Dispersion-Managed Optical Solitons with Higher-Order Nonlinearity

FIO 24(6) #124046

Fiber and Integrated Optics, 24:537–548, 2005Copyright © Taylor & Francis Inc.ISSN: 0146-8030 print/1096-4681 onlineDOI: 10.1080/01468030500240635

Dispersion-Managed Optical Solitons withHigher-Order Nonlinearity

S. KONARMANOJ MISHRAS. JANA

Department of Applied PhysicsBirla Institute of TechnologyMesra, Ranchi, India

In this paper we have investigated the propagation characteristics of optical solitonsin dispersion managed optical communication systems taking into account of the ef-fect of quintic nonlinearity. Using variational formalism, several ordinary differentialequations have been established for pulse parameters. These equations have beensolved numerically to investigate the propagation characteristics. It has been noticedthat stable periodic pulse propagation is possible over long distance. Numerical sim-ulation has been undertaken to show that parabolic nonlinearity reduces collisiondistance between neighbouring pulses of the same channel.

Keywords dispersion-managed soliton, quintic nonlinearity, optical fiber, variationalmethod, split-step Fourier method

Introduction

Since their first introduction [1] in 1996, dispersion managed (DM) optical commu-nication systems have attracted a great deal of attention [1–12] as they promise veryhigh bit rate of transmission of data over conventional soliton based systems. In adispersion-managed system, fiber dispersion varies alternately between anomalous andnormal values. This variation is maintained periodically and the average dispersion overa period could be positive, negative, or even zero. The advantage of DM solitons overconventional one are mainly three fold. These are enhanced signal to noise ratio at thereceiver end, reduced Gordon Haus timing jitter, and the possibility of suppression of thephase matched four wave-mixing processes which lead to substantial performance penal-ties in the wavelength division multiplexing systems. The last factor could be achievedwith considerable success by allowing large local dispersion [13, 14]. Unlike in the con-stant dispersion system where the pulse propagates maintaining its shape unchanged, inthe dispersion-managed system the pulse width oscillates periodically and the shape ofthe pulse ranges from hyperbolic secant to Gaussian to flattop, depending on the strengthof the dispersion management [4, 15]. A particularly interesting point is that sufficientlystrong periodic dispersion management allows for stationary propagation of nonlinear

Received 30 September 2004; accepted 5 February 2005.Address correspondence to S. Konar, Department of Applied Physics, Birla Institute of Tech-

nology, Mesra, Ranchi-835215, India. E-mail: swakonar@yahoo.com

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538 S. Konar et al.

return to zero (RZ) pulse with finite energy when the average dispersion is close to zeroor even zero [2, 3].

Although a tremendous amount of work has been reported in the literature on thepropagation characteristics of RZ pulses in DM systems, many issues related to DMsolitons remain unanswered. For example, since optical fibers are made of glass whereKerr nonlinearity arises due to electrostriction, soliton propagation in the DM system hasbeen studied based on Kerr nonlinearity [1–12]. However, this is true only at low power.Transmission at very high bit rate requires pulses of lower temporal width and higherpeak power. At high peak power, higher order nonlinearity may influence the propagationdynamics [16, 17]. In optical fibers, the most important higher order nonlinearity is thequintic nonlinearity, which needs to be incorporated in the DM soliton systems designedfor high bit rate of transmission. Thus, the main aim of this article is to investigate thebreathing soliton propagation in DM systems under the influence of Kerr and quinticnonlinearity. It should be pointed out that a somewhat identical investigation was carriedout by Mandal and Choudhary [18] in multimode optical fibers. Our investigation differsfrom earlier ones on two counts. First, we have considered single-mode fiber since weare interested in long-haul DM systems. Second, the influence of quintic nonlinearity onthe pulse dynamics has been investigated explicitly.

Different methods have been developed to study the propagation of nonlinear pulsesin a dispersion-managed system. These are the guiding center theory [19, 20], the vari-ational theory [21], the multiscale theory [22], and the numerical averaging method [2].All these methods are approximate and have demonstrated their usefulness in success-fully determining the pulse dynamics. Since the Lagrangian method is also applicable tononintegrable cases, we use this method to demonstrate the DM soliton scheme.

