As shown in Fig. 1, the Er:fiber femtosecond laser source has a ring cavity including four wave plates, a polarization beam splitter (PBS), two collimators, a WDM with isolator, an EOM, and fibers. A 2-cm PZT mounted on a stage is employed to control the cavity length. The cavity contains 39-cm SMF-28 fiber, 8-cm HI-1060 fiber, and 39-cm highly erbium-doped fiber (Liekki ER110-4/125). The net dispersion of the cavity is assumed to be slightly negative, considering all components and fibers in the cavity. The laser is pumped with two single-mode 980-nm pigtailed diode lasers, whose powers are both up to 0.6 W. By setting wave plates at proper angles, the laser can start mode-locking automatically while the pumping power is 1 W. Unfortunately, the mode-locked laser works in a multi-pulse state in this condition. In order to avoid this situation, we reduce the pump power to around 350 mW. The output power is ∼ 50 mW, and the duration of the pulse emitted from PBS is ∼ 83 fs with a 3-dB optical spectrum width of ∼ 50 nm.

Two feedback terminals, an EOM and a PZT, are employed to control f_r. The EOM is an 8-mm long lithium niobate crystal with ∼ 800 fs² dispersion at a wavelength of 1550 nm. To measure the response bandwidth of the EOM, we set up a Mach– Zehnder interferometer which contains the EOM in one arm. Then, we drive the EOM with an external radio-frequency (RF) source and monitor the phase response of the EOM at the output of the interferometer using a 2-GHz bandwidth photodetector (PD). The 3-dB bandwidth in magnitude is about 10 MHz, while the phase delay is close to π /2. The monotonic change of f_r caused by EOM is about 60 Hz while tuning the offset voltage of EOM from − 200 V to 200 V. The tuning range of the PZT (from 0 V to 200 V) is measured to be around 3 kHz with a response bandwidth of hundreds of Hertz.

To detect the repetition rate, we send a fraction of the output from the mode-locked laser into an InGaAs PD (EOT-3000A) with a bandwidth of 2 GHz. At the output of the PD, we obtain a current pulse train exhibiting m × f_r in frequency domain, where m is an integer. Usually, we do not stabilize f_r directly, but stabilize its high order harmonics, so as to reduce technical noise induced by the phase-locking process. However, no high frequency microwave synthesizer is available for the moment in our laboratory. Eventually, we lock the phase of the fourth harmonics of the repetition rate (838 MHz) to a reference signal.

The repetition rate is stabilized with a standard phase-locked loop technique. The phase detector is an RF mixer which produces an error signal indicating phase difference between 4f_r signal and the reference signal. Following the mixer, a home-made loop filter converts the error signal into a negative feedback signal for driving EOM with high bandwidth and PZT with long range. Although the EOM has a response bandwidth of about 10 MHz, the control bandwidth of EOM servo-loop is estimated to be 100 kHz– 200 kHz according to the response measurement results of the home-made loop-filter and high voltage amplifier. The PZT control-loop has a bandwidth lower than 100 Hz for compensating only the temperature drift effect.

In addition, f_r keeps being phase-locked while the box of the laser is hit, because EOM control-loop cancels the vibration influence efficiently. The peak-to-peak temperature fluctuation is about 1 ° C in the laboratory, and well below 1 ° C in the laser box, so that PZT control-loop can eliminate the thermal drift effect independently. Extra thermal control is not required. As a result, f_r of the laser can be continuously phase-locked onto the reference signal over a week.

We use f– 2f interference method to produce f_ceo signal; and lock f_ceo to an extra-reference (radio-frequency RF) signal in phase by controlling the ring laser’ s pump power.

In order to perform f– 2f interference, we first prechirp the seed pulses for reducing the pulse peak’ s power, aiming to reduce the nonlinear effect in amplification stage. Secondly, the prechirped pulses are delivered into an erbium-doped fiber amplifier with a selective optimum gain of ∼ 14 dB. Thirdly, laser pulse width is compressed by dechirping the amplified pulses with an SMF-28 fiber, thereby yielding ∼ 50-fs pulses as shown in Fig. 2(a). Fourthly, these narrow pulses are fed into a short highly nonlinear fiber (HNLF), resulting in an octave-spanning supercontinuum (SC) as shown in Fig. 2(b). This figure gives the optical spectrum of the SC spanning from 950 nm to 2250 nm for 25-dB bandwidth. Most of the energy shifts into two solitons with 1-μ m and 2-μ m wavelengths respectively. It is worth noting that these two solitons have exactly the right wavelength for f– 2f interference. After that, all optical powers go through a 1-mm thick PPLN crystal for frequency-doubling. Finally, we pick-up 1-μ m-wavelength light with a 10-nm-wavelength bandpass filter and obtain a beatnote signal of f– 2f interference with an InGaAs PD.

