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Semi-analytical simulation for temperature of material irradiated by CW laser considering the effect of atmospheric thermal blooming
Accepted Manuscript Title: Semi-analytical simulation for temperature of material irradiated by CW laser considering the effect of atmospheric thermal blooming Author: Guibo Chen Juan Bi PII: DOI: Reference:
S0030-4026(16)31257-8 http://dx.doi.org/doi:10.1016/j.ijleo.2016.10.071 IJLEO 58339
To appear in: Received date: Accepted date:
22-8-2016 24-10-2016
Please cite this article as: Guibo Chen, Juan Bi, Semi-analytical simulation for temperature of material irradiated by CW laser considering the effect of atmospheric thermal blooming, Optik - International Journal for Light and Electron Optics http://dx.doi.org/10.1016/j.ijleo.2016.10.071 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Semi-analytical simulation for temperature of material irradiated by CW laser considering the effect of atmospheric thermal blooming Guibo Chen Juan Bi* [email protected] School of Science, ChangChun University of Science and Technology, ChangChun 130022, PR China *Corresponding author
Abstract In this paper, the temperature of material irradiated by continuous wave (CW) laser considering the effect of atmospheric thermal blooming is simulated and analyzed. By using the analytical solution of the atmosphere thermal blooming for axial symmetry, the heat conduction model of CW laser irradiating surface of material after the atmosphere thermal blooming is established. Separation of variables is used to solve the heat conduction equation and its semi-analytical solution is obtained. We studied the effect of laser propagation distance and the intensity of undistorted laser beam on the thermal blooming, and on the temperature distribution of the material. The results show that the greater the laser propagation distance in the atmosphere, the more obvious the effect of the thermal blooming, which leads to the greater distortion of the temperature distribution of the material.
Keywords: Thermal blooming; Laser irradiation; Temperature; Semi-analytical solution
1 Introduction Mathematical modeling is an important method to find physical laws and characteristics of the laser matter interaction. It is able to reduce the experimental cost and minimizes the experimentation time [1,2]. Moreover, modeling studies can provide results of laser heating for sufficient conditions even if in the environment that traditional experiment cannot achieve. The simulation of physics problems can be divided into numerical modeling and analytical modeling. Analytical modeling establishes a direct functional relation between the parameters and the laser heating process, which can provide very useful information for revealing mechanism of the irradiation effects and parameters optimization of the laser [3-10]. Considerable research studies were carried out to solve the laser heating using analytical modeling. The absorption mechanism of the material on the laser energy is studied and 1D analytical solution to the temperature rise problem of the laser heating is given by Ready[11]. El-Adawi (1986, 1995) studied on the 1D analytical solutions of heat conduction in homogeneous material and two layered material respectively[12,13]. A closed form solution for temperature rise inside solid substrate due to time exponentially varying laser pulse was obtained using Laplace transform by Yilbas [14]. Gospavic (2004) studied the 2D axisymmetric analytical solution of the thermal effect problem of laser irradiation[8]. Yilbas (2012) introduced analytical simulation methods and the related results of laser heating in the application field systematically [1]. At present, most of studies on laser heating material are to ignore the influence of atmosphere. In many cases, the laser is propagating in the atmosphere. Laser propagation in the atmosphere is a very complicated physical process, especially the thermal blooming induced by laser heating air. When the laser propagates in the atmosphere, the nonlinear effects will cause the distortion of the laser beam, which will lead to the thermal blooming[15-21]. Thermal blooming can affect the quality of
the laser beam, which leads to the instability of the laser propagation, and affects the interaction between the laser and the material. Thermal blooming is very important for the study of laser propagation in the atmosphere, and the effective application of high energy laser. At present, the research on the thermal blooming is mainly focused on the thermal blooming effects and phase compensation, and few studies have aimed at the interaction between the distorted laser and material. In this paper, a CW laser is used to study the thermal blooming effect and the distribution of temperature in the material of the distorted laser irradiating. It is necessary to point out that only the axisymmetric thermal blooming of no crosswind is considered in this paper. The thermal blooming considering crosswind will be introduced in another paper.
