Preprint for submission to Journal of Applied Physics

Thermal and Fluid Processes of a Thin Melt Zone during

Femtosecond Laser Ablation of Glass: The Formation of Rims by

Single Laser Pulses

Adela Ben-Yakar,1, ∗ Anthony Harkin,2 Jacqueline

Ashmore,2 Robert L. Byer,1 and Howard A. Stone2

1Applied Physics Department, Ginzton Lab,

Stanford University CA, USA 94305

2Division of Engineering and Applied Sciences,

Harvard University, Cambridge, MA, USA 02138

(Dated: September 29, 2003)

Abstract

We study the formation mechanism of rims created around femtosecond laser-ablated craters on

glass. Experimental studies of the surface morphology reveal that a thin rim is formed around the

smooth craters and is raised above the undamaged surface by about 50-100 nm. To investigate the

mechanism of rim formation following a single ultrafast laser pulse, we perform a one-dimensional

theoretical analysis of the thermal and fluid processes involved in the ablation process. The results

indicate the existence of a very thin melted zone below the surface and suggest that the rim is

formed by the high pressure plasma producing a pressure-driven fluid motion of the molten material

outwards from the center of the crater. The heat transfer calculations show a melt thickness of

the order of 1 µm and a melt lifetime of the order of 1 µs. The numerical solution of fluid motion

of the thin melt, on the other hand, demonstrate that the time for the melt to flow to the crater

edge and to form a rim is about 1 µs, which is of the same order of magnitude as the melt lifetime.

The possibility of controlling or suppressing the rim formation is also discussed.

∗email: adela@stanford.edu; tel: (650) 723-0161, fax: (650) 723-2666

1

I. INTRODUCTION

Laser micromachining has been widely studied for many materials and has recently been

extended to ultrashort laser pulses that can machine any material to very high precision at

the micron scale [1]. The key benefits of an ultrashort pulse include its ability to produce

a very high peak intensity (> 1016 W/cm2) and rapid deposition of energy into the mate-

rial. While high peak intensities allow energy delivery even into transparent high band-gap

materials such as glass, the rapid absorption of energy allows material removal, or ablation,

before significant heating of the substrate occurs.

Ultrashort pulses interact with the material by a nonlinear mechanism that is very dif-

ferent from that of conventional longer pulse lasers. The high energy density during the

ultrashort laser pulses can initiate nonlinear processes such as multiphoton absorption [2, 3],

which allows machining of any material independent of the laser wavelength. For example,

it is possible to micromachine glass with very high precision using near infrared wavelengths,

around 800 nm, at which the linear absorption properties of glass are very weak.

While the nonlinear effects and short time scales associated with ultrashort laser ablation

are believed to provide a non-thermal micromachining of glass materials, in some cases

complicated surface morphologies result [4, 5]. Detailed experimental studies have shown

that the laser ablation process might actually be thermal in nature, even in the femtosecond

operating range [5]. Our preliminary estimates show that a very small amount of energy, less

than 3% of the incident energy, might be deposited in the undamaged part of the borosilicate

glass as thermal energy and so create a transient shallow molten zone below the ablated area

(where there is the expanding plasma) [5]. The molten material rearranges (flows) due to

forces acting at the surface of the melt. Another recent study also measured that about 8%

of the incoming energy was thermalized and transmitted to the undamaged part of a quartz

material when irradiated with an ultrafast laser pulse [6].

In order to improve the quality and precision of ultrafast laser micromachining, it is

important to understand, model and quantify the thermal nature of the process and any

material rearrangement that occurs. In this paper, we provide scaling arguments for the

depth of the melted layer and model its motion during the lifetime of the melt. If the melt

lifetime is long enough, forces acting on the liquid will be able to drive the molten material

from the center to the edges of the crater, which creates an elevated rim around the ablated

2

crater. Multiple laser pulses show that this rim formation causes a surface roughness and

therefore reduces the precision of an ultrafast laser micromachining process.

Two main forces might affect the flow of a molten layer below the expanding plasma:

1) thermocapillary forces (Marangoni flow) [7, 8] and 2) forces exerted by the pressure of

the plasma above the surface [9, 10]. Thermocapillary flow is induced by the temperature

gradient on the surface which is expected to follow the Gaussian beam intensity profile of

the laser. In studies of laser texturing of silicon surfaces in the absence of ablation, the rim

formation was attributed to the thermocapillary flow in thin films created by nanosecond

laser pulse heating [7]. The temperature gradient on the surface creates surface tension

gradients that drives material from the hot center to the cold periphery. This response is

expected in most materials where the surface tension, γ, decreases as the fluid gets hotter

(dγ/dT < 0). However, in the case of glass dγ/dT is positive [11]. Consequently, such a

thermocapillary flow in laser irradiated glass surfaces would actually drive fluid from the cold

periphery to the hot center of the melt in contrast to what was observed in our experiments.

In addition, the effect of thermocapillary flow in glass is expected to be negligible because

of its high viscosity, which leads to a flow time scale much longer than the typical melt time

scale, as detailed further below.

On the other hand, a hydrodynamic force due to the pressure gradients exerted by the

plasma onto the molten material might play an important role in the rim formation. A

gradient of ablation pressure on the molten surface can induce a lateral melt flow to the

periphery. The pressure gradients are expected to be particularly large at the edges of

the ablated crater, which should be close the plasma/air interface. Because of these large

pressure gradients, we might expect a melt flow to the periphery and rise of a thin rim at

the edges of the melted surface much like a splash of a fluid. Although some of aspects

of this mechanism have been discussed qualitatively in the literature for nanosecond laser

irradiation of metals [9], in this paper we provide a detailed model of the flow processes

associated with the first pulse and so give the time evolution for the profile of the melt

surface. The mathematical model of the physical processes gives rise to a rim topography,

though has one significant unknown which is the pressure gradient along the surface that

results from the plasma. This uncertainty primarily affects time scales and not the physics.

