Quantum emitters such as NV-centers or quantum dots can be used as single-photon sources. To improve their performance, they can be coupled to microcavities or nano-antennas. Plasmonic antennas offer an appealing solution as they can be used with broadband emitters. When properly designed, these antennas funnel light into useful modes, increasing the emission rate and the collection of single-photons. Yet, their inherent metallic losses are responsible for very low radiative efficiencies. Here, we introduce a new design of directional, metallo-dielectric, optical antennas with a Purcell factor of 150, a total efficiency of 74% and a collection efficiency of emitted photons of 99%.
© 2014 Optical Society of America
Since the pioneering proposal of Dieter Pohl [1–3], there have been several works using the concept of antennas in the optical range. Indeed, this concept holds great promises to improve the efficiency of both light-emitting (single-photon sources, LED) and light-absorbing devices (photovoltaics, spectroscopy, photodetection) [3–8]. A review of the applications of single-photon sources can be found in . Coupling them to nano-antennas allows increasing their performance: i) by controlling the emission direction in order to increase the collection efficiency, ii) by reducing the decay time in order to increase the repetition rate of a source of single-photons on demand. The spectral bandwidth of the antenna is an important property. If indistinguishable photons are needed, the antenna should serve as a monochromatic filter. By contrast, when dealing with broadband emitters such as nitrogen-vacancy (NV) centers or quantum dots at ambient temperature, broadband antennas are desirable. Regarding applications to single-photon sources, plasmonic nanoantennas have been introduced by many groups as a way to efficiently couple quantum emitters with propagating modes, in terms of excitation enhancement [10, 11], emitted power [11–21] and directivity [11,19–30]. Recent advances with NV centers have been reported . In order to reduce the decay time, the local density of optical states (LDOS) has to be increased. It has been shown some time ago that metal-insulator-metal (MIM) structures are very efficient for this purpose [32, 33]. Yet, metals introduce losses so that the total efficiency of the antenna is significantly reduced.
Plasmonic patch antennas (see Fig. 1(a)) were designed to take advantage of the large LDOS of MIM structures. They have been investigated both theoretically and experimentally [20, 21]. The size of the metallic disk controls the radiation pattern, and thus the directivity. This allows for an increase in the collection efficiency of radiated photons. Due to their highly confined plasmonic modes, MIM cavities have been shown to produce spontaneous emission acceleration on the order of 100, with broadband operations [33, 34]. This acceleration factor is given by the ratio of LDOS with and without the antenna. Here we refer to it as the Purcell factor [35–37]. By choosing a proper dielectric thickness, the quantum emitter can simultaneously access those high Purcell factors and avoid quenching [7, 21]. Hence, one can convert almost all quantum emitter excitations into the plasmonic modes of the antenna with high efficiency. The major drawback of the plasmonic patch antenna is that more than 90% of the energy of the plasmonic modes is then converted into heat instead of being radiated. Let us summarize the efficiency issue. The emission process by a quantum emitter in the presence of a plasmonic nanoantenna consists in two steps: decay of the quantum emitter into plasmonic antenna modes and (radiative) decay of the antenna modes into photons. The efficiency of the first step can be reduced due to competition with two other processes: either a non-unity internal quantum efficiency or quenching which is a non-radiative coupling into losses in the metal. The former will not be considered here. The latter can also be viewed as Joule losses induced by the 1/r3 near-field component of the electric field produced by the quantum emitter. This quenching can be virtually suppressed by keeping the emitter at a distance larger than 10 nm from the surface . Finally, the plasmonic antenna modes can either radiate photons or decay into heat in the antenna. We define the antenna radiative efficiency as the radiated power normalized by the sum of the absorbed and radiated power. In this paper, we introduce a novel design that allows increasing the radiative efficiency of optical patch antennas by more than one order of magnitude.
By using an in-situ optical lithography technique, C. Belacel et al.  fabricated patch antennas of diameters ranging from 1.46 μm to 2.1 μm coupled with CdSe/CdS quantum dots emitting at 630 nm. The antennas were shown to have the expected properties of a high, broadband Purcell factor as well as directivity. However, the radiative efficiency was reported to be on the order of a few percent. Indeed, once the plasmonic mode of the antenna is excited by the quantum emitter, most of its energy is absorbed in the metal instead of being radiated by the patch. The purpose of this paper is to analyze the origin of this low radiative efficiency and introduce new ideas to overcome this limitation.
