Wavelength modulation spectroscopy by employing the first harmonic phase angle method

Chenguang Yang; Liang Mei; Hao Deng; Zhenyu Xu; Bing Chen; Ruifeng Kan

doi:10.1364/OE.27.012137

1. Introduction

As the rapid development of infrared diode laser technologies, tunable diode laser absorption spectroscopy (TDLAS) technology has been widely applied in greenhouse gas and air pollution monitoring, combustion diagnostics, industrial processes control, etc [1–6]. Traditionally, there are mainly two different approaches to accomplish the TDLAS technology, namely the direct absorption spectroscopy (DAS) [5] that recovers the absorption line-shape by scanning the laser wavelength across the absorption peak of the target gas, and the wavelength modulation spectroscopy (WMS) [6] that detects the gas absorption harmonics by scanning and modulating the laser frequency simultaneously. The WMS technique can greatly improve the detection sensitivity up to 10⁻⁵-10⁻⁶ Hz^-1/2 as the 1/f noise is significantly suppressed by shifting the detection band to high frequency [2]. During the past decades, the WMS technique has been extensively employed as an effective tool for high-sensitivity gas sensing [7–21].

In the WMS technique, the second harmonic (2f) signal is often employed for gas concentration retrieval [8–13]. In order to eliminate the dependency on the laser power, Hanson et al. [9,10] utilized the first harmonic normalized second harmonic (2f/1f) signal for gas concentration retrieval based on the quadrature demodulation method. In order to achieve the maximize signal intensity, the optimized modulation index (m), which is the ratio between the frequency modulation (FM) amplitude and the half-width at half-maximum (HWHM) of the absorption feature, should be equal to about 2.2 for 2f signal [9] and 1.1 for 2f/1f signal [10]. In high pressure conditions, where the absorption profile will be greatly broadened, large FM amplitude is thus required to achieve the optimum modulation index for the 2f method, which can cause strong nonlinear laser response and increase the complexity of signal analysis [11]. In strong turbulence or high time-resolution measurements, requiring high modulation frequencies, large FM amplitude is very difficult to achieve due to the limited FM response ratio of laser diodes [12]. Thus, the optimum modulation index for the 2f method is very difficult to achieve in extreme conditions due to the limitations on the injection current of laser diodes.

In this work, a gas concentration retrieval approach based on the first harmonic phase angle (1f-PA) method, which is immune to the laser power and the demodulation phase, has been developed and validated. The relationship between the 1f-PA of WMS and the gas absorption is, to our knowledge, for the first time investigated. Theoretical analysis as well as numerical simulation has been carried out and the phenomenological explanation is given by the phasor decomposition method. Moreover, the 1f-PA method has been studied experimentally by measuring carbon dioxide (CO₂) absorption line around 6362.5 cm⁻¹, and the inter-comparison between the traditional 2f/1f method and the 1f-PA method has also been performed.

2. Principles of the 1f-PA method

As mentioned in several early studies [13–16], WMS adds the high-frequency modulation current on the linearly increased injection current of the diode laser, which causes the modulations of the light intensity ( $I_{0}$ ) as well as the frequency ( $ν$ ) simultaneously

I_{0} = {\bar{I}}_{0} (1 + i_{1} \cos (2 π f_{m} t + ψ)),

ν = ν_{0} + ν_{1} \cos (2 π f_{m} t) .

Here f_m is the modulation frequency,

{\bar{I}}_{0}

and

ν_{0}

are the average laser intensity and the center frequency, i₁ is the relative intensity modulation (IM) amplitude (normalized by

{\bar{I}}_{0}

), ν₁ is the relative frequency modulation (FM) amplitude, ψ is the phase shift between the IM and the FM (IM-FM phase shift).

According to the Beer-Lambert Law, the transmitted laser intensity I can be expressed as

I \approx I_{0} (1 - α) \approx {\bar{I}}_{0} (1 + i_{1} \cos (2 π f_{m} t + ψ) - α (ν) - i_{1} \cos (2 π f_{m} t + ψ) α (ν)) .

The absorption coefficient α(ν) can be expanded in the Fourier cosine series

α (ν) = α (ν_{0} + ν_{1} \cos (2 π f_{m} t)) = \sum_{k = 0}^{\infty} H_{k} (ν_{0}, ν_{1}) \cos (2 π k f_{m} t) .

