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Abstract
Dual-comb spectroscopy (DCS) is an emerging and promising spectrometric technique with high resolution, high sensitivity, broad spectral range, and fast acquisition speed. For the recovery of the information encoded on comb modes without resolution loss, two continuous wave lasers are commonly utilized as optical intermedia to track the real-time jitter of dual-comb interferograms. This paper presents a simplified error correction method based on single optical intermedium for quasi-free-running fiber DCS. This method combines the strengths of conventional optical referencing and self-referencing error correction. We acquired whole P branch H13C14N transmittance spectra in the near infrared as a demonstration. In contrast to that of conventional dual intermedium error correction, the standard deviation of our method was merely 0.01 over the 4 THz spectral range. Our method provides a balanced and practical postprocessing routine for high-performance mode-resolved DCS applications.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical frequency comb (OFC) is a kind of coherent laser source that consists of large amounts of single-mode lasers across a broad spectral range [1–4]. The absolute frequency of each discrete optical mode ${ν_n}$ can be determined by repetition frequency ${f_r}$ and offset frequency ${f_0}$ as ${\upsilon _n} = {f_0} + n{f_r}$. In the time domain, the coherent superposition of the optical modes corresponds to femtosecond pulse sequences with repetition period $1/{f_r}$ and periodic carrier envelope phase slip $2\pi {f_0}/{f_r}$. Given the equidistant and dense distribution of optical modes, high-resolution optical response information can be encoded over a broad spectral range, making OFC a superior light source for spectroscopic applications [5].
The key of frequency comb spectroscopy is the separation of each optical mode and recovery of mode-resolved information. The virtually imaged-phased-array (VIPA) method [6] uses two vertical gratings with different free spectral ranges to disperse OFC modes onto a two-dimensional plane. Although this method enables parallel and real-time spectral measurement, it requires an expensive high-speed detector array and its resolution is limited to hundreds of megahertz due to the current dispersion capability of the grating [5]. Although Fourier transform spectroscopy (FTS) [7,8] can simultaneously measure broad-band spectral elements by using a single-pixel photodetector through scanning the delay arm of a Michelson interferometer, its acquisition speed and instrument size are limited by its mechanical scanning configuration. Dual-comb spectroscopy (DCS) [9–11] is an emerging scan-less spectrometric technique that uses a single-pixel photodetector. By introducing a second local oscillator (LO) comb with a slightly asynchronous repetition frequency, DCS eliminates the need for a dispersive device or mechanical scanning component. Time-domain cross-correlation scanning is automatically performed through asynchronous optical sampling. From the frequency domain perspective, DCS is a frequency multiplexed multiheterodyne interference method that compresses two OFCs with frequency spacings ${f_r}$ and ${f_r}\textrm{ + }\Delta {f_r}$ into a radio frequency (RF) sub-comb with frequency spacing $\Delta {f_r}$. Given its unique potential capability for obtaining high resolution, high sensitivity, broad spectral range, and fast measurement speed, DCS has undergone rapid development in recent years. Its spectral measurement range has been expanded from the near-infrared to the mid-infrared, terahertz and ultraviolet regions via second-order and third-order nonlinear conversion [5]. The practical applications of DCS include molecular absorption spectroscopy [12], remote sensing [13], coherent Raman anti-Stokes spectroscopy [14], microscopy [15], ellipsometry [16], time-resolved spectroscopy [17], multidimensional spectroscopy [18], photoacoustic spectroscopy [19], and hyperspectral imaging [20].
In contrast to that of VIPA and FTS, the dispersion capability of DCS is not limited by device performance or mechanical scanning configuration. However, it requires extremely high relative phase stability between dual combs. The repetition frequency and offset frequency of OFCs are time-dependent variables as a result of environmental perturbations, such as vibration, temperature drift, and pump noise [21]. Moreover, the relative linewidth between the optical modes of two combs is transformed into the RF domain following multiheterodyne interference. For typical DCS configurations, such as solid or fiber DCS, the relative linewidth of the RF sub-comb considerably exceeds mode spacing $\Delta {f_r}$, causing mutual aliasing between adjacent modes [22,23]. In this case, long-term coherent averaging for increased sensitivity is invalid due to RF interferogram (IGM) distortion and mutual coherence degradation. In addition, mode-resolved spectral measurement is unavailable because of resolution loss.
