# Sensitivity Improvement in Dual Comb Spectroscopy With Time-Programmed Electro-Optic Frequency Combs

Debanuj Chatterjee, Eve-Line L Bancel, Matteo Conforti, Siddharth

Sivankutty, Hervé Rigneault, Pascal Szriftgiser, Steven Cundiff, Arnaud
Mussot

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## ► To cite this version:

Debanuj Chatterjee, Eve-Line L Bancel, Matteo Conforti, Siddharth Sivankutty, Hervé Rigneault, et al. Sensitivity Improvement in Dual Comb Spectroscopy With Time-Programmed Electro-Optic Frequency Combs. Journal of Lightwave Technology, 2025, 43 (10), pp.4648 - 4658. 10.1109/jlt.2025.3544461. hal-05371343

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Distributed under a Creative Commons Attribution 4.0 International License
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# Sensitivity Improvement in Dual Comb Spectroscopy With Time-Programmed Electro-Optic Frequency Combs

Debanuj Chatterjee, Eve-Line Bancel, Matteo Conforti, Siddharth Sivankutty, Hervé Rigneault, Pascal Szriftgiser, Steven Cundiff, Fellow, IEEE, and Arnaud Mussot

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**Abstract**—We demonstrate the signal-to-noise ratio (SNR) and/or the recording speed improvement of dual comb spectroscopy (DCS) with time-programmed electro-optically modulated (EOM) frequency combs, in a simple all-fiber architecture. Unlike mode-locked laser based combs, where time-programming requires complex locking mechanisms, time-programming in EOM combs is simple and robust. Thus by directly engineering time-shifts into the pulse train of an EOM comb, the dead time of the interferogram signal in DCS is reduced. The utilized time-shifting technique is theoretically analyzed and the results are experimentally verified. A SNR improvement of 11.6 dB or a 12 times reduction in recording speed is demonstrated experimentally with this simple method. These results open new possibilities for improvement of SNR and/or recording speed of EOM-based multi-comb spectroscopic techniques.

**Index Terms**—Frequency comb, dual comb spectroscopy, time programmable comb, electro-optically modulated comb, nonlinear optics, time shifted comb, time multiplexed interferogram.

## I. INTRODUCTION

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DUAL comb spectroscopy (DCS) is a well-known technique for performing fast and high-resolution

Received 1 November 2024; revised 30 January 2025; accepted 16 February 2025. This work was supported in part by the Agence Nationale de la Recherche (Programme Investissements d'Avenir, FARCO Projects), Ministry of Higher Education and Research, in part by the Hauts de France Council (GPEG Project), in part by European Regional Development Fund (Photonics for Society P4S), and in part by CNRS (IRP LAFONI). The work of Debanuj Chatterjee and Arnaud Mussot was supported by Project FRESCOS (Marie Curie Postdoctoral Fellowship). (*Corresponding author: Debanuj Chatterjee*.)

Debanuj Chatterjee, Matteo Conforti, Siddharth Sivankutty, and Pascal Szriftgiser are with the Laboratoire de Physique des Lasers, Atomes et Molécules, PhLAM, UMR 8523 at the University of Lille, CNRS, Lille, France (e-mail: debanuj.chatterjee@univ-lille.fr; matteo.conforti@univ-lille.fr; siddharth.sivankutty@univ-lille.fr; pascal.szriftgiser@univ-lille.fr).

Eve-Line Bancel is with the Laboratoire de Physique des Lasers, Atomes et Molécules, PhLAM, UMR 8523 at the University of Lille, CNRS, Lille, France, and also with the Office national d'études et de recherches aérospatiales, 91120 Palaiseau, France (e-mail: eve-line.bancel@hotmail.fr).

Hervé Rigneault is with the Aix Marseille Univ, CNRS, Centrale Med, Institut Fresnel, Marseille, France (e-mail: herve.rigneault@fresnel.fr).

Steven Cundiff is with the Department of Physics, University of Michigan, Ann Arbor, MI 48109 USA (e-mail: cundiff@umich.edu).

Arnaud Mussot is with the Laboratoire de Physique des Lasers, Atomes et Molécules, PhLAM, UMR 8523 at the University of Lille, CNRS, Lille, France, and also with the Institut Universitaire de France, France (e-mail: arnaud.mussot@univ-lille.fr).

Color versions of one or more figures in this article are available at https://doi.org/10.1109/JLT.2025.3544461.

Digital Object Identifier 10.1109/JLT.2025.3544461

spectroscopic measurements [1], [2], [3]. Demonstrations of DCS in several spectral windows ranging from ultra-violet [4] to IR [5], [6], probing electronic transitions in atoms or molecules or the vibrational transitions of gas-phase molecules respectively, have been reported, making this technique applicable to a large range of materials and analyses. Recent advances in light sources for DCS offer the opportunity to observe the dynamics of ultra-fast irreversible non-equilibrium and stochastic physico-chemical processes, in chemistry and biology [7], photolysis [8], or supersonic jets [9], [10] to name a few.

DCS relies on the interference between two combs with slightly different repetition rates, $f_1$ and $f_2$. These time domain interferogram traces exhibit bursts separated by the inverse of the difference in repetition rates $\Delta f$. The temporal interferograms, collected in the radio frequency domain, are converted to spectra by numerical Fourier transformation. The factor to map between optical and radio frequency domains is $f_1/\Delta f$. Data sampling requirements impose that it must be high enough to avoid spectral aliasing between parts of the fundamental RF beating comb with high-order beating ones [1], [2]. In conventional DCS systems with low repetition rate frequency combs, sampling often extends far beyond the interferogram centerburst and the exponentially decaying free induction decay signal [11], resulting in acquisition of data points with no useful information. Thus, numerous techniques have been developed in the past to minimize the so called dead-time of an interferogram signal, or in other words enhance the SNR and/or detection speed of the system [3], [12], [13], [14], [15], [16], [17], [18], [19], [20].

