## Full-vectorial whispering-gallery-mode cavity analysis |

Optics Express, Vol. 21, Issue 19, pp. 22012-22022 (2013)

http://dx.doi.org/10.1364/OE.21.022012

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### Abstract

We present a full-vectorial three-dimensional whispering-gallery-mode microcavity analysis technique. With this technique, optical properties such as resonance wavelength, quality factor, and electromagnetic field distribution of a microcavity in the presence of individual nanoparticle adsorption can be simulated with high accuracy, even in the presence of field distortion from plasmon effects at a wavelength close to plasmon resonance. This formulation is applicable to a wide variety of whispering-gallery related problems, such as waveguide to cavity coupling and full wave propagation analysis of a general whispering-gallery-mode microcavity where axisymmetry along the azimuthal direction is not required.

© 2013 OSA

## 1. Introduction

1. K. Vahala, “Optical microcavities,” Nature **424**, 839–846 (2003). [CrossRef] [PubMed]

8. M. R. Lee and P. M. Fauchet, “Nanoscale microcavity sensor for single particle detection,” Opt. Lett. **32**, 3284–3286 (2007). [CrossRef] [PubMed]

9. 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]

10. S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold raman laser using a spherical dielectric microcavity,” Nature **415**, 621–623 (2002). [CrossRef] [PubMed]

1. K. Vahala, “Optical microcavities,” Nature **424**, 839–846 (2003). [CrossRef] [PubMed]

11. M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. **21**, 453–455 (1996). [CrossRef] [PubMed]

12. S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in microspheres by protein adsorption,” Opt. Lett. **28**, 272–274 (2003). [CrossRef] [PubMed]

13. 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. U. S. A. (2011). [CrossRef]

14. B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface plasmon polariton whispering gallery microcavity,” Nature **457**, 455–458 (2009). [CrossRef] [PubMed]

17. M. A. Santiago-Cordoba, M. Cetinkaya, S. V. Boriskina, F. Vollmer, and M. C. Demirel, “Ultrasensitive detection of a protein by optical trapping in a photonic-plasmonic microcavity,” J. Biophotonics **5**, 629–638 (2012). [CrossRef]

31. P. Bienstman and R. Baets, “Optical modelling of photonic crystals and vcsels using eigenmode expansion and perfectly matched layers,” Opt. Quantum. Electron. **33**, 327–341 (2001). [CrossRef]

34. J. Zheng and M. Yu, “Rigorous mode-matching method of circular to off-center rectangular side-coupled waveguide junctions for filter applications,” IEEE Trans. Microwave Theory Tech. **55**, 2365–2373 (2007). [CrossRef]

## 2. Theoretical formulations

### 2.1. Ideal whispering-gallery-mode microcavities

*ñ*(

*ρ*,

*z*,

*ϕ*) independent of the azimuthal angle

*ϕ*(ie.

*ñ*(

*ρ*,

*z*,

*ϕ*) =

*ñ*(

*ρ*,

*z*)) in a concentric cylindrical coordinate system, as shown in Fig. 1(a). The electric field

**Ẽ**(

*ρ*,

*z*,

*ϕ*) of light at wavelength

*λ*

_{0}that propagates inside the cavity has the form where

*ϕ̂*direction. That is,

*Ã*(

*ϕ*)

^{*}·

*Ã*(

*ϕ*) represents the power at

*ϕ*in units of Watts. Here

*η*

_{0}= 377 Ohms is the free space impedance and

*ñ*(

_{r}*ρ*,

*z*) is the real part of the refractive index at

*ϕ. m̃*=

*m̃*+

_{r}*jm̃*is a constant complex number whose real part

_{i}*m̃*encompasses the phase change of the wave-front and the imaginary part

_{r}*m̃*characterizes the loss of the wave along the propagation direction. In the case of an ideal WGM cavity,

_{i}*Ã*(

*ϕ*) =

*Ã*(

*ϕ*= 0)

