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Abstract
We propose a new simple approximate solution to the two-state rate equation model for analyzing decay signals of saturation cavity ring-down spectrometry in the adiabatic and low-saturation regime. It helps obtain baseline-immune Doppler-free spectra for hyperfine transitions and linear absorption coefficients of a gas in the saturation regime. To demonstrate it, a baseline-immune Lamb dip spectrum of the R1A2 transitions in the 2v2 + v3 band of methane was recorded. The line position was determined to be 6 076.108 457 7(11) cm−1, the relative uncertainty being 1.8 × 10−10.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
High-accuracy laser spectroscopy has a wide range of applications in physics, chemistry and other areas of science. Some examples from physics are the study of molecular hyperfine structure [1], radioisotope detection [2], testing the cosmological variation of the proton-to-electron mass ratio [3], and medical diagnostics [4]. A powerful means to overcome Doppler broadening present in linear absorption spectroscopy, without having to cool the sample, is saturated absorption (or saturation) spectroscopy [5]. Its basic method is implemented by using a pump beam to pump a considerable fraction of atoms or molecules from the ground state to the excited state, while a probe beam is used to measure the fraction of ground-state population. Provided the gas pressure is not too high, the linewidth of the Doppler-free spectrum (arising from homogenous collisional and spontaneous emission broadening) is roughly two to three orders of magnitude smaller than the inhomogeneous Doppler width (proportional to the velocity dispersion of the absorbers). However, while the basic method works well on strong transition lines, it is less well suited to the study of weak ones, where sensitivity is limited by the lack of available optical power density.
By measuring the dynamical relaxation process of the photon absorption enclosed in an optical resonator, saturation cavity ring-down spectroscopy (SCRDS) [6] has proven its potential for saturation spectroscopy of weak lines [1,7,8]. The understanding of dynamical relaxation process is important for SCRDS [9] and the measurements of low pressure gas concentrations [1]. Several models [9–13] have been proposed to describe time-dependent gas saturation. Giusfredi et al. [1,10] derived the theoretical approximate differential model by starting from ab initio calculations in the framework of the density matrix formalism, within the adiabatic approximation regime. This model shows the powerful ability of decoupling empty-cavity background and absorption signals. Even so, concise and satisfactory models are still wanted. Here we propose a novel approximate solution, which can effectively treat the dynamical absorption process in the adiabatic and low saturation regime, thereby enabling one to obtain baseline-immune Doppler-free spectra and linear absorption coefficients of a gas.
The rest of the paper is as follows. In section 2 we present the new model to describe population dynamics in SCRDS. The experimental apparatus is summarized in section 3. Some results are shown in section 4, together with a discussion. The final section, 5, contains a conclusion and perspectives for applications.
2. Dynamical model
To describe the dynamic absorption process in an optical resonator, we consider a closed two-level (Eg, Ee) system of absorbing particles with a population density difference N = Ng - Ne at thermal equilibrium without a radiation field [9]. Following the approach of Lee and Hahn [9], to simplify the problem, we neglect the influence of the standing-wave intra-cavity optical field and spatially average the different saturation levels of molecules interacting with the radiation over the length of the resonator. The rate equations for the system can then be written as [9]:
where ρ is the intra-cavity photon energy density, and n = Ng−Ne the population density difference of a sample. The variable τc = (αcc)−1 denotes the decay time of the cavity loss, h the Planck constant, the Einstein coefficient, f the optical frequency and c the speed of light in vacuum. The overall relaxation rate of the upper state population, due to spontaneous emission, molecular collisions, transit-time, etc [14]., is denoted by R. In the absence of saturation, the population density difference n lies very close to N. In the saturation regime, however, the value of n differs markedly from N due to the pumping rate 2Bfρ being comparable with the relaxation rate R. In such a case, the population density difference n and the intra-cavity photon energy density ρ are both time-dependent and mutually coupled. This also makes the total absorption coefficient in SCRDStime-dependent. If we assume the population density of the two-level system is in steady state at the start of each decay event (dn/dt = 0), the initial population density difference n(0) is given bySince the absorption coefficient is a linear function of the population density difference (Eq. (2)), we propose a new approximate solution based on numerical simulation of the time variation of the population density difference n(t). The simulation result by using Eq. (1) when R/γc = 100 (an approximate adiabatic regime) is shown in Fig. 1. The evolution curve of n follows the exponential behavior of
when the initial saturation parameter S(0) is small. The characteristic constant time τs is defined by 1 - n(τs)/N = [1 - n(0)/N]/e. The limit of application of our model is the adiabatic and low saturation regime. This leads straightforwardly towhere I(t) is the optical power leaking from the cavity at time t, αu is given byandis the decrement of the absorption coefficient due to the saturation effect at the start of a decay event.
Fig. 1 The numerical simulation results of the population density difference n in different initial saturation parameters.
Put simply, in this new model, the total absorption coefficient α(t) = αu - Δαsexp(-t/τs) decays exponentially with time. The model contains five parameters: I(0), αu, ∆αs, τs, and the background parameter Ib.
