The temperature profile in the molten layer of a semi-infinite target induced by irradiation using a pulsed laser

Optics & Laser Technology 33 (2001) 539–551

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The temperature pro"le in the molten layer of a semi-in"nite target induced by irradiation using a pulsed laser S.E.-S. Abd El-Ghany Department of Physics, Faculty of Science, Benha University, Benha, Egypt Received 3 October 2000; received in revised form 2 January 2001; accepted 3 January 2001

Abstract The two-dimensional Laplace integral transform technique has been used to compute the temperature pro"le of the molten layer on the surface of a semi-in"nite target when irradiated by a pulsed laser. Mathematical expressions for the temperature distribution in the molten layer thickness and the solid part of the target, taking cooling and the temperature-dependent absorption coe3cient of the irradiated surface into account, were obtained. As an illustrative example computations were carried out on a semi-in"nite aluminum (Al) target. c 2001 Elsevier Science Ltd. All rights reserved.
Keywords: Laser damage; Laser melting; Laser heating

1. Introduction The application of technology based on laser radiation progressively covers more and more areas of interest in the "elds of science, engineering and medicine. The advantages of the laser as a light source with a very high-power density and realizable extremely small focus spots, which localize the melting process, are of considerable interest in material processing such as spot welding, cutting, drilling of holes, etc. [1–5]. Due to this, the theoretical study of the thermal e8ects induced by the laser in a solid target is of great interest since it gives information on how to control the melting process or how to avoid it, as in the case of laser mirrors where it is necessary to extend their lifetime. Several authors have considered di8erent aspects of this problem [6 –13]. El-Nicklawy et al. [14] have studied the problem of melting a semi-in"nite target by using a laser pulse, considering the constant temperature distribution in the molten layer thickness to be equal to the constant melting temperature (Tm ). Nevertheless, the full analysis of the problem requires further computation in order to extend the results to cover the thermal behavior of other materials and for di8erent laser=material geometries. The present study looks at the problem of melting a semi-in"nite target with a pro"le in the molten layer induced by surface absorption of a time-dependent laser pulse pro"le [5]. It is aimed at de"n-

ing mathematical expressions for the temperature distribution in both the liquid part and the solid part of the target. By knowing the temperature distribution in the molten layer thickness, values of the molten layer thickness x(tl ) can be achieved. 1.1. Theory Assume a laser pulse of arbitrary relative temporal intensity distribution g(t) to be incident perpendicular to a semi-in"nite target. The laser pulse, which possesses enough energy to initiate the melting process, is subdivided into two parts. The "rst part is responsible for the heating process up to the time at which the surface of the target reaches the melting temperature [15]. The second part deals with the laser=material interaction at times greater than the melting time. Moreover to avoid any chemical reaction with the heated material it is assumed that the front surface of the target is subjected to an inert gas stream. Neglecting the temperature dependence of the material parameters except that of the absorption coe3cient which is assumed to be linearly dependent on the surface temperature during the heating and melting process, non-linear e8ects such as multi-photon absorption and plasma formation in front of the irradiated surface, the relations governing the temperature distribution inside the molten layer and the solid

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PII: S 0 0 3 0 - 3 9 9 2 ( 0 1 ) 0 0 0 0 6 - 8

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S.E.-S. Abd El-Ghany / Optics & Laser Technology 33 (2001) 539–551

part of the target are given for a one-dimensional problem by the following equations: (1) The heat di8usion equation in the liquid part is @2 Tl (x; tl ) 1 @Tl (x; tl ) − = 0; 2 @x l @tl

(1)

where Tl (x; tl ) is the excess temperature distribution above ambient in the liquid part of the target, l is the thermal di8usivity of the liquid material, tl is the time measured after initiation of melting and x is the spatial coordinate in the liquid, with origin at the surface of the liquid part of the target. (2) The di8erential heat balance equation at the front surface of the liquid is q0 g(tl + tm )[A01 + A02 Tl (0; tl )] − hTl (0; tl )
@Tl (x; tl )

+kl
= 0; @x x=0

(2)

where q0 is the maximum laser power density, g(tl + tm ) is the relative temporal intensity distribution of the laser pulse pro"le with maximum value one, tm is the time of initiation of the melting process, A01 and A02 are the temperature-independent and temperature-dependent parts of the surface absorption coe3cient, respectively, h is the heat transfer coe3cient resulting from the gas spreading out the molten layer, kl is the thermal conductivity of the liquid part and Tl (0; tl ) is the excess surface temperature distribution above ambient of the liquid part of the target. (3) The heat di8usion equation in the solid part of the target is @2 Ts ( ; tl ) 1 @Ts ( ; tl ) − = 0; @ 2 s @tl

