Abstract

As Q factor is boosted in microscale optical resonant systems there will be a natural tendency for these systems to experience a radiation-pressure induced instability. The instability is manifested as a regenerative oscillation (at radio frequencies) of the mechanical modes of the microcavity. The first observation of this radiation-pressure-induced instability is reported here. Embodied within a microscale, chip-based device reported here this mechanism can benefit both research into macroscale quantum mechanical phenomena [1] and improve the understanding of the mechanism within the context of LIGO [2]. It also suggests that new technologies are possible which will leverage the phenomenon within photonics.

© 2005 Optical Society of America

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References

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  1. S. Mancini, V. Giovanetti, D. Vitali, and P. Tombesi, �??Entangling macroscopic oscillators exploiting radiation pressure,�?? Phys. Rev. Lett. 88, 120401 (2002).
    [CrossRef] [PubMed]
  2. V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, �??Parametric oscillatory instability in Fabry-Perot interferometer,�?? Phys. Lett. A. 287, 331-338 (2001).
    [CrossRef]
  3. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala K. J, �??Ultra-high-Q toroid microcavity on a chip,�?? Nature 421, 925-929 (2003).
    [CrossRef] [PubMed]
  4. V. B. Braginsky, I. I. Minakovä, and P.M. Stepunin, �??Absolute measurement of energy and power in optical spectrum according to electromagnetic pressure,�?? Instrum. Exper. Tech-U. 3, 658-663 (1965).
  5. V. B. Braginsky, and A. B. Manukin, �??Ponderomotive effects of electromagnetic radiation,�?? Sov. Phys. JETP-USSR. 25, 653-655 (1967).
  6. A. Dorsel, J. D. Mccullen, P. Meystre, et al. �??Optical bistability and mirror confinement induced by radiation pressure,�?? Phys. Rev. Lett. 51, 1550-1553 (1983).
    [CrossRef]
  7. V. B. Braginsky, A. B. Manukin, and M. Y. Tikhonov, �??Investigation of dissipative Ponderomotive effects of electromagnetic radiation,�?? Sov. Phys. JETP-USSR. 31, 829-830 (1970).
  8. The characteristics of the overall waveguide-resonator system can be viewed as an optical modulator that is driven by this oscillation. This modulator has a nonlinear transfer function that manifests itself (in the modulated pump power) through the appearance of harmonics of the characteristic mechanical eigen-frequencies. These harmonics are easily observed upon detection of the modulated pump (see Fig. 2).
  9. For f (d) <0 , i.e. a red shift of the pump frequency with respect to the cavity mode, the phase of the radiation pressure variations actually damps or �??cools�?? the vibrations. Note that no external feedback system is necessary here to damp the vibrations or �??cool�?? the resonator. The feedback system is inherent to the coupling mechanism. Due to the high quality factor of our cavities (Q ~ 10 million) the �??red shifted�?? tail of the optical mode is not thermally stable (see H. Rokhsari et. al. "Loss characterization in micro-cavities using the thermal bistability effect. Applied Physics Letters 85, 3029-3031 (2004)). Replacing the cavity material (silica) with a negative thermo-optic coefficient material would stabilize the red shifted tail and cavity-cooling induced by radiation pressure effects could be observable.
    [CrossRef]
  10. M. Cai, O. Painter, and K. J. Vahala, �??Observation of critical coupling in a Fiber taper to a silica microsphere whispering-gallery mode system,�?? Phys. Rev. Lett. 85, 74-77 (2000).
    [CrossRef] [PubMed]
  11. For the sample tested, it is calculated that radial variations of about 10 picometers will shift the resonant frequency of the excited optical mode by its linewidth.
  12. M. Zalalutdinov, et a, �??Autoparametric optical drive for micromechanical oscillators,�?? Appl. Phys. Lett. 79, 695-697. (2001).
    [CrossRef]
  13. C. H. Metzger, K. Karrai, �??Cavity cooling of a microlever,�?? Nature. 432, 1002-1005, (2004).
    [CrossRef] [PubMed]
  14. T. J. Kippenberg, H. Rokhsari, T. Carmon, and K. J. Vahala, accepted by Phys. Rev. Lett.
  15. H. Rokhsari, .S. M. Spillane, and K. J. Vahala, �??Observation of Kerr nonlinearity in microcavities at room temperature,�?? Opt. Letts. 30, 427-429 (2005).
    [CrossRef]
  16. V. S. Ilchenko, and M. L. Gorodetsky, �??Thermal nonlinear effects in optical whispering gallery microresonators,�?? Laser Phys. 2, 1004-1009 (1992).
  17. We note that as evident in the renderings provided in Figs. 1 and 2, the n=3 mechanical mode has a strong radial component to its motion and hence understanding of its excitation by way of radiation pressure (which itself is primarily radial in direction) is straightforward. In contrast, the n=1 mode motion is transverse, requiring a different method of force transduction. The details here, including threshold calculations, will be presented in a forthcoming article where it is shown that minute offsets of the optical mode from the equatorial plane provide a moment arm for action of radiation pressure. The resulting torque induces the transverse motion associated with the n=1 mode. Modelling, including an SEM measurement of the offset, confirms this mechanism.
  18. X. M. H. Huang, C. A. Zorman, M. Mehregany, and M. L. Roukes, �??Nanodevice motion at microwave frequencies,�?? Nature. 421, 496 (2003).
    [CrossRef]
  19. W. Kells, and E. D'Ambrosio, �??Considerations on parametric instability in Fabry-Perot interferometer,�?? Phys. Lett. A. 299, 326-330 (2002).
    [CrossRef]
  20. V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, �??Analysis of parametric oscillatory instability in power recycled LIGO interferometer,�?? Phys Lett. A. 305, 111-124 (2002).
    [CrossRef]
  21. S. W. Schediwy, C. Zhao, L. Ju, et al, �??An experiment to investigate optical spring parametric instability,�?? Classica Quant. Grav. 21, S1253-S12587 (2004).
    [CrossRef]
  22. I. Tittonen, et al, �??Interferometric measurements of the position of a macroscopic body: Towards observation of quantum limits,�?? Phys. Rev. A. 59, 1038 (1999);
    [CrossRef]
  23. B. Julsgaard, A. Kozhekin, E. S. Polzik, �??Experimental long-lived entanglement of two macroscopic objects,�?? Nature (London), 413, 400 (2001).
    [CrossRef]
  24. V. Giovanetti, S. Mancini, P. Tombesi, �??Radiation pressure induced Einstein-Podolsky_Rosen paradox,�?? Europhys. Lett. 54, 559-565, (2001).
    [CrossRef]
  25. S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, �??Continuous-variable entanglemet and quantum-state teleportation between optical and macroscopic vibrational modes through radiation pressure,�?? Phys. Rev. A., 68, 062317, (2003); W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, �??Towards Quantum Superpositions of a Mirror,�?? Phys.Rev. Lett. 91, 130401, SEP (2003).
    [CrossRef]

