Compensation of nonlinear absorption in a soliton communication system

Chaos, Solitons and Fractals 35 (2008) 151–160 www.elsevier.com/locate/chaos

Compensation of nonlinear absorption in a soliton communication system G. Tsigaridas *, I. Polyzos, V. Giannetas, P. Persephonis Department of Physics, University of Patras, GR-26500 Patras, Greece Accepted 23 May 2006

Abstract In the present work, the design parameters of a soliton communication system exhibiting both linear and nonlinear absorption are discussed. Initially, the evolution of the soliton parameters during propagation is calculated using perturbation theory. Then, the propagation of solitary pulses under continuous or localized amplification is studied and the conditions for quasi-stable propagation are determined. The effects of higher-order nonlinear absorption are also investigated. Finally, numerical simulations are performed, verifying the analytical results. Ó 2006 Elsevier Ltd. All rights reserved.

1. Introduction In recent years, optical solitons constitute a field of intense research because of their applications in telecommunications [1,2] and optical data processing [3,4]. Their stability renders them ideal candidates for signal transmission through optical fibers [1,2]. They are also utilized in all-optical switching and optical data processing [3,4], used as bits of information. In most cases, fiber losses are assumed to be linear, which is a valid approximation for ordinary silica fibers. However, nonlinear absorption is likely to become important for new types of fibers, as semiconductor-doped and lead-silicate optical fibers [5,6]. Moreover, in many applications it is desirable to reduce the pulse power required to form a fundamental soliton. A simple way for achieving this target is to increase the nonlinear refractive index of the material used for fiber fabrication. However, in many cases the increased nonlinear refractive index is accompanied by an enhancement of the nonlinear absorption coefficient of the material [5]. Consequently, the effects of nonlinear absorption on the propagation of optical solitons are an interesting subject for investigation. Several efforts have been attempted towards this direction [7–13], mainly concerning dark [8,1,13] and spatial solitons [7,9,12], or fiber amplifiers [10]. Further, some theoretical treatments considering the effects of two-photon absorption on soliton propagation have been presented, principally in the framework of the Ginzburg–Landau equation [14,15]. However, the design parameters of a soliton communication system exhibiting nonlinear losses and especially the compensation of nonlinear absorption – both in the cases of continuous and localized amplification – have not yet been adequately studied.

*

Corresponding author. Tel.: +30 2610 997488; fax: +30 2610 997470. E-mail address: [email protected] (G. Tsigaridas).

0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.05.086

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In the present work, we first present the mathematical formulation of optical pulse propagation in a single-mode fiber characterized by both linear and nonlinear absorption. Then, the evolution of the soliton parameters is calculated using perturbation theory. Based on these results, the propagation of optical solitons under continuous or localized amplification is studied and the conditions for quasi-stable propagation are determined for both cases. The effects of higher-order nonlinear absorption are also included in our consideration. Finally, numerical simulations are performed, verifying the analytical results.

2. Mathematical formulation The propagation of a picosecond optical pulse in a single-mode fiber characterized by both linear and nonlinear losses is described by the equation [16] oA 1 oA i o2 A a0 a2 þ þ b  icjAj2 A ¼  A  jAj2 A oz tg ot 2 2 ot2 2 2

ð1Þ

where A is the slowly varying envelope of the electric field, tg the group velocity of the pulse, b2 the group-velocity-dispersion (GVD) coefficient, and c = k0n2/Aeff a factor describing the nonlinearity of the fiber Here, k0 is the wavenumber, n2 the nonlinear refractive index and Aeff the effective core area of the optical fiber [16]. a0, a2 are the linear and nonlinear (two-photon) absorption coefficients respectively. If we introduce the normalized variables Z¼

z ; LD

T ¼

t  z=tg Tp

ð2Þ

where LD ¼ T 2p =jb2 j is the dispersion length and Tp the pulse duration, Eq. (1) becomes i

ou 1 o2 u þ þ juj2 u ¼ iC1 u  iC2 juj2 u oZ 2 oT 2

ð3Þ

where u¼

pffiffiffiffiffiffiffiffi cLD A;

C1 ¼

a0 LD ; 2

C2 ¼

a2 2c

ð4Þ

In deriving Eq. (3) we have also taken into account that in order to form solitons, the group-velocity-dispersion coefficient b2 should be negative. Eq. (3) is the well-known nonlinear Scro¨dinger (NLS) equation with linear and nonlinear loss terms, which can be treated as perturbations.

