Study of self-absorption in laser induced breakdown spectroscopy

Spectrochimica Acta Part B 101 (2014) 51–56

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Study of self-absorption in laser induced breakdown spectroscopy M. Burger, M. Skočić, S. Bukvić ⁎ University of Belgrade, Faculty of Physics, POB 368, 11000 Belgrade, Serbia

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 30 March 2014 Accepted 15 July 2014 Available online 24 July 2014

We present a simple analytical expression for self-absorption correction of a spectrum recorded in the image mode of a CCD camera. It is assumed that two spectra are available, F2 recorded with a back mirror and F1 recorded without. The corrected spectrum F0, free of self-absorption, is given by the following simple

Keywords: LIBS Self-absorption correction Abel inversion

illustrating proposed method for self-absorption correction and Abel inversion is given in details. © 2014 Elsevier B.V. All rights reserved.

expression F 0 ¼ 1þ2FF2 1− F1 . We discuss the influence of noise on subsequent inverse Abel transform. An example, G F1

1. Introduction

2. Self-absorption issue

It is well known that plasma created by strong laser pulse interacting with a metal surface is a source of very clean metal spectra [1,2]. The plasma plume has axial symmetry in respect to the axis defined by the laser beam with plasma parameters varying along the radius. Electron density and electron temperature usually achieve their maximum values in the vicinity of the axis. Boundary of the plasma plume is difficult to determine since optical emission at the periphery gradually approaches values indistinguishable from the noise. In typical arrangements spectroscopic observations of the plasma are accomplished side-on. The detector collects light emitted by different regions of the plasma plume along the line of sight. In this way measured intensity represents ‘integral value’ of the light emission along the line of sight, see Fig. 1. If an imaging CCD is applied as a detector, then one can measure simultaneous emission along numerous lines of the sight, each at different wavelength λ and offset y in respect to the axis of the plume. Set of intensities captured by the CCD at the same wavelength λ but different y values (different lines of the sight) is commonly referred as the lateral profile, F(λ, y)1. Starting with the lateral profile and applying inverse Abel transform one can infer form of the radial emission function ε(λ, r). At a first glance it appears to be a straightforward, easy to automate process. In practice, however, one has to overcome number of problems in order to obtain reliable radial emission function ε(λ, r) starting from rough data F(λ, y). Within this paper we propose a simple way to correct lateral profiles affected by selfabsorption applying ‘back mirror’ approach.

We suppose that plasma plume is symmetric in respect to the axis defined by the laser beam. In consequence, emission function ε(λ, r) depends only on the radial coordinate r for given λ. If the plasma is free of self-absorption (at wavelength λ) the lateral profile F(λ, y) registered by the detector corresponds to the forward Abel transform of the emission function, i.e.

⁎ Corresponding author. 1 In fact F(λ, y) is a matrix representing captured image; every column of the matrix (constant λ) represents a single lateral profile Fλ(y).

http://dx.doi.org/10.1016/j.sab.2014.07.007 0584-8547/© 2014 Elsevier B.V. All rights reserved.

Z F ðλ; yÞ ¼ 2

r¼∞ r¼y

εðλ; rÞrdr pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ≡ r2 −y2

Z

x¼∞ x¼−∞

εðλ; r Þdx:

ð1Þ

The above relation is a definition of the forward Abel transform [3], under assumption that ε(λ, r) approaches zero more quickly than 1/r. The last equivalence in Eq. (1) relies on the following identities: r 2 = rdr . When lateral profile F(λ, y) is known, (recorded x2 + y2, dx ¼ pffiffiffiffiffiffiffiffiffiffi r −y by a CCD detector, for example) the emission function ε(λ, r) can be evaluated by the inverse Abel transform 2

εðλ; r Þ ¼ −

1 π

Z

2

y¼∞ y¼r

dF ðλ; yÞ dy pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : dy y2 −r 2

ð2Þ

If the plasma is not optically thin then recorded lateral profile F(λ, y) is affected by self-absorption and inverse Abel transform is meaningless. Therefore, we need a way to check whether the plasma is free of selfabsorption or not. A suitable approach is based on the use of duplicating or back mirror. This technique is presented and discussed in [4,5]. The idea of the method is to record two images, F2(λ, y) with back mirror and F1(λ, y) without. The ‘back mirror’ is in fact a system of a flat mirror and concave lens, properly positioned see Fig. 2, in order to provide perfect overlapping of the plasma plume with reflected image of the plume. In this manner reflected light is directed again through the plasma providing the way for estimating the amount of self-absorption.