Three major technical routes are widely adopted for improving the relative phase stability between dual combs to solve the above problem. The first route uses sophisticated resonant cavity designs, such as a dual-comb shared single cavity scheme, to eliminate common-mode noise and improve natural coupling between dual combs [24–26]. However, in contrast to independent dual-comb configuration, this scheme sacrifices tuning flexibility, stability, and robustness. The second route suppresses the relative linewidth of DCS via tight locking. The performance of the tight locking scheme [27,28] relies on the bandwidth of the servo actuator, requiring a fussy control strategy to optimize several parallel phase lock loops. As the third route, digital error correction [23,29] requires only modest feedback control to ensure long-term stable operation. Real-time or offline calculation can completely replace the tight feedback loop and break the coherence time limit. Therefore, digital error correction has become an important routine for DCS.
The conventional digital error correction method has already reduced the requirement for field-deployed DCS configurations with high performance. However, two free-running continuous wave (CW) lasers and two extra sampling channels are still required to obtain complete jitter information for broadband DCS measurements. These requirements limit further reductions in setup complexity and cost. Considering that IGMs already contain jitter information, several novel self-referencing error correction schemes based on Kalman filter [30,31], ambiguity function [25], or Fourier transform [32,33] have been proposed. These methods are effective for high-repetition frequency IGMs [30–32] or specially designed common-mode noise-immune DCS configuration [25,32]. Nonetheless, when pursuing high resolution and broad spectral coverage with low ${f_r}$ and $\Delta{f_r}$ (e.g., solid or fiber DCS), the rigorous criteria for self-correctable DCS [34] cannot be met, causing severe phase unwrap error and interpolation error far from the centerbursts. Hence, periodical apodization should be adopted [33], and physical resolution loss is inevitable.
In this paper, we propose a hybrid error correction method based on single optical intermedium. Our method combines the advantages of conventional dual intermedium method and the self-referencing error correction method. By introducing a single CW laser, the unwrapped phase noise of a RF mode can be correctly obtained and, with the addition of the time jitter extracted from IGMs themselves, complete jitter information can be reconstructed via phase extrapolation. In contrast to previously demonstrated single intermedium error correction methods [35,36], our scheme eliminates the need for the bulk f–2f offset frequency lock system while still maintaining high coherence across a broad spectral coverage. Our scheme circumvents the rigorous criteria for self-correctable IGMs, making it applicable for strongly mode aliasing scenarios, such as quasi-free-running fiber DCS with low repetition frequency. The complexity and cost of the error measurement configuration of this scheme are reduced by half compared with those of the conventional optical referencing error correction method, thus making our method an attractive tool for compact mode-resolved DCS applications with high resolution, broad spectral range, and high cost-effectiveness.
2. Principle
The principle of DCS is illustrated in Fig. 1(a). The combination of dual combs with slightly different repetition frequencies leads to multiheterodyne interference. The spectral information encoded on discrete optical modes are down-converted into the corresponding RF sub-comb with the scale stretched factor $k = {f_r} / \Delta{f_r}$. The RF sub-comb can be detected by using a photodetector, and time-domain IGMs are generated by the coherent superposition of each RF mode as
Fig. 1. Principle of DCS error correction. (a) Multiheterodyne interference principle of DCS. Optical modes of dual OFCs (blue and green) beat reciprocally to generate the RF sub-comb (purple). Two free-running CW lasers can be used as intermedia to isolate and track the real-time phase of fp1 and fp2, enabling conventional digital error correction. (b) Carrier envelope pulse sequence of time-domain IGMs. Centerbursts appear periodically with the time interval being inversely proportional to the repetition frequency difference. Each frame around a specific centerburst can be Fourier transformed to obtain a short time spectrum. (c) RF spectrum of a single frame. The discrete time jitter and wrapped carrier envelope phase jitter can be extracted frame by frame.
