Abstract

To describe the temporal evolution of the mode amplitude of a spherical microcavity, a nonlinear equation is developed by considering loss and Kerr nonlinear effect as perturbations. In order to study the impact of Kerr nonlinearity, the tensor components of χ(3) in spherical coordinates are calculated. To describe the impact of Kerr nonlinearity, effective mode volume and effective nonlinear coefficient are defined. We found that the resonant modes undergo a negative frequency shift proportional to the injected energy, consistant with the reported experimental observations.

© 2015 Optical Society of America

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References

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    [Crossref] [PubMed]
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  22. B. Min, T. J. Kippenberg, and K. J. Vahala, “Compact, fiber-compatible, cascaded Raman laser,” Opt. Lett. 28 (17), 1507–1509 (2003).
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  23. T. J. Kippenberg, S. M. Spillane, B. Min, and K. J. Vahala, “Theoretical and experimental study of stimulated and cascaded Raman scattering in ultrahigh-Q optical microcavities,” IEEE J. Sel. Topics Quantum Electron. 10(5), 1219–1228 (2004).
    [Crossref]
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    [Crossref]
  26. A. Kobyakov, S. Kumar, D. Chowdhury, A. B. Ruffin, M. Sauer, S. Bickham, and R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Optics Express,  13(14), 5338–5346 (2005).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]

2014 (2)

M. D. Baaske, M. R. Foreman, and F. Vollmer, “Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform,” Nat. Nanotechnol. 9(11), 933–939 (2014).
[Crossref] [PubMed]

S. K. Ozdemir, J. Zhu, X. Yang, B. Penq, L. He, F. Monifi, S. H. Huang, G. L. Long, and L. Yang, “Highly sensitive detection of nanoparticles with a self-referenced and self-heterodyned whispering-gallery Raman microlaser,” Proc. Natl. Acad. Sci. 111(37), E3836–E3844 (2014).
[Crossref] [PubMed]

2013 (1)

L. He, S. K. zdemir, and L. Yang, “Whispering gallery microcavity lasers,” Laser Photonics Rev. 7(1), 60–82 (2013).
[Crossref]

2011 (2)

L. He, S. K. zdemir, J. Zhu, W. Kim, and L. Yang, “Detecting single viruses and nanoparticles using whispering gallery microlasers,” Nat. Nanotechnol. 6(7), 428–432 (2011).
[Crossref] [PubMed]

T.J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332(6029), 555–559 (2011).
[Crossref] [PubMed]

2010 (1)

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics,  4(1), 46–49 (2010).
[Crossref]

2008 (3)

2007 (2)

P. DelHaye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450(7173), 1214–1217 (2007).
[Crossref]

T. Carmon and K. J. Vahala, “Visible continuous emission from a silica microphotonic device by third-harmonic generation,” Nat. Phys. 3(6), 430–435 (2007).
[Crossref]

2006 (1)

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering gallery modes I: basics,” IEEE J. Sel. Top. Quantum Electron,  12(3), 3 (2006).
[Crossref]

2005 (2)

A. Kobyakov, S. Kumar, D. Chowdhury, A. B. Ruffin, M. Sauer, S. Bickham, and R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Optics Express,  13(14), 5338–5346 (2005).
[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 71(1), 013817 (2005).
[Crossref]

2004 (3)

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93(8), 083904 (2004).
[Crossref] [PubMed]

Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82(3), 033801 (2004).
[Crossref]

T. J. Kippenberg, S. M. Spillane, B. Min, and K. J. Vahala, “Theoretical and experimental study of stimulated and cascaded Raman scattering in ultrahigh-Q optical microcavities,” IEEE J. Sel. Topics Quantum Electron. 10(5), 1219–1228 (2004).
[Crossref]

2003 (2)

2002 (2)

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature,  415(6872), 621–623 (2002).
[Crossref] [PubMed]

A. N. Oraevsky, “Whispering-gallery waves,” Quantum Electron. 32(5), 377–400 (2002).
[Crossref]

1998 (1)

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1(3), 235–238 (1998).
[Crossref]

1993 (1)

B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt. Soc. Am. B 10(2), 343–352 (1993).
[Crossref]

1976 (1)

G. Mie, “Contributions to the optics of turbid media particularly of colloidal metal solutions,” Ann. Phys.(Leipzig) 25(3), 377–445 (1976).

1910 (1)

L. Rayleigh, “The problem of the whispering gallery,” Philos. Mag. J. Sci. 20(120), 1001–1004 (1910).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic, 2001).

Arcizet, O.

P. DelHaye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, “Full stabilization of a microresonator-based optical frequency comb,” Phys. Rev. Lett. 101(5), 053903 (2008).
[Crossref]

P. DelHaye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450(7173), 1214–1217 (2007).
[Crossref]

Baaske, M. D.

M. D. Baaske, M. R. Foreman, and F. Vollmer, “Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform,” Nat. Nanotechnol. 9(11), 933–939 (2014).
[Crossref] [PubMed]

Bickham, S.

A. Kobyakov, S. Kumar, D. Chowdhury, A. B. Ruffin, M. Sauer, S. Bickham, and R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Optics Express,  13(14), 5338–5346 (2005).
[Crossref] [PubMed]

S. Bickham, D. Chowdhury, S. Kumar, B. Ruffin, and S. Ten, “High Stimulated Brillouin Scattering (SBS) threshold fiber,” United States Patent 6952519 (May24, 2005).

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2003).

Carmon, T.

T. Carmon and K. J. Vahala, “Visible continuous emission from a silica microphotonic device by third-harmonic generation,” Nat. Phys. 3(6), 430–435 (2007).
[Crossref]

Chembo, Y. K.

Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82(3), 033801 (2004).
[Crossref]

Chen, D. R.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics,  4(1), 46–49 (2010).
[Crossref]

Chipouline, A.

Chowdhury, D.

A. Kobyakov, S. Kumar, D. Chowdhury, A. B. Ruffin, M. Sauer, S. Bickham, and R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Optics Express,  13(14), 5338–5346 (2005).
[Crossref] [PubMed]

S. Bickham, D. Chowdhury, S. Kumar, B. Ruffin, and S. Ten, “High Stimulated Brillouin Scattering (SBS) threshold fiber,” United States Patent 6952519 (May24, 2005).

Deen, M. J.

S. Kumar and M. J. Deen, Fiber Optic Communications: Fundamentals and Applications (John Wiley and Sons, 2014), Appendix B.
[Crossref]

DelHaye, P.

P. DelHaye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, “Full stabilization of a microresonator-based optical frequency comb,” Phys. Rev. Lett. 101(5), 053903 (2008).
[Crossref]

P. DelHaye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450(7173), 1214–1217 (2007).
[Crossref]

Deych, L.

Diddams, S. A.

T.J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332(6029), 555–559 (2011).
[Crossref] [PubMed]

Dong, C. H.

