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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 11 — May. 21, 2012
  • pp: 12622–12630
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Critical coupling control of a microresonator by laser amplitude modulation

Jong H. Chow, Michael A. Taylor, Timothy T-Y. Lam, Joachim Knittel, Jye D. Sawtell-Rickson, Daniel A. Shaddock, Malcolm B. Gray, David E. McClelland, and Warwick P. Bowen  »View Author Affiliations


Optics Express, Vol. 20, Issue 11, pp. 12622-12630 (2012)
http://dx.doi.org/10.1364/OE.20.012622


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Abstract

We present a laser amplitude modulation technique to actively stabilize the critical coupling of a microresonator by controlling the evanescent coupling gap from an optical fiber taper. It is a form of nulled lock-in detection, which decouples laser intensity fluctuations from the critical coupling measurement. We achieved a stabilization bandwidth of ∼ 20 Hz, with up to 5 orders of magnitude displacement noise suppression at 10 mHz, and an inferred gap stability of better than a picometer/√Hz.

© 2012 OSA

1. Introduction

One common way to couple light into these resonators is via an optical fiber taper, where the taper waist diameter is reduced to below the wavelength of the interrogating laser. The optical field then propagates mainly evanescently outside the glass material of the fiber. When this waist is placed close enough to a microresonator, light can be evanescently coupled into the resonator with very high efficiency. An example of such a setup is shown in Fig. 1, where we display the microscope image of the microtoroid resonator and taper used in our experiment.

Fig. 1 Top view microscope image of the microtoroid cavity used in our experiment, with a fiber taper for evanescent coupling. Ein is the input optical field, Ecirc is the circulating field, while Eout the output field from the optical system.

To optimize coupling efficiency, the taper must be placed at a precise distance from the cavity, such that the cavity is critically coupled, resulting in maximum circulating power and minimum transmitted output power. This coupling efficiency is sensitive to any displacement fluctuations due to mechanical and acoustic noise, or thermal drifts in the opto-mechanical mounts.

Due to the small volume and high Q, micro resonators are highly susceptible to to a range of nonlinear effects. These include photothermal, Kerr nonlinearity, and intra-cavity opto-mechanical radiation pressure effects. The photothermal effect describes the process of intra-cavity material heating due to photon absorption [6

6. T. Carmon, L. Yang, and K. J. Vahala, “Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express , 124742–4750, (2004). [CrossRef] [PubMed]

, 7

7. H. Rokhsari, S. M. Spillane, and K. J. Vahala, “Loss characterization in microcavities using the thermal bistability effect,” Appl. Phys. Lett. 85, 3029–3031 (2004). [CrossRef]

]. Hence fluctuations in optical power causes optical path length changes. This is the largest of the nonlinear effects and dominates over long time scales. The Kerr nonlinearity is a χ3 effect, which yields an intensity dependent refractive index change over short time scales [3

3. 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, 083904 (2004). [CrossRef] [PubMed]

, 8

8. H. Rokhsari and K. J. Vahala, “Observation of Kerr nonlinearity in microcavities at room temperature,” Opt. Lett. 30, 427–429 (2005). [CrossRef] [PubMed]

]. The high intensity in microresonators can also exert a non-negligible force on the waveguide due to radiation pressure [9

9. H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala, “Theoretical and experimental study of radiation pressure-induced mechanical oscillations (parametric instability) in optical microcavities,” IEEE J. Sel. Top. Quantum Electron. 12, 96–107 (2006). [CrossRef]

,10

10. M. Hossein-Zadeh, H. Rokhsari, A. Hajimiri, and K. J. Vahala, “Characterization of a radiation-pressure-driven micromechanical oscillator,” Phys. Rev. A 74, 023813 (2006). [CrossRef]

]. Once again, this couples intensity fluctuations to cavity length.

