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To deduce the optimum values of the parameters, we then compared the simulation results for Q WGM of spherical cavities with the analytical ones. The simulation results show that a linear i. We have also used these optimum values for the simulations of the microtoroidal cavities. The location of PML is also important to obtain accurate results. After running series of simulations we find the following as optimum values w.

In order to incorporate the PML, we have reformulated Eq. It should be noted that Eq. In our comparison for the analytical results, we have applied Eq. Figure 1 shows a fundamental TE mode of a silica spherical cavity in air. We have plotted the quality factor due to the fundamental TE whispering gallery mode radiation for various sphere diameters at nm.

Similar results are obtained for the TM mode. Figure 2 shows the comparison of the FEM simulation results and results obtained by using analytical expressions presented in section 2. The slight difference between the analytical and the FEM values is attributed to discretization of the computation domain. The accuracy will improve with finer mesh elements see section 3.

One of the possible reasons for this discrepancy may be that while deriving these bounds, in order to simplify the derivation, Oxborrow assumes that the modes are transverse but in reality the modes of these axisymmetric resonators are not perfectly transverse [ 16 ]. Figure 3 shows the results for tunneling distance for both microsphere and microtoroidal cavities. The tunneling distance results are also in agreement with experimental results presented in [ 11 ].

Tunneling distance of a fundamental TE mode as a function of the microcavity geometries. Inset of Fig. Our model is also appropriate for other axisymmetric resonators where analytical solutions are not available. In order to test this, we have used our FEM model to determine the total quality factor of microtoroidal cavities [ 31 ] and have compared the simulation results with experimental measurement of the quality factor.

In order to validate the model with reasonable range of quality factors, it is necessary to ensure suitable experimental conditions in which to observe these quality factors.

We have performed the experiments with the microtoroids immersed in ethanol refractive index: 1. We have preferred ethanol over water as ethanol not only provides low refractive index contrast between the silica cavity and the surroundings but also has low absorption as compared to water at nm [ 33 ]. This will allow us to select a wide range of microtoroidal cavities whose Q total is mainly limited by the Q wgm.

The resonant peaks are observed on a high resolution optical spectrum analyzer Apex AP B, resolution: 0. The tapered fiber is positioned using 10 nm resolution nanostages along the equator of the cavity to couple light into fundamental transverse modes. It is ensured that the tapered fiber does not touch the cavity during the measurements. The comparison between simulation and experimental results is shown in Fig. WGM Quality factors of fundamental TM mode for various silica microtoroidal cavity diameters immersed in ethanol.

For the results presented in Figs. To check numerical accuracy of the model, we have run a convergence test. Convergence of the solution is of order 2 which is expected for the Galerkin method for quadratic Lagrange elements [ 34 ].

It can be seen in the Fig. The results presented in section 3 show that our finite element model is both physically and numerically accurate. The quality factors for all the modes fundamental and higher order modes are also obtained by one single simulation rather than multiple simulations, which was the case for the original Oxborrow model. Moreover, no prior knowledge of any of the mode frequencies is required to obtain the quality factors.

The model also gives fast results and without any fitting algorithms. It should be noted that we have neglected dispersion for estimating the quality factors and tunneling distances.

Whilst dispersion must be considered for very high Q applications where an equidistant modal spectrum is required [ 35 ], it has negligible effect for biosensing applications. We have also applied our model to the PS-CRDS microtoroidal cavity sensor [ 7 ] in order to determine the optimum parameters for an application of refractometric sensing.

This change in refractive index will influence the Q total of the cavity and can easily be measured via PS-CRDS microtoroidal cavity sensor [ 7 ]. Figure 6 shows the modeling results for change in Q total of the cavity as a function of D. Since Q total varies with the wavelength, the optimum geometry will be different for each of the wavelength.

Figure 7 shows that the optimum region exists close to the point of inflection of the Q total curve. Results in Fig. Various quality factors Eq. In the optimum region the Q wgm is close to Q total. Currently the standard experimental approach to determine the total quality factor of a microcavity is either by cavity ring down spectroscopy or Lorentzian fitting of a resonant peak.

However, tunneling distance is also related to WGM radiative quality factor i. This suggests that with model provided here coupled with the experimental procedure outlined in [ 11 ], one can also extract Q wgm even when Q total is greater than 10 8.

Such a measurement can possibly open up the way to a new sensing method for microcavity sensors. We have formulated an empirical expression Eq. This formulation is not surprising as these equations are true only for fundamental modes which lie along the equatorial region of a microcavity.

In summary, our finite element model, without any approximation in its master equation and coupled with a PML, not only gives accurate quality factors and tunneling distances, but can also determine the other important parameters e. We thank Prof. We are grateful to Prof. Andrea Armani at University of Southern California for her invaluable suggestions for improving the manuscript.

We appreciate the efforts of Ashley Maker at University of Southern California, for fabricating the microtoroidal cavities used in this work. We are also grateful to Prof. Tal Carmon at University of Michigan for pointing us to tunneling distance calculations. Vollmer and L. Fan, I. White, S. Shopoua, H. Zhu, J. Suter, and Y. Maleki and V. Nadeau, V.

Ilchenko, D. Kossakovski, G. Bearman, and L. SPIE , 72 Armani and K. Barnes, B. Carver, J. Fraser, G. Gagliardi, H. Loock, Z. Tian, M. Wilson, S. Yam, and O. Express 16 , — [ CrossRef ] [ PubMed ]. Cheema, S. Mehrabani, A. Hayat, Y. Peter, A. Armani, and A. Express 20 , — [ CrossRef ] [ PubMed ].

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