Optical frequency comb comparison between optical clock mode and optical frequency synthesizer mode Eok Bong Kim 1, Chang Yong Park, Dai-Hyuk Yu, Sang Eon Park, and Won-Kyu Lee* Korea Research Institute of Standards and Science, 1 Doryong-Dong, Yuseong-Gu, Daejeon 305-340, Korea *
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Abstract. We report on an accuracy limit test result of the optical frequency comb technique by comparing the frequencies of two optical frequency combs that operate in two different modes. These two modes are distinguished by the reference frequency for the comb frequency stabilization; an optical frequency (in an optical clock mode) or a microwave frequency (in an optical frequency synthesizer mode). The comparison of the two combs was carried out by measuring an absolute frequency of an acetylene-stabilized diode laser with both combs simultaneously. The frequencies of the two combs agreed with each other within the uncertainty of 16 Hz at 194 THz. The uncertainty is reduced by more than an order compared with a previous report.
Subject terms: optical frequency comb; optical clock; optical frequency synthesizer; frequency comb comparison. 1
Current Address: Department of Physics and Coherent X-ray Research Center, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea 1
1
Introduction
Thanks to the unprecedented accuracy and the ability to measure the absolute optical frequency in one step, optical frequency combs (OFCs) have been widely used in the past decade. OFCs nowadays play a key role in a variety of applications such as absolute frequency measurements [1], optical atomic clocks [2,3], direct comb spectroscopy [4,5], and determination of fundamental physical constants [6]. As these applications demand very high accuracy levels, there have been continuing efforts to investigate the accuracy limits of the OFC technique by comparisons of OFC frequencies [7-15]. The OFC frequency comparison results up to now can be divided into two groups according to the frequency reference, that is used for the frequency stabilization: a comparison between OFCs referenced to a microwave frequency (which we will call an optical frequency synthesizer mode; an OFS mode [16]) and that between OFCs referenced to an optical frequency (which we will call an optical clock mode; an OC mode [17]) The OFC comparison in an OFS mode is performed by measuring the frequency of a common optical radiation by two OFCs referenced to a common microwave frequency such as that of a hydrogen maser (H-maser), a cesium atomic clock, and a quartz oscillator [16]. R. Holzwarth et. al. [7] compared two Ti:sapphire-based OFCs referenced to a microwave signal provided by a quartz oscillator with an uncertainty of 5.1×10-16 (1σ). J. Ye et. al. [8] measured the frequency of an Nd:YAG laser quasi-simultaneously at JILA and at NIST by using an optical fiber network connecting these two institutes. The two measurements agreed in 0.74 Hz with a standard deviation of 9.1 Hz (3.2×10-14) at 10-s gate time. The frequency reference for the JILA OFC was provided by a fiber-delivered H-maser signal. L.-S. Ma et. al. [9] made the first international comparison of OFCs using a H-maser as the frequency reference. The agreement 2
among the three OFCs participated in that comparison was found to be on the subhertz level with an uncertainty of 0.69 Hz (1.2×10-15, 1σ) at 563 THz. P. Kubina et. al. [10] compared two modelocked erbium-doped fiber OFCs referenced to an H-maser. The two measurements agreed within 6×10-16 (1σ). Summarizing these results, the OFC comparisons in an OFS mode, in which a typical stability of a reference frequency is of an order of 10-13 at 1 s, reach the uncertainty level of 10-14 ~ 10-15 in averaging time of 1000 s ~ 10000 s. The OFC comparison in an OC mode is performed by measuring the frequency of a common optical radiation by two OFCs referenced to an optical frequency in an optical clock configuration [17]. In this case, the frequency stability of the reference provided by a CW laser at optical frequency (locked to a supercavity) can be as good as 10-15 at 1 s. Referencing to this optical standard provides improved stability allowing shorter averaging times, leading to lower uncertainty of OFC frequency comparison. S. A Diddams et. al. [11] compared two octavespanning Ti:sapphire-based OFCs that have the same repetition rate and are phase-locked to a low-frequency-noise diode laser. They demonstrated the intrinsic fractional frequency noise of an OFC is ≤ 6.3×10-16 with 1 s of averaging time. L.-S. Ma et. al. [12-14] compared optically referenced Ti:sapphire laser OFCs at the relative frequency uncertainty level as low as 8×10-20 (95% confidence level determined from a χ2 analysis). To reach this accuracy level, they averaged the data over 76000 s, enclosed the beam path, and arranged the optical path of two OFCs to have better common path rejection, because the mechanical and thermal fluctuations along the light path begin to play a role at this level of accuracy. Although these comparisons have verified the consistency of the OFC technique either in the OFS mode or in the OC mode, there has seldom been a report that compares the frequencies of the OFCs in these two modes explicitly. For all that this kind of comparison is required for a
