Chapter 9 Soliton Systems The word soliton was coined in 1965 to describe the particle-like properties of pulses propagating in a nonlinear medium [1]. The pulse envelope for solitons not only prop- agates undistorted but also survives collisions just as particles do. The existence of solitons in optical fibers and their use for optical communications were suggested in 1973 [2], and by 1980 solitons had been observed experimentally [3]. The potential of solitons for long-haul communication was first demonstrated in 1988 in an experi- ment in which fiber losses were compensated using the technique of Raman amplifica- tion [4]. Since then, a rapid progress during the 1990s has converted optical solitons into a practical candidate for modern lightwave systems [5]–[9]. In this chapter we fo- cus on soliton communication systems with emphasis on the physics and design of such systems. The basic concepts behind fiber solitons are introduced in Section 9.1, where we also discuss the properties of such solitons. Section 9.2 shows how fiber solitons can be used for optical communications and how the design of such lightwave systems differs from that of conventional systems. The loss-managed and dispersion-managed solitons are considered in Sections 9.3 and 9.4, respectively. The effects of amplifier noise on such solitons are discussed in Section 9.5 with emphasis on the timing-jitter issue. Section 9.6 focuses on the design of high-capacity single-channel systems. The use of solitons for WDM lightwave systems is discussed in Section 9.7. 9.1 Fiber Solitons The existence of solitons in optical fibers is the result of a balance between the group- velocity dispersion (GVD) and self-phase modulation (SPM), both of which, as dis- cussed in Sections 2.4 and 5.3, limit the performance of fiber-optic communication systems when acting independently on optical pulses propagating inside fibers. One can develop an intuitive understanding of how such a balance is possible by following the analysis of Section 2.4. As shown there, the GVD broadens optical pulses during their propagation inside an optical fiber except when the pulse is initially chirped in the right way (see Fig. 2.12). More specifically, a chirped pulse can be compressed during the early stage of propagation whenever the GVD parameter β 2 and the chirp parameter 404 Fiber-Optic Communications Systems, Third Edition. Govind P. Agrawal Copyright  2002 John Wiley & Sons, Inc. ISBNs: 0-471-21571-6 (Hardback); 0-471-22114-7 (Electronic) 9.1. FIBER SOLITONS 405 C happen to have opposite signs so that β 2 C is negative. The nonlinear phenomenon of SPM imposes a chirp on the optical pulse such that C > 0. Since β 2 < 0 in the 1.55- µ m wavelength region, the condition β 2 C < 0 is readily satisfied. Moreover, as the SPM-induced chirp is power dependent, it is not difficult to imagine that under certain conditions, SPM and GVD may cooperate in such a way that the SPM-induced chirp is just right to cancel the GVD-induced broadening of the pulse. The optical pulse would then propagate undistorted in the form of a soliton. 9.1.1 Nonlinear Schr ¨ odinger Equation The mathematical description of solitons employs the nonlinear Schr¨odinger (NLS) equation, introduced in Section 5.3 [Eq. (5.3.1)] and satisfied by the pulse envelope A(z,t) in the presence of GVD and SPM. This equation can be written as [10] ∂ A ∂ z + i β 2 2 ∂ 2 A ∂ t 2 − β 3 6 ∂ 3 A ∂ t 3 = i γ |A| 2 A − α 2 A, (9.1.1) where fiber losses are included through the α parameter while β 2 and β 3 account for the second- and third-order dispersion (TOD) effects. The nonlinear parameter γ = 2 π n 2 /( λ A eff ) is defined in terms of the nonlinear-index coefficient n 2 , the optical wavelength λ , and the effective core area A eff introduced in Section 2.6. To discuss the soliton solutions of Eq. (9.1.1) as simply as possible, we first set α = 0 and β 3 = 0 (these parameters are included in later sections). It is useful to write this equation in a normalized form by introducing τ = t T 0 , ξ = z L D , U = A √ P 0 , (9.1.2) where T 0 is a measure of the pulse width, P 0 is the peak power of the pulse, and L D = T 2 0 /| β 2 | is the dispersion length. Equation (9.1.1) then takes the form i ∂ U ∂ξ − s 2 ∂ 2 U ∂τ 2 + N 2 |U| 2 U = 0, (9.1.3) where s = sgn( β 2 )=+1or−1, depending on whether β 2 is positive (normal GVD) or negative (anomalous GVD). The parameter N is defined as N 2 = γ P 0 L D = γ P 0 T 2 0 /| β 2 |. (9.1.4) It represents a dimensionless combination of the pulse and fiber