The different approaches in this pursuit can be broadly classified into two categories : 1. apodization based approach and 2. approach based on altering *f* and/or *Δf* of the system. The apodization approach simply flips the sign of *Δf* dynamically, such that the pulse trains keep slewing across each other back and forth, each time generating an interferogram peak [3], [21], [22], [23]. On the other hand, the second approach is based on increasing either *f* or *Δf* of the system using techniques like spectral filtering [14], [17], introducing delay lines [19], [20], [24], [25], [26], etc. It is worthwhile to mention here that while increasing the comb repetition rate might seem the easiest route towards increasing the SNR, there are scenarios where a lower repetition rate is preferable. These include studies of samples where photobleaching occurs due to the population

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being trapped in dark excited states. In addition, if the DCS is used in nonlinear experiment, such as a pump-probe [27] or tricomb spectroscopy [28], [29] where higher pulse energies are desirable, increasing the repetition rate is problematic as it lowers the pulse energy for a given average power.

The idea of dead-time reduction in DCS through altering the pulse-to-pulse time gap have been implemented with several different architectures, however, all these different approaches come with their own advantages and disadvantages. Techniques for apodized DCS, based on cavity length modulation [3], [22], [23], often also referred to as programmable combs [30], although extremely robust, involve complex phase locking techniques for precise control of the MLL repetition rates. Also, the techniques based on delay lines and spectral filters are either lossy or constrained with respect to frequency tunability, or both. The EOM comb technology on the other hand can avoid these issues, owing to its inherent frequency agility and simplicity of implementation. Thus in our approach, by applying periodic time shifts on one of the pulse trains of an EOM based DCS setup, we can access an extra degree of freedom on the generated interferogram. These time shifts, that introduce a dynamic delay, are produced simply by programming a time shifted modulation signal on an arbitrary waveform generator and modulating the EOMs with that signal. The length and periodicity of the delay can be tailored to modify (increase in our case) the number of interferograms in a given time duration, and hence enhance the SNR-detection speed ratio of the system.

## II. PRINCIPLE OF TIME MULTIPLEXED INTERFEROGRAMS

As discussed above, DCS requires two coherent optical frequency combs (or pulse trains) with a slightly different comb line spacing (or repetition rate). Due to the difference in the repetition rates of the two pulse trains, the pulses in the train with a lower period repetitively scan through the pulses of the other train at a rate $\Delta f$. The interferograms are produced only when the pulses from both the pulse trains line up perfectly in time. Thus the interference signal shows periodic short bursts of amplitude (when the two pulses line up), and a near-zero amplitude at all other times. Hence most of the interference signal is practically devoid of useful information [as long as the sample's free induction decay time scale is much shorter than the time period of the pulse train(s) probing the sample]. Therefore to utilize the available time window of the interference signal more efficiently, we utilize the idea of folding or multiplexing the interferograms in time, similar to the apodization technique [21].

A numerical example of the interferogram time multiplexing principle is illustrated in Fig. 1(a). The box in Fig. 1(a) shows two optical pulse trains labeled 1 and 2 corresponding to the comb probing the sample (called signal comb) and the reference comb (called LO comb) respectively in a standard DCS setup. The signal pulse train has a shorter time period, $T_1$, compared to that of the LO, which has a period $T_2$. The times when the two pulse trains coincide are indicated with dashed vertical lines and enumerated with encircled numbers 1 and 2. These are the times when the interferogram peaks are obtained in the RF signal when the combined optical pulse trains are incident on a photodetector.

<figure><img src="image_3.png" /><figcaption>Fig. 1. (a) Illustration of temporal optical pulse trains (1 and 2) corresponding to signal and LO combs in a DCS setup. A standard case is shown in the box and a time shifted version of the LO pulse train is shown in the bottom. Dashed lines with circled numbers are provided to aid the eye in following the times when the two pulse trains coincide. (b) Numerical simulation of time domain and (c) numerical simulation of frequency domain interferogram signal for standard (red) and multiplexed (blue) case in a DCS. Parameters are provided in text.</figcaption></figure>

The natural time period of such an interferometric signal is $1/\Delta f$, where $\Delta f$ is the difference between the repetition rates of the two pulse trains. Contrary to the standard DCS case, the bottom most pulse train in Fig. 1(a) (in orange) shows a time shifted version of the standard LO pulse train, where due to the time shift, the pulses in the signal train and the time shifted LO train line up more often compared to the standard case (see encircled numbers 1, 2 and 3). Therefore, by periodically providing a time shift on the LO pulse train, we can realize a larger number of interferogram bursts in a given time as compared to the standard case. That in turn leads to an increase in the SNR.

Correspondingly in Fig. 1(b), (c) we also show the numerical simulation of an interference signal produced from combs 1 (frequency $f_1 = 500$ MHz) and 2 (frequency $f_2 = 499$ MHz) with (blue) and without (red) the time shifting. The time shifting rate $r$, defined as the rate at which the periodic time shifts are provided, was set to 2 MHz, which was twice the frequency ($\Delta f = f_1 - f_2 = 1$ MHz) of the interferogram signal. Thus, in this case, the multiplexing factor $M$ is defined as

$$M = \frac{r}{\Delta f} \qquad (1)$$

It becomes 2 in that example, indicating a two-fold multiplexing of the interference signal. We clearly see from Fig. 1(b) that the standard case (without time shifts) has a larger time period between the interferograms compared to the time shifted case, where the interferogram time period is halved. We should also note here that a successful multiplexing is critically dependent on the rate of time shifting $r$ (or conversely its inverse $w$, which is the time period of the time shifting) to ensure the overlap of the pulses after a time shift. In fact it is easy to see that for a precise multiplexing, $w$ should be an integer multiple of the time period of the signal pulse train $T_1$.
Equivalently, we can see from Fig. 1(c), due to the multi-
plexing, the increase in the number of intergerogram pulses
in the time domain leads to an increase in the SNR in the
frequency domain, for the multiplexed comb compared to the
standard case. However, it is also important to note that the
multiplexed interferogram comb has fewer (half) the number
of teeth compared to the standard case, leading to a reduction
of spectral resolution. Additionally, although the multiplexing
technique increases the number of interferogram peaks, it does
not increase the total RF bandwidth, hence mitigating problems
of spectral aliasing.

