Thermodynamic damage mechanism of transparent films caused by a low-power laser ZhiLin Xia, JianDa Shao, ZhengXiu Fan, and ShiGang Wu
A new model for analyzing the laser-induced damage process is provided. In many damage pits, the melted residue can been found. This is evidence of the phase change of materials. Therefore the phase change of materials is incorporated into the mechanical damage mechanism of films. Three sequential stages are discussed: no phase change, liquid phase change, and gas phase change. To study the damage mechanism and process, two kinds of stress have been considered: thermal stress and deformation stress. The former is caused by the temperature gradient and the latter is caused by high-pressure drive deformation. The theory described can determine the size of the damage pit. © 2006 Optical Society of America OCIS codes: 310.3840, 140.3330, 140.3440.
1. Introduction
The interaction between a laser pulse and a transparent solid has been extensively studied for many years.1– 4 Many experiments have been conducted to study the damage mechanism. A useful method is to deduce damage mechanisms from damage morphologies. When the films are irradiated by a laser beam with a long pulse width, damage beginning with crack formation is observed. When the films are irradiated by a laser beam with an ultrashort pulse width, damage occurs due to ablation. It was thought that the change in the type of damage was caused by the change in the mechanism. In Refs. 1– 6, the authors thought that the damage produced by a nanosecond pulse was determined by an extrinsic absorption mechanism such as absorption by foreign inclusions. The most frequently employed extrinsic mechanism is that of spherical absorbing particles embedded in a host material. The particles absorb incident radiation energy and their temperatures rise, which produce melting, vaporization, or stress fracture of the host around them, while the damage produced by the irradiation of the femtosecond pulse
The authors are with the Shanghai Institution of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China. Z. L. Xia (
[email protected]) and S. G. Wu are also with the Graduate Schools, Chinese Academy of Sciences, Beijing 100080, China. Received 17 January 2006; revised 26 May 2006; accepted 27 June 2006; posted 10 July 2006 (Doc. ID 67286). 0003-6935/06/328253-09$15.00/0 © 2006 Optical Society of America
was caused by an intrinsic absorption mechanism such as avalanche ionization or photoionization or their combination. The physical process of avalanche ionization is that when the initial free electrons have been sped up to the energy level, which is larger than the bandgap energy of the material, they will ionize some other bound electrons by colliding with them. This process will produce a large number of free electrons and will rapidly and ultimately cause film damage, whereas for the mechanism of photoionization, the bound electrons will be ionized only when they absorb enough laser energy. This mechanism evidently occurs only when the laser power density is very high. The damage process can be divided into sequential stages: generation of free electrons, energy absorption by free electrons, energy transference from conductive electrons to a lattice, and development of mechanical damage. If the laser pulse width is longer than a picosecond, we think that the average temperature of the free electrons is the same as that of the lattice. For the femtosecond pulse width, a large number of electrons will be ionized, and the average temperature of the free electrons is different from that of the lattice. Electrons are superheated, so plasma will form and a Coulombic blast may occur.7–9 The mechanical damage process caused by a laser with a pulse width in the nanosecond range is complicated, and it cannot be interpreted by a single model. In the range of the nanosecond pulse width, the phase change of materials was usually ignored. It will lose some useful information neglecting the phase change. This is the main reason that the experimen10 November 2006 兾 Vol. 45, No. 32 兾 APPLIED OPTICS
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tal data cannot conform to theoretical evaluation. In this paper, we will introduce new theoretical models for analyzing the mechanical damage process. The absorption mechanism is inclusion absorption. Three sequential stages (these stages occur sequentially as the material temperature increases) were discussed: Neither the inclusion nor the host have any phase change, liquid phase change occurs in the inclusion and part of the host, and the inclusion and part of the host are in a vaporized state when laser irradiation is over. In Section 2, the model will be described in detail. In this section, two kinds of stress (thermal stress and deformation stress) are considered to cause film damage. Three possible terminal stages are discussed. In Section 3, some results and discussion are provided. At last, in Section 4, some conclusions are drawn. 2. Theoretical Model A.
