Measurement of acetylene-d absorption lines with a ... - OSA Publishing

Jiang et al.

Vol. 24, No. 10 / October 2007 / J. Opt. Soc. Am. B

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Measurement of acetylene-d absorption lines with a self-referenced fiber laser frequency comb Jie Jiang,1,* John E. Bernard,2 Alan A. Madej,2 Andrzej Czajkowski,3 Sibyl Drissler,1 and David J. Jones1 1

Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, V6T 1Z1, Canada 2 Frequency and Time Group, Institute for National Measurement Standards, National Research Council of Canada, 1200 Montreal Road, Ottawa, Ontario, K1A 0R6, Canada 3 Department of Physics, University of Ottawa, 150 Louis Pasteur, Ottawa, Ontario, K1N 6N5, Canada *Corresponding author: [email protected] Received June 20, 2007; revised August 14, 2007; accepted August 14, 2007; posted August 16, 2007 (Doc. ID 84324); published September 25, 2007 A self-referenced optical frequency comb, derived from a mode-locked fiber laser, and a cavity-enhanced diodelaser-based saturated absorption spectrometer were employed to measure the absolute frequency of 55 absorption lines of the P and R branches of the 2␯1 band of 12C2HD at 1.5 ␮m with a one standard deviation uncertainty of better than 2.5 kHz for all lines measured. The shift sensitivities of the P(16) line to changes in power, pressure, and modulation amplitude have been studied in detail. Improved values of the ground-state molecular constants are obtained. © 2007 Optical Society of America OCIS codes: 140.4050, 140.7090, 300.6320, 120.3940.

1. INTRODUCTION The introduction of femtosecond frequency combs (FFCs) [1–3] into optical frequency metrology has enabled straightforward absolute measurements of optical frequencies with low measurement uncertainties. In part, this technology has led to improved frequency references with uncertainties approaching 10−15 both in ion-based [4,5] and neutral-atom-based systems [6]. These high performance references typically have transition frequencies in the visible or UV wavelength regions. In recent years, there has also been significant activity in developing portable and small-scale gas-cell-stabilized laser systems in the 1.5 ␮m region. These references can be used in various applications in the 1.5 ␮m optical telecommunications band. In particular, reference frequencies for test equipment associated with dense wavelength division multiplexing (DWDM) are required, and metrology labs around the world have been called upon for measurements traceable to the basic SI units. With the large variety and relatively low cost of laser sources and optical components in this wavelength region, it can be anticipated that work at these wavelengths will continue to be very active. The application of frequency comb technology as a means to measure and calibrate these stabilized laser systems is therefore of significant interest. An important step in the development of frequency and wavelength standards in the 1.5 ␮m spectral region has been the observation of saturated absorption resonances in weak overtone molecular bands of acetylene. Since its first demonstration by de Labachelerie et al. in 1994 [7], a number of teams, including ours at the National Research Council (NRC), have used Fabry–Perot (FP) cavities for power and effective path-length enhancements and have observed saturated absorption resonances in the ␯1 + ␯3 overtone bands of 13C2H2 and 12C2H2 [8–17]. Recently, 0740-3224/07/102727-9/$15.00

the advent of FFC technology has allowed the direct measurement of these transition frequencies with an uncertainty of the order of 10 kHz, limited by the laser stabilized standards [12–17]. The acetylene transitions provide uniform spectral coverage, relatively low sensitivity to perturbation, and comparatively large absorption strengths. For these reasons, the International Committee of Weights and Measures (CIPM) has adopted the P(16) transition of 13C2H2 as a recognized reference for absolute optical frequency [18]. Similar tabulations of other lines in the 13C2H2 and 12C2H2 isotopes are recognized as equivalent references [13–17]. Given that the line spacing of these lines is several tens of gigahertz, it is of interest to obtain new line values for other molecular systems covering this region. The 2␯1 band of acetylene is symmetry forbidden for the 13C2H2 and 12C2H2 isotopic species [19], but is allowed in the 12C2HD system, which has a small dipole moment through the presence of two isotopes of hydrogen. The 2␯1 band of 12C2HD is comparable in strength to the ␯1 + ␯3 overtone band and extends from 1510 nm to beyond 1535 nm, therefore providing very good coverage of the telecommunications C band. Given the large interfrequency line spacing of a particular acetylene molecule isotope band 共50– 90 GHz兲 and the utility of having more lines for calibration in closer proximity to the reference frequency grid as specified by the International Telecommunication Union (ITU), the addition of complementary lines in 12C2HD to the existing known frequencies in 13 C2H2 and 12C2H2 isotopes was considered to be of significant utility. To date, absorption resonances of the 2␯1 band of 12C2HD have only been explored at moderate resolution using laser spectroscopy on its Doppler broadened resonances [20,21] and the best measurements of the line frequencies are limited in uncertainty to 10 MHz. © 2007 Optical Society of America

