Download Wavelength dependence of femtosecond laser

JOURNAL OF APPLIED PHYSICS 117, 223103 (2015)
Wavelength dependence of femtosecond laser-induced damage threshold
of optical materials
2
,1 G. Batavičiu
_ 2 E. Pupka,2 M. Sčiuka,
te,
L. Gallais,1,a) D.-B. Douti,1 M. Commandre
L. Smalakys,2 V. Sirutkaitis,2 and A. Melninkaitis2
1
Aix-Marseille Universit
e, CNRS, Centrale Marseille, Institut Fresnel UMR 7249, 13013 Marseille, France
Laser Research Center, Vilnius University, Saul
etekio al
eja 10, LT-10223 Vilnius, Lithuania
2
(Received 17 March 2015; accepted 29 May 2015; published online 10 June 2015)
An experimental and numerical study of the laser-induced damage of the surface of optical material
in the femtosecond regime is presented. The objective of this work is to investigate the different
processes involved as a function of the ratio of photon to bandgap energies and compare the results
to models based on nonlinear ionization processes. Experimentally, the laser-induced damage
threshold of optical materials has been studied in a range of wavelengths from 1030 nm (1.2 eV) to
310 nm (4 eV) with pulse durations of 100 fs with the use of an optical parametric amplifier system.
Semi-conductors and dielectrics materials, in bulk or thin film forms, in a range of bandgap from 1
to 10 eV have been tested in order to investigate the scaling of the femtosecond laser damage
threshold with the bandgap and photon energy. A model based on the Keldysh photo-ionization
theory and the description of impact ionization by a multiple-rate-equation system is used to
explain the dependence of laser-breakdown with the photon energy. The calculated damage fluence
threshold is found to be consistent with experimental results. From these results, the relative
importance of the ionization processes can be derived depending on material properties and irradiation conditions. Moreover, the observed damage morphologies can be described within the framework of the model by taking into account the dynamics of energy deposition with one dimensional
C 2015
propagation simulations in the excited material and thermodynamical considerations. V
AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4922353]
I. INTRODUCTION
Damage of optical materials induced by intense laser
pulses is a subject of great interest, both for fundamental
physics and for applications such as material laser structuring or designing laser resistant optics. The damage event is
the result of a sequence of complex physical processes
involving free carriers generation from nonlinear ionization,
absorption of the laser energy by the free electrons, energy
transfer from the electron system to the lattice, and eventually irreversible modifications of the material through thermal or mechanical mechanisms. For pulses shorter than
10 ps, particular attention has been given to the description
of the first ionization processes. Indeed, in this temporal regime, the energy is absorbed faster by the electrons than it is
transferred to the lattice and the evolution of the free electron
density drives the laser-induced damage. The description of
the evolution of free electron density through models is then
of main interest, and different theoretical works have studied
those processes: photo-ionization is often modeled using
Keldysh theory that describes in a simple formalism photoionization tunneling and multiphoton ionization,1 while different models can be applied to describe free carrier absorption leading to impact ionization and avalanche.2–4 The
approach of using the Single Rate Equation (SRE)5 or
Multiple Rate Equations (MRE)6 is particularly useful to
describe this free electron density evolution under intense
a)
[email protected]
0021-8979/2015/117(22)/223103/14/$30.00
irradiation, because it provides a practical and simple tool
for theoretical and experimental investigations, without considering the details of the microscopic processes. The MRE
model is a particularly accurate description, because it takes
into account the distribution of electron energies in the conduction band and gives similar results as the one that can be
obtained with a full kinetic approach,7 provided that the calculation parameters are chosen judiciously. Indeed, the different physical input quantities and therefore the
contribution of each process depends on material properties,
pulse duration, wavelength and only comparison to experimental data can provide some validation of the models. The
measurement of laser-induced damage threshold (LIDT) as a
function of a particular laser or material parameter is a commonly used approach. Many studies have dealt with the analysis of the temporal pulse dependence of the LIDT, because
the pulse duration strongly influences the evolution of the
process,5,8–11 particularly the relative importance of the avalanche ionization. Additionally, this parameter can be easily
modified without critical modification of the experimental
configuration. We can note that the majority of these studies
were conducted in the Near InfraRed (NIR) (800 nm).
Another interesting theoretical and experimental investigation is to explore the influence of the photon energy in the
different excitation channels and damage process. Indeed,
the processes are strongly dependent on wavelength: in the
UltraViolet (UV), for instance, few photons are required to
excite electrons from the valence to the conduction band as
opposed to the NIR. Therefore, significant differences in the
117, 223103-1
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Gallais et al.
damage process are expected when changing the wavelength; and the comparison of theoretical and experimental
results should allow optimizing, improving, and verifying
the excitation models that have been often compared only to
NIR results. There have been only few studies dedicated to
such a topic. Simanovskii et al.12 have studied the optical
breakdown threshold dependence with wavelength, for wide
and narrow gap materials, in the case of mid-IR picosecond
pulses (4.7–7.8 lm). In these conditions and depending on
the bandgap of the material, tunnel ionization (TI) alone or
TI with subsequent avalanche was reported as the cause for
dense plasma formation leading to laser absorption and damage. Jia et al.13 have also studied the wavelength dependence
of high bandgap materials (fused silica and CaF2 crystals) in
the range 250–2000 nm with 150 fs pulses. The threshold fluences were observed to be nearly constant in the NIR region,
and to decrease rapidly with laser wavelength in the visible
and Near UltraViolet (NUV) regions. From the comparison
of the experimental results and model the authors conclude
that conduction band electron absorption, via subConduction Band (CB) transitions, and subsequent impact
ionization plays an important role in the damage of dielectrics irradiated by the visible and NUV fs lasers. Recently,
Grojo et al.14 have studied various band-gap materials with
tightly focused femtosecond laser pulses with wavelengths
in the range 1200–2200 nm. In these conditions, they show
that nonlinear absorption is independent of the wavelength,
except for low bandgap materials.
