Chapter 8'Nonlinearity Management

1
Chapter 8.Nonlinearity Management

• 8.1 Role of Fiber Nonlinearity
• 8.1.1 System Design Issues
• 8.1.2 Semianalytic Approach
• 8.1.3 Soliton and Pseudo-linear Regimes
• 8.2 Solitons in Optical Fibers
• 8.2.1 Properties of Optical Solitons
• 8.2.2 Loss-Managed Solitons
• 8.3 Dispersion-Managed Solitons
• 8.3.1 Dispersion-Decreasing Fibers
• 8.3.2 Periodic Dispersion Maps
• 8.3.3 Design Issues of DM Solitons

2
Chapter 8.Nonlinearity Management

• 8.4 Pseudo-linear Lightwave Systems
• 8.4.1 Intrachannel Nonlinear Effects
• 8.4.2 Intrachannel XPM
• 8.4.3 Intrachannel FWM
• 8.5 Control of Intrachannel Nonlinear Effects
• 8.5.1 Optimization of Dispersion Maps
• 8.5.2 Phase-Alternation Techniques
• 8.5.3 Polarization Bit Interleaving

3
Chapter 8.Nonlinearity Management

• The use of dispersion compensation solves the
  dispersion problem, just as optical amplifiers
  solve the loss problem for lightwave systems.
• However, noise added by optical amplifiers forces
  one to launch into a fiber link an average
  power level close to 1 mW or more for each
  channel.
• At such power levels, nonlinear effects will
  impact considerably on the performance of a
  long-haul lightwave system.
• In fact, along with amplifier noise, the
  nonlinear nature of optical fibers is the
  ultimate limiting factor for such systems.

4
8.1 Role of Fiber Nonlinearity

• The use of dispersion management in combination
  with optical amplifiers can extend the
  transmission distance to several thousand
  kilometers.
• If the optical signal is regenerated
  electronically every 300 to 400 km, such a system
  works well as the nonlinear effects do not
  accumulate over long lengths.
• In contrast, if the signal is maintained in the
  optical domain by cascading many amplifiers,
  several nonlinear effects, such as self-phase
  modulation (SPM), cross-phase modulation (XPM),
  and four-wave mixing (FWM), would ultimately
  limit the system performance.

5
8.1.1 System Design Issues

• In the absence of nonlinear effects, the use of
  dispersion management ensures that each pulse is
  confined to its bit slot when the optical signal
  arrives at the receiver, even if pulses have
  spread over multiple slots during their
  transmission.
• Any dispersion map can be used as long as the
  accumulated group-velocity dispersion (GVD)
• at
  the end of a link of length L.
• Different dispersion maps can lead to different
  Q factors at the receiver end even when
  da(L) 0 for all of them.

6
8.1.1 System Design Issues

• The dispersive and nonlinear effects do not act
  on the signal independently. As a result,
  degradation induced by the nonlinear effects
  depends on the local value of da(z) at any
  distance z within the fiber link.
• The major nonlinear phenomenon affecting the
  performance of a single channel is the SPM.
• And the propagation of an optical bit stream
  inside a dispersion-managed system is
  governed by the nonlinear Schrodinger (NLS)
  equation.

7
8.1.1 System Design Issues

• From Eq. (6.1.1), it can be written as
• where we have ignored the noise term to
  simplify the discussions.
• In a dispersion-managed system the three fiber
  parameters (b2, g, and a) are functions of z
  because of their different values in two or
  more fiber sections used to form the dispersion
  map.

8
8.1.1 System Design Issues

• The gain parameter g0 is also a function of z
  because of loss management. Its functional
  form depends on whether a lumped or a distributed
  amplification scheme is employed.
• In general, Eq. (8.1.1) is solved numerically to
  study the performance of dispersion-managed
  systems.
• It is useful to eliminate the gain and loss terms
  in this equation with the transformation
  and write it in
  terms of U(z, t) as

9
8.1.1 System Design Issues

• where P0 is the input peak power and p(z) governs
  variations in the peak power of the signal along
  the fiber link through
• If losses are compensated in a periodic fashion,
  p(zm) 1, where zm mLA is the location of the
  m-th amplifier and LA is the amplifier spacing.

10
8.1.1 System Design Issues

• In the case of lumped amplifiers, g0 0 within
  the fiber link, and
  . Equ. (8.1.2) shows that the
  effective nonlinear parameter ge(z) gp(z) is
  also z-dependent because of changes in the signal
  power induced by fiber losses and optical
  amplifiers.
• In particular, when lumped amplifiers are used,
  the nonlinear effects are strongest just
  after signal amplification and become negligible
  in the tail end of each fiber section
  between two amplifiers.

11
8.1.1 System Design Issues

• There are two major design issues for any
  dispersion-managed system What is the optimum
  dispersion map and which modulation format
  provides the best performance. Both of these
  issues have been studied by solving the NLS
  equation (8.1.2) numerically.
• Figure 8.1 shows the numerical results for the
  (a) NRZ and (b) RZ formats by plotting the max.
  transmission distance L at which eye opening is
  reduced by 1 dB at the receiver of a 40-Gb/s
  system as average launched power is increased.

12
8.1.1 System Design Issues

• Figure 8.1 Maximum transmission distance as a
  function of average input power for a 40- Gb/s
  dispersion-managed system designed with the (a)
  NRZ and (b) RZ formats. The filled and empty
  symbols show numerical data obtained with and
  without amplifier noise, respectively.

13
8.1.1 System Design Issues

• The periodic dispersion map consisted of 50 km of
  standard fiber with D 16 ps/(km-nm), a 0.2
  dB/km , and g 1.31W-1/km, followed by 10 km of
  dispersion-compensating fiber (DCF) with D -80
  ps/(km-nm), a 0.5 dB/km, and g 5.24
  W-1/km.
• Optical amplifiers with 6-dB noise figure were
  placed 60 km apart and compensated total fiber
  losses within each map period. The duty cycle was
  50 in the case of the RZ format.

14
8.1.1 System Design Issues

• As evident from Figure 8.1, distance can be
  continuously increased in the absence of
  amplifier noise by decreasing the launched power
  (open squares).
• However, when noise is included, an optimum power
  level exists for which the link length is
  maximum.
• This distance is lt 400 km when the NRZ format is
  employed but becomes 3 times larger when the RZ
  format is implemented with a 50 duty cycle.
• The reason behind this improvement can be
  understood by noting that the dispersion length
  is relatively small (lt5 km) for RZ pulses
  propagating inside a standard fiber.