(3)

where Ts ( ; tl ) is the excess temperature distribution above ambient in the solid part of the target, s is the thermal di8usivity of the solid part, is the spatial coordinate of the solid part, with origin at the solid=liquid interface. (4) The point where the temperature in the solid target is ambient is assumed to be far away from the front surface. At this point the excess temperature above ambient is considered to be zero: lim Ts ( ; tl ) = 0:

→∞

(4)

The two-dimensional Laplace integral transform technique applied on the above-cited equations requires the temperature distribution within the target at the time of initiation of the melting process Tl (x; 0) to be calculated: It is equal to the temperature distribution at the end of the heating process, which is given according to the following convolution

equation [15]:  √  t  exp(−(x2 =4(t − )))  G() Tb:m (x; t) = k 0
(t − )   √ xh h2  h  exp + 2 (t − ) − k k k   √  h  (t − ) x  × Erfc d; + k 2 (t − ) (5) with G() = q0 g()[A01 + A02 T (0; )]. x = xl in the liquid phase and x = x(tl ) + in the solid part of the target. t is a parameter given by t = tm + tl with tm 6 t 6 ∞, T (0; ) is the time-dependent surface temperature during the heating process and  is a time between 0 and tm . This technique applied on Eq. (1) leads to algebraic equations in the p and s domains for the temperature distribution in the liquid part of the target given by Tb:m (x; t) pT˜ l (0; s) − T˜˜ l (p; s) = 2 (p − s=l ) l p(p2 − s=l )
@T˜ l (x; s)

1 + 2 ; (p − s=l ) @x
x=0

(6)

where s; p are the complex temporal and spatial angular frequency, respectively. The inverse Laplace transform w.r.t. p of the last equation gives  T˜ l (x; s) = T˜ l (0; s) cosh ( s=l )x   Tb:m (x; t) cosh ( s=l )x − 1 − al s=l 
sinh ( s=l )x @T˜ l (x; s)

 + : (7) @x
x=0 s=l Applying the initial value theorem given by lim sT˜ l (x; s) = Tb:m (x; tm + tl )

s→∞

tl →0

on Eq. (7) one gets


Tb:m (x; tm + tl ) − sT˜ l (0; s) @T˜ l (x; s)

: = √ @x
sl x=0

(8)

T˜ l (0; s) in Eq. (7) can be obtained by using Eq. (8) from the Laplace transform of Eq. (2) w.r.t. s. By substituting from Eq. (8) into Eq. (7) and using the inverse Laplace transform technique w.r.t. s, one gets  √  tl l exp(−(x2 =4l (tl − )))  Tl (x; tl ) = G() kl 0
(tl − )   xh −a:exp + a2 (tl − ) kl   √ x  × Erfc + a tl −  d 2 l (tl − )

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Table 1 The optical and thermal parameters of aluminuma Element

l (W m−1 K −1 )

 (kg=m3 )

cp (J=kg: K −1 )

A01

A02 (K −1 )

Tm (K −1 )

Tboi: (K −1 )

ql (J=kg)

Al-slab 238 2707 896 0.056 3:05 × 10−5 633 2440 4 × 105 the density of the aluminum target, cp , the speci"c heat of the aluminum by constant pressure, Tboi: , the boiling temperature of aluminum, ql , latent heat in the aluminum target.

a ,



 x + Tb:m (0; tm + tl ) exp a2 tl + a √ l   √ x ×erfc a tl + √ 2 l tl 

 x √ + Tb:m (x; tm + tl ) 1 − erfc ; 2 l tl

2. Computations

(9)

√ where a = h l =kl . Eq. (9) gives also information about the molten layer thickness at di8erent times, through calculating the spatial temperature distribution in the molten layer up to the point at which Tl (x; tl ) = Tm . Applying the same procedure on Eq. (3) one gets, after "nding the inverse Laplace transform w.r.t. p for the temperature distribution in the solid part of the target, the following mathematical expression    Tm exp( s=s ) + exp(− s=s ) ˜ T s ( ; s) = s 2  1 − Tb:m (; tm + tl ) s=s 0  ×sinh ( s=s ( − )) d 
 1 @T˜ s ( ; s)

