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We propose a method to dynamically control the Fano resonance of a ring/crescent-ring gold nanostructure by spatially changing the phase distribution of a probe Bessel beam. We demonstrate that a highly tunable Fano interference between the quadrupole and bright dipole modes can be realized in the near-infrared range. Even though a complex interference between a broad resonance and a narrower resonance lead to these observations, we show that a simple coupled oscillator model can accurately describe the behavior, providing valuable insights into the dynamics of the system. A further analysis of this structure uncovers a series of interesting phenomena such as anticrossing, sign changing of coupling and the spectral inversion of quadrupole and bright dipole modes. We further show that near field enhancement at Fano resonance can be actively controlled by modulating the phase distribution of the exciting incident Bessel beam.
© 2014 Optical Society of America
1. Introduction
Metallic nanostructures are endowed with remarkable properties for manipulating light at subwavelength level, owing to their ability to support surface plasmon resonances. The resulting strong light-matter interaction allows people to readily probe the fundamental physical processes [1–3] and gives birth to novel functional nanodevices [4–7]. Plasmon resonances of individual nanostructures are quite sensitive to their morphologies and sizes, which provide a wide set of parameters for engineering the local field distribution [8] and spectrum response of the nanostructures [9]. Owing to this ability, the plasmonic particles have found applications in areas such as surface enhanced spectroscopy [10,11], molecular sensing [12,13] and biomedical diagnostics [14,15]. Plasmon resonances of two or more closely placed nanostructures can couple to each other through near field interactions, leading to the hybridization of plasmonic modes. It therefore offers an ideal platform for observing quantum interferences such as Fano resonance or electromagnetically induced transparency in classical optical systems [16].
Fano resonance in a plasmonic nanostructure arises from the interference between the bright and the dark modes. In the widely used quasi-static approximation, the nanostructures have a dominated dipole-like response named bright mode, which has a large scattering cross-section and a broad linewidth. In systems with symmetry breaking, the high order multipole resonances, such as the quadrupole mode, can be excited. However, normally the quadrupole mode is weakly radiative and has a narrow linewidth, known as dark mode. In a properly designed nanostructure, the narrow dark mode and broad bright mode can overlap, giving rise to a pronounced asymmetric spectral lineshape. This is analogous to the Fano resonance in the atomic system [17,18], which results from the interference between two competing pathways, one associated with discrete states and the other with a continuum of states. Over the past few years, Fano resonances have been extensively studied in plasmonic nanostructures, including metallic nanoclusters [19,20], dolmen [2,21], ring/disk cavities [22] and split rings [23,24]. The most intriguing characteristics of a plasmonic Fano resonance may be its tunability in spectrum, the so-called “lineshape engineering”. Spectral control of Fano resonance is of central importance, both in view of quantum inference physics [25,26] and the chemical or bio-sensing applications [27,28]. Conventionally, the Fano lineshape can be tuned by carefully altering the geometry of a nanostructure [29–31]. However, it is inevitable to introduce a geometry deviation during fabrication procedure, thereby rendering a substantial difficulty for symmetrically tailoring the Fano resonance. Moreover, the dynamical control of Fano resonance is also highly desirable, and is not yet easily attained with geometry changing way. In this work, we demonstrate a tunable Fano resonance in a plasmonic nanostructure composed of a crescent-ring and a symmetric ring. In contrast to conventional approaches, the high tunability of the Fano resonance is realized solely by changing the phase structure of the Bessel-like excitation beam. As Bessel beams can be produced and remodulated by a spatial light modulator (SLM), it is very flexible to change the phase structure of beam and control the associated Fano resonance. We further elucidate the fundamental physics of this tunable Fano resonance with a coupled oscillator system, providing valuable insights into the operation of the device.
