1. Introduction

Mechanical vibration measurement has always been a focus of industrial production, structural health monitoring, MEMS (Micro-Electro-Mechanical system), rotating machinery, hearing and acoustics. The optical non-contact measuring device claimed particular advantages over traditional instrumentation, such as accelerometers or strain gauges etc., particular for measurements on hot, light, or rotating structures where traditional contacting instrumentation would change dynamics or be difficult to attach. Laser Doppler Vibrometry (LDV) is the most common optical vibration measurement technique [1,2]. It detects the Doppler frequency shift that occurs when the light is scattered by the moving surface which is directly proportional to the surface velocity. Usually, the frequency shift caused by the velocity of 0.1 m/s is almost 9 orders of magnitude lower than the frequency of laser [1]. However, this poses a high demand for the monochromatic nature of the laser source and the stability of the system. In our present work, we proposed Optical Coherence Velocimeter (OCV) which is a new optical non-contact velocity measuring technology based on the principle of broadband optical coherence which achieve spatial resolution from interference cancellation or enhancement of different components of the broadband light. Compared with the LDV, the OCV velocity measurement range and the working distance of the OCV are smaller, however the OCV has the advantages of low cost, small size, and high precision of velocity measurement, which makes the performance better for micro structure vibration measurement and acoustic measurement.

In fact, the OCV is developed from the Optical Coherence Tomography (OCT) technology. OCT is a powerful 3-D inner structure optical imaging technology proposed in the 90s of last century. It has been rapidly used in ophthalmology field [3] and other medical fields [4] because it can provide an inner observation of micron-structure of organ tissue. The development trend of OCT technology is higher imaging spatial resolution (~1$\mu m$), higher sensitivity (more than 90 dB) and faster imaging speed (real-time imaging), which mainly depends on the improvement of the system’s principle [5] and the application of better performance devices [6,7]. With the improvement of the system performance, the application fields of system are no longer limited in the field of medical imaging, such as industrial composites material performance testing [8], film coating thickness on-line monitoring [9], car paint monitoring [10], art heritage protection [11], and paper industry [12] and so on. In recent years, the OCT is also beginning to be applied in the field of displacement measurement. In our previous work, our research group enhanced the spatial resolution of OCT to nanoscale level by combining OCT with spectrum correction method [13] and PZT (Piezoelectric ceramic transducer)’s hysteresis curve was measured [14]. It shows great potential for displacement measurement of optical coherence technology. Even more, we have proposed an optical coherence vibration tomography (OCVT) technology [15,16] which can obtain both the inner structure and the vibration parameters of the sample simultaneously. But it is a pity that just some simple attempts and only low-speed (about 0.5 mm/s) and low-frequency (about 0~500 Hz) motion can be measured, limiting the further application of OCVT. It is noted here that the Doppler OCT [17,18] also a velocity measurement technology which can measure fluid flow in biological tissues. However, the technology is an approximate estimation of fluid flow and the maximum measurable velocity estimation limited by phase-shift measurements between axial scans, regardless of the method (color Doppler or velocity variance). In the present work, we propose the OCV technology which is based on the principle of the broadband optical coherence. The phenomenon that the interference fringes become blurred or disappear when measuring a high velocity moving object, was introduced and analyzed. Some solutions to improve the velocity measurement performance of the system, are given. The system developed in the work can provide a large velocity measurement range from static to 1.06 m/s with very high precision. The measured maximum vibration frequency can be 50 kHz. OCV technology has excellent velocity measurement performance and can enrich the optical non-contact velocity measurement method together with LDV technology.

2. Principle and set-up of optical coherence velocimeter (OCV)

The traditional OCT system is based on the broadband light interference principle and its basic structure is the Michelson interference structure. The back-scattered light from the target is mixed with a mutually coherent reference beam to produce interference. The result of the interference is that the intensity of the merged beam changes periodically in the spectral domain. This is so-called frequency domain optical coherence tomography (FD-OCT). The frequency of the interference fringes corresponds to the spatial position information of the target, so OCT provides a spatial resolution accuracy based on wavelength. The OCV system has the same structure as the typical OCT system, consisting of four parts: sample, interference structure, spectrometer, and signal analysis device (PC). There are two interference structures: free space structure and optical fiber structure. A simplest OCV system will include a 2*2 optical fiber as the interference structure, with a commercial spectrometer and signal processor (e.g. computer or DSP). The free-space system is slightly more complicated than the optical fiber one, but the bandwidth of the light is not limited by optical fiber, and the system has a better signal than the optical fiber one.

