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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 12 — Jun. 4, 2012
  • pp: 12742–12751
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Cavity optoelectromechanical regenerative amplification

Michael A. Taylor, Alex Szorkovszky, Joachim Knittel, Kwan H. Lee, Terry G. McRae, and Warwick P. Bowen  »View Author Affiliations


Optics Express, Vol. 20, Issue 12, pp. 12742-12751 (2012)
http://dx.doi.org/10.1364/OE.20.012742


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Abstract

Cavity optoelectromechanical regenerative amplification is demonstrated. An optical cavity enhances mechanical transduction, allowing sensitive measurement even for heavy oscillators. A 27.3 MHz mechanical mode of a microtoroid was linewidth narrowed to 6.6 ± 1.4 mHz, 30 times smaller than previously achieved with radiation pressure driving in such a system. These results may have applications in areas such as ultrasensitive optomechanical mass spectroscopy.

© 2012 OSA

1. Introduction

High quality factor (Q), low linewidth mechanical oscillators have many applications, such as highly sensitive spin [1

1. D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature 430, 329–332 (2004). [CrossRef] [PubMed]

] or charge sensing [2

2. A. Cleland and M. Roukes, “A nanometre-scale mechanical electrometer,” Nature 392, 160–162 (1998). [CrossRef]

], frequency standards in clocks [3

3. F. G. Major, The Quantum Beat: Principles and Applications of Atomic Clocks (Springer, 1998).

], mass sensing with subzeptogram sensitivity [4

4. K. Jensen, K. Kim, and A. Zettl, “An atomic-resolution nanomechanical mass sensor,” Nat. Nanotechnol. 3, 533–537 (2008). [CrossRef] [PubMed]

], characterizing surface diffusion processes [5

5. Y. T. Yang, C. Callegari, X. L. Feng, and M. L. Roukes, “Surface adsorbate fluctuations and noise in nanoelectromechanical systems,” Nano Lett. 11, 1753–1759 (2011). [CrossRef] [PubMed]

] and measuring forces with attonewton resolution [6

6. J. D. Teufel, T. Donner, M. A. Castellanos-Beltran, J. W. Harlow, and K. W. Lehnert, “Nanomechanical motion measured with an imprecision below that at the standard quantum limit,” Nat. Nanotechnol. 4, 820–823 (2009). [CrossRef] [PubMed]

]. The linewidth of a mechanical mode is determined by the energy dissipation rate of the mode, which without feedback is dominated by friction forces [7

7. G. Anetsberger, R. Rivière, A. Schliesser, O. Arcizet, and T. J. Kippenberg, “Ultralow-dissipation optomechanical resonators on a chip,” Nat. Photonics 2, 627–633 (2008). [CrossRef]

]. A common technique to reduce the linewidth is to apply a feedback force to amplify the oscillator motion and bring it into the regenerative oscillation regime. When the feedback force overcomes friction, the oscillation becomes coherent and self-sustained. This reduces the mechanical linewidth and increases the mechanical energy by several orders of magnitude, which facilitates extremely precise monitoring of mechanical frequency-shifts.

The past few years have seen the rapid development of cavity optomechanical systems, which combine mechanical oscillators with a confined optical field. In these systems, the strong interaction between the mechanical motion of the resonator and the cavity-enhanced optical field allows both control and measurement of the motion. Cavity optomechanical systems have many promising applications, such as on-chip phononic information processing [8

8. Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4, 236–242 (2010). [CrossRef]

, 9

9. A. H. Safavi-Naeini and O. Painter, “Proposal for an optomechanical traveling wave phonon-photon translator,” New J. Phys. 13, 013017 (2011). [CrossRef]

], displacement sensing at the standard quantum limit [10

10. G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Rivière, A. Schliesser, E. M. Weig, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009). [CrossRef]

, 11

11. A. Schliesser, G. Anetsberger, R. Rivière, O. Arcizet, and T. J. Kippenberg, “High-sensitivity monitoring of micromechanical vibration using optical whispering gallery mode resonators,” New J. Phys. 10, 095015 (2008). [CrossRef]

], and ultrasensitive force measurement [12

12. E. Gavartin, P. Verlot, and T. J. Kippenberg, “A hybrid on-chip opto-nanomechanical transducer for ultra-sensitive force measurements,” arXiv:1112.0797v1 (2011).

]. Ground-state cooled optomechanical oscillators are also proposed to probe exotic problems such as macroscopic quantum behavior [13

13. M. Arndt, M. Aspelmeyer, and A. Zeilinger, “How to extend quantum experiments,” Fortschr. Phys. 57, 1153–1162 (2009). [CrossRef]

], quantum gravity [14

14. W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2003). [CrossRef] [PubMed]

] and microscale gravity [15

15. L. Haiberger, M. Weingran, and S. Schiller, “Highly sensitive silicon crystal torque sensor operating at the thermal noise limit,” Rev. Sci. Instrum. 78, 025101 (2007). [CrossRef] [PubMed]

].

Radiation pressure driven regenerative amplification was demonstrated in an early cavity optomechanical experiment [16

16. 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, 033901 (2005). [CrossRef] [PubMed]

]. However, the process is constrained by the low magnitude of available radiation pressure forces and the inability to modify the force spectrum independently from optical parameters. This both limits the achievable mechanical line narrowing, and results in unavoidable mechanical frequency pulling which introduces noise in sensing applications [17

17. A. Mirzaei and A. A. Abidi, “The spectrum of a noisy free-running oscillator explained by random frequency pulling,” IEEE Trans. Circuits Syst., I: Regul. Pap. 57, 642–653 (2010). [CrossRef]

]. These constraints are overcome here using electrical gradient forces and ultra-sensitive optomechanical transduction within a feedback loop, rather than radiation pressure, to control the mechanical motion. In the context of cavity optomechanics, such a control system was first implemented in a recent demonstration of optoelectromechanical feedback cooling of a microtoroid [18

18. K. H. Lee, T. G. McRae, G. I. Harris, J. Knittel, and W. P. Bowen, “Cooling and control of a cavity optoelectromechanical system,” Phys. Rev. Lett. 104, 123604–123607 (2010). [CrossRef] [PubMed]