In this paper, we present a radio frequency laser amplitude modulation (AM) technique to control and stabilize the critical coupling of a microtoroid resonator, with a significantly enhanced bandwidth of ∼ 20 Hz. In contrast to Ref. [15

15. 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]

], our technique offers a zero-crossing correction signal at critical coupling, and over two orders of magnitude higher stabilisation bandwidth. A zero crossing error signal has distinct advantages, providing optimum signal-to-noise ratio while decoupling the critical coupling measurement from laser intensity fluctuations [18

18. P. R. Saulson, Fundamentals of Interferometric Gravitational Wave Detectors, 1st ed. (World Scientific, Singapore, 1994). [CrossRef]

]. A higher stablization bandwidth mitigates high frequency displacement noise, and ensures tighter control of the evanescent coupling gap. We demonstrate inferred gap displacement stabilization of < 1 picometer/√Hz within the Fourier frequency range of 10 mHz to 10 Hz, with optimum performance of ∼ 70 femtometers/√Hz between 10 mHz and 1 Hz.

2. Microresonator transmission characteristics and the critical coupling error signal

Following the discussion in Ref. [19

19. J. H. Chow, I. C. M. Littler, D. S. Rabeling, D. E. McClelland, and M. B. Gray, “Using active resonator impedance matching for shot-noise limited, cavity enhanced amplitude modulated laser absorption spectroscopy,” Opt. Express 16, 7726–7738 (2008). [CrossRef] [PubMed]

], when the microtoroid is interrogated by a laser with amplitude modulated sidebands of frequencies much greater than the resonance full-width half-maximum, the output optical power on the photodetector can be described by |Eout|2, where
Eout=Aeiω0t[at1ta+β2e+iωmt+β2eiωmt].
(2)
In Eq. (2), A is the input laser field amplitude, ω0 is the laser carrier frequency, ωm is the amplitude modulation frequency, and β is the modulation depth. The RF output signal from the photodetector is then demodulated by the local oscillator at ωm. It is straight-forward to show, with some algebra, that after low-pass filtering we obtain an error signal which can be approximated by
SA2at1ta.
(3)
For small signals near critical coupling when at, this can be further reduced to
SQA2(at),
(4)
where Q is the optical quality factor of the microcavity. From Eq. (4) we see that the error signal is enhanced with increased Q, and its polarity depends on whether it is under- or over-coupled. Since it has a zero crossing when critically coupled, this error signal can be used by a servo for active feedback control to zero this error signal. This can be done by actuating on the taper-toroid gap, and thereby adjusting the coupling coefficient t.

Equation 4 also indicates that the slope of the error signal is dependent on the input optical power A2. When the system is locked, however, at = 0 and hence S is insensitive to fluctuations in A2. This “nulled” readout essentially decouples laser intensity noise from intra-cavity loss measurements.

3. Experimental technique

Our technique requires a Pound-Drever-Hall (PDH) frequency locking loop to keep the laser resonant, and a second critical coupling locking loop to optimize the circulating power in the cavity. The PDH frequency locking loop is a phase sensitive technique which detects optical path length changes in the cavity, and then adjusts the frequency of the laser to keep it resonant. The critical coupling locking loop, on the other hand, is an amplitude sensitive technique which matches the evanescent coupling coefficient to the loss in the cavity by adjusting the gap between the fiber taper and the microresonator.

PDH locking [16

16. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983). [CrossRef]

] was enabled by laser phase modulation (PM), with feedback to the piezoelectric transducer (PZT) of a tunable fiber laser at around 1560 nm. The critical coupling locking loop uses AM radio-frequency (RF) sidebands to sense the coupling condition of the cavity [19

19. J. H. Chow, I. C. M. Littler, D. S. Rabeling, D. E. McClelland, and M. B. Gray, “Using active resonator impedance matching for shot-noise limited, cavity enhanced amplitude modulated laser absorption spectroscopy,” Opt. Express 16, 7726–7738 (2008). [CrossRef] [PubMed]

], with feedback to a translation actuator on the microtoroid mount and hence the taper-toroid gap distance. Both control loops, including modulation and demodulation, were implemented using digital signal processing and Field Programmable Gate Arrays (FPGA).

