## Abstract

The traceable absolute distances network with multiple global targets for multilateration is developed with a femtosecond pulse laser. It is aiming to enhance the ability and flexibility of the coordinate measurement, especially to monitor the positions of distributed stations in real time for some critical industrial environments. Here, multi-target absolute distances are determined by the temporal coherence method simultaneously with the pulse-to-pulse interferometer. Besides, the performance of the proposed system is evaluated in detail by comparing with a conventional interferometer. The experimental results indicate that the accuracy of distances measurement could all reach the sub-micron level and could be traceable to the length standard. Furthermore, a simple scheme of multilateration is presented based on the developed network. The coordinate of the initial point of multiple beams is measured by cooperation with a laser tracker. The results of coordinate measurement show that these methods have the potential for further industrial applications.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Spatial coordinate measurement is a competitive and frontier field in the large-scale metrology, which has numerous applications in some technological advances, such as geodesy and industrial engineering [1, 2]. Normally, typical dimensions of large-size objects are tens to hundreds of meters. Meanwhile, coordinate measurements are required to be realized in a simple, flexible way with excellent accuracy.

As innovative ways are being developed to harvest the enormous potential for large-scale space, distribution coordinate measurement systems such as theodolite networks [3], indoor global position system (iGPS) [4] and multilateration [5] have emerged. The typical configuration is shown in Fig. 1. The global control network consists of multiple targets in the three-dimension (3-D) workspace, which coordinates are always measured by a precise tool previously. Meanwhile, positions of distributed stations are determined by the geometric constraint with the cooperation of the global targets to maintain the overall accuracy and performance. Because the angular measurement would produce a greater contribution to the measurement uncertainty, multilateration techniques based on the lengths measurement have been successfully used in to enhance the coordinate measurement ability [5]. In such cases, the traceable absolute distances network is always demanded instead of the conventional interferometric methods, and the measuring components must have the ability to monitor multiple targets in space simultaneously. In addition, positions of distributed stations should have the ability to be self-monitored in real time, in particular in some critical industrial environments.

Currently, femtosecond pulse laser, also called optical frequency comb, is a great invention, which brings a great breakthrough in the absolute distance measurement [6,7]. When its two key parameters, repetition frequency and carrier envelope offset frequency, locked to an atomic clock, the absolute distance measurement with the femtosecond pulse laser could be accomplished with high precision, and it could also be traced to the SI definition of the meter. Based on the superb frequency and time characteristics, the absolute distance measurement using femtosecond pulse laser could be divided into several groups: inter-mode beats and phase measurements [8, 9], pulse-to-pulse interferometry based on coherence methods [10–16], heterodyne interferometry [17, 18], multi-wavelength interferometry [19, 20], dispersive interferometry [21, 22] and dual-comb interferometry [23, 24]. In these methods, the dual-comb interferometer could realize an arbitrary and absolute distance measurement with high update rate and high accuracy. Nevertheless, the dual-comb system is complex and expensive, and it also requires the extra synchronous circuit for two combs. It means it is not a simple and compact system for further industrial applications. The pulse-to-pulse interferometer based on temporal coherence method using a single comb could realize a long absolute distance measurement with sub-wavelength accuracy [10–16]. Although the measured range is limited by the range of movement in reference arm or the tuning range of repetition frequency. But it could be solved by introducing a long fiber with optical sampling and cavity tuning method [25, 26]. Therefore, the pulse-to-pulse interferometer with a femtosecond pulse laser has the potential to be a more economical and flexible system for engineering environment.