, 19

19. T. G. McRae, K. H. Lee, G. I. Harris, J. Knittel, and W. P. Bowen, “Cavity optoelectromechanical system combining strong electrical actuation with ultrasensitive transduction,” Phys. Rev. A 82, 23825–23831 (2010). [CrossRef]

], and has also been demonstrated in silicon disk resonators [20

20. S. Sridaran and S. A. Bhave, “Electrostatic actuation of silicon optomechanical resonators,” Opt. Express 19, 9020–9026 (2011). [CrossRef] [PubMed]

]. Some applications require large oscillators, such as mass sensing of large samples. However, transducing the motion of large oscillators can be difficult as mechanical amplitude scales inversely with the square root of resonator mass. In this regime, cavity enhanced optical readout is particularly relevant as it offers extremely sensitive detection. For applications requiring high sensitivity and large size, cavity optoelectromechanical oscillators are a promising class of resonators. Note that, similar to regenerative amplification, mass sensing can be achieved via mechanical parametric amplification [21

21. W. Zhang, R. Baskaran, and K. L. Turner, “Effect of cubic nonlinearity on auto-parametrically amplified resonant MEMS mass sensor,” Sens. Actuators A, Phys. 102, 139–150 (2002). [CrossRef]

]. This is a fundamentally different process based on electrically induced mechanical nonlinearities, which by contrast is phase sensitive and requires no feedback [22

22. D. Rugar and P. Grutter, “Mechanical parametric amplification and thermomechanical noise squeezing, Phys. Rev. Lett. 67, 699–702 (1991). [CrossRef] [PubMed]

].

In this work, a model of the regenerative amplification process is derived which includes both electrical and radiation pressure forces. We experimentally demonstrate regenerative oscillations in a silica microtoroidal optoelectromechanical system, observing mechanical linewidths as low as 6.6±1.4 mHz with a corresponding effective mechanical quality factor of 4×109. For comparison, the best optically driven oscillators published have achieved 200 mHz linewidth in a microtoroid [23

23. M. Hossein-Zadeh, H. Rokhsari, A. Hajimiri, and K. J. Vahala, “Characterization of a radiation-pressure-driven micromechanical oscillator,” Phys. Rev. A 74, 23813–23827 (2006). [CrossRef]

], and recently 20 mHz in a silicon nitride ring oscillator [24

24. S. Tallur, S. Sridaran, and S. A. Bhave, “A monolithic radiation-pressure driven, low phase noise silicon nitride opto-mechanical oscillator,” Opt. Express 19, 24522–24529 (2011). [CrossRef] [PubMed]

]. Recent experiments in a silicon oscillator have also demonstrated cavity optoelectromechanical regenerative amplification, as a means to filter and modulate the optical fields [25

25. S. Sridaran and S. A. Bhave, “Opto-acoustic oscillator using silicon MEMS optical modulator,” in Proceedings of 16th International Solid-State Sensors, Actuators and Microsystems Conference (IEEE, 2011), pp. 2920–2923. [CrossRef]

]. The higher optical quality factor of reflown silica used in the work reported here, combined with externally controllable electrodes allow 15 dB lower phase noise to be achieved at similar offset frequencies. This is an enabling step towards a new range of applications, including ultrasensitive mass spectroscopy [26

26. X. L. Feng, C. J. White, A. Hajimiri, and M. L. Roukes, “A self-sustaining ultrahigh-frequency nanoelectromechanical oscillator,” Nat. Nanotechnol. 3, 342–346 (2008). [CrossRef] [PubMed]

, 27

27. A. K. Naik, M. S. Hanay, W. K. Hiebert, X. L. Feng, and M. L. Roukes, “A self-sustaining ultrahigh-frequency nanoelectromechanical oscillator,” Nat. Nanotechnol. 4, 445–450 (2009). [CrossRef] [PubMed]

], photonic clocks [23

23. M. Hossein-Zadeh, H. Rokhsari, A. Hajimiri, and K. J. Vahala, “Characterization of a radiation-pressure-driven micromechanical oscillator,” Phys. Rev. A 74, 23813–23827 (2006). [CrossRef]

] and a range of nonlinear radio-frequency (RF) processes [28

28. M. Hossein-Zadeh and K. J. Vahala, “An optomechanical oscillator on a slicon chip,” IEEE J. Sel. Top. Quantum Electron. 16, 276–287 (2010). [CrossRef]

].

2. Theory

The motion of a mechanical oscillator under the action of thermal FT, optical Fopt, and feedback Ffb forces can be described by the equation of motion
m[x¨(t)+Γ0x˙(t)+ωm2x(t)]=Ffb(t)+Fopt+FT(t),
(1)
where m, Γ0 and ωm are respectively the effective mass, dissipation rate and natural frequency [29

29. 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, 223902 (2005). [CrossRef] [PubMed]

]. When the feedback force is proportional to the velocity, it opposes the dissipation, and if large enough, will cause regenerative oscillations. In the recent work of Kippenberg et al. [16

16. 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, 033901 (2005). [CrossRef] [PubMed]

], optical radiation pressure was used to drive an oscillator to regenerative amplification. The radiation pressure force due to intracavity optical power PIC is given by Fopt=2πncmPIC, with n the index of refraction [30

30. H. Rokhsari, M. Hossein-Zadeh, A. Hajimiri, and K. J. Vahala, “Brownian noise in radiation-pressure-driven micromechanical oscillators,” Appl. Phys. Lett. 89, 261109 (2006). [CrossRef]

]. With the optical power below the threshold for regenerative amplification, this modifies the equation of motion to
m[x¨(t)+Γrx˙(t)+ωr2x(t)]=Ffb(t)+FT(t),
(2)
where the action of the optical force is to decrease the mechanical decay rate to Γr=Γ0(1PoptPthresh) and shift the mechanical resonance frequency to ωr = ωm(1 + ηpPopt) [23

23. M. Hossein-Zadeh, H. Rokhsari, A. Hajimiri, and K. J. Vahala, “Characterization of a radiation-pressure-driven micromechanical oscillator,” Phys. Rev. A 74, 23813–23827 (2006). [CrossRef]

]. The frequency pulling is unavoidable with optical driving, and could limit sensitivity in some applications such as mass sensing. The regenerative oscillation regime is entered when the incident optical power Popt exceeds the threshold power Pthresh, such that the decay rate becomes negative. The degree of frequency pulling is determined by the constant ηp, which depends on the optical detuning and physical parameters of the system.

