Abstract

We simulate light propagation in perturbed whispering-gallery mode microcavities using a two-dimensional finite-difference beam propagation method in a cylindrical coordinate system. Optical properties of whispering-gallery microcavities perturbed by polystyrene nanobeads are investigated through this formulation. The light perturbation as well as quality factor degradation arising from cavity ellipticity are also studied.

© 2013 Optical Society of America

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  1. M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett.21, 453–455 (1996).
    [CrossRef] [PubMed]
  2. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421, 925–928 (2003).
    [CrossRef] [PubMed]
  3. F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80, 4057–4059 (2002).
    [CrossRef]
  4. F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: labelfree detection down to single molecules,” Nat. Methods5, 591–596 (2008).
    [CrossRef] [PubMed]
  5. T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. U. S. A.108, 5976–5979 (2011).
    [CrossRef] [PubMed]
  6. 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).
  7. J. Knittel, T. G. McRae, K. H. Lee, and W. P. Bowen, “Interferometric detection of mode splitting for whispering gallery mode biosensors,” Appl. Phys. Lett.97, 1–3 (2010).
    [CrossRef]
  8. Y. Sun and X. Fan, “Optical ring resonators for biochemical and chemical sensing,” Anal. Bioanal.Chem.399, 205–211 (2011).
    [CrossRef]
  9. S. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold raman laser using a spherical dielectric mcirocavity,” Nature415, 621–623 (2002).
    [CrossRef] [PubMed]
  10. B. Min, T. J. Kippenberg, and K. J. Vahala, “Compact, fiber-compatible, cascaded raman laser,” Opt. Lett.28, 1507–1509 (2003).
    [CrossRef] [PubMed]
  11. M. Cai and K. J. Vahala, “Highly efficient hybrid fiber taper coupled microsphere laser,” Opt. Lett.26, 884–886 (2001).
    [CrossRef]
  12. A. Polman, B. Min, J. Kalkman, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold erbium-implanted toroidal microlaser on silicon,” Appl. Phys. Lett.84, 1037–1039 (2004).
    [CrossRef]
  13. T. Lu, L. Yang, R. V. A. van Loon, A. Polman, and K. J. Vahala, “On-chip green silica upconversion microlaser,” Opt. Lett.34, 482–484 (2009).
    [CrossRef] [PubMed]
  14. T. Lu, L. Yang, T. Carmon, and B. Min, “A narrow-linewidth on-chip toroid raman laser,” IEEE J. Quantum Electron.47, 320–326 (2011).
    [CrossRef]
  15. S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A At. Mol. Opt. Phys.71, 013817 (2005).
    [CrossRef]
  16. 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]
  17. J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express12, 90–103 (2004).
    [CrossRef] [PubMed]
  18. J. Hong, W. P. Huang, and T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol.10, 1860–1868 (1992).
    [CrossRef]
  19. X. Du, S. Vincent, and T. Lu, “Full-vectorial whispering-gallery-mode cavity analysis,” Opt. Express21, 22012–22022 (2013).
    [CrossRef] [PubMed]
  20. J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A5, 53 (2003).
    [CrossRef]
  21. 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. Express19, 15669–15678 (2011).
    [CrossRef] [PubMed]
  22. M. D. Feit and J. J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt.17, 3990–3998 (1978).
    [CrossRef] [PubMed]
  23. D. Yevick and B. Hermansson, “Efficient beam propagation techniques,” IEEE J. Quantum Electron.26, 109–112 (1990).
    [CrossRef]
  24. J. Saijonmaa and D. Yevick, “Beam-propagation analysis of loss in bent optical waveguides and fibers,” J. Opt. Soc. Am.73, 1785–1791 (1983).
    [CrossRef]
  25. W. Huang, C. Xu, S.-T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method: Analysis and assessment,” J. Lightwave Technol.10, 295–305 (1992).
    [CrossRef]
  26. J. V. Roey, J. van der Donk, and P. E. Lagasse, “Beam-propagation method: analysis and assessment,” J. Opt. Soc. Am.71, 803–810 (1981).
    [CrossRef]
  27. W. Huang, C. Xu, and S. Chaudhuri, “A finite-difference vector beam propagation method for three-dimensional waveguide structures,” IEEE Photonics Technol. Lett.4, 148–151 (1992).
    [CrossRef]
  28. B. Hermansson and D. Yevick, “Propagating-beam-method analysis of two-dimensional microlenses and three-dimensional taper structures,” Opt. Soc. Am. A1, 663–671 (1984).
    [CrossRef]
  29. H. Rao, R. Scarmozzino, and R. M. Osgood, “A bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photonics Technol. Lett.11, 830–832 (1999).
    [CrossRef]
  30. R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron.6, 150–162 (2000).
    [CrossRef]
  31. A. Ahmed, R. Koya, O. Wada, M. Wang, and R. Koga, “Eigenmode analysis of whispering gallery mode of pillbox-type optical resonators utilizing the FE-BPM formulation,” IEICE Trans. Electron.E78-C, 1638–1645 (1995).
  32. W. Yang and A. Gopinath, “A boundary integral method for propagation problems in integrated optical structures,” IEEE Photonics Technol. Lett.7, 777–779 (1995).
    [CrossRef]
  33. T. Lu and D. Yevick, “Boundary element analysis of dielectric waveguides,” J. Opt. Soc. Am. A19, 1197–1206 (2002).
    [CrossRef]
  34. T. Lu and D. Yevick, “Comparative evaluation of a novel series approximation for electromagnetic fields at dielectric corners with boundary element method applications,” J. Lightwave Technol.22, 1426–1432 (2004).
    [CrossRef]
  35. T. Lu and D. Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” J. Light-wave Technol.21, 1793–1807 (2003).
    [CrossRef]
  36. H. Deng and D. Yevick, “The nonunitarity of finite-element beam propagation algorithms,” IEEE Photonics Technol. Lett.17, 1429–1431 (2005).
    [CrossRef]
  37. S.-T. Chu and S. Chaudhuri, “A finite-difference time-domain method for the design and analysis of guided-wave optical structures,” J. Lightwave Technol.7, 2033–2038 (1989).
    [CrossRef]
  38. M. Reed, T. M. Benson, P. C. Kendall, and P. Sewell, “Antireflection-coated angled facet design,” Proc. Inst. Electr. Eng.143, 214–220 (1996).
  39. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys.114, 185–200 (1994).
    [CrossRef]
  40. W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photonics Technol. Lett.8, 649–651 (1996).
    [CrossRef]
  41. G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis I: Uniform regions and dielectric interfaces,” J. Lightwave Technol.20, 1210–1218 (2002).
    [CrossRef]
  42. G. R. Hadley, “Wide-angle beam propagation using pade approximant operators,” Opt. Lett.17, 1426–1428 (1992).
    [CrossRef] [PubMed]
  43. S. Lidgate, P. Sewell, and T. Benson, “Conformal mapping: limitations for waveguide bend analysis,” IEE Proc. Sci. Meas. Technol.149, 262–266 (2002).
    [CrossRef]
  44. M. Rivera, “A finite difference BPM analysis of bent dielectric waveguides,” J. Lightwave Technol.13, 233 (1995).
    [CrossRef]
  45. H. Deng, G. H. Jin, J. Harari, J. P. Vilcot, and D. Decoster, “Investigation of 3-D semivectorial finite-difference beam propagation method for bent waveguides,” J. Lightwave Technol.16, 915–922 (1998).
    [CrossRef]
  46. M. Krause, “Finite-difference mode solver for curved waveguides with angled and curved dielectric interfaces,” J. Lightwave Technol.29, 691–699 (2011).
    [CrossRef]
  47. K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis (John Wiley, 2001).
  48. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am.55, 1205–1208 (1965).
    [CrossRef]
  49. R. Kitamura, L. Pilon, and M. Jonasz, “Optical constants of silica glass from extreme ultraviolet to far infrared at near room temperature,” Appl. Opt.46, 8118–8133 (2007).
    [CrossRef] [PubMed]
  50. G. M. Hale and M. R. Querry, “Optical constants of water in the 200-nm to 200-m wavelength region,” Appl. Opt.12, 555–563 (1973).
    [CrossRef] [PubMed]
  51. K. S. Chiang, “Performance of the effective-index method for the analysis of dielectric waveguides,” Opt. Lett.16, 714–716 (1991).
    [CrossRef] [PubMed]
  52. M. A. C. Shirazi, W. Yu, S. Vincent, and T. Lu, “Whispering-gallery mode propagation simulations,” http://youtu.be/SJpEIkmsfMs (2013).
  53. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2 (Cambridge University, 1992), chap. 16, pp. 718–725.
  54. 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.48, 4165–4172 (2003).
    [CrossRef]
  55. M. A. Waldron, “Perturbation theory of resonant cavities,” Proc. IEE Part C Monogr.107, 272–274 (1960).
    [CrossRef]
  56. H. G. L. Schwefel, H. E. Tureci, D. A. Stone, and R. K. Chang, “Progress in asymmetric resonant cavities: Using shape as a design parameter in dielectric microcavity lasers,” in Optical Microcavities, K. Vahala, ed. (World Scientific, 2005).
  57. Y.-F. Xiao, C.-L. Zou, Y. Li, C.-H. Dong, Z.-F. Han, and Q. Gong, “Asymmetric resonant cavities and their applications in optics and photonics: a review,” Front. Optoelectron. China3, 109–124 (2010).
    [CrossRef]

