## Abstract

The simultaneous and independent measurements of in-plane and out-of-plane displacements are significant issues to be solved in research. Here a novel system to realize single-spot two-dimensional (2D) displacement measurement of a noncooperative target is reported. The performance of the system is tested in the displacement measurement of an aluminum target with a rough surface. 2D random movement and 2D movement with different parameters of Lissajous figures are measured by the system. The ranges of the 2D displacement measurement reach 500 μm and the accuracies reach the submicron scale. The resolutions of the two dimensions are all better than 5 nm. The measurement system is based on laser heterodyne self-mixing interferometry with frequency multiplexing, which has advantages such as noncontact, nondestruction, nanometer-scale resolution and high sensitivity. The method is promising to be applied in 2D deformation tests of materials, 2D rotor vibration measurement, 2D positioning of particles, thermal expansion coefficient measurement, and other applications.

© 2017 Optical Society of America

## 1. INTRODUCTION

Precise three-dimensional (3D) measurements of dynamic displacement, target position, material deformation, and rotor vibration play important roles in modern technology [1–3]. Taking the target surface plane as a reference, 3D measurement can be decomposed into two-dimensional (2D) in-plane measurement and one-dimensional (1D) out-of-plane measurement. Due to the symmetry of the 2D in-plane measurement, only 1D in-plane measurement is necessary. Then the other 1D in-plane measurement can be easily obtained by relatively rotating the light beam and the target at 90 deg. Thus, 2D measurement (1D for in-plane and 1D for out-of-plane) are actually required, which is exactly the main aim of this paper.

The optical interferometer is widely used for the out-of-plane displacement measurement because of the advantages such as noncontact, high resolution, and wide dynamic measurement range [4]. Although the 2D displacement measurement can be actualized using a pair of laser interferometers mounted perpendicular to each other or one interferometer beam split into two components, the two beams do not irradiate on the same spot and any mechanical or optical adjustment error will cause a measurement error [5]. Moreover, target mirrors or retroreflectors are needed for the laser interferometer, which limits the application occasions.

Therefore, the simultaneous and independent measurements of in-plane and out-of-plane displacements are significant issues to be solved in research. The most common methods developed are grating interferometry [6], speckle pattern interferometry [7], and laser Doppler distance sensing [8]. Grating interferometry is a valuable tool for displacement measurement. The measurement resolution is a few nanometers and the accuracy can be down to the submicron scale. However, the 2D grating needs to be etched as the measurement target in this method, and it is difficult to be applied in the noncontact measurement field. Speckle pattern interferometry is widely used in industrial nondestructive detection because of the advantages such as noncontact and full-field measurement. This method is based on the speckle-gram obtained and image processing algorithm. The measurement range and accuracy are related to the speckle size and come to a compromise in the method. The laser Doppler distance sensor can realize precise and dynamic position measurements of fast moving objects. The resolution is only on the submicron order and the uncertainty is on the micrometer order.

Recently, laser feedback (self-mixing) interferometry has been attracting much attention because of its compactness, auto alignment, and high sensitivity [9,10]. In this scheme, the laser beam interacts with the target to be measured, and then returns back to the resonator, which modulates the laser power, frequency, or polarization. From the modulation, the information of the target can be precisely recovered. Specifically, when the laser beam is frequency shifted before it returns back into the laser resonator, this feedback system presents ultra-high sensitivity and good performance in the detection of weak scattering light [11,12]. Thus, laser feedback interferometry is versatile for displacement measurement [13], velocimetry [14], vibrometry [15], profilometry [16], and microscopy [17]. However, it was hardly applied in the in-plane displacement measurement. In this paper, we present a single-spot 2D displacement measurement system based on the frequency-shifting and multiplexing feedback effect, with the advantages of nondestruction, noncontact, and high sensitivity.

