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Abstract
The rate of spontaneous emission from an optical emitter can be greatly enhanced using a metallic optical antenna at the penalty of efficiency. In this paper we propose a metal-dielectric antenna that eliminates the tradeoff between spontaneous emission enhancement and radiative efficiency by using nanoscopic dielectric structures at the antenna tips. This tradeoff occurs due to Ohmic loss and is further exacerbated by electron surface collisions. We find that our metal-dielectric antenna can enhance spontaneous emission by a factor 5 × 105 with efficiency = 70%, greatly exceeding the radiative efficiency of a purely metallic antenna with similar enhancement. Moreover, the metal-dielectric antenna design strategy is naturally amenable to short-distance optical communications applications. We go on to discuss the Purcell effect within the context of antenna enhancement. Metallic optical antennas are best analyzed with conventional antenna circuit models, but if the Purcell enhancement were to be employed, we provide the effective mode volume, Veff = (3/4π2)2 d2λ(λ/l)5, that would be needed.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Enhancing the rate of decay and spontaneous light emission from nanoscale optical sources using antennas has been the subject of considerable classical and contemporary research [1–8], with potential applications in spectroscopy [9–12], single-photon sources [13–16], and efficient on-chip optical data communications [17–23]. Metallic optical antennas are well-suited for spontaneous emission enhancement because electromagnetic fields are naturally confined to sharp metallic tips, thereby boosting the radiative transition rate of excited molecules near the tips by the increased electric dipole interaction potential [24,25]. However, one finds that large enhancement factor comes at the expense of inefficiency in metallic optical antennas [17]. In Section 2, we discuss this tradeoff of enhancement versus efficiency, which occurs because of Ohmic loss and is further exacerbated by nonlocal surface collision effects [17,26–28]. To alleviate loss, metal-dielectric antennas have been proposed [29–35]. These antennas typically use lossless dielectrics to reduce Ohmic loss by pulling the highest field regions away from the lossy metal, which may come at the penalty of reduced antenna enhancement. In Section 3, we propose a novel metal-dielectric antenna that leverages dielectrics for extreme near-field light focusing–inspired by the purely dielectric cavities in [36,37], but also benefitting from the presence of metal. The proposed antenna improves radiative efficiency and maintains the ultra-high spontaneous emission enhancement usually attained by purely metallic antennas. Section 4 demonstrates that the metal-dielectric antenna design principle is applicable to electrically-injected antenna-enhanced light-emitting diodes (antenna-LEDs), which can be used for on-chip optical communications. In Section 5, we derive a new antenna effective mode volume formula, which permits continued use of the Purcell effect for describing antenna enhancement. We compare the effective mode volume formula to a full electromagnetic numerical analysis in Section 6.
2. Tradeoff of enhancement versus efficiency in metallic optical antennas
Optical antenna-enhanced spontaneous emission can be regarded as the increase in steady-state radiated power from an oscillating dipole when coupled to an optical antenna:
where $P_{\textrm {o}}$ is the nominal radiated power from the light source without the antenna present and $P_{\textrm {rad}}$ is the radiated power with the antenna. For consistency, the reference source power $P_{\textrm {o}}$ is chosen to be a point dipole emitting into free space. Note that in Eq. (1), $P_{\textrm {rad}}$ includes only the radiated power, not the power that goes into Ohmic heating. To account for these additional metal losses, the antenna efficiency is defined as: where $P_{\textrm {loss}}$ is synonymous with Ohmic loss. Neither antenna directivity nor waveguide mode-matching efficiency will be considered here.Consider the metallic dipole antenna in Fig. 1(a). The optical antenna consists of two cylindrical silver wires with 25 nm radii. At the center feedgap the antenna includes sharp cone-shaped metallic tips that are adjacent to an optical point dipole source, which could represent a dye molecule or other atomically sized emitter. Importantly, the tips are separated by a vacuum gap of width $d$. In the limiting case where $d=1$ nm the radius of curvature at the tips is 1 nm, but the radius of curvature increases as $d$ increases. Practically speaking, a 1 nm tip is technologically difficult to achieve, with at least one recent report claiming experimentally fabricated metallic tips of this dimension [12] to the authors’ knowledge. Nevertheless, in this report we will examine several antennas with very sharp nanoscale tips in order to investigate their limiting behavior.