2013

2011

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. U. S. A.108, 5976–5979 (2011).
[CrossRef] [PubMed]

Y. Sun and X. Fan, “Optical ring resonators for biochemical and chemical sensing,” Anal. Bioanal.Chem.399, 205–211 (2011).
[CrossRef]

T. Lu, L. Yang, T. Carmon, and B. Min, “A narrow-linewidth on-chip toroid raman laser,” IEEE J. Quantum Electron.47, 320–326 (2011).
[CrossRef]

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. Express19, 15669–15678 (2011).
[CrossRef] [PubMed]

M. Krause, “Finite-difference mode solver for curved waveguides with angled and curved dielectric interfaces,” J. Lightwave Technol.29, 691–699 (2011).
[CrossRef]

2010

Y.-F. Xiao, C.-L. Zou, Y. Li, C.-H. Dong, Z.-F. Han, and Q. Gong, “Asymmetric resonant cavities and their applications in optics and photonics: a review,” Front. Optoelectron. China3, 109–124 (2010).
[CrossRef]

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).

J. Knittel, T. G. McRae, K. H. Lee, and W. P. Bowen, “Interferometric detection of mode splitting for whispering gallery mode biosensors,” Appl. Phys. Lett.97, 1–3 (2010).
[CrossRef]

2009

2008

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: labelfree detection down to single molecules,” Nat. Methods5, 591–596 (2008).
[CrossRef] [PubMed]

2007

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]

R. Kitamura, L. Pilon, and M. Jonasz, “Optical constants of silica glass from extreme ultraviolet to far infrared at near room temperature,” Appl. Opt.46, 8118–8133 (2007).
[CrossRef] [PubMed]

2005

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A At. Mol. Opt. Phys.71, 013817 (2005).
[CrossRef]

H. Deng and D. Yevick, “The nonunitarity of finite-element beam propagation algorithms,” IEEE Photonics Technol. Lett.17, 1429–1431 (2005).
[CrossRef]

2004

2003

B. Min, T. J. Kippenberg, and K. J. Vahala, “Compact, fiber-compatible, cascaded raman laser,” Opt. Lett.28, 1507–1509 (2003).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421, 925–928 (2003).
[CrossRef] [PubMed]

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A5, 53 (2003).
[CrossRef]

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.48, 4165–4172 (2003).
[CrossRef]

T. Lu and D. Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” J. Light-wave Technol.21, 1793–1807 (2003).
[CrossRef]

2002

G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis I: Uniform regions and dielectric interfaces,” J. Lightwave Technol.20, 1210–1218 (2002).
[CrossRef]

S. Lidgate, P. Sewell, and T. Benson, “Conformal mapping: limitations for waveguide bend analysis,” IEE Proc. Sci. Meas. Technol.149, 262–266 (2002).
[CrossRef]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80, 4057–4059 (2002).
[CrossRef]

S. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold raman laser using a spherical dielectric mcirocavity,” Nature415, 621–623 (2002).
[CrossRef] [PubMed]

T. Lu and D. Yevick, “Boundary element analysis of dielectric waveguides,” J. Opt. Soc. Am. A19, 1197–1206 (2002).
[CrossRef]

2001

2000

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron.6, 150–162 (2000).
[CrossRef]

1999

H. Rao, R. Scarmozzino, and R. M. Osgood, “A bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photonics Technol. Lett.11, 830–832 (1999).
[CrossRef]

1998

H. Deng, G. H. Jin, J. Harari, J. P. Vilcot, and D. Decoster, “Investigation of 3-D semivectorial finite-difference beam propagation method for bent waveguides,” J. Lightwave Technol.16, 915–922 (1998).
[CrossRef]

1996

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

M. Reed, T. M. Benson, P. C. Kendall, and P. Sewell, “Antireflection-coated angled facet design,” Proc. Inst. Electr. Eng.143, 214–220 (1996).