Similar to the generation of the mode-locked laser, the coherent superposition of the equal spacing RF sinusoidal wave generates a carrier envelope pulse consequence that can be expressed as
In order to compensate for above-mentioned time jitter and carrier envelope phase jitter, the conventional digital error correction method [23,29] requires two CW lasers to isolate and track the phase jitter of the two separate RF modes. As shown in Fig. 1(a), each CW laser beats with the closest mode of dual combs, and the relative noise between the modes of dual combs is extracted in the succeeding electronic mixing process. Given that the relative beat is immune to the external CW laser phase noise, free-running CW laser is already sufficient for use as the optical intermedium. After sampling the isolated sinusoidal signal of the ${f_{p1}}$ and ${f_{p2}}$ modes with two analog-to-digital converter (ADC) channels, Hilbert transform can be utilized to obtain the real-time phase jitter $\delta {\phi _{p1}}(t)$ and $\delta {\phi _{p2}}(t)$ with the ADC sampling rate. According to the noise mechanism of mode-locked lasers [21], time jitter $\delta {\tau _0}(t)$ can be calculated by using
Instead of using the optical intermedium to isolate a specific mode, discrete jitter information can also be extracted from IGMs via self-referencing error correction. As is shown in Fig. 1(b), the ideal IGM stream appears frame by frame with the update period exactly equal to ${T_r}$. However, for real IGMs distorted by the time jitter as shown in Eq. (2), the centerburst within the Nth frame is located at
Therefore, the time jitter can be extracted frame by frame by comparing the real and ideal centerburst position of the pulse envelope. In our previous work [33], we used Fourier transform to obtain the short time spectrum of each frame, and the accurate time interval between different frames can be derived by applying a linear fit to the phase spectrum. This time jitter extraction method is inspired by time-of-flight dual-comb ranging [37–39]. As shown in Fig. 1(c), the slope difference between the phase spectrum of the measurement frame and reference frame corresponds to the time jitter at ${t_N}$. Meanwhile, the wrapped carrier envelope phase jitter can also be obtained as an appendant intercept of linear fit.
Self-referencing error correction dispenses with the CW laser intermedium and extra measurement channel at the cost of reducing the jitter refresh rate from the ADC sampling rate to the RF repetition frequency $\Delta {f_r}$. Thus, an additional numerical interpolation algorithm (e.g., cubic spline interpolation) should be adopted to enable point-by-point error extraction. However, interpolation for the continuous carrier envelope phase jitter $\delta {\varphi _0}(t)$ requires that the measured $\delta {\varphi _0}({t_N})$ be correctly unwrapped. In other words, the phase difference between two consecutive frames should be smaller than π. This requirement is a very rigorous limitation for DCS configurations. Although the center frequency jitter $\delta {f_c}({t_N})$ extracted from the amplitude spectrum (shown in Fig. 1[(c)]) can be obtained and help estimate the unwrapped carrier envelope phase jitter at ${t_N}$ [25,40], the frequency estimation error should be less than $\Delta {f_r}$ in accordance with the self-correctable criteria [34]. In actuality, the precision of discrete frequency jitter estimation is merely at the kHz level [33] because of the limitation of photodetector amplitude noise and numerical calculation accuracy. In addition, extra error is generated in interpolation considering the low error sampling rate. Hence, the existing self-referencing error correction method limits the possible minimum value of $\Delta {f_r}$ corresponding to limited frequency multiplexed density and spectral measurement range.