 + @
=0 s=s    exp( s=s ) − exp(− s=s ) × : (10) 2 Eq. (10) requires, according to Eq. (4), that the sum of the  coe3cients of exp( s=s ) must be set equal to zero; this gives 
  ∞ @T˜ s ( ; s)

1 1  =√ Tb:m (; tm + tl ) @
=0 ss 0 s=s  Tm exp(− s=s ) d − : (11) s Thus Eq. (10) can be written after being Laplace transformed as    ∞ 1

+ √ Tb:m (; tm + tl ) Ts ( ; tl ) = Tm Erfc √ 2 tl s 2
tl s 0

 exp[ − ( − )2 =(4s tl )]− × @: (12) exp[ − ( + )2 =(4s tl )]

Eqs. (9) and (12) have been calculated for a semi-in"nite aluminum target subjected to the relative distribution of the Ready laser pulse pro"le [5] approximated by the following relation [15]: (n+1)n+1 t t n for 0 6 t 6 t ; nn t (1 − t ) (13) g(t) = 0 for t ¿ t where n is given the value 3 and t is the pulse duration. It is the time interval from the moment of initiation of the pulse up to its end. The optimal and thermal parameters of aluminum shown in Table 1. Figs. 1–3 represent the time dependence of the molten layer thickness x(tl ) for the combinations of h and A02 listed in Table 2. Consider q0 to be 2 × 1010 W m−2 and t to have the values (1:5 × 10−3 , 3 × 10−3 , 5 × 10−3 s), respectively. The "gures show that the molten layer thickness increases with increasing the time up to the point at which the absorbed laser energy is equal to the sum of the losses resulting from the cooling and conduction in the liquid at which point the molten layer thickness exhibits a maximum. The maximum of the liquid thickness is found to shift towards smaller times as h=105 W m−2 K −1 and A02 =0 K −1 . This behavior is due to the fact that the losses for h = 105 W m−2 K −1 are increased and the absorption for A02 = 0 K −1 is decreased, since the maximum value of x(tl ) is obtained as the losses are compensated by the absorption; thus, to ful"l this condition according to the chosen laser pro"le the maximum will shift towards shorter times. It is also found that the molten layer thickness becomes greater as the pulse duration increases and=or h = 0 W m−2 K −1 , A02 = 3:05 × 10−5 K −1 . This behavior is due to the greater absorbed laser energy. The comparison between the obtained values of the molten layer thickness and those obtained by constant temperature in the molten layer [14], shows that the present values are smaller. This is because of the dissipated heat energy for raising the temperature of the molten layer above the melting temperature. The calculation of the integrated heat balance equation in the case of A02 = 0 K −1 and h = 0 W m−2 K −1 given by  x(tl )  ∞ cp Tl (x; tl ) d x + ql x(tl ) + cp Ts ( ; tl ) d 0

(n + 1)n t =q0 A01 nn n+2

0

shows that (RHS − LHS)=RHS ∼ = 15%.

(14)

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S.E.-S. Abd El-Ghany / Optics & Laser Technology 33 (2001) 539–551

Fig. 1. The time dependence of x(tl ) calculated during the irradiation of a semi-in"nite Al-target with the Ready pulse pro"le g(t) curve 5. The calculations were carried out considering q0 = 2 × 1010 W m−2 and t = 1:5 × 10−3 s for the cases listed in Table 2.

Fig. 3. The time dependence of x(tl ) calculated during the irradiation of a semi-in"nite Al-target with the Ready pulse pro"le g(t) curve 5. The calculations were carried out considering q0 = 2 × 1010 W m−2 and t = 5 × 10−3 s for the cases listed in Table 2.

Table 2 The combinations of A02 and h used for calculating the molten layer thickness and the temperature distributions Curve no.a

A02 (K −1 )

1 0 2 0 3 3:05 × 10−5 4 3:05 × 10−5 a Curve 5 represents the laser pulse pro"le.

h (W m−2 K −1 ) 0 105 0 105

Fig. 2. The time dependence of x(tl ) calculated during the irradiation of a semi-in"nite Al-target with the Ready pulse pro"le g(t) curve 5. The calculations were carried out considering q0 = 2 × 1010 W m−2 and t = 3 × 10−3 s for the cases listed in Table 2.