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M. Burger et al. / Spectrochimica Acta Part B 101 (2014) 51–56

Detector at x=

8

Target

dx

Lateral profile x y

Offset=y

r Offset=0

Line of sight at y=0

y

Fig. 2. Optical layout provides perfect overlapping of the plasma plume with reflected image of the plume. The electric bulb imitates plasma plume.

x

The total intensity over the line of the sight, at the offset y is

z

Laser Beam

Z

Fig. 1. Concentric circles mimic radial emission function ε(r). Each point, F(y), of the lateral profile corresponds to the integral emission along the line of sight at specific y. It is assumed that detector is placed at large distance from the plasma plume, at a region where emission function drops to zero. A plasma slice of the thickness dx at a position (x, y) contributes to the lateral profile as dF(y) = ε(r)dx if the plasma is optically thin. See Section 2, Eq. (3) for optically thick plasma.

In addition to the emission function ε(λ, r) we introduce axially symmetric absorption function, k(λ, r), which is responsible for selfabsorption. We consider the amount of light originating from a plasma slice, having thickness dx and lying on the line of sight y at the position x, coming to the detector, see Fig. 3, Z

∞

−

dF 1 ðλ; yÞ ¼ εðλ; r Þdx  e

x

kðλ; r Þdx:

ð3Þ

∞

−∫ kðλ; r Þdx quantifies absorption of the light emitted by the Term e x slice at the position (x, y), passing through the absorber k(λ, r) and reaching the detector at the position (x → ∞, y). To handle the ∞ −∫ kðλ; r Þdx x

term e

Z urf ðλ; x; yÞ ¼

x 0

Z urfcðλ; x; yÞ ¼

x

−∞

Z

þ

d F 1 ðλ; yÞ ¼ ½1−C ðλ; yÞ ∞

−∞

∞ −∞

εðλ; r Þdx

εðλ; r Þ  urf ðλ; x; yÞdx: ∞

The last integral ∫−∞ε(λ, r) ⋅ urf(λ, x, y)dx ≡ 0 since the argument is an odd function as the product of even ε and odd urf functions. Finally, the above equation simplifies to the following form F 1 ðλ; yÞ ¼ F 0 ðλ; yÞ½1−C ðλ; yÞ

ð5Þ

where F1 is the measured intensity over the line of the sight without ∞ back mirror, while F0(λ, y) = ∫−∞ε(λ, r)dx is an intensity that we would obtain in the case of negligible self-absorption, see Eq. (1). With back mirror in the place a certain portion of the light is directed back trough the plasma. Due to self-absorption just transmitted amount of the reflected light comes to the detector F 2 ðλ; yÞ ¼ F 1 ðλ; yÞ þ G  F 1 ðλ; yÞ  T ðλ; yÞ:

ð6Þ

The value G b 1 quantifies reflected fraction of the light, taking in account reflectivity of the mirror, transmission of the lens, solid angle etc., while

we introduce two auxiliary functions

kðλ; rÞdx

∞

Z

∞

F 1 ðλ; yÞ ¼

Z −

kðλ; r Þdx:

T ðλ; yÞ ¼ e

∞ −∞

kðλ; r Þdx

The urf function determines the optical depth measured from 0 to x, while the urfc function measures the optical depth from x to infinity. The main properties of these functions are discussed in the Appendix A. Therefore, Z

∞

−

e

x

kðλ; rÞdx¼e−urfcðλ;x;yÞ :

The maximum value of the function urfc is 2C, where constant ∞ C(λ, y) = ∫ k(λ, r)dx depends on the offset y and specific form of 0

k(λ, r).2 If C ≪ 1, the case when self-absorption correction is possible, expression (3) becomes dF 1 ðλ; yÞ≈εðλ; r Þdx  ½1−urfcðλ; x; yÞ

ð4Þ

¼ εðλ; r Þdx  ½1−fC ðλ; yÞ−urf ðλ; x; yÞg:

2 2C is the total optical depth of the plasma plume along the line of the sight at the offset y.