parameters. The phys- ical significance of N will become clear later. The NLS equation is well known in the soliton literature because it belongs to a special class of nonlinear partial differential equations that can be solved exactly with a mathematical technique known as the inverse scattering method [11]–[13]. Although the NLS equation supports solitons for both normal and anomalous GVD, pulse-like solitons are found only in the case of anomalous dispersion [14]. In the case of normal dispersion ( β 2 > 0), the solutions exhibit a dip in a constant-intensity background. Such solutions, referred to as dark solitons, are discussed in Section 9.1.3. This chapter focuses mostly on pulse-like solitons, also called bright solitons. 406 CHAPTER 9. SOLITON SYSTEMS 9.1.2 Bright Solitons Consider the case of anomalous GVD by setting s = −1 in Eq. (9.1.3). It is common to introduce u = NU as a renormalized amplitude and write the NLS equation in its canonical form with no free parameters as i ∂ u ∂ξ + 1 2 ∂ 2 u ∂τ 2 + |u| 2 u = 0. (9.1.5) This equation has been solved by the inverse scattering method [14]. Details of this method are available in several books devoted to solitons [11]–[13]. The main result can be summarized as follows. When an input pulse having an initial amplitude u(0, τ )=N sech( τ ) (9.1.6) is launched into the fiber, its shape remains unchanged during propagation when N = 1 but follows a periodic pattern for integer values of N > 1 such that the input shape is recovered at ξ = m π /2, where m is an integer. An optical pulse whose parameters satisfy the condition N = 1 is called the fun- damental soliton. Pulses corresponding to other integer values of N are called higher- order solitons. The parameter N represents the order of the soliton. By noting that ξ = z/L D , the soliton period z 0 , defined as the distance over which higher-order soli- tons recover their original shape, is given by z 0 = π 2 L D = π 2 T 2 0 | β 2 | . (9.1.7) The soliton period z 0 and soliton order N play an important role in the theory of optical solitons. Figure 9.1 shows the pulse evolution for the first-order (N = 1) and third-order (N = 3) solitons over one soliton period by plotting the pulse intensity |u( ξ , τ )| 2 (top row) and the frequency chirp (bottom row) defined as the time derivative of the soliton phase. Only a fundamental soliton maintains its shape and remains chirp-free during propagation inside optical fibers. The solution corresponding to the fundamental soliton can be obtained by solving Eq. (9.1.5) directly, without recourse to the inverse scattering method. The approach consists of assuming that a solution of the form u( ξ , τ )=V( τ )exp[i φ ( ξ )] (9.1.8) exists, where V must be independent of ξ for Eq. (9.1.8) to represent a fundamental soliton that maintains its shape during propagation. The phase φ can depend on ξ but is assumed to be time independent. When Eq. (9.1.8) is substituted in Eq. (9.1.5) and the real and imaginary parts are separated, we obtain two real equations for V and φ . These equations show that φ should be of the form φ ( ξ )=K ξ , where K is a constant. The function V ( τ ) is then found to satisfy the nonlinear differential equation d 2 V d τ 2 = 2V (K −V 2 ). (9.1.9) 9.1. FIBER SOLITONS 407 Figure 9.1: Evolution of the first-order (left column) and third-order (right column) solitons over one soliton period. Top and bottom rows show the pulse shape and chirp profile, respectively. This equation can be solved by multiplying it by 2 (dV /d τ ) and integrating over τ . The result is given as (dV /d τ ) 2 = 2KV 2 −V 4 +C, (9.1.10) where C is a constant of integration. Using the boundary condition that both V and dV /d τ should vanish at | τ | = ∞ for pulses, C is found to be 0. The constant K is de- termined using the other boundary condition that V = 1 and dV /d τ = 0 at the soliton peak, assumed to occur at τ = 0. Its use provides K = 1 2 , and hence φ = ξ /2. Equa- tion (9.1.10) is easily integrated to obtain V ( τ )=sech( τ ). We have thus found the well-known “sech” solution [11]–[13] u( ξ , τ )=sech( τ )exp(i ξ /2) (9.1.11) for the fundamental soliton by integrating the NLS equation directly. It shows that the input pulse acquires a phase shift ξ /2 as it propagates inside the fiber, but its amplitude remains unchanged. It is this property of a fundamental soliton that makes it an ideal candidate for optical communications. In essence, the effects of fiber dispersion are exactly compensated by the fiber nonlinearity when the input