1. Introduction

Absolute distance measurement with large range has been paid increased attention in the applications of high-end equipment manufacturing, space mission, large-scale metrology and so on [1–5]. Laser interferometry has been the focus of research for distance measurement because of its high precision and traceability to the definition of meter [6,7]. However, the inherent ambiguity of single-wavelength interferometry limits the measuring range to half of the used wavelength. To overcome the ambiguity, multi-wavelength interferometry (MWI) was proposed by C. R. Tilford [8] and it has been rapidly developed in recent years [9–11]. In MWI, multiple laser wavelengths are used to generate a chain of decreasing synthetic wavelengths, where the maximum one covers the measuring range and the minimum one determines the resolution. To realize high precision absolute distance measurement with large range, the key issues in MWI are the generation of the required synthetic wavelength chain, the stability of the synthetic wavelengths, and the accuracy of phase demodulation corresponding to the synthetic wavelength.

To generate the synthetic wavelength chain, multiple laser sources with fixed wavelength can be employed in parallel [12], while the discrete spectra of the available laser sources make it difficult to generate required synthetic wavelengths to cope with different measuring distances. A wide and continuous spectral range can be acquired using an external cavity diode laser (ECDL), and it can be employed in sequence to obtain multiple wavelengths. For instance, C. Y. Yin proposed a changeable synthetic wavelength chain (CSWC) method for absolute length measurement in optical probes [13]. Using one ECDL, desired synthetic wavelengths determined by the measuring distance and the measurement uncertainty were created. S. H. Lu presented the variable synthetic wavelength interferometry (VSWI) to measure large step heights, and one ECDL was employed to synthesize a series of synthetic wavelengths in descending order [14]. However, the achieved measurement relative uncertainties are not better than 10⁻⁶ in CSWC and VSWI, and the maximum error source comes from the synthetic wavelength uncertainty which is determined by the individual uncertainty of the used wavelengths. That is because the frequency stability of a free-running ECDL is not better than 10⁻⁷, making it become the main contribution to the measurement uncertainty, and this contribution increases with the measuring distance. Therefore, it is necessary to take certain frequency stabilization methods to ensure the stability of the synthetic wavelength.

In MWI, the synthetic wavelength Λ_S is described as Λ_S=λ₁λ₂/|λ₁-λ₂|=c/Δf, where λ₁ and λ₂ are laser wavelengths for generating the synthetic wavelength, Δf is the frequency difference between the two used wavelengths, and c is the speed of light in vacuum. To get synthetic wavelength with high stability, one way is stabilizing the individual used wavelengths to certain frequency references. It can be realized by locking the ECDL to the Fabry-Perot cavity [15] or the absorption lines of selected atoms or molecules [16]. However, the Fabry-Perot cavity with high stability requires precise control of the environment parameters, like temperature, sound and vibration, and it cannot trace to the definition of meter. The available absorption lines in nature are isolated discrete points, making it difficult to generate arbitrary synthetic wavelengths we need in MWI. Recently, optical frequency comb (OFC) has been used as a precision wavelength ruler for it can provide a few hundreds of thousands of narrow line width wavelengths over a broad optical spectral range [17,18]. S.W. Kim et al. employed an ECDL locked to the OFC for absolute length measurement in MWI, and the required multiple wavelengths were provided by sequentially tuning and phase-locking the ECDL to a set of pre-selected optical modes of the OFC [19,20]. Such methods will take a longer time for unlocking and locking ECDL to different modes. In addition, the uncertainty of the synthesized wavelength will be deteriorated by 1–2 orders of magnitude by stabilizing the individual used wavelengths to the frequency references. According to the definition of the synthetic wavelength, the uncertainty is only determined by the frequency difference between the used wavelengths, so directly stabilizing the frequency difference is a potential way to further reduce the uncertainty of the synthetic wavelength.

