Dual-comb spectroscopy with spectral acquisition rate surpassing Nyquist-limited trade-off constraint


 Optical frequency comb with evenly spaced lines over a broad bandwidth has revolutionized the fields of optical metrology and spectroscopy. Despite dual-comb spectroscopy (DCS) offers a superb overall performance on spectral resolution, measurement bandwidth and refresh rate, these parameters are still constrained by the Nyquist-limited trade-off. Here, we propose a novel DCS system to overcome this limitation with tens of spectral acquisition rate improvement, which is built with two electro-optic frequency combs seeded by an ultralinearly swept light source. The proposed scheme records a spectrum spanning 304 GHz with a spectral resolution of 10 kHz in 1.6 ms, which achieves a spectral sampling-rate of 1.9×1010 points/s. We also demonstrate the 100-averaged results with 28.9 dB signal-to-noise ratio for high sensitivity measurement. The demonstration shows great prospect for precise measurement with significant performance breakthrough.

Beating two OFCs with slightly different repetition rates, DCS retrieves each frequency component in radio frequency (RF) domain, which may fully exploit the spectral resolution and measurement bandwidth determined by OFCs. The spectral resolution can be further reduced to kHz less than line-spacing by time-consuming spectral interleave techniques 13,18,19 . In recent years, novel approaches for OFC generation have been explored, which facilitate DCS performed at different wavebands with versatility 9, 11, 12 . Mutual coherence establishment between two independent combs requires phase-locking circuits 10,20 or phase correction [21][22][23] . OFCs may also be generated by electro-optic modulation from a single seed laser to form a DCS system with intrinsic mutual coherence [24][25][26][27][28][29][30][31] , which significantly reduces the system complexity. Electro-optic frequency combs (EOFCs) 32,33 have been demonstrated with nonlinearly broadened bandwidth 34 or ultra-dense line-spacing 35,36 , which promotes flexible electro-optic DCS for various spectroscopic applications. Laser swept spectroscopy (LSS) is another widely used method to obtain the optical spectrum point by point by sweeping the wavelength in different ways 37,38 . Internally swept laser may scan a wide spectrum over 100 nm range with poor nonlinearity. Relatively, external modulation 39-41 using the chirped electrical signal is an effective way to realize the theoretical kHz-resolution with tens of GHz frequency range limited by modulators and RF generators.
The overall performance of spectral resolution, measurement bandwidth and measurement speed is still limited by the analog-to-digital converter (ADC) sampling rate for all FTS including DCS to obey the equation expressed by Eq. (1) 42 , where F S is the spectral sampling-rate, F ADC is the ADC sampling rate, B is the measurement bandwidth, δν is the spectral resolution and T 0 is the single-shot measurement time. Similarly, we can find LSS must obey the sampling limitation equation expressed by Eq. (2), to realize the performance that each sampling point represents one spectral point. The proposed method has the potential to break this kind of limitation to realize the spectral sampling rate to be K times of the ADC sampling rate, as expressed by Eq. (3), where K is the number of the comb teeth.
In this paper, we propose a novel DCS technique to simultaneously have high spectral resolution, large bandwidth and fast measurement speed, which realizes an ultra-high spectral acquisition rate surpassing Nyquist-limitation existed in traditional DCS. A linearly swept light source serves as the seed of two EOFCs, which have a slight repetition rate difference to build a dual-comb interferometer. Each comb-line pair records a spectrum with a high resolution and a limited range. The sweep range is in accordance with the repetition rate of the probe comb to realize the full coverage of total bandwidth with simultaneous sweeping of all comb teeth. The dual-comb interferometer locates each comb-line pair to different frequencies in electrical domain. So the spectrum information loaded on each comb tooth can be separated by using a digital filter, and the total spectrum can be recovered from all the EOFC teeth. In the experiment, a spectrum covering 304 GHz bandwidth with 10 kHz spectral resolution is recorded. Considering the measurement time of 1.6 ms, the demonstration shows an ultra-high spectral sampling-rate of 1.9 × 10 10 points/s to be 19 (number of comb teeth) times of the ADC sampling rate (1 GS/s), thanks to the capability of simultaneously recording all channels. The operation principle is illustrated in Fig. 1. This method consists of two primary components, which are respectively an ultra-linearly frequency-swept optical source generation system and an electro-optic dual-comb interferometer system. A frequency stabilized laser seeds the swept optical source by using external modulation, which utilizes the ultra-high an interval of ∆f = f p − f l . Therefore, the recorded spectrum of each channel can be retrieved by using a digital filter to distinguish. Since each sampling point obtained by the ADC represents K spectral points after the demodulation process, the upper limitation of spectral sampling rate is increased to be K times of the ADC sampling rate.