2. Results and discussion
The design under consideration is a gold ring/crescent-ring nanostructure (RCRN), shown schematically in Fig. 1(a). The geometry is defined by the parameters: the inner and outer radii of the crescent-ring r1 = 260 nm and r2 = 300 nm, the central offset of the crescent-ring Δ = 30 nm, the inner and outer radii of the small ring r3 = 160 nm and r4 = 200 nm, and the height of the system H = 50 nm. The optical response of our system is numerically calculated by using the finite element method. In the simulation, we use a perfectly matched layer boundary condition. The nanostructure is swept meshed with the minimum size of 9 nm. The permittivity of gold is described by the Drude model ε(ω) = ε∞-ωp2/(ω2 + iγω) with ε∞ = 9.1, ωp = 1.38 × 1016s−1 and γ = 1.08 × 1014s−1 [32]. We start our discussion with the scattering spectrum of the RCRN excited by a horizontal polarized plane wave, as shown in the middle panel of Fig. 1(b). The scattering spectrum can be qualitatively interpreted within the plasmon hybridization framework [33,34], illustrated in Fig. 1(b). The left and right panels of Fig. 1(b) correspond to the normalized scattering spectrum as well as the labeled resonance modes of the isolated small- and big-ring, respectively. It is clear from these results that in isolation, both rings exhibit a dominated dipole resonance with plane wave excitation but with different spectral widths. When they start interacting with each other, these two dipole modes form two new hybridization modes, i.e., a low energy narrow dark mode and a high energy broad bright mode. The dark mode results from the out-of-phase oscillation of two dipole resonances, which has a small total dipole moment and a weak far-field radiation. In contrast to this, the bright mode has a large total dipole moment due to in-phase oscillation of two dipole resonances, thus it is highly radiative. Owing to the symmetry breaking in the big ring, a narrow quadrupole mode also appears at the high energy part of the scattering spectrum. This mode overlaps with the broad bright mode both in spatially and spectrally, leading to the Fano resonance. We also examine the scattering spectrum excited by a vertical polarized plane wave [see the red curve in Fig. 1(c)]. It is seen that the scattering spectrum excited by the vertical polarization beam has a similar but a redshift lineshape compared with the horizontal one [see the blue curve in Fig. 1(c)]. Similar results are also observed in a periodical split ring structure within terahertz range [35].
Fig. 1 (a) Schematic view of the ring/crescent-ring nanostructure (RCRN). The geometry parameters r1, r2, Δ, r3, r4, and H are 260, 300, 30, 160, 200 and 50 nm, respectively. (b) Hybridization diagram for horizontal polarized plane wave excited Fano resonance of the RCRN. Insets: near field intensity distributions at resonance energies. (c) Comparison between the scattering spectrum of RCRN excited by a horizontal (blue) and vertical (red) polarized plane wave.
Figure 2 shows the optical response of the RCRN when excited by an incident Bessel beam. Bessel beams are among a class of non-diffraction beams, whose amplitude is defined by Bessel function of the first kind. In the top of Fig. 2(a), we depict the complex amplitude distribution of a Bessel beam with the form of J0(βr), where r = (x2 + y2)1/2 and β is a constant that determines the central spot diameter (CSD) of the Bessel beam. The middle of Fig. 2(b) represents the corresponding phase distribution of the Bessel beam, which exhibits an annular staggered π phase discontinuity. Figure 2(b) illustrates the scattering spectrum of the RCRN excited by the Bessel beam shown in Fig. 2(a), which is concentric with the small ring of the RCRN [shown in the bottom of Fig. 2(a)] and has a CSD of 525 nm (defined by full wave at half maximum). The polarization direction of the Bessel beam remains horizontal, as depicted in Fig. 2(a). It reveals that a clear Fano dip locates at the broad profile of the bright dipole resonance. The insets of Fig. 2(b) show the associated near field distributions at Fano (green) and dark dipole (red) resonances.
Fig. 2 (a) Complex amplitude (top) and phase (middle) distribution of Bessel beam as well as its field distribution illuminating on the RCRN (bottom). (b) Scattering spectrum of RCRN excited by a Bessel beam. The insets correspond to the associated near field intensity distributions at Fano (green) and dark dipole (red) resonances.
