CN109696416B - Method for measuring gas absorption coefficient based on cavity ring-down technology - Google Patents

Method for measuring gas absorption coefficient based on cavity ring-down technology Download PDF

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CN109696416B
CN109696416B CN201910059718.0A CN201910059718A CN109696416B CN 109696416 B CN109696416 B CN 109696416B CN 201910059718 A CN201910059718 A CN 201910059718A CN 109696416 B CN109696416 B CN 109696416B
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彭志敏
丁艳军
王振
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Abstract

A method for measuring a gas absorption coefficient based on a cavity ring-down technology comprises the steps of obtaining ring-down time with periodic change by adopting a laser frequency scanning mode, then carrying out fast Fourier transform on the ring-down time, selecting Fourier coefficients of characteristic frequency, and substituting the Fourier coefficients into Fourier series to reconstruct the ring-down time; and measuring the laser frequency by adopting an interferometer, and reestablishing the relation between the ring-down time and the laser frequency to further obtain the gas absorption coefficient. The invention can selectively eliminate various natural frequency noises and reduce the gas absorption coefficient with higher signal-to-noise ratio; the interferometer can measure the laser wavelength once, and is not limited by the measurement precision and the measurement speed of the wavelength meter; the laser working in a scanning mode does not need a high-precision laser controller, and the influence of random jitter of laser wavelength is reduced.

Description

Method for measuring gas absorption coefficient based on cavity ring-down technology
Technical Field
The invention relates to a method for measuring a gas absorption coefficient based on a cavity ring-down technology, which is suitable for measuring gas temperature, pressure, concentration and spectral parameters and belongs to the technical field of measurement.
Background
The gas absorption coefficient may determine the temperature, pressure, concentration, and many spectral parameters of the gas. Cavity ring-down spectroscopy (CRDS) has been proven to be an accurate method for characterizing gas absorption coefficients, and is a promising method for realizing accurate measurement of spectral parameters. One of the advantages is that the equivalent optical length of the CRDS can easily reach several kilometers or even several tens of kilometers, so that the measured spectrum has a high signal-to-noise ratio. Another advantage is that the cavity ring-down spectroscopy (CW-CRDS) using a Continuous wave laser with a narrow linewidth has a high spectral resolution. In short, this laser-based absorption spectroscopy technique determines the absolute value of the optical loss in a highly accurate stable cavity from the measured photon lifetime or ring-down time; these losses include gas absorption, specular transmission, rayleigh scattering, etc. In a narrow wavelength range, only the absorption of gas changes with the wavelength, and the rest losses are constants; therefore, the absorption coefficient can be reproduced by measuring the ring-down time.
In actual measurements, there are still some problems with CW-CRDS. Firstly, when the ring-down time is measured, the wavelength meter is required to measure the laser wavelength, the measurement interval is hundreds of milliseconds each time due to the limitation of the measurement delay of the wavelength meter, and therefore the whole absorption spectrum needs to be measuredA time of several minutes or even tens of minutes; secondly, the process puts high requirements on the measurement accuracy of the wavelength meter and the stability of the laser, and the inherent noise of the system in the process cannot be ignored. For the former, we can use a more accurate wavemeter and a more stable, narrower linewidth laser; or more complex techniques such as optical feedback frequency-stabilized cavity ring-down spectroscopy and Pound-Drever-Hall (PDH) cavity ring-down spectroscopy are used to stabilize the laser frequency; or we can increase the cavity ring down time and average the spectra over a longer time to improve the signal-to-noise ratio. For the latter, since the system noise frequency is fixed, the fine tuning of the system is the best way to reduce the system noise at present. In the CW-CRDS technique, considering that system noise is of a fixed frequency, such noise can be eliminated in the frequency domain. Therefore, the ring-down time can be made a time-varying periodic signal by means of wavelength scanning. Since the ring-down time acquisition rate can reach above 20Hz, this scheme is not very difficult to implement. There are two problems to be solved here: firstly, the typical measurement rate of the wavemeter is 1-10 Hz, and the measurement precision is 0.001cm-1The wavelength scan rate must therefore not exceed the measurement rate of the wavemeter at the highest, thereby limiting the wavelength scan rate; secondly, it is difficult to directly obtain the correspondence between the wavelength and the ring-down time without using a wavelength meter. Therefore, in order to eliminate system noise and further improve the spectral signal-to-noise ratio, it is necessary to design a cavity ring-down spectroscopy measurement method based on wavelength scanning, extract a characteristic frequency through fourier transform to reconstruct ring-down time, eliminate the system noise from a frequency domain, and further obtain a gas absorption coefficient with higher precision.