The experimental schematic of our critical coupling control technique is shown in Fig. 2. The fiber laser was phase modulated at 15 MHz and amplitude modulated at 12.5 MHz simultaneously with an electro-optic modulator. The output from the modulator was then used to interrogate the microtoroid via the tapered fiber. The microtoroid used in this experiment, as shown in Fig. 1, was approximately 65 microns in diameter, with an optical quality factor Q of ∼ 1 × 107. It was mounted on a translation stage with a piezo-electric transducer (PZT) for displacement actuation. This PZT was used to control the coupling gap between the fiber taper and the microtoroid cavity. The output from the fiber taper was received by a New Focus 1811 photodetector, which converted the light to an electronic signal. The photodetector provided a DC output for monitoring the transmitted optical power and an AC output for the RF signal.

Fig. 2 Experimental schematic for critical coupling control of a microtoroid cavity, showing two control loops: 1) The Pound-Drever-Hall locking using PM modulation; and 2) the critical coupling feedback using AM modulation.

The RF electronic signal was demodulated at 15 MHz in the FPGA to generate the PDH feedback control signal. This signal was applied to the laser frequency tuning PZT to keep the laser resonant while the critical coupling stabilisation was active. Meanwhile, the RF electronic signal was simultaneously demodulated at 12.5 MHz in the FPGA to provide the coupling control feedback signal. The coupling control feedback signal was applied to the PZT attached to the microtoroid mount, to compensate for any displacement noise in the gap between the taper and the microtoroid.

Before closing the critical coupling control loop, we varied the gap between the fiber taper and the microtoroid, by injecting a voltage ramp to the microtoroid translation PZT using a function generator with a 0.1 Vp–p signal while the PDH frequency feedback control was active. This voltage was acquired with a digital oscilloscope and displayed as Fig. 3, Trace (a), in green. The voltage ramp was amplified by a factor of 15 before being applied to the PZT, resulting in a peak-to-peak displacement of ∼ 0.4 μm. This displacement was measured by applying a DC voltage to the PZT of the microtoroid mount, and observing the translation of the microtoroid as a fraction of its 65 μm diameter. This was found to be consistent with the specifications of the PZT at ∼ 0.27 μm/V.

Fig. 3 The system behavior when the evanescent gap was varied while the PDH frequency locking was active: (a) the input voltage to the translation stage amplifier, which linearly varied the evanescent gap; (b) the output power from the microtoroid; (c) the demodulated critical coupling error signal. The PZT varied the coupling gap by 0.4 μm peak-to-peak.

The corresponding output optical power was observed with the DC output of the photodetector on the oscilloscope, and shown as Fig. 3, Trace (b), in blue. We see that there was a minimum in the output power as the coupling gap was decreased. This turning point occured at optimum power throughput, where the cavity was critically coupled, or impedance matched.

The coupling control error signal was generated digitally by the FPGA, and output via the digital-to-analog electronics by converting a signed 32-bit number to a ± 20 V signal. When the coupling gap was varied with the PZT, this voltage was also recorded by the oscilloscope, and is plotted as Fig. 3, Trace (c), in red. We note that this error voltage crosses zero at the transmitted power minimum, and has opposing polarity depending on whether the gap was too large or too small. For small displacements, the size and polarity of this error signal was then directly proportional to the PZT translation required to keep the cavity critically coupled. With the known displacement gain of the PZT, the 32-bit digital error signal could be calibrated to infer an equivalent displacement. This calibration was then used to measure the performance of our evanescent gap stabilization.

4. Critical coupling performance

An 18 Hz dither, near the edge of our control bandwidth, was injected into the coupling gap to observe the closed-loop suppression of the cavity output optical power. The locking servo used in our experiment had cascaded proportional integrators. The lock acquisition process is demonstrated by Fig. 4, where we monitored the DC output optical power with the photodetector. At the start, the evanescent gap was free running. After 8.5 seconds, the first integrator was engaged and we observed a large dip in output optical power, but only modest reduction of short term power modulation. When the second stage integrator was turned on at 10.8 seconds, the average output optical power was further reduced to approach zero. At the same time, the short term power fluctuations were also significantly suppressed. At 18 Hz, the critical coupling servo had approximately unity gain and was not responsible for the observed suppression seen in Fig. 4. From Fig. 3 Trace (b), we observe that when close to critical coupling, output power as a function of gap displacement is at a turning point. When close to critical coupling, therefore, fluctuations in output power due to gap displacement is a secondary effect.