15
8.1.1 System Design Issues

• As a result, RZ-format pulses spread quickly and
  their peak power is reduced considerably compared
  with the NRZ case. This reduction in the peak
  power lowers the impact of SPM.
• Figure 8.1 also shows how the buildup of
  nonlinear effects within DCFs affects the system
  performance.
• In the case of RZ format, maximum distance is
  below 900 km at an input power level of -4 dBm
  because of the DCF-induced nonlinear degradation
  (filled squares).

16
8.1.1 System Design Issues

• Not only DCFs have a larger nonlinear parameter
  because of their smaller core size, pulses are
  also compressed inside them to their original
  width, resulting in much higher peak powers.
• If the nonlinear effects can be suppressed within
  DCF, maximum distance can be increased close to
  1,500 km by launching higher powers.
• This improvement can be realized in practice by
  using an alternate dispersion compensating device
  requiring shorter lengths (such as a two-mode DCF
  or a fiber grating).
• In the case of NRZ format, the link length is
  limited to below 500 km even when nonlinear
  effects are negligible within DCFs.

17
8.1.1 System Design Issues

• The nonlinear effects play an important role in
  dispersion-managed systems whenever a DCF is used
  because its smaller core size enhances optical
  intensities (manifested through a larger value of
  the g parameter).
• Placement of the amplifier after the DCF helps
  since the signal is then weak enough that the
  nonlinear effects are less important in spite of
  a small core area of DCFs.
• The optimization of system performance using
  different dispersion maps has been the subject of
  intense study.

18
8.1.1 System Design Issues

• Because of cost considerations, most laboratory
  experiments employ a fiber loop in which the
  optical signal is forced to recirculate many
  times to simulate a long-haul lightwave
  system.
• Two optical switches determine how long a
  pseudo-random bit stream circulates inside the
  loop before it reaches the receiver.
• The loop length and the number of round trips set
  the total ransmission distance. The loop shown in
  Fig. 8.2 contains two 102-km sections of standard
  fiber and two 20-km DCFs. A filter with a 1-nm
  bandwidth reduces the buildup of broadband ASE
  noise.

19
8.1.1 System Design Issues

• Figure 8.2 Recirculating fiber loop used to
  demonstrate the transmission of a 10-Gb/s signal
  over 2,040 km of standard fiber. Two
  acousto-optic (AO) switches control the timing of
  signal into and out of the loop. BERTS stands for
  bit-error-rate test set.

20
8.1.1 System Design Issues

• The 10-Gb/s signal could be transmitted over
  2,040 km with both the RZ and NRZ formats when
  launched power was properly optimized.
• However, it was necessary to add a 38-km section
  of standard fiber in front of the receiver in the
  NRZ case so that dispersion was not fully
  compensated.
• Perfect compensation of GVD in each map period is
  not generally the best solution in the presence
  of nonlinear effects.
• A numerical approach is generally used to
  optimize the design of dispersion-managed
  lightwave systems.

21
8.1.1 System Design Issues

• A systematic study based on the NLS equ. (8.1.2)
  shows that although the NRZ format can be used at
  10 Gb/s, the RZ format is superior in most
  practical situations for lightwave systems
  operating at bit rates of 40 Gb/s or higher.
• Even at 10 Gb/s, the RZ format can be used to
  design systems that are capable of transmitting
  data over a distance of up to 10,000 km over
  standard fibers.

22
8.1.2 Semianalytic Approach

• Considerable insight can be gained by adopting a
  semi-analytic approach in which the dispersive
  and nonlinear effects are considered for a single
  optical pulse of 1 bit.
• In this case, NLS equation (8.1.2) can be reduced
  to solving a set of two ordinary differential
  equations using a variational approach or the
  moment method.
• Both methods assume that each optical pulse
  maintains its shape even though its amplitude,
  width, and chirp may change during propagation.

23
8.1.2 Semianalytic Approach

• A chirped Gaussian pulse maintains its functional
  form in the linear case (g 0). If the nonlinear
  effects are relatively weak in each fiber section
  locally compared with the dispersive effects, the
  pulse is likely to retain its Gaussian shape
  approximately even when nonlinear effects are
  included.
• At a distance z inside the fiber, the envelope of
  a chirped Gaussian pulse has the form
• where a is the amplitude, T is the width, C
  is the chirp, and f is the phase. All four
  parameters vary with z.

24
8.1.2 Semianalytic Approach

• The variational or the moment method can be used
  to obtain four ordinary differential equations
  governing the evolution of these four parameters
  with z.
• The phase equation can be ignored as it is not
  coupled to the other three equations.
• The amplitude equation can be integrated to find
  that the product a2T does not vary with z and is
  related to the input pulse energy E0 as
  as a(0) 1.

25
8.1.2 Semianalytic Approach

• Thus, we only need to solve the following two
  coupled equations
• Details of loss and dispersion managements appear
  in these equations through the z dependence of
  three parameters b2, g, and p.

26
8.1.2 Semianalytic Approach

• Eqs. (8.1.5) and (8.1.6) require values of three
  pulse parameters at the input end, namely the
  width T0 , chirp C0 , and energy E0 , before they
  can be solved.
• The pulse energy E0 is related to the average
  power launched into the fiber link through the
  relation Pav (1/2)BE0 (vp/2)P0(T0/Tb), where
  Tb is the duration of bit slot at the bit rate B.

27
8.1.2 Semianalytic Approach

• Consider first the linear case by setting g(z)
  0. In this case, E0 plays no role because
  pulse-propagation details are independent of the
  initial pulse energy.
• Eqs. (8.1.5) and (8.1.6) can be solved
  analytically in the linear case and have the
  following general solution
• where details of the dispersion map are included
  through b2(z).
• This solution looks complicated but it is easy to
  perform integrations for a two-section dispersion
  map.

28
8.1.2 Semianalytic Approach

• The values of T and C at the end of the map
  period z Lmap are given by
• where the dimensionless parameter d is
  defined as
• and b2 is the average value of the dispersion
  parameter over the map period Lmap.

29
8.1.2 Semianalytic Approach

• As is evident from Eq. (8.1.8), the final pulse
  parameters depend only on the average dispersion,
  and not on details of the dispersion map, when
  nonlinear effects are negligible.
• If the dispersion map is designed such that b2
  0, both T and C return to their input values at
  z Lmap.
• In the case of a periodic dispersion map, each
  pulse would recover its original shape after each
  map period if d 0.
• However, when the average GVD of the
  dispersion-managed link is not zero, T and C
  change after each map period, and pulse
  evolution is not periodic.