In this article we include electrical feedback while keeping the optical power below the threshold for optical regenerative amplification. Assuming the mechanical oscillations shift the optical resonant frequency much less than the optical linewidth, the mechanical transduction is linear. Then the applied force is Ffb(t) = mΓrGx(tτ), where τ is a small delay such that x(tτ) ∝ (t), G = G(x,G0) is the steady-state feedback gain and G0 is the small-signal gain. The gain G is normalized here so that regenerative amplification occurs for G0 > 1. Below this threshold there is no saturation in the feedback, and the gain is given by G = G0.

In the regenerative amplification regime, the feedback exceeds the dissipation, and the motion grows exponentially. The feedback force also grows exponentially as it is proportional to the position, until a component of the feedback electronics saturate, causing the steady-state gain G to take a value which is very close to but smaller than 1. To determine the narrowed mechanical linewidth, we substitute the expression for the electrical feedback force into Eq. (1), Fourier transform and rearrange to find
x(ω)=FT(ω)m(ωr2ω2+iΓrω)imωmΓrG.
(3)
In the high Q limit, the frequency range of interest lies very near the mechanical resonance frequency. We therefore take ω = ωr + Δ, with Δ ≪ ωm, and assume perfect feedback phase such that eiωτi + Δτ. With these approximations, Eq. (3) may be re-expressed as
x(Δ)=1mωmFT(ω)2Δ+iΓr(1G).
(4)
The feedback narrowed full-width half-maximum linewidth Γ is then easily found to be
Γ=Γ0(1PoptPthresh)(1G).
(5)
We see that the linewidth Γ goes to zero as the gain G approaches 1 or the optical power Popt approaches the threshold Pthresh. The steady state gain G can be found in terms of experimentally measurable parameters by calculating the total mechanical energy Eosc
Eosc=12πmωm2|x(ω)|2dω=12kT(1PoptPthresh)(1G),
(6)
where |FT (ω)|2 = mΓ0kT due to the fluctuation-dissipation theorem, with thermal amplitude fluctuations freezing out in the high power limit [30

30. H. Rokhsari, M. Hossein-Zadeh, A. Hajimiri, and K. J. Vahala, “Brownian noise in radiation-pressure-driven micromechanical oscillators,” Appl. Phys. Lett. 89, 261109 (2006). [CrossRef]

, 31

31. W. A. Edson, “Noise in oscillators,” Proc. IRE 48, 1454–1466 (1960). [CrossRef]

]. In the absence of driving (G = 0 and Popt = 0) these fluctuations do not freeze out, doubling the thermal contribution and reducing this expression to the expected thermal energy ET = kT. Equation (6) allows the steady-state gain G to be established from measurements of the intrinsic mechanical decay rate and the oscillator energy with and without feedback. G approaches unity in the limit of regenerative amplification, with 1 − G ≈ 10−5 in the experiments reported here. The oscillator energy can then be expressed simply as ETEosc=2(1G)(1PoptPthresh). Substituting this expression into Eq. (5) finally gives the expected narrowed mechanical linewidth.
Γ=Γ02ETEosc.
(7)
For optical powers below the threshold for radiation pressure driven regenerative oscillations, as in our experiment, optical driving is indistinguishable from electrical feedback in this expression. This expression for the linewidth is identical to that obtained from a more complex analysis of radiation pressure driving [30

30. H. Rokhsari, M. Hossein-Zadeh, A. Hajimiri, and K. J. Vahala, “Brownian noise in radiation-pressure-driven micromechanical oscillators,” Appl. Phys. Lett. 89, 261109 (2006). [CrossRef]

], and is of a similar form to those for other regenerative amplifiers, such as RLC electronic oscillators [32

32. W. P. Robins, Phase Noise in Signal Sources (Peter Peregrinus Ltd, 1982), pp. 48–53.

], optoelectronic oscillators [33

33. X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. B 13, 1725–1735 (1995). [CrossRef]

], masers [34

34. J. P. Gordon, H. J. Zeiger, and C. H. Townes, “The Maser–new type of microwave amplifier, frequency standard, and spectrometer,” Phys. Rev. 99, 1264–1274 (1955). [CrossRef]

], and the Schawlow-Townes limit for laser linewidth [35

35. A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958). [CrossRef]

,36

36. R. Paschotta, A. Schlatter, S. C. Zeller, H. R. Telle, and U. Keller, “Optical phase noise and carrier-envelope offset noise of mode-locked lasers,” Appl. Phys. B 82, 265–273 (2006). [CrossRef]

]. Note that an equivalent expression could be derived in terms of the mechanical output power rather than the oscillator energy, similar to the expressions common for lasers and optoelectronic oscillators. Here, however, the oscillator energy is more generally relevant, since in contrast to those situations, the output phonon field is not readily accessible. It is instead enhancements in the oscillator energy upon which applications typically depend.

3. Experiment and results

3.1. Experimental system

Fig. 1 A schematic of the experiment. FPC, fiber polarization control. The Feedback Control includes filtering and control over both feedback phase and amplitude. The network analyzer was used to characterize the driven response of the system, and all other results were taken with the signal analyzer. The subplots show the motion of the mechanical mode investigated here, (a) with driving from the network analyzer, and (b) the thermal motion measured on the spectrum analyzer. Shown in light blue is a fit to the data shown in green, which determines the intrinsic decay rate Γ0 and frequency ωm. The vertical axis of each subplot is in dB against an arbitrary reference.
Fig. 2 Oscillator mechanical energy as a function of small-signal feedback gain. Green trace: measured data. Below saturation the theoretical model (blue dashed line) is given by Eq. (6). To estimate the above-threshold mechanical energy, Eq. (1) is numerically solved for sinusoidal motion with a limit applied to the magnitude of the feedback force Ffb(t). Two fitting parameters are used; one normalizes the gain such that saturation occurs at G0 = 1, and another defines the mechanical energy at which saturation occurs. Inset: A finite-element model of the mechanical mode being amplified.