Fig. 4 The time series of the microtoroid output power during the lock acquisition process. The gap was injected with an 18 Hz dither, and was initially free-running with no feedback control. The first integrator of the locking servo was turned on at 8.5 seconds, and the second was engaged at 10.8 seconds.

Figure 4 demonstrates that stable control of the evanescent gap at critical coupling provides two benefits: it maximizes the DC circulating power while significantly suppressing AC power fluctuations in the cavity due to gap displacement noise, even at Fourier frequencies near unity gain and beyond. The fact that the cavity output power fluctuations were so well suppressed indicates that the DC error of our feedback system was minimized, and we were locking extremely close to the critical coupling point.

Fig. 5 The spectra of the critical coupling error signal calibrated for equivalent evanescent gap displacement for (a) when coupling stabilisation was not active; and (b) closed-loop performance of the critical coupling control.

The bottom red trace (b) in Fig. 5 is the calibrated Fourier spectrum of the closed-loop AM error signal when feedback control was engaged. The comparison between open-loop and closed-loop traces shows the displacement noise suppression in the evanescent coupling gap, and demonstrates that we achieved approximately 5 orders of magnitude suppression at 10 mHz, and better than two orders of magnitude suppression for all frequencies below 5 Hz. We see that the closed-loop gap displacement noise was around 70 fm/√Hz, and remained relatively flat up to Fourier frequency of 1 Hz, before rolling up towards the unity gain bandwidth of the control loop. The unity gain occured when the two traces coincide, showing we had about 20 Hz of stabilization bandwidth. This bandwidth was limited by the mechanical resonances of the mountings for the tapered fiber and microtoroid.

The long term coupling stabilization performance over two hours is displayed in Fig. 6, where we observed the transmitted power on resonance on the photodetector with an oscilloscope, while critical coupling feedback control was active. This is displayed as a fraction of total power. We see that the short term fluctuations in transmitted power was ∼ 0.1%, with a longer term variation envelope of less than 2 %. The short term fluctuations was due to displacement noise above our control bandwidth of 20 Hz. The slowly varying envelope was mainly due to polarization wander, causing some of the light to be non-resonant and hence transmitted directly to the photodetector. We will discuss this further in Sect. 5.

Fig. 6 The long term transmitted power over two hours, as a percentage fraction of total input power.

5. System limitations

In an ideally critically coupled system, we expect the laser carrier to be completely coupled into the resonator, such that the only transmitted light is due to the non-resonant sidebands. Typically, the power in the sidebands is negligible. In both Figs. 3 and 6, however, we see that the minimum transmitted power through the fiber taper was non-zero. In Fig. 3 Trace (b), the turning point of the transmitted power was above zero when the critical coupling error signal crossed zero as the evanescent gap was reduced. In our experiment, there was a residual transmitted power of just above 2% through the fiber taper, even when the system was optimized. This implies that the maximum coupling achievable was just below 98%. The residual transmitted power was attributed to the waist diameter of the fiber taper, which was not sufficiently small to cut off the second evanescently guided transverse mode. This second transverse mode was not resonant in the microtoroid, and was therefore promptly transmitted through the fiber taper. We note that this results in a small constant offset in the zero-crossing position relative of the critical coupling error signal when compared to the output power minimum.

Another limit to our system performance was polarization wander in the optical fiber. Since the fiber used to fabricate the taper was SMF-28 and hence circularly symmetric in geometry, the laser polarization was not maintained as it propagated along its length. Although a polarization controller was used to match the polarisation eigenmode of the microtoroid at the start of each data run, ordinary laboratory environmental perturbations, including temperature drift and mechanical creep, caused the polarization in the fiber to drift. Since the polarization eigen-modes of the microtoroid were non-degenerate, due to a differential group index between two polarizations, this reduced the coupling efficiency at long time scales, resulting in the long term variation envelope of the transmitted power shown in Fig. 6. This polarization wander also caused an offset in the critical coupling error signal zero crossing position. Unlike the residual second transverse mode, however, this offset drifted with time, and in severe cases, can pull the critical coupling system out of lock. In our experiment, polarization wander was the main cause for degradation in coupling efficiency between the fiber taper and microtoroid. This effect can be reduced by better thermal and mechanical isolation of the fiber, or by replacing it with a waveguide geometry which maintains input polarization. Alternatively, an active polarization stabilization scheme can be implemented with some added complexity in the overall system architecture.