*e*. When

^{jm̃ϕ}*λ*

_{0}coincides with the cavity resonance wavelength

*λ̃*,

_{res}*m̃*becomes an integer

_{r}*M*whose value determines the azimuthal order of the mode. The loss of the cavity can be estimated by the quality factor

*Q*=

_{tot}*M*/2

*m̃*with the value contributed by cavity material absorption (

_{i}*Q̃*=

_{abs}*M*/2

*m̃*), radiation loss (

_{abs}*Q̃*=

_{rad}*M*/2

*m̃*), surface roughness (

_{rad}*Q̃*=

_{surf}*M*/2

*m̃*), and cavity to tapered waveguide coupling (

_{surf}*Q̃*=

_{couple}*M*/2

*m̃*). Evidently,

_{couple}### 2.2. Non-ideal whispering-gallery-mode microcavities

*ϕ*-dependent refractive index profile

*n*(

*ρ*,

*z*,

*ϕ*) if particle binding occurs, surface imperfections exist, a waveguide taper is placed close to the cavity, or other geometrical disturbances are introduced. For simplicity, we drop ∼ symbols for all the physical quantities to distinguish them from those defined in an ideal WGM.

*ϕ*

_{0}, the electrical field distribution at the cross section

**E**(

*ρ*,

*z*,

*ϕ*

_{0}) can be expanded onto the normalized WGM mode

**ê**(

*ρ*,

*z*,

*ϕ*

_{0}) at

*ϕ*

_{0}according to Note that by analogy to the MMM in Cartesian coordinates, we may obtain a complete set of orthogonal modes at the azimuthal angle

*ϕ*

_{0}that satisfies the mode equation of a perfect WGM cavity where both the mode profile and

*m*have become

*ϕ*-dependent. Additionally, given that (5) is identical to (2) for any fixed angle

*ϕ*

_{0}, one may find the

*M*

^{th}order resonant wavelength

*λ*(

_{r}*ϕ*) at

*ϕ*

_{0}by replacing

*k*

_{0}with 2

*π*/

*λ*,

_{r}*m*with

*M*, and solving the mode equation above using a two-dimensional mode solver for ideal WGM’s (such as that in [20

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

*λ*(

_{r}*ϕ*) is also a

*ϕ*-dependent quantity. After propagating an infinitesimal azimuthal angle

*δϕ*(Fig. 1(b)), we obtain the electrical field at

*ϕ*

_{0}+

*δϕ*Under the approximation that, on average, photons in the mode travel the same optical path length and experience the same loss as those at the resonance wavelength

*λ*(

_{r}*ϕ*

_{0}), we can obtain the real

*m*and imaginary

_{r}*m*part of

_{i}*m*(

*ϕ*

_{0}) after infinitesimal rotation

*δϕ*via Here, we ignore

*m*and

_{couple}*m*for a purpose explained in a later section. On the other hand,

_{surf}**E**(

*ϕ*

_{0}+

*δϕ*) can be expanded onto the normalized eigen mode

**ê**(

*ρ*,

*z*,

*ϕ*

_{0}+

*δϕ*), defined at

*ϕ*

_{0}+

*δϕ*,

*ϕ*

_{0}+

*δϕ*, we obtain the evolution of

*A*(

*ϕ*

_{0}+

*δϕ*) according to

*m*(

_{m}*ϕ*

_{0}) characterized by arising from the mode mismatch between

*ê*(

*ϕ*

_{0}+

*δϕ*) and

*ê*(

*ϕ*

_{0}) in addition to the absorption and radiation loss derived from

*m*. For simplicity, we redefine

_{i}*m*(

_{i}*ϕ*

_{0}) =

*m*(

_{m}*ϕ*

_{0}) +

*m*(

_{abs}*ϕ*

_{0}) +

*m*(

_{rad}*ϕ*

_{0}) and define a mode order detuning term

*δm*(

*ϕ*) = [

*m*(

_{r}*ϕ*)−

*M*]+

*jm*(

_{i}*ϕ*). Consequently, the field

**E**(

*ρ*,

*z*, 0) =

*A*(0)

