Nanobeam Resonator Research Articles

Lithium niobate photonic-crystal electro-optic modulator

Modern advanced photonic integrated circuits require dense integration of high-speed electro-optic functional elements on a compact chip that consumes only moderate power. Energy efficiency, operation speed, and device dimension are thus crucial metrics underlying almost all current developments of photonic signal processing units. Recently, thin-film lithium niobate (LN) emerges as a promising platform for photonic integrated circuits. Here, we make an important step towards miniaturizing functional components on this platform, reporting high-speed LN electro-optic modulators, based upon photonic crystal nanobeam resonators. The devices exhibit a significant tuning efficiency up to 1.98 GHz V−1, a broad modulation bandwidth of 17.5 GHz, while with a tiny electro-optic modal volume of only 0.58 μm3. The modulators enable efficient electro-optic driving of high-Q photonic cavity modes in both adiabatic and non-adiabatic regimes, and allow us to achieve electro-optic switching at 11 Gb s−1 with a bit-switching energy as low as 22 fJ. The demonstration of energy efficient and high-speed electro-optic modulation at the wavelength scale paves a crucial foundation for realizing large-scale LN photonic integrated circuits that are of immense importance for broad applications in data communication, microwave photonics, and quantum photonics.

Vibration of nonlocal perforated nanobeams with general boundary conditions

This article presents a comprehensive model to investigate a free vibration and resonance frequencies of nanostructure perforated beam element as nano-resonator. Nano-scale size dependency of regular square perforated beam is considered by using nonlocal differential form of Eringen constitutive equation. Equivalent mass, inertia, bending and shear rigidities of perforated beam structure are developed. Kinematic displacement assumptions of both Timoshenko and Euler-Bernoulli are assumed to consider thick and thin beams, respectively. So, this model considers the effect of shear on natural frequencies of perforated nanobeams. Equations of motion for local and nonlocal elastic beam are derived. After that, analytical solutions of frequency equations are deduced as function of nonlocal and perforation parameters. The proposed model is validated and verified with previous works. Parametric studies are performed to illustrate the influence of a long-range atomic interaction, hole perforation size, number of rows of holes and boundary conditions on fundamental frequencies of perforated nanobeams. The proposed model is supportive in designing and production of nanobeam resonator used in nanoelectromechanical systems NEMS.

Application of nonlocal continuum theory to the primary resonance analysis of an axially loaded nano beam under time delay control

The paper is concerned with the nonlinear primary resonance of nano beams with axial load under the velocity time delay control. In order to have a deep insight into the system, the amplitude frequency response curve of the system is firstly obtained using the multiple scales method. The effects of the control gains and time delays on the system stability are then investigated. The analyses illustrated that both delay feedback gain coefficient and velocity time delay control can mitigate the system vibrations properties (e.g. hardening nonlinearity, resonance amplitude and the corresponding width) to an excellent level. The nonlinear primary resonance of nano beam is also discussed with the influences of small scale effect, axial initial load, wave number, Winkler foundation modulus and the ratio of the length to the diameter. This paper establishes the relationship between the time delay controller and vibration properties of a nano beam system which provides the guidance for applying time delayed active control for different types of nano devices in engineering practices.

Frequency Stabilization of Nanomechanical Resonators Using Thermally Invariant Strain Engineering.

Microfabricated mechanical resonators enable precision measurement techniques from atomic force microscopy to emerging quantum applications. The resonance frequency-based physical sensing combines high precision with long-term stability. However, widely used Si3N4 resonators suffer from frequency sensitivity to temperature due to the differential thermal expansion vs the Si substrates. Here we experimentally demonstrate temperature- and residual stress-insensitive 16.51 MHz tuning fork nanobeam resonators with nonlinear clamps defining the stress and frequency by design, achieving a low fractional frequency sensitivity of (2.5 ± 0.8) × 10-6 K-1, a 72× reduction. On-chip optical readout of resonator thermomechanical fluctuations allows precision frequency measurement without any external excitation at the thermodynamically limited frequency Allan deviation of ≈7 Hz/Hz1/2 and (relative) bias stability of ≈10 Hz (≈ 0.6 × 10-6) above 1 s averaging, remarkably, on par with state-of-the-art driven devices of similar mass. Both the resonator stabilization and the passive frequency readout can benefit a wide variety of micromechanical sensors.