Egorov, O.

Foreman, M. R.

M. D. Baaske, M. R. Foreman, and F. Vollmer, “Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform,” Nat. Nanotechnol. 9(11), 933–939 (2014).
[Crossref] [PubMed]

Gaddam, V.

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 71(1), 013817 (2005).
[Crossref]

Hare, J.

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1(3), 235–238 (1998).
[Crossref]

Haroche, S.

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1(3), 235–238 (1998).
[Crossref]

S. Haroche, “Cavity Quantum Electrodynamics,” in Proceedings of the Les Houches Summer School of Theoretical Physics (1992).

He, L.

S. K. Ozdemir, J. Zhu, X. Yang, B. Penq, L. He, F. Monifi, S. H. Huang, G. L. Long, and L. Yang, “Highly sensitive detection of nanoparticles with a self-referenced and self-heterodyned whispering-gallery Raman microlaser,” Proc. Natl. Acad. Sci. 111(37), E3836–E3844 (2014).
[Crossref] [PubMed]

L. He, S. K. zdemir, and L. Yang, “Whispering gallery microcavity lasers,” Laser Photonics Rev. 7(1), 60–82 (2013).
[Crossref]

L. He, S. K. zdemir, J. Zhu, W. Kim, and L. Yang, “Detecting single viruses and nanoparticles using whispering gallery microlasers,” Nat. Nanotechnol. 6(7), 428–432 (2011).
[Crossref] [PubMed]

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics,  4(1), 46–49 (2010).
[Crossref]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2012).
[Crossref]

Holzwarth, R.

T.J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332(6029), 555–559 (2011).
[Crossref] [PubMed]

P. DelHaye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, “Full stabilization of a microresonator-based optical frequency comb,” Phys. Rev. Lett. 101(5), 053903 (2008).
[Crossref]

P. DelHaye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450(7173), 1214–1217 (2007).
[Crossref]

Huang, S. H.

S. K. Ozdemir, J. Zhu, X. Yang, B. Penq, L. He, F. Monifi, S. H. Huang, G. L. Long, and L. Yang, “Highly sensitive detection of nanoparticles with a self-referenced and self-heterodyned whispering-gallery Raman microlaser,” Proc. Natl. Acad. Sci. 111(37), E3836–E3844 (2014).
[Crossref] [PubMed]

Ilchenko, V. S.

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering gallery modes I: basics,” IEEE J. Sel. Top. Quantum Electron,  12(3), 3 (2006).
[Crossref]

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1(3), 235–238 (1998).
[Crossref]

Imoto, N.

Johnson, B. R.

B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt. Soc. Am. B 10(2), 343–352 (1993).
[Crossref]

Kim, W.

L. He, S. K. zdemir, J. Zhu, W. Kim, and L. Yang, “Detecting single viruses and nanoparticles using whispering gallery microlasers,” Nat. Nanotechnol. 6(7), 428–432 (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 71(1), 013817 (2005).
[Crossref]

Kippenberg, T. J.

P. DelHaye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, “Full stabilization of a microresonator-based optical frequency comb,” Phys. Rev. Lett. 101(5), 053903 (2008).
[Crossref]

P. DelHaye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450(7173), 1214–1217 (2007).
[Crossref]

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 71(1), 013817 (2005).
[Crossref]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93(8), 083904 (2004).
[Crossref] [PubMed]

T. J. Kippenberg, S. M. Spillane, B. Min, and K. J. Vahala, “Theoretical and experimental study of stimulated and cascaded Raman scattering in ultrahigh-Q optical microcavities,” IEEE J. Sel. Topics Quantum Electron. 10(5), 1219–1228 (2004).
[Crossref]

B. Min, T. J. Kippenberg, and K. J. Vahala, “Compact, fiber-compatible, cascaded Raman laser,” Opt. Lett. 28 (17), 1507–1509 (2003).
[Crossref] [PubMed]

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature,  415(6872), 621–623 (2002).
[Crossref] [PubMed]

T. J. Kippenberg, “Nonlinear optics in ultra-high-Q whispering-gallery optical microcavities,” Ph.D. Thesis, California Institute of Technology (2004).

Kippenberg, T.J.

T.J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332(6029), 555–559 (2011).
[Crossref] [PubMed]

Kobyakov, A.

A. Kobyakov, S. Kumar, D. Chowdhury, A. B. Ruffin, M. Sauer, S. Bickham, and R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Optics Express,  13(14), 5338–5346 (2005).
[Crossref] [PubMed]

Kumar, S.

A. Kobyakov, S. Kumar, D. Chowdhury, A. B. Ruffin, M. Sauer, S. Bickham, and R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Optics Express,  13(14), 5338–5346 (2005).
[Crossref] [PubMed]

S. Bickham, D. Chowdhury, S. Kumar, B. Ruffin, and S. Ten, “High Stimulated Brillouin Scattering (SBS) threshold fiber,” United States Patent 6952519 (May24, 2005).

S. Kumar and M. J. Deen, Fiber Optic Communications: Fundamentals and Applications (John Wiley and Sons, 2014), Appendix B.
[Crossref]

Lederer, F.

Lefevre-Seguin, V.

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1(3), 235–238 (1998).
[Crossref]

Li, L.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics,  4(1), 46–49 (2010).
[Crossref]

Long, G. L.

S. K. Ozdemir, J. Zhu, X. Yang, B. Penq, L. He, F. Monifi, S. H. Huang, G. L. Long, and L. Yang, “Highly sensitive detection of nanoparticles with a self-referenced and self-heterodyned whispering-gallery Raman microlaser,” Proc. Natl. Acad. Sci. 111(37), E3836–E3844 (2014).
[Crossref] [PubMed]

Matsko, A. B.

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering gallery modes I: basics,” IEEE J. Sel. Top. Quantum Electron,  12(3), 3 (2006).
[Crossref]

Mie, G.

G. Mie, “Contributions to the optics of turbid media particularly of colloidal metal solutions,” Ann. Phys.(Leipzig) 25(3), 377–445 (1976).

Min, B.

T. J. Kippenberg, S. M. Spillane, B. Min, and K. J. Vahala, “Theoretical and experimental study of stimulated and cascaded Raman scattering in ultrahigh-Q optical microcavities,” IEEE J. Sel. Topics Quantum Electron. 10(5), 1219–1228 (2004).
[Crossref]

B. Min, T. J. Kippenberg, and K. J. Vahala, “Compact, fiber-compatible, cascaded Raman laser,” Opt. Lett. 28 (17), 1507–1509 (2003).
[Crossref] [PubMed]

Mishra, R.

A. Kobyakov, S. Kumar, D. Chowdhury, A. B. Ruffin, M. Sauer, S. Bickham, and R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Optics Express,  13(14), 5338–5346 (2005).
[Crossref] [PubMed]

Monifi, F.

