Simple approach for extending the ambiguity-free-range of dual-comb ranging

Dual-comb (DC) ranging is an established method for high-precision and high-accuracy distance measurements. It is, however, restricted by an inherent length ambiguity and the requirement for complex control loops for comb stabilization. Here, we present a simple approach for expanding the ambiguity-free measurement length of dual-comb ranging by exploiting the intrinsic intensity modulation of a single-cavity dualcolor DC for simultaneous time-of-flight a nd D C distance measurements. This measurement approach enables the measurement of distances up to several hundred km with the precision and accuracy of a dualcomb interferometric setup while providing a high data acquisition rate (~2 kHz) and requiring only the repetition rate of one of the combs to be stabilized.

precise, fast, robust, and inexpensive ranging technologies which are applicable outside the laboratory frame.
Interferometric distance-measurement techniques based on homodyne or heterodyne detection offer high precision [1]. In their most basic form, however, they require continuous tracking and unidirectional motion of the target to clearly interpret relative phase changes during the measurement. Otherwise, absolute measurements can only be obtained modulo the ambiguity-free range given by λ/2, where λ is the optical wavelength. Alternatively, these ambiguity limitations can be overcome by using modulated or pulsed light. In these so-called time-of-flight (TOF) measurements the distance is measured by analyzing the temporal delay between the pulse coming back from the target and the pulse that has travelled along a known reference path [7,5]. However, such systems typically lack the high resolution of an interferometric system.
Since their invention in the late 1990s, optical frequency combs (OFCs) have become an indispensable tool in optical metrology [17,5]. Initially used for counting the cycles of optical clocks, they quickly revolutionized many other fields, including applications in spectroscopy and ranging [8]. Various ranging methods based on OFCs have been reported so far, including optical cross-correlation [10], synthetic-wavelength interferometry [16], and multiple wavelengths referenced to a stabilized comb [19], among others.
Dual-comb ranging systems [2] enable an ambiguity-free measurement range of c/frep (typically a few meters) in a single measurement via analysis of the beating signal between two optical frequency combs with slightly different repetition rates, i.e. different frequency spacings of the comb lines. A first comb is sent along a known reference path and along a signal path. At the end of both paths, the light is reflected back. The signal and reference reflections are sampled with a second comb. By evaluating the temporal shift of the respective interferograms the precision of the established TOF technique is improved by a factor of f rep,2 ∆frep [2]. Here, frep,2 is the repetition rate of the comb used for the remote sensing and ∆frep is the difference to the repetition rate frep,1 of the local reference comb. As these systems depend on a complex light source (two phase-stable frequency combs), different methods have been presented to simplify them while further improving their accuracy and precision [23,25,12]. Via additional evaluation of the interferogram carrier phases, for example, it is possible to achieve wavelength resolution. A detailed review on the DC ranging technique is provided by Zhu et al. [24].
Different approaches have been reported to overcome the remaining ambiguity limitation, e.g. by using the Vernier-effect [2]. The Vernier-effect method can be implemented by switching the role of the signal and reference path [2], by adjusting the repetition rate of the signal [22] or by adding an extra measurement path [14]. However, all these approaches either introduce moving parts [2], or increase the measurement time and experimental complexity. Besides the Vernier-effect, one can exploit nonlinear optics to enable absolute distance ranging [21], however, this method demands high peak powers.
An alternative high-precision ranging technique relies on using a single frequency comb and an electro-optical modulator (EOM) [11]. This elegant method offers high-precision and long-distance measurement, but depends on the detection of weak side-bands, requiring high reflectivity of the target.
Liu et al. used a single-cavity dual-comb [14] to reduce the complexity of the light source. However, they could only overcome the ambiguity limitation by exploiting the Vernier-effect.
This letter presents a new approach to extend the ambiguity-free range of DC ranging based on a single-cavity dual-color DC [4] without the need to use the Verniereffect or nonlinear optics. The ambiguity-free range is increased to c/∆frep (150 km for the laser parameters used in the presented experiment) within a single measurement, while allowing for data acquisition in the millisecond range.
Pulse trains emitted by single-cavity dual-color lasers are known to exhibit intensity modulations with a periodicity of ∆frep caused by intracavity pulse collisions [20,13]. We exploit this periodic modulation imprinted on the individual pulse trains as a marker to perform a secondary long-range TOF measurement, which allows us to extend the non-ambiguity range. Additionally, we use the shift of the dual-comb interferograms to measure the distance with high precision. The system presented in this letter is based on an all-polarization-maintaining fiber laser using Ytterbium as a gain material and a nonlinear amplifying loop mirror in combination with a non-reciprocal phase bias for mode-locking [9,6]. For a detailed analysis of how to operate and mode-lock these lasers, see [15]. A sketch of the dual-color dual-comb can be seen in Fig. 1.