Cavity electromechanics with parametric mechanical driving

Microwave optomechanical circuits have been demonstrated to be powerful tools for both exploring fundamental physics of macroscopic mechanical oscillators, as well as being promising candidates for on-chip quantum-limited microwave devices. In most experiments so far, the mechanical oscillator is either used as a passive element and its displacement is detected using the superconducting cavity, or manipulated by intracavity fields. Here, we explore the possibility to directly and parametrically manipulate the mechanical nanobeam resonator of a cavity electromechanical system, which provides additional functionality to the toolbox of microwave optomechanics. In addition to using the cavity as an interferometer to detect parametrically modulated mechanical displacement and squeezed thermomechanical motion, we demonstrate that this approach can realize a phase-sensitive parametric amplifier for intracavity microwave photons. Future perspectives of optomechanical systems with a parametrically driven mechanical oscillator include exotic bath engineering with negative effective photon temperatures, or systems with enhanced optomechanical nonlinearities.

Hearing distributed mass in nanobeam resonators

One-dimensional vibrating nanostructures show remarkable performance in detecting small adherent masses added to a referential configuration. The mass sensing principle is based on measuring the resonant frequency shifts caused by the unknown attached masses. In spite of its important application in several fields, few studies have been devoted to this inverse eigenvalue problem. In this paper we have developed a distributed mass reconstruction method for initially uniform nanobeams based on measurements of the first lower resonant frequencies of the free bending vibration. Two main inverse problems are addressed. In the first problem, the mass variation is determined by using the first lower eigenfrequencies of a supported nanobeam, under the a priori assumption that the mass variation has support contained in half of the axis interval. In the second problem, we show that the a priori assumption can be removed, provided that the spectral input data include an additional set of first lower eigenfrequencies belonging to a second spectrum associated to different end conditions. The nanobeam is modelled using the modified strain gradient elasticity accounting for size effects. The reconstruction is based on an iterative procedure which takes advantage of a closed-form solution when the mass change is small, and shows to be convergent under this assumption and for smooth mass variation. The accuracy of the reconstruction deteriorates in presence of discontinuous mass variation. For these cases, a constrained least-squares optimization filtering shows to be very effective to reduce the spurious oscillations around the target coefficient. Numerical simulations show that the identification method performs well even for not necessarily small mass changes and it is stable in presence of errors on the data. An experimental validation of the method has provided encouraging results, despite the fact that only the first four eigenfrequencies under cantilever end conditions were used.

Nonlinear secondary resonance of FG porous silicon nanobeams under periodic hard excitations based on surface elasticity theory

To impart desirable material properties, functionally graded (FG) porous silicon has been produced in which the porosity changes gradually across the material volume. The prime objective of this work is to predict the influence of the surface free energy on the nonlinear secondary resonance of FG porous silicon nanobeams under external hard excitations. On the basis of the closed-cell Gaussian-random field scheme, the mechanical properties of the FG porous material are achieved corresponding to the uniform and three different FG patterns of porosity dispersion. The Gurtin–Murdoch theory of elasticity is implemented into the classical beam theory to construct a surface elastic beam model. Thereafter, with the aid of the method of multiple time-scales together with the Galerkin technique, the size-dependent nonlinear differential equations of motion are solved corresponding to various immovable boundary conditions and porosity dispersion patterns. The frequency response and amplitude response associated with the both subharmonic and superharmonic hard excitations are obtained including multiple vibration modes and interactions between them. It is found that for the subharmonic excitation, the nanobeam is excited within a specific range of the excitation amplitude, and this range shifts to higher excitation amplitude by incorporating the surface free energy effects. For the superharmonic excitation, by taking surface stress effect into account, the excitation amplitude associated with the peak of the vibration amplitude enhances. Moreover, in the subharmonic case, it is demonstrated that by increasing the porosity coefficient, the value of the excitation frequency at the joint point of the two branches of the frequency-response curve reduces. In the superharmonic case, it is revealed that an increment in the value of porosity coefficient leads to decrease the peak of the oscillation amplitude and the associated excitation frequency.

