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The effect of asymmetry caused by oblique line-shaped laser ablation on the generation of ultrasonic waves in metal, especially the effect of transverse component of the ablation force source on the ultrasonic waves is analyzed. Due to the oblique force source, the displacements of shear wave increase obviously by the enhanced shear force, the energy concentration area of longitudinal wave deflects to the small range centered on the incident direction while that of shear wave is approximately perpendicular to incident direction. In addition, surface wave enhances in the direction of transverse power flow. Furthermore, some ultrasonic characteristics under vortex laser ablation condition are inferred.
© 2014 Optical Society of America
1. Introduction
The well characterized ultrasonic signals can be used for monitoring the process and tracking the change of vaporization front position etc in laser-material interaction area. And the ultrasonic beam could be regulated and controlled after knowing the generation and evolution characteristics of ultrasonic waves. The explicit regulation of ultrasonic waves can provide credible basics to optimize ultrasonic signal generation of the experiment in material evaluation and laser processing. The obvious ultrasonic signal, which is easy to be detected, is important to the study of laser ultrasonic. Under thermoelastic mechanism, the excitation efficiency and displacement would be improved by means of ring-shaped laser irradiation [1,2] and coating transparent film [3] etc. While under ablation condition, laser ultrasonic signals can be significantly enhanced, and this leads to a direct increase in signal-to-noise ratio of laser-based systems for the remote generation and detection of ultrasonic. Being one of the most promising methods for material processing [4,5], the laser ablation techniques have been studied widely. The in-depth study of mathematical model and dynamics of laser ablation has developed prosperously [6–10]. Besides the study of ablation models, the characteristics of ultrasonic wave in ablative regime have also been analyzed extensively. Murray et al [11] studied the effect of laser ablation on ultrasonic signals in aluminum in 1999. Then Bao et al [12] investigated the time and frequency domains of ultrasonic caused by laser ablation. Reese et al [13] analyzed the epicentral waveform of laser ultrasound in melting regime. In addition, the surface wave induced by laser ablation also been studied by Shen et al [14]. The core vaporization model to be presented has been widespread used in the study of various laser ablation phenomena, including the laser-induced plasma and plume [15,16], and the surface microstructure molding [17] etc.
There are two aspects being the main research issues of laser ablation kinetics study, one is gasification and ablation of the material induced by laser irradiation, another is the reactive pressure in material produced by vaporization and plasma splashing expansion. As the laser intensity increases to a certain level, the vaporization rate is large. The vaporization induces a high pressure with short duration in the solid and a thin gas layer adjacent to the surface known as a Knudsen layer. The physical origin of the normal force is assumed to be the momentum transfer from the evaporating species. In this study, it is assumed that vaporization takes place in vacuum, hence the gas dynamic process which is outside of the Knudsen layer is not considered. The time of pressure pulse applied to the material is short due to the narrow duration of high power laser pulse. Therefore, the macroscopic mechanical effects can be characterized with impulse which produce force source in material. The study of laser ablation pressure and its impulse coupling with material have many important applications, such as inertial confinement fusion, laser-induced shock wave in material and laser-driven high speed slapper. Many modeling efforts have relied on replacing the laser source with an equivalent set of stress boundary conditions [18,19], which have proven to be extremely useful in describing many features of laser generated ultrasonic waveforms [20,21].
Ultrasonic signals based laser ablation can be enhanced relative to thermoelastic mechanism, whereas under normal incident condition, longitudinal wave is obvious while shear wave is relatively weak. For some practical application, not only the wave amplitudes should be highly enhanced but also strong signals of both the longitudinal and shear waves be detected simultaneously at the same point. Under thermoelastic condition, the longitudinal wave and shear wave can be both improved for oblique incidence of laser source in semi-transparent materials [22], it is shown that the loss of symmetry of the acoustic source cause asymmetrical behavior of acoustic waves. The asymmetrical ultrasonic waves induced by oblique incident laser are meaningful for comprehensive understanding of the ultrasonic characteristics, and then, it help to control beam by regulating conditions. Under ablation condition, the direction of force source produced by vaporization can be regulated by changing the laser incident angle. The force source can be decomposed to normal and transverse component to the surface, which the effect of former is same with the normal incident condition while the latter may have some extra impact on the generation of ultrasonic waves.
