1. Introduction
Owing to provision of direct traceability for the definition of the meter, laser interferometers have become one of the most major apparatus in the precision mechanical engineering industry while further measuring characteristics, including high resolution, contactless measurement, and capability for various measuring ranges, can be met simultaneously [
1,
2,
3,
4,
5,
6,
7,
8]. In this section, there are several issues to discuss, including optical alignment before measurement procedure, optical design, and the installation of laser interferometers.
The alignment between optical axes of the laser interferometer and the movement axis of the linear stage is a critical procedure. The purpose is to be capable of performing linear displacement measurement over the whole measuring range. If misalignment or poor alignment quality occurs, a cosine error will be induced and then the measurement accuracy will also be reduced. For this reason, the essential point of optical alignment consists of the operational mode and the available sensing range being performed in fine alignment. For the common commercial laser interferometer system e.g., XL-80 laser measurement system, Keysight Technology 5530 Dynamic Calibrator and API XD LASER [
9,
10,
11], one alignment target is utilized to implement optical alignment by a manual operation. The laser spot incident on this target should be located at the same position either at the start-point or at the endpoint of the whole measuring range. However, the alignment accuracy is limited by the perceived resolution of the human eye. In the newer high-cost interferometer product (Renishaw XM- 60 multi-axis calibrator), a visual interface is introduced to implement fine alignment [
12]. Although the operating convenience can be improved, the available sensing range is only within ± 0.25 mm, which means that quite accurate coarse alignment is required before carrying out fine alignment. Furthermore, the entire process is manually operated and the accuracy and the efficiency of this alignment still depends on operational experience. Then in the correlated study [
13,
14,
15], detectors e.g., a position-sensitive detector (PSD) and four-quadrant photodiode detectors placed in the linear stage, are employed as an alignment reference for optical axes. Nevertheless, one detector contains multi-signal and power cables and the corresponding cable should be connected with the power supply, the data acquisition board, or the signal processing module over the whole scale. Therefore, the detector and the corresponding cables will all be pulled together regardless of aligning in any dynamic range, meaning the risk of misalignment and a broken cable may be likely to occur. Once the above-mentioned problem occurs, the alignment procedure should be carried out again, which means that the alignment efficiency will be greatly reduced. In summary, the conventional alignment method is limited by the perceived resolution of the human eye and the problem of pulling between the detector and the corresponding cables. In addition, the alignment quality for optical axes depends on operational experience. A high-quality alignment procedure can be obtained only by an experienced operator and an untrained operation may take more time to implement and may induce manual error.
In accordance with the optical design, laser interferometers can be summarized as two types that contain a non-common path and a common path structure. Commercial laser interferometers are mainly based on the Michelson interferometer with the non-common path structure. The measured displacement is determined by the optical path difference between the reference and the measurement beam [
16,
17]. Except for the measurement beam, the condition of a reference optical path will also be introduced to the measurement result. In contrast to the Fabry–Pérot interferometer based on a common path structure, there is no additional reference beam in the optical design and laser beams are transmitted under the same optical path [
18,
19,
20]. Therefore, the measured displacement can be precisely defined by its optical cavity and this kind of interferometer has a high resistance of thermal expansion, vibration, and micro-flow gradient [
21]. The conventional Fabry–Pérot interferometer (
Figure 1) was presented by Charles Fabry and Alfred Pérot in 1897 [
22]. The laser beam with a tiny incident angle (α) spreads into the optical cavity composed of two coated plane mirrors. Laser beams are transmitted forward and backward reciprocally and then they are divided into numerous reflected or transmitted beams that are detected by the photodetector. The equation of the light intensity for the interferometric signal is indicated in Equation (1), where A
0 is the amplitude of the incident beam, R is the reflectance of the plane mirror, respectively.
