## Abstract

A stabilized interferometric displacement measurement system, which is suitable for on-line measurement and is endowed with large measurement range and high resolution, is proposed. The system is stabilized by a feedback loop which compensates the influences induced by the environmental disturbances and makes the system stabile enough for on-line measurement. Two different wavelengths are working simultaneously in the system. The measurement range which is determined by the synthetic-wavelength interferometric signal is expanded to the order of millimeter, while the measurement resolution which is determined by one of the single-wavelength interferometric signal is the order of sub-nanometer.

© 2010 OSA

## 1. Introduction

Optical fiber interferometers have been widely used in metrology due to their prominent advantages such as non-contact, compactness, light weight, immunity to electromagnetic interference, high resolution, low cost. Hence there is a great amount of interest to exploit optical fiber interferometers for the measurement of a large variety of parameters such as displacement, vibration, velocity, strain, pressure and temperature [1–8].

For a fiber interferometric displacement measurement system, the fiber should be used just for transmitting the light and the phase change in the interferometric signal should be just induced by the measured displacement. However, the length of the fiber configuring the interferometric arms will change randomly because of temperature fluctuations and other types of the environmental disturbances, thus will induce low frequency random phase drifts in the interferometric signal, which will decrease the measurement precision, and in the worst circumstances, will even make the measurement system unfunctioning. Although techniques such as common-path interferometric arms [1] and feedback compensating methods [2,3] are able to compensate the random phase drifts in interferometric signals, however, because of the phase ambiguity in interferometric signal, the measurement range of [1] which is limited to a half-wavelength is only several hundreds of nanometers. And this limitation makes the systems unable to be used in many fields.

We propose a highly stabilized fiber Michelson interferometric displacement measurement system which is endowed with large measurement range and high resolution. A feedback loop compensates the random phase drifts in the interferometric signal which is induced by the environmental disturbances and stabilizes the measurement system for on-line measurement. The light from two temperature-stabilized DFB laser sources with different wavelengths are coupled simultaneously into the interferometric measurement system. The two wavelengths are guided into proper optical routes by employing a fiber Bragg grating(FBG)as an in-fiber reflective mirror. A photo detector detects the single-wavelength interferometric signal while another photo detector detects the synthetic-wavelength interferometric signal at the same time. By combining and processing the two signals properly, the measurement range of the system which is determined by the synthetic-wavelength interferometric signal can be expanded to the order of millimeter, while the resolution which is determined by the single-wavelength interferometric signal can be the order of sub-nanometer.

## 2. The principle of the system

#### 2.1 The proposed fiber interferometer measurement system

The proposed fiber interferometer measurement system is shown in Fig. 1 . Two temperature-stabilized DFB laser sources with outputs at ${\lambda}_{1}=1558.52$nm and ${\lambda}_{2}=1557.32$nm are used in the system. The lasers give an output power about 2.5mW with 20dB spectral bandwidth 0.2nm. The light emitted from the two lasers passes through 3dB-coupler 1, circulator 1, 3dB-coupler 2 and is divided into two beams. The two beams are collimated by two grin lenses respectively and are reflected back into the system by measurement mirror and reference mirror. The two reflected beams, which are configurated with two different wavelengths, are combinated again at 3dB-coupler 2 and interfere with each other and then the synthetic-wavelength interferometric signal is produced. The synthetic-wavelength interferometric signal from one port of 3dB-coupler 2 is guided by circulator 1 and is detected by photo detector 1 (PD1). The interferometric signal from the other port of 3dB-coupler 2 is guided by circulator 2 and reaches a fiber Bragg grating (FBG). As the Bragg wavelength of FBG is chosen to be $1557.32$nm with 3dB bandwidth 0.3nm, the light with the wavelength of $1557.32$nm will reflected by the FBG while the light with the wavelength of $1558.52$nm will transmit FBG and is guided out of the system. The light reflected by FBG, which is a single-wavelength interferometric signal with the wavelength of $1557.32$nm, is guided by circulator 2 again and is detected by photo detector 2 (PD2). The signals detected by PD1 and PD2 are processed by the electronic processor simultaneously, and the signal detected by PD2 is input into an electronic feedback loop at the same time. An information derived from this electronic feedback loop is used as a correction signal and is applied on a stacked piezoelectric transducer (PZT) on which the reference mirror is mounted. The feedback signal drives PZT to tune the optical path of the reference arm in order to keep the interferometer in quadrature state (the phase difference is $\pi /2$). The random phase drift with low frequency in the fiber interferometer, which is induced by the environmental disturbances, is effectively eliminated and therefore the fiber interferometer measurement system is stabilized. By the means of this, the environmental disturbances will not influence the measurement system anymore.

