Development of a Full-Field Interferometric System to Detect Low-Frequency Vibrations in Micro-Acoustic Components

Master's thesis conducted at Qualcomm RF360

Topics covered:

Theory MIMO → RF Vector Signal Generators → Demonstration Application

MEMS Characterization

Common Examples of Resonant MEMS Devices
  • MEMS Oscillators → Silicon beam oscillates at a stable resonant frequency (e.g.\ 32.768 kHz).
  • Mass Sensors → Resonant frequency shifts when adsorbed mass increases (e.g., molecule binding).
  • Resonant Pressure Sensors → Membrane tension changes with pressure, shifting resonance frequency.
  • BAW/SAW Filters → Piezoelectric excitation generates acoustic waves with frequency-selective response.

Micro-electro-mechanical systems (MEMS) integrate micrometer-scale mechanical structures with electrical functionality on a common substrate, typically using semiconductor-compatible batch fabrication.

Resonant and acoustic MEMS are especially attractive because of their small footprint, low power consumption, and wafer-scale manufacturability.

Accurate dynamic characterization is essential for evaluating device performance and validating analytical or numerical models.

Central Figure of Merit
The efficiency with which a resonator stores mechanical energy is described by the quality factor: \[ Q = 2\pi \frac{\text{energy stored}}{\text{energy dissipated per cycle}} \]

Choosing a Measurement Technique

Measurement Requirements
  1. Operating frequencies mainly below approximately 1 MHz
  2. Sub-nanometer displacement sensitivity
  3. Practical usability in a laboratory environment
Stroboscopic white-light interferometry system. The full-field system developed by Lipainen, Kokkonen, and Kaivola achieved noise floors of approximately 30 pm.

For dynamic MEMS characterization, optical non-contact techniques are particularly attractive.

Commercial turnkey instuments are availale but custom platforms offer more room for flexibility.

Two techniques stand out:

Scanning Laser Vibrometry
Offers extremely high sensitivity and a very wide frequency range, but typically requires specialized and comparatively complex hardware.
Full-Field Stroboscopic Interferometry
Requires less hardware effort and is better suited to cost-aware laboratory setups, although the per-pixel sensitivity is lower because the available optical power is distributed across many pixels.
Decision
Despite its lower per-pixel sensitivity, modern full-field interferometric systems can still achieve sub-nanometer resolution and are therefore well matched to the requirements of this project.

Wave Interference

Optical fields can be conveniently represented using complex phasors.

The time independent (spatial) part of the E-field of a monochromatic wave can be written as \[ \psi = A e^{i(kz + \phi)} \]

For two overlapping waves of the same optical frequency, the total field is the superposition \[ \begin{aligned} \psi(z) &= \psi_1(z) + \psi_2(z)\\ &= A_1 e^{i(kz + \phi_1)} + A_2 e^{i(kz + \phi_2)}\\ &=\underbrace{\left(A_1 e^{i\phi_1} + A_2 e^{i\phi_2}\right)}_{Ae^{i\phi}} e^{ikz} \end{aligned} \]

The combined field depends on the relative phase difference

\[ \Delta\varphi = \phi_1 - \phi_2 . \]
Key Idea
While optical frequency is unchanged, the amplitude and phase of the combined field vary with \( \Delta\varphi \).
Phasor representation: (Right) at fixed $k\,z = const.$, where the complex amplitudes of $ψ_1$are added vectorially to yield the resultant field $ψ$. Real part: (Left) Corresponding real parts of $ψ_1$as functions of $k\,z$.

Constructive interference occurs when the phase difference is an integer multiple of \(2\pi\): \[ \Delta\varphi = 2\pi n, \qquad n \in \mathbb{Z}. \]

Destructive interference occurs for odd multiples of \( \pi \): \[ \Delta\varphi = (2n+1)\pi, \qquad n \in \mathbb{Z}. \]

In between, patially constructive/destructive interference occurs.

Optical detectors measure intensity, not the complex field directly ($I \propto |\psi|^2$):

\[ I = \bigl|\left(A_1 e^{i\phi_1} + A_2 e^{i\phi_2}\right)\bigr|^2 = |A_1|^2 + |A_1|^2 + 2A_1A_2\cos(\Delta\varphi) . \]

Interference Intensity Relation
Expressing the individual beam intensities as $I_1 = A_1^2$ and $I_2 = A_2^2$ leads to \[ I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos(\Delta\varphi) . \] The intensity therefore directly encodes the phase difference between the two beams.