Fig. 1. The efficiency of metallic antennas suffers due to spreading resistance and surface collisions. (a) Metallic dipole antenna. An optical point source resides in a vacuum gap of length $d$ between sharp metallic tips (minimum radius of curvature = 1 nm, cone angle = 90$^\circ$). (b) The simplified circuit model of metallic optical antenna shows the antenna radiation resistance in series with a parasitic spreading resistance. (c) The spontaneous emission enhancement of the metallic antenna versus the vacuum gap $d$ at a wavelength of $\lambda =1550$ nm calculated using both a circuit model (black line) [17] and full 3D FDTD simulations (red squares). (d) The efficiency of the metallic antenna versus the vacuum gap $d$, calculated by circuit model (black line) and FDTD (red squares). For small $d$, the efficiency falls off dramatically due to spreading resistance. Also shown is the antenna efficiency that includes an estimate of the surface collision effect in the sharp tips (dashed line), which further exacerbates the spreading resistance effect.
Eggleston et al. [17] demonstrated a circuit model for a metallic dipole antenna similar to that shown in Fig. 1(a). We have provided a condensed explanation of the antenna circuit model in Appendix A and a simplified illustration of the antenna circuit model is presented in Fig. 1(b). The point dipole source is modeled as a current source ($J_{\textrm {source}}$) in series with radiation resistance in the antenna arms ($R_{\textrm {radiation}}$, which accounts for radiated light) and a parasitic spreading resistance ($R_{\textrm {spread}}$, which accounts for most Ohmic loss). The enhancement predicted by the antenna circuit model is plotted in Fig. 1(c), which agrees with full 3D Finite-Difference Time-Domain (FDTD) Maxwell simulations (black curve and red points, respectively).
When $d$ is small, very large antenna enhancement is accompanied by a severe drop in antenna efficiency, as revealed in Fig. 1(d). Antenna efficiency decreases dramatically because of spreading resistance [38,39], which is inversely proportional with the vacuum gap width; $R_{\textrm {spread}}=2\cdot \textrm {resistivity}/d$. Both the antenna circuit model and the full wave FDTD simulation correctly account for spreading resistance, as shown in the solid black curve and red points respectively.
However, there are additional losses associated with electron surface collisions that are not captured by Maxwell simulators, sometimes called the anomalous skin effect. This is a nonlocal effect that does not appear in optical material data handbooks due to its contingency on the specific geometrical structure of an optical material and motion of free electrons in the confined geometry. In fact, it is mentioned as one of the main sources of error for metals in Palik’s Handbook [26]. Consequently, this effect is not included in Maxwell simulators and some prior investigators have been overoptimistic with regard to efficiency.
Eggleston et al. [17] modeled electron surface collisions in the dipole antenna, which we have reproduced here in the dashed line of Fig. 1(d). Surface collisions increase the effective spreading resistance in the concentrated current region near the center antenna feedgap region by the factor $(1+l_\infty /\beta d)$, where $l_\infty$ is the bulk electron mean free path in silver, $d$ is the vacuum gap width (which also defines the radius of curvature at the antenna tips), and $\beta$ is a numerical parameter that requires an intricate non-local electrodynamic calculation. Note that the factor $(l_\infty /\beta d)$ multiplying the spreading resistance can be regarded as a term that corrects the mean free path of electrons from the nominal bulk mean free path $l_e\approx l_\infty$, to a net mean free path that is contingent upon the radius of curvature in the confined metallic tips [27,28], $l_e\approx \beta d$. In Fig. 1(d) we plotted an estimate of this surface collision effect with $l_\infty =50$ nm [27,28] and $\beta =0.5$. With our chosen parameter $\beta =0.5$, the surface collision effect bounds the expected antenna efficiency to $\approx$50% for a practical antenna gap, $d=10$ nm. Note that if a shorter bulk mean free path ($l_\infty$) were to be used in our circuit model, the surface collision effect would be less severe because the factor $l_\infty /\beta d$ would be reduced. Thus, the use of $l_\infty$=50 nm in our calculation can be regarded as a worst-case modification to the antenna efficiency compared to FDTD simulation.
