Method for measuring the concentration of trace gases by SCAR spectroscopy

Abstract

The present invention is relative to a method of ring-down spectroscopy in saturated-absorption condition, for measuring a first concentration of a gas through a measurement of the spectrum of a molecular transition of said gas.

Claims

1. A method of ring-down spectroscopy in saturated-absorption condition, for measuring a first concentration of a gas through a measurement of the spectrum of a molecular transition of said gas, the method comprising the steps of: inserting said gas whose first concentration is to be measured in a resonant cavity comprising two or more reflecting mirrors arranged so as to form a closed optical path for an electromagnetic radiation emitted by a laser source; tuning a frequency of said electromagnetic radiation emitted by said laser source so as to fix it to a value .sub.i within a range of frequencies [.sub.min, .sub.max] including the resonance frequency of said molecular transition .sub.0; fixing an intensity of said electromagnetic radiation in the cavity at a value much greater than a saturation intensity I.sub.s of the molecular transition to be detected; irradiating said gas by means of said electromagnetic radiation beam emitted by said laser source having said fixed frequency .sub.i and intensity in said resonant cavity; coupling said electromagnetic radiation to said cavity so as to obtain a laser-cavity resonance condition; changing the frequency of the electromagnetic radiation emitted by the laser so as to switch off the laser-cavity resonance; detecting an electromagnetic radiation beam in output from said cavity after the laser-cavity resonance has been switched off; recording a plurality of data representative of said output, obtaining a decay signal for the fixed frequency; considering a fitting curve S(t, .sub.i) for the recorded decay signal which depends on the following parameters: B(.sub.i) is a detection background, with (.sub.i=.sub.i.sub.0); A.sub.d(.sub.i) is an amplitude of the decay signal at the beginning of the decay event; .sub.c(.sub.i) is a cavity decay rate due to non-resonant and non-saturable losses (empty cavity decay rate); .sub.g(.sub.i) is contribution of the targeted molecular transition to the decay signal; g is a peak normalized line profile g(.sub.o,w.sub.R) centered at the molecular resonance frequency .sub.o and w.sub.R is the HWHM width of the resonance, and w.sub.R=w.sub.L for a Lorentzian shape, w.sub.R=w.sub.G for a Gaussian shape, w.sub.R={w.sub.L, w.sub.G} for a Voigt shape; $Z_{o} = \frac{CP (0)}{{CP}_{s}} = A_{d} Z_{1 U}$ is the saturation parameter at the beginning of the decay event and at the frequency of the targeted molecular transition .sub.0, P(0) is the intracavity power at the beginning of the decay signal; and $P_{s} = \frac{w^{2}}{2} I_{s}$ is a saturation power, where w is the spot size radius of the laser beam, i.e. a radius for which the amplitude of the field is 1/e times that of the axis and I.sub.s the saturation intensity; replacing Z.sub.1Ug(.sub.o,w.sub.R) in the function S(t, .sub.i) with a constant value Z.sub.1ueff=constant of a predetermined value; fitting said recorded data with a function S.sup.repl(t, .sub.i) in which Z.sub.1Ueff=constant replaces Z.sub.1Ug(.sub.o,w.sub.R) in the fitting function S(t, .sub.i).

2. The method according to claim 1, including the step of: obtaining from said fit the following parameters: B(.sub.i) is the detection background, with (.sub.i=.sub.i.sub.0); A.sub.d(.sub.i) is the amplitude of the decay signal at the beginning of the decay event; .sub.c(.sub.i) is the cavity decay rate due to non-resonant and non-saturable losses (empty cavity decay rate); .sub.g(.sub.i) is contribution of the targeted molecular transition to the decay signal.

3. The method according to claim 1, including the step of parametrizing said S.sup.repl(t, .sub.i) as:
S.sup.repl(t;p(.sub.i)=B(.sub.i)+A.sub.d(.sub.i)e.sup..sup.c.sup.(.sup.i.sup.)t(t;.sub.c(.sub.i),A.sub.d(.sub.i),.sub.g(.sub.i),Z.sub.1Ueff) where p(.sub.i) is the set of parameters free during the fit
p(.sub.i)={B(.sub.i),A.sub.d(.sub.i).sub.c(.sub.i),.sub.g(.sub.i)} B(.sub.i) is the detection background, .sub.g(.sub.i) is contribution of the targeted molecular transition to the decay signal, A.sub.d(.sub.i) is the amplitude of the decay signal, .sub.c(.sub.i) is the cavity decay rate due to non-resonant and non-saturable losses, while the following are determined before the fitting: Z.sub.1Ueff is the effective saturation parameter, fixed during the fit and equal to a constant; is the non-linear function that follows one of the below rate equations depending of the gas conditions: homogeneous regime (w.sub.Lw.sub.G) $\frac{df}{dt} = -_{g} (v_{i}) \frac{\ln [1 + A_{d} (v_{i}) Z_{1 Ueff} e^{-_{c} (v_{i}) t} f]}{A_{d} (v_{i}) Z_{1 Ueff}}$ inhomogeneous regime still diffusive gas (w.sub.L<w.sub.G) $\frac{df}{dt} = -_{g} (v_{i}) \frac{2}{\sqrt{1 + A_{d} (v_{i}) Z_{1 Ueff} e^{-_{c} (v_{i}) t} f} + 1} f$ non-diffusive gas (w.sub.L<w.sub.G) $\frac{df}{dt} = -_{g} (v_{i}) \frac{1.4256}{\sqrt{1 + 0.5 A_{d} (v_{i}) Z_{1 Ueff} e^{-_{c} (v_{i}) t} f} + 0.4256} f .$

4. The method according to claim 1, wherein the peak normalized line profile g(.sub.o,w.sub.R) centered at the resonance frequency .sub.0 of the molecular transition has either a Lorentzian shape, a Gaussian shape, or a Voigt shape depending on the experimental conditions.

5. The method according to claim 1, wherein said fitting is a least squares fitting.

6. The method according to claim 5, wherein said least square fitting uses the Levenberg-Marquardt (L-M) algorithm.

7. The method according to claim 6, including: obtaining for each of a plurality of m frequencies .sub.j with .sub.j from a .sub.min to a .sub.max a value of .sub.g(.sub.j), and fitting said m values of .sub.g(.sub.j) so as to obtain a value of the first concentration of the target gas.

