## Abstract

The impact of photodetector nonlinearity on dual-comb spectrometers is described and compared to that of Michelson-based Fourier transform spectrometers (FTS). The optical sampling occurring in the dual-comb approach, being the key difference with FTS, causes optical aliasing of the nonlinear spectral artifacts. Measured linear and nonlinear interferograms are presented to validate the model. Absorption lines of H^{13}CN are provided to understand the impact of nonlinearity on spectroscopic measurements.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The impact of the nonlinear response of a photodetector has been extensively studied with conventional Fourier transform spectrometers (FTS) [1–6]. However, its impact on dual-comb spectrometers (DCS) has yet to be fully investigated. FTS and DCS are similar spectroscopic approaches since both methods yield an interferogram (IGM) related to field correlation functions. In the FTS case, the IGM results from the beating of a source signal with a delayed version of itself (autocorrelation) while in DCS, the IGM results from the beat of two pulsed lasers with detuned repetitions rate (cross-correlation).

In this paper, we study photodetection nonlinearity (NL) in DCS by highlighting similarities and differences with conventional FTS. Understanding these, and the validity limits of the nonlinear model, is mandatory to correct acquisition chain NL, as done in [7].

The fact that sensitivity and signal-to-noise ratio (SNR) scale linearly with detected power in additive noise-limited regime, but only with the square root of measurement time is a very strong incentive to use all available light. This often forces operation of the acquisition chain outside its linear regime. This is mostly why NL characterization and correction algorithms were developed for conventional FTS. The situation is even more critical for DCS: more power is available and comes in the form of very short pulses with enormous peak intensities. Photodetectors are thus rarely used in the linear regime in DCS. This limits SNR and introduces systematic errors.

There can be several sources of NL in a photodetection chain. For this demonstration, we chose commercial balanced detectors (Thorlab’s PDB series) because they are widely used in the DCS community [8–10]. For these, saturation of the final amplifying stage is the main culprit but even unamplified detectors show similar modifications of their impulse responses under high peak powers due to defects and transit times in the junction [11].

Regardless of the physical source and type of NL, the model developed here only assumes a static NL. That is, there is no dynamic or "memory" effect intermingled with the nonlinear behaviour. It has been shown that this condition is respected if the nonlinear impulse responses are well separated, that is their width is smaller than the pulse repetition period [12].

## 2. Nonlinearity and optical sampling

Since the dual-comb IGM is the result of the interference between two pulse trains, one pulse train samples the other in a way that is similar to equivalent-time sampling used by digital oscilloscopes. This operation known as optical sampling is a key difference between FTS and DCS. As a result of optical sampling, the dual-comb spectrum becomes periodic and photodetector NL manifests itself differently.

One key hypothesis in classical FTS NL studies is that NL is assumed to be static. That is to say there is, as shown in the top panel of Fig. 1, a one to one relation between the linear and nonlinear interferograms. It is supposed that there is no dynamic or memory effects occurring in the process. A continuous linear IGM (green in Fig. 1) is transformed at any given time into the measured nonlinear IGM (red in Fig. 1). A static nonlinear transformation can be represented as an instantaneous increase or decrease in responsivity.

Optical sampling in DCS occurs in the detector and is thus intrinsically linked to NL. Depending upon where NL occurs, for instance in the photodiode or in the amplifying chain, it can be conceptualized as occurring during or after optical sampling. Figure 1 shows that inverting the order of theses operations produces similar results providing there is no in-between bandwidth limiting filters. The pulses on the continuous IGMs on Fig. 1 illustrate the detector’s impulse response sampling the IGMs. Provided that the detector’s impulse response stays the same, sampling the linear IGM and applying the static NL transformation shown on Fig. 1 yields the same result as carrying the NL transformation on the continuous signal and sampling afterwards. To some extent, changes in the sampling function properties can be expressed with a constant sampling function and a modified continuous NL relation.

The end result is that the order in which optical sampling and NL occur or are conceptualized is not a significant matter. NL introduces spectral artifacts as in the case of conventional FTS (details provided with the mathematical model in next section). Optical sampling will however periodize the full spectrum, including the artifacts, regardless if it happens before or after NL. The periodisation of spectral artifacts is the key difference between NL in FTS and in DCS.

The order in which optical sampling and NL occur in the detector may not be relevant in general, but it is worth describing the overall order of operations occurring for the specific case considered here. In amplified balanced detectors commonly used in DCS experiments, it was demonstrated [12] that NL occurs mainly in the final amplification step, where the second operational amplifier is close to saturation. So, the operations assumed here, including the current generation in the diodes, the alternate current (AC)-coupling step often used in detectors for DCS experiment and the amplification, are shown in Fig. 2. The exact location where NL occurs in the detection chain and the presence of any filter in the chain will influence how NL arises. If NL happens before a filter whose impulse response spreads across pulses, a dynamic description of NL becomes required. Here, the photodiodes and AC-coupling steps are occurring before NL generation in the amplifier. Moreover, as the bandwidths of the diode and of the amplifier are chosen to be much larger than the repetition rate of the laser, the detector’s impulse responses are well separated [11,12] and do not overlap one another. As a result, dynamic NL effects are minimized.