pulse has a “sech” shape and its width and peak power are related by Eq. (9.1.4) in such a way that N = 1. An important property of optical solitons is that they are remarkably stable against perturbations. Thus, even though the fundamental soliton requires a specific shape and 408 CHAPTER 9. SOLITON SYSTEMS Figure 9.2: Evolution of a Gaussian pulse with N = 1 over the range ξ = 0–10. The pulse evolves toward the fundamental soliton by changing its shape, width, and peak power. a certain peak power corresponding to N = 1 in Eq. (9.1.4), it can be created even when the pulse shape and the peak power deviate from the ideal conditions. Figure 9.2 shows the numerically simulated evolution of a Gaussian input pulse for which N = 1 but u(0, τ )=exp(− τ 2 /2). As seen there, the pulse adjusts its shape and width in an attempt to become a fundamental soliton and attains a “sech” profile for ξ  1. A similar behavior is observed when N deviates from 1. It turns out that the Nth-order soliton can be formed when the input value of N is in the range N − 1 2 to N + 1 2 [15]. In particular, the fundamental soliton can be excited for values of N in the range 0.5 to 1.5. Figure 9.3 shows the pulse evolution for N = 1.2 over the range ξ = 0–10 by solving the NLS equation numerically with the initial condition u(0, τ )=1.2sech( τ ). The pulse width and the peak power oscillate initially but eventually become constant after the input pulse has adjusted itself to satisfy the condition N = 1 in Eq. (9.1.4). It may seem mysterious that an optical fiber can force any input pulse to evolve toward a soliton. A simple way to understand this behavior is to think of optical solitons as the temporal modes of a nonlinear waveguide. Higher intensities in the pulse center create a temporal waveguide by increasing the refractive index only in the central part of the pulse. Such a waveguide supports temporal modes just as the core-cladding index difference led to spatial modes in Section 2.2. When an input pulse does not match a temporal mode precisely but is close to it, most of the pulse energy can still be coupled into that temporal mode. The rest of the energy spreads in the form of dispersive waves. It will be seen later that such dispersive waves affect the system performance and should be minimized by matching the input conditions as close to the ideal requirements as possible. When solitons adapt to perturbations adiabatically, perturbation theory developed specifically for solitons can be used to study how the soliton amplitude, width, frequency, speed, and phase evolve along the fiber. 9.1. FIBER SOLITONS 409 Figure 9.3: Pulse evolution for a “sech” pulse with N = 1.2 over the range ξ = 0–10. The pulse evolves toward the fundamental soliton (N = 1) by adjusting its width and peak power. 9.1.3 Dark Solitons The NLS equation can be solved with the inverse scattering method even in the case of normal dispersion [16]. The intensity profile of the resulting solutions exhibits a dip in a uniform background, and it is the dip that remains unchanged during propagation inside the fiber [17]. For this reason, such solutions of the NLS equation are called dark solitons. Even though dark solitons were discovered in the 1970s, it was only after 1985 that they were studied thoroughly [18]–[28]. The NLS equation describing dark solitons is obtained from Eq. (9.1.5) by changing the sign of the second term. The resulting equation can again be solved by postulating a solution in the form of Eq. (9.1.8) and following the procedure outlined there. The general solution can be written as [28] u d ( ξ , τ )=( η tanh ζ −i κ )exp(iu 2 0 ξ ), (9.1.12) where ζ = η ( τ − κξ ), η = u 0 cos φ , κ = u 0 sin φ . (9.1.13) Here, u 0 is the amplitude of the continuous-wave (CW) background and φ is an internal phase angle in the range 0 to π /2. An important difference between the bright and dark solitons is that the speed of a dark soliton depends on its amplitude η through φ .For φ = 0, Eq. (9.1.12) reduces to u d ( ξ , τ )=u 0 tanh(u 0 τ )exp(iu 2 0 ξ ). (9.1.14) The peak power of the soliton drops to zero at the center of the dip only in the φ = 0 case. Such a soliton is called the black soliton. When φ = 0, the intensity does not drop to zero at the dip center; such solitons are referred to as the gray soliton. Another 410 CHAPTER 9. SOLITON SYSTEMS Figure 9.4: (a) Intensity and (b) phase profiles of dark solitons for several values of the internal phase φ . The intensity drops to zero at the center for black solitons. interesting