will come at the expense of difficulties to place emitters at the corresponding hotspots.

Because of the directional radiation, the dimer nanoantenna and similar antennas using particles cluster represent promising candidates for integrated photonic applications, such as single photon emission/detection through waveguide structures.

See supplementary material for more information about COMSOL Multiphysics simulation details.

This work was supported by the Office of Naval Research Multi-University Research Initiative (N00014-13-1-0678), the NSF Career Award (1554021), the Darpa/Ziva Award No. 20144161, and the Hellman Fellowship.