Modeling optical breakdown in dielectrics during ultrafast

laser processing

Ching-Hua Fan and Jon P. Longtin

Laser ablation is widely used in micromachining, manufacturing, thin-film formation, and bioengineering
applications. During laser ablation the removal of material and quality of the features depend strongly
on the optical breakdown region induced by the laser irradiance. The recent advent of amplified
ultrafast lasers with pulse durations of less than 1 ps has generated considerable interest because of the
ability of the lasers to process virtually all materials with high precision and minimal thermal damage.
With ultrashort pulse widths, however, traditional breakdown models no longer accurately capture the
laser–material interaction that leads to breakdown. A femtosecond breakdown model for dielectric
solids and liquids is presented that characterizes the pulse behavior and predicts the time- and position-
dependent breakdown region. The model includes the pulse propagation and small spatial extent of
ultrashort laser pulses. Model results are presented and compared with classical breakdown models for
1-ns, 1-ps, and 150-fs pulses. The results show that the revised model is able to model breakdown
accurately in the focal region for pulse durations of less than 10 ps. The model can also be of use in
estimating the time- and position-resolved electron density in the interaction volume, the breakdown
threshold of the material, shielding effects, and temperature distributions during ultrafast processing.
© 2001 Optical Society of America
OCIS codes: 320.0320, 140.3390, 140.3440, 140.7090, 320.2250, 320.7120.

1. Introduction over, short laser pulses can efficiently heat transpar-

Short-pulse lasers have become increasingly impor- ent solids and liquids, e.g., water, resulting in rapid
tant in recent years for many technologies, including vaporization.9,10 In addition, laser ablation of
microfabrication, thin-film formation, laser cleaning, highly reflective materials, e.g., metals,11,12 thin
and medical and biological applications. For exam- metal films,13 and semiconductors,14,15 has also been
ple, short-pulse lasers are powerful tools for surface investigated. Pulse lasers are also widely used in
patterning and micromachining because of the non- medical and biological fields, e.g., laser lithotripsy,
contact nature of processing. Also, structures with microsurgery, and dentistry,16 –19 in addition to ma-
high aspect ratios, which are usually difficult to terials processing.
achieve with traditional microfabrication techniques Rapid progress in laser technology has brought a
such as chemical etching, can be fabricated with new set of opportunities and challenges in the past
pulse laser processing.1 few years with the development of high-intensity
Furthermore, many of the technologies developed femtosecond-pulse lasers. These lasers are
for microfabrication are restricted to use with sili- uniquely characterized by their ultrashort pulse du-
con2; however, laser ablation can be used on a wide ration 共from 10 fs to 1 ps兲 and extremely high inten-
variety of materials. Examples of such materials sity 共1012–1015 W兾cm2兲. Compared with their
include dielectrics such as oxide ceramics,3 optical relatively long-pulse counterparts, ultrashort-pulse
glasses,4 – 6 polymers,7 and silica aerogels.8 More- lasers have many advantages for laser material pro-
cessing, including negligible heat diffusion effects,
absence of the liquid phase during material removal,
minimal plasma absorption and shielding effects, and
C. H. Fan and J. P. Longtin 共jlongtin@ms.cc.sunysb.edu兲 are
smaller laser fluences for processing, which make
with the Department of Mechanical Engineering, State University
of New York at Stony Brook, Stony Brook, New York 11794-2300. them capable of producing high-quality features with
Received 7 August 2000; revised manuscript received 12 Febru- high spatial resolution. In addition, optical break-
ary 2001. down becomes deterministic with femtosecond laser
0003-6935兾01兾183124-08$15.00兾0 pulses, rather than statistical, as is the case in the
© 2001 Optical Society of America nanosecond regime;20 thus one can more readily con-

