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we report on significant mode splitting in plasmonic resonators induced by intracavity resonance. In contrast to traditional dielectric resonators where only picometer range of splitting was achieved, splitting over several hundred nanometers can be obtained without using ultrahigh quality resonators. We show that by appropriately choosing the coupling length, minute reflection is sufficient to establish intracavity resonance, which effectively lifts the degeneracy of the counterpropagating modes in the resonator. The mode splitting provides two self-referenced channels enabling simultaneous monitoring of the position and the polarizability of nano-scatterers in the resonator.
©2012 Optical Society of America
1. Introduction
Whispering gallery modes (WGMs), featured by their high quality factors and small mode volume, are ideal for the investigation of nonlinear optics [1–3], quantum electrodynamics [4], and bio-sensing applications [5–7]. Conventional dielectric microspheres and ring resonators with excellent surface finishes exhibit ultrahigh quality factors rendering them potential platforms for ultra-narrow band filtering, low threshold switching, and quantum electromagnetic dynamics. In analogy to the case of dielectrics, subwavelength resonator side coupled to waveguide utilizing metal-dielectric-metal (MDM) structures have attracted extensive attention due to the great potential for the realization of ultra-dense optical circuits [8]. In long wavelength limit, nearly zero bending loss can be achieved [9, 10]. A variety of waveguide-resonator coupled structures are therefore proposed and their transmission characteristics analyzed [11–14]. Besides, it is widely known that perturbations such as roughness, inhomogeneity, and surface adsorbates may break the degeneracy of the counterpropagating resonant modes [15, 16], turning traveling wave resonance into standing-wave characteristics. The corresponding spectral response manifests shifting, splitting, and broadening with respect to the otherwise unperturbed spectrum. As a result, label-free single-molecule sensing is capable [5, 6, 17]. Recently, single nanoparticle detection and sizing by mode splitting was successfully demonstrated [18]. The reported sensitivity is crucially dependent on the ratio of the splitting to the variation of the linewidth. Due to the relatively short focal length, so far, the resolution of the handheld spectrometer is still too low to resolve the linewidth of the high-Q resonant modes. Consequently, enlarge the splitting is of significant importance that may effectively relax either the stringent requirement of a high resolution spectrometer or the necessity of a high Q resonator.
In the present study, we analyze the spectral responses based on plasmonic waveguide side coupled to resonators with various geometries. Of particular interest, we found that giant mode splitting can be achieved by minute reflections due to bent corners of resonators with moderate Q factors. Under critical coupling condition, it is found that the transmission spectra exhibit singlet or doublet dips depending on whether the Fabry-Pérot (FP) resonance was established in the coupling zone. The result calculated by finite difference time domain (FDTD) method is compared with coupled mode theory (CMT) and the discrepancy is well justified by considering the structural dispersion in bent corners. The time averaged Poynting vector was mapped out showing that with the absorption takes into account, nonzero power flow of the standing wave type resonance was formed. It should be noted that the power flow exhibits universal distributions of the power source, sink or saddle in regardless of the mode order rendering the system a promising platform for nano-object trapping, sorting, and sizing applications. A practical example where simultaneous detection of the position and the polarizability of a nano-scatterer exploiting the mode splitting as two monitor channels is illustrated.