In Fig. 3, we show the simulation results of Fano resonance tuned by the Bessel beam with horizontal polarization. Figure 3(a) reveals that as the CSD decreasing the illuminations on inner and outer ring of RCRN gradually evolve from in-phase to out-of-phase, which effectively introduces a varying phase retardation that affects the plasmonic modes participating in the Fano resonance. To determine the influence of phase retardation on the Fano resonance, it is more intuitive to examine the absorption spectrum of individual consisting rings, as illustrated in Figs. 3(b) and 3(c). The quadrupole mode, displayed as the shorter wavelength peak in Fig. 3(b), exhibits less sensitive to the phase retardation effect [22], remaining a robust resonance frequency but the intensity decreases if the CSD size decreases. However, smaller CSD results in the blue-shift of dipole modes in both rings. When the two rings are assembled, the two blue-shifted dipole modes of the rings anti-bond together and interact with quadrupole mode. As a result, the scattering spectrum of RCRN varies with the CSD decreasing, exhibiting a tunable Fano resonance as shown in Fig. 3(d). There are two features emerging in this tunable Fano resonance. (1) When the CSD decreases, the bright dipole mode resulting from the two anti-bonded individual dipole modes gets weaker. This is mainly because the increased phase retardation effect will quench the charges forming the anti-bonded dipole mode, leading to a reduced polarizability. (2) Both the bright dipole and the quadrupole modes have more obvious blue-shift than their uncoupled counterparts. However, the dipole mode is more dispersive than the quadrupole mode as the CSD decreases, resulting in the Fano dip shifts from one side of the bright dipole profile to the other.
Fig. 3 (a) Bessel beams with varying central spot diameter (CSD) from 535 to 325 nm. (b) Normalized absorption spectra of individual outer and (c) inner rings with decreasing of the CSD of Bessel beam. (d) Controlling Fano resonance with the CSD of Bessel beam.
To provide a deeper insight into the system, a coupled oscillator model, shown schematically in Fig. 4(a), is proposed to describe the Bessel beam tuned Fano resonance. The two oscillators are referred to as the quadrupole and bright dipole modes, which are characterized by the resonance frequencies ωq, ωd and nonradiative damping γq, γd. When the dipole oscillator is driven by an external force F, it can couple with the quadrupole oscillator through the near field interaction. At the meantime, the dipole oscillator also dissipates energy via the radiative damping, which can exert a radiation reaction force on itself [36]. In contrast to the radiation of the dipole mode, that of the quadrupole mode is very weak. Therefore, the radiation reaction force is negligible for the quadrupole mode. Then, the equation governing the motions of the two oscillators can be written as [30]
where the coupling constant and the radiation reaction force are described by g and Γd3x/dt3, respectively. The excitation beam E = J0(βr) has a nonuniform value over the nanostruture, which cannot be easily included in the one-dimensional coupled oscillator model. So, without loss of generality, we equivalently represented the Bessel excitation by a plane wave E = E0e iωt with a conversation efficiency of η related to F [30]. At steady state, the motion of the two oscillators are harmonics with forms of xd,q = Ad,qe iωt, where Ad,q can be derived asThe scattering coefficient of the system can be defined by σsca = |Ad + Aq|2, which is used to fit the scattering spectrum of the Bessel beam excited RCRN in Fig. 3(d). The fitting parameters of coupled oscillator model are summarized in Table 1. A good agreement between theory and simulation is achieved, which is representatively shown in Figs. 4(b)–4(d). Here, the dotted and solid lines are respectively for the simulation and fitting results.Fig. 4 (a) Schematic of the coupled oscillator model proposed for the fitting. The two oscillators correspond to the dipole and quadrupole modes, with frequency of ωd, ωq and nonradiative damping γq, γd. (b)-(d) Fano resonance of RCRN excited by Bessel beams with CSD of 535, 360 and 325 nm, respectively. The dotted and solid lines correspond to the simulation and fitting results, respectively.