In the present work, the optical breakdown threshold of
a large range of materials, from semiconductors to high
bandgap dielectrics, is explored: Silicon (Si), Zinc Selenide
(ZnSe), Niobium Oxide (Nb2O5), Tantalum Oxide (Ta2O5),
Hafnium Oxide (HfO2), Scandium Oxide (Sc2O3),
Aluminum Oxide (Al2O3), Silica (SiO2), and Calcium
Fluoride (CaF2). Such a set of samples provide materials
with bandgap ranging from 1 to 10 eV. An optical parametric
amplifier system delivering 100 fs pulses was used to measure the breakdown threshold in a range of wavelengths from
the UV (310 nm) to the NIR (1030 nm) thus providing photon energies from 1.2 to 4 eV. In order to investigate the scaling of the femtosecond laser damage threshold with the
bandgap and photon energy, increments of 0.5 eV were chosen to set the different photon energies of the tests thus providing a two-dimensional parametric study (bandgap/photon
energy). In order to calculate the optical breakdown, we
have modeled the evolution of the free-electron density in
the conduction band of the laser-irradiated materials. The
Keldysh theory is used to obtain the photoionisation coefficient and the multiple rate equation introduced by Rethfeld6
to describe collisional excitation and avalanche. This
approach has been indeed shown to be valid in broad range
of time scales. However, the applicability at different wavelengths has not been studied. Additionally, another pathway
for direct electronic excitation has been added to the model
in order to take into account linear absorption when going
down to the UV. Eventually, a time and space dependence of
the optical properties of the materials with the electron density has been introduced in an electromagnetic wave equation model, thus providing a complete description of the
J. Appl. Phys. 117, 223103 (2015)
excitation, particularly for the case of thin films for which interference effects have to be taken into account.
Let us note that in the present work, we investigate only
the single pulse damage threshold dependence on wavelength. The case of multiple pulses should also be of very
high interest but should be a complete study in itself. Indeed,
the incubation behavior under successive irradiations
involves native or laser-induced electronic defect states that
require specific models to address15 and are beyond the
scope of this paper.
The paper is organized as follows: in a first part, we analyze the wavelength dependence of the different ionization
mechanisms, second, we describe experiments and models,
then the experimental and numerical results are reported and
eventually a discussion is conducted.
II. IONIZATION MECHANISMS AND THEIR
DEPENDENCE ON WAVELENGTH
A phenomenological description of light-matter interaction at high intensities can be described as follows. In materials for which there exists an energy gap between the valence
and conduction band, i.e., semiconductors and dielectrics, it
is necessary to supply enough energy to bridge this gap to
promote electrons to the conduction band. Depending on the
photon energy, the absorption can be either linear or nonlinear. In each case, the result of the absorption process is
the creation of free electrons in the material. Because both
absorption mechanisms initiate the excitation process by creating seed electrons for the avalanche, we will refer to it as
the “early ionization mechanism.” The resulting free electrons may further absorb single photons and gain energy, a
process that we will refer as “free electron heating.” Once
electrons have gained sufficient energy, they can transfer
energy to bound electrons by collisional excitation which
will generate other free electrons. The last mechanism gives
the potential for an avalanche process. Eventually, an important effect that has to be taken into account is the modification of the material properties during the irradiation that
influences the pulse propagation and absorption in the material. In order to apprehend the influence of the photon energy
on the excitation processes, we analyze in the following its
influence on each separate mechanism.
A. Early ionization mechanisms
In the case of a transparent material at the wavelength of
irradiation, the initial excitation of carriers involves nonlinear processes due to the strong electric field. This process
can be described by the Keldysh theory,1 and we will use its
complete formulation for solids in this work. An important
parameter for this description is the so-called Keldysh parameter, c, given as
pffiffiffiffiffiffiffiffiffiffiffi
c ¼ x m Eg =eE;
(1)
with x being the laser frequency, m* being the electron
reduced mass, Eg being the bandgap of the material under
consideration, e being the electron charge, and E being the
applied electric field.
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Gallais et al.
J. Appl. Phys. 117, 223103 (2015)
Depending on the value of this parameter, multiphoton
absorption (MPA) or tunneling effect (TE) is the prevailing
mechanism: in the limit of c 1, MPA is the dominant
mechanism. Correspondingly, TE is dominant for the case c
1. For a given material, the Keldysh parameter increases
with the photon energy and MPA is the dominant mechanism
at high photon energy. In order to evaluate the limit between
these 2 regimes as a function of material properties and irradiation conditions, we have plotted on Figure 1 the Keldysh
parameter limit value (c ¼ 1) for different photon energies
and material bandgap values.
In the case of near infrared irradiation, and considering
the measured LIDT values in the experiments, TE is the
main mechanism of photo-ionization. Alternatively, in the
case of UV irradiation, MPA is the main mechanism. In
the visible, it is not possible to distinguish between one or
the other effect so that they both contribute to the ionization
mechanism. Figure 2 presents the evolution of the photoionization rate calculated with the Keldysh theory, as a
function of photon energy for two different materials. The
calculations are done for fluences representative of experimental observations, and for two different materials: Niobia
and Silica. In that case, the Photo Ionization (PI) rate is two
decades higher in the low bandgap material (Niobia) as compared to the high bandgap one (Silica).
B. Free electron heating
When electrons have been excited to the conduction
band, they linearly gain energy from the laser field. The
absorption rate can be described in a first approach by simple
Joule heating. The energy gain rate of the conduction band
electrons (CBE) in this case is2
d 1
¼ rjEj2 ;
dt 3
(2)
with E being the applied electric field, and r being the
absorption cross section that can be described with a Drude
model
FIG. 2. Comparison of PI and II rates as a function of photon energy in SiO2
and Nb2O5. Calculations are done for a fluence of 1.5 J/cm2 in the case of
silica, and 0.25 J/cm2 for the case of Niobia. N is the free electron density.
r¼
e2 s D
;
m 1 þ x2 s2D
(3)
with e being the electron charge, m* being the conduction
band electron reduced mass, x being the laser pulsation, and
sD being the CBE mean scattering time. Obviously, the
absorption rate should decrease with the wavelength if
x2 s2D > 1 and be independent of wavelength if x2 s2D 1. In
the case of k ¼ 1 lm, the limit is obtained for sD ¼ 0.5 fs.