We first present typical experimental results of the surface morphology of craters ablated

using single and multiple femtosecond laser pulses with an emphasis on the rim formation

3

around the craters (Section II). We then discuss the level of the material heating and

estimate the variation of melt depth with time for a single laser pulse (Section IIIA). Finally,

we present a thin film model to estimate the characteristic time scales of various mechanisms

causing the melt to flow outwards to create the rim and show representative numerical

simulations of fluid flow in a thin molten layer (Section III B). We conclude with some ideas

for suppressing formation of the rim (Section IV).

II. EXPERIMENTAL RESULTS

Our experimental studies focus on the surface morphology of borosilicate glass

(BorofloatTM) ablated using single femtosecond laser pulses. We irradiated the glass samples

with 800-nm 100-fs pulses from a regeneratively amplified Ti:sapphire laser. The laser was

focused with a 250-mm focal-length lens to a spot size of about 30µm. The surface of the

sample was positioned to be normal to the direction of the incident beam. We performed

these experiments in air at atmospheric pressure. Following irradiation, the samples were

analyzed with a scanning electron microscope (SEM). The basic results were reported re-

cently [5] and we summarize them here with some additional features as they form the basis

for the theoretical and modeling considerations given in Section III.

Figure 1 presents three SEM images of crater rims produced by an average laser fluence of

34 J/cm2. The first image (Fig. 1a) shows a thin circular rim around a nearly smooth crater

following a single pulse of the laser on a flat glass substrate. It is the resemblance of this

rim to a “resolidified splash” of a molten layer that originally motivated our investigation

of the dynamical processes that result in the formation of the rim.

The second image (Fig. 1b) shows that when a second pulse irradiates a previously formed

rim, a new rim is formed inside the original rim. The distance between the two rims is

approximately equal to the wavelength-λ of the light, which suggests that diffraction of

light plays an important role [12]. Upon close inspection of the area between the pulses,

there is a wave pattern apparently due to the diffraction-induced modulation of the second

laser pulse.

When a third pulse irradiates two previously overlapping craters (Fig. 1c), micrometer-

scale organized features appear along the rim. The smaller scale ripple-like features that are

evident circumferentially along the rim, which is basically semicircular in cross-section, are

4

presumably a manifestation of a Rayleigh-capillary instability familiar from the disintegra-

tion of fluid filaments. This aspect of the surface evolution has not been studied. As shown

in Fig. 2, the interplay between rim formation and diffraction results in a perceptible sur-

face roughness when attempting to micromachine channels in a glass substrate by scanning

a pulsed laser across the surface.

A detailed AFM study of a single crater is shown in Fig. 3. The rim is raised by about

50 − 100 nm above the surface and the maximum depth of the ablated crater is about 400-

600 nm depending on the laser fluence [13].

In order to understand and control the micromachining process using ultrafast lasers, it is

necessary to investigate the formation mechanism of the surface microfeatures. To address

these issues, we first have to examine the rim formation of a single laser pulse, which is the

main focus of this paper.

III. THEORETICAL MODELLING AND DISCUSSION

A theoretical model is described to characterize the thermal and flow processes associated

with the first laser pulse. The goal is to identify the nature of the rim formation around the

laser ablated craters and suggest ways (such as modification of the laser profile) to eliminate

it. The laser deposits energy to the substrate which melts the glass. To estimate the melt

thickness, and its variation in time, the optical light penetration and heat conduction into

the material are studied (Section IIIA). A thin film model of the molten surface layer is

then introduced to explore the temporal evolution of the melt surface (Section III B).

A. Heat Transfer Calculations

In this section, we estimate the thickness and the lifetime of the molten layer and present

a model for the time dependent variations of the melt thickness. To achieve this, we first

calculate the fraction of the laser energy deposited in the bulk of the material as heat by

analyzing 1) the fraction of the incoming energy that is absorbed by the material and 2) how

this energy dissipates from the illuminated regime (i.e., partition sequence of the absorbed

laser energy).

In dielectric materials, the incident laser beam is absorbed by electrons through nonlinear

5

processes (multiphoton and avalanche ionization) [2, 3] and then transferred to the lattice

on the time scale of few picoseconds. At this stage, a high pressure plasma is formed above

the surface; the plasma expands and removes the ablated material away from the surface.

Dissipation of the energy deposited in the substrate begins only after the laser pulse

is gone and when the energy is transferred from the high energy electrons to the lattice.

Numerical simulations of the fluid and thermal dynamics of a laser-induced plasma on an

aluminum target irradiated with 100 picosecond laser pulses show that a few microseconds

after irradiation, ∼ 70 % of the absorbed energy is used by the expanding plasma to move

the ambient gas [14]. About 20 % of the absorbed energy is lost in radiation and only 10 %

of it remains in the target as thermal energy.

In the case of femtosecond laser ablation of glass, a rather complicated and time consum-

ing numerical solution is required to estimate the partition of the absorbed laser energy. In

this paper, we propose to use the optical light penetration depth to estimate the fraction of

the incoming laser energy deposited in the glass as heat (thermal energy) [13].

1. Initial Energy Deposition Depth

We assume that the absorbed laser energy is deposited uniformly in a layer defined by the

penetration of light as illustrated in Fig. 4. The light intensity is attenuated exponentially

with depth z according to the Beer-Lambert law, which provides a convenient means to

quantify the penetration depth of the absorbed laser energy. Then the absorbed laser fluence

as a function of depth, Fa(z), is given by

Fa(z) = AF0 exp

(

− z

α−1eff

)

, (1)

where A is the surface absorptivity, α−1eff is the effective optical penetration depth, and F0 is

the average laser fluence (energy per unit area). Let us next summarize how we determine

the values of A and α−1eff from experimental measurements.

The absorptivity of glass depends on the intensity of the laser irradiation and varies

with time during the duration of the laser pulse. When glass is exposed to high intensity

ultrashort pulses, its reflectivity rises with time as the plasma density increases [2]. Once

the critical surface plasma density is formed, any further incident laser energy is reflected

back from the surface due to an induced skin effect. Perry et al. [2] showed that when

6

the incident laser fluence is high above threshold (F > 5 − 10Fth where Fth ≈ 2 J/cm2 for

fused silica), a plasma with a critical density can be achieved early in the pulse and a large

portion of the energy is reflected back from the sample. At a laser irradiance of 1014 W/cm2

they calculate the reflectance of fused silica to be around 70%. We used a similar laser

irradiance, and therefore we assume that on average 30% of the incident energy is absorbed

by the borosilicate glass, namely A = 0.3.

