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Correction of the self-absorption for reversed spectral lines: application to two resonance lines of neutral aluminium
Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 365 – 372
www.elsevier.com/locate/jqsrt
Correction of the self-absorption for reversed spectral lines: application to two resonance lines of neutral aluminium Hssaine Amamoua;∗ , Andre Boisa , Belkacem Ferhatb , Roland Redona , Bruno Rossettoa , Marc Riperta a
Laboratoire d’Optique Appliquee (LOA), Universite de Toulon et du Var, BP 132, 83957 La Garde Cedex, France b Institut de Physique USTHB, B.P. No. 32, EL-Alia-Dar El Beida-Alger, Algeria Received 18 April 2002; accepted 9 July 2002
Abstract A new method for the correction of reversed spectral lines, in consequence of self-absorption, is proposed in this article. This method was applied to two resonance lines of neutral aluminium (Al I) obtained by a laser pulse on an aluminium solid placed in air. Several spectra of these lines were corrected for various delays after the laser pulse. The evolution of several parameters of these lines was computed as a function of this delay after the correction of self-absorption. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Plasma laser spectroscopy; Self-absorption; Optical thickness; Transition probabilities; Resonance lines; Reversed spectral lines; Stark e
1. Introduction Several methods [1–3] were already used to ?t the spectral lines. For a broadening by Stark e
Corresponding author. Tel.: +33-494-14-25-16; fax: +33-494-14-21-68. E-mail address: [email protected] (H. Amamou).
0022-4073/03/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 0 2 ) 0 0 1 6 3 - 2
366
H. Amamou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 365 – 372
Thus, we propose a model of plasma divided into two distinct parts: the center of the plasma and periphery. These two parts are characterized by di
(3)
j=1
with fap being the instrumental function; ⊗, convolution product; Ij theor , the pro?le of line “j” and a0 + a1 + a2 2 + a3 3 a polynomial ?t of the continuous spectrum. The convolution product is due to the fact that the observed spectrum is a convolution of the self-absorbed pro?le by the instrumental function [8–10], which is experimentally determined. The resulting pro?le is given by Eq. (3). The Fast Fourier Transform was used to calculate the required convolutions. The continuous spectrum was ?tted by a polynomial of degree 3. The theoretical pro?le g∗ () used to ?t the experimental spectrum g() is thus n i theor ∗ i=1 I : (4) g () = n n theor i=1 I This pro?le is normalized, as the experimental spectra that we ?t are not calibrated in intensity (they are given in a number of counts). It is thus necessary to normalize the experimental and theoretical spectra in order that they have the same dimension. One should note that the number of parameters is equal to 8n + 3, for n spectral lines in the spectrum.
H. Amamou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 365 – 372
367
3. Fitting method A minimization of a criteria Fcr quali?es the ?t of the experimental spectrum g() by the pro?le model g∗ (). In several works [1,3,6] this criteria Fcr is taken as the correlation factor fcor with the least mean square error. The correlation factor compares the forms of the experimental and theoretical pro?les, and the least mean square error compares the surfaces of these pro?les. The minimization criteria we used is [1] ∗ 2 Fcr = [1 + (1 − fcor )] 1 + (5) |g() − g ()|
with fcor being the correlation factor between the signals g() and g∗ (). The ?tting method is a “synthese” method [1], the minimization uses the simplex algorithm [11]. This algorithm was already used to ?t emission spectra [12]. The program of this method was written under Matlab. Initial parameters and the instrumental function are needed for this program, thus, a ?rst theoretical spectrum is computed from relation (4). At each iteration, the algorithm computes a theoretical pro?le which is compared to the experimental spectrum, until the minimization criteria becomes lower than a preset value. 4. Application to the lines of neutral aluminium The resonance lines, centered on 394.40 and 396:15 nm, of neutral aluminium emitted by a plasma created by the interaction of a laser pulse (Nd:Yag wavelength 1064 nm) with a solid aluminium target in air, were recorded. For details of the experimental device, see Refs. [2,12]. It should be noted that the line centered at 394:40 nm is the right spectral line of the spectrum (the wavelength axis is reversed from right to left). Fig. 1 presents the experimental spectrum of these two resonance lines and the ?t obtained with our theoretical pro?le. As it can be seen, the ?t is quite excellent in spite of the strong deformation of the lines. Fig. 2 shows the emission pro?le of the center of the plasma after correction of the self-absorption. We then ?tted several spectra of spectral lines recorded at various delays after the laser pulse. These delays are, respectively, of 0.55, 0.6, 0.8, 1, 1.5, 2, 2.5, 3 and 3:5 s. The ?ts are almost good except when the delay is rather large. The signi?cant noise of the corresponding spectrum makes the ?t worse, although it remains still acceptable, see Fig. 3. Fig. 4 presents the evolution of the width (in pixels) of the emission lines of the central part of the plasma. It is to be noted that the width decreases when the delay increases. As the Stark e
368
H. Amamou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 365 – 372
Fig. 1. Experimental spectrum “points” and ?t “continuous line”, for resonance lines, centered to 394.40 and 396:15 nm, of Al I.