2 Theoretical model 2.1 Thermal blooming model We assume that D is the laser propagation distance in the atmosphere and x,y are the variables in the transverse plane. From the reference [15] and [16], the scalar wave equation in parabolic approximation is, n 2 x, y, D, t 2 2ik 0 k0 1 0 z n02 2 T
where, n is refractive index of atmosphere, T2
(1)
2 2 , n0 is static refractive x 2 y 2
index, is the field amplitude. The structural equation can be written as: n2 1 2 n0 1 1 2 n0 0
(2)
where, 0 is the density of the atmosphere with no distorting, and 1 is the change of the density of atmosphere and it yields, 1 1 2 I p t cs
(3)
where,
cp cv
, c p and cv is specific heat capacity at constant pressure and constant
volume respectively, cs2
p is the sound velocity, p is the atmospheric pressure,
and is the laser absorption in atmosphere. When the laser is a collimated beam, the diffraction effect can be neglected due to the large Rayleigh distance. Form the reference [22], the expression of the thermal blooming laser intensity with axial symmetry can be written as:
I p r , D I 0 e D e r
2
/ a2
exp N D g1 D 2e r
2
/ a2
1
where, a is the waist radius of Gaussian beam, g1 D
(4) 2 1 eD 1 , and N D D D
is the distortion parameter and it yields, dn D 2 I 0 ND dT 2k 0
where,
(5)
dn is the temperature change ratio of the refractive index of the atmosphere, dT
and k 0 is the wave number of free space. The distortion parameter N D is the key parameter to describe the intensity of thermal blooming. We will discuss the effect of distortion parameter on thermal blooming in the third part of this paper. 2.2 Laser heating model It is assumed that the intensity of thermal blooming laser beam on the surface of the material can be described as: I r, D, t I r , Dut
(6)
where, u t is the Step function, t is the laser irradiation time, and I r , D is spatial distribution function of thermal blooming laser intensity. When the laser irradiates the surface of material, laser energy was absorbed to lead to rise of surface temperature, and the internal temperature of the material is increased by heat conduction with the surface of material. Assumed laser irradiates at the center of the material surface as shown in Fig.2, we can establish
a two-dimensional axisymmetric physical problem. It only consider that in the case of conduction limited heating process and the material remains in solid phase, the governing equation of temperature can be written as, 2T r , z, t 1 T r , z, t 2T r , z, t T r , z, t mc k 2 t r r z 2 r
(7)
T r, z, t t 0 T0
(8)
k
(9)
T r , z, t AI r , D, t z z 0
where m is the material density, c is the heat capacity, k is the thermal conductivity, and A is the absorption coefficient of the surface to the laser. It is assumed that the boundary condition is adiabatic as
k
T r, z, t T r, z, t k 0 r z rR z h
(10)
Laser beam
0
r
h GaAs material
R
z
Axis of symmetry
Fig. 1
Schematic diagram of laser irradiation material surface
2.3 Semi-analytical solution method An analytical method is used to solve above problems of temperature rise. Assuming that the time function of laser interaction is Dirac induction (t ) , the temperature rise of the material with respect to the initial moment is Gr , z, t , then Gr, z, t is equivalent to the Green function and it satisfies the equation that,
mc
2Gr , z, t 1 Gr , z, t 2Gr , z, t Gr , z, t k t r 2 r r z 2
The initial condition of equation (11) is G t 0 0 . And the boundary conditions of equation (11) are
(11)
(12)
k
G z
AI r , D (t ) ， z 0
G r
rR
G z
0
(13)
z h
The temperature rise distribution can be describe by convolution integral for arbitrary time dependences of laser beam intensity, t
T (r , z, t ) u (t t ' ) G(r , z, t ' )dt '
(14)
0
To solve equation （11）to（13）, we introduce the method of separation of variables, and G(r , z, t ) can be written as, G(r, z, t ) Gr (r )Gz ( z, t )
(15)
Then equations of Gr and Gz can be obtained from equation （11）and（15）:
d 2Gr (r ) 1 dGr (r ) (16) 2Gr (r ) 0 dr 2 r dr 1 Gz ( z, t ) 2Gz ( z, t ) (17) 2 Gz ( z , t ) 2 t z where k c is the thermal diffusivity. Equation (16) is zero-order Bessel equation, its solution is zero order Bessel function J 0 ( r ) because of its limited value at r=0. Using the boundary condition
G r
0 , one can obtain r R
(18) J 1 ( R) 0 Equation (17) has infinite number of positive solutions n (n 1,2,) because the first order Bessel function has infinite number of zeros, and every solution n satisfies the equation (17). Using Laplace transform, equation (17) in Laplace domain can be written as: * 2Gzn ( z, s) 2 s * n Gzn ( z, s) 0 z 2 The solution of the equation (19) is given by:
* Gzn ( z, s) B1ein z / h B2 e in z / h
where
n2
condition
s
(19)
(20)
in . According to Laplace transform of the boundary h
G 0 , one can obtain, z z h
B2 B1e2in * Then the expression of Gzn ( z, s) can be written as:
(21)
* Gzn ( z, s) B1 ein z / h ein 2 z / h
(22)
In the Laplace domain, G* (r , z, s) and the boundary condition (13) can be written as,
* G* (r , z, s) anGzn ( z, s) J 0 ( n r )
(23)
n1
k
G * (r , z, s)
AI r , D
z
(24)
z 0
Substituting the equation (23) into (24), and using the Bessel function expanding formula of right hand of equation (24), one can obtain that A * a G ( z , s ) J ( r ) Cn J 0 ( n r ) n zn 0 n z n1 k n1 where, Cn satisfies that
rI r, D J ( r )dr 2 rI r, D J ( r )dr R J R J R rJ ( r )dr R
Cn
(25)
R
0
0
R
2 0
0
n
0
0 2
2 0
n
n
n
2 1
n
(26)
n
Substituting the equation (22) into (25), one can obtain: an B1
AC n h
(27)
ikn e 2in 1
Substituting the equation (27) into (20), expression of G* (r , z, s) can be obtained as:
* G * (r , z, s) an Gzn ( z, s) J 0 ( n r ) n 1
AC n J 0 ( n r ) h e in z / h ein 2 z / h k in e 2in 1 n 1 Using the Euler's formula, equation (28) can be simplified as:
(28)
* G * (r , z , s ) an Gzn ( z, s) J 0 ( n r ) n 1
AC n J 0 ( n r ) cos n 1 z/h i n k n 1 sin n h Using the inverse Laplace transform, equation (29) can be written as:
2AI 0 Cn J 0 ( n r ) nm cosnm 1 z/h exp snmt sin nm nm cos nm kh n 1 m 1
(29)
G(r , z, t )
(30)
where, nm m 1,2,3, are the roots of transcendent equation expressed by
n sin n 0 , they are also the poles of functions (29). In equation (30), snm can be
written as
snm
2 nm 2 (31) n h Substitute the expression of Gr, z, t into the equation (14), the temperature
rise generated by laser interaction can be expressed by the following equations: T (r , z, t )
2AI 0 kh
C J n 1 m 1
n
0
( n r )
nm cosnm 1 z/h 1 e s snm sin nm nm cos nm
nm
t
(32)
3 Results and discussions 3.1 Thermal blooming results We assume that the laser is a collimated Gaussian beam and its waist radius is 5cm, and the power density of undistorted laser is 30000 W/m2. In the expression (4) and (5), we assume that 105 m1 ,
dn 10 6 K 1 and k0 0.0267m 1 . dT
The intensity of the laser beam for different propagation distances in the atmosphere is calculated, as shown in Fig.2. We can see that the laser transmission distance in the atmosphere has a significant effect on the laser thermal blooming. The greater the propagation distance of laser in atmosphere, the stronger the laser thermal blooming effect. When the D is equal to zero, the laser intensity distribution is in the shape of Gauss. With the increase of the laser propagation distance, the beam distortion becomes more and more obvious. When the distance is 500m, the distribution of the beam is no longer at the center point. We can explain this by formula (5). Distortion parameters for D=0,100,200,300,400,500m is 0.056, 0.22, 0.5, 0.9, 1.4 respectively. And the larger the distortion parameter, the stronger the thermal blooming effect is.