To insure the validity of the applied equations, after switching o8 the laser pulse, the molten layer thickness for A02 = 0 K −1 and h = 0 W m−2 K −1 was calculated. The results obtained are represented in Fig. 3, curve 1. The curve shows, as expected, a decrease in the molten layer thickness on increasing the time. Fig. 4 shows the maximum molten layer thickness for different pulse durations by keeping the laser pulse pro"le unchanged and considering, q0 = 2 × 1010 W m−2 , A02 and h to have the values given in Table 2. The "gure shows a monotone increase of the molten layer thickness with increasing pulse duration. This is due to the increase of the absorbed energy.

Fig. 4. The dependence of the maximum molten layer thickness on the pulse duration obtained from the irradiated semi-in"nite Al-target with a Ready pulse pro"le of q0 = 2 × 1010 W m−2 . The calculations are carried out for the cases listed in Table 2.

S.E.-S. Abd El-Ghany / Optics & Laser Technology 33 (2001) 539–551

Fig. 5. The temperature distribution in a semi-in"nite Al-target irradiated with Ready pulse pro"le of q0 = 2 × 1010 W m−2 and t = 5 × 10−3 s calculated at t = 4 × 10−6 s after solidi"cation of the molten layer for the case A02 = 0 K −1 , h = 0 W m−2 K −1 .

Fig. 6. The temperature distribution in a semi-in"nite Al-target irradiated with Ready pulse pro"le of q0 = 2 × 1010 W m−2 and t = 5 × 10−3 s calculated at t = 2 × 10−5 s after solidi"cation of the molten layer for the case A02 = 0 K −1 , h = 0 W m−2 K −1 .

Figs. 5 and 6 represent the temperature distributions in the target, at times (4 × 10−6 , 2 × 10−5 s), after switching o8 the laser pulse and solidi"cation of the molten layer thickness, respectively. It is calculated from the solution of the heat transfer equation for h = 0 W m−2 K −1 considering the initial temperature distribution to be equal to that of the solid part when the surface temperature is equal to the melting temperature. The "gures show a decrease of the temperature gradient in moderate depths by increasing the time. Both curves have a gradient equal to zero at the front

543

Fig. 7. The time dependence of the surface temperature of a semi-in"nite Al-target irradiated with Ready pulse pro"le of q0 = 2 × 1010 W m−2 and t = 5 × 10−3 s for the case of A02 = 0 K −1 , h = 0 W m−2 K −1 calculated at t = 1:013 × 10−3 s after solidi"cation of the molten layer.

surface which is due to −kl (@T (x; tl )=@x|x=0 = 0. Since in both curves the stored energy is the same and the penetration depth increases with increasing time, the temperature distribution for the shorter time must take a more or less rectangular shape. Fig. 7 represents the time dependence of the surface temperature at time equal to 1:013 × 10−2 s after switching o8 the laser pulse and solidi"cation of the molten layer thickness. The "gure shows, as expected, that the surface temperature shrinks to zero as the time tends to in"nity. Figs. 8–10 represent the temperature distribution in the molten layer calculated under the conditions given in Figs. 1–3. The "gures show that the temperature distribution increases as t increases. This is due to the fact that the times at which these curves are calculated are, respectively, given by (t − tm )=5. t and tm are given in Table 3. Since the time of calculation increases as t increases, leading to an increase in the absorbed energy, thus the temperature distribution has to increase with increasing t and that it attains its greatest values at h=0 W m−2 K −1 , A02 =3:05×10−5 K −1 and its smallest values for the case of A02 = 0 K −1 , h = 105 W m−2 K −1 . The "gures indicate that the e8ect of A02 is greater than that of h. This behavior can be attributed to the greater absorption and small losses as A02 and h are given the values 3:05 × 10−5 K −1 , 0 W m−2 K −1 respectively. The crossing of the curve 3 with curve 4 in Fig. 8 is due to the negative slope of the cooled curve at the irradiated front surface which leads to a shift in its maximum value at a point inside the molten layer thickness. Since the di8erence between the two curves is small the curves cross. Table 3 illustrates the melting time, for each pulse duration of Ready pro"le, for the four possible combinations of h and A02 .

