1. Introduction

Radio frequency (RF) counter based on the modern electronic devices enabling frequency counting up to about 60 GHz have been widely commercialized. A wavelength meter that utilizing interference technique by comparing with a reference light source is also well implemented for optical frequency (or wavelength) measurement. However, the measurement uncertainty is approximately 10–100 MHz. A commercial optical frequency counter has been developed with a minimum resolution 100 kHz, based on an optical frequency comb generated by modulating a continuous-wave of laser light with an electro-optical modulator inside a Fabry-Perot cavity [1]. However, the measurable wavelength range is only from 1530–1565 nm. Mode-locked (ML) femtosecond lasers, which can serve as ultraboradband optical frequency combs, have been employed in optical frequency metrology over the past few years [2]. The frequency of a laser under measurement (LUM) when adopting an ML laser comb to measure the beat frequency f_b can be expressed as f_L=nf_r±f_o±f_b, where n indicates the mode number (ordinal number) of the beating comb line; f_r denotes the repetition rate of the pulse train, and f_o represents the carrier-envelope offset (CEO) frequency. To realize an optical frequency counter based on an ML laser, both the signs of the CEO frequency and the beat frequency and the value of n need to be determined. The sign of the CEO frequency and beat frequency can be determined uniquely by observing the corresponding beat frequency variations while altering the repetition frequency or the CEO frequency of the laser comb [3, 4].

Ma et al. developed a scheme for obtaining the mode number without applying a wavelength meter [5]. They changed the repetition rate smoothly, while counting the total number of shifts in mode. The mode number was then obtained by measuring the beat frequencies at different repetition rates. For low repetition rate and noisy LUM the shift in mode number can be larger than 100, which means that counting the total mode number shifted is not practical. Peng et. al. observed that the mode number shifted is linear to the difference in repetition rate for repetition rate change of some tens of kHz [3]. Therefore, they only needed to measure the repetition rate change required to shift one comb mode while slowly scanning the repetition rate, and did not need to count the total mode number shifted. Based on this technique, a semi-automatic optical frequency counter using two fiber laser combs has been demonstrated [6]. However, the process for the determination of one comb mode shifted is still complex. Zhang et al. determined the mode number of a laser comb with a comb spacing of 200 MHz in two steps [7]. First, an estimated mode number with certain accuracy is obtained by operating the laser comb at two different repetition rates with the difference being small enough to maintain the same mode number. The laser comb is then operated at larger different repetition rate. The mode number change is then calculated from the estimated mode number, and the exact mode number is derived from the mode number difference. No mode number change is required in the first step only if the frequency instability of the LUM is less than approximately 10 kHz. Otherwise, the method processed by scanning the repetition rate, and counting the mode number change by monitoring the RF spectrum analyzer, which is still not practical for an LUM with instability larger than 10 kHz.

This work proposes an optical frequency counter based on two fiber laser combs operated at different repetition rates. Monochromators are applied to offer an approximate frequency of the LUM, thus yielding an estimated mode number. The mode number difference of the beating comb line between the two laser combs operated at different repetition rate is first obtained from the measured beat frequencies and the estimated mode number. The exact mode number is then derived from the mode number difference, the difference in repetition frequency and in offset frequency, and the measured beat frequencies. The determination of the mode number using this two-comb technique is independent of the frequency fluctuation of the LUM. Both laser combs with octave-spanning spectrum can measure frequency spans of up to an octave. The process for measuring the mode number and the frequency of the LUM, except the frequency stabilization of the laser combs and the optimization of the beat signal-to-noise ratio (S/N), are controlled by a computer to demonstrate an optical frequency counter.

2. Theory

The LUM is beaten with the two laser combs operated at different repetition rates, f_r1 and f_r2, with the CEO frequencies of f_o1 and f_o2 and corresponding beat frequencies of f_b1 and f_b2. The mode numbers of the beating comb lines were n and n+m for f_r1 and f_r2, respectively. For simplicity, the sign of the CEO frequency of the two laser combs and the beat frequencies is assumed to be determined to be positive by observing the beat frequency variation while slightly changing the repetition rate and the CEO frequency [4]. Thus, the frequency of the LUM is written as:

The mode number difference m and the comb mode number n can then be derived by Eqs. (1) and (2) as

In Ref. 3, m is obtained by further measuring the value of f_r1/n, which equals the repetition rate change expected to shift one comb mode, while smoothly varying the repetition frequency. This work utilizes a monochromator to measure the approximate wavelength of the LUM, and thus obtain an estimated value of n (denoted as n_est), i.e. n_est=f_L/f_r=(c/λ)(1/f_r), where the refractive index of air is assumed to be 1. The relative uncertainty of the refraction index in air contributed wavelength uncertainty is <10^-5, which is far smal ler than that affected by the monochromator measurement of δλ/λ>1×10^-3. Therefore, the refractive-index-induced uncertainty can be neglected for the wavelength measurement. This n_est is substituted into Eq. (3) to obtain the mode number difference m. The exact n is then calculated from Eq. (4). If the two beating frequencies f_b1 and f_b2 are measured simultaneously, and two laser combs are phase locked to the same reference, then the frequency fluctuation of the LUM can be canceled from the f_b2-f_b1 in Eqs. (3) or (4). Therefore, the determination of the mode number using this two-comb technique is independent of the frequency fluctuation of the LUM.

