1. Introduction

Dual comb interferometry has found applications in spectroscopy [1–3], OCT [4], hyperspectral lidar [5,6] and distance metrology [7–9]. In most demonstrated applications involving distance measurements, a mirror is used as a target, as opposed to a diffusely reflective surface. When using dual comb systems for distance measurements or vibrometric applications, the rapidity of the vibration that can be measured is limited by the interferogram repetition rate and the poor duty cycle of the measurements inherent to such systems.

Recently, a new approach for comb-based interferometry measurements, named optical sampling by cavity tuning (OSCAT), has been demonstrated [8,10,11]. This new approach uses only one comb source whose repetition rate is periodically varied and a delay line to replicate the dual-comb experiment. This has the advantage of enabling a near unity measurement duty cycle for spectroscopic applications. With such a configuration it is also possible, by keeping the repetition rate constant, to maintain both interfering pulses trains at a chosen mutual delay. This trait is particularly suited for vibrometric measurements, where relatively fast sampling rates (compared to typical dual-comb interferogram repetition rate) and continuous position measurements can be useful.

In this paper, we demonstrate the use of a comb system in the OSCAT configuration to make vibrometric measurements on remote diffusely reflective targets. First, the setup configuration is shown and the technique is described. Then the correction algorithm, which is adapted from the algorithm used in optical filter based referencing [12], is presented, as well as a noise analysis for the additive noise limited case. The results are finally presented, where a human voice sample is recovered from the demodulated vibrations of an expanded polystyrene foam plate and of a diffusely reflective office wall. The range resolving capabilities of the system are also shown by independently measuring two different targets which are vibrating simultaneously in the path of the measuring beam.

2. Description of the technique

Coherent vibrometric measurements are done by making the target part of one arm of a two-beam interferometer. Figure 1 shows a schematic of the measurement setup. The comb is split, as in a typical OSCAT experiment [10]. For one of the two vibrometry approaches described in this paper an acousto-optic modulator (AOM) is also inserted in the short delay arm. Both paths are then pulse-picked and one of the paths is amplified, as described in [5]. One arm is sent directly to the balanced photodiode to serve as the local oscillator. The second arm is sent to the target through a launching collimator, which is also used to collect the retro-reflected signal. The reflected pulse is then sent to the photodetector using a fiber coupler, where it is coherently mixed with the local oscillator pulse to generate interferometric modulation that contains information on the target position as a function of time. The referencing signals, used to track and correct for the repetition rate and phase fluctuations of the comb as in [12], are tapped just before the amplification and pulse picking chains.

 

Fig. 1 Schematic of the experimental setup. The comb is split into two parts. The delay between the two arms is 300 m. This allows the generation of an inter-pulse delay with a change of repetition frequency. The other part is sent to an optional acousto-optic modulator (AOM), which resolves phase ambiguity without the need to sweep the interferogram. After amplification and pulse picking, one arm is used as a local oscillator, while the other one is sent to the target using a beam expander. The backscattered signal is combined with the local oscillator on a balanced photodiode (BPD).

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Since the local oscillator is a pulse, and not a continuous wave laser, it is possible to choose specific positions in the target path for which to extract phase information. Indeed, by changing the repetition rate of the measuring comb, the relative delay between the local oscillator pulse and the target signal can be swept. Thus, the local oscillator can be aligned to interfere only with a specific part of the target’s impulse response. This makes the measurement range-resolved, so that unwanted parts of the target’s reflection do not contribute to the measured signal. It is thus possible, for example, to extract the position of a weakly reflecting target from behind a stronger reflecting object. The range resolution of the system is given by the length of the cross-correlation between the target pulse and the local oscillator pulse. In the case where there is no differential chirp between both pulses, it is roughly equal to the inverse of the optical bandwidth of the most spectrally narrow pulse. For 10 nm wide pulses at 1550 nm, the lower limit on range resolution is of the order of 250 µm. Differential chirp between both interfering pulses results in worse range resolution.

If the response from the scene has a longer duration than the repetition period of the probing comb, there will be overlap between the responses to successive probing pulses. By changing the pulse-picking factor, it is possible to increase the repetition period of the probing comb to any multiple of its fundamental period. This can be used to remove this overlap.

