1. Introduction

Detector and acquisition chain non-linearity (NL) is a problem in Fourier-transform based spectroscopic approaches because the time domain interferogram (IGM) is usually a very high dynamic range signal [1,2]. This was well studied for conventional Fourier-transform spectrometers (FTS) where methods were devised to reduce non-linearity by proper detector and electronic design [3–5] or with balanced detection [6]. Ultimately, it was also sometimes necessary to post-correct for residual non-linearity [7–9]. A comprehensive review for NL and its correction in FTS is provided in [10]. This NL problem is exacerbated in dual-comb spectroscopy (DCS) due to the large laser power available, but also to its short pulsed nature, at least for frequency combs based on mode-locked lasers. Some similarities can be noted for NL between FTS and DCS. For example, significant NL can arise in the electronics after photodetection if the acquisition chain is not properly designed. There are however key differences. Most notably, the laser’s short pulses induce time-varying electrical impulse responses such that NL can in general no longer be assumed static in DCS [11,12]. Further, even if DCS commonly uses balanced detection, this does not provide the relief seen in FTS [6] since the intensity-dependent impulse responses of both photodiodes are different hence the balanced pair is not matched for all pulse intensities [11].

Until recently very little attention was however devoted to NL in DCS even if it was well-known in the community that $100$ $\mu$W of comb power was enough to saturate an amplified balanced detector [2] and even if spectral line distortions attributed to NL were indeed observed [13,14]. With milliwatts of comb power often available, amplifier NL thus severely limits the signal-to-noise ratio (SNR) of DCS measurements. In recent papers, we demonstrated that, as long as the detector’s time-varying impulse response is maintained shorter than the pulse repetition period, NL in DCS can be supposed static [12] and software correction based on the minimisation of out-of-band spectral artifacts can be used to improve linearity [14] and reduce systematic spectral errors. Even if a posteriori correction can help, a proper design optimizing the detection and acquisition chain to use all the available power with minimal NL will undoubtedly lead to better performances, always keeping the option to post-correct for any leftover NL if needed.

In this paper, we show that using a non-amplified balanced detector with a properly managed impulse response produces nonlinear spectral artifacts at least 40 dB under the spectral baseline when 23.5 mW of average power is used for each comb, a $1000$ fold NL reduction for a $400$ fold increase in usable power when compared to typical $10$ dB artifacts produced by amplified detectors using only $50$ $\mu$W [12,14]. We argue here that the amplification stage is in fact doing more harm than good since a photocurrent over only $I=1$ mA ($\sim 1$ mW optical power at $1550$ nm) is expected to produce shot noise dominating the thermal noise ($2 q I > K_b T/R$) when the $R=50~\Omega$ resistance of the acquisition electronics is used as the trans-impedance gain element. It is shown that the balanced detectors impulse responses areas have a linear relation with the input power up to 35 mW, making the retrieval of linear dual-comb interferograms possible even if the shape of the impulse responses changes as a result of nonlinear photodetection. This achievement is possible by removing the amplifier NL as well as by properly managing the detector nonlinear behavior giving rise to time-varying impulse responses.

The demonstration is carried here using near-infrared fiber-based combs for the availability of high bandwidth balanced detectors, but also because there is interest in improving knowledge of carbon dioxide line intensities which are subject to recent debate [15–17] and are important for green house gasses quantification. An HITRAN comparison of $\mbox {CO}_2$ spectral transmittance measurements is presented with residuals smaller than 0.1%. From a signal processing point of view, the ideas introduced here can easily be transposed to any spectral range, most notably in the mid-infrared where spectroscopic interest is arguably more important [2]. The work however highlights the need to develop fast InSb or HgCdTe Mid-infrared balanced detectors. On the other hand, this can provide further arguments to electro-optic sampling approaches that bring the signal back to a spectral range where optimized detectors are available [18].

