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Wang -xiangmin Int. Journal of Engineering Research and Applications
ISSN: 2248-9622, Vol. 5, Issue 12, (Part - 2) December 2015, pp.177-181
RESEARCH ARTICLE
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Mathematical Model for Dynamic Damage Probability of the
Repetition Pulse Rate High-energy Laser System
Wang xiangmin, wang jun
*(School of Automation,, Nanjing University of Science and Technology, Nanjing 210094， Jiangsu, China)
ABSTRACT
Aimed at the high-energy laser system, under the assumption that the tracking error is the normal stochastic
process of mean square differentiability and ergodicity, the Series Expression of the dynamic damage probability
was given. The example demonstrate some characteristics of the dynamic damage probability as follow. The
system proposed indicates the quantitative relationship between dynamic damage probability and transfer function
of the tracking error system, which offers theoretical and technological support on proofing, designing and testing
the dynamic damage probability of laser system.
Keywords: high power laser system; tracking error; stochastic passage characteristic; damage probability
I. INTRODUCTION
causing ruin time changes induced by intense laser;
With the advantages of high reliability, small
destroy target time is very short, due to temperature
size and high efficiency, the repetition rate pulsed
rise caused destruction can be approximated a
laser system, which is a representative of the diode
thermal process, namely in a limited time, causing
pumped solid state laser, has become an important
ruin time can be decomposed into multiple periods
direction of the development of high energy laser
followed by irradiation, as long as the irradiation
system. It can make the laser energy density in a
time, the damage effect will remain unchanged. In
small area within a relatively small area of the target
this paper, the dynamic damage probability is derived,
surface in a short time by the dense emission of high
and the characteristics of the dynamic damage
power laser pulse. When a stationary target is directly
probability are analyzed by a numerical example.
irradiated with a stationary target, the different
characteristics of the high energy laser in different
II. PROBLEM DESCRIPTION
environments, the damage ability of different
When using repetition rate pulsed laser device
materials, and the static damage characteristics of the
irradiates a target, it is reasonable to use circular area
high energy laser, are pointed. A lot of research
irradiated as the shooting gate. Then it is obviously
literature about this characteristic, the literatures[1-5]
that the shooting progress is a discrete time series
on the theoretical analysis and a large number of
composed of several pulses. The cross situation of
reliable test results generalize a universal, the laser
direction tracking error to shooting gate with radius
 is shown in Fig.1.
damage characteristics of static: in the uncertain
environment of direct irradiation target specific
material a certain time (or time of ruin causing
damage caused by pulse number), can lead to the
damage; if the conditions change is deterministic,
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Wang -xiangmin Int. Journal of Engineering Research and Applications
ISSN: 2248-9622, Vol. 5, Issue 12, (Part - 2) December 2015, pp.177-181
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Once it matches the conditions listed above,
Y(k)
when the quantity of pulses in one laser shoot
Z(k)
process is fixed, the target will be damaged or injured
t0
t2

t7
while the total numbers of irradiation exceeds
Damage Threshold, this paper mainly discussed the
t1
o
t6
o
t5
dynamic damage probability under such condition.
x(k)
t4
III. THE STOCHASTIC PASSAGE
CHARACTERISTICS OF THE TRACKING
t3
ERROR
The density function of Z  k  can be marked
Fig 1.the tracking error randomly crosses the
respectively as follows[6]:
shooting gate
f [ Z (k )]  f [ x(k )] f [ y (k )]
Where the symbols “×” marked in black are
samples of the laser pulses. Time series composed of
samples within the circuit area is the irradiation
period, moreover, samples out of circuit area
compose the non-irradiation period . The two periods
alternately appear during the whole shooting process
which is a normal stationary ergodic Markov process
with discrete parameter[6].

1
2 x
1
2 y
exp[
exp[
( x(k )) 2
]
2 x2
(0.1)
( y (k ))2
]
2 y2
Where  x2 and  y2  y are the variances of x(k )
and y (k ) .
To dynamically destroy or injury a target, laser
As shown in Fig. 1, there are two situations in a
system should match 3 damage conditions as below :
stochastic passage period. One is that it starts from
1> The total number of irradiation pulses when laser
beam covers a fixed circular area (the point O
the irradiation point with probability of
 0 , the
is the center in the Fig 1.) on the target exceeds
other is that it starts from non-irradiation point with
the Damage Threshold( Th ), then the target can
probability of
 1 [7],they can defined as
be judged to be destroyed.
0 (Z ) 