Due to the potentially large power imbalance between the local oscillator pulse and the collected backscattered target signal, balanced photodetection is necessary to obtain reasonable signal-to-noise ratio by exploiting the full detection range of the photodetector. Balanced detection removes the unmodulated contribution from the local oscillator, leaving the interferometrically modulated beating term to fill the photodetector's dynamic range.

The most advantageous strategy, in the additive noise limited case, is to keep the local oscillator at the maximum power which still conserves photocurrent linearity, which for the Thorlabs PDB130C used here this is 400 µW, and launching as much power as possible towards the target without encountering important optical nonlinearities or the beating term exceeding the dynamic range of the detector.

The system, shown in Fig. 1, is essentially, for a constant repetition rate of the probing comb and stationary target, a constant optical path difference (OPD) interferometer. Compared to a dual-comb based experiment, this has the advantage of having almost no measurement dead time, since the local oscillator pulse and the retro-reflected target pulse can overlap at all times, provided the delay between the local pulse and the desired target is adjusted properly. This is especially important for vibrometric measurements. Indeed, in the case of time domain spectroscopic dual-comb measurements, either the impulse response or the autocorrelation of the sample is measured every time a new interferogram is generated. As long as one can assume the sample to be time invariant, no important information is lost during the down time between interferograms. In this case, all that is lost is the opportunity to take more samples during that down time, resulting in a suboptimal measurement signal-to-noise ratio. However, in the vibrometric measurement case, what is measured is a constantly varying parameter of the target, namely its position. Downtime during the measurement therefore results not only in reduced signal-to-noise ratio, but also in lost information. OSCAT eliminates most of that downtime and thus enables continuous measurements of the target position.

Stationary OSCAT, while enabling continuous target position measurements, has the downside of generating a baseband interferogram, which results in phase extraction ambiguity. This can be solved in many ways. Optical hybrids can be used to obtain in-phase and quadrature samples, which lifts phase ambiguity by measuring a complex exponential instead of a sine wave. Alternatively, the interferogram can be made band-pass, for example by using an AOM that frequency-shifts the signal to a desired center frequency, enabling continuous unambiguous phase extraction at almost constant OPD.

One other option is to sweep the comb’s repetition rate, generating forwards and backwards OPD sweeps in succession. This generates interferograms around the desired range of interest that have a nonzero average frequency linked to the sweep speed. This enables vibrometric measurements with the least amount of equipment, since an AOM or quadrature detection is not needed. This sweeping approach has the obvious drawback of introducing some dead measurement time at turnarounds, as well as some other less obvious issues that will be discussed in the next section. The dead time is due to the turnaround of the repetition rate sweeping element inside the cavity, when its speed goes to zero before changing direction and interference fringes disappear. Ideally, with perfect triangular sweeping, this dead time would be negligible, but the limited bandwidth of the piezoelectric element in the cavity causes a small portion of the signal, approximately ten percent, to be unusable.

This paper presents results of using those two methods, that is, OSCAT sweeping and modulation of the interferogram using an AOM.

3. Phase extraction and modified referencing algorithm

Contrary to the traditional comb interferometry case, where the stationary impulse response or the autocorrelation of the impulse response of a system is characterized as a function of lag, the goal of the vibrometry experiment is to identify the non-stationary delay in one of the interferometer arms. Since the nature of the system being measured is different, it is expected that the referencing algorithm used to compensate for comb fluctuations has to be slightly modified to work for vibrometry measurements.

As the target vibrates, the optical path difference (OPD) between the stationary arm and the target arm fluctuates accordingly. Let us call this target vibration-induced delay$\delta$. This delay contribution between the pulses coming from both arms is added to the varying delay and phase caused by comb fluctuations as well as to the repetition frequency variations due to the OSCAT system. The measured complex interferogram for each interfering pulse pair, as measured on the balanced photodiode, is given by

To remove their effect on the measured signals, $\tau$and $\Delta \varphi$ must first be measured. This is done by bandpass filtering the combined comb outputs by using a pair of uniform fiber Bragg gratings at different frequencies [12]. The setup used to generate a single reference signal is shown in Fig. 2. The phase obtained from the first reference channel is given by

 

Fig. 2 Schematic of one referencing experiment. The two trains to be referenced are combined and sent to a fiber Bragg grating. The beating signal collected on a photodetector through a circulator.