S. K. Ozdemir, J. Zhu, X. Yang, B. Penq, L. He, F. Monifi, S. H. Huang, G. L. Long, and L. Yang, “Highly sensitive detection of nanoparticles with a self-referenced and self-heterodyned whispering-gallery Raman microlaser,” Proc. Natl. Acad. Sci. 111(37), E3836–E3844 (2014).
[Crossref] [PubMed]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2012).
[Crossref]

Oraevsky, A. N.

A. N. Oraevsky, “Whispering-gallery waves,” Quantum Electron. 32(5), 377–400 (2002).
[Crossref]

Ozdemir, S. K.

S. K. Ozdemir, J. Zhu, X. Yang, B. Penq, L. He, F. Monifi, S. H. Huang, G. L. Long, and L. Yang, “Highly sensitive detection of nanoparticles with a self-referenced and self-heterodyned whispering-gallery Raman microlaser,” Proc. Natl. Acad. Sci. 111(37), E3836–E3844 (2014).
[Crossref] [PubMed]

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics,  4(1), 46–49 (2010).
[Crossref]

Penq, B.

S. K. Ozdemir, J. Zhu, X. Yang, B. Penq, L. He, F. Monifi, S. H. Huang, G. L. Long, and L. Yang, “Highly sensitive detection of nanoparticles with a self-referenced and self-heterodyned whispering-gallery Raman microlaser,” Proc. Natl. Acad. Sci. 111(37), E3836–E3844 (2014).
[Crossref] [PubMed]

Pertsch, T.

Raimond, J. M.

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1(3), 235–238 (1998).
[Crossref]

Rayleigh, L.

L. Rayleigh, “The problem of the whispering gallery,” Philos. Mag. J. Sci. 20(120), 1001–1004 (1910).
[Crossref]

Roch, J. F.

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1(3), 235–238 (1998).
[Crossref]

Ruffin, A. B.

A. Kobyakov, S. Kumar, D. Chowdhury, A. B. Ruffin, M. Sauer, S. Bickham, and R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Optics Express,  13(14), 5338–5346 (2005).
[Crossref] [PubMed]

Ruffin, B.

S. Bickham, D. Chowdhury, S. Kumar, B. Ruffin, and S. Ten, “High Stimulated Brillouin Scattering (SBS) threshold fiber,” United States Patent 6952519 (May24, 2005).

Sauer, M.

A. Kobyakov, S. Kumar, D. Chowdhury, A. B. Ruffin, M. Sauer, S. Bickham, and R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Optics Express,  13(14), 5338–5346 (2005).
[Crossref] [PubMed]

Schliesser, A.

P. DelHaye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, “Full stabilization of a microresonator-based optical frequency comb,” Phys. Rev. Lett. 101(5), 053903 (2008).
[Crossref]

P. DelHaye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450(7173), 1214–1217 (2007).
[Crossref]

Schmidt, C.

Schutz, B.

B. Schutz, A First Course in General Relativity (Cambridge University Press, 2009).
[Crossref]

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 71(1), 013817 (2005).
[Crossref]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93(8), 083904 (2004).
[Crossref] [PubMed]

T. J. Kippenberg, S. M. Spillane, B. Min, and K. J. Vahala, “Theoretical and experimental study of stimulated and cascaded Raman scattering in ultrahigh-Q optical microcavities,” IEEE J. Sel. Topics Quantum Electron. 10(5), 1219–1228 (2004).
[Crossref]

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature,  415(6872), 621–623 (2002).
[Crossref] [PubMed]

Ten, S.

S. Bickham, D. Chowdhury, S. Kumar, B. Ruffin, and S. Ten, “High Stimulated Brillouin Scattering (SBS) threshold fiber,” United States Patent 6952519 (May24, 2005).

Tnnermann, A.

Treussart, F.

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1(3), 235–238 (1998).
[Crossref]

Vahala, K. J.

T. Carmon and K. J. Vahala, “Visible continuous emission from a silica microphotonic device by third-harmonic generation,” Nat. Phys. 3(6), 430–435 (2007).
[Crossref]

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 71(1), 013817 (2005).
[Crossref]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93(8), 083904 (2004).
[Crossref] [PubMed]

T. J. Kippenberg, S. M. Spillane, B. Min, and K. J. Vahala, “Theoretical and experimental study of stimulated and cascaded Raman scattering in ultrahigh-Q optical microcavities,” IEEE J. Sel. Topics Quantum Electron. 10(5), 1219–1228 (2004).
[Crossref]

B. Min, T. J. Kippenberg, and K. J. Vahala, “Compact, fiber-compatible, cascaded Raman laser,” Opt. Lett. 28 (17), 1507–1509 (2003).
[Crossref] [PubMed]

K. J. Vahala, “Optical microcavities,” Nature,  424(6950), 839–846 (2003).
[Crossref] [PubMed]

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature,  415(6872), 621–623 (2002).
[Crossref] [PubMed]

Vollmer, F.

M. D. Baaske, M. R. Foreman, and F. Vollmer, “Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform,” Nat. Nanotechnol. 9(11), 933–939 (2014).
[Crossref] [PubMed]

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 71(1), 013817 (2005).
[Crossref]

Wilken, T.

P. DelHaye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450(7173), 1214–1217 (2007).
[Crossref]

Xiao, Y. F.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics,  4(1), 46–49 (2010).
[Crossref]

Y. F. Xiao, S. K. zdemir, V. Gaddam, C. H. Dong, N. Imoto, and L. Yang, “Quantum nondemolition measurement of photon number via optical Kerr effect in an ultra-high-Q microtoroid cavity,” Opt. Express,  16(26), 21462–21475 (2008).
[Crossref] [PubMed]

Yang, L.

S. K. Ozdemir, J. Zhu, X. Yang, B. Penq, L. He, F. Monifi, S. H. Huang, G. L. Long, and L. Yang, “Highly sensitive detection of nanoparticles with a self-referenced and self-heterodyned whispering-gallery Raman microlaser,” Proc. Natl. Acad. Sci. 111(37), E3836–E3844 (2014).
[Crossref] [PubMed]

L. He, S. K. zdemir, and L. Yang, “Whispering gallery microcavity lasers,” Laser Photonics Rev. 7(1), 60–82 (2013).
[Crossref]

L. He, S. K. zdemir, J. Zhu, W. Kim, and L. Yang, “Detecting single viruses and nanoparticles using whispering gallery microlasers,” Nat. Nanotechnol. 6(7), 428–432 (2011).
[Crossref] [PubMed]

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics,  4(1), 46–49 (2010).
[Crossref]

Y. F. Xiao, S. K. zdemir, V. Gaddam, C. H. Dong, N. Imoto, and L. Yang, “Quantum nondemolition measurement of photon number via optical Kerr effect in an ultra-high-Q microtoroid cavity,” Opt. Express,  16(26), 21462–21475 (2008).
[Crossref] [PubMed]

Yang, X.