3. Calculation of the soliton parameters in the simultaneous presence of linear and nonlinear absorption In the absence of perturbations (C1 = C2 = 0) Eq. (3) admits the well-known one-soliton solution in the general form [1] uðT ; ZÞ ¼ g sech½gðT  T C Þ expðijT þ irÞ

ð5Þ

dT C ¼ j; dZ

ð6Þ

where dr 1 2 ¼ ðg  j2 Þ dZ 2

The parameters j, g are related to the real and imaginary part of the soliton eigenvalue f = (j + ig)/2 and correspond to the velocity (relative to tg) and the amplitude of the solitary wave respectively. TC is the location of the soliton center and r the time-independent phase shift. When the perturbation terms iC1u, iC2juj2u are present, it is assumed that as far as C1, C2  1 the soliton maintains its shape, but the parameters g, j, T, r are functions of the propagation distance Z (adiabatic approximation [1]). Within the limits of this approximation we find that the equations describing the evolution of the soliton parameters are the following: dg 4 ¼ 2C1 g  C2 g3 dZ 3 dj dT C ¼ j ¼ 0; dZ dZ

ð7Þ ð8Þ

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153

Fig. 1. The evolution of the soliton amplitude and phase for the initial condition u(T, 0) = sech T. In all cases C1 = 0.1.

dr 1 2 ¼ ðg  j2 Þ dZ 2

ð9Þ

It should be noted that the loss terms (iC1u, iC2juj2u) affect only the amplitude and phase of the soliton, and not its velocity (j). The solutions of the above equations are  1=2 2 C2 2 g0 ½1  expð4C1 ZÞ expð2C1 ZÞ ð10Þ gðZÞ ¼ g0 1 þ 3 C1 jðZÞ ¼ j0 ;

T C ðZÞ ¼ j0 Z þ T 0   3 2 C2 2 1 rðZÞ ¼ ln 1 þ g0 ½1  expð4C1 ZÞ  j20 Z þ r0 16C2 3 C1 2

ð11Þ ð12Þ

where g0, j0, T0, r0 are the initial (Z = 0) values of the soliton amplitude, velocity, center location and time-independent phase respectively. The evolution of the soliton amplitude and phase for the initial condition u(T, 0) = sech T is shown in Fig. 1. It is evident that nonlinear absorption does not alter significantly the shape of the curves.

4. Study of soliton propagation under continuous or localized amplification Because in an optical communication system fiber losses are inevitable, long-distance soliton propagation is not possible without amplification. There are two kinds of amplification setups. The first kind is continuous amplification, where the gain is distributed along the fiber compensating for the losses at each point. In practice this can be achieved using Raman amplification or by lightly doping the fiber with an active material, creating a distributed amplifier. This

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way, an almost constant amplification factor along the fiber can be achieved [17]. The second kind is localized amplification where the pulse is amplified at certain points in the propagation line. In practice, this can be achieved using fiber amplifiers (usually Erbium doped fiber amplifiers, EDFA’s) at certain points along the fiber. 4.1. Continuous amplification In this case, Eq. (3) takes the form i

ou 1 o2 u þ þ juj2 u ¼ iGu  iC1 u  iC2 juj2 u oZ 2 oT 2

ð13Þ

where G = gLD/2, with g being the gain per unit length. The evolution of the soliton amplitude g is then described by the equation dg 4 ¼ 2Gg  2C1 g  C2 g3 dZ 3