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photonics Technol. Lett.8, 649–651 (1996).
[CrossRef]

1995

M. Rivera, “A finite difference BPM analysis of bent dielectric waveguides,” J. Lightwave Technol.13, 233 (1995).
[CrossRef]

A. Ahmed, R. Koya, O. Wada, M. Wang, and R. Koga, “Eigenmode analysis of whispering gallery mode of pillbox-type optical resonators utilizing the FE-BPM formulation,” IEICE Trans. Electron.E78-C, 1638–1645 (1995).

W. Yang and A. Gopinath, “A boundary integral method for propagation problems in integrated optical structures,” IEEE Photonics Technol. Lett.7, 777–779 (1995).
[CrossRef]

1994

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys.114, 185–200 (1994).
[CrossRef]

1992

J. Hong, W. P. Huang, and T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol.10, 1860–1868 (1992).
[CrossRef]

W. Huang, C. Xu, S.-T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method: Analysis and assessment,” J. Lightwave Technol.10, 295–305 (1992).
[CrossRef]

W. Huang, C. Xu, and S. Chaudhuri, “A finite-difference vector beam propagation method for three-dimensional waveguide structures,” IEEE Photonics Technol. Lett.4, 148–151 (1992).
[CrossRef]

G. R. Hadley, “Wide-angle beam propagation using pade approximant operators,” Opt. Lett.17, 1426–1428 (1992).
[CrossRef] [PubMed]

1991

1990

D. Yevick and B. Hermansson, “Efficient beam propagation techniques,” IEEE J. Quantum Electron.26, 109–112 (1990).
[CrossRef]

1989

S.-T. Chu and S. Chaudhuri, “A finite-difference time-domain method for the design and analysis of guided-wave optical structures,” J. Lightwave Technol.7, 2033–2038 (1989).
[CrossRef]

1984

B. Hermansson and D. Yevick, “Propagating-beam-method analysis of two-dimensional microlenses and three-dimensional taper structures,” Opt. Soc. Am. A1, 663–671 (1984).
[CrossRef]

1983

1981

1978

1973

1965

1960

M. A. Waldron, “Perturbation theory of resonant cavities,” Proc. IEE Part C Monogr.107, 272–274 (1960).
[CrossRef]

Ahmed, A.

A. Ahmed, R. Koya, O. Wada, M. Wang, and R. Koga, “Eigenmode analysis of whispering gallery mode of pillbox-type optical resonators utilizing the FE-BPM formulation,” IEICE Trans. Electron.E78-C, 1638–1645 (1995).

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421, 925–928 (2003).
[CrossRef] [PubMed]

Arnold, S.

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: labelfree detection down to single molecules,” Nat. Methods5, 591–596 (2008).
[CrossRef] [PubMed]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80, 4057–4059 (2002).
[CrossRef]

Benson, T.

S. Lidgate, P. Sewell, and T. Benson, “Conformal mapping: limitations for waveguide bend analysis,” IEE Proc. Sci. Meas. Technol.149, 262–266 (2002).
[CrossRef]

Benson, T. M.

M. Reed, T. M. Benson, P. C. Kendall, and P. Sewell, “Antireflection-coated angled facet design,” Proc. Inst. Electr. Eng.143, 214–220 (1996).

Berenger, J.-P.

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys.114, 185–200 (1994).
[CrossRef]

Bowen, W. P.

J. Knittel, T. G. McRae, K. H. Lee, and W. P. Bowen, “Interferometric detection of mode splitting for whispering gallery mode biosensors,” Appl. Phys. Lett.97, 1–3 (2010).
[CrossRef]

Braun, D.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80, 4057–4059 (2002).
[CrossRef]

Brock, R. S.

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.48, 4165–4172 (2003).
[CrossRef]

Cai, M.

Carmon, T.

T. Lu, L. Yang, T. Carmon, and B. Min, “A narrow-linewidth on-chip toroid raman laser,” IEEE J. Quantum Electron.47, 320–326 (2011).
[CrossRef]

Chang, R. K.

H. G. L. Schwefel, H. E. Tureci, D. A. Stone, and R. K. Chang, “Progress in asymmetric resonant cavities: Using shape as a design parameter in dielectric microcavity lasers,” in Optical Microcavities, K. Vahala, ed. (World Scientific, 2005).

Chaudhuri, S.

W. Huang, C. Xu, and S. Chaudhuri, “A finite-difference vector beam propagation method for three-dimensional waveguide structures,” IEEE Photonics Technol. Lett.4, 148–151 (1992).
[CrossRef]

S.-T. Chu and S. Chaudhuri, “A finite-difference time-domain method for the design and analysis of guided-wave optical structures,” J. Lightwave Technol.7, 2033–2038 (1989).
[CrossRef]

Chaudhuri, S. K.

W. Huang, C. Xu, S.-T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method: Analysis and assessment,” J. Lightwave Technol.10, 295–305 (1992).
[CrossRef]

Chen, T.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. U. S. A.108, 5976–5979 (2011).
[CrossRef] [PubMed]

Chiang, K. S.

Chu, S.-T.

W. Huang, C. Xu, S.-T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method: Analysis and assessment,” J. Lightwave Technol.10, 295–305 (1992).
[CrossRef]

S.-T. Chu and S. Chaudhuri, “A finite-difference time-domain method for the design and analysis of guided-wave optical structures,” J. Lightwave Technol.7, 2033–2038 (1989).
[CrossRef]

Decoster, D.

H. Deng, G. H. Jin, J. Harari, J. P. Vilcot, and D. Decoster, “Investigation of 3-D semivectorial finite-difference beam propagation method for bent waveguides,” J. Lightwave Technol.16, 915–922 (1998).
[CrossRef]

Deng, H.