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Equation (3) can be used to remove the effect of $\Delta \varphi$ on the measured target phase, while Eq. (4) can be used to remove the effect of $\tau$. The modified referencing algorithm is thus as follows: first, the phase from Eq. (1) is extracted from the measured interferogram by using a Hilbert transform. Then, as is the case for the traditional referencing algorithm, $R_1$ is subtracted from ${\varphi }_M$, resulting in

For the second correction step, instead of the usual interpolation of the interferogram on an even OPD grid, we wish to remove as much of the OPD contribution as possible from the measured phase. This is done in two substeps. First, the contributions from ${\tau }_0$ are removed. Both the sweeping profile and the baseline phase profile, ${\tau }_0$ and $\theta \left({\tau }_0\right)$, are assumed known. When using an AOM, they are approximately constant. In the OSCAT sweeping case, this assumption relies mainly on the repeatability of the response of the in-cavity piezoelectric element. Multiple sweeps can be corrected using the first step and then averaged to obtain a fluctuation-free sweeping profile, given by $\theta \left({\tau }_0\right)-2\pi f_1{\tau }_0$ . All terms depending on ${\tau }_0$ in Eq. (5) can thus be removed, at least approximately. This results in

The only undesired terms remaining in Eq. (6) are linearly dependant on $\Delta \tau .$ By the same procedure that led to Eq. (6), the baseline OPD profile can be removed from Eq. (4), resulting in $2\pi \Delta f\Delta \tau .$ This centered version of Eq. (4) can then be added to Eq. (6) with the correct gain, equal to $\left(2\pi f_1-D\right)/2\pi \Delta f$, thus removing all terms depending on $\Delta \tau$. The end result is then

The correct gain to apply in the last step is dependent on the value of $D,$which is not always readily estimated. However, when the optimal gain is used, the standard deviation of the corrected signal should be minimized. The last correction step is thus done by sweeping the gain applied to the centered OPD reference signal and using the gain that minimizes the resulting corrected signal. Figure 3 shows a plot of the standard deviation of the corrected phase as a function of the applied gain.

 

Fig. 3 Standard deviation of corrected phase as a function of the gain applied to the second reference signal. The optimal gain in this case is a small positive value, and results in a marginal reduction of noise caused by inter-pulse jitter.

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There are some considerations specific to the sweeping method. When sweeping a chirped interferogram, its instantaneous frequency, and thus the value of $D$, changes throughout the sweep. Although this was not done for this paper, it might be useful to compute a variable gain by applying the optimization locally on zones where the small $\Delta \tau$ approximation is valid. Care would however need to be taken to ensure continuity of the corrected phase. When the sweep has low enough amplitude, this is not necessary, since $D$ is nearly constant.

Slow OPD drifts caused, for example, by temperature variations resulting in effective delay line drifts, also affect the optimal gain, as well as the repeatability of the sweeping profile. Since the sweeping phase profile is in most cases slow, this lack of repeatability results in a slowly varying baseline on the corrected waveform for each sweep, as will be shown in the next section.

When using the AOM method, where ${\tau }_0$ is assumed constant, the correction algorithm can be summed up as