Such an increase of useful comb power should lead to a commensurate increase in the figure of merit (FOM) used to quantify DCS performance [2]. However the commonly used FOM formulation opens the way to using widely varying definitions for the spectral width effectively making difficult the comparison of current numbers. We remind that the FOM in its original formulation [19] shall not depend upon the spectral shape and we explicitly spell out a formulation that allows straightforward unambiguous comparison. We also show that, at the shot noise limit, the FOM is just proportional to the square root of the detected photons number during a normalized 1 s measurement. Under that formulation, the FOM attained here is $7.2\times 10^7 \, \mbox {Hz}^{1/2}$ and is limited by the laser relative intensity noise. This would correspond to an ideal, shot noise limited, FOM for 0.85 mW of average power per comb.

2. Detector characterization

A non-amplified balanced detector (Thorlabs BDX1BA) quoted with a $5$ GHz bandwidth is used for all measurements in this paper. The detector’s positive and negative photodiodes impulse responses for various incident optical powers were first characterized and are shown in Fig. 1. Pulses from a single femtosecond laser with a $160$ MHz repetition rate were sent to a variable optical attenuator before reaching either photodiode. Impulse responses were digitized for average optical powers up to 35 mW, which is 25 mW above the maximum input power specified by the manufacturer. The impulse responses were acquired with a 10 GHz bandwidth oscilloscope.

 

Fig. 1. (top and middle panels) Impulse responses for the positive and the negative photodiodes of Thorlabs’ balanced detector BDX1BA up to 35 mW. (bottom panel) Maximum repetition rate as a function of average pulse power.

Download Full Size | PDF

The impulse responses shape is broadening with increased power as it has been previously demonstrated in [20]. This effect is explained by the excess of photocarriers which causes a screening of the electric field that slows down the carriers and reduces the photodetector’s bandwidth [21]. It is important to understand that this implies the detector’s impulse responses are varying with power and that this can introduce a dynamic non-linearity if successive pulses are overlapped. The value of an IGM point will be affected by the amplitude of the preceding pulse. Therefore, non-linearity can be handled as static in a dual-comb measurement by first making sure that the photodetector’s impulse responses do not overlap such that the area of each pulse can be individually retrieved, regardless of their shape.

Another way of seeing this is that although the small signal detector bandwidth is indeed $5$ GHz as quoted by the manufacturer and as seen from the $1$ mW pulses in Fig. 1 having a $0.2$ ns rise time, the real bandwidth for high intensity signals is actually smaller than $500$ MHz, as seen from the tails of the larger responses extending beyond $2$ ns. The argument here is thus that one must make sure to use a detector that has sufficient bandwidth for all the signal levels of interest. Using a detector whose bandwidth is insufficient to produce separate impulse response would require the development of a proper dynamic nonlinear photodetection model or correction algorithm.

Once this condition for the separation of time-varying pulses is respected, any properly designed linear and stationary low pass filter keeping only the first spectral alias can be then used to obtain the area in each repetition period. Any ripple in the impulse responses are taken into account in the filter. In practice, the width of the non-stationary photodetector impulse responses must always be smaller than the duration of the stationary filter to properly mitigate the dynamic NL.

The bottom panel of Fig. 1 shows the maximum allowable repetition rate to maintain pulse separation as a function of optical power. Pulse separation has been arbitrary defined here as having 99% of the area separated from the next pulse. The nonlinear spectral artifacts later discussed will be indicative of the required pulse separation. Depending on the level of artifacts suppression required, higher or lower pulse separation constraints might apply. It is seen in bottom panel of Fig. 1 that the detector can allow up to approximately 200 MHz of repetition rate at high power.

The area of the impulse response is computed and plotted against the incident power in Fig. 2. For both photodiodes, the relation appears somewhat linear with regression coefficient of 0.9987 and 0.9990 for positive and negative areas respectively, thus hinting that the detector’s NL will minimally affect the measurement as long as the area of each pulse is properly obtained. The different slopes in the figure indicates that the matching of photodiodes is imperfect, but this can be corrected by adjusting the power sent to each photodiode in a balanced measurement. Any residual difference between the photodiodes’ behavior may impact the nonlinear performance of the measurement.