III. THEORETICAL MODEL

In this section, we introduce an approximate theoretical model
to investigate the implications of generating a pulse train with
periodic time shifts and capturing its physical insights. Recall
from the previous section’s discussion that imparting periodic
time shifts in the time domain LO pulse train led to the multi-
plexing of the resulting interferogram. To understand the origins
of this effect, we aim to express the signal and LO pulse trains in
the time and frequency domains and analyze the effect of time
shifting on the characteristics of the same.

We start with defining the two optical frequency combs (la-
belled 1 and 2) in the two arms of the dual comb spectrometer
with comb line separation f₁ and f₂ = f₁ - Δf respectively.
Without loss of generality, we assume the pulse duration is much
shorter compared to the comb period. Thus, for simplicity, we
can fairly assume the time domain optical signal, which is the
electric field amplitude (neglecting the fast oscillating carrier)
Sⱼ(t) in arm j (j = 1, 2) is a pulse train Pⱼ(t) that can be
approximated as a Dirac comb

$$
S_j(t) = P_j(t) = \sum_{k=-\infty}^{\infty} \delta(t - kT_j - \phi_j), \quad \text{where } j = 1, 2, \qquad (2)
$$

where $\delta(t)$ is the Dirac delta function, $T_j = 1/f_j$ and $\phi_j$ are the
pulse train time period and input phase in arm $j$ respectively [see
Fig. 2(a)]. In this case, the frequency domain representation of
$S_j(t)$ or its Fourier transform $\mathcal{F}\{S_j(t)\}$ is [31]

$$
\begin{align}
\mathcal{F}\{S_j(t)\} &= \mathcal{F}\{P_j(t)\} \nonumber \\
&= \frac{1}{T_j} \sum_{k=-\infty}^{\infty} \delta\left(\nu - \frac{k}{T_j}\right) \exp\left(-2\pi i \frac{k}{T_j}\phi_j\right), \tag{3}
\end{align}
$$

where $\nu$ represents frequency. We have neglected the possibility
that the pulse train has a pulse-to-pulse carrier-envelope phase
shift in the time domain, which results in a rigid shift of the
comb in the frequency domain [32]. Since in our investigation,
the pulses in both the arms of the DCS are generated from the
same CW source, they experience the same fluctuations in the
carrier-envelope phase shift, justifying the above assumption.

In the time shifted case, while the signal in the arm 1 remains
the same, the signal in arm 2, i.e. $\tilde{S}_2(t)$ can be thought of as
a periodically repeated “leitmotif” pattern $L(t)$ [see Fig. 2(b)],
with a period $w$ (which is the inverse of the time shifting rate
$r$), where $w$ is not an integer multiple of $T_2$ [see Fig. 2(c)]. If
$w$ were an integer multiple of $T_2$, the time shift would leave the

<figure><img src="image_4.png" /><figcaption>Fig. 2. Illustration of (a) Dirac comb (blue) and gate signal (green), (b) leitmotif signal (magenta) and (c) time shifted pulse train (red) in DCS.</figcaption></figure>

pulse train unchanged. Thus in this case the total time domain
signal in arm 2 becomes

$$
\tilde{S}_2(t) = \sum_{l=-\infty}^{\infty} L(t - wl). \quad (4)
$$

Further, the leitmotif $L(t)$ is constructed by multiplying the pulse
train $P_2(t)$ with a gate function $G(t)$ that has a width $w$ and is
centered at $w/2$. $G(t)$ is

$$
G(t) = \begin{cases} 1 & \text{if } |t - \frac{w}{2}| \le \frac{w}{2}, \\ 0 & \text{if } |t - \frac{w}{2}| > \frac{w}{2}. \end{cases} \quad (5)
$$

Therefore the time domain signal can be written

$$
\tilde{S}_2(t) = \sum_{l=-\infty}^{\infty} P_2(t - wl)G(t - wl). \quad (6)
$$

Correspondingly, in the frequency domain, $\mathcal{F}\{\tilde{S}_2(t)\}$ is

$$
\mathcal{F}\{\tilde{S}_2(t)\} = \sum_{l=-\infty}^{\infty} \mathcal{F}\{P_2(t - wl)\} \otimes \mathcal{F}\{G(t - wl)\}, \quad (7)
$$

where $\otimes$ is the convolution operator and we used the convolution
theorem to convert the Fourier transform of product of two
functions into the convolution of their corresponding Fourier
transforms. Here we can see that the resulting frequency domain
signal is a comb with $1/w$ periodicity, convolved with a sinc
functions, since the Fourier transform of a gate function is a
sinc function.

By inserting the expressions for $P_2(t - wl)$ and $G(t - wl)$
in (7), and evaluating the convolution (see Appendix for the
detailed derivation), we find a simplified expression

$$
\mathcal{F}\{\tilde{S}_2(t)\} = \frac{1}{T_2} \sum_{k=-\infty}^{\infty} \exp\left[-2\pi i \left(\frac{k\phi_2}{T_2} + \frac{w\nu}{2} - \frac{kw}{2T_2}\right)\right] \\
\times \text{sinc}\left[\pi w \left(\nu - \frac{k}{T_2}\right)\right] \sum_{l=-\infty}^{\infty} \delta\left(\nu - \frac{l}{w}\right). \qquad (8)
$$
<figure><img src="image_5.png" /><figcaption>Fig. 3. Plot of amplitudes (in arbitrary units) of optical combs: gate comb (green), standard comb (blue), sinc envelope (cyan) and the time shifted comb (red) following (8) for (a), (b) and (c), (d) from numerics as a function of the frequency relative to the carrier. Plots are shifted vertically to visualize the overlaps. For (a), (c), $\frac{1}{w}$ = 200 MHz is not divisible by $\frac{1}{T_2}$ = 1.1 GHz and for (b), (d) $\frac{1}{w}$ = 200 MHz is divisible by $\frac{1}{T_2}$ = 1 GHz. Black dotted lines are provided to highlight the positions of the comb lines dropped due to multiplexing. Inset in (a) zoom on the sinc function superposed with other curves.</figcaption></figure>