Thermal Process
The thermal process is the basic process of laserinduced damage, and the temperature distribution is necessary for evaluating the stress distribution, so we must evaluate the temperature distribution around the inclusion first. Inclusions embedded in films will enhance the local electric field strength (or we can say that an inclusion will cause a focusing effect; an inclusion is like a lens), and inclusions with different geometric shapes will cause different enhancement behaviors. Details about the field enhancement can be found in Ref. 10. For simplification, we only consider the shape of a sphere. Inclusions with the other shapes can be converted to spherical inclusions using a modification coefficient according to Ref. 10. Like the usual method, we consider a homogeneous spherical inclusion of radius a embedded in infinite homogeneous films. Laser energy is absorbed by the inclusion when t ⬎ 0 at the rate A per unit time per unit volume. The thermal conductive functions are
冉 冊 冉 冊
⭸Ti i ⭸ 2 ⭸Ti i Ai共Ti兲 r ⫹ , ⫽ 2 ⭸t ⭸r Ki r ⭸r
when 0 ⱕ r ⬍ a, t ⬎ 0,
⭸Tf f ⭸ 2 ⭸Tf ⫽ 2 r , ⭸t ⭸r r ⭸r
when r ⬎ a, t ⬎ 0, (1)
in which T, , K are the temperature, the thermal diffusion coefficient, and the thermal conductivity, respectively. The suffix i denotes inclusion and f denotes a host of films. The initial conditions and boundary conditions are
Ki
Ti ⫽ Tf ⫽ 0,
when t ⫽ 0,
Ti ⫽ Tf,
when r ⫽ a,
⭸Ti ⭸Tf ⫽ Kf , when r ⫽ a, ⭸r ⭸r
Ti|r⫽0 and Tf|r⫽⬁ are finite values. 8254
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Fig. 1. Temperature distribution in films when an intensive absorbing inclusion is included. The power density of the laser pulse is 1 GW兾cm2; the pulse width is 12 ns; the depth that the inclusions are embedded is 2000 nm; and the radii of the inclusions are 50, 100, 200, and 400 nm, respectively.
Supposing Ai ⫽ const, Goldenberg and Tranter11 had obtained an analytic solution of function group (1). In this paper, the temperature distribution in the films was evaluated using finite-time-difference numerical methods,12 since we considered the temperature dependence of the absorption coefficient. For most dielectrics, the bandgap energy will decrease when the temperature increases; accordingly, three sequential absorption mechanisms will occur: multiphoton absorption, single-photon absorption, and metallic absorption. As a result, the absorption coefficient increases as the temperature increases. The theoretical details about the absorption coefficient have been summed up in other papers.13–17 According to Mie scattering theory, the energy absorption is determined by the Mie absorption cross section.18,19 We did not use the Mie absorption cross section. Instead, we used the inclusion absorption volume, which is derived from the absorption depth. The absorption depth is obtained according to the absorption coefficient. In this treatment, energy absorption is determined by the inclusion radius. There are many characters we can use to classify the inclusions. For example, we can distinguish metal inclusions and dielectric inclusions based on the materials of the inclusion. Different kinds of inclusion correspond to different absorbing mechanisms. The details of this can be found in Ref. 20. We have evaluated the temperature distribution in films when intensive absorption inclusions exist. The results are plotted in Fig. 1 and the parameters used in the evaluation are listed in Table 1. In evaluation, we assume the inclusion is platinum because the raw material 共HfO2兲, which is used for depositing films in our experiments, abounds in platinum.21 For dielectric films, the low thermal conductivity of the host allows the absorbed incident energy to be stored
Parameters Density Melting point Boiling point Specific heat Latent heat of fusion Latent heat of vaporization Thermal conductivity Coefficient of lineal expansion Young’s modulus Poisson ratio Laser pulse width Laser wavelength
Inclusion Material (Pt)
Host Material (HfO2) 9.7 g兾cm3 3080 K 4602 K 144 J兾(Kg K) 113,400 J兾Kg
2615,400 J兾Kg
2,990,000 J兾Kg
67.2 W兾(m K)
1.1 W兾(m K)
9.0 ⫻ 10⫺6 K
5.8 ⫻ 10⫺6 K
170 GPa
0.33 12 ns
0.27 12 ns
1064 (nm)
1064 (nm)
⭸ur , ⭸r
⫽ ⫽
(4)
ur , r
(5)
where E is Young’s modulus, and ur is the radial component of the displacement vector. We concentrate only on the radial component of the displacement and the circumferential component of the stress. The circumferential thermal stress can be deduced from Eqs. (3)–(5), and it has the form
(2)
where max is the maximal circumferential tensile stress existing in the films and th is the tensile strength for a crack to form. In this paper, we concentrate on the stress distribution in the films. The characteristic time of thermoelasticity stress established in the region of local heating of size ⬃1 m is of the order of 10⫺9 s. The characteristic time of the crack formation is of the order of 10⫺8 s.5 We think that the stress is established synchronously with the temperature rise. B.