Fig. 2. Corrected spectrum for resonance lines of Al I.
H. Amamou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 365 – 372
369
Fig. 3. Experimental spectrum “points” and ?t ‘continuous line” for resonance lines of Al I.
Fig. 4. Width of the emission line for the center of the plasma corrected of self-absorption as a function of the delay after the laser pulse.
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H. Amamou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 365 – 372
Fig. 5. Distance between the centers of the emissions lines of the center and the periphery of the plasma as a function of the delay after the laser pulse.
in Fig. 5 is due to the decrease in the electronic density with time and due to the homogenization of the plasma which occurs. We determined the ratios of the transition probabilities from these two lines. As these resonance lines have almost identical higher energy levels, their widths are equal and the ratio of their intensities is independent of the temperature. This ratio of intensities enabled us to calculate the ratio of transition probabilities A2 =A1 : A2 02c gs1 I02c = A1 01c gs2 I01c
(6)
with 0i c being the line-center wavelength of the line i; gsi the statistical weight of the line i and I0i c the height of the emission line of the center of plasma. Fig. 6 shows these ratios of transition probabilities for various delays after the laser pulse. This ratio of transition probabilities for the two lines is theoretically about 1.99. The values of this ratio we obtained are included between 1.55 and 2.29. These values are relatively in good agreement with the theoretical value, except when the delay is higher than 2 s. Fig. 7 presents the evolution of the width (in pixels) of the emission lines of the periphery of the plasma. It is noted that these widths behave in a very similar way as the corresponding widths for the center of the plasma.
H. Amamou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 365 – 372
371
Fig. 6. Ratios of transition probabilities of resonance lines of Al I.
Fig. 7. Width of the emission line for the periphery of the plasma corrected of self-absorption as a function of the delay after the laser pulse.
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H. Amamou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 365 – 372
5. Conclusion We established a new method to ?t the spectral lines strongly deformed by self-absorption. These spectral lines present a central valley. We propose a model of plasma made up of a hot central plasma surrounded by a colder peripheral plasma. We applied this model to two resonance lines, centered to 394.40 and 396:15 nm, of neutral aluminium. We treated several spectra recorded at various delays after the laser pulse. The ?t of these spectra enabled us to compute the pro?le of the emission lines of the center of the plasma. We computed the evolution of the parameters of these lines according to this delay. We also determined the ratio of transition probabilities from the two lines and we obtained a ratio close to the theoretical ratio. Acknowledgements The author would like to thank all members of LOA for their assistances in various ?elds. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
Redon R, Ferhat B, Richou J. JQSRT 1997;58:511. Escarguel A, Ferhat B, Lesage A, Richou J. A single laser spark in aqueous medium. JQRST 2000;64:353–61. Amamou H, Bois A, Ferhat B, Redon R, Rossetto B, Matheron P. JQSRT 2002;75(6):747–763. Lorentz HA. Proc Roy Acad Sci 1906;8:591. Karabourniotis D, Karras C, Drakakis M, Damenlincourt JJ. J Appl Phys 1982;53(11):7259. Skudra A, Revalde G. JQSRT 1999;61(6):717–28. Cowan RD, Dieke GH. Rev Mod Phys 1948;20:418. Griem HR. Principles of plasma spectroscopy. Cambridge: Cambridge University Press, 1997. Lochte-Holgreven W. Plasma diagnostics. Amsterdam: North-Holland, 1968. Kastner SO. On expressions for line optical thickness. JQRST 1998;60(4):515–21. Nelder JA, Mead R. A simplex method for function minimization. Comput J 1965;7:308–13. Matheron P, Escarguel A, Redon R, Lesage A, Richou J. SiII transition probabilities measurements in a laser induced plasma. JQSRT 2001;69:535–41. [13] Griem HR. Spectral line broadening by plasmas. New York: Academic Press, 1974.
