A novel integrated fiber-optic interferometer model and its application in micro-displacement measurement

Optics and Lasers in Engineering 86 (2016) 125–131

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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng

A novel integrated fiber-optic interferometer model and its application in micro-displacement measurement Chi Wang a,b,c, Long-long Xu a, Jun Zhu b,n, Zhi-wen Yuan b, Ying-jie Yu a, Anand K. Asundi d a

Department of Precision Mechanical Engineering, Shanghai University, Shanghai 200072, China Science and Technology on Near-surface Detection Laboratory, Wuxi 214035, China c State Key Lab of Precision Measuring Technology and Instruments, Tianjin University, Tianjin 300072, China d School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Singapore b

art ic l e i nf o

a b s t r a c t

Article history: Received 7 March 2016 Received in revised form 23 April 2016 Accepted 17 May 2016

We conducted an investigation in a novel integrated fiber-optic interferometer model based on ultrasmall self-focusing optical fiber probe and the method of its application in micro-displacement measurement. Firstly, we proposed the structure model of integrated fiber-optic interferometer and established its input–output mathematical model applied in micro-displacement measurement. Secondly, we established the hardware system of the integrated fiber-optic interferometer. Finally, we analyzed the fitting result of experimental data of micro-displacement measurement and some error factors and defined the linear working range. The experimental results indicate that, under the given experimental conditions, the linear measurement range, linearity and sensitivity of the integrated fiber-optic interferometer were 10 μm, 1.36% and 8.8 mv/μm respectively. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Self-focusing lens Fiber optical probe Interferometer Micro-displacement

1. Introduction In recent years, with the development of science and technology and modern industrial manufacturing, the development of precision measuring instruments gradually tended towards miniaturization, integration and intelligence. Thanks to its advantages such as non-contact measurement, small size, light weight, resistance to electromagnetic interference, high resolution and low cost, interferometers which transmit light based on optical fiber are widely used in the field of precision measurement, and they can be used to measure parameters such as displacement, vibration, speed, strain, pressure and temperature, etc. [1–4]. The organic combination of ultra-small optical probe and fiber-optic interferometer is an important technical way to achieve miniaturization, integration and high-precision of precision measuring instruments. Among them, the self-focusing lens (gradient index lens), due to its self-focusing properties and small size, are widely used in fiber-optic sensing system. For example, Tan et al. proposed the confocal measurement system based on self-focusing lens [5]; Xie et al. developed the fiber-optic Michelson interferometer based on GRIN lens, achieving vibration and displacement measurement successfully [6,7]. However, the different n Corresponding author at: Science and Technology on Near-surface Detection Laboratory, Wuxi 214035, China. E-mail address: [email protected] (J. Zhu).

http://dx.doi.org/10.1016/j.optlaseng.2016.05.012 0143-8166/& 2016 Elsevier Ltd. All rights reserved.

external dimensions of GRIN lens and optical fiber, increase the package size of probe and make the bonding process relatively complicated and the interface transmission quality unstable. As a matter of fact, ultra-small and highly integrated fiber-optic interferometer cannot be really realized. Due to ultra-small structural dimensions and superior focusing performance of the ultra-small self-focusing optical fiber probe (Gradient index fiber probe, GRIN fiber probe), an all-fiber optical probe consisting of single-mode fiber, non-core fiber and self-focusing fiber [8,9], can be integrated with the signal arm of fiberoptic interferometer through the fusing process directly, and this type of probes have been favored by scholars in recent years. For example, in the research area of OCT (optical coherence tomography) systems, Mao et al. studied the production of GRIN optical fiber probe and the method to detect light transmission performance [10,11]; Fang et al. studied the feasibility to use OCT systems based on such probe in measurement of lungs, muscle and other biological tissue [12,13]. As far as we know, the measurement systems based on ultra-small GRIN fiber probe are more concentrated in the field of OCT technology, only limited systematic analysis and theoretical study can be found regarding integrated interferometer based on such probe. In the precision industrial measurement, Schmitt [14,15] studied the Fizeau interferometer based on an ultra-small GRIN optical fiber lens and discussed the feasibility to detect micro-bore in the engine nozzle. However, the optical probe is composed of GRIN fiber and singlemode fiber which are welded together, and the working distance is

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relatively small due to the limited mode field diameter of singlemode fiber. Based on the existing research results of ultra-small GRIN fiber optical probe, this paper develops an integrated fiber-optic interferometer based on the ultra-small GRIN fiber optical probe, establishes its physical model, analyzes its application methods in micro-displacement measurement and builds fiber optical sensing experiment system for micro-displacement measurement. The micro-displacement measurement results show that, under the given experimental conditions, the sensitivity and the linear measurement range of the integrated fiber optical interferometer are 8.8 mv/μm and 10 μm respectively, which demonstrates the characteristics of miniaturization, integration and high precision of the integrated fiber-optic interferometer. In addition, it is of great application potential in the field of detecting microdeep hole and microvibration.

