Dynamic modulation of modal coupling in microelectromechanical gyroscopic ring resonators

Understanding and controlling modal coupling in micro/nanomechanical devices is integral to the design of high-accuracy timing references and inertial sensors. However, insight into specific physical mechanisms underlying modal coupling, and the ability to tune such interactions is limited. Here, we demonstrate that tuneable mode coupling can be achieved in capacitive microelectromechanical devices with dynamic electrostatic fields enabling strong coupling between otherwise uncoupled modes. A vacuum-sealed microelectromechanical silicon ring resonator is employed in this work, with relevance to the gyroscopic lateral modes of vibration. It is shown that a parametric pumping scheme can be implemented through capacitive electrodes surrounding the device that allows for the mode coupling strength to be dynamically tuned, as well as allowing greater flexibility in the control of the coupling stiffness. Electrostatic pump based sideband coupling is demonstrated, and compared to conventional strain-mediated sideband operations. Electrostatic coupling is shown to be very efficient, enabling strong, tunable dynamical coupling.

Supplementary Note 1. The ring resonator and degenerate wineglass modes The top view false-colour SEM picture of the ring resonator structure before encapsulating is shown in Figure 1a. The orange part is the resonator structure, and the blue parts are capacitive electrodes. A cross-section view of the resonator is provided in Figure 1b. The resonator is fabricated using a SOI wafer. The resonator and electrodes are patterned and separated by slots etched by deep reactive ion etching process. The flexure nested rings are released by etching the SiO 2 layer underneath. This "Epi-seal" process used to fabricate the devices is described in detail in ref. [1].
The in-plane deformation of the ideal ring structure can be expressed by the radial displacement W . ; t / and tangential displacement V . ; t/ of its mid surface, both of which are functions of circumferential position and time t. The displacements of the n-th degenerate in-plane modes are given by W n1 . ; t/ D W 1;0 cos .n / cos .! n t/; V n1 . ; t/ D V 1;0 sin .n / cos .! n t/, and W n2 . ; t/ D W 2;0 sin .n / cos .! n t/; V n2 . ; t/ D V 2;0 cos .n / cos .! n t/. Thus, the antinodal axes of the n-th degenerate modes have an angular interval of 90 ı =n. The order-2 degenerate modes with antinodal axes interval of 45 ı can be equivalent to a reduced-order two-degree-of-freedom lumped parameter system, in which the antinodal axes interval is 90 ı . The angle in the equivalent coordinates is the double of that in the real setup coordinates. In this paper, unless otherwise specified, all the specified angles are based on equivalent coordinates. Further introducing drive into EOMs leads to where F cos ! d t is the mass normalized driving force induced by˙V d cos.! d t/ in push-pull form, A d is the area of single drive electrode, and m II is the effective mass of the n D 2 normal modes. The normalized stiffness matrix in equation (4) is nondiagonal, which indicates that states detected along x and y directions are coupled. Those two hybrid states are combinations of normal modes II-1 and II-2.
The schematic of electrostatic tuning by varying V t1 is depicted by Figure 2a. The stiffness of each modes is affected by the DC voltage V 0 before the tuning voltage is applied. The resonant frequencies of the bare mechanical resonators excluding the V 0 influence are ! D q ! 2 C A t 0 V 2 0 =.d 3 0 m II /, where D II-1 or II-2, A t is the area of tuning electrode. The tuning is applied along x direction. The system can be equivalent to introducing a stiffness (normalized by mass m II ) t1 D A t 0 V t1 .2V 0 V t1 /=.d 3 0 m II / along x direction to original system. Thus, the V t1 tuning process can be described by the following equations, The V t1 tuning process can be simulated by numerically solving above equations using slowly varying envelope approximation [2]. By introducing slowly varying complex amplitudes A.t/ and B.t/, the displacements in equations (6,7) can be approximated by where cc denotes the complex conjugate of the preceding terms. Then the equations (6,7) can be rewritten as The second time derivative of the complex amplitudes have been neglected. For the steady-state analysis, P A D P B D 0. The complex amplitudes can be approximated to be Frequency responses can be obtained by calculating the frequency-dependent amplitudes jAj and jBj ( Fig. 2d,e).
The misalignment Â can be adjusted by tuning voltages V t1 and V t2 . Wherein V t2 is applied along 45 ı off-axis electrodes, which can be equivalent to introducing a stiffness (normalized by mass m II ) t2 D A t 0 V t2 .2V 0 V t2 /=.d 3 0 m II / along 45 ı off-axis direction to original system (Fig. 2b). If both V t2 and V t2 are applied, the primary orientation of normal modes change to x 0 ! -o-y 0 ! , the misalignment of new normal modes with electrode axes become Â 0 . The stiffness matrix of the tuned system along By using the condition of M stiffness being diagonal, we obtain Thus, the adjusted Â 0 by applying tuning voltages V t1 and V t2 is given by V t1 and V t2 may also influence the eigenfrequencies ! II-1 and ! II-2 . The stiffness matrix of the tuned system along x-o-y is given by Calculating the eigenvalues 0˙o f M 0 stiffness by solving det .M 0 stiffness 0˙I / D 0, we obtain the tuned eigenfrequencies, The tuned frequency difference ! 0 D ! 0 II-2 ! 0 II-1 can be further obtained.
Supplementary Note 3. Dynamical sideband coupling based on hybrid state coupling

