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We demonstrate that fiber-based frequency combs with multi-branch configurations can transfer both linewidth and frequency stability to another wavelength at the millihertz level. An intra-cavity electro-optic modulator is employed to obtain a broad servo bandwidth for repetition rate control. We investigate the relative linewidths between two combs using a stable continuous-wave laser as a common reference to stabilize the repetition rate frequencies in both combs. The achieved energy concentration to the carrier of the out-of-loop beat between the two combs was 99% and 30% at a bandwidth of 1 kHz and 7.6 mHz, respectively. The frequency instability of the comb was 3.7×10−16 for a 1 s averaging time, improving to 5-8×10−19 for 10000 s. We show that the frequency noise in the out-of-loop beat originates mainly from phase noise in branched optical fibers.
©2010 Optical Society of America
1. Introduction
Optical clocks have been developed competitively worldwide since the emergence of widespan femtosecond laser combs to relate an optical frequency directly to the existing cesium primary standard [1, 2]. An ultra-stable laser with a narrow linewidth and high frequency stability are essential to the operation of optical clocks, and the heart of a stable laser system is a high finesse optical cavity made from two highly reflective mirrors in optical contact with a spacer that has an axial bore hole [3–5]. To achieve a high finesse cavity (typically more than 100,000), highly reflective mirrors must be specially designed for an appropriate clock laser wavelength with the result that each atomic species requires its own cavity. In general, highly reflective coatings for visible wavelengths are more difficult to achieve than those for infrared wavelengths, and therefore a high finesse cavity at an infrared wavelength is preferable to that at a visible wavelength, if it is possible to use it. Since these cavities have a relatively large free spectral range of several GHz, and the resonant frequency of the cavity cannot be scanned, frequency shifters such as acoustic-optical modulators have to be employed to bridge the frequency between the clock transition and the cavity resonance. An attractive way to overcome these difficulties would be to transfer the linewidth and frequency stability to another frequency using optical frequency combs, and the development of such systems has been the subject of many recent comb studies.
Octave-spanning combs with narrow relative linewidths have been reported [6–9]. In these studies, Ti:sapphire lasers [6, 8, 9], erbium-doped fiber lasers [7], and ytterbium-doped fiber lasers [8] were used as master oscillators for the narrow linewidth comb. High-speed control of the group index and the cavity length are required if we are to reduce the comb mode linewidth, because phase locking without frequency division is efficient. Fast control (> 200 kHz) of the group index can be achieved by changing the current injected into the pump laser diode (LD) in the mode-locked fiber laser [10]. On the other hand, to change the cavity length, conventional combs [6–9] have employed a piezoelectric transducer (PZT) with the result that the servo bandwidth is only several tens of kHz. Therefore, these combs require frequency division to lock the mode of a comb to a continuous-wave (CW) laser. Meanwhile, the high-speed control of the effective cavity length in a fiber laser has been achieved by employing an intra-cavity electro-optical modulator (EOM) [11, 12]. As a result, the servo bandwidth of these lasers is reported to range from several hundred kHz to a few MHz.
In recent years, erbium-doped fiber based frequency combs (fiber combs) have become widely used owing to their robustness [13], cost effectiveness, and user friendliness [14]. Another important advantage of fiber combs is their flexible construction with a multi-branch configuration [15]. In each branch, the amplifier, polarization, and highly nonlinear fibers (HNLFs) [16] can be optimized for the proper purpose [17, 18]. Consequently, the multi-branch configuration enables us to obtain a beat signal between the comb and a CW laser with a high signal to noise ratio (S/N), and to use the comb for plural applications. The relative timing jitter between the two parallel pulse trains of a two-branch fiber laser was investigated [19]. However, the degree to which the multi-branch configuration affects the linewidth and stability of the octave-spanning comb has yet to be studied.
In this study, to consider the above issues, we employ a mode-locked erbium-doped fiber laser with an EOM as a comb source. The laser output is launched into three parallel amplifiers to detect the carrier envelope offset (CEO) beat [14, 20] and heterodyne beat notes between the comb and the CW lasers. We achieve a narrow relative-linewidth fiber comb with an EOM without frequency division for phase locking. Furthermore, we show that amplified spontaneous emission (ASE) from erbium-doped fiber amplifiers (EDFAs) and nonlinear optical effects in HNLFs are not dominant sources of phase noise in the fiber combs at the millihertz level.