30
8.1.2 Semianalytic Approach

• To study how the nonlinear effects governed by
  the g term in Eq. (8.1.8) affect the pulse
  parameters, we can solve Eqs. (8.1.5) and (8.1.6)
  numerically.
• Fig. 8.3 shows the evolution of pulse width and
  chirp over the first 60-km span for an isolated
  pulse in a 40-Gb/s bit stream using the same
  two-section dispersion map employed for Figure
  8.1 (50-km standard fiber followed with 10 km of
  DCF). Solid lines represent 10-mW launched power.

31
8.1.2 Semianalytic Approach

• Figure 8.3 (a) Pulse width and (b) chirp at the
  end of successive amplifiers for several values
  of average input power for the 40-Gb/s system
  with a periodic dispersion map used in Figure 8.1.

32
8.1.2 Semianalytic Approach

• Dotted lines show the low-power case for
  comparison. In the first 50-km section, pulse
  broadens by a factor of about 15, but it is
  compressed back in the DCF because of dispersion
  compensation.
• Although the nonlinear effects modify both the
  pulse width and chirp, changes are not large even
  for a 10-mW launched power. In particular, the
  width and chirp are almost recovered after the
  first 60-km span.

33
8.1.2 Semianalytic Approach

• Figure 8.4 shows the pulse width and chirp after
  each amplifier (spaced 60-km apart) over a
  distance of 3,000 km (50 map periods).
• At a relatively low power level of 1 mW, the
  input values are almost recovered after each map
  period as dispersion is fully compensated.
• As the launched power is increased beyond 1 mW,
  the nonlinear effects start to dominate, and the
  pulse width and chirp begin to deviate
  considerably from their input values, in spite of
  dispersion compensation.
• Even for Pav 5 mW, pulse width becomes larger
  than the bit slot after a distance of 1,000 km,
  and the situation is worse for Pav 10 mW.

34
8.1.2 Semianalytic Approach

• Figure 8.4 (a) Pulse width and (b) chirp at the
  end of successive amplifiers for three values of
  average input power for a 40-Gb/s system with the
  periodic dispersion map used in Figure 8.1.

35
8.1.3 Soliton and Pseudo-linear Regimes

• When the nonlinear term in Eq. (8.1.6) is not
  negligible, pulse parameters do not return to
  their input values after each map period even for
  perfect dispersion compensation (d 0).
• Eventually, the buildup of nonlinear distortion
  affects each pulse within the optical bit stream
  so much that the system cannot operate beyond a
  certain distance.
• As seen in Figure 8.1, this limiting distance can
  be under 500 km depending on the system design.

36
8.1.3 Soliton and Pseudo-linear Regimes

• The parameters associated with a dispersion map
  (length and GVD of each section) can be
  controlled to manage the nonlinearity
  problem.
• Two main techniques have evolved, and systems
  employing them are said to operate in the
  pseudo-linear and soliton regimes.

37
8.1.3 Soliton and Pseudo-linear Regimes

• A nonlinear system performs best when GVD
  compensation is only 90 to 95 so that some
  residual dispersion remains after each map
  period.
• In fact, if the input pulse is initially chirped
  such that b2C lt 0, the pulse at the end of the
  fiber link may even be shorter than the input
  pulse.
• This behavior is expected for a linear system and
  follows from Eq. (8.1.8) for C0d lt 0. It also
  persists for weakly nonlinear systems.
• This observation has led to the adoption of the
  chirped RZ (CRZ) format for dispersion-managed
  fiber links.

38
8.1.3 Soliton and Pseudo-linear Regimes

• To understand how the system and fiber parameters
  affect the evolution of an optical signal inside
  a fiber link, consider a lightwave system in
  which dispersion is compensated only at the
  transmitter and receiver ends.
• Since fiber parameters are constant over most of
  the link, it is useful to introduce the
  dispersion and nonlinear length scales as

39
8.1.3 Soliton and Pseudo-linear Regimes

• Introducing a normalized time t as t t /T0 ,
  the NLS equation (8.1.2) can be written in the
  form
• where s sign(b2) 1, depending on the
  sign of b2.
• If we use g 2 W-1/km as a typical value, the
  nonlinear length LNL 100 km at peak-power
  levels in the range of 2 to 4 mW.
• In contrast, the dispersion length LD can vary
  over a wide range (from 1 to 10,000 km),
  depending on the bit rate of the system and the
  type of fibers used to construct it.

40
8.1.3 Soliton and Pseudo-linear Regimes

• If LD gtgt LNL and link length L lt LD, the
  dispersive effects play a minor role, but the
  nonlinear effects cannot be ignored when L gt LNL.
  This is the situation for systems operating at a
  bit rate of 2.5 Gb/s or less.
• For example, LD exceeds 1,000 km at B 2.5 Gb/s
  even for standard fibers with b2 -21 ps2/km and
  can exceed 10,000 km for dispersion-shifted
  fibers.
• Such systems can be designed to operate over long
  distances by reducing the peak power and
  increasing the nonlinear length accordingly. The
  use of a dispersion map is also helpful for this
  purpose.

41
8.1.3 Soliton and Pseudo-linear Regimes

• If LD and LNL are comparable and much shorter
  than the link length, both the dispersive and
  nonlinear terms are equally important in the NLS
  equation (8.1.11).
• This is often the situation for10-Gb/s systems
  operating over standard fibers because LD becomes
  100 km when T0 is close to 50 ps. The use of
  optical solitons is most beneficial in the regime
  in which LD and LNL have similar magnitudes.
• A soliton-based system confines each pulse
  tightly to its original bit slot by employing the
  RZ format with a low duty cycle and maintains
  this confinement through a careful balance of
  frequency chirps induced by GVD and SPM.

42
8.1.3 Soliton and Pseudo-linear Regimes

• If LD ltlt LNL, we enter a new regime in which
  dispersive effects dominate locally, and the
  nonlinear effects can be treated in a
  perturbative manner.
• This situation is encountered in lightwave
  systems whose individual channels operate at a
  bit rate of 40 Gb/s or more.
• The bit slot is only 25 ps at 40 Gb/s. If T0 is
  lt10 ps and standard fibers are employed, LD is
  reduced to below 5 km.
• A lightwave system operating under such
  conditions is said to operate in the
  pseudo-linear regime.

43
8.1.3 Soliton and Pseudo-linear Regimes

• In such systems, input pulses broaden so rapidly
  that they spread over several neighboring bits.
  The extreme broadening reduces their peak power
  by a large factor.
• Since the nonlinear term in the NLS equation
  (8.1.2) scales with the peak power, its impact is
  considerably reduced.
• Interchannel nonlinear effects are reduced
  considerably in pseudo-linear systems because of
  an averaging effect that produces nearly constant
  total power in all bit slots.
• In contrast, overlapping of neighboring pulses
  enhances the intrachannel nonlinear effects.

