Onedimensional model of discharge pumped excimer lasersU. Krause and J. Kleinschmidt Citation: Journal of Applied Physics 72, 1237 (1992); doi: 10.1063/1.351729 View online: http://dx.doi.org/10.1063/1.351729 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/72/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in One-dimensional bubble model of pulsed discharge in water J. Appl. Phys. 102, 063302 (2007); 10.1063/1.2783848 Differential pumping scheme for discharge pumped excimer lasers Rev. Sci. Instrum. 77, 045105 (2006); 10.1063/1.2188335 One-dimensional model of valveless pumping in a closed loop and a numerical solution Phys. Fluids 18, 017106 (2006); 10.1063/1.2165780 A onedimensional model of dc glow discharges J. Appl. Phys. 71, 5783 (1992); 10.1063/1.350472 Onedimensional diffusion model for extended solid solution in laser cladding J. Appl. Phys. 61, 2645 (1987); 10.1063/1.337895

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One-dimensional model of discharge pumped excimer lasers U. Krause -- Jenoptik GmbH, Gtischwitzer Strasse 33, 690.5 Jena, Germany

J. Kleinschmidt Friedrich-Schiller-UniversitCt Jena, Physikalisch-Astronomische FakultiEt, Institut ftir Optik und Quantenelektronik, ‘Max- Wien-Platz I, 6900 Jena, Germany

(Received 23 October 1991; accepted for publication 5 May 1992)

For many.excimer laser applications for example, in medicine and semiconductor processing, the long-term stability of the output pulse energy and the laser-beam profile are very important. The spatial intensity distribution of an excimer laser with a Fabry-Perot cavity depends on many discharge conditions (e.g., the loading voltage, the total gas pressure, the concentrations of the gas components, the repetition rate, the electrode profiles, and the spatial distribution of the preionization intensity). A one-dimensional model of a discharge pumped excimer laser is described. It can be shown that the laser output power of a KrF laser depends on the spatial variations of the discharge parameters to a higher degree than does a XeCl laser. The double structure of the transverse intensity distribution for long-term KrF laser operation can be explained as a consequence of the depletion of Fz concentration within the gas mixture after many discharges. ’

I. INTRODUCTION

Excimer lasers are used as intense UV radiation sources in many industrial and scientific applications. For some applications,, for example, in semiconductor process- ing and medicine, long-term stability of the output pulse energy and the laser-beam uniformity are very important. It is known that the spatial intensity distribution of an excimer laser with a Fabry-Perot cavity depends on many discharge conditions, e.g., load voltage, total gas pressure, gas mixture, repetition rate, electrode profiles, and preion- ization. The experiments of Ref. 1 show that the near- and far-field beam patterns have a double structure when the laser is operated for several lo5 shots, although this effect did not appear just after a new gas exchange. The authors explain this phenomena by the dissociation attachment of the electrons by I-IF created in the center of the discharge region alItter many laser pulses. This is a very special aspect in our opinion.

The influences of the discharge conditions on the spa- tial beam profile cannot be explained by the so-called zero- dimensional models of discharge pumped lasers. These models include only the time dependencies of all the ex- cited species and the time dependence of the-energy distri- bution of the electrons within the discharge.2-5 In the present paper we present a one-dimensional discharge model for an excimer laser discharge.

An analogous model for transverse avalanche electric discharge has been described by Kushner et al. 6 This model has been applied to investigate geometrical effects within the discharge of a HgBr laser.

In the present paper we discuss the effects of spatial variations of different discharge parameters ( preionization intensity, concentrations of the gas components, electrode separation) on the output and the discharge properties of a XeCl and ‘a KrF laser. Furthermore we examine the vari-

ations of the laser intensity distribution by halogen deple- tion.

A possible explanation for the measured double struc- ture of the beam profile’ during long-term operation (after 10’ pulses) as a consequence of the depletion of F2 concentration will be given

II. MODEL

The one-dimensional laser model presented in this pa- per is an extension of the zero-dimensional model given in Ref. 5. The calculations5 are based on an analytical approx- imate solution of the Boltzmann equation. The correctness and the limits of the model were proved by comparison with the results of the full numerical Boltzmann code cal- culations.7 The excimer laser model’ was proved further- more by means of comprehensive measurements of the dis- charge current, the discharge voltage, the laser output energies, the laser pulse form, the time delay between the trigger point of time of the thyratron and the maximum of the laser pulse, the fluorescence of transitions between states, and the excited-state absorptions on a ns time scale.7’8 Detailed information about the measurements and the experimental results will be given in another publica- tion.

