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Simultaneous measurement of gas absorption and path length based on the dual-sideband heterodyne phase-sensitive detection of dispersion spectroscopy

Mingli Zou, Liqun Sun, and Shaomin Li

Open Access Open Access

Abstract

We present a novel approach based on dual-sideband heterodyne phase-sensitive detection of dispersion spectroscopy to realize simultaneous measurement of the gas absorption signal and corresponding path length. The details of heterodyne phase-sensitive detection of dispersion spectroscopy are derived. A standard Mach–Zehnder intensity modulator (MZM) is adopted to generate a spectrum of a carrier and two sidebands. Phase shift of the beatnote signal generated by the two sidebands is detected to retrieve the path length as well as the gas absorption signal. The measurement range of the path length can be adapted by changing the modulation frequency. Proof-of-principle experiments are conducted with methane (CH4) as the absorber which is filled into a gas cell with a variable path length. We also utilize this approach to evaluate the path length of a White cell and meanwhile calibrate the experimental system with different concentrations of methane. The proposed method has a great potential for detecting the path length and gas absorption in multipass cells and the open path environment.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Nowadays accurate measurement of gas concentration is of great significance in a variety of applications including scientific research, industry and medical care. Methods based on laser absorption spectroscopy (LAS), such as tunable diode laser absorption spectroscopy (TDLAS) [1,2], quartz-enhanced photoacoustic spectroscopy [3,4], cavity ring-down spectroscopy (CRDS) [5] and chirped laser dispersion spectroscopy (CLaDS) [6,7] etc., are commonly used for gas concentration detection. While in LAS, the detected gas absorption signal is always proportional to the product of gas concentration and absorption path length. For absolute gas concentration measurement, both the absorption signal and the path length should be precisely determined. Traditionally, the absorption path length is measured prior to absorption signal by two kinds of methods: one is to calibrate the path length with known gas concentration [8]; the other is to measure the path length directly with mechanical measurements, time of flight methods, or interferometric techniques [9]. For the former, the accuracy of path length detection always depends on the concentration precision of the gas sample; as for the latter, principles of path length calibration and concentration measurement are usually different, so the path lengths determined by the two methods may not be exactly the same, and there are still some conditions where the absorption path length is difficult to be pre-calibrated. For example, when the gas cell is a sophisticated multipass cell, the path length may change because of structural deformation or mechanical vibration during long-term measurements; if the detection is for gas in scattering media absorption spectroscopy (GASMAS) [10], the path length will be uncertain; and for remote sensing in open path, the path length may vary with the change of target position. Therefore, it is necessary to develop methods for measuring path length and gas absorption simultaneously.

Recently, there have been some studies on simultaneous measurement of gas absorption and path length with LAS. For these methods, because the measured path length is inversely proportional to the modulation bandwidth which may can’t be changed, the measurement range is limited. Lou and co-workers use the optical frequency-modulated continuous-wave (FMCW) interferometry to realize simultaneous detection of the gas absorption spectrum and optical length in a White cell [9]. For this method, the measurement range of path length is mainly limited by the linewidth of the tunable laser, and to acquire the beat signal, a data acquisition card with relatively high sampling frequency is needed. In GASMAS, path length of the scattering porous media is uncertain, Mei et al. combine TDLAS and FMCW to obtain the gas absorption signal and path length respectively [10]. Although a high precision can be achieved, the system is relatively complex, and the path length measurement range also depends on the linewidth of the laser, and it cannot be used under high absorbance conditions. For remote sensing, Yang et al. utilize the phase angle of the first harmonic signal in the wavelength modulation spectroscopy (WMS) technique to measure the gas absorption and path length simultaneously [11]. In WMS technique, the sinusoidal modulation frequency is in kHz level, so the absorption path length should be large enough (hundreds of meters) to get obvious phase angle change.

In this paper, we have developed a new approach to realize simultaneous detection of the gas absorption signal and path length based on dual-sideband heterodyne phase-sensitive detection of dispersion spectroscopy. Dual-sideband heterodyne of dispersion spectroscopy is a kind of phase demodulation chirped laser dispersion spectroscopy (PD-CLaDS) [6,7,12], which is also independent of the intensity of the laser and can be used under high absorbance conditions. We modify the principle of the original PD-CLaDS, and find that the phase shift of the beatnote signal generated by the two sidebands contains both gas absorption and absorption path length information. In the theory, the absorption path length is obtained from the non-absorption region of the phase shift, and the difference between the maximum and minimum of phase change is used to get the gas absorption signal. The measurement range of path length can be adjusted by changing the modulation frequency adopted to MZM. Proof-of-principle experiments are conducted to demonstrate the theory.