44
8.1.3 Soliton and Pseudo-linear Regimes

• As nonlinear effects remain important, such
  systems are called pseudo-linear.
• Of course, pulses must be compressed back at the
  receiver end to ensure that they occupy their
  original time slot before the optical signal
  arrives at the receiver.
• This can be accomplished by compensating the
  accumulated dispersion with a DCF or another
  dispersion-equalizing filter.

45
8.2 Solitons in Optical Fibers

• The existence of solitons in optical fibers is
  the result of a balance between the chirps
  induced by GVD and SPM, both of which limit the
  system performance when acting independently.
• The GVD broadens an optical pulse during its
  propagation inside an optical fiber, except when
  the pulse is initially chirped in the right way
  (see Figure 3.3).

46
8.2 Solitons in Optical Fibers

• A chirped pulse can be compressed during the
  early stage of propagation whenever b2 and the
  chirp parameter C happen to have opposite signs
  so that b2C is negative.
• SPM imposes a chirp on the optical pulse such
  that C gt 0. If b2 lt 0, the condition b2C lt
  0 is readily satisfied.
• Under certain conditions, SPM 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.

47
8.2.1 Properties of Optical Solitons

• To find the conditions under which solitons can
  form, we use s -1 in Eq. (8.1.11), assuming
  that pulses are propagating in the region of
  anomalous GVD, and set p(z) 1, a condition
  requiring perfect distributed amplification.
• Introducing a normalized distance as x z/LD ,
  Eq. (8.1.11) can be written as
• where the parameter N is defined as

48
8.2.1 Properties of Optical Solitons

• It represents a dimensionless combination of the
  pulse and fiber parameters. Even the single
  parameter N appearing in Eq. (8.2.1) can be
  removed by introducing u NU as a renormalized
  amplitude.
• With this change, the NLS equation takes on its
  canonical form
• The NLS equation (8.2.3) 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.

49
8.2.1 Properties of Optical Solitons

• The main result can be summarized as follows
• When an input pulse having an initial amplitude
• is launched into the fiber, its shape remains
  unchanged during propagation when N 1 but
  follows a periodic pattern for integer values of
  N gt 1 such that the input shape is recovered at x
  mp/2, where m is an integer.

50
8.2.1 Properties of Optical Solitons

• An optical pulse whose parameters satisfy the
  condition N 1 is called the fundamental
  soliton.
• Pulses corresponding to other integer values of N
  are called higher order solitons.
• The parameter N represents the order of the
  soliton.
• Noting that x z/LD, the soliton period z0,
  defined as the distance over which higher-order
  solitons recover their original shape, is given by

51
8.2.1 Properties of Optical Solitons

• The soliton period z0 and soliton order N play an
  important role in the theory of optical solitons.
  
• Figure 8.5 shows the evolution of a 3rd-order
  soliton over one soliton period by solving the
  NLS equation (8.2.1) numerically with N 3.
• The pulse shape changes considerably but returns
  to its original form at z z0.
• Only a fundamental soliton maintains its shape
  during propagation inside optical fibers.

52
8.2.1 Properties of Optical Solitons

• Figure 8.5 Evolution of a third-order soliton
  over one soliton period. The power profile u2
  is plotted as a function of z/LD .

53
8.2.1 Properties of Optical Solitons

• The solution corresponding to the fundamental
  soliton can be obtained by solving Eq. (8.2.3)
  directly, without recourse to the inverse
  scattering method.
• The approach consists of assuming that a solution
  of the form
• exists,
• where V must be independent of x for Eq. (8.2.6)
  to represent a fundamental soliton that maintains
  its shape during propagation. The phase f can
  depend on x but is assumed to be time-independent.

54
8.2.1 Properties of Optical Solitons

• When Eq. (8.2.6) is substituted in Eq. (8.2.3)
  and the real and imaginary parts are separated,
  we obtain two real equations for V and f.
• These equations show that f should be of the form
  f(x) Kx, where K is a constant.
• The function V(t) is then found to satisfy the
  nonlinear differential equation
• This equation can be solved by multiplying it
  with
• 2(dV/dt) and integrating over t. The result is

55
8.2.1 Properties of Optical Solitons

• where C is a constant of integration. Using the
  boundary condition that both V and dV/dt should
  vanish for any optical pulse at , C
  can be set to zero.
• The constant K in Eq. (8.2.8) is determined using
  the boundary condition that V 1 and dV/dt
  0 at the soliton peak, assumed to occur at t
  0.
• Its use provides K 1/2, resulting in f x/2.

56
8.2.1 Properties of Optical Solitons

• Equ. (8.2.8) is easily integrated to obtain
  V(t)sech(t). We have thus found the well-known
  sech solution
• for the fundamental soliton by integrating
  the NLS equation directly.
• It shows that the input pulse acquires a phase
  shift x/2 as it propagates inside the fiber, but
  its amplitude remains unchanged.
• 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. (8.2.2)
  in such a way that N 1.

57
8.2.1 Properties of Optical Solitons

• Optical solitons are remarkably stable against
  perturbations. Even though the fundamental
  soliton requires a specific shape and a certain
  peak power corresponding to N 1 in Eq. (8.2.2),
  it can be created even when the pulse shape and
  the peak power deviate from the ideal conditions.
• Figure 8.6 shows the numerically simulated
  evolution of a Gaussian input pulse for which N
  1 but u(0,t).exp(-t2/2).

58
8.2.1 Properties of Optical Solitons

• Figure 8.6 Evolution of a Gaussian pulse with N
  1 over the range x 0 to 10. The pulse evolves
  toward the fundamental soliton by changing its
  shape, width, and peak power.

59
8.2.1 Properties of Optical Solitons

• As seen there, the pulse adjusts its shape and
  width as it propagates down the fiber in an
  attempt to become a fundamental soliton and
  attains a sech profile for x gtgt 1.
• Similar behavior is observed when N deviates from
  1. It turns out that the N-th-order soliton can
  form when the input value of N is in the range N
  - 1/2 to N 1/2.
• In particular, the fundamental soliton can be
  excited for values of N in the range of 0.5 to
  1.5.

60
8.2.1 Properties of Optical Solitons

• 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 leads to
  spatial modes of optical fibers.

61
8.2.1 Properties of Optical Solitons

• When the input pulse does not match a temporal
  mode precisely but it is close to it, most of the
  pulse energy can still be coupled to 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 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.