再
Err ⫽ rr ⫺ 共 ⫹ 兲 ⫹ E␣T, E ⫽ ⫺ 共rr ⫹ 兲 ⫹ E␣T, E ⫽ ⫺ 共 ⫹ rr兲 ⫹ E␣T, rr ⫽
mainly in the inclusion. As a result, the temperature of the inclusion is high.13 If inclusions exist on the surface of the films, they will cause film damage when their temperatures reach the melting point. This kind of damage is thermal process dominated. If inclusions exist inside the films, the thermal process is not the main process, and the mechanical mechanism dominates the damage process. In this paper, we mainly discuss the latter kind of damage. If the condition described below [expression (2)] is satisfied, a crack will form in the film and will cause it to be damaged: max ⱖ th,
(3)
where ur is the radial component of the displacement vector, ␣ is the linear thermal expansion coefficient, and is the Poisson ratio. For the radial symmetry system, the relationships between strains and stress have the form22
21.45 g兾cm3 2041 K 4098 K 132.6 J兾(Kg K) 122,000 J兾Kg
95 GPa
册
冋
1 ⫹ ⭸T共r, t兲 ⭸ ⫺2 ⭸ 2 r r ur兲 ⫽ ␣ , 共 ⭸r ⭸r 1⫺ ⭸t
Table 1. Main Parameters Used in Evaluation
No Phase Change
In the stage where no phase change occurred, thermal stress dominates the circumferential stress. We can evaluate thermal stress from the thermoelasticity theory. When the temperature of the film is low, the displacement of the atoms in the lattice is very small. Therefore the analysis of the stress and strain distributions can be performed using the equation of the quasi-stationary thermoelasticity theory.5 For the spherically symmetrical system, the equation has the form
冋冕
␣E 1 共r, t兲 ⫽ 1 ⫺ r3
r
册
T共r⬘, t兲r⬘2dr⬘ ⫺ T共r, t兲 .
0
(6)
If this stage is the last stage and Eq. (2) has been satisfied, the damage morphology will be crack or buckle. See Fig. 2(b). This case will happen when the tensile strength of the films and the power density of the laser pulse are low. In addition, from the evaluating results discussed in Subsection 3.A, the radius of inclusion cannot be too little. From the temperature plotted in Fig. 1, we can see that the temperatures of the inclusion and the host zone around it are larger than 5000 K. According to the melting points and boiling points of the inclusion and the host, which are listed in Table 1, we can draw the conclusion that the phase change of the inclusion and part of the host cannot be ignored. From Fig. 2(c), we can see that the bottom of the damage pit is rough, with the remainder of the melted materials. When the materials are boiled, the spray speed (the speed in which the sprayed materials flee from the films) is very high and little remainder can be observed in the damage pit (see Fig. 11 below). When the phase change occurred, the thermal stress is not enough, and the stress distribution is different from that of no phase change. C.
