Whispering gallery mode (WGM) microtoroids [Fig. 1(a)] are excellent sensors due to their ultra-high-quality (Q) factors and their ability to be integrated on chip ^[1–9]. These sensors when combined with frequency-locking, balanced detection, and data processing techniques are able to detect wavelength shifts (<∼0.005fm) that correspond to the detection of a biomolecule a radius of 2 nm binding to the microtoroid ^[1,2,4]. Coupling plasmonic particles to WGM cavities is one way to improve the sensitivity of the systems ^[10]. These particles increase the frequency shift upon molecular binding by providing enhanced interaction between the optical field and the analyte due to localized surface plasmon resonance (LSPR) ^[10,11]. The scattering and absorption losses of the metal, however, cause degradation of the Q factor of the resonance, that is, the broadening of the linewidth. Dark mode plasmonic resonances can help solve this problem ^[12–16], but simulation is necessary to understand the specific impact of a given plasmonic structure on the system.

Currently, due to limitations in computing performance and time consumption, a wedge-shaped model with perfect electric conductor (PEC) boundary conditions is widely used in plasmonically enhanced WGM toroid simulations ^[17]. However, such boundary conditions act as mirrors, effectively replicating the plasmonic particle multiple times, which results in an inaccurate calculation of the coupled system Q factor for a single particle (see Appendix A for more details).

Here, we use a three-dimensional (3D) eigenmode simulation model of a whole microtoroid with a major diameter of 10 μm to explore the interaction of a single cavity with one or several nanostructures. The model is implemented in COMSOL. Due to simulation constraints in both time and memory, we cannot accurately simulate larger whole toroids in 3D. The schematic of our 3D model and the field distribution of the fundamental TE mode are shown in Fig. 1(a). The polarization of the TE mode is perpendicular to the equatorial plane of the toroid cavity. Nanorods are placed perpendicularly to the equatorial plane for maximum coupling and excitation, and their near-field enhancement is shown in Fig. 1(c).

We define a figure of merit known as a combined enhancement factor (f_C) to quantify the limit of detection and resolution of our sensing system. With this figure of merit, we can predict that slightly off-resonant coupling between the plasmon and WGM provides better performance than a direct match of the WGM and plasmon resonances. Further, we discuss the improvements that a lateral dark mode supported by a plasmonic trimer structure brings to the system and verify its stability.

Two critical parameters affect the limit of detection of a WGM sensing system. One is the Q factor mentioned above ^[18], which determines the linewidth of the resonance peak in the output spectrum. Higher Q factors correspond to narrower resonances, whose central resonance frequency can be more precisely determined and tracked ^[1–3,19]. The other parameter is the electric (E)-field enhancement factor. The magnitude of the shift in resonance frequency of the WGM upon interaction with an analyte is proportional to this enhancement factor. According to perturbation theory ^[20], the frequency shift caused by a particle of volume ΔV with (possibly anisotropic) material parameters Δε↔ (r) and Δμ↔ (r) relative to the background is given by ^[20–22] where ε0 and μ0 are the permittivity and permeability of free space, ε and μ are relative permittivities and permeabilities, εb is the permittivity of the background media (in this case, vacuum), E0 and H0 are the E and magnetic fields for an unperturbed WGM, † represents conjugate-transpose, E and H are the electric and magnetic fields within the perturbation volume, |E0|max2 is the maximum intensity within the unperturbed toroid, Vm=∫Vεr|E0|2dV/(εr|E0|max2) is the mode volume of the resonant mode ^[19], εr is the dielectric constant of silica ^[23], and α↔(ω) is the polarizability tensor of the particle. In the numerator, we have assumed Δμ↔=0, and in the denominator, we have assumed that the magnetic field contributes an equal amount of energy as the electric field and that the perturbation has a negligible impact on the total energy stored in the cavity. The polarizability tensor for a metallic nanorod can be expressed as ^[24–26] where α‖ and α⊥ correspond to the longitudinal and transverse modes of the nanorod, respectively. The nanorods are small enough that they can be accurately modeled based on the LSPR of prolate spheroids ^[25]: where ΔV=43π(d2)3+π(d2)2(L−d) is the volume of the cylindrical nanorod, εm is the relative permittivity of the metal, εb is the dielectric constant of the surrounding environment, and the G‖,⊥ are geometrical factors that can be, respectively, written as where e=1−(dL)2 is the eccentricity of the prolate spheroid, and Rs is a parameter that describes the particle shape. Here, we use Rs=0.88 for a spherocylinder ^[27]. In the simulation, we put the rod parallel to the polarization of the fundamental TE mode and study only the influence of the longitudinal plasmon mode and its corresponding α‖. It is apparent that α‖ is greatly influenced by the nanorod aspect ratio. This in turn affects the shift of the unperturbed WGM.

