Isolation of optomechanical nonlinearities in nanomechanical cantilevers

In this paper we study the nonlinearities present in a nano-optomechanical system consisting of a cantilever coupled to a racetrack resonator. We show that higher-order harmonics in the mechanical frequency spectrum are indicative of a readout nonlinearity and we connect these harmonics to the optical cavity transmission with detuning function and its derivatives. We also show that there are optical force nonlinearities that cause a Duffing-type response in the driven mechanical signal. The Duffing-type nonlinearity produces a nonlinear softening when the probe laser is blue detuned, and a nonlinear stiffening when the probe laser is red detuned. The optical force and readout nonlinearities are difficult to separate using a purely experimental approach. We demonstrate the use of a numerical model of the pump-and-probe system to analyze the nonlinearities independently. The numerical model also allows us to estimate the amplitude of the system, predicting the exact detuning where nonlinearities cancel each other to produce higher critical amplitude, and ultimately gives a prospective mass sensitivity level of $83\mathrm{yg}/{\mathrm{Hz}}^{1/2}$. The numerical model provides a rich understanding of the complexities of the nano-optomechanical system and is much needed for the development of sensors with increased dynamic range and overall performance.

Figure 1

Concept of mechanical motion induced nonlinearity in large-amplitude optomechanical systems. (a) A generic Lorentzian cavity transmission function with respect to optical detuning from cavity resonance is considered. A fixed optical frequency interacts with the cavity, initially located at the red star. The position of the star on the Lorentzian curve changes with mechanical position via optomechanical coupling. (b) For small-amplitude motion, the cavity movement under the fixed frequency samples the local function slope. (c) For large-amplitude motion (comparable to the cavity linewidth), the output is nonlinear with harmonics. The input in parts (b) and (c) arises from sinusoidal mechanical motion.

Figure 2

Scanning electron microscope (SEM) image of a nanomechanical cantilever adjacent to an optical racetrack resonator.

Figure 3

(a) Sweep of the optical cavity response over a wide wavelength range. Three resonances are observed. The two lowest-wavelength resonances exhibit peak splitting, but the highest-wavelength response has no splitting and shows good extinction. For the following experiments, the probe is set to the side of the slope in the middle cavity. The pump is set to the bottom of the third resonance unless stated otherwise. (b) Coarse sweep of the probe optical cavity. Peak splitting is observed. (c) Fine sweep of the probe optical cavity at $77\mu \mathrm{W}$ (blue circles, left curve) and the corresponding curve fit (solid line). The experimental cavity data do not reach 0 dB on the red side because of the peak splitting. A second sweep at $108\mu \mathrm{W}$ (red circles, right curve) is also shown, with a theoretical model approximating the behavior of the optical cavity (dashed line). The input probe-laser power (in mW) is converted to find power at the bus waveguide (in $\mu \mathrm{W}$) considering 65% losses. Zero dB transmission level is with respect to the highest power level in the bus waveguide after the calculated 65% loss.

Figure 4

(a) Experimental measurement of the thermomechanical noise of the cantilever, smoothed to eliminate noise introduced by vacuum pump vibrations. (b) Optomechanical spring effect, peak noise level, and SNR vs detuning. A set of thermomechanical noise peak-frequency shifts is fit to Eq. (10) in order to extract the optomechanical coupling coefficient $G$.

Figure 5

The frequency response of the cantilever in apparent nanometers as the driving voltage is increased with the probe laser positioned on the (a) red and (b) blue sides of the optical cavity. The vertical dashed line is the same arbitrary frequency reference in both graphs and allows a quick visual comparison of optomechanical spring effects on frequency. The apparent amplitude is calculated using the thermomechanical noise calibration and assumes there is no readout nonlinearity. The power of the probe laser in the bus waveguide is $154\mu \mathrm{W}$. The DC power of the pump laser in the bus waveguide is $170\mu \mathrm{W}$. For (a), the AC optical power is increased from 1.13 to $39.7\mu \mathrm{W}$. For (b), the AC optical power is increased from 1.13 to $7.93\mu \mathrm{W}$. The pump-laser frequency is held constant for both measurements. When the probe laser is positioned on the blue side, nonlinear softening is observed. When the probe laser is positioned on the red side, nonlinear stiffening is observed. This is opposite to the expected result in optomechanical systems.

Figure 6

The frequency response of the cantilever as the probe laser is swept through the optical cavity. In the bus waveguide, the probe power is $108\mu \mathrm{W}$, the DC pump power is $258\mu \mathrm{W}$, and the AC pump power is $34\mu \mathrm{W}$. (a) The response at the fundamental resonance frequency. (b)–(d) The harmonic components at $2\mathrm{\Omega }/2\pi$, $3\mathrm{\Omega }/2\pi$, and $4\mathrm{\Omega }/2\pi$, respectively. The harmonics are indicative of a readout nonlinearity. The labels “amplitude (mV)” at the top of (a)–(d) denote the colorbar units. (e)–(g) The fundamental response is replotted to show the changing nonlinearity as the detuning is decreased. In (e), little nonlinearity or slight softening is observed for large positive (blue) detuning. In (f), as the detuning approaches zero, a clearer softening nonlinearity is observed and the peaks begin to have dips. The last peak without a dip is at $\mathrm{\Delta }/2\pi =2.0$ GHz and the first peak with a dip is at $\mathrm{\Delta }/2\pi =1.88$ GHz. The frequency difference between these peaks (i.e., $\mathrm{\Delta }/2\pi =2$) and $\mathrm{\Delta }/2\pi =0$ can be used to estimate that the amplitude is $12.3\pm 0.4$ nm. In (g), the nonlinearity switches from softening to stiffening for detuning changing from zero to negative values. In (g) (inset) five peaks indicate the shape of peaks changes with detuning. From left to right, the detuning values of the inset plots are $-0.63$, $-0.50$, $-0.38$, $-0.25$, and 0 GHz, respectively.