2. Modeling
The plasmonic waveguide coupled to resonator with various geometries considered in this study are shown schematically in Fig. 1 , where w represents the waveguide width, Cl and Gp denote the coupling length and the gap, respectively. The dielectric permittivity of the silver is modeled by Drude model ε(ω) = 1-ωp2/ω2 + iωνp, where ωp stands for the plasma frequency and νp represents for the collision frequency. These values are taken from reference [19] with ωp = 1.38 × 1016 rad/s and νp = 2.73 × 1013 rad/s at the wavelength λ = 1550 nm. Following Haus’s approach [20], critical coupling occurs when the intrinsic quality factor Qin equals to the external quality factor Qex [16, 20, 21], and the transmission can be expressed as Eq. (1),
ω0 is the resonant frequency of the unperturbed resonator. The intrinsic quality factor of the resonator was dominated by the propagation loss of the plasmonic waveguide and can be expressed as Qin = nR/2nI, where nR and nI are the real and imaginary part of the modal index, respectively. These parameters can be obtained by solving the characteristic equation of the plasmonic waveguide for the TM mode. On the other hand, the external quality factor can be represented by Qex = 2πN/η, where N denotes the order of the resonant mode, and η represents the external coupling efficiency. At the wavelength of 1550nm, the modal index of the fundamental mode of the Ag/Air(w = 100nm)/Ag waveguide is calculated to be n = 1.200 + i0.00212, corresponding to Qin = 283. To maintain the resonance at λ~1550 nm, the total length of the resonator is kept at 3864 nm (for N = 3 modes) in all cases. Therefore, the length in x-direction has to be adjusted correspondingly for different coupling lengths. To find the critical coupling condition, resonators with different coupling length Cl and gap Gp are considered. Similar to the case of directional couplers, the external coupling efficiency η in the coupling zone was calculated by the finite-different time-domain (FDTD) method. The resulting Qex can therefore be determined. In the simulation, an otherwise infinitely extended space is truncated by fifteen-layer convolution perfectly matched layer (CPML) [22]. To transform the dispersive permittivity of silver in time domain, the recursive convolution method was applied [23]. The spatial and temporal domain was meshed in grids with unit size 2 nm × 2nm and 6.67 × 10−18 sec, respectively. To reach convergence, a total time steps of 3 × 105 is acquired.Figure 2 shows the critical coupling achieved with a variety of combinations of structural parameters. As expected, the Qex decreases monotonically with the increase of the perturbation, for instance, the coupling length. Normally, the larger the gap, the weaker the perturbation, and a longer coupling length is required to transfer the energy completely into the resonator.
When the condition Qex = Qin is satisfied, according to Eq. (1), the spectral response should exhibit a singlet anti-Lorentzian shape. However, our FDTD result shows spectral splitting under certain conditions, as shown in Fig. 3 . Although similar phenomena were observed by researchers in the case of dielectric resonators, it is widely attributed to the effect of scattering due to roughness or adsorbate on the surface of the resonator, and the maximization of the splitting has rarely been addressed. In the present study, we found that the splitting may arise from the intrinsic feedback of the resonator upon coupling with the waveguide. In particular, it is found that when the intra-cavity resonance Cl = mλ0/nR = mλeff (m = integer) is established in the coupling zone, the splitting is maximized. As shown in Fig. 3(a)-3(h), the spectral response evolved from singlet to doublet and back to singlet again. To quantify the degree of splitting, we define a modal splitting factor Ms = cos2(2πCl/λeff), corresponding to a measure of the detuning from the resonant condition. With the increase of the Cl, the FP resonance is gradually established. When Cl = 1300 nm (~λeff), the splitting is maximized as predicted by the splitting factor Ms. In general, the closer to the resonant condition, the larger the splitting results. It should be also noted that to achieve a mode splitting as large as 140nm, only minute backward reflection as low as 5% is needed. This is analyzed in section 5.
Fig. 3 Modal evolution of the plasmonic waveguide side-coupled to a rectangular resonator under condition Qex = Qin. The full vectorial computation result is in close agreement with that obtained by the coupled mode theory.