www.elsevier.com/locate/jqsrt
Correction of the self-absorption for reversed spectral lines: application to two resonance lines of neutral aluminium Hssaine Amamoua;∗ , Andre Boisa , Belkacem Ferhatb , Roland Redona , Bruno Rossettoa , Marc Riperta a
Laboratoire d’Optique Appliquee (LOA), Universite de Toulon et du Var, BP 132, 83957 La Garde Cedex, France b Institut de Physique USTHB, B.P. No. 32, EL-Alia-Dar El Beida-Alger, Algeria Received 18 April 2002; accepted 9 July 2002
Abstract A new method for the correction of reversed spectral lines, in consequence of self-absorption, is proposed in this article. This method was applied to two resonance lines of neutral aluminium (Al I) obtained by a laser pulse on an aluminium solid placed in air. Several spectra of these lines were corrected for various delays after the laser pulse. The evolution of several parameters of these lines was computed as a function of this delay after the correction of self-absorption. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Plasma laser spectroscopy; Self-absorption; Optical thickness; Transition probabilities; Resonance lines; Reversed spectral lines; Stark e
1. Introduction Several methods [1–3] were already used to ?t the spectral lines. For a broadening by Stark e
Corresponding author. Tel.: +33-494-14-25-16; fax: +33-494-14-21-68. E-mail address: [email protected] (H. Amamou).
0022-4073/03/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 0 2 ) 0 0 1 6 3 - 2
366
H. Amamou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 365 – 372
Thus, we propose a model of plasma divided into two distinct parts: the center of the plasma and periphery. These two parts are characterized by di
(3)
j=1
with fap being the instrumental function; ⊗, convolution product; Ij theor , the pro?le of line “j” and a0 + a1 + a2 2 + a3 3 a polynomial ?t of the continuous spectrum. The convolution product is due to the fact that the observed spectrum is a convolution of the self-absorbed pro?le by the instrumental function [8–10], which is experimentally determined. The resulting pro?le is given by Eq. (3). The Fast Fourier Transform was used to calculate the required convolutions. The continuous spectrum was ?tted by a polynomial of degree 3. The theoretical pro?le g∗ () used to ?t the experimental spectrum g() is thus n i theor ∗ i=1 I : (4) g () = n n theor i=1 I This pro?le is normalized, as the experimental spectra that we ?t are not calibrated in intensity (they are given in a number of counts). It is thus necessary to normalize the experimental and theoretical spectra in order that they have the same dimension. One should note that the number of parameters is equal to 8n + 3, for n spectral lines in the spectrum.
H. Amamou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 365 – 372
367
3. Fitting method A minimization of a criteria Fcr quali?es the ?t of the experimental spectrum g() by the pro?le model g∗ (). In several works [1,3,6] this criteria Fcr is taken as the correlation factor fcor with the least mean square error. The correlation factor compares the forms of the experimental and theoretical pro?les, and the least mean square error compares the surfaces of these pro?les. The minimization criteria we used is [1] ∗ 2 Fcr = [1 + (1 − fcor )] 1 + (5) |g() − g ()|
with fcor being the correlation factor between the signals g() and g∗ (). The ?tting method is a “synthese” method [1], the minimization uses the simplex algorithm [11]. This algorithm was already used to ?t emission spectra [12]. The program of this method was written under Matlab. Initial parameters and the instrumental function are needed for this program, thus, a ?rst theoretical spectrum is computed from relation (4). At each iteration, the algorithm computes a theoretical pro?le which is compared to the experimental spectrum, until the minimization criteria becomes lower than a preset value. 4. Application to the lines of neutral aluminium The resonance lines, centered on 394.40 and 396:15 nm, of neutral aluminium emitted by a plasma created by the interaction of a laser pulse (Nd:Yag wavelength 1064 nm) with a solid aluminium target in air, were recorded. For details of the experimental device, see Refs. [2,12]. It should be noted that the line centered at 394:40 nm is the right spectral line of the spectrum (the wavelength axis is reversed from right to left). Fig. 1 presents the experimental spectrum of these two resonance lines and the ?t obtained with our theoretical pro?le. As it can be seen, the ?t is quite excellent in spite of the strong deformation of the lines. Fig. 2 shows the emission pro?le of the center of the plasma after correction of the self-absorption. We then ?tted several spectra of spectral lines recorded at various delays after the laser pulse. These delays are, respectively, of 0.55, 0.6, 0.8, 1, 1.5, 2, 2.5, 3 and 3:5 s. The ?ts are almost good except when the delay is rather large. The signi?cant noise of the corresponding spectrum makes the ?t worse, although it remains still acceptable, see Fig. 3. Fig. 4 presents the evolution of the width (in pixels) of the emission lines of the central part of the plasma. It is to be noted that the width decreases when the delay increases. As the Stark e
368
H. Amamou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 365 – 372
Fig. 1. Experimental spectrum “points” and ?t “continuous line”, for resonance lines, centered to 394.40 and 396:15 nm, of Al I.
