# Three-Dimensional Modeling of the Heat-Affected Zone in  Laser Machining Applications

Authors: Martynas Beresna; Titas Gertus; Rolandas Tomašiūnas; Hiroaki Misawa; Saulius Juodkazis
Journal: Laser Chemistry
 (2008)

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## Abstract

Thermal load as well as its three-dimensional (3D) spatial distribution
has been estimated inside representative materials: glass (low thermal
diffusion), silicon (semimetal properties), and sapphire (a crystalline dielectric
of a high thermal conductivity) for typical laser processing and
direct laser writing applications. The 3D temperature distribution allows
to calculate thermal stress around the focal region. This provides an
assessment tool for optimization of laser microprocessing conditions for
controlled laser dicing and cutting applications.


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## Body

## 1. Introduction

Laser processing is expanding fast into micro- and
nanostructuring of complex multilayered and composite materials,
heterostructures, and MEMS devices. Hence, precise account of energy delivery,
heat-affected zone (HAZ), and thermal stress is of a paramount importance for
high-precision fabrication and optical waveguide recording which usually have a
narrow parameter space for optimization [1–9]. A high speed is required for efficient material
processing by the direct laser writing (0.1–1 m/s for a linear scan), and it recently
becomes achievable due to improvements in femtosecond laser sources which
delivers large (∼10 W) average power, at ∼MHz repetition rate with a tens-of-μJ pulse energy.Picosecond and femtosecond lasers are the most
prospective in the nano-microdrilling, dicing, and cutting applications where
micrometer precision is required. At high repetition rates, thermal effects are
pronounced and are usually responsible for crack formation in the case of
in-bulk fabrication when nanosecond lasers are employed [10, 11]. For the waveguide
recording, there are remarkable differences in defect and stress generation
depending on pulse duration, scanning speed, and repetition rate [12, 13]. In the case of shorter
pulses, the size and positioning of microcracks inside the bulk of transparent
materials can be achieved with high precision avoiding damage of the entrance
and exit surfaces. This allows to realize a, so-called, stealth dicing when the
cracks inside workpiece are large enough for successful cleaving of the sample
while there is no optical damage on both surfaces
after laser irradiation [14, 15]. Similarly, cleaving of glasses can be controlled by
designing a temperature field for a creation of a tensile stress which, first,
forms and then guides a crack on the surface. There is a considerable interest
in modeling of thermal fields inside the bulk of materials since they can be
used for a controlled recording of waveguides, laser joining, and splitting
applications in material processing.Here, we explore numerically the temperature
distribution inside bulk of a workpiece in different materials at realistic
conditions of microprocessing by ultrafast (sub-1 ps) lasers. Combination of
several laser pulses separated in time and space can be also modeled by the
same approach for practical 3D laser dicing and writing (photo-modification)
applications.


## 2. Model

Recently, experimental results on laser welding and
joining of glass [16, 17] have been quantitatively modeled by an in-bulk
scanning of a rectangular heat source created at the focus of femtosecond laser
beam [16]. Such a
seemingly approximate description of the focal volume in the model nevertheless
delivered good fit to the experimental data [16]. Hence, we use this model of the rectangular heat
source for simulation of thermal field in different materials. Limitations of
this approach are expected for focal spots of few micrometers in cross-section
and at the very early stages of thermal diffusion, which were not considered
here. It is also important that the model provides analytical solution, hence,
the physical description and scaling rules of the laser processing can be
qualitatively predicted for the more complex sample composition and geometry.
For the quantitative modeling, however, the finite-difference time-domain (FDTD)
simulations should be carried out for the actual sample geometry.We use this model and calculate the extension of HAZ
in glass, silicon, and sapphire. The dimensions of the heating source
corresponds to tight focusing with numerical apertureNA≃0.5,
which is a practical choice in laser microfabrication. We simulate the
temperature field for a heat-source: a rectangular with a base area (2a)2 and height 2h (in z-direction), scanned along x-direction at
speed v inside infinite material (Figure 1(a)). This
model is realistic for actual laser processing, waveguide recording, and
welding experiments when the 3D dimensions of the workpiece are much larger
than dimensions of the heat-source (focal volume).(a) Schematic presentation of the heat source
with a square cross-section,2a×2a,
on the xy-plane and 2h high. (b) The 3D temperature field in glass
calculated by (1) with the same parameters as in Figure 2 at time moment t=0.1T (T=1/f where f is the repetition rate) for z=0.
(a)(b)The temperature rise (θ) at the time moment (t) at the location (x,y,z) in quasi-steady state is [16]
(1)θ(x,y,z,t)=AQ64cpρr∑i=0∞(Erfcξ+iυΞ2/2−δτiΞ2/τ2+1−Erfcξ+iυΞ2/2+δτiΞ2/τ2+1)×(Erfcψ−δτiΞ2/τ2+1−Erfcψ+δτiΞ2/τ2+1)×(Erfcζ−ητiΞ2/τ2+1−Erfcζ+ητiΞ2/τ2+1),where erfc(U)=(2/π)∫U∞exp(−u2)du, A is absorbance of the laser pulse, Q is the instantaneous laser energy, 2a is the diameter of the focus, 2h is the length of a heat source along z-axis, cp is specific heat, and ρ is the mass density. The following
nondimensional parameters are used in (1): r3=a2h, ξ=x/r, ψ=y/r, ζ=z/r, δ=a/r=1/n3, η=h/r=n2/3, τ=4χt/r, Ξ=4χT/r,
and υ=vr/(2χ) (here χ is the temperature diffusivity).


