## Abstract

Mutual phase coherence is the utmost crucial factor in an asynchronous optical sampling system. The enhancement is commonly achieved by phase-locking dual femtosecond lasers to a shared narrow-linewidth cavity-stabilized laser. Here we report such a system with a low residual optical phase jitter at a level of 0.04 rad in a Fourier frequency band from 1 Hz to 5 MHz around 1.55 µm, corresponding to a timing jitter of 30 as. The residual phase jitter reaches 1 rad at a Fourier frequency between 0.06 Hz and 0.1 Hz, from which the phase-coherence time is inferred to be at least 10 s. The multi-heterodyne beats experimentally reveal a hardware-limited phase coherence time of ∼8.155 s throughout the direct lasing spectral band.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Asynchronous optical sampling systems established by dual femtosecond lasers possess very attractive features like high temporal resolution, broadband spectral detection and rapid signal acquisition for applications, for instance, ranging, spectroscopy, time distribution [1–7]. Theoretically, pairs of optical pulses with slightly detuned repetition rates imprinted with femtosecond-resolution linear-scanning time delays, and periodically generate interferometric patterns at the detuning frequency. By doing Fourier Transform on this pattern, a comb structure of evenly-spaced multi-heterodyne beats appears in radio-frequency band. After linear frequency mapping, features in optical-frequency region could be obtained realizing massive parallel sensing. In practice, the prerequisites of ‘linear scanning’, ‘even spacing’, and perfect ‘comb structure’ that origin from Fourier Transform principle is mutual phase coherence. Generally, the advance in precision measurements with asynchronous optical sampling systems outlines the progress in its mutual phase coherence.

Taking advantages of shorter temporal duration than picosecond pulses, time-of-flight measurements with femtosecond pulses promote time interval measurement precision to sub-femtosecond level [1,2]. This method drops the carrier phase allowing a relaxation on mutual coherence. Therefore, the system could be simply established by dual femtosecond lasers, whose repetition rates are locked to a common radio-frequency standard, leaving their carrier-envelop offset frequencies free-running. Indeed, mutual phase coherence in optical region is low due to the large frequency leveraging factor of femtosecond laser, always ranging from 10^{5} to 10^{6}. In several applications, self-seeded difference frequency generation or similar process could avoid the coherence degradation resulted from the free-running carrier-envelop offset frequencies, making the system still works well [3–5]. In order to exploit the carrier phase of femtosecond pulses to further enhance precision, highly coherent optical oscillations have to be maintained within one-shot interrogation duration. In this case, optical longitudinal modes of the dual femtosecond lasers are fully stabilized by optical phase locking and self-referencing. The optical phase locking to a common phase-coherent optical frequency reference dramatically decreases the frequency leveraging factor to ∼1, keeping the comb modes across the whole spectral bandwidths in phase with an ultra-low phase jitter. As a result, a successive phase-stable interferometric pattern could be observed in a relatively long time duration. Its carrier-phase leads a time interval measurement precision to attosecond level [6,7]. Benefited from the recent development of laser frequency comb in the past decade [8–12], asynchronous optical sampling system could exhibit a more intriguing performance and come into its own as an extremely powerful tool in the field of precision measurements.

Here we present a phase-coherent asynchronous optical sampling system. A narrow-linewidth cavity-stabilized laser is employed as a phase-coherent optical frequency reference to manipulate dual self-referenced femtosecond lasers. To characterize the phase coherence, we investigate its phase noise performance and dispersive interferogram. In addition, we experimentally validate the coherence time by observing the linewidth of its single-shot long-term multi-heterodyne beats.

## 2. System architecture

The phase-coherent asynchronous optical sampling system follows our previous work on coherent narrow-linewidth optical frequency synthesis [13]. As presented in Fig. 1, (a) narrow-linewidth cavity-stabilized laser at 1542.14 nm serves as an optical frequency reference. Since the reference laser and dual femtosecond lasers are settled on separate optical tables, phase-coherent optical oscillation from the reference laser is separately transferred through ∼15 m fiber links to both femtosecond lasers. Along each link, additive phase noise is suppressed by a homebuilt compact all-fiber-device-based fiber noise canceller, of which additional broadening linewidth and fractional frequency instability are below 1.5 mHz and 4.0×10^{−17} at 1 s, respectively, exhibiting one order of magnitude improvement than our previous bulky-sized ensembles with free-space optics [13].

