review on laser drilling i. fundamentals, modeling, and simulation

Review on laser drilling I. Fundamentals, modeling, and simulation

Wolfgang SchulzFraunhofer Institute for Laser Technology, ILT, Aachen, Germany and Nonlinear Dynamics of LaserProcessing, RWTH Aachen University, 52074 Germany

Urs EppeltNonlinear Dynamics of Laser Processing, RWTH Aachen University, 52074 Germany

Reinhart PopraweFraunhofer Institute for Laser Technology, ILT, Aachen, Germany and Nonlinear Dynamics of LaserProcessing, RWTH Aachen University, 52074 Germany

(Received 19 April 2012; accepted for publication 17 December 2012; published 23 January 2013)

High peak power lasers have been used for years for ablating matter. The most relevant application

of this process is laser marking. Marking meets the demands of applications although the quality of

ablation has potential to be further improved. However, the qualitative results of the ablation process

especially for highly efficient removal of matter in the liquid phase like drilling have not met the

standards of alternative processes—the latter is only the case in niches. On the other hand, the

ablation by ultrafast lasers in the pulse regime of ps or below, which might meet the quality demands

in terms of geometric precision, was too slow for economically feasible application because of the

lack of average power. In fact, both process domains have been developed substantially and thus lead

to a technological level which make them ready for industrial innovations. In a series of three articles

on laser drilling—from fundamentals to application technology—the results of more than a decade of

research and development are summarized with the purpose of displaying the bright application

future of this laser process. This present part I deals with fundamentals, modeling, and simulation of

laser drilling. Part II covers processing techniques, whereas part III is dedicated to systems and

application technology. Fundamentals, modeling, and simulation: Theoretical analysis of the process

from fs- to ls-pulses involves three inputs: numerical simulation, relevant analytic modeling, and as

an important input for understanding, process analysis. The reduction of the models guided by

experimental input leads to descriptions and knowledge of the process, which allows for strategic

improvement of the applicability. As a consequence, process strategies can be derived, meeting the

challenges of the application related to shape and accuracy of the surface free of recast as well as the

economical demand for high speed processing. The domains of “cold ablation,” “hot ablation,” and

“melt expulsion” are differentiated. Especially, the formation of recast up to closure of the drill

is quantified. Tailoring the process parameters toward the individual application according to the

know-how reached by the state of the art modeling and simulation leads to sound innovations and

shorter innovation cycles. VC 2013 Laser Institute of America.

Key words: laser drilling, modeling, simulation

I. INTRODUCTION

New generations of capable radiation sources have been

demonstrated and are increasingly available on an industrial

level. Diffraction-limited high beam quality fiber lasers, slab

lasers, disk lasers, and still rod lasers are available. In particu-

lar, high-energy and high-power applications like generation

of x-rays1 become feasible in an industrial environment. From

a scientific point of view, these achievements introduce new

opportunities to probe deeper into a wider variety of the phe-

nomena encountered in laser–matter interaction. The phenom-

ena are related to physical models describing propagation and

absorption of radiation, ionization, evaporation, and nonlinear

transport of mass, momentum, and energy. The variety of in-

triguing physical phenomena spans from recast formation well

known from the action of ls-pulses, via formation of cracks

typical for ns- to ps-pulse duration, toward homogeneous

expansion, phase explosion, and spallation characteristic for

ps- to fs-pulses. Technical achievements like lasers emitting

short pulses in the sub-ps range at averaged power levels of

400 W in 2009 (Ref. 2) is climbing up to 1 kW today. Related

experimental observations introduce the future need for simu-

lations to cope also with kinetic properties of beam–matter

interaction. In particular, beam aberrations instead of a free

running or multiple reflected beam pattern are encountered in

modeling independent of pulse duration. Beam aberrations are

not only introduced by the action of beam guiding and form-

ing optics but also by spatially distributed feedback from the

dynamical shape of the ablated material surface. Effects

changing the phase distribution of the incident laser radiation

are incorporated in the models for the first time: for example,

some temporal and spatial changes of the density in the gase-

ous phase. Temperatures approaching the critical state during

ablation with pulse durations in the range from a few ps to a

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few hundred ns raise the question whether equation of state

phenomena is contributing to the overall appearance in dril-

ling. In drilling, the dynamic phenomena governing the shape

of the drilled hole are identified experimentally and can be

related to the processing parameters theoretically.

The state of the art in short pulse high-power laser devel-

opment is summarized in Fig. 1. Diffraction-limited high beam

quality fiber lasers, slab lasers, disk lasers, and rod lasers are

available. This new generation of radiation sources has been

demonstrated and is increasingly available at an industrial

level. From a scientific point of view, these achievements are

exciting, since high-power, high-beam quality laser radiation

gives new opportunities to probe deeper into a wider variety of

the phenomena encountered in laser–matter interaction. To

approach the upper right part of Fig. 1 is the goal for future

high-precision high-rate ablation applications. Nevertheless,

many of the commercially available lasers are applied in niche

markets today and are about to expand into new areas.

The advent of promising technical applications3 lies today

in (a) ns and ls drilling of shaped holes by helical drilling,4 (b)

drilling of extreme aspect ratios in dielectrics/glass by ns-slab

lasers,5 (c) nm-size periodic structuring of polymers by inter-

ferometric approaches, (d) ablation by ns- and ps-pulses for

metal moulds, and (e) generation of waveguide structures in

glass by fs-pulses.6 On the laboratory scale, a next generation

of diffraction-limited short pulse lasers is on the horizon.7 In

particular, ps-lasers at multihundred watts of average power

with repetition rates of several MHz, fs-lasers at 400 W

(Ref. 5) average power, and green frequency-doubled lasers at

200 W are under construction. At the short pulse end, attosec-

ond lasers have been demonstrated and will themselves open a

new domain in the interaction of light and matter.8

Modeling and simulation, relevant to developing, improved

technical applications by extending the scientific basis from

macroapplications into the field of shorter pulses of about a few

nanoseconds, and smaller length scales down to the wavelength

of the radiation is considered. Some key physical phenomena

become apparent and more dominant on smaller scales. As a

result, there is also feedback from improved models on the

microscale to simulations describing macroapplications.

The analysis deals with the interaction of physical quantities

(Sec. II) including photon and wave properties of radiation,

energy flow density, and photon energy, via evaporation dy-

namics changing from quasisteady to kinetic behavior with

shorter time scales and changing material properties while

approaching the critical state at large temperatures, toward non-

linear transport phenomena due to the fast movement of the

ablation front in drilling with ls and ns-pulses (Sec. III).

Research directions in the field of pulsed beam–matter

interaction are twofold. While diagnostic methods are

extended and refined to reveal additional phenomena and to

improve understanding of them,9,10 the investigation of phe-

nomena that are already known is also detailed to achieve a

mapping of the dominant phenomena to the causal radiation

properties (Sec. III). To detail the relevance of the different

physical phenomena, we divide the parameter space spanned

by the radiation properties and the processing parameters

into subspaces, which are called processing domains. Differ-

ent processing domains can be distinguished either by the

different properties of the final processing result or by the

corresponding path the system undergoes in the physical

phase space during the interaction. As a result, a single proc-

essing domain is then characterized by a unique subset of

selected physical phenomena, which are found to be domi-

nant. Their typical time scales and intensity regimes are indi-

cated (Sec. II). The theoretical goal is to reveal the specific

model structure which reproduces the characteristics of the

different processing domains. The approach consists of itera-

tive extension, refinement, and reduction of the specific

physical models so that finally the essential physics is

included, but there is no unnecessary complexity.

