distribution of the intensity absorbed by the keyhole wall in laser processing

Journal of Materials Processing Technology 145 (2004) 303–310

Distribution of the intensity absorbed by thekeyhole wall in laser processing

C.Y. Hoa,∗, M.Y. Wenb,1

a Department of Mechanical Engineering, Hwa Hsia College of Technology and Commerce, Taipei 235, Taiwan, ROCb Graduate Institute of Mechatronics Engineering, Cheng Shiu Institute of Technology, Kaohsiung 833, Taiwan, ROC

Received 17 December 2002; accepted 28 July 2003

Abstract

The distribution of the energy flux absorbed by the wall is investigated after the laser beam undergoes multiple diffuse and specularreflections in a hemispherical cavity. The laser beam is characterized by different energy distribution parameter, wavelength, and polar-ization. Inverse Bremsstrahlung absorption within the plasma and Fresnel absorption on the cavity wall are taken into account. Scatteringwithin the plasma, radiation absorption due to surface temperature and alternation of polarization in the process of multiple reflectionsare assumed to be negligible. The analytic model is derived and compared with the Monte Carlo method. The influences of wavelength,polarization, inverse Bremsstrahlung absorption coefficient and material property on the absorbed intensity are discussed.© 2003 Published by Elsevier B.V.

Keywords: Multiple reflections; Fresnel absorption; Inverse Bremsstrahlung absorption

1. Introduction

A keyhole is produced in the workpiece by a high-power-density laser beam during drilling, welding and cutting. Thedistribution of the intensity absorbed by the keyhole wall isquite different from that of the incident flux[1,2] and de-pends on some factors such as the energy distribution pa-rameter, wavelength, and polarization of the laser beam andthe refractive index, absorptive index, and diffuse reflectiv-ity of material.

The cavity wall is not smooth, since surface waves appearon the liquid layer around the cavity[3]. For small angle ofincidence large ratio values of the surface roughness to thelaser wavelength lead to the limiting cases of diffuse reflec-tion but at intermediate and large angles of incidence thereflected intensity distribution exhibits its maximum valueat the specular angle approximately to the incident angleas the surface roughness-to-the laser wavelength ratio in-crease[4–6]. In some cases, the reflected angle is largerthan the specular angle[7]. Since reflectivities of many ma-terials have a pronounced peak in the direction of specularreflection, the reflectance can be approximated by a simple

∗ Corresponding author. Fax:+886-2-29478732.E-mail addresses: [email protected] (C.Y. Ho),[email protected] (M.Y. Wen).

1 Fax: +886-7-7337100.

summation of specular and diffuse components, as proposedby Eckert and Sparrow[8] and Seban[9]. This reflectivityconcept was better envisioned by Sarofim and Hottel[10].

The importance of polarization on laser cutting was firstnoted by Olson[11]. He demonstrated that the cuttingprofiles became worse when the cutting direction was notaligned with the principal polarization axis. Wallace andCopley [12] observed that material removal rates duringCO2 laser machining Si3N4 can be increased by aligning thepolarization direction with groove direction or focusing ator slightly above the surface. Nuss and Biermann[13] andLu et al. [14] observed the bending of the etched groovesand proposed it to be due to asymmetric light absorptionand reflections in the groove. Petring et al.[15] developeda three-dimensional model to predict energy absorbed byaccounting for laser beam parameters including power,mode, polarization, focal location and beam divergence. Anenergy balance analysis was provided to compare exper-imental results of the minimal laser power versus cuttingspeed by choosing an average absorptivity of the idealizedcutting front. Bang and Modest[16,17] provided an elabo-rate three-dimensional heat conduction model by includingmultiple specular reflections associated with polarization topredict the evaporative cutting process. The computed re-sults showed that the reflections of a polarized beam play animportant role not only in increasing the material removalrate but also in forming different final groove shapes. Cai

0924-0136/$ – see front matter © 2003 Published by Elsevier B.V.doi:10.1016/j.jmatprotec.2003.07.009

