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Self-corrected frequency modulation spectroscopy immune to phase random and light intensity fluctuation

Jian Chen, Zhenhui Du, Tao Sun, Jinyi Li, and Yiwen Ma

Open Access Open Access

Abstract

Light intensity fluctuations and phase randomness in quadrature demodulation disturb the accuracy of frequency modulation spectroscopy. The proposed self-corrected method eliminated these effects: the profile of correctly demodulated signals identified whether a demodulation was phase matched and corrected a phase-mismatched demodulated signal; we extracted the measured signal’s direct current component and corrected the light intensity fluctuation. We conducted theoretical analysis and experimental verification to reduce light-intensity errors by 16.8% under different intensity conditions and obtained spectral features by phase difference corrections under the same measurement conditions. We reduced the effect of light intensity fluctuation and demodulated signals were freed from phase stability limitations.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Frequency modulation spectroscopy (FMS) is a heterodyne detection method characterized with high sensitivity and short response time [1]. Modulation of hundreds of MHz or higher reduce the effects of 1/f noise and thus make the detection limits pretty low. FMS has succeeded in physicochemical [2], photochemical [3], and time-resolved spectroscopy [4]. The commonly used quadrature demodulation, however, is interrupted by the phase randomness between the measured signal and reference signal [5] and light intensity fluctuations always affect the accuracy of FMS.

The phase randomness of a measured signal universally existed in FMS. Research always addressed this issue under stable conditions. North et al. [5] simulated a family of possible FM line shapes to illustrate the effect of phase difference θ, which combines the absorption component AFM and the dispersion component DFM. The correct phase difference was determined by a specific relationship between AFM and DFM. Wynands et al. [6] reconstructed directly the inverted line shape to determine the correct phase by observing whether there was a characteristic jump. Hall et al. [7] thought that simultaneous phase determination and phase correction were necessary for the measurement of arbitrary Doppler line shapes. Whittaker et al. [8] suggested that the commonly used Kramers-Kronig (KK) relationship should be replaced by Hilbert transform to reduce computation time by using signal-processing software packages. Suter et al. [9] recommended recording the signals on both channels simultaneously and analyzing the combined data to determine the demodulation phase. In previous studies, the dispersion usually is obtained using finite difference and KK relationship (or Hilbert transform). Noise is accumulated in the process of finite integration, however, because there is noise in the measured signal [5]. This noise is not conducive to the subsequent data analysis. In addition, the phase difference θ is random, and its value is not fixed.

The light intensity fluctuation always deteriorates the accuracy of optical spectroscopy, in particular for modulation spectroscopy. Hanson’s group [10,11] from Stanford University proposed effective calibration-free technology, which normalized the second harmonic with the first harmonic of wavelength-modulation spectroscopy (WMS), a low-frequency modulation with sub-megahertz or kilohertz. WMS benefits from these advantages through on-site deployment and harsh environmental applications. No such a calibration method exists for FMS to eliminate fluctuations in light intensity.

The phase difference θ between the measured signal and reference signal is susceptible to external randomness, which is caused by a great uncertainty. For example, the optical path length may change according to various applications, temperature variations, and even the instability of system configuration [12–14]. Phase randomization has different effects on the modulation of different frequencies. In quadrature demodulation, the in-phase represents the amplitude component of the signal and the quadrature represents the phase component of the signal. For low-frequency modulated signals, most of the demodulated signals are amplitude component of signals, and the phase component is negligible. Therefore, the phase difference can be easily obtained by using the demodulated signals. For the RF signal, however, the phase component in the demodulated signal cannot be ignored, and the phase difference cannot be directly obtained by the demodulated signals. Therefore, the phase difference needs to be solved by other means. In addition to the phase factor, the instability of the output of the laser intensity or the scattering on the transmission path causes the fluctuation of light intensity during the detection process, which also affects the measured result.

In this paper, we demonstrated a novel method for deriving the accurate absorption and dispersion of FMS from demodulation signals. We eliminated the influence of light intensity fluctuation by the direct current (DC) component in the measured signal, and we determined the phase between the reference signal and the measured signal different by means of the profile of correctly demodulated signals. In addition, we avoided the finite difference processing of the demodulated signals, so noise was not accumulated.