4
D=0 D=100m D=200m D=300m D=400m D=500m
3.0x10
4
2.5x10
4
I/ W/m
2
2.0x10
4
1.5x10
4
1.0x10
3
5.0x10
0.0 0
2
4
6
8
10
r/cm Fig. 2 Laser intensity for different propagation distance in the atmosphere
Next we assume that the laser propagation in the atmosphere is 200m and remains constant, laser intensity after the action of the atmosphere for different undistorted laser is calculated, as shown in Fig.3. We can see that the undistorted laser has a significant effect on the atmosphere thermal blooming. The greater the undistorted laser, the stronger the thermal blooming effect. When the intensity undistorted laser is 200000W/m2, the maximal distribution of the beam is no longer at the center point. Similar to above, we can explain this by formula (5). Distortion parameters for I0=20000, 40000, 60000, 80000, 100000, 150000, 200000 W/m2 is 0.15, 0.3, 0.5, 0.6, 0.75, 1.1236, 1.5 respectively. And the larger the undistorted laser, the stronger the thermal blooming effect is.
5
1.2x10
2
I0=20000 W/m
2
I0=40000 W/m
5
1.0x10
2
I0=60000 W/m
2
I0=150000 W/m
2
I0=200000 W/m
2
2
I0=80000 W/m
4
8.0x10
4
I/ W/m
2
I0=100000 W/m
6.0x10
4
4.0x10
4
2.0x10
0.0 0
2
4
6
8
10
r/cm Fig. 3 Laser intensity for different undistorted laser
3.2 Temperature results In this part, we will calculate temperature in solid GaAs induced by thermal blooming CW laser using above analytical solutions. GaAs is widely used in the optoelectronic devices because of its wide band gap and direct band gap. When the optoelectronic devices work under the intense laser environment, they often face performance degradation and failure even induced damage due to laser heating. So it is important to study the interaction of laser with GaAs and its heating mechanisms for destruction and protection of the optoelectronic devices. The thermo-physical and geometric parameters of GaAs bulk are given in Table 1. We assume that the laser spot radius on the material surface is 1mm and the initial temperature is 300K. All of the following calculated examples are using computational parameters in Table 1. Table 1
Thermo-physical and geometric parameters of GaAs
Parameters
Values 3
density /(kg/m ) specific heat /(J/kg·K) coefficient of heat conductivity /(W/m·K) optical absorptivity thickness of material /cm radius of material /cm
5320 318 42.5 0.61 1 10
In order to validate the above semi-analytical solution, we calculate some examples using the finite element method (FEM) of our published papers [2] and the semi-analytical algorithm of this paper. The results of two methods are in good agreement, so as to verify the validity of the semi-analytical solution. Due to limited space, we have not given the specific calculation results. We assume that the laser is a collimated Gaussian beam and its waist radius is 5cm, and the power density of undistorted laser is 30000 W/m2. First, the time of laser irradiation is 10s. The relative temperature T /T0 of different transmission distances is calculated, as shown in Fig.4. It can be seen from the Fig.4 and Fig.2 that the distribution curves of the relative temperature are similar to that of the laser intensity distribution curves. When there is no atmospheric effect, the curve is close to the Gaussian distribution. With the increase of the laser propagation distance in the atmosphere, the distortion of the relative temperature distribution becomes more and more strong. This is due to the influence of laser propagation distance on the thermal blooming, which affects the laser intensity distribution at the material surface, and leads to the distortion of the relative temperature distribution of the material.
1.08
D=0 D=100m D=200m D=300m D=400m D=500m
T/T0
1.06
1.04
1.02
1.00
0
2
4
6
8
10
r/cm
Fig. 4
Relative temperature T /T0 for different propagation distance in the atmosphere（laser irradiation time is 10s）
We consider the laser propagation distance (D=500m) of the most obvious thermal blooming effect, the temperature at different positions in the z direction of the material is calculated, as shown in Fig.5. We can see that the temperature difference is not obvious in the thickness direction of the material, this is mainly because the skin depth of GaAs is very small, and the temperature rise is mainly from the action of heat conduction in the thickness direction of the material.
1.05
z=0 z=2mm z=5mm z=10mm
1.04
T/T0
1.03
1.02
1.01
1.00
0
2
4
6
8
10
r/cm
Fig. 5 Relative temperature T /T0 for different positions in the z direction（laser irradiation time is10s, laser propagation distance is 500m）
Next, we assume that the time of laser irradiation is 60s. The relative temperature T /T0 of different transmission distances is calculated, as shown in Fig.6. It can be
seen from Fig.6 that, with the increase of the laser irradiation time, the material surface temperature rise gradually increased. The thermal blooming effect is also enhanced with the increase of the laser propagation distance in the atmosphere, but it is not as obvious as the thermal blooming effect of 10s as shown in Fig.4. This is due to the increase in the time of laser irradiation, the heat conduction at high temperature to low temperature gradually increased, thus reducing the temperature difference between the highest point and the lowest point.
