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Abstract
A new laser confocal vibration measurement method (LCVM) is proposed to meet the requirements of high precision and high dynamic range measurements in micro and nano electromechanical systems. The proposed method uses different measurement modes to ensure that the amplitude solution interval of the out-of-plane is always in the optimal test interval of a confocal curve with the highest sensitivity to axial displacement, and thereby achieving the high-precision extraction of large-scale frequency and the high-precision measurement of large-scale amplitude. Using a 100×, NA=0.9 objective lens with a working distance of 1 mm, the theoretical analysis and preliminary experimental results indicate that the maximum measurable amplitude is 500 µm, the displacement resolution of the amplitude is 4 nm, and the measurable frequency range limited by electrical design is 0-120 MHz. The LCVM provides a novel approach for out-of-plane vibration measurements.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Various kinds of sensors based on micro-electromechanical systems (MEMS) have been widely used in many fields including biology, medicine, aviation, and communications, and in various industrial and military applications [1–7]. MEMS devices have gradually become a research hotspot in the fields of micro and nano engineering. Out-of-plane vibration MEMS devices realize their system functions through mechanical movement or deformation. Their performance is determined by the dynamic characteristics of the resonant elements. The characterization of the out-of-plane vibration parameters can provide feedback for the simulation and design of MEMS and other micro structures, which is crucial for analyzing their performances [8–9].
Optical vibration measurement methods are widely used in out-of-plane vibration parameter tests because of their advantages of being non-contact and non-destructive [10–17]. Computer micro-vision systems (CMVs) [10] can obtain “stop action” images of the targets through optical microscopes, CCD cameras, and stroboscopic illuminators, but they suffer from small measurable frequency ranges of 100KHz and low measurement resolution of 5nm. The stroboscopic interference system [11–13] has a high measurement frequency of 1MHz and a resolution of 1nm. Its minimum resolvable time interval is controlled by the light pulse width, rather than the integral time of the detector. However, its out-of-plane vibration measurement range is small, and depends on the depth of focus. The vibration frequency of the tested sample must be known to synchronize the stroboscopic light with the movement of the tested sample. The stroboscopic interference system, hence, cannot meet the requirements of a large amplitude measurement and time aperiodic transient motion detection. Traditional laser Doppler vibrometer (LDV) [14–16] technology cannot measure tiny objects such as MEMS devices because of the large spot sizes. Microscopic laser Doppler vibration measurement [17] realizes vibration measurement and positional positioning of small objects through laser heterodyne interference technology and microscopy technology combined with circuit demodulation. Because of the need to couple the laser heterodyne interference and microscopy optical paths, the system structure is complicated, difficult to troubleshoot, and expensive. In summary, the achievement of high dynamic range out-of-plane vibration measurement with a simple and versatile method is a challenging topic in current research.
This paper therefore proposes a new laser confocal vibration measurement method (LCVM) with a high dynamic measurement range. The method is not only capable of confocal micro-area measurement, but also has nano-scale amplitude resolution, a measurable amplitude range far greater than the focal depth, large measurable frequency range, simple structure, and low development costs. Using the method proposed in this paper, the frequency and amplitude of the out-of-plane vibration can be extracted from the vibration response curve. This provides a new solution for the precision measurement of the out-of-plane vibration parameters in micro and nano structures.
2. Principles and methods
As shown in Fig. 1, the LCVM adopts the optical path structure of the confocal microscope system. After the laser beam passes through the beam expander, the reflected light is used as the reference beam, and is received by the photodetector (Detector_B) to monitor the change of the laser source in the intensity. The transmitted light is used as the measurement beam. The measuring beam is reflected by a beam splitter (BS2), and is focused on the sample by the objective lens. The reflected light modulated by the vibration of the surface is transmitted by the BS2. The light is then converged by the collecting lens onto a pinhole on the lens focal plane, received by the photodetector (Detector_A) placed close to the pinhole, and finally converted into an electrical signal [18–19]. In the process of vibration measurement, the monitoring beam monitors the change of laser power in real time, to eliminate the interference caused by laser power fluctuation from the vibration measurement results.