Appl. Phys. Lett. (1)

M. Zalalutdinov, et a, �??Autoparametric optical drive for micromechanical oscillators,�?? Appl. Phys. Lett. 79, 695-697. (2001).
[CrossRef]

Applied Physics Letters (1)

For f (d) <0 , i.e. a red shift of the pump frequency with respect to the cavity mode, the phase of the radiation pressure variations actually damps or �??cools�?? the vibrations. Note that no external feedback system is necessary here to damp the vibrations or �??cool�?? the resonator. The feedback system is inherent to the coupling mechanism. Due to the high quality factor of our cavities (Q ~ 10 million) the �??red shifted�?? tail of the optical mode is not thermally stable (see H. Rokhsari et. al. "Loss characterization in micro-cavities using the thermal bistability effect. Applied Physics Letters 85, 3029-3031 (2004)). Replacing the cavity material (silica) with a negative thermo-optic coefficient material would stabilize the red shifted tail and cavity-cooling induced by radiation pressure effects could be observable.
[CrossRef]

Classica Quant. Grav. (1)

S. W. Schediwy, C. Zhao, L. Ju, et al, �??An experiment to investigate optical spring parametric instability,�?? Classica Quant. Grav. 21, S1253-S12587 (2004).
[CrossRef]

Europhys. Lett. (1)

V. Giovanetti, S. Mancini, P. Tombesi, �??Radiation pressure induced Einstein-Podolsky_Rosen paradox,�?? Europhys. Lett. 54, 559-565, (2001).
[CrossRef]

Instrum. Exper. Tech-U. (1)

V. B. Braginsky, I. I. Minakovä, and P.M. Stepunin, �??Absolute measurement of energy and power in optical spectrum according to electromagnetic pressure,�?? Instrum. Exper. Tech-U. 3, 658-663 (1965).