This scattering time sD is energy dependent and its value can
be in the range from 10 to 0.1 fs in silica, for instance,
depending on the CBE energy.3 Therefore, for low energy
CBE, and high sD value, a linear dependence of absorption
with wavelength should be expected, this dependence
becoming weak when electrons gain energy.
C. Collisional excitation
After the sequential absorption of photons, the CBE can
gain sufficient energy to collisionally ionize another electron
from the valence band. The result of the collisional ionization is two electrons near the conduction band minimum,
each of which can absorb energy through free electron
absorption and subsequently ionize additional other valence
band electron (VBE). As long as laser intensity is applied,
the electron density will exponentially grow through this
process. Only CBE with or above a critical energy are able
to perform collisional excitation. Assuming the electron
mass in the conduction band to be equal to the hole mass in
the valence band, the critical energy for impact ionization is
given by4
c ¼
FIG. 1. Fluence necessary to obtain a Keldysh parameter value (c) of 1 as a
function of material bandgap. The values are calculated for 3 different wavelengths, at 100 fs. The obtained lines show the transition between the TE
and the MPA.
3
Eg þ hosc i ;
2
(4)
with Eg being the material bandgap and osc ¼ e2 E2 =
ð4m x2 Þ is the mean oscillation energy of the electrons in the
laser field. In this expression, the corresponding oscillation
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Gallais et al.
J. Appl. Phys. 117, 223103 (2015)
energy has to be provided in addition to the bandgap when
an electron is to be shifted from the VB into the CB. Its
contribution is a squared dependence with wavelength.
For instance, in the case of a 100 fs pulse with 1 J/cm2,
which is a typical laser-induced breakdown value in this regime, osc 1.5 eV in the NIR (1 lm)and 0.1 eV in the UV
(300 nm).
Considering the different elements discussed above and
in order to estimate the impact ionization rate evolution with
wavelength, we can in a first approach calculate the mean
rate of energy absorbed by the electrons, which is the rate of
energy absorbed by one electron (rI, with I being the intensity) multiplied by the number of electrons. The rate of
impact ionization can then be estimated by
dN rIN
¼
:
dt
c
(5)
We have plotted on Figure 2 this ionization rate. We
have chosen to plot it for electronic densities in the range of
magnitude of what can be experimentally measured at the
damage threshold:16 1020 cm–3. It is plotted on the same
graph with the photo-ionization rate, for the sake of
comparison.
In the case of a low bandgap material, such as Niobia in
the example shown on Fig. 2, PI should be the main ionization mechanism leading to the generation of a high density
of free carriers from the UV to the visible: indeed even in the
case of a very high carrier density as plotted on Fig. 2, the
Impact Ionization (II) rate is of the same order of magnitude
or lower than the PI rate and no avalanche should occur.
Only in the NIR the effect of II could have a significant contribution. However, in the case of high bandgap materials
such as silica in the present case, the PI should start the excitation, but during a 100 fs pulse the impact ionization should
be the dominating excitation mechanism from the NIR to the
visible. In the UV (for photon energies greater than 3 eV),
only PI should be involved.
These considerations are based on an oversimplified first
order estimation of the II rate. A more accurate description
takes into account the CBE energy distribution as discrete
levels, and only CBE that have reached a critical energy can
transfer their energy to a VBE. This approach is described in
detail by Rethfled7 and shows that II is overestimated with
the approach we have based on discussions. We will go further on these points in Sec. IV by applying the MRE model
for interpretation of results.
D. Energy deposition into the material
Once a high density of free carriers has been generated
in the material, the optical properties change dramatically.
This effect has consequences on the pulse propagation in the
material. The refractive index (~
n ) dependence on the electronic density can be described based on the Drude model
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ne2
1
;
n~ð N Þ ¼ n20 2
m 0 x þ ix=sD
(6)
where n0 is the refractive index of the unexcited material.
Absorption in the electron plasma is the main mechanism for
energy transfer from the light field to the solid leading to
damage if sufficient energy is deposited. A critical electron
density Nc can be defined when the material becomes
strongly absorbing, which occurs when x ¼ xp
Nc ¼
x2 0 m
:
e2
(7)
The critical density value is increasing when the wavelength
is decreasing: a higher density of electrons is required in the
UV to induce a strong absorption. Once the onset of absorption is reached the material acquires metal-like properties
such as a finite penetration depth that can be defined as
n Þ. The energy is then deposited in a thin
ls ¼ k=ð4pÞimgð~
layer and the higher the photon energy, the thinner the layer.
Considering this effect, Gamaly et al.18 have derived a rough
estimate of the threshold dependence with wavelength
3
Fth ¼ ðb þ Ji Þknc ;
p
(8)
with b being the binding energy of ions in the lattice and Ji
being the ionization potential, that depend on the material,
and Nc being the critical density that is used to define threshold. A linear relationship of the LIDT with wavelength is
therefore expected from these considerations.
III. EXPERIMENTS
The objective of the experiments was to study the laser
damage resistance of optical materials as a function on the
photon energy. For this purpose, a femtosecond laser facility
with a tunable output wavelength from DUV to Infrared was
used in order to explore the defined photon energies in the
maximum range as possible. Because the photon energy is
the relevant parameter in the models, we tried to adjust linearly this parameter and a step target of 0.5 eV was defined
with a minimum of 1 eV and a maximum of 5 eV. The different wavelengths/photon energies that have been used in practice in the experiments are given in Table I.
Different materials were under investigation in order to
study the different processes involved as a function of the ratio of photon to bandgap energies. Materials with bandgap in
the range from 1 eV (silicon) to 10 eV (CaF2) were chosen
with a targeted step of 1 eV. Therefore, LIDT of the materials can be investigated in a two-dimensional space.
A. Samples
Thin films and surface of bulk samples have been investigated. The films were oxides (Nb2O5, Ta2O5, HfO2, Sc2O3,
Al2O3, and SiO2) deposited on fused silica substrates.
Nb2O5, Ta2O5, HfO2, and SiO2 films were deposited by
TABLE I. The different photon energies (wavelengths) explored in the
experiment.
Photon energy (eV)
Wavelength (nm)
1.2
1030
1.5
800
1.9
650
2.5
500
3.1
400
3.5
357
3.6
343
4
310
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Gallais et al.
