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Abstract
Development of a computational technique for the analysis of quasi-normal modes in hybrid-plasmonic resonators is the main goal of this research. Because of the significant computational costs of this analysis, one has to take various symmetries of these resonators into account. In this research, we consider cylindrical symmetry of hybrid-plasmonic ring resonators and implement a body-of-revolution finite-difference time-domain (BOR-FDTD) technique to analyze these resonators. We extend the BOR-FDTD method by proposing two different sets of auxiliary fields to implement multi-term Drude-Lorentz and multi-term Lorentz models in BOR-FDTD. Moreover, we utilize the filter-diagonalization method to accurately compute the complex resonant frequencies of the resonators. This approach improves numerical accuracy and computational time compared to the Fourier transform method used in previous BOR-FDTD methods. Our numerical analysis is verified by a 2D axisymmetric solver in COMSOL Multiphysics.
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1. Introduction
Hybrid-plasmonic resonators show high Q-factor and are able to trap light in very small spatial volumes [1–4]. These characteristics render them useful in emerging technologies including nano-tweezers [5] and single-photon sources [6]. Since hybrid-plasmonic cavities play an important role in modern photonics, computing their modal properties such as resonant modes and Q-factors is important. The exact calculation of quasi-normal modes (QNMs) in plasmonic cavities is challenging owing to the fine spatial and temporal resolution required to model deep subwavelength plasmonic modes. Several numerical full-wave methods have been implemented to address this challenge. Ge et al [7] used 3D-FDTD analysis based on the QNMs to efficiently calculate the QNMs in metallic resonators. Others have used the finite-element method (FEM) to calculate the complex eigen-frequencies in dispersive cavities. This can be faster than the FDTD method but requires solving a nonlinear eigenvalue problem. Several QNMs solvers based on FEM, which linearize the nonlinear eigenvalue problem, have been presented so far [8,9,10,11]. Moreover, for FEM, a contour integral method based on Riesz projections is introduced in [12] which does not require solving a nonlinear eigenvalue problem. These approaches lend themselves for frequency-domain numerical analyses.
In this research, we present a body-of-revolution FDTD or BOR-FDTD which enables both QNMs calculations and dynamic analysis for cylindrically symmetric hybrid-plasmonic resonators. To our knowledge, this represents the first calculation of QNMs in hybrid-plasmonic cavities using BOR-FDTD. To accomplish this, we build on previous work which used Debye dispersive media in the BOR-FDTD method to calculate scattering cross-sections of azimuthally symmetric objects [13,14]. We extend their model by proposing two different sets of auxiliary fields to implement a multi-term Drude-Lorentz and multi-term Lorentz model in BOR-FDTD.
In this work, we also implement a filter-diagonalization method (FDM) to calculate the QNMs of the hybrid-plasmonic resonators. As will be mentioned, this technique converges four times faster compared to the Fourier transform method used in the previously introduced research articles known to the authors. As an illustrative example of the advantages of the FDM approach, we demonstrate a Q-factor of greater than 1000 for a hybrid-plasmonic ring resonator with a ring radius of only $2\mu m$. To verify the accuracy of our approach, we compare our calculation results with a 2D-axisymmetric (BOR-FEM) method implemented in COMSOL Multiphysics and show good agreement.
In BOR-FDTD, 3D Yee cells in cylindrical coordinates $({\rho ,\varphi ,z} )$ are reduced to 2D cells in the $\rho - z$ plane [15]. As a result, the memory requirements and computational costs are significantly reduced in comparison with 3D FDTD. This method has been used in several applications, such as modeling of light propagation through dielectric optical lenses [16], calculating spontaneous emission lifetimes in dielectric microdisks, [17] and calculating the decay rates of emitters coupled to gold nano-antennas [18].
To obtain spectral information of the QNM’s, a Fourier transform method or by the Prony method can be implemented as noted above. For high Q-factor resonators, however, calculation of exact eigenvalues using the Fourier transform method increases the computation time and memory requirements. On the other hand, the Prony method requires less memory, but leads to an ill-conditioned, nonlinear fitting problem which makes it less efficient. To overcome these limitations, we have utilized FDM [19,20] to accurately calculate modal properties including eigen-frequencies, decay rates, Q-factors, and effective refractive indices of a ring resonator. FDM is an approach based on linear algebra which keeps memory usage at a minimum. In addition, its resolution is not limited by the time-frequency uncertainty principle imposed by the Fourier transform.
In the following section, we first introduce typical structures to be analyzed in this research work. In Section 3, we present our numerical analysis and discuss the obtained results.
