1. Introduction

Recent progress in the field of optomechanics has led to the development of novel and complex systems. These systems generally consist of an optical cavity coupled to an element with vibrational degrees of freedom. The vibrational modes can be breathing modes of disks [1] and photonic crystals [2], or flexural modes of doubly clamped beams [3,4] and cantilevers [5,6]. Much of the current work in optomechanics focuses on developing systems for use in quantum computing [7–9]. However, one of the early applications of nanomechanical devices was for mass sensing [10]. To date, the application of optomechanical systems to mass sensing has been fairly limited. Some work has been done to develop optomechanical systems for sensing in liquid [11,12]. Gas sensing using optomechanical doubly clamped beams has recently been demonstrated [13] and sensitivity even been shown to improve with higher damping [4].

With optomechanical mass sensors, providing efficient actuation can be a challenge. In [13] and [4], a piezoelectric disk was used to drive the nanomechanical devices. Piezoelectric disks are limited to frequencies in the range of tens of megahertz. Higher actuation frequencies are required for many optomechanical devices. In [11], the thermomechanical noise is used and the device is not actuated. In [12], the device undergoes self-oscillation induced by a blue-detuned laser, a common approach in optomechanics [14,15]. Furthermore, the nanomechanical mass sensor should have the largest amplitude possible (for example, driven at its onset of nonlinearity) to provide the highest dynamic range for mass sensing experiments [4,16], and so a relatively high-power actuation technique is necessary. The optical pump and probe technique described in [17] is able to meet all of these requirements. Actuation using optical pumping is a high-bandwidth technique, is compatible with phase lock loop operation, and has tunable power to reach the upper limits of the dynamic range of the mass sensor.

The pump-probe approach uses two lasers, a weak probe laser and a strong pump laser. The probe laser is positioned at the side of one optical resonance to obtain the most sensitive readout. The pump is positioned on the optical resonance ($\omega _0$) of a different cavity mode and is amplitude-modulated at the mechanical resonance frequency. Amplitude modulation of the pump laser delivers an oscillatory optical force to the nanomechanical resonator, and therefore the nanomechanical device behaves as a driven damped harmonic oscillator with a coherent phase signal that can be tracked in a phase lock loop. The amplitude modulation uses an electro-optic modulator (EOM) designed for telecommunications applications, and so the possible actuation frequency of this technique is in the range of tens of gigahertz.

To explore this promising technique further, we examine the design of optical cavities optimized for pump-probe actuation. Intuitively, an optimized design would maximize the optical power delivered to the nanomechnical device, which in turn would apply the maximal driving force to the resonator. This implies that an optical cavity with excellent extinction and high quality factor is required. Optical cavities with these characteristics often suffer from coherent backscattering effects [18], which may affect device actuation. There are additional complications when considering devices that vibrate with large amplitudes. An example of large-amplitude devices are cantilevers, which are known to have a higher mass sensitivity than doubly clamped beams due to their lower effective mass [16] and are therefore desirable for mass sensing. For a device with a large optomechanical coupling coefficient ($G = \partial \omega / \partial x$) and a large amplitude of oscillation, the cavity will experience a significant shift in the detuning ($\Delta = \omega - \omega _0$) during one period of oscillation. Therefore, the pump laser is not positioned at zero detuning for the entire period of oscillation, and the maximum power is not consistently applied to the nanomechanical device.

At first glance, a straightforward way to address this issue is to create a feedback loop to modulate the pump laser frequency so it is oscillating in phase with the mechanical device, and therefore the detuning is always equal to zero. However, the large bandwidth required by this feedback loop is very difficult to implement experimentally. The swings in the cavity response caused by a large-amplitude vibration of a cantilever, and therefore the bandwidth required, may be on the order of several GHz. The state of the art for locking to these high finesse optical cavities is currently on the order of 3.5 MHz [19], and so an improvement of three orders of magnitude is a daunting task.

Instead of a feedback loop, we can design optical cavities optimized for actuation of nanomechanical devices that compensate for the pump laser moving off the optical cavity resonance. In Section 2, we derive a model for optimal actuation using a pump-probe technique. The model takes into account the optomechanical coupling, optical cavity loss rate, coupling loss rate, and the effect of the mechanical motion on the pumping efficiency. It is applicable to any optomechanical system where the mechanical device can be modelled as a driven damped harmonic oscillator. We show the numerical solution to the model for the specific example of a simple cantilever coupled to an optical racetrack resonator. In Section 3, the fabrication and experimental measurement of several different devices will be discussed. The experimental results are compared to the model. A modification to the model that can be used when the pump optical cavity suffers from coherent backscattering is also verified experimentally. We will demonstrate that for optomechanical sensors, coherent backscattering affects only the actuation of the devices using an optical pump. Readout of the devices is unaffected by coherent backscattering.