For the effective optical penetration depth, we use experimental data from our previous

measurements [13]. We determined α−1eff = 224 nm by calculating the slope of the linear

relationship between the ablation depth, ha, and the logarithm of the incident laser fluence,

F0, expressed by

ha = α−1eff ln

(F0

Fth

)

. (2)

Here, Fth = 1.7 J/cm2 is the measured threshold fluence for borosilicate glass when 200-fs

and 780-nm laser pulses are used. Since at laser fluences below Fth no ablation occurs, the

material is ablated to a depth, z = ha, where the laser fluence drops to Fth. Therefore, by

substituting Fa(z = ha) = AFth in Eq. (1) we can obtain the expression given in Eq. (2).

The distance, where the laser fluence decreases to 1/e of its value, is interpreted as the

“effective optical penetration depth” (α−1eff ) in accordance with the Beer-Lambert absorption

law. Using Eqs. (1) and (2), we can write the variation of the absorbed laser fluence with

depth, Fa(z), in terms of Fth and ha,

Fa(z) = AFth exp

(

−z − ha

α−1eff

)

, (3)

where z = 0 corresponds to the location of the flat surface away from the irradiated zone.

We assume that the absorbed laser energy that penetrates beyond the ablation depth,

ha, remains in the material as thermal energy. Since the laser fluence at the ablation

depth is equal to AFth, the thermalized energy remaining in the material as heat is Eheat =

AFth(πw20), where w0 is the 1/e2 laser beam radius at the surface. Thus, we find that a fixed

amount of laser fluence, approximately Fheat = AFth = 0.5 J/cm2, heats the undamaged part

of the material independent of the incident laser fluence when it is well above the threshold.

This means that, in our experiments, presented in Fig. 1, less than 2% of the total incident

laser pulse energy goes into heating of the material (the value Fheat = 0.5 J/cm2 is about

1.5% of the incident laser fluence of Fheat = 34 J/cm2).

7

2. Initial Temperature Distribution

The initial energy deposition described by Eq. (3) produces an initial temperature distri-

bution in the glass, T (z, 0), with an exponential profile

T (z, 0) − T∞ =Fa

ρCpα−1eff

=Fheat

ρCpα−1eff

exp

(

−z − ha

α−1eff

)

(4)

where Fheat = AFth is the portion of the absorbed laser fluence that goes into heating, T∞

is the initial temperature, ρ is the density of the substrate, and Cp is the heat capacity.

Melting occurs when the local temperature exceeds the temperature, Tm ≈ 1500K [15]. The

value Tm used here is the working point temperature of glass, defined as the temperature at

which the glass can readily be formed, which corresponds to a viscosity of 103 Pa.s. Since

the glass materials do not have a latent heat of melting, all of the absorbed energy goes

into melting. We can therefore calculate the initial melting depth, hm(t = 0) = zm − ha,

from hm = α−1eff ln(Fheat/(Tm −T∞)ρCpα

−1eff ) where Cp = 1250 J/kgK (see the thermophysical

properties of glass in Table I); we have chosen to evaluate all physical parameters at the

mean temperature of 900K. This calculation yields an initial molten layer thickness of

hm(t = 0) ≈ 424 nm.

In the above discussion of energy deposition, we assume that the incident laser beam

and the resulting initial temperature distribution inside the glass material attenuate with

depth in accordance with the Beer-Lambert law (see the illustration in Fig. 4). When the

material is illuminated with an ultrashort laser pulse, a high-temperature and high-pressure

plasma is formed above the surface, presumably down to the ablation depth. The ablated

material is removed by the expansion of the plasma. Below the plasma there is a thin zone

of a molten glass with an initial thickness of hm = z(Tm) − ha that can be calculated using

Eq. (4). The molten region grows, then decays according to conduction process and the

shape of the molten layer changes. Both these effects are described in more detail below.

3. Heat Conduction and Variation of Melt Thickness

Following the cessation of ultrafast energy input, the melting process continues as the

heat flows out of the region where the initial energy is deposited. Diffusion of the thermal

energy determines the movement of the melting front and therefore the variation of the melt

thickness.

8

We calculate the heat flow out of the region where the initial energy is deposited by solv-

ing a one-dimensional (1-D) heat conduction equation with an initial temperature profile

described by Eq. (4). The cooling at the top of the melt zone is assumed to be negligi-

ble because of the presence of the high temperature plasma. During the expansion, the

plasma cools in tens of microseconds from some very high initial temperature to the ambi-

ent temperature. So all of the heat loss is assumed to take place through the solid, as the

rearrangement of the molten layer takes place on shorter time scales.

The one-dimensional heat conduction model can then be described by

∂T

∂t=

∂

∂z(D

∂T

∂z), z > ha (5)

∂T

∂z= 0 at z = ha (6)

T = T∞ as z → ∞ (7)

T (z, 0) = T∞ +Fheat

ρCpα−1eff

e−(z−ha)/α−1

eff (8)

where ha is the ablation depth or, in other words, the depth where the molten region begins.

As a first approximation, we assume that the heat capacity, Cp, the thermal conductivity, k,

the density, ρ, and therefore the thermal diffusivity, D = k/ρCp, are constants at an average

temperature of 900K.