2. Method

2.1 Basic principle

For FMS, laser can be expressed after transmitting a gas sample, as follows [15]:

$${I_{\bmod }}(t) = \overbrace{{{I_0}\exp ( - 2{\delta _0})}}^{{\textrm{DC component}}} + \overbrace{{{I_0}\exp ( - 2{\delta _0})[({\delta _{ - 1}} - {\delta _1})M\cos {\omega _m}t + ({\phi _{ - 1}} + {\phi _1} - 2{\phi _0})M\sin {\omega _m}t]}}^{{\textrm{AC component}}},$$
where Imod(t) is modulated light intensity after transmitting a gas sample; I0 is the incident light intensity; t is time in s; ω0 is the line center frequency; δ−1 is the amplitude attenuation at ω0-ωm; δ1 is the amplitude attenuation at ω0+ωm; ϕ−1 is the phase shift at ω0-ωm; ϕ1 is the phase shift at ω0+ωm; ϕ0 is the phase shift at ω0; ωm is the modulation frequency in Hz; M is the modulation index; δ(ω) =αω/2 and φ(ω) = ηωLω/c; αω is the absorbance at ω; ηω is the refractive index at ω; c is the light speed in m / s; and L is the propagation path in m.

Imod consists of a DC component and an alternate current (AC) component. In the AC component, δ(ω) and ϕ(ω) are affected by the temperature T, the pressure P, the gas concentration X, and the propagation path L.

After the measured signal is quadrature demodulated by the demodulator, it can be expressed as follows [15]:

$${I_{FM}}(\omega ) = {I_0}\exp ( - 2{\delta _0})\gamma M[\cos \theta ({\delta _{ - 1}} - {\delta _1}) + \sin \theta ({\phi _{ - 1}} + {\phi _1} - 2{\phi _0})],{\textrm{and}}$$
$${Q_{FM}}(\omega ) = {I_0}\exp ( - 2{\delta _0})\gamma M[\sin \theta ({\delta _{ - 1}} - {\delta _1}) - \cos \theta ({\phi _{ - 1}} + {\phi _1} - 2{\phi _0})].$$
where γ is the loss in the circuit, related to the device, wire, and filter.

We obtained the amplitude and phase of the demodulated signals by quadrature demodulation. Amplitude of demodulated signal is in-phase and represents the absorption component. Similarly, the phase of demodulated signal is quadrature and represents the dispersion component.

The absorption and dispersion components reflect different physical phenomena. Therefore, correct demodulation is essential to study physical processes. The demodulated signals, IFM(ω) and QFM(ω), are the output signals of the two channels of demodulator, which have a 90-degree phase difference between them. Equation (2) and Eq. (3) contain the demodulation phase θ. So, the obtained IFM(ω) and QFM(ω) are linear combinations of absorption and dispersion components with the demodulation phase θ. The absorption and dispersion components are obtained when θ = ± nπ / 2. Simultaneously, it can be seen from the expression that IFM(ω) and QFM(ω) contain the I0 component, that is, the incident light intensity. Therefore, any fluctuations of I0 can cause a direct influence on the demodulated signal.

2.2 Eliminate the influence of light intensity fluctuations

Fluctuations in incident light intensity I0 can cause unexpected errors in measurement. We eliminated the influence of light intensity fluctuations by the DC component. Specifically, the demodulated signal was divided by the DC component, which can be expressed as follows:

$${I_{eli}}(\omega ) = {I_{FM}}(\omega )\textrm{/[}{I_0}\exp ( - 2{\delta _0})] = \gamma M[\cos \theta ({\delta _{ - 1}} - {\delta _1}) + \sin \theta ({\phi _{ - 1}} + {\phi _1} - 2{\phi _0})],{\textrm{and}}$$
$${Q_{eli}}(\omega ) = {Q_{FM}}(\omega )/[{I_0}\exp ( - 2{\delta _0})] = \gamma M[\sin \theta ({\delta _{ - 1}} - {\delta _1}) - \cos \theta ({\phi _{ - 1}} + {\phi _1} - 2{\phi _0})].$$
The effect of light intensity fluctuation was eliminated, which meant that the fluctuation as well suppressed. Because the DC component was easily extracted from the measured signal using a low-pass filter, this method can be used for most detectors.