2. Model of integrated fiber-optic interferometer and principle of micro-displacement measurement Fig. 1 shows the physical model diagram of integrated fiber optical interferometer studied in this paper, in which the signal arm of the interferometer is welded onto an ultra-small GRIN fiber optical probe composed of single-mode fiber, non-core fiber and GRIN fiber lens as shown in Fig. 2 [16,17]. In the model of ultrasmall GRIN fiber probe shown in Fig. 2, as a special fiber with uniform index of refraction, the non-core fiber can overcome the problem of small mode field diameter of single-mode fiber through expanding light beams, thus improving the focusing performance of the probe. Due to the flat surface, GRIN fiber lens, which boast the self-focusing performance for the constantly changing index of refraction, can be integrated with other optical element with a flat surface easily by means of fusion welding. This is beneficial to improve the mechanical strength and stability of the probe. Boasting the advantages such as ultra-small size, superior focusing performance and weldability with other components, the ultra-small GRIN fiber probe can be integrated into a traditional fiber-optic interferometer, so as to realize miniaturization and integration of measuring head of the fiber-optic interferometer. According to the model of all-fiber integrated Michelson interferometer based on the ultra-small GRIN fiber probe shown in Fig. 1, its working principle is: the light beam emitted from laser injects into the optical fiber, passes through the fiber-optic isolator, 3 dB coupler and then is divided into two beams, of which one passes through signal arm and then focuses on the target object by the ultra-small GRIN fiber probe, and the other one passes through the reference arm and is collimated on the reference mirror by the fiber-optic collimator. The two beams reflected back by the target

object and the reference mirror respectively pass through the ultra-small GRIN fiber probe and fiber optical collimator again, inject into the signal arm and reference arm respectively. The two reflected beams are combined again at the 3 dB coupler and interfere with each other. The interferometric signal outputted from the output terminal of the 3 dB coupler is received by the photoelectric detector and converted into electrical signal. Finally, the electrical signal is transmitted into the signal collecting and processing unit, and the captured data will be transmitted into the PC to demodulate and analyze the physical quantities to be measured. According to the model of integrated fiber-optic interferometer shown in Fig. 1, the theoretical basis in micro-displacement measurement will be analyzed in the following part. According to Fig. 1, the light emitted from laser source injects into optical fiber, in which the light field of incident light can be expressed as:

E = E0 ei (ωt − k 0 nl)

(1)

where E0 is the amplitude of light waves, ω the frequency of light waves, k0 the wave number when light waves propagate in vacuum, n the refractive index of fiber core, l the optical path through the propagation process of light wave. Suppose I0 is the light intensity injected into the optical fiber, and I is the interferometric light intensity received by the photodetector, the following (Eqs. (2) and 3) can be obtained according to the literature [18]:

I0 = E02

(2)

I = I0 αRf ⎡⎣ ξ 2 + (1 − ξ )2 + 2ξ (1 − ξ ) cos Δφ⎤⎦

(3)

where ξ is the coupling ratio of the coupler (coupling coefficient), α the same optical attenuation coefficient of the sensing arm and reference arm of the interferometer, and Rf the reflectance product of reference mirror and target object. Suppose the refractive index of air is 1, the target object and reference mirror are both placed in the air, the distance from the target object to the output end of the fiber probe is l1, that from the reference mirror to the output end of the fiber-optic collimator is l2, the length of the signal arm is ls, and that of the reference arm is lr, then the phase difference Δφ can be expressed as:

Δφ = 2k 0 nls + 2k 0 l1 − 2k 0 nlr − 2k 0 l2 If the coupling coefficient (3) can be transformed into:

I=

(4)

ξ of the 3 dB coupler is 0.5, then Eq.

I0 αRf (1 + cos Δφ) 2

(5)

Substituting Eq. (4) into Eq. (5), we can get the following form:

I=

I0 αRf {1 + cos [2k 0 (l1 + nls − nlr − l2 )]} 2

Fig. 1. Model of integrated fiber-optic interferometer.

(6)

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Fig. 2. Model of ultra-small GRIN fiber-optic probe.