To find an analytical solution for this heat conduction problem, nondimensionalize vari-

ables as:

T =ρCpα

−1eff

Fheat(T − T∞), z =

z − ha

α−1eff

, t =D

(α−1eff )2

t. (9)

The nondimensional problem statement is then

∂T

∂t=

∂2T

∂z2, z > 0 (10)

∂T

∂z= 0 at z = 0 (11)

T = 0 as z → ∞ (12)

T (z, 0) = f(z) = e−z, (13)

which has the solution

T (z, t) =1√4πt

∫∞

0

f(ζ)[

e−(z+ζ)2/4t + e−(z−ζ)2/4t]

dζ. (14)

9

The solution for f(ζ) = e−ζ can be expressed in terms of the complementary error function,

erfc(s), as:

T (z, t) =et

2

[

ezerfc

(z + 2t√

4t

)

+ e−zerfc

(−z + 2t√4t

)]

. (15)

We can now calculate the temperature distribution in the material that is not ablated (i.e.

below the plasma) using the analytical solution given in Eq. (15). The variation of the melt

depth as a function of time follows from hm(t) = zm −ha = α−1eff z as where T (zm, t) = Tm. A

plot of the melt depth as a function of time is shown in Fig. 5 for two different laser fluences

thermalized in the material as heat; 1)Fheat = AFth = 0.5 J/cm2 and 2)Fheat = 1.0 J/cm2

since it is possible that radiation from the high temperature plasma also contributes to

the melting of the material. The numerical results show that for the material parameters

representative of the experiment, the melting process continues for 0.25-0.75µs and then

solidification begins. These calculations provide us estimates for the order of magnitude of

two important characteristic scales of the melt zone:

• An average melt depth, 〈hm〉, of the order of 1µm.

• An average melt lifetime, tm, of the order of 1µs.

With these estimates in hand, we will discuss in Section B which hydrodynamic forces acting

on the melt have enough time to rearrange the liquid to form a rim within the melt lifetime.

B. Fluid Dynamics Calculations

1. Thin Film Model

In order to examine hydrodynamic conditions under which a crater partially filled with

molten glass can form rims reaching the heights measured in experiments, we develop a

one-dimensional model of the fluid motion as illustrated in Fig. 6. It is natural to work with

cylindrical coordinates. The free surface of the molten glass is described by z = h(r, t) and

the boundary between the liquid and the solid substrate is given by the time-independent

profile z = b(r). I don’t know quite how to state the justification for this. - Tony

The shape of the thin liquid film evolves owing to variations in the surface tension γ along

the free surface and because of the very high pressure ppl(r) of the plasma above the free

surface. Because the melted region has a typical height much less than the typical width of

10

the laser ablated craters, the lubrication approximation (see Appendix A) can be used to

obtain an evolution equation for the time-dependent profile of the free surface h(r, t):

∂h

∂t+

1

r

∂

∂rr

(h − b)2

2µ

dγ

dr︸ ︷︷ ︸

− (h − b)3

3µ

dppl

dr︸ ︷︷ ︸

+(h − b)3

3µ

∂

∂r

(γ

r

∂

∂r

(

r∂h

∂r

))

︸ ︷︷ ︸

= 0, (16)

Marangoni Pressure Curvature

where µ is the liquid viscosity and the different flux contributions have been labelled. The

first term in brackets accounts for motion caused by the surface tension gradients due to

the uneven heating of the surface (thermocapillary or Marangoni-driven flow). The second

term accounts for the motion due to the pressure gradients exerted by the plasma onto the

molten material, and the third term in (16) represents surface tension effects, which enter

as the product of surface tension and surface curvature (the latter has been linearized).

Also, because of the very complicated dynamics of the plasma, for which the details are

unknown, we assume in our model that the plasma pressure field, ppl, at the surface of the

molten material is independent of time and therefore we use a time-averaged plasma pressure

distribution, i.e. ppl(r) only.

There are three assumptions needed to justify the use of Eq. (16): (i) The film must

be thin, i.e. 〈hm〉/L � 1, where 〈hm〉 is the average melt depth and L is typical of the

radial dimensions of the molten region. (ii) The Reynolds number Re for the film flow (see

Appendix A) must be small, Re = ρ〈hm〉3〈ppl〉/(µ2L) � 1. (iii) The flow must be quasi-

steady which requires the time for “viscous diffusion” across the thin layer, ρ〈hm〉2/µ, to be

small compared to the time scale for film evolution. We verify these assumptions below.

When a surface temperature distribution, Ts(r), is imposed, then the surface tension

varies according todγ

dr=

dγ

dTs

dTs

dr. (17)

We assume γT = dγ/dTs is constant and neglect the effect of temperature variations on

the viscosity. Using Eq. (16) we can estimate characteristic time scales associated with the

Marangoni flow (τM), and the pressure-driven flow (τp),

Marangoni flow: τM ≈ µL2

γTTm〈hm〉(18)

Pressure-driven flow: τp ≈µL2

〈ppl〉〈hm〉2(19)

11

where 〈hm〉 is an average melt depth, L is a typical radial dimension, and 〈ppl〉 is an average

plasma pressure. The thermophysical properties of borosilicate glass are summarized in

Table I. For an average melt depth of 〈hm〉 ≈ 1µm, as estimated in the heat transfer

calculation in Section A, a typical crater radius of L = 5µm, and an average plasma pressure

of 〈ppl〉 ≈ 1000 atm (the typical plasma pressure drops from millions of atmospheres to about

10 atm during the first microsecond of its expansion [16]), we obtain τp ∼ 10−4 sec and

τM ∼ 10−1 sec. It should be noted that using L = 5µm gives timescales for τp and τM that

are too slow by one or two orders of magnitude in cases where the plasma pressure gradient,

ppl/L, is very large only near the edge of the crater. In such cases, the characteristic length

scale L would better be described by the much shorter distance over which the pressure

gradient changes. In any event, we have

τM

τp

= O

(〈ppl〉〈hm〉γT Tm

)

≈ 103 (20)

and hence the characteristic time scale for Marangoni flow is two to three orders of magnitude

longer than that of pressure-driven flow, and τp � τM even if the peak pressure is lowered

more than a factor of ten. It is clear from this estimate that the large plasma pressure above

the free surface acts to move the fluid much more quickly than do surface tension gradients.

This result suggests that once an expanding plasma is formed above the glass surface during

laser ablation [? ], the spatially varying plasma pressure will control the evolution of the

free surface at the interface.