2. Theory

To optimize the optomechanical system design, the optical force acting on the mechanical device must be quantified. A dispersive optical force generated by the gradient of the optical cavity field can be described by Eq. (1), where a negative force indicates the force is attractive and pulls the mechanical element towards the racetrack resonator [20]:

To maximize the optical drive, the applied force at the mechanical resonance frequency should be as large as possible which points to a large AC power. In other words, the peak-to-peak amplitude of the optical force should be maximized. However, the force described by Eq. (1) is a function of the detuning and, for large amplitudes of motion, can be dependent on the position of the mechanical resonator ($x(t)$) over the oscillation cycle. Explicitly, the instantaneous detuning takes on the form $\Delta (t) = G x(t) = GB\cos {\Omega t}$, where $B$ is the vibration amplitude of oscillation. Since the cavity shifts as the mechanical device vibrates, we cannot consider the pump laser to be constantly positioned at the optical resonance. In the extreme case where the amplitude of the mechanical resonator is very large, and the cavity linewidth is very small, the pump laser is rarely positioned near the optical resonance. Clearly, this will not produce efficient actuation of the mechanical resonator. For efficient actuation, we must optimize the interaction of the mechanical device and the optical cavity. This will result in the largest amplitude mechanical oscillations, which as established earlier, are most desirable for mass sensing applications.

To derive a model to optimize the cavity performance for efficient actuation, we integrate the force given by Eq. (1) over an oscillation cycle. Equation (1) can be rewritten as a function of time as:

To find the harmonic content of the force at the angular frequency of $1\Omega$ we take the first term in the Fourier series:

To demonstrate the use of our numerical model in the design of an optomechanical system optimized for efficient actuation, we input the following parameters to Eq. (8). The mechanical device is a cantilever with length = 3 $\mu$m, width = 130 nm, and thickness = 220 nm. The cantilever is side-coupled to a racetrack resonator [17]. The racetrack resonator parameters are a radius of 5 $\mu$m and a 3 $\mu$m straight section. The cantilever has effective mass $m = 48$ fg, spring constant $k = 0.8$ N/m, mechanical resonant frequency $\Omega /2\pi = 21.1956$ MHz, and $Q = 10,000$. We set $P_{\mathrm {bus}}$ equal to 260 $\mu$W, a typical experimental value. The EOM has a $V_\pi = 1.5$ V. The intrinsic loss rate $\gamma _0/2\pi$ is fixed at 2.61 GHz, which is an average measured value of several nominally identical optical cavities from the same foundry. For a sufficiently large racetrack resonator, $\gamma _0$ is set only by the material loss in the waveguide as there is negligible bending loss with high contrast waveguides, such as those fabricated in an silicon-on-insulator platform. Therefore, the intrinsic loss rate will remain fairly constant for the same cavity dimensions so long as the same fabrication process is used. The coupling loss from the bus waveguide into the racetrack resonator ($\gamma _{\mathrm {ex}}$) is used to tune the cavity linewidth $\gamma$, since $\gamma = \gamma _0 + \gamma _{\mathrm {ex}}$. The optomechanical coupling can be adjusted by varying the distance between the cantilever and the racetrack resonator. Given these parameters, we can solve Eq. (8) for a wide range of optomechanical coupling and cavity linewidth values to obtain a map of the possible cantilever amplitudes created by these specific conditions. This result is the design amplitude.