Figure 2(a) illustrates the frequency shift of the cavity mode around on-resonance (λ≈774nm) coupling obtained in the simulation, which is consistent with both experimental data and perturbation theory ^[28]. The data for the perturbation theory is obtained by numerical calculation of Eqs. (1)–(4), where only the longitudinal polarizability α‖ is necessary, and the unperturbed E fields and mode volumes are determined from a simulation of the bare cavity. In our calculations for different aspect ratios, the nanorod diameter is held fixed, while the length is varied, so at large aspect ratios, the assumption is that the nanorod size ≪λ starts to fail, and the theory deviates from the numerical simulation results ^[29]. At the same time, the linewidth of the coupled system also varies with the resonance frequency of a single rod, as shown in Fig. 2(b).

The inset of Fig. 2(c) shows how we calculate the intensity enhancement at the particle, which is done by adding a sphere in the near field of the nanorod to represent an analyte molecule. Here, we use the volume integration of the E-field intensity over the volume of an analyte particle as the enhancement factor instead of the maximum E-field intensity at the hot spot of the nanoparticle (NP) ^[10,30]. This is more accurate because the maximum E-field hot spot intensity is sensitive to the mesh of the simulation model and does not take into account the frequency shift caused by the size of the measured particle. The ratio of the integrated field involving a coupled plasmonic particle to that of the evanescent field of the bare toroid can be used to define a field enhancement factor as follows: where |Ec(r)| is the E-field magnitude of the near field of the coupled system. The enhancement factor corresponding to the E-field intensity directly affects the magnitude of the frequency shift when a particle binds to the cavity. This enhancement factor closely follows the same trend as the linewidth broadening caused by the nanorod [Fig. 2(b)]. It is evident that introducing a particle with a lossy material such as gold will significantly decrease the Q factor of the system, even with strong local E-field enhancement, so there is a trade-off between Q and the intensity enhancement factor, as shown in Fig. 2(c). At a wavelength of λ≈774nm, the maximum 2500× intensity enhancement is achieved when the rod aspect ratio equals 5.4. In this case, the peak LSPR wavelength coincides with the particular WGM resonance being simulated. The unperturbed, bare toroid exhibits a Q factor of 7×106, which drops to 1.87×105 when coupled to the nanorod with an aspect ratio 5.4.

The lower Q of the coupled system involving resonant nanorods may make it difficult to precisely lock to resonance peaks in experiments involving tracking the wavelength shift of the mode. Therefore, a balance is desired between the Q factor and the intensity enhancement factor.

From Eqs. (1) and (3), it can be shown that the frequency shift is directly proportional to the E-field intensity times the volume of the detected particle: Δω∝E2ΔV, for scalar α. Then, the volume of the smallest detectable particle follows, ΔVmin∝Δωmin/|E|2∝(Q|E|2)−1, assuming that the smallest experimentally measurable shift, Δωmin, is inversely proportional to the Q factor because high-Q resonances correspond to narrow peaks that can be tracked with greater precision than broad resonances. So, the volume of the smallest detectable particle is inversely proportional to a figure of merit that accounts for both the E-field and Q-factor effects, which we call the combined enhancement factor (fC): where Qc and Q0 represent the Q factors of the coupled and bare toroid cavities, respectively. Higher fC values correspond to better sensitivity and/or more well-defined resonances. Unlike the peak in fE, the highest fC does not occur when the LSPR and WGM resonances coincide. Instead, as shown in Fig. 2(d), the position of the peak is shifted significantly to a smaller aspect ratio than that of resonant coupling. We can state that the improvement to the system from a single rod with an aspect ratio of 5.0 is ideal, which corresponds to a 874 field enhancement factor and a Q factor of 6.8×105. Using fC as a figure of merit is helpful in selecting the best plasmonic structure for experimental use.