When the measured displacement varies, the signals detected by PD1 and PD2 will vary correspondently. The period of the signal detected by PD2 is half the wavelength ${\lambda}_{2}$, while the period of the signal detected by PD1 is half the synthetic wavelength of ${\lambda}_{1}$ and ${\lambda}_{2}$, that is${\lambda}_{1}{\lambda}_{2}/2({\lambda}_{2}-{\lambda}_{1})$, which is much bigger than ${\lambda}_{1}/2$ and ${\lambda}_{2}/2$. By employing these characteristics, the present system uses the signal from PD1 to determine the measurement range and the signal from PD2 to determine the measurement resolution. So the system is capable of maintaining the resolution as high as the order of sub-nanometer while expanding the measurement range to the order of millimeter.

#### 2.2 The principle of the electronic feedback loop

During the experiments, the optical power emitted from the two DFB lasers has been adjusted to be the same, both are 2.5mW. The signals detected by PD1 and PD2 can be expressed as Eq. (1) and Eq. (2) respectively.

*k*is the interferometric fringe visibility, ${\varphi}_{d1}$ and ${\varphi}_{d2}$ are the static differential phase between the two arms for the two wavelengths respectively, and ${\varphi}_{s1}$ and ${\varphi}_{s2}$are the differential phase induced by environmental disturbances for the two wavelengths respectively.

After processed by current-to-voltage converters respectively, the signals shown in Eq. (1) and Eq. (2) have the forms that are shown in Eq. (3) and Eq. (4).

The interferometric signal from PD2 is also used to eliminate the random phase drifts with an electronic feedback loop that is resulted from the environmental disturbances. The schematic diagram of the circuit of electronic feedback loop is shown in Fig. 2 . PD2 is connected to current-to-voltage converter U1. After passing through the electronic differentiator U2, the DC part in ${u}_{PD2}^{(1)}$ will be erased, and the output from U2 will be shown as Eq. (5).

where*K*is the conversion gain of the interferometric signal from U2. After passing through the electronic integrator U3, the output from U3 is the value shown in Eq. (6).where ${K}_{1}$ is the conversion gain of U3. In quadrature position, there will be ${\varphi}_{d2}+{\varphi}_{s2}=\frac{\pi}{2}$, it obtains ${u}_{PD2}^{(3)}=0$ and the feedback loop voltage applied on PZT is zero. In the vicinity of quadrature position, Eq. (6) can be rewritten as Eq. (7).where ${K}_{2}$ is the conversion gain of the interferometric signal from U3 in the vicinity of quadrature position, and$\Delta \varphi =({\varphi}_{d2}+{\varphi}_{s2})-\frac{\pi}{2}$. ${u}_{PD2}^{(4)}$ is linear to the value the interferometer excursion to quadrature state and is used as a correction signal applied on PZT. As soon as the interferometer drifts away from the quadratrue state because of the influences from the environmental disturbances, the correction signal will drive PZT to tune the optical path in the reference arm and the interferometer will be drawn back to the quadratrue state again. The interferometer is kept at the quadrature state and the measurement system is stabilized. The feedback loop is a first order system with a frequency bandwidth ranging from 0 to 21.65Hz which is verified by the experiments sufficient to eliminate the environmental disturbances.

In the qudrature state, the signals from PD1 and PD2 have the following forms that are shown in Eq. (8) and Eq. (9) respectively.

It can be known from Eq. (8) and Eq. (9) that both the signals detected by PD1 and PD2 are stabilized at constant values when the interferometer is locked in quadrature state by the signal from PD2.

## 3. The experimental results

#### 3.1 The validation of the feedback control loop

With the feedback loop out of operation and with measurement mirror and reference mirror in static state, the interferometric signals detected by PD1 and PD2 are shown in Fig. 3 . Curve 1 is the signal detected by PD1 which is synthetic-wavelength interferometric signal, while curve 2 is the signal detected by PD2 which is single-wavelength interferometric signal. It can be seen that the interferometric signals are fluctuating at all the time even though measurement mirror and reference mirror are not moving. The variation in the interferometric signals are resulting from temperature fluctuations and other types of the environmental disturbances. However, as soon as the feedback loop is turned on, the interferometric signals detected by PD1 and PD2 are stabilized at constant values, just as shown in Fig. 4 . In Fig. 4, the feedback loop is out of operation during the first part of the recorded time, it can be seen that the interferometric signals detected by PD1 and PD2 are fluctuating during this period. And during the second part of the recorded time in Fig. 4, the feedback loop is turned on and the signals detected by PD1 and PD2 are stabilized respectively at constant values promptly. These results are confirmed precisely to the theoretical analysis that are shown in Eq. (8) and Eq. (9). We had experimented continuously to turn on the feedback loop for more than 4 hours, it was shown that the interferometer had been stabilized robustly during the experiments.

#### 3.2 The measurement results

When there is displacement $\Delta L$ in the position of measurement mirror, the signals from PD1 and PD2 are shown in Eq. (10) and Eq. (11).

In order to realize the displacement measurement, the interferometric signals have been modulated linearly. A ramp voltage with low frequency is applied on an one-dimension translation stage M (PI product, P-611.1S) to tune linearly the optical path difference in the measurement interferometer. The displacement of M is amplified 8 times linearly by a level made of aluminum alloy to meet the needed scanning distance. The interferometric signals detected by PD1 and PD2 will vary correspondently, which have the forms that are shown in Eq. (12) and Eq. (13) respectively.

where c is the scanning velocity of M and t is the scanning time.