Fig. 3. For the purpose of simplicity emission area (red) and absorption area (green) are separated. With dark gray we indicate region of absorption for direct ray emitted by the plasma ∞ slice at (x, y). Corresponding optical depth is ∫x k(λ, r)dx. Light gray is the marked region of ∞ absorption for reflected beam, corresponding optical depth is given by ∫−∞k(λ, r)dx where k(λ, r) is the absorption function. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

M. Burger et al. / Spectrochimica Acta Part B 101 (2014) 51–56

53

Fig. 4. The right panel: image of the plasma plume recorded 70 ns after the laser pulse. Pressure of air in the chamber was 7000 Pa. Vertical resolution is set to 128 points — each point is an average value of the four camera pixels. In this way the noise is reduced approximately two times while resolution is still sufficient. Vertical line indicates position at which the lateral profile is calculated. The lateral profile, presented in the left panel, consists of 128 points. The abscissa is rescaled so that the maximum of the lateral profile corresponds to the x = 0.

Fig. 5. Cu II 240.01 nm and 240.33 nm spectral lines recorded without back mirror, the leftmost panel, and with the back mirror, the middle panel. In both cases the background is subtracted employing numerical procedure [6]. Purple color indicates points with negative values after the background removal. The rightmost panel depicts a net effect of the back mirror; the picture is obtained subtracting the image recorded without back mirror from the one recorded with the back mirror. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

is the transmission function. For low absorption Z T ðλ; yÞ≈1−

∞ −∞

2

kðλ; rÞdx ¼ 1−urfcðλ; x→−∞; yÞ ¼ 1−2C ðλ; yÞ:

F 2 ðλ; yÞ ¼ F 1 ðλ; yÞ þ G  F 1 ðλ; yÞ  ½1−2C ðλ; yÞ:

ð7Þ

Relying on Eqs. (5) and (7) one can evaluate image F0 which would be recorded by the detector in the case of negligible self-absorption 2F 1 ðλ; yÞ : Þ− F ðλ;yÞ 1 þ F ðλ;y G F ðλ;yÞ 2

ð8Þ

1

1

The term F G−F F is in fact transmission, see Eq. (6). For optically thin plasma, T = 1, corrected image F0 is equal to the image F1, recorded without back mirror. If self-absorption is present, intensity of the corrected image will be increased by a certain amount since T b 1. Eq. (8) holds for each pixel of the CCD image. It should be emphasized that background must be removed from the images F1 and F2 before applying Eq. (8). A convenient method for background removal is discussed in [6]. We need to evaluate constant G, an effective fraction of the light reflected by the back mirror, before proceeding with correction given by Eq. (8). This quantity is commonly estimated at the line F . Continuum wings, where absorption is negligible, as the ratio G ¼ F − F radiation is also a suitable choice for evaluating G because it is not affected by self-absorption in typical experiments. For noisy images spectrum obtained by the full vertical binning method, due to better signal to noise ratio, can help one to calculate G more accurately. It is essential to estimate the amount of self-absorption before applying Eq. (8), since correction is meaningful only if the optical depth C(λ, y) ≪ 1. A simple method is to calculate absorption Aðλ; yÞ ¼ 2

1

1

Thus for image F2, recorded with back mirror, we can write

F 0 ðλ; yÞ ¼

Þ− F ðλ;yÞ for lateral profile at the line pick. If the plas1−T ðλ; yÞ ¼ 1− F ðλ;y G F ðλ;yÞ ma is optically thin, the value of the absorption should be constant across the lateral profile and close to zero. A bad estimate of the G results in constant absorption across the lateral profile, but with value shifted from zero. Clear maximum of the absorption coefficient signifies the presence of the self-absorption. If the maximum is less than 0.5 it is safe to calculate corrected image3 according to Eq. (8). Underlying assumption of the self-absorption correction is that both, emission and absorption functions are radially symmetric and that both drop to zero more quickly than 1/r. Magnification of the optical system must be adjusted so that lateral profile F(y) attains zero for y → ± ∞; in other words recorded image of the spectrum must have dark area at the top and bottom. A different approach to self-absorption correction is presented in Cvejić et al. [7]. This approach is based on mathematical expressions given in Konjević [4], valid for homogenous plasma column with a fixed length. A laser induced plasma is neither homogenous nor it has clearly determined radius; however, correction proposed by Cvejić et al. [7] can be considered as a first approximation improvement applicable if the absorption function is nearly homogenous in the bulk of the plasma.