304 C.Y. Ho, M.Y. Wen / Journal of Materials Processing Technology 145 (2004) 303–310

Nomenclature

dA differential area, d̂A/4πr̂2odF differential view factorFij transfer factorI intensity of incident radiation,̂Ir̂2o/Q̂Io intensity first impinging on the

cavity wallk2 absorptive index of workpiecel path length between any two successive

reflectionsn1 refractive index of plasman2 refractive index of workpieceNb photo numbersNe number of subdivisionNi total number of energy bundles emitted

by area element dAiNij number of energy bundles absorbed by

area element dAi from dAjNr number of ring-elementPθ probability distribution function of bundle

emission for the angleθPϕ probability distribution function of bundle

emission for the angleϕPθr probability distribution function of bundle

reflection for the angleθrPϕr probability distribution function of bundle

reflection for the angleϕrqw intensity absorbed by the keyhole wallr radial coordinate,̂r/r̂or̂o dimensional hemispherical radius or

radius at the cavity openingW diffuse radiation

Greek lettersα absorptivityβ inverse Bremsstrahlung absorption

coefficient,β̂r̂oεi mean hemispherical emissivityθ cone angle measured from normal lineθr cone angle of reflection directionλ wavelength,λ/r̂oξ angle between horizon through center

and line from center to dAiρ reflectivityσ energy distribution parameter,σ̂/r̂oφ angle between incident bundle and line

from center to the impinging pointϕ circumferential angle onx–y planeϕr circumferential angle of reflection

directionψ angle between horizon through center

and line from center to dAj

Subscriptsi positionij positionj(dAi–dAj)t exchange factor between dAi and dAj(dAi–dAj)k k surface contacts between dAi and dAjn n surface contacts

Superscriptss, d diffuse and specular reflectionsd specular reflection in exchange factor∧ dimensional quantity

and Sheng[18] also predicted the evaporative and fusionlaser cutting processes by including not only polarizationbut also plasma absorption. The predicted and measuredgeometries of the cutting front showed agreement in cuttingpolymethyl-methacrylate and aluminum.

In contrast to cutting or etching, the effect of polariza-tion on laser drilling and welding have been a subject oflimited number of analytical investigations. Nolte et al.[19]observed that a use of a polarized laser with ultra shortpulses for drilling resulted in non-uniform intensity distri-butions and non-circular cavity shapes after a certain depth.A rotation of the direction of a linearly polarized lasercan significantly improve the circular geometry of producedmicro-holes. In welding with high-intensity laser beam un-der He gas shielding, Beyer et al.[20] originally foundthe effects of polarization on the depth of the fusion zonefor welding speeds less than 5 cm/s were insignificant. Itwas proposed to be due to plasma absorption by inverseBremsstrahlung[21]. However, a further increase in weld-ing speed beyond a threshold, penetration becomes greaterfor a parallel polarization than that for perpendicular polar-ization. The threshold was proposed to depend on the beampower and beam focus characteristics. Sato et al.[22] alsostudied the effects of polarization of a laser beam of TEM00mode on the welding of aluminum alloy in Ar and He gasshielding. Similarly, deeper penetration can be obtained withparallel polarization than perpendicular polarization.

In the present work Monte Carlo method and analyticalmethod will be used to investigate the energy absorption ina hemispherical cavity irradiated by a polarized laser beamwith the intensity of Gaussian or uniform distribution. Byconsidering Fresnel absorption, diffuse reflection, and ab-sorption within the plasma, the computed results will showthe effect of the polarization associated with multiple diffuseand specular reflections on energy absorption.

2. System model and assumptions

As illustrated inFig. 1, a collimated polarized laser beamirradiates into a hemispherical cavity. The incident flux canbe partially absorbed and the remainder diffusely and spec-

C.Y. Ho, M.Y. Wen / Journal of Materials Processing Technology 145 (2004) 303–310 305

jdA

idA

ξ

ψ

sρ

dρ

α

Gaussian distribution of incident beam

sd3l

Fig. 1. Physical model and schematic diagram of orientation angle.

ularly reflected by the keyhole wall. In order to simplify theanalysis, the primary assumptions made are the following:

(i) The cavity is stationary even though drilling or weldingis unsteady. The time scale for energy redistributioncan be estimated to be around 10−10 s, which is muchsmaller than 10−3 s for drilling a cavity, as measuredby Miyazaki [23].