To analyze the effect of oblique force source on the asymmetry of ultrasonic wave in metal caused by oblique laser in the ablative regime, simulation models of ultrasonic wave generation by line-shaped laser source are presented in this paper. The pressure exerted on the surface, which employ equivalent stress boundary conditions, is calculated through a finite element method. In addition, taking into account the material properties dependence of temperature, the transient temperature fields and ultrasonic displacement fields are obtained. The influences of oblique force source on directivity patterns of bulk waves and energy distribution of ultrasonic waves are analyzed in detail.
2. Vaporization model
Under laser ablation condition, when ablation products eject quickly and scatter and do not affect the subsequent laser incident, that process is described as full ablation. The pulse duration is longer than the electro-acoustic relaxation time in nanosecond laser ablation, so the corresponding process is thermal equilibrium ablation which can be depicted by Clausius-Clayperon equation. The physical mechanism of laser induced gasification can be depicted by phase-change equation, equilibrium gasification model and Knudsen layer model. For the present analysis, in order to calculate the pressure applied to the sample surface, it is only necessary to know the pressure jump across the Knudsen layer under vacuum conditions.
The propagation of ultrasonic waves in material is related to the mechanical effect produced by laser ablation. The influence of Knudsen layer on ablated regime, which contributes to equivalent mechanical effect as impulse, is considered in simulations. The size of ablated pool is 1/80 of the sample, so the impulse can be regarded as acting on the material surface. The impulse pressure exerted on the surface is assumed to act as a force source with an amplitude and time dependence which determined by the vaporization model. Phipps et al [23] concluded the calibration relationship between ablation pressure and incident laser intensity from a large number of experiment results, which can be described as
where P is ablation pressure, n = – 0.3 ± 0.03,b0 = 5.6 ~6.5, the specific value depend on the material, λ and τ are laser wavelength and pulse width respectively, I0 is the peak value of incident laser intensity. After determining the peak value of pressure on the basis of calibration relationship, the spatial distribution and temporal distribution are given, it is equivalent to define the boundary conditions which is different with thermoelastic mechanism. Thereby we can solve the wave equation and draw the corresponding ultrasonic displacement field distribution.The impulse coupling coefficient Cm is defined as
where P0 is ambient gas pressure. The above calibration law was verified as a superior one after collecting experimental data of various laser wavelengths and duration which act on all kinds of aluminum and CH compounds [24]. According to ablation experiments of aluminum in [24], the impulse coupling coefficientis adopted in our numerical simulations.In the process of laser ablation, the material physical parameters change with temperature in the ablation regime among gas, liquid and solid states. The equivalent heat capacity which contains the latent heat of phase transition is adopted due to a large amount of latent heat released in ablation process. During the phase transition process, the total released heat per unit mass of metal is manifested by the change of enthalpy. Therefore, equivalent heat capacity can be substituted as, is enthalpy,
where Tm is phase transition temperature(melting point and gasification point), define the half width of the curve which represent duration of the transition temperature. The changed amount of thermal capacity isThe change of heat capacity exists during phase transition process, so we apply smooth function to make the calculation convergence.The geometry model of laser ablation is schematically shown in Fig. 1(a). Two-dimensional numerical model is built in rectangular coordinate system. In this study, energy expenditure is dominated by the latent heat of vaporization while the enthalpy of the vapor and the kinetic energy of the vapor are quite small that can be neglected. Thus the distribution of transient temperature field in bulk aluminum is determined by thermal conduction equation, which can be described as [25]
where T(x, y, t) is the spatial and temporal distribution of temperature; ρ and k are the density and thermal conductive coefficient respectively; L is latent heat of phase transition. The second term of right side of Eq. (6) is the heat source provided by line-shaped laser, where A is the optical absorptivity of the specimen surface, I0 is the peak intensity of the incident laser, β is the optical absorption coefficient, f(x) and g(t) are the spatial and temporal distributions of the pulsed laser, these two functions arewhere x0 is radius of the laser spot, t0 and τ are the arrival time of peak value and half width of the pulsed laser.
Fig. 1 Schematic diagram of system induced by line-shaped laser under different conditions: (a) normal incident condition; (b) oblique incident condition.