This mechanism for rim formation contrasts with that in laser texturing of silicon surfaces

in the absence of ablation, which is attributed to the Marangoni flow in thin films created by

nanosecond laser pulse heating [7]. The idea that thermocapillary effects do not contribute to

the observed formation of a rim at the edge of the melt zone in our experiments is supported

by the observation that the surface tension coefficient of borosilicate glass is positive [11] in

contrast to the usual negative values of most pure liquids. Therefore, as discussed earlier,

thermocapillary (Marangoni) flow in laser irradiated glass surfaces would be expected to

drive fluid from the cold periphery to the hot center of the melt, which is not what is

observed in the experiments shown in Fig. 1.

We close by verifying the three assumptions used to obtain Eq. (16). First, the molten

region must be thin 〈hm〉/L � 1. Since 〈hm〉 ≈ 1µm and L ≈ 5µm, this condition is

reasonably well satisfied. In fact, errors in using Eq. (16) are O ((〈hm〉/L)2), which further

12

justifies the use of the lubrication approximation for this thin film flow. Second, we require

that the effective Reynolds number, Re, for the film flow must be small. Using typical

parameter values and viscosity at T = 2000K we obtain

Re = O

(ρ〈hm〉3〈ppl〉

µ2L

)

≈ 10−4. (21)

Note that even if the pressure gradient that drives the flow was increased three orders of

magnitude, the low-Reynolds-number assumption, equation (21), is still satisfied. Finally,

the third assumption requires that the viscous effects must act quickly to establish the

velocity profile across the thin region. This time is O(ρ〈hm〉2/µ) ≈ 10−11 s, which is much

faster than the time scale, τp, of fluid motion. Hence, the three principle assumptions in the

fluid dynamics calculation are verified.

2. Numerical Results and Discussion for Pressure-Driven Flow

To perform numerical simulations of the evolution of the molten glass we first nondimen-

sionalize the time, length, height, and plasma pressure by characteristic values

S = t/τp, R = r/L, H = h/〈hm〉, B = b/〈hm〉, Ppl = ppl/〈ppl〉.

Since the key estimate of the previous section, Eq. (20), indicates plasma pressure gradients

primarily drive the fluid flow, then we take γ = constant (the value is given in Table I),

neglect the Marangoni term from Eq. (16), and arrive at the following nondimensional

evolution equation for the free surface height, H(R, S),

∂H

∂S− 1

R

∂

∂RR

[(H − B)3

3

dPpl

dR+ Λ

(H − B)3

3

∂

∂R

(1

R

∂

∂R

(

R∂H

∂R

))]

= 0. (22)

The nondimensional parameter Λ is given by

Λ =γ〈hm〉〈ppl〉L2

≈ 2.8 × 10−5.

Even though Λ is small, we will keep the curvature term in the numerical simulations. We

also assume for simplicity that the melt-solid boundary is time independent, i.e., B(R) only.

In Fig. 7 and Fig. 8 we numerically explore the behavior of solutions of Eq. (22). The

rim height is the feature of the solutions in which we are most interested. More specifically,

we investigate how changing the plasma pressure profile, Ppl(R), or modifying the substrate

13

shape, B(R), affect the rim height. Since the plasma pressure and bottom substrate are not

well characterized experimentally, then it is important to at least understand how changes

in these quantities affect the crater and rim formation.

Figure 7 shows the numerical solutions of pressure-driven melt flow for two different

plasma pressure profiles. It is not well understood how the plasma pressure distribution

depends upon the laser beam profile, and because of this uncertainty, we studied the effect

of two different pressure profiles, one with a Gaussian distribution and another with a steeper

gradient at the edges. The solid curve in the top plot describes a pressure profile of Ppl(R) =

exp(−R10) and the dashed curve is the gentler distribution Ppl(R) = exp(−R2). For the

free surface and bottom substrate profiles used in the numerical simulations, we require that

the initial thickness of molten glass in the center of the crater reflects a reasonable value

from the melt depth plot of Fig. 5. Therefore, the free surface shape of the molten glass

in the numerical simulations is initially described as H(R, 0) = −0.5 exp(−15R10), and the

bottom substrate is given by B(R) = −0.01 − 0.9 exp(−R10). At the center of the crater,

R = 0, the melt depth is H(0, 0) − B(0) ≈ 0.4, or roughly 0.4µm, which based on Fig. 5

is a reasonable melt depth. Solving Eq. (22) numerically for each of the plasma pressure

profiles to a dimensionless time of S = 0.08 gives the two solutions shown in the bottom

plot of Fig. 7 (solid and dashed curves). The rim height of the solid curve solution is larger

than that of the dashed curve solution. These rim heights are consistent with the plasma

pressure gradient of the solid curve in the top plot being larger than that of the dashed

pressure curve. Converting the rim heights to dimensional values gives heights of roughly

210nm for the solid rim and 65nm for the dashed rim after a time of roughly 2µs. There

is consistency between this range of rim heights, the time-scale of their formations, and

rim heights observed experimentally, though it should be noted that the plasma pressure

is not known with a large degree of certainty. We believe that the order of magnitude is

reasonable and so the qualitative and quantitative trends offer insight into the details of the

film rearrangement.

In Fig. 8 we explore how changing the shape of the bottom substrate, B(R), while

keeping all other parameters the same, affects the rim height. A plasma pressure profile of

Ppl(R) = exp(−R10) was chosen for the two simulations to be performed. The initial shape

of the free surface is again given by H(R, 0) = −0.5 exp(−15R10). For the first simulation

(dashed curve), the bottom shape was chosen to be B(R) = −0.01 − 0.9 exp(−R10), which

14

reproduces the solid curve solution in Fig. 7. The bottom shape for the second simulation

(solid curve) is given by B(R) = −0.01 − 0.9 exp(−0.3R20), which provides more fluid at

the edges of the domain than in the first simulation. The solutions are shown at S = 0.04,

or roughly 1µs. The solution corresponding to the solid curve has a taller rim height than

that of the dashed curve. The reason for this is that the fluid flux is proportional to

(H − B)3, as seen in Eq. (22), and there was more fluid available to be transported in the

second simulation. So, although we do not know exactly how the melt thickness changes

in the experiments, these simulations show that a rim of a reasonable height may still be

formed within the order of magnitude of the melt lifetime and that the typical results are

independent of the precise forms of the plasma and bottom profiles.

