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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Optical investigation of movable microsystem components using time-averaged holography is investigated in this paper. It is shown that even a harmonic excitation of a non-linear microsystem may result in an unpredictable chaotic motion. Analytical results between parameters of the chaotic oscillations and the formation of time-averaged fringes provide a deeper insight into computational and experimental interpretation of time-averaged MEMS holograms.

Interferometry is a powerful experimental technique for the analysis of profiles of surfaces, detection of deflections, motion and structural vibrations of microsystems, where the amplitudes of those vibrations are in the range of nanometers to a few micrometers [

Holographic interferometry is a powerful optical method for mapping changes in the shape of three-dimensional objects with high accuracy. Digital holography is used for nondestructive testing, strain analysis and analysis of vibrations [

Time average holography is an experimental method for the quantitative registration of surface oscillations, which has been widely applied to the investigation of microsystems. The application of this method has been employed in dynamic micro-metrology [

One of the assumptions made when applying the time-average holography method is that oscillations are harmonic, which might not be the case in real life applications of microsystems. It is well known that even a periodic excitation of non-linear system may result in unpredictable chaotic behavior. Nonlinear and chaotic effects in microsystems are widely investigated in [

Computation and plotting of patterns of time average holographic fringes in virtual numerical environments involves such tasks as modeling of the optical measurement setup, geometrical and physical characteristics of the investigated structures and the dynamic response of analyzed microsystems [

A fixed-fixed paradigmatic fixed-fixed beam model is used to illustrate the formation of time-averaged holographic fringes when the beam performs complex transient oscillations. A typical MEMS device comprising a deformable fixed-fixed beam over a fixed ground electrode is modeled. Finite element method (FEM) is used for the simulation of this MEMS device using a COMSOL Multiphysics package.

This article is organized as follows. Optical background of the formation of time-averaged holographic interference fringes is given in Section 2. A description of the model for the fixed-fixed beam is given in Section 3. Numerical results for the fixed-fixed beam with different conditions are given in Section 4. The description of the process of computational reconstruction of time-averaged holographic fringes and discussion on various problems when beam performs chaotic oscillations is given in Section 5. The details of the experiment are given in Section 6. Conclusions are given in Section 7.

The basic principle of the formation of time-averaged holographic interference fringes can be illustrated by the harmonically oscillating cantilever beam example (_{T}_{0} is the zero order Bessel function of the first kind. Then, the resulting intensity

The ability to enumerate time-averaged holographic fringes and to identify their centerlines results into accurate reconstruction of the field of amplitudes of harmonic oscillations. Really, the amplitude of harmonic oscillations at point _{k}_{k}

All schematic diagrams in figure are based on the assumption that the illumination and the observation directions are perpendicular to the plane of the beam in the state of equilibrium (in the par-axial model). In general case the distribution of intensity of the laser beam ^{2} (_{L}_{d}_{s}

As mentioned in the introduction we will use fixed-fixed beam model for the computational experiment illustrating the formation of time-averaged holographic fringes when the beam performs complex transient oscillations. A schematic diagram of a typical MEMS device comprising a deformable fixed-fixed beam over a fixed ground electrode is shown in

Finite element method (FEM) is used for the simulation of this MEMS device. Time-dependent simulation is implemented in COMSOL Multiphysics package;

A material used for the beam structure is Silicon with relative permittivity ^{3}, Young's modulus

Electromechanics physics Comsol interface is used for computational simulation. It combines solid mechanics and electrostatics with a moving mesh to model the deformation of electrostatically actuated structures:
_{υ}

An electrostatic force caused by the potential difference between the two electrodes bends the beam toward the grounded plane beneath it. This model calculates the electric field in the surrounding air in order to compute the electrostatic force. As the beam bends, the geometry of the air gap changes continuously, resulting in a change in the electric field between the electrodes. The coupled physics is handled automatically by the Comsol electromechanics interface.

The electrostatic field in the air and in the beam is governed by Poisson's equation:

A condition of the zero charge on the boundary is used as this is the default boundary condition at exterior boundaries.

The initial values for electric potential and solid displacement field are:

The fixed constraint condition that makes the geometric entity fixed (the displacements are zero in all directions) are applied to the edges of the beam. Ground boundary condition is applied to the bottom plain which gives zero potential on the boundary. The boundary condition is applied to fixed-fixed beam of an electric potential _{0}.

Computational simulation is performed for the model of fixed-fixed beam (the model is described in details in Section 3) with three different electric potential boundary conditions. In order to simplify the visualization of transient processes we select a single point at the center of the beam (where the displacements are largest) and plot time signals and phase diagrams for all three different potential boundary conditions. The illustration of the deformed fixed-fixed beam is given in

During the first computational experiment (i) a DC (direct current) bias of _{DC}_{DC}

The second computational experiment (ii) employs a DC bias on top of a sinusoidal AC voltage:
_{DC}_{AC}_{DC}_{AC}

The results of computational experiments are presented in _{dM}

The results of the second computational experiment are different compared to the periodic oscillation caused by the short DC transient in the first computational experiment. The harmonic AC component interacts with the natural frequency of the beam - the resultant process of oscillations is no longer periodic (

Time-averaged holographic fringes illustrated in

The characteristic function defining the complex amplitude of the laser beam _{t}

Note that when _{s}_{t}

But when oscillations are defined by a non-harmonic function _{s}^{2} (

In practice one needs to compute arithmetic averages of _{c}_{s}

It is well known that a non-linear system with a harmonic load may exhibit complex chaotic solutions.