Two multi-branch Erbium-fiber femtosecond lasers provide broad lasing spectra at ∼1550 nm directly and octave continuums ranging from 1100 nm to 2200 nm via nonlinear optical process. We phase locked the optical beats (*f*_{beat,i}, *i*=1,2) between the transferred optical frequency reference and the adjacent mode of both femtosecond lasers. The carrier-envelop offset frequencies (*f*_{ceo,i}) are stabilized via self-referencing [14]. Subsequently, system characterizations were implemented in the following cases listed in the Table 1. The phase noise spectra could give quantitative phase coherence times and residual linewidths at separate spectral regions of interest, while the dispersive interferogram exhibits the coherent band. The spectrum of its multi-heterodyne interferogram presents a straight view of the residual linewidths at other wavelengths besides those in phase noise characterization, and experimentally validate the first two characterizations. During the measurement, all the phase lock loops are referenced to a hydrogen maser. Since the fibers out of the phase lock loops still bring additive phase noise, they are shortened as possible and enclosed with sponges firmly fixed on the optical tables.

## 3. Phase noise characterization

In the phase noise characterization, we set the frequency offsets of *f*_{beat,i} and *f*_{ceo,i} to be equal, i.e. *f*_{ceo,1} - *f*_{ceo,2} = *f*_{beat,1} - *f*_{beat,2} =Δ*f*. In this configuration, the repetition rates are forced to be equal, i.e. *f*_{rep,1} = *f*_{rep,2}. Pairs of the longitudinal modes of the dual femtosecond lasers beat at an identical frequency of Δ*f*. Several groups of heterodyne beats at 1530.33, 1542.14, 1550.11, 1563.05, 1564.68 nm with a spectral bandwidth of 0.7 nm are filtered to remove noise from other bands as possible. This results in ∼400 pairs of optical longitudinal modes to be simultaneously detected in each passband.

Figure 2(a) gives the phase noise power spectrum densities in the five separate spectral regions. The integral RMS phase noise from 5 MHz to 1 Hz at 1530.33 nm is 36.6 mrad, indicating a residual timing jitter approaching to 30 as. This metrics at the other wavelengths maintains this level as well. The coherent peak contains 99.9% of the total power within a 10-MHz bandwidth at 1 s observing time. The integral RMS phase noise increase dramatically below 0.3 Hz due to environment variation. It reaches 1 rad at Fourier frequency between 0.06 Hz and 0.1 Hz, from which we could estimate a coherence time in a range from 10 s and ∼16 s. Figure 2(b) illustrates the RF spectra of frequency down-mixed heterodyne beats at 1530.33 nm and 1564.68 nm taken with a resolution bandwidth of 31.25 mHz. The linewidths being in the range from 0.06 Hz to 0.1 Hz validate the phase noise characterization.

Figure 3(a) gives the phase evolution at 1530.33 nm within 60 s. The phase evolution shows that the system owns a frequency stability less than 1×10^{−16}/s in a long timescale. In most time, the system maintains a frequency stability of less than 5×10^{−17}/s. We obtained time deviation for the phase evolution data in Fig. 3(b). At an averaging time of 1 s, the time deviation is 30 as that is exactly same with the integral RMS timing jitter in the above phase noise characterization. Correspondingly, this performance leads to a special precision of ∼9 nm at 1s, which is close to the work in Ref. [6,7]. The bump at averaging times centered at 0.02 s is probably caused by power supply and mechanical vibration, corresponding to the glitches at 50 Hz and bump from ∼6 Hz to ∼ 40 Hz in the phase noise spectrum in Fig. 2(a). Above 0.2 s, the time deviation drift is mainly resulted from the differential fiber path variations out of phase lock loops suffering ambient temperature variation. Figure 3(c) exhibits the fractional frequency stability in term of modified Allan deviation derivative from the time deviation. It is beyond 1×10^{−16}/s below an averaging time of 0.1 s, which is inferred by the drastic phase evolution in a short term depicted in the inset of Fig. 3(a). At 1 s, the fractional frequency stability is 5.2×10^{−17}/s, corresponding to the trend of the phase evolution in Fig. 3(a).