1

1

2

1

1

3. The inverse Abel transform Once the image F0, corrected for self-absorption is available, one can evaluate the emission function ε(λ, r) by applying (favorite) inverse Abel transform. The Abel inversion is a well defined mathematical method, however its application to the real-world data is known as a 3 For absorption less than 0.5 error made substituting exponential function with first two terms in expansion is under 10% which is value commonly lower than uncertainty introduced due to noise and unperfect symmetry of the plume.

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M. Burger et al. / Spectrochimica Acta Part B 101 (2014) 51–56

Fig. 6. The left panel: absorption along the lateral profile evaluated at the position indicated on the image, the right panel. Constant G, which quantifies the amount of the light reflected by the back mirror, is estimated to be G = 0.15. Exact fit to the horizontal line in the region of interest gives value for average absorption of 0.021 ± 0.11. The blue graph shown on the image is in fact actual lateral profile plotted in the ‘natural’ way in respect to the image. The lateral profile intensity is normalized so that the maximum is always in the middle of the image. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

technically difficult problem. The noise, inevitable present in practice, makes numerical realizations of Eq. (2) prone to instabilities. To overcome this obstacle a number of algorithms were proposed in specialized literature [8–11]. As a background for our numerical procedure we adopted an approach proposed by [11]. It is an elegant technique adequate for plasmas with undetermined radius, the typical case of the laser-induced plasma. The underlying idea of the method relies on the well known property of the Gaussian function. Suppose that emission 2 2 function ε is of the Gaussian type, ε ðr Þ ¼ be−α r . The lateral profile, the forward Abel transform Eq. (1), will be: Z F ðyÞ ¼ b

∞

e

−α 2 r 2

−∞

Z dx ¼ b

pffiffiffi π −α 2 y2 : ¼b e α

∞

−α 2 ðx2 þy2 Þ

e

−∞

dx ¼ be

−α 2 y2

Z

∞ −∞

−α 2 x2

e

dx

ð9Þ

Therefore, Gaussian function transforms into the Gaussian function, the reciprocal width (parameter α) remains the same while the magnipffiffi tude b changes value to a ¼ b απ. In fact we assume that recorded lateral profile, corrected for self-absorption if necessary, can be approximated by the sum of the Gaussian functions4 " −α 2 ðy−cÞ2

F ðyÞ ¼ a0 þ a1 e

# n X −ði−1Þβ2 ðy−cÞ2 : 1þ ai e

ð10Þ

procedure. The form of the emission function ε(r) is already given by Eq. (11), but we need just a simple arithmetic to calculate its coefficients bi. In this way problem of the inverse Abel transform is reduced to common data fitting procedure. A question regarding the number of Gaussians, n, in Eq. (10) needs additional attention. The choice of n reflects our prior knowledge regarding spatial distribution of emitters, also it depends on the quality of the rough data, especially amount of noise. With larger n shape the lateral profile is reproduced more precisely. On the other hand, a lot of noise usually hides details in shape making large n useless; the best-fit function tends to reproduce noise structure. With n = 2 one can handle noisy data with good prospects to detect deep in the center, even if it is not obvious at first glance. Use of larger n is justified if we expect complex shape (more than one deep) of the lateral profile while scattering of the data is sufficiently low and permits revealing of details. After Abel inversion is done for all available values of λ index, i.e. for all lateral profiles F0(λ, y), one can calculate spectral line, Lr(λ), emitted by the plasma at the radius r relying on the emission function ε(λ, r). For given r = r0 spectral line is defined by the set of points Lr0 ðλÞ ¼ εðλ; r 0 Þ for all values of λ.

