Ranging is a crucial foundation for advancements in modern science and technology, with widespread applications in various fields, including object morphology measurement, damage detection, and the sensing of velocity and acceleration [1–5]. It is prevalent in various fields of human production activities and daily life, including infrastructure, industry, transportation, and healthcare [6–10]. The advancement of modern science has significantly improved the precision and accuracy of distance measurement, as the concept of distance is inherently linked to fundamental physical constants such as the speed of light and the wavelength of radiation [11]. With the continuous progress of science and technology, the application of ultrafast pulsed laser technology across various fields has garnered considerable attention [12–15]. Ultrafast pulsed lasers, with their extremely short pulse widths, enable precise measurements and analyses on remarkably short time scales, playing a crucial role in disciplines such as physics, materials science, and biology. Among these, the length measurement techniques using ultrafast pulsed lasers have emerged as a significant focus in research and application due to their high-precision measurement capabilities at minute time scales [16–18].

In recent years, there has been increasing interest in the application of ultrafast pulsed lasers in interference measurement, particularly when the laser’s mode and frequency are locked. This enables a direct connection between optical frequency and easily controllable radio frequency signals, allowing for rapid absolute distance measurements with sub-wavelength precision, as opposed to the traditional incremental measurement methods based on phase shifts [19,20]. This development represents a revolutionary advance with profound implications for precision measurement. For a long time, researchers have overcome numerous challenges to achieve nanometer-level precision in remote length measurement [3,21–25]. Among various distance measurement methods based on ultrafast pulsed lasers (including optical frequency combs) are techniques such as dual-comb ranging [5,23,26–28], multi-wavelength interferometry [29,30], synthetic wavelength interferometry [31,32], and dispersion interferometry [21,24,33–36]. These ranging technologies feature relatively simple system architectures, utilizing multiple longitudinal modes in laser interferometry while being combined with homodyne interferometry (HDI) to achieve nanometer-level length measurement precision [21,30]. However, despite these advantages, significant and insurmountable challenges remain regarding measurement range and directional measurement.

We have made considerable efforts in this regard. For instance, we utilized soliton microcomb (SMC)-based dispersive interferometry (DPI) for long-distance measurements [24]. However, achieving long-distance measurements required an auxiliary ranging system, making the system relatively complex. Additionally, the demodulation algorithm relied on FFT transformation and peak tracking, resulting in limited direct measurement accuracy. Besides, we have explored the adjustment of the pose of the grating array in the reference arm of the DPI system to produce a group delay dispersion that approaches linearity [37]. This approach enables full-range length measurement based on linear group delay [38]. This method effectively eliminates the ambiguity range during the DPI ranging process; however, the measurement range for a single cycle remains limited. To further eliminate the non-measurable range and directional ambiguity associated with the DPI method, we recently proposed a phase saltation tracking (PST) method. This approach refines the DPI structure by generating asymmetric interference spectra, significantly improving measurement precision while eliminating non-measurable range-related issues [39]. Although frequency-domain measurements provide high precision, their measurement speed is slow, preventing the capture and measurement of dynamic displacements. Furthermore, due to the limitations imposed by the repetition frequency of the light source, the measurement range for a single cycle remains very constrained. Achieving a large dynamic measurement range and ultrafast measurement speed while ensuring high precision is an urgent requirement for ranging.

Here, we propose a dual-swept laser based on a novel unbalanced Michelson interferometer for the precise measurement of absolute distance at any position, without non-measurable regions or directional ambiguity, and maintaining the high precision advantage of the previous work. Firstly, we utilize a large dispersive element to stretch ultrafast laser pulses, significantly expanding the measurement range within a single cycle. Secondly, we introduce precisely controlled linear group delay dispersion in the reference arm, ensuring that the oscillation zero frequency point accurately appears in the time-domain interference signals. This facilitates distance measurements at any position across the full range, enabling timely and accurate measurement of dynamic displacements as well. In the subsequent demodulation algorithm, we extensively leverage the characteristics of the pulsed light source for filtering and noise reduction. Ultimately, the demodulated interference phase is used to accurately locate the specific position of the oscillation zero frequency point, thereby achieving higher measurement precision and improved resistance to interference. We experimentally demonstrated the capability for precise measurement of static distances and the ability to capture dynamic processes, with a large dynamic measurement range (56 mm). Compared to the calibrated motorized displacement platform, the residual error for full-range distance measurement is within 10 μm, and the error in the average speed during dynamic processes is 0.46%. This indicates that, with the assistance of a simple low-precision ranging device, our system can achieve high-precision measurements of dynamic displacements over arbitrary distances. Furthermore, the system demonstrates excellent long-term stability, achieving a minimum Allan deviation of 4.25 nm over an average duration of approximately 4 ms.