ð14Þ

It is evident that the soliton amplitude remains constant if the gain factor G is equal to C1 + (2/3)C2g2. In practice, if the system is designed in a way that the gain factor G is equal to C1 þ ð2=3ÞC2 g20 , where g0 is the initial soliton amplitude, then, within the limits of the adiabatic approximation, the soliton will propagate without distortion. 4.2. Localized amplification In this case, the extension of the guiding center soliton theory [1] leads to the conclusion that under certain conditions, solitons can propagate with negligible distortion. Specifically, setting uðT ; ZÞ ¼ ~ aðZÞqðT ; ZÞ, where ~ aðZÞ and is the rapidly varying pulse amplitude and q(T, Z) represents the ‘‘average’’ (slowly varying) form of the pulse, Eq. (3) can be written in the form i

oq 1 o2 q þ þ ~a2 jqj2 q ¼ iCB1 q  iC2 ~a2 jqj2 q oZ 2 oT 2

ð15Þ

The evolution of the pulse amplitude ~aðZÞ is described by the equation d~aðZÞ ¼ CA aðZÞ; for ðn  1ÞZ A < Z < nZ A 1~ dZ ~aðnZ A þ 0Þ ¼ G0 ~aðnZ A  0Þ; at Z ¼ nZ A

ð16Þ

where ZA is the normalized propagation distance between two successive amplifiers and G0 the gain ratio at each amplifier. The symbolism ~aðnZ A  0Þ; ~aðnZ A þ 0Þ denotes the pulse amplitude exactly before and after the nth amplifier. The A B parameters CB1 ; CA 1 are related to the linear loss factor C1 through the equation C1 ¼ C1 þ C1 and the purpose of their use will be clear later. The solution of Eq. (16) is ( for ðn  1ÞZ A < Z < nZ A an1 expfCA 1 ½Z  ðn  1ÞZ A g; ~aðZÞ ¼ ð17Þ G0 an1 expðCA Z Þ  a ; at Z ¼ nZ A n 1 A where an is the a~ðZÞ value exactly after the nth amplifier. If G0 ¼ expðCA aðZÞ will retain its 1 Z A Þ then the pulse amplitude ~ initial value after each amplification stage. Further, if the average value of ~ a2 ðZÞ during propagation between two successive amplifiers is equal to unity, then Eq. (15) is equivalent to Eq. (3). Using the definition [1] Z ZA 1 ~a2 ðZÞ dZ ð18Þ h~a2 ðZÞi ¼ ZA 0 and demanding h~a2 ðZÞi ¼ 1, it follows that the initial pulse must be amplified by the factor  1=2 2CA 1 ZA a0 ¼ 1  expð2CA 1 ZAÞ

ð19Þ

In this case the evolution of the ‘‘average’’ pulse q(T, Z) is still described by Eqs. (7)–(9). Consequently, if we set CB1 ¼ 2C2 g20 =3 or equivalently 2 2 CA 1 ¼ C1 þ C2 g0 3

ð20Þ

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where g0 is the initial soliton amplitude, then dg/dZ = 0 and – within the limits of the adiabatic approximation – q(T, Z) will propagate undistorted. Under these conditions, the original pulse described by u(T, Z) undergoes only a periodic variation of its amplitude, while its ‘‘average’’ form remains unchanged. It should be noted that in the above formalism CA 1 and consequently C1 needs not be much less than unity [1]. On the other hand, the restrictions on the C2 value are much tighter. Indeed, as C2 increases the distortions of the pulse shape due to nonlinear absorption are more intense and the validity of the adiabatic approximation gradually breaks. Another interesting remark is that continuous amplification can be regarded as a special case of localized amplification, where the amplifier spacing ZA tends to zero. Consequently, the linear loss parameter C1 needs not be much less than unity, even in the case of continuous amplification. 4.3. Numerical simulations In order to verify our results we have numerically solved Eq. (3) using the split-step Fourier method [16] and calculated the evolution of the peak amplitude ju(0, Z)j2 during propagation under continuous or localized amplification. The results are shown in Fig. 2. It is evident that in both cases the pulse propagates with negligible distortion (62%), provided that the nonlinear loss parameter is quite low (C2 6 0.04). This is further verified by the evolution of the pulse profiles shown in Figs 3 and 4, where it is clear that apart from a slight fluctuation of the pulse amplitude, the width and shape of the pulse remain practically unaltered during propagation. For higher values of C2, the distortion of the pulse shape due to nonlinear absorption increases, and the validity of the adiabatic approximation gradually

Fig. 2. The evolution of the peak amplitude ju(0, Z)j2 as a function of the normalized propagation distance Z in the case of continuous (a) and localized (b) amplification. In the second case the peak pulse amplitude has been calculated at the half distance between two successive amplifiers. The linear loss parameter C1 = 1.0 and the distance between two successive amplifiers is ZA = 0.1.