The DCS configuration based on dual separate fiber mode-locked lasers remains the most widely used scheme, especially for field-deployed industrial applications, due to its mature fabrication technology, high robustness, and high tuning flexibility [13,27]. We circumvent the rigorous criteria required for self-referencing error correction by introducing an external free-running CW laser intermedium to obtain a broadband mode-resolved spectrum with a small $\Delta {f_r}$ (normally smaller than kHz level) on this platform. In our single intermedium method, the correctly unwrapped real-time mode phase jitter $\delta {\phi _{p1}}(t)$ can directly be extracted with the ADC sampling rate similar to conventional digital error correction method. Meanwhile, the continuous time jitter $\delta {\tau _0}(t)$ can be calculated by using self-referencing discrete error extraction and cubic spline interpolation. Then, Eq. (4) can be used directly to extrapolate the correct carrier envelope phase jitter $\delta {\varphi _0}(t)$ without a second auxiliary optical intermedium. After the IQ demodulation of the sampled IGM U(t), the carrier envelope phase jitter within the analytical signal A(t) can be eliminated by phase rotation as
Then, the time jitter can be eliminated via time axis resampling (e.g., cubic spline interpolation) as
Ucor2(t) is the phase-aligned IGM stream without any phase jitter, which enables long-time coherent averaging for enhanced sensitivity without resolution loss.3. Experimental results
Figure 2 shows the experimental setup of our DCS configuration. Two separate home-made OFCs based on nonlinear polarization rotation are combined together and interrogate a 16.5 cm-long fiber-coupled gas cell filled with 100 Torr (13.2 kPa) H13C14N gas. Another reference arm without a gas cell is utilized to obtain the background comb spectrum for real-time transmission spectrum acquisition. The spectra of dual combs are located at approximately 1560 nm with an approximately 20 nm full width at half maximum. Therefore, the repetition frequency difference of DCS is set to 200 Hz under the limitation of the band-pass sampling theorem. Bulk f–2f offset frequency lock systems are not required in our scheme. The repetition frequencies of comb1 and comb2 are loosely locked to 57.2002 and 57.2000 MHz by using low bandwidth phase-lock loops. Introducing modest repetition frequency control (<10 Hz bandwidth) does not change the free-running characteristics of OFCs but instead guarantees long-term stability and improves tuning flexibility. Our method only uses a single free-running CW laser intermedium to track the continuous time jitter and carrier envelope phase jitter of IGMs (shown in the green dashed box). As a comparison, an extra set of optical reference error detection module is included in the experiment for simultaneous dual optical intermedium error correction (shown in the blue dashed box). Moreover, a parallel self-referencing correction without any intermedium is also performed. In the electrical path, electronic low-pass filters are used to obtain the pure IGMs without any periodical replica or independent comb response. Then, IGMs and relative beat signals are digitized with a 57.2 MHz sampling rate, and further computational postprocessing error correction algorithms are exploited to retrieve ideal IGMs without distortion.
Fig. 2. DCS experimental setup for H13C14N gas analysis. The all-fiber optical measurement configuration is represented by red solid lines, and the following electrical wires after photodetection are represented by black dashed lines. The part in the green dashed box is the error detection module for our single optical intermedium error correction method. Whereas the part in the blue dashed box is the error detection module for dual optical intermedium error correction method as a simultaneous comparison. OC: optical coupler; DWDM: dense wavelength division multiplexer; LPF: low-pass filter; PD: photodetector; BPD: balance photodetector.
The time jitter extraction algorithm is the key difference between our single intermedium error correction method and the conventional dual intermedium error correction method. The conventional error correction method utilizes two CW lasers to isolate the phase jitter of two separate RF modes. Continuous time jitter can then be derived by using Eq. (3). By contrast, our method extracts discrete time jitter frame by frame from IGMs themselves. Continuous time jitter can then be reconstructed via cubic spline interpolation. Given that the time jitter of IGMs is dominated by slowly changing random walking noise and the unwrap problem is absent from time jitter extraction, the results of the two algorithms are highly consistent as shown in Fig. 3(a). The range of the original time jitter drift is ∼15 µs in 1 s laboratory time, whereas the slowly drifting trend is completely eliminated with ±200 ns residual time jitter and 46 ns standard deviation after correction (shown in Fig. 3[(b)]). This result proves that our method can effectively track slowly changing time jitter drift, and the standard deviation of the residual time jitter is reduced by two orders of magnitude compared with that of the raw IGMs. Correspondingly, the unwrapped carrier envelope phase jitter without slowly changing drift can then be calculated using Eq. (4). The precision of our time jitter extraction method is limited by photodetector amplitude noise and numerical interpolation error. It can be further enhanced if the signal-to-noise ratio (SNR) can be improved or the repetition frequency difference can be increased.