Dispersion Map and Variational Analysis

The dynamics of an optical pulse propagating in an optical fiber under the influenceof Kerr and quintic nonlinearity and periodically varying dispersion is governed by themodified nonlinear Schrödinger equation (MNSE) [16],

i∂A

∂z− β2(z)

2

∂2A

∂t2+ kn2

n0α0|A|2A+ kn4

n0β0|A|4A = 0 (1)

where A is the slowly varying electrical field envelope, β2 is the group velocity dispersionvarying periodically with z, t is the retarded time, and k is the wave number. n0, n2,and n4 are, respectively, the linear refractive index and third- and fifth-order nonlinearrefractive index coefficient. α0 and β0 are two constants arising due to the confinementof electric field in the fiber, α0 ∼ 0.5 and β0 ∼ 0.3 for Gaussian pulses. Fiber lossesare neglected since the length of a unit DM cell is much smaller than the amplifyingdistance. At this stage we introduce following normalization: Propagation distance z isnormalized with 2.5 times dispersion map length L in km, time is normalized to T0 (inpicoseconds), the electric field envelope A is normalized with a characteristic power P0such that A = √

P0u, and normalized dispersion is written as

d(z) = −2.5β(j)2 L

T 20

(2)

Dispersion-Managed Optical Solitons 539

With the above normalization, Eq. (1) takes the form

i∂u

∂z+ d(z)

2

∂2u

∂τ 2+ |u|2u+ γ |u|4u = 0 (3)

where γ = n4β0P0n2α0

and the characteristic power P0 = n0(2.5Lkn2α0)

. In fiber, quintic non-linearity is defocusing, thus giving negative value and generally 0 < |γ | < 1. Thedispersion map is symmetrical as shown in Figure 1 and the fiber dispersion is the sumof a locally varying part and an average dispersion 〈d〉 and is given by

d(z) =∣∣∣∣∣∣d1 + 〈d〉 if 0 ≤ Z ≤ L/4d2 + 〈d〉 if L/4 ≤ Z ≤ 3L/4d1 + 〈d〉 if 3L/4 ≤ Z ≤ L

(4)

where the normalized locally varying part of the dispersion and the average dispersionare given by

dj = −2.5β(j)2 L

T 20

, 〈d〉 = 2.5〈β2〉LT 2

0

= d1 + d2

2(5)

Strength of the dispersion map is defined as [7]

S =∣∣∣∣∣β(1)2 L1 − β

(2)2 L2

T 2FWHM

∣∣∣∣∣ (6)

where L1 and L2 are, respectively, the lengths of fibers with anomalous and normaldispersion in the map.

At this point, it will be appropriate to comment on the power level and transmis-sion speed at which the fifth-order nonlinearity becomes important. First we consider a200 Gb/s return-to-zero (RZ) soliton transmission system through a constant dispersionfiber with duty cycle of 1:4 and operating at λ = 1.55 µm. Using a standard formula [23],the peak power required for soliton formation in a fiber with constant anomalous disper-sion β2 = −0.835 ps2/km and possessing only Kerr nonlinearity is ∼180 mW. Standardsingle-mode fiber has an effective area ∼80 µm2. Thus, the power density of the electricfield |E|2 ∼ 2.25 × 109 W/m2. For standard silica fiber [23, 25] n2 ∼ 2.0 × 10−20 m2/Wand n4 ∼ 10−29 m4/W2, giving n2|E|2/n4|E|4 ∼ 5.0. On the other hand, consider the

Figure 1. Periodic dispersion map with average anomalous dispersion.

540 S. Konar et al.

same system for 25 Gb/s transmission rate. The required soliton peak power is ∼3 mW,giving n2|E|2/n4|E|4 ∼ 140. Therefore, it is clear from the above discussion that Kerrnonlinearity is sufficient to investigate pulse propagation if single channel transmissionspeed is lower than 100 Gb/s, whereas for a system with a transmission rate larger than100 Gb/s, fifth-order nonlinearity should be taken into account to capture the essentialfeatures of pulse dynamics in a dispersion-managed soliton-based optical communicationsystem.