2.3 Phase difference determination

Correct demodulation was affected by the phase difference θ. For a particular absorption line shape, the profile of demodulated signals reflected whether there was a phase difference θ. When the phase difference was θ = 0, the demodulated signals were the absorption component AFM and the dispersion component −DFM. When the phase difference was θ ≠ 0, the demodulated signals were linear combinations of the absorption component AFM and the dispersion component DFM. When the phase difference was different, the profile of demodulated signals was different from those of AFM and DFM, as shown in Fig. 1. Figure 1 shows the demodulated signals when the phase difference between the reference signal and the measured signal was different. The zero crossings and extreme points of the AFM and the DFM are marked in Fig. 1. When the phase difference was θ ≠ 0, the frequencies of extreme points changed and the number of extreme points were not the same. Therefore, phase difference θ influenced the distribution and the number of the extreme points. Bjorklund’s [15] studies have shown that the number of extreme points of the absorption component AFM were not affected by the modulation index, and its number was always two. The number of extreme points of the dispersion component DFM was affected by the modulation index, and the number of extreme points could be 2, 4, or 6.

 figure: Fig. 1.

Fig. 1. Demodulated signals with different phase differences. The vertical line is the extreme point frequency of each signal, and the auxiliary line is the equal frequency line.

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In addition, by solving the extremum of line shapes function, we obtained the quantitative relationship, which included the extreme points frequencies of the absorption component AFM, the modulation frequency, and the absorption linewidth (see Eq. (6)). Equation (6) shows that when line shape was determined, the extreme point frequencies were related only to the modulation frequency and line width. According to the KK relationship, we easily proved that the extreme points of the dispersion component DFM also had a quantitative relationship with the modulation frequency and the absorption linewidth. Because we used shape line function instead of demodulated signals, no errors were accumulated (for further discussion, see Section 2.3.2).

$$\left\{ \begin{array}{l} U ={\pm} \sqrt {\frac{{2{n^2} - 2 - 4\sqrt {{n^4} + {n^2} + 1} }}{{ - 6}}} ,\textrm{n} = \frac{{\frac{{\Delta {v_L}}}{2}}}{{{\omega_m}}}\\ \exp \{ 16\ln 2\frac{{{\omega_m}}}{{\Delta {v_D}^2}}\ast U\} = \frac{{U + {\omega_m}}}{{U - {\omega_m}}} \end{array} \right.$$
where $\varDelta$υL is Lorentz full width at half maximum, $\varDelta$υD is Gaussian full width at half maximum, and U is the extreme point frequency ωex of the absorption component that deviates from the line center frequency ω0.

For a known absorption line, we determined the AFM and DFM line shapes after modulation frequency and modulation index were determined. We obtained a prior profile of the AFM and DFM by numerical calculation [15]. When there was a phase difference θ between the reference signal and the measured signal, the line shapes of AFM and DFM were obtained by inverting the line shapes of Ie1i(ω) and Qe1i(ω) by the phase difference θ. When there was a deviation in the phase difference θ, however, the line shapes of AFM and DFM inverted through the linear inversion of Ie1i(ω) and Qe1i(ω) deviated, which resulted in erroneous A'FM and D'FM line shapes. To reduce ambiguity, in this paper, we referred to the absorption and dispersion components depicted by the line shapes function as A'FM and D'FM, respectively. We referred to the absorption and dispersion components depicted by the demodulated signals as AFM and DFM, respectively. The incorrect A'FM and D'FM had frequencies of extreme points, so the line shapes calculated by the extreme points frequencies were not the same as the line shapes of AFM and DFM. With this characteristic, we were able to judge whether or not the phase difference θ deviated by calculating the correlation coefficient R (AFM, A'FM) between the absorption components and the correlation coefficient R (DFM, D'FM) between the dispersion components. Figure 2 is a flow chart for the phase correction algorithm. It references the flow chart of the phase correction section summarized by Hall et al. [7], to make the readers aware of the difference between this method and the commonly used phase correction method. On the basis of this idea, all of the possible values (0–2π) of the phase difference θ were calculated to obtain the sum of the correlation coefficient R (AFM, A'FM) and R (DFM, D'FM) at all of the phase differences θ. By comparing the sum of the correlation coefficients, we obtained the correct phase difference θ. Detailed information is described in the following subsections.

 figure: Fig. 2.