Dynamical sideband coupling model
Applying periodical signal V p cos ! p t will cause complex modification of the stiffness along y axis. Similar to the applying of static tuning in Figure 2a, a parametric pump p is introduced into equations (4), Transforming above equation (22) into normal mode coordinates x ! -o-y ! using coordinate-transformation matrix p leads to where A d and A p are the area of drive and pump electrodes, respectively.

Dynamical sideband coupling simulation based on rotating-frame approximation
Suppose the solutions of equations (24,25) can be written as [3] x where A m and B m are slowly varying complex amplitudes of the m-th order idler resonance, and m is any integer. Substituting them into equations (24,25), we obtain upplementary Figure 3. The drive frequency ! d and pump frequency ! p responses along x axis detected at  when Â is tuned to be 36 ı is shown in Figure 2g of the main file. The simulations of drive and pump frequency responses when Â 45 ı and V p D 1.5 V, 3 V, or 6 V are shown in Figure 3b, d, f, respectively. The corresponding experimental results are shown in Figure 3a, c, e. The V p dependence of the first-order mode splitting can be simulated by setting ! p D ! II-2 ! II-1 and varying V p .

Dynamical sideband coupling interpretation based on multiple-scale analysis
Here we consider the 1-order harmonic pump. the DC and 2-order harmonic terms in the pump are neglected. The dynamics are governed by where i (i D 1; 2) and ƒ are the coefficients of intra-and inter-modal coupling, respectively. 1 D The multiple-scale analysis [4] is used to interpretation the coupling process. First, we introduce dimensionless variables t D ! 0 t , , d 0 is the initial capacitive gap. By then introducing small parameter D II =! 0 , equations (32,33) can be nondimensionalized as We define multiple times scales T 0 D t; T 1 D t; T 2 D 2 t; The solutions can be expressed as Using chain rule, we have The higher order terms are neglected. As described previously, D n m denotes the n-th order differential operator with respect to T m , (m D 0; 1; 2: n D 1; 2). Substituting (36,37) into (34,35), and equating the coefficients of like power of , we obtain Order 0 The general solutions of (38) and (39) can be written as Substituting (44) and (45) into (40) and (41) leads to Homogeneous secular term For the first-order dynamical coupling case, p ! 2 ! 1 . We introduce pump detuning parameter p , which is defined by p D ! 2 ! 1 C p . The ! j˙ p (j D 1; 2) tones (sidebands) are generated by wave mixing in (46) and (47). Apart from the homogeneous secular terms, the inter-modal coupling term will cause inhomogeneous secular terms B=2 exp . i p T 1 / in (46) and and ! p .! II-2 ! II-1 /=2. Source data are provided as a Source Data file.
A=2 exp .i p T 1 / in (47), indicating that sidebands 2 and 3 in Figure 2d of the main file will generate resonances. Of course, external forces will also induce inhomogeneous secular terms if d ! 1 or ! 2 . Here, we also introduce drive detuning parameter d that is defined by d D ! i C d , 1 or 2). The inter-modal coupling induced secular terms will actuate one mode using another mode's displacement. When the system is externally excited, the coupling term will cause energy exchange between the two modes. Suppose the drive force is actuated at mode II-1, which means p ! 2 ! 1 and d ! 1 . Annihilating the secular terms leads to The experimentally observed avoided crossing can be reproduced by transforming normal modes amplitudes into hybrid modes amplitudes using coordinate-transformation matrix p (Fig.4c). The steady-state frequency response of the first-order dynamical coupling when p ! 2 ! 1 and d ! 2 can also be obtained similarly (Fig.4d).
In order to demonstrate the interaction between the external force and sidebands, we assume d D p D 0 and apply the steady-state condition D 1 A D D 1 B D 0. Solving equations (48,49), we obtain B D f 1 =. 4! 1 ! 2 C / and A D 2i! 2 f 1 =.4! 1 ! 2 C 2 /, which indicate that displacement of mode II-2 is in phase with external force, while displacement of mode II-1 is in quadrature with external force. Sideband 2 in (48) is in antiphase with external force. In other words, sideband 2 is destructive to the external force for red-detuned pump condition, thus will cause cooling or avoided crossing (mode splitting).
To calculate the mode splitting, we make p D 0 in (50). The coupling rate g 1 can be obtained by finding the value of d that makes the denominator of juj the minimum, which is given by d0 D p 2 =16! 1 ! 2 1=4. g 1 is given by For the second-order dynamical coupling case, p .! 2 ! 1 /=2. The pump will not produce secular term in (46) and (47). Thus, we turn to higher order equations (42) and (43). Suppose the external force is driving at II-1 mode, d D ! 1 C d (The analysis of driving at II-2 mode are similar). The particular solutions of (46) and (47) after eliminating the secular terms are Substituting (44), (45), (53), and (54) into (42) and (43), we can find that wave mixing will produce ! j˙ p˙ p (j D 1; 2) tones (sidebands), which could produce additional secular terms.
where 1 denotes the non-secular terms. The steady-state frequency responses of the second-order dynamical coupling can be obtained by eliminating all the secular terms in (46), (47), (55), and (56) (Fig.4e). The second-order coupling strength g 2 can be obtained by finding the value of d that makes the frequency-response amplitude the maximum, which is given by Then, g 2 is obtained, Likewise, the second-order coupling when external force is driving at II-2 mode ( d ! 2 ) can also be simulated (Fig.4f). When the forces are not driving at resonance condition, but driving at a frequency that is half the frequency difference away from one of the resonance frequencies, they will also provide inhomogeneous secular terms in (55) and (56). Combined with mode coupling induced secular terms, avoided crossing could take place at those frequencies ( Fig.4g-i). Here we only analyse the first-and second-order approximations. For higher-order approximations, the ! j˙ p˙ p˙ p˙ (j D 1; 2) tones will be generated, which will provide higher-order couplings.
The sideband processes of the first-and second-order dynamical coupling can be revealed by the spectrum graphs. When mode II-1 is actuated without pump, the spectrum line 0 reveals the drive tone ( Fig.5a). When drive at mode II-1 with pump frequency ! p !, the sideband driven spectrum lines are shown in Figure 5b. When drive at mode II-1 with pump frequency ! p !=2, the sideband driven spectrum lines are shown in Figure 5c.