Another key issue in MWI is to demodulate the synthetic phase corresponding to the synthetic wavelength accurately. A common solution is demodulating the interference phase of the used wavelengths individually, and then make a difference to get the synthetic phase. To demodulate the interference phase of every laser wavelength with high precision, L. Hartmann et al. adopted the homodyne detection technique using a single frequency laser, and the measured phase was demodulated from a pair of quadrature interference signals obtained by a Fresnel rhombi [21]. While it is vulnerable to low frequency disturbances, and the laser power drift, unequal gain of the detectors and non-quadrature interference signals will result in nonlinear errors. N. Schuhler et al. adopted the heterodyne detection technique which uses a dual-frequency laser with two linearly polarized orthogonal beams, and the measured phase shift is carried on the beat signal of the dual-frequency laser [22,23]. The technique has advantages of fast response, immune to noise and inherent freedom from drift, however, the nonlinear errors contributed by frequency mixing and polarization mixing limit the precision of the phase demodulation. In our previous work, the phase generated carrier (PGC) homodyne detection technique which has simple optical configuration was used for absolute distance measurement [24]. In this technique, a high-frequency carrier phase signal is used to convert the measured phase signal onto the sidebands of the carrier frequency and its harmonics, making the phase demodulation have less sensitivity to the optical and electronic noises. Even so, no matter which phase detection technique is adopted, nonlinear errors in phase demodulation are always inevitable. Then, the operation of subtraction for getting the synthetic interference phase will amplify the nonlinear error, and the amplitude of the nonlinear error varies periodically when the synthetic interference phase changes in 0∼2π.

In this paper, we establish a principle of laser interferometric wavelength leverage (LIWL) to measure large distance with respect to synthetic wavelengths by detecting nanometer displacement with respect to a single wavelength. And we further propose an electro-optical modulator (EOM) based dynamic-sideband locking method to generate variable synthetic wavelengths with high stability. The principle of LIWL is deduced, its realization is revealed, and experiments are performed for verification.

2. Principle of the laser interferometric wavelength leverage

Figure 1 shows the principle schematic of LIWL. In Fig. 1(a), two orthogonal linearly polarized beams with fixed wavelength of λ₁ and tunable wavelength of λ₂ are combined at PBS₁. The output beam of PBS₁ is divided into two beams by BS. The reflected beam is incident on PBS₂, where beam λ₂ is reflected, beam λ₁ transmits PBS₂, directs to the reference M₁, and then is reflected. The transmitted beam of BS is incident on the measurement M₂ and then reflected. The reflected beams recombine at BS and generate interference signals of λ₁ and λ₂, which are split by PBS₃ and detected by PD₁ and PD₂, respectively.

 

Fig. 1. Principle and schematic of the laser interferometric wavelength leverage. (a) Optical layout. (b) Phase difference with the tuning of λ₂. (c) Conception of wavelength leverage.

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In Fig. 1(b), the phase difference Δφ between the interference signals of λ₁ and λ₂ can be expressed as

When wavelength λ₂ is tuned to λ₂+Δλ₂, the phase difference will change to Δφ'

To determine the distance difference L, M₁ is moved a very small displacement Δl to make the phase difference equal to the initial Δφ again. That is

When Δφ " =Δφ, a one-to-one correspondence can be established between L and Δl

The expression of Eq. (4) is similar to the mechanical leverage in mechanics, as shown in Fig. 1(c). Here, we call it the laser interferometric wavelength leverage (LIWL). Similar to the definitions of the five components of fulcrum, resistance, resistance arm, effort, and effort arm in the mechanical lever, in LIWL, L corresponds to the resistance, 1/Λ_SS to the resistance arm, Δl to the effort, 1/λ₁ to the effort arm, and phase difference Δφ keeping constant to the fulcrum.

Therefore, by moving M₁ to keep Δφ constant and detecting the moved displacement Δl, the absolute long distance to be measured can be obtained

Equation (5) indicates that a measured distance L in the range of Λ_SS/2 can be achieved by detecting the displacement Δl in the range of λ₁/2. For instance, when the frequency of laser 2 is changed by 1 MHz, Λ_SS will be about 300 m, and when Δλ₂ reaches 6 nm, Λ_SS will decrease to 100 µm. Therefore, by tuning λ₂ to construct a gradually decreasing variable self-synthetic wavelength, absolute distance measurement in hundreds of meters with micrometer-level accuracy can be realized.

