## Abstract

A measurement system with three degrees of freedom (3 DOF) that compensates for errors caused by incident beam drift is proposed. The system’s measurement model (i.e. its mathematical foundation) is analyzed, and a measurement module (i.e. the designed orientation measurement unit) is developed and adopted to measure simultaneously straightness errors and the incident beam direction; thus, the errors due to incident beam drift can be compensated. The experimental results show that the proposed system has a deviation of 1 μm in the range of 200 mm for distance measurements, and a deviation of 1.3 μm in the range of 2 mm for straightness error measurements.

© 2015 Optical Society of America

## 1. Introduction

The linear guides used in precision machinery are expected to travel along a straight line during operation. However, the actual path deviates from a straight line – a phenomenon referred to as “straightness error” – due to geometric errors. Accordingly, certain technology is required to calibrate the straightness of precision machines such as X-Y tables and coordinate measuring machines (CMMs) [1]. Current technology employs the laser interferometer, a length measurement device; in this manner, a single geometric error can be measured during each adjustment. However, it is quite time-consuming to use the laser interferometer for measuring displacement and straightness errors. Consequently, multi-degree-of-freedom (DOF) measuring technology [2–5 ] has recently been developed.

An early work by Ni *et al.* is regarded at the most advanced system for calibrating CMM geometric errors. This work focuses on combining laser interference and collimation [1, 6
]. Much research has been carried out in recent years based on the principle of complex optical configurations [7–9
], though these systems and methods are not convenient for workshop measurements and *in situ* measurements. Recently, some multi-DOF systems have appeared [10, 11
], though these systems have limited measurement ranges, which in turn limits their applicability. There is in fact one kind of commercially available 6DOF measuring system from Automated Precision Inc. (API) [12]. However, this equipment is not only expensive and complex, but also contains cable connections between moving parts. Recently, Feng *et al.* proposed a compact 6 DOF measurement system which is convenient for *in situ* measurements [3]; more specifically, the pitch, yaw, and roll of the retro-reflector can all be obtained, and the straightness errors can be compensated. Nevertheless, the incident beam direction is not obtained by this system, and the error of the displacement measurement is not provided.

In this paper, a compact 3 DOF measurement system that compensates for errors caused by incident beam drift is proposed. The system measurement model (i.e. its mathematical foundation) is presented and analyzed, and a measurement module (i.e. the designed orientation measurement unit) is developed and adopted in order to implement the measurement model. The module has the ability to measure simultaneously straightness errors and the incident beam direction. In this way, errors of 3 DOF measurements due to incident beam drift can be compensated.

## 2. Measurement model

Figure 1 shows the method used for measuring straightness error. The measurement of the CC lateral displacement can be performed simultaneously with the interferometer displacement measurement by evaluating the lateral position of the returning beam; this constitutes a simple way to calibrate straightness error for CMMs [13–17 ]. In this way, the 3 DOF measurements of the CC along the X-, Y-, and Z-axes are all obtained. A position-sensitive detector (PSD) is a standard device used for measuring lateral displacements. The lateral displacement along the Y- or Z-axis constitutes the straightness error measurement.

If the incident beam doesn’t produce drift, the lateral displacement Δ*Y* of the retro-reflector would have to be measured by the PSDs. On the other hand, if beam drift occurs, the measured value 2Δ*Y* will be influenced by the deflection angle caused by beam drift, which will lead to a straightness error. Laser beam drift cannot be changed under actual conditions owing to adjustments of the fixed laser device, as well as changes in environmental parameters [3]. In a good environment, the incident beam drift is mainly due to the laser source. In this paper, we consider the incident beam drift only due to the laser source. Therefore, the PSD in Fig. 1 ought to be replaced by a developed measurement module that has the capability to measure straightness errors while simultaneously obtaining the incident beam direction. In this way, the errors in straightness measurements due to the incident beam drift can be compensated.