To enable sub-Hz linewidth measurements it was critical to suppress low frequency noise in the apparatus. The primary sources of low frequency noise were motion of the tapered fiber due to air currents and thermal fluctuations in the toroid due to absorption of background light. Air currents were eliminated by placing the experiment in a hermetically sealed box, and background light levels were minimized. The use of a shot noise limited laser and low noise electronics ensured that both the radiation pressure back-action on the mechanical oscillator and the electronic feedback were at the quantum noise limit. The low noise performance allows high sensitivity measurements with low optical intensity, thus minimizing optical back-action. The detected photocurrent was band-pass filtered to suppress other mechanical modes. Electronic components controlled the phase (JSPHS-26) and amplitude (ZX73-2500) of this signal before it was amplified and fed back to an electrode. This amplified signal was fed back to a sharp stainless steel electrode with a 2 μm diameter tip. This allowed electrical gradient forces to be applied to the microtoroid following the approach of Ref. [18

18. K. H. Lee, T. G. McRae, G. I. Harris, J. Knittel, and W. P. Bowen, “Cooling and control of a cavity optoelectromechanical system,” Phys. Rev. Lett. 104, 123604–123607 (2010). [CrossRef] [PubMed]

,19

19. T. G. McRae, K. H. Lee, G. I. Harris, J. Knittel, and W. P. Bowen, “Cavity optoelectromechanical system combining strong electrical actuation with ultrasensitive transduction,” Phys. Rev. A 82, 23825–23831 (2010). [CrossRef]

]. This force was intensified by placing the electrode at the end of a coaxial half-wave transmission line resonator tuned to the mechanical resonance frequency. This enhanced the applied force by the resonator’s Q factor of 25 compared to the work of Ref. [18

18. K. H. Lee, T. G. McRae, G. I. Harris, J. Knittel, and W. P. Bowen, “Cooling and control of a cavity optoelectromechanical system,” Phys. Rev. Lett. 104, 123604–123607 (2010). [CrossRef] [PubMed]

, 19

19. T. G. McRae, K. H. Lee, G. I. Harris, J. Knittel, and W. P. Bowen, “Cavity optoelectromechanical system combining strong electrical actuation with ultrasensitive transduction,” Phys. Rev. A 82, 23825–23831 (2010). [CrossRef]

]. Further to this, a constant voltage of 120 V was applied to the electrode increasing the polarization of the silica microtoroid structure and thus enhancing its response to the applied electric field. This constant polarization had a negligible effect on the mechanical mode properties.

The electrode was positioned 5–20 μm vertically above the microtoroid rim. At this location, it produced an electric field gradient which was not azimuthally symmetric, and therefore able to excite the mechanical mode. While the overlap between the applied force and the mechanical mode was small, it was sufficient to allow regenerative amplification of the mechanical mode. If the electric field were tailored to the mechanical mode, the same applied forces could be generated with much a lower electrode voltage. This could be important in applications for which power consumption is a limiting factor.

By adjusting the phase of the feedback, the gradient force was optimized to maximally amplify the mechanical motion of the 27.3 MHz mode, and the feedback gain was varied by adjusting the variable attenuator. As the small signal gain was increased toward threshold, the energy in the oscillator increased as expected from Eq. (6), and the mechanical resonance narrowed as described by Eq. (5). Above threshold the feedback overcame the mechanical damping, with the mechanical energy growing until the feedback amplifier saturated. This clamped the energy to a fairly constant level, typically 4–5 orders of magnitude greater than the thermal energy. Figure 2 shows the increase in oscillator energy as the small signal gain was increased across the threshold. Spectra of the mechanical resonance are shown in Fig. 3 both without amplification, and in the regime of regenerative amplification. The amplified signal has a peak power spectral density which is over 50 dB higher and a sub-Hz linewidth, which could not be resolved on our spectrum analyzer (shown in inset).

Fig. 3 Blue and green traces respectively show the mechanical power spectrum with and without amplification. These were measured by sending the signal from the optical detector directly into a spectrum analyzer. Note that this mode is composed of two degenerate modes which are distinguished only by the azimuthal position of the nodes and antinodes. Only the mode with greatest overlap with the driving field is amplified, although both are present in the measurements. Inset: A near resonance spectrum with amplification and a 1 Hz resolution bandwidth (right).

3.2. Determining the mechanical linewidth

Because the mechanical linewidth was unresolvable, it was determined via phase noise analysis. The finite linewidth of the mechanical signal causes a floor in phase noise power which is given by
10(ΔΩ)/10=ΓΔΩ2,
(8)
where ΔΩ is the detuning from the carrier frequency and 𝒧 (ΔΩ) is the phase noise power measured at the detuning normalized to the power of the central peak, in units of dBc/Hz [30

30. H. Rokhsari, M. Hossein-Zadeh, A. Hajimiri, and K. J. Vahala, “Brownian noise in radiation-pressure-driven micromechanical oscillators,” Appl. Phys. Lett. 89, 261109 (2006). [CrossRef]

, 36

36. R. Paschotta, A. Schlatter, S. C. Zeller, H. R. Telle, and U. Keller, “Optical phase noise and carrier-envelope offset noise of mode-locked lasers,” Appl. Phys. B 82, 265–273 (2006). [CrossRef]

]. An example of a measured phase noise trace is shown in Fig. 4(a). The results match the predicted ΔΩ−2 dependence over a wide range of offset frequencies. At several frequencies the phase noise exceeds the prediction from the expression above, with other noise sources dominant. Much of the additional phase noise seen in the trace at low offset frequencies are contributed by harmonics of one low Q mechanical resonance at 108 Hz in the tapered optical fiber. A high noise floor is present partly because we use a signal analyzer to extract phase noise rather than a dedicated phase noise analyzer. This noise floor was found to be well below the measured noise between 20 Hz and 1 kHz offset, where we perform our analysis. The minimum phase noise achieved in our system was determined to be −84.4 ± 1.1 dBc/Hz at an offset frequency of 500 Hz.