Fig. 2. Algorithm for phase determination with Ieli and Qeli data.

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2.3.1 Absorption and dispersion signals transformation

We obtained the absorption AFM and dispersion DFM components by demodulating signals Ieli(ω) and Qeli(ω). They can be derived from Eq. (4) and Eq. (5) and can be expressed as follows:

$${A_{FM}} = \gamma M({\delta _{ - 1}} - {\delta _1}) = \cos \theta {I_{eli}} + \sin \theta {Q_{eli}}\textrm{,and}$$
$${D_{FM}} = \gamma M({\phi _{ - 1}} + {\phi _1} - 2{\phi _0}) = \sin \theta {I_{eli}} - \cos \theta {Q_{eli}}.$$
As noted, Eq. (7) and Eq. (8) show that the changes in AFM and DFM are determined by the phase difference θ.

2.3.2 Line shape inversion by extreme point frequency

Gas absorption line shape affects the correction of phase difference. In practice, the absorption line shape usually is considered to have a Gaussian [5], Lorentzian [1,3,15], or Voigt [16] line shape. When we determined the gap absorption line shape, we also determined the profile of the demodulated signals. We found that the extreme points frequencies of the absorption and dispersion components were related only to the profile of demodulated signals and the phase difference. So the phase difference could be corrected by the profile of demodulated signals. Equation (6) shows the fixed numerical relationships of the profiles of Gaussian and Lorentzian line shapes. The Vogt shape is a convolution of a Gaussian line shape and a Lorentz line shape. Therefore, when we determined the frequencies of the extreme points of the Gaussian line shape and the Lorentz line shape, we also determined the frequencies of the extreme point of the Voigt line shape. By using those good characteristics, the line shape could be inverted by the extreme point frequency. By comparing the line shape with the profiles of demodulated, we determined the final phase difference.

2.3.3 Discrimination basis

In this paper, we determined the phase difference θ by discriminating the sum of the correlation coefficient R (AFM, A'FM) and R (DFM, D'FM), and R is expressed as follows:

$$R(X,Y) = \frac{{Cov(X,Y)}}{{\sqrt {Var(X)Var(Y)} }}$$
where Cov(X,Y) is the covariance of X and Y, Var(X) is the variance of X, and Var(Y) is the variance of Y.

The correlation coefficient can be used not only to indicate the magnitude of the variance but also to reflect the influence of the change in the number of extreme points in the case of any phase difference θ. Therefore, sum of the correlation coefficient is more suitable as a basis for discrimination.

3. Experimental setup

We based the experimental setup constructed on FMS, including near-infrared laser, laser controller, high-frequency modulation module, self-made gas cell, high-speed detector, demodulation module, data acquisition card, and host computer, as shown in Fig. 3.

 figure: Fig. 3.

Fig. 3. Schematic diagram of experiment setup based on FMS.

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The injection current and temperature of the laser (NLK1E5EAAA, NEL, Japan) was driven by a laser controller (LDC-3908, ILX Lightwave, USA). The high-frequency modulation module included a voltage-controlled oscillator and a voltage-controlled attenuator. The laser beam passed through a self-made gas cell and was detected by a high-speed detector (1611, New Focus, USA). The AC output signal of the high-speed detector was sent to the RF (signal) port of the IQ demodulator (ZAMIQ-895D, Mini-Circuits, USA). The reference signal was fed through phase shifter to the LO (carrier) terminal of the demodulator. The I/Q demodulated signal was acquired by a data acquisition card (PCI-4474, NI, USA) through a 1 kHz low-pass filter. The data acquisition card simultaneously acquired the DC output signal outputted by the high-speed detector.