Obviously, constructing a variable synthetic wavelength with high stability and providing a small displacement Δl with nanometer accuracy are essential for LIWL. Here, it should be emphasized that the key innovative of LIWL is to keep Δφ constant, which can eliminate the influence of the nonlinear error of single-wavelength phase demodulation on the phase of synthetic wavelength in most MWI.

3. Realization of LIWL for absolute distance measurement

3.1 Synthetic wavelength generation with EOM- based dynamic-sideband locking

Figure 2 shows the principle of proposed synthetic wavelength generation with dynamic- sideband locking. In Fig. 2(a), the first external cavity diode laser (ECDL₁) is referenced to an optical frequency comb (OFC). Its output beam is split into two beams by a fiber splitter (FS₁). The first beam of ECDL₁ passes through a high-frequency EOM (EOM_H) modulated by a sinusoidal signal and generates the corresponding sidebands with frequency of f_sideband

 

Fig. 2. Principle of synthetic wavelength generation with dynamic-sideband locking. (a) Optical configuration of laser sources. (b) Schematic of the changes of f₂ and f_b with f_r.

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Output beam of another external cavity diode laser (ECDL₂) is also split into two parts by FS₂. Sidebands of ECDL₁ and one beam of ECDL₂ are combined by a fiber combiner (FC) and detected by an avalanche photodiode (APD). The beat frequency signal is filtered, amplified, and then phase locked to the Rubidium clock signal (REF) by using a phase-locked loop (PLL). As Fig. 2(b) shown, the frequency of ECDL₂ is offset-frequency locked to Nth sideband of ECDL₁ and expressed as

The second beams of ECDL₁ and ECDL₂ are combined into one orthogonal linearly polarized beam at a fiber polarization combiner (FPC) and acts as the light source for LIWL.

According to Eq. (6), if we adjust the modulating frequency f_r, the frequencies of sidebands will change dynamically (Here, we call them dynamic sidebands). As Fig. 2(b) shown, once ECDL₂ keeps locked to the Nth sideband of ECDL₁ (e.g. N=5), f₂ will change with the adjustment of f_r. Denoting the ith modulating frequency as f_r[i], then f_2[i] and f_b[i] can be expressed as

Equations (8) and (9) indicate that both f_2[i] and f_b[i] can be adjusted by dynamically locking to the sideband. Therefore, by adopting the wavelength of λ₂ before and after adjusting f_r, we can construct a variable self-synthetic wavelength Λ_SS[i] traced to Rubidium clock with high stability.

According to Eq. (10), it is possible to flexibly construct a Λ_SS[i] from tens of kilometer to millimeter by only setting different Δf_r[i]. For example, the non-ambiguity range (NAR) of absolute distance measurement, i.e. Λ_SS[i]/2, can be up to 300 m when Δf_r[i] is set to 100 kHz, further to 3 km when Δf_r[i] is set to 10 kHz or to 30 km when Δf_r[i] is set to 1 kHz. While by setting Δf_r[i] to 1GHz, NAR will decrease to 30 mm.

In addition, by adopting the frequency difference between ECDL₁ and ECDL₂, another variable synthetic wavelength Λ_S[i] can be obtained

By comparing Eq. (10) and Eq. (11), we can see that Λ_S[i] will be much smaller than Λ_SS[i] because f_r[i] is much larger than Δf_r[i]. Therefore, Λ_SS[i] and Λ_S[i] are used for coarse distance measurements and fine distance measurements, respectively.

Here, it should be emphasized that when constructing different synthetic wavelengths, ECDL₂ always remains locked to the same sideband. This avoids the unlocking and locking operation of ECDL₂ and will greatly reduce the adjusting time of synthetic wavelength.