When the sample vibrates near the focal plane of the objective lens, the axial response curve will change with the vibration. In the proposed method, an optimal test interval is defined within the oblique section of the confocal response curve. The maximum positive or negative out-of-plane position of the vibration surface can be ensured to lie within the optimal testing interval by measuring three different ranges of vibration amplitudes with different measuring modes to obtain the axial response curve of the vibrating surface. In the vibration measurement, the amplitude measurement range is greatly broadened by the combined use of the different measurement modes. The amplitude is calculated by using the correspondence between the intensity and the axial position of the vibration surface. At the same time, frequency information of the sample can also be obtained from the system response curve. Out-of-plane vibration measurement with high dynamic range is hence achieved.
2.1 Definition of optimal test interval
In the vibration measurement, the intensity response curve has a one-to-one correspondence with the sample position. To obtain nanometer scale axial resolution with different vibration amplitudes, an optimal test interval is defined as follows.
According to the principle of confocal microscopic imaging [20–22], under the assumption that the system is an ideal confocal coherent imaging system, the confocal response curve I(u) can be expressed as
The derivative of the confocal response curve I(u) is expressed as
The simulated curves of Eq. (1) and Eq. (3) are shown in Fig. 2.
It can be seen from Fig. 2 that the slope of the confocal curve at the focal point is 0, i.e., the intensity is insensitive to changes in the sample defocus. The confocal response curve has a large slope on the oblique section of the main lobe. To meet the measurement requirements at different amplitude ranges, two optimal test intervals are defined on the oblique sections on both sides of the main lobe of the confocal response curve, as shown in Fig. 2. The two optimal test intervals need to satisfy the conditions that the interval lengths and absolute values of the slopes at the endpoints should be equal, and that the spacing between the intervals is equal to the length of each interval. The conditions can be expressed as
According to Eq. (4), u0 = 1.3355. The absolute values of the slopes k at the endpoints of the optimal confocal response curve test intervals [−3u0, -u0] and [u0, 3u0] at −3u0, -u0, u0, and 3u0 are all 0.1973. When u=±2.6, the slopes have the largest absolute value of 0.2701 in the intervals. The maximum position error at the endpoint of an optimal test interval is given by Eq. (5),
where SNR is the signal-to-noise ratio of the system. High axial resolution can be achieved within the optimal test interval by using a large NA objective and high SNR detector.2.2 Dynamic measurement range analysis
The zero point of the z coordinate is taken to be the focal point, and the downward direction is taken to be positive. When the sample performs simple harmonic motion, the position of the measured point on the sample is
where z0 is the normalized value of the absolute position in the object space when the sample surface is static. A0 is the normalized vibration amplitude of the sample, ω is the angular frequency, and t is the time. To simplify the analysis, we set the initial phase φ0=0°.A0 is restricted by the working distance (W.D) of the objective, so the possible values that A0 can be divided into three ranges.
To achieve both large amplitude measurement range and nanometer scale axial resolution at different amplitudes, this work combines the previous analysis of the optimal test interval of the confocal response curve with the analysis of the sample amplitude range. Three measurement modes are used to obtain the system vibration response curve of the vibrating surface to extract the out-of-plane vibration parameters such as frequency and amplitude.
In the first measurement mode, which is used when the vibration amplitude of the measured point A0<u0, the relative position of the objective focal plane to the sample is adjusted along the optical axis direction so that the equilibrium position of the sample is at the midpoint of the optimal test range of the confocal response curve I(u). To simplify the analysis, only the position z0>0 is considered here, that is, the measured sample is adjusted so that the vibration range is located within [z0-A0, z0+A0] ⊆ [u0, 3u0]. The equation for the vibration response can be obtained as follows:
The simulated curve of the intensity response to the vibration of the measured point is shown as I1(t) in Fig. 3(a). The frequency information can be extracted from I1(t). The vibration function z(t) and the actual amplitude of the measured point can be solved from the intensity response signal I1(t) using Eq. (2) and Eq. (8).