H. Deng and D. Yevick, “The nonunitarity of finite-element beam propagation algorithms,” IEEE Photonics Technol. Lett.17, 1429–1431 (2005).
[CrossRef]

H. Deng, G. H. Jin, J. Harari, J. P. Vilcot, and D. Decoster, “Investigation of 3-D semivectorial finite-difference beam propagation method for bent waveguides,” J. Lightwave Technol.16, 915–922 (1998).
[CrossRef]

Dominguez-Juarez, J.

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).

Dong, C.-H.

Y.-F. Xiao, C.-L. Zou, Y. Li, C.-H. Dong, Z.-F. Han, and Q. Gong, “Asymmetric resonant cavities and their applications in optics and photonics: a review,” Front. Optoelectron. China3, 109–124 (2010).
[CrossRef]

Du, X.

Fan, X.

Y. Sun and X. Fan, “Optical ring resonators for biochemical and chemical sensing,” Anal. Bioanal.Chem.399, 205–211 (2011).
[CrossRef]

Feit, M. D.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2 (Cambridge University, 1992), chap. 16, pp. 718–725.

Fleck, J. J. A.

Fraser, S. E.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. U. S. A.108, 5976–5979 (2011).
[CrossRef] [PubMed]

Goh, K. W.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A At. Mol. Opt. Phys.71, 013817 (2005).
[CrossRef]

Gong, Q.

Y.-F. Xiao, C.-L. Zou, Y. Li, C.-H. Dong, Z.-F. Han, and Q. Gong, “Asymmetric resonant cavities and their applications in optics and photonics: a review,” Front. Optoelectron. China3, 109–124 (2010).
[CrossRef]

Gopinath, A.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron.6, 150–162 (2000).
[CrossRef]

W. Yang and A. Gopinath, “A boundary integral method for propagation problems in integrated optical structures,” IEEE Photonics Technol. Lett.7, 777–779 (1995).
[CrossRef]

Gorodetsky, M. L.

Guo, G.-C.

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. Express19, 15669–15678 (2011).
[CrossRef] [PubMed]

Hadley, G. R.

G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis I: Uniform regions and dielectric interfaces,” J. Lightwave Technol.20, 1210–1218 (2002).
[CrossRef]

G. R. Hadley, “Wide-angle beam propagation using pade approximant operators,” Opt. Lett.17, 1426–1428 (1992).
[CrossRef] [PubMed]

Hale, G. M.

Han, Z.-F.

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. Express19, 15669–15678 (2011).
[CrossRef] [PubMed]

Y.-F. Xiao, C.-L. Zou, Y. Li, C.-H. Dong, Z.-F. Han, and Q. Gong, “Asymmetric resonant cavities and their applications in optics and photonics: a review,” Front. Optoelectron. China3, 109–124 (2010).
[CrossRef]

Harari, J.

H. Deng, G. H. Jin, J. Harari, J. P. Vilcot, and D. Decoster, “Investigation of 3-D semivectorial finite-difference beam propagation method for bent waveguides,” J. Lightwave Technol.16, 915–922 (1998).
[CrossRef]

Helfert, S.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron.6, 150–162 (2000).
[CrossRef]

Herchak, S.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. U. S. A.108, 5976–5979 (2011).
[CrossRef] [PubMed]

Hermansson, B.

D. Yevick and B. Hermansson, “Efficient beam propagation techniques,” IEEE J. Quantum Electron.26, 109–112 (1990).
[CrossRef]

B. Hermansson and D. Yevick, “Propagating-beam-method analysis of two-dimensional microlenses and three-dimensional taper structures,” Opt. Soc. Am. A1, 663–671 (1984).
[CrossRef]

Hong, J.

J. Hong, W. P. Huang, and T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol.10, 1860–1868 (1992).
[CrossRef]

Hu, X.-H.

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.48, 4165–4172 (2003).
[CrossRef]

Huang, W.

W. Huang, C. Xu, S.-T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method: Analysis and assessment,” J. Lightwave Technol.10, 295–305 (1992).
[CrossRef]

W. Huang, C. Xu, and S. Chaudhuri, “A finite-difference vector beam propagation method for three-dimensional waveguide structures,” IEEE Photonics Technol. Lett.4, 148–151 (1992).
[CrossRef]

Huang, W. P.

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photonics Technol. Lett.8, 649–651 (1996).
[CrossRef]

J. Hong, W. P. Huang, and T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol.10, 1860–1868 (1992).
[CrossRef]

Huang, Y.

Ilchenko, V. S.

Jacobs, K. M.

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.48, 4165–4172 (2003).
[CrossRef]

Jin, G. H.

H. Deng, G. H. Jin, J. Harari, J. P. Vilcot, and D. Decoster, “Investigation of 3-D semivectorial finite-difference beam propagation method for bent waveguides,” J. Lightwave Technol.16, 915–922 (1998).
[CrossRef]

Jonasz, M.

Kalkman, J.

A. Polman, B. Min, J. Kalkman, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold erbium-implanted toroidal microlaser on silicon,” Appl. Phys. Lett.84, 1037–1039 (2004).
[CrossRef]

Kawano, K.

K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis (John Wiley, 2001).

Kendall, P. C.

M. Reed, T. M. Benson, P. C. Kendall, and P. Sewell, “Antireflection-coated angled facet design,” Proc. Inst. Electr. Eng.143, 214–220 (1996).

Khoshsima, M.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80, 4057–4059 (2002).
[CrossRef]

Kim, J.-H.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. U. S. A.108, 5976–5979 (2011).
[CrossRef] [PubMed]

Kimble, H. J.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A At. Mol. Opt. Phys.71, 013817 (2005).
[CrossRef]

Kippenberg, T. J.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A At. Mol. Opt. Phys.71, 013817 (2005).
[CrossRef]

A. Polman, B. Min, J. Kalkman, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold erbium-implanted toroidal microlaser on silicon,” Appl. Phys. Lett.84, 1037–1039 (2004).
[CrossRef]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421, 925–928 (2003).
[CrossRef] [PubMed]

B. Min, T. J. Kippenberg, and K. J. Vahala, “Compact, fiber-compatible, cascaded raman laser,” Opt. Lett.28, 1507–1509 (2003).
[CrossRef] [PubMed]

S. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold raman laser using a spherical dielectric mcirocavity,” Nature415, 621–623 (2002).
[CrossRef] [PubMed]

Kitamura, R.

Kitoh, T.

K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis (John Wiley, 2001).

Knittel, J.

