Sha TaoAdvanced Optowave Corporation,

Ronkonkoma, NY 11779

Benxin Wu1

Department of Mechanical, Materials,

and Aerospace Engineering,

Illinois Institute of Technology,

10 W. 32nd Street,

Engineering 1 Building, Room 207 A,

Chicago, IL 60616

e-mail: bwu11@iit.edu

Yun ZhouElectro Scientific Industries, Inc.,

Fremont, CA 94538

Gary J. ChengSchool of Industrial Engineering,

Purdue University,

West Lafayette, IN 47906

The Investigation of PlasmaProduced by Intense NanosecondLaser Ablation in Vacuum UnderExternal Magnetic Field Usinga Two-Stage ModelIn this paper a two-stage physics-based model has been applied to study the evolution ofplasma produced by high-intensity nanosecond laser ablation in vacuum under externalmagnetic field. In the early stage (Stage I), the laser-induced plasma generation and itsshort-term evolution are described through one-dimensional (1D) hydrodynamic equa-tions. An equation of state (EOS) that can cover the density and temperature range in thewhole physical domain has been applied to supplement the hydrodynamic equations. Inthe later stage (Stage II), the plasma long-term evolution is simulated by solving 2D gasdynamic equations. The two-stage model can predict the spatial distributions and tempo-ral evolutions of plasma temperature, density, velocity, and other parameters. The modelis used to study and discuss the effects of external magnetic field on the plasma evolution.It provides a useful tool for related fundamental studies and practical applications.[DOI: 10.1115/1.4025685]

Keywords: laser ablation, laser-induced plasma

1 Introduction

Plasma generation and evolution during laser-material interac-tions may be a very important physical process in many laser-based applications, such as pulsed laser deposition (PLD), laserinduced breakdown spectrometry (LIBS), and laser ablation[1–4]. Due to the existing free electrons, external magnetic fieldmay have a significant effect on laser-induced plasma evolution,which has been experimentally studied in literatures (e.g., in Refs.[5–7]). The experimental study has provided very useful scientificinformation to understand the effect of magnetic field on laser-produced plasma. However, it is also highly desirable to have aphysics-based model in this area, because the model can providea useful scientific tool to help better understand the relevantprocesses.

During nanosecond (ns) laser ablation at sufficiently high laserbeam intensities, the target material may be driven above the ther-modynamic critical temperature, where the dominant mechanismfor plasma generation is expected to be hydrodynamic expansion[8–10], instead of surface vaporization across a sharp liquid–vaporinterface. The plasma early-stage evolution may be approximatelyone-dimensional due to the small plasma expansion length abovethe target, yet its long-term evolution will become multidimen-sional. Therefore, for a physics-based model for the effect ofexternal magnetic field on plasma produced by intense ns laserablation, it is very desirable to consider both the hydrodynamicexpansion mechanism for plasma generation, and the plasmalong-term multidimensional evolution.

However, this kind of model has been rarely reported in litera-tures. For example, in Ref. [11], a model has been developed forlaser ablated carbon plume flow under magnetic field. However,the model does not simulate the initial plasma generation process.In Ref. [12], a model has been used to study the effect of magnetic

field on laser-induced plasma, and the authors of Ref. [12] men-tion in the paper that the laser and material settings in Ref. [12]are similar to those in Ref. [13]. In Ref. [12], the initial plasmageneration process is not simulated through physics-based model-ing. In Ref. [14], a one-dimensional (1D) model has been devel-oped to study the effect of magnetic field on plasma induced byintense ns laser ablation based on the hydrodynamic expansionmechanism. However, the 1D model cannot well describe thelong-term, multidimensional evolution of the plasma.

This paper will employ a physics-based model to study plasmageneration and evolution under an external magnetic field due tointense ns laser ablation of a metal target in vacuum. The modelhas considered both the involved hydrodynamic expansion mech-anism and the long-term, multidimensional expansion of theplasma. Due to the complexity of the physical processes and theinvolved numerical challenges, a two-stage model is employed,which is an approximate, but reasonable approach that shows rea-sonably good agreements with some of the experimental measure-ments taken from the literature as shown later. The model is usedto study how an external magnetic field affects the evolutions ofimportant plasma parameters, such as temperature, density, andvelocities. The study provides useful information and also a usefultool (the model) for future fundamental research work and practi-cal applications in related areas.