## 4. Dispersive interferometry

The dispersive interferometry gives an intuitional clue that how broad the phase coherent spectral band is. It is substantially a sort of homodyne interferometry, where the *f*_{beat,i} and *f*_{ceo,i} must be identical, i.e. *f*_{ceo,1} = *f*_{ceo,2} and *f*_{beat,1} = *f*_{beat,2}, resulting in identical repetition rates, i.e. *f*_{rep,1} = *f*_{rep,2}. Since both the femtosecond laser pulse trains are guided via optical fibers, pulse chirp always exists if delicate dispersion compensation has not been done in advance. Next, we briefly introduce the theoretical model of dispersive interferometric pattern generated by chirped pulses. The Gaussian pulses from both lasers are defined as below

*E*

_{0}is electrical field,

*a*is the attenuation factor of Gaussian pulse,

*a*= 2ln2/

*τ*

_{0}

^{2},

*τ*

_{0}is pulse duration,

*b*is chirp rate,

*τ*is time delay between the pulses,

*ω*

_{c}is the angular frequency of optical carrier,

*φ*

_{0}is a constant phase. By doing Fourier Transform, one can obtain the optical spectra of both pulses.

The dispersive interferometric pattern generated by the chirped pulses is written by

*b*

_{1}=

*b*

_{2}=0, Eq. (5) could be simplified into the common dispersive interferometric pattern.

We recorded the spectral features of both femtosecond lasers plotted in Fig. 4(a), and the dispersive interferometric pattern in Fig. 4(b) with a resolution bandwidth of 0.033 nm in 1 s. Following the guidance in Ref. [15], we adjusted the parameters in Eq. (5) to make the calculation match the measurement. According to the Eq. (2) in Ref. [15], the contrast of the interferometric pattern reaches 99.98% throughout the direct lasing band, therefore inferring a high phase coherence within this band.

## 5. Multi-heterodyne interferometry

In order to evaluate the coherence at other wavelengths besides those in phase noise characterization quantitatively, multi-heterodyne interferometry was done with frequency detuned repetition rates, i.e. *f*_{rep,1} ≠ *f*_{rep,2}, resulted from *f*_{ceo,1} = *f*_{ceo,2} and *f*_{beat,1} = -*f*_{beat,2}. In this case, Pairs of the asynchronous pulses own linearly increasing time delays, and generate pulse-like interferometric peaks periodically with an update rate at the detuned frequency of the repetition rates. Due to a prior expectation about coherence time at least 10 s in phase noise characterizations, we continuously digitized this periodical interferometric signal for a long time as possible. The raw data are recorded for ∼8.155 s by a 16-bit high-speed data acquisition system clocked to one of the femtosecond lasers.

After zeropadding to smooth spectral outline, the spectral profile is obtained by Fourier Transform algorithm and presented as Fig. 5(a). The outline agrees with the total constructive interferometric pattern in Fig. 4(b). The multi-heterodyne beats are finely revolved, or so-called comb-tooth resolved, shown as the zoomed spectrum around 1564.68 nm in Fig. 5(b). In Fig. 5(c)–5(f), single comb tooth at 1590.00, 1564.68, 1530.33 and 1515.00 nm are displayed, respectively. Due to the truncation introduced by data acquisition, side lobes are clearly observed around the teeth. Correspondingly, all the teeth are fitted by absolute value function of sinc waveform. Note that even at a spectral regime with a low signal-to-noise ratio, see Fig. 4(f), the fitting functions possessing an identical linewidth match these teeth pretty well. The reason is that the limited sampling duration mainly by the memory of our workstation does not reach the resolution requirement of the coherence time. Taking the linewidth broadening coefficient of the truncation window of 1.207 into consideration [16], the measured linewidth in RF domain is 0.123 Hz, which exactly corresponds to the sampling duration of ∼8.155 s according to Nyquist principle. Therefore, the coherence time of at least ∼8.155 s is confirmed and agrees with the phase noise analysis. As an expectation, the excellent coherence could maintain across the spectral coverage from 1 µm to 2 µm via nonlinear spectral broadening due to the ultra-low noise feature of the femtosecond lasers [13,17,18].

## 6. Conclusion

We have present a phase-coherent asynchronous optical sampling system. The phase noise characterization indicates a residual linewidth of less than 0.1 s, and a corresponding coherence time of at least 10 s. The obtained high-contrast dispersive interferometric pattern gives a general view of high phase coherence throughout the direct lasing band. The multi-heterodyne spectral analysis gets access to the phase coherence at arbitrary wavelength quantitatively. It infers a hardware-limited phase coherence time of ∼8.155 s across the lasing spectral band, further validating the phase noise characterization.

## Funding

State Key Laboratory of Precision Measurement Technology and Instruments (DL18-02); National Natural Science Foundation of China (11704037).

## Disclosures

The authors declare no conflicts of interest.

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