Figures 1 and 2 show some calculated time histories in comparison with the corresponding measured results for a XeCl laser and a KrF laser. To avoid spatial inhomogene- ities, the investigations were carried out after a new gas fill, at sufficient high-load voltage and with low repetition rates ( < 10 Hz).

The one-dimensional laser model presented here is based on the following assumptions:

(i) The variations of the discharge parameters along the x axis (see Fig. 3) are much stronger than the varia- tions of these parameters along the other axis. The follow- ing discharge parameters vary along the x axis: the dis-

1237 J. Appl. Phys. 72 (4), 15 August 1992 0021-8979/92/l 61237-07$04.00 @ 1992 American Institute of Physics 1237 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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no 10 5 !O 3 IO 0 .I0 -20 -30 .40

FIG. 1. Comparison of the experimental data (solid line) with the cal- culated results (dashed line) for the excimer laser EMG 1003i MSC (f&=23 kV, pa=60 mbar, pHa= 4 mbar, pm=2436 mbar). 12: t ime history of the discharge current; U,. . the voltage over the peaking capac- itor; Nr,& time history of the output power (a.u.); Xe I: the relative f luorescence intensity (a.u.) of the transition Xe I (a=467.1 nm).

tance between the discharge electrodes; the electrical-field strength with the consequence that there is x variation of the mean electron energy, the rates of the dissociation at- tachment, the rates of the ionization, and so on; the partial pressures of the gas components and the total pressure especially at high repetition rates due to shock waves. The variations of the discharge parameters per length until along the z axis (see Fig. 3) will be neglected in compar- ison to the x variations of these parameters. This is a rather good approximation.

FIG. 3. Modeled discharge geometry. Each discharge region i is charac- terized by the following parameters. Ai: discharge area; di: electrode sep- aration; PI; preionization intensity; Nb, Nx, Nh; number density of the buffer gas, the rare gas, and the halogen. S is the source of the preioniza- tion radiation.

(ii) In high-pressure transverse gas discharges increas- ing values of the parameters E/p and pd (E is the applied electrical-field strength, p the total pressure, and d the elec- trode separation) lead to the development of an inhomo- geneous discharge mode. The discharge is observed to break up into individual spark channels (so-called stream- ers). The streamers are formed in times of about 50-100 ns after the beginning of the discharge. Up to these times the laser process is over or is terminated by the streamer dis- charge. The streamer mechanism is not included in our model. This means that we discuss the discharge and the laser process for times shorter than the streamer forming time.

electrical field, see Fig. 3). For times shorter than the streamer forming time (50-100 ns) the cathode fall causes only a reduction of the effective distances between the elec- trodes.

(iv) The discharge along the x axis is divided in dis- charge regions i (i=1,2,... ). The discharge within each region is assumed to be homogenous. Each discharge re- gion is characterized by the electrode separation (ii, the discharge area A, the number densities of the buffer gas Nbi, the rare gas iVX, the halogens Nhi, and the intensity of the preionization PI? The preionization intensity is as- sumed to be proportional to the electrical power of the preionization sparks,

PIi= Q,JTR,, (iii) The cathode fall influences spatial variations of

the plasma parameters along the y axis (parallel to the where R, is the t ime-dependent resistance of the sparks5 and I1 is the charge-transfer current (see Fig. 4). The

, , , , I a . 7-.,

z a0 5 60

LO , ei

-40

/A, f? x-

-_I -60 I

220 240 260 280 300 320 t bl

FIG. 2. Comparison of the experimental data (solid line) with the cal- culated results (dashed line) for the excimer laser EMG 203 MSC ( Uo ~24 kV, pKr= 120 mbar, pFZ = 4 mbar,pu,=2376 mbar). 12: t ime history of the discharge current; U,: the voltage over the peaking capacitor; Nph: time history of the output power (a.u.); Kr I: the relative f luorescence intensity (a.u.) of the spectral line Kr I (/2=431.96 run)