Additionally, we solve Eq. (8) for three different driving voltages, $V_{\mathrm {drive}}=200$ mV, 500 mV, and 700 mV. Although we increase the driving voltage in these simulations, the same effect is achieved if the $P_{\mathrm {bus}}$, $Q$, or $1/k$ is increased. To increase $Q$ or $1/k$, the initial mechanical design must be altered. We can increase $V_{\mathrm {drive}}$ or $P_{\mathrm {bus}}$ easily during the experiment. Increasing $V_{\mathrm {drive}}$ is preferred in this case to reduce any potential optical nonlinearities that may be introduced with higher DC input power. $V_{\mathrm {drive}}$ should also be kept well below $V_\pi$ to avoid generating higher drive harmonics [6]. The results of these calculations are shown in Fig. 1, which shows the range of optomechanical couplings and cavity linewidths where the maximum amplitude can be achieved. As the optomechanical coupling increases, the range of optimal cavity linewidth also increases. In a system with large optomechanical coupling, it is acceptable to have a so-called “bad” cavity to achieve maximum mechanical amplitude. As the optomechanical coupling increases, the cavity will shift more for a same amplitude of vibration. This means that the pump laser will spend correspondingly less time near the bottom of the cavity where the power injected into the cavity is the greatest. This means less optical force is applied to the mechanical device as the average power enhancement is reduced off-resonance. However, if the cavity has a wider linewidth, the power drop as the cavity moves off-resonance is reduced. For this reason, a wider cavity linewidth is beneficial as the optomechanical coupling increases. This trend is emphasized as the driving voltage increases. For increased driving voltage, there is a higher amplitude of AC force applied, and a correspondly higher amplitude achieved. Additionally, the region where the amplitude is near its maximum grows larger as the driving voltage increases, including wider cavities for all optomechanical coupling values.

 

Fig. 1. Optimization of the driven amplitude of a cantilever for a range of optical cavity linewidth and optomechanical coupling values. There is a triangular shaped region of maximum amplitude for a range of both optomechanical coupling and cavity linewidth. This region becomes wider as the driving voltage is increased from (a) 200 mV to (b) 500 mV to (c) 700 mV. Single colors also provide visual amplitude contours

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This simple model is adequate to understand the interplay between the optical cavity linewidth, the optomechanical coupling, and the peak amplitude of the nanomechanical resonator. We can use this model to design optomechanical systems for efficient actuation. In the next section, we will demonstrate the design of an optomechanical system using this model, and show the amplitude can be approximated by Eq. (8).

3. Experimental validation

3.1 Device parameters

To experimentally validate our model, we designed and measured several cantilevers coupled to optical racetrack cavities with the same dimensions described in Section 2. The devices were fabricated on silicon-on-insulator substrates at imec, a silicon photonics foundry, via CMC Microsystems. A timed buffered oxide etch was completed to release the cantilever from the sacrifical oxide layer.

To study the effect of the optomechanical coupling on the driven amplitude, two gaps between the racetrack resonator and the cantilever were fabricated to obtain different optomechanical couplings. The gaps were 90 nm and 145 nm. To achieve the highest amplitude possible in Fig. 1, the optomechanical coupling should be made as high as possible. Given the practical restrictions of photolithography, 90 nm is the smallest gap we were able to fabricate. Simulations showed the optomechanical coupling for a 90 nm gap was $G/2\pi \approx$ 188 MHz/nm. The wider gap of 145 nm produced a simulated optomechanical coupling of 51 MHz/nm. The simulations were done using commercial eigenmode solver software [22] to calculate the change in the effective index for a change in the gap spacing between the racetrack and the cantilever, $\partial n_{\mathrm {neff}} / \partial x$. From this value, the optomechanical coupling can be calculated using Eq. (9) [17], where $l_{\mathrm {m}}$ is the length of the mechanical resonator. $\beta = 0.394$ is the average normalized displacement of the cantilever due to its mode shape [23]. $n_{\mathrm {g}}$ is the waveguide group index obtained from mode solver simulations, and $L_{\mathrm {c}}$ is the length of the optical cavity.

The width of the waveguides was 500 nm. The coupling gap between the bus waveguide and the racetrack resonator was designed to achieve the desired $\gamma$ to optimize the optical cavity for efficient actuation. The intrinsic cavity loss is known to be approximately $\gamma _0/2\pi \approx 2.61$ GHz. We then designed the coupling loss to be approximately equal to 1.6-3.2 GHz in order to obtain a cavity linewidth in the range of 4.0-5.6 GHz. This range of linewidths and estimated optomechanical coupling places our device close to the maximum amplitude area of Fig. 1 for all actuation voltages. The coupling gap required to achieve this linewidth was 250 nm, as calculated from mode solver simulations [22]. To obtain optical cavities without optimized linewidths for comparison, several smaller coupling gaps, down to 130 nm, were also fabricated. These overcoupled cavities had larger cavity linewidths, up to 15.9 GHz, well away from the maximum amplitude region in Fig. 1.

