## Abstract

We report on the use of the acousto-optic frequency combs generated by frequency shifting loops as compact and versatile optical waveforms generators for pulse compression systems in the optical coherent domain. The high degree of tunability and mutual coherence of these sources permits an efficient use of the available detection bandwidth, and represent simple alternatives to broadband lasers that do not require fast electronics. The full, complex optical field is retrieved using heterodyne measurements in bandwidths as high as 20 GHz. Compression ratios up to 150 at 80-MHz repetition rate, with autocorrelation peak-to-sidelobe ratios in excess of 28 dB, are demonstrated. In a proof-of-concept ranging experiment, we obtain resolutions of 4 mm in free space at meter scales, limited by detection bandwidth. Systems based on frequency shifting loops thus enable compact implementations of the pulse compression concept in the optical coherent domain, for its use in general optical metrology systems.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Pulse compression radar systems are based on the increase of the time-bandwidth product (TBP) of a probe wave which, after matched-filter detection, is compressed to a single virtual pulse carrying the energy of the initial probe waveform. For a given probe power, the waveform may show the bandwidth, and thus the resolution, of shorter pulses; and also the energy, and thus the range, of longer pulses. Typical implementations involve a continuous wave modulated in frequency to increase its bandwidth, or modulated with a number of amplitude or phase levels according to a sequence with good correlation properties [1].

Intensity-coded pulse compression systems have been extensively used in the optical domain, usually in the form of pseudorandom bit sequences [2–4], noise [5], or of single or complementary pairs of unipolar sequences [6, 7], for its use in laser ranging, reflectometry, and sensor interrogation. These systems benefit from the fact that the optical wave merely acts as a non-coherent carrier of an intensity code, so that only direct detection and subsequent correlation in the electrical domain are necessary. This way, optically incoherent information, namely the range-resolved reflectivity, can be retrieved. Within the coherent optical domain, the most successful approach has been the use of Frequency-Modulated Continuous Waves (FMCW) [8–11], since range and reflectivity can be extracted from the Fourier transform of the interference between probe and echo signals. Correlation techniques are also employed in the distributed analysis of Brillouin gain spectra [12–14], where the stimulated acoustic wave is proportional to a weighted correlation between pump and signal fields.

Genuine coherent pulse compression systems based on matched-filter detection, analogous in concept to their radar counterparts, are, however, scarce, and limited to sub-GHz bandwidths. In [11] it was presented a FMCW ranging system based on heterodyne detection, and not on the standard probe/echo interference, showing km range and 50-cm spatial resolution. In [15], compression from a complementary pair of bipolar Golay phase codes was used to characterize, after I/Q detection, dynamic Brillouin gratings generated in polarization maintaining fibers, attaining resolutions of 20 cm in a 30-m range. However, and despite their advantage in accessing in real time the full, complex response of optical circuits even at low signal-to-noise ratios, the widespread use of coherent sytems based on pulse coding seems to be hampered by the necessity of using phase diversity receivers [15, 16].

In this paper we report, for the first time to the best of our knowledge, on the use of CW injection-seeded Frequency Shifting Loops (FSL) [17] as compact sources for pulse compression ranging systems in the optical coherent domain. The main advantage of FSL within this context is its versatility in the generation of optical waveforms suitable for pulse compression, with a high degree of tunability and mutual coherence, and without requiring a broadband laser, nor fast electronics. This versatility allows for the compact generation of different optical fields, ranging from chirped pulses to noise-like, periodic waveforms, at selectable bandwidths that reach 20 GHz in our experiments. Moreover, once given an available detection bandwidth, the periodic character of the emitter allows for the extension of the TBP by enlarging the probe’s duration by concatenation with additional amplitude or phase codes, so that the attainable range can be, in principle, arbitrary. Finally, the single-sidedcharacter of the FSL optical spectrum permits, as an additional advantage in terms of simplicity, the use of simple heterodyne receivers for the reconstruction of the complex electric field. In summary, the use of FSL represents a simple and versatile solution leading to compact coherent pulse compression architectures, only limited in resolution by the available detection bandwidth.