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Therefore, we decided to apply the NRC 1.5 ␮m diodelaser-based saturated absorption spectrometer system [11,15,17] and a recently developed fiber-laser-based optical frequency comb to the traceable measurement of these resonances. Previous work has demonstrated that FFCs based on fiber lasers are appropriate for optical frequency metrology in the 1.5 ␮m region [22–25]. In addition to direct spectral coverage of the 1 – 2 ␮m region, FFCs generated by fiber lasers offer advantages over FFCs derived from Ti:sapphire laser systems including low cost, portability, established technological infrastructure, lower power consumption, and a small equipment size. In Sections 2–4, we describe the construction and operation of the fiber laser comb system and give details of the cavity-enhanced, saturation-stabilized diode laser that was locked to lines in the 2␯1 band of 12C2HD. We present the results of measurements of the sensitivity of the line frequency to changes in the operating conditions as well as measurements of the frequencies of a total of 55 lines.

2. EXPERIMENTAL SETUP A. Fiber-Based Femtosecond Frequency Comb The experimental setup used in the frequency measurements is shown in Fig. 1. The femtosecond oscillator of our FFC is based on a stretched-pulse erbium-doped fiber (EDF) oscillator employing nonlinear polarization rotation as the mode-locking mechanism [26]. It is similar to the system first developed by Tauser et al. [27]. The oscil-

Jiang et al.

lator generates a pulse train with 90 MHz repetition rate 共frep兲 and 8 mW output power from the 20% output coupler. The (chirped) output pulse duration is 1 ps and the spectral bandwidth is 40 nm. After leaving the oscillator, the pulses pass through a fiber coupler and are equally distributed into two parallel branches, each with a fiberbased chirped-pulse amplifier (CPA). The first CPA branch is designed to generate a oneoctave spectrum for detection and control of the carrierenvelope offset frequency, fceo. Prior to amplification, the pulse is stretched via a piece of SMF-28 fiber whose length is chosen to generate a maximum spectral bandwidth (approximately 80 nm) at the output of the amplifier. The amplifier consists of a 1.2 m long EDF that is pumped by a 400 mW, 980 nm pump diode. It delivers an average output power of 100 mW. A second piece of singlemode fiber is used to recompress the pulse width to approximately 70 fs before the highly nonlinear fiber (HNLF). The HNLF is 25 cm long with a core diameter of 3.7 ␮m and a zero dispersion wavelength of 1.52 ␮m. It is directly spliced to the output of the previously described fiber-based CPA. The output continuum generated in the HNLF covers one octave from approximately 1.0– 2.0 ␮m at the −30 dB points. A common-path f-to-2f interferometer [23] is used for fceo beat detection. In this approach, the entire one octave from the HNLF is collimated and focused into a 5 mm beta-barium borate (BBO) crystal [cut for second-harmonic generation (SHG) at 2 ␮m] and the fundamental and second-harmonic components (centered at 960 nm) are focused onto a silicon avalanche photodi-

Fig. 1. Schematic of the experimental setup for absolute frequency measurement of the acetylene-d absorption lines. EDF, erbiumdoped fiber; WDM, wavelength division multiplexer; OC, output coupler; CL, collimated lens; PBS, polarization beam splitter; ␭ / 2, halfwave plate; ␭ / 4, quarter-wave plate; BF, birefringent filter; OI, optical isolator; PC, polarization controller; HNLF, highly nonlinear fiber; PZT, piezoelectric transducer; M, mirror; L, lens; G, grating; APD, avalanche photodiode; BBO, ␤-barium borate.