Disclosure of Invention
The invention aims to eliminate the noise of a CW-CRDS system so as to further improve the spectral signal-to-noise ratio, and provides a method for measuring a gas absorption coefficient based on a cavity ring-down technology. The core of the method is as follows: 1, acquiring ring-down time with periodic change by adopting a wavelength scanning mode; performing Fourier transform on the ring-down time, extracting Fourier coefficients of characteristic frequencies, and reconstructing the ring-down time by utilizing Fourier series; 3, measuring the laser frequency by adopting an interferometer to obtain a laser frequency calibration coefficient; and 4, obtaining the relation between ring-down time and wavelength according to a derived formula, and further obtaining the absorption coefficient of the gas.
In order to achieve the purpose, the technical scheme of the invention is as follows:
1) selecting corresponding absorption spectrum lines from a spectrum database according to the type of the gas to be detected; the center frequency of which is v0The linear strength is S;
2) collecting ring-down time τ: as shown in fig. 1, a tunable semiconductor laser is used as a light source, when the laser frequency is fixed, a laser beam output by the tunable semiconductor laser sequentially passes through an optical isolator, an acousto-optic modulator and an optical collimator, the devices are connected by optical fibers, the light beam reaches a first plano-concave reflector after being collimated, passes through a ring-down cavity and a second plano-concave reflector, and is focused on a photoelectric detector through a convex lens; the signal of the photoelectric detector is divided into two paths, one path is collected by a collecting card and input into a main control computer, and the other path generates a pulse signal by a digital delay generator and inputs the pulse signal into an acousto-optic modulator. The piezoelectric ceramic is tightly attached to the second plano-concave reflecting mirror, a low-frequency triangular wave signal is generated by the function signal generator and is input into the high-voltage driver, and the piezoelectric ceramic is driven to scan the cavity length of the ring-down cavity; the scanning length is slightly longer than the laser wavelength so as to realize the mode matching of the incident laser wavelength and the eigenmode of the ring-down cavity, at the moment, the incident laser can be effectively coupled into the ring-down cavity, and the light intensity received by the photoelectric detector is the maximum; when the signal of the photoelectric detector reaches 0.9V, the digital delay generator can generate a pulse signal, the rising edge of the pulse signal is in the nanosecond level, and the acousto-optic modulator can be instantly turned off; because the light source is cut off, the light intensity reaching the photoelectric detector is attenuated in a single exponential manner; collecting the attenuation signal by a collection card and fitting the attenuation signal by a computer to calculate the corresponding ring-down time tau under the wavelength;
3) the temperature and current of the tunable semiconductor laser are controlled by a computer, so that the laser frequency is at the spectral line central frequency v0Scanning nearby in a triangular wave form in a stepping mode, wherein the scanning frequency is f; phase acquisition at each step intervalObtaining the ring-down time tau (t) with periodic variation by the corresponding ring-down time tau; carrying out fast Fourier transform on tau (t) to obtain Fourier coefficients, and taking out the Fourier coefficients corresponding to characteristic frequency kf from the Fourier coefficients, wherein the real part of the Fourier coefficients is AkImaginary part of Bk(ii) a Then A is mixedkAnd BkSubstituting into Fourier series to obtain reconstructed ring-down time tau' (t) of periodic variation:
Figure GDA0002627193370000021
where τ' (t) is the reconstructed periodically varying ring-down time, AkAnd BkRespectively a real part and an imaginary part of a Fourier coefficient corresponding to the characteristic frequency; omega is the angular frequency of the triangular wave, f is the scanning frequency of the triangular wave, and then omega is 2 pi f; p is the total number of the characteristic frequencies, k is 0,1,2 …, and p and t are scanning time;
