Unlocking a lower shot noise limit in dual-comb interferometry

Dual-comb interferometry (DCI) uses the beat note between two envelope periodic optical fields for metrology. Approaches using mode-locked lasers, micro-resonators, or electro-optically modulated sources are now employed in an increasing application range, for instance, in spectroscopy,¹ time transfer,² and for the characterization of optical waveforms.^3,4

In the frequency domain, DCI can be viewed as the multi-heterodyne detection of frequency comb teeth pairs. As shown in Fig. 1, the beating of adjacent optical modes (top) produces a radio-frequency (RF) comb (bottom) near base band. The heterodyne beating between further spaced optical comb lines yields two spectral copies, or aliases, of the RF comb in each repetition frequency interval (f_r).

In spectroscopy, only the base band RF comb is normally used.¹ For waveform characterization, several groups have, however, exploited the spectral aliases to independently determine the modulus and phase for all spectral lines of both combs^5–8 and retrieve the temporal phase and amplitude for both sources, yielding results similar to frequency resolved optical gating (FROG).⁹ 

Here, we use a wide detection bandwidth to acquire and average N_a spectral aliases to improve the DCI signal-to-noise ratio (SNR). This approach does not, however, apply equally to all noise sources. Additive noise in the photodetection chain directly averages down with the square root of the number of aliases. When using short pulse comb sources, the fundamental shot noise, however, does not average with N_a. Spectral correlations a priori prevent surpassing the shot noise limit set by the semi-classical photodetection model yielding an SNR equal to the square root of the average photon number per pulse. This is the Poissonian shot noise limit when only the base band alias is used, or equivalently when the detector integration time is set to the pulse repetition period. Chirping the pulses of one comb allows reducing the shot noise spectral correlations between aliases. In that case, the multiple aliases can be phase-corrected and averaged to reduce the shot-noise limited SNR below that of standard, single copy, DCI. Using only N_a = 5 partially decorrelated aliases, we demonstrate DCI-based spectroscopy of CO₂ with 1.65× improved sensitivity, corresponding to an almost three-fold reduction in acquisition time.

The setup, shown in Fig. 2, is typical for DCI experiments where a single comb probes the sample of interest, here a CO₂ gas cell. Distinctive features include the 50 m SMF-28 fiber before the cell to avoid Fano line asymmetries¹⁰ and a 700 MHz detection bandwidth (Thorlabs BDX1BA 5 GHz unamplified balanced detector, RF BAY LNA-725 amplifier). The local oscillator (LO) arm can optionally include a 2500 m SMF-28 fiber both to mitigate dynamic range issues in the acquisition chain without modifying the fundamental DCI comb tooth resolution¹¹ and to alter the spectral shot noise correlations, as discussed below. In spectral ranges where fiber dispersion can not be used to conveniently chirp LO pulses, a grating pair can be used, as in any dispersion management strategy, such as chirped pulse amplification.¹² Availability of high bandwidth detectors for certain wavelength ranges can be an issue that may be circumvented by using electro-optical sampling.¹³ 

The sources are based on the design published in Ref. 14. Both the LO and probe repetition frequencies (⁠$frLO$⁠, $frp$⁠) are around 160 MHz with a df_r = 1.3 kHz difference. Each comb includes a referencing setup to stabilize the repetition rate and carrier envelope offset as in Ref. 15. Referencing signals are acquired to allow software corrections on measured dual-comb beat notes. A separate detector records the LO pulse train independently via a 10% pickup before the mixing coupler. This allows following the LO repetition rate in signal processing.

A measured multi-copy spectrum is shown in Fig. 3, displaying the N_a = 5 spectral aliases that will be used to prove the discussed concepts. In the absence of significant LO chirp, these copies can, in principle, be folded around $frLO/2$ and, thus, summed by making sure the electrical sampling is synchronous with the LO pulses. As discussed below, a strong LO chirp is, however, critical to reduce the shot noise contribution. In that case, aliases are individually filtered, downshifted to base band, phase corrected, and numerically summed.