2. Methods

When the laser (whose wavelength scans over the absorption line of the studied gas) passes through the gas sample, there will be not only absorption but also dispersion. The refractive index n(ω) and absorption coefficient α(ω) obey the Kramers–Kronig relation [6,13]:

$$n(\omega ) = 1 + \frac{c}{\pi }\int_0^{ + \infty } {\frac{{\alpha ({\omega ^{\prime}})}}{{{\omega ^{\prime}}^2\textrm{ - }{\omega ^2}}}} d{\omega ^{\prime}},$$
where ω is the optical angular frequency, c is light speed in vacuum. At room temperature and under the atmospheric pressure, Lorentz function is usually used to describe the gas absorption line-shape, then the corresponding refractive index can be expressed as [14]
$$n(\omega ) = {n_0} + s\frac{{\omega - {\omega _c}}}{{{{(\omega - {\omega _c})}^2} + \frac{{\Delta {\omega ^2}}}{4}}},$$
where ${n_0}$ is the refractive index of non-absorbing region which usually equals to 1, s is a variable dependent of the absorption line, ωc is central optical angular frequency of the target absorption line, and $\Delta \omega$ is the full width at half maximum (FWHM) of the absorption line.

The intensity of the laser with an optical angular frequency of ω0 from a continuous wave tunable laser source will be modulated at an angular frequency Ω after passing through a standard MZM. As a result, an optical spectrum composed of a carrier (E1) and two sidebands (E2 and E3) will generate, which can be expressed as

$${E_1} = {A_1}\cos ({\omega _0}t),$$
$${E_2}\textrm{ = }{A_2}\cos [{({\omega_0} - \Omega )t} ],$$
$${E_3}\textrm{ = }{A_3}\cos [{({\omega_0}\textrm{ + }\Omega )t} ],$$
where A1, A2, A3 are the amplitudes of E1, E2, and E3.

From Eq. (2), it is known that the refractive index is a function of optical angular frequency. Thus, when the laser beams propagate through the gas sample, the three components will travel at slightly different speeds producing changes in their optical phases. If the absorption path length is L, the three optical signals can be described as

$${E_1} = {A_1}\cos ({\omega _0}t\textrm{ - }{\varphi _1}),$$
$${E_2}\textrm{ = }{A_2}\cos [{({\omega_0} - \Omega )t\textrm{ - }{\varphi_2}} ],$$
$${E_3}\textrm{ = }{A_3}\cos [{({\omega_0}\textrm{ + }\Omega )t\textrm{ - }{\varphi_3}} ],$$
the three optical phases are
$${\varphi _1}\textrm{ = }\frac{{{\omega _0}L}}{c}n({{\omega_0}} ),$$
$${\varphi _2}\textrm{ = }\frac{{({\omega _0} - \Omega )L}}{c}n({{\omega_0} - \Omega } ),$$
$${\varphi _3}\textrm{ = }\frac{{({\omega _0} + \Omega )L}}{c}n({{\omega_0} + \Omega } ).$$
Here we redefine the three optical phases compared with original PD-CLaDS [7,12]. In original PD-CLaDS, for ${\omega _0} \gg \Omega $, Ω in Eq. (10) and Eq. (11) are usually omitted, and the three refractive indices in three optical phases are subtracted by 1, which means that the optical phases are phase differences of the case with gas absorption and without gas absorption. To achieve this result, a gas-free reference channel with the same absorption path length is needed, but this point has not been mentioned in original PD-CLaDS. Therefore, in the theory we put forward here, the three phases in Eqs. (9), (10) and (11) are phase shifts just brought by gas absorption without a reference gas cell.

