We present a full three-dimensional (3D) power flow analysis of an emitter-nanoantenna system. A conventional analysis, based on the total Poynting vector, calculates only the coupling strength in terms of the Purcell enhancement. For a better understanding of the emitter-nanoantenna system, not only the Purcell enhancement but also complete information on the energy transfer channels is necessary. The separation of the pure scattering and emitter output Poynting vectors enables the quantification of the individual energy transfer channels. Employing the finite-difference time-domain method (FDTD), we examine a nanodisk antenna that supports the bright dipole and dark quadrupole resonance modes for which the power flow characteristics are completely distinct, and we analyze the power flow enhancements to the energy transfer channels with respect to the wavelength, polarization, and position of the emitter coupled to the antenna. The 3D power flow analysis reveals how the constructive or destructive interference between the emitter and the antenna resonance mode affects the power flow enhancements and the far-field radiation pattern. Our proposed power flow analysis should play a critical role in characterizing the emitter-antenna system and customizing its energy transfer properties for desired applications.
© 2015 Optical Society of America
Optical nanoantennas interface free-propagating light and a localized electromagnetic field by receiving or transmitting optical radiation [1–4]. In particular, plasmonic resonances supported by metallic nanoantennas enable strong modification of the local density of optical states (LDOS) beyond the diffraction limit and the tailoring of the emission properties of various quantum emitters, including radiative recombination efficiency, modulation speed, spectral behavior, and polarization state, based on the Purcell effect [5–8]. The interaction between a metallic nanoantenna and a quantum emitter is quite complex due to high radiation leakage, absorption losses in metal, and spectrally broad resonances. When its size exceeds the Rayleigh criterion, the metallic nanoantenna supports multiple resonances with different Q factors and mode volumes, and their spectra overlap . Thus, a continuum of the resonance modes contributes to the modified LDOS. Non-negligible retardation also significantly influences emitter-antenna interaction, depending on the geometry of the nanoantenna and the position of the emitter with respect to the antenna [10–12].
The typical normal-mode expansion method provides the exact value of the coupling strength between an emitter and a simple nanoantenna with a spherical, cylindrical, or planar geometry [13–15]. For a nanoantenna with a complex geometry, numerical simulations can be used to calculate the coupling strength with the emitter and the total energy decay rate enhancement of the emitter based on the total Poynting vector [16–18]. However, the coupling strength and enhancement of the total energy decay rate do not provide complete information on the emitter-antenna system. To completely understand and engineer the emitter-antenna interactions, it is necessary to characterize each energy transfer channel and quantify the power transfer from the emitter into the channels [19,20].
A recent study reported on the energy transfer pathways in a two-dimensional (2D) emitter-antenna system based on a Poynting vector power flow analysis in which the total Poynting vector was separated into the emitter output and antenna scattering components . However, most practical optical antennas have three-dimensional (3D) structures and have more complicated emitter-antenna interactions, the details of which cannot be replicated by 2D simulations . For example, subwavelength-scale nanoantennas support not only fundamental bright-mode resonance but also higher-order dark-mode resonance [9,10]. While the normal incidence of free-space illumination excites only the bright mode, a 3D point-source emitter seamlessly couples to both bright and dark modes. Because of its capacity for strong light absorption and low radiation production, the dark-mode resonance has completely different power flow characteristics than the bright-mode resonance. The phase retardation and subsequent interference between the antenna’s scattering and the emitter’s emission strongly modify the power flow. 3D nanoantennas can produce not only constructive but also destructive interference. The destructive interference can remarkably suppress the radiative power flow, even via the bright mode. In addition, the dipole emitter in a 2D simulation is actually an infinite chain of coherent point dipoles, which cannot be achieved in practice.
In this study, we present for the first time a 3D Poynting vector power flow analysis using FDTD simulations and quantitatively analyze the energy transfer channels of a single-dipole emitter coupled to a gold nanodisk antenna. The far-field distributions of the total radiation and its enhancement are also investigated and correlated with the power flow at the near-field region. We aim to provide a design strategy to customize emitter-antenna systems for use in many applications, including nanooptical sensing, active metamaterials, nonlinear signal generation, and quantum plasmonics [23–26].