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more complete proof of the consistency of the OFC technique, only one result [15] has been reported with the frequency comparison uncertainty of 0.24 kHz (1σ). The uncertainty in Ref. [15] was limited by the repeatability of the stabilized optical frequency source, of which frequency was measured by an OFC, alternatively stabilized either in an OFS mode or in an OC mode. In this paper, we report on the frequency comparison results between the comb in the OFS mode and that in the OC mode with a remarkably smaller measurement uncertainty than the previous result. This improvement was possible by the simultaneous measurement of the common optical frequency source by the two OFCs, which was stabilized by the OFS mode and the OC mode, respectively. In addition, the two OFCs used in this comparison are based on different mode-locking mechanisms; one is a Ti:sapphire femtosecond laser, and the other is a fiber femtosecond laser, which also provides the consistency check between the OFCs with quite different structure and mode-locking mechanism.
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Optical frequency synthesizer mode and optical clock mode
The definition and the basic difference of the OC mode and the OFS mode were explained briefly in our introduction. This will be treated in detail in this section. The absolute frequency of the optical radiation (f) measured by an OFC is given by [16]
f = N frep ± fceo ± fb
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(1)
where N is the longitudinal mode number, frep is the repetition rate of the OFC, fceo is the carrierenvelope-offset frequency of the OFC, and fb is the beat frequency between the OFC and the optical radiation. In the OFS mode, we stabilize frep and fceo by using a microwave frequency reference, and measure fb by a frequency counter to obtain the absolute frequency f. When we speak of the phase noise of the OFS, the frep is critical because it is multiplied by N, which is typically 105 ~ 106. If we phase-locked frep by using a local oscillator, the phase noise spectral density would be multiplied by N
2
in optical frequency range [15]. On the other hand, in the OC mode, the
frequency of the OFC is stabilized to an optical frequency standard by phase locking fb and fceo using a microwave synthesizer, and we measure frep by a high resolution frequency counter to obtain the absolute frequency f. Because the phase-lock of the OFC was done in optical frequency range in this case, there is no multiplication of the phase noise density like in the case of the OFS mode. The expectation that the two OFCs respectively in these two modes will yield exactly the same frequency measurement result is not self-evident, which is the motivation of our research. This is because these two schemes have quite different characteristics, especially in the phase noise properties. The uncertainty limit for the OFC comparison basically depends on the stability of the reference frequency for the OFC stabilization, the frequency stability of the optical source measured with the OFC, and the total data acquisition time. Thus, the ultimate uncertainly limits are greatly different by orders according as two OFCs to be compared are in the OC mode or in the OFS mode. As the total data acquisition time increases, the statistical uncertainty (the standard deviation of the estimated mean of the frequency difference) can be reduced. However, there is a practical limit for this due to the possibility of phase slips or unstable counting. In this
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work, we compare the comb in the OC mode and that in the OFS mode. The uncertainty of the OFC comparison in this case is limited by the frequency measurement in the OFS mode, which is referenced to a microwave frequency. Although an ultra-precise comparison as in the OC mode is intrinsically impossible in this case, we expect the uncertainty level of 10-14 would be possible.