3.2 Realization of an equivalent displacement Δl with an EOM

In Fig. 1(a), the purpose of moving M₁ is to adjust the interference phase of wavelength λ₁ to keep Δφ constant. The accuracy and stability of the displacement Δl is very important because its error will be magnified by a factor of Λ_SS/λ₁ when calculating the measured distance L. Generally, as Fig. 1(a) shows, PZT stage equipped with capacitive feedback sensors can provide a sub-micrometer displacement with repeatability of ±1nm. However, in the configuration of Fig. 1(a), partial optical path of beams λ₁ and λ₂ is not in common, which will make the tiny drift between PBS₂ and M₁ be magnified and result in larger measurement error. In addition, the mechanical movement of PZT stage might introduce some other errors, such as vibration, and thermal deformation.

Here, an EOM made of LiNbO3 crystal is adopted to generate an equivalent nanometer displacement. As shown in Fig. 3, due to the birefringent and linear electro-optic effects, when the orthogonal linearly polarized beam passes through the EOM, an induced phase difference between λ₁ and λ₂ will be generated

 

Fig. 3. Realization of the equivalent displacement with an EOM. (a) Optical layout. (b) Schematic of transverse electro-optic effect of crystal.

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As beam λ₁ is p-polarization and beam λ₂ is s-polarization, we define ${l_\textrm{P}} = {{n_e^3{\gamma _{33}}WV} / {2d}}$ and ${l_\textrm{S}} = {{n_o^3{\gamma _{13}}WV} / {2d}}$. Then, we can get

Usually, l_S/l_P is about 0.3∼0.5. Equation (13) indicates that the adjustment of Δφ_e (i.e. phase difference between λ₁ and λ₂) can be realized by changing the applied voltage V of EOM. In LIWL, the range of Δφ is from -π to π, which means l_P and l_S are in several hundred nanometers. Therefore, for distance measurement with Λ_S larger than 1mm, the first term in Eq. (13) is small enough to be neglected. Then, (l_P-l_S) of the second term is equivalent to the nanometer displacement Δl provided by PZT.

4. Experiments and results

4.1 Experimental setup

Figure 4 shows the optical configuration and experimental setup of constructed LIWL for absolute distance measurement. As Fig. 4(a) shows, LIWL includes two sinusoidal phase modulating interferometers. The first one consisting of BS₁, M₁ and M₂ is used for absolute distance measurement. The second one consisting of BS₁, M₁, BS₃, BS₂ and BS₄ is used to measure the equivalent Δl and monitor the drift of optical path of BS₁-M₁. In the two interferometers, wavelengths λ₁ and λ₂ share the same optical path of BS₁-M₁ and their phases are modulated sinusoidally by EOM₁ and EOM₂. EOM₃ is adopted to provide the equivalent displacement Δl to adjust the phase difference Δφ for LIWL. The interference phases of four detectors PD₁, PD₂, PD₃, and PD₄ are denoted as φ_m1, φ_m2, φ_r1, and φ_r2, respectively. The four phases are demodulated simultaneously with PGC algorithm realized by a FPGA-based (XC7K160T, Xilinx) high-speed signal processing board [25].

 

Fig. 4. Experimental setup of LIWL for absolute distance measurement. (a) Optical configuration of LIWL. (b) Constructed experimental setup.