62
8.2.1 Properties of Optical Solitons

• The NLS equation can be solved with the inverse
  scattering method even when an optical fiber
  exhibits normal dispersion.
• 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 an optical fiber.
• For this reason, such solutions of the NLS
  equation are called dark solitons.

63
8.2.2 Loss-Managed Solitons

• Solitons use SPM to maintain their width even in
  the presence of fiber dispersion. However, this
  property holds only if soliton energy is
  maintained inside the fiber.
• It is not difficult to see that a decrease in
  pulse energy because of fiber losses would
  produce soliton broadening simply because a
  reduced peak power weakens the SPM effect
  necessary to counteract the GVD.
• When optical amplifiers are used periodically for
  compensating fiber losses, soliton energy changes
  in a periodic fashion. Such energy variations are
  included in the NLS equation (8.1.11) through the
  periodic function p(z).

64
8.2.2 Loss-Managed Solitons

• In the case of lumped amplifiers, p(z) decreases
  exponentially between two amplifiers and can vary
  by 20 dB or more over each period.
• Solitons can remain stable over long distances,
  provided amplifier spacing LA is kept much
  smaller than the dispersion length LD.
• Large rapid variations in p(z) can destroy a
  soliton if its width changes rapidly through
  the emission of dispersive waves.

65
8.2.2 Loss-Managed Solitons

• The concept of the path-averaged soliton makes
  use of the fact that solitons evolve little
  over a distance that is short compared with the
  dispersion length (or soliton period).
• Thus, when LA ltlt LD, the width of a soliton
  remains virtually unchanged even if its peak
  power varies considerably in each section between
  two amplifiers.
• In effect, one can replace p(z) by its average
  value p in Eq. (8.1.11) when LA ltlt LD . Noting
  that p is just a constant that modifies gP0, we
  recover the standard NLS equation.

66
8.2.2 Loss-Managed Solitons

• From a practical viewpoint, a fundamental soliton
  can be excited if the input peak power Ps, (or
  energy) of the path-averaged soliton is chosen to
  be larger by a factor of 1/p.
• If we introduce the amplifier gain as Gexp(aLA)
  and use , the
  energy enhancement factor for loss-managed
  solitons is given by

67
8.2.2 Loss-Managed Solitons

• Soliton evolution in lossy fibers with periodic
  lumped amplification is identical to that in
  lossless fibers provided (1). amplifiers are
  spaced such that LA ltlt LD (2). the launched
  peak power is larger by a factor fLM.
• As an example, G 10 and fLM 2.56 when LA
  50 km and a 0.2 dB/km.
• The condition LA ltlt LD is somewhat vague for
  designing soliton systems. The question is how
  close LA can be to LD before the system may
  fail to work.
• The semi-analytic approach can be extended to
  study how fiber losses affect the evolution of
  solitons.

68
8.2.2 Loss-Managed Solitons

• However, we should replace Eq. (8.1.4) with
• to ensure that the sech shape of a soliton
  is maintained.
• Using the variational or the moment method,
  we obtain the following two coupled
  equations
• where E0 2P0T0 is the input pulse energy.

69
8.2.2 Loss-Managed Solitons

• A comparison with Eqs. (8.1.5) and (8.1.6)
  obtained for Gaussian pulses shows that the width
  equation remains unchanged the chirp equation
  also has the same form but different
  coefficients.
• As a simple application, let us use the moment
  method for finding the soliton formation
  condition in the ideal case of p(z) 1.
• If the pulse is initially unchirped, both
  derivatives in Eqs. (8.2.12) and (8.2.13)
  vanish at z 0 if b2 is negative and the pulse
  energy is chosen to be E0 2b2/(gT0)

70
8.2.2 Loss-Managed Solitons

• Under such conditions, the width and chirp of the
  pulse will not change with z, and the pulse will
  form a fundamental soliton.
• Using E0 2P0T0, it is easy to see that this
  condition is equivalent to setting N 1 in Eq.
  (8.2.2).
• Consider now what happens when p(z) exp(-az) in
  each fiber section of length LA in a periodic
  fashion.
• Figure 8.7 shows how the soliton width changes at
  successive amplifiers for several values of LA in
  the range 25 to 100 km, assuming LD 100 km.
• Such values of dispersion length are realized for
  a 10-Gb/s soliton system, for example, when T0
  20 ps and b2 -4 ps2/km.

71
8.2.2 Loss-Managed Solitons

• Figure 8.7 Evolution of pulse with T and chirp C
  along the fiber length for three amplifier
  spacing (25, 50, and 75 km) when LD 100 km.

72
8.2.2 Loss-Managed Solitons

• When amplifier spacing is 25 km, both the width
  and chirp remain close to their input values. As
  LA is increased to 50 km, they oscillate in a
  periodic fashion, and oscillation amplitude
  increases as LA increases.
• For example, the width can change by more than
  10 when LA 75 km. The oscillatory behavior can
  be understood by performing a linear stability
  analysis of Eqs. (8.2.12) and (8.2.13).
• However, if LA/LD exceeds 1 considerably, the
  pulse width starts to increase exponentially in a
  monotonic fashion.

73
8.2.2 Loss-Managed Solitons

• Figure 8.7 shows that LA/LD ? 0.5 is a reasonable
  design criterion when lumped amplifiers are used
  for loss management.
• The variational equations such as Eqs. (8.2.12)
  and (8.2.13) only serve as a guideline, and their
  solutions are not always trustworthy, because
  they completely ignore the dispersive radiation
  generated as solitons are perturbed.
• For this reason, it is important to verify their
  predictions through direct numerical simulations
  of the NLS equation itself.

74
8.2.2 Loss-Managed Solitons

• Figure 8.8 shows the evolution of a loss-managed
  soliton over a distance of 10,000 km, assuming
  that solitons are amplified every 50 km.
• When the input pulse width corresponds to a
  dispersion length of 200 km, the soliton is
  preserved quite well even after 10,000 km because
  the condition LA ltlt LD is well satisfied.
• However, if the dispersion length is reduced to
  25 km, the soliton is unable to sustain itself
  because of the excessive emission of dispersive
  waves.

75
8.2.2 Loss-Managed Solitons

• Figure 8.8 Evolution of loss-managed solitons
  over 10,000 km for (a) LD 200 km and (b) 25 km
  with LA 50 km, a 0.22 dB/km, and b2 -0.5
  ps2/km.

76
8.2.2 Loss-Managed Solitons

• The condition LA lt LD can be related to the width
  T0 through LD T02/b2. The resulting condition
  is
• The pulse width T0 must be a small fraction of
  the bit slot Tb 1/B to ensure that the
  neighboring solitons are well separated.
• Mathematically, the soliton solution in Eq.
  (8.2.9) is valid only when a single pulse
  propagates by itself.