1.30 z=0 z=100m z=200m z=300m z=400m z=500m
1.25
T/T0
1.20 1.15 1.10 1.05 1.00
0
2
4
6
8
10
r/cm
Fig. 6 Relative temperature T /T0 for different propagation distance in the atmosphere（laser irradiation time is 60s）
We also consider the laser propagation distance (D=500m) of the most obvious thermal blooming effect when laser irradiation time is 60s, the temperature at different positions in the z direction of the material is calculated, as shown in Fig.7. We can see that the temperature difference is not obvious in the thickness direction of the material, this is also because of that the temperature rise is mainly from the action of heat conduction in the thickness direction of the material.
1.20
z=0 z=2mm z=5mm z=10mm
1.18 1.16
T/T0
1.14 1.12 1.10 1.08 1.06
0
2
4
6
8
10
r/cm
Fig. 7 Relative temperature T /T0 for different positions in the z direction（laser irradiation
time is 60s, laser propagation distance is 500m）
In the following studies, we investigated the effect of the undistorted laser intensity on the thermal blooming and the temperature distribution when the laser propagation distance in the atmosphere is 200m. First, we assume the time of laser irradiation is 10s. The relative temperature T /T0 of different undistorted laser intensity is calculated, as shown in Fig.8.
It can be seen that, the distortion of the
relative temperature distribution becomes more and more strong with the increase of the undistorted laser intensity, this is due to the influence of undistorted laser intensity on the thermal blooming, which affects the laser intensity distribution at the material surface, and leads to the distortion of the relative temperature distribution on the surface of material.
1.30 1.25
2
I0=20000 W/m
2
I0=80000 W/m
I0=40000 W/m
2
I0=100000 W/m
I0=60000 W/m
2
I0=150000 W/m
2 2 2
I0=200000 W/m
T/T0
1.20 1.15 1.10 1.05 1.00 0
2
4
6
8
10
r/cm
Fig.8 Relative temperature T /T0 for different undistorted laser intensity（laser irradiation time is 10s）
We consider the undistorted laser intensity ( I 0 200000W/m 2 ) of the most obvious thermal blooming effect, the temperature at different positions in the z direction of the material is calculated, as shown in Fig.9. Similar to Fig.5, we can see that the temperature difference is not obvious in the thickness direction of the material, this is mainly because the skin depth of GaAs is very small, and the temperature rise
is mainly from the action of heat conduction in the thickness direction of the material.
z=0 z=2mm z=5mm z=10mm
1.28 1.24
T/T0
1.20 1.16 1.12 1.08 1.04
0
2
4
6
8
10
r/cm
Fig.9 Relative temperature T /T0 for different positions in the z direction（laser irradiation time 2
is 10s, the undistorted laser intensity is 200000W/m ）
Next, we assume that the time of laser irradiation is 60s. The relative temperature T /T0 for different undistorted laser intensity is calculated, as shown in Fig.10. It can
be seen that, with the increase of the laser irradiation time, the material surface temperature rise gradually increased. With the increase of the undistorted laser intensity, the thermal blooming effect is also enhanced, but it is not as obvious as the thermal blooming effect of 10s as shown in Fig.8. This is due to the increase in the time of laser irradiation, the heat conduction at high temperature to low temperature gradually increased, thus reducing the temperature difference between the highest point and the lowest point.
2
I0=100000 W/m
2.2
2
2
I0=150000 W/m
2
I0=200000 W/m
I0=20000 W/m
2.0
2
I0=40000 W/m 2
1.8
I0=60000 W/m
T/T0
2
I0=80000 W/m
1.6 1.4 1.2 1.0
0
2
4
6
8
10
r/cm
Fig.10
Relative temperature T /T0 for different undistorted laser intensity（laser irradiation time is 60s）
We also consider the laser propagation distance (D=500m) of the most obvious thermal blooming effect when laser irradiation time is 60s, the temperature at different positions in the z direction of the material is calculated, as shown in Fig.7. We can see that the temperature difference is not obvious in the thickness direction of the material, this is also because of that the temperature rise is mainly from the action of heat conduction in the thickness direction of the material. We consider the undistorted laser intensity ( I 0 200000W/m 2 ) of the most obvious thermal blooming effect when laser irradiation time is 60s, the temperature at different positions in the z direction of the material is calculated, as shown in Fig.11. Similar to Fig.7, we can see that the temperature difference is not obvious in the thickness direction of the material, this is also because of that the temperature rise is mainly from the action of heat conduction in the thickness direction of the material.
2.2
T/T0
2.0
1.8
z=0 z=2mm z=5mm z=10mm
1.6
1.4 0
2
4
6
8
10
r/cm
Fig.11 Relative temperature T /T0 for different positions in the z direction（laser irradiation time is 60s, the undistorted laser intensity is 200000W/m 2 ）
4 Conclusions （ 1 ）Based on the analytical solution of axisymmetric thermal blooming, the semi-analytical solution of the temperature field of the material under the condition of atmosphere thermal blooming is obtained by using the method of separation of variables and Laplace transform. （2）The laser propagation distance in the atmosphere affects the laser intensity distribution at the material surface due to the thermal blooming, it leads to the distortion of the relative temperature distribution of the material. With the increase of the laser propagation distance in the atmosphere, the distortion of the relative temperature distribution becomes more and more strong. （3）The undistorted laser intensity affects the laser intensity distribution at the material surface due to the thermal blooming, it leads to the distortion of the relative temperature distribution of the material. With the increase of the undistorted laser intensity, the distortion of the relative temperature distribution becomes more and more strong.