S. K. Ozdemir, J. Zhu, X. Yang, B. Penq, L. He, F. Monifi, S. H. Huang, G. L. Long, and L. Yang, “Highly sensitive detection of nanoparticles with a self-referenced and self-heterodyned whispering-gallery Raman microlaser,” Proc. Natl. Acad. Sci. 111(37), E3836–E3844 (2014).
[Crossref] [PubMed]

Yu, N.

Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82(3), 033801 (2004).
[Crossref]

zdemir, S. K.

L. He, S. K. zdemir, and L. Yang, “Whispering gallery microcavity lasers,” Laser Photonics Rev. 7(1), 60–82 (2013).
[Crossref]

L. He, S. K. zdemir, J. Zhu, W. Kim, and L. Yang, “Detecting single viruses and nanoparticles using whispering gallery microlasers,” Nat. Nanotechnol. 6(7), 428–432 (2011).
[Crossref] [PubMed]

Y. F. Xiao, S. K. zdemir, V. Gaddam, C. H. Dong, N. Imoto, and L. Yang, “Quantum nondemolition measurement of photon number via optical Kerr effect in an ultra-high-Q microtoroid cavity,” Opt. Express,  16(26), 21462–21475 (2008).
[Crossref] [PubMed]

Zhu, J.

S. K. Ozdemir, J. Zhu, X. Yang, B. Penq, L. He, F. Monifi, S. H. Huang, G. L. Long, and L. Yang, “Highly sensitive detection of nanoparticles with a self-referenced and self-heterodyned whispering-gallery Raman microlaser,” Proc. Natl. Acad. Sci. 111(37), E3836–E3844 (2014).
[Crossref] [PubMed]

L. He, S. K. zdemir, J. Zhu, W. Kim, and L. Yang, “Detecting single viruses and nanoparticles using whispering gallery microlasers,” Nat. Nanotechnol. 6(7), 428–432 (2011).
[Crossref] [PubMed]

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics,  4(1), 46–49 (2010).
[Crossref]

Ann. Phys.(Leipzig) (1)

G. Mie, “Contributions to the optics of turbid media particularly of colloidal metal solutions,” Ann. Phys.(Leipzig) 25(3), 377–445 (1976).

Eur. Phys. J. D (1)

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1(3), 235–238 (1998).
[Crossref]

IEEE J. Sel. Top. Quantum Electron (1)

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering gallery modes I: basics,” IEEE J. Sel. Top. Quantum Electron,  12(3), 3 (2006).
[Crossref]

IEEE J. Sel. Topics Quantum Electron. (1)

T. J. Kippenberg, S. M. Spillane, B. Min, and K. J. Vahala, “Theoretical and experimental study of stimulated and cascaded Raman scattering in ultrahigh-Q optical microcavities,” IEEE J. Sel. Topics Quantum Electron. 10(5), 1219–1228 (2004).
[Crossref]

J. Opt. Soc. Am. B (1)

B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt. Soc. Am. B 10(2), 343–352 (1993).
[Crossref]

Laser Photonics Rev. (1)

L. He, S. K. zdemir, and L. Yang, “Whispering gallery microcavity lasers,” Laser Photonics Rev. 7(1), 60–82 (2013).
[Crossref]

Nat. Nanotechnol. (2)

L. He, S. K. zdemir, J. Zhu, W. Kim, and L. Yang, “Detecting single viruses and nanoparticles using whispering gallery microlasers,” Nat. Nanotechnol. 6(7), 428–432 (2011).
[Crossref] [PubMed]

M. D. Baaske, M. R. Foreman, and F. Vollmer, “Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform,” Nat. Nanotechnol. 9(11), 933–939 (2014).
[Crossref] [PubMed]

Nat. Photonics (1)

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics,  4(1), 46–49 (2010).
[Crossref]

Nat. Phys. (1)

T. Carmon and K. J. Vahala, “Visible continuous emission from a silica microphotonic device by third-harmonic generation,” Nat. Phys. 3(6), 430–435 (2007).
[Crossref]

Nature (3)

K. J. Vahala, “Optical microcavities,” Nature,  424(6950), 839–846 (2003).
[Crossref] [PubMed]

P. DelHaye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450(7173), 1214–1217 (2007).
[Crossref]

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature,  415(6872), 621–623 (2002).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Optics Express (1)

A. Kobyakov, S. Kumar, D. Chowdhury, A. B. Ruffin, M. Sauer, S. Bickham, and R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Optics Express,  13(14), 5338–5346 (2005).
[Crossref] [PubMed]

Philos. Mag. J. Sci. (1)

L. Rayleigh, “The problem of the whispering gallery,” Philos. Mag. J. Sci. 20(120), 1001–1004 (1910).
[Crossref]

Phys. Rev. A (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 71(1), 013817 (2005).
[Crossref]

Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82(3), 033801 (2004).
[Crossref]

Phys. Rev. Lett. (2)

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93(8), 083904 (2004).
[Crossref] [PubMed]

P. DelHaye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, “Full stabilization of a microresonator-based optical frequency comb,” Phys. Rev. Lett. 101(5), 053903 (2008).
[Crossref]

Proc. Natl. Acad. Sci. (1)

S. K. Ozdemir, J. Zhu, X. Yang, B. Penq, L. He, F. Monifi, S. H. Huang, G. L. Long, and L. Yang, “Highly sensitive detection of nanoparticles with a self-referenced and self-heterodyned whispering-gallery Raman microlaser,” Proc. Natl. Acad. Sci. 111(37), E3836–E3844 (2014).
[Crossref] [PubMed]

Quantum Electron. (1)

A. N. Oraevsky, “Whispering-gallery waves,” Quantum Electron. 32(5), 377–400 (2002).
[Crossref]

Science (1)

T.J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332(6029), 555–559 (2011).
[Crossref] [PubMed]

Other (9)

S. Bickham, D. Chowdhury, S. Kumar, B. Ruffin, and S. Ten, “High Stimulated Brillouin Scattering (SBS) threshold fiber,” United States Patent 6952519 (May24, 2005).

T. J. Kippenberg, “Nonlinear optics in ultra-high-Q whispering-gallery optical microcavities,” Ph.D. Thesis, California Institute of Technology (2004).

S. Haroche, “Cavity Quantum Electrodynamics,” in Proceedings of the Les Houches Summer School of Theoretical Physics (1992).

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2012).
[Crossref]

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic, 2001).