When the three optical signals reach a square law photodetector, there will be three heterodyne signals which can be defined as

$${I_{1,2}} \propto {A_1}^2 + {A_2}^2 + 2{A_1}{A_2}\cos [{\Omega t - ({{\varphi_1} - {\varphi_2}} )} ],$$
$${I_{1,3}} \propto {A_1}^2 + {A_3}^2 + 2{A_1}{A_3}\cos [{\Omega t - ({{\varphi_3} - {\varphi_1}} )} ],$$
$${I_{2,3}} \propto {A_2}^2 + {A_3}^2 + 2{A_2}{A_3}\cos [{2\Omega t - ({{\varphi_3} - {\varphi_2}} )} ].$$
Substituting Eq. (2), Eqs. (10) and (11) into Eq. (14), the phase shift of the beatnote signal generated by the two sidebands equal to
$${\varphi _0}\textrm{ = }{\varphi _2}\textrm{ - }{\varphi _3}\textrm{ = }\frac{{sL}}{c}\left[ {\frac{{({{\omega_0} - \Omega } )({\omega_0} - \Omega - {\omega_c})}}{{{{({\omega_0} - \Omega - {\omega_c})}^2} + \frac{{\Delta {\omega^2}}}{4}}} - \frac{{({{\omega_0} + \Omega } )({\omega_0} + \Omega - {\omega_c})}}{{{{({\omega_0} + \Omega - {\omega_c})}^2} + \frac{{\Delta {\omega^2}}}{4}}}} \right] - \frac{{2\Omega }}{c}L.$$
From Eq. (15), we can know that the first term of the right-hand side of the equation is the same as the whole phase shift defined by original dual-sideband PD-CLaDS and proportional to the product of gas concentration and absorption path length. While the second one is an extra offset (marked as ${\varphi _{\textrm{offset}}}$ below) and only related to the absorption path length.

Therefore, in this paper, we detect the phase of the beatnote signal generated by the two sidebands with an angular frequency of 2Ω. The bias term ${\varphi _{\textrm{offset}}}$ (non-absorbing part) of the phase shift is used to obtain the absorption path length and the relationship between the distance and phase offset can be described as

$$L = \frac{{c{\varphi _{offset}}}}{{4\pi f}},$$
where f is the frequency of the modulated signal applied to the MZM. The maximum measurement range of the absorption path length equals to c/2f. Meanwhile, the absorption signal is evaluated by the peak-to-peak value of the total phase shift, which can be expressed as
$$\begin{aligned} {\varphi _{\textrm{absorption}}} & = {\varphi _{\max }} - {\varphi _{\min }} \\ &= \frac{{sL}}{c}\textrm{\{ }\left[ {\frac{{({{\omega_{\textrm{p}\max }} - \Omega } )({\omega_{\textrm{p}\max }} - \Omega - {\omega_c})}}{{{{({\omega_{\textrm{p}\max }} - \Omega - {\omega_c})}^2} + \frac{{\Delta {\omega^2}}}{4}}} - \frac{{({{\omega_{\textrm{p}\max }} + \Omega } )({\omega_{\textrm{p}\max }} + \Omega - {\omega_c})}}{{{{({\omega_{\textrm{p}\max }} + \Omega - {\omega_c})}^2} + \frac{{\Delta {\omega^2}}}{4}}}} \right]\\ \textrm{ } & \quad - \left[ {\frac{{({{\omega_{\textrm{p}\min }} - \Omega } )({\omega_{\textrm{p}\min }} - \Omega - {\omega_c})}}{{{{({\omega_{\textrm{p}\min }} - \Omega - {\omega_c})}^2} + \frac{{\Delta {\omega^2}}}{4}}} - \frac{{({{\omega_{\textrm{p}\min }} + \Omega } )({\omega_{\textrm{p}\min }} + \Omega - {\omega_c})}}{{{{({\omega_{\textrm{p}\min }} + \Omega - {\omega_c})}^2} + \frac{{\Delta {\omega^2}}}{4}}}} \right]\textrm{\} }\textrm{.} \end{aligned}$$
${\varphi _{\max }}$ and ${\varphi _{\min }}$ are the maximum and minimum value of total phase shift, respectively, ${\omega _{\textrm{p}\max }}$ and ${\omega _{\textrm{p}\min }}$ are the corresponding angular frequencies. It should also be mentioned that the absorption signal is proportional to the gas absorption in full range.