8. The method according to claim 7, wherein said fitting includes: selecting as a free parameters in the fitting a parameter which takes into account the presence of other molecular absorptions l=1 . . . n in addition to the target molecular resonance l=0.

9. The method according to claim 7, wherein said fitting includes: selecting as a free parameters in the fitting a parameter which takes into account the presence of a polynomial background around the resonance frequency of the target transition .sub.o.

10. The method according to claim 7, wherein said fitting is at least squares fitting.

11. The method according to claim 10, wherein said least square fitting uses the Levenberg-Marquardt (L-M) algorithm.

12. The method according to claim 7, wherein a result of said fitting is multiplied by a correcting factor R in order to obtain the concentration of the target gas, said correcting factor taking into account the line profile modifications due to the Z.sub.1Ueff=constant approximation.

13. The method according to claim 12, wherein the value R is calculated performing a first and a second measurement of a concentration of the same target gas at the same temperature and pressure, the first measurement in saturation absorption and at a second concentration, wherein said second concentration is at least 10 times said first concentration, and said second measurement in linear absorption at said second concentration.

14. The method according to claim 1, including: changing the frequency of the electromagnetic radiation emitted by the laser to a frequency .sub.i+d where .sub.i+d belongs to [.sub.min, .sub.max] and repeating the steps of: fixing the intensity of said electromagnetic radiation in the cavity at a value much greater than the saturation intensity I.sub.s of the molecular transition to be detected; irradiating said gas by means of said electromagnetic radiation beam emitted by said laser source having said fixed frequency .sub.i+d and intensity in said resonant cavity; coupling said electromagnetic radiation to said cavity so as to obtain a laser-cavity resonance condition; changing the frequency of the electromagnetic radiation emitted by the laser so as to switch off the laser-cavity resonance; detecting an electromagnetic radiation beam in output from said cavity after the laser-cavity resonance has been switched off; recording a plurality of data representative of said output which has the form of a decay signal; fitting said recorded data with a function S.sup.repl(t, .sub.i+d) in which Z.sub.1Ueff=constant replaces Z.sub.1Ug(.sub.o,w.sub.R) in the fitting function S(t, .sub.i+d).

15. The method according to claim 14, including repeating the steps of claim 7, for a frequency .sub.i+2d=.sub.i+d of the electromagnetic radiation emitted by the laser, as long as the frequency of the electromagnetic radiation is included in [.sub.min, .sub.max].

16. The method according to claim 1, including the step of: selecting the value of Z.sub.1Ueff constant to be introduced in the fitting by means of the following step: inserting said gas at a second concentration, wherein said second concentration is at least 10 times said first concentration in the resonant cavity; tuning the frequency of said electromagnetic radiation emitted by said laser source so as to fix it to a value .sub.i within a range of frequencies [.sub.min, .sub.max] including said molecular transition .sub.0; fixing the intensity of said electromagnetic radiation in the cavity at a value much greater than the saturation intensity I.sub.s of the molecular transition to be detected; irradiating said gas by means of said electromagnetic radiation beam emitted by said laser source having said fixed frequency .sub.i and intensity in said resonant cavity; coupling said electromagnetic radiation to said cavity so as to obtain a laser-cavity resonance condition; changing the frequency of the electromagnetic radiation emitted by the laser so as to switch off the laser-cavity resonance; detecting an electromagnetic radiation beam in output from said cavity after the laser-cavity resonance has been switched off; recording a plurality of data representative of said output which has the form of a decay signal; fitting the data of the recorded decay with a curve S(t, .sub.i) which depends on the following parameters: B.sup.high conc(.sub.i) is the detection background, with (.sub.i=.sub.i.sub.0) A.sub.d.sup.high conc(.sub.i) is the amplitude of the decay signal at the beginning of the decay event, g .sub.c.sup.high conc(.sub.i) is the cavity decay rate due to non-resonant and non-saturable losses (empty cavity decay rate); .sub.g.sup.high conc(.sub.i) is contribution of the targeted molecular transition to the decay signal; Z.sub.1Ueff.sup.high conc=Z.sub.1Ug(.sub.i,w.sub.R), where g is the peak normalized line profile g(.sub.o,w.sub.R) centered at the molecular resonance frequency .sub.0 and w.sub.R is the HWHM width of the resonance, and w.sub.R={w.sub.L, w.sub.G} for a Lorentzian shape, with w.sub.R=w.sub.G for a Gaussian shape, w.sub.R={w.sub.L, w.sub.G} for a Voigt shape; $Z_{o} = \frac{CP (0)}{{CP}_{s}} = A_{d} Z_{1 U}$ is the saturation parameter at the beginning of the decay event and at the frequency of the targeted molecular transition .sub.0, P(0) is the intracavity power at the beginning of the decay signal; and $P_{s} = \frac{w^{2}}{2} I_{s}$ is the saturation power, where w is the spot size radius of the laser beam, i.e. the radius for which the amplitude of the field is 1/e times that of the axis and I.sub.s the saturation intensity; selecting as Z.sub.1Ueff.sup.high conc=Z.sub.1Ueff=value obtained from the above fitting.

17. The method according to claim 1, wherein said electromagnetic radiation emitted by said laser is in the Infrared range.