Effect of the DC term in the pump
As depicted in equation (23), there is a DC term Ä V 2 p 2 in the pump, which is applied along y direction. Based on the electrostatic tuning theory in Section Supplementary Note 2 of the Supplementary Information, we can see that this DC term will tune the hybrid coupling condition, thus affect resonant frequencies and misalignment angle Â.
Repeating the sweeps depicted in Figure 3f by applying non-resonance pumps with different values of V p . In this condition only the DC term in the pump signal would affect the system. The effect of DC term on the order-2 modes is obtained, as shown in Figure 6. The dashed lines are theoretical values obtained using the electrostatic tuning theory.  The ring resonator in this resonator is dominated by stiffness-softening electrostatic nonlinearity, which is confirmed by applying different values of DC voltage V 0 while keep the displacement amplitude almost constant (Fig. 7, Fig. 8). The constant displacement amplitude can be approximately  Fig. 7 and Fig. 8, respectively. This resonator shows stiffness-softening Duffing nonlinear frequency responses, and the nonlinearity strength is highly DC voltage-dependent. The electrostatic nonlinearity is sensitive to the DC voltage whereas other nonlinearities are not. Based on the results depicted in Figure 7 and Figure   8), we can draw a conclusion that the ring resonator used in this paper is dominated by electrostatic nonlinearity when V 0 is set to be 30 V.