To our knowledge, there is still little research about the evaluation of absolute distance measurement using femtosecond pulse laser in spatial coordinate measurement. In 2015, Han et al. proposed a multiple targets measurement method combined nonlinear optical cross-correlation technique with dual-comb interferometry, it was applied for angular measurement of a rigid body which is located at a nominal distance of ~3.7 m and the deviation was 0.073 arcsec with the averaging time of 0.5 s [27]. In the same year, Weimann et al. presented a distance measurement based on the phase evaluation of inter-mode beats of a frequency comb. They combined a MEMS steering mirror and a four-quadrant (4-Q) photodiode, and it could track a retroreflector with a volumetric accuracy of 24.1 *µm* [28]. In addition, there are some other absolute distance measurement methods aiming to realize spatial localization. In 2013, Hughes et al. presented a multilateration method of coordinate measurement using frequency scanning interferometry and a couple of collimating lenses, and they realized a coordinate measurement with uncertainties of the order of 40 *µm* in a measurement volume of 10 m × 5 m × 2.5 m [29]. In 2016, Meiners-Hagen et al. presented a tracking interferometer, also called “3D-Lasermeter” with an intrinsic compensation of the refractive index of air, showing an asymptotic length dependent uncertainty in the order of 0.1 *µm*/m for distances over 10 m [30].

In this paper, a traceable absolute distances network for multilateration is developed with a femtosecond pulse laser, which aims to enhance the ability and flexibility of distributed coordinate measurement. In particular, in some harsh industrial environments, positions of the measurement stations should be self-monitored in real-time. Here, multi-target absolute distances in the global frame are determined by the temporal coherence method using a single femtosecond pulse laser. Compared with dual-comb systems, there is no nonlinear crystal in this system and no synchronous circuit module for two combs. With the key parameters of the femtosecond pulse laser locked to an atomic clock, this method could finish the task of traceability in the industrial site for coordinate measurement. Firstly, the principle of multiple absolute distances measurement in a 3-D space with a femtosecond pulse laser is proposed, and the method for generation of multiple beams is also introduced. After that, the theoretical model and experimental performance of multi-target absolute distances measurement based on temporal coherence method are evaluated in detail by comparing with a conventional interferometer. Furthermore, a simple scheme for evaluating the traceable absolute distances network using a femtosecond pulse laser is firstly demonstrated in the multilateration. In our experiments, the coordinate of the initial point of multiple beams is performed as the position of measurement station and measured by multilateration with a femtosecond pulse laser and a laser tracker. Finally, the experimental results show that it has the potential to be applied for further industrial applications.

This paper is organized as follows. In Section 2, the principle of multi-target absolute distances measurement using a single femtosecond pulse laser is introduced. In Section 3, the performance of this experimental set up is evaluated in detail by comparing with a conventional interferometer. In Section 4, the multilateration for spatial positioning and a simple experiment for coordinate measurement are demonstrated. Finally, in Section 5, the conclusions and future work are summarized.

## 2. Measurement principle using a single femtosecond pulse laser

The experimental set up for multi-target absolute distances measurement is described in Fig. 2. The system of femtosecond pulse laser consists of a fiber laser, a rubidium frequency standard, repetition rate synchronization electronics and an erbium-ytterbium-doped fiber amplifier (EYDFA). The rubidium frequency standard (8040C, Symmetricom) is used as a frequency standard. A 10-MHz signal is generated from the rubidium frequency standard and is sent to the repetition rate synchronization electronics (RRE-SYNCRO, Menlosystems), which controls the repetition frequency of a mode-locked femtosecond pulse laser (C-Fiber, Menlosystems). The repetition rate of the femtosecond pulse laser is 100 MHz, and the width of the pulse is 50 fs. The central wavelength is 1582 nm and the spectral width is 101 nm. The average output power of the laser is 33 mW, then, the output power is amplified by an EYDFA.

