Abstract

The theory of slow acoustic modes propagating along the optical fiber and being controlled by the nanoscale variation of the effective fiber radius (analogous to the theory of slow optical whispering gallery modes) is developed. Surprisingly, it is shown that, in addition to acoustic bottle resonators (which are similar to optical bottle resonators), there exist antibottle resonators, the neck-shaped deformations of the fiber that can fully confine acoustic modes. It is also shown that an eigenfrequency of the mechanical vibrations of a silica parabolic bottle resonator can match the separation between the eigenfrequencies of a series of its optical modes, thereby enabling the resonant mechanical excitation of these series. The developed theory paves the groundwork for slow-mode optomechanics in an optical fiber.

© 2017 Optical Society of America

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References

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  1. M. Sumetsky, Opt. Lett. 29, 8 (2004).
    [Crossref]
  2. Y. Louyer, D. Meschede, and A. Rauschenbeutel, Phys. Rev. A 72, 031801(R) (2005).
    [Crossref]
  3. M. Pöllinger and A. Rauschenbeutel, Opt. Express 18, 17764 (2010).
    [Crossref]
  4. A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, Nat. Photonics 5, 293 (2011).
    [Crossref]
  5. V. Dvoyrin and M. Sumetsky, Opt. Lett. 41, 5547 (2016).
    [Crossref]
  6. M. N. M. Nasir, G. S. Murugan, and M. N. Zervas, J. Opt. Soc. Am. B 33, 1963 (2016).
    [Crossref]
  7. J. Volz, M. Scheucher, C. Junge, and A. Rauschenbeutel, Nat. Photonics 8, 965 (2014).
    [Crossref]
  8. M. Sumetsky, Phys. Rev. Lett. 111, 163901 (2013).
    [Crossref]
  9. M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, Laser Photon. Rev. 10, 603 (2016).
    [Crossref]
  10. M. Sumetsky, Nanophotonics 2, 393 (2013).
    [Crossref]
  11. N. A. Toropov and M. Sumetsky, Opt. Lett. 41, 2278 (2016).
    [Crossref]
  12. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).
    [Crossref]
  13. A. W. Snyder and J. Love, Optical Waveguide Theory (Chapman & Hall, 1983).
  14. M. Sumetsky and J. M. Fini, Opt. Express 19, 26470 (2011).
    [Crossref]
  15. L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Pergamon, 1970).
  16. These modes are of special importance for optomechanical applications, since they can be excited by the axially symmetric radiation pressure of optical WGMs modulated with the frequency of acoustic vibrations.
  17. M. Onoe, H. D. McNiven, and R. D. Mindlin, Trans. ASME. J. Appl. Mech. 29, 729 (1962).
    [Crossref]
  18. J. Zemanek, J. Acoust. Soc. Am. 51, 265 (1972).
    [Crossref]
  19. G. N. Greaves, A. L. Greer, R. S. Lakes, and T. Rouxel, Nat. Mater. 10, 823 (2011).
    [Crossref]
  20. M. Fukuhara, A. Sanpei, and K. Shibuki, J. Mater. Sci. 32, 1207 (1997).
    [Crossref]
  21. X. Sun, K. Y. Fong, C. Xiong, W. H. P. Pernice, and H. X. Tang, Opt. Express 19, 22316 (2011).
    [Crossref]
  22. R. O. Pohl, X. Liu, and E. Thompson, Rev. Mod. Phys. 74, 991 (2002).
    [Crossref]
  23. A. D. Fefferman, R. O. Pohl, A. T. Zehnder, and J. M. Parpia, Phys. Rev. Lett. 100, 195501 (2008).
    [Crossref]
  24. M. Sumetsky, Sci. Rep. 5, 18569 (2015).
    [Crossref]
  25. S. Kohler, J. Lehmann, and P. Hänggi, Phys. Rep. 406, 379 (2005).
    [Crossref]
  26. M. Y. Sumetsky and M. L. Felshtyn, J. Exp. Theor. Phys. Lett. 53, 24 (1991).
  27. M. Y. Sumetskii, Sov. Phys. J. Exp. Theor. Phys. 62, 355 (1985).
  28. B. I. Ivlev and V. I. Mel’nikov, Sov. Phys. J. Exp. Theor. Phys. 63, 1295 (1986).