The schematic diagram of free-space and fiber-optic OCV system are shown in Fig. 1(a) and Fig. 1(b) respectively. The basic principle of these two systems are similarly the Michelson interference structure, therefore we take free-space OCV system as an example to explain the system in detail. The output of a super-luminescent diode (SLD, EBS300003-01, Exalos) with a central wavelength of 870 nm and bandwidth of 100 nm corresponding to a theoretical axial resolution of 3.3$\mu m$ in air, was collimated by lens L1 (f = 25 mm). The source was split into a reference beam and a detection beam by a beam-splitter BS (50:50). In this experiment, the detected object was a silicon mirror that was driven by a loudspeaker which was controlled by the signal from a function signal generator. Both reflection beams of reference beam and detection beam were combined to a signal beam after passing through the beam-splitter BS again. In particular, all components of the OCV system (not including the PC and function signal generator) are fixed on an optical platform with vibration isolation to eliminate the interference of environmental vibrations. In the meanwhile, the OCV system is installed in a room at constant temperature and humidity condition to ensure the sub-nanometer displacement accuracy of the OCV system. A homemade spectrometer consisting of grating (2400 line/mm), lens (f = 50 mm) and linear CCD camera (P4-CM-04K10D-00-R, 100k fps, 4k*2 pixel, DALSA) was employed to detect the signal beam. The components of the spectrometer are encapsulated in an opaque box to block the interference of ambient light source. Interference occurs when both the reference beam and detection beam are spatially matched in orientation and their optical lengths are matched within the coherence length of the light source. Generally, the detection range of the system is determined by the spectral resolution of the spectrometer and the central wave-length of the spectrum, so the detection range of this system was 7.2 mm corresponding to the spectral resolution of 0.025 nm of the homemade spectrometer when the center wavelength is 870 nm. The traditional OCT acquires the spatial position information by doing the fast Fourier transform (FFT) to the spectral interferogram. However, due to signal leakage effects, the calculation of amplitude, phase and frequency by the FFT of the measured finite-length spectral interferogram is normally different from the real one. It means that the OCT system can only identify the integral change of the interferogram fringes, and the original accuracy of the spatial resolution will be several microns. A spectrum correction theory for sparse spectrum was used as an enhanced signal processing method for OCV system. It could improve the spatial resolution of the system from micron to nanometer level of 0.05 nm. The relevant principles and conclusion have been reported in the previous work [14,15]. OCV is the derivation of displacement with respect to time to obtain the velocity of the target. The electronic clock provides an extremely high time accuracy, so the OCV system possess a very high velocity measurement accuracy because of its sub-nanometer displacement measurement accuracy.

 

Fig. 1 Schematic diagram of optical coherence velocimeter (OCV) with free space reference structure (a) and fiber-optic reference structure (b). The vibration structure (LS, loud speaker) was controlled by a function signal generator (FSG). SLD, Super luminescent diode; PC, personal computer; BS, Beam-splitter; L, Lens; M, Mirror; SM, silicon mirror; FO, 2*2 fiber optic; G, Grating; Detector, Linear array camera

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3. Analysis of interference fringes becoming blurred at high velocity

In the traditional applications, OCT technology is used for static or quasi-static object imaging. The spectral interferogram of the interferometer can be expressed as:

In the OCV system, the detected object is no longer static but dynamic. For a vibrating object, the optical path difference $\Delta L$ is no longer constant. The optical path difference $\Delta L$ becomes a function of time$t$. Since the signal collected by the camera, the signal receiver, is the result of light intensity integral in the exposure time, the spectral interferogram of OCV system can be expressed as:

where$t_1$,$t_2$ represent the beginning and the end time of the camera exposure. ${\nu }_{(t)}$means the velocity of the vibration object. In order to simplify the analysis process, we assume that the object only has the surface layer along the axial direction and does not consider the auto-correlation term. In the process of the object vibration, for a sample with multi-layered structure, the distance between layers is constant, so making the above assumption does not affect the subsequent analysis results.