--- S

/

-

i+l

/

‘UO

FIG. 4. Electrical circuit modeled in the calculation. Uo: loading voltage; R,: resistance of the thyratron; R,: t ime-dependent resistance of the preionization sparks; R,,: t ime-dependent discharge resistance of the dis- charge region i; CT,=70 nF; Cz=40 nF; L,= 100 nH; I,,=5 nH, &=50 H; I,: charge-transfer current; 12: discharge current (1z= 21,).

1238 J. Appl. Phys., Vol. 72, No. 4, 15 August 1992 U. Krause and J. Kleinschmidt 1238

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proportional factor Qi characterizes the x dependence of the preionization intensity.

(v) Diffusion of the heavy particles along the x axis will be neglected in the time scale of interest ( < 100 ns), that means none of the heavy particles enter the neighbor- ing discharge regions. If the width of the smallest discharge region is greater than the characteristic lateral diffusion length ( ~~0.1 mm), the same assumption is true for the electrons. Since electron diffusion is ignored, each dis- charge region i is in effect a tiny separate discharge itself. This assumption is not correct if the electron density shows considerable spatial variation. This can occur in x-ray pre- ionized excimer lasers, where the electrons are produced by preionization within a sharp limited region.’

(v) Radiation transport between different discharge regions i will be neglected.

All the transport processes [neglected in assumptions (v) and (vi)] smooth out some of the differences between neighboring regions with regard to the electron density, the photon density, and the densities of excited species. The calculation given here must be seen as a marginal case.

We now consider how all the discharge elements i cou- ple together to form the total discharge. Each discharge element i is characterized by the time-dependent discharge resistance. Rdip

Rdi= d/A#,N,, e,

where bi is the mobility of electrons and N,, is the number density of electrons.

The discharge resistance Rd of the total discharge is given by

The discharge elements i couple together only via the ex- ternal electrical circuit. The electrical circuit modeled in the calculation is a.charge-transfer circuit (see Fig. 4). For calculation of the mobility and the number density of the electrons see Ref. 5. It must be noted, that the discharge voltage is equal for all different discharge regions by virtue of the transverse conducting electrode and given by I,R, where I2 is the total current flowing through the discharge region.

Ill. RESULTS AND DISCUSSION

A. The influence of spatial inhomogeneities on the discharge and laser properties

First we will discuss the influence of the spatial inho- mogeneities on the discharge and laser properties. To this end we divided the discharge in two discharge regions, i=1,2. The standard parameters P, used for the calcula- tions are tabulated in Table I. If these parameters are used for both discharge regions, we obtain the so-called homo- geneous case. For the so-called inhomogeneous case we use the standard parameters (Table I) only for the discharge region 1; the appropriate parameters for the region 2 are varied. The following discussions are based on the relative deviations ( P2- PI )/PI between the discharge parameters

TABLE I. Parameters of the discharge regions used for the calcuIations.

Pl A (cm*) d (cm)

Q (cme3 W-‘) Nb (cme3) Nx (cmw3) Nh (cmM3)

XeCl KrF

23.5 23.5

;$I ;;?a

6x 1Ol9 6~10’~ 2x 10’8 3x10’8

1.2x 10” 1 x IO”

v*-P,)/P, (%I

0 +0.5

+50; +500 +0.5 $0.5 -5.0

of region 1 (PI> and region 2 (P2) tabulated in the fourth column of the Table I. The chosen values of the parameter variations seem to be typical for the spatial variations of the discharge parameters within an excimer laser plasma.

As an example Fig. 5 shows the calculated time histo- ries of the electron densities Nei, the photon number den- sities Nphi> as well as the discharge currents IZi (i- 1,2) when the electrode separations of the two regions are dif- ferent, d,= 1.005dt.