To increase the radiative efficiency of the patch antenna, there are several options. One possibility is to shift the emission wavelength from the optical to the near-infrared range [20, 21] in order to take advantage of the larger propagation length of plasmonic modes in the IR. To further increase the radiative efficiency, it is necessary to increase the radiative losses. This can be done by using a resonant half-wavelength dipole (see next section). Dipolar patchs operating at 1.55 μm can achieve radiative efficiencies ranging from 20% to 40%. Hence, this efficiency can be significantly increased, albeit at the cost of a lower directivity.
Another approach to increase the radiative efficiency is based on the concept of metallo-dielectric structures. Previous theoretical studies reported designs of antennas with high radiative efficiencies and high Purcell factors by combining a metallic and a dielectric antenna [11, 17]. A hybrid metallo-dielectric antenna has also been used to combine large Purcell factor and spectral narrowing of a broadband emitter . In this article, we follow this route by introducing a novel patch antenna consisting of a dipolar metallic patch (MP) and a dielectric resonator antenna (DRA) (see Fig. 1(b)). The DRA is a high-refractive-index cylinder of roughly twice the emission wavelength in diameter and half this wavelength in height. By using a genetic algorithm as a global optimization scheme  and a cylindrical aperiodic Fourier Modal method [21, 39] for the numerical simulations of the system, we design a metallic patch - dielectric resonator antenna (MP-DRA) exhibiting a radiative efficiency of 75% with a Purcell factor of 150 and a collection efficiency of 99% in a microscope objective with a numerical aperture of 0.85. The dipole source emits at telecom wavelength (λ = 1.55 μm) and is polarized perpendicularly to the patch. Furthermore, the antenna is broadband (with a spectral width exceeding 100 nm) and relatively insensitive to the off-centering of the emitter. In what follows, we first report the analysis of the role of the metallic patch size and of the surrounding dielectric index. We then introduce the concept of coupled metallic patch and dielectric resonator antenna, MP-DRA, and design a structure with optimized performances.
2. Small metallic patches : a plasmonic strategy
It has been shown previously that metallic patch antennas (MP, Fig. 1(a)) enhance the spontaneous emission rate over a large bandwidth. Furthermore, a directional radiation pattern [20, 21] is obtained if the antenna has a typical size larger than twice the emission wavelength. However, for these sizes, the radiative efficiency of the antenna never exceeds a few percents for emission wavelengths in the visible range . We studied the Purcell factor and radiative efficiency of a gold MP at 1.55 μm which is the relevant wavelength regarding applications to quantum telecommunications. We use an algorithm based on a cylindrical aperiodic Fourier modal method (a-FMM) [21, 39] for the numerical simulations of the system. In this article, the spectral dependence of the refractive index of gold has been taken from , giving a value of nAu = 0.5605 + 9.7810i at λ = 1.55 μm. We fix the thickness of silica to 30 nm (nSiO2 = 1.45), as it ensures large Purcell factors while preventing quenching . We also fixed the thickness of the gold disk to 38 nm, as this is the value found in the optimized, final system (see last section).
The dependence on the diameter of the metallic disk is reported in Figs. 2(a) and 2(b). It is seen that there is an optimum value of the disk diameter which maximizes both the Purcell factor and the radiative efficiency. At an emission wavelength of 1.55 μm, this optimum value is approximately half this wavelength. It corresponds to the dipolar plasmonic resonance of the antenna. The existence of a joint optimum is due to the fact that the effective wavelength of the main plasmonic mode only differs from the emission wavelength by a few percent. Hence, the patch cavity is both a resonant cavity for the plasmonic mode, ensuring a large Purcell factor, as well as a half-wavelength antenna regarding the coupling with propagating plane waves. Purcell factors as high as 500 can be obtained while the radiative efficiency reaches 24%. Figure 2(b) shows that this efficiency is twice as good as the maximum of the radiative efficiency reachable at a wavelength of 0.63 μm. This confirms the advantage of working in the near-infrared range.