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Fig. 3. Evolution of the pulse profile in the case of continuous amplification for (a) C2 = 0.02 and (b) C2 = 0.04.

breaks. However, we have found through numerical simulations that for a propagation distance of 50LD the pulse distortion remains below 10%, for C2 values up to 0.07. Finally, we have considered the case where the initial pulse is not a pure soliton, but has the more general form u(T, 0) = A sech T, where A is a real constant close to unity. The results of the numerical simulations for this initial condition are shown in Fig. 5. It is clear that the action of nonlinear absorption tends to destabilize the oscillations of the peak amplitude as it tends to its ultimate value, corresponding to the emerging soliton [18] (shown by the straight line in the figure). This indicates that special care must be taken regarding the initial shape of the pulses, in order to achieve quasi-stable propagation throughout the optical link.

5. Effects of higher-order nonlinear absorption Sometimes, two-photon absorption is accompanied by higher-order effects, as three-photon or excited state absorption [19]. In this case an additional nonlinear loss term must be added to the right hand side of Eq. (1), or equivalently Eq. (3), which takes the form i

ou 1 o2 u þ juj2 u ¼ iC1 u  iC2 juj2 u  iC3 juj4 u þ oZ 2 oT 2

ð21Þ

C3 is the normalized higher-order nonlinear loss parameter, given by the relation C3 ¼

a3 2c2 LD

ð22Þ

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Fig. 4. Evolution of the pulse profile in the case of localized amplification for (a) C2 = 0.02 and (b) C2 = 0.04. The profiles have been calculated at the half-distance between two successive amplifiers.

where a3 is the higher-order nonlinear absorption coefficient of the fiber. Then, Eq. (7), describing the evolution of the soliton amplitude, takes the form dg 4 16 ¼ 2C1 g  C2 g3  C3 g5 dZ 3 15

ð23Þ

Thus, in the case of continuous amplification, the condition for stable propagation becomes 2 8 G ¼ C1 þ C2 g20 þ C3 g40 3 15

ð24Þ

where G is the normalized constant gain factor and g0 the initial value of the soliton amplitude. In the case of localized amplification an additional iC3 ~ a4 jqj4 q term appears on the right hand side of Eq. (15), which takes the form i

oq 1 o2 q þ ~a2 jqj2 q ¼ iCB1 q  iC2 ~a2 jqj2 q  iC3 ~a4 jqj4 q þ oZ 2 oT 2

ð25Þ

In this case it is not possible to set h~a2 i ¼ h~a4 i ¼R1 simultaneously, in order Eq. (25) to be equivalent with Eq. (21). Z However, if CA a4 ðZÞi ¼ 0 A ~a4 ðZÞdZ=Z A  1, even when the initial amplification factor a0 is 1 Z A  1 it follows that h~ given by Eq. (19). Bearing also in mind that usually the C3 values are quite low (C3  C2, C1), we are allowed to accept

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Fig. 5. Evolution of the peak amplitude ju(0, Z)j2 for a disturbed soliton of the form u(T, 0) = A sech T with (a) A = 0.95 and (b) A = 1.05. The straight line corresponds to the amplitude of the emerging soliton. The results were obtained for continuous amplification. Similar curves are produced in the case of localized amplification.

that Eq. (25) is approximately equivalent to Eq. (21). Then, the evolution of the soliton amplitude can still be described by Eq. (23). Thus, setting CB1 ¼ 2C2 g20 =3 þ 8C3 g40 =15 or equivalently 2 8 2 C3 g40 CA 1 ¼ C1 þ C2 g0 þ 3 15

ð26Þ

it follows that dg/dZ = 0 and consequently quasi-stable soliton propagation is still possible. Eqs. (24) and (26) indicate that both in the cases of continuous and localized amplification the conditions for quasistable soliton propagation can still be described by the formulas of the previous section, if C2 is replaced by ðeffÞ C2 ¼ C2 þ ð4=5ÞC3 g20 . This has also been verified by numerical simulations shown in Fig. 6. It is clear that the evolution of the soliton pulse during propagation is not altered as higher-order nonlinear absorption is involved.