The femtosecond pulses are collimated by a collimator and introduced into an ordinary Michelson interferometer. In the reference arm, a precision translation stage (P-521.DD1, PI) is used for scanning the temporal coherence patterns. On the translation stage, there are two back-to-back retro-reflector, one is used as reference mirror M_{R}_{1} (PS975M-C, Thorlabs), and the other is a retro-reflector M_{R}_{2} of a laser Doppler displacement interferometer (MCV-500, Optodyne). The temporal coherence patterns are detected by a balanced amplified photodetector (PDB450C, Thorlabs) and recorded by an oscilloscope (MDO4054C, Tektronix). The I/O trigger output of the controller of translation stage (C-863 Mercury Controller, PI) and the displacement of Doppler interferometer are used for the monitor signal to find the position of measurement pulse and reference pulse overlapped completely. Besides, a neutral density filter (NDC-50C-4M, Thorlabs) is placed in the reference arm. It would be used to adjust the intensity of reference pulse and the intensity of measurement pulse approximately equal.

Here, a diffractive optical element (DOE) is used for generating multiple beams to measure multiple targets. Compared with collimating lenses pairs, the efficiency of utilization of light power is higher. Although the measurement range is limited and the measurement targets only can be arranged along the first-order diffraction lines of the DOE, it is enough for us to evaluate the absolute distances measurement by temporal coherence method using a single femtosecond pulse laser. In our experiments, the measured retro-reflector are noted as Target 1, Target 2, Target 3 *…* and Target *n*. The geometric relationship of the optical beams and position of targets is shown in the inset of Fig. 2. The initial point of multiple beams is termed as point *O*. The length of multiple optical beams between DOE and targets are noted as *l*_{1}, *l*_{2}, *l*_{3} *…* and *l _{n}*. However, in our system, a silver D-shaped mirror

*M*0 (PFD10-03-P01, Thorlabs) is placed in the femtosecond interferometer and the DOE. It is used for determining the initial point of absolute distances measurement. Therefore, it could bring an extra value

*l*

_{0}in the distances measurement. The measured absolute distances here is expressed as

*L*

_{1},

*L*

_{2},

*L*

_{3}

*…*and

*L*. The measured absolute distances could also be expressed as

_{n}*L*=

_{n}*l*+

_{n}*l*

_{0}. Besides, the separation angle of two beams after DOE is recorded as

*θ*. For further analysis, one is assumed that the length of multiple beams are same and noted as

*l*

_{1}=

*l*

_{2}

*…*=

*l*=

_{n}*l*. In addition, the distance between the initial point

*O*and every two targets is recorded as

*d*. Meanwhile, the distance between the Target 1 and Target 2 is called as

*l*

_{12}. So, the

*d*(

*θ, l*) =

*lcos*(

*θ/*2) and

*l*

_{12}(

*θ, l*) = 2

*lsin*(

*θ*/2). This means the arrangement of global targets is determined by the measured absolute distances and the separation angle. For example, when the measured absolute distance is 100 m and the separation angle is 20°, the distance between two targets is around 34.73 m.

## 3. Temporal coherence method for multi-target absolute distances measurement

For evaluating the measurement principle in Section 2, some experiments about absolute distances measurement based on temporal coherence method with a DOE have been demonstrated in this section. Firstly, the theoretical model of distances measurement based on temporal coherence method is introduced. After that, the generation of multiple optical beams and the detected method are analyzed with four targets. Besides, the simultaneous distances measurement of four targets is presented and the process of pulse-to-pulse alignment is also proposed. Furthermore, the distance measurement of one of four targets is evaluated by comparing with a conventional interferometer.

#### 3.1. Theoretical model of temporal coherence method

In time domain, the stable femtosecond pulses generated from the femtosecond pulse laser can be expressed as

*E*(

_{train}*t*) is the electric field of a pulse train in the time domain.

*A*(

*t*) is the pulse envelope,

*ω*

_{0}is the central angular frequency and

*φ*

_{0}is the initial phase of carrier pulse. The repetition frequency of the comb is

*f*and the offset frequency is

_{rep}*f*from the initial frequency. Correspondingly, the pulse repetition period is

_{ceo}*T*= 1/

_{r}*f*, the carrier phase slips is Δ

_{rep}*φ*= 2

_{ce}*π f*/

_{ceo}*f*, and

_{rep}*m*is an integer.