Interferometry

Point detection: Configuration determines $f(\lambda)$.
Full-field detection: Spatial OPD variations result in fringes.
Michelson type interferometer: A beam splitter divides the incident light into reference arm and a sample arm. The reference arm is terminated by a fixed mirror while in the sample arm, the optical path length may vary. The two beams recombine at the beam splitter, and the resulting interference intensity is recorded by a photodetector. → $f(\lambda) = \lambda/2$
Measurement Principle
Interferometry converts a change in optical path length into a measurable change in intensity. The fundamental principle involves splitting an incident light beam into two or more paths that traverse different optical path lengths before being recombined

In point detection, the interference signal is measured at a single location.

In full-field interferometry, the pattern is recorded simultaneously across an extended area using a camera.

Various interferometric configurations exist—including Michelson, Mach–Zehnder, and Fabry–Pérot interferometers—each offering specific advantages.

Michelson Type Interferometer (relevant to this work)
Light traverses the sample arm twice. A physical displacement \( \Delta z \) therefore produces a round-trip optical path difference of \(2\Delta z\). The corresponding phase shift is \[ \Delta\varphi = \frac{4\pi}{\lambda}\Delta z. \]

System Architecture

Measurement system: interferograms are captured using stroboscopic LED illumination synchronized to the DUT excitation.
System components components Michelson interferometer LED – low-coherence   illumination – λ ≈ 618 nm BS – non-polarizing   beamsplitter – splits into   reference/sample arms REF – reference mirror – piezo-mounted sample stage – holds DUT – alignment and tilt   adjustment MO – magnification optics – long working distance camera – monochrome digital   camera – captures   interferograms signal generator – DUT excitation – LED strobe pulses voltage controller – drives reference piezo – fine z-adjustment controlled by custom software
System components
Key Idea
The system combines stroboscopic full-field Michelson interferometry with software-controlled reference-arm stabilization.

Software-Based Stabilization

Measurement system: interferograms are captured using stroboscopic LED illumination synchronized to the DUT excitation.

Continuous stabilization is required because the interferometer operating point can drift due to mechanical creep, thermal expansion, or external disturbances.

The system uses a software-based PID controller whose feedback is derived directly from the camera image.

Control Loop
  1. Measure mean image intensity inside static reference ROI
  2. Compare with intensity setpoint
  3. Compute PID correction
  4. Adjust reference mirror position
Constraint
The achievable closed-loop bandwidth is limited first by the camera frame rate. At higher frame rates, acquisition and processing latency becomes the dominant bottleneck.

Operating Principle

Measurement principle: vibration-induced path-length changes produce phase and intensity modulation.

A surface displacement \(z(t)\) changes the optical path length in the sample arm and therefore modulates the interference phase.

\[ \Delta\varphi(t) = \Delta\varphi_0 + \frac{4\pi}{\lambda_0} z(t), \]

where \( \Delta\varphi_0 \) is the static operating point phase.

Linearizing the interference signal around this operating point gives

\[ I(t) \approx I\!\left(\Delta\varphi_0\right) + \left.\frac{dI}{d\varphi}\right|_{\Delta\varphi_0} \frac{4\pi}{\lambda_0} z(t). \]
Key Idea
Near a fixed operating point, small displacements are encoded as intensity modulations.

Calibration and Operating-Point Selection

Calibration sweep and operating-point selection from representative measurement data.
Practical Note
The PID setpoint must be sufficiently far from the extrema. Otherwise drift may push the operating point beyond an extremum, the slope changes sign and the controller can relock to a neighboring quadrature point one fringe away.

The small-signal sensitivity depends on the operating point and is maximal at quadrature:

\[ \Delta\varphi_0 = \frac{(2m+1)\pi}{2}, \qquad m \in \mathbb{Z}. \]

A piezo sweep of the reference mirror is used to record the mean intensities in a reference ROI and a vibration ROI.

A difference in surface height between these ROIs results in phase-shifted intensity curves.

By fitting sinusoidal models to both curves the optimal measurement operating points are identified and provide the corresponding PID setpoints.

Purpose of Calibration
Calibration identifies operating points with high sensitivity and maps them to stabilization setpoints for closed-loop operation.