This paper is organized as follows. In Section 2 we present an analytical model of the FSL field and its periodic autocorrelation. This model is based on the assumption of a uniform spectral decay between the lines within the single-sided FSL frequency comb [18]. Section 3 explains the proposed approach to field reconstruction using heterodyne measurements, and in Section 4 we present our experimental results describing the different alternatives for waveform selection and their compression properties. Finally, we describe in Section 5 the results of a proof-of-concept ranging experiment reaching mm resolution in a meter-scale, composite fiber/free space interferometer, and end in Section 6 with our conclusions.

## 2. FSL field autocorrelation

A CW injection-seeded Frequency Shifting Loop (FSL), schematically shown in Fig. 1(a), is composed of a CW laser, here at 1550 nm, that feeds a fiber loop. Inside the loop it is placed an acousto-optic frequency shifter (AOFS) followed by an EDFA that partially compensates for the losses within the loop. The loop is completed with an isolator and an optical tunable bandpass filter (TBPF) that defines the optical width of the FSL output and also blocks the amplified spontaneous emission from the amplifier. In the following, we assume that the filter transmission function is an ideal flat-top profile (*i.e.*, the transmission is constant over the filter bandwidth). The loop can thus sustain a number of recirculating frequencies, as shown in Fig. 1(b), in the form of a frequency comb and with a tunable spectral separation determined by the AOFS.

In our setup, a second EDFA boosts the power level of the FSL, and self-heterodyne measurements are obtained after mixing a portion of the seed CW laser with the loop’s output, followed by a high-bandwidth photodiode (PD) and a digital sampling oscilloscope (DSO), not shown in the figure. This scheme permits the introduction of a device under test (DUT) after the amplified output and prior to detection, and its analysis through the digital signal recorded by the scope.

Assuming that the AOFS imparts a positive frequency shift ${f}_{s}=1/T$, with *T* the modulation period, the FSL output is described by an electric field $\mathcal{E}\left(t\right)=E\left(t\right)\mathrm{exp}\phantom{\rule{0.2em}{0ex}}(j{\omega}_{0}t)$, with ${\omega}_{0}=2\pi {\nu}_{0}$ the seed frequency and $E\left(t\right)$ the optical envelope. Up to a global phase and a constant delay, this envelope is a one-sided comb of harmonics spaced by *f _{s}* [17, 19], as is schematically shown in Fig. 1(b),

*E*is the complex amplitude of the

_{n}*n*-th harmonic and, in particular,

*E*

_{0}is the complex amplitude of the first line in the comb at the seed frequency

*ν*

_{0}.

*N*is the number of lines, selected by the TBPF, $\rho <1$ is the power decay between consecutive lines [18], and

*τ*is the loop’s round-trip time. We assume here that the transmission function of the loop (gain and losses) is constant over the TBPF bandwidth. The spectral power of the

_{c}*n*-th line is denoted as${P}_{n}=|{E}_{n}{|}^{2}={P}_{0}{\rho}^{n}$, and the spectral width is $B=\left(N-1\right){f}_{s}\simeq N{f}_{s}$. For future use we define the effective number of spectral lines, ${N}_{eff}$, as the inverse of the power distribution within the comb,

Since ${N}_{eff}\le N$, this is a measure of the number of lines weighted by their power. ${N}_{eff}$ can also be interpreted as an effective TBP provided that we define an effective bandwidth as ${N}_{eff}{f}_{s}$. In our model, Eq. (1), the effective number of lines is related to *N* and *ρ* as:

The compression properties of $E\left(t\right)$ are described by its periodic autocorrelation function (PACF) [1], as defined in the first part of the following equation:

*k*= 1 and $k=N$. The PACF can thus be tailored by controlling the spectral decay and the number of lines in the comb, as is exemplified in Fig. 2 for different values of

*ρ*and a typical value $N=250$. For $\rho \le 0.99$, the coefficient

*C*can be neglected, and the functional form of Eq. (6) becomes that of the square root of the periodic reflectivity of a Fabry-Pérot filter, resulting in a sidelobe-free PACF with a decay ∼1$/\tau $. At

_{N}*ρ*= 1, Eq. (6) reads $\left|\mathcal{R}\left(\tau \right)\right|={E}_{p}|\mathrm{sin}\phantom{\rule{0.2em}{0ex}}(\pi N{f}_{s}\tau )/\left(N\mathrm{sin}\phantom{\rule{0.2em}{0ex}}(\pi {f}_{s}\tau \right))|$, and corresponds to the PACF of an ideal comb of lines with equal power. However, it is difficult to attain in practice values of $\rho >0.99$, equivalent to decays lower than $0.05$ dB per line, and the PACF shown by the FSL output will be mostly of Fabry-Pérot type. In either case, the PACF width is defined as an equivalent width:

## 3. FSL field autocorrelation using heterodyne detection

In our experiments, we determined the envelope $E\left(t\right)$ using a heterodyne receiver. This is enabled by the one-sided character of the FSL optical spectrum, as described as follows. The heterodyne signal at the detector output in Fig. 1(a) is given by:

*I*is the intensity of the local oscillator (LO), $I\left(t\right)=|E\left(t\right){|}^{2}$ the FSL intensity,

_{LO}*Re*stands for real part, and

*φ*is the LO phase. Let us first assume that we use balanced detection instead of the single-branch heterodyne receiver of Fig. 1(a). The output would be twice the last term in Eq. (8), $4\sqrt{{I}_{LO}}\phantom{\rule{0.2em}{0ex}}Re\left[E\left(t\right){e}^{-j\phi}\right]$. Use of Eq. (1) in this expression shows that, blocking dc and Hilbert transforming (

*i. e.*, detaching the negative harmonics and multiplying by a factor of two the positive ones), yields the complex signal $2\sqrt{{I}_{LO}}\phantom{\rule{0.2em}{0ex}}{E}_{+}\left(t\right){e}^{-j\phi}$, where ${E}_{+}\left(t\right)$ is twice the dc-subtracted envelope, ${E}_{+}\left(t\right)=2\left(E\left(t\right)-{E}_{0}\right)$. Field ${E}_{+}\left(t\right)$ shows equivalent compression properties to those of $E\left(t\right)$, since they are related by the removal of only the first line in the comb. In fact, it is immediate to show, using Eq. (5), that the autocorrelation ${\mathcal{R}}_{+}\left(\tau \right)$ of ${E}_{+}\left(t\right)$ is related to that of $E\left(t\right)$ as

The same procedure –dc block followed by Hilbert transform– can be applied to the single-detector heterodyne signal, Eq. (8), with the result

where ${I}_{+}\left(t\right)$ is the analytic signal of the dc-subtracted intensity,*i. e.*, the complex signal associated to the positive harmonics of the intensity. We observe, finally, that ${E}_{c}\left(t\right)$ is a good approximation to ${E}_{+}\left(t\right)$ at high values of the coherent gain

*I*, and so to the true envelope $E\left(t\right)$.

_{LO}## 4. Experimental waveforms and autocorrelations

The aforementioned procedure was applied to a number of FSL outputs, following the scheme presented in Fig. 1(a). In our setup, the loop’s round-trip time is ${\tau}_{c}=$ 73.202 ns, the AOFS (AA Optoelectronic) imparts frequency shifts at about 80 MHz, and the total optical spectral width is $\mathrm{\Delta}\nu \simeq $ 20 GHz ($N\simeq 250$). This figure coincides with the electrical bandwidth *B* of the detection stage, composed of a 25-GHz photodiode (PD, Alphalas UPD-15-IR2-FC) and the DSO (Lecroy-Teledyne Wavemaster 8, depth of 8 bits, sampling rate 40 GS/s). All the setup is built from polarization-maintaining fibers, so that the previous scalar model is suitable for the analysis of the system. The required conditions of high LO gain were experimentally achieved by a procedure described below.

Though all FSL waveforms share the same PACF for a given spectral decay and number of lines, the FSL waveforms depend on the spectral phases carried by the frequency comb which, according to Eq. (1), are given by $\mathrm{exp}\phantom{\rule{0.2em}{0ex}}(-j\pi {f}_{s}{\tau}_{c}{n}^{2})$. They can be classified by use of the theory of temporal Talbot effect [20]. When the shifting frequency *f _{s}* is adjusted so that the product ${f}_{s}{\tau}_{c}$ is an integer

*p*, the spectral phases are either in phase or show a consecutive

*π*shift. The output is simply a sequence of transform-limited pulses at a rate

*f*. When the product ${f}_{s}{\tau}_{c}$ equals an irreducible fraction of the form $p/q$, the output is composed of

_{s}*q*transform-limited pulses with equal energy and equally spaced within the period. Neither of these waveforms, however, are suitable for our purposes, because the pulse width is of the order of the minimum time feature resolvable by the detector. Nonetheless, when the shifting frequency is slightly detuned by an amount $\delta {f}_{s}$ from a certain Talbot condition, the

*q*pulses pulses per period are stretched by a controllable group velocity dispersion ${\varphi}^{\u2033}={\tau}_{c}\delta {f}_{s}/\left(2\pi {f}_{s}^{2}\right)$ [17], thus allowing for a proper recording with the available bandwidth.

