Figure 1a illustrates the conceptual design of our DFT based dual comb vernier rangefinder. To measure the TOF of optical pulses, we utilize two phase-locked soliton fiber laser combs, one as the signal comb and the other as the local comb. Among them, the signal comb is divided into two paths, one is used as a probe, and the other as a reference. At the receiving terminal, we analyze the synthesized signal carrying ranging information through the DFT. The two laser combs have slightly different repetition rate *Δf**r*, which enables us to realize a maximum NAR = *c*/2*Δf**r* by conveniently swapping the signal comb with repetition rate *f**r1* and local comb with repetition rate *f**r2* [24], here *c* is the light velocity. Figure 1b shows the operation principle. In a dual comb ranging system, the TOF *Δτ* is enlarged to be *Δt* = *mΔτ*, here *m* = *f**r1**/Δf**r*, also known as the zooming factor. In this operation, the sampling step *ΔT* equals to *|1/f**r1* *– 1/f**r2**| = Δf**r**/f**r1**f**r2*, while the sampling period *T**update* is 1/*Δf**r*. The measured distance *L* = *cΔτ/*2.

In conventional dual comb ranging methods, the *Δτ* was typically estimated via fitting the interferometric pulse envelope. In this case, for meeting the Nyquist sampling theorem, there is an inherent limitation *Δf**r* < *f**r1**f**r2*/2*B*, here *B* is the 3-dB spectral bandwidth of the dual comb. This is a main reason that typical fiber dual comb system cannot measure rapidly. On the other hand, in practice, *f**r1* and *f**r2* are not necessarily integer multiples of their repetition frequency difference (i.e. *m* is not an integer), thus their pulses are difficult to be strictly aligned in the time domain. Specifically, the pulse-to-pulse offsets could be *τ**1* to *τ**4*, here *τ**1* + *τ**2* = *τ**3* + *τ**4* = *ΔT*. As Fig. 1b shows, the left side (blue box) shows the region that the reference pulses overlap with the local pulses, while the right region (red box) shows the region that the returned probe pulses with a TOF delay overlap with the local pulses, the accurate TOF *Δτ = T**r1* – *N(T**1**-T**2**)* + *τ**1* – *τ**3*, or *Δτ = T**r2* – *N(T**1**-T**2**)* + *τ**2* – *τ**4*. More calculations are shown in supplementary note S1. Therefore, the distance to be measured can be expressed as:

$$L=\frac{{({T_{r1}} - N\Delta {T_r}+{\tau _1} - {\tau _3})c}}{2}=\frac{{({T_{r2}} - N\Delta {T_r}+{\tau _2} - {\tau _4})c}}{2}$$

1

In our scheme, we break the Nyquist’s limitation and realize the measurement of *τ**1* to *τ**4* with extremely high resolution. In dual comb ranging system, to identify *τ**1* to *τ**4* is not easy, as commonly they are too small (e.g., down to fs level) to well detect, but the DFT offers a way to obtain high quality pulse-to-pulse interferograms, as Fig. 1c shows. According to the space-time duality, the demonstration of spectral information in time domain can be analogized to the spatial Fourier transform based on Fraunhofer’s diffraction [39]. Large group velocity dispersion adds a linear frequency chirp and broadens the pulses. A broadened pulse can have an enlarged temporal width *δ*,

$$\delta =\left| {\frac{{ - 2\pi c{\beta _2}}}{{{\lambda ^2}}}} \right|L\Delta \lambda$$