Fig. 3. Vibration measurement at different amplitudes. (a), (b), and (c) are the simulated vibration measurements when the amplitude of the measured point A0≤u0, u0<A0≤3u0, and 3u0<A0≤0.5W.D, respectively. The normalized axial coordinates of points A, B, C, and D are uA=−3u0, uB=-u0, uC=u0, and uD=3u0, respectively.
In the second measurement mode, which is used when the vibration amplitude of the measured point u0<A0≤3u0, the relative position of the objective focal plane and the sample is adjusted along the optical axis direction so that the equilibrium position of the sample is on the focal plane of the confocal system, i.e., z0=0. The maximum out-of-plane position of the vibration surface is located inside the optimal test interval on both sides of the main lobe of the confocal response curve. Under these conditions, the intensity response equation of the tested point can be expressed as
The simulated curve of the vibration intensity response is shown as I2(t) in Fig. 3(b). The frequency information of the sample can be extracted from the intensity response signal I2(t). The frequency of I2(z) is twice the frequency of the measured point. The vibration function z(t) and the amplitude of the measured point can be solved from the intensity response signal I2(t) using Eq. (2) and Eq. (9).
Under the measurement methods of Fig. 3(a) and Fig. 3(b), the maximum out-of-plane position of the measured point is always within an optimal test interval of the confocal response curve. The intensity value I1(t) measured in Fig. 3(a) is always in the CD segment. In Fig. 3(b), the intensity value corresponding to the maximum out-of-plane position is always in the AB segment or the CD segment. According to the previous discussions, the system has a high axial resolution when solving for amplitudes in the AB and CD segments. The instantaneous velocity and acceleration of the measured point at different times can be obtained by calculating the first and second derivatives of the intensity response signal of the measured vibration point with respect to time t.
It is known from Section 2.1 that the confocal response curve has the maximum absolute slope at u = uup=-2.6 and u = udown=2.6 in the optimal test intervals. When the amplitude of the measured point is 3u0<A0≤0.5W.D, two groups of intensity response curves of the vibration sample, shown as Iup(t) and Idown(t) in Fig. 3(c), can be obtained by the following two-step scanning process.
In the first step, the objective lens is driven by the axial displacement platform piezoelectric transducer (PZT) from the farthest positive distance in the negative direction as the sample vibrates. When the maximum value of the received intensity signal is equal to I(uup) for the first time, the normalized position of the PZT in space zup, the intensity response curve Iup(t), and its maximum value I(uup) are recorded, where uup<0.
In the second step, the objective lens is driven by the axial displacement platform PZT from the farthest negative distance in the positive direction as the sample vibrates. When the maximum received intensity signal is equal to I(udown) for the first time, the normalized position of the PZT in the object space zdown, the intensity response curve Idown(t), and its maximum value I(udown) are recorded, where udown>0.
In the ideal case, where there are no errors introduced by the axial displacement platform PZT, the amplitude of the measured point in the object space can be calculated using the maximum values I(uup) and I(udown) of the intensity response curves, Eq. (1) and Eq. (10). The amplitude can be expressed as
In this case, the solved amplitude has a high axial resolution because uup and udown are located in the optimal test ranges of the confocal response curve. From the intensity signal Iup(t) or Idown(t), the frequency of the vibration sample can be extracted. In order to avoid collision between the sample and the objective during scanning, the maximum amplitude should not exceed 0.5W.D. The displacement measurement range of amplitude measurement is (−0.5W.D, 0.5W.D).
