Simulating robust far-field coupling to traveling waves in large three-dimensional nanostructured high-Q microresonators

Lei Chen; Cheng Li; Yu-Min Liu; Judith Su; Euan McLeod

doi:10.1364/PRJ.7.000967

Abstract

Ultra-high quality (

Q

) whispering gallery mode (WGM) microtoroid optical resonators have demonstrated highly sensitive biomolecular detection down to the single molecule limit; however, the lack of a robust coupling method has prevented their widespread adoption outside the laboratory. We demonstrate through simulation that a phased array of nanorods can enable free-space coupling of light both into and out of a microtoroid while maintaining a high

Q

. To simulate large nanostructured WGM resonators, we developed a new approach known as FloWBEM, which is an efficient and compact 3D wedge model with custom boundary conditions that accurately simulate the resonant Fano interference between the traveling WGM waves and a nanorod array. Depending on the excitation conditions, we find loaded

Q

factors of the driven system as high as

2.1 \times 10^{7}

and signal-to-background ratios as high as 3.86%, greater than the noise levels of many commercial detectors. These results can drive future experimental implementation.

Ultra-high-quality ( $Q$ ) factor whispering gallery mode (WGM) optical resonators such as microtoroids and spheres have demonstrated sensitive biomolecular detection down to the single molecule level. To enable widespread use of these devices outside the laboratory and to enable portable point of care systems, robust light coupling strategies are needed ^[1]. Most often, light is evanescently coupled into a microtoroid optical resonator using optical fibers that have been tapered to hundreds of nanometers in diameter ^[2]. These tapers, however, are fragile, sensitive to environmental vibrations such as fluid flow fluctuations, and require precise alignment to phase match with the WGM with high efficiency ^[3–6]. Other coupling approaches include prism coupling, direct illumination from embedded light emitters in a resonator, angled fiber illumination, and polished half-block couplers, but these methods suffer from either $Q$ degradation due to poor coupling or difficulty in maintaining robust coupling to on-chip WGM resonators ^[7]. An efficient and high $Q$ free-space coupling scheme is chaos-assisted momentum transformation, which can couple free-space light into deformed, nonrotationally-symmetric microresonators without external couplers ^[8–13]. This method is broadband and free from a phase-match condition as a result of its pump-induced nonresonant dynamical tunneling nature ^[8–10]. However, the spectra and mode field distribution are irregular in these devices, which can limit their use in applications such as frequency comb generation or evanescent biosensing, where a predictable and regular response is advantageous. Previously, randomly positioned polystyrene nanospheres on the microtoroid surface have been used for incoupling of light ^[14]; however, tapered fibers were still used to couple the light out of the microtoroid. Furthermore, the random positioning of these particles prevented efficient coupling. Finally, individual nanoscale scatterers placed on the surface of microspheres have been used in conjunction with nanopositioners in laboratory settings to couple in free-space light with a high coupling efficiency of 16.8% ^[15,16]. However, this method relied on using the microsphere itself as a ball lens for focusing the light onto the scatterer, which would not work for microtoroid-shaped resonators.

Precisely positioned photonic nanostructures have the potential to alleviate these problems through their unique ability to trap and direct light scattering via surface plasmon resonances (SPRs), antenna resonances ^[17], and/or grating resonances ^[18]. The design of efficient coupling structures can be accomplished by full three-dimensional (3D) finite-element method simulations of smaller (lower- $Q$ ) nanostructure-microresonator hybrid systems ^[8,19]. For example, in Ref. ^[8], a 12 μm major radius and 2 μm minor radius microtoroid is simulated, which is significantly smaller in volume than the microtoroid simulated here, which has major radius $R = 45 μm$ and minor radius $r = 2 μm$ . These toroid dimensions and an operating wavelength near 633 nm are chosen to match previous experiments ^[2,20]. Conventional 3D simulations remain intractable for such large high- $Q$ WGM microresonators. For WGM resonators with complete azimuthal symmetry (e.g., bare microtoroids), a 2D axisymmetric simulation method is often used, while for resonators with axial translation symmetry, such as cylindrical microcapillaries ^[21], a planar 2D simulation can be used, but neither approach can simulate isolated nanoparticles on a resonator surface.