For multiple gold nanoantennas, we investigate the intrinsic properties of the coupled system. The single gold nanorod can be approximated as a nanocavity. According to coupled mode theory ^[21,31–33], the equations of several coupled modes can be written as where bWGM and bn represent the mode amplitude of the WGM mode and LSPR mode of the nth nanorod, ΩWGM and Ωn correspond to their resonance frequencies, and ΓWGM and Γn represent their corresponding losses, respectively, which can be calculated from COMSOL using the eigenfrequency solver. The coupling coefficient is defined as a complex number κ, which takes into account the frequency shift of the mode and the degradation of the Q. Because the rods in the array are spaced relatively far apart, we only consider the coupling of each LSPR mode with the cavity and no direct interparticle coupling. Therefore, the above equations for the coupled system can be written as a matrix equation M ˙=HM, where M=[bWGM,b1,…,bn]T, and The intrinsic Q factor of the coupled system can be calculated from the eigenvalues of H, corresponding to the steady state of the system. Through the simulation data, the coupling coefficient κ (∼1.43×1011Hz) can be obtained from a 2×2 matrix generated by a system in which a cavity is coupled to a single rod (see Appendix B). Further, the Q factor of a (n+1)×(n+1) matrix corresponding to a multiple (n) rod coupled system can be calculated numerically. We compare these values with the Q factors obtained from COMSOL simulation of the multi-rod system [Fig. 3(a)]. The coupled mode theory can then be used to predict the interaction between the WGM and additional particles without having to run new COMSOL simulations.

Figure 3(a) also shows the full simulation results of coupling several rods to the cavity, where each rod is placed at an antinode of the standing wave WGM [Fig. 3(b)]. Here, the nanorods are sufficiently far apart such that direct coupling between neighboring nanorods is negligible compared to nanorod-WGM coupling, as verified in Fig. 3(a) by the agreement between the numerical simulation results and coupled mode theory [eigenvalues of H, Eq. (9)].

The dipole modes induced in isolated antennas lead to large energy radiation, which adversely limit the Q factors of the coupled systems. Dark modes, an example of which is the antibonding interaction of dipolar resonances, have a longer lifetime and lower loss compared to bright dipolar modes, but their excitation is not easily feasible from the far field ^[12,13,16]. Due to the nearly zero net dipole moment of dark plasmons, it is almost impossible for their out-of-phase state to interact with light in free space.

Here, we designed a plasmonic trimer consisting of three isolated rods arranged in a symmetric triangle ^[34,35] [Fig. 4(a)] such that the microtoroid WGM excites one of the nanorods in the trimer, which in turn excites a dark mode resonance (lateral dark mode) of the other two nanorods ^{[12,15,36–38]}. Coupling of the dark mode and WGM standing wave can significantly reduce degradation of the Q factor. The superposition of rod near-fields at the center of the trimer can also bring greater intensity enhancement factors. Figure 4(b) shows the spectrum of the lateral dark mode of a trimer and three individual rods in free space. When the ends of three rods exhibit the same charge distribution, we call it a breathing dark mode. Only when two antennas of the trimer generate an antibonding mode [Fig. 4(b)] do we refer to this as a lateral dark mode. Although the trimer spectrum exhibits a peak near 800 nm upon far-field or WGM excitation, we still call its mode a lateral dark mode because the laterally oriented lower two rods of the trimer are excited in an antibonding, or dark, mode. The apparent resonance peak in Fig. 4(b) is due to the excitation of the vertical rod, which remains bright. Because of this simultaneous combination of bright and dark modes, it is possible to excite a plasmon resonance with minimal scattering and absorption losses. Although the breathing dark mode has much lower loss, and its corresponding resonance peak is almost covered by the tail of the lateral dark mode, its coupling to the WGM is very weak. Therefore, this paper mainly studies the lateral dark mode instead of the breathing dark mode.