Signal-independent additive noise sources will contribute noise that is uncorrelated across all frequencies in our spectral range. It follows immediately that stationary additive noise sources, such as thermal noise, will average down as $Na$⁠. The time domain interpretation is that only additive noise near the pulse locations, within the detection impulse response width, is integrated into the measurement. This provides a rejection set by the ratio between the repetition period and the electrical impulse response duration (≈N_a) as noise temporally away from the pulses is ignored.

Spectral correlations in comb shot noise can be understood using the semi-classical photodetection model (classical fields with quantized detection). In that context, for an integration time equal to the pulse repetition period, the detected photo-electrons follow Poisson statistics with an average and variance of N_ph yielding an average current I = qN_phf_r, and a white photocurrent power spectral density (PSD) equal to 2qI(A²/Hz), q being the electron charge.

When widening the detection bandwidth, the integration time drops below the pulse repetition period. One can then still consider Poissonian detection statistics, but now with a time-varying average rate of arrival conditioned by the pulse amplitude. The photocurrent noise is concentrated near the pulse arrival times and is zero between pulses. The low-frequency PSD level remains the same but now the shot noise PSD is also measured at larger frequencies. Because the shot noise variance is periodic with 1/f_r in the time domain, the PSD must be correlated on f_r intervals.^16,17

To observe this correlation, light from only the LO is sent to the balanced detector. Figure 4 (top) shows that the noise PSD is close to white over the 400 MHz bandwidth with the expected 2qI level for the I = 5.5 mA average photocurrent (black dashed line). To visualize the frequency domain correlations, color coded slices for each alias are downshifted to base band by an integer $frLO$ multiple and shown around 24 MHz with vertical offsets in the two bottom panels.

Figure 4 (top and middle) shows the expected correlation when the 2.5 km fiber is absent from the LO path. Optical pulses are not chirped and are far shorter than the electrical impulse response. The correlation between all aliases was computed to be above 0.9 using a centered 60 MHz range for each copy. The agreement between the measured and calculated PSD levels and the strong spectral correlations provide convincing evidence that the measurement is shot noise limited, with only 10% additive noise from the detection chain, that the PSD level is properly calibrated, and that the assumed time-varying Poisson statistics provide an accurate model for comb shot noise in these measurements.

Adding the 2.5 km fiber chirps pulses to a significant fraction (here ∼48%) of the repetition period, which greatly reduces the variations in the pulse envelope intensity and, hence, of the mean rate of arrival for the photocurrent statistics. One can, thus, expect reduced shot noise correlation between spectral aliases.^18,19 Figure 4 (bottom) shows the same spectral slices for pulses chirped by the 2.5 km fiber. Correlation between aliases is visibly reduced and computed to be 0.7, 0.33, 0.18, and 0.03 for the nearest to the furthest neighboring copies. These correlations are consistent with the fact the pulses are chirped to ∼3 ns. The LO signal is then still zero over half of the 6.25 ns repetition period so shot noise still displays some non-stationarity. The Fourier transform relation between the shape of noise variance temporal non-stationarity and the spectral correlations¹⁸ implies that chirping over only half of the repetition period (τ_c ≈ 3 ns) will leave spectral correlations over a frequency range of f_c ≈ ±1/2τ_c = 166 MHz, meaning that only first neighboring copies will retain appreciable correlation, as observed experimentally. Quinlan et al. provide¹⁶ an exact expression for the spectral correlation as a function of pulse duration in the case of Gaussian pulses.

To increase the SNR in multi-copy DCI, one can then strongly chirp the LO comb, thereby reducing its shot noise correlations between spectral aliases. With chirp, each alias, however, now contains a distinct parabolic phase that prevents their direct averaging. This is what hampers synchronous electrical sampling: LO pulses are no longer well-defined in the time domain. A correction removing the spectral phase²⁰ for each copy must, therefore, first be applied. This operation retrieves the same time-domain DCI signal as would be measured with compressed pulses, within the detector bandwidth. The uncorrelated noise is, however, unaffected by this spectral phase correction; the parabolic phase does not change the noise variance in any spectral bin, nor can it induce correlations across frequencies. Upon removing the parabolic phase for each alias, one can, thus, coherently average spectral copies, and the LO shot noise contribution is expected to be reduced by as much as $Na$⁠, depending on the chirp.

