Abstract:
In a wavelength modulation spectroscopy method for measuring the concentration of a gas component in a gas sample a portion of the light of a tunable light source is passed through a reference gas comprising the gas component in a constant concentration. Afterwards the light is detected by a reference detector. Another portion of the light is passed through the gas sample and thereafter to a measuring detector. The light emitted by the light source is modulated with a frequency f m , while the wavelength is swept over a molecular absorption line of the gas component. Demodulation of the detector outputs is made at a higher harmonic Nf m .  
     In order to compensate for variations of the modulation parameters of the light source ( 2 ) in real time, a mathematical description of the demodulated reference detector output (S(υ) N,Ref ) based on Fourier analysis of the modulated light ( 1 ) and on a mathematical expression for the absorption line is provided, said mathematical description comprising unknown modulation parameters with respect to the modulation of the light ( 1 ). Said modulation parameters are determined from the demodulated reference detector output (S(υ) N,Ref ) and its mathematical description. In a further step the concentration (c Meas ) is determined from the demodulated measuring detector output (S(υ) N,Meas ), a corresponding mathematical description of it and the modulation parameters.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS  
       [0001]     This application claims priority to the European application No. 03029102.5, filed Dec. 17, 2003 and which is incorporated by reference herein in its entirety.  
       FIELD OF THE INVENTION  
       [0002]     The invention relates to a wavelength modulation spectroscopy method for measuring the concentration of a gas component of interest in a gas sample.  
       BACKGROUND OF THE INVENTION  
       [0003]     In wavelength modulation spectroscopy (WMS) for measuring the concentration of a gas component in a gas sample a portion of the light of a tunable light source, usually a continuously tunable laser such as a diode laser, is passed through a reference gas comprising the known gas component or another suitable gas component of constant concentration. Afterwards the light is detected by a reference detector. Another portion of the light is directed to a monitor detector for normalization purposes. Yet another portion of the light is passed through the gas sample and thereafter to a measuring detector. The light emitted by the light source is modulated with a frequency f m , while the wavelength is swept over a molecular absorption line of the gas component. As the light propagates through the reference gas or gas sample, respectively, wavelength dependent absorption converts some of the wavelength modulation into an intensity modulation of the light. Thus, the light will have an overtone spectrum generated by the absorption process, the harmonic content of the spectrum being dependent on the width and shape of the molecular absorption line in the gas and the etalons in the spectroscopy system. When the light then impinges onto the reference detector or measuring detector, respectively, the detector outputs contain AC components at the modulation frequency f m  and its higher harmonics Nf m  (N=2, 3, 4, etc.). Demodulating the respective detector outputs at one of said higher harmonics Nf m  shifts the measurement from frequencies near DC, where the light source is noisy, into a higher frequency range, where the noise is lower, thus improving the measurement sensitivity.  
         [0004]     The modulation of the emitted light can most conveniently be accomplished by modulation of the injection current of the diode laser, which imposes modulation on the wavelength and to some extend on the intensity of the emitted light. As the demodulated Nf m  absorption signal depends not only on the concentration of the measured gas but also on the modulation parameters of the light source, variations of these modulation parameters can affect the accuracy of the measurement.  
       SUMMARY OF THE INVENTION  
       [0005]     Therefore, the invention seeks to provide a wavelength modulation spectroscopy method, which automatically compensates for variations of the modulation parameters of the light source in real time.  
         [0006]     According to the invention this is achieved by the claims.  
         [0007]     Preferred embodiments of the method and the system according to the invention are specified in the dependent claims.  
         [0008]     The approach in this invention is to provide a mathematical description of the demodulated reference detector output based on Fourier analysis of the modulated light and on a mathematical expression for the absorption line, said mathematical description comprising the unknown modulation parameters of the light source, and determining said modulation parameters from the demodulated reference detector output and its mathematical description.  
         [0009]     In a further step the concentration of the gas component in the gas sample can be determined by providing a further equivalent mathematical description of the demodulated measuring detector output based on Fourier analysis of the modulated light and on a mathematical expression for the absorption line, said further mathematical description comprising said modulation parameters and the unknown concentration of the gas component of interest in the gas sample, and determining said concentration of the gas component from the demodulated measuring detector output, its mathematical description and the determined modulation parameters. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0010]     The invention will be now described by way of a preferred example and with reference to the accompanying drawing, wherein  
         [0011]      FIG. 1  shows a block diagram of a spectroscopy system in accordance with the invention, and  
         [0012]      FIG. 2  is a schematic block diagram of the calculating means of the system of  FIG. 1 . 
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0013]     For a better understanding of the following description, reference is made to Applied Optics, 38 (1999) 5803-5815, where a theoretical description of the wavelength-modulation (WM) spectrometry technique is given. In the following description the optical frequency υ is used instead of the wavelength λ, which are inversely proportional to each other.  
         [0014]     As  FIG. 1  shows, the light  1  of a tunable light source  2 , here a diode laser, is split by means of beam splitters  3  and  4  into a measurement path  5 , a monitor path  6  and a reference path  7 . Passing through the measurement path  5  the light  1  interacts with a sample  8 , here a weakly absorbing gas sample, and is attenuated exponentially according to the Beer-Lambert law:  
                   I   =       ⁢       T   Meas     ⁢       I   L     ·     exp   ⁡     [       -     α   ⁡     (     T   ,   p   ,   v   ,     γ   Meas       )         ⁢     L   Meas       ]                       =       ⁢       T   Meas     ⁢       I   L     ·     exp   [       -       A   Meas     ⁡     (   T   )         ⁢           ⁢     c   Meas     ⁢     p   Meas                               ⁢       1     π   ⁢           ⁢     γ   Meas         ⁢     χ   ⁡     (     v   ,     γ   Meas       )       ⁢     L   Meas       ]     ,                 (     Equation   ⁢           ⁢   1     )             
 
