Systems And Methods Resulting From The Parametric Coupling Of Mechanical And Electrical Resonator Assemblies And Systems And Methods To Parametrically Couple The Assemblies

Abstract

A resonator-based comb generation system configured for stable frequency comb generation in a media environment across a range of media environment densities. A system configured for frequency comb generation in a media environment across a range of media environment densities can include a resonant mechanical assembly and a resonant electrical assembly, wherein the assemblies are non-linearly coupled. A microelectromechanical (MEM) resonator can be parametrically coupled to a resonant electrical circuit to serve as an electromechanical comb generation system.

Claims

1. A resonator-based comb generation system configured for stable frequency comb generation in a media environment across a range of media environment densities.

2. The system of claim 1, wherein the system is configured for stable frequency comb generation in the media environment having a density ranging from about 900 to about 1100.

3. The system of claim 1 comprising a coupled resonant mechanical assembly and resonant electrical assembly.

4. The system of claim 1 comprising a non-linearly coupled resonant mechanical assembly and resonant electrical assembly.

5. The system of claim 1 comprising a resonant mechanical assembly coupled to two or more resonant electrical assemblies.

6. The system of claim 1, wherein the system is driven to parametric resonance.

7. The system of claim 1, wherein the system is driven to parametric resonance by an input selected from the group consisting of a mechanical input, an audio input, and a combination thereof.

8. The system of claim 3 further comprising a driving mechanism configured to drive the system into parametric resonance.

9. The system of claim 6, wherein an initiation threshold for parametric resonance is variable.

10. The system of claim 6, wherein an initiation threshold for parametric resonance is lowered by reducing an electrical resistance of the system.

11. A system configured for frequency comb generation in a media environment across a range of media environment densities comprising: a resonant mechanical assembly; and a resonant electrical assembly; wherein the assemblies are non-linearly coupled.

12. The system of claim 11, wherein the system is driven to parametric resonance; and wherein an initiation threshold for parametric resonance is selected form the group consisting of: being variable; being lowered by reducing an electrical resistance of the system; and a combination thereof.

13. The system of claim 11, wherein the system is driven to parametric resonance by an input selected from the group consisting of a mechanical input, an audio input, and a combination thereof.

14. The system of claim 11 further comprising a driving mechanism configured to drive the system into parametric resonance.

15. The system of claim 11 further comprising a second resonant electrical assembly; wherein each resonant electrical assembly is non-linearly coupled to the resonant mechanical assembly.

16. The system of claim 11, wherein the frequency combs are selected from the group consisting of acoustic frequency combs, mechanical frequency combs, phononic frequency combs, and a combination thereof.

17. The system of claim 11, wherein the resonant mechanical assembly comprises a microelectromechanical (MEM) resonator; and wherein the resonant electrical assembly comprises a resonant electrical circuit.

18. An electromechanical system for frequency comb generation comprising a coupled resonant mechanical assembly and resonant electrical assembly; wherein the system is configured to generate stable frequency combs while operating in a media environment across a range of media environment densities; and wherein the system is driven to parametric resonance by one or more inputs.

19. The system of claim 18, wherein the system is configured to generate stable frequency combs while operating in the media environment having a media environment density greater than about 1.2 kg/m.sup.3.

20. The system of claim 18, wherein the system is configured to generate stable frequency combs while operating in the media environment having a media environment density greater than about 10 kg/m.sup.3.

21. The system of claim 18, wherein the system is configured to generate stable frequency combs while operating in the media environment having a media environment density greater than about 50 kg/m.sup.3.

22. The system of claim 18, wherein the system is configured to generate stable frequency combs while operating in the media environment having a density ranging from about 900 kg/m.sup.3 to about 1100 kg/m.sup.3.

23. The system of claim 18, wherein the resonant electrical assembly comprises an RLC circuit.

24. The system of claim 18, wherein one or more of the inputs drive an electrical parameter of the resonant electrical assembly.

25. The system of claim 18, wherein one or more of the inputs drive a voltage of the resonant electrical assembly.

26. The system of claim 18, wherein one or more of: the resonant mechanical assembly and resonant electrical assembly are non-linearly coupled; the system further comprises a second resonant electrical assembly, wherein each resonant electrical assembly is coupled to the resonant mechanical assembly; a mechanical resonance frequency of the resonant mechanical assembly is approximately equal to twice an electrical resonance frequency of the resonant electrical assembly; the system is configured to generate stable frequency combs in a liquid; at least one of the inputs is selected from the group consisting of a mechanical input, an audio input, and a combination thereof; an initiation threshold for parametric resonance is selected form the group consisting of: being variable; being lowered by reducing an electrical resistance of the system; and a combination thereof; and the frequency combs are selected from the group consisting of acoustic frequency combs, mechanical frequency combs, phononic frequency combs, and a combination thereof.

