Accurate determination of the quality factor and tunneling distance of axisymmetric resonators for biosensing applications

Due to ultra high quality factor ($10^6-10^9$), axisymmetric optical microcavities are popular platforms for biosensing applications. It has been recently demonstrated that a microcavity biosensor can track a biodetection event as a function of its quality factor by using phase shift cavity ring down spectroscopy (PS-CRDS). However, to achieve maximum sensitivity, it is necessary to optimize the microcavity parameters for a given sensing application. Here, we introduce an improved finite element model which allows us to determine the optimized geometry for the PS-CRDS sensor. The improved model not only provides fast and accurate determination of quality factors but also determines the tunneling distance of axisymmetric resonators. The improved model is validated numerically, analytically, and experimentally.


Introduction
Perfectly matched layers (PML) act as artificial boundaries that truncate the computation domain of open region scattering problems in the finite element method. The whispering gallery modes (WGM) of an open micro-cavity radiate into the surroundings and a PML is required in order to block the unwanted reflections from the boundaries of the computation domain. There have been previous attempts of developing finite element models (FEM) of axisymmetric cavities along with the PML. In 2005, Chinellato et al. [1] showed a FEM model which was implemented in MATLAB. However, their model failed to simulate the resonators with large size. In their work they solved the numerical examples of dielectric spheres of sizes less than 3µm. In 2009, Karl et al. developed a FEM model for studying a micro-pillar cavity [2]. But in their model they did not consider the suppression of false solutions, a well known problem in finite element formulations [3]. Moreover, they estimated the quality factor of the modes by fitting the Lorentzian peek to the calculated spectrum of the cavity, thus an extra approximation had been introduced.
In 2007, Oxborrow [4] developed a FEM for open axisymmetric resonators in COMSOL. In his work, he showed that the model could simulate an arbitrary cross section resonators in optical and microwave regimes, thus removing the size limitation in [1]. Another difference from [1] is in terms of suppression of false modes; Oxborrow used a simple penalty term in his master equation whereas Chinellato et al. used Nédélec edge and modified Lagrange nodal element functions to avoid the spurious modes. However in Oxborrow's model no PML was implemented and as a result the WGM quality factor could not be determined accurately. The quality factor due to the WGM radiation was estimated by placing a bound on its minimum and maximum possible values. These maximum and minimum values were determined by executing the model multiple times with different boundary conditions. Moreover, the model lacks the capability to estimate the quality factor of multiple modes simultaneously.
Determination of the quality factor with high accuracy is important in certain applications such as cavity ring down spectroscopy where decay time depends upon the quality factor. In order to provide accurate determination of the WGM quality factor, we have improved Oxborrow's model by modifying its master equation and implementing the PML along the boundaries of the computation domain. The modified model does not have any of the drawbacks of the previous model. Moreover, we have computed the quality factors of all the modes without using any fitting algorithms.
In our model, we treat the PML as an anisotropic absorber and implement it in the cylindrical coordinate system. Our model is applicable to any axisymmetric resonator geometry but due to the availability of analytical expressions for spherical resonators, we have tested the model by determining the quality factors of a silica micro-sphere in air. We have found that our simulation results are in excellent agreement with the analytical results. We also apply our model to micro-toroids and show that our results are consistent with those obtained by experiment. By using PML, our model has also overcome all discrepancies present in the previous models.

Mathematical Description
Applying Galerkin's method to the wave equation and after using the boundary conditions for open resonators, one can arrive at the FEM equation in the weak form [4]: where H represents the magnetic field of the resonator and H represents the test magnetic field, an essential component of the weak form. The second term of the equation 1 represents a penalty term to suppress false solutions. None of the field components will depend upon the azimuthal coordinate φ in the axisymmetric resonators, resulting in reduction of the 3D problem to a 2D problem.