In Fig. 3, left, we show the heterodyne signals of the four waveforms used in our experiments. In the first three waveforms on the left hand side, the linear chirps generated by dispersion from a single pulse per period (integer Talbot with *p* = 1) or from four or twelve pulses per period (fractional Talbot with $p/q=1/4$ and $1/12$) are clearly observable, since the overlap with adjacent pulses is negligible. Chirped pulses within the same period show different phases following a Gauss sequence of length *q* [21, 22]. The characterization of these phases will be the object of a separate study, as they do not influence the compression properties of the waveforms. In Fig. 3(d), the waveform was generated by imparting a large dispersion to the fractional Talbot condition $p/q=1/21$, resulting in a noise-like, but periodic, waveform encompassing the interference of 21 dispersed pulses per period.

These traces were retrieved using the entire DSO memory of 20 $\mu \mathrm{s}$, which amounts to ∼1600 periods of the waveform and permits a clear observation, after Fourier transform, of the spectral comb structure. The heterodyne spectra are shown in Fig. 3, right. Sufficient coherent gain was provided as follows. We first notice that the spectrum of both the heterodyne signal, Eq. (8), and its complex version, Eq. (10), are composed of the contribution from $\sqrt{{I}_{LO}}{E}_{+}\left(t\right){e}^{-j\phi}$, which shows a smooth decay, and that from ${I}_{+}\left(t\right)$, which is concentrated in the harmonics of $q{f}_{s}$. In the experiment, we simply modified the EDFA gains so that the smooth contribution dominates over the spikes at $q{f}_{s}$. As a result, the four traces in Fig. 3, right, are essentially equivalent, as they show the same smooth decay and only differ by small differences in *f _{s}*. Due to the roll-off in bandwidth and to the spectral dependence of the TBPF transmission function, the spectral decay is, however, not uniform, in contrast to what was assumed in our model, Eq. (1). A fit of the spectral power below $\le $12 GHz shows a good linearity in the decay, permitting the estimation of the spectral decay

*ρ*. The effective number of lines, in turn, can be obtained directly from the heterodyne spectrum and its definition, Eq. (2). In all cases, the total number of lines is $N\simeq 250$, the effective number of lines is ${N}_{eff}\simeq 150$, and the spectral decay is $\rho \simeq 0.99$.

In the first part of our analysis, we studied the PACF of field ${E}_{c}\left(t\right)$ for the Talbot 1/4 configuration at different FSL amplifier gains, following the procedure described in the previous section. In Fig. 4, we plot the heterodyne spectra at three different pump currents of the EDFA inside the loop, where it is observed the decrease in the spectral decay *ρ*, and thus the decrease in the occupied bandwidth with decreasing EDFA gain. The maximum bandwidth of 20 GHz, however, is ultimately determined by the TBPF.

In Fig. 5 we show the PACF in dB scale, $20\underset{10}{\mathrm{log}}\left|{\mathcal{R}}_{c}\left(\tau \right)\right|$, of the heterodyne field ${E}_{c}\left(t\right)$ together with the expected PACF $\mathcal{R}\left(\tau \right)$ generated from the effective number ${N}_{eff}$ of spectral lines and the spectral decay *ρ* extracted from Fig. 4. The experimental traces were generated by computing the autocorrelation of the fist 10 periods of the waveforms with a portion of the waveform located at approximately the middle position in the 20-*μ*s run. The agreement is good near the correlation peak, and clearly shows the tunability of the PACF in bandwidth or resolution through the control of the spectral decay. However, the PACF also presents three small sidelobes at the same positions within the interval as the original Talbot 1/4 pulses. They are clearly visible in Fig. 5(b) and also perceptible in Fig. 5(a), where the loop is operated with higher amplification. However, they are not visible in Fig. 5(c) where the amplification level is lower and the autocorrelation peakwider. The presence of these sidelobes is ascribed to the imparted LO gain, which was not sufficient to totally suppress the ${I}_{+}\left(t\right)$ contribution to Eq. (10). This way, the PACF of ${E}_{c}\left(t\right)$ contains mixing terms that correlate with the analytic intensity ${I}_{+}\left(t\right)$ and thus reflect the positions of the underlying pulses within the period. The peak-to-sidelobe power levels depend on the FSL/LO relative phase *φ*, as implied by Eq. (10), which was not controlled in our experiments. In Fig. 5(c), the sidelobes are likely to be hidden beneath the wide autocorrelation peak. Notice that, since this effect is originated by the presence of the intensity $I\left(t\right)$ in Eq. (8), and therefore by the single-branch character of our heterodyne receiver, these low-amplitude sidelobes would be absent if a balanced heterodyne receiver is used instead.