27. The system of claim 18 further comprising a driving mechanism configured to drive the system into parametric resonance; wherein the resonant mechanical assembly comprises a MEM resonator; wherein the resonant electrical assembly comprises a resonant electrical circuit comprising electrical elements; wherein the MEM resonator terminates with one or more of the electrical elements of the resonant electrical circuit; and wherein the driving mechanism drives one or more of the electrical elements with one or more electrical tones.

28. The system of claim 18, wherein the resonant mechanical assembly has a mechanical Q-factor in the media environment; wherein the resonant electrical assembly has an electrical Q-factor in the media environment; and wherein the system is configured to generate the stable frequency combs while operating in the media environment in a range of mechanical Q-factor from about 25 to about 200.

29. The system of claim 28, wherein the system is configured to generate the stable frequency combs while operating in the media environment in a range of mechanical Q-factor lower than 100.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0101] The following detailed description of specific embodiments of the disclosure will be better understood when read in conjunction with the appended drawings. For the purpose of illustrating the disclosure, specific embodiments are shown in the drawings. It should be understood, however, that the disclosure is not limited to the precise arrangements and instrumentalities of the embodiments shown in the drawings.

[0102] FIG. 1: 1-D lumped parameter representation of the present electromechanical frequency comb generation system according to an exemplary embodiment. The capacitive mechanical resonator is approximated by a baffled parallel plate piston and electrically terminated in series with an inductor, resistor and a voltage source.

[0103] FIG. 2A: Top-view of the MEM array used in the experiments with the zoomed inset (FIG. 2B) showing the arrangement of the individual membrane resonators.

[0104] FIG. 3: The real and imaginary parts of the impedance measured by connecting two elements in parallel shows a mechanical resonance at 2.6 MHz when biased at 10 VDC.

[0105] FIG. 4: Simulated response of the 1-D frequency comb generation model when the input voltage is below and above the critical voltage required for comb generation.

[0106] FIG. 5: A 2-D map representing the locations of frequency comb formation in the frequency-forcing space. Here, States 1, 2 and 3 represent stable combs, higher order combs and chaotic combs respectively.

[0107] FIG. 6: A 2-D plot obtained using the semi-analytical solution displaying the dependance of comb generation on input voltage (V.sub.in) and mechanical quality-factor (Q.sub.m), where the lighter shaded region represents the existence of frequency combs.

[0108] FIG. 7A: Variation of V.sub.in-crit as a function of (Q.sub.m) for decreasing values of Q.sub.el.

[0109] FIG. 7B: Frequency spacing between comb lines as a function of V.sub.in for different Q.sub.m.

[0110] FIGS. 8A-8B: Voltage measured across the resonator and the corresponding frequency domain representation of combs when V.sub.in>V.sub.crit for MEM resonator operating in air.

[0111] FIG. 9: Increasing spacing between spectral lines observed for larger values of input drive V.sub.in.

[0112] FIG. 10: Comparison of the numerical and experimentally obtained change in spectral line spacing as a function of input voltage V.sub.in. Note that no combs are formed below a drive voltage of 0.4 V in the experiment.

[0113] FIG. 11: Experimental set up used to observe frequency combs in liquid.

[0114] FIG. 12: Impulse response of the MEM array in FC-70 as recorded by the measurement hydrophone.

[0115] FIG. 13: Increasing number of spectral lines spaced apart by 60 kHz for increasing input voltage levels.

[0116] FIG. 14: Demonstration of the tunability of generated frequency comb spacing by changing the frequency difference Aw between the two input drive tones.

[0117] FIG. 15: Experimental setup to simulate boundary conditions of an enclosed microfluidic channel.

[0118] FIG. 16: The effect of channel height d, on the location and spacing between standing wave resonances.

[0119] FIG. 17: Electromechanical comb spectrum for increasing input voltage levels when the MEM resonator is operated inside a liquid-filled channel.

[0120] FIGS. 18A-18B: Numerical simulation comparing the linear frequency response (FIG. 14A) of an electromechanical resonator in a fluid channel to its frequency comb response when the density of the fluid is reduced by 5%. The change in the frequency spacing between the combs can be seen more clearly as we measure further away from the central spectral line.

DETAILED DESCRIPTION

[0121] Although preferred embodiments of the disclosure are explained in detail, it is to be understood that other embodiments are contemplated. Accordingly, it is not intended that the disclosure is limited in its scope to the details of construction and arrangement of components set forth in the following description or illustrated in the drawings. The disclosure is capable of other embodiments and of being practiced or carried out in various ways. Also, in describing the preferred embodiments, specific terminology will be resorted to for the sake of clarity.

[0122] It must also be noted that, as used in the specification and the appended claims, the singular forms “a,” “an” and “the” include plural referents unless the context clearly dictates otherwise.