A GN-based modified model for size-dependent coupled thermoelasticity analysis in nano scale, considering nonlocality in heat conduction and elasticity: An analytical solution for a nano beam with energy dissipation

This investigation deals with a size-dependent coupled thermoelasticity analysis based on Green-Naghdi (GN) theory in nano scale using a new modified nonlocal model of heat conduction, which is based on the GN theory and nonlocal Eringen theory of elasticity. In the analysis based on the proposed model, the nonlocality is taken into account in both heat conduction and elasticity. The governing equations including the equations of motion and the energy balance equation are derived using the proposed model in a nano beam resonator. An analytical solution is proposed for the problem using the Laplace transform technique and Talbot technique for inversion to time domain. It is assumed that the nano beam is subjected to sinusoidal thermal shock loading, which is applied on the one of beam ends. The transient behaviors of fields\' quantities such as lateral deflection and temperature are studied in detail. Also, the effects of small scale parameter on the dynamic behaviors of lateral deflection and temperature are obtained and assessed for the problem. The proposed GN-based model, analytical solution and data are verified and also compared with reported data obtained from GN coupled thermoelasticity analysis without considering the nonlocality in heat conduction in a nano beam.

Theoretical analysis of thermoelastic damping of silver nanobeam resonators based on Green–Naghdi via nonlocal elasticity with surface energy effects

The vibration of a silver nanobeam resonator, considering the surface effect as well as the thermal effect has been investigated in this paper. The emerged bending in nanobeam resonator causes the surface effects to appear in nanobeam. The governing equations for nanobeam are obtained considering the surface and thermal effects and using the nonlocal elasticity theory. The temperature effects based on the Green–Naghdi thermoelasticity theory, and considering the thermoelastic damping, are taken into account. The vibration governing equations are derived by the coupled Green–Naghdi thermoelastic, nonlocal elasticity theory, and surface effect for Euler–Bernoulli beam model. The dynamic and temperature responses of the nanobeam are obtained in the Laplace domain using the Laplace method. The technique of inverse Laplace, called a Talbot method, is utilized to calculate the dynamic and thermal responses of the nanobeam in the time domain. To investigate the results, the effects of the various parameters, such as the surface effects, nonlocal parameter, and the initial temperature conditions, on the dynamic and temperature responses of the microbeam are scrutinized.

Thermoelastic damping in micro and nano-mechanical resonators utilizing entropy generation approach and heat conduction model with a single delay term

Analysis of thermoelastic damping (TED) of micro and nano-beam resonators plays a very important role in designing the resonator with high-quality factors. TED of micro and nano-beam resonators has therefore been an interesting and challenging area of research in recent years. Prediction of TED is more challenging under the effect of non-Fourier heat conduction. In the past, several works have been dedicated to TED modeling with the influence of non-Fourier heat conduction in view of the fact that non-Fourier heat conduction model is more relevant for analysis of TED for small scale devices. The present work aims to investigate TED in micro and nano-mechanical resonators utilizing non-Fourier heat conduction model with a single delay term introduced by Quintanilla in 2011. We derive an explicit formula of the quality factor for TED based on the entropy generation approach. With the help of numerical results, we present the influence of TED in the context of normalized frequency as well as beam thickness. Further, the effects of the time delay parameter and the material constants on TED have been discussed in detail. The results of the present model are compared to those obtained for GN-III model. It has been observed that the current model offers a high-quality factor as compared to GN-III model.

Two-Temperature Thermoelastic Damping of a Gold Nano-Beam Resonator with Variable Young's Modulus

This paper deals with the thermoelastic damping (Q-factor) of a gold nano-beam resonator and is based on twotemperature thermoelasticity models. An explicit formula of the Q-factor has been derived when Young’s modulus is variable as a function of the room temperature. The length of the beam and Young’s modulus has been studied with comparison being made between the Biot model and the Lord-Shulman model (L-S). The numerical results show that the values of beam length, the relaxation time parameter, and the two-temperature parameter have a strong influence on the thermoelastic damping quality factor.

Nanophotonic Quantum Storage at Telecommunication Wavelength

Quantum memories for light are important components for future long distance quantum networks. We present on-chip quantum storage of telecommunications band light at the single photon level in an ensemble of erbium-167 ions in an yttrium orthosilicate photonic crystal nanobeam resonator. Storage times of up to 10 $\mu$s are demonstrated using an all-optical atomic frequency comb protocol in a dilution refrigerator under a magnetic field of 380 mT. We show this quantum storage platform to have high bandwidth, high fidelity, and multimode capacity, and we outline a path towards an efficient erbium-167 quantum memory for light.

