Introduction

Due to their broad emission spectrum and potential for frequency accuracy, there is a significant interest in using frequency combs for spectroscopic applications. Measurement techniques involving a single comb such as direct frequency comb spectroscopy (DFCS) [1], VIPA-based spectroscopy (for Virtually Imaged Phased Array) [2,3] and optical sampling by cavity tuning (OSCAT) [4,5] have been developed. The use of two combs with slightly detuned repetition rates was also found to be an efficient way to measure impulse responses and, consequently, optical spectra [6–8]. This is commonly known as dual-comb spectroscopy or multiheterodyne spectroscopy.

The latter approach imposes some restrictions on the choice of frequency combs. In general, one must use two lasers with nearly identical repetition rates (f_r) adapted to the duration of the impulse response of interest. Indeed, when considering the dual-comb interferometer as an optical equivalent-time sampler, one realizes that the comb probing the sample is used to generate a periodic version of its impulse response, while the second comb is used as an optical sampler, taking one point on each repetition of the impulse response. With this technique, the spectral point spacing is limited by the chosen repetition rate for the probing comb, while the repetition rate of the sampling comb remains a free parameter. This is equivalent in the time domain to a maximum measurable optical time delay, which is fixed by the probe comb repetition period (1/f_r).

It is important here to distinguish the spectral resolution from the spectral point spacing. In comb spectroscopy, the ultimate resolution that can be reached is defined by the linewidth of the comb modes over the measurement time [9]. However, the dual comb spectrometer normally samples the spectrum of interest with the same density as the comb modes, which are spaced by f_r. In this manuscript, we describe this with the expression “spectral point spacing” to avoid any confusion with the spectral resolution.

In cases where impulse responses are longer than the comb repetition period, such as with high-quality-factor microresonators [10–12], the generated impulse responses overlap and temporal aliasing occur, preventing the retrieval of information at smaller spectral intervals than the repetition rate. Using a pair of combs with low repetition rates is usually the simplest solution. However, a reduction of f_r inherently implies an equivalent reduction of the optical sampling rate and of the impulse response repetition rate. Therefore, the acquisition time for a complete interferogram is expected to be increased by a factor N² when both f_r are reduced N times, because the interferograms are longer and are measured at a slower rate.

An approach that allows reducing the spectral point spacing while bypassing the f_r reduction has already been proposed [9,13,14]. Several traces with slight changes in the comb repetition rates (or the carrier-envelope offset frequency) are acquired to generate multiple interleaved spectra that are afterwards recombined. Therefore, spectral features narrower than the comb mode spacing can be measured simply by shifting the comb modes between acquisitions. That way, the f_r reduction is avoided and the acquisition time is only increased by a factor N, which corresponds to the number of interleaved spectra generated. Because this approach requires N measurements with precisely controlled frequency shifts, it can be more cumbersome to use than typical dual-comb setups.

In this paper, we demonstrate a generalized method for dual-comb spectroscopy, allowing the use of two combs with repetition rates having any integer ratio to obtain temporally interleaved measurements. A degree of freedom is added to the conventional dual-comb method, which enables a partial decoupling of the impulse response repetition rate from the optical sampling rate. Thus, it offers the possibility to accomplish dual-comb spectroscopy with an N-fold reduction of the spectral point spacing while reducing the repetition rate of only one of the two combs N times. Again, the acquisition time of an interferogram is only increased by a factor N as opposed to N² with conventional dual-comb.

While theoretically equivalent to the spectral interleaving technique regarding the resulting spectral point spacing, this new approach is potentially easier to use, more robust and can be better adapted to applications where low repetitions rates are needed, such as hyperspectral lidar [15,16]. When allowing the use of post-referencing algorithms [17] with both the spectral and temporal interleaving methods, the combs are subject to the same stability requirement. This requirement is obviously determined by the targeted spectral resolution. As for the signal-to-noise ratio (SNR), both methods can lead to similar performance if we consider probe combs having the same average power. In the temporal interleaving case, the slow probe comb can be seen as N spectrally interleaved combs whose mode amplitudes are √N lower than those of a faster comb with equal power used for spectral interleaving. By measuring all the interleaved spectra simultaneously, each resolved spectral element can be measured N times longer than in the spectral interleaving case, which results in an SNR increase that compensates exactly for the reduction in signal amplitude. This conclusion is valid for the additive noise limited case and can be extended to the shot noise limited case as well. Since the power spectral density for the shot noise photocurrent, in [A²/Hz], is proportional to the average optical power, a fixed parameter, the same noise level is expected for both methods.