[0123] Also, in describing the preferred embodiments, terminology will be resorted to for the sake of clarity. It is intended that each term contemplates its broadest meaning as understood by those skilled in the art and includes all technical equivalents which operate in a similar manner to accomplish a similar purpose.

[0124] Ranges can be expressed herein as from “about” or “approximately” one particular value and/or to “about” or “approximately” another particular value. When such a range is expressed, another embodiment includes from the one particular value and/or to the other particular value.

[0125] By “comprising” or “containing” or “including” is meant that at least the named compound, element, particle, or method step is present in the composition or article or method, but does not exclude the presence of other compounds, materials, particles, method steps, even if the other such compounds, material, particles, method steps have the same function as what is named.

[0126] It is also to be understood that the mention of one or more method steps does not preclude the presence of additional method steps or intervening method steps between those steps expressly identified. Similarly, it is also to be understood that the mention of one or more components in a device or system does not preclude the presence of additional components or intervening components between those components expressly identified.

[0127] The examples disclosed herein illustrate devices, systems and methods for electromechanical frequency comb generation in fluid media with a parametrically driven capacitive microresonator. The present invention relates to the generation of electromechanical frequency combs in both air and liquid environments using a capacitive microresonator array.

[0128] In contrast to frequency comb generation in purely mechanical resonators, the present invention recognizes that the damping dependent threshold for comb generation can be reduced by parametrically coupling a resonant electrical circuit to the mechanical resonator.

[0129] A 1-D lumped parameter model of an exemplary embodiment is herein presented and semi-analytical solutions developed to investigate the parameters influencing frequency comb formation under various operating conditions. The results obtained with numerical simulations were experimentally validated using a commercially available MEM resonator and frequency combs with a repetition rate sensitive to the force on the mechanical resonator are generated with a single electrical drive in air and in a liquid-filled microfluidic channel.

[0130] In inventive contrast to prior work on electromechanical frequency combs, the preset invention represents a robust approach to generating stable combs, thereby enabling its practical use in applications such as gas sensing and microfluidics.

[0131] A simplified 1-D lumped parameter model of the present electromechanical frequency comb generation system 100 is illustratively shown in FIG. 1. The present system 100 comprises the parametrical coupling of a mechanical resonating assembly 200 and an electrical resonating assembly 300, driven by a driving mechanism 400.

[0132] The mechanical resonating assembly 200 transfers energy repeatedly from kinetic to potential form and back again. Simple mechanical systems include weights-and-springs assemblies, pendulums, and in FIG. 1, specifically a parallel-plate baffled piston. The piston has an equivalent mass m, stiffness k and damping b, and is radiating into a fluid half-space 220. The damping term only includes the radiation losses in the fluid, while the structural losses in the mechanical resonating assembly 200 are ignored as they are negligible in comparison. The parallel-plate piston, which acts as a time-varying capacitor with static capacitance C.sub.0 in the electrical domain, is connected to an inductor L and resistor R of the electrical resonating assembly 300 (representing the total electrical resistance in the circuit including parasitic resistance) to form a series RLC circuit driven by the driving mechanism 400, here shown as voltage source. Note (that the mechanical resonance frequency of the parallel plate piston is (ω.sub.m=√{square root over (k/m)}) the mechanical Q-factor is (Q.sub.m=ω.sub.mm/b), the electrical resonance frequency of the RLC circuit is (ω.sub.el=1/√{square root over (LC.sub.0)}), and the electrical Q-factor is (Q.sub.m=ω.sub.elL/R) Mathematically, the 1-D model can be expressed by the following coupled second-order ordinary differential equations (ODEs):

[00001]$\begin{array}{cc}\left[\frac{d^2}{d{t}^2}+\frac{b}{m}\frac{d}{dt}+\frac{k}{m}\right]x=\frac{{\epsilon }_{0}A}{2m}\frac{V_c^2}{{\left(d_0-x\right)}^2}& \left(\mathrm{Equation}1\right)\end{array}$$\begin{array}{cc}\left[\frac{d^2}{d{t}^2}+\frac{R}{L}\frac{d}{dt}+\frac{d_0-x}{LA{\epsilon }_0}\right]V_c=\frac{d_0-x}{LA{\epsilon }_0}{V}_{in}\sin \left({\omega }_{in}t\right)& \left(\mathrm{Equation}2\right)\end{array}$

[0133] where Equation 1 represents the dynamics of the mechanical resonator and Equation 2 represents the electrical resonator. d.sub.0 represents the initial gap between the parallel plates of the piston, x represents and the displacement of the piston, V.sub.c is the voltage across the variable capacitor, A is the area of the piston, and ε.sub.0 is the permittivity of free space.