Due to the reference and measurement beams traveling over the same optical path, the conventional Fabry–Pérot interferometer can be performed for precise length measurement which indicates that phase errors induced by environmental effects can be corrected by this optical arrangement. Nevertheless, the parallelism between two plane mirrors is hard to maintain which leads to measurement limitation over a large range and the signal processing based on the fringe counting method is not capable of determining the moving direction of the tested target. Consequently, this interferometer is rarely employed in dynamic displacement measurements. The quadrature phase-shift fiber-optic Fabry–Pérot interferometer was proposed by Kent A. Murphy et al., in 1999 [
23]. The signal processing is according to the spatial phase adjustment to obtain the quadrature interferometric signals. The phase-shift depends on the position of two sensor probes, so the optomechanical arrangement and alignment are essential points in this structure. If further thermal expansion and mechanical vibration appear, the quadrature phase-shift is difficult to maintain. For this reason, the dynamic length measurement within the large measuring range is also hard to realize. The polarized Fabry–Pérot interferometer was presented in our previous study [
24,
25,
26,
27,
28,
29]. By utilizing the polarized phase-shifted method, one octadic-wave plate (λ/8) is introduced into the optical cavity. Hence, the orthogonal phase difference between interferometric signals can be achieved and then the measurement signal can be acquired by photodiodes (PDs). In this structure, signal processing with complex feedback control is adopted in order to avoid signal drift and decline. Furthermore, because each measurement signal possesses its own gain and offset control channels, a data acquisition board with high resolution, sampling rate, and multi-channel is needed. In the whole measurement procedure, the signal processing should be carried out in real-time by the conversion of analog to digital and digital to analog. Therefore, the cost of the systematic construction is raised while the measurement velocity of this structure cannot exceed 4 mm/s.
With respect to the installation of the commercial laser interferometer system, the corresponding optical components should first be arranged and then additional optical adjustment is also required so that the reference and measurement beams can be superimposed for displacement counting after optical alignment. For this reason, there is still room for improvement in convenience of installation.
From the outcome of the above description, a novel calibration system with integration of the multi-beam Fabry–Pérot interferometer and the auto-alignment module is proposed and the correlated features are as follows. A convenient and effective coarse alignment method is proposed to reduce effectively the initial angular deviation between the optical axes and the movement axis and then auto-alignment can be performed within the measuring range of millimeter-scale, not limited by the operational experience and the perceived resolution of the human eye. The entire alignment process can be accomplished in the short-term. Through optimization of the optical design, the linear displacement measurement can be directly implemented after the auto-alignment procedure without extra placement and installation of optical components. For this reason, compact optical arrangement and convenient installation can be realized. The detector for optical alignment is arranged inside the interferometric sensor head which means that cable pulling does not exist in the linear stage. Hence, the risk of optical misalignment and broken cable can be avoided. The signal response also can be enhanced with no existing complex feedback control and advanced data acquisition board by optimization of the signal processing module. Therefore, the proposed system is beneficial for the linear displacement calibration of machine tools in the precision mechanical engineering industry.
2. Measurement Principle and Optomechatronic Design
The proposed linear displacement calibration system consists of two chief modules including the linear displacement and the auto-alignment module. Through the optimization and the integration of these two modules, the linear displacement calibration of the linear stage in the machine tool could be realized conveniently and effectively.
2.1. Measurement Module for Linear Displacement
The folded Fabry–Pérot interferometer for determining linear displacement is composed of three measurement units including the laser light source, the sensor head, and the signal processing unit demonstrated in
Figure 2. According to this optical structure, the laser beam reflected by the non-polarizing beam-splitter (BS) is not yet employed in other measurements. In order to enhance the measurement function of the proposed system, the unused laser beam will be used for the optical alignment revealed in
Section 2.3.
The stabilized He–Ne laser spreads through the isolator, single-mode optical fiber, the BS and into the optical cavity. The optical cavity consists of a plane mirror and a corner cube retro-reflector (CCR) while a λ/8 is arranged in the cavity to produce the orthogonal phase shift between interferometric signals. The backward reflected beams emerging from the optical cavity are reflected by the BS. Afterward, the interference beams are divided by the polarizing beam-splitter (PBS) and obtained by two PDs. According to the proposed structure based on multi-beam interference, the laser beam passes through the optical cavity repeatedly, so the transmittance of the intensity loss from the cavity needs to be considered. In order to obtain the continuous change of the signal, the reflectance of the plane mirror is adjusted for further signal processing [
30,
31]. The theoretical formula of the interferometric light intensity is illustrated in Equation (2) to Equation (5), where A
0 is the amplitude of the laser, R and T are the reflectance and transmittance of the plane mirror, L is the transmittance evaluated by the intensity loss in the optical cavity.
is the phase difference which equals to 8πnd/λ
0, where λ
0 is the vacuum wavelength, d is the distance of the optical cavity, and n is the refractive index while m is the order number of the backward reflected laser beam, w is the frequency of the electric field, x is the initial traveling path, and k is the wave number.
The electric field of s-type and p-type can be denoted as Equations (2) and (3).