Liquid Phase Change
As mentioned above, if the material is melted, a new treatment is needed. Before the new model is introduced, some characteristics of the liquid should be emphasized. Liquid can transfer pressure without attenuation to any part inside it, so the pressure is uniform in every part of it. Besides, liquid will not shrink when external pressure presses on it. We can evaluate the radius of the melted zone by assuming 10 November 2006 兾 Vol. 45, No. 32 兾 APPLIED OPTICS
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Fig. 2. Damage morphology of films under different laser fluence density. In (a) and (b), every small grid denotes 10 m and every large grid denotes 50 m. In (c) the bottom of the damage pit is rough and it is formed because of the remainder of the melted materials.
that the surface of the melted zone is free. From Eq. (3), the radius of the melted zone can be written as 1 ⫹ r Rm ⫽ R ⫹ 1 ⫺ r
冕
R
␣rT共r, t兲dr,
(7)
0
where
再
␣r ⫽ ␣i, r ⫽ i when r ⱕ a . ␣r ⫽ ␣h, r ⫽ h when r ⬎ a
Rm is the radius of the melted zone after it is melted. R is the original radius of the zone (after laser irradiation, this zone will be melted and will expand to the radius of Rm) before laser irradiation. In the melted zone, the circumferential stress is zero, whereas in the other part, which is in solid phase, Eq. (6) is still valid for evaluating the circumferential stress. In this stage, if the relation described in formulation (2) is satisfied, the films will be damaged. The damage morphology is a crater with a 8256
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rough inner surface [see Fig. 2(c)]. Most often, cracks originating from the bottom of the damage crater are visible. If the relation described in formulation (2) cannot be satisfied all the time, and the stage of the liquid phase change is the last stage, bubbles will form [see Fig. 2(a)]. If the power density of a laser pulse is high enough, just as in the case of Fig. 1, the liquid phase change is only a transition stage of the damage process, and the gas phase change needs to be considered. D.
Gas Phase Change
In the stage of the gas phase change, hydrodynamic pressure of the boiled zone (or the superheated zone) is a liquid with a temperature beyond the boiling point. We think it (as a gas) needs to be considered. Like a liquid, the circumferential stress in the boiled zone is zero. However, the pressure the boiled zone endured is self-determined; its volume is significant only when the pressure and the temperature are given. It can be compressed to high density if the
Fig. 3. Sketch map of the inclusion in films (left); sketch map of the spherical shell model (right). The depth that the inclusion embedded is D0; the radius of the inclusion is a.
external pressure is superhigh and will expand infinitely if the pressure is removed. If the gas phase change occurs, we can analyze the evolution of the damage process by solving function (8), which describes the dynamic condition during the development of mechanical damage. A model of a spherical shell, which endured inner pressure, is used (see Fig. 3): P ⫽ P1 ⫹ P2,
(8)
where P is the hydrodynamic pressure of the boiled materials and the driving inner pressure of the shell, P1 is the instantaneous static inner pressure of the shell, and P2 is the inertial pressure that relates to the expansion inertia of the shell. The hydrodynamic pressure of the boiled materials (P) contains two parts: one from the lattice and the other from the electrons. In the case of the laserinduced damage of the films, the density of the boiled zone is initially very large, so the state function of real gas should be used. The relation usually used is the Van der Waals equation:
冉
P⫹
B1 V2
冊
共V ⫺ B2兲 ⫽ RT,
(9)
where B1 ⫽ 27R2Tc2兾64Pc and B2 ⫽ RTc兾8Pc. V and T are the volume and the temperature of the boiled materials, respectively. B1 and B2 are the Van der Waals constants of the boiled materials. R is the gas constant [not the radius of the melted or boiled zone in Eq. (7)]. Pc, Vc, and Tc are the critical pressure, volume, and temperature of the boiled materials, respectively. Because of the partial ionization of the boiled zone, hydrodynamic pressure sourced from thermal electrons should also be included. It has the form nekBT, in which ne is the number density of the thermal electrons and kB is the Boltzmann constant. As for P1, which is the instantaneous static inner pressure of the shell, under the driving action of inner pressure P, the inner radius of the shell changes
starting from Rm and the deformation pressure has the form23
共D3 ⫺ Rm3 兲共R⬘m ⫺ Rm兲
P1 ⫽
3 m
R
冉
Rm D3 ⫹ 2 3 ⫹ 2 4 · Rm
冊
.
