† Corresponding author. E-mail:

Project supported by the State Key Laboratory of Precision Measurement Technology & Instruments of Tsinghua University and the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 61205147).

Dual-comb spectrometry suffers the fluctuations of parameters in combs. We demonstrate that the repetition rate is more important than any other parameter, since the fluctuation of the repetition rate leads to a change of difference in the repetition rate between both combs, consequently causing the conversion factor variation and spectral frequency misalignment. The measured frequency noise power spectral density of the repetition rate exhibits an integrated residual frequency modulation of 1.4 Hz from 1 Hz to 100 kHz in our system. This value corresponds to the absorption peak fluctuation within a root mean square value of 0.19 cm^{−1} that is verified by both simulation and experimental result. Further, we can also simulate spectrum degradation as the fluctuation varies. After modifying misaligned spectra and averaging, the measured result agrees well with the simulated spectrum based on the GEISA database.

An optical frequency comb,^{[1]} providing hundreds of thousands of even spaced and mutual coherent longitudinal modes which are traced to frequency standard, pushes laser metrology forward by a tremendous step including molecular spectroscopy.^{[2]} Based on multi-heterodyne detection that is similar to conventional Fourier transform spectrometry,^{[3]} a dual-comb spectrometer employs two comb lasers with slightly different repetition rates to mimic scanning sampling. In this scenario, detuned pulse trains act as asynchronous optical sampling probes for each other.^{[4,5]} Thus, a dual-comb system can also be applied to monitor instantaneous phenomenon^{[6]} or remote, fast sensing.^{[7]} Because of the unique measurement principle, it has many advantages, such as high-speed acquisition, high resolution, and high-frequency accuracy.

To a remarkable extent, the three ‘High’ advantages are confined by comb laser performance. In one comb, there are two freedoms, i.e., repetition rate (*f*_{rep}) and carrier-envelope offset frequency (*f*_{ceo}). The stabilities of *f*_{rep} and *f*_{ceo} directly affect the performance of the measurement system. In order to record non-distorted interferograms by a dual-comb spectrometer, one method is to use highly stabilized laser combs via phase-locking both freedoms to optical frequency standards.^{[8,9]} Owing to the precise control of laser combs, the radio frequency (RF) beats generated by pairs of nearby comb modes are fixed. Alternatively, a raw distorted interferogram of free-running laser combs can be corrected by mixing adaptive signals which contain phase error information on real-time compensation hardware^{[10]} or by a posteriori correction algorithm.^{[11,12]} No frequency standards in these systems cause a little trouble in mapping spectral features along the horizontal axis. However, these studies show the solution of laser fluctuation, but they do not include how the spectrum degrades since laser performance deteriorates.

In this paper, the frequency stabilities of *f*_{rep} and *f*_{ceo} are evaluated. According to the principle, the repetition rate fluctuation is the main reason that causes spectral frequency misalignment compared with carrier-envelope offset frequency variation. By establishing a dual-comb system traced to a Rubidium clock, we measure the P branch in the 2v_{3} rotational–vibrational band of hydrogen cyanide (H^{13}C^{14}N). The measured spectra show absorption peak jitter that is in agreement with unstable pulse interference simulation. The signal-to-noise ratio of the absorption spectrum is effectively improved by averaging modified spectra in the frequency domain. Finally, discussion and further improvement are presented.

A well-stabilized mode-locked laser provides a train of ultrafast pulses at a constant repetition rate and with equal carrier-envelope phase shift in the time domain. In the optical domain, the comb structure of laser modes extends hundreds of nanometers in the optical domain. The output field can be described as

*A*is the amplitude of the longitudinal mode and

*n*is the mode number. If a gas cell is inserted into the path, this beam will be attenuated and phase shifted since some components resonate with rotational-vibrational energy levels of gas under test. Equation (

*α*is the amplitude attenuation coefficient of the electric field and

*φ*is the phase shift induced by the gas sample. Similarly, the secondary laser comb is expressed as

*f*

_{rep}is the difference between the repetition rates and Δ

*f*

_{ceo}is the difference between both carrier-envelope offset frequencies. Then the superposition field is obtained by a square-law photodiode as

*f*

_{rep}are filtered. In this case,

*n*is equal to

*m*. So equation (

*f*(

*ω*) =

*∫ f*(

*t*) · exp (−i

*ω*

*t*)d

*t*, the interferometric signal can be analyzed in the frequency domain as follows:

*α*(

*ν*) in the optical region. Simultaneous, parallel detection can be achieved. Continuous and periodical pulses result in a repeated interferometric signal. From Eqs. (

*β*is the conversion factor that equals

*f*

_{rep}/Δ

*f*

_{rep}. In the other configuration, i.e., only one beam interrogates the gas sample; the dispersion spectrum

*φ*(

*ν*) is a byproduct.