Previously, eigenfrequency analysis was used to simulate a thin 3D wedge of a microresonator coupled to nanoparticles with perfect electrical/magnetic conductor boundary conditions and perfectly matched layers ^[22,23]. While these simulations provided some insight, they do not accurately model typical traveling-wave WGM experiments, because the perfect conductor boundary conditions can only simulate standing waves. When the WGM is a traveling wave, a nanoparticle would experience the same time-average field intensity regardless of its azimuthal coordinate on the resonator, while in the case of a standing wave, the nanoparticle experiences significantly different fields depending on whether it is located at a node, an antinode, or somewhere in-between. As a result, it is unlikely that standing-mode simulations would accurately predict the interaction between a traveling wave WGM and plasmonic nanostructures, a case which is common in biosensing experiments ^[23].

Sign up for Photonics Research TOC. Get the latest issue of Photonics Research delivered right to you！Sign up now

Here we design a gold nanorod (NR) grating as an experimentally feasible alternative for robust coupling of free-space light to a microtoroid resonator [Fig. 1(a)], and numerically demonstrate a compact and computationally efficient 3D model that simulates the performance of the driven system. The WGM backaction-mediated reflection spectrum is characterized by a Fano-like optical interference between the WGM and the plasmonic grating resonance [Fig. 1(b)]. All simulations are carried out using the software package COMSOL Multiphysics (wave optics module).

Figure 1.Free-space coupling to a WGM microtoroid optical resonator via a phased array of gold nanorods. (a) Schematic view. (b) Collection of far-field scattering from the grating shows the Fano resonance corresponding to the interference between the grating and WGM resonances. (c) Established 2D axisymmetric simulations identify a resonance of the bare WGM at

λ_{2} \approx 633 nm

, corresponding to an azimuthal mode number

m = 660

. (d) The FloWBEM simulation of the same bare toroid as in panel (c). Surfaces

S_{1}

and

S_{2}

are simulated with Floquet boundary conditions, and

S_{3}

is simulated with scattering boundary conditions. (e) For simulating the driven system, a far-field domain (

S_{4}

) is added at the circumference, replacing the Floquet and scattering boundary conditions for that region. (f) Nanorods are placed in the equatorial plane, between the light source, which is incident at 45°, and the silica toroid. A field continuity condition is applied between the light source and the domain surrounding the toroid.

We are able to simulate the large microtoroid in 3D by simulating a small (five wavelengths in arclength) wedge using periodic Floquet-wedge boundary conditions (see Appendix B for detailed methods descriptions) with a beam envelope method solver, a method we call FloWBEM. The beam envelope method (not to be confused with the beam propagation method ^[24]) has been previously used to simulate ring resonators in 2D ^[25] but to the best of our knowledge has never been used to simulate a 3D WGM. In FloWBEM, the Floquet boundary conditions are applied to the faces of the wedge, except for a small far-field port at the outer circumference of the domain when simulating externally driven systems [Figs. 1(d) and 1(e)]. In this way, the Floquet boundary conditions are applied to the toroid and its surrounding evanescent zone. The remaining outer surface of the wedge uses scattering boundary conditions. This approach allows the simulation to accurately capture the periodic nature of the wedge to form a toroid structure without imposing mirror boundary conditions, which prevent simulation of traveling waves.

To validate the FloWBEM approach, we first compare it to the established 2D axisymmetric model of a bare toroid ^[26] [Figs. 1(c) and 1(d)]. Both models are run using an eigenfrequency solver, which finds the normal modes of the system without considering any specific type of excitation. As expected, both simulations show the same mode field pattern and exhibit similar $Q$ ( $7.4 \times 10^{7}$ using a 2D axisymmetric simulation and $7.3 \times 10^{7}$ using FloWBEM). Following previous approaches ^[14], we assume an imaginary part of the refractive index equal to $10^{- 8}$ to limit the $Q$ factor to values similar to what is seen in experiments.

Using an external light source to drive the system and a linear array of nanorods acting as a grating, we find that it is possible to couple from the far-field into a high- $Q$ WGM (Fig. 2). We choose a grating periodicity ( $Λ = 264.8 nm$ ) that provides partial phase-matching (see Appendix B Fig. 7) between the incident light and the WGM to study the simultaneous excitation of degenerate clockwise (CW) and counterclockwise (CCW) modes. In the grating, the nine individual nanorods are modeled as nanocylinders with a length-to-diameter aspect ratio of 1.87 and a radius of 5 nm. Our simulation results show that the two modes remain uncoupled due to the lack of any major perturbations (internal defects, quantum emitters, etc.) that would lift the degeneracy ^[27–30]. Mode splitting due to a lifted degeneracy becomes sizeable only if the frequency splitting is larger than the linewidth of the WGM resonance ^[29]. Here, we find that is not the case and coupling the grating to the WGM results in only a frequency shift and broadening of the linewidth (see Appendix B Fig. 8) ^[31].

