American Journal of Computational Mathematics, 2012, 2, 331-335
http://dx.doi.org/10.4236/ajcm.2012.24045 Published Online December 2012 (http://www.SciRP.org/journal/ajcm)
Modelling the Cavity of Continuous Chemical Lasers
(CCLs) Using Matlab Applications
Mohammedi Ferhat, Zergui Belgacem, Bensaada Said
Laboratory Larhyss University, Biskra, Algeria
Received January 5, 2012; revised August 23, 2012; accepted September 5, 2012
ABSTRACT
The subject that concerns us in this work is the numerical simulation and optimal control of equilibrium of the continu-
ous chemical lasers (CCLs). Laser Chemistry: Spectroscopy, Dynamics and Applications are a carefully structured in-
troduction to the basic theory and concepts of this subject. In this paper we present the design and discuss the perfor-
mances of a continuous DF chemical laser, based on the exothermic reaction:

,

FD2DFvj D.
Keywords: Chemical Laser; Spectroscopy Vibrational; Rotational Processes; Lasers Cavity
1. Introduction
The rapid developments in new laser techniques and ap-
plications have extended the field of laser chemistry into
many other scientific fields, such as biology, medicine,
and environmental science, as well as into modern tech-
nological processes. This “natural” invasion is a result of
the multidisciplinary character of modern laser chemistry.
The success of simple models to simulate the kinetic be-
haviour of gas lasers has amused a lot of interest in the
numerical and theoretical study of such systems. Chemi-
cal lasers find their origin in the study [1-3] of the radia-
tion emitted from chemical reactions. In many cases such
radiation is of thermal origin. Energy levels and transi-
tions; A laser chemical or otherwise, requires a mecha-
nism to populate an excited state at a sufficiently fast rate
such that at some time point there are more molecules in
an upper (higher energy) state than in a lower. Under
these conditions the number of photons produced by
stimulated emission cexceed those absorbed, and optical
amplification or gain will result (See Figures 1 and 2).
Under equilibrium conditions the ratio [4,5] of the num-
ber of molecules in an upper and lower state are given by
the Boltzmann distribution:

exp
uu
LL
Ng EKT
Ng


(1)
This then leads to the concept of partial inversions cha-
racterized by vibrational and rotational “temperatures”.
For a diatomic molecule, in the harmonic oscillator-rigid
rotator approximation the energy levels are given by:
 
,12
er
EvJvBJ J   
The number of molecules in a vibrational state is given
by:

12
exp
V
VV
Vhc
N
NQKT
 


(3)
In any given vibrational level the rotational level
populations are given by:

221
Nn ggn 1 (3a)
2. Model Formulation
In general the pumping chemical is using a very difficult
and complexity; exigent primary cavity (chamber of re-
active) and secondary cavity (the laser chamber for reac-
tions). The mixing of flows is accompanied by the exo-
thermic reaction:
FD2DF*D; DF2DF*F


The reacting system can be presented in simplified
forms as follows process us initiation, pumping reactions,
energy exchange reactions [6,7], (see Figure 3) termina-
tion and stimulated emission. The usual one dimensional
flow equations govern the jet, wherein the energy equa-
tion includes a loss term due to lasing. The rate equations
for HF(v) have the form of radiative transport equation,
where ρ is density, u is flow velocity, and n(v)is mole-
mass ratio of HF(v). The quantity γch(v) is a summation
over all chemical reactions, including energy transfer
reactions, That affect HF(v):


d1
dch radrad
nv
uvv
x

v

 (4)

1 (2)
C
opyright © 2012 SciRes. AJCM

M. FERHAT ET AL. 332
the surfaces of caity in 3D
-1
-0.5
0
0.5
1
-1 -0.5 00.5 1
-1
-0.5
0
0.5
1
x
y
z
Figure 1. Principle processes in laser chemistry [Telle &
All].
-0.8
-0.8
-0.6
-0.6
-0.6
-0.6
-0.4
-0.4
-0.4
-0.4
-0.2
-0.2
-0.2
-0.2
-0.2
-0.2
00
0
00
0
00
0.2
0.2
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0
.8
0.8
0
.8
0.8
The level curves of reactions E(v,J) = we(V+1/2).B.J(J+1)
x
y
510 1520 25 3035 40
5
10
15
20
25
30
35
40
Figure 2. The map surfaces of cavity and curves levels.
 