These observations points to the possibility of selecting specific FSL waveforms with enhanced peak-to-sidelobe ratios in the PACF reconstructed from single-branch heterodyne detection. Since the sidelobes are originated by the existence an underlying repetitive pulse structure within the period, they can be avoided using different strategies. For instance, one may select waveforms where there is a single pulse, such as the chirped pulse of Fig. 3(a), or noise-like waveforms with low peak-to-average power ratio, such as that of Fig. 3(d), since in this last case $I\left(t\right)$ approaches its dc value and thus enhances the approximation ${E}_{c}\left(t\right)\sim {E}_{+}\left(t\right)$ in Eq. (10).

In Fig. 6 we show the PACF of the rest of waveforms in Fig. 3. In Fig. 6(a), the chirped pulse does not originate any sidelobes; in Fig. 6(b), the remnants of the 12-pulse structure appear as 11 sidelobes in the PACF of the dispersed 1/12 Talbot field. In the last example of Fig. 6(c), the PACF of the noise-like waveform does not show any pulse structure, as expected. In all cases, the PACF peak to sidelobe level is higher than 28 dB.

## 5. Coherent pulse-compression ranging

In a final experiment, we used this approach in a proof-of-concept ranging system. A Michelson interferometer based on a polarization-maintaining fiber 3dB splitter was inserted in the scheme, as shown in Fig. 1(c), after the FSL output and before the heterodyne receiver. One of the interferometer’s arms, with a fiber length ∼2 m, was ended by a fiber mirror. The other arm was partially built with fiber (∼1 m) followed by a collimator (Col) and a free-space mirror whose location can be varied in a 2-m optical rail. The reference field envelope, ${E}_{c,ref}\left(t\right)$ was recorded by blocking the free-space arm. Afterwards, this reference envelope was digitally cross-correlated with the envelope ${E}_{c}\left(t\right)$ describing the complete interferometer.

In this experiment we seek the optical path difference between the fiber mirror, used as reference, and three positions of the movable free-space mirror. The experiment was performed with the chirped waveform of Fig. 3(a) and with the noise-like waveform of Fig. 3(d), by first approximately balancing the interferometer and then reducing the length of the free-space arm at three definite positions. The obtained cross-correlation for two of these positions is presented in Fig. 7. Here we used a 5-GHz photodiode (Thorlabs, DET08CFC). Its frequency response has two notches at 4 GHz and 10 GHz, which are responsible of satellite pulses in its impulse response and therefore of the sidelobes observed in the cross-correlation peaks. The spectral decay of the waveform was *ρ* = 0.960, with a PACF width $\mathrm{\Delta}{\tau}_{eq}=$ 250 ps, which corresponds to 330 ps FWHM, in good agreement with the value measured in Fig. 7. However, the range resolution in this experiment is determined by the accuracy in the determination of the PACF peak, rather than by the PACF width. This resolution can thus be conservatively ascribed to the 25-ps DSO sampling interval, which amounts to 4 mm in free space. The results for three different mirror positions are presented in Table 1. They were also compared with the relative measurement provided by the ruler in the optical rail and with a conventional FMCW ranging measurement using the chirped waveform of Fig. 3(a). The results are compatible within the 4-mm resolution, and show the feasibility of the proposed procedure.

## 6. Conclusions

In this paper, we have demonstrated the use of the optical frequency combs generated by FSL in coherent systems based on optical waveform compression. Approximate reconstruction of the complex FSL field have been experimentally achieved using single-detector, polarization-maintaining heterodyne receivers, using the single-sided character of the FSL spectrum. The exact field, in turn, can be determined by balanced heterodyne detection.

