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Abstract
We report the two-dimensional (2D) measurement of resonance in MEMS resonators using stroboscopic differential interference contrast (DIC) microscopy, for the investigation of the linear and nonlinear oscillations of MEMS resonators. The DIC microscopy measures the interference of two sheared illumination light beams reflected from the sample surface to determine the differential surface deflection. By modulating the illumination light at the resonance frequency, the DIC image of the MEMS resonator periodically change its brightness and contrast with the sweeping illumination phase, which have been used to derive the oscillation amplitude and the resonance mode shape of the MEMS resonator. Comparing with conventional interference microscopy, the DIC microscopy can observe the surface deflection larger than the wavelength of the illumination light, enabling the measurement of nonlinear oscillations with a large oscillation amplitude. We demonstrate that the stroboscopic DIC microscopy can measure the 2D mechanical resonance with a high vertical resolution at the nanometer(nm)-scale, and a large measurement range of ∼1 µm, which is very promising for the investigation of linear and nonlinear oscillations of MEMS resonators.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Microelectromechanical-system (MEMS) resonators [1–3] have attracted considerable interests for realizing high sensitivity sensing devices. Owing to the high quality(Q)-factors, MEMS resonators can detect very small shift in the resonance frequency, which has been utilized to measure the changes in mass [4–6], charge [7,8], electron spin [9,10], temperature [11,12], and infrared [13]/terahertz radiations [14,15]. In general, a MEMS resonator is often modeled as a spring-mass resonator to estimate the device performance. However, important parameters of the spring-mass model, such as the mass, equivalent spring constant, and nonlinearity coefficient are determined by not only the geometry of the MEMS resonator, but also its distributed deformation in the resonance, i.e., the resonance mode shapes. Furthermore, the mode shapes are also crucial for studying the internal coupling in MEMS resonators [16–18], which has been attracting lots of research interests from both fundamental physics and advanced device applications. Therefore, the measurement of mode shapes is essential for studying the device physics of MEMS resonators.
Laser Doppler vibrometer (LDV) is commonly used for the high sensitivity measurement of the resonances of MEMS resonators. However, since LDV can only measure the vibrations of a single point, mode shape measurement with LDV requires a two-dimensional (2D) scan, making the measurement very slow. Another technique has been proposed for mode shape measurement is the stroboscopic interference microscopy [19–25]. In the interference microscopy, two illumination light beams reflected by the MEMS device surface and a reference plane, interfere with each other to generate a contrast-modulated image, as schematically shown in Fig. 1(a). The brightness of the image Iu,v is expressed as a function of the 2D distributed vertical surface deflection xu,v, as
Fig. 1. (a) Schematic of stroboscopic Michaelson-type interference microscopy for the measurement of resonance mode shapes of MEMS resonators. Vertical surface deflection x is measured by the interference of a measurement light and a reference light. (b) Schematic of stroboscopic differential interference contrast (DIC) microscopy. Differential surface deflection Δx is measured by the interference of two measurement light beams with a small shear.
To correctly derive the oscillation motion, it is preferable to have the image brightness Iu,v monotonic with respect to the surface deflection xu,v (see Eq. (1)). Then the peak-to-peak surface deflection in the oscillation must be smaller than λ/4, limiting the measurement range of the oscillation amplitude to be typically ∼100 nm. Such a small measurement range is good for studying the linear oscillations of MEMS resonators, but not sufficient for the investigating the nonlinear oscillations, which have been attracting lots of research interests such as nonlinear mode coupling [16–18], mode synchronization [26], and nonlinear chaos oscillations [27,28] and Moreover, for conventional interference microscopy, the wavefront of measurement light must exact match that of the reference light. This brings the requirements of very precious optical alignment and strict anti-vibration protection, which increase the complexity of the measurement system.