 where I is the intensity of the light  1  after passing through the measurement path  5 , I L  is the intensity of the light  1  emitted from the light source  2 , T Meas  is a transmission factor over the measurement path  5 , which transmission factor stands for the wavelength independent transmission of the optical system, L Meas  is the length of the measurement path  5 , α is the wavelength dependent absorption coefficient of the gas sample  8 , A and χ represent the intensity and the peak-normalized shape of a molecular absorption line of a gas component of interest in the gas sample  8 , respectively, c Meas  is the concentration (mole fraction) of the absorbing gas component, p Meas  is the total pressure in the measurement path  5  and γ Meas  is the half width at half maximum (HWHM) of the absorption line. At atmospheric pressure the shape χ of the molecular absorption line is typically given by the Lorentzian line-shape function:  
                 χ   ⁡     (     v   ,   γ     )       =       1     1   +       (       (     v   -     v   c       )     /   γ     )     2         =     1     1   +       (       v   _     -       v   _     c       )     2             ,           (     Equation   ⁢           ⁢   2     )             
 
 where υ c  is the line center frequency and {overscore (ν)}=υ/γ and {overscore (ν)} c =ν c /γ are the halfwidth-(HWHM-)normalized frequency and line center frequency, respectively. 
 
         [0015]     As exp x≈(1+x) for small x and the gas sample  8  is only weakly absorbing, Equation 1 can be written as:  
             I   =         T   Meas     ⁢     I   L       -       T   Meas     ⁢     I   L     ⁢       A   Meas     ⁡     (   T   )       ⁢     c   Meas     ⁢     p   Meas     ⁢     1     π   ⁢           ⁢     γ   Meas         ⁢     χ   ⁡     (     v   ,     γ   Meas       )       ⁢       L   Meas     .                 (     Equation   ⁢           ⁢   3     )             
 
         [0016]     The light  1  of the diode laser  2  is modulated through its injection current i, which imposes modulation on the optical frequency υ L  and to some extend on the intensity I L  of the emitted light  1 . The modulation is performed by a first modulation means  9  generating a sinusoidal signal at a frequency f m  and a second modulation means  10  generating a periodic slow sweep function, which may be part-wise linear in time or of an arbitrary shape. The signals of said first and second modulation means  9  and  10  are summed in adding means  11  and fed to a modulation input of the diode laser  2 . Thus, the injection current i of the diode laser  2  is given by: 
 
 i=i   0 ( t )+ i   a ( t )cos(2 πf   m   t )  (Equation 4), 
 
 where i 0 (t) includes a bias and a slow current function, for example a slow current ramp, and i a (t) is the modulation amplitude at the modulation frequency f m . 
 
         [0017]     The modulation of the injection current i of the diode laser  2  results in a modulation of the optical frequency υ L  of the emitted light  1 : 
 
ν=ν 0 ( t )+ν a  cos(2π f   m   t )  (Equation 5), 
 
 where υ 0 (t) represents a sweep of the optical frequency over the absorption line of interest and υ a  is the modulation amplitude of the optical frequency υ L  at the modulation frequency f m . For simplicity it is assumed that the modulation of the optical frequency υ L  follows the modulation of the injection current i without phase shift. 
 