Table 1. Extracted fitting parameters from coupled oscillator model for varying central spot diameter (CSD) of Bessel beam
The coupled oscillator model allows us to further analysis the main features in Fig. 3(d). In Fig. 5(a), we show the dependence of the frequencies of the quadrupole (blue) and bright dipole (red) resonances on the CSD. It yields two main characteristics. The first one is a spectral inversion of quadrupole and bright dipole resonance appears for small CSD values. When CSD is large, the quadrupole resonance resides at the blue side of the bright dipole resonance. However, it shifts to the red side of the dipole resonance when CSD is reduced below 423 nm. This result confirms the tunable Fano resonance observed in Fig. 3(d) and can be attributed to the change of the coupling sign and strength [37], as shown in Fig. 5(b). Another interesting feature is that the quadrupole and dipole curves in Fig. 5(a) have no intersection point but show up a discontinue jump when spectral inversion occurs, exhibiting an anticrossing behavior. The anticrossing behavior is a consequence of strong coupling effect [37] that can be directly examined in Fig. 5(b), where the maximum coupling strength just locates the same CSD as the discontinue jump in Fig. 5(a). The unusual sign change of coupling constant can be qualitatively interpreted by considering the phase structure of the Bessel beam. When CSD decreases to 423 nm, the illumination on the outer of the crescent-ring begins to be out of phase with that on the other parts of the RCRN. Such a phase jump affects the charge distributions of the RCRN, and consequently the coulomb interaction between the quadrupole and bright dipole resonances changes from repulsive to attractive. The similar coupling sign reversion phenomenon also exists in other tunable Fano systems [30]. Note also that the change of coupling sign can influence the spectral structure of relative phase between the bright dipole and quadrupole resonances, which can be mathematically deduced from Eq. (2)
Therefore, when the CSD decreases, an inversion of relative phase structure companying with the change of coupling sign is observed, as shown in Fig. 5(c).Fig. 5 (a) Dipole (red) and quadrupole (blue) resonance frequencies extracted from the coupled oscillator model versus CSD. (b) Coupling constant g varying with the CSD. (c) Dependence of relative phase Δφ on the CSD.
As shown clearly in Fig. 5(b), the coupling constant is closely dependent on the CSD of the Bessel beam. Thus, by varying the CSD, it is possible to alter the energy transfer between the bright dipole mode and the quadrupole mode at the Fano resonance. As a consequence, the near field distribution of the RCRN alters accordingly at the Fano resonance. Figures 6(a)–6(f) illustrate the near field amplitude distributions (normalized to incident amplitude over the structure surface) of RCRN at the Fano resonances when the CSD decreases from 535 to 325 nm. It reveals that the near field amplitude strongly relies on the CSD of the Bessel beam. In Fig. 6(g), we examine the dependence of the near field enhancement, which is defined by the ratio between the maximum near field amplitude Eloc and the averaged amplitude of the incident beam over the RCRN Ein, on the CSD. It is clear that compared with Fig. 5(b), the near field enhancement follows the evolution trend of coupling strength and has a maximum value at the CSD of 420 nm. The spatial tuning of Bessel beam actually provides a method to actively control the near field response of the plasmonic nanostructure that can be beneficial for applications in biosensing and enhanced near-infrared spectroscopy.
Fig. 6 (a)-(f) Near field amplitude distributions (normalized to the average incident amplitude over the structure surface) at Fano resonance with the CSD decreasing from 535 to 325 nm. (g) Dependence of near field enhancement Eloc/ Ein on the CSD.
3. Conclusion
In this paper, we have presented a scheme to dynamically control the Fano resonance in a gold nanostructure consisting of a crescent-ring and a symmetric ring. The robust control of Fano resonance within near-infrared range was achieved solely by changing the phase distribution of an excitation beam which determines the coupling strength between two interacting rings (e.g. here we used a Bessel beam). A coupled oscillator model considering the radiation damping was introduced to fit the simulation results. A good agreement between the model and the simulations suggests that the functional format of the model accurately represents the spectral interference process. Most importantly, the model projects significant insights into the operation of the device which is hardly achieved just by merely carrying out a full-blown simulation. Further analysis uncovers remarkable features of the system, such as anticrossing behavior, sign changing of coupling, and inversion of quadrupole and bright dipole spectral position. Moreover, the near field enhancement of the nanostructure can be actively controlled via the modulation of CSD, which can have potential applications in biosensing and enhanced near-infrared spectroscopy.
Acknowledgments
The work of F. Xiao and J. Zhao was supported by the National Basic Research Program (973 Program) of China (2012CB921900), NSFC (61308031, 61377035), the Technology Innovation Foundation (2011KJ01011) of Northwestern Polytechnical University. The work of W. Zhu and M. Premaratne was supported by the Australian Research Council through its Discovery Grant scheme under Grant No. DP110100713.
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