In the measurement of micro-displacement, when the target object moves at minor displacement Δl, the intensity of output interference light can be expressed as:

I=

⎡ ⎤⎫ I0 αRf ⎧ ⎨ 1 + cos ⎢ 2k 0 (l1 + Δl + nls − nlr − l2 ) ⎥ ⎬ ⎣ ⎦⎭ 2 ⎩

(7)

Setting l1 ¼ l2 þnlr nls, that is to say the optical paths of the signal arm and the reference arm are equal when the target object does not produce minor displacement. Then the intensity of output interference light shown in Eq. (7) can be expressed as:

I=

⎤ I0 αRf ⎡ ⎢ 1 + cos (2k 0 Δl) ⎥ ⎦ 2 ⎣

(8)

Expanding the cosine function in Eq. (8) into a power series, we can get the following form:

I=

I0 αRf I0 αRf + 2 2

∞

∑ n= 0

( − 1)n4nk 02n Δl2n (2n)!

(9)

According to (Eqs. (8) and 9), when the relative parameters of the interferometer and target object are determined, the intensity of interference light is only related to minor displacement Δl, and the change in the intensity of interference light reflects the change of minor displacement. Therefore, the measurement of minor displacement can be converted into the measurement of the intensity of interference light. If the change of minor displacement Δl is caused by the periodic vibration of the surface of target object, the intensity of output light will also show a corresponding periodic change. It is noted that, when n is 0 in Eq. (9), the first item of cosine function expansion in Eq. (8) is 1. That is to say, when the minor displacement Δl is 0 in Eq. (8), it means that n is 0 in Eq. (9).

3. Experiment equipment system According to the mathematical model of (Eqs. (8) and 9), the system model and experiment equipment of the integrated fiberoptic interferometer based on GRIN optical fiber probe in the micro-displacement measurement are shown in Figs. 3 and 4 respectively. The laser source of this system is the low-coherence light source named SLD-1310-18 made by Fiberlabs Inc. from

Japan. Considering the non-visible light used, the red laser pointer with the center wavelength of 650 nm is added in the system as an indicating light source to adjust the output beam angle of the reference arm and the signal arm so as to ensure the output beam is vertical to target object and reference mirror. The piezoceramic vibrator fixed vertically to vibration isolation platform and piezoceramic actuator are made by Beijing Force Technology Limited Company, and they can achieve micro-vibration of tens to hundreds Hertz. The reference arm is formed by a single-mode fiber pigtail welded with a fiber collimator that is fixed to one-dimensional precision linear guide rail for adjusting the optical path difference of two arms. The signal arm is formed by a single-mode fiber pigtail welded with an ultra-small GRIN fiber-optic probe that is fixed to a five-dimensional adjustment stage with thread through the magnetic fixture. The fiber-optic interferometer module named INT-MSI-1300B made by Thorlabs Inc. consists of FC/APC interfaces, 50/50 fusion coupler, WDM coupler, circulator and balanced detector. The data acquisition card is the NI-PXI multi-channel data acquisition system made by National Instruments (NI) Corporation. Firstly, according to (Eqs. (8) and 9), the axial distances from the fiber collimator to the reference mirror and from the GRIN fiber probe to the piezoceramic vibrator are adjusted respectively before the experiment, in order to acquire an optical path difference close to zero between the reference arm and the signal arm. Then, the piezoceramic actuator is started to make the piezoceramic vibrator vibrate, and the signal records acquired through the data acquisition card will be shown in computer. Fig. 5 shows the time-frequency signals acquired by this experiment system when the set frequency of piezoceramic actuator is 200 Hz. In Fig. 5(a), Curve 1 is the time-domain vibration signal acquired by the integrated fiber-optic interferometer, and Curve 2 is the excitation signal of the piezoceramic actuator in time domain; Fig. 5 (b) is a corresponding frequency-domain vibration signal of Fig. 5 (a). According to Fig. 5, the excitation signal is a triangular wave with the frequency of 200 Hz, and the detected vibration signal of the piezoceramic vibrator is quasi triangle wave or sine wave with the frequency of 200 Hz, reflecting a good tracking property and indicating the feasibility of experimental test system. Its errors result from inertia and damping of piezoceramic vibrator and noise etc. When the position of measured piezoceramic vibrator changes within the coherence length of the broadband light

Fig. 3. Schematic diagram for integrated fiber-optic interferometer and micro-displacement measurement equipment.

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Fig. 4. Picture of experimental system.

Fig. 5. The experimental signal.