feature of dark solitons is related to their phase. In contrast with bright solitons which have a constant phase, the phase of a dark soliton changes across its width. Figure 9.4 shows the intensity and phase profiles for several values of φ .For a black soliton ( φ = 0), a phase shift of π occurs exactly at the center of the dip. For other values of φ , the phase changes by an amount π −2 φ in a more gradual fashion. Dark solitons were observed during the 1980s in several experiments using broad optical pulses with a narrow dip at the pulse center. It is important to incorporate a π phase shift at the pulse center. Numerical simulations show that the central dip can propagate as a dark soliton despite the nonuniform background as long as the back- ground intensity is uniform in the vicinity of the dip [18]. Higher-order dark solitons do not follow a periodic evolution pattern similar to that shown in Fig. 9.1 for the third- order bright soliton. The numerical results show that when N > 1, the input pulse forms a fundamental dark soliton by narrowing its width while ejecting several dark-soliton pairs in the process. In a 1993 experiment [19], 5.3-ps dark solitons, formed on a 36-ps wide pulse from a 850-nm Ti:sapphire laser, were propagated over 1 km of fiber. The same technique was later extended to transmit dark-soliton pulse trains over 2 km of fiber at a repetition rate of up to 60 GHz. These results show that dark solitons can be generated and maintained over considerable fiber lengths. Several practical techniques were introduced during the 1990s for generating dark solitons. In one method, a Mach–Zehnder modulator driven by nearly rectangular elec- trical pulses, modulates the CW output of a semiconductor laser [20]. In an extension of this method, electric modulation is performed in one of the arms of a Mach–Zehnder in- terferometer. A simple all-optical technique consists of propagating two optical pulses, with a relative time delay between them, in the normal-GVD region of the fiber [21]. The two pulses broaden, become chirped, and acquire a nearly rectangular shape as they propagate inside the fiber. As these chirped pulses merge into each other, they interfere. The result at the fiber output is a train of isolated dark solitons. In another all-optical technique, nonlinear conversion of a beat signal in a dispersion-decreasing 9.2. SOLITON-BASED COMMUNICATIONS 411 fiber was used to generate a train of dark solitons [22]. A 100-GHz train of 1.6-ps dark solitons was generated with this technique and propagated over 2.2 km of (two soliton periods) of a dispersion-shifted fiber. Optical switching using a fiber-loop mirror, in which a phase modulator is placed asymmetrically, can also produce dark solitons [23]. In another variation, a fiber with comb-like dispersion profile was used to generate dark soliton pulses with a width of 3.8 ps at the 48-GHz repetition rate [24]. An interesting scheme uses electronic circuitry to generate a coded train of dark solitons directly from the nonreturn-to-zero (NRZ) data in electric form [25]. First, the NRZ data and its clock at the bit rate are passed through an AND gate. The re- sulting signal is then sent to a flip-flop circuit in which all rising slopes flip the signal. The resulting electrical signal drives a Mach–Zehnder LiNbO 3 modulator and converts the CW output from a semiconductor laser into a coded train of dark solitons. This technique was used for data transmission, and a 10-Gb/s signal was transmitted over 1200 km by using dark solitons. Another relatively simple method uses spectral filter- ing of a mode-locked pulse train through a fiber grating [26]. This scheme has also been used to generate a 6.1-GHz train and propagate it over a 7-km-long fiber [27]. Numerical simulations show that dark solitons are more stable in the presence of noise and spread more slowly in the presence of fiber losses compared with bright solitons. Although these properties point to potential application of dark solitons for optical communications, only bright solitons were being pursued in 2002 for commercial ap- plications. 9.2 Soliton-Based Communications Solitons are attractive for optical communications because they are able to maintain their width even in the presence of fiber dispersion. However, their use requires sub- stantial changes in system design compared with conventional nonsoliton systems. In this section we focus on several such issues. 