2016 (4)

2015 (1)

M. Sumetsky, Sci. Rep. 5, 18569 (2015).
[Crossref]

2014 (2)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

J. Volz, M. Scheucher, C. Junge, and A. Rauschenbeutel, Nat. Photonics 8, 965 (2014).
[Crossref]

2013 (2)

M. Sumetsky, Phys. Rev. Lett. 111, 163901 (2013).
[Crossref]

M. Sumetsky, Nanophotonics 2, 393 (2013).
[Crossref]

2011 (4)

G. N. Greaves, A. L. Greer, R. S. Lakes, and T. Rouxel, Nat. Mater. 10, 823 (2011).
[Crossref]

M. Sumetsky and J. M. Fini, Opt. Express 19, 26470 (2011).
[Crossref]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, Nat. Photonics 5, 293 (2011).
[Crossref]

X. Sun, K. Y. Fong, C. Xiong, W. H. P. Pernice, and H. X. Tang, Opt. Express 19, 22316 (2011).
[Crossref]

2010 (1)

2008 (1)

A. D. Fefferman, R. O. Pohl, A. T. Zehnder, and J. M. Parpia, Phys. Rev. Lett. 100, 195501 (2008).
[Crossref]

2005 (2)

S. Kohler, J. Lehmann, and P. Hänggi, Phys. Rep. 406, 379 (2005).
[Crossref]

Y. Louyer, D. Meschede, and A. Rauschenbeutel, Phys. Rev. A 72, 031801(R) (2005).
[Crossref]

2004 (1)

2002 (1)

R. O. Pohl, X. Liu, and E. Thompson, Rev. Mod. Phys. 74, 991 (2002).
[Crossref]

1997 (1)

M. Fukuhara, A. Sanpei, and K. Shibuki, J. Mater. Sci. 32, 1207 (1997).
[Crossref]

1991 (1)

M. Y. Sumetsky and M. L. Felshtyn, J. Exp. Theor. Phys. Lett. 53, 24 (1991).

1986 (1)

B. I. Ivlev and V. I. Mel’nikov, Sov. Phys. J. Exp. Theor. Phys. 63, 1295 (1986).

1985 (1)

M. Y. Sumetskii, Sov. Phys. J. Exp. Theor. Phys. 62, 355 (1985).

1972 (1)

J. Zemanek, J. Acoust. Soc. Am. 51, 265 (1972).
[Crossref]

1962 (1)

M. Onoe, H. D. McNiven, and R. D. Mindlin, Trans. ASME. J. Appl. Mech. 29, 729 (1962).
[Crossref]

Asano, M.

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, Laser Photon. Rev. 10, 603 (2016).
[Crossref]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

Chen, W.

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, Laser Photon. Rev. 10, 603 (2016).
[Crossref]

Dvoyrin, V.

Fefferman, A. D.

A. D. Fefferman, R. O. Pohl, A. T. Zehnder, and J. M. Parpia, Phys. Rev. Lett. 100, 195501 (2008).
[Crossref]

Felshtyn, M. L.

M. Y. Sumetsky and M. L. Felshtyn, J. Exp. Theor. Phys. Lett. 53, 24 (1991).

Fini, J. M.

Fong, K. Y.

Fukuhara, M.

M. Fukuhara, A. Sanpei, and K. Shibuki, J. Mater. Sci. 32, 1207 (1997).
[Crossref]

Greaves, G. N.

G. N. Greaves, A. L. Greer, R. S. Lakes, and T. Rouxel, Nat. Mater. 10, 823 (2011).
[Crossref]

Greer, A. L.