(10)
R⬘m is the radius of the boiled zone, which changes due to high hydrodynamic pressure. D is the deepness from the center of the inclusion to the surface of the films. and are Lamé elasticity constants. The dynamic inertial pressure has the form P2 ⫽
共D3 ⫺ R3兲 3r2
r¨.
is the ratio between the partial mass of the shell, which is sprayed out, and the mass of the entire spherical shell. is the density of the boiled zone before laser irradiation. r¨ is the radial expansion acceleration of the spherical shell. r is the radial coordinate of the front of the boiled zone. From the instantaneous static inner pressure, we can obtain the deformation stress distribution according to the following formation23: ⫽
3 Rm P1 3 D3 ⫺ Rm
冉
1⫹
D3 2r3
冊
.
(11)
When the gas phase change occurs, the circumferential stress in the films is the incorporative stress of the thermal stress and the deformation stress. When the thermal stress dominates, the maximal tensile stress appears in the cold zone. When the deformation stress dominates, the maximal stress appears in the zone near the boiled zone. If the relation described in formulation (2) is satisfied, the films will be damaged. The damage morphology is a crater with a smooth inner surface. Around it, few accumulations of the sprayed materials can be found. This is because the sprayed speed of the boiled materials is very high, 10 November 2006 兾 Vol. 45, No. 32 兾 APPLIED OPTICS
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Fig. 4. Circumferential stress in films that have inclusions with different radii included. The laser power density is 1 GW兾cm2.
and small parts of them can return and deposit around the crater. See Fig. 11 below. However, if the relation described in formulation (2) cannot be satisfied, bubbles will also form. In this stage, the expansion process of the boiled zone is taken as an adiabatic process. If the adiabatic index is ␥, the relationship between temperature and volume has the form T共V ⫺ B2兲␥⫺1 ⫽ const.
(12)
We use the leap-frog method to solve Eq. (8), which has the discrete formation rin⫹1 ⫽ rin ⫹ hin ⫹ h2Fin兾共2m兲,
Fi ⫽ 共P ⫺ P1兲 ⫻ 4r 兾共m兲, 2
the inclusion is 50 nm, compressive stress exists. If a crack forms near the boiled zone, this compressive stress can prevent the crack from propagating to the shallow surface of the film. It also can be seen that the maximal tensile stress initialized by a small inclusion is less than that initialized by a large inclusion, so the small inclusion is safer. It also can be seen that the circumferential stress decreases rapidly when radial distance increases. If the inclusion is embedded very deeply, the crack formed around it is difficult to propagate to the shallow surface because of the relatively smaller circumferential stress in there, so the deeply embedded inclusions are also safer. B. Stress Caused by Laser Pulses with Different Power Densities
in⫹1 ⫽ in ⫹ h共Fin⫹1 ⫹ Fin兲兾共2m兲, n
Fig. 5. Sketch of the thermal stress distribution and the dynamic stress distribution.
(13)
The circumferential stress distribution in films, when a laser with different power densities irradiates on films, has been plotted in Fig. 6. The radius of the
in which n is the time step, i is the space step, h is the interval in time, m is the mass of the shell, F is the inertial force, and is the expansion velocity of the front of the boiled zone. 3. Results and Discussion
Based on the theory described in Section 2, we have studied the distribution and evolution of circumferential stress in films. A. Stress Initialized from Inclusions with a Different Radius
The circumferential stress distributions in films that have inclusions with different radii included have been plotted in Fig. 4. The laser power density is 1 GW兾cm2. It can be seen that near the boiled zone, deformation stress dominates. When the radial distance (radial distance defined as the distance from the studied spot to the center of the inclusion) increases, distributions of thermal stress and deformation stress are plotted in Fig. 5. When the radius of 8258
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Fig. 6. Circumferential stress in a film when it is irradiated by different laser power densities. The radius of the inclusion is 200 nm.