The Eq. (2) can be rewritten as:

The blurry coefficient item $\sin (\beta )/\beta$is a classical mathematical expression. The result of this function oscillates like a sine function, but the amplitude of the oscillation decreases with the increment of the variable$\beta$ . The variable$\beta$ is related to the wavelength$\lambda$, exposure time$t$ and the vibration velocity$\nu$. For a certain system, the wavelength of light source is constant, so the variable$\beta$ is proportional to the exposure time $t$ and vibration velocity$\nu$. The relationship between blurry coefficient and the vibration velocity is shown in Fig. 2. The position and negative signs of velocity represent the direction of motion. When the velocity is close to 0, the value of the blurry coefficient is close to 1. At this situation, we can omit the blurry coefficient, when the interferogram signal expression is for static or quasi-static OCT system. When the vibration velocity of the object increases, the blurry coefficient begins to decay rapidly. At some velocity value, e.g. the parameter $\beta =n\pi$, the blurry coefficient becomes zero, which means that the contrast of interference fringe is zero, and the system cannot obtain the desired information. Over that velocity, the blurry coefficient starts rising up to a peak and then falling to zero again. The peaks are getting smaller and smaller and finally become to zero. It should be noted here that the sign of $\sin (\beta )/\beta$does not represent whether the interference fringes are enhanced or canceled; only the magnitude of $\sin (\beta )/\beta$ represents the attenuation coefficient of the interference. Therefore, the blurry coefficient defined as the absolute value of the $\sin (\beta )/\beta$. The velocities when blurry coefficients become 0 are called blurry velocities: first blurry velocity, second blurry velocity and so on.

 

Fig. 2 The relationship between blurry coefficient and vibration velocity when the center wavelength is 800nm.

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When the velocity of moving object is close to the blurry velocities, the interference fringes obtained by system become blurred and the system loses measurement ability. For real vibration measurement application, high velocity measurement capability is an important parameter of the system. It means that the larger the blurry velocity, the better the performance of the OCV system. According to Eq. (4), we could know that increasing the wavelength or reducing the exposure time could increase the system’s blurry velocities. However, changing the wavelength of the light source has a limited increase to blurry velocities, so the most effective way to increase the blurry velocities is to reduce the camera’s exposure time. In general, the system with better performance needs more high-speed, more sensitive camera and stronger optical density light source. Figure 2, the dash line is the relationship between blurry coefficient and velocity when exposure time is 100$\mu s$, and the solid line corresponds to 40$\mu s$ . It can be seen that the smaller exposure time corresponds to greater blurry velocity of the system which allows to measure the object vibrating at higher velocity.

In the above analysis, the system will lose its measurement ability when the velocity of moving object is close to the first blurry velocity. In the experiment, we found that even if the velocity of vibration object is first-order blurry velocity, the interference fringes obtained by the developed OCV system will still not be completely blurred. The reason for such phenomena is due to the polychromatism of the broadband light source we used. According to Eq. (4), the blurry velocity is related to the wavelength. Assuming that the velocity of the vibration object $\nu$ is the first-order blurry velocity of the wavelength${\lambda }_{\text{1}}$, the velocity $\nu$ is not first-order blurry velocity for the other wavelengths (e.g.${\lambda }_{\text{2}}$)., Furthermore, the farther the wavelength ${\lambda }_{\text{2}}$ is away from the wavelength${\lambda }_{\text{1}}$, the farther the vibration velocity deviates from the blurry velocity of the wavelength${\lambda }_{\text{1}}$. The relationship between blurry coefficient and vibration velocity for different wavelength is shown in Fig. 3. We could find that different wavelengths correspond to different blurry velocities; also, large wavelength interval corresponds to large blurry velocity difference.

 

Fig. 3 The velocity-independent blurry coefficient of different wavelengths.

