Romanian Reports in Physics, Vol. 65, No. 4, P. 1420–1430, 2013

OPTICS

PROPAGATION OF THE RAMAN SOLITON IN OPTICAL FIBERS*

A. FAZACAŞ1,2, P. STERIAN1 1 Academic Center for Optical Engineering and Photonics, Department of Applied Sciences,

“Politehnica” University, 313 Splaiul Independenţei Street, RO – 060042 Bucharest, Romania 2 Institute of Atomic Physics, 407 Atomiştilor Street, Magurele, Ilfov, RO-077125 Romania,

E-mail: a.fazacas@ifa-mg.ro

Received September 14, 2012

Abstract. In this paper, we present the effects of Raman soliton in optical fibers. To achieve this objective we used, in our simulations, the Split-Step Fourier Method. This method is widely used to simulate soliton-like numerical solutions of the nonlinear Schrödinger equation (NLSE). Consequently, this is a powerful method with a very high speed in calculations. During their propagation in optical fiber, soliton pulses are affected by the Raman scattering. Raman scattering is a process that has a considerable role in the higher order nonlinear effects. The Intrapulse Stimulated Raman Scattering (ISRS) is a phenomenon that appears for pulses with a wide spectrum. For these pulses the energy is transferred from the high-frequency components of the same pulse to the low-frequency components. The result of this process is that the pulse spectrum shifts toward the red side of the frequencies. The shift toward the red side of the spectra is called self-frequency shift and its physical origin is related to the delayed nature of the Raman response. The ISRS is useful to generate Raman solitons whose carrier wavelength can be tuned by changing fiber length or input peak power. Also, this phenomenon is responsible for soliton decay. The Raman scattering process can limit the peak power in fiber optical devices and can cause problems in high-power fiber lasers and amplifiers. Also, this process can turn optical fibers into Raman amplifiers or lasers. In this paper we describe the effects induced by Intrapulse Stimulated Raman Scattering in optical fibers by using the NLSESolver Software. Also, the propagation of Raman soliton was studied in the presence of normal and third order dispersion. Key words: soliton, split step method, optical fibers, Raman soliton.

* Paper presented at the 8th General Conference of Balkan Physical Union, July 5–7, 2012,

Constanţa, Romania.

2 Propagation of the Raman soliton in optical fibers 1421

1. INTRODUCTION

Nonlinear Schrödinger Equation (NLSE) has been studied for over three decades. Split-step Fourier method it is a mathematical algorithm used to solve analytically the nonlinear Schrödinger equation. In this algorithm it is considered that the fiber is subdivided into small sections in which the dispersion and the nonlinearity can be taken into account separately, in the frequency and time domains respectively [1]. Split-step Fourier method it is widely used because of its simplicity, good accuracy and high speed in simple simulations. There are numerous studies that use this algorithm, especially in the field of photonics. These studies [1–5] refer to the effects of second and third-order dispersion in optical fiber, the soliton propagation on birefringent optical fibers near the zero-dispersion wavelength, the pulse compression in dispersion-decreasing-like fibers. There are also studies that propose new enhanced numerical algorithm of the Split-step Fourier Method [7, 8].

Stimulated Raman Scatering (SRS) is an important nonlinear process that can turn optical fibers into broadband Raman amplifiers and tunable Raman lasers. It can also severely limit the performance of multichannel lightwave systems by transferring energy from one channel to the neighboring channels [9].

2. THEORY

The generalized NLSE used in our simulations is:

22 3i 232 i i2 32 2 6

,RA A A

A A Az T T

AT A

T

ββ∂ α ∂ ∂+ + − = γ γ

∂ ∂ ∂

∂−

∂ (1)

where A (z,T) is the complex envelope of the pulse, T = t – β1z represents the normalized time, β1 the inverse of the group velocity, β2 is the quadric dispersion coefficient which is responsible for second order dispersion effects, β3 is the cubic dispersion coefficient which is responsible for third order dispersion effects, α is the fiber loss coefficient, γ is the nonlinear parameter and TR is the Raman resonant time constant.