Acknowledgement This work was supported by the CUST Young Scientists Project “Numerical modeling and inverse problems of heat transfer in laser induced materials” under Grant No. XQNJJ-2014-03.
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S0030-4026(16)31257-8 http://dx.doi.org/doi:10.1016/j.ijleo.2016.10.071 IJLEO 58339
To appear in: Received date: Accepted date:
22-8-2016 24-10-2016
Please cite this article as: Guibo Chen, Juan Bi, Semi-analytical simulation for temperature of material irradiated by CW laser considering the effect of atmospheric thermal blooming, Optik - International Journal for Light and Electron Optics http://dx.doi.org/10.1016/j.ijleo.2016.10.071 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Semi-analytical simulation for temperature of material irradiated by CW laser considering the effect of atmospheric thermal blooming Guibo Chen Juan Bi* [email protected] School of Science, ChangChun University of Science and Technology, ChangChun 130022, PR China *Corresponding author
Abstract In this paper, the temperature of material irradiated by continuous wave (CW) laser considering the effect of atmospheric thermal blooming is simulated and analyzed. By using the analytical solution of the atmosphere thermal blooming for axial symmetry, the heat conduction model of CW laser irradiating surface of material after the atmosphere thermal blooming is established. Separation of variables is used to solve the heat conduction equation and its semi-analytical solution is obtained. We studied the effect of laser propagation distance and the intensity of undistorted laser beam on the thermal blooming, and on the temperature distribution of the material. The results show that the greater the laser propagation distance in the atmosphere, the more obvious the effect of the thermal blooming, which leads to the greater distortion of the temperature distribution of the material.
Keywords: Thermal blooming; Laser irradiation; Temperature; Semi-analytical solution
1 Introduction Mathematical modeling is an important method to find physical laws and characteristics of the laser matter interaction. It is able to reduce the experimental cost and minimizes the experimentation time [1,2]. Moreover, modeling studies can provide results of laser heating for sufficient conditions even if in the environment that traditional experiment cannot achieve. The simulation of physics problems can be divided into numerical modeling and analytical modeling. Analytical modeling establishes a direct functional relation between the parameters and the laser heating process, which can provide very useful information for revealing mechanism of the irradiation effects and parameters optimization of the laser [3-10]. Considerable research studies were carried out to solve the laser heating using analytical modeling. The absorption mechanism of the material on the laser energy is studied and 1D analytical solution to the temperature rise problem of the laser heating is given by Ready[11]. El-Adawi (1986, 1995) studied on the 1D analytical solutions of heat conduction in homogeneous material and two layered material respectively[12,13]. A closed form solution for temperature rise inside solid substrate due to time exponentially varying laser pulse was obtained using Laplace transform by Yilbas [14]. Gospavic (2004) studied the 2D axisymmetric analytical solution of the thermal effect problem of laser irradiation[8]. Yilbas (2012) introduced analytical simulation methods and the related results of laser heating in the application field systematically [1]. At present, most of studies on laser heating material are to ignore the influence of atmosphere. In many cases, the laser is propagating in the atmosphere. Laser propagation in the atmosphere is a very complicated physical process, especially the thermal blooming induced by laser heating air. When the laser propagates in the atmosphere, the nonlinear effects will cause the distortion of the laser beam, which will lead to the thermal blooming[15-21]. Thermal blooming can affect the quality of
the laser beam, which leads to the instability of the laser propagation, and affects the interaction between the laser and the material. Thermal blooming is very important for the study of laser propagation in the atmosphere, and the effective application of high energy laser. At present, the research on the thermal blooming is mainly focused on the thermal blooming effects and phase compensation, and few studies have aimed at the interaction between the distorted laser and material. In this paper, a CW laser is used to study the thermal blooming effect and the distribution of temperature in the material of the distorted laser irradiating. It is necessary to point out that only the axisymmetric thermal blooming of no crosswind is considered in this paper. The thermal blooming considering crosswind will be introduced in another paper.
2 Theoretical model 2.1 Thermal blooming model We assume that D is the laser propagation distance in the atmosphere and x,y are the variables in the transverse plane. From the reference [15] and [16], the scalar wave equation in parabolic approximation is, n 2 x, y, D, t 2 2ik 0 k0 1 0 z n02 2 T
where, n is refractive index of atmosphere, T2
(1)
2 2 , n0 is static refractive x 2 y 2
index, is the field amplitude. The structural equation can be written as: n2 1 2 n0 1 1 2 n0 0
(2)
where, 0 is the density of the atmosphere with no distorting, and 1 is the change of the density of atmosphere and it yields, 1 1 2 I p t cs
(3)
where,
cp cv
, c p and cv is specific heat capacity at constant pressure and constant
volume respectively, cs2
p is the sound velocity, p is the atmospheric pressure,
and is the laser absorption in atmosphere. When the laser is a collimated beam, the diffraction effect can be neglected due to the large Rayleigh distance. Form the reference [22], the expression of the thermal blooming laser intensity with axial symmetry can be written as:
I p r , D I 0 e D e r
2
/ a2
exp N D g1 D 2e r
2
/ a2
1
where, a is the waist radius of Gaussian beam, g1 D
(4) 2 1 eD 1 , and N D D D
is the distortion parameter and it yields, dn D 2 I 0 ND dT 2k 0
where,
(5)
dn is the temperature change ratio of the refractive index of the atmosphere, dT