The criteria of the required wavelength measurement uncertainty and the difference of the repetition rate are determined by the requirement that the uncertainty of the measured m and n should be much less than 1. The uncertainty of m and n can be derived from Eqs. (3) and (4), and be expressed as

, where δ() denotes the uncertainty. The uncertainty of f_o2-f_o1 is ignored, since f_o2 and f_o1 are generally phase-locked to a highly stable RF standard. The uncertainty of mf_r2 in Eq. (6) is also disregarded, because f_r2 is phase-locked to the RF standard, and m≪n (m≈30, n≈10⁶ in this experiment). The uncertainty of the estimated comb number n_est is computed mainly from the monochromator, and can be written as

where c is the velocity of light; λ is the wavelength of the LUM, and δλ indicates the uncertainty of the measured wavelength λ. Since the frequency fluctuation of the LUM can be subtracted for the two-comb technique, the uncertainty of the beat frequency difference f_b2-f_b1 depends only on the relative instability between the comb lines, i.e. δ(f_b2-f_b1)=δ((n+m)f_r2-nf_r1)≅nδ(f_r2-f_r1), where the uncertainty of the CEO frequency, and that from the microwave counter, are neglected. Because f_r2 and f_r1 are phase-locked to the same source, they are correlated as δ(f_r1-f_r2)≅2σ(τ)f_r, where σ(τ) represents the tracking instability of the repetition rate f_r at an integration time of τ, and f_r=f_r1 or f_r2. Therefore,

For a repetition rate of 100 MHz with a tracking instability of 2×10^-13 @ 1s and n≈3×10⁶ for the wavelength of the LUM of ~1 µm employed in this experiment, f_r1-f_r2≫120 Hz is necessary to satisfy δn≪1. In this experiment, f_r1-f_r2 was set to be around 1 kHz, which can be easily achieved by tuning a PZT mounted on the laser cavity. The term (2nσ(τ))² in Eq. (8) can be ignored, since it is quite small (~10^-12), and (δλ/λ²)≪3.3×10⁴ (where λ is in meters) is stipulated to satisfy δm≪1 for sufficient accuracy to determine m. If the wavelength of the LUM is ~1 µm, then the uncertainty of the measured wavelength needs to be much smaller than 33 nm. This can be easily achieved by a general monochromator.

Once the comb number difference m has been accurately obtained, the accurate value of n is then calculated from Eq. (4), and the frequency of the LUM f_L1 or f_L2 can be computed from Eqs. (1) or (2).

3. Experimental setup and results

Figure 1 schematically illustrates the construction of a semi-automatic optical frequency counter using two ML Erbium-doped fiber lasers. The two ML Er:fiber lasers were home-made ring lasers based on polarization additive pulse mode-locking (P-APM) [8]. Detailed construction of the two fiber laser combs has been described elsewhere [3]. Each laser comb has two branches of octave-spanning supercontinuum ranging from 1050 nm to 2100 nm. One branch was utilized to detect the CEO frequency, and the other was applied to create a beat signal with the LUM. Both the repetition rate and the offset frequency were stabilized to synthesizers with the time base referenced to a 10 MHz low-noise oven-controlled quartz oscillator, which was phase-locked to a global positioning system receiver disciplined Rb clock. The 10 MHz reference signal had an instability value of less than 2×10^-12 for an integration time of over 1s, and a relative uncertainty of 2×10^-12. The stabilized repetition frequency had an out-of-loop tracking instability of 2×10^-13 @ 1s, and the fluctuation of the CEO frequency was of the order of mHz. The frequency stabilization of the repetition rate and the CEO frequency are described in detail elsewhere [9].

The LUM was an iodine-stabilized Nd:YAG laser at wavelength near 1064nm, which was coupled into a single mode fiber and split into two beams by a 3 dB coupler. The second harmonic generation of the Nd:YAG laser was locked to the a ₁₀ component of the R(56)32-0 transition in the iodine molecule. The two beams were then combined with two fiber laser combs with corresponding 3 dB couplers. Polarization controllers were employed to ensure that the polarization of both fiber laser combs matched that of the Nd:YAG laser. The beat signals were detected using InGaAs photodiodes after they had been filtered through corresponded monochromators. The monochromator was calibrated by a Neon lamp with a wavelength uncertainty below 5nm, which is sufficiently accurate for estimating the comb number difference m as described in the theory section. The detected beat signals had a signal-to-noise ratio of 30 dB in a 100 kHz resolution bandwidth. Two frequency counters were triggered externally to measure the beat frequencies simultaneously, and the gate time was set to 1s for all measurements.