G. N. Greaves, A. L. Greer, R. S. Lakes, and T. Rouxel, Nat. Mater. 10, 823 (2011).
[Crossref]

Hänggi, P.

S. Kohler, J. Lehmann, and P. Hänggi, Phys. Rep. 406, 379 (2005).
[Crossref]

Ikuta, R.

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, Laser Photon. Rev. 10, 603 (2016).
[Crossref]

Ilchenko, V. S.

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, Nat. Photonics 5, 293 (2011).
[Crossref]

Imoto, N.

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, Laser Photon. Rev. 10, 603 (2016).
[Crossref]

Ivlev, B. I.

B. I. Ivlev and V. I. Mel’nikov, Sov. Phys. J. Exp. Theor. Phys. 63, 1295 (1986).

Junge, C.

J. Volz, M. Scheucher, C. Junge, and A. Rauschenbeutel, Nat. Photonics 8, 965 (2014).
[Crossref]

Kippenberg, T. J.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

Kohler, S.

S. Kohler, J. Lehmann, and P. Hänggi, Phys. Rep. 406, 379 (2005).
[Crossref]

Lakes, R. S.

G. N. Greaves, A. L. Greer, R. S. Lakes, and T. Rouxel, Nat. Mater. 10, 823 (2011).
[Crossref]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Pergamon, 1970).

Lehmann, J.

S. Kohler, J. Lehmann, and P. Hänggi, Phys. Rep. 406, 379 (2005).
[Crossref]

Liang, W.

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, Nat. Photonics 5, 293 (2011).
[Crossref]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Pergamon, 1970).

Liu, X.

R. O. Pohl, X. Liu, and E. Thompson, Rev. Mod. Phys. 74, 991 (2002).
[Crossref]

Louyer, Y.

Y. Louyer, D. Meschede, and A. Rauschenbeutel, Phys. Rev. A 72, 031801(R) (2005).
[Crossref]

Love, J.

A. W. Snyder and J. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

Maleki, L.

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, Nat. Photonics 5, 293 (2011).
[Crossref]

Marquardt, F.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

Matsko, A. B.

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, Nat. Photonics 5, 293 (2011).
[Crossref]

McNiven, H. D.

M. Onoe, H. D. McNiven, and R. D. Mindlin, Trans. ASME. J. Appl. Mech. 29, 729 (1962).
[Crossref]

Mel’nikov, V. I.

B. I. Ivlev and V. I. Mel’nikov, Sov. Phys. J. Exp. Theor. Phys. 63, 1295 (1986).

Meschede, D.

Y. Louyer, D. Meschede, and A. Rauschenbeutel, Phys. Rev. A 72, 031801(R) (2005).
[Crossref]

Mindlin, R. D.

M. Onoe, H. D. McNiven, and R. D. Mindlin, Trans. ASME. J. Appl. Mech. 29, 729 (1962).
[Crossref]

Murugan, G. S.

Nasir, M. N. M.

Onoe, M.

M. Onoe, H. D. McNiven, and R. D. Mindlin, Trans. ASME. J. Appl. Mech. 29, 729 (1962).
[Crossref]

Özdemir, S. K.

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, Laser Photon. Rev. 10, 603 (2016).
[Crossref]

Parpia, J. M.

A. D. Fefferman, R. O. Pohl, A. T. Zehnder, and J. M. Parpia, Phys. Rev. Lett. 100, 195501 (2008).
[Crossref]

Pernice, W. H. P.

Pohl, R. O.

A. D. Fefferman, R. O. Pohl, A. T. Zehnder, and J. M. Parpia, Phys. Rev. Lett. 100, 195501 (2008).
[Crossref]

R. O. Pohl, X. Liu, and E. Thompson, Rev. Mod. Phys. 74, 991 (2002).
[Crossref]

Pöllinger, M.

Rauschenbeutel, A.

