Experimental and theoretical investigationof a self-absorbed spectral line emitted

from laser-induced plasmas

Jalloul Ben Ahmed1 ,* and John Cowpe2,3

1Laboratoire de Spectroscopie Atomique, Moléculaire et Applications, Département de Physique,Faculté des Sciences de Tunis, 2092 Tunis, Tunisie

2Institute for Materials Research, University of Salford, Salford M5 4WT, UK3j.s.cowpe@salford.ac.uk

*Corresponding author: jbenahmed@yahoo.fr

Received 4 January 2010; revised 4 May 2010; accepted 21 May 2010;posted 24 May 2010 (Doc. ID 121825); published 18 June 2010

Using well-known expressions describing radiative transfer, we have established an expression predict-ing the spectral profile of a self-absorbed Caþþ 393:4nm emission line as emitted by a transientlaser-induced plasma. In this approach, the plasma was approximated as comprising five distinct layers,each of thickness 0:5mm, and each characterized by a unique uniform electron density, electron tempera-ture, and optical depth. The validity of the theoretical model was confirmed by successful comparisonwith experimental data. Inhomogeneous laser-induced plasmas were produced on the surface of anaqueous CaCl2 (0:01mol=l) solution using a frequency-doubled Nd:YAG laser. Optical emissionspectra were collected in such a way as to allow for temporal and spatial diagnostics of the plasmaplumes. © 2010 Optical Society of AmericaOCIS codes: 020.3690, 140.3440, 300.2140, 300.6365.

1. Introduction

Optical emission spectroscopy (OES) has been em-ployed successfully to characterize plasma sourcesacross a diverse range of disciplines; correlation be-tween certain plasma properties and the profile ofthe emitted optical spectrum forms the basis of thistechnique. Generally, a spectrograph or monochro-mator is used to disperse the optical emission of theplasma, which is then subsequently captured using aCCD array, photomultiplier tube, etc., and analyzed.The dispersed emission spectrum can yield informa-tion regarding the elemental composition of the plas-ma, the fractional ionization of the plasma, electronand ionic species temperatures, and electron density.The main advantage of OES plasma analysis is thatthe measurements are performed remotely; the plas-

ma is not disturbed during themeasurement process.The major disadvantage of OES plasma analysis isthat the profiles of the emission spectra may be cor-rupted due to physical processes occurring within theplasma.

The spectral profile of emission lines is the resultof several complex physical phenomena. For a givenplasma, the peak intensities are a function of tem-perature. For most laser-induced plasmas, the line-width broadening is dominated by the Stark effectand is proportional to the charged particle density;Doppler and resonance broadening effects are gener-ally considered negligible in typical laser-inducedplasmas. To draw the correct conclusions from ex-perimentally obtained spectroscopic data, it is neces-sary to take into account all the complex processesoccurring in the plasma. One process that is wellknown to alter the spectrum profile is emission lineself-absorption; currently, there is a concerted effortfrom many research groups to study emission line

0003-6935/10/183607-06$15.00/0© 2010 Optical Society of America

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self-absorption and its effect on the resulting emis-sion spectra [1–7].

A plasma may be described as optically thin if theradiation emitted from the plasma is without signif-icant absorption or scattering. If a plasma is opticallythick then self-absorption, and in extreme casesreversal, of emission lines can occur; this self-absorption leads to a lower perceived emission lineintensity, which will corrupt spectroscopic plasma di-agnostics. Self-absorption and reversal of emissionlines is generally attributed to temperature gradi-ents across the plasma volume. In general, the outerlayers of a laser-induced plasma are comprised ofcooler species than those radiating in the core. Underthis circumstance, the cooler outer layers absorbthe radiation emitted from the core of the plasma,leading to an observed drop in line intensity. Self-absorption is generally encountered in emission lineswhere the lower energy level of the transition is theground state, or close to the ground state. As energytransitions are element specific, emitting photons ofa corresponding specific wavelength, a given specieshas the highest probability of reabsorbing a photonfrom that same species.