In (8), the last summation term results in a dense Dirac comb with a periodicity of $1/w$ (smaller than $1/T_2$), which we refer to as the gate comb since it originates from the gate function. This gate comb is shaped by a comb with lines at $k/T_2$ where each line has a sinc function profile. Note that since the sinc function has a width of $\frac{1}{w}$, which is equal to the gate comb's line spacing, a single sinc envelope can effectively remove only one line from the gate comb. For clarity, in Fig. 3 we plot the absolute values of the different terms in (8), viz. gate comb (solid green lines), the sinc envelope (solid grey lines), and the final time shifted comb (solid red lines). For reference, we also show the standard comb [as in (3)] in blue. Two cases were considered, (a) when $w$ is not a multiple of $T_2$ ($1/w$ = 200 MHz, $1/T_2$ = 1.1 GHz) and (b) when $w$ is a multiple of $T_2$ ($1/w$ = 200 MHz, $1/T_2$ = 1 GHz). As we expect, from Fig. 3(b) we see that when $w$ is a multiple of $T_2$, the time shifting effectively has no effect and the resulting comb converges with the standard comb (see blue and red curves). However, from Fig. 3(a) we see that when $w$ is not a multiple of $T_2$, each alternating tooth in the standard comb is dropped for the time shifted comb. This is due to the lack of overlap between the gate comb and the sinc envelope. As a result, the time shifted comb effectively becomes a comb with a larger comb line spacing, increasing the repetition rate of the RF interferograms without additional loss [13], [15]. The dropping of the comb lines are often not exactly perfect due to the presence of the sidelobes of the sinc envelope which cause small sidebands to emerge in the spectrum.

To validate this theoretical analysis, we also numerically simulated the pulse trains in the time domain with a pulse width of 1 ns, and the same parameters as used in Fig. 3(a) and (b). The numerical results after moving to the frequency domain are shown in Fig. 3(c) and (d), where we see that alternating teeth in the comb gets dropped as predicted from the theory (since in this case $M = 2$). The larger amplitude of the sinc sidebands in the numerical case, compared to the theory, can be attributed to the fact that the numerical case represents a convolution of the sinc

function with a comb with finite linewidth for each comb line, which is a consequence of the finite length of the signal in the time domain. Apart from this slight difference in amplitude, we do not see a major difference between the theory and simulation, indicating our simplified model enables accurate predictions.

From (8), we can deduce that when the leitmotiv duration equals the period of the LO ($w = T_2$), the time shifting converges to the standard DCS case for arm 2, we get back (3) from (8). Furthermore, after approximating the sinc function in (8) as a Dirac delta centered at $\frac{k}{T_2}$ (since the sinc is non-negligible only in the vicinity of $\frac{k}{T_2}$), and dropping a few phase terms (see Appendix for details), we arrive at an even simpler expression:

$$ 
\begin{aligned}
\mathcal{F}\{\tilde{S}_2(t)\} &\approx \frac{1}{T_2} \sum_{k=-\infty}^{\infty} \delta\left(\nu - \frac{k}{q}\right) \exp\left(-\frac{2\pi ipk\phi_2}{T_2}\right) \\
&= \frac{1}{T_2} \sum_{k=-\infty}^{\infty} \delta\left(\nu - M\frac{k}{T_2}\right) \exp\left(-\frac{2\pi ipk\phi_2}{T_2}\right),
\end{aligned}
\quad (9) $$

where $\frac{1}{q}$ is the LCM (least common multiple) of $\frac{1}{T_2}$ and $\frac{1}{w}$, and $p$ is the quotient of LCM($\frac{1}{w}$, $\frac{1}{T_2}$) divided by $\frac{1}{T_2}$. $M$ is the multiplexing factor, that is redefined here assuming that $M$ is an integer. Note that unlike (1), here the dependence of $M$ on $T_1$ is implicit, since the assumption that $M$ must be an integer already makes $M$ dependent on $T_1$. Thus from (9), we find a Dirac comb with fewer (reduced by a factor of $M$) number of teeth than a standard case [see Fig. 3(a)], making a similar effect as spectral filtering [14] but, importantly, without additional loss. Thanks to the frequency agility of the EOM comb, where the number of dropped teeth can be adjusted easily by adjusting the parameter $w$, thereby changing the value of $q$. On the other hand, when $w$ is an exact multiple of $T_2$, the LCM of $1/w$ and $1/T_2$ is always $1/T_2$. Therefore, the time shifted pulse train converges to the standard DCS case [see Fig. 3(b)].

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<figure><img src="image_6.png" /><figcaption>Fig. 4. Experimental scheme of a DCS setup with time multiplexed interferogram. A cross section of the utilized tri-core fiber (only two cores used), from scanning electron microscope is shown in the inset. AOM: acousto optic modulator, AWG: arbitrary waveform generator, EDFA: Erbium doped fiber amplifier, IM: intensity modulator, RFS: radio frequency synthesizer, DCF: dual core fiber (since only two cores of a tri-core fiber are used), VOA: variable optical attenuator, PD: photodetector, DUT: device under test.</figcaption></figure>

## IV. EXPERIMENT

In this section, we outline the experimental demonstration of the time multiplexing technique discussed above, in a realistic setting and quantitatively investigate the system's performance enhancement resulting from multiplexing.

