## Abstract

We demonstrate a significant improvement in the performance of a fiber-based frequency comb when a GPS-disciplined Rb clock is replaced with an acetylene-stabilized laser as the frequency reference. We have developed a compact, maintenance-free acetylene-stabilized fiber laser with a sub-kHz short-term linewidth and an Allan deviation below 3×10^{−13} for integration times above 1 s. Switching the comb reference from the Rb clock to the acetylene-stabilized laser improves both comb tooth linewidth and Allan deviation by about two orders of magnitude. Furthermore, long-term measurements of the acetylene-stabilized laser frequency with reference to the GPS-disciplined clock indicate a potential relative frequency uncertainty of 2 × 10^{−12}.

© 2017 Optical Society of America

## 1. Introduction

Recent years have seen a vast increase in the use of femtosecond frequency combs [1] for a range of different applications in e.g. time and frequency metrology [2–4], length metrology [5], frequency-comb spectroscopy [6], low-noise microwave generation [7], and spectral calibration [8]. This evolution has been especially aided by the development of fiber-based combs which offer ease of use and hands-off operation. However, as opposed to the Ti:Sapphire-based frequency comb, the fiber-based comb does not have an intrinsically narrow linewidth of the comb’s teeth, and for applications where a high accuracy and stability is required, the comb must be stabilized to an external frequency reference [4,9,10]. This reference is often provided either by an ultra-stable optical cavity [11–13] - which provides excellent short-term stability, but lacks long-term reliability and accuracy - or an RF reference [10], such as a Cs or Rb oscillator, which provides the long term accuracy but lacks sufficient short term stability for most frequency sensitive applications [14].

This paper explores the frequency comb performance that can be achieved when the reference is a laser with an intrinsic narrow linewidth, and this laser is stabilized to a molecular transition in a relatively simple setup. Erbium-doped fiber lasers can achieve a very narrow linewidth at wavelengths near 1.5 µm, which naturally coincides with the center wavelength of fiber-based frequency combs. Hence, an Erbium-doped fiber laser appears as an obvious source for a frequency comb reference. Acetylene absorption lines provide some of the best absolute frequency references at this wavelength range [15,16]. Iodine lines can be reached after frequency tripling 1.5 µm radiation, and iodine generally provides stronger absorption and better frequency stability than acetylene [17]. However, the performance increase with iodine comes with additional complexity and cost when frequency tripling is required.

This paper demonstrates the performance of an acetylene-stabilized narrow-line fiber laser, which is developed specifically as a compact and reliable frequency comb reference. The device fits in a 19 inch rack with dimensions (26 × 52 × 50) cm^{3}, features turn-key operation, and remains locked to the acetylene line for at least several months. Due to the maturity of the narrow-line fiber lasers and the simplicity of the acetylene stabilization scheme providing long-term stability and absolute accuracy, the acetylene stabilized fiber laser may in some cases provide a cost effective and reliable alternative to e.g. a laser stabilized to an ultra-stable optical cavity. The acetylene-stabilized fiber laser provides a comb performance that is sufficient for a range of applications and experiments, including dual comb spectroscopy [18], references for stabilization and line narrowing of lasers for e.g. spectroscopy or laser cooling on narrow-line atomic or molecular transitions [14], as well as references for toroidal microresonator-based combs [19,20].

This paper focuses on three performance parameters for the frequency comb locked to an external reference; the linewidth of a single comb tooth, the frequency stability of a comb tooth, and the absolute frequency accuracy of the comb. For each of those key features, the performance of the comb locked to the rubidium clock is compared to its performance when locked to the acetylene-stabilized laser. To this end, the beat note between one tooth of the comb and an uncorrelated frequency reference is investigated. Three identical stabilized lasers are available as frequency references for the experiments and they are refered to as Sλ_{1}, Sλ_{2} and Sλ_{3}.

## 2. Setup

The frequency comb is a MenloSystems FC1500-250-WG operated at Aarhus University. The output of the comb is determined by its two degrees of freedom, the carrier-envelope offset frequency *f*_{ceo} and the repetition rate *f*_{rep}. The frequency of the *n*^{th} comb tooth is given by

*n*is the mode number. The frequency

*f*

_{rep}is locked either to an RF reference or an optical reference. In the first case, it is phase-locked to a rubidium oscillator (Stanford Research Systems FS725) via a feedback loop locking the fourth harmonic of

*f*

_{rep}to the 1 GHz output of the rubidium oscillator, yielding

*f*

_{rep,Rb}= 250 MHz. To reduce the long-term drift of the rubidium oscillator (FS725 has a specified monthly aging of up to 5×10