The periodic autocorrelation function of the dc-blocked, Hilbert transformed heterodyne signals has been experimentally determined at different bandwidths, showing peak-to-sidelobe power ratios in excess of 28 dB. The autocorrelation’s functional form is similar to that of the Fabry-Pérot filter reflectivity, and can be tailored, at the expense of resolution, using different approaches, including optical filtering of the FSL field or digital techniques such as weighting-on-receipt [1]. Compression of FSL waveforms from periods of 12.5 ns down to 80 ps have been demonstrated, resulting in effective TBP or compression ratios up to 150. Using the same or related technologies, this figure can be extended by several means. On the one hand, TBPs in excess of 1000 have been demonstrated at low spectral shifts of 500 kHz in multi-heterodyne spectroscopy systems [23], and also using high-bandwidth (>100 GHz) detection [17]. Use of electro-optic frequency shifting is envisioned to push this figure well into the THz bandwidth range [24], and therefore comparable to FMCW systems employing specific frequency-sweep linearization techniques [25–27].

On the other, the TBP can be enlarged, for a given bandwidth, by concatenation of the periodic FSL output with additional phase or amplitude codes implemented by subsequent modulation. Systems operating at the typical 100-MHz frequency shift offered by AOFS, even allowing for a substantial decrease in scan rate for periodic averaging, can operate at MHz repetition rate, suitable for real-time monitoring and detection tasks. In our proof-of-concept ranging experiment, the procedure has been applied to the static determination of optical path differences in a fiber/free space interferometer. The attained sub-cm resolution can be slightly improved by standard peak determination techniques, such as interpolation.

In summary, the proposed use of FSL represents a compact and versatile means to incorporate pulse compression radar concepts to coherent optical systems, for its use in different application domains such as laser ranging, reflectometry, optical vector network analysis, or optical fiber sensors.

## Funding

Agence Nationale de la Recherche, France (ANR) (ANR-14-CE32-0022); Ministerio de Ciencia, Innovación y Universidades, Spain, Agencia Estatal de Investigaciń, Spain, and European Union (FEDER, EU) (TEC2017-89688-P); Generalitat Valenciana, Spain (ACIF/2016/214).

## References

**1. **N. Levanon and E. Mozeson, *Radar signals* (John Wiley & Sons, 2004). [CrossRef]

**2. **N. Takeuchi, N. Sugimoto, H. Baba, and K. Sakurai, “Random modulation cw lidar,” Appl. Opt. **22**, 1382–1386 (1983). [CrossRef] [PubMed]

**3. **R. Matthey and V. Mitev, “Pseudo-random noise-continuous-wave laser radar for surface and cloud measurements,” Opt. Lasers Eng. **45**, 557–571 (2005). [CrossRef]

**4. **X. Ai, R. Nock, J. G. Rarity, and N. Dahnoun, “High-resolution random-modulation cw lidar,” Appl. Opt. **50**, 4478–4488 (2011). [CrossRef] [PubMed]

**5. **D. Mermelstein, M. Biton, S. Sternklar, and E. Granot, “Fiber-optic range sensing based on amplified spontaneous emission noise radar with Kramers-Kronig phase retrieval,” in *CLEO 2011 - Laser Applications to Photonic Applications, OSA Technical Digest* (Optical Society of America, 2011), paper JThB135. [CrossRef]

**6. **M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, and S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. **7**, 24–38 (1989). [CrossRef]

**7. **N. Arbel, L. Hirschbrand, S. Weiss, N. Levanon, and A. Zadok, “Continuously operating laser range finder based on incoherent pulse compression: noise analysis and experiment,” IEEE Photonics J. **8**, 1–11 (2016). [CrossRef]

**8. **D. Derickson, *Fiber optics test and measurement*(Prentice Hall, 1998).

**9. **B. Culshaw and I. P. Giles, “Frequency modulated heterodyne optical Sagnac interferometer,” IEEE J. Quantum Electron. **18**, 690–693 (1982). [CrossRef]

**10. **J. Zheng, “Analysis of optical frequency-modulated continuous-wave interference,” Appl. Opt. **43**, 4189–4198 (2004). [CrossRef] [PubMed]

**11. **W. Zou, S. Yang, X. Long, and J. Chen, “Optical pulse compression reflectometry: proposal and proof-of-concept experiment,” Opt. Express **23**, 512–522 (2015). [CrossRef] [PubMed]

**12. **K. Hotate and T. Hasegawa, “Measurement of Brillouin gain spectrum distribution along an optical fiber using a correlation-based technique-proposal, experiment and simulation,” IEICE Trans. Electron. **E83-C**, 405–412 (2000).