For larger patches, multipolar plasmonic resonances appear. Due to the finite propagation length of lossy plasmonic modes, their quality factor tends to decrease when the patch diameter increases as seen in Fig. 2(a). Hence, at an emission wavelength of 1.55 μm the global optimum value corresponds to the dipolar plasmonic resonance.
From the previous analysis, it is seen that both the radiative efficiency and the total normalized decay rate are maximum for a resonant dipolar patch antenna. In the example, the radiative efficiency reaches values of about 24% with large Purcell factors of several hundreds. Yet, a higher radiative efficiency is needed for single-photon applications. In addition, the lack of directivity of the angular emission pattern of a dipole on a mirror results in a collection efficiency less than 50% when using a numerical aperture of 0.85. Thus, we discuss a different strategy in the next section.
3. High-refractive-index medium : a dielectric strategy
It is well known  that locating a dipole close to a high-index medium has two effects: the power emitted increases and most of the power is radiated into the high refractive medium. This property has been used to enhance the collection of light for sensitive detection of fluorophores  and to increase the efficiency of light emitting devices [43, 44]. This idea has been revisited and optimized recently to obtain almost unity collection efficiency [27, 28]. Here, we illustrate this idea by considering a dipole source in the middle of a 30 nm-thick silica layer on a gold substrate and covered by a high-index material as shown in Fig. 3(a). Here again, the silica layer is used as a spacer to prevent quenching. We have studied the changes in the Purcell factor and the radiative efficiency of the antenna when varying the superstrate refractive index. We consider a dipole perpendicularly polarized to the stratified system. We use a plane wave expansion to derive an explicit form of the radiative and total decay rate [33, 45, 46]. Numerical results are given in Figs. 3(b) and 3(c).
It is seen that both the Purcell factor and the radiative efficiency increase when increasing the superstrate refractive index. The onset of this phenomenon is when the refractive index becomes higher than the silica refractive index (nSiO2 = 1.45). Here, the increase of radiation is a near-field effect. It is due to the coupling of evanescent modes of the dipolar field into propagating modes in the dielectric. The higher the refractive index of the superstrate, the more evanescent modes of the source can couple with propagating modes of the superstrate by optical tunneling [41, 47]. This phenomenon takes place only if the superstrate refractive index is larger than the silica refractive index. As a result, the power radiated increases with an increase in the refractive index of the superstrate. Thus, the total decay rate of the source and the radiative efficiency of the antenna follow the same trend, as shown in Figs. 3(b) and 3(c).
We now consider a system consisting of a gold substrate, a 30 nm-thick silica layer containing the emitter, a 38 nm-thick gold film and a high-index medium as depicted in Fig. 3(d). The thickness of the metallic film is the value obtained in the next section when optimizing the system. We report a similar study: Purcell factor and radiation efficiency as a function of the refractive index of the upper medium. The behavior observed in Figs. 3(e) and 3(f) is an abrupt increase of radiation for n > 2.43. This value corresponds to the real part of the effective index (neff = 2.43 + 0.0539i) of a mode propagating in the metal-silica-metal waveguide. For a superstrate index larger than 2.43, the guided mode becomes leaky, so that part of the decay of the mode becomes radiative. This result illustrates the fact that the radiation process can be viewed as a two-steps process: i) the dipole couples with the metal-silica-metal mode, ii) this mode couples with propagating waves in the superstrate. Finally, we note that the transition width is given by the imaginary part of the effective index. When comparing the two cases shown in Fig. 3, it is clearly seen that the confinement of the field produced by the MIM structure is necessary to attain large values of the decay rate while the high refractive index is required to extract radiation. However, this planar structure introduces a new difficulty since one needs to extract all the emitted photons from the high-refractive-index superstrate. It has been pointed out [27, 28] that collecting radiation is a difficult task as light is coupled into grazing angles. This issue was avoided [27, 28] by increasing the distance between the emitter and the interface. In what follows we introduce a different strategy.