Jiang et al.

ode. The resulting beat signal 共fceo兲 had a signal-to-noise (S/N) ratio of approximately 35 dB at 100 kHz resolution. It is bandpass filtered, amplified, and then sent to a digital frequency divider (divide-by-36), and a digital phaselocked loop (PLL) circuit for phase locking with a reference signal from a synthesizer (Stanford Research DS345). The fceo is stabilized to the reference by controlling the pump power of the oscillator. The repetition rate of the pulse train is detected from the rejection port of the polarizing beam splitter by an InGaAs p-i-n diode and stabilized to a reference signal, generated by a second synthesizer (Anritsu MG3641A), by controlling the voltage applied to the piezoelectric transducer (PZT) mounted on one of the collimators. A simplified version of the fiber oscillator was employed for some of the later frequency measurements. The freespace optics in this cavity were arranged in a linear fashion and consisted of a quarter-, and a half-wave plate on each side of the optical isolator. Both the repetition rate 共frep兲 and offset 共fceo兲 signals were detected by the avalanche photodiode in the f-to-2f arm. The S/N ratio for the fceo signal was reduced to 20 to 25 dB, and we attribute this reduction in the S/N ratio to changes in the oscillator output pulse characteristics as the overall dispersion of this cavity differed from that of the cavity shown in Fig. 1. In this case, an rf tracking oscillator was locked to the fceo signal and acted as the input to the digital divider circuit. The second CPA branch amplifier in Fig. 1 is used to measure the absolute frequency of the acetylene-d absorption lines. Although the direct output of the CPA had enough spectral coverage for this investigation, for future flexibility, this branch also utilized HNLF to generate an octave bandwidth (1 – 2 ␮m). One quarter- and one halfwave plate are employed to control the polarization of the output. This output is combined with the beam from the acetylene-d stabilized laser by a 50/ 50 beam splitter. The overlapped beams are dispersed by a 600 line/ mm grating and focused onto an InGaAs photodiode for heterodyne beat measurements. S/N ratios of the heterodyne beat between the diode laser and the comb system (at frequency, fB) of 35– 40 dB at a resolution bandwidth of 300 kHz were obtained as shown in Fig. 2 and were of sufficient stability and spectral purity to be counted directly. B. Saturation Stabilized 1.5 ␮m Diode Laser System The construction and operation of the cavity-enhanced acetylene-stabilized laser system has been described in detail in a number of other reports where the system was

Fig. 2. Typical beat note between the acetylene-d stabilized laser (locked to enhancement cavity only) and the fiber comb. The S/N ratio is ⬃40 dB at 300 kHz RBW.


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used in the study of the 12C2H2 [17] and 13C2H2 [11,15] isotopic species. Briefly, the stabilized laser system consisted of a commercial Littman configuration extendedcavity diode laser system (New Focus Model 6328 H), which was servo stabilized to a Fabry–Perot cavity located in an evacuated chamber. Stabilization to the cavity was obtained using the Pound–Drever–Hall technique [28] with electro-optic phase modulation of the input beam at 10 MHz. The demodulated servo control signal was fed to three different transducers depending on the frequency range of the control signal. Servo error signals in the range of dc to 2 kHz were fed to the piezotranslator, which moved the mirror of the diode laser external cavity while a 600 kHz bandwidth signal 共2 – 600 kHz兲 was fed into the current modulation input of the laser controller. A 2 MHz bandwidth control signal 共50 kHz to 2 MHz兲 was directly applied the diode. With the servo loops closed, the diode laser linewidth was reduced from approximately 1 – 2 MHz to a short term 共⬍1 s兲 linewidth of less than 50 kHz. The evacuated reference cavity had a free spectral range of 470 MHz and a finesse of 380. The cavity mode TEM00 radius on the flat input mirror was ␻o = 0.48 mm while that on the 1 m radius-of-curvature exit mirror was ␻ = 0.58 mm. The resonant cavity was placed within a sealed vacuum chamber whose absolute pressure was monitored by an absolute capacitance pressure gauge. The standard one-way intracavity power used in the current measurements was 0.20± 0.10 W. Isotopically enriched 12C2HD at greater than 90% isotopic purity (Icon Isotopes) was introduced into the chamber. The major impurity to the 12C2HD was 12C2H2, with negligible presence of other contaminants. After initial studies to determine the optimal operating condition, a total pressure of P = 2.0± 0.7 Pa was employed in all the measurements of the line frequencies. Differences between the total measured pressure as determined with a calibrated capacitance based gas manometer and the actual partial pressure of 12C2HD are considered small. The cavitystabilized diode laser system was locked to the desired acetylene reference line by modulating the FP cavity through a sinusoidal voltage at 1.01 kHz, with a modulation excursion of 1.8± 0.2 MHz (peak to peak), applied to one of the cavity mirror PZTs. Linewidth measurements of the saturation dips of the transitions were obtained and gave a value of 0.98± 0.13 MHz, similar to that obtained for the other isotopes of acetylene [15,17]. In the present experimental configuration, the majority of the linewidth broadening occurs due to transit time effects of the molecules traversing the saturating beams [11]. The transmitted signal through the cavity was demodulated at the third harmonic 共3f兲 of the applied modulation frequency with the lock-in amplifier phase adjusted for the maximum dispersion signal. The output from the lock-in amplifier was fed into an integrator-based servo amplifier of 100 Hz bandwidth whose output was applied to the mirror PZT of the FP cavity to maintain the cavity (and thus the laser) in resonance with the center of the acetylene absorption line. With the servo loops closed, the stabilized laser system could remain locked to the acetylene line for several hours, limited primarily by the range of PZT travel of the enhancement cavity. In all measurements, more than 5 mW of power was available from the

