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Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 6, Iss. 8 — Aug. 26, 2011
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Surface optomechanics: calculating optically excited acoustical whispering gallery modes in microspheres

John Zehnpfennig, Gaurav Bahl, Matthew Tomes, and Tal Carmon  »View Author Affiliations


Optics Express, Vol. 19, Issue 15, pp. 14240-14248 (2011)
http://dx.doi.org/10.1364/OE.19.014240


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Abstract

Stimulated Brillouin scattering recently allowed experimental excitation of surface acoustic resonances in micro-devices, enabling vibration at rates in the range of 50 MHz to 12 GHz. The experimental availability of such mechanical whispering gallery modes in photonic-MEMS raises questions on their structure and spectral distribution. Here we calculate the form and frequency of such vibrational surface whispering gallery modes, revealing diverse types of surface vibrations including longitudinal, transverse, and Rayleigh-type deformations. We parametrically investigate these various modes by changing their orders in the azimuthal, radial, and polar directions to reveal different vibrational structures including mechanical resonances that are localized near the interface with the environment where they can sense changes in the surroundings.

© 2011 OSA

1. Introduction

The acoustical density wave (ρ˜) and the optical waves (E˜) that are circulating in a sphere can be written in spherical coordinate system as
ρ˜=Aa(t)Ta(θ,r)ei(Mφaφϖat),E˜p=Ap(t)Tp(θ,r)ei(Mφpφϖpt),E˜S=AS(t)TS(θ,r)ei(MφSφϖSt).
(1)
Here A stands for the wave amplitudes, T for the wave distributions in the plane transverse with propagation, and ϖ for the angular frequencies. Subscript a, p, and S represent acoustical density, optical pump, and optical Stokes modes respectively. Similar to [15

15. P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of orbital angular momentum transfer between acoustic and optical vortices in optical fiber,” Phys. Rev. Lett. 96(4), 043604 (2006). [CrossRef] [PubMed]

], Mϕ corresponds to the angular momentum of the azimuthally circulating wave.

2. Calculated Rayleigh, transverse, and longitudinal whispering gallery modes

3. High-order whispering gallery modes

3.1 Increasing mode order in a direction transverse to propagation

In Fig. 3 we can see that mode [Mr,Mθ,Mφ] = [3,1,20] has minimal deformation on the interface and is hence attractive for applications where minimal dissipation by air is required. On the contrary, the [Mr,Mθ,Mφ] = [1,7,20] is extending into a large area on the interface; this fact might be useful for sensing changes in the surroundings.

3.2 Increasing mode order in directions parallel to propagation

The azimuthal order of the experimentally excited modes for a typical 100-micron radius sphere varies from Mϕ = 600 for backward Brillouin scattering excitation [11

11. M. Tomes and T. Carmon, “Photonic micro-electromechanical systems vibrating at X-band (11-GHz) rates,” Phys. Rev. Lett. 102(11), 113601 (2009). [CrossRef] [PubMed]

] and down to Mϕ~10 [13

13. G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Stimulated cavity-optomechanics,” arXiv Archive http://arxiv.org/abs/1106.2582(2011).

] when the excitation is by forward scattering. In what follows, we accordingly change Mϕ = along this span of azimuthal wavelengths.

In order to give a scale for the resonant frequencies and their dependency on Mϕ, we will now present the results from Fig. 4 in terms of frequencies for a r = 100-micron silica sphere. Figure 5
Fig. 5 Vibration frequencies for the various modes in a r = 100 micron silica sphere as a function of their azimuthal mode order. Left, with Mϕ typical to forward Brillouin excitation. Right, with Mϕ typical to backward Brillouin excitation. The shadowed regions estimate how high resonance frequencies can go for each of these modes via relying on high order transverse members of this mode family. The shadowed region is bounded in the Mϕ direction as estimation from momentum conservation consideration. We assume excitation with 1.5-micron telecom pump.
shows these vibrational frequencies. Regions where resonances were experimentally optically excited in silica spheres [11

11. M. Tomes and T. Carmon, “Photonic micro-electromechanical systems vibrating at X-band (11-GHz) rates,” Phys. Rev. Lett. 102(11), 113601 (2009). [CrossRef] [PubMed]

,13

13. G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Stimulated cavity-optomechanics,” arXiv Archive http://arxiv.org/abs/1106.2582(2011).