4. The metallic patch - dielectric resonator antenna: best of both worlds
From the results of the last section, it can be construed that inserting a patch antenna in a highly refractive medium drastically improves its radiative efficiency into the high index material. We are facing now the problem of extracting light from the high index material. To address this issue, we use a small dielectric cylinder as a dielectric resonator antenna (DRA ). By tailoring a high-refractive-index dielectric resonator antenna added on top of the patch antenna, the coupling between patch plasmonic modes and DRA modes can be optimized. Note that the DRA does not absorb, but radiates light. If the characteristic lengths of the DRA are of the order of a multiple of half the emission wavelength, the excited eigenmodes of the DRA and propagating modes of the free space can couple efficiently. As a result, the coupled system Metallic Patch - Dielectric Resonator Antenna (MP-DRA, see Fig. 1(b)) may have a better efficiency than the MP alone, while keeping a large Purcell factor due to the confinement of the mode in the metal-dielectric-metal. Regarding directivity, radiation is mostly due to the DRA, so that the DRA diameter is the relevant parameter. One can thus make the most of the radiation properties of a resonant dipolar metallic patch and use a DRA in order to improve the radiative efficiency, while collecting most of the emitted single-photons into a microscope objective.
The concept of DRA is standard in microwaves. DRAs working in the optical range have recently been studied theoretically  and experimentally . For our simulations, we chose to work with a refractive index of 2.58 for the DRA, which is the value corresponding to amorphous SiC at 1.55 μm . The thickness of the silica layer is again chosen to be 30 nm. Four parameters are left unknown : the thicknesses and diameters of both the metallic disk and the DRA. Using a genetic algorithm  as a global optimization scheme, these parameters were optimized to get the highest total efficiency of the antenna and the highest Purcell factor possible. The total efficiency is defined in this article as the product of the radiative efficiency and the collection efficiency in a standard microscope objective of numerical aperture 0.85. The a-FMM was used to perform the numerical simulations of the system.
The result of the optimization is shown in Fig. 4 where we have plotted the intensity (square modulus of the electric field) in the structure. The resulting optimized MP-DRA is composed of a 38 nm-thick metallic disk of 764 nm in diameter and a 738 nm-thick DRA of 2447 nm in diameter. For a centered dipole source polarized perpendicularly to the patch, results of the numerical simulations give a Purcell factor of 150 with a 74% total efficiency. The total efficiency is composed of a collection efficiency of 99% and a radiative efficiency of 75%. This is to be compared with a centered dipole source polarized in the plane of the patch, for which numerical results give a Purcell factor of 2.5 with a total efficiency below 0.1%. These latter results ensure that the radiation properties from a randomly oriented dipole source is governed by the vertical component of this dipole. The key to understanding the physical mechanism of the increased efficiency lies in the field distribution plotted in Fig. 4. It is clearly observed that a hybrid mode of the metallo-dielectric structure is excited. This hybrid mode is composed of a resonantly excited plasmonic mode in the metal-silica-metal region and a resonantly excited mode in the dielectric cylinder. As the spatial extension of the mode is much larger in the dielectric, a large part of the hybrid-mode energy is in a non-lossy region so that the nonradiative damping is reduced. Conversely, the large spatial extension of the resonator gives rise to an increase of radiation losses: the DRA is a better radiator than the patch antenna. Both factors are responsible for the increased efficiency of the antenna. An important factor for enhancing the radiation is the resonant character of the excitation of the DRA. In order to assess the role of the large index of the dielectric versus the role of the resonance in the DRA, we have optimized the total efficiency with a silica cylinder instead of a SiC cylinder (not shown here). Interestingly, we found that it is possible to obtain 60% efficiency without using a large index material.
An important issue for fabrication is the sensitivity of the antenna performances on the geometrical parameters. As shown in Figs. 5(a) and 5(b), the enhancement of the spontaneous emission rate and the total efficiency of the optimized MP-DRA are relatively insensitive to the lateral off-centering of the dipole source. We note that the in-situ optical lithography technique used to fabricate patch antennas  has an accuracy of 25 nm in the dipole centering so that virtually no degradation is expected from an alignement error. We have also checked that the performances are not strongly dependent on the heights and diameters of the dielectric cylinder and the metallic disk. Indeed, a variation over a 10% range on either the optimized heights or diameters leads to a total efficiency varying from 64% to 74%. This is not surprising as resonators with low quality factor (on the order of 10) are not expected to be very sensitive to their size dimensions. Furthermore, we have checked that the air gap between the DRA and the silica layer is not crucial. We have optimized the total efficiency for a metallic disk embedded in the DRA and found that it is possible to obtain more than 65% total efficiency with acceleration factor of the spontaneous emission rate on the order of 100. Hence, even without this air gap, the metallo-dielectric antenna shows good performances.