J. Volz, M. Scheucher, C. Junge, and A. Rauschenbeutel, Nat. Photonics 8, 965 (2014).
[Crossref]

M. Pöllinger and A. Rauschenbeutel, Opt. Express 18, 17764 (2010).
[Crossref]

Y. Louyer, D. Meschede, and A. Rauschenbeutel, Phys. Rev. A 72, 031801(R) (2005).
[Crossref]

Rouxel, T.

G. N. Greaves, A. L. Greer, R. S. Lakes, and T. Rouxel, Nat. Mater. 10, 823 (2011).
[Crossref]

Sanpei, A.

M. Fukuhara, A. Sanpei, and K. Shibuki, J. Mater. Sci. 32, 1207 (1997).
[Crossref]

Savchenkov, A. A.

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, Nat. Photonics 5, 293 (2011).
[Crossref]

Scheucher, M.

J. Volz, M. Scheucher, C. Junge, and A. Rauschenbeutel, Nat. Photonics 8, 965 (2014).
[Crossref]

Seidel, D.

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, Nat. Photonics 5, 293 (2011).
[Crossref]

Shibuki, K.

M. Fukuhara, A. Sanpei, and K. Shibuki, J. Mater. Sci. 32, 1207 (1997).
[Crossref]

Snyder, A. W.

A. W. Snyder and J. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

Sumetskii, M. Y.

M. Y. Sumetskii, Sov. Phys. J. Exp. Theor. Phys. 62, 355 (1985).

Sumetsky, M.

Sumetsky, M. Y.

M. Y. Sumetsky and M. L. Felshtyn, J. Exp. Theor. Phys. Lett. 53, 24 (1991).

Sun, X.

Takeuchi, Y.

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, Laser Photon. Rev. 10, 603 (2016).
[Crossref]

Tang, H. X.

Thompson, E.

R. O. Pohl, X. Liu, and E. Thompson, Rev. Mod. Phys. 74, 991 (2002).
[Crossref]

Toropov, N. A.

Volz, J.

J. Volz, M. Scheucher, C. Junge, and A. Rauschenbeutel, Nat. Photonics 8, 965 (2014).
[Crossref]

Xiong, C.

Yamamoto, T.

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, Laser Photon. Rev. 10, 603 (2016).
[Crossref]

Yang, L.

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, Laser Photon. Rev. 10, 603 (2016).
[Crossref]

Zehnder, A. T.

A. D. Fefferman, R. O. Pohl, A. T. Zehnder, and J. M. Parpia, Phys. Rev. Lett. 100, 195501 (2008).
[Crossref]

Zemanek, J.

J. Zemanek, J. Acoust. Soc. Am. 51, 265 (1972).
[Crossref]

Zervas, M. N.

J. Acoust. Soc. Am. (1)

J. Zemanek, J. Acoust. Soc. Am. 51, 265 (1972).
[Crossref]

J. Exp. Theor. Phys. Lett. (1)

M. Y. Sumetsky and M. L. Felshtyn, J. Exp. Theor. Phys. Lett. 53, 24 (1991).

J. Mater. Sci. (1)

M. Fukuhara, A. Sanpei, and K. Shibuki, J. Mater. Sci. 32, 1207 (1997).
[Crossref]

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

Laser Photon. Rev. (1)

M. Asano, Y. Takeuchi, W. Chen, Ş. K. Özdemir, R. Ikuta, N. Imoto, L. Yang, and T. Yamamoto, Laser Photon. Rev. 10, 603 (2016).
[Crossref]

Nanophotonics (1)

M. Sumetsky, Nanophotonics 2, 393 (2013).
[Crossref]

Nat. Mater. (1)

G. N. Greaves, A. L. Greer, R. S. Lakes, and T. Rouxel, Nat. Mater. 10, 823 (2011).
[Crossref]

Nat. Photonics (2)

J. Volz, M. Scheucher, C. Junge, and A. Rauschenbeutel, Nat. Photonics 8, 965 (2014).
[Crossref]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, Nat. Photonics 5, 293 (2011).
[Crossref]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rep. (1)

