Open Access
Add to BibSonomy
Abstract
Real-time vibration measurement with a high accuracy is described in which a He-Ne laser interferometer and a feedback control system are used. The measured vibration waveform is calculated as the sum of two measured waveforms obtained from the controller output signal and the feedback signal in the feedback system, respectively. This calculation method provides an exact measured vibration amplitude with error less than about 8 nanometers at the vibration frequency of 100 Hz. The vibration amplitude measured from only the controller output signal has an error less than about 21 nanometers. The maximum measurable amplitude decreases from 1935nm to 903 nm when the vibration frequency increases from 50 Hz to 500 Hz.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Laser interferometer is one of the most sensitive methods for small displacement measurement in scientific and industrial applications. A displacement or movement quantity is detected as a phase change of cosine function in the interference signal. In order to extract this phase change from the cosine function, a phase modulation term is introduced in the cosine function. Linear phase modulation or heterodyne modulation has been widely used, but more simple modulation is sinusoidal phase modulation (SPM) [1]. This modulation was easily applied to an interferometer using an optical fiber for displacement measurement [2]. Recently SPM has been utilized for measurement of vibration amplitude [3,4]. In these measurements the sinusoidally phase-modulated signal is processed by a computer to calculate the phase change. On the other hand, the phase change can be extracted by incorporating a feedback control system into an interferometer. The interference signal with the SPM was processed with electric circuits to obtain an electric signal proportional to the phase change. The feedback control system was applied for eliminating the phase change in the interference signal caused by external disturbances [5]. This technique using a feedback control system was also applied for measuring vibration displacement of an object [6]. In Ref. 6 a laser diode was used as a light source because it is easily carried out to change the phase of the interference signal by changing the injection current of the laser diode. The controller output signal in the feedback system is almost proportional to the phase change caused by the vibration of an object. The measured vibration displacement is calculated from the controller output signal. However, since the relation between the injections current change and the phase change is sensitivity to temperature and the bias value of the injection current, the stability and the resolution of the controller output is not so high in a laser diode interferometer.
In this paper a He-Ne laser is used as a light source insensitive to a small temperature change. The phase change of the interference signal is provided by giving a displacement to a reference mirror with a piezoelectric transducer (PZT). The movement of the PZT generates an exact phase change in the interference signal. Moreover by considering the fact that a feedback signal or an error signal which is the input signal of the feedback controller is not zero value, the measured vibration of the object is calculated by adding the feedback signal and the controller output signal. This new calculation method using the two signals leads to an exact measurement of the sinusoidal vibration amplitude.
2. Principle
Figure 1 shows an interferometer for real-time measurement of vibration or movement by using a feedback control system. A light beam of wavelength λ from a He-Ne laser source is collimated with lens L1 and L2, and the collimated beam is divided into an object beam and a reference beam by a beam splitter (BS). The object beam is incident onto an object, and the reference beam is incident onto a reference mirror. The object is a mirror attached on a piezoelectric transducer 3 (PZT3). A voltage applied to the PZT3 is VD(t), which produces a displacement d(t) of the mirror. The reference mirror is attached on the PZT1 which is attached on the PZT2. Since the PZT1 is used to generate a sinusoidal phase-modulation in the reference beam, its applied voltage is VZcos(ωct) which produces a vibration of acos(ωct). A voltage applied to the PZT2 is VC(t) which is an output of a feedback controller (FBC). This applied voltage produces a displacement dC(t)=KCVC(t), where KC is a constant value. The displacements of d(t) and dC(t) produce phase changes of αD(t)=(4π/λ)d(t) and αC(t)=(4π/λ)dC(t), respectively. Then an interference signal detected with a photodiode (PD) is expressed as
where A is a constant value and B is amplitude of the phase-modulated signal. And the phase modulation amplitude Z and the phase α(t) are given by where phase α0 is the initial phase when d(t)=dC(t) = 0. For sake of simplicity it is assumed that α0=0. The interference signal S(t) can be expressed as
Fig. 1. Laser interferometer for vibration measurement with a feedback system. L1 and L2: lens, PZT: piezoelectric transducer, FBSG; feedback signal generator, FBC: feedback controller, BS: beam splitter, PD: photodiode, OSCSCO: oscilloscope.
