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Научни трудове на Съюза на учените в България - Пловдив Серия В. Техника и технологии, том XIII., Съюз на учените, сесия 5 - 6 ноември 2015 Scientific Works of the Union of Scientists in Bulgaria-Plovdiv, series C. Technics and Technologies, Vol. XIII., Union of Scientists, ISSN 1311-9419, Session 5 - 6 November 2015.
APPROXIMATE SOLUTION OF T HE NONLINEAR AMPLITUDE EQUATION INCLUDING THE RAMAN EFFECT V. SLAVCHEV1, D. DAKOVA2, L. KOVACHEVS A. DAKOVA1
'Institute of Electronics,BulgarianAcademy of Sciences, 72 Tzarigradsko shossee, 1784 Sofia, Bulgaria 2Faculty of Physics, University of Plovdiv "Paisii Hilendarski",
244 f Tsar Asen" Str., 4000 Plovdiv, Bulgaria
Abstract
In the present work is considered the evolution of ultra-short optical pulses in nonlinear dispersive media. For these pulses the influence of Raman effect cannot be neglected. The most of the well-known theoretical models, describing the behavior of laser pulses, are based on the use of the nonlinear Schrödinger equation in which is added a term responsible for the Raman effect. The experiments confirmed the numerical results for nanosecond and picoseconds pulses, but in the femtosecond and attosecond region a significant deviations are observed.
In our work is found an approximate solution of the nonlinear amplitude equation in which is included a term that governs the Raman effect. It was used the small parameter method. For a small parameter we assume the coefficient tr, which gives the ratio of the nonlinear response of the medium and the initial pulse duration.
The results are important for ultrafast optics and technologies based on the Raman effect. Keywords: Raman effect, modified nonlinear Schrödinger equation, approximate solution
1. Introduction
In present paper is investigated the propagation of ultra-short laser pulses in nonlinear dispersive medium. For optical pulses with widths below 100 ps the influence of Raman effect can be easily observed. Most of the nowadays theoretical models, used for describing the behavior of such pulses, are based on the nonlinear Schrodinger equation (NSE) in which is included additional term responsible for the Raman effect [1]:
du 1 d2u | ,2 dial2
+ u u — "
I — + —
ÔÇ
2 дт2
дт
(1)
where tr, gives the ratio of the nonlinear response of the medium and the initial pulse duration. The numerical simulations for equation (1) are shown in figure 1.
I
-Aéé
Figure 1. (a) Temporal and (b) spectral evolutions of a second order (N=2) soliton when tr = 0.01, depicting soliton's fission induced by intrapulse Raman scattering. [1- G. P. Agrawal, Nonlinear fiber optics, Academic Press, INC, New York (2007)]
R
The nonlinear Schrodinger equation is derived for laser pulses with narrow-band spectrum and works very well in the nano and picosecond region, but in the frames of femto and attosecond optics it is necessary to be used the nonlinear amplitude equation (NAE) in which is added a term responsible for the Raman effect. We found an approximate solution of NAE by applying the small parameter method. For a small parameter we assume the coefficient tr where y is a parameter connected with the nonlinearity of the medium. p characterizes the second order of linear dispersion of the medium. We are interested in a media with anomalous dispersion (k" <0).
2. Basic equation
It is investigated the longitudinal evolution of short optical pulse in single-mode fiber. We work with NAE [2] in local time coordinate system which differs from NSE with two additional non-paraxial terms (in the brackets):
i d2u ■ ■ 2 d , ,2 (2)
—- + nu\u = su—u W
2a dz dz
du 1 i— +—
df 2a
d2u „ d2u
"T - 2-
df dfdr
where U =A* =T=I' T =t-Z'Vgr' Z = (3)
a = k0z0, \p\= k0|k"\v2ir, y = an2Po, s = 7Tr < 1
In the equation above u is the amplitude function of the envelope of the pulse; t is time; k0, vgr, n2 are respectively the wave number, group velocity and the nonlinear refractive index of the medium; z0 and T0 are the initial longitudinal length and duration of the pulse; P0 is its peak power. The constant a (a>1) characterizes the number of harmonic oscillations at level 1/e from the maximum of the pulse amplitude.
3. Finding an approximate solution of NAE
When e=0, the solution of equation (2) is of the kind [3]:
u = uo(f t) = ® o( *)exp(iXf t)) where O 0(r) = ^sec h(nr),
t) = af + br,
a = b b =
1
1 1 -0t(1 -H)
(4)
(5)
The dimensionless parameter n describes the amplitude of the pulses. For a fundamental soliton n=1; a and b are constants. In the case when e£0 we search for a solution of equation (2) in first approximation by using the small parameter method [4, 5]. Thus, the next substitution is made:
u = [O o(T) + £<51(r)]exp( iW(4,z)), (6)
where 01(z) is a complex function.
It is clearly seen that to obtain an approximate solution of equation (2), we have to find the real and imaginary parts of the unknown function:
®1(t)= ®m(T)+i ®1Y(T)- (7)
To define them it is used the next algorithm [6]:
• The expression (6) is substituted in NAE (2).
• The coefficients in front of the same degrees of s on both sides of the obtained equation are equalized. Having in mind that s<<1, it is assumed that the terms multiplied by s2, s3, ..., are too small and can be neglected.