II. PHENOMENA OF BEAM–MATTER INTERACTION

In optics, generally photons or electromagnetic radiation

is considered to interact with matter. The laser matured into

a versatile light source and now continues to open up new

opportunities for probing deeply into a variety of optical phe-

nomena related to both photons and waves. Although numer-

ous optical phenomena have already been found and

researchers have been exploiting their potential for technical

applications for decades, it remains a challenge to set up a

structuring or mapping (Sec. II B) of the relation between

well-known phenomena and the radiation properties which

cause them. Experimental diagnosis as well as theoretical

analysis suffer from the complication that the properties of

laser radiation as well as the response of the material are de-

pendent, at the very least, on time, space and energy. Effects

on long time scales and their causes on short time scales are

indeed intermediated by a subsequent temporal evolution but

as result can be closely related without detailed knowledge

of the intermediating physical processes. Regarding fs-pulse

durations, the typical time scales of electron dynamics are

FIG. 1. Laser systems and industrial availability. With chirped pulse amplifi-

cation (with CPA) a factor 103 for the peak power is achievable. Yb:

Disclaser (dotted): 1—Jenoptik, 2—time-bandwidth (without CPA), 3—

TRUMPF (6.5 ps without CPA), Yb:fiber laser (dashed): 4—Corelase, 5—

IAP FSU Jena, 6—PolarOnyx, 7—University of Michigan UoM,

Yb:tungstate laser (solid bold), 8—amplitude systems, 9—light conversion,

Ti:sapphire-laser (without borderline), 10—APRC Japan, 11—coherent,

12—MBI Berlin, 13—spectra physics, 14—Thales 100/30, glass laser

(dashed-dotted) 15—Nova LLNL/Phelix GSI/Vulcan/RAP, 16—POLARIS

FSU Jena, Yb:Innoslab (solid) current fs/ps-research (white bar) with speci-

fications: sp � 1 ps (Yb:YAG), sp � 200 fs (Yb:KLG), �rep 2 f10 kHz, 50

MHzg, Pav¼ 200 W at k¼ 1030 nm.

012006-2 J. Laser Appl., Vol. 25, No. 1, February 2013 Schulz, Eppelt, and Poprawe


similar to the pulse duration. It is worth to mention that in

dielectrics, the criteria for processes taking place on much

longer time scales like modification of the index of refraction,

damage by cracking, and ablation are well approximated by

the density of electrons generated in the conduction band in

the fs-range, while the phonon system remains cold. To

emphasize the effect of time or pulse duration on both elec-

tron density and long time effects like modification, damage,

and ablation, the regime referred to as cold ablation (Fig. 2) is

introduced. Moreover, pulse parameters such as pulse energy

and pulse duration, as well as the spatial and temporal shape

of the pulse, cannot be chosen independently but depend on

the technical set-up of the radiation source and the dynamics

of laser light generation. One of the numerous examples

where details of the radiation properties become relevant to

the interpretation of laser–matter interaction is the so-called

pedestal pulse following the main laser pulse.11 As a conse-

quence of the pedestal pulse, the action of the main fs-pulse,

which might be related to the processing domain referred to

as cold ablation (Fig. 2) characterized by spallation, is

obscured by the action of the longer ps-pedestal related to the

processing domain known as hot ablation which mainly

causes evaporation.

Bearing in mind, however, that the different phenomena

take place simultaneously in different spatial regions as well

as overlapping with respect to time and energy, it remains a

challenge to distinguish the particular contribution of the dif-

ferent phenomena to the overall action of the laser pulse. In

spite of that, there are selected parameter settings where only

a small number of physical phenomena contribute; they can

be distinguished by only four parameters, namely the photon

energy, the energy flow density � [J m s�1 m�3] often called

intensity I [W m�2], and the temporal and spatial extent of

interaction often addressed as pulse duration and beam

radius. It is therefore worth mentioning that there are indica-

tions of the existence of different physical domains of beam–

matter interaction which can be classified by pulse duration

(Fig. 2) and intensity (Fig. 4).

A. Short pulse ablation and drilling by melt expulsion

The microscopic mechanisms of laser ablation are

delineated in various publications since the 1990s. The laser

induced pressure build up and the phase explosion due to over-

heating of the irradiated material are identified as the key proc-

esses that determine the dynamics of laser ablation.12–17 The

fluence threshold behavior of material response in the form of

laser ablation is predicted. The physical phenomena below

and above threshold are analyzed.13,15,16 A series of large scale

simulations have been performed in the regime of thermal con-

finement that is characteristic for spallation of metals as well

as removal of large fragile biomolecules. The physical rela-

tions between laser ablation and heavy ion beam nanostructur-

ing of dielectrics18 introduce new insight into the ablation

mechanisms beyond the irradiation regimes like stress con-

finement and thermal confinement. In particular, the action of

interference is present in laser ablation and absent in ablation

by particles. The interference patterns generated by diffraction

of radiation at the evolving ablation cavity subsequently

induce spatial structures of the electron density in transparent

materials and finally lead to corresponding damage structures.

Molecular dynamic simulations of laser ablation and dam-

age reveal two different irradiation regimes, stress confine-

ment and thermal confinement. An analysis of the microscopic

FIG. 2. Mapping of the three different physical domains—cold ablation, hot ablation and melt expulsion—with respect to the corresponding time scales. The

ablation threshold (solid line), the maximum available pulse energy (green), and the threshold for the critical state (dashed) of matter are indicated qualita-

tively. For long pulse durations, the radiation also interacts with the ablated material in the gaseous phase (plasma threshold, dotted line). The atomic time

scale tB for an electron roundtrip and typical time scales for thermalization of electrons se, relaxation of phonon temperature tph, for kinetics in the Knudsen

layer approaching a quasisteady state tKn and the onset of fully developed melt flow tin limited by inertia are indicated.

J. Laser Appl., Vol. 25, No. 1, February 2013 Schulz, Eppelt, and Poprawe 012006-3


mechanisms of laser ablation is combined with a systematic

investigation of the interrelation between the different parame-

ters of the ablation process accessible for experimental investi-

gation, including the fluence dependence of the total and

monomer yields,14 cluster composition of the ablation plume,

velocity,19 and angular distributions of ejected molecules, and

profiles of the acoustic waves propagating from the absorption

region.16 Desorption and ablation are identified as the two dif-

ferent regimes of molecular ejection. Ejection of volatile sol-

utes dominates in the desorption regime, whereas nonvolatile

solutes are ejected only in the ablation regime. The volatile

molecules are mainly ejected as monomers, and the nonvola-

tile solutes tend to eject as part of molecular clusters.20

Historically until 1990s, the investigations on fundamen-

tal experiments21,22 and modeling are carried out to reveal

the different physical mechanisms contributing to the techni-

cal process. The physical foundations and their possible

application to thermal processing are given by Boven-

siepen23 for short pulse ablation mechanisms and by van All-

men24 for macroscopic processing as well as melt expulsion.

In the monograph of Steen,25 the dominant physical proc-

esses and their technical relevance for laser applications are

discussed. A collection of the model equations and their

application to machining is summarized by Chryssolouris.26

The main modeling activities until 1994 are reviewed by

Steen.27 The reports of 16 German research partners are pre-

sented by Geiger28 and Sepold.29 Comprehensive sur-

veys30,31 of the practical tasks for laser cutting dominated by

melt expulsion are available. The physical mechanisms dom-

inant in thermal processing such as the global balance of the

thermal energy32,33 heating up the material to be drilled, the

two phase transitions melting and evaporation,34,35 the melt

flow and its hydrodynamical stability,36 as well as the flow

of the assist gas in drilling related to momentum and heat

transfer37,38 from the inert gas jet to the melt are revealed by

experimental evidence and discussed by the early models.

The thin melt film is mainly driven by evaporation exerting a

strong pressure on the top of the free moving liquid surface.

Depending on the position and slope of the liquid surface

within the laser beam and depending on the temperature at

the melt film surface forces from evaporation can change

direction and magnitude dynamically.