3124 APPLIED OPTICS 兾 Vol. 40, No. 18 兾 20 June 2001

trol ultrafast breakdown by changing the irradiated The model includes the pulse propagation and the
laser intensity. finite spatial extent of ultrashort pulses. Also, with
Material processing of high-bandgap materials, in- further development, the new model can be useful for
cluding diamond, glass, plastics, oxide ceramics, and estimating the time- and space-resolved electron den-
liquids, usually requires an optical breakdown pro- sity in the interaction volume, the breakdown thresh-
cess to deposit sufficient energy in the material, as old of a material, shielding effectiveness, the energy
these materials generally exhibit minimal absorption deposition, and temperature increase in the material.
at low laser intensities. 共Such is not the case for
metals because of their large intrinsic free-electron
density.兲 During optical breakdown, a very high 2. Breakdown Models
density of free electrons, i.e., a plasma, of the order of
1018–1020 electrons兾cm3, is produced. Two mecha-
nisms, multiphoton absorption and avalanche ioniza- A. Classical Breakdown Model
tion, also called cascade or impact ionization, are
responsible for laser-induced optical breakdown. A moving breakdown model was proposed by Raizer32
Avalanche ionization is initiated when free electrons and subsequently modified by Docchio et al.30 For
that are present in the material absorb the laser optical breakdown by nanosecond and longer pulses,
energy through inverse bremsstrahlung absorp- the breakdown region tends to expand back up to the
tion.19,20 During inverse bremsstrahlung absorp- beam path toward the laser, a phenomenon known as
tion, seed electrons absorb laser photons by favorable moving breakdown.27,30 For optical breakdown by
collisions with heavier particles, i.e., molecules and femtosecond pulses, however, it is found that break-
ions. If they sustain enough favorable collisions, down starts up the beam path and then propagates
they will eventually gain sufficient energy to ionize toward the focus.31 The plasma produced by the
other molecules on impact, freeing new electrons to breakdown process can effectively absorb laser en-
repeat the process, which results in a geometric in- ergy and heat the bulk material, resulting in a conical
crease in the free-electron density. Multiphoton ab- hole in solids and a cylindrical cavitation region in
sorption occurs when a molecule absorbs enough liquids.8,19
photons simultaneously to ionize the molecule.19,20 Owing to the complexity of interactions between
For long pulses, e.g., in the nanosecond regime, laser pulses and materials, the classical model in-
breakdown is caused primarily by avalanche cludes the following assumptions for simplicity27,30:
ionization, whereas multiphoton absorption domi-
nates breakdown in the low-femtosecond regime 共1兲 Breakdown occurs independently at each loca-
共⬍100 fs兲.20 tion.
Both multiphoton absorption and avalanche ion- 共2兲 The time required for breakdown to occur is
ization depend on the laser intensity in the interac- negligible.
tion volume, and both require a minimum—though 共3兲 The breakdown threshold is independent of the
different—threshold intensity before breakdown is beam diameter.
initiated. Free electrons in the plasma then absorb 共4兲 The threshold is constant in time; i.e., plasma
a large fraction of the incident laser energy by means formed early during the laser pulse arrival does not
of inverse bremsstrahlung,19,20 by which they are influence the breakdown threshold at other points in
heated to a high temperature. At the same time, the the medium.
newly formed electrons transfer energy to the ions
and the lattice and the energy heats the material, The notation used in the balance of this paper is
resulting in vaporization in the interaction volume. defined in Appendix A.
This is the process of laser ablation. In general, a In Raizer’s original model it is assumed that optical
material can be effectively ablated when the incom- breakdown takes place as soon as the incoming laser
ing laser irradiance exceeds the material breakdown irradiance exceeds the threshold; hence the charac-
threshold. Theoretical and experimental investiga- teristics of the laser intensity dominate this model
tions to determine breakdown thresholds in and the related material interactions. Docchio et
solids,4,5,20 –24 liquids,19,25–28 and gases22,29 have been al.30 revised the moving breakdown model by taking
reported. The threshold depends on the bandgap or into account the spatial and temporal characteristics
ionization potential of the material, the pulse dura- of nanosecond laser pulses. For a Gaussian laser
tion, and the wavelength of the incident laser pulse. pulse, the temporal pulse shape has the form33
Although an existing breakdown model by Docchio

冋 冉 冊册
et al.30 has been successfully applied in the
2
nanosecond-and-longer-pulse regime, disagreement t
between experiments and predictions for low picosec- P共t兲 ⫽ P max exp 共⫺4 ln 2兲 , (1)
␶p
ond 共⬍10-ps兲 and femtosecond pulses has been re-
ported.19,28,31 In this paper we present a revised
breakdown model to characterize optical breakdown where Pmax is the maximum pulse power and ␶p is the
initiated by ultrashort laser pulses in terms of the full width at half-maximum pulse duration. The
time- and position-dependent breakdown region. maximum laser power Pmax can be related to the

20 June 2001 兾 Vol. 40, No. 18 兾 APPLIED OPTICS 3125

measured pulse energy Ep by the following expres-
sion:

E p ⫽ P max
兰 冋
⬁

⫺⬁
exp 共⫺4 ln 2兲 冉 冊册
t
␶p
2

dt ⫽ 1.064P max␶ p.