To this end we now proceed to estimate value of γ for the DM systems with200 Gb/s transmission rate. We use standard fiber available for the system [24] withβ(1)2 = −0.835 ps2/km and β

(2)2 = 0.81 ps2/km. Required pulse duration TFWHM =

1.25 ps, the dispersion length LD ∼ 1.87 km. We take L1 = L2 = 2 km. For fiber lossof 0.22 dB/km, an amplifier spacing of 40 km is realistic. Therefore, our assumptionof loss less system is justified. Inserting values of all parameters, we find characteristicpower P0 ∼ 0.897 × 109 W/m2 and |γ | ∼ 0.08, a value that is sufficient to make someimpact on soliton dynamics.

Most of the DM systems with single channel transmission speed lower than 100 Gb/shave been designed and investigated with conventional single-mode fibers. These fibers(could be Corning SMF-28) possess large dispersion (β2 ≈ −22 ps2/km) and low at-tenuation α ∼ 0.22 dB/km at 1550 nm. However, these fibers are not suitable for DMsystems for single channel transmission larger than 100 Gb/s. For such high bit ratesystems, dispersion-shifted fibers with very low β2 value such as β2 ≈ −0.835 ps2/kmand reverse dispersion of β2 ∼ 0.8 ps2/km fibers are preferred. Moreover, since the non-linearity induced self-phase modulation is utilized for breathing soliton formation, theeffective cross-sectional area of these fibers is small (∼80 µm2), and larger effective areafibers (LEAF) in which the cross-sectional area is increased intentionally to reduce theimpact of fiber nonlinearity are not suitable for DM soliton-based optical communicationsystems.

We now proceed to solve Eq. (3) using variational formalism. In order to solveEq. (3), we note that the MNSE can be derived from the variation of the Lagrangiandensity �

δ

∫ ∞

−∞

∫ ∞

−∞�dτdZ = 0 (7)

with

� = i

2

(u∗ ∂u∂Z

− u∂u∗

∂Z

)− d(Z)

2

∣∣∣∣ ∂u∂T∣∣∣∣2

+ 1

2|u|4 + 1

3|u|6 (8)

In order to analytically estimate the periodic variation of pulse parameter in the dispersion-managed system, it is convenient to postulate a simple Ansatz

u(Z, τ) = A(Z) exp�−p2(Z)[τ − To(Z)]2 + iC(z)[τ − To(Z)] − iκ(Z)τ + iσ (Z)�(9)

where A(z), p(Z), T0(X), C(Z), κ(Z), and σ(Z) are real amplitude, inverse of the pulsewidth, temporal pulse position, chirp, frequency, and phase of the pulse, respectively.

Dispersion-Managed Optical Solitons 541

Using the pulse profile (9), we define a finite dimensional Lagrangian

L

(A,p, To, C, κ, σ,

dA

dZ,dp

dZ,dT0

dZ,dC

dZ,dκ

dZ,dσ

dZ

),

such that δ

∫ ∞

−∞LdZ = 0

(10)

The Lagrangian is given by

L =∫ ∞

−∞�dτ

= −√π

2A

[1

4p3

dC

dZ− T0

p

dκ

dZ+ 1

p

dσ

dZ

+ A2

2√

2p+ d(Z)p

2+ d(Z)C2

2p3+ d(z)κ2

2p− γA4

3√

3p

](11)

By applying a variational method we find a set of first-order ordinary differential equationsobeyed by pulse width parameters. These are as follows:

dA

dZ= −Ad(z)C (12a)

dp

dZ= −pd(z)C (12b)

dC

dZ= −A2p2

√2

+ 2d(z)p4 − 2d(Z)C2 − 2γp2A2

3√

3(12c)

dσ

dZ= 5

√2

8A2 − d(z)p2 + 4γA4

3√

3− d(Z)κ2

2(12d)

dκ

dZ= 0 (12e)

dT0

dZ= −κd(Z) (12f)

Additionally, the pulse amplitude and the inverse width p are related through the nor-malized pulse energy

E0 = A2

p(13)

The dispersion-managed solitons are a solution of Eq. (12) with the periodic boundaryconditions C(0) = C(L) and p(0) = p(L). Interestingly, unlike in the case of constantdispersion system, in the DM system periodic solution over a large range of input energyE0 is possible.

We have solved set of ODEs (12) to find out allowed values of p and C of periodicDM solitons for a given pulse energy E0. The periodic behavior of p and C is shown in

542 S. Konar et al.

Figure 2. Phase space trajectories of inverse pulse width p and chirp C. Solid line for γ = 0.0and broken line for γ = −0.1. (a) d1 = 10, d2 = −9, 〈d〉 = 0.5; and (b) d1 = 15, d2 = −14,〈d〉 = 0.5.