There are numerous effects present in laser cutting and

also in laser drilling, actually the presence of oxygen as

assist gas is of particular interest in laser drilling with domi-

nant melt ejection, too.39–41 Performing numerical simula-

tions of the supersonic gas flow and using the Schlieren

method,42–47 the characteristics of the complex gas jet like

supersonic domains, shocks, expansion waves, and flow sep-

aration have become more transparent. Turbulent effects of

the gas flow were discussed.48 The main interest then

changed to the following question: How to give a closed for-

mulation of the process and what are the additional effects

introduced by coupling of the physical processes? It became

essential to distinguish between the independent parameters

of the process like speed of feeding, laser power, intensity

distribution, etc., and the quantities which evolve during the

process slope of the cutting front, the melt film thickness, the

surface temperature, the width of the cut kerf and has to be

calculated as part of the solution. The additional effects of

the reactive gas cutting process are investigated with respect

to oxygen concentration within the gas jet which is changed

along the molten iron (low alloyed or carbon steels) surface

within the boundary layer of the gas jet due to reaction with

the iron and diffusion through the metal-oxide layer. The ox-

ygen and iron concentrations within the oxide layer are

investigated49–51 which yield the reaction rate and hence the

energy released by the exothermal reaction. By comparison

of the action of the oxygen gas jet in laser cutting and in

autogenous gas cutting, the high performance laser gas cut-

ting process referred to as burning stabilized gas cutting is

invented52 and later applied in industries.

Available models for the simulation of laser drilling with

the dominant characteristics of the generation and ejection of

melt describe the physical processes of heat conduction and

convection in solid and liquid phase, the flow of melt, the

evaporation of the melt on the melt film surface, as well as

the expansion and ionisation of the metal vapor that is gener-

ated. There are analytical24,53,54 reduced models as well as

numerical models55 that have been developed since the

1970s. An analytical representation, where there is an analyti-

cal solution at hand,56 is limited to a few nonlinear subpro-

blems with a few restrictions (e.g., asymptotic limit cases),

especially in the case, where more than one nonlinear system

is under consideration and those systems are coupled.57

Completely numerical models (e.g., finite-element mod-

els) are extremely computationally expensive and assume a

numerical solution with controllable error. Reduced models

(e.g., constructed by a singular pertubation series) reproduce

fundamental specifications of the full solution and are much

easier to handle concerning their solvability. This is why

modeling58 is based on a reduced representation of the pro-

cess together with validation by means of completely numer-

ical models. Reduced models are mostly based on integral

balancing equations, which are balancing nominal values

and fluxes of mass, momentum, and energy in a proper bal-

ancing volume. Aside from theoretical models, which are

based on physical relationships, that can be written or

simulated, also so-called metamodels (also called Response-

Surface-Models) are at hand, which are encapsulating exper-

imental data. As one result from a metamodel, physical

modeling and model reduction can be guided efficiently by

data from completely numerical solutions and experimental

data. Metamodels are also used for implementing the process

know-how into self-optimizing production systems.59

Numerical models listed in literature cannot describe

any physical phenomena involved in the drilling process at

the same time. Mostly, the fundamental first phase transition

from solid to liquid is considered. Based on this fundamental

phase transition, the second phase transition namely evapora-

tion is added in terms of estimates about vapor expansion or

plasma absorption. The influence from process gas flow is

considered in some of the numerical laser drilling models.60

These reduced models are based on assumptions for the gas

flow (e.g., an assumed isentropic expansion of the gas). Spa-

tially resolving simulations of the gas flow have also been

presented.60 Some models include the dynamics of vapor

and plasma, e.g., the absorption of radiation.40,55 For the



drilling of deep drills (in comparison to the Rayleigh-length

of the laser radiation used), the contribution of multiple

reflections of the laser radiation inside the drill has been

demonstrated.61–63

Therefore, raytracing methods as well as wave-optical

methods55 [e.g., finite difference time domain (FDTD)] have

been used.64–66 A self-consistent simulation of the radiation

field and the material processing while taking into account

the interaction between both seems to be inefficient with

continuum methods like FDTD because of the demanding

spatial resolution of those methods. Raytracing based meth-

ods considering the transport of power as well that of phase

rather show the opportunity of coupling between beam prop-

agation and ablation simulations.

B. Physical domains—Map of intensity and pulseduration

The interaction of radiation with matter in general takes

place via excitation of electrons leading to polarization and

ionization of atoms, molecules, or condensed matter.

Three physical domains distinguished by the correspond-

ing time scales of their activation are indicated in Fig. 2,

namely cold ablation, hot ablation and melt expulsion. Cold

ablation is characterized by beam–matter interaction taking

place dominantly in the condensed state and the dynamics of

subsequent ablation is governed by pure kinetic properties

(Sec. II F). Thermalization of the photon energy absorbed by

the electrons which leads to a thermodynamic equilibrium in

the electron system and which establishes an electron tem-

perature Te, takes place. The wording cold ablation makes

sense, although a subsequent heating of the phonon system

and changes of material state will take place, too, and can be

diminished or avoided by external cooling, only. However, it

is worth to mention that in dielectrics, the criteria for proc-

esses taking place on much longer time scales like modifica-

tion, damage, and ablation are well approximated by the

density of electrons generated in the conduction band in the

fs-range, while the phonon system remains cold. One impor-

tant method of describing kinetic properties is molecular

dynamic simulation. Although cold ablation is analysed by

molecular dynamic calculations67,68 of the heavy particles,

neglecting charged particles like electrons and ions, the

results give remarkable insight into the dynamics of ablation:

Very fast propagation 4000 m/s of the melting front in metals

cannot be explained by heterogeneous melting front propa-

gation only,69 the combination of the two-temperature model

for the fast relaxation of electrons and a molecular dynamic

approach for the slower heavy particles improves the under-

standing of spallation type removal70 and overheating before

homogeneous melting.71 Analyzing the competition between

heat conduction by fast electrons and the electron–phonon

interaction with subsequent heat conduction by phonons

reveals a general behavior of metal targets changing from

photomechanical to photothermal damage at the electron–

phonon relaxation time.72

Hot ablation sets in while the phonon system approaches

local thermodynamical equilibrium with the initially heated

electron system. The radiation interacts with both condensed

and gaseous matter. In hot ablation, the condensed and gase-

ous phases do not reach a common equilibrium state, since

they are separated by the evaporation front, which has the

short time kinetic properties of a dynamical Knudsen layer.

Melt expulsion is characterized by reaching a fully devel-

oped continuum limit, where the long time limit of the ki-

netic Knudsen layer73 across the evaporation front is also

established. For ns-pulses, evaporation takes a dominant role

in the absorption of energy, but only a small amount of mo-

mentum is transferred to the melt and melt ejection sets in.

With increasing pulse duration approaching the ls time

scale, the energy is finally used mainly for melting, while

evaporation acts as a powerful driving force introducing

large momentum to the molten material via the recoil pres-

sure at the liquid surface. Even in the range of dominant

melting with increasing pulse duration, a characteristic tran-

sition takes place, namely, from an inertia-dominated onset

of melt flow at time scale tin typical for sub-ls-pulses to fully

developed melt ejection for ls-pulses and longer pulse dura-

tions. For ls-pulses, melt flow is governed by the equilib-

rium between spatial changes of the recoil pressure and

friction of the liquid at the solid melting front only. Further

details are discussed in Sec. III.

1. Cold ablation

Some more detailed relations and physical scales to sup-

port the understanding of the different physical domains can

be given. For pulse durations tp < 100 fs, absorption of radia-

tion initiates the processing domain called cold ablation

(Fig. 2). In cold ablation, the absorption of laser light by elec-

trons and thermalization—first between the electrons and

later in the phonon system—is almost terminated while the

matter remains in the condensed state. A few picoseconds

later the phonon temperature rises and ablation of hot matter

starts to take place. Typical atomic time scales for changes of

the binding forces initiated by electron excitation or emission

are comparable to the time rB=ve � 2:5 � 10�17 s, the electron

needs for one round trip of the Bohr orbit with radius

rB ¼ 5:3 � 10�11 m and velocity ve ¼ 2:2 � 106 m s�1.74 In

metals characterized by a large electron density, a pulse dura-

tion in the range of a few femtoseconds is also a long time

compared with the atomic scale tB ¼ rB=ve. Absorption and

subsequent cold ablation can therefore happen without ther-

malization of energy. Subsequent thermalization of electrons

by electron–electron collisions at se � 10 fs is much faster

than the thermalization of phonons, and thermalization takes

place separately with respect to time. It is worth mentioning

that the time scale se for relaxation of energy in the electron

system

se ¼ ðs�1e�e þ s�1

e�phÞ�1 ¼ ðA T2

e þ B TphÞ�1(1)

can be related to the final temperatures Te and Tph in which

the subsystems of electrons and phonons are approaching

during thermalization.75,76 The constants A and B are related

to the electron–electron and electron–phonon collision fre-

quencies, �e�e ¼ A T2e , �e�ph ¼ B Tph. Values for the con-

stants A and B are collected together in Table I.