## 3. Results and Discussion

We compare here axial temperature distributions for a
high-power and high-repetition-rate heat source which models laser cutting,
dicing, and waveguide recording when a focal spot is set inside the processed
material. Materials with different thermal conductivity (thermal diffusivity)
were modeled. The formulae given in Section2 can be used for estimations of
both axial and lateral cross-sections of the temperature field. In the
following part, we present the temperature profiles along the beam scanning.
The extension of lateral cross-section is closely matching that of the axial
around the maximum temperature (see, Figure 1(b)). In order to compare thermal
fields in very different materials, we used the same absorbance A=0.5 which corresponds to a strong ionization of
material, for example, at the optical breakdown conditions of dielectrics such
as silica or sapphire [18, 19], and is relevant for laser processing by ultrashort
pulses at high irradiance. For more quantitative estimations, the absorbance of
material and its temporal evolution during the pulse of high irradiance should
be known at the focal volume.


### 3.1. Temperature Field

Figure2 shows the thermal profiles along the scan
direction with heat source at x=0 at different time moments within the period of
laser pulsing inside glass. Similar conditions are created in glass-welding
applications where high-temperature melts are essential for good quality joints
[16, 17]. The spatial extension of
the heated zone does not change significantly beyond a ±5μm range, only the maximum temperature is
changing at the origin of the heat source. Trailing temperature distribution is
recognizable at negative x values.Figure 2
The axial temperature profile in
glass at different time moments. Conditions:Q=0.2 W at f=0.1 MHz, A=0.5 absorbance, 10 cm/s scan speed, the
rectangular heat source is modeled by 2×2×20 μm3 volume (a×a×h). Typical borosilicate glass parameters were
taken from [16].Silicon differs from glass mainly by the thermal
diffusivity which isχ=0.8cm2/s (for a glass 4.6×10−3cm2/s), while the specific heat and mass density are almost the same.
This makes almost impossible to effectively heat and localize the hot spot
inside silicon as it is shown in Figure 3. Geometrical
parameters of the heat source (laser-heated focal
volume) were the same as for glass at the same scanning speed of 10 cm/s. The
heating power was increased 50 times to Q=10 W and repetition rate was 1 or 10 MHz in order
to heat up the focal volume (Figure 3). When repetition rate is larger (see
Figure 3, curves (2,3)), there is a significant spread of temperature
beyond the acceptable ±5μm range. This would be critical for dicing of
wafers with microelectronic chips at narrow street width. It is noteworthy that
the heat capacity of solid and molten silicon is almost the same. This entitles
a quantitative comparison of theoretical simulations and experiments in terms
of thermal field. It should be noted that the optical properties
change upon melting: the complex refractive index
is changing from n+ik=3.69+6×10−3i (crystalline)
silicon to (3.9+5.5i) (liquid) silicon (for the silica native oxide,
the refractive index is 1.45+4.6×10−6i). The phase transition, the latent heat
necessary to melt the focal volume, was not included in the used simple model.
Hence, the obtained thermal field close to the melting point is only
approximate (Figure 3) since the increased absorbtion is not taken into
account.Figure 3
The axial temperature profile in Si.
Conditions:Q=10 W at f=1 MHz (profiles: 1, 4) and 10 MHz (2, 3), A=0.5 absorbance, 10 cm/s scan speed, time moment t=0.01T (1, 2, 4) and 0.1T (3), and dimensions of a rectangular
heat source were a=2,h=20μm (1–3), and a=5μm (4). Typical Si parameters: cp=0.7 J/(g K), ρ=2.329g/cm3, χ=0.8cm2/s, and Tm=1412°C is the melting temperature of Si.The in-bulk structuring of silicon remains a
challenging task due to required high power and repetition rate; moreover, the