(ii) The radiation is considered on the basis of geometricaloptics [24]. This is because the ratio between the cavityopening radius and wavelength of the laser beam isusually much greater than unity.

(iii) The reflectance is approximated by a simple summa-tion of specular and diffuse components [8,9]. How-ever, this representation is inadequate to describe thereflectivity of real surfaces under certain conditions[7].

(iv) Each component of polarization is not changed afterstriking the cavity wall for a clearer study. The changeof polarization after striking the cavity wall was ac-counted by Waluschka [25] and Bang and Modest [17].

(v) Beer–Lambert law describes the absorption within theplasma. Length of inverse Bremsstrahlung absorptionwere around 1 m for plasmas of aluminum and titaniumirradiated by pulsed Nd:YAG laser [26].

(vi) Radiation in the plasma is neglected. Shui et al. [27]calculated emission from a layer of vapor over alu-minum and results indicated that radiation from theplasma thus is only about 1% of the incident flux.

(vii) Scattering between the incident flux and ultra fine par-ticles resulting from either aggregation from the vaporor blown off the liquid layer on the cavity wall is ne-glected. Matsunawa and Ohnawa [26] and Matsunawaet al. [28] experimentally confirmed attenuation due toRayleigh scattering for a YAG or CO2 laser is less thaninverse Bremsstrahlung absorption. A typical value ofthe absorption length due to Mie scattering was around0.01 m, as estimated by Miyamoto et al. [29].

3. Dimensionless incident flux

The laser beam is assumed to have uniform or Gaussianintensity distributions. For Gaussian beam the intensity perunit area normal to the propagating direction is given by

I(ψ) = 3

πσ2exp

[−3

(cosψ

σ

)2]

(1)

where σ is the energy distribution parameter. Constant 3assures 95% of energy to be located within the radius σ [1].The power of uniform distribution is obtained by dividingtotal energy on the area normal to the beam.

4. Monte Carlo method

This method is to trace energy bundles and determinewhat fraction of the emitted energy has been absorbed andreflected by a surface or escaped from the cavity throughan opening. If Ni rays are imaged to leave dAi, all withenergy εi/Ni where εi is the mean hemispherical emissivityof surface dAi, the sum of energies of the rays absorbed bysurface dAj is the transfer factor Fij [30]:

Fij = εi limNi→∞

(Nij

Ni

)(2)

where Ni and Nij represent the total number of energy bun-dles emitted by surface dAi, and the number of energy bun-dles emitted by surface dAi, which are eventually absorbedat surface dAj , respectively.

The next step is to set up equations that express the proba-bility that an energy bundle will be emitted or reflected fromthe surface in a given direction. It is convenient to expressthe direction of bundle emission or reflection in terms ofthe cone angle θ and the polar angle ϕ of a spherical coor-dinate system centered at the emitting or reflected location.The probability function can be normalized so that it takeson values between zero and unity. For a diffuse surface, theprobability distribution functions of bundle emission and re-flection for the angles θ and ϕ yield, respectively:

Pϕ = ϕ

2π, Pθ = sin2 θ (3)

Pϕr = ϕ

2π, Pθr = sin2 θ (4)

Once the probability functions are established, the path ofthe energy bundle can be traced. The direction of departure(θ, ϕ) is found by drawing a pair of random numbers Pθ andPϕ from a uniformly distributed set between zero and one.

Suppose that the surface location on which the bundle isincident has an absorptivity α. In the present work α dependson the angle of impingement relative to the surface normaland the complex refractive index of workpiece material. Arandom number Pα is drawn from a uniformly distributed setbetween zero and one. If 0 ≤ Pα ≤ α, the incident bundle is


Fig. 2. A flow chart of Monte Carlo method.

absorbed; the absorption is tallied and attention is redirectedto the next energy bundle leaving the point of emission. Onthe other hand, if α ≤ Pα ≤ 1, the bundle is reflected.