2 Model

Figure 1 shows the model schematic diagram. It is assumed thatthe target (aluminum) is in the region of z< 0 and its original sur-face is located at z¼ 0. The ambient vacuum is in the region ofz> 0. The ns laser pulse (which starts at t¼ 0 and ends at t¼ tp)

propagates along the �z direction. An external magnetic field (~B)is assumed to be applied as shown in Fig. 1, which exists in thedomain above the target.

The laser-induced plasma generation and evolution process canbe approximately divided into two stages [10,15,16]. Stage I

1Corresponding author.Manuscript received March 21, 2013; final manuscript received October 5, 2013;

published online November 7, 2013. Assoc. Editor: Yung Shin.

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(from t¼ 0 to the end of laser pulse, t¼ tp) is the laser-inducedplasma generation and its early-stage evolution. Due to the shortduration of ns laser pulse, the process in Stage I can be approxi-mately assumed to be one-dimensional (1D). Stage II (t> tp) isthe long-term plasma evolution, which is a multidimensionalprocess.

Therefore, an approximate, yet reasonable two-stage model isemployed. The hydrodynamic expansion process in Stage I isdescribed through 1D hydrodynamic equations supplemented by awide range EOS. Due to the short laser pulse duration, the effectof the external magnetic field is not very significant in Stage Iunder the studied conditions as implied by the results in Ref. [5].Figure 4 of Ref. [5] shows that during the early stage the plumefront locations with and without the external magnetic field areclose. For example, Fig. 4 of Ref. [5] shows that the plume frontlocation is at approximately around �0.7 mm at the time ofaround �10 ns both with and without the external magnetic field.Therefore, it is assumed that the magnetic field can be approxi-mately neglected in Stage I. In Stage II, the plasma long-termevolution in vacuum under the applied external magnetic field issimulated by solving 2D axisymmetric gas dynamic equations thatinclude the effect of the magnetic field. Stage I ends and Stage IIstarts at the end of the laser pulse. For example, if an 8-ns (fullwidth at half maximum) laser pulse is used in the simulation, thenthe laser pulse completes at approximately t¼�16 ns, at whichStage I ends and Stage II starts.

Based on the simulation result at the end of Stage I, at thebeginning of Stage II, the plasma total mass and energy are firstdetermined, based on which the plasma initial conditions (whichare assumed to be spatially uniform) in Stage II are determined.As shown in Fig. 1, the plasma plume resulted from the 1D calcu-lation in Stage I is in the shape of a cylinder (with the same radiusas the laser spot), which appears as a rectangle with two sharp cor-ners at its top in the 2D r-z plane. At the beginning of Stage II, itis assumed that the two sharp corners can be changed into twoquarter-circles (with the same radius as the laser spot), which hasbeen found to yield more reasonable long-term plasma profiles inthe simulation in Stage II. Although this two-stage model is an ap-proximate approach, its predictions agree reasonably well withsome of the experimental measurements taken from the literatureas shown later.

2.1 Stage I: Plasma Generation and Early-StageEvolution. Ns laser pulse at sufficiently high intensities can drivethe target material above the thermodynamic critical temperature,where the dominant mechanism for plasma generation is expectedto be hydrodynamic expansion [8–10]: driven by pressure gradi-ent, the material in the target condensed phase moves into theplasma plume above the target across a narrow transition layer ataround the target surface, where the material density decreasesquickly but continuously.

Following the same approach as the corresponding author andShin’s previous work in Ref. [10], the laser-induced plasma gener-ation and its early stage evolution in Stage I are described through

1D hydrodynamic equations [10,17,18] in the whole physicaldomain including both the target condensed phase and the plasmaplume above the target. The hydrodynamic equations are supple-mented by the quotidian equation of state model [19], which cancover the entire wide range of the involved material temperaturesand densities. The material thermal conductivity is obtainedthrough the Lee-More model [20,21], while the optical propertiesare obtained using the Drude model [22–24] combined with theLee-More model. The hydrodynamic equations are numericallysolved with a finite difference method [25–27]. Please seeRef. [10] for more details on the model in Stage I.