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stabilized diode laser system to be sent to the frequency comb apparatus for heterodyning with a nearby mode of the comb.

3. EXPERIMENTAL RESULTS For the measurements of the optical frequencies of the acetylene transitions, the repetition and offset signals of the fiber-based frequency comb were phase-locked to signals from the synthesizers, and the heterodyne beat frequency between the output from the stabilized diode laser system and a single comb element was counted. All of the synthesizers and counters used in the measurements were referenced to a 10 MHz signal provided by a hydrogen maser. The offset of this maser signal from the SI second, as realized at the NRC, was monitored throughout the measurements. In all the current measurements, the maser offset from the International Atomic Time (TAI) realization of the second was within 2 parts in 1014 for the entire period of the measurements. No correction was thus applied for reference frequency offsets in the data. To guarantee the reliability of the measurements, three counters were used to record the heterodyne beat frequencies. The fceo heterodyne signal was amplified and sent to a counter (HP5385A), which was used to monitor the integrity of the offset 共fceo兲 lock. When the simplified fiber oscillator was used, the output from the tracking oscillator was sent to the counter since the S/N ratio of the fceo signal was too weak to be counted directly. The fluctuations in the counted frequency were typically ±0.5 Hz and limited by the resolution of the counter. Two counters (HP5385A and HP5342A) were used to measure the frequency of the heterodyne beat between the acetylenestabilized laser and the optical comb 共fB兲. These counters responded differently when the S/N ratio of the fB signal dropped below approximately 30 dB, permitting counter errors to be easily identified. As the counters were triggered through software, the gate periods were not completely synchronous. A measurement was ruled valid if the average reading from the two counters agreed to within approximately 100 Hz and the counted fceo frequency did not fluctuate by more than a few hertz. The frequency of the acetylene-stabilized laser is given by flaser = n ⫻ frep ± fceo ± fB ,

has confirmed that the frequencies reported in [21] are correct to within the quoted uncertainty of 10 MHz. A. Systematic Shifts of the Acetylene-d P(16) Line To study the performance of the diode-laser system and to investigate the shift sensitivities of the output frequency to changes in the operating parameters, detailed measurements were made with the system locked to the P(16) line of the 2␯1 band of 12C2HD. Studies were performed of changes in the laser frequency due to changes in the ambient gas pressure, intracavity power, and modulation excursion. The P(16) line was chosen for these studies because it is relatively strong and located near the center of the band. Its shift sensitivities should therefore be representative of the other lines. Asymmetry in the reference absorption lines can lead to shifts in the operating frequency of the stabilized laser as the modulation amplitude, used in the 3f lock to the line centre, is varied. Studies of this shift were made by varying the amplitude of the modulation of the laser from 0.5 to 4.3 MHzpp. The observed shift in the lock frequency exhibited a linear dependence with a fitted slope of −840± 40 Hz/ MHzpp 共1␴兲, as shown in Fig. 3. This value is smaller than our previous measurement of the modulation shift for the P(16) line of the ␯1 + ␯3 band of 13C2H2 of 4.7± 0.3 kHz/ MHzpp [15] and indicates that the C2HD lines exhibit good symmetry when not perturbed. Given our current uncertainty of the modulation amplitude of 0.2 MHz, the associated systematic uncertainty in the optical frequency is 0.2 kHz. Studies of the variation of the P(16) stabilized frequency as a function of the intracavity power were performed by changing the input power into the enhancement cavity using a set of calibrated neutral density filters. The intracavity power was calculated from the transmitted optical power exiting the cavity and the known reflectivity of the cavity mirror. Measurements could be performed for intracavity powers ranging from 0.05 to 0.56 W. Due care was taken during these measurements to ensure that the relevant signal levels on the detectors and servos were maintained near their optimal values. The studies showed a weak dependence on intracavity power that was roughly linear and had a slope of

