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

Optics Express, Vol. 21, Issue 7, pp. 8724-8735 (2013)

http://dx.doi.org/10.1364/OE.21.008724

Acrobat PDF (892 KB)

### Abstract

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.

© 2013 OSA

## 1. Introduction

2. X. Fan, I. M. White, S. I. Shopoua, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Analytica Chimica Acta **620**, 8–26 (2008) [CrossRef]

5. A. M. Armani and K. J. Vahala, “Heavy water detection using ultra-high-q microcavities,” Opt. Lett. **31**, 1896–1898 (2006) [CrossRef] [PubMed] .

6. J. Barnes, B. Carver, J. M. Fraser, G. Gagliardi, H. P. Loock, Z. Tian, M. W. B. Wilson, S. Yam, and O. Yastrubshak, “Loss determination in microsphere resonators by phase-shift cavity ring-down measurements,” Opt. Express **16**, 13158–13167 (2008) [CrossRef] [PubMed] .

7. M. I. Cheema, S. Mehrabani, A. A. Hayat, Y. A. Peter, A. M. Armani, and A. G. Kirk, “Simultaneous measurement of quality factor and wavelength shift by phase shift microcavity ring down spectroscopy,” Opt. Express **20**, 9090–9098 (2012) [CrossRef] [PubMed] .

*pM*−

*fM*concentrations, and proteins, viruses etc), it is important to design a cavity with the optimum parameters for high sensitivity. One of the goals of the current work, is to rapidly estimate the quality factors of the microcavities with high accuracy to address these applications.

8. C. Spielmann, R. Szipöcs, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. **73**, 2308–2311, (1994) [CrossRef] [PubMed] .

9. P. Pereyra, “Closed formulas for tunneling time in superlattices,” Phys. Rev. Lett. **84**, 1772–1775 (2000) [CrossRef] [PubMed] .

10. V. A. Podolskiy and E. E. Narimanov, “Chaos-assisted tunneling in dielectric microcavities,” Opt. Lett. **30**, (2005) [CrossRef] [PubMed] .

11. M. Tomes, K. J. Vahala, and T. Carmon, “Direct imaging of tunneling from a potential well,” Opt. Express **17**, 19160 (2009) [CrossRef] .

12. O. Chinellato, P. Arbenz, M. Streiff, and A. Witzig, “Computation of optical modes in axisymmetric open cavity resonators,” Future Gener. Comp. Sy. **21**, 1263–1274 (2005) [CrossRef] .

*μm*) it was not adequate for resonators possessing the dimensions typically used in experiments. In 2009, Karl et al. [13] developed a 3D FEM model in JCMsuite for studying a micro-pillar cavity. However, this model does not consider the suppression of false solutions, a well known problem in finite element formulations [14

14. M. Koshiba, K. Hayata, and M. Suzuki, “Improved finite element formulation in terms of the magnetic field vector for dielectric waveguides,” IEEE Trans. on Microwave Theory Tech. **33**, 227–233, (1985) [CrossRef] .

16. M. Oxborrow, “Traceable 2-D finite element simulation of the whispering gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. on Microwave Theory Tech. **55**, 1209–1218 (2007) [CrossRef] .

18. M. Soltani, Q. Li, S. Yegnanarayanan, and A. Adibi, “Toward ultimate miniaturization of high Q silicon traveling-wave microresonators,” Opt. Express **18**, 19541–19557 (2010) [CrossRef] [PubMed] .

12. O. Chinellato, P. Arbenz, M. Streiff, and A. Witzig, “Computation of optical modes in axisymmetric open cavity resonators,” Future Gener. Comp. Sy. **21**, 1263–1274 (2005) [CrossRef] .

12. O. Chinellato, P. Arbenz, M. Streiff, and A. Witzig, “Computation of optical modes in axisymmetric open cavity resonators,” Future Gener. Comp. Sy. **21**, 1263–1274 (2005) [CrossRef] .

**21**, 1263–1274 (2005) [CrossRef] .

16. M. Oxborrow, “Traceable 2-D finite element simulation of the whispering gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. on Microwave Theory Tech. **55**, 1209–1218 (2007) [CrossRef] .