### A. *Proof of Concept*

The experimental scheme is shown in Fig. 4. We used two EOM-based frequency combs that were generated by modulating an incident CW light from a diode laser from NKT Photonics (wavelength: 1549.6 nm, linewidth < 1 kHz), with an intensity modulator (IM). The IMs were modulated with RF signals produced from two different programmable arbitrary waveform generators (AWG) from Tektronix. These AWGs were fed with slightly different clock signals generated from a common multichannel RF synthesizer (RFS) from Holzworth, to create a small difference in their sampling rates. Therefore, the same RF pattern played on the two AWGs produces pulse trains with slightly different repetition rates. We generated pulses with 80ps pulsewidth at full-width half maximum as shown in Fig. 5(a) (measured using a fast photodetector with 60 GHz bandwidth and a fast oscilloscope with 70 GHz bandwidth). An acousto-optic modulator (AOM) driven at 100 MHz was used in one of the arms to offset the center of the RF interferogram comb from the DC, for non-ambiguous detection [1]. The EOM combs generated in the two arms of DCS were amplified with an Erbium doped fiber amplifier (EDFA) and subsequently spectrally broadened (to about 3 nm) via propagation through two different cores of a 1 km nonlinear tri-core fiber [here we refer as a dual core fiber (DCF) since we use only two of the three cores] with nonlinear coefficient $5 (\text{W.km})^{-1}$. The DCF ensures a high degree of mutual coherence between the combs [33]. The cross section of the fiber face is shown in inset of Fig. 4. The spectra of the two combs labeled 1 and 2 recorded using an optical spectrum analyzer (OSA) with 0.02 nm resolution are shown in Fig. 5(b). The undulating shape of the spectra is a typical characteristic of SPM broadening for optical pulses [33]. It is however possible to achieve a flatter spectrum by reducing the RF noise in the system, improving the modulator extinction

ratio [34], using pulse shaping techniques prior to the nonlinear spectral broadening [35] or by ensuring a similar shape for the two broadened comb spectra.

After broadening, at the output of the DCF, the two combs were combined and then split into two arms, one contributing to a reference interferogram, and the other generating the measurement interferogram, where the combs passed through the test absorption medium or device under test (DUT). In our case, we used a commercial programmable filter (Wave-shaper) as the DUT. On the reference arm, a variable optical attenuator (VOA) was inserted to balance out the insertion loss of the programmable filter in the detection arm. We note that this configuration, where both the combs pass through the DUT is known as a symmetric or collinear configuration, as opposed to an asymmetric configuration, where only one of the combs passes through the absorption medium, and is more sensitive to the sample's absorption amplitude response [1]. A Lorentzian absorption profile with 50 GHz full width at half maximum (FWHM) and 10 dB peak attenuation level was generated on the programmable filter.

In the DCS setup, both the reference and detection arms were detected using a photodetector (1.6 GHz bandwidth) and their electrical signals were fed into two independent channels of an oscilloscope (8 GHz bandwidth) making simultaneous measurements at 1 G samples/sec, with a record length of 1 ms. A simultaneous recording scheme was chosen to avoid measurement errors from the slow drift of the comb envelope due to the fluctuations in the system arising from the IMs and the EDFs.

In this experiment, the comb line spacing of comb 1, $f_1 = 500$ MHz and comb 2, $f_2 = 500$ MHz–50 kHz, meaning $\Delta f = 50$ kHz was set. A sample time and frequency domain interferogram from the experiment with multiplexing rate $M = 1.56$ is shown in Fig. 6. We clearly see from Fig. 6(a), that the interferogram time period decreases due to multiplexing, and as a consequence, in the frequency domain, the SNR increases [see Fig. 6(b)]. The smaller peaks on either sides of the interferogram burst in Fig. 6(a), which we also refer to as sidelobe peaks, originate due to the presence of sidelobes in the temporal pulses after spectral broadening. The additional secondary peaks in the multiplexed case (blue curve) in Fig. 6(a) arise due to the uneven spacing of the pulses at each time shift location. The PD, having a longer averaging time than the time period of the optical pulses, perceives this time shift as a change in the average incident optical power, and produces a peak. Furthermore, the strengths of these peaks can be reduced by dropping the first pulse after the time shift (to reduce the increase of average power on the PD) and also by optimizing the time point of the shift. However, these peaks do not influence the qualitative nature of the analysis. Thus after acquiring the radio frequency spectra in the reference and detection arms, in a digital post-processing step, the frequency and amplitudes of the individual comb lines were extracted and the waveshaper transfer function was recovered. The final results of the proof-of-concept DCS experiment with interferogram time multiplexing are shown in Fig. 7. We can see from the frequency spectra in the reference [Fig. 7(a)], and detection arms [Fig. 7(d)], that the RF comb shows a dip in the center due

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<figure><img src="image_7.png" /><figcaption>Fig. 5. (a) Temporal pulses recorded using a fast photodetector (60 GHz bandwidth) and fast oscilloscope (70 GHz bandwidth). (b) Optical spectra of the two combs used for DCS after spectral broadening, recorded using an optical spectrum analyzer.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6. Interferogram from experiments in (a) time domain and (b), (c) frequency domain for standard (red) and multiplexed (blue) case in DCS with a multiplexing rate $M = 1.56$. (c) is the zoomed version of (b). The dashed line at 81.95 MHz in (b) and (c) indicates the position of the peak on one of the edges of the spectrum.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7. Interferogram spectra from experiments (a), (d) with (blue) and without (red) multiplexing ($M = 2$). The recovered filter function considering a single trace and 20 traces after post-processing of the interferogram data are shown in (b) and (c) respectively. Red for standard case and blue for multiplexed, and the programmed filter function is shown in a green solid line. (e) and (f) shows the residue of the recovered function compared to the programmed function. Inset of (f) shows the same plot with an enlarged vertical axis. Pow.: power, WS: waveshaper function.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Fig. 8. (a)–(h) Sample radio frequency spectra (only the outermost lobe of the whole spectrum is shown) for different values of multiplexing rate *M*. Green dotted lines indicate the position of the highest point in the considered lobe. Yellow dashed-dot line shows the noise floor. Black dotted horizontal lines provide aid to the eye to show the levels of the lowest (*M* = 1) and highest (*M* = 3.3) SNRs. (i) Evolution of SNR increase (compared to the standard case) as a function of *M*. Red circles are from the experiment and dashed blue curve is the theoretical prediction.</figcaption></figure>

to the presence of the programmed Lorentzian filter function on the waveshaper. The programmed Lorentzian function as a function of wavelength is depicted in green solid line in Fig. 7(c), which we considered as the true filter transfer function and used for the evaluation of residue. This assumption was validated by comparing the OSA spectra of a broadband source, with and without the waveshaper, and experimentally obtaining its transfer function that converged to the programmed function with a residue less than 2%. The residue expressed in % was calculated as

$$ \text{residue} = \left| \frac{\text{measured power}}{\text{programmed power}} - 1 \right| \times 100. \quad (10) $$