6. Conclusions

In conclusion, we have demonstrated an active critical coupling stabilization technique with a zero-crossing sensing signal, for evanescent coupling of laser light into a microtoroid resonator via a fiber taper. We achieved 5 orders of magnitude suppression of gap displacement fluctuations, attaining an inferred displacement noise of well below a picometer/√Hz from 10 mHz to 10 Hz, with optimum performance of ∼ 70 fm/√Hz between 10 mHz to 1 Hz.

Acknowledgments

The authors acknowledge the Australian Research Council for funding support of this research under the Linkage Project scheme (LP10020064) and Discovery Project scheme (DP0987146).

References and links

1.

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

2.

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008). [CrossRef] [PubMed]

3.

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, 083904 (2004). [CrossRef] [PubMed]

4.

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

5.

J. Knittel, T. G. McRae, K. H. Lee, and W. P. Bowen, “Interferometric detection of mode splitting for whispering gallery mode biosensors,” Appl. Phys. Lett. 97, 123704 (2010). [CrossRef]

6.

T. Carmon, L. Yang, and K. J. Vahala, “Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express , 124742–4750, (2004). [CrossRef] [PubMed]

7.

H. Rokhsari, S. M. Spillane, and K. J. Vahala, “Loss characterization in microcavities using the thermal bistability effect,” Appl. Phys. Lett. 85, 3029–3031 (2004). [CrossRef]

8.

H. Rokhsari and K. J. Vahala, “Observation of Kerr nonlinearity in microcavities at room temperature,” Opt. Lett. 30, 427–429 (2005). [CrossRef] [PubMed]

9.

H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala, “Theoretical and experimental study of radiation pressure-induced mechanical oscillations (parametric instability) in optical microcavities,” IEEE J. Sel. Top. Quantum Electron. 12, 96–107 (2006). [CrossRef]

10.

M. Hossein-Zadeh, H. Rokhsari, A. Hajimiri, and K. J. Vahala, “Characterization of a radiation-pressure-driven micromechanical oscillator,” Phys. Rev. A 74, 023813 (2006). [CrossRef]

11.

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

12.

J. D. Swaim, J. Knittel, and W. P. Bowen, “Detection limits in whispering gallery biosensors with plasmonic enhancement,” Appl. Phys. Lett. 99, 243109 (2011). [CrossRef]

13.

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, 023825 (2010). [CrossRef]

14.

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 (2010). [CrossRef] [PubMed]

15.

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]

16.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983). [CrossRef]

17.

E. D. Black, “An introduction to Pound-Drever-Hall laser frequency stabilization,” Am. J. Phys. 69, 79–87 (2001). [CrossRef]

18.

P. R. Saulson, Fundamentals of Interferometric Gravitational Wave Detectors, 1st ed. (World Scientific, Singapore, 1994). [CrossRef]

19.

J. H. Chow, I. C. M. Littler, D. S. Rabeling, D. E. McClelland, and M. B. Gray, “Using active resonator impedance matching for shot-noise limited, cavity enhanced amplitude modulated laser absorption spectroscopy,” Opt. Express 16, 7726–7738 (2008). [CrossRef] [PubMed]

20.

D.S. Rabeling, J.H. Chow, M.B. Gray, and D.E. McClelland, “Experimental demonstration of impedance match locking and control of coupled resonators,” Opt. Express 18, 9314–9323 (2011). [CrossRef]

21.

J. E. Heebner and R. W. Boyd, “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt. Lett. 24, 847–849 (1999). [CrossRef]

22.