Fig. 2.

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	Fig. 2. (a) Measured intensity autocorrelation of the compressed pulse; (b) octave-spanning supercontinuum produced with HNLF.

Figure 3(a) shows the spectrum of the f– 2f interference with 3-MHz RBW, showing an SNR of f_ceo to be 40 dB. The linewidth of the free-running f_ceo beat is about 500 kHz observed in fine RBW. The SNR is about 20-dB higher than that of the previously reported Er:fiber domestic system.^[22] High SNR enables the system to generate a clean feedback signal for controlling the pump power and stabilizing f_ceo on a reference frequency provided with a synthesizer. A 20× frequency divider is used to improve the phase locking tolerance, making the phase-locked loop more robust. The control bandwidth of the loop is about 25 kHz indicated by the gain-bump as shown in Fig. 3(b). The accumulated noise power in a frequency range of 50 kHz– 10 Hz from the carrier is about 1%, indicating that the linewidth of frequency-divided f_ceo should be well below 1 Hz. More accurate evaluation will be shown later with frequency stability.

Fig. 3.

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	Fig. 3. (a) Spectrum of the f– 2f interference in 3-MHz RBW; (b) spectrum of the frequency-divided fceo signal in 910-Hz RBW.

The longest continuous phase-locking time of f_ceo is more than one day. However, some unknown noises coupled from electric ground (introduced by other events) disturb the phase-locked loop from time to time. We suppose that f_ceo can be stabilized much longer if the electric ground turns clean.

As shown in Fig. 1, down-converted frequency of 4f_r at 1 MHz is recorded with a frequency counter (Agilent 53230A). Figure 4(a) shows the frequency fluctuation of 4f_r, measured over a week. The result exhibits a root-mean-square (RMS) value of about 0.1 mHz. The frequency stability (Δ f_r/f_r) shown in Fig. 4(b) is about 1.6× 10^{− 13} in an integration time of 1 s. However, this result is limited by the noise floor as shown with solid squares, which is measured by using the same counter recording a 1-MHz reference signal. On the other hand, the frequency stability is inversely proportional to τ ^1/2, where τ is the integration time, because the dead-time decouples the data relationship in measurement system and converts the white phase noise into a white frequency noise.^[24] For a phase-locked loop, the stabilized signal should exhibit the white phase noise, and its frequency stability should be inversely proportional to τ . This is verified by eliminating dead-time effect with different gate times as shown with a dashed line in Fig. 4(b). Note that the first point of calculated Allan deviations has no dead-time effect. Therefore, the frequency stability of f_r is about 1.6× 10^{− 13} in an integration time of 1 s and rolls down with a slope of 1/τ for short-term stability.

Fig. 4.

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	Fig. 4. (a) Residual fluctuations of the 4th harmonic of the repetition rate. The gate time is 1 s and the dead-time between samplings is about 0.2 s; (b) in-loop frequency stability of 4fr, measured with different gate (solid red triangle for 1 s, empty round for 2 s, 4 s, 8 s, and 16 s) times and measurement noise floor (solid black square).

The RMS of f_ceo fluctuation is about 0.6 mHz in an integration time of 1 s, and the peak-to-peak fluctuation is about 10 mHz as shown in Fig. 5(a). Considering the fact that the optical frequency is about 200 THz, the corresponding optical comb frequency stability induced by in-loop f_ceo fluctuation is about 4× 10^{− 18}/τ ^1/2 as shown in Fig. 5(b).

Fig. 5.

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	Fig. 5. (a) Residual fluctuation of the stabilized in-loop CEO frequency; (b) optical comb frequency instability induced by in-loop fceo fluctuation.

It is clear that the frequency stability induced by in-loop f_ceo fluctuation is negligible even compared with the highest stable atomic clock with a stability of 3.2 × 10^{− 16}/τ ^1/2.^{[25, 26]}