3. Selection of modulation frequency

In section 2, we can know that the detection range of the path length equals to c/2f, then, the detection range is inversely proportional to the modulation frequency. While the peak-to-peak value of total phase shift also depends on the modulation frequency, there has to be a trade-off between the measurement range of path length and absorption signal. Based on Eq. (1), Eq. (2) and Eq. (15), the peak-to-peak value of total phase shift is simulated when the modulation frequency changes. The result is shown in Fig. 1. To display more intuitively, peak-to-peak value of total phase shift is normalized by the maximum value. As indicated in Fig. 1, the maximum peak-to-peak value of total phase shift is achieved at a modulation frequency of 0.5824FMCW, for the absorption peak of CH4 near 1653.7 nm at atmospheric pressure, using the absorption coefficient from Hitran database to estimate [15], the best modulation frequency is about 5.59 GHz. Hence, in the subsequent experimental system, in order to expand the length measurement range and increase the peak to peak phase shift, we use two different modulation frequencies, the smaller one is used to measure the path length, and the larger one is for obtaining the absorption signals. Different measurement ranges also can be achieved by changing the modulation frequencies. In our experiments, we use four different frequencies, 15 MHz, 25MH, 50 MHz and 730 MHz, the first three are applied for path length detection and the last one is for absorption detection due to the limited bandwidth of the available components (1.5 GHz).

 figure: Fig. 1.

Fig. 1. The normalized peak-to-peak value of phase shift when modulation frequency changes.

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4. Experiments and results

4.1 Experiment arrangement

The proposed method for simultaneous measurement of gas absorption and path length is validated by measuring CH4 in different gas cells. The experiment setup is depicted in Fig. 2. The system is designed with differential detection to eliminate the possible phase error caused by modulation frequency fluctuation or the use of electronic components. Laser light emitted from a fiber-coupled tunable distributed feedback (DFB) diode laser (NLK1U5EAAA, NEL, Japan) is modulated by a ramp signal for scanning over the absorption line of CH4 near 1653.7 nm. Temperature and injection current of the laser are controlled with a laser temperature controller (TED200C, Thorlabs, USA) and a current driver (LDC205C, Thorlabs, USA), respectively. The central injection current is set to 160 mA, and the ramp signal is generated by the LabVIEW controlled DAQ card (USB6341, National Instruments, USA) with a frequency of 5 Hz and an amplitude of 40 mA.

 figure: Fig. 2.

Fig. 2. Experiment arrangement for our dual-sideband PD-CLaDS sensing. DFB, distributed feedback laser; MZM, Mach–Zehnder intensity modulator; S1, S2, RF signal generators; BV, Bias voltage; FS, fiber beam splitter; GC, gas cell; PD1, PD2, photodetector; M1, M2, mixers with low-pass filter; LIA, lock in amplifier; DAQ, data acquisition card; C&T, Current and temperature controller.

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A standard Mach–Zehnder intensity modulator (MZM, PowerBit F10, Oclaro) driven at frequency f is applied to the laser beam to generate a dual-sideband spectrum. In the following experiments, four different frequencies are used, 15 MHz, 25MH, 50 MHz and 730 MHz. To enhance the sidebands, the MZM is controlled by a bias voltage of 3.5 V. After that, the multicolor beams are divided into two beams, one passes through the gas cell as the measurement channel and the other is as the reference channel. Then, heterodyne signals from the two channels are detected by two high-speed photodetectors (DET08CL/M, DET08CFC/M, Thorlabs). In order to separate the 2f signals and enhance their amplitudes, signals from the detectors are processed with the same bandpass filters and RF amplifiers, in subsequent steps, the electronic devices used in both channels are also the same. Since the frequency of the 2f signals is relatively high which is beyond the maximum input frequency of lock in amplifier (LIA, LI5640), we used mixers (ZX05-5-S+, Mini-Circuits) and low pass filters (SLP-1.9+, Mini-Circuits) to downshift the frequencies. The amplified signal from the measurement channel is fed to the RF input port of the mixer, while the LO input of the mixer is connected with RF signal from S2 at frequency f1 (f1-2f=100kHz). Then the mixed signal is processed by the low pass filter and a low frequency signal at 100kHz is obtained. Amplified signal of the reference channel is processed in the same way. After that, signals from the measurement channel and the reference channel are fed to the LIA, then the phase difference of the two channels can be obtained and recorded by the DAQ card.