 

Fig. 2. Area of the impulse response for both photodiodes of the BDX1BA detector as a function of the averaged incident optical power.

Download Full Size | PDF

Nonlinear behavior from amplifier saturation, from unmatched photodiodes or from impulse response broadening is typically expected with photodetectors as is was shown in previous experiments [11]. In those cases, the area-to-power relation will show significant deviation from a linear slope. As a result, the measured gas transmission spectrum will be distorted and lead to the retrieval of incorrect line intensities [12]. In this work, the results presented are a specific case where a detector operated in a nonlinear regime produces linear interferograms. A less ideal photodetector than the one used in this work will have nonlinear area-to-power relation that will require a static non-linearity correction method [14].

3. Dual-comb interferograms

Dual-comb interferograms were acquired using two semiconductor saturable absorbant mirror (SESAM) lasers based on the design in [22]. The lasers operate at 1550 nm with a repetition rate of 160 MHz. The experimental setup is shown in Fig. 3. A $400$ m SMF-28 fiber was placed in the setup to chirp the few hundred femtoseconds pulses to make sure the signal dynamic range is smaller than allowed by the acquisition card, as in [23–26]. It has also been specifically placed in the gas cell arm to avoid any Fano asymmetry in the absorption lines caused by delta-like excitation of gas molecules [27]. A CO₂ gas cell was used to measure absorption lines around $1579$ nm. A multi-pass cell with a total path length of 80 cm and a pressure of 100 Torr was used at room temperature. Not shown to simplify Fig. 3, the output of each laser is split with 90/10 couplers to measure the beating with reference lasers and phase correct the comb as was done in [26]. In addition to the phase-correction with reference lasers, a self-correction [28] is performed on the data to remove out-of-loop phase noise.

 

Fig. 3. Block diagram of the dual-comb setup. OC: 50/50 Output coupler, PD : Balanced non-amplified photodetector

Download Full Size | PDF

A chirped interferogram centerburst for 23.5 mW of average optical power per comb is shown on the left of Fig. 4. This is the maximum power for our laser sources taking into account the losses introduced by the referencing setup and other components such as the gas cell and the chirping fiber.

 

Fig. 4. (left) Interferogram’s centerburst at the zero path difference ZPD. (right) Low resolution dual-comb spectrum in black showing nonlinear artifact at 30 and 45 MHz suppressed to at least 40 dB below the signal and (inset) dechirped interferogram centerburst. Simulation of nonlinearity is shown in red to highlight where artifacts are expected.

Download Full Size | PDF

To quantify the presence of non-linearity in this high power signal, one can look for the nonlinear spectral artifacts. Since an interferogram with static NL ($\text{IGM}_{\text{NL}}$) can be written as the series expansion of the linear interferogram ($\text{IGM}_{\text{L}}$) [8–10,14] as:

A low 120 Hz repetition rate difference ($\Delta f_r$) was set to compress the optical spectrum, allowing to observe spectral artifacts. The interferogram’s spectrum was placed around 15 MHz by tuning the frequency locks with the optical reference to allow a clear visualization of potential 2nd and 3rd order artifacts at 30 and 45 MHz respectively.

The impact of non-linearity is exacerbated in the interferogram centerburst where the signal explores the largest dynamic range. To quantify small amounts of non-linearity against noise, it is therefore advantageous to compute a short Fourier transform around the centerburst before looking for spectral content at the 2nd and 3rd harmonics of the signal. Furthermore, one can remove the interferogram chirp in post-processing by evaluating the parabolic spectral phase and correcting for it. This has been done here to remove as much noise as possible. The spectrum of the dechirped interferogram’s central portion is shown in the right panel of Fig. 4 as well as the dechirped interferogram in the inset. The Fourier transform has been performed on a 6.4 $\mu$s window as shown in the inset. It can readily be seen on this 23.5 mW low resolution spectrum that artifacts are not visible at a 40 dB level below the signal of interest. Simulation of nonlinearity of second and third order has been added in red to observe where nonlinear artifacts are expected. As a comparison, artifacts were visible at a 10 dB level below the signal in a 50 $\mu$W previous measurement with an amplified detector [12]. The hump centered at 30 MHz in the experimental results could be interpreted as second order non-linearity, but this is not the case. This is actually a contribution from the laser wake mode [29,30]. This has been confirmed by shifting the interferogram’s spectrum and showing that the hump is always at a constant frequency offset from the signal rather than being at an harmonic of the signal as nonlinear spectral artifacts would. This means that nonlinear artifacts are actually below the $-40$ dB level. The second order artifact at low frequencies (near DC) is expected to be smaller than its counterpart around $40$ MHz, because the chirping fiber creates an IGM having less and slower variations in average intensity. This is readily seen in Fig. 4 where the artifact near DC does not appear in the simulated nonlinear interferogram. The DC artifact is thus harder to quantify over the higher low frequency noise level, specially since $\Delta f_r$ multiples fall over the line voltage harmonics.