J. Knittel, T. G. McRae, K. H. Lee, and W. P. Bowen, “Interferometric detection of mode splitting for whispering gallery mode biosensors,” Appl. Phys. Lett.97, 1–3 (2010).
[CrossRef]

Koga, R.

A. Ahmed, R. Koya, O. Wada, M. Wang, and R. Koga, “Eigenmode analysis of whispering gallery mode of pillbox-type optical resonators utilizing the FE-BPM formulation,” IEICE Trans. Electron.E78-C, 1638–1645 (1995).

Koya, R.

A. Ahmed, R. Koya, O. Wada, M. Wang, and R. Koga, “Eigenmode analysis of whispering gallery mode of pillbox-type optical resonators utilizing the FE-BPM formulation,” IEICE Trans. Electron.E78-C, 1638–1645 (1995).

Kozyreff, G.

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).

Krause, M.

M. Krause, “Finite-difference mode solver for curved waveguides with angled and curved dielectric interfaces,” J. Lightwave Technol.29, 691–699 (2011).
[CrossRef]

Lagasse, P. E.

Lee, H.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. U. S. A.108, 5976–5979 (2011).
[CrossRef] [PubMed]

Lee, K. H.

J. Knittel, T. G. McRae, K. H. Lee, and W. P. Bowen, “Interferometric detection of mode splitting for whispering gallery mode biosensors,” Appl. Phys. Lett.97, 1–3 (2010).
[CrossRef]

Li, Y.

Y.-F. Xiao, C.-L. Zou, Y. Li, C.-H. Dong, Z.-F. Han, and Q. Gong, “Asymmetric resonant cavities and their applications in optics and photonics: a review,” Front. Optoelectron. China3, 109–124 (2010).
[CrossRef]

Libchaber, A.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80, 4057–4059 (2002).
[CrossRef]

Lidgate, S.

S. Lidgate, P. Sewell, and T. Benson, “Conformal mapping: limitations for waveguide bend analysis,” IEE Proc. Sci. Meas. Technol.149, 262–266 (2002).
[CrossRef]

Lu, J. Q.

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.48, 4165–4172 (2003).
[CrossRef]

Lu, T.

X. Du, S. Vincent, and T. Lu, “Full-vectorial whispering-gallery-mode cavity analysis,” Opt. Express21, 22012–22022 (2013).
[CrossRef] [PubMed]

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. U. S. A.108, 5976–5979 (2011).
[CrossRef] [PubMed]

T. Lu, L. Yang, T. Carmon, and B. Min, “A narrow-linewidth on-chip toroid raman laser,” IEEE J. Quantum Electron.47, 320–326 (2011).
[CrossRef]

T. Lu, L. Yang, R. V. A. van Loon, A. Polman, and K. J. Vahala, “On-chip green silica upconversion microlaser,” Opt. Lett.34, 482–484 (2009).
[CrossRef] [PubMed]

T. Lu and D. Yevick, “Comparative evaluation of a novel series approximation for electromagnetic fields at dielectric corners with boundary element method applications,” J. Lightwave Technol.22, 1426–1432 (2004).
[CrossRef]

T. Lu and D. Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” J. Light-wave Technol.21, 1793–1807 (2003).
[CrossRef]

T. Lu and D. Yevick, “Boundary element analysis of dielectric waveguides,” J. Opt. Soc. Am. A19, 1197–1206 (2002).
[CrossRef]

Lui, W.

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photonics Technol. Lett.8, 649–651 (1996).
[CrossRef]

Ma, X.

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.48, 4165–4172 (2003).
[CrossRef]

Makino, T.

J. Hong, W. P. Huang, and T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol.10, 1860–1868 (1992).
[CrossRef]

Malitson, I. H.

Martorell, J.

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).

McRae, T. G.

J. Knittel, T. G. McRae, K. H. Lee, and W. P. Bowen, “Interferometric detection of mode splitting for whispering gallery mode biosensors,” Appl. Phys. Lett.97, 1–3 (2010).
[CrossRef]

Min, B.

T. Lu, L. Yang, T. Carmon, and B. Min, “A narrow-linewidth on-chip toroid raman laser,” IEEE J. Quantum Electron.47, 320–326 (2011).
[CrossRef]

A. Polman, B. Min, J. Kalkman, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold erbium-implanted toroidal microlaser on silicon,” Appl. Phys. Lett.84, 1037–1039 (2004).
[CrossRef]

B. Min, T. J. Kippenberg, and K. J. Vahala, “Compact, fiber-compatible, cascaded raman laser,” Opt. Lett.28, 1507–1509 (2003).
[CrossRef] [PubMed]

Mookherjea, S.

Osgood, R. M.

H. Rao, R. Scarmozzino, and R. M. Osgood, “A bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photonics Technol. Lett.11, 830–832 (1999).
[CrossRef]

Oxborrow, M.

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]

Paloczi, G. T.

Pilon, L.

Polman, A.

T. Lu, L. Yang, R. V. A. van Loon, A. Polman, and K. J. Vahala, “On-chip green silica upconversion microlaser,” Opt. Lett.34, 482–484 (2009).
[CrossRef] [PubMed]

A. Polman, B. Min, J. Kalkman, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold erbium-implanted toroidal microlaser on silicon,” Appl. Phys. Lett.84, 1037–1039 (2004).
[CrossRef]

Poon, J. K. S.

Pregla, R.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron.6, 150–162 (2000).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2 (Cambridge University, 1992), chap. 16, pp. 718–725.

Querry, M. R.

Rao, H.

H. Rao, R. Scarmozzino, and R. M. Osgood, “A bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photonics Technol. Lett.11, 830–832 (1999).
[CrossRef]

Reed, M.

M. Reed, T. M. Benson, P. C. Kendall, and P. Sewell, “Antireflection-coated angled facet design,” Proc. Inst. Electr. Eng.143, 214–220 (1996).

Rivera, M.

M. Rivera, “A finite difference BPM analysis of bent dielectric waveguides,” J. Lightwave Technol.13, 233 (1995).
[CrossRef]

Roey, J. V.

Saijonmaa, J.

Savchenkov, A. A.

Scarmozzino, R.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron.6, 150–162 (2000).
[CrossRef]

H. Rao, R. Scarmozzino, and R. M. Osgood, “A bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photonics Technol. Lett.11, 830–832 (1999).
[CrossRef]

Scheuer, J.

