in Optical Communication

Rakesh Kumar∗ Department of Electrical Engineering, University of New Mexico,Albuquerque, New Mexico, 87131-1156, USA

Mukesh Tiwari† Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131-1156,USA (Dated: May 5, 2005) In recent years Optical technologies have attracted much interest in the academic and industrial worlds. Today the potential of soliton engineering is recognised world-wide with reasearch groups actively working on this topic in every part of the globe. In this report, we present a brief overview of the fundamental concepts of temporal solitons and its potential applications in the eld of ber optic communications. We also present an overview of the research carried out in the last 25 years and the current challenges in the eld.

PACS numbers:

INTRODUCTION starting in 1965 by Zabrusky and Kruskal, Lax, Zakharov and Shabat, and Mieura et al. and the developement of lasers. The most important soliton from the stand- In the last 30 years, soliton has been an active area point of potential technological applications are Fiber of research and its study has been conducted in elds optic solitons and these form an essential part of con- as diverse as particle physics, molecular biology, quan- temporary non linear . In the context of nonlinear tum mechanics, geology, cosmology, oceanography, as- optics, solitons are classied as being either temporal or trophysics and . The history of solitons, spatial, depending on whether the connement of light however, can be traced back to 1834 (report published occurs in time or space during propogation. Both type in 1844) when James Scott Russel observed that a heap of solitons evolve from a nonlinear change in the refrac- of water in a canal propogated undistorted over several tive index of a material induced by the light intensity kilometers. Boussinesq's equation (1872) and Korteweg- (optical kerr eect). A spatial soliton is formed when deVries (KDV) equation (1895) proved that these soli- the self focussing of an optical beam balances its nat- tary waves were indeed possible theoretically. But it was ural diraction-induced spreading. Self phase modula- only 70 years later that these ndings were taken any tion (SPM) counteracts the natural- broad- further. The word soliton was rst used in a paper by ening of an optical pulse and leads to the formation Zabusky and Kruskal in 1965 when they solved the KDV of temporal solitons. The earliest example of tempo- equation numerically for periodic boundary conditions. ral solitons is related to the discovery of the self-induced They also found out that even though the equations were transperancy (SIT) in 1967 [2]. Optical soliton in a ber nonlinear and two solitary waves propogating at dier- forms a solitary wave as the envelope satises Nonlin- ent speeds should interact strongly with each other, the ear Schro¨dinger equation, was shown theoretically for the interaction is only temporary and the waves quickly re- rst time in 1973[1]. It was discovered that optical pulses cover their original shapes and velocities. This was sim- can propogate in an optical ber without changing their ilar to elastic collision of elementary particles and hence shapes if they experience anamolous dispersion. Experi- the word 'soliton' for the particle like behaviour. In 1967, mentally this was observed for the rst time in 1980[3]. Gardner et al found an exact solution to the KDV equa- Since then, there has been an enormous interest in the tion by what is now known as inverse scattering trans- eld of ber soliton and has promising potential in long- form. The area of soliton research in forefront right now haul ber-optic communication system. is the study of optical solitons, where the goal is to use these nonlinear pulses as the information carrying "bits" in optical bers. Optical solitons are dierent from the KDV solitons as TEMPORAL OPTICAL SOLITONS KDV solitons describe the solitary wave of a wave where as optical soliton in a ber describes that of an envelope. In order to understand ber solitons we should under- Optical solitons are localized electromagnetic waves that stand how optical pulses propogate inside single mode propogate steadily in nonlinear media resulting from a bers in the presence of dispersion(chromatic) and non- robust balance between nonlinearity and linear broaden- linearity (intensity dependence of ).The ing due to dispersion and/or diraction and have their pulse envelope is time dependent and is written as: roots in two very important scientic advances in 1960s: the developement of the mathematical theory of solitons E(r, t) = A(Z, t)F (X,Y ) exp(iβ0z) (1) 2