[0134] The term on the right-hand side of Equation 2 represents the sinusoidal input voltage applied to the circuit having an amplitude V.sub.in and frequency ω.sub.in (Note that in order to obtain Equation 2, it is assumed that the change in capacitance due to the motion of the plate is much smaller than the static capacitance, i.e., ΔC<<C.sub.0). When the system parameters are selected such that the electrical resonance frequency is approximately half the mechanical resonance frequency (ω.sub.el≈ω.sub.m/2), the two resonators can be parametrically coupled to each other to enable non-linear modal interactions under certain input excitation conditions. These interactions result in the formation of frequency combs in the electrical as well as in the mechanical domain.

[0135] The transient response of the 1-D lumped parameter model is analyzed by solving the coupled equations numerically using a commercial solver such as MATLAB. However, this method of solution can be both non-intuitive and time consuming, especially when performing extensive parametric studies.

[0136] Alternately, the coupled ODEs can be re-written using new scaled and normalized parameters to make explicit a timescale separation, and approximated using an improved averaging theory (see, for example, J. A. Sanders, F. Verhulst, and J. Murdock, Averaging Methods in Non-linear Dynamical Systems, Vol. 59 (Springer, 2007); M. Tao, Simply Improved Averaging for Coupled Oscillators and Weakly Non-linear Waves, Communications in Non-linear Science and Numerical Simulation 71, 1 (2019)) to obtain approximate semi-analytical solutions. The semi-analytical solutions provide more insight into the operation of the proposed comb generation system by providing expressions for critical drive voltage and frequency comb spacing. These expressions can be used to explore the dependency of comb generation on a range of environmental and system parameters such as mechanical damping, electrical resistance, and operating frequency. Thus, analysis of the 1-D model using a combination of the complete numerical solution and the approximate semi-analytical techniques can be used to guide the selection of optimal operating parameters to generate stable frequency combs in different media.

Mechanical Resonator Array

[0137] A capacitive MEM resonator array is used as a testbed to demonstrate the proposed approach to frequency comb generation. Such MEM resonator arrays have previously been used for chemical and biological sensing in both fluid and gas environments. Here, we use a commercially available (Phillips innovations) MEM array (FIGS. 2A-2B) consisting of 128 elements, with each element composed of 42 electrostatically actuated membrane-based drumhead resonators connected in parallel. The circular membranes measure 120 μm in diameter and are covered by a thin layer of metal to form a top electrode (FIG. 3).

[0138] All 128 elements are separated from a common bottom electrode by a vacuum-filled cavity measuring approximately 450 nm. This vacuum gap corresponds to the full range of mechanical motion of the membranes when they are electrostatically actuated using the top and bottom electrodes. Note that each membrane operates in its fundamental or first mode for the selected frequency range of operation. Furthermore, the operating frequency is selected such that it is far away from the band in which acoustic crosstalk is dominant, thereby ensuring that all the membranes in the array oscillate in-phase and higher order array modes are suppressed. As each individual metallized membrane forms a mini-capacitor with the bottom electrode, the MEM array also behaves as a time-varying capacitor and can be terminated with an inductor to realize an inductor-capacitor (LC) electrical resonator.

[0139] Two adjacent elements of the MEM array (highlighted by the outlined box in FIG. 2B) are connected in parallel to form the active mechanical resonator for the experiment. The corresponding mechanical and electrical parameters are characterized using a vector network analyzer (Agilent 8753ES) and shown in FIG. 3. The mechanical resonance frequency in air is measured to be 2.60 MHz when a 10 V DC bias is applied (note that the unbiased mechanical frequency of the resonator is closer to 2.61 MHz as the application of bias voltage leads to a reduction in resonance frequency due to spring softening).

[0140] The mechanical Q-factor estimated from the bandwidth of the resonance peak is approximately 200, reflecting the low damping experienced by the resonator in air. The static capacitance of the active elements can also be extracted by fitting a curve to the imaginary impedance and is found to be roughly 42 pF. The parameters extracted from the experimental device are used to fully model the electromechanical comb generation system in the numerical simulations and to inform the selection of electrical components in the experiment.

Numerical Results

[0141] The 1-D model of the comb generation system described above is solved numerically using the parameters listed in TABLE 1 with the aim of investigating the influence of various system parameters and operating conditions on the formation of mechanical frequency combs.