Laser Phys. (1)

V. S. Ilchenko, and M. L. Gorodetsky, �??Thermal nonlinear effects in optical whispering gallery microresonators,�?? Laser Phys. 2, 1004-1009 (1992).

Nature (3)

X. M. H. Huang, C. A. Zorman, M. Mehregany, and M. L. Roukes, �??Nanodevice motion at microwave frequencies,�?? Nature. 421, 496 (2003).
[CrossRef]

C. H. Metzger, K. Karrai, �??Cavity cooling of a microlever,�?? Nature. 432, 1002-1005, (2004).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala K. J, �??Ultra-high-Q toroid microcavity on a chip,�?? Nature 421, 925-929 (2003).
[CrossRef] [PubMed]

Nature (London) (1)

B. Julsgaard, A. Kozhekin, E. S. Polzik, �??Experimental long-lived entanglement of two macroscopic objects,�?? Nature (London), 413, 400 (2001).
[CrossRef]

Opt. Letts. (1)

H. Rokhsari, .S. M. Spillane, and K. J. Vahala, �??Observation of Kerr nonlinearity in microcavities at room temperature,�?? Opt. Letts. 30, 427-429 (2005).
[CrossRef]

Phys Lett. A. (1)

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, �??Analysis of parametric oscillatory instability in power recycled LIGO interferometer,�?? Phys Lett. A. 305, 111-124 (2002).
[CrossRef]

Phys. Lett. A. (2)

W. Kells, and E. D'Ambrosio, �??Considerations on parametric instability in Fabry-Perot interferometer,�?? Phys. Lett. A. 299, 326-330 (2002).
[CrossRef]

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, �??Parametric oscillatory instability in Fabry-Perot interferometer,�?? Phys. Lett. A. 287, 331-338 (2001).
[CrossRef]

Phys. Rev. A. (2)

I. Tittonen, et al, �??Interferometric measurements of the position of a macroscopic body: Towards observation of quantum limits,�?? Phys. Rev. A. 59, 1038 (1999);
[CrossRef]

S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, �??Continuous-variable entanglemet and quantum-state teleportation between optical and macroscopic vibrational modes through radiation pressure,�?? Phys. Rev. A., 68, 062317, (2003); W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, �??Towards Quantum Superpositions of a Mirror,�?? Phys.Rev. Lett. 91, 130401, SEP (2003).
[CrossRef]

Phys. Rev. Lett. (4)

S. Mancini, V. Giovanetti, D. Vitali, and P. Tombesi, �??Entangling macroscopic oscillators exploiting radiation pressure,�?? Phys. Rev. Lett. 88, 120401 (2002).
[CrossRef] [PubMed]

M. Cai, O. Painter, and K. J. Vahala, �??Observation of critical coupling in a Fiber taper to a silica microsphere whispering-gallery mode system,�?? Phys. Rev. Lett. 85, 74-77 (2000).
[CrossRef] [PubMed]

A. Dorsel, J. D. Mccullen, P. Meystre, et al. �??Optical bistability and mirror confinement induced by radiation pressure,�?? Phys. Rev. Lett. 51, 1550-1553 (1983).
[CrossRef]

T. J. Kippenberg, H. Rokhsari, T. Carmon, and K. J. Vahala, accepted by Phys. Rev. Lett.

Sov. Phys. JETP-USSR. (2)

V. B. Braginsky, A. B. Manukin, and M. Y. Tikhonov, �??Investigation of dissipative Ponderomotive effects of electromagnetic radiation,�?? Sov. Phys. JETP-USSR. 31, 829-830 (1970).

V. B. Braginsky, and A. B. Manukin, �??Ponderomotive effects of electromagnetic radiation,�?? Sov. Phys. JETP-USSR. 25, 653-655 (1967).