Schwefel, H. G. L.

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. Express19, 15669–15678 (2011).
[CrossRef] [PubMed]

H. G. L. Schwefel, H. E. Tureci, D. A. Stone, and R. K. Chang, “Progress in asymmetric resonant cavities: Using shape as a design parameter in dielectric microcavity lasers,” in Optical Microcavities, K. Vahala, ed. (World Scientific, 2005).

Sewell, P.

S. Lidgate, P. Sewell, and T. Benson, “Conformal mapping: limitations for waveguide bend analysis,” IEE Proc. Sci. Meas. Technol.149, 262–266 (2002).
[CrossRef]

M. Reed, T. M. Benson, P. C. Kendall, and P. Sewell, “Antireflection-coated angled facet design,” Proc. Inst. Electr. Eng.143, 214–220 (1996).

Spillane, S.

S. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold raman laser using a spherical dielectric mcirocavity,” Nature415, 621–623 (2002).
[CrossRef] [PubMed]

Spillane, S. M.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A At. Mol. Opt. Phys.71, 013817 (2005).
[CrossRef]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421, 925–928 (2003).
[CrossRef] [PubMed]

Stone, D. A.

H. G. L. Schwefel, H. E. Tureci, D. A. Stone, and R. K. Chang, “Progress in asymmetric resonant cavities: Using shape as a design parameter in dielectric microcavity lasers,” in Optical Microcavities, K. Vahala, ed. (World Scientific, 2005).

Sun, F.-W.

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. Express19, 15669–15678 (2011).
[CrossRef] [PubMed]

Sun, Y.

Y. Sun and X. Fan, “Optical ring resonators for biochemical and chemical sensing,” Anal. Bioanal.Chem.399, 205–211 (2011).
[CrossRef]

Teraoka, I.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80, 4057–4059 (2002).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2 (Cambridge University, 1992), chap. 16, pp. 718–725.

Tureci, H. E.

H. G. L. Schwefel, H. E. Tureci, D. A. Stone, and R. K. Chang, “Progress in asymmetric resonant cavities: Using shape as a design parameter in dielectric microcavity lasers,” in Optical Microcavities, K. Vahala, ed. (World Scientific, 2005).

Vahala, K.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. U. S. A.108, 5976–5979 (2011).
[CrossRef] [PubMed]

Vahala, K. J.

T. Lu, L. Yang, R. V. A. van Loon, A. Polman, and K. J. Vahala, “On-chip green silica upconversion microlaser,” Opt. Lett.34, 482–484 (2009).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A At. Mol. Opt. Phys.71, 013817 (2005).
[CrossRef]

A. Polman, B. Min, J. Kalkman, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold erbium-implanted toroidal microlaser on silicon,” Appl. Phys. Lett.84, 1037–1039 (2004).
[CrossRef]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421, 925–928 (2003).
[CrossRef] [PubMed]

B. Min, T. J. Kippenberg, and K. J. Vahala, “Compact, fiber-compatible, cascaded raman laser,” Opt. Lett.28, 1507–1509 (2003).
[CrossRef] [PubMed]

S. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold raman laser using a spherical dielectric mcirocavity,” Nature415, 621–623 (2002).
[CrossRef] [PubMed]

M. Cai and K. J. Vahala, “Highly efficient hybrid fiber taper coupled microsphere laser,” Opt. Lett.26, 884–886 (2001).
[CrossRef]

van der Donk, J.

van Loon, R. V. A.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2 (Cambridge University, 1992), chap. 16, pp. 718–725.

Vilcot, J. P.

H. Deng, G. H. Jin, J. Harari, J. P. Vilcot, and D. Decoster, “Investigation of 3-D semivectorial finite-difference beam propagation method for bent waveguides,” J. Lightwave Technol.16, 915–922 (1998).
[CrossRef]

Vincent, S.

Vollmer, F.

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: labelfree detection down to single molecules,” Nat. Methods5, 591–596 (2008).
[CrossRef] [PubMed]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80, 4057–4059 (2002).
[CrossRef]

Wada, O.

A. Ahmed, R. Koya, O. Wada, M. Wang, and R. Koga, “Eigenmode analysis of whispering gallery mode of pillbox-type optical resonators utilizing the FE-BPM formulation,” IEICE Trans. Electron.E78-C, 1638–1645 (1995).

Waldron, M. A.

M. A. Waldron, “Perturbation theory of resonant cavities,” Proc. IEE Part C Monogr.107, 272–274 (1960).
[CrossRef]

Wang, M.

A. Ahmed, R. Koya, O. Wada, M. Wang, and R. Koga, “Eigenmode analysis of whispering gallery mode of pillbox-type optical resonators utilizing the FE-BPM formulation,” IEICE Trans. Electron.E78-C, 1638–1645 (1995).

Wiersig, J.

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A5, 53 (2003).
[CrossRef]

Wilcut, E.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A At. Mol. Opt. Phys.71, 013817 (2005).
[CrossRef]

Xiao, Y.-F.

Y.-F. Xiao, C.-L. Zou, Y. Li, C.-H. Dong, Z.-F. Han, and Q. Gong, “Asymmetric resonant cavities and their applications in optics and photonics: a review,” Front. Optoelectron. China3, 109–124 (2010).
[CrossRef]

Xu, C.

W. Huang, C. Xu, S.-T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method: Analysis and assessment,” J. Lightwave Technol.10, 295–305 (1992).
[CrossRef]

W. Huang, C. Xu, and S. Chaudhuri, “A finite-difference vector beam propagation method for three-dimensional waveguide structures,” IEEE Photonics Technol. Lett.4, 148–151 (1992).
[CrossRef]

Xu, C. L.

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photonics Technol. Lett.8, 649–651 (1996).
[CrossRef]

Yang, L.

T. Lu, L. Yang, T. Carmon, and B. Min, “A narrow-linewidth on-chip toroid raman laser,” IEEE J. Quantum Electron.47, 320–326 (2011).
[CrossRef]

T. Lu, L. Yang, R. V. A. van Loon, A. Polman, and K. J. Vahala, “On-chip green silica upconversion microlaser,” Opt. Lett.34, 482–484 (2009).
[CrossRef] [PubMed]

Yang, P.

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.48, 4165–4172 (2003).
[CrossRef]

Yang, W.

W. Yang and A. Gopinath, “A boundary integral method for propagation problems in integrated optical structures,” IEEE Photonics Technol. Lett.7, 777–779 (1995).
[CrossRef]

Yariv, A.