2> The tracking error of laser system,
Assume that
2
Z  k    x  k  , y  k 
T
x ( k ) y
, is a
mutually independent, mean square differentiability,
ergodicity two dimensional Gaussian process, whose
mean value is zero.
3> The laser launches at t  t0 and ends at
Ts  Th  0 , where Ts is the total number of laser
pulses during a shooting process.
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2
1
2 x y
( k ) 
exp
(0.2)
( x(k ))2 ( y (k )) 2
[

]dx(k )dy (k )
2 x2
2 y2
1 ( Z )  1   0 ( Z ) if  x   y , then

2 
2 
 2 x 
 0 ( Z )  1  exp  
(0.3)
Then, the probability transition matrix is[7]
 P ( Z ) P01 ( Z ) 
P ( Z )   00

 P10 ( Z ) P11 ( Z ) 
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Wang -xiangmin Int. Journal of Engineering Research and Applications
ISSN: 2248-9622, Vol. 5, Issue 12, (Part - 2) December 2015, pp.177-181
The first element in the matrix is defined as
P00 ( Z )  P{Z (k  1)   | Z (k )  }
the irradiation time series are defined as
Ti  T1 , T2 ,...i  1, 2,... , which are positive,
f [ Z (k  1) | Z (k )] f [ Z (k )]dZ (k  1)dZ (k )
1

 0 ( Z ) x2 ( k ) y2 ( k )  x2 ( k ) y 2 ( k ) 
4 2 x2 y2

( x(k ))
exp[

2 x2
(1  rx2 )(1  ry2 )
From Fig 2. ，the irradiation time series are
defined as Si  S1 , S2 ,...i  1, 2,... ,which are
positive, independent, identically , distributed,
random variables with distribution function F  x  ;
1

 0 ( Z ) x2 ( k ) y2 ( k )  x2 ( k ) y 2 ( k ) 
1
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2
independent, identically , distributed, random
variables with distribution function G  x  .there is
defined as
n
S n   Si , n  1
S0  0,
(0.4)
i 1
and
( x(k  1)  rx x(k )) 2 ( y (k )) 2


2 x2 (1  rx2 )
2 y2
n
Tn   Ti , n  1
i 1
The non-irradiation period obeys the geometric
( y (k  1)  ry y (k )) 2
distribution
2 y2 (1  ry2 )
[7]
F (x  k)
 P{Ti  kT | Z (k )  }
]dx(k  1)dy (k  1)dx(k )dy (k )
Where rx and ry represent the correlation
coefficients of the adjacent two sampling points for
x(k ) and y (k ) respectively. The other elements
(0.5)
 P11k 1 ( Z ) P10 ( Z )
The irradiation period also obeys the geometric
distribution
G( x  k )
 P{Si  kT | Z (k )  }
can also be calculated uniformly.
IV. THE DISTRIBUTIONS FUNCTION OF
IRRADIATION PERIOD AND NONIRRADIATION PERIOD
Consider Z (k ) to be a zero mean, ergodic
stationary random Gaussian process, let  be the
 P ( Z ) P01 ( Z )
Where k  1 ， T is the sample time。
From formula (0.6), the distribution of n
irradiation periods is
Fn 
area of shooting gate, the distribution of Z (k ) in
and outside the shooting gate is illustrated in Fig.2.
(0.6)
k 1
00
 n  k  1! k 1 n
P00 P01
k ! n  1!
(0.7)
From formula (0.5), the distribution of n
non-irradiation periods is
Gn 
 n  k  1! n 1 k 1
P10 P11
k ! n  1!
(0.8)
Z(k)
S1
0
S2
T1
T2
t
V. THE DYNAMIC DAMAGE PROBABILITY OF
THE LASER SYSTEM
Fig 2 Z (k ) showing as irradiation period and nonirradiation period
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Because the shooting moment of laser system
start at t0 (Fig 1), so the first period is irradiation
period, then the stochastic period will be renewal
alternatively between irradiation period and
non-irradiation period. Since every period is
randomized, any random period is possibly ended at
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Wang -xiangmin Int. Journal of Engineering Research and Applications
ISSN: 2248-9622, Vol. 5, Issue 12, (Part - 2) December 2015, pp.177-181
www.ijera.com
the time when stops shooting.
If the pulse counts of the laser belonged to the
number is Ts , the probability of the irradiation pulse
irradiation periods beyond Th , the target will be
number more tan Th equivalent to the probability of
damaged. Assume Ts is the total pulse counts of the
the irradiation pulse number less than Ts  Th :
laser in once shooting process, then the dynamic
damage probability of the laser system can be
calculated as follow.
P( Sn  Th )  P(Tn  Ts  Th )
(0.12)
So the probability is
Case 1: if the target is damaged in the first irradiation