4) the interferometer 18 is used to measure the laser frequency v (t) so that the laser frequency is at the spectral line center frequency v0Scanning nearby in a triangular wave form, wherein the scanning frequency is f, and fitting the rising edge and the falling edge of the laser frequency by using an nth-order polynomial:
Figure GDA0002627193370000031
wherein, vOn the upper part(t) laser frequency at rising edge, vLower part(t) represents the laser frequency of the falling edge; a isiThe ith order coefficient of the rising edge, biThe coefficient of the ith time of the falling edge is 0,1,2 …, n, and n is a polynomial order; l represents the period number, T is the period of the triangular wave, and T is 1/f;
5) establishing a relation between v (t) and tau' (t): let θ be ω (t-lT) ω t-2 pi l, where 0 < θ < pi, and l represents the number of cycles; replacing t with theta according to the expression of the laser frequency in the step 4) to obtain the following formula:
Figure GDA0002627193370000032
Figure GDA0002627193370000033
obtaining the ring-down time tau ' with the rising edge and the falling edge reconstructed and periodic change according to the expression of tau ' (t) in the step 3) 'On the upper partAnd τ'Lower part：
Figure GDA0002627193370000034
Figure GDA0002627193370000035
By an intermediate variable theta, let vOn the upper partAnd τ'On the upper partSimultaneous, vLower partAnd τ'Lower partSimultaneous to obtain reconstructed cyclically varying τ'On the upper part(v) and τ'Lower part(v); averaging the two to obtain reconstructed ring-down time tau' (ν) with periodic variation along with laser frequency variation;
6) obtaining a gas absorption coefficient alpha (v): substituting τ' (v) into the following equation:
Figure GDA0002627193370000036
obtaining the gas absorption coefficient alpha (v); wherein R is the reflectivity of the plano-concave reflector, L is the cavity length of the ring-down cavity, and c is the speed of light.
Compared with the prior art, the method has the following advantages and prominent technical effects: the ring-down time obtained by the method is a periodic signal which changes along with time, and the ring-down time is reconstructed by extracting characteristic frequency through Fourier transform, so that various inherent frequency noises can be selectively eliminated, and an absorption coefficient with a higher signal-to-noise ratio is reduced. Secondly, a wavelength meter is not needed in the measuring process, only the interferometer is needed to measure the laser wavelength once, and the measuring error and the measuring speed of the wavelength meter are not limited. Compared with the CW-CRDS method, the method scans the frequency of the laser, so that the laser frequency is not required to be kept stable for a long time, a high-precision laser controller is not required, and the influence of random jitter of the laser wavelength on measurement is reduced.
Drawings
Fig. 1 is a schematic diagram of the system architecture of the present invention.
FIG. 2 is a schematic diagram of the system structure of the interferometer for measuring the laser frequency in the present invention.
Fig. 3 shows the laser current, laser frequency, ring down time of the periodic variation and fourier transform coefficients thereof measured by the experiment of the present invention.
Fig. 4 is a graph of the absorption coefficients and their fitted residuals measured experimentally for the present invention.
In the figure: 1-a tunable laser; 2-an optical isolator; 3-an acousto-optic modulator; 4-an optical fiber; 5-a light collimator; 6-a first plano-concave mirror; 7-ring down cavity; 8-a second plano-concave mirror; 9-piezoelectric ceramics; 10-a high voltage driver; 11-function signal generator, 12-convex lens; 13-a photodetector; 14-high speed acquisition card; 15-a computer; 16-a laser controller; 17-a digital delay generator; 18-interferometer.
Detailed Description
The invention will be further described with reference to the accompanying drawings.
1) Selecting corresponding absorption spectrum lines from the spectrum database according to the type of the gas to be detected, wherein the center frequency of the absorption spectrum lines is v0The line intensity is S.
