Abstract

Quantitative measurements of the vibrational eigenmodes in ultrahigh-Q silica microspheres are reported. The modes are excited via radiation-pressure-induced dynamical backaction of light confined in the optical whispering-gallery modes of the microspheres (i.e., via the parametric oscillation instability). Two families of modes are studied and their frequency dependence on sphere size investigated. The measured frequencies are in good agreement both with Lamb’s theory and numerical finite-element simulation and are found to be proportional to the sphere’s inverse diameter. In addition, the quality factors of the vibrational modes are studied.

© 2007 Optical Society of America

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References

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2006 (3)

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, Nature 444, 67 (2006).
[CrossRef] [PubMed]

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, Nature 444, 71 (2006).
[CrossRef] [PubMed]

A. Schliesser, P. Del'Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, Phys. Rev. Lett. 97, 243905 (2006).
[CrossRef]

2005 (3)

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef] [PubMed]

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, Phys. Rev. Lett. 94, 223902 (2005).
[CrossRef] [PubMed]

H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala, Opt. Express 13, 5293 (2005).
[CrossRef] [PubMed]

2003 (4)

K. J. Vahala, Nature 424, 839 (2003).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, Nature 421, 925 (2003).
[CrossRef] [PubMed]

M. H. Kuok, H. S. Lim, S. C. Ng, N. N. Liu, and Z. K. Wang, Phys. Rev. Lett. 90, 255502 (2003).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef] [PubMed]

2001 (1)

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, Phys. Lett. A 287, 331 (2001).
[CrossRef]

1982 (1)

A. Tamura, K. Higeta, and T. Ichinokawa, J. Phys. C 15, 4975 (1982).
[CrossRef]

1981 (1)

N. Nishiguchi and T. Sakuma, Solid State Commun. 38, 1073 (1981).
[CrossRef]

1882 (1)

H. Lamb, Proc. London Math. Soc. 13, 189 (1882).
[CrossRef]

J. Phys. C (1)

A. Tamura, K. Higeta, and T. Ichinokawa, J. Phys. C 15, 4975 (1982).
[CrossRef]

Nature (4)

K. J. Vahala, Nature 424, 839 (2003).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, Nature 421, 925 (2003).
[CrossRef] [PubMed]

S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, Nature 444, 67 (2006).
[CrossRef] [PubMed]

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, Nature 444, 71 (2006).
[CrossRef] [PubMed]

Opt. Express (1)

Phys. Lett. A (1)

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, Phys. Lett. A 287, 331 (2001).
[CrossRef]

Phys. Rev. Lett. (5)

A. Schliesser, P. Del'Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, Phys. Rev. Lett. 97, 243905 (2006).
[CrossRef]

M. H. Kuok, H. S. Lim, S. C. Ng, N. N. Liu, and Z. K. Wang, Phys. Rev. Lett. 90, 255502 (2003).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef] [PubMed]

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, Phys. Rev. Lett. 95, 033901 (2005).
[CrossRef] [PubMed]

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, Phys. Rev. Lett. 94, 223902 (2005).
[CrossRef] [PubMed]

Proc. London Math. Soc. (1)

H. Lamb, Proc. London Math. Soc. 13, 189 (1882).
[CrossRef]

Solid State Commun. (1)

N. Nishiguchi and T. Sakuma, Solid State Commun. 38, 1073 (1981).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Radio-frequency spectrum of the photocurrent induced by the transmission of the fiber taper when coupled to a silica microsphere with a diameter of 49 μ m . Two families of spheroidal mechanical modes could be driven regeneratively. Specifically, the modes are identified as ν ( 1 , 2 ) (a quadrupole mode) and ν ( 1 , 0 ) (a radial breathing mode). The launched power in this experiment was 600 μ W and was sufficient to exceed the threshold for parametric oscillation instability. The inset shows the scanning electron microscope image of the microsphere.

Fig. 2
Fig. 2

Experimentally measured frequencies of the spheroidal modes, with stars denoting the ν ( 1 , 2 ) mode and dots denoting the ν ( 1 , 0 ) mode. Numerically calculated eigenfrequencies of these modes are shown as the solid line ( ν ( 1 , 2 ) ) and the dashed line ( ν ( 1 , 0 ) ) . The spheroidal mode ν ( 1 , 1 ) has a frequency that lies between ν ( 1 , 2 ) and ν ( 1 , 0 ) and is not experimentally observed (dotted line). The inset shows the relationship between the frequencies of the eigenmodes and the inverse diameter of the silica microspheres.

Fig. 3
Fig. 3

Finite-element modeling of three spheroidal modes, ν ( 1 , 2 ) (left), and ν ( 1 , 0 ) (right) of a silica microsphere with its von Mises stress (color coded), and strain (greatly exaggerated for clarity).

Equations (3)

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ρ u ̈ = ( λ + 2 μ ) ( u ) μ × ( × u ) ,
ϕ k ( r , t ) = l , m A k ( l , m ) j l ( 2 π ν n , l , m r V k ) Y l m ( θ , ψ ) e 2 π i ν n , l , m t ,
tan ( h R ) h R 1 1 1 4 ( k 2 h 2 ) h 2 R 2 = 0 ,

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