B. Schutz, A First Course in General Relativity (Cambridge University Press, 2009).
[Crossref]

R. W. Boyd, Nonlinear Optics (Academic, 2003).

S. Kumar and M. J. Deen, Fiber Optic Communications: Fundamentals and Applications (John Wiley and Sons, 2014), Appendix B.
[Crossref]

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Figures (2)

Fig. 1
Fig. 1 Distribution of electric field intensity |Rl(r)|2 with radius 15 μm and angular momentum mode number l = 80 for first three mode numbers (ν = 1, 2, 3). The field shows slower decay for higher order radial numbers (ν).
Fig. 2
Fig. 2 Observation of phase change and frequency shift. Parameters: radius a = 15 μm and angular momentum mode number l = 80 for first three radial mode numbers (ν = 1, 2, 3) for different injected energy Einj in the microsphere.

Tables (1)

Tables Icon

Table 1 Effective mode volumes and effective nonlinear coefficients for the first three radial mode numbers.

Equations (105)

Equations on this page are rendered with MathJax. Learn more.

Δ × Δ × E k 2 n 2 ( r ) 2 E t 2 = 0 .
E ( r ) = E θ θ + E ϕ ϕ ,
E θ = 1 2 [ Aq r X θ ( θ , ϕ ) exp ( im ϕ ) R l ( r ) exp ( i ω t ) + c . c . ] ,
E ϕ = 1 2 [ Aq r sin θ X ϕ ( θ , ϕ ) exp ( im ϕ ) R l ( r ) exp ( i ω t ) + c . c . ] .
X θ ( θ , ϕ ) = im sin θ P l m ( cos θ ) ,
X ϕ ( θ , ϕ ) = θ P l m ( cos θ ) .
[ d 2 d r 2 + k 2 n 2 ( r ) l ( l + 1 ) r 2 ] r R l ( r ) = 0 ,
R l ( r ) = { j l ( k 1 r ) for r a B h l ( 1 ) ( k r ) for r > a
j l ( k 1 a ) = B h l ( 1 ) ( k a ) ,
j l ( k 1 r ) k 1 = B h l ( 1 ) ( k r ) k .
j l ( k 1 a ) k 1 j l ( k 1 a ) = h l ( 1 ) ( k a ) k h l ( 1 ) ( k a ) .
2 E 1 c 2 2 E t 2 = μ 0 2 P L t 2 + μ 0 2 P NL t 2 ,
P L ( r , t ) = ε 0 χ ( 1 ) E ( r , t ) ,
P NL ( r , t ) = ε 0 χ ( 3 ) E ( r , t ) E ( r , t ) E ( r , t ) .
2 E n 2 ( r ) c 2 2 E t 2 = 1 c 2 2 P NL t 2 ,
E = E θ θ + E ϕ ϕ ,
n 2 ( r ) c 2 2 E t 2 = n 2 ( r ) c 2 1 2 [ ( 2 q t i ω q ω 2 ) f ( r ) exp ( i ω t ) + c . c . ] ,
f ( r ) = f θ ( r ) θ + f ϕ ( r ) ϕ .
f θ ( r ) = A R l ( r ) im r sin θ P l m ( cos θ ) exp ( im ϕ ) ,
f ϕ ( r ) = A R l ( r ) r sin θ [ θ P l m ( cos θ ) ] exp ( im ϕ ) .
2 E + ω 2 c 2 n r 2 ( r ) E = 0 ,
n r 2 ( r ) = Re [ n 2 ( r ) ] = ε r ( r ) ,
n i 2 ( r ) = Im [ n 2 ( r ) ] = ε i ( r ) ,
2 E n 2 ( r ) c 2 2 E t 2 = [ ω 2 c 2 q 2 n r 2 ( r ) f ( r ) n 2 ( r ) c 2 1 2 ( 2 q t i ω q ω 2 ) f ( r ) ] exp ( i ω t ) + c . c . , = [ n 2 ( r ) c 2 q t i ω q 2 i ε i ( r ) c 2 ω 2 ] f ( r ) exp ( i ω t ) + c . c . .
2 E n 2 ( r ) c 2 2 E t 2 = ω 2 c 2 ( P NL θ θ + P NL ϕ ϕ ) ,
P NL θ = | q | 2 q χ x x x ( 3 ) x { [ 3 8 r 2 | f θ ( r ) | 2 f θ ( r ) + 1 4 r 2 sin 2 θ | f ϕ ( r ) | 2 f θ ( r ) + 1 8 r 2 sin 2 θ ( f ϕ ( r ) ) 2 f θ * ( r ) ] } exp ( i ω t ) + c . c . ,
P NL ϕ = | q | 2 q χ x x x ( 3 ) x { [ 3 8 r 2 sin 2 θ | f ϕ ( r ) | 2 f ϕ ( r ) + 1 4 r 2 | f θ ( r ) | 2 f ϕ ( r ) + 1 8 r 2 ( f θ ( r ) ) 2 f ϕ * ( r ) ] } exp ( i ω t ) + c . c .
i [ q t + ω q 2 ε i ( r ) n 2 ( r ) ] f ( r ) = ω | q | 2 q χ x x x ( 3 ) x n 2 ( r ) ( P NL ( 1 ) θ θ + P NL ( 1 ) ϕ ϕ ) ,
P NL ( 1 ) θ = [ 3 8 r 2 | f θ ( r ) | 2 f θ ( r ) + 1 4 r 2 sin 2 θ | f ϕ ( r ) | 2 f θ ( r ) + 1 8 r 2 sin 2 θ ( f ϕ ( r ) ) 2 f θ * ( r ) ] + c . c . ,
P NL ( 1 ) ϕ = [ 3 8 r 2 sin 2 θ | f ϕ ( r ) | 2 f ϕ ( r ) + 1 4 r 2 | f θ ( r ) | 2 f ϕ ( r ) + 1 8 r 2 ( f θ ( r ) ) 2 f ϕ * ( r ) ] + c . c . .
