## Abstract

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, C_{l} and G_{p} denote the coupling length and the gap, respectively. The dielectric permittivity of the silver is modeled by Drude model ε(ω) = 1-ω_{p}^{2}/ω^{2} + iων_{p}, where ω_{p} stands for the plasma frequency and ν_{p} represents for the collision frequency. These values are taken from reference [19] with ω_{p} = 1.38 × 10^{16} rad/s and ν_{p} = 2.73 × 10^{13} rad/s at the wavelength λ = 1550 nm. Following Haus’s approach [20], critical coupling occurs when the intrinsic quality factor Q_{in} equals to the external quality factor Q_{ex} [16, 20, 21], and the transmission can be expressed as Eq. (1),

_{0}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 Q

_{in}= n

_{R}/2n

_{I}, where n

_{R}and n

_{I}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 Q

_{ex}= 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 Q

_{in}= 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 C

_{l}and gap G

_{p}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 Q

_{ex}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 × 10

^{5}is acquired.

Figure 2
shows the critical coupling achieved with a variety of combinations of structural parameters. As expected, the Q_{ex} 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 Q_{ex} = Q_{in} 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 C_{l} = mλ_{0}/n_{R} = mλ_{eff} (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 M_{s} = cos^{2}(2πC_{l}/λ_{eff}), corresponding to a measure of the detuning from the resonant condition. With the increase of the C_{l}, the FP resonance is gradually established. When C_{l} = 1300 nm (~λ_{eff}), the splitting is maximized as predicted by the splitting factor M_{s}. 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.

## 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 ${a}_{cw}$ and ${a}_{ccw}$ are the complex amplitude of the CW and CCW modes, respectively. 1/τ_{in} and 1/τ_{ex} are the decay rate due to the intrinsic and external loss associated with the quality factor Q = ω_{0}τ/2 at the resonant frequency ω_{0}. 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/τ_{ex}, 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/τ_{ex}, 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 ω_{0} splits into doublets and shifts to ω_{0} + β and ω_{0}-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 $\overrightarrow{S}$ are calculated. As shown in Fig. 4(a) , when the coupling zone is on-resonance, $\overrightarrow{S}$ 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).

## 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 δφ_{c} 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 δφ_{p} in the coupling zone which counteracts the accelerated phase front around corners. The net phase shift per round trip $\delta \varphi $ is consequently equal to 4δφ_{c}- δφ_{p}, resulting in the blue shift of the resonant wavelength calculated by FDTD. Following the calculation of the modal index of directional coupler, δφ_{p} can be determined and discriminated from the total phase shift$\delta \varphi $. 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 δφ_{c} = 2π × 0.016 rad and δφ_{p} = 2π × 0.006 rad are obtained, as shown in Fig. 5(b). In the calculation, structural parameters (C_{l,} G_{p}) = (966, 24) subject to critical coupling condition of the traveling wave resonance was applied.

## 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 R_{120}≅0.001, R_{90}≅0.012 and
R_{60}≅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 (R_{60}/ R_{90})^{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.

The on-resonance time averaged magnetic field intensity |H_{y}|^{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.

## 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 (α_{s} = 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 δλ_{1} (δλ_{2}), 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 δλ_{2}/(δλ_{1} + δλ_{2}). 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 (δλ_{1} + δλ_{2}), 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 δα_{s} = (δλ_{1} + δλ_{2})/50 and δz = 324 × δλ_{2}/(δλ_{1} + δλ_{2}), respectively.

## 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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