Experimental setup
A specific experimental setup of the DCS system is depicted in Fig swept sideband is generated after the modulation, and then injected into an isolator-removed distributed feedback laser diode (DFB-LD) which serves as a slave laser via an optical circulator.
Due to the injection locking effect, the 1st-order positive sideband is selected and amplified to 10 mW, and the carrier and all other sidebands are suppressed with an extinction ratio of over 20 dB.
Therefore, a linearly-swept lightwave with low power fluctuation is generated. We characterize the property of the swept lightwave by using an unbalanced Mach-Zehnder interferometer with 1 km delay fiber. As shown in Fig. 3   A reflectance spectrum of a fiber Fabry-Perot interferometer (FFPI) is measured by using the proposed system. The FFPI is composed of a pair of high reflection fiber Bragg grating (FBG) with reflectivity of 99 %. The electrical spectrum of the reflected probe comb is shown in Fig. 4(c). A digital filter is used to select the 3-rd channel for demodulation, and the zoom-in electrical spectrum is shown in Fig. 4(d). The temporal waveform of the 3-rd channel is a sinusoidal signal, whose varying intensity in 1.6 ms represents the optical spectrum of the FFPI in a bandwidth of 16 GHz.  Fig. 6(a), containing the absorption lines of P 10, P 11, P 12, and P 13 in the 2v 3 band. The averaged results are respectively shown in Fig. 6(b), 6(c) and 6(d) with an average time of 6, 25 and 100. A fitting line of the P 11 branch based on Voigt function is shown in Fig. 6(e) together with the experimental results, and the relative residual for all spectral points are shown in Fig. 6(f). The ratios of the standard deviation are calculated to be 0.66%, 0.37%, 0.22% and 0.13% for single, 6-averaged, 25-averaged, and 100-averaged results. The signal-to-noise ratio (SNR) are improved thanks to the average process. It reaches 28.9 dB for 100 times average, which also enables high sensitivity measurement.

Conclusions
In conclusion, we propose a novel DCS technique with ultra-high spectral acquisition rate implemented by EOFCs seeded by a linearly frequency-swept light source. This method retrieves a spectrum in 1.6 ms with a spectral resolution of 10 kHz and a bandwidth of 304 GHz. The realized spectral sampling-rate of 1.9 ×10 10 points/s shows the capability to break the limitation of ADC sampling rate (1GS/s in the experiments) for all FTS and LSS. The spectroscopic results of a HCN transmittance spectrum and a FFPI reflectance spectrum validate the novel method, and the results also show the capability of SNR improvement by using averages. This paper provides an effective method based on EOFCs for ultra-high spectral resolution measurement applications such as measuring high-Q cavity, electro-magnetically induced transparency, or physical and biochemical sensing requiring hyperfine spectrum measurement.

Methods
Mathematical proof for the validity of the proposed DCS is provided. The optical field of the stabilized laser output can be expressed as: where E 0 is the amplitude, f 0 is the center optical frequency. The frequency of the RF driven signal is ν(t) = ν 0 + γt(0 ≤ t ≤ T 0 ), where ν 0 and γ are the initial frequency and frequency sweeping rate of the RF driven signal, and T 0 is the swept time. Therefore the output signal of the MZM is expressed as: where a ±1 (t) and a 0 are respectively the amplitude of the ±1 st order sidebands and the residual carrier. The +1 st order sideband is selected by using injection locking technique to be where a sl (t) is the intensity of the slave laser and f c is the sum of f 0 and ν 0 . The optical field of the generated probe and the local combs generated from DD-MZMs can be expressed as where m is the index of the comb line, A pm and A lm are the complex coefficients of the m-th comb line, f p and f l are the line spacing of the probe and local combs, k is the number of one-sided comb lines (2k + 1 = K, K is the number of total lines). The optical field of the local comb is frequency-upshifted by the AOM to be where f AOM is the driving frequency applied to the AOM. The probe comb goes through the DUT, and the output can be expressed as where S m (t) = H(f c +mF p +γt) and H(f ) are the complex transfer function of the DUT. Both of the probe combs in the measurement and reference paths interfere with the local comb. Considering the effective detection bandwidth, the generated photocurrent of the BPDs for measurement part and reference part can be written as where R is the photoelectric conversion efficiency and ∆f is the difference between f p and f l .
These two signals become combs in RF domain centered at f AOM with a line-spacing of ∆f . For demodulation process, the same lines are digitally filtered to obtain the envelop expressed as The results for one line is generated by combining Eqs. (13) and (14) to be e resm (t) = | S m (t)| = | H(f c + mf p + γt)| This result is a spectrum covering from f c + f p to f c + 2f p (γT 0 is set to be f p in experiments). So, the resolved spectrum for the m-th line can be expressed as D m (n) = e resm ( n F ADC ) = | H(f c + f p (m + n T 0 F ADC )| where F ADC is the ADC sampling rate and n = tF ADC (0 ≤ n ≤ T 0 F ADC ) is the index of the spectral point. The whole spectrum is simultaneously obtained, which can be written as Here, the number of points for each D m (m = −k, −k + 1, ..., k − 1, k) is T 0 F ADC , same as the points sampled by ADC. Therefore the spectral sampling rate F S reaches KF ADC , which surpasses the Nyquist-limitation.