Pixel-Wise Sensitivity and Displacement Mapping

Calibration Steps
  1. Select vibration ROI
  2. Select reference ROI
  3. Sweep the reference piezo over one or more fringes
  4. Record the ROI intensity curves
  5. Fit sinusoidal models
  6. Choose quadrature operating points
  7. Compute the PID setpoint and pixel-wise gain
Interpretation
The illumination wavelength acts as the scale that links measured intensity modulation to nanometer-scale displacement.

Since fringe contrast and phase offset vary spatially, the calibration frames are saved, and later a pixel-wise operation point identification is performed.

The local slope at the selected operating point defines the pixel sensitivity

\[ s(x,y) = \left.\frac{dI}{d\varphi}(x,y)\right|_{\Delta\varphi_0(x,y)}. \]

For displacement reconstruction, it is convenient to define the local gain

\[ G(x,y) = \frac{4\pi}{\lambda_0}\,s(x,y), \]
Intensity-to-Displacement Mapping
Small intensity modulations are converted to displacement via \[ z(t) = \frac{\Delta I(t)}{G(x,y)}, \qquad \Delta I(t) = I(t) - I\!\left(\Delta\varphi_0\right). \]

Measurement Modalities

Acquisition modalities: continuous acquisition with shifted frequencies (left) and phase stepping with averaging (right).
Continuous with Shifted Frequencies
The DUT and LED are driven at slightly different frequencies, \[ f_{\mathrm{DUT}} = f_{\mathrm{LED}} + \Delta f. \] The relative phase then evolves at the beat frequency \( \Delta f \), so the vibration appears in apparent slow motion with synthetic period \[ T_{\mathrm{beat}} = \frac{1}{\Delta f}. \]
Phase Stepping with Averaging
The LED and DUT are driven at the same frequency, \[ f_{\mathrm{LED}} = f_{\mathrm{DUT}}. \] A full vibration cycle is reconstructed by stepping the relative phase in \[ \Delta\theta = \frac{2\pi}{M} \] increments and averaging \(N\) frames per step, which improves SNR by approximately \( \sqrt{N} \).
Use Case
Shifted-frequency acquisition is useful for rapid frequency sweeps and immediate visual feedback. For quantitative measurements, the preferred modality is phase stepping with averaging.

The Complete Measurement Workflow

Measurement workflow Presentation-friendly workflow diagram for setup, calibration, and measurement. Setup Calibration Measurement 1) Place and align sample Set tilt and z-position for clear fringes. 2) Configure camera Frame rate, exposure, gain. 3) Configure LED pulsing Set drive voltage, frequency, pulse width. 4) Select ROIs Choose reference and vibration regions. 5) Run z-sweep Record mean intensities in reference and vibration ROIs Enough modulation? No Yes 6) Fit and set QP Find quadrature point and PID setpoint. 7) Warm-up and baseline Allow thermal settling; optional baseline ROI. 8) Stabilize at setpoint Camera-based PID locks the operating point. Acquisition mode? Live view 9B) Detuned mode Use Δf for slow-beat visualization. Quantitative 9A) Phase stepping M phase steps, N frames/step, average for SNR. 10) Convert and export Intensity → s(x,y) and gain G(x,y) → post proccessing
Measurement workflow
Key Considerations
  • Measurement area must be reasonably flat and vibration amplitudes small (compared to $\lambda_0$)
  • Camera frame rate and exposure time must be aligned with drive frequency
  • Fit at most one fringe (piezo non-linearity)
  • Camera sensor temperature dependence (dark currents)
  • Currently, lateral operation point drift is not compensated

Amplitude Extraction Procedure

Acquisition Details
  • \(M\): phase steps per period
  • \(N\): averaged frames per step
  • \(n_p\): repeated periods
  • Frames with large PID error are rejected
  • LED duty cycle: \(10\%\)
Output
A phase-resolved image stack is reduced to a per-pixel harmonic amplitude map and phase map at the drive frequency.
Linear Least Squares Fit as DFT Alternative
By fitting \[ I = A\cos(\phi) + B\sin(\phi), \] amp. and phase can be computed from \(A_1=\sqrt{A^2+B^2}\) and \(\varphi=\arctan(B/A)\)

For each excitation frequency, a phase-stepped image sequence is acquired using \(M\) equally spaced phase steps per vibration period. At each step, \(N\) frames are averaged, and the sequence is repeated over \(n_p\) periods for robustness.