i q t + i q ω 2 [ ε i ( r ) f ( r ) f * ( r ) ] d v / n 2 ( r ) f ( r ) . f * ( r ) d v = ω | q | 2 q χ x x x ( 3 ) x [ ( P NL ( 1 ) θ θ + P NL ( 1 ) ϕ ϕ ) . f * ( r ) ] d v / n 2 ( r ) f ( r ) . f * ( r ) d v
i ( q t + q 2 τ p ) = γ | q | 2 q ,
τ p = f ( r ) . f * ( r ) d v ω [ ε i ( r ) f ( r ) . f * ( r ) ] d v / n 2 ( r ) .
γ = χ x x x ( 3 ) x ω c V eff ,
V eff = c f ( r ) . f * ( r ) d v [ ( P NL ( 1 ) θ θ + P NL ( 1 ) ϕ ϕ ) . f * ( r ) ] d v / n 2 ( r ) .
V eff = c 2 I θ + I ϕ ,
I θ = π 4 m 2 | A | 4 ( 3 m 2 k 1 + 2 k 2 ) r 2 | R l ( r ) | 4 n 2 ( r ) d r ,
I ϕ = π 4 | A | 4 ( 3 k 3 + 2 m 2 k 2 ) r 2 | R l ( r ) | 4 n 2 ( r ) d r ,
k 1 = 1 1 [ p l m ( x ) ] 4 ( 1 x 2 ) 2 d x ,
k 2 = 1 1 [ p l m ( x ) ] 2 [ ( 1 x 2 ) x p l m ( x ) x p l m ( x ) ] 2 ( 1 x 2 ) 2 d x ,
k 3 = 1 1 [ ( 1 x 2 ) x p l m ( x ) x p l m ( x ) ] 4 ( 1 x 2 ) 2 d x .
q ( t ) = X ( t ) e i θ ( t ) .
d X d t = X 2 τ p ,
X ( t ) = X ( 0 ) exp ( t / 2 τ p ) ,
d θ d t = γ X 2 ( t ) = γ exp ( t / τ p ) X 2 ( 0 ) .
θ ( t ) = θ ( 0 ) γ [ 1 exp ( t / τ p ) ] τ p X 2 ( 0 ) .
Δ f ( t ) = 1 2 π d θ d t = γ 2 π E inj exp ( t / τ p ) ,
q ( t ) = q ( 0 ) e t / 2 τ p i γ E inj [ 1 exp ( t / τ p ) ] τ p ,
1 exp ( t / τ p ) t / τ p ,
θ ( t ) γ E inj t ,
Δ f = 1 2 π d θ d t = γ E inj / 2 π .
ϕ NL = γ P av L R ,
P NL n = χ k l m ( 3 ) n E k E l E m , k , l , m , n = x , y , or z .
χ y z z ( 3 ) y = χ z y y ( 3 ) z = χ z x x ( 3 ) z = χ x z z ( 3 ) x = χ x y y ( 3 ) x = χ y x x ( 3 ) y ,
χ z y z ( 3 ) y = χ x y x ( 3 ) y = χ y z y ( 3 ) z = χ x z x ( 3 ) z = χ z x z ( 3 ) x = χ y x y ( 3 ) x ,
χ z z y ( 3 ) y = χ x x y ( 3 ) y = χ y y z ( 3 ) z = χ x x z ( 3 ) z = χ x z z ( 3 ) x = χ y y x ( 3 ) x ,
χ x x x ( 3 ) x = χ y y y ( 3 ) y = χ z z z ( 3 ) z = χ x y y ( 3 ) x + χ y x y ( 3 ) x + χ y y x ( 3 ) x .
χ x x y ( 3 ) x = χ x x z ( 3 ) x = χ y y z ( 3 ) x = = 0 .
r = ( x 2 + y 2 + z 2 ) 1 / 2 ,
θ = tan 1 ( ( x 2 + y 2 ) 1 / 2 z ) ,
ϕ = tan 1 ( y x ) .
Λ = ( r x r y r z θ x θ y θ z ϕ x ϕ y ϕ z ) = ( sin θ cos ϕ sin θ sin ϕ cos θ cos θ cos ϕ r cos θ sin ϕ r sin θ r sin ϕ r sin θ cos ϕ r sin θ 0 ) .
Δ r = Δ r r + Δ θ θ + Δ ϕ ϕ , = Δ x x + Δ y y + Δ z z ,
Δ r = r x Δ x + r y Δ y + r z Δ z , = Λ i j ( Δ x ) i , j = r .
( Δ r ) j = Λ i j ( Δ x ) i , j = r , θ or ϕ .
A j = Λ i j A , i = x , y or z and j = r , θ or ϕ .
χ k l m ( 3 ) j = Λ j j Λ k k Λ l l Λ m m χ k l m ( 3 ) j .
χ θ θ θ ( 3 ) θ = Λ j θ Λ θ k Λ θ l Λ θ m χ k l m ( 3 ) j = Λ x θ Λ θ x Λ θ x Λ θ x χ x x x ( 3 ) x + Λ x θ Λ θ x Λ θ y Λ θ y χ x y y ( 3 ) x + Λ x θ Λ θ x Λ θ z Λ θ z Λ x z z ( 3 ) x + Λ x θ Λ θ y Λ θ x Λ θ y χ y x y ( 3 ) x + Λ x θ Λ θ y Λ θ y Λ θ x χ y y x ( 3 ) x + Λ x θ Λ θ z Λ θ x Λ θ z χ z x z ( 3 ) x + Λ x θ Λ θ z Λ θ z Λ θ x χ z z x ( 3 ) x + Λ y θ Λ θ y Λ θ x Λ θ x χ y x x ( 3 ) y + Λ y θ Λ θ y Λ θ y Λ θ y χ y y y ( 3 ) y + Λ y θ Λ θ y Λ θ z Λ θ z χ y z z ( 3 ) y + Λ y θ Λ θ x Λ θ x Λ θ y χ x x y ( 3 ) y + Λ y θ Λ θ x Λ θ y Λ θ x χ x y x ( 3 ) y + Λ y θ Λ θ z Λ θ y Λ θ z χ z y z ( 3 ) y + Λ y θ Λ θ z Λ θ z Λ θ y χ z z y ( 3 ) y + Λ z θ Λ θ x Λ θ x Λ θ z χ x x z ( 3 ) z + Λ z θ Λ θ x Λ θ z Λ θ x χ x z x ( 3 ) z + Λ z θ Λ θ y Λ θ y Λ θ z χ y y z ( 3 ) z + Λ z θ Λ θ y Λ θ z Λ θ y χ y z y ( 3 ) z , + Λ z θ Λ θ z Λ θ x Λ θ x χ z x x ( 3 ) z + Λ z θ Λ θ z Λ θ y Λ θ y χ z y y ( 3 ) z + Λ z θ Λ θ z Λ θ z Λ θ z χ z z z ( 3 ) z , = [ cos θ cos ϕ r r 3 cos 3 θ cos 3 ϕ + cos θ sin ϕ r r 3 cos 3 θ sin 3 ϕ + ( sin θ ) r ( r 3 sin 3 θ ) + cos θ cos ϕ r r cos θ cos ϕ r 2 cos 2 θ sin 2 ϕ + cos θ sin ϕ r r cos θ sin ϕ r 2 cos 2 θ cos 2 ϕ + cos θ cos ϕ r r cos θ cos ϕ r 2 sin 2 θ + cos θ sin ϕ r r cos θ sin ϕ r 2 sin 2 θ + ( sin θ ) r ( r sin θ ) r 2 cos 2 θ cos 2 ϕ + ( sin θ ) r ( r sin θ ) r 2 cos 2 θ sin 2 θ ] χ x x x ( 3 ) x , r 2 χ x x x ( 3 ) x ,
