Thermal fields in laser–multi-layer structures interaction

ARTICLE IN PRESS

Optics & Laser Technology 39 (2007) 796–799 www.elsevier.com/locate/optlastec

Thermal fields in laser–multi-layer structures interaction M. Oanea,, F. Scarlata,b, Shyh-Lin Tsaoc, I.N. Mihailescua a

National Institute for Laser, Plasma and Radiation Physics, P.O.Box MG 36 Magurele R-76900 Bucharest, Romania b Department of Physics, Valahia State University of Targoviste, Targoviste, Romania c Institute of Electro-optical Science and Technology, National Taiwan Normal University, 88, Sec.4, Ting-Chou RD., Taipei, Taiwan, 116, ROC Received 4 April 2005; received in revised form 31 January 2006; accepted 31 January 2006 Available online 2 May 2006

Abstract In the present paper, we are dealing with the thermal fields for laser–periodic multilayer structures interaction. Our point of view is originally that we consider any order transverse laser beams, like heating sources. We consider that the laser beam acts in IR (Nd:YAG or CO2 laser beam) and it is in one transverse mode or in only a few decoupled modes. In order to solve this problem, we will use the Green function method. Specific results are presented for a laser beam (Nd:YAG laser) operating in the mode TEM01 and a two-layer structure. r 2006 Elsevier Ltd. All rights reserved. Keywords: Heat equation; Green function method; Laser processing

1. Introduction Light has always played a central role in the study of physics, chemistry and biology. In the last century, laser light has provided important contributions to medicine, industrial material processing, data storage, printing and defense [1]. In all these areas of applications, the laser–solid interaction has played a crucial role. The theory of heat conduction was well studied long time ago [2–9]. For describing this interaction, the classical heat equation was used in many applications. Apart of some criticism [10], the heat equation still remains a powerful tool in describing thermal effects in laser–solid interaction [11–15]. The heat equation can be used for describing both: interaction with homogeneous [16–19] and inhomogeneous solids [20]. Special attention was given to multi-layered samples and thin films. In this paper the authors would like to describe the interaction of IR laser, which is operated in one or a few transverse decoupled modes. The theory is valid mostly

Corresponding author.

E-mail address: [email protected] (M. Oane). 0030-3992/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2006.01.011

for high power laser and it could be useful to the study of laser processing for different materials.

2. The statement of the problem Let us consider that a laser beam acting in a few decoupled modes is heating one sample consisting of a few layered media. Because the heat equation is a linear one (in the sense that if T 1 and T 2 are solutions of the heat equation, the sum T 1 þ T 2 is also a solution of the heat equation) , we can consider that if we do not lose from generalite just one transverse mode. The temperature variation depends on physical-thermal parameters of the target as well as the characteristics of incident laser beam (for instance, transverse mode structure, power, wavelength, and beam width). The temperature in general P case P can be obtained by applying the formula: T ¼ m n T mn , where T mn is the temperature given by the transverse mode ðm; nÞ. We will use the Green function method and generalize the model found in literature [4]. We consider the case that the direction of heat transfers is perpendicular to the layers in series. We can introduce the effective thermal conductivity for

ARTICLE IN PRESS M. Oane et al. / Optics & Laser Technology 39 (2007) 796–799

"

this multilayer medium as [4] 2

l1 þ l2 þ    þ ln l 1 =k1 þ l 2 =k2 þ    þ l n =kn l ¼ , l 1 =k1 þ l 2 =k2 þ    þ l n =kn

y  w ln H n

K? ¼

þ

Kk ¼

ð2Þ

where A is the total area of the near or far faces. For most materials, the heat flux in one direction is only caused by the temperature gradient in that direction. In this case, the heat equation in equilibrium can be written as        q qT q qT q qT K x ðTÞ K y ðTÞ K z ðTÞ þ þ qx qX qy qy qz qz ¼ Sðx; y; zÞ,