2) When the laser frequency is fixed, as shown in fig. 1, a laser beam output by the tunable semiconductor laser 1 sequentially passes through an optical isolator 2, an acousto-optic modulator 3 and an optical collimator 5, the devices are connected by adopting an optical fiber 4, the light beam reaches a first plano-concave reflector 6 after being collimated, passes through a ring-down cavity 7 and a second plano-concave reflector 8, and is focused on a photoelectric detector 13 through a convex lens 12; the signal of the photoelectric detector 13 is divided into two paths, one path is collected by the collection card 14 and input into the main control computer 15, and the other path generates a pulse signal by the digital delay generator 17 and inputs the pulse signal into the acousto-optic modulator 4. The piezoelectric ceramic 9 is tightly attached to the second plano-concave reflecting mirror 8, a low-frequency triangular wave signal is generated by the function signal generator 11 and is input into the high-voltage driver 10, and the piezoelectric ceramic 9 is driven to scan the cavity length of the ring-down cavity; the scanning length is slightly longer than the laser wavelength so as to realize the mode matching of the incident laser wavelength and the eigenmode of the ring-down cavity, at the moment, the incident laser can be effectively coupled into the ring-down cavity, and the light intensity received by the photoelectric detector 13 is the maximum; when the signal of the photoelectric detector 13 reaches 0.9V, the digital delay generator 17 can generate a pulse signal, the rising edge of the pulse signal is in the nanosecond level, and the acousto-optic modulator 3 can be instantly turned off; since the light source is switched off, the light intensity reaching the photodetector 13 decays in a single exponential manner; the attenuation signal is collected by the acquisition card 14 and fitted by the computer 15 to calculate the corresponding ring-down time tau at the wavelength.
3) The temperature and current of the tunable semiconductor laser are controlled by a computer, so that the laser frequency is at the spectral line central frequency v0Scanning nearby in a triangular wave form in a stepping mode, wherein the scanning frequency is f; collecting corresponding ring-down time tau at each stepping interval to obtain ring-down time tau (t) with periodic change; carrying out fast Fourier transform on tau (t) to obtain Fourier coefficient, and extracting characteristic frequency k from the Fourier coefficientfThe real part of the corresponding Fourier coefficient is AkImaginary part of Bk(ii) a Then A is mixedkAnd BkSubstituting into Fourier series to obtain reconstructed ring-down time tau' (t) of periodic variation:
Figure GDA0002627193370000051
where τ' (t) is the reconstructed periodically varying ring-down time, AkAnd BkRespectively a real part and an imaginary part of a Fourier coefficient corresponding to the characteristic frequency; omega is the angular frequency of the triangular wave, f is the scanning frequency of the triangular wave, and then omega is 2 pi f; p is the total number of the characteristic frequencies, k is 0,1,2 …, and p and t are scanning time;
4) the laser frequency v (t) is measured with the interferometer 18,make the laser frequency at the center frequency v of the spectral line0Scanning nearby in a triangular wave form, wherein the scanning frequency is f, and fitting the rising edge and the falling edge of the laser frequency by using an nth-order polynomial:
Figure GDA0002627193370000052
wherein, vOn the upper part(t) laser frequency at rising edge, vLower part(t) represents the laser frequency of the falling edge; a isiThe ith order coefficient of the rising edge, biThe coefficient of the ith time of the falling edge is 0,1,2 …, n, and n is a polynomial order; l represents the period number, T is the period of the triangular wave, and T is 1/f;
5) establishing a relation between v (t) and tau' (t): let θ be ω (t-lT) ω t-2 pi l, where 0 < θ < pi, and l represents the number of cycles; replacing t with theta according to the expression of the laser frequency in the step 4) to obtain the following formula:
Figure GDA0002627193370000053
Figure GDA0002627193370000054
obtaining the ring-down time tau ' with the rising edge and the falling edge reconstructed and periodic change according to the expression of tau ' (t) in the step 3) 'On the upper partAnd τ'Lower part：
Figure GDA0002627193370000061
Figure GDA0002627193370000062
By an intermediate variable theta, let vOn the upper partAnd τ'On the upper partSimultaneous, vLower partAnd τ'Lower partSimultaneous, i.e. obtaining a reconstructed periodicityOf varying τ'On the upper part(v) and τ'Lower part(v); averaging the two to obtain reconstructed ring-down time tau' (ν) with periodic variation along with laser frequency variation;
6) obtaining a gas absorption coefficient alpha (v): substituting τ' (v) into the following equation:
Figure GDA0002627193370000063
obtaining the gas absorption coefficient alpha (v); wherein R is the reflectivity of the plano-concave reflector, L is the cavity length of the ring-down cavity, and c is the speed of light.