In this paper, we report the 2D measurement of resonance in MEMS resonators using stroboscopic differential interference contrast (DIC) microscopy [29], for the investigation of the linear and nonlinear oscillations of MEMS resonators. The DIC microscopy measures the interference of two sheared illumination light beams reflected from the sample surface to determine the differential surface deflection of the sample. By modulating the illumination light at the resonance frequency, the DIC image of the MEMS resonator periodically change its brightness and contrast with the sweeping illumination phase, which is used to derive the oscillation amplitude and the resonance mode shape of the MEMS resonator. Comparing with conventional interference microscopy, the DIC microscopy can observe the surface deflection larger than the wavelength of the illumination light, enabling the measurement of nonlinear oscillations with a large oscillation amplitude. We demonstrate that the stroboscopic DIC microscopy can measure the two dimensional mechanical resonance with a high vertical resolution at the nanometer(nm)-scale, and a large measurement range of ∼1 µm, which is very promising for the investigation of linear and nonlinear oscillations of MEMS resonators.
2. Concept and measurement setup
Figure 1(b) shows the schematic diagram of stroboscopic DIC microscopy for 2D resonance measurement of MEMS resonators. We use a commercial DIC microscope (Olympus BX60M), with its illumination unit replaced by a home-made pulse-driven LED illuminator. The illumination light goes through a collimator and a polarizer, and then reflected by a beamsplitter in the DIC microscope. A Nomarski prism splits the illumination light into two parallel beams of different polarizations, and a small shear and additional phase shift are added to one of the light beams. The two light beams illuminate the sample surface through a 10× objective lens (Olympus LMPLFLN 10×), and the reflected light is combined again by the prism and is monitored through the beamsplitter and an analyzer by a monochrome CMOS camera. Since there is a small shear between the two illumination light beams, the DIC image gives the differential vertical deflection of the sample surface, as schematically shown in Fig. 1(b). The amount of shear is determined by the Nomarski prism and the objective lens, and in our system it is ∼3 µm, and decreases inverse proportional with the magnification of the objective lens.
The use of a DIC microscope instead of a conventional Michelson-type interferometry brings two advantages in the measurement of MEMS oscillations. First, since the DIC microscope measures the differential deflection of sample surface, it allows to observe samples with a large surface deflection (see Fig. 1(b)). Thus in the measurement of MEMS resonance, the allowed oscillation amplitude is much larger than that for conventional interferometry system. In general, the measurement range is limited only by the depth of field of the objective lens, which is typically ∼1 µm to several micrometers. The LMPLFLN 10× objective lens we used in this research has a depth of field of 4.4 µm, and a resolution power of 1.34 µm. The second advantage is about the stability. Since both light beams in the interference are from the sample surface, the common-mode mechanical noise is very much suppressed, and no precise optical adjustments or anti-vibration facilities are needed, which greatly decreases the total cost of the system.
3. Results and discussions
3.1 Stroboscopic DIC sampling of MEMS oscillations
The samples we used for the mode shape measurement are GaAs doubly-clamped MEMS beam resonators. This sample structure has been widely studied in the field of bit operation [30], phonon squeezing [31], nonlinear chaos oscillation [28], mode coupling effect [32], and THz sensing [14,15], thus it provides a good example showing the function and characteristics of the stroboscopic DIC system. We used a modulation doped AlGaAs/GaAs heterojunction structure for the fabrication of the MEMS resonators. Two piezoelectric capacitors were formed on both ends of the MEMS beam to drive the MEMS resonator. We formed 100/130(L)×30(W)×1.2(t)µm3 MEMS beam by selectively etching the sacrificial layer with dilute hydrofluoric acid. The MEMS resonator is driven by an AC voltage applied to one of two piezoelectric capacitors. The schematic structure and microscope image of the fabricated MEMS beam resonator are shown in Figs. 2(a) and 2(b), respectively. All measurements were performed in a vacuum (∼10−2 torr) at room temperature.
Fig. 2. (a) Schematic structure of GaAs doubly-clamped MEMS beam resonators utilized in this paper. (b) A DIC microscope image of the fabricated GaAs MEMS beam resonator, which has a geometry of 100(L)×30(W)×1.2(t)µm3. (c) The schematic diagram of sampling the fast oscillation motions of MEMS resonator with a pulsed light illumination. (d) The image brightness of a small area of ∼10 µm × 10 µm on the MEMS beam marked by a red square in Fig. 2(b), plotted as a function of the phase difference between the illumination light and the driving voltage (VD = 40 mV) of the MEMS resonator. The red square area covers ∼20×20 pixels of the COMS camera. The light intensity was readout by the internal 12-bit ADC of the CMOS camera, and the averaged light intensity of the red square area was used as I. The integration time of the CMOS camera was 1/24 second, and an average of 12 frames were taken at each phase to improve the signal to noise ratio.