The static non-linearity model states that errors arise from terms proportional to the powers of the IGM, thus producing spectral artifacts. Here the second and third order artifacts are characterized to be under $40$ dB and from the convergence of power series, we can expect higher order terms to be even smaller. Systematic errors arising from NL will therefore come from spectral artifacts falling in the electrical bandwidth of the linear IGM. The most obvious offender is the third order artifact falling directly in the range of interest, but the second harmonic may also overlap the first one if the electrical spectrum is wide enough, or if it is folded by sampling.

In any event, because spectral artifacts are additive terms, it is expected that NL systematic errors on un-calibrated spectra should be of the same order than the relative amplitude of the artifacts, here $40$ dB or $0.01$ % of the spectral baseline. Spectral calibration should reduce those systematic errors even further, potentially to $0.01$ % of a line’s depth, because NL is expected to affect gas and reference spectra in a similar manner as long as experimental conditions are maintained.

This independent quantification of the upper bound for NL systematic errors is important because improving spectroscopic knowledge on some transitions requires confidence that observed residuals are from model errors, not instrumental imperfections. Here we can be safely state that any error above 0.01% is not from non-linearity.

4. Figure of merit

Being linear up to 23.5 mW of average power per comb, the measurement shows great potential to reach an unprecedented figure of merit. The FOM initially proposed by Connes [1,19] for Fourier transform spectrometers is applied to dual-comb spectrometers since 2010, following Newbury et al. [31] who chose to express it in its customary form describing the spectral SNR obtained in a normalized duration $\tau$ multiplied by the number of resolved spectral elements $N$:

Using the same metric, a figure of merit FOM$_{\text{box}}$ between $7\times 10^7$ and $2.4\times 10^8$ Hz$^{1/2}$ can be computed for the $2\times 23.5$ mW data from the non amplified balanced detector. This is respectively using a strict $3$ dB or a $30$ dB bandwidth. Using the FOM$_{\text{box}}$ formulation for non-boxcar spectra compares the peak spectrum value to the spectral noise, normalizing by an arbitrarily chosen spectral width. This leaves room for choices that change the estimated FOM by at least a factor 3.

In its initial proposal, Connes however introduced the FOM as "the ratio between the total energy available in the spectral range and the minimum detectable energy in the spectrum, that is, the energy of a single line whose intensity is equal to the root mean square (RMS) noise". This formulation is remarkably independent of the signal’s spectral shape.

To obtain a formulation that one can straightforwardly and uniformly apply to any experimental dataset, let us assume that a single unchirped interferogram having a power at the zero path difference $P_{z}$ is measured at $M$ points in a time $\tau$. Using the most common definition for the inverse discrete Fourier transform (DFT) [32], summing $N$ double-sided unitary spectral components will lead to a maximal interferogram value of $2N/M$, meaning that a power $P_z$ will lead to a spectral level $h_\text{spc}= \frac {M P_z}{2N}$. It must be noted that with this definition, $P_z$ is in fact the value at zero path difference of a zero mean interferogram, such that for a single detector $P_{z_s} = \eta P_\text{tot}/2$ where $\eta$ is the modulation efficiency and $P_\text{tot}$ is the total average power (sum of both combs). If balanced detection is used, then $P_{z_\text{bal}} = \eta P_\text{tot}$.