J. Appl. Phys. 117, 223103 (2015)
TABLE II. Optical properties (bandgap energy and refractive index) of the samples.
Material
Form
Bandgap energy (eV)
n at 1240 nm
n at 800 nm
n at 650 nm
n at 500 nm
n at 400 nm
n at 350 nm
n at 310 nm
n at 361 nm
ZnSe
Nb2O5
Ta2O5
2.7a
2.4a
2.5a
2.58a
2.75 þ 0.1ia
2.8 þ 0.4ia
2.9 þ 0.5ia
3 þ 0.7ia
3.2 þ 0.9ia
3.46b
2.21b
2.24b
2.27b
2.27b
2.35b
2.69 þ 0.015ib
3 þ 0.27ib
2.95 þ ib
4.11b
2.1b
2.12b
2.13b
2.18b
2.25b
2.33b
2.44b
3 þ 0.3ib
Si
<–Bulk–>
1.06a
3.75a
3.73 þ 0.005ia
3.9 þ 0.02ia
4.32 þ 0.07ia
5.2 þ 0.26ia
5.6 þ 3ia
5 þ 3.6ia
2 þ 4ia
HfO2
Sc2O3
Al2O3
<–Thin film–>
SiO2
SiO2
CaF2
<–Bulk–>
5.55b
1.98b
1.99b
2.00b
2.02b
2.05b
2.08b
2.12b
2.22b
7.5c
1.50b
1.50b
1.50b
1.51b
1.51b
1.52b
1.53b
1.55b
8.5a
1.45a
1.45a
1.46a
1.46a
1.47a
1.48a
1.49a
1.50a
5.66b
1.94b
1.95b
1.96b
1.98b
2.02b
2.05b
2.09b
2.21b
6.12b
1.66b
1.67b
1.67b
1.67b
1.67b
1.67b
1.67b
1.68b
10a
1.43a
1.43a
1.43a
1.44a
1.44a
1.45a
1.45a
1.46a
Value extracted from Handbook of optical materials.36
Data obtained from UV-visible spectrophotometric measurements and fit of data with Tauc-Lorentz model.
c
Data obtained from DUV spectrophotometric measurements and Tauc-gap plot.
a
b
electron beam deposition with ion assistance and Sc2O3,
Al2O3 films were deposited with Ion Beam Sputtering. These
two techniques can produce dense, smooth, and stoichiometric layers. Their optical properties (refractive index, absorption coefficient, and band gap energy) have been determined
from UV-visible spectrophotometric measurements and fitting algorithms.19 The bulk materials come from different
vendors as windows for laser applications. The optical properties of the samples are given in Table II.
B. Experimental set-up
Experimental set-up used for characterization of LIDTs
was developed at Vilnius University (Fig. 3). High peak
power Ti:Sapphire laser system (Coherent, “Libra”) was
combined with commercial Optical Parametric Amplifier
system—OPA (Light Conversion, “Topas 4–800”) in order
to produce light pulses at desired wavelength. The irradiation
of primary laser source was centered at 800 nm wavelength
with pulse duration of 100 fs and a repetition rate of 500 Hz.
After nonlinear light conversion in OPA system, a set of
dichroic separator mirrors was used to extract spectrum of
testing pulses from residual irradiation (either from pump or
one of signal-idler wavelengths). The laser pulse spectrum
was characterized by a fiber spectrometer (Avantes,
“AvaSpec series”). The number of incident laser pulses was
set by a computer controlled mechanical shutter. Pulse
energy was adjusted by motorized k/2 plate and broadband
polarizer. A photodiode calibrated with a pyro-electric
energy meter head (Ophir, “PE10-C”) was employed to
determine and record the energies of the incident laser
pulses. Depending on the available energy at various wavelengths, the beam was focused into the target plane sample
by one of fused silica lens (f ¼ 4.5, 7.5, 10, and 50 cm). The
effective beam diameter was specified at Imax/e level of intensity for the different wavelengths. Spectral widths and
beam diameters used at particular wavelength are summarized in Table III. The maximal output power available from
secondary OPA source was strongly dependent on the wavelength and for some samples, having higher LIDTs, maximum fluence was not enough to cause damage. Note that
additional tests were conducted at 1030 and 343 nm on
another experiment, described in Ref. 20, for comparison of
results and additional data.
C. Test procedures
All the samples have been measured with the same experimental configuration and with the same procedure. The
samples have been tested at normal incidence, with the surface to be tested facing the incoming beam (front face testing), with a linear polarization. The damage process was
found to be very deterministic, and therefore, measurements
were performed in 1-on-1 mode by gradually increasing laser
fluence in fresh unexposed sites. Since measurements for all
wavelengths were done on the same samples with very limited area budget at least 50 sites were irradiated for each
sample and wavelength. The increment of fluence step was
dependent on the relative threshold of the sample at particular wavelength and varied from 0.02 J/cm2 to 0.07 J/cm2 for
low and high LIDT samples, respectively. This resulted in at
least 10% or better measurement accuracy. The state of
TABLE III. Beam diameters and spectral widths used for testing at different
wavelengths.
FIG. 3. Schematic of the experimental configuration. OPA: Optical
Parametric Amplifier; SHG/THG: Second/Third Harmonic Generation.
Central wavelength (nm)
Spectral width (FWHM) (nm)
Beam diameter (1/e) (lm)
310
2.7
22.7
357
3.5
35.3
400
n.a.
32.6
500
6
28.2
650
13.9
36
800
14.7
69
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Gallais et al.
J. Appl. Phys. 117, 223103 (2015)
irradiated sample was determined from post mortem
Nomarski microscopy (Olympus, BX51). Damage was
defined as any modification observed in the irradiated area
with Nomarski mode. The damage threshold is calculated to
be between the highest fluence with no damage and the lowest fluence at which damage occurred.
The experimental LIDT data have been rescaled in order
to take into account of the electric field distribution in the
samples (see, for instance, Fig. 4). The internal, or intrinsic
LIDT is then derived from the experimental data and is the
relevant value for comparisons21
LIDTinternal ¼ jEmax =Einc j2 LIDTmeasured ;
(9)
800 nm and 1030–1053 nm are plotted in Fig. 5, along with
the same results rescaled at 100 fs with the scaling law
described in Ref. 10.