共1兲

where the value of the integer n and the proper signs must be determined through a fitting to a previously measured value of flaser. For these nominal values, we used the results of Hardwick et al. [21], which had an uncertainty of approximately 10 MHz or ⬃1 / 10 of the comb spacing. Therefore, care had to be taken to find the correct solution for n and the signs of the heterodyne beats. The signs found in the analysis were verified by noting immediately after each run how the frequency of the fB signal changed when the frequencies of frep and fceo were changed. To confirm that the analysis had found the correct value of n, measurements for each line were performed with at least two different values of frep, which differed by at least 100 kHz and in some cases by several megahertz. Only if the correct value of n is found for each frep will the values of flaser be in agreement. Our analysis

Fig. 3. Observed systematic shift dependence of the laser frequency stabilized to the P(16) line as a function of the modulation amplitude. The plotted frequency is the observed measured frequency relative to the value of f = 195, 903, 630, 363.6 kHz obtained under the stated operating conditions given in Table 1.

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−1.6± 0.6 kHz/ W 共1␴兲. The systematic uncertainty in the optical frequency associated with our control of intracavity power is 0.3 kHz. These results are comparable to those found in previous power shift studies using other acetylene isotopes [15,17]. The laser system design permitted the acetylene gas pressure in the optical cavity to be varied, and therefore investigations of pressure-related changes in the laser system’s performance and its optical frequency were possible. Studies were made by starting at low acetylene pressures and measuring the stabilized laser frequency as additional gas was introduced in the chamber. For each pressure setting, the gas pressure was allowed to equilibrate and the intracavity power was adjusted to the nominal value. A small change in the optical frequency within the statistical uncertainty of the measurements was observed for a range of pressures from 0.8 to 6.8 Pa. The laser frequency was observed to increase with a fitted linear slope of +200± 60 Hz/ Pa 共1␴兲. The magnitude of the shift is comparable to that observed for the 13C2H2 ␯1 + ␯2 band but is of opposite sign [11,29]. The shift may thus be not purely pressure related but may be of a more complex origin. With our current uncertainty of setting the pressure in the cell of 0.7 Pa, a one standard deviation in uncertainty in the optical frequency of 0.2 kHz is expected. Two other sources of systematic shifts exist due to limitations in the setting of the offsets of the electronic locks. The first arises due to an offset in the Pound–Drever–Hall lock to the enhancement cavity caused by residual amplitude modulation of the incident light by the phase modulator and etalon effects between the modulator, cavity, and detection systems. This offset results in a shift of the lock point on the cavity-enhanced acetylene absorption profile. Based on our current level of adjustment and control, it is estimated that such offsets should be below the level of 0.7 kHz. A second electronic offset occurs due to any residual voltage offsets in the demodulation of the 3f signal. These offsets are estimated to be on the level of 0.7 kHz, based on the possible voltage offsets and the discriminator slope for the acetylene lines utilized in the current study. Table 1 summarizes the estimated systematic shifts. Table 1. Summary of Estimated Systematic Uncertainties (1␴ Confidence Interval) Associated with the Absolute Frequency Measurement of the 12 C2HD Reference Lines

Source of Systematic Shift Uncertainty in modulation amplitude Uncertainty in intracavity power Uncertainty in operating cell pressure Electronic offsets of laser lock to cavity and etalon effects Electronic offsets of servo (shift of the lockpoint to the acetylene line profile) Uncertainty in the absolute frequency measurement due to the maser reference Total uncertainty

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Table 2. Operating Parameters and Characteristics of the Stabilized Laser System C2HD pressure (Pa)

2.0± 0.7

One-way intracavity saturation power 200± 100 (mW) Peak-to-peak modulation amplitude 1.8 MHz± 0.2 MHz (MHz) Cavity coupling efficiency 60% Empty cavity finesse (calculated from 380 the mirror reflectivities) Cavity waist (mm) on input mirror 0.47 Modulation frequency used in lock to cavity 10 MHz Modulation frequency used in lock 1.01 kHz to absorber line