Equation (12) can be rewritten as Eq. (14).

It is known from Eq. (14) that the amplitude ${u}_{0}k$ is modulated by the factor $\mathrm{cos}\frac{1}{2}\left[\frac{{\lambda}_{2}-{\lambda}_{1}}{{\lambda}_{1}}\cdot \frac{\pi}{2}+\frac{{\lambda}_{2}-{\lambda}_{1}}{{\lambda}_{1}{\lambda}_{2}}\cdot 4\pi \left(\Delta L-ct\right)\right]$. The amplitude will be the maximum when,

where m is an integral.

The scanning distance that the peak of the signal from PD1 occurring is shown in Eq. (16).

It can be seen from Eq. (17) that peak position is shifting linearly with the value of $\Delta L$ rigorously.

The variation in the signals from PD1 and PD2 during a scanning period of M is shown in Fig. 5 . Curve 1 is the synthetic-wavelength interferometric signal detected by PD1 while curve 2 is the single-wavelength interferometric signal detected by PD2. As the frequencies of the modulated interferometric signals from PD1 and PD2 are much higher than 21.65Hz, the compensating action of the feedback loop will not influence the interferometric signals. The amplitude of the ramp voltage is tuned to correspond to one period change in the synthetic-wavelength interferometric signal. When the displacement of the measurement mirror changes, the peak of curve 1 will shift linearly. And it is known from Eq. (13) that the phase change of curve 2 between the beginning of scanning action and the position of the peak on curve 1 is linear precisely with the displacement of the measurement mirror. In order to measure the phase change of curve 2 and then calculate the displacement of the measurement mirror, curve 1 and curve 2 are converted simultaneously into digital signals by an A/D converter and the data is processed by a software in a PC.

The position of the peak on curve 1 can be determined easily by the software, and the correspondent phase change of curve 2 between the beginning of the scanning action and the position of the peak on curve 1 can be also measured by software. The software calculates the amplitude of the signle-wavelength interferometric signal first, and decides sequentially each data’s phase based on its value and finally calculates the total phase change during the addressed period. The displacement can therefore be calculated. Figure 6
is the zoom of Fig. 5 in the area of the top of synthetic-wavelength interferometric signal. The measurement range of the system is determined by the synthetic-wavelength interferometric signal which is a half synthetic-wavelength. Determined by the specific wavelengths used in the system, the measurement range of the system is calculated to be about 1mm. The used A/D converter is 16 bits. The amplitude of the input voltage of A/D converter is 10V and so its voltage resolution is 10V/2^{16} = 0.15 mV. The amplitude of the single-wavelength interferometric signal is 2.48V during the experiments. The corresponding resolution displacement of A/D converter could be calculated to be $\frac{180\mathrm{deg}}{2480mV}\times \frac{0.15mV}{360\mathrm{deg}}\times \frac{1557.32nm}{2}=0.02$nm. The scanning translation stage M is a nanopositioner which is made by PI company, of which the normal displacement resolution is 0.2nm. The resolution of the system is decided by the resolution of A/D converter, the resolution of the scanning translation stage and the noise in the electronic circuit. In order to determine the real resolution of the system, the single-wavelength interferometric signal from PD2 has been Fourier transformed with Fast Fourier Transform (FFT). The amplitude of the signal from PD2 is more than 53dB times bigger than the noise, as shown in Fig. 7
. Because the amplitude of the signal from PD2 is 2.48V, the maximum amplitude of the noise is calculated to be 5.55mV, corresponding to 0.87nm variation in displacement. So the resolution of the system is experimentedly to be 0.87nm.

During the experiments, another one-dimension translation stage is used to move the measurement mirror. The experimental displacement step is 10$\mu m$. The experimental results over 200$\mu m$ range is given in Fig. 8 , in which it can be seen that the points are almost on a straight line with very little scatter. After linear fitting of the experiment data by the least-squares method, it is obtained that the corresponding linear regression coefficient is 0.9998, which implies that the system has good linearity.

## 4. Conclusion

The highly stabilized fiber Michelson interferometer system with large range and high resolution is presented. An electronic feedback loop compensates the influences from the environmental disturbances, which stabilizes the interferometer and endows the measurement system with high stability for on-line measurement. Two wavelengths from two DFB lasers are working simultaneously in the measurement system. The measurement range of the system, which is determined by the synthetic-wavelength interferometric signal, is the order of millimeter. And the resolution, which is determined by the single-wavelength interferometric signal, can be maintained as the order of sub-nanometer. The linear regression coefficient of the displacement measurement results is 0.9998.

## Acknowledgments

The authors are grateful that this research is supported by National Natural Science Foundation of China (50975022) and the Undergraduate Innovating Project at Beijing Jiaotong University, China.

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