TABLE-US-00001 TABLE 1 System Parameters Used In The Numerical Simulations Parameter Symbol Value Equivalent Mass m (Kg) 1.1155 × 10.sup.−8 Equivalent Stiffness k (N/m)   .sup. 3 × 10.sup.6 Equivalent Damping b (N-s/m) 9.1468 × 10.sup.−4 Area of Plates A (m.sup.2)  2.08 × 10.sup.−6 Gap Between Plates d.sub.0(m)   450 × 10.sup.−9 Resistance R(Ω) 60 Inductance L (H) 3.6343 × 10.sup.−4 Permittivity ε.sub.0 (F/m)   .sup. 8.854 × −10.sup.−12

[0142] The input drive frequency and electrical circuit parameters are selected such that ω.sub.in=ω.sub.m/2=ω.sub.el. This 2:1 resonance frequency relationship is intentionally selected to enable parametric coupling between the mechanical and electrical resonator. Equation 1 and Equation 2 are solved simultaneously using the ODE45 package in MATLAB, and the simulations results obtained are shown in FIG. 5. It is observed that for small values of input voltage V.sub.in, the solutions for the variables V.sub.c and x i.e., the voltage across the capacitor and piston displacement, are purely harmonic oscillations at ω.sub.in and 2ω.sub.in respectively.

[0143] However, when V.sub.in exceeds a critical input threshold V.sub.in-crit, a bifurcation appears and equally spaced spectral lines or frequency combs with spacing Δω are generated on either side of ω.sub.in and 2ω.sub.in, in both the mechanical and electrical domains. Additionally, the temporal response of the two oscillators takes the form of a pulse train having a beat frequency equal to Δω, which is characteristic of frequency comb generation systems.

[0144] The effect of detuning the input drive frequency from the electrical resonance frequency is studied by sweeping ω.sub.in/2π from 1.29 MHz to 1.32 MHz (steps of 1 kHz) at different input drive levels (steps of 0.1 V). The coupled equations are solved for a fixed time of 1 ms with a step size of 25 ns at each discrete frequency and voltage value.

[0145] It can be seen in FIG. 5 that V.sub.in-crit increases as ω.sub.in is detuned either side of w.sub.el on the frequency axis. The shaded regions where frequency combs are generated are slightly asymmetric about web similar to an instability tongue seen in parametrically excited systems. The types of combs generated within this instability can be classified into three states, where State 1 represents the most commonly occurring stable frequency combs. The squares labelled as State 2 occur at higher levels of forcing and represent regions where higher-order frequency combs are generated, caused by the non-linear interaction of State 1 combs with each other. Finally State 3 represents frequency combs that demonstrate chaotic behavior. The transient response of the system at these frequencies is highly unstable and frequently resulted in the ODE45 solver crashing.

[0146] The semi-analytical solutions developed for Equation 1 and Equation 2 can be used to determine the impact of external operating conditions on frequency comb generation. The critical threshold value V.sub.in-crit, can be numerically determined for a given set of system parameters and is found to be dependent on the mechanical damping b i.e., proportional to the reciprocal of the mechanical Q-factor.

[0147] A map of the regions where frequency combs exist in the V.sub.in−Q.sub.m space when Q.sub.el=50 is shown in FIG. 6, with the lighter shaded region indicating the existence of combs. Here, the boundary separating the lighter region from the rest of the map represents V.sub.in-crit, which increases dramatically with reduced Q.sub.m. While light damping might not impede the ability to generate frequency combs in air, the dependence of V.sub.in-crit on b introduces a significant challenge while trying to generate combs using purely mechanical resonators in heavily damped media such as water and biofluids.

[0148] Fortunately, it is found that in the case of the parametrically coupled electrical-mechanical resonator, V.sub.in-crit is a function of both mechanical damping and the resistance R in the electrical circuit. FIG. 7A shows how the critical input voltage required for parametric resonance reduces with increasing value of electrical quality factor Q.sub.el, for a fixed value of Q.sub.m. Hence maintaining a low value of R can partially mitigate the effect of mechanical damping on V.sub.in-crit thereby allowing for practically achievable values of V.sub.in-crit even at large values of b.

[0149] The semi-analytical solutions can also be used predict the spacing between the spectral lines, both as a function of the input drive voltage as well as environmental conditions. The frequency spacing Δω is plotted against V.sub.in for different values of Q.sub.m in figure FIG. 7B.

[0150] While Δω is initially zero at low values of input drive, a sudden jump or discontinuity in the curve is observed when V.sub.in=V.sub.in-crit, indicating the onset of frequency comb generation. The spacing between the comb lines initially increases with increasing V.sub.in, before it reaches a maximum value and then gradually begins to decrease. Furthermore, the spacing between the combs also increases as the mechanical Q-factor is reduced, indicating that frequency combs spanning a larger bandwidth can be obtained in heavily damped environments. It is important to note here that the semi-analytical solutions are approximate solutions to Equation 1 and Equation 2 and as such are not accurate, especially for V.sub.in>>V.sub.in-crit. They are better suited to qualitatively inform trends as opposed to quantitative predictions which require one to fully solve the coupled equations.

