## Abstract

Vibration is one of the confused problems in many fields. To give a comprehensive analysis of vibration, an electro-optical heterodyne interferometry with temporal intensity analysis method that can track the trajectory of the vibration dynamically has been built in this paper. The carrier frequency is introduced by the electrically controlled electro-optical frequency shifter. The trajectory is obtained by using temporal evolution of the light intensity in heterodyne interferometry. The instantaneous displacement of the vibration is extracted with spectral analysis technique. No target mirror and moving parts are required in our self-developed system. The principle and system configuration are described. The simulations and the preliminary experiments have been performed and the results show that this trajectory tracking system is high-efficiency, low-cost, jamproof, robust, precise and simple.

© 2014 Optical Society of America

## 1. Introduction

Vibration is one of the most important factors to be detected in many fields, such as auto industry, aviation, mechanical industry, construction industry, manufacturing industry, etc. Most of the vibration measurement systems can only output the amplitude or the frequency of the vibration [1–3]. However, along with the developments and the requirements of techniques, especially in the field of mechanical control, such as numerical control machine tools [4], trajectory is served as a real-time feedback to compensate errors dynamically. Moreover, in multi-degree-of-freedom measurement [5], trajectory can be served as a continuous position information for straightness errors measurement. By obtaining the motion trajectory, the measured object can be analyzed comprehensively, so that the motion information, such as instantaneous displacement, acceleration, amplitude, frequency, etc., can be obtained at any point in time domain.

In traditional non-contact optical vibration measurement system, holographic interferometry [6] and Laser Doppler [7, 8] are commonly used. Compared with holographic interferometry, Laser Doppler has a better noise immunity as heterodyne interferometry is adopted in most of the Laser Doppler Velocimetry (LDV). On the other hand, in terms of imaging, holographic interferometry can achieve the full field measurement, while Laser Doppler requires scanning techniques to fulfill this goal. This is because the carrier frequency in Laser Doppler is usually too high for the image acquisition devices. Hence, in this paper, one of the goals is to ensure the carrier frequency can be changed in a large dynamic range. Thus, in certain case, such as creep strain measurement, a low carrier frequency can provide a potential possibility for the image acquisition devices to collect the heterodyne signal without scanning, while retaining the superior noise immunity of LDV.

Various methods of introducing the carrier frequency in heterodyne interferometry systems have been reported. Among these, rotating wave-plate [9], dual-frequency laser [10] and acousto-optic modulators [11] are commonly used. The rotating wave-plate can generate a low enough carrier frequency for cameras, but it requires a motor to drive the rotation which is vulnerable because of the mechanical parts (bearing, gear, etc.) of the rotation and the upper limit of the carrier frequency are restricted by the frequency of the mechanical rotation. The dual-frequency laser generates the carrier frequency by using Zeeman Effect, and the carrier frequency is around 2 MHz, which is considered too high for most industrial cameras and cannot be changed once the equipment is produced. The fixed carrier frequency (around 2 MHz) limits the application of the heterodyne interferometry in high speed vibration measurement. Besides, as everyone knows, the cost of a dual-frequency laser is usually pretty high. As for the acousto-optic modulators, by using ultrasonic grating in acousto-optic medium which is driven by the piezoelectric transducer, the carrier frequency is generated by the first-order diffraction and it is usually about tens of megahertz, which is also considered too high for cameras. Moreover, the structure of the acousto-optic modulator is complex and the efficiency of frequency shift is low, because only the first-order diffraction beam is accessible. Besides, the Bragg diffraction angle between the zero- and the first-order beams is quite small which makes it difficult to separate the two beams.

Hence, in this paper, a vibration trajectory tracking system based on electro-optical heterodyne interferometry (**EOHI**) with temporal intensity analysis method is established. The carrier frequency is introduced by using a lithium niobate crystal (LiNbO_{3}) which is located in the center of two transversely electric fields [12]. The carrier frequency is electrically controlled, and it can be turned up and down in a large dynamic range which is depended on the rotating frequency of the electric fields. By setting a property voltage of each field, the incident light can be shifted into a new frequency. The system is able to measure the amplitude and frequency of the vibration as well as the vibration trajectory. The trajectory can be tracked nondestructively by using Fourier transformation method of the light intensity recorded in a sequence as the object is vibrating. No target and moving parts are required in our self-developed system. The preliminary experiment shows that our self-developed system is high-efficiency, low-cost, jamproof, robust, precise and simple.