20. M. Oxborrow, J. D. Breeze, and N. M. Alford, “Room-temperature solid-state maser,” Nature **488**, 353–356 (2012) [CrossRef] [PubMed] .

23. Y. F. Xiao, C. L. Zou, B. B. Li, Y. Li, C. H. Dong, Z. F. Han, and Q. Gong, “High-Q exterior whispering-gallery modes in a metal-coated microresonator,” Phys. Rev. Lett. **105**, 153902, (2010) [CrossRef] .

## 2. Mathematical description

16. M. Oxborrow, “Traceable 2-D finite element simulation of the whispering gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. on Microwave Theory Tech. **55**, 1209–1218 (2007) [CrossRef] .

*H⃗*represents the magnetic field of the resonator and

*ϕ*in the axisymmetric resonators, resulting in reduction of the 3D problem to a 2D problem.

### 2.1. Perfectly matched layer formulation

26. F. L. Teixeira and W. C. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microwave Guided Wave Lett. **7**, 371–373, (1997) [CrossRef] .

*t*,

_{rpml}*t*,

_{upml}*t*are the PML thicknesses in the radial, +z and −z directions respectively and

_{lpml}*r*,

_{pml}*z*,

_{upml}*z*are the locations of the start of PML in the radial, +z and −z directions respectively.

_{lpml}*n*is refractive index of the medium,

_{medium}*N*is order of the PML, and

*G*is a positive integer.

*s*,

_{r}*s*,

_{z}*r̃*), 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*of spherical cavities with the analytical ones. The simulation results show that a linear (i.e.

_{WGM}*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 microtoroidal cavities. The location of PML is also important to obtain accurate results. The radial PML should be greater than the tunneling distance (t) of the microcavity and the z PML should be greater than FWHM (

*w*) of WGM along the z axis. After running series of simulations we find the following as optimum values (w.r.t. (

_{z}*r*,

*z*) = (0,0), see Fig. 1):

### 2.2. Finite element method equation with perfectly matched layer

*f*) of all the modes can easily be determined. Quality factor due to the WGM radiation can be calculated as [16

_{r}**55**, 1209–1218 (2007) [CrossRef] .

### 2.3. Analytical expressions of the spherical resonator

#### 2.3.1. Quality factor

*q*root of equation

^{th}*m*can be calculated using the characteristic equation for WGM frequencies [28]: where

*j*,

*h*are Bessel functions,

*k*

_{0}is the wave number (2

*π/λ*

_{0}), and

*a*is the radius of a microsphere.

*Q*of the spherical resonator which requires

_{wgm}*m*≫ 1, however the error is less than 1% for

*m*≥ 19 [29

29. J. R. Buck and H. J. Kimble, “Optimal sizes of dielectric microspheres for cavity QED with strong coupling,” Phys. Rev. A **67**033806 (2003) [CrossRef] .

#### 2.3.2. Tunneling distance

30. B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,”J. Opt. Soc. Am. A **10**, 2, (1993) [CrossRef] .

*E*=

*k*

^{2}is the total energy and Ψ

*is a position probability function of a photon. The potential (*

_{r}*V*) is given by where

_{r}*p*is ratio of relative permittivity of the sphere and the surrounding medium in radial coordinates. Analytically, tunneling distance for a fundamental mode of a microsphere can be written as [30

_{r}30. B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,”J. Opt. Soc. Am. A **10**, 2, (1993) [CrossRef] .

## 3. Results

### 3.1. Comparison between simulations and analytical results

*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 2.3. We have also calculated the minimum and maximum

*Q*values using Oxborrow’s model [16

_{wgm}**55**, 1209–1218 (2007) [CrossRef] .

**55**, 1209–1218 (2007) [CrossRef] .

*D*and

*d*are major and minor diameter of a microtoroid and are shown in Fig. 3(b). The tunneling distance results are also in agreement with experimental results presented in [11

11. M. Tomes, K. J. Vahala, and T. Carmon, “Direct imaging of tunneling from a potential well,” Opt. Express **17**, 19160 (2009) [CrossRef] .

### 3.2. Comparison between simulations and experimental results

31. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature **421**, 925–928 (2003) [CrossRef] [PubMed] .