(2)

Typically, the laser beam is focused to a small spot to

increase the intensity above the breakdown thresh-
old. After the focusing optics, beam spot size w is a
function of axial position z, and it is characterized by
beam waist w0 at the focal point and at the Rayleigh
range or focal region zR as follows33:

冉 z2
w共 z兲 ⫽ w 0 1 ⫹ 2
zR
冊 1兾2

, (3)
Fig. 1. Comparison of focal region and laser pulse length 共not to
2 scale兲: 共a兲 length of the focal region 共Rayleigh range兲, 共b兲 1-ns
n␲w 0 pulse, 共c兲 150-fs pulse.
zR ⫽ , (4)
␭

where w共z兲 is the beam radius at position z and ␭ is Values of Nz ⬎⬎ 1 imply that the laser pulse is
the laser wavelength. Thus the irradiance of laser much larger than the focal region, and the spatial
pulses I共t, z兲 can be expressed as variation in the pulse shape can be ignored. For
Nz ⬍⬍ 1, however, the pulse length is relatively
I共t, z兲 ⫽
P max
␲w 共 z兲
2 冋
exp 共⫺4 ln 2兲
t
␶p
冉 冊册
2

, (5)
small, and hence propagation of the pulse within
the focal volume must be considered in the break-
down model. From the preceding example, a 1-ns
where ␲w2共z兲 is the irradiated area at axial location pulse 共␭ ⫽ 800 nm兲 with a 20-␮m focused beam
z and t is time, with t ⫽ 0 corresponding to the pulse’s waist yields Nz ⬇ 190, whereas a 150-fs pulse pro-
peak intensity. In this expression, laser power P is duces Nz ⬇ 0.03.
treated as uniform in the focal volume and depends To account for the small spatial extent and propa-
only on time, whereas the intensity depends on both gation of ultrashort pulses, a femtosecond breakdown
the time-dependent power and the irradiated area. model is presented in which the incoming laser power
Note that the spatial propagation of the laser pulse in becomes both time and position dependent. Thus
the classical model is neglected. the laser intensity takes the form

冋 冉 冊册
B. Femtosecond Breakdown Model 2
P max t ⫺ z兾c
For a laser pulse, the spatial extent of the pulse can I共t, z兲 ⫽ exp 共⫺4 ln 2兲 . (7)
be characterized by the pulse length, lp ⫽ c␶p, where ␲w 共 z兲
2
␶p
c is the speed of light in the medium. For a long
laser pulse, e.g., ␶p ⬎ 1 ns, the pulse length is much Compared with Eq. 共5兲, the additional z兾c term in the
larger than the focal region. Thus, compared with exponential agrument represents the pulse propaga-
the spatial extent of the pulse, the focal region can be tion time in the medium, which becomes significant
treated as a point and the incoming laser power be- in the femtosecond regime. Point z ⫽ 0 represents
comes time dependent only, which is the basis for the the position of the peak pulse intensity when t ⫽ 0.
classical breakdown model.30 For example, a laser
pulse with ␭ ⫽ 800 nm and w0 ⫽ 20 ␮m yields zR ⬃ 3. Results and Discussion
1.6 mm. As the pulse duration approaches the fem- The existing breakdown model of Docchio et al.30 suc-
tosecond regime, however, the pulse will not fill the cessfully described the interaction between nanosec-
entire focal region but rather will occupy only a small ond laser pulses in an aqueous medium. However,
fraction of the focal region at any given instant, e.g., disagreement between experiment and predictions
lp ⬃ 45 ␮m for a 150-fs pulse 共Fig. 1兲, and the laser was recently reported when this model was applied in
intensity varies dramatically as a function of position the low picosecond 共⬍10-ps兲 and femtosecond re-
in the focal region. gimes.19,28,31
The parameter Nz is used to characterize the ratio Referring to the classical model, i.e., Eq. 共5兲, we plot
of the pulse length to the focal region 共Rayleigh in Fig. 2 the variation in laser intensity in the focal
range兲19: volume, with Pmax ⫽ 1010 W and w0 ⫽ 20 ␮m, for
several times. The top of the figure shows the beam
l p c␶ p profile near the focal point located at z ⫽ 0, and the
Nz ⫽ ⫽ . (6)
zR zR time-varying intensity distribution in the corre-

3126 APPLIED OPTICS 兾 Vol. 40, No. 18 兾 20 June 2001

 Fig. 3. Time-dependent breakdown region.