^{−11}), it is itself locked to a one-pulse-per-second GPS signal, which corrects the long-term drift via a PID loop with an 8-hour integration time. Although this GPS locking improves the stability of the Rb-locked comb for time scales longer than 10

^{5}s, it has no effect in the short and middle term. This is the domain where locking

*f*

_{rep}to an optical reference can enhance the performance of the frequency comb. When stabilizing the comb to an optical reference at frequency

*f*

_{Sλ1}, the frequency of one specific mode of the comb, here

*n*

_{0}= 777478, is fixed by locking the beat note between

*n*

_{0}and and the output of the optical reference to the value

*f*

_{beat}= 60 MHz. Equation (1) then becomes

*f*

_{ceo}= −20 MHz is set by a GPS-slaved RF reference.

Figure 1 presents the setup used for characterizing the performance of the frequency comb. The output of the comb is coupled into free space, and a grating is used to select a narrow part of the wide comb spectrum. The comb light is subsequently launched into a fiber and mixed with the Sλ_{3} output in a fiber coupler. The half-wave plate is used to match the polarization of the comb field and the Sλ_{3} output. An optical attenuator and various RF filters and amplifiers are used to maximize the signal-to-noise ratio of the beat note under investigation at frequency *f*_{BN}. The photo detector output eventually goes to either a frequency counter for measurements of frequency instability or a mixer followed by a digital storage oscilloscope for spectral analysis and linewidth measurements. The mixer shifts the beat frequency down to about 1 MHz for a better match with the digital sampling rate.

The design of the stabilized laser, which is sketched in Fig. 1, is based on saturated absorption spectroscopy of acetylene in a retroreflection configuration as demonstrated in [16, 21]. The Erbium-doped fiber laser is the new Koheras BASIK X15 module from NKT Photonics [22]. An acousto-optic modulator (AOM) is used to frequency shift and frequency modulate one part of the X15 output with a shift of about 40 MHz, a modulation frequency of about 1 kHz, and a peak-to-peak frequency modulation width of about 1 MHz. The modulated output pass through an Erbium Doped Fiber Amplifier (EDFA) before free space coupling into the spectroscopy setup. The signals from the reference and signal photo detectors are digitized, filtered and demodulated to obtain an error signal, which is fed back to the X15 fiber laser [16]. The other part of the X15 output provides the unmodulated output of the acetylene-stabilized fiber laser, see Fig. 1.

The three acetylene-stabilized lasers (Sλ* _{i}*) are significantly improved with respect to the setup described in [16]. The new X15 laser has much less frequency noise in the 1 Hz to 1 kHz range, which is obtained by active stabilization to a frequency reference integrated in the laser module. This noise reduction improves the short-term frequency stability and linewidth of the acetylene-stabilized laser. In addition to improvements of the laser source, the control and signal processing for the acetylene stabilization scheme has been digitized, and the use of two independent photo detectors allows for individual signal processing and a simpler setup. The number of free space optical components has been reduced, and the remaining free space components are fixed in a monolithic design, which improves mechanical stability. The overall size is significantly reduced and fits in a single box, including all electronics.

## 3. Performance of the acetylene-stabilized fiber lasers

#### 3.1. Frequency instability

The frequency instability of the individual stabilized lasers is characterized by using the so-called three cornered hat (TCH) method [23], where the three lasers Sλ_{1}, Sλ_{2} and Sλ_{3} are compared. The TCH method is a well-known tool to extract frequency instabilities by cross beating three or more laser sources [24]. With the assumption that the three frequencies behave like mutually independent variables, the additivity of their variances enables us to extract the individual Allan variances. The short-term instabilities (averaging time *τ* ≲ 50 s) are mainly due to uncorrelated residual intensity noise [16]. On a longer timescale (*τ* ≳ 50 s), the instability is caused by temperature dependent residual etalon effects and imperfect modulation in each device. As discussed below, the temperature induced frequency fluctuations are uncorrelated when averaged over a long time.

The beat notes between the devices Sλ_{1} – Sλ_{2} and Sλ_{1} – Sλ_{3} are measured simultaneously by two synchronized frequency counters. The AOM induced frequency shifts noted *f*_{AOM} in Fig. 1 shifts the frequency of the locked lasers with respect to the acetylene resonance frequency *f*_{ac} by 43.7 MHz (Sλ_{1}), −40.4 MHz (Sλ_{2}) and −40.0 MHz (Sλ_{3}), and the measured beat note frequencies are in the MHz range.

