Education of the republic of uzbekistan tashkent state technical university named after islam karimov

№3/2021 year 
Technical science and innovation
176 
Present time high contract and intense laser light is used to scan for observations on 
deforming areas of open mining objects [10], for laser cutting in many human industries [11] 
for communication by fiber-optic lines [12]. 
One of the main requirements for the implementation of the OPA process is its high 
efficiency. High-energy laser beams are often used for this. This is due to the fact that; highly 
efficient nonlinear optical processes are directly proportional to the laser intensities. It is 
known, that coherent high-energy sources are present in short laser pulses. In turn, the use of 
short high-energy laser pulses leads to two undesirable effects. The first is the enhancement of 
the effect of dispersion of the optical medium, the second is the appearance of third-order 
nonlinear optical processes (Kerr nonlinearity). The first nonlinear optical process leads to time 
delays of pulses and an increase in pulse durations. The second effect leads to an additional 
change in the refractive index of the medium. In this study, the impact of these undesirable 
processes on the OPA was analyzed using numerical calculations. The calculation results 
showed that both the dispersion of the medium and the influence of the third-order input were 
observed after the pulse duration and intensity exceeded a certain limit. 
THE THEORETICAL PART. 
It is known that the OPA is a process in which the 
input of the crystal falls 
3

(
p

"pumping" wave) and 
2

(
s

"Signal" wave) weak radiation 
and free radiation 
1

(
i

"idle" wave) frequency where they are connected
3
2
1
  


or 
p
s
i

 


... If in this case the phase is synchronous
3
1
2
k k
k
k
   
is satisfied, then both 
waves
2

and 
1

amplified by getting energy 
3

waves. Under conditions of nonstationarity of 
the process and taking into account high-order nonlinear effects, the OPA can be written as 
follows [13-15]: 
 
 
2
2
2
2
*
3
3
3
1
1
1
1
1
2
1
2
3
1
1111
1
1212
2
1313
3
2
1 1
1 1
4
3
2
2
2
eff
i kz
id
A
A
A
i
i
D
D
A A e
A
A
A
A
z
t
t
n
n









 







 








 
 
2
2
2
2
*
3
3
3
2
2
2
1
2
2
2
1
3
2
1111
1
1212
2
1313
3
2
2 2
2 2
4
3
2
2
2
eff
i kz
id
A
A
A
i
i
D
D
A A e
A
A
A
A
z
t
t
n
n









 







 








 
 
2
2
2
2
3
3
3
3
3
3
1
3
2
3
1 2
3
1111
1
1212
2
1313
3
2
3 3
3 3
4
3
2
2
2
eff
i kz
id
A
A
A
i
i
D
D
A A e
A
A
A
A
z
t
t
n
n

















 








(1) 
with boundary conditions: 
2
2
4
3
2
3
3
(
0, )
exp(
/
);
(
0, ) 10
(
0, );
(
0, ) 0
o
A z
t
A
t
A z
t
A z
t
A z
t











(2) 
Here;
3
A
- complex pump,
2
A
signal and 
1
A
"idle" amplitudes. where applicable
3

2

and 
1

pump, signal and idle wavelengths, respectively; 
 
1
1
1
n
n


 
2
1
2
n
n


 
3
3
3
n
n


- 
refractive indexes (the dependence of the refractive index on the wavelength is taken from the 
article [14]);
2
/
c

 

radiation frequency (
c
-speed of light, 

-wavelength), 
eff
d
and 
 
 
 
3
3
3
1111
1212
1313
,
,



nonlinear coupling coefficients of the second and third orders.
As mentioned above, (1) the system of differential-differential equations with boundary 
conditions (2) were solved using numerical calculations. For this, the Runge-Kutta method of 

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Electrical and Computer Engineering
177 
the 4
th
order was applied to the nonlinear part of the system of equations, and the fast Fourier 
transform method was applied to the linear part (that is, the dispersion part). To improve the 
accuracy of the numerical method of calculation, the method of a symmetric scheme of steps 
along the nonlinear and linear parts of the equations was used [16]. 
To analyze the above OPA, the following parameters of the real state of laser and 
crystal were selected: nonlinear photonic crystal Mg: PPLN, 
16
2
3
10
/
I
W m

,
12
2
1
10
/
I
W m

,
2
2
0
/
I
W m

, 
1
3, 4
m



, 
3
1,064
m



, 
2
1,5486
m



, 
12
4,7 10
/
eff
d
m V



, 
 
 
 
3
3
3
21
2
2
1111
1212
1313
2,94 10
/
m V








[13,14,17]. We calculated the duration of the main 
laser pulse at several values:
100
fs


,
50
fs


and
20
fs


. 
It should be noted that the reason for obtaining a nonlinear photonic crystal Mg: PPLN 
is that its linear and nonlinear optical properties are well studied and calculated. In numerical 
calculations, it is necessary to set values close to real ones.