Figures 6(a) and 6(b) show the radiation pattern of the MP-DRA for a centered dipole and for a 140 nm-laterally-off-centered one. Numerical results indicate that 99% of the emitted photons can be collected in a microscope objective of numerical aperture 0.85. Those results ensure that almost all the single-photons emitted can be collected. Finally, one key feature of plasmonic antennas is their large bandwidth. We have seen that the dielectric antenna reduces the ohmic losses but increases the radiation losses. Numerical results (see Figs. 7(a) and 7(b)) show that the optimized MP-DRA preserves the broadband behavior. Purcell factors ranging from 75 to 160 and total efficiencies ranging from 60% to 74% can be obtained over a 100 nm bandwitdth.
In summary, we have introduced a metallo-dielectric patch antenna for single-photon emission. This new design combines the large Purcell factor due to the confinement of the field in the metal-dielectric-metal structure with a large collection efficiency and large emission efficiency which is characteristic of dielectric antenna resonators. An important feature of this design is that it is based on relatively simple technological steps for fabrication. It is fully compatible with the in-situ lithography technique already used for fabricating patch antennas . We anticipate Purcell factors above 100, a collection efficiency of 99% with a numerical aperture of 0.85 and an overall efficiency exceeding 70%. This type of hybrid antennas should be useful for many applications.
F. Bigourdan acknowledges the support of the french Direction Generale de l’Armement. The authors thank C. Sauvan for fruitful discussions.
References and links
1. D. W. Pohl, “Near-field optics: comeback of light in microscopy,” Solid State Phenomena , 63–64, 251–256 (1998). [CrossRef]
2. D. W. Pohl, “Near-field optics seen as an antenna problem,” in Near-Field Optics - Principles and Applications: The Second Asia-Pacific Workshop on Near-Field Optics, X. Zhu and M. Ohtsu, eds. (World Scientific, 2000), pp. 9–21. [CrossRef]
5. L. Novotny and N. F. van Hulst, “Antennas for light,” Nat. Photonics 5, 83–90 (2011). [CrossRef]
7. N. P. de Leon, M. D. Lukin, and H. Park, “Quantum plasmonics circuits,” IEEE J. Sel. Top. Quantum Electron. 18, 1781–1791 (2012). [CrossRef]
9. B. Lounis and M. Orrit, “Single-photon sources,” Rep. Prog. Phys. 68, 1129 (2005). [CrossRef]
12. S. Kühn, U. Hakanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97, 017402 (2006). [CrossRef]
13. J. P. Hoogenboom, G. Sanchez-Mosteiro, G. Colas des Francs, D. Heinis, G. Legay, A. Dereux, and N. F. van Hulst, “The single molecule probe: nanoscale vectorial mapping of photonic mode density in a metal nanocavity,” Nano Lett. 9, 1189–1195 (2009). [CrossRef]
14. I. S. Maksymov, M. Besbes, J.-P. Hugonin, J. Yang, A. Beveratos, I. Sagnes, I. Robert-Philip, and P. Lalanne, “Metal-coated nanocylinder cavity for broadband nonclassical light emission,” Phys. Rev. Lett. 105, 180502 (2010). [CrossRef]
15. S. Derom, R. Vincent, A. Bouhelier, and G. Colas des Francs, “Resonance quality, radiative/ohmic losses and modal volume of Mie plasmons,” Europhys. Lett. 98, 47008 (2012). [CrossRef]
16. N. P. de Leon, B. J. Shields, C. L. Yu, D. E. Englund, A. V. Akimov, M. D. Lukin, and H. Park, “Tailoring light-matter interaction with a nanoscale plasmon resonator,” Phys. Rev. Lett. 108, 226803 (2012). [CrossRef]
18. M. P. Busson, B. Rolly, B. Stout, N. Bonod, and S. Bidault, “Accelerated single-photon emission from dye molecule-driven nanoantennas assembled on DNA,” Nat. Commun. 3, 962 (2012). [CrossRef]
21. C. Belacel, B. Habert, F. Bigourdan, F. Marquier, J.-P. Hugonin, S. Michaelis de Vasconcellos, X. Lafosse, L. Coolen, C. Schwob, C. Javaux, B. Dubertret, J.-J. Greffet, P. Senellart, and A. Maitre, “Controlling spontaneous emission with plasmonic optical patch antennas,” Nano Lett. 13, 1516–1521 (2013). [PubMed]