For convenience, the envelope of pulse is simplified as a Gaussian pulse model. So, in the unbalanced dispersive interferometer, the intensity of first-order temporal coherence function can be expressed as

*I*

_{1}is the DC component,

*I*

_{2}is the AC component, and

*x*is the optical length of measured distance. Then,

*ξ*′ is the substituting variable and proportional to the measurement optical path length

*x*.

*v*and

_{g}*v*are group velocity and phase velocity in the space medium, respectively.

_{ϕ}*d*is the equivalent scanning length of the stage in reference arm when we acquire the interference fringes. Eqs. (1)–(4) and their simulations are described in detail in [31].

In the traceable absolute distances network, the DOE is a key component and performs as a 2-D diffractive beam splitter. It is employed for generating multiple beams with separation angle in 3-D space, and the intensity of every optical spot is all the same. So, after the femtosecond pulses propagate through the DOE, the intensity of the pulses can be expressed as

where*I*(

_{train}*t*) is the intensity of femtosecond pulses before passing through the DOE,

*η*is the transmission efficiency of the DOE,

*n*is the number of spots generating by the DOE.

From Eq. (5), the process of generation of multiple beams only influences the intensity of measured pulses, and it does not influence the position of the reference pulse and measured pulse overlap completely. In this case, the absolute distance between the initial point and measured targets can be calculated from

where*N*is the positive integer and Δ is the fraction.

*N*could be determined by a distance meter with lower resolution. And Δ

*can be expressed as Δ*

_{n}*=*

_{n}*mod*(2

*L*),

_{n}, N × C_{n}× T_{r}*C*is the light velocity in air, it could be calculated by the Edlen formula with the measured environmental parameters [32].

_{n}#### 3.2. Analysis of multiple beams and detection signals

For evaluating the ability of multiple distances measurement in this system, the distances measurement of four targets is proposed here. The multiple optical beams are generated by a DOE (MS-354-G-Y-A, Holo-Or), and the 1-st order diffraction lines are used for detecting measured targets. The separation angle of this DOE is 0.73°, and the diameter is 11 mm. Please note that the beam diameter of our collimator (F280APC-1550, Thorlabs) is 3.6 mm. The theoretical full-angle beam divergence is 0.032°. For our measured volume, it could promise that the optical beam could propagate through the DOE twice for detecting by the photodetector. If the measured distance becomes larger, the collimator and DOE should be carefully selected and designed. In our system, a silver D-shaped mirror *M*_{0} (PFD10-03-P01, Thorlabs) is performed as the initial point of the absolute distances measurement. There would be an extra value *l*_{0}, the difference distance between the reference mirror and DOE, added into the measured distance. But it does not influence the evaluation of distance measurement based on temporal coherence patterns. In the multilateration, the distance *l*_{0} could be calibrated by redundancy measurement with more targets. The four measured targets are retro-reflectors of the laser tracker (Two of them are Red-Ring Reflector 1.5” Ball, Leica, and the others are LTBP-A-Z-RS-SHA, MetrologyWorks). In addition, the environmental parameters are monitored by an environmental compensator of the interferometer (XC-80, Renishaw) and a compact USB temperature and humidity logger (TSP01, Thorlabs).

In Fig. 3, the principle of detection with four targets is presented. As shown in Fig. 3(a), the multiple optical beams are generated after the femtosecond pulses passing through the DOE. When the optical beams are reflected by the targets, they would pass through the DOE twice. After that, each reflected beam would generate four diffractive beams for detection. However, only one of the four beams of each target has the same path after passing through the DOE twice. The detected spots in front of the photodetector are shown in Fig. 3(b). The detected area is the region of diffractive optical spots overlapped. In Fig. 3(c), a schematic of detected coherence patterns is shown in the time domain. These coherence patterns could be distinguished by the time division method using the optical shuttle in front of each target.