The overall estimated systematic uncertainty in the frequency measurements is 1.1 kHz 共1␴兲. This uncertainty is added in quadrature with the statistical uncertainties of the frequency measurements to determine the final total uncertainty. In summary, the studies of systematic shifts illustrate that the C2HD lines studied in the current work appear to be of comparable metrological quality to those measured for the two isotopes of C2H2 and should serve as useful references for precision measurements. B. Stability of the Acetylene Stabilized Laser System The fiber-based FFC was used to study the stability of the laser system while it was locked to P(16) for periods of up to 1 h. The duration of the measurements was limited by thermal drifts in the fiber laser cavity that eventually exceeded the (limited) travel range of the PZT controlling the repetition rate of the oscillator. The instability of the locked diode laser was measured to be 5 ⫻ 10−12 at 1 s averages and decreased as the inverse square root of the averaging time to 7 ⫻ 10−13 at 60 s. A stability floor of 5 ⫻ 10−13 was observed for 100 s averages, which may be due to drifts in the lock point of the Pound–Drever–Hall lock to the cavity as described in Subsection 3.A. C. Absolute Optical Frequency Measurement of the Acetylene-d Lines in the 1.5 ␮m 2␯1 Band Measurements of the absolute frequency of the P and R branch lines in the 2␯1 band of C2HD were performed during the period of March 1–27th, 2007. A total of 55 lines were measured, ranging from R0 to R27 and P1 to

Estimated Shift Uncertainty [kHz] 0.2 0.3 0.2 0.7 0.7 0.002 1.1

Fig. 4. Frequency of the diode laser system locked to the P(16) line of the 2␯1 band of 12C2HD. The error bars reflect the standard deviation of the individual 1 s readings of each run.

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Table 3. Observed Line Frequency and Vacuum Wavelengths for the Transitions in the 2␯1 Band of 12C2HDa Line

Frequency (kHz)

Uc (kHz)

Vacuum Wavelength (nm)

Uc (nm)

P27 P26 P25 P24 P23 P22 P21 P20 P19 P18 P17 P16 P15 P14 P13 P12 P11 P10 P9 P8 P7 P6 P5 P4 P3 P2 P1 R0 R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16 R17 R18 R19 R20 R21 R22 R23 R24 R25 R26 R27

195,083,584,556.3 195,161,449,714.5 195,238,655,952.4 195,315,202,226.7 195,391,087,966.7 195,466,309,716.0 195,540,867,837.3 195,614,760,668.5 195,687,985,368.1 195,760,540,273.7* 195,832,422,907.6 195,903,630,363.6 195,974,159,502.3 196,044,006,223.8 196,113,166,245.0 196,181,634,239.2 196,249,404,477.3 196,316,469,424.0 196,382,821,147.6 196,448,450,320.2 196,513,346,479.1 196,577,498,143.4 196,640,893,107.2 196,703,518,963.8 196,765,363,847.5 196,826,417,376.6 196,886,671,625.8 197,004,767,626.3 197,062,611,544.6 197,119,660,022.9 197,175,921,813.3 197,231,407,144.6 197,286,126,795.2 197,340,091,336.2 197,393,310,617.9 197,445,793,469.0 197,497,547,587.3 197,548,579,273.4 197,598,894,431.8 197,648,497,165.4 197,697,391,167.2 197,745,579,093.3 197,793,063,418.1 197,839,845,664.7 197,885,927,073.1 197,931,308,538.2 197,975,990,083.8 198,019,972,926.1 198,063,257,107.1 198,105,840,644.7 198,147,725,370.0 198,188,910,238.6 198,229,394,674.8 198,269,179,255.0 198,308,261,613.7

1.7 1.3 1.2 1.3 1.2 1.2 1.1 1.2 1.1 1.7 1.1 1.1 1.5 1.1 1.1 1.4 1.1 1.2 1.4 1.2 1.1 1.3 1.4 1.1 2.4 1.2 1.5 1.3 1.3 1.5 1.3 1.1 1.3 1.1 1.1 1.1 1.1 1.2 1.3 1.2 1.1 1.1 1.2 1.1 1.1 1.1 1.1 1.8 1.4 1.1 1.1 1.4 1.1 1.3 2.0

