Laser Pulse Phenomena and Applications Part 3 doc

The Effect of the Time Structure of Laser Pulse on Temperature Distribution and Thermal Stresses in Homogeneous Body with Coating 51 For greater pulse rise times r τ the dimensionless time c τ , connected with the change of stresses type from compressive to tensile one also increases (Fig. 4). This dependence is approximately described by the equation 2 0.8173 0.075 0.1457 crr τττ =++. In thermal processing of brittle materials, it is the sign change of superficial stresses what plays key role in controlled thermal splitting (Dostanko et al., 2002). The beginning of superficial cracks generation is accompanied with the monotonic increase of tensile lateral stresses and makes the controlled evolution of the crevices possible. By equating the relation (45) to 0 at cs τ ττ =>, 0 ζ = and taking into account the equations (42), (48), (49), one obtains: (0) (0) 00 2 () ( ) ( ) ccs ccs QQ τ ττ τ ττ π −−= −−, (70) where the function (0) 0 ()Q τ has the form (66). With the absolute inaccuracy, smaller than 3%, the solution of nonlinear equation (70) can be approximated by the function 32 1.133 1.172 0.604 0.052 csss τ τττ =− + + + (Fig. 5). 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 τ c τ s Fig. 5. Dimensionless time c τ of the sign change of lateral stress y σ ∗ on the irradiated surface 0 ζ = versus dimensionless duration s τ of the rectangular laser pulse (Yevtushenko et al., 2007). The forced cooling of the surface in the moment time c tt = would cause the jump of temperature (0, 0) (0, 0) cc TT t T tΔ= − − + in thin superficial layer. From the equations (45) and (46) it follows, that the dimensionless lateral deformation of the plate y ε ∗ is determined by the integral characteristic of temperature only. For this reason, the rapid cooling of the thin film, practically, does not change the surface deformation (0, 0) (0, 0) yc c c tTt ε α + =− but Laser Pulse Phenomena and Applications 52 at the same moment it produces the increase of the normal stresses (0, 0) yc c tT σ α +=Δ. Finally, the development of the superficial crack can be described as a series of the following phases: 1. due to local short heating a surface of the sample in it the field of normal lateral stresses is formed; 2. tensile lateral stresses occur near the subsurface cooled region and are proportional to the temperature jump observed before and after the cooling agent is applied; 3. when the stresses exceeds the tensile strength of the material, the surface undergoes tear; 4. development of the crack into the material is limited by the regions of lateral compressive stress, which occur beneath the cooled surface. As mentioned in the introduction, there is considerable interest (for scientific and practical reasons) in thermal processing of ceramic coatings from zirconium dioxide 2 ZrO . Authors presented the numerical examinations of thermal stresses distribution for the system consisting of 2 ZrO ceramic coating ( 2.0 W/(mK) c K = , 62 0.8 10 m /s c k − =⋅ ), deposited on the 40H steel substrate ( 41.9 W/(mK) s K = , 62 10.2 10 m /s s k − =⋅ ) (Fig. 6). The coefficient of thermal activity for this system is equal 5.866 ε = and the parameter λ found from the equation (22), has the value 0.708 λ = − . Thermal diffusivity of zirconium dioxide is small when compared with the value for steel. That difference is the cause of high temperatures on the processed surface and considerably higher than for the homogeneous half-space (one order of magnitude) lateral tensile stresses generated in the superficial layer when the heating is finished. So, the thermal processing of the coating from zirconium dioxide leads to the generation of superficial cracks, which divide the surface into smaller fragments. Of course the distribution of cracks at different depths depends on the heat flux intensity, the diameter of the laser beam, pulse duration and other parameters of the laser system. But when using dimensionless variables and parameters the