## 3. Static nonlinear model

Pursuing with the static NL hypothesis, the transformation between the linear and measured nonlinear IGM can be written as a polynomial expansion :

Redefining the y-axis in Fig. 1 such that the measured IGM is also zero-mean allows us to pose $A_0=0$. Another way to write Eq. (2) that provides insight with $A_0=0$ and factorising the linear IGM is:

With this notation, the term in parenthesis acts as a "gain" for the linear IGM. The linear coefficient $A_1$ acts as a constant gain that usually reduces the level of the linear signal, a consequence of the generation of higher harmonics. This leads to a signal-to-noise degradation for the frequencies of interest, but this shall provide no systematic error upon calibration for instance in a transmittance measurement, provided the signal level is similar for the reference measurement. Subsequent terms provide a gain that varies with signal strength. This is readily apparent on the second order term, where the "gain" on $\textrm {IGM}_{\textrm {0L}}$ varies with $\textrm {IGM}_{\textrm {0L}}$. This means that the baseline low resolution spectrum linked to the shape of the IGM peak through the Fourier transform does not experience the same gain as small signals in the wings of the IGM, such as the free induction decay caused by molecular absorption features. This can be seen on Fig. 1 where the small and large signal gains are identified to be different

In the spectral domain, the n-th NL order is construed as the (n-1)-times convolution of the spectrum with itself. For instance, the second order nonlinear term is the auto-convolution of the spectrum. Figure 3 illustrates the spectral artifacts for the first 5 orders for a uniform spectral distribution generated around 1$f$ = 20 MHz. The second order term has a spectral contribution at twice the frequency of the signal (2$f$), but also around DC. The third order term has a spectral contribution at three times the frequency (3$f$) of the linear signal, but also generates content overlapping at $1f$. This process can be generalized to any NL order.

It is worth emphasizing that applying a band-pass filter at $1f$ is not sufficient to get a linear signal. As explained above, nonlinear artifacts create a gain that varies with signal strength so features having different widths translating to different spreads across the IGM will experience a different gain. Moreover, odd NL artifacts generate spectral content that additively combine in the band of interest, a definitive source of irregular systematic spectral errors.

Now that the NL generation of spectral artifacts has been described, the effect of optical sampling can be better understood. Sampling the interferogram periodises the spectrum and creates a spectral alias at each repetition rate $(f_r)$ multiple, limited by the bandwidth of the sampling function. Whether the optical sampling occurs before or after the NL, the result is similar given the assumptions made here: the spectrum is periodised and NL artifacts are aliased. The resulting spectrum for the case of a FTS IGM is shown in the top of Fig. 4 while the IGM is shown in bottom of the same figure for the DCS case where the repetition rate of the two lasers is 160 MHz. In that case, the fourth and fifth order spectral artifacts are aliased as their spectral location exceeds half of the lasers’ rep rates. One can see this as folding the spectral information that is found above 80 MHz. This explains why the green curve no longer overlaps the yellow one at 60 MHz and why the spectral fourth order artifact at 80 MHz becomes twice as important.

Even if only the first alias of a DCS measurement is kept by filtering the signal above half the repetition rate of the laser, optical aliasing has occurred and the fourth and fifth order term have already been summed with the signal. One can see that this can become problematic when the $1f$ signal of interest is spread over a large fraction of the $f_r/2$ band and that all the nonlinear artifacts are folded and overlapped. The presented example is simplified in a sense that the artifacts are mostly separated and that only the fourth and the fifth order are aliased.

## 4. Experimental methods

In order to experimentally observe the effect of static NL coupled with optical sampling, the bandwidth of the detector was chosen to allow separation of the detector’s impulse responses. Moreover, the repetition rate difference between the two frequency combs was precisely tuned to wisely place the nonlinear artifacts in a configuration that minimizes optical aliasing and overlap. This condition is the one presented in Fig. 4. The objective here was to observe NL spectral artifacts to validate the model. By doing so, the experiment brings insights into the impact of photodetector NL on dual-comb experiments.

The setup is shown on Fig. 5 where two custom-made passively mode-locked lasers based on an erbium-doped fiber were used [13]. The lasers’ central wavelength is 1550 nm and their repetition rate is 160 MHz. The lasers followed by an optical variable attenuator were combined by a 50/50 optical coupler and sent on a balanced photodetector (Thorlabs PDB480C). The lasers repetitions rates were adjusted so that the repetition rate difference is about 150 Hz. The carrier-offset frequency of the lasers ($f_{\textrm {CEO}}$) was adjusted so that the signal of interest is centered at 20 MHz. Interferograms were measured for low intensity pulses as well as for a signal clearly saturating the detector at the IGM centerburst.