2

Here *β**2* is the group velocity dispersion parameter, *λ* is the central wavelength of the comb pulses, *L* is the length of dispersive element, and *Δλ* is the 3-dB spectral bandwidth of the pulse. After DFT, the temporal shape of the broadened pulse is the same as its original spectral shape. Figure 1d shows the calculated result. For a 1550 nm band comb laser, *δ* linearly increases with *|β**2**|LΔλ.* On the other hand, for avoiding pulse-to-pulse overlap, *|β**2**|L* should be smaller than *λ**2**/(2πcΔλf**r**)*, here *f**r* is the larger one of the *f**r1* and *f**r2*. Besides, due to the chirping effect, once two stretched pulses overlap, they would interfere with each other in a photodetector. The frequency of their interferometric fringes is given by:

$${f_i}=\frac{\tau }{{2\pi |{\beta _2}|L}}$$

3

Here *τ* is the delay of the two pulses. Figure 1e maps the parametric space of the |*β**2*|*L*, the *τ* and the *f**i*. To satisfy the far-field diffraction condition [44], *|β**2**|L* must be much larger than *τ**0**2**/(2π)*, here *τ**0* is the original pulse duration. In experiment, we can control the dispersion parameters to get a proper coefficient for converting *τ* to *f**i*. In a nutshell, by using the DFT, we successfully convert the absolute offset measurement of two pulse separations in time domain into a dispersion-related interferometric measurement, which offers a much higher accuracy.

Figure 1f demonstrates our experimental setup. The dual comb source consists of two stabilized mode-locked fiber laser combs. The comb 1 and comb 2 respectively have repetition rates of 24.465 MHz and 24.55 MHz, offering a *Δf**r* = 85 kHz. They are phase locked on two radio-frequency clocks through two digital feedback loops. After averaging, the minimum uncertainty of the dual comb repetition rates is smaller than 0.25 mHz (10− 10). More characterizations of the dual comb source are shown in supplementary note S2. The comb 1 is divided into two paths, one works as the probe and the other serves as the reference. The probe light is used for ranging based on the TOF mechanism. The returned probe pulse and the reference pulse (comb 1) are coupled together and launched into a dispersion compensating fiber (DCF) section, while the local pulse (comb 2) is launched into the same DCF section in opposite directions. The DCF provides a *β**2* = 110.94 ps2/km. All the pulses are time-stretched in this process. By using 30.39 km DCF (*β**2**L* = 3371.47 ps2), our combs with a 3-dB spectral bandwidth ≈ 6.84 nm (368 fs duration) can be stretched to 18 ns. Finally, the local-probe-reference interferogram is detected by a fast photodetector (25 GHz) and analyzed in a high-speed oscilloscope (16 GHz analog bandwidth).

For a target with a fixed position, the interferometric trace of the synchronized pulse trains (including the returned probe, the reference and the local pulses) is shown in Fig. 2a. In a macroscopic view, it appears similar to the results based on conventional dual-comb ranging method, one can roughly estimate the distance by *T**update* and the enlarged TOF. Within the TOF duration, we see *N* = 171. But in a microscopic view, we can see more detailed information, as every pulse has been temporally stretched due to the large dispersion, and interferences occur in the pulse-to-pulse overlapping regions. When zooming the time in, we demonstrate the interferometric details in Fig. 2b. Due to the DFT, one can observe both the ‘local-reference’ fringes plotted in the blue boxes, and the ‘local-probe’ fringes plotted in the orange boxes. These fringes reflect the slight pulse-to-pulse mismatches (*τ*). Specifically, the fringe frequencies corresponding to *τ**1* and *τ**2* are *f**i,1* = 3.66 GHz and *f**i,2* = 3.02 GHz; meanwhile the fringe frequencies corresponding to *τ**3* and *τ**4* are *f**i,3* =1.77 GHz and *f**i,4* = 4.91 GHz. Referring the Eq. (3), we can accurately obtain *τ**1* = 77.52 ps, *τ**2* = 64 ps, *τ**3* = 37.48 ps, and *τ**4* = 104.04 ps. These values meeting will with the period difference of the comb 1 and the comb 2 (141.52 ps). Therefore, referring Eq. (1), the measured distance is 2.507189 m.