Figure 2.Frequency domain simulations of the driven system. (a) Intracavity energy of the coupled TE WGM with a grating spaced

d = 1000 nm

from the toroid. Zoom-in:

Δ λ_{FWHM} = 0.14 pm

. (b) Intracavity energy of the coupled TM WGM with a grating-toroid spacing

d = 1000 nm

. Zoom-in:

Δ λ_{FWHM} = 0.22 pm

. (c) TE WGM backaction-mediated reflection spectrum corresponding to the same simulation as panel (a). (d) TM WGM backaction mediated reflection spectrum corresponding to the same simulation as panel (b). In all panels, the wavelength step is 0.5 nm for the broad spectrum, and between 0.005 pm and 2 pm in the vicinity of the resonance (insets). The SBR and

δ λ_{FWHM}

values for panels (c) and (d) are given in Table 1.

From a series of frequency-domain simulations, the resonance of the driven system can be identified and characterized by the energy stored in the resonator as a function of wavelength [Figs. 2(a) and 2(b)]. Owing to the perturbation induced by the grating on the WGM, the TE resonance shifts slightly from $λ_{2} \approx 632.58 nm$ for a bare resonator to $λ_{res} \approx 633.74 nm$ in the coupled system (see Appendix C). The loaded $Q$ of this driven TE mode is $λ_{res} / Δ λ_{FWHM} = 4.53 \times 10^{6}$ , which is only degraded by approximately 1 order of magnitude relative to the intrinsic $Q$ of the bare toroid due to the scattering and absorption by the grating. Because we have specified a particular azimuthal mode number $m$ through our Floquet boundary conditions, we find only a single resonance despite the free spectral range of order 1 nm for a microtoroid of this size. Other resonances can be identified by selecting a different $m$ . As a result, we expect the spectrum to be accurate only within a small neighborhood of the resonance and not over the full background. As a validation of our simulation approach, we find very similar loaded $Q$ factors by performing eigenfrequency simulations of the coupled grating-toroid system, where $Q$ is evaluated as $Re {f_{res}} / (2 Im {f_{res}})$ and $f_{res}$ is the complex eigenfrequency (Table 1). The advantage of eigenfrequency simulations is that they can be run significantly faster than a series of frequency domain simulations.

Table 1. Calculation Methods for Loaded

Q

of the Driven WGM Coupled to a Grating^a

In an experiment, it would be difficult to measure the intracavity energy, and instead one would track the resonance by measuring the power reflected (scattered) by the grating coupler [Figs. 2(c) and 2(d)]. Mediated by the WGM backaction, the reflection is characterized by a Fano lineshape due to the interference of the ultra-narrow WGM resonance with the broadband plasmonic grating resonance, corresponding to a radiative continuum of states. In general, Fano lineshapes result from the coherent interference between narrow and broad spectral features ^[32]. The different lineshapes in Figs. 2(c), 2(d), 5, and 6 are commonly encountered in Fano interference and are due to differences in phase between the WGM and plasmonic grating resonance ^[33–36]. To track any resonant shifts with the greatest precision during sensing experiments, it is important that the linewidth should be narrow and the line amplitude be easily detectable above the background. The polarization of the light and the spacing between the grating and toroid influence the reflected spectrum linewidth and signal-to-background ratio (SBR). As shown in Figs. 2(c) and 2(d) and Table 1, we can achieve ultra-narrow linewidths while maintaining SBR levels of the order of 1%, which is greater than the noise levels of many commercial detectors. In Table 1, we also report the intracavity energy $W$ normalized by the intensity $I_{0} = P_{in} \cos 45 ° / A$ of the incident plane wave, where $P_{in} = 1 mW$ is the input optical power through the input side surface of the far-field domain [see Fig. 1(e)], which has an area of $A = 2.72 \times 10^{- 12} m^{2}$ .