chriri r
r
v  
L (5)
ri ri
rr ir i
ii
LK CKC





(6)
3. Analytical Model
Before you begin to format your paper, first write and
save the content as a separate text file. We simulate the
-2
-1
0
1
2
-2
-1
0
1
2
-1
0
1
x( J)
Evoltuion chemicals reactions exothermic i n cavity DF
y (v)
E(v,J
-2
0
2
-2
-1
0
1
2
-0.5
0
0.5
1
1.5
the map cavi ty in 3D
Figure 3. Reactions chemicals in primary and secondary
reactions cavity.
differential equations for the density of inversion of laser
populations with 3 or on 4 levels, the designations used
in this diagram are: ω the probability of transitions sti-
mulated in the channel from oscillation (channel 2-1); ωp
for the radiation of pumping in the channel (1-3); 1/τ, the
probability of spontaneous transitions in channels 1-3
and 2-3, ni(z,t) the density of population of the ith level
and by n’ the total number of active centre per unit of
volume. The equations of the assessment for the popula-

M. FERHAT ET AL. 333
tion’s n1, n2, n3 have the following equations form Ode’s
(Ordinary differential equations) [8,9];



3 12
21 13
31
3 22
21
32
333
1
32 31
3
()
p
p
n nn
nn nn
t
n nn
nn
t
nnn
n
tn





 

 



(7)
With condition
123
0ntn tn t (8)
If
123
nnn n

  (9)
Then
2
nt nt
1
(10)
3.1. ODE’s Solvers in MATLAB
A list of ODE solvers and of other routines that act on
functions is returned by help funfun, and a documenta-
tion window is opened by doc funfun. The two main rou-
tines of interest are ODE45, an explicit single-step inte-
grator, and ODE15s, an implicit multistep integrator that
works well for stiff systems. We demonstrate the use of
ODE45 and ODE15s for a simple batch reactor with the
two elementary reactions
ABC and CBD 
The following set of differential equations describes
the change in concentration three species in a tank. The
reactions occur within the tank. The con-
stants k1, and k2 describe the reaction rate for
and respectively. The following ODE’s are ob-
tained:
ABC
C
AB
B
d1
d
d12
d
d2
d
Ca KCa
t
Cb
K
CaK Cb
t
Cc KCb
t



(11)
where k1 = 1 mn–1 and k2 = 2 mn–1 and at time t = 0, Ca
= 5 mol and Cb = Cc = 0 mol. Solve the system of equa-
tions and plot the change in concentration of each species
over time (Figure 4). Select an appropriate time interval
for the integration. We wish to use a general notation
system for IVPs, and so define a state vector, x, that
completely describe the state of the system at any time
sufficiently well to predict its future behavior; The
UF6-H2 chemical laser takes advantage of the low mo-
ment of inertia of HF and the effectiveness of UF6 as a
photolytic fluorine atom source 15. Two reactions could
-5 05
-5
0
5
-1
-0.5
0
0.5
1
x
shock waves relaxation-dissociation
y
Figure 4. The cavity shock waves relaxation-dissociation.
contribute to the population inversions observed;
(v = 2→ 1 for HF and both 3 → 2 and 2 → 1 for DF).
65
UFF UFh