Molecular dynamic calculations for a fs-pulse67 and a

ps-pulse6 showed that matter passes through one of at least

four different possible paths in the physical phase space dur-

ing beam–matter interaction [Fig. 3(a)]. Typically, the mate-

rial is modeled by Newton’s equations of motion using a

2D-Lennard Jones potential with three parameters, namely

binding energy � of a few eV, atomic length scale r of the

order of a few Angstrom, and a cut-off radius. The calculated

quantities such as kinetic energy and density are averaged

with respect to an initially selected material volume and

scaled by the binding energy � and the initial density 1=r2,

respectively. Absorption of photons is approximated by an

instantaneous energy transfer to the heavy particles with a

spatial distribution given by the Lambert-Beer law. The most

interesting and plausible result from the molecular dynamic

calculations is that the pulse energy is a dominant parameter

for the selection of the path in phase space. The ablation

threshold depending on pulse energy and pulse duration is

investigated in Ref. 77 and the literature cited. At large

energy density of the pulse far above the ablation threshold,

different partial processes contribute their particular charac-

teristics to the overall appearance of ablation. Even for small

pulse duration in the range of tenths of femtoseconds, the

larger pulse energy leads to the temporal and spatial coexis-

tence of different states of the gaseous phases as in the case

of vaporization (4) and fragmentation (3) as well as nuclea-

tion (2) of ablated particles [Fig. 3(a)]. Near the ablation

threshold, spallation (1) is observed in the calculations not

only for fs-pulses67 but also for ps-pulses.68 In spallation, no

gaseous phase is involved, while the thermo-mechanical load

exceeds the yield stress of the material and removal like

spalling of a thin metal disk is observed.

The nucleation (2) path goes through the liquid phase as

well as through the coexistence of a liquid–solid phase and

ends in the coexistence of a liquid–gas phase. Employing a

phenomenological interpretation, nucleation is characterized

by the development of vapor bubbles in homogeneously

boiling molten material. For the last two ablation mecha-

nisms, fragmentation (3) and vaporization (4), ablation

occurs outside the liquid–solid and liquid–vapor-coexistence

regions, having the effect that the mass is separated from the

surface just before relaxation to thermodynamic equilibrium

is noticeable. The two paths are differentiated by cluster for-

mation,78,79 which takes place in fragmentation (3) and is

absent in vaporization (4).

2. Hot ablation

For pulse durations around tp¼ 100 ps, absorption of

radiation initiates the processing domain called hot ablation

(Fig. 2). Electron Te and phonon Tph temperatures are well

defined thermodynamical parameters. Relaxation toward a

common temperature T ¼ Tph ¼ Te for electrons and pho-

nons as well as the formation of a dissipative heat flux are

the main physical effects encountered. In general, the initial

heat flux Qe of the electrons is much smaller than the spatial

TABLE I. Parameters of the hyperbolic two-temperature model at Te ¼ Tph

¼ 300 K.

Au Al Cu Pb

A K�2s�1 1:2 � 107 2 � 107 1 � 107 0:9 � 107

B K�1s�1 1:2 � 1011 7 � 1011 1:08 � 1011 47 � 1011

sQefs 26 4.6 31 0.7

te ¼ Ce=hex ps 0.579 0.067 0.467 0.377

tph ¼ Cph=hex ps 70.6 4.27 57.5 11.2

FIG. 3. (a) The physical path in the phase space of a preselected material volume during beam–matter interaction for a fs-pulse related to the processing do-

main cold ablation. The physical path is projected onto a subspace spanned by the kinetic energy and the density. The kinetic energy is scaled by the Lennard

Jones potential ULJ energy scale �: ULJðrÞ ¼ 4�½ðr=rÞ12 � ðr=rÞ6 � ðr=rcÞ12 þ ðr=rcÞ6� if r � rc and ULJ ¼ 0 else. Thereby � is the energy scale and r is the

length scale of the Lennard Jones potential. The density is scaled by r�2. For reference, the binodal lines of the state of matter diagram for thermodynamical

equilibrium are indicated (dashed). Depending on the energy absorbed, the dynamics of the material volume follows four different paths in the physical phase

space indicated by (1) spallation, (2) nucleation, (3) fragmentation and (4) vaporization. (b) A typical path in the phase space for pulses longer than tp� 100 ps

is related to the processing domains called melt expulsion. The temperature is given by the ratio �=kB of the energy scale from the Lennard Jones potential ULJ

and the Boltzmann constant kB. G: Gaseous, V: Vapor, L: Liquid, S: Solid, TP: Triple Point, CP: Critical Point.



change Qe � �kerUe=Ce / rTe of the energy Ue absorbed

by the electrons. A detailed discussion of the dynamics

involved by analysing the hyperbolic two-temperature model

@Ue

@tþ @Qe

@z¼ I0A0f ðtÞae�az þ hexðTph � TeÞ;

sQe

@Qe

@tþ Qe ¼ �ke

@Te

@z; (2)

@Uph

@tþ @Qph

@z¼ hexðTe � TphÞ;

sQph

@Qph

@tþ Qph ¼ �kph

@Tph

@z(3)

is given by Kostrykin et al.75 Here, the absorption of the

laser radiation is described by the term I0A0f ðtÞae�az, where

I0f ðtÞ is the time-dependent intensity of the radiation, A0

¼ 1� R0 is the degree of transmission at the surface related

to the degree of reflection R0, and a�1 is the penetration

depth of the laser radiation. The thermal diffusivity of the

phonons can be neglected compared to the electron diffusiv-

ity, kph=Cph � ke=Ce. The main questions arising from Eqs.

(2) and (3) are related to the time scales for relaxation of

electron temperature te ¼ Ce=hex and phonon temperature

tph ¼ Cph=hex toward a common value, as well as the relaxa-

tion time sQeof the electron heat flux. The values of these

coefficients at Te ¼ Tph¼ 300 K are presented in Table I.

Time scale separation for relaxation in the electron and

phonon systems is pronounced. The long time limit for the

heat flux Qe of the electrons, which is given by Fourier’s

Law Qe ¼ �kerUe=Ce, is established during the same very

short time sQe� se � 10 fs that the thermalization of the

electron system requires.75,76 For pulse durations much lon-

ger than sQe, the relaxation of the heat flow is negligible,

thus showing that the standard parabolic two-temperature

model of Anisimov is adequate, where the idealized assump-

tion of instantaneous heat propagation is assumed.80 As a

result of time scale separation, for ps-pulses and the material

data in Table I, the electron temperature Te is the fast vari-

able with time scale te instantaneously following the dynam-

ics of the phonon temperature, whose time scale turns out to

be the slow variable with time scale tph � te governing the

dynamics during absorption and relaxation to a common

temperature in metals.

3. Melt expulsion

Kinetics in the Knudsen layer involved in evaporation

and the numerous effects of the physical domain referred to

as melt expulsion [Fig. 3(b)] are discussed in subsequent

Secs. II B–II F and Sec. III.

Ionisation and excitation depend not only on photon

energy compared to atomic transitions energies but also on

the number density of photons, so that the intensity of the

radiation induces quite different types of beam–matter inter-

action in metals. At least four domains can be differentiated

with respect to intensity, which are characterized by heating

of matter, the effects of subatomic and superatomic field

strengths, as well as the properties of relativistic particles

(Fig. 4).

At intensities lower than 1011 W cm�2, absorption of

infrared radiation in metals basically behaves optically lin-

ear, and elevated temperatures are established via Joule heat-

ing of the atoms by collisions with excited electrons from the

conduction band. For pulse durations greater than 10 ps, the

energy is dissipated in local thermal equilibrium. Thermal

emission of electrons from condensed matter and collisional

ionization in metal vapors by the Richardson effect dominate

the formation of ionized vapor or plasma. Corresponding

thermal processes such as drilling and microprocessing are

widely used in applications today.

