Ultrafast and precise distance measurement via real-time chirped pulse interferometry

Since the first demonstration in 2000 [8], frequency comb-based absolute distance measurement have developed for over 20 years, and various schemes have been investigated in great depth, such as inter-mode beat [8, 9, 10], pulse cross correlation [11, 12, 13, 14, 15, 16, 17, 18], dual comb interferometry [19, 20, 21, 22, 23, 24, 25, 26, 27, 77], multi-heterodyne interferometry [28], pulse-to-pulse alignment [29, 30], and stationary phase evaluation [31, 32]. Additionally, frequency comb can also work as the reference source when using one or more continuous wave lasers to measure the absolute distances. The problem may be that the whole system involves multi optical sources to increase the system cost and complexity [33, 34, 35]. Very recently, optical ranging with microcombs has attracted the interest of the scientists due to their small footprint and low power consumption. All the methods mentioned above can be used to measure the distances using the soliton microcombs, and the attempts of the pulse cross correlation [18], dual-comb interferometry [36, 37], and dispersive interferometry have already been reported [38, 39]. To date, the physics of the microcombs is becoming clear, but the long-term reliable operation of the soliton state is still relatively challenging. In addition, the commercialization of the microcombs needs to be accelerated. For any kind of measurement methods, there are several parameters to evaluate their performances, including precision/accuracy, ambiguity range, measurement speed, and system cost [20, 30]. Generally speaking, improvement in one parameter is achieved at the exchange of degradation in other parameters.

One important group among these methods is the spectral interferometry, also known as dispersive interferometry, in which the spectrum will be modulated when the pulses interferes with each other [40]. In this case and often, an optical spectrum analyzer (OSA) is exploited to measure and hold the spectrograms. The following data process including the Fourier transform and the phase unwrapping can extract the distance information. Yet, limited by the resolution of the optical spectrum analyzer, the spectrograms can not be reconstructed any more when the pulse-to-pulse interval is too large [58, 59]. This means that, there are dead zones along the measurement path, as shown in Figure 1(a). To overcome this limitation, virtual imaging phase array VIPA and a grating can be used to disperse the light in a two-dimensional plane, and a CCD camera can consequently image each individual mode [41]. This scheme can be considered as homodyne interferometry with a number of wavelengths, and the dead zones can be therefore removed. But, frequency comb with larger repetition frequency (e.g., 1 GHz) is needed, otherwise the individual mode can not be resolved. It is necessary to mention that the problem of the dead zone can be relaxed if using the electro-optic combs [4, 25] or the micro combs [36, 37], since the repetition frequencies of these combs are inherently large over tens of GHz. Therefore, the pulse-to-pulse interval is relatively small. Additionally, for spectral interferometry, the small distances of +$l$ and -$l$ in the vicinity of $N$×$L_{pp}$ can not be directly distinguished [40]. + and – represent the positions before and after the reference pulse, $N$ is an integer, and $L_{pp}$ is the pulse-to-pulse length of the frequency comb.

Another problem of using the CCD-based OSA is that the measurement speed is limited by the frame rate of the camera, which is only about kfps in general. Similarly, the speed of the OSA based on the long translation stage is determined by the moving speed of the mechanical stage [31]. In recent years, a new kind of real-time OSA has found a number of applications, which relies on the theory of dispersive Fourier transform DFT [42, 43], also referred to as time stretch. In DFT, the femtosecond pulses are greatly stretched, and then detected by a fast photodetector and a fast AD card. Now, technique of time stretch has been used in the fields of optical ranging [44, 60], imaging [45], spectroscopy [61, 62, 63], intelligent control of the pulses [64], and ultrafast phenomenon observation [46, 65, 66, 67, 68, 69]. In the case of long distance measurement using time stretch-based dispersive interferometry, the spectrograms in the frequency domain can be translated into the time domain, and more important the measurement speed is directly determined by the repetition frequency of the pulsed laser, which can be as high as several nanoseconds per measurement [70, 71]. However, limited by the bandwidth of the photodetector and the AD card, the spectrograms can not be reconstructed either for this new type of OSA when the distance is too large. This means that, the dead zones still exist in the measurement path, as shown in Figure 1(a).

