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Научни трудове на Съюза на учените в България-Пловдив. Серия В. Техника и технологии, т. XV, ISSN 1311 -9419 (Print), ISSN 2534-9384 (On- line), 2017. Scientific Works of the Union of Scientists in Bulgaria-Plovdiv, series C. Technics and Technologies, Vol. XV., ISSN 1311 -9419 (Print), ISSN 2534-9384 (On- line), 2017.
CNOIDAL WAVES IN SINGLE-MODE FIBERS A. Dakova13, D. DakovaEV. N avchev2, L. Kovachev3,M. Kolev1 AFacultyof P Dysics, Univers Vyof Plovdiv"Paisii ffilef d arski",
Plovdiv, Bulgaria
2Faculty of Pharmacy, Medical Uniuersity - Plovdiv, Plovdiv, Bulgaria 3Institutf of Etecttonics iBA0, So0"1 Bodgcria
Abstract
In the present paper analytically and numerically is investigated the evolution of cnoidal waves in silica single-mode fibers. A new exact analytical solution of the one-dimensional nonlinear amplitude equation, describing the propagation of optical pulses in nonlinear dispersive media, has been obtained. The solution is presented in the form of elliptic delta function and describes cnoidal waves. It is shown that at certain values of the parameter rthe solution is reduced to sech-soliton.
Key words: cnoidal waves, soliton solution, nonlinear amplitude equation
1. Introduction
In recent decades, a considerable attention attracts the investigation of the nonlinear regime of propagation of ultrashort light pulses in dispersive single-mode fibers, or more specifically the possibility of formation of optical solitons. There are a number of publications which, with their experimental and theoretical contribution, complete the models, associated with the behavior of this type of pulses [1-4]. On the other hand, the development of modern laser systems and the growing needs of ultrafast optics increased the interest of scientists in studying the effects of the evolution of broadband light pulses [5-9]. Nowadays, it is not difficult to obtain phase-modulated broadband pulses or to reach the attosecond region where (Aa&ao). For this reason, it is important to study the behavior of ultrashort optical pulses in different waveguides.
The well-known (1+1D) nonlinear Schrodinger equation (NSE) describes very well the propagation of laser pulses in optical fibers and other one-dimensional structures [10-13]. It is derived for narrowband pulses (Aa<<ao). However, when we examine the evolution of femtosecond and attosecond light pulses, it is necessary to use the more general nonlinear amplitude equation (NAE) [14]. It differs from the standard NSE with two additional terms. The dimensionless analysis shows that these terms play a significant role in the dynamics of broadband pulses. For this reason, we have directed our attention to NAE.
The main task of the present paper is to find an exact analytical soliton solution of NAE in the form of cnoidal waves.
2. Basic equation
We are interested in the evolution of one-dimensional optical pulses in nonlinear singlemode waveguides. It is assumed that the Oz axis of the coordinate system matches with the
geometry axis of the fiber. In "local time" coordinate system the propagation of narrowband and broadband laser pulses is described by the nonlinear amplitude equation (NAE), written in dimensionless form [6,14,15]:
.dU 1 i-+ —
3Ç 2a
fd 2U
- 2
d2U ^
dÇôr
+
ß d 2U 2a dz2
+
y\U\2 U = 0
(1)
where ß = k0v2\k"\ < 1, Y = an2\ A0\ / 2,
« = k0 Z0 = k0T0Vgr
In the equation above U is the amplitude function of the pulse envelope, the constant a (a>1) determines the number of harmonic oscillations at level 1/e from the maximum of the pulse amplitude. The coefficient fi characterizes the second order of the linear dispersion. We examine the propagation of the pulses in medium with anomalous dispersion. The parameters y is connected with the nonlinearity.
It is important to mention that the coefficient (1/a) is inversely proportional to the initial pulse duration T0. For optical pulses with A0=1,5jum and different T0 , its magnitude is presented in Table 1. For nanosecond and picosecond laser pulses this term is quite small and can be neglected. Then equation (1) is reduced to the well-known standard NSE. That is way, in the narrowband case NSE describes well the evolution of optical pulses in single-mode fibers. Obviously, for femtosecond and attosecond optical pulses the expression in brackets must be taken into account.
Table 1.
To 1/a
5fs 6,4.10-'
15 fs 7,6.10-'
35 fs 3,2.10-2
50 fs 6,4.10-2
70 fs 1,6.10-2
250fs 4.10-3
70 ps 1,6.10-4
70 ns 1,6.10-7
Having in mind that, we search for a solution of (1) in the form of [16]:
U(t, g) = O(t) exp(ia% + ib t) , (2)
where a and b are constants about to be defined. &(T) is a new real amplitude function. After substituting (2) in (1), it is obtained the following complex ordinary differential equation:
^ ib| y^L b2| p\ a ^ ia ah
J—1-O"-aO+—■—i-O'--LJ-O--O--0' +—O + 7O3 = 0 (3)
2a a 2a 2a a a
As a next step, in order to find the solution of equation (1) and the unknown constants a and b, we equalize the real and imaginary part on both sides of (3):
(5)
Re: 0''-n20 + 2$3 = 0, i
where: V =
2a + a2 + b2
P
- 2ab
IP
= const
m — (b P\- a) = 0
a
Using (5) and (6) a connection between the constants a and b is found: a = b\\\
(6)
b1,2 =
a
1 -IP
1 ±\ 1 ^-IP)
(7)
Let's now consider equation (4). By substitution = 2-k2, the equation (4) can be presented as follow [17]:
/'-(2-K2 )y + 2 y3 = 0, y(0) = 1, 0 <k< 1
y - \2 — K )y + 2y = 0, y(0) = 1, 0 SKS 1, (8)
where k is a constant. For random values of k the solution of (8) is the elliptic delta function dn(t,k) of second order. Its periods are 2k and 4ik, where k =(1-k)1/2. The graphics of this function is presented on Figure 1.
Figure 1. Graphics of function dn(t, k) for different k
The solution of (8) is of the kind: O(r) = dn(z,k), k2 = 2 — r/2.
Using that result, the solution of NAE (1) can be presented in the form of cnoidal waves:
U (£,t) = dn(t, k) exp(i /) (10)
where /the phase of the pulses.
It is well-known that for k=1 the dn(TK) function can be written in the form of sech(T)
[18:]
dn(t,\) = = sec h(t) (11) Having in mind (7), (9), (10) and (11),
the solution (2) of NAE (1) in "local time" coordinate system takes the form:
U(£ t) = n sech[n(t - to )]exp(i /) (12)
where the phase is written as follow: a
/ =
1
1 ±J1 -5(1
№\+T]
(13)
The solution of NAE has the same sech form as that of NSE. The difference is in the phase of the pulse, which is more complex than that observed in the frames of NSE. The expressions (12) and (13) describe the evolution of broadband as well as narrowband fundamental solitons in singlemode fiber. 3. Conclusion
In present paper is reviewed the evolution of optical pulses with broadband and narrowband spectrum in nonlinear dispersive medium. It is found an exact analytical soliton solution of NAE (1) in the form of cnoidal waves. The expressions (12) and (13) differ from the classical soliton solution of NSE - the constant n depends on the coefficient a, characterizing the number of harmonic oscillations at level 1/e from the maximum of the pulse amplitude. The graphics of the elliptic delta function dn(t k) show that the soliton solution with standard sech-form is observed when the constant Kis equal to 1.
Acknowledgments: The present work is supported by project FP17-FF-010 to the Scientific Research Fund at Plovdiv University "Paisii Hilendarski", Plovdiv, Bulgaria.
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