And fitting alpha (v) by adopting a proper linear function (Voigt, Galatry and other linear functions) to obtain the concentration, pressure, temperature and spectral line physical parameters of the gas.
Example (b):
1) with CO2And N2Taking the mixed gas as an example, the absorption spectrum line is selected from a HITRAN spectrum database, and the center frequency v of the absorption spectrum line is0Is 6526.374cm-1Line intensity of 8X 10-25cm-1/(molec·cm-2) (ii) a The temperature of the mixed gas is 23 ℃, the pressure is 20kPa, and CO is2The concentration is 1%;
2) when the laser frequency is fixed, a laser beam output by the tunable semiconductor laser 1 sequentially passes through an optical isolator 2, an acousto-optic modulator 3 and an optical collimator 5, the devices are connected by adopting an optical fiber 4, the light beam reaches a first plano-concave reflector 6 after being collimated, passes through a ring-down cavity 7 and a second plano-concave reflector 8, and is focused on a photoelectric detector 13 through a convex lens 12; the signal of the photoelectric detector 13 is divided into two paths, one path is collected by the collection card 14 and input into the main control computer 15, and the other path generates a pulse signal by the digital delay generator 17 and inputs the pulse signal into the acousto-optic modulator 4. The piezoelectric ceramic 9 is tightly attached to the second plano-concave reflecting mirror 8, a low-frequency triangular wave signal is generated by the function signal generator 11 and is input into the high-voltage driver 10, and the piezoelectric ceramic 9 is driven to scan the cavity length of the ring-down cavity; the scanning length is slightly longer than the laser wavelength so as to realize the mode matching of the incident laser wavelength and the eigenmode of the ring-down cavity, at the moment, the incident laser can be effectively coupled into the ring-down cavity, and the light intensity received by the photoelectric detector 13 is the maximum; when the signal of the photoelectric detector 13 reaches 0.9V, the digital delay generator 17 can generate a pulse signal, the rising edge of the pulse signal is in the nanosecond level, and the acousto-optic modulator 3 can be instantly turned off; since the light source is switched off, the light intensity reaching the photodetector 13 decays in a single exponential manner; the attenuation signal is collected by the acquisition card 14 and fitted by the computer 15 to calculate the corresponding ring-down time at the wavelength. Wherein, the cavity length of the ring-down cavity is L equal to 50cm, and the reflectivity R of the plano-concave mirror is 0.999975; the cavity length scanning range exceeds 2 mu m, and the scanning period is 50 Hz;
3) the temperature and current of the laser are controlled by a computer program to make the laser frequency be at the center frequency v of the spectral line0The vicinity is scanned stepwise in the form of a triangular wave, the laser frequency is shown by a solid line in fig. 3(b), and the laser current is shown by a broken line in fig. 3 (b); the corresponding ring down time is also changed gradually, and the process obtains a ring down time with periodic change, as shown by the solid line in FIG. 3(a), the ring down time is about 38 μ s at the center frequency, and the ring down time at the baseline is about 51 μ s. Wherein the triangular wave scanning frequency f is 16Hz, the laser current scanning range is +/-15 mA, the central current is 80mA, the temperature is stabilized at 36.632 ℃, and the corresponding wavelength scanning range is about +/-0.2 cm-1(ii) a Carrying out fast Fourier transform on tau (t) to obtain Fourier coefficients, and taking out the Fourier coefficients corresponding to characteristic frequency kf from the Fourier coefficients, wherein the real part of the Fourier coefficients is AkImaginary part of Bk(ii) a Then A is mixedkAnd BkSubstituting into Fourier series to obtain reconstructed ring-down time tau' (t) of periodic variation:
Figure GDA0002627193370000071
where τ' (t) is the reconstructed periodically varying ring-down time, AkAnd BkRespectively a real part and an imaginary part of a Fourier coefficient corresponding to the characteristic frequency; omega is the angular frequency of the triangular wave, f is the scanning frequency of the triangular wave, and then omega is 2 pi f; p is the total number of characteristic frequencies, whereTaking p as 100, namely k as 0,1,2 …, 100, and t as scanning time;
4) the interferometer 18 is used to measure the laser frequency v (t) so that the laser frequency is at the spectral line center frequency v0Scanning nearby in a triangular wave form, wherein the scanning frequency is f ═ 16Hz, and fitting the rising edge and the falling edge of the laser frequency respectively by using an nth-order polynomial:
Figure GDA0002627193370000072
wherein, vOn the upper part(t) laser frequency at rising edge, vLower part(t) represents the laser frequency of the falling edge; a isiThe ith order coefficient of the rising edge, biThe coefficient of the ith term of the falling edge is 0,1,2 …, n, n is 2; l represents the period number, T is the period of the triangular wave, and T is 1/f; obtaining a calibration coefficient a by fitting2＝9.00×10-4，a1＝0.0637，a0＝0.0198，b2＝8.98×10-4，b1＝-0.0925，b0＝1.2683。
5) Establishing a relation between v (t) and tau' (t): let θ be ω (t-lT) ω t-2 pi l, where 0 < θ < pi, and l represents the number of cycles; according to the expression of the laser frequency in the formula (2), the formula can be obtained by replacing t with theta:
Figure GDA0002627193370000073
Figure GDA0002627193370000074
obtaining the ring-down time tau ' with the rising edge and the falling edge reconstructed and periodic change according to the expression of tau ' (t) in the step 3) 'On the upper partAnd τ'Lower part：
Figure GDA0002627193370000081
Figure GDA0002627193370000082
Due to vOn the upper partAnd τ'On the upper partAre all in one-to-one correspondence with the intermediate variable theta, and are combined with (3) and (4) to obtain reconstructed periodically varying tau'On the upper part(v), analogously to obtain reconstructed cyclically varying τ'Lower part(v); averaging the two to obtain reconstructed ring-down time tau' (ν) with periodic variation along with laser frequency variation; the intermediate variable θ gradually increases from 0 to 1 at an interval of 0.025, and the total number of points is 401.
6) Obtaining a gas absorption coefficient alpha (v): substituting τ' (v) into the following equation:
Figure GDA0002627193370000083
obtaining the gas absorption coefficient alpha (v); wherein, R is the reflectivity of the reflector, R is 0.999975, L is the cavity length of the ring-down cavity, L is 50cm, c is the speed of light, and c is 3 × 108m/s; as shown in fig. 4, the black open boxes are experimentally measured absorption coefficients, as described in step 5), the total number of points 401; the solid line is a Voigt linear fitting result, and fitting residual errors of the solid line have obvious W-shaped structures and are large in root mean square error; this is because of the background gas N2To CO2The collision of (2) narrows the two wings of the spectrum, while the Voigt line type does not eliminate the inherent structure of the spectrum itself. This shows that the method has high spectral signal-to-noise ratio, and can measure the fine structure of the spectrum. For more accurate results, Galatry line fitting based on soft collision model was used, as shown by the dashed line in FIG. 4; the fitting residual error of the method does not have a W-shaped structure, and the root mean square error is obviously reduced. Fitting by using a Galatry linear function, wherein the difference between the obtained spectral line intensity and the HITRAN2016 database is 4%, and the spectral line intensity is far smaller than the given error range (more than 20%) of the database; the difference of the air broadening coefficients is only 1.31%, and is smaller than the given error range (2-5%) of the database.