The probe comb will also contribute shot noise. However, its noise is correlated over intervals of $frP$⁠. Therefore, this noise does not perfectly align upon folding or downshifting aliases by multiples of $frLO$ to average them. The probe shot noise will, thus, average with $Na$ regardless of the chirp in the probe comb.

The use of balanced detection removes the direct contribution from relative intensity noise (RIN) for each comb. However, the dual-comb signal still contains RIN cross terms that do not balance out at detection. These can be seen as the pseudo-periodic amplified spontaneous emission (ASE),²¹ or the wake mode,²² from one comb beating with the pulses of the other. Fluctuations in the probe comb sampled by the LO pulses will produce non-stationary noise at $frLO$ while fluctuations in the LO comb sampled by the probe pulses will have a time domain variance periodic with $frp$⁠. From the arguments above, it is expected that only the latter will average down with $Na$⁠.

We combine this frequency-domain averaging with phase-corrected time domain averaging as follows. A 1 s interferogram (IGM) stream is acquired with a bandwidth preserving N_a = 5 spectral aliases. In post-processing, each copy is separately bandpass filtered and downshifted by an integer multiple of $frLO$ such that all spectral aliases are superimposed in base band. At this point, one has N_a similar signals, as if N_a interferograms streams were measured simultaneously. Fast software post-correction is performed as in Ref. 23 but independently on each of the N_a downshifted interferogram streams. Out-of-loop fluctuations are removed with the self-correction algorithm presented in Ref. 24. Corrections parameters extracted from the low frequency interferogram stream (first copy) are applied in base band to each interferogram stream individually. The self-correction output consists in N_a IGM trains that are summed into N_a average IGMs, one for each spectral copy. Over any allowed measurement duration, this can be seen as simultaneously measuring N_a interferograms rather than only one.

The spectral phase is removed before coherently averaging the N_a aliases. This is analogous to Mertz’s²⁰ phase correction in classical Fourier-transform spectroscopy: a smooth, low resolution spectral phase is fitted to the data and removed. It is applied here by fitting a different spline to each copy’s parabolic phase. The end result is a multi-copy electrical comb without a dispersive phase as if an unchirped LO sampled an unchirped probe comb from the signal perspective, but with spectrally uncorrelated noise averaged over the N_a copies.

Figure 5 shows a multi-copy DCI measurement for the case of a strong LO and a weak probe comb with powers of 6.40 and 0.64 mW, respectively. To remove the DCI signal and better show noise contributions, PSDs (bottom panel) are computed for an interferogram section away from the centerburst, as shown in the top panel. The gray region in the PSD display indicates the spectral band of spectroscopy interest for the CO₂ gas cell.

For the standard single copy DCI approach, the PSD (blue curve in Fig. 5) has an average level two times above the single copy shot noise limit (black dashed line), highlighting that broadband noise contributions are ∼50% RIN and 50% shot noise. Humps at 30 and 60 MHz correspond to the cross-RIN terms mentioned earlier, where the wake mode of one comb is sampled by the other comb’s pulses. In contrast, the PSD level after averaging of the N_a = 5 spectral aliases (red curve) shows much lower noise levels. At low RF frequencies, the noise drops to the dashed green curve, which shows the true shot noise limit when managing the signal non-stationarity for N_a = 5 copies.

While being below the single copy shot noise level in the grayed spectral range of interest, the noise floor of the frequency-domain averaged signal is still limited by remaining cross-RIN noise at frequencies above around 10 MHz. This warrants further investigation but seems to be related to the fact that the wake modes are, in fact, optical signals that can’t be modeled as spectrally uncorrelated noise and are, thus, affected by the chirp and dechirp operations.

Finally, as discussed above, only one of the wake mode humps is averaged out, suggesting that each of the humps is associated with a particular comb wake mode. We confirmed this by switching the probe and LO combs, resulting in a change in which the hump averaged down. A proper choice of which comb shall be used to probe the sample and which to use as the chirped LO can thus be made to optimize the performance in a given frequency range of interest.