18. A method of ring-down spectroscopy in saturated-absorption condition, for measuring a first concentration of a target gas through a measurement of a spectrum of a molecular transition of said target gas, the target gas being in a mixture together with other gasses the method comprising the steps of: repeating for a number m of fixed frequencies .sub.i spaced each other by a of a frequency step in a range of frequencies including a frequency of the molecular transition of the target gas, the gases in the mixture having l=0, . . . ,n, absorptions in the measured spectral range, where l=0 is a target absorption, the following steps: repeating d times at the same frequency the following steps: inserting said gas whose first concentration is to be measured in a resonant cavity comprising two or more reflecting mirrors arranged so as to form a closed optical path for an electromagnetic radiation emitted by a laser source; tuning a frequency of said electromagnetic radiation emitted by said laser source so as to fix it to a value .sub.i within a range of frequencies [.sub.min, .sub.max] including said molecular transition .sub.0; fixing an intensity of said electromagnetic radiation in the cavity at a value much greater than a saturation intensity I.sub.s of the molecular transition to be detected; irradiating said gas by means of said electromagnetic radiation beam emitted by said laser source having said fixed frequency .sub.i and intensity in said resonant cavity; coupling said electromagnetic radiation to said cavity so as to obtain a laser-cavity resonance condition; changing the frequency of the electromagnetic radiation emitted by the laser so as to switch off the laser-cavity resonance; detecting an electromagnetic radiation beam in output from said cavity after the laser-cavity resonance has been switched off; and recording a plurality of data representative of said output, obtaining a decay signal for the fixed frequency; collecting the d*m SCAR decay signals obtained; fitting at the same time the d*m decay signals with d*m fitting curves considering a fitting curve s(t; p())=[S(t; p(.sub.i).sub.j)].sub.j=1.sup.d*m for the recorded decay by signals which depends on the following parameters p()=[p(.sub.i).sub.j].sub.j=1.sup.d*m: $p (v) = {[{p (v_{i})}_{j}]}_{j = 1}^{d * m} = {[B_{j}, A_{d_{j}},_{c_{j}}, {[{_{l} (v_{il}), Z_{l 1 U} {\overline{g}}_{l} (v_{il}, w_{Rl})}_{j}]}_{l = 0}^{n}]}_{j = 1}^{d * m}, i = 1 .Math. m$ [B.sub.j(.sub.il)].sub.j=1.sup.d*m are the detection backgrounds of the j=1 . . . d*m recorded SCAR decay signals; and .sub.il=.sub.i.sub.l is the detuning of the i (=1 . . . m) scanned frequency of the laser radiation with respect to the resonance frequency of the l resonant transition, [A.sub.d.sub.j(.sub.il)].sub.j=1.sup.d*m are amplitudes of the j=1 . . . d*m recorded SCAR decay signals, [.sub.c.sub.j(.sub.il)].sub.j=1.sup.d*m are cavity decay rates of the j=1 . . . d*m recorded SCAR decay signals due to non-resonant and non-saturated gas absorption losses, [[.sub.l(.sub.il).sub.j].sub.l=0.sup.n].sub.j=1.sup.d*m are arrays of the absorption decay rates of the l=0 . . . n saturated transitions at the .sub.il detuning for the j=1 . . . d*m recorded SCAR decays, ${[{[{Z_{1 U l} {\overline{g}}_{l} (v_{i} - v_{l}, w_{Rl})}_{j}]}_{l = 0}^{n}]}_{j = 1}^{d * m}$ are arrays of the l=0 . . . n saturation parameters of the j=1 . . . d*m recorded SCAR decay signals; g.sub.l(.sub.i.sub.l,w.sub.Rl) is the peak normalized line profile centered at the molecular resonance frequency .sub.l and w.sub.Rl is the HWHM width of the l-resonance; $Z_{o} = \frac{CP (0)}{{CP}_{s}} = A_{d} Z_{1 U}$ is a saturation parameter at the beginning of the decay event and at the frequency of the targeted molecular transition .sub.0, P(0) is an intracavity power at the beginning of the decay signal; and $P_{s} = \frac{w^{2}}{2} I_{s}$ is a saturation power, where w is a spot size radius of the laser beam, i.e. a radius for which a amplitude of the field is 1/e times that of the axis and I.sub.s the saturation intensity; dividing the above mentioned parameters in a first and a second group; and fitting said recorded data with a function S(t; p())=[S(t; p(.sub.i).sub.j)].sub.j=1.sup.d*m by keeping the first parameter group fixed and equal to a set of pre-determined values and performing a first fitting only considering the second group as free parameters in a first fitting step; and keeping the second parameter group fixed to a value given in the first fitting step and performing a second fitting considering only the second group as free parameters in a second fitting step.

19. The method according to claim 18, including the step of calculating the first concentration of the target gas N.sub.0 by: $N_{0} = \frac{_{0}^{fit} (0) g_{0}^{fit} (0)}{c^{2} L_{S} (T)}$ where c is the speed of light in vacuum and L.sub.s(T) is the line-strength of the absorbent molecular transition at the temperature T.

20. The method according to claim 18, including the step of parametrizing said S(t; p())=[S(t; p(.sub.i)j)].sub.j=1.sup.d*m as: $S (t; p (v)) = {[S (t; {p (v_{i})}_{j})]}_{j = 1}^{d * m} = {[B_{j} + A_{d_{j}} e^{-_{c_{j}} t} f_{j} (t; A_{d_{j}},_{c_{j}}, {[{_{l} (v_{il}), Z_{l 1 U} {\overline{g}}_{l} (v_{il}, w_{Rl})}_{j}]}_{l = 0}^{n})]}_{j = 1}^{d * m}, i = 1 .Math. m$ with =[.sub.j].sub.j=1.sup.d*m are the non-linear functions that follows one of the below rate equations depending of the gas conditions: homogeneous regime (w.sub.Lw.sub.G) ${[\frac{{df}_{j}}{dt} = - {.Math.}_{l = 0}^{n} {_{l} (v_{il})}_{j} \frac{\ln [1 + A_{d_{j}} Z_{l 1 U} {\overline{g}}_{l} (v_{il}, w_{Rl}) e^{-_{c_{j}} t} f_{j}]}{A_{d_{j}} Z_{l 1 U} {\overline{g}}_{l} (v_{il}, w_{Rl}) e^{-_{c_{j}} t}}]}_{j = 1}^{d * m}$ inhomogeneous regime still diffusive gas (w.sub.L<w.sub.G) ${[\frac{{df}_{j}}{dt} = - {.Math.}_{l = 0}^{n} {_{l} (v_{il})}_{j} \frac{2}{\sqrt{1 + A_{d_{j}} Z_{l 1 U} e^{-_{c_{j}} t} f_{j}} + 1} f_{j}]}_{j = 1}^{d * m}$ non-diffusive gas (w.sub.L<<w.sub.G) ${[\frac{{df}_{j}}{dt} = - {.Math.}_{l = 0}^{n} {_{l} (v_{il})}_{j} \frac{1.4256}{\sqrt{1 + 0.5 A_{d_{j}} Z_{l 1 U} e^{-_{c_{j}} t} f_{j}} + 0.4256} f_{j}]}_{j = 1}^{d * m} .$