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S.E.-S. Abd El-Ghany / Optics & Laser Technology 33 (2001) 539–551

Table 3 The melting time tm versus pulse duration t for all combinations of h and A02 t (s)

h = 0; A02 = 0 curve 1, tm

h = 105 ; A02 = 0 curve 2, tm

h = 0; A02 = 3:05 × 10−5 curve 3, tm

h = 105 ; A02 = 3:05 × 10−5 curve 4, tm

1:5 × 10−3 3 × 10−3 5 × 10−3

3:521 × 10−4 5:259 × 10−4 8:0712 × 10−4

4:544 × 10−4 5:940 × 10−4 9:08999 × 10−4

2:531 × 10−4 3:890 × 10−4 5:57777 × 10−4

3:2999 × 10−4 4:649 × 10−4 6:91234 × 10−4

Fig. 8. The temperature distribution in the liquid part of a semi-in"nite Al-target irradiated with Ready pulse pro"le of q0 = 2 × 1010 W m−2 and t = 1:5 × 10−3 s calculated at t = ((t − tm )=5) s for the cases listed in Table 2.

Fig. 10. The temperature distribution in the liquid part of a semi-in"nite Al-target irradiated with Ready pulse pro"le of q0 = 2 × 1010 W m−2 and t = 5 × 10−3 s calculated at t = ((t − tm )=5) s for the cases listed in Table 2.

Fig. 11. The temperature distribution in the liquid part of a semi-in"nite Al-target irradiated with Ready pulse pro"le of q0 = 2 × 1010 W m−2 calculated at t = (1:5 × 10−3 − tm ) s for the cases listed in Table 2. Fig. 9. The temperature distribution in the liquid part of a semi-in"nite Al-target irradiated with Ready pulse pro"le of q0 = 2 × 1010 W m−2 and t = 3 × 10−3 s calculated at t = ((t − tm )=5) s for the cases listed in Table 2.

At the end of the pulses, at which Figs. 11–13 are respectively calculated, the same behavior as found in Figs. 8–10 is observed. Except for A02 = 0 K −1 in Fig. 11 the molten layer thickness was not found. This is due to the small pulse

S.E.-S. Abd El-Ghany / Optics & Laser Technology 33 (2001) 539–551

Fig. 12. The temperature distribution in the liquid part of a semi-in"nite Al-target irradiated with Ready pulse pro"le of q0 = 2 × 1010 W m−2 calculated at t = (3 × 10−3 − tm ) s for the cases listed in Table 2.

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Fig. 14. The time dependence of the surface temperature of the molten layer of a semi-in"nite Al-target, calculated during the irradiation with the Ready pulse pro"le. The calculations were carried out considering q0 = 2 × 1010 W m−2 and t = 1:5 × 10−3 s for the cases listed in Table 2.

Fig. 13. The temperature distribution in the liquid part of a semi-in"nite Al-target irradiated with Ready pulse pro"le of q0 = 2 × 1010 W m−2 calculated at t = (5 × 10−3 − tm ) s for the cases listed in Table 2.

Fig. 15. The time dependence of the surface temperature of the molten layer of a semi-in"nite Al-target, calculated during the irradiation with the Ready pulse pro"le. The calculations were carried out considering q0 = 2 × 1010 W m−2 and t = 3 × 10−3 s for the cases listed in Table 2.

duration and absorption. The "gures have generally smaller values for Tl (x; tl ) than the corresponding curves given in Figs. 8–10. This due to the vanishing absorbed power at the end of the laser pulse. Two exceptions are found to exist for t equal to (3 × 10−3 , 5 × 10−3 s) for the case h=0 W m−2 K −1 , A02 =3:05×10−5 K −1 . This behavior can be attributed to the fact that the previously absorbed laser energy for these cases is much greater compared with the corresponding curves given in Figs. 9 and 10. Figs. 14 –16 describe the surface temperature of the molten layer as a function of time. All the curves exhibit a maximum occurring at times greater than the maximum of g(t). This behavior can be attributed to the fact that, at the

beginning of the melting process, the absorbed laser energy is greater than the penetrating heat energy. By the end of the pulse the earlier absorbed energy di8uses out of the laser spot and the incoming energy is not enough to keep the temperature up. It is also observed that the maximum temperature shifts toward shorter times as A02 and h are given the values 0 K −1 and 105 W m−2 K −1 , respectively. This behavior is due to the fact that the maximum occurs as the absorbed laser power covers the losses. As A02 and=or h are given the values 0 K −1 and 105 W m−2 K −1 , respectively, the need for a greater power to ful"l the above-cited condition occurs at smaller times. When t is given the value 5 × 10−3 s, q0 = 2 × 1010 W m−2 it is observed that for

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S.E.-S. Abd El-Ghany / Optics & Laser Technology 33 (2001) 539–551

Fig. 16. The time dependence of the surface temperature of the molten layer of a semi-in"nite Al- target, calculated during the irradiation with the Ready pulse pro"le. The calculations were carried out considering q0 = 2 × 1010 W m−2 and t = 5 × 10−3 s for the cases listed in Table 2.