2. Experimental setup for narrow-linewidth combs
In this experiment, we use two similar octave-spanning fiber combs (comb #1 and comb #2), and the system is shown schematically in Fig. 1 . The oscillator in each system is an erbium-doped fiber based mode-locked laser with an intra-cavity EOM. An erbium fiber oscillator is a ring resonator that employs nonlinear polarization rotation as the mode-locking mechanism [21]. The oscillators are pumped by a 1480 nm laser diode via a wavelength division multiplexing (WDM) coupler. The erbium-doped fibers in the oscillators are 150 cm (comb #1) and 50 cm long (comb #2), respectively. The peak absorptions of the erbium-doped fibers at 1530 nm are approximately 40 dB/m (comb #1) and 150 dB/m (comb #2), respectively. The repetition rates are approximately 44.3 MHz (comb #1) and 83.5 MHz (comb #2), respectively. The total output power of each oscillator is approximately 7 mW (comb #1) and 13 mW (comb #2), respectively. In each system, 30% of the circulating power is coupled out of the cavity and distributed equally to three parallel amplifiers. The optical chirp of each optical pulse is optimally managed in each EDFA to broaden the spectrum with each HNLF[22]. The EDFAs are pumped by 1480 nm laser diodes from their output side at a power of 200-300 mW. The output power from each amplifier is 50-90 mW. A 15-20 cm non-polarization-maintaining HNLF is spliced to the output of each amplifier. In each system, these three branches are used to detect a CEO beat, a heterodyne beat note with a reference laser (578 nm), and a beat note with a 1064 nm CW laser (slave laser), respectively. The CEO frequency of each comb is individually detected using f –2f interferometers. The full width at half maximum of the CEO beat was 10-30 kHz under a free-run condition. The CEO beat was stabilized by controlling the pump power for the oscillator. On the other hand, an EOM (a 4 cm long, 2 mm thick piece of LiNbO3) is inserted into the free-space section of the cavity to change the effective cavity length quickly [11, 12]. The servo bandwidths of all the repetition rates and CEO frequencies were estimated to be more than 200 kHz from the frequency at the servo bumps, and all the phase locking was achieved. Figure 2 shows the experimental setup we used to evaluate relative linewidths and frequency stability of the fiber combs. A 578 nm CW laser (reference laser) stabilized to a high-finesse, ultra-low-expansion (ULE) cavity is used as a common frequency reference. Each comb is stabilized to the reference laser, and this is accomplished by phase locking the heterodyne beat note between one of the comb modes and the CW laser to a microwave reference. A second CW laser operating at 1064 nm (slave laser) is phase locked to comb #2 and a fraction of its power is used for beat measurement with comb #1. The heterodyne beat note between the 1064 nm laser and comb #1 is then recorded with an FFT analyzer to obtain a linewidth measurement and a frequency counter to determine the relative stability of the two combs. Figure 3(a) shows a phase-locked CEO beat note of comb #1, and Fig. 3(b) shows a phase-locked heterodyne beat note between comb #1 and the 578 nm reference laser. We succeeded in phase locking all of the beats (“Servo” in Fig. 2) without frequency division because of their sufficiently broad servo bandwidth. Consequently, we were able to obtain very clean in-loop beat spectra. These linewidths were limited by the resolution of the spectrum/FFT analyzer, which shows that all the phase-lock-loops work properly.
Fig. 1 Setup for a comb with an intra-cavity EOM and multi-branch configuration. Thick solid lines and curves represent optical fiber; thin solid lines represent beams in space, and broken lines represent electric wire. PD, photo detector; ISO, optical isolator; PBS, polarization beam splitter; Q, quarter wave plate; H, half wave plate; TEC, thermo-electric cooler; BPF, optical bandpass filter.
Fig. 2 Experimental setup used to observe an out-of-loop beat between two combs referring a common CW laser. The CEO frequency in each comb is narrowed by fast phase locking. A narrow linewidth CW laser is used for repetition rate locking (578 nm) and a stabilized CW slave laser used for out-of-loop phase characterization (1064 nm). The slave laser is phase-locked to the 578 nm laser using a self-referenced fiber comb.
Fig. 3 (a) Phase-locked CEO beat (in-loop beat) note of comb #1. Inset: High-resolution spectrum of the beat note. (b) Phase-locked heterodyne beat (in-loop beat) note between comb #1 and the 578 nm reference laser. We were able to obtain very clean in-loop beat spectra. These linewidths were 30 mHz, which was the resolution limit of the FFT analyzer. RBW, resolution bandwidth; VBW, video bandwidth.