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In the experiment, two same ECDL lasers (TLB6700, New Focus) with a tunable range of 765–781 nm are adopted. ECDL₁ is locked to the OFC (FC1500-250, Menlo Systems) and then modulated by a high-frequency EOM (NIR-MPX800-LN-10, Ixblue) with modulating frequency about 12 GHz. This high-frequency modulating signal is provided by a programmable digital signal generator (LMS-183CX, Vaunix) with a frequency sweeping range of 6–18 GHz. ECLD₂ is locked to the −4th sideband of ECDL₁ with a beat frequency f_o=60 MHz. The sinusoidal modulating signals of EOM₁ and EOM₂, the DC modulating signal of EOM₃ are generated by FPGA board and amplified to a range of ±400 V with high-voltage amplifiers (ATA-2161, Aigtek). A 10 MHz Rubidium clock (STM-Rb, Sync-tech) with stability of 3.3×10⁻¹² (relative Allan deviation in 1s) provides a global reference for the whole experimental system. By sweeping the frequency of the high-frequency EOM, four self-synthetic wavelengths Λ_SS[i] from hundred meters to tens of millimeter and five synthetic wavelengths Λ_S[i] with very small difference are generated. Table 1 shows one group of selected parameters and the generated synthetic wavelengths in the experiments.

Table 1. Selected parameters and generated synthetic wavelengths

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Figure 4(b) shows the experimental setup constructed on an 80 m underground granite air-bearing guide rail in the laboratory of National Institute of Metrology (NIM), China. To evaluate the performance of the proposed LIWL system, three He-Ne interferometers (Agilent 5519B, Keysight) configured in triangular were adopted to measurement the displacement of guide rail. The displacement results of three interferometers are used to calculate a corrected displacement with the smallest Abbe error, which corresponds to a virtual interferometer at the center of triangle [26]. Therefore, for accurate comparison, our LIWL was placed at position of the virtual interferometer. The target mirrors of our system and Agilent interferometers were mounted on the same slider of the guide rail, whose actual moving range is about 72 m. In the experiments, the ambient parameters (temperatures along the guide rail, pressure, relative humidity, and CO₂ concentration) were measured to compensate the refractive index of air (n_air) for our system and Agilent interferometers. The n_air was determined by the modified Edlén equation [27]. Several experiments were carried out.

4.2 Resolution and stability of the equivalent displacement Δl

From Figs. 4(a) and 3(a), we can see that, the change in the phase difference Δφ_r=φ_r1-φ_r2 represents the phase change Δφ_e caused by the applied voltage change of EOM₃. Therefore, the equivalent displacement can be obtained by Δl=δ(Δφ_r) × λ₁/2π when changing the applied voltage. Here, δ(Δφ_r) is the variation of Δφ_r. Figure 5 shows the resolution and stability evaluation results of Δl. An equivalent displacement Δl with step of 0.5 nm can be achieved by increasing the applied voltage. And after the voltage stops increasing, the recorded fluctuation of Δl is less than ±0.21 nm in 140 s, with a standard deviation of 0.073 nm. This result shows that an accurate Δl can be provided for LIWL by adjusting the applied voltage of EOM.

 

Fig. 5. Resolution and stability evaluation results of the equivalent displacement Δl

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4.3 Stability of the frequency difference f_b with dynamic-sideband locking

The stability of synthetic wavelength is determined by the stability of frequency difference f_b between ECDL₂ and the sideband of ECDL₁. According to Eq. (9), the stability of f_b is related to that of f_r and f_o, where f_r can be further expressed as f_r = K × f_Rb. Here, f_Ru is the frequency of Rubidium atomic clock, and K is the ratio of f_r to f_Rb (for example, K=1200 for f_r = 12 GHz). Therefore, by simultaneously recording f_o and f_Rb, f_b can be obtained. Fig. 6 shows the recorded experimental results in 4 hours. The fluctuation of f_b is less than ±25 Hz, with a standard deviation of 5.97 Hz. And the relative Allan deviation of f_b reaches 1.47×10⁻¹⁰ in 1s averaging time. These results indicate that the proposed dynamic-sideband-locked synthetic wavelength generation method can provide a variable synthetic wavelength with high stability.

 

Fig. 6. Stability experimental results of the frequency difference fb. (a) Fluctuation of f_b. (b) Relative Allan standard deviation of f_b.