3. Correspondence with coupled mode theory
To explain the splitting induced by intermodal coupling, coupled mode theory Eq. (2) was applied for the calculation of the net energy transferring between the clockwise (CW) and counter-clockwise (CCW) modes in time domain [15, 16, 20]. The and are the complex amplitude of the CW and CCW modes, respectively. 1/τin and 1/τex are the decay rate due to the intrinsic and external loss associated with the quality factor Q = ω0τ/2 at the resonant frequency ω0. β denotes the backward coupling coefficient, and κ represents the coupling strength between the incident wave and the cavity modes. In steady state, the transmittance and reflectance subject to critical coupling condition can be represented by Eq. (3). If β<<1/τex, Eq. (3) can be reduced to Eq. (1), corresponding to the off-resonance condition in the coupling zone and traveling wave resonance occurs in the resonator. On the contrary, if β>>1/τex, intracavity resonance occurs in the coupling zone, and standing wave resonance results in the resonator. Due to the strong coupling between the CW and CCW waves, the unperturbed singlet resonant frequency ω0 splits into doublets and shifts to ω0 + β and ω0-β. The results are compared with our full vectorial FDTD simulation in Fig. 3.
To verify the type of resonance, distributions of time-average Poynting vector are calculated. As shown in Fig. 4(a) , when the coupling zone is on-resonance, is composed of the CW and CCW modes. Due to the intercoupling between the two modes, a power source forms at the center of the lower side, and a power saddle forms at the center of the upper side. The power flow exhibits a standing wave resonance. On the other hand, when the coupling zone is off-resonance, a purely traveling wave resonance is obtained, as shown in Fig. 4(b).
Fig. 4 Time-average Poynting vector in plasmonic rectangular resonator. (a) Standing wave resonance, which is composed of two counterpropagating traveling waves on resonance. (b) Traveling wave resonance.
4. Phase front acceleration
Although similar spectral responses were obtained in Fig. 3, there is a significant difference between the resonant wavelengths. The blue-shifted resonant wavelength in the FDTD calculation needs to be justified. From the FDTD calculation of the time averaged Poynting vector around the corner of the resonator, it is found that the real optical path for the plasmonic wave deviates from the one assumed in the calculation. As shown by the red-dashed line in Fig. 5(a) , the real optical path is reduced as oppose to the presumed path (the blue-dashed line) in the CMT. As a result, the phase front is effectively accelerated by δφc per corner. In contrast to this, the coupling between the plasmonic waveguide and the resonator slightly raises the modal index, corresponding to an increased optical path. This leads to a slowed down phase front δφp in the coupling zone which counteracts the accelerated phase front around corners. The net phase shift per round trip is consequently equal to 4δφc- δφp, resulting in the blue shift of the resonant wavelength calculated by FDTD. Following the calculation of the modal index of directional coupler, δφp can be determined and discriminated from the total phase shift. The results have been confirmed by analyzing the shift of the resonant wavelength for different order of modes (N = 2 and N = 3), and the phase shift δφc = 2π × 0.016 rad and δφp = 2π × 0.006 rad are obtained, as shown in Fig. 5(b). In the calculation, structural parameters (Cl, Gp) = (966, 24) subject to critical coupling condition of the traveling wave resonance was applied.
Fig. 5 (a) Time-average Poynting vector around the bent corner, where the red arrow represents the real optical path and the blue arrow represents the presumed path. The effective acceleration of the phase front is represented by δφc. (b) The shift of the resonant wavelength and the total accumulated optical phase in a round trip.
5. Angle dependent reflectivity and mode splitting
Based on our analysis, the maximum mode splitting depends on the backward coupling coefficient (more precisely, the backward coupling ideality [20]), which is associated with the bent angle of the plasmonic resonator. In order to clarify the interplay between the bent angle, backward reflectivity and the associated mode splitting, full vectorial FDTD calculation was applied to three bent angles: 120°, 90°, and 60°. The mode splitting for resonators constructed by the abovementioned bents subject to a common resonator length are calculated, as shown in Fig. 6(a) . And the reflectivity for the bents at the wavelength of 1550nm are all referenced to the straight waveguide and calculated to be R120≅0.001, R90≅0.012 and R60≅0.051 respectively, as shown in Fig. 6(b)-6(e). As in our illustration, significant mode splitting ≅141nm was achieved in triangular-shaped resonator. Compared to the rectangular-shaped resonator, the mode splitting is twice larger which is attributed to the doubled ratio of the reflected amplitude (R60/ R90)0.5≅2. While for parallelogram-shaped resonator with uneven reflectivity at both ends of the coupling zone, the mode splitting valued in between the two extremes.