*Q*) can be represented by:

_{total}*nm*[32

32. S. Scheurich, S. Belle, R. Hellmann, S. So, I.J.G. Sparrow, and G. Emmerson, “Application of a silica-on-silicon planar optical waveguide Bragg grating sensor for organic liquid compound detection,” Proc. SPIE **7356** (2009) [CrossRef]

7. M. I. Cheema, S. Mehrabani, A. A. Hayat, Y. A. Peter, A. M. Armani, and A. G. Kirk, “Simultaneous measurement of quality factor and wavelength shift by phase shift microcavity ring down spectroscopy,” Opt. Express **20**, 9090–9098 (2012) [CrossRef] [PubMed] .

*nm*[33

33. M. Gallignani, S. Garrigues, and M. Guardia, “Direct determination of ethanol in all types of alcoholic beverages by near-infrared derivative spectrometry,”Analyst **118**, (1993) [CrossRef] .

*Q*is mainly limited by the

_{total}*Q*.

_{wgm}*nm*) is used to couple the light into microtoroids via a tapered optical fiber (taper waist: ≤ 1

*μm*). The resonant peaks are observed on a high resolution optical spectrum analyzer (Apex AP 2443B, resolution: 0.16

*pm*) and quality factor is determined by Lorentz curve fitting of the peaks (

*Q*=

*λ*/Δ

*λ*). 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. In order to minimize the effect of

*Q*on

_{coupling}*Q*, the power at the peak wavelength of the source is set around −48

_{total}*dbm*(Optical spectrum analyzer sensitivity: −70

*dbm*) and all the measurements are taken in highly undercoupled regime. The comparison between simulation and experimental results is shown in Fig. 4.

### 3.3. Computational speed and numerical accuracy

*μm*and the number of degrees of freedom of order 10

^{5}. As an example, a microtoroidal cavity model (Computation domain size:25

*μm*× 16

*μm*,

*D*= 30

*μm*,

*d*= 5

*μm*) with the aforementioned statistics took only 35

*s*to find the first 20 eigenvalues on a quad core 64 bit operating system PC.

## 4. Discussion and application to PS-CRDS microcavity sensor

35. V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Dispersion compensation in whispering-gallery modes,” J. Opt. Soc. Am. A **20**, 157–162, (2003) [CrossRef] .

7. M. I. Cheema, S. Mehrabani, A. A. Hayat, Y. A. Peter, A. M. Armani, and A. G. Kirk, “Simultaneous measurement of quality factor and wavelength shift by phase shift microcavity ring down spectroscopy,” Opt. Express **20**, 9090–9098 (2012) [CrossRef] [PubMed] .

*δn*) is introduced. This change in refractive index will influence the

*Q*of the cavity and can easily be measured via PS-CRDS microtoroidal cavity sensor [7

_{total}**20**, 9090–9098 (2012) [CrossRef] [PubMed] .

*Q*of the cavity as a function of

_{total}*D*. It can be clearly seen that to achieve maximum sensitivity there exists an optimum geometry for both

*λ*= 633

*nm*and

*λ*= 1530

*nm*. Due to high water absorption at 1530

*nm*, Δ

*Q*is lower than the one at 633

*nm*. Since

*Q*varies with the wavelength, the optimum geometry will be different for each of the wavelength.

_{total}*Q*measurement, we have plotted the individual terms of Eq. (22) as a function of the cavity geometry (Fig. 7). We have assumed

*Q*= 0 which is a reasonable assumption as experimentally, contribution of

_{coupling}*Q*can be minimized by taking measurements at multiple input power levels [36

_{coupling}36. A. M. Armani, D. K. Armani, B. Min, K. J. Vahala, and S. M. Spillane, “Ultra-high-Q microcavity operation in *H*_{2}*O* and *D*_{2}*O*, Appl. Phys. Lett. **87**, (2005) [CrossRef] .

*Q*curve. Results in Fig. 7 show that as diameter increases, the WGM is better confined within the microcavity (see Fig. 1(a)) and so

_{total}*Q*is less influenced by the external environment, with the result that the total quality factor is limited by silica absorption.