Here H_k is the k^th order Fourier coefficient of absorption, given by

\begin{matrix} H_{0} (ν_{0}, ν_{1}) = \frac{1}{2 π} \int_{- π}^{π} α (ν_{0} + ν_{1} \cos θ) d θ \\ H_{k} (ν_{0}, ν_{1}) = \frac{1}{π} \int_{- π}^{π} α (ν_{0} + ν_{1} \cos θ) \cos (k θ) d θ \end{matrix}} .

Demodulating the transmitted laser intensity with the in-phase reference signal and the out-of-phase reference signal, the first harmonic absorption (1f) signals, namely X₁_f and Y₁_f, can be described by

\begin{matrix} X_{1 f} = \frac{{\bar{I}}_{0}}{2} (i_{1} \cos ψ - H_{1} - i_{1} H_{0} \cos ψ - \frac{1}{2} i_{1} H_{2} \cos ψ) \\ Y_{1 f} = - \frac{{\bar{I}}_{0}}{2} (i_{1} \sin ψ - i_{1} H_{0} \sin ψ + \frac{1}{2} i_{1} H_{2} \sin ψ) \end{matrix}} .

The first harmonic phase angle (1f-PA), denoted as θ_1f, can be described by

θ_{1 f} = arc \tan (\frac{Y_{1 f}}{X_{1 f}}) = arc \tan (\tan (- ψ) \frac{1 - H_{0} + \frac{1}{2} H_{2}}{1 - \frac{1}{i_{1} \cos (- ψ)} H_{1} - H_{0} - \frac{1}{2} H_{2}}) .

As shown in Eq. (7), θ_1f is the function of the Fourier coefficients of absorption (H_k), relative IM amplitude (i₁) and the IM-FM phase shift (ψ), but is independent of the laser average intensity

{\bar{I}}_{0}

. The background 1f-PA without any absorption equals to –ψ.

As illustrated in Fig. 1(a), the 1f-PA has the similar shape with the first derivative of the gas absorption curve. The waveform distortion of the 1f-PA increases with the increasing of the integral absorbance. The linearity between the 1f-PA and the integral absorbance is beyond 0.998 with an integral absorbance in the region from 10⁻⁴ to 0.1, as shown in Fig. 1(b).

Fig. 1 1f-PA with different integral absorbances (a) and the relationship between the integral absorbance and the peak-to-peak value of 1f-PA (b). FM amplitude (ν₁) is supposed to be equal to the absorption half width. The IM-FM phase shift (ψ) is supposed to be 1.25π.

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In order to better illustrate the 1f-PA, the harmonic signals is further analyzed by the phasor decomposition method [17–21]. According to Eq. (6), the first harmonic signal vector ${\vec{R}}_{1 f}$ , consisting of the X₁_f and Y₁_f, can be expressed as the sum of a series of vectors, as shown in Fig. 2.

{\vec{R}}_{1 f} = R_{1 f} e^{i θ_{1 f}} = \frac{{\bar{I}}_{0}}{2} i_{1} e^{i (- ψ)} + \frac{{\bar{I}}_{0}}{2} H_{1} e^{i π} + \frac{{\bar{I}}_{0}}{2} i_{1} H_{0} e^{i (π - ψ)} + \frac{{\bar{I}}_{0}}{2} i_{1} H_{2} e^{i (π + ψ)} .

The background subtracted 1f-PA, namely θ₁_f-₀, which is the angle difference between the 1f signal vector

{\vec{R}}_{1 f}

and the 1f RAM vector

{\bar{I}}_{0} i_{1} e^{i (- ψ)} / 2

is dependent on the absorption and equals to zero under the circumstance of no absorption. As can be seen from Eq. (8), the average laser intensity (

{\bar{I}}_{0}

) appears in all vectors, which implies that the background subtracted 1f-PA (θ₁_f-₀) is independent of

{\bar{I}}_{0}

. As shown in Fig. 3(a), the 1f signal vector can be decomposed to X_1f and Y_1f. The background subtracted 1f-PA (θ₁_f-₀) is independent of the demodulation reference phase.

Fig. 2 Phasor diagram of 1f signal vector with the coordinate axes in-phase with the FM response. The IM-FM phase shift (ψ) varies from π to 1.5π.