At higher intensities above 1011 W cm�2, the multiphoton

absorption or multiphoton ionisation (MPI) in the infrared spec-

trum are observable.81 The threshold intensity I ¼ n � h� � c for

FIG. 4. Mapping of the different physical domains—heating, subatomic, and superatomic field strength as well as relativistic particles—with respect to the cor-

responding intensity for metals.



MPI of about 1011 W cm�2 corresponds to a photon density

n ¼ 4=3pDs3. The spatial distance Ds ¼ cDtvirtual=N between

the photons moving at the speed of light c has to be as short as

the photon travel distance c Dtvirtual during the lifetime Dtvirtual

of virtual states in the atom divided by the number N of pho-

tons, which are needed to overcome the ionization potential.

Inserting Dtvirtual¼ 10�15 s, N¼ 5, and the photon energy

h�¼ 1.5 eV, then the spatial distance Ds¼ 6 � 10�8 m, the pho-

ton density n �1021 m�3, and the threshold intensity I¼ 7

� 1010 W cm�2 for MPI give plausible measures for the experi-

mental evidence. The response of metals shows intensity-

dependent optical properties. The nonlinear effects are useful

for increasing temporal and spatial resolution or selectivity in

diagnosis and processing. Notable examples for increased tem-

poral resolution are the colliding pulse mode-laser and two-

photon spectroscopy or subwavelength modification which is

based on increased spatial resolution. The data for threshold

intensity IMPI in MPI of aluminium, copper, and iron for the

KrF- and TiSa-laser have been thoroughly investigated82 and

are collected in Table II.

Above-threshold ionisation (ATI) is a kind of beam–

matter interaction in which atoms subject to intense laser

fields at intensities above 1012 W cm�2 absorb more than the

minimum number of photons required for ionization. Since

the first experimental discovery of ATI for Xenon atoms by

Agostini in 1979,83 ATI has been an important subject in

ultrafast laser–atom interactions. Xenon with the ionization

potential of 12.27 eV absorbs 12 photons emitted from a Nd–

glass laser, each having a photon energy of 1.17 eV at

1013 W cm�2 of laser intensity. Both MPI and ATI are also

involved in the formation of ionized processing gases.

The effect of field ionisation (FI) is revealed by the ob-

servation that fewer photons are necessary to ionize an atom

than are required to overcome the energy Uion for ionization.

FI and tunnel ionisation come into play at intensities where

the laser field E changes the atomic structure by ponderomo-

tive forces. At intensities I ¼ ð�0c=2ÞE2atom above I > 1016 W

cm�2, the electrical field strength becomes comparable with

typical atomic fields Eatom ¼ ð4p�0Þ�1e=r2B �5 � 109 Vcm�1

depending on the Bohr radius rB¼ 5.3 � 10�11 m.84,85 In

particular, the potential UðxÞ ¼ U0ðxÞ � eEx for an electron

with charge e becomes asymmetric compared with the

unperturbed potential U ¼ U0ðxÞ. The energy Uion for ioni-

zation of Rydberg electrons and continuum electronic states

is decreased by the ponderomotive energy Up

Up ¼1

me

eE

2x

� �; c ¼ Uion

2Up

; (4)

which for free electrons in the laser field depends on electron

mass me and charge e as well as the electric field amplitude

E and photon energy �hx. The transition from dominant Field

Ionization to tunneling is characterized by c ¼ 1, where c¼ ðUion=2UpÞ is the well-known Keldysh-parameter.

At 1018–1019 W cm�2, the travel distance an electron

needs to pick up its own rest energy mec2 is only one wave-

length k of the incident visible light eEk ¼ mec2 and the

motion of the electron becomes relativistic. The energy of

the wave becomes so large that the electron “surfs” on the

wave. The wave exerts direct light pressure on the electron.

The plasma and the particles become relativistic, the mecha-

nisms lead to concepts and realisation for particle accelera-

tion and to free electron lasers. If the field is a factor of 103

higher (I¼ 1024 W cm�2), the same effect occurs not only for

electrons—they continue surfing at the bottom of this

wave—but also for heavy protons with mass mp. The proton

motion becomes relativistic eEk ¼ mpc2. Matter is directly

compressed by the radiation in accordance with the fast igni-

tor concept of the Lawrence Livermore National Ignition Fa-

cility. At about 1029 W cm�2, the distance an electron needs

to pick up its own rest energy is only its own de Broglie

wavelength kdB. Thus, the field comes into resonance with

the matter wave. A change from collective absorption of

electromagnetic radiation by electrons to a direct localized

interaction of the photon with the particle has occurred, and

electron positron pairs are generated.

C. Beam propagation

The effects of the wave-like, moving absorption front

leading to dynamical aberrations of the laser radiation have

to be analysed in comparison with free propagation in vac-

uum. The effects of the moving absorption front, where the

laser is partially absorbed and reflected, on radiation propa-

gation are rarely discussed beyond the level of geometric

optics.

Modeling and calculation of radiation propagation is fea-

sible as long as well-known model equations, such as Max-

well’s equations, vector valued wave equations, and higher

order slowly varying-envelope (SVE)-approximations or the

geometrical optics approximation remain tractable by the

available numerical methods. Numerical methods such as

the FDTD-method, the boundary-element-method (BEM),

and the beam-propagation-method BPM are sufficiently well

developed in the literature to be applicable.64–66

At the absorption front (the drill wall, the surface of the

capillary in welding, or the cutting front), the laser radiation

is reflected, absorbed, or diffracted. Reflection and diffraction

introduce changes in the laser beam. More precisely, the ori-

entation and the absolute value of the Poynting vector are

changed. The beam-forming effect of the absorption front has

to be investigated in more detail by taking into account the

wave properties of the radiation and the boundary conditions

for strongly absorbing media. To solve Maxwell’s equations

directly the FDTD-method can be applied.64–66 The material

TABLE II. Threshold intensity IMPI, number of photons n involved and ioni-

zation potential Uion for MPI of atoms. KrF-laser: wavelength 248 nm, pho-

ton energy 5 eV. TiSa-laser: wavelength 800 nm, photon energy 1.5 eV.

Atom Uion ½eV� laser n IMPI ½W cm�2�

Cu 7:73 KrF 2 1:0 � 109

TiSa 5 5:0 � 1011

Al 5:99 KrF 2 2:3 � 1010

TiSa 4 9:5 � 1011

Fe 7:87 KrF 2 1:0 � 1010

TiSa 7 1:0 � 1013



equations of Drude and Lorentz are used to describe the elec-

tromagnetic properties of the metal or dielectric. Absorbing

boundary conditions of so-called “perfectly matched layer”

(PML)-type as well as periodic conditions work well at the

external boundaries. Reflection, transmission, and absorption

of a plane wave incident on a plane surface are known

analytically and serve as a test case. The accuracy of the nu-

merical solution can be determined by comparison with the

well-known Fresnel Formulae (Fig. 5). Although the required

spatial resolution in the range of 1/30-times the radiation

wavelength is rather high, it is worth the cost of the calcula-

tion in order to gain a qualitative understanding.

Analysis of the structural stability of the physically

approximated model equations for radiation propagation is

carried out by revealing the changes in the solution depend-

ing on different physical model equations (Maxwell’s

equations, vector valued wave equations, the higher order

SVE-approximation, the geometrical optics approximation).

As a matter of fact, the accuracy of the different numerical

methods such as the FDTD-method, the BEM, and the BPM

also have to be investigated.