It is often necessary to test multiple points in the vibration measurement. When the amplitude of the measured point is unknown, the amplitude may be first tested by the method shown in Fig. 3(b). If the maximum out-of-plane position of the vibration is inside the optimal test intervals [−3u0,-u0] and [u0, 3u0], the vibration information can be solved for directly. If the maximum out-of-plane position of vibration is within [-u0, u0], the vibration information can be obtained by the method in Fig. 3(a), which has a higher amplitude resolution. If the maximum out-of-plane position of vibration is not within the range of [−3u0, 3u0], the method in Fig. 3(c) can be adopted to obtain the vibration parameters. Although this method needs two scans, it has a potentially larger amplitude measurement range than the previous two methods.
The laser confocal vibration measurement method is not limited by the focal depth of the objective lens when measuring the vibration amplitude due to the cooperative use of the above three cases. The method can not only ensure that the solution of the amplitude is in the optimal test range, but can also achieve high axial resolution for the measurement of different amplitudes. It can also extract out-of-plane vibration parameters, such as the frequency of the sample, from the response curves.
3. Experimental system and experimental analysis
An experimental setup for laser confocal vibration measurement was built according to the principle shown in Fig. 1. The axial resolution, amplitude measurement capability, frequency measurement capability, spectrum scanning capability, and MEMS surface amplitude distribution measurement capability of the measurement method were experimentally verified. In the experimental system, a 532 nm laser was used as the laser source, and an Olympus 100× flat field achromatic objective was used as the microscope objective, which has the focal length of f0=1.8 mm, the numerical aperture of NA=0.90, and the W.D of 1 mm. The detector used for the light intensity detection is photodiode S5973 (Hamamatsu Co. Ltd.) with cut-off frequency of 1GHz. The transimpedance amplifier used is LTC6268-10 (Linear technology Inc.) with Gain Bandwidth Product of 4GHz. The passband of the detection module with S5973 as the detector is 0Hz to 120MHz, the signal to noise ratio of the detector is greater than 200:1. Tektronix's oscilloscope MSO44 used for signal acquisition has ADC resolution of 12bits@3.125GS/s. The data acquisition card NI6353 (National Instrument Company) was used to control the PZT and acquire trigger signals, which has the ADC resolution of 16bits and DAC resolution of 16bits. Function generator F120 is used to generate sinusoidal voltage signal to drive MEMS samples. The PZT (PI Co. Ltd.) used to drive the objective lens with a driving resolution of 0.75 nm. A high-precision nanopositioning stage (PI Co. Ltd.) with closed-loop resolution of 2 nm is used as a lateral scanner. A MEMS that generates out-of-plane vibration by voltage driving was used as the test sample.
3.1 Axial resolution test
To verify the axial resolution of the method, a plane mirror was used as a sample for testing, and scanning was performed at a step size of 4 nm. The measured axial response curve is shown in Fig. 4.
It can be seen from the Fig. 4 that the laser confocal out-of-plane vibration measurement system can clearly distinguish the intensity signals when the axial feed of the sample is 4 nm, i.e. the axial displacement resolution of amplitude measurement can achieve 4 nm.
3.2 Single point vibration test with different amplitudes
To verify the feasibility of the proposed method, different vibration amplitudes of the MEMS device were tested. In the experimental setup, an air-coupled transformer with a resonant frequency of 872 kHz was mounted onto the piezoelectric ceramic two-dimensional drive translation platform horizontally under the confocal microscope objective. The contour map of the MEMS device and the confocal response curve of the center point A are shown in Fig. 5(a) and Fig. 5(b), respectively.
Fig. 5. (a) Schematic of the MEMS. (b) Confocal response curve of point A. (c) – (e) are the intensity response curves when the amplitude A0≤u0, u0<A0≤3u0, and 3u0<A0≤0.5W.D, respectively.
