1. Introduction

The advent of the self-referenced optical frequency comb [1,2] almost half a decade ago marks a breakthrough in the measurement of optical frequencies. The pulse train emitted by a mode-locked femtosecond (fs) laser can be seen as a comb of equally spaced modes in the frequency domain [3]. This frequency comb is fully determined by two frequencies f_rep and f_CEO. The pulse repetition rate f_rep corresponds to the spacing of the comb lines and the carrier-envelope-offset (CEO) frequency f_CEO to the comb’s offset from zero [4, 5]. The CEO frequency is due to the phase slippage of the electrical field versus pulse envelope per cavity round trip. As both f_rep and f_CEO are in the radio frequency regime, they may be detected and counted with established radio frequency techniques. Optical frequencies are measured by observing a beat frequency f_x with the nearest mode of the frequency comb. Therefore the optical frequency ν_opt is expressed as ν_opt=mf_rep+f_CEO+f_x.

Up to now, most frequency comb systems are based on Ti:Sapphire femtosecond lasers. Recently, systems based on mode-locked Er-doped fiber lasers attracted a lot of attention [6–8] due to their ease of use and the capability to be operated continuously over weeks. Thus they are excellent candidates for routinely facilitated long-term metrology and permanently running optical clocks. Other advantages of using frequency combs based on Er-doped fiber lasers are cost-effectiveness and a compact setup, as well as their wavelength around 1.55µm. Matching the telecommunication bands, a time standard as realized by an optical clock may be broadcasted via existing terrestrial fiber transmission networks.

Here we present to our knowledge the first direct comparison between two fiber based frequency combs. The same optical frequency has been measured with two independent fiber frequency combs over more than 10hours and their results are compared. This frequency is derived from a cw fiber laser at 1543nm (194THz) stabilized to an optical reference at 657 nm.

 

Fig. 1. Experimental setup. rf-signals are denoted by blue lines and the letter f. Optical signals are denoted by red lines and the letter ν. Not shown in this figure: Frequency counters are referenced onto the rf-reference. ECDL: external cavity diode laser, SMF: single-mode fiber.

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2. Setup

The fiber frequency comb operated at Physikalisch-Technische Bundesanstalt (PTB) was developed in a collaboration between PTB and the University of Konstanz and is described in detail in [9]. It is laid out on an optical breadboard of approximately one square meter. The second system was developed in a collaboration between the Max-Planck-Institute for Quantum Optics (MPQ) and Menlo Systems GmbH. This setup is enclosed in an easily transportable box with a 40×65cm² footprint. The frequency comb systems are referred to as “PTB fiber comb” and “MPQ fiber comb” respectively.

Both fiber frequency comb systems used in this experiment were similarly built: The frequency comb generators are erbium-doped, polarization-mode-locked, femtosecond fiber-ring lasers. The repetition rate f_rep of both systems is about 100MHz. The signal from the laser oscillator is amplified in an erbium-doped fiber amplifier and spectrally broadened in a highly nonlinear fiber (HNLF) to cover a spectrum of one octave in frequency space. f_CEO then is detected with the f-2f interferometer setup [4, 5].

The power of the PTB fiber comb oscillator is split into two branches and amplified in independent amplifiers followed by HNLFs. One branch serves to measure the CEO frequency. The other branch is optimized to generate a comb in the visible by frequency doubling, thus reaching the frequency of an extended cavity diode laser (ECDL) at 657nm. This ECDL is stabilised to an optical reference cavity and serves as an optical reference in our experiment. As a suitable source for a comparison of the two comb systems a cw fiber laser at 1543nm (194THz) has been selected. As the stability of the free running cw fiber laser is not sufficient for the experiment, the cw fiber laser is stabilised on the ECDL. Using the transfer technique described in [10], the stability of the ECDL is transferred to the cw fiber laser without introducing additional noise from the frequency comb. The PTB comb merely serves to cover the gap in wavelength between the cw fiber laser at 1543nm and the ECDL at 657nm, but is otherwise free to move against these lasers. Alternatively, the cw fiber laser may be referenced to a microwave source.

Figure 1 shows an overview of the experimental setup. Both fiber combs are locked to rf-reference signals derived from a hydrogen maser. While the repetition rate of the MPQ fiber comb is locked at the 10^th harmonic of f_rep, the 114^th harmonic is used for the PTB fiber comb. Furthermore the repetition rate of the PTB fiber comb is counted at 128×114(f_rep-100 MHz) with a harmonic tracker that generates the 128^th harmonic of 114(f_rep-100 MHz). The beat signals between the cw fiber laser and the frequency comb (f_NIR) at 1543nm and between the ECDL and the frequency comb (f_opt.ref) at 657nm are processed together with f_CEO to derive a rf “transfer beat” signal $f_{\mathit{tb}}=\frac{1}{2}\left(v_{\mathit{fiberlaser}}-v_{\mathit{ECDL}}(m_{\mathit{fiberlaser}}⁄m_{\mathit{ECDL}})\right)$ where m_fiberlaser and m_ECDL are the mode numbers of the comb modes that beat with the ECDL and the cw fiber laser respectively. The ratio (m_fiberlaser/m_ECDL) is processed by a direct digital synthesizer (DDS).