Fig. 17. The temperature distribution in the solid part of a semi-in"nite Al-target irradiated with Ready pulse pro"le of q0 = 2 × 1010 W m−2 calculated at t = ((1:5 × 10−3 − tm )=5) s for the cases listed in Table 2.

times greater than (1:4 × 10−3 s), the temperature exceeds the evaporation temperature, for which the above-cited equations are not valid. Figs. 17–19 show the temperature distributions in the solid layer at times equal to (t −tm )=5 for t equal to (1:5×10−3 , 3×10−3 , 5×10−3 s), respectively, and q0 =2×1010 W m−2 . Hear it is observed that the di8erences between the curves having di8erent combinations of A02 and h are practically negligible. The curve 2, for A02 =0 K −1 , h=105 W m−2 K −1 has the greatest penetration depth while curve 3, for A02 = 3:05 × 10−5 K −1 , h = 0 W m−2 K −1 , has the smallest penetration depth. This is due to the time elapsed between the initiation of the melting process which is greatest for the "rst case and smallest for the second one. Other calculations

Fig. 18. The temperature distribution in the solid part of a semi-in"nite Al-target irradiated with Ready pulse pro"le of q0 = 2 × 1010 W m−2 calculated at t = ((3 × 10−3 − tm )=5) s for the cases listed in Table 2.

Fig. 19. The temperature distribution in the solid part of a semi-in"nite Al-target irradiated with Ready pulse pro"le of q0 = 2 × 1010 W m−2 calculated at t = ((1:5 × 10−3 − tm )=5) s for the cases listed in Table 2.

for the temperature distribution in the solid part of the slab at di8erent t values yielded no new information. Figs. 20, 3 and 21 represent the time dependence of the molten layer thickness for the case of t = 5 × 10−3 s and q0 equal to (1 × 1010 , 2 × 1010 , 3 × 1010 W m−2 ), respectively. The "gures show the same behavior as that obtained previously in Figs. 1–3. The calculated temperature distribution in the molten layer showed also the same behavior as that obtained previously in Figs. 8–13. The calculation of the surface temperature of the molten layer thickness as well as the temperature distribution in the solid part of the slab showed the same behavior as previously obtained in Figs. 14 –19, respectively. Fig. 22 shows the maximum of the molten layer thickness versus q0 for t = 5 × 10−3 s and the considered values of

S.E.-S. Abd El-Ghany / Optics & Laser Technology 33 (2001) 539–551

Fig. 20. The time dependence of x(tl ) calculated during the irradiation of a semi-in"nite Al-target with the Ready pulse pro"le g(t) curve 5. The calculations were carried out considering t = 5 × 10−3 s and q0 = 1 × 1010 W m−2 for the cases listed in Table 2.

Fig. 21. The time dependence of x(tl ) calculated during the irradiation of a semi-in"nite Al-target irradiated with Ready pulse pro"le of q0 = 3 × 1010 W m−2 calculated at t = 5 × 10−3 s for the cases listed in Table 2.

h and A02 . The "gure shows a smaller than linear monotone increase of xmax: (tl ) with increasing q0 . This behavior is due to the greater increase of the surface temperature by increasing q0 values and therefore in the stored energy in the molten layer, such that less energy will contribute to increase the molten layer thickness. As the pulse pro"le was changed, to be square, to take a distribution de"ned by Ready [5] or to have a Gaussian distribution, by keeping the energy under the temporal distribution to be constant, it was found that, for the considered values in Table 4 and q0 =4:5×1010 W m−2 , the surface temperature achieves its greatest value when the square pulse was considered. This distribution is followed by Gaussian distribution and ends with the Ready pro"le as shown in

547

Fig. 22. The power dependence of the maximum molten layer thickness obtained from the irradiated semi-in"nite Al-target with a Ready pulse pro"le of t = 5 × 10−3 s. The calculations were carried out for the cases listed in Table 2.