All of the measurements were performed using a confocal microscope setup [24]. Briefly, the mechanical motion of the device was probed using a tunable diode laser (Santec TSL-510). A tunable pump laser (Photonetics) was directed through an EOM for amplitude modulation (Lucent), and the output was amplified with an erbium-doped fibre amplifier (EDFA). The pump and probe lasers were coupled together into a free-space confocal microscope setup, where the laser spot was imaged onto the grating couplers through a microscope objective. The light was coupled by a grating coupler into the photonic integrated circuit, consisting of the racetrack resonator and the nanomechanical device. The output signal was coupled out of the circuit with a second grating coupler, and directed back through the confocal microscope setup to a photodiode (New Focus), where the AC and DC components of the signal were separated. The AC component of the signal was sent to a lock-in amplifier (Zurich UHF) to measure the mechanical response. The lock-in amplifier was also used to apply an AC voltage at the desired frequency to the EOM. The DC component was used to measure the optical cavity properties.

As will be discussed later, optomechanical readout nonlinearities do come into play in our test system. In addition to these, there can also be mechanical nonlinearities resulting from the optomechanical force nonlinearity [25,26]. These mechanical nonlinearities arise at relatively high driving powers and are omitted from the present analysis. We intend a forthcoming publication with a detailed analysis of optomechanical force nonlinearities [25,27]. Finally, the intrinsic mechanical nonlinearity of the cantilever can also arise at large enough amplitude [28]. This nonlinearity arises at amplitudes of order the cantilever length divided by square root of Q, which is above 30 nm for the present device.

3.2 Effects of coherent backscattering

Five optical cavities with different coupling conditions were characterized by sweeping the probe laser through the appropriate wavelength range while the voltage was measured from the photodiode. Figure 2 is the response from one optical cavity, and shows two optical cavity resonances. Figure 2(a) is an example of peak splitting caused by coherent backscattering. The coherent backscattering effect in the all-pass microring resonator system can be modelled using the power coupling method [29]. By introducing a lumped element reflector in the transfer matrix for the optical cavity, we can model the backscattering in the device and perform a fit of the transmission in Fig. 2(a) to obtain the coupling parameters of our system. Equation (10) shows the equation used to fit the power through the bus waveguide, where the subscript b indicates the equation is for a backscattered cavity:

 

Fig. 2. Optical response of one racetrack resonator. The resonance shown in (a) is used as the probe cavity, and suffers from coherent backscattering. The resonance shown in (b) is used as the pump cavity, and is not backscattered.

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We experimentally determined the optomechanical coupling by measuring the thermomechanical noise of the cantilever as a function of cavity detuning (Fig. 3(a)). For this measurement, only the probe laser is activated, and zero detuning is set to the center of the optical cavity shown in Fig. 2(a), the highest point between the two resonance dips. The shift in the mechanical resonance frequency as the probe wavelength is changed from red to blue cavity detunings is caused by the optical spring effect [31]. The optical spring effect can be used to directly measure the optomechanical coupling, as described in [32]. The resonance frequency shift is extracted from each mechanical sweep in Fig. 3(a). Plotting the shift versus the detuning results in Fig. 3(b). We can modify the equations from [32] to account for the split cavity by regarding the single split cavity as two uncoupled cavities. By adding the optical spring equation for each cavity together and including an offset to account for the detuning of each cavity from zero, we can extract the optomechanical coupling coefficient for each dip in the split cavity [25,33]. This is shown by Eq. (13), where $\delta \Omega _0$ is the shift in the mechanical resonance frequency, with $\Omega _0 = 2 \pi f_0$. $g_0$ is the single-photon optomechanical coupling rate, and $\delta$ is the offset of each individual resonance dip from zero detuning. The subscripts 1 and 2 indicate whether the dip is the red or blue detuned optical cavity resonance, respectively.

 

Fig. 3. (a) Sweep of the thermomechanical noise along the full range of cavity detuning. The input power to the cavity ($P_{bus}$) was 77 $\mu$W. (b) Change in mechanical resonance frequency as a function of detuning caused by the optical spring effect. The circles are measured data and the black line is the fit with Eq. (13).