D. Refraction and reflection

The relevance of probing deeper into the details of radia-

tion propagation becomes important if the optical path con-

tains nonideal optical objects such as density variations or

particles within the gaseous phase. The cost of modification

and adaptation of the model equations as well as the appropri-

ate numerical methods are least in so far as ray-tracing can be

used to describe the phenomena in question. For every addi-

tional order of reflection, the cost increases in proportion to

the number N of grid points. Calculating diffraction integrals

means an increase to N2. Finite-Difference approaches are far

more expensive, since the temporal evolution of the wave

field has to be resolved, too. The aberration or beam forming

due to metal vapor/plasma or a supersonic gas flow is investi-

gated in detail. Radiation propagation in media with a

spatially distributed index of refraction can be calculated by

ray-tracing. The eikonal equation of geometrical optics is

solved allowing for a spatially distributed index of refraction

nðrÞ, where the arc length s along the ray r is introduced

d

dsnðrðsÞÞ drðsÞ

ds

� �¼ rnðrðsÞÞ: (5)

In inhomogeneous media rnðrðsÞÞ 6¼ 0, the light rays

become curved instead of being straight lines. The intensity

is calculated solving the hierarchically ordered irradiance

transport equations for the field amplitude.86 To cope with

the property of the solution that the wave fronts do not

remain single valued but interfere with each other, the power

of the refracted rays is projected and stored on a background

grid. Methods for interpolation of point-wise defined quanti-

ties and adaptive approximation of functions by adaptive

grid refinement are applied (Fig. 6).

It is worth mentioning that via the Cauchy dispersion

relation, the index of refraction n

n� 1 ¼ A 1þ B

k2

� �(6)

depends on the material parameters A and B (A¼ 87.80 � 10�5

and B¼ 22.65 � 10�11 cm2 for mercury vapor, A¼ 29.06 � 10�5

and B¼ 7.7 � 10�11 cm2 for nitrogen, on radiation wavelength

k and is also related to the density q of the gaseous phase,

where n�1q ¼

n0�1q0

which indeed changes in supersonic flow.

Measured reference values n0 and q0 are available.87–89 Anal-

ysis of refraction in inhomogeneous media combined with

multiple reflections is used to reveal parameter ranges—

so-called processing domains—where these effects show a

dominant effect on the dynamical stability of the process.

Improved understanding of radiation propagation is applied

for a process-adapted design with beam-forming optics.

E. Absorption and scattering in the gaseous phase

Absorption of infrared laser radiation in the metal vapor

is governed by accelerated electrons colliding with atoms or

ions. For the typical situation of drilling steel with ls-pulses,

FIG. 5. Comparison of the degree of reflection for a metal (complex refraction index n¼ 1.42 þ i � 1.59) calculated by the FDTD-method (circle) and the exact

solution (solid line) from the Fresnel formulae as a function of the angle of incidence. Spatial resolution becomes important at glancing incidence. Amplitude

of a plane wave field emitted from boundary 1 incident on a plane surface (boundary 2) of a dielectric with periodic boundary conditions at the left and right

boundaries calculated by the FDTD-method. The PML is located above boundary 3.



which is characterized by a pulse duration tp > 100 ls, pulse

energy Ep < 400 Mj, and focal diameter 2 w0¼ 40 lm of the

laser beam, the metal plasma remains in local thermody-

namic equilibrium and the number of multiple ionized ions

is considerably smaller. The coefficient of absorption a of

the metal vapor can therefore be well approximated by the

well-known Drude-model

a ¼x2

p �e

cðx2 þ �2eÞ; xp ¼

ffiffiffiffiffiffiffiffiffiffine e2

�0 me

s; ne ¼ neðTg; qgÞ;

�e ¼ �eðTg; qgÞ; (7)

where �e is the collision frequency of electrons, x the laser

frequency, and me the electron mass. The coefficient of

absorption a depends on electron density ne and on electron

temperature Te via the frequency �e of electron collisions. In

consequence, the gas-dynamic state of the vapor enters the

absorption coefficient a via the density qg and temperature

Tg. These quantities have to be estimated by calculations

using the drilling model of Sec. III, for instance. The electron

collision frequency �e is composed of two parts �e ¼ �e;g

þ �e;i, since electrons are dominantly colliding with atoms

�e;g (Ref. 90) and ions �e;i (Ref. 91)

�e;g ¼ c1

ffiffiffiffiffiTe

png;

c1 ¼8 ð2pÞ3=2r2

Fe

ffiffiffiep

3ffiffiffiffiffiffimep 2:8 10�13 Hz

m3ffiffiffiffiffiffieVp

� �; (8)

�e;i ¼ c2

i2 ni lnðgÞT

3=2e

;

c2 ¼e4ð2peÞ�3=2

3e20

ffiffiffiffiffiffimep 1:6 10�12 eV3=2

m3 s

� �; (9)

g ¼ 12p ne k3Db; kDb ¼

ffiffiffiffiffiffiffiffiffiffie0 Te

ne e2

r; (10)

where the atomic radius rFe for iron, the neutral atom density

ng, the degree of ionization i, and the particle density ni of

ions, the Debye length kDb, the number g of electrons within

the Debye sphere, and the dielectric constant e0 are intro-

duced. From Eqs. (9) and (10), it turns out that for a small

degree of ionization i, the contribution of the electron–ion

collisions remains small compared to electron–atom colli-

sions. As a result, the electron density neðTg; qgÞ and the

absorption coefficient aðTg; qgÞ can be calculated using the

gas-dynamic state fTg; qgg resulting from the evaporation

model.

The coefficient of absorption a for the parameter range

encountered in high speed drilling with Nd:yttrium alumi-

num garnet (YAG) lasers k¼ 1 lm dominated by melt ejec-

tion with intensities up to I0 < 1012 W m�2 can be estimated

from Eqs. (7)–(10) and is below a< 2 m�1. As a result, the

absorption length is expected to be of the order of half a me-

ter, much longer than the typical ablation depth or depth of a

drilled hole. In consequence, plasma absorption in steel

vapor produced at k ¼ 1 lm, I0 < 1012 W m�2 can be

excluded from the list of dominant phenomena.

F. Kinetics and equation of state

Energy and momentum is transported over a given dis-

tance L by the repeated movement of particles along a free

path length k between two subsequent collisions. Depending

on the Knudsen number Kn ¼ k=L, either the continuum

assumption of fluid mechanics Kn� 1 holds or the kinetic

properties of matter become essential Kn �1. It is worth men-

tioning that the kinetic properties of matter are present not

only for evaporation by fs-pulse duration at intensities larger

than 1016 W cm�2 but also play a dominant role in continuous

evaporation at moderate intensities. Fluid mechanics fails as

long as the distance L from the condensed surface remains

comparable with the mean free path kg � n�1=3g in the gase-

ous phase whose particle density is ng. For laser drilling

within the condensed matter having a particle density nc, the

Knudsen number Kn� 1 is small, since the mean free path

kc � n�1=3c is small compared with the optical penetration

depth a�1. Immediately on the gaseous side of the condensed

surface, there is a layer formed known as the Knudsen layer,

where the Knudsen number jumps to large values Kn�1,

while the distance L� kg remains small in the vicinity the

condensed surface. The Knudsen number again approaches

small values at distances L � nkg, which are n-times the

mean free path kg ¼ n�1=3g in the gaseous phase. In conse-

quence, kinetic items are involved in any kind of condensed-

gaseous phase change and the Navier-Stokes equations are

FIG. 6. Multiple reflections calculated by forward 3D-ray-tracing account for curvature and redistribution of power into a triangular surface grid. Ray-tracing

extended to inhomogeneous media and reconstruction of the multiply overlapping rays leading to caustic formation. A circular wave travelling outward from

the white circular area is diffracted by a spatially circular distribution of the index of refraction located left of the wave-emitting center (white circle).



not applicable. The task is to find the kinetic details of the

Knudsen layer, like analyzing the quasisteady state of the

Knudsen layer, for example, and determining the time tKn, the

transition from a kinetic to a continuum state of the vapor

takes. The transient time tKn for the build-up of a quasisteady

Knudsen layer (Fig. 2) has to be compared with pulse dura-

tion tp. Separation of the time scales tKn and tp is expected to

change the overall result of evaporation.