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If a broadband light source is employed in the OCV system, the interference fringe contrast will be obvious even if the interference fringes of a part wavelength range will be blurred due to the corresponding velocity being located at blurry velocity. Hence, there is always some part of the entire wavelength range that will have obvious interference fringes if a broadband light source is used. For a narrow-band source (for example ${\lambda }_{width}<0.025{\lambda }_0$) OCV system, the maximum velocity-measuring capability is the first blurry velocity. However, the velocity-measuring capability of broadband system can pass through the blurry velocities. Therefore, increasing the bandwidth of the light source is an alternative effective way to increase the velocity-measuring capability of the system. In the broadband OCV system, the interference signal collected is the result of a combination of all the wavelength components. The wavelength-independent blurry coefficient can be obtained by integrating the blurry coefficient by wavelength. For broadband source system, the blurry coefficient $C_{blur}$ can be calculated as:

It is difficult to get the numerical solution of an integral formula like Eq. (5). But we still have a way to get an approximate solution of the equation. We can discuss it in two situations.

The first case, when the wavelength $\lambda$ varies within the bandwidth and the value of$\beta$ changes a little, the situation corresponds to low vibration velocity. The change of the value of$\beta$within entire bandwidth range of light source $\Delta \beta \approx n\pi {\lambda }_{width}/{\lambda }_0$at each order blurry velocity, where$n$ is the order number of blurry velocity. When $\Delta \beta$ is small (for example $\Delta \beta <\pi /6$), the below formula can be established.

The blurry coefficient $C_{blur}$ of each wavelength of nth order blurry velocity can calculated as

From the above result, we can infer that in order to increase the blurry coefficient value at each order blurry velocity the effective way is to increase the ratio of the bandwidth and the central wavelength of the light source. Another interesting conclusion is that the blurry coefficient for each order blurry velocity is equal according to Eq. (7). However, in reality, as the order increases, the blurry coefficient decreases. The reason is that Eq. (7) is based on Eq. (6) to calculate the blurry coefficient. As the order of blurry velocity increases, that is, the $\Delta \beta$ (the change of the value of $\beta$ within entire bandwidth range of light source) becomes larger, Eq. (6) will be false. The left part of Eq. (6) will gradually become smaller than the right part. This will result in that the calculated result of blurry coefficient $C_{blur}$ become smaller. After performing simulations using light sources of various bandwidths, we find that this reduction of blurry coefficient of each order blurry velocity is only related to the change of the value of $\beta$ within entire bandwidth range of light source. When $\Delta \beta$ equal to $\pi /2$, the value of blurry coefficient $C_{blur}$decreases to 87% of the value calculated using Eq. (7) whilst when $\Delta \beta$equal to$\pi$, the value of $C_{blur}$ decreases to 60% of the value calculated using Eq. (7).

The second case, when the wavelength$\lambda$ varies within the bandwidth and the value of $\beta$ changes by more than$\pi$, this situation corresponds to a large vibration velocity. At this situation, the value of $\beta$is much bigger than $\Delta \beta$. The value of $\sin (\beta )$ changes very large in whole wavelength range. But the value of $\beta$ changes slowly compared to $\sin (\beta )$ and it can be replaced by the $\beta$ value at central wavelength${\lambda }_0$. The light source can be considered as a rectangular spectrum light source whose normalized power spectral density function is: $S_{(\lambda )}=\frac{1}{{\lambda }_{width}}$,${\lambda }_0-\frac{{\lambda }_{width}}{2}\le \lambda \le {\lambda }_0+\frac{{\lambda }_{width}}{2}$ . The blurry coefficient $C_{blur}$can be simply approximated as

In this case, as the velocity of object increases, the value of blurry coefficient $C_{blur}$ will become steady and the oscillation of the value will disappear quickly. As can be seen from Eq. (8), the value of blurry coefficient $C_{blur}$ is proportional to the central wavelength of the light source and inversely proportion to the vibration velocity and integration time of the spectrometer.