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By fitting the data in Fig. 3(b) with Eq. (13), we are able to extract the single-photon optomechanical coupling parameter for both the red and blue detuned cavities. This method has the advantage of being able to account for asymmetry in the case of slightly different cavity widths. Indeed, we note that the blue cavity is slightly narrower than the red (which can also be noticed in the data in Fig. 2(a)). As we use the blue-detuned cavity in the pump-and-probe experiments, we are interested in $g_{0,2}$, which is equal to $2 \pi \times 14.3$ MHz. To obtain $G$, we divide by the zero-point fluctuations of our cantilever, which gives $G/2\pi = 0.158 \pm 0.005$ GHz/nm. This experimental value is slightly less than the simulated value.

The good agreement between the simulated and measured optomechanical coupling shows that we can use an optical resonance that is split due to coherent backscattering to probe the optomechanical system. The external and intrinsic loss rates can be extracted from the backscattered cavity using Eq. (10), and these values can be used with the standard optomechanical equations with minimal modification. From Fig. 3, we can see that this split optical resonance essentially behaves as two separate optical cavities when probing the nanomechanical device. For this reason, we can choose one side of the split cavity resonance as the probe laser position for our driven measurements. For driven measurements, the probe is set to the blue slope of one optical cavity, and the pump laser is set to the transmission minimum of the optical cavity resonance of another optical cavity. Setting the laser to the optical cavity minimum ensures maximum force is delivered to the nanomechanical cantilever, since the power enhancement ($PE$) is maximized on resonance. The power enhancement of a typical all-pass racetrack resonator [25,34] is defined as the power within the optical cavity to the ratio of the input power and is equal to $PE = |\kappa /(\tau - a_{\mathrm {rt}} \exp (-i \phi ))|^2$, where $\kappa = (1-\tau ^2)^{1/2}$.

If the optical resonance used for pumping is backscattered, the power enhancement is more complex. We can derive the expression for the amplitude within the optical cavity using the same transfer matrix method with a lumped reflector element used to derive Eq. (10), where the subscript b is again used to indicate the equation is specific to a backscattered cavity:

The position in radians of the lumped reflector element is defined as $\theta$, and the reflection of the lumped reflector is defined as $r = (1-t^2)^{1/2}$. The top plot in Fig. 4(a) shows the measurement of a second, backscattered, racetrack resonator used as the pump cavity to actuate a second cantilever. The bottom half of Fig. 4(a) plots the power enhancement corresponding to the same cavity as a function of wavelength and the position $\theta$ of the reflector element. It is clear from this figure that the power enhancement can vary significantly depending on the location of the reflector element. By plotting the power enhancement at the pump laser position, shown by the dashed line in Fig. 4(a), it can be shown that the power enhancement is a sinusoidal function of reflector position (Fig. 4(b)). Although the lumped element model implies a single reflection at a one location, experimentally the scattering can occur at several locations within the racetrack. The power enhancement is therefore an average value of the sinusoid in Fig. 4(b), shown by the blue line. This approach has been previously experimentally verified [34].

 

Fig. 4. (a) Experimental measurement of a backscattered racetrack resonator cavity is shown in the top plot, the the corresponding power enhancement is shown in the bottom plot. The power enhancement is plotted versus the wavelength and the reflector position $\theta$. The dashed line indicates the pump laser position. (b) The black dashed line shows the power enhancement at the pump laser position, plotted versus the reflector position. The overall power enhancement for the device at the cavity minima is shown by the solid blue line, which is the average value of the sinusoid.

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To include backscattered cavities in the model derived in Eq. (8), the amplitude calculated from Eq. (8) should be multiplied by the ratio of the average power enhancement of the backscattered cavity to the power enhancement of the same optical cavity with no backscattering. That is:

The overall force provided to the nanomechanical device is proportional to the power enhancement, and the amplitude is directly proportional to the force via $F = k B / Q$. From Eq. (15), we expect that the amplitude of a nanomechanical device driven with a backscattered cavity will be reduced compared to a typical optical cavity. This result will be verified experimentally later in the section. For now, we will conclude our discussion on coherent backscattering by stating that a backscattered cavity may be used as the probe cavity with no additional modeling of optomechanical system. However, placing the pump at an optical cavity resonance that suffers from coherent backscattering degrades the driven amplitude of the nanomechanical device, and requires additional modeling. This is an important observation, as coherent backscattering is present in many photonic systems. Previously, backscattering has been examined in both membrane-in-the-middle systems [35] and sideband-resolved regimes [36]. In both cases, including the effects of backscattering was necessary to accurately model the optomechanical systems.