As a final remark, we note that the formation of transient rim-like structures occur in

other fluid dynamics problems, such as the splash of drop on a solid surface or on a liquid

layer (e.g. the famous Edgerton photograph of a splashing drop of milk). In particular,

in the case of drop splash, numerical calculations for thin-film dynamics in the absence of

viscous effects, which is very different from the dynamical situation we discuss, are reported

by, among others, Yarin and Weiss [17], and experiments with drops splashing following

their impact with thin layers are reported in Shin and McMahon [18].

IV. HOW TO SUPPRESS THE RIM FORMATION?

We can now answer the question of “how to suppress the rim formation for clean laser

processing” or in other words seek “how to achieve clean borders of the irradiated spot”.

If the pressure-driven flow time, τp, is long enough, the pressure induced forces acting

on the fluid will not be able to drive the molten material from the center to the edges of

the crater during the melt lifetime. From the parametric dependence shown in Eq. (19), τp

depends on the viscosity, µ, the average thickness of the molten layer, 〈hm〉, and the average

pressure above the surface, 〈ppl〉. Smaller melt thickness or smaller average pressures above

the surface lead to longer times for melt flow. The melt lifetime tm ∝ (α−1eff )2/D and therefore

a thinner melt thickness can be achieved by reducing the optical penetration depth, α−1eff (e.g.

using shorter laser wavelengths or/and shorter pulse durations). Both the melt thickness

and lifetime will be further reduced by the decreased amount of the absorbed thermal energy

(Fheat ∝ Fth) because the threshold fluence decreases with both laser wavelength and pulse

15

duration [19, 20]. In addition, materials with higher thermal diffusivities can reduce the

melt lifetime by more rapidly removing the absorbed thermal energy.

Another way of suppressing the rim formation may be modifying the plasma pressure

profile. As shown in Fig. 7, with a less steep Gaussian pressure profile the formation of a

rim takes longer time. It may be possible that for even a less steep pressure gradient near

the edge, a rim cannot be formed during the lifetime of the molten material. Therefore, if

one can modify the pressure profile, by modifying for example the laser spatial beam profile,

it may be possible to achieve a cleaner border of the irradiated spot. Although it is easy to

control the intensity distribution of the laser beam profile, it is not clear, however, how this

may impact the pressure distribution. This remains a subject for further investigation.

V. CONCLUSIONS

The morphology of the single-shot ablated areas revealed a smooth and shallow crater

surrounded by an elevated rim. From these experimental observations, conclusions could be

drawn about the ablation mechanism of borosilicate glass. We argued that a very thin melt

zone existed during the ablation process and calculated the thermal and flow properties of

this thin melt zone. In these calculations, several characteristic time constants associated

with ablation, melting, and flow processes were determined. The comparative values revealed

that a flow of fluid driven by a plasma-induced pressure gradient localized near the radius

of the laser pulse, with reasonable values of the plasma pressure, would have enough time

to move melted material towards the edge and so deposit a thin rim around the ablated

area. Physically based estimates of the melt lifetime and the pressure-driven flow process

then suggest ways to suppress this rim formation.

Acknowledgments

The work has been supported by the TRW research fund and the Harvard NSEC. The

authors gratefully acknowledge the contributions of Prof. Eric Mazur, Dr. Mengyan Shen,

and Dr. Catherine Crouch to this investigation.

16

VI. APPENDIX

APPENDIX A: DERIVATION OF THE EQUATION OF MOTION FOR THE

THIN FILM MODEL

In this appendix we present a short derivation of the partial differential equation describ-

ing viscous fluid flow in a thin film [21–23]. Suppose that the position of the free surface of

the fluid is denoted z = h(x, y, t), and the shape of the time-independent bottom substrate

is denoted z = b(x, y). We assume that the flow is incompressible and described by the

Navier-Stokes equations. Then, the fluid velocity u and pressure p satisfy

∂u

∂t+ u · ∇u = −1

ρ∇p +

µ

ρ∇2u, (A1)

∇ · u = 0. (A2)

For some of the estimates below, it is convenient to denote the mean film thickness hm =

O(h−b) and the average, or typical, fluid pressure by 〈p〉. Note that in (A1) we are neglecting

the gravitational body force since for the small length scales characteristic of the rims in the

experiments, ρghm/〈p〉 � 1. The boundary conditions to be satisfied are: no slip on the

solid substrate, the normal and tangential stress balances across the free surface, and the

kinematic boundary condition on the free surface.

For pressure-driven flows on the scale L typical of the flow direction, we expect a typical

velocity along the film to have magnitude u = O(h2m〈p〉/(µL), in which case we define

the Reynolds number for the thin-film flow as Re = ρh3m〈p〉/(µ2L). Let us suppose first

that the velocity field is u(x, y, z, t) = (u, v, w). Then, under the thin-film (lubrication)

approximation we assume hm/L � 1 and inertial effects are negligible, which is equivalent

to the requirement that the Reynolds number is small, Re � 1, so that the Navier-Stokes

and continuity equations reduce to

∇2p = µ∂2u2

∂z2(A3)

∂p

∂z= 0 (A4)

∇2 · u2 +∂w

∂z= 0 (A5)

where ∇2 = (∂x, ∂y) and u2 = (u, v). Within the lubrication approximation, the boundary

conditions are

17

p = ppl − γκ on z = h (normal stress) (A6)

µ∂u

∂z= ∇2γ on z = h (tangential stress) (A7)

u = v = w = 0 on z = b (no slip) (A8)

∂h

∂t+ u2 · ∇2h − w = 0 on z = h (kinematic condition) (A9)

where ppl is the plasma pressure above the free surface, γ is the surface tension of the

interface, and κ is twice the mean curvature of the interface. We have assumed ppl to be

time-independent in the numerical simulations reported in this paper.