Figure 4 shows such noise measurements when using the parked OSCAT with AOM method, with the AOM being driven at 40 MHz. Subfigure (a) shows the voltage power spectral densities measured on a reference detector with and without the reference signal fiber being plugged in. The voltage PSD of the noise-only trace, combined with the time domain amplitude of the reference signal, can be used to estimate the phase PSD contribution of that reference channel. Subfigure (b) shows power spectral densities on the vibrometry detector with the local oscillator pulses aligned and misaligned from the target pulse. The shown noise power spectral density thus contains noise contribution from the fiber leading up to the launcher, which backscatters some power back to the detector. The peak at 40 MHz on the noise trace is a strongly attenuated backscattered beating signal, as will be shown later. On subfigure (c), the raw vibrometry and corrected signal power spectral densities are shown. It can be seen that by performing only the first correction step, and thus using only one reference, the result is almost indistinguishable from the one obtained by using two references, apart from a peak at approximately 10 kHz, where the two references correction is beneficial. This peak comes from the resonant vibration of the piezoelectric element inside the comb cavity. The teal line is the sum of all noise contributions from the reference and target detectors, when the voltage to phase conversion is applied. It thus corresponds to the lowest achievable post-correction phase noise contribution for the detectors and optical power conditions used for this measurement. It can be seen that this floor is reached with two-reference correction, where the peak of the piezoelectric element disappears completely. The sharp power spectral density (PSD) wall at 100 kHz is due to a 200 kHz wide filter that is used prior to phase extraction to ensure that additive noise is low enough for valid extraction. Converted back to vibration amplitude, this noise floor results in sensitivity on the order of the nanometer in a 4 kHz bandwidth, which is used for the voice demodulations examples in the next section.

 

Fig. 4 a): Voltage power spectral density of the vibrometry detector with the local oscillator pulse aligned and misaligned from the target pulse. Optical noise, including backscattered power from the fiber, is thus included. b): Voltage power spectral density on a reference detector with and without the reference signal fiber plugged in. c): Phase power spectral densities of the raw vibrometry signal, along with corrected signals using one and two references for correction. The sum of the noise contributions from the detectors, shown on a) and b), is converted to phase noise and shown on the light blue curve. It corresponds to the best case post-correction noise floor.

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On Fig. 4(c), a peak can be seen at DC on the curve corresponding to the sum of all noise contributions. This peak can be found on the vibrometry channel at the expected interferogram frequency, even when the system is not aligned to the target. This spurious interferogram is generated by the coherent beating between the local oscillator pulse and the backscattered signal coming from the fiber leading to the launcher. The fiber backscatters some of the pulse signal at every position along its length, so that there is always some backscattered signal available to interfere with the local oscillator, regardless of the range at which the system is tuned. To verify that backscatter is the cause of the spurious interferogram, a spool of 50 m of fiber was added before the launcher. As Fig. 5 shows, adding the fiber spool increases the variance of the spurious signal by almost a factor 10, which closely matches the fiber length increase. This backscattered signal should, however, have a negligible impact on measurement quality. Indeed, since its phase contributions, assuming time invariance of the backscattering fiber, will be highly correlated with comb fluctuations, they should be mostly removed by the correction algorithm. Note that the red and green traces being higher than the backscatter peak on Fig. 4(c) indicates that low frequency vibrations of higher phase PSD than the backscatter contributions are being measured, and not necessarily that the correction algorithm does not remove backscatter fluctuations.

 

Fig. 5 Measured noise power spectral density from the signal channel, with both no extra fiber and 50 m of extra fiber added before the launcher. Adding fiber increases the measured signal amplitude at the AOM frequency, which confirms that the peak is due to backscatter in the fiber before the launcher.

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4. Results

In this section, human voice samples are extracted from various targets, namely an expanded polystyrene foam cafeteria plate, a beige wall and a glass slab. Range resolution and target separation is demonstrated by applying a strong interfering signal on one of the targets. Results with and without an AOM are shown.

The comb used in the experiment is a Menlo c-comb laser, with a base repetition rate of 100 MHz, or a non-overlapping range of 1.5 m. A Gooch and Housego M040-8J-F2S 40 MHz shift AOM was used for the spectral shifting of the interference pattern. INO FAD C-Band EDFAs, with a maximum output power of 40 mW, were used in both interferometer arms. The target pulse was launched using a 10 mm focal length collimator, with a 4.5 mm output beam. The beating signals are detected with Thorlabs PDB130C balanced photodetectors and acquired using a GaGe CS14G8 for the target interferogram and a synchronized GaGe CS8349 for the referencing interferograms. To maximize single-shot acquisition time, the cards were externally clocked with a signal derived from the repetition rate of the probing comb, so that only one sample was taken per base repetition period. The single-shot acquisition time was thus limited to ten seconds by the sample memory of 1 GS of the CS14G8 when sampling at 100 MHz.