Fig. 4 (a) An example of a phase noise trace. The fitted noise floor is for a linewidth Γ = 13.3 ± 3.7 mHz. A low Q mechanical resonance at 108 Hz is clearly visible, and electronic noise from the signal analyzer also appears for frequency offsets above 1 kHz. The dotted line indicates the phase noise typically achievable with optomechanical driving only [23]. (b) The measured linewidth as a function of oscillator energy. Fitting to the data (green) shows shows a dependence of ΓEosc0.91±0.12, while theory predicts ΓEosc1.

The mechanical linewidth was inferred from phase noise data by fitting a noise floor proportional to ΔΩ−2 to the phase noise spectrum, as in Fig. 4(a). Since the measured phase noise must always be greater than or equal to the noise floor of Eq. (8), this fit gives an upper bound on linewidth. Due to the clear ΔΩ−2 trend observed over a significant frequency range in all phase noise traces analyzed, we are confident this corresponds closely to the actual linewidth. To verify this we experimentally confirmed the predicted dependence of linewidth on mechanical energy.

To test this dependence, the phase noise was analyzed for data with a range of mechanical energies. Mechanical energy was controlled by varying the electrode-microtoroid gap while keeping the feedback above threshold, which adjusts the force applied to the mechanical mode. The energy was measured on a spectrum analyzer for each data point, and normalized against measurements of the thermal motion to give the ratio Eosc/ET. Each energy measurement was made with a bandwidth larger than the linewidth, so that the spectral peak height would contain essentially all of the mechanical energy. The measured thermal motion was from two degenerate modes, so had an energy of 2kT. The extracted linewidths are shown against oscillator energy in Fig. 4(b). The linewidth is found to follow a trend of ΓEosc0.91±0.12, which to within error matches the predicted relation from Eq. (7) of ΓEosc1. The linewidth achieved for maximum oscillator energy was 6.6 ± 1.4 mHz, giving an effective Q factor of 4 × 109, compared to the smallest published linewidth achieved by optomechanical driving in a microtoroid of 200 mHz [23

23. M. Hossein-Zadeh, H. Rokhsari, A. Hajimiri, and K. J. Vahala, “Characterization of a radiation-pressure-driven micromechanical oscillator,” Phys. Rev. A 74, 23813–23827 (2006). [CrossRef]

], with an equivalent Q factor of 2.5 × 108.

3.3. Application to mass sensing

This system has potential application for mass sensing, with the regeneratively narrowed mechanical linewidth providing high sensitivity to mechanical resonance frequency shifts due to the deposition of small masses on the oscillator. The mass sensitivity is then limited by the accuracy with which the mechanical resonance frequency can be determined within the detection bandwidth Δf. In our current experiments, the limiting factor is drift in the position of both the optical taper, and hence coupling condition, and electrode cause feedback phase fluctuations which in turn result in fluctuations of the mechanical resonance frequency. However, it is possible to stabilize the position of both taper [37

37. C. Junge, S. Nickel, D. O’Shea, and A. Rauschenbeutel, “Bottle microresonator with actively stabilized evanescent coupling,” Opt. Lett. 36, 3488–3490 (2011). [CrossRef] [PubMed]

] and electrode using active and/or passive techniques. Assuming that the drift in mechanical resonance frequency over the oscillation lifetime can be made smaller than the narrowed linewidth, thermomechanical fluctuations place a fundamental sensitivity limit to mass sensing [38

38. K. L. Ekinci, Y. T. Yang, and M. L. Roukes, “Ultimate limits to inertial mass sensing based upon nanoelectromechanical systems,” J. Appl. Phys. 95, 2682–2689 (2004). [CrossRef]

]
δmmin=2mmode(ETEosc)1/2(Γ2πQωm)1/2,
(9)
for a mode mass mmode and detection bandwidth Δf = Γ/2π. The mode mass here is the mass which moves in the oscillations, mmode = ∫ρ |u(r)|2dr, with ρ the density and u(r) the spatial shape of the mode [39

39. M. Pinard, Y. Hadjar, and A. Heidmann, “Effective mass in quantum effects of radiation pressure,” Eur. Phys. J. D 7, 107–116 (1999).

]. Using the spatial profile calculated with finite-element modeling this gives a mode mass of 5 × 10−9 g. Using this, along with EoscET=105, Γ = 2π × 0.01 Hz, Q = 600 and ωm = 2π × 27.3 MHz, an optimum sensitivity of δmmin ≈ 10−17 g is predicted at room temperature and atmospheric pressure. The predicted mass sensitivity is comparable to the the best sensitivities achieved in room temperature mechanical sensors. Sensitivity at this level has been achieved in microfluidic channels embedded within cantilevers [40

40. J. Lee, W. Shen, K. Payer, T. P. Burg, and S. R. Manalis, “Toward attogram mass measurements in solution with suspended nanochannel resonators,” Nano Lett. 10, 2537–2542 (2010). [CrossRef] [PubMed]

], and cantilevers in air [41

41. J. Verd, A. Uranga, G. Abadal, J. Teva, F. Torres, F. Pérez-Murano, J. Fraxedas, J. Esteve, and N. Barniol, “Monolithic mass sensor fabricated using a conventional technology with attogram resolution in air conditions,” Appl. Phys. Lett. 91, 013501 (2007). [CrossRef]

]. Both of these systems have significant differences to our setup. The microfluidic integrated cantilevers measure sample masses within fluid, whereas the microtoroid would operate on samples in air. Cantilever operating in air perform a similar measurement, but are much smaller; the device in Ref. [41

41. J. Verd, A. Uranga, G. Abadal, J. Teva, F. Torres, F. Pérez-Murano, J. Fraxedas, J. Esteve, and N. Barniol, “Monolithic mass sensor fabricated using a conventional technology with attogram resolution in air conditions,” Appl. Phys. Lett. 91, 013501 (2007). [CrossRef]

] is 3 orders of magnitude lighter than our microtoroid. Performing mass measurement with the larger structure of a microtoroid may enable measurements with comparative sensitivity on substantially larger samples. This could allow, for example, sensitive characterization of single-cell dynamical mass changes in processes such as photosynthetic growth. Introducing cavity enhanced transduction to mechanical mass sensors could be a promising technique to extend the range of ultrasensitive measurements to larger samples.