Figure 2(c) shows the method of sampling the fast oscillation of the MEMS resonator with a pulsed light illumination. When an AC driving voltage is applied to the MEMS resonator, mechanical oscillation is excited, which modulates the brightness of DIC image of the MEMS resonator. To capture the fast oscillation motions (sub-MHz and MHz range) by a simple slow CMOS camera, we used a home-made pulse-driven illuminator which consists of a LED with a wavelength of ∼515 nm to illuminate the MEMS resonator. The LED illuminator is driven by a function generator to provide pulsed illumination that is synchronous with the driving AC voltage of the MEMS resonator, hence is also synchronous with the MEMS oscillation. The duty cycle of the light pulse used in this measurement was set to be ∼15%. Since the illumination pulses always come at the same phase of the mechanical oscillation, the fast MEMS oscillations become stationary images, which can be captured by a simple CMOS camera. When the phase difference θ between the illumination light and the driving voltage of the MEMS resonator is modulated, the captured DIC image changes the contrast owing to the modulated surface deflection in the oscillation. Figure 2(d) shows the brightness I as a function of θ of a small area of ∼10 µm × 10 µm on the sample surface marked by the red square in Fig. 2(b), which covers ∼20×20 pixels of the COMS camera. The light intensity was readout from the internal 12-bit ADC of the CMOS camera, and the averaged light intensity of the red square area was used as I. The integration time of the CMOS camera was 1/24 second, and an average of 12 frames at a single phase were taken to improve the signal to noise ratio. Note that the orientation of the MEMS beam has been adjusted to match the shear direction of the DIC microscope. The position of the measured red square area has been chosen to have a large differential surface deflection in the oscillation of the first bending mode. As seen, when θ was modulated, the brightness changed periodic, corresponding to periodic mechanical oscillations of the MEMS beam.
3.2 Resonance spectrum measurement
The red curves in Fig. 3 plot the normalized amplitudes of the brightness change, ΔI (see Fig. 2(d)), as a function of driving frequencies at various driving voltages (VD = 20-100 mV), which give the resonance spectra of the first bending mode of the MEMS resonator. ΔI was calculated with I at 4 different phases, i.e., 0, π/2, π, π3/2, 2π. As seen, the resonance frequency was ∼695.7 kHz, and the quality(Q)-factor was ∼3000. When the driving voltage VD exceeded 60 mV, the resonance frequency increased at large oscillation amplitude, indicating that the MEMS resonator entered the hardening nonlinear oscillation regime [33]. To estimate the vertical resolution of this measurement, we used a LDV to measure the resonance spectra at the same driving conditions, and the measured resonance spectra are shown as the black curves in Fig. 3(a). As seen, the measured resonance feature agrees well with those measured by our DIC system. The slight change in the resonance frequency is owing to the frequency drift with room temperature. Since the driving conditions were identical, we assumed that two resonance spectra took the same resonance amplitudes. Thus, we can estimate that 1% light intensity change corresponded to the oscillation amplitude of ∼20 nm. Note that such a correspondence may change with respect to different mode shapes. The light intensity noise for the measurement above is shown in Fig. 3(b), which was obtained with the same measurement conditions as that of Fig. 3(a), but the driving voltage of the MEMS resonator is set to be VD = 0. As seen, the light intensity noise <∼0.00015, and the corresponding oscillation amplitude noise was ∼0.3 nm. Such a low noise enables the high-sensitivity measurement of oscillations in MEMS resonators at the nm-scale.