3. Metal-dielectric antenna
In the previous section we demonstrated that purely metallic antennas suffer from poor efficiency at small gap width $d$ due to Ohmic losses. In this section, we will show that by including dielectrics in the antenna design we can greatly boost the antenna efficiency without significantly compromising the antenna enhancement at small $d$. This metal-dielectric antenna performs the best balance between all-metal and all-dielectric antenna designs when combining the two metrics of enhancement and efficiency.
Work from Vanderbilt and MIT demonstrated that the effective mode volume (i.e. the spatial light field concentration) of photonic crystal cavities is drastically improved by using sharp dielectric tips [36,37]. This electromagnetic enhancement effect surpasses the anticipated enhancement associated with simple dielectric boundary conditions. Furthermore, dielectrics are effectively lossless compared to metals so this field concentration can be achieved with no series resistance limitation. We will demonstrate that metallic antennas augmented with dielectric tips can improve antenna efficiency while maintaining large enhancement.
Consider the metal-dielectric antenna in Fig. 2(a). This antenna is similar to the all-metal antenna in Fig. 1(a) except that the sharp metal tips have been replaced by sharp dielectric tips (refractive index $n=3.4$) covering hemispherical metal tips, indicated by the white dashed lines. The dielectric tips are separated by vacuum gap $d$, while the larger metal-to-metal distance at the metallic hemisphere tips is fixed to 20 nm. In Figs. 2(b) and 2(c) we compare the enhancement (Eq. (1)) and efficiency (Eq. (2)) of the metal-dielectric antenna versus the all-metal antenna from Fig. 1. The enhancement factor (Eq. (1)), was determined by direct FDTD computation, which is reliable for enhancement factor but not for efficiency. The efficiency was obtained by FDTD with a correction provided by the surface collision effect circuit model given by [17] using numerical coefficient $\beta =0.5$ and 20 nm metal-to-metal spacing. We find that the metal-dielectric antenna maintains ultra-high efficiency with little compromise to the peak enhancement at small dielectric tip spacing $d$. For $d=1$ nm, the metal-dielectric antenna reaches a peak radiation enhancement of $5\times 10^5$. Although 4 times less enhancement than the all-metal antenna with the same dimensions, the corresponding metal-dielectric antenna efficiency is 70% versus 2% for the all-metal antenna. Note that if the dielectric cones in Fig. 2(a) are removed (e.g. refractive index $n$=1), the efficiency of the resulting antenna is approximately unchanged, but the enhancement is reduced dramatically. Indeed, compared to the metal-dielectric antenna with $d=1$ nm, the enhancement is reduced by over $100\times$ in simulation. Therefore, from an alternative perspective, the metal-dielectric antenna tips boost enhancement without changing the antenna efficiency.
Fig. 2. The metal-dielectric antenna uses sharp dielectric tips to maintain high efficiency with little compromise to the enhancement factor. (a) Metal-dielectric dipole antenna. This antenna is similar to the all-metal antenna in Fig. 1(a) except the sharp metal tips have been replaced with sharp dielectric tips of refractive index $n=3.4$ (minimum radius of curvature = 1 nm, cone angle = 90$^\circ$). (b) FDTD calculation of the enhancement of the metal-dielectric antenna (black line) compared to the all-metal antenna (silver line) as a function of $d$ at a wavelength of $\lambda =1550$ nm. (c) Efficiency of the metal-dielectric antenna compared to the all-metal antenna as function of $d$. The efficiency of the all-metal antenna was calculated using the circuit model including the surface collision effect (Fig. 1(c)). The efficiency of the metal-dielectric antenna was calculated in FDTD with a correction for surface collisions.
Given the clear efficiency improvement provided by nanoscale dielectrics, to what degree is some metal required for optimal light concentration? To address this question, we investigated the enhancement offered by an all-dielectric antenna in comparison to the all-metal and metal-dielectric antennas in Figs. 1(a) and 2(a). Consider the all-dielectric bowtie antenna in Fig. 3(a). This antenna consists of two opposing dielectric cones of refractive index $n=3.4$ (cone angle = 90$^\circ$) with a small vacuum gap of width $d$ at the center. The length of the bowtie is chosen to be 680 nm in order to tune the fundamental resonance wavelength to 1550nm. The antenna enhancement (Eq. (1)) and efficiency (Eq. (2)) are shown in Figs. 3(b) and 3(c) respectively. As depicted here, the enhancement of the dielectric antenna increases with decreasing vacuum gap $d$, similar to the all-metal and metal-dielectric antennas. For $d=1$ nm, the antenna enhancement peaks at $1.8\times 10^4$, which is $25\times$ times smaller than the metal-dielectric antenna enhancement. The corresponding all-dielectric bowtie’s efficiency improves to 100% versus 70% for the metal-dielectric antenna. From this analysis, we can conclude that although the all-dielectric antenna can provide high efficiency, some metal in the antenna design drastically improves the enhancement and is therefore beneficial. Conversely, the all-metal antenna provides high enhancement, but at very poor efficiency.