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4.4 Absolute distance measurement

In the absolute distance measurement experiments, the modulating frequency is adjusted in sequence according to the values listed in Table 1, and the demodulated phases (normalized to 0–1) of four detectors are updated in real time. The distance measurement includes coarse results by LIWL method with four second-order Λ_SS[i] and fine results by MWI with five first-order Λ_S[i]. The detail measurement procedure is as following:

Step 1. Set the initial modulating frequency of high-frequency EOM_H as f_r[i], reset EOM₃'s applied voltage, i.e. V=0, calculate Λ_S[i] with Eq. (11), record the initial phase difference Φ_S[i]=φ_m1[i]- φ_m2[i].

Step 2. Sweep the modulating frequency from f_r[i] to f_r[i+1], the phase difference will change to Φ_S[i+1]=φ_m1[i+1]- φ_m2[i+1]. Calculate Λ_SS[i] with Eq. (10), record the reference phase difference Δφ_r[i]=φ_r1[i]-φ_r2[i].

Step 3. Increase the applied voltage of EOM₃ to make the phase difference Φ' _S[i+1] =φ '_m1[i+1]- φ '_m2[i+1] equal to the original Φ_S[i], record the new Δφ ' _r[i]=φ ' _r1[i]-φ ' _r2[i] when Φ ' _S[i+1]=Φ_S[i]. Then, obtain the equivalent displacement as Δl_[i] =(Δφ ' _r[i]- Δφ_r[i]) × λ₁/2. According to LIWL method, the fractional part of measured distance can be calculated as L_ɛ[i] = Δl_[i] ×Λ_SS[i]/λ₁. Then, the coarse absolute distance can be determined as

Step 4. If i<2, update i = i+1 and repeat steps 1–3. Otherwise, stop and determine five fine absolute distance measurement results as

Step 5. Because the differences in five Λ_S[j] are very small, we determine the final distance measurement result as $L\textrm{ = }\frac{1}{5}\sum\limits_{j = 0}^4 {{L_{\textrm{f[}j\textrm{]}}}}$.

Distance measurement repeatability, resolution, and comparison experiments were implemented to evaluate the performance of the constructed LIWL system. Firstly, the target retroreflector M₂ was placed at a distance ∼52 m and 30 measurements were repeated. For each measurement, four coarse results L_c[i] and one final result L were recorded and shown in Fig. 7. From Figs. 7(a)–7(d) with the Λ_SS[i] decreasing from ∼214 m to ∼61 mm, the distance measurement results become more and more accurate with standard deviations (STDs) decreasing from 120.1 mm to 27.9 µm. As expected, the measurement accuracy of the final result shown in Fig. 7(e) has been further improved, with fluctuation about 6 µm and STD about 1.8 µm. This result is reasonable for the five Λ_S[j] being in a range of 6.7 mm to 6.0 mm.

 

Fig. 7. Repeat measurement results of a distance ∼52 m. (a)–(d) four coarse results. (e) the final results.

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Secondly, at a distance ∼52 m, M₂ was driven to move forth and back with a moving step of 15 µm by a PZT stage (P-753.1CD, PI). After the PZT stage stops completely, five absolute distance measurements from coarse to fine were implemented. The experimental results shown in Fig. 8 indicate that the distance step of 15 µm can be obviously distinguished, and the distance resolution is better than 7.5 µm.

 

Fig. 8. Experimental results of distance resolution evaluation at a distance ∼52 m.

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At last, long distance measurement with a moving step of 5 m was performed. This experiment includes two part, first 0–70 m and second 60–100 m. In the first part, the offset distance is ∼3.3 m, and in the second part, the offset distance is ∼60 m by folding a fixed path of 20 m. The absolute distance results obtained by LIWL system (L_ADM) are compared with displacement results obtained by Agilent interferometer (L_RDM) after removing the offset. Each experiment was conducted twice, and at each position, 15 consecutive data were recorded in 1.5 seconds. Figure 9 shows the difference of the averaged data between LIWI system and interferometer. It can be seen that, in the range of 0–70 m and the ranges of 60–100 m, the maximum residual errors are approximately 12.2 µm and 14.2 µm, respectively. All the experiments show that our LIWL system has realized a distance measurement range of 100 m with a residual error of less than 15 µm.