21. The method according to claim 18, including: approximating the parameters B.sub.j, A.sub.d.sub.j and .sub.c.sub.j as having small variations from decay to decay:
B.sub.j=B+B.sub.j B is the mean value of the background of the j=0 . . . d*m recorded SCAR decays, in U {B.sub.j}.sub.j=1.sup.d*m are small variations around B for each j=0 . . . d*m recorded SCAR decay, in U
A.sub.d.sub.j=.sub.d+A.sub.d.sub.j .sub.d is the mean value of the amplitude of the j=0 . . . d*m recorded SCAR decays, in U ${A_{d_{j}}}_{j = 1}^{d * m}$ are small variations around .sub.d for each j=0 . . . d*m recorded SCAR decay, in U
.sub.c.sub.j=.sub.c+.sub.c.sub.j .sub.c is the mean value of the cavity decay rate of the j=0 . . . d*m recorded SCAR decays due to non-absorbent gas and non-saturated absorption losses, in s.sup.1 {.sub.c.sub.j}.sub.j=1.sup.d*m are small variations around .sub.c for each j=0 . . . d*m recorded SCAR decay, in s.sup.1 selecting ${[p_{{lc}_{j}}]}_{j = 1}^{d * m} = {[B_{j}, A_{d_{j}},_{c_{j}}]}_{j = 1}^{d * m}$ as the first group of parameters.

22. The method according to claim 21, including: introducing two further parameters to the second group of parameters, said parameters being: d is the contribution to the cavity decay rate due to residual detection non-linearities, in s.sup.1 Z.sub.d is the equivalent-saturation parameter at 1U due to residual detection non-linearities, in U.sup.1 so that the second group of parameters is equal to:
p.sub.gb()=[B,.sub.d,.sub.c,Z.sub.d,d,[Z.sub.1Ul,.sub.l(0),g.sub.l(0),g.sub.l(.sub.il,w.sub.Rl)].sub.l=0.sup.n] where .sub.l(.sub.il).sub.j=.sub.l(0)g.sub.l(0)g.sub.l(.sub.il,w.sub.Rl) with {.sub.l(0)}.sub.l=0.sup.n{c.sub.l(0)}.sub.l=0.sup.n are the absorption decay rates for each transition l=0 . . . n at the resonance frequency .sub.i=.sub.l, in s.sup.1, {g.sub.l(0)}.sub.l=0.sup.n are the area normalization factors for each transition l=0 . . . n at the resonance frequency .sub.i=.sub.l, {g.sub.l(.sub.i.sub.l,w.sub.Rl)}.sub.l=0.sup.n are the peak normalized line profiles for each transition l=0 . . . n, centered at the resonance frequency .sub.l and linewidth w.sub.Rl.

23. The method according to claim 22, including a further second parameter [C.sub.l.sup.].sub.l=0.sup.n, which is the amplitude factor of a function G.sub.j(.sub.i; [C.sub.l.sup.].sub.l=0.sup.n), proportional, to the spectrum profile [g.sub.l(.sub.il,w.sub.Rl)].sub.l=0.sup.n G.sub.j(.sub.i;[C.sub.l.sup.].sub.l=0.sup.n)=.sub.l=0.sup.nC.sub.l.sup.g.sub.l(.sub.il,w.sub.Rl), which is used to update the local parameters ${[_{c_{j}}]}_{j = 1}^{d * m} .$

24. The method according to claim 22, including the step of parametrizing the non-linear function as: homogeneous regime (w.sub.Lw.sub.G) ${[\begin{matrix} \frac{{df}_{j}}{dt} = - {.Math.}_{l = 0}^{n}_{l} (0) g_{l} (0) \frac{\ln [\begin{matrix} 1 + ({\overline{A}}_{d} + A_{d_{j}}) \\ Z_{l 1 U} {\overline{g}}_{l} (v_{il}, w_{Rl}) e^{- {({\overline{}}_{c} +_{c_{j}})}^{t}} f_{j} \end{matrix}]}{({\overline{A}}_{d} + A_{d_{j}}) Z_{l 1 U} e^{- {({\overline{}}_{c} +_{c_{j}})}^{t}}} - \\ d \frac{\ln [1 + ({\overline{A}}_{d} + A_{d_{j}}) Z_{d} e^{- {({\overline{}}_{c} +_{c_{j}})}^{t}} f_{j}]}{({\overline{A}}_{d} + A_{d_{j}}) Z_{d} e^{- {({\overline{}}_{c} +_{c_{j}})}^{t}}} \end{matrix}]}_{j = 1}^{d * m}$ and g.sub.l(.sub.il,w.sub.Rl) is a Voigt function inhomogeneous regime still diffusive gas (w.sub.L<w.sub.G) ${[\frac{{df}_{j}}{dt} = [\begin{matrix} - {.Math.}_{l = 0}^{n} \frac{2_{l} (0) g_{l} (0) {\overline{g}}_{l} (v_{il}, w_{Rl})}{\sqrt{1 + ({\overline{A}}_{d} + A_{d_{j}}) Z_{l 1 U} e^{- {({\overline{}}_{c} +_{c_{j}})}^{t}} f_{j}} + 1} - \\ \frac{2 d}{\sqrt{1 + ({\overline{A}}_{d} + A_{d_{j}}) Z_{d} e^{- {({\overline{}}_{c} +_{c_{j}})}^{t}} f_{j}} + 1} \end{matrix}] f_{j}]}_{j = 1}^{d * m}$ and g(.sub.il,w.sub.Rl) is a Gaussian function non-diffusive gas (w.sub.L<<w.sub.G) ${[\frac{{df}_{j}}{dt} = [\begin{matrix} - {.Math.}_{l = 0}^{n} \frac{1.4256_{l} (0) g_{l} (0) {\overline{g}}_{l} (v_{il}, w_{Rl})}{\sqrt{1 + 0.5 ({\overline{A}}_{d} + A_{d_{j}}) Z_{l 1 U} e^{- {({\overline{}}_{c} +_{c_{j}})}^{t}} f_{j}} + 0.4256} - \\ \frac{1.4256 d}{\sqrt{1 + 0.5 ({\overline{A}}_{d} + A_{d_{j}}) Z_{d} e^{- {({\overline{}}_{c} +_{c_{j}})}^{t}} f_{j}} + 0.4256} \end{matrix}] f_{j}]}_{j = 1}^{d * m}$ and g.sub.l(.sub.il,w.sub.Rl) is a Gaussian function.