The individual frequency Allan variance of laser *S*λ_{i} is derived from the following equation,

*f*(

_{ij}*t*) between Sλ

*and Sλ*

_{i}*. Figure 2 shows the individual frequency instabilities of the three stabilized lasers obtained via the TCH method.*

_{j}The data presented in Fig. 2 have been acquired during 53 hours, for which the signal-to-noise ratio (SNR) for the two beat notes was typically 40 dB at a 1 MHz resolution bandwidth. The middle range data (1 s ≤ *τ* ≤ 10^{3} s) have been confirmed by another independent TCH acquisition with a 45 dB SNR, measuring the beat notes *f*_{13} and *f*_{23}.

Figure 2 shows a short-term white-noise fractional frequency instability of about
$3\times {10}^{-13}{\left(\tau /\mathrm{s}\right)}^{-\frac{1}{2}}$ for integration times from 150 ms to 30 s. The best laser reaches *σ _{y}* (1 s) = 2.9 × 10

^{−13}and

*σ*(10 s) = 8.5 × 10

_{y}^{−14}. The small performance differences are due to imperfect acetylene pressure (Sλ

_{2}) [16], less efficient intensity noise cancellation with an older software version (Sλ

_{1}), and variations in residual etalon effects.

The bumps around 200 s − 1000 s visible in Fig. 2 for all three lasers are due to imperfect modulation and residual etalon effects in the optical setups. The residual etalon effects give rise to sinusoidal variations in frequency with increasing temperature. Since each of the lasers will be at different and changing positions on this sine curve, the correlation between laser frequencies will average out after a while. Hence, the three lasers can be considered independent. This argument is supported by a calculation of the Pearson’s correlation coefficient between the laboratory temperature and the beat note frequencies, which gave |*ρ*(*temperature*(*t*), *f _{ij}*(

*t*))| < 0.1. The peak-peak fluctuations in temperature were around 0.5 °C with a characteristic time scale of 4 × 10

^{3}s. A temperature change of 0.5 °C results in about five periods of the sinusoidal frequency variation due to the path length in the optical setup and its thermal expansion coefficient. This explains why the bumps in Fig. 2 occur at a shorter time scale than the characteristic time for the temperature fluctuations. These residual etalon effects may be effectively reduced by implementing techniques for interference cancellation [25].

#### 3.2. Line profile

For short time scales it is often more relevant to investigate the spectral properties than the Allan deviation. The frequency noise spectral density provides a complete spectral characterization, but the spectral linewidth is often a more convenient parameter which is easier to interpret. The spectral linewidth is derived from a recording of the beat note between two stabilized lasers on a digital storage oscilloscope. The beat note signal is first down-mixed to about 1 MHz so that the Nyquist frequency of the data acquisition is at least twice the beat note frequency. Fast Fourier Transforms (FFT) are then applied to the acquired data to derive the frequency spectra. With this method it is possible to investigate how the linewidth depends on the sampling time. An almost identical technique for laser linewidth analysis was published very recently in [26]. Figure 3 shows a frequency spectrum obtained this way.

The beat note spectrum features a narrow line emerging from a flatter, wider pedestal. A Voigt profile fit of this narrow line suggests a dominant Lorentzian contribution. Therefore, the spectrum is fitted with a model of a narrow Lorentzian peak on top of a wide Gaussian pedestal. The FWHM linewidth from the fit in Fig. 3 is 1020 Hz. The Gaussian pedestal has a FWHM linewidth of about 11.2 kHz, and the Lorentzian contribution has a linewidth of 880 Hz. A similar analysis for a single data set with 10 ms sampling time gives a FWHM linewidth of 650 Hz and a similar Gaussian pedestal as for 100 ms sampling time.

We also performed a complementary analysis on the data to extract the linewidth in a different way. This “direct time series method” allows us to follow the evolution of the linewidth as a function of the sampling time, see inset in Fig. 3. Here, the acquired time series of the beat note is first divided into intervals corresponding to the chosen sampling time. An FFT analysis is performed for each interval, and the individual FFT spectra are then smoothed over a number of points equal to one tenth of the expected linewidth. The linewidth for each interval is derived via an interpolation of the points surrounding the half-height value. The linewidths are averaged over all the intervals, yielding the FWHM linewidth for the chosen sampling time for a single acquisition. The linewidths from up to ten individual acquisitions obtained under identical conditions have been averaged, and the average FWHM linewidths and corresponding standard deviations (error bars) are plotted in the inset in Fig. 3. This analysis is repeated for various choices of the sampling time using the same data acquisitions. The standard deviation describes the variations between the individual acquisitions, which include noise in the measurement process as well as fluctuations in the actual laser linewidth [26]. The smallest possible linewidth after the Fourier transform is given by 1/*T*, where *T* is the sampling time [26]. This Fourier-limited linewidth is plotted in Fig. 3 (green line), and it is seen that the measured linewidth is only about two times larger than this fundamental limit for the shortest sampling time. The algorithm used for deriving the linewidth as a function of sampling time is tested with a pure sine wave test data set. The algorithm reproduces the Fourier-limited linewidth for these test data, which confirms that the numerical analysis does not introduce additional broadening. Furthermore, when the beat note signal is replaced with the output from a signal generator, the analysis gives linewidths close to the Fourier-limited linewidth and well below the linewidths in Fig. 3. Therefore, the time base stability of the oscilloscope is not a limitation for the linewidth measurements.