results can be compared and the conclusion is that for the heating duration 0.15 s τ = , penetration depth of cracks for coating–substrate system ( 2 ZrO –40H steel) is, more than two times greater than for the homogeneous material (one can compare Figs. 3а and 6). The opposite, to the discussed above, combination of thermo-physical properties of the coating and the substrate is represented by the copper–granite system, often used in ornaments decorating interiors of the buildings like theatres and churches. For the copper coating 402 W/(mK) c K = , 62 125 10 m /s c k − =⋅ , while for the granite substrate 1.4 W/(mK) s K = , 62 0.505 10 m /s s k − =⋅ , what means that the substrate is practically thermal insulator and the coating has good thermal conductivity (see Figs. 7 and 9). The distribution of lateral thermal stresses for copper–granite system is presented in Fig. 8. In this situation, when the thickness of the coating increases, the temperature on the copper surface decreases. Therefore the effective depth of heat penetration into the coating is greater for the better conducting copper than for thermally insulating zirconium dioxide ( 2 ZrO ) (see Figs. 6 and 8). We note that near to the heated surface 0 ζ = lateral stresses y σ are compressive not only in the heating phase 0 0.15 τ < < but also during relaxation time, when the heat source is off. Considerable lateral tensile stresses occur during the cooling phase close to the interface of the substrate and the coating, 1 ζ = . This region of the tensile stresses on the copper-granite interface can destroy their contact and in effect the copper coating exfoliation can result. The Effect of the Time Structure of Laser Pulse on Temperature Distribution and Thermal Stresses in Homogeneous Body with Coating 53 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Fig. 6. Isolines of dimensionless lateral stress y σ ∗ for 2 ZrO ceramic coating and 40H steel substrate at rectangular laser pulse duration 0.15 s τ = (Yevtushenko et al., 2007). 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Fig. 7. Isotherms of dimensionless temperature T ∗ for 2 ZrO ceramic coating and 40H steel substrate at rectangular laser pulse duration 0.15 s τ = (Yevtushenko et al., 2007). Laser Pulse Phenomena and Applications 54 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Fig. 8. Isolines of dimensionless lateral stress y σ ∗ for copper coating and granite substrate at rectangular laser pulse duration 0.15 s τ = (Yevtushenko et al., 2007). 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Fig. 9. Isotherms of dimensionless temperature T ∗ for copper coating and granite substrate at rectangular laser pulse duration 0.15 s τ = (Yevtushenko et al., 2007). The Effect of the Time Structure of Laser Pulse on Temperature Distribution and Thermal Stresses in Homogeneous Body with Coating 55 6. Effective absorption coefficient during laser irradiation The effective absorption coefficient A in the formula (1) and (12) is defined as the ratio of laser irradiation energy absorbed on the metal’s surface and the energy of the incident beam (Rozniakowski, 2001). This dimensionless parameter applies to the absorption on the metal’s surface, on the very sample surface (so called “skin effect”). The absorption coefficient A can be found in book (Sala, 1986) or obtained on the basis of calorimetric measurements (Ujihara, 1972). The mixed method of effective absorption coefficient determination for some metals and alloys was presented by Yevtushenko et al., 2005. This method is based on the solution of axisymmetric boundary-value heat conduction problem for semi-space with circular shape line of division in the boundary conditions and on the metallographic measurements of dimensions of laser induced structural changes in metals. The calculations in this method are very complex because, in particular, the numerical calculation of the Hankel’s integrals has to be done. Therefore, we shall try to use with this purpose obtained above the