4.2 Experiments with a variable-length gas cell

The proof-of-principle experiments are conducted by measuring CH4 in a homemade single-pass variable-length gas cell with a total path length of 1 m at 1 atm and 298.25 K. The gas cell consists of two optical fiber collimators sealed in a stainless-steel box, one is fixed and the other is mounted on a slider placed on a ball screw, a distance sensor is fixed on the outside of slider to determine the distance between two collimators with a measurement accuracy of 0.005 mm. Two fibers are connected with the fiber collimators to input and output laser beams.

As discussed in section 2, the measurement range of path length is inversely proportional to the modulation frequency, while large peak-to-peak value is obtained with relatively high modulation frequencies, so we utilize relatively low modulation frequencies for path length measurement and high modulation frequencies for absorption assessment. In our experiments, three different modulation frequencies of 15 MHz, 25 MHz, 50 MHz are applied to the MZM for path length detection, corresponding to measurement ranges of 10 m, 6 m and 3 m, and a modulation frequency of 730 MHz is utilized for absorption assessment. The gas pool is filled with 5% CH4 gas sample, and the absorption path length varies from 0 to 0.7 m. Phase shifts detected for path lengths of 0.5 m and 0.495 m with modulation frequency of 730 MHz are depicted in Fig. 3. The fitting result based on Eq. (15) at path length of 0.5 m is also shown, the R2 of the fitting result is 0.998. There is an obvious phase offset in the nonabsorbent region, and the proposed theory shows good agreement with the experimental result. Difference between the phase offset obtained at path lengths of 0.5 m and 0.495 m is also close to that calculated with Eq. (15).

 figure: Fig. 3.

Fig. 3. Detected phase shifts and fitting result. (a), detected phase shifts with absorption of path lengths of 0.5 m and 0.495 m at modulation frequency of 730 MHz; (b) detected phase shift with absorption of path length of 0.5 m at modulation frequency of 730 MHz together with fitting result.

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Phase shifts at different absorption path lengths with three different relatively low modulation frequencies are also detected. Because the modulation frequency is too small, the phases of absorption region and non-absorption region is very close. Average values of detected phases at the nonabsorbent region of different path lengths at three modulation frequencies are shown in Fig. 4. The initial phases are not zero mainly caused by the input and output fibers of the gas cell. We also make linear fitting analysises about them. It can be found that there are good linear relationships between the phase shifts and absorption path length. The slopes also coincide with that inferred from Eq. (17) which are 36 degrees, 60 degrees and 120 degrees at modulation frequencies of 15 MHz, 25 MHz and 50 MHz, respectively. Therefore, the proposed method is consistent with the experimental results, this approach can be used to obtain gas absorption and path length simultaneously.

 figure: Fig. 4.

Fig. 4. Phase shifts of the nonabsorbent region at different absorption path lengths with different modulation frequencies and the linear fitting curves. Red triangles, phase shifts at different absorption path lengths; black curves, the linear fitting relation between the phase shifts and the absorption path length. (a), phase shifts at modulation frequency of 15 MHz; (b), phase shifts at modulation frequency of 25 MHz; (c), phase shifts at modulation frequency of 50 MHz.

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The measurement process can be summarized as follows: (1) absorption path length detection, set the frequencies of S1 and S2 to relatively low frequencies, keep the gas cell alone and record the initial phase, which only needs to be detected once under a certain modulation frequency; (2) connect the gas cell and record the phase value, the path length can be determined with the phase difference between the recorded phase value and the initial phase; (3) set the frequencies of S1 and S2 to relatively high frequencies, record the phases and obtain absorption information.

4.3 Path length detection and concentration calibration with a White cell

After verifying the theory, we use this method to determine the path length and absorption signal with a White cell whose nominal length is 8 m. Modulation frequencies of 15 MHz and 730 MHz are used to obtain the path length and absorption signal, respectively.

4.3.1 Path length detection with the White cell

Due to the long optical path length, the absorption signal may can be measured with a low modulation frequency. We measure the phase shifts when the White cell is full of methane with concentration of 100.9 ppm and 0.697% at 1 atm and 298.25 K, respectively. Modulation frequency applied to the MZM is 15 MHz. The phases obtained are depicted in Fig. 5. It can be found that the phase shift of absorption region and non-absorption region is very close under low methane concentration with low modulation frequency, the phases of non-absorption region are also similar at the two gas concentrations. The peak-to-peak value is relatively large for gas concentration at 0.697% which is suitable for path length and concentration assessment simultaneously in theory, while the concentration is relatively high which may not applicable in reality. The absorption measurement of the system also has an upper limit. When the absorption is too strong, the amplitude of the measurement channel may be too small for the system to detect.

 figure: Fig. 5.