F(X,Y) is the transverse eld distribution associated with N = 1 (called fundamental solitons) but follows a peri- the fundamental mode of a single mode ber β0 is the odic pattern for integer values of N > 1 (called higher propogation constant in terms of optical wavelength β0 = order temporal solitons). The parameter N is related to 2πn0/λ. Here A being a function of time implies that all the input pulse parameters as spectral components of the pulse may not propagate at 2 2 (6) the same speed because of chromatic dispersion. The N = γP0LD = γP0T0 /|β2| refractive index is given as :

where, P0 is the pulse peak power. Only a fundamental 2 n˜ = n(ω) + n2|E| (2)

The frequency dependence of n(ω) plays an important role in the formation of temporal solitons and leads to broadening of pulses in the absence of nonlinear eects. The model equation which describes the envelope soliton propogation in a ber is known as nonlinear, or cubic schro¨dinger (NLSE) equation and is expressed as:

2 ∂u 1 ∂ u 2 (3) i − sgn(β2) 2 ± |u| u = 0 ∂z 2 ∂τ FIG. 1: Evolution of the rst order (left column) and third- In eqn(3), z represents the distance along the direction order (right column) temporal solitons over one soliton period. The bottom row shows the chirp prole of propogation, and τ represents the time (in the frame). The second term originates from group velocity dispersion and the third incorporates the nonlin- soliton maintains its shape and remains chirp-free during ear eect. The dierent terms in eqn(3) are as follows: propagation inside optical bers. This property makes fundamental soliton an ideal candidate for optical com- p munications. The eects of ber dipersion are exactly τ = (t − β1z)/T0, z = Z/LD, u = |γ|LDA, m balanced by ber nonlinearity when the input pulse has d β  βm = , γ = n2ω0/(cAeff ) a sech shape and its width and peak power are related dωm ω=ω0 by eqn(5) in such a way that N = 1. It has also been Here, T0 is the temporal scaling factor (generally the observed experimentally that optical solitons are remark- input pulse width), LD is the dispersion length, β1 = ably stable against perturbations i.e fundamental solitons 1/vg where vg is the group velocity associated with the can also be created when pulse shape and peak power pulse and β2 is the group velocity dispersion parameter deviates from the ideal conditions. Fig(2) shows the nu- (GVD) whose sign can be positive (normal) or negative merically simulated evolution of a gaussian input pulse (anamolous), depending onthe wavelength. From eqn(3) for which N=1. As seen there, the pulse adjusts its shape the necessary and sucient condition required for soliton and width to become a fundamental soliton and attains solution is that GVD and nonlinear term should have a sech prole for z >>1. This might seem counterintu- opposite signs. For anamolous dispersion the NLSE for itive but can be understood by thinking optical solitons bright solitons is hence given as: as temporal modes of a nonlinear . If the in- put pulse doesnot match the temporal mode precisly but is close to it, then also most of the energy is coupled to ∂u 1 ∂2u 2 (4) the temporal mode. The rest of the energy spreads as i + 2 + |u| u = 0 ∂z 2 ∂τ dispersive wave. Eqn (4) belongs to a special class of completely inte- grable systems and can be solved exactly by using the inverse scattering method and it has been shown that the solutions are solitons for the case when reection co- ecient vanishes for initial potential u(0,τ). A special role is played by those solitons whose initial amplitude at z = 0 is given by

u(0, τ) = Nsech(τ) (5) FIG. 2: Evolution of a gaussian input pulse with N = 1. The Doing the analysis for input pulse having an initial pulse evolves toward the fundamental soliton by changing its amplitude given by eqn(5), it is found that its shape re- shape, width and peak power. mains unchanged during propagation in the ber when 3