ð3Þ

where K x ðTÞ; K y ðTÞ; K z ðTÞ are temperature dependent thermal conductivity along x, y, z directions. Sðx; y; zÞ is the heat source of the equation. Because we are dealing, in the present paper, with a solid without melting, we can neglect the blackbody radiation of the sample. It was calculated that it is possible to reach the temperature 1350  C in a silicon substrate when the incident laser beam has a power of 4 W and a radius of 10 mm [21]. The absorbed power and the energy loss due to blackbody radiation are estimated to be 2.3 W and 0.12 mW respectively (only 0.005% as much as the absorbed power). In this case, the media layer is parallel with the substrate surface. The thermal conductivity of the substrate has a value of K k along x and y directions as well as a value of K ? along z direction. Then, the heat equation becomes:        q qT q qT q qT K k ðTÞ K k ðTÞ K ? ðTÞ þ þ qx qX qy qy qz qz ¼ Sðx; y; zÞ.

ð4Þ

In the general case, for the mode ðm; nÞ we have [3] " pffiffiffi ! pffiffiffi !  #2 2x 2y ðx2 þ y2 Þ I mn ¼ I 0mn H m . Hn exp  w w w2 (5) and Sðx; y; zÞ ¼ 2I 0mn f1  R½TðyÞg " 2 8 pffiffiffi !# > 2x > 2 2 >x  w ln H m w 6 > 6 <  exp6 6> w2 4 > > > :

w2

ð1Þ

where l is the total thickness of this medium structure. Similarly, the effective thermal conductivity in the case of the heat flows is in the parallel direction: K 1 A1 þ K 2 A2 þ    þ K n An A1 þ A2 þ    þ An K 1 A1 þ K 2 A2 þ    þ K n An , ¼ A

2

797

pffiffiffi !#932 2y > > > > =7 w 7 7 dðzÞ. 7 > > 5 > > ;

ð6Þ

By definition: pffiffiffi ! 2x x  x  w ln H m w pffiffiffi ! 2y 2 . y0  y2  w2 ln H n w 02

2

2

and

ð7Þ

Using this definition we have "

#2 ðx0 2 þ y0 2 Þ dðzÞ. Sðx; y; zÞ ¼ 2I 0mn f1  RðTðyÞÞg exp  w2 (8) One can notice that formula (8) is similar to that for the Gaussian source, and therefore we can use the Green function for the Gaussian form source. Here w represents the width of the beam ðw ¼ wx ¼ wy Þ. In order to continue our calculation, we introduce the linearized temperature like: Z T yðTÞ ¼ yðT 0 Þ þ ð1=KðT 0 ÞÞ KðT 0 Þ dT 0 , T0

where in above equation yðTÞ and yðT 0 Þ are the linearized temperature at a temperature T and at a substrate temperature T 0 [3,21]. One can observe easily that the variation of linearized temperature is equal to the variation of the usual temperature when K is independent of T. This is the case for Al and Ag as we will see later in the present paper. The linearized temperature for the heat equation is obtained as [4] Z 1 P½1  R? ðTÞ y? ¼ 1=2 3=2 f ? ðxÞ dx where ðaÞ ðpÞ KðT 0 Þ 0 K? a¼ and K k ¼ kðTÞ. ð9Þ kk Here P ¼ p=w represents the normalized incident power, R? is the surface reflectivity when the incident beam is perpendicular to the layer structure of the substrate and T 0 is the original substrate temperature before laser irradiation. The function f ? ðxÞ is given by f ? ðxÞ ¼

exp ½½X 2 ðx2 þ 1Þ þ ½Y 2 =ðx2 þ 1Þ þ ðZ 2 =ax2 Þ fðx2 þ 1Þg (10) 0

0

with X ¼ x =w; Y ¼ y =w, and Z ¼ z=w, which are the normalized coordinates of the system.