Before the devised measurement module is proposed, we first employ spatial geometry to analyze the straightness measurement model. It is assumed that the unit vector of the incident beam that arrives at the retro-reflector is described by: (*m _{i}*,

*n*,

_{i}*q*), the retro-reflector’s corner joint has coordinates (

_{i}*x*,

_{c}*y*,

_{c}*z*) and some fixed point in the laser beam that arrives at the retro-reflector has coordinates (

_{c}*x*,

_{i}*y*,

_{i}*z*). We can therefore express the beam reflected by the retro-reflector as follows:

_{i}*x*

^{′}

*,*

_{i}*y*

^{′}

*,*

_{i}*z*

^{′}

*) is generated by rotating the point (*

_{i}*x*−

_{i}*x*,

_{c}*y*−

_{i}*y*,

_{c}*z*−

_{i}*z*) 180° around the unit vector (

_{c}*m*,

_{i}*n*,

_{i}*q*):

_{i}*x*,

_{cf}*y*,

_{cf}*z*) and the latter sample point to be (

_{cf}*x*,

_{ch}*y*,

_{ch}*z*), where the subscript “

_{ch}*c*” refers to the corner joint, “

*f*” refers to the first sample point, and “

*h*” refers to the latter sample point. This can be obtained through Eqs. (1) and (3) :

From Eq. (4), it is known that *x _{ch}* −

*x*=

_{cf}*L*is the displacement along the X-axis, where

_{M}m_{i}*L*is the interferometer’s measured displacement; thus, the lateral displacements (i.e. the straightness errors) of

_{M}*y*−

_{ch}*y*and

_{cf}*z*−

_{ch}*z*can be written as:

_{cf}
Equation (5) represents the measurement model of our measurement system. In this equation, (*y _{yozh}* −

*y*)/2 and (

_{yozf}*z*−

_{yozh}*z*)/2 are standard terms of straightness errors, whereas

_{yozf}*L*and

_{M}n_{i}*L*are terms for modifying the straightness error due to beam drift. Because

_{M}q_{i}*L*is the interferometer’s measured displacement, as long as we know the unit vector (

_{M}*m*,

_{i}*n*,

_{i}*q*) of the incident beam,

_{i}*y*−

_{yozh}*y*, and

_{yozf}*z*−

_{yozh}*z*, the straightness errors of

_{yozf}*y*−

_{ch}*y*and

_{cf}*z*−

_{ch}*z*can be obtained through Eq. (5). The function of the devised measurement module is therefore to obtain measurements of (

_{cf}*m*,

_{i}*n*,

_{i}*q*),

_{i}*y*−

_{yozh}*y*, and

_{yozf}*z*−

_{yozh}*z*. Along with the straightness error measurement, we also require the displacement measurement

_{yozf}*x*−

_{ch}*x*along the X-axis. Therefore, it is necessary to know the value of

_{cf}*m*. In addition, in order to guarantee the repeatability of the motion of the linear guide, the value of

_{i}*x*−

_{ch}*x*between the same sampling points should be identical at each measurement. Because of incident beam drift, the same

_{cf}*x*−

_{ch}*x*value corresponds to different

_{cf}*L*values. Thus, in order to obtain the same

_{M}*x*−

_{ch}*x*value at each measurement, the value of

_{cf}*m*must be obtained. Regardless of the incident beam drift that occurs, as long as

_{i}*m*is known, any values of displacement along the X-axis can be obtained.

_{i}## 3. The developed measurement module

#### 3.1 Dual PSD-based orientation measurement unit and its improvement

As shown in Fig. 2(a)
, a dual PSD-based orientation measurement unit is used to obtain angular measurements related to the laser sensing and tracking technique [18]. The dual PSD-based unit can measure the incident light’s angular variation, though Ref [18]. only gave the resulting formulas of the angles between the incident beam and the X-, Y-, and Z-axes. Here, we analyze the underlying model of the dual PSD-based unit in order to employ the dual PSD-based unit as the module for obtaining the direction vector (*m _{i}*,

*n*,

_{i}*q*) of the incident light beam,

_{i}*y*−

_{yozh}*y*, and

_{yozf}*z*−

_{yozh}*z*in Eq. (5).

_{yozf}The CC joint locates at the origin of the rectangular coordinate system in Fig. 2(a). The cube’s center line (i.e. the X-axis) and the beam splitter’s (BS’s) reflecting surface are fixed at an angle of 45°. *L*
_{1} is the distance between the CC joint and the center point of the BS (i.e. the point of intersection between the X-axis and the BS’s reflecting surface). *L*
_{2} and *L*
_{3} are the distances between the X-axis and PSD I and PSD II, respectively. The BS’s reflecting surface and PSDs are all perpendicular to the XOY plane. Assuming the unit vector of the incident beam to be (*m*, *n*, *q*), the point of intersection between the incident beam and the BS’s reflecting surface can be written as (*x*
_{0}, *y*
_{0}, *z*
_{0}).

