The potential of fiber-laser-based sub-Doppler noise-immune cavity-enhanced optical heterodyne molecular spectrometry for trace gas detection is scrutinized. The non-linear dependence of the on-resonance sub-Doppler dispersion signal on the intracavity pressure and power is investigated and the optimum conditions with respect to these are determined. The linearity of the signal strength with concentration is demonstrated and the dynamic range of the technique is discussed. Measurements were performed on C2H2 at 1531 nm up to degrees of saturation of 100. The minimum detectable sub-Doppler optical phase shift was 5 × 10-11 cm-1 Hz-1/2, corresponding to a partial pressure of C2H2 of 1 × 10-12 atm for an intracavity pressure of 20 mTorr, and a concentration of 10 ppb at 400 mTorr.
©2008 Optical Society of America
Various types of laser-based absorption spectrometric (AS) techniques have, during the last decades, been successfully applied to trace gas analysis. Two of the major issues regarding their performance are sensitivity and selectivity, which both affect the ability to detect a given species in the presence of others . Over the years, the sensitivity has been significantly increased, either by the use of modulation techniques for reduction of noise (wavelength modulation spectrometry, WMS, or frequency modulation spectrometry, FMS), as in tunable diode laser absorption spectrometry (TDLAS) , or by the use of external cavities for enhanced interaction length , as, e.g., in cavity ringdown spectroscopy (CRDS)  and integrated cavity output spectroscopy (ICOS) . The selectivity has, however, only slightly been improved, either by multi-spectral fitting or reduction of the sample pressure to decrease the linewidth from the pressure broadened regime (a few GHz) to the Doppler limit (often in the hundreds of MHz range) . This might still be insufficient for accurate assessment of trace gases when concomitant species or isotopologues with overlapping transitions exist in large concentrations.
The spectral resolution can be significantly improved with sub-Doppler techniques, in which the linewidth is given solely by the homogenous broadening mechanisms and can, at low pressure, be in the low or even sub-MHz range. Although sub-Doppler spectroscopy has a great potential to resolve possible spectral interferences, it has not yet found wide application in the field of trace gas detection; it has so far mostly been used for studies of fundamental phenomena (e.g. collision processes and light-matter interactions through analysis of lineshapes) or for frequency standard applications (since the position of a transition can be determined to a fraction of its linewidth, i.e. with a kHz or sub-kHz accuracy). The reason is that sub-Doppler signals from molecules can be obtained only when a transition is saturated, which requires specific conditions regarding transition dipole moment, laser power, and sample pressure.
Simple pump-and-probe techniques in direct absorption spectrometry (DAS) under low pressure conditions are capable of saturating only the molecular transitions that have strong dipole moments (> tens of mD), which lie primarily in the electronic and the fundamental vibration bands, corresponding to the UV, mid- and far-infrared ranges. Sub-Doppler spectroscopy has been performed by this simplest methodology on species such as CO, OCS, and NO at ~5 µm (see Ref.  and references therein) and CH4 at ~3.2 µm . Also strong transitions in combination bands can be addressed by this methodology, as Castrillo et al. demonstrated by detection of CO2 at ~4.3 µm . However, in order to saturate overtone and weaker combination band transitions, which have significantly smaller dipole moments and lie mostly in the near infrared range, much higher laser intensities are required and other techniques have to be used. A common means of enhancing the intensity is an optical cavity.
Sub-Doppler spectroscopy from saturated transitions in Fabry-Perot (FP) cavities was first demonstrated by de Labachelerie et al., who obtained a narrow (2 MHz wide) sub-Doppler response from acetylene at ~1.5 µm using a low-power diode laser and a FP cavity with a finesse of 120 . Later, Gagliardi et al. reported on the observation of sub-Doppler signals from water in the transit-time-limited regime at combination bands at ~1.4 µm using a sub-mW laser and an external FP cavity with a finesse of 500 . Using a transition in the fundamental bands of CH4 at ~3.4 µm, Anzai et al. demonstrated that Lamb dips can be detected in an optical cavity with as little as 3.9 µW of input power . This clearly shows that it is possible to perform sub-Dopler spectroscopy in external cavities with low-power lasers. However, since direct cavity-enhanced (CE) AS techniques require locking of the laser frequency to a mode of the cavity, they are susceptible to laser-frequency noise, which often limits their sensitivity.
Sub-Doppler signals have also been obtained with the CRDS technique . However, since the light intensity decreases during a ringdown event, the sub-Doppler feature changes shape and size during a measurement, which implies that the CRDS technique is not optimal for quantifiable sub-Doppler gas detection.