The final displacement is calculated as the superposition of displacement induced by temperature gradient and ablative force source. The former induced by the transient temperature field in themoelastic mechanism satisfies the Navier-Stokes equation [2], and the latter is decided by the boundary condition. The spatial distribution and temporal distribution of force induced by vaporization, is obtained from thermal conduction equation. The system has no initial heat source, stress and mechanical displacement, so the initial temperature is taken as 300K, the initial stress and displacement both are zero, respectively.
2.2. Numerical model parameters
The spatial mode of the line-shaped laser is assumed to be a Gaussian distribution. The peak power intensity of laser is 8e9W/cm2 which under full ablation condition. The laser duration is 1ns. The wavelength and radius of the pulse laser are taken to be 1064nm and 2.5μm. Corresponding to laser wavelength, optical absorption coefficient of aluminum is 0.06. The thermal conductivity coefficient, density, heat capacity etc of aluminum used in the calculation change with temperature among gas, liquid and solid state in the ablation regime. The half of the transition temperature is defined as 15K. Latent heat of melting is 397KJ/Kg and of vaporization is 10900KJ/Kg. The length and height of material are 80μm and 40μm, respectively. Mesh generation based on Fig. 1 is that the element size of grid is 0.1μm in the vicinity of the laser affected area for accuracy while the element size is 0.6μm outside the heat-affected zone.
Normal incident laser condition and oblique incident condition which the oblique angle θ = 45° like Fig. 1(b) (The direction of transverse component of oblique force source is positive) are simulated respectively. Due to the same densities of laser energy, the impulse acted on the sample, which is force source that stimulate ultrasonic waves, are equal. Therefore, we can compare the effect of oblique force source on ultrasonic waves with normal condition conveniently.
3. Numerical results and discussion
The temperature in the laser incident point in ablative regime is shown in Fig. 2. There are two temperature platforms in 933K and 2740K, which represent the absorption of latent heat of melting and vaporization, respectively. When the temperature is above the gasification point, the vaporization occurs. Therefore, the vaporization pressure action time can be obtained from the gasification duration.
Figure 3 shows the dependences of displacement of the bulk waves at epicenter on time stimulated by normal force source and oblique force source, respectively. In Fig. 3, we can clearly see that the shear wave (S-wave) increases significantly while longitudinal wave (L-wave) decreases under oblique force source condition. We speculate it is the transverse component of force source that affects the ultrasonic characteristics.
Amplitudes of the bulk wave at different positions on the rear surface along transverse direction under different conditions are shown in Fig. 4. The displacements on the rear present asymmetric distortion under oblique force source condition. Under normal force source condition, the displacement is relatively stronger in center for longitudinal wave while that in the place deflect center for shear wave, that is to say, the longitudinal wave and shear wave have different directivity distribution. Under oblique incident condition, both the longitudinal wave and shear wave are asymmetric and shear wave enhances greatly on the rear surface relative to the normal incident condition. The displacements of shear wave increase obviously owing to enhanced shear force produced by the oblique force source. The total energy is a certain value, so the energy of longitudinal wave reduces and the displacements decrease. In briefly, the oblique force source leads the displacement loss symmetry and change the energy distribution.
Fig. 4 Amplitudes of (a) longitudinal wave and (b) shear wave at different positions on the rear surface of specimen along the transverse direction.
The displacements in Fig. 5 have been normalized and the data above certain degree are abandoned for avoiding the interference of surface wave to bulk wave displacement. Figures 5(a) and 5(b) represent that longitudinal wave and shear wave amplitudes are symmetric with respect to θ = 0° in the normal force source condition. The energy of longitudinal wave centralize to the normal direction while that of shear wave exists in the two symmetrical wide lobes centered at 35°~50° to the incident direction. The result in Figs. 5(c) and 5(d) show that the maximum displacement of longitudinal wave deflects 45° to the normal direction, whereas shear wave energy concentrate to −25°~-45°. Longitudinal wave is strongest in the force source direction, thus the energy concentrate area deflects to the direction which corresponding to oblique degree of incident laser. Whereas, there is large change of the displacement distribution of shear wave mainly owing to the transverse component of oblique force source. In Fig. 5(d), the displacement for θ<0° is relative larger than θ>0°. For a thermoelastic source in an elastically isotropic sample, shear waves are generated by mode conversion at the free surface of longitudinal waves emanating from subsurface sources. For ablation regime, there is a small melting pool located in the laser irradiation area. In this pool, longitudinal waves emanating from the subsurface cannot convert mode because the molten material has no shear rigidity, and therefore, cannot sustain shear waves [19–21]. In this area, shear waves emanate only from an adjacency thermoelastic source surrounding the melt pool. Under oblique incident condition, the melting pool is larger in the positive direction than negative direction, it make the shear wave stronger in the negative position due to the above theory. As expected, amplitudes of shear waves generated by mode conversion upon oblique incidence are larger for θ<0° than that for θ>0° region as shown in Fig. 5(d). Combining the analysis of Figs. 4(b) and 5(d), we know the energy of shear wave is enhanced great in the most area while the displacement in the side within the force source component is relative weaker that the other side. Therefore, the oblique force source could help enhancing and controlling shear wave generation. In summary, the oblique force source not only makes the energy between longitudinal wave and shear wave transferred, but also makes the directivity patterns of longitudinal wave and shear wave changed.