3. Method for evaluating relative linewidth of multi-branch frequency combs
The frequency of the n-th mode of a multi-branch frequency comb is given by
where n is an integer, and f r and f 0 are the repetition rate and the CEO frequency, respectively. The term ε n, k represents the frequency noise of the n-th mode that is generated in the k-th branch. In this paper, the comb imperfection described in [9] is involved in ε.For example, we can assume that there is amplitude-to-phase noise conversion from the optical amplification and highly nonlinear process, and frequency noise from the fibers (fiber noise) in each branch can be the source of ε. To reduce the linewidths of all the comb modes, we need to suppress the phase noise of f r and f 0. The noise of f 0 can be directly suppressed by phase locking to a reference frequency. However, it is not easy to obtain a sub-hertz linewidth in the optical frequency region by directly suppressing the noise of f r because such a low-noise microwave oscillator is not currently available, and shot noise in the photo detection process for f r interferes with the noise suppression [23, 24]. Therefore, phase locking a heterodyne beat note between an ultra-stable laser with a narrow linewidth and a comb is a practical method for reducing the linewidths in the comb [6–9].
We have developed an Yb optical lattice clock [25], and a 578 nm ultrastable laser [26] is now available as a reference laser for our combs. Our fiber combs broaden in the near infrared region. Therefore, the combs have to be frequency-doubled to the visible region to detect a beat note with the 578 nm laser. However, in this section, we assume for simplicity that the wavelength of the reference laser is 1156 nm (578 nm×2). For further simplification, we also assume that the noise is zero in the branches used to detect CEO beats, and consider that other branches have relative frequency noise against the noise generated in the branches used to detect the CEO beats (See Fig. 2). Figure 4 is a schematic diagram of our experimental setup in the frequency domain. The beat frequency between the reference laser and the nearest mode in comb #1, f 1156, 1, is given by
where ν m, 1 and ν 1156 are the frequencies of the nearest mode (the m-th mode in this paper) in comb #1, and the reference laser, respectively. f r, 1 and f 0, 1 are the repetition rate and the CEO frequency in comb #1, respectively. ε m, 1 is the frequency noise of the m-th mode in branch #1, which is used to detect the beat note between comb #1 and the 1156 nm reference laser. From Eqs. (1) and (2), the frequency of the n-th mode in comb #1 (ν n, 1) can be described aswhere ε n, 2 is the frequency noise of the n-th mode in branch #2 used to detect the beat note between comb #1 and the 1064 nm reference laser. α = n/m and is approximately 1.1 in this case. It generally has a value between 0.5 and 2 when an octave-spanning comb is employed. From Eq. (3), we understand that if the noise term ε is sufficiently small, the linewidths of the n-th comb component (ν n, 1) should be reduced to the linewidth of the reference laser by narrowing f 0, 1 and f 1156, 1, which means that all comb components are reduced by narrowing f 0, 1 and f 1156, 1.Fig. 4 Schematic of the experimental setup in the frequency domain. Each comb is stabilized to a reference laser by phase locking f 0, 1, f 0, 2, f 1156, 1, and f 1156, 2 to microwave references. The slave laser is stabilized to comb #1 by phase locking f 1064, 1. f 1064, 2 is “out-of-loop”. Each comb component at 1156 nm is frequency-doubled to detect a beat note with the 578 nm reference laser.
Now, we phase lock the q-th mode in comb #2 to the reference frequency ν 1156. As with the frequency in comb #1, the frequency of the p-th mode in comb #2 is given by
where β = p/q. Next, we observe two heterodyne beat notes between the two combs and another CW laser (the frequency is ν 1064 in this paper). f 1064, 1 is the beat note between the laser and the nearest mode (n-th mode) in comb #1, and f 1064, 2 is the beat note between the laser and the nearest mode (p-th mode) in comb #2. They have the following relationship.If ν n, 1, f 1064, 2 and ν p, 2 are phase locked to reference frequencies, f 1064, 1 represents the “out-of-loop beat” between the two combs, indicating the relative linewidth and the frequency instability of the combs. From Eqs. (3), (4), and (5), the out-of-loop beat frequency f out of loop is
For simplification, we assume that none of the fluctuations are correlated, and <(δε n, 2)2> = <(δε q, 4)2> = <(δε m, 1)2> = <(δε p, 3)2> = <ε2>. ε corresponds to the frequency noise in each branched fiber. The frequency fluctuation of the out-of-loop beat is given by
The frequency fluctuations and the linewidths of f 0, 1, f 0, 2, f 1156, 1, f 1156, 2 and f 1064, 2 are directly observed as the in-loop beat signals. The term (α - β)· ν 1156 originates in the fact that the frequency in each comb is synthesized with a different mixture ratio of optical and microwave frequencies. Although (α − β) is usually less than 10−7, the frequency of the out-of-loop beat is affected by the frequency fluctuation of the reference laser. In this experiment, this fluctuation is negligible because we use a sufficiently stable laser [26] as the reference laser. Therefore, we are able to evaluate the frequency noise of the combs <ε 2> by observing the linewidth and frequency stability of the out-of-loop beat.