Electrostatic parametric coupling
The coupling is described by two mechanical modes sharing a biased capacitor. When one resonator oscillates, the capacitance is alternatively changed, which will also alternatively modifying the other modes effective stiffness. The equations of motion can be obtained using Lagrangian method.
where k , m , and X ( D II, III) are the stiffness, mass and the displacement respect to initial equilibrium position, respectively. V is the bias voltage applied on the capacitor. A c and d 0 are the area and initial gap of the capacitor, respectively. Substituting them into Lagrangian function, d dt @T @ P X @T @X C @U @X D 0; . D II; III/: The equations of motion can be obtained Expanding the nonlinear restoring force in to Taylor series respect to X II and X III leads to where x D X X ;0 , ( D II, III). If X II;0 and X III;0 are the new equilibrium positions, which satisfy and denote d 1 D d 0 C X II;0 C X III;0 , the EOMs can be simplified as where damping rate is additionally introduced, and ; . D II, III/: If both mode II and III are simultaneously actuated, the dispersive frequency shift can be simulated by solving the following coupling equations with external forces using multiple-scale analysis.
where F and ! d-( D II, III) are the amplitudes and frequencies of the external forces. The calculation process will be reported elsewhere. Here, we just give the results. The amplitudes of the modes jx II j and jx III j are decided by the following equations, (78) (79) (80) The approximation processes of M j and N j (j D 1; 2) are based on the facts that ! 1 anď ( D II; III). Thus, we can conclude that the third-order nonlinearity coefficients are dominant in this system.
The dispersive frequency shift of mode II caused by actuating mode III can be simulated by solving jx II j using equations (75,76) (Fig.9). In this simulation, mode III suffers from stiffness-softening nonlinearity, but it has not been driven into the bifurcation condition.
The sign (direction) of the frequency shift is determined by that of the third-order nonlinearity coefficients II and III . In fact, the experimentally observed frequency shift of mode II caused by actuation of mode III is mostly caused by the 3 II x 2 III x II term in expanded equation (73).
Supplementary Note 5. Coupling-abundant multiple-mode system The experimentally observed skewed "#" configuration can be simulated by separately modeling the sequential dynamical coupling of III-1 to II-1 and II-2 and that of III-2 to II-1 and II-2. Here, we give the theoretical model for the sequential dynamical coupling of one higher mode III (refering to III-1 or III-2) to order-two modes. The drive and pump signals are applied along H-1 and H-2 hybrid states, respectively. The higher mode III is used as the phonon cavity. Before the drive and pump are applied. The initial electrical potential energy is given by.
where, x, y, and z are the displacements of hybrid states H-1, H-2, and mode III, respectively. This initial electrical potential energy is caused by the bias V 0 applied on the resonator body, which results in the frequency shift for the mechanical resonator. We'd rather include this static potential energy as part of the initial mechanical potential energy. Thus, the potential energy of the system when drive and pump are applied is given by where k ( DII-1, II-2, III) is the effective stiffness of normal modes II-1, II-2, and III, in which the initial electrical potential energy induced stiffness change has been considered. The displacements x, y of hybrid states H-1, H-2 can be transformed into displacements x ! , y ! of normal modes II-1, II-2 using coordinate-transformation matrix p. By using Lagrange method, the equations of motion can be obtained as Here, d 1 D d 0 C x !;0 cos Â y !;0 sin Â C z 0 and d 2 D d 0 C x !;0 sin Â C y !;0 cos Â C z 0 . x !;0 , y !;0 , and z 0 are new equilibrium positions, which satisfy Denoting u D x ! x !;0 , v D y ! y !;0 , and w D z z 0 , the equations of motion can be rewritten as sin Â cos .! p t/.u sin Â C v cos Â C w/ D 0; cos .! p t/.u sin Â C v cos Â C w/ D 0: Neglecting the non-resonance driving and pumping terms, and additionally introducing damping terms, the EOMs can be simplified as R u C II P u C ! 2 II-1 u C˛1u Cˇ1v C 1 w C ƒ 1 cos .! p t/.u sin Â C v cos Â C w/ D g 1 cos .! d t/; (105) R v C II P v C ! 2 II-2 v C˛2u Cˇ2v C 2 w C ƒ 2 cos .! p t/.u sin Â C v cos Â C w/ D g 2 cos .! d t/; (106) R w C III P w C ! 2 III w C˛3u Cˇ3v C 3 w C ƒ 3 cos .! p t/.u sin Â C v cos Â C w/ D 0; where a m , b m , and c m are slowly varying complex amplitudes of the m-th order idler resonance, and m is any integer. Substituting them and their first-and second-order derivative respect to time into equations (105,106,107), we obtain .a m 1 sin Â C b m 1 cos Â C c m 1 / C ƒ 2 2 .a mC1 sin Â C b mC1 cos Â C c mC1 / D g 1 2 ı.m/; (126) The second time derivatives of the complex amplitudes have been neglected. The frequency responses of normal modes can be obtained by applying steady-state condition P a m D P b m D P c m D 0 and calculating ja 0 j and jb 0 j. The experimental frequency response of the modes II can be reproduced by transforming amplitudes of x ! o y ! to x o y using coordinate-transformation matrix p.
Be simulating the coupling of modes III-1 to II-1 and II-2, the lower half of the skewed "#" in Figure   4g is obtained. Likewise, the higher half of the skewed "#" can also be obtained by simulating the coupling of modes III-2 to II-1 and II-2.