2. Theoretical methods
Quantitative analysis of the power flow channels starts by separating the total electric and magnetic fields and into the incidence fields and and the scattered fields and . We perform two separate simulations, with and without the antenna, to obtain the total and incident fields, respectively, and subtract the incident field from the total field to obtain the scattered field at each simulation time step. Then, the total Poynting vector can be decomposed into the emitter output Poynting vector and the pure scattering Poynting vectorFig. 1(a), we calculate the power flow into each energy transfer channel of interest.
In this article, we concentrate on the following energy transfer channels and their power flow enhancements: total energy decay enhancement , total radiation enhancement , nonradiative transfer enhancement by antenna absorption , antenna scattering enhancement , and direct energy transfer enhancement between the emitter and the free space . The enhancement factors are obtained by normalizing the time-averaged power flow into the energy transfer channels to the emission power of the emitter without the antenna . An enhancement factor less than 1 means that the power flow to the corresponding energy transfer channel is suppressed. is the ratio of the emission power of the emitter with and without the nanoantenna:
One of the most important consequences of the power flow analysis is that provides quantitative information on not only how much power from the emitter flows into the antenna but also how much couples directly to the free space without going through the antenna depending on the choice of the integration surface. First,
3. Power flow analysis of an emitter-antenna system
We analyze a gold nanodisk antenna coupled to a single-electric-dipole emitter as a model 3D system [Fig. 1(b)]. The nanodisk antenna is placed on the glass substrate with a refractive index of 1.5. The thickness and diameter of the nanoantenna are 40 nm and 120 nm, respectively. The nanoantenna supports two localized surface plasmon resonance modes, dipole and quadrupole, at ~650 and ~550 nm, respectively. The inset in Fig. 1(b) shows the surface charge density distribution on the top surface of the nanodisk antenna for the two resonance modes excited by a y-polarized dipole emitter that is a distance d of 20 nm from the nanodisk and 10 nm above the substrate. The charge distribution is slightly asymmetric about y-axis because of the retardation effect between the near side and the far side of the nanodisk in relation to the emitter. The dipole mode is naturally radiative and supports a bright resonance. On the other hand, because of its well-balanced fourfold symmetry of the field and charge distributions, the quadrupole mode has a near-zero net dipole moment at the far-field and thus supports a dark resonance. The normally incident plane wave cannot excite the quadrupole-mode resonance. However, an obliquely incident plane wave or a point-like emitter, e.g., quantum dots or fluorescence molecules, couples to the dark quadrupole mode remarkably with completely different power flow properties from those of the bright dipole mode [9,10].
In the 3D FDTD simulations, we modeled the dielectric function of gold using the Drude critical points (DCP) model, which gives
We use two critical points to account for the interband transitions in gold. The DCP dielectric function fits the experimentally measured dielectric constants of gold in the spectral range 340-1,000 nm ; the fitting parameters are given in Table 1. The domain and grid sizes were 400 × 400 × 400 nm3 and 1 nm, respectively. The convolutional perfectly matched layer (CPML) is used as the absorbing boundary condition. The intrinsic quantum efficiency of the emitter is assumed to be unity. The emission power of the emitter without the nanoantenna, which normalizes the enhancement factors, is calculated with the presence of the glass substrate. This enables the separation of the effects of the nanoantenna from those of the substrate on the power flow channels and their enhancement factors.