Figure 1 shows a schematic overview of our designed experimental setup. The ultrafast pulsed laser is emitted from a coupler and first is stretched using a segment of dispersion compensation fiber. Subsequently, the optical pulse undergoes power amplification through an erbium-doped fiber amplifier. The stretched optical signal is directed through a beamsplitter; one path is received by a photodetector for time reference on an oscilloscope, while the other path enters a measurement system based on a Michelson interferometer. In the measurement arm of this system, a segment of dispersion compensation fiber is used to control the slopes of the frequency-sweeping curves, as the two interference arms differ, thereby introducing asymmetry into the interference signal to accurately locate the frequency-sweeping information. In the time domain, the signals from the measurement arm and the reference arm can be respectively expressed as $$E_1(t)=A_1(t)e^{-i\left[{\omega }_{0}t+2\pi {\int }_0^t{f}_1(t)t\mathrm{d}t\right]},$$(1)$$E_2(t)=A_2(t)e^{-i[{\omega }_{0}t+2\pi {\int }_0^t{f}_2(t)t\mathrm{d}t]},$$(2)where $A_1(t)$ and $A_2(t)$ represent the optical fields incident on the measurement arm and reference arm, respectively, with ${\omega }_0$ being the central angular frequency, and $f_1(t)$ and $f_2(t)$ denoting the dispersion parameters of the measurement and reference arms, respectively. After linearly tuning the dispersion in both arms, we consider only the second-order dispersion. Thus, the signals from the two arms can be expressed as follows: $$E_1(t)=A_1(t)e^{-i({\omega }_{0}t+\pi k_1{t}^2)},$$(3)$$E_2(t)=A_2(t)e^{-i({\omega }_{0}t+\pi k_2{t}^2)}\mathrm{.}$$(4)Here $k_1=1/({\beta }_1{L}_1)$, $k_2=1/({\beta }_1{L}_1+{\beta }_2{L}_2)$, where ${\beta }_1$ and ${\beta }_2$ represent the second-order dispersion coefficients of DCF1 and DCF2, and $L_1$ and $L_2$ denote the lengths of DCF1 and DCF2, respectively. When the measurement arm signal is used for distance measurement, after introducing the corresponding delay, the signal from the measurement arm becomes $E_1(t-\tau )$. Upon interference with the reference arm signal, the actual detected interference signal is expressed as $$I(t)={|E_1(t-\tau )+E_2(t)|}^2={|E_1(t-\tau )|}^2+{|E_2(t)|}^2+2A_1(t-\tau )A_2(t)\cos \phi (t),$$(5)where the phase difference of the delayed measurement pulse relative to the reference pulse is described by the equation $\phi (t)=\pi (k_1-k_2)t^2-2\pi k_1\tau t+(\pi k_1{\tau }^2-{\omega }_0\tau )$; here $\tau =2n_{\mathrm{air}}L/c$ represents the time delay introduced by the spatial distance $L$ to be measured, where $c$ is the speed of light in vacuum and $n_{\mathrm{air}}$ is the refractive index of air. The acquired interference signal exhibits oscillations in the time domain that transition from fast to slow and back to fast due to the introduced dispersion, as illustrated in Fig. 3. We define the point with the lowest oscillation frequency in the interference signal, which also corresponds to the minimum phase shift, as the zero-frequency point of the oscillation (ZPO). At the ZPO, the condition $\partial \phi (t)/\partial t=2\pi (k_1-k_2)t-2\pi k_1\tau =0$, $t=[k_1/(k_1-k_2)]$$\tau$ holds; we define $N_{\text{amplifying}}=k_1/(k_1-k_2)$ as the amplification factor of the time delay $\tau$. The intersection of the two frequency-sweep curves simultaneously corresponds to the ZPO, as illustrated in Figs. 1(b) and 2(d). Based on the above relationship, it is evident that the time delay $\tau$ in the measurement arm exhibits a linear correlation with the ZPO $t$ of the interference signal. Then, determine the $N_{\text{amplifying}}$ using a known set of distances and their corresponding ZPO positions. Finally, based on the $N_{\text{amplifying}}$, the time delay $\tau$ corresponding to each ZPO can be calculated, which further allows for the determination of the measured distance $L$. As the target distance changes, this oscillation null point shifts linearly in response, with the degree of linearity being influenced by the dispersion introduced by the system.