Interestingly, several groups have already reported dual-comb experiments with different repetition rates. Some past attempts involved integer-ratio repetition rates [18–20], but did not take advantage of the potential spectral point spacing reduction or acquisition speed gain. Totally arbitrary detuning between the lasers was also reported, but only to carry incoherent pump-probe measurements [21]. Maintaining or tracking the coherent phase relations with an arbitrary detuning between the combs would be a significant challenge.

In this paper, we therefore restrict ourselves to a quasi-integer relation between the comb rates, generating periodic impulse responses with the slower comb and sampling many points on each repetition with the faster comb. The methodology is first described, including the experimental setup and the algorithms allowing to retrieve the interferograms and to minimize the crosstalk between adjacent samples. The technique is then demonstrated experimentally by probing a simple resonator that has a long (~60 ns) impulse response.

1. Methodology

The proposed modification to the standard dual-comb method is based on the use of two frequency combs with repetition rates that are integer multiples. In our case, a first 16.67 MHz probe comb is used to periodically generate the impulse response of an optical device under test (DUT) while a second 100 MHz comb from Menlo Systems is used as a sampler. For this paper, a ring cavity made from a fiber loop between one output and one input of a 50/50 coupler serves as a DUT. Both sources are mode-locked fiber lasers operating around 1560 nm. The lasers are tuned until their repetition rates are almost exactly in a 6:1 ratio. A slight effective detuning (Δf_r) is intentionally left between the two combs for the pulses to slide against one another as time goes by. As shown in Fig. 1, both signals are afterwards combined in a 50/50 coupler and sent to a balanced photodetector (D) with 350 MHz of bandwidth (BW). A fast analog-to-digital converter (ADC) synchronized with the sampling comb finally acquires the detected signal with a sampling rate (f_s) of 100 MHz. As the optical field is already optically sampled by the comb, the information lies temporally coincident with the sampling pulses on the detector. Synchronizing the ADC to the sampling pulses then gives a single data point per optical sample. Rejecting the unwanted signal between two optical samples also improves the SNR. An explanation of this effect is given in the time and frequency domains in [22]. A 225 m fiber spool is placed at the output of the sampling comb, but it could also have been placed in the probe comb’s path. This serves to chirp the pulses and spread their energy over a larger period of time. The resulting interferogram has a reduced dynamic range and more power can be sent to the detector for an improved signal-to-noise ratio [17].

 

Fig. 1 Experimental setup. The slow probe comb is sent through the DUT, which is a fiber ring cavity in that case, while the faster sampling comb goes through a fiber spool to chirp its pulses to maximize SNR. Both signals are aligned in polarization with a polarization controller (PC), combined in a 50/50 coupler, sent to a balanced photodetector and acquired by an ADC. A referencing stage keeps track of the combs’ fluctuations. Solid lines represent optical fibers and dashed lines are electrical links.

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The interest of the method relies on the fact that a slow-repetition-rate comb is used to generate the DUT’s impulse response or, equivalently, to probe its spectrum on a fine grid, while a comb six times faster serves as a sampler. Thus, a small spectral point spacing can be reached without sacrificing the sampling rate. The 100 MHz comb can be seen as a set of 6 slower combs operating at 16.67 MHz, as depicted on Fig. 2(a). On Fig. 2(b), each one of these slower combs asynchronously samples the impulse response of interest, as in conventional dual-comb spectroscopy. Because the electrical sampling is synchronized with the sampling comb, each data point corresponds to an optical sample coming from one of the six slow combs. As a result, the system is said to have six time-multiplexed acquisition channels acquiring six offset interferograms simultaneously, which can be sliced at the signal processing stage, as shown on Fig. 2(c).