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Fig. 3 Phasor analysis of the 1f signal vector: (a) decomposition of the 1f vector in different coordinate system and (b) 1f vector with different integral absorbance.

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The approximate linear relationship between the background subtracted 1f-PA (θ₁_f-₀) and the integral absorbance for weak absorption can be explained by Fig. 3(b). The absorption-related vectors, ${\bar{I}}_{0} H_{1} e^{i (- π)} / 2$ , ${\bar{I}}_{0} i_{1} H_{0} e^{i (π - ψ)} / 2$ and ${\bar{I}}_{0} i_{1} H_{2} e^{i (π + ψ)} / 4$ , are linearly proportional to the integral absorbance and much smaller than the 1f RAM vector ${\bar{I}}_{0} i_{1} e^{i (- ψ)} / 2$ . Thus, the sum of these absorption-related vectors has the same properties. As a result, the modulus of the 1f signal vector (R₁_f) approximately equals to the modulus of the 1f RAM vector ( ${\bar{I}}_{0} i_{1} / 2$ ), and the background subtracted 1f-PA (θ₁_f-₀) can be approximately estimated by the ratio between the modulus of the sum of these absorption-related vectors and the modulus of the 1f RAM vector, which is linearly dependent on the integral absorbance.

As the contribution of the nonlinear response of the diode laser is much smaller than the 1f RAM vector ${\bar{I}}_{0} i_{1} e^{i (- ψ)} / 2$ and other absorption-related vectors in Eq. (8) [9,11], the nonlinear amendment of 1f-PA would not affect the properties of this method.

3. Experiments and results

3.1 Experimental setup

Absorption measurement of CO₂ has been carried out to verify the feasibility and performance of the 1f-PA method. The diagram of the experiment setup is shown in Fig. 4. The absorption line of CO₂ at 6362.498 cm⁻¹ and several other nearby but relatively weaker absorption lines are chosen to compare the 1f-PA method with the traditional 2f/1f method. An 100 Hz, 1.4 V_pp (peak-to-peak voltage) sawtooth signal plus a sine wave, generated by the multifunction I/O device (NI Corporation, USB-6363) with 2 MS/s sample rate, is sent to a homemade laser controller to drive the near infrared DFB diode laser (NTT, NLK1L5EAAA). The bandwidth of the homemade laser controller is 2 MHz, which is much higher than the upper limit (100 kHz) of the modulation frequency in the following experiments. The laser beam is divided by the fiber splitter into two parts, the measurement path and the reference path. The measurement path is addressed to the 50 cm gas cell and the transmitted laser beam is detected by a photodiode (PD1, Thorlabs, PDA20CS-EC). The detected signal is acquired by the same multifunction I/O device mentioned above with 2 MS/s sample rate. The reference signal detected by the other photodiode (PD2, Thorlabs, PDA10DT-EC) is acquired by an oscilloscope. The free spectral range of the homemade quartz etalon is 0.0173 cm⁻¹. The limited vacuum and vacuum leak rate of the gas cell are 280 Pa and 5 Pa/h, respectively. The pressure in the gas cell is controlled at the near atmosphere pressure (105500 ± 500 Pa) by the metering valve.

Fig. 4 Diagram of the experiment setup for the first harmonic detection. DFB-distributed feedback diode laser, AO-Analog Output, AI-Analog Input, PD-photo diode, PC-Personal Computer

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3.2 First harmonic phase angle signal

Figure 5 shows the experimental results measured with pure CO₂ and pure N₂. The frequency and amplitude of the modulation signal are 30 kHz and 350 mV corresponding to 28 mA modulation current, respectively. The laser FM amplitude, measured by the etalon, equals to 0.1038 cm⁻¹. The modulation index m is 1.059 for 1 atm pure CO₂. The quadrature 1f signals of absorption (X₁_f, Y₁_f) and background (X⁰₁_f, Y⁰₁_f), shown in Fig. 5(c), are obtained by digitally demodulating the original signals (Figs. 5(a) and 5(b)) with 2 kHz low-pass filter and arbitrary demodulation phase, i.e., 0° in this case. As can be seen in Fig. 5(d), the absorption 1f-PA (θ₁_f) has explicit waveform distortions around the absorption peaks, while the background 1f-PA ( $θ_{1 f}^{0}$ ) is nearly a straight line. The background subtracted 1f-PA (θ₁_f_-0) is a baseline-free curve with phase distortions only around the absorption peaks.