9.2.1 Information Transmission with Solitons As discussed in Section 1.2.3, two distinct modulation formats can be used to generate a digital bit stream. The NRZ format is commonly used because the signal bandwidth is about 50% smaller for it compared with that of the RZ format. However, the NRZ format cannot be used when solitons are used as information bits. The reason is easily understood by noting that the pulse width must be a small fraction of the bit slot to ensure that the neighboring solitons are well separated. Mathematically, the soliton solution in Eq. (9.1.11) is valid only when it occupies the entire time window (−∞ < τ < ∞). It remains approximately valid for a train of solitons only when individual solitons are well isolated. This requirement can be used to relate the soliton width T 0 to the bit rate B as B = 1 T B = 1 2q 0 T 0 , (9.2.1) where T B is the duration of the bit slot and 2q 0 = T B /T 0 is the separation between neighboring solitons in normalized units. Figure 9.5 shows a soliton bit stream in the 412 CHAPTER 9. SOLITON SYSTEMS Figure 9.5: Soliton bit stream in RZ format. Each soliton occupies a small fraction of the bit slot so that neighboring soliton are spaced far apart. RZ format. Typically, spacing between the solitons exceeds four times their full width at half maximum (FWHM). The input pulse characteristics needed to excite the fundamental soliton can be obtained by setting ξ = 0 in Eq. (9.1.11). In physical units, the power across the pulse varies as P(t)=|A(0,t)| 2 = P 0 sech 2 (t/T 0 ). (9.2.2) The required peak power P 0 is obtained from Eq. (9.1.4) by setting N = 1 and is related to the width T 0 and the fiber parameters as P 0 = | β 2 |/( γ T 2 0 ). (9.2.3) The width parameter T 0 is related to the FWHM of the soliton as T s = 2T 0 ln(1 + √ 2)  1.763T 0 . (9.2.4) The pulse energy for the fundamental soliton is obtained using E s =  ∞ −∞ P(t)dt = 2P 0 T 0 . (9.2.5) Assuming that 1 and 0 bits are equally likely to occur, the average power of the RZ signal becomes ¯ P s = E s (B/2)=P 0 /2q 0 . As a simple example, T 0 = 10 ps for a 10-Gb/s soliton system if we choose q 0 = 5. The pulse FWHM is about 17.6 ps for T 0 = 10 ps. The peak power of the input pulse is 5 mW using β 2 = −1ps 2 /km and γ = 2W −1 /km as typical values for dispersion-shifted fibers. This value of peak power corresponds to a pulse energy of 0.1 pJ and an average power level of only 0.5 mW. 9.2.2 Soliton Interaction An important design parameter of soliton lightwave systems is the pulse width T s .As discussed earlier, each soliton pulse occupies only a fraction of the bit slot. For practical reasons, one would like to pack solitons as tightly as possible. However, the presence of pulses in the neighboring bits perturbs the soliton simply because the combined optical field is not a solution of the NLS equation. This phenomenon, referred to as soliton interaction, has been studied extensively [29]–[33]. 9.2. SOLITON-BASED COMMUNICATIONS 413 Figure 9.6: Evolution of a soliton pair over 90 dispersion lengths showing the effects of soliton interaction for four different choices of amplitude ratio r and relative phase θ . Initial spacing q 0 = 3.5 in all four cases. One can understand the implications of soliton interaction by solving the NLS equa- tion numerically with the input amplitude consisting of a soliton pair so that u(0, τ )=sech( τ −q 0 )+r sech[r( τ + q 0 )]exp(i θ ), (9.2.6) where r is the relative amplitude of the two solitons, θ is the relative phase, and 2q 0 is the initial (normalized) separation. Figure 9.6 shows the evolution of a soliton pair with q 0 = 3.5 for several values of the parameters r and θ . Clearly, soliton interaction depends strongly both on the relative phase θ and the amplitude ratio r. Consider first the case of equal-amplitude solitons (r = 1). The two solitons at- tract each other in the in-phase case ( θ = 0) such that they collide periodically along the fiber length. However, for θ = π /4, the solitons separate from each other after an initial attraction stage. For θ = π /2, the solitons repel each other even more strongly, and their spacing increases with distance. From the standpoint of system design, such behavior is not acceptable. It would lead to jitter in the arrival time of solitons because the relative phase of neighboring solitons is not likely to remain well controlled. One way to avoid soliton interaction is to increase q 0 as the strength of interaction depends on soliton spacing. For sufficiently large q 0 , deviations in the soliton position are ex- pected to be small enough that the soliton remains at its initial position within the bit slot over the entire transmission distance. [...]