25. The method according to claim 18, wherein said first fitting step is a least squares fitting.

26. The method according to claim 25, wherein said least square fitting uses the Levenberg-Marquardt (L-M) algorithm.

27. The method according to claim 18, wherein said second fitting step is a least squares fitting.

28. The method according to claim 27, wherein said least square fitting uses the Levenberg-Marquardt (L-M) algorithm.

29. The method according claim 18, including the step of: dividing the second group of parameter in a first sub-group of free parameters and a second sub-group of calibration parameters which are set equal to calibration values which remain constant during the first and second fitting steps.

30. The method according to claim 29, wherein the sub-group of calibration parameters includes:
p.sub.gb.sup.cal()=[Z.sub.d,[Z.sub.1Ul,g.sub.l(.sub.il,w.sub.Rl)].sub.l=0.sup.n].sup.cal where:{g.sub.l(.sub.i.sub.l,w.sub.Rl)}.sub.l=0.sup.n are the peak normalized line profiles for each transition l=0 . . . at, centered at the resonance frequency .sub.l and linewidth w.sub.Rl, and Z.sub.d is the equivalent-saturation parameter at 1U due to residual detection non-linearities, in U.sup.1.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The invention will be now described in a non-limiting manner with reference to the appended drawing, in which:

(2) FIGS. 1-4 and 5A-5B are different method steps of the method of the invention according to its first aspect;

(3) FIGS. 6A-6B, 7A-7B, 8 and 9 are different method steps of the method of the invention according to its second aspect;

(4) FIG. 10 is a schematic diagram of an apparatus to perform the method of the invention according to the first or to the second aspect;

(5) FIG. 11 is a first measurement performed and fitted according to the first aspect of the method of the invention;

(6) FIG. 12 is a fitting of a measurement similar to the one in FIG. 11, but with a larger frequency scan, in CRD conditions performed according to a prior art method;

(7) FIG. 13 is a fitting of the same measurement of the one of FIG. 12 in SCAR conditions but according to a method not of the invention;

(8) FIG. 14 is the fitting of the same measurement of the one of FIG. 12 with a fitting according to the invention;

(9) FIG. 15 is a second measurement performed and fitted according to the first aspect of the method of the invention;

(10) FIG. 16 is a fitting of the same measurement of the one in FIG. 11 in CRD conditions performed according to a prior art method;

(11) FIG. 17 is a fitting of the same measurement of the one of FIG. 11 in SCAR conditions but according a method not of the invention;

(12) FIG. 18 is the fitting of FIG. 11 with a fitting according to the invention;

(13) FIG. 19 is a third measurement performed on a highly enriched sample and fitted according to the second aspect of the method of the invention; and

(14) FIG. 20 is a fitting of the same measurement of the one in FIG. 19 in CRD conditions performed according to a prior art method.

DETAILED DESCRIPTION OF AN EMBODIMENT

(15) With initial reference to FIG. 10, reference numeral 1 indicates an apparatus for SCAR spectroscopy including a laser source 2, for example a continuous wave (CW) coherent laser source generated by a frequency difference tunable over a predetermined range.

(16) Preferably, the radiation emitted by laser source 2 has a wavelength in the mid-infrared, however other wavelengths may be used. The mid-infrared has the advantage of having the strongest molecular absorption.

(17) The type of laser source 2 used in the present invention is for example described in the article written by Galli et al., Opt. Lett. 35, 3616 (2010). Other types of laser sources may be used, provided that the intensity of radiation I inside the cavity is much greater than the intensity of saturation Is of the molecular transition to be detected, i.e. I>>I.sub.s.

(18) For example, in a range of wavelengths of 4-5 m, for the transitions of CO.sub.2, which have an Einstein coefficient A of about 200 s.sup.1, at a pressure of about 12 mbar, Voigt enlargement condition, the power emitted by the laser must be greater than 20 mW, preferably greater than 100 mW.

(19) Device 1 further includes a resonant cavity 3, for example, a cavity having a length of 1 m, provided at opposite ends thereof with two reflecting mirrors 4a and 4b. Preferably, the reflectivity of the mirrors is greater than 99.9%, even more preferably it is greater than 99.99%.

(20) The gas of which the concentration has to be measured is introduced into cavity 3, for example through a duct 5 which connects cavity 3 to a suitable container, such as a cylinder 6.

(21) Apparatus 1 further includes a photodetector 7 suitably arranged for detecting the radiation beam outgoing from cavity 3 as well as a diffuser element 8 interposed between cavity 3 and photodetector 7.

(22) The diffuser element 8 is adapted to diffuse the laser beam exiting cavity 3 before it impinges on photodetector 7.

Experiment 1

(23) The SCAR apparatus is composed of an IR laser emitting around 4.5 m, where .sup.14CO.sub.2 absorbs, a high-finesse Fabry-Perot cavity and an IR-detector followed by amplifying and digitalizing electronics.