#### 3.3. Simultaneous distances measurement of multiple targets

Firstly, the four targets are settled at a nominal distance of ∼ 3 m. In this case, the measured volume here is around 30 cm × 4 cm × 4 cm as introduced in Section 2. The four targets are settled with the previous offset in the distances to avoid overlapping. After that, the coherence patterns could be obtained by scanning the reference mirror in the reference arm. The fringe acquisition time is about 100 seconds, and the scanning velocity of the translation stage is 1 mm/s. In our experiments, the measurement resolution of distance measurement is about 10 nm theoretically. But it is also influenced by the resolution of the translation stage. The typical image of interference fringes in one measurement with four targets obtained by the oscilloscope (MDO4054C, Tektronix) is displayed in Fig. 4. And Fig. 5 shows the zoomed coherence patterns of the initial point and three targets.

From Fig. 4, the fractional measured distances Δ* _{n}* could be determined by the relative positional relationship in the interference fringes between the

*M*

_{0}and the target. The integer

*N*in Eq. (6) is determined by a tape measure with millimeter resolution. In Fig. 4, there are some small noise fringes after the detected fringes, these might be brought by the process of amplifying the light intensity by the EYDFA. But it does not influence the process to find the fringes signal from the targets. In Fig. 5, the coherence patterns of

*M*

_{0}and three targets are displayed. Compared with the coherence pattern of

*M*

_{0}, the coherence patterns of targets are a little bit chirped and broadened. This is mainly due to the dispersion of air and DOE. Figure 6 shows the pulse-to-pulse alignment for determining the position of reference pulses and measurement pulses reflected by the targets overlap completely. The black circles in Fig. 6 represent the peaks of fringes in the coherence patterns. The envelope of the coherence patterns is obtained by the optimized modulated Gaussian curve fitting method. And this method has been introduced in detail in the [31]. The peaks of the envelope represent the positions of reference pulses and measurement pulses reflected by the targets overlap completely. And it is used for determining the fractional measured distances Δ

*accurately.*

_{n}#### 3.4. Distances measurement by comparing with an interferometer

Here, the performance of distances measurement based on temporal coherence method is evaluated. Some experiments are proposed to evaluate the repeatability and linearity of this method. Firstly, one retro-reflector target (LTBP-A-Z-RS-SHA, MetrologyWorks) is settled along one of the diffraction beams. For the limited experimental environment, the measured absolute distances are selected at *L*_{1} =∼ 1.5 m and *L*_{2} = ~3 m, respectively. At each position, the distances are measured for 25 times. The variations of fractional part of distances Δ_{1.5} and Δ_{3.0} are shown in Fig. 7. In Fig. 7(a), the standard deviation of the distance measurements is 172.96 nm at the path length difference of 1.5 m. In addition, at the position of 3 m, the standard deviation of distance measurements is 335.88 nm in Fig. 7(b).

On the other hand, one of the optical beams is collimated and one target is used for evaluating the distance measurement by comparing with a counting laser interferometer (XL80, Renishaw). The measured target in the femtosecond laser interferometer and the measurement mirror in the conventional interferometer are firmly attached together on a PZT (TPZNFL5/M, Thorlabs). The target mirror is moved along this PZT with a step of 5 *µm* for a short distance of 20 *µm* at the path length difference around 1.5 m and 3 m, respectively. At each position, the distances are measured 10 times. And the experimental results are acquired and shown in Figs. 8 and 9. During the distance measurement, the temperature, humidity, and air pressure are recorded for calculation of the refractive index of air.

