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
Whispering-gallery-mode microresonators are emerging as an important class of devices for a broad range of applications, including resonant optomechanics, cavity quantum electrodynamics, and cavity enhanced optical sensing [1, 2]. These devices combine high finesse with small volumes to enhance circulating optical power and intensity, thereby accentuating the dynamic optical , opto-mechanical  and light-matter effects  observed.
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.
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, 7]. 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, 8]. The high intensity in microresonators can also exert a non-negligible force on the waveguide due to radiation pressure [9,10]. Once again, this couples intensity fluctuations to cavity length.
The evanescent gap displacement due to acoustic and mechanical noise cause fluctuations in the circulating power and, as a consequence of intensity driven nonlinearities, causes perturbations in cavity optical phase, which can be detrimental to a range of opto-mechanical and sensing measurements [11, 12]. In microresonator electromechanical systems, these intensity driven nonlinearities are also caused by the external electric field that actuates the resonator [13, 14], which causes displacement in the fiber taper. In addition, the evanescent gap displacement causes changes in the group velocity of the whispering gallery mode, causing its resonance frequency to shift, resulting in phase detuning. To mitigate these effects, the gap between the fiber taper and the microresonator can be stabilized using active feedback control. Recently, Junge et al  demonstrated active stabilization of evanescent coupling in a bottle microresonator using a variant of the Pound-Drever-Hall (PDH) technique [16, 17], with a control bandwidth of approximately 0.1 Hz.
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. , 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 . 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
The use of RF laser amplitude modulation to impedance match a cavity was previously proposed for absorption spectroscopy  and gravitational wave detection instrumentation . Here, we discuss using a similar active feedback technique for critical coupling of a microresonator. For the whispering gallery mode microtoroid shown in Fig. 1, consider the special case where the laser is resonant. Its transmittance can be described by [21, 22]
Following the discussion in Ref. , 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, whereEq. (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 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, a − t = 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  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 , 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.
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.
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.
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.
Figure 5 quantifies the coupling gap stabilization performance of our system. The blue Trace (a) is the Fourier spectrum of the open loop AM error signal with neither the 18 Hz injected displacement dither nor feedback control. Since this error signal can be inferred as a measure of equivalent displacement from critical coupling, Fig. 5 was calibrated as gap displacement between the taper and the microtoroid. As mentioned in the previous Section, this calibration was enabled by injecting a known gap displacement via the translation stage PZT. The top Trace (a) shows that, without feedback, there was approximately 10 nm/√Hz of displacement noise at Fourier frequency of 10 mHz, rolling down to around 10 pm/√Hz at 10 Hz.
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.
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.
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.
Finally, we note that with both frequency locking and critical coupling control systems in place, we are offered a readout system which senses both amplitude and phase changes in a microresonator. It can be used to completely describe the complex optical response of the device in the presence of any absorption and loss, or dynamic changes in optical path length of the whispering gallery mode.
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
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]
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]
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]
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]