4. Relative linewidth and frequency stability of fiber combs
Figure 5 shows the details of a heterodyne beat note between the slave laser stabilized to comb #2 and the nearest mode in comb #1, which corresponds to the out-of-loop beat of two independent combs. Figure 5(a) shows that we achieved successful phase locking without frequency division. The linewidth of the out-of-loop beat was 7.6 mHz, as shown in Fig. 5(b), which is the resolution limit of the FFT analyzer. The out-of-loop beats contain approximately 99% and 30% of the RF power within the coherent carrier at a bandwidth of 1 kHz and 7.6 mHz, respectively. Although some jitter and drift components remain, these components contain approximately 90% of the RF power within 1 Hz. We do not believe that the noise degrades the quality of ultrastable lasers. In addition, we succeeded in observing the out-of-loop beat continuously for more than 16 hours as shown in Fig. 5(c). In this experiment, the phase locking of the slave laser and the Pound-Drever-Hall locking of the reference laser to a ULE cavity are occasionally broken. The locking of the combs themselves can continue for longer periods of time. Therefore, we consider that the long-term operation of the systems for a period of several days is feasible. The frequency stability of one comb calculated from the frequencies counted with dead-time-free counters was 3.7×10−16 with an averaging time of 1 s as shown in Fig. 5(d), and the stability improved to 5 - 8×10−19 after 10000 s. This relative stability is similar to the result reported in [27], and can be improved by using the fiber comb’s robustness to obtain a longer measurement period.
Fig. 5 Evaluation of the out-of-loop beat. (a) The RF spectrum of an out-of-loop beat observed with a spectrum analyzer. The spectrum is clean and its energy concentration to the coherent carrier is more than 99% at a bandwidth of 1 kHz. (b) The RF spectrum of an out-of-loop beat observed with a high-resolution FFT analyzer. The linewidth is less than the resolution limit of the FFT analyzer (7.6 mHz). The energy concentration to the coherent carrier is approximately 30% at a bandwidth of 7.6 mHz. (c) Recorded frequency of the out-of-loop beat for a measurement period of more than 16 hours. The averaging time per point is 0.1 s. (d) Allan deviations of the recorded frequency of the out-of-loop beat and all in-loop beats.
5. Effects of multi branch in comb systems
The relative linewidth and the frequency stability shown in Fig. 5(b) and (d), respectively, contain little residual instability. We believe that the instability originates in the multi-branch configuration of our combs. In other words, the frequency noise <ε 2> in Eq. (7) is essentially generated by the multi-branch configuration of our comb. Since plural optical fiber lines including EDFAs and HNLFs are independent, the branched configuration adds phase noise such as fiber noise caused by both acoustic noise and temperature variations [28] and/or the amplitude-to-phase noise conversion from the amplification and nonlinear optical process in the EDFAs and/or HNLFs. The linewidths of a multi-branch fiber comb would not be reduced to the millihertz level if the additional phase noise were large. We carried out the following experiments to determine the dominant noise source.
- 1. We evaluated the frequency instability of the out-of-loop beat with the fibers covered with paper after branching in each comb, and with the fibers uncovered. The EDFAs and the HNLFs were covered in both cases. (This was to determine the effect of fiber noise on frequency stability.)
- 2. We evaluated the frequency instability of the out-of-loop beat while increasing the injection current of the pump laser for the EDFA in the fiber comb within the range in which the beat note can be detected properly. (This was to determine the effect of ASE from an EDFA on frequency instability.)
- 3. We used one comb synchronized to the reference laser to evaluate the frequency instability of the beat note between the reference laser and an identical comb in another branch. (This was to determine whether the residual instability of the out-of-loop beat originated in the fiber noise in each branch rather than in the comb imperfection. In other words, this was to determine whether or not ε n, k in Eq. (1) depends more on the mode number n than on the branch number k.)
Figure 6 shows the results of the above experiments. The result of experiment 1 shows that covering the fibers has a strong effect on the frequency stability, and experiment 2 shows that the ASE has no obvious effect on the stability. In addition, the result of experiment 3 shows that the frequency noise of the out-of-loop beat does not originate in the wavelength conversion process in the comb.
Fig. 6 Relative frequency stability (Allan deviation) of the combs. The black circles represent the best stability of the out-of-loop beat of the combs. The red circles represent the stability when the fiber covers in the comb systems are partly removed, which represents the effect of fiber noise. The blue circles show the frequency stability when the pump power for the EDFA of the branch used to detect the CEO beat in comb #1 is decreased from 290 to 180 mW, which represents the effect of the ASE from an EDFA. These (black, red, and blue circles) are translated to the stability of one comb (divided by square root of 2). The green circles indicate the stability of the beat note between the reference laser and an identical comb in another branch, which shows whether or not the wavelength conversion in each comb is the dominant noise source.
As a result, we conclude that the dominant noise source of our combs is fiber noise after branching. The previous work [29] reports fiber-optic links degrade the optical frequency stability. The 1-s instability of the comparison was approximately an order of magnitude less than that reported in this paper. We expect that the difference is originated in the multi-branch configuration of our combs. This noise can be reduced by carefully covering fibers, and the linewidth and stability are already sufficient for current optical clocks. In addition, branching is a key technique for obtaining high S/N signals between combs and CW lasers. Therefore, we consider the branched comb to be suitable for application to ultra-stable lasers for optical clocks provided sufficient care and testing is taken.
6. Discussion and conclusion
The relative linewidth and frequency stability of the combs reveal how finely each comb can transfer the performance of a laser to other wavelengths. These results show that the combs described above can transfer a linewidth at the millihertz level, and a frequency stability of 3.7×10−16 at an averaging time of 1 s. This is better than the performance of most ultrastable lasers. An evaluation of the relative linewidth and the frequency stability of the branched comb provides information about the phase noise source in the fiber combs. For example, when the CEO beat is narrowed in a branch, the CEO beat would not be narrowed in another branch if the EDFAs and HNLFs in these branches add a large phase noise. On this basis, we consider that the dominant source of the phase noise is the oscillators from the fact that we can obtain a narrow out-of-loop beat at the millihertz level. In other words, EDFAs and HNLFs do not add significant phase noise at this level as regards the linewidth and stability, although strong nonlinear processes such as adiabatic narrowing [30, 31] occur in each branch when we manage the dispersion of the EDFAs and the HNLFs [22]. As previously discussed using computer simulations, the experimental evaluation and verification of phase noise generation in EDFAs and HNLFs are possible by using a branched comb [32, 33].
In conclusion, we have developed practical, high-speed-controllable and multi-branched fiber combs by employing erbium-doped fiber-based oscillators and intra-cavity EOMs. High-speed phase locking enables us to stabilize the repetition rate and CEO frequency without any frequency division. As a result, we were able to obtain the highest reported energy concentration (99% at a bandwidth of 1 kHz) and a sufficiently narrow relative linewidth (7.6 mHz). In addition, the relative frequency stability was 3.7×10−16 at a 1 s averaging time and 5-8×10−19 at a 10000 s averaging time. The dominant noise source was considered to be fiber noise after the branching, but the noise is less than that of currently available ultra-stable lasers. In addition, we have succeeded in observing an out-of-loop beat continuously for more than 16 hours.
Fiber combs were found to have relatively large phase noise in early research [34]. But they are no longer noisy and we can obtain high reliability and user friendliness with high levels of performance. We believe such practical and high-performance combs will have an impact on the field of frequency metrology. For example, we can imagine the realization of such functions as the regular evaluation of ultra-stable lasers, and the narrowing of lasers at plural wavelengths to obtain magneto-optical traps and/or clock lasers.
Acknowledgements
We thank M. Onishi, T. Sasaki, T. Okuno and M. Hirano of Sumitomo Electronics Inc. for providing us with excellent highly nonlinear fibers. Part of this work was supported by the “Grant for Industrial Technology Research (financial support to young researchers)” program of the New Energy and Industrial Technology Development Organization.
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