The linewidth increases with sampling time for sampling times up to about 200 ms. Short sampling times give a “snapshot-like” measurement of the laser spectrum. Integrating this over longer time will average the fluctuating spectrum into a spectrum with a wider linewidth, thus giving the increasing linewidth with sampling time. Above approximately 200 ms the linewidth is basically constant due to the acetylene lock.

With this complementary analysis we get a (695 ± 150) Hz FWHM linewidth for a sampling time of 10 ms, which matches the analysis using a Gaussian plus Lorentzian fit as described above. For a sampling time of 100 ms, the obtained FWHM linewidth of (1170 ± 115) Hz is close to the value for the single acquisition in Fig. 3.

Assuming the two lasers Sλ_{1} and Sλ_{3} have equal linewidths dominated by a Lorentzian contribution, the individual laser linewidths (FWHM) range from ∼ 300 Hz to ∼ 600 Hz on timescales from 5 ms to 100 ms.

## 4. Performance of the frequency comb

#### 4.1. Frequency instability

The stabilization methods for the frequency comb use either Sλ_{1} or a Rb clock as reference for the comb repetition rate. The comb is locked using the MenloSystems FC1500-250-WG standard interface with feedback to an intra-cavity piezo (RF lock) or to an intra-cavity electro optical modulator plus intra-cavity piezo (optical lock). We measure the beat note between Sλ_{3} and the comb tooth located approximately 2*f*_{rep} from
${f}_{\mathrm{S}{\mathrm{\lambda}}_{3}}$. This comb tooth was chosen for investigation as it provided the best signal-to-noise ratio with the available RF components. When the comb is locked to Sλ_{1}, the comb-Sλ_{3} beat note has a frequency of
${f}_{\mathrm{S}{\mathrm{\lambda}}_{1}}+2{f}_{\text{rep},\text{opt}}-{f}_{\text{beat}}-{f}_{\mathrm{S}{\mathrm{\lambda}}_{3}}=523.7\phantom{\rule{0.2em}{0ex}}\text{MHz}$. When the comb is locked to the Rubidium oscillator, the comb-Sλ_{3} beat note has a frequency of
${f}_{\mathrm{S}{\mathrm{\lambda}}_{3}}-(({n}_{0}-2){f}_{\text{rep},\text{Rb}}+{f}_{\text{ceo}})=549.4\phantom{\rule{0.2em}{0ex}}\text{MHz}$. For either of these two beat notes the obtained SNR for the following analysis is typically 30 dB at a resolution bandwidth of 100 kHz.

Figure 4 shows a comparison of Allan deviations for long term acquisitions of the beat notes between Sλ_{3} and the comb locked to either the Rb clock or the Sλ_{1} optical reference. The red curve in Fig. 4 is an average over three different acquisitions of the comb stability when locked to Sλ_{1}, yielding very convincing results in the short and medium range as compared to the Rb reference (blue curve). At 1 s the Allan deviation of the comb locked to Sλ_{1}, *σ*_{Sλ}(1 s) = 4.3 × 10^{−13}, is almost 50 times smaller than the fractional instability at one second with the Rb reference, *σ*_{Rb}(1 s) = 2.0 × 10^{−11}. This enhancement in the frequency stability remains above one order of magnitude until *τ* ≃ 100 s. The instability of the Sλ_{1}-locked comb compares well to the Allan deviation measured directly between the Sλ_{1} and Sλ_{3} (gray curve) when small environmental changes between measurements are taken into account.