In addition, it has recently been shown that substantial intensities can build up over long absorption lengths in hollow core fibers . Sub-Doppler signals from a variety of species have been demonstrated lately (see Ref.  and references therein). However, the technique suffers from a slow gas exchange, given by the diffusion time of gas in the fiber material, wherefore its use for trace gas analysis is severely restricted .
A highly sensitive CE technique for detection of sub-Doppler signals is noise-immune cavity-enhanced optical heterodyne molecular spectrometry (NICE-OHMS). Due to the unique combination of cavity-enhanced absorption, for prolonged interaction length with the sample, with frequency modulation (fm), for reduction of 1/f noise, the technique is immune to laser frequency noise [15–18]. NICE-OHMS was originally developed in its Doppler-free mode of operation for frequency standard applications [19–21] and sensitivities down to 10-14 cm-1, close to the shot noise limit, were demonstrated using fixed frequency lasers and cavities with finesse up to 105 . It has since then been used for studies of weak molecular transitions and for trace gas applications by detection of both Doppler-broadened [22–26] and sub-Doppler [27, 28] signals, and sensitivities in the 10-10–10-11 cm-1 range have been routinely obtained using tunable lasers and cavities with finesse in the 103–104 range. A review of all realizations of NICE-OHMS, as well as a comparison with the performance of other techniques for trace gas detection has been given recently .
Taubman et al. , who used a quantum cascade laser for chemical sensing of N2O at 8.5 µm by various CE techniques, including NICE-OHMS, concluded that the sub-Doppler mode of detection is a promising alternative to Doppler-broadened detection. However, they did not provide any systematic study of sub-Doppler NICE-OHMS signals. Therefore, this paper addresses some of the key issues of the Doppler-free mode of operation of the NICE-OHMS technique, aiming for assessing its potential for trace gas detection. In particular, it investigates the influence of the degree of saturation, the intracavity pressure and power on the sub-Doppler signal strength and shape. The optimum pressure, i.e. the one at which the signal is maximized for a given intracavity power, is discussed. The analysis is restricted to the on-resonance carrier-carrier interaction, which only gives rise to a dispersion signal. The ability to reduce and/or to eliminate the influence of spectral interference from concomitant species is, however, not investigated here and will be the subject of a future work.
The theoretical modeling is adopted from Ref. , which provides a detailed description of sub-Doppler NICE-OHMS dispersion signals for both low and high degrees of saturation. The experiments were performed on C2H2 with a fiber-laser-based (FLB) NICE-OHMS system, whose performance in the Doppler-broadened mode of operation has been assessed previously [25, 26, 30]. The main results and conclusions should, however, be valid also for NICE-OHMS systems based on other types of lasers.
2. Signal strength and shape
The strength and shape of the sub-Doppler NICE-OHMS signals in the low saturation regime (for G 0<1, where G 0 is the degree of saturation induced by the carrier) were studied by Ye . Recently, Axner et al.  extended the theory to arbitrary degrees of saturation. It was shown that the center sub-Doppler fm-NICE-OHMS dispersion signal, originating from the carrier-carrier interaction, is well described by a Lorentzian dispersion function up to high degrees of saturation (G 0 = 100), and that its peak-to-peak value increases monotonically with the degree of saturation as 0.45G 0(1+G 0)-1 α 0/2 up to a value of 0.45α 0/2, where α 0 is the unsaturated Doppler-broadened peak absorption.
The sub-Doppler wm-NICE-OHMS dispersion signal, detected at the first harmonic of the wm dither frequency, can be described as a product of a signal strength and the first Fourier coefficient of a Lorentzian dispersion function , i.e. as
The signal strength, SwmDF,0(cA,G 0), can be written as
where ηwm is an instrumentation factor (V/W), F the cavity finesse, Jj(β) the Bessel function of order j, β the fm modulation index, P 0 the power incident on the detector (W), S the transition linestrength (cm-2/atm), χ0 the peak value of the area-normalized Gaussian function (cm), given by , where δνD is the Doppler width of the transition (Hz), cA the relative concentration of the analyte, p the total intracavity pressure of the gas (atm, wherefore cAp is the partial pressure of the analyte, pA), L the cavity length (cm), and where Φ(G 0) is a degree-of-saturation-dependent function that relates the peak-to-peak sub-Doppler optical phase shift to (half of) the single-pass unsaturated Doppler-broadened absorption, α 0 = Sχ 0 cApL, and accounts for the Gaussian intensity distribution of the cavity mode with a radius w 
The single-pass sub-Doppler optical phase shift is therefore given by α 0 Φ(G 0)/2.