Yevick, D.

H. Deng and D. Yevick, “The nonunitarity of finite-element beam propagation algorithms,” IEEE Photonics Technol. Lett.17, 1429–1431 (2005).
[CrossRef]

T. Lu and D. Yevick, “Comparative evaluation of a novel series approximation for electromagnetic fields at dielectric corners with boundary element method applications,” J. Lightwave Technol.22, 1426–1432 (2004).
[CrossRef]

T. Lu and D. Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” J. Light-wave Technol.21, 1793–1807 (2003).
[CrossRef]

T. Lu and D. Yevick, “Boundary element analysis of dielectric waveguides,” J. Opt. Soc. Am. A19, 1197–1206 (2002).
[CrossRef]

D. Yevick and B. Hermansson, “Efficient beam propagation techniques,” IEEE J. Quantum Electron.26, 109–112 (1990).
[CrossRef]

B. Hermansson and D. Yevick, “Propagating-beam-method analysis of two-dimensional microlenses and three-dimensional taper structures,” Opt. Soc. Am. A1, 663–671 (1984).
[CrossRef]

J. Saijonmaa and D. Yevick, “Beam-propagation analysis of loss in bent optical waveguides and fibers,” J. Opt. Soc. Am.73, 1785–1791 (1983).
[CrossRef]

Yokoyama, K.

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photonics Technol. Lett.8, 649–651 (1996).
[CrossRef]

Zou, C.-L.

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. Express19, 15669–15678 (2011).
[CrossRef] [PubMed]

Y.-F. Xiao, C.-L. Zou, Y. Li, C.-H. Dong, Z.-F. Han, and Q. Gong, “Asymmetric resonant cavities and their applications in optics and photonics: a review,” Front. Optoelectron. China3, 109–124 (2010).
[CrossRef]

Anal. Bioanal.Chem.

Y. Sun and X. Fan, “Optical ring resonators for biochemical and chemical sensing,” Anal. Bioanal.Chem.399, 205–211 (2011).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

A. Polman, B. Min, J. Kalkman, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold erbium-implanted toroidal microlaser on silicon,” Appl. Phys. Lett.84, 1037–1039 (2004).
[CrossRef]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80, 4057–4059 (2002).
[CrossRef]

J. Knittel, T. G. McRae, K. H. Lee, and W. P. Bowen, “Interferometric detection of mode splitting for whispering gallery mode biosensors,” Appl. Phys. Lett.97, 1–3 (2010).
[CrossRef]

Front. Optoelectron. China

Y.-F. Xiao, C.-L. Zou, Y. Li, C.-H. Dong, Z.-F. Han, and Q. Gong, “Asymmetric resonant cavities and their applications in optics and photonics: a review,” Front. Optoelectron. China3, 109–124 (2010).
[CrossRef]

IEE Proc. Sci. Meas. Technol.

S. Lidgate, P. Sewell, and T. Benson, “Conformal mapping: limitations for waveguide bend analysis,” IEE Proc. Sci. Meas. Technol.149, 262–266 (2002).
[CrossRef]

IEEE J. Quantum Electron.

T. Lu, L. Yang, T. Carmon, and B. Min, “A narrow-linewidth on-chip toroid raman laser,” IEEE J. Quantum Electron.47, 320–326 (2011).
[CrossRef]

D. Yevick and B. Hermansson, “Efficient beam propagation techniques,” IEEE J. Quantum Electron.26, 109–112 (1990).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron.6, 150–162 (2000).
[CrossRef]

IEEE Photonics Technol. Lett.

W. Yang and A. Gopinath, “A boundary integral method for propagation problems in integrated optical structures,” IEEE Photonics Technol. Lett.7, 777–779 (1995).
[CrossRef]

W. Huang, C. Xu, and S. Chaudhuri, “A finite-difference vector beam propagation method for three-dimensional waveguide structures,” IEEE Photonics Technol. Lett.4, 148–151 (1992).
[CrossRef]

IEEE Photonics Technol. Lett.

H. Deng and D. Yevick, “The nonunitarity of finite-element beam propagation algorithms,” IEEE Photonics Technol. Lett.17, 1429–1431 (2005).
[CrossRef]

H. Rao, R. Scarmozzino, and R. M. Osgood, “A bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photonics Technol. Lett.11, 830–832 (1999).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photonics Technol. Lett.8, 649–651 (1996).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

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]

IEICE Trans. Electron.

A. Ahmed, R. Koya, O. Wada, M. Wang, and R. Koga, “Eigenmode analysis of whispering gallery mode of pillbox-type optical resonators utilizing the FE-BPM formulation,” IEICE Trans. Electron.E78-C, 1638–1645 (1995).

J. Lightwave Technol.

G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis I: Uniform regions and dielectric interfaces,” J. Lightwave Technol.20, 1210–1218 (2002).
[CrossRef]

J. Comput. Phys.

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys.114, 185–200 (1994).
[CrossRef]

J. Light-wave Technol.

T. Lu and D. Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” J. Light-wave Technol.21, 1793–1807 (2003).
[CrossRef]

J. Lightwave Technol.

S.-T. Chu and S. Chaudhuri, “A finite-difference time-domain method for the design and analysis of guided-wave optical structures,” J. Lightwave Technol.7, 2033–2038 (1989).
[CrossRef]

H. Deng, G. H. Jin, J. Harari, J. P. Vilcot, and D. Decoster, “Investigation of 3-D semivectorial finite-difference beam propagation method for bent waveguides,” J. Lightwave Technol.16, 915–922 (1998).
[CrossRef]

M. Krause, “Finite-difference mode solver for curved waveguides with angled and curved dielectric interfaces,” J. Lightwave Technol.29, 691–699 (2011).
[CrossRef]

J. Lightwave Technol.

M. Rivera, “A finite difference BPM analysis of bent dielectric waveguides,” J. Lightwave Technol.13, 233 (1995).
[CrossRef]

J. Hong, W. P. Huang, and T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol.10, 1860–1868 (1992).
[CrossRef]

W. Huang, C. Xu, S.-T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method: Analysis and assessment,” J. Lightwave Technol.10, 295–305 (1992).
[CrossRef]

T. Lu and D. Yevick, “Comparative evaluation of a novel series approximation for electromagnetic fields at dielectric corners with boundary element method applications,” J. Lightwave Technol.22, 1426–1432 (2004).
[CrossRef]

J. Opt. A

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A5, 53 (2003).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nat. Commun.