The incident beam arrives at the PSD I at the point (*x*
_{1}, *y*
_{1}, *z*
_{1}) is:

*x*

_{2},

*y*

_{2},

*z*

_{2}):

*x*

^{′′}_{0},

*y*

^{′′}_{0},

*z*

^{′′}_{0}) is the point of intersection between the beam reflected by the CC and the BS, and is written as:

*x*

^{′}_{0},

*y*

^{′}_{0},

*z*

^{′}_{0}) is generated by rotating the point (

*x*

_{0},

*y*

_{0},

*z*

_{0}) 180° around the unit vector (

*m*,

*n*,

*q*):

*x*

^{′′}_{0},

*y*

^{′′}_{0},

*z*

^{′′}_{0}) can be expressed in terms of (

*x*

_{0},

*y*

_{0},

*z*

_{0}) and (

*m*,

*n*,

*q*). Because the point (

*x*

_{0},

*y*

_{0},

*z*

_{0}) lies in the BS’s reflecting surface, we can ultimately obtain the unit vector of the incident beam in Fig. 2(a) with Eqs. (6) and (7) as follows:

In Eq. (10), (*x*
_{1}, *z*
_{1}) and (*x*
_{2}, *z*
_{2}) can be measured by PSD I and II, respectively. Moreover, *L*
_{1}, *L*
_{2}, and *L*
_{3} are known parameters, and thus the direction of the incident beam can be obtained by the dual PSD-based unit. Here, the measured result (−*m*, −*n*, −*q*) is equal to the incident beam (*m _{i}*,

*n*,

_{i}*q*) in Eq. (5). In addition,

_{i}*y*−

_{yozh}*y*and

_{yozf}*z*−

_{yozh}*z*can be obtained by PSD I or II. Therefore, the straightness errors of

_{yozf}*y*−

_{ch}*y*and

_{cf}*z*−

_{ch}*z*in Eq. (5) can be obtained with the dual PSD-based unit.

_{cf}Concerning the original pattern, it is important to note that (*x*
_{1}, *z*
_{1}) and (*x*
_{2}, *z*
_{2}) are absolute coordinate values, whereas PSD only measures its own values along its own X- and Z-axes. If the centers of the PSDs are set on the XOY plane, then the values of *z*
_{1} and *z*
_{2} measured by the PSDs are absolute coordinate values, though *x*
_{1} and *x*
_{2} should actually be *x*
_{1}
* _{P}* +

*L*

_{1}and

*x*

_{2}

*+*

_{P}*L*

_{1}in Eq. (10), respectively, where

*x*

_{1}

*and*

_{P}*x*

_{2}

*are values measured by the PSDs.*

_{P}In this paper, it is necessary to have two beams that are separated in space owing to the design of our measurement system. However, the PSD’s detection area is limited, and we therefore modified the original dual PSD-based unit into the improved pattern shown in Fig. 2(b). In the improved pattern, because the positions of the PSDs vary, we need to add two parameters: *L*
_{4} and *L*
_{5}, which represent the distances between the CC joint and the center point of the BS for PSD I and II, respectively. Therefore, *x*
_{1}
* _{P}* +

*L*

_{1}and

*x*

_{2}

*+*

_{P}*L*

_{1}have to be changed into

*x*

_{1}

*+*

_{P}*L*

_{4}and

*x*

_{2}

*+*

_{P}*L*

_{5}in Eq. (10). In addition to the parameters mentioned above, two other parameters must be considered. More specifically, the centers of PSD I and II can deviate from the XOY plane; here they are expressed as

*L*

_{6}and

*L*

_{7}, respectively. In other words, the values of

*z*

_{1}and

*z*

_{2}should be transformed into

*z*

_{1}

*+*

_{P}*L*

_{6}and

*z*

_{2}

*+*

_{P}*L*

_{7}in Eq. (10), where

*z*

_{1}

*and*

_{P}*z*

_{2}

*are values measured by the PSDs.*

_{P}#### 3.2 Practical design of the improved dual PSD-based orientation measurement unit and the correction of its parameters

The analysis of the dual PSD-based orientation measurement unit in Section 3.1 did not consider refraction, which means that the CC must be of the hollow variety, and that the BS must have no thickness; the former condition can be realized, though the latter cannot. Otherwise, this will lead measurement errors and will cause the rectangular coordinate system to change owing to alterations in the direction of incident beam. Here, a new type of the dual PSD-based orientation measurement unit (the modified dual PSD-based unit) is proposed, which acts as the measurement module of the proposed system.