If we consider that 0( , )U A z T P= , where P0 is the peak value power of initial pulse, equation (4) can be written as follows:

2

2 3 2322 3

ii

i ,2 2 6 R

UU U UU U U T Uz T T T

γ∂ββ∂ α ∂ ∂

+ + − = γ −∂ ∂ ∂ ∂

(2)

where the intensity is defined as 2I U= .

A. Fazacaş, P. Sterian 3 1422

In anomalous dispersion regime (β2 < 0) of an optical fiber, the interplay between the Group velocity dispersion (GVD) and Self-phase modulation (SPM) leads to the propagation of optical solitons [10].

We define the nth order dispersion coefficient in normalized form as:

2

0 2

.!

nn nn T −

βδ =

β (3)

The evolution of pulses is influenced by the numerical sign of the third order dispersion coefficient (TOD), where n from equation (3) is equal with three. To obtain a soliton formed from blue components of the spectra it is necessary to assign negative values for the TOD coefficient. In this case we have an unusual dynamics because the Raman soliton pulse its counterpart in the red region of the spectrum [12]. For positive values of TOD, the dispersive wave is generated on the red side. Consequently, the dispersive wave will travel faster than the Raman soliton.

Pulses with soliton order N 2 =1 are attractive for optical communications systems, due to their ability to maintain their shape during the propagation inside optical fibers. The soliton order (N) is given by the following equation:

2

0 02

2

,D

NL

P TLN

Lγ

= =β

(4)

where LD is the dispersive length, LNL is the non-linear length, T0 is the pulse width and P0 is the peak power of the input pulse. Such soliton propagation is possible for an ideal lossless fiber, which is practicably impossible to achieve [10]. The dispersive length and the non-linear length are given by the following equations:

1

0

2 0

1;D NLT

L LP

= =β γ

(5)

Pulses corresponding to other integer values of N represent higher-order temporal solitons. Noting that / Dz Z L= , the soliton period Z0, defined as the distance over which higher-order soliton recover their original shape, is given by

2

00

2

.2 2D

TZ L

ππ= =

β (6)

The soliton period Z0 and soliton order N play an important role for quantifying temporal solitons. Raman scattering has the most important role in higher-order nonlinear effects, effects that are governed by te last term of equation

4 Propagation of the Raman soliton in optical fibers 1423

(2). This term was included when, in 1986, it was observed the effect called soliton self-frequency shift. This effect has been explained using the delayed nature of Raman response. SRS leads to a continuous downshift of the soliton carrier frequency when the pulse spectrum becomes so broad that the high-frequency components of a pulse can transfer energy to the low-frequency components of the same pulse through Raman amplification. The Raman-induced frequency shift is negligible for T0 > 10 ps, but becomes important for short solitons with T0 < 5 ps [11]. To isolate the effects of intrapulse Raman scattering β3 must be equal to zero s3/m. Also, if we consider that losses are equal to zero then equation (2) will become:

2

2 222

.2 Ri

UiU U i U U T Uz T T

γ∂β∂ ∂

+ = γ −∂ ∂ ∂

(7)

The Raman-induced frequency shift is given by the following relation:

2

40

8( ) .

15R

R

T zz

Tβ

∆ω = − (8)

The negative sign shows that the carrier frequency is reduced, meaning that the soliton spectrum shifts toward longer wavelengths. We can see that the frequency shift increases linearly along the fiber and it becomes large for short pulses. The shift toward longer wavelengths can be understood in terms of SRS, i.e., for pulse widths ~ 1 ps or shorter, the spectral width of the pulse is large enough that the Raman gain can amplify the low-frequency (red) spectral components of the pulse, with high-frequency (blue) components of the same pulse acting as a pump. The process continues along the fiber, and the energy from blue components is continuously transferred to red components [11]. In 1986, using 0.5 ps pulses, it was observed the Raman-induced red shift soliton. For this a passively mode-locked color-center laser was used and it was found that the pulse spectrum shifts with 8 THz for a fiber length under 0.4 km. This shift was induced by the soliton itself [11].