2. Typical structures
Figure 1(a) and (b) show the 3D geometry and cross-sectional view of a typical hybrid-plasmonic ring resonator, respectively. Here, to maintain the analysis as general as possible, we consider a multiple ring configuration. It comprises a high-index dielectric ring pair composed of a silicon core of dimension ${w_1} \times {H_1}$ and a thin silicon stripe of dimension ${w_2} \times {H_2}$. The high-index pair is embedded within a silica layer above a silver substrate. The ring radius R is equal to the mid-radius of the ring resonators, and g denotes the gap between the two rings. A similar configuration has been exploited previously for designing optical waveguides [21,22].
Fig. 1. (a) 3D view and (b) the cross-sectional view of the typical hybrid-plasmonic ring resonator to be analyzed.
3. Numerical analysis
The hybrid-plasmonic ring resonator to be analyzed is a rotationally symmetric structure. In the cylindrical coordinate system, any component of the electromagnetic field $\mathrm{\Psi }$ can be expressed as an infinite Fourier series of the form:
In the BOR-FDTD method, the azimuthal variation, is treated analytically, whereupon discretization and analysis are reduced to an arbitrary cross-section of the ring resonator, i.e., to the $\rho - z\; $ plane. For a given m, eigenvalues are thereafter calculated from a transverse resonance condition. To this end, Maxwell’s equations are discretized, for instance, by uniform and fine meshes with the increments of $\Delta = \textrm{min}\left\{ {\frac{{{\lambda_{min}}}}{{{N_\lambda }}},\frac{{{d_{min}}}}{{{N_d}}}} \right\}$. Here, ${\lambda _{min}}$ denotes the minimum wavelength in the desired frequency range, ${N_\lambda }$ stand for the number of sample points per wavelength which can be as large as 500 for achieving accurate results, ${d_{min}}$ denotes the minimum structural dimension, and ${N_d}$ stands for the number of sample points per smallest feature which should be ${N_d} \ge 1$.
On the other hand, the silver substrate has to be modelled with high accuracy. For this purpose, the dielectric function of silver $({{{\tilde{\varepsilon }}_{Ag}}} )$ is described by a multi-pole Drude-Lorentz model [10]:
In which ${\varepsilon _{\infty \; }} = 1,{\omega _d} = 1.3911 \times {10^{11}}\frac{{rad}}{s},{\gamma _d} = 1.8672 \times {10^{13}}\frac{{rad}}{s},{\omega _1} = 3.2670 \times {10^{15}}\frac{{rad}}{s},{\omega _2} = 7.7721 \times {10^{15}}\frac{{rad}}{s},{\gamma _1} = 1.16432 \times {10^{15}}\frac{{rad}}{s},{\gamma _2} = 3.471 \times {10^{14}}\frac{{rad}}{s},\mathrm{\Delta }{\varepsilon _1} = 0.092,\mathrm{\Delta }{\varepsilon _2} = 2.082$. These parameters are calculated using an optimization procedure and are consistent with the reported results in [23]. To implement the dispersion relation given by Eq. (2), we first introduce the auxiliary fields for each Drude and Lorentz poles in the frequency domain according to
Then, we perform the inverse Fourier transform of these auxiliary fields. Since the time excitation is assumed to be of the form ${e^{j\omega t}}$, the time derivative $\frac{\partial }{{\partial t}}$ can be replaced by $j\omega $ in inverse Fourier transform. Therefore, Eq. (3) can be expressed by
To discretize the multi-pole Drude-Lorentz model, we follow these steps: First, the magnetic field components are updated at time steps $n + \frac{1}{2}$ using the equation $\vec{\nabla } \times \vec{E}({\vec{r},t} )={-} {\mu _0}\frac{{\partial \vec{H}({\vec{r},t} )}}{{\partial t}}$. Second, the electric flux density components are updated at time steps $n + 1$ using the equation $\vec{\nabla } \times \vec{H}({\vec{r},t} )= \frac{{\partial \vec{D}({\vec{r},t} )}}{{\partial t}}$. Third, the electric field components are discretized at time steps $n + 1$ using the equation $\vec{E}({\vec{r},t} )= \frac{1}{{{\varepsilon _0}{\varepsilon _\infty }}}\left( {\vec{D}({\vec{r},t} )- {{\vec{J}}_d}({\vec{r},t} )- \mathop \sum \limits_{k = 1}^2 {{\vec{Q}}_k}({\vec{r},t} )} \right)$. Finally, the auxiliary fields ${\vec{J}_d}({\vec{r},t} )$ and ${\vec{Q}_k}({\vec{r},t} )$ are updated at time steps $n + 1$ using the Eqs. (4) and (5). The details of discretization of Drude-Lorentz model are presented in Supplement 1.