The frequency of the LUM was obtained through the following processes. Except that the frequency stabilization of the laser combs and the optimization of the beat signal S/N were manually operated, which were indicated in the following steps 1, 2, and the later part of step 4, all the other measurement processes were controlled by a computer.

 

Fig. 1. Schematic diagram of the semi-automatic optical frequency counter using two mode-locked Er:fiber combs. Each laser has two branches of octave-spanning supercontinuum. One branch is for the frequency stabilization of the repetition rate and the CEO frequency (not shown), and the other branch is for beating with the Nd:YAG laser.

Download Full Size | PDF

All the calculations are fulfilled by the computer and the measured frequencies are shown on the display after the calculation. Since the whole measurement is not yet fully computer controlled, we claim that it is only a semi-automatic optical frequency counter.

 

Figs. 2. (a) and (b) The measured beat frequencies for f_r1=100 MHz and f_r2=99.9989 MHz.

Download Full Size | PDF

 

Fig. 3. The mode number n determined by Eq. (4).

Download Full Size | PDF

The wavelength of the LUM measured by the monochromator was 1066.4nm±5.3nm, which corresponded to an estimated comb number n_est=2811621±13876. The signs of the beat frequencies were found to be negative according to the aforementioned methods. The signs of the offset frequency of 140 MHz in the calculation of the frequency of the LUM were also found to be negative for both laser combs. Figures 2(a) and 2(b) show the measured beat frequencies f_b1 and f_b2 for f_r1 and f_r2. The difference between f_b1 and f_b2 was 2.032491 MHz±106 Hz. The fluctuations of the beat frequency difference are smaller than that of the individual beat frequency, revealing that the frequency variations of the Nd:YAG laser were subtracted in the difference of the beat frequencies. The calculated value of m using Eq. (3) was 30.95±0.15, and clearly m=31. The uncertainty mainly comes from the estimated n_est, which has a relative uncertainty of 0.005, and is accurate enough to calculate m. The mode number n was computed as depicted in Fig. 3. Clearly, n=2816303. The frequencies of the LUM measured by the two combs were equal to 281630091745.8(0.9) kHz and 281630091746.0(0.9) kHz, which agrees with the previously measured value within the uncertainty [3].

4. Discussion

Zhang et al. [7] used one laser comb operating at two similar repetition rates to estimate the mode number n_est first, and then shifted to a larger repetition rate to determine the accurate comb number n. However, they indicated that the mode number can be kept the same only for an LUM with instability no larger than 10 kHz for a laser comb with 200 MHz comb spacing. Otherwise, increasing the accuracy of the estimated mode number requires scanning the repetition rate and counting the changes in mode number. Zhang et al. describe in detail the one-comb technique using three different repetition rates [7]. The proposed monochromator-based detection technique can be also applied with a one-fiber laser comb operating successively at two different repetition frequencies. However, since the beat frequencies cannot be measured simultaneously, the frequency fluctuation of the LUM cannot be canceled. In the one-comb case, δ(f_b2-f_b1)=√2δν, where δν indicates the frequency instability of the LUM. The uncertainty of m still follows the expression of Eq. (8), because the last term of Eq. (8) is significantly less than 1. However, the uncertainty of n is then modified as

For the criteria of δm≪1 and δn≪1, the repetition rate change of f_r1-f_r2 should thus obey the following criteria:

For f_r=100 MHz and λ=1 µm, the left hand side of Eq. (11) can be written as 3.3×10^-5/δλ. If δλ=0.1 nm, then the criterion 330 kHz≫f_r1-f_r2≫1.4δν is required. The slit of the monochromator needs to be small enough to reach δλ=0.1 nm. In such case, the beat signal-to-noise ratio maybe not large enough for frequency counting due to low light level. Furthermore, the measurable frequency instability of the LUM is limited to some tens of kHz. Therefore, the two-comb technique is more adequate to realize an optical frequency counter and is immune to the frequency instability of the LUM.

5. Conclusions

An optical frequency counter with octave-spanning counting capability using two erbium-doped fiber laser combs is demonstrated as semi-automatically independent of the fluctuation of the LUM. Monochromators are utilized to offer an approximate frequency of the LUM to obtain an estimated mode number, and the mode number difference between the two laser combs was first simply calculated from the two measured beat frequencies and the estimated mode number without scanning the repetition rates of the laser combs. The exact mode number of the beating comb line is then obtained from the mode number difference, the difference in repetition frequency and in offset frequency, and the measured beat frequencies. The frequency measurement of an Iodine-stabilized Nd:YAG laser was demonstrated entirely with the proposed program-controlled, semi-automatic optical frequency counter, although the frequency stabilization of the laser combs, and the optimization of the beat signal S/N, were performed manually. The uncertainty of the measured frequency of the Iodine-stabilized Nd:YAG laser is ~1 kHz. The frequency stabilization and the optimization of the beat signal S/N can be fulfilled automatically in principle. We believe a fully automatic optical frequency counter can be implemented using program-controlled electronics for the frequency stabilization of the laser combs, and automatic polarization controllers for the optimization of beating signals.