The first Fourier coefficient of the Lorentzian dispersion function, χdispL ,1(Δν,νa,δνL), can be calculated by numerical integration as
where Δν is the detuning from the center of the transition, νa the modulation amplitude, δνL the homogenous linewidth (HWHM) and fm the modulation frequency (all in Hz). The peak value of this function depends on the modulation amplitude and is maximized to a value of 0.60 for a modulation amplitude equal to 1.27δνL
It should be noted that the signal strength, as defined in Eq. (2), is not the peak value of the sub-Doppler wm-NICE-OHMS signal, but is rather proportional to the peak-to-peak value of the sub-Doppler fm-NICE-OHMS signal and independent of the wm modulation amplitude.
3. Experimental setup and procedure
The experimental setup, shown in Fig. 1, which has been described in detail elsewhere [26, 30], was based on a distributed-feedback-laser-pumped erbium-doped fiber laser (EDFL, Koheras, Adjustik E15) with a free-running linewidth of 1 kHz over 120 µs, working in the 1530.8–1531.8 nm range, and a cavity with a length of 39.48 cm, a free-spectral-range (FSR) of 379.9 MHz, a finesse of 4800 and power buildup of 1300. The only modifications to the setup used in the previous works were that the fiber-coupled electro-optic modulator (EOM) was replaced with a device of the same kind (Photline, MPZ-LN-10, bandwidth of 30 kHz – 10 GHz) with shorter input and output fibers (25 cm instead of 1.5 m) and that the optical fiber connecting the fiber coupled polarizer and the EOM was temperature stabilized. This was done to reduce the temperature induced drift of the refractive index of the fiber in front of the EOM, which caused varying fm background signal. The fiber was wound around a copper block, whose temperature was measured with an AD590 sensor (Analog Devices) and stabilized with a laser temperature driver (Newport, 300B) with an accuracy of ca 0.1°C through two Peltier elements (Supercool) connected in series.
The EOM was used to create the 20 MHz sidebands (with a modulation index of 0.30) for Pound-Drever-Hall (PDH) locking of the laser to the cavity mode , as well as the 379.9 MHz sidebands (β = 0.334) for the NICE-OHMS detection. The fm modulation frequency was locked to the cavity FSR by the deVoe-Brewer technique . Although the cavity length did not change much during a scan over the sub-Doppler profile, the active lock was crucial to avoid manual adjustments of the modulation frequency when measurements at different intracavity pressures were performed (the FSR changes with the intracavity pressure).
A scan over a sub-Doppler signal was achieved by applying a triangular low frequency (40–100 mHz) ramp from a function generator (HP, 33120A) to the cavity input piezo. The signal was detected in cavity transmission (New Focus, 1611, PD2), amplified (Mini-Circuits, ZRL-700) and demodulated at 379.9 MHz with a double-balanced mixer (Mini-Circuits, ZLW-1). The fm detection phase was set with a phase shifter in the reference arm (Mini-Circuits, JSPHS-446) to dispersion phase by maximizing the sub-Doppler signal. A wm dither at 125 Hz was applied to the cavity output piezo and the signal was further demodulated with a lock-in amplifier (Stanford Research Systems, SR830). The maximum dither amplitude at this frequency was 5.4 MHz, limited by the gain of the laser-to-cavity locking servo. Both the scan and the dither were fed forward to the laser piezos (after proper adjustment of the gain and phase) in order to remove some load from the locking servo, which was particularly important for the dither.
For frequency calibration purposes, a frequency scale was recorded after each measurement series with the laser unlocked by scanning and dithering the cavity with the same voltage as during the signal acquisition and recording the 1f wm signal in direct cavity transmission. The output from a function generator was applied to the EOM to create the first and second order sidebands at low RF frequencies, typically around 10 MHz, which, together with the carrier, provided 5 frequency markers. The dither amplitude was calibrated by fitting a second Fourier coefficient of a Lorentzian lineshape function  to the 2f wm signals obtained by scanning the laser around a cavity mode, whose width was known (40 kHz).
The laser output power was 14 mW, which resulted in a maximum power of 3.5 mW in front of the cavity (mostly due to power losses in the EOM and in the fiber polarizer). The input power to the cavity was adjusted with a variable neutral density filter (Reynard Corporation, variable attenuator, VA1) using the same procedure as in Ref. . The power reaching the detector in reflection was kept constant with the help of VA2 (in order to keep the gain in the locking servos constant). For each measurement, the cavity transmitted power, Pt, which was equal to 86% of the cavity input power, was measured and recalculated to intracavity power through the relation Pc = 1530 Pt . A third attenuator, VA3, was used in front of the transmission detector to avoid saturation when high optical powers were transmitted through the cavity and to provide a constant DC level for all degrees of saturation used (P 0 = 0.25 mW).