Moreover, the dispersive characteristics of silicon $({\tilde{\varepsilon }_{si}})$ is described by a multi-pole Lorentz model [24]. Note that the details of discretization are reported in Supplement 1.
As we know QNMs are the resonance modes of the ring resonator with complex eigen-frequencies. To excite these modes, several z-directed short-time modulated Gaussian sources are located inside the core and the stripe to excite fundamental quasi-TM photonic and surface plasmon polariton (SPP) modes. To excite the desired mode, multiple excitations are preferred to a single excitation to avoid excitation at a location close to a null of the desired mode. We have chosen two dipole sources inside the core and the stripe to excite the photonic and pure plasmonic modes. A dipole source is located inside the core to excite the fundamental TM-like mode. Then, the plasmonic mode can be excited at the interface of the metal and the stripe by the evanescent tails of the dielectric mode, i.e., by Otto coupling method [25]. However, for large gap thicknesses, the plasmonic mode may not be excited by Otto coupling configuration. Therefore, we can use another dipole source inside the stripe, to make sure that the plasmonic mode is excited. In this case, the plasmonic mode will be excited by the Kretschmann coupling method [25]. Moreover, to satisfy one of Maxwell’s equations, i.e.,
We record the time evolution of the dominant field component ${E_z}({\rho ,z,t} )$ on the $\rho - z{\; }$ plane. The data is recorded at several points instead of a single point. The ${E_z}({\rho ,z,t} )$ electric field component is calculated at three points, i.e., inside the core, the gap, and the stripe. We performed the spectral analysis at different points to ensure that the QNMs calculations lead to identical results at all chosen points. For a given m, if the data is merely recorded at a single point, the recording point may be near a mode-field zero; due to very low field intensity at this point, the eigenvalues can only be determined with great uncertainty.
To eliminate the field singularities arising as $\rho $ approaches 0, we truncate the computational domain at $\rho = {\rho _{min}}$ as shown in Fig. 1(b). Here, the computational domain is bounded by Perfect Matched Layer (PML) boundary conditions.
Figure 2(a) shows a typical time evolution of ${E_z}(t )$ for mode number $m = 23$, where the dimensions of the structure are chosen as follows $:\; \; {\textrm{w}_1} = 200\; nm,\; {\textrm{H}_1} = 150\; nm,\; {\textrm{w}_2} = 200\; nm\; ,\; {\textrm{H}_2} = 15\; nm,\; and\; g = 40\; nm.$ Moreover, the radius of the ring resonator is $R = 1.6\; \mu m$ and the spatial increment is chosen to be 2 $nm$. For the following computations we have chosen ${N_\lambda } = 500$ and ${N_d} = 1$.
Fig. 2. (a) Typical time evolution of ${E_z}(t )$ at the center of the gap and (b) the discrete Fourier transform of ${E_z}(t )$.
Here, the data is recorded at the center of the gap where the field intensity is high. We terminate the simulation after the amplitude of the time signal decays sufficiently, i.e., when all the spectral information can be efficiently resolved using the recorded data.
The QNMs are the solution of eigenvalue problem of source-free Maxwell’s equations. After the excitation source is turned off and upon achieving a steady-state solution, time-domain signal is being processed using FDM and discrete Fourier transform (DFT).
First, we utilized FDM to compute the modal properties. The main objective of FDM is to extract eigen-frequencies, line widths, and complex amplitudes of the time-domain data by fitting the signals to a sum of exponentially damped sinusoids, i.e.,
We have excluded the effects of the initial short-lived transients in the spectral analysis. The recorded data starts at 20 femtoseconds. Moreover, to reduce the effects of noise in the calculation, we search for the eigenvalues in narrow spectral ranges. For the structure under study, we search for the eigenvalues in the frequency range $({300 - 350} )\; \textrm{THz}$. The results of FDM calculations are given in Table 1.
Table 1. Calculation of amplitudes ($|{{d_k}} |$), resonance frequencies (${f_k}^{\prime}$), and decay rates (${f_k}^{^{\prime\prime}}$) of eigenmodes using FDM.