In this work, the time-dependent saturation parameter S(t), which is used to characterize the degree of saturation. The time-dependent saturation parameter S(t) [14] can be expressed as
From Eqs. (2) and (6–8), the saturation parameter at t = 0 is equal to S(0) = ∆αs/(αu - ∆αs - αc).3. Experimental apparatus
The performance of the new approach was tested using our existing experimental set-up [8], inspired by the designs of Hodges et al. [15,16] and Lin et al. [17]. Details of the apparatus can be found in our earlier article [8]. The sample cell (the ring-down cavity) consists of a 1.3 m long hollow invar cylinder and two high-reflectivity mirrors (99.998%) mounted in custom-designed Invar flexture mounts. An extended-cavity diode laser (Sacher, LION), a booster optical amplifier (Thorlabs, BOA1082P), and two digital gate and delay generators (SRS, DG645) are used to produce decay signals [15,17]. Decay signals are recorded using an InGaAs photodetector with an adjustable gain amplifier (New Focus, 2053-FS) and a 16-bit analog-digital converter (Gage, CSE1622) of 5 Msamples/s. The rearm time of the trigger is 1.5 ms. The optical frequency of the probe laser is measured using a wavemeter (Bristol, 621A) and an optical frequency comb (Menlo Systems, FC1500-250-WG). The cavity length is finely scanned using the thermal expansion of the cavity [8].
4. Result and discussion
In this experiment, we selected pure methane gas as an example to test the performance of the new model since methane has strongly overlapped absorption spectra in the infrared region. The degenerate energy level information encoded in methane is essential for an understanding of the subtle structure of this spherical top molecule [18]. Figure 2(a) shows the decay curves (averaging 150 decays per curve) recorded for wavenumbers near the R1A2 transition (6076.108 cm−1) of the 2v2 + v3 band. We changed the intra-cavity photon energy density by setting different gains of the PD (Eq. (3) in [8]). The flat tails of the decay signals correspond to the level of dark current in the PD. The solid and dashed curves in Figs. 2(b)–2(e) are the residuals obtained respectively from fits using the new model (Eq. (5)) and the exponential-fit approach
Fig. 2 (a) Decay signals recorded in different saturation states. (b)–(e) Comparisons between the fitting residuals (experiment-fit) obtained with the new model (‘N’) and the exponential-fit approach (‘E’). (f)–(k) Comparisons between the fitting residuals obtained with the new model (‘N’) and the model in [10] (‘G’).
In all four tests, the residuals produced by the new model are much smaller than those given by the exponential one. Moreover, the residuals of fits using the new model are almost flat for S(0) = 0.12, 0.22 and 0.81. The relatively large residuals for S(0) = 69.9 arise from the strong correlation between the intra-cavity photon energy density ρ(t) and the population density difference n(t), which causes a non-exponential decay of the population density difference. Figures 2(f)–2(k) show the fitting residuals by our model (red line) and the model Eq. (37) in [10] (blue line). In the cases of S(0) = 0.22 and 0.81, the two models give the same level of residuals. In the case of S(0) = 0.12, our model obtains smaller residuals. As expected, in the case of S(0) = 69.9, our model produces much larger residuals than those by the model in [10]. Table 1 lists the values of fitted parameters obtained by the two models. γc = cαc is the empty cavity decay rate and γg = c(αu - αc) is the gas absorption rate without saturation. The correlation between the parameters of γc and γg obtained by the model is observed in the cases of S(0) = 0.12,0.22, and 0.81 as discussed in [12], nevertheless, the values of the total decay rate γu = γc + γg are steady. The values of the total decay rate γu obtained by our model are consistent with those by the model in [10] in the cases of S(0) = 0.12 and 0.22. Our model gives a more reasonable initial saturation parameter of S(0) = 0.12 than the S(0) = 0.0013 obtained by the model in [10], there are obvious fitting residuals induced by saturation (curve ‘E’ in Fig. 2(b)). The above comparison indicates that the model in [10] is a powerful model to decouple the empty-cavity background and absorption signals, especially in the strong saturation regime. Moreover, the model in [10] can obtain steady total decay rates γu in all the tests. By contrast, our model can be used to obtain steady total decay rates γu and more reasonable initial saturation parameter in the low saturation regime. It can also help one to retrieve baseline-immune Doppler-free spectra described in the next paragraph. We also done experiments in the unsaturated condition of S(0) = 0. The fitting parameters (αu and -△αs) of our model are quite correlated, nevertheless, the value of (αu - △αs) is exactly equal to the value of the absorption obtained by the exponential-fit approach. In summary, our solution provides a new feasible way for SCRDS when a gas in the adiabatic regime and initial saturation parameter S(0) ≤ 0.22.