Fig. 3. All-dielectric bowtie antenna provides insufficient enhancement compared to the all-metal and metal-dielectric variants. (a) All-dielectric bowtie antenna. The antenna consists of two opposing cones with a center vacuum gap of width $d$ (minimum radius of curvature = 1 nm, cone angle = 90$^\circ$). (b) Comparison of the antenna enhancement provided by the all-dielectric bowtie (blue line) with the all-metal (silver line) and metal-dielectric (black line) antennas as a function of $d$. (c) Efficiency of the all-dielectric bowtie with comparison to the all-metal and metal-dielectric antennas as a function of $d$. The dielectric antenna is lossless.
Up to this point we have demonstrated that (1) purely metallic antennas suffer from an efficiency versus enhancement tradeoff due to Ohmic losses, which are worse than typically predicted because of the anomalous skin effect; (2) this tradeoff can be mitigated by including dielectrics in the antenna design, thus enabling high efficiency and high antenna enhancement simultaneously; and (3) a purely dielectric antenna design is efficient, but does not offer comparable enhancement to the metal-dielectric and all-metal designs. Going forward, we will show how the metal-dielectric antenna design strategy can be applied to semiconductor spontaneous emission enhancement.
4. Metal-dielectric antenna-LED
The antennas discussed in the previous sections use a light source in vacuum, but communications applications require an electrically-injected light source such as a semiconductor. The optical antenna-enhanced light emitting diode (antenna-LED) emits from a semiconductor [19,22,40]. Consider the metal-dielectric antenna-LED depicted in Fig. 4(a). The structure is similar to the metal-dielectric antenna given in Fig. 2(a) except now the two inner tips are connected by a semiconductor bridge of width $b$ at the center. The bridge is composed of a material with refractive index $n=3.4$, which is similar to the refractive index of many III-V semiconductors. By contrast, we have also investigated the antenna-LED in Fig. 4, denoted “lithographically patterned” in the sense that this structure could be fabricated using two-dimensional top-down processing. Both antennas assume the optical point source is located at the center of the bridge.
Fig. 4. Continuous semiconductor bridge antenna provides efficient enhancement for electrically-injected semiconductor devices. (a) Metal-dielectric antenna-LED with cylindrical symmetry. The structure is similar to that in Fig. 2(a) except the sharp dielectric tips have been connected by a bridge of width $b$. Perspective view ((b), upper graphic) and top view ((b), lower graphic) of a metal-dielectric antenna-LED that is compatible with top-down semiconductor fabrication. This antenna has the same cross-section as the antenna in (a), but the cross-section is extruded 50 nm in depth. (c) Peak spontaneous emission enhancement as a function of bridge width $b$ calculated in FDTD. (d) Efficiency of the antennas as a function of bridge width $b$ calculated in FDTD and corrected for the anomalous skin effect.
As shown in Figs. 4(c)–4(d), there is a marked increase in antenna enhancement as the $b$ dimension is reduced below 10nm without compromise to the antenna efficiency; this is yet another version of dielectric focusing, similar to that shown before in Figs. 2 and 3. Note that the enhancement reference according to Eq. (1) is a point dipole source radiating into vacuum. To obtain the enhancement of a reference dipole source in a homogeneous semiconductor, divide the value in Fig. 4(c) by $n=3.4$. Furthermore, the antenna enhancement and efficiency were calculated at a resonance wavelength of $\lambda =1550$ nm for all values of $b$. For small $b$, real semiconductor optical emitters may undergo a blueshift from their nominal bandgap energy. The antenna resonance can be tuned to accommodate this effect by decreasing the total antenna length.