4. Results and discussion

We verified the method by measuring water absorbance in two air samples: S1 from indoor air and S2 from outdoor air with a higher water vapor concentration. After we detected the S1 air, the receiving angle of the detector changed. The phase and light intensity of the measured signal changed at the same time after changing the settings, but the propagation path remained the same or the change could be ignored. In this experiment, the output wavenumber of the laser was 7184.9–7186.4 cm−1, which was used to measure water vapor at 7185.58 cm−1, and the scanning frequency was 10 Hz. The modulation frequency was 0.88 GHz and the modulation index was 2. The modulation index was measured by a Scanning Fabry Perot Interferometer (SA200, Thorlabs, USA) in advance. The SA200 was a high finesse Spectrum Analyzer used to examine the fine structures of the spectral characteristics of CW lasers. From the spectrum, the size of the main peak and the first order sideband was obtained. Therefore, the modulation index is determined according to the ratio of the main peak and the first order sideband [17], thereby avoiding to measure the frequency characteristic of the laser modulation efficiency. The temperature in the gas cell was 297.05 K, pressure was 1 atm, and the effective optical path of gas absorption was 40 cm.

The results are shown in Fig. 4. On average, we sampled each signal 10 times. We obtained the absorbance of the sample from the DC term signal shown in Fig. 4(a). The absorbance of S1 was 0.0369, the absorbance of S2 was 0.0541, and their ratio was 1.466. Figure 4(a) also shows that the signal amplitude of S1 was greater than that of S2. This signal represented that the light intensity of S1 entering the detector was greater than that of the light intensity of S2 entering the detector. Figures 4(b) and 4(c) are the demodulated signals received from the IQ demodulator after passing through the low-pass filter. As can be seen, the vertical positions of demodulated signals were not at the 0 scale. This phenomenon was caused by the DC offset of the IQ demodulator output signal and amplitude modulation [18]. Therefore, we had to perform data processing on the demodulated signals in advance to remove the influence of this offset. We obtained the DC offset of the IQ demodulator and amplitude modulation by quadratic fitting the nonabsorptive portion of the demodulated signals. The real demodulated signals included the original data minus the DC offset of the IQ demodulator and amplitude modulation. For convenience, the demodulated signals described later are all real demodulated signals.

 figure: Fig. 4.

Fig. 4. The recorded (a) DC terms (b), demodulated I, and (c) demodulated Q of atmospheric water absorbance around 7185.58cm−1 by FMS descript in Fig. 3. The scanning frequency is 10 Hz, the modulation frequency is 0.88 GHz, the modulation index is 2, T = 297.05 K, P = 1 atm, and the effective optical path L = 40 cm, 10 times average sampling.

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After obtaining the demodulated signals, we first corrected the demodulated signals for light intensity to eliminate the influence of the intensity fluctuation. This process is described in Section 2.2 and the results are shown in Fig. 5. As shown in Fig. 5, the ratio of the signals before and after correction changed. The detailed data are listed in Table 1. According to Table 1, the ratio of the absorbance calculated by the uncorrected I-channel signal was 1.111, which was 24.2% less than the absorbance ratio calculated by direct absorption. The ratio of the absorbance calculated by the corrected I-channel signal was 1.356, which was 7.5% less than the absorbance ratio calculated by direct absorption. Thus, the error was reduced. The ratio of the absorbance calculated by the uncorrected Q-channel signal was 1.138, which was 22.4% less than the absorbance ratio calculated by direct absorption. The ratio of the absorbance calculated by the corrected Q-channel signal was 1.385, which was 5.5% less than the absorbance ratio calculated by direct absorption. The error in IQ was reduced by an average of 16.8%. As shown from the previous ratio, the corrected signal error was smaller than the uncorrected signal error. This showed that it effectively reduced the adverse effects of light intensity fluctuations after we corrected the demodulated signals.

 figure: Fig. 5.