_{wgm}*D*+

*d*). This formulation is not surprising as these equations are true only for fundamental modes which lie along the equatorial region of a microcavity. However, this treatment will not be true for higher order modes (non equatorial modes) as the curvature for a microsphere of diameter

*D*+

*d*will be quite different from a microtoroid with

*D*and

*d*as its dimensions (see Fig. 3(b)).

## Acknowledgments

## References and links

1. | F. Vollmer and L. Yang, “Label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophotonics 267–291 (2012). |

2. | X. Fan, I. M. White, S. I. Shopoua, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Analytica Chimica Acta |

3. | L. Maleki and V. S. Ilchenko, “Techniques and devices for sensing a sample by using a whispering gallery mode resonator,” California Institute of Technology, US Patent 6,490,039, (2002). |

4. | J. L. Nadeau, V. S. Ilchenko, D. Kossakovski, G. H. Bearman, and L. Maleki, “High-Q whispering-gallery mode sensor in liquids,” Proc. SPIE |

5. | A. M. Armani and K. J. Vahala, “Heavy water detection using ultra-high-q microcavities,” Opt. Lett. |

6. | J. Barnes, B. Carver, J. M. Fraser, G. Gagliardi, H. P. Loock, Z. Tian, M. W. B. Wilson, S. Yam, and O. Yastrubshak, “Loss determination in microsphere resonators by phase-shift cavity ring-down measurements,” Opt. Express |

7. | M. I. Cheema, S. Mehrabani, A. A. Hayat, Y. A. Peter, A. M. Armani, and A. G. Kirk, “Simultaneous measurement of quality factor and wavelength shift by phase shift microcavity ring down spectroscopy,” Opt. Express |

8. | C. Spielmann, R. Szipöcs, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. |

9. | P. Pereyra, “Closed formulas for tunneling time in superlattices,” Phys. Rev. Lett. |

10. | V. A. Podolskiy and E. E. Narimanov, “Chaos-assisted tunneling in dielectric microcavities,” Opt. Lett. |

11. | M. Tomes, K. J. Vahala, and T. Carmon, “Direct imaging of tunneling from a potential well,” Opt. Express |

12. | O. Chinellato, P. Arbenz, M. Streiff, and A. Witzig, “Computation of optical modes in axisymmetric open cavity resonators,” Future Gener. Comp. Sy. |

13. | M. Karl, B. Kettner, S. Burger, F. Schmidt, H. Kalt, and M. Hetterich, “Dependencies of micropillar cavity quality factors calculated with finite element methods,” Opt. Express |

14. | M. Koshiba, K. Hayata, and M. Suzuki, “Improved finite element formulation in terms of the magnetic field vector for dielectric waveguides,” IEEE Trans. on Microwave Theory Tech. |

15. | R. A. Osegueda, J. H. Pierluissi, L. M. Gil, A. Revilla, G. J. Villalva, G. J. Dick, D. G. Santiago, and R. T. Wang, “Azimuthally dependent finite element solution to the cylindrical resonator,”10th annual review of progress in applied computational electromagnetics |

16. | M. Oxborrow, “Traceable 2-D finite element simulation of the whispering gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. on Microwave Theory Tech. |

17. | S. M. Spillane, “Fiber-coupled ultra-high-Q microresonators for nonlinear and quantum optics,” PhD. thesis, CALTECH (2004) |

18. | M. Soltani, Q. Li, S. Yegnanarayanan, and A. Adibi, “Toward ultimate miniaturization of high Q silicon traveling-wave microresonators,” Opt. Express |

19. | |

20. | M. Oxborrow, J. D. Breeze, and N. M. Alford, “Room-temperature solid-state maser,” Nature |

21. | C. Shi, H. S. Choi, and A. M. Armani, “Optical microcavities with a thiol-functionalized gold nanoparticle polymer thin film coating,” Appl. Phys. Lett. |

22. | T. Lu, H. Lee, T. Chen, S. Herchak, J. H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. (2011) [CrossRef] . |

23. | Y. F. Xiao, C. L. Zou, B. B. Li, Y. Li, C. H. Dong, Z. F. Han, and Q. Gong, “High-Q exterior whispering-gallery modes in a metal-coated microresonator,” Phys. Rev. Lett. |