The different values are found to be in the range of 2–5
J cm–2, with a mean value of 3.35 J cm–2. The discrepancy
in the results of the groups may be due to experimental differences: the LIDT can depend on the damage test procedure, damage criterion, damage detection, and laser
stability. Another possible explanation may be that, since
the tested fused silica samples were not the same, depending on the fabrication and polishing processes used, the
samples may have different properties. In our case, the
measured damage threshold of the fused silica sample at
800 nm/100 fs is 3.68 J cm–2 (not corrected with internal
Electric Field value).
with Emax/Einc being the ratio of maximum of the standing
wave electric field distribution in the film to the incident one.
This correction factor is applied to the coating samples, since
there are interference effects in the film, and also to the bulk
samples to compare all samples on the same basis (in this
case, Emax is the value on the surface). This ratio is calculated numerically based on the refractive index and film
thickness given in Table II.
It has to be pointed out that due to this rescaling factor,
the experimental data are not directly comparable to other
published data. For instance, in the case of silicon, a LIDT
of 0.04 J cm–2 is obtained at 800 nm which corresponds to a
measured LIDT of 0.25 J cm–2. To compare the LIDT values reported in this paper with other published data, we
have investigated the results of several LIDT measurements
that were made on the surface of fused silica, the most
widely studied dielectric material in the literature. Only
experiments involving single pulses were selected, since
incubation effects occur for multiple pulses, which can
make the results non comparable. Also, only results
between 50 fs and 500 fs were selected, closed to our experimental conditions. The results found by different groups at
In order to describe the generation of free electrons in
the conduction band and the evolution of their density under
short laser irradiation, we used the MRE model.6 This model
takes into account the non-stationary electron distribution in
FIG. 4. Normalized electric field distribution in a Ta2O5 film at different
wavelengths. The thickness of the film is 451 nm, and the refractive index is
given in Table II. E is the electric field in the sample and Einc is the incident
electric field. Also plotted is the normalized electric field distribution in the
case of a fused silica sample irradiated at 800 nm. The fill circles are indication of the values taken to rescale the LIDT data with Eq. (9).
FIG. 5. Comparison of the laser-induced damage threshold measured by different groups on the surface of fused silica samples at 800 or 1053 nm.
Measurements at the pulse duration of the experiments are given (dashed
bars) and also rescaled values with the scaling law given in the text for comparing the results at 100 fs (gray bars). Data are extracted from Refs. 11, 13,
and 22–32 The present measurement is given on the right (dark gray bar)
and compared to the mean value of bibliographic data (dotted line).
IV. NUMERICAL SIMULATIONS
The model employed to simulate the LIDT dependence
with wavelength is described. Photo-ionization is calculated
with the Keldysh theory, as described in the first section. The
evolution of the free-electron density in the conduction band
is calculated with a multiple rate equation, taking into
account the kinetic energy of photo-ionized electrons. The
pulse propagation in the excited material is described with a
one-dimensional electromagnetic wave equation using a
Drude model for the dielectric permittivity. We solve
numerically the coupled equations and obtain the wavelength
dependence of threshold fluences and a description of energy
deposition in the material.
A. Rate equation
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the CB, by discretizing the conduction band into discrete
energy levels separated by one photon energy. It provides an
accurate estimation of the role of the impact ionization and
avalanche process with a simple formalism. The single rate
equation results indeed in an overestimation of the avalanche
and with it the free electron density.7 As a consequence, a
lower LIDT should be found with the SRE compared to the
MRE. In the MRE model, electrons in the valence band are
excited to the conduction band by strong field ionization. In
the present work, based on discussions in Sec. II A, we used
the complete Keldysh theory for calculation of this term.
Strong-field excitation generates electrons with low energy.
In the original paper,6 the model implicitly assumes that
electrons excited by multiphoton ionization are injected in
the CB with zero kinetic energy; i.e., above threshold ionization is neglected. This is a limitation, particularly for low
bandgap materials or high photon energies: indeed in this
case, this kinetic energy is not negligible as compared to the
critical energy for impact ionization. However, this cannot
be included in a simple way in the model, because it is based
on finite and discrete energy levels. Another pathway has
been included in these equations to account for possible linear absorption that occurs for UV irradiation of low band
gap materials, based on the absorption coefficient a0 value.
This coefficient is obtained from the imaginary part of the refractive index (Table II). In case of photon energy greater
than bandgap value, we used this measured absorption coefficient in the equation and set Keldysh expression to zero.
The process is then followed by one-photon excitation leading to an increased energy of the conduction-band electrons.
Only the most energetic electrons are able to induce the collisional excitation, with a critical energy given in Eq. (4).
Consequently, two electrons with a small kinetic energy are
generated and the process can continue until the end of the
pulse. These processes are described with the following set
of ordinary differential equations:
dN1
¼ WPI W1pt N1 þ 2~a Nkþ1 þ a0 IðtÞ=Ep ;
dt
dN2
¼ W1pt N1 W1pt N2 ;
dt
:::
dNk
¼ W1pt Nk1 W1pt Nk ;
dt
dNkþ1
¼ W1pt Nk ~a Nkþ1 ;
dt
(10)
with WPI being the photo-ionization rate, W1pt is the intraband
single photon absorption probability, ~a is the impact ionization probability, and k þ 1 is the number of discrete energy
states in the CB (k þ 1 ¼ bc =Ep þ 1c), with Ep the photon
energy. We used Eq. (5) for the calculation of ~a . In order to
obtain the intraband single photon absorption probability, for
the different wavelengths and materials under consideration,
we used the asymptotic value of the avalanche parameter
given in the original description of the MRE model,7 i.e.,
aasymp ¼
pffiffiffi k
21 W1pt :
(11)
The asymptotic value was estimated from a fixed cross
section derived from Drude absorption (Eq. (5)), taking into
account the ln(2) factor mentioned in Ref. 7. Consequently,
W1pt ¼
r
1
ffiffiffi I:
p
k
ð
Þ
ln 2 c 21
(12)
a0I(t)/Ep is a term that takes into account the contribution
of linear absorption for a0 6¼ 0, which need to be taken
into account particularly in the UV for low bandgap
materials.