         [0018]     The modulation of the injection current i of the diode laser  2  also results in modulation of the intensity I L  of the emitted light  1 : 
 
 I   L (ν 0 ,ν a   ,t )= I   L,0 (ν 0 )+κ 1 ν a  cos(2 πf   m   t +φ)  (Equation 6), 
 
 where the slow intensity variation due to the sweep of the optical frequency of the light  1  is taken as the DC term I L,0 (υ 0 ) and κ 1  is defined as the linear intensity modulation coefficient. The term κ 1 υ a =I L,1 (υ 0 )=m represents the intensity modulation amplitude, i.e. the first Fourier component of the intensity modulation, whereas φ stands for the phase shift between the intensity and frequency modulation. In Equation 5 possible nonlinear terms of the intensity modulation of the emitted light  1  are neglected. 
 
         [0019]     According to the slow sweep function of the second modulation means  10  the optical frequency of the emitted light  1  sweeps over the molecular absorption line of interest of the gas sample  8  in the measurement path  5 , while the light  1  is modulated with the frequency f m . Due to the nonlinear wavelength dependent absorption the light  1  will have an overtone spectrum, the harmonic content of the spectrum being dependent on the width and shape of the molecular absorption line.  
         [0020]     After passing through the measurement path  5  the light  1  impinges onto a measuring detector  12 , the output of which is given by:  
                 S   Meas     =         η   Meas     ⁢   I     =         η   Meas     ⁢     T   Meas     ⁢     I   L       -       η   Meas     ⁢     T   Meas     ⁢     I   L     ⁢       A   Meas     ⁡     (   T   )       ⁢     c   Meas     ⁢     p   Meas     ⁢       χ   ⁡     (     v   ,     γ   Meas       )         πγ   Meas       ⁢     L   Meas             ,           (     Equation   ⁢           ⁢   7     )             
 
 where η Meas  is an instrument factor of the measurement path  5 . 
 
         [0021]     The portion of the light  1  diverted into the monitor path  6  impinges onto a monitor detector  13 . Since there is no molecular absorption in the monitor path  6 , the monitor detector output is given by: 
 
 S   Mon =η Mon   I=η   Mon   T   Mon   I   L   =G   Mon   I   L   (Equation 8), 
 
 where η Mon  and T mon  are the instrument factor and the transmission factor of the monitor path  6 , respectively, and G Mon =η Mon T Mon  is a constant gain. The monitor detector output S Mon  is fed via an analog-to-digital converter  14  and a low-pass filter  15  to a calculating means  16  of the spectroscopy system. The monitor detector output S Mon  is further used for correcting any transmission changes in the measurement path  5  and is therefore fed to an automatic gain control unit  17  together with the measuring detector output S Meas . In the automatic gain control unit  17  the measuring detector output S Meas  is controlled so as to maintain the condition: 
 
η Meas   T   Meas =η Mon   T   Mon   G   Mon   (Equation 9). 
 
         [0022]     Both the intensity I of the light  1  impinging on the measuring detector  12  and the line-shape function χ are periodic functions of time, so that they can be expressed in terms of a Fourier series:  
                 I   ⁡     (       v   0     ,     v   a     ,   t     )       =         ∑     n   =   0     ∞     ⁢         I   n   e     ⁡     (       v   0     ,     v   a       )       ⁢     cos   ⁡     (     2   ⁢   π   ⁢           ⁢     nf   m     ⁢   t     )           +       ∑     n   =   0     ∞     ⁢         I   n   o     ⁡     (       v   0     ,     v   a       )       ⁢     sin   ⁡     (     2   ⁢   π   ⁢           ⁢     nf   m     ⁢   t     )               ,           (     Equation   ⁢           ⁢   10     )                   χ   ⁡     (         v   _     0     ,       v   _     a     ,   t     )       =       ∑     n   =   0     ∞     ⁢         χ   n   e     ⁡     (         v   _     0     ,       v   _     a     ,   t     )       ⁢     cos   ⁡     (     2   ⁢   π   ⁢           ⁢     nf   m     ⁢   t     )             ,           (     Equation   ⁢           ⁢   11     )             
 