Fig. 5. Detected phase shifts of methane with concentrations of 100.9 ppm and 0.697%.

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The initial phase detected without gas cell at the modulation frequency of 15 MHz is 173.43 degree, then the length of gas cell is calculated as 8.6132 m. The gas cell is input with a single mode fiber (G652) of 40 cm, if the core refractive index is 1.45, the detected path length of the White cell is 8.0332 m. In order to evaluate the stability for path length measurement of the system, we analyze the Allen deviation of path length detection with a period longer than 30 minutes. The White cell is full of methane with a concentration of 100.9 ppm, the Allen deviation is given in Fig. 6. As a result, the fluctuation of the measured absorption path length for the White cell is less than 30.38 μm, the optimum integration time of the path-length measurement is about 10 seconds. The trend of Allan deviation after 10 seconds could be attributed to the drift of the diode laser and other electronic devices, variation of environmental temperature and pressure, etc.

 figure: Fig. 6.

Fig. 6. Allen deviation of the detected absorption path length.

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4.3.2 Absorption calibration with the White cell

A higher modulation of 730 MHz is utilized to obtain the absorption information of methane in the White cell. We measure CH4 gas samples of seven different concentrations (20.2 ppm, 49.9 ppm, 100.9 ppm, 500.8 ppm, 1002 ppm, 1499 ppm, 2996 ppm) to calibrate the relationship between concentrations and peak-peak values of phase shifts with the White cell at 1 atm and 298.25 K. For each concentration, we take samples for 1 second, and the obtained peak-peak values of phase shifts are then averaged. The relationship between the detected concentrations and peak-peak values of phase shifts is shown in Fig. 7. The linear relation (R-square value: 0.9996) between the peak-peak values of phase shifts (P/ degree) and the concentrations (C, ppm) can be fitted as:

$$C = 371.2 \times P - 59.97(\textrm{ppm}).$$
Take the absorption path length of the White cell into consideration, the linear relation between the peak-peak values of phase shifts (P/ degree) and the product of gas concentration and absorption path length (CL, ppm·m) can be described as
$$CL = (371.2 \times P - 59.97) \times 8.0332\textrm{ }(\textrm{ppm}\cdot \textrm{m}).$$

 figure: Fig. 7.

Fig. 7. Peak-to-peak phase shifts vs. gas concentrations.

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The Alan variance analysis of the stability of gas concentration measurement is also carried out. The White cell is full of CH4 under a concentration of 20.2 ppm, then we record the peak-to-peak values of phase shifts for more than 30 minutes and the sampling interval is one second. The calculated Allen deviation is depicted in Fig. 8. It can be found that a detection limit of 17.09 ppb is obtained with a 65 s averaging time.

 figure: Fig. 8.

Fig. 8. Allen deviation curve of CH4 absorption signal.

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5. Conclusion

In this paper, dual-sideband heterodyne phase-sensitive detection of dispersion spectroscopy is used to determine the gas absorption and path length simultaneously. The absorption path length is obtained from the non-absorption region of the phase shift generated by the beatnote signal, and the peak-to peak value of phase shift is used to get the gas absorption signal. In order to increase the length measurement range and enhance the absorption signal, we use different modulation frequencies. Experiments with a variable-length gas cell full of CH4 prove the theory well. Allen deviation analysis shows that the system has a detection limit of 14.65 μm at 10 s for path length measurement and 17.09 ppb for gas concentration at 65 s. With advantages of low cost and simple configuration, this method has a promising potential for path length and gas absorption detection in multipass cells and open path environment. Since both the laser and MZM are fiber-optic integrated devices, our method also can be easily used for the calibration of some complex gas cells which are integrated in some compound measurement system.

Funding

National Key Research and Development Program of China (2018YFF0109600).

Disclosures

The authors declare no conflicts of interest.

References

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Figures (8)
Fig. 1.
Fig. 1. The normalized peak-to-peak value of phase shift when modulation frequency changes.

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Fig. 2.
Fig. 2. Experiment arrangement for our dual-sideband PD-CLaDS sensing. DFB, distributed feedback laser; MZM, Mach–Zehnder intensity modulator; S1, S2, RF signal generators; BV, Bias voltage; FS, fiber beam splitter; GC, gas cell; PD1, PD2, photodetector; M1, M2, mixers with low-pass filter; LIA, lock in amplifier; DAQ, data acquisition card; C&T, Current and temperature controller.

