All-optically stabilized frequency comb

We present an all-optically stabilized, erbium-doped mode-locked fiber laser with an optically pumped ytterbium-doped fiber. The mode-locked fiber laser has two frequency-control actuators that are pump laser powers for erbium-doped and ytterbium-doped fibers. We investigate the frequency-control characteristics of the mode-locked laser and find that the fixed points for the two actuators are sufficiently apart from each other, realizing the simultaneous phase locking of the repetition and carrier envelope offset frequencies. We describe a long-term frequency measurement of an acetylene-stabilized laser at 1542 nm using an all-optically stabilized frequency comb.

M ode-locked fiber-laser-based optical frequency combs (fiber combs) have already been practically utilized for frequency metrology. The number of fields employing the fiber comb is increasing and now includes length measurement, 1) astronomy, 2) and dual-comb spectroscopy. 3) Comb-mode frequencies originate from a comb source, which is a mode-locked laser that has two degrees of freedom for frequency controls. For example, comb-mode frequencies are determined by repetition rate ( f rep ) and carrier envelope offset frequency ( f ceo ). Therefore, two independent actuators are needed to control the comb mode frequencies. Specifically, fast control using the two actuators is necessary to achieve comb modes with narrow linewidths.
One of the two actuators is the pump power into the gain medium, which is suitable for stabilizing either f rep or f ceo , and fast enough since it has a servo bandwidth of several hundred kHz. On the other hand, a piezoelectric transducer (PZT), 4) an electro-optic phase modulator (EOM), 5) or an acousto-optic modulator (AOM) 6,7) has been used as the second actuator. The PZT and=or EOM change the laser cavity length, mainly controlling f rep . EOMs are particularly powerful actuators, because they have a fast servo bandwidth beyond 1 MHz. Although AOMs have a faster servo bandwidth than PZTs, they are only able to change f ceo . In addition, f ceo beat detection using an optical pulse train diffracted by an AOM has not been reported for fiber combs.
Recently, an erbium-doped mode-locked fiber laser with an optically pumped ytterbium-doped fiber as the second optical control for changing f ceo and=or f rep has been proposed by Hellwig et al. 8) They pointed out the possibility of locking f ceo and f rep simultaneously by using all-optical controls. The all-optically controlled configuration may be realized as a robust and cost-effective system while retaining easy self-mode locking, and a broad servo bandwidth. In their work, the f ceo behavior was observed when the pump power into the erbium-doped fiber (EDF) in the cavity of their mode-locked fiber laser was modulated by 1 Hz, while f rep was phase-locked by controlling the pump power into the ytterbium-doped fiber (YDF). As a result, f ceo modulation synchronized with the pump power modulation to the EDF was indirectly observed, which indicates the possibility that f ceo and f rep can be phase-locked simultaneously by employing all-optical controls. Although this is of great interest, the simultaneous locking of f rep and f ceo is not selfevident because 1 Hz modulation is rather low and the available control bandwidth may be very slow for phase locking.
In this study, we first investigated the ratios of f ceo deviation to f rep deviation when changing the power to each fiber to find appropriate pump power regions in which to phaselock f rep and f ceo simultaneously. As a result, we found that the fixed points for the two actuators are sufficiently apart from each other and succeeded in phase-locking f rep and f ceo simultaneously. Furthermore, we measured the frequencies of a 1542 nm acetylene-stabilized laser by using the all-optically controlled comb synchronized with universal coordinated time (UTC). Figure 1 shows an all-optically controlled mode-locked fiber laser. The cavity design is based on that reported in Ref. 8. An EDF-based oscillator is a ring resonator that employs nonlinear polarization rotation as the mode-locking mechanism. [9][10][11] This oscillator is pumped by a 1480 nm laser diode with an optical isolator via a wavelength division multiplexing (WDM) coupler. The EDF in the oscillator is 60 cm long and commercially available (OFS, EDF 80). An YDF is inserted into the ring resonator, and pumped by a 976 nm laser diode with an external fiber Bragg grating for stabilizing the wavelength via another WDM coupler. There is neither an optical isolator nor a 980=1030 WDM coupler between the pump laser and the YDF for simplicity in this study. The YDF in the oscillator is approximately 11 cm long and commercially available (CorActive YB 125). The repetition rate is about 60 MHz. An output coupler consumes 30% of the intracavity power, and the output power in the single-pulse regime is approximately 3.3-3.6 mW at an EDF pump power of 132-152 mW. The spectral width is approximately 55 nm. We consider that the mode-locked laser operates in the stretched-pulse 11) regime from the spectral shape and roughly estimated total cavity dispersion. The mode-locked laser behaves stably, and the mode locking starts without any initiating triggers.