3.3 Amplitude calibration

To complete the driven measurements and compare the theoretical result to experiment for both the backscattered and non-backscattered cavities, six measurements were taken of five different mechanical devices. To obtain the sixth measurement, the position of the pump and probe lasers were switched for one nanomechanical cantilever such that the pump was acting on a backscattered cavity. The power in the bus waveguide due to the pump laser was initially set to 260 $\mu$W for the first measurement, then reduced to 100 $\mu$W for the remaining measurements after a readout nonlinearity was observed. The applied AC voltage at the EOM was 200 mV. The DC bias of the EOM was optimized to provide the highest input amplitude. The pump laser was filtered out immediately before the photodiode by a bandpass filter centered at the probe wavelength. The probe laser power in the bus waveguide was 100 $\mu$W.

3.3.1 Linear optomechanical readout

For measurements with linear readout, the amplitude of the cantilevers was recorded in volts and converted to metres using the transduction coefficient of the optomechanical system, defined as $\mathrm {d} T / \mathrm {d} x$ in units of W/m [17], where $T$ is the transmission of the optical cavity:

The expression $\mathrm {d} T/\mathrm {d} \lambda$ in Eq. (16) is the slope of the optical cavity where the probe laser is positioned. This can be recorded directly, making all of the parameters in Eq. (16) straightforward to obtain experimentally. The measured voltages are first converted to optical powers using the conversion factor of the photodiode. Next, the amplitudes in W are divided through by the transduction coefficient to give an amplitude in metres. This provides an experimental result to compare to the model, and to determine whether we have successfully optimized the optical cavities for efficient actuation.

3.3.2 Nonlinear optomechanical readout

For devices where large amplitude is predicted, such as the device designed for optimal actuation where the expected amplitude is greater than 10 nm for a high input power, nonlinearities arise in the readout of the mechanical amplitude. These nonlinearities are due to the Lorentzian lineshape of the optical cavity [25]. The slope of the optical cavity is no longer linear throughout the entire period of vibration of the cantilever, and therefore we cannot use the transduction coefficient to calibrate the amplitude of the cantilever. However, a form of calibration can be done by observing the distortion of the mechanical resonance as the optical detuning is changed [25].

A distortion in the mechanical peak shape as the probe laser detuning approaches zero is visible in Fig. 5(a). This distortion, or dip, can be attributed to the bottom of the optical resonance crossing through the probe laser wavelength for a portion of the vibration cycle [14], as illustrated in Fig. 5(b). Since the dip in the mechanical response is related to the magnitude of the cavity shift, it allows for experimental estimation of the cantilever’s amplitude. We can estimate the amplitude by noting the nominal detuning of the probe before a dip is observed, and the nominal detuning after a dip is observed. These detunings give a proxy measure for the mechanical amplitude represented as a detuning excursion of the cavity via the optomechanical coupling $G$. The first appearance of the dip is at a detuning of $2 \pi \times 1.88$ GHz. Since the optomechanical coupling is equal to $2 \pi \times 0.158$ GHz/nm, this detuning corresponds to an amplitude of 11.95 nm. This means the amplitude is equal to or greater than 11.95 nm. The last detuning before the dip is observed is $2 \pi \times 2.00$ GHz and therefore an amplitude of 12.75 nm. Averaging these two amplitudes gives $12.3 \pm 0.4$ nm, and an amplitude that we can use to compare with the model from Eq. (8).

 

Fig. 5. Illustration of amplitude calibration procedure for non-linear optomechanical readout. (a) The mechanical response near the fundamental frequency as the cavity detuning oscillates through the fixed probe laser frequency. The pump laser provides amplitude modulated drive at 200 mV applied at the EOM. Visible in (a) are dips that develop in the response as the nominal detuning level of the probe laser is decreased. These dips are the result of the optical cavity moving under the probe laser during the oscillation cycle; effectively, the dips appear when the probe laser crosses the minimum of the optical resonance, as illustrated in (b). (b) Optical cavity resonance with probe laser nominal detunings. The colored ticks on the blue side of the resonance curve represent the nominal value of each setting of probe detuning corresponding with the colored curves in (a); the arrows represent the extent of the cavity excursion due to the optomechanical coupling, given an amplitude of 12.3 nm.