SOLITON BASED FIBER OPTIC Loss-Managed Solitons COMMUNICATION SPM eect is power dependent. As the pulse propa- An important application of solitons is the transmis- gates through the ber, it loses power due to ber losses, sion of information through optical bers. In this sec- there by weakening the SPM eect. As a result, the tion the structure of optical transmission systems will be SPM eect is not strong enough to counteract dispersion, considered and the concept of solitons as carriers of infor- which results in broadening of the pulse. This brings the mation will be introduced. The main obstacles related to need for ampliers in the system. There are two kinds soliton based transmission systems will be considered and of amplication techniques: Lumped- and Distributed- the ways to overcome this problem will be discussed. The amplication techniques. question how to improve the characteristics of existing Lumped-Amplication technique: In the Lumped optical networks by means of solitons will be answered. amplication scheme, optical ampliers are placed pe- riodically along the ber link such that the ber losses between two ampliers are exactly compensated by the amplier gain. An important design parameter is the am- Information Transmission with solitons plier spacing it should be as large as possible to mini- mize the cost. For non-soliton systems, it is typically 80- The basic idea is to use a soliton in each bit slot rep- 100km. For soliton systems using Lumped-amplication resenting 1 in a bit stream as shown in g(3).The neig- scheme, amplier spacing is restricted to much smaller bouring solitons in this scheme should be well seperated values. The reason being optical ampliers boost soliton and so the spacing between two solitons exceeds a few energy to the input level over a length of few meters with- out allowing for gradual recovery of fundamental soliton. times their (FWHM). The soliton width T0 is related to the B as: The amplied soliton adjusts its width dynamically in the ber section following the amplier. However, it also 1 1 sheds a part of its energy as dispersive waves during this B = = (7) adjustment phase, which can accumulate to signicant TB 2q0T0 levels over a large number of amplier stages and must be avoided. It can be shown that this eect can be mini- TB is the duration of the bit slot and 2q0 = TB/T0 is the distance between neighbouring solitons in normalised mized by keeping the amplier spacing LA much smaller units. than the dispersion length LD of the pulse. Distributed Amplication: The condition LA << LD, imposed on lumped amplied soliton systems, be- comes increasingly dicult to satisfy in practice as bit rates exceed 10Gbps. This condition can be relaxed con- siderably when Distributed amplication scheme is used. In this technique, signal is amplied locally at every point FIG. 3: Soliton bit stream in RZ format along the ber length.

Dispersion-Managed solitons

Soliton Interaction Dispersion management is employed commonly for modern WDM systems. It turns out that soliton sys-

The presence of solitons in the neighbouring bits per- tems benet considerably if the GVD parameter β2 varies turbs a soliton simply because the combined optical eld along the ber length. An interesting scheme proposed is not a solution to the NLSE. Nieghbouring solitons ei- in 1987 [6] relaxes completely the restriction LA << LD, ther come closer together or move apart because of non- imposed normally on loss-managed solitons, by decreas- linear interaction between them which introduces error ing the GVD along the ber length. Such bers are called in the data. However, a relatively large soliton spacing, dispersion-decreasing bers (DDFs) and are designed in necessary to avoid soliton interaction, limits the bit rate such a way that the decreasing GVD counteracts the re- of soliton communication system. The phenomenon of duced SPM experienced by solitons weakened from ber soliton interaction has been studied extensively[4][5]. It losses. The soliton would then remain unperturbed. Such has been found that this interaction can be reduced by systems are called Dispersion-Managed Solitons. Quali- using unequal amplitudes for neighbouring solitons. This tatively, it can be understood that since the soliton peak interaction can be modied by other factors, such as ini- power decreases exponentially in a lossy ber, the GVD tial frequency chirp imposed on input pulse. parameter also has to decrease exponentially with the 4 same loss coecient to counteract the reduced SPM. Rig- orous mathematical analysis gives the same result. The fundamental soliton then maintains its shape and width even in a lossy ber. Fabricating exponentially decreasing DDF bers: Fibers with a nearly exponential GVD prole have been fabricated. A practical technique for making such DDFs consists of reducing the core diameter along the ber FIG. 4: First experimental observation of soliton by Mol- lenauer et al. Theoretically derived peak power for the for- length in a controlled manner during the ber draw- mation of fundamental soliton was 1.2W ing process. Variations in the ber diameter change the waveguide contribution to the GVD parameter and re- duce its magnitude. Propagation of solitons in DDFs 4000-km using Raman amplication. The 55-ps pulses have been demonstrated in several experiments[7][8]. In could travel 4000-km without signicant change in their a 40km DDF, solitons preserved their width and shape pulse width, indicating soliton recovery over 4000-km. inspite of energy losses of more than 8dB[7]. In a recir- The experimental set up is shown in Fig(5). The suc- culating loop made using DDFs, a 6.5ps soliton train at cess of this experiment was quite promising to the idea 10Gbps could be transmitted over 300km[8]. of transoceanic soliton communication channel. Ra-