(24) The IR radiation is provided by an Optical-Frequency-Comb (OFC) assisted Difference-Frequency-Generation (DFG) continuous-wave (CW) coherent source (Ti:sapphire laser intracavity difference-frequency generation of 30 mW cw radiation around 4.5 m, I. Galli et al, Opt. Lett. 35, 3616 (2010)). The DFG process occurs inside the cavity of a Ti:Sapphire laser operating around 850 nm (pump laser), single-mode controlled by an injected extended-cavity diode-laser (ECDL). A Nd:YAG laser at 1064 nm, amplified up to 10 W using an Yb-doped fiber amplifier provides the DFG signal laser. It is mixed with the intracavity Ti:Sapphire radiation through a periodically-poled lithium niobate non-linear crystal. The frequency of the ECDL is phase-locked to the Nd:YAG frequency by direct digital synthesis, using the OFC to cover the frequency gap (about 70 THz) between the two CW lasers (Ultra-stable, widely tunable and absolutely linked mid-IR coherent source, I. Galli et al., Opt. Express 17, 9582 (2009)). In this way, the linewidth of the IR generated radiation is given by a fraction of the narrow Nd:YAG linewidth (about 5 kHz in 1 ms integration time), thus allowing a highly efficient coupling of the IR radiation to the high-finesse Fabry-Perot cavity. Moreover, the frequency chain used to cover the frequency difference between both signal and pump lasers of the DFG process, includes a microwave synthesizer. In this way, frequency of the generated IR radiation is scanned and tuned in a synthesized way by changing the microwave frequency. In addition, the Nd:YAG frequency is stabilized against the nearest tooth of the OFC. As a consequence, the IR frequency is absolutely traceable against the primary frequency standard with a precision of 610.sup.13 in 1 s and an accuracy of 210.sup.12. Moreover, the intracavity DFG boosts the generated IR power up to 30 mW around 4.5 m wavelength, which provides the required power to saturate the .sup.14CO.sub.2 transitions.

(25) The measurement cell is a cylindrical vacuum chamber 1.2 m long and 10 cm in internal diameter. It is enclosed in a polystyrene box, which can be filled with dry-ice pellets for cooling the chamber down to 195 K. The chamber houses a Fabry-Perot optical resonator resting inside of it on 4 cantilevered legs, which dampen vibrations (for frequencies >20 Hz) in all 3 spatial directions. The mechanical frame of the Fabry-Perot resonator is made of 3 Invar bars connected by 2 circular flanges, leaving 8 L internal volume available for the gas. At both ends of the frame, properly machined flanges house the mirrors, with high reflectivity dielectric coatings at 4.5 m wavelength and a 6-m of radius of curvature. The total losses for each mirror (transmission plus absorption/scattering) amount to .sup.270 parts per million (ppm) and the achieved optical finesse is higher than 11,000. The mirror mounting flanges have screws for coarse alignment and PZT for fine adjustment of both the alignment and the cavity length. The mirror spacing is 1 m and the corresponding free spectral range is 150 MHz.

(26) A N.sub.2-cooled InSb detector is used to detect the radiation transmitted by the cavity. It is followed by a transimpedance amplifier (Z=32000 Ohm) with a final bandwidth of about 1 MHz. A 18-bit digitizing oscilloscope with a sampling rate of 10 Ms/s is used to digital convert the analogical detected signal for further process and analysis.

(27) For the SCAR experiment, the IR radiation is efficiently coupled (about 86% as expected) in the high-finesse Fabry-Perot cavity by using a couple of lenses to achieve the TEM.sub.00 mode propagation inside of the cavity. In this conditions, taking into account the mirror losses and an available power of about 20 mW before the cavity input, we estimate an inside cavity power in resonance of about 40 W.

(28) Then, the frequency of the IR coherent source is changed to be in resonance with the cavity. OFC-assisted frequency stability of the IR radiation and the high mechanical stability of the cavity length allow to maintain a long term resonance condition without need of any active lock of the cavity length to the IR frequency of the laser.

(29) A double-pass acusto-optic modulator placed at the output of Nd:YAG laser is used to quickly switch off resonance the input IR light when a threshold coupling level (about 3V at the output of the amplified detector) is reached, and thus the transmission cavity decay is detected by a N2-cooled InSb detector during 100 s. The output tension of the amplified detector is digitalized, and thus recording the single SCAR-decay-event. In order to improve S/N ratio of the detected decay, compatible with digitalization resolution, 128 consecutive SCAR-decay-events are averaged, giving the SCAR-decay-signals to be analyzed by the fit routines, as described in this invention.

(30) For the SCAR spectroscopy of .sup.14C.sup.16O.sub.2, the (00.sup.01-00.sup.00) P(20) rovibrational transition around 4.5 m was targeted. It is a quasi-isolated absorption line of this molecule with respect to other absorptions from other CO.sub.2 isotopologue. Nevertheless, the (05.sup.51-05.sup.50) P(19)e line of .sup.13C.sup.16O.sub.2, blue-shifted by .sup.230 MHz with respect to the frequency of the target transition, is an interference line that must be taken into account in the analysis to get an accurate concentration measurement of radiocarbon dioxide.

(31) The Fabry-Perot cell is filled with a CO.sub.2 sample at total pressure P=12 mbar and temperature T=195 K (corresponding to .sup.0.15 L volume at standard thermodynamic conditions). At this temperature, the intensity of the nearby .sup.13C.sup.16O.sub.2 interference line is decreased by more than three orders of magnitude, minimizing its interference effect. The CO.sub.2 sample is a natural mix of almost all carbon dioxide isotopologues including .sup.14C.sup.16O.sub.2, at the present natural abundance (about 1.210.sup.12). Other gases must be avoided and in particular .sup.14N.sub.2.sup.16O, which has absorption line almost at the same frequency of the target transition. N.sub.2O concentration below 0.3 ppb in the measured CO.sub.2 sample is required to produce a negligible interference with the target line.

(32) In these conditions of temperature and pressure, the deformation from a pure exponential produced by radiocarbon dioxide absorption along decay signals is of the order of 1 V out of 3 V. This set very stringent limits on the residual non-linearity, which can be born. At this respect particular attention was putted to get a linear response of the detection system for the more than 6 decades as require radiocarbon-dioxide detection at ppt level or better. An optical diffuser was placed between the cavity and the detector, in order to uniform the way to illuminate the detector area. In addition, a numerical calibration of residual non-linearities is allowed in the SCAR decays obtained in vacuum conditions. In this case, the decay must be a perfect exponential, and deviations are taken into account by adding a Fourier function to the expected exponential behavior. This correction is then applied to all recorded SCAR decays.

(33) On the other side, the measurement is possible thanks to improve the noise present in the signal to be captured, which, in good approximation, has zero, or at least constant, average. In addition to the 128 consecutive SCAR decays averaged by the digitation electronics as described above, ten of these acquisitions for each laser frequency are again averaged for a total of 1280 consecutive decay events. The decay resulting of the total average, which is stored in memory for further analysis as described in this invention, increases the resolution of digitization by approximately 35 times.