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3.4 Comparison of experimental amplitudes to theoretical model

The experimental amplitudes are shown graphically in Fig. 6. The large amplitude calibrated using the dip in the mechanical spectrum is shown, in addition to the other six measurements calibrated using Eq. (16). Figure 6(a) shows the results for devices with higher optomechanical coupling, and Fig. 6(b) shows the results for devices with lower optomechanical coupling. The experimental measurements are shown by the stars. Vertical error bars account for errors in determining the optomechanical coupling and the cavity slope. The error bars are large because the optomechanical coupling was directly measured only for one device. From that measurement, the optomechanical coupling was calculated based on the exponential relationship between the optomechanical coupling and the gap between the device and waveguide (Eq. (9)), and the gaps were measured directly from scanning electron microscope images. Horizontal error bars account for errors in determining the cavity linewidth from curve fitting techniques. The lines refer back to the design amplitudes calculated in Fig. 1, which were used to initially design the optomechanical couplings and cavity linewidths. In Fig. 6(a), the black line is the design calculated using the high input power of 260 $\mu$W, and the red and blue lines are calculated using the low input power of 100 $\mu$W, as the power during the experiment was reduced to eliminate readout nonlinearities for the remaining devices. In Fig. 6(b), all lines were calculated using the lower input power of 100 $\mu$W.

 

Fig. 6. Comparison of the experimental amplitude (stars, with error bars) to the original design models (lines). (a) shows the results for high optomechanical coupling, close to $2\pi \times 0.16$ GHz/nm. The black line shows the design values for high input power, and the red and blue lines show the design values for low input power. (b) shows the results for low optomechanical coupling, approximately $2\pi \times 0.05$ GHz/nm. The design values are calculated at low input power. (b) also includes one device where the pump was placed on a backscattered peak (green).

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First we examine the correlation for the case where the optomechanical coupling is high, close to the ideal case of $2\pi \times 0.16$ GHz/nm. For the device with a low cavity linewidth, on the order of $2\pi \times 4$ GHz, the optimal design has very nearly been achieved. A large amplitude of $12.3 \pm 0.4$ nm is the result, demonstrating that we have successfully designed to achieve a high output amplitude. The drawback is that the device suffers from readout nonlinearities due to the high amplitude, however, the close agreement confirms that our model can also be used to estimate the amplitude of a nanomechanical device when the readout is nonlinear. Another important aspect of experiment is that it demonstrates that the model presented in Eq. (8) is accurate despite the presence of coherent backscattering in the probe optical cavity. The other two points in Fig. 6(a) are below the predicted amplitude, are either within error or close to within error, and have low amplitudes as expected due to the large $\gamma$ and low input power, despite the higher $G$.

For the case of low optomechanical coupling (Fig. 6(b)), the experimental values are mostly within error (three of the four) of the values calculated using Eq. (8). We note that the narrowest cavity linewidth had the pump laser placed on the backscattered cavity resonance shown in Fig. 4(a). The power enhancement reduction, from the expression in Eq. (15), is implemented to reduce the amplitude generated from Eq. (8) and produce the green line plot. If the power enhancement reduction had not been used, the green line would have been too high by a factor of 2. This demonstrates that it is necessary to account for the power enhancement reduction in a backscattered cavity to correctly estimate the amplitude.

4. Conclusions

We have derived a model to design optical cavities for efficient actuation. Efficient actuation is necessary in order to reach the maximum dynamic range of the optomechanical system for sensing applications. For both cases of low and high optomechanical coupling, the simple model produces resonably good estimates for the amplitudes. Most importantly, the measured amplitudes follow the trend predicted by the model, which indicates the model is suitable for designing optical cavities to maximize the optical force delivered to a nanomechancal device. The experiment presented here shows a high driving amplitude can be achieved with an optimized optical cavity and optomechanical coupling. Using this optimized optical actuation, a nanomechanical cantilever can be driven to sufficiently high amplitude such that the readout of the cantilever becomes nonlinear. In the case of nonlinear readout, the model can be used to estimate the amplitude of a nanomechanical device. We have also demonstrated that optical cavities suffering from coherent backscattering can be used for optomechanical sensing purposes. If the probe cavity is backscattered, no further modeling is needed, as the split optical cavity behaves as two independent cavities. For the case of the pump laser, we have demonstrated that additional modeling is needed to account for coherent backscattering, and that coherent backscattering reduces the amount of driving force delivered to the cantilever. Therefore, the pump laser should be positioned at a non-backscattered peak if possible.