4. Considering the noise

i¼2

Coefficient a0 corresponds to the base line of the lateral profile, c is center of symmetry of the lateral profile while coefficients ai are magnitudes of the corresponding Gaussians. Coefficient α defines asymptotic behavior of the F(y) while (i − 1)β2, i N 1, defines the width of the Gaussian functions in the sum. Emission function ε(r), the inverse Abel transform of the Eq. (10), is of the same form " −α 2 r 2

εðr Þ ¼ b1 e

n X −ði−1Þβ 2 r2 1þ bi e

# ð11Þ

i¼2

where

α b1 ¼ a1 pffiffiffi π ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α 2 þ ði−1Þβ2 bi ¼ ai α2

iN1:

The lateral profile recorded on experiment is typically given as the data set { yj, Fj} of the length m, ( j = 1, 2,.., m). If we consider Eq. (10) as the model function of the lateral profile, its coefficients ai(i = 0, 1, 2,, n), α, β, c may be found by the ordinary least-squares fitting 4 We omit index λ in F as well as in coefficients {ai}, α, and β to make notation more compact.

Noise present in recorded profile of the spectral line translates in errors of the line parameters. Recording spectra in the image mode of the CCD camera provides data necessary to evaluate spatial distribution of the emitters, however a drawback is an inevitable increase of noise in respect to the spectra recorded in the full vertical binning (FVB) mode5. Another source of errors in spectral line parameters is an imperfect symmetry of the image. Sometimes authors apply smoothing of the data and image symmetrization [7] to minimize effects of the data scattering and imperfect symmetry. This method simplifies subsequent numerical procedure with expense that spectral line parameters could be affected by this approach, while corresponding errors are almost certainly underestimated. The function given by Eq. (10) is an even function — it is a symmetric representation of the whole lateral profile recorded in experiment. The best fit parameters of this function provide the ‘best agreement’ of the function and entire lateral profile in the sense of the least-squares method. In this way possible asymmetry in experimental data reflects in the best fit parameters as well as in their uncertainties. Thus, smoothing and symmetrization of the data are unnecessary. The final result is a realistic 5 Full vertical binning is a phrase related to the spectrum recorded by the imaging CCD camera, equivalent to the spectrum which would be recorded by a linear array CCD. This can be accomplished by appropriate camera hardware setting or, after the spectrum F(λ, y) is captured in the image mode, by the following transform: FVB = F(λ) = ∑ iF(λ, yi).

M. Burger et al. / Spectrochimica Acta Part B 101 (2014) 51–56

55

Fig. 7. Image corrected according to the Eq. (8). Blue line indicates a lateral profile, with magenta is shown the best fit profile, model function with three Gaussians is given by Eq. (10), while with yellow color we present Abel-inverted profile, Eq. (11). In the left panel the profile of Cu 240.01 nm spectral line is shown, after the inverse Abel transform is performed. Line profile, Lr0 ¼5;R¼5 ðλÞ corresponds to the integral emission from the ring with the inner radius of 0.26 mm (five pixels) and outer radius 0.52 mm (10 pixels). The excessively intense point (blue color) is most likely an artifact caused by noise. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

scattering in the Abel-inverted profiles, proportional to the noise and other imperfections in the rough data. There is a specific way to reduce noise after the Abel inversion is done. Instead of defining spectral line emitted from the plasma at the radius r0 as a set of values ε(λ, r0) for all λ, we introduce the following alternative definition Z

r 0 þΔr

ε ðλ; r Þdr r  Lr0 ;Δr ðλÞ ¼  0 π ðr0 þ Δr Þ2 −r 20

for all λ:

ð12Þ

The intensity of the line is proportional to the integral emission from the ring with inner radius r0 and outer radius r0 + Δr, normalized to the area of the ring. In this way changes in the emission function within the range r0, r0 + Δr are integrated resulting in smaller scattering of the data. Spectral line evaluated as Lr0 ¼0;R ðλÞ, where r0 + Δr ≡ R is sufficiently large, is proportional to the full vertical binning profile. If everything is correctly calculated the shape of both profiles and amount of data scattering will be almost identical. Sometimes corrected image contains excessive amount of noise which makes inverse Abel transform numerically very difficult. The