2.2 Stage II: Plasma Long-Term Evolution. At the end ofStage I, the conditions of laser-induced plasma can be predictedby the model in Stage I, which will be used to determine theplasma initial conditions (in the way discussed earlier) for themodel calculation in Stage II, where the plasma long-term evolu-tion in vacuum under the applied external magnetic field isdescribed by solving 2D gas dynamic equations in the plasmaplume region [11,12,14,25]

@ L!

@tþ 1

r

@M!

@rþ @ N

!

@z¼ R!

(1a)

L!¼

q

qVr

qVz

Etotal

2666664

3777775 M

!¼

rqVr

rqV2r

rqVrVz

rðEtotal þ PÞVr

2666664

3777775

N!¼

qVz

qVrVz

qV2z

ðEtotal þ PÞVz

2666664

3777775 R

!¼

0

� @P=@r

�@P=@zþ ð~J � ~BÞ � ~az

1

r~J�� ��2þð~J � ~BÞ � ~V

26666664

37777775

(1b)

where r and z are spatial coordinates, q and P are the material den-sity and pressure, respectively, Vr and Vz represents velocity com-ponents in r and z directions, respectively, t is time, Etotal is thevolumetric total energy, including both the internal energy andthe kinetic energy, ~az denotes the unit vector in z direction, r isthe plasma electrical conductivity, given by r ¼ 1=qr (where qr is

the electrical resistivity), ~V is the velocity vector, and ~J is theelectric current density induced by electromagnetic field and canbe calculated as [11]

~J ¼ rð�ruþ ~V � ~BÞ (2)

where u represents the electrostatic potential, and the term ru inEq. (2) is neglected in this study. Also, following Ref. [11], it isassumed that the magnetic field in plasma is equal to the exter-nally applied magnetic field.

The plasma electrical resistivity, qr , can be calculated using thefollowing equation [28]:

qr ¼1

cE

p3=2m1=2e Ze2c2 ln K

2 2kBTð Þ3=2(3)

where cE is a parameter whose value depends on the ionic chargeZ [28], me is the electron mass, e is the electron charge, c is thespeed of light in vacuum, kB is the Boltzmann constant, T is theplasma temperature, and K can be calculated by [29]

K ¼ 12pZe3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffie3

0k3BT3

ne

s(4)

Fig. 1 The schematic diagram of the two-stage model (sizesnot drawn to scale)

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where e0 is the permittivity of free space, and ne denotes the elec-tron number density, which is obtained through Saha equations[18] based on the aluminum ionization potentials [30,31]. Basedon the electron number density, the plasma EOS can also beobtained [18].

In Stage II, it is assumed that at the target surface, the adiabaticand no-slip boundary conditions can be applied. It should be noted

that the external magnetic field will induce the current density ~J

and hence the electromagnetic (EM) force of ~F ¼ ~J � ~B [11] inthe plasma. The force may confine the plasma expansion in zdirection and its expansion in any horizontal plane (a plane where

z¼ constant), as long as the plasma material velocity ~V is not par-

allel to ~B (see Eq. (2)). However, because the plasma overallexpansion velocity in r direction is often smaller than the overallvelocity in z direction [5,32], the model in this paper mainlyfocuses on the magnetic field confinement effect on the plasmaz-direction expansion, and the confinement effect in any horizon-tal plane is neglected in the model. In addition, in the studied timerange in Stage II the plasma velocity can go up to the order of�104 m/s, but the plasma overall mass density is low due to theexpansion. Therefore, the convection process is expected to bedominant over the thermal conduction process, and hence thelatter is neglected in the model in Stage II.

The finite-difference essentially non oscillatory scheme fromRef. [27] is applied to solve Eq. (1). Due to the very high expan-sion speed of laser-induced plasma in vacuum, a relatively largecomputational domain in r and z directions is needed. This resultsin a large computational cost, and therefore parallel computingwith message passing interface (MPI) [33] is performed on amulti-core computer for this study.