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Fig. 3.
Fig. 3. Detected phase shifts and fitting result. (a), detected phase shifts with absorption of path lengths of 0.5 m and 0.495 m at modulation frequency of 730 MHz; (b) detected phase shift with absorption of path length of 0.5 m at modulation frequency of 730 MHz together with fitting result.

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Fig. 4.
Fig. 4. Phase shifts of the nonabsorbent region at different absorption path lengths with different modulation frequencies and the linear fitting curves. Red triangles, phase shifts at different absorption path lengths; black curves, the linear fitting relation between the phase shifts and the absorption path length. (a), phase shifts at modulation frequency of 15 MHz; (b), phase shifts at modulation frequency of 25 MHz; (c), phase shifts at modulation frequency of 50 MHz.

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Fig. 5.
Fig. 5. Detected phase shifts of methane with concentrations of 100.9 ppm and 0.697%.

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Fig. 6.
Fig. 6. Allen deviation of the detected absorption path length.

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Fig. 7.
Fig. 7. Peak-to-peak phase shifts vs. gas concentrations.

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Fig. 8.
Fig. 8. Allen deviation curve of CH4 absorption signal.

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Equations (19)

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n ( ω ) = 1 + c π ∫ 0 + ∞ α ( ω ′ ) ω ′ 2  -  ω 2 d ω ′ ,
n ( ω ) = n 0 + s ω − ω c ( ω − ω c ) 2 + Δ ω 2 4 ,
E 1 = A 1 cos ⁡ ( ω 0 t ) ,
E 2  =  A 2 cos ⁡ [ ( ω 0 − Ω ) t ] ,
E 3  =  A 3 cos ⁡ [ ( ω 0  +  Ω ) t ] ,
E 1 = A 1 cos ⁡ ( ω 0 t  -  φ 1 ) ,
E 2  =  A 2 cos ⁡ [ ( ω 0 − Ω ) t  -  φ 2 ] ,
E 3  =  A 3 cos ⁡ [ ( ω 0  +  Ω ) t  -  φ 3 ] ,
φ 1  =  ω 0 L c n ( ω 0 ) ,
φ 2  =  ( ω 0 − Ω ) L c n ( ω 0 − Ω ) ,
φ 3  =  ( ω 0 + Ω ) L c n ( ω 0 + Ω ) .
I 1 , 2 ∝ A 1 2 + A 2 2 + 2 A 1 A 2 cos ⁡ [ Ω t − ( φ 1 − φ 2 ) ] ,
I 1 , 3 ∝ A 1 2 + A 3 2 + 2 A 1 A 3 cos ⁡ [ Ω t − ( φ 3 − φ 1 ) ] ,
I 2 , 3 ∝ A 2 2 + A 3 2 + 2 A 2 A 3 cos ⁡ [ 2 Ω t − ( φ 3 − φ 2 ) ] .
φ 0  =  φ 2  -  φ 3  =  s L c [ ( ω 0 − Ω ) ( ω 0 − Ω − ω c ) ( ω 0 − Ω − ω c ) 2 + Δ ω 2 4 − ( ω 0 + Ω ) ( ω 0 + Ω − ω c ) ( ω 0 + Ω − ω c ) 2 + Δ ω 2 4 ] − 2 Ω c L .
L = c φ o f f s e t 4 π f ,
φ absorption = φ max − φ min = s L c {  [ ( ω p max − Ω ) ( ω p max − Ω − ω c ) ( ω p max − Ω − ω c ) 2 + Δ ω 2 4 − ( ω p max + Ω ) ( ω p max + Ω − ω c ) ( ω p max + Ω − ω c ) 2 + Δ ω 2 4 ]   − [ ( ω p min − Ω ) ( ω p min − Ω − ω c ) ( ω p min − Ω − ω c ) 2 + Δ ω 2 4 − ( ω p min + Ω ) ( ω p min + Ω − ω c ) ( ω p min + Ω − ω c ) 2 + Δ ω 2 4 ] }  .
C = 371.2 × P − 59.97 ( ppm ) .
C L = ( 371.2 × P − 59.97 ) × 8.0332   ( ppm ⋅ m ) .
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