A 32 mW fiber laser is stabilized to the 13C2H2 P(16) (ν 1 + ν3) transition at 1542 nm using saturated absorption. The short-term shot-noise limited fractional frequency instability is 5.0 × 10−13(τ/s)−½ for averaging times τ up to about 100 s. The relative lock-point repeatability over 2½ month is 4.3 × 10−13 corresponding to 83 Hz. The simple setup includes a 21 cm long gas cell, but it does not require an enhancement cavity or external modulators. The spectroscopic lineshape is analyzed with respect to optical power and acetylene pressure. Narrow linewidths of 300 kHz FWHM are observed with a signal to noise ratio of 35 dB in a 9 Hz bandwidth.
©2011 Optical Society of America
Frequency-stable lasers have a number of applications, including standards for length and time metrology, interferometric sensors, coherent optical communication , and high-resolution spectroscopy. In the 1550 nm telecommunication band the 13C2H2 P(16) (ν 1 + ν3) line at 1542.3837 nm is the preferred frequency reference with an entry in the Mise en pratique for the definition of the metre . Advances in optical communication techniques, including high-density wavelength division multiplexing (DWDM), were driving some of the initial work on stabilized lasers in the telecom band . More recently, acetylene-stabilized lasers and Rb stabilized lasers at 1556 nm with similar performance have been developed for the Terrestial Planet Finder Coronograph project (TPF-C)  and for the optical synchronization signal in the Atacama Large Millimeter Array (ALMA) .
Most reported realizations of high-stability optical frequency standards based on the P(16) line use an external enhancement cavity, an external optical modulator, and multiple servo loops in order to boost the signal-to-noise ratio for the relatively weak P(16) combination band transition [6–8]. Cavity-enhanced setups achieve high stability and repeatability, but are likely too complicated for operation outside the laboratory environment. Hence, simplifications of the frequency standards are desirable for more practical applications.
P. Balling et al.  have shown that it is possible to stabilize a DFB diode laser to the P(16) line using a 50 cm gas cell without an enhancement cavity and without external modulators. They pay a penalty in terms of frequency stability , which is about 5 times worse than demonstrated with the cavity-enhanced setups . Further developments in the frequency standard setup are investigated with gas filled hollow-core fibers replacing conventional gas cells. However, these devices are still inferior in terms of frequency stability and experimental complexity [10,11].
The intrinsic linewidth of a fiber laser can be very narrow, reaching a short-term linewidth below 1 kHz . This implies that the fiber laser should be an ideal source for an optical frequency standard in the telecom band as recognized 20 years ago by Gilbert . More recent developments of stabilized fiber lasers in the telecom band include stabilization to Doppler broadened absorption lines in acetylene  and saturated absorption in acetylene . None of these implementations achieve a frequency stability comparable to the cavity-enhanced setups.
This paper shows that the use of a highly stable fiber laser source, balanced detection, and optimized experimental parameters can lead to a simple and relatively compact acetylene-stabilized laser surpasssing more complex cavity-enhanced setups in performance.
2. Experimental configuration
Two independent laser systems named P (primary) and S (secondary) are developed for evaluation of the frequency standard performance. A schematic layout for laser P is shown in Fig. 1 . The core of the setup is a Koheras E15 BasiK ModuleTM fiber laser manufactured in 2009 with type number K80-151-14. The laser output power is 32 mW delivered through a polarization maintaining single mode fiber. The laser is temperature tuned to the acetylene P(16) line at 1542.3837 nm. A piezo tuning input is used for fast fine-tuning and modulation. The piezo tuning range is about 3 GHz. The laser output is focused with aspheric lens L1 to a minimum spot size (radius) of w 0 = 580 μm at the beam waist located at mirror M1. The spot size is a compromise between high optical intensity (small spot size) and small beam divergence as well as small transit time linewidth (large spot size). The waist location is chosen for minimum variation in spot size through the gas cell without adding more lenses to the setup. Lens L2 has a focal length of 1000 mm and the optical path length from M1 to L2 is 280 mm. Mirror M2 retroreflects the beam, which gives a total of four passes through the gas cell. The gas cell has Brewster angled windows and a length of 21 cm. The cell is sealed with a PTFE valve, which allows refilling of the cell. A wedged UV fused silica substrate with an anti-reflection coating on one surface is used as a beamsplitter. A small fraction of the beam before and after the passage through the gas cell is reflected off the beamsplitter and provides the reference and signal beams for the balanced detection. A New Focus Nirvana 2017 photoreceiver is used for auto-balanced detection. The beamsplitter angle is adjusted to a reflectivity of 1.72%, and the power on each of the photodiodes is close to the 500 μW saturation limit. Lenses L3 and L4 are used to focus the beams onto the relatively small photodiodes. The optical components are selected and aligned to minimize possible interference effects from stray light. The optical setup including laser module, photoreceiver, gas cell and bulk optics fits easily on a 30 cm x 60 cm breadboard.