Fig. 3. (a) Measured resonance spectra of the MEMS beam resonator by using a LDV (black curves), and stroboscopic DIC microscopy (red curves). The MEMS resonator was driven at various VD = 20/40/60/80/100 mV. The amplitude of the light intensity change, ΔI, was calculated with I at 4 different phases, i.e., θ = 0, π/2, π, π3/2, 2π. The settings of CMOS camera were the same with that shown in the caption of Fig. 2(d). (b) Light intensity noise in the DIC microscopy measurement, measured by setting the driving voltage VD= 0 mV.
3.3 Detection sensitivity
To have a quantitative understanding how image brightness changes with the differential surface deflection of the MEMS resonator, we used a simple interference model, that two plane light waves, E0sinωt and E0sin(ωt+θ+ϕ), interfere with each other, where θ = 2π/λ×2Δx, and ϕ expresses the modulatable additional phase from the Nomarski prism. Then the light intensity is expressed by,
The black dots in Fig. 4(b) show the measured light intensity as a function of Δx by using a buckled MEMS beam [34]. Δx of the buckled MEMS beam was calibrated by a confocal laser scanning microscope (Olympus LEXT OLS4000). Although the confocal laser scanning microscope has a relative lower vertical resolution (∼10 nm), it provides the absolute surface height information with a large measurement range, that is necessary for the calculation of the Δx used in Fig. 4(b). Another note is that the confocal laser scanning microscope cannot measure the dynamical surface deflection change caused by mechanical resonance at high frequencies, and it was only used to measure the static surface profile of the buckled MEMS beam. Nevertheless, the measurement result agrees with theoretical prediction of Eq. (2) very well, as shown in Fig. 4(b). To investigate the effect of illumination light wavelength on the sensitivity, we used another red LED with a wavelength of ∼627 nm to illuminate the MEMS beam. The illumination spectrum of the red LED is plotted as the red curve is Fig. 4(a). The red curve and red dots in Fig. 4(b) show the calculated and measured light intensities as a function of Δx, respectively. As seen, with a longer illumination wavelength, the light intensity becomes less sensitive with the Δx, thus shorter wavelength is preferable for high sensitivity measurement.
Fig. 4. (a) The black and red curves plot the spectra of the illumination green LED (wavelength: ∼515 nm), and a red LED (wavelength: ∼627 nm) for comparison, respectively. (b) The calculated (black curve) and experimental (black dots) light intensities as a function of the differential surface deflection Δx, respectively, when the green LED was used as the illumination light. Similarly, the red curve and red dots plot calculated and experimental light intensities as a function of Δx when the red LED was used as the illumination light. The additional phase was adjusted to be ϕ= ∼π/2 in the measurement, and the detailed adjustment process is shown in Supplement 1.
Although Eq. (2) has been sufficient to derive Δx from the measured light intensity, it is a nonlinear equation that hinders an intuitive understanding about how sensitively the light intensity changes with Δx. Note that at a small $|{\Delta x} |< \sim \frac{\lambda }{{20}}$, the light intensity keeps approximately a linear relation with Δx, as seen in Fig. 4(b). Therefore, we define the responsivity R at small vibration amplitude as,
We can estimate that with the green LED (515 nm) or red LED (627 nm) as the illumination light, R ≈ 0.024, or 0.020 nm-1, respectively. Thus with the measured brightness change of the DIC image, we can obtain the differential surface deflection Δx as well as the surface deflection x, by using Eq. (3).