Due to the plasmonic coupling between closely placed nanorods, the resonance position of the lateral dark mode shifts with respect to the spacing and position of the rods. In Fig. 4(b), we compare the extinction cross sections when three rods with the same aspect ratio of 5.4 are placed separately or placed as a trimer with an 8 nm central gap. The extinction cross sections are normalized by that of the three isolated rods. The maximum extinction of the trimer is about half of that obtained from three individual rods. At many wavelengths, including 774 nm, corresponding to the WGM resonance, the loss from the trimer is even lower than that of a single rod. Such a low-dissipation mode improves the WGM sensing system.

We also adjust the aspect ratio of the three rods in the trimer to optimize the improvement of the trimer coupling system. Figure 4(c) shows how the Q factor and intensity enhancement vary with the rod aspect ratio. The biggest intensity enhancement factor here corresponds to the case where the spherical analyte particle is placed in the center of the trimer. At an aspect ratio of five, the maximum field enhancement factor (∼4300) is much larger than that of a single rod (∼2500), as mentioned above. At the same time, the Q factor of 1.31×105 is similar to that of the single rod case (∼1.87×105) at on-resonance coupling and is twice as large as the Q for the three rod case (∼5.98×104) presented in Fig. 3(a). Even the relatively dark region just below the center of the trimer in the inset of Fig. 4(c) can bring dozens of times of intensity enhancement, which also demonstrates that the trimer structure is beneficial to increasing the capture area for biomolecule detection. The maximum fC of the system is ∼105, which is achieved using the trimer with a rod aspect ratio of 4.5. This fC is also significantly larger than that of a single rod (∼84).

To test the robustness of our system, we studied the effect of tilted angle perturbation, length perturbation, and inter-rod spacing on fC. Here, we chose the trimer with a rod aspect ratio of 4.5, mentioned above as a reference. As shown in the insets in Fig. 5, we observe the difference of fC in the system by introducing different geometric perturbations. Considering the need to bind biomolecules at the center of the trimer in experiments, the spacing between the rods should not be too small. Although different perturbations will impact the Q factor or enhancement factor of the system, the overall fC changes little from its initial value of 105. The spacing in Fig. 5(c) has little effect on the loss caused by the structure, so the corresponding Q value is almost unchanged, and the observed variation is primarily due to E-field enhancement.

Rotations of the entire structure are evaluated for single nanorods and trimer structures in Figs. 5(d) and 5(e). The gold nanorods in current plasmonic-enhanced WGM systems are randomly attached on the surface of the cavity, and their orientation is challenging to control. Here, we use a rod with an aspect ratio of five (Fig. 2) to study the influence of the rotation angle on fC. When the rotation angle is greater than 45 deg, the fC decreases as the rotation angle increases [Fig. 5(d)] due to mismatch with the polarization direction of the WGM. The system has only 4× gain when rotated 90 deg. In comparison, the trimer is robust to orientation changes. Due to its rotational symmetry, we only rotate the whole trimer by 0–60 deg [Fig. 5(e)]. The 0 deg point corresponds to the idealized trimer composed of three rods with an aspect ratio of 4.5. The weakest fC at a 30 deg angle is 92, which is only 11.5% lower than the maximum enhancement of 104. The trimer system is much less affected by the rotation angle than the single rod system. The inconsistent signal amplification caused by the rotation of a single rod can add uncertainty to particle detection experiments.

In this work, we present a lateral dark mode of a symmetric trimer structure generated using WGM excitation and define a combined enhancement factor to analyze the improvements it brings to the system. The trimer exhibits a ∼105 combined enhancement factor versus 84 for a single rod. We also demonstrate the stability and robustness of the trimer structure. We verify the correctness of our simulation from the frequency shift and linewidth changes caused by detuning and the effect on the Q factor. Compared to the existing single rod plasmonic hybrid WGM system, the trimer system exhibits higher enhancement, stability, and greater contact area for particle detection. In future work, we plan to use high-precision optical tweezers ^[39,40] to position trimer microresonator systems for biological sensing experiments.