Fig. 23. The time dependence of the surface temperature of the molten layer of a semi-in"nite Al-target, calculated during the irradiation of an Al-target with a square pulse pro"le. The calculations were carried out considering q0 = 4:5 × 1010 W m−2 and t = 7:12345 × 10−5 s for the cases listed in Table 2.

Figs. 23–25, respectively. It is due to the greater absorbed energy in a small time which leads to smaller di8used heat inside the con"guration and therefore to increase the surface temperature. The di8erence between the surface temperature for h = 0; 105 W m−2 K −1 for the cases A02 = 0; 3:05 × 10−5 K −1 is found to be greatest for the Gaussian distribution rather than that of Ready and the square laser pulse. This phenomenon is the result of the greater pulse duration, which allows a lot of energy to be dissipated either through conduction in the cooled material or through cooling the front surface. This makes the surface temperature more sensitive to h and A02 . Due to this explanation it is found that

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S.E.-S. Abd El-Ghany / Optics & Laser Technology 33 (2001) 539–551

Fig. 24. The time dependence of the surface temperature of the molten layer of a semi-in"nite target, calculated during the irradiation of an Al-target with the Gaussian pulse pro"le. The calculations were carried out considering q0 = 4:5 × 1010 W m−2 and t = 1:7248 × 10−4 s for the cases listed in Table 2.

Fig. 25. The time dependence of the surface temperature of the molten layer of a semi-in"nite target, calculated during the irradiation of an Al-target with the Ready pulse pro"le. The calculations were carried out considering q0 = 4:5 × 1010 W m−2 and t = 1:5026 × 10−4 s for the cases listed in Table 2.

the smallest temperature rise is given in the case of operating with a square pulse. As can be seen from the "gures, the cases illuminated with a laser pulse having a maximum power density, possess a maximum surface temperature for the given values of h and A02 . Figs. 26 –28 represent the temperature distribution within the molten layer thickness for the three di8erent illuminating pro"les and the combinations of A02 and h calculated at times equal to 2 × (t − tm )=5. The "gures show that the highest temperature distribution is achieved for the case of Gaussian illumination, with A02 and h given the values

Fig. 26. The temperature distribution in the molten layer of a semi-in"nite Al-target irradiated with square pulse pro"le of q0 =4:5×1010 W m−2 and energy density equal to (3:2×106 J m−2 ) at the time equal to 2(t −tm )=5 and t = 7:12345 × 10−5 s calculated for the cases listed in Table 2.

Fig. 27. The temperature distribution in the molten layer of a semi-in"nite Al-target irradiated with Ready pulse pro"le of q0 =4:5×1010 W m−2 and energy density equal to (3:2×106 J m−2 ) at the time equal to 2(t −tm )=5 and t = 1:5026 × 10−4 s calculated for the cases listed in Table 2.

3:05 × 10−5 K −1 and 0 W m−2 K −1 , respectively. This behavior is because of the greater time at which these curves are calculated. Also, here the same di8erences between the curves having di8erent A02 and h values as in the case of surface temperature were observed. The penetration depth is also due to the di8erent illumination times, i.e. greatest for the case of Gaussian distribution and smallest for the case of square pulse. Figs. 29 –31 represent the time dependence of the molten layer thickness for the considered combinations of A02 and h. Due to the smallest illumination time, by keeping the energy under illuminating curve to be constant which leads to small

S.E.-S. Abd El-Ghany / Optics & Laser Technology 33 (2001) 539–551

Fig. 28. The temperature distribution in the molten layer of a semi-in"nite Al-target irradiated with Gaussian pulse pro"le of q0 = 4:5 × 1010 W m−2 and energy density equal to (3:2 × 106 J m−2 ) at the time equal to 2(t − tm )=5 and t = 1:7248 × 10−4 s calculated for the cases listed in Table 2.