] are marked by shadows. The shadows extend to higher frequencies representing the effects of the high order transverse modes in accordance with observation.

4. Conclusion

References and links

1.

L. Rayleigh, “The problem of the whispering gallery,” Philos. Mag. 20, 1001–1004 (1910).

2.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003). [CrossRef] [PubMed]

3.

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

4.

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]

5.

L. Yang, T. Carmon, B. Min, S. M. Spillane, and K. J. Vahala, “Erbium-doped and Raman microlasers on a silicon chip fabricated by the sol-gel process,” Appl. Phys. Lett. 86(9), 091114 (2005). [CrossRef]

6.

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]

7.

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]

8.

E. Ippen and R. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21(11), 539–541 (1972). [CrossRef]

9.

R. Chiao, C. Townes, and B. Stoicheff, “Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,” Phys. Rev. Lett. 12(21), 592–595 (1964). [CrossRef]

10.

H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala, “Radiation-pressure-driven micro-mechanical oscillator,” Opt. Express 13(14), 5293–5301 (2005). [CrossRef] [PubMed]

11.

M. Tomes and T. Carmon, “Photonic micro-electromechanical systems vibrating at X-band (11-GHz) rates,” Phys. Rev. Lett. 102(11), 113601 (2009). [CrossRef] [PubMed]

12.

I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin lasing with a CaF2 whispering gallery mode resonator,” Phys. Rev. Lett. 102(4), 043902 (2009). [CrossRef] [PubMed]

13.

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Stimulated cavity-optomechanics,” arXiv Archive http://arxiv.org/abs/1106.2582(2011).

14.

M. J. Damzen, V. Vlad, A. Mocofanescu, and V. Babin, Stimulated Brillouin Scattering: Fundamentals and Applications, 1st ed. (Taylor & Francis, 2003).

15.

P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of orbital angular momentum transfer between acoustic and optical vortices in optical fiber,” Phys. Rev. Lett. 96(4), 043604 (2006). [CrossRef] [PubMed]

16.

N. Shibata, A. Nakazono, N. Taguchi, and S. Tanaka, “Forward Brillouin scattering in holey fibers,” IEEE Photon. Technol. Lett. 18(2), 412–414 (2006). [CrossRef]

17.

R. M. Shelby, M. D. Levenson, and P. W. Bayer, “Resolved forward Brillouin scattering in optical fibers,” Phys. Rev. Lett. 54(9), 939–942 (1985). [CrossRef] [PubMed]

18.

M. Grech, G. Riazuelo, D. Pesme, S. Weber, and V. T. Tikhonchuk, “Coherent forward stimulated-brillouin scattering of a spatially incoherent laser beam in a plasma and its effect on beam spray,” Phys. Rev. Lett. 102(15), 155001 (2009). [CrossRef] [PubMed]

19.

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Optomechanics with surface-acoustic-wave whispering-gallery modes,” Phys. Rev. Lett. 103(25), 257403 (2009). [CrossRef] [PubMed]

20.

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “Self-referenced stabilization of temperature of an optomechanical microresonator,” Phys. Rev. A 83(2), 021801 (2011). [CrossRef]

21.

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 2008), pp. xix, 613 p.

22.

F. V. Hunt, “Symmetry in the equations for electromechanical coupling,” in Thirty-Ninth Meeting of the Acoustical Society of America, (Acoustic Society of America, The Pennsylvania State College, 1950), p. 671.

23.

M. Oxborrow, “How to simulate the whispering-gallery modes of dielectric microresonators in FEMLAB/COMSOL,” in Laser Resonators and Beam Control IX (SPIE, 2007).

24.

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett. 95(3), 033901 (2005). [CrossRef] [PubMed]

25.

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94(22), 223902 (2005). [CrossRef] [PubMed]

26.

M. L. Povinelli, M. Loncar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett. 30(22), 3042–3044 (2005). [CrossRef] [PubMed]

27.

Q. Lin, X. Jiang, M. Eichenfield, R. Camacho, P. Herring, K. Vahala, and O. Painter, “Opto-mechanical oscillations in a double-disk microcavity,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference (OSA, 2009), p. CMKK1.

28.

X. Jiang, Q. Lin, J. Rosenberg, K. Vahala, and O. Painter, “High-Q double-disk microcavities for cavity optomechanics,” Opt. Express 17(23), 20911–20919 (2009). [CrossRef] [PubMed]

29.