The distance between the grating and the toroid also impacts the loaded $Q$ (Fig. 3) and stored energy in the cavity (Appendix B Table 2). Experimentally, different grating-toroid spacings could be obtained by mounting the nanoparticles on a flat transparent substrate that is independently positioned ^[36], by constructing a scaffold out of a dielectric material with a relatively low refractive index ^[37] or by using nanoparticles that are coated with a dielectric shell ^[38]. In general, a stronger coupling at shorter distances between the grating and WGM decreases the loaded $Q$ due to increased scattering and absorption losses. However, we also find a peak in loaded $Q$ at $d = 700 nm$ , due to strong destructive interference between the incident light scattered by the grating and the circulating WGM light also scattered by the grating (see Appendix A Figs. 5 and 6). As a further validation of our system, we see good agreement in Fig. 3 for three different methods of calculating $Q$ : the previously discussed driven-system frequency-domain $λ_{res} / Δ λ_{FWHM}$ and eigenfrequency $Re {f_{res}} / (2 Im {f_{res}})$ , as well as another method: eigenfrequency $2 π f_{res} W / (P_{abs} + P_{rad})$ , where $W$ is the intracavity energy and $P_{abs} + P_{rad}$ the total power loss ^[39].

Figure 3.Effect of grating-WGM separation on linewidths. Loaded

Q

factors for both TE and TM polarizations depend on grating-toroid separation.

Q

factors are evaluated using three methods: eigenfrequency

Re {f_{res}} / (2 Im {f_{res}})

(solid circles and lines), eigenfrequency

2 π f_{res} W / (P_{abs} + P_{rad})

(hollow circles), and frequency domain

λ_{res} / Δ λ_{FWHM}

(red stars). Owing to the increased computational costs of frequency domain simulations, only four points are shown.

Table 2. Intracavity Energy of the Frequency Domain Driven TE CCW and CW Modes for Different Coupling Separation

The FloWBEM approach enables visualization of the magnitudes of the traveling-mode WGM fields. In the systems presented here, the CCW mode [Figs. 4(a) and 4(b)] is preferentially excited due to the partial phase matching of the $x$ component of the incident light wavevector, the grating period, and the WGM wavenumber (see Appendix B Fig. 7 and Appendix B Table 2). The CW mode [Fig. 4(c)] is also excited at the same wavelength (see Appendix B Fig. 8) but with a lower amplitude than the CCW mode. Thus, the net electric field within the toroid consists of a partially traveling/partially standing mode, manifested by the ripples seen in Fig. 4(d). The ripples could be reduced and a purer traveling mode excited by fine-tuning the phase matching of the grating to account for the precise distance between the grating and toroid mode, by shaping or tapering the grating to further promote a particular mode direction ^[40] and/or by increasing the number of nanorods in the grating to minimize backward scattering due to edge effects.

Figure 4.Mode field distributions of the driven coupled system. (a) 3D frequency domain simulation of the microtoroid coupled to the grating with various separation distances. The CCW mode is shown, which is selected through appropriate choice of the Floquet boundary conditions. The incident light is

s

-polarized, which drives a TE WGM. Field distributions are plotted as field amplitude

| E |

. (b) Same as panel (a), but with a separation of

d = 1000 nm

. (c) Same as panel (b), but where the CW solution has been selected through the choice of Floquet boundary conditions with opposite sign. (d) The superposition of the field distributions in (b) and (c).

In summary, a major barrier to widespread commercialization of microtoroid resonator sensors is the need for evanescent light coupling using fragile fiber tapers. As an alternative to these fragile tapers, we designed a gold NR grating that eliminates the tapered optical fiber, while maintaining a high $Q$ factor and sufficient SBR ratio. Ultimately, these designs could be fabricated using pick-and-place nanomanufacturing approaches or through directed self-assembly ^[41,42]. Simulations of this structure were made possible by a novel finite-element 3D beam-envelope model with custom boundary conditions called FloWBEM that can model interactions among traveling waves within the microtoroid, nanostructures on its surface, and far-field radiation. Both eigenfrequency and frequency-domain solvers are reliable in studying the driven system. Frequency-sweep simulations run slower than eigenfrequency simulations but allow the Fano optical response to be determined. We anticipate that our proposed modeling approach can solve a variety of other nanostructure-microcavity coupled systems in the future, including single-photon resonator-atom interactions ^[43,44].