A new mechanism of energy dissipation in nanomechanical resonators due to the Casimir force

In this study, we report a new energy dissipation mechanism of nanomechanical resonators due to the Casimir effect originating from quantum fluctuation of the vacuum electromagnetic field at the nanoscale. An analytical study on the evaluation of the Casimir effect-induced energy loss in nanobeam resonators undergoing in-plane flexural vibration is presented. Two-dimensional elastic wave theory is employed to determine the energy transmission from the vibrating resonator to the substrate. Fourier transform and Green's function technique are adopted to solve the problem of wave motions on the surface of the substrate excited by the Casimir force. Analytical expressions of the Casimir effect-induced energy loss in terms of the quality factor, taking into account both pressure wave propagation in the noncontact substrate and shear wave propagation in the supporting substrate, as well as linear and nonlinear terms of time-varying Casimir force, have been derived. Effects of beam geometry, initial separation gap, and structural boundary conditions on energy loss are examined. Results of the present study demonstrate that the Casimir effect-induced energy loss plays an important role in the dissipation of the nanobeam resonators, in which the influence of shear wave propagation is remarkable. Also, as reflected by our results, the influence of nonlinear terms of time-varying Casimir force on the energy dissipation cannot be neglected for large-amplitude vibration, which is obviously a feature of nonlinear damping. Furthermore, we propose a possible way to experimentally measure the Casimir force by using the energy dissipation mechanism due to the Casimir force.

Nonlinear vibration analysis of an electrostatic functionally graded nano-resonator with surface effects based on nonlocal strain gradient theory

In this paper, a comprehensive analysis of the nonlinear vibration of an electrostatic nanobeam resonator is presented based on the nonlocal strain gradient theory (NSGT) and by incorporating the Gurtin-Murdoch surface elasticity theory. The Von-Kármán geometrical nonlinearity and inter-molecular dispersion forces, i.e., van der Waals and Casimir forces are included in the equation of motion. Both DC and AC components of the electrostatic actuation are regarded as the excitation terms. The nanobeam is considered to be composed of a power-law functionally graded (FG) material. Utilizing Hamilton’s principle, the size-dependent nonlinear equation of motion of the system is derived. Multiple scales technique in conjunction with the differential quadrature method (DQM) is adopted to analytically obtain the solution. Results obtained are shown to be in good agreement against available literature. Static deflection and fundamental natural frequency are obtained for different size-dependent and volume fraction index parameters. Meanwhile, the variation of the oscillation amplitude by the quality factor, excitation magnitude, and frequency is determined near the primary resonance. The acquired results revealed that the nonlocal and strain gradient parameters can significantly displace the multi-valued portions and instability thresholds of the dynamical response diagrams. It is shown that the increment of the volume fraction index reduces the pull-in voltage while increasing any of the size-dependent parameters enlarge the instability voltage. Moreover, the surface effects induce the stiffness hardening behavior, whereas the inter-molecular forces impose the stiffness softening effect.

Nonlinear secondary resonance of nanobeams under subharmonic and superharmonic excitations including surface free energy effects

• Size-dependent nonlinear hard oscillation of nanobeams. • Surface free energy effects on the nonlinear secondary resonance of nanobeam with various end supports. • Size-dependent frequency-response of a nanobeam for the both of subharmonic and superharmonic excitations. • Size-dependent amplitude-response of a nanobeam for the both of subharmonic and superharmonic excitations. The main objective of this study is to predict both the subharmonic and superharmonic resonances of the nonlinear oscillation of nanobeams in the presence of surface free energy effects. To this purpose, Gurtin–Murdoch elasticity theory is adopted to the classical beam theory in order to consider the surface Lame constants , surface mass density, and residual surface stress within the differential equations of motion. The Galerkin method together with the method of multiple scales is utilized to investigate the size-dependent response of nanobeams under hard excitations corresponding to various boundary conditions. A parametric analysis is carried out to indicate the influence of the surface elastic parameters on the frequency-response as well as amplitude-response of the nonlinear secondary resonance including multiple vibration modes and interactions between them. It is seen that for the superharmonic excitation, except for the clamped–free boundary condition, the jump phenomenon is along the hardening direction, while in the clamped–free end supports, it is along the softening direction. Moreover, it is revealed that for the subharmonic excitation, within a specific range of the excitation amplitude, the nanobeam is excited, and this range shifts to lower external force by incorporating the surface free energy effects. It is found that in the case of superharmonic excitation, the value of the excitation frequency associated with the bifurcation point at the peak of the frequency-response curve increases by taking the surface free energy effect into consideration.