and k 0 is the wave number of free space. The distortion parameter N D is the key parameter to describe the intensity of thermal blooming. We will discuss the effect of distortion parameter on thermal blooming in the third part of this paper. 2.2 Laser heating model It is assumed that the intensity of thermal blooming laser beam on the surface of the material can be described as: I r, D, t I r , Dut
(6)
where, u t is the Step function, t is the laser irradiation time, and I r , D is spatial distribution function of thermal blooming laser intensity. When the laser irradiates the surface of material, laser energy was absorbed to lead to rise of surface temperature, and the internal temperature of the material is increased by heat conduction with the surface of material. Assumed laser irradiates at the center of the material surface as shown in Fig.2, we can establish
a two-dimensional axisymmetric physical problem. It only consider that in the case of conduction limited heating process and the material remains in solid phase, the governing equation of temperature can be written as, 2T r , z, t 1 T r , z, t 2T r , z, t T r , z, t mc k 2 t r r z 2 r
(7)
T r, z, t t 0 T0
(8)
k
(9)
T r , z, t AI r , D, t z z 0
where m is the material density, c is the heat capacity, k is the thermal conductivity, and A is the absorption coefficient of the surface to the laser. It is assumed that the boundary condition is adiabatic as
k
T r, z, t T r, z, t k 0 r z rR z h
(10)
Laser beam
0
r
h GaAs material
R
z
Axis of symmetry
Fig. 1
Schematic diagram of laser irradiation material surface
2.3 Semi-analytical solution method An analytical method is used to solve above problems of temperature rise. Assuming that the time function of laser interaction is Dirac induction (t ) , the temperature rise of the material with respect to the initial moment is Gr , z, t , then Gr, z, t is equivalent to the Green function and it satisfies the equation that,
mc
2Gr , z, t 1 Gr , z, t 2Gr , z, t Gr , z, t k t r 2 r r z 2
The initial condition of equation (11) is G t 0 0 . And the boundary conditions of equation (11) are
(11)
(12)
k
G z
AI r , D (t ) ， z 0
G r
rR
G z
0
(13)
z h
The temperature rise distribution can be describe by convolution integral for arbitrary time dependences of laser beam intensity, t
T (r , z, t ) u (t t ' ) G(r , z, t ' )dt '
(14)
0
To solve equation （11）to（13）, we introduce the method of separation of variables, and G(r , z, t ) can be written as, G(r, z, t ) Gr (r )Gz ( z, t )
(15)
Then equations of Gr and Gz can be obtained from equation （11）and（15）:
d 2Gr (r ) 1 dGr (r ) (16) 2Gr (r ) 0 dr 2 r dr 1 Gz ( z, t ) 2Gz ( z, t ) (17) 2 Gz ( z , t ) 2 t z where k c is the thermal diffusivity. Equation (16) is zero-order Bessel equation, its solution is zero order Bessel function J 0 ( r ) because of its limited value at r=0. Using the boundary condition
G r
0 , one can obtain r R
(18) J 1 ( R) 0 Equation (17) has infinite number of positive solutions n (n 1,2,) because the first order Bessel function has infinite number of zeros, and every solution n satisfies the equation (17). Using Laplace transform, equation (17) in Laplace domain can be written as: * 2Gzn ( z, s) 2 s * n Gzn ( z, s) 0 z 2 The solution of the equation (19) is given by:
* Gzn ( z, s) B1ein z / h B2 e in z / h
where
n2
condition
s
(19)
(20)
in . According to Laplace transform of the boundary h
G 0 , one can obtain, z z h
B2 B1e2in * Then the expression of Gzn ( z, s) can be written as:
(21)
* Gzn ( z, s) B1 ein z / h ein 2 z / h
(22)
In the Laplace domain, G* (r , z, s) and the boundary condition (13) can be written as,
* G* (r , z, s) anGzn ( z, s) J 0 ( n r )
(23)
n1
k
G * (r , z, s)
AI r , D
z
(24)
z 0
Substituting the equation (23) into (24), and using the Bessel function expanding formula of right hand of equation (24), one can obtain that A * a G ( z , s ) J ( r ) Cn J 0 ( n r ) n zn 0 n z n1 k n1 where, Cn satisfies that
rI r, D J ( r )dr 2 rI r, D J ( r )dr R J R J R rJ ( r )dr R
Cn
(25)
R
0
0
R
2 0
0
n
0
0 2
2 0
n
n
n
2 1
n
(26)
n
Substituting the equation (22) into (25), one can obtain: an B1
AC n h
(27)
ikn e 2in 1
Substituting the equation (27) into (20), expression of G* (r , z, s) can be obtained as:
* G * (r , z, s) an Gzn ( z, s) J 0 ( n r ) n 1
AC n J 0 ( n r ) h e in z / h ein 2 z / h k in e 2in 1 n 1 Using the Euler's formula, equation (28) can be simplified as:
(28)
* G * (r , z , s ) an Gzn ( z, s) J 0 ( n r ) n 1
AC n J 0 ( n r ) cos n 1 z/h i n k n 1 sin n h Using the inverse Laplace transform, equation (29) can be written as:
2AI 0 Cn J 0 ( n r ) nm cosnm 1 z/h exp snmt sin nm nm cos nm kh n 1 m 1
(29)
G(r , z, t )
(30)
where, nm m 1,2,3, are the roots of transcendent equation expressed by
n sin n 0 , they are also the poles of functions (29). In equation (30), snm can be
written as
snm
2 nm 2 (31) n h Substitute the expression of Gr, z, t into the equation (14), the temperature
rise generated by laser interaction can be expressed by the following equations: T (r , z, t )
2AI 0 kh
C J n 1 m 1
n
0
( n r )
nm cosnm 1 z/h 1 e s snm sin nm nm cos nm
nm
t
(32)
3 Results and discussions 3.1 Thermal blooming results We assume that the laser is a collimated Gaussian beam and its waist radius is 5cm, and the power density of undistorted laser is 30000 W/m2. In the expression (4) and (5), we assume that 105 m1 ,
dn 10 6 K 1 and k0 0.0267m 1 . dT
The intensity of the laser beam for different propagation distances in the atmosphere is calculated, as shown in Fig.2. We can see that the laser transmission distance in the atmosphere has a significant effect on the laser thermal blooming. The greater the propagation distance of laser in atmosphere, the stronger the laser thermal blooming effect. When the D is equal to zero, the laser intensity distribution is in the shape of Gauss. With the increase of the laser propagation distance, the beam distortion becomes more and more obvious. When the distance is 500m, the distribution of the beam is no longer at the center point. We can explain this by formula (5). Distortion parameters for D=0,100,200,300,400,500m is 0.056, 0.22, 0.5, 0.9, 1.4 respectively. And the larger the distortion parameter, the stronger the thermal blooming effect is.