4. Conclusion

Regenerative oscillation was achieved in a cavity optoelectromechanical system using feedback which combined cavity enhanced optical transduction with electrical actuation. The optical cavity enhances the mechanical transduction over standard optical or electrical techniques, and the electrical forces allow stronger force with improved control when compared to optomechanical driving of such an optomechanical cavity. A theoretical model of this system was formulated to include both radiation-pressure and electrical feedback. Linewidth narrowing from 46 kHz to 6.6±1.4 mHz was achieved at a frequency of 27.3 MHz, corresponding to an effective quality factor of 4 × 109. This linewidth is smaller than that achieved in similar radiation pressure driven regenerative oscillators, which have been reported to reach 200 mHz. The linewidth was predicted and experimentally confirmed to scale inversely with mechanical energy. This opens new possibilities for sensitive mass spectroscopy at room pressure and temperature.

Acknowledgments

This research was funded by the Australian Research Council Centre of Excellence CE110001013 and Discovery Project DP0987146. Device fabrication was undertaken within the Queensland Node of the Australian National Fabrication Facility (ANFF).

References and links

1.

D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature 430, 329–332 (2004). [CrossRef] [PubMed]

2.

A. Cleland and M. Roukes, “A nanometre-scale mechanical electrometer,” Nature 392, 160–162 (1998). [CrossRef]

3.

F. G. Major, The Quantum Beat: Principles and Applications of Atomic Clocks (Springer, 1998).

4.

K. Jensen, K. Kim, and A. Zettl, “An atomic-resolution nanomechanical mass sensor,” Nat. Nanotechnol. 3, 533–537 (2008). [CrossRef] [PubMed]

5.

Y. T. Yang, C. Callegari, X. L. Feng, and M. L. Roukes, “Surface adsorbate fluctuations and noise in nanoelectromechanical systems,” Nano Lett. 11, 1753–1759 (2011). [CrossRef] [PubMed]

6.

J. D. Teufel, T. Donner, M. A. Castellanos-Beltran, J. W. Harlow, and K. W. Lehnert, “Nanomechanical motion measured with an imprecision below that at the standard quantum limit,” Nat. Nanotechnol. 4, 820–823 (2009). [CrossRef] [PubMed]

7.

G. Anetsberger, R. Rivière, A. Schliesser, O. Arcizet, and T. J. Kippenberg, “Ultralow-dissipation optomechanical resonators on a chip,” Nat. Photonics 2, 627–633 (2008). [CrossRef]

8.

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4, 236–242 (2010). [CrossRef]

9.

A. H. Safavi-Naeini and O. Painter, “Proposal for an optomechanical traveling wave phonon-photon translator,” New J. Phys. 13, 013017 (2011). [CrossRef]

10.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Rivière, A. Schliesser, E. M. Weig, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009). [CrossRef]

11.

A. Schliesser, G. Anetsberger, R. Rivière, O. Arcizet, and T. J. Kippenberg, “High-sensitivity monitoring of micromechanical vibration using optical whispering gallery mode resonators,” New J. Phys. 10, 095015 (2008). [CrossRef]

12.

E. Gavartin, P. Verlot, and T. J. Kippenberg, “A hybrid on-chip opto-nanomechanical transducer for ultra-sensitive force measurements,” arXiv:1112.0797v1 (2011).

13.

M. Arndt, M. Aspelmeyer, and A. Zeilinger, “How to extend quantum experiments,” Fortschr. Phys. 57, 1153–1162 (2009). [CrossRef]

14.

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2003). [CrossRef] [PubMed]

15.

L. Haiberger, M. Weingran, and S. Schiller, “Highly sensitive silicon crystal torque sensor operating at the thermal noise limit,” Rev. Sci. Instrum. 78, 025101 (2007). [CrossRef] [PubMed]

16.

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, 033901 (2005). [CrossRef] [PubMed]

17.

A. Mirzaei and A. A. Abidi, “The spectrum of a noisy free-running oscillator explained by random frequency pulling,” IEEE Trans. Circuits Syst., I: Regul. Pap. 57, 642–653 (2010). [CrossRef]

18.

K. H. Lee, T. G. McRae, G. I. Harris, J. Knittel, and W. P. Bowen, “Cooling and control of a cavity optoelectromechanical system,” Phys. Rev. Lett. 104, 123604–123607 (2010). [CrossRef] [PubMed]

19.

T. G. McRae, K. H. Lee, G. I. Harris, J. Knittel, and W. P. Bowen, “Cavity optoelectromechanical system combining strong electrical actuation with ultrasensitive transduction,” Phys. Rev. A 82, 23825–23831 (2010). [CrossRef]

20.

S. Sridaran and S. A. Bhave, “Electrostatic actuation of silicon optomechanical resonators,” Opt. Express 19, 9020–9026 (2011). [CrossRef] [PubMed]

21.

W. Zhang, R. Baskaran, and K. L. Turner, “Effect of cubic nonlinearity on auto-parametrically amplified resonant MEMS mass sensor,” Sens. Actuators A, Phys. 102, 139–150 (2002). [CrossRef]

22.

D. Rugar and P. Grutter, “Mechanical parametric amplification and thermomechanical noise squeezing, Phys. Rev. Lett. 67, 699–702 (1991). [CrossRef] [PubMed]

23.

M. Hossein-Zadeh, H. Rokhsari, A. Hajimiri, and K. J. Vahala, “Characterization of a radiation-pressure-driven micromechanical oscillator,” Phys. Rev. A 74, 23813–23827 (2006). [CrossRef]

24.

S. Tallur, S. Sridaran, and S. A. Bhave, “A monolithic radiation-pressure driven, low phase noise silicon nitride opto-mechanical oscillator,” Opt. Express 19, 24522–24529 (2011). [CrossRef] [PubMed]

25.