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3. Simulations

2K Ag K Al ðK Al þ K Ag Þ

and

Kk ¼

200 150 100 50

ðK Ag þ K Al Þ . 2

0.02

0.04

0.06

0.08

0.1

z [mm]

The bulk thermal conductivities for Ag and Al are presented in Figs. 1 and 2, respectively. All the figures are considered after the steady state is established; in other words it is understood that we are dealing with a cw Nd:YAG laser with w ¼ wx ¼ wy ¼ 50 mm. Fig. 3 shows the temperature rise at the origin ðx ¼ y ¼ 0Þ when the power of the laser beam is 2 W (incident power). The same plot is presented in Fig. 4, but the power of the laser beam is 1 W. It is easily noticed that the temperature increases in proportion to the power. One can also notice that the thermal field along the direction of the beam propagation ðzÞ is very small, merely about 0.02 mm. What is interesting is that the propagation

Fig. 3. The temperature rise versus z coordinate when the incident laser power is 2 W ðx ¼ y ¼ 0Þ.

140 120 100 dT [K]

K? ¼

250

dT [K]

We consider a system consisting of equal parts of Al and Ag: l=2 ¼ l 1 ¼ l 2 ¼ 15 mm and A ¼ A1 ¼ A2 ¼ 4 mm2 . We have the following dependence of the thermal conductivity: for Al [22]: KðtÞ ¼ 1:692 þ 2:5  103  T  4:3  106  T 2 þ 2  109  T 3 and for Ag: KðTÞ ¼ 4:5 expð1:9 times104  TÞ, where KðTÞ is expressed in Wcm1 K1 (these formulas are valid for Nd:YAG laser). From the bulk thermal conductivity, one can calculate K ? and K k by using the relations [4]

80 60 40 20 0.02

0.04

0.06

0.08

0.1

z [mm] 2.1

K [W/cmK]

Fig. 4. The temperature rise versus z coordinate when the incident laser power is 1 W ðx ¼ y ¼ 0Þ.

2.05 dT [K]

200

400

600

800

1000

250

dT [K]

1.95

200

1.9 150

Fig. 1. The bulk thermal conductivity for Al. 100

4.4

50

4.3 -0.1

K [W/cmK]

4.2

0.05

0.1

x [mm]

4.1

Fig. 5. The temperature rise versus x coordinate when the incident laser power is 2 W ðy ¼ z ¼ 0Þ.

200 3.9

-0.05

400

600

800

dT [K]

3.8

Fig. 2. The bulk thermal conductivity for Ag.

1000

of thermal field along z is still very small if we increase the incident power, but we have a significant increase in temperature for the first layer. Fig. 5 shows the temperature rise along x direction, when the incident power is 2 W. One can easily notice that the

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References 0.25

dT [K]

0.2 0.15 0.1 0.05

0.02

0.04 y [mm]

0.06

0.08

Fig. 6. The temperature rise versus y coordinate when the incident laser power is 2 W ðx ¼ z ¼ 0Þ.

maximum temperature rise is in the center of the sample, where the intensity of the laser beam is zero (in Figs. 1–6 it was supposed that the laser beam is acting in the TEM01 mode). The same behavior was noticed in the laser–solid interaction which is shown in Ref. [17]. Fig. 6 also shows the same temperature rise but along y axis. We can notice that the temperature increases to the same value as shown in Fig. 5. 4. Conclusions We have studied the bulk properties of Al, Ag and Au during laser irradiation in a different paper [23]. In this paper, a model has been derived to analyze the temperature rise by a cw laser beam which acts in a transverse mode ðm; nÞ or in a few decoupled modes for multilayer structure. The crux of the present paper is Eq. (7) which transforms the transverse mode into a Gaussian one. Our results are in concordance with the simulations from Refs. [4] to [21], where the authors considered substrate materials like Si, GaAs, and Mn–Zn ferrite and a Gaussian laser beam, especially we consider metals like Al and Ag, whose thermal conductivity is almost constant with temperature. In conclusion, the main achievement of the present paper is that we have obtained the exact analytical expression (formulas 9 and 10) of the temperature rise, when the laser beam is propagating in a TEMmn mode or in a sum of few decoupled modes.

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