Similarly, since the DFT is a sum of M independent temporal samples, the spectral variance is simply $\sigma ^2_\text{spc} = \frac {M}{2} \sigma ^2_\text{igm}$, where $\sigma _\text{igm}$ is the noise RMS deviation for each measured interferogram sample, and the factor 2 here reflects that the spectral noise is equally divided between the real and imaginary parts of the spectrum, if noise is assumed uncorrelated and stationary. When the spectrum’s absolute value is used, an extra factor $\frac {(4-\pi )}{2}$ derived from the Rayleigh distribution reduces the FOM.

This allows writing the FOM as:

This formulation clearly shows the FOM depends solely upon the total optical power contributing to the signal, and upon the monolateral noise power spectral density.

For a given measured discrete spectrum $\text{SPC}(k)$ , $P_z = \sum _k \frac {2}{M} |\text{SPC}(k)|$, where the absolute value is taken to ignore any spectral phase, and the factor of two taking into account that one customarily sums only the positive frequency bins. The PSD can also be expressed in terms of $\sigma _{\text{spc}}$ so that:

This provides a straightforward way to numerically evaluate the FOM, provided the SNR is sufficient so that the spectral sum is not significantly biased by the rectified noise. The FOM formulation of Eq. (5) is now expressed independently of the spectral shape. For flat, boxcar shaped, spectra it is entirely equivalent to formulation of Eq. (2). The spectral noise standard deviation $\sigma _{\text{spc}}$ includes terms such as optical shot noise and photodetection thermal noise, but also excess noise sources such as amplifier noise as quantified by its noise figure.

If the measurement is limited by shot-noise as defined by the semi-classical photodetection model, then $\text{PSD}_\text{sn} = 2 q P_\text{avg}/\mathcal{R}$, where $q$ is the electron charge, $\mathcal{R}$ is the detector responsivity and $P_\text{avg}$ is the average power reaching the detector(s). For a single detector, $P_{\text{avg}_\text{s}} = P_\textrm {tot}/2$ while balance detection independently sums two such term so that $P_{\text{avg}_\text{bal}}= P_\textrm {tot}$. The shot-noise limited figure of merit for a single detector is thus:

Ultimately, the responsivity is limited to $\mathcal{R}=q/h\nu$. This allows expressing the FOM in terms of the photon rate $N_\text{ph} = P_\text{tot}/h\nu$:

5. CO₂ spectroscopy

A second round of measurements with an higher repetition rate difference was performed. Using a higher $\Delta f_r$ of about 800 Hz to improve the performance of the self-correction algorithm [28] and to reduce the impact of low frequency noise, the signal of interest is spread across the $f_r/2$ band and the nonlinear artifacts, if any below 40 dB, are aliased and folded on the signal.

The measured spectrum was fitted to the HITRAN model using a sum of Voigt profiles. An optimization procedure is performed to retrieve the cell parameters (pressure, length and temperature) as well as the optical point spacing and the absolute frequency of the dataset. The optimization procedure includes a ninth-order polynomial for baseline calibration and a smoothing spline that remove any slow baseline residual fluctuations. The inteferograms were digitized with a Gage CS8389 acquisition card that has a 125 MS/s sampling rate and thus required a hardware low-pass filter at 62.5 MHz (Minicircuits BLP-70+). This filter is the one used to keep only one spectral alias and thus provides the area of each pulse. Interferograms acquired for 15.2 s are phase-corrected and then averaged.

The transmittance spectrum for a $P_\text{tot} = 2\times 23.5$ mW balanced measurement is shown in Fig. 5. The transmittance curves for the experimental data are shown in the top panel while the bottom panel shows the residuals between the HITRAN model and the data. The residuals show spikes directly on the lines while the rest of the residuals is limited by random noise and some residuals etalons. These errors on the lines are in the 0.05% level and are at least five times greater than the NL error level, expected between $0.01\%$ and $5$ ppm ($0.01\%$ of the lines depth).