The photonic-like, the SPP-like, and the hybrid-plasmonic modes are the possible QNM modes that can be excited within the resonator. In this research, we only analyze the hybrid-plasmonic modes. To determine the hybrid-plasmonic modes among the eigen-frequencies which are obtained in Table 1, we have repeated the simulation with long-duration Gaussian sources whose central frequency are $\; {f_k}^{\prime}$. This is used to obtain the electric field profile at the corresponding resonance frequency. The hybrid-plasmonic mode is the one with strong confinement inside the gap region. After investigation of various field profiles, the eigen-frequency ${\tilde{f}_2} = 3.149 \times {10^{14}}\; -{-}j1.8201 \times {10^{11}}\; \textrm{Hz}$ has the character of a hybrid-plasmonic mode. Its Q-factor $\; \left( {Q ={-} \frac{{f^{\prime}}}{{2f^{\prime\prime}}}} \right)$ amounts to 864. For the hybrid-plasmonic mode, the enhanced $z$-polarized electric field due to its discontinuity at the interface between high-contrast-index dielectrics leads to a nanoscale mode-field confinement.
Then, we performed the DFT of ${E_z}(t )$ as shown in Fig. 2(b). The resonance frequency and the Q-factor amount to 315 THz and 300, respectively. The Q-factor is calculated using $\frac{{{f_r}}}{{FWHM}}$. Where ${f_r}$ is the resonance frequency of the hybrid-plasmonic mode, and FWHM is the full-width half-maximum of the resonance peak.
The calculated resonance frequency and Q-factor using FDM and DFT are compared with a BOR-FEM [26]. The results are presented in Table 2.
Table 2. Comparison of resonance frequency and Q-factor calculated by BOR-FDTD and BOR-FEM method.
According to the given comparison in Table 2, the calculated resonance frequency and the Q-factor by BOR-FDTD (FDM), and BOR-FEM are in good agreement. On the other hand, BOR-FDTD (DFT) gives the resonance frequency accurately, but its Q-factor is inaccurate because computation of FWHM of the resonance requires long computational time. To obtain an accurate Q-factor using BOR-FDTD (DFT), the time-domain signal should completely decay before calculation of the Fourier transform.
The normalized field profile $({|{\overrightarrow {\tilde{E}} ({\rho ,z} )} |} )$ of the fundamental hybrid-plasmonic mode is shown in Fig. 3(a) in the ring cross-section.
Fig. 3. Normalized 2D electric field intensity distribution in the ring cross-section for (a) $g = 40\; nm$ and (b)$\; g = 6\; nm$. We have assumed $\; \; m = 23$, ${w_1} = 200\; nm,\; {H_1} = 150\; nm,\; {w_2} = 200\; nm\; ,\; {H_2} = 15\; nm,\; and\; ,R = 1.6\; \mu m$.
In addition, Fig. 3(b) shows the normalized field profile of the electric field for $g = 6\; nm$. As depicted in Fig. 3(a) and 3(b), the hybrid mode can be strongly confined in low-index gap. The enhancement in the gap is caused by hybridization of photonic-like and SPP-like modes. An extra comparison between the results of BOR-FDTD (FDM) and BOR-FDTD (DFT) is presented in Supplement 1.
In Fig. 4, we have compared the resonance frequencies and Q-factors calculated by BOR-FDTD (FDM) and BOR-FDTD (DFT) for different time-windowing. Note that we have observed that the amplitude of various field components vanishes approximately after 2000 femtoseconds. This explains why the BOR-FDTD (DFT) does not converge up to 2000 femtoseconds, whereas after almost 500 femtoseconds the calculated resonance frequencies and Q-factors using BOR-FDTD (FDM) converge to the exact values. For instance, in Supplement 1, the DFT of the signal ${E_z}(t )$ is shown for several number of time steps.
Fig. 4. (a) Resonance frequency and (b) Q-factor calculated using FDM and DFT at different number of time samples. The simulation parameters are chosen as follows:$\; \; m = 23$, ${w_1} = 200\; nm,\; {H_1} = 150\; nm,\; {w_2} = 200\; nm\; ,\; {H_2} = 15\; nm,\; and\; ,R = 1.6\; \mu m$.
The convergence performance of BOR-FDTD is shown in Fig. 5. We have also investigated the effect of the resolution of spatial increment on the calculation of the QNMs. The relative error for real and imaginary parts of the eigen-frequency is computed for different spatial increments. The relative error $\delta $ is defined as $\delta (\Delta )= |{f(\Delta )- {f_0}|/ |{f_0}} |$, where $f(\Delta )$ is the numerical value calculated for the spatial increment $\Delta $, and ${f_0}$ is the numerical value calculated for the highest resolution, i.e., $\Delta = 1nm$.
Fig. 5. Relative error for (a) real part and (b) imaginary part of eigen-frequency of QNM for $\; \; m = 23$, ${w_1} = 200\; nm,\; {H_1} = 150\; nm,\; {w_2} = 200\; nm\; ,\; {H_2} = 15\; nm,\; and\; ,R = 1.6\; \mu m$.