**ê**(

*ρ*,

*z*, 0) propagating from

*ϕ*= 0 to an azimuthal angle

*ϕ*

_{0}can be expressed as

*π*azimuthal angle should be 2

*Mπ*:

*λ*by simply taking the arithmetic mean of the resonance wavelengths

_{res}*λ*(

_{r}*ϕ*) at each cross section of the cavity, where

*λ*(

_{r}*ϕ*

_{0}) at each

*ϕ*

_{0}can be obtained from the WGM mode solver for the case of an ideal WGM whose cross-sectional refractive index profile is identical to that at

*ϕ*

_{0}.

*δm*as

*Q*factor from the inclusion of the mode mismatch

_{tot}*Q*.

_{m}## 3. Application and discussion

### 3.1. The ideal WGM cavity

*λ*is identical at any azimuthal angle

_{r}*ϕ*and the cavity resonance wavelength

*λ*=

_{res}*λ*. Therefore, an ideal WGM cavity can be treated as a special case under the current MMM formulation.

_{r}*ϕ*

_{0}and

*ϕ*

_{0}+ Δ

*ϕ*, we obtain a new formulation of coupling Q as By treating the tapered waveguide as part of an asymmetric WGM cavity, one may easily obtain a resonance wavelength shift Δ

*λ*=

*λ*−

_{res}*λ̃*due to the perturbation of the waveguide taper to the cavity according to

_{res}11. M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. **21**, 453–455 (1996). [CrossRef] [PubMed]

*π*azimuthal angle must be performed. Evidently, the surface roughness could perturb optical path length and yield an addition shift of cavity resonance wavelength. This shift, however, would usually be small in magnitude.

### 3.2. Nanoparticles on a WGM cavity

*N*nanoparticles land on the surface of a WGM cavity. Here we assume each nanoparticle

*i*occupies a space lying between

*ϕ*and

_{i}*ϕ*+ Δ

_{i}*ϕ*. From (13) and (14) we derive that the particle induced resonance wavelength shift Δ

_{i}*λ*and quality factor Q change follows the expression

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

*μ*m and a minor radius of 5

*μ*m. The surrounding environment is filled with water. Polystyrene (PS) beads and gold (Au) beads are individually placed on the toroid equator and the refractive indices of silica (1.457+(6.95 × 10

^{−11})

*j*), water (1.33168+(1.47×10

^{−8})

*j*), PS (1.583+(5.29×10

^{−4})

*j*), and Au (0.1834+3.433

*j*) are taken from literature [35

35. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. **55**, 1205–1208 (1965). [CrossRef]

38. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B: Condens. Matter Mater. Phys. **6**, 4370–4379 (1972). [CrossRef]

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

25. A. Kaplan, M. Tomes, T. Carmon, M. Kozlov, O. Cohen, G. Bartal, and H. G. L. Schwefel, “Finite element simulation of a perturbed axial-symmetric whispering-gallery mode and its use for intensity enhancement with a nanoparticle coupled to a microtoroid,” Opt. Express **21**, 14169–14180 (2013). [CrossRef] [PubMed]

25. A. Kaplan, M. Tomes, T. Carmon, M. Kozlov, O. Cohen, G. Bartal, and H. G. L. Schwefel, “Finite element simulation of a perturbed axial-symmetric whispering-gallery mode and its use for intensity enhancement with a nanoparticle coupled to a microtoroid,” Opt. Express **21**, 14169–14180 (2013). [CrossRef] [PubMed]

25. A. Kaplan, M. Tomes, T. Carmon, M. Kozlov, O. Cohen, G. Bartal, and H. G. L. Schwefel, “Finite element simulation of a perturbed axial-symmetric whispering-gallery mode and its use for intensity enhancement with a nanoparticle coupled to a microtoroid,” Opt. Express **21**, 14169–14180 (2013). [CrossRef] [PubMed]