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Experimental setup
In order to perform the OFC comparison, we used two kinds of OFCs as shown in Fig. 1. One OFC (OFC-OC) is based on a Kerr-lens mode-locked femtosecond Ti:sapphire laser in the OC mode. The spectrum of the OFC-OC is centered at 800 nm with the spectral width (the full width at the half maximum) of 57 nm and with the repetition rate frequency (frep,OC) of about 200 MHz. The beat signal (fb, OC) between one component of the OFC-OC and the frequency-doubled radiation of an acetylene-stabilized laser (LD/13C2H2) at 1542 nm was phase-locked using a frequency synthesizer referenced to an H-maser, which is linked to the SI second via a global positioning system. The relative frequency stability of this H-maser and the acetylene-stabilized laser was about 2 × 10 −13 and 3 × 10 −12 at 1 s in terms of Allan deviation [18], respectively. To
obtain the second harmonic generation signal at 771 nm, we amplified the output of the acetylene-stabilized laser diode by an erbium-doped fiber amplifier (EDFA) and a periodically poled lithium niobate (PPLN) waveguide was used for the frequency doubling. The frequency of this laser diode was locked to the P(16) transition line of 13C2H2, which is one of the reference transition recommended by the International Committee of Weights and Measures (CIPM) [19]. Using the f-2f interferometer setup [7], the carrier-envelope-offset frequency (fceo,
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OC)
was
stabilized to 20 MHz using another frequency synthesizer, which was referenced to the same Hmaser. The octave spectral broadening of the OFC-OC output was obtained using a photonic crystal fiber for the f-2f interferometer. After measuring the OFC-OC repetition rate (frep,OC) by a dead-time-free Π-estimator frequency counter [20], with a gate time of 1 s, that is referenced to the same H-maser, we can obtain the absolute frequency of the acetylene-stabilized laser in the OC mode, fOC, which is given by
2fOC = NOC frep,OC ± fceo, OC ± fb, OC
(2)
where the comb mode number NOC can be determined uniquely by using the previously known frequency value of the acetylene-stabilized laser [18]. The factor of 2 in equation (2) is due to the second harmonic generation. The signal-to-noise ratios (SNRs) of the beat signals of frep,OC, fceo, OC,
and fb, OC were above 30 dB in a resolution bandwidth (RBW) of 300 kHz, which is sufficient
for correct frequency counting and stable phase-locking. The other OFC is based on a mode-locked erbium-doped fiber laser (OFC-OFS) in the OFS mode. The fiber laser spectrum is centered at 1550 nm and the repetition rate (frep, OFS) is about 250 MHz. The output of the fiber femtosecond laser was amplified by an EDFA and was spectrally broadened to the wavelength range from 1050 nm to 2100 nm using a highly nonlinear fiber in order to measure and stabilize the carrier-envelope-offset frequency (fceo, OFS). The frep, OFS and the fceo, OFS of the OFC-OFS were phase-locked to 250 MHz and 20 MHz, respectively, by using a frequency synthesizer, of which frequency is phase-locked to the same H-maser as was used in the OC mode case. In order to measure the absolute frequency fOFS of the common acetylene-stabilized laser, the optical heterodyne beat frequency (fb, OFS) at 1542 nm between the
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acetylene-stabilized diode laser and the OFC-OFS was measured by the same frequency counter as the OC mode case using another channel, which was referenced to the same H-maser. The frequency relation for the absolute frequency in the OFS mode, fOFS, is written as
fOFS = NOFS frep,OFS ± fceo,OFS ± fb,OFS
(3)
where the comb mode number NOFS can be determined uniquely like in the OC mode case. The SNRs of the beat signals of frep,OFS, fceo, OFS, and fb, OFS were above 30 dB in an RBW of 300 kHz, which is sufficient for correct frequency counting and stable phase-locking. The comb frequency comparison was done by measuring the common acetylene-stabilized laser with both OFCs simultaneously.