Figure 5 (a) the original absorption signal with high purity CO₂; (b) the original absorption signal with high purity N₂; (c) the absorption and background quadrature first-harmonic signals; (d) the absorption and background 1f-PAs and the background subtracted 1f-PA.

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As shown in Fig. 6, the peak and valley of the absorption and the background 1f-PAs periodically change as the demodulation phase varies from 0° to 360°. Except the hopping of the 1f-PA signal at 30°, 40°, 210° and 220° demodulation phase due to the arc-tangent function, the peak-to-peak values of the 1f-PA (θ₁_f) and the background subtracted 1f-PA (θ₁_f_-0) remain unchanged in this case, i.e., 0.344 rad and 0.343 rad, respectively. In order to reduce the phase jitter and baseline due to the arbitrary demodulation phase, the background subtracted 1f-PA (θ₁_f_-0) will be used in the following section.

Fig. 6 The peak, valley, and peak-to-peak values with different demodulation phases.

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3.3 First harmonic phase angles with different modulation parameters

The peak-to-peak values of the background subtracted 1f-PA (θ_1f-0) under different modulation amplitudes and frequencies were measured with pure CO₂ to investigate the feasibility of the 1f-PA method for gas concentration retrieval. The measurement results were also compared with the peak values of the root-sum-square magnitude of the background subtracted 2f/1f signals (R_2f/1f-0). Demodulation parameters are the same with those used in Section 3.2. The maximum modulation amplitude is limited by the ramp current and the maximum inject current of the laser, and the maximum modulation frequency is limited by the sample rate of the I/O device.

The FM amplitude and IM-FM phase shift of the DFB laser diode used in this paper are measured with the etalon. The modulation index m is calculated by dividing the FM amplitude by the self-broadened HWHM of CO₂ under 1 atm pressure. As shown in Fig. 7(a), the FM amplitude linearly increases with the modulation current, but decreases with the increasing of the modulation frequency. This implies that the best modulation index (m = 1.1) for 2f/1f signals cannot be achieved when the modulation frequency is higher than 40 kHz. The IM-FM phase shift increases with the increasing of the modulation frequency, and almost remains constant for different modulation amplitudes.

Fig. 7 Frequency amplitude (a) and IM-FM phase shift (b) of the DFB laser diode with different modulation currents and frequencies.

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The peak-to-peak values of θ_1f-0 and the peak values of R_2f/1f-0 are plotted in Fig. 8. The local maximum of the peak values of R_2f/1f-0 appears at 16 mA (200 mV) modulation amplitude for 10 kHz modulation frequency. However, the modulation amplitude of the R_2f/1f-0 local maximum continuously increases with the increasing of the modulation frequency. Unlike the peak value of R_2f/1f-0, the peak-to-peak value of θ_1f-0 decreases with the modulation amplitude monotonously, especially at low modulation frequencies. At high modulation frequencies, the peak-to-peak value of θ_1f-0 is not sensitive to modulation amplitude, because of the low FM response ratio. As mentioned in Section 2, θ_1f-0 can be approximately estimated by the ratio between the modulus of the sum of absorption-related vectors and the modulus of the 1f RAM vector, which is linearly proportional to the IM amplitude. Therefore, θ_1f-0 decreases with the increasing of the modulation amplitude. However, this does not imply that small modulation amplitude should be chosen, as the signal-to-noise ratio of the absorption signals also deteriorates with the decreasing of the modulation amplitude due to the presence of various noises. After all, the modulation amplitude should be optimized to achieve the best SNR for θ_1f-0 rather than the maximum value. On the other hand, the influence of the FM-IM phase shift is notable at small modulation amplitude. At 4 mA (50 mV) modulation amplitude, the peak-to-peak value of θ_1f-0 decreases with the modulation frequency due to the increasing of the FM-IM phase shift.

Fig. 8 Peak-to-peak values of θ_1f-0 (a) and peak values of R_2f/1f-0 (b) for pure CO₂ with different modulation currents and frequencies.