Acknowledgment. This project was supported by the Defense Threat Reduction Agency-Joint Science and Technology Office for Chemical and Biological Defense (Grant #HDTRA11810044). L. C. is supported by the National Key R&D Program of China, NSFC, Natural Science Foundation of Beijing, CSC Foundation, and Beijing University of Posts and Telecommunications Excellent Ph.D. Students Foundation. C. L. is supported by a DeMund Foundation Graduate Student Endowed Scholarship in Optical and Medical Sciences and a Friends of Tucson Optics (FOTO) Scholarship from the University of Arizona. L. C. designed and performed the simulations. C. L. assisted with the simulations. E. M. and J. S. conceived the idea and supervised the project. All authors (L. C., C. L., Y. L., J. S., and E. M.) analyzed the data, discussed the results, and wrote the paper. This material is based upon High Performance Computing (HPC) resources supported by the University of Arizona TRIF, UITS, and RDI and maintained by the UA Research Technologies department.

Figure 5.Reflected spectrum exhibiting a Lorentzian lineshape, corresponding to the third column in Table 1 of the main text (

d = 700 nm

). The SBR is 0.05%.

Figure 6.Reflected spectrum exhibiting a Fano lineshape, corresponding to the fifth column in Table 1 of the main text (

d = 700 nm

). The SBR is 1.18%.

In all simulations, the refractive indices of the various materials are

n_{{SiO}_{2}} = 1.45 + i 10^{8}

for the silica toroid, and

n_{water} = 1.33

for the water background; the optical parameters of the gold are described by an interpolation function from the COMSOL Material Library [45].

As an additional validation step and to better understand the wavelength shift induced on the WGM by the nanostructure, we performed a series of TE-mode simulations using only a single nanorod and not a full grating. We validated the results against the prediction from the Bethe–Schwinger cavity perturbation (BSCP) formula [19,47,48], with good agreement, as shown in Fig. 9. The equation used for BSCP is

\frac{Δ ω}{ω} = \frac{n_{water}^{2} α_{∥} (AR) {| E_{0} |}^{2}}{2 ε_{0} n_{{S i o}_{2}}^{2} V_{m} {| E_{0} |}_{\max}^{2}},

(C1)

where

{| E_{0} |}^{2} = 2.00 \times 10^{15} V^{2} m^{2}

is the magnitude-squared electric field of the unperturbed cavity at the location of the nanorod,

{| E_{0} |}_{\max}^{2} = 1.91 \times 10^{16} V^{2} m^{2}

is the maximum magnitude-squared electric field inside the toroid,

V_{m} = 0.2 {μm}^{3}

is the mode volume of the field in the wedge segment, and

α_{| |} (AR)

is the polarizability of the nanorod as a function of aspect ratio [19,49]:

α_{∥} = Δ V ε_{0} \frac{ε_{A u} n_{water}^{2}}{G_{∥} ε_{A u} + (1 G_{∥}) n_{water}^{2}},

(C2)

where

Δ V = 1470 {nm}^{3}

is the nanorod volume, and

G_{∥} = R_{s} \frac{1 e_{c}^{2}}{e_{c}^{2}} [1 + \frac{1}{2 e_{c}} \ln (\frac{1 + e_{c}}{1 e_{c}})],

(C3)

where

R_{s} = 0.74

is a shape correction to account for the differences between a cylinder and a prolate spheroid [50], and

e_{c} = \sqrt{1 d^{2} / L^{2}}

is the eccentricity.

Figure 9.WGM frequency shift for a single nanorod as a function of nanorod aspect ratio (AR).

Although the fit is good between FloWBEM and BSCP for a single rod, BSCP is not expected to be a good model for the grating, as it is known that systems that are designed to efficiently radiate are not well-described by BSCP [35].

	Driven TE		Driven TM
Coupling distance d	1000 nm	700 nm	1000 nm	700 nm
Frequency domain driven Q (λres/ΔλFWHM)	4.53×106	6.34×106	2.88×106	2.10×107
Eigenfrequency coupled Q(Re{fres}2 Im{fres})	4.59×106	6.04×106	2.91×106	1.80×107
Reflected spectrum linewidth (δλFWHM)	0.16 pm	0.1 pm	0.28 pm	0.04 pm
SBR (from reflected spectrum)	3.86%	0.05%	1.33%	1.18%
Normalized intracavity energy (W/I0)	9.77×10−23 m2·s	4.36×10−23 m2·s	3.81×10−23 m2·s	2.60×10−22 m2·s

d (nm)	Energy of the Driven TE CCW (J)	Energy of the Driven TE CW (J)	Energy Ratio for the TE CCW/CW
400	4.67×10−16	4.05×10−16	1.15
700	1.13×10−14	2.71×10−15	4.18
1000	2.50×10−14	1.20×10−14	2.08