Lets denote intensity level of interference bands as stochastic time series defined by process _{i}^{2} then the intensity level of interference bands decrease exponentially.

The importance of this derivation cannot be overestimated—time-averaged holographic fringes do not form when the oscillation of the investigated object is chaotic. One should make sure that chaotic oscillations do not affect the components of the experimental setup before questioning the operability of the optical system—if only time-averaged hologram does not produce any fringes.

So far the dynamics of the central point of the fixed-fixed beam is discussed (where the deflections from the state of equilibrium are largest). In order to visualize the dynamic of the whole surface of the beam we will use whole field Time-Averaged holography simulation techniques.

FEM modeling yields complete data on the dynamics of the fixed-fixed beam-distances between the beam plane and the ground plane are available from the simulation results for all time steps. The method described in the Sections 2 and 5.1 will be applied for the data produced by all three computational experiments. _{L}_{d}_{s}

Numerically reconstructed time-averaged holograms generated by the fixed-fixed beam according to the computational experiments (i), (ii) and (iii) are illustrated in

As mentioned previously

Harmonic oscillations around the state of equilibrium (according to the first eigenmode of the beam) are illustrated in

An important conclusion follows from the above observations. One should not use straightforward techniques for the identification of the process of the oscillation from the available image of time-averaged hologram. This inverse problem may become very complex if the motion law governing the dynamics of the analyzed structure is not known before the experiment.

The situation becomes even more complex when the oscillation is not periodic but is chaotic. As mentioned previously chaotic oscillations do not generate time-averaged holographic fringes. However, this result holds when exposure time tends to infinity. Finite exposure times result into interpretable patterns of fringes even for chaotic oscillations.

We use the same data produced by computational experiment (iii) but construct time-averaged holograms for different exposure times (_{1}, _{2}, _{3} and _{4} are illustrated in

The methods of holographic interferometry allow to get much more information about the deformable surface comparing with other experimental methods. Because of this holographic interferometry methods are widely applied in mechanics, biology, non-destructive control, automotive industries and microsystem engineering and others areas.

Digital holography technology uses real-time, 3D, full field-of view surface measurements to eliminate point-by-point data gathering. Holograms are used to measure the surface of any component. Superb resolution allows for monitoring shape changes that are smaller than 20 nanometers for superb accuracy and resolution of the results

Measurements of vibration amplitudes of surface of MEMS, as functions of operating conditions of the MEMS were performed using the optoelectronic laser interferometric microscope. The methodology of investigation using optical micro-holographic system for recording resonance frequencies of vibrating mechatronic system is presented in paper [

The general view of PRISM setup is presented in

Success in deformation and vibration measurement requires a fast affordable solution. PRISM offers a high-speed holographic technique for production measurement of vibration and deformation without surface contact and minimal sample preparation. PRISM can access complex geometries (deep recesses or curves) difficult or impossible for other techniques and can be configured for specific needs in experimental investigation of mechatronics systems. The PRISM system videohead were modified in order to make registration microscale objects possible.

In

The wave fronts are planar in the setup that used in this study, thus parallel interference fringes result when the proximal beam splitter is introduced into the path. Single frame images captured by the CCD and proceeded by computer give us a valuable insight in the motion of investigated MEMS.

Time averaged holography is a powerful optical experimental technique for the investigation of microsystem components. The inverse problem for the interpretation of time-averaged holographic fringes is a well defined and straightforward task, if only the oscillations of the investigated structures are harmonic. It is demonstrated that complex chaotic motions can be generated even in rather simple micro-electromechanical systems. Therefore, a straightforward interpretation of time-averaged holographic interferograms of micro-mechanical components can be not only be misguiding but the experimental holographic images may not reveal any interpretable fringes at all.

A relationship between the decay of intensity in a time-averaged holographic interferogram and the intensity of chaotic motion is explicitly derived. It is shown that chaotic oscillations do not generate time-averaged holographic fringes. Thus one needs to make sure that the response of the excited electro-mechanical system is not chaotic before questioning the functionality of the optical experimental setup. The derived analytical results yield a deeper insight into the dynamical properties of MEMS components.

This research work was funded by EU Structural Funds Project “Microsensors, microactuators and controllers for mechatronics systems (Go-Smart)” (No. VP1-3.1-MM-08-K-01-015).

Financial support from the Lithuanian Science Council under project No. MIP-100/2012 is acknowledged.

The authors declare no conflict of interest.

A schematic diagram illustrating the formation of time-averaged holographic fringes: the one-dimensional structure oscillating according to its first eigenform is shown in part (

Contrast enhancement of time-averaged holographic fringes: the solid line represents the decay of intensity at increasing amplitude

The schematic diagram of the MEMS device—a fixed-fixed beam with an applied voltage

Fixed-fixed beam MEMS model geometry and its FEM mesh.

The shape of the fixed-fixed beam in the state of maximum deformation while performing periodic oscillations according to the first computational experiment.

Results of simulations illustrating the dynamics of central node of the fixed-fixed beam. Parts (

The formation of interference fringes for: (

The formation of interference fringes for periodic and harmonic oscillation laws. Time graphs of three different points on the surface of the beam are shown in (

Computational experiment (iii) divided into time intervals (T1), (T2), (T3), (T4).

The formation of interference fringes for time intervals: (

PRISMA system setup: (1) videohead; (2) control block; (3) illumination head of the object; (4) investigated object.

Schematic diagram of the microscopic holographic system: (1) He-Ne laser; (2)

Holographic interferogram of vibrating MEMS, when fixed-fixed beam is performing: (