Fig. 5. The real demodulated signals and the corrected demodulated signals by using DC term. (a) I-channel (b) Q-channel. S1: Laboratory air, S2: Laboratory outside air; I: I-channel, Q: Q-channel; Real: The real demodulated signals (left vertical axis); Corrected: the corrected demodulated signals by using DC term (right vertical axis)

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Tables Icon

Table 1. Absorbance of water vapor, intensity of the demodulated signal, and corrected intensity of the demodulated signal of laboratory air (S1) and laboratory outside air (S2)

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After eliminating the influence of light intensity, we obtained the phase difference θ by the corrected demodulated signals. First, we determined the line shape of the spectral feature. For water vapor in the air at normal temperature and pressure, the spectral feature was Lorentzian or near-Lorentzian line shape. Therefore, this experiment used the Lorentzian line shape as a fitted line shape. By using Eqs. (6)–(9) in Section 2.3, we solved the sums of the correlation coefficient for the corrected demodulated signals, as shown in Fig. 6. The closer the sum of the correlation coefficient was to 2, the more linear the correlation between the corrected experimental value and the theoretical value would be. Conversely, the closer the sum of the correlation coefficient of the IQ channel was to −2, the less linear the correlation between the corrected experimental value and the theoretical value would be. According to Fig. 6, the phase difference of S1 was finally 93.3°, and the phase difference of S2 was 91.6°. The linewidths calculated from the respective phase differences θ were both 2.6915 GHz. This result was exactly the same as expected because the linewidth of the two samples theoretically should be the same under the same measuring conditions. Therefore, the phase determination was effective.

 figure: Fig. 6.

Fig. 6. Sums of correlation coefficient of the corrected demodulated signals using a Lorentzian line shape fitting. The step interval of the phase difference θ is 0.1°.

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After eliminating of influence of the intensity fluctuation and the correction of phase difference, we restored the absorption and dispersion components of the demodulated signals. As shown in Fig. 7, the absorption and dispersion components still had an asymmetry phenomenon caused by amplitude modulation. This phenomenon demonstrated that the original information of the amplitude modulation of the demodulated signal was retained. From the absorption and dispersion components, we obtained many spectral features of interest. This experiment provided proof of the advantages of the algorithm in eliminating the effects of light intensity and phase correction.

 figure: Fig. 7.

Fig. 7. The (a) absorption and (b) dispersion components of the demodulated signals were restored.

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5. Conclusion

The FMS demodulated signals were self-corrected from two aspects: the influence of the intensity fluctuation, and the phase difference between the measured signal and the reference signal. We eliminated effect of light intensity fluctuations by using the DC component of the measured signal. We corrected the phase difference determination by using the profile of a prior demodulated signal. The resulting line shapes retained the characteristics of the original signal. This method made full use of the measured signal, which did not additionally measure other parameters to correct the influence of light intensity fluctuation. It also made FMS signals more accurate and stable. At the same time, based on the line shapes of the FMS signal, we used the profile of the correctly demodulated signals as a reference to identify whether a demodulation was phase matched. This method avoided the effects of error accumulation from the finite integration and corrected mismatched demodulated signals. Although the method avoided noise accumulation, it failed when the signal-to-noise ratio is too low to determine extreme point frequency of the signal. The self-corrected algorithm was immune to light intensity fluctuation and was phase random, which reduced the dependence of FMS on light intensity and phase. This FMS also has a wider application in nonlaboratory measurement environments and in harsh environments. Therefore, this algorithm broke the limitations of FMS in practical applications and made FMS better reflect its advantages.

Funding

National Natural Science Foundation of China (61505142); National Key Scientific Instrument and Equipment Development Projects of China (2012YQ06016501).

References

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5. S. W. North, X. S. Zheng, R. Fei, and G. E. Hall, “Line shape analysis of Doppler broadened frequency-modulated line spectra,” J. Chem. Phys. 104(6), 2129–2135 (1996). [CrossRef]  

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8. K. A. Whittaker, J. Keaveney, I. G. Hughes, and C. S. Adams, “The Hilbert transform: Applications to atomic spectra,” Phys. Rev. A 11, 1–16 (2014).