Therefore, controlling the linearity of the dispersion introduced by the system can enhance the measurement accuracy. In the time domain, we determine the target distance by pinpointing the location of the ZPO. Given the rapid acquisition speed of the oscilloscope, the proposed measurement system is also capable of simultaneously measuring dynamic displacements, even the instantaneous velocities of the target. We designate the proposed method as the dispersion-controlled dual-swept laser (DCDSL) method.

We conducted a simulation of the proposed distance measurement method using the nonlinear Schrodinger equation (NLSE), $$\frac{\partial A}{\partial z}=-\frac{\alpha }{2}A-i\frac{{\beta }_2}{2}\frac{{\partial }^{2}A}{\partial T^2}+\frac{{\beta }_3}{6}\frac{{\partial }^{3}A}{\partial T^3}+i\gamma {|A|}^{2}A,$$(6)where the value of $\alpha$ is set to zero. Based on dispersion control, the relevant parameters were set as follows: the pulse width of the ultrafast laser was 5 ps, and the length of the dispersion compensating fiber in the first segment for pulse stretching was 7400 m. For DCF1, its second-order and third-order dispersion coefficients (${\beta }_2$, ${\beta }_3$) are $0.2246\times {10}^{-24}$ and $-0.1486\times {10}^{-39}$, respectively. The temporal evolution of the measurement arm’s signal optical pulse as it propagates through the 7400-m-long DCF1 is shown in Fig. 2(a). The length of the dispersion compensating fiber introduced in the reference arm was 105 m. For DCF2, its second-order and third-order dispersion coefficients (${\beta }_2$, ${\beta }_3$) are $0.3846\times {10}^{-24}$ and $-0.1456\times {10}^{-39}$, respectively. The temporal evolution of the reference arm’s signal optical pulse as it propagates through the 7400-m-long DCF1 and the 105-m-long DCF2 is shown in Fig. 2(b). The pulse shapes of the two interferometric arms’ optical signals after dispersion control are shown in Fig. 2(c); the two pulses exhibit a slight difference in pulse widths. Additionally, to simulate realistic experimental conditions, we incorporated noise arising from the fiber and the detector into the simulation. The interferometric signals and the sweep frequency curves of the two interferometric arms’ optical signals when the measured distance is zero are shown in Fig. 2(d). It is clearly evident that the intersection point of the two frequency-swept curves coincides with the ZPO. Firstly, we varied the target distance $L$ in increments of $d=2\mathrm{mm}$ and collected 25 sets of corresponding interference signals, with a portion of the signal shown in Fig. 3. Figure 3 illustrates the movement of the ZPO across the stretched broadband interference signal as the target distance changes. The algorithm for locating the position of the ZPO through demodulation of the interference phase is depicted in Fig. 3. Figure 3(a) presents the collected interference signal. After applying envelope filtering, we obtained the AC component shown in Fig. 3(b). Subsequently, the phase derived from the Hilbert transform is depicted in Fig. 3(c). By further unwrapping this phase, we arrive at the phase curve illustrated in Fig. 3(d). To accurately locate the ZPO, we divide the phase curve shown in Fig. 3(d) into left and right segments and perform polynomial fitting on each segment separately. Finally, we compute the derivative zero points for both segments, which approximately coincide [as indicated by the blue line in Fig. 3(e)]. This common point corresponds to the ZPO.

The results of our simulation experiments are shown in Fig. 4. As previously mentioned, we introduced corresponding random noise in the simulation considering the actual detection noise of the experimental system, allowing us to adjust the system’s dispersion based on feedback from the computational results. As illustrated in Fig. 4(a), significant noise can be observed at the ZPO. Using the algorithm illustrated in Fig. 3 to solve for the ZPO, we obtain the linear results shown in Fig. 4(b). The slope of this ZPO’s time-distance curve is 0.285 ns/mm. Based on the identified position of the ZPO, the calculated target distance is presented in Fig. 4(c). We performed a linear fitting on the data, achieving a linearity of one and an $R^2$ value of 0.9999. To investigate the impact of system noise on the precision of locating the ZPO, we conducted nine sets of repeatability experiments at the same target position, each consisting of 1000 interference signals with the varying signal-to-noise ratios (SNRs). Figure 4(e) illustrates the error fluctuations in the distance measurement results from these nine repeatability experiments, while Fig. 4(d) presents the standard deviations of the results from the 1000 trials at different SNRs. It is evident that the precision of the algorithm is significantly influenced by the level of noise. As the SNR increases from 10 to 90, the standard deviation of the measurement results decreases from 178.3 nm to 0.057 nm, resulting in an improvement in precision by four orders of magnitude. This underscores the accuracy of the proposed method, demonstrating that our positioning algorithm performs exceptionally well under controlled environmental and system noise conditions.