3. RESULTS

In this paper we will use NLSE Solver program developed by Godvind P. Agrawal. As an input field we will consider that the shape of our pulse is defined as:

1 i

00

Secant Sech ,C

TPT

+

= ×

(9)

A. Fazacaş, P. Sterian 5 1424

where P0 is the peak power in W, T = t – β1z represents the normalized time, T0 is the width of the pulse and C is the chirp parameter, which will be equal to zero in our case. In Fig. 1 presents the decay of a second order soliton (N = 2) with TR = 0.01, γ = 4 W-1m-1, β2 = –2 s2/m, T0 = 1 s and P0 = 1 W. The graphical representation was made in frequency and time domain, respectively.

The decay of the third order soliton (N = 3) is shown in Fig. 3. For this simulation we used the following parameters: TR = 0.01, γ = 4 W-1m-1, β2 = –1 s2/m, T0 = 1.5 s and P0 = 1 W. We can see an important red shift of the soliton spectrum. It is well known that the shifting toward the right corresponds with a red-shifted spectral peak, while the shifting toward the left to the blue-shifted spectral peak.

Fig. 1 – Evolution of input and output profiles for a second order soliton

in time and frequency domains (Color on line).

Fig. 2 – Evolution of a second order soliton in time and frequency domains.

6 Propagation of the Raman soliton in optical fibers 1425

Fig. 3 – Evolution of input and output profiles for a third order soliton in time and frequency domains

(Color on line).

The evolution of the third order soliton is presented in Fig. 4. The blue components travel faster than the red components.

Fig. 4 – Evolution of a third order soliton in time and frequency domains.

Fig. 5 – Evolution of a second order soliton in normal dispersion ( 2 0.5δ = )

and in third order dispersion ( 3 0.16δ = − ).

A. Fazacaş, P. Sterian 7 1426

Fig. 6 – Evolution of a second order soliton in normal dispersion ( 2 0.5δ = )

and in third order dispersion ( 3 0.16δ = ).

Figures 5 and 6 present the evolution of the second order soliton in normal and third order dispersion. Due to the influence of these effects and the presence of Raman effects the second order soliton decreases. The following parameters were used in our simulation: TR = 0.01, γ = 4 W-1m-1, β2 = 1 s2/m, β3 = –1 s3/m and 1 s3/m, T0 = 1 s and P0 = 1 W. The graphical representation was made in frequency and time domain, respectively.

In Fig. 5 we represented the evolution of the second order soliton where we considered that the third order dispersion coefficient has negative value. The blue components of the pulse experience anomalous group dispersion velocity and form a higher order soliton. In the case from Fig. 6 we can see that the spectrum shifts towards the red side. This means that the optical soliton is formed by the red components of the pulse.

Figure 7 reveals that the temporal trajectory bends with the increase of the order of the soliton. In our simulations the bending of the trajectory depends on the width of the input pulse. If we increase the order of the soliton, we can see that a multiple fission process occurs. The soliton break-up play an essential role in the supercontinuum generation in photonic crystal fibers and in soliton fiber lasers.

The signal splits into several parts as we can see in Fig. 8. The break-up of the input signal into fundamental solitons is described as a result of the energy loss due to the nonlinear absorption. When we increase the order of the soliton, we can observe that the bending increases too. Consequently, the red shifts will be higher, meaning that the speed will increase. The decay process is also present.

Although in the case of positive values of TOD, the soliton is produced by the red components of the spectra, it is well known that a substantial part of the pulse is present in the blue part of the spectra. Our simulations are valid for TelCom based pulse propagation.

8 Propagation of the Raman soliton in optical fibers 1427

Fig. 7 – Temporal and frequency pulse evolution for

22 1s / mβ = − , N = 4, 6, 10 and 3 0.08, 0.05, 0.03δ = − − − .