A fraction of the output signal of the cw fiber laser is transmitted via 10m of single-mode fiber into the neighboring laboratory where the beat signal with the MPQ fiber comb is generated. The MPQ comb has its repetition rate locked to exactly 100MHz at the 10^th harmonic. A stepper motor is used to coarsly adjust the repetition rate and compensate temperature drifts which proved unnecessary in the state of the art climatised PTB laboratory. The offset frequency is locked to 20MHz derived from the 10MHz radio frequency reference with a synthesizer. As both offset and repetition rate are very reliably locked and the present counter setup has only four channels available, we skipped the cycle slip detection technique usually employed for precision measurements [2]. The frequency counters used for this measurement count continuously without dead time. The four channels are read out synchronously every second. Since the frequencies of the MPQ comb are tightly phase-locked to the microwave reference only the beat signal of the MPQ comb with the cw fiber laser is counted. The remaining three channels are used to count f_rep, f_CEO and f_NIR derived from the PTB comb. Because of the lack of a dead time, the measured data can be divided into samples with integration times that are integer multiples of 1s but arbitrary otherwise. This is the prerequisite to be able to specify the Allan standard deviation and the standard error of the mean for integration times other than 1s later in this article. The continuous locking time for both systems exceeds several days of operation.

 

Fig. 2. Upper plot: difference between the 1s frequency measurements of the cw fiber laser performed by the PTB and the MPQ fiber comb. Lower plot: the two independent frequency measurements.

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3. Results

Both fiber frequency comb systems as well as the remaining phase locks worked very reliably. The continuous measurement time of 39467s was only limited by the need of time sharing the ECDL with the calcium-group the next morning.

The lower plot in Fig. 2 shows the two independent frequency measurements of the cw fiber laser frequency with 1s integration time. The ordinate shows the absolute optical frequency minus 194.329 THz. The blue plot is the frequency measured by the PTB fiber comb, the red plot is the one measured by the MPQ fiber comb which shows slightly more noise. A small drift of the measured frequencies which changes sign after some 20000s can be seen and is attributed to the optical reference cavity. The brief, larger frequency deviations in the lower plot are caused by transient excursions of the cw fiber laser. The upper plot shows the difference between the two measurements. The drift and the larger frequency deviations cancel out in real time as they are detected by both systems. The mean of the frequency difference is 5mHz with a standard error of 0.25Hz (1 s integration time, 39467 samples). Figure 3 shows the frequency difference with the data combined in 1000 s samples. The standard error of the mean decreases to 23 mHz in this case (39 samples). A linear fit gives a drift of 5±2µHz/s of the two frequency combs against each other. The maximum deviations of the linear fit from zero are +94mHz and -103mHz. Because of the drift, the standard error of the mean of 23mHz is no longer a suitable measure for the uncertainty of the measurement. Instead, the disagreement between the two combs is given by the maximum deviation of 103mHz of the linear fit from zero, which corresponds to a relative accuracy of 6×10^-16. To explain the drift, we consider the following: The experiment was carried out at PTB in a state of the art climatized laboratory with temperature deviations of less than 200 mK. If the rate of temperature change was 0.2°K/h, we estimate the single-mode fiber that transmits the cw laser signal between the adjacent laboratories to cause a constant frequency deviation of 12 mHz which can consequently not explain the observed drift. One other candidate that might cause such a drift is a clean-up quartz oscillator used to separate the 100MHz reference signals in both laboratories. A small temperature change in one of the labs can change the locking point of the quartz due to a baseline drift of the phase detector. Since f_rep as well as f_CEO of the MPQ comb are not counted, this phase change would not be compensated. At the present state of our knowledge this effect would adequately explain the observed drift. Figure 4 shows the Allan standard deviation derived from the data shown in Fig. 2. The Allan deviation of the independent frequency measurements does not drop below 10^-14. The rise for integration times larger than 1000s is caused by the drift of the reference cavity. The Allan deviation of the difference between the two frequency measurements starts at 3×10^-13 for 1s integration time and drops as 1/T as is expected for phase-locked signals. It deviates from 1/T for integration times larger than 512s and drops down to 2×10^-16 for integration times of 8192s. This value perfectly agrees with the drift of 5µHz/s which would add up to 41 mHz (2.1×10^-16) in 8192 s.

 

Fig. 3. Frequency difference of 1000s samples with linear fit.

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Fig. 4. Allan standard deviation of the independent measurements (PTB, MPQ) and the difference of the independent measurements (MPQ-PTB).

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4. Summary

We measured the same optical frequency of 194 THz with two fiber frequency combs referenced to the same rf-reference over a period of nearly 4×10⁴ s. The average frequencies measured agreed within 6×10^-16. Because the agreement was limited by a drift in the measurement setup and not by any fundamental limitation of the frequency comb technique, future measurements could prove that fiber frequency combs are able to transfer the projected accuracy of 10^-18 of optical atomic clocks from the optical into the radio frequency domain. The trouble-free long term operation demonstrated here makes them promising candidates for such clockworks. This work was supported by DFG through SFB 407.