In Fig. 3 , the original BS is replaced by a custom BS, which consists of two BSs; both reflected surfaces are coplanar. The incident light reflected by the custom BS will go through a refraction off-set piece which is as thick as the first BS, and then arrives at the PSD I. The transmitted portion of the incident light is reflected by the CC and received by PSD II. The modified dual PSD-based unit practically realizes the model of the dual PSD-based unit proposed in Section 3.1. Here, Eq. (10) can be rewritten as:

*L*

_{4}+

*L*

_{5}=

*P*

_{1},

*L*

_{6}+

*L*

_{7}=

*P*

_{2}, 2

*L*

_{1}+

*L*

_{3}−

*L*

_{2}=

*E*, 2

*L*

_{1}−

*P*

_{1}=

*F*,

*x*

_{1}

*+*

_{P}*x*

_{2}

*=*

_{P}*M*

_{1}and

*z*

_{1}

*+*

_{P}*z*

_{2}

*=*

_{P}*M*

_{2}.

In Eq. (11), *M _{1}* and

*M*are values measured by the PSDs. The parameters are

_{2}*E*,

*F*, and

*P*, which play an important role in the performance of the modified dual PSD-based unit. However, the three parameters are determined by the precision of mechanical processing: e.g., flatness error, parallelism error, 45° angle error, and positioning error. Obviously, a correction process for these parameters is required to guarantee the accuracy of the modified dual PSD-based unit. Its design is relative small; the high accuracies of flatness, parallelism, and angle processing can easily be assured, though the magnitude of the positioning error is not related to the size of the dual PSD-based unit in the mechanical processing. Aside from the mechanical a processing error, PSDs need to be welded in a printed circuit board, which also easily generates positioning error, and therefore has a significant effect on the accuracy of these three parameters. Here, we take advantage of the simple manner in which the fiber-coupled laser can be collimated to correct the three parameters.

_{2}It is known from Eq. (11) that if we can guarantee a laser beam to be parallel to the X-axis and it arrives at the modified dual PSD-based unit, then (*m*, *n*, *q*) will take on certain values: i.e. *m =* −1, *n =* 0, and *q =* 0. If *M*
_{1} and *M*
_{2} are measured and *F = M*
_{1} and *P*
_{2}
*=* −*M*
_{2}, then *E* will be obtained owing to the fact that *m =* −1. Therefore, the three parameters *E*, *F*, and *P*
_{2} are all completely corrected. In order to conveniently fix the laser source, a fiber-coupled laser is adopted in Fig. 4
. Obviously, the accuracy of the correction of the parameters is mainly determined by the degree to which the laser is parallel to the X-axis. Because the fiber-coupled laser is easily adjusted and driven, the auto-reflection alignment method could be used [19]. Moreover, there are many ways of ensuring that the fiber-coupled laser beam lies parallel to the X-axis, which forms the basis of our future work. The significance of the parameter correction technique, however, is to reduce the positioning error to the level of other mechanical processing errors.

## 4. Measurement system and experiment

In Fig. 5 , after the dual-frequency beam passes through the polarizing beam splitter I (PBS I), one frequency beam is reflected to the retro-reflector, and another frequency beam arrives at the reference cube corner (RCC). After the first frequency beam arrives at the retro-reflector, it enters the modified dual PSD-based unit for measurements of straightness error and incident beam direction; next, it enters PBS II, and interference occurs with the second frequency beam reflected by the RCC. At this location, the plane mirror (PM) prevents the leaked beam from PBS II from entering PBS I, thereby causing measurement errors in the system. Here, an Agilent 5530 laser calibration system is used as a reference for the calibration effectiveness of the 3 DOF measurement system.

#### 4.1 3 DOF measurement experiment

The retro-reflectors of the proposed system and Agilent 5530 are placed as shown in Figs. 6(a)–6(c) , which correspond to measurements along the X-, Y-, and Z-axes, respectively. Figures 7(a), 8(a) and 9(a) show the measurement results before laser beam direction variation. Figures 7(b), 8(b) and 9(b) show the measurement results after laser beam direction variation. The results in Figs. 7(b), 8(b) and 9(b) are achieved by slightly changing the direction of the dual-frequency laser beam.