We consider the case where the emitter and the nanodisk antenna are strongly coupled to each other. The y-polarized dipole emitter, oscillating parallel to the radial direction of the nanodisk, couples to the antenna resonance modes more strongly than does the x-polarized dipole emitter. Figure 2(a) shows the spectra of the power flow enhancement factors for the y-polarized emitter. The total energy decay rate enhancement clearly shows that the emitter couples strongly to the dipole and quadrupole modes [the black curve in Fig. 2(a)]. The excitation of the resonance modes is confirmed by the time-averaged intensity distribution of the electric field, which depends on the emission wavelength as shown in Fig. 2(b). As expected, the two resonance modes exhibit completely different behaviors. The dipole mode dominates the spectrum of [the red curve of Fig. 2(a)]. The bright dipole mode reradiates a considerable portion of the power transferred from the emitter, which leads to a high radiation extraction efficiency of ~0.78. The ratio of the antenna scattering to the total radiation, is ~0.88. On the other hand, the dark quadrupole mode absorbs most of the transferred power in the metal, where ~0.93, and plays a major role in the spectrum of [the yellow curve of Fig. 2(a)]. The radiation extraction efficiency of the quadrupole mode is only ~0.07 despite the strong coupling strength. Obviously, a strong Purcell enhancement does not guarantee a large enhancement of the final radiation from the emitter-antenna system.
For a wavelength of <640 nm, <1 or even negative [the green curve of Fig. 2(a)] and thus the radiation enhancement is limited. First, the strong absorption in metal is responsible for the limitation of the radiation enhancement around the quadrupole mode resonance. Second, in the spectral range between the quadrupole- and dipole-mode resonances, the radiation enhancement is limited by the destructive interference between the antenna scattering and the emitter’s emission. The plots in Fig. 2(c) show the distribution of the phase differences of the scattered Ey fields from the phase of the emitter. As expected, at the 600-nm wavelength, the scattered Ey field at the region, where its intensity is strong [see Fig. 2(b)], oscillates almost in antiphase with respect to the emitter, as seen in Fig. 2(c), which causes destructive interference of the far-field radiation. On the other hand, at the wavelength that is red-detuned from the dipole-mode resonance, the scattered Ey field oscillates almost in-phase with the emitter and leads to constructive interference of the far-field radiation. The constructive interference allows to remain as high as ~6.5 and still results in a large , even out of resonance.
The x-polarized emitter exhibits completely different power flow properties and spectra of the power flow enhancement factors from the y-polarized emitter. Figure 3(a) shows that the x-polarized dipole emitter excites the quadrupole mode more strongly than the dipole mode, while the y-polarized emitter actively couples with both modes because the induced surface charges of the quadrupole mode are distributed closer to the dipole emitter than the surface charges of the dipole mode, as seen in Fig. 3(b). In addition, the strength of the coupling of the x-polarized emitter with the antenna modes is weak overall and the emitted power does not transfer to the antenna effectively. Thus, most of the field is located near the emitter [Fig. 3(b)]. On the other hand, for the y-polarized emitter, the field is concentrated mainly around the nanodisk antenna due to efficient energy transfer [Fig. 2(b)].
Despite its radiative properties, the dipole mode coupled with the x-polarized emitter has a of only ~1.1 at the resonance condition. At the wavelength that is red-detuned from the dipole resonance, decreases to ~0.19 at 750 nm and the total radiation from the emitter-antenna system is suppressed, far from being enhanced. The radiation suppression originates from the destructive interference between the emitter and the dipole resonance mode rather than from the metal absorption that dominates in the case of the quadrupole mode coupled to the y-polarized emitter. As shown in Fig. 3(c), at a wavelength of >650 nm, the scattered Ex field, where its intensity is strong [see Fig. 3(b)], oscillates almost in antiphase with respect to the emitter because of the spatial retardation effects on the antenna volume. The antiphase oscillation leads to destructive interference and negative , which cancels out the contribution of to Radiative and nonradiative properties of the antenna resonance modes and their interference characteristics with the emitter are crucial to the design and engineering of 3D emitter-antenna systems. The enhancement or suppression of radiation based on the interaction between the emitter and the antenna scattering correlates with the Poynting vector power flow at the near-field and the radiation distribution at the far-field, which is discussed in detail below. At the quadrupole-mode resonance, most of the power was absorbed into the antenna with a ratio of to of ~0.83, and the total radiation power is almost equal to the emission power of the original emitter.