## 2. Description of EOHI system

#### 2.1 The theoretical basis of heterodyne interferometry

In heterodyne interferometry system, a crucial step is to introduce a carrier frequency in the measurement system. Normally, the carrier frequency appears as a beat frequency signal which is generated by two light beams with a small difference in frequency. For each beam, different frequency shifts are introduced by frequency shifter or using dual-frequency laser directly. During the measurement, the measured vibration signal is carried with a carrier frequency which can be regarded as signal modulation. The vibration information can be demodulated by common frequency analysis techniques, such as Fourier analysis. Since the frequency shift is known, the relative phase of the measured beat frequency can be measured very precisely even if the intensity levels of beams are slowly drifting.

In our self-developed system, the carrier frequency is introduced by linear electro-optic, or Pockels effect. A piece of LiNbO_{3} crystal is placed in the center of two transversely electric fields which are alternating in sinusoid with phase delay π/2. When a beam of monochromatic linearly polarized light passes through the crystal, the emergent light contains left- and right-circularly polarized light components with different frequency shift. The amount of frequency shift is determined by the rotating frequency of the electric fields which are electrically controlled. By turning the rotating frequency of the electric fields up or down, the carrier frequency can be changed in a very large range. The phase which is converted from the vibration trajectory is retrieved by inverse Fourier transform of a filtered spectrum obtained by Fourier transformation of the time series intensity signal.

#### 2.2 The properties of LiNbO_{3} crystal in two transversely electric fields

LiNbO_{3} belongs to trigonal system. In the absence of electric field, the indicatrix of LiNbO_{3} is a rotating ellipsoid with the three-fold axis as its rotation axis. As shown in Fig. 1, Cartesian coordinate system is built along the three-fold axis (z-axis). The x- and the y-axis are the coordinate axes in the crystal. The profile, perpendicular to z-axis, is a circle at z = 0, which means n_{x} = n_{y} = n_{0}. In this case, there is no intrinsic birefringence for a beam of light passing through the crystal along the three-fold axis which is called degeneracy state.

When two transversely electric fields are applied to the crystal, the electric fields can be expressed as

*x*and

*y*are the same directions as they are in the crystal (see Fig. 1),

*E*is the amplitude of the applied electric fields, and

_{m}*ω*is the rotating frequency of the electric fields. The linear electro-optic or Pockels effect is taken into account and according to the symmetry of LiNbO

_{m}_{3}(trigonal system), the properties of the electro-optic tensors are [13]Thus, under the action of the electric fields, in the principle axis system, the perpendicular profile at z = 0 is described as

*n*is the refractive index of the x- and the y-axis in the absence of the electric fields,

_{0}**represents the synthetic electric field with the components expressed in Eq. (1). From Eq. (3), it is clear that the effect of two transversely electric fields is rotating the three-fold axis with an angle**

*E**θ*(Fig. 2) given by

_{3}crystal in the center of two transversely electric fields can be regarded as a rolling wave-plate with phase delay

*Г*. The phase delay between the fast and the slow directions of polarization is [14]where

*d*is the length of the crystal along the three-fold axis,

*E*is the amplitude of the electric fields and

_{m}*λ*is the wavelength of the incident light.

Based on the analysis above, it can be seen that when light passes through LiNbO_{3} along its three-fold axis, under the action of two transversely electric fields, the crystal can be regarded as a rolling wave-plate whose angular frequency is *ω _{m}/2* and phase delay is

*Г*.

#### 2.3 Method to introduce carrier frequency

As shown in Fig. 3, our self-developed frequency shifter consists of a piece of LiNbO_{3} crystal with two transversely electric fields and a stationary quarter-wave plate (QWP). The two electric fields are alternating in sinusoid with phase delay π/2.