Fig. 7. Evolution of the circumferential stress distribution in films. The laser power density is 1 GW兾cm2; the radius of the inclusion is 200 nm.
inclusion is 200 nm. It can be seen that when the power density of the laser pulse is 0.5 GW兾cm2, no deformation stress exists and the maximal tensile stress is very small. In these cases, inclusions cannot initialize damage under this power density unless the tensile strength of the host is low enough. When the power density increases, the maximal tensile stress increases, and the contribution of the deformation stress increases. In addition the radius of the boiled zone increases. A larger boiled zone will cause a larger damage pit. C. Stress Development during Laser Irradiation
The evolution of circumferential stress distribution in films has been plotted in Fig. 7. The laser power density is 1 GW兾cm2, and the radius of the inclusion is 200 nm. In the first 9 ns, the circumferential stress is dominated by thermal stress. The inclusion is melted after 3 ns and boiled after 7 ns. Before it is boiled, compressive stress is the main formation. We
Fig. 8. Evolution of the radial expansion velocity and acceleration of the liquid–solid interface. The radius of the inclusion is 200 nm; the power density of the laser pulse is 1 GW兾cm2.
Fig. 9. Radial expansion velocity of the liquid–solid interface.
can see that under this laser power density, for an inclusion with a radius less than 200 nm, damage will not occur at the stage of the liquid phase change because the maximal tensile stress is very small. It takes only approximately 2 ns for the deformation stress to obtain domination. From Fig. 8, we can see that the dynamic process is intensive. It takes only approximately 0.2 ns for the expansion velocity of the solid–liquid interface to accelerate to a relatively stable value. During the expansion stage, the hydrodynamic pressure will decrease and the deformationresistant pressure will increase, so the acceleration will return to zero. After that, the inclusion continues absorbing energy to preserve expanding and the acceleration fluctuates around zero. When the tensile stress reaches the tensile strength, damage occurs. From Fig. 9, we can see that the velocity and the acceleration will be larger if the radius of the inclusions or the power density of the laser pulse increases. This is because the temperatures are high, and correspondingly, much energy density is con-
Fig. 10. Strain of the melted zone initialized by the inclusions. 10 November 2006 兾 Vol. 45, No. 32 兾 APPLIED OPTICS
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Fig. 11. Size of the damage pit in films initialized by the inclusion. The power density of the laser pulse is 1 GW兾cm2; the host material is HfO2.
served in the boiled zone. From Figs. 6 and 7, it can be seen that we can study the evolution of stress distribution by analyzing the final stress distributions, that are caused by the laser pulse with different power densities. D. Strain Caused by a Laser Pulse
We define strain as a ratio that is the radials expansive quantity of the melted zone divided by its original radius. If the film is damaged by laser irradiation, this strain relates to the damage crater in size. The strains initialized from the inclusions with different radii under irradiation by a laser pulse with different power densities are plotted in Fig. 10. It can be seen that the strain will decrease if the laser power density decreases or the inclusion radius increases. It is easy to comprehend that the strain increases when the laser power density increases. If we have acknowledged the fact that the thermal linear expansion coefficient of the inclusion is larger than that of the host, compared with the melted zone initialized from the inclusion with a large radius, it has a larger proportion of inclusion materials when it is initialized from the inclusion with a small radius in the melted zone. So it is also not difficult to comprehend the trend that the strain decreases when the radius of the inclusion increases. E.