3.4 2D mode shape measurement
Figure 5(a) shows a DIC microscope image of a MEMS resonator with a geometry of 130 (L)×30 (W) ×1.2(t)µm3. The MEMS beam rotation is adjusted to align with the shear direction of two illumination light beams, as indicated by the arrow in Fig. 5(a). The green LED (515 nm) was used in this measurement. A numerical vertical surface deflection in the resonance, i.e., the mode shape for the first bending mode of the MEMS resonator is plotted in Fig. 5(b), which is calculated by using Finite Element Method. Note that in the simulation only the resonant part, i.e., the doubly-clamped MEMS beam was considered. The derived normalized differential surface deflection is plotted in Fig. 5(c), where the shear direction was along the MEMS beam. Then, we drove the MEMS resonator in its first bending mode (f = 326 kHz) with a driving voltage VD = 360 mV. Owing to the oscillation of the MEMS resonator, the DIC image of the MEMS resonator changes the brightness and contrast with the sweeping of the illumination phase (see Visualization 1). The brightness change of the DIC image has been used to derive the oscillation amplitude of the 2D differential surface deflection Δx by using Eq. (3), as plotted in Fig. 5(d). As seen, the measured Δx distribution agrees very well with the shape of the numerical result shown in Fig. 5(c). Here we chose to use Eq. (3) rather than Eq. (2) to calculate Δx because the light intensity change in this measurement was small, and Eq. (3) gave a sufficiently good linear approximation. Furthermore, by integrating Δx along the shear direction and dividing the sum with a scale factor of Δu/d=∼6 (Δu: shear distance, d: pixel size), we can obtain the oscillation amplitude of the surface deflection x of the MEMS resonator. The obtained x is shown in Fig. 5(d). As seen, the obtained surface deflection image agrees well with the numerical mode shape in Fig. 5(b), and the oscillation amplitude is ∼270 nm, which is in the deep nonlinear oscillation regime that the oscillation amplitude is nearly 10 times that of the linear oscillation regime. Such results demonstrate the effectiveness of the DIC microscopy for the investigation of linear and nonlinear oscillations of MEMS resonators.
Fig. 5. (a) A DIC image of the MEMS beam resonantor used in the mode shape measurement. The beam has a geometry of 130(L)×30(W)×1.2(t)µm3. The red arrow indicates the shear direction of DIC microscopy in the measurement. (b) Calculated surface deflection x in the resonance, i.e., the resonance mode shape, of the MEMS beam using Finite Element Method. (c) Calculated differential surface deflection Δx using x shown in Fig. 5(b). The shear direction is along the beam direction. (d) Measured differential surface deflection Δx using the stroboscopic DIC microscopy. The MEMS resonator was driven in its first bending mode (f =∼326 kHz) with a driving voltage VD = 360 mV. The brightness and contrast change of the DIC image with the sweeping illumination phase can be seen in Visualization 1. (e) Surface deflection x obtained by integrating the measured Δx along the shear direction, and dividing the sum with a scale factor of Δu/d=∼6 (Δu: shear distance, d: pixel size).
Finally, it should be noted that the simple relation between ΔI and Δx shown in Eqs. (2) and (3) holds only for flat MEMS beams, the fine structures on the beam surface, such as the thin metal films, however, will affect the measurement result at the edge part of each flat area. In the analysis shown above, the influence from the surface structures of the MEMS beam was neglected to simplify the calculation. Reasonable agreement between experimental and simulated results was obtained because the edge parts of the flat areas took a small percentage of the beam area. However, error analysis of the measurement result with surface fine structures is required in the future research, for applying the proposed stroboscopic DIC microscopy in measuring MEMS devices of various surface conditions.
4. Conclusion
In summary, we have investigated the measurement of 2D resonance of MEMS resonators using the stroboscopic DIC microscopy, for the investigation of the linear and nonlinear resonances of MEMS resonators. The DIC microscopy measures nm-scale differential surface deflection of the MEMS resonators, and a pulsed illumination is utilized for sampling the fast motions of MEMS resonator. By modulating the illumination light at the resonance frequency, the DIC image of the MEMS resonator periodically change its brightness and contrast with the sweeping illumination phase, which has been used to derive the oscillation amplitude and the resonance mode shape of the MEMS resonator. Comparing with conventional Michaelson interference microscopy, the DIC microscopy can observe the surface deflection larger than the wavelength of the illumination light, enabling the measurement of nonlinear MEMS resonances of a large oscillation amplitude. We have demonstrated that the stroboscopic DIC microscopy can measure the 2D resonance with a high vertical resolution at the nm-scale, and a large measurement range of ∼1 µm, which is very promising for the investigation of linear and nonlinear oscillations of MEMS resonators.
Funding
Japan Science and Technology Agency (A-step); Japan Society for the Promotion of Science (21K04151).
Acknowledgments
We thank K. Hirakawa for his fruitful discussion and continuous encouragement. We thank I. Morohashi, B. Q. Qiu, T. Y. Niu for the support in MEMS device fabrication processes.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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