 where {overscore (ν)} 0 ν 0 /γ and {overscore (ν)} a /γ represent the halfwidth-(HWHM-) normalized sweep and the modulation amplitude of the optical frequency υ L , respectively. As the line-shape function χ(υ L ,t) follows the modulation of the frequency without phase delay, only the cosine terms in the series expansion are needed. By inserting Equations 6 and 11 into Equation 7 one obtains an optical-frequency-dependent expression for measuring detector output S(υ) Meas . The gained measuring detector output S(υ) Meas  containing AC components at the modulation frequency f m  and its higher harmonics 2f m , 3f m , 4f m , etc. is demodulated at a higher harmonic Nf m , most commonly at 2f m , in a first demodulation means  18  comprising an analog-to-digital converter  19  and a lock-in amplifier  20  for digitizing the gained measuring detector output S(υ) Meas  and converting it to base band. The demodulation at Nf m  shifts the measurement from frequencies near DC, where the light source  2  is noisy, into a higher frequency range, where the noise is lower, thus improving the measurement sensitivity by approximately an order of 10 2 -10 3  compared to a direct unmodulated absorption measurement. The in-phase component of the measuring detector output S(υ) Meas  demodulated at Nf m  can be written as:  
                 S   ⁡     (   v   )         N   ,   Meas     e     ≈       -     G   Mon       ⁢       A   Meas     ⁡     (   T   )       ⁢     c   Meas     ⁢     p   Meas     ⁢     1     π   ⁢           ⁢     γ   Meas         ⁢         L   Meas     ⁡     (                 I     L   ,   0     e     ⁡     (     v   0     )       ⁢       χ   N   e     ⁡     (         v   _     0     ,       v   _     a       )         +                     κ   1     ⁢     v   a     ⁢   cos   ⁢           ⁢   φ     2     ⁢     (         χ     N   -   1     e     ⁡     (         v   _     0     ,       v   _     a       )       +       χ     N   +   1     e     ⁡     (         v   _     0     ,       v   _     a       )         )             )       .               (     Equation   ⁢           ⁢   12     )             
 
         [0023]     As the phase difference p between the intensity modulation and the frequency modulation of the light  1  at the modulation frequency f m  is close to n and consequently cos φ≈−1, S(υ) Meas  can be rewritten as:  
                   S   ⁡     (   v   )         N   ,   Meas     e     =             c   Meas     ·     G   Mon       ⁢       A   Meas     ⁡     (   T   )       ⁢     p   Meas     ⁢     L   Meas         ︸     par   ⁡     (     T   ,   p     )           ·     
     ⁢           ⁢         1     π   ⁢           ⁢     γ   Meas         ⁢     (                 I     L   ,   0     e     ⁡     (     v   0     )       ⁢       χ   N   e     ⁡     (         v   _     0     ,       v   _     a       )         -                 m   2     ⁢     (         χ     N   -   1     e     ⁡     (         v   _     0     ,       v   _     a       )       +       χ     N   +   1     e     ⁡     (         v   _     0     ,       v   _     a       )         )             )         ︸       Γ   Meas     ⁡     (       v   0     ,     v   a     ,   m   ,     γ   Meas       )               ,           (     Equation   ⁢           ⁢   13     )             
 
 where m=κ 1 υ a  is the intensity modulation amplitude. As shown in Equation 13 the demodulated measuring detector output S(υ) Meas  can be presented as a product of the is the concentration (mole fraction) c Meas  of the absorbing gas component, a known pressure and temperature dependent parameter par(T,p) and a function Γ Meas ({overscore (ν)} 0 , {overscore (ν)} a , m, γ Meas ) dependent on laser modulation parameters and the width of the molecular absorption line of interest. According to Journal of Quantitative Spectroscopy &amp; Radiative Transfer, 68 (2001) 299-317, which is incorporated herein by reference, the Nth Fourier component of a wavelength modulated Lorentzian line-shape function χ N  can be expressed by:  
                 χ   N     ⁡     (         v   _     0     ,       v   _     a       )       =           A   N         v   _     a   N       ⁡     [       B   N     +           C   N     ⁢     S   +       +       D   N     ⁢     S   -             2     ⁢   R         ]       .             (     Equation   ⁢           ⁢   14     )             
 
         [0024]     For N=2, the Nth, (N−1)th and (N+1)th Fourier components of the line-shape function χ are needed and the factors of Equation 15 are as follows: 
    A 1 =2−δ 1,0 , A 2 =2−δ 2,0 , A 3 =2−δ 3,0 , where δ n,0  is the Kronecker delta,     B 1 =0, B 2 =2, B 3 =−8{overscore (ν)} 0 ,     C 1 =−{overscore (ν)} 0 , C 2 =[(2+{overscore (ν)} a   2 )−2{overscore (ν)} 0   2 ], C 3 ={overscore (ν)} 0 [3(4+{overscore (ν)} a   2 )−4{overscore (ν)} 0   2 ],     D 1 =sign 2 ({overscore (ν)} 0 ), D 2 =−sign 2 ({overscore (ν)} 0 )4{overscore (ν)} 0 , D 3 =−sign 2 ({overscore (ν)} 0 )[(4+3{overscore (ν)} a   2 )−12{overscore (ν)} 0   2 ],     R={square root}{square root over (M 2 +4ν)} 0   2 , S + {square root}{square root over (R+M)} and S − ={square root}{square root over (R−M)}, where M=1+{overscore (ν)} a   2 −{overscore (ν)} 0   2 .    
 