When considering emission from ionic species,with the degree of plasma ionization exceedingapproximately 3%, the Doppler line broadening me-chanism is negligible compared to the Stark-effect-induced broadening of the line profile [8,9], and thereversal of spectral lines is no longer attributed pri-marily to temperature gradients across the volume ofthe plasma, but to electron density inhomogeneities.The Stark broadening of linewidths is correlated to aredshift of the line center wavelength [10], leading toasymmetrical self-absorption profiles [11].

In this paper, we propose a theoretical treatment ofthe self-absorption of an ionic Caþþ 393:4nm emis-sion line as emitted by a transient laser-inducedplasma and compare the findings to experimentaldata.

2. Theoretical Background

It is known that light may be emitted through twophenomena: spontaneous emission and stimulatedemission. If IωðrÞ is taken to represent the line inten-sity at a radial position r, where r ¼ 0 defines theplasma central axis normal to the sample surface,then the variation of IωðrÞ may be described by

Iωðrþ drÞ − IωðrÞ ¼ ðlight emitted-light absorbedÞ:

For the case of spontaneous emission, the variation ofline intensity may be represented by

dIωðrÞdr

¼�ℏω4π AkiNk

�PðωÞ; ð1Þ

where Aki, Nk, and PðωÞ are, respectively, the prob-ability of transition, the population of the level k,and the emission line profile corresponding to that

particular energy level transition. The emission coef-ficient ∈ may be defined as ∈¼ ðℏω=4πÞAkiNk.

The variation of line intensity due to stimulatedemission is given by

dIωðrÞdr

¼ IωðrÞ�ℏω4π BkiNk

�PðωÞ; ð2Þ

where Bki is the Einstein coefficient of stimulatedemission. Similarly, the variation in line intensitydue to absorption phenomena may be described by

dIωðrÞdr

¼ IωðrÞ�ℏω4π BikNi

�PðωÞ; ð3Þ

where Bik represents the Einstein coefficient of ab-sorption i → k. Combination of Eqs. (1)–(3) yields

dIωðrÞdr

¼ ℏω4π ðAkiNk þ IωðrÞðBkiNk − BikNiÞÞPðωÞ;

ð4Þ

dIωðrÞdr

¼∈ω −KωIωðrÞ; ð5Þ

where ∈ω¼∈ PðωÞ and Kω is the absorption coeffi-cient, defined here as

Kω ¼ ℏω4π BikNi

�1 −

giNk

gkNi

�PðωÞ;

with gi and gk representing the statistical weights ofthe energy levels i and k, respectively.

To resolve Eq. (5), we must multiply all terms byexpðKωrÞ:

dIωðrÞdr

expðKωrÞ ¼∈ω expðKωrÞ − KωIωðrÞ expðKωrÞ;ð6Þ

ddr

ðIωðrÞ expðKωrÞÞ ¼∈ω expðKωrÞ: ð7Þ

The total optical emission intensity Iω of a homoge-neous plasma slice expanded between r ¼ li and r ¼li þΔr may be determined:

ZliþΔr

li

�ddr

ðIωðrÞ expðKωrÞÞ�dr

¼Z

liþΔr

li

∈ω expðKωrÞdr; ð8Þ

Iωðli þΔrÞ ¼ IωðliÞ expð−KωΔrÞ þ εωKω

× ð1 − expð−KωΔrÞÞ; ð9Þ

3608 APPLIED OPTICS / Vol. 49, No. 18 / 20 June 2010

Iωðli þΔrÞ ¼ IωðliÞ expð−τωðΔrÞÞ

þ IeωτωðΔrÞ ð1 − expð−τωðΔrÞÞÞ; ð10Þ

where τωðΔrÞ ¼ KωΔr represents the optical depth.IωðliÞ represents the intensity of the light radiated

by an emitting species at r ¼ li; the intensity is di-minished as it traverses the volume of the plasma ex-panded between li and li þΔr due to absorptioneffects. The intensity of this emission is attenuatedaccording to IωðliÞ expð−τωðΔrÞÞ at the edge, wherer ¼ lþΔr (see Fig. 1). Taking into account the auto-absorption phenomenon, the intensity of the lightemitted by the plasma expanded between li and li þΔr is given by

IeωτωðΔrÞ ð1 − expð−τωðΔrÞÞÞ;

if this phenomenon is negligible, then the observedintensity will be Ieω ¼∈ω Δr.