6. Conclusions In this work we have studied the propagation of optical solitons in the presence of both linear and nonlinear absorption. The evolution of the soliton parameters in the absence of amplification has been calculated. The propagation of the solitary waves under continuous or localized amplification has also been studied and the conditions for quasi-stable soliton propagation were determined. The case of higher-order nonlinear absorption has also been considered and similar conditions for quasi-stable soliton propagation were found. Our results have been verified by numerical simulations.

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Fig. 6. Evolution of the peak amplitude ju(0, Z)j2 for the case of higher-order nonlinear absorption. Both continuous (a) and localized ðeffÞ (b) amplification are considered. The linear loss parameter and the amplifier spacing is the same as in Fig. 2. In all cases C2 ¼ 0:04.

Acknowledgements We thank the European Social Fund (ESF), Operational Program for Educational and Vocational Training II (EPEAEK II), and particularly the Program PYTHAGORAS II, for funding the above work.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

Hasegawa A, Kodama Y. Solitons in optical communications. Oxford: Oxford University Press; 1995. Hasegawa A. Soliton based optical communications: an overview. IEEE J Sel Top Quant 2000;6:1161–72. Haus H. Optical-fiber solitons, their properties and uses. P IEEE 1993;81:970–83. Chan VWS, Hall KL, Modiano E, Rauschenbach KA. Architectures and technologies for high-speed optical data networks. J Lightwave Technol 1998;16:2146–68. Newhouse MA, Weldman DL, Hall DW. Enhanced-nonlinearity single-mode lead silicate optical fiber. Opt Lett 1990;16:1185–7. Okuno T, Onishi M, Kashiwada T, Ishikawa S, Nishimura M. Silica-based functional fibers with enhanced nonlinearity and their applications. IEEE J Sel Top Quant 1999;5:1385–91. Silberberg Y. Solitons and two-photon absorption. Opt Lett 1990;15:1005–7. Chen Y, Atai J. Absorption and amplification of dark solitons. Opt Lett 1991;16:1933–5. Aceves AB, Moloney JV. Effect of two-photon absorption on bright spatial soliton switches. Opt Lett 1992;17:1488–90. Agrawal GP. Effects of two-photon absorption on the amplification of ultrashort optical pulses. Phys Rev E 1993;48:2316–8. Kivshar YS, Yang X. Perturbation induced dynamics of dark solitons. Phys Rev E 1994;49:1657–70.

160

G. Tsigaridas et al. / Chaos, Solitons and Fractals 35 (2008) 151–160

[12] Afanasjev VV, Aitchison JS, Kivshar YS. Splitting of high-order spatial solitons under the action of two-photon absorption. Opt Commun 1995;116:331–8. [13] Lashkin VM. Perturbation theory for dark solitons: Inverse scattering transform approach and radiative effects. Phys Rev E 2004;70:066620. [14] Biswas A, Aceves AB. Dynamics of solitons in optical fibers. J Mod Opt 2001;48:1135–50. [15] Soto-Crespo JM, Akhmediev NN, Afanasjev VV. Stability of the pulselike solutions of the quintic complex Ginzburg–Landau equation. J Opt Soc Am B 1996;13:1439–49. [16] Agrawal GP. Nonlinear fiber optics. San Diego (CA): Academic Press; 1989 [chapter 2]. [17] Liao ZM, Agrawal GP. Role of distributed amplification in designing high-capacity soliton systems. Opt Express 2001;9:66–70. [18] Tsigaridas G, Fragos A, Polyzos I, Fakis M, Ioannou A, Giannetas V, et al. Evolution of near-soliton initial conditions in nonlinear wave equations through their Backlund transforms. Chaos, Solitons & Fractals 2005;23:1841–54. [19] Fakis M, Tsigaridas G, Polyzos I, Giannetas V, Persephonis P, Spiliopoulos I, et al. Intensity dependent nonlinear absorption of pyrylium chromophores. Chem Phys Lett 2001;342:155–61.