In this study, we changed the pump power into the YDF from 27 to 95 mW since we could observe the f ceo signal in this power range. We observed a significant change in spectral shape when we increased the pump power into the YDF. The spectral shape changed from the typical shape of a stretched pulse to the distinct shape of a similariton, 12) which suggests that the pump power changes the cavity dispersion. Figure 1 shows the experimental setup that we used to observe f rep , f ceo , and the beat note between the comb and a 1542 nm acetylene-stabilized laser using the above-mentioned mode-locked laser. We used 1% of the laser output to monitor the laser spectrum, and we divided the other 99% of the output into two equal parts. One of the two parts was further divided into two branches. The first branch was used to detect the f ceo signal by employing an f-2 f interferometer, and the second branch was used to detect a beat note with the 1542 nm laser. The power to the EDF was approximately 670 µW and it was optimally amplified to 90 mW at the first branch. 13) The amplified pulse train was spectrally broadened with a highly nonlinear fiber (HNLF), 14) and guided to the f-2 f common-path interferometer. As a result, we obtained a signal-to-noise ratio of 35 dB for the f ceo signal at a 300 kHz resolution bandwidth. In this experiment, we found no negative effect on the f ceo detection in terms of obtainable S=N when we inserted the YDF into the oscillator.
The spectrum of the mode-locked laser contained sufficiently strong 1542 nm components, and we obtained a signal-to-noise ratio of 40 dB at the resolution bandwidth (RBW) of 300 kHz for a heterodyne beat note between the laser and the comb without the HNLF.
We evaluated the independence of the two frequency control actuators of the frequency comb to determine whether it is possible to stabilize two frequency degrees of freedom simultaneously.
The N-th mode frequency in the comb is well known as When we adjust the frequency-control actuator, there is a fixed point 15) that results from canceling the changes in f rep and f ceo . At the fixed point, Eq. (1) is We obtain the following equation by differentiating both components of Eq.
(2) by f rep : Therefore, −df ceo =df rep corresponds to the mode number at the fixed point. Here, we consider the two fixed points N Er fix and N Yb fix when we change the pump powers to the EDF and YDF, respectively. To characterize the independence of the two frequency control actuators, we consider the changing ratio N Yb fix =N Er fix , which corresponds to the ratio of fixed-point frequencies. The changing ratio should not be close to 1 if we are to control both frequencies independently.
First, to measure the N Yb fix =N Er fix ratio, we investigate a tunable pump-power range for which we can observe the f ceo signal by increasing the pump power to the EDF and YDF individually. We then choose 6 individual pump-power points as typical powers within the tunable power ranges to the EDF and YDF. Next, we observe f rep visually with a frequency counter and f ceo with a spectrum analyzer while increasing the pump power to the EDF from minimum to maximum (6 points) with the pump power to the YDF fixed (measurement #1). We also measure f rep and f ceo while increasing the pump power to the YDF with the pump power to the EDF fixed (measurement #2). Figure 2 shows the f ceo deviation versus the f rep deviation when each of the pump powers to the EDF and YDF is changed. To reduce the longterm frequency-drift effect during the observation, we use data of measurements #1 and #2 for the plots in Figs. 2(a) and 2(b), respectively. N Er fix and N Yb fix at different pump powers are calculated from Figs. 2(a) and 2(b), respectively. Furthermore, fixed points in the optical frequency range can be derived from N Er fix or N Yb fix using Eqs. (2)-(4). Figures 3(a) and 3(b) show maps of the fixed points in the optical frequency range considered, and Fig. 3(c) indicates the ratio of the fixed points for the two actuators, calculated from the experimental results shown in Fig. 2. The ratio of the fixed points does not indicate a strong dependence on the pump powers to the EDF and YDF. The calculated ratios are always sufficiently smaller than that obtained when we adjust the pump powers to the EDF and YDF in this experiment. We consider the entire pump power region to be appropriate for the simultaneous phase locking of two degrees of freedom of the optical frequency comb.