analytical solution of the transient one-dimensional heat conduction problem for homogeneous semi-space in the form 0 (,) ( ,)Tzt ATT ζ τ ∗ ′ = , 0, 0zt≥≥, (71) where, taking the formula (12) into account, the coefficient 00 /TTA ′ = and the dimensionless temperature ( , )T ζ τ ∗ is defined by formulae (41) and (60). It should be noticed that the temperature on the irradiated surface has maximum value at the moment of laser switching off, for s tt = ( s τ τ = ), and in the superficial layers the maximum is reached for hs tt t t≡=+Δ(in dimensionless units, for hs τ ττ = +Δ , 2 Δkt/d τ Δ= , d is the radius of the irradiated zone). The parameter t Δ ( τ Δ ) is known as “the retardation time” (Rozniakowska & Yevtushenko, 2005). The time interval, when the temperature T reaches its maximum in the point h zz = beneath the heated surface, can be found from the condition: (,) 0 h Tz t t ∂ = ∂ , 0 s tt>>. (72) By taking into account the solutions (71), (41) and (60), the equation (72) can be rewritten as: 22 (,) 11 exp exp 0 44() () hh h s s T ζτ ζ ζ ττ ττ πτ π τ τ ∗ ⎛⎞ ⎡ ⎤ ∂ = −− − = ⎜⎟ ⎢⎥ ⎜⎟ ∂− − ⎢⎥ ⎝⎠ ⎣ ⎦ , 0 s τ τ >>, (73) where / hh zd ζ = . After substituting hs τ ττ τ ≡ =+Δ in equation (73), one gets 2 exp 4( ) hs ss ζτ τ τ ττττ ⎡ ⎤ Δ =− ⎢ ⎥ +Δ Δ +Δ ⎢ ⎥ ⎣ ⎦ . (74) From the equation (74) for the known dimensionless hardened layer depth h ζ and the pulse duration s τ , the dimensionless retardation time τ Δ can be found. On the other hand, at known τ Δ from equation (74) we find the dimensionless hardened layer depth h ζ of maximum temperature can be found: Laser Pulse Phenomena and Applications 56 0 0.02 0.04 0.06 0.08 0.1 0 0.0002 0.0004 0.0006 0.0008 0.001 Δτ ζ h St. 45: τ s =0.0732 (a) 0 0.1 0.2 0.3 0 0.002 0.004 0.006 0.008 0.01 Δτ ζ h cobalt: τ s =0.672 (b) Fig. 10. Dimensionless hardened layer depth h ζ from the heated surface versus dimensionless retardation time τ Δ , for the dimensionless laser pulse duration: а) 0.0732 s τ = (St.45 steel sample); b) 0.672 s τ = (Co monocrystal sample) (Yevtushenko et al., 2005). The Effect of the Time Structure of Laser Pulse on Temperature Distribution and Thermal Stresses in Homogeneous Body with Coating 57 parameter metal laser type d mm s t ms 9 0 10q − × 2 W/m K 11 Wm K − − 5 10k × m 2 s 1 − h T K h z μm St. 45 Nd:YAG 0.64 2 0.58 33.5 1.5 1123 40 Co QUANTUM15 0.35 4.5 4.62 70.9 1.83 693 100 Table 1. Input data needed for the calculations of the effective absorption coefficients for the St.45 steel and Co monocrystal samples. 21 ln1 s h s τ τ ζτ τ τ ⎛⎞ Δ ⎛⎞ =Δ + + ⎜⎟ ⎜⎟ ⎜⎟ Δ ⎝⎠ ⎝⎠ . (75) The curves which represent the dependency of h ζ on the parameter τ Δ , for fixed values 0.0732 s τ = and 0.672 s τ = are shown in Figs. 10a and 10b, respectively. The dimensionless retardation time quickly increases with the distance increase from the heated surface. The dimensionless pulse durations s τ were calculated from equation (12) with the use of material constants characteristic for St. 45 steel and Co monocrystals (Table 1), which were presented by Yevtushenko et al., 2005. Assuming the temperature h T of the structural phase transition, characteristic for the material, is achieved to a depth h z from the heated surface at the moment h t . It should be noticed that for steel the region of structural phase transitions is just the hardened layer, while for cobalt – it is the region where, as a result of laser irradiation, no open domains of Kittel’s type are observed. It can be assumed that the thickness of these layers h z , is known – it can be found in the way described by Rozniakowski, 2001. Then, from the condition (,) hh h Tz t T = , (76) the following formula, which can be used for the determination of the effective absorption coefficient, is obtained: 1 0 (,) h hh T AT T ζτ − ∗ ⎡ ⎤ = ⎣ ⎦ ′ , (77) where dimensionless temperature T ∗ is expressed by equations (41) and (60), the dimensionless retardation time τ Δ can be found from the equation (75) and the constant 00 /TqdK ′ = . The input data needed for the calculations