To obtain dynamic information, the MEMs was driven by a time-varying sinusoidal signal. At the driving frequency of 887 kHz and driving amplitude 1.5 V, the relative axial position of the objective and point A on the sample surface was adjusted so that the amplitude satisfied A0≤u0. The intensity response curve is shown in Fig. 5(c). The measurement is repeated for 10 times. The measured results of the peak-to-peak value of the amplitude are shown in Fig. 6(a). The average peak-to-peak amplitude value A=132.75 nm, and the standard deviation of the repeated measurements is σA=4.13nm. The average frequency is f = 887.08 kHz, and the standard deviation of the repeated frequency measurements is σf = 0.04 kHz.
A sinusoidal drive signal with a frequency of 862.0 KHz and an amplitude of 6.5 V was then applied to the MEMS, and the relative axial position of the objective and point A are adjusted so that the amplitude satisfied u0<A0≤3u0. The intensity response curve is shown in Fig. 5(d). The measurement is repeated for 10 times. The measured results of the peak-to-peak value of the amplitude are shown in Fig. 6(b). The average peak-to-peak amplitude is A=346.76 nm, the standard deviation of the repeated measurements is σA=3.57 nm. The average frequency is f = 862.05 kHz, the standard deviation of the repeated frequency measurements is σf = 0.09 kHz.
A sinusoidal drive signal with a frequency of 0.872 MHz and an amplitude of 13 V was finally applied to the MEMS and the two intensity response curves, Iup(t) and Idown(t), of the sample vibration shown in Fig. 5(e) are obtained by two scans. The measured results of the peak-to-peak value of the amplitude are shown in Fig. 6(c). The average peak-to-peak amplitude in 10 measurements is A=1.4970µm, and the standard deviation is σA=12 nm. The average frequency of 10 measurements is f = 872.12 kHz, and the standard deviation of the repeated frequency measurements σf = 0.16 kHz.
Therefore, under the experimental configuration in this paper, the precision of the repeated measurement of the peak-to-peak amplitude can reach 12 nm, and the precision of the repeated measurement of the frequency can reach 0.16 kHz.
3.3 Resonance frequency test
The resonance frequency of the vibrating sample can also be obtained by frequency sweep and the method proposed in this paper. The sample position was first adjusted to align the optical axis with the sample center A. The sample was then driven by a sinusoidal voltage signal with amplitude of 6.5 V. The amplitude change of point A was measured as the driving signal frequency was varied from 0.1 MHz to 1.3 MHz. The obtained curve is shown in Fig. 7.
It can be seen from Fig. 7 that the maximum peak-to-peak amplitude of the measured sample is 406.5 nm, and the corresponding frequency of 872.5 kHz is the resonance frequency of the sample.
3.4 Measurement of amplitude distribution on MEMS surface
A sinusoidal drive signal with a frequency of 0.887 MHz and a peak-to-peak value of 3 V was applied to the MEMS. Using the method proposed in this paper, the vibration of the MEMS was measured along the radial direction. The peak-to-peak amplitude distribution of the MEMS surface is shown in Fig. 8.
The peak-to-peak value of the amplitude is 133.1 nm at the center point A and less than 10 nm at the edge. The frequency is about 886.9 kHz.
4. Conclusions
In this paper, a laser confocal vibration measurement method that can precisely measure the out-of-plane vibration parameters was proposed. This method has the advantages of high precision, simple structure, and easy and non-contact implementation. The method can measure samples vibrating with different amplitudes and frequencies with nanometer-scale amplitude axial resolution. This will provide precise dynamic response data for the development and verification of MEMS devices and other micro structures, which will in turn allow better control of the MEMS manufacturing process in MEMS research. Combined with the relevant parameters of the sample, the method can even be used to evaluate the modal parameters and mechanical properties of the micro scale structure, which will help to improve and shorten the MEMS design cycle, simplify troubleshooting, and improve the yield. The method allows characterization of numerous kinds of devices, although we are currently mostly interested in microstructures.
Funding
National Key Research and Development Program of China (NO. 2018YFF01012001); National Science Fund for Distinguished Young Scholars (NO. 51825501); National Natural Science Instrumental Foundation of China (NO. 61827826).
Disclosures
The authors declare no conflicts of interest.
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