The laser is frequency modulated via the reference output from a Stanford Research Systems SR850 lock-in amplifier. The modulation frequency is fixed at 1.20 kHz while the amplitude is varied in the analysis below. The lock-in amplifier detects the third harmonic content in the photoreceiver output. The lock-in amplifier has discrete settings for the output low pass filter bandwidth. A bandwidth of 31 Hz is used when locking the laser; this is sufficiently low for filtering out the modulation frequency and high enough for efficient locking. A 9.4 Hz bandwidth setting is used for analyzing the saturated absorption signals at various experimental parameters. This setting improves the signal-to-noise ratio in measurements that do not require a higher bandwidth since the signal-to-noise ratio scales as B −½, where B is the filter bandwidth. The third harmonic signal is integrated in a Stanford Research Systems SIM960 PID controller. The PID output is amplified in a piezo driver and then added to the modulation signal using a simple three-terminal resistor configuration before entering the laser piezo input.
Laser S uses a principle similar to laser P. The gas cell is 50 cm long, and the laser is based on a Koheras fiber laser manufactured in 2007. Laser S uses an older design of the laser substrate, which has more friction between fiber and substrate during piezo modulation. As a consequence, laser S has a lower piezo-response bandwidth and more harmonic distortion as compared to laser P. Variations in the degree of harmonic distortion in the laser frequency modulation can limit the laser frequency stability as explained later. Such harmonic distortion variations over time have been observed, and they seem to be caused by changes in the piezo response with temperature. Therefore, an acousto-optic modulator (AOM) provides the frequency modulation in the laser S setup. A Rohde & Schwarz SML01 signal generator drives the AOM with an average frequency of 35 MHz. The SML01 output is frequency modulated with a frequency of 400 Hz and an amplitude of 405 kHz. After the AOM, the laser output is amplified to 75 mW using an erbium-doped fiber amplifier (EDFA). The signal-to-noise ratio for laser S is 2.5 times larger than for laser P due to the longer gas cell and the increased optical power in the gas cell.
The gas cells are filled with 13C2H2 acetylene at various pressure levels. The acetylene is purchased from Cambridge Isotope Laboratories and has a specified purity of 99%. A pressure gauge is used for measuring cell pressure with a relative standard uncertainty of about 4% in the range from 0.5 Pa to 10 Pa.
Each fiber laser has a monitor output with a power level of about 100 μW. The two monitor outputs are mixed on a beamsplitter, and the beat note frequency is detected with a high frequency photoreceiver and recorded with a frequency counter. The AOM in the laser S setup gives a beat note frequency around 35 MHz when both lasers are locked to the P(16) line. The monitor output of laser P is modulated in frequency due to its piezo modulation, whereas laser S provides a modulation free monitor output.
3. Lineshape analysis
The error signal for the frequency standard is derived by lock-in detection of the saturated absorption Lamb dip at the third harmonic frequency (3f) when the laser frequency is modulated at frequency f. Analysis of the lineshape identifies the optimum parameters for the frequency standard. The 3f signal detected when a frequency-modulated laser is scanned across a Lamb dip is given by [14,15]:Equation (1) is theoretically valid in the limit where , and for homogenous broadening. In practice we find that Eq. (1) fits all the data acquired at various pressure and power levels, although at high optical power and at low pressure the fitted parameter s deviates from the saturation parameter s 0.
The absorption coefficient of the P(16) line is α = 6.93 × 10−3 m−1 Pa−1 . For laser P with l = 4 × 21 cm the calculated absorption at 1.0 Pa is 0.58% in agreement with the measured absorption of (0.57 ± 0.02) %, where the value following ± is the standard uncertainty. The modulation amplitude m is related to the amplitude of the voltage modulation applied to the laser piezo input, and m can be measured independently with a relative standard uncertainty of 1%. The saturated absorption linewidth Γ and the parameter s are obtained by fitting a measured 3f signal to Eq. (1). In general, Γ and s are functions of gas pressure, optical beam radius and optical intensity.