A flow chart of the Monte Carlo method used to obtain en-ergy absorption is plotted in Fig. 2. The surface of the cavityis divided into Nr ring-elements. In view of the curvature of

the cavity, each ring-element is further divided into Ne sub-divisions. A random generator with a cycle 231 − 1 gener-ates photo numbers Nb. Convergence was tested by changingring-elements, subdivisions and photon numbers. ChoosingNr = 100–200, Nb = 1000–2000, and Ne = 20–30 a rela-tive error of intensity absorbed was found to be less than 1%.


5. Analytical method

Considering diffuse reflections, Fresnel and inverseBremsstrahlung absorption, the integral equation describingthe radiant exchange in the hemispherical cavity is

Wdj (ψ)= ρd

{Io(ψ) sinψ +

∫Ai

Wdj (ξ) dF(dAj−dAi)t

+∞∑n=1

[exp(−βlsk)ρsn]nIo(φn) cosφn

dAndAj

}(5)

The first term in the right hand side of Eq. (5) representsthe radiation directly arriving at dAj from outside the cav-ity through its opening. The second integral term gives thediffuse irradiation received by dAj from all the surface ofthe cavity. The last summation term is the energy receivedby the elements dAj which has undergone only specular re-flections on the hemispherical surface before reaching dAjand the exponential term stands for inverse Bremsstrahlungabsorption.

The incoming rays enter the keyhole full of plasma andthen reflect in it. While passing through the plasma, eachray can be considered to release energy to the plasma. TheBeer–Lambert law can describe the absorption within theplasma, thus the intensity of the beam initially impinging onthe keyhole wall is

Io(ψ) = I(ψ) exp(−β̂r̂o sinψ) (6)

where r̂o sinψ is the path length of the incident beamfrom the opening to the first point of impingement onthe keyhole wall. The inverse Bremsstrahlung absorptioncoefficient β̂ is, in general, a complicated function of thedensity of electrons in the keyhole, the degree of ionizationand the temperature [31]. In the experimental literaturekeyhole temperatures extending over the range of values9000–19 000 K have been reported [32]. These temperaturesare associated with a range of values of β̂ from 100 m−1 fortemperatures of order 20 000 K to 300 m−1 for temperatureof order 9000 K. The effect of the wavelength on inverseBremsstrahlung absorption coefficient was proposed [33].At the CO2 wavelength, β̂ is 100 times greater than atthe Nd:YAG wavelength. The value of β̂ for the Nd:YAGwavelength is about 3 m−1 at 9000 K.

All the angles of incidence in the process of multiplereflections are equal to the initial angle of emission in ahemispherical cavity with the aid of Fig. 1. For the rayarriving at dAj from dAi after making k reflections, the angleof incidence at any point of the wall collision follows as

φ(dAi−dAj)k = (k + 1)π − |ξ − ψ|2(k + 1)

(7)

The exchange factor, the fraction of radiation originatingdiffusely from dAi which reaches the area element dAj byall possible paths of successive specular surface reflectionswas found to be [34,35]

dF(dAi−dAj)t = dAi

∞∑K=0

(ρsdk )k[exp(−βlsd

k )]k+1

(k + 1)2(8)

where lsdk is the non-dimensional path length between the

successive surface collisions for the ray arriving at dAj fromdAi after k reflections and given by

lsdk = 2 cos[φ(dAi−dAj)k ] (9)

Neglecting the factor of off-specular peaks [36], the specularreflectivity ρsd

k is taken in the following form:

ρsdk = g [n1 cosφ(dAi−dAj)k − p]2 + q2

[n1 cosφ(dAi−dAj)k + p]2 + q2

for s-polarization (10)

ρsdk = g

{[n1 cosφ(dAi−dAj)k − p]2 + q2


}

×{

[p− n1 sin φ(dAi−dAj)k tan φ(dAi−dAj)k ]2 + q2

[p+ n1 sin φ(dAi−dAj)k tan φ(dAi−dAj)k ]2 + q2

}

for p-polarization (11)