Eq. 共5兲, we can obtain the time for breakdown occur-

rence as a function of position, tB共z兲:
t B共 z兲
␶p
⫽⫾ 再 1
4 ln 2 冋冉z2
ln ␤ 1 ⫹ 2
zR
冊 册冎
⫺1 1兾2

. (9)

Equivalently, the time-dependent breakdown loca-

Fig. 2. Time-dependent intensity in the focal region. tion zB共t兲 takes the form

sponding focal volume is shown at the bottom. Be-

cause the nanosecond-pulse length is much larger
z B共t兲
zR 再 冋
⫽ ⫾ ␤ exp ⫺4 ln 2
t
␶p
冉 冊册 冎 2

⫺1
1兾2

. (10)

than the focal region, the power variation is time The location for breakdown formation is plotted in
dependent only, with a peak value at t ⫽ 0, whereas Fig. 3 by use of Eqs. 共9兲 and 共10兲. Also, the maxi-
the spatial variations in intensity are due simply to mum extent of the breakdown region can be obtained
the changes in irradiated area of the laser beam. as t ⫽ 0:

z max ⫽ z R 冑 ␤ ⫺ 1.
The ratio of peak pulse intensity to the breakdown
(11)
threshold of a material, ␤, is defined as
It should be noted here that absorption by the
P max I max plasma, i.e., the shielding effect, is not considered in
␤⫽ ⫽ . (8) this model.
P th I th For an ultrashort laser pulse 共⬍1 ps兲, however,
both time- and position-dependent breakdown will
For a given laser power Pmax, a larger value of ␤ take place because of the small spatial extent of the
represents a lower breakdown threshold. Break- pulse and its finite propagation speed, and the exist-
down is assumed to occur once the breakdown thresh- ing breakdown model will no longer accurately cap-
old is exceeded. In Fig. 2 we have chosen two ture the breakdown. In the femtosecond regime,
arbitrary breakdown thresholds to demonstrate the breakdown starts up the beam path and then propa-
occurrence of breakdown; both of them are plotted as gates toward the focus at a later time,19,31 a process
dashed lines: ␤ ⫽ 2 and ␤ ⫽ 6. For ␤ ⫽ 6, the that is not captured with the existing breakdown
breakdown starts at the focal point 共z ⫽ 0兲 with model. Noack and Vogel28 presented a rate equation
t兾␶p ⫽ ⫺0.8 and then expands outward. The extent model to predict the breakdown threshold of water by
of the breakdown region can be determined by the applying a time-dependent laser intensity from
portion of intensity above the threshold value 共the which lower breakdown predictions are obtained for
dashed line兲. As can be seen, the peak intensity is femtosecond pulses. Also, less-dense plasma and
always located at the focal point 共 z ⫽ 0兲, resulting in lower temperature in the breakdown region have
breakdown that starts from the focus and expands been reported.19,34 The proposed model, however,
away from this region. Such behavior of a long- provides an alternative description of these findings.
pulse moving breakdown has been experimentally For an ultrafast laser system, the pulse duration
verified as well.30 generally is less than 1 ps, yielding Nz ⬍ 0.2; thus the
For a higher breakdown threshold, e.g., ␤ ⫽ 2, the laser pulse no longer completely fills the focal volume,
breakdown region starts at t兾␶p ⫽ ⫺0.5, which is and the propagation of the pulse must be taken into
later than that with ␤ ⫽ 6. Additionally, the break- account. By applying the modified intensity expres-
down region becomes smaller. Taking I共t, z兲 ⫽ Ith in sion in Eq. 共7兲 one can obtain the pulse propagation in

20 June 2001 兾 Vol. 40, No. 18 兾 APPLIED OPTICS 3127

Fig. 4. 共a兲 Time- and position-dependent intensity of a 1-ps pulse. Fig. 5. 共a兲 Time- and position-dependent intensity of a 150-fs
共b兲 Time- and position-dependent breakdown region. pulse. 共b兲 Time- and position-dependent breakdown region.

the focal volume. As shown in Fig. 4共a兲, for ␤ ⫽ 2, a even shorter pulse, e.g., ␶p ⫽ 150 fs 共Nz ⫽ 0.03兲, this
1-ps pulse with a 20-␮m beam waist initiates break- effect becomes even more significant; the breakdown
down at z ⬇ ⫺1.5 mm when t兾␶p ⬇ ⫺5, which con- starts at t兾␶p ⬇ ⫺35 and z ⬇ ⫺1.5 mm, respectively,
tradicts the prediction from the classical model30 that as shown in Fig. 5. As can be seen, the spatial ex-
the breakdown will begin at the focus 共z ⫽ 0兲 at t兾␶p ⫽ tent of the femtosecond pulse in Fig. 5共a兲 is much
⫺0.5 共Figs. 2 and 3兲. Because no pulse propagation shorter than the focal region. Such a short pulse
in the medium is considered, the classical model al- length localizes the electron formation and produces
ways predicts an identical result, no matter how a less-dense plasma, and thus less plasma absorp-
short the laser pulse. tion, in the breakdown region. Less absorption by
To relate the time and position of the onset of the plasma, in turn, leads to a lower temperature in
breakdown, one sets I共t, z兲 ⫽ Ith in Eq. 共7兲; the the breakdown region, as has been reported in the
position-dependent time for the femtosecond-pulse literature.19,34
breakdown model becomes Once breakdown occurs, the absorption of the laser