4
D=0 D=100m D=200m D=300m D=400m D=500m
3.0x10
4
2.5x10
4
I/ W/m
2
2.0x10
4
1.5x10
4
1.0x10
3
5.0x10
0.0 0
2
4
6
8
10
r/cm Fig. 2 Laser intensity for different propagation distance in the atmosphere
Next we assume that the laser propagation in the atmosphere is 200m and remains constant, laser intensity after the action of the atmosphere for different undistorted laser is calculated, as shown in Fig.3. We can see that the undistorted laser has a significant effect on the atmosphere thermal blooming. The greater the undistorted laser, the stronger the thermal blooming effect. When the intensity undistorted laser is 200000W/m2, the maximal distribution of the beam is no longer at the center point. Similar to above, we can explain this by formula (5). Distortion parameters for I0=20000, 40000, 60000, 80000, 100000, 150000, 200000 W/m2 is 0.15, 0.3, 0.5, 0.6, 0.75, 1.1236, 1.5 respectively. And the larger the undistorted laser, the stronger the thermal blooming effect is.
5
1.2x10
2
I0=20000 W/m
2
I0=40000 W/m
5
1.0x10
2
I0=60000 W/m
2
I0=150000 W/m
2
I0=200000 W/m
2
2
I0=80000 W/m
4
8.0x10
4
I/ W/m
2
I0=100000 W/m
6.0x10
4
4.0x10
4
2.0x10
0.0 0
2
4
6
8
10
r/cm Fig. 3 Laser intensity for different undistorted laser
3.2 Temperature results In this part, we will calculate temperature in solid GaAs induced by thermal blooming CW laser using above analytical solutions. GaAs is widely used in the optoelectronic devices because of its wide band gap and direct band gap. When the optoelectronic devices work under the intense laser environment, they often face performance degradation and failure even induced damage due to laser heating. So it is important to study the interaction of laser with GaAs and its heating mechanisms for destruction and protection of the optoelectronic devices. The thermo-physical and geometric parameters of GaAs bulk are given in Table 1. We assume that the laser spot radius on the material surface is 1mm and the initial temperature is 300K. All of the following calculated examples are using computational parameters in Table 1. Table 1
Thermo-physical and geometric parameters of GaAs
Parameters
Values 3
density /(kg/m ) specific heat /(J/kg·K) coefficient of heat conductivity /(W/m·K) optical absorptivity thickness of material /cm radius of material /cm
5320 318 42.5 0.61 1 10
In order to validate the above semi-analytical solution, we calculate some examples using the finite element method (FEM) of our published papers [2] and the semi-analytical algorithm of this paper. The results of two methods are in good agreement, so as to verify the validity of the semi-analytical solution. Due to limited space, we have not given the specific calculation results. We assume that the laser is a collimated Gaussian beam and its waist radius is 5cm, and the power density of undistorted laser is 30000 W/m2. First, the time of laser irradiation is 10s. The relative temperature T /T0 of different transmission distances is calculated, as shown in Fig.4. It can be seen from the Fig.4 and Fig.2 that the distribution curves of the relative temperature are similar to that of the laser intensity distribution curves. When there is no atmospheric effect, the curve is close to the Gaussian distribution. With the increase of the laser propagation distance in the atmosphere, the distortion of the relative temperature distribution becomes more and more strong. This is due to the influence of laser propagation distance on the thermal blooming, which affects the laser intensity distribution at the material surface, and leads to the distortion of the relative temperature distribution of the material.
1.08
D=0 D=100m D=200m D=300m D=400m D=500m
T/T0
1.06
1.04
1.02
1.00
0
2
4
6
8
10
r/cm
Fig. 4
Relative temperature T /T0 for different propagation distance in the atmosphere（laser irradiation time is 10s）
We consider the laser propagation distance (D=500m) of the most obvious thermal blooming effect, the temperature at different positions in the z direction of the material is calculated, as shown in Fig.5. We can see that the temperature difference is not obvious in the thickness direction of the material, this is mainly because the skin depth of GaAs is very small, and the temperature rise is mainly from the action of heat conduction in the thickness direction of the material.
1.05
z=0 z=2mm z=5mm z=10mm
1.04
T/T0
1.03
1.02
1.01
1.00
0
2
4
6
8
10
r/cm
Fig. 5 Relative temperature T /T0 for different positions in the z direction（laser irradiation time is10s, laser propagation distance is 500m）
Next, we assume that the time of laser irradiation is 60s. The relative temperature T /T0 of different transmission distances is calculated, as shown in Fig.6. It can be
seen from Fig.6 that, with the increase of the laser irradiation time, the material surface temperature rise gradually increased. The thermal blooming effect is also enhanced with the increase of the laser propagation distance in the atmosphere, but it is not as obvious as the thermal blooming effect of 10s as shown in Fig.4. This is due to the increase in the time of laser irradiation, the heat conduction at high temperature to low temperature gradually increased, thus reducing the temperature difference between the highest point and the lowest point.