Equation (*f*_{rep} and *f*_{ceo} variation, would happen if laser modes could not keep fixed during measurement. Between these two unstable parameters, the former is more critical than the latter, because a small offset in Δ*f*_{rep} makes a large change in *β*, further resulting in spectral frequency misalignment. What is more, since the stabilization strategy of *f*_{rep} directly locks to the RF frequency standard, the phase noise of the comb mode in the optical region is *n*^{2} times larger than that of *f*_{rep}. Considerable phase noise of the mode in the optical region reduces the signal-to-noise ratio of the interferometric signal. Given this, some methods mentioned in the previous section come into use.

The schematic of the experimental setup is illustrated in Fig. *f*_{rep} ∼ 250 MHz provide pulse trains with an average power of ∼ 20 mW and a duration of ∼ 80 fs. The 20-dB output bandwidths stretch for 100 nm around 1.55 μm just as indicted in the inset in Fig. *f*–2*f* interferometers.^{[13]} All phase lock loops are referred to as an Rb clock (Symmetricom, 8040C). With the help of quarter wave plates, half wave plates and polarized beam splitters, the polarization and power of each beam are fine-adjusted to improve modulation depth and avoid detector saturation. A region of interest, i.e., the 2v_{3}-P rotational–vibrational band of H^{13}C^{14}N, is selected by an optical band-pass filter (Thorlabs, FB1550-12). The combined beam, outgoing from a 200 Torr (1 Torr = 1.33322×10^{2} Pa), 7cm-long H^{13}C^{14}N gas cell, is aligned to an InGaAs photodetector with a bandwidth of 125 MHz (New Focus, 1811-FS). A high-speed data acquisition system based on a field programmable gate array (National Instruments) is synchronized to one of the laser combs.

The stabilities of *f*_{ceo} and *f*_{rep} are evaluated first. Achieved by a sequence of amplification, nonlinear spectrum boardening and second harmonic generation, the linewidth of *f*_{ceo} is on the order of several hundred kHz. It is better to evaluate this parameter via Allan deviation. We use a Π-type frequency counter with no dead time at a gate time of 1 s to record locked *f*_{ceo} for 4000 s. The result is shown in Fig. *f*_{ceo} is ∼ 0.66 Hz. The Allan deviation comes down as the average time increases. The power spectral density of the phase noise is suitable to locked *f*_{rep}. Figure *f*_{rep} with Fourier frequency from 1 Hz to 100 kHz. The integrated RMS residual frequency modulation d *f*_{rep} is 1.4 Hz.

To obtain the interferometric signal, laser combs are locked at different repetition rates, *f*_{rep,1} = 250.000 MHz and *f*_{rep,2} = 250.002 MHz. Thus, the difference Δ*f*_{rep} is 2 kHz, and interferograms are repeatedly generated with a time interval of 0.5 ms. 25 interferograms are recorded within 12.5 ms. Figure ^{[6]} can be seen on both sides of the centerburst as both laser beams interact with the gas sample. According to the above parameters, we simulate dual-comb interferometric signals. Figure

Figure

The nominal conversion factor *β* is 1.25× 10^{5}. As indicated in the previous analysis, a small fluctuation in *f*_{rep} leads to a considerable change in *β*, which further causes significant misalignment in up-conversion to the optical domain. With the parameters of the experimental setup, the RF comb is located around 46 MHz. Therefore, the RMS spectral frequency jitter is estimated from

^{−1}. Along the horizontal axis in Fig.

^{−1}is within about ±0.3 cm

^{−1}due to laser fluctuation during measurement. The standard deviation of the distribution is 0.20 cm

^{−1}. Meanwhile, the frequency misalignment range in the simulation is slightly smaller than that in the measurement result, and the standard deviation of the simulation is 0.15 cm

^{−1}. The relative intensity noise of the laser combs contributing to the amplitude-to-frequency noise probably results in the difference.

^{[14]}Further, we can simulate spectrum degradation according to laser performance. Along the vertical axis of Fig.

Focusing on a narrow region of interest, the frequency jitter of each absorption peak in one spectrum could be approximately treated as a constant. Therefore, the measured spectra are dynamically modified one by one with respect to the absorption peak at 6446.16 cm^{−1}. In Fig. ^{−1}.

In this study, we demonstrate that the repetition rate fluctuation has a significant effect on spectral frequency misalignment by considering the inevitable phenomenon of laser fluctuation. By measuring the frequency noise of the repetition rate, the RMS residual frequency modulation of locked *f*_{rep} is 1.4 Hz in our system. According to the experimental setup, we simulate absorption peak jitter which is proven experimentally. By modifying the spectra with respect to the reference peak and averaging in the frequency domain, the signal-to-noise ratio is effectively improved. The averaged spectrum agrees well with the simulation result. However, this kind of modification is limited to the case of a low-resolution spectrometer. When it comes to high-resolution, more attention should be paid to the phase noises of both laser combs.

Further, the relative linewidth between laser combs is the most important parameter that seriously restricts the application of interferometric spectrometry. Next, a narrow linewidth cw laser is expected. By utilizing a phase-lock loop with a large control bandwidth, the phase noise of the comb mode that is offset-locked to the cw laser could be suppressed.^{[15,16]} It will benefit the coherent averaging in the time domain.

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