A Concise Linear Algebra Model for Metal Heating by Ultra-Short Laser Pulses
Benxin Wu
Abstract: The heat transfer process of metal heating by ultra-short laser pulses, where the pulse duration is
less than 1 nanosecond, is governed by parabolic two-step (PTS) heat conduction equations. In this paper,
we proposed, for the first time to our knowledge, a concise linear algebra model based on PTS heat
conduction equations for the modeling of one-dimensional metal heating by ultra-short laser pulses through
an approximate explicit finite difference method. The simulation results by this model are compared with
experimental data and a good agreement has been obtained.
The heat transfer process of metal heating by ultra-short laser pulses, where the pulse
duration is less than 1 nanosecond, is governed by parabolic two-step (PTS) heat
conduction equations, whose one-dimensional form is as follows [1]:
Te
 2Te
T
Ce
 ke
 G (Te  Tl )  S , (1); Cl l  G (Te  Tl ) , (2); S  I 0 exp( x) , (3);
2
t
t
x
Please see appendix 1 for the nomenclature. We assume, without causing obvious errors,
that the thermal properties are temperature independent. If we neglect the heat loss at the
boundary, the initial conditions and boundary conditions are:
Te ( x, t initial )  Tl ( x, t initial )  T0 ,and
Te Tl

 0 at x  0 and x  L (4)
x
x
By applying an approximate explicit finite difference method to equations (1) to (4),
we have the following finite difference equations:
Te
Te
Te
n 1
(i)  a2Te (i)  a1Te (i  1)  a1Te (i  1)  a3Tl (i)  S (i) , ( 2  i  N 0  1 )
(5)
n 1
( N 0 )  a2Te ( N 0 )  2a1Te ( N 0  1)  a3Tl ( N 0 )  S ( N 0 )
(6)
n 1
(1)  a2Te (1)  2a1Te (2)  a3Tl (1)  S (1) , (7);
a1 
n
n
n
n
n
n
n
n
n
n
Tl
n 1
(i)  a4Tl (i)  a5Te (i) , (8)
n
Gt
G t
t
k ; a3 
; a2  1  2a1  a3 ; a 5 
; a 4  1  a5 .
2 e
Ce (x)
Ce
Cl
n
(9a)
S (i )  I 0
t
[exp(  (i  1 / 2)x)  exp(  (i  1 / 2)x)] , 2  i  N 0  1 ;
C e (x)
(9b)
S (1)  I 0
2t
[1  exp( x / 2)] ,
C e (x)
(9c)
S (N0 )  I 0
2t
[exp(  ( N 0  1 / 2)x)  exp( N 0 x)]
C e (x)
(9d)
where Te (i) and Tl (i) are electron and lattice temperatures at x  (i  1)x and
n
n
t  (n  1)t , where i  1,2 N 0 , and n  1,2  , and L  ( N 0  1)x . From the finite
difference equations above, we set up a concise linear algebra model as follows:
Tenew=BTeold+a3Tlold+S (10) ;
Tlnew= a5Teold+a4Tlold
(11)
where Tenew, Teold, Tlnew, and Tlold  R N0 1 , and are electron temperatures at (n+1)th, and nth
time step, and the lattice temperatures at (n+1)th, and nth time step respectively. That is,
Tenew (i,1)  Te
n1
(i) , and Teold (i,1)  Te (i) , and Tln ew (i,1)  Tl
n
n 1
(i) , and Tlold (i,1)  Tl (i) .
n
S  R N0 1 , and it is defined by equations (9b) to (9d). The matrix B  R N0 N0 , and its
definition is as follows:
B(1,2)  B( N 0 , N 0  1)  2a1 ;
B(i, j )  a2 , when i  j ;
B(i, i  1)  B(i, i  1)  a1 , for 2  i  N 0  1 . All the other entries of B are zeros.
This linear algebra model is very concise and easy to use. A comparison of
experimental data [1] with the simulation result by this model is shown in figure 1 in
appendix 2, and the agreement is very good. See appendix 3 for the program code.
Reference:
1. T.Q. Qiu, T. Juhasz, C. Suarez, W.E. Bron, and C.L. Tien, Femtosecond Laser
Heating of Multi-Layer Metals-II Experiments, Int. J. Heat Mass Transfer 37, pp.