[0151] Hence the 1-D model allows for the investigation of the operational characteristics of the present electromechanical comb generation system. The semi-analytical solutions shed light on how different operating conditions including mechanical and electrical damping affect frequency comb generation and the comb spacing, while suggesting that careful selection of parameters can lower the critical drive voltage required for comb generation. Since the electrical resistance in a circuit can be easily minimized by various active and passive methods, the present system can potentially be utilized for mechanical frequency comb generation in hitherto inaccessible damped fluid environments.

Experimental Results

[0152] The results of the numerical analysis are experimentally verified by operating the MEM array in different media, starting with air. The MEM array is terminated with wire-wound inductor of suitable inductance such that the electrical resonance frequency of the LC circuit is half the unbiased mechanical frequency, i.e., ω.sub.el≈ω.sub.m/2=1.305 MHz. Note that circuit components with the lowest series resistance are chosen to minimize the total loss in the electrical resonator, such the electrical Q-factor is approximately 50.

[0153] The circuit is then driven by a tone-burst (10 ms ON, 2% duty cycle) from a signal generator (Agilent 33250a) through an RF amplifier for

[00002]$1.295\mathrm{MHz}\le \frac{{\omega }_{in}}{2\pi }\le 1.308\mathrm{MHz}$

and the voltage across the MEMS array is recorded. It is found that at low values of V.sub.in that frequency spectrum of V.sub.c consists primarily of the ω.sub.in/2π component. However, as the level of V.sub.in is increased and crosses the critical threshold (V.sub.in-crit=0.4 V), spectral components at (ω.sub.in∓nΔω)/2π begin to appear on either side of the drive tone where n is an integer representing the number of sidebands.

[0154] As expected, the time domain signal takes the form of a pulse train as seen in FIGS. 8A and 8B. It is observed that further increasing V.sub.in beyond V.sub.in-crit results in two phenomena—first, the number of sidebands increases as the input voltage is increased. Secondly it is seen that the frequency spacing between the spectral lines increases with V.sub.in as predicted by the numerical simulations (FIG. 9). This increase in spacing is not linear, with the sharpest change being observed closer to V.sub.in-crit, followed by a plateau before eventually reducing at high input voltage levels.

[0155] Further increasing V.sub.in causes non-linear interactions between the lines, leading to the formation of higher order combs that results in a highly unstable signal. In order to verify the relation between line spacing and V.sub.in, the coupled equations are numerically solved using ODE45 and the results are compared with the experimental data (FIG. 10). It is found that simulations and experimental data showed a close match, where the offset between the simulated and experimental Δω can be attributed to non-uniformities in fabrication and parasitic capacitance in the experimental device.

[0156] Thus, a single drive tone with V.sub.in>V.sub.in-crit is sufficient to generate stable frequency combs in the proposed system under lightly damped conditions. The spacing between the spectral lines is directly dependent on V.sub.in, which essentially acts as an electrostatic force on the mechanical resonator. Hence, by operating the system at a fixed V.sub.in, the change in line spacing can be used to directly determine an increment in force or mass acting on the mechanical resonator. Unlike conventional force sensing resonators that require complex electronic feedback loops to track changes in frequency, the present system can be monitored by simply tracking the beat frequency using a frequency counter.

[0157] The same MEM array is next immersed in non-conductive water-like liquid (Fluorinert-FC 70, Sigma Aldrich) to study the effect of increased mechanical damping on the generation of frequency combs. The experimental setup used in the investigation is shown in FIG. 11.

[0158] The frequency response of the mechanical resonator in immersion is first characterized by applying a 50 ns unipolar pulse from a signal generator to the array. The impulse response of the resonator is recorded using a broadband hydrophone (Onda Corp.) and from the FFT of the signal (FIG. 12), it is seen that the 3-dB bandwidth is 3.15 MHz and the mechanical Q-factor is less than 1 when operating in immersion.

[0159] The additional mass loading and increased damping experienced by the resonator, in contrast to operation in air, leads to a reduction in the mechanical resonance frequency to approximately 1.7 MHz, so the value of the series inductor is adjusted such the new electrical resonance frequency is 850 kHz (ω.sub.el≈ω.sub.m/2).

[0160] The system is first driven by a single electrical drive tone at ω.sub.in/2π=850 kHz and the spectrum of the signal received by the hydrophone is monitored for increasing drive amplitude levels. It is observed that for all input drive levels, the output spectrum consists purely of the ω.sub.in/π component and no additional spectral lines are observed. This can be explained by the fact that when the resonator operates in immersion, the increased damping causes V.sub.in-crit to exceed the maximum operating voltage of the resonator—as a result, frequency combs are not generated when the resonator is driven by a single drive tone in an open liquid domain.

[0161] Alternately, the case of a resonator operating in a fluid with a rigid boundary is considered, as many biosensing applications consist of an enclosed microfluidic channel through which a fluid of interest is flown and the sensing element is placed inline along the channel.