Figure 6 shows the signal measured from an expanded polystyrene foam cafeteria plate, placed approximately 3 m from the beam launcher. The OSCAT sweep method was used for this measurement. A piecewise linear voltage signal was sent to the piezoelectric transducer at 10 Hz, resulting in one approximately linear sweep every 50 ms. The piezoelectric element has a stroke of 15 µm, which, coupled with the 300 m delay line, results in a 2.3 mm sweep. The delay line provides enough differential chirp between the target and the LO pulse for the interferogram to be longer than the sweep length, resulting in no loss of signal.

 

Fig. 6 a) Zoom on the extracted phase from several OSCAT sweeps. Some slowly varying phase is left over from the correction on each sweep due to the drifts between the measurement acquisitions and the reference acquisition. (Media 1) b) Twice differentiated phase, proportional to the voltage signal which would generate the measured displacement waveform. The slowly varying error from a) has been attenuated by the differentiation (Media 2).

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The extracted audio signal is from a male human voice counting from one to eight, in French. The person speaking was in the same room as the target, approximately 5 m away from it. Figure 6(a) (Media 1) shows the extracted waveforms from consecutive OSCAT sweeps, with some turnaround dead time between each one. The slowly varying phase contributions coming from the chirp in the interferogram have been removed by subtracting a second silent measurement. This is equivalent to the baseline OPD removal step described in the preceding section when the $\Delta \tau$ contributions can be neglected. OPD drifts, caused mainly by variations in the effective length of the delay line with temperature, between the first and second measurements are most likely responsible for the remaining parabolic baseline on each sweep. The roughly 1 radian phase fluctuations that are observed correspond to vibrations on the order of the micron for a wavelength of 1550 nm. The resulting audio signal is intelligible, although the chopping effect caused by the dead time between sweeps is very noticeable. It can be shown [13] that the relationship between the sound pressure level and the resulting displacement is a double integral on a mass-limited target. Figure 6(b) (Media 2) shows the signal obtained by twice differentiating the phase from each sweep. It results in a somewhat more natural sounding voice, although, of course, the chopping effect remains. Since differentiation is a high-pass process, the drift-induced correction errors are greatly attenuated. Further filtering and processing the sound would of course yield better speech intelligibility.

Figure 7 (Media 3 and Media 4) shows the extracted phase PSD of a human male voice recovered from a beige-painted office drywall, using the stationary OSCAT and AOM method. As expected, the chopping effect cause by the sweeping dead time is absent from the resulting audio signal, resulting in a continuous sound. However, since a wall is much less responsive to voice than a polystyrene plate, the system noise is more present than before. However, in this measurement, the dominant noise source is not electronic or optical. The equipment used for this experiment have vibrating parts, such as cooling fans, transformers and power supplies. These vibrations are coupled to the measured phase through the fiber that is used to transport the optical signals. The blue curve (Media 3) shows the measured phase PSD when all devices are on the same table as the signal carrying fiber. The green curve (Media 4) is the result of the same measurement when the nosiest devices are removed from the optical table. The noise floor is reduced by a factor of more than ten near 1 kHz. Thus, mechanical coupling to optical phase has a surprisingly significant effect on the quality of the vibrometric measurements. In this case, removing the noisiest sources turns an unintelligible recovered voice sample (Media 3) into a barely intelligible one (Media 4). This mechanical coupling to the fiber can raise concerns about the source of the recovered voice sample: the measured voice could plausibly be picked up by the fibers instead of the intended target. This concern was put to rest by making a measurement using a more rigid white sample, which has a good reflectivity, but a weak response to sound waves. By supressing the response of the target, a lack of recovered voice signal confirms that the fibers in the systems are not picking up any voice signals. The sidewall of a white plastic kettle was used as a target. In that case, no voice signal was recovered, confirming that the retrieved voice signal is indeed caused by vibrations of the wall.