The laser source used for this experiment is similar to the system presented in [4], however, the repetition rate of the laser is increased to around 92 MHz. A tuneable mechanical spectral filter [3] allows the all-PM laser to emit two independent pulse trains with different center wavelengths and slightly different repetition rates. As shown in [4], the two pulse trains may run in a different net intracavity dispersion regime each. The output spectrum of the laser is measured using an optical spectrum analyzer (Ando AQ6315A), see Fig. 2(a). One of the two pulse trains has a center wavelength of 1010 nm (running in the normal dispersion regime), an output power of 1.5 mW and a repetition rate of 92.455 MHz, while the other one has a center wavelength of 1085 nm (running in the anomalous dispersion regime), an output power of 2 mW and a repetition rate of 92.457 MHz. Additionally, we analyzed the laser output using a fast photodiode (Thorlabs DET08CL/M) connected to a microwave spectrum analyzer (Keysight PXA N9030B). Figure 2(b) shows a zoom into the two repetition rates of the two pulse trains operating inside the single laser cavity with ∆frep ≈ 1.95 kHz. The large spectral separation of ≈ 75 nm enables a clean spectral separation of two combs at the output of laser cavity by using a standard dichroic mirror with a cut-on wavelength of 1050 nm (Edmund Optics #87-041). We generate a spectral overlap between the two combs and thus enable a dual-comb beating signal via amplification and nonlinear broadening: first, we amplify comb 1 inside a PM-Yb single-mode fiber-amplifier to approximately 30 mW. The output of the amplifier is temporally compressed using a grating compressor consisting of two gratings (II −V I Transmission Grating: T-1000-1040-31.8x12.3-94) used in double-pass configuration. The temporally compressed (≈ 250 fs) light is coupled into a highly nonlinear fiber, similar to the one used in [18], with a coupling efficiency of ≈ 65%. Inside the fiber, comb 1 is spectrally broadened to overlap with comb 2 (see Fig. 2(c)) and subsequently sent through a long-pass filter with a cut-on wavelength of 1050 nm (Thorlabs FELH1050).
A detailed sketch of the complete ranging experiment can be seen in Fig. 3. The cross-phase modulation signal is imprinted on both combs and is visible as an intensity modulation on both pulse trains, see Fig. 4. For the TOF measurement, only comb 2 is required. In our implementation, the distance L to be measured is the path delimited by the reflective side of the partial reflector (50:50RBS, turquoise optic in Fig 3) and the reflective side of the end-mirror (SM, blue optic in Fig 3).
The light reflected by the 50:50RBS (starting point of L), is sent onto a fast photodiode (PD3) (Thorlabs DET08CL/M) and used as a reference. The light reflected back by the end-mirror is sent onto a second fast photodiode (PD2, Thorlabs DET08CL/M). The photodiodes are connected to an oscilloscope (Teledyne LeCroy WavePro 7Zi 6 GHz-40GS/s) to evaluate the time traces. The length is evaluated by measuring the temporal shift of the cross-phase modulation signal between the reference (PD3) and the signal (PD2). A typical measurement is shown in Fig. 4. Note that the detection of the two beams on different photodiodes is not strictly required, but simplifies the analysis of the two pulse trains by separating them directly. However, to compensate for the different distances of the two photodiodes from the reflector 50:50RBS, a seperate TOF calibration measurement needs to be performed. This calibration is easily implemented by sending the reference beam onto both photodiodes. The accuracy of the TOF measurement must only be on the order of a single cavity length (≈ 3.2 m), which is easily achieved.
Taking into account the calibration and the bandwidth of the two photodiodes (5 GHz), we achieve a precision of a few millimeters with the TOF measurement. The ambiguity-free length for this measurement is given by c/∆frep, which corresponds to 150 km for ∆frep = 2 kHz. Hence, we use the TOF measurement to calculate the integer division factor n = int L Lc where Lc = c/frep,2 is the optical cavity length. To additionally measure the distance with high precision, we simultaneously implement the dual-comb ranging-principle [2]. We use comb 1 to sample the signal beam (coming back from the end mirror, blue optic) as well as the reference beam (coming back from the partial reflector, turquoise optic) and measure the optical beating using photodiode PD1 (Thorlabs PDA05CF2). The fact that the reference beam is also able to reach PD1 despite being vertically polarized is due to the low extinction ratio of the PBS when used in reflection. To avoid optical aliasing, we filter the light in front of PD1 using a spectral filter (a grating and slit) and focus it on the photodiode.