77
8.2.2 Loss-Managed Solitons

• This requirement can be used to relate the
  soliton width T0 to the bit rate B using Tb
  2q0T0, where 2q0 is a measure of separation
  between two neighboring pulses in an optical bit
  stream.
• Typically, q0 exceeds 4 to ensure that pulse
  tails do not overlap significantly. Using T0
  (2q0B)-1 in Eq. (8.2.14), we obtain the following
  design criterion

78
8.2.2 Loss-Managed Solitons

• Choosing typical values, b2 -2 ps2/km, LA 50
  km, and q0 5, we obtain T0 gt 10 ps and B lt 10
  GHz.
• Clearly, the use of path-averaged solitons
  imposes a severe limitation on both the bit rate
  and the amplifier spacing for soliton
  communication systems.
• To operate even at 10 Gb/s, one must reduce
  either q0 or LA if b2 is kept fixed.
• Both of these parameters cannot be reduced much
  below the values used in obtaining the preceding
  estimate.

79
8.2.2 Loss-Managed Solitons

• One could prechirp the soliton to relax the
  condition LA ltlt LD, even though the standard
  soliton solution in Eq. (8.2.9) has no chirp.
• The basic idea consists of finding a periodic
  solution of Eqs. (8.2.12) and (8.2.13) that
  repeats itself at each amplifier using the
  periodic boundary conditions

80
8.2.2 Loss-Managed Solitons

• The input pulse energy E0 and input chirp C0 can
  be used as two adjustable parameters.
• A perturbative solution of Eqs. (8.2.12) and
  (8.2.13) shows that the pulse energy must be
  increased by a factor close to the energy
  enhancement factor fLM in Eq. (8.2.10).
• At the same time, the input chirp that provides a
  periodic solution is related to this factor as

81
8.2.2 Loss-Managed Solitons

• Numerical results based on the NLS equation show
  that with a proper prechirping of input solitons,
  amplifier spacing can exceed 2LD.
• However, dispersive waves eventually destabilize
  a soliton over long fiber lengths when LA is
  made significantly larger than the dispersive
  length.
• The condition LA ltlt LD can also be relaxed
  considerably by employing distributed
  amplification.
• A distributed-amplification scheme is superior to
  lumped amplification because its use provides a
  nearly lossless fiber by compensating losses
  locally at every point along the fiber link.

82
8.3 Dispersion-Managed Solitons

• Dispersion management is employed commonly for
  modern WDM lightwave systems as it helps in
  suppressing FWM among channels.
• It turns out that solitons can form even when the
  GVD parameter b2 varies along the link length but
  their properties are quite different.
• This section is devoted to such
  dispersion-managed solitons. We first consider
  dispersion-decreasing fibers and then focus on
  fiber links with periodic dispersion maps.

83
8.3.1 Dispersion-Decreasing Fibers

• An interesting scheme relaxes completely the
  restriction LA ltlt LD imposed normally on
  loss-managed solitons by employing a new kind of
  fiber in which GVD varies along the fiber
  length.
• Such fibers are called dispersion-decreasing
  fibers (DDFs) and are designed such that the
  decreasing GVD counteracts the reduced SPM
  experienced by solitons weakened from fiber
  losses.

84
8.3.1 Dispersion-Decreasing Fibers

• Soliton evolution in a DDF is governed by Eq.
  (8.1.2) except that b2 is a continuous function
  of z.
• Introducing the normalized distance and time
  variables as
• we can write this equation in the form
• where

85
8.3.1 Dispersion-Decreasing Fibers

• If the GVD profile is chosen such that
  b2(z)b2(0)p(z),
• N becomes a constant, and Eq. (8.3.2) reduces
  the standard NLS equation obtained earlier with
  p(z) 1.
• As a result, fiber losses have no effect on a
  soliton in spite of its reduced energy when DDFs
  are used.
• More precisely, lumped amplifiers can be placed
  at any distance and are not limited by the
  condition LA ltlt LD , provided the GVD decreases
  exponentially in the fiber section between two
  amplifiers as

86
8.3.1 Dispersion-Decreasing Fibers

• This result can be understood by noting from Eq.
  (8.2.2) that the requirement N 1 can be
  maintained, in spite of power losses, if both
  b2 and g decrease exponentially at the same
  rate.
• A practical technique for making such DDFs
  consists of reducing the core diameter along the
  fiber length in a controlled manner during the
  fiber-drawing process.
• Variations in the fiber diameter change the
  waveguide contribution to b2 and reduce its
  magnitude.

87
8.3.1 Dispersion-Decreasing Fibers

• GVD can be varied by a factor of 10 over a length
  of 20 to 40 km. The accuracy realized by the
  use of this technique is estimated to be better
  than 0.1 ps2/km.
• The exponential GVD profile of a DDF can be
  approximated with a staircase profile by splicing
  together several constant-dispersion fibers with
  different b2 values.
• It was found that most of the benefits of DDFs
  can be realized using as few as four fiber
  segments.

88
8.3.1 Dispersion-Decreasing Fibers

• Several methods on selecting the length and the
  GVD of each fiber used for emulating a DDF have
  been proposed.
• In one approach, power deviations are minimized
  in each section.
• In another approach, fibers of different GVD
  values Dm , and different lengths Lmap chosen
  such that the product DmLmap is the same for each
  section.
• In a third approach, Dm and Lmap are selected to
  minimize the shading of dispersive waves.
• Advantages offered by DDFs for soliton systems
  include a lower timing jitter and a reduced
  noise level.

89
8.3.2 Periodic Dispersion Maps

• The main disadvantage of DDFs from the standpoint
  of system design is that the average dispersion
  along the link is relatively large for them.
• Dispersion maps consisting of alternating-GVD
  fibers are attractive because their use lowers
  the average dispersion of the entire link,
• while keeping the GVD of each section large
  enough that the FWM crosstalk remains negligible
  in WDM systems.

90
8.3.2 Periodic Dispersion Maps

• The use of dispersion management forces each
  soliton to propagate in the normal dispersion
  regime of a fiber during each map period.
• At first sight, such a scheme should not even
  work because the normal-GVD fibers do not support
  solitons and lead to considerable broadening and
  chirping of the pulse.
• Why should solitons survive in a
  dispersion-managed fiber link? An intense
  theoretical effort devoted to this issue has led
  to the discovery of dispersion-managed (DM)
  solitons.