wavelength of irradiation should be in the transparency region (a cutoff is at
approximately 1.1 eV or 1120 nm in Si). Typical fs-lasers are usually operating
at slightly shorter wavelengths. This makes them useful only for cutting and
dicing of silicon by ablation.Sapphire stands out from transparent dielectric
materials due to its high thermal conductivity (though, inferior to diamond
which has a higher thermal conductivity than many metals). In terms of
temperature diffusivity relevant in this study, sapphire has aχ=0.16cm2/s value. Thus, it is expected to be able to localize heat more
similar to silicon than glass. Figure 4 summarizes data, which show fast
relaxation of maximum temperature.Figure 4
The axial temperature profile in sapphire at
different time moments. Conditions:Q=5 W at f=1 MHz, A=0.5 absorptivity, 10 cm/s scan speed, the
rectangular heat source dimensions were a=2,h=20μm. Typical sapphire parameters [20]: cp=0.761 J/(g K), ρ=3.9g/cm3, χ=0.16cm2/s, and Tm>2050°C is the melting temperature of sapphire.The estimations given above are of qualitative nature
since we are not considering here nonlinear effects of light propagation,
filamentation, and self-focusing. However, general tendencies are depicted and
can be used as first estimate of the thermal load to regions in close proximity
of the irradiated laser spot. The most challenging task in laser processing of
transparent materials is the evaluation of the
amount of absorbed energy and the volume of its deposition. Recent success of
this model [16] in
description of glass welding encourages to apply it to other laser-processing
tasks such as dicing, cutting, and waveguide recording in different materials.


### 3.2. Stress Confinement

With the known thermal field calculated numerically or
analytically, one can estimate the stress acting inside materials due to
thermal expansion. We discuss here a model where material is described by its
average thermomechanical properties, hence, the model is better suited for
glasses and amorphous materials rather than crystals. However, a qualitative
mechanism of the stress confinement should be valid for crystals as well. It is
noteworthy that the crystalline properties become more. important in crack
propagation discussed in section3.3.For example, for a linear heat source scanned alongx-direction inside isotropic medium, the stress along y-axis is given by
[14](2)σy(x)=σmax(e(x/l)K1(|x|l)signx−lx),where σmax=qμα(1+ν)/(2πdκ) [Pa] is the maximum stress for the heating
power q [W], K1 is the modified Bessel function of the first
kind and order, ν is the Poisson ratio, l=κ/(cv) is the length of thermal diffusion expressed
via the specific heat capacity density c and the thermal conductivity κ, v is the velocity of the heat source, d is the thickness of sample (in this case equal
to the length of the heat source). This expression (2) links the absorbed heat
to the generated stress. In the case of a heating source of finite dimensions,
the temperature field can be calculated by (1) and then integrated by the same
way as in the case of (2) according to [14](3)σy(x)=μ(1+ν)α[∫0lT(x,ξ)dξ−T(x)].It should be noted that
analytical result is not available in the case of a
heat source of finite dimensions and (3) should be integrated numerically.The stress field around waveguides recorded by the
femtosecond direct laser writing inside Nd-doped yttrium aluminium garnet (YAG)
can be engineered to exert compressive and tensile stresses on the order of±1 kbar, respectively (up to +5 kbar for compression). Such photomodification
can create the following: (i) waveguiding regions due to augmented
refractive index Δn=((n2−1)(n2+2)/2n)(P/E),
here P is the pressure and E is the Young modulus, and, (ii) the
spectral properties of Nd transitions can be effectively modified via pressure
[21, 22]. The combined effects of refractive index and emission changes can be utilized for the creation of efficiently lasing
waveguide laser-written in Nd:YAG ceramics [23].