Equation (A4) and the normal stress balance give the local pressure in the liquid to be

p = ppl − γκ, b(x, y) ≤ z ≤ h(x, y, t) (A10)

and we will use a linearized expression for the mean curvature term, κ = ∇22h. Substituting

the pressure into Eq. (A3), integrating the result twice with respect to z, and using the

no-slip and tangential stress boundary conditions yields

u2 =1

µ

(z2

2− b2

2+ bh − zh

)(∇2ppl −∇2(γ∇2

2h))

+1

µ(z − b)∇2γ. (A11)

Integrating the continuity equation shows that

∂h

∂t+ ∇2 · q = 0 with q =

∫ z=h

z=b

u2 dz, (A12)

where q is the flux vector. Substituting (A11) for u2 to calculate the depth-averaged flux,

q, we then obtain the evolution equation for the height, h, of the thin film:

∂h

∂t+ ∇2 ·

[(h − b)2

2µ∇2γ − (h − b)3

3µ

(∇2ppl −∇2(γ∇2

2h))]

= 0. (A13)

In cylindrical coordinates and assuming an axisymmetric shape, h(r, t), Eq. (A13) becomes

∂h

∂t+

1

r

∂

∂rr

[(h − b)2

2µ

dγ

dr− (h − b)3

3µ

dppl

dr+

(h − b)3

3µ

∂

∂r

(γ

r

∂

∂r

(

r∂h

∂r

))]

= 0. (A14)

18

[1] J. Meijer, K. Du, A. Gillner, D. Hoffmann, V. S. Kovalenko, T. Masuzawa, and A. Ostendorf,

Annals of the CIRP 51/2/2002, 1 (2002).

[2] M. D. Perry, B. C. Stuart, P. S. Banks, M. D. Feit, V. Yanovsky, and A. M. Rubenchik, J. of

Appl. Phys. 85, 6803 (1999).

[3] C. B. Schaffer, A. Brodeur, and E. Mazur, Meas. Sci. Technol. 12, 1784 (2001).

[4] S. Ameer-Beg, W. Perrie, S. Rathbone, W. Wright, J. Weaver, and H. Champoux, Appl.

Surface Phys. 127-129, 875 (1998).

[5] A. Ben-Yakar, R. L. Byer, A. Harkin, J. Ashmore, H. A. Stone, M. Shen, and E. Mazur,

submitted to Appl. Phys. Lett. (2003).

[6] F. Ladieu, P. Martin, and S. Guizard, Appl. Phys. Lett. 81, 957 (2002).

[7] T. Schwarz-Selinger, D. G. Cahill, S. C. Chen, S. J. Moon, and C. P. Grigoropoulos, Phys.

Rev. B 64, 155323 (1999).

[8] D. A. Willis and X. Xu, J. Heat Transfer 122, 763 (2000).

[9] V. N. Tokarev and A. F. H. Kaplan, J. Phys. D: Appl. Phys. 32, 1526 (1999).

[10] A. Ben-Yakar, A. Harkin, J. Ashmore, M. Shen, E. Mazur, R. L. Byer, and H. A. Stone,

Proceedings of the SPIE, The International Society for Optical Engineering 4977 (2003).

[11] W. D. Kingery, J. Am. Ceram. Soc. 42, 6 (1959).

[12] M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999), 7th ed.

[13] A. Ben-Yakar and R. L. Byer, Proceedings of the SPIE, The International Society for Optical

Engineering 4637, 212 (2002), also submitted to J. of Appl. Phys.

[14] F. Vidal, S. Laville, B. Le Drogoff, T. W. Johnston, M. Chaker, O. Barthelemy, J. Margot,

and M. Sabsabi (Annual meeting of OSA - Optical Society of America, 2001).

[15] R. H. Doremus, Glass Science (John Wiley and Sons Inc., New York, 1994), 2nd ed.

[16] F. Vidal, S. Laville, T. W. Johnston, O. Barthelemy, M. Chaker, B. Le Drogoff, J. Margot,

and M. Sabsabi, Spectrochimica Acta Part B 56, 973 (2001).

[17] A. L. Yarin and D. A. Weiss, J. Fluid Mech. 283, 141 (1995).

[18] J. Shin and T. A. McMahon, Phys. Fluids 2, 1312 (1990).

[19] B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, Phys.

Rev. B 53, 1746 (1996).

19

[20] T. Q. Jia, Z. Z. Xu, R. X. Li, B. Shai, and F. L. Zhao, Appl. Phys. Lett. 82, 4382 (2003).

[21] A. Oron, S. H. Davis, and S. G. Bankoff, Rev. Mod. Phys. 69, 931 (1997).

[22] T. G. Myers, SIAM Rev. 40, 441 (1998).

[23] H. A. Stone, Nonlinear PDE’s in Condensed Matter and Reactive Flows (2002), p. 297.

[24] R. W. Cahn, P. Haasen, and E. J. Kramer, Materials Science and Technology: A Comprehen-

sive Treatment (John Wiley and Sons Inc., 1998).

[25] R. L. David, ed., CRC Handbook of Chemistry and Physics (CRC Press, 1999-2000), 80th ed.

20

APPENDIX B: LIST OF TABLES

Table I Thermophysical properties of borosilicate glass (BorofloatTM). The chemical compo-

sition includes 81% SiO2, 13% B2O3, 2% Al2O3, 4% Na2O.

21

APPENDIX C: LIST OF FIGURES

FIG. 1 SEM images of crater rims generated by (a) one laser pulse (b) two overlapping laser

pulses, and (c) three overlapping laser pulses of 800-nm and 100-fs. The laser fluence

was F0 = 34 J/cm2. The numbers correspond to the order of the incident laser pulses.

FIG. 2 An SEM image of a microchannel created using 200-fs and 780-nm laser pulses of

F0 = 23 J/cm2 focused to a spot size of about 12µm. The image shows the microscale

surface roughness created by the rims of overlapping laser pulses when scanning the

laser across the surface.

FIG. 3 An AFM study of a single crater ablated with a 200-fs and 780-nm laser pulse of

F0 = 12.6 J/cm2. (a) The AFM image. b) The ablation profile at the center-line of

the crater. The ablation depth, ha, is normalized by an average melt depth of 〈hm〉 =

1µm and the radial distance, r, is normalized by a characteristic radial dimension of

L = 5 µm.