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11. G. B. Rieker, J. B. Jeffries, and R. K. Hanson, “Calibration-free wavelength-modulation spectroscopy for measurements of gas temperature and concentration in harsh environments,” Appl. Opt. 48(29), 5546–5560 (2009). [CrossRef]  

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Figures (7)
Fig. 1.
Fig. 1. Demodulated signals with different phase differences. The vertical line is the extreme point frequency of each signal, and the auxiliary line is the equal frequency line.

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Fig. 2.
Fig. 2. Algorithm for phase determination with Ieli and Qeli data.

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Fig. 3.
Fig. 3. Schematic diagram of experiment setup based on FMS.

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Fig. 4.
Fig. 4. The recorded (a) DC terms (b), demodulated I, and (c) demodulated Q of atmospheric water absorbance around 7185.58cm−1 by FMS descript in Fig. 3. The scanning frequency is 10 Hz, the modulation frequency is 0.88 GHz, the modulation index is 2, T = 297.05 K, P = 1 atm, and the effective optical path L = 40 cm, 10 times average sampling.

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Fig. 5.
Fig. 5. The real demodulated signals and the corrected demodulated signals by using DC term. (a) I-channel (b) Q-channel. S1: Laboratory air, S2: Laboratory outside air; I: I-channel, Q: Q-channel; Real: The real demodulated signals (left vertical axis); Corrected: the corrected demodulated signals by using DC term (right vertical axis)

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Fig. 6.
Fig. 6. Sums of correlation coefficient of the corrected demodulated signals using a Lorentzian line shape fitting. The step interval of the phase difference θ is 0.1°.

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Fig. 7.
Fig. 7. The (a) absorption and (b) dispersion components of the demodulated signals were restored.

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Tables (1)

Tables Icon

Table 1. Absorbance of water vapor, intensity of the demodulated signal, and corrected intensity of the demodulated signal of laboratory air (S1) and laboratory outside air (S2)

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Equations (9)

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I mod ( t ) = I 0 exp ⁡ ( − 2 δ 0 ) ⏞ DC component + I 0 exp ⁡ ( − 2 δ 0 ) [ ( δ − 1 − δ 1 ) M cos ⁡ ω m t + ( ϕ − 1 + ϕ 1 − 2 ϕ 0 ) M sin ⁡ ω m t ] ⏞ AC component ,
I F M ( ω ) = I 0 exp ⁡ ( − 2 δ 0 ) γ M [ cos ⁡ θ ( δ − 1 − δ 1 ) + sin ⁡ θ ( ϕ − 1 + ϕ 1 − 2 ϕ 0 ) ] , and
Q F M ( ω ) = I 0 exp ⁡ ( − 2 δ 0 ) γ M [ sin ⁡ θ ( δ − 1 − δ 1 ) − cos ⁡ θ ( ϕ − 1 + ϕ 1 − 2 ϕ 0 ) ] .
I e l i ( ω ) = I F M ( ω ) /[ I 0 exp ⁡ ( − 2 δ 0 ) ] = γ M [ cos ⁡ θ ( δ − 1 − δ 1 ) + sin ⁡ θ ( ϕ − 1 + ϕ 1 − 2 ϕ 0 ) ] , and
Q e l i ( ω ) = Q F M ( ω ) / [ I 0 exp ⁡ ( − 2 δ 0 ) ] = γ M [ sin ⁡ θ ( δ − 1 − δ 1 ) − cos ⁡ θ ( ϕ − 1 + ϕ 1 − 2 ϕ 0 ) ] .
{ U = ± 2 n 2 − 2 − 4 n 4 + n 2 + 1 − 6 , n = Δ v L 2 ω m exp ⁡ { 16 ln ⁡ 2 ω m Δ v D 2 ∗ U } = U + ω m U − ω m
A F M = γ M ( δ − 1 − δ 1 ) = cos ⁡ θ I e l i + sin ⁡ θ Q e l i ,and
D F M = γ M ( ϕ − 1 + ϕ 1 − 2 ϕ 0 ) = sin ⁡ θ I e l i − cos ⁡ θ Q e l i .
R ( X , Y ) = C o v ( X , Y ) V a r ( X ) V a r ( Y )
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