S. Sridaran and S. A. Bhave, “Opto-acoustic oscillator using silicon MEMS optical modulator,” in Proceedings of 16th International Solid-State Sensors, Actuators and Microsystems Conference (IEEE, 2011), pp. 2920–2923. [CrossRef]

26.

X. L. Feng, C. J. White, A. Hajimiri, and M. L. Roukes, “A self-sustaining ultrahigh-frequency nanoelectromechanical oscillator,” Nat. Nanotechnol. 3, 342–346 (2008). [CrossRef] [PubMed]

27.

A. K. Naik, M. S. Hanay, W. K. Hiebert, X. L. Feng, and M. L. Roukes, “A self-sustaining ultrahigh-frequency nanoelectromechanical oscillator,” Nat. Nanotechnol. 4, 445–450 (2009). [CrossRef] [PubMed]

28.

M. Hossein-Zadeh and K. J. Vahala, “An optomechanical oscillator on a slicon chip,” IEEE J. Sel. Top. Quantum Electron. 16, 276–287 (2010). [CrossRef]

29.

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, 223902 (2005). [CrossRef] [PubMed]

30.

H. Rokhsari, M. Hossein-Zadeh, A. Hajimiri, and K. J. Vahala, “Brownian noise in radiation-pressure-driven micromechanical oscillators,” Appl. Phys. Lett. 89, 261109 (2006). [CrossRef]

31.

W. A. Edson, “Noise in oscillators,” Proc. IRE 48, 1454–1466 (1960). [CrossRef]

32.

W. P. Robins, Phase Noise in Signal Sources (Peter Peregrinus Ltd, 1982), pp. 48–53.

33.

X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. B 13, 1725–1735 (1995). [CrossRef]

34.

J. P. Gordon, H. J. Zeiger, and C. H. Townes, “The Maser–new type of microwave amplifier, frequency standard, and spectrometer,” Phys. Rev. 99, 1264–1274 (1955). [CrossRef]

35.

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958). [CrossRef]

36.

R. Paschotta, A. Schlatter, S. C. Zeller, H. R. Telle, and U. Keller, “Optical phase noise and carrier-envelope offset noise of mode-locked lasers,” Appl. Phys. B 82, 265–273 (2006). [CrossRef]

37.

C. Junge, S. Nickel, D. O’Shea, and A. Rauschenbeutel, “Bottle microresonator with actively stabilized evanescent coupling,” Opt. Lett. 36, 3488–3490 (2011). [CrossRef] [PubMed]

38.

K. L. Ekinci, Y. T. Yang, and M. L. Roukes, “Ultimate limits to inertial mass sensing based upon nanoelectromechanical systems,” J. Appl. Phys. 95, 2682–2689 (2004). [CrossRef]

39.

M. Pinard, Y. Hadjar, and A. Heidmann, “Effective mass in quantum effects of radiation pressure,” Eur. Phys. J. D 7, 107–116 (1999).

40.

J. Lee, W. Shen, K. Payer, T. P. Burg, and S. R. Manalis, “Toward attogram mass measurements in solution with suspended nanochannel resonators,” Nano Lett. 10, 2537–2542 (2010). [CrossRef] [PubMed]

41.

J. Verd, A. Uranga, G. Abadal, J. Teva, F. Torres, F. Pérez-Murano, J. Fraxedas, J. Esteve, and N. Barniol, “Monolithic mass sensor fabricated using a conventional technology with attogram resolution in air conditions,” Appl. Phys. Lett. 91, 013501 (2007). [CrossRef]

OCIS Codes
(230.1040) Optical devices : Acousto-optical devices
(140.3948) Lasers and laser optics : Microcavity devices
(230.4685) Optical devices : Optical microelectromechanical devices

ToC Category:
Optical Devices

History
Original Manuscript: March 6, 2012
Revised Manuscript: May 9, 2012
Manuscript Accepted: May 10, 2012
Published: May 22, 2012

Citation
Michael A. Taylor, Alex Szorkovszky, Joachim Knittel, Kwan H. Lee, Terry G. McRae, and Warwick P. Bowen, "Cavity optoelectromechanical regenerative amplification," Opt. Express 20, 12742-12751 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-12742