18. I. Teraoka, S. Arnold, and F. Vollmer, “Perturbation approach to resonance shifts of whispering-gallery modes in a dielectric microsphere as a probe of a surrounding medium,” J. Opt. Soc. Am. B **20**, 1937–1946 (2003). [CrossRef]

*ẑ*direction displayed for the TE mode). Due to the coupling of the WGM and the SP, the field intensity outside the cavity is stronger than that inside the cavity. Such significant distortion to the WGM precludes the treatment of such problems with first order perturbation theory. On the other hand, MMM intrinsically incorporates the change in mode field distribution and yields accurate evaluations of the perturbed system.

*m*,

_{r}*m*, and

_{m}*m*arising from the PS (Fig. 3(a)) and gold (Fig. 3(b)) nanoparticles. It is observed that the mode mismatch loss dominates the degradation of the Q factor over material absorption in both cases.

_{abs}*δϕ*to the expected result obtained through the Richardson extrapolation procedure. One can observe that the simulated resonance wavelength shift converges on the order of

*O*(

*δϕ*

^{2}) (black line in Fig. 4(a)) while the convergence rate of Q is

*O*(

*δϕ*

^{0.9})(black line in Fig. 4(b)). A higher convergence rate is possible by implementing higher order integration procedures to evaluate (17) and (18).

^{9}. The resonance shifts predicted by MMM (red solid square markers) are compared with those predicted by a first order perturbation method (black plus symbol markers). For PS beads, MMM predicts wavelength shifts which are in good agreement with those predicted by the first order perturbation method. The quality factor (blue solid circle markers) of the toroid is unaffected when a bead with radius smaller than 5 nm is attached to the surface, yet there is a drop to approximately 10

^{7}if a 50-nm radius bead is attached instead. The calculated wavelength shift and quality factor degradation is in line with the experimental observation reported in [13

13. 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. U. S. A. (2011). [CrossRef]

*α*= 4

*πR*

^{3}

*ε*(

_{m}*ε*−

_{p}*ε*)/(

_{m}*ε*+ 2

_{p}*ε*) [39] where

_{m}*ε*and

_{p}*ε*are the permittivities of the particle and the medium, respectively. MMM predicts resonance shifts from 2×10

_{m}^{−2}to 359 fm and Q factor degradation from 3 × 10

^{8}to 7.5 × 10

^{4}when the attached gold bead radius increases from 2.5 nm to 50 nm. As expected, at an off-plasmon resonance wavelength of 633 nm, the perturbation method matches the MMM results when the bead size is below 20 nm yet for large beads yields greater errors due to the non-negligible field distortion from the bead.

19. M. R. Foreman and F. Vollmer, “Theory of resonance shifts of whispering gallery modes by arbitrary plasmonic nanoparticles,” New J. Phys. **15**, 083006 (2013). [CrossRef]

*μ*m radius microsphere with a 4 nm inter-particle gap as reported in [16

16. M. Santiago-Cordoba, S. Boriskina, F. Vollmer, and M. Demirel, “Nanoparticle-based protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. **99**, 073701 (2011). [CrossRef]

17. M. A. Santiago-Cordoba, M. Cetinkaya, S. V. Boriskina, F. Vollmer, and M. C. Demirel, “Ultrasensitive detection of a protein by optical trapping in a photonic-plasmonic microcavity,” J. Biophotonics **5**, 629–638 (2012). [CrossRef]

16. M. Santiago-Cordoba, S. Boriskina, F. Vollmer, and M. Demirel, “Nanoparticle-based protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. **99**, 073701 (2011). [CrossRef]

^{6}was reported in the same paper, we estimated that the quality factor will drop to 5.8 × 10

^{5}by the adsorption of 10 beads. This value is in good agreement with the reported experimental value on the order of 10

^{5}.