### 3.3. Crack Propagation

Laser dicing, splitting, and cutting applications
rely on a seeding of photo-modification which
alters material strength and serves as a crack generator. The critical stress
required for the surface or in-bulk crack formation is discussed in what
follows.The theoretical strength of material, in terms of
stress, can be estimated from the work necessary to increase the atomic intraplane
distance by 25% at which cleaving of a crystal occurs [24]:(4)σth=γEa0≃E2π,where E is the Young modulus, γ is the surface energy per area, and a0 is the atomic intraplane distance. When crack
is present inside material, the stress concentration at the crack tip depends
on the crack shape and size [24]:(5)σmax=2σlclρ,where 2lc is the length of the crack (assumed to be
elliptical), σ is the applied stress, and lρ is the radius at the crack tip. Crack
propagation ensues at the fracture stress when σth=σmax,
or [24](6)σF=2γElρ3πa0lc.Since the material failure (strength limit) for the
tensile stress is approximately ten times lower as compared with that for a
compressive load (4) scanning of two or several heat sources, the laser foci,
spatially separated across (or along) the scanning direction is prospective for
the controlled cleaving and dicing with a literarily zero street width. The
absorbed laser power (the heat source) depends on the linear and nonlinear
absorption at the focus, that is, on irradiance and ionization level at the
focus. The cooling of the two or more spatially and temporally separated laser
foci can generate tensile stresses for the controlled cleaving at the optimized
conditions of laser machining.The thermal diffusion lengthLD=χt (t is an effective irradiation duration or pulse
duration) and the absorption depth (∼1/α(λ)) defines a typical feature size of laser
micromachining via the absorption at the wavelength λ (linear and nonlinear absorption with the
absorption coefficient α which is intensity-/irradiance-dependent)
[25]. The minimum
feature sizes of the in-bulk modification or ablation smaller than both 1/α and LD are achievable via control of processing
parameters: velocity of scan, laser repetition rate, and pulse energy.The laser heating necessary to achieve fracture
strength limit in compression requires very high laser power and a high
temperature gradient [19], hence, it is considered impractical for the laser processing
of multilayered and multicomponent workpieces. Ultrashort pulses
favor creation of a fast heating due to strong
nonlinear absorption and rapid thermal quenching [18]. Both factors
facilitate a stress generation. Generation of
compressive and tensile stresses with a particular patterns made of several
spatially (e.g., by using Gauss-Bessel beams/pulses [26]) and temporally shaped beams/pulses
is expected to deliver practical solutions for the surface and in-bulk laser
machining as well as for other direct laser writing applications. Since the
temperature field is not symmetrical around its maximum along the scanning
direction (Figure 1), it may provide an explanation to the recently observed
nonreciprocity effect in femtosecond laser recording in a non-centrosymmetric
lithium niobate [27].