An hydrogen cyanide H$^{13}$CN gas cell (Wavelength References) having absorption features in the spectral bandwidth of the lasers was added in one arm before the output coupler.

Due to the requirement of high sampling rate, but also because of memory limitation, only the central portions of the interferograms have been acquired [14]. Consequently, spectral resolution is limited, but since the need of this experiment requires only a relative comparison between linear and nonlinear measurements, this does not affect the interpretation of the results. To properly assess the impact of NL on the spectral region of interest here, the absorption lines of H$^{13}$CN for the two datasets shown in Fig. 6 are compared. For spectral transmittances, 80 high power (50 $\mu$W) and 1000 low power (10 $\mu$W) IGMs were digitized at their centerburst (1 ms at ZPD). Once digitized, the IGMs have been aligned, phase-corrected [15,16] and averaged to limit the noise at a level allowing to clearly distinguish the impact of NL. The spectrum is normalized by fitting a eighth-order polynomial to the smooth spectral baseline instead of using an independent reference measurement [17].

## 5. Results

Linear and nonlinear interferograms were measured having respectively 10 $\mu$W and 50 $\mu$W average powers at the photodetector. The waveforms are shown in Fig. 6. It can readily be seen that the nonlinear IGM saturates the amplified photodetector to its rail of $\pm$2 V. It is worth noting that 50 $\mu$W is sufficient to saturate the photodetector’s amplifier even though its continuous wave saturation level is nearly an order of magnitude higher at 400 $\mu$W. This is explained by the fact that short pulses have a much greater peak power for a given continuous wave equivalent power.

The spectra of the low power and high power IGMs are shown in Fig. 7 where many aliases are displayed in the top panel to highlight the periodisation of spectral artefacts. Only the central portion of the IGMs as shown on Fig. 6 has been used to compute the spectrum. This allows a better visualization of the spectral artifacts since most of the NL occurs near zero path difference (ZPD) and since reducing the observation window reduces the noise floor level. The middle panel focuses on the first alias where the low intensity interferogram produces spectral content centered around 20 MHz and is otherwise limited by the oscilloscope additive noise. The high power measurement shows nonlinear artifacts that are generated at the expected locations. The second order term is visible at DC and at 40 MHz, while the third order NL is visible at 60 MHz and is known to also have a contribution at 20 MHz. The fourth and fifth order are not clearly not visible and might not have a sufficient SNR to stand out of the noise. An inset in the middle panel is showing the spectrum of an interferogram in similar experimental conditions acquired using a PDB130C detector (350 MHz bandwidth). This second measurement is meant as an additional proof of the model’s validity as the spectral artifacts are clearly visible at the expected locations.

In order to better evaluate the distortions produced on a spectroscopic measurement, absorption lines of H$^{13}$CN are measured. On Fig. 8, it can be seen that the depth of the lines is greater for the high power measurement. This is a distortion of the signal that is explained by NL: as the IGM saturates, the IGM portion at ZPD, the baseline of the spectrum, is mostly affected while the small signal such as the free induction decay in the wings of the IGM remains mostly intact. This translates to a baseline level underestimated for the same absorption lines strength and, thus absorption lines get deeper when normalized to provide transmittance data. The transmittance slightly exceeds one since the eighth-order polynomial does not fully capture the slow baseline variation.

It is worth mentioning that the depth of the absorption lines may not match line-resolved spectroscopic analysis here [18] for the gas cell parameters as having measured only the IGM centerbursts with a single point phase correction induces an instrument lineshape (ILS) that reduces the depth of the lines [19]. Nevertheless, as the ILS is the same for both measurements, the comparison remains valid.

## 6. Conclusion

As a conclusion, a greater understanding of the impact of photodetector NL has been provided for dual-comb spectroscopy. A model of static NL used in classical FTS has been adapted for DCS where differences mostly due to optical sampling were highlighted. Experimental data in a configuration of clear NL display was provided to support the model and further the understanding of NL on spectroscopic measurements. The NL distorts the interferogram and the spectroscopic lines were deepened, resulting in incorrect line intensities. With a proper NL model, these systematic errors can be taken into account properly in the data processing chain.

## Funding

King Abdullah University of Science and Technology (OSR-CRG2019-4046); Natural Sciences and Engineering Research Council of Canada; Fonds de recherche du Québec – Nature et technologies.

## Acknowledgments

The authors thank Ian Coddington at NIST for providing the dual-comb system.

## Disclosures

The authors declare no conflicts of interest

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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