## 4. Conclusion

## References and links

1. | K. Vahala, “Optical microcavities,” Nature |

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

3. | S. I. Shopova, R. Rajmangal, Y. Nishida, and S. Arnold, “Ultrasensitive nanoparticle detection using a portable whispering gallery mode biosensor driven by a periodically poled lithium-niobate frequency doubled distributed feedback laser,” Rev. Sci. Instrum. |

4. | J. Dominguez-Juarez, G. Kozyreff, and J. Martorell, “Whispering gallery microresonators for second harmonic light generation from a low number of small molecules,” Nat. Commun. |

5. | H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics |

6. | Y. Sun, J. Liu, G. Frye-Mason, S.-j. Ja, A. K. Thompson, and X. Fan, “Optofluidic ring resonator sensors for rapid dnt vapor detection,” Analyst |

7. | G. Bahl, X. Fan, and T. Carmon, “Acoustic whispering-gallery modes in optomechanical shells,” New J. Phys. |

8. | M. R. Lee and P. M. Fauchet, “Nanoscale microcavity sensor for single particle detection,” Opt. Lett. |

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

10. | S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold raman laser using a spherical dielectric microcavity,” Nature |

11. | M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. |

12. | S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in microspheres by protein adsorption,” Opt. Lett. |

13. | 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. U. S. A. (2011). [CrossRef] |

14. | B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface plasmon polariton whispering gallery microcavity,” Nature |

15. | S. I. Shopova, R. Rajmangal, S. Holler, and S. Arnold, “Plasmonic enhancement of a whispering-gallery-mode biosensor for single nanoparticle detection,” Appl. Phys. Lett. |

16. | M. Santiago-Cordoba, S. Boriskina, F. Vollmer, and M. Demirel, “Nanoparticle-based protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. |

17. | M. A. Santiago-Cordoba, M. Cetinkaya, S. V. Boriskina, F. Vollmer, and M. C. Demirel, “Ultrasensitive detection of a protein by optical trapping in a photonic-plasmonic microcavity,” J. Biophotonics |

18. | I. Teraoka, S. Arnold, and F. Vollmer, “Perturbation approach to resonance shifts of whispering-gallery modes in a dielectric microsphere as a probe of a surrounding medium,” J. Opt. Soc. Am. B |

19. | M. R. Foreman and F. Vollmer, “Theory of resonance shifts of whispering gallery modes by arbitrary plasmonic nanoparticles,” New J. Phys. |

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

21. | J. D. Swaim, J. Knittel, and W. P. Bowen, “Detection limits in whispering gallery biosensors with plasmonic enhancement,” Appl. Phys. Lett. |

22. | J. Y. Lee, X. Luo, and A. W. Poon, “Reciprocal transmissions and asymmetric modal distributions in waveguide-coupled spiral-shaped microdisk resonators,” Opt. Express |

23. | X. Luo and A. W. Poon, “Coupled spiral-shaped microdisk resonatorswith non-evanescent asymmetric inter-cavitycoupling,” Opt. Express |

24. | A. Massaro, V. Errico, T. Stomeo, R. Cingolani, A. Salhi, A. Passaseo, and M. De Vittorio, “3-d fem modeling and fabrication of circular photonic crystal microcavity,” J. Lightwave Technol. |

25. | A. Kaplan, M. Tomes, T. Carmon, M. Kozlov, O. Cohen, G. Bartal, and H. G. L. Schwefel, “Finite element simulation of a perturbed axial-symmetric whispering-gallery mode and its use for intensity enhancement with a nanoparticle coupled to a microtoroid,” Opt. Express |

26. | J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A: Pure Appl. Opt. |

27. | C.-L. Zou, Y. Yang, Y.-F. Xiao, C.-H. Dong, Z.-F. Han, and G.-C. Guo, “Accurately calculating high quality factor of whispering-gallery modes with boundary element method,” J. Opt. Soc. Am. B |