Amplier Noise and Timing Jitter

To take care of ber losses, optical ampliers are intro- duced in soliton systems. But these ampliers add noise originating from amplied spontaneous emission (ASE). The eect of ASE can be studied as a perturbation to the system and nding how soliton parameters in the per- turbed NLSE are aected. The eect of ASE is to change randomly these parameters at the output of each ampli- er. Variances of such uctuations for the soliton para- FIG. 5: Schematic diagram of the experiment of optical soli- ton transmission beyond 4000 km by repeated Raman ampli- meters can be calculated using the perturbation meth- cations ods. The results of this analysis shows that variances of both amplitude and frequency uctuations increase lin- early along the ber link because of cumulative eects of ASE. Amplitude uctuations degrade the SNR of the soliton bit stream. The SNR degradation, although unde- sirable, is not the most limiting factor. In fact, frequency uctuations aect system performance much more dras- tically by inducing the timing jitter. This can be under- stood by noting that as the soliton frequency uctuates randomly, its transit time through the ber link becomes random. ASE-induced uctuations in the arrival time of a soliton is referred to as the Gordon-Haus timing jitter. This puts an upper limit on the bit rate-length product.

EXPERIMENTAL PROGRESS FIG. 6: Measured eye diagram of 40Gbps soliton pulses af- The successful experimental demonstration of the exis- ter 10000-km of transmission. Upper plot is of input signal tence of optical solitons[3] g(4) sparked a lot of interest and bottom represents output signal. The opening of the eye in the eld. One revolutionary idea was the possibility indicates error free transmission. of an all-optical transmission system based only on op- tical ampliers instead of regenrative repeaters, which man ampliers required very high power (500mW of CW) were considered standard until late 80s. In particluar, which was not possible at that time from semiconductor using the Raman eect of transmission ber itself for lasers and the color-center lasers were too bulky for prac- optical amplication[9]. This was successfully demon- tical purposes. The situation changed with the advent of strated in 1988 [10] when solitons were transmitted over Erbium Doped Fiber Ampliers (EDFAs) around 1989. 5

Several experiments were performed on this during the discovery of optical solitons in bers and its experimen- early 1990s. These experiments can be divided into two tal conrmation. However, attempts for its practical im- categories, depending on whether a linear ber link or plementation have been a real challenge for the soliton a recirculating ber loop is used. In 1991 experiment, community for more than twenty years because of the 2.5Gb/s solitons were transmitted over 12,000km using ever-increasing demands for higher bit rate and longer a 75-km ber loop containing three EDFAs. The con- distance transmission as well as the successful improve- cept of Dispersion-Managed soliton which showed up in ment of linear transmission quality owing to the continu-

1987 helped in relaxing the condition of LA<< LD on ous improvements in bers and ampliers. The potential amplier spacing. DMSs have been experimented within still exists and poses a challenge to the soliton commu- both single-channel ultra-high speed transmissions and in nity. wavelength division multiplexed (WDM) transmissions. Moria et al[11] succeeded in achieving a record transmis- sion of 40Gbit/s single-channel DMS transmission over 10000km. Fukuchi et al[12] succeeded in the transmis- sion of 1.1Tbit/s (20 Gbit/s times 55 channels) WDM DMSs over 3000km.