As shown in Fig. 5, the extremely fine meshes in the size of $1\; nm$ or $2\; nm$ are necessary to achieve an accurate result.
By tuning the dimensions of silicon nanowires and the gap, hybridization between photonic-like and SPP-like modes can be controlled. For this purpose, we now present a parameter analysis. Figure 6(a)–(d) show the resonance frequencies, the Q-factors, the real part of effective refractive indices $\left( {{n_{eff}} = Re\left( {\frac{{m{c_0}}}{{2\pi R\tilde{f}}}} \right)} \right)$, and the decay rates ($f^{\prime\prime}$) as a function of the gap thickness. The parameters are chosen as follows: $m = 18$, $,\; {\textrm{H}_2} = 15\; nm$, $R = 2\; \mu m$, and the cross-section of the silicon core is fixed at $200\; nm \times 150\; nm$. A Q-factor close to 1200 is obtained by increasing the gap thickness from $6\; nm\; $ to $95\; nm$.
Fig. 6. (a) Resonance frequency (b) decay rate (c) real part of effective refractive index and (d) Q-factor as a function of the gap
Then, we have compared the computed data by BOR-FDTD (FDM) with the standard COMSOL Multiphysics data. We have also used the cylindrical symmetry for modeling the resonator in COMSOL (BOR-FEM). As depicted in Fig. 6, an acceptable agreement has been achieved between the results of these two methods.
For the gap thicknesses larger than $95\; nm,$ the coupling between photonic-like and SPP-like modes is weak. Consequently, the hybrid mode cannot be created. Figure 7 shows the normalized electric field profile for $g = 110\; nm$, where $m = 23$, ${\textrm{w}_1} = 200\; nm,\; {\textrm{H}_1} = 150\; nm,\; {\textrm{w}_2} = 200\; nm\; ,\; {\textrm{H}_2} = 15\; nm,\; and\; R = 1.6\; \mu m.$ Obviously, the created mode is the fundamental photonic TM-like mode. The mode field profile is calculated with a cell size of $10 \times 10\; n{m^2}$. The computation time for ${N_t} = 3 \times {10^9}$ time samples amount to 2.7 hours for our MATLAB code. After postprocessing (DFT and FDM), the equivalent computation time for 500 frequency points (in the frequency range of 300 THz -350 THz) is 25 seconds for each frequency point. The simulation is performed on a laptop system with the following characteristics: CPU: 12th Generation Intel, Dell, Core i7-12700H, and RAM: 32GB.
Fig. 7. Normalized 2D electric field intensity distribution in the ring cross-section for $g = 110\; nm$, where $m = 23$, ${\textrm{w}_1} = 200\; nm,\; {\textrm{H}_1} = 150\; nm,\; {\textrm{w}_2} = 200\; nm\; ,\; {\textrm{H}_2} = 15\; nm,\; and\; R = 1.6\; \mu m$
The resonance frequency and Q-factor as a function of ring radius for $m = 23$ are shown in Fig. 8. The dimensions of the structure are chosen as follows: ${w_1} = 200\; nm,\; {H_1} = 150\; nm,\; {w_2} = 200\; nm\; ,\; {H_2} = 15\; nm,\; and\; g = 40\; nm.$ For a ring radius of $1.8\; \mu m$, a Q-factor close to 1200 is obtained.
Fig. 8. (a) Resonance frequency and (b) Q-factor as a function of ring radius. The dimensions of the structure are chosen as follows: ${w_1} = 200\; nm,\; {H_1} = 150\; nm,\; {w_2} = 200\; nm\; ,\; {H_2} = 15\; nm,\; and\; g = 40\; nm.$
Numerical calculations show that the guided modal properties of the plasmonic resonators are improved by the combination of hybrid-plasmonic and dielectric slot modes.
In conclusion, we have generalized a BOR-FDTD in MATLAB to calculate the QNMs of circular coupled resonators. We have shown how BOR-FDTD in combination with FDM can be used to accurately calculate the complex eigenvalues. To verify the precision reached by our method, we have analyzed a coupled dielectric-plasmonic ring resonator. Then, we have tested the accuracy of the obtained results, by comparing the computed data with those obtained with a BOR-FEM approach in COMSOL Multiphysics solver. Our numerical method is applicable to the modal analysis of high Q-factor dielectric-plasmonic cylindrical cavities.
Funding
National Science Foundation CAREER Award (1846477); Brigham Young University (HIGRA AWARD).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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