(34) To obtain the SCAR decay signals for .sup.14CO.sub.2 concentration measurements, the following steps have been performed: 1) The cavity-cell is filled with the CO.sub.2 sample at the thermodynamic conditions described above. The IR laser is operated at the required power for saturation of the target transition (see above) and coupled to the cavity as described above. 2) The frequency of the IR coherent source is set to a value 66 227 000 MHz, which is 400 MHz apart of the known frequency of the target P(20) transition of radiocarbon dioxide. This is achieved by setting the proper OFC repetition rate and taking into account the lock frequency chain described above. 3) The Fabry-Perot is bringing in resonance with the IR radiation by changing the cavity length by means of the tension applied to the PZT transducers. The transmitted intensity is detected with the N.sub.2-cooled InSb detector. 4) When the transimpedance tension at the output of the amplified detector reach the 3 V threshold condition, the laser frequency is quickly and automatically switched-off the cavity resonance by means of the AOM as described above. 5) The SCAR decay is detected by 100 s, digitized and averaged with consecutive decay signals. The averaged signal is temporally stored in the digitizing electronics. 6) The laser is quickly and automatically switched-on the cavity resonance by means of the AOM as described above, waiting to threshold condition be again reached. 7) The steps 4 to 6 are repeated by 128 consecutive decay events. This step is repeated by ten times, where the result of each 128-average decay is averaged with the consecutive one. At the end, a SCAR decay signal for the present frequency of the spectrum (i.e. the laser frequency), resulting of the average of 1280 consecutive events is stored for the further analysis described in this invention. 8) The laser frequency is changed-up by a frequency step of 10 MHz, and the steps from 3 to 7 are repeated up to the laser frequency is scanned-up by 740 MHz. (i.e. 73 frequency steps in the upward direction). 9) The laser frequency is changed-down by a frequency step of 10 MHz, and the steps from 3 to 7 are repeated up to the laser frequency is scanned-down by 740 MHz. (i.e. 73 frequency steps in the downward direction). 10) As a final result, 146 SCAR decay signals, two for each scanned IR frequency, are stored to be analyzed by the methods, subject of this invention.

(35) The results of this experiment and the fit according to the first aspect of the invention can be seen in FIG. 11. A spectrum of the .sup.14C.sup.16O.sub.2 P(20) transition (black trace) recorded at the present natural abundance (named 2010) is shown. In this figure, two more spectra are shown, corresponding to a .sup.14C-enriched sample (green trace) and a .sup.14C-depleted sample (blue trace). .sup.14C-enriched spectra were used to accurately measure the center frequency of the P(20) transition, as well as the most favorable thermodynamic conditions of the gas sample for .sup.14C detection in subnatural abundance.

(36) More quantitatively, the fit of the 2010 CO.sub.2 spectrum to the expected Voigt profile yields an area of 1:50(8) ms.sup.1 MHz. Assuming for the P(20) the calculated line strength, S=3.10(15)*10.sup.18 cm, a .sup.14C.sup.16O.sub.2 natural abundance concentration of 1.24(10) ppt is measured.

(37) For the calibration it has been used the target gas at a second concentration of: 58 times the first concentration. Further, in the calibration the target gas was at a temperature: 195K, and pressure 11.6 mbar. The parameter obtained from the calibration are:

(38) line center: 66227382.3 MHz

(39) FWHM Gaussian width: 123.54 MHz

(40) FWHM Lorentzian width: 74.32 MHz

(41) Z.sub.1Ueff=8.2 V.sup.1

(42) Ratio R of the spectral area: (effective area)=(area measured by SCAR)0.925.

(43) FIG. 12 shows a similar measurement, but with a larger frequency scan, performed with a standard CRD routine, that is, not in saturation condition, and a fitting of the result. It is clear that the variations (drift and oscillation) of the curve for a standard CRD measurement is much higher. The FIG. 12 shows two different measurements which are partially overlapped, however they do not match.

(44) FIG. 13 shows SCAR measurements of the same sample in which, instead of using the method according to the first aspect, the line profile has been selected a priori. The tails of the curve as shown are not correct and they move upwards.

(45) FIG. 14 shows the SCAR fitting method according to the first aspect of the invention, which has a much more correct behaviour.

Experiment 2

(46) The experimental set up is the same as in experiment 1.

(47) Sub-Doppler measurements on low-pressure .sup.17O.sup.12C.sup.16O at natural abundance (7.5*10.sup.4) have been performed. FIG. 15 shows the resolved hyperfine structure of the (00.sup.01-00.sup.00) R(0) transition of .sup.17O.sup.12C.sup.16O, due to interaction between the .sup.17O electric quadrupole (which is non-null for a nuclear spin I=5/2) and the electric field gradient at the nucleus position. This spectrum was recorded in about 3 h with 11 forward-backward frequency scans in 20-kHz steps.

(48) In Table I the fit results for the line centers are reported.

(49) TABLE-US-00001 TABLE I Measured absolute frequencies .sub.F of the hyperfine triplet with 1 uncertainties and relative intensities. F .sub.F (kHz) rel. intens. 1 70 174 358 594.9 (6.2) 0.207 (11) 0 70 174 357 409.8 (3.0) 0.345 (9) +1 70 174 358 229.8 (3.6) 0.448 (13)

(50) The fitted FWHM of each Lorentzian dip is =217.1(5:5) kHz.

(51) The measured line-center frequency .sub.c=70 174 358 037.3 (3:9) kHz is consistent with the value 70 174 368 MHz reported by HITRAN, within the declared 3-30 MHz uncertainty. Thus, we improved the frequency accuracy by more than 3 orders of magnitude.

(52) FIG. 16 shows for comparison the same measurement of the one in FIG. 11 according to CRD, that is in a linear regime, and with its fitting.

(53) FIG. 17 shows SCAR fitting of the same measurement of the one of FIG. 11, in which, instead of using the method according to the first aspect, the line profile has been imposed a priori. The tails of the curve as shown are not correct and they move upwards.

(54) FIG. 18 shows the SCAR fitting method according to the first aspect of the invention, which has a much more correct behaviour.

Experiment 3

(55) In this experiment, the fitting is performed according to the second aspect of the invention.