It should be noted that although this paper follows the sameapproach as the corresponding author and Shin’s previous work inRef. [10] for the model in Stage I, the model in Stage II is signifi-cantly different from that in Ref. [10]. In Ref. [10], the effect ofexternal magnetic field is not considered and the 2D gas dynamicequations (Eq. (1)) are not numerically solved in Stage II. Instead,in Ref. [10] a semi-analytical model has been applied to simulatethe process in Stage II. This has shown clear differences of thework in this paper from the previous work in Ref. [10].

3 Results and Discussions

The prediction of the model will be compared with some ofthe experimental measurements in Ref. [5] to test the model. InRef. [5], an 8-ns laser beam (wavelength: 1.06 lm and intensity:4 GW/cm2) is incident onto an aluminum target, and the generatedplasma expands in vacuum under a transverse magnetic field(B¼�0.64 T) [5]. An 8-ns laser pulse has also been used in thesimulation.

Figure 2(a) compares the model-predicted plasma front loca-tions with those measured in Ref. [5], with and without the exter-nal magnetic field. Both the experiments and the modelcalculations in the figure show that the external magnetic field hasa confinement effect on the plasma expansion. Figure 2(b) showsthe comparison of model-predicted electron number density at1 mm from an aluminum target surface, with the experimentallymeasured electron density at 1 mm from an aluminum target sur-face taken from Ref. [5]. It can be seen from Fig. 2 that the overallagreements between the model predictions and the experimentalmeasurements given in the figure are reasonably good consideringthat the involved physical process is very complicated.

Figure 3 shows the model-predicted spatial distributions ofplasma mass density with and without the external magnetic field.For better comparisons, the density scale bars are the same for thetwo plots. The plasma can expand without any confinement invacuum when there is no external magnetic field; however, whenthe external magnetic field is applied, the plasma expansion willbe confined due to the induced EM force by the magnetic field inthe plasma. As a result, when there is no external magnetic field,

the plasma has a much larger relative low-density region behindits expanding top front than that with the external magnetic field,which can be seen from Fig. 3. It can also be seen from Fig. 3 thatdue to the confinement effect of the EM force induced by the mag-netic field, the plasma density distribution is relatively more uni-form than that without the magnetic field.

Figure 4 shows the spatial distributions of model-predictedplasma temperature at different times, with and without the exter-nal magnetic field. When the magnetic field is absent, the plasmapeak temperature decreases quickly with time, and the percentageof the plasma relative high-temperature (red) region is alsosmaller than that when the external magnetic field exists (pleaseread the online electronic version of the paper for colorfulfigures). This is because when there is no magnetic field, theplasma expands freely without any confinement, during whichpart of the plasma internal energy converts into its kinetic energy.When the external magnetic field is applied, it can be seen that att¼ 40 ns a high temperature (red) region exists right behind the

Fig. 2 Comparisons of (a) model-predicted plasma front loca-tions with experimental measurements taken from Ref. [5],where the plasma is produced by ns laser ablation of an alumi-num target in vacuum with or without a transverse magneticfield (in Ref. [5], laser pulse duration: 8 ns, laser wavelength:1.06 lm, and intensity: 4 GW/cm2; magnetic field: B 5�0.64 T),and (b) model-predicted electron number density at 1 mm froman aluminum target surface, with the experimentally measuredelectron density at 1 mm from an aluminum target surface takenfrom Ref. [5] (see Ref. [5] for experimental details and the mea-surement data error bar information).

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plasma top front and the overall plasma temperature is muchhigher than that without the magnetic field. As time goes on, therelative high-temperature region becomes larger and by t¼ 70 nsmost of the plasma region becomes red (above 35,000 K). In addi-tion, it can be seen that when the external magnetic field is appliedthe plasma peak temperature drop from t¼ 40 ns to t¼ 70 ns ismuch smaller. This is because under the magnetic field the plasmaexpansion is strongly confined by the EM force, which will slowdown the plasma peak temperature drop.