For discussion of the influence of the inhomogeneous discharge on the relevant laser properties, we calculated the ratios

where

E,Ij(hom.,inhom.) = s

m I$& dt, 0

and ELi(hom.,inhom.) are the energy inserted into the plasma and the output pulse energy, respectively for the case of a homogeneous (horn. ) and an inhomogeneous (in- horn.) discharge. The separate investigation of the energy inserted into the plasma and the laser pulse energy gives information about the effects of plasma inhomogeneities on the impedance matching on one side and on the laser ki- netics on the other side more clearly.

The laser output energy EL1 is given by

ELi= hv 5 In

L L-- I 8 3 I b

40 60 80 100 120 140 160 180

thd

FIG. 5. Calculated time histories of the discharge currents I,, the num- ber densities of the electrons Nei, and the number densities of the photons within the cavity N&i= 1,2), where the two discharge regions are dif- ferent with respect to the electrode separation (d,-d,)/d,= +0.5%. U, is the calculated voltage across the peaking capacitors.

1239 J. Appl. Phys., Vol. 72, No. 4, 15 August 1992 U. Krause and J. Kleinschmidt 1239 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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2.5

2.0

.

a.l Q Q

RLi

d Nb Nh Nx, 0

Q Q d

(a)

Nb Nh Nx

1.5 - FZZi RI1 0 Rlz

Rli

@!Rplr lI.Il Rplz

Q Q d Nb Nh Nx

1.5 t

m Rlr 0 Rl2

Q Q d Nb Nh Nx

Ib)

FIG. 6. R,,, and R,, for different inhomogeneities. The two discharge regions are different with respect to one of the following parameters: the preionization intensity characterized by the factor Q, the electrode separation d, and the number densities N,,, IV,, N, of the buffer gas, the halogens and the rare gas. The relative deviations of the parameters of regions 1 and 2 (Pa-P,)/P, are tabulated in the fourth column of Table I. (a) XeCl laser; (b) KrF laser.

where hv is the photon energy, 1 is the length of the reso- nator, R is the reflectivity of the output coupling mirror, and Vi is the, discharge volume of the region i.

The box heights in Fig. 6 indicate the values of the ratios R,,[ and RLi for a XeCl and a KrF laser if one of the parameters Q, d, NL, NH, Nx of region 2 is different from its value in region 1. The relative parameter deviations (PZ--Pl)/Pl based on the calculation are tabulated in the fourth column of Table I. It can be seen that the energy inserted into the plasma of a KrF laser discharge especially depends more sensitively on the variation of the electrode separation, the buffer gas pressure, and the gas mix than that of a XeCl discharge. We examine the effect of the inhomogeneous preionization for two different deviations of the preionization intensities between the two discharge regions (Qz-Qt)/Qt= +50% and +SOO%. These strong variations of the preionization intensities can be quite re- alistic, because the UV radiation of the preionization sparks can be strongly absorbed by impurities within the gas or can be shadowed by opaque constructive elements within the discharge chamber. It can be seen from Fig. 6 that for the very inhomogeneous preionization [ ( Q2- Q1 ) / Q,= +500%] no laser emission will be obtained from re- gion 1 for both the XeCl and KrF lasers. The results of Fig. 6 show furthermore that variations of the discharge

1240 J. Appl. Phys., Vol. 72, No. 4, 15 August 1992 U. Krause and J. Kleinschmidt 1240

conditions in region 2. influence the- discharge in’ region 1 and vice versa.

We have calculated the analogous relations for the la- ser peak power, the maximum electron density, and the maximum density of the excited rare-gas-atoms, too. But, it must be noted, that the laser pulse energy ELI is nearly proportional to the laser peak power, and the electron den- sity as well as the density of the excited rare-gas atoms depends on the discharge parameters similar to Epli, so no new information can be obtained.

The ratios Rpu and RLi characterize the plasma ener- gies and the laser output energies of the two discharge regions separately. Now we will discuss the-discharge laser system as a whole. The discharge consists of two regions, i= 1,2. The total energy inserted into the plasma Epl and the output pulse energy EL of this laser system are given by

Ep~=Epll +&z and EL=ELI+ELZ,

respectively, where E,,i and ELI are the corresponding en- ergy values of the particular regions. E,t(hom.) and EL (horn. ) mean the plasma energy and laser pulse energy of the laser system as a whole, when the two regions are characterized by the same plasma parameters P, (see Table I). I E,t(inhom.)- and E,(inhom.) mean these energies when region 1 is characterized by the parameter set PI and

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. TABLE II. Variations of the output pulse energy EL and the plasma energy Epl caused by the spatial variations of the individual parameters of a XeCi laser discharge.