28. | C.-L. Zou, H. G. L. Schwefel, F.-W. Sun, Z.-F. Han, and G.-C. Guo, “Quick root searching method for resonances of dielectric optical microcavities with the boundary element method,” Opt. Express |

29. | T. Lu and D. Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” J. Light-wave Technol. |

30. | T. Lu and D. Yevick, “Boundary element analysis of dielectric waveguides,” J. Opt. Soc. Am. A: |

31. | P. Bienstman and R. Baets, “Optical modelling of photonic crystals and vcsels using eigenmode expansion and perfectly matched layers,” Opt. Quantum. Electron. |

32. | K. Jiang and W.-P. Huang, “Finite-difference-based mode-matching method for 3-d waveguide structures under semivectorial approximation,” J. Lightwave Technol. |

33. | J. Mu and W.-P. Huang, “Simulation of three-dimensional waveguide discontinuities by a full-vector mode-matching method based on finite-difference schemes,” Opt. Express |

34. | J. Zheng and M. Yu, “Rigorous mode-matching method of circular to off-center rectangular side-coupled waveguide junctions for filter applications,” IEEE Trans. Microwave Theory Tech. |

35. | I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. |

36. | G. M. Hale and M. R. Querry, “Optical constants of water in the 200-nm to 200-μm wavelength region,” Appl. Opt. |

37. | X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. |

38. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B: Condens. Matter Mater. Phys. |

39. | S. A. Maier, |

**OCIS Codes**

(040.1880) Detectors : Detection

(230.3990) Optical devices : Micro-optical devices

(230.5750) Optical devices : Resonators

**ToC Category:**

Optical Devices

**History**

Original Manuscript: July 23, 2013

Revised Manuscript: August 28, 2013

Manuscript Accepted: August 29, 2013

Published: September 11, 2013

**Citation**

Xuan Du, Serge Vincent, and Tao Lu, "Full-vectorial whispering-gallery-mode cavity analysis," Opt. Express **21**, 22012-22022 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-19-22012

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### References

- K. Vahala, “Optical microcavities,” Nature424, 839–846 (2003). [CrossRef] [PubMed]
- D. Armani, T. Kippenberg, S. Spillane, and K. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421, 925–928 (2003). [CrossRef] [PubMed]
- S. I. Shopova, R. Rajmangal, Y. Nishida, and S. Arnold, “Ultrasensitive nanoparticle detection using a portable whispering gallery mode biosensor driven by a periodically poled lithium-niobate frequency doubled distributed feedback laser,” Rev. Sci. Instrum.81, 103110 (2010). [CrossRef] [PubMed]
- J. Dominguez-Juarez, G. Kozyreff, and J. Martorell, “Whispering gallery microresonators for second harmonic light generation from a low number of small molecules,” Nat. Commun.2, 1–8 (2010).
- H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics6, 369–373 (2012). [CrossRef]
- Y. Sun, J. Liu, G. Frye-Mason, S.-j. Ja, A. K. Thompson, and X. Fan, “Optofluidic ring resonator sensors for rapid dnt vapor detection,” Analyst134, 1386–1391 (2009). [CrossRef] [PubMed]
- G. Bahl, X. Fan, and T. Carmon, “Acoustic whispering-gallery modes in optomechanical shells,” New J. Phys.14, 115026 (2012). [CrossRef]
- M. R. Lee and P. M. Fauchet, “Nanoscale microcavity sensor for single particle detection,” Opt. Lett.32, 3284–3286 (2007). [CrossRef] [PubMed]
- 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]
- S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold raman laser using a spherical dielectric microcavity,” Nature415, 621–623 (2002). [CrossRef] [PubMed]
- M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett.21, 453–455 (1996). [CrossRef] [PubMed]
- S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in microspheres by protein adsorption,” Opt. Lett.28, 272–274 (2003). [CrossRef] [PubMed]
- 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. U. S. A.(2011). [CrossRef]
- B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface plasmon polariton whispering gallery microcavity,” Nature457, 455–458 (2009). [CrossRef] [PubMed]
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