... solitons based on these two amplification schemes CHAPTER 9 SOLITON SYSTEMS 420 Figure 9.10: (a) Lumped and (b) distributed amplification schemes for compensation of fiber losses in soliton communication systems 9.3.2 Lumped Amplification The lumped amplification scheme shown in Fig 9.10 is the same as that used for nonsoliton systems In both cases, optical amplifiers are placed periodically along the fiber link... only 83% of the input energy is converted into a soliton for the case C = 0.5 shown in Fig 9.7, and this fraction reduces to 62% when C = 0.8 416 CHAPTER 9 SOLITON SYSTEMS 9.2.4 Soliton Transmitters Soliton communication systems require an optical source capable of producing chirpfree picosecond pulses at a high repetition rate with a shape as close to the “sech” shape as possible The source should operate... erbium-doped fiber amplifiers (EDFAs) can be used for compensating them Semiconductor lasers, commonly used for nonsoliton lightwave systems, remain the lasers of choice even for soliton systems Early experiments on soliton transmission used the technique of gain switching for generating optical pulses of 20–30 ps duration by biasing the laser below threshold and pumping it high above threshold periodically [38]–[40]... were simultaneously amplified and compressed inside an EDFA after first passing them through a narrowband optical filter [40] It was possible to generate 17-ps-wide, nearly chirp-free, optical pulses at repetition rates in the range 6–24 GHz Mode-locked semiconductor lasers are also suitable for soliton communications and are often preferred because the pulse train emitted from such lasers is nearly chirpfree... modulator, driven by an electrical data stream in the NRZ format, to convert the CW output of a DFB laser into an optical bit stream in the RZ format [53] Although optical pulses launched from such transmitters typically do not have the “sech” shape of a soliton, they can be used for soliton systems because of the soliton-formation capability of the fiber discussed earlier 9.3 Loss-Managed Solitons As... capacity The design of such systems would, however, require attention to many details Soliton interaction can also be modified by other factors, such as the initial frequency chirp imposed on input pulses 9.2.3 Frequency Chirp To propagate as a fundamental soliton inside the optical fiber, the input pulse should not only have a “sech” profile but also be chirp-free Many sources of short optical pulses have a... between amplifiers—it should be as large as possible to minimize the overall cost For nonsoliton systems, L A is typically 80–100 km For soliton systems, L A is restricted to much smaller values because of the soliton nature of signal propagation [57] The physical reason behind smaller values of L A is that optical amplifiers boost soliton energy to the input level over a length of few meters without allowing... chaotic behavior [60] For this reason, distributed amplification is used with ξA < 4π in practice [62]–[66] Modeling of soliton communication systems making use of distributed amplification requires the addition of a gain term to the NLS equation, as in Eq (9.3.4) In the case of soliton systems operating at bit rates B > 20 Gb/s such that T0 < 5 ps, it is also necessary to include the effects of third-order... 1988, and the color-center lasers used in the experiment were too bulky to be useful for practical lightwave systems The situation changed with the advent of EDFAs around 1989 when several experiments used them for loss-managed soliton systems [38]–[40] These experiments can 426 CHAPTER 9 SOLITON SYSTEMS Figure 9.13: Setup used for soliton transmission in a 1990 experiment Two EDFAs after the LiNbO3 modulator... Nevertheless, its parameters repeat from period to period at any location within the map For this reason, DM solitons can be used for optical communications in spite of oscillations in the pulse width Moreover, such solitons perform better from a system standpoint CHAPTER 9 SOLITON SYSTEMS 432 5 4 4 4 3 3 3 3 2 2 2 2 1 1 1 1 0 0 0 0 -1 -1 -1 -1 -2 10 -2 -2 0 1 2 3 4 5 6 7 8 9 Chirp 0 1 2 Distance (km) 3 4 . converted optical solitons into a practical candidate for modern lightwave systems [5]–[9]. In this chapter we fo- cus on soliton communication systems with. solitons can be used for optical communications and how the design of such lightwave systems differs from that of conventional systems. The loss-managed