In addition to improved light concentration, a small semiconducting bridge can provide favorable polarization selection rules. Electron-to-heavy hole (C-HH) radiative transitions are preferentially stimulated by an electric field polarized along the long direction of a semiconducting quantum wire, perpendicular to the confinement direction [24,25]. Thus, the polarization of band-edge light emission from an unstrained semiconductor bridge is in favorable alignment with the metal-dielectric antenna-LED mode polarization [40]. The combination of all these benefits indicates that antenna-enhanced, efficient, electrically-injected antenna LED devices are feasible. Such a device could serve as a nanoscale optical source for ultra-fast, low-power, on chip data communications.
5. Antenna enhancement versus Purcell enhancement
In Sections 2–4 we showed metal-dielectric antennas that can provide high efficiency and high antenna enhancement of both atoms and semiconductors. Up to this point we have only employed the antenna enhancement metric defined in Eq. (1) without invoking the Purcell Effect [41], which has been emphasized by many prior investigators. While we advocate that the Purcell Effect is not needed to describe antenna properties, we show how it may be employed in the context of antenna enhancement.
An antenna can concentrate zero-point electromagnetic energy into a sub-wavelength volume, thus enhancing the spontaneous emission rate over the normal vacuum emission. This fact is reflected in the Purcell enhancement factor, defined as,
To estimate antenna enhancement, an engineer would use the circuit representation of a dipole antenna [17]; a simple version of the metallic dipole antenna was shown in Figs. 1(a)–1(b). In Appendix A, we show that the voltage between the distant antenna tips (separated by antenna length $l$) ends up concentrated in the narrow antenna gap $d$. Thus, the antenna enhancement factor $(P_{\textrm {rad}}/P_{\textrm {o}})$ can be rewritten as,
A consequence of Eq. (5) is that the maximum antenna power is attained for the longest single-mode resonant antenna (namely, the half-wave antenna). Note that Eq. (5) applies generally to one-dimensional metallic antennas with high conductivity. Antennas with arbitrary geometrical configurations, including the metal-dielectric antenna discussed in Fig. 2, require a more detailed treatment.In contrast with Eq. (5), the Purcell Factor, Eq. (3), is repeated here:
By equating Eqs. (5) and (6) in Eq. (7), we may obtain the effective mode volume: Rearranging to solve for $V_{\textrm {eff}}$ and combining terms, we find: where the only remaining unknown is the quality factor, $Q$. If the antenna loss is limited primarily by radiation and not resistance, the $Q$ may be obtained from the well-established Wheeler-Chu Limit [46,47], where $l$ is the size of the longest antenna dimension (in this case, the antenna length) and $\lambda$ is the wavelength. Notably, the quality factor increases rapidly when the antenna length is very small, but in antennas we want a low $Q$ representing efficient radiation. Combining Eq. (8) with the lower bound of Eq. (9) we may obtain the effective mode volume of a Wheeler-Chu limited antenna: which has no unknowns. An interesting special case of Eq. (10) is the half-wave dipole antenna; plugging in $l\rightarrow \lambda /2$: which represents a bound on the single-mode antenna effective mode volume. In principle, the vacuum gap width $d$ can be as small as 1nm, and therefore the effective mode volume of antennas may be extremely small. Note that based on our derivation, Eq. (10) and Eq. (11) apply to one-dimensional purely metallic antennas, such as that depicted in Fig. 1(a). Antennas of arbitrary geometry may require a full numerical electromagnetic analysis. In the next section we will check the half-wave dipole antenna effective mode volume formula, Eq. (11), against full numerical calculations using the customary formula (Eq. (4)).6. Electromagnetic numerical calculations of antenna effective mode volume
The effective mode volume that is used for electromagnetic numerical calculation was given above in Eq. (4) and is reproduced here:
We considered three antennas for numerical calculation, depicted in Fig. 5. The all-metal, metal-dielectric, and all-dielectric antennas refer to the antennas from Figs. 1(a), 2(a) and 3(a) respectively with vacuum gap widths of $d=1$ nm between respective metallic or dielectric tips and minimum radius of curvature of 1 nm. The antenna effective mode volume (Fig. 5, $x$-axis), is normalized by wavelength and inverted $(\lambda ^3/V_{\textrm {eff}})$ so that it may easily be plugged into the Purcell factor (Eq. (3)). Antenna efficiency (Fig. 5, $y$-axis) was obtained previously in Sections 2–4. The electromagnetically calculated effective mode volume values of the all-dielectric, metal-dielectric, and all-metal antennas were $5.6\times 10^{-6}\lambda ^3$, $7.8\times 10^{-7} \lambda ^3$, and $1.5\times 10^{-7} \lambda ^3$ at $\lambda =1550$ nm respectively. Note that if we were to consider a larger or more practical tip parameter $d$ in our calculation, the efficiency of the metallic antenna and the effective mode volume of all three antennas would increase. For example, if we used $d=2$ nm, the effective mode volume for the three antennas would increase by approximately $4\times$.