LATEST ISSUES REFERENCES

Despite the remarkable success of DMSs, there remain a number of unsolved problems, one of them is their ap- plication to dense WDM systems. In dense WDM sys- tems, in order to maximize the bandwidth eciency, a ∗ Electronic address: [email protected] large number of channels are packed into a limited band- † Electronic address: [email protected] width. It has been found that DMSs could be destroyed [1] A.Hasegawa and F.D.Tappert, Appl.Phys.Lett.23, 142 if channel spacing were too narrow. In addition, when (1973). two solitons at neighbouring channels overlap in time [2] S.L.McCall and E.L.Hahn, Phys.Rev.Lett.18,908(1967). at the input, large temporal position shifts results. An- [3] L.F.Mollenauer, R.H.Stolen and J.P.Gordon, other unsolved problem is that of polarization mode dis- Phys.Rev.Lett.45, 1095 (1980). [4] F.M.Mitschke and L.F.Mollenauer,Opt.Lett12,355(1987) persion (PMD). In optical bers, the degeneracy of two [5] Y.Kodama and K.Nozaki,Opt.Lett.12,1038(1987) orthogonal polarizations is generally broken down be- [6] K.Tajima,Opt.Lett.12,54(1987) cause these two polarization states generally have dif- [7] A.J.Stenz, R.Boyd and ferent propagation characteristics. As a result the NLSE A.F.Evans,Opt.Lett.20,1770(1995) becomes a coupled equation that represents the two po- [8] D.J.Richardson, L.Dong, R.P.Chamberlain, larizations having two dierent group velocities and dis- A.D.Ellis,T.Widdowson and W.A.Pender,Electron persions. Simulation results based on the numerical solu- Lett.32,373(1996) [9] A.Hasegawa,Opt.Lett.8,650(1983) tion of these coupled equation have shown that the soli- [10] L.F.Mollenauer and K.Smith,Opt.Lett.13,675(1988) ton pulse should broaden and its broadening is propor- [11] I.Morita, K.Tanaka, N.Edagawa and M.Suzuki, Proceed- tional to the square-root of distance z. The challenge for ings of the 1998 European Conference on Optical com- the soliton community is to demonstrate the merit of soli- munication 3,47-52,(1998) ton for a WDM system. This could be achieved through [12] K.Fukuchi, M.Kakui,A.Sasaki,T.Ito,Y.Inada,T.Suzuki,T.Shitomi,K.Fuji,S.Shikii,H.Sugahara the reduction of non-linear cross-talk either by means of and A.Hasegawa, Proceedings of European conference a strong local dispersion, by densely managed dispersion, on Optical Communication (ECOC99),paper PD2-10 [13] L.du Mouza, E.Seve, H.Mardoyan, S.Wabnitz, P.Sillard, by a doubly periodic dispersion management, or by tak- and P.Nouchi, OPt.Lett.26,1128-30(2001) ing advantage of the intrinsic nature of a soliton. In this [14] A.Hasegawa,Optical Solitons in Fibers, volume116, regard the recent experimental demonstration of superior Springer-Verlag, Berlin, Heidelberg (1989) transmission quality of high order DMSs in dense WDM [15] Govind P. Agrawal Contemporary Nonlinear Optics, Aca- by du Mouza et al[13]deserves special attention. A clear demic Press (1992) demonstration of DMSs withstanding PMD is also de- [16] Yuri S. Kivshar and Govind P. Agrawal,Optical Solitons, sired. Elsevier Science & Technology Books, San Diego, USA, (2003) CONCLUSION [17] Robert W. Boyd, Nonlinear Optics, 2nd edn.,Academic Press (2002) Soliton has promised a lot of potentiality since its the- [18] Ajoy Ghatak and K. Thyagarajan, Introduction to Fiber ory was put forward. Seven years elapsed between the Optics,1st edn., Cambridge University Press (1998)