24. | M. I. Cheema and A. G. Kirk, “Implementation of the perfectly matched layer to determine the quality factor of axisymmetric resonators in COMSOL,” in COMSOL conference,Boston, Oct 8 2010. |

25. | A. Arbabi, Y. M. Kang, and L. L. Goddard, “Analysis and Design of a Microring Inline Single Wavelength Reflector,” in FIO , October24, 2010. |

26. | F. L. Teixeira and W. C. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microwave Guided Wave Lett. |

27. | V. V. Datsyuk, “Some characteristics of resonant electromagnetic modes in a dielectric sphere,”Appl. Phys. |

28. | L. A. Weinstein, |

29. | J. R. Buck and H. J. Kimble, “Optimal sizes of dielectric microspheres for cavity QED with strong coupling,” Phys. Rev. A |

30. | B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,”J. Opt. Soc. Am. A |

31. | D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature |

32. | S. Scheurich, S. Belle, R. Hellmann, S. So, I.J.G. Sparrow, and G. Emmerson, “Application of a silica-on-silicon planar optical waveguide Bragg grating sensor for organic liquid compound detection,” Proc. SPIE |

33. | M. Gallignani, S. Garrigues, and M. Guardia, “Direct determination of ethanol in all types of alcoholic beverages by near-infrared derivative spectrometry,”Analyst |

34. | A. Quarteroni and A. Valli, |

35. | V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Dispersion compensation in whispering-gallery modes,” J. Opt. Soc. Am. A |

36. | A. M. Armani, D. K. Armani, B. Min, K. J. Vahala, and S. M. Spillane, “Ultra-high-Q microcavity operation in |

**OCIS Codes**

(280.1415) Remote sensing and sensors : Biological sensing and sensors

(280.1545) Remote sensing and sensors : Chemical analysis

(140.3945) Lasers and laser optics : Microcavities

(140.3948) Lasers and laser optics : Microcavity devices

(280.4788) Remote sensing and sensors : Optical sensing and sensors

**ToC Category:**

Sensors

**History**

Original Manuscript: January 11, 2013

Revised Manuscript: March 12, 2013

Manuscript Accepted: March 17, 2013

Published: April 2, 2013

**Virtual Issues**

Vol. 8, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

M. Imran Cheema and Andrew G. Kirk, "Accurate determination of the quality factor and tunneling distance of axisymmetric resonators for biosensing applications," Opt. Express **21**, 8724-8735 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-8724