ρsdk = 0.5g

1+[p− n1 sin φ(dAi−dAj)k tan φ(dAi−dAj)k ]

2

+q2

[p+ n1 sin φ(dAi−dAj)k tan φ(dAi−dAj)k ]2

+q2

×{

[n1 cosφ(dAi−dAj)k − p]2 + q2


}

for c-polarization (12)

where g is constant and

p2 = 0.5

{√[n2

2 − k22 − (n1 sin φ(dAi−dAj)k )

2]2 + 4n22k

22

+ [n22 − k2

2 − (n1 sin φ(dAi−dAj)k )2]

}(13)

q2 = 0.5

{√[n2

2 − k22 − (n1 sin φ(dAi−dAj)k )

2]2 + 4n22k

22

− [n22 − k2

2 − (n1 sin φ(dAi−dAj)k )2]

}(14)

As illustrated in Fig. 3, for the collimated incoming raysreaching dAj after making n specular surface reflectionsthe angle of incidence at each contact point and thenon-dimensional path length between the successive specu-lar surface collisions yield, respectively:

φn = nπ − (ψ − π/2)2n+ 1

(15)

lsn = 2 cosφn (16)


jdA

1dA

2dA1φ

2φ ψ

s2l

s1l

Fig. 3. Tracks of specularly reflected beam from a light source.

The specular reflectivity ρsn at every point of impingement

for the collimated incoming rays reaching dAj after makingn surface contacts is

ρsn = g(n1 cosφn − p)2 + q2

(n1 cosφn + p)2 + q2for s-polarization (17)

ρsn = g

{(p− n1 sin φn tan φn)2 + q2

(p+ n1 sin φn tan φn)2 + q2

}

×{(n1 cosφn − p)2 + q2

(n1 cosφn + p)2 + q2

}for p-polarization (18)

ρsn = 0.5g

{1 + (p− n1 sin φn tan φn)2 + q2

(p+ n1 sin φn tan φn)2 + q2

}

×{(n1 cosφn − p)2 + q2

(n1 cosφn + p)2 + q2

}for c-polarization (19)

where

p2 = 0.5

{√[n2

2 − k22 − (n1 sin φn)2]2 + 4n2

2k22

+ [n22 − k2

2 − (n1 sin φn)2]

}(20)

q2 = 0.5

{√[n2

2 − k22 − (n1 sin φn)2]2 + 4n2

2k22

− [n22 − k2

2 − (n1 sin φn)2]

}(21)

6. Results and discussion

In this study the distribution of the intensity absorbedby the keyhole wall is governed by dimensionless parame-ters such as the energy distribution parameter, wavelength,and polarization of a laser beam and the complex refrac-tive index and diffuse reflectivity of workpiece material.

0 .0 0 .5 1 .0r

0 .0

0 .8

1 .6

q

Incident Gaussian powerAnalytical solutionMonte Carlo method

w

Fig. 4. A comparison of radial distributions for dimensionless absorbedenergy per unit area of cavity wall predicted by Monte Carlo methodwith analytical model.

The specular reflectivity is determined by Fresnel’s rela-tion as a function of the incident angle and complex refrac-tive index. Typical parameters chosen for the cavity openingradius, energy distribution parameter, wavelength, inverseBremsstrahlung absorption coefficient, polarization, com-plex refractive index, workpiece material and diffuse re-flectivity are 600 �m, 600 �m, 1.06 �m and 3 m−1, circularpolarization, 3.26 + 4.34i, iron and 0.2, respectively.

The predicted dimensionless energy absorbed per unitarea of the cavity wall versus radius by using the MonteCarlo method is compared with the analytical solution, asindicated in Fig. 4. It can be seen that agreement is verygood. Incident flux of Gaussian distribution is also sketchedin this figure. Evidently, the absorbed intensity is quite dif-ferent from the incident flux, which is very high at the cavitybase and rapidly decreases with the distance from the sym-metry axis. This is attributed to the high ratio of areas fromwhich the light is reflected towards the center to the centralarea and the fact that all incoming rays get there both di-rectly and by multiple specular reflections. Fig. 5 indicatesthat this effect is more pronounced for uniform irradiationbecause the intensity reflected from the wings towards thecentral area is higher.