Assume the matrix of the incident linearly polarized light is

where*ω*is the frequency of the incident light. Based on the analysis in Section 2.2, the LiNbO

_{3}crystal in the center of two transversely electric fields can be regarded as a rolling wave plate. The effect of this crystal to the incident light can be denoted as Jones matrix

*ω*is the rotating frequency of the electric fields. Hence, when the incident linearly polarized light passes through this crystal along its three-fold axis, the matrix of the emergent light is

_{m}*ω*and the third item is a left-circularly polarized light with frequency shift

_{m}*-ω*(see Fig. 3). Then, the light is incident on the QWP after passing through the crystal. The QWP is placed with an angle 45° between the directions of its optical axis and y-axis. The Jones matrix of this QWP isThen, the light represented by Eq. (8) passes through the QWP, and the emergent light is

_{m}*2ω*as shown in Fig. 3.

_{m}After the light passes through the frequency shifter, a pair of lights with frequency difference *2ω _{m}* is obtained. The first item of Eq. (10) is considered as background light. By setting the voltage of the two electric fields to the crystal’s half-wave voltage, the first item can be zero. However, this is not a necessary procedure, because the carrier frequency is far away from the frequency of the incident light (

*ω>>2ω*). By means of spectral analysis, the first item can be easily removed.

_{m}#### 2.4 System configuration and signal processing procedure

After the incident light passes through the frequency shifter, the emergent light is put into Michelson interferometer as shown in Fig. 4. The light is divided into two parts equally by the beam splitter (BS). One of them is incident into the reference arm, while the other is incident into the measuring arm. Two polarizers are placed in front of each arm to separate the two linearly polarized lights whose polarization directions are perpendicular. The allowed polarization directions of the polarizers are shown in Fig. 4. In front of the detector, a polarizer is placed with allowed polarization direction 45° to make the two reflected beams interfered and the intensity is collected by the detector in time sequence.

The signal from the measuring arm is modulated by the vibration in time domain. In the detection plane, for the general case, the time series intensity function which is collected by the detector can be expressed as

*I*and

_{1}*I*are the intensities of the two linearly polarized lights respectively, and

_{2}*I*is the intensity of the left-circularly polarized light,

_{3}*φ*is the initial phase difference between each two lights according to the subscripts,

*2ω*is the carrier frequency which is generated by the frequency difference between the two perpendicular linearly polarized lights. The two perpendicular linearly polarized lights are reflected from the reference arm and the measuring arm respectively. The last two items in Eq. (11), which are generated by the frequency shifted light and the original frequency light, are considered as the influence of the beam with frequency of

_{m}*ω*. It is clear that the carrier frequency item in Eq. (11) and the other items are in different frequency band, thus, the carrier frequency item can be easily obtained from the frequency spectrum by band-pass filtering. Hence, to make it easy to understand, Eq. (11) is simplified aswhere

*I*is the sum of the intensities which are irrelevant to the carrier frequency,

_{0}*I*is the amplitude of the carrier frequency and

_{c}*Ф*is the initial phase which usually can be ignored.

_{12}When the out-of-plane vibration is introduced by the vibration source, the optical path difference between the reference arm and the measuring arm is changed. As the object vibrating the intensity fluctuation is recorded as a function of time. Let *Δz(t)* denote the instantaneous vibration displacement, then the phase change which is caused by the displacement *Δz(t)* can be expressed as

*λ*is the wavelength of the incident light and the time series intensity function which contains the vibration can be written as

*f*is the carrier frequency whose value is twice the rotating frequency of the electric field (

_{0}*f*). By using of Euler’s formula, Eq. (14) can be expanded as

_{0}= ω_{m}/π***represents the complex conjugate and

*c*can be expressed aswhere

*Ф(t)*is the total phase in the cosine function in Eq. (14) as expressed in Eq. (17)To calculate the phase

*Ф(t)*, Fourier transform is acted on Eq. (15) and the result iswhere capital letters represent the frequency spectrum of each item in Eq. (15) correspondingly. In frequency domain, since the three items in Eq. (18) are in different frequency band, a proper filter, whose center frequency is the carrier frequency, is adopted to screen out