FIG. 4 Absorbed laser energy deposition according to the Beer-Lambert law. A sketch of

the resulting initial temperature distribution inside the material is shown along with

definitions of several variables used in the analysis.

FIG. 5 Melt depth variation with time for two different laser fluences using the approximation

that 1.5-3% of the incident laser energy remains in the glass as thermal energy. The

calculations are performed assuming constant thermophysical properties at an average

temperature of 900K.

FIG. 6 Description of parameters used in the thin film model to calculate the evolution of the

surface of the melt.

FIG. 7 Numerical solutions of pressure-driven melt flow, described by Eq. (22), for two differ-

ent pressure profiles. The plasma pressures are shown in the top plot where the solid

curve is given by Ppl(R) = exp(−R10), and the dashed curve is Ppl(R) = exp(−R2).

The bottom plot shows the evolution of the free surface for each of the pressure profiles.

FIG. 8 Numerical solutions of pressure-driven melt flow for two different bottom substrate

shapes. The bottom plot shows the evolution of the free surface for each of the

22

substrate profiles (solid and dashed curves).

23

TABLE I: Thermophysical properties of borosilicate glass (BorofloatTM). The chemical composi-

tion includes 81% SiO2, 13% B2O3, 2% Al2O3, 4% Na2O.

Property Symbol Units Values

Density [15] ρ kg/m3 2.23 × 103

Melting Tm K 1500

temperature [15]a

Viscosity [15] µ Pa s ≈ 103 (at 1500 K)

≈ 102 (at 2000 K)

Surface tension [24] γ N/m 0.28 (at 300 K)

Temperature coefficient γT N/mK 3.4 × 10−5

of surface tension [11]

Thermal k W/mK 1.25 (at 300 K)

conductivity [24]b 1.6 (at 600 K)

4.5 (at 900 K)

42.0 (at 1500 K)

Specific heat [25]b Cp J/kg K 746 (at 300 K)

1000 (at 600 K)

1250 (at 900 K)

1320 (at 1500 K)

Thermal D m2/s 0.75 × 10−6 (300 K)

diffusivityb 0.72 × 10−6 (600 K)

1.60 × 10−6 (900 K)

14.3 × 10−6 (1500 K)

Ben-Yakar et al.

aTm is the working point temperature of glass, defined as the temperature at which the glass can readily

be formed and has a viscosity of approximately 103 Pa.s.bIn the heat conduction calculations, we used data at an average temperature of 900K, which is an average

between T∞ = 300K and Tm = 1500K.

24

12

1

2

3

4

(a)

(c)

(b)

10 mmx 3,000

10 mmx 4,000

10 mmx 4,000

2 pulsesoverlapping3 pulses

overlapping

FIG. 1: SEM images of crater rims generated by (a) one laser pulse (b) two overlapping laser pulses,

and (c) three overlapping laser pulses of 800-nm and 100-fs. The laser fluence was F0 = 34 J/cm2.

The numbers correspond to the order of the incident laser pulses.

Ben-Yakar et al.

25

5 mmx 5,000

FIG. 2: An SEM image of a microchannel created using 200-fs and 780-nm laser pulses of F0 =

23 J/cm2 focused to a spot size of about 12 µm. The image shows the microscale surface roughness

created by the rims of overlapping laser pulses when scanning the laser across the surface.

Ben-Yakar et al.

26

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

Abla

tion

pro

file,h

a/<

hm>

Radial position, x/L

(a)

(b)

FIG. 3: An AFM study of a single crater ablated with a 200-fs and 780-nm laser pulse of F0 =

12.6 J/cm2. (a) The AFM image. b) The ablation profile at the center-line of the crater. The

ablation depth, ha, is normalized by an average melt depth of 〈hm〉 = 1µm and the radial distance,

r, is normalized by a characteristic radial dimension of L = 5 µm.

Ben-Yakar et al.

27

plasma

melt

solid

ha

hm

z

Incident beam, F0

AFth

F (z)a

z

T(z,0)

ha hm

melt frontTth

Tm

ablationdepth

AF0

FIG. 4: Absorbed laser energy deposition according to the Beer-Lambert law. A sketch of the

resulting initial temperature distribution inside the material is shown along with definitions of

several variables used in the analysis.

Ben-Yakar et al.

28

0 0.5 1 1.5 20

0.5

1

1.5

time (µs)

h m (

µm)

Fheat

= 0.5 J/cm2

Fheat

= 1 J/cm2

FIG. 5: Melt depth variation with time for two different laser fluences using the approximation

that 1.5-3 % of the incident laser energy remains in the glass as thermal energy. The calculations

are performed assuming constant thermophysical properties at an average temperature of 900 K.

Ben-Yakar et al.

29

p(x) - plasma pressure

x

x

L

z=h(x,t) - interface height

h(x,0)=<h >m

Thin melted layer

plasma

liquid

FIG. 6: Description of parameters used in the thin film model to calculate the evolution of the

surface of the melt.

Ben-Yakar et al.

30

−1 −0.5 0 0.5 1

0

0.5

1

Ppl

−1 −0.5 0 0.5 1−1

−0.5

0

0.2

R

H

S = 0.08

plasma

molten glass

FIG. 7: Numerical solutions of pressure-driven melt flow, described by Eq. (22), for two different

pressure profiles. The plasma pressures are shown in the top plot where the solid curve is given

by Ppl(R) = exp(−R10), and the dashed curve is Ppl(R) = exp(−R2). The bottom plot shows the

evolution of the free surface for each of the pressure profiles.

Ben-Yakar et al.

31

−1 −0.5 0 0.5 1

0

0.5

1

Ppl

−1 −0.5 0 0.5 1−1

−0.5

0

0.2

0.4

R

H

Ppl

= exp(−R10)

S = 0.04

plasma

molten glass

FIG. 8: Numerical solutions of pressure-driven melt flow for two different bottom substrate shapes.

The bottom plot shows the evolution of the free surface for each of the substrate profiles (solid and

dashed curves).

Ben-Yakar et al.

32