by formula (77) are included in Table 1. Experimental results obtained by Rozniakowski, 1991, 2001 as well as the solutions for the axisymmetric (Yevtushenko et al., 2005) and one-dimensional model are presented in Laser Pulse Phenomena and Applications 58 parameter metal 5 0 10T − ′ × , K 1 − h ζ , - s τ , - 3 10 τ Δ× - A exp. A axisym. A one-dimen. St. 45 0.111 0.0625 0.0732 0.36 0.3 ÷ 0.5 0.42 0.41 Co 0.228 0.286 0.672 9.8 0.1 0.112 0.045 Table 2. Values of the effective absorption coefficient for the St.45 steel and Co monocrystal samples. Values of the effective absorption coefficient for the St.45 steel sample irradiated with pulses of short duration 0.0732 s τ = , found on the basis of the solutions for axisymmetric (0.42A = ) and one-dimensional ( 0.41A = ) transient heat conduction problem are nearly the same, and correspond to the middle of the experimentally obtained values range 0.3 0.5A =÷ (Table 2). The cobalt monocrystal samples were irradiated with pulses of much longer duration 0.672 s τ = . In this case, there is more than twofold difference of A values found on the basis of the solutions for axisymmetric ( 0.112A = ) and one-dimensional ( 0.045A = ) transient heat conduction problem. Moreover, only the value of effective absorption coefficient obtained from the axisymmetric solution of transient heat conduction problem corresponds to the experimental value 0.1A = . In that manner, it was proved that the solution of one-dimensional boundary heat conduction problem of parabolic type for the semi-space can be successfully applied in calculations of the effective absorption coefficient only for laser pulses of dimensionless short duration 1 s τ < < . Otherwise, the solution of axisymmetric heat conduction problem must be used. 7. Conclusions The analytical solution of transient boundary-value heat conduction problem of parabolic type was obtained for the non-homogeneous body consisting of bulk substrate and a thin coating of different material deposited on its surface. The heating of the outer surface of this coating was realised with laser pulses of the rectangular or triangular time structure. The dependence of temperature distribution in such body on the time parameters of the pulses was examined. It was proved that the most effective, from the point of view of the minimal energy losses in reaching the maximal temperature, is irradiation by pulses of the triangular form with flat forward and abrupt back front. Analysis of the evolution of stresses in the homogeneous plate proves that when it is heated, considerable lateral compressive stresses occur near the outer surface. The value of this stresses decreases when the heating is stopped and after some time the sign changes – what means that the tensile stresses takes place. The time when it happens increases monotonously with increase of a thermal pulse duration (for rectangular laser pulses) or with increase of rise time (for triangular laser pulses). When the lateral tensile stresses exceed the strength of the material then a crack on the surface can arise. The region of lateral compressive stresses, which occur beneath the surface, limits their development into the material. The Effect of the Time Structure of Laser Pulse on Temperature Distribution and Thermal Stresses in Homogeneous Body with Coating 59 The presence of the coating (for example, ZrO 2 ) with thermal conductivity lower than for the substrate results in considerably higher than for the homogeneous material, lateral tensile stresses in the subsurface after the termination of heating. The depth of thermal splitting is also increased in this case. When the material of the coating (for example, copper) has greater conductivity than the substrate (granite) then the stresses have compressive character all the time. The coating of this kind can protect from thermal splitting. The region that is vulnerable for damage in this case is close to the interface of the substrate and the coating, where considerable tensile stresses