Witnessing Optomechanical Entanglement with Photon Counting.

The ability to coherently control mechanical systems with optical fields has made great strides over the past decade, and now includes the use of photon counting techniques to detect the nonclassical nature of mechanical states. These techniques may soon be used to perform an optomechanical Bell test, hence highlighting the potential of cavity optomechanics for device-independent quantum information processing. Here, we propose a witness which reveals optomechanical entanglement without any constraint on the global detection efficiencies in a setup allowing one to test a Bell inequality. While our witness relies on a well-defined description and correct experimental calibration of the measurements, it does not need a detailed knowledge of the functioning of the optomechanical system. A feasibility study including dominant sources of noise and loss shows that it can readily be used to reveal optomechanical entanglement in present-day experiments with photonic crystal nanobeam resonators.

Nonlinear frequency conversion in one dimensional lithium niobate photonic crystal nanocavities

We demonstrate flexible nonlinear frequency up-conversion in high-Q lithium niobate photonic crystal nanobeam resonators. The high optical Q together with strong optical mode confinement allows us to observe clear second harmonic generation and sum frequency generation with an optical power around only tens of microWatts. These demonstrations show that high-Q lithium niobate photonic crystal nanoresonators are of great promise for nonlinear photonic applications.

Mach-Zehnder crossbar switching and tunable filtering using N-coupled waveguide Bragg resonators.

This theoretical modeling-and-simulation paper presents designs and projected performance of ~1500-nm silicon-on-insulator 2 x 2 Mach-Zehnder interferometer (MZI) optical crossbar switches and tunable filters that are actuated by thermo-optical (TO) means. A TO heater stripe is assumed to be on the top of each waveguided arm in the interferometer. Each strip-waveguide arm contains an inline set of N-fold coupled, phase-shifted Bragg-grating resonators. To implement accurate and realistic designs, a mixed full-vectorial mathematical model based upon the finite-element, coupled-mode, and transfer-matrix approaches was employed. The Butterworth-filter technique for grating length and weighting was used. The resulting narrowband waveguide-transmission spectral shape was better-than-Lorentzian because of its steeper sidewalls (faster rolloff). The metrics of crossbar switching, insertion loss (IL) and crosstalk (CT), were evaluated for choices of grating strength and TO-induced change in the grating-waveguide refractive index. The predicted ILs and CTs were quite superior to those cited in the literature for experimental and theoretical MZI devices based upon silicon nanobeam resonators. This was true for the Type-I and Type-II resonator addressing discussed here. Finally, we examined the TO-tunable composite filter profiles that are feasible by connecting two or more Type-I MZIs in an optical series arrangement. A variety of narrow filter shapes, tunable over ~2 nm, was found.

Encapsulated Silicon Nitride Nanobeam Cavity for Hybrid Nanophotonics

Most existing implementations of silicon nitride photonic crystal cavities rely on suspended membranes due to the low refractive index of silicon nitride. Such floating membranes are not mechanically robust, making them suboptimal for developing a hybrid optoelectronic platform where new materials, such as layered 2D materials, are transferred on a pre-existing optical cavity. To address this issue, we propose a silicon nitride nanobeam resonator design where the silicon nitride membrane is encapsulated by material with a refractive index of ~1.5, such as silicon dioxide or PMMA. The theoretically calculated quality factor of the cavities can be as large as 100,000 , with a mode-volume of 2.5 times the cubic wavelength. We fabricated the cavity, and measured the transmission spectrum with highest quality factor of 7000. We also successfully transferred monolayer tungsten diselenide on the encapsulated silicon nitride nanobeam, and demonstrated coupling of the cavity with the monolayer exciton and the defect emissions.

Size effect of a uniformly distributed added mass on a nanoelectromechanical resonator

We develop a new analytical formulation to determine the size effect of an added mass on the natural frequencies of an electrostatically actuated resonator. A clamped–clamped nanobeam is used as a mass sensor and modeled using Euler–Bernoulli beam theory. The added mass is uniformly distributed and its size is modeled by a Unit-Step function. The mathematical problem is solved numerically using Galerkin approximation. In order to validate the analytical model, a finite element method (FEM) is used to determine the natural frequencies of the nanobeam resonator and the induced frequency shifts due to the attached particles modeled with different size ratios. The effect of the mesh size on the natural frequencies is investigated. A good agreement is demonstrated between the analytical approach and the FEM technique.