If the reference pulses are chirped by a dispersive element (e.g., a piece of fiber, a pair of gratings, or a piece of glass), chirped pulse interferometry will occur [47, 48]. Different from the classical spectral interferometry, the modulation frequency of the spectrograms for chirped pulse interferometry is not constant any more [72, 73]. There is a widest fringe in the spectrograms, where the arm length difference is balanced at the corresponding wavelength [49]. Compared to the spectral interferometry, the measurable range is expanded. However, the dead zones still exist due to the relatively small dispersion introduced in the reference arm, as shown in Figure 1(c). If the dispersion amount is sufficiently large, the measurement pulse can always meet the reference pulse in space, and the dead zones can be thus removed in principle, as shown in Figure 1(e). Considering the measurement speed, we can also use the technique of DFT in chirped pulse interferometry. Nevertheless, the limitation of the bandwidth of the photodetector and the AD card is not very strict, since a widest fringe exists in the spectrograms, whose width keeps constant for different distances. In contrast, for the spectral interferometry, the modulation frequency linearly increases with increasing the distance. Here, we summarize the recent progress for the field of distance measurement in Figure 2, mostly focusing on the spectral interferometry and chirped pulse interferometry. Some reviews can be found in [74, 75, 76].

In this work, we describe a method based on real-time chirped pulse interferometry for absolute distance measurement. A highly dispersive fiber is used to broaden the reference pulse. To make sure that the measurement pulses can always overlap with the reference pulse, we change the repetition frequency to adjust the relative delay between the measurement and the reference pulses due to the difference of the pulse index. Therefore, the spectrograms can be always observed along the whole measurement path, and the dead zones can be removed. The distances can be determined via two steps of phase measurements, which are successively the flight time of the pulses and the phase of the synthetic wavelength. In the detection of the spectrograms, we use the highly dispersive fiber to stretch the pulse, and real-time detect the spectrograms with the help of fast photodetectors and oscilloscope. In this case, the measurement speed is directly linked to the repetition frequency, up to about 250 MHz (4 ns for one single measurement). The experimental results show that the distances can be measured with high accuracy and precision, high measurement speed, and high resolution. This paper is organized as follows: Section 2 introduces the measurement principle. The distances can be measured via the slope of the phase difference between the two spectrograms, even if the phase of one spectrogram is not linear any more. The experimental setup is described in Section 3. Section 4 demonstrates the experimental results, in the long distance measurement and ultrafast measurement, respectively. Uncertainty budget is discussed in Section 5. Finally, we summarize this work in Section 6.

Assume that, the spectrum of the frequency comb is $E(\omega)$. The reference pulse $E_{ref}(\omega)$ can be expressed as $\alpha$$E(\omega)$ in the frequency domain, and $\alpha$ is the power factor. The measurement pulse $E_{meas}(\omega)$ can be written as $\beta$$E(\omega)exp(-i\tau\omega)$, and $\beta$ is the power factor corresponding to the measurement pulse. $\tau$ is the time delay between the reference pulse and measurement pulse.

The local oscillator $E_{LO}(\omega)$ can be expressed as $\gamma$$E(\omega)\cdot{exp(i\psi)}$, where $\gamma$ is the power factor, and $\psi$ is the phase change caused by the long dispersive fiber, which can be Taylor expanded as:

where $\beta_{m}$ is the coefficient of the group velocity dispersion at different orders, $m$ is an integer, $\omega_{c}$ is the center angular frequency, and $z$ is the fiber length. The spectrogram corresponding to the reference pulse can be expressed as:

where $E_{ref}(\omega)$ is the spectrum of the reference pulse, and $\phi_{r}$ is the initial phase difference between the reference pulse and the local oscillator. Based on Equations (1) and (2), we find that, different from the classical spectral interferometry, the spectral phase in the chirped pulse interferometry is not linearly related to $\omega$, which means that the modulation frequency of the spectrograms is not constant. There will be a “widest fringe” in the spectrograms. Similarly, the spectrogram corresponding to the measurement pulse can be written as:

where $E_{meas}(\omega)$ is the spectrum of the measurement pulse. Similar with the principle of dispersive interferometry in Ref. [40], the phase change from Equation (2) to Equation (3) can be given by:

Therefore, the distances can be determined by:

where $c$ is the light speed in vacuum, and $n_{g}$ is the group refractive index of air. From Equation (5), we finally reach an expression related to the phase slope, exactly the same as spectral interferometry and dual-comb interferometry. This step is actually to measure the distance by time-of-flight method. As mentioned before, we try to efficiently remove the dead zones in the measurement path, and the spectrograms with one widest fringe shall be observed at any distances, as long as the dispersion is sufficiently large to cover the pulse-to-pulse length of the comb source (i.e., $c/(n_{g}{f_{rep}})$). In addition, the ambiguity range can be easily expanded to km level by slightly changing the repetition frequency. To make this idea more clearly, the comparison between the spectral interferometry and the chirped pulse interferometry with $1.1$×$10^{24}$ $\rm{rad/s^{2}}$ chirp rate is shown in Figure 4. We find that, the method based on the phase slope can be universally used in the comb-based distance measurement, e.g., spectral interferometry, dual-comb interferometry, and chirped pulse interferometry.

In principle, the pulse width should be broadened to about 1/$f_{rep}$ to make sure that the pulses can always meet each other in space, to further remove the dead zones. Please note that, the pulse width here does not mean the full width at half maximum, but means the total duration time of the pulse. In this case, extremely long fiber is required, and the stabilization of such a long fiber is not easy (In fact, we used the 300 m long fiber to perform the distance measurement in our experiments, see APPENDIX). Although the long fiber can be stabilized by using cavity-stabilized laser, the system would become complicated and more expensive. In this work, we use a relatively short fiber, and change the repetition frequency of the frequency comb to ensure that the pulses can always overlap at arbitrary distances [50]. Consequently, the measurement principle will be slightly updated. The spectrogram corresponding to the reference mirror can be expressed as Equation (2) when the current repetition frequency is $f_{rep1}$. Here, the length difference between the reference pulses and the local oscillator can be calculated as:

where $M_{ref}$ is the pulse index difference between the pulses, which is related to the fiber length and the index. In Equation (6), the first term at the right side is the integer part of the pulse-to-pulse interval, i.e., $c/(n_{g}\times f_{rep1})$, due to the pulsed nature of comb laser. The second term is the fractional part $d_{ref}$.

In the case of the target mirror, the repetition frequency shall be increased to generate the spectrograms again (expressed as Equation (3)), and the updated repetition frequency is $f_{rep2}$. Consequently, we can calculate the length difference between the measurement pulses and the local oscillator as:

where $d_{meas}$ is the fractional part.

Therefore, the measured distance (i.e., the difference between $L_{meas}$ and $L_{ref}$) can be calculated as:

We measure the distance by two steps. At the first step, the distances can be coarsely measured with a larger uncertainty based on Equation (8). Further, we use the synthetic wavelength interferometry to measure the distance. Since laser frequency comb is composed of numerous wavelengths, two wavelengths can be picked up to form a virtual synthetic wavelength, and the distances can be expressed as:

where $r$ is the integer part, $e$ is the fractional part of the synthetic wavelength, and ${n_{g^{\mathrm{{}^{\prime}}}}}$ is the group refractive index. The synthetic wavelength $\varLambda$ can be calculated as:

where $\lambda_{1}$ and $\lambda_{2}$ are the chosen wavelengths. ${n_{g^{\mathrm{{}^{\prime}}}}}$ can be written as:

where $n_{1}$ and $n_{2}$ are the corresponding phase refractive index of air. Please note that, the access condition from the first step to the second step is the measurement uncertainty of the first step is smaller than half the synthetic wavelength to obtain the right integer $r$. Finally, the distances can be finely determined by the two cascaded steps.