In order to recover the filter function, first the time domain interferogram data obtained from the two (reference and measurement) oscilloscope channels were Fourier transformed. After that, each comb line powers were determined and the ratio of the powers in the corresponding lines in the measurement and reference arms were computed. After this post-processing of the interferogram data, the recovered function in log scale is shown in red (standard) and blue (multiplexed, *M* = 2) circular symbols in Fig. 7(b). From Fig. 7(b) we can see that the time multiplexed DCS method recovered the programmed filter function very precisely. In fact, it slightly outperformed the traditional method, as can be seen from the residue data [see Fig. 7(e)]. The tendency of a larger residue near the absorption center is expected due to the low SNR of the absorbed spectrum at the center. Note for long averaging times, as in standard DCS setups, an almost perfect recovery of the filter function [(less than 10% residue, see inset of Fig. 7(f))] is reported for the standard and multiplexed techniques [see Fig. 7(c), (f)]. These observations reiterate the potential of the time multiplexing technique to improve the SNR of a DCS system at short recording timescales.

## B. Signal-to-Noise Ratio (SNR)

The signal-to-noise ratio (SNR) of a DCS setup depends on several different parameters of the system [36], [37]. In this

work, we focused on improving the SNR through temporal folding of the interferogram signal. However, to develop a better understanding of the dynamics of the system, we scanned the multiplexing rate *M*, and plotted the SNR increase (compared to a standard case) on *M* in Fig. 8(i). For each case, the SNR was evaluated by finding the ratio of the maximum power in the outermost lobe (near 97 MHz RF frequency) of the comb spectrum [green dashed-dotted line in Fig. 8(a)–(h)] and the noise power level [yellow dashed-dot lines in Fig. 8(a)–(h)]. We varied *M* from a value of 1, which corresponds to the standard DCS case, to a maximum of 3.3, where SNR increase was about 11.6 dB [see black dotted lines in Fig. 8(a) marking the highest and lowest values of the spectrum peaks with varying values of *M*]. Above this value of *M*, the DCS interferogram structure degraded, due to the temporal overlap between consecutive interferograms for large *M*.

Since on every doubling of *M*, there is a 2 × 3 = 6 dB (the factor of 2 comes from the fact that the RF signal is generated from the product of two optical wave amplitudes) increase in the SNR of the RF interferogram, the increase in SNR (in dB) in terms of *M* is

$$ \text{SNR increase} = 6 \log_2 M. \quad (11) $$

Indeed in Fig. 8(i) we see that the SNR increase correctly follows the predicted logarithmic *M*-dependence [38]. We also see from Fig. 8, that the gap (in the x-axis) between subsequent experimental data points increases with *M*. For an efficient multiplexing, the system should always satisfy the condition

$$ s_2 \left( \frac{k_1 c_1}{s_1} - \frac{k_2 c_2}{s_2} \right) = \alpha, \quad (12) $$

where *α* must be an integer, and *k*<sub>*i*</sub> is the number of cycles of pulse train *i* before a time shift of pulse train 2 occurs, *s*<sub>*i*</sub> is the sampling rate of AWG in the arm *i* and *c*<sub>*i*</sub> is the length of one cycle (in samples) of the pulse train in arm *i*. In Fig. 10 we show an illustrative example of the encoded pulse train patterns in the two AWGs, following (12). For the two pulse trains labeled

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<figure><img src="image_11.png" /><figcaption>Fig. 9. Plot of (a) RF spectrum, (b) recovered waveshaper function and (c) residual of the measurement, for a multiplexed (blue) and standard (red) DCS. The programmed Lorentzian (10 dB maximum attenuation) function on the waveshaper is shown in green in (b).</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 10. Illustrative explanation of (12) showing the two pulse train patterns coded in the two AWGs with sampling rates $s_1$ and $s_2$. Vertical dashed lines form a grid demarcating the samples in the pulse train pattern. Each cycle in the pulse trains are outlined with a rectangle with thick black borders.</figcaption></figure>

1 and 2, the cycle lengths (in samples) were same ($c_1 = c_2 = 11$) and the sample rates were $s_1$ and $s_2$ respectively. In such a case, for obtaining a clean multiplexing, the time difference between $k_1$ cycles of train 1 and $k_2$ cycles of train 2 should be an integer multiple ($\alpha$) of $\frac{1}{s_2}$, the sample duration of train 2. Note that this constraint of $\alpha$ being an integer holds only for specific values of the system parameters and hence all possible values of $M$ could not be explored in this investigation. As a perspective, it is in fact possible to get relax this limitation by cascading EOMs to produce shorter pulses up to hundred fs [34], and hence a lower temporal overlap between consecutive pulses. Thus extrapolating the theoretical curve in Fig. 8, a SNR increase (compared to the standard case) of 18 dB would be achievable with this multiplexing technique for $M = 8$ and 100 fs pulse duration. In fact, for a given system, the maximum value of $M$, i.e. $M_{max}$ is given by the ratio of the deadtime (for the standard case, $M = 1$) to the pulsewidth of the interferogram. Increasing $M$ beyond $M_{max}$ would lead to successive interferogram pulses overlapping with each other.

To emphasize the importance of SNR improvement through multiplexing, we experimentally investigated the system while placing the absorption dip of the DUT over one of the side lobes of the combs, which has a lower SNR compared to the center [see Fig. 9(a)]. For this experiment, we used $\Delta f = 5$ kHz, and a signal recording time of 1ms, meaning only 5 interferogram pulses were recorded in each time domain signal for the standard case. However, in the multiplexed case, about 13 pulses were

recorded in the same time, leading to an increase in the SNR. As can be seen from Fig. 9(b), with a Lorentzian function with 25 GHz linewidth, and a peak attenuation of 10 dB, programmed on the DUT, and 50 traces being averaged, the standard DCS technique was unable to recover the true transfer function near the peak, since the signal hit the noise floor level where there was strong attenuation by the DUT [see dashed line in Fig. 9(a)]. On the other hand, the multiplexed DCS case shows a clean recovery of the programmed DUT function owing to its higher SNR. This advantage is also reflected in the computed residue of the function recovery, which is typically below 10% for the multiplexed case and rises to a high value of 100% near the peak center in the standard case [see Fig. 9(c)]. This means, that for use cases with a transfer function of the DUT having a large extinction ratio, multiplexed DCS will perform better compared to a standard one in terms of the fidelity of the transfer function reconstruction.