Fig. 2. Corrected spectrum for resonance lines of Al I.
H. Amamou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 365 – 372
369
Fig. 3. Experimental spectrum “points” and ?t ‘continuous line” for resonance lines of Al I.
Fig. 4. Width of the emission line for the center of the plasma corrected of self-absorption as a function of the delay after the laser pulse.
370
H. Amamou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 365 – 372
Fig. 5. Distance between the centers of the emissions lines of the center and the periphery of the plasma as a function of the delay after the laser pulse.
in Fig. 5 is due to the decrease in the electronic density with time and due to the homogenization of the plasma which occurs. We determined the ratios of the transition probabilities from these two lines. As these resonance lines have almost identical higher energy levels, their widths are equal and the ratio of their intensities is independent of the temperature. This ratio of intensities enabled us to calculate the ratio of transition probabilities A2 =A1 : A2 02c gs1 I02c = A1 01c gs2 I01c
(6)
with 0i c being the line-center wavelength of the line i; gsi the statistical weight of the line i and I0i c the height of the emission line of the center of plasma. Fig. 6 shows these ratios of transition probabilities for various delays after the laser pulse. This ratio of transition probabilities for the two lines is theoretically about 1.99. The values of this ratio we obtained are included between 1.55 and 2.29. These values are relatively in good agreement with the theoretical value, except when the delay is higher than 2 s. Fig. 7 presents the evolution of the width (in pixels) of the emission lines of the periphery of the plasma. It is noted that these widths behave in a very similar way as the corresponding widths for the center of the plasma.
H. Amamou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 365 – 372
371
Fig. 6. Ratios of transition probabilities of resonance lines of Al I.
Fig. 7. Width of the emission line for the periphery of the plasma corrected of self-absorption as a function of the delay after the laser pulse.
372
H. Amamou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 365 – 372
5. Conclusion We established a new method to ?t the spectral lines strongly deformed by self-absorption. These spectral lines present a central valley. We propose a model of plasma made up of a hot central plasma surrounded by a colder peripheral plasma. We applied this model to two resonance lines, centered to 394.40 and 396:15 nm, of neutral aluminium. We treated several spectra recorded at various delays after the laser pulse. The ?t of these spectra enabled us to compute the pro?le of the emission lines of the center of the plasma. We computed the evolution of the parameters of these lines according to this delay. We also determined the ratio of transition probabilities from the two lines and we obtained a ratio close to the theoretical ratio. Acknowledgements The author would like to thank all members of LOA for their assistances in various ?elds. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
Redon R, Ferhat B, Richou J. JQSRT 1997;58:511. Escarguel A, Ferhat B, Lesage A, Richou J. A single laser spark in aqueous medium. JQRST 2000;64:353–61. Amamou H, Bois A, Ferhat B, Redon R, Rossetto B, Matheron P. JQSRT 2002;75(6):747–763. Lorentz HA. Proc Roy Acad Sci 1906;8:591. Karabourniotis D, Karras C, Drakakis M, Damenlincourt JJ. J Appl Phys 1982;53(11):7259. Skudra A, Revalde G. JQSRT 1999;61(6):717–28. Cowan RD, Dieke GH. Rev Mod Phys 1948;20:418. Griem HR. Principles of plasma spectroscopy. Cambridge: Cambridge University Press, 1997. Lochte-Holgreven W. Plasma diagnostics. Amsterdam: North-Holland, 1968. Kastner SO. On expressions for line optical thickness. JQRST 1998;60(4):515–21. Nelder JA, Mead R. A simplex method for function minimization. Comput J 1965;7:308–13. Matheron P, Escarguel A, Redon R, Lesage A, Richou J. SiII transition probabilities measurements in a laser induced plasma. JQSRT 2001;69:535–41. [13] Griem HR. Spectral line broadening by plasmas. New York: Academic Press, 1974.