S. S. Li, Y. W. Lin, Y. Xie, Z. Ren, and C. T. C. Nguyen, “Micromechanical 'hollow-disk' ring resonators,” in 17th IEEE International Conference on Micro Electro Mechanical Systems (IEEE, 2004), pp. 821–824.

30.

S. Nikolaou, N. D. Kingsley, G. E. Ponchak, J. Papapolymerou, and M. M. Tentzeris, “UWB elliptical monopoles with a reconfigurable band notch using MEMS switches actuated without bias lines,” IEEE Trans. Antenn. Propag. 57(8), 2242–2251 (2009). [CrossRef]

31.

M. Thompson and H. Stone, Surface-Launched Acoustic Wave Sensors: Chemical Sensing and Thin-Film Characterization, Chem. Analysis: A Series of Monographs on Analytical Chemistry and its Applications (John Wiley & Sons, 1997), p. 216.

32.

T. Carmon and K. J. Vahala, “Modal spectroscopy of optoexcited vibrations of a micron-scale on-chip resonator at greater than 1 GHz frequency,” Phys. Rev. Lett. 98(12), 123901 (2007). [CrossRef] [PubMed]

33.

T. Carmon, M. C. Cross, and K. J. Vahala, “Chaotic quivering of micron-scaled on-chip resonators excited by centrifugal optical pressure,” Phys. Rev. Lett. 98(16), 167203 (2007). [CrossRef] [PubMed]

34.

L. Rayleigh, “On waves propagated along the plane surface of an elastic solid,” Proc. Lond. Math. Soc. s1-17(1), 4–11 (1885). [CrossRef]

35.

T. Franke, A. R. Abate, D. A. Weitz, and A. Wixforth, “Surface acoustic wave (SAW) directed droplet flow in microfluidics for PDMS devices,” Lab Chip 9(18), 2625–2627 (2009). [CrossRef] [PubMed]

36.

COMSOL, COMSOL Multiphysics with Matlab, COMSOL Group, Burlington, MA, 2008–2010.

37.

T. Carmon, H. G. L. Schwefel, L. Yang, M. Oxborrow, A. D. Stone, and K. J. Vahala, “Static envelope patterns in composite resonances generated by level crossing in optical toroidal microcavities,” Phys. Rev. Lett. 100(10), 103905 (2008). [CrossRef] [PubMed]

38.

A. Savchenkov, A. Matsko, V. Ilchenko, D. Strekalov, and L. Maleki, “Direct observation of stopped light in a whispering-gallery-mode microresonator,” Phys. Rev. A 76(2), 023816 (2007). [CrossRef]

39.

L. Kinsler, A. Frey, A. Coppens, and J. Sanders, Fundamentals of Acoustics, 3rd ed. (John Wiley & Sons, Inc., 1999).

40.

D. S. Ballantine, R. M. White, S. J. Martin, A. J. Ricco, E. T. Zellers, G. C. Frye, and H. Wohltjen, Acoustic Wave Sensors: Theory, Design, and Physico-Chemical Applications, App. of Modern Acoustics (Academic Press, 1997).

41.

P. G. Malischewsky and P. C. Vinh, “Improved approximations of the Rayleigh wave velocity,” J. Thermoplastic Compos. Mater. 21(4), 337–352 (2008). [CrossRef]

42.

V. Ilchenko, P. Volikov, V. Velichansky, F. Treussart, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Commun. 145(1-6), 86–90 (1998). [CrossRef]

43.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(6), 066611 (2002). [CrossRef] [PubMed]

OCIS Codes
(240.4350) Optics at surfaces : Nonlinear optics at surfaces
(240.6690) Optics at surfaces : Surface waves
(290.1350) Scattering : Backscattering
(290.2558) Scattering : Forward scattering
(280.4788) Remote sensing and sensors : Optical sensing and sensors

ToC Category:
Optics at Surfaces

History
Original Manuscript: May 16, 2011
Revised Manuscript: June 15, 2011
Manuscript Accepted: June 20, 2011
Published: July 11, 2011

Virtual Issues
Vol. 6, Iss. 8 Virtual Journal for Biomedical Optics

Citation
John Zehnpfennig, Gaurav Bahl, Matthew Tomes, and Tal Carmon, "Surface optomechanics: calculating optically excited acoustical whispering gallery modes in microspheres," Opt. Express 19, 14240-14248 (2011)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-19-15-14240


