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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    [Crossref]

2013 (1)

2011 (5)

M. Krause, “Finite-difference mode solver for curved waveguides with angled and curved dielectric interfaces,” J. Lightwave Technol. 29, 691–699 (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. Express 19, 15669–15678 (2011).
[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]

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]

2010 (3)

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]

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. China 3, 109–124 (2010).
[Crossref]

2009 (1)

2008 (1)

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: labelfree detection down to single molecules,” Nat. Methods 5, 591–596 (2008).
[Crossref] [PubMed]

2007 (2)

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 (2)

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 (3)

2003 (5)

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,” Nature 421, 925–928 (2003).
[Crossref] [PubMed]

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A 5, 53 (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]

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]

2002 (5)

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,” Nature 415, 621–623 (2002).
[Crossref] [PubMed]

T. Lu and D. Yevick, “Boundary element analysis of dielectric waveguides,” J. Opt. Soc. Am. A 19, 1197–1206 (2002).
[Crossref]

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]

2001 (1)

2000 (1)

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 (1)

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 (1)

1996 (3)

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]

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).

1995 (3)

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]

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

1994 (1)

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

1992 (4)

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

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]

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]

1991 (1)

1990 (1)

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

1989 (1)

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 (1)

B. Hermansson and D. Yevick, “Propagating-beam-method analysis of two-dimensional microlenses and three-dimensional taper structures,” Opt. Soc. Am. A 1, 663–671 (1984).
[Crossref]

1983 (1)

1981 (1)

1978 (1)

1973 (1)

1965 (1)

1960 (1)

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,” Nature 421, 925–928 (2003).
[Crossref] [PubMed]

Arnold, S.

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: labelfree detection down to single molecules,” Nat. Methods 5, 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.

Deng, H.

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. China 3, 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).
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Helfert, S.

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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).
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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).
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Jonasz, M.

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

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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).
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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).
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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).
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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).
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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).
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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).
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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).
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T. Lu and D. Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” J. Light-wave Technol. 21, 1793–1807 (2003).
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T. Lu and D. Yevick, “Boundary element analysis of dielectric waveguides,” J. Opt. Soc. Am. A 19, 1197–1206 (2002).
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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).
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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).
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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).
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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).
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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).
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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).
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B. Min, T. J. Kippenberg, and K. J. Vahala, “Compact, fiber-compatible, cascaded raman laser,” Opt. Lett. 28, 1507–1509 (2003).
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Osgood, R. M.

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

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Saijonmaa, J.

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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).
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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).
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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. Express 19, 15669–15678 (2011).
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S. Lidgate, P. Sewell, and T. Benson, “Conformal mapping: limitations for waveguide bend analysis,” IEE Proc. Sci. Meas. Technol. 149, 262–266 (2002).
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M. Reed, T. M. Benson, P. C. Kendall, and P. Sewell, “Antireflection-coated angled facet design,” Proc. Inst. Electr. Eng. 143, 214–220 (1996).

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S. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold raman laser using a spherical dielectric mcirocavity,” Nature 415, 621–623 (2002).
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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).
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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).
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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).

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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).
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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).
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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).
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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).
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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).
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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).
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B. Min, T. J. Kippenberg, and K. J. Vahala, “Compact, fiber-compatible, cascaded raman laser,” Opt. Lett. 28, 1507–1509 (2003).
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S. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold raman laser using a spherical dielectric mcirocavity,” Nature 415, 621–623 (2002).
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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. China 3, 109–124 (2010).
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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).
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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).
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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).
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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]

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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).
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T. Lu and D. Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” J. Light-wave Technol. 21, 1793–1807 (2003).
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T. Lu and D. Yevick, “Boundary element analysis of dielectric waveguides,” J. Opt. Soc. Am. A 19, 1197–1206 (2002).
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D. Yevick and B. Hermansson, “Efficient beam propagation techniques,” IEEE J. Quantum Electron. 26, 109–112 (1990).
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B. Hermansson and D. Yevick, “Propagating-beam-method analysis of two-dimensional microlenses and three-dimensional taper structures,” Opt. Soc. Am. A 1, 663–671 (1984).
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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).
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Zou, C.-L.

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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.

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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.

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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.

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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.

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Fig. 5
Fig. 5 Resonance wavelength of the ring resonator and its relevant error vs. grid size in the ρ̂ direction.

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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.

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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.

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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).

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Fig. 9
Fig. 9 Quality factor vs. ellipticity. The insets show the field intensities at the second round of propagation for (a) R0° = 45 μm and R90° = 37.5 μm as well as (b) R0° = 45 μm and R90° = 32.5 μm taken from Media 2 and Media 3, respectively.

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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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