Figure 2 shows a flowchart of the feedback system associated with Fig. 1. Regarding the FBC, only a box of “FBC” is provided without depicting coefficients such as proportional gain and integral time. The values of these coefficients are adjusted so that the SF(t) is reduced to zero value.
Amplitudes of sinusoidal signals of VC(t) and SF(t) are denoted by VCA and SFA, respectively. In the condition that the time difference between VC(t) and SF(t) is small, the amplitude aD to be measured is given by
where aC= KCVCA and aF= (λ/4π)(SFA/K). The values of KC and K can be decided beforehand as shown in the next section.3. Experiments
The experimental setup shown in Fig. 1 was built. The wavelength λ of the He-Ne laser interferometer was 632.8 nm. The sinusoidal phase modulation was given by vibrating the reference mirror with the PZT1. The frequency ωc/2π was 10KHz, and amplitude Z of the modulation was about 1.8 rad at which J1(Z) has a maximum value. The vibrating object was an optical mirror vibrated sinusoidally with the PZT3. The interference signal S(t) detected with a photodiode (PD) was transferred to the FBSG to generate the feedback signal SF(t). A low-pass filter (LPF) with a cutoff frequency of 400 Hz worked well to extract fundamental frequency components of sinαJ1(Z)cos(ωct) in the interference signal.
First, the feedback control system was not used to obtain the coefficients of K, KC=aC/VCA of PZT2, and KD=aD/VDA of PZT3, where VDA is the voltage amplitude applied to PZT3. The object or PZT3 was vibrated with the waveform of aDcos(ωDt) by the amplitude VDA. Then the feedback signal SF(t)=Ksin[(4π/λ)aDcos(ωDt)] was observed as shown in Fig. 3, where ωD/2π=100 Hz. Since the peak-to-peak value of the signal was 1540 mV, the value of K was determined to be 770 mV.
The waveform of Fig. 3 indicated that the phase of (4π/λ)aDcos(ωDt) changed by 4π while aDcos(ωDt) changed from aD to -aD. Since these changes led to an equation of (4π/λ)2aD4π, the amplitude aD was measured to be λ/2 = 316 nm. This amplitude measurement method was applied to obtain a constant ratio of a PZT between the vibration amplitude and the applied voltage amplitude. A PZT was vibrated by an applied voltage of Vcos(ωt) and the interference signal of S(t)=cos[(4π/λ)acos(ωt)] was observed to measure the amplitude of a. By changing the voltage amplitude V, a constant ratio of a/V was obtained in the vibration frequency range from 30 Hz to 700 Hz for PZT2 and PZT3. The measured values of KC=aC/VCA of PZT2 and KD=aD/VDA of PZT3 were 236 nm/V and 129 nm/V, respectively. Since the feedback signal without the feedback control system is very complicated and its waveform is not sinusoidal wave, it is difficult to measure the vibration exactly in real-time from the feedback signal. Therefore a PI control system is required to measure the vibration of the object in real-time.
Second, a PI feedback control system was operated to get the output voltage VC(t) of the FBC. The vibration frequency ωD/2π of the object was 100 Hz, and the applied voltage VDA was 14 V. Figure 4 shows the vibration dC(t) calculated from the detected output signal VC(t) with the relation of the dC(t)=KCVC(t). Since the maximum and minimum voltages of VC(t) were 7.6 V and −7.54 V, respectively, the value of aC=KCVCA was 236 × 7.57 = 1786.5 nm. The detected waveform of the feedback signal SF(t) was shown in Fig. 5, where the values of the vertical axis were converted from voltage to nm with the relation of dF(t)=(λ/4π)[SF(t)/K]. Since the maximum and minimum voltages of SF(t) were 0.416 V and −0.248 V, respectively, the value of aF=(λ/4π)(SFA/K) was 50.357×(332/770) = 21.7 nm. Figure 6 shows the measured result of d(t)=dC(t)+dF(t) whose amplitude was 1803.4 nm. On the other hand another measured amplitude of aD=aC+aF was 1808.2 nm. Because there was a time difference between dC(t) and dF(t), the amplitude of d(t) was smaller than aD. Since the difference of these two measured amplitudes was very small as 4.8 nm, the measured amplitude aD was used to examine the characteristics of this vibration measurement by the feedback control system.