*C*. Simultaneously, the influence of the beam with frequency of

*ω*and any other undesired frequency can be removed together. Then inverse Fourier transform is conducted on the filtered spectrum. Thus, in time domain,

*c(t)*is obtained and the total phase

*Ф(t)*in Eq. (17) can be denoted as [15]

*Ф(t)*which is expressed in Eq. (17). Thus, the truncated phase of Eq. (17) is obtained. Phase unwrapping technique is used to obtain the total phase. Then the carrier frequency (

*2πf*) is removed, the remained part is the phase converted from vibration displacement. Finally, according to Eq. (13) and the incident wavelength, the instantaneous vibration displacement

_{0}t*Δz*can be measured. The signal processing flowchart is shown in Fig. 5.

## 3. Simulations and experiments

#### 3.1 Simulations

A series of simulations have been conducted based on the theory described above. Three types of simple harmonic vibrations are taken into the simulations and the vibration function can be described as following

where S1, S2, and S3 represent the three types of simple harmonic vibrations respectively. The carrier frequency and the sampling frequency are set individually to fit the properties of each vibration. The system simulation parameters are shown in Table 1.According to the parameters in Table 1, the time series intensities are generated and the signal processing is conducted on them. The frequency spectrums and the filtered spectrums are shown in Figs. 6(a), 6(b), and 6(c) and Figs. 6(d), 6(e), and 6(f) respectively. The vibrations which are obtained by the simulation system are shown in Figs. 6(g), 6(h), and 6(i). To figure out the systematic errors, the trajectories which are obtained by the simulation system are subtracted from the vibration function (Eq. (20), Eq. (21), and Eq. (22)). The systematic errors are shown in Figs. 6(j), 6(k), and 6(l).

From Figs. 6(j), 6(k), and 6(l), it can be seen that the results of the simulations are coincident with the vibration functions perfectly. The absolute errors and the relative errors are small enough to be ignored. Hence, our vibration trajectory tracking system is proved theoretically. The systematic errors are mainly caused by two reasons. One is the frequency spectrum filtering. The higher harmonic of the measured signal has a broader distribution in the spectrum. Due to the filter settings, the higher harmonic which is far away from the carrier frequency is removed from the spectrum. The other one is the digital discretization which leads to discrete signal.

#### 3.2 Experiments

To verify the theory and the performance of our self-developed mechanical vibration trajectory tracking system, the preliminary experiments have been performed. The system arrangement is shown in Fig. 7.

The light source is provided by semiconductor laser, with a central wavelength of 0.532 μm, and the maximum output power is 120 mW which is tunable. The frequency shifter consists of a piece of LiNbO_{3} crystal and a QWP. The size of the crystal is 5 mm × 5 mm × 30 mm and the length of the three-fold axis is 30 mm. The sinusoidal alternating electric fields with a frequency of 5,903 Hz are generated between every two opposite 5 mm × 30 mm surfaces. The voltage between one pair of electrodes is 171 Volt. The phase difference between two electric fields is π/2 which is realized by the drive power supply. Thus, the frequency of the carrier is 11,806 Hz according to Eq. (10). The angles of QWP and polarizers are adjusted as the directions shown in Fig. 3 and Fig. 4. A Si-biased detector DET36A (U.S. THORLABS Company) is adopted at a sampling frequency of 50,000 Hz to collect the intensity in time sequence. The sampling time is 20.9715 s.

The measured out-of-plane vibration is imported by the mechanical vibrator. The direction of the vibration is along the wave vector of the incident light. The mechanical vibrator is driven by signal generator. To connect the vibrator and the signal generator, a power converter is added between them. The signal generator output signal with frequency of 0.2 Hz and voltage of 20 Volt. Thus, the mechanical vibrator vibrates at a frequency of 0.2 Hz.

When the mechanical vibrator vibrates under the control of the signal generator, the optical path difference will be changed and the time series intensity can be expressed by Eq. (14). Then the signal processing is acted on the time series intensity to extract the phase represented by Eq. (17). The frequency spectrum is obtained by fast Fourier transform and band-pass filtering is conducted to screen out the spectrum around the carrier frequency. The frequency spectrums are shown in Fig. 8.