再 冋冉 冊 册冎
pulse by the plasma attenuates the pulse power that
⫺1 1兾2
z 1 z2 is responsible for further breakdown behind the ini-
t B共 z兲 ⫽ ⫾ ␶ p ln ␤ 1 ⫹ 2 . (12) tial position; this is the so-called distributed shielding
c 4 ln 2 zR
effect.35 For nanosecond laser pulses, the shielding
Unfortunately, no explicit expression for the time- effect blocks the incoming laser behind the focus and
dependent breakdown position zB共t兲 exists, although leads to an asymmetric expansion of the breakdown
one can readily use numerical procedures to deter- region with respect to the focal point, with a shorter
mine zB共t兲. expansion occurring behind the focal region. Exper-
For ␤ ⫽ 2 and ␶p ⫽ 1 ps, the dimensionless time for imental results by Docchio et al.30,35 indicate that
breakdown occurrence, tB兾␶p, is plotted in Fig. 4共b兲 approximately half of the breakdown elongation pre-
versus axial position z. In this figure the solid and dicted by Eq. 共11兲 is due to plasma shielding. How-
dashed curves represent the breakdown region pre- ever, in the femtosecond regime, breakdown occurs in
dicted by the femtosecond breakdown and the classi- front of the focus, and the intensity increases as the
cal models, respectively. The enclosed regions pulse approaches the focus owing to the reduction in
indicate the time and location when breakdown oc- beam diameter expressed in Eq. 共3兲, which may offset
curs; however, the classical model 共dashed curve兲 is intensity decreases by plasma absorption. There-
no longer able to determine accurately the break- fore the classical model for characterization of the
down process induced by ultrashort pulses. For an shielding effect is no longer feasible. The present

