Subwavelength FabryPerot resonator: a pair of quantum dots incorporated with gold nanorod
 JiunnWoei Liaw^{1, 2},
 ChunHui Huang^{3},
 BaeRenn Chen^{3} and
 MaoKuen Kuo^{3}Email author
DOI: 10.1186/1556276X7546
© Liaw et al.; licensee Springer. 2012
Received: 23 August 2012
Accepted: 24 September 2012
Published: 2 October 2012
Abstract
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 twolevel emitters are simulated as two oscillating dipoles (bidipole) with some possible phase differences. Our results show that the maximum radiative and nonradiative powers of the bidipole occur at the longitudinal plasmon dipole, quadrupole, sextupole, and octupole modes of GNR. Moreover, the strongest emissions are induced by the inphase bidipole coupled to the odd modes and the 180° outofphase 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 FabryPerot type. In addition, the correlation between these modes of the GNR and the dispersion relation of gold nanowire is also discussed.
Keywords
Gold nanorod Quantum dot Longitudinal plasmon mode FabryPerot resonator Radiative power Nonradiative power Gold nanowire BidipoleBackground
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 FabryPerot 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 nanooptics [15, 16]. Because the lowerorder plasmon modes of an elongated metallic nanorod are within the nearinfrared (NIR) regime [14], 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 plasmonenhanced fluorescence of a fluorophore endlinked to GNR has also been demonstrated [21]. In addition, the excitonplasmon structure of two identical QDs coupling to gold nanoparticle has been studied theoretically [22].
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 farfield radiation patterns and the nearfield 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.
Methods
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) n_{sub} + β n_{air}, where the value of β is taken as 0.5, hence n = 1.25. The permittivity of gold is referred in [23]. 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 S_{c} is the surface of GNR [25, 26].
where J_{0} and J_{1} are Bessel functions of the first kind of order 0 and 1, respectively, and H_{0}^{(1)} and H_{1}^{(1)} are Hankel functions of the first kind of order 0 and 1. Here, ζ_{1} and ζ_{2} are related to the wavenumber k as ${\zeta}_{i}^{2}=\mu {\epsilon}_{i}{\omega}^{2}{k}^{2}$, where ε_{1} and ε_{2} are the permittivity of the surrounding medium and gold, respectively, and μ is the permeability. The complex roots $k={k}^{\prime}+i{k}^{\u2033}$ are found numerically to satisfy Equation 3 under the conditions, ${k}^{\prime}\ge 0$ and $\omega =2\pi \phantom{\rule{0.1em}{0ex}}c/\lambda $, for a given angular frequency $\omega =2\pi \phantom{\rule{0.1em}{0ex}}c/\lambda $, where c is the light speed in vacuum. The phase velocity of the surface plasmon wave in GNW is ${v}_{\text{p}}=\omega /k\text{'}$, and the group velocity ${v}_{\text{g}}=\partial \omega /\partial {k}^{\prime}$.
Results and discussion
Conclusions
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: bidipole with phase difference. The radiative power of the bidipole 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 higherorder modes (e.g. the third and fourth modes) of GNR show the darkmode behavior. Moreover, the odd modes are easily induced by the inphase bidipole, but fully suppressed by the 180° outofphase one. On the contrary, the even modes are induced by the 180° outofphase bidipole, but suppressed by the inphase one. Moreover, the strongest emissions are induced by the inphase bidipole coupled to the odd modes, and the 180° outofphase 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 FabryPerot resonator [10] 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].
Abbreviations
 ACS:

absorption cross section
 AR:

aspect ratio
 ECS:

extinction cross section
 EELS:

electric energy loss spectroscopy
 GNR:

gold nanorod
 GNW:

gold nanowire
 MMP:

multiple multipole
 NIR:

nearinfrared
 QD:

quantum dot
 SCS:

scattering cross section.
Declarations
Acknowledgments
This work was supported by the National Science Council of Taiwan under grant numbers NSC 992221E182030MY3, NSC 1002221E002041MY2 and Chang Gung Memorial Hospital of Taiwan under grant CMRPD 290042.
Authors’ Affiliations
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