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References

  1. D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature430, 329–332 (2004). [CrossRef] [PubMed]
  2. A. Cleland and M. Roukes, “A nanometre-scale mechanical electrometer,” Nature392, 160–162 (1998). [CrossRef]
  3. F. G. Major, The Quantum Beat: Principles and Applications of Atomic Clocks (Springer, 1998).
  4. K. Jensen, K. Kim, and A. Zettl, “An atomic-resolution nanomechanical mass sensor,” Nat. Nanotechnol.3, 533–537 (2008). [CrossRef] [PubMed]
  5. Y. T. Yang, C. Callegari, X. L. Feng, and M. L. Roukes, “Surface adsorbate fluctuations and noise in nanoelectromechanical systems,” Nano Lett.11, 1753–1759 (2011). [CrossRef] [PubMed]
  6. J. D. Teufel, T. Donner, M. A. Castellanos-Beltran, J. W. Harlow, and K. W. Lehnert, “Nanomechanical motion measured with an imprecision below that at the standard quantum limit,” Nat. Nanotechnol.4, 820–823 (2009). [CrossRef] [PubMed]
  7. G. Anetsberger, R. Rivière, A. Schliesser, O. Arcizet, and T. J. Kippenberg, “Ultralow-dissipation optomechanical resonators on a chip,” Nat. Photonics2, 627–633 (2008). [CrossRef]
  8. Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics4, 236–242 (2010). [CrossRef]
  9. A. H. Safavi-Naeini and O. Painter, “Proposal for an optomechanical traveling wave phonon-photon translator,” New J. Phys.13, 013017 (2011). [CrossRef]
  10. G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Rivière, A. Schliesser, E. M. Weig, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys.5, 909–914 (2009). [CrossRef]
  11. A. Schliesser, G. Anetsberger, R. Rivière, O. Arcizet, and T. J. Kippenberg, “High-sensitivity monitoring of micromechanical vibration using optical whispering gallery mode resonators,” New J. Phys.10, 095015 (2008). [CrossRef]
  12. E. Gavartin, P. Verlot, and T. J. Kippenberg, “A hybrid on-chip opto-nanomechanical transducer for ultra-sensitive force measurements,” arXiv:1112.0797v1 (2011).
  13. M. Arndt, M. Aspelmeyer, and A. Zeilinger, “How to extend quantum experiments,” Fortschr. Phys.57, 1153–1162 (2009). [CrossRef]
  14. W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett.91, 130401 (2003). [CrossRef] [PubMed]
  15. L. Haiberger, M. Weingran, and S. Schiller, “Highly sensitive silicon crystal torque sensor operating at the thermal noise limit,” Rev. Sci. Instrum.78, 025101 (2007). [CrossRef] [PubMed]
  16. 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, 033901 (2005). [CrossRef] [PubMed]
  17. A. Mirzaei and A. A. Abidi, “The spectrum of a noisy free-running oscillator explained by random frequency pulling,” IEEE Trans. Circuits Syst., I: Regul. Pap.57, 642–653 (2010). [CrossRef]
  18. K. H. Lee, T. G. McRae, G. I. Harris, J. Knittel, and W. P. Bowen, “Cooling and control of a cavity optoelectromechanical system,” Phys. Rev. Lett.104, 123604–123607 (2010). [CrossRef] [PubMed]
  19. T. G. McRae, K. H. Lee, G. I. Harris, J. Knittel, and W. P. Bowen, “Cavity optoelectromechanical system combining strong electrical actuation with ultrasensitive transduction,” Phys. Rev. A82, 23825–23831 (2010). [CrossRef]
  20. S. Sridaran and S. A. Bhave, “Electrostatic actuation of silicon optomechanical resonators,” Opt. Express19, 9020–9026 (2011). [CrossRef] [PubMed]
  21. W. Zhang, R. Baskaran, and K. L. Turner, “Effect of cubic nonlinearity on auto-parametrically amplified resonant MEMS mass sensor,” Sens. Actuators A, Phys.102, 139–150 (2002). [CrossRef]
  22. D. Rugar and P. Grutter, “Mechanical parametric amplification and thermomechanical noise squeezing, Phys. Rev. Lett.67, 699–702 (1991). [CrossRef] [PubMed]
  23. M. Hossein-Zadeh, H. Rokhsari, A. Hajimiri, and K. J. Vahala, “Characterization of a radiation-pressure-driven micromechanical oscillator,” Phys. Rev. A74, 23813–23827 (2006). [CrossRef]
  24. S. Tallur, S. Sridaran, and S. A. Bhave, “A monolithic radiation-pressure driven, low phase noise silicon nitride opto-mechanical oscillator,” Opt. Express19, 24522–24529 (2011). [CrossRef] [PubMed]
  25. S. Sridaran and S. A. Bhave, “Opto-acoustic oscillator using silicon MEMS optical modulator,” in Proceedings of 16th International Solid-State Sensors, Actuators and Microsystems Conference (IEEE, 2011), pp. 2920–2923. [CrossRef]
  26. X. L. Feng, C. J. White, A. Hajimiri, and M. L. Roukes, “A self-sustaining ultrahigh-frequency nanoelectromechanical oscillator,” Nat. Nanotechnol.3, 342–346 (2008). [CrossRef] [PubMed]
  27. A. K. Naik, M. S. Hanay, W. K. Hiebert, X. L. Feng, and M. L. Roukes, “A self-sustaining ultrahigh-frequency nanoelectromechanical oscillator,” Nat. Nanotechnol.4, 445–450 (2009). [CrossRef] [PubMed]
  28. M. Hossein-Zadeh and K. J. Vahala, “An optomechanical oscillator on a slicon chip,” IEEE J. Sel. Top. Quantum Electron.16, 276–287 (2010). [CrossRef]
  29. 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, 223902 (2005). [CrossRef] [PubMed]
  30. H. Rokhsari, M. Hossein-Zadeh, A. Hajimiri, and K. J. Vahala, “Brownian noise in radiation-pressure-driven micromechanical oscillators,” Appl. Phys. Lett.89, 261109 (2006). [CrossRef]
  31. W. A. Edson, “Noise in oscillators,” Proc. IRE48, 1454–1466 (1960). [CrossRef]
  32. W. P. Robins, Phase Noise in Signal Sources (Peter Peregrinus Ltd, 1982), pp. 48–53.
  33. X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. B13, 1725–1735 (1995). [CrossRef]
  34. J. P. Gordon, H. J. Zeiger, and C. H. Townes, “The Maser–new type of microwave amplifier, frequency standard, and spectrometer,” Phys. Rev.99, 1264–1274 (1955). [CrossRef]
  35. A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev.112, 1940–1949 (1958). [CrossRef]
  36. R. Paschotta, A. Schlatter, S. C. Zeller, H. R. Telle, and U. Keller, “Optical phase noise and carrier-envelope offset noise of mode-locked lasers,” Appl. Phys. B82, 265–273 (2006). [CrossRef]
  37. C. Junge, S. Nickel, D. O’Shea, and A. Rauschenbeutel, “Bottle microresonator with actively stabilized evanescent coupling,” Opt. Lett.36, 3488–3490 (2011). [CrossRef] [PubMed]
  38. K. L. Ekinci, Y. T. Yang, and M. L. Roukes, “Ultimate limits to inertial mass sensing based upon nanoelectromechanical systems,” J. Appl. Phys.95, 2682–2689 (2004). [CrossRef]
  39. M. Pinard, Y. Hadjar, and A. Heidmann, “Effective mass in quantum effects of radiation pressure,” Eur. Phys. J. D7, 107–116 (1999).
  40. J. Lee, W. Shen, K. Payer, T. P. Burg, and S. R. Manalis, “Toward attogram mass measurements in solution with suspended nanochannel resonators,” Nano Lett.10, 2537–2542 (2010). [CrossRef] [PubMed]
  41. J. Verd, A. Uranga, G. Abadal, J. Teva, F. Torres, F. Pérez-Murano, J. Fraxedas, J. Esteve, and N. Barniol, “Monolithic mass sensor fabricated using a conventional technology with attogram resolution in air conditions,” Appl. Phys. Lett.91, 013501 (2007). [CrossRef]

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