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References

  1. L. Rayleigh, “The problem of the whispering gallery,” Philos. Mag. 20, 1001–1004 (1910).
  2. D. K. Armani, T. J. Kippenberg, S. M. Spillane, K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003). [CrossRef] [PubMed]
  3. K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003). [CrossRef] [PubMed]
  4. S. M. Spillane, T. J. Kippenberg, K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415(6872), 621–623 (2002). [CrossRef] [PubMed]
  5. L. Yang, T. Carmon, B. Min, S. M. Spillane, K. J. Vahala, “Erbium-doped and Raman microlasers on a silicon chip fabricated by the sol-gel process,” Appl. Phys. Lett. 86(9), 091114 (2005). [CrossRef]
  6. T. J. Kippenberg, S. M. Spillane, K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93(8), 083904 (2004). [CrossRef] [PubMed]
  7. T. Carmon, K. J. Vahala, “Visible continuous emission from a silica microphotonic device by third-harmonic generation,” Nat. Phys. 3(6), 430–435 (2007). [CrossRef]
  8. E. Ippen, R. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21(11), 539–541 (1972). [CrossRef]
  9. R. Chiao, C. Townes, B. Stoicheff, “Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,” Phys. Rev. Lett. 12(21), 592–595 (1964). [CrossRef]
  10. H. Rokhsari, T. J. Kippenberg, T. Carmon, K. J. Vahala, “Radiation-pressure-driven micro-mechanical oscillator,” Opt. Express 13(14), 5293–5301 (2005). [CrossRef] [PubMed]
  11. M. Tomes, T. Carmon, “Photonic micro-electromechanical systems vibrating at X-band (11-GHz) rates,” Phys. Rev. Lett. 102(11), 113601 (2009). [CrossRef] [PubMed]
  12. I. S. Grudinin, A. B. Matsko, L. Maleki, “Brillouin lasing with a CaF2 whispering gallery mode resonator,” Phys. Rev. Lett. 102(4), 043902 (2009). [CrossRef] [PubMed]
  13. G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Stimulated cavity-optomechanics,” arXiv Archive http://arxiv.org/abs/1106.2582(2011) .
  14. M. J. Damzen, V. Vlad, A. Mocofanescu, and V. Babin, Stimulated Brillouin Scattering: Fundamentals and Applications, 1st ed. (Taylor & Francis, 2003).
  15. P. Z. Dashti, F. Alhassen, H. P. Lee, “Observation of orbital angular momentum transfer between acoustic and optical vortices in optical fiber,” Phys. Rev. Lett. 96(4), 043604 (2006). [CrossRef] [PubMed]
  16. N. Shibata, A. Nakazono, N. Taguchi, S. Tanaka, “Forward Brillouin scattering in holey fibers,” IEEE Photon. Technol. Lett. 18(2), 412–414 (2006). [CrossRef]
  17. R. M. Shelby, M. D. Levenson, P. W. Bayer, “Resolved forward Brillouin scattering in optical fibers,” Phys. Rev. Lett. 54(9), 939–942 (1985). [CrossRef] [PubMed]
  18. M. Grech, G. Riazuelo, D. Pesme, S. Weber, V. T. Tikhonchuk, “Coherent forward stimulated-brillouin scattering of a spatially incoherent laser beam in a plasma and its effect on beam spray,” Phys. Rev. Lett. 102(15), 155001 (2009). [CrossRef] [PubMed]
  19. A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, L. Maleki, “Optomechanics with surface-acoustic-wave whispering-gallery modes,” Phys. Rev. Lett. 103(25), 257403 (2009). [CrossRef] [PubMed]
  20. A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, L. Maleki, “Self-referenced stabilization of temperature of an optomechanical microresonator,” Phys. Rev. A 83(2), 021801 (2011). [CrossRef]
  21. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 2008), pp. xix, 613 p.
  22. F. V. Hunt, “Symmetry in the equations for electromechanical coupling,” in Thirty-Ninth Meeting of the Acoustical Society of America, (Acoustic Society of America, The Pennsylvania State College, 1950), p. 671.
  23. M. Oxborrow, “How to simulate the whispering-gallery modes of dielectric microresonators in FEMLAB/COMSOL,” in Laser Resonators and Beam Control IX (SPIE, 2007).