Figure 5 shows the vector plots for plasma velocity distribu-tions at t¼ 90 ns, with and without the external magnetic field.The same length scale is used for the two vector plots, so that

the arrow length in the two plots can be compared to reveal therelative velocity magnitude. When there is no external magneticfield, the plasma expands freely in vacuum, and has much largeroverall velocity magnitudes than those under the magnetic field,particularly at locations near the plasma top front. In addition,when the magnetic field is applied, the spatial distribution ofplasma velocity magnitude becomes relatively more uniform thanthat in the absence of magnetic field, which should be due to theconfinement effect of the EM force.

High-intensity ns laser-induced plasma evolution involves verycomplicated physical processes, and the application of an externalmagnetic field makes the processes even more difficult to model.

Fig. 3 Model-predicted spatial distributions of plasma density at t 5 90 ns with and without thetransverse external magnetic field (the plasma is induced by ns laser ablation of an aluminumtarget in vacuum at 4 GW/cm2, and the density scale bars are the same for the two plots)

Fig. 4 Model-predicted spatial distributions of plasma temperature at different times with andwithout the transverse external magnetic field (the plasma is induced by ns laser ablation of analuminum target in vacuum at 4 GW/cm2)

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It should be kept in mind that the employed two-stage model isbased on simplified assumptions. Although the model calculationshows reasonably good agreements with the experimentalmeasurements given in Fig. 2, there still exists lots of good (butcertainly also very challenging) work to do in the future toimprove the model. For example, for laser ablation under certainlaser conditions phase explosion may occur at certain delay timeand spatial locations [34–39]. It may be a good work in the futureto also include the possible effect of the phase explosion process(if occurs) in the model. Another possible work in the future is toextend the model to 3D so that it can better describe the plasmalateral expansion and can also consider the magnetic field effecton the lateral expansion. Certainly, the involved numerical chal-lenge is expected to be huge. It is also among the possible futurework to consider in the model the possibly different behaviors ofelectrons, neutral particles, and/or different ionic species in theplasma and to consider the possible effect of plasma on the exter-nally applied magnetic field.

Although previous studies exist in the literature (e.g.,Refs. [5–7,40–45]) about the effects of magnetic field on laser-induced plasma, the employed model in this paper adds a usefultool for related scientific studies and practical applications.

4 Conclusion

The evolution of plasma produced by high-intensity ns laser abla-tion of a metal target in vacuum under an external magnetic fieldhas been studied through a two-stage physics-based model. In StageI (from t¼ 0 to the end of laser pulse), the laser-induced plasma gen-eration and its early-stage evolution are described through 1Dhydrodynamic equations coupled by a wide-range EOS. Then basedon the model-predicted plasma conditions at the end of Stage I, themodel in Stage II is applied to simulate the plasma long-term evolu-tion by solving 2D gas dynamic equations. The model-predictionsagree reasonably well with some of the experimental measurementstaken from the literature [5] as shown in Fig. 2.

Based on the model, the study has revealed some of the majoreffects of the magnetic field on plasma evolution under the inves-tigated conditions: (i) the magnetic field has confinement effect onthe plasma expansion due to its induced EM force, and the mate-rial velocity in the plasma is reduced; (ii) as a result, the conver-sion of the plasma internal energy into its kinetic energy isdecreased, and the plasma overall temperature is higher than thatwithout the magnetic field; and (iii) the magnetic field has madethe spatial distributions of plasma parameters (e.g., density, tem-perature, and velocity) relatively more uniform.

It is good work in the future to improve and further test themodel. For many laser-based technologies, such as laser ablation,

pulsed laser deposition, and laser-induced breakdown spectros-copy [1–4,46], laser-induced plasma evolution may be an impor-tant process and may be affected by an externally appliedmagnetic field. Therefore, the model may provide useful informa-tion for the relevant investigations about the above technologies ifan external magnetic field is also applied.

Acknowledgment

This material is based upon work supported by the NationalScience Foundation under Grant Nos. CMMI 0970079 and1000226. Any opinions, findings, and conclusions or recommen-dations expressed in this material are those of the author(s) and donot necessarily reflect the views of the National ScienceFoundation.

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061009-6 / Vol. 135, DECEMBER 2013 Transactions of the ASME

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