Measurements were performed on a strong acetylene transition in the ν 1+ν 3 band at 1531.588 nm [Pe(11)] with a linestrength, S, of 0.29 cm-2/atm , a dipole moment, µ, of 7.40 mD  and a Doppler width (HWHM), δνD, of 236.5 MHz at room temperature (296 K). The pressure dependence of the saturation power for this transition is given by
where Cs = 3cε 0 h 2/(2µ 2), and where the transit-time broadening, Γtt, and the pressure broadening coefficient, B, were determined experimentally from measurements of optically saturated Doppler-broadened NICE-OHMS signals to be 94 kHz and 5.55 kHz/mTorr, respectively . The beam radius was taken as the average over the cavity length (0.54 mm).
Acetylene was available as a premixed gas of 1000 ppm of C2H2 in N2. In order to obtain lower relative concentrations this gas was diluted with pure N2 by volumetric mixing. The pressure was measured with two capacitive sensors (Leybold, Ceravac CRT 90), which covered a pressure range from 10-5 to 1 Torr, one at each side of the valve that separated the cavity from the vacuum system. The intracavity pressure increased at a rate of 0.05 mTorr/min, wherefore no long-term quantitative measurements were performed at intracavity pressures below 10 mTorr. The upper pressure limit was determined by the working range of the cavity piezos, which was 1 Torr.
4.1 Typical sub-Doppler NICE-OHMS signals
Typical examples of strongly saturated fm-NICE-OHMS absorption and dispersion signals are shown in Fig. 2. The signals were taken at an intracavity pressure, p, of 20 mTorr, with a C2H2 concentration, cA, of 500 ppm. The intracavity power, Pc, was 4.6 W, resulting in a degree of saturation for the carrier and the fm sidebands, G 0 and G ±1, of 50 and 1.5, respectively. The amplitude of the Doppler-broadened absorption signal, Fig. 2(a), is reduced by a factor of (1+G +1)-1/2, while the Doppler-broadened dispersion signal, Fig. 2(b), is basically unaffected by optical saturation in the Doppler limit [30, 35]. Nine narrow sub-Doppler features can be seen on top of the Doppler-broadened background, four at the absorption and five at the dispersion detection phase, each originating from interaction of various pairs of fm light modes with specific velocity groups of molecules. The center sub-Doppler feature, originating mainly from the carrier-carrier interaction, is missing in the absorption signal, due to the insensitivity of FMS to the absorption of the carrier . The outermost signals, at a detuning equal to the fm modulation frequency, i.e. at 380 MHz, are created mainly by sideband-sideband interactions, while the signals at a detuning equal to half the fm modulation frequency, i.e. at 190 MHz, originate from carrier-sideband interactions.
The center dispersion signal is the largest of all the sub-Doppler NICE-OHMS signals, and therefore an obvious choice for trace gas detection. The lack of the corresponding absorption signal implies that incorrect setting of the experimental phase will affect only the size, and not the shape, of the signal. Examples of this signal obtained under different intracavity pressure and power conditions are shown in Fig. 3. The upper panels show the fm-NICE-OHMS signals, while the lower panels display the corresponding wm signals. All signals were taken with the same partial pressure of C2H2 (pA = 10 µTorr). The three vertical pairs of panels correspond to intracavity pressures of 10, 100 and 500 mTorr, respectively, and the two curves in each panel represent different intracavity powers (0.49 and 4.1 W, respectively). The fm-NICE-OHMS signals were recorded with an electronic bandwidth of 180 Hz and an acquisition time of 1.25 s. For acquisition of the wm-NICE-OHMS signals the time constant of the lock-in amplifier was set to 10 ms and each scan was recorded over 12.5 s. The modulation amplitudes were optimized for the high power signals (wherefore the low power signals were overmodulated) and equal to (d) 2.3 MHz, (e) 3.6 MHz, and (f) 5.4 MHz, respectively. Equation (1) was fitted to the wm-NICE-OHMS signals in Figs 3(d)–(f), with the center frequency, linewidth and signal strength as the fitting parameters. The theoretical lineshapes fit well and overlap the measured curves almost completely, as can be seen in the residues shown below the panels. The power- and pressure-broadened widths returned by the fits were, for the two powers used, (d) 0.85 MHz and 1.9 MHz, (e) 1.7 MHz and 3.0 MHz, and (f) 4.1 MHz and 5.0 MHz, respectively. The figure also shows that the slope of the Doppler-broadened signal is removed by the wm process, which facilitates the quantification of the signal and provides good rejection of drifts of the background.