Figure 2c shows the fast Fourier transform spectra of these fringes (*f**i1* to *f**i4*). Thanks to the high stability of the interference, the linewidth of each Fourier peak is < 3 MHz, while the spectrally detectable resolution of the frequency of fringes is < 90 kHz, suggesting a temporal resolution < 0.533 fs for identifying the *τ*. For ranging, such a performance suggests an in-principle resolution on 240 nm level in single shot measurement. In practice, the system performance is mainly limited by the stability of the dual comb source. More discussion is shown in supplementary note S3. Moreover, the signal-to-noise ratio (SNR) of the lines is commonly higher than 30 dB, suggesting that our measurement can effectively resist the intensity jitter of the pulses. We also note that in our measurement, *τ**1* + *τ**2* = *τ**3* + *τ**4* = *ΔT* is a fixed number. In extreme case, the maximum *τ* may equal to *ΔT* (141.52 ps). Referring Eq. (3), the bandwidth of our photodetector & oscilloscope must be higher than 6.684 GHz.

Then we record 1041 repeated ranging results in Fig. 2d. For a single-shot detection, the maximum error is ± 1.3 µm, majorly limited by the instability of the fiber links. Here we also analyze the distribution of the data, which follows a Gaussian distribution. Standard root-mean-square-error (RMSE) of these points is 272.5 nm. Figure 2e plots the statistical Allan deviation of our ranging measurements. For single shot measurement, the typical detection limit is 262 nm. After 1041 averages, the detection limit approaches 2.8 nm. Figure 2f shows the ranging distance. For a single comb with a repetition rate of 24.465 MHz, the NAR is about 6.1 m, which is limited by its pulse period. But after exchanging the role of dual comb, the NAR extends to 1.75 km, determined by the dual comb repetition rate difference *Δf**r*. These results show that the measured linearity and consistence of the dual comb based Vernier ranging system are good. In supplementary note S3, we compare our ranging system with the conventional TOF method, using the same dual comb light source.

Moreover, the DFT based dual comb ranging can uniquely eliminate the “dead zones”, which has been a widely-recognized challenge in any ranging methods rely on TOF. For a typical TOF measurement, when the probe pulse and the reference pulse are close to each other (the distance approaching 0 or an integer multiple of the *L**NAR*), one cannot distinguish them due to the Rayleigh criterion. We show the dead zones schematically in Fig. 3a. As a consequence, pulse fitting won’t make sense. To solve this problem, orthogonal polarization multiplexing was applied, but introduced addition complexity [43]. In addition, if the detected pulse has deformation (i.e., not in a perfect sech2 or Gaussian shape), it will be more difficult to deterministically fit its peak position.

However, in our scheme, by analyzing the TOF via DFT based interference, we smartly address this problem. Figure 3b plots a case in experiment, here we set the real distance *L**R* = 2 mm, resulting in a temporal offset of the probe-reference pulses of 13.3 ps. Typically, the small offset between the ‘local-reference’ peak and the ‘local-probe’ peak is difficult to retrieve. But our DFT scheme can distinguish the minor pulse-to-pulse offset clearly, since we can measure the interferometric fringes in the broadened pulses. By zooming the above measured traces in, we demonstrate that the result based on the DFT demonstrates clear interferometric fringes, illustrating a *f**i* = 628.2 MHz. This verifies *L**R* = 2 mm accurately.

Specifically, Fig. 3c plots the fast Fourier transform spectra (in log scale) of more cases, where the frequencies of fringes increase when the ranging distance scales from 10 µm to 10 mm. Most of these cases are in the dead zone of a conventional TOF based dual-comb ranging system. In our scheme, the minimum detectable *τ* is mainly determined by the frequency resolution of the Fourier transform peak (hundreds of nanometers). Therefore, the DFT based dual comb ranging is in-principle ‘dead zone free’. Finally, we show the measured distance versus the real distance in Fig. 3d. The DFT based dual comb ranging enables linear and reliable measurements down to sub µm level.