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Title: 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
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