4.2 Power, pressure, and concentration dependence of the center dispersion signal
The strength and linewidth of the sub-Doppler NICE-OHMS dispersion signal depend strongly on the experimental conditions, mainly the degree of saturation, which, in turn, is affected by the intracavity pressure and power. Figure 4(a) shows, by the solid markers, the strength of the sub-Doppler wm-NICE-OHMS signal, SwmDF,0, as a function of degree of saturation for a partial pressure of C2H2 of 10 µTorr retrieved from fits to signals similar to those shown in Fig. 3. The large range of degrees of saturation (from 0.03 to 90) was obtained by using various combinations of cavity input power and intracavity buffer pressure. The power incident on the transmission detector was kept constant to remove the influence of P 0 on the signal strength. The wm modulation amplitude was equal to 3.5 MHz. The solid curve shows a fit of Eq. (2), verifying the validity of the theory for sub-Doppler NICE-OHMS presented in Ref. . Since the signal strength increases with the degree of saturation, and since there is virtually no background in the sub-Doppler wm-NICE-OHMS signal, increasing the power is, in principle, always beneficial, although the signal reaches 90% of its strength for a degree of saturation of 10 .
Figure 4(b) shows the same data as Fig. 4(a), this time plotted as a function of intracavity pressure for different intracavity powers [for clarity only the data for every second power used in Fig. 4(a) are shown]. For all intracavity powers the signal strength decreases with increasing pressure, although the decrease is much faster for the lower powers (a factor of 15, i.e. from 11.5 to 0.75 V, for an increase of the intracavity pressure by a factor of 50, from 10 to 500 mTorr, for an intracavity power of 0.54 W) than for the higher (only a factor of 3.5, i.e. from 14 to 4 V, for the same increase in pressure but a power of 4.38 W). The reason is that the degree of saturation is highest for the lowest intracavity pressures, wherefore a certain change in power (which gives rise to a corresponding change in the degree of saturation) affects the signal strength less than at higher pressures, where the degree of saturation is smaller and the slope of the signal–strength-vs.–degree-of-saturation dependence is larger.
A practically relevant question is which pressure an atmospheric pressure sample should be pumped down to in order to yield the maximum signal strength. Figure 5(a) shows the pressure dependence of the strength of the sub-Doppler wm-NICE-OHMS signal from a sample with a constant concentration (mixing ratio) of C2H2 (20 ppm) for different intracavity powers. For each intracavity power there is an intracavity pressure at which the sub-Doppler NICE-OHMS signal is maximized (for smaller pressures the signal decreases because of a reduced partial pressure of the analyte, whereas for larger pressures the signal decreases because of a reduced degree of saturation). Since a higher degree of saturation can be obtained with a higher laser power, the maximum of the signal strength moves to higher pressures for higher powers. This means that the higher the intracavity power the less one has to pump an atmospheric pressure sample down in order to obtain the largest analytical signal. For example, for the highest intracavity power of 4.45 W the maximum sub-Doppler NICEOHMS signal is obtained at a pressure of ca 400 mTorr, whereas for an intracavity power of 0.55 W the maximum signal is obtained at a pressure of 140 mTorr. Moreover, the maximum signal for the highest intracavity power (4.45 W) is ~4.5 times larger than for the lowest power (0.55 W). This shows that for the highest intracavity power used in this study an atmospheric pressure sample should be pumped down approximately three orders of magnitude to yield maximum signal strength.
It has been shown previously that the Doppler-broadened NICE-OHMS signal strength is linear with analyte concentration up to concentrations at which the double pass absorption is a significant fraction of the empty cavity losses [20, 21, 25, 26]. The sub-Doppler wm-NICE-OHMS signal has a similar dependence, as is shown in Fig. 5(b), where its strength is plotted as a function of concentration for three different intracavity pressures. The signals were measured with the maximum intracavity power and a modulation amplitude optimized with respect to the linewidth [the same as in Figs 3(d)–(f)]. A straight line was fitted to the data recorded at the lowest pressure, 10 mTorr, whereas a polynomial fit was used for the data recorded at 100 and 500 mTorr in order to account for the nonlinear dependence at the highest concentrations. The slopes of the three calibration curves at low concentrations are different since the strength of the signal from a given concentration of the analyte depends on the intracavity pressure, as was shown in Fig. 5(a). The nonlinearity is significant (> 5%) when the double pass absorption of C2H2 exceeds 5% of the empty cavity losses, sses, 2π/F, equal to 1.3 × 10-3, which happens e.g., above 4000 ppm at 10 mTorr [outside the range of Fig. 5(b)] and above 80 ppm at 500 mTorr. Since the decrease of finesse is a fully predicable phenomenon, it does not pose a limitation to the dynamic range of the technique; it only restricts its linear dynamic range. It should be noted that optical saturation reduces the double pass absorption due to the analyte, wherefore the linear dynamic range increases with the degree of saturation (and thereby laser power). For a given intracavity pressure, the upper limit of the dynamic range of sub-Doppler NICE-OHMS is estimated to be the same as that of Doppler-broadened NICE-OHMS, which is roughly 2 orders of magnitude above that of the linear dynamic range.