A. Yariv, “Critical coupling and its control in optical wave-guide-ring resonator systems,” IEEE Photon. Technol. Lett. 14483–485 (2002). [CrossRef]

OCIS Codes
(230.5750) Optical devices : Resonators
(140.3945) Lasers and laser optics : Microcavities
(240.3990) Optics at surfaces : Micro-optical devices

ToC Category:
Optical Devices

History
Original Manuscript: February 28, 2012
Revised Manuscript: May 11, 2012
Manuscript Accepted: May 15, 2012
Published: May 18, 2012

Citation
Jong H. Chow, Michael A. Taylor, Timothy T-Y. Lam, Joachim Knittel, Jye D. Sawtell-Rickson, Daniel A. Shaddock, Malcolm B. Gray, David E. McClelland, and Warwick P. Bowen, "Critical coupling control of a microresonator by laser amplitude modulation," Opt. Express 20, 12622-12630 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-11-12622


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References

  1. K. J. Vahala, “Optical microcavities,” Nature424, 839–846 (2003). [CrossRef] [PubMed]
  2. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science321, 1172–1176 (2008). [CrossRef] [PubMed]
  3. 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, 083904 (2004). [CrossRef] [PubMed]
  4. H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala, “Radiation-pressure-driven micro-mechanical oscillator,” Opt. Express13, 5293–5301 (2005). [CrossRef] [PubMed]
  5. J. Knittel, T. G. McRae, K. H. Lee, and W. P. Bowen, “Interferometric detection of mode splitting for whispering gallery mode biosensors,” Appl. Phys. Lett.97, 123704 (2010). [CrossRef]
  6. T. Carmon, L. Yang, and K. J. Vahala, “Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express, 124742–4750, (2004). [CrossRef] [PubMed]
  7. H. Rokhsari, S. M. Spillane, and K. J. Vahala, “Loss characterization in microcavities using the thermal bistability effect,” Appl. Phys. Lett.85, 3029–3031 (2004). [CrossRef]
  8. H. Rokhsari and K. J. Vahala, “Observation of Kerr nonlinearity in microcavities at room temperature,” Opt. Lett.30, 427–429 (2005). [CrossRef] [PubMed]
  9. H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala, “Theoretical and experimental study of radiation pressure-induced mechanical oscillations (parametric instability) in optical microcavities,” IEEE J. Sel. Top. Quantum Electron.12, 96–107 (2006). [CrossRef]
  10. M. Hossein-Zadeh, H. Rokhsari, A. Hajimiri, and K. J. Vahala, “Characterization of a radiation-pressure-driven micromechanical oscillator,” Phys. Rev. A74, 023813 (2006). [CrossRef]
  11. M. Hossein-Zadeh and K. J. Vahala, “An optomechanical oscillator on a silicon chip,” IEEE J. Sel. Top. Quantum Electron.16, 276–287 (2010). [CrossRef]
  12. J. D. Swaim, J. Knittel, and W. P. Bowen, “Detection limits in whispering gallery biosensors with plasmonic enhancement,” Appl. Phys. Lett.99, 243109 (2011). [CrossRef]
  13. 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, 023825 (2010). [CrossRef]
  14. 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 (2010). [CrossRef] [PubMed]
  15. 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]
  16. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B31, 97–105 (1983). [CrossRef]
  17. E. D. Black, “An introduction to Pound-Drever-Hall laser frequency stabilization,” Am. J. Phys.69, 79–87 (2001). [CrossRef]
  18. P. R. Saulson, Fundamentals of Interferometric Gravitational Wave Detectors, 1st ed. (World Scientific, Singapore, 1994). [CrossRef]
  19. J. H. Chow, I. C. M. Littler, D. S. Rabeling, D. E. McClelland, and M. B. Gray, “Using active resonator impedance matching for shot-noise limited, cavity enhanced amplitude modulated laser absorption spectroscopy,” Opt. Express16, 7726–7738 (2008). [CrossRef] [PubMed]
  20. D.S. Rabeling, J.H. Chow, M.B. Gray, and D.E. McClelland, “Experimental demonstration of impedance match locking and control of coupled resonators,” Opt. Express18, 9314–9323 (2011). [CrossRef]
  21. J. E. Heebner and R. W. Boyd, “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt. Lett.24, 847–849 (1999). [CrossRef]
  22. A. Yariv, “Critical coupling and its control in optical wave-guide-ring resonator systems,” IEEE Photon. Technol. Lett.14483–485 (2002). [CrossRef]

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