To evaluate our results, a commercial distance meter (Renishaw XL80) is used to measure the same distance simultaneously in the HeNe interferometer, as depicted in Figure 5. The target corner retroreflector of the commercial distance meter is fixed on the same carriage. Since the cw counting interferometer can only measure the incremental distances, we use the distance variations measured by our system to compare to the reference values. At the initial position, the reference distance meter is reset to zero, and we can measure an absolute distance of $L_{0}$. Then, the target mirror is moved far away from the beam splitter. The reference distance meter can measure the distance increment $\Delta$$L$, and meanwhile the updated absolute distance of $L_{1}$ can be determined by our system. Consequently, $(L_{1}-L_{0})/2$ can be used to compare with the reference value of $\Delta$$L$. The environmental conditions are well controlled and measured, which can be used to calculate the air refractive index based on the empirical equations. To suppress the possible cosine error in long distance measurement, both the measurement beams of the HeNe interferometer and the frequency comb interferometer are carefully aligned to be parallel to the optical rail. The Abbe offset between the targets of the HeNe interferometer and the frequency-comb interferometer is less than 100 mm.

In Equation (14), the first term at the right side is related to the synthetic wavelength $\varLambda$. In our experiments, laser frequency comb is referenced to the Hydrogen maser with 7.5×$10^{-14}$ stability at 1 s, and each frequency marker holds the same frequency stability. We use a well-calibrated optical spectrum analyser to measure the wavelengths. Therefore, it is reasonable and sufficient that the uncertainty due to the clock stability can be evaluated to be $10^{-13}L$. The precise mapping from the frequency domain to the time domain can also make contribution to the first term, which is related to the stability of the long fiber and the stability of the clock source for the oscilloscope. The long fibers used in the detection unit are well vibration-isolated and temperature controlled, and the temperature drift can achieve 1 mK over 2 days. In addition, the oscilloscope is locked to the Hydrogen maser. We measure the mapping for 2 days continuously, and for 1565 nm the stability of the corresponding time can reach 0.013 fs, which corresponds to 4.1×$10^{-9}L$ uncertainty. The second term is related to the group refractive index ${n_{g^{\mathrm{{}^{\prime}}}}}$. We measure the air refractive index using Ciddor formula based on the environmental sensor network. The inherent uncertainty of the empirical formula is 2×$10^{-8}$, and the measurement of the environmental conditions also plays an important role. With consideration of the uncertainty of the sensors, the environmental stability, and the environmental inhomogeneity, the uncertainties of the temperature, air pressure, and humidity are 28 mK, 12 Pa, and 1.6$\%$, which respectively correspond to 2.6×$10^{-8}$, 3.1×$10^{-8}$, and 1.7×$10^{-8}$ uncertainty of air refractive index. The combined uncertainty of the air refractive index can thus reach 9.7×$10^{-8}$ (k=2), which is 9.7×$10^{-8}L$ for the distance uncertainty. The third term is due to the measurement of the fractional part, which can be estimated by the standard deviation. At each position, we measure the distance by 1000 times with ultrafast measurement speed, and the standard deviation is below 0.6 $\upmu$m. We use twice the standard deviation to evaluate this part, which is less than 1.2 $\upmu$m. Since the frequency comb laser and the reference distance meter do not share the same target mirror, Abbe error could exist. The Abbe error is always below 2 $\upmu$m along the 50 m travel, corresponding to a 20 $\upmu$rad yaw error of the rail. In addition, the stability of the translation stage is below 0.2 $\upmu$m over 2 hours. Finally, the combined uncertainty of the distance measurement is $\rm[(2.4\upmu{m})^{2}+(9.7\times 10^{-8}\cdot{L})^{2}]^{1/2}$, which can well cover the results in Figure 11(b). The uncertainty evaluation is summarized in Table I.

Further, we would like to give an analysis about the measurement resolution, which is very important in the practical distance measurement. In general and basically, the measurement resolution is related to the linewidth of the optical source, the phase resolution of the measurement system, and the short-term stability of the environment. Right now, the optical source can be locked to a reference cavity or an atomic clock (the Hydrogen maser in this work), and the resulting linewidth of the optical source can reach Hz or even sub Hz [52], corresponding to much better than pm resolution. Here, the phase resolution of the phasemeter could make the main contribution. In our work, we use the oscilloscope and the post data process to measure the phases. The oscilloscope is locked to the Hydrogen maser, and the digital noise and the ADC (AD card) noise can be neglected. In this case, we can use the value provided by the manufacturer, and the phase resolution can reach 0.01 degree. Therefore, the length resolution can achieve 0.01/360×244.9$\upmu$m (synthetic wavelength), and that is 6.8 nm. We find, smaller synthetic wavelength can lead to better length resolution. If we use the single wavelength to measure the distances, the measurement resolution can be expected to be better. However, generally speaking, coherent averaging is needed to reach such a high performance for high-speed measurement. Of course, there are the other methods (e.g., optical method) for the phase detection. Finally, the stability of the environment could affect the measurement resolution. In the case of the long distance measurement, it is difficult to maintain the environment homogeneity and stability. Despite this, the nanometric resolution has been reported in the previous reports [29, 30].