1.30 z=0 z=100m z=200m z=300m z=400m z=500m
1.25
T/T0
1.20 1.15 1.10 1.05 1.00
0
2
4
6
8
10
r/cm
Fig. 6 Relative temperature T /T0 for different propagation distance in the atmosphere（laser irradiation time is 60s）
We also consider the laser propagation distance (D=500m) of the most obvious thermal blooming effect when laser irradiation time is 60s, the temperature at different positions in the z direction of the material is calculated, as shown in Fig.7. We can see that the temperature difference is not obvious in the thickness direction of the material, this is also because of that the temperature rise is mainly from the action of heat conduction in the thickness direction of the material.
1.20
z=0 z=2mm z=5mm z=10mm
1.18 1.16
T/T0
1.14 1.12 1.10 1.08 1.06
0
2
4
6
8
10
r/cm
Fig. 7 Relative temperature T /T0 for different positions in the z direction（laser irradiation
time is 60s, laser propagation distance is 500m）
In the following studies, we investigated the effect of the undistorted laser intensity on the thermal blooming and the temperature distribution when the laser propagation distance in the atmosphere is 200m. First, we assume the time of laser irradiation is 10s. The relative temperature T /T0 of different undistorted laser intensity is calculated, as shown in Fig.8.
It can be seen that, the distortion of the
relative temperature distribution becomes more and more strong with the increase of the undistorted laser intensity, this is due to the influence of undistorted laser intensity on the thermal blooming, which affects the laser intensity distribution at the material surface, and leads to the distortion of the relative temperature distribution on the surface of material.
1.30 1.25
2
I0=20000 W/m
2
I0=80000 W/m
I0=40000 W/m
2
I0=100000 W/m
I0=60000 W/m
2
I0=150000 W/m
2 2 2
I0=200000 W/m
T/T0
1.20 1.15 1.10 1.05 1.00 0
2
4
6
8
10
r/cm
Fig.8 Relative temperature T /T0 for different undistorted laser intensity（laser irradiation time is 10s）
We consider the undistorted laser intensity ( I 0 200000W/m 2 ) of the most obvious thermal blooming effect, the temperature at different positions in the z direction of the material is calculated, as shown in Fig.9. Similar to Fig.5, we can see that the temperature difference is not obvious in the thickness direction of the material, this is mainly because the skin depth of GaAs is very small, and the temperature rise
is mainly from the action of heat conduction in the thickness direction of the material.
z=0 z=2mm z=5mm z=10mm
1.28 1.24
T/T0
1.20 1.16 1.12 1.08 1.04
0
2
4
6
8
10
r/cm
Fig.9 Relative temperature T /T0 for different positions in the z direction（laser irradiation time 2
is 10s, the undistorted laser intensity is 200000W/m ）
Next, we assume that the time of laser irradiation is 60s. The relative temperature T /T0 for different undistorted laser intensity is calculated, as shown in Fig.10. It can
be seen that, with the increase of the laser irradiation time, the material surface temperature rise gradually increased. With the increase of the undistorted laser intensity, the thermal blooming effect is also enhanced, but it is not as obvious as the thermal blooming effect of 10s as shown in Fig.8. This is due to the increase in the time of laser irradiation, the heat conduction at high temperature to low temperature gradually increased, thus reducing the temperature difference between the highest point and the lowest point.
2
I0=100000 W/m
2.2
2
2
I0=150000 W/m
2
I0=200000 W/m
I0=20000 W/m
2.0
2
I0=40000 W/m 2
1.8
I0=60000 W/m
T/T0
2
I0=80000 W/m
1.6 1.4 1.2 1.0
0
2
4
6
8
10
r/cm
Fig.10
Relative temperature T /T0 for different undistorted laser intensity（laser irradiation time is 60s）
We also consider the laser propagation distance (D=500m) of the most obvious thermal blooming effect when laser irradiation time is 60s, the temperature at different positions in the z direction of the material is calculated, as shown in Fig.7. We can see that the temperature difference is not obvious in the thickness direction of the material, this is also because of that the temperature rise is mainly from the action of heat conduction in the thickness direction of the material. We consider the undistorted laser intensity ( I 0 200000W/m 2 ) of the most obvious thermal blooming effect when laser irradiation time is 60s, the temperature at different positions in the z direction of the material is calculated, as shown in Fig.11. Similar to Fig.7, we can see that the temperature difference is not obvious in the thickness direction of the material, this is also because of that the temperature rise is mainly from the action of heat conduction in the thickness direction of the material.
2.2
T/T0
2.0
1.8
z=0 z=2mm z=5mm z=10mm
1.6
1.4 0
2
4
6
8
10
r/cm
Fig.11 Relative temperature T /T0 for different positions in the z direction（laser irradiation time is 60s, the undistorted laser intensity is 200000W/m 2 ）
4 Conclusions （ 1 ）Based on the analytical solution of axisymmetric thermal blooming, the semi-analytical solution of the temperature field of the material under the condition of atmosphere thermal blooming is obtained by using the method of separation of variables and Laplace transform. （2）The laser propagation distance in the atmosphere affects the laser intensity distribution at the material surface due to the thermal blooming, it leads to the distortion of the relative temperature distribution of the material. With the increase of the laser propagation distance in the atmosphere, the distortion of the relative temperature distribution becomes more and more strong. （3）The undistorted laser intensity affects the laser intensity distribution at the material surface due to the thermal blooming, it leads to the distortion of the relative temperature distribution of the material. With the increase of the undistorted laser intensity, the distortion of the relative temperature distribution becomes more and more strong.
Acknowledgement This work was supported by the CUST Young Scientists Project “Numerical modeling and inverse problems of heat transfer in laser induced materials” under Grant No. XQNJJ-2014-03.
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