 

Fig. 2 Schematic representation of dual-comb spectroscopy with integer-ratio repetition rates. (a) A slow comb (black) responsible for generating the DUT’s impulse response and a faster sampling comb (multicolored). The fast comb can be seen as 6 slow combs. (b) Asynchronous sampling of the impulse response. The signal is composed of 6 time-multiplexed measurements of the impulse response. (c) Zoom out on the demultiplexed signal.

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Due to the interlaced nature of the measurement, the photodetector must have a sufficient bandwidth to prevent the electrical pulses from spreading and causing crosstalk between the time-multiplexed acquisition channels. This issue results from the rapid signal variation between two consecutive data points, unlike conventional dual-comb interferometry where the signal varies smoothly between consecutive samples. In our specific case, the 350 MHz photodetector has just enough bandwidth to make this demonstration possible. Still, a slight crosstalk can be observed between the channels. Measuring and deconvolving the photodetector’s electrical impulse response from the data easily fixes this. Note in Fig. 1 that we avoid placing an antialiasing filter before the ADC. This would make the requirement of having a fast photodetector pointless.

A good way to measure the photodetector’s electrical impulse response is to use the exact same configuration as for the DUT’s impulse response, but without the DUT, to obtain calibration data. After demultiplexing and averaging the data coming from the six acquisition channels, the result consists of the photodetector’s response to the probe comb being sampled at 100 MHz by the sampling comb. Figure 3(a) shows an example of such a measurement. In the ideal case, where a detector with a much shorter impulse response than the sampling comb’s repetition period was used, a single interferogram peak would be seen. However, since each electrical pulse generated by the sampling comb has a non-negligible value when the next sampling pulses arrive, each sample contains information about the previous pulses, which is manifested by additional peaks in the signal. On Fig. 3(a), the first burst corresponds with the arrival of an actual light pulse on the detector, while the subsequent bursts result from the leakage of the detected light on each of the 10 ns measurement increments. The amplitude of the peaks relative to the first peak at zero path difference (ZPD) gives a measurement of the inter-channel crosstalk. To get a more accurate estimate of these numbers, the first peak at ZPD can be taken separately and cross-correlated with the entire data from Fig. 3(a). Since the cross-correlation estimates the similarity between two signals, cross-correlating the first peak with the subsequent peaks, which all have the same shape, yields discrete values corresponding to the amplitude ratios between them. The result is shown on Fig. 3(b).

 

Fig. 3 (a) Photodetector’s response to the 16.67 MHz comb being sampled by the 100 MHz comb. The first burst is an actual light pulse arriving on the detector. The subsequent bursts result from the leakage of the detected light on each of the 10 ns measurement increments. (b) Cross-correlation of the ZPD peak with the complete set of data from (a). The noise floor is reduced by a factor exceeding 10. Note that the impulse response is null at a 50 ns delay.

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The combs used for this measurement are not actively stabilized. For that reason, a referencing setup measuring the beat notes between the combs and 2 continuous-wave (CW) lasers keeps track of the fluctuations [17]. These fluctuations are afterwards removed by a post-correction algorithm. However, the beat signals must be aliased through a 6-fold downsampling step before being suitable for the algorithm. This step ensures that they undergo the same downsampling treatment as the signal at the demultiplexing stage so they can properly do their referencing task.

Figure 4 depicts this downsampling step in the frequency domain for the beat references produced by the two combs and a single CW laser. The blue curve shows a simulated spectrum obtained when mixing only the 16.67 MHz comb with the 100 MHz comb. Notice that flipped copies of the spectrum are found every multiple of f_N,1 = 8.33 MHz, the Nyquist frequency of the slow comb. After demultiplexing the data, a single copy of the spectrum positioned between direct current (DC) and f_N,1 remains. Because the slow comb has a 16.67 MHz mode spacing, the reference beat obtained from a CW laser and the comb (green) is necessarily inferior to f_N,1, and can be successfully used for referencing purposes. Downsampling that reference beat 6 times to make sure it has the same number of points as the demultiplexed signal does not affect its position relative to the first copy of the spectrum. This is because its frequency was already below the post-downsampling Nyquist frequency.