Visualizing , , and is also important to the investigation of the energy transfer channels and understanding their power flow characteristics. Figure 4 shows the magnitude and streamline distributions of the time-averaged components of the Poynting vector laid on the cross-sectional plane. The magnitude distributions are normalized to their maximum values. The streamlines directed into the nanoantenna and toward the free space are white and black, respectively. The streamlines of from the free space into the nanoantenna occur when the quadrupole mode is coupled to the y-polarized emitter and when the dipole mode is coupled to the x-polarized emitter. flowing from the free space represents the destructive interference between the incident and scattered fields and the consequent reduction of the total radiation to the far-field, which is seen in the spectrum of . always flows outward from the emitter-antenna system at the far-field region. However, at the near-field region, strong absorption by the metal produces that flows directly from the emitter into the nanoantenna, e.g., in the case of the quadrupole mode coupled to the y-polarized emitter. , the streamlines of which always start from the antenna’s surface, shows how the nanoantenna reradiates the captured power. The magnitude distribution of clearly reflects the modal characteristics of the nanoantenna at the near-field.
The far-field radiation pattern correlates with the emission power flow distribution at the near-field and includes the information of the power flow channels and their enhancement factors. We calculated the far-field radiation patterns on the northern hemispheres with respect to wavelength, as shown in upper row of Figs. 5(b) and 5(c) for the y- and x-polarized emitters, respectively. The emitter is 20 nm from the nanoantenna along the + y-axis and each far-field radiation pattern is normalized to its maximal value. Figure 5(a) shows the far-field radiation pattern of the emitter on the glass substrate and the far-field pattern mapping scheme. The near-to-far-field (NTFF) transformation, obtained by using the reciprocity theorem and the transfer matrix method, is employed to take account of the substrate . To evaluate the interference between the antenna scattering and the emitter’s emission, we also calculated the far-field distribution of the total radiation enhancement, which is the ratio of the far-field radiation intensities with and without the nanoantenna [bottom row of Figs. 5(b) and 5(c)]. The calculated far-field radiation patterns can be compared to the cathodoluminescence emission patterns of single gold nanodisks nanodisk . In particular, the combination of the far-field radiation pattern analysis and the multipole component decomposition method [5, 29] can be useful for understanding not only the emitter-antenna coupling but also other emission phenomena involving multipole component contributions, such as cathodoluminescence.
For the y-polarized emitter, the dipole mode leads to strong and fairly uniform radiation enhancement in all directions based on the constructive interference. At the dipole-mode resonance (λ = 650 nm), the far-field radiation enhancement is ~20 on average, which is identical to the value of obtained in near-field region. The in-phase oscillation of the emitter and the dipole mode, which is observed in the near-field distribution, produces a far-field radiation pattern that is nearly same as the original pattern of the emitter, even at the wavelength of 700 nm. On the other hand, the quadrupole mode (λ = 550 nm) considerably modifies the far-field radiation depending on the direction. Both destructive and constructive interference appear in the far-field-radiation enhancement pattern. The positions of the destructive and constructive interference correlate with the power flow distribution at the near-field.
For the x-polarized emitter, there is strong suppression of the far-field radiation at the dipole-mode resonance due to destructive interference. In particular, at the wavelength of 700 nm, radiation suppression is maximized at ~0.06 near the pole of the northern hemisphere, since the dipole mode and the emitter oscillate almost in antiphase and the net dipole momentum to the vertical direction is cancelled out. Even though the dipole mode is radiative in nature, its antiphase interaction with the emitter causes a quadrupolar radiation pattern at the far-field. At the quadrupole-mode resonance, the shape and intensity of the far-field radiation are similar to those of the original radiation from the x-polarized emitter. As we see in the spectra of and , the dark quadrupole mode hardly affects the radiative power flow and far-field radiation pattern. Although the emitter is located close to the nanoantenna and its total energy decay rate is enhanced many times, all traces of the emitter-antenna interaction are covered up in the subsequent far-field radiation. Consequently, when manipulating the radiation characteristics of a quantum emitter as needed, it is necessary to design a nanoantenna that strongly modifies LDOS and to combine the nanoantenna and the emitter with a proper phase difference and interference.