H n   P (Tn  (Ts  Th ))
period, the probability is
(0.13)
n2
H 0 ( x  ThTm )  1  P00Th 1 ( Z ) P01 ( Z )
(0.9)
 P(Sn  Th )  P(Sn 1  Th )
Case 2:if the target is damaged needing more than
Taking into account that the total laser pulse
one irradiation period, the probability is
number is limited by Ts ,the Maximum value of

H n  P(Tn  Ts  Th ) 
(0.10)
n2
 P(Sn  Th )  P(Sn1  Th )
N (t ) is
N (t )  min Th , Ts  Th 
(0.14)
From the two cases, the total dynamic damage
s1
0
probability of the laser system is
T
s
Th
P  1  ( P00 )Th 1 (1  P00 )
N ( t )  Ts Th
 n  k  1! P k 1 (1  P )n  

   
 11 

11
n2 
 k 1 k ! n  1!

s2
T1
T2
t
Fig 3.The damage case experiencing multiple
irradiation periods
 N (t )  n  k  1! k 1
P00 (1  P00 ) n

 k 1 k ! n  1!
N (t )

Proof From Fig 3, the dashed vertical lines indicate
k 1
 n  k ! P
k !n !
k 1
00
(0.15)
 
(1  P00 ) n 1  
 
the possible time when the irradiation pulse number
VI. SIMULATION RESULT
has reached the Damage Threshold( Th ).Let N (t )
Assumed the laser system emission frequency
be the renewal process generated by S1 , S2 ... . Using
is fr , the damage threshold of the target: Th  150 ,
a renewal process property, it follows instantly that
the
P  N (t )  n 
P  N (t )  n  P N (t )  n  1
laser
emission
P01  1  P00  0.2
,
time
is
5
seconds,
P10  1  P11  0.3
,the
(0.11)
 Fn  t   Fn 1  t  , t  0, n  1,2,...
relationship between the
fr and the dynamic
damage probability is shown below:
In once shooting process, the total laser pulse
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Wang -xiangmin Int. Journal of Engineering Research and Applications
ISSN: 2248-9622, Vol. 5, Issue 12, (Part - 2) December 2015, pp.177-181
www.ijera.com
instability with realistic adaptive optics
P( dynami c damage pr obabi l i t y)
1
geometry, J. Opt. Soc. Am. B, 1992, 9(10):
0. 9
1803-1812.
0. 8
[3]
0. 7
Robert K. Tyson. Principles of Adaptive
Optics. Boston: Academic Press, 1991.
0. 6
0. 5
[4]
0. 4
Schmidt J. Numerical simulation of optical
wave
0. 3
propagation.
proc
SPIE,
2010:
149-183.
0. 2
[5]
0. 1
0
50
100
fr
Boley C D, Cutter K P, Fochs S N, et al.
Interaction of a high-power laser beam with
150
metal sheets, Journal of Applied Physics,
Fig 5 Relationship between fr and the damage
probability
2010, 107(4):043106.
[6]
Wang Jun, Li Yushan, Guozhi. Analysis on
Stochastic Passage Characteristics for gun
From Fig 5 we can see that in one shoot process,
Barrel
the total number of laser pulse needs to exceed the
damage threshold, will achieve the goal of the
in
Shooting
Domain.
Journal
of China Ordance. 2010, 6(4): 247-252.
[7]
Wang Jun, Guo Zhi. First Round Hitting
damage, that is, the laser pulse frequency has a
Probability and Firing Delay time for Tank
minimum requirements; greater than the lowest value
with Shooting Gate. Proc. 2010 IEEE
of the laser emission frequency and damage
International Conference on Mechatronics
probability
and
is
now
a
monotonic
increasing
relationship.
Automation,
Xi'an,
China,
2010:
973-978.
[8]
VII. CONCLUSION
Wang Jun, Liu Chao, Zhang Chunxian, Guo
Zhi. The Estimation for the Distribution of
Because of the support of stochastic thesis
Residence Time under Interval Objective
during formulate deductions, it is reliable when laser
Domain. Proc.2008 China Control and
system satisfies the conditions proposed in this paper,
Decision Conference, Yantai, China, 2008:
the conclusions given are inerrant. Beyond that, since
5154-5158.
original data required in calculating the dynamic
damage probability contains the tracking error and
relevant variances of the derivative of the laser
system, it can either be seen as an evidence of the
probability, or directly combined with the transfer
function or state equation of laser system to optimize
control strategy, while treat the probability as the
objective function.
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