3128 APPLIED OPTICS 兾 Vol. 40, No. 18 兾 20 June 2001

 within a short length, resulting in low measured
power after the breakdown region. This phenome-
non of plasma absorption has been used as a sign of
optical breakdown in the measurement of threshold
intensity or fluence.8,38 By taking into account the
plasma evolution and its absorption, the femtosecond
breakdown model is able to evaluate the shielding
effect for an ultrashort laser pulse. Moreover, the
maximum extent of the breakdown region can be de-
termined. Thereafter, the so-called ablation rate,
which is the volume of removed material per single
pulse, can be obtained as well.
共3兲 Predicting the energy deposition and the temper-
ature increase within the breakdown region. Laser
Fig. 6. Regime map: femtosecond breakdown model versus clas-
energy absorbed by the excited electrons can be re-
sical model.
leased in radiative and nonradiative fashions. For
those transited in nonradiative fashion, the energy
can be stored in the bulk material as thermal energy,
model neglects shielding effects to illustrate the fun-
which gives rise to a sharp temperature increase
damental differences between long 共nanosecond-兲 and
ultrashort 共picosecond- and femtosecond-兲 pulse within the breakdown region. The measured
breakdown in dielectrics. Including shielding in the plasma temperature was 6,000 to 15,000 K,20,39 – 41
model is the next logical step in refining this model. depending on the energy of the incoming laser pulse;
however, few studies have investigated energy tran-
4. Applications and Future Research sition and storage within the material and the corre-
The revised moving breakdown model developed here sponding temperature rise during femtosecond laser–
explores the importance of pulse propagation in the matter interactions. By considering the plasma
focal volume as the pulse duration is reduced to a absorption and energy transition mechanism we can
femtosecond regime. The appropriate choice of mod- use the present breakdown model to predict the de-
els to use as a function of ␶p and Nz is shown in the posited energy in the material and, in turn, the tem-
regime map in Fig. 6. As can be seen, for ␶p ⬍ 10 ps perature rise in the interaction volume.
and Nz ⬍ 1 the new femtosecond-pulse model should
be used instead of the classical model.
In addition to being of scientific interest, the fem- 5. Conclusions
tosecond breakdown model also has several potential A time- and position-resolved moving breakdown
applications, including model for ultrafast laser–material interactions that
includes pulse propagation in the focal volume has
共1兲 Determining the breakdown threshold of a ma- been presented. In the nanosecond regime the spa-
terial by ultrashort laser pulses. Inasmuch as the
tial extent of a pulse is much larger than the focal
threshold intensity is a fundamental parameter in
region. This long pulse overfills the focal volume,
laser material processing, several research methods
and thus the entire focal region can be treated as a
have been presented to estimate the breakdown
threshold of dielectrics and the corresponding elec- point with uniform laser power. Consequently the
tron density, including solid, liquids, and gases, are laser-induced breakdown in the nanosecond regime
presented. A time-dependent intensity 共or electric becomes time dependent only. However, in the case
field兲, i.e., I ⫽ I共t兲, is applied in all these models to of an ultrashort pulse, i.e., ⬍1 ps, the pulse occupies
produce a time-dependent electron density ␳共t兲 in the only a small fraction of the focal volume as it propa-
breakdown region. However, disagreement with gates, which results in both time- and position-
the measured thresholds for femtosecond pulses was dependent moving breakdown. The ultrafast
reported.4,25,28,36,37 Because of the small spatial ex- breakdown model presented here simultaneously
tent and the finite propagation time of femtosecond characterizes the temporal and the spatial behavior
pulses, a modified expression of laser intensity that of a pulse as it interacts with a material, by which the
takes into account pulse propagation is absolutely time- and position-dependent breakdown region can
required. By applying the present breakdown be determined. This model is also capable of ex-
model one can obtain both time- and position- plaining the lower plasma temperature and the less-
dependent electron density in the breakdown region, dense plasma in the breakdown region, and the
by which an accurate electron density and the corre- evolution of a moving breakdown in the femtosecond
sponding prediction of breakdown threshold can be regime. The model shows promise for estimating
determined. the time- and space-resolved electron density in the
共2兲 Evaluating the shielding effect and the maxi- focal region of a pulse, the breakdown threshold of a
mum extent of the breakdown region. Strong absorp- material, the shielding effectiveness of plasmas, and
tion by plasma can attenuate the laser intensity energy deposition in the material.