**13. **A. Motil, A. Bergman, and M. Tur, “State of the art of Brillouin fiber-optic distributed sensing,” Opt. Laser Technol. **78**, 81–103 (2016). [CrossRef]

**14. **A. Zadok, E. Preter, and Y. London, “Phase-coded and noise-based Brillouin optical correlation-domain analysis,” Appl. Sci.-Basel **8**, 1482 (2018). [CrossRef]

**15. **A. Bergman, T. Langer, and M. Tur, “Coding-enhanced ultrafast and distributed Brillouin dynamic gratings sensing using coherent detection,” J. Lightwave Technol. **34**, 5593–5600 (2016). [CrossRef]

**16. **R. Goldman, A. Agmon, and M. Nazarathy, “Direct detection and coherent optical time-domain reflectometry with Golay complementary codes,” J. Lightwave Technol. **31**, 2207–2222 (2013). [CrossRef]

**17. **H. Guillet de Chatellus, L. Romero Cortés, C. Schnébelin, M. Burla, and J. Azaña, “Reconfigurable photonic generation of broadband chirped waveforms using a single CW laser and low-frequency electronics,” Nat. Commun. **9**, 2438 (2018). [CrossRef] [PubMed]

**18. **K. Nithyanandan, L. Djevarhidjian, V. Bosch, C. Schnébelin, S. Kassi, G. Méjean, D. Romanini, and H. Guillet de Chatellus, “Optimization of acousto-optic frequency combs for multi-heterodyne spectroscopy,” in *Frontiers in Optics/Laser Science, OSA Technical Digest* (Optical Society of America, 2018), paper LTh1F.3.

**19. **H. Guillet de Chatellus, E. Lacot, W. Glastre, O. Jacquin, and O. Hugon, “Theory of Talbot lasers,” Phys. Rev. A **88**, 033828 (2013). [CrossRef]

**20. **J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron ., **7**, 728–744 (2001). [CrossRef]

**21. **L. Romero-Cortés, H. Guillet de Chatellus, and J. Azaña, “On the generality of the Talbot condition for inducing self imaging effects on periodic objects,” Opt. Lett. **41**, 340–343 (2016). Erratum, Opt. Lett. 41, 5748 (2016). [CrossRef]

**22. **C. R. Fernández-Pousa, “On the structure of quadratic Gauss sums in the Talbot effect,” J. Opt. Soc. Am. A **34**, 732–742 (2017). [CrossRef]

**23. **V. Durán, C. Schnébelin, and H. Guillet de Chatellus, “Coherent multi-heterodyne spectroscopy using acousto-optic frequency combs,” Opt. Express **26**, 13800–13809 (2018). [CrossRef] [PubMed]

**24. **L. Wang and S. LaRochelle, “Talbot laser with tunable GHz repetition rate using an electro-optic frequency shifter,” in *CLEO 2017 - Conference on Lasers and Electro-Optics, OSA Technical Digest* (Optical Society of America, 2017), paper JW2A.66.

**25. **P. A. Roos, R.R. Reibel, T. Berg, B. Kaylor, Z. W. Barber, and Wm. Randall Babbitt, “Ultrabroadband optical chirp linearization for precision metrology applications,” Opt. Lett. **34**, 3692–3694 (2009). [CrossRef] [PubMed]

**26. **E. Baumann, F. R. Giorgetta, J.-D. Deschênes, W. C. Swann, I. Coddington, and N. R. Newbury, “Comb-calibrated laser ranging for three-dimensional surface profiling with micrometer-level precision at a distance,” Opt. Express **22**, 24914–24928 (2014). [CrossRef] [PubMed]

**27. **T. Hariyama, P. A. M. Sandborn, M. Watanabe, and M. C. Wu, “High-accuracy range-sensing system based on FMCW using low-cost VCSEL,” Opt. Express **26**, 9285–9297 (2018). [CrossRef] [PubMed]