The signal of PD1 is electronically filtered with a 70 MHz low pass filter (to filter away the repetition rate) and analyzed using an oscilloscope. Looking at the filtered time trace, we get four signals, each signal occurring periodically with a repetition Figure 4: Time-of-flight measurement using the intensity modulation on the pulse train as a marker. The turquoise trace (top) shows the reference measured at PD3 (Fig. 3). The dark blue trace (bottom) shows the signal measured at PD2 (Fig. 3). Due to the path length difference, the modulation is shifted. The time-delay between the modulation signals is used for coarse length calculation.
frequency of ∆frep. The signals correspond to the modulation signals occurring on pulse train 1 and pulse train 2, the dual-comb interferogram corresponding to the reference beam, and the dual-comb interferogram corresponding to the signal beam. A typical measurement trace is shown in Fig. 5.
Since PD1 is an amplified detector, power levels of a few µW for the reference and signal beam are sufficient to measure the interferograms. However, PD2 and PD3 are biased detectors (low cost and high bandwidth), requiring a power of around 50 µW to achieve an adequate signal-to-noise ratio for evaluation of the time traces. The signal-to-noise ratio could be improved by using amplified fast detectors. In this case, a few µW signal would be sufficient to perform the ranging measurement, making it a promising candidate for applications outside the laboratory frame. The time-trace of PD1 is evaluated by measuring the time delay between the reference and signal interferogram. The time delay between the two interferograms is given by where Li is the short-scale measurement length as defined below. Li depends on the temporal position of the signal interferogram relative to the reference interferogram.
To identify the reference and signal interferogram, we set different power levels, i.e. different amplitudes, of the interferogram. We must distinguish between three cases: • Case 1: The signal is delayed compared to the reference interferogram (this is the case for the time-trace plotted in Fig. 5): Li = (L mod Lc).
• Case 2: The reference is delayed compared to the signal interferogram: Li = Lc − (L mod Lc). • Case 3: The reference is exactly at the same position as the signal interferogram: The total length of the distance measured is then given by:  To achieve sufficient accuracy, frep,2 and ∆frep,2 can either be tracked or stabilized.
Here, we stabilized frep,2 (by stabilizing the 15th harmonic frep,2 ≈ 1.386 GHz) to a frequency generator (Rohde&Schwarz SMF 100A) using a piezo-controlled end mirror inside the laser resonator, and we tracked ∆frep by measuring the periodicity of the cross-phase modulation signal in the time trace. We verified the capability of our system to perform single-shot measurements over a large ambiguity-free range by measuring the path lengths of different multi-pass cell configurations. We varied the measurement distance 2L (to the end mirror and back) from less than a single cavity length (measurement 1 in Fig. 6) to over more then 9 cavity lengths (measurement 2 in Fig. 6) and up to more than 24 cavity lengths (measurement 3 in Fig. 6). Using the presented approach, we were able to measure the different distances without any ambiguity in a single sweep (measurement time = 1/∆frep = 500 µs). In each configuration we changed the path-length in increments of 2 mm (by scanning the end-mirror in 1-mm steps using a translation stage). For each measurement point, we took 15 measurements and got an error of σ < 100 µm. This precision is limited by the dual comb parameters. However, it can be significantly improved by increasing frep while decreasing ∆frep at the expense of an increase in measurement time.
The presented measurement principle enables single-shot high-precision absolutedistance ranging over an ambiguity-free measurement range of around 150 km by combining the advantages of TOF measurements and dual-comb ranging. The system is based on a single light source with much less stringent requirements on stabilization (only frep,2 needs to be either stabilized or continuously measured) compared to two mutually phase-stable (locked) frequency combs. In addition, the measurement does not rely on nonlinear processes. Hence, it only needs µW-level reflected powers returning to the sensor. The high precision, large ambiguity-free measurement range and fast data Using both signals, we are not limited by the ambiguity range of traditional dualcomb ranging. The dark blue section is measured at a starting distance L1 within a single cavity length (Lc = c/frep), which is the maximum ambiguity range for conventional dual-comb-ranging. The light blue section is measured with a distance which is L2 > 9Lc and the turquoise section is measured with a distance L3 > 24Lc.
To verify the precision of the measurement, we changed the end-mirror position using a micrometer translation stage.
acquisition in combination with the high sensitivity, make the presented measurement approach an ideal system for field applications.