FH2HFFH32K calmol

 
65
HUFHFUFH46Kcalmol

 
Systematic Approach: The following reaction data has
been obtained from a simple decay reaction; ,
Use MATLAB to plot the concentration of component A
in mol/L against the reaction time, t, in minutes. Title the
plot, label the axes, and obtain elementary statistics for
the data. The idea of the simulation part was to form a
basic setup for a future test stand for chemical cavities
lasers within a numerical routine. The framework for the
numerical routine [10,11] has been MATLAB7, a nu-
merical computing environment by “The MATH-
WORKS”, (Figure 5) since it offers a large range of in-
built functions and visualization tools. In principle, the
very simulations could have been done with any other
sophisticated programming environment or language, but
the MATLAB language offered the quite comfortable
advantage of vector based variables useful for this prob-
3.2. Simulation with MATLAB
AB

M. FERHAT ET AL. 334
Figure 5. Propagations shock waves in cylindrical lasers
cavity.
lem. Cavity parameter variation: Thus, we simulated the
work of a continuous chemical laser with
simultaneous lasing on the V; , j (j = 14 -
13) and (j = 17 - 16) transitions. Sup-
pression of the lasing on vibrational and rotational transi-
tions increased the energy extraction for purely rotational
transitions Parameters of simulation cavity: Equations (7),
(8) and (11) governing physical phenomena in DF che-
mical laser cavity is to be discretized. There exist a lot of
numerical methods for solving this type of system. Time
stepping method, Boundary conditions and the resonator
is equipped with two flat mirrors of which separate dis-
tances (Fabry-Perrot).
H2 F2
1jv 1
, , 1vj vj
The usual initial conditions for the translation tem-
perature, pressure, density, and Mach number immedi-
ately behind the shock are given the Rankine-Hugoniot
[12-14] relations (Figure 6), State-resolved vibrational
kinetics are simulated by the master equation:
 

 
 
,,
2
2
d
d
,, ,,
d 1
d
mat
atiatm
iat
molijmolm n
ijn
mol mol
m
mol mol
atmol at
atm at
at molat
at mol
molm mol
at mol
fx
kimfkm if
t
kijmnffkmnijff
xx
fxt
xx
km fkmx
x
km fkmx



















  










0,,, ,
mol
mol
diss
x
ijmni m









(13)
-2 0246810 12
-2
0
2
4
6
8
10
12 reaction s vi b rationals dir ec ti ons
Figure 6. Oscillations shock waves lasers modes in cavity.
The concentration of atoms is given by the equation
 


2
1
11
1
1
2
1
d
d
21 1
21
1
at
mol at
at atat
mol at
at mol
at mol
at mol
at mol
mol
at mol
x
t
xx
kf k
xx
kf
xx
k











 






 

 





(14)
The last two terms in Equation (13), simulate the vi-
bration-dissociation coupling. Evib(T) and Evib(t) are vi-
brational energies:

11
vibv veev
EEf vxv  f

 


 (14)
where fv can either the equilibrium or a nonequilibrium
distribution function.

M. FERHAT ET AL. 335
4. Conclusion
We are now ready to treat more complex problems of
greater relevance to chemical engineering practice. We
begin with the study of initial value problems (IVPs) of
ordinary differential equations (ODEs), in which we
compute the trajectory in time of a set of N variables xi(t)
governed by the set of first-order ODEs. We start the
simulation, usually at t0 = 0, at the initial condition, x(t0)
= x[0]. Such problems arise commonly in the study of
chemical kinetics or process dynamics. While we have
interpreted above the variable of integration to be time
[15-19], it might be another variable such as a spatial
coordinate. The following curves is produced upon exe-
cution, to be able to solve higher order ODE’s in MAT-
LAB, they must be written in terms of a system of first
order ordinary differential equations. We have success-
fully designed, built and tested a in Military applications
DF generator for a chemical H-F or DF laser A laser
powered by an array of μSOG chips would be useful for
a variety of industrial applications. The comparison of
the results of numerical simulations performed with the
use of our model with the results of experimental studies.
5. Acknowledgements
This work has been supported by the Project CNEPRU
under Grant No. ID 0142009011 and the Laboratory
LARHYSS University of Biskra through Project We
should like to acknowledge the substantial research sup-
port provided by the M-E-S-R-S Office of Scientific Re-
search for the laser study conducted at the University of
Biskra represented by references._PNR-LARHYSS.
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