Intensity (rel. units)

Corrected Uncorrected

4

2

only thing we can do in this case is to skip Abel inversion and reduce corrected image to the full vertical binning spectrum. In this way we lose radial distribution of the intensities, but we still have FVB spectrum properly corrected for self-absorption. 5. Example Fig. 4 presents an image of the plasma plume, recorded in the zeroth order of the spectrograph. The left panel depicts a lateral profile at the position indicated on the image. A certain amount of asymmetry, noticeable in the lateral profile, reflects a shot per shot fluctuation in the form of the plasma plume. Imperfections in alignment of the optical system also may contribute to the asymmetry of the lateral profile. All images are recorded with 128 points in vertical direction as a reasonable choice between resolution and noise level. In Fig. 5 we present 240.01 nm (4s–4p transition) and 240.33 nm (4p–5s transition) Cu II spectral lines [12] recorded without back mirror, leftmost panel, and with the back mirror, the second panel. Image shown in the third panel is obtained after subtracting the first from the second image; it illustrates a net effect of the back mirror. The left panel in Fig. 6 shows absorption profile along the y-axis (vertical direction) evaluated at the position indicated on the spectral line image depicted in the right panel. One can see that absorption is almost constant with the value very close to zero in the whole interval. Relying on this fact we can proceed with self-absorption correction. Corrected image, obtained applying Eq. (8) is presented in Fig. 7, the right panel. According to our analysis it is safe to apply the inverse Abel transform to the lateral profiles in the corrected image. Spectral line in the left panel corresponds to the emission from the ring with the inner radius of 0.26 mm (five pixels) and outer radius of 0.52 mm (10 pixels). Table 1 FWHM and Height of the lines are evaluated by fitting the line shapes to the Lorentzian profile (all coefficients free). Cu II 240.01 nm

0 240.0

240.2

240.4

Wavelength (nm) Fig. 8. Spectrum of 240.01 nm and 240.33 nm Cu II spectral lines in FVB mode. Both lines are recorded in the image mode of the ICCD camera. With black we present uncorrected profiles, while corrected spectrum is plotted in light magenta. A certain amount of hardly observable self-absorption is corrected applying equation Eq. (8). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Uncorrected Corrected

FWHM (pm)

Height (rel. units)

18.55 ± 0.26 18.50 ± 0.50

4.306 ± 0.035 4.994 ± 0.078

Cu II 240.33 nm

Uncorrected Corrected

FWHM (pm)

Height (Rel. Units)

54.32 ± 0.7 51.72 ± 1.1

2.190 ± 0.016 2.559 ± 0.030

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M. Burger et al. / Spectrochimica Acta Part B 101 (2014) 51–56

The amount of the proposed correction is illustrated relying on the full vertical binning spectrum. In this way we avoid any influence of the inverse Abel transform, while intensities and profiles of both spectra are easy to compare. In Fig. 8 we present FVB spectrum (240.01 nm and 240.33 nm Cu II spectral lines) before and after correction. Tiny difference is caused by the self-absorption, in average about 0.02 and certainly less than 0.11. Table 1 outlines the values of the width and height for investigated lines after correction. One can notice that for Cu II 240.01 nm line self-absorption is negligible while for Cu II 240.33 nm width of the line is reduced for ∼5% due to correction. More pronounced self-absorption in 240.33 nm spectral line can be attributed to the significantly higher transition probability A240.33 = 1 × 108 1/s in respect to the A240.01 = 0.078 × 108 1/s [12]. 6. Conclusion

We suppose that a radially symmetric function, k(r), that drops to zero more quickly than 1/r is given6. The radial coordinate r relates to pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the x and y as follows r ¼ x2 þ y2 , thus the function k(r) depends on x and y in the symmetric way. We define two auxiliary functions 7 Z x urf ðx; yÞ ¼ kðrÞdx 0

Z

∞

urfcðx; yÞ ¼

kðr Þdx:

x ∞

It follows that integral ∫0 k(r)dx exists and its value C depends only on y, namely Z ∞ urf ð∞; yÞ ¼ kðr Þdx ¼ C ðyÞ: 0