### C. Acquisition Time

Since the time multiplexing technique increases the number of pulses in the time domain interferogram signal, for a given value of SNR (or a fixed number of interferogram peaks), the time multiplexing technique allows a significant reduction in the acquisition time. To emphasize the remarkable advantage of this technique, in Fig. 11 we show the evolution of the SNR as we increase the acquisition time window, both for a standard

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<figure><img src="image_13.png" /><figcaption>Fig. 11. (a) Time domain interferograms with (blue) and without (red) multiplexing for M = 3.33 with a 10 ms time window. (b) SNR evolution with the length of time window, with (blue) and without (red) multiplexing. Circular symbols indicate experimental data whereas solid lines represent corresponding theoretical trends. Dotted grey lines are used as visual cues to show the time advantage in the multiplexed case (considering same SNR for a standard case).</figcaption></figure>

and multiplexed (M = 3.33) case. In this experiment $f_1 = 250$ MHz, $f_2 = 250$ MHz–2.5 kHz, meaning $\Delta f = 2.5$ kHz. From Fig. 11(a) we can see that the multiplexed signal (blue) has about 3.3 times more interferograms per unit of time compared to the standard case (red). We also extracted the SNR of the system following a similar method as described in the previous subsection and plotted in Fig. 11(b). We can clearly see from the figure that the SNR (in dB) of both the multiplexed and standard cases follow a logarithmic distribution

$$SNR = 5\log_{10}(kt) + C, \quad (13)$$

where t is the length of the time window and k and C are fit parameters. The multiplexed case is shifted by about 12 dB from the standard case as expected from (11) with M = 3.3. The experimental data follows the expected square root evolution with the number of interferograms [36], [39]. This also means, that for a given value of SNR, the multiplexed case uses much less acquisition time compared to a standard case. For example, following the dotted line in Fig. 11(b), we can see that the multiplexed case can reduce the acquisition time from 8 ms to about 0.6 ms (about 13 times reduction), while maintaining the same SNR value.

## V. CONCLUSION AND PERSPECTIVES

To conclude, we reported a simple, loss less and robust technique to enhance the sensitivity and/or the recording speed of an EOM comb based DCS, with a trade-off for the spectral resolution. This technique is based on time multiplexing the interferogram signal through precise control of the comb pulse trains enabled by the agility of the EOM comb technique.

We report a significant SNR improvement of 11.6 dB with a 3.3-fold time multiplexing of the interferogram signal. Additionally, we demonstrated that for a given SNR, the multiplexed case can achieve a 13 times shorter acquisition time compared to the standard case. The proposed method however might encounter practical limitations in terms of its usability. For example, in the spectroscopy of gaseous samples with a long free induction decay time, the multiplexing technique might lead to distortions in the recovered spectral signature due to cutting off the full free induction decay. Another disadvantage of EOM based combs is

its limited bandwidth of operation. Although in our case, the two combs were broadened within a nonlinear dual-core fiber (DCF), for particular applications, more sophisticated techniques to attain a flatter and wider EOM comb [34], [35], [40], [41], [42], [43] might be necessary. Also, the proposed time-programming technique might demand additional AWG and EOM bandwidth while operating at high comb repetition rates. However, optimization of the system parameters, for example the position of the time shift, with respect to a particular target case such limitations could be relaxed.

These results open new possibilities for improving the sensitivity and/or recording speed of dual-comb spectroscopic techniques, as well as for more complex multidimensional spectroscopic techniques [28], [29], [33], [44], involving more than two combs. Future work might include an added layer of time multiplexing by multiplexing the measurement and reference arm interferograms as in [12] or implementing time programming in not one but both the arms of the DCS setup, leading to a more robust control over multiplexing or further optimization of the EOM pulse generation for obtaining a flatter spectrum after multiplexing.

Finally, this multiplexing technique is suitable for applications requiring high data-flow recording, such as dual-comb hyper-spectral holography [45] or gas sensing in rapidly changing environments like a compression engine [7] or the atmosphere [4].

## APPENDIX

To find out the Fourier transform of $\tilde{S}_2(t)$, we start from (7) and apply the phase shifting property of Fourier transforms to get

$$ \begin{aligned} \mathcal{F}\{\tilde{S}_2(t)\} &= \sum_{l=-\infty}^{\infty} \mathcal{F}\{P_2(t - wl)\} \otimes \mathcal{F}\{G(t - wl)\} \\ &= \sum_{l=-\infty}^{\infty} e^{-2\pi iwlv} \mathcal{F}\{P_2(t)\} \otimes e^{-2\pi iwlv} \mathcal{F}\{G(t)\}. \end{aligned} \quad (14) $$
563 Using the phase shifting property of Fourier transforms once
564 more, we can write $\mathcal{F}\{G(t)\}$ as

$$
\mathcal{F}\{G(t)\} = \mathcal{F}\left\{G\left(t - \frac{w}{2}\right)\right\} = e^{-\pi i w \nu} \mathcal{F}\{G(t)\}, \quad (15)
$$

565 where $\mathcal{G}(t)$ is a gate function of width $w$, but centered at 0. We
566 know that the Fourier transform of such a function is given by

$$
\mathcal{F}\{\mathcal{G}(t)\} = w \sinc(\pi w \nu). \quad (16)
$$

567 Also, from (3) we have

$$
\mathcal{F}\{P_2(t)\} = \frac{1}{T_2} \sum_{k=-\infty}^{\infty} \delta\left(\nu - \frac{k}{T_2}\right) \exp\left(-2\pi i \frac{k}{T_2} \phi_2\right). \quad (17)
$$