The intensity distribution of s-type and p-type can be described with Equations (4) and (5). Then the simulation of the orthogonal signal is shown in
Figure 3, where A
0, R, and L are equal to 1, 0.25, and 0.86 respectively. In this theoretical simulation, focus is on observing the curve distribution of the orthogonal interferometric signal and not on analyzing the displacement measurement. Therefore, the cavity refractive index is not considered in the analysis. In the experiment, although the wavelength in the air (λ) is affected by the refractive index (Equation (6)), this refractive index can be corrected by Edlén theoretical formula in Equation (7), where P, T, and RH are atmospheric pressure (kPa), temperature (°C), and relative humidity (%) respectively [
32].
The pre-amplifier for two interferometric signals is introduced in the sensor head unit in order to reduce the noise effect, and then signals can be transmitted to the signal processing unit. The purpose is to eliminate the DC offset and to retain the signal amplitude during the measurement process. The signal processing components and procedure are as follows.
The pre-amplifier, secondary amplifier, and differential amplifier are based on the low offset operational amplifier of LF412. The Butterworth filter is adopted and then in the automatic gain control (AGC) circuit, it is mainly integrated with a variable gain amplifier VCA810 and a high-speed comparator AD8561. The elimination of the DC offset can be obtained by further signal processing with a Butterworth filter and a differential amplifier. In the AGC circuit, the gain amplifier VCA810 relies on the control voltage obtained by feedback to control the magnification. The comparator AD8561 is utilized to compare the output signal with the setup voltage. After passing through the detection circuit, amplification can be performed by the gain amplifier VCA810. Therefore, the signal magnification can be auto-adjusted in accordance with the output signal, so that the signal amplitude remains almost the same during the displacement measurement. Compared to previous signal processing, the need of the elimination of the DC offset and the retention of the signal amplitude can be realized only by a correlated circuit without complex feedback control and an advanced multi-channel data acquisition board. Therefore, the signal response can be enhanced according to the optimized signal processing.
2.2. Auto-Alignment Module for Optical Axes
The alignment procedure including coarse alignment and fine alignment is to align the optical axes of the laser interferometer and the movement axis of the linear stage to a parallel state. After coarse alignment, the initial angular deviation between the optical axes of the interferometer and the movement axis of the linear stage can be reduced and then the laser spot will be at the sensing area of the detector within the whole alignment distance. This indicates that the fine alignment can be performed automatically.
For CNC machine tools, geometric errors e.g., the straightness of a single axis, and the parallelism and squareness between multi-axes, should be detected before delivery. If large geometric errors occur, they must be corrected and minimized during the assembly process. In view of the above-mentioned description, a compact fixture with high squareness could be utilized as a reference for the two angular directions (pitch, yaw) of the linear stage. In the coarse alignment procedure, the fixture with a mirror is placed in the linear stage and aligned to one side of it, as shown in
Figure 4. Then the laser beam from the sensor head is incident to this mirror, meeting the orthogonal incident situation between the integrated fixture and the laser beam. Hence, the initial angular deviation between the optical axes and the movement axis can be easily and quickly reduced.
When the coarse alignment procedure is accomplished, a CCR is placed on the linear stage and then the laser beam incident to it, will be retro-reflected into the two-dimensional PSD (2D-PSD) fixed in the sensor head to carry out the fine alignment procedure illustrated in
Figure 5.
When there is an angular deviation between the optical axes and the movement axis over the whole measuring distance, a triangular geometric relationship can be obtained through their corresponding positions. It contains ΔACF and ΔDEF, where
is the placement distance of the sensor head (D
1) and
is the measuring distance (D
2). From Equations (8) and (9), the fine aligned distance (
) and the tilt angle (θ) between two axes can be acquired, respectively.
The mechanism design for auto-alignment, composed of two step motors and an adjustment platform, is shown in
Figure 6. The laser interferometer is fixed on the adjustment platform and then the two rotary axes of it are adjusted by two motors, respectively. The cosine error can be determined by the pitch and yaw angle during the alignment process.
2.3. Linear Displacement Calibration System
By integrating two modules, including a measurement module for linear displacement and an auto- alignment module for the optical axes, a linear displacement calibration system was proposed. The unused laser beam revealed in
Section 2.1 was employed to align the optical axes by further optical arrangement. Then the two major laser beams which travel along the inner and outer edge of the CCR can be utilized to measure the linear displacement (red line) and align the optical axes (purple line) respectively, shown in
Figure 7. Namely, the need of optical alignment and linear displacement measurement can be achieved simultaneously. Therefore, by the proposed novel optical design, laser beams are fully arranged to improve the usability and measurement function of the proposed system and optical misalignment and broken cable will not occur. Moreover, further straightness measurements could also be implemented in this structure.