Figure 7

The peak value of the mechanical signal is shown for the fundamental frequency (blue solid circles) and second harmonic (red open circles). These peaks are extracted from Figs. 6 and 6. The first (blue, dashed line) and second (red, dotted line) derivatives of the nonlinear optical cavity are shown as well as a simple moving average of the first (blue, solid line) and second (red, long-dashed line) derivatives with a window size equal to the peak-to-peak amplitude of the nanomechanical cantilever. There is excellent agreement between the experimental peaks and the moving average of the first and second derivatives. This demonstrates that the harmonics in the mechanical frequency spectrum are directly related to the shape of the optical cavity resonance curve and are a result of a readout nonlinearity. The large window of the moving average indicates that the readout is also directly affected by the large amplitude of the cantilever.

Figure 8

Example output of the numerical simulation in the time domain with comparison to the reconstructed experimental time-domain signal. Simulation data are for $\xi =0.999925$ and $\mathrm{\Delta }/2\pi =0$ GHz where the simulation has its highest second-harmonic signal. Experimental data are for $\mathrm{\Delta }/2\pi =0$ GHz and $\xi =0.99997$ where the experiment has its highest second-harmonic signal. (a) Numerical simulation output [Eq. (17)] $P_{\text{out},1}$ (open circles) over a few mechanical oscillations showing the nonlinear shape of the response, reminiscent of Fig. 1. (b) Experimental time trace reconstructed from measured harmonics showing nonlinear response similar to part (a).

Figure 9

Comparing the results from the numerical model (dashed lines with open symbols) to the experimental data (solid lines) for the fundamental mechanical resonance frequency, where the model includes a displacement-dependent $G$ caused by the evanescent force. Values for detuning are (a) $\mathrm{\Delta }/2\pi =-0.63$ GHz, (b) $\mathrm{\Delta }/2\pi =0$ GHz, (c) $\mathrm{\Delta }/2\pi =1.13$ GHz, and (d) $\mathrm{\Delta }/2\pi =2$ GHz. The correct stiffening and softening behavior is identified by the numerical model for each detuning. The simulated peak shapes agree well with the experimental shapes, including at zero detuning. The slight dip in the result at $\mathrm{\Delta }/2\pi =1.13$ GHz is also predicted by the simulation. The difference in the size of the dip may be explained by the asymmetry in the optical cavity, which is not modeled by these simulations.

Figure 10

Comparing the numerically modeled results (dashed lines with open symbols) to the experimental data (solid or dotted lines) for the second, third, and fourth harmonics, where the model includes a displacement dependent $G$ caused by the evanescent force. The results for $\mathrm{\Delta }/2\pi =-0.63$ and 0 GHz are shown for (a) the second harmonic, (c) the third harmonic, and (e) the fourth harmonic. The results for $\mathrm{\Delta }/2\pi =1.13$ and 2 GHz are shown for (b) the second harmonic, (d) the third harmonic, and (f) the fourth harmonic. The correct directions for the stiffening and softening are predicted for all detunings.

Figure 11

Monitoring the sensitivity of the simulation to changes in detuning. Numerical modeling (dashed lines with open symbols) is done at three nearby values of detuning: $\mathrm{\Delta }/2\pi =1.0$ GHz (red open circles), 1.13 GHz (green open triangles), and 1.25 GHz (blue open squares), and compared to the experimental detuning at 1.13 GHz (solid lines). Results are shown at (a) first harmonic, (b) second harmonic, (c) third harmonic, and (d) fourth harmonic.

Figure 12

Calculated amplitudes obtained from simultaneous solutions of Eqs. (14, 15, 16) (a) without and (b) with the exponential form of the optomechanical coupling included. The dashed lines are a guide to the eye. The importance of including the exponential optomechanical coupling is verified by the zero-detuning case which in (a) shows no nonlinearity while in (b) shows a softening nonlinearity: the latter is consistent with the experimental observations.

Figure 13

(a)–(f) Modeled backbone nonlinear response curves for varying probe-laser detuning values ranging from $-0.63$ to 0 GHz. Resonance peak amplitude and frequencies are plotted for increasing simulated AC pump power. The optical cavity data input to the simulation were the same as the experimental results in Fig. 6; the pump laser was positioned at the highest-wavelength optical cavity resonance (Table 2). Fits of Eq. (4) with 95% confidence interval allow extracting the total nonlinearity value $\alpha$ and ultimately the critical amplitude. (g) Simulated backbone curves plotted on a single-frequency axis for comparison. (h) Critical amplitude and total nonlinearity plotted as a function of detuning. The detuning value of $-0.37$ GHz provides a near null in the nonlinearity and a peak in the critical amplitude near 70 nm. The spline fitting for critical amplitude is based on the application of Eq. (2) to a spline fit of the total nonlinearity.