1536.738514836 1536.125389715 1535.517935921 1534.916148780 1534.320020016 1533.729564116 1533.144765674 1532.565625291 1531.992152896 1531.424349263 1530.862221633 1530.305780672 1529.755038936 1529.210016540 1528.670735067 1528.137224275 1527.609517076 1527.087660447 1526.571704429 1526.061709886 1525.557746440 1525.059891551 1524.568228220 1524.082840913 1523.603809827 1523.131203604 1522.665071863 1521.752298750 1521.305617794 1520.865336138 1520.431375408 1520.003646175 1519.582055109 1519.166510819 1518.756927788 1518.353228665 1517.955345078 1517.563219653 1517.176798291 1516.796040949 1516.420910918 1516.051379629 1515.687420071 1515.329012681 1514.976140214 1514.628788210 1514.286949003 1513.950605942 1513.619751482 1513.294393666 1512.974511518 1512.660106154 1512.351175222 1512.047707699 1511.749715118

1.3⫻ 10−8 9.9⫻ 10−9 9.0⫻ 10−9 9.7⫻ 10−9 9.0⫻ 10−9 9.1⫻ 10−9 8.6⫻ 10−9 9.0⫻ 10−9 8.5⫻ 10−9 1.3⫻ 10−8 8.5⫻ 10−9 8.6⫻ 10−9 1.2⫻ 10−8 8.5⫻ 10−9 8.5⫻ 10−9 1.1⫻ 10−8 8.5⫻ 10−9 8.9⫻ 10−9 1.1⫻ 10−8 9.5⫻ 10−9 8.5⫻ 10−9 1.0⫻ 10−8 1.1⫻ 10−8 8.8⫻ 10−9 1.8⫻ 10−8 9.1⫻ 10−9 1.2⫻ 10−8 9.7⫻ 10−9 9.9⫻ 10−9 1.1⫻ 10−8 1.0⫻ 10−8 8.6⫻ 10−9 9.8⫻ 10−9 8.6⫻ 10−9 8.5⫻ 10−9 8.5⫻ 10−9 8.6⫻ 10−9 9.3⫻ 10−9 9.8⫻ 10−9 9.2⫻ 10−9 8.5⫻ 10−9 8.5⫻ 10−9 9.5⫻ 10−9 8.5⫻ 10−9 8.5⫻ 10−9 8.7⫻ 10−9 8.5⫻ 10−9 1.4⫻ 10−8 1.1⫻ 10−8 8.5⫻ 10−9 8.5⫻ 10−9 1.1⫻ 10−8 8.7⫻ 10−9 9.8⫻ 10−9 1.6⫻ 10−8

a The quoted total uncertainties are given for a confidence interval of one standard deviation 共1␴兲. Line P共18兲 marked with an asterisk is within 1.5 MHz of another transition and thus may be perturbed.

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Table 4. Comparison of Ground State Molecular Constant Results for the 2␯1 Band of

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12

C2HD

Molecular Constant 共cm 兲

This Work

Reference [21]

Reference [30]

B⬙ D⬙ 共⫻106兲 H⬙ 共⫻1012兲

0.99152748195(34) 1.13506796(93) 1.16547(76)

0.99152629(130) 1.13568(104) 1.24(34)

0.991527586(80) 1.13531(27) 1.17(11)

−1

P27. All lines were measured at least twice on each of at least two separate days with independent gas fills and often with different operators of the comb and laser equipment. Each measurement run typically comprised a total of 100 1 s counter readings. The final value for the transition frequency for each line was determined by taking the mean of the daily average values. The day-to-day reproducibility of the stabilized laser frequency was typically on the order of 1 kHz and was larger than the observed run-to-run repeatability of the laser on a particular day and the stability floor due to drifts. The scatter in the day-to-day average values was then used to estimate the statistical uncertainty of the frequency measurement. The statistical uncertainty was combined in quadrature with the estimated systematic shifts for the apparatus of 1.1 kHz (see Subsection 3.A) to yield the total uncertainty for each reference line. The absolute optical frequency of the stabilized diode laser system, while it was locked to the P(16) line, was measured a total of 11 times on 4 days over a period of 2 weeks with the standard operating parameters shown in Table 2. The results of these measurements are shown in Fig. 4. The reproducibility of the laser locked to the P(16) line was better than 0.5 kHz. The results of the frequency measurements of all the lines in the 2␯1 band of C2HD are listed in Table 3. Almost all lines studied of the 2␯1 band were sufficiently well spaced from other nearby transitions that perturbation from overlapped lines are considered to be small. The exception is the P(18) line. It was observed that a nearby weaker line located at f = 195, 760, 541, 658± 2 kHz was located within 1.5 MHz of the P(18) line. This line has been assigned as the P(19) line of the ␯1 + ␯3 + ␯5 band of 12 C2HD [21]. Given the close proximity of this other line to the P(18) transition of the 2␯1 band, the immunity of the P(18) line to asymmetry, collisional, or other perturbation may be of concern. In summary, the results obtained for the 2␯1 band of C2HD represent an improvement of more than 103 in accuracy over previous measurements, and therefore contribute to a significant improvement in our knowledge of the line frequencies.