Finally, we investigate how the power flow channels and their enhancements depend on the distance d between the emitter and the nanoantenna [Figs. 6(a) and 6(b)]. The quadrupole mode has a shorter evanescent field tail and thus fades from the spectra of the enhancement factors faster than the dipole mode as the distance increases. Similarly, the spectrum of , which the quadrupole-mode resonance dominates, loses its resonant behavior and becomes flat faster than the spectrum of , which the bright dipole mode dominates. This dependence on distance is common to both the x- and y-polarized emitters. When the antenna and the emitter are finally decoupled, , , and converge to unity and and approach zero over the entire wavelength range. However, perturbation of the near-field, even minute scattering, results in significant changes in the far-field radiation pattern. Figures 6(c) and 6(d) show the far-field radiation patterns for different distances d between the emitter and the antenna at the dipole-mode resonance (λ = 650 nm). The far-field radiation patterns seen in Figs. 6(c) and 6(d) are distinctly different from those of the single-dipole emitter [Fig. 5(a)], even for d = 150 nm, where the dipole-mode resonance nearly fades from the spectra of the enhancement factors and, in particular, is only ~0.1. The effects of the nanoantenna on the far-field radiation pattern are diminished when the distance of the nanoantenna from the emitter is at least >400 nm. The far-field radiation pattern is modified by the interference of the emitter’s emission and the antenna scattering that accompanies the changes in and its spectrum. can provide a criterion for characterizing the emitter-antenna coupling at both the near- and far-field regions.
We presented 3D Poynting vector power flow analysis to quantitatively characterize the emitter-nanoantenna system. Separation of the total Poynting vector into pure scattering and emitter output components enables us to quantify and visualize the individual energy transfer channels, including antenna scattering reradiation, nonradiative absorption, and direct energy transfer between the emitter and the free space, while conventional analysis, which focuses on the total Poynting vector, provides only the coupling strength and the Purcell enhancement factor. The analysis of the individual power flow channels successfully identifies the distinct resonance modes of the 3D nanodisk antenna, i.e., the bright-dipole and dark-quadrupole modes, and reveals their different coupling behaviors. In addition, we examined how the constructive or destructive interference between the emitter and the antenna influences the power flow properties at the near- and far-field regions. We expect that the 3D power flow analysis will provide optimized strategies for designing emitter-antenna systems suitable for a desired application.
The authors acknowledge support for this work by the National Research Foundation of Korea (NRF) (2013R1A2A2A01014224, 2014M3C1A3052537, and 2014M3A6B3063709).