Fig. 5 The directivity patterns of bulk wave under different conditions (normalized): (a) the longitudinal wave induced by normal force source; (b) the shear wave induced by normal force source; (c) the longitudinal wave induced by oblique force source and (d) the shear wave induced by oblique force source.
Amplitudes of surface wave under normal and oblique force source condition are shown in Fig. 6. It demonstrates that the displacements of surface wave lost symmetry due to the oblique force source, and displacements of positive direction enhance while negative direction reduce. It is the transverse component of force source makes the surface wave enhanced in the force direction.
The full field numerical solutions at different conditions are presented in Fig. 7, which makes it intuitive to understand the evolution characteristics of ultrasonic waves within the specimen. We can clearly observe the wavefronts of both the longitudinal and shear waves propagating away from the heated area in the form of cylindrical wave. Compare with Fig. 7(a), the shear wave in Fig. 7(b) enhances greatly, and which is more evident than longitudinal wave in most area. The longitudinal wave is distinct in the area which concentrates to the incident direction while the energy concentration area of shear wave is approximately perpendicular to incident direction, in addition, the surface wave is obvious in the positive direction, and these views consist with Figs. 4 and 6. Based on the analysis of Figs. 5 and 6, we know that the transverse component of force source makes ultrasonic waves changed significantly and we conclude that the oblique force source not only changes the energy distribution of bulk waves and enhances the shear wave, but also affects energy distribution of surface wave.
Fig. 7 Ultrasonic displacement fields at 7ns under different conditions: (a) line-shaped laser normal incident condition; (b) line-shaped laser oblique incident.
4. Further discussion
Optical vortex has been intensively investigated and attracted much attention due to the orbital angular momentum (OAM) carried by spiral phase [26–28]. Vortex laser ablation gives possibilities of annular ablation processing fabricating structures and provides another key to applications of laser plasma employing vortex laser. The experiment shows that the rotational motion of laser-induced plasma forms smoother surfaces in the ablated zone due to the OAM [29]. The further laser ablation experiment demonstrates that vortex light can twist metal to control the chirality of metal nanostructures, which illustrates the mechanical characteristics of OAM [16]. Since OAM could change direction and speed of the ablated particles, the OAM may have some effect on the generation and propagation of ultrasonic waves in the material.
For a plane wave, the normals of wavefront parallels to the propagation axis and the linear momentum are coaxial with the direction of propagation, so it is no component in transverse plane. Whereas, for optical vortex, the normals of wavefront are no longer parallel to the propagation axis, therefore, the linear momentum not only exists in the direction of propagation axis, but also exists in the transverse plane which is perpendicular to propagation direction. Owing to the spiral phase of optical vortex, the transverse linear momentum density of points in optical vortex field can be taken advantaged to characterize the characteristics of OAM.