## 3.1. Temperature Field

Figure2 shows the thermal profiles along the scan
direction with heat source at x=0 at different time moments within the period of
laser pulsing inside glass. Similar conditions are created in glass-welding
applications where high-temperature melts are essential for good quality joints
[16, 17]. The spatial extension of
the heated zone does not change significantly beyond a ±5μm range, only the maximum temperature is
changing at the origin of the heat source. Trailing temperature distribution is
recognizable at negative x values.Figure 2
The axial temperature profile in
glass at different time moments. Conditions:Q=0.2 W at f=0.1 MHz, A=0.5 absorbance, 10 cm/s scan speed, the
rectangular heat source is modeled by 2×2×20 μm3 volume (a×a×h). Typical borosilicate glass parameters were
taken from [16].Silicon differs from glass mainly by the thermal
diffusivity which isχ=0.8cm2/s (for a glass 4.6×10−3cm2/s), while the specific heat and mass density are almost the same.
This makes almost impossible to effectively heat and localize the hot spot
inside silicon as it is shown in Figure 3. Geometrical
parameters of the heat source (laser-heated focal
volume) were the same as for glass at the same scanning speed of 10 cm/s. The
heating power was increased 50 times to Q=10 W and repetition rate was 1 or 10 MHz in order
to heat up the focal volume (Figure 3). When repetition rate is larger (see
Figure 3, curves (2,3)), there is a significant spread of temperature
beyond the acceptable ±5μm range. This would be critical for dicing of
wafers with microelectronic chips at narrow street width. It is noteworthy that
the heat capacity of solid and molten silicon is almost the same. This entitles
a quantitative comparison of theoretical simulations and experiments in terms
of thermal field. It should be noted that the optical properties
change upon melting: the complex refractive index
is changing from n+ik=3.69+6×10−3i (crystalline)
silicon to (3.9+5.5i) (liquid) silicon (for the silica native oxide,
the refractive index is 1.45+4.6×10−6i). The phase transition, the latent heat
necessary to melt the focal volume, was not included in the used simple model.
Hence, the obtained thermal field close to the melting point is only
approximate (Figure 3) since the increased absorbtion is not taken into
account.Figure 3
The axial temperature profile in Si.
Conditions:Q=10 W at f=1 MHz (profiles: 1, 4) and 10 MHz (2, 3), A=0.5 absorbance, 10 cm/s scan speed, time moment t=0.01T (1, 2, 4) and 0.1T (3), and dimensions of a rectangular
heat source were a=2,h=20μm (1–3), and a=5μm (4). Typical Si parameters: cp=0.7 J/(g K), ρ=2.329g/cm3, χ=0.8cm2/s, and Tm=1412°C is the melting temperature of Si.The in-bulk structuring of silicon remains a
challenging task due to required high power and repetition rate; moreover, the
wavelength of irradiation should be in the transparency region (a cutoff is at
approximately 1.1 eV or 1120 nm in Si). Typical fs-lasers are usually operating
at slightly shorter wavelengths. This makes them useful only for cutting and
dicing of silicon by ablation.Sapphire stands out from transparent dielectric
materials due to its high thermal conductivity (though, inferior to diamond
which has a higher thermal conductivity than many metals). In terms of
temperature diffusivity relevant in this study, sapphire has aχ=0.16cm2/s value. Thus, it is expected to be able to localize heat more
similar to silicon than glass. Figure 4 summarizes data, which show fast
relaxation of maximum temperature.Figure 4
The axial temperature profile in sapphire at
different time moments. Conditions:Q=5 W at f=1 MHz, A=0.5 absorptivity, 10 cm/s scan speed, the
rectangular heat source dimensions were a=2,h=20μm. Typical sapphire parameters [20]: cp=0.761 J/(g K), ρ=3.9g/cm3, χ=0.16cm2/s, and Tm>2050°C is the melting temperature of sapphire.The estimations given above are of qualitative nature
since we are not considering here nonlinear effects of light propagation,
filamentation, and self-focusing. However, general tendencies are depicted and
can be used as first estimate of the thermal load to regions in close proximity
of the irradiated laser spot. The most challenging task in laser processing of
transparent materials is the evaluation of the
amount of absorbed energy and the volume of its deposition. Recent success of
this model [16] in
description of glass welding encourages to apply it to other laser-processing
tasks such as dicing, cutting, and waveguide recording in different materials.


## 3.2. Stress Confinement

With the known thermal field calculated numerically or
analytically, one can estimate the stress acting inside materials due to
thermal expansion. We discuss here a model where material is described by its
average thermomechanical properties, hence, the model is better suited for
glasses and amorphous materials rather than crystals. However, a qualitative
mechanism of the stress confinement should be valid for crystals as well. It is
noteworthy that the crystalline properties become more. important in crack
propagation discussed in section3.3.For example, for a linear heat source scanned alongx-direction inside isotropic medium, the stress along y-axis is given by
[14](2)σy(x)=σmax(e(x/l)K1(|x|l)signx−lx),where σmax=qμα(1+ν)/(2πdκ) [Pa] is the maximum stress for the heating
power q [W], K1 is the modified Bessel function of the first
kind and order, ν is the Poisson ratio, l=κ/(cv) is the length of thermal diffusion expressed
via the specific heat capacity density c and the thermal conductivity κ, v is the velocity of the heat source, d is the thickness of sample (in this case equal
to the length of the heat source). This expression (2) links the absorbed heat
to the generated stress. In the case of a heating source of finite dimensions,
the temperature field can be calculated by (1) and then integrated by the same
way as in the case of (2) according to [14](3)σy(x)=μ(1+ν)α[∫0lT(x,ξ)dξ−T(x)].It should be noted that
analytical result is not available in the case of a
heat source of finite dimensions and (3) should be integrated numerically.The stress field around waveguides recorded by the
femtosecond direct laser writing inside Nd-doped yttrium aluminium garnet (YAG)
can be engineered to exert compressive and tensile stresses on the order of±1 kbar, respectively (up to +5 kbar for compression). Such photomodification
can create the following: (i) waveguiding regions due to augmented
refractive index Δn=((n2−1)(n2+2)/2n)(P/E),
here P is the pressure and E is the Young modulus, and, (ii) the
spectral properties of Nd transitions can be effectively modified via pressure
[21, 22]. The combined effects of refractive index and emission changes can be utilized for the creation of efficiently lasing
waveguide laser-written in Nd:YAG ceramics [23].