No recombination term was used in the set of rate equations that were used. Indeed considering the very short pulse
duration, 100 fs, and the lack of knowledge on material in
thin film form we made the assumption of neglecting this
effect. In silica, for instance, that has a relatively short
recombination time due to self-trapped-excitons, it is of the
order of 150 fs.37 It is above our 100 fs pulse duration but if
one considers this trapping effect, the calculated LIDT
should be slightly higher than the ones calculated in this
work.
B. Transient optical properties of the material
When free electron density in the CB is increasing due
to excitation, the optical properties of the material change
dramatically, up to the point when it acquires a metal-like
behavior, i.e., high reflectivity and finite penetration depth.
These changes in optical properties lead to modifications
of the electric field distribution during irradiation, particularly in the case of thin films where interference effects
take place.33 It has to be taken into account for an accurate
calculation of the LIDT and to determine the deposited
energy in the material. We have applied the Drude model
to relate the dielectric function to the free electron density,
as described in Eq. (6). In this simple model, the electron
scattering time is a main parameter that requires evaluation. In many publications, it is taken as a constant. In
other approaches, this term is evaluated by calculating the
different contributions of its components: electron-phonon,
electron-ion, electron-electron scattering. An overview of
these approaches can be found, for instance, in the work of
Balling and Schou.38 The proposed approach in this work
is to describe the dependency of this term with intrinsic
properties of the material, mainly characterized by its
bandgap, and the electronic excitation of this material. We
have taken into account both electron-electron (se–e) and
electron-ion (se–i) scattering times using simple models as
described below.
se–e is described with the approach developed by
Christensen and Balling,39 employing a scattering rate from
kinetic gas theory with a Debye length for the electrons
“radius.” The expression is
rffiffiffiffi
e2
m
ðkB Te Þ3=2 ;
see ¼
(13)
4p0 6
where Te is the electronic temperature. It can be rewritten as
a function of the electron kinetic energy (Ekin ¼ 3/2kBTe)
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see
J. Appl. Phys. 117, 223103 (2015)
pffiffiffiffi
3e2 m 3=2
¼
E
:
16p0 kin
(14)
For se–i the simple model derived by Starke et al. is
employed17
sei ¼
pffiffiffiffi 3=2
16p20 mEkin
pffiffiffi
:
2e4 N
(15)
C. Pulse propagation
The overall scattering time time is given as
1
1
s1
D ¼ see þ sei :
of the energy absorption as it will be described latter. The
increase of the complex refractive index will lead to strong
coupling of the energy in the material. The two parameters
are connected, and transient calculations of the intensity distribution are necessary for a complete description of energy
deposition in the material.
(16)
It can be seen from these equations that sD is dependent on the electron density and the excitation level of the
electrons. The mean kinetic energy of the conduction band
electrons is necessary for calculating the collision time.
We roughly estimated it to Ekin ¼ 0.5Eg. Typical values
obtained for fused silica with the previous expressions are
se–e ¼ 0.2 fs and se–i ¼ 1.6 fs (with N ¼ 1021 cm3). A more
accurate estimation would be to take the mean energy
value taking into account the contribution of electrons at
each energy level in the conduction band, as defined in the
MRE model. However, given the assumptions used for the
estimation of sD, it seems as an unnecessary and complicated refinement. Fig. 6 shows how the optical properties
(refractive index and extinction coefficient) depend on free
electron density evolution. The calculation is conducted
for a low bandgap (niobia) and high bandgap (silica)
material.
The optical properties are strongly affected when the
electronic density is increasing, the materials gradually acquire a metal-like behavior. The changes are more pronounced when the material bandgap is higher, because the
kinetic energy of the electrons can reach higher levels in the
conduction band. The modification of the refractive index
will induce phase shifts that will have an effect on the electric field distribution in the sample, particularly in optical interference coatings. This effect can change the localization
FIG. 6. Evolution of the real and imaginary part of the complex refractive
index as a function of the free electron density, as described by the presented
model. Calculations done at a wavelength of 1030 nm.
The electric field distribution in the sample is calculated
in the frequency domain, i.e., under stationary assumption,
with the matrix method that is classically used in the field of
optical thin films:40 it offers the ability to take into account
the polarization and the incidence angle of the incoming
wave. An assumption of negligible spectral dispersion is
done: calculation is done at the central wavelength of the
laser which we have considered as reasonable given the
spectral widths reported in Table III. Practically, the system
is divided in thin slices, typically 1 nm, each slice being
characterized by constant material properties (bandgap) and
time dependent properties (electronic density and refractive
index). The system is then equivalent to a multilayer stack in
which the electric field has to be recalculated for each time
iteration. At each of these iterations, the optical properties at
the end of the previous step are taken as an input. A
Gaussian temporal shape is considered. This approach
involves coupled differential equations that are solved
numerically. An example of calculation is given in Fig. 7 for
FIG. 7. Calculations of the normalized electric field distribution and resulting free electron density in a Ta2O5 film (deposited on a Fused silica substrate) irradiated with a 500 fs pulse at 1030 nm. The incident fluence is 1.1 J
cm2. The pulse is coming from the air, it has a Gaussian temporal profile
centered at t ¼ 0. Calculations are done with the model described in the text,
which take into account transient optical properties of the material. The
black curve is the E-field distribution at the beginning of the pulse
(t ¼ 1500 fs). The free electron density in the material in shown at time
t ¼ þ200 fs in red and t ¼ þ1500 fs in blue (end of the pulse) as the E-field
distribution at these times. The film has been discretized in 100 sub-layers
for this calculation.
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the case of an optical thin film under investigation in the
experiments.
The modifications are mainly induced at the location of
the peaks of standing electric-field. In the particular case
under investigation in Fig. 7, they occur at the film/substrate
interface and in the middle of the film. At these locations,
the optical response of the dielectric material is strongly
modified due to the high free electron density. This results in
local absorption and a decrease of the electric field in the
film. At the end of the pulse, the maximum E-field occurs at
the air/film interface and a new peak of electronic density
appears. The correlation between these localized modifications and the observed morphologies will be established in
Sec. V B.