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).

Nat. Methods

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: labelfree detection down to single molecules,” Nat. Methods5, 591–596 (2008).
[CrossRef] [PubMed]

Nature

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421, 925–928 (2003).
[CrossRef] [PubMed]

S. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold raman laser using a spherical dielectric mcirocavity,” Nature415, 621–623 (2002).
[CrossRef] [PubMed]

Opt. Express

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. Express19, 15669–15678 (2011).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Opt. Soc. Am. A

B. Hermansson and D. Yevick, “Propagating-beam-method analysis of two-dimensional microlenses and three-dimensional taper structures,” Opt. Soc. Am. A1, 663–671 (1984).
[CrossRef]

Phys. Med. Biol.

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.48, 4165–4172 (2003).
[CrossRef]

Phys. Rev. A At. Mol. Opt. Phys.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A At. Mol. Opt. Phys.71, 013817 (2005).
[CrossRef]

Proc. IEE Part C Monogr.

M. A. Waldron, “Perturbation theory of resonant cavities,” Proc. IEE Part C Monogr.107, 272–274 (1960).
[CrossRef]

Proc. Inst. Electr. Eng.

M. Reed, T. M. Benson, P. C. Kendall, and P. Sewell, “Antireflection-coated angled facet design,” Proc. Inst. Electr. Eng.143, 214–220 (1996).

Proc. Natl. Acad. Sci. U. S. A.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. U. S. A.108, 5976–5979 (2011).
[CrossRef] [PubMed]

Other

K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis (John Wiley, 2001).

M. A. C. Shirazi, W. Yu, S. Vincent, and T. Lu, “Whispering-gallery mode propagation simulations,” http://youtu.be/SJpEIkmsfMs (2013).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2 (Cambridge University, 1992), chap. 16, pp. 718–725.

H. G. L. Schwefel, H. E. Tureci, D. A. Stone, and R. K. Chang, “Progress in asymmetric resonant cavities: Using shape as a design parameter in dielectric microcavity lasers,” in Optical Microcavities, K. Vahala, ed. (World Scientific, 2005).

Supplementary Material (3)

» Media 1: AVI (14325 KB)     
» Media 2: AVI (13400 KB)     
» Media 3: AVI (13400 KB)     

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Figures (9)

Fig. 1
Fig. 1

A cylindrical coordinate system.

Fig. 2
Fig. 2

Reflectivity in dB vs. different σ0 values. The insets are the field intensity for three different σ0 values: (a) 0, (b) 2.5 × 1016, and (c) 1020. The white lines indicate the inner 2-μm PML edges. The σ0 values of inset (a) and (c) are not in the plot σ0 range.

Fig. 3
Fig. 3

(a) Field intensity in logarithmic scale, where the radiation is observable. To reiterate, the solid green lines are showing the resonator edges and the dashed green lines are showing the PML edges. (b) Partial intensity distribution frame (from Media 1) for propagation in a 1-mm diameter microdisk, wherein the PML effectiveness is confirmed by the lack of artificial reflection.

Fig. 4
Fig. 4

(a) Quality factor vs. grid spacings, (b) its relative error vs. grid spacings, (c) variations for the cross section at Δρ = 3.1 nm, and (d) variations for the cross section at Δϕ = 0.0015 radians. In (c), the last two points are omitted for the line of best fit.

Fig. 5
Fig. 5

Resonance wavelength of the ring resonator and its relevant error vs. grid size in the ρ̂ direction.

Fig. 6
Fig. 6

Magnitude of the overlap factor between the output profile and mode profile as well as its relative error vs. round trip number. The mode profile (blue) and output profile (red) at the 1st, 25th, and 125th rounds of propagation are plotted in insets from left to right.

Fig. 7
Fig. 7

(a) Power at the output of the ring after each round trip normalized to the first round power for resonance (blue) and for λ res ( 1 + 1 2 Q ) (green). The red curve represents the relative deviation of the power from the saturation power. (b) Normalized saturation power and its relative error vs. grid size in the ρ̂ direction.

Fig. 8
Fig. 8

(a) Field intensity (in logarithmic scale) over a slice of the ring, where the 400-nm bead is located. The inset shows the field distribution inside the bead. (b) Resonance wavelength shift vs. nanobead radius, for λ = 970 nm, calculated by the BPM method (red) and perturbation method (black).

Fig. 9
Fig. 9

Quality factor vs. ellipticity. The insets show the field intensities at the second round of propagation for (a) R = 45 μm and R90° = 37.5 μm as well as (b) R = 45 μm and R90° = 32.5 μm taken from Media 2 and Media 3, respectively.

Equations (12)

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[ 2 ϕ 2 + ρ 2 2 ρ 2 + ρ ρ + n 2 ( ρ , ϕ ) k 0 2 ρ 2 ] E = 0
E ( ρ , ϕ ) = A ψ ^ m r ( ρ ) exp ( j m ϕ )
[ ρ 2 2 ρ 2 + ρ ρ + n 2 ( ρ ) k 0 2 ρ 2 ] ψ ^ m r = m 2 ψ ^ m r
E ( ρ , ϕ ) = ψ ( ρ , ϕ ) exp ( j m ¯ ϕ )
| 2 ψ ϕ 2 | 2 | m ¯ ψ ϕ |
j ϕ ψ = ρ 2 2 m 2 ψ ρ 2 + ρ 2 m ψ ρ + ( ρ 2 k 0 2 n 2 ( ρ , ϕ ) 2 m m 2 ) ψ
a p ψ p 1 , l + 1 + ( 1 j Δ ϕ + b p ) ψ p , l + 1 c p ψ p + 1 , l + 1 = a p ψ p 1 , l + ( 1 j Δ ϕ b p ) ψ p , l + c p ψ p + 1 , l
a p = p ( 1 2 p ) 8 m b p = p 2 ( 2 Δ ρ 2 k 0 2 n p , l + 1 2 ) 4 m + m 4 c p = p ( 1 + 2 p ) 8 m
H ˜ | ψ l + 1 = D ˜ | ψ l
ρ 1 1 + j σ ( ρ ) ω ρ
σ ( ρ ) = { σ 0 ( ρ ρ 0 ) 2 Inside the PML 0 Elsewhere
e = 1 R 90 ° R 0 °

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