occur during the cooling phase. The method for calculation of the effective absorption coefficient during high-power laser irradiation based on the solution of one-dimensional boundary problem of heat conduction for semi-space, when heating is realised with short pulses, was proposed, too. 8. References Coutouly, J. F. et al. (1999). Laser diode processing for reducing core-loss of gain-oriented silicon steels, Lasers Eng., Vol. 8, pp. 145-157. Dostanko, A. P. et al. (2002). Technology and technique of precise laser modification of solid-state structures, Tiechnoprint, Minsk (in Russian). Duley, W. W. (1976). CO 2 Lasers: Effects and Applications, Acad. Press, New York. Gureev, D. М. (1983). Influence of laser pulse shape on hardened coating depth, Kvant Elektron (in Russian), Vol. 13, No. 8, pp. 1716-1718. Hector, L. G. & Hetnarski, R. B. (1996). Thermal stresses in materials due to laser heating, In: R. B. Hetnarski, Thermal Stresses IV, pp. 1-79, North–Holland. Kim, W. S. et al. (1997). Thermoelastic stresses in a bonded coating due to repetitively pulsed laser radiation, Acta Mech., Vol. 125, pp. 107-128. Li, J. et al. (1997). Decreasing the core loss of grain-oriented silicon steel by laser processing, J. Mater. Process. Tech., Vol. 69, pp. 180-185. Loze, M. K. & Wright, C. D. (1997). Temperature distributions in a semi-infinite and finite- thickness media as a result of absorption of laser light, Appl. Opt., Vol. 36, pp. 494- 507. Luikov, A. V. (1986). Analytical Heat Diffusion Theory, Academic Press, New York. Ready, J. F. (1971). Effects of high-power laser radiation, Academic Press, New York. Rozniakowska, М. & Yevtushenko, A. A. (2005). Influence of laser pulse shape both on temperature profile and hardened coating depth, Heat Mass Trans., Vol. 42, pp. 64- 70. Rozniakowski, K. (1991). Laser-excited magnetic change in cobalt monocrystal, J. Materials Science, Vol. 26, pp. 5811-5814. Rozniakowski, K. (2001). Application of laser radiation for examination and modification of building materials properties, (in Polish), BIGRAF, Warsaw. Rykalin, N. N. et al. (1985). Laser and electron-radiation processing of materials, (in Russian), Mashinostroenie, Moscow. Said-Galiyev, E. E. & Nikitin, L. N. (1993). Possibilities of Modifying the Surface of Polymeric Composites by Laser Irradiation, Mech. Comp. Mater., Vol. 29, pp. 259- 266. Sala, A. (1986). Radiant properties of materials, Elsevier, Amsterdam. Laser Pulse Phenomena and Applications 60 Sheng, P. & Chryssolouris, G. (1995). Theoretical Model of Laser Grooving for Composite Materials, J. Comp. Mater., Vol. 29, pp. 96-112. Timoshenko, S. P. & Goodier, J. N. (1951). Тheory of Elasticity, McGraw-Hill, New York. Ujihara, K. (1972). Reflectivity of metals at high temperatures, J. Appl. Phys., Vol. 43, pp. 2376-2383. Welch, A. J. & Van Gemert, M. J. C. (1995). Optical-thermal response of laser-irradiated tissue, Plenum Press, New York and London. Yevtushenko, A. A. et al. (2005). Evaluation of effective absorption coefficient during laser irradiation using of metals martensite transformation, Heat Mass Trans., Vol. 41, pp. 338-346. Yevtushenko, A.A. et al. (2007). Laser-induced thermal splitting in homogeneous body with coating, Numerical Heat Transfer, Part A., Vol. 52, pp. 357-375. [...]... that the path and kinetic energy of the electron is controlled by the phase of the electric field (Eq 7 – Eq 9) If phase is ~18° the kinetic energy of electron is maximized and its value is ~3. 17 ⋅ Up (Pfeifer et al., 2006)1 Thus the energy of the highest harmonic order is given by: ν = 3. 