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### References

- F. Vollmer and L. Yang, “Label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophotonics267–291 (2012).
- X. Fan, I. M. White, S. I. Shopoua, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Analytica Chimica Acta620, 8–26 (2008) [CrossRef]
- L. Maleki and V. S. Ilchenko, “Techniques and devices for sensing a sample by using a whispering gallery mode resonator,” California Institute of Technology, US Patent 6,490,039, (2002).
- J. L. Nadeau, V. S. Ilchenko, D. Kossakovski, G. H. Bearman, and L. Maleki, “High-Q whispering-gallery mode sensor in liquids,” Proc. SPIE4629, 72 (2002).
- A. M. Armani and K. J. Vahala, “Heavy water detection using ultra-high-q microcavities,” Opt. Lett.31, 1896–1898 (2006). [CrossRef] [PubMed]
- J. Barnes, B. Carver, J. M. Fraser, G. Gagliardi, H. P. Loock, Z. Tian, M. W. B. Wilson, S. Yam, and O. Yastrubshak, “Loss determination in microsphere resonators by phase-shift cavity ring-down measurements,” Opt. Express16, 13158–13167 (2008). [CrossRef] [PubMed]
- M. I. Cheema, S. Mehrabani, A. A. Hayat, Y. A. Peter, A. M. Armani, and A. G. Kirk, “Simultaneous measurement of quality factor and wavelength shift by phase shift microcavity ring down spectroscopy,” Opt. Express20, 9090–9098 (2012). [CrossRef] [PubMed]
- C. Spielmann, R. Szipöcs, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett.73, 2308–2311, (1994). [CrossRef] [PubMed]
- P. Pereyra, “Closed formulas for tunneling time in superlattices,” Phys. Rev. Lett.84, 1772–1775 (2000). [CrossRef] [PubMed]
- V. A. Podolskiy and E. E. Narimanov, “Chaos-assisted tunneling in dielectric microcavities,” Opt. Lett.30, (2005). [CrossRef] [PubMed]
- M. Tomes, K. J. Vahala, and T. Carmon, “Direct imaging of tunneling from a potential well,” Opt. Express17, 19160 (2009). [CrossRef]
- O. Chinellato, P. Arbenz, M. Streiff, and A. Witzig, “Computation of optical modes in axisymmetric open cavity resonators,” Future Gener. Comp. Sy.21, 1263–1274 (2005). [CrossRef]
- M. Karl, B. Kettner, S. Burger, F. Schmidt, H. Kalt, and M. Hetterich, “Dependencies of micropillar cavity quality factors calculated with finite element methods,” Opt. Express2, (2009).
- M. Koshiba, K. Hayata, and M. Suzuki, “Improved finite element formulation in terms of the magnetic field vector for dielectric waveguides,” IEEE Trans. on Microwave Theory Tech.33, 227–233, (1985). [CrossRef]
- R. A. Osegueda, J. H. Pierluissi, L. M. Gil, A. Revilla, G. J. Villalva, G. J. Dick, D. G. Santiago, and R. T. Wang, “Azimuthally dependent finite element solution to the cylindrical resonator,”10th annual review of progress in applied computational electromagnetics1, 159–170, (1994).
- M. Oxborrow, “Traceable 2-D finite element simulation of the whispering gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. on Microwave Theory Tech.55, 1209–1218 (2007). [CrossRef]
- S. M. Spillane, “Fiber-coupled ultra-high-Q microresonators for nonlinear and quantum optics,” PhD. thesis, CALTECH (2004)
- M. Soltani, Q. Li, S. Yegnanarayanan, and A. Adibi, “Toward ultimate miniaturization of high Q silicon traveling-wave microresonators,” Opt. Express18, 19541–19557 (2010). [CrossRef] [PubMed]
- https://sites.google.com/site/axisymmetricmarkoxborrow/home
- M. Oxborrow, J. D. Breeze, and N. M. Alford, “Room-temperature solid-state maser,” Nature488, 353–356 (2012). [CrossRef] [PubMed]
- C. Shi, H. S. Choi, and A. M. Armani, “Optical microcavities with a thiol-functionalized gold nanoparticle polymer thin film coating,” Appl. Phys. Lett.100, 013305 (2012). [CrossRef]
- T. Lu, H. Lee, T. Chen, S. Herchak, J. H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. (2011). [CrossRef]
- Y. F. Xiao, C. L. Zou, B. B. Li, Y. Li, C. H. Dong, Z. F. Han, and Q. Gong, “High-Q exterior whispering-gallery modes in a metal-coated microresonator,” Phys. Rev. Lett.105, 153902, (2010). [CrossRef]
- M. I. Cheema and A. G. Kirk, “Implementation of the perfectly matched layer to determine the quality factor of axisymmetric resonators in COMSOL,” in COMSOL conference,Boston, Oct 8 2010.
- A. Arbabi, Y. M. Kang, and L. L. Goddard, “Analysis and Design of a Microring Inline Single Wavelength Reflector,” in FIO, October24, 2010.
- F. L. Teixeira and W. C. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microwave Guided Wave Lett.7, 371–373, (1997). [CrossRef]
- V. V. Datsyuk, “Some characteristics of resonant electromagnetic modes in a dielectric sphere,”Appl. Phys.B54, 184–187, (1992).
- L. A. Weinstein, Open Resonators and Open Waveguides (The Golem Press, Boulder, Colorado, USA, 1969) pp. 298.
- J. R. Buck and H. J. Kimble, “Optimal sizes of dielectric microspheres for cavity QED with strong coupling,” Phys. Rev. A67033806 (2003). [CrossRef]
- B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,”J. Opt. Soc. Am. A10, 2, (1993). [CrossRef]
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