In Figs. 8 and 9, the linear fitting method is applied for evaluating the linearity of measured distances. The final fitting results show that the measured distance is in good accordance with the results of the reference interferometer. In Fig. 8, at the position of around 1.5 m, the fitting slope is 0.9775 and the correlation coefficient (*R*^{2}) is 0.9991. The residuals are ranged from -252 nm to 333 nm. The maximum standard deviation and minimum standard deviation is 139.91 nm and 59.90 nm, respectively. When the path length difference is around 3 m, as shown in Fig. 9, the fitting slope is 0.9871 and the correlation coefficient (*R*^{2}) is 0.9993. The residuals are scattered from −152 nm to 360 nm. And the standard deviations vary from 51.20 nm to 240.66 nm. The measurement errors might be brought from several aspects. Firstly, for further industrial application, our experimental environment is not strictly controlled, the fluctuation of the air would increase the measurement error. Although the environment parameters are monitored and the refractive index of air is calculated by the Edlen formula, it also would bring several tens nanometers deviation. Besides, the vibration of the translation stage and electrical noise would also affect the measurement.

From experimental results, the accuracy of repeatability and linearity all reach the sub-micron level. This would meet the demand of coordinate measurements by multilateration with the accuracy of several tens microns for industrial application. In section 4, the multilateration for coordinate measurement based on the distances measurement using temporal coherence method would be discussed for the first time. And a coordinate of the initial point of multiple beams is measured and evaluated by combining a laser tracker.

## 4. Multilateration positioning methods based on the traceable absolute distances network with a femtosecond pulse laser

Multilateration techniques based on interferometric length measurements have been successfully employed in the coordinate measurement to enhance the measurement ability [5]. However, the interferometric length measurement could not be realized in the harsh industrial environment. Here, we are supposed to apply the high-precision multi-target absolute distances measurement using a femtosecond pulse laser to the multilateration positioning technique. The frequency comb could perform as a reference frequency ruler when locking to an atomic clock. In industrial environments, the work of traceability for coordinate measurement could be straightforwardly accomplished with the cooperation of a femtosecond pulse laser. Meanwhile, positions of stations in the distributed coordinate measurement network could be self-monitored in real time. In this section, a simple multilateration is evaluated based on the proposed experimental set up introduced above. The initial point of multiple beams is regarded as the position of the measurement station, and its coordinate is measured by the multilateration with a femtosecond pulse laser and a laser tracker.

In Fig. 10, the multilateration scheme using a femtosecond pulse laser and a laser tracker is presented. The coordinates of the four global targets are measured by a laser tracker (AT960-LR, Leica), which are denoted as (*x _{n}, y_{n}, z_{n}*).

*n*is an integer. The measured point is the initial point

*O*of the multiple beams at DOE, which is expressed as (

*x*

_{0},

*y*

_{0},

*z*

_{0}). And the coordinate of the measured point could be calculated from the equation by multilateration

In Eq. (7), the absolute distances of the targets and the initial point *O* of multiple beams is noted as *l _{n}*. Here, the

*L*is the absolute distance measured by the temporal coherence method using a femtosecond pulse laser. Hence, there is a length offset

_{n}*l*

_{0}introduced, common to all distance measurements using the femtosecond pulse laser. It could be considered as a system parameter and it could be calculated and calibrated by the redundancy measurement. In our case, the known measured target points are four, it is enough to solve the Eq. (7) analytically to obtain the coordinate of the initial point of multiple beams (

*x*

_{0},

*y*

_{0},

*z*

_{0}) and the initial length

*l*

_{0}between the D-shaped mirror and DOE. Here, the measured volume is around 30 cm × 4 cm × 4 cm. The coordinates of targets determined by the laser tracker are listed in Table 1.

In our experiment, the four absolute distances are measured simultaneously by a femtosecond pulse laser. The acquisition time is around 100 seconds. The distances of four targets in space are measured 8 times. To calculate the refractive index of air, the humidity of air is measured as 42.22 % and the atmospheric pressure is acquired as 1011.6 mbar. Besides, the temperature of air recorded as 20.27 °C is used. The initial point of the multiple beams is calculated by the Eq. (7) and Eq. (8) using the least square method. The initial length *l*_{0} is measured by a tape as the initial iterative value. The experimental results of absolute distances measurement and the calculated coordinate of the initial point are shown in Table 2 and Table 3. And the 3-D distribution of the coordinate of four targets and calculated initial point *O* is displayed in Fig. 11.