Fig. 29. The time dependence of x(tl ) calculated during the irradiation of a semi-in"nite Al-target with a square pulse pro"le g(t) curve 5. The calculations were carried out considering q0 = 4:5 × 1010 W m−2 , t = 7:12345 × 10−5 s for the cases listed in Table 2.

losses, it is found, as expected, that the greatest x(tl ) is seen by the square pulse. The Gaussian distribution exhibits the second-greatest value, although its illumination time is greater than that of Ready. This can be attributed to the fact that its half width is smaller. Due to the linear increase of the absorbed laser radiation in the case of a square pulse a monotone increase of the molten layer thickness is achieved. As the Gaussian or Ready pro"le is taken into consideration a molten layer thickness exhibits a maximum which appears after the illumination has reached its maximum value. This behavior is due to the fact that at the beginning the absorbed laser radiation overcompensates the losses while at the end,

549

Fig. 30. The time dependence of x(tl ) calculated during the irradiation of a semi-in"nite Al-target with the Ready pulse pro"le g(t) curve 5. The calculations were carried out considering q0 = 4:5 × 1010 W m−2 , t = 1:5026 × 10−4 s for the cases listed in Table 2.

Fig. 31. The time dependence of x(tl ) calculated during the irradiation of a semi-in"nite Al-target with the Gaussian pulse pro"le g(t) curve 5. The calculations were carried out considering q0 = 4:5 × 1010 W m−2 , t = 1:7248 × 10−4 s for the cases listed in Table 2.

the losses cannot be covered from radiation leading to the reduction of the molten layer thickness. Table 4 gives the melting time for square, Gaussian, Ready pulse pro"les for the considered values of t, h, A02 at q0 = 4:5 × 1010 W m−2 . Figs. 32–34 represent, for the four combinations of h and A02 , the temperature distributions in the solid layer at times equal to (t − tm )=5 for each case. The "gures show practically the same penetration depths, although the times at which the penetration depths are calculated are different given by (4:8 × 10−5 s) for square, (8:1 × 10−5 s) for Ready and (10:7×10−5 s) for Gaussian pro"le. From these times one can conclude that the largest penetration

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S.E.-S. Abd El-Ghany / Optics & Laser Technology 33 (2001) 539–551

Table 4 The melting time tm versus pulse duration t for all combinations of h and A02 h = 0; A02 = 0 curve 1, tm

h = 105 ; A02 = 0 curve 2, tm

h = 0; A02 = 3:05 × 10−5 curve 3, tm

h = 105 ; A02 = 3:05 × 10−5 curve 4, tm

Square 7:12345 × 10−5

4:274 × 10−5

4:5744 × 10−5

2:849 × 10−5

3:11442 × 10−5

Ready 1:5026 × 10−4

6:3 × 10−5

6:31 × 10−5

4:51 × 10−5

4:85000 × 10−5

Gauss 1:7248 × 10−4

9:548 × 10−5

1:0348 × 10−4

8:624 × 10−5

9:0500 × 10−5

t (s)

Fig. 32. The temperature distribution in the solid layer of a semi-in"nite Al-target irradiated with square pulse pro"le of q0 =4:5×1010 W m−2 and energy density equal to (3:2 × 106 J m−2 ) at the time equal to (t − tm )=5 and t = 7:12345 × 10−5 s calculated for the cases listed in Table 2.

Fig. 33. The temperature distribution in the solid part of a semi-in"nite Al-target irradiated with Ready pulse pro"le of q0 = 4:5 × 1010 W m−2 and energy density equal to (3:2 × 106 J m−2 ) at t equal to (t − tm )=5 and t = 1:5026 × 10−4 s calculated for the cases listed in Table 2.

Fig. 34. The temperature distribution in the solid part of a semi-in"nite Al-target irradiated with Gaussian pulse pro"le of q0 = 4:5 × 1010 W m−2 and energy density equal to (3:2 × 106 J m−2 ) at t equal to (t − tm )=5 and t = 1:7248 × 10−4 s calculated for the cases listed in Table 2.

depth must be given for the Gaussian distribution while the smallest one has to be obtained for the case of illuminating by the square pulse. Since the molten layer thickness at these times is greatest in the Gaussian case (4 × 10−5 m) and smallest in the case of the square pulse (2 × 10−5 m), it can be concluded that the molten layer thickness in the Gaussian case has reduced the thermal penetration depth in the solid part, leading to the observed behavior. The "gures show in accordance with the surface temperature that, the smallest deviation between the di8erent temperature distributions, for the considered combinations of h and A02 , within the solid part is for the case of illuminating with a square pulse. Due to the shielding behavior of the molten layer thickness, the di8erence between the curves is much smaller than that of the surface temperature and that of temperature distribution in the molten layer.

S.E.-S. Abd El-Ghany / Optics & Laser Technology 33 (2001) 539–551

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