In discussion, first the distances can be precisely determined via the spectrograms shown in Figure 8 actually. In this case, the mapping from the frequency domain to the time domain is not required. Therefore, the calibration step in Figure 7 is not needed any more. In general, the schemes based on the chirped pulse interferometry involve one calibration step, so that the distance can be measured via the position of the widest fringes. We would like to show that, the distance can be also measured by the slope of the phase difference in the chirped pulse interferometry, even if the phase of one spectrogram is not linear, but quadratic. The main advantage of the real-time chirped pulse interferometry is that the measurement speed can be improved to be record high, i.e., 4 ns per measurement. Second, in our experiments, one long fiber link is involved to stretch the femtosecond pulses in the local oscillator. The active stabilization of the fiber in the local oscillator is not strictly required. The two spectrograms used to measure the distances always share the same fiber in local oscillator. Therefore, the fiber fluctuations can be cancelled out in real time. Finally, we use 90-m dispersion compensation fiber in the local oscillator, and therefore the repetition frequency should be changed so as to remove the dead zones. In fact, at the very beginning we used 300-m dispersion compensation fiber in the local oscillator, and the adjustment of the repetition frequency is not needed in that case. The problem is that less fringes can be obtained on the oscilloscope if with such a long fiber because the sampling rate of the oscilloscope is limited. Another is that the stabilization of such a long fiber is not easy, and our experimental results also show that shorter fiber can lead to a better performance of distance measurement. Please find more details in APPENDIX. In addition, we use a fully-stabilized frequency comb in our experiments, and the locking circuits indeed make the whole system bulky and expensive. We consider that, the stabilization of the repetition frequency is needed. If the repetition frequency is free, the pulse-to-pulse interval will be free, and the spectrograms can not be sufficiently stable. The position of the widest fringe will fluctuate, leading to the performance degradation. The $f_{ceo}$ locking could not be strictly required, because the spectrograms in chirped pulse interferometry is related to the temporal positions of the wave packets. If the optical phase of one wavelength is used in the distance measurement, the $f_{ceo}$ locking is also suggested. We would like to discuss the implementation of the soliton microcombs in the real-time chirped pulse interferometry. The repetition frequency of microcomb is often high at tens of GHz because of the small footprint. If with 10 GHz repetition frequency, the pulse period is 100 ps. Consequently, the sample number is 8 with 80 GSa/s sampling rate, which makes the observation of the spectrograms inaccessible. Therefore, to observe the birth of microcomb solitons is rather difficult.

To date, the technique of frequency combs is becoming mature, and has seen a wealth of applications in both science and technology. However, most of the applications based on the combs are only at the level of laboratory demonstration, and there is still a long way to go to realize the practical applications in the industry. In fact, the limitation of the comb source itself has been solved, and portable and cost-efficient combs are available. Scientists have tried hard to develop the frequency-comb instruments, e.g., distance meter, 3D scanner, space-borne frequency combs, and spectrometer, etc [2]. But, the market share is not comparable to the traditional metrology instruments. We focus on the technique of distance metrology, and precision measurement of the physical quantity related to the distance. Considering the future field measurement using laser frequency combs, frequency comb Lidar (laser-based light detection and ranging) using the technique demonstrated in this work is feasible, which can realize 3D measurement at long range with the weak light detection [53]. If out of the lab, the air refractive index is going to make great contribution to the final uncertainty. The two-color or three-color method can be used to correct the air refractive index in real time in this case [54, 55]. Our work could provide a solution for the practical applications using frequency combs in future.