Figure 2 for different values of dispersion and quintic nonlinearity. Like Kerr medium,quintic medium supports DM solitons. Naturally, a p-C phase diagram is marginallymodified with the introduction of finite γ . Hence, equal energy DM solitons will have adifferent p-C relationship in Kerr and quintic media. Quintic nonlinearity only marginallymodifies the p-C relationship, still admitting periodic stable soliton. However, for higherd(Z) values, the modification is very small. The variation of C and p with the normalized

Figure 3. Variation of chirp with the distance of propagation in a DM cell. Solid line for γ = 0.0and broken line for γ = −0.1. (a) d1 = 10, d2 = −9, 〈d〉 = 0.5; (b) d1 = 10, d2 = −8.5,〈d〉 = 0.75; (c) d1 = 15, d2 = −14, 〈d〉 = 0.5; and (d) d1 = 15, d2 = −13.5, 〈d〉 = 0.75.

Dispersion-Managed Optical Solitons 543

Figure 4. Variation of inverse of pulse width with distance of propagation in a DM cell. Solidline for γ = 0.0 and broken line for γ = −0.1. (a) d1 = 10, d2 = −9, 〈d〉 = 0.5; (b) d1 = 10,d2 = −8.5, 〈d〉 = 0.75; (c) d1 = 15, d2 = −14, 〈d〉 = 0.5; and (d) d1 = 15, d2 = −13.5,〈d〉 = 0.75.

propagation distance showing periodic variation of both parameters are shown in Figures 3and 4, respectively.

We have undertaken full numerical simulation of Eq. (3) using a split-step Fouriermethod [23]. Initial values of p and C are chosen from the allowed periodic solution ofODEs as predicted by the variational method. The behavior of the pulse profile in the firstDM cell is shown in Figure 5. In the first cell, pulse propagates as a stable periodic pulsewith no visible emission of radiation. We have investigated pulse dynamics over 250 DMcells and the pulse sample has been taken after each 10th cell. Figures 6 and 7 show the

Figure 5. Pulse dynamics within a DM cell. d1 = 10, d2 = −9, 〈d〉 = 0.5. (a) γ = 0.0 and(b) γ = −0.1.

544 S. Konar et al.

Figure 6. Shape of pulse through 250 DM cells. Normalized energy E0 = √2. (a) d1 = 10,

d2 = −9, 〈d〉 = 0.5, γ = 0.0; (b) d1 = 10, d2 = −9, 〈d〉 = 0.5, γ = −0.1; (c) d1 = 10,d2 = −8.5, 〈d〉 = 0.75, γ = 0.0; and (d) d1 = 10, d2 = −8.5, 〈d〉 = 0.75, γ = −0.1.

Figure 7. Shape of pulse through 250 DM cells. Normalized energy E0 = √2. (a) d1 = 15,

d2 = −14, 〈d〉 = 0.5, γ = 0.0; (b) d1 = 15, d2 = −14, 〈d〉 = 0.5, γ = −0.1; (c) d1 = 15,d2 = −13.5, 〈d〉 = 0.75, γ = 0.0; and (d) d1 = 15, d2 = −13.5, 〈d〉 = 0.75, γ = −0.1.

Dispersion-Managed Optical Solitons 545

pulse dynamics with two different d(Z). It is obvious that as predicted by the variationalformalism the stable periodic pulses are possible in fibers with quintic nonlinearity. Incase of single pulses, the noticeable difference between pulse propagation in Kerr andparabolic media is the minor variation of pulse width and amplitude of equal energypulses.

Soliton Interaction

We next consider soliton interaction in the same channel in the DM systems incorporatingthe effect of quintic nonlinearity. For simplicity we take two adjacent in phase solitonswith the following initial pulse waveform.

q(0, τ ) = A1 exp�−p2(Z)[τ − To(Z)]2� + A2 exp�−p2(Z)[τ + To(Z)]2�We further assume that A1 = A2 and A1 is determined through the energy width re-lationship. The contour plot of pulse interaction is shown in Figure 8 for different γvalues. An important feature of pulse interaction is the reduction of collapse distancewith the increase in the value of the strength of the parabolic nonlinearity |γ |. Obviouslythe reduction of collapse distance is less affected in a system with higher d.