 

Fig. 4 Simulated transmission spectrum (blue) obtained with a 16.67 MHz comb and a 100 MHz comb with its beat signals generated for referencing purposes. A beat signal between a CW laser and the slow comb (solid green) is found below the corresponding Nyquist frequency f_N,1 (dashed green). A beat signal between the same CW laser and the fast comb (solid red) is found below the Nyquist frequency f_N,2 (dashed red). A 6-fold downsampled and aliased version of the latter beat is also shown.

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For the fast comb, the beat signal with a CW laser can however be found anywhere between DC and f_N,2 = 50 MHz, the Nyquist frequency of the fast comb. The 6-fold downsampling aliases the beat to map it below f_N,1, around the remaining copy of the spectrum. This ensures that the referencing information is mapped to the closest 16.67 MHz tooth and can thus be used to correct the combs’ fluctuations. The same treatment is performed for the second CW laser beating with both combs.

2. Experimental demonstration

To highlight the capabilities of the method, a device with a long enough impulse response is required. For this paper, a ring cavity made from a polarization maintaining (PM) 50/50 coupler with an output spliced with one of its inputs is used. Such a cavity can be easily adjusted to generate impulse responses of any desired duration. An incoming pulse is divided in the two arms of the coupler; one half escapes from the device while the other half completes one lap in the cavity. After the first lap, the remaining of the pulse is once again divided in two and the process is repeated. Therefore, the expected response is an infinite number of equally spaced bursts decreasing in amplitude by a 1/√2 factor. The length of the cavity determines the spacing between the bursts. In our case, a loop of approximately 80 cm is used in order to benefit from the full delay range provided by the 16.67 MHz probe comb while minimising temporal aliasing.

We also show here that averaging can be used with this technique to increase the final SNR. The ring cavity is thus temperature stabilized to ensure that the system is time-invariant. While achieving long-term averaging, we also have to keep in mind that the post-referencing algorithm requires that the CW lasers beat with the same comb modes during the whole measurement in order to achieve coherent co-adding. For that reason, slow servo loops with no more than a few Hz of bandwidth are used to lock the relative positions of the combs and the CW lasers. Because the only way of controlling the probe comb’s f_r is via the slow process of thermal tuning, it is easier to leave this source free-running. The pump currents of both CW lasers are controlled to follow the probe comb’s drifts. In addition, the sampling comb is also locked to the probe comb by slowly controlling its pump current and the intracavity piezoelectric element. The effective detuning between the repetition rates of the combs is fixed around Δf_r = 5 Hz, based on the slow comb’s repetition rate, for the optical spectrum to be properly downconverted to the radio frequency range comprised between DC and f_N,1.

Ideally, both combs would be locked to the CW lasers, which are more stable than the probe comb. However, because the probe comb is difficult to control, we opt for simplicity and work in the opposite direction. We thus accept that the CW lasers can drift over a few MHz while following the comb. For short measurements, these drifts are slow enough to be ignored. However, when taking multiple measurements spaced by a significant period of time, these drifts can prevent the interferograms to be co-added correctly. Assuming that both CW lasers suffer from the same frequency shift because of the lock, we obtain sets of measurements that are frequency shifted. In the time domain, this is equivalent to a residual phase ramp. These shifts become an issue only when they are comparable to the spectral point spacing. They could obviously be tracked with a third reference at least as stable as the desired point spacing. However, the slow linear phase drifts between two sets of data is very easy to track and correct at the signal processing stage. One only has to cross-correlate two different portions of the temporal signal, extract the phase from these two correlations, fit a linear phase ramp and subtract it from the data.