Fig. 5. The efficiency and effective mode volume of three antennas from Figs. 1(a), 2(a), and 3(a) with vacuum gap widths of $d=1$ nm and radius of curvature $=\,1$ nm are plotted.
The numerically calculated effective mode volume of the all-metal antenna ($1.5\times 10^{-7} \lambda ^3$) agrees within a factor 2 to the predicted value of the half-wave antenna effective mode volume from Eq. (11) using $d$=1nm: $V_{\textrm {eff}} = 0.185\cdot d^2\lambda = 7.7\times 10^{-8} \lambda ^3$. The small disagreement between the two values may be attributed to inaccuracy in the numerical $V_{\textrm {eff}}$ calculation (discussed in Appendix B) or to our choice of the canonical lower bound of the Wheeler-Chu limited quality factor from Eq. (9) corresponding to $Q$=0.6 for a half-wave antenna, whereas a more realistic antenna quality factor is $Q$=1. Furthermore, compare the effective mode volume of the all-dielectric antenna ($5.6\times 10^{-6}\lambda ^3$) to MIT’s photonic crystal cavity from [36] with $V_{\textrm {eff}} = 7.0\times 10^{-5} \lambda ^3$ at $\lambda =1550$ nm, which reportedly uses a similar dielectric tip geometry chosen here except their structure was optimized for both high quality factor and low effective mode volume. These results indicate that antennas are capable of both extreme concentration of electromagnetic energy and good radiation. Notably, the metal-dielectric antenna provides the best balance of efficient radiation (70%) and effective mode volume ($7.8\times 10^{-7} \lambda ^3$), complimentary to our results from prior sections.
7. Conclusion
In this work we have introduced a metal-dielectric antenna that eliminates the tradeoff of enhancement versus efficiency present in purely metallic optical antennas. We proposed a feasible structure for a metal-dielectric antenna-enhanced light-emitting diode that could lead to improvements in the speed and efficiency of nanoscale light sources for optical communications. By comparing antenna enhancement and the Purcell effect, we have introduced a new formula for antenna effective volume that permits continued to use of the Purcell enhancement factor for metallic antennas.
Appendix A: enhancement and efficiency of a variable length antenna from antenna circuit theory
The power radiated by a Hertzian dipole antenna (electrically small antenna) can be modeled as a lumped-element AC circuit model [17,45–48]. In this Appendix, we will consider a series RLC circuit with a current source, similar to Fig. 1(b) from the text. On resonance the antenna reactive impedance (capacitance and inductance) is minimized, and therefore will be ignored. An extensive discussion of these elements in optical antennas can be found in [20]. The antenna radiated power on resonance is then given by,
where $|I|$ is the peak antenna current amplitude and $R_{\textrm {rad}}$ is the antenna radiation resistance. $R_{\textrm {rad}}$ is fundamental and can be derived [48]. For dipole antennas $R_{\textrm {rad}}$ is given by, where $Z_{\textrm {o}}=1/\varepsilon _{\textrm {o}}c\approx 377\Omega$ is the impedance of free space, $l$ is the antenna length, and $\lambda$=1550nm is the wavelength. Note that long dipole antennas (such as a typical half-wave antenna) will effectively have a reduced antenna radiation resistance than what this formula suggests because the antenna current will vary spatially along its length [17]. The reduction is not considered in this analysis for simplicity and generality.The Hertzian dipole, representing an atomic point dipole, can be shown to have an antenna current $|I|=q\omega$ where $\omega$ is the oscillation frequency and $q$ is the elementary charge. Furthermore, we may take an effective antenna length of $x_{\textrm {o}}$, where $x_{\textrm {o}}$ is the amplitude of the dipole motion. Hence, the total power radiated of the Hertzian dipole is,
To find the antenna enhancement factor, we take the ratio $(P_{\textrm {rad}}/P_{\textrm {o}})$:
The enhancement factor in Eq. (18) may also be obtained by an intuitive argument, namely that the total voltage between the two antenna ends (separated by $l$) will drop completely across the antenna vacuum gap (with width $d$) because there is negligible voltage drop in the conductive metal. The spontaneous emission factor may then be estimated by the ratio of the electric field intensity in the antenna gap to that in free space. Observe,