Equation (5) shows the model for measuring straightness errors along the Y-and Z-axes. If the retro-reflector has no movement along the X-axis (i.e. if *L _{M}* is equal to 0), then Eq. (5) is the standard form of the straightness measurement, which does not include the term

*L*(

_{M}n_{i}*L*). However, in actual applications, the retro-reflector moves, and

_{M}q_{i}*L*must be non-zero. In order to verify the effectiveness of the model contained in Eq. (5) for straightness error measurements, we induce random, tiny motions along the X-axis between two measurement sample points, thereby making

_{M}*L*take on a non-zero value during straightness error measurements (see Figs. 8 and 9 ). Deviations of the calibration of displacement measurements along the X-axis are both about 1μm within a measurement range of 200 mm (see Figs. 7(a) and 7(b)). Deviations in the calibration of straightness error measurements along the Y- and the Z-axes are all less than 1.3 μm within a measurement range of 2 mm (see Figs. 8 and 9 ). The resulted deviations before laser beam direction variation are almost the same as those after laser beam direction variation. It demonstrates that the modified dual PSD-based unit can compensate errors for different directions of the incident beam.

_{M}The deviations observed between the proposed system’s displacement measurement and those obtained using Agilent 5530 in Figs. 7(a) and 7(b) can be explained by the fact that the incident beams of Agilent 5530 are not completely parallel with the X-axis of our proposed system. In Figs. 8 and 9 , the deviations observed between the straightness error measurement results obtained using our proposed system and those obtained using Agilent 5530 are due to the measurement error inherence in the PSDs of the modified dual PSD-based unit, which leads to systematic error in the proposed system. In addition, parameter-correction processing only decreases the error for measuring the direction of the incident beam, and cannot eliminate all mechanical processing errors of the modified dual PSD-based unit. Accordingly, manufacturing and mechanical processing errors in the dual PSD-based unit also cause systematic errors in our system.

#### 4.2 Repeatability experiment

To verify the repeatability of the proposed system, measurements are performed by moving the linear stage over a distance of 1200 mm with an increment of 120 mm for three runs. The three runs are performed to three different directions of the incident beam, and each run time lasts 1000s, the stationary time after each step motion is about 100s, each actual motion time is neglected because it is too short, therefore, each run contains 10 stationary points except the starting point. The measurement results are shown in Figs. 10(a) and 10(b) . And the Figs. 11 (a) and 11(b) are the plots of straightness errors versus travel time at each stationary point in 1.2m linear stage for the three runs. At each stationary point, the average of measured straightness errors in Figs. 11(a) and 11(b) is equal to the straightness errors value in Figs. 10(a) and 10(b). In Figs. 11(a) and 11(b), the fluctuations of straightness errors in two lateral directions are all less than 1.6µm at each stationary point. In general, Figs. 10 and 11 all exhibit a good repeatability of the proposed system during the three runs. The difference among the three results is mainly due to the fact that it is impossible to achieve the same sample positions for the three runs in the linear stage.

Figures 12(a) and 12(b) are the plots of beam drifts versus travel distances (time) along the Y and Z axis for the three runs. The laser beam generates about 6-7 arc-seconds beam drift in two lateral directions during each run. The good repeatability and little fluctuation at each stationary point for the three beam drifts demonstrate the compensation of straightness errors is always effective. The experiment proves that even if the incident beam drift is varied, the modified dual PSD-based unit still can compensate errors due to the incident beam drift.

## 5. Conclusion

In this paper, a compact 3 DOF measurement system which compensates for errors caused by the incident beam drift, is proposed. Because the retro-reflector of the proposed system is wireless, which make it convenient for *in situ* measurements. The proposed system’s measurement model is presented and analyzed, and a measurement module is developed and adopted in order to implement the proposed system’s measurement model. The measurement module has the ability to measure simultaneously straightness errors and the direction of the incident beam. In this manner, errors of the 3 DOF measurements due to incident beam drift can be compensated. The experimental results show that the proposed system has a deviation of 1 μm in the range of 200 mm for displacement measurements, and deviations of 1.3 μm in the range of 2 mm for straightness error measurements. The experimental results show even if the incident beam drift is varied, the modified dual PSD-based unit still can compensate errors due to the incident beam drift.

## Acknowledgments

This work was financially supported by the National Natural Science Foundation of China (NSFC) (Grant No. 51105114) and the Fundamental Research Funds for the Central Universities (Grant No. HIT.NSRIF.20168).

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