Fig. 6 (a) Mode splitting of plasmonic resonators with rectangular, triangular, and parallelogram shapes. The dashed lines refer to the transmittance and the dotted lines refer to the reflectance. (b)-(e) The instantaneous distribution of the magnetic field intensity for straight, 120°, 90° and 60° bent waveguides, respectively.
The on-resonance time averaged magnetic field intensity |Hy|2 of the plasmonic resonator is shown in Fig. 7 . It is found that the energy distributions between the two standing waves are orthogonal, i.e., the nodes of energy density for the symmetric mode lie at the antinodes of antisymmetric mode.
Fig. 7 The spatial distribution of the time-averaged magnetic field intensity |Hy|2 for various geometries at resonant condition.
6. Illustration of practical applications
The significant mode splitting provides two self-referenced channels which may facilitate the detection of the polarizability and the position of a nano-object simultaneously and independently. To illustrate how it works, a nano-object with size 100 nm × 100 nm and the refractive index of n is placed at the position z in the coupling zone, as shown in Fig. 8(a) . To generalize the sensing capability, the index variation can be correlated to the polarizability by [24]. Essentially the mode splitting is proportional to the spatial overlapping between the resonant mode and the nano-object, the position can therefore be determined. Since the field distribution of the symmetric and the antisymmetric modes form complementary set in spatial domain, presumably, the position of the nano-object can be monitored linearly and continuously. Here, the following cases are considered: when the nano-object with fixed normalized polarizability (αs = 0.18) shifts from the node (antinode) at z = 0 to the anti-node (node) at z = 324 of the symmetric (anti-symmetric) mode, the resonant wavelength of the symmetric (anti-symmetric) mode exhibits blue (red) shift δλ1 (δλ2), as shown in Fig. 8(b). The position of the nano-object can be determined by the normalized shift of the resonant wavelength defined by δλ2/(δλ1 + δλ2). On the other hand, when the nano-object is placed at z = 0 and the polarizability is varied, the electric field of the symmetric mode is disturbed and the corresponding resonant wavelength shifted, as shown in Fig. 8(c). It should be noted that the position can be independently determined in regardless of the polarizability of the nano-object which is merely a function of the total wavelength shift (δλ1 + δλ2), as shown in the contour map in Fig. 8(b) and 8(c). Thus by measuring the wavelength shift at the two individual resonances, the sensitivity of the polarizability and the position were estimated to be δαs = (δλ1 + δλ2)/50 and δz = 324 × δλ2/(δλ1 + δλ2), respectively.
Fig. 8 (a) Schematic diagram of a nano-object in the rectangular ring cavity. (b) Spectrum shift of the nano-object with fixed normalized polarizability αs = 0.18 at various positions in the resonator. (c) Spectrum shift of the nano-object located at z = 0 with various polarizabilities. Note that the position and the polarizability can be determined simultaneously and independently, when the polarizability and the position are both variables, as shown in the contour map of (b) and (c).
7. Conclusion
We show that significant mode splitting can be achieved by plasmonic resonators without the necessity of ultrahigh quality factors. By tailoring the coupling length so as to establish the intracavity resonance in the coupling zone, mode splitting can be maximized. The mode splitting as large as 140 nm is achieved by the triangular resonator with a moderate quality factor Q = 120 in combination of corner reflections as low as 5%. Simultaneous detection of the position and the polarizability of a nano-object exploiting the splitted self-referenced resonances are illustrated. It is expected that with the orders of magnitude improvement of the mode splitting, the stringent demand for highly spectral resolution can be much relaxed, and low-cost handheld spectrometers will be sufficient to cope with sensing applications.
Acknowledgments
This work was sponsored by the National Science Council, Taiwan (R.O.C.). The authors would like to thank for the grant support under contract number NSC100-2112-M-008-009-MY3.
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