Figure 6 shows a measured transmittance for a CO₂ gas cell between 1568 and 1586 nm. Here, the total 7.53 mW average power is equally split between both combs. The motivation for this different power allocation is to highlight that the method is working in diverse conditions. Here, RIN cross terms prevent breaking the single copy shot limit, but using several spectral aliases still improves measurements. We compare the standard DCI single copy measurement (blue) to N_a = 5 aliases averaged with a chirped LO comb (red). Both are fitted to HITRAN using a cepstral approach²⁵ to estimate and remove the spectral baseline. The fit residuals from 1568 to 1586 nm are reduced by a factor of 1.65 in standard deviation when averaging N_a = 5 copies. This improvement is close to the 1.82 ratio computed by constructing the covariance matrix for the chirped LO shot noise from the correlation coefficients experimentally measured in Sec. III B and summing its 25 terms to retrieve the sum of five copies variance. In this calculation, other noise terms, including the probe comb shot noise, are assumed to have a variance scaling linearly with the number of copies. The improvement is limited by the finite number of copies, by the non-averaging RIN cross term, and by systematic errors arising from back scattered light in the long chirping fiber, as in Ref. 26. This factor of 1.65 improvement can translate to a 1.65² = 2.7 faster measurement rate for the same SNR, greatly helping fast measurements of non-recurring events such as combustion reactions.

For this 1 s measurement, the figure of merit²⁷ is 4.8 × 10⁷ for the single copy case and 6.5 × 10⁷ for N_a = 5, comparable to the 7.2 × 10⁷ $(1/s)$ reported in Ref. 11 for a single alias at the much higher average comb power of 47 mW. The ratio of the figure of merit for single copy vs frequency-averaged spectra is smaller (1.4) than for the spectral residuals because it is more impacted by the cross-RIN noise peak at 1550 nm (see Fig. 5).

The results presented in this paper show that the use of a large detection bandwidth in dual-comb interferometry improves the shot noise limited SNR beyond the single copy limit. The potential improvement scales with the number of acquired spectral aliases, provided that the stationarity properties of the local oscillator comb shot noise, are properly managed. Here, this is achieved using by chirping the local oscillator pulses to a duration that is a significant fraction of the repetition period. Shot noise is, therefore, less correlated between spectral aliases. The technique is applicable to any comb technology and in any spectral region.

The number of acquired aliases is determined by the ratio between the acquisition bandwidth and the comb repetition rates. To simplify the processing, unlock the full potential of the technique and average more than a handful of spectral copies, a hardware folding scheme must be devised, as it was shown possible for the case of the beat note between a single comb and a continuous wave laser.¹⁷ 

It shall be noted that none of the measurements presented herein exhibit a shot noise PSD below the expected 2qI level. However, by properly exploiting the signal and noise stationarity properties in the context of time-varying Poisson statistics, one can realize a lower shot noise limit that is reduced with the number of copies.

The technique is also useful in situations where other noise sources are dominant. Additive noise will, in general, also average with the number of aliases, as will at least one of the RIN cross terms. By wisely choosing the LO and probe combs, it is, thus, possible to significantly reduce noise in the region of interest compared to standard DCI that uses only a frequency-domain copy. The results could be further improved by increasing the LO chirp to further de-correlate shot noise between nearest neighbors or by averaging more aliased copies.

This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors thank D. Plusquellic and A. J. Fleisher for lending the CO₂ gas cell. They also thank N. Newbury, I. Coddington, and S. Diddams for comments.

The authors have no conflicts to disclose.

M. Walsh: Conceptualization (equal); Data curation (lead); Formal analysis (equal); Investigation (lead); Methodology (equal); Software (lead); Writing – original draft (supporting); Writing – review & editing (equal). P. Guay: Methodology (supporting); Software (supporting). J. Genest: Conceptualization (equal); Formal analysis (equal); Funding acquisition (lead); Methodology (equal); Software (supporting); Supervision (lead); Writing – original draft (lead); Writing – review & editing (equal).

A dataset and sample code illustrating all the processing steps to filter, downshift and average several spectral aliases of a wide band interferogram is available on Borealis,²⁸ complete data is available on request.