         [0030]     As mentioned above, yet another portion of the light  1  of the diode laser  2  is passed through the reference path  7 , which contains in a reference cell of known length L Ref  a reference gas  21  comprising the gas component to be detected in the gas sample  6  in a known concentration. After passing through the reference path  7  the light  1  impinges onto a reference detector  22 . The reference detector output S(υ) Ref  is demodulated at the higher harmonic Nf m  in a second demodulation means  23  comprising an analog-to-digital converter  24  and a lock-in amplifier  25 . As the reference detector output S(υ) Ref  is processed in the same way as the measuring detector output S(υ) Meas , the in-phase component of the reference detector output S(υ) Ref  demodulated at Nf m  can be written by using Equation 13 as:  
                 S   ⁡     (   v   )         N   ,   Ref     e     =           n   Ref     ⁢     T   Ref     ⁢       A   Ref     ⁡     (   T   )       ⁢     c   Ref     ⁢     p   Ref     ⁢     L   Ref     ⁢     1     π   ⁢           ⁢     γ   Ref             ︸     cons   ⁢           ⁢   tant         ·         (                 I     L   ,   0     e     ⁡     (     v   0     )       ⁢       χ   N   e     ⁡     (         v   _     0     ,       v   _     a       )         -                 m   2     ⁢     (         χ     N   -   1     e     ⁡     (         v   _     0     ,       v   _     a       )       +       χ     N   +   1     e     ⁡     (         v   _     0     ,       v   _     a       )         )             )       ︸       Γ   Ref     ⁡     (         v   _     0     ,       v   _     a     ,   m     )           .               (     Equation   ⁢           ⁢   15     )             
 
         [0031]     Since the product η Ref T Ref A Ref (T)c Ref p Ref L Ref  is constant, the demodulated reference detector output S(υ) N,Ref  can be written as a product of a constant value and a function Γ Ref ({overscore (ν)} 0 , {overscore (ν)} a , m), which is solely dependent on laser modulation parameters, since the half width γ Ref  of the reference absorption line is also constant.  
         [0032]     The demodulated measuring detector output S(υ) N,Meas  and reference detector output S(υ) N,Ref  and the low-pass filtered monitor detector output S MOn,LP  are fed to the calculating means  16  for calculating the concentration of the gas component in the gas sample  8  and for automatically correcting any changes of the FM/AM parameters of the diode laser  2  in real time.  
         [0033]      FIG. 2  shows a functional block diagram of the calculating means  16 . In block  26  the average value I L,0 (υ) of the intensity of the modulated light  1  is calculated from the low-pass filtered monitor detector output S Mon,LP  and the known constant gain G Mon  by using Equation 8. In block  27  Equation 15 is applied to the demodulated reference detector output S(υ) N,Ref . Since I L,0 (υ) is provided and the width γ Ref  of the reference absorption line is constant, the laser modulation parameters, i.e. the intensity modulation amplitude m and the frequency modulation amplitude υ a  can be extracted. It should be noted that the gas in the reference path  7  do not have to be the same as the gas component to be measured in the measurement path  5 . What is crucial is that the concentration, temperature and pressure of the gas in the reference path  7  are kept constant, thus assuring a constant width γ Ref  of the reference absorption line. The parameters υ a  and m are then used for determining the concentration c Meas  of the gas component of interest in the measurement path  5  by fitting Equation 13 to the demodulated measuring detector output S(υ) N,Meas  in block  28  and dividing the result c Meas par(T,p) by the known parameter par(T,p) in block  29 . This method allows real time monitoring of any changes in FM/AM laser characteristics in the frequency band around f m  and any drifts of the sine amplitude generated in the first modulation means  9 .  
         [0034]     For correcting any FM changes in the slow sweep function from the second modulation means  10  the width γ Ref  of the reference absorption line is extracted from the fit of Equation 15 to the demodulated reference detector output S(γ) N,Ref  in block  27  and afterwards compared to an initial recorded value γ Ref,initial  in block  30 . The ratio is then fed to a sweep control unit  31 , which controls the amplitude of the slow sweep function generated by the second modulation means  10 .