To validate the capabilities of our fast distance measurement system, we demonstrated experiments for long-distance measurement of static targets and for small-range dynamic displacement measurement.

The proposed fast distance measurement system was first employed for the measurement of static targets. The experimental system diagram is shown in Fig. 1. This system utilizes an ultrafast pulsed laser with a spectral range of approximately 11.6 nm and a repetition rate of about 7.75 MHz. The spectral profiles of the pulses before and after amplification and stretching were acquired using an optical spectrum analyzer (OSA), as shown in Figs. 5(a) and 5(b), respectively. Figure 5(c) illustrates the time-domain pulse acquired by the oscilloscope, with an interval of approximately 129 ns. Figure 5(d) shows the result after the laser pulse is stretched and amplified through DCF1, resulting in a pulse width of approximately 20.9 ns. The amplified and stretched pulse is split into two paths by a beamsplitter with a splitting ratio of 10:90. The weaker path is received by a photodetector to serve as a time reference in the oscilloscope, while the stronger path is further split into measurement light and reference light using a beamsplitter with a splitting ratio of 30:70. The measurement light is emitted through a collimator after passing through a ring resonator. It travels through air, is reflected off the target, and is collimated again before being sent to a coupler by the ring resonator. This reference light interferes with the measurement light that has passed through DCF2, forming an interference signal. Finally, the signal is received by a photodetector with a bandwidth of approximately 20 GHz and is collected by the oscilloscope, which has a sampling rate of 50 GS/s.

During the experiment, the target was fixed on a motorized displacement platform with a full-range accuracy of 0.1 μm. We initially performed a full-range distance measurement with a step size of 1 mm over one pulse cycle, with a portion of the collected interference signals shown in Fig. 5(e). From the four selected interference signals, it is evident that as the target distance varies, the position of the oscillation zero frequency point shifts from left to right, exhibiting a directional movement. We measured a total distance of 56 mm across the entire signal range, with the positioning results of the oscillation zero frequency point indicated by the blue dots in Fig. 6(a) and the calculated distances shown by the red circles. The measured slope of the ZPO’s time-distance curve is 0.286 ns/mm, which is consistent with the simulation results. To further validate the accuracy of the measurement system, we adjusted the movement step size to 20 μm, collecting 153 sets of interference signals at each position. The distance measurement results and their corresponding standard deviations are illustrated in Fig. 6(b). Finally, we conducted repeatability experiments at a target distance of 1136 mm, collecting a total of 3070 sets of interference signals, whose superimposed eye maps are shown in Fig. 5(f). It is apparent that these interference signals are nearly overlapping, indicating good repeatability of the system. The statistical characterization of the calculated results is presented in Figs. 6(c) and 6(d), with the Allan variance reaching 4.25 nm shown in Fig. 6(e), which comprehensively reflects the measurement accuracy and stability of the system.

According to the distance calculation equation $\mathit{L}=\tau c/2n_{\mathrm{air}}$, we further evaluate the uncertainty of the proposed method in ranging: $$u_L=\sqrt{{\left(\frac{c}{2n}{u}_{\tau }\right)}^2+{\left(\frac{L}{n}{u}_n\right)}^2}.$$(7)

Based on Eq. (7), we can identify the main factors influencing the measurement uncertainty: the calculation of time delay $\tau$ and the air refractive index $n_{\mathrm{air}}$. Table 1 presents the individual contributions of various sources. The calculation of time delay $\tau$ is primarily related to the phase noise of the light source and the noise of PD. According to Fig. 6(d), the uncertainty associated with the time delay $\tau$ is 4.227 μm (the standard deviation). The air refractive index $n_{\mathrm{air}}$ is mainly influenced by environmental conditions such as temperature, atmospheric pressure, and water vapor pressure. The uncertainty caused by the air refractive index can be calculated using the corrected Edlen formula, where $u_n$ can be expressed as $$u_n=\sqrt{{(c_t{u}_t)}^2+{(c_p{u}_p)}^2+{(c_w{u}_w)}^2},$$(8)where $u_t$, $u_p$, and $u_w$ represent the uncertainties associated with temperature, atmospheric pressure, and water vapor pressure, respectively, while $c_t$, $c_p$, and $c_w$ denote the corresponding calculation coefficients. According to the corrected Edlen formula, $c_t$, $c_p$, and $c_w$ are $9.310\times {10}^{-7}$, $2.644\times {10}^{-9}$, and $3.786\times {10}^{-10}$, respectively. The uncertainty related to the refractive index is $1.12\times {10}^{-8}·L$. In the case of short-range measurements, the impact of the refractive index can be neglected. However, for long-distance precision measurements, variations in the refractive index will affect the final measurement accuracy. Based on the actual measurement conditions, we adopt the coverage factor $k=2$. Therefore, for a measurement distance of 1 m, the corresponding combined uncertainty is approximately 8.454 μm.Table 1.