 

Fig. 7 Phase power spectral density for the same measurement before and after removing the noisy devices (Media 3 and Media 4). The PSD is significantly lower with the devices off the optical tables, which shows that the coupling from mechanical vibrations to optical phase is significant. The remaining peaks on the green curves are mostly multiples of 60 Hz, coming from the power supplies of the devices which were not removed from the table.

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The remaining peaks in the green curve on Fig. 7 are mostly multiples of 60 Hz, which can be traced back to the transformers and power supplies of the devices that remain on the optical table. These harmonics can be notched out with a digital filter; the result of this filtering is shown on Fig. 8 (Media 4 and Media 5). After notch filtering, the resulting voice sample is clearly intelligible.

 

Fig. 8 Phase PSD before (Media 4) and after (Media 5) notching out multiples of 60 Hz.

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By doing more involved signal processing, it is possible to improve the results further. Using a technique called spectral noise gating [14], a form of nonlinear filter, most of the background noise can be removed from the waveform. The basic principle of the technique consists of taking the discrete Fourier transform of the isolated background noise to get its spectral footprint. The signal is then Fourier transformed in sections. For every section, each signal frequency bin is compared to the noise footprint. If the signal amplitude is not significantly higher than the noise footprint in a bin, it is strongly attenuated. In the other case, it is left untouched. The implementation of spectral noise gating used for this paper is the noise removal feature of Audacity [15], an open source sound editing application. Figure 9 shows the waveform before (Media 5) and after (Media 6) spectral noise gating. The background noise is attenuated significantly, as can also be heard in the sound files.

 

Fig. 9 Voice waveform before (Media 5) and after (Media 6) spectral noise gating. A substantial amount of noise it removed, to the point where individual words can be seen on the waveform.

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The last measurement showcases the range-resolved nature of the technique. As was the case for the last measurement, a human voice sample is still recovered from a wall, but a second signal is introduced, stronger both in optical power and in mechanical vibration amplitude. Approximately one meter in front of the wall, a glass slab was introduced in the path of the optical beam, resulting in a strong first reflection which was aimed back at the launcher. The slab was mounted on a piezoelectric actuator that was fed a chirped sine wave between 300 Hz and 3 kHz. By changing the repetition rate of the comb, it is possible to tune the system on only one of the reflections, thus completely ignoring the other. Figure 10 (Media 7 and Media 8) shows the independently recovered amplitudes, on a logarithmic scale, from both reflectors. The human voice sample, shown on the green curve [noise gated audio file in Media 7], is clearly weaker than the chirped signal, shown on the blue curve (Media 8). However, due to the range resolved nature of the measurement, the desired voice sample is completely isolated from the stronger parasite signal.

 

Fig. 10 Signal amplitudes recovered from the piezoelectric actuated glass slab (Media 7) and from the wall (Media 8). Although the vibration amplitude of the glass slab is much higher than that of the wall, the human voice sample can be recovered in isolation from the chirp signal on the glass slab.

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5. Conclusion

In this paper, we have demonstrated techniques that enable vibrometric measurements of diffuse surfaces using a single frequency comb in the OSCAT configuration. Two different methods for performing the measurements, either by sweeping the repetition rate or by frequency-shifting the interferogram using an acousto-optic modulator where shown. An adapted version of the correction algorithm used in conventional comb-based spectroscopy was also presented, along with a noise analysis.

Results were presented using both the OSCAT sweeping method and the stationary AOM assisted method. The voice signal obtained from a diffusely reflective office wall was intelligible, provided appropriate care was taken to move mechanical noise sources away from the signal carrying fibers and appropriate post-processing filtering was done. Furthermore, the range resolution capabilities of the system were demonstrated by measuring both a weak human voice sample from a wall and a strong interference signal from a glass slab in isolation. The system has a thermal noise-limited sensitivity on the order of the nanometer in the 4 kHz bandwidth used for the presented voice measurements.

Although the measurements were done at a relatively short distance of approximately 3 m, it is foreseeable that longer distances could be achieved using this method, given the fact that the coherent gain from the local oscillator results in a decay of interferogram amplitude proportional to the distance and the facts that a small aperture launcher was used. Additionally, at 40 mW of output power, there is still some amplification headroom to increase the power sent to the target.