Fig. 3. Comparison of the time jitters extracted by using the conventional dual optical intermedium error correction method and our single optical intermedium error correction method in 1 s laboratory time. (a) Red curve, time jitter calculated by combining the phase jitters of two isolated RF modes in accordance with Eq. (3). Blue curve, time jitter calculated via our single optical intermedium error correction method. (b) Difference between the time jitters extracted via the conventional dual optical intermedium error correction method and our single optical intermedium error correction method.
In our experiment, 1 s corrected IGMs were processed through fast Fourier transform to obtain mode-resolved RF spectra to verify the mutual coherence of DCS. As shown in Fig. 4, more than 75000 individual RF modes across 15 MHz can be clearly distinguished without mutual aliasing. The spacing of adjacent RF modes is 200 Hz, corresponding to repetition frequency resolution after conversion into the optical frequency domain. This resolution is extremely high for Doppler-limited gas detection applications. Each RF mode has an approximately 1 Hz Fourier transform limited linewidth, even at the edge of comb spectrum with low power, achieving analogous relative linewidth suppression in contrast to tight stabilization [10,28] and other error correction methods [23,26]. By benefitting from the inherent strengths of digital error correction, our method can break the coherence time limitation for extended spectroscopy measurement. The spectrum of the coherent averaged IGM frame is mathematically equivalent to the mode-resolved spectrum Fourier transformed by the entire IGM stream [11]. Hence, for IGMs over 1 s acquisition time, the time domain coherent averaging can be imposed on each frame to save ADC memory and computational cost.
Fig. 4. Mode-resolved RF spectra of 1 s corrected IGMs by our method. (a) Complete spectrum of corrected IGMs containing more than 75000 individual RF modes. (b) View under 100× magnification showing the details of the H13C14N absorption lineshape. (c) View under 10000× magnification showing the dozens of resolved RF modes with 200 Hz spacing. (d) View under 1000000× magnification showing the single RF mode realizing the ∼1 Hz Fourier transform limited linewidth.
Similar to other postprocessing correction methods without an accurate optical Ref. [22,25,29], the calibration of the absolute optical frequency axis can be operated by using priori knowledge, such as the known absorption line, which has been proven sufficient for DCS frequency up-conversion from RF frequency to optical frequency. After 9 s of coherent averaging, as shown in Fig. 5(a), our transmittance 100 Torr H13C14N spectrum is compared with the one obtained via self-referencing correction and conventional dual intermedium correction. Owing to the severe unwrap error under the current noise level and repetition frequency setting of our quasi-free-running fiber DCS configuration, self-referencing correction introduces distinct resolution loss. By contrast, our modified single intermedium method achieves consistent transmittance spectrum compared to the mode-resolved dual intermedium correction method. All weak absorption lines and affiliated hot-band transitions in the P branch of H13C14N can be clearly distinguished. As shown in Fig. 5(b), the residuals between our single intermedium method and the dual intermedium method are within ±0.02 over the 2.5 THz spectral range (191.2–193.7 THz) with ∼0.004 standard deviation, wherein the SNR of the comb mode exceeds 150. At the edge of the comb spectrum, the SNR is hampered by low mode power, which causes large amplitude noise. However, a ±0.1 residual range with merely ∼0.01 standard deviation is still achieved across over the 4 THz spectral range and can be further improved by broadening the spectral coverage of the dual mode-locked lasers. The residual is close to zero at the common optical intermedium frequency (∼191.6 THz) but gradually increases when moving away from the common frequency due to the deviation of time jitter estimation. Notably, the residual is also greatly affected by transmittance spectrum noise level. Thus, the results of the two methods approach each other with the increase in coherent averaging time and spectral SNR.