N = 4

N = 6

N = 10

N = 4

N = 6

N = 10

A. Fazacaş, P. Sterian 9 1428

Fig. 8 – Temporal and frequency pulse evolution for

22 2s / mβ = − , N = 4, 6, 10 and 3 0.16, 0.11, 0.06δ = − − − .

N = 4

N = 6 N = 6

N = 10

N = 4

N = 10

10 Propagation of the Raman soliton in optical fibers 1429

4. CONCLUSIONS

Soliton pulses inside optical fibers are affected by numerous types of nonlinear effects. In this paper we studied the pulses that are affected by Raman scattering in conjunction with second order dispersion and third order dispersion. We saw that this process can limit the peak power in optical devices and the process is responsible for soliton decay. The nonlinear processes were studied by using the Split-step Fourier method, a mathematical algorithm used to solve analytically the nonlinear Schrödinger equation. By using this method we saw a shift toward the red side of the spectra, or toward the blue side of the spectra. By introducing the third order dispersion in the nonlinear Schrödinger equation we have noticed, in our simulation, that in the case of negative values, the components of the spectra are shifted toward the blue side, while in the case of positive values, the components are shifted toward the red side. This leads to the conclusion that the optical soliton is formed either by the red components of the pulse, either by the blue components of the pulse. When we increased the order of the soliton, break-up and decay processes occurred. The obtained results reveal the effects of different interesting nonlinear processes, processes that can be used in system design.

Acknowledgements. This research is supported by the Academic Center for Optical

Engineering and Photonics, Faculty of Applied Sciences, University “Politehnica” Bucharest, Romania.

REFERENCES

1. Xiangyang Zhang, Xiangzhao Wang, Soliton propagation in birefringent optical fibers near the zero-dispersion wavelength, Optik, 115, 36–42 (2004).

2. Yuntuan Fang, Jun Zhou, Effects of third-order dispersion on soliton switching in fiber nonlinear directional couplers, Optik, 119, 86–89 (2008).

3. Hongjun Zheng, Shanliang Liu, Xin Li, Chongqing Wu, Pulse compression in dispersion-decreasing-like fibers, Optics & Laser technology, 43, 1321–1324 (2011).

4. Andrei D. Petrescu, Andreea Rodica Sterian, Paul Sterian, Soliton Propagation in Optical Computer Experiments for Students Training, Computational Science and its Applications, Springer Proceedings International Conference Kuala Lumpur, Malaysia, 2007, p. 450.

5. Z.B. Wang, H. Y. Yang, Z Q Li, The Numerical Analysis of Soliton propagation with Plit-Step Fourier Transform Method, Journal of Physics: Conference Series, 48, p. 878 (2006).

6. Xinrong Shi, Qi Guo, Wei Hu, Propagation proprieties of spatial optical solitons in different nonlocal nonlinear media with arbitrary degree of nonlocaly, Optik, 119, 503–510 (2008).

7. Patrick L. Nash, A new fourth-order Fourier-Bessel split-step method for the extended nonlinear Schrödinger equation, Journal of Computational Physics, 227, 2073–2082 (2008).

8. Xueming Liu, Adaptive higher-order split-step Fourier algorithm for simulating lightwave propagation in optical fiber, Optics Communications, 282, 1435–1439 (2009).

A. Fazacaş, P. Sterian 11 1430

9. Govind P.Agrawal, Nonlinear Fiber Optics, Third Edition, Academic Press, San Diego, 2001. 10. Dipankar Ghosh, Mousumi Basu, Propagation of short soliton pulses through a parabolic index

fiber with dispersion decreasing along length, Optics Communications, 281, 3361–3368 (2008).

11. Yuri S. Kivshar, Godvind P. Agraval, Optical Solitons From Fibers to Photonic Crystals, Academic Press, Elsevier Science, San Diego, 2003.

12. Samudra Roy, Shyamal K. Bhadra, Kunimasa Saitoh, Masanori Koshiba, Godvind P. Agrawal, Dynamics of Raman soliton during supercontinuum generation near the zero-dispersion wavelength of optical fibers, Optics Express, 19, 11 (2011).