[0162] Instead, two input drive tones of equal amplitude ω.sub.in1 and ω.sub.in2 (ω.sub.in1/π=820 kHz, ω.sub.in2/π=880 kHz, Δω=60 kHz) are applied to the system and the input drive strength is gradually increased. At low input drive levels, the output spectrum contains components at 2ω.sub.in1, 2ω.sub.in2 and ω.sub.in1+ω.sub.in2, that are generated due to the voltage-squared dependence of the MEM resonator (see Equation 1).

[0163] At higher input drive levels, an increasing number of spectral lines spaced apart by 60 kHz are generated on either side of at 2ω.sub.in1 and 2ω.sub.in2 (FIG. 13). A similar approach of using two drive tones to generate electromechanical frequency combs in a resonator has been previously reported and is governed by a parametric 4-wave mixing process. However, the combs generated in earlier demonstrations are confined to a narrow bandwidth within the vibrational mode of a high-Q resonator.

[0164] The broadband characteristics of our MEM array, coupled with the reduced electrical resistance in the LC circuit, enables the generation of equally spaced spectral lines spanning over an octave in the frequency scale. It can be observed in FIG. 14 that the repetition rate of the combs can be also controlled by simply changing the frequency spacing Δω between the input drive tones. More significantly, Δω can be made arbitrarily large or small, as long as the input drive level meets the required conditions to enable the parametric mode coupling required for comb generation. The ability to easily generate tunable, broadband and coherent frequency combs in a damped environment potentially allows the proposed system to be used for precise distance measurement in the short to medium range in fluid.

[0165] In addition to being able to generate frequency combs with a fixed repetition rate using two input drive tones, it would be tremendously useful to generate combs in liquid using a single input drive, in which the spacing between the spectral lines are a function of the force acting on the mechanical resonator. Many biosensing applications consist of an enclosed microfluidic channel through which a fluid of interest is flown, and the sensing element is placed inline along the channel.

[0166] To simulate similar boundary conditions, a MEM array is fixed to the bottom of a container filled with FC-70 and a piezoelectric transducer is placed above at a distance d, with its flat face parallel to the array surface (FIG. 15). The face of the transducer acts as a hard wall or reflector mimicking the walls of a microfluidic channel and enables the formation of acoustic standing waves between the MEM array and the piezo transducer. These high-Q factor standing wave resonances can be exploited as the primary vibrational mode of the mechanical resonator to generate frequency combs with a single drive tone in liquid.

[0167] As the vertical distance d can be adjusted by a screw gauge micrometer, the position and frequency spacing between the standing waves can be manually tuned to a desired value as seen in FIG. 16. The screw gauge is adjusted to obtain a spacing of d=1.8 mm and a standing wave resonance (Q.sub.m≈13) is generated at 1.15 MHz (c.sub.FC-70=687 m/s). The electrical circuit is tuned to half of this frequency (ω.sub.el=575 kHz) and the system is driven with a single tone at ω.sub.in=ω.sub.el.

[0168] Similar to the behavior in air, it is observed in FIG. 17, that additional evenly spaced spectral lines begin to appear on either side of ω.sub.el as the input drive is increased above V.sub.in-crit=46 V. The spacing between the spectral lines also increases with higher input voltage levels, confirming the dependency of line spacing on the force experienced by the mechanical resonator.

[0169] Despite the larger damping experienced in immersion, the confinement of energy within a high-Q standing wave mode enables the generation of electromechanical frequency combs using a single drive tone, in an enclosed fluid-filled channel. The change in spectral line spacing with increasing forcing further indicates that the generated combs can be used to sense fluid properties or mass of particles in solution in a simple manner without fabrication of complex resonator arrays.

[0170] To highlight the advantage of comb-based sensing over conventional resonance spectroscopy when interrogating bulk fluid properties in microfluidics, we simulate and compare the frequency response of a linear resonator to a frequency comb generating resonator, when operating in a 200 μm high, fluid-filled channel. The methods of solution described above cannot be used to solve this particular problem due to the non-trivial boundary conditions—instead we employ a previously used technique that makes use of SIMULINK.

[0171] The sensitivity of the resonator to a 5% change in fluid density is evaluated when operating in these two distinct modes, and the results are shown in FIGS. 18A-18B. It is observed that in the linear case, the density change produces a negligible shift in both the amplitude and frequency the resonant peak, partly as the low-Q frequency response makes it challenging to accurately detect the peak shift. The time of flight or spacing between the standing wave resonances cannot be used either as the speed of sound of the medium is held constant. However, a clear and distinguishable separation between the spectral lines for the two different fluid densities can be observed when the resonator is configured to generate frequency combs (the recording length or gate time here is 5 ms).