Fig. 5. Transmittance spectra of 100 Torr H13C14N after 9 s of coherent averaging. (a) Comparison among the transmittance spectra obtained by our single intermedium correction method, the self-referencing correction and conventional dual intermedium correction method. Self-referencing correction introduces distinct resolution loss under the current noise level and repetition frequency difference setting of the quasi-free-running fiber DCI configuration. Whereas our modified single intermedium correction method obtains consistent transmittance spectra compared to the mode-resolved dual intermedium correction method. (b) Residual between the transmittance spectra obtained with our method and the conventional dual intermedium method. The standard deviation is ∼0.004 over the 2.5 THz (∼20 nm) spectral range and increases to ∼0.01 over the 4 THz (∼33 nm) spectral range.
Our DCS method is also subjected to detailed molecular lineshape study. Twelve absorption lines in the H13C14N 2ν3 rotational–vibrational band are taken into account and fitted to the numerical Voigt model [41]. Considering that the Doppler width at room temperature can be calculated in advance, line intensity, line center frequency, and Lorentzian width participate in the nonlinear iterations as the free variables. As is shown in Fig. 6, the Voigt function fit residuals of three representative lines (P5, P14, and P23) range from −0.04 to 0.04 even at the edge of comb spectrum with low SNR, and the standard deviations are 0.013, 0.011, and 0.006. These results indicate remarkable fit accuracy. The calculated Lorentzian widths of these three representative absorption lines are 7.867, 7.241, and 2.402 GHz. Moreover, the fit uncertainties, defined as 95% confidence interval (2σ), of the lines are 21.6, 25.0, and 13.3 MHz. Compared with other stronger lines, the P23 absorption line lying at the edge of spectral coverage has larger fit residuals due to its considerably smaller SNR (∼63), which can be further enhanced by prolonged coherent averaging. According to the precise H13C14N model calibrated by NIST [42], the pressure of the gas cell can be derived by using the known pressure broadening coefficient of specific absorption lines. The average pressures of 12 absorption lines in the P branch is 99.5369 Torr, which lies within the uncertainty of Voigt fit and gas cell fabrication. Furthermore, the fit uncertainty of line intensity is also small, allowing gas concentration measurement in the future if a precise H13C14N database containing standard spectral line intensity information can be proposed. The result of lineshape analysis proves the effectiveness of our method, which thus provides an alternative scheme for compact mode-resolved DCS applications with high resolution, broad spectral range, and high cost-effectiveness.
Fig. 6. Absorption lineshape analysis results of P5, P14, and P23 lines in the H13C14N 2ν3 rotational–vibrational band. (a) (b) and (c) Comparison between the measured absorbance spectrum and the Voigt fit result around three representative absorption lines. The Voigt function fit residuals of three representative lines range from −0.04 to 0.04, and the standard deviations are 0.013, 0.011, and 0.006. Mea: measured absorbance lineshape.
4. Conclusions
This work demonstrates a digital error correction method that is based on single optical intermedium for retrieving mode-resolved spectra for quasi-free-running fiber dual-comb configuration. By combining the mode phase jitter extracted via an external optical intermedium with the time jitter extracted from the IGMs themselves, the correctly unwrapped carrier envelope phase jitter can be continuously obtained. After phase rotation and resampling steps, the whole P branch H13C14N transmittance spectrum has been acquired in the near-infrared with a standard deviation of merely 0.01 over the 4 THz spectral range. Our method reduces the complexity of conventional optical referencing error correction setup by 50%, only single set of optical reference error detection module is required. And it circumvents the rigorous criterion required for self-referencing error correction. Our method can be further combined with the state-of-art comb miniaturization technique [43,44], providing a cost-effective way for field-deployed DCS applications requiring high resolution, high sensitivity and broad spectral range.
Funding
Shenzhen Fundamental Research Funding (JCYJ201704121715 35171, JCYJ20200109143008165); National Natural Science Foundation of China (51835007); Tsinghua Shenzhen International Graduate School cross-research and innovation funding (JC20170003).
Disclosures
The authors declare no conflicts of interest.
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