Controllable multichannel acousto-optic modulator and frequency synthesizer enabled by nonlinear MEMS resonator

Nonlinear physics-based harmonic generators and modulators are critical signal processing technologies for optical and electrical communication. However, most optical modulators lack multi-channel functionality while frequency synthesizers have deficient control of output tones, and they additionally require vacuum, complicated setup, and high-power configurations. Here, we report a piezoelectrically actuated nonlinear Microelectromechanical System (MEMS) based Single-Input-Multiple-Output multi-domain signal processing unit that can simultaneously generate programmable parallel information channels (> 100) in both frequency and spatial domain. This significant number is achieved through the combined electromechanical and material nonlinearity of the Lead Zirconate Titanate thin film while still operating the device in an ambient environment at Complementary-Metal–Oxide–Semiconductor compatible voltages. By electrically detuning the operation point along the nonlinear regime of the resonator, the number of electrical and light-matter interaction signals generated based on higher-order non-Eigen modes can be controlled meticulously. This tunable multichannel generation enabled microdevice is a potential candidate for a wide variety of applications ranging from Radio Frequency communication to quantum photonics with an attractive MEMS-photonics monolithic integration ability.


Resonator Design
In a MEMS system that is supplied with a continuous driving force or signal, there occurs a steady conversion of energy from one form to another. The rate of energy transfer is frequency dependent and it reaches its local maxima at a certain operational point called the resonant frequency of the device and each resonant frequency corresponds to a unique vibration pattern known as the mode shape. COMSOL Finite Element Method (FEM) software was utilized to run Eigen-frequency analysis to identify the modes of interest and carry out frequency domain studies to extract the dynamic parameters. The flexural mode discussed in this work is the flapping mode 1 as shown in Fig. S1(a). Analytically 1 the resonant frequency of device A is 670kHz. The material property of PZT used in the COMSOL simulation are quoted below.
(1 At the target resonant mode, the resonator has a three segmented displacement profile along the width (W) of the device and its edges and central section are always out of phase. The z-displacement, phase and also strain across the AA′ cross-section plane is shown in Fig. S1(a). As the pseudo nodal line due to the mode shape passes along the side electrode regions, the amplitude of vibration would be lesser along segment I and segment III compared to segment II (central electrode). Polytec LDV measurement of the out-of-plane vibration of the side and central electrode area corroborates the simulated trend of z-axis displacement of the resonator. The resonant frequency is majorly a function of structural dimension along with the material property. The resonator used for the HHG study is a rectangular shaped TPoS structure with anchors at the central point of the width dimension. The major structural volume is composed of Single Crystal Silicon oriented along the <100> axis. The resonant frequency deviation trend with changes in the resonator's lateral dimension and also the material thickness is shown in Fig. S1(b) and (c) respectively. Generally, as the percentage volume of high acoustic velocity material increases, the resonant frequency moves to a higher value 2 . Nonlinear behavior of the resonator system is also thickness dependent as increasing the silicon thickness improves the power handling of the device 3 since the maximum energy (Emax) stored by the device is a function of stiffness as shown in the equation.
where k and x are resonator's stiffness and vibration amplitude respectively. For a linear system, the stiffness is a unity powered; however, for a nonlinear system, the stiffness expression comprises higher order terms 4 .
Butterworth Van Dyke Model of the resonator A mass-spring-damper lumped model is a traditional method to represent a vibratory mechanical system. A resonator is a mass and spring system and to account for the energy losses, a damper is included as shown in Fig. S2(a). For an input force F, the resultant displacement x and its time derivatives can be represented as where meff, beff, and keff represent the effective mass, damping, and stiffness of the system. An electrical equivalency can be developed from the mass-spring-damper system 5 . Hence, the output response for an applied voltage V(t) can be expressed as where L, R, C, and i(t) are the electrical inductance, resistance, capacitance, and the generated current respectively. A lumped element electrical circuit that characterizes a MEMS resonator is commonly known as the Butterworth Van-Dyke (BVD) model 6 . The BVD model has two branches namely motional and static arms as shown in Fig. S2(b). Apart from the constituent materials, the motional arm components are generally mode dependent, unlike the static arm element which is majorly a function of the area of the active electrode. The feedthrough capacitance (Cf) is caused by the interaction between the input and output electrodes. The capacitances (CO1 and CO2) are the individual port capacitance between the top and bottom electrodes. The two-arm BVD model transfer function fits for a single resonant frequency and the feedthrough level around it. At the resonant frequency, effective impedance falls to a minimum value of motional resistance (Rm), as the motional capacitance (Cm) and inductance (Lm) impedances cancel out each other. The resonant frequency is hence computed by equating both motional capacitance impedance (XCm) and motional inductance impedance (XLm). The equations used for the calculation of the BVD model elements are also listed below.
where Z0, I.L., Q, and ωr are the termination impedance, insertion loss at resonance, quality factor, and the resonant frequency in radian respectively. Fig. S2(c) shows the fitted frequency response of the BVD model and the resonator measurement in the air for a driving power of -25 dBm. The parameters are tuned around the values attained using the above equations to fit the measurement data.