D. Energy deposition in the material
When the material is irradiated with the laser pulse a
fraction of the laser energy is reflected, another is transmitted
and a certain percentage is absorbed. Except in the UV for
low band-gap materials, there is no initial absorptivity. The
interaction and subsequent transmission, reflection, and
absorption of laser energy into the material can be described
with the previous numerical model. Since we can calculate
the time-dependent optical properties and hence absorptivity
linked to the evolution of the free electron density, we
directly obtain the absorbed energy in the material with time
integration of the absorbed intensity. An example that corresponds to the previous case of Fig. 7 is given in Fig. 8. The
energy is absorbed in this case at the trailing end of the
pulse, after sufficient free electrons have been generated by
the avalanche process for the material to be locally highly
absorbing. For a fluence of 6.4 J cm2, the absorbed fluence
is 0.5 J cm2.
E. Damage threshold
The described model has the ability to evaluate in
space and time the energy absorbed in the material, this
FIG. 8. Same case as Fig. 7. The incident intensity on the structure is plotted
as a function of time, as the reflected, transmitted (through the substrate) and
absorbed part. The absorbed fluence can be obtained by time integration of
the absorbed intensity (red filled area).
J. Appl. Phys. 117, 223103 (2015)
energy being in the form of excited electrons. Subsequent
to the laser irradiation, the laser energy has to be transferred
from the heated electronic system to the lattice. Without
going in the detail of this transfer mechanism, one can
assume that damage of the material takes place when the
deposited energy is larger than a critical energy that is sufficient to cause material modifications (for instance, melting,
vaporization). In order to obtain the theoretical damage
threshold, it is possible to use a criterion based on the total
absorbed energy density at the end of the pulse. However,
because the material becomes strongly absorbing when the
critical density (Eq. (7)) is reached, the criteria of the critical density are often used to evaluate the damage threshold.
Recently, it was demonstrated that the energy criteria was
more appropriate than the critical density one.41 An example of calculation is given in Fig. 9 for the case of a Nb2O5
film. The specific energy (energy per unit of mass) deposited in the material at the end of the irradiation is calculated
as a function of the fluence, as the maximum electronic
density that is reached. Taking into account an order of
magnitude of 1 kJ/cm3 for the melting threshold and 1021
electrons/cm3 for the critical electron density, the respective theoretical damage thresholds are 0.6 J/cm2 and 0.35 J/
cm2. A significant difference is obtained because the critical density is only a criteria for the onset of absorption
whereas the energy criteria takes into account the onset of
absorption and the quantity of deposited energy in the
material.
V. RESULTS AND DISCUSSION
A. Comparison between experimental and numerical
results
For all samples and wavelengths, a deterministic behavior has been observed: there is a clear transition between the
fluences for which damage and no damage occurs. The measured damage thresholds, rescaled according to Eq. (9), are
shown in Figs. 10–13. The data are also available plotted as
a function of wavelength in the Appendix (Figs. 18–21).
FIG. 9. Case of Nb2O5 film irradiated with 500 fs pulses at 1030 nm. The
energy per unit of mass is calculated as a function of the incident fluence
and compared to the electronic density reached. Taking into account a critical specific energy of 1 kJ/cm3 and a critical density of 1021 cm3, two thresholds can be defined.
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FIG. 10. Experimental and calculated laser-induced damage threshold versus wavelength for silicon and zinc selenide.
FIG. 12. Experimental and calculated laser-induced damage threshold versus wavelength for hafnium oxide, scandium oxide and aluminum oxide.
Firstly, based on the previous detailed model, we have
compared the experimental results to numerical simulations
on the same figures. We have calculated the LIDT as a function of the photon energy for the studied materials. The comparison of the simulations to the experimental results is
given in Figs. 11–13. The different parameters used in the
simulation have been described previously. The effective
electron mass was set to 0.5me in the simulations, except for
ZnSe and Si: in these cases an effective mass of 0.2me was
used. As explained previously, a critical absorbed energy
density has been applied as the damage threshold criterion.
This damage criterion has been set as the energy needed to
reach the melting temperature of the material. It can be calculated from the specific heat capacity and density of the
materials. Taking into account tabulated values in handbooks,36 values range from 2 to 5 kJ/cm3 depending of the
material, with higher values for the high bandgap materials
(melting temperature increases with the bandgap). Because
materials in thin film form can have very different properties
from bulk materials,40 and because also the specific heat
capacity is temperature dependent, the damage threshold criterion can be only roughly approximated. Therefore, only the
order of magnitude was considered in our simulations: 2 kJ/
cm3 for all materials. Indeed using such a criterion, it will be
shown below that a relatively good description of the results
can be obtained.
The results show that the trends are well reproduced
with the model used for the simulations. However, some
large steps are predicted with the model, which correspond
to transition from N to N þ 1 photon absorption that cannot
be really compared to experiments because of the lack of
precision in LIDT determination and the low number of
tested wavelengths. One can note however that such transitions were observed experimentally on TiO2 on the same experimental set-up.34 Additionally and as discussed before,
FIG. 11. Experimental and calculated laser-induced damage threshold versus wavelength for niobium oxide and tantalum oxide.
FIG. 13. Experimental and calculated laser-induced damage threshold versus wavelength for silicon oxide (in film and bulk form) and calcium
fluoride.
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FIG. 14. Single pulse LIDT of optical materials measured at 100 fs/800 nm.
Results of numerical simulations are compared to experimental data.
the MRE model does not account for the kinetic energy of
electrons injected in the CB, therefore, the model should not
be accurate at the N N þ 1 transitions and when near the
linear absorption band.
Second, we have calculated the evolution of the LIDT
with the bandgap of the material for specific wavelengths.
These calculations were done at 800 nm and 400 nm for
which a large number of different materials were tested
(because of the high fluence available at these 2 wavelengths). The results of the simulations are compared to the
experimental data on Figs. 14 and 15. It should be noted that
with the exception of the wavelength, exactly the same simulations parameters were used in both cases.