Uncertainty Evaluation of the Distance Measurement

Following the measurement of static target distances, we conducted experimental validation of the proposed measurement method during a small-range dynamic displacement process. The target mirror was fixed on a motorized displacement platform, which was controlled by a host computer to move at a speed of 23.7 mm/s. The oscilloscope was set to a sampling rate of 50 GS/s, with each sampling session comprising $2\times {10}^8$ data points. The 30,700 sets of interference signals collected during a single dynamic experiment within 4 ms are shown in Fig. 7(a). It is evident that as the displacement platform moves at a constant speed, the oscillation zero frequency point drifts within one cycle, and the light intensity at the ZPOs exhibits a regular oscillatory behavior. Using the proposed DCDSL method, the measured dynamic displacement results are presented in Fig. 7(b), with the displacement over time illustrated in Fig. 7(c). Based on these measurement results, the average velocity of the target movement was calculated to be 23.59 mm/s, resulting in a relative error of 0.46% compared to the actual speed of the displacement platform. Then we record 30 repeated velocity measurement results in Fig. 7(d) and Fig. 7(e). According to the measurement results, the maximum error is $\pm 0.14\mathrm{mm}/\mathrm{s}$, majorly limited by the phase noise of the laser and noise of the PD. Here we also analyze the distribution of the data, which follows a Gaussian distribution. Standard deviation (STD) of these points is 0.09 mm/s. In the experimental process, we processed the interference signals collected within a 4-ms window and reconstructed the corresponding distance variation of the target. Therefore, the data processing is not performed in real-time. However, by incorporating deep learning algorithms into the data processing in the future, processing efficiency can be significantly improved, potentially enabling real-time dynamic displacement monitoring.

When measuring the displacement and velocity of moving targets, the reflected measurement signal experiences Doppler frequency shifts. To investigate its effects on the measurement results, we conducted simulation experiments based on the equation $\omega ={\omega }_0{[(c-v)/(c+v)]}^{1/2}$, where ${\omega }_0$ is the frequency of emitted light, $\omega$ is the frequency of reflected light, and $v$ is the velocity of the moving target. The velocity of the moving target was set to 20 mm/s, and 300 velocity measurement experiments were performed. The experimental results are shown in Fig. 8. Figures 8(a), 8(b), and 8(d) present the randomly acquired interference signals, the signals after envelope removal, and the phase curves under both conditions with and without Doppler frequency shifts. It can be observed that the signal curves in both cases almost completely coincide. The results of 300 velocity measurements also indicate that the maximum error caused by Doppler frequency shifts in velocity measurement is 5 μm/s, with an average relative error of 0.0115%, as shown in Fig. 8(c). Therefore, in the previous dynamic measurement experiments, the influence of Doppler frequency shifts was neglected. However, for future measurements of targets with higher velocities, it is essential to consider the effects of Doppler frequency shifts.

In summary, we have introduced a novel and effective method for achieving fast, full-range precise distance measurement without the complex systems or adjustable components. Its simple and effective system architecture demonstrates significant potential for practical applications, and its remarkable ability to measure the dynamic motion states of targets paves the way for the enhancement and realization of traditional DPI distance measurement systems, further broadening their application fields. Our method is also applicable to the measurement and sensing of other physical quantities based on length, such as velocity and acceleration, and can be integrated with deep learning algorithms for real-time output of target motion states. Due to the powerful learning ability of deep neural network models, by constructing some data labels, training neural networks can be used to detect the ZPOs. Furthermore, utilizing a light source with greater dispersion to stretch pulse widths can maximize the repetition frequency between the pulses, thereby expanding the measurement range of a single cycle. Additionally, employing improved linear group delay devices can further enhance system performance. These aspects warrant further investigation.

Acknowledgment. We thank the National Natural Science Foundation for help identify collaborators for this work.