Third, various voltages VDA were applied to PZT3, and the vibration amplitude measurement was carried out to obtain the amplitude value aD. The measurement results are shown in Table 1. The given amplitude was calculated as aDG =KDVDA. The measured values of aC and aF were obtained from the signals of VC(t) and SF(t), respectively, as described above. The amplitudes measured by the proposed method were the values of aD=aC+aF. The differences of aDG-aC and aDG - aD indicate that the measured values of aD were closer to the given values of aDG than the measured values of aC. It was made clear by these measurements that the use of the feedback signal provides the exact vibration measurement with an error less than 8 nm.
Table 1. Experimental result at different applied voltages.
Finally, the vibration frequency of fd=ωD/2π was changed from 50 Hz to 500 Hz. At a fixed vibration frequency the voltage VDA applied to PZT3 or the vibration amplitude of the object was increased. At an increased vibration amplitude the waveform of the control signal VC(t) became unstable with time. At a vibration amplitude slightly smaller than the increased amplitude the waveform of the control signal VC(t) returned back to be stable. This amplitude was defined as the maximum measurable amplitude aDmax. Figure 7 shows the distribution of aDmax which was about from 1935nm to 903 nm in the range from 50 Hz to 500 Hz.
4. Conclusions
The real-time vibration measurement with a high accuracy has been described in which a He-Ne laser interferometer and a PI feedback control system were used. The measured vibration waveform was calculated as the sum of two measured waveforms obtained from the controller output signal and the feedback signal, respectively. This calculation method provided exact measured vibration amplitudes which agreed quite well with the given ones. When the sinusoidal vibration amplitude of the object was about from 200 nm to 1810nm at the vibration frequency of 100 Hz, the measurement error was less than about 8nanometers. The main sources of this measurement error were random phase and amplitude changes in the feedback signal caused by a quite large external disturbance. The vibration amplitude measured from only the controller output signal had an error less than about 21 nanometers. When the vibration frequency was from 50 Hz to 500 Hz, the maximum measurable amplitude was from 1935nm to 903 nm. In the future, the measurement range of the vibration amplitude and frequency will be enlarged, and vibrations of non-sinusoidal waveforms will be measured with this feedback type interferometer.
Funding
Japan International Cooperation Agency (JICA) (2017 Year).
Acknowledgement
The authors would like to thank JICA for supporting this research through the project for enhancement of engineering higher education in Myanmar.
References
1. O. Sasaki and H. Okazaki, “Sinusoidal phase modulating interferometry for surface profile measurement,” Appl. Opt. 25(18), 3137–3140 (1986). [CrossRef]
2. O. Sasaki and K. Takahashi, “Sinusoidal phase modulating interferometer using optical fibers for displacement measurement,” Appl. Opt. 27(19), 4139–4142 (1988). [CrossRef]
3. M. J. Connelly, J. H. Galeti, and C. Kitano, “Michelson interferometer vibrometer using self-correcting synthetic-heterodyne demodulation,” Appl. Opt. 54(18), 5734–5738 (2015). [CrossRef]
4. J. H. Galeti, C. Kitano, and J. Connelly, “Improved synthetic-heterodyne Michelson interferometer vibrometer using phase and gain control feedback,” Appl. Opt. 54(35), 10418–10424 (2015). [CrossRef]
5. O. Sasaki, K. Takahashi, and T. Suzuki, “Sinusoidal phase modulating laser diode interferometer with a feedback control system to eliminate external disturbance,” Opt. Eng. 29(12), 1511–1515 (1990). [CrossRef]
6. T. Suzuki, T. Okada, O. Sasaki, and T. Maruyama, “Real-time vibration measurement using a feedback type of laser diode interferometer with an optical fiber,” Opt. Eng. 36(9), 2496–2502 (1997). [CrossRef]