In order to have an idea about real world values, we take as an example a DMsystem with 200 Gb/s transmission speed. Relevant parameters for this system are asfollows: β(1)2 = −0.835 ps2/km, β(2)2 = 0.778 ps2/km, the dispersion length LD ∼ 1.87km, we choose L1 = L2 = 2 km. Required pulse duration TFWHM = 1.25 ps, |γ | ≈ 0.08,and 〈d〉 = 0.5. We find that two adjacent solitons can travel 502 km before collision

Figure 8. Soliton interaction in the DM system with quintic nonlinearity. For the first row: d1 = 10,d2 = −9, 〈d〉 = 0.5, (a) γ = 0.0, (b) γ = −0.05, (c) γ = −0.1. For the second row: d1 = 15,d2 = −14, 〈d〉 = 0.5, (d) γ = 0.0, (e) γ = −0.05, (f) γ = −0.1.

546 S. Konar et al.

if we ignore fifth-order nonlinearity, whereas they can travel 410 km before collision ifwe take into account fifth-order nonlinearity, which is about 18% less. Thus, for designof ultrahigh speed DM systems one should take into account the effect of fifth-ordernonlinearity.

Conclusion

In conclusion, we have investigated ultra high bit rate pulse transmission through dis-persion-managed optical communication systems talking into account quintic nonlinear-ity, which arises due to high power density of input pulses. Following a variationalanalysis, we have derived a set of ordinary differential equations. These equations havebeen solved for a symmetric map. We have noticed that parabolic nonlinearity signif-icantly modifies pulse width-chirp (p-C) relationship achievable in Kerr media. It hasbeen noticed that parabolic nonlinearity has marginal influence on the overall propa-gation characteristics of a single isolated pulse. Through full numerical simulation wehave shown that the parabolic nonlinearity influences significantly the collapse distancebetween neighboring pulses of the same channel. We have noticed that the collapse dis-tance decreases with the increase in the strength of the parabolic nonlinearity. However,with the increase in the value of dispersion, the influence of quintic nonlinearity on col-lapse distance decreases. Thus, for very a high bit rate system, the influence of quinticnonlinearity should be taken into account in actual design.

Acknowledgement

We thank an anonymous referee for valuable suggestions. This work is supported by theDepartment of Science and Technology (DST), government of India through the R&Dgrant SP/S2/L-21/99. The authors, Manoj Mishra and S. Jana, would like to thank DSTfor providing a junior research fellowship.

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Biographies

S. Konar received his M.Sc. degree in nuclear physics from Kalyani University,West Bengal, India, in 1982. He received an M.Tech in energy management from DeviAhilya University, Indore, India. He received his M.Phil. and Ph.D. degrees from Jawa-harlal Nehru University, India, in 1987 and 1990, respectively. At present he is workingas a reader in the Department of Applied Physics, Birla Institute of Technology, Mesra,India. His current research interest is in the field of photonics and optoelectronics, partic-ularly soliton propagation in dispersion-managed optical communication systems, non-linear optical waveguide, induced focusing, and all-optical switching. He has also beeninvestigating waves and instabilities in cluster plasmas. Dr. Konar has so far published50 research papers in international journals.

Mr. Manoj Mishra was born in Azamgarh, Uttar Pradesh, India, in 1977. He re-ceived the B.Sc. degree in math science from Govt (Auto) Science College in 1998 andan M.Sc. degree in applied physics from Jabalpur Engg. College in 2002; both are a partof Rani Durgawati University, Jabalpur, Madhya Pradesh, India. At present he is pursuing

548 S. Konar et al.

his Ph.D. degree from the Department of Applied Physics, Birla Institute of Technology,Mesra, Ranchi, India. His research interests are in optical solitons, dispersion-managedbreathers and all-optical switching.

Mr. S. Jana was born in Kaukundu, West Bengal, India, in 1977. He obtained aB.Sc. degree in physics from Midnapore College, under Vidyasagar University, WestBengal, India, in 1998. In 2001 he received his M.Sc. degree in physics from VidyasagarUniversity, India. Presently he is pursuing his Ph.D. degree as a senior research fellow inthe Department of Applied Physics, Birla Institute of Technology, Mesra, Ranchi, India.His current research areas are centered on optical solitons, induced focusing, and spectralswitching.