Other (3)

The characteristics of the overall waveguide-resonator system can be viewed as an optical modulator that is driven by this oscillation. This modulator has a nonlinear transfer function that manifests itself (in the modulated pump power) through the appearance of harmonics of the characteristic mechanical eigen-frequencies. These harmonics are easily observed upon detection of the modulated pump (see Fig. 2).

For the sample tested, it is calculated that radial variations of about 10 picometers will shift the resonant frequency of the excited optical mode by its linewidth.

We note that as evident in the renderings provided in Figs. 1 and 2, the n=3 mechanical mode has a strong radial component to its motion and hence understanding of its excitation by way of radiation pressure (which itself is primarily radial in direction) is straightforward. In contrast, the n=1 mode motion is transverse, requiring a different method of force transduction. The details here, including threshold calculations, will be presented in a forthcoming article where it is shown that minute offsets of the optical mode from the equatorial plane provide a moment arm for action of radiation pressure. The resulting torque induces the transverse motion associated with the n=1 mode. Modelling, including an SEM measurement of the offset, confirms this mechanism.

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Figures (5)

Fig. 1.
Fig. 1.

Panel A illustrates the “below threshold” behavior where the optical pump wave at frequency ω is not strong enough to induce mechanical oscillations of the micro-toroid. Panel B illustrates the “above threshold” case for the n=3 vibrational mode. Mechanical oscillation at frequency Ω creates optical stokes (ω-Ω) and anti-stoked sidebands (ω+Ω) in the transmitted pump wave. Inset of panel B shows the exaggerated cross-section of the third order mode and variation of the toriod radius as a result of these oscillations.

Fig. 2.
Fig. 2.

The measured, spectral content of pump-power (at 1550 nm) transmission as observed on an electrical spectrum analyzer (bandwidth set at 100kHz). Two families of frequencies are observed. Those at lower frequency are driven by oscillation of an n=1 vibrational mode and those at higher frequency by an n=3 mode. Harmonics of the fundamental mechanical frequency are caused by the nonlinear transfer characteristic of the waveguide resonator system (see footnote 8). The inset shows the numerically modelled cross-sectional plot (exaggerated for clarity) of the strain field for the first and third vibrational eigen-modes of a toroidal silica micro-cavity on a silicon post. The stress field is superimposed (color coded). As evident from the modelling, the mechanical oscillations cause a displacement of the toroidal periphery and thereby induce a shift in the whispering gallery mode resonant frequency.

Fig. 3.
Fig. 3.

Numerical calculation (solid line) and measured (points) dispersion relation for the fundamental (n=1) and third order (n=3) flexural modes of a micro-toroid as a function of the free hanging length of the disk structure. The inset shows the agreement between the numerical and measured frequencies.

Fig. 4.
Fig. 4.

Measured amplitude response (points) of the mechanical vibrations of an n=1 mechanical mode as a function of driving-force frequency (modulation frequency of the pump power). Circles (green), triangles (red), and stars (blue) represent the data for 2µW, 5µW and 9µW of average pump power. The inset shows the effect of the optical power on the linewidth of the mechanical oscillator inferred from the theoretical fits (such as the solid lines in the main Fig.). A linear fit shows a threshold of 11µW for the mechanical oscillations and an intrinsic quality factor of 630 for the measured mechanical mode of the toroidal structure.

Fig. 5.
Fig. 5.

Measured mechanical oscillator displacement as a function of the optical pump power showing threshold behaviour. Oscillations initiate at about 20µW of input power and start to saturate for higher values of pump power. This saturation is associated with the lower optical-mechanical coupling at displacements large enough to shift the resonant frequency of the optical mode by greater than its linewidth. Inset shows the dependence of measured threshold power on optical quality factor. Data taken from Ref. [14]. These data were obtained using different optical modes in a single micro-toroid (so that only optical Q would be varied). The slope of the fit in a log-log scale is 3±0.3, which agrees well with inverse cubic behaviour expected for radiation-pressure-induced regenerative oscillations (as opposed to inverse quadratic behaviour for photothermal effects).

Equations (1)

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γ = γ 0 ( 1 P P threshold ) , P threshold = K opt− mesh Q c Q total 4 Q mech f ( d )

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