## 2. SCHEMATIC DIAGRAM AND THEORETICAL ANALYSIS

The system schematic diagram is shown in Fig. 1. The microchip laser works in single-longitudinal mode with a wavelength of 1064 nm. Its output laser is divided into two beams by the beam splitter ${\mathrm{BS}}_{1}$. The reflected one is used as the detecting beam and transferred into an electrical signal by the photodetector (PD). The transmitted one is collimated via the lens L, and divided into three beams. The first measuring beam marked in blue passes through ${\mathrm{BS}}_{2}$, reflector ${R}_{2}$, ${\mathrm{BS}}_{3}$, and acousto-optic modulator ${\mathrm{AOM}}_{1}$ and gets a frequency shift of ${\omega}_{1}=70.9\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{MHz}$; the second measuring beam, marked in red, passes through ${\mathrm{BS}}_{2}$, ${R}_{2}$, ${\mathrm{BS}}_{3}$, ${R}_{1}$, and ${\mathrm{AOM}}_{2}$ and gets a frequency shift of ${\omega}_{2}=71.6\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{MHz}$. The two measuring beams are incident on the same spot on the target by adjusting the reflectors ${R}_{3}$ and ${R}_{4}$. The angles of the beams do not need to be adjusted strictly to certain values in the experiment, but they need to be calibrated, which will be introduced in the following section. The sizes of the beams are about 2 mm; thus, the diameter of the intersection spot is limited to millimeters. This should be considered during the experiment design because the out-of-plane measurement range is limited to millimeters as well. The target tested in the system is made of aluminum with a rough surface. The beam scattered by the target marked in purple passes through ${\mathrm{AOM}}_{3}$ and gets a frequency shift of ${\omega}_{3}=70\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{MHz}$. Thus, the beams returning back to the laser resonator get a frequency shift ${\mathrm{\Omega}}_{1}=|{\omega}_{1}-{\omega}_{3}|=0.9\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{MHz}$ and ${\mathrm{\Omega}}_{2}=|{\omega}_{2}-{\omega}_{3}|=1.6\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{MHz}$. The symmetrical structure designed can reduce the environmental influence for the in-plane direction, such as the air disturbance and the thermal effect of AOMs.

The detected signal of the PD contains two carrier waves at the frequencies of ${\mathrm{\Omega}}_{1}$ and ${\mathrm{\Omega}}_{2}$. By sending it to the designed bandpass filters, two signals will be separated, and then the separated signals are sent to the lock-in amplifier (LIA) as the measurement channels. The reference signals are generated by the driving signal of the AOMs with stable ${\mathrm{\Omega}}_{1}$-frequency and ${\mathrm{\Omega}}_{2}$-frequency sinusoidal signals. Finally, the phase variations of the two measuring beams are demodulated by the LIA. The in-plane and out-of-plane displacements are measured simultaneously and independently according to Eq. (7) as follows.

When the measuring beams return to the laser cavity, the laser power is modulated by the two beams at ${\mathrm{\Omega}}_{1}$ and ${\mathrm{\Omega}}_{2}$, respectively, as [11,12]

For a microchip laser, the amplification coefficient $G$ can be up to ${10}^{6}$, which means even the feedback strength $\kappa $ is as low as ${10}^{-6}$; a 100% modulation can be obtained. Thus, the frequency-shift laser feedback interferometry has a high sensitivity and is a great advantage for detecting the weak scattering light signal.

From Eq. (1), it is obvious that the method of the laser frequency shift feedback is the heterodyne detection. The information of the external cavity length $({\phi}_{1},{\phi}_{2})$ is embedded in the two carrier waves, which can be modulated by the LIA. For the 1D displacement measurement, the optical phase change is similar to the traditional interferometry as $\mathrm{\Delta}L=(c/2n\omega )\mathrm{\Delta}P$ [11]. In the 2D displacement measurement system, the relationship between the phases and displacements needs to be analyzed first. The specific light-path of the system can be simplified as follows.

The directions of the in-plane and out-of-plane displacement are defined by beam ${B}_{3}$, the direction of which is parallel to the out-of-plane direction and perpendicular to the in-plane direction, as shown in Fig. 2. When the target moves with the velocity $v$, the measuring beams ${B}_{1}$ and ${B}_{2}$ are scattered by the surface of the target. According to the Doppler frequency shift formula, the frequencies of the detected measuring beams are varied as

Generally, the velocity $v$ of the target is much lower than the velocity of light, therefore the frequency changes of the two beams can be expressed as

The change in the frequencies is essentially the change in the phases. By computing the integral of the left side of Eq. (3), we can obtain

For the right side of Eq. (3), the integral of velocity $v$ is the displacement of target $S$. Thus, the additional phases of the carrier waves are related to the displacement of target $S$ according to Eqs. (3),(4) as

The in-plane/out-of-plane displacement is related to the target displacement $S$ by the geometric relationship

By linking the phase change with the external cavity length change, the 2D displacement can be deduced from Eqs. (5),(6) as

In Eq. (7), the wavelength $\lambda $ can be considered as a constant. ${\phi}_{1}$ and ${\phi}_{2}$ are the measured values in the experiment demodulated from the LIA. Therefore, the 2D displacement can be calculated only if the parameters $\theta $ and ${\theta}^{\prime}$ are foreknown in the system.