Table 1. The fitted parameters of saturated decay signals obtained by the model in [10] and our modela
According to Eq. (7), the decrement of the absorption coefficient ∆αs is only related to the initial population density difference n(0) and so baseline-immune. Figure 3 shows a comparison between Doppler-free spectra from repeated measurements with the exponential-fit approach and the new model of this work. The chronological order of the measurements of spectra is 2→1→3→4. Whereas the exponential-fit approach gives spectra with distinct baseline differences (Fig. 3(a)) due to different instrumental parameters of the four measurements, the new model gives almost overlapping spectra (Fig. 3(b)). The S(0) was ≈0.50. The baseline of the concave function arises because that the decrement of absorption coefficient ∆αs (or saturation degree) is maximum at the peak of an absorption spectrum. The three hyperfine transitions of the E1, A1, F1 in the 2v3 Q12 multiplet near 6038 cm−1 of 12CH4 were observed by Lamb-dip spectroscopy for the first time. But the correspondence relationships of the transitions and their names are not clear [19,20].
Fig. 3 Repeated measurements of Doppler-free spectra by the exponential-fit approach (a) and the new model (b).
In a further study, the Lamb-dip spectrum of the R1A2 transition of the 2v2 + v3 band of methane at S(0) ≈0.10 was recorded. Figure 4(b) shows plots of the linear absorption spectrum (αu) and the baseline-immune Doppler-free spectrum (-∆αs) retrieved by the new model, and the spectrum obtained using the exponential-fit approach. The baseline of spectrum of -∆αs is flat because the Lamb dip coincides with the absorption peak of the isolated transition. The spectrum of the linear absorption coefficient αu is baseline-dependent, and the Lamb dip feature barely discernable from the background. The absorption coefficients of the spectrum obtained using the exponential-fit approach are significantly lower than the linear absorption coefficients obtained by the new model. This difference is illustrated by the schematic of Fig. 4(a) which is a plot of the linear absorption spectra obtained with the exponential-fit approach and with the new model. Whereas the exponential-fit approach yields an underestimate of gas concentration in the case of saturation, using the new model more accurate values are obtained. The noise levels (1 × standard deviation) for retrieved fitting parameters of , αu, and -△αs are 1.5 × 10−11 cm−1, 1.4 × 10- 10 cm- 1, and 1.2 × 10−10 cm−1, respectively. Due to the correlation between parameters of the fit, the noise levels for the obtained unsaturated absorption and for the absorption variation due to saturation, appear to be higher than for the absorption by the exponential-fit approach.
Fig. 4 (a) The schematic absorption spectra of a gas in saturation regime by the exponential-fit approach and the new model. (b) Observation of the line R1A2 of the 2v2 + v3 band of methane. Upper: Linear absorption. Middle: the exponential-fit approach. Lower: Baseline-immune Doppler-free spectrum.
The Lamb dips (‘a’ and ‘b’) were fitted by a Lorentz profile. Results and fitting uncertainties are listed in Table 2. The linewidth given by the exponential-fit approach is 12% smaller than that given by the new model because the power broadening [14] at the start is larger than the average value for a decay event. Because the degree of saturation at the start of a decay event is also greater than its mean value, the area of the Lamb dip given by the new model is roughly twice that for the exponential-fit approach.
Table 2. The fitted parameters of Lamb dips given by the exponential-fit approach and the new modela
The average unperturbed position of the 2v2 + v3 R1A2 transition found from a series of 10 repeated measurements is 182 157 148 961 kHz with an overall uncertainty of 36 kHz. The overall uncertainty includes four main uncertainty components (Table 3): a) the Type-A uncertainty of 35 kHz; b) an uncertainty of 1.8 kHz in the absolute position due to the relative uncertainty of less than 1 × 10−11 of the 10 MHz GPS time base; c) a positional uncertainty induced by the shift of the time base of the oscilloscope of less than 4 kHz; d) a pressure-measurement-related positional uncertainty of about 4 kHz related to the self-pressure shifting coefficient of 4.3 kHz/Pa [22]. We assumed the aforementioned contributions are statistical independent. Our frequency uncertainty of this transition is larger than those results [23] of other strong methane transitions lying close-by due to the performance difference of SCRDS apparatuses [8,23]. Nevertheless, compared with the results of Zolot et al. [21] and the HITRAN2016 database [20] (Table 4), this work represents roughly a two orders of magnitude improvement in uncertainty.
Table 3. The main uncertainty sources of the absolute frequency measurement of the 2v2 + v3 R1A2 transition
Table 4. Position of the 2v2 + v3 band R1A2 transition of methane obtained here and elsewhere.
5. Conclusion
We proposed a new approximate solution to the two-state rate equation model, which is valid in the adiabatic regime in the adiabatic and low saturation regime. As an application, we recorded baseline-immune Doppler-free spectra of the three hyperfine transitions of methane near 6038 cm−1, resolving them for the first time. The transition frequency of the 2v2 + v3 R1A2 transition has been determined with an uncertainty of 36 kHz. In addition, the new method could help in obtaining linear absorption spectra of a gas in the saturation regime, which could be useful for low concentration measurements, e. g.,14C16O2 [1,2].
Funding
National Key R&D Program of China (2016YFF0200101); Program of International Science & Technology Cooperation of China (2015DFG71880).
Acknowledgments
The authors would like to express their sincere thanks to Dr. J. T. Hodges of NIST for his ongoing interest in this work and constant encouragement. They are most grateful to Yuxin Tang of the Chinese Academy of Sciences for her help with mathematics.
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