(56) The apparatus used is the one described according to the first experiment.

(57) SCAR spectroscopy in a .sup.14C.sup.16O.sup.2-enriched CO.sub.2 sample as in the first experiment. The laser frequency was scanned across the P(20) line, and the SCAR decay signals where fitted by using the above described fit function.

(58) The sample was much more enriched than in the first experiment (about 6400 times with respect to natural abundance) in order to minimize the effects of the S/N ratio in the fitted parameters uncertainty and to calibrate the SCAR procedure. Moreover, to check the linearity of the detection, this very high radiocarbon abundance has been changed by reducing it with successive dilutions. Another set of measurements was done with gas at natural abundance, obtained by fermentation of the same cane sugar. All these new measurements were done at temperature T=195 K and pressure P=12 mbar.

(59) In this procedure, at the highest abundance, the line center, the Lorentzian and Gaussian width, the spectral area c.sub.o of the line profile and the saturation parameter Z.sub.1U were fitted as global variables, while the amplitude, the background and the cavity decay constant of each single decay signals were fitted as local variables.

(60) The values obtained for Z.sub.1U, line widths and line center at the highest abundance where then used as fixed parameter for the fit of subsequent measured signals at lower abundances.

(61) The decay signals were detected at various laser frequencies, covering a total span of about 650 MHz around the P(20) line frequency, at 10 MHz steps. Due to technical reasons of our experimental setup, continuous scans larger than 400 MHz are not allowed, and such 650 MHz span is the result of the partially overlapping of two consecutive 400 MHz scans at the blue and red sides of the line center. At this frequency span, the interfering line (05.sup.51-05.sup.50) P(19) of .sup.13C.sup.16O.sub.2 on the blue side of the P(20) line must be considered. Its interfering effects at 195 K may be neglected for highly-enriched samples, nevertheless they have been considered.

(62) The interfering line parameters .sub.o, w.sub.L, w.sub.G and Z.sub.1U were fixed to a value determined with the following procedure: first preliminary values for the global parameters have been calculated by using the enriched-sample SCAR spectrum without considering the interfering line. Then, a SCAR spectrum recorded with a CO.sub.2 sample with radiocarbon dioxide at natural abundance has been fitted by fixing all global parameters except the P(20) line area, d, and all interfering line parameters. Indeed, at this .sup.14C.sup.16O.sub.2 abundance and the values of T=195K and P=12 mbar, the area of the interference line is almost equal to the P(20) one, and hence its effects are better taken into account. Therefore, with these first estimations of the parameters of the interfering line, the global parameters of the enriched spectrum are again calculated. This procedure is repeated several times, till final convergence.

(63) In FIG. 19, one of the three SCAR spectra recorded for the about 6400 times enriched sample is shown. The fitted global parameters for the three spectra were all self-consistent within the errors. The averaged values are shown in Table 2.

(64) These results have been checked by comparing them with those measured for the spectra recorded with the same CO.sub.2 enriched sample at the same thermodynamic conditions, but near the linear-absorption regime.

(65) The laser power was not as low as to decrease too much the S/N ratio. In order to fit the decay signals, we used the same global procedure. However, because, at low power, the two decay parameters .sub.c and .sub.g are quite correlated, the signal is essentially obtained by reducing the SCAR procedure to the standard CRD procedure, where the overall =.sub.c+.sub.g is considered, and by subtracting an averaged value for the cavity decay rate out of resonance. In this way, the non-linearity due to a residual saturation was corrected and an experimental linear absorption line profile was generated. A successive Voigt fit also takes into account, in this case, optical fringes that modulate the .sub.c contribution.

(66) FIG. 20 shows the average of three profiles obtained with successive measurements and the corresponding fit. The residuals do not allow to shed light on the real shape of the profile, alternative to the Voigt profile. Indeed, the residuals obtained in the left and right scans have different values in the common frequency range: likely they are dominated by an incomplete neutralization of the fringes. The main discrepancy from a Voigt profile is an asymmetric residual in the wings. Also in this regime the values of the fitted Voigt profile obtained from the three independent measurements are consistent within the errors.

(67) The averaged values are shown in Table 3.

(68) These values do not coincide with those obtained at high power. For the linewidths, the discrepancies can be ascribed to the local approximation of the laser field-molecules interaction done in the theory. Indeed, the residuals for the saturated case (FIG. 19) clearly show a systematic small discrepancy from a Voigt profile. Anyway, such discrepancies are small and, for the calibration purpose, its effect will be included in the ratio between the areas estimated in the two regimes. The ratio between the area obtained in the linear regime and the area obtained at high power with the SCAR procedure is 1.0284(8).

(69) Again, this is the correction factor to get the true radiocarbon dioxide concentration from the SCAR measurements in the cell at our experimental conditions with this new analysis procedure. Its discrepancy can be considered as a measure of the goodness of the present approach.

(70) TABLE-US-00002 TABLE 2 Fitted parameters of the SCAR spectrum of the P(20) transition of .sup.14C.sup.16O.sub.2 in a 6375 times enriched CO.sub.2 sample (p = 12 mbar, T = 195 K) Line center, .sub.o - 66227000 MHz 382.1143(47) MHz Spectral Area 8968.44(32) ms.sup.1MHz Lorentzian HWHM, w.sub.L 49.927(25) MHz Gaussian HWHM, w.sub.G 41.6905(42) MHz Saturation parameter Z.sub.1V 9.030(24) V.sup.1

(71) TABLE-US-00003 TABLE 3 Fitted parameters of the linear-CRD spectrum of the P(20) transition of .sup.14C.sup.16O.sub.2 in a 6375 times enriched CO.sub.2 sample (p = 12 mbar, T = 195 K). Line center, .sub.o- 66227000 MHz 382.34 (7) MHz Spectral Area 9224 (7) ms.sup.1MHz Lorentzian HWHM, w.sub.L 53.30 (9) MHz Gaussian HWHM, w.sub.G 39.22 (13) MHz

Method for measuring the concentration of trace gases by SCAR spectroscopy

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G01N2201/129

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G01N2201/06113

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G01N21/3504

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G01N2021/391

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G01J3/42

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International classification

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G01N21/3504

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