We utilized UTC (NMIJ), a frequency standard based on a hydrogen maser at NMIJ, as a frequency reference for the all-optically controlled frequency comb. We used a universal signal generator (Keysight 33622A) to synthesize the reference frequencies for f rep and f ceo . We employed a frequency mixer and an 8-bit digital phase comparator 16) as the phase discriminators to phase-lock f rep and f ceo , respectively. The 8-bit phase comparator can discriminate a phase difference of up to ±128π, which facilitates appropriate phase locking without cycle slips. As a result of trials at every pump power level, we succeeded in phase-locking f rep and f ceo simultaneously.
We have measured the optical frequencies of an acetylenestabilized laser 17) using our all-optically stabilized frequency comb. Figure 4 shows measurement results for f rep and f ceo and the beat note between the comb and the stabilized laser. The Allan deviation of f rep improves with the inverse of the averaging time, which is the resolution limit of the frequency counter we used. The deviation of f ceo also improves with the inverse of the averaging time, which is limited by residual phase noise, indicating its appropriate phase locking. We consider that the Allan deviation of the beat frequency between the comb and the acetylenestabilized laser is limited by the frequency stability of the stabilized laser in the averaging time range above 10 s since the stabilities are adequate for the stabilized laser, and the stability improves with the inverse of the square root of the averaging time. In the regime below 10 s, the relative frequency stability of the function generator (Keysight 33622A) used as the reference for f rep might constitute a frequency stability bottleneck. The averaging frequency calculated from Eq.  19) These results show that the controllability of the all-optically stabilized comb does not limit the measurement. In addition, we phase-locked the comb for more than 3 d, although we measured the frequencies for only approximately 10000 s, which indicates that the observed robustness is promising.
We achieved an all-optically stabilized frequency comb. Table I shows the frequency control characteristics of the frequency control mechanisms. We do not consider the above-mentioned mechanisms to have any serious problems for long-term durability, severe environments, or mechanical stability as actuators. Nevertheless, a silica-based YDF is notably superior to other actuators in terms of robustness. In particular, the wide tunability of f ceo using the new actuator (pump power to the YDF) is significant. In this experiment, we tuned f ceo more than two times than f rep at least. Further optimizations (e.g., total dispersion of the oscillator cavity, pump current setting, and polarization setting) will enable us to tune f ceo over an even greater range. The f rep tunability is sufficiently wide, although long-term f rep stabilization requires temperature control. The bandwidth of the new actuator is of great interest, and is yet unclear because this experimental setup is not specialized for fast servo. We expect the servo bandwidth to be several hundred kHz since the f ceo phase locking of an ytterbium-doped fiber-based comb is easily achieved with a similar servo bandwidth. 20) The relationship between the fixed point N Yb fix and the YDF length should be investigated as the next step, although it is outside the scope of this study. The fixed point may be able to be managed by changing the YDF length.  An all-optically stabilized frequency comb is realized. It is cost-effective and robust, and has wide f ceo tunability. It is an evolutionary configuration of the mode-locked fiber laser for an optical frequency comb and might become a mainstream comb if a narrow-linewidth comb can be realized by alloptical stabilization in the near future.  a) When using a fiber-attached PZT, the fiber is mechanically stretched and becomes slow; when using a free-space PZT, independent PZTs are needed to obtain a broad bandwidth and a passable tunability simultaneously. b) A fast high-voltage amplifier is needed to obtain a broad bandwidth and a passable tunability simultaneously.