The measured impulse response of the ring cavity is shown on Fig. 5(a). This is the result obtained after averaging 15,000 interferograms acquired in 500 s (500 batches of 1 s each), each one of the 6 measurement channels acquiring 2,500 interferograms during that time interval. The dispersion introduced by the 225 m fiber spool is compensated by deconvolving the chirp measured on the first burst as in [17]. After this compensation, a peak temporal SNR of 100 dB is measured considering the squared signal to work with power values. As predicted, the impulse response is composed of a series of pulses decreasing each time in amplitude. It takes approximately 3.825 ns for a pulse to complete one lap in the cavity, which is in good agreement with the approximate length of 80 cm, assuming an effective refractive index for the fiber of 1.46. However, the decay is slightly higher than expected, since we find an amplitude ratio between two successive pulses of approximately 0.57 as opposed to the projected value of 0.707. This is due to a combination of intracavity dispersion, which spread successive impulses, and round-trip losses. The weakest bursts that don’t belong to the main pulse train correspond to residual crosstalk that was not perfectly compensated by the calibration process described earlier. Furthermore, since the last pulse is still quite high, one could wonder why the measure is free of temporal aliasing. This is easily explained by the fact that the phase ramp used to correct the slow drifts is ineffective for aliased signal, which prevents it from co-adding coherently when averaging. The successful co-adding observed over the full delay range however proves that the algorithm provides a coherence time at least as long as the measurement time of a single interferogram, which is around 200 ms when an effective detuning of 5 Hz is used. Figure 5(b) shows an enlarged view of the second burst.

 

Fig. 5 (a) Impulse response of the ring cavity after correction, averaging and dispersion compensation. The full delay range covered by the probe comb is displayed. The weakest bursts correspond to residual crosstalk. (b) Enlarged view of the second burst. A carrier frequency of 5 THz is added to the signal.

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The power spectrum corresponding to the entire impulse response is shown on Fig. 6(a). A peak spectral SNR of 55 dB is measured, computed as the ratio between the maximum value and the standard deviation of the out-of-band noise. The coarse features in the spectrum, including the small dips on either side of the maximum, come from the combs’ emission spectra. A slight frequency-dependent oscillation is also noticeable on the right side of the curve when zooming in. This is caused by interference between pulses propagating at different speeds in the two polarization axes of the PM fiber. Efforts have been made to launch the light on a single axis of the coupler to minimize the interference displayed by an optical spectrum analyser.

 

Fig. 6 (a) Power spectrum of the measured impulse response. (b) Enlarged view of a few cavity modes caused by the lossy ring cavity.

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The spectral features of interests are found at a much smaller scale. Figure 6(b) is a close-up view of the top of the spectrum and reveals numerous cavity modes. They have a free spectral range of 260 MHz, which is related to the cavity length. The depth of the lines is linked to coupling losses.

3. Conclusion

We successfully carried out an interferometry experiment with two frequency combs having quasi-integer-ratio repetition rates. The proposed method was used to accomplish spectroscopy with a small spectral point spacing while maintaining a high acquisition rate. Using a slow comb only in the probing arm of the interferometer allows a reduction of the point spacing with an unchanged sampling rate. When the point spacing is already small enough, this method can be used the other way around to reduce the acquisition time by increasing the sampling comb’s rate. Since any integer-ratio works fine, one can simply take advantage of the proposed technique to gain some freedom when choosing a pair of sources devoted to dual-comb experiments. As a demonstration, we measured the impulse response of an 80-cm-long ring cavity using a 16.67 MHz comb and a 100 MHz comb. After some signal processing steps including deconvolution of the photodectector’s impulse response, demultiplexing, post-referencing and averaging, a high-quality measurement was retrieved. This technique can be useful with microresonators, which usually have long impulse responses. It can also be used in cases where low repetition rates are needed, such as with hyperspectral lidar. This temporal interleaving approach is believed to be similar to spectral interleaving in terms of stability requirements and SNR, although the implementation of temporal interleaving is more straightforward.