91
8.3.2 Periodic Dispersion Maps

• If the dispersion length associated with each
  fiber section used to form the map is a fraction
  of the nonlinear length, the pulse would evolve
  in a linear fashion over a single map period.
• On a longer length scale, solitons can still form
  if the SPM effects are balanced by the average
  dispersion.
• As a result, solitons can survive in an average
  sense, even though not only the peak power but
  also the width and shape of such solitons
  oscillate periodically.

92
8.3.2 Periodic Dispersion Maps

• Consider a simple dispersion map consisting of
  two fibers with opposite GVD characteristics.
• Soliton evolution is governed by Eq. (8.1.2) in
  which b2 is a piecewise continuous function of z
  taking values b2a and b2n , in the anomalous and
  normal GVD sections of lengths la and ln ,
  respectively.
• The map period Lmap la ln can be different
  from the amplifier spacing LA .
• As is evident, the properties of DM solitons will
  depend on several map parameters even when only
  two types of fibers are used in each map period.

93
8.3.2 Periodic Dispersion Maps

• The variational equations (8.1.5) and (8.1.6)
  should be solved with the periodic boundary
  conditions given in Eq. (8.2.16) to ensure that
  the DM soliton recovers its initial state after
  each amplifier.
• The periodic boundary conditions fix the values
  of the initial width T0 and the chirp C0 at z 0
  for which a soliton can propagate in a periodic
  fashion for a given value of pulse energy E0.
• A new feature of the DM solitons is that the
  input pulse width depends on the dispersion map
  and cannot be chosen arbitrarily. In fact, T0
  cannot fall below a critical value that is set by
  the map itself.

94
8.3.2 Periodic Dispersion Maps

• Figure 8.9 shows how the pulse width T0 and the
  chirp C0 of allowed periodic solutions vary with
  input pulse energy for a specific dispersion
  map.
• The map is suitable for 40-Gb/s systems and
  consists of alternating fibers with GVD of -4
  and 4 ps2/km and lengths la ln 5 km such
  that the average GVD is -0.01 ps2/km.
• The solid lines show the case of ideal
  distributed amplification for which p(z) 1 in
  Eq. (7.1.5).
• The lumped-amplification case is shown by the
  dashed lines in Figure 8.9, assuming 80-km
  amplifier spacing and 0.25 dB/km losses in each
  fiber section.

95
8.3.2 Periodic Dispersion Maps

• Figure 8.9 (a) Changes in T0 (upper curve) and
  Tm (lower curve) with input pulse energy E0 for a
  0 (solid lines) and 0.25 dBkm (dashed lines).
  The inset shows the input chirp C0 in the two
  cases. (b) Evolution of the DM soliton over one
  map period for E0 0.1 pJ and LA 80 km.

96
8.3.2 Periodic Dispersion Maps

• Several conclusions can be drawn from Figure 8.9.
  
• First, both T0 and Tm decrease rapidly as pulse
  energy is increased.
• Second, T0 attains its minimum value at a certain
  pulse energy Ec , while Tm keeps decreasing
  slowly.
• Third, T0 and Tm , differ by a large factor for
  E0 gtgt Ec . This behavior indicates that pulse
  width changes considerably in each fiber section
  when this regime is approached.
• An example of pulse breathing is shown in Figure
  8.9(b) for E0 0.1 pJ in the case of lumped
  amplification. The input chirp C0 is relatively
  large (C0 1.8) in this case.

97
8.3.2 Periodic Dispersion Maps

• The most important feature of Figure 8.9 is the
  existence of a minimum value of T0 for a specific
  value of the pulse energy. The input chirp C0
  1 at that point.
• It is interesting to note that the minimum value
  of T0 does not depend much on fiber losses
  and is about the same for the solid and
  dashed curves, although the value of Ec is
  much larger in the lumped amplification case
  because of fiber losses.

98
8.3.2 Periodic Dispersion Maps

• As seen from Figure 8.9, both the pulse width and
  the peak power of DM solitons vary considerably
  within each map period.
• Figure 8.10(a) shows the width and chirp
  variations over one map period for the DM soliton
  of Fig. 8.9(b).
• The pulse width varies by more than a factor of 2
  and becomes minimum nearly in the middle of each
  fiber section where frequency chirp vanishes.
• The shortest pulse occurs in the middle of the
  anomalous-GVD section in the case of ideal
  distributed amplification in which fiber losses
  are compensated fully at every point along the
  fiber link.

99
8.3.2 Periodic Dispersion Maps

• Figure 8.10 Variations of pulse width and chirp
  (dashed line) over one map period for DM solitons
  with the input energy (a). E0 0.1 pJ and
  (b). E0 close to Ec.

100
8.3.2 Periodic Dispersion Maps

• For comparison, Figure 8.10(b) shows the width
  and chirp variations for a DM soliton whose input
  energy is close to Ec where the input pulse is
  shortest. Breathing of the pulse is reduced
  considerably together with the range of chirp
  variations.
• In both cases, the DM soliton is quite different
  from a standard fundamental soliton as it does
  not maintain its shape width, or peak power.
  Nevertheless, its parameters repeat from period
  to period atany 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.

101
8.3.3 Design Issues of DM Solitons

• Figures 8.9 and 8.10 show that Eqs. (8.1.5) and
  (8.1.6) permit periodic propagation of many
  different DM solitons in the same map by choosing
  different values of E0, T0, and C0.
• How should one choose among these solutions when
  designing a soliton system?
• Pulse energies much smaller than Ec
  (corresponding to the minimum value of T0)
  should be avoided because a low average power
  would then lead to rapid degradation of SNR as
  amplifier noise builds up with propagation.

102
8.3.3 Design Issues of DM Solitons

• On the other hand, when E0 gtgt Ec , large
  variations in the pulse width in each fiber
  section would induce XPM-induced interaction
  between two neighboring solitons if their tails
  begin to overlap considerably.
• For this region, the region near E0 Ec is most
  suited for designing DM soliton systems.
• The 40-Gb/s system design used for Figs. 8.9 and
  8.10 was possible only because the map period
  Lmap was chosen to be much smaller than the
  amplifier spacing of 80 km, a configuration
  referred to as the dense dispersion management.

103
8.3.3 Design Issues of DM Solitons

• When Lmap is increased to 80 km using la lb
  40 km, while keeping the same value of average
  dispersion, the minimum pulse width supported
  by the map increases by a factor of 3. The bit
  rate is then limited to below 20 Gb/s.
• It is possible to find the values of T0 and Tm by
  solving Eqs. (8.1.5) and (8.1.6) approximately.
  Equ. (8.1.6) shows that
  at any point within the map.