During the signal processing, in order to separate the carrier frequency and the phase introduced by the vibration (see Eq. (17)), linear fitting is used to figure out the carrier frequency and then it is removed from Eq. (17). Finally, according to Eq. (13), by replacing the parameter λ = 0.532 μm, the vibration trajectory can be obtained (as shown in Fig. 9).

From Fig. 9, it can be seen clearly that the signal-to-noise ratio of the vibration curve is pretty high. The frequency and the amplitude of the vibration can be seen from the vibration trajectory clearly and directly. The average amplitude of the trajectory is 207.4 μm. The amplitude is compared with the measured values of a grating ruler (Czech ESSA Company, SM 50.61-1.0-CA9) attached to the mechanical vibrator. The measurement accuracy of the grating ruler is 0.1 μm. The measured amplitude from grating ruler is 208.2 μm. Therefore, the measured error is 0.8 μm and the relative error is 0.38%.

## 4. Discussion

By comparing Fig. 8(a) with Figs. 6(a), 6(b), and 6(c), there are significant differences in the two frequency spectrums. No electric frequency signals ω_{m} (whose value is half of the carrier frequency) exist in the simulation spectrum, whereas the electric frequency signals (ω_{m}) are remarkable in the measured spectrum. These differences are generated due to the voltage settings. In the simulation, the voltage is assumed to be the half-wave voltage. In this case, the first item of Eq. (10) vanishes. As a result of the interference between the second (ω + ω_{m}) and the third (ω-ω_{m}) items in Eq. (10), there is only the carrier frequency (2ω_{m}) in the spectrum. In terms of the intensity, the third, the fifth, and the sixth items in Eq. (11) vanish. On the contrast, when the voltage is not the half-wave voltage, the first item of Eq. (10) cannot be ignored. And the electric frequency signals ω_{m} has the influence on the intensity captured by the detector, as described in Eq. (11). As shown in the experimental results, besides the carrier frequency (2ω_{m}), the electric frequency signals (ω_{m}) is appeared due to the existing of the frequency shifted lights (ω + ω_{m} or ω-ω_{m}) and the original frequency light (ω). In our experiment, the high modulation degree of the carrier frequency signal (2ωm) can be obtained due to the amplitude of the frequency shifted light (ω + ω_{m} and ω-ω_{m}) can be controlled to be equal. As a comparison, the frequency signals ω_{m} is not suitable to be used as the carrier frequency, because the amplitudes of the frequency shifted lights (ω + ω_{m} or ω-ω_{m}) and the original frequency light (ω) are not controllable.

In the aspect of the signal processing, Fig. 9 is a direct result of the measurement system based on the theory described above. Neither noise reduction nor the signal subdivision circuit is added in the system. From Fig. 9, the baseline (equilibrium position) of the vibration is found drifting, because the frequency of the crystal drive power supply is not absolutely stable and when the carrier frequency (in Eq. (17) shown as *2πf _{0}t*) is subtracted from the total phase (

*Ф(t)*), the linear fitting may not figure out the drift of the carrier frequency. In the low speed vibration measurement, linear fitting can provide an acceptable result. But in the high speed vibration measurement, the carrier frequency need to be pretty high, so frequency stabilization of the power supply and real-time feedback on the carrier frequency are two proper methods to be taken in our self-developed system and these will be studied in our next work.

## 5. Conclusion

A new vibration trajectory tracking system based on electro-optical heterodyne interferometry has been built. The carrier frequency can be changed in a large dynamic range which is depended on the rotating frequency of the crystal drive electric fields. The system can measure the instantaneous vibration trajectory in a high-efficiency, low-cost, jamproof, robust, precise and simple way, thanks to the electrically controlled electro-optical frequency shifter and the heterodyne interferometry measurement. A vibration curve with high signal-to-noise ratio is obtained by using the temporal intensity analysis method. The relative error of the amplitude is 0.38% at present. The preliminary experimental results show that our self-developed electro-optical heterodyne interferometry system can achieve the trajectory tracking effectively.

## Acknowledgment

The authors acknowledge the financial support from Chinese National Natural Science Foundation 51275033.

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