XeCl i&=20 kV Uc=20 kV uc=25 kV t&=25 lcv

Parameter A&,I& (%I A&/EL (%)- AE~,/E~, (%) AEJ& (%)

Q +0.001 -0.23 -0.016 -0.28 d -0.15 -9.8 -0.16 -9.9 Nb -0.15 -4.0 +0.16 - 3.0 Nh -0.04 -0.6 -0.07 -0.67 NX +0.1 -0.7 -0.16 -0.36

region 2 is characterized by a parameter set P2 ( Q&,Nrn,NHz,Nx2) where one of these parameters is dif- ferent from the corresponding parameter of region 1. The question is, now, how sensitively this laser system reacts when the excimer laser plasma goes from a homogeneous discharge (the same plasma parameters PI for the two re-’ gions 1 and 2) to an inhomogeneous discharge. To this end we calculate the ratios

AEpl E,t(inhom.) -E,t(hom.) -= EPl E,l(hom.)

and

AEL EL(inhom.) -EL(hom.) -= EL EL(hom.) ’

The results for the XeCl and the KrF lasers for load volt- ages of 20 and 25 kV are tabulated in Tables II and III for a fixed relative parameter variation of (P2- P,)/P, = 1%. The results of Tables II and III cannot be generalized. The values relate to the concrete laser device assumed here (electrical circuit from Fig. 4 and discharge parameter set P, >. But, it must be remarked, that the geometrical dimen- sions and the parameters PI are typical for many commer- cial laser devices. The signs of the quantities in Tables II and III show that inhomogeneities lead to decreasing laser energy and decreasing power deposition. Only in a few cases, especially in the KrF laser at a load voltage of 20 kV, are the quantities AEr, positive. This means that the laser system was not optimized with respect to the determined parameters for the case of a homogeneous discharge at the present load voltage. It must be noted that the discharge impedance depends on the power deposition and varies with the load voltage. The quantities in-Tables II and III

TABLE III. Variations of the output pulse energy EL and the plasma energy EPl caused by the spatial variations of the individual parameters of a KrF laser discharge.

KrF Uc=20 kV t&=20 kV Uc=25 kV lJ,= 25 kV

Parameter AEpv~pl (%I AEJEL (%b) AEpl/Epl (%) AEJEL {%I

Q =l- 0.03 -0.18 -0.01 -0.42 d $4.0 -29.3 -1.8 -54.8 Nb f2.2 - 15.4 -0.6 -5.7 Nh +0.6 -3.3 -0.24 -5.7 NX -to.2 -1.3 -t 0.28 -2.4

give information about the sensitivity of the laser output on the spatial variations of the different discharge parameters, as follows.

The output energy of both the XeCl and the KrF lasers are most sensitive to spatial variations of the electrode sep- aration d (caused, for example, by burndown).

The dependence of the XeCl and the KrF lasers on the relative spatial variations of the preionization intensity are weak comparatively; but, it must be noted, that the abso- lute spatial variations of the preionization intensity in prac- tice is about 10-100 times greater than the variations of the other discharge parameters.

Spatial variations of the plasma parameters influence the laser output energy much stronger than the energy inserted into the plasma. This is an indication that radia- tionless deactivation processes (especially caused by elec- tron collisions) are very sensitive to the plasma parame- ters.

The laser output energy of a KrF laser depends on the spatial variations of the discharge parameters to a higher degree than that of a XeCl laser. The main reason for this seems to be the differences of the electron attachment pro- cesses within the two gas mixtures. The electron attach- ment coefficient of F, is about 10 times larger than the attachment of HCl. The electron attachment in the KrF laser gas is a one-step process,

F2+e-F+F-;

The electron attachment of HCl molecules in the vibra- tional ground state is very unlikely. The main contribution to the electron loss is given by the dissociative attachment of the HCl (U = 2) molecules Electron loss takes place via a two-step process,

HCl+e+HCl(u>O)+e,

That means that in a XeCl laser discharge the electron loss increases with increasing electron density. This is not the case within a KrF laser plasma.. Because of this negative feedback the XeCl laser is less sensitive to local variations of the discharge parameters. Furthermore, the main quenching process of the upper laser state is electron col- lision quenching and so the stability of the spatial electron distribution leads to a greater stability of the laser-beam intensity distribution in the XeCl laser.