PML Formulation
A PML can be treated as an anisotropic absorber in which the diagonal permittivity and permeability tensors of the absorber are modified according to equation 2 [5].
The radial and axial modification factors are represented byΛ, which is given by equation 3Λ where t rpml , t upml , t lpml are the PML thicknesses in the radial, +z and -z directions respectively and r pml , z upml , z lpml are the locations of the start of PML in the radial, +z and -z directions respectively. n medium is refractive index of the medium, N is order of the PML, and G is a positive integer.
In the PML expressions (sr, sz,r), the imaginary component contributes to the attenuation of waves in the PML but at the same time, due to the discrete nature of the FEM mesh, a large imaginary component will introduce reflections at the interface between the PML and the medium. In order to determine the optimal value for the imaginary component, we have investigated linear, quadratic, and cubic PML of different thicknesses for various values of G by running many simulations for various sphere diameters. To deduce the optimum values of the parameters, we then compared the simulation results for Q W GM of spherical cavities with the analytical ones [10]. The simulation results show that a linear (i.e. N = 1), and λ/4 thick PML with a G value of 5 is optimum. We have also used these optimum values for the simulations of the disc and the toroidal cavities.

FEM equation with PML
In order to incorporate the PML, we have reformulated equation 1 in the following way: By casting equation 4 into the FEM software COMSOL, a full vectorial finite element model of a silica sphere in air can be obtained. By using the eigenvalue solver in COMSOL, resonant frequencies of all the modes can easily be determined. Since the PML introduces losses in the computational domain, the resonant frequency (fr) will be a complex number and the quality factor due to the WGM radiation can be calculated as:

Analytical expressions of the spherical resonator
The quality factor due to WGM radiation losses in a spherical micro-cavity can be written as [6]: where m = azimuthal mode number m can be calculated using the characteristic equation for WGM frequencies [7]: where j, h are Bessel functions, k 0 is the wave number (2π/λ 0 ), and a is the radius of a microsphere.
It should be noted that equation 6 is an asymptotic solution for the Qwgm of the spherical resonator which requires m 1, however the error is less than 1% for m ≥ 19 [9]. In our comparison for the analytical results, we have applied equation 6 to silica spheres with mode numbers (m) ranging from 27 − 98. Fig. 1 shows the TE and TM fundamental modes of a silica spherical cavity in air. We have plotted the quality factor due to the TE/TM fundamental whispering gallery mode radiation for various sphere diameters at 850nm. Fig. 2 shows the comparison of the FEM simulation results and results obtained by using analytical expressions presented in section 2.3. We have also calculated the minimum and maximum Qwgm values using Oxborrow's model [4] for each sphere diameter and those are also shown in Fig. 2.

Results
From the simulation results, it is clear that the analytical results confirm the accuracy of our finite element model. The quality factors for all the modes (TE, TM fields of the fundamental and higher order modes) are also obtained by one single simulation rather than multiple simulations. Moreover, no prior knowledge of any of the mode frequencies is required to obtain the quality factors.

Discussion and application to other axisymmetric resonator geometries
The excellent agreement between the simulation and analytical results shows that the PML is absorbing the radiated waves effectively. However, the simulation results indicate that there are less reflections from the PML for the TM modes and values of quality factors for the TM modes are more accurate than the TE modes. This suggests that the PML is absorbing TM waves better than the TE waves. One possible explanation is that since the TM component is parallel to the radial PML so it will remain continuous along its boundary which will result in reflections only due to the discretization of the computation domain. Our PML model is appropriate for other axisymmetric resonators where analytical solutions are not available. In order to test this, we have applied the FEM model to micro-toroid and micro-disc resonators, employing the same PML parameter values that were determined in section 2.1.
Armani et. al [8] demonstrated toroidal micro-cavities (160 − 240µm diameter and 5 − 10µm toroid thickness) which can exhibit quality factors in excess of 10 8 . We tested our model for toroidal micro-cavities of the same geometry and found that the overall quality factor was also greater than 10 8 .
Oxborrow [4] estimated the Qwgm in the range of 1.31 − 3.82 × 10 7 of a certain microdisc resonator. We also calculated the Qwgm of the same micro-disc and found the value of 7.98 × 10 6 which lies within an order of magnitude of Oxborrow's estimate. We believe that our value is a better estimate as Oxborrow's upper and lower bounds are also less accurate for the micro-spheres (Fig. 2). In addition, our finite element model, without any approximation in its master equation and coupled with a PML, does not only give accurate quality factors,   [4] but also determines the important parameters accurately for wide range of applications based on the axisymmetric micro-cavities such as biosensing, non-linear processes, and lasers.