At each pixel \((x,y)\), the intensity sequence is modeled as

\[ I_{x,y}(\phi_m) \approx I_0(x,y) + A_1(x,y)\cos\!\bigl[\phi_m-\varphi(x,y)\bigr]. \]

The complex first-harmonic phasor is then estimated using an \(M\)-point DFT:

\[ \hat{C}_1(x,y) = \frac{2}{M}\sum_{m=0}^{M-1} I_{x,y}(\phi_m)\,e^{-j\phi_m}. \]
Harmonic Extraction
\[ A_1(x,y)=\bigl|\hat{C}_1(x,y)\bigr|, \qquad \varphi(x,y)=\arg\!\bigl\{\hat{C}_1(x,y)\bigr\}. \] The first harmonic therefore provides the local vibration amplitude and phase in camera units.

Displacement Reconstruction

Interpretation
The calibrated gain map provides the conversion factor from intensity modulation to nanometer-scale displacement.
Practical Note
Phase correction is necessary because the same displacement can produce opposite intensity changes depending on the local operating slope.

Using the calibrated displacement-to-intensity gain \(G(x,y)\), the measured harmonic amplitude is converted to out-of-plane displacement amplitude via

\[ A_z(x,y) = \frac{A_1(x,y)}{|G(x,y)|}. \]

The phase map is corrected according to the sign of \(G(x,y)\).

\[ \varphi_z(x,y)= \begin{cases} \varphi(x,y)+\pi, & \text{if } G(x,y)<0,\\[4pt] \varphi(x,y), & \text{otherwise.} \end{cases} \]

The corrected phase is subsequently wrapped to the interval \( (-\pi,\pi] \).

Final Measurement Output
The processing pipeline yields the displacement amplitude map \(A_z(x,y)\) and phase map \( \varphi_z(x,y) \) at the excitation frequency.

Sample Introduction

Measured MEMS cantilever array. One beam with comparatively small static curvature was selected for the following measurements.
Practical Measurement Constraint
Interferometric sensitivity is best on locally flat surfaces. Because some cantilevers exhibit static curvature or tilt, a beam with minimal out-of-plane deformation was selected and aligned as closely as possible to the optical axis.
Micrograph. Screenshot taken with reference mirror blocked.
Sample Features
  • Piezoelectrically actuated cantilever array on a single chip
  • Nominally identical beam geometry and material stack
  • Released by anisotropic backside etching
  • Spatial sampling: \(\sim 1\,\mathrm{µm}\) per pixel
  • Typical beam size in the image: \(\sim 400 \times 80\) px
  • All beams driven simultaneously via the integrated piezoelectric layer

Frequency Response Measurement

Single-pixel frequency response and Lorentzian peak fits. Out-of-plane vibration amplitude at pixel \((x,y)=(200,60)\) versus excitation frequency.
Interpretation
The increase in \(Q\) with mode order is qualitatively consistent with reduced anchor-loss contribution and weaker effective air damping at higher frequencies.

A coarse frequency sweep from \(1\) to \(200\,\mathrm{kHz}\) in \(1\,\mathrm{kHz}\) steps was performed using \(M=18\) phase steps, \(n_p=4\) periods, and \(N=4\) averaged frames per step.

A pixel in the central span of the beam, \((x,y)=(200,60)\), was monitored to generate a representative response curve.

Three prominent resonance peaks were identified and refined using Lorentzian fits. \[ A(f) \;=\; A_0\,\frac{1}{\sqrt{1 + \left(\frac{2Q}{f_n}\right)^2 (f-f_n)^2}}, \]

Extracted Resonances
  1. \(f_1 \approx 8.8\,\mathrm{kHz}, \qquad Q_1 \approx 4\)
  2. \(f_2 \approx 55.9\,\mathrm{kHz}, \qquad Q_1 \approx 30\)
  3. \(f_3 \approx 157.5\,\mathrm{kHz}, \qquad Q_1 \approx 48\)

Validation Against Euler–Bernoulli Beam Theory

Measured full-field amplitude and phase at the first three bending modes.
Validation Assessment
The measured resonance frequencies agree well with Euler–Bernoulli beam theory. In addition, the phase maps exhibit the expected nodal structure of the first three clamped–free bending modes.