104
8.3.3 Design Issues of DM Solitons

• The chirp equation cannot be integrated
  analytically but the numerical solutions show
  that C(z) varies almost linearly in each fiber
  section.
• As seen in Fig. 8.10, C(z) changes from C0 to -C0
  in the 1st section and then back to C0 in the 2nd
  section.
• Noting that the ratio (1 C2)/T2 is related to
  the spectral width that changes little over one
  map period when the nonlinear length is much
  larger than the local dispersion length,
• we average it over one map period and obtain the
  following relation between T0 and C0

105
8.3.3 Design Issues of DM Solitons

• where Tmap is a parameter with dimensions of time
  involving only the four map parameters.
• It provides a time scale associated with an
  arbitrary dispersion map in the sense that the
  stable periodic solutions supported by it have
  input pulse widths that are close to Tmap
  (within a factor of 2 or so).
• The minimum value of T0 occurs for T0min
  (v2)Tmap and is given by C0 1 .

106
8.3.3 Design Issues of DM Solitons

• Equ. (8.3.4) can also be used to find the
  shortest pulse within the map. Recalling that the
  shortest pulse occurs at the point at which the
  pulse becomes unchirped, we obtain
• When the input pulse corresponds to its minimum
  value (C0 1), Tm is exactly equal to Tmap . The
  optimum value of the pulse stretching factor is
  equal to under such conditions.
• These conclusions are in agreement with the
  numerical results shown in Figure 8.10 for a
  specific map for which Tmap 3.16 ps.

107
8.3.3 Design Issues of DM Solitons

• If dense dispersion management is not used for
  this map and Lmap equals LA 80 km, this value
  of Tmap increases to 9 ps.
• Since the FWHM of input pulses then exceeds 21
  ps, such a map is unsuitable for 40-Gb/s
  soliton systems.
• In general, the required map period becomes
  shorter as the bit rate increases, as is evident
  from the definition of Tmap in Eq. (8.3.4).

108
8.3.3 Design Issues of DM Solitons

• It is useful to look for other combinations of
  the four map parameters that may play an
  important role in designing a DM soliton system.
• Two parameters that are useful for this purpose
  are defined as
• where TFWHM 1.665Tm is the FWHM at the
  location where pulse width is minimum in the
  anomalous-GVD section.

109
8.3.3 Design Issues of DM Solitons

• Physically, b2 represents the average GVD of the
  entire link, while the map strength Smap is a
  measure of how much GVD changes abruptly between
  two fibers in each map period.
• The solutions of Eqs. (8.1.3) and (8.1.6) as a
  function of map strength S for different values
  of b2 reveal the surprising feature that DM
  solitons can exist even when the average GVD is
  normal, provided the map strength exceeds a
  critical value Scr .

110
8.3.3 Design Issues of DM Solitons

• Figure 8.11 shows periodic DM-soliton solutions
  as contours of constant Smap by plotting peak
  power as a function of the dimensionless
  ratio b2 /b2a .
• The map strength is zero for the straight line
  (the case of a constant-dispersion fiber).
• It increases in steps of 2 for the next 10 curves
  and takes a value of 25 for the leftmost curve.
• Periodic solutions in the normal-GVD regime exist
  only when Smap exceeds a critical value of 4.8,
  indicating that pulse width for such solutions
  changes by a large factor in each fiber section.

111
8.3.3 Design Issues of DM Solitons

• Figure 8.11 Peak power of DM solitons as a
  function of b2/b2a .
• The map strength is zero for the straight
  line, increases in step of 2 until 20, and
  becomes 25 for the leftmost curve.

112
8.3.3 Design Issues of DM Solitons

• Moreover, when Smap gt Scr , a periodic solution
  can exist for two different values of the input
  pulse energy in a small range of positive values
  of b2 gt 0.
• Numerical solutions of Eq. (8.3.2) confirm these
  predictions, except that the critical value of
  the map strength is found to be 3.9.

113
8.4 Pseudo-linear Lightwave Systems

• Pseudo-linear lightwave systems operate in the
  regime in which the local dispersion length is
  much shorter than the nonlinear length in all
  fiber sections of a dispersion-managed link.
• This approach is most suitable for systems
  operating at bit rates of 40 Gb/s or more and
  employing relatively short optical pulses that
  spread over multiple bits quickly as they
  propagate along the link.
• This spreading reduces the peak power and lowers
  the impact of SPM on each pulse.

114
8.4 Pseudo-linear Lightwave Systems

• There are several ways one can design such
  systems. In one case, pulses spread throughout
  the link and are compressed back at the
  receiver end using a dispersion-compensating
  device.
• In another, pulses are spread even before the
  optical signal is launched into the fiber link
  using a DCF (pre-compensation) and they compress
  slowly within the fiber link, without requiring
  any post-compensation.

115
8.4 Pseudo-linear Lightwave Systems

• One can employ in-line compensation, the
  dispersion map is made such that the pulse
  broadens by a large factor in the first section
  and is compressed in the following section with
  opposite dispersion characteristics.
• An optical amplifier restores the signal power
  after the second section, and the whole process
  repeats itself.
• Often, a small amount of dispersion is left
  uncompensated in each map period.
• This residual dispersion per span can be used to
  control the impact of intrachannel nonlinear
  effects in combination with the amounts of pre-
  and post-compensation.

116
8.4 Pseudo-linear Lightwave Systems

• The spreading of bits belonging to different WDM
  channels produces an averaging effect that
  reduces the interchannel nonlinear effects
  considerably.
• At the same time, an enhanced nonlinear
  interaction among the 1 bits of the same channel
  produces new intrachannel nonlinear effects that
  limit the system performance, if left
  uncontrolled.
• The pulse spreading helps to lower the overall
  impact of fiber nonlinearity and allows higher
  launched powers into the fiber link.

117
8.4.1 Intrachannel Nonlinear Effects

• The main limitation of pseudo-linear systems
  stems from the nonlinear interaction among the
  neighboring overlapping pulses.
• In a numerical approach, one solves the NLS
  equation (8.1.2) for a pseudo-random bit stream
  with the input
• where tj jTb , Tb is the duration of the
  bit slot,
• M is the total number of bits included in
  numerical simulations, and Um 0 if the m-th
  pulse represents a 0 bit. In the case of 1
  bits, Um governs the shape of input pulses.

118
8.4.1 Intrachannel Nonlinear Effects

• Although numerical simulations are essential for
  a realistic system design, considerable physical
  insight can be gained with a semianalytic
  approach that focuses on three neighboring
  pulses.
• If we write the total field as U U1 U2 U3
  in Eq. (8.1.2), it reduces to the following set
  of three coupled NLS equations

119
8.4.1 Intrachannel Nonlinear Effects

• The first nonline