17U p + I p 1 18° = ~31 4 mrad (11) 68 Laser Pulse Phenomena and Applications Fig 7 Summary of the 3- step model of... K., Toleikis, S., Tschentscher, T., Wabnitz, H., and Zastrau, U., (2007) Characteristics of focused soft X-ray free-electron laser beam determined by ablation of organic molecular solids, Opt Express, Vol 15, 6 036 80 Laser Pulse Phenomena and Applications Christov, P., Murnane, M M., and Kapteyn, H C., (1997) High-Harmonic Generation of Attosecond Pulses in the “Single-Cycle” Regime, Phys Rev Lett.,... (Mocek et al., 2009) The experimental setup is shown in Fig 9 For strong HHG a two-color laser field, consisting of fundamental and second harmonic (SH) of a femtosecond laser pulse, was applied to a gas jet of He (Kim et al., 2005) Femtosecond laser pulses at 820 nm with an energy of 2.8 mJ and pulse duration of 32 fs were focused by a spherical mirror (f =600 mm) into a He gas jet For SH generation,... in the range of single optical cycle (~ 3. 3 fs at ~810 nm central wavelength) and shifting the laser pulse central wavelength to the mid-infrared spectral range (MIR) in around 2 -3 μm Besides, the lasers’ repetition rates have been significantly increased typically to a few kHz (and energy ~mJ per pulse; e.g Schultze et al., 2007) Another recent achievement of particular interest is carrier-envelope... W / cm2 ] product For λ=1 µm and intensities below 1018 W/cm2 Lorentz force can be approximated only by component coming from electric field, i.e by e ⋅ Ez However, if intensity becomes comparable to 1018 W/cm2 the relativistic effects bring increasingly significant input to Lorentz force from magnetic field component 74 Laser Pulse Phenomena and Applications For laser pulse intensities below the... interaction of NIR femtosecond laser pulses generated by Ti: Al2O3 laser system (Δt =32 fs, E=2.8 mJ, λc=820 nm) and the second harmonic, combined with extreme ultraviolet (XUV) high-order harmonics with the strongest spectral line at 21.6 nm High-order Harmonic Generation 75 Fig 9 Schematic of the experimental setup for surface modification by dual action of XUV and Vis-NIR ultrashort pulses (Mocek et al.,... about 1014 - 1015 W/cm2 and time durations in Fig 1 Ms Röntgen's hand First medical imaging with X-rays (December 22, 1985; source: wikipedia.org) 62 Laser Pulse Phenomena and Applications range of femtoseconds (1 fs = 10-15 s) is applied to the gas, a plateau of equally intense harmonics of very high order can be observed The atom is ionized when the absolute electric field of the laser is close to its... 21.6 nm, 14.7 mJ/cm2 at 820 nm, and 6 .3 mJ/cm2 at 410 nm per shot, respectively As all these values lie far below the ablation threshold for PMMA by infrared (2.6 J/cm2 for single-shot and 0.6 J/cm2 for 100 shots – Baudach et al., 2000) as well as by XUV (2 mJ/cm2 – Chalupsky et al., 2007) radiation, no damage of target surface was expected 76 Laser Pulse Phenomena and Applications Fig 10 AFM images... improvements in femtosecond laser technology - especially in terms of pulse intensity and pulse contrast For instance, efficient HHG from plasma at the interface of the vacuum and solid target intensities higher than 1017 W/cm2 are required with the pulse contrast greater than 106 Moreover, generation of single attosecond pulses requires carrier-envelope stabilization of the driving laser field Finally, in... Ultashort pulse laser ablation of polycarbonate and polymethylmethacrylate, Appl Surf Sci., Vol 154-155, pp 555560 Becker, U., and Shirley, D A., (1996) VUV and Soft X-Ray Photoionization, Plenum, New York Bezzerides, B., Jones, R D., and Forslund, D W., (1982) Plasma Mechanism for Ultraviolet Harmonic Radiation Due to Intense CO2 Light, Phys Rev Lett., Vol 49, No 3, pp 202 - 205 (1982) Boyd, R W., (20 03) . 2007). Laser Pulse Phenomena and Applications 54 0.05 0.10 0.15 0.20 0.25 0 .30 0 .35 0.40 0.45 0.50 0.05 0.10 0.15 0.20 0.25 0 .30 0 .35 0.40 0.45 0.50 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0 .3 0.2 0.1 0.0 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0 .3 0.2 0.1 0.0 . of Laser Pulse on Temperature Distribution and Thermal Stresses in Homogeneous Body with Coating 53 0.05 0.10 0.15 0.20 0.25 0 .30 0 .35 0.40 0.45 0.50 0.05 0.10 0.15 0.20 0.25 0 .30 0 .35 . highest harmonic order is given by: 3. 17 pp UI ν = += (11) 1 18° = ~31 4 mrad. Laser Pulse Phenomena and Applications 68 Fig. 7. Summary of the 3- step model of HHG. The first step