taneously via a coupled Hamiltonian. We therefore only extract the pertinent state information for the ground vibrational state of 12C2HD. The energy levels of the ground state of 12C2HD can be expressed as [19] E共J兲 hc

= BJ共J + 1兲 − DJ2共J + 1兲2 + HJ3共J + 1兲3 + LJ4共J + 1兲4 共2兲

+ ... ,

where J is the rotational quantum number. The results presented in Table 3 were fitted by the method of combination differences [19] to yield the ground-state molecular constants. The data were fitted using the relation ⌬2F⬙ = ␯R共J−1兲 − ␯P共J+1兲

冉

= 4B⬙ − 6D⬙ +

27 4

H⬙ +

27 4

冊

L⬙ 共J + 1/2兲

+ 共− 8D⬙ + 34H⬙ + 75L⬙兲共J + 1/2兲3 + 共12H⬙ + 100L⬙兲共J + 1/2兲5 + 16L⬙共J + 1/2兲7 , 共3兲 where ␯R共J−1兲 and ␯P共J+1兲 are the wavenumbers of the lines R共J − 1兲 and P共J + 1兲, respectively. The results yield the ground-state constants as given in Table 4. The observed difference between the measured combination differences and those obtained by the fitted constants is given in Fig. 5. Fitting to the term L⬙ did not improve the fit to the data and thus only terms up to H⬙ were included in the final quoted result. The results of Table 4 are in very good agreement with the two best previous determinations of the ground-state constants, given in studies by Hardwick et al. [21] and Fusina et al. [30], but improve upon the accuracy of these values by several orders of magnitude. These results, together with those obtained in recent studies of other isotopic species of acetylene, may further

4. ANALYSIS AND COMPARISON OF GROUND STATE MOLECULAR CONSTANT RESULTS OF THE 2␯1 BAND OF 12C2HD The very accurate results obtained in this work can be employed for the better understanding of the molecular potentials of the 12C2HD molecule. Unfortunately, the excited state of the 2␯1 band lies within another transition band, namely, the ␯1 + ␯3 + ␯5 vibration, and it has been shown by others [21,30] that analysis of the excited state constants requires the treatment of the two bands simul-

Fig. 5. Observed difference between the measured combination differences based on the absolute frequency measurements and those obtained using the fitted molecular constants determined in this paper.

2734


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our in-depth understanding of the structure of this molecule. 7.

5. CONCLUSIONS A fiber-laser-based, self-referenced femtosecond frequency comb and a cavity-enhanced saturated absorption spectrometer were used to measure the absolute frequency of 55 absorption lines of the P and R branches of the 2␯1 band of 12C2HD at 1.5 ␮m with an uncertainty of less than 2.5 kHz. These results represent a reduction in uncertainty of more than three orders of magnitude over previous studies. Measurements of the optical frequency of the P(16) line for a range of gas pressures, intracavity optical powers, and modulation amplitudes have revealed that the transition frequencies show little sensitivity to the operating conditions of the saturated absorption spectrometer. Our measurements have yielded a significant improvement in the accuracy of the calculated groundstate molecular constants, and thus provide an illustration of the utility of frequency comb technology to the detailed understanding of the spectra of molecules in the optical region. The measurements presented here should be useful to workers who require reference frequencies in the 1.5 ␮m region of the spectrum with uncertainties at the kilohertz level.

8.

9.

10.

11.

12.

13.

ACKNOWLEDGMENTS The essential contributions of Raymond Pelletier and Bill Hoger in the development and construction of the electronics are gratefully acknowledged. We thank JeanSimon Boulanger and Stan Cundy for providing the reference maser signal together with a determination of its offset. We also acknowledge helpful discussions with F. Adler, A. Leitenstorfer, and H. Inaba. The HNF fiber used in the FFC was kindly provided by M. Hirano, M. Onishi, and T. Okuno of Sumitomo Electric Industries Ltd. This work has received partial support from the Natural Sciences and Engineering Research Council (NSERC), Canadian Foundation for Innovation (CFI), and a Canadian Institute for Photonic Innovations TEN grant.

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