χ ϕ ϕ ϕ ( 3 ) ϕ = j k l m Λ j ϕ Λ ϕ k Λ ϕ l Λ ϕ m χ k l m ( 3 ) j , = [ Λ x ϕ ( Λ ϕ x ) 3 + Λ y ϕ ( Λ ϕ y ) 3 + Λ z ϕ ( Λ ϕ z ) 3 + ( Λ x ϕ ) ( Λ ϕ x ) ( Λ ϕ y ) 2 ) + ( Λ y ϕ ) ( Λ ϕ y ) ( Λ ϕ x ) 2 + ( Λ x ϕ ) ( Λ ϕ x ) ( Λ ϕ z ) 2 + ( Λ y ϕ ) ( Λ ϕ y ) ( Λ ϕ z ) 2 + ( Λ z ϕ ) ( Λ ϕ z ) ( Λ ϕ x ) 2 + ( Λ z ϕ ) ( Λ ϕ z ) ( Λ ϕ y ) 2 ] χ x x x ( 3 ) x , = [ ( sin ϕ ) sin θ ( r 3 sin 3 θ sin 3 ϕ + cos ϕ r sin θ r 3 sin 3 θ cos 3 ϕ + cos ϕ r sin θ r sin θ cos ϕ ) r 2 sin 2 θ sin 2 ϕ + sin ϕ r sin θ ( r sin θ sin ϕ ) r 2 sin 2 θ cos 2 ϕ ] χ x x x ( 3 ) x , = ( r 2 sin 2 θ sin 4 ϕ + r 2 sin 2 θ cos 4 ϕ + r 2 cos 2 ϕ sin 2 θ sin 2 ϕ + r 2 cos 2 ϕ sin 2 θ sin 2 ϕ ) χ x x x ( 3 ) x , = r 2 sin 2 θ χ x x x ( 3 ) x ,
χ θ ϕ ϕ ( 3 ) θ = j k l m Λ j θ Λ θ k Λ ϕ l Λ ϕ m χ k l m ( 3 ) j , = Λ x θ Λ θ x Λ ϕ x Λ ϕ x χ x x x ( 3 ) x + Λ x θ Λ θ x Λ ϕ y Λ ϕ y χ x y y ( 3 ) x + Λ x θ Λ θ x Λ ϕ z Λ ϕ z χ x z z ( 3 ) x + Λ x θ Λ θ y Λ ϕ x Λ ϕ y χ y x y ( 3 ) x + Λ x θ Λ θ y Λ ϕ y Λ ϕ x χ y y x ( 3 ) x + Λ x θ Λ θ z Λ ϕ x Λ ϕ z χ z x z ( 3 ) x + Λ x θ Λ θ z Λ ϕ z Λ ϕ x χ z z x ( 3 ) x + Λ y θ Λ θ y Λ ϕ x Λ ϕ x χ y x x ( 3 ) x + Λ y θ Λ θ y Λ ϕ y Λ ϕ y χ y y y ( 3 ) y + Λ y θ Λ θ y Λ ϕ z Λ ϕ y χ y z z ( 3 ) y + Λ y θ Λ θ x Λ ϕ x Λ ϕ y χ x x y ( 3 ) y + Λ y θ Λ θ x Λ ϕ y Λ ϕ x χ x y x ( 3 ) y + Λ y θ Λ θ z Λ ϕ y Λ ϕ z χ z y z ( 3 ) y + Λ y θ Λ θ z Λ ϕ z Λ ϕ y χ z z y ( 3 ) y + Λ z θ Λ θ x Λ ϕ x Λ ϕ z χ x x z ( 3 ) z + Λ z θ Λ θ x Λ ϕ z Λ ϕ x χ x z x ( 3 ) z + Λ z θ Λ θ y Λ ϕ y Λ ϕ z χ y y z ( 3 ) z + Λ z θ Λ θ y Λ ϕ z Λ ϕ y χ y z y ( 3 ) z + Λ z θ Λ θ z Λ ϕ x Λ ϕ x χ z x x ( 3 ) z + Λ z θ Λ θ y Λ ϕ y Λ ϕ z χ z y y ( 3 ) z + Λ z θ Λ θ z Λ ϕ z Λ ϕ z χ z z z ( 3 ) z , = 1 3 r 2 sin 2 θ χ x x x ( 3 ) x .
χ ϕ ϕ θ ( 3 ) θ = 1 3 r 2 sin 2 θ χ x x x ( 3 ) x ,
χ ϕ θ ϕ ( 3 ) θ = 1 3 r 2 sin 2 θ χ x x x ( 3 ) x ,
χ ϕ θ θ ( 3 ) ϕ = 1 3 r 2 χ x x x ( 3 ) x ,
χ θ ϕ θ ( 3 ) ϕ = 1 3 r 2 χ x x x ( 3 ) x ,
χ θ θ ϕ ( 3 ) ϕ = 1 3 r 2 χ x x x ( 3 ) x ,
χ θ θ θ ( 3 ) ϕ = χ ϕ ϕ ϕ ( 3 ) θ = 0 .
P NL n = χ k l m ( 3 ) n E k E l E m , k , l , m , n = r , θ or ϕ .
P NL θ = χ θ θ θ ( 3 ) θ ( E θ ) 3 + [ χ ϕ ϕ θ ( 3 ) θ + χ ϕ θ ϕ ( 3 ) θ + + χ θ ϕ ϕ ( 3 ) θ ] ( E ϕ ) 2 E θ , = r 2 [ ( E θ ) 3 + 1 3 sin 2 θ ( E ϕ ) 2 E θ + 1 3 sin 2 θ ( E ϕ ) 2 E θ + 1 3 sin 2 θ ( E ϕ ) 2 E θ ] χ x x x ( 3 ) x , = r 2 [ ( E θ ) 3 + sin 2 θ ( E ϕ ) 2 E θ ] χ x x x ( 3 ) x , = [ α 1 ( E θ ) 2 + α 2 ( E ϕ ) 2 ] E θ ,
α 1 = r 2 χ x x x ( 3 ) x ,
α 2 = r 2 sin 2 θ χ x x x ( 3 ) x ,
E θ = 1 2 [ q ( t ) f θ ( r ) exp ( i ω t ) + c . c . ] ,
E ϕ = 1 2 [ q ( t ) f ϕ ( r ) exp ( i ω t ) + c . c . ] .