S. Kohler, J. Lehmann, and P. Hänggi, Phys. Rep. 406, 379 (2005).
[Crossref]

Phys. Rev. A (1)

Y. Louyer, D. Meschede, and A. Rauschenbeutel, Phys. Rev. A 72, 031801(R) (2005).
[Crossref]

Phys. Rev. Lett. (2)

M. Sumetsky, Phys. Rev. Lett. 111, 163901 (2013).
[Crossref]

A. D. Fefferman, R. O. Pohl, A. T. Zehnder, and J. M. Parpia, Phys. Rev. Lett. 100, 195501 (2008).
[Crossref]

Rev. Mod. Phys. (2)

R. O. Pohl, X. Liu, and E. Thompson, Rev. Mod. Phys. 74, 991 (2002).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

Sci. Rep. (1)

M. Sumetsky, Sci. Rep. 5, 18569 (2015).
[Crossref]

Sov. Phys. J. Exp. Theor. Phys. (2)

M. Y. Sumetskii, Sov. Phys. J. Exp. Theor. Phys. 62, 355 (1985).

B. I. Ivlev and V. I. Mel’nikov, Sov. Phys. J. Exp. Theor. Phys. 63, 1295 (1986).

Trans. ASME. J. Appl. Mech. (1)

M. Onoe, H. D. McNiven, and R. D. Mindlin, Trans. ASME. J. Appl. Mech. 29, 729 (1962).
[Crossref]

Other (3)

L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Pergamon, 1970).

These modes are of special importance for optomechanical applications, since they can be excited by the axially symmetric radiation pressure of optical WGMs modulated with the frequency of acoustic vibrations.

A. W. Snyder and J. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

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Figures (3)

Fig. 1.
Fig. 1.

Illustrations of (a) optical bottle resonator, (b) acoustic bottle resonator, and (c) acoustic antibottle resonator.

Fig. 2.
Fig. 2.

Dimensionless parameters Λnr02 as a function of α=VT/VL for modes with n=0,1,2 and 3 (LP01, LP02, and LP03). The vertical dashed line corresponds to α=0.631 for the silica. Left: cross-sectional distribution of mode amplitudes for a silica fiber.

Fig. 3.
Fig. 3.

(a) Radius variation of parabolic bottle resonator with the axial radius R0,1=824  m. The acoustic cutoff frequency ν0,1(ac)=255  MHz of this resonator is equal to the separation of its optical eigenfrequencies along the axial quantum number q. (b) Axial distribution of optical and acoustic modes in the same resonator.

Equations (13)

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

2Ut2V2[2Uz2+1ρρ(ρUρ)+1ρ22Uφ2]=0,
ΔQm,n+(2πνm,n/V)2Qm,n=0,
2Ψm,nz2+βm,n2(z)Ψm,n=0.
βm,n2(z)=Λm,n(Δνν+Δr(z)r0),
Λm,n=Λm,n(op)=2(km,n(op))2,km,n(op)=2πνm,n(op)nr/c,
Δr(z)=z2/(2R0).
Ψm,n,q(z)=exp(z22ζm,n2)Hq(zζm,n),ζm,n=(r0R0Λm,n)1/4.
νm,n,q=νm,n[1+(2π)1(Λm,nr0R0)1/2(q+12)].
[2β2(2πν/VT)2]2ϒ(hr)+4  h2β2ϒ(kr)2(2πν/VT)2h2=0,h=[α2Ω2β2]1/2,k=[Ω2β2]1/2,
ΩnJ0(αΩn)2αJ1(αΩn)=0.
U0,n(L)(z,ρ)=J1(αΩnρ/r0)Ψ0,n(z)
Λ0,n=Λ0,n(ac)=1r022α2Ωn2(α)(44α2Ωn2(α))(Ωn2(α)12α2+8α2ϒ(Ωn(α)),
ζm,n(op)/ζ0,1(ac)=(Λ0,1(ac)/Λm,n(op))1/40.12.

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