3. Experimental Results

Laser-induced plasmas were induced on the surfaceof a 0:01mol=L CaCl2 solution; this weak concentra-tion of CaCl2 was chosen to minimize the effects ofemission line self-absorption. The variation of the io-nic Caþþ 393:4nm emission line profile PðωÞ with ra-dial position l was characterized. The laser-inducedplasmas were generated using a frequency-doubledNd:YAG laser producing 10ns duration pulses ofwavelength 532nm and energy 75mJ at a fre-quency of 2Hz. Optical emission from the plasmaswas collected at right angles to the direction of laserbeam propagation, dispersed using a 60 cm focallength monochromator equipped with a grating1200 line=mm (its instrumental broadening ≈ 1:7Å)and detected with a photomultiplier tube with a risetime of 2ns. The experimental apparatus and meth-odology are presented in greater detail elsewhere[12–16]. In the aforementioned works, it was demon-strated that plasmas induced by high-power ns dura-

tion laser pulses exhibit pronounced spatiotemporalinhomogeneities.

Figure 2 shows IðrÞ of the Caþþ 393:4nm emissionline measured at a delay of 500ns after instigation ofthe laser pulse; we note that the line intensity variessignificantly with distance from the plasma centralaxis. At r ¼ 2:5mm, the line intensity has droppedto 24% of that recorded at the central axis of the plas-ma, r ¼ 0. This observation leads us to conclude thatthe visible radiation is emitted primarily from thenarrow central region of the plasma. The emissionline intensity is much weaker toward the outerboundaries of the plasma volume; therefore, in thetheoretical treatment presented in this paper, wehave assumed that the visible emission from theplasma volume, where r > 2:5mm, may be consid-ered negligible.

Figure 3 illustrates the variation of the Caþþ392:4nm emission line profile with radial distancer. We note that a Lorentzian function can be em-ployed to simulate the emission line profiles. TheLorentzian fit used here is I0PðωÞ, where PðωÞ isthe normalized line profile:

PðωÞ ¼ 1

1þ 4�ω−ω0Δω

�2 ;

where ω0,Δω, and I0 represent the emission line cen-ter wavelength, the full width at half-maximum, andthe peak intensity at the line center wavelength,respectively.

Using these experimental data, and those pre-sented in previous work [14], we may approxi-mate and consider the plasma as composed of fiveconcentric distinct uniform regions, designated 1through 5, where layer 1 is the outermost layer, andlayer 5 is the closest to the plasma central axis. Eachlayer is of 0:5mm thickness, and each layer is char-acterized by a distinct uniform electron density anduniform electron temperature. These five definedplasma regions shall be characterized by the corre-sponding optical depths τ1, τ2, τ3, τ4, and τ5.

Fig. 1. (Color online) Schematic of radiative transfer in plasma. Fig. 2. Variation of line intensity with space.

20 June 2010 / Vol. 49, No. 18 / APPLIED OPTICS 3609

For a plasma considered as exhibiting local ther-mal equilibrium, the line profiles display identicalforms for both emission and absorption phenomena.The line intensity I1 emitted from plasma region 1 isgiven by I1 ¼ I10P1ðωÞ, and the corresponding opticaldepth ∈1 is given by τ1 ¼ τ10P1ðωÞ; here I10 representsthe intensity at the emission line center wavelengthemitted by region 1, and τ10 is the optical depth at theline center. As an approximation in our treatment,we will suppose the following relationship:

τi0τj0

¼ Ii0Ij0

: ð11Þ

4. Investigation of the Self-Absorbed Caþþ 393:4nmEmission Line

Plasma volume is represented in Fig. 4, and plasmaemission was collected from the right to the left in thediagram. We define the emission intensity at theplasma “edge” (2:5mm) as negligible and, hence,equal to zero. From Eq. (10), the intensity emittedfrom region 1, which is designated by Iðr ¼ 2mmÞ,is equal to ðI1=τ1Þð1 − expðτ1ÞÞ.