To assess physical consistency, the measured resonances are compared with the Euler–Bernoulli model for a uniform clamped–free beam:

\[ f_n = \frac{\beta_n^2}{2\pi L^2}\sqrt{\frac{EI}{\rho A}}, \] with \(\beta_1=1.875,\;\beta_2=4.694,\;\beta_3=7.855,\ldots \)

Frequency ratios eliminate dependence on unknown composite parameters:

\[ \frac{f_n}{f_1}=\left(\frac{\beta_n}{\beta_1}\right)^2, \qquad \frac{f_2}{f_1}\approx 6.27, \qquad \frac{f_3}{f_1}\approx 17.55.\\ \]
Validation Results
  1. \(f_2/f_1 = 6.3, \qquad \varepsilon = 1.0\%\)
  2. \(f_3/f_1 = 17.8, \qquad \varepsilon = 1.6\%\)

Mode-Shape Fits Along the Beam Centerline

Euler–Bernoulli mode-shape fits along the beam centerline.
Fit Assessment
Modes 1 and 2 yield mutually consistent fit parameters, while mode 3 deviates more strongly. Nevertheless, the inferred beam lengths remain close to the nominal \(\sim 400\) px scale, and the analytical shapes agree well with the measured profiles outside excluded regions.

For a more quantitative comparison, the measured centerline displacement profile is fit to the analytical clamped–free Euler–Bernoulli beam shape.

\[ \begin{align} &\phi_n(x)\\ &=A_n\Bigl[\cosh(\beta_n x)-\cos(\beta_n x) -\eta_n\bigl(\sinh(\beta_n x)-\sin(\beta_n x)\bigr)\Bigr], \end{align} \] with \[ \eta_n= \frac{\cosh(\beta_n L)+\cos(\beta_n L)} {\sinh(\beta_n L)+\sin(\beta_n L)}. \]

To account for uncertainty in clamp position and pixel origin, horizontal and vertical offsets are included as fit parameters.

Regions with outliers or low sensitivity are excluded from the fit.

Fit Parameter Results
$n$ $A_n$ (arb.) $L$ (px) $x_0$ (px) $y_0$ (arb.)
1 0.78 430.0 -17.0 0.06
2 0.27 423.0 -19.5 -0.01
3 0.33 388.8 10.1 0.03

Noise Analysis

Profile line (at $y = 70$ for different averaging numbers $N$ of the beam at the third bending mode.
Interpretation
Ideally, averaging improves SNR by approximately \( \sqrt{N} \). From \(N=9\) to \(N=30\), this would predict an improvement factor of \[ \sqrt{30/9} \approx 1.8. \] The weaker observed gain suggests that slow lateral drift becomes the dominant limitation.

A direct noise-floor measurement would require strict pixel correspondence across repeated acquisitions, which is currently limited by uncorrected lateral drift.

Measurement Parameters
As a qualitative alternative, measurements were compared at \(f=158\,\mathrm{kHz}\) for
  • \(N=1\)
  • \(N=9\)
  • \(N=30\)
averaged frames per phase step, with all other parameters held fixed.

In the third bending mode, the profile line at \(y=70\) shows that averaging produces cleaner nodal transitions and deeper minima. The improvement from \(N=9\) to \(N=30\), however, is only marginal.

Practical Noise Floor
The smallest detectable out-of-plane vibration amplitude is estimated to be on the order of \[ \sim 200\,\mathrm{pm}. \]

Limitations and Future Directions

In addition to the MEMS cantilever sample, the system was tested on similarly dimensioned micro-acoustic samples yielding similar results in terms of sensitivity for frequencies up to 1 MHz.

The current system fulfils the requirements formulated at the outset of this project, but three main limitations remain.

Lateral Stability and Noise Reduction
Lateral drift and creep currently limit how much noise can be reduced through frame averaging. Brighter pulsed illumination would also help by increasing photon flux and reducing the relative impact of shot noise.

Future direction: improve mechanical stability, add drift compensation, and use a dedicated high-current LED driver.
Low-Frequency Measurements
Very low-frequency vibrations have not yet been addressed. The present stabilization loop depends on camera-based feedback and becomes increasingly ineffective as the relevant dynamics approach DC.

Future direction: develop a dedicated low-frequency acquisition mode or redesign the stabilization strategy.
Sample Topography Robustness
The current calibration assumes that the local operating point remains close to quadrature. On curved or tilted surfaces, this assumption becomes less accurate and reconstruction errors increase.

Future direction: extend the calibration with operating-point sweeps or multiple reference regions for larger surface-height variations.

Any Questions?