We present a simple and numerically efficient method for selfabsorption correction of a spectrum recorded in the image mode of a CCD camera. The proposed method needs two spectra — the first one, F1, is recorded in a common way, and the second spectrum, F2, is recorded with the back mirror. Corrected spectrum, free of self-absorption, is given by the simple relation F 0 ¼ 1þ2 F . It is not necessary to symmetrize or to smooth spectra F1 and F2 before evaluating corrected spectrum F0. A noise issue, regarding subsequent inverse Abel transform, is discussed in details. 1

F2 − F1 G F1

In the same way Z urf ð−x; yÞ ¼

−x 0

Z kðrÞdx ¼ −

0 −x

kðr Þdx ¼ −urf ðx; yÞ

therefore, function urf(x) is an odd function. Having in mind definition of the urfc one can express this function via urf, i.e. urfcðx; yÞ ¼ C ðyÞ−urf ðx; yÞ: Obviously, function urfc is neither odd nor even, with the following characteristic values:

Acknowledgments This work is part of the “Determination of atomic parameters on the basis of spectral line profiles” (ON 171008) project supported by the Ministry of Education and Science of the Republic of Serbia.

urfcðx→∞; yÞ ¼ 0 urfcðx ¼ 0; yÞ ¼ C ðyÞ urfcðx→−∞; yÞ ¼ 2C ðyÞ:

Appendix A

References [1] A. Alonso-Medina, C. Colón, Measured stark widths of several Sn I and Sn II spectral lines in a laser-induced plasma, Astrophys. J. 672 (2008) 1286–1291. [2] M. Skočić, M. Burger, Z. Nikolić, S. Bukvić, S. Djeniže, Stark broadening in the laserinduced Cu I and Cu II spectra, J. Phys. B: At. Mol. Opt. Phys. 46 (2013) 185701, http://dx.doi.org/10.1088/0953-4075/46/18/185701. [3] R.N. Bracewell, The Fourier Transform and its Applications, McGraw-Hill, New York, 1978. [4] N. Konjević, Plasma broadening and shifting of non-hydrogenic spectral lines: present status and applications, Phys. Rep. 316 (1999) 339–401. [5] Heh-Young Moon, K.K. Herrera, N. Omenetto, B.W. Smith, J.D. Winefordner, On the usefulness of a duplicating mirror to evaluate self-absorption effects in laser induced breakdown spectroscopy, Spectrochim. Acta Part B 64 (2009) 702–713. [6] S. Bukvić, Dj. Spasojević, An alternative approach to spectrum base line estimation, Spectrochim. Acta Part B 60 (2005) 1308–1315. [7] M. Cvejić, M.R. Gavrilović, S. Jovićević, N. Konjević, Stark broadening of Mg I and Mg II spectral lines and Debye shielding effect in laser induced plasma, Spectrochim. Acta Part B 85 (2013) 20–33. [8] H. Nestor, H. Olsen, Numerical methods for reducing line and surface probe data, SIAM Rev. 2 (1960) 200–207. [9] J.A. Aguilera, C. Aragón, J. Bengoechea, Spatial characterization of laser-induced plasmas by deconvolution of spatially resolved spectra, Appl. Opt. 42 (2003) 5938–5946. [10] S. Djurović, Fitting and Abel inversion of experimental data using Jacobi polynomials, J. Res. Phys. 28 (1999) 155–164. [11] Lj.M. Ignjatović, A.A. Mihajlov, The realization of Abel's inversion in the case of discharge with undetermined radius, JQSRT 72 (2002) 677–689. [12] A. Kono, S. Hattori, Lifetimes and transition probabilities in Cu II, J. Opt. Soc. Am. 72 (1982) 601–605.

6

Within this section we omit index λ, i.e. k(r) ≡ k(λ, r). If k(r) is Gaussian, function urf becomes error function, erf, while urfc becomes complementary error function, erfc. Introduced properties of the urf and urfc are a subset of the well known properties of the error functions. 7

∞

Fig. 9. Flow of the functions urf(x, y) = ∫ x0k(r)dx and urfc(x, y) = ∫ x k(r)dx for arbitrary radially symmetric function k(r).