Figure 8(a) shows that the intensity distribution of optical vortex assembles to the nonvortex ring-shaped beam. The larger topological charge of optical vortex is, the narrower ring width of intensity distribution area is. From the three-dimension graph of spiral phase of optical vortex in Fig. 8(b), we know the linear momentum p, which is dependence with electric field intensity E and magnetic flux density B, can be decomposed as pz along the propagation axis and pϕ in the transverse plane. The oblique angle θ (the angle of wavefront direction and propagation axis) of the spiral phase satisfied
where m is the topological charge of optical vortex, r is the distance of a point in the wavefront to propagation axis, λ is the laser wavelength.Figure 9(a) describes the three-dimensional shear wave field induced by vortex laser ablation in bulk aluminum. Due to the small radius of the laser, irradiation area of vortex laser can be regarded as point source stimulation in the far field, and the induced ultrasonic waves propagate in the form of spherical wave. Based on the above analysis about directivity pattern of shear wave, we know the energy of shear wave induced by line-shaped laser exists in the two symmetrical wide lobes centered at 35°~50° in the cross-section as Fig. 7(b) shown. Therefore, in the three-dimension condition, the maximum value of displacements of shear wave could form a hollow cone in the specimen, and the interface of maximum value and α plane is a circle as colored area shown in Fig. 9(a). In the laser irradiation area, the infinitesimal element dS of optical vortex resemble to the cross-section of line-shaped laser in the transverse planes which perpendicular to dS, despite the limited length of dS of vortex laser leads the induced ultrasonic wave beam broadening in certain area during the process of propagation. Both of them have the oblique wavefront and component in the transverse plane, thus, the effect of transverse linear momentum of former is similar to the transverse force source of latter. Therefore, in the far-field, some characteristics of generation and propagation of ultrasonic wave stimulated by vortex laser could be speculated. The direction of transverse component is decided by the direction of the helicity of optical vortex, as right-hand vortex in Fig. 9, the direction of transverse component is perpendicular to dS which rotates counter-clockwise in β plane. Based on the analysis of the effect of oblique line-shaped ablation on ultrasonic waves, we can infer that the transverse linear momentum of infinitesimal element make the shear wave strengthen in the A1A2 and B1B2 due to the enhanced shear force, while the shear wave in B1B2 is enhanced greater than A1A2 which resembles to oblique line-shaped laser condition. The area A1A2 is called the major enhanced area and B1B2 minor enhanced area.
Fig. 9 The schematic diagram of shear wave field induced by vortex laser: (a) three-dimensional diagram; (b) two- dimensional diagram from the view of propagation direction.
Figure 9(b) is the two-dimensional diagram of shear wave field induced by vortex laser ablation from the view of propagation direction. By rotating the infinitesimal element dS of a small angle along the ring, the corresponding major enhanced area C1C2 and minor enhanced area D1D2 can be superimposed with A1A2 and B1B2 respectively, which makes shear wave in the area A2C1 and D1B2 stronger. In the far field, the ultrasonic produced by infinitesimal elements dSs of optical vortex makes the characteristics of shear waves more obvious. As surface wave can be enhanced in the direction of shear force for line-shaped laser, it could also be enhanced for the vortex laser in the direction of the transverse linear momentum component. In addition, according to the analysis about oblique line-shaped laser, the energy distribution of bulk waves induced by vortex laser may have been changed. Owing to the transverse component, the energy concentration area of longitudinal wave deflects off propagation axis while the specific direction dependence with the oblique angle of the spiral phase. Meanwhile, the direction of energy concentration area of shear wave is approximately perpendicular to that of longitudinal wave.
5. Conclusions
This paper demonstrates the asymmetry effects of ultrasonic wave stimulated by oblique incidence of line-shaped laser ablation in metal, and analyzes the effect of transverse component of the ablation force source on the generation of ultrasonic waves. The simulation results show that the oblique force source not only change the energy distribution of bulk waves and enhance the shear wave, but also affect energy distribution of surface wave. The displacements of shear wave increase obviously owing to enhanced shear force, the energy concentrate area of longitudinal wave deflects to the small range centered on the incident direction while that of shear wave is approximately perpendicular to the incident direction. In addition, surface wave is enhanced in the direction of transverse power flow. It promotes the feasibility of the selection of specific bulk acoustic modes by tuning adequately the incidence angle of the source. Furthermore, based on the analysis of effect of oblique line-shaped laser ablation on ultrasonic waves, some characteristics of ultrasonic waves under vortex laser ablation condition are speculated. Knowing the explicit characteristic of ultrasonic waveform under various ways of laser irradiation is essential and it is an important part of the further study of beam control.
Acknowledgments
This work was supported by National Basic Research Program (973 Program) of China under Grant No. 2012CB921900.
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