6. Calculation of the transverse beam profile

The present discharge model allows the calculation of the transverse beam profile in dependence on electrode form and separation, on gas pressure and mixture, and on the preionization intensity distribution.

The calculation is based, for example, on the following laser design and discharge conditions.

The discharge chamber contains two identical elec- trodes with Chang profile k=0.2.9 The minimum electrode separation is 20 mm. The total discharge region is divided

1241 J. Appl. Phys., Vol. .72, No. 4, 15 August 1992 U. Krause and J. Kleikchmidt 1241 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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400

- -2 300 ” 1; E

200 aI E 5

= 100

-10 -8 -6 -4 -2 0 2 4 6 8 IO x (mm1

FIG. 7. Calculated transverse intensity profiles for the XeCl laser and the KrF laser with homogeneous preionization intensity (solid lines) and x-dependent preionization intensity [dashed lines, the preionization inten- sity increases linearly with 1x1, Qi=Q(i-2)/5 for 07 and Q,=Q(ll --i)/5 for i<6; Q is given in Table I].

in 12 discharge regions (i= 1,...,12). The difference of the electrode separation in neighboring discharge regions is 0.1 mm.

The electrical circuit is shown in Fig. 4. The load volt- age is Ua=25 kV.

The values for the number densities of the gas compo- nents and the proportional factor Q characterizing the preionization intensity are given in Table I (Pi).

The results are shown in Fig. 7. The fluence distribu- tion of a XeCl laser is generally wider than the transverse beam profile of a KrF laser for the same electrode coniig- uration. This is in agreement with the experimental find- ings. The steep slope of the transverse fluence profile of a KrF laser at 1 x I=4-6 mm shows the very sensitive depen- dence of the KrF laser output energy on the spatial varia- tion of the electric-field strength (the electriofield strength varies from x=0 to x=6 mm by about 1% ) . The solid lines give the fluence distribution of the KrF and the XeCl lasers when the preionization intensity is uniform (Qj=Q for i= l,..., 12). The dashed lines give the fluence profiles of these lasers for the case of x-dependent preionization in- tensity. This is caused, for instance, by the absorption of the preionization radiation within the excimer laser gas. We assumed that the preionization intensity decreases from the outside to the inside of the discharge regions linearly,

Qj=Q(i--210 for i>7

and

Qj=Q( 11 -i)/5, for i<6.

That means, the preionization intensity decreases from the outer regions to the inner ones by a factor of 2.

C. The variations of the transverse b6am profile by the effect of the depletion of F2 concentration

The experiments of Ref. 1 show that the near- and far-field patterns have a structure with two peaks and a minimum in the middle of the discharge region when the laser operated for several lo5 shots, although this effect did not appear just after a new gas fill. The distance between the peaks is about 8 mm measured at the position of the output coupler. Furthermore, when the number of the dis- charges increased to 7~ lo’, the laser output power (and the output pulse energy also) decreased to half of the be- ginning level, while the F, concentration reduced from 0.24% to 0.135%.

The authors of Ref. 1 explain this phenomena by the dissociation attachment of vibrational excited HF mole- cules [HF(v> l)] created in the center of the discharge region. We think that the analysis of Ref. 1 is not correct: (i) The authors of Ref. 1 calculated the x dependence of the electron-density distribution and not the laser intensity profile, but there exists no sample connection between the electron density in the discharge and the laser output power; (ii) under the assumption that the dominant pro- cess for the explanation of this effect is the dissociative attachment by the molecules HF( u > 1) and the concen- tration of these excited molecules is proportional to the electron density (this was the assumption of Ref. 1 ), no minimum of the electron density in the center of the dis- charge can be calculated in our opinion.