Claims (1)

1. A method for measuring a gas absorption coefficient based on a cavity ring-down technology is characterized by comprising the following steps:
1) selecting corresponding absorption spectrum lines from a spectrum database according to the type of the gas to be detected; the center frequency of which is v0The spectral line intensity is S;
2) collecting ring-down time τ: the tunable semiconductor laser (1) is used as a light source, when the laser frequency is fixed, the laser passes through an optical isolator (2), an acousto-optic modulator (3), an optical collimator (5) and a ring-down cavity (7) in sequence, and is received by a photoelectric detector (13); scanning the cavity length of the ring-down cavity by utilizing piezoelectric ceramics to match a laser mode with a cavity mode, wherein the received light intensity of the photoelectric detector is the maximum; turning off the laser to obtain a single-exponential attenuated light intensity signal, and fitting the signal to obtain ring-down time tau;
3) the temperature and current of the tunable semiconductor laser are controlled by a computer, so that the laser frequency is at the spectral line central frequency v0Scanning nearby in a triangular wave form in a stepping mode, wherein the scanning frequency is f; collecting corresponding ring-down time tau at each stepping interval to obtain ring-down time tau (t) with periodic change; carrying out fast Fourier transform on tau (t) to obtain Fourier coefficients, and taking out the Fourier coefficients corresponding to characteristic frequency kf from the Fourier coefficients, wherein the real part of the Fourier coefficients is AkImaginary part of Bk(ii) a Then A is mixedkAnd BkSubstituting into Fourier series to obtain reconstructed ring-down time tau' (t) of periodic variation:
Figure FDA0001953734460000011
where τ' (t) is the reconstructed periodically varying ring-down time, AkAnd BkRespectively a real part and an imaginary part of a Fourier coefficient corresponding to the characteristic frequency; omega is the angular frequency of the triangular wave, f is the scanning frequency of the triangular wave, and then omega is 2 pi f; p is the total number of the characteristic frequencies, k is 0,1,2, and p and t are scanning time;
4) measuring the laser frequency v (t) by an interferometer (18) so that the laser frequency is at the spectral line central frequency v0Scanning nearby in a triangular wave form, wherein the scanning frequency is f, and fitting the rising edge and the falling edge of the laser frequency by using an nth-order polynomial:
Figure FDA0001953734460000012
wherein, vOn the upper part(t) laser frequency at rising edge, vLower part(t) represents the laser frequency of the falling edge; a isiThe ith order coefficient of the rising edge, biThe coefficient of the ith time of the falling edge is 0,1,2, n is a polynomial order; l represents the period number, T is the period of the triangular wave, and T is 1/f;
5) establishing a relation between v (t) and tau' (t): let θ be ω (t-lT) ω t-2 pi l, where 0 < θ < pi, and l represents the number of cycles; replacing t with theta according to the expression of the laser frequency in the step 4) to obtain the following formula:
Figure FDA0001953734460000021
Figure FDA0001953734460000022
obtaining the ring-down time tau ' with the rising edge and the falling edge reconstructed and periodic change according to the expression of tau ' (t) in the step 3) 'On the upper partAnd τ'Lower part：
Figure FDA0001953734460000023
Figure FDA0001953734460000024
By an intermediate variable theta, let vOn the upper partAnd τ'On the upper partSimultaneous, vLower partAnd τ'Lower partSimultaneous reaction to obtainReconstructed periodically varying ring-down time τ'On the upper part(v) and τ'Lower part(v); averaging the two to obtain reconstructed ring-down time tau' (ν) with periodic variation along with laser frequency variation;
6) obtaining a gas absorption coefficient alpha (v): substituting τ' (v) into the following equation:
Figure FDA0001953734460000025
obtaining the gas absorption coefficient alpha (v); wherein R is the reflectivity of the plano-concave reflector, L is the cavity length of the ring-down cavity, and c is the speed of light.
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