The above two cases are connected when $\Delta \beta$ within entire bandwidth of light source equal to$\pi$. At this time, $\beta =\pi *{\lambda }_0/{\lambda }_{width}$. So,

 

Fig. 4 The velocity dependent blurry coefficient of system with different bandwidths.

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Here, we discuss the low-cost OCV system design, including choosing the bandwidth of light source and the achievable measurement capability of the system. It is related to the SSNR (Static Signal to Noise Ratio) which means the SNR of interference signal when the vibrator remains stationary. In order to achieve low-cost OCV system, the criteria for OCV system design are that the SNR reduction caused by blurry coefficient must be smaller than the SSNR; the bandwidth used by the system is as small as possible; the achievable velocity measurement capability is as large as possible. Therefore, it is generally required that the blurry coefficient of first-order blurry velocity should be greater than twice of SSNR, that is,

In this way, even if the blurry coefficient is close to the steady state and decreases to 60% of the first-order blurry coefficient, the interference signal still exists. The system has a high velocity measurement capability. If the SSNR of the system is large enough, using a narrow-bandwidth light source can also achieve high velocity measurement capability. Therefore, smoothing and brightening the surface of the vibration object to improve the SSNR is also an effective way to improve the system’s velocity measurement capability.

In the real engineering applications, sometimes the SSNR is unknown in advance, we generally recommend that the bandwidth of the light source should be greater than 1/10 of the central wavelength. In this case, the value of $\beta$ changes$\pi /10$ at first-order blurry velocity in the entire bandwidth range, and the value of blurry coefficient is 0.03388 corresponding 30 dB reduction of interference intensity. Even at tenth-order blurry velocity, the value of blurry coefficient is 0.02 corresponding 34 dB reduction. It satisfies most of the engineering applications. Hence, we could define the tenth-order blurry velocity as the maximum velocity measurement capability of the OCV system.

4. Experiment and discussion

We Use the home-made OCV system to measure the vibration of loudspeaker which was excited by the signal generated from a function signal generator (Agilent 33220). The spectrum correction theory was used to improve the spectral resolution of the OCV system. The signal processing method based on spectrum correction method can be referred to reference [14] and [19]. In our previous work in Reference [14], we have experimentally verified the measurement accuracy of the OCV system by measuring the dynamic curves of a PZT stack actuator (Thorlabs) compared with a commercial STIL MICROMESURE system. MICROMESURE is a modular measurement system dedicated to high resolution 3D microtopography and to shape and texture analysis. The axial resolution of the MICROMESURE system is 8 nm. The results measured by the two systems show that the performance OCV system is much higher than the one of MICROMESURE system. It has been demonstrated that the developed OCV system has sub-nanometer measurement accuracy of 0.05 nm [14].The Fig. 5 (a) is OCV measured results with and without spectrum correction method for the sinusoidal vibration with amplitude in micrometers. From the figure, we could find that the spectrum-corrected OCV had better resolution. Furthermore, Fig. 5 (b) shows the he OCV measured results with spectrum correction method for the vibration amplitudes in nanometers. Combined with spectrum correction method, the displacement resolution of OCV system has been greatly improved to nanometer level resolution.

 

Fig. 5 (a) The comparison of OCV measured results with and without spectrum correction method for the vibration with amplitude in micrometers; (b) the OCV measured results with spectrum correction method for the vibration amplitude in nanometers. For clarity, the curves are offset vertically.

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We choose a sinusoidal vibration to verify the high-velocity measurement performance of the developed OCV system. Firstly, a sinusoidal signal with amplitude of 6 v and frequency of 25 Hz generated by function signal generator is used to excite the loudspeaker vibration. The diaphragm of the loudspeaker vibrated in sinusoidal form under the driven of signals. The advantage of choosing this form of motion is that the velocity of object changes smoothly and the velocity at each time can be simply calculated by the current time and the maximum velocity; also, no additional velocity measuring device is needed. Figure 6 (a) is the loudspeaker’s vibration measured by the developed OCV system. The dashed line is the displacement of the vibrating loudspeaker diaphragm. The amplitude of vibration is 318$\mu m$, and the vibration period is 0.04 s corresponding to the frequency of 25 Hz. The velocity of vibration could be obtained by taking the derivative of the displacement with respect to time. The solid line in Fig. 6 (a) is the velocity curve of the vibration. The velocity curve is in cosine form with a maximum velocity of 25.2 mm/s.