• By equalizing the coefficients in front of e1 it is obtained an ordinary differential
2
a
equation, in which are included the real and imaginary parts of the unknown function
0i(r):
_ 1 2ia ,, a , 2 ,. 2ife _, 2ia ,, 2ab , iO« - aO. +—01«« +-O«--O.--O).--O«--O) +-O.
1 2a « 2a « 2a 1 2a « 2a 1 2a t 2a
\\ \\ \\ 2 2 2 2
+ 2jO"T + a2m'T- a Oi + 3YoO0O1R + /YQOOOIY = 2O0O0)
(8)
where Oj (),£) = Oj (x), x = ) - ^^ .
After equalizing the coefficients in front s1 we obtain the following expression:
2 2
„.. > „.. p „..» 2iap a , 2 p ,,, 2ibp , , 2ia 2ab ,
-/po;-aO[ + 2-oj---o; -t Oj + --o;'---o; + _o;+—01 (9)
2a 2- 2- 2- 2- 2- 2- v"
H H H 2 2 2 2
+V-o;'+^2bo; -^-Lb Oj+3yfio„01R + YoOoOiy = 20„o^ 2- 2- 2-
Next, the real and imaginary parts on both sides of the equality are equalized. The following two linear ordinary nonhomogeneous differential equations of second order, corresponding to the functions 01R ()) and 01Y ()) are obtained:
°!R O1R + 6OoOiR ^T-JooO0
4a ^2^, \O2<°0
O1v -n2Oly + 2O0O1Y = 0 The general solutions of equations (10) and (11) are of the kind:
oir = ciii(t) + c2i2 ()) + çr (t)
(10) (11)
(j2)
OjY = C3I3()) + C414 ()) (13)
In the expressions above Ih I2, I3 and I4 are respectively linearly independent partial solutions of the corresponding homogeneous equations (10) and (11) and Cb C2, C3, C4 are constants. The function qR is partial solution of equation (12). We have shown that:
i1t) = O0(t), I 3(t) = O o(t),
3 1
I 2(t) = -(O 0 + tO0) - - ch(t),
I 4(t) = |(t® 0 + sh(t)),
(14)
2a
<pr (t) = -5\Oo
2 2
1 — sh (t)-2lnchT 3
(15)
(16)
From the initial and boundary conditions it is defined that:
0ir(û)=0iy(O)=O and 0'ir(O)=0 'm(0)=0, and the constants are found:
C1=2a/5\b \ and C2=C3=C4=0 (17)
Using the algorithm above the following expression for the unknown function 01(t)= 01r(t) + i01Y(t) is obtained:
sh2)
4 a
O1(T) = O1R ()) = 5\O'O
3
- + lnch t
o1y (t) = 0
(18)
As a result of the applied method, we have found a solution in first approximation of the scalar one-dimensional NAE (2). It is of the kind:
u «,)) =
„s 4 ,,
O 0 + 5TR O0
^ sh 2t 3
+ lnch)
w(«,t)
(19)
It is clearly seen that Raman parameter )R has a significant impact on the amplitude of the optical pulses.
2
4. Numerical results
The numerical simulations of our approximate solution (19) are presented in figures 2 and 3.
i.i
] mHm I *n Lh 11 Hi)
-I
i - B i - It-" IJ.№-:1
isa* 'J-a -tf U - I'
Figure 2. The initial soliton pulse, with parameters To=30fs and 1=1,55 ¡im for z=0, tr= 0, s=0.
Figure 3. The intensity profile of the pulse with parameters T0=30 fs and 1=1,55 ¡im for z= z0/2, tr= 0.1, s -f- 0.
In figure 2 is shown the initial ultra-short soliton pulse with a typical sech-shape. The evolution of the pulse under the influence of the Raman effect is presented in figure 3. Thus, it is generated a Stoke wave. Any deviations from the experimental results are caused by the used approximation model.
5. Conclusion
In present work is being reviewed the propagation of ultra-short optical pulses under the influence of Raman effect. It is found a solution in first approximation of the nonlinear amplitude equation (2) which describes the propagation of one-dimensional light pulses in silica single-mode optical fibers. It is used the small parameter method. The expression (4) is the typical solution of equation (2) when s=0 and gives the initial soliton pulse (see fig. 2). The numerical simulation of expression (19) is presented in figure 3. It is shown the temporal evolution of the pulse under the effect of the intrapulse Raman scattering when #0. It is observed the typical growth of the intensity of the Raman pulse. The results are important for ultrafast optics and technologies based on the Raman effect.
Acknowledgments: The present work is supported by project NI15-FFIT005, Faculty of Physics, Plovdiv University "Paisii Hilendarski".
References
[1] G. P. Agrawal, Nonlinear fiber optics, Academic Press, INC, New York, (2007).
[2] R. W. Boyd, Nonlinear optics, Third Edition, Academic Press, Orlando, (2003).
[3] A. Dakova, D. Dakova, L. Kovachev, Comparison of soliton solutions of the nonlinear Schrodinger equation and the nonlinear amplitude equation, Proc. SPIE 9447, 18th International School on Quantum Electronics: Laser Physics and Applications, 94471A doi: 10.1117/12.2177906, (2015).
[4] D. Dakova, Method of small parameter approximation in analyzing the propagation and interaction of soliton-like pulses, Proc. SPIE 6604, 660410(5), (2007).
[5] N. N. Bogolubov, Yu. A. Mytropolski, Asymptotical Method in non-linear vibration theory, GITTL, Moskow, (1955).