Although the spectral resolution of the sub-Doppler NICE-OHMS technique is not covered in any detail in this work, it is of interest to scrutinize the linewidth of the signals, mainly its dependence on intracavity power and pressure. Figure 6 shows the linewidth, δνL, evaluated from the fits to the sub-Doppler wm-NICE-OHMS signals whose strength was shown in Fig. 4, as a function of intracavity power for all intracavity pressures used. Both the pressure and the power broadening are clearly visible. It was recently shown by Axner et al.  that the widths of the sub-Doppler NICE-OHMS dispersion signals cannot be modeled by the conventional expression, δν 0 L(1+G 0)1/2, where δν 0 L is the pressure-broadened linewidth . The reason is that the molecules experience different light intensities depending on the trajectory through the beam. The linewidth would therefore need to be evaluated by an integration over all paths through the beam (and thereby degrees of saturation) if a proper description should be given. Since this is outside the scope of this work, no fits are shown in the figure. As has already been shown in Ref. , for high degrees of saturation the power broadening is actually smaller than predicted by the conventional expression (e.g., at 10 mTorr the linewidth increases by a factor of 2.2, from 0.87 to 1.9 MHz, for a change of degree of saturation from 11 to 90, whereas the conventional expression predicts a power broadening by a factor of 2.75) . Since the risk for spectral interference on the MHz scale is small, the power broadening of the sub-Doppler NICE-OHMS signal is of little concern for trace gas applications.
4.3 Sub-Doppler FLB-NICE-OHMS sensitivity
The long term stability of the fiber-laser-based NICE-OHMS spectrometer in the Doppler-free mode of detection was evaluated by the use of Allan variance. Measurements of C2H2 were performed with the maximum possible intracavity power (4.7 W) at two different intracavity pressures: the lowest at which the influence of the leak in the vacuum system was negligible (20 mTorr), and the pressure at which the signal from a given concentration was maximized [400 mTorr, according to Fig. 5(a)]. For each intracavity pressure the modulation amplitude was optimized with respect to the linewidth to yield the largest peak value of the sub-Doppler wm-NICE-OHMS signal (2.6 MHz at 20 mTorr and 5.4 MHz at 400 mTorr) and thus the maximum signal-to-noise ratio. The lock-in time constant was set to 3 ms, and a scan over a sub-Doppler signal was acquired every 5 seconds. The concentration was retrieved from fits to the experimental lineshapes. The results of these long-term measurements are shown in the upper panels of Fig. 7, and the plots of the corresponding square root of Allan variance of the sub-Doppler optical phase shift, σ(τ), are shown in the lower panels.
The solid lines in the lower panels show the τ -1/2 dependence characteristic for white noise. The data taken at 20 and 400 mTorr follow this behavior for integration times up to ~200 and ~70 s, respectively, and the gray curves in the upper panels show the concentration averaged over these integration times. The drift at longer integration times originates mainly from the residual amplitude modulation (RAM) from the EOM, which results in a dc offset in the fm-NICE-OHMS signal and destroys the ‘noise immunity’ to a certain extent. In the presence of RAM, the noise in the fm-NICE-OHMS signal is a copy of the laser amplitude noise, with an amplitude proportional to the magnitude of the dc offset. The latter depends e.g. on the temperature of the fiber components (see discussion in the next section), especially the polarization maintaining (PM) fibers and the EOM, wherefore a drift in the background signal can be induced by a change in the room temperature. This causes, in turn, a varying level of noise in the wm-NICE-OHMS signal. In this experiment the PM fiber in front of the EOM was temperature stabilized to adjust the initial dc offset close to zero. The remaining drift, which is visible in Fig. 7, was caused mainly by temperature changes of the EOM and the PM fiber after the EOM. The reason why the drift starts at shorter averaging times for measurements at higher intracavity pressure is not yet fully understood. A possible explanation is that a larger frequency scanning range (70 MHz at 400 mTorr as compared to 40 MHz at 20 mTorr) and a larger modulation amplitude were needed, which put more stress on the laser-to-cavity lock.