2799-2808 (1994).
Appendix 1 Nomenclature
a1, a2, a3, a4, a5.
Constants in finite difference equations
Ce
Volumetric heat capacity of electrons
Cl
Volumetric heat capacity of the lattice
G
Electron-phonon coupling factor
i
Grid point index in x direction
I0
Absorbed laser power density
ke
Effective electron thermal conductivity
L
Length of heated metal
n
Time step index
N0
Maximum grid point index in x direction
S
Laser heating source
t
Time
t
Temporal step size
tinitial
Pulse starting time
T0
Initial temperature
Te
Temperature of electrons
Tl
Temperature of the lattice
n
Temperature of electrons at x  (i  1)x and t  (n  1)t ,
Tl (i)
n
Temperature of the lattice at x  (i  1)x and t  (n  1)t ,
x
Spatial step size ( grid size )

Absorption coefficient
Te (i)
Appendix 2: Comparison of experimental data [1] with our linear algebra model
Our linear algebra model
Experimental data
Fig. 1 Comparison of experimental data with our linear algebra model simulation result
( Pulse duration: tp = 100 fs ; Pulse starts at t = -200 fs; Pulse energy density: 10 J/m2; L
= 100 nm. Material: gold. The normalized electron temperature change is at rear surface,
and is defined as: (Te - T0 )/(Te,max – T0). See reference 1 for materials properties and the
temporal shape of the laser pulse).
Appendix 3: MatLab Program Code
G=2.6e016;
Ce=21000;
ke=315;
Clt=2.5e006;
tp=100e-015;
tst=-2.*tp;
tend=2000e-015;
tN=44001;
deltt=(tend-tst)./(tN-1);
J=10;
afa=0.065e009;
L=0.1e-006;
N0=21;
deltx=L./(N0-1);
a1=deltt./(Ce.*deltx.*deltx).*ke;
a3=G.*deltt./Ce;
a2=1-2.*a1-a3;
a5=G.*deltt./Clt;
a4=1-a5;
for i=1:N0
for j=1:N0
B(i,j)=0;
end
end
B(1,2)=2.*a1;
B(N0,N0-1)=2.*a1;
for i=1:N0
B(i,i)=a2;
end
for i=2:N0-1
B(i,i+1)=a1;
B(i,i-1)=a1;
end
for i=1:N0
T0(i)=300;
end
Tenew=T0';
Teold=T0';
Tltnew=T0';
Tltold=T0';
Tefront(1)=300;
Terear(1)=300;
for n=2:tN
t=tst+(n-1).*deltt;
%*******************
I0=0.94.*J./tp.*exp(-2.77.*(t./tp).^2);
SS(1)=I0.*2.*deltt./(Ce.*deltx).*(1-exp(-afa.*deltx./2));
for i=2:N0-1
SS(i)=I0.*deltt./(Ce.*deltx).*( exp(-afa.*(i-0.5).*deltx)-exp(-afa.*(i+0.5).*deltx) );
end
SS(N0)=I0.*2.*deltt./(Ce.*deltx).*( exp(-afa.*(N0-0.5).*deltx)-exp(-afa.*N0.*deltx) );
S=SS';
%*******************
Tenew=B*Teold+a3.*Tltold+S;
Tltnew=a5.*Teold+a4.*Tltold;
Tefront(n)=Tenew(1,1);
Terear(n)=Tenew(N0,1);
Teold=Tenew;
Tltold=Tltnew;
end
n=1:tN;
figure(1);
plot((tst+(n-1).*deltt).*10.^12, (Tefront(n)-300)./(max(Tefront(n))-300) );
figure(2);
plot((tst+(n-1).*deltt).*10.^12, (Terear(n)-300)./(max(Terear(n))-300) );
hold on;
expt=[0.05,0.1,0.11,0.15,0.29,0.39,0.50,0.60,0.72,0.86,1,1.2,1.5,1.64];
expTe=[0.36,0.50,0.62,0.80,1.00,0.93,0.80,0.71,0.61,0.54,0.51,0.43,0.36,0.32];
for i=1:14
figure(2);
plot(expt(i)', expTe(i)','o');
hold on;
end