The optical emission from region 1 must success-fully traverse regions 2, 3, 4, 5, 6, 7, 8, and 9 to beobserved. Considering Fig. 4, and assuming thatthe plasma is symmetrical about the axis of laser pro-

pagation, regions 4 and 6 are identical; similarly,regions 3 and 7, 2 and 8, and 1 and 9 are identical.

When the light emitted from region 1 traversesregion 2, it will become attenuated and its in-tensity will be diminished according to ðI1=τ1Þð1 − expð−τ1ÞÞ × expð−τ2Þ. When the optical emissionsubsequently traverses region 3, it will be further at-tenuated, and the intensity is now given by ðI1=τ1Þð1 − expð−τ1ÞÞ × expð−τ2Þ × expð−τ3Þ.

It follows that, for the emission from region 1traversing the entire breadth of the plasma, the ob-served emission intensity will be given by

Fig. 3. (Color online) Variation of line profile with space: the fit by Lorentzian functionPðωÞ ¼ 1½1þ 4ðω − ω0=ΔωÞ2� givesΔω ¼ 3, 2.8, 2.5,and 2Å for, respectively, r ¼ 0, 0.5, 1, and 2mm.

Fig. 4. (Color online) Simplest approach of plasma.

3610 APPLIED OPTICS / Vol. 49, No. 18 / 20 June 2010

I1τ1

ð1 − expð−τ1ÞÞYi>1

expð−τiÞ

→

I1τ1

ð1 − expð−τ1ÞÞ expXi>1

ð−τiÞ:

Taking into account the light emitted from all regionsof plasma, the total observed intensity I is given by

I ¼Xi

�Iiτið1 − expð−τiÞÞ exp

�Xj>i

ð−τjÞ��

; ð12Þ

where i and j represent the region numbersconsidered.

Equation (12) was used to simulate the profile ofthe self-reversed Caþþ 393:4nm emission line;Fig. 5 compares the experimental spectrum of thisline and the fit obtained with Eq. (12). From Fig. 5,it can be seen that there is a good agreement betweenthe observed emission line profile and that generatedby our theoretical treatment of the plasma emission.The simulated line profile does not take into accountthe Stark-effect-induced shift of the emission linecenter wavelength, leading to the discrepancy in themodeling of the asymmetrical line profile.

5. Summary and Conclusion

Using well-known expressions describing radiativetransfer, we have successfully established an ex-pression predicting the spectral profile of the self-absorbed Caþþ 393:4nm emission line as emitted bya transient laser-induced plasma. By considering thelaser-induced plasma as comprised of distinct con-centric layers, it was possible to model the attenua-tion of optical emission by complex self-absorptionprocesses occurring within the plasma. By combiningexpressions describing the optical emission intensityof each plasma layer, and expressions governing theemission attenuation of each layer, a model describ-ing the observed emission profile for the entire plas-ma volume was developed. It was shown that the

theoretical model predicted a good fit to experimen-tally observed data for the spectral profile of theCaþþ 392:4nm emission line as emitted by a plasmainduced by a laser on the surface of an aqueous CaCl2(0:01mol=l) solution. The problems due to emissionline self-absorption can corrupt the accuracy of opti-cal plasma diagnostics; indeed, this subject has beenthe focus of a great many researchers attempting toimprove the accuracy of optical emission plasmaspectroscopy techniques. The work presented hereprovides a valid starting point from which to expandand develop the model more fully, enabling a morethorough understanding of the complex processes oc-curring within laser-induced plasmas and the devel-opment of more accurate spectroscopic techniques.

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Fig. 5. (Color online) Experimental spectrum and fit by expres-sion 12 (continuous line) of resonance line 393:4nm of CaII.

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