The Allan variance of white noise is equivalent to the statistical variance, thus its square root can be used to determine the detection limit . The minimum detectable single-pass sub-Doppler optical phase shift, Φ(G 0)α 0/2, was found to be 2.2 × 10-9 Hz-1/2 at 20 mTorr and 1.9 × 10-9 Hz-1/2 at 400 mTorr. The small difference between these numbers originates from the slightly different experimental conditions during the two measurement. Since it is not suitable to quote a sensitivity in terms of an integrated absorbance for a technique that measures a signal created by optical saturation, we have chosen to express the sensitivity normalized with respect to the cavity length. The two noise-equivalent sub-Doppler optical phase shifts then become 5.7 × 10-11 cm-1 Hz-1/2 and 4.9 × 10-11 cm-1 Hz-1/2, respectively. The corresponding minimum detectable single-pass sub-Doppler absorption, given by α 0[(1+G 0)-1/2-(1+2G 0)-1/2] , was 1.0 × 10-11 cm-1 Hz-1/2 at 20 mTorr (G 0 = 54) and 8.0 × 10-11 cm-1 Hz-1/2 at 400 mTorr (G 0 = 0.42). The minimum detectable concentration of C2H2 at 20 mTorr was 39 ppb for an integration time of 200 s, corresponding to a partial pressure of C2H2 of 0.8 nTorr and a single-pass sub-Doppler optical phase shift of 1.6 × 10-10. At 400 mTorr the minimum detectable concentration was 10 ppb for an integration time of 70 s, corresponding to a partial pressure of 4 nTorr, and a single-pass sub-Doppler optical phase shift of 2.3 × 10-10. The detection limit in terms of concentration at 400 mTorr is thus only a factor of 4 lower than at 20 mTorr [and not a factor of 7 as could be expected from Fig. 5(a), which is due to the shorter optimum integration time].
The shot-noise-limited sub-Doppler optical phase shift, ϕ min, can be deduced from Eqs (26) and (27) in Ref.  and be written as
where χdispL ,1(0,νa,δνL) accounts for the wavelength modulation. For the conditions of the measurements performed at 20 mTorr, i.e. β of 0.334, an electronic bandwidth, Bw, of 5 mHz (averaging time 200 s), η of 1 A/W, P 0 of 0.25 mW, νa of 2.6 MHz, δνL of 2.3 MHz, and G 0 equal to 54, ϕ min takes a value of 6.1 × 10-12. This implies that the sensitivity is ~26 times above the shot noise limit.
5. Discussion and conclusions
The advantage of NICE-OHMS over other sub-Doppler techniques is that it can detect dispersion signals due to the use of FMS. The Lamb dip detected in pump-and-probe and direct CEAS techniques has a depth of at most 13% of the Doppler-broadened absorption, while in NICE-OHMS, the sub-Doppler dispersion signal can have a size equal to 22.5% of the Doppler-broadened absorption . Moreover, the size of the Lamb dip decreases for degrees of saturation above 1.4, whereas the strength of the sub-Doppler NICE-OHMS dispersion signal increases monotonically with increasing degree of saturation . Therefore it is in principle always beneficial for sub-Doppler NICE-OHMS to maximize the intracavity power, which can be done either by using a higher laser power or a cavity with a larger finesse. A higher intracavity power results also in a higher optimum intracavity pressure, which implies that an atmospheric pressure sample needs to be pumped down less, which improves the minimum detectable concentration in an atmospheric pressure sample.
The width of the sub-Doppler signal is determined by pressure, power, and transit-time broadening, which all are significantly smaller than a typical Doppler-broadened linewidth (the power and pressure broadenings are of the order of a few MHz even for the highest powers and at Torr pressures, whereas transit-time broadening is in the sub-MHz range). This implies that the width of the sub-Doppler signal is not an important issue for the detectability of the technique, since it is sufficiently small for interference-free trace gas detection.
Increasing the modulation index should in principle increase the signal strength, as the product J 0(β)J 1(β) is maximized for β = 1.08. However, it has also other consequences. First of all, the center sub-Doppler NICE-OHMS dispersion signal has contributions from the sideband-sideband interactions, which alter the signal strength and shape and cannot be neglected for higher modulation indices. Secondly, the power in the carrier decreases with increasing modulation index, wherefore the degree of saturation decreases. Thirdly, second order sidebands appear, which further affect the signal shape and strength. Finally, when a larger modulation index is used, the RAM from the EOM increases, which implies that the detectability would not be improved unless some active measures are taken to reduce the background signal. The optimum modulation amplitude is therefore below the one that maximizes the J 0(β)J 1(β) product, and is, to a certain extent, system dependent.