20 June 2001 兾 Vol. 40, No. 18 兾 APPLIED OPTICS 3129

Appendix A. Notation Used in This Paper 10. J. P. Longtin, “Using multiphoton absorption with high-
intensity lasers to heat transparent liquids,” Chem. Eng. Tech-
c Speed of light in a medium 共⫽c0兾n兲, m兾s; nol. 22, 77– 80 共1999兲.
Ep single pulse energy, J; 11. S. Preuss, A. Demchuk, and M. Stuke, “Sub-picosecond UV
laser ablation of metals,” Appl. Phys. A 61, 33–37 共1995兲.
I laser intensity兾irradiance, W兾m2;
12. B. N. Chichkov, C. Momma, S. Nolte, F. von Alvensleben, and
lp pulse length, m; A. Tünnermann, “Femtosecond, picosecond and nanosecond
Nz ratio of pulse propagation length to diffraction length; laser ablation of solids,” Appl. Phys. A 63, 109 –115 共1996兲.
n refractive index; 13. J. Jandeleit, G. Urbasch, H. D. Hoffmann, H. G. Treusch, and
P power, W; E. W. Kreutz, “Picosecond laser ablation of thin copper films,”
t time, s; Appl. Phys. A 63, 117–121 共1996兲.
w beam radius, m; 14. G. Herbst, M. Steiner, G. Marowsky, and E. Matthias, “Abla-
w0 beam waist, m; tion of Si and Ge using UV photoablation,” in Advanced Laser
z axial location, m; Processing of Materials–Fundamentals and Applications, Ma-
zR Rayleigh range, m. ter. Res. Soc. Symp. Proc. 397, 69 –74 共1996兲.
15. I. Zergioti and M. Stuke, “Short pulse UV laser ablation of solid
and liquid gallium,” Appl. Phys. A 67, 391–395 共1998兲.
Greek symbols 16. A. A. Oraevsky, L. B. Da Silva, A. M. Rubenchik, M. D. Feit,
M. E. Glinsky, M. D. Perry, B. M. Mammini, W. Small, and
␤ Ratio of pulse peak intensity to threshold inten- B. C. Stuart, “Plasma mediated ablation of biological tissues
sity; with nanosecond-to-femtosecond laser pulses: relative role of
␭ wavelength, m; linear and nonlinear absorption,” IEEE J. Sel. Top. Quantum
␳ electron density, m⫺3; Electron. 2, 801– 809 共1996兲.
␶p pulse duration 共FWHM兲, s. 17. R. Esenaliev, A. Oraevsky, S. Rastegar, C. Frederickson, and
M. Motamedi, “Mechanism of dye-enhanced pulsed laser ab-
Subscripts lation of hard tissues: implications for dentistry,” IEEE J.
Sel. Top. Quantum Electron. 2, 836 – 846 共1996兲.
18. D. X. Hammer, G. D. Noojin, R. J. Thomas, C. E. Clary, B. A.
0 Free space, Rockwell, C. A. Toth, and W. P. Roach, “Intraocular laser
B breakdown, surgical probe for membrane disruption by laser-induced
max peak, maximum, breakdown,” Appl. Opt. 36, 1684 –1693 共1997兲.
th threshold. 19. P. K. Kennedy, D. X. Hammer, and B. A. Rockwell, “Laser-
induced breakdown in aqueous media,” Prog. Quantum. Elec-
The authors gratefully acknowledge support of this tron. 21, 155–248 共1997兲.
research by the Division of Chemical and Transport 20. X. Liu, D. Du, and G. Mourou, “Laser ablation and microma-
Systems, National Science Foundation, under con- chining with ultrashort laser pulses,” IEEE J. Quantum Elec-
tract CTS-9702644. tron. 33, 1706 –1716 共1997兲.
21. M. Sparks, D. L. Mills, R. Warren, T. Holstein, A. A. Maradu-
References din, L. J. Sham, E. Loh, Jr., and D. F. King, “Theory of electron-
1. B. C. Stuart, M. D. Perry, M. D. Feit, L. B. Da Silva, A. M. Nev avalanche breakdown in solids,” Phys. Rev. B 24, 3519 –3536
Rubenchik, S. Herman, H. Nguyen, and P. Armstrong, “Fem- 共1981兲.
tosecond laser processing,” in Conference on Laser and Electro- 22. Y. R. Shen, The Principles of Nonlinear Optics 共Wiley, New
Optics, Vol. 11 of 1997 OSA Technical Digest Series 共Optical York, 1984兲, Chap. 27.
Society of America, Washington, D.C., 1997兲, pp. 159 –160. 23. S. C. Jones, P. Braunlich, R. T. Casper, X. A. Shen, and P.
2. T. R. Hsu, MEMS: Design, Manufacture and Processing Kelly, “Recent progress on laser-induced modifications and
共McGraw-Hill, New York, 2001兲, Chap. 4. intrinsic bulk damage of wide-gap optical materials,” Opt. Eng.
3. J. Ihlemann, A. Scholl, H. Schmidt, and B. Wolff-Rottke, 28, 1039 –1068 共1989兲.
“Nanosecond and femtosecond excimer-laser ablation of oxide 24. M. D. Perry, B. C. Stuart, P. S. Banks, M. D. Feit, V. Yanovsky,
ceramics,” Appl. Phys. A 60, 411– 417 共1995兲. and A. M. Rubenchik, “Ultrashort-pulsed laser micromachin-
4. B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. ing of dielectric materials,” J. Appl. Phys. 85, 6803– 6810
Shore, and M. D. Perry, “Optical ablation by high-power short- 共1999兲.
pulse lasers,” J. Opt. Soc. Am. B 13, 459 – 468 共1996兲. 25. P. K. Kennedy, “A first-order model for computation of laser-
5. B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. induced breakdown thresholds in ocular and aqueous media. I.
Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser- Theory,” IEEE J Quantum Electron. 31, 2241–2249 共1995兲.
induced breakdown in dielectrics,” Phys. Rev. B 53, 1749 –1761 26. D. X. Hammer, R. J. Thomas, G. D.Noojin, B. A. Rockwell, P. K.
共1996兲. Kennedy, and W. P. Roach, “Experimental investigation of
6. J. Ihlemann, B. Wolff, and P. Simon, “Nanosecond and femto- ultrashort pulse laser-induced breakdown thresholds in aque-
second excimer laser ablation of fused silica,” Appl. Phys. A 54, ous media,” IEEE J Quantum Electron. 32, 670 – 677 共1996兲.
363–368 共1992兲. 27. A. Vogel, K. Nahen, D. Theisen, and J. Noack, “Plasma forma-
7. S. Küper and M. Stuke, “Ablation of polytetrafluoroethylene tion in water by picosecond and nanosecond Nd:YAG laser
共Teflon兲 with femtosecond UV excimer laser pulses,” Appl. pulses. I. Optical breakdown at threshold and superthreshold
Phys. Lett. 54, 4 – 6 共1989兲. irradiance,” IEEE J. Sel. Top. Quantum Electron. 2, 847– 859
8. J. Sun, J. P. Longtin, and P. M. Norris, “Ultrafast laser micro- 共1996兲.
machining of silica aerogels,” J. Non-Cryst. Solids 281, 39 – 47 28. J. Noack and A. Vogel, “Laser-induced plasma formation in
共2001兲. water at nanosecond to femtosecond time scales: calculation
9. J. P. Longtin and C. L. Tien, “Efficient laser heating of trans- of thresholds, absorption coefficients, and energy density,”
parent liquids using multiphoton absorption,” Int. J. Heat IEEE J. Quantum Electron. 35, 1156 –1167 共1999兲.
Mass Transfer 40, 951–959 共1997兲. 29. J. R. Bettis, “Correlation among the laser-induced breakdown