Table 1 Location-voltage experiment data. Location/um

First/V

Second/V

Third/V

Average/V

Location/um

First/V

Second/V

Third/V

Average/V

 30  25  20  15  10 5 0

0.00529 0.00847 0.01721 0.05330 0.08740 0.12547 0.14005

0.00316 0.01300 0.03744 0.05285 0.08094 0.13852 0.12939

0.00194 0.01133 0.02343 0.06160 0.09193 0.13946 0.14438

0.00347 0.01093 0.02602 0.05592 0.08675 0.13448 0.13794

5 10 15 20 25 30

0.11317 0.06431 0.02918 0.00706 0.00350 0.00230

0.10623 0.06399 0.01995 0.00614 0.00267 0.00225

0.11208 0.06365 0.03176 0.00888 0.00311 0.00206

0.11049 0.06398 0.02697 0.00736 0.00309 0.00220

source, the interference light intensity will vary correspondingly, showing that the root-mean-square value of a time-domain waveform represented by Curve 1 will also vary. The displacement changes of piezoceramic vibrator can be calculated by recording the corresponding changes of output interference intensity with piezoceramic vibrator in different positions.

4. The experiment results and analysis According to the system model and the experimental equipment of integrated fiber-optic interferometer based on GRIN fiber optical probe used in the micro-displacement measurement shown in Figs. 3 and 4, the experiment of micro-displacement measurement is conducted as follows: Based on the SLD light source with the coherence length of 60–70 μm, the length of the reference arm is adjusted to make the optical path difference between the reference optical path and the signal optical path less

than the coherence length of the light source so as to satisfy the interference requirements. Piezoceramic vibrator starts to oscillate with the driving main frequency of piezoceramic actuator set at 200 Hz, and then the five-dimensional adjustment stage with thread is adjusted to produce the microdisplacement movement along the axis. Meanwhile, the voltage value of the interference intensity acquired every movement is recorded by computer when the five-dimensional adjustment stage with thread is adjusted from the starting point when interference occurs. And then the ultra-small GRIN fiber probe is moved in opposite directions in the same way, and the three groups of different displacement-voltage data from the experiments repeated three times are recorded as shown in Table 1. In order to minimize errors existing in the experiment such as the mechanical error of device and the vertical error between probe and target object, etc., which can result in the misalignment of the input–output fitting curve in the course that the input value of micro-displacement changes from small to large (positive stroke) and from large to small (reverse stroke), the data

0.14 0.12 0.1 0.08

Average values Fitting curve

Voltage

Voltage

C. Wang et al. / Optics and Lasers in Engineering 86 (2016) 125–131

0.06 0.04 0.02 -30

-20

-10

0 10 20 Displacement

0.14 0.12 0.1 0.08 0.06 0.04

-30

-20

-10 0 10 Displacement

0.14 0.12 0.1 0.08

20

30

Average values Fitting curve

0.06 0.04 0.02

0.02 0

0

Average values Fitting curve

Average values Fitting curve

0.06 0.04 0.02

30

Voltage

Voltage

0

0.14 0.12 0.1 0.08

129

-30

-20

-10

0 10 20 Displacement

0

30

-30

-20

-10 0 10 Displacement

20

30

Fig. 6. The input–output polynomial curve fitting (a) second order polynomial fitting curve, (b) fourth order polynomial fitting curve, (c) sixth order polynomial fitting curve and (d) eighth order polynomial fitting curve.

Table 2 Coefficient of determination and variance of several kinds of fitting. Second order fitting R-square 0.7173 SSE 0.0088

Fourth order fitting

Sixth order fitting

Eighth order fitting

0.9227 0.0024

0.9537 0.0013

0.9610 0.0012

Table 3 Linearity, sensitivity and coefficient of determination in different measurement range. Range (μm)

Linearity/%

Sensitivity (v/μm)

R-square

9–11 8–12 7–13 6–14 5–15 4–16

0.013 0.210 0.380 0.790 1.360 2.110

0.0094 0.0093 0.0092 0.0090 0.0088 0.0086

1.0000 1.0000 0.9999 0.9997 0.9993 0.9986

collected from the three experiments are processed averagely and the average input–output curves are re-fitted. In the method of curve fitting, taking into account the advantages including small deviation, high fitting accuracy and low linearity of least squares curve fitting, and the even polynomial relation between the output signal and the input signal as known by Eq. (9), the average input–output values of experimental data are fitted by second order, fourth order, sixth order and eighth order polynomial fitting respectively, which only retain even orders in the least square method. In order to do simplified

Fig. 7. The linear fitting in the range from 5 to 15 μm.

computation with the curve fitting toolbox of MATLAB, the original testing data is processed by way of normalization in the MATLAB. Eq. (10) is the normalization processing formula for original displacement measuring data Δl, which are normalized by the mean μ and the standard deviation s.