 

Fig. 9. Comparison experimental results. (a) Distance measurement from 3–73 m. (b) Distance measurement from 60–100 m.

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5. Discussions

According to Eq. (10), the measurement range (NAR) is determined by Δf_r[i] and N, where Δf_r[i] is limited by the frequency sweeping resolution of the signal generator driving the high-frequency EOM. Generally, 100Hz frequency step is available for most signal generator, which means that it is possible to construct a Λ_SS[i] of 300 m, 3 km or more lager by setting N=5 and Δf_r[i]=100 kHz, 10 kHz, etc. However, in such case, the coherence length of the laser source might be the key factor limiting the measurement distance. In our experiment, the linewidth of the ECDL is less than 200 kHz, corresponding to a maximal measurement distance of 497 m.

For the measurement uncertainty of the result L, it only depends on the fractional measurement of fine results L_f[i] because the integral fringe number of the synthetic wavelength Λ_S[i] can be correctly determined. Therefore, the uncertainty of L is given according to Eq. (15) as

We can see that the first two terms are length-dependent, the third term is determined by the value of synthetic wavelength and phase demodulation error. Combining the uncertainties of the environmental parameters and the uncertainty of Edlén equation, the contribution of u(n)/n to u(L) is 2.3 × 10⁻⁸L[26]. In addition, the stability of the synthetic wavelength (equal to the stability of f_b) has been tested and achieved a relative uncertainty of 1.47×10⁻¹⁰. Therefore, the contribution of u(Λ_S)/ Λ_S to u(L) is insignificant. Lastly, with the influences of electronic noise, crosstalk in multi-channel high-speed ADC, and the phase difference calculation, the uncertainty of Φ_S is about 0.0005 (corresponding to 0.18°) in the experiment, which contributes 1.7 µm to u(L) for Λ_S=6.7 mm. Assuming the measurement error is Gaussian distribution, the measurement expanded uncertainty U(L) is obtained with a coverage factor of k=2 as

According to Eq. (17), the proposed LIWL method can achieve a measurement uncertainty of 4.1 µm at a measurement distance of 52 m, which agrees with the obtained results of measurement fluctuation being 6 µm and STD being 1.8 µm. In addition, a relative uncertainty of 5.7 × 10⁻⁸ at a measurement distance of 100 m can be achieved with LIWL method.

The main factor affecting the measurement uncertainty is the first term in Eq. (16), which is limited by the uncertainty of Edlén equation. Even if the uncertaintiy of environmental parameters is improved, this term will not be less than 1 × 10⁻⁸L. For the last term, the minimum synthetic wavelength Λ_S plays a main role on it. In our method, if we increase the laser power of ECDL₁ or the modulation depth of EOM_H to generate more sidebands to lock ECDL₂ to a higher one (e.g. N=8) and increase the modulating frequency f_r (e.g. 20 GHz), a synthetic wavelength Λ_S of 1.87 mm is theoretically attainable. In such case, the second term in Eq. (17) can be decreased to 0.47 µm and the final accuracy can be further improved.

6. Conclusion

In summary, we have demonstrated the principle of laser interferometric wavelength leverage (LIWL) for absolute distance measurement with the dynamic-sideband-locked synthetic wavelength generation. The high-frequency EOM-based dynamic-sideband locking method can flexibly generate variable synthetic wavelengths from hundreds meter to millimeter with a relative uncertainty of 1.47×10⁻¹⁰. Theoretical analysis and experiments have verified the feasibility of the proposed LIWL method. The results show that: 1) the residual error is less than 15 µm between the results of the LIWL system and those of Agilent interferometers in a measurement range of 100 m; 2) at the distance of 52 m, the measurement fluctuation and STD are 6 µm and 1.8 µm, respectively, and the distance resolution is better than better than 7.5 µm. These results indicate the proposed method has a great potential application in the fields of precision equipment manufacturing, space mission, length metrology, and so on.