The quasisteady properties of the Knudsen layer are cal-

culated using the Boltzmann equation for the kinetic distri-

bution function U ¼ Uðx;VGÞ. The function U describes the

distribution of particles at position x with the hydrodynami-

cal velocity VG at the gaseous side of the Knudsen layer

whose thickness is Dvap. The long time limit

VGrU ¼ JðU;UÞ; Ujx2Dvap¼ /ðVGÞ;

Ujx!1 ¼ /MBðVGÞ (11)

has been solved numerically by Aoki,92 where the operator

JðU;UÞ is called the collision integral for relaxation of the

distribution function U toward the Maxwell-Boltzmann equi-

librium distribution /MBðVGÞ present on the gaseous side of

the Knudsen layer. The result of the kinetic calculation gives

a relation between the continuum properties of the gaseous

phase outside the Knudsen layer (gas pressure PG, tempera-

ture TG, Mach number Ma ¼ VG=cðTGÞ) and the temperature

TL at the condensed or liquid side of the Knudsen layer

PG ¼ G1ðMaÞ � PsatðTLÞ; (12)

TG ¼ G2ðMaÞ � TL; (13)

where Psat is the saturation pressure. For strong evaporation,

referred to as the Chapman-Jouguet case

x 2 Dvap : Ma ¼ VG

cðTGÞ¼ 1; (14)

the Mach number Ma ¼ 1 is already known. Here the gas-

dynamics, in particular the gas pressure, do not change the

evaporation process. It is worth mentioning that, for moder-

ate evaporation, the Mach number Ma < 1 has to be deter-

mined and the additional information has to be taken from

the gas flow outside the Knudsen layer

x 2 Dvap : PG ¼ PGðTG;VGÞ: (15)

These model equations for quasisteady evaporation corre-

spond to the system path in the state diagram in Fig. 3(b) for

large pulse duration. While the quasisteady properties of the

Knudsen layer are well known from the literature,73 the analysis

related to the transient time tKn for the build-up of quasisteady

properties of the Knudsen layer remains an open question.

Moreover, relations depending on the state model, such

as the ideal gas equation PG ¼ qG Rs TG and other material

properties like the enthalpies for phase transitions, become

functions of temperature and pressure. Approaching the criti-

cal temperature, the difference between the density of con-

densed matter and of the gas vanishes. Idealized state models

are not applicable while approaching the critical temperature

and more refined models have to be involved. In addition to

molecular dynamic calculations, there are promising thermo-

dynamical approaches like the quotidian equation of state

developed by More et al.93

III. MELT EXPULSION: DRILLING MODEL FORMICROSECOND AND NANOSECOND PULSES

In drilling with ls-pulses, the dynamical phenomena

governing the shape of the drilled hole are identified experi-

mentally and can be related to the processing parameters the-

oretically. Technical concern is focused on the build-up of

recast at the drilled wall. Advances are made by relating the

borders of the processing domain to dominant phenomena

(Fig. 7) encountered in drilling such as ablation at the drill

base, widening of the drill by melt flow and narrowing by

recast at the drill wall, shadowing of the beam from the drill

base near the entrance of the drill by the melt film and recast

layer, recondensation of vapor at the drill wall, and absorp-

tion by the plasma.

After reaching, a typical drilled depth recast formation

sets in at the entrance of the drill hole [Fig. 9(f)]. Recast for-

mation leads to deviation from a cylindrical drill and tends

to narrow or even to close the entrance. Shadowing effects

and strong aberration of the radiation take place. The distri-

bution of the aberrated radiation determines the hole shape.

Mathematical model reduction is carried out using the

existence of an inertial manifold, which is present in distrib-

uted dynamical systems dominated by diffusion.94,95 The

corresponding long time limit of the underlying physical

model is derived.7,96,97 In principle, a spatially distributed

dynamical system consists of an infinite set of dynamical pa-

rameters. Approaching the inertial manifold, a substantial

reduction in the number of degrees of freedom takes place.

While the fast quantities undergo relaxation on the shortest

time scales, they are adiabatically linked to the small number

of slowly varying quantities. Detailed exploitation of the

time scale separation allows the analysis to be based on a

FIG. 7. Ablation at the drill base, widening by melt flow and narrowing by

recast at the drill wall, shadowing the beam from the drill base, and recon-

densation of vapor.



model which shows no unnecessary complexity. Subsequent

extensions including shorter time scales and the correspond-

ing additional dynamical parameters can be carried out step-

wise and with controlled error.

Experimental evidence (Fig. 8) shows that the final

shape of the drill base is established during the first few

tenths of microseconds, which reinforces the mathematical

arguments for a reduced model describing the dynamics of

the drill base in the long time limit. Although temporal

changes of the processing parameters on time scales larger

than a few microseconds result in a dynamical evolution of

the drill hole, the dynamical system does not leave the iner-

tial manifold.

Applying pulse durations in the range up to a few tenths

of ns, the drill terminates in a transient regime in which the

melt flow is not fully developed. The appearance of such a

transient regime is seen to be a consequence of the time

scales [Eq. (16)] of the model (Fig. 11) and can be observed

in experiments (Sec. III E).

A. Initial heating and relaxation of melt flow

Time scales in the parameter range available for ls-

pulses can be derived from the dynamical model. The time

tm to melting and to evaporation, tv, are small compared with

the time for relaxation trelax of the melt flow. The time trelax

of a few microseconds is small compared with the time

tl ’100 ls, the flow takes for moving up the drilled depth ‘B

¼ 1 mm. With the dependent quantities melt film thickness

dm, drilling velocity vp—as well as the material properties:

inverse Stefan-number hm, kinematic viscosity �, and evapo-

ration temperature Tv—the time scales read

tm ¼p

4js

k2ðTm � TaÞ2

A2pI2

0

; tv ¼jl

v2p

ln 1þ

Tv � Tm

Tm � Ta

1þ hm

0B@

1CA;

trelax ¼5

12

dm

vp

1

1þ 5

8

�

vpw0

; tl ¼‘B

w0

dm

vp

: (16)

The time tm until melting is tm 0.1 ls (Tm¼ 1800 K,

js¼ 10�5 m2 s�1, Ap¼ 0.4). Performing experiments with

pulses truncated by a Pockels cell, the evolution of the drill

hole can be observed [Figs. 9(a)–9(c)]. As a result, the time

scales fit well with the experimental evidence.

B. Widening of the drill by convection

There is no simple relation between the width of the

laser beam and the drill, depending on the material properties

and the laser radiation. The integrated model, which covers

evaporation at the drill base and melt flow along the wall as

well as recast formation, shows the widening of the drill

base and gives fairly good results when compared with dril-

ling experiments [Figs. 9(d)–9(f)].

Global energy balance considerations yield the ratio

FðPelÞ ¼ PK=Pl of convective PK and dissipative power Pl

depends on the P�eclet-number Pel ¼ ðV0 dmÞ=j only, Pl is

comparable to PK, and hence the ablated volumes per unit

time inside and outside the w0-environment around the beam

axis are of the same order of magnitude. Energy balance and

the Stefan condition at the melting front yield the widening

Dr of the drill

Dr ¼ expPel

2

� �� 1

� �� w0: (17)

As an example, for a drilling velocity V0¼ 6 ms�1 and a

laser beam radius w0¼ 20 lm, the widening Dr 0:82

w0¼ 16 lm of the drill corresponds fairly well to the experi-

mental evidence [Figs. 9(d)–9(f)]. The measured values for

the width are mostly above the predicted values of this

model, which suggests that there are additional causes for

widening such as recondensation of the evaporated material,

plasma heating of the drill wall, or beam aberration.

C. Narrowing of the drill by recast formation

During drilling, the laser beam dominantly heats the drill

base and the melt cools down while flowing upward along

the walls resulting in recast formation on the additional time

scale trec.

To slow down the build-up of a recast layer (Fig. 10),

heating at the melt film surface along the drill wall has to

balance the heat release into the solid. As a result, typically a

power of a few hundred Watts per mm drilled depth is suffi-

cient to avoid recast formation. The spatially resolved model

reproduces the onset of recast formation on the analytical

FIG. 8. Shape of the single pulse drills. Cross-sections for truncated pulse durations ttr 2 f5, 75gls (CMSX-4, Nd:YAG laser, tp¼ 300 ls, Ep¼ 325 mJ).



scales and its tendency to narrow the drill and even to

shadow the beam from the drill base [Figs. 10(c) and 10(d)].