 

Fig. 6 (a) The displacement and velocity of the sinusoidal vibration measured by the OCV system. (b) The change of blurry coefficient in a motion cycle under different exposure time conditions.

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Figure 6 (b) is the blurry coefficient of the sinusoidal motion measured by OCV system with different exposure time. In order to keep the consistency of the experimental conditions, we inserted an attenuator with different extinction ratio into light path and adjusted the exposure time of the camera to make the same intensity of light received. The extinction ratios of 50%, 30%, 7.5% and 3.5% corresponded to the exposure time of 11.5$\mu s$, 19.5$\mu s$, 80$\mu s$and 165$\mu s$respectively. In the present work, the central wavelength of the light source is 850 nm. The bandwidth is 100 nm slightly larger than 1/10 of central wavelength. According to Eq. (4), the first order blurry velocity of OCV system at 11.5 $\mu s$and the exposure time is 36.9 mm/s, this is greater than the maximum velocity 25.2 mm/s of sinusoidal motion. Therefore, in the whole motion cycle, the blurry coefficient of the system is less than 10 dB, the interference fringes collected by system are still clear. When the exposure time is 19.5$\mu s$, the first-order blurry velocity of the system is 21.8 mm/s. Considering that the velocity varies according to cosine form with a maximum velocity of 25.2 mm/s, the first-order blurry velocity occurs at 0.0035 s, 0.0165 s, 0.0235 s, 0.0365 s for the four exposure times. The blurry coefficient was reduced to about −33dB at first order blurry velocity and increased rapidly when it exceeded the first-order blurry velocity. When the exposure time is 80$\mu s$, the first-order blurry velocity of system is 5.3 mm/s. The motion cycle crosses four order blurry velocity, the blurry coefficient correspondingly had four decreasing minimum values. When the exposure time is 165$\mu s$, the first-order blurry velocity is 2.58 mm/s. The maximum velocity of motion is close to the 10th-order blurry velocity.It can be seen from the figure that the coefficient does have nine minimum values. It is proved that the OCV system has the measuring ability at 10th-order blurry velocity. It is noted that as the exposure time increases, the number of sampling points of a motion cycle is reduced. The exposure time of 11.5$\mu s$corresponds to 7844 points. For the exposure time of 165$\mu s$, the sampling point is only 480 points. Therefore, when the exposure time is large, the value of blurry coefficient is smoothed because there are not enough sampling points to describe the rapid change of blurry coefficient.

In summary, the velocity measurement capability of the system is inversely proportional to the exposure time; the shorter the exposure time, the higher the velocity measurement performance; also, when the bandwidth is wide enough, the velocity measurement performance of the system can reach 10th order blurry velocity.

In order to give a better understanding of the high-velocity measurement performance of the homemade OCV system, high-frequency vibration measurement results are shown in Fig. 7. Figure 7 (a) is the OCV measurement result of the vibration excited by the periodic swept frequency signal which sweep from 10 Hz to 500 Hz in 0.5 s. From the OCV measurement results, it can be seen that the vibration structure is basically vibrated according to the frequency of the swept signal. However, it still had residual oscillation in the low-frequency part which is originated from the last swept period. Figure 7 (b) is the OCV measurement results of vibration excited at 5 kHz, 10 kHz and 20 kHz. The measured frequencies by the OCV system are 4.9998 kHz, 9.9999 kHz and 19.9997 kHz, and the corresponding amplitudes are 680 nm, 70 nm and 46 nm. It is noted here that there a low frequency component in the vibration signal measured. That is due to the resonance of the loud speaker. These results demonstrated the excellent performance of the OCV system in high-velocity measurement field.

 

Fig. 7 (a) The OCV measurement result of vibration excited by swept frequency signal with a range of 10-500 Hz (b) The OCV measurement result of 5 kHz, 10 kHz and 20 kHz acoustic signal.