The analysis of the problem needs the knowledge of the exact electrode profile. The authors of Ref. 1 assumed elec- trodes with cylindrical surfaces (radius of 25 mm). This assumption does not seem to be meaningful, because the electric-field strength falls in this case over 4 mm from the center by more than 2.5%. That means, the occurrence of a discharge for 1 x I>4 mm is very unlikely.

For want of knowledge of the exact discharge geome- try and the laser design of Ref. 1 we choose electrodes with Chang profile (k=0.2) and an electrical circuit shown in Fig. 4. The load voltage is U0=25 kV. The preionization intensity will be assumed to be homogeneously in agree- ment with the assumption of Ref. 1, from which we ob- tained the partial pressures of gas components used in our calculation. For want of the knowledge of the exact exper- imental discharge conditions in Ref. 1 we can only com- pare our calculated relative values of the output energy for different F, concentrations with the experimental results given in Ref. 1 (Fig. 8).

Figure 9 shows the calculated fluence profiles for dif- ferent F2 concentrations. The calculated distance between the peaks is about 6.5 mm. It must be noted that this distance will be determined mainly by the electrode profile.

The present calculation explains the double structure of the beam profile as a consequence of the depletion of F2 concentration without the assumption of electron attach- ment by HF. This effect can also play a role. In our opin- ion, the dominant effect is the quenching of the upper laser state by the electrons. The number density of electrons is higher in the center of the discharge due to the higher

1242 J. Appl. Phys., Vol. 72, No. 4, 15 August 1992 U. Krause and J. Kleinschmidt 1242

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I \o 0.4 ’

0.24 0.22 0.20 038 0.16 0.14

F, concentration ( % 1

FIG. 8. Laser output energy of a KrF laser dependence on the F2 con- centration, normalized to the energy for a Fz concentration of 0.24%. --X--: new gas filling (Ref. 1); --+--: long-term operation (Ref. 1); --0--: present work.

electric-field strength. This erect will be pronounced with decreasing F2 concentration.

The present calculation shows that the depletion of F, concentration influenced the transverse beam profile. But, we think that for understanding of the observed long-term behavior of the excimer laser design, the discharge condi- tions and the plasma chemistry must be considered in de- tail. A very special assumption of the calculations is the uniformity of the preionization, for example. This is not correct for the long-term operation of the excimer laser. Many impurities are generated within the discharge and it is known that some of these impurities absorb the preion- ization radiation very effectively. Since the preionization spark gaps adjust outside of the discharge region (see Fig. 3), the preionization intensity in the center of the dis- charge will be much lower than in discharge regions near the sparks. The results of Fig. 6 show that in the regions with comparatively small preionization intensity the laser intensity will be very weak.

IV. CONCLUSION

A one-dimensional model for transverse electric dis- charges in excimer laser gases has been described. The model has been applied to study the influence of discharge inhomogeneities caused by preionization, electrode separa- tion, and form, by buffer gas concentration and laser gas mixture variations at various power deposition levels on the laser output energy and the laser beam uniformity.

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1243 J. Appl. Phys., Vol. 72, No. 4, 15 August 1992

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FIG. 9. Transverse intensity profile of a KrF laser dependence on the F, concentration. pJ=O.24% (solid line), [FzJ=0.2% (dashed line), and [F2] =O. 14% (dashed-dotted line).

ergy inserted into the plasma. This is an indication that the radiationless deactivation processes (especially caused by electron collisions) are very dependent on the plasma pa- rameters.

Spatial variation of the discharge parameters influences the laser output energy (and the energy inserted into the plasma, also) of a KrF laser much more strongly than those of a XeCl laser. The main reasons for this are the differences of the electron attachment processes within the two gas mixtures and the important role of the radiation- less deactivation of the upper laser state by electron colli- sions.

It has been demonstrated in agreement with the exper- imental findings that the depletion of F2 in the KrF laser leads to a change in the laser-beam intensity distribution. It must be remarked that the present model allows one to optimize electrode profile and preionization distribution for a laser output with maximum beam uniformity in con- sideration of discharge instabilities caused by halogen de- pletion or pressure inhomogeneities in high-repetition-rate laser systems.

U. Krause and J. Kleinschmidt 1243 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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