( E θ ) 3 = 1 8 [ q f θ ( r ) exp ( i ω t ) + q * f θ * ( r ) exp ( i ω t ) ] [ q f θ ( r ) exp ( i ω t ) + q * f θ * ( r ) exp ( i ω t ) ] [ q f θ ( r ) exp ( i ω t ) + q * f θ * ( r ) exp ( i ω t ) ] , = 1 8 [ q 2 ( f θ ( r ) ) 2 exp ( 2 i ω t ) + 2 | q | 2 | f θ ( r ) | 2 + q * f θ * ( r ) q * f θ * ( r ) exp ( 2 i ω t ) [ q f θ ( r ) exp ( i ω t ) + q * f θ * ( r ) exp ( i ω t ) ] , = 1 8 [ q 3 ( f θ ( r ) ) 3 exp ( 3 i ω t ) + 2 | q | 2 q | f θ ( r ) | 2 f θ ( r ) exp ( i ω t ) + | q | 2 | f θ ( r ) | 2 q * f θ * ( r ) exp ( i ω t ) + | q | 2 | f θ ( r ) | 2 q f θ ( r ) exp ( i ω t ) + 2 | q | 2 | f θ ( r ) | 2 q * f θ * ( r ) exp ( i ω t ) + q * f θ * ( r ) q * f θ * ( r ) f θ * ( r ) q * exp ( 3 i ω t ) .
( E θ ) 3 = | q | 2 | q 8 [ 3 | f θ ( r ) | 2 f θ ( r ) exp ( i ω t ) + c . c . ] .
( E ϕ ) 2 E θ = | q | 2 | q 8 { [ 2 | f ϕ ( r ) | 2 f θ ( r ) + ( f ϕ * ( r ) ) 2 q f θ * ( r ) ] exp ( i ω t ) + c . c . } .
P NL θ = [ α 1 ( E θ ) 2 + α 2 ( E ϕ ) 2 ] E θ , = | q | 2 q χ x x x ( 3 ) x [ ( 3 8 r 2 | f θ ( r ) | 2 f θ ( r ) + 1 4 r 2 sin 2 θ | f ϕ ( r ) | 2 f θ ( r ) + 1 8 r 2 sin 2 θ ( f ϕ ( r ) ) 2 f θ * ( r ) ) exp ( i ω t ) + c . c . ] .
P NL ϕ = χ ϕ ϕ ϕ ( 3 ) ϕ ( E ϕ ) 3 + [ χ ϕ θ θ ( 3 ) ϕ + χ θ θ ϕ ( 3 ) ϕ + χ θ ϕ θ ( 3 ) ϕ ] ( E θ ) 2 E ϕ , = r 2 [ sin 2 θ ( E ϕ ) 3 + 1 3 ( E θ ) 2 E ϕ + 1 3 ( E θ ) 2 E ϕ + 1 3 ( E θ ) 2 E ϕ ] χ x x x ( 3 ) x , = r 2 [ sin 2 θ ( E ϕ ) 3 + ( E θ ) 2 E ϕ ] χ x x x ( 3 ) x , = [ α 2 ( E ϕ ) 2 + α 1 ( E θ ) 2 ] E ϕ ,
( E ϕ ) 3 = | q | 2 | q 8 [ 3 | f ϕ ( r ) | 2 f ϕ ( r ) exp ( i ω t ) + c . c . ] ,
( E θ ) 2 E ϕ = | q | 2 | q 8 [ ( 2 | f θ ( r ) | 2 f ϕ ( r ) exp ( i ω t ) + c . c . ) + ( ( f θ ( r ) ) 2 f ϕ * ( r ) exp ( i ω t ) + c . c . ) ] .
P NL ϕ = [ α 2 ( E ϕ ) 2 + α 1 ( E θ ) 2 ] E ϕ , = | q | 2 q χ x x x ( 3 ) x [ ( 3 8 r 2 sin 2 θ | f ϕ ( r ) | 2 f ϕ ( r ) + 1 4 r 2 | f θ ( r ) | 2 f ϕ ( r ) + 1 8 r 2 ( f θ ( r ) ) 2 f ϕ * ( r ) ) exp ( i ω t ) + c . c . ] .
g θ θ = θ . θ = r 2 ,
g ϕ ϕ = ϕ . ϕ = r 2 sin 2 θ ,
g θ ϕ = g ϕ θ = 0 .
1 c ( f ( r ) . f * ( r ) ) d v = 1 c ( | f θ ( r ) | 2 g θ θ + | f ϕ ( r ) | 2 g ϕ ϕ ) d v , = 1 ,
f θ ( r ) = A R l ( r ) im r sin θ P l m ( cos θ ) exp ( im ϕ ) ,
f ϕ ( r ) = A R l ( r ) r sin θ [ θ P l m ( cos θ ) ] exp ( im ϕ ) .
| A | 2 = c [ r 2 | R l ( r ) | 2 m 2 sin θ [ p l m ( cos θ ) ] 2 + r 2 | R l ( r ) | 2 [ θ p l m ( cos θ ) ] 2 sin θ ] d r d θ d ϕ .
V eff = c 2 I θ + I ϕ ,
I θ = P NL ( 1 ) θ f θ * ( r ) g θ θ d v n 2 ( r ) ,
I ϕ = P NL ( 1 ) ϕ f ϕ * ( r ) g ϕ ϕ d v n 2 ( r ) .
I θ = [ 3 8 r 2 | f θ ( r ) | 4 n 2 ( r ) g θ θ + 1 4 r 2 | f ϕ ( r ) | 2 | f θ ( r ) | 2 n 2 ( r ) sin 2 θ g θ θ ] d v , = 3 8 | A | 4 r 2 | R l ( r ) | 4 n 2 ( r ) m 4 sin 3 θ [ p l m ( cos θ ) ] 4 d r d θ d ϕ + 1 4 | A | 4 r 2 | R l ( r ) | 4 n 2 ( r ) m 2 sin θ [ p l m ( cos θ ) ] 2 [ θ p l m ( cos θ ) ] 2 d r d θ d ϕ , = π 4 m 2 | A | 4 ( 3 m 2 k 1 + 2 k 2 ) r 2 | R l ( r ) | 4 n 2 ( r ) d r ,
k 1 = 1 1 [ p l m ( x ) ] 4 ( 1 x 2 ) 2 d x ,
k 2 = 1 1 [ p l m ( x ) ] 2 [ ( 1 x 2 ) x p l m ( x ) x p l m ( x ) ] 2 ( 1 x 2 ) 2 d x .
I ϕ = [ 3 8 r 2 | f ϕ ( r ) | 4 n 2 ( r ) sin 2 θ g ϕ ϕ + 1 4 r 2 | f ϕ ( r ) | 2 | f θ ( r ) | 2 n 2 ( r ) sin 2 θ g ϕ ϕ ] d v , = 3 8 | A | 4 r 2 | R l ( r ) | 4 n 2 ( r ) [ θ p l m ( cos θ ) ] 4 sin θ d r d θ d ϕ + 1 4 | A | 4 r 2 | R l ( r ) | 4 n 2 ( r ) m 2 sin θ [ p l m ( cos θ ) ] 2 [ θ p l m ( cos θ ) ] 2 d r d θ d ϕ , = π 4 | A | 4 ( 3 k 3 + 2 m 2 k 2 ) r 2 | R l ( r ) | 4 n 2 ( r ) d r ,
k 3 = 1 1 [ ( 1 x 2 ) x p l m ( x ) x p l m ( x ) ] 4 ( 1 x 2 ) 2 d x .

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