References and links
1. L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5(2), 83–90 (2011). [CrossRef]
3. 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(5994), 930–933 (2010). [CrossRef] [PubMed]
5. A. G. Curto, T. H. Taminiau, G. Volpe, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Multipolar radiation of quantum emitters with nanowire optical antennas,” Nat. Commun. 4, 1750 (2013). [CrossRef] [PubMed]
6. M. K. Schmidt, S. Mackowski, and J. Aizpurua, “Control of single emitter radiation by polarization- and position-dependent activation of dark antenna modes,” Opt. Lett. 37(6), 1017–1019 (2012). [CrossRef] [PubMed]
7. C. Gruber, A. Trügler, A. Hohenau, U. Hohenester, and J. R. Krenn, “Spectral Modifications and Polarization Dependent Coupling in Tailored Assemblies of Quantum Dots and Plasmonic Nanowires,” Nano Lett. 13(9), 4257–4262 (2013). [CrossRef] [PubMed]
8. D. Chang, A. Sørensen, P. Hemmer, and M. Lukin, “Strong coupling of single emitters to surface plasmons,” Phys. Rev. B 76(3), 035420 (2007). [CrossRef]
9. M. Liu, T.-W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of Dark Plasmons in Metal Nanoparticles by a Localized Emitter,” Phys. Rev. Lett. 102(10), 107401 (2009). [CrossRef] [PubMed]
10. R. Esteban, R. Vogelgesang, J. Dorfmüller, A. Dmitriev, C. Rockstuhl, C. Etrich, and K. Kern, “Direct Near-Field Optical Imaging of Higher Order Plasmonic Resonances,” Nano Lett. 8(10), 3155–3159 (2008). [CrossRef] [PubMed]
11. J. Dorfmüller, R. Vogelgesang, W. Khunsin, C. Rockstuhl, C. Etrich, and K. Kern, “Plasmonic Nanowire Antennas: Experiment, Simulation, and Theory,” Nano Lett. 10(9), 3596–3603 (2010). [CrossRef] [PubMed]
12. D. Vercruysse, X. Zheng, Y. Sonnefraud, N. Verellen, G. Di Martino, L. Lagae, G. A. E. Vandenbosch, V. V. Moshchalkov, S. A. Maier, and P. Van Dorpe, “Directional Fluorescence Emission by Individual V-Antennas Explained by Mode Expansion,” ACS Nano 8(8), 8232–8241 (2014). [CrossRef] [PubMed]
14. 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(23), 237401 (2013). [CrossRef] [PubMed]
17. E.-K. Lee, J.-H. Song, K.-Y. Jeong, and M.-K. Seo, “Design of plasmonic nano-antenna for total internal reflection fluorescence microscopy,” Opt. Express 21(20), 23036–23047 (2013). [CrossRef] [PubMed]
18. J. Barthes, G. Colas des Francs, A. Bouhelier, J. C. Weeber, and A. Dereux, “Purcell factor for a point-like dipolar emitter coupled to a two-dimensional plasmonic waveguide,” Phys. Rev. B 84(7), 073403 (2011). [CrossRef]
19. C. F. Bohren, “How can a particle absorb more than the light incident on it?” Am. J. Phys. 51(4), 323 (1983). [CrossRef]
20. H. T. Miyazaki and Y. Kurokawa, “How Can a Resonant Nanogap Enhance Optical Fields by Many Orders of Magnitude?” IEEE J. Sel. Top. Quantum Electron. 14(6), 1565–1576 (2008). [CrossRef]
21. K. C. Y. Huang, Y. C. Jun, M.-K. Seo, and M. L. Brongersma, “Power flow from a dipole emitter near an optical antenna,” Opt. Express 19(20), 19084–19092 (2011). [PubMed]
22. J. Y. Suh, M. D. Huntington, C. H. Kim, W. Zhou, M. R. Wasielewski, and T. W. Odom, “Extraordinary Nonlinear Absorption in 3D Bowtie Nanoantennas,” Nano Lett. 12(1), 269–274 (2012). [CrossRef] [PubMed]
23. A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009). [CrossRef]
25. H. Kollmann, X. Piao, M. Esmann, S. F. Becker, D. Hou, C. Huynh, L.-O. Kautschor, G. Bösker, H. Vieker, A. Beyer, A. Gölzhäuser, N. Park, R. Vogelgesang, M. Silies, and C. Lienau, “Toward Plasmonics with Nanometer Precision: Nonlinear Optics of Helium-Ion Milled Gold Nanoantennas,” Nano Lett. 14(8), 4778–4784 (2014). [CrossRef] [PubMed]
26. M. S. Tame, K. R. McEnery, Ş. K. Özdemir, J. Lee, S. A. Maier, and M. S. Kim, “Quantum plasmonics,” Nat. Phys. 9(6), 329–340 (2013). [CrossRef]
27. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]
28. K. Demarest, Z. Huang, and R. Plumb, “An FDTD near- to far-zone transformation for scatterers buried in stratified grounds,” IEEE Trans. Antenn. Propag. 44(8), 1150–1157 (1996). [CrossRef]