568 Thus using (15), (16) and (17) in (14) and applying the convo-
569 lution formula, we find

$$
\begin{equation}
\begin{split}
\mathcal{F}\{\tilde{S}_2(t)\} ={}& \frac{w}{T_2} \sum_{l,k=-\infty}^{\infty} \int_{\tau=-\infty}^{\infty} \delta\left(\tau - \frac{k}{T_2}\right) \\
& \times \exp\left[-2\pi i \left(\tau\phi_2 + wl\nu + \frac{w\nu}{2} - \frac{\tau w}{2}\right)\right] \text{sinc}[\pi w(\nu - \tau)] d\tau.
\end{split}
\tag{18}
\end{equation}
$$

570 Using the filtering property of Dirac delta functions, the above
571 integral can be evaluated as

$$
\begin{equation}
\mathcal{F}\{\tilde{S}_2(t)\} = \frac{w}{T_2} \sum_{l,k=-\infty}^{\infty} \exp\left[-2\pi i \left(\frac{k\phi_2}{T_2} + wl\nu + \frac{w\nu}{2} - \frac{kw}{2T_2}\right)\right] \\
\qquad \times \text{sinc}\left[\pi w \left(\nu - \frac{k}{T_2}\right)\right]. \tag{19}
\end{equation}
$$

572 Now separating out the terms that depend only on $l$, we get

$$
\begin{equation}
\mathcal{F}\{\tilde{S}_2(t)\} = \frac{w}{T_2} \sum_{k=-\infty}^{\infty} \exp\left[-2\pi i \left(\frac{k\phi_2}{T_2} + \frac{w\nu}{2} - \frac{kw}{2T_2}\right)\right] \nonumber \\
\qquad \times \sin\left[\pi w \left(\nu - \frac{k}{T_2}\right)\right] \sum_{l=-\infty}^{\infty} e^{-2\pi i w l \nu}. \tag{20}
\end{equation}
$$

573 Note that the last summation term in the above equation can
574 be represented in terms of the Fourier series of a Dirac comb.
575 Therefore we have

$$
\begin{equation}
\begin{split}
\mathcal{F}\{\tilde{S}_2(t)\} ={}& \frac{1}{T_2} \sum_{k=-\infty}^{\infty} \exp\left[-2\pi i \left(\frac{k\phi_2}{T_2} + \frac{w\nu}{2} - \frac{kw}{2T_2}\right)\right] \\
& \times \sin\left[\pi w \left(\nu - \frac{k}{T_2}\right)\right] \sum_{l=-\infty}^{\infty} \delta\left(\nu - \frac{l}{w}\right).
\end{split}
\tag{21}
\end{equation}
$$

576 For a sanity check, we can consider the case of $w = T_2$. In
577 that case, the sinc term reduces to sinc[$\pi(l-k)$]. We know that
578 sinc[$\pi(l-k)$] is non-zero only if $l=k$. Thus replacing $l$ by $k$,
579 replacing $\nu$ by $\frac{l}{w}$ due to the Dirac delta, and using $w = T_2$, we
580 arrive at

$$
\begin{align}
\mathcal{F}\{\tilde{S}_2(t)\} &= \frac{1}{T_2} \sum_{k=-\infty}^{\infty} \delta\left(\nu - \frac{k}{T_2}\right) \exp\left(-2\pi i \frac{k}{T_2} \phi_2\right) \nonumber \\
&= \mathcal{F}_{\nu}\{S_2(t)\}, \tag{22}
\end{align}
$$

which is nothing but the equation for a standard comb [(3)]. 581

Further regrouping the complex phase term in (21), we get 582

$$
\begin{equation}
\begin{split}
\mathcal{F}\{\tilde{S}_2(t)\} ={}& \frac{1}{T_2} \sum_{k=-\infty}^{\infty} \\
& \times \exp\left(\frac{-2\pi i k \phi_2}{T_2}\right) \exp\left[-i\pi w \left(\nu - \frac{k}{T_2}\right)\right] \\
& \times \sinc\left[\pi w \left(\nu - \frac{k}{T_2}\right)\right] \sum_{l=-\infty}^{\infty} \delta\left(\nu - \frac{l}{w}\right).
\end{split}
\tag{23}
\end{equation}
$$

Noting that the sinc function is non-negligible (≈ 1) only in the vicinity of $\nu = \frac{k}{T_2}$, we can approximate it as a Dirac delta at $\frac{k}{T_2}$ and the associated phase term $\exp[-i\pi w(\nu - k/T_2)] \approx 1$. Thus we arrive at a simplified expression for $\mathcal{F}_{\nu}\{\tilde{S}_2(t)\}$ as 586

$$
\begin{align}
\mathcal{F}\{\tilde{S}_2(t)\} ={}& \frac{1}{T_2} \sum_{k,l=-\infty}^{\infty} \delta\left(\nu - \frac{k}{T_2}\right) \nonumber \\
& \times \delta\left(\nu - \frac{l}{w}\right) \exp\left(-\frac{2\pi i k \phi_2}{T_2}\right), \nonumber \\
& = \frac{1}{T_2} \sum_{k=-\infty}^{\infty} \delta\left(\nu - \frac{k}{q}\right) \exp\left(-\frac{2\pi i p k \phi_2}{T_2}\right), \tag{24}
\end{align}
$$

where $\frac{1}{q}$ is the LCM (least common multiple) of $\frac{1}{T_2}$ and $\frac{1}{w}$, and 587
$p$ is the quotient of LCM($\frac{1}{w}$, $\frac{1}{T_2}$) divided by $\frac{1}{T_2}$. 588

ACKNOWLEDGMENT 589

DC, AM, MC and HR acknowledge Agence Nationale de
la Recherche (Programme Investissements d'Avenir, FARCO
projects), Ministry of Higher Education and Research; Hauts
de France Council (GPEG project); European Regional De-
velopment Fund (Photonics for Society P4S) and the CNRS
(IRP LAFONI, (AM)). Support from project FRESCOS (Marie
Curie Postdoctoral Fellowship) is also acknowledged by DC
and AM.
597

Disclosures: The authors declare no conflicts of interest. 598

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