From the experimental results, the standard deviations of the simultaneous absolute distances measurement are all sub-micron level, which are 548 nm, 839 nm, 995 nm, and 840 nm. The experimental data are basically consistent with the analysis above. Meanwhile, the standard deviations of initial point in X coordinate, Y coordinate and Z coordinate are 9.71 *µm*, 31.94 *µm*, and 41.27 *µm*. The mean calculated *l*_{0} is 84.85 mm. It is observed that the errors in Y coordinate and Z coordinate are higher than X coordinate. It is mainly because that the targets are compressed in the arrangement along the direction of Y and Z. It also means that the distribution of known measured targets is another key influencing factor in multilateration. Besides, the number of known measured targets and the location algorithm would also influence the results of accuracy in multilateration. However, from the experimental results, it could be observed that this method could be utilized in the industrial applications where coordinates measurement requires several tens microns accuracy, such as aerospace industry. In addition, the ability of positioning could be better with further optimization.

## 5. Conclusions and future work

This paper has demonstrated a traceable absolute distances network for multilateration with a femtosecond pulse laser, which expects to enhance the coordinate measurement ability in the large-scale space. Especially in some critical industrial environments, positions of distributed measurement stations should be self-monitored in real time. In our system, the multi-target absolute distances were determined simultaneously by the temporal coherence method with the compact and economical pulse-to-pulse interferometer. Here, the repetition frequency of the femtosecond pulse laser was locked to an atomic clock. Therefore, the adjacent pulse repetition interval length could be traceable to the time standard. Although the process of pulse-to-pulse alignment was not traced to the same time standard as the repetition interval length, this process ensured the absolute distances measurement with high precision. In addition, the performance of repeatability and linearity was evaluated in detail by comparing with a conventional interferometer. The experimental results of the distances measurement all reached the sub-micron level, which is sufficient for some industrial applications where require several tens microns accuracy, such as the aerospace industry. Although the measured range was limited in our experiments, this problem could be further solved by increasing the tuning range of repetition rate, such as optical sampling by cavity tuning with a long fiber, or increasing the scanning range of the reference arm. Furthermore, a simple scheme of multilateration has been presented for evaluating the ability of coordinate measurement based on the proposed system. The initial point of multiple optical beams has been performed as the position of the measurement station, and its coordinate has been measured by the multilateration with a femtosecond pulse laser and a laser tracker. The standard deviations in X coordinate, Y coordinate and Z coordinate were 9.71 *µm*, 31.94 *µm*, and 1.27 *µm*, respectively. The experimental results might be limited by the imperfect arrangement of the targets, and it could be better with further optimization.

It is of course that our proposed traceable absolute distances network was limited in space by the separation angle of the diffractive optical elements. However, it does not influence the evaluation of multilateration based on temporal coherence method with a femtosecond pulse laser. Therefore, in future work, this problem should be solved by exploring other methods for generating multiple beams in a wider range of space, such as the collimating lenses system [29]. In addition, the study for finding a find a compact, simple and flexible arbitrary absolute distance measurement method using the femtosecond pulse laser for engineering environment would be continuing. Finally, the multilateration would be applied for the coordinate measurement for higher accuracy with the cooperation of multiple measurement stations.

## Funding

National Natural Science Foundation of China (51835007, 51705360, 51775380), Foundation for Innovative Research Groups of the National Natural Science Foundation of China (51721003) and the Young Elite Scientists Sponsorship Program by CAST (2016QNRC001).

## Acknowledgments

We deeply thank Associate Prof. Fumin Zhang at Tianjin university and Dr. Changyu Long for their experimental support.

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