Figure 2 shows the measured 3f signal, i.e. V 3(Δ)/V0, for an acetylene pressure of 0.90 Pa, a modulation amplitude m set to 400 kHz, and at maximum available optical power. The signal is measured with laser S locked to the saturated absorption line center and laser P offset locked to laser S with a tunable offset . The offset frequency delivered by an RF signal generator is tuned in steps of 10 kHz at a rate of 200 ms per step. The frequency uncertainty of laser P is about 2 kHz for each step, and one data point is sampled from the lock-in amplifier in each step with a bandwidth setting of 9.4 Hz. The measured lineshape is fitted to Eq. (1), and the residual multiplied by 10 is shown as the red curve in Fig. 2. The fit gives a linewidth Γ = 470 kHz and an amplitude s = 0.16. Linewidths of 300 kHz are observed with a signal to noise ratio of 35 dB in a 9.4 Hz bandwidth at 0.13 Pa and 15 mW as shown by the blue curve in Fig. 2. This is, to our knowledge, by far the narrowest lines observed with acetylene in the near infrared region, including both 3f measurements [6–9] and recent direct measurements of the Lamb dip for the P(16) line .
The measurements are repeated for various values of pressure, modulation amplitude, and optical power level. The saturation parameters s 0 can be determined from the measured optical powers P and the derived fitting parameters s. The general relation between the saturation parameter and the optical power is: s 0 = P/P sat, where P sat is the saturation power. As described previously, s = s 0 in the limit where. The saturation power depends on the linewidth and hence on the pressure. The fitting parameter s is plotted as a function of P/P sat in Fig. 3 , where the saturation power is determined for each pressure level by requiring s = P/P sat for . The data in Fig. 3 show a unique relation between s and s 0 independent of pressure.
Figure 4 presents the linewidths (Γ), amplitudes (s parameters) and saturation parameters (s 0) derived from the measurement data. The figure includes the linewidths obtained at maximum optical power as well as the linewidths Γ0 obtained by extrapolating to zero optical power. The linewidth for small s 0 is given approximately by [14,15] . Γtransit is the transit time linewidth, Γp is the pressure-broadening coefficient, and p is the pressure. The linear expression agrees with the exact theory to within 2% for the pressure range 0.6 Pa to 10 Pa . The linear fit to Γ0 data above 0.6 Pa in Fig. 4 gives Γtransit = 167 kHz ± 22 kHz and ΓP = 234 kHz/Pa ± 10 kHz/Pa, where the uncertainty includes the contribution from the pressure measurements. The theoretical expression for Γtransit is given by . The pressure-broadening coefficient is almost 3 times larger than for Doppler-limited spectroscopy at higher pressure . Similar discrepancies have been observed for CO2 and CO . A possible explanation for the difference in the pressure-broadening coefficient between Doppler-limited spectroscopy and saturation spectroscopy has been suggested to be due to a velocity dependent collision rate .
In the transit-time regime at low pressure, the saturation parameter depends on the optical power rather than the optical intensity . The saturation parameter in the transit-time regime is s 0 = 0.75 (see Fig. 4) at an optical power of 30 mW. This corresponds to a transit-time saturation power of 40 mW. Only a few published values exist for the transit-time saturation power for acetylene lines of similar strength. The saturation power for the P(12) line in the transit-time regime is estimated to be 25 mW , and the transit-time saturation power for the P(9) line of 12C2H2 is measured in a hollow-core fiber to be 23 mW . The transit-time saturation power is a key parameter when selecting an absorption line for a saturated absorption stabilized laser.
The optimum conditions for the frequency standard are generally obtained when the slope of the error signal at zero crossing normalized to the off-resonant noise is maximized. It can be shown directly from Eq. (1) that the optimum modulation amplitude is always given by m = 0.818 Γ. The optimum pressure is a compromise between the low pressure regime, where the linewidth is narrow and the saturation parameter is high, and the high pressure regime where the absorbance is high. The experimentally measured error signal slopes divided by the off-resonant rms-noise are plotted in Fig. 5 together with the signal-to-noise ratio defined as the amplitude of the 3f signal divided by the off-resonant rms-noise. The data shows that the optimum performance of the frequency standard is realized at a pressure of 1.1 Pa. The voltage signal-to-noise ratio is not influenced by pressure broadening of the linewidth, and this ratio reaches a maximum of 610 (corresponding to 55.7 dB) at 2.6 Pa for a 9.4 Hz detection bandwidth.
An optimal optical power level is expected as well, since strong saturation does not increase the error signal amplitude V 3(Δ)/V0 but adds to the linewidth due to power broadening. The optical noise level is constant as long as the power on the photoreceiver is adjusted to be close to the receiver saturation power. We cannot quite reach this optimum with the available power, but extrapolation of measurement data indicates that we are a few percent below the maximum error signal slope with respect to power when the pressure is below 1 Pa. At higher optical power the beam size may be expanded to reduce the transit time broadening, and detectors allowing higher power levels without saturation may be used to increase the shot-noise limited signal-to-noise ratio.