Looking at Fig. 14 that corresponds to 800 nm, one can
observe that the experimental LIDT values are increasing
with the bandgap of the material in a regular and continuous way. These results can be compared to the ones of
Mero et al.10 that were obtained at 800 nm at different subpicosecond pulse durations: different oxides (TiO2, Ta2O5,
HfO2, Al2O3, and SiO2) with band gaps ranging from 3.3 to
J. Appl. Phys. 117, 223103 (2015)
FIG. 15. Single pulse LIDT of optical materials measured at 100 fs /
400 nm. Results of numerical simulations are compared to experimental
data.
8.3 eV were investigated and their findings suggest a phenomenological formula Fth ¼ (c1 þ c2Eg)tk, with k a material and pulse duration independent constant (k 0.3). The
threshold fluence is then determined by the band gap of the
material only. This linear behavior was explained by taking
into account the physical processes involved in the ionization processes at the initiation of the damage event (photo
and impact ionization) and their dependency with the
bandgap. A similar behavior was also observed at 1030 nm/
500 fs for oxide films.35 A comparison to the phenomenological model of Mero et al. is done on Fig. 14, with the
coefficients taken from the original publication, including
the error bars. The linear behavior is observed to be in good
agreement with LIDT values in a limited range of values.
In the case of high bandgap materials (Calcium Fluoride in
our case), there is a deviation from this linear relationship.
Indeed, the avalanche process requires seed electrons that
are generated by photo-ionization, and as the bandgap is
increasing this generation is obviously less efficient. The
FIG. 16. Right: SEM observations of a single layer HfO2 film deposited on a fused silica substrate and irradiated at 1030 or 343 nm with different fluences
(increasing fluences from the left to right). Left: simulation of the electron density for irradiating fluences above the damage threshold. The positions of the
peaks of electron density are correlated to the locations of damage initiation.
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FIG. 17. Right: SEM observations of a single layer Ta2O5 film deposited on a fused silica substrate and irradiated at 1030 or 343 nm with different fluences
(increasing fluences from the left to right). Left: simulation of the electron density for irradiating fluences above the damage threshold. The positions of the
peaks of electron density are correlated to the locations of damage initiation.
non-linear behavior is explained very well with the model.
Additionally, the simulations are in accordance with experimental data at both wavelengths. In the case of 400 nm,
large steps are predicted due to the predominant contribution of PI at this wavelength and the low number of photons
involved in the process: 2 photons from 3 to 6 eV, 3 photons
from 6 to 9 eV. However, these steps cannot be really evidenced on the experimental data.
B. Analysis of spatial energy deposition
The localization of the deposited energy and therefore
the localization of the damage initiation are related to the
wavelength of irradiation. In the case of the surface of bulk
materials, the absorption in the ionized dielectric occurs in
a skin layer near the surfaces that scales linearly with the
wavelength.18 In the case of thin films, that were investigated in our study, the energy deposition process is however more complex because of interference effects that
modulate the intensity distribution in space, but also in
time due to the transient optical effects described in
Sec. IV. As an example of such complex deposition processes, observation of damage sites by SEM are displayed
in Figs. 16 and 17.
In these two figures, it can be observed that on these two
samples damage occurs at different locations depending on
the irradiation fluence and wavelength: localized at the surface, inside the film at different depth, or at the film/substrate
interface. Multiple damage initiations at different depths in
the films are evidenced by the ring structures on the images.
These effects have also been observed on the other sample
materials (Niobia and silica films). A really interesting point
that is illustrated for the case of Hafnia is that the integrity of
the film can be preserved when removed from the substrate,
meaning that the damage process is very localized.
Simulations have been conducted based on the model developed in this work and are reported on the left hand side of
Figs. 16 and 17.
In the case of the Ta2O5 film irradiated at 1030 nm,
energy should be deposited preferentially at 200 nm below
the air/film interface, and at the film/substrate interface for
higher fluences. The SEM observation that has been completed with optical profilometry measurements (not
reported here) reveals that damage occurs first at this
200 nm depth in the film, which is separated in two parts.
At higher fluence, the film is totally removed because of
damage at the film substrate interface (black area on the
last figure). This is very well correlated with what can be
expected from simulations. Similar observations and interpretations can be done in the case of UV irradiation: damage occurs at the location of the peaks of electronic
density. The same agreement between calculations and experimental observations is evidenced in the case of the
HfO2 film.
VI. CONCLUSIONS
The laser-induced damage threshold of optical materials
has been studied experimentally and numerically in a range of
wavelengths from the NIR to the UV with pulse durations of
100 fs. Experimental results reveal a decrease of the damage
threshold with the wavelength with a dependence on the
bandgap of the material, bringing new data for the community
in this field. Numerical models based on nonlinear ionization
processes have been applied using Keldysh photo-ionization
theory and the description of impact ionization by a multiplerate-equation system. By taking into account the change in the
dielectric function of the materials during the laser pulse, its
consequence on the intensity distribution in the material, and
applying a critical energy level for the theoretical damage
threshold, a fair description of the experimental data could be
obtained, both for the localization of the damage initiation
and for the quantitative evolution of the damage threshold
with photon energy and material properties.
Therefore, the main conclusion of this paper is that
using a simple and generic model, taking into account only
few unknown parameters that were estimated based on reasonable physical assumptions, the wavelength and material
dependence of the laser damage threshold can be obtained.
This is of main interest particularly in the case of materials
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such as thin films because of the difficulty to measure or estimate their physical properties that can be very different from
bulk materials.
This work was based on the study of wavelength and
bandgap dependence of the laser damage threshold. To go
further, the influence of a third parameter such as the pulse
duration should be analyzed to explore the range of validity
of the approach and its limitations.
ACKNOWLEDGMENTS
The research leading to these results has received
funding from LASERLAB-EUROPE (Grant Agreement No.
284464, EC’s Seventh Framework Programme).
APPENDIX: EXPERIMENTAL RESULTS: LIDT VERSUS
WAVELENGTH
FIG. 18. Laser-induced damage threshold versus wavelength for silicon and
zinc selenide.
FIG. 20. Laser-induced damage threshold versus wavelength for hafnium
oxide, scandium oxide and aluminum oxide.
FIG. 21. Laser-induced damage threshold versus wavelength for silicon oxide (in film and bulk form) and calcium fluoride.
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FIG. 19. Laser-induced damage threshold versus wavelength for niobium
oxide and tantalum oxide.
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