## 3. EXPERIMENT AND RESULT

#### A. Calibration of the Parameters

From Eq. (7), the parameters $\theta $ and ${\theta}^{\prime}$ need to be measured in the system. However, it is difficult to measure the actual angle directly because the beams are virtual and invisible. Thus, we assume that the displacement of the translation stage from Physik Instrumente (PI) is foreknown, and that the parameters $\theta $ and ${\theta}^{\prime}$ can be derived reversely according to Eq. (7). To simplify the calibration process, the sum of the angles $\theta $ and ${\theta}_{1}$ is set to 90 deg; therefore, the out-of-plane displacement is equal to zero. Eq. (7) can be simplified as

Specifically, the $x$ axis is the displacement of the PI platform, and the $y$ axis is the measurement result related to the phase as $d1=\lambda {\phi}_{1}/(2\pi )$ and $d2=\lambda {\phi}_{2}/(2\pi )$ in Fig. 3. The slopes calculated are $\mathrm{sin}\text{\hspace{0.17em}}\theta =0.22606$ and $-\mathrm{sin}\text{\hspace{0.17em}}{\theta}^{\prime}=-0.23408$. Their standard errors are on the order of ${10}^{-5}$.

#### B. Experimental Analysis

After the calibration of the parameters $\theta $ and ${\theta}^{\prime}$, the 2D displacement can be calculated according to Eq. (7). To illustrate the transcendental ability of the system in the 2D displacement measurement field, a 2D motion control system is designed and the movement in the shape of the traditional Lissajous figures is implemented in the experiment.

The 2D translation stages are set to move in the same amplitude, the same frequency, and at a 90 deg phase difference first in the experiment as follows:

The in-plane and out-of-plane displacement can be measured in the system simultaneously and independently, and the resultant movement can be obtained, as shown in Fig. 4.

The experimental results are obviously different from the simulated results shown in Fig. 4. The measured circle seems to be a little out of shape compared with the simulation. To explain the measurement error in the experiment, the mechanical system and the simulation process are analyzed below.

A Lissajous figure is produced by taking two sine waves and displaying them at right angles to each other. It can be easily realized on an oscilloscope in the XY mode, but it is difficult in the real case, i.e., 2D mechanical movement, because of the precision of the mechanical installation in the system. As Fig. 5 shows, the two translation stages are not strictly perpendicular to each other. When onlyStage_1 moves, ${S}_{\text{out}}$ is equal to zero; but when only Stage_2 moves, ${S}_{\text{in}}$ is not equal to zero and there is a component weight in the ${S}_{\text{in}}$ direction. The positioning accuracy of Stage_1 and Stage_2 is 50 nm and 5 nm, respectively; the travel distance of Stage_1 and Stage_2 is 150 mm and 500 um, respectively.

The deviation between the experiment and simulation can be explained. It is not the error of the system but the wrong simulation of the movement that causes the deviation in Fig. 4. The angle between the two translation stages in the system can be calculated when only Stage_2 is set to move:

The simulation can be modified considering the angle error of the mechanical installation,

The results of the experiment and modified simulation are plotted, as shown in Fig. 6. Compared with Fig. 4, the measured circle is now in good agreement with the simulation, proving that the theoretical calculation above is valid.

To estimate the accuracy of the measurement, the radial error is defined as the deviation between the polar radius of the simulation and that of the experiment in the polar coordinate system.

The relationship between the orthogonal coordinates and the polar coordinate system is

The simulation of the Eq. (11) can be expressed in the polar coordinate system as

The deviation of the polar radius with the change of the polar angle is shown in Fig. 7.

The standard deviation can be calculated as

#### C. Experimental Results of Various 2D Displacements

Then the movements in the shape of various Lissajous figures are measured in the system. Considering the resolution of the translation stage and the limitation of the travel distance, the amplitude is set to 5 μm and 240 μm, respectively. The results are shown in Fig. 8.

All the results above verify that the system has a commendable performance in the 2D displacement measurement field. The range of the displacement measurement in the experiment can be as large as 480 μm. Because of the limitation of the translation stage travel distance, the larger displacement is not tested in the system.

Finally, irregular random 2D motions are applied to the system. The two translation stages are controlled by the program on a computer, generating a series of random points step by step as the motion point. The random displacement range of the two translation stages is from 0 to 80 μm. The results are shown in Fig. 9, proving that the system has the potential to be used in a much broader application field such as the deformation of materials, thermal expansion, and so on.

## 4. DISCUSSION ON THE EXPERIMENTAL SYSTEM

#### A. Resolution of the System

Theoretically, the resolution of the 2D displacement measurement is limited by the precision of the phase discrimination. However, the actual resolution is usually smaller than the theoretical result because of the environmental influence.

Here, the step test experiment is designed in a general laboratory environment to evaluate the resolution of the system. The PI platform named Stage_3, with the positioning accuracy of 1 nm and the travel distance of 100 um, is set to move 5 nm back and forth, respectively, for the two dimensions. The results are shown in Fig. 10, which illustrate that the resolutions of the two dimensions can all reach 5 nm.

#### B. Measurement Range of the System

In part 3 of the experiment, the travel distance is limited to 500 μm because Stage_2 utilized. Here, the measurement ranges of the in-plane and out-of-plane are analyzed, respectively, using Stage_1.