Supplementary Note 2
Nonlinear Piezoelectricity Theory For a resonator operating in the low amplitude drive non-hysteresis region, the device follows the linear piezoelectricity principle. Following the first law of thermodynamics, conservation of energy for the linear piezoelectric continuum can be represented as 7 where U is the stored energy density for the piezoelectric continuum and T, S, E and D are the stress, strain, electric field, and electric displacement vector component respectively. The general constitutive equations for piezoelectricity in the linear regime can be expressed as where c, e, and ε are elastic, piezoelectric, and dielectric constants, respectively and superscript t indicates transpose of the matrix. The abovementioned equations presented in the IEEE Standard of Piezoelectricity are valid only for low amplitude drive conditions. However, when the device moves into the nonlinear regime the piezoelectric constitutive equations have to be modified. N. Aurelle et al. provides equations that extend into the nonlinear domain 8 . Two nonlinear coefficients, ζ, and η are introduced to accommodate the extent of variation of Duffing and output amplitude respectively. Hence a non-matrix form of equation (8) can be written as As the output displacement spectra exhibit harmonics, the displacement x(t) can be expressed as where β is driving pulsation. Using equations (9) and (10) in the mass damper equation of a transducer, the following equation can be deduced 7 where M, A, and l are the mass, area and length respectively of the transducer, ω is the pulsation-related to the linear system's natural resonant frequency and λ is the damping factor.

Frequency Response
An alternating electric field is applied to the input electrode of the device over a range of frequencies. The indirect piezoelectric effect converts the input drive signal to mechanical displacement. At resonant frequencies of various flexural or bulk modes, the effective displacement is enhanced by a direct proportionality relation to the quality factor of the mode. The displacement-induced strain results in charge generation due to the direct piezoelectric effect, which is then collected by the output electrode. A wide frequency sweep of Design A for different drive amplitude is presented in Fig. S3(a) using the measurement setup shown in Fig. 7(a) of the main text. The extent of nonlinearity is stronger for flexural modes than the higher frequency bulk mode. The out-of-plane device deformation of flexural and bulk modes measured using the LDV for linear operation drive amplitude is shown in Fig. S3(b). The mode of interest exhibits a hardening phenomenon i.e., maximum amplitude overhangs to the higher frequency side with an increase in driving amplitude. Fig. 4(b) in the main text, presents the HHG result in the electrical domain for Design B when driven at 2Vpp. To corroborate the result, mechanical displacement readouts were performed for the same driving configuration for a range of input signal amplitude including 2Vpp AC input as shown in Fig. S4. As the drive amplitude increases, enhancement in the HHG phenomenon is seen as anticipated. The highest frequency peak corresponding to the 77 th harmonics can be observed at 24.81MHz. As the maximum measurable frequency in the LDV facility is 25MHz, harmonics beyond 25MHz cannot be recorded. For the high amplitude drive of 2Vpp, the fundamental mode displacement gets into the over-range of the LDV sensor head. Hence to capture the mode shapes of the fundamental, second and third harmonic modes of the resonator, the device is driven at 200mVpp and the mode shapes thereby captured is shown in Fig. 3(c) in the main text. Movies S1, S2, and S3 are the animation files of the out-of-plane displacement of Device B.

HHG Measurement
As a confirmatory test for the fact that the highly nonlinear resonant mode of the released MEMS PZT TPoS resonator is the source of the HHG phenomenon, two schemes were followed. Firstly, a sample of Design A is manually broken such that only the Ground-Signal-Ground pads of the two-port measurement scheme are left. A drive signal with an amplitude of 2Vpp was provided to the probing pads. As expected, there were no harmonics generated as can be seen in Fig. S5(a). The low output power shows that no signal is transmitted between the input and the output ports in the absence of the resonator and that the nonlinear phenomenon observed earlier solely comes from the released TPoS MEMS resonator. Next, to validate that the existence of a highly nonlinear resonant mode is quintessential for the HHG, Design A is driven at a far frequency offset from its flapping mode resonance. The output spectrum corresponding to the faded red region in Fig. S5(b) shows that no higher harmonics were generated as the output amplitude was very low, which shows it is difficult to drive the device into nonlinearity. Whilst when driven along the resonance region (faded blue region), a strong HHG phenomenon can be observed. The above two sets of measurements emphasize the fact that HHG comes from released TPoS MEMS and that it is necessary to drive the resonator at or in the close vicinity of the mode which has a high degree of nonlinearity.   Mass-spring-damper schematic of a mechanical vibratory system. The lumped arrangement of m, b, and k represents the resonator's mass, system losses, and the restoring force respectively. b. An electrical equivalency of the TPoS microelectromechanical resonator comprises Motional arm: Rm, Lm, and Cm and Static arm: Cf. The termination impedance in this work is 50Ω. c. The measured and fitted two-port transmission plot of the resonator operating in air for a driving power of -25 dBm. The phase information of the resonator measured using the Network Analyzer shows an overall 180º phase change at resonance.   The resonator is configured in a low-stiffness drive configuration for an AC input signal of 2Vpp and is subjected to different ranges of frequency sweep signal. a. The drive frequency is swept around the flapping mode resonant frequency and the output frequency span is set from 0.5-10MHz. The absence of peaks in the transmission spectrum of the Spectrum Analyzer with no released TPoS MEMS resonator between the driving and sensing ports indicates that there is no HHG phenomenon for this scenario. The inset shows the optical microscope image of the device under test. b. Spectrum Analyzer output power spectrum shows HHG when the resonator is driven along the nonlinear regime of the flapping mode, and no HHG when driven significantly far from resonance. The inset shows the Device A under measurement and different drive regions highlighted in the frequency response of the resonator measured using the Network Analyzer.