The intensity absorbed reduces with increasing the wave-length of the laser beam, as illustrated in Fig. 6. This is ex-plained by the fact that the laser beam with the shorter wave-length possesses the higher energy and absorptivity. Inten-sities absorbed by cavity walls of iron and aluminum com-monly used in welding and drilling are plotted in Fig. 7. Highreflectivity of aluminum reduces energy absorption. The ef-fects of different polarizations on the intensity absorbed bythe cavity wall are presented in Fig. 8. It is found that theabsorbed intensity is the highest for a p-polarization but thelowest for s-polarization.

The absorbed intensity for different diffuse reflectivitiesis shown in Fig. 9. An increase of diffuse reflectivity resultsin a decrease of the energy absorption except for the vicinity


0 .0 0 .5 1 .0r

0 .0

1 .5

3 .0

4 .5

q

Uniform Gaussian

w

Fig. 5. Radial distributions of dimensionless absorbed energy per unitarea of cavity wall for different intensity distributions of laser beams.

0 .0 0 .5 1 .0r

0 .0

0 .8

1 .6

q w

= 10.6 m= 1.06 m

Fig. 6. Radial distributions of dimensionless absorbed energy per unitarea of cavity wall for different wavelengths of laser beams.

0.0 0.5 1.0r

0.0

0.8

1.6

qw

n = 1.24, k = 10.36 (Al)2 2n = 3.26, k = 4.34 (Fe)2 2

Fig. 7. Radial distributions of dimensionless absorbed energy per unitarea of cavity wall for Fe and Al.

0.0 0.5 1.0r

0.0

0.8

1.6

q

p - polarizationc - polarizations - polarization

w

Fe

Al

Fig. 8. Radial distributions of dimensionless absorbed energy per unitarea of cavity wall for different polarizations of laser beams.

0 .0 0 .5 1 .0r

0 .0

1 .0

2 .0

q w

d= 0.3

d= 0.1

Fig. 9. Radial distributions of dimensionless absorbed energy per unitarea of cavity wall for different diffuse reflectivities.

0 .0 0 .5 1 .0r

0 .0

0 .8

1 .6

q

=0.0018=0.18

w

Fig. 10. Radial distributions of dimensionless absorbed energy per unit areaof cavity wall for different inverse Bremsstrahlung absorption coefficients.


0 .0 0 .5 1 .0r

0.0

0.5

1.0

1.5

2.0

q

=1=2

w

Fig. 11. Radial distributions of dimensionless absorbed energy per unitarea of cavity wall for different energy distribution parameters of laserbeams.

of the opening where the energy absorption is dominatedby diffuse reflection. The absorbed intensity accounting fordifferent inverse Bremsstrahlung absorption coefficient isrevealed by Fig. 10. The intensity absorbed by the cavitywall is lower for the case of β = 0.18 than β = 0.0018 sincethe laser beam releases much more energy to the plasmafor β = 0.18. With reference to Fig. 11 the large energydistribution parameter makes the incident flux tend to be auniform distribution so that the intensity profile for the largeenergy distribution parameter is similar to that for uniformtop-hat distribution.

7. Conclusions

The intensity absorbed in a hemispherical cavity pro-duced during laser materials processing is investigated.The predicted result from the analytical model agrees wellwith the solution obtained by using a Monte Carlo method.The results show that the absorbed intensity is remark-ably increased at the bottom of a hemispherical cavity andquite different from the incident flux. The incident beamwith a shorter wavelength enhances the absorbed intensity.Iron absorbs much more energy than aluminum, whichleads to more efficient materials processing for iron. En-ergy absorption is the highest for p-polarization but theleast for s-polarization. Moreover, the absorption withinthe plasma reduces the intensity absorbed by the keyholewall.

Acknowledgements

Support for this work by National Science Council NSC90-2212-E-146-001 is gratefully acknowledged.

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distribution of the intensity absorbed by the keyhole wall in laser processing

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