4. Noise analysis
The off-resonant noise seen in Fig. 2 reflects a variation over time within the measurement bandwidth. With the laser frequency fixed off resonance, the noise in the 3f-signal is measured as a function of optical power. The optical power is adjusted with a variable neutral density filter after lens L1. The relative intensity noise (RIN) in dBc units is given by:Fig. 6 for the lock-in amplifier bandwidth at 31 Hz. The theoretical expression for the shot-noise limited RIN in dBc units is:Fig. 6 as the dashed line. The lock-in amplifier only measures the noise in one quadrature, whereas RIN is generally defined as the total noise. Furthermore, the balanced detection means that the measured noise includes contributions from both the signal and reference photodiodes. These two effects cancel since the power levels on each photodiode are close to equal.
The measured RIN data in Fig. 6 have the power dependence characteristic for shot-noise limited RIN, which indicates a shot-noise limited detection of saturated absorption signals. The 2 dB offset in Fig. 6 is probably due to inaccuracy in the specified lock-in amplifier bandwidth and quantum efficiency.
5. Repeatability and stability
The performance of the stabilized laser is evaluated over a period of 11 weeks. During this period the beat frequency between the locked lasers is measured over 48 hours over consecutive weekends. Before each measurement, the gas cells in both setups are refilled to a pressure between 1.0 Pa and 1.1 Pa. Realignments and minor adjustments are applied to the setups during the 11 weeks to check for systematic dependencies. The average beat frequencies, corrected for the AOM frequency in the laser S setup, are shown in Fig. 7 . The data in Fig. 7 (excluding the single measurement with piezo-modulation of laser S) give an average frequency difference of 197 Hz and a lock-point repeatability (standard deviation) of 83 Hz. This corresponds to a difference of 1.0 × 10−12 and a repeatability of 4.3 × 10−13 relative to the laser frequency.
The average relative Allan deviation for the 8 runs with filled symbols in Fig. 7 is shown in Fig. 8 . The error bars in Fig. 8 represents the sample standard deviation of the Allan deviation over the 8 runs. The 3 runs with open symbols in Fig. 7 are excluded because of a poor Allan deviation at short averaging times; probably due to interference with scattered light caused by imperfect alignment (week 30 & 37) or piezo-modulation of laser S (week 39). The dependence of the Allan deviation is close to 5.0 × 10−13(τ/s) −½ for averaging times below 100 seconds. For longer averaging times, the Allan deviation increases to about 2.5 × 10−13; close to the relative repeatability of 4.3 × 10−13 over 11 weeks shown in Fig. 7. For comparison, the measured relative Allan deviation with laser S locked and laser P free running has a minimum of 9.5 × 10−12 for an averaging time of 15 ms.
The expected short-term Allan deviation can be estimated from the peak error signal slope in Fig. 5 as , where ν 0 is the optical frequency, β is the slope normalized to the off-resonant rms noise, and B is the bandwidth used when measuring β. Although this estimate ignores details in the bandwidth definition, it is close to the observed short-term stability in Fig. 8, and it supports the assumption that the noise observed off-resonance indeed limits the frequency stability. Further measurements of the short-term Allan deviations with various attenuations of the optical beams using filters in front of the photoreceiver have confirmed that shot noise limits the short-term Allan deviation.
The Allan deviations for averaging times above 100 seconds show some variation between each run, which is evident from the increased size of the error bars in Fig. 8. This variation is correlated to the temperature variations during each run. In particular, the run with the smallest overall Allan deviation (open circles in Fig. 8) are obtained during the weekend with the smallest temperature variations in the laboratory (0.10°C peak to peak). The coupling between the temperature and the frequency of the locked laser may be related to the fiber laser piezo response. With piezo-modulation of the S laser frequency instead of the normal AOM modulation, the beat frequency variations increase on the time scale around 1000 seconds (see Fig. 8, triangles) and a minor shift in the average frequency is observed (see Fig. 7, triangle). Preliminary measurements of harmonic distortion in the realized frequency modulation of the lasers show a larger second harmonic contribution for the piezo-modulated S laser than for the piezo-modulated P laser. Furthermore, this distortion typically has variations in amplitude and phase on a 1000 seconds time scale. These variations become much faster when the laser module temperature is changed, indicating that the harmonic distortion depends on laser module temperature. A second harmonic contribution in the frequency modulation influence the zero crossing of the error signal as seen by replacing msint with msint + m 2sin(2t + ϕ2) in Eq. (1), where m 2 and ϕ 2 are the amplitude and phase of the second harmonic distortion term. The zero crossing displacement depends only on Γ, m, m 2, and ϕ 2 and is independent of the error signal signal-to-noise ratio. As an example, for Γ = 500 kHz, m = 400 kHz, and a second harmonic distortion at –60 dB we calculate frequency shifts between ± 365 Hz depending on ϕ 2. Hence nonlinearity in the fiber laser piezo response may contribute to the average frequency difference seen in Fig. 7, and temperature induced variations in this nonlinearity may explain the Allan deviations at averaging times above 100 seconds in Fig. 8.