For the in-plane displacement measurement, there is no limit to the in-plane motion, in theory. However, the surface property of the target influences the measurement range. The measuring signals in the system are generated by the scattering of the target. The random structure or surface smoothness of the target can all lead to a different degree of scattering, thus affecting the signal-to-noise ratio (SNR) in the experiment. To ensure the accuracy of the in-plane displacement measurement, a sufficient SNR should be maintained in the whole measuring process, in case of signal loss. In addition to the aluminum, paper, carbon fiber, and iron are also tested in the system. The experiment results show that the system can measure different kinds of targets and the range of in-plane displacement is several millimeters. The relationship between the measurement range and the roughness of the target surface will be researched in future work.

For the out-of-plane displacement measurement, the surface property of the material has little influence because the beams are incident on the same spot of the material. In other words, the size of the intersection spot limits the measurement range, which is dependent on the experimental adjustment. If the out-of-plane displacement deviates large enough from the intersection zone, the single-spot measurement cannot be ensured. Therefore, the method is appropriate for the micro-displacement measurement, and the out-of-plane range usually reaches the millimeter order.

#### C. Limitation on the Targets to be Measured

The previous discussions in Section 4.B are all based on the properties of the target tested. In fact, there is a limitation on the property of the target surface. For example, when the surface roughness is on the order of the laser wavelength, the displacement measurement may not be right because the phase jump can be over $2\pi $ during the motion. Generally, the surface roughness should be smaller than the laser wavelength.

On the other hand, the surface roughness should be smaller than the spot size; otherwise, the micro structure can influence the 2D displacement motion measurement. In the system, the sizes of the measuring beams are about 2 mm. As the phase measured is the average value of the whole spot incident on the target, and the diameter of the spot is much larger than the roughness, it can prevent the measurement error caused by the micro structure. This is the reason why we do not implement convergent beams but the collimated beam in the system.

#### D. Limitation of the Target Velocity

The method is based on heterodyne detection, and the signal processing imposes a limitation on the target velocity to be measured.

From Eq. (3), the Doppler shifts need to be detected by the LIA. For the in-plane velocity, the Doppler shifts of the two measuring beams can be deduced as

For the out-of-plane velocity, the Doppler shifts of the two measuring beams can be deduced as

The wavelength is 1064 nm and the $\mathrm{sin}\text{\hspace{0.17em}}\theta $ and $\mathrm{sin}\text{\hspace{0.17em}}{\theta}^{\prime}$ are known according to the experiment. In other words, the bandwidth of the signal processing determines the maximum measurable velocity.

Considering only the measurement speed, the bandwidth can be set large enough. However, the measurement accuracy is decreased with the increase of the bandwidth. There is a trade-off between measurement speed and measurement accuracy. In order to measure fast events, it is necessary to increase the bandwidth, allowing more noise in the measurement result. The opposite is to decrease the bandwidth, which increases the SNR, but limits the capability to detect the changes in the signal of interest.

In the experiment, the SNR is generally maintained above 10 dB. Then the bandwidth can be set to 1 kHz to ensure the measurement accuracy and decrease the influence of noise according to the simulation and experiment. Therefore, the maximum velocities of the two dimensions are

In summary, the target’s measurable velocities are related to the parameters set in the LIA. Generally, the maximum out-of-plane velocity is on the order of $1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}/\mathrm{s}$ and the maximum in-plane velocity is about nine times that of the former.

#### E. Summary of the System Performance

The performance of the translation stage determines the motion of the target, so the PI platform with a high positioning accuracy is utilized in the experiment. Therefore, the travel distance of the system is limited to 500 μm as well. Taking into account all the factors above, the system reaches a resolution of 5 nm, a measurement range of 500 μm, an accuracy better than 0.1 μm, and a maximum velocity on the order of $1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}/\mathrm{s}$ in the 2D displacement measurement experiment.

## 5. CONCLUSION

In conclusion, we propose a new method for single-spot two-dimensional displacement measurement of the noncooperative target using laser heterodyne self-mixing interferometry with frequency shifting and multiplexing. The method realizes noncontact in-plane and out-of-plane displacement measurement with high sensitivity simultaneously and independently. Various movements in the track of Lissajous figures and random motion are measured in the experiments. The results prove that the resolutions of the two dimensions are all better than 5 nm and the standard deviation can be better than 0.1 μm. If the environmental disturbance is considered and compensated, the accuracy can be further improved. In this work, the method can be applied to 2D deformation of materials, 2D thermal expansion measurement fields, etc. In our future work, we would like to apply this method to 2D particle measurement, 3D shape measurement, and so on.

## Funding

Beijing Municipal Science and Technology Commission (Z151100002415027); National Natural Science Foundation of China (NSFC) (61475082).

## Acknowledgment

We thank Dr. Jiyang Li for his help in the paper writing and his advice in the figure drawing.

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