22. J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: a Yagi-Uda nanoantenna in the optical domain,” Phys. Rev. B 76, 245403 (2007). [CrossRef]
23. T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2, 234–237 (2008). [CrossRef]
24. T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Single emitters coupled to plasmonic nano-antennas: angular emission and collection efficiency,” New J. Phys. 10, 105005 (2008). [CrossRef]
25. A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329, 930–933 (2010). [CrossRef]
26. J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J.-M. Gerard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire,” Nat. Photonics 4, 174–177 (2010). [CrossRef]
27. K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Götzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics 5, 166–169 (2010). [CrossRef]
29. T. Shegai, V. D. Miljkovic, K. Bao, H. Xu, P. Nordlander, P. Johansson, and M. Kall, “Unidirectional broadband light emission from supported plasmonic nanowires,” Nano Lett. 11, 706–711 (2011). [CrossRef]
31. J. T. Choy, B. J. M. Hausmann, T. M. Babinec, I. Burlu, M. Khan, P. Maletinsky, A. Yacoby, and M. Loncar, “Enhanced single-photon emission from a diamond silver aperture,” Nat. Photonics 5, 738–743 (2011). [CrossRef]
32. G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195 (1984). [CrossRef]
33. Y. C. Jun, R. D. Kekatpure, J. S. White, and M. L. Brongersma, “Nonresonant enhancement of spontaneous emission in metal-dielectric-metal plasmon waveguide structures,” Phys. Rev. B 78, 153111 (2008). [CrossRef]
34. Y. C. Jun, R. Pala, and M. L. Brongersma, “Strong modification of quantum dot spontaneous emission via gap plasmon coupling in metal nanoslits,” J. Phys. Chem. C 114, 7269–7273 (2010). [CrossRef]
35. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
37. C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. 110, 237401 (2013). [CrossRef]
38. J. H. Holland, Adaptation in Natural and Artificial Systems (The University of Michigan, 1975).
39. A. Armaroli, A. Morand, P. Benech, G. Bellanca, and S. Trillo, “Three-dimensional analysis of cylindrical microresonators based on the aperiodic Fourier modal method,” J. Opt. Soc. Am. A 25, 667–675 (2008). [CrossRef]
40. E. Palik, Handbook of Optical Constants of Solids (Academic, 1985).
41. W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane interface. I. total radiated power,” J. Opt. Soc. Am. A 67, 1607–1615 (1977).
42. H. Choumane, N. Ha, C. Nelep, A. Chardon, G. O. Reymond, C. Goutel, G. Cerovic, F. Vallet, C. Weisbuch, and H. Benisty, “Double interference fluorescence enhancement from reflective slides: application to bicolor microarrays,” Appl. Phys. Lett. 87, 031102 (2005). [CrossRef]
45. R. R. Chance, A. H. Miller, A. Prock, and R. Silbey, “Fluorescence and energy transfer near interfaces: the complete and quantitative description of the Eu+3/mirror systems,” J. Chem. Phys. 63, 1589–1595 (1975). [CrossRef]
46. J. E. Sipe, “New green-function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481–489 (1987). [CrossRef]
47. L. Novotny, “Allowed and forbidden light in near-field optics. I. A single dipolar light source,” J. Opt. Soc. Am. A 14, 91–104 (1997). [CrossRef]
48. R. K. Mongia and P. Bhartia, “Dielectric resonator antennas - A review and general design relations for resonant frequency and bandwidth,” Int. J. Microwave Mill. 4, 230–247 (1994).
49. G. N. Malheiros-Silveira, G. S. Wiederhecker, and H. E. Hernandez-Figueroa, “Dielectric resonator antenna for applications in nanophotonics,” Opt. Express 21, 1234–1239 (2013). [CrossRef]
50. L. Zou, W. Withayachumnankul, C. M. Shah, A. Mitchell, M. Bhaskaran, S. Sriram, and C. Fumeaux, “Dielectric resonator nanoantennas at visible frequencies,” Opt. Express 21, 1344–1352 (2013). [CrossRef]