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Results
Fig. 2 shows the absolute frequency of the acetylene-stabilized laser measured by the OFC-OFS (squares) and by the OFC-OC (circles) simultaneously in four measurement sets. The fCIPM (194 369 569 385 kHz) represents the recommended frequency of the P(16) transition line
of 13C2H2 by CIPM [19]. The error bars are given by the standard deviation divided by the square root of the number of data in each measurement. The average frequency of all these measurement sets is higher than fCIPM by about 1.4 kHz, but this is in a good agreement with the recommended value within the recommended frequency uncertainty of 5 kHz [19]. The error bars of the measurement in the OC mode is slightly larger than that in the OFS mode due to the counter resolution limit in measuring frep,OC in our experiment. As this error bar is dominantly
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given by the stability of the acetylene-stabilized laser, one can expect that they are almost the same in both modes. If we had counted a down-converted frequency for improved counter resolution, the error bars would have been the same in both modes, which was impossible for the counter used in this experiment due to its limited frequency measurement range. Although the acetylene-stabilized laser has a frequency variation in a range of about 0.1 kHz due to the repeatability, the simultaneous measurement agrees with each other fairly good, as can be seen in Fig 2. For the OFC comparison, we calculated the difference frequency between the absolute frequencies measured by the OFC-OFS and by the OFC-OC using the equations (2) and (3). This difference frequency was measured with the counter gate time of 1 s, the result of which is plotted in Fig. 3. The fluctuation of the difference frequency is mainly due to the stability of the acetylene-stabilized laser. We have calculated the weighted mean of the whole data sets to obtain the expectation value for the frequency difference between the OFC-OFS and the OFC-OC. With the total acquisition time of 3224 s, the frequency of the OFC-OFS and that of the OFC-OC agree within 7 Hz (3.8×10-14) with the uncertainty of 16 Hz (8.2×10-14, 1σ) at 194 THz, which means the two modes for the OFC stabilization show no systematic discrepancy at this level of the OFC-comparison uncertainty. It is noted that this result has smaller comparison uncertainty than the previous report [15] by more than an order. Fig. 4 shows the stability (Allan deviation [18]) of the frequency difference between these two combs. The Allan deviation of the frequency difference between the OFC-OFS and the OFC-OC starts at 5.5×10-12 for 1 s averaging time and decreases as the inverse of the averaging time, which is expected for phase-locked signals.
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Conclusion
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We compared the frequency of the OFC in an optical frequency synthesizer mode and that in an optical clock mode to investigate the accuracy limits of the OFC technique. Although there have been many OFC comparison results using two OFCs in the same mode, there has seldom been a report that compares the frequencies of the OFCs in these two modes explicitly. The OFC comparison was performed by the simultaneous measurement of the absolute frequency of an acetylene-stabilized diode laser with an OFC in an OFS mode and with an OFC in an OC mode. The frequencies of the two OFCs agree with each other within 7 Hz (3.8×10-14) with the uncertainty of 16 Hz (8.2×10-14, 1σ) at 194 THz. It is noted that the uncertainty for the comparison was decreased by more than an order compared with the previous report [15]. As we used a mode-locked Ti:sapphire laser and a mode-locked fiber laser for the OFC comparison, it can be concluded that the OFC technique shows no systematic discrepancy to this level of uncertainty regardless of the reference frequency (optical frequency or microwave frequency) and the mode-locking mechanism of the OFCs (Ti:sapphire laser or fiber laser).
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Figure Captions:
Fig. 1. Experimental setup for the comparison of two OFCs; one in an OC mode and the other in an OFS mode. The microwave signals are drawn by the dotted lines and the optical signal by the solid lines. OFC-OC is phase-locked to an acetylene-stabilized laser and OFC-OFS is phaselocked to a microwave reference (H-maser). LD/13C2H2: acetylene-stabilized laser diode, EDFA: erbium-doped fiber amplifier, PPLN: periodically-poled lithium niobate waveguide, PD: photodiode.
Fig. 2. Results of the simultaneous absolute frequency measurement at 1542 nm of P(16) transition line of 13C2H2 by the OFC-OC (circles) and by the OFC-OFS (squares). The error bars are given by the standard deviation divided by the square root of the number of data in each measurement.
Fig. 3. The difference between the absolute frequency measured by the OFC-OC and that measured by the OFC-OFS. The gate time of the frequency counter is 1 s.
Fig. 4. Allan deviation of the frequency difference between the OFC-OC and the OFC-OFS.
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