 

Fig. 5. (top panel) Transmittance spectrum of CO₂ for a 23.5 mW measurement (red) and as modeled with Voigt line shapes computed with parameters from HITRAN 2020 database (black). (bottom panel) Residuals between the experimental data and the theoretical modeling.

Download Full Size | PDF

Using Eq. (5), a figure of merit of $7.2\times 10^7$ Hz$^{1/2}$ is calculated for this measurement. The area of the spectrum is calculated from the spectrum shown in the top panel of Fig. 6. The power spectral density is calculated from an IGM section away from the centerburst to allow the estimation of noise in the spectral range from 30 to 36 MHz where the absorption lines are located. This is shown in the bottom panel of Fig. 6. The humps at 10 MHz and 38 MHz are on the edges of the signal bandwidth and are attributed to laser’s filtered amplified spontaneous emission or wake mode [30]. The prominence around 25 MHz is actually backscattered signal present everywhere in the IGM that would lead to systematic errors in that spectral region rather than a FOM degradation [26].

 

Fig. 6. (top panel) Fourier transform of the averaged interferogram from which is estimated the power $P_z$. (bottom panel) Power spectral density of an interferogram section away from ZPD to estimate the noise level $\text{PSD}_n$ below the signal.

Download Full Size | PDF

The theoretical value in shot-noise limited regime using Eq. (7) for a balanced detector would be $3.5 \times 10^8$ Hz$^{1/2}$. This value is not reached with the DCS system since the measurement is not shot-noise limited as shown in the bottom panel of Fig. 6. Assuming it was shot-noise limited, the FOM reached would correspond to having 0.85 mW of power per comb. As it will be seen, the measurement is limited by the relative intensity noise (RIN) of the lasers.

Table 1 shows the FOMs computed using equation (5) for the results presented in [12] and [14]. Experimentally achieved FOMs are given, as well as what is theoretically possible at the shot noise limit for each power level. The $10$ $\mu$W case corresponds to the maximal power level with the PDB480C amplified detector before NL artifacts are significant and $50$ $\mu$W is the maximal power where amplifier NL could be adequately post-corrected. The table clearly shows the improvement in achieved and shot-noise limited FOM with increasing power. It further shows that amplified detector measurements were limited by constant amplifier or acquisition noise levels as the experimental FOM improves almost linearly with increasing optical power while in the non-amplified $23.5$ mW case, the FOM improves by a factor of $39$ short of the $\sqrt {2350}=48$, highlighting that RIN is becoming a dominant contribution. This is also seen on the bottom panel of Fig. 6, from the fact that wake mode humps are visible in the power spectral density, which are telltale RIN contributions.

Table 1. Experimentally achieved and shot-noise limited FOMs, as calculated from Eq. (5) for an amplified balanced detector operated in the linear regime (2 $\times$ 10 $\mu$W), with software NL correction (2 $\times$ 50 $\mu$W) [12,14] and for the non-amplified detector with impulse response management (2 $\times$ 23.5 mW ). Improvement factors in parentheses are from the $10~\mu$W baseline. Cases with amplified detectors are denoted with an asterisk.

View Table

Given that measurements with lasers using polarization nonlinear rotation [30], saturable absorber mirrors [22] and figure 9 cavities [33] yielded similar RIN levels and humps, our results indicate that further efforts should be made to reduce RIN [34,35] such as to take full advantage of the laser’s power and the detector’s linear behavior in this regime.

6. Conclusion

As a conclusion, linear dual-comb interferograms can be obtained from photodetectors operated in a nonlinear regime. To that effect, one has to ensure the power dependence of the detector impulse response does not affect the estimation of each pulse area. This usually implies that the photodetector has a quoted small signal bandwidth larger than $f_r$. In the experimental demonstration, 23.5 mW of continuous power is sent to each photodiode of an unamplified balanced detector pair, yielding interferograms with nonlinear 2nd and 3rd harmonics under 40 dB below the signal level, enabling measuring absorption lines of CO₂ with precision that could further the line modeling theory.