## 3.3. Crack Propagation

Laser dicing, splitting, and cutting applications
rely on a seeding of photo-modification which
alters material strength and serves as a crack generator. The critical stress
required for the surface or in-bulk crack formation is discussed in what
follows.The theoretical strength of material, in terms of
stress, can be estimated from the work necessary to increase the atomic intraplane
distance by 25% at which cleaving of a crystal occurs [24]:(4)σth=γEa0≃E2π,where E is the Young modulus, γ is the surface energy per area, and a0 is the atomic intraplane distance. When crack
is present inside material, the stress concentration at the crack tip depends
on the crack shape and size [24]:(5)σmax=2σlclρ,where 2lc is the length of the crack (assumed to be
elliptical), σ is the applied stress, and lρ is the radius at the crack tip. Crack
propagation ensues at the fracture stress when σth=σmax,
or [24](6)σF=2γElρ3πa0lc.Since the material failure (strength limit) for the
tensile stress is approximately ten times lower as compared with that for a
compressive load (4) scanning of two or several heat sources, the laser foci,
spatially separated across (or along) the scanning direction is prospective for
the controlled cleaving and dicing with a literarily zero street width. The
absorbed laser power (the heat source) depends on the linear and nonlinear
absorption at the focus, that is, on irradiance and ionization level at the
focus. The cooling of the two or more spatially and temporally separated laser
foci can generate tensile stresses for the controlled cleaving at the optimized
conditions of laser machining.The thermal diffusion lengthLD=χt (t is an effective irradiation duration or pulse
duration) and the absorption depth (∼1/α(λ)) defines a typical feature size of laser
micromachining via the absorption at the wavelength λ (linear and nonlinear absorption with the
absorption coefficient α which is intensity-/irradiance-dependent)
[25]. The minimum
feature sizes of the in-bulk modification or ablation smaller than both 1/α and LD are achievable via control of processing
parameters: velocity of scan, laser repetition rate, and pulse energy.The laser heating necessary to achieve fracture
strength limit in compression requires very high laser power and a high
temperature gradient [19], hence, it is considered impractical for the laser processing
of multilayered and multicomponent workpieces. Ultrashort pulses
favor creation of a fast heating due to strong
nonlinear absorption and rapid thermal quenching [18]. Both factors
facilitate a stress generation. Generation of
compressive and tensile stresses with a particular patterns made of several
spatially (e.g., by using Gauss-Bessel beams/pulses [26]) and temporally shaped beams/pulses
is expected to deliver practical solutions for the surface and in-bulk laser
machining as well as for other direct laser writing applications. Since the
temperature field is not symmetrical around its maximum along the scanning
direction (Figure 1), it may provide an explanation to the recently observed
nonreciprocity effect in femtosecond laser recording in a non-centrosymmetric
lithium niobate [27].


## 4. Conclusions

The knowledge of the 3D temperature field provides a
possibility to estimate stresses inside the workpiece under laser fabrication.
The 3D localization of temperature inside glass, silicon, and sapphire has been
calculated and discussed for practical irradiation conditions in direct laser
writing/dicing/cutting applications. Strong thermal gradients can be exploited
for generation of tensile stress created by several heating sources
(irradiation spots/patterns) and are prospective for the controlled laser
cleaving. Thermal stress sources can be created by either of the following or
their combination: by a cumulative fast repetition laser irradiation, by usage
of ultra-short laser pulses, or by implementation of fast laser-spot (sample)
scanning. The femtosecond direct laser writing can be used for the in-bulk
localization of stresses and photo-modification for photonic applications
[23, 27].



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*Source: 976205-2008-11-09.xml*