For a given laser power, a cavity mode with a smaller diameter would provide a higher intensity and thereby a larger sub-Doppler signal. Reducing the beam size would therefore be beneficial for sub-Doppler NICE-OHMS, since the transit-time broadening is of little importance under the conditions that normally prevail for trace gas applications. However, the size of the beam is determined by the geometry of the cavity, i.e., the cavity length and the curvature of the mirrors, which should be considered already at the construction stage.
The major limitation of the performance of the sub-Doppler FLB-NICE-OHMS instrumentation used in this work was the RAM created in the fiber-coupled EOM, which gave rise to fm-NICE-OHMS background signal. As was discussed by Wong and Hall , this RAM, which is present only at the dispersion phase, can be caused by a misalignment between the linear polarization direction of the light entering the EOM and the slow axis (the RF modulation direction) of the EOM, or by an ellipticity of the input light polarization. Both cases give a projection on the fast axis of the EOM and therefore different phase shifts for the fast-axis carrier and the slow-axis sidebands after the e EOM, ϕ o and ϕ e, respectively. Unless the output polarizer is fully aligned with the slow axis of the EOM and removes the fast-axis component, the beat signal between these components creates a dc offset proportional to sin(ϕe-ϕ o) at dispersion phase . Moreover, the polarization of light in the PM fiber in front of the EOM can be elliptical, which introduces an initial phase difference, ϕPM, between the slow-axis and fast-axis components of the light in the EOM, making the amplitude of the dc offset proportional to sin(ϕPM+ϕe-ϕo). Since all these phase shifts are temperature dependent, they can give rise to a drifting dc-offset. This offset can be removed by controlling the temperature of the PM fiber and the EOM. In this experiment the PM fiber in front of the EOM was temperature stabilized, which reduced the offset in the fm-NICE-OHMS signal, and the associated noise in the wm-NICE-OHMS signal, by 2 orders of magnitude. The remaining drift, which is considered to be the limiting factor for the present instrument, can be eliminated by the use of an active stabilization scheme. As suggested by Wong and Hall , this could be done by applying a dc voltage to the EOM and thus modifying the refractive index along the slow axis. However, this was not implemented here but will be the subject of a future work.
As long as the specific saturation conditions required to obtain the sub-Doppler signal are fulfilled, NICE-OHMS can provide low detection limits and a large dynamic range (5–6 orders of magnitude). The sub-Doppler sensitivity presented in this work, ~5 × 10-11 cm-1 Hz-1/2, is better than that of direct CEAS (e.g. ~3000 times better than that of Gagliardi et al. ), as well as wm-CEAS , and comparable to those of more established CE techniques used for detection of Doppler-broadened signals, e.g. CRDS  and ICOS . Moreover, the sensitivity of the present instrumentation can be improved, not only by the factor of 20 that remains to the shot noise limit, but also further, e.g. by using cavities with higher finesse, as was demonstrated by the founders of the technique .
Although only one species was detected with sub-Doppler NICE-OHMS in this work, there exist a number of other molecules with transitions within the wavelength tuning range of the fiber laser, e.g. CO2, N2O, CH4, NH3, H2S, and CH2O. Moreover, narrow linewidth fiber lasers exist in several wavelength ranges, presently corresponding to the emission windows of Yb, Er, and Tm doped fibers, i.e., from 1020 to 1200 nm, from 1530 to 1600 nm, and from 1700 to 2000 nm, respectively. The application of sub-Doppler NICE-OHMS can also be extended to other wavelength ranges by the use of different laser sources, e.g. quantum cascade lasers , external cavity diode lasers [22, 23, 27], and presumably also distributed feedback lasers. This implies that it should be possible to construct NICE-OHMS instrumentation for sub-Doppler detection of a variety of species.
In conclusion, sub-Doppler NICE-OHMS is a technique with high sensitivity and a wide dynamic range. It is significantly less affected by etalon effects than Doppler-broadened techniques, and most importantly, it provides a signal with a very narrow linewidth, which opens up possibilities for spectral-interference-free detection. The main drawback is that the sample has to be pumped down to a pressure at which the transition can be saturated, which might reduce the detectability with respect to the Doppler-broadened mode of detection. Nevertheless, the technique has a large potential for trace gas detection when sub-Doppler resolution is needed.
This work was supported by the Swedish Research Council under the project 621-2005-4919. The authors would also like to acknowledge the Kempe foundations, Carl Trygger’s foundation, and Stiftelsen Johan Gustaf Richert for support of this project.
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