[0172] Inherently, the system acts as an amplitude to frequency converter, where the change in operating point on the linear frequency response curve is expressed as a shift in the spectral line spacing. Furthermore, the frequency shift Δω is multiplied by a factor n (where n represents the number of spectral lines formed on either side of the driving frequency) as we move away from the central spectral line, resulting in improved sensitivity with a greater number of frequency combs.

[0173] For example, a frequency spacing of 3 kHz can be observed at the 2.sup.nd spectral line from the center, whereas a spacing of 6 kHz is observed at the 4.sup.th spectral line, for the same change in density.

[0174] It is important to point out that the spectral line width can be further reduced by increasing the gate width or acquisition time. The collection of a larger number of data points in the time domain improves the resolution of the combs in the frequency domain, thus enabling higher sensor resolution at the cost of a longer acquisition time. The sensitivity of the comb-based system can also be improved by generating a larger number of equidistant spectral lines on either side of the drive tone, as shown in the simulation. Increasing the strength of the drive tone increases the signal-to-noise ratio (SNR) of the sidebands, however, there is an upper limit on the drive strength beyond which the generated combs become unstable.

[0175] Alternately, tuning the bandwidth of the input excitation signal or modifying the non-linearity that mediates frequency mixing could potentially enable broadband spectral lines as seen in OFCs. Thus, the potential of mechanical comb-based techniques for fluidic measurements and spectroscopy is demonstrated. Further improvements in resolution and sensitivity can be made possible by optimizing the driving voltage and other system parameters, with the ultimate system resolution eventually limited by effects such as thermal fluctuations, external mechanical vibrations and phase noise.

[0176] As disclosed herein, we demonstrate the generation of stable electromechanical frequency combs in air and liquid by using a MEM array parametrically coupled to an electrical resonator. A 1-D lumped parameter model of the proposed system is presented, and approximate semi-analytical solutions are developed, that allow the investigation of system parameters influencing frequency comb generation. Numerical simulations reveal that the critical input voltage required to generate frequency combs is a function of both the mechanical Q-factor and electrical Q-factor.

[0177] In contrast to purely mechanical resonator-based frequency comb generation methods, the initiation threshold in the proposed system can be lowered by reducing the electrical resistance using passive or active methods, thereby enabling frequency comb formation even in highly damped environments. The results obtained by numerical simulations are experimentally validated using a commercially available MEM resonator terminated with a wire-wound inductor. Frequency combs with a repetition rate sensitive to the force on the mechanical resonator are generated with a single electrical input drive in air and in an enclosed fluid-filled microfluidic channel.

[0178] The advantage of our approach in sensing applications is highlighted by comparing the performance of a comb-based sensor with conventional resonance spectroscopy when interrogating bulk fluid properties. Thus, the ability to generate stable and tunable electromechanical combs in fluids enables several applications previously inaccessible to optical combs. Future work will be focused on optimizing the proposed system for applications in microfluidic particle detection. The generation of frequency combs in an open fluid domain using a single drive tone will also be explored.

[0179] In this description, numerous specific details have been set forth. It is to be understood, however, that implementations of the disclosed technology can be practiced without these specific details. In other instances, well-known methods, structures and techniques have not been shown in detail in order not to obscure an understanding of this description. References to “one embodiment,” “an embodiment,” “some embodiments,” “example embodiment,” “various embodiments,” “one implementation,” “an implementation,” “example implementation,” “various implementations,” “some implementations,” etc., indicate that the implementation(s) of the disclosed technology so described can include a particular feature, structure, or characteristic, but not every implementation necessarily includes the particular feature, structure, or characteristic. Further, repeated use of the phrase “in one implementation” does not necessarily refer to the same implementation, although it can.

[0180] As used herein, unless otherwise specified the use of the ordinal adjectives “first,” “second,” “third,” etc., to describe a common object, merely indicate that different instances of like objects are being referred to, and are not intended to imply that the objects so described must be in a given sequence, either temporally, spatially, in ranking, or in any other manner.

[0181] While certain implementations of the disclosed technology have been described in connection with what is presently considered to be the most practical and various implementations, it is to be understood that the disclosed technology is not to be limited to the disclosed implementations, but on the contrary, is intended to cover various modifications and equivalent arrangements included within the scope of the appended claims. Although specific terms are employed herein, they are used in a generic and descriptive sense only and not for purposes of limitation.

[0182] This written description uses examples to disclose certain implementations of the disclosed technology, including the best mode, and also to enable any person skilled in the art to practice certain implementations of the disclosed technology, including making and using any devices or systems and performing any incorporated methods. The patentable scope of certain implementations of the disclosed technology is defined in the claims, and can include other examples that occur to those skilled in the art. Such other examples are intended to be within the scope of the claims if they have structural elements that do not differ from the literal language of the claims, or if they include equivalent structural elements with insubstantial differences from the literal language of the claims.