D. Melt closures of the drill

Experimental observation—showing resolidified drop-

lets beneath a closure—gives a hint that, starting at a critical

depth Lkrit, the drill hole can be closed by the melt. The melt

flow slows down [Eq. (22)] at the entrance of the drill, the

melt film thickness increases [Eq. (20)] with distance from

the drill base, and finally starts to shadow the laser beam

[Fig. 10(c)]. Radiation heats the closure to evaporation tem-

perature thus re-ejecting part of the molten closure [Fig.

10(d)]. This so-called lift-up of the melt then happens peri-

odically and leads to pronounced changes in the drill shape.

The critical depth Lcrit at which the drill starts to close can be

estimated from the reduced model. The melt flow is driven

by the gradient of the pressure Pl and shear stress Rg. The

reduced model for melt flow reads is

Re@m

@sþ 6

5

@

@bm2

h

� �� ¼ � @Pl

@bhþ 3

2Rg � 3

m

h2;

(18)

@h

@sþ @m

@b¼ vpðb; sÞ; (19)

where m ¼ mðb; sÞ and h ¼ hðb; sÞ denotes the mass flow

and the melt film thickness, respectively.

The curvature K of the melt film surface enters the

boundary condition for the pressure, when the melt flow

slows down. The corresponding dimensionless group is

called the Weber number We, scaling the ratio of hydrody-

namic (q V20), and capillary pressure (r K). For We < 1, the

surface tension r tends to form droplets and hinders separa-

tion. The pressure Pl is then given by

Pl ! Pl þWe�1 @2h

@b2; We ¼ qV2

0

rK: (20)

The resulting equation of motion is known as the Kuramoto–

Sivashinsky equation, which describe unstable periodic

orbits and the transition to turbulence of thin film flows.

For small Reynolds numbers (Re! 0), a separation of

time scales of the dependent variables can be observed.

From Eqs. (18) and (19), the slow manifold reads

m ¼ @Pl

@bh3

3þ Rg

h2

2: (21)

FIG. 10. Simulation of the integrated model (a,b). Gaussian intensity profile

without (a) and with offset (b) heating the wall: (a) closure by recast (dark

grey), (b) heating the wall prevents recast formation, melt film (black).

Lengths scale w0¼ 20 lm. Simulation of the spatially resolved model (c,d):

Temperature in the solid [ambient (grey) to melting (light-grey)] and liquid

[melting (grey) to evaporation (light-grey)] (c) no shadowing, without offset

(d) geometric optical shadowing, with offset.

FIG. 9. Cross-sections of truncated single pulse drills. Pockels cell with jitter Dtp ¼ 66 ls. (a)–(c) Time increment Dt¼ 5 ls for simulation output. Truncated

pulse duration fttr 2 f30, 60, 90 ls gg(X5CrNi18-10, tp¼ 180 ls, Ep¼ 200 mJ).



The dynamics of melt flow is dominated by the melt film

thickness as is well known from lubrication theory

@h

@sþ @

@b@Pl

@bh3

3þ Rg

h2

2

� �¼ vpðb; sÞ: (22)

Three major phenomena decelerate the melt at the drill wall:

friction, resolidification at the wall, and recondensation of

the vapor. As a result, Eqs. (18) and (19) show that the posi-

tion Lkrit ¼ bkritw0 at which the drill is sealed by the melt

Lkrit ¼V0 w2

0

�

2

5m0ð1� eh0Þ (23)

depends only on the inflow m0 of mass and the melt film

thickness h0 from the drill base (e ¼ dm=w0). Shadowing of

the radiation from the drill base by a large melt film thick-

ness is also reproduced by the spatially resolved model

[Figs. 10(c) and 10(d)] and a pronounced feedback of shad-

owing onto the shape of the drill base and the acceleration of

the melt is observed.

E. Drilling with inertial confinement—helical drilling

Drilling with ns-pulse duration intensities up to I0

¼ 1010 W cm�2 can be achieved, but the molten material is

not accelerated to high velocities as in drilling with ls-pulses.

The well-known piston model and early finite element

method-calculations55 already showed that the acceleration of

the melt flow might dominate over the ablation results.

However, advanced models can handle the pronounced

sensitivity of the melt flow to the details of the intensity dis-

tribution (Fig. 11) and the nonlinear evaporation process. At

the time scale tin of submicroseconds, the melt flow is domi-

nated by the inertia of the liquid and does not escape far

away from the laser beam axis. This behavior suggests that a

moving beam can catch the recast from the preceding pulse

and explains why helical drilling works with high-precision

and almost no recast. During the pulse duration tp¼ 14 ns of

one single pulse in helical drilling, the momentum trans-

ferred to the molten material by the recoil pressure of evapo-

ration is too small to overcome the inertia of the melt

significantly. If the drill walls reach to a depth greater than

the distance the melt can move, this strategy works well and

almost all material is evaporated.

To improve precision, the pulse duration can be reduced

using short and ultrashort laser pulses.98,99 Alternatively, the

laser beam can be moved as in helical drilling. The effect of

rotational movement is demonstrated by comparing helical

and percussion drilling with identical laser parameters.

To get a circular drill profile independent of the symme-

try of the laser beam cross-section, the beam movement has

to consist of two synchronised rotations. One rotation moves

the beam axis across the work piece and another rotates the

beam itself (Fig. 12). Like the cutting edges of a mechanical

spiral bit, always the same part of the laser beam is in contact

with the wall during one rotation. If beam rotation is absent,

the cross-section of the shape of the hole reproduces the

beam profile. Both rotational movements can be realized

FIG. 11. Simulation of single pulse drilling in steel: The melt flow is domi-

nated by inertia of the liquid. Most of the molten material is evaporated and

the rest does not escape far away from the beam axis. For pulse durations (a)

tp¼ 14 ns, (b) 21 ns, (c) 27 ns the melt flow is transient and widening is not

yet pronounced. (Ep¼ 0.5 mJ, w0¼ 11 lm).

FIG. 12. Helical drilling: Principle, optical setup and beam rotation movement.



with an image rotator such as the Dove prism. During one

cycle of the prism, the beam itself is rotated two times. If the

beam is shifted parallel to the optical axis, the outgoing beam

axis moves on a circular path with a rotational frequency twice

the frequency of the Dove prism rotation. By tilting the beam

with respect to the rotational axis, the outgoing beam describes

a cone. The angle of the cone defines the entrance diameter

and the shift results in the taper of the drill. By changing the

two parameters, tilt and shift, cylindrical as well as conically

shaped holes with positive or negative taper can be produced.

If the parameters are changed during the drilling process,

counterbores are also possible. The technical implementation

of helical drilling optics has been developed by Fraunhofer-

ILT.100 A Dove prism is mounted into a high speed hollow

shaft motor (Fig. 12). The shift of the laser beam is done by

motorized movement of a laser mirror, and the tilt is done by a

motorized wedge prism. All motors are controlled by an exter-

nal control unit, which provides an interface for changing the

parameters. A half-wave plate is also integrated into the hol-

low shaft motor, which rotates the polarization synchronized

to the laser beam. The drilling optics allow rotational frequen-

cies of up to 20 000 rpm. The helical diameter has a range of

0–400 lm, and aspect ratios of the drillings up to 1:40 for

2 mm material thickness can be achieved. Conical holes with

an expansion ratio from entrance to exit in a range from 1:4 to

2:1 are possible. High-precision helical drilling is achieved

with the Dove prism optics (Fig. 13).

ACKNOWLEDGMENTS

The support of diagnosis and simulation applied to high

speed drilling under Contract No. Kr 516/30-3 and the inves-

tigations related to nonlinear coupling of gas and vapor flow

with the condensed phase under contract no. SCHU 1506/1-1

EN116/4-1 by the German Research Foundation is gratefully

acknowledged. The research of pulse burst ablation was

funded by the German Research Foundation DFG under Con-

tract No. Gi 265/4. The depicted research related to ablation

and structuring was funded by the German Research Founda-

tion DFG as part of the Cluster of Excellence “Integrative

Production Technology for High-Wage Countries”.

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review on laser drilling i. fundamentals, modeling, and simulation

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