#### 4.2. Linewidth

The short-term comb tooth linewidth is derived from the beatnote between a single comb line and Sλ_{3} as described in section 3.2. The spectrum for the Rb clock referenced comb displays an almost pure Gaussian profile. Figure 5 presents a single measured line profile with a Gaussian fit as well as the linewidth data from the direct time series analysis plotted as a function of sampling time. Both methods yield consistent results at 10 ms; a 270 kHz linewidth from the Gaussian fit and (274 ± 13) kHz from the direct time series method. According to the measurements, the line broadens from 120 kHz to 270 kHz for sampling times from 200 *µ*s to 5 ms, where it reaches a plateau for longer sampling times. These results are comparable to other measurements on an Er:fiber laser frequency comb locked to an RF reference [10]. Using the *β*-separation method described in [27], the measured phase noise in [10] results in a short term linewidth of 115 kHz. It has been noted that the *β*-separation approach may underestimate the actual linewidth [26].

For the comb locked to the optical reference of Sλ_{1}, we use the same fitting model as described in Section 3.2. The result is shown in Fig. 6. The FWHM linewidth from the fit to a single measurement is 900 Hz for a sampling time of 100 ms. The direct time series method gives a linewidth of (1.05 ± 0.09) kHz. The measured Sλ_{1}-locked comb tooth linewidth appears to be slightly smaller than observed in section 3.2 when beating Sλ1 and Sλ2, although the difference is comparable to the measurement uncertainties of about 100 Hz.

## 5. Long-term absolute accuracy

The potential accuracy of the frequency comb locked to Sλ_{1} is evaluated by a long-term measurement of the repetition rate *f*_{rep,opt} using the GPS disciplined Rb oscillator as reference. This results in a direct measurement of the absolute frequency of Sλ_{1} according to,

*f*

_{beat}= 60 MHz is the fixed beat note in the locking process,

*f*

_{ceo}= −20 MHz, and

*n*

_{0}= 777478.

The optical frequency of Sλ_{1} is plotted in Fig. 7. The optical frequency averaged over all measurements is found to be 〈*f _{Sλ}*

_{1}〉 =

*f*

_{ac}+

*f*

_{AOM}+ 4.9 kHz, where

*f*

_{ac}=

*c*/

*λ*= 194 369 569 384 kHz is the CIPM recommended value for the acetylene transition frequency [15] and

_{ac}*f*

_{AOM}= 43 700.0 kHz is the AOM frequency used in Sλ

_{1}. The statistical uncertainty on the average frequency is 9 Hz. The three acetylene-stabilized lasers operate according to the CIPM recomendation with a

^{13}C

_{2}H

_{2}pressure of 1.0 Pa, a peak-to-peak frequency modulation width of 0.9 MHz, and a one-way beam power density of 7 W cm

^{−2}. The 4.9 kHz difference between

*f*

_{ac}and the output from Sλ

_{1}, after correction for the AOM frequency shift, is consistent with the 5 kHz standard uncertainty (2.6 × 10

^{−11}relative standard uncertainty) for an acetylene-stabilized laser operated as a primary standard for realization of the meter [15].

The linear fit in Fig. 7 gives an upper drift limit of about 1 Hz/day corresponding to 2 × 10^{−12} relative change over one year. This indicates that a relative frequency uncertainty at the level of 2 × 10^{−12} can be achieved for the Sλ_{1}-stabilized frequency comb with annual calibration against e.g. a GPS disciplined oscillator.

## 6. Conclusion

The performance of a compact acetylene-stabilized fiber laser has been characterized in terms of linewidth (Δ*ν* < 1 kHz), stability (relative Allan deviation *σ*_{y} (*τ* > 1 s) < 3 × 10^{−13}), and potential relative frequency uncertainty (≈ 2 × 10^{−12}) when calibrated annually against e.g. a GPS disciplined oscillator. The efficiency of this device as an optical reference for the stabilization of a frequency comb has been demonstrated. When a fiber-based frequency comb is referenced to the stabilized laser, the comb tooth linewidth is reduced by more than two orders of magnitude as compared to the linewidth with a rubidium RF reference. Furthermore, the acetylene-stabilized fiber laser reference provides an enhancement of more than one order of magnitude of the frequency stability between 1 and 100 s and a potential relative frequency uncertainty at the level of 2 × 10^{−12}.

The acetylene-stabilized laser system used in this work has been advanced to the point of being made into a commercially available product known as the Stabiλaser 1542 [28].

## Funding

Danish Agency for Science, Technology and Innovation (6114-00011B); EU FP7 (31748, 607491); Innovation Fund Denmark (5150-00004); The Danish Council for Independent Research, Sapere Aude DFF-Advanced Grant (12-133348).

## Acknowledgments

The authors acknowledge the Danish Center for Laser Infrastructure, LASERLAB.DK, for support and access to the fs frequency comb.

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