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3.4 Linearity and sensitivity of first harmonic phase angles

Experiments with a series of concentrations of CO₂ balanced by N₂ that varies with two orders of magnitude from 0 to 80% were performed to verify the linearity of the 1f-PA method. The peak-to-peak values of θ_1f-0 and the peak values of R_2f/1f-0 show good correlations with the CO₂ concentration particularly at the high modulation frequency, as illustrated in Fig. 9. The correlation coefficient is generally better than 0.99 for both methods. The stability of the 1f-PA method has also been compared with the traditional 2f/1f method by employing the Allen deviation. Figure 10 shows the Allen deviations of CO₂ mole fraction retrieved by peak-to-peak values of θ_1f-0 and peak values of R_2f/1f-0. For the low modulation frequency (10 kHz), the two techniques have similar stabilities. However, the Allen deviation of θ_1f-0 at the high modulation frequency (100 kHz) has similar performance as that measured at the low modulation frequency, while the Allen deviation of R_2f/1f-0 significantly deteriorates with the increasing of the modulation frequency.

Fig. 9 Peak-to-peak values of θ_1f-0 and peak values of R_2f/1f-0 as a function of CO₂ concentration with different modulation amplitudes and frequencies.

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Fig. 10 Allen deviation of retrieved CO₂ mole fraction by peak-to-peak values of θ_1f-0 and peak values of R_2f/1f-0.

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Table 1 shows the correlation coefficient and the Allen Deviation measured with different modulation parameters and 1-s integration time. As has been discussed above, the correlation coefficient of θ_1f-0 slightly improves with the increasing of the modulation frequency. The Allen deviations of θ_1f-0 and R_2f/1f-0 decrease simultaneously with the increasing of the modulation current. By increasing the modulation frequency from 10 kHz to 100 kHz, the Allen deviation of θ_1f-0 keeps the same level, about 2 ppm within 1-s integration time, while the Allen deviation of R_2f/1f-0 is one order of magnitude worse.

Table 1. Correlation coefficient and Allen variance of θ_1f-0 and R_2f/1f-0

View Table

4. Conclusion

In this paper, the 1f-PA method, which is immune to the variances of signal intensity and demodulation phase, is proposed and validated for WMS measurements. The 1f-PA is approximately linearly dependent on the integral absorption under weak absorption conditions. The principle has been verified by measuring the CO₂ absorption around 6362.5 cm⁻¹ with a near infrared DFB diode laser. The relationship between the peak-to-peak value of θ₁_f_-0 and modulation parameters is carefully studied. The peak-to-peak value of θ₁_f_-0 decreases with the modulation amplitude monotonously. The relationship between the peak-to-peak value of θ₁_f_-0 and the modulation frequency is complicated due to the influence of the laser FM response ratio and the IM-FM phase shift. Finally, the linearity and sensitivity of the 1f-PA method are compared with the traditional 2f/1f method for different modulation conditions. The correlation coefficients with the CO₂ concentrations are generally better than 0.998 for these two methods and their Allen Variances are also similar for low modulation frequencies. Nevertheless, the sensitivity of the 1f-PA method nearly stays the same with the increasing of the modulation frequency, while the sensitivity of the traditional 2f/1f method deteriorates significantly. Overall, the 1f-PA method shows better performance than the traditional 2f/1f method under high modulation frequencies or modulation depth limited conditions, such as strong turbulence or high pressure environments.

Funding

National Key R&D Program of China (2017YFC0804900, 2016YFC0201100, 2016YFC0200600); National Natural Science Foundation of China (NSFC) (61805286, 61705030).

References

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Modulation condition	Correlation coefficient		Allen deviation(ppm@1s)
Modulation condition	θ_1f-0	R_2f/1f-0	θ_1f-0	R_2f/1f-0
10 kHz 250 mV	0.9983	0.9991	2.17	1.88
10 kHz 500 mV	0.9985	0.9989	1.91	1.36
50 kHz 250 mV	0.9997	0.9995	1.99	5.62
50 kHz 500 mV	0.9993	0.9999	1.12	2.04
100 kHz 250 mV	0.9998	0.9998	3.85	72.49
100 kHz 500 mV	0.9999	1.0000	1.83	14.85

Wavelength modulation spectroscopy by employing the first harmonic phase angle method

Abstract

1. Introduction

2. Principles of the 1f-PA method

3. Experiments and results

3.1 Experimental setup

3.2 First harmonic phase angle signal

3.3 First harmonic phase angles with different modulation parameters

3.4 Linearity and sensitivity of first harmonic phase angles

4. Conclusion

Funding

References

Cited By

Figures (10)

Tables (1)

Equations (8)

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