The frequency of the stabilized laser also depends on gas pressure, modulation amplitude, optical power, as well as small electronic offsets in the feedback loop and wavefront curvature. The frequency shifts are measured from changes in the beat frequency while changing the parameter under investigation for one of the lasers. A pressure-induced shift of (−315 ± 25) Hz/Pa is measured by varying the pressure in the P laser gas cell. Previously reported measurements of pressure shifts for the P(16) were much higher at 1.7 kHz/Pa [7,16]. We have no explanation for this discrepancy, but an independent measurement from the fitted 3f lineshapes used for the data in Fig. 4 gives a pressure shift of (−300 ± 50) Hz/Pa. With this value the 0.04 Pa variation in gas fillings between each measurement in Fig. 7 results in a 13 Hz contribution to the overall repeatability. The measured modulation shifts are (600 ± 120) Hz/MHz for laser P and (330 ± 150) Hz/MHz for laser S. For comparison, the modulation shift measured in  is (9.4 ± 0.6) kHz/MHz. The modulation shift may come from asymmetric lineshapes as well as from harmonic distortion. For Γ = 500 kHz, ϕ 2 = −π/2 and a second harmonic distortion at −60 dB the calculated modulation shift is 360 Hz/MHz near the optimum modulation amplitude. Hence a nonlinear piezo response resulting in harmonic distortion of the frequency modulation may explain differences observed in modulation shifts [7,8]. The power shifts are (8 ± 4) Hz/mW for laser P and (−16 ± 3) Hz/mW for laser S, and power shifts have negligible influence on the repeatability in Fig. 7. Variations in electronic offsets in laser S contribute with about 30 Hz to the repeatability in Fig. 7, whereas the variations in offsets in laser P are estimated to contribute with less than 10 Hz to the repeatability. Frequency instability from wavefront curvature variations have not been investigated, but it is expected to be a small effect since the laser output is delivered through polarization maintaining single mode fibers. The average frequency difference between the two lasers is likely caused by harmonic distortions, power shifts and electronic offsets.
A frequency standard based on the P(16) line of 13C2H2 is constructed by locking a narrow-linewidth fiber laser to a shot-noise limited saturated absorption signal. A lock-in amplifier bandwidth of just 31 Hz is used in the feedback loop. This is made possible by the high intrinsic frequency stability of the fiber laser. The applied scheme is simple and compact, and does not require external modulators, external enhancement cavities or multiple servo loops as used in most acetylene-stabilized lasers [7–9].
The saturated absorption lineshapes are analysed, and optimum working parameters are identified. A FWHM linewidth of 300 kHz is observed at a pressure of 0.13 Pa with a high signal-to-noise ratio of 35 dB in a 9.4 Hz detection bandwidth; to our knowledge the narrowest observed absorption lines for acetylene in this wavelength region.
Pressure broadening and pressure shift are measured. The pressure broadening is significantly larger than previously observed in Doppler-limited spectroscopy at higher pressure . The pressure shift is considerably smaller than in previous measurements , which is advantageous for a frequency standard.
The noise normalized error signal slope has a maximum at 1.1 Pa and suggests an achievable shot-noise limited fractional frequency instability of 4.5 × 10−13(τ/s)−½ for the stabilized laser. The measured short-term fractional instability of 5.0 × 10−13(τ/s)−½ and the long-term relative repeatability of 4.3 × 10−13, which were measured through the comparison of two acetylene-stabilized fibre lasers, are to our knowledge superior to previously published results for acetylene-stabilized lasers in the telecom range. The average fractional frequency instability reaches a minimum of 6 × 10−14 at an integration time of 150 s and increases to a long-term fractional instability of about 2.5 × 10−13, possibly caused by variations over time in the fiber laser piezo response and harmonic distortion in the applied frequency modulation.
This work was financially supported by the European Community's 7th Framework Programme, ERA-NET +, under the iMERA + project-grant agreement No. 217257.
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