  24. T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett. 95(3), 033901 (2005). [CrossRef] [PubMed]
  25. T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94(22), 223902 (2005). [CrossRef] [PubMed]
  26. M. L. Povinelli, M. Loncar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett. 30(22), 3042–3044 (2005). [CrossRef] [PubMed]
  27. Q. Lin, X. Jiang, M. Eichenfield, R. Camacho, P. Herring, K. Vahala, and O. Painter, “Opto-mechanical oscillations in a double-disk microcavity,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference (OSA, 2009), p. CMKK1.
  28. X. Jiang, Q. Lin, J. Rosenberg, K. Vahala, O. Painter, “High-Q double-disk microcavities for cavity optomechanics,” Opt. Express 17(23), 20911–20919 (2009). [CrossRef] [PubMed]
  29. S. S. Li, Y. W. Lin, Y. Xie, Z. Ren, and C. T. C. Nguyen, “Micromechanical 'hollow-disk' ring resonators,” in 17th IEEE International Conference on Micro Electro Mechanical Systems (IEEE, 2004), pp. 821–824.
  30. S. Nikolaou, N. D. Kingsley, G. E. Ponchak, J. Papapolymerou, M. M. Tentzeris, “UWB elliptical monopoles with a reconfigurable band notch using MEMS switches actuated without bias lines,” IEEE Trans. Antenn. Propag. 57(8), 2242–2251 (2009). [CrossRef]
  31. M. Thompson and H. Stone, Surface-Launched Acoustic Wave Sensors: Chemical Sensing and Thin-Film Characterization, Chem. Analysis: A Series of Monographs on Analytical Chemistry and its Applications (John Wiley & Sons, 1997), p. 216.
  32. T. Carmon, K. J. Vahala, “Modal spectroscopy of optoexcited vibrations of a micron-scale on-chip resonator at greater than 1 GHz frequency,” Phys. Rev. Lett. 98(12), 123901 (2007). [CrossRef] [PubMed]
  33. T. Carmon, M. C. Cross, K. J. Vahala, “Chaotic quivering of micron-scaled on-chip resonators excited by centrifugal optical pressure,” Phys. Rev. Lett. 98(16), 167203 (2007). [CrossRef] [PubMed]
  34. L. Rayleigh, “On waves propagated along the plane surface of an elastic solid,” Proc. Lond. Math. Soc. s1-17(1), 4–11 (1885). [CrossRef]
  35. T. Franke, A. R. Abate, D. A. Weitz, A. Wixforth, “Surface acoustic wave (SAW) directed droplet flow in microfluidics for PDMS devices,” Lab Chip 9(18), 2625–2627 (2009). [CrossRef] [PubMed]
  36. COMSOL, COMSOL Multiphysics with Matlab, COMSOL Group, Burlington, MA, 2008–2010.
  37. T. Carmon, H. G. L. Schwefel, L. Yang, M. Oxborrow, A. D. Stone, K. J. Vahala, “Static envelope patterns in composite resonances generated by level crossing in optical toroidal microcavities,” Phys. Rev. Lett. 100(10), 103905 (2008). [CrossRef] [PubMed]
  38. A. Savchenkov, A. Matsko, V. Ilchenko, D. Strekalov, L. Maleki, “Direct observation of stopped light in a whispering-gallery-mode microresonator,” Phys. Rev. A 76(2), 023816 (2007). [CrossRef]
  39. L. Kinsler, A. Frey, A. Coppens, and J. Sanders, Fundamentals of Acoustics, 3rd ed. (John Wiley & Sons, Inc., 1999).
  40. D. S. Ballantine, R. M. White, S. J. Martin, A. J. Ricco, E. T. Zellers, G. C. Frye, and H. Wohltjen, Acoustic Wave Sensors: Theory, Design, and Physico-Chemical Applications, App. of Modern Acoustics (Academic Press, 1997).
  41. P. G. Malischewsky, P. C. Vinh, “Improved approximations of the Rayleigh wave velocity,” J. Thermoplastic Compos. Mater. 21(4), 337–352 (2008). [CrossRef]
  42. V. Ilchenko, P. Volikov, V. Velichansky, F. Treussart, V. Lefevre-Seguin, J. M. Raimond, S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Commun. 145(1-6), 86–90 (1998). [CrossRef]
  43. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(6), 066611 (2002). [CrossRef] [PubMed]

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