From the size of the damage pit shown in Fig. 11, we simplified the pit to be in the shape of part of a taper. When the pit is formed, a new surface is also formed. Using the energy conservation law, we can evaluate the maximal area of this new surface. The following formula can be used:
(14)
where is the average density of the melted zone; Mi is the average atomic mass of the melted zone; ␥ is the surface energy of the host; S ⫽ 共Ri ⫹ Rd兲 ⫻ 8260
age pit, where Ri is the radius of the melted zone when the crack appears, Rd is the maximal radius of damage pit, and D is the embedded depth of the inclusions. Rd in Fig. 11 is ⬃130 m. It is difficult to obtain a precise value for the surface energy, and we assume it to be the usual value for films, which is ⬃1 J兾cm2. Using Eq. (14), we can evaluate that Ri is ⬃180 nm. From the strain plotted in Fig. 10, we can deduce that the initial radius of the inclusion is ⬃75 nm. From Fig. 11, we can see that the inclusion with an initial radius of 75 nm is reasonable. If we do not consider the phase change of materials, the initial radius of the inclusion should be larger than 400 nm. This is because, under a laser power density of 1 GW兾cm2, when only thermal stress is considered, the maximal tensile stress initialized from the inclusions with radii of 400 nm is ⬃300 MPa, which is less than the tensile strength (we do not know the exact tensile strength of HfO2, but from Ref. 24 we can infer that it is ⬃750 MPa. The residual stress of the samples used in this paper is ⬃300 MPa, so the nonresidual stress, which is larger than 450 MPa can induce film damage). The inclusion with an initial radius of 400 nm is not reasonable for this case. 4. Conclusion
Application for Interpreting a Damage Pit in Size
4 k T ⫽ S␥, Ri3 3 Mi B
冑共Rd ⫺ Ri兲2 ⫹ D2 is the inner surface area of the dam-
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From the theoretical results, we can see that the inclusions with larger radii are more likely to induce damage. If the laser power density is high enough, the gas phase change will occur. In these cases, the inclusions with larger radii can cause larger deformation and higher expansion velocity. If the tensile strength of the host is very high, the film will not be damaged until the temperature of the inclusion in the illuminating region is beyond the boiling point. If the inclusion is very small, it can initialize damage to the films when the laser power density is very high and the dominating circumferential stress is deformation stress, whereas large inclusions initialize damage mainly by thermal stress. During the damage process, the circumferential stress is thermal
stress initially; deformation stress will dominate after the materials are boiled. A model that considered the phase change of the inclusions has been put forward. This model is valid for the case of inclusion-initialized damage. Moreover, this model is valid only for bulk inclusion, which is included in the films, so the inclusion radius should be less than the films’ thickness. It has three parts, which correspond to three stages: no phase change, liquid phase change, and gas phase change. Not every phase will be sure to happen, and the damage process in each phase has its corresponding damage morphology. From this model, we can evaluate the size of the damage pit. It is a pity that we cannot demonstrate direct experimental evidence for the onset temperature well above the melting point. But in Ref. 25, from the morphology of the bubble in the film surface, which is caused by a laser pulse with a power density well below the threshold, we have deduced that the inclusion has been melted. The authors thank Hong Ji Qi for some useful discussions and Chaoyang Wei for amending our writing style. References 1. B. C. Stuart, M. D. Feit, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Laser-induced damage in dielectrics with nanosecond to subpicosecond pulses,” Phys. Rev. Lett. 74, 2248 –2251 (1995). 2. D. Du, X. Liu, G. Korn, J. Squier, and G. Mourou, “Laserinduced breakdown by impact ionization in SiO2 with pulse widths from 7 ns to 150 fs,” Appl. Phys. Lett. 64, 3071–3073 (1994). 3. R. W. Hopper and D. R. Uhlman, “Mechanism of inclusion damage in laser glass,” J. Appl. Phys. 41, 4023– 4037 (1970). 4. M. Lenzner, “Femtosecond laser-induced damage of dielectrics,” Int. J. Mod. Phys. B 13, 1559 –1578 (1999). 5. M. F. Koldunov, A. A. Manenkov, and L. L. Pokotilo, “Mechanical damage in transparent solids caused by laser pulse of different durations,” Quantum Electron. 32, 335–340 (2002). 6. M. F. Koldunov, A. A. Manenkov, and I. L. Pokotilo, “Efficiency of various mechanisms of the laser damage in transparent solids,” Quantum Electron. 32, 623– 628 (2002). 7. M. Lezius, S. Dobosz, D. Normand, and M. Schmidt, “Explosion dynamics of rare gas clusters in strong laser fields,” Phys. Rev. Lett. 80, 261–264 (1998). 8. T. Ditmire, J. W. G. Tisch, E. Springate, M. B. Mason, N. Hay, J. P. Marangos, and M. H. R. Hutchinson, “High energy ion explosion of atomic clusters: transition from molecular to plasma behavior,” Phys. Rev. Lett. 78, 2732–2735 (1997). 9. P. Mora, “Plasma expansion into a vacuum,” Phys. Rev. Lett. 90, 185002 (2003).
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10 November 2006 兾 Vol. 45, No. 32 兾 APPLIED OPTICS
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