3130 APPLIED OPTICS 兾 Vol. 40, No. 18 兾 20 June 2001

 thresholds in solids, liquids, and gases,” Appl. Opt. 31, 3448 – M. D. Perry, “Laser-induced damage in dielectrics with nano-
3452 共1992兲. second to subpicosecond pulses,” Phys. Rev. Lett. 74, 2248 –
30. F. Docchio, P. Regondi, M. R. C. Capon, and J. Mellerio, “Study 2251 共1995兲.
of the temporal and spatial dynamics of plasmas induced in 37. Q. Feng, J. V. Moloney, A. C. Newell, E. M. Wright, K. Cook,
liquids by nanosecond Nd:YAG laser pulses. 1. Analysis of the P. K. Kennedy, D. X. Hammer, B. A. Rockwell, and C. R.
plasma starting times,” Appl. Opt. 27, 3661–3668 共1988兲. Thompson, “Theory and simulation on the threshold of water
31. D. X. Hammer, E. D. Jansen, M. Frenz, G. D. Noojin, R. J. breakdown induced by focused ultrashort laser pulses,” IEEE
Thomas, J. Noack, A. Vogel, B. A. Rockwell, and A. J. Welch, J Quantum Electron. 33, 127–137 共1997兲.
“Shielding properties of laser-induced breakdown in water for 38. D. Du, X. Liu, G. Korn, J. Squier, and G. Mourou, “Laser-
pulse durations from 5 ns to 125 fs,” Appl. Opt. 36, 5630 –5640 induced breakdown by impact ionization in SiO2 with pulse
共1997兲. widths from 7 ns to 150 fs,” Appl. Phys. Lett. 64, 3071–3073
32. Y. P. Raizer, “Breakdown and heating of gases under the in- 共1994兲.
fluence of a laser beam,” Sov. Phys. Usp. 8, 650 – 673 共1966兲. 39. D. J. Stolarski, J. Hardman, C. M. Bramlette, G. D. Noojin,
33. A. E. Siegman, Lasers 共University Science, Mill Valley, Calif., R. J. Thomas, B. A. Rockwell, and W. P. Roach, “Integrated
1986兲, Chap. 17. light spectroscopy of laser induced breakdown in aqueous me-
34. A. Penzkofer, “Parametrically generated spectra and optical dia,” in Laser–Tissue Interaction VI, S. L. Jacques, ed., Proc.
breakdown in H2O and NaCl,” Opt. Commun. 11, 265–269 SPIE 2391, 100 –109 共1995兲.
共1974兲. 40. R. J. Thomas, D. X. Hammer, G. D. Noojin, D. J. Stolarski,
35. F. Docchio, P. Regondi, M. R. C. Capon, and J. Mellerio, “Study B. A. Rockwell, and W. P. Roach, “Time-resolved spectroscopy
of the temporal and spatial dynamics of plasmas induced in of laser induced breakdown in water,” Laser–Tissue Interac-
liquids by nanosecond Nd:YAG laser pulses. 2. Plasma lumi- tion VII, S. L. Jacques, ed., Proc. SPIE 2681, 402– 410 共1996兲.
nescence and shielding,” Appl. Opt. 27, 3669 –3674 共1988兲. 41. M. Shirk and P. A. Molian, “A review of ultrashort pulsed laser
36. B. C. Stuart, M. D. Feit, A. M. Rubenchik, B. W. Shore, and ablation of materials,” J. Laser Appl. 10, 18 –28 共1998兲.

20 June 2001 兾 Vol. 40, No. 18 兾 APPLIED OPTICS 3131