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When the OCV system is applied in vibration measurement, it is more concerned with the maximum measurable frequency that the system can achieve. The displacement function of the vibration can be simply assumed as $s=A*\sin (2\pi *f*t)$, where A means the amplitude of vibration and the$f$ means the frequency of the vibration. The maximum velocity of the vibration is:${\nu }_{\max }=A*2\pi f$. At the same time, the sampling of the system should satisfy Nyquist sampling law. Therefore, the maximum measurable frequency of the system should satisfy the following formula:

In this work, the central wavelength of the OCV system is 850 nm, the minimum exposure time is 4$\mu s$, and the maximum sampling frequency is 100 kHz. It can be calculated that the maximum measurable velocity of the OCV system is 1.06 m/s in theory. Under this condition, the maximum measurable frequency of the system is related to the amplitude of the vibration. The relationship can be shown in Fig. 8. The solid curve is the relation between maximum measurable frequency and amplitude calculated by Eq. (12). From this curve, we can obtain the maximum measurable frequency of the system from the amplitude of vibration. In real engineering applications, we can’t accurately know the amplitude of vibration, but we can estimate the approximate order of vibration amplitude. A so-called conservative curve was proposed (shown as the dashed line in Fig. 8) to estimate the maximum measurable frequency of vibration if the approximate amplitude’s order is known. Take an example, if we know that the amplitude order of vibration is a few microns, we can assume that the amplitude of vibration is 10 microns to calculate the maximum measurable frequency. Based on these two curves, the OCV system offers three application prospects: The first application is acoustic measurement, the frequency range of sound is from 20 Hz to 20 kHz, the measurable range of the amplitude of sound vibration can cover from 1.0e-9 m to 1.0e-5 m. The second application is mechanical structure vibration measurement, the frequency of mechanical structure is usually from several hertz to several thousand hertz, the measurable range of the amplitude of mechanical structure vibration can cover from sub-micron to several millimeters. The third application is displacement measurement, which is used to measure the displacement of low-velocity objects. Furthermore, the measurement range of the OCV system is very wide, from nanometers to centimeters.

 

Fig. 8 The relationship between maximum measurable frequency and amplitude of vibration

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5. Conclusions

The optical coherence velocimeter (OCV) is an optical non-contact vibration measurement technology which is based on the broadband optical interference. It can provide nanometer level displacement measurement accuracy and the corresponding velocity measurement accuracy. When the target is moving at high speed, the interference fringes will be blurred and the OCV will lose the velocity measurement ability. Theoretical analysis showed that the exposure time of the camera, the central wavelength and the bandwidth of light source are the main factors that affect the maximum velocity measurement ability of the OCV system. The shorter the exposure time and the larger the central wavelength, the greater the maximum measurable velocity of the OCV system. The blurry coefficient was proposed to describe the contrast reduction of the interference fringes changes with oscillatory decay form as the velocity of the target increasing. It was found that the blurry coefficient changed with oscillatory decay form and reached the minimum at each order blurry velocity. The bandwidth of the light source can increase the trough of each order blurry velocity. It is strongly recommended that the bandwidth of the light source exceeds the 1/10 of the central wavelength, so that the maximum measurable velocity of the system can reach 10th order blurry velocity. A loudspeaker driven by function signal generator was used to verify the above conclusion. Subsequently, we analyzed the relationship between vibration amplitude, vibration frequency and the maximum measurable velocity. The experiment demonstrated that the OCV system developed in the work can provide a large velocity measurement ranges from static to 25.2 mm/s with nanometer-level precision and the maximum measurable vibration frequency of up to 50 kHz. Due to the limit of our current experimental facility, we cannot provide larger velocity vibration for the validation of the maximum measurable velocity of our OCV system. However, in theory, the maximum measurable velocity of the developed OCV system can be up to 1.06 m/s for current configuration (e.g., the central wavelength of 850 nm, the minimum exposure time of 4$\mu s$, and the maximum sampling frequency of 100 kHz). The optical coherence velocimeter has high precision, large dynamic range, and high-velocity measurement capability, making it attractive for applications in the field of mechanical structure vibration monitoring and acoustic measurement.