x=

Δl − μ σ

(10)

Where, m ¼0, s¼ 19.47, x is the value of Δl after normalization. The following equations from (Eqs. (11)–14) acquired with the curve fitting toolbox of MATLAB are the second order, fourth order, sixth order and eighth order polynomial fitting respectively, which

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Fig. 8. Fitting results comparison of three groups of experimental values and average values (a) first group of fitting results, (b) second group of fitting results, (c) third group of fitting results and (d) fitting results of average values.

only retain even orders.

y= −

0.0507x2

+ 0.0983

(11)

y = 0.04116x 4 − 0.1464x2 + 0.1241

(12)

y = − 0.02938x6 + 0.1459x 4 − 0.2368x2 + 0.1351 y=

0.02063x 8

−

0.1246x6

+

0.2789x 4

−

0.2953x2

+ 0.1389

(13) (14)

In the above equations, y and x represent the output amount of voltage signal and the input amount of displacement respectively. From the second, fourth, sixth and eighth orders polynomial fitting curve in Fig. 6 we can see that in the polynomial fitting, the higher the order of polynomial fitting, the closer the experimental value to the fitting curve. The fitting curve begins to approach the actual value comparatively starting from the sixth order fitting curve shown in Fig. 6(c). However, a too high order will make the fitting curve unsmooth, for example in Fig. 6(d), the eighth order polynomial fitting curve begins to become unsmooth at the position about 20 μm. In addition, the coefficient of determination and variance of several kinds of fitting are compared in Table 2. Taking into consideration both Fig. 6 and Table 2, the sixth order polynomial fitting is the best in the cases of fitting curves, its coefficient of determination (R-square) can reach 0.9537, and the sum of squared error (SSE) is 0.0013. According to Fig. 6(c), the fitting curve in the range from 5 to 15 μm is a linear area approximately. In order to determine the linear working range of the micro-displacement measurement more precisely, several points are set to both sides, respectively at the interval of 1 μm, from the position at 10 μm of the average fitting curve in the method, and linear fitting is done for these points to calculate linearity. As shown in Table 3, in the range from

5 to 15 μm, the maximum of linearity is 1.36%, the coefficient of determination can reach 99.93%, which meets the requirements of the micro-displacement measurement, and the sensitivity is 8.8 mv/μm after calculation. Fig. 7 shows the linear fitting in the range from 5 to 15 μm. The linear fitting formula is shown in the following Eq. (15).

y = kΔl + b

(15)

In the equation, k ¼  0.0088, expressing the slop (sensitivity); b¼0.157, expressing the intercept. While in the actual experiment of micro-displacement measurement, the friction and clearance between and unequal energy released and absorbed by mechanical components can lead to hysteresis, which may result in the misalignment of the fitting curve at positive and reverse stroke. Fig. 8 shows the fitting results of experimental data, but on the whole, the trend of the fitting curve in the three groups of experimental data boasts a good consistency.

5. Conclusion Following the trend of development of precision measurement technology toward miniaturization and integration, in this paper, we investigated a kind of integrated fiber-optic interferometer model based on the ultra-small GRIN fiber probe on the basis of existing research results, analyzed the method of its application in the micro-displacement, and verified the feasibility of the measurement method by building an experiment system. Results show that under a given condition, in the case of linearity within 1.36%,

C. Wang et al. / Optics and Lasers in Engineering 86 (2016) 125–131

the linear measurement range is 10 μm and the sensitivity is 8.8 mv/μm, thus verifying the model of integrated fiber-optic interferometer and the method of its application in the micro-displacement measurement. The integrated fiber-optic interferometer based on ultra-small GRIN fiber probe developed in this paper boasts the advantages of ultra-small probe and highly integrated system compared to the traditional fiber-optic interferometer, and combines the characteristics of low coherent fiberoptic interferometer and superior focusing performance of ultrasmall GRIN fiber-optic probe, with a broad prospect of application in the areas of microvibration and endoscopic examination of micro-deep holes. It is noted that, the content of this paper is at the initial stage of the investigation of the GRIN fiber probe based integrated fiber-optic interferometer system. By combining it with other techniques or systems, such as those described in Refs. [19– 28], relevant issues can be explored in different applications in the future.

Acknowledgments The project is supported, in part, by the Natural Science Foundation of Shanghai (Grant no. 16ZR1411700), the Science and Technology on Near-Surface Detection Laboratory (Grant no. TCGZ2015A005), the State Key Laboratory of Precision Measuring Technology and Instruments (Grant no. PIL1402), and the High-end Foreign Experts Program of China (Grant no. GDW20153100099).

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