Subwavelength Fabry-Perot resonator: a pair of quantum dots incorporated with gold nanorod
© Liaw et al.; licensee Springer. 2012
Received: 23 August 2012
Accepted: 24 September 2012
Published: 2 October 2012
The two apexes of an elongated gold nanorod (GNR) irradiated by a plane wave are shown to be the hotspots at the longitudinal plasmon modes. This phenomenon implies that a pair of quantum dots (QDs) located at these apexes might be excited simultaneously if the excitation band of QDs coincides with one of these modes. Consequently, a coherent emission of the two emitters could happen subsequently. In the following coherent emission, these two-level emitters are simulated as two oscillating dipoles (bi-dipole) with some possible phase differences. Our results show that the maximum radiative and nonradiative powers of the bi-dipole occur at the longitudinal plasmon dipole, quadrupole, sextupole, and octupole modes of GNR. Moreover, the strongest emissions are induced by the in-phase bi-dipole coupled to the odd modes and the 180° out-of-phase one to the even modes, respectively. The excitation and emission behaviors of a pair of QDs incorporated with GNR demonstrate the possibility of using this structure as a subwavelength resonator of Fabry-Perot type. In addition, the correlation between these modes of the GNR and the dispersion relation of gold nanowire is also discussed.
KeywordsGold nanorod Quantum dot Longitudinal plasmon mode Fabry-Perot resonator Radiative power Nonradiative power Gold nanowire Bi-dipole
Single photon of a quantum dot (QD) coupling with the surface plasmon polaritons of metallic nanowire has attracted wide attentions recently [1–4]. In addition, the dispersion relations of the surface plasmon polaritons (or waves) along gold or silver nanowire [5–9] and the longitudinal plasmon modes of gold or silver nanorods [10, 11] have been studied extensively. The nanoantenna effect and Fabry-Perot resonator of gold nanorod (GNR) through the longitudinal plasmon modes for the emission of nanoemitters (e.g., QD and molecule) have also been studied in the past decade [12–14]. The correlation between the surface plasmon polaritons of metallic nanowire and the plasmon modes of nanorod is an important pivot in linking the submicron and the nano-optics [15, 16]. Because the lower-order plasmon modes of an elongated metallic nanorod are within the near-infrared (NIR) regime , it is particularly worth for study. Recently, these longitudinal plasmon modes of nanorods and nanowires have been investigated using the electric energy loss spectroscopy (EELS) [17–20]. Moreover, the plasmon-enhanced fluorescence of a fluorophore end-linked to GNR has also been demonstrated . In addition, the exciton-plasmon structure of two identical QDs coupling to gold nanoparticle has been studied theoretically .
In this paper, the longitudinal plasmon modes of an elongated GNR irradiated by a plane wave will be studied first to illustrate that the apexes of GNR are the hotspots at these modes. This phenomenon implies that a pair of QDs at these areas might be excited simultaneously with the aid of the plasmon modes of GNR. Once the two QDs are excited and start to emit photon coherently, they are modeled as two electric dipoles with a phase difference in our analysis. To clarify the transition roles from metallic nanorod to nanowire, we investigate the plasmonic enhancement of an elongated GNR with a higher aspect ratio (AR), e.g., AR = 8, on the luminescence of nearby QDs. The far-field radiation patterns and the near-field distributions of the system will be analyzed, particularly at longitudinal plasmon modes of the GNR. In addition, the correlation between these modes and the dispersion relation of a gold nanowire (GNW) will be addressed.
We assume that the GNR is placed on a glass substrate in air. The effective refractive index of the surrounding medium is denoted by n; n = (1−β) nsub + β nair, where the value of β is taken as 0.5, hence n = 1.25. The permittivity of gold is referred in . Note that the wavelength of light, λ, throughout this paper is referred to that in vacuum; the corresponding wavelength in the surrounding medium is then λ/n. We employed the multiple multipole (MMP) method to analyze the electromagnetic field of the problem, based on the Maxwell’s equations [24, 25].
where J0 and J1 are Bessel functions of the first kind of order 0 and 1, respectively, and H0(1) and H1(1) are Hankel functions of the first kind of order 0 and 1. Here, ζ1 and ζ2 are related to the wavenumber k as , where ε1 and ε2 are the permittivity of the surrounding medium and gold, respectively, and μ is the permeability. The complex roots are found numerically to satisfy Equation 3 under the conditions, and , for a given angular frequency , where c is the light speed in vacuum. The phase velocity of the surface plasmon wave in GNW is , and the group velocity .
Results and discussion
Our analysis shows that the two apexes of GNR are hotspots, as an elongated GNR is irradiated by a plane wave at the plasmon modes. The phenomenon can increase the probability of the simultaneous excitation of a pair of QDs at these apexes. Consequently, the coherent emission of the two excited QDs may occur subsequently. They were modeled as two emitters: bi-dipole with phase difference. The radiative power of the bi-dipole at the apexes of the GNR shows the efficient nanoantenna effect for the emission of QDs at the first and second longitudinal plasmon modes which correspond to the dipole and quadrupole modes. Because the first and second modes of an elongated GNR are in the NIR regime, these modes can be used for the optical communication. On the other hand, the higher-order modes (e.g. the third and fourth modes) of GNR show the dark-mode behavior. Moreover, the odd modes are easily induced by the in-phase bi-dipole, but fully suppressed by the 180° out-of-phase one. On the contrary, the even modes are induced by the 180° out-of-phase bi-dipole, but suppressed by the in-phase one. Moreover, the strongest emissions are induced by the in-phase bi-dipole coupled to the odd modes, and the 180° out-of-phase one to the even modes, respectively. Summarily, the plasmon modes of GNR can enhance the simultaneous excitation and coherent emission of a pair of QDs.
These longitudinal plasmon modes of GNR are tunable by adjusting the AR as well as the permittivity of the surrounding medium. In addition, these modes of GNR are consistent with the dispersion relation of GNW. Our preliminary study shows the possibility of using an elongated GNR associated with two QDs at the ends as a subwavelength Fabry-Perot resonator  and might provide further insights for the nanorod spaser [16, 30] and quantum optics [2, 3]. Our analysis could be useful for the plasmonic applications in a variety of rapidly growing fields, e.g., surface enhanced fluorescence [25, 26, 31–33].
absorption cross section
extinction cross section
electric energy loss spectroscopy
scattering cross section.
This work was supported by the National Science Council of Taiwan under grant numbers NSC 99-2221-E-182-030-MY3, NSC 100-2221-E-002-041-MY2 and Chang Gung Memorial Hospital of Taiwan under grant CMRPD 290042.
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