The antenna efficiency is given by the ratio of the radiated power to the total power lost to radiation and heat. On resonance, this is given by,
Appendix B: numerical effective mode volume calculation
There are several methods in the literature to numerically calculate the effective mode volume of optical antennas using forms similar to Eq. (12) [42,50–53]. For this paper we used a variation on the method proposed originally in [42]. It should be remarked that calculating the effective mode volume of antennas is more nuanced than doing so for high-Q dielectric resonators, and our calculation involves heuristics that may be less accurate than strongly-mathematically motivated solutions in the literature [50,51]. Nevertheless, we will briefly outline the challenges of performing this calculation, and how they were addressed in this work. All simulations of effective mode volume were performed in Lumerical FDTD with 0.25nm resolution in the antenna feedgap, 1nm resolution of the rest of the antenna, and 10nm resolution of the rest of the simulation domain which was $(2\mu\textrm{m})^3$ in total.
There are four points to address in calculation of the numerator of Eq. (12):
- 1. Antennas are leaky, or in other words, it is difficult to distinguish the mode that is bound to the antenna and radiation in the far-field.
- 2. The energy of the simulation source (e.g. a plane wave) must also be distinguished from the antenna mode energy.
- 3. The excited antenna may consist of both dark (nonradiative) and bright (radiative) modes.
- 4. Metals are dispersive.
Point (1) was addressed originally in [42] and is out of the scope of this appendix. Essentially, one may find a transition between the local antenna mode and the far-field radiation by self-consistently choosing the radius of integration, $r'$, in Eq. (12). Since far-field antenna radiation intensity falls off as $1/r^2$, there is a well-defined transition.
Point (2) was addressed in two ways: (a) A total-field scattered-field source (TFSF) source was used, which minimizes the total footprint of the incident plane wave source in simulation. (b) Time-apodization in the Fourier transform of the FDTD data was employed. Time-apodization is essentially a long-pass filter for time, which allows one to effectively filter out an incident broadband source pulse. Thus, the simulation will only capture electric field data from the resonant antenna mode that continues to oscillate after the source has died out. Because antennas have very low quality factor, this process is not perfect and it is difficult to completely decouple the source from the antenna response. Nevertheless, testing with different apodization cutoffs tended to provide convergence of the energy integral for sufficiently long cutoff after the source pulse.
Point (3) is believed to be addressed by the same procedures as Point (2) above. Because an incident plane wave (TFSF) source was used, dark modes should not have been excited in simulation. This was confirmed by measuring the antenna scattering efficiency, which was much greater than the radiative efficiency of the antennas as excited by point dipole sources. Furthermore, dark modes most likely have smaller $Q$ than the antenna radiative mode, and therefore time-apodization filtered them out. Since we have removed dark modes from simulation, the effective mode volume values reported in Section 6 correspond only to the antenna radiative mode, and therefore may be properly compared to the antenna effective mode volume formula derived in Eqs. (10) and (11).
Point (4) is addressed by using the dispersion correction term, $\textrm {Re}[\partial (\varepsilon \omega )/\partial \omega ]$, in Eq. (12) [43,44]. Palik’s data for silver [26] was interpolated to obtain this term.
The peak energy density term in the denominator of Eq. (12) was calculated at the center of the antenna feedgap, after time-apodization. This is the correct choice because we were interested in the enhanced radiation of dipole point sources from this location. This differs slightly from the “true” location of peak energy density, which is very close to the metal boundary.
Funding
National Science Foundation (ECCS-0939514).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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