Novel phase-locking schemes for the carrier envelope offset frequency of an optical frequency comb

We propose simple schemes to phase-lock the carrier envelope offset frequency (fceo) referring to the repetition rate (frep) of an optical frequency comb. We demonstrate the locking of fceo such that fceo = (1/2)frep, (1/3)frep, and (2/3)frep. The Allan deviation and signal-to-noise ratio of the coherent δ-function peak for the in-loop beat signal are 5.3 × 10−17/τ and 80–85 dB·Hz, respectively, where τ is the averaging time of the frequency measurement. These new locking schemes simplify the sign and mode-number determination in frequency measurements.

where ν n is the optical frequency of the n-th mode of the frequency comb. The repetition rate and carrier envelope offset frequency are presented as f rep and f ceo , respectively. To measure the optical frequencies using the frequency comb, it is generally important to stabilize the two parameters, f rep and f ceo , to a reference frequency standard. f rep can be easily monitored using a photodetector by measuring the pulse train of the comb and can be stabilized by the servo control of the effective optical length of the comb oscillator cavity. On the other hand, f ceo is usually observed by the f -2 f self-referencing technique. 2,9) Figure 1 schematically shows the observed f ceo signal on an RF spectrum analyzer. In a conventional stabilizing scheme [ Fig. 1(a)], one signal is selected using a bandpass filter from a pair of f ceo -related signals ( f ceo and f rep − f ceo ). The filtered signal is mixed with a frequency signal generated by a synthesizer phase-locked to the reference frequency standard. An error signal is generated to phase-lock f ceo and is fed back to the injection current of the pump laser for the comb oscillator.
In this paper, we propose simple and robust schemes to lock f ceo referring to the f rep of an optical frequency comb. With the new schemes, we can decrease the parameters of the frequency comb from two to one. In one sample scheme, we use the frequency range where two f ceo -related signals overlap around f rep =2 [see Fig. 1 We set our bandpass filter at f rep =2 and adjust it so that two signals would overlap inside the passband of the filter. We split the filtered signals into two parts by a simple power splitter and mix them again by a double-balanced mixer (DBM). From the intermediate frequency (IF) output of the DBM, we obtained a signal from the product of the two components of the beat note, mixer output Here, φ is an additional phase difference between the two signals at the DBM inputs induced by the length difference of the two cables between the splitter and the DBM in Fig. 1(b). We assume that both the phase differences to f ceo and f rep − f ceo are φ because the two frequencies are almost the same. The mixer output was appropriately time-averaged by a low-pass filter after the DBM to drop the fast varying terms, i.e., 3rd-5th terms in Eq. (2). Consequently, we showed that the time-averaged signal (〈⋯〉 t ) of the mixer output has the form where Φ is an arbitrary phase obtained when 2 f ceo approaches f rep . If φ = π=2, the amplitude of the time-averaged signal drops to zero. Since the time-averaged signal of Eq.
(3) has an offset, we need to subtract a stable bias to obtain a direction-sensitive error signal for controlling f ceo , as shown in Fig. 2. Points A and B reflect the lock points of f ceo with an opposite sign in the servo system. The error signal is fed back to the injection current of the pump laser for the comb oscillator. Then, f ceo is phase-locked to be just Here, we also propose a similar scheme for this new method for f ceo locking in such a way that f ceo = (1=3) f rep or (2=3) f rep . In this case, we compare the doubled frequency of the f ceo signal with the frequency of the ( f rep − f ceo ) signal using a DBM, as shown in Fig. 1(c). The output signal from the DBM is similar to the case we have discussed above, As described in the previous case, the time-averaged signal is calculated as where ΦA is an arbitrary phase obtained when 3 f ceo approaches f rep . In this case, we need not add a bias voltage in order to have a zero-cross error signal. When we set the phase ΦA to be π=2 or 3π=2, we can lock f ceo . The relationship between f ceo and f rep is f ceo = (1=3) f rep . If we change the definition of the f ceo -related beat signals in Fig. 1(c) [namely, ( f rep − f ceo ) and f ceo from the left-hand side to the right-hand side], the time-averaged signal becomes In this situation, we lock f ceo such that the relationship between f ceo and f rep is f ceo = (2=3) f rep . Generally, this scheme can be extended to the case of f ceo = (q=p) f rep , where p and q are integers. Here, we compare the ( p − q) times multiplied frequency of the f ceo signal with the q times multiplied frequency of the ( f rep − f ceo ) signal. Figure 3 shows a schematic of the experimental setup. We implemented a fiber-based frequency comb with three output branches. 10,11) The f rep of the comb was 44 MHz. Every branch was amplified and spectrally broadened by an erbiumdoped fiber amplifier and a highly nonlinear fiber. We used one of the branches to observe an f-2 f interference beat note for f ceo stabilization, and another to observe a beat note ( f beat ) between one of the comb components and a 1064 nm Nd:YAG reference laser. The third branch is to be used for any applications.
f beat was used for locking one vicinity mode of the comb to the reference laser. The error signal was fed back to an electrooptic modulator and a thermoelectric cooler in the comb oscillator with different time constants to control the effective length of the comb oscillator. The frequency of the reference laser was stabilized to a high-finesse Fabry-Perot cavity. 12,13) We note that the frequency of the reference laser varies at a rate of 80 mHz=s owing to the drift of the cavity.
The observed f ceo beat signal can be monitored by an RF spectrum analyzer, as conceptually shown in Fig. 1(b). The beat signal is filtered and amplified with an appropriate gain.
In this branch, f ceo and f rep were simultaneously measured by frequency counters. In the f ceo = (1=2) f rep locking case, we used a bandpass filter with a 3 dB passband of 17.9-25.3 MHz to filter the beat signals. The filtered signals were split  Fig. 1(b).   Fig. 3. Schematic of the experimental setup. LD: laser diode for pumping, EOM: electrooptic modulator, TEC: thermoelectric cooler, EDFA: erbium-doped fiber amplifier, HNLF: highly nonlinear fiber, and PD: photodetector. into two equal portions by a simple power splitter and immediately the signals from the two portions were combined by a DBM [see Fig. 1(b)]. Here, we used two identical 150-mm-long RF cables for connecting the splitter and DBM to set φ to zero. If the lengths of the two cables are different, the amplitude of the IF signal depends on the difference according to Eq. (3). The DBM used inverts the phase of the input signal, and the sign of the mixer output described in Eqs. (2) and (3) is also inverted. The obtained IF signal was biased by a proper DC voltage and was fed back to the injection current of the pump laser diode for the comb oscillator through a proportional-integral-derivative servo controller. It acts as a low-pass filter because its bandwidth is not so large that the fast varying term in Eq. (2) drops. To lock f ceo , we adjusted f ceo to be sufficiently close to ( f rep − f ceo ) ≃ (1=2) f rep by varying the bias current to the pump laser diode and then closing the feedback loop. Consequently, we could stabilize f ceo referring to f rep such that f ceo = (1=2) f rep . In this experiment, we need not use an extra RF synthesizer for f ceo locking.
When f ceo was locked using the new scheme, we measured the in-loop f ceo frequency, as shown in Fig. 4(a). To evaluate the locking performance of the scheme without the effect of the slow cavity drift, we calculated f ceo =f 1064 and converted it to the Allan deviation [see Fig. 4(b)], where f 1064 ≃ n × f rep was approximately considered the optical frequency of the Nd:YAG reference laser. This trace shows the locking performance more quantitatively than the trace in Fig. 4(a). The instability of 5.3 × 10 −17 =τ is much less than that of f rep and hence does not contribute to that of the frequency comb. The observed instability was limited by the signal-to-noise ratio of the f ceo signal measured by the frequency counter. If we use a tracking oscillator for determining f ceo , the instability of f ceo may be further reduced. Figure 5 shows the RF spectrum of the in-loop f ceo signal. The servo bandwidth was more than 300 kHz, which was estimated on the basis of the bump of the in-loop f ceo spectrum. We obtained a coherent δ-function peak with a signal-to-noise ratio of 80-85 dB·Hz. The results show that the new locking scheme induces very few additional phase noises to the comb.
In the f ceo = (1=3) f rep locking case, it was difficult to arrange the beat measurement as shown in Fig. 1(c) because of the difficulty in finding appropriate bandpass filters due to the relatively small f rep of 44 MHz in this study. Instead, we compared the doubled frequency of the ( f rep − f ceo ) signal with the frequency of the ( f rep + f ceo ) signal, as shown in Fig. 6. In this arrangement, we used twice of f rep and arranged the beat measurements using appropriate bandpass filters experimentally. We actually locked f ceo such that f ceo = (1=3) f rep and verified the locking performance. Similar Allan deviation and phase noise characteristics were observed as in the case of f ceo = (1=2) f rep .
In the case of f ceo = (1=2) f rep locking, we realized a "halfinteger comb". In this comb system, the f ceo -related signals, f ceo and ( f rep − f ceo ), are identical. The frequency of the n-th mode is expressed as The ( f rep − f ceo ) signal is also recognized as a − f ceo signal as we can see when we substitute ( f rep − f ceo ) into Eq. (1) and shift the integer from n to n + 1. Therefore, in the halfinteger comb, we need not care about the sign of f ceo . Conventionally, in order to lock f ceo , we need to check and determine the sign of the beat frequency. For simplicity of these procedures, we would like to reduce these sign   ambiguities in each comb. If we apply this new method also to f beat stabilization, the frequency of the reference laser can be expressed as Hence, all the comb modes can be used as an optical frequency ruler referring to the reference laser without any microwave synthesizers. The new locking schemes in this study provide a type of f ceo -free optical frequency comb, [14][15][16][17] which simplifies the sign and mode-number determination in frequency measurements.
In the conventional f ceo stabilizing scheme [ Fig. 1(a)], we can use a frequency divider to widen the capture range of phase locking when the servo bandwidth of f ceo is not sufficiently large. In the case of the half-integer comb, since we use the square of the sum of the f ceo -related signals, f ceo and ( f rep − f ceo ), as described in Eq. (2), each signal cannot be individually frequency-divided. Therefore, a large servo bandwidth of f ceo locking is necessary for phase locking. On the other hand, when f ceo = (q=p) f rep , each f ceo -related signal can be individually frequency-divided. Thus, we can phaselock f ceo without a large servo bandwidth by paying the penalty of having a larger residual phase noise than in the case of not using a divider.
In conclusion, we have proposed simple locking schemes for f ceo referring to f rep and achieved good locking performance characteristics with our fiber-based optical frequency comb. The performance of the locking schemes has been evaluated by measuring the Allan deviation and phase noise characteristics